LMFDB - Embedded newform 31.7.b.c.30.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [31,7,Mod(30,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.30"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	31.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.13167659222$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 7208 x^{10} + 19859688 x^{8} + 26566749360 x^{6} + 17884354852944 x^{4} + \cdots + 59\!\cdots\!00 $ x^12 + 7208x^10 + 19859688x^8 + 26566749360x^6 + 17884354852944x^4 + 5570285336959680*x^2 + 590986232936064000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			30.12
Root			$-37.0208i$ of defining polynomial
Character	$\chi$	$=$	31.30
Dual form			31.7.b.c.30.11

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+13.4661 q^{2} +37.0208i q^{3} +117.337 q^{4} -79.2253 q^{5} +498.527i q^{6} +281.109 q^{7} +718.240 q^{8} -641.539 q^{9} -1066.86 q^{10} +871.860i q^{11} +4343.90i q^{12} -1472.34i q^{13} +3785.45 q^{14} -2932.99i q^{15} +2162.36 q^{16} -9057.43i q^{17} -8639.05 q^{18} +11694.6 q^{19} -9296.05 q^{20} +10406.9i q^{21} +11740.6i q^{22} -2258.34i q^{23} +26589.8i q^{24} -9348.34 q^{25} -19826.7i q^{26} +3237.86i q^{27} +32984.4 q^{28} -35365.9i q^{29} -39496.0i q^{30} +(-12692.3 + 26952.0i) q^{31} -16848.7 q^{32} -32276.9 q^{33} -121969. i q^{34} -22271.0 q^{35} -75276.2 q^{36} +14322.6i q^{37} +157482. q^{38} +54507.2 q^{39} -56902.8 q^{40} -29502.1 q^{41} +140141. i q^{42} +74210.9i q^{43} +102301. i q^{44} +50826.2 q^{45} -30411.1i q^{46} -68983.7 q^{47} +80052.4i q^{48} -38626.6 q^{49} -125886. q^{50} +335313. q^{51} -172760. i q^{52} +233111. i q^{53} +43601.5i q^{54} -69073.4i q^{55} +201904. q^{56} +432945. i q^{57} -476242. i q^{58} +248597. q^{59} -344147. i q^{60} -82420.2i q^{61} +(-170916. + 362939. i) q^{62} -180343. q^{63} -365278. q^{64} +116647. i q^{65} -434646. q^{66} -152766. q^{67} -1.06277e6i q^{68} +83605.5 q^{69} -299904. q^{70} +180795. q^{71} -460779. q^{72} -430086. i q^{73} +192869. i q^{74} -346083. i q^{75} +1.37221e6 q^{76} +245088. i q^{77} +734001. q^{78} +601085. i q^{79} -171314. q^{80} -587550. q^{81} -397279. q^{82} -973523. i q^{83} +1.22111e6i q^{84} +717578. i q^{85} +999334. i q^{86} +1.30928e6 q^{87} +626205. i q^{88} -172788. i q^{89} +684432. q^{90} -413888. i q^{91} -264986. i q^{92} +(-997784. - 469878. i) q^{93} -928944. q^{94} -926512. q^{95} -623752. i q^{96} +1.15730e6 q^{97} -520151. q^{98} -559332. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{2} + 122 q^{4} - 146 q^{5} + 6 q^{7} + 1142 q^{8} - 5668 q^{9} - 3004 q^{10} - 1312 q^{14} - 1102 q^{16} - 16382 q^{18} + 10430 q^{19} - 8052 q^{20} + 29618 q^{25} + 136504 q^{28} + 90076 q^{31}+ \cdots + 408810 q^{98}+O(q^{100}) $ 12 * q - 2 * q^2 + 122 * q^4 - 146 * q^5 + 6 * q^7 + 1142 * q^8 - 5668 * q^9 - 3004 * q^10 - 1312 * q^14 - 1102 * q^16 - 16382 * q^18 + 10430 * q^19 - 8052 * q^20 + 29618 * q^25 + 136504 * q^28 + 90076 * q^31 + 38406 * q^32 - 166248 * q^33 - 198898 * q^35 + 26982 * q^36 + 448196 * q^38 + 208224 * q^39 - 354932 * q^40 - 116690 * q^41 - 363818 * q^45 - 221120 * q^47 - 580182 * q^49 - 60318 * q^50 + 179784 * q^51 + 750496 * q^56 + 799510 * q^59 - 73070 * q^62 - 469874 * q^63 - 322286 * q^64 - 511884 * q^66 - 13744 * q^67 + 475128 * q^69 - 71312 * q^70 + 640678 * q^71 - 118870 * q^72 + 2044196 * q^76 + 1373868 * q^78 + 1356020 * q^80 - 678724 * q^81 - 488056 * q^82 - 1051200 * q^87 - 2598532 * q^90 - 2260440 * q^93 - 5045848 * q^94 + 3828206 * q^95 + 3708966 * q^97 + 408810 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	13.4661			1.68327			0.841633	−	0.540049i	$-0.181594\pi$
$2$	13.4661			1.68327			0.841633	+	0.540049i	$0.181594\pi$
$3$			37.0208i			1.37114i	0.728006	+	0.685570i	$0.240447\pi$
$3$			37.0208i			1.37114i	−0.728006	+	0.685570i	$0.759553\pi$
$4$	117.337			1.83339
$5$	−79.2253			−0.633803			−0.316901	−	0.948458i	$-0.602642\pi$
$5$	−79.2253			−0.633803			−0.316901	+	0.948458i	$0.602642\pi$
$6$			498.527i			2.30800i
$7$	281.109			0.819560			0.409780	−	0.912184i	$-0.365605\pi$
$7$	281.109			0.819560			0.409780	+	0.912184i	$0.365605\pi$
$8$	718.240			1.40281
$9$	−641.539			−0.880027
$10$	−1066.86			−1.06686
$11$			871.860i			0.655041i	0.944844	+	0.327521i	$0.106213\pi$
$11$			871.860i			0.655041i	−0.944844	+	0.327521i	$0.893787\pi$
$12$			4343.90i			2.51383i
$13$		−	1472.34i		−	0.670159i	−0.942190	−	0.335080i	$-0.891237\pi$
$13$		−	1472.34i		−	0.670159i	0.942190	−	0.335080i	$-0.108763\pi$
$14$	3785.45			1.37954
$15$		−	2932.99i		−	0.869033i
$16$	2162.36			0.527921
$17$		−	9057.43i		−	1.84356i	−0.387709	−	0.921782i	$-0.626733\pi$
$17$		−	9057.43i		−	1.84356i	0.387709	−	0.921782i	$-0.373267\pi$
$18$	−8639.05			−1.48132
$19$	11694.6			1.70501			0.852503	−	0.522722i	$-0.175084\pi$
$19$	11694.6			1.70501			0.852503	+	0.522722i	$0.175084\pi$
$20$	−9296.05			−1.16201
$21$			10406.9i			1.12373i
$22$			11740.6i			1.10261i
$23$		−	2258.34i		−	0.185612i	−0.995684	−	0.0928059i	$-0.970416\pi$
$23$		−	2258.34i		−	0.185612i	0.995684	−	0.0928059i	$-0.0295836\pi$
$24$			26589.8i			1.92345i
$25$	−9348.34			−0.598294
$26$		−	19826.7i		−	1.12806i
$27$			3237.86i			0.164500i
$28$	32984.4			1.50257
$29$		−	35365.9i		−	1.45008i	−0.688708	−	0.725039i	$-0.741822\pi$
$29$		−	35365.9i		−	1.45008i	0.688708	−	0.725039i	$-0.258178\pi$
$30$		−	39496.0i		−	1.46281i
$31$	−12692.3	+	26952.0i	−0.426044	+	0.904702i
$31$	−12692.3	+	26952.0i	−0.426044	+	0.904702i
$32$	−16848.7			−0.514181
$33$	−32276.9			−0.898154
$34$		−	121969.i		−	3.10321i
$35$	−22271.0			−0.519440
$36$	−75276.2			−1.61343
$37$			14322.6i			0.282758i	0.989956	+	0.141379i	$0.0451537\pi$
$37$			14322.6i			0.282758i	−0.989956	+	0.141379i	$0.954846\pi$
$38$	157482.			2.86998
$39$	54507.2			0.918883
$40$	−56902.8			−0.889106
$41$	−29502.1			−0.428056			−0.214028	−	0.976828i	$-0.568658\pi$
$41$	−29502.1			−0.428056			−0.214028	+	0.976828i	$0.568658\pi$
$42$			140141.i			1.89154i
$43$			74210.9i			0.933388i	0.884419	+	0.466694i	$0.154555\pi$
$43$			74210.9i			0.933388i	−0.884419	+	0.466694i	$0.845445\pi$
$44$			102301.i			1.20094i
$45$	50826.2			0.557763
$46$		−	30411.1i		−	0.312434i
$47$	−68983.7			−0.664436			−0.332218	−	0.943203i	$-0.607797\pi$
$47$	−68983.7			−0.664436			−0.332218	+	0.943203i	$0.607797\pi$
$48$			80052.4i			0.723854i
$49$	−38626.6			−0.328321
$50$	−125886.			−1.00709
$51$	335313.			2.52779
$52$		−	172760.i		−	1.22866i
$53$			233111.i			1.56580i	0.622150	+	0.782898i	$0.286260\pi$
$53$			233111.i			1.56580i	−0.622150	+	0.782898i	$0.713740\pi$
$54$			43601.5i			0.276898i
$55$		−	69073.4i		−	0.415167i
$56$	201904.			1.14969
$57$			432945.i			2.33780i
$58$		−	476242.i		−	2.44087i
$59$	248597.			1.21043			0.605216	−	0.796062i	$-0.293087\pi$
$59$	248597.			1.21043			0.605216	+	0.796062i	$0.293087\pi$
$60$		−	344147.i		−	1.59327i
