LMFDB - Embedded newform 8039.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.1917381849$
Analytic rank:	$0$
Dimension:	$391$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78421 q^{2} +2.45333 q^{3} +5.75180 q^{4} +2.41719 q^{5} -6.83056 q^{6} +1.41777 q^{7} -10.4458 q^{8} +3.01881 q^{9} +O(q^{10})$	\(q-2.78421 q^{2} +2.45333 q^{3} +5.75180 q^{4} +2.41719 q^{5} -6.83056 q^{6} +1.41777 q^{7} -10.4458 q^{8} +3.01881 q^{9} -6.72994 q^{10} +4.47290 q^{11} +14.1110 q^{12} -0.0690116 q^{13} -3.94737 q^{14} +5.93015 q^{15} +17.5796 q^{16} +6.39154 q^{17} -8.40498 q^{18} -0.961522 q^{19} +13.9032 q^{20} +3.47826 q^{21} -12.4535 q^{22} +7.43108 q^{23} -25.6269 q^{24} +0.842792 q^{25} +0.192143 q^{26} +0.0461415 q^{27} +8.15474 q^{28} +1.15632 q^{29} -16.5107 q^{30} +7.65464 q^{31} -28.0536 q^{32} +10.9735 q^{33} -17.7954 q^{34} +3.42702 q^{35} +17.3636 q^{36} -5.55557 q^{37} +2.67707 q^{38} -0.169308 q^{39} -25.2494 q^{40} +8.12499 q^{41} -9.68418 q^{42} +10.4792 q^{43} +25.7272 q^{44} +7.29702 q^{45} -20.6897 q^{46} -0.711758 q^{47} +43.1285 q^{48} -4.98992 q^{49} -2.34651 q^{50} +15.6805 q^{51} -0.396941 q^{52} -10.8485 q^{53} -0.128467 q^{54} +10.8118 q^{55} -14.8097 q^{56} -2.35893 q^{57} -3.21944 q^{58} -2.92814 q^{59} +34.1090 q^{60} +7.87866 q^{61} -21.3121 q^{62} +4.27998 q^{63} +42.9479 q^{64} -0.166814 q^{65} -30.5524 q^{66} -4.88448 q^{67} +36.7628 q^{68} +18.2309 q^{69} -9.54153 q^{70} +8.71125 q^{71} -31.5338 q^{72} +4.37166 q^{73} +15.4679 q^{74} +2.06764 q^{75} -5.53048 q^{76} +6.34156 q^{77} +0.471388 q^{78} +2.28633 q^{79} +42.4932 q^{80} -8.94322 q^{81} -22.6216 q^{82} -0.0850534 q^{83} +20.0062 q^{84} +15.4495 q^{85} -29.1763 q^{86} +2.83684 q^{87} -46.7229 q^{88} -16.4557 q^{89} -20.3164 q^{90} -0.0978428 q^{91} +42.7421 q^{92} +18.7793 q^{93} +1.98168 q^{94} -2.32418 q^{95} -68.8247 q^{96} -15.8136 q^{97} +13.8930 q^{98} +13.5028 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) $ 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.78421	−1.96873	−0.984365	−	0.176140i	$-0.943639\pi$
$2$	−2.78421	−1.96873	−0.984365	+	0.176140i	$0.943639\pi$
$3$	2.45333	1.41643	0.708214	−	0.705998i	$-0.249501\pi$
$3$	2.45333	1.41643	0.708214	+	0.705998i	$0.249501\pi$
$4$	5.75180	2.87590
$5$	2.41719	1.08100	0.540499	−	0.841344i	$-0.318235\pi$
$5$	2.41719	1.08100	0.540499	+	0.841344i	$0.318235\pi$
$6$	−6.83056	−2.78857
$7$	1.41777	0.535868	0.267934	−	0.963437i	$-0.413659\pi$
$7$	1.41777	0.535868	0.267934	+	0.963437i	$0.413659\pi$
$8$	−10.4458	−3.69314
$9$	3.01881	1.00627
$10$	−6.72994	−2.12820
$11$	4.47290	1.34863	0.674315	−	0.738444i	$-0.264439\pi$
$11$	4.47290	1.34863	0.674315	+	0.738444i	$0.264439\pi$
$12$	14.1110	4.07351
$13$	−0.0690116	−0.0191404	−0.00957019	−	0.999954i	$-0.503046\pi$
$13$	−0.0690116	−0.0191404	−0.00957019	+	0.999954i	$0.503046\pi$
$14$	−3.94737	−1.05498
$15$	5.93015	1.53116
$16$	17.5796	4.39490
$17$	6.39154	1.55018	0.775088	−	0.631854i	$-0.217706\pi$
$17$	6.39154	1.55018	0.775088	+	0.631854i	$0.217706\pi$
$18$	−8.40498	−1.98107
$19$	−0.961522	−0.220588	−0.110294	−	0.993899i	$-0.535179\pi$
$19$	−0.961522	−0.220588	−0.110294	+	0.993899i	$0.535179\pi$
$20$	13.9032	3.10884
$21$	3.47826	0.759018
$22$	−12.4535	−2.65509
$23$	7.43108	1.54949	0.774744	−	0.632275i	$-0.217879\pi$
$23$	7.43108	1.54949	0.774744	+	0.632275i	$0.217879\pi$
$24$	−25.6269	−5.23107
$25$	0.842792	0.168558
$26$	0.192143	0.0376823
$27$	0.0461415	0.00887994
$28$	8.15474	1.54110
$29$	1.15632	0.214724	0.107362	−	0.994220i	$-0.465760\pi$
$29$	1.15632	0.214724	0.107362	+	0.994220i	$0.465760\pi$
$30$	−16.5107	−3.01444
$31$	7.65464	1.37481	0.687407	−	0.726272i	$-0.258749\pi$
$31$	7.65464	1.37481	0.687407	+	0.726272i	$0.258749\pi$
$32$	−28.0536	−4.95923
$33$	10.9735	1.91024
$34$	−17.7954	−3.05188
$35$	3.42702	0.579272
$36$	17.3636	2.89393
$37$	−5.55557	−0.913330	−0.456665	−	0.889639i	$-0.650956\pi$
$37$	−5.55557	−0.913330	−0.456665	+	0.889639i	$0.650956\pi$
$38$	2.67707	0.434279
$39$	−0.169308	−0.0271110
$40$	−25.2494	−3.99228
$41$	8.12499	1.26891	0.634455	−	0.772960i	$-0.281225\pi$
$41$	8.12499	1.26891	0.634455	+	0.772960i	$0.281225\pi$
$42$	−9.68418	−1.49430
$43$	10.4792	1.59807	0.799034	−	0.601286i	$-0.205345\pi$
$43$	10.4792	1.59807	0.799034	+	0.601286i	$0.205345\pi$
$44$	25.7272	3.87853
$45$	7.29702	1.08778
$46$	−20.6897	−3.05052
$47$	−0.711758	−0.103821	−0.0519103	−	0.998652i	$-0.516531\pi$
$47$	−0.711758	−0.103821	−0.0519103	+	0.998652i	$0.516531\pi$
$48$	43.1285	6.22506
$49$	−4.98992	−0.712846
$50$	−2.34651	−0.331846
$51$	15.6805	2.19571
$52$	−0.396941	−0.0550458
$53$	−10.8485	−1.49016	−0.745078	−	0.666977i	$-0.767588\pi$
$53$	−10.8485	−1.49016	−0.745078	+	0.666977i	$0.767588\pi$
$54$	−0.128467	−0.0174822
$55$	10.8118	1.45787
$56$	−14.8097	−1.97903
$57$	−2.35893	−0.312447
$58$	−3.21944	−0.422734
$59$	−2.92814	−0.381211	−0.190606	−	0.981667i	$-0.561045\pi$
$59$	−2.92814	−0.381211	−0.190606	+	0.981667i	$0.561045\pi$
$60$	34.1090	4.40345
$61$	7.87866	1.00876	0.504379	−	0.863482i	$-0.331721\pi$
$61$	7.87866	1.00876	0.504379	+	0.863482i	$0.331721\pi$
$62$	−21.3121	−2.70664
$63$	4.27998	0.539227
$64$	42.9479	5.36849
$65$	−0.166814	−0.0206907
$66$	−30.5524	−3.76074
$67$	−4.88448	−0.596735	−0.298367	−	0.954451i	$-0.596442\pi$
$67$	−4.88448	−0.596735	−0.298367	+	0.954451i	$0.596442\pi$
$68$	36.7628	4.45815
$69$	18.2309	2.19474
$70$	−9.54153	−1.14043
$71$	8.71125	1.03384	0.516918	−	0.856035i	$-0.327079\pi$
$71$	8.71125	1.03384	0.516918	+	0.856035i	$0.327079\pi$
