LMFDB - Embedded newform 112.6.p.b.47.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} + 691 x^{12} - 8602 x^{11} + 416261 x^{10} - 3521447 x^{9} + 66162087 x^{8} + \cdots + 17213603549184 $ x^14 - 3x^13 + 691x^12 - 8602x^11 + 416261x^10 - 3521447x^9 + 66162087x^8 - 31847546x^7 + 3819865057x^6 + 14320412589x^5 + 280505627703x^4 + 1315235310354x^3 + 6574080917124x^2 + 11537944725888*x + 17213603549184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{6}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.3
Root			$-1.05545 + 1.82809i$ of defining polynomial
Character	$\chi$	$=$	112.47
Dual form			112.6.p.b.31.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.47608 - 7.75280i) q^{3} +(-62.1336 - 35.8729i) q^{5} +(-115.750 - 58.3859i) q^{7} +(81.4294 - 141.040i) q^{9} +(132.963 - 76.7665i) q^{11} +891.047i q^{13} +642.279i q^{15} +(-1317.23 + 760.503i) q^{17} +(884.013 - 1531.15i) q^{19} +(65.4523 + 1158.73i) q^{21} +(3323.54 + 1918.85i) q^{23} +(1011.22 + 1751.49i) q^{25} -3633.31 q^{27} -2658.88 q^{29} +(3089.59 + 5351.33i) q^{31} +(-1190.31 - 687.226i) q^{33} +(5097.50 + 7780.02i) q^{35} +(-4264.01 + 7385.48i) q^{37} +(6908.10 - 3988.39i) q^{39} +18788.9i q^{41} -19232.2i q^{43} +(-10119.0 + 5842.21i) q^{45} +(-3540.78 + 6132.81i) q^{47} +(9989.16 + 13516.4i) q^{49} +(11792.1 + 6808.14i) q^{51} +(-12581.5 - 21791.7i) q^{53} -11015.3 q^{55} -15827.6 q^{57} +(-645.772 - 1118.51i) q^{59} +(20737.2 + 11972.6i) q^{61} +(-17660.2 + 11571.0i) q^{63} +(31964.4 - 55363.9i) q^{65} +(-14254.9 + 8230.08i) q^{67} -34355.6i q^{69} -25824.7i q^{71} +(9044.00 - 5221.56i) q^{73} +(9052.64 - 15679.6i) q^{75} +(-19872.6 + 1122.53i) q^{77} +(14892.4 + 8598.11i) q^{79} +(-3524.35 - 6104.36i) q^{81} -53986.4 q^{83} +109126. q^{85} +(11901.4 + 20613.8i) q^{87} +(-26565.6 - 15337.7i) q^{89} +(52024.6 - 103139. i) q^{91} +(27658.5 - 47906.0i) q^{93} +(-109854. + 63424.1i) q^{95} -70219.1i q^{97} -25004.2i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 9 q^{3} + 33 q^{5} - 28 q^{7} - 538 q^{9} + 333 q^{11} + 801 q^{17} - 2135 q^{19} + 2017 q^{21} + 2667 q^{23} + 5434 q^{25} + 17910 q^{27} + 684 q^{29} + 3119 q^{31} + 29013 q^{33} - 2247 q^{35}+ \cdots - 124833 q^{95}+O(q^{100}) $ 14 * q - 9 * q^3 + 33 * q^5 - 28 * q^7 - 538 * q^9 + 333 * q^11 + 801 * q^17 - 2135 * q^19 + 2017 * q^21 + 2667 * q^23 + 5434 * q^25 + 17910 * q^27 + 684 * q^29 + 3119 * q^31 + 29013 * q^33 - 2247 * q^35 + 3431 * q^37 + 2424 * q^39 - 28086 * q^45 + 8409 * q^47 - 24706 * q^49 + 24657 * q^51 + 1707 * q^53 - 56622 * q^55 + 68450 * q^57 - 49305 * q^59 - 21843 * q^61 + 164142 * q^63 + 17448 * q^65 - 73263 * q^67 - 77043 * q^73 - 147834 * q^75 + 22287 * q^77 + 206523 * q^79 - 145567 * q^81 - 305400 * q^83 - 176454 * q^85 - 95034 * q^87 + 171381 * q^89 + 174816 * q^91 + 292873 * q^93 - 124833 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−4.47608	−	7.75280i	−0.287141	−	0.497342i	0.685985	−	0.727615i	$-0.259371\pi$
$3$	−4.47608	−	7.75280i	−0.287141	−	0.497342i	−0.973126	+	0.230273i	$0.926038\pi$
$4$		0			0
$5$	−62.1336	−	35.8729i	−1.11148	−	0.641713i	−0.172268	−	0.985050i	$-0.555109\pi$
$5$	−62.1336	−	35.8729i	−1.11148	−	0.641713i	−0.939212	+	0.343337i	$0.888443\pi$
$6$		0			0
$7$	−115.750	−	58.3859i	−0.892845	−	0.450363i
$7$	−115.750	−	58.3859i	−0.892845	−	0.450363i
$8$		0			0
$9$	81.4294	−	141.040i	0.335101	−	0.580411i
$10$		0			0
$11$	132.963	−	76.7665i	0.331322	−	0.191289i	−0.325106	−	0.945678i	$-0.605400\pi$
$11$	132.963	−	76.7665i	0.331322	−	0.191289i	0.656428	+	0.754389i	$0.272067\pi$
$12$		0			0
$13$			891.047i			1.46232i	0.682207	+	0.731159i	$0.261020\pi$
$13$			891.047i			1.46232i	−0.682207	+	0.731159i	$0.738980\pi$
$14$		0			0
$15$			642.279i			0.737048i
$16$		0			0
$17$	−1317.23	+	760.503i	−1.10545	+	0.638232i	−0.937647	−	0.347588i	$-0.887001\pi$
$17$	−1317.23	+	760.503i	−1.10545	+	0.638232i	−0.167803	+	0.985821i	$0.553667\pi$
$18$		0			0
$19$	884.013	−	1531.15i	0.561791	−	0.973050i	−0.435550	−	0.900165i	$-0.643446\pi$
$19$	884.013	−	1531.15i	0.561791	−	0.973050i	0.997340	−	0.0728852i	$-0.0232207\pi$
$20$		0			0
$21$	65.4523	+	1158.73i	0.0323874	+	0.573367i
$22$		0			0
$23$	3323.54	+	1918.85i	1.31003	+	0.756346i	0.982101	−	0.188357i	$-0.0603161\pi$
$23$	3323.54	+	1918.85i	1.31003	+	0.756346i	0.327929	+	0.944703i	$0.393649\pi$
$24$		0			0
$25$	1011.22	+	1751.49i	0.323592	+	0.560478i
$26$		0			0
$27$	−3633.31			−0.959165
$28$		0			0
$29$	−2658.88			−0.587089			−0.293544	−	0.955945i	$-0.594835\pi$
$29$	−2658.88			−0.587089			−0.293544	+	0.955945i	$0.594835\pi$
$30$		0			0
$31$	3089.59	+	5351.33i	0.577427	+	1.00013i	0.995773	+	0.0918454i	$0.0292766\pi$
$31$	3089.59	+	5351.33i	0.577427	+	1.00013i	−0.418346	+	0.908288i	$0.637390\pi$
$32$		0			0
$33$	−1190.31	−	687.226i	−0.190272	−	0.109854i
$34$		0			0
$35$	5097.50	+	7780.02i	0.703375	+	1.07352i
$36$		0			0
$37$	−4264.01	+	7385.48i	−0.512052	+	0.886900i	0.487850	+	0.872927i	$0.337781\pi$
$37$	−4264.01	+	7385.48i	−0.512052	+	0.886900i	−0.999902	+	0.0139727i	$0.995552\pi$
$38$		0			0
$39$	6908.10	−	3988.39i	0.727273	−	0.419891i
$40$		0			0
$41$			18788.9i			1.74559i	0.488090	+	0.872793i	$0.337694\pi$
$41$			18788.9i			1.74559i	−0.488090	+	0.872793i	$0.662306\pi$
$42$		0			0
$43$		−	19232.2i		−	1.58620i	−0.609090	−	0.793101i	$-0.708465\pi$
$43$		−	19232.2i		−	1.58620i	0.609090	−	0.793101i	$-0.291535\pi$
$44$		0			0
$45$	−10119.0	+	5842.21i	−0.744915	+	0.430077i
$46$		0			0
$47$	−3540.78	+	6132.81i	−0.233805	+	0.404963i	−0.958925	−	0.283660i	$-0.908451\pi$
$47$	−3540.78	+	6132.81i	−0.233805	+	0.404963i	0.725119	+	0.688623i	$0.241785\pi$
$48$		0			0
$49$	9989.16	+	13516.4i	0.594345	+	0.804210i
$50$		0			0
$51$	11792.1	+	6808.14i	0.634839	+	0.366525i
$52$		0			0
$53$	−12581.5	−	21791.7i	−0.615236	−	1.06562i	−0.990343	−	0.138638i	$-0.955728\pi$
$53$	−12581.5	−	21791.7i	−0.615236	−	1.06562i	0.375107	−	0.926981i	$-0.377606\pi$
$54$		0			0
$55$	−11015.3			−0.491011
$56$		0			0
$57$	−15827.6			−0.645252
$58$		0			0
$59$	−645.772	−	1118.51i	−0.0241518	−	0.0418321i	0.853697	−	0.520770i	$-0.174355\pi$
$59$	−645.772	−	1118.51i	−0.0241518	−	0.0418321i	−0.877849	+	0.478938i	$0.841022\pi$
$60$		0			0
$61$	20737.2	+	11972.6i	0.713552	+	0.411970i	0.812375	−	0.583135i	$-0.198174\pi$
$61$	20737.2	+	11972.6i	0.713552	+	0.411970i	−0.0988225	+	0.995105i	$0.531508\pi$
$62$		0			0
$63$	−17660.2	+	11571.0i	−0.560589	+	0.367300i
$64$		0			0
$65$	31964.4	−	55363.9i	0.938389	−	1.62534i
$66$		0			0
$67$	−14254.9	+	8230.08i	−0.387952	+	0.223984i	−0.681272	−	0.732030i	$-0.738573\pi$
$67$	−14254.9	+	8230.08i	−0.387952	+	0.223984i	0.293320	+	0.956014i	$0.405240\pi$
$68$		0			0
$69$		−	34355.6i		−	0.868710i
$70$		0			0
