LMFDB - Embedded newform 4034.2.a.c.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4034,2,Mod(1,4034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4034, 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("4034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4034 = 2 \cdot 2017 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4034.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:	$32.2116521754$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.600384 q^{3} +1.00000 q^{4} +0.151444 q^{5} +0.600384 q^{6} -2.68719 q^{7} -1.00000 q^{8} -2.63954 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.600384 q^{3} +1.00000 q^{4} +0.151444 q^{5} +0.600384 q^{6} -2.68719 q^{7} -1.00000 q^{8} -2.63954 q^{9} -0.151444 q^{10} -3.19669 q^{11} -0.600384 q^{12} -3.37983 q^{13} +2.68719 q^{14} -0.0909244 q^{15} +1.00000 q^{16} -5.05388 q^{17} +2.63954 q^{18} -3.19526 q^{19} +0.151444 q^{20} +1.61335 q^{21} +3.19669 q^{22} -4.99371 q^{23} +0.600384 q^{24} -4.97706 q^{25} +3.37983 q^{26} +3.38589 q^{27} -2.68719 q^{28} -1.95000 q^{29} +0.0909244 q^{30} -1.23811 q^{31} -1.00000 q^{32} +1.91924 q^{33} +5.05388 q^{34} -0.406958 q^{35} -2.63954 q^{36} +10.9509 q^{37} +3.19526 q^{38} +2.02920 q^{39} -0.151444 q^{40} -3.13433 q^{41} -1.61335 q^{42} -6.28846 q^{43} -3.19669 q^{44} -0.399742 q^{45} +4.99371 q^{46} -9.20442 q^{47} -0.600384 q^{48} +0.220992 q^{49} +4.97706 q^{50} +3.03427 q^{51} -3.37983 q^{52} -7.20538 q^{53} -3.38589 q^{54} -0.484119 q^{55} +2.68719 q^{56} +1.91838 q^{57} +1.95000 q^{58} +14.7601 q^{59} -0.0909244 q^{60} -3.02915 q^{61} +1.23811 q^{62} +7.09294 q^{63} +1.00000 q^{64} -0.511855 q^{65} -1.91924 q^{66} +7.68856 q^{67} -5.05388 q^{68} +2.99814 q^{69} +0.406958 q^{70} +15.5477 q^{71} +2.63954 q^{72} +4.86285 q^{73} -10.9509 q^{74} +2.98815 q^{75} -3.19526 q^{76} +8.59012 q^{77} -2.02920 q^{78} -9.72905 q^{79} +0.151444 q^{80} +5.88579 q^{81} +3.13433 q^{82} +3.42807 q^{83} +1.61335 q^{84} -0.765379 q^{85} +6.28846 q^{86} +1.17075 q^{87} +3.19669 q^{88} -11.8101 q^{89} +0.399742 q^{90} +9.08226 q^{91} -4.99371 q^{92} +0.743340 q^{93} +9.20442 q^{94} -0.483903 q^{95} +0.600384 q^{96} +0.970934 q^{97} -0.220992 q^{98} +8.43780 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 49 q^{2} + 8 q^{3} + 49 q^{4} - 8 q^{5} - 8 q^{6} + 18 q^{7} - 49 q^{8} + 59 q^{9}+O(q^{10}) $ 49 * q - 49 * q^2 + 8 * q^3 + 49 * q^4 - 8 * q^5 - 8 * q^6 + 18 * q^7 - 49 * q^8 + 59 * q^9	\( 49 q - 49 q^{2} + 8 q^{3} + 49 q^{4} - 8 q^{5} - 8 q^{6} + 