$72$	−31.5338	−3.71629
$73$	4.37166	0.511664	0.255832	−	0.966721i	$-0.417651\pi$
$73$	4.37166	0.511664	0.255832	+	0.966721i	$0.417651\pi$
$74$	15.4679	1.79810
$75$	2.06764	0.238751
$76$	−5.53048	−0.634389
$77$	6.34156	0.722687
$78$	0.471388	0.0533742
$79$	2.28633	0.257232	0.128616	−	0.991694i	$-0.458947\pi$
$79$	2.28633	0.257232	0.128616	+	0.991694i	$0.458947\pi$
$80$	42.4932	4.75088
$81$	−8.94322	−0.993691
$82$	−22.6216	−2.49814
$83$	−0.0850534	−0.00933582	−0.00466791	−	0.999989i	$-0.501486\pi$
$83$	−0.0850534	−0.00933582	−0.00466791	+	0.999989i	$0.501486\pi$
$84$	20.0062	2.18286
$85$	15.4495	1.67574
$86$	−29.1763	−3.14617
$87$	2.83684	0.304141
$88$	−46.7229	−4.98068
$89$	−16.4557	−1.74430	−0.872150	−	0.489239i	$-0.837274\pi$
$89$	−16.4557	−1.74430	−0.872150	+	0.489239i	$0.837274\pi$
$90$	−20.3164	−2.14154
$91$	−0.0978428	−0.0102567
$92$	42.7421	4.45617
$93$	18.7793	1.94733
$94$	1.98168	0.204395
$95$	−2.32418	−0.238456
$96$	−68.8247	−7.02439
$97$	−15.8136	−1.60563	−0.802813	−	0.596231i	$-0.796664\pi$
$97$	−15.8136	−1.60563	−0.802813	+	0.596231i	$0.796664\pi$
$98$	13.8930	1.40340
$99$	13.5028	1.35709
$100$	4.84757	0.484757
$101$	−17.2553	−1.71696	−0.858482	−	0.512844i	$-0.828592\pi$
$101$	−17.2553	−1.71696	−0.858482	+	0.512844i	$0.828592\pi$
$102$	−43.6578	−4.32277
$103$	−3.90151	−0.384428	−0.192214	−	0.981353i	$-0.561567\pi$
$103$	−3.90151	−0.384428	−0.192214	+	0.981353i	$0.561567\pi$
$104$	0.720880	0.0706881
$105$	8.40760	0.820498
$106$	30.2045	2.93372
$107$	−14.5399	−1.40563	−0.702814	−	0.711374i	$-0.748073\pi$
$107$	−14.5399	−1.40563	−0.702814	+	0.711374i	$0.748073\pi$
$108$	0.265397	0.0255378
$109$	16.2879	1.56010	0.780049	−	0.625719i	$-0.215194\pi$
$109$	16.2879	1.56010	0.780049	+	0.625719i	$0.215194\pi$
$110$	−30.1024	−2.87015
$111$	−13.6296	−1.29367
$112$	24.9239	2.35508
$113$	−2.72498	−0.256344	−0.128172	−	0.991752i	$-0.540911\pi$
$113$	−2.72498	−0.256344	−0.128172	+	0.991752i	$0.540911\pi$
$114$	6.56773	0.615125
$115$	17.9623	1.67499
$116$	6.65095	0.617525
$117$	−0.208333	−0.0192604
$118$	8.15254	0.750502
$119$	9.06175	0.830689
$120$	−61.9450	−5.65478
$121$	9.00684	0.818804
$122$	−21.9358	−1.98597
$123$	19.9332	1.79732
$124$	44.0280	3.95383
$125$	−10.0487	−0.898787
$126$	−11.9164	−1.06159
$127$	−22.2141	−1.97118	−0.985590	−	0.169153i	$-0.945897\pi$
$127$	−22.2141	−1.97118	−0.985590	+	0.169153i	$0.945897\pi$
$128$	−63.4685	−5.60988
$129$	25.7090	2.26355
$130$	0.464444	0.0407345
$131$	−16.2734	−1.42181	−0.710906	−	0.703287i	$-0.751715\pi$
$131$	−16.2734	−1.42181	−0.710906	+	0.703287i	$0.751715\pi$
$132$	63.1173	5.49365
$133$	−1.36322	−0.118206
$134$	13.5994	1.17481
$135$	0.111533	0.00959921
$136$	−66.7646	−5.72502
$137$	−11.6418	−0.994629	−0.497315	−	0.867570i	$-0.665681\pi$
$137$	−11.6418	−0.994629	−0.497315	+	0.867570i	$0.665681\pi$
$138$	−50.7585	−4.32085
$139$	17.9123	1.51930	0.759650	−	0.650332i	$-0.225370\pi$
$139$	17.9123	1.51930	0.759650	+	0.650332i	$0.225370\pi$
$140$	19.7115	1.66593
$141$	−1.74618	−0.147054
$142$	−24.2539	−2.03534
$143$	−0.308682	−0.0258133
$144$	53.0694	4.42245
$145$	2.79505	0.232116
$146$	−12.1716	−1.00733
$147$	−12.2419	−1.00970
$148$	−31.9545	−2.62665
$149$	−17.8871	−1.46537	−0.732683	−	0.680570i	$-0.761732\pi$
$149$	−17.8871	−1.46537	−0.732683	+	0.680570i	$0.761732\pi$
$150$	−5.75675	−0.470036
$151$	−4.72471	−0.384491	−0.192246	−	0.981347i	$-0.561577\pi$
$151$	−4.72471	−0.384491	−0.192246	+	0.981347i	$0.561577\pi$
$152$	10.0438	0.814663
$153$	19.2948	1.55989
$154$	−17.6562	−1.42278
$155$	18.5027	1.48617
$156$	−0.973826	−0.0779684
$157$	−14.8690	−1.18668	−0.593339	−	0.804953i	$-0.702191\pi$
$157$	−14.8690	−1.18668	−0.593339	+	0.804953i	$0.702191\pi$
$158$	−6.36560	−0.506420
$159$	−26.6149	−2.11070
$160$	−67.8109	−5.36092
$161$	10.5356	0.830320
$162$	24.8998	1.95631
$163$	−6.91586	−0.541692	−0.270846	−	0.962623i	$-0.587303\pi$
$163$	−6.91586	−0.541692	−0.270846	+	0.962623i	$0.587303\pi$
$164$	46.7333	3.64926
$165$	26.5250	2.06497
$166$	0.236806	0.0183797
$167$	17.8076	1.37799	0.688996	−	0.724765i	$-0.258052\pi$
$167$	17.8076	1.37799	0.688996	+	0.724765i	$0.258052\pi$
$168$	−36.3331	−2.80316
$169$	−12.9952	−0.999634
$170$	−43.0147	−3.29908
$171$	−2.90265	−0.221971
$172$	60.2744	4.59588
$173$	4.92561	0.374487	0.187244	−	0.982314i	$-0.440045\pi$
$173$	4.92561	0.374487	0.187244	+	0.982314i	$0.440045\pi$
$174$	−7.89835	−0.598772
$175$	1.19489	0.0903250
$176$	78.6318	5.92709
$177$	−7.18368	−0.539958
$178$	45.8160	3.43405
$179$	−24.5610	−1.83578	−0.917888	−	0.396840i	$-0.870107\pi$
$179$	−24.5610	−1.83578	−0.917888	+	0.396840i	$0.870107\pi$
$180$	41.9710	3.12833
$181$	−13.1086	−0.974355	−0.487178	−	0.873303i	$-0.661974\pi$
$181$	−13.1086	−0.974355	−0.487178	+	0.873303i	$0.661974\pi$
$182$	0.272414	0.0201927
$183$	19.3289	1.42883
$184$	−77.6234	−5.72248
$185$	−13.4289	−0.987309
$186$	−52.2855	−3.83376
$187$	28.5887	2.09061
$188$	−4.09389	−0.298578
$189$	0.0654182	0.00475847
$190$	6.47099	0.469455
$191$	−0.459794	−0.0332695	−0.0166348	−	0.999862i	$-0.505295\pi$
$191$	−0.459794	−0.0332695	−0.0166348	+	0.999862i	$0.505295\pi$
$192$	105.365	7.60408
$193$	9.83531	0.707961	0.353981	−	0.935253i	$-0.384828\pi$
$193$	9.83531	0.707961	0.353981	+	0.935253i	$0.384828\pi$
$194$	44.0283	3.16105
$195$	−0.409249	−0.0293069
$196$	−28.7010	−2.05007
$197$	−18.2287	−1.29874	−0.649372	−	0.760471i	$-0.724968\pi$
$197$	−18.2287	−1.29874	−0.649372	+	0.760471i	$0.724968\pi$
$198$	−37.5946	−2.67173
$199$	−4.66302	−0.330553	−0.165276	−	0.986247i	$-0.552852\pi$