$71$		−	25824.7i		−	0.607980i	−0.952675	−	0.303990i	$-0.901681\pi$
$71$		−	25824.7i		−	0.607980i	0.952675	−	0.303990i	$-0.0983190\pi$
$72$		0			0
$73$	9044.00	−	5221.56i	0.198634	−	0.114681i	−0.397384	−	0.917652i	$-0.630082\pi$
$73$	9044.00	−	5221.56i	0.198634	−	0.114681i	0.596018	+	0.802971i	$0.296749\pi$
$74$		0			0
$75$	9052.64	−	15679.6i	0.185833	−	0.321872i
$76$		0			0
$77$	−19872.6	+	1122.53i	−0.381969	+	0.0215760i
$78$		0			0
$79$	14892.4	+	8598.11i	0.268470	+	0.155001i	0.628192	−	0.778058i	$-0.283795\pi$
$79$	14892.4	+	8598.11i	0.268470	+	0.155001i	−0.359722	+	0.933060i	$0.617128\pi$
$80$		0			0
$81$	−3524.35	−	6104.36i	−0.0596852	−	0.103378i
$82$		0			0
$83$	−53986.4			−0.860180			−0.430090	−	0.902786i	$-0.641518\pi$
$83$	−53986.4			−0.860180			−0.430090	+	0.902786i	$0.641518\pi$
$84$		0			0
$85$	109126.			1.63825
$86$		0			0
$87$	11901.4	+	20613.8i	0.168577	+	0.291984i
$88$		0			0
$89$	−26565.6	−	15337.7i	−0.355504	−	0.205250i	0.311603	−	0.950213i	$-0.399134\pi$
$89$	−26565.6	−	15337.7i	−0.355504	−	0.205250i	−0.667107	+	0.744962i	$0.732468\pi$
$90$		0			0
$91$	52024.6	−	103139.i	0.658575	−	1.30562i
$92$		0			0
$93$	27658.5	−	47906.0i	0.331606	−	0.574358i
$94$		0			0
$95$	−109854.	+	63424.1i	−1.24884	+	0.721017i
$96$		0			0
$97$		−	70219.1i		−	0.757750i	−0.925448	−	0.378875i	$-0.876311\pi$
$97$		−	70219.1i		−	0.757750i	0.925448	−	0.378875i	$-0.123689\pi$
$98$		0			0
$99$		−	25004.2i		−	0.256404i
$100$		0			0
$101$	−55272.0	+	31911.3i	−0.539140	+	0.311273i	−0.744730	−	0.667365i	$-0.767422\pi$
$101$	−55272.0	+	31911.3i	−0.539140	+	0.311273i	0.205590	+	0.978638i	$0.434089\pi$
$102$		0			0
$103$	−53856.0	+	93281.2i	−0.500197	+	0.866366i	0.499803	+	0.866139i	$0.333406\pi$
$103$	−53856.0	+	93281.2i	−0.500197	+	0.866366i	−1.00000		0.000226934i	$0.999928\pi$
$104$		0			0
$105$	37500.1	−	74343.9i	0.331939	−	0.658070i
$106$		0			0
$107$	−114226.	−	65948.6i	−0.964509	−	0.556860i	−0.0669515	−	0.997756i	$-0.521327\pi$
$107$	−114226.	−	65948.6i	−0.964509	−	0.556860i	−0.897558	+	0.440896i	$0.854661\pi$
$108$		0			0
$109$	−47778.9	−	82755.5i	−0.385186	−	0.667161i	0.606609	−	0.795000i	$-0.292529\pi$
$109$	−47778.9	−	82755.5i	−0.385186	−	0.667161i	−0.991795	+	0.127839i	$0.959196\pi$
$110$		0			0
$111$	76344.2			0.588124
$112$		0			0
$113$	−37554.2			−0.276670			−0.138335	−	0.990385i	$-0.544175\pi$
$113$	−37554.2			−0.276670			−0.138335	+	0.990385i	$0.544175\pi$
$114$		0			0
$115$	−137669.	−	238450.i	−0.970714	−	1.68133i
$116$		0			0
$117$	125673.	+	72557.4i	0.848746	+	0.490024i
$118$		0			0
$119$	196872.	−	11120.6i	1.27443	−	0.0719881i
$120$		0			0
$121$	−68739.3	+	119060.i	−0.426817	+	0.739269i
$122$		0			0
$123$	145666.	−	84100.6i	0.868154	−	0.501229i
$124$		0			0
$125$			79103.3i			0.452814i
$126$		0			0
$127$			358280.i			1.97112i	0.169318	+	0.985561i	$0.445844\pi$
$127$			358280.i			1.97112i	−0.169318	+	0.985561i	$0.554156\pi$
$128$		0			0
$129$	−149104.	+	86085.0i	−0.788885	+	0.455463i
$130$		0			0
$131$	−83435.9	+	144515.i	−0.424790	+	0.735758i	−0.996401	−	0.0847668i	$-0.972985\pi$
$131$	−83435.9	+	144515.i	−0.424790	+	0.735758i	0.571611	+	0.820525i	$0.306319\pi$
$132$		0			0
$133$	−191722.	+	125617.i	−0.939818	+	0.615773i
$134$		0			0
$135$	225751.	+	130337.i	1.06609	+	0.615509i
$136$		0			0
$137$	127404.	+	220670.i	0.579937	+	1.00448i	0.995486	+	0.0949100i	$0.0302563\pi$
$137$	127404.	+	220670.i	0.579937	+	1.00448i	−0.415548	+	0.909571i	$0.636410\pi$
$138$		0			0
$139$	190751.			0.837393			0.418696	−	0.908126i	$-0.362487\pi$
$139$	190751.			0.837393			0.418696	+	0.908126i	$0.362487\pi$
$140$		0			0
$141$	63395.3			0.268540
$142$		0			0
$143$	68402.5	+	118477.i	0.279725	+	0.484499i
$144$		0			0
$145$	165206.	+	95381.6i	0.652537	+	0.376743i
$146$		0			0
$147$	60077.3	−	137944.i	0.229307	−	0.526514i
$148$		0			0
$149$	43686.8	−	75667.7i	0.161207	−	0.279219i	−0.774095	−	0.633070i	$-0.781795\pi$
$149$	43686.8	−	75667.7i	0.161207	−	0.279219i	0.935302	+	0.353851i	$0.115128\pi$
$150$		0			0
$151$	−158556.	+	91542.4i	−0.565901	+	0.326723i	−0.755511	−	0.655136i	$-0.772611\pi$
$151$	−158556.	+	91542.4i	−0.565901	+	0.326723i	0.189610	+	0.981860i	$0.439278\pi$
$152$		0			0
$153$			247709.i			0.855488i
$154$		0			0
$155$		−	443330.i		−	1.48217i
$156$		0			0
$157$	−259967.	+	150092.i	−0.841724	+	0.485969i	−0.857850	−	0.513901i	$-0.828200\pi$
$157$	−259967.	+	150092.i	−0.841724	+	0.485969i	0.0161260	+	0.999870i	$0.494867\pi$
$158$		0			0
$159$	−112631.	+	195083.i	−0.353318	+	0.611965i
$160$		0			0
$161$	−272666.	−	416154.i	−0.829023	−	1.26529i
$162$		0			0
$163$	514813.	+	297227.i	1.51768	+	0.876233i	0.999784	+	0.0207905i	$0.00661829\pi$
$163$	514813.	+	297227.i	1.51768	+	0.876233i	0.517897	+	0.855443i	$0.326715\pi$
$164$		0			0
$165$	49305.5	+	85399.6i	0.140989	+	0.244200i
$166$		0			0
$167$	−336959.			−0.934946			−0.467473	−	0.884007i	$-0.654835\pi$
$167$	−336959.			−0.934946			−0.467473	+	0.884007i	$0.654835\pi$
$168$		0			0
$169$	−422671.			−1.13838
$170$		0			0
$171$	−143969.	−	249362.i	−0.376513	−	0.652139i
$172$		0			0
$173$	−65047.1	−	37555.0i	−0.165239	−	0.0954008i	0.415100	−	0.909776i	$-0.363747\pi$
$173$	−65047.1	−	37555.0i	−0.165239	−	0.0954008i	−0.580339	+	0.814375i	$0.697080\pi$
$174$		0			0
$175$	−14786.8	−	261777.i	−0.0364989	−	0.646154i
$176$		0			0
$177$	−5781.05	+	10013.1i	−0.0138699	+	0.0240234i
$178$		0			0
$179$	−160495.	+	92661.7i	−0.374393	+	0.216156i	−0.675376	−	0.737473i	$-0.736019\pi$
$179$	−160495.	+	92661.7i	−0.374393	+	0.216156i	0.300983	+	0.953630i	$0.402685\pi$
$180$		0			0
$181$			302350.i			0.685984i	0.939338	+	0.342992i	$0.111440\pi$
$181$			302350.i			0.685984i	−0.939338	+	0.342992i	$0.888560\pi$
$182$		0			0
$183$		−	214362.i		−	0.473173i
$184$		0			0
$185$	529877.	−	305925.i	1.13827	−	0.657181i
$186$		0			0
$187$	−116762.	+	202238.i	−0.244174	+	0.422921i
$188$		0			0
$189$	420556.	+	212134.i	0.856386	+	0.431973i
$190$		0			0
$191$	779856.	+	450250.i	1.54679	+	0.893039i	0.998384	+	0.0568265i	$0.0180982\pi$
$191$	779856.	+	450250.i	1.54679	+	0.893039i	0.548405	+	0.836213i	$0.315235\pi$
$192$		0			0
$193$	308039.	+	533539.i	0.595268	+	1.03103i	0.993509	+	0.113753i	$0.0362873\pi$
$193$	308039.	+	533539.i	0.595268	+	1.03103i	−0.398242	+	0.917281i	$0.630379\pi$
$194$		0			0
$195$	−572301.			−1.07780
$196$		0			0
$197$	−698830.			−1.28294			−0.641469	−	0.767149i	$-0.721675\pi$
$197$	−698830.			−1.28294			−0.641469	+	0.767149i	$0.721675\pi$
$198$		0			0
$199$	−364105.	−	630648.i	−0.651769	−	1.12890i	−0.982693	−	0.185240i	$-0.940694\pi$
$199$	−364105.	−	630648.i	−0.651769	−	1.12890i	0.330924	−	0.943657i	$-0.392639\pi$
$200$		0			0
$201$	127612.	+	73677.0i	0.222793	+	0.128630i
$202$		0			0