18 q^{7} - 49 q^{8} + 59 q^{9} + 8 q^{10} + q^{11} + 8 q^{12} + 9 q^{13} - 18 q^{14} + 15 q^{15} + 49 q^{16} - 27 q^{17} - 59 q^{18} + 27 q^{19} - 8 q^{20} + 13 q^{21} - q^{22} + 16 q^{23} - 8 q^{24} + 71 q^{25} - 9 q^{26} + 29 q^{27} + 18 q^{28} - 7 q^{29} - 15 q^{30} + 75 q^{31} - 49 q^{32} - 3 q^{33} + 27 q^{34} - 16 q^{35} + 59 q^{36} + 36 q^{37} - 27 q^{38} + 24 q^{39} + 8 q^{40} - 12 q^{41} - 13 q^{42} + 22 q^{43} + q^{44} + 5 q^{45} - 16 q^{46} + 26 q^{47} + 8 q^{48} + 107 q^{49} - 71 q^{50} + 35 q^{51} + 9 q^{52} - 10 q^{53} - 29 q^{54} + 76 q^{55} - 18 q^{56} - 10 q^{57} + 7 q^{58} + 9 q^{59} + 15 q^{60} + 87 q^{61} - 75 q^{62} + 68 q^{63} + 49 q^{64} - 6 q^{65} + 3 q^{66} + 46 q^{67} - 27 q^{68} + 70 q^{69} + 16 q^{70} + 40 q^{71} - 59 q^{72} + 6 q^{73} - 36 q^{74} + 69 q^{75} + 27 q^{76} - 12 q^{77} - 24 q^{78} + 76 q^{79} - 8 q^{80} + 77 q^{81} + 12 q^{82} - 32 q^{83} + 13 q^{84} + 19 q^{85} - 22 q^{86} + 36 q^{87} - q^{88} + 34 q^{89} - 5 q^{90} + 119 q^{91} + 16 q^{92} - 5 q^{93} - 26 q^{94} - 2 q^{95} - 8 q^{96} + 52 q^{97} - 107 q^{98} + 26 q^{99}+O(q^{100}) \) 49 * q - 49 * q^2 + 8 * q^3 + 49 * q^4 - 8 * q^5 - 8 * q^6 + 18 * q^7 - 49 * q^8 + 59 * q^9 + 8 * q^10 + q^11 + 8 * q^12 + 9 * q^13 - 18 * q^14 + 15 * q^15 + 49 * q^16 - 27 * q^17 - 59 * q^18 + 27 * q^19 - 8 * q^20 + 13 * q^21 - q^22 + 16 * q^23 - 8 * q^24 + 71 * q^25 - 9 * q^26 + 29 * q^27 + 18 * q^28 - 7 * q^29 - 15 * q^30 + 75 * q^31 - 49 * q^32 - 3 * q^33 + 27 * q^34 - 16 * q^35 + 59 * q^36 + 36 * q^37 - 27 * q^38 + 24 * q^39 + 8 * q^40 - 12 * q^41 - 13 * q^42 + 22 * q^43 + q^44 + 5 * q^45 - 16 * q^46 + 26 * q^47 + 8 * q^48 + 107 * q^49 - 71 * q^50 + 35 * q^51 + 9 * q^52 - 10 * q^53 - 29 * q^54 + 76 * q^55 - 18 * q^56 - 10 * q^57 + 7 * q^58 + 9 * q^59 + 15 * q^60 + 87 * q^61 - 75 * q^62 + 68 * q^63 + 49 * q^64 - 6 * q^65 + 3 * q^66 + 46 * q^67 - 27 * q^68 + 70 * q^69 + 16 * q^70 + 40 * q^71 - 59 * q^72 + 6 * q^73 - 36 * q^74 + 69 * q^75 + 27 * q^76 - 12 * q^77 - 24 * q^78 + 76 * q^79 - 8 * q^80 + 77 * q^81 + 12 * q^82 - 32 * q^83 + 13 * q^84 + 19 * q^85 - 22 * q^86 + 36 * q^87 - q^88 + 34 * q^89 - 5 * q^90 + 119 * q^91 + 16 * q^92 - 5 * q^93 - 26 * q^94 - 2 * q^95 - 8 * q^96 + 52 * q^97 - 107 * q^98 + 26 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.600384	−0.346632	−0.173316	−	0.984866i	$-0.555448\pi$
$3$	−0.600384	−0.346632	−0.173316	+	0.984866i	$0.555448\pi$
$4$	1.00000	0.500000
$5$	0.151444	0.0677278	0.0338639	−	0.999426i	$-0.489219\pi$
$5$	0.151444	0.0677278	0.0338639	+	0.999426i	$0.489219\pi$