$199$	−4.66302	−0.330553	−0.165276	+	0.986247i	$0.552852\pi$
$200$	−8.80362	−0.622510
$201$	−11.9832	−0.845232
$202$	48.0422	3.38024
$203$	1.63940	0.115064
$204$	90.1912	6.31465
$205$	19.6396	1.37169
$206$	10.8626	0.756834
$207$	22.4330	1.55920
$208$	−1.21320	−0.0841200
$209$	−4.30079	−0.297492
$210$	−23.4085	−1.61534
$211$	7.53923	0.519022	0.259511	−	0.965740i	$-0.416439\pi$
$211$	7.53923	0.519022	0.259511	+	0.965740i	$0.416439\pi$
$212$	−62.3984	−4.28554
$213$	21.3715	1.46435
$214$	40.4821	2.76730
$215$	25.3303	1.72751
$216$	−0.481984	−0.0327949
$217$	10.8525	0.736718
$218$	−45.3488	−3.07141
$219$	10.7251	0.724735
$220$	62.1875	4.19268
$221$	−0.441090	−0.0296710
$222$	37.9477	2.54688
$223$	25.5530	1.71116	0.855578	−	0.517674i	$-0.173202\pi$
$223$	25.5530	1.71116	0.855578	+	0.517674i	$0.173202\pi$
$224$	−39.7737	−2.65749
$225$	2.54423	0.169615
$226$	7.58690	0.504673
$227$	19.2076	1.27485	0.637426	−	0.770511i	$-0.279999\pi$
$227$	19.2076	1.27485	0.637426	+	0.770511i	$0.279999\pi$
$228$	−13.5681	−0.898567
$229$	20.1742	1.33315	0.666573	−	0.745440i	$-0.267761\pi$
$229$	20.1742	1.33315	0.666573	+	0.745440i	$0.267761\pi$
$230$	−50.0108	−3.29761
$231$	15.5579	1.02363
$232$	−12.0787	−0.793006
$233$	−8.97571	−0.588018	−0.294009	−	0.955803i	$-0.594990\pi$
$233$	−8.97571	−0.588018	−0.294009	+	0.955803i	$0.594990\pi$
$234$	0.580041	0.0379185
$235$	−1.72045	−0.112230
$236$	−16.8421	−1.09633
$237$	5.60910	0.364350
$238$	−25.2298	−1.63540
$239$	12.6156	0.816034	0.408017	−	0.912974i	$-0.366220\pi$
$239$	12.6156	0.816034	0.408017	+	0.912974i	$0.366220\pi$
$240$	104.250	6.72928
$241$	11.0376	0.710997	0.355499	−	0.934677i	$-0.384311\pi$
$241$	11.0376	0.710997	0.355499	+	0.934677i	$0.384311\pi$
$242$	−25.0769	−1.61200
$243$	−22.0791	−1.41637
$244$	45.3164	2.90109
$245$	−12.0616	−0.770586
$246$	−55.4982	−3.53844
$247$	0.0663562	0.00422214
$248$	−79.9587	−5.07738
$249$	−0.208664	−0.0132235
$250$	27.9778	1.76947
$251$	11.1330	0.702712	0.351356	−	0.936242i	$-0.385721\pi$
$251$	11.1330	0.702712	0.351356	+	0.936242i	$0.385721\pi$
$252$	24.6176	1.55076
$253$	33.2385	2.08969
$254$	61.8485	3.88072
$255$	37.9028	2.37356
$256$	90.8135	5.67585
$257$	−14.3573	−0.895583	−0.447791	−	0.894138i	$-0.647789\pi$
$257$	−14.3573	−0.895583	−0.447791	+	0.894138i	$0.647789\pi$
$258$	−71.5791	−4.45632
$259$	−7.87654	−0.489424
$260$	−0.959481	−0.0595045
$261$	3.49072	0.216070
$262$	45.3084	2.79916
$263$	2.44684	0.150879	0.0754393	−	0.997150i	$-0.475964\pi$
$263$	2.44684	0.150879	0.0754393	+	0.997150i	$0.475964\pi$
$264$	−114.627	−7.05478
$265$	−26.2229	−1.61086
$266$	3.79548	0.232716
$267$	−40.3712	−2.47067
$268$	−28.0946	−1.71615
$269$	−11.3803	−0.693868	−0.346934	−	0.937889i	$-0.612777\pi$
$269$	−11.3803	−0.693868	−0.346934	+	0.937889i	$0.612777\pi$
$270$	−0.310530	−0.0188982
$271$	−30.5699	−1.85699	−0.928496	−	0.371343i	$-0.878897\pi$
$271$	−30.5699	−1.85699	−0.928496	+	0.371343i	$0.878897\pi$
$272$	112.361	6.81286
$273$	−0.240040	−0.0145279
$274$	32.4133	1.95816
$275$	3.76973	0.227323
$276$	104.860	6.31185
$277$	−3.40667	−0.204687	−0.102343	−	0.994749i	$-0.532634\pi$
$277$	−3.40667	−0.204687	−0.102343	+	0.994749i	$0.532634\pi$
$278$	−49.8715	−2.99109
$279$	23.1079	1.38343
$280$	−35.7979	−2.13933
$281$	−22.2667	−1.32832	−0.664161	−	0.747589i	$-0.731211\pi$
$281$	−22.2667	−1.32832	−0.664161	+	0.747589i	$0.731211\pi$
$282$	4.86171	0.289511
$283$	13.1839	0.783702	0.391851	−	0.920029i	$-0.371835\pi$
$283$	13.1839	0.783702	0.391851	+	0.920029i	$0.371835\pi$
$284$	50.1054	2.97321
$285$	−5.70196	−0.337755
$286$	0.859434	0.0508194
$287$	11.5194	0.679968
$288$	−84.6885	−4.99032
$289$	23.8518	1.40304
$290$	−7.78200	−0.456975
$291$	−38.7959	−2.27425
$292$	25.1449	1.47149
$293$	−5.75375	−0.336138	−0.168069	−	0.985775i	$-0.553753\pi$
$293$	−5.75375	−0.336138	−0.168069	+	0.985775i	$0.553753\pi$
$294$	34.0840	1.98782
$295$	−7.07786	−0.412089
$296$	58.0323	3.37306
$297$	0.206386	0.0119758
$298$	49.8013	2.88491
$299$	−0.512831	−0.0296578
$300$	11.8927	0.686624
$301$	14.8572	0.856353
$302$	13.1546	0.756960
$303$	−42.3328	−2.43196
$304$	−16.9032	−0.969463
$305$	19.0442	1.09047
$306$	−53.7207	−3.07101
$307$	−2.71134	−0.154744	−0.0773721	−	0.997002i	$-0.524653\pi$
$307$	−2.71134	−0.154744	−0.0773721	+	0.997002i	$0.524653\pi$
$308$	36.4754	2.07838
$309$	−9.57169	−0.544514
$310$	−51.5153	−2.92587
$311$	16.8505	0.955507	0.477753	−	0.878494i	$-0.341451\pi$
$311$	16.8505	0.955507	0.477753	+	0.878494i	$0.341451\pi$
$312$	1.76855	0.100125
$313$	−4.66624	−0.263751	−0.131876	−	0.991266i	$-0.542100\pi$
$313$	−4.66624	−0.263751	−0.131876	+	0.991266i	$0.542100\pi$
$314$	41.3984	2.33625
$315$	10.3455	0.582904
$316$	13.1505	0.739773
$317$	15.7805	0.886319	0.443160	−	0.896443i	$-0.353857\pi$
$317$	15.7805	0.886319	0.443160	+	0.896443i	$0.353857\pi$
$318$	74.1014	4.15540
$319$	5.17212	0.289583
$320$	103.813	5.80333
$321$	−35.6712	−1.99097
$322$	−29.3332	−1.63468
$323$	−6.14560	−0.341950
$324$	−51.4396	−2.85776
$325$	−0.0581625	−0.00322627
$326$	19.2552	1.06644
$327$	39.9595	2.20977
$328$	−84.8718	−4.68626
$329$	−1.00911	−0.0556341
$330$	−73.8509	−4.06536
$331$	33.4834	1.84041	0.920207	−	0.391433i	$-0.128020\pi$
$331$	33.4834	1.84041	0.920207	+	0.391433i	$0.128020\pi$
$332$	−0.489210	−0.0268489
$333$	−16.7712	−0.919056
$334$	−49.5800	−2.71290
$335$	−11.8067	−0.645070
$336$	61.1464	3.33581
$337$	28.4910	1.55200	0.776001	−	0.630731i	$-0.217245\pi$
$337$	28.4910	1.55200	0.776001	+	0.630731i	$0.217245\pi$