$203$	307766.	+	155241.i	0.524179	+	0.264403i
$204$		0			0
$205$	674011.	−	1.16742e6i	1.12017	−	1.94018i
$206$		0			0
$207$	541268.	−	312501.i	0.877983	−	0.506904i
$208$		0			0
$209$		−	271450.i		−	0.429857i
$210$		0			0
$211$		−	504855.i		−	0.780657i	−0.920676	−	0.390329i	$-0.872361\pi$
$211$		−	504855.i		−	0.780657i	0.920676	−	0.390329i	$-0.127639\pi$
$212$		0			0
$213$	−200214.	+	115593.i	−0.302374	+	0.174576i
$214$		0			0
$215$	−689915.	+	1.19497e6i	−1.01789	+	1.76303i
$216$		0			0
$217$	−45178.1	−	799806.i	−0.0651297	−	1.15302i
$218$		0			0
$219$	−80963.4	−	46744.2i	−0.114072	−	0.0658594i
$220$		0			0
$221$	−677644.	−	1.17371e6i	−0.933299	−	1.61652i
$222$		0			0
$223$	−1.06011e6			−1.42755			−0.713774	−	0.700377i	$-0.753015\pi$
$223$	−1.06011e6			−1.42755			−0.713774	+	0.700377i	$0.753015\pi$
$224$		0			0
$225$	329374.			0.433743
$226$		0			0
$227$	−463410.	−	802650.i	−0.596899	−	1.03386i	−0.993276	−	0.115772i	$-0.963066\pi$
$227$	−463410.	−	802650.i	−0.596899	−	1.03386i	0.396377	−	0.918088i	$-0.370267\pi$
$228$		0			0
$229$	894784.	+	516604.i	1.12753	+	0.650982i	0.943314	−	0.331903i	$-0.107691\pi$
$229$	894784.	+	516604.i	1.12753	+	0.650982i	0.184221	+	0.982885i	$0.441024\pi$
$230$		0			0
$231$	97654.2	+	149044.i	0.120410	+	0.183774i
$232$		0			0
$233$	338884.	−	586964.i	0.408941	−	0.708307i	−0.585830	−	0.810434i	$-0.699231\pi$
$233$	338884.	−	586964.i	0.408941	−	0.708307i	0.994771	+	0.102127i	$0.0325648\pi$
$234$		0			0
$235$	440003.	−	254036.i	0.519740	−	0.300072i
$236$		0			0
$237$		−	153943.i		−	0.178029i
$238$		0			0
$239$		−	447964.i		−	0.507281i	−0.967299	−	0.253640i	$-0.918372\pi$
$239$		−	447964.i		−	0.507281i	0.967299	−	0.253640i	$-0.0816280\pi$
$240$		0			0
$241$	290776.	−	167880.i	0.322490	−	0.186190i	−0.330012	−	0.943977i	$-0.607053\pi$
$241$	290776.	−	167880.i	0.322490	−	0.186190i	0.652502	+	0.757787i	$0.273719\pi$
$242$		0			0
$243$	−472998.	+	819257.i	−0.513859	+	0.890029i
$244$		0			0
$245$	−135793.	−	1.19816e6i	−0.144531	−	1.27526i
$246$		0			0
$247$	1.36433e6	+	787696.i	1.42291	+	0.821517i
$248$		0			0
$249$	241648.	+	418546.i	0.246993	+	0.427804i
$250$		0			0
$251$	−702313.			−0.703633			−0.351816	−	0.936069i	$-0.614436\pi$
$251$	−702313.			−0.703633			−0.351816	+	0.936069i	$0.614436\pi$
$252$		0			0
$253$	589212.			0.578722
$254$		0			0
$255$	−488455.	−	846029.i	−0.470408	−	0.814770i
$256$		0			0
$257$	−264047.	−	152448.i	−0.249372	−	0.143975i	0.370104	−	0.928990i	$-0.379322\pi$
$257$	−264047.	−	152448.i	−0.249372	−	0.143975i	−0.619477	+	0.785015i	$0.712655\pi$
$258$		0			0
$259$	924768.	−	605912.i	0.856610	−	0.561255i
$260$		0			0
$261$	−216511.	+	375008.i	−0.196734	+	0.340753i
$262$		0			0
$263$	−1.64193e6	+	947969.i	−1.46375	+	0.845094i	−0.999182	−	0.0404463i	$-0.987122\pi$
$263$	−1.64193e6	+	947969.i	−1.46375	+	0.845094i	−0.464563	+	0.885540i	$0.653789\pi$
$264$		0			0
$265$			1.80533e6i			1.57922i
$266$		0			0
$267$			274610.i			0.235743i
$268$		0			0
$269$	−1.34017e6	+	773746.i	−1.12922	+	0.651955i	−0.943738	−	0.330693i	$-0.892718\pi$
$269$	−1.34017e6	+	773746.i	−1.12922	+	0.651955i	−0.185481	+	0.982648i	$0.559384\pi$
$270$		0			0
$271$	783413.	−	1.35691e6i	0.647989	−	1.12235i	−0.335613	−	0.942000i	$-0.608944\pi$
$271$	783413.	−	1.35691e6i	0.647989	−	1.12235i	0.983602	−	0.180350i	$-0.0577231\pi$
$272$		0			0
$273$	−1.03248e6	+	58321.0i	−0.838446	+	0.0473607i
$274$		0			0
$275$	268912.	+	155256.i	0.214426	+	0.123799i
$276$		0			0
$277$	938471.	+	1.62548e6i	0.734888	+	1.27286i	0.954772	+	0.297338i	$0.0960988\pi$
$277$	938471.	+	1.62548e6i	0.734888	+	1.27286i	−0.219884	+	0.975526i	$0.570568\pi$
$278$		0			0
$279$	1.00634e6			0.773985
$280$		0			0
$281$	−1.48022e6			−1.11831			−0.559155	−	0.829063i	$-0.688874\pi$
$281$	−1.48022e6			−1.11831			−0.559155	+	0.829063i	$0.688874\pi$
$282$		0			0
$283$	186755.	+	323469.i	0.138614	+	0.240086i	0.926972	−	0.375130i	$-0.122402\pi$
$283$	186755.	+	323469.i	0.138614	+	0.240086i	−0.788358	+	0.615216i	$0.789069\pi$
$284$		0			0
$285$	983429.	+	567783.i	0.717184	+	0.414067i
$286$		0			0
$287$	1.09701e6	−	2.17482e6i	0.786149	−	1.55854i
$288$		0			0
$289$	446801.	−	773883.i	0.314681	−	0.545043i
$290$		0			0
$291$	−544395.	+	314306.i	−0.376861	+	0.217581i
$292$		0			0
$293$		−	1.51194e6i		−	1.02888i	−0.857525	−	0.514442i	$-0.827999\pi$
$293$		−	1.51194e6i		−	1.02888i	0.857525	−	0.514442i	$-0.172001\pi$
$294$		0			0
$295$			92662.7i			0.0619941i
$296$		0			0
$297$	−483098.	+	278917.i	−0.317793	+	0.183478i
$298$		0			0
$299$	−1.70978e6	+	2.96143e6i	−1.10602	+	1.91568i
$300$		0			0
$301$	−1.12289e6	+	2.22613e6i	−0.714368	+	1.41623i
$302$		0			0
$303$	494804.	+	285675.i	0.309618	+	0.178758i
$304$		0			0
$305$	−858986.	−	1.48781e6i	−0.528733	−	0.915792i
$306$		0			0
$307$	1.02264e6			0.619267			0.309634	−	0.950856i	$-0.399794\pi$
$307$	1.02264e6			0.619267			0.309634	+	0.950856i	$0.399794\pi$
$308$		0			0
$309$	964254.			0.574507
$310$		0			0
$311$	−531329.	−	920288.i	−0.311503	−	0.539539i	0.667185	−	0.744892i	$-0.267499\pi$
$311$	−531329.	−	920288.i	−0.311503	−	0.539539i	−0.978688	+	0.205353i	$0.934166\pi$
$312$		0			0
$313$	309475.	+	178676.i	0.178552	+	0.103087i	0.586612	−	0.809868i	$-0.300461\pi$
$313$	309475.	+	178676.i	0.178552	+	0.103087i	−0.408060	+	0.912955i	$0.633795\pi$
$314$		0			0
$315$	1.51238e6	−	85428.8i	0.858785	−	0.0485096i
$316$		0			0
$317$	−216990.	+	375838.i	−0.121281	+	0.210064i	−0.920273	−	0.391277i	$-0.872033\pi$
$317$	−216990.	+	375838.i	−0.121281	+	0.210064i	0.798992	+	0.601341i	$0.205367\pi$
$318$		0			0
$319$	−353534.	+	204113.i	−0.194515	+	0.112304i
$320$		0			0
$321$			1.18076e6i			0.639588i
$322$		0			0
$323$			2.68918e6i			1.43421i
$324$		0			0
$325$	−1.56066e6	+	901048.i	−0.819597	+	0.473194i
$326$		0			0
$327$	−427724.	+	740840.i	−0.221205	+	0.383138i
$328$		0			0
$329$	767916.	−	503142.i	0.391132	−	0.256272i
$330$		0			0
$331$	2.85104e6	+	1.64605e6i	1.43032	+	0.825795i	0.997145	−	0.0755155i	$-0.0240602\pi$
$331$	2.85104e6	+	1.64605e6i	1.43032	+	0.825795i	0.433174	+	0.901310i	$0.357394\pi$
$332$		0			0
$333$	694432.	+	1.20279e6i	0.343178	+	0.594401i
$334$		0			0
$335$	1.18095e6			0.574934
$336$		0			0
$337$	1.52754e6			0.732684			0.366342	−	0.930480i	$-0.380610\pi$
$337$	1.52754e6			0.732684			0.366342	+	0.930480i	$0.380610\pi$
$338$		0			0
$339$	168096.	+	291150.i	0.0794433	+	0.137600i
$340$		0			0
$341$	821606.	+	474355.i	0.382629	+	0.220911i
$342$		0			0
$343$	−367082.	−	2.14775e6i	−0.168472	−	0.985706i
$344$		0			0
$345$	−1.23243e6	+	2.13464e6i	−0.557463	+	0.965554i
$346$		0			0
$347$	1.14392e6	−	660441.i	0.510001	−	0.294449i	−0.222833	−	0.974857i	$-0.571530\pi$
$347$	1.14392e6	−	660441.i	0.510001	−	0.294449i	0.732834	+	0.680407i	$0.238197\pi$