$61$		−	82420.2i		−	0.363115i	−0.983380	−	0.181557i	$-0.941886\pi$
$61$		−	82420.2i		−	0.363115i	0.983380	−	0.181557i	$-0.0581138\pi$
$62$	−170916.	+	362939.i	−0.717146	+	1.52286i
$63$	−180343.			−0.721235
$64$	−365278.			−1.39342
$65$			116647.i			0.424749i
$66$	−434646.			−1.51183
$67$	−152766.			−0.507929			−0.253965	−	0.967214i	$-0.581735\pi$
$67$	−152766.			−0.507929			−0.253965	+	0.967214i	$0.581735\pi$
$68$		−	1.06277e6i		−	3.37997i
$69$	83605.5			0.254500
$70$	−299904.			−0.874356
$71$	180795.			0.505140			0.252570	−	0.967579i	$-0.418724\pi$
$71$	180795.			0.505140			0.252570	+	0.967579i	$0.418724\pi$
$72$	−460779.			−1.23451
$73$		−	430086.i		−	1.10557i	−0.833323	−	0.552786i	$-0.813565\pi$
$73$		−	430086.i		−	1.10557i	0.833323	−	0.552786i	$-0.186435\pi$
$74$			192869.i			0.475958i
$75$		−	346083.i		−	0.820345i
$76$	1.37221e6			3.12594
$77$			245088.i			0.536846i
$78$	734001.			1.54672
$79$			601085.i			1.21914i	0.792731	+	0.609572i	$0.208659\pi$
$79$			601085.i			1.21914i	−0.792731	+	0.609572i	$0.791341\pi$
$80$	−171314.			−0.334598
$81$	−587550.			−1.10558
$82$	−397279.			−0.720533
$83$		−	973523.i		−	1.70260i	−0.524681	−	0.851299i	$-0.675815\pi$
$83$		−	973523.i		−	1.70260i	0.524681	−	0.851299i	$-0.324185\pi$
$84$			1.22111e6i			2.06024i
$85$			717578.i			1.16846i
$86$			999334.i			1.57114i
$87$	1.30928e6			1.98826
$88$			626205.i			0.918900i
$89$		−	172788.i		−	0.245101i	−0.992462	−	0.122550i	$-0.960893\pi$
$89$		−	172788.i		−	0.245101i	0.992462	−	0.122550i	$-0.0391073\pi$
$90$	684432.			0.938864
$91$		−	413888.i		−	0.549236i
$92$		−	264986.i		−	0.340298i
$93$	−997784.	−	469878.i	−1.24047	−	0.584167i
$94$	−928944.			−1.11842
$95$	−926512.			−1.08064
$96$		−	623752.i		−	0.705015i
$97$	1.15730e6			1.26803			0.634015	−	0.773321i	$-0.281406\pi$
$97$	1.15730e6			1.26803			0.634015	+	0.773321i	$0.281406\pi$
$98$	−520151.			−0.552651
$99$		−	559332.i		−	0.576454i
$100$	−1.09690e6			−1.09690
$101$	−1.64413e6			−1.59577			−0.797887	−	0.602806i	$-0.794049\pi$
$101$	−1.64413e6			−1.59577			−0.797887	+	0.602806i	$0.794049\pi$
$102$	4.51537e6			4.25494
$103$	−724173.			−0.662721			−0.331360	−	0.943504i	$-0.607508\pi$
$103$	−724173.			−0.662721			−0.331360	+	0.943504i	$0.607508\pi$
$104$		−	1.05749e6i		−	0.940108i
$105$		−	824489.i		−	0.712225i
$106$			3.13911e6i			2.63565i
$107$	−1.80317e6			−1.47193			−0.735963	−	0.677022i	$-0.763270\pi$
$107$	−1.80317e6			−1.47193			−0.735963	+	0.677022i	$0.763270\pi$
$108$			379920.i			0.301593i
$109$	18284.2			0.0141188			0.00705938	−	0.999975i	$-0.497753\pi$
$109$	18284.2			0.0141188			0.00705938	+	0.999975i	$0.497753\pi$
$110$		−	930152.i		−	0.698837i
$111$	−530232.			−0.387701
$112$	607860.			0.432663
$113$	1.76280e6			1.22171			0.610853	−	0.791744i	$-0.290827\pi$
$113$	1.76280e6			1.22171			0.610853	+	0.791744i	$0.290827\pi$
$114$			5.83009e6i			3.93515i
$115$			178918.i			0.117641i
$116$		−	4.14972e6i		−	2.65855i
$117$			944564.i			0.589758i
$118$	3.34764e6			2.03748
$119$		−	2.54613e6i		−	1.51091i
$120$		−	2.10659e6i		−	1.21909i
$121$	1.01142e6			0.570921
$122$		−	1.10988e6i		−	0.611219i
$123$		−	1.09219e6i		−	0.586925i
$124$	−1.48927e6	+	3.16246e6i	−0.781104	+	1.65867i
$125$	1.97852e6			1.01300
$126$	−2.42852e6			−1.21403
$127$			1.63403e6i			0.797715i	0.917013	+	0.398857i	$0.130593\pi$
$127$			1.63403e6i			0.797715i	−0.917013	+	0.398857i	$0.869407\pi$
$128$	−3.84057e6			−1.83132
$129$	−2.74735e6			−1.27981
$130$			1.57078e6i			0.714966i
$131$	1.06827e6			0.475190			0.237595	−	0.971364i	$-0.423641\pi$
$131$	1.06827e6			0.475190			0.237595	+	0.971364i	$0.423641\pi$
$132$	−3.78727e6			−1.64666
$133$	3.28747e6			1.39736
$134$	−2.05717e6			−0.854980
$135$		−	256521.i		−	0.104261i
$136$		−	6.50541e6i		−	2.58617i
$137$			151915.i			0.0590798i	0.999564	+	0.0295399i	$0.00940420\pi$
$137$			151915.i			0.0590798i	−0.999564	+	0.0295399i	$0.990596\pi$
$138$	1.12584e6			0.428391
$139$			708344.i			0.263755i	0.991266	+	0.131877i	$0.0421005\pi$
$139$			708344.i			0.263755i	−0.991266	+	0.131877i	$0.957900\pi$
$140$	−2.61320e6			−0.952334
$141$		−	2.55383e6i		−	0.911035i
$142$	2.43461e6			0.850286
$143$	1.28367e6			0.438982
$144$	−1.38724e6			−0.464584
$145$			2.80188e6i			0.919063i
$146$		−	5.79160e6i		−	1.86097i
$147$		−	1.42999e6i		−	0.450174i
$148$			1.68056e6i			0.518405i
$149$	1.32685e6			0.401108			0.200554	−	0.979683i	$-0.435726\pi$
$149$	1.32685e6			0.401108			0.200554	+	0.979683i	$0.435726\pi$
$150$		−	4.66040e6i		−	1.38086i
$151$			2.16558e6i			0.628989i	0.949259	+	0.314494i	$0.101835\pi$
$151$			2.16558e6i			0.628989i	−0.949259	+	0.314494i	$0.898165\pi$
$152$	8.39956e6			2.39180
$153$			5.81070e6i			1.62239i
$154$			3.30039e6i			0.903655i
$155$	1.00555e6	−	2.13528e6i	0.270028	−	0.573403i
$156$	6.39570e6			1.68467
$157$	−1.59984e6			−0.413407			−0.206703	−	0.978404i	$-0.566273\pi$
$157$	−1.59984e6			−0.413407			−0.206703	+	0.978404i	$0.566273\pi$
$158$			8.09430e6i			2.05214i
$159$	−8.62996e6			−2.14693
$160$	1.33484e6			0.325889
$161$		−	634840.i		−	0.152120i
$162$	−7.91203e6			−1.86099
$163$	98835.9			0.0228219			0.0114109	−	0.999935i	$-0.496368\pi$
$163$	98835.9			0.0228219			0.0114109	+	0.999935i	$0.496368\pi$
$164$	−3.46168e6			−0.784792
$165$	2.55715e6			0.569252
$166$		−	1.31096e7i		−	2.86593i
$167$			1.44983e6i			0.311292i	0.987813	+	0.155646i	$0.0497460\pi$
$167$			1.44983e6i			0.311292i	−0.987813	+	0.155646i	$0.950254\pi$
$168$			7.47464e6i			1.57639i
$169$	2.65902e6			0.550886
$170$			9.66300e6i			1.96682i
$171$	−7.50257e6			−1.50045
$172$			8.70767e6i			1.71126i
$173$	4.34117e6			0.838434			0.419217	−	0.907886i	$-0.362305\pi$
$173$	4.34117e6			0.838434			0.419217	+	0.907886i	$0.362305\pi$
$174$	1.76309e7			3.34677
$175$	−2.62791e6			−0.490338
$176$			1.88528e6i			0.345810i
$177$			9.20326e6i			1.65967i
$178$		−	2.32679e6i		−	0.412570i
$179$			2.62033e6i			0.456874i	0.973559	+	0.228437i	$0.0733614\pi$
$179$			2.62033e6i			0.456874i	−0.973559	+	0.228437i	$0.926639\pi$
$180$	5.96378e6			1.02260
$181$		−	994173.i		−	0.167659i	−0.996480	−	0.0838293i	$-0.973285\pi$
$181$		−	994173.i		−	0.167659i	0.996480	−	0.0838293i	$-0.0267151\pi$
$182$		−	5.57348e6i		−	0.924511i
$183$	3.05126e6			0.497881
$184$		−	1.62203e6i		−	0.260379i
$185$		−	1.13471e6i		−	0.179213i
$186$	−1.34363e7	−	6.32745e6i	−2.08805	−	0.983308i
$187$	7.89681e6			1.20761
$188$	−8.09433e6			−1.21817
$189$			910193.i			0.134818i
$190$	−1.24765e7			−1.81900
$191$	−9.22097e6			−1.32336			−0.661678	−	0.749788i	$-0.730155\pi$
$191$	−9.22097e6			−1.32336			−0.661678	+	0.749788i	$0.730155\pi$
$192$		−	1.35229e7i		−	1.91058i
$193$	6.85415e6			0.953414			0.476707	−	0.879062i	$-0.341830\pi$
$193$	6.85415e6			0.953414			0.476707	+	0.879062i	$0.341830\pi$
$194$	1.55843e7			2.13443
$195$	−4.31835e6			−0.582390
$196$	−4.53232e6			−0.601939