$338$	36.1814	1.96801
$339$	−6.68526	−0.363093
$340$	88.8627	4.81925
$341$	34.2385	1.85412
$342$	8.08157	0.437001
$343$	−16.9990	−0.917859
$344$	−109.464	−5.90189
$345$	44.0674	2.37251
$346$	−13.7139	−0.737265
$347$	−12.3748	−0.664313	−0.332156	−	0.943224i	$-0.607776\pi$
$347$	−12.3748	−0.664313	−0.332156	+	0.943224i	$0.607776\pi$
$348$	16.3169	0.874680
$349$	12.5656	0.672624	0.336312	−	0.941751i	$-0.390820\pi$
$349$	12.5656	0.672624	0.336312	+	0.941751i	$0.390820\pi$
$350$	−3.32681	−0.177826
$351$	−0.00318430	−0.000169965 0
$352$	−125.481	−6.68817
$353$	−11.9695	−0.637071	−0.318536	−	0.947911i	$-0.603191\pi$
$353$	−11.9695	−0.637071	−0.318536	+	0.947911i	$0.603191\pi$
$354$	20.0008	1.06303
$355$	21.0567	1.11757
$356$	−94.6498	−5.01643
$357$	22.2314	1.17661
$358$	68.3829	3.61415
$359$	−13.8611	−0.731560	−0.365780	−	0.930701i	$-0.619198\pi$
$359$	−13.8611	−0.731560	−0.365780	+	0.930701i	$0.619198\pi$
$360$	−76.2231	−4.01731
$361$	−18.0755	−0.951341
$362$	36.4971	1.91824
$363$	22.0967	1.15978
$364$	−0.562772	−0.0294973
$365$	10.5671	0.553108
$366$	−53.8157	−2.81299
$367$	−23.0792	−1.20472	−0.602362	−	0.798223i	$-0.705774\pi$
$367$	−23.0792	−1.20472	−0.602362	+	0.798223i	$0.705774\pi$
$368$	130.635	6.80984
$369$	24.5278	1.27686
$370$	37.3887	1.94375
$371$	−15.3807	−0.798527
$372$	108.015	5.60031
$373$	3.26904	0.169265	0.0846323	−	0.996412i	$-0.473028\pi$
$373$	3.26904	0.169265	0.0846323	+	0.996412i	$0.473028\pi$
$374$	−79.5968	−4.11585
$375$	−24.6529	−1.27307
$376$	7.43487	0.383424
$377$	−0.0797998	−0.00410990
$378$	−0.182138	−0.00936815
$379$	−7.38199	−0.379187	−0.189594	−	0.981863i	$-0.560717\pi$
$379$	−7.38199	−0.379187	−0.189594	+	0.981863i	$0.560717\pi$
$380$	−13.3682	−0.685774
$381$	−54.4983	−2.79203
$382$	1.28016	0.0654987
$383$	10.3313	0.527903	0.263951	−	0.964536i	$-0.414974\pi$
$383$	10.3313	0.527903	0.263951	+	0.964536i	$0.414974\pi$
$384$	−155.709	−7.94599
$385$	15.3287	0.781224
$386$	−27.3835	−1.39378
$387$	31.6348	1.60809
$388$	−90.9566	−4.61762
$389$	−8.92607	−0.452570	−0.226285	−	0.974061i	$-0.572658\pi$
$389$	−8.92607	−0.452570	−0.226285	+	0.974061i	$0.572658\pi$
$390$	1.13943	0.0576975
$391$	47.4960	2.40198
$392$	52.1236	2.63264
$393$	−39.9239	−2.01389
$394$	50.7526	2.55688
$395$	5.52648	0.278067
$396$	77.6655	3.90284
$397$	3.97033	0.199265	0.0996325	−	0.995024i	$-0.468233\pi$
$397$	3.97033	0.199265	0.0996325	+	0.995024i	$0.468233\pi$
$398$	12.9828	0.650769
$399$	−3.34442	−0.167430
$400$	14.8159	0.740797
$401$	−26.7496	−1.33581	−0.667906	−	0.744246i	$-0.732809\pi$
$401$	−26.7496	−1.33581	−0.667906	+	0.744246i	$0.732809\pi$
$402$	33.3638	1.66403
$403$	−0.528259	−0.0263145
$404$	−99.2488	−4.93781
$405$	−21.6174	−1.07418
$406$	−4.56444	−0.226529
$407$	−24.8495	−1.23174
$408$	−163.795	−8.10907
$409$	−12.7816	−0.632009	−0.316004	−	0.948758i	$-0.602341\pi$
$409$	−12.7816	−0.632009	−0.316004	+	0.948758i	$0.602341\pi$
$410$	−54.6807	−2.70049
$411$	−28.5612	−1.40882
$412$	−22.4407	−1.10558
$413$	−4.15143	−0.204279
$414$	−62.4581	−3.06965
$415$	−0.205590	−0.0100920
$416$	1.93603	0.0949215
$417$	43.9447	2.15198
$418$	11.9743	0.585681
$419$	18.5518	0.906313	0.453157	−	0.891431i	$-0.350298\pi$
$419$	18.5518	0.906313	0.453157	+	0.891431i	$0.350298\pi$
$420$	48.3588	2.35967
$421$	29.2915	1.42758	0.713789	−	0.700360i	$-0.246977\pi$
$421$	29.2915	1.42758	0.713789	+	0.700360i	$0.246977\pi$
$422$	−20.9908	−1.02181
$423$	−2.14866	−0.104471
$424$	113.321	5.50336
$425$	5.38674	0.261295
$426$	−59.5027	−2.88292
$427$	11.1701	0.540561
$428$	−83.6307	−4.04244
$429$	−0.757298	−0.0365627
$430$	−70.5247	−3.40100
$431$	5.45937	0.262968	0.131484	−	0.991318i	$-0.458026\pi$
$431$	5.45937	0.262968	0.131484	+	0.991318i	$0.458026\pi$
$432$	0.811149	0.0390264
$433$	−3.85838	−0.185422	−0.0927109	−	0.995693i	$-0.529553\pi$
$433$	−3.85838	−0.185422	−0.0927109	+	0.995693i	$0.529553\pi$
$434$	−30.2157	−1.45040
$435$	6.85717	0.328776
$436$	93.6847	4.48668
$437$	−7.14515	−0.341799
$438$	−29.8609	−1.42681
$439$	−19.6480	−0.937747	−0.468873	−	0.883265i	$-0.655340\pi$
$439$	−19.6480	−0.937747	−0.468873	+	0.883265i	$0.655340\pi$
$440$	−112.938	−5.38411
$441$	−15.0636	−0.717315
$442$	1.22809	0.0584141
$443$	30.6708	1.45721	0.728607	−	0.684932i	$-0.240168\pi$
$443$	30.6708	1.45721	0.728607	+	0.684932i	$0.240168\pi$
$444$	−78.3949	−3.72046
$445$	−39.7765	−1.88559
$446$	−71.1448	−3.36880
$447$	−43.8828	−2.07559
$448$	60.8904	2.87680
$449$	−17.3339	−0.818038	−0.409019	−	0.912526i	$-0.634129\pi$
$449$	−17.3339	−0.818038	−0.409019	+	0.912526i	$0.634129\pi$
$450$	−7.08365	−0.333927
$451$	36.3423	1.71129
$452$	−15.6735	−0.737220
$453$	−11.5912	−0.544604
$454$	−53.4779	−2.50984
$455$	−0.236504	−0.0110875
$456$	24.6408	1.15391
$457$	29.7428	1.39131	0.695656	−	0.718375i	$-0.255114\pi$
$457$	29.7428	1.39131	0.695656	+	0.718375i	$0.255114\pi$
$458$	−56.1690	−2.62460
$459$	0.294915	0.0137655
$460$	103.316	4.81712
$461$	2.37811	0.110760	0.0553799	−	0.998465i	$-0.482363\pi$
$461$	2.37811	0.110760	0.0553799	+	0.998465i	$0.482363\pi$
$462$	−43.3164	−2.01526
$463$	29.1508	1.35475	0.677377	−	0.735636i	$-0.263117\pi$
$463$	29.1508	1.35475	0.677377	+	0.735636i	$0.263117\pi$
$464$	20.3277	0.943690
$465$	45.3932	2.10506
$466$	24.9902	1.15765
$467$	−13.0869	−0.605588	−0.302794	−	0.953056i	$-0.597919\pi$
$467$	−13.0869	−0.605588	−0.302794	+	0.953056i	$0.597919\pi$
$468$	−1.19829	−0.0553909
$469$	−6.92509	−0.319771
$470$	4.79009	0.220951
$471$	−36.4786	−1.68084
$472$	30.5867	1.40787
$473$	46.8726	2.15520
$474$	−15.6169	−0.717308