$348$		0			0
$349$		−	196028.i		−	0.0861498i	−0.999072	−	0.0430749i	$-0.986285\pi$
$349$		−	196028.i		−	0.0861498i	0.999072	−	0.0430749i	$-0.0137154\pi$
$350$		0			0
$351$		−	3.23745e6i		−	1.40261i
$352$		0			0
$353$	−3.86551e6	+	2.23176e6i	−1.65109	+	0.953257i	−0.674463	+	0.738308i	$0.735625\pi$
$353$	−3.86551e6	+	2.23176e6i	−1.65109	+	0.953257i	−0.976625	+	0.214948i	$0.931042\pi$
$354$		0			0
$355$	−926406.	+	1.60458e6i	−0.390149	+	0.675758i
$356$		0			0
$357$	−967431.	−	1.47653e6i	−0.401744	−	0.613158i
$358$		0			0
$359$	−2.97445e6	−	1.71730e6i	−1.21806	−	0.703250i	−0.253561	−	0.967319i	$-0.581602\pi$
$359$	−2.97445e6	−	1.71730e6i	−1.21806	−	0.703250i	−0.964504	+	0.264069i	$0.914935\pi$
$360$		0			0
$361$	−324907.	−	562756.i	−0.131217	−	0.227275i
$362$		0			0
$363$	1.23073e6			0.490226
$364$		0			0
$365$	−749249.			−0.294370
$366$		0			0
$367$	71351.3	+	123584.i	0.0276527	+	0.0478958i	0.879521	−	0.475861i	$-0.157863\pi$
$367$	71351.3	+	123584.i	0.0276527	+	0.0478958i	−0.851868	+	0.523757i	$0.824530\pi$
$368$		0			0
$369$	2.64998e6	+	1.52997e6i	1.01316	+	0.584947i
$370$		0			0
$371$	183975.	+	3.25698e6i	0.0693942	+	1.22851i
$372$		0			0
$373$	994539.	−	1.72259e6i	0.370126	−	0.641077i	−0.619458	−	0.785029i	$-0.712648\pi$
$373$	994539.	−	1.72259e6i	0.370126	−	0.641077i	0.989585	+	0.143952i	$0.0459811\pi$
$374$		0			0
$375$	613272.	−	354073.i	0.225203	−	0.130021i
$376$		0			0
$377$		−	2.36919e6i		−	0.858511i
$378$		0			0
$379$		−	2.27226e6i		−	0.812568i	−0.913747	−	0.406284i	$-0.866824\pi$
$379$		−	2.27226e6i		−	0.812568i	0.913747	−	0.406284i	$-0.133176\pi$
$380$		0			0
$381$	2.77767e6	−	1.60369e6i	0.980323	−	0.565989i
$382$		0			0
$383$	752460.	−	1.30330e6i	0.262112	−	0.453991i	−0.704691	−	0.709514i	$-0.748914\pi$
$383$	752460.	−	1.30330e6i	0.262112	−	0.453991i	0.966803	+	0.255523i	$0.0822478\pi$
$384$		0			0
$385$	1.27503e6	+	643141.i	0.438397	+	0.221133i
$386$		0			0
$387$	−2.71251e6	−	1.56607e6i	−0.920649	−	0.531537i
$388$		0			0
$389$	−1.41991e6	−	2.45936e6i	−0.475759	−	0.824039i	0.523856	−	0.851807i	$-0.324493\pi$
$389$	−1.41991e6	−	2.45936e6i	−0.475759	−	0.824039i	−0.999614	+	0.0277686i	$0.991160\pi$
$390$		0			0
$391$	−5.83715e6			−1.93090
$392$		0			0
$393$	1.49386e6			0.487898
$394$		0			0
$395$	−616878.	−	1.06846e6i	−0.198933	−	0.344562i
$396$		0			0
$397$	−2.79588e6	−	1.61420e6i	−0.890312	−	0.514022i	−0.0162674	−	0.999868i	$-0.505178\pi$
$397$	−2.79588e6	−	1.61420e6i	−0.890312	−	0.514022i	−0.874045	+	0.485846i	$0.838512\pi$
$398$		0			0
$399$	1.83205e6	+	924112.i	0.576110	+	0.290598i
$400$		0			0
$401$	3.02149e6	−	5.23337e6i	0.938340	−	1.62525i	0.169772	−	0.985483i	$-0.445697\pi$
$401$	3.02149e6	−	5.23337e6i	0.938340	−	1.62525i	0.768567	−	0.639769i	$-0.220970\pi$
$402$		0			0
$403$	−4.76829e6	+	2.75297e6i	−1.46251	+	0.844382i
$404$		0			0
$405$			505715.i			0.153203i
$406$		0			0
$407$			1.30933e6i			0.391800i
$408$		0			0
$409$	2.50759e6	−	1.44776e6i	0.741221	−	0.427944i	−0.0812923	−	0.996690i	$-0.525905\pi$
$409$	2.50759e6	−	1.44776e6i	0.741221	−	0.427944i	0.822513	+	0.568746i	$0.192571\pi$
$410$		0			0
$411$	1.14054e6	−	1.97547e6i	0.333047	−	0.576855i
$412$		0			0
$413$	9442.91	+	167172.i	0.00272415	+	0.0482267i
$414$		0			0
$415$	3.35437e6	+	1.93665e6i	0.956073	+	0.551989i
$416$		0			0
$417$	−853815.	−	1.47885e6i	−0.240449	−	0.416471i
$418$		0			0
$419$	−3.80628e6			−1.05917			−0.529585	−	0.848257i	$-0.677652\pi$
$419$	−3.80628e6			−1.05917			−0.529585	+	0.848257i	$0.677652\pi$
$420$		0			0
$421$	−4.30950e6			−1.18501			−0.592504	−	0.805567i	$-0.701861\pi$
$421$	−4.30950e6			−1.18501			−0.592504	+	0.805567i	$0.701861\pi$
$422$		0			0
$423$	576648.	+	998783.i	0.156697	+	0.271406i
$424$		0			0
$425$	−2.66403e6	−	1.53808e6i	−0.715430	−	0.413053i
$426$		0			0
$427$	−1.70130e6	−	2.59660e6i	−0.451556	−	0.689183i
$428$		0			0
$429$	612350.	−	1.06062e6i	0.160641	−	0.278238i
$430$		0			0
$431$	2.33697e6	−	1.34925e6i	0.605981	−	0.349863i	−0.165410	−	0.986225i	$-0.552895\pi$
$431$	2.33697e6	−	1.34925e6i	0.605981	−	0.349863i	0.771391	+	0.636362i	$0.219561\pi$
$432$		0			0
$433$			3.11101e6i			0.797410i	0.917079	+	0.398705i	$0.130540\pi$
$433$			3.11101e6i			0.797410i	−0.917079	+	0.398705i	$0.869460\pi$
$434$		0			0
$435$		−	1.70774e6i		−	0.432712i
$436$		0			0
$437$	5.87610e6	−	3.39257e6i	1.47192	−	0.849816i
$438$		0			0
$439$	3.34202e6	−	5.78854e6i	0.827651	−	1.43353i	−0.0722248	−	0.997388i	$-0.523010\pi$
$439$	3.34202e6	−	5.78854e6i	0.827651	−	1.43353i	0.899876	−	0.436146i	$-0.143657\pi$
$440$		0			0
$441$	2.71976e6	−	308242.i	0.665938	−	0.0754736i
$442$		0			0
$443$	896344.	+	517505.i	0.217003	+	0.125287i	0.604562	−	0.796558i	$-0.293348\pi$
$443$	896344.	+	517505.i	0.217003	+	0.125287i	−0.387559	+	0.921845i	$0.626682\pi$
$444$		0			0
$445$	1.10041e6	+	1.90597e6i	0.263424	+	0.456264i
$446$		0			0
$447$	−782181.			−0.185156
$448$		0			0
$449$	5.42168e6			1.26917			0.634583	−	0.772855i	$-0.281172\pi$
$449$	5.42168e6			1.26917			0.634583	+	0.772855i	$0.281172\pi$
$450$		0			0
$451$	1.44236e6	+	2.49824e6i	0.333912	+	0.578352i
$452$		0			0
$453$	1.41942e6	+	819502.i	0.324986	+	0.187631i
$454$		0			0
$455$	−6.93236e6	+	4.54211e6i	−1.56983	+	1.02856i
$456$		0			0
$457$	−1.38367e6	+	2.39658e6i	−0.309914	+	0.536786i	−0.978343	−	0.206989i	$-0.933634\pi$
$457$	−1.38367e6	+	2.39658e6i	−0.309914	+	0.536786i	0.668429	+	0.743776i	$0.266967\pi$
$458$		0			0
$459$	4.78591e6	−	2.76315e6i	1.06031	−	0.612170i
$460$		0			0
$461$			414792.i			0.0909029i	0.998967	+	0.0454515i	$0.0144726\pi$
$461$			414792.i			0.0909029i	−0.998967	+	0.0454515i	$0.985527\pi$
$462$		0			0
$463$			3.28204e6i			0.711526i	0.934576	+	0.355763i	$0.115779\pi$
$463$			3.28204e6i			0.711526i	−0.934576	+	0.355763i	$0.884221\pi$
$464$		0			0
$465$	−3.43705e6	+	1.98438e6i	−0.737146	+	0.425591i
$466$		0			0
$467$	−1.65003e6	+	2.85793e6i	−0.350105	+	0.606400i	−0.986268	−	0.165155i	$-0.947187\pi$
$467$	−1.65003e6	+	2.85793e6i	−0.350105	+	0.606400i	0.636163	+	0.771555i	$0.280521\pi$
$468$		0			0
$469$	2.13053e6	−	120346.i	0.447255	−	0.0252638i
$470$		0			0
$471$	2.32727e6	+	1.34365e6i	0.483386	+	0.279083i
$472$		0			0
$473$	−1.47639e6	−	2.55718e6i	−0.303423	−	0.525544i
$474$		0			0
$475$	3.57574e6			0.727164
$476$		0			0
$477$	−4.09801e6			−0.824663
$478$		0			0
$479$	−1.16966e6	−	2.02592e6i	−0.232928	−	0.403444i	0.725740	−	0.687969i	$-0.241497\pi$
$479$	−1.16966e6	−	2.02592e6i	−0.232928	−	0.403444i	−0.958669	+	0.284525i	$0.908164\pi$
$480$		0			0
$481$	−6.58081e6	−	3.79943e6i	−1.29693	−	0.748783i
$482$		0			0
$483$	−2.00588e6	+	3.97667e6i	−0.391235	+	0.775624i
$484$		0			0
$485$	−2.51896e6	+	4.36297e6i	−0.486258	+	0.842224i
$486$		0			0