$197$		−	1.00221e7i		−	1.31088i	−0.755248	−	0.655439i	$-0.772484\pi$
$197$		−	1.00221e7i		−	1.31088i	0.755248	−	0.655439i	$-0.227516\pi$
$198$		−	7.53205e6i		−	0.970325i
$199$		−	1.38162e7i		−	1.75319i	−0.481227	−	0.876596i	$-0.659809\pi$
$199$		−	1.38162e7i		−	1.75319i	0.481227	−	0.876596i	$-0.340191\pi$
$200$	−6.71435e6			−0.839294
$201$		−	5.65553e6i		−	0.696442i
$202$	−2.21401e7			−2.68611
$203$		−	9.94169e6i		−	1.18843i
$204$	3.93446e7			4.63441
$205$	2.33731e6			0.271303
$206$	−9.75181e6			−1.11554
$207$			1.44881e6i			0.163343i
$208$		−	3.18373e6i		−	0.353791i
$209$			1.01961e7i			1.11685i
$210$		−	1.11027e7i		−	1.19886i
$211$	510177.			0.0543092			0.0271546	−	0.999631i	$-0.491355\pi$
$211$	510177.			0.0543092			0.0271546	+	0.999631i	$0.491355\pi$
$212$			2.73525e7i			2.87071i
$213$			6.69318e6i			0.692618i
$214$	−2.42818e7			−2.47764
$215$		−	5.87938e6i		−	0.591584i
$216$			2.32556e6i			0.230763i
$217$	−3.56792e6	+	7.57645e6i	−0.349169	+	0.741458i
$218$	246217.			0.0237656
$219$	1.59221e7			1.51589
$220$		−	8.10485e6i		−	0.761162i
$221$	−1.33356e7			−1.23548
$222$	−7.14018e6			−0.652605
$223$			8.29587e6i			0.748078i	0.927413	+	0.374039i	$0.122027\pi$
$223$			8.29587e6i			0.748078i	−0.927413	+	0.374039i	$0.877973\pi$
$224$	−4.73632e6			−0.421403
$225$	5.99733e6			0.526515
$226$	2.37380e7			2.05646
$227$	1.38617e7			1.18505			0.592527	−	0.805551i	$-0.298130\pi$
$227$	1.38617e7			1.18505			0.592527	+	0.805551i	$0.298130\pi$
$228$			5.08003e7i			4.28610i
$229$			1.92385e7i			1.60201i	0.598659	+	0.801004i	$0.295700\pi$
$229$			1.92385e7i			1.60201i	−0.598659	+	0.801004i	$0.704300\pi$
$230$			2.40933e6i			0.198022i
$231$	−9.07335e6			−0.736091
$232$		−	2.54012e7i		−	2.03419i
$233$	1.03937e6			0.0821676			0.0410838	−	0.999156i	$-0.486919\pi$
$233$	1.03937e6			0.0821676			0.0410838	+	0.999156i	$0.486919\pi$
$234$			1.27196e7i			0.992720i
$235$	5.46526e6			0.421121
$236$	2.91696e7			2.21919
$237$	−2.22527e7			−1.67162
$238$		−	3.42865e7i		−	2.54327i
$239$			8.58950e6i			0.629179i	0.949228	+	0.314589i	$0.101867\pi$
$239$			8.58950e6i			0.629179i	−0.949228	+	0.314589i	$0.898133\pi$
$240$		−	6.34218e6i		−	0.458780i
$241$			2.20696e7i			1.57668i	0.615239	+	0.788341i	$0.289059\pi$
$241$			2.20696e7i			1.57668i	−0.615239	+	0.788341i	$0.710941\pi$
$242$	1.36199e7			0.961012
$243$		−	1.93912e7i		−	1.35140i
$244$		−	9.67092e6i		−	0.665730i
$245$	3.06021e6			0.208091
$246$		−	1.47076e7i		−	0.987951i
$247$		−	1.72185e7i		−	1.14263i
$248$	−9.11611e6	+	1.93580e7i	−0.597660	+	1.26913i
$249$	3.60406e7			2.33450
$250$	2.66430e7			1.70515
$251$		−	3.11207e7i		−	1.96801i	−0.178132	−	0.984007i	$-0.557005\pi$
$251$		−	3.11207e7i		−	1.96801i	0.178132	−	0.984007i	$-0.442995\pi$
$252$	−2.11608e7			−1.32230
$253$	1.96896e6			0.121583
$254$			2.20040e7i			1.34277i
$255$	−2.65653e7			−1.60212
$256$	−2.83398e7			−1.68918
$257$	−1.57258e7			−0.926429			−0.463214	−	0.886246i	$-0.653304\pi$
$257$	−1.57258e7			−0.926429			−0.463214	+	0.886246i	$0.653304\pi$
$258$	−3.69961e7			−2.15426
$259$			4.02620e6i			0.231737i
$260$			1.36869e7i			0.778729i
$261$			2.26886e7i			1.27611i
$262$	1.43855e7			0.799871
$263$			1.91504e7i			1.05271i	0.850263	+	0.526357i	$0.176443\pi$
$263$			1.91504e7i			1.05271i	−0.850263	+	0.526357i	$0.823557\pi$
$264$	−2.31826e7			−1.25994
$265$		−	1.84683e7i		−	0.992406i
$266$	4.42695e7			2.35212
$267$	6.39676e6			0.336067
$268$	−1.79251e7			−0.931231
$269$			1.82164e6i			0.0935848i	0.998905	+	0.0467924i	$0.0148999\pi$
$269$			1.82164e6i			0.0935848i	−0.998905	+	0.0467924i	$0.985100\pi$
$270$		−	3.45434e6i		−	0.175499i
$271$			2.30717e7i			1.15923i	0.814889	+	0.579617i	$0.196798\pi$
$271$			2.30717e7i			1.15923i	−0.814889	+	0.579617i	$0.803202\pi$
$272$		−	1.95855e7i		−	0.973256i
$273$	1.53225e7			0.753080
$274$			2.04571e6i			0.0994470i
$275$		−	8.15045e6i		−	0.391907i
$276$	9.81000e6			0.466597
$277$		−	3.51879e6i		−	0.165559i	−0.996568	−	0.0827796i	$-0.973620\pi$
$277$		−	3.51879e6i		−	0.165559i	0.996568	−	0.0827796i	$-0.0263798\pi$
$278$			9.53866e6i			0.443969i
$279$	8.14260e6	−	1.72908e7i	0.374930	−	0.796162i
$280$	−1.59959e7			−0.728677
$281$	−4.05218e6			−0.182629			−0.0913146	−	0.995822i	$-0.529107\pi$
$281$	−4.05218e6			−0.182629			−0.0913146	+	0.995822i	$0.529107\pi$
$282$		−	3.43903e7i		−	1.53352i
$283$	1.19218e7			0.525996			0.262998	−	0.964796i	$-0.415289\pi$
$283$	1.19218e7			0.525996			0.262998	+	0.964796i	$0.415289\pi$
$284$	2.12139e7			0.926117
$285$		−	3.43002e7i		−	1.48171i
$286$	1.72861e7			0.738924
$287$	−8.29330e6			−0.350818
$288$	1.08091e7			0.452493
$289$	−5.78995e7			−2.39873
$290$			3.77305e7i			1.54703i
$291$			4.28440e7i			1.73865i
$292$		−	5.04649e7i		−	2.02694i
$293$	−3.24505e6			−0.129009			−0.0645043	−	0.997917i	$-0.520547\pi$
$293$	−3.24505e6			−0.129009			−0.0645043	+	0.997917i	$0.520547\pi$
$294$		−	1.92564e7i		−	0.757762i
$295$	−1.96952e7			−0.767175
$296$			1.02870e7i			0.396657i
$297$	−2.82296e6			−0.107755
$298$	1.78675e7			0.675172
$299$	−3.32504e6			−0.124390
$300$		−	4.06083e7i		−	1.50401i
$301$			2.08614e7i			0.764968i
$302$			2.91619e7i			1.05876i
$303$		−	6.08669e7i		−	2.18803i
$304$	2.52881e7			0.900108
$305$			6.52977e6i			0.230143i
$306$			7.82476e7i			2.73091i
$307$	3.69785e7			1.27801			0.639004	−	0.769203i	$-0.279347\pi$
$307$	3.69785e7			1.27801			0.639004	+	0.769203i	$0.279347\pi$
$308$			2.87578e7i			0.984246i
$309$		−	2.68095e7i		−	0.908683i
$310$	1.35409e7	−	2.87540e7i	0.454529	−	0.965190i
$311$	−3.70510e7			−1.23174			−0.615869	−	0.787848i	$-0.711195\pi$
$311$	−3.70510e7			−1.23174			−0.615869	+	0.787848i	$0.711195\pi$
$312$	3.91493e7			1.28902
$313$			5.13011e7i			1.67299i	0.547973	+	0.836496i	$0.315400\pi$
$313$			5.13011e7i			1.67299i	−0.547973	+	0.836496i	$0.684600\pi$
$314$	−2.15436e7			−0.695874
$315$	1.42877e7			0.457121
$316$			7.05294e7i			2.23516i
$317$	1.81320e7			0.569204			0.284602	−	0.958646i	$-0.408139\pi$
$317$	1.81320e7			0.569204			0.284602	+	0.958646i	$0.408139\pi$
$318$	−1.16212e8			−3.61385
$319$	3.08341e7			0.949860
$320$	2.89393e7			0.883156
$321$		−	6.67549e7i		−	2.01822i
$322$		−	8.54884e6i		−	0.256059i
$323$		−	1.05923e8i		−	3.14329i
$324$	−6.89413e7			−2.02696
$325$			1.37639e7i			0.400952i
$326$	1.33094e6			0.0384153
$327$			676896.i			0.0193588i
$328$	−2.11896e7			−0.600482
$329$	−1.93920e7			−0.544546
$330$	3.44350e7			0.958203
$331$			3.03116e7i			0.835843i	0.908483	+	0.417921i	$0.137241\pi$
$331$			3.03116e7i			0.835843i	−0.908483	+	0.417921i	$0.862759\pi$
$332$		−	1.14230e8i		−	3.12152i
$333$		−	9.18848e6i		−	0.248835i
$334$			1.95236e7i			0.523988i
$335$	1.21030e7			0.321927
$336$			2.25035e7i			0.593242i
$337$		−	2.12669e7i		−	0.555667i	−0.960629	−	0.277834i	$-0.910384\pi$
$337$		−	2.12669e7i		−	0.555667i	0.960629	−	0.277834i	$-0.0896164\pi$
$338$	3.58068e7			0.927289
$339$			6.52601e7i			1.67513i