$6$	0.600384	0.245106
$7$	−2.68719	−1.01566	−0.507831	−	0.861457i	$-0.669553\pi$
$7$	−2.68719	−1.01566	−0.507831	+	0.861457i	$0.669553\pi$
$8$	−1.00000	−0.353553
$9$	−2.63954	−0.879846
$10$	−0.151444	−0.0478908
$11$	−3.19669	−0.963839	−0.481920	−	0.876215i	$-0.660060\pi$
$11$	−3.19669	−0.963839	−0.481920	+	0.876215i	$0.660060\pi$
$12$	−0.600384	−0.173316
$13$	−3.37983	−0.937397	−0.468699	−	0.883358i	$-0.655277\pi$
$13$	−3.37983	−0.937397	−0.468699	+	0.883358i	$0.655277\pi$
$14$	2.68719	0.718182
$15$	−0.0909244	−0.0234766
$16$	1.00000	0.250000
$17$	−5.05388	−1.22575	−0.612873	−	0.790182i	$-0.709986\pi$
$17$	−5.05388	−1.22575	−0.612873	+	0.790182i	$0.709986\pi$
$18$	2.63954	0.622145
$19$	−3.19526	−0.733043	−0.366522	−	0.930409i	$-0.619451\pi$
$19$	−3.19526	−0.733043	−0.366522	+	0.930409i	$0.619451\pi$
$20$	0.151444	0.0338639
$21$	1.61335	0.352061
$22$	3.19669	0.681537
$23$	−4.99371	−1.04126	−0.520630	−	0.853783i	$-0.674303\pi$
$23$	−4.99371	−1.04126	−0.520630	+	0.853783i	$0.674303\pi$
$24$	0.600384	0.122553
$25$	−4.97706	−0.995413
$26$	3.37983	0.662840
$27$	3.38589	0.651614
$28$	−2.68719	−0.507831
$29$	−1.95000	−0.362105	−0.181053	−	0.983473i	$-0.557950\pi$
$29$	−1.95000	−0.362105	−0.181053	+	0.983473i	$0.557950\pi$
$30$	0.0909244	0.0166005
$31$	−1.23811	−0.222371	−0.111185	−	0.993800i	$-0.535465\pi$
$31$	−1.23811	−0.222371	−0.111185	+	0.993800i	$0.535465\pi$
$32$	−1.00000	−0.176777
$33$	1.91924	0.334097
$34$	5.05388	0.866733
$35$	−0.406958	−0.0687885
$36$	−2.63954	−0.439923
$37$	10.9509	1.80031	0.900157	−	0.435566i	$-0.143452\pi$
$37$	10.9509	1.80031	0.900157	+	0.435566i	$0.143452\pi$
$38$	3.19526	0.518340
$39$	2.02920	0.324932
$40$	−0.151444	−0.0239454
$41$	−3.13433	−0.489500	−0.244750	−	0.969586i	$-0.578706\pi$
$41$	−3.13433	−0.489500	−0.244750	+	0.969586i	$0.578706\pi$
$42$	−1.61335	−0.248945
$43$	−6.28846	−0.958982	−0.479491	−	0.877547i	$-0.659179\pi$
$43$	−6.28846	−0.958982	−0.479491	+	0.877547i	$0.659179\pi$
$44$	−3.19669	−0.481920
$45$	−0.399742	−0.0595900
$46$	4.99371	0.736282
$47$	−9.20442	−1.34260	−0.671301	−	0.741185i	$-0.734264\pi$
$47$	−9.20442	−1.34260	−0.671301	+	0.741185i	$0.734264\pi$
$48$	−0.600384	−0.0866579
$49$	0.220992	0.0315703
$50$	4.97706	0.703863
$51$	3.03427	0.424882
$52$	−3.37983	−0.468699
$53$	−7.20538	−0.989735	−0.494867	−	0.868968i	$-0.664783\pi$
$53$	−7.20538	−0.989735	−0.494867	+	0.868968i	$0.664783\pi$
$54$	−3.38589	−0.460761
$55$	−0.484119	−0.0652787
$56$	2.68719	0.359091
$57$	1.91838	0.254096