$487$	−2.65756e6	+	1.53434e6i	−0.507763	+	0.293157i	−0.731914	−	0.681398i	$-0.761373\pi$
$487$	−2.65756e6	+	1.53434e6i	−0.507763	+	0.293157i	0.224151	+	0.974554i	$0.428039\pi$
$488$		0			0
$489$		−	5.32165e6i		−	1.00641i
$490$		0			0
$491$		−	4.39524e6i		−	0.822770i	−0.911462	−	0.411385i	$-0.865045\pi$
$491$		−	4.39524e6i		−	0.822770i	0.911462	−	0.411385i	$-0.134955\pi$
$492$		0			0
$493$	3.50236e6	−	2.02209e6i	0.648997	−	0.374699i
$494$		0			0
$495$	−896972.	+	1.55360e6i	−0.164538	+	0.284988i
$496$		0			0
$497$	−1.50780e6	+	2.98921e6i	−0.273812	+	0.542832i
$498$		0			0
$499$	−6.73309e6	−	3.88735e6i	−1.21049	−	0.698879i	−0.247627	−	0.968855i	$-0.579651\pi$
$499$	−6.73309e6	−	3.88735e6i	−1.21049	−	0.698879i	−0.962867	+	0.269976i	$0.912984\pi$
$500$		0			0
$501$	1.50826e6	+	2.61238e6i	0.268461	+	0.464988i
$502$		0			0
$503$	5.37075e6			0.946488			0.473244	−	0.880931i	$-0.343083\pi$
$503$	5.37075e6			0.946488			0.473244	+	0.880931i	$0.343083\pi$
$504$		0			0
$505$	4.57900e6			0.798991
$506$		0			0
$507$	1.89191e6	+	3.27688e6i	0.326874	+	0.566162i
$508$		0			0
$509$	1.89203e6	+	1.09236e6i	0.323693	+	0.186884i	0.653037	−	0.757326i	$-0.273494\pi$
$509$	1.89203e6	+	1.09236e6i	0.323693	+	0.186884i	−0.329344	+	0.944210i	$0.606828\pi$
$510$		0			0
$511$	−1.35171e6	+	76353.2i	−0.228998	+	0.0129352i
$512$		0			0
$513$	−3.21189e6	+	5.56316e6i	−0.538850	+	0.933316i
$514$		0			0
$515$	6.69253e6	−	3.86393e6i	1.11192	−	0.641965i
$516$		0			0
$517$			1.08725e6i			0.178898i
$518$		0			0
$519$			672396.i			0.109574i
$520$		0			0
$521$	−4.30421e6	+	2.48504e6i	−0.694703	+	0.401087i	−0.805371	−	0.592771i	$-0.798034\pi$
$521$	−4.30421e6	+	2.48504e6i	−0.694703	+	0.401087i	0.110669	+	0.993857i	$0.464701\pi$
$522$		0			0
$523$	5.88630e6	−	1.01954e7i	0.940997	−	1.62985i	0.177422	−	0.984135i	$-0.443224\pi$
$523$	5.88630e6	−	1.01954e7i	0.940997	−	1.62985i	0.763575	−	0.645719i	$-0.223442\pi$
$524$		0			0
$525$	−1.96331e6	+	1.28637e6i	−0.310879	+	0.203689i
$526$		0			0
$527$	−8.13941e6	−	4.69929e6i	−1.27663	−	0.737065i
$528$		0			0
$529$	4.14576e6	+	7.18067e6i	0.644118	+	1.11564i
$530$		0			0
$531$	−210339.			−0.0323731
$532$		0			0
$533$	−1.67418e7			−2.55260
$534$		0			0
$535$	4.73153e6	+	8.19525e6i	0.714689	+	1.23788i
$536$		0			0
$537$	1.43677e6	+	829522.i	0.215007	+	0.124134i
$538$		0			0
$539$	2.36580e6	+	1.03035e6i	0.350756	+	0.152761i
$540$		0			0
$541$	−3.23323e6	+	5.60011e6i	−0.474945	+	0.822628i	−0.999588	−	0.0286937i	$-0.990865\pi$
$541$	−3.23323e6	+	5.60011e6i	−0.474945	+	0.822628i	0.524644	+	0.851322i	$0.324199\pi$
$542$		0			0
$543$	2.34406e6	−	1.35334e6i	0.341169	−	0.196974i
$544$		0			0
$545$			6.85586e6i			0.988715i
$546$		0			0
$547$			1.14808e7i			1.64061i	0.571927	+	0.820305i	$0.306196\pi$
$547$			1.14808e7i			1.64061i	−0.571927	+	0.820305i	$0.693804\pi$
$548$		0			0
$549$	3.37724e6	−	1.94985e6i	0.478224	−	0.276103i
$550$		0			0
$551$	−2.35048e6	+	4.07116e6i	−0.329821	+	0.571267i
$552$		0			0
$553$	−1.22178e6	−	1.86474e6i	−0.169895	−	0.259301i
$554$		0			0
$555$	−4.74354e6	−	2.73869e6i	−0.653688	−	0.377407i
$556$		0			0
$557$	4.53151e6	+	7.84880e6i	0.618877	+	1.07193i	0.989691	+	0.143220i	$0.0457455\pi$
$557$	4.53151e6	+	7.84880e6i	0.618877	+	1.07193i	−0.370814	+	0.928707i	$0.620921\pi$
$558$		0			0
$559$	1.71368e7			2.31953
$560$		0			0
$561$	2.09055e6			0.280449
$562$		0			0
$563$	1.32480e6	+	2.29462e6i	0.176148	+	0.305098i	0.940558	−	0.339633i	$-0.110303\pi$
$563$	1.32480e6	+	2.29462e6i	0.176148	+	0.305098i	−0.764410	+	0.644731i	$0.776969\pi$
$564$		0			0
$565$	2.33338e6	+	1.34718e6i	0.307513	+	0.177543i
$566$		0			0
$567$	51535.5	+	912353.i	0.00673207	+	0.119181i
$568$		0			0
$569$	−4.54125e6	+	7.86567e6i	−0.588023	+	1.01849i	0.406468	+	0.913665i	$0.366760\pi$
$569$	−4.54125e6	+	7.86567e6i	−0.588023	+	1.01849i	−0.994491	+	0.104821i	$0.966573\pi$
$570$		0			0
$571$	−1.33094e6	+	768420.i	−0.170832	+	0.0986299i	−0.582978	−	0.812488i	$-0.698113\pi$
$571$	−1.33094e6	+	768420.i	−0.170832	+	0.0986299i	0.412146	+	0.911118i	$0.364779\pi$
$572$		0			0
$573$		−	8.06142e6i		−	1.02571i
$574$		0			0
$575$			7.76153e6i			0.978989i
$576$		0			0
$577$	3.66985e6	−	2.11879e6i	0.458890	−	0.264940i	−0.252688	−	0.967548i	$-0.581314\pi$
$577$	3.66985e6	−	2.11879e6i	0.458890	−	0.264940i	0.711577	+	0.702608i	$0.247981\pi$
$578$		0			0
$579$	2.75761e6	−	4.77633e6i	0.341851	−	0.592103i
$580$		0			0
$581$	6.24894e6	+	3.15205e6i	0.768008	+	0.387394i
$582$		0			0
$583$	−3.34575e6	−	1.93167e6i	−0.407682	−	0.235376i
$584$		0			0
$585$	−5.20568e6	−	9.01651e6i	−0.628909	−	1.08930i
$586$		0			0
$587$	1.20181e7			1.43960			0.719799	−	0.694182i	$-0.244234\pi$
$587$	1.20181e7			1.43960			0.719799	+	0.694182i	$0.244234\pi$
$588$		0			0
$589$	1.09250e7			1.29757
$590$		0			0
$591$	3.12802e6	+	5.41789e6i	0.368384	+	0.638059i
$592$		0			0
$593$	−1.24383e7	−	7.18125e6i	−1.45253	−	0.838617i	−0.453902	−	0.891052i	$-0.649968\pi$
$593$	−1.24383e7	−	7.18125e6i	−1.45253	−	0.838617i	−0.998624	+	0.0524350i	$0.983302\pi$
$594$		0			0
$595$	−1.26313e7	−	6.37141e6i	−1.46270	−	0.737807i
$596$		0			0
$597$	−3.25952e6	+	5.64566e6i	−0.374299	+	0.648304i
$598$		0			0
$599$	−9.17016e6	+	5.29439e6i	−1.04426	+	0.602905i	−0.921038	−	0.389473i	$-0.872657\pi$
$599$	−9.17016e6	+	5.29439e6i	−1.04426	+	0.602905i	−0.123225	+	0.992379i	$0.539324\pi$
$600$		0			0
$601$			2.68606e6i			0.303340i	0.988431	+	0.151670i	$0.0484652\pi$
$601$			2.68606e6i			0.303340i	−0.988431	+	0.151670i	$0.951535\pi$
$602$		0			0
$603$			2.68068e6i			0.300229i
$604$		0			0
$605$	8.54205e6	−	4.93175e6i	0.948797	−	0.547788i
$606$		0			0
$607$	4.01156e6	−	6.94823e6i	0.441918	−	0.765425i	−0.555914	−	0.831240i	$-0.687631\pi$
$607$	4.01156e6	−	6.94823e6i	0.441918	−	0.765425i	0.997832	+	0.0658153i	$0.0209648\pi$
$608$		0			0
$609$	−174030.	−	3.08092e6i	−0.0190143	−	0.336617i
$610$		0			0
$611$	−5.46462e6	−	3.15500e6i	−0.592185	−	0.341898i
$612$		0			0
$613$	−4.11201e6	−	7.12221e6i	−0.441980	−	0.765533i	0.555856	−	0.831279i	$-0.312391\pi$
$613$	−4.11201e6	−	7.12221e6i	−0.441980	−	0.765533i	−0.997836	+	0.0657460i	$0.979057\pi$
$614$		0			0
$615$	−1.20677e7			−1.28658
$616$		0			0
$617$	1.44777e7			1.53104			0.765519	−	0.643414i	$-0.222482\pi$
$617$	1.44777e7			1.53104			0.765519	+	0.643414i	$0.222482\pi$
$618$		0			0
$619$	−5.90572e6	−	1.02290e7i	−0.619507	−	1.07302i	−0.989576	−	0.144014i	$-0.953999\pi$
$619$	−5.90572e6	−	1.02290e7i	−0.619507	−	1.07302i	0.370068	−	0.929004i	$-0.379334\pi$
$620$		0			0
$621$	−1.20755e7	−	6.97177e6i	−1.25653	−	0.725460i
$622$		0			0
$623$	2.17947e6	+	3.32639e6i	0.224973	+	0.343363i
$624$		0			0
$625$	5.99774e6	−	1.03884e7i	0.614168	−	1.06377i
$626$		0			0
$627$	−2.10450e6	+	1.21503e6i	−0.213786	+	0.123430i