$340$			8.41983e7i			2.14223i
$341$	−2.34984e7	−	1.10659e7i	−0.592617	−	0.279077i
$342$	−1.01031e8			−2.52566
$343$	−4.39305e7			−1.08864
$344$			5.33012e7i			1.30937i
$345$	−6.62368e6			−0.161303
$346$	5.84588e7			1.41131
$347$			3.57102e6i			0.0854681i	0.999086	+	0.0427341i	$0.0136068\pi$
$347$			3.57102e6i			0.0854681i	−0.999086	+	0.0427341i	$0.986393\pi$
$348$	1.53626e8			3.64525
$349$	6.94250e7			1.63320			0.816600	−	0.577204i	$-0.195856\pi$
$349$	6.94250e7			1.63320			0.816600	+	0.577204i	$0.195856\pi$
$350$	−3.53877e7			−0.825370
$351$	4.76724e6			0.110242
$352$		−	1.46897e7i		−	0.336810i
$353$		−	3.55636e7i		−	0.808504i	−0.914648	−	0.404252i	$-0.867532\pi$
$353$		−	3.55636e7i		−	0.808504i	0.914648	−	0.404252i	$-0.132468\pi$
$354$			1.23932e8i			2.79367i
$355$	−1.43236e7			−0.320159
$356$		−	2.02744e7i		−	0.449364i
$357$	9.42597e7			2.07167
$358$			3.52856e7i			0.769040i
$359$	−3.71713e7			−0.803387			−0.401693	−	0.915774i	$-0.631578\pi$
$359$	−3.71713e7			−0.803387			−0.401693	+	0.915774i	$0.631578\pi$
$360$	3.65054e7			0.782437
$361$	8.97187e7			1.90705
$362$		−	1.33877e7i		−	0.282214i
$363$			3.74436e7i			0.782813i
$364$		−	4.85643e7i		−	1.00696i
$365$			3.40737e7i			0.700714i
$366$	4.10887e7			0.838067
$367$		−	317655.i		−	0.00642624i	−0.999995	−	0.00321312i	$-0.998977\pi$
$367$		−	317655.i		−	0.00642624i	0.999995	−	0.00321312i	$-0.00102277\pi$
$368$		−	4.88335e6i		−	0.0979884i
$369$	1.89267e7			0.376701
$370$		−	1.52801e7i		−	0.301663i
$371$			6.55297e7i			1.28327i
$372$	−1.17077e8	−	5.51340e7i	−2.27427	−	1.07100i
$373$	−9.81348e7			−1.89102			−0.945511	−	0.325590i	$-0.894437\pi$
$373$	−9.81348e7			−1.89102			−0.945511	+	0.325590i	$0.894437\pi$
$374$	1.06339e8			2.03273
$375$			7.32465e7i			1.38897i
$376$	−4.95469e7			−0.932079
$377$	−5.20707e7			−0.971783
$378$			1.22568e7i			0.226935i
$379$	−3.41768e7			−0.627789			−0.313895	−	0.949458i	$-0.601634\pi$
$379$	−3.41768e7			−0.627789			−0.313895	+	0.949458i	$0.601634\pi$
$380$	−1.08714e8			−1.98123
$381$	−6.04929e7			−1.09378
$382$	−1.24171e8			−2.22756
$383$			4.31721e7i			0.768435i	0.923243	+	0.384218i	$0.125529\pi$
$383$			4.31721e7i			0.768435i	−0.923243	+	0.384218i	$0.874471\pi$
$384$		−	1.42181e8i		−	2.51100i
$385$		−	1.94172e7i		−	0.340254i
$386$	9.22989e7			1.60485
$387$		−	4.76092e7i		−	0.821406i
$388$	1.35793e8			2.32479
$389$		−	6.64150e7i		−	1.12828i	−0.825679	−	0.564140i	$-0.809208\pi$
$389$		−	6.64150e7i		−	1.12828i	0.825679	−	0.564140i	$-0.190792\pi$
$390$	−5.81515e7			−0.980318
$391$	−2.04548e7			−0.342187
$392$	−2.77432e7			−0.460572
$393$			3.95482e7i			0.651552i
$394$		−	1.34960e8i		−	2.20656i
$395$		−	4.76212e7i		−	0.772697i
$396$		−	6.56303e7i		−	1.05686i
$397$	−1.04736e7			−0.167388			−0.0836939	−	0.996492i	$-0.526672\pi$
$397$	−1.04736e7			−0.167388			−0.0836939	+	0.996492i	$0.526672\pi$
$398$		−	1.86051e8i		−	2.95109i
$399$			1.21705e8i			1.91597i
$400$	−2.02145e7			−0.315852
$401$			1.99007e7i			0.308627i	0.988022	+	0.154314i	$0.0493166\pi$
$401$			1.99007e7i			0.308627i	−0.988022	+	0.154314i	$0.950683\pi$
$402$		−	7.61581e7i		−	1.17230i
$403$	3.96825e7	+	1.86874e7i	0.606295	+	0.285518i
$404$	−1.92917e8			−2.92567
$405$	4.65489e7			0.700720
$406$		−	1.33876e8i		−	2.00044i
$407$	−1.24873e7			−0.185218
$408$	2.40835e8			3.54601
$409$			5.67061e7i			0.828819i	0.910090	+	0.414410i	$0.136012\pi$
$409$			5.67061e7i			0.828819i	−0.910090	+	0.414410i	$0.863988\pi$
$410$	3.14745e7			0.456676
$411$	−5.62401e6			−0.0810067
$412$	−8.49721e7			−1.21502
$413$	6.98830e7			0.992022
$414$			1.95099e7i			0.274950i
$415$			7.71277e7i			1.07911i
$416$			2.48070e7i			0.344583i
$417$	−2.62235e7			−0.361645
$418$			1.37302e8i			1.87996i
$419$	1.09740e8			1.49184			0.745918	−	0.666038i	$-0.232011\pi$
$419$	1.09740e8			1.49184			0.745918	+	0.666038i	$0.232011\pi$
$420$		−	9.67429e7i		−	1.30578i
$421$	4.48395e7			0.600917			0.300459	−	0.953795i	$-0.402860\pi$
$421$	4.48395e7			0.600917			0.300459	+	0.953795i	$0.402860\pi$
$422$	6.87011e6			0.0914169
$423$	4.42558e7			0.584721
$424$			1.67430e8i			2.19652i
$425$			8.46720e7i			1.10299i
$426$			9.01313e7i			1.16586i
$427$		−	2.31691e7i		−	0.297595i
$428$	−2.11578e8			−2.69861
$429$			4.75226e7i			0.601906i
$430$		−	7.91726e7i		−	0.995794i
$431$	7.06269e7			0.882141			0.441071	−	0.897472i	$-0.354599\pi$
$431$	7.06269e7			0.882141			0.441071	+	0.897472i	$0.354599\pi$
$432$			7.00144e6i			0.0868432i
$433$		−	1.10210e8i		−	1.35756i	−0.734341	−	0.678780i	$-0.762509\pi$
$433$		−	1.10210e8i		−	1.35756i	0.734341	−	0.678780i	$-0.237491\pi$
$434$	−4.80461e7	+	1.02026e8i	−0.587745	+	1.24807i
$435$	−1.03728e8			−1.26016
$436$	2.14541e6			0.0258851
$437$		−	2.64105e7i		−	0.316469i
$438$	2.14409e8			2.55165
$439$	8.57473e7			1.01351			0.506754	−	0.862091i	$-0.330845\pi$
$439$	8.57473e7			1.01351			0.506754	+	0.862091i	$0.330845\pi$
$440$		−	4.96113e7i		−	0.582401i
$441$	2.47805e7			0.288931
$442$	−1.79579e8			−2.07965
$443$	−5.49974e7			−0.632602			−0.316301	−	0.948659i	$-0.602441\pi$
$443$	−5.49974e7			−0.632602			−0.316301	+	0.948659i	$0.602441\pi$
$444$	−6.22157e7			−0.710806
$445$			1.36892e7i			0.155345i
$446$			1.11713e8i			1.25922i
$447$			4.91209e7i			0.549976i
$448$	−1.02683e8			−1.14200
$449$		−	3.78986e7i		−	0.418682i	−0.977843	−	0.209341i	$-0.932868\pi$
$449$		−	3.78986e7i		−	0.418682i	0.977843	−	0.209341i	$-0.0671319\pi$
$450$	8.07609e7			0.886265
$451$		−	2.57217e7i		−	0.280394i
$452$	2.06841e8			2.23986
$453$	−8.01714e7			−0.862432
$454$	1.86663e8			1.99476
$455$			3.27905e7i			0.348107i
$456$			3.10958e8i			3.27950i
$457$		−	7.79215e7i		−	0.816411i	−0.912890	−	0.408206i	$-0.866155\pi$
$457$		−	7.79215e7i		−	0.816411i	0.912890	−	0.408206i	$-0.133845\pi$
$458$			2.59068e8i			2.69661i
$459$	2.93267e7			0.303267
$460$			2.09936e7i			0.215682i
$461$		−	1.69250e8i		−	1.72753i	−0.503894	−	0.863765i	$-0.668100\pi$
$461$		−	1.69250e8i		−	1.72753i	0.503894	−	0.863765i	$-0.331900\pi$
$462$	−1.22183e8			−1.23904
$463$		−	9.40006e7i		−	0.947082i	−0.880772	−	0.473541i	$-0.842976\pi$
$463$		−	9.40006e7i		−	0.947082i	0.880772	−	0.473541i	$-0.157024\pi$
$464$		−	7.64740e7i		−	0.765526i
$465$	7.90498e7	+	3.72263e7i	0.786216	+	0.370246i
$466$	1.39962e7			0.138310
$467$	3.28395e7			0.322437			0.161219	−	0.986919i	$-0.448458\pi$
$467$	3.28395e7			0.322437			0.161219	+	0.986919i	$0.448458\pi$
$468$			1.10832e8i			1.08125i
$469$	−4.29440e7			−0.416279
$470$	7.35959e7			0.708860
$471$		−	5.92273e7i		−	0.566839i
$472$	1.78552e8			1.69801
$473$	−6.47015e7			−0.611408
$474$	−2.99657e8			−2.81378
$475$	−1.09325e8			−1.02009
$476$		−	2.98754e8i		−	2.77009i
$477$		−	1.49550e8i		−	1.37794i
$478$			1.15667e8i			1.05908i
$479$	−2.36154e7			−0.214877			−0.107438	−	0.994212i	$-0.534265\pi$
$479$	−2.36154e7			−0.214877			−0.107438	+	0.994212i	$0.534265\pi$
$480$			4.94170e7i			0.446840i