$475$	−0.810363	−0.0371820
$476$	52.1213	2.38898
$477$	−32.7496	−1.49950
$478$	−35.1243	−1.60655
$479$	−26.7094	−1.22038	−0.610192	−	0.792254i	$-0.708908\pi$
$479$	−26.7094	−1.22038	−0.610192	+	0.792254i	$0.708908\pi$
$480$	−166.362	−7.59336
$481$	0.383399	0.0174815
$482$	−30.7311	−1.39976
$483$	25.8472	1.17609
$484$	51.8055	2.35480
$485$	−38.2244	−1.73568
$486$	61.4726	2.78846
$487$	−10.6584	−0.482979	−0.241490	−	0.970403i	$-0.577636\pi$
$487$	−10.6584	−0.482979	−0.241490	+	0.970403i	$0.577636\pi$
$488$	−82.2987	−3.72549
$489$	−16.9668	−0.767267
$490$	33.5819	1.51708
$491$	20.5488	0.927356	0.463678	−	0.886004i	$-0.346529\pi$
$491$	20.5488	0.927356	0.463678	+	0.886004i	$0.346529\pi$
$492$	114.652	5.16891
$493$	7.39069	0.332860
$494$	−0.184749	−0.00831226
$495$	32.6389	1.46701
$496$	134.565	6.04217
$497$	12.3506	0.553999
$498$	0.580963	0.0260336
$499$	39.5780	1.77176	0.885878	−	0.463918i	$-0.153557\pi$
$499$	39.5780	1.77176	0.885878	+	0.463918i	$0.153557\pi$
$500$	−57.7984	−2.58482
$501$	43.6878	1.95183
$502$	−30.9967	−1.38345
$503$	−21.7116	−0.968070	−0.484035	−	0.875049i	$-0.660829\pi$
$503$	−21.7116	−0.968070	−0.484035	+	0.875049i	$0.660829\pi$
$504$	−44.7077	−1.99144
$505$	−41.7092	−1.85604
$506$	−92.5428	−4.11403
$507$	−31.8816	−1.41591
$508$	−127.771	−5.66891
$509$	−3.16882	−0.140456	−0.0702278	−	0.997531i	$-0.522373\pi$
$509$	−3.16882	−0.140456	−0.0702278	+	0.997531i	$0.522373\pi$
$510$	−105.529	−4.67291
$511$	6.19802	0.274184
$512$	−125.906	−5.56433
$513$	−0.0443661	−0.00195881
$514$	39.9736	1.76316
$515$	−9.43069	−0.415566
$516$	147.873	6.50974
$517$	−3.18362	−0.140016
$518$	21.9299	0.963544
$519$	12.0841	0.530434
$520$	1.74250	0.0764138
$521$	−16.8916	−0.740034	−0.370017	−	0.929025i	$-0.620648\pi$
$521$	−16.8916	−0.740034	−0.370017	+	0.929025i	$0.620648\pi$
$522$	−9.71888	−0.425384
$523$	−18.8691	−0.825088	−0.412544	−	0.910938i	$-0.635360\pi$
$523$	−18.8691	−0.825088	−0.412544	+	0.910938i	$0.635360\pi$
$524$	−93.6012	−4.08899
$525$	2.93145	0.127939
$526$	−6.81251	−0.297039
$527$	48.9249	2.13120
$528$	192.909	8.39530
$529$	32.2210	1.40091
$530$	73.0098	3.17134
$531$	−8.83949	−0.383601
$532$	−7.84096	−0.339949
$533$	−0.560719	−0.0242874
$534$	112.402	4.86409
$535$	−35.1457	−1.51948
$536$	51.0222	2.20383
$537$	−60.2561	−2.60024
$538$	31.6851	1.36604
$539$	−22.3194	−0.961366
$540$	0.641514	0.0276063
$541$	−2.83889	−0.122054	−0.0610268	−	0.998136i	$-0.519438\pi$
$541$	−2.83889	−0.122054	−0.0610268	+	0.998136i	$0.519438\pi$
$542$	85.1130	3.65592
$543$	−32.1597	−1.38010
$544$	−179.306	−7.68768
$545$	39.3709	1.68646
$546$	0.668321	0.0286015
$547$	−30.9374	−1.32279	−0.661393	−	0.750039i	$-0.730034\pi$
$547$	−30.9374	−1.32279	−0.661393	+	0.750039i	$0.730034\pi$
$548$	−66.9615	−2.86045
$549$	23.7841	1.01508
$550$	−10.4957	−0.447538
$551$	−1.11183	−0.0473656
$552$	−190.436	−8.10548
$553$	3.24149	0.137842
$554$	9.48486	0.402973
$555$	−32.9454	−1.39845
$556$	103.028	4.36935
$557$	3.64149	0.154295	0.0771474	−	0.997020i	$-0.475419\pi$
$557$	3.64149	0.154295	0.0771474	+	0.997020i	$0.475419\pi$
$558$	−64.3371	−2.72361
$559$	−0.723189	−0.0305876
$560$	60.2456	2.54584
$561$	70.1374	2.96120
$562$	61.9952	2.61511
$563$	19.9992	0.842866	0.421433	−	0.906860i	$-0.361527\pi$
$563$	19.9992	0.842866	0.421433	+	0.906860i	$0.361527\pi$
$564$	−10.0436	−0.422914
$565$	−6.58678	−0.277108
$566$	−36.7067	−1.54290
$567$	−12.6795	−0.532487
$568$	−90.9958	−3.81810
$569$	11.8162	0.495362	0.247681	−	0.968842i	$-0.420331\pi$
$569$	11.8162	0.495362	0.247681	+	0.968842i	$0.420331\pi$
$570$	15.8754	0.664949
$571$	8.83428	0.369703	0.184851	−	0.982766i	$-0.440820\pi$
$571$	8.83428	0.369703	0.184851	+	0.982766i	$0.440820\pi$
$572$	−1.77548	−0.0742365
$573$	−1.12802	−0.0471239
$574$	−32.0723	−1.33867
$575$	6.26286	0.261179
$576$	129.651	5.40214
$577$	9.08042	0.378023	0.189011	−	0.981975i	$-0.439472\pi$
$577$	9.08042	0.378023	0.189011	+	0.981975i	$0.439472\pi$
$578$	−66.4082	−2.76222
$579$	24.1292	1.00278
$580$	16.0766	0.667544
$581$	−0.120586	−0.00500277
$582$	108.016	4.47739
$583$	−48.5243	−2.00967
$584$	−45.6654	−1.88965
$585$	−0.503579	−0.0208204
$586$	16.0196	0.661765
$587$	−18.6768	−0.770873	−0.385437	−	0.922734i	$-0.625949\pi$
$587$	−18.6768	−0.770873	−0.385437	+	0.922734i	$0.625949\pi$
$588$	−70.4130	−2.90378
$589$	−7.36010	−0.303268
$590$	19.7062	0.811292
$591$	−44.7211	−1.83958
$592$	−97.6647	−4.01399
$593$	27.5662	1.13201	0.566005	−	0.824402i	$-0.308488\pi$
$593$	27.5662	1.13201	0.566005	+	0.824402i	$0.308488\pi$
$594$	−0.574622	−0.0235770
$595$	21.9039	0.897974
$596$	−102.883	−4.21424
$597$	−11.4399	−0.468204
$598$	1.42783	0.0583882
$599$	29.9481	1.22365	0.611824	−	0.790994i	$-0.290436\pi$
$599$	29.9481	1.22365	0.611824	+	0.790994i	$0.290436\pi$
$600$	−21.5982	−0.881741
$601$	44.2626	1.80551	0.902755	−	0.430156i	$-0.141541\pi$
$601$	44.2626	1.80551	0.902755	+	0.430156i	$0.141541\pi$
$602$	−41.3654	−1.68593
$603$	−14.7453	−0.600476
$604$	−27.1756	−1.10576
$605$	21.7712	0.885126
$606$	117.863	4.78787
$607$	−7.72212	−0.313431	−0.156716	−	0.987644i	$-0.550091\pi$
$607$	−7.72212	−0.313431	−0.156716	+	0.987644i	$0.550091\pi$
$608$	26.9742	1.09395
$609$	4.02199	0.162979
$610$	−53.0229	−2.14684
$611$	0.0491196	0.00198717
$612$	110.980	4.48610
$613$	−28.8858	−1.16669	−0.583343	−	0.812226i	$-0.698256\pi$
$613$	−28.8858	−1.16669	−0.583343	+	0.812226i	$0.698256\pi$
$614$	7.54892	0.304649
$615$	48.1824	1.94290
$616$	−66.2425	−2.66899
$617$	−4.08078	−0.164286	−0.0821431	−	0.996621i	$-0.526176\pi$