$628$		0			0
$629$		−	1.29712e7i		−	1.30723i
$630$		0			0
$631$			1.17328e7i			1.17308i	0.809920	+	0.586540i	$0.199510\pi$
$631$			1.17328e7i			1.17308i	−0.809920	+	0.586540i	$0.800490\pi$
$632$		0			0
$633$	−3.91404e6	+	2.25977e6i	−0.388254	+	0.224158i
$634$		0			0
$635$	1.28525e7	−	2.22613e7i	1.26490	−	2.19086i
$636$		0			0
$637$	−1.20437e7	+	8.90081e6i	−1.17601	+	0.869122i
$638$		0			0
$639$	−3.64231e6	−	2.10289e6i	−0.352879	−	0.203735i
$640$		0			0
$641$	380293.	+	658688.i	0.0365573	+	0.0633190i	0.883725	−	0.468006i	$-0.155028\pi$
$641$	380293.	+	658688.i	0.0365573	+	0.0633190i	−0.847168	+	0.531325i	$0.821694\pi$
$642$		0			0
$643$	−6.37262e6			−0.607842			−0.303921	−	0.952697i	$-0.598296\pi$
$643$	−6.37262e6			−0.607842			−0.303921	+	0.952697i	$0.598296\pi$
$644$		0			0
$645$	1.23525e7			1.16911
$646$		0			0
$647$	3.94167e6	+	6.82717e6i	0.370185	+	0.641180i	0.989594	−	0.143889i	$-0.0459608\pi$
$647$	3.94167e6	+	6.82717e6i	0.370185	+	0.641180i	−0.619408	+	0.785069i	$0.712628\pi$
$648$		0			0
$649$	−171728.	−	99147.3i	−0.0160040	−	0.00923994i
$650$		0			0
$651$	−5.99851e6	+	3.93025e6i	−0.554742	+	0.363470i
$652$		0			0
$653$	609680.	−	1.05600e6i	0.0559524	−	0.0969124i	−0.836693	−	0.547673i	$-0.815514\pi$
$653$	609680.	−	1.05600e6i	0.0559524	−	0.0969124i	0.892645	+	0.450761i	$0.148847\pi$
$654$		0			0
$655$	1.03683e7	−	5.98617e6i	0.944292	−	0.545187i
$656$		0			0
$657$		−	1.70075e6i		−	0.153719i
$658$		0			0
$659$		−	7.56715e6i		−	0.678764i	−0.940649	−	0.339382i	$-0.889782\pi$
$659$		−	7.56715e6i		−	0.678764i	0.940649	−	0.339382i	$-0.110218\pi$
$660$		0			0
$661$	−3.69564e6	+	2.13368e6i	−0.328993	+	0.189944i	−0.655394	−	0.755287i	$-0.727497\pi$
$661$	−3.69564e6	+	2.13368e6i	−0.328993	+	0.189944i	0.326401	+	0.945231i	$0.394164\pi$
$662$		0			0
$663$	−6.06637e6	+	1.05073e7i	−0.535976	+	0.928338i
$664$		0			0
$665$	1.64187e7	−	927431.i	1.43974	−	0.0813256i
$666$		0			0
$667$	−8.83689e6	−	5.10198e6i	−0.769103	−	0.444042i
$668$		0			0
$669$	4.74515e6	+	8.21885e6i	0.409907	+	0.709979i
$670$		0			0
$671$	3.67639e6			0.315221
$672$		0			0
$673$	−1.12574e7			−0.958075			−0.479038	−	0.877794i	$-0.659014\pi$
$673$	−1.12574e7			−0.958075			−0.479038	+	0.877794i	$0.659014\pi$
$674$		0			0
$675$	−3.67410e6	−	6.36372e6i	−0.310378	−	0.537591i
$676$		0			0
$677$	1.59919e7	+	9.23295e6i	1.34100	+	0.774228i	0.986954	−	0.161000i	$-0.0514719\pi$
$677$	1.59919e7	+	9.23295e6i	1.34100	+	0.774228i	0.354047	+	0.935228i	$0.384805\pi$
$678$		0			0
$679$	−4.09981e6	+	8.12787e6i	−0.341263	+	0.676554i
$680$		0			0
$681$	−4.14852e6	+	7.18545e6i	−0.342788	+	0.593726i
$682$		0			0
$683$	−1.37655e6	+	794751.i	−0.112912	+	0.0651898i	−0.555392	−	0.831588i	$-0.687432\pi$
$683$	−1.37655e6	+	794751.i	−0.112912	+	0.0651898i	0.442480	+	0.896778i	$0.354099\pi$
$684$		0			0
$685$		−	1.82814e7i		−	1.48861i
$686$		0			0
$687$		−	9.24944e6i		−	0.747694i
$688$		0			0
$689$	1.94174e7	−	1.12107e7i	1.55828	−	0.899671i
$690$		0			0
$691$	−2.78270e6	+	4.81977e6i	−0.221703	+	0.384000i	−0.955325	−	0.295557i	$-0.904495\pi$
$691$	−2.78270e6	+	4.81977e6i	−0.221703	+	0.384000i	0.733623	+	0.679557i	$0.237828\pi$
$692$		0			0
$693$	−1.45989e6	+	2.89424e6i	−0.115475	+	0.228929i
$694$		0			0
$695$	−1.18520e7	−	6.84277e6i	−0.930745	−	0.537366i
$696$		0			0
$697$	−1.42890e7	−	2.47493e7i	−1.11409	−	1.92966i
$698$		0			0
$699$	−6.06748e6			−0.469694
$700$		0			0
$701$	−1.20878e7			−0.929078			−0.464539	−	0.885553i	$-0.653780\pi$
$701$	−1.20878e7			−0.929078			−0.464539	+	0.885553i	$0.653780\pi$
$702$		0			0
$703$	7.53888e6	+	1.30577e7i	0.575332	+	0.996504i
$704$		0			0
$705$	−3.93898e6	−	2.27417e6i	−0.298477	−	0.172326i
$706$		0			0
$707$	8.26091e6	−	466629.i	0.621555	−	0.0351094i
$708$		0			0
$709$	7.90276e6	−	1.36880e7i	0.590423	−	1.02264i	−0.403752	−	0.914868i	$-0.632294\pi$
$709$	7.90276e6	−	1.36880e7i	0.590423	−	1.02264i	0.994175	−	0.107775i	$-0.0343725\pi$
$710$		0			0
$711$	2.42535e6	−	1.40028e6i	0.179929	−	0.103882i
$712$		0			0
$713$			2.37138e7i			1.74694i
$714$		0			0
$715$		−	9.81517e6i		−	0.718014i
$716$		0			0
$717$	−3.47297e6	+	2.00512e6i	−0.252292	+	0.145661i
$718$		0			0
$719$	1.05303e7	−	1.82391e7i	0.759661	−	1.31577i	−0.183362	−	0.983045i	$-0.558698\pi$
$719$	1.05303e7	−	1.82391e7i	0.759661	−	1.31577i	0.943023	−	0.332726i	$-0.107969\pi$
$720$		0			0
$721$	1.16801e7	−	7.65288e6i	0.836778	−	0.548260i
$722$		0			0
$723$	−2.60308e6	−	1.50289e6i	−0.185200	−	0.106925i
$724$		0			0
$725$	−2.68872e6	−	4.65701e6i	−0.189977	−	0.329050i
$726$		0			0
$727$	−1.07408e7			−0.753701			−0.376851	−	0.926274i	$-0.622993\pi$
$727$	−1.07408e7			−0.753701			−0.376851	+	0.926274i	$0.622993\pi$
$728$		0			0
$729$	6.75587e6			0.470828
$730$		0			0
$731$	1.46262e7	+	2.53333e7i	1.01237	+	1.75347i
$732$		0			0
$733$	−9.60458e6	−	5.54520e6i	−0.660265	−	0.381204i	0.132113	−	0.991235i	$-0.457824\pi$
$733$	−9.60458e6	−	5.54520e6i	−0.660265	−	0.381204i	−0.792378	+	0.610030i	$0.791157\pi$
$734$		0			0
$735$	−8.68127e6	+	6.41583e6i	−0.592741	+	0.438061i
$736$		0			0
$737$	−1.26359e6	+	2.18860e6i	−0.0856914	+	0.148422i
$738$		0			0
$739$	−8.76469e6	+	5.06030e6i	−0.590372	+	0.340851i	−0.765244	−	0.643740i	$-0.777382\pi$
$739$	−8.76469e6	+	5.06030e6i	−0.590372	+	0.340851i	0.174873	+	0.984591i	$0.444049\pi$
$740$		0			0
$741$		−	1.41032e7i		−	0.943563i
$742$		0			0
$743$		−	1.69668e7i		−	1.12753i	−0.825934	−	0.563766i	$-0.809352\pi$
$743$		−	1.69668e7i		−	1.12753i	0.825934	−	0.563766i	$-0.190648\pi$
$744$		0			0
$745$	−5.42883e6	+	3.13434e6i	−0.358357	+	0.206897i
$746$		0			0
$747$	−4.39609e6	+	7.61424e6i	−0.288247	+	0.499258i
$748$		0			0
$749$	9.37123e6	+	1.43028e7i	0.610368	+	0.931569i
$750$		0			0
$751$	662389.	+	382431.i	0.0428562	+	0.0247430i	0.521275	−	0.853389i	$-0.325457\pi$
$751$	662389.	+	382431.i	0.0428562	+	0.0247430i	−0.478419	+	0.878132i	$0.658790\pi$
$752$		0			0
$753$	3.14361e6	+	5.44489e6i	0.202042	+	0.349946i
$754$		0			0
$755$	1.31356e7			0.838650
$756$		0			0
$757$	−1.42566e7			−0.904223			−0.452112	−	0.891961i	$-0.649329\pi$
$757$	−1.42566e7			−0.904223			−0.452112	+	0.891961i	$0.649329\pi$
$758$		0			0
$759$	−2.63736e6	−	4.56804e6i	−0.166175	−	0.287823i
$760$		0			0
$761$	4.44249e6	+	2.56487e6i	0.278077	+	0.160548i	0.632552	−	0.774518i	$-0.282007\pi$
$761$	4.44249e6	+	2.56487e6i	0.278077	+	0.160548i	−0.354476	+	0.935065i	$0.615341\pi$
$762$		0			0
$763$	698655.	+	1.23686e7i	0.0434462	+	0.769145i
$764$		0			0
$765$	8.88604e6	−	1.53911e7i	0.548978	−	0.950857i
$766$		0			0
$767$	996644.	−	575413.i	0.0611719	−	0.0353176i
$768$		0			0
$769$			2.22364e7i			1.35596i	0.735079	+	0.677982i	$0.237145\pi$
$769$			2.22364e7i			1.35596i	−0.735079	+	0.677982i	$0.762855\pi$
$770$		0			0