$481$	2.10877e7			0.189493
$482$			2.97193e8i			2.65397i
$483$	2.35023e7			0.208578
$484$	1.18677e8			1.04672
$485$	−9.16872e7			−0.803680
$486$		−	2.61124e8i		−	2.27477i
$487$		−	9.78087e7i		−	0.846819i	−0.905938	−	0.423410i	$-0.860833\pi$
$487$		−	9.78087e7i		−	0.846819i	0.905938	−	0.423410i	$-0.139167\pi$
$488$		−	5.91975e7i		−	0.509382i
$489$			3.65898e6i			0.0312920i
$490$	4.12091e7			0.350272
$491$			9.95424e7i			0.840937i	0.907307	+	0.420469i	$0.138134\pi$
$491$			9.95424e7i			0.840937i	−0.907307	+	0.420469i	$0.861866\pi$
$492$		−	1.28154e8i		−	1.07606i
$493$	−3.20324e8			−2.67331
$494$		−	2.31866e8i		−	1.92334i
$495$			4.43133e7i			0.365358i
$496$	−2.74453e7	+	5.82800e7i	−0.224918	+	0.477611i
$497$	5.08232e7			0.413993
$498$	4.85328e8			3.92959
$499$		−	5.67673e7i		−	0.456875i	−0.973559	−	0.228437i	$-0.926638\pi$
$499$		−	5.67673e7i		−	0.456875i	0.973559	−	0.228437i	$-0.0733616\pi$
$500$	2.32153e8			1.85723
$501$	−5.36740e7			−0.426826
$502$		−	4.19075e8i		−	3.31269i
$503$	2.38069e7			0.187068			0.0935338	−	0.995616i	$-0.470184\pi$
$503$	2.38069e7			0.187068			0.0935338	+	0.995616i	$0.470184\pi$
$504$	−1.29529e8			−1.01176
$505$	1.30257e8			1.01141
$506$	2.65142e7			0.204657
$507$			9.84392e7i			0.755343i
$508$			1.91731e8i			1.46252i
$509$		−	6.75845e7i		−	0.512500i	−0.966611	−	0.256250i	$-0.917513\pi$
$509$		−	6.75845e7i		−	0.512500i	0.966611	−	0.256250i	$-0.0824870\pi$
$510$	−3.57732e8			−2.69679
$511$		−	1.20901e8i		−	0.906083i
$512$	−1.35831e8			−1.01202
$513$			3.78656e7i			0.280474i
$514$	−2.11765e8			−1.55943
$515$	5.73729e7			0.420034
$516$	−3.22365e8			−2.34638
$517$		−	6.01442e7i		−	0.435233i
$518$			5.42174e7i			0.390076i
$519$			1.60714e8i			1.14961i
$520$			8.37803e7i			0.595843i
$521$	−2.66667e8			−1.88563			−0.942813	−	0.333321i	$-0.891831\pi$
$521$	−2.66667e8			−1.88563			−0.942813	+	0.333321i	$0.891831\pi$
$522$			3.05528e8i			2.14803i
$523$		−	6.93321e7i		−	0.484651i	−0.970195	−	0.242326i	$-0.922090\pi$
$523$		−	6.93321e7i		−	0.484651i	0.970195	−	0.242326i	$-0.0779102\pi$
$524$	1.25347e8			0.871206
$525$		−	9.72872e7i		−	0.672323i
$526$			2.57882e8i			1.77200i
$527$	2.44116e8	+	1.14959e8i	1.66788	+	0.785440i
$528$	−6.97945e7			−0.474154
$529$	1.42936e8			0.965548
$530$		−	2.48697e8i		−	1.67048i
$531$	−1.59485e8			−1.06521
$532$	3.85741e8			2.56189
$533$			4.34371e7i			0.286866i
$534$	8.61396e7			0.565691
$535$	1.42857e8			0.932911
$536$	−1.09723e8			−0.712529
$537$	−9.70065e7			−0.626438
$538$			2.45304e7i			0.157528i
$539$		−	3.36770e7i		−	0.215064i
$540$		−	3.00993e7i		−	0.191151i
$541$	−1.38080e8			−0.872048			−0.436024	−	0.899935i	$-0.643614\pi$
$541$	−1.38080e8			−0.872048			−0.436024	+	0.899935i	$0.643614\pi$
$542$			3.10686e8i			1.95130i
$543$	3.68051e7			0.229884
$544$			1.52606e8i			0.947926i
$545$	−1.44857e6			−0.00894851
$546$	2.06335e8			1.26763
$547$	1.62204e8			0.991061			0.495530	−	0.868591i	$-0.334974\pi$
$547$	1.62204e8			0.991061			0.495530	+	0.868591i	$0.334974\pi$
$548$			1.78252e7i			0.108316i
$549$			5.28758e7i			0.319551i
$550$		−	1.09755e8i		−	0.659684i
$551$		−	4.13592e8i		−	2.47239i
$552$	6.00488e7			0.357016
$553$			1.68971e8i			0.999162i
$554$		−	4.73844e7i		−	0.278680i
$555$	4.20078e7			0.245726
$556$			8.31148e7i			0.483564i
$557$		−	1.89356e8i		−	1.09576i	−0.836558	−	0.547878i	$-0.815436\pi$
$557$		−	1.89356e8i		−	1.09576i	0.836558	−	0.547878i	$-0.184564\pi$
$558$	1.09649e8	−	2.32840e8i	0.631108	−	1.34015i
$559$	1.09264e8			0.625519
$560$	−4.81579e7			−0.274223
$561$			2.92346e8i			1.65580i
$562$	−5.45672e7			−0.307414
$563$	−1.03763e8			−0.581459			−0.290729	−	0.956805i	$-0.593898\pi$
$563$	−1.03763e8			−0.581459			−0.290729	+	0.956805i	$0.593898\pi$
$564$		−	2.99659e8i		−	1.67028i
$565$	−1.39658e8			−0.774320
$566$	1.60541e8			0.885392
$567$	−1.65166e8			−0.906090
$568$	1.29854e8			0.708617
$569$			1.02000e8i			0.553686i	0.960915	+	0.276843i	$0.0892882\pi$
$569$			1.02000e8i			0.553686i	−0.960915	+	0.276843i	$0.910712\pi$
$570$		−	4.61891e8i		−	2.49411i
$571$			9.68561e7i			0.520258i	0.965574	+	0.260129i	$0.0837651\pi$
$571$			9.68561e7i			0.520258i	−0.965574	+	0.260129i	$0.916235\pi$
$572$	1.50622e8			0.804824
$573$		−	3.41368e8i		−	1.81451i
$574$	−1.11679e8			−0.590520
$575$			2.11117e7i			0.111050i
$576$	2.34340e8			1.22625
$577$	8.81237e7			0.458739			0.229369	−	0.973339i	$-0.426334\pi$
$577$	8.81237e7			0.458739			0.229369	+	0.973339i	$0.426334\pi$
$578$	−7.79682e8			−4.03770
$579$			2.53746e8i			1.30727i
$580$			3.28763e8i			1.68500i
$581$		−	2.73666e8i		−	1.39538i
$582$			5.76943e8i			2.92661i
$583$	−2.03240e8			−1.02566
$584$		−	3.08905e8i		−	1.55091i
$585$		−	7.48334e7i		−	0.373790i
$586$	−4.36983e7			−0.217156
$587$			3.61838e8i			1.78896i	0.447112	+	0.894478i	$0.352453\pi$
$587$			3.61838e8i			1.78896i	−0.447112	+	0.894478i	$0.647547\pi$
$588$		−	1.67790e8i		−	0.825343i
$589$	−1.48432e8	+	3.15194e8i	−0.726408	+	1.54252i
$590$	−2.65218e8			−1.29136
$591$	3.71028e8			1.79740
$592$			3.09706e7i			0.149274i
$593$	−2.88359e8			−1.38283			−0.691417	−	0.722456i	$-0.743013\pi$
$593$	−2.88359e8			−1.38283			−0.691417	+	0.722456i	$0.743013\pi$
$594$	−3.80144e7			−0.181380
$595$			2.01718e8i			0.957620i
$596$	1.55688e8			0.735386
$597$	5.11487e8			2.40387
$598$	−4.47755e7			−0.209381
$599$	−4.83307e7			−0.224876			−0.112438	−	0.993659i	$-0.535866\pi$
$599$	−4.83307e7			−0.224876			−0.112438	+	0.993659i	$0.535866\pi$
$600$		−	2.48571e8i		−	1.15079i
$601$			1.46400e8i			0.674402i	0.941433	+	0.337201i	$0.109480\pi$
$601$			1.46400e8i			0.674402i	−0.941433	+	0.337201i	$0.890520\pi$
$602$			2.80922e8i			1.28765i
$603$	9.80056e7			0.446991
$604$			2.54102e8i			1.15318i
$605$	−8.01302e7			−0.361851
$606$		−	8.19642e8i		−	3.68304i
$607$	1.18744e8			0.530942			0.265471	−	0.964119i	$-0.414473\pi$
$607$	1.18744e8			0.530942			0.265471	+	0.964119i	$0.414473\pi$
$608$	−1.97039e8			−0.876682
$609$	3.68049e8			1.62950
$610$			8.79307e7i			0.387392i
$611$			1.01568e8i			0.445278i
$612$			6.81808e8i			2.97446i
$613$		−	1.77223e8i		−	0.769377i	−0.923047	−	0.384688i	$-0.874309\pi$
$613$		−	1.77223e8i		−	0.769377i	0.923047	−	0.384688i	$-0.125691\pi$
$614$	4.97957e8			2.15123
$615$			8.65291e7i			0.371995i
$616$			1.76032e8i			0.753094i
$617$	2.72434e8			1.15986			0.579930	−	0.814666i	$-0.303080\pi$
$617$	2.72434e8			1.15986			0.579930	+	0.814666i	$0.303080\pi$
$618$		−	3.61020e8i		−	1.52956i
$619$		−	1.26366e8i		−	0.532791i	−0.963864	−	0.266395i	$-0.914167\pi$
$619$		−	1.26366e8i		−	0.532791i	0.963864	−	0.266395i	$-0.0858327\pi$
$620$	1.17988e8	−	2.50547e8i	0.495066	−	1.05127i
$621$	7.31219e6			0.0305332
$622$	−4.98933e8			−2.07334
$623$		−	4.85724e7i		−	0.200875i
$624$	1.17864e8			0.485097
$625$	−1.06812e7			−0.0437502
$626$			6.90828e8i			2.81609i
$627$	−3.77467e8			−1.53136
$628$	−1.87720e8			−0.757934