$617$	−4.08078	−0.164286	−0.0821431	+	0.996621i	$0.526176\pi$
$618$	26.6495	1.07200
$619$	−7.12162	−0.286242	−0.143121	−	0.989705i	$-0.545714\pi$
$619$	−7.12162	−0.286242	−0.143121	+	0.989705i	$0.545714\pi$
$620$	106.424	4.27408
$621$	0.342881	0.0137594
$622$	−46.9154	−1.88113
$623$	−23.3304	−0.934714
$624$	−2.97637	−0.119150
$625$	−28.5037	−1.14015
$626$	12.9918	0.519255
$627$	−10.5512	−0.421376
$628$	−85.5237	−3.41277
$629$	−35.5086	−1.41582
$630$	−28.8040	−1.14758
$631$	−25.2452	−1.00499	−0.502497	−	0.864579i	$-0.667585\pi$
$631$	−25.2452	−1.00499	−0.502497	+	0.864579i	$0.667585\pi$
$632$	−23.8825	−0.949993
$633$	18.4962	0.735158
$634$	−43.9361	−1.74492
$635$	−53.6955	−2.13084
$636$	−153.084	−6.07016
$637$	0.344363	0.0136441
$638$	−14.4003	−0.570112
$639$	26.2976	1.04032
$640$	−153.415	−6.06427
$641$	35.5405	1.40377	0.701883	−	0.712292i	$-0.252343\pi$
$641$	35.5405	1.40377	0.701883	+	0.712292i	$0.252343\pi$
$642$	99.3158	3.91968
$643$	−19.3171	−0.761791	−0.380896	−	0.924618i	$-0.624384\pi$
$643$	−19.3171	−0.761791	−0.380896	+	0.924618i	$0.624384\pi$
$644$	60.5986	2.38792
$645$	62.1434	2.44689
$646$	17.1106	0.673208
$647$	14.8480	0.583736	0.291868	−	0.956459i	$-0.405723\pi$
$647$	14.8480	0.583736	0.291868	+	0.956459i	$0.405723\pi$
$648$	93.4189	3.66984
$649$	−13.0973	−0.514113
$650$	0.161936	0.00635166
$651$	26.6248	1.04351
$652$	−39.7786	−1.55785
$653$	−17.4133	−0.681434	−0.340717	−	0.940166i	$-0.610670\pi$
$653$	−17.4133	−0.681434	−0.340717	+	0.940166i	$0.610670\pi$
$654$	−111.255	−4.35043
$655$	−39.3358	−1.53698
$656$	142.834	5.57673
$657$	13.1972	0.514872
$658$	2.80957	0.109529
$659$	−23.8033	−0.927246	−0.463623	−	0.886032i	$-0.653451\pi$
$659$	−23.8033	−0.927246	−0.463623	+	0.886032i	$0.653451\pi$
$660$	152.566	5.93863
$661$	16.3561	0.636177	0.318089	−	0.948061i	$-0.396959\pi$
$661$	16.3561	0.636177	0.318089	+	0.948061i	$0.396959\pi$
$662$	−93.2246	−3.62328
$663$	−1.08214	−0.0420268
$664$	0.888449	0.0344785
$665$	−3.29515	−0.127781
$666$	46.6945	1.80937
$667$	8.59274	0.332712
$668$	102.426	3.96297
$669$	62.6898	2.42373
$670$	32.8723	1.26997
$671$	35.2404	1.36044
$672$	−97.5778	−3.76415
$673$	−30.2599	−1.16643	−0.583217	−	0.812317i	$-0.698206\pi$
$673$	−30.2599	−1.16643	−0.583217	+	0.812317i	$0.698206\pi$
$674$	−79.3247	−3.05547
$675$	0.0388877	0.00149679
$676$	−74.7460	−2.87485
$677$	3.50541	0.134724	0.0673618	−	0.997729i	$-0.478542\pi$
$677$	3.50541	0.134724	0.0673618	+	0.997729i	$0.478542\pi$
$678$	18.6131	0.714833
$679$	−22.4201	−0.860403
$680$	−161.382	−6.18874
$681$	47.1225	1.80574
$682$	−95.3269	−3.65026
$683$	7.00256	0.267945	0.133973	−	0.990985i	$-0.457227\pi$
$683$	7.00256	0.267945	0.133973	+	0.990985i	$0.457227\pi$
$684$	−16.6955	−0.638367
$685$	−28.1405	−1.07519
$686$	47.3287	1.80702
$687$	49.4938	1.88831
$688$	184.221	7.02335
$689$	0.748673	0.0285222
$690$	−122.693	−4.67083
$691$	24.3228	0.925284	0.462642	−	0.886545i	$-0.346902\pi$
$691$	24.3228	0.925284	0.462642	+	0.886545i	$0.346902\pi$
$692$	28.3311	1.07699
$693$	19.1439	0.727218
$694$	34.4539	1.30785
$695$	43.2973	1.64236
$696$	−29.6330	−1.12324
$697$	51.9312	1.96703
$698$	−34.9853	−1.32421
$699$	−22.0203	−0.832886
$700$	6.87275	0.259766
$701$	−34.9576	−1.32033	−0.660166	−	0.751120i	$-0.729514\pi$
$701$	−34.9576	−1.32033	−0.660166	+	0.751120i	$0.729514\pi$
$702$	0.00886575	0.000334616 0
$703$	5.34180	0.201470
$704$	192.102	7.24011
$705$	−4.22083	−0.158966
$706$	33.3255	1.25422
$707$	−24.4640	−0.920065
$708$	−41.3191	−1.55287
$709$	−41.6006	−1.56234	−0.781172	−	0.624316i	$-0.785378\pi$
$709$	−41.6006	−1.56234	−0.781172	+	0.624316i	$0.785378\pi$
$710$	−58.6262	−2.20020
$711$	6.90198	0.258844
$712$	171.892	6.44194
$713$	56.8823	2.13026
$714$	−61.8968	−2.31643
$715$	−0.746142	−0.0279041
$716$	−141.270	−5.27951
$717$	30.9501	1.15585
$718$	38.5921	1.44025
$719$	−19.2292	−0.717129	−0.358565	−	0.933505i	$-0.616734\pi$
$719$	−19.2292	−0.717129	−0.358565	+	0.933505i	$0.616734\pi$
$720$	128.279	4.78066
$721$	−5.53146	−0.206002
$722$	50.3258	1.87293
$723$	27.0789	1.00708
$724$	−75.3981	−2.80215
$725$	0.974541	0.0361936
$726$	−61.5218	−2.28329
$727$	37.6513	1.39641	0.698205	−	0.715898i	$-0.253983\pi$
$727$	37.6513	1.39641	0.698205	+	0.715898i	$0.253983\pi$
$728$	1.02204	0.0378795
$729$	−27.3375	−1.01250
$730$	−29.4210	−1.08892
$731$	66.9784	2.47729
$732$	111.176	4.10918
$733$	10.3394	0.381894	0.190947	−	0.981600i	$-0.438844\pi$
$733$	10.3394	0.381894	0.190947	+	0.981600i	$0.438844\pi$
$734$	64.2572	2.37178
$735$	−29.5910	−1.09148
$736$	−208.469	−7.68427
$737$	−21.8478	−0.804775
$738$	−68.2904	−2.51380
$739$	50.7434	1.86663	0.933314	−	0.359061i	$-0.116903\pi$
$739$	50.7434	1.86663	0.933314	+	0.359061i	$0.116903\pi$
$740$	−77.2401	−2.83940
$741$	0.162793	0.00598036
$742$	42.8231	1.57208
$743$	−32.5058	−1.19252	−0.596261	−	0.802790i	$-0.703348\pi$
$743$	−32.5058	−1.19252	−0.596261	+	0.802790i	$0.703348\pi$
$744$	−196.165	−7.19175
$745$	−43.2364	−1.58406
$746$	−9.10169	−0.333236
$747$	−0.256760	−0.00939435
$748$	164.437	6.01239
$749$	−20.6143	−0.753230
$750$	68.6386	2.50633
$751$	22.1273	0.807436	0.403718	−	0.914884i	$-0.367718\pi$
$751$	22.1273	0.807436	0.403718	+	0.914884i	$0.367718\pi$
$752$	−12.5124	−0.456281
$753$	27.3130	0.995341
$754$	0.222179	0.00809129
$755$	−11.4205	−0.415635
$756$	0.376272	0.0136849
$757$	−47.5164	−1.72701	−0.863506	−	0.504339i	$-0.831736\pi$
$757$	−47.5164	−1.72701	−0.863506	+	0.504339i	$0.831736\pi$
$758$	20.5530	0.746517
$759$	81.5449	2.95989