$771$			2.72947e6i			0.165365i
$772$		0			0
$773$	−2.52779e7	+	1.45942e7i	−1.52157	+	0.878480i	−0.521896	+	0.853009i	$0.674775\pi$
$773$	−2.52779e7	+	1.45942e7i	−1.52157	+	0.878480i	−0.999676	+	0.0254706i	$0.991892\pi$
$774$		0			0
$775$	−6.24855e6	+	1.08228e7i	−0.373701	+	0.647270i
$776$		0			0
$777$	−8.83685e6	−	4.45743e6i	−0.525103	−	0.264869i
$778$		0			0
$779$	2.87687e7	+	1.66096e7i	1.69854	+	0.980654i
$780$		0			0
$781$	−1.98247e6	−	3.43374e6i	−0.116300	−	0.201437i
$782$		0			0
$783$	9.66054e6			0.563115
$784$		0			0
$785$	2.15369e7			1.24741
$786$		0			0
$787$	7.68832e6	+	1.33166e7i	0.442481	+	0.766399i	0.997873	−	0.0651893i	$-0.0207651\pi$
$787$	7.68832e6	+	1.33166e7i	0.442481	+	0.766399i	−0.555392	+	0.831589i	$0.687432\pi$
$788$		0			0
$789$	1.46988e7	+	8.48637e6i	0.840601	+	0.485321i
$790$		0			0
$791$	4.34690e6	+	2.19264e6i	0.247024	+	0.124602i
$792$		0			0
$793$	−1.06682e7	+	1.84778e7i	−0.602431	+	1.04344i
$794$		0			0
$795$	1.39964e7	−	8.08081e6i	0.785412	−	0.453458i
$796$		0			0
$797$			1.57429e7i			0.877887i	0.898515	+	0.438943i	$0.144647\pi$
$797$			1.57429e7i			0.877887i	−0.898515	+	0.438943i	$0.855353\pi$
$798$		0			0
$799$		−	1.07711e7i		−	0.596888i
$800$		0			0
$801$	−4.32645e6	+	2.49787e6i	−0.238259	+	0.137559i
$802$		0			0
$803$	801681.	−	1.38855e6i	0.0438746	−	0.0759930i
$804$		0			0
$805$	2.01309e6	+	3.56385e7i	0.109490	+	1.93834i
$806$		0			0
$807$	1.19974e7	+	6.92670e6i	0.648490	+	0.374406i
$808$		0			0
$809$	1.71161e7	+	2.96460e7i	0.919461	+	1.59255i	0.800235	+	0.599687i	$0.204708\pi$
$809$	1.71161e7	+	2.96460e7i	0.919461	+	1.59255i	0.119227	+	0.992867i	$0.461959\pi$
$810$		0			0
$811$	−2.08905e7			−1.11531			−0.557656	−	0.830072i	$-0.688299\pi$
$811$	−2.08905e7			−1.11531			−0.557656	+	0.830072i	$0.688299\pi$
$812$		0			0
$813$	−1.40265e7			−0.744256
$814$		0			0
$815$	−2.13248e7	−	3.69356e7i	−1.12458	−	1.94783i
$816$		0			0
$817$	−2.94475e7	−	1.70015e7i	−1.54345	−	0.891114i
$818$		0			0
$819$	−1.03103e7	−	1.57361e7i	−0.537110	−	0.819760i
$820$		0			0
$821$	−1.25323e7	+	2.17066e7i	−0.648894	+	1.12392i	0.334494	+	0.942398i	$0.391435\pi$
$821$	−1.25323e7	+	2.17066e7i	−0.648894	+	1.12392i	−0.983387	+	0.181519i	$0.941899\pi$
$822$		0			0
$823$	1.38348e7	−	7.98750e6i	0.711987	−	0.411066i	−0.0998094	−	0.995007i	$-0.531823\pi$
$823$	1.38348e7	−	7.98750e6i	0.711987	−	0.411066i	0.811796	+	0.583941i	$0.198490\pi$
$824$		0			0
$825$		−	2.77976e6i		−	0.142191i
$826$		0			0
$827$			8.71482e6i			0.443093i	0.975150	+	0.221546i	$0.0711104\pi$
$827$			8.71482e6i			0.443093i	−0.975150	+	0.221546i	$0.928890\pi$
$828$		0			0
$829$	1.71534e7	−	9.90349e6i	0.866887	−	0.500498i	0.000574741	−	1.00000i	$-0.499817\pi$
$829$	1.71534e7	−	9.90349e6i	0.866887	−	0.500498i	0.866313	+	0.499502i	$0.166484\pi$
$830$		0			0
$831$	8.40134e6	−	1.45515e7i	0.422033	−	0.730982i
$832$		0			0
$833$	−2.34373e7	−	1.02074e7i	−1.17029	−	0.509684i
$834$		0			0
$835$	2.09365e7	+	1.20877e7i	1.03917	+	0.599967i
$836$		0			0
$837$	−1.12255e7	−	1.94431e7i	−0.553848	−	0.959293i
$838$		0			0
$839$	−2.72849e6			−0.133819			−0.0669094	−	0.997759i	$-0.521314\pi$
$839$	−2.72849e6			−0.133819			−0.0669094	+	0.997759i	$0.521314\pi$
$840$		0			0
$841$	−1.34415e7			−0.655327
$842$		0			0
$843$	6.62560e6	+	1.14759e7i	0.321112	+	0.556182i
$844$		0			0
$845$	2.62621e7	+	1.51624e7i	1.26528	+	0.730511i
$846$		0			0
$847$	1.49080e7	−	9.76779e6i	0.714021	−	0.467830i
$848$		0			0
$849$	1.67186e6	−	2.89574e6i	0.0796032	−	0.137877i
$850$		0			0
$851$	−2.83432e7	+	1.63640e7i	−1.34161	+	0.774577i
$852$		0			0
$853$		−	1.50708e7i		−	0.709194i	−0.935019	−	0.354597i	$-0.884618\pi$
$853$		−	1.50708e7i		−	0.709194i	0.935019	−	0.354597i	$-0.115382\pi$
$854$		0			0
$855$			2.06584e7i			0.966453i
$856$		0			0
$857$	−1.45058e7	+	8.37495e6i	−0.674669	+	0.389521i	−0.797844	−	0.602864i	$-0.794026\pi$
$857$	−1.45058e7	+	8.37495e6i	−0.674669	+	0.389521i	0.123174	+	0.992385i	$0.460693\pi$
$858$		0			0
$859$	−4.71493e6	+	8.16649e6i	−0.218018	+	0.377618i	−0.954202	−	0.299163i	$-0.903292\pi$
$859$	−4.71493e6	+	8.16649e6i	−0.218018	+	0.377618i	0.736184	+	0.676781i	$0.236626\pi$
$860$		0			0
$861$	−2.17712e7	+	1.22978e6i	−1.00086	+	0.0565351i
$862$		0			0
$863$	−5.29346e6	−	3.05618e6i	−0.241943	−	0.139686i	0.374127	−	0.927378i	$-0.377943\pi$
$863$	−5.29346e6	−	3.05618e6i	−0.241943	−	0.139686i	−0.616069	+	0.787692i	$0.711276\pi$
$864$		0			0
$865$	2.69441e6	+	4.66685e6i	0.122440	+	0.212072i
$866$		0			0
$867$	−7.99967e6			−0.361430
$868$		0			0
$869$	2.64019e6			0.118600
$870$		0			0
$871$	−7.33339e6	−	1.27018e7i	−0.327536	−	0.567309i
$872$		0			0
$873$	−9.90370e6	−	5.71790e6i	−0.439807	−	0.253922i
$874$		0			0
$875$	4.61852e6	−	9.15621e6i	0.203931	−	0.404293i
$876$		0			0
$877$	9.54078e6	−	1.65251e7i	0.418875	−	0.725513i	−0.576951	−	0.816779i	$-0.695758\pi$
$877$	9.54078e6	−	1.65251e7i	0.418875	−	0.725513i	0.995827	+	0.0912652i	$0.0290911\pi$
$878$		0			0
$879$	−1.17218e7	+	6.76758e6i	−0.511707	+	0.295434i
$880$		0			0
$881$			1.21256e7i			0.526338i	0.964750	+	0.263169i	$0.0847676\pi$
$881$			1.21256e7i			0.526338i	−0.964750	+	0.263169i	$0.915232\pi$
$882$		0			0
$883$			2.06429e7i			0.890980i	0.895287	+	0.445490i	$0.146971\pi$
$883$			2.06429e7i			0.890980i	−0.895287	+	0.445490i	$0.853029\pi$
$884$		0			0
$885$	718395.	−	414766.i	0.0308323	−	0.0178010i
$886$		0			0
$887$	−1.87403e6	+	3.24591e6i	−0.0799774	+	0.138525i	−0.903240	−	0.429136i	$-0.858818\pi$
$887$	−1.87403e6	+	3.24591e6i	−0.0799774	+	0.138525i	0.823263	+	0.567661i	$0.192151\pi$
$888$		0			0
$889$	2.09185e7	−	4.14710e7i	0.887722	−	1.75991i
$890$		0			0
$891$	−937220.	−	541104.i	−0.0395501	−	0.0228343i
$892$		0			0
$893$	6.26019e6	+	1.08430e7i	0.262699	+	0.455009i
$894$		0			0
$895$	1.32962e7			0.554841
$896$		0			0
$897$	3.06125e7			1.27033
$898$		0			0
$899$	−8.21486e6	−	1.42286e7i	−0.339001	−	0.587167i
$900$		0			0
$901$	3.31454e7	+	1.91365e7i	1.36023	+	0.785326i
$902$		0			0
$903$	2.22849e7	−	1.25879e6i	0.909476	−	0.0513730i
$904$		0			0
$905$	1.08462e7	−	1.87861e7i	0.440205	−	0.762457i
$906$		0			0
$907$	3.62087e6	−	2.09051e6i	0.146149	−	0.0843790i	−0.425143	−	0.905126i	$-0.639776\pi$
$907$	3.62087e6	−	2.09051e6i	0.146149	−	0.0843790i	0.571291	+	0.820747i	$0.306443\pi$
$908$		0			0
$909$			1.03941e7i			0.417231i
$910$		0			0
$911$		−	1.56891e6i		−	0.0626329i	−0.999510	−	0.0313165i	$-0.990030\pi$
$911$		−	1.56891e6i		−	0.0626329i	0.999510	−	0.0313165i	$-0.00996997\pi$
$912$		0			0
$913$	−7.17822e6	+	4.14435e6i	−0.284997	+	0.164543i
$914$		0			0
$915$	−7.68978e6	+	1.33191e7i	−0.303641	+	0.525922i
$916$		0			0
$917$	1.80954e7	−	1.18562e7i	0.710631	−	0.465608i
$918$		0			0