$58$	1.95000	0.256047
$59$	14.7601	1.92160	0.960801	−	0.277237i	$-0.0894188\pi$
$59$	14.7601	1.92160	0.960801	+	0.277237i	$0.0894188\pi$
$60$	−0.0909244	−0.0117383
$61$	−3.02915	−0.387843	−0.193922	−	0.981017i	$-0.562121\pi$
$61$	−3.02915	−0.387843	−0.193922	+	0.981017i	$0.562121\pi$
$62$	1.23811	0.157240
$63$	7.09294	0.893627
$64$	1.00000	0.125000
$65$	−0.511855	−0.0634878
$66$	−1.91924	−0.236242
$67$	7.68856	0.939307	0.469653	−	0.882851i	$-0.344379\pi$
$67$	7.68856	0.939307	0.469653	+	0.882851i	$0.344379\pi$
$68$	−5.05388	−0.612873
$69$	2.99814	0.360934
$70$	0.406958	0.0486408
$71$	15.5477	1.84517	0.922586	−	0.385792i	$-0.126072\pi$
$71$	15.5477	1.84517	0.922586	+	0.385792i	$0.126072\pi$
$72$	2.63954	0.311073
$73$	4.86285	0.569154	0.284577	−	0.958653i	$-0.408147\pi$
$73$	4.86285	0.569154	0.284577	+	0.958653i	$0.408147\pi$
$74$	−10.9509	−1.27301
$75$	2.98815	0.345042
$76$	−3.19526	−0.366522
$77$	8.59012	0.978935
$78$	−2.02920	−0.229761
$79$	−9.72905	−1.09460	−0.547302	−	0.836935i	$-0.684345\pi$
$79$	−9.72905	−1.09460	−0.547302	+	0.836935i	$0.684345\pi$
$80$	0.151444	0.0169319
$81$	5.88579	0.653976
$82$	3.13433	0.346129
$83$	3.42807	0.376280	0.188140	−	0.982142i	$-0.439754\pi$
$83$	3.42807	0.376280	0.188140	+	0.982142i	$0.439754\pi$
$84$	1.61335	0.176030
$85$	−0.765379	−0.0830170
$86$	6.28846	0.678102
$87$	1.17075	0.125517
$88$	3.19669	0.340769
$89$	−11.8101	−1.25187	−0.625934	−	0.779876i	$-0.715282\pi$
$89$	−11.8101	−1.25187	−0.625934	+	0.779876i	$0.715282\pi$
$90$	0.399742	0.0421365
$91$	9.08226	0.952079
$92$	−4.99371	−0.520630
$93$	0.743340	0.0770808
$94$	9.20442	0.949363
$95$	−0.483903	−0.0496474
$96$	0.600384	0.0612764
$97$	0.970934	0.0985834	0.0492917	−	0.998784i	$-0.484304\pi$
$97$	0.970934	0.0985834	0.0492917	+	0.998784i	$0.484304\pi$
$98$	−0.220992	−0.0223236
$99$	8.43780	0.848030
$100$	−4.97706	−0.497706
$101$	−5.66965	−0.564151	−0.282075	−	0.959392i	$-0.591023\pi$
$101$	−5.66965	−0.564151	−0.282075	+	0.959392i	$0.591023\pi$
$102$	−3.03427	−0.300437
$103$	0.648425	0.0638912	0.0319456	−	0.999490i	$-0.489830\pi$
$103$	0.648425	0.0638912	0.0319456	+	0.999490i	$0.489830\pi$
$104$	3.37983	0.331420
$105$	0.244331	0.0238443
$106$	7.20538	0.699848
$107$	−4.46203	−0.431360	−0.215680	−	0.976464i	$-0.569197\pi$
$107$	−4.46203	−0.431360	−0.215680	+	0.976464i	$0.569197\pi$
$108$	3.38589	0.325807
$109$	8.59001	0.822774	0.411387	−	0.911461i	$-0.365045\pi$
$109$	8.59001	0.822774	0.411387	+	0.911461i	$0.365045\pi$
$110$	0.484119	0.0461590
$111$	−6.57473	−0.624046