$629$	1.29726e8			0.521283
$630$	1.92400e8			0.769456
$631$			2.64442e7i			0.105255i	0.998614	+	0.0526274i	$0.0167596\pi$
$631$			2.64442e7i			0.105255i	−0.998614	+	0.0526274i	$0.983240\pi$
$632$			4.31724e8i			1.71023i
$633$			1.88872e7i			0.0744655i
$634$	2.44168e8			0.958121
$635$		−	1.29456e8i		−	0.505594i
$636$	−1.01261e9			−3.93615
$637$			5.68715e7i			0.220027i
$638$	4.15217e8			1.59887
$639$	−1.15987e8			−0.444537
$640$	3.04270e8			1.16070
$641$			1.54497e8i			0.586607i	0.956019	+	0.293303i	$0.0947546\pi$
$641$			1.54497e8i			0.586607i	−0.956019	+	0.293303i	$0.905245\pi$
$642$		−	8.98930e8i		−	3.39720i
$643$		−	4.60112e8i		−	1.73074i	−0.501136	−	0.865368i	$-0.667084\pi$
$643$		−	4.60112e8i		−	1.73074i	0.501136	−	0.865368i	$-0.332916\pi$
$644$		−	7.44901e7i		−	0.278895i
$645$	2.17659e8			0.811145
$646$		−	1.42638e9i		−	5.29099i
$647$			2.25487e8i			0.832547i	0.909239	+	0.416274i	$0.136664\pi$
$647$			2.25487e8i			0.832547i	−0.909239	+	0.416274i	$0.863336\pi$
$648$	−4.22002e8			−1.55092
$649$			2.16742e8i			0.792882i
$650$			1.85347e8i			0.674910i
$651$	−2.80486e8	−	1.32087e8i	−1.01664	−	0.478760i
$652$	1.15971e7			0.0418414
$653$	3.15330e8			1.13247			0.566235	−	0.824244i	$-0.308400\pi$
$653$	3.15330e8			1.13247			0.566235	+	0.824244i	$0.308400\pi$
$654$			9.11517e6i			0.0325860i
$655$	−8.46340e7			−0.301176
$656$	−6.37942e7			−0.225980
$657$			2.75917e8i			0.972932i
$658$	−2.61135e8			−0.916615
$659$	−4.05029e7			−0.141524			−0.0707619	−	0.997493i	$-0.522543\pi$
$659$	−4.05029e7			−0.141524			−0.0707619	+	0.997493i	$0.522543\pi$
$660$	3.00048e8			1.04366
$661$	4.74397e7			0.164262			0.0821311	−	0.996622i	$-0.473827\pi$
$661$	4.74397e7			0.164262			0.0821311	+	0.996622i	$0.473827\pi$
$662$			4.08180e8i			1.40695i
$663$		−	4.93695e8i		−	1.69402i
$664$		−	6.99223e8i		−	2.38843i
$665$	−2.60451e8			−0.885648
$666$		−	1.23733e8i		−	0.418855i
$667$	−7.98683e7			−0.269152
$668$			1.70119e8i			0.570719i
$669$	−3.07120e8			−1.02572
$670$	1.62980e8			0.541889
$671$	7.18588e7			0.237855
$672$		−	1.75342e8i		−	0.577802i
$673$		−	4.53411e8i		−	1.48746i	−0.668478	−	0.743732i	$-0.733054\pi$
$673$		−	4.53411e8i		−	1.48746i	0.668478	−	0.743732i	$-0.266946\pi$
$674$		−	2.86383e8i		−	0.935336i
$675$		−	3.02687e7i		−	0.0984197i
$676$	3.12001e8			1.00999
$677$			1.31924e8i			0.425167i	0.977143	+	0.212583i	$0.0681877\pi$
$677$			1.31924e8i			0.425167i	−0.977143	+	0.212583i	$0.931812\pi$
$678$			8.78801e8i			2.81969i
$679$	3.25327e8			1.03923
$680$			5.15393e8i			1.63912i
$681$			5.13170e8i			1.62488i
$682$	−3.16432e8	−	1.49015e8i	−0.997533	−	0.469760i
$683$	1.74520e8			0.547751			0.273876	−	0.961765i	$-0.411694\pi$
$683$	1.74520e8			0.547751			0.273876	+	0.961765i	$0.411694\pi$
$684$	−8.80327e8			−2.75091
$685$		−	1.20355e7i		−	0.0374449i
$686$	−5.91574e8			−1.83247
$687$	−7.12224e8			−2.19658
$688$			1.60471e8i			0.492755i
$689$	3.43219e8			1.04933
$690$	−8.91953e7			−0.271516
$691$	−1.22138e8			−0.370184			−0.185092	−	0.982721i	$-0.559258\pi$
$691$	−1.22138e8			−0.370184			−0.185092	+	0.982721i	$0.559258\pi$
$692$	5.09379e8			1.53717
$693$		−	1.57234e8i		−	0.472439i
$694$			4.80879e7i			0.143866i
$695$		−	5.61188e7i		−	0.167168i
$696$	9.40374e8			2.78916
$697$			2.67213e8i			0.789149i
$698$	9.34886e8			2.74911
$699$			3.84781e7i			0.112663i
$700$	−3.08350e8			−0.898980
$701$	−2.91548e8			−0.846363			−0.423181	−	0.906045i	$-0.639087\pi$
$701$	−2.91548e8			−0.846363			−0.423181	+	0.906045i	$0.639087\pi$
$702$	6.41962e7			0.185566
$703$			1.67497e8i			0.482105i
$704$		−	3.18471e8i		−	0.912751i
$705$			2.02328e8i			0.577417i
$706$		−	4.78905e8i		−	1.36093i
$707$	−4.62180e8			−1.30783
$708$			1.07988e9i			3.04282i
$709$		−	1.42479e8i		−	0.399771i	−0.979819	−	0.199886i	$-0.935943\pi$
$709$		−	1.42479e8i		−	0.399771i	0.979819	−	0.199886i	$-0.0640571\pi$
$710$	−1.92883e8			−0.538913
$711$		−	3.85620e8i		−	1.07288i
$712$		−	1.24103e8i		−	0.343830i
$713$	6.08667e7	+	2.86635e7i	0.167923	+	0.0790789i
$714$	1.26931e9			3.48718
$715$	−1.01700e8			−0.278228
$716$			3.07460e8i			0.837626i
$717$	−3.17990e8			−0.862692
$718$	−5.00554e8			−1.35231
$719$		−	6.09122e8i		−	1.63877i	−0.573246	−	0.819384i	$-0.694316\pi$
$719$		−	6.09122e8i		−	1.63877i	0.573246	−	0.819384i	$-0.305684\pi$
$720$	1.09905e8			0.294455
$721$	−2.03572e8			−0.543140
$722$	1.20816e9			3.21007
$723$	−8.17035e8			−2.16185
$724$		−	1.16653e8i		−	0.307383i
$725$			3.30613e8i			0.867573i
$726$			5.04221e8i			1.31768i
$727$	6.85888e8			1.78505			0.892524	−	0.451000i	$-0.148933\pi$
$727$	6.85888e8			1.78505			0.892524	+	0.451000i	$0.148933\pi$
$728$		−	2.97271e8i		−	0.770475i
$729$	2.89553e8			0.747386
$730$			4.58841e8i			1.17949i
$731$	6.72160e8			1.72076
$732$	3.58025e8			0.912809
$733$	3.55965e8			0.903849			0.451924	−	0.892056i	$-0.350738\pi$
$733$	3.55965e8			0.903849			0.451924	+	0.892056i	$0.350738\pi$
$734$		−	4.27758e6i		−	0.0108171i
$735$			1.13291e8i			0.285321i
$736$			3.80501e7i			0.0954381i
$737$		−	1.33191e8i		−	0.332715i
$738$	2.54870e8			0.634088
$739$			3.43102e8i			0.850138i	0.905161	+	0.425069i	$0.139750\pi$
$739$			3.43102e8i			0.850138i	−0.905161	+	0.425069i	$0.860250\pi$
$740$		−	1.33143e8i		−	0.328567i
$741$	6.37442e8			1.56670
$742$			8.82432e8i			2.16008i
$743$			4.66125e8i			1.13641i	0.822886	+	0.568207i	$0.192363\pi$
$743$			4.66125e8i			1.13641i	−0.822886	+	0.568207i	$0.807637\pi$
$744$	−7.16648e8	−	3.37485e8i	−1.74015	−	0.819476i
$745$	−1.05120e8			−0.254223
$746$	−1.32150e9			−3.18309
$747$			6.24553e8i			1.49833i
$748$	9.26586e8			2.21402
$749$	−5.06889e8			−1.20633
$750$			9.86347e8i			2.33801i
$751$	−6.58688e8			−1.55511			−0.777553	−	0.628817i	$-0.783539\pi$
$751$	−6.58688e8			−1.55511			−0.777553	+	0.628817i	$0.783539\pi$
$752$	−1.49168e8			−0.350770
$753$	1.15211e9			2.69842
$754$	−7.01191e8			−1.63577
$755$		−	1.71569e8i		−	0.398655i
$756$			1.06799e8i			0.247174i
$757$		−	8.19859e8i		−	1.88996i	−0.327134	−	0.944978i	$-0.606083\pi$
$757$		−	8.19859e8i		−	1.88996i	0.327134	−	0.944978i	$-0.393917\pi$
$758$	−4.60229e8			−1.05674
$759$			7.28923e7i			0.166708i
$760$	−6.65458e8			−1.51593
$761$			3.30476e8i			0.749869i	0.927051	+	0.374935i	$0.122335\pi$
$761$			3.30476e8i			0.749869i	−0.927051	+	0.374935i	$0.877665\pi$
$762$	−8.14606e8			−1.84112
$763$	5.13986e6			0.0115712
$764$	−1.08196e9			−2.42622
$765$		−	4.60355e8i		−	1.02827i
$766$			5.81362e8i			1.29348i
$767$		−	3.66020e8i		−	0.811182i
$768$		−	1.04916e9i		−	2.31611i
$769$	6.02370e8			1.32460			0.662299	−	0.749240i	$-0.269581\pi$
$769$	6.02370e8			1.32460			0.662299	+	0.749240i	$0.269581\pi$
$770$		−	2.61474e8i		−	0.572739i
$771$		−	5.82180e8i		−	1.27026i
$772$	8.04244e8			1.74798
$773$			4.69335e7i			0.101612i	0.998709	+	0.0508059i	$0.0161790\pi$
$773$			4.69335e7i			0.101612i	−0.998709	+	0.0508059i	$0.983821\pi$