$760$	24.2778	0.880650
$761$	37.0882	1.34445	0.672224	−	0.740348i	$-0.265339\pi$
$761$	37.0882	1.34445	0.672224	+	0.740348i	$0.265339\pi$
$762$	151.735	5.49676
$763$	23.0925	0.836006
$764$	−2.64464	−0.0956798
$765$	46.6392	1.68624
$766$	−28.7643	−1.03930
$767$	0.202076	0.00729653
$768$	222.795	8.03943
$769$	2.65083	0.0955913	0.0477956	−	0.998857i	$-0.484780\pi$
$769$	2.65083	0.0955913	0.0477956	+	0.998857i	$0.484780\pi$
$770$	−42.6783	−1.53802
$771$	−35.2231	−1.26853
$772$	56.5707	2.03603
$773$	38.2533	1.37587	0.687937	−	0.725770i	$-0.258516\pi$
$773$	38.2533	1.37587	0.687937	+	0.725770i	$0.258516\pi$
$774$	−88.0778	−3.16589
$775$	6.45127	0.231737
$776$	165.185	5.92980
$777$	−19.3237	−0.693234
$778$	24.8520	0.890988
$779$	−7.81235	−0.279906
$780$	−2.35392	−0.0842838
$781$	38.9646	1.39426
$782$	−132.239	−4.72885
$783$	0.0533546	0.00190674
$784$	−87.7208	−3.13289
$785$	−35.9412	−1.28280
$786$	111.156	3.96481
$787$	−1.68649	−0.0601169	−0.0300584	−	0.999548i	$-0.509569\pi$
$787$	−1.68649	−0.0601169	−0.0300584	+	0.999548i	$0.509569\pi$
$788$	−104.848	−3.73506
$789$	6.00290	0.213709
$790$	−15.3868	−0.547439
$791$	−3.86340	−0.137367
$792$	−141.048	−5.01191
$793$	−0.543719	−0.0193080
$794$	−11.0542	−0.392299
$795$	−64.3332	−2.28166
$796$	−26.8207	−0.950636
$797$	10.5350	0.373168	0.186584	−	0.982439i	$-0.440258\pi$
$797$	10.5350	0.373168	0.186584	+	0.982439i	$0.440258\pi$
$798$	9.31155	0.329625
$799$	−4.54923	−0.160940
$800$	−23.6434	−0.835920
$801$	−49.6766	−1.75523
$802$	74.4764	2.62985
$803$	19.5540	0.690045
$804$	−68.9251	−2.43080
$805$	25.4665	0.897575
$806$	1.47078	0.0518061
$807$	−27.9196	−0.982815
$808$	180.245	6.34099
$809$	32.0448	1.12664	0.563318	−	0.826240i	$-0.309524\pi$
$809$	32.0448	1.12664	0.563318	+	0.826240i	$0.309524\pi$
$810$	60.1874	2.11477
$811$	−28.8743	−1.01392	−0.506958	−	0.861971i	$-0.669230\pi$
$811$	−28.8743	−1.01392	−0.506958	+	0.861971i	$0.669230\pi$
$812$	9.42953	0.330912
$813$	−74.9980	−2.63030
$814$	69.1862	2.42497
$815$	−16.7169	−0.585568
$816$	275.657	9.64993
$817$	−10.0760	−0.352515
$818$	35.5866	1.24426
$819$	−0.295369	−0.0103210
$820$	112.963	3.94484
$821$	42.0098	1.46615	0.733076	−	0.680147i	$-0.238084\pi$
$821$	42.0098	1.46615	0.733076	+	0.680147i	$0.238084\pi$
$822$	79.5203	2.77359
$823$	−2.35803	−0.0821958	−0.0410979	−	0.999155i	$-0.513086\pi$
$823$	−2.35803	−0.0821958	−0.0410979	+	0.999155i	$0.513086\pi$
$824$	40.7544	1.41975
$825$	9.24837	0.321987
$826$	11.5584	0.402170
$827$	22.9317	0.797414	0.398707	−	0.917078i	$-0.369459\pi$
$827$	22.9317	0.797414	0.398707	+	0.917078i	$0.369459\pi$
$828$	129.030	4.48411
$829$	21.0364	0.730624	0.365312	−	0.930885i	$-0.380962\pi$
$829$	21.0364	0.730624	0.365312	+	0.930885i	$0.380962\pi$
$830$	0.572405	0.0198685
$831$	−8.35766	−0.289924
$832$	−2.96390	−0.102755
$833$	−31.8933	−1.10504
$834$	−122.351	−4.23667
$835$	43.0443	1.48961
$836$	−24.7373	−0.855557
$837$	0.353197	0.0122083
$838$	−51.6519	−1.78429
$839$	20.3097	0.701167	0.350584	−	0.936531i	$-0.385983\pi$
$839$	20.3097	0.701167	0.350584	+	0.936531i	$0.385983\pi$
$840$	−87.8239	−3.03021
$841$	−27.6629	−0.953894
$842$	−81.5535	−2.81052
$843$	−54.6276	−1.88147
$844$	43.3642	1.49266
$845$	−31.4119	−1.08060
$846$	5.98231	0.205676
$847$	12.7697	0.438770
$848$	−190.712	−6.54909
$849$	32.3444	1.11006
$850$	−14.9978	−0.514420
$851$	−41.2839	−1.41519
$852$	122.925	4.21133
$853$	−23.5790	−0.807330	−0.403665	−	0.914907i	$-0.632264\pi$
$853$	−23.5790	−0.807330	−0.403665	+	0.914907i	$0.632264\pi$
$854$	−31.1000	−1.06422
$855$	−7.01624	−0.239950
$856$	151.881	5.19118
$857$	−50.1986	−1.71475	−0.857375	−	0.514692i	$-0.827906\pi$
$857$	−50.1986	−1.71475	−0.857375	+	0.514692i	$0.827906\pi$
$858$	2.10847	0.0719821
$859$	48.4135	1.65185	0.825924	−	0.563781i	$-0.190654\pi$
$859$	48.4135	1.65185	0.825924	+	0.563781i	$0.190654\pi$
$860$	145.695	4.96814
$861$	28.2608	0.963126
$862$	−15.2000	−0.517714
$863$	1.71231	0.0582877	0.0291438	−	0.999575i	$-0.490722\pi$
$863$	1.71231	0.0582877	0.0291438	+	0.999575i	$0.490722\pi$
$864$	−1.29444	−0.0440377
$865$	11.9061	0.404820
$866$	10.7425	0.365045
$867$	58.5161	1.98731
$868$	62.4216	2.11873
$869$	10.2265	0.346911
$870$	−19.0918	−0.647272
$871$	0.337086	0.0114217
$872$	−170.140	−5.76166
$873$	−47.7382	−1.61569
$874$	19.8936	0.672909
$875$	−14.2468	−0.481631
$876$	61.6886	2.08427
$877$	−40.7693	−1.37668	−0.688341	−	0.725387i	$-0.741661\pi$
$877$	−40.7693	−1.37668	−0.688341	+	0.725387i	$0.741661\pi$
$878$	54.7040	1.84617
$879$	−14.1158	−0.476115
$880$	190.068	6.40718
$881$	4.94899	0.166736	0.0833678	−	0.996519i	$-0.473432\pi$
$881$	4.94899	0.166736	0.0833678	+	0.996519i	$0.473432\pi$
$882$	41.9402	1.41220
$883$	−15.3702	−0.517249	−0.258624	−	0.965978i	$-0.583269\pi$
$883$	−15.3702	−0.517249	−0.258624	+	0.965978i	$0.583269\pi$
$884$	−2.53706	−0.0853307
$885$	−17.3643	−0.583694
$886$	−85.3938	−2.86886
$887$	52.5169	1.76335	0.881673	−	0.471861i	$-0.156417\pi$
$887$	52.5169	1.76335	0.881673	+	0.471861i	$0.156417\pi$
$888$	142.372	4.77769
$889$	−31.4945	−1.05629
$890$	110.746	3.71221
$891$	−40.0021	−1.34012
$892$	146.976	4.92111
$893$	0.684371	0.0229016
$894$	122.179	4.08627
$895$	−59.3685	−1.98447
$896$	−89.9839	−3.00615
$897$	−1.25814	−0.0420081
$898$	48.2611	1.61050
$899$	8.85125	0.295206
$900$	14.6339	0.487796
$901$	−69.3386	−2.31000
$902$	−101.184	−3.36907
$903$	36.4495	1.21296
$904$	28.4645	0.946715
$905$	−31.6860	−1.05328
$906$	32.2724	1.07218
$907$	−43.3380	−1.43902	−0.719508	−	0.694484i	$-0.755633\pi$