$919$	−3.21820e7	−	1.85803e7i	−1.25697	−	0.725710i	−0.284483	−	0.958681i	$-0.591822\pi$
$919$	−3.21820e7	−	1.85803e7i	−1.25697	−	0.725710i	−0.972484	+	0.232971i	$0.925155\pi$
$920$		0			0
$921$	−4.57743e6	−	7.92835e6i	−0.177817	−	0.307988i
$922$		0			0
$923$	2.30110e7			0.889061
$924$		0			0
$925$	−1.72475e7			−0.662783
$926$		0			0
$927$	8.77092e6	+	1.51917e7i	0.335232	+	0.580639i
$928$		0			0
$929$	−2.02938e7	−	1.17166e7i	−0.771479	−	0.445414i	0.0619229	−	0.998081i	$-0.480277\pi$
$929$	−2.02938e7	−	1.17166e7i	−0.771479	−	0.445414i	−0.833402	+	0.552667i	$0.813610\pi$
$930$		0			0
$931$	2.95262e7	−	3.34633e6i	1.11643	−	0.126530i
$932$		0			0
$933$	−4.75654e6	+	8.23857e6i	−0.178890	+	0.309847i
$934$		0			0
$935$	1.45097e7	−	8.37720e6i	0.542788	−	0.313379i
$936$		0			0
$937$			1.07872e7i			0.401383i	0.979654	+	0.200692i	$0.0643189\pi$
$937$			1.07872e7i			0.401383i	−0.979654	+	0.200692i	$0.935681\pi$
$938$		0			0
$939$		−	3.19907e6i		−	0.118402i
$940$		0			0
$941$	4.34621e6	−	2.50928e6i	0.160006	−	0.0923795i	−0.417859	−	0.908512i	$-0.637219\pi$
$941$	4.34621e6	−	2.50928e6i	0.160006	−	0.0923795i	0.577865	+	0.816132i	$0.303886\pi$
$942$		0			0
$943$	−3.60530e7	+	6.24456e7i	−1.32027	+	2.28677i
$944$		0			0
$945$	−1.85208e7	−	2.82672e7i	−0.674653	−	1.02968i
$946$		0			0
$947$	6.30773e6	+	3.64177e6i	0.228559	+	0.131959i	0.609907	−	0.792473i	$-0.291207\pi$
$947$	6.30773e6	+	3.64177e6i	0.228559	+	0.131959i	−0.381348	+	0.924432i	$0.624540\pi$
$948$		0			0
$949$	4.65265e6	+	8.05863e6i	0.167701	+	0.290466i
$950$		0			0
$951$	3.88506e6			0.139298
$952$		0			0
$953$	−3.16792e6			−0.112991			−0.0564953	−	0.998403i	$-0.517993\pi$
$953$	−3.16792e6			−0.112991			−0.0564953	+	0.998403i	$0.517993\pi$
$954$		0			0
$955$	−3.23035e7	−	5.59514e7i	−1.14615	−	1.98519i
$956$		0			0
$957$	3.16489e6	+	1.82725e6i	0.111707	+	0.0644938i
$958$		0			0
$959$	−1.86299e6	−	3.29812e7i	−0.0654128	−	1.15803i
$960$		0			0
$961$	−4.77661e6	+	8.27333e6i	−0.166844	+	0.288983i
$962$		0			0
$963$	−1.86028e7	+	1.07403e7i	−0.646415	+	0.373208i
$964$		0			0
$965$		−	4.42009e7i		−	1.52796i
$966$		0			0
$967$		−	4.48807e7i		−	1.54345i	−0.635954	−	0.771727i	$-0.719393\pi$
$967$		−	4.48807e7i		−	1.54345i	0.635954	−	0.771727i	$-0.280607\pi$
$968$		0			0
$969$	2.08486e7	−	1.20370e7i	0.713294	−	0.411820i
$970$		0			0
$971$	−1.70288e7	+	2.94948e7i	−0.579611	+	1.00392i	0.415912	+	0.909405i	$0.363462\pi$
$971$	−1.70288e7	+	2.94948e7i	−0.579611	+	1.00392i	−0.995524	+	0.0945115i	$0.969871\pi$
$972$		0			0
$973$	−2.20794e7	−	1.11372e7i	−0.747662	−	0.377131i
$974$		0			0
$975$	1.39713e7	+	8.06633e6i	0.470679	+	0.271747i
$976$		0			0
$977$	1.84492e7	+	3.19549e7i	0.618359	+	1.07103i	0.989785	+	0.142567i	$0.0455357\pi$
$977$	1.84492e7	+	3.19549e7i	0.618359	+	1.07103i	−0.371426	+	0.928463i	$0.621131\pi$
$978$		0			0
$979$	−4.70967e6			−0.157049
$980$		0			0
$981$	−1.55624e7			−0.516303
$982$		0			0
$983$	−2.30167e7	−	3.98660e7i	−0.759729	−	1.31589i	−0.942989	−	0.332825i	$-0.891998\pi$
$983$	−2.30167e7	−	3.98660e7i	−0.759729	−	1.31589i	0.183260	−	0.983065i	$-0.441335\pi$
$984$		0			0
$985$	4.34208e7	+	2.50690e7i	1.42596	+	0.823279i
$986$		0			0
$987$	−7.33801e6	−	3.70139e6i	−0.239765	−	0.120941i
$988$		0			0
$989$	3.69037e7	−	6.39190e7i	1.19972	−	2.07797i
$990$		0			0
$991$	−164564.	+	95010.9i	−0.00532292	+	0.00307319i	−0.502659	−	0.864485i	$-0.667645\pi$
$991$	−164564.	+	95010.9i	−0.00532292	+	0.00307319i	0.497336	+	0.867558i	$0.334312\pi$
$992$		0			0
$993$		−	2.94713e7i		−	0.948477i
$994$		0			0
$995$			5.22459e7i			1.67300i
$996$		0			0
$997$	1.67476e7	−	9.66925e6i	0.533599	−	0.308074i	−0.208881	−	0.977941i	$-0.566982\pi$
$997$	1.67476e7	−	9.66925e6i	0.533599	−	0.308074i	0.742481	+	0.669867i	$0.233649\pi$
$998$		0			0
$999$	1.54925e7	−	2.68338e7i	0.491142	−	0.850683i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.p.b.47.3	yes	14
4.3	odd	2		112.6.p.c.47.5	yes	14
7.2	even	3		784.6.f.d.783.10		14
7.3	odd	6		112.6.p.c.31.5	yes	14
7.5	odd	6		784.6.f.c.783.5		14
28.3	even	6	inner	112.6.p.b.31.3	✓	14
28.19	even	6		784.6.f.d.783.9		14
28.23	odd	6		784.6.f.c.783.6		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.p.b.31.3	✓	14	28.3	even	6	inner
112.6.p.b.47.3	yes	14	1.1	even	1	trivial
112.6.p.c.31.5	yes	14	7.3	odd	6
112.6.p.c.47.5	yes	14	4.3	odd	2
784.6.f.c.783.5		14	7.5	odd	6
784.6.f.c.783.6		14	28.23	odd	6
784.6.f.d.783.9		14	28.19	even	6
784.6.f.d.783.10		14	7.2	even	3

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.47608 - 7.75280i) q^{3} +(-62.1336 - 35.8729i) q^{5} +(-115.750 - 58.3859i) q^{7} +(81.4294 - 141.040i) q^{9} +(132.963 - 76.7665i) q^{11} +891.047i q^{13} +642.279i q^{15} +(-1317.23 + 760.503i) q^{17} +(884.013 - 1531.15i) q^{19} +(65.4523 + 1158.73i) q^{21} +(3323.54 + 1918.85i) q^{23} +(1011.22 + 1751.49i) q^{25} -3633.31 q^{27} -2658.88 q^{29} +(3089.59 + 5351.33i) q^{31} +(-1190.31 - 687.226i) q^{33} +(5097.50 + 7780.02i) q^{35} +(-4264.01 + 7385.48i) q^{37} +(6908.10 - 3988.39i) q^{39} +18788.9i q^{41} -19232.2i q^{43} +(-10119.0 + 5842.21i) q^{45} +(-3540.78 + 6132.81i) q^{47} +(9989.16 + 13516.4i) q^{49} +(11792.1 + 6808.14i) q^{51} +(-12581.5 - 21791.7i) q^{53} -11015.3 q^{55} -15827.6 q^{57} +(-645.772 - 1118.51i) q^{59} +(20737.2 + 11972.6i) q^{61} +(-17660.2 + 11571.0i) q^{63} +(31964.4 - 55363.9i) q^{65} +(-14254.9 + 8230.08i) q^{67} -34355.6i q^{69} -25824.7i q^{71} +(9044.00 - 5221.56i) q^{73} +(9052.64 - 15679.6i) q^{75} +(-19872.6 + 1122.53i) q^{77} +(14892.4 + 8598.11i) q^{79} +(-3524.35 - 6104.36i) q^{81} -53986.4 q^{83} +109126. q^{85} +(11901.4 + 20613.8i) q^{87} +(-26565.6 - 15337.7i) q^{89} +(52024.6 - 103139. i) q^{91} +(27658.5 - 47906.0i) q^{93} +(-109854. + 63424.1i) q^{95} -70219.1i q^{97} -25004.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 9 q^{3} + 33 q^{5} - 28 q^{7} - 538 q^{9} + 333 q^{11} + 801 q^{17} - 2135 q^{19} + 2017 q^{21} + 2667 q^{23} + 5434 q^{25} + 17910 q^{27} + 684 q^{29} + 3119 q^{31} + 29013 q^{33} - 2247 q^{35}+ \cdots - 124833 q^{95}+O(q^{100}) \) 14 * q - 9 * q^3 + 33 * q^5 - 28 * q^7 - 538 * q^9 + 333 * q^11 + 801 * q^17 - 2135 * q^19 + 2017 * q^21 + 2667 * q^23 + 5434 * q^25 + 17910 * q^27 + 684 * q^29 + 3119 * q^31 + 29013 * q^33 - 2247 * q^35 + 3431 * q^37 + 2424 * q^39 - 28086 * q^45 + 8409 * q^47 - 24706 * q^49 + 24657 * q^51 + 1707 * q^53 - 56622 * q^55 + 68450 * q^57 - 49305 * q^59 - 21843 * q^61 + 164142 * q^63 + 17448 * q^65 - 73263 * q^67 - 77043 * q^73 - 147834 * q^75 + 22287 * q^77 + 206523 * q^79 - 145567 * q^81 - 305400 * q^83 - 176454 * q^85 - 95034 * q^87 + 171381 * q^89 + 174816 * q^91 + 292873 * q^93 - 124833 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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