$774$		−	6.41112e8i		−	1.38265i
$775$	1.18652e8	−	2.51956e8i	0.254900	−	0.541278i
$776$	8.31216e8			1.77881
$777$	−1.49053e8			−0.317745
$778$		−	8.94353e8i		−	1.89920i
$779$	−3.45016e8			−0.729838
$780$	−5.06701e8			−1.06775
$781$			1.57628e8i			0.330888i
$782$	−2.75446e8			−0.575993
$783$	1.14510e8			0.238538
$784$	−8.35247e7			−0.173327
$785$	1.26748e8			0.262018
$786$			5.32561e8i			1.09674i
$787$		−	9.16247e7i		−	0.187970i	−0.995574	−	0.0939850i	$-0.970039\pi$
$787$		−	9.16247e7i		−	0.187970i	0.995574	−	0.0939850i	$-0.0299606\pi$
$788$		−	1.17597e9i		−	2.40335i
$789$	−7.08963e8			−1.44342
$790$		−	6.41273e8i		−	1.30065i
$791$	4.95538e8			1.00126
$792$		−	4.01735e8i		−	0.808656i
$793$	−1.21351e8			−0.243345
$794$	−1.41039e8			−0.281758
$795$	6.83712e8			1.36073
$796$		−	1.62115e9i		−	3.21428i
$797$		−	3.87232e7i		−	0.0764885i	−0.999268	−	0.0382443i	$-0.987824\pi$
$797$		−	3.87232e7i		−	0.0764885i	0.999268	−	0.0382443i	$-0.0121765\pi$
$798$			1.63889e9i			3.22509i
$799$			6.24815e8i			1.22493i
$800$	1.57507e8			0.307631
$801$			1.10850e8i			0.215695i
$802$			2.67985e8i			0.519502i
$803$	3.74975e8			0.724195
$804$		−	6.63602e8i		−	1.27685i
$805$			5.02954e7i			0.0964142i
$806$	5.34370e8	+	2.51646e8i	1.02056	+	0.480602i
$807$	−6.74385e7			−0.128318
$808$	−1.18088e9			−2.23857
$809$		−	6.09625e8i		−	1.15138i	−0.817669	−	0.575688i	$-0.804734\pi$
$809$		−	6.09625e8i		−	1.15138i	0.817669	−	0.575688i	$-0.195266\pi$
$810$	6.26834e8			1.17950
$811$	−5.59001e8			−1.04797			−0.523987	−	0.851726i	$-0.675556\pi$
$811$	−5.59001e8			−1.04797			−0.523987	+	0.851726i	$0.675556\pi$
$812$		−	1.16653e9i		−	2.17884i
$813$	−8.54132e8			−1.58947
$814$	−1.68155e8			−0.311772
$815$	−7.83031e6			−0.0144646
$816$	7.25069e8			1.33447
$817$			8.67870e8i			1.59143i
$818$			7.63612e8i			1.39512i
$819$			2.65526e8i			0.483342i
$820$	2.74252e8			0.497404
$821$		−	5.51307e7i		−	0.0996241i	−0.998759	−	0.0498121i	$-0.984138\pi$
$821$		−	5.51307e7i		−	0.0996241i	0.998759	−	0.0498121i	$-0.0158622\pi$
$822$	−7.57337e7			−0.136356
$823$		−	5.26309e8i		−	0.944150i	−0.881558	−	0.472075i	$-0.843505\pi$
$823$		−	5.26309e8i		−	0.944150i	0.881558	−	0.472075i	$-0.156495\pi$
$824$	−5.20130e8			−0.929673
$825$	3.01736e8			0.537360
$826$	9.41053e8			1.66984
$827$		−	2.63685e8i		−	0.466196i	−0.972453	−	0.233098i	$-0.925114\pi$
$827$		−	2.63685e8i		−	0.466196i	0.972453	−	0.233098i	$-0.0748864\pi$
$828$			1.69999e8i			0.299472i
$829$			9.18588e8i			1.61234i	0.591682	+	0.806171i	$0.298464\pi$
$829$			9.18588e8i			1.61234i	−0.591682	+	0.806171i	$0.701536\pi$
$830$			1.03861e9i			1.81643i
$831$	1.30268e8			0.227005
$832$			5.37813e8i			0.933817i
$833$			3.49858e8i			0.605280i
$834$	−3.53129e8			−0.608744
$835$		−	1.14863e8i		−	0.197298i
$836$			1.19638e9i			2.04762i
$837$	−8.72668e7	−	4.10959e7i	−0.148824	−	0.0700845i
$838$	1.47777e9			2.51116
$839$	−9.01102e8			−1.52577			−0.762884	−	0.646536i	$-0.776217\pi$
$839$	−9.01102e8			−1.52577			−0.762884	+	0.646536i	$0.776217\pi$
$840$		−	5.92181e8i		−	0.999118i
$841$	−6.55926e8			−1.10272
$842$	6.03815e8			1.01150
$843$		−	1.50015e8i		−	0.250410i
$844$	5.98625e7			0.0995698
$845$	−2.10662e8			−0.349153
$846$	5.95954e8			0.984242
$847$	2.84320e8			0.467904
$848$			5.04071e8i			0.826617i
$849$			4.41355e8i			0.721215i
$850$			1.14020e9i			1.85663i
$851$	3.23452e7			0.0524833
$852$			7.85356e8i			1.26984i
$853$	4.06635e8			0.655175			0.327588	−	0.944821i	$-0.393764\pi$
$853$	4.06635e8			0.655175			0.327588	+	0.944821i	$0.393764\pi$
$854$		−	3.11998e8i		−	0.500931i
$855$	5.94394e8			0.950990
$856$	−1.29511e9			−2.06484
$857$	−2.18408e8			−0.346998			−0.173499	−	0.984834i	$-0.555507\pi$
$857$	−2.18408e8			−0.346998			−0.173499	+	0.984834i	$0.555507\pi$
$858$			6.39946e8i			1.01317i
$859$			6.47929e8i			1.02223i	0.859513	+	0.511114i	$0.170767\pi$
$859$			6.47929e8i			1.02223i	−0.859513	+	0.511114i	$0.829233\pi$
$860$		−	6.89868e8i		−	1.08460i
$861$		−	3.07025e8i		−	0.481021i
$862$	9.51071e8			1.48488
$863$		−	9.84008e8i		−	1.53097i	−0.643455	−	0.765484i	$-0.722499\pi$
$863$		−	9.84008e8i		−	1.53097i	0.643455	−	0.765484i	$-0.277501\pi$
$864$		−	5.45537e7i		−	0.0845830i
$865$	−3.43931e8			−0.531402
$866$		−	1.48411e9i		−	2.28514i
$867$		−	2.14348e9i		−	3.28899i
$868$	−4.18648e8	+	8.88996e8i	−0.640162	+	1.35938i
$869$	−5.24062e8			−0.798589
$870$	−1.39681e9			−2.12119
$871$			2.24924e8i			0.340393i
$872$	1.31324e7			0.0198060
$873$	−7.42451e8			−1.11590
$874$		−	3.55647e8i		−	0.532702i
$875$	5.56181e8			0.830217
$876$	1.86825e9			2.77922
$877$	−7.18675e8			−1.06545			−0.532726	−	0.846288i	$-0.678832\pi$
$877$	−7.18675e8			−1.06545			−0.532726	+	0.846288i	$0.678832\pi$
$878$	1.15468e9			1.70600
$879$		−	1.20134e8i		−	0.176889i
$880$		−	1.49362e8i		−	0.219175i
$881$			3.72631e8i			0.544944i	0.962164	+	0.272472i	$0.0878411\pi$
$881$			3.72631e8i			0.544944i	−0.962164	+	0.272472i	$0.912159\pi$
$882$	3.33697e8			0.486348
$883$		−	1.14811e8i		−	0.166764i	−0.996518	−	0.0833821i	$-0.973428\pi$
$883$		−	1.14811e8i		−	0.166764i	0.996518	−	0.0833821i	$-0.0265722\pi$
$884$	−1.56476e9			−2.26512
$885$		−	7.29132e8i		−	1.05190i
$886$	−7.40602e8			−1.06484
$887$	−5.26691e8			−0.754719			−0.377360	−	0.926067i	$-0.623168\pi$
$887$	−5.26691e8			−0.754719			−0.377360	+	0.926067i	$0.623168\pi$
$888$	−3.80834e8			−0.543872
$889$			4.59340e8i			0.653775i
$890$			1.84341e8i			0.261488i
$891$		−	5.12262e8i		−	0.724200i
$892$			9.73410e8i			1.37152i
$893$	−8.06740e8			−1.13287
$894$			6.61468e8i			0.925756i
$895$		−	2.07596e8i		−	0.289568i
$896$	−1.07962e9			−1.50088
$897$		−	1.23096e8i		−	0.170556i
$898$		−	5.10348e8i		−	0.704753i
$899$	9.53182e8	+	4.48874e8i	1.31189	+	0.617797i
$900$	7.03707e8			0.965305
$901$	2.11139e9			2.88665
$902$		−	3.46371e8i		−	0.471979i
$903$	−7.72305e8			−1.04888
$904$	1.26611e9			1.71382
$905$			7.87637e7i			0.106263i
$906$	−1.07960e9			−1.45170
$907$	3.14153e8			0.421037			0.210518	−	0.977590i	$-0.432485\pi$
$907$	3.14153e8			0.421037			0.210518	+	0.977590i	$0.432485\pi$
$908$	1.62648e9			2.17266
$909$	1.05477e9			1.40432
$910$			4.41561e8i			0.585958i
$911$			3.98182e8i			0.526655i	0.964706	+	0.263328i	$0.0848200\pi$
$911$			3.98182e8i			0.526655i	−0.964706	+	0.263328i	$0.915180\pi$
$912$			9.36184e8i			1.23417i
$913$	8.48776e8			1.11527
$914$		−	1.04930e9i		−	1.37424i
$915$	−2.41737e8			−0.315559
$916$			2.25738e9i			2.93710i
$917$	3.00300e8			0.389447
$918$	3.94917e8			0.510480
$919$	2.46305e8			0.317342			0.158671	−	0.987332i	$-0.449279\pi$
$919$	2.46305e8			0.317342			0.158671	+	0.987332i	$0.449279\pi$
$920$			1.28506e8i			0.165029i
$921$			1.36897e9i			1.75233i
$922$		−	2.27914e9i		−	2.90790i
$923$		−	2.66192e8i		−	0.338524i
$924$	−1.06464e9			−1.34954
$925$		−	1.33892e8i		−	0.169173i
$926$		−	1.26582e9i		−	1.59419i
$927$	4.64585e8			0.583212