$907$	−43.3380	−1.43902	−0.719508	+	0.694484i	$0.755633\pi$
$908$	110.478	3.66635
$909$	−52.0903	−1.72773
$910$	0.658477	0.0218283
$911$	−4.34539	−0.143969	−0.0719847	−	0.997406i	$-0.522933\pi$
$911$	−4.34539	−0.143969	−0.0719847	+	0.997406i	$0.522933\pi$
$912$	−41.4690	−1.37317
$913$	−0.380435	−0.0125906
$914$	−82.8102	−2.73912
$915$	46.7216	1.54457
$916$	116.038	3.83399
$917$	−23.0719	−0.761903
$918$	−0.821105	−0.0271005
$919$	53.3055	1.75838	0.879192	−	0.476467i	$-0.158083\pi$
$919$	53.3055	1.75838	0.879192	+	0.476467i	$0.158083\pi$
$920$	−187.630	−6.18599
$921$	−6.65179	−0.219184
$922$	−6.62115	−0.218056
$923$	−0.601177	−0.0197880
$924$	89.4859	2.94387
$925$	−4.68219	−0.153950
$926$	−81.1619	−2.66715
$927$	−11.7779	−0.386838
$928$	−32.4391	−1.06487
$929$	5.16564	0.169479	0.0847395	−	0.996403i	$-0.472994\pi$
$929$	5.16564	0.169479	0.0847395	+	0.996403i	$0.472994\pi$
$930$	−126.384	−4.14429
$931$	4.79792	0.157245
$932$	−51.6265	−1.69108
$933$	41.3399	1.35341
$934$	36.4365	1.19224
$935$	69.1043	2.25995
$936$	2.17620	0.0711313
$937$	−10.1817	−0.332623	−0.166311	−	0.986073i	$-0.553186\pi$
$937$	−10.1817	−0.332623	−0.166311	+	0.986073i	$0.553186\pi$
$938$	19.2809	0.629543
$939$	−11.4478	−0.373585
$940$	−9.89570	−0.322762
$941$	31.3497	1.02197	0.510985	−	0.859590i	$-0.329281\pi$
$941$	31.3497	1.02197	0.510985	+	0.859590i	$0.329281\pi$
$942$	101.564	3.30913
$943$	60.3775	1.96616
$944$	−51.4755	−1.67538
$945$	0.158128	0.00514390
$946$	−130.503	−4.24301
$947$	18.2802	0.594027	0.297014	−	0.954873i	$-0.404009\pi$
$947$	18.2802	0.594027	0.297014	+	0.954873i	$0.404009\pi$
$948$	32.2624	1.04784
$949$	−0.301695	−0.00979344
$950$	2.25622	0.0732013
$951$	38.7146	1.25541
$952$	−94.6570	−3.06785
$953$	−16.1799	−0.524117	−0.262059	−	0.965052i	$-0.584401\pi$
$953$	−16.1799	−0.524117	−0.262059	+	0.965052i	$0.584401\pi$
$954$	91.1815	2.95211
$955$	−1.11141	−0.0359643
$956$	72.5622	2.34683
$957$	12.6889	0.410174
$958$	74.3644	2.40261
$959$	−16.5055	−0.532990
$960$	254.687	8.22000
$961$	27.5935	0.890114
$962$	−1.06746	−0.0344163
$963$	−43.8932	−1.41444
$964$	63.4863	2.04476
$965$	23.7738	0.765305
$966$	−71.9640	−2.31540
$967$	−22.8986	−0.736370	−0.368185	−	0.929753i	$-0.620021\pi$
$967$	−22.8986	−0.736370	−0.368185	+	0.929753i	$0.620021\pi$
$968$	−94.0835	−3.02396
$969$	−15.0772	−0.484348
$970$	106.425	3.41709
$971$	−16.4979	−0.529442	−0.264721	−	0.964325i	$-0.585280\pi$
$971$	−16.4979	−0.529442	−0.264721	+	0.964325i	$0.585280\pi$
$972$	−126.994	−4.07335
$973$	25.3955	0.814144
$974$	29.6753	0.950856
$975$	−0.142692	−0.00456979
$976$	138.504	4.43339
$977$	33.5622	1.07375	0.536876	−	0.843661i	$-0.319604\pi$
$977$	33.5622	1.07375	0.536876	+	0.843661i	$0.319604\pi$
$978$	47.2392	1.51054
$979$	−73.6047	−2.35241
$980$	−69.3757	−2.21613
$981$	49.1700	1.56988
$982$	−57.2122	−1.82571
$983$	28.6965	0.915277	0.457639	−	0.889138i	$-0.348695\pi$
$983$	28.6965	0.915277	0.457639	+	0.889138i	$0.348695\pi$
$984$	−208.218	−6.63775
$985$	−44.0623	−1.40394
$986$	−20.5772	−0.655312
$987$	−2.47568	−0.0788017
$988$	0.381667	0.0121425
$989$	77.8721	2.47619
$990$	−90.8733	−2.88814
$991$	1.71417	0.0544524	0.0272262	−	0.999629i	$-0.491333\pi$
$991$	1.71417	0.0544524	0.0272262	+	0.999629i	$0.491333\pi$
$992$	−214.741	−6.81802
$993$	82.1457	2.60681
$994$	−34.3865	−1.09067
$995$	−11.2714	−0.357327
$996$	−1.20019	−0.0380295
$997$	−7.57071	−0.239767	−0.119883	−	0.992788i	$-0.538252\pi$
$997$	−7.57071	−0.239767	−0.119883	+	0.992788i	$0.538252\pi$
$998$	−110.193	−3.48811
$999$	−0.256343	−0.00811032

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.b.1.7	✓	391

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.b.1.7	✓	391	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78421 q^{2} +2.45333 q^{3} +5.75180 q^{4} +2.41719 q^{5} -6.83056 q^{6} +1.41777 q^{7} -10.4458 q^{8} +3.01881 q^{9} +O(q^{10})\)	\(q-2.78421 q^{2} +2.45333 q^{3} +5.75180 q^{4} +2.41719 q^{5} -6.83056 q^{6} +1.41777 q^{7} -10.4458 q^{8} +3.01881 q^{9} -6.72994 q^{10} +4.47290 q^{11} +14.1110 q^{12} -0.0690116 q^{13} -3.94737 q^{14} +5.93015 q^{15} +17.5796 q^{16} +6.39154 q^{17} -8.40498 q^{18} -0.961522 q^{19} +13.9032 q^{20} +3.47826 q^{21} -12.4535 q^{22} +7.43108 q^{23} -25.6269 q^{24} +0.842792 q^{25} +0.192143 q^{26} +0.0461415 q^{27} +8.15474 q^{28} +1.15632 q^{29} -16.5107 q^{30} +7.65464 q^{31} -28.0536 q^{32} +10.9735 q^{33} -17.7954 q^{34} +3.42702 q^{35} +17.3636 q^{36} -5.55557 q^{37} +2.67707 q^{38} -0.169308 q^{39} -25.2494 q^{40} +8.12499 q^{41} -9.68418 q^{42} +10.4792 q^{43} +25.7272 q^{44} +7.29702 q^{45} -20.6897 q^{46} -0.711758 q^{47} +43.1285 q^{48} -4.98992 q^{49} -2.34651 q^{50} +15.6805 q^{51} -0.396941 q^{52} -10.8485 q^{53} -0.128467 q^{54} +10.8118 q^{55} -14.8097 q^{56} -2.35893 q^{57} -3.21944 q^{58} -2.92814 q^{59} +34.1090 q^{60} +7.87866 q^{61} -21.3121 q^{62} +4.27998 q^{63} +42.9479 q^{64} -0.166814 q^{65} -30.5524 q^{66} -4.88448 q^{67} +36.7628 q^{68} +18.2309 q^{69} -9.54153 q^{70} +8.71125 q^{71} -31.5338 q^{72} +4.37166 q^{73} +15.4679 q^{74} +2.06764 q^{75} -5.53048 q^{76} +6.34156 q^{77} +0.471388 q^{78} +2.28633 q^{79} +42.4932 q^{80} -8.94322 q^{81} -22.6216 q^{82} -0.0850534 q^{83} +20.0062 q^{84} +15.4495 q^{85} -29.1763 q^{86} +2.83684 q^{87} -46.7229 q^{88} -16.4557 q^{89} -20.3164 q^{90} -0.0978428 q^{91} +42.7421 q^{92} +18.7793 q^{93} +1.98168 q^{94} -2.32418 q^{95} -68.8247 q^{96} -15.8136 q^{97} +13.8930 q^{98} +13.5028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more