$928$			5.95870e8i			0.745602i
$929$		−	2.93178e8i		−	0.365666i	−0.983144	−	0.182833i	$-0.941473\pi$
$929$		−	2.93178e8i		−	0.365666i	0.983144	−	0.182833i	$-0.0585268\pi$
$930$	1.06450e9	+	5.01294e8i	1.32341	+	0.623223i
$931$	−4.51724e8			−0.559789
$932$	1.21956e8			0.150645
$933$		−	1.37166e9i		−	1.68889i
$934$	4.42221e8			0.542748
$935$	−6.25627e8			−0.765387
$936$			6.78424e8i			0.827320i
$937$	−1.37821e9			−1.67531			−0.837655	−	0.546199i	$-0.816074\pi$
$937$	−1.37821e9			−1.67531			−0.837655	+	0.546199i	$0.816074\pi$
$938$	−5.78290e8			−0.700708
$939$	−1.89921e9			−2.29391
$940$	6.41276e8			0.772079
$941$		−	2.38576e8i		−	0.286324i	−0.989699	−	0.143162i	$-0.954273\pi$
$941$		−	2.38576e8i		−	0.286324i	0.989699	−	0.143162i	$-0.0457270\pi$
$942$		−	7.97563e8i		−	0.954140i
$943$			6.66257e7i			0.0794523i
$944$	5.37557e8			0.639012
$945$		−	7.21104e7i		−	0.0854481i
$946$	−8.71279e8			−1.02916
$947$		−	1.60709e8i		−	0.189230i	−0.995514	−	0.0946150i	$-0.969838\pi$
$947$		−	1.60709e8i		−	0.189230i	0.995514	−	0.0946150i	$-0.0301620\pi$
$948$	−2.61106e9			−3.06472
$949$	−6.33233e8			−0.740909
$950$	−1.47219e9			−1.71709
$951$			6.71261e8i			0.780458i
$952$		−	1.82873e9i		−	2.11953i
$953$			1.43500e9i			1.65796i	0.559276	+	0.828981i	$0.311079\pi$
$953$			1.43500e9i			1.65796i	−0.559276	+	0.828981i	$0.688921\pi$
$954$		−	2.01386e9i		−	2.31945i
$955$	7.30534e8			0.838746
$956$			1.00786e9i			1.15353i
$957$			1.14150e9i			1.30239i
$958$	−3.18009e8			−0.361695
$959$			4.27047e7i			0.0484194i
$960$			1.07135e9i			1.21093i
$961$	−5.65316e8	−	6.84165e8i	−0.636973	−	0.770886i
$962$	2.83969e8			0.318967
$963$	1.15681e9			1.29533
$964$			2.58958e9i			2.89067i
$965$	−5.43022e8			−0.604277
$966$	3.16485e8			0.351093
$967$			4.43002e8i			0.489921i	0.969533	+	0.244960i	$0.0787750\pi$
$967$			4.43002e8i			0.489921i	−0.969533	+	0.244960i	$0.921225\pi$
$968$	7.26443e8			0.800895
$969$	3.92137e9			4.30989
$970$	−1.23467e9			−1.35281
$971$	4.46135e8			0.487313			0.243657	−	0.969862i	$-0.421653\pi$
$971$	4.46135e8			0.487313			0.243657	+	0.969862i	$0.421653\pi$
$972$		−	2.27530e9i		−	2.47765i
$973$			1.99122e8i			0.216163i
$974$		−	1.31711e9i		−	1.42542i
$975$	−5.09552e8			−0.549762
$976$		−	1.78222e8i		−	0.191696i
$977$	9.10448e8			0.976273			0.488136	−	0.872767i	$-0.337677\pi$
$977$	9.10448e8			0.976273			0.488136	+	0.872767i	$0.337677\pi$
$978$			4.92724e7i			0.0526728i
$979$	1.50647e8			0.160551
$980$	3.59075e8			0.381510
$981$	−1.17300e7			−0.0124249
$982$			1.34045e9i			1.41552i
$983$		−	1.04694e9i		−	1.10221i	−0.834437	−	0.551103i	$-0.814207\pi$
$983$		−	1.04694e9i		−	1.10221i	0.834437	−	0.551103i	$-0.185793\pi$
$984$		−	7.84454e8i		−	0.823346i
$985$			7.94008e8i			0.830838i
$986$	−4.31353e9			−4.49989
$987$		−	7.17906e8i		−	0.746649i
$988$		−	2.02036e9i		−	2.09488i
$989$	1.67593e8			0.173248
$990$			5.96729e8i			0.614995i
$991$			1.76900e9i			1.81764i	0.417191	+	0.908819i	$0.363015\pi$
$991$			1.76900e9i			1.81764i	−0.417191	+	0.908819i	$0.636985\pi$
$992$	2.13848e8	−	4.54106e8i	0.219064	−	0.465181i
$993$	−1.12216e9			−1.14606
$994$	6.84392e8			0.696860
$995$			1.09459e9i			1.11118i
$996$	4.22889e9			4.28004
$997$	−8.93969e8			−0.902063			−0.451031	−	0.892508i	$-0.648944\pi$
$997$	−8.93969e8			−0.902063			−0.451031	+	0.892508i	$0.648944\pi$
$998$		−	7.64437e8i		−	0.769042i
$999$	−4.63745e7			−0.0465139

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.7.b.c.30.12	yes	12
3.2	odd	2		279.7.d.f.154.1		12
4.3	odd	2		496.7.e.c.433.3		12
31.30	odd	2	inner	31.7.b.c.30.11	✓	12
93.92	even	2		279.7.d.f.154.2		12
124.123	even	2		496.7.e.c.433.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.7.b.c.30.11	✓	12	31.30	odd	2	inner
31.7.b.c.30.12	yes	12	1.1	even	1	trivial
279.7.d.f.154.1		12	3.2	odd	2
279.7.d.f.154.2		12	93.92	even	2
496.7.e.c.433.3		12	4.3	odd	2
496.7.e.c.433.10		12	124.123	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	31.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+13.4661 q^{2} +37.0208i q^{3} +117.337 q^{4} -79.2253 q^{5} +498.527i q^{6} +281.109 q^{7} +718.240 q^{8} -641.539 q^{9} -1066.86 q^{10} +871.860i q^{11} +4343.90i q^{12} -1472.34i q^{13} +3785.45 q^{14} -2932.99i q^{15} +2162.36 q^{16} -9057.43i q^{17} -8639.05 q^{18} +11694.6 q^{19} -9296.05 q^{20} +10406.9i q^{21} +11740.6i q^{22} -2258.34i q^{23} +26589.8i q^{24} -9348.34 q^{25} -19826.7i q^{26} +3237.86i q^{27} +32984.4 q^{28} -35365.9i q^{29} -39496.0i q^{30} +(-12692.3 + 26952.0i) q^{31} -16848.7 q^{32} -32276.9 q^{33} -121969. i q^{34} -22271.0 q^{35} -75276.2 q^{36} +14322.6i q^{37} +157482. q^{38} +54507.2 q^{39} -56902.8 q^{40} -29502.1 q^{41} +140141. i q^{42} +74210.9i q^{43} +102301. i q^{44} +50826.2 q^{45} -30411.1i q^{46} -68983.7 q^{47} +80052.4i q^{48} -38626.6 q^{49} -125886. q^{50} +335313. q^{51} -172760. i q^{52} +233111. i q^{53} +43601.5i q^{54} -69073.4i q^{55} +201904. q^{56} +432945. i q^{57} -476242. i q^{58} +248597. q^{59} -344147. i q^{60} -82420.2i q^{61} +(-170916. + 362939. i) q^{62} -180343. q^{63} -365278. q^{64} +116647. i q^{65} -434646. q^{66} -152766. q^{67} -1.06277e6i q^{68} +83605.5 q^{69} -299904. q^{70} +180795. q^{71} -460779. q^{72} -430086. i q^{73} +192869. i q^{74} -346083. i q^{75} +1.37221e6 q^{76} +245088. i q^{77} +734001. q^{78} +601085. i q^{79} -171314. q^{80} -587550. q^{81} -397279. q^{82} -973523. i q^{83} +1.22111e6i q^{84} +717578. i q^{85} +999334. i q^{86} +1.30928e6 q^{87} +626205. i q^{88} -172788. i q^{89} +684432. q^{90} -413888. i q^{91} -264986. i q^{92} +(-997784. - 469878. i) q^{93} -928944. q^{94} -926512. q^{95} -623752. i q^{96} +1.15730e6 q^{97} -520151. q^{98} -559332. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} + 122 q^{4} - 146 q^{5} + 6 q^{7} + 1142 q^{8} - 5668 q^{9} - 3004 q^{10} - 1312 q^{14} - 1102 q^{16} - 16382 q^{18} + 10430 q^{19} - 8052 q^{20} + 29618 q^{25} + 136504 q^{28} + 90076 q^{31}+ \cdots + 408810 q^{98}+O(q^{100}) \) 12 * q - 2 * q^2 + 122 * q^4 - 146 * q^5 + 6 * q^7 + 1142 * q^8 - 5668 * q^9 - 3004 * q^10 - 1312 * q^14 - 1102 * q^16 - 16382 * q^18 + 10430 * q^19 - 8052 * q^20 + 29618 * q^25 + 136504 * q^28 + 90076 * q^31 + 38406 * q^32 - 166248 * q^33 - 198898 * q^35 + 26982 * q^36 + 448196 * q^38 + 208224 * q^39 - 354932 * q^40 - 116690 * q^41 - 363818 * q^45 - 221120 * q^47 - 580182 * q^49 - 60318 * q^50 + 179784 * q^51 + 750496 * q^56 + 799510 * q^59 - 73070 * q^62 - 469874 * q^63 - 322286 * q^64 - 511884 * q^66 - 13744 * q^67 + 475128 * q^69 - 71312 * q^70 + 640678 * q^71 - 118870 * q^72 + 2044196 * q^76 + 1373868 * q^78 + 1356020 * q^80 - 678724 * q^81 - 488056 * q^82 - 1051200 * q^87 - 2598532 * q^90 - 2260440 * q^93 - 5045848 * q^94 + 3828206 * q^95 + 3708966 * q^97 + 408810 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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