$112$	−2.68719	−0.253916
$113$	−4.09107	−0.384855	−0.192428	−	0.981311i	$-0.561636\pi$
$113$	−4.09107	−0.384855	−0.192428	+	0.981311i	$0.561636\pi$
$114$	−1.91838	−0.179673
$115$	−0.756266	−0.0705222
$116$	−1.95000	−0.181053
$117$	8.92120	0.824765
$118$	−14.7601	−1.35878
$119$	13.5807	1.24494
$120$	0.0909244	0.00830023
$121$	−0.781157	−0.0710143
$122$	3.02915	0.274247
$123$	1.88180	0.169676
$124$	−1.23811	−0.111185
$125$	−1.51097	−0.135145
$126$	−7.09294	−0.631890
$127$	−8.74345	−0.775856	−0.387928	−	0.921690i	$-0.626809\pi$
$127$	−8.74345	−0.775856	−0.387928	+	0.921690i	$0.626809\pi$
$128$	−1.00000	−0.0883883
$129$	3.77549	0.332413
$130$	0.511855	0.0448927
$131$	10.4005	0.908696	0.454348	−	0.890824i	$-0.349872\pi$
$131$	10.4005	0.908696	0.454348	+	0.890824i	$0.349872\pi$
$132$	1.91924	0.167049
$133$	8.58628	0.744525
$134$	−7.68856	−0.664190
$135$	0.512772	0.0441324
$136$	5.05388	0.433367
$137$	−13.5036	−1.15369	−0.576847	−	0.816852i	$-0.695717\pi$
$137$	−13.5036	−1.15369	−0.576847	+	0.816852i	$0.695717\pi$
$138$	−2.99814	−0.255219
$139$	−9.47559	−0.803709	−0.401855	−	0.915703i	$-0.631634\pi$
$139$	−9.47559	−0.803709	−0.401855	+	0.915703i	$0.631634\pi$
$140$	−0.406958	−0.0343943
$141$	5.52618	0.465388
$142$	−15.5477	−1.30473
$143$	10.8043	0.903500
$144$	−2.63954	−0.219962
$145$	−0.295315	−0.0245246
$146$	−4.86285	−0.402452
$147$	−0.132680	−0.0109433
$148$	10.9509	0.900157
$149$	−7.31620	−0.599366	−0.299683	−	0.954039i	$-0.596881\pi$
$149$	−7.31620	−0.599366	−0.299683	+	0.954039i	$0.596881\pi$
$150$	−2.98815	−0.243981
$151$	−1.15435	−0.0939393	−0.0469697	−	0.998896i	$-0.514956\pi$
$151$	−1.15435	−0.0939393	−0.0469697	+	0.998896i	$0.514956\pi$
$152$	3.19526	0.259170
$153$	13.3399	1.07847
$154$	−8.59012	−0.692212
$155$	−0.187504	−0.0150607
$156$	2.02920	0.162466
$157$	18.1408	1.44779	0.723896	−	0.689910i	$-0.242350\pi$
$157$	18.1408	1.44779	0.723896	+	0.689910i	$0.242350\pi$
$158$	9.72905	0.774002
$159$	4.32599	0.343074
$160$	−0.151444	−0.0119727
$161$	13.4190	1.05757
$162$	−5.88579	−0.462431
$163$	1.24873	0.0978079	0.0489039	−	0.998803i	$-0.484427\pi$
$163$	1.24873	0.0978079	0.0489039	+	0.998803i	$0.484427\pi$
$164$	−3.13433	−0.244750
$165$	0.290657	0.0226277
$166$	−3.42807	−0.266070
$167$	−22.6773	−1.75482	−0.877412	−	0.479738i	$-0.840732\pi$
$167$	−22.6773	−1.75482	−0.877412	+	0.479738i	$0.840732\pi$
$168$	−1.61335	−0.124472
$169$	−1.57673	−0.121287
$170$	0.765379	0.0587019
$171$	8.43402	0.644966
$172$	−6.28846	−0.479491
$173$	−2.25725	−0.171616	−0.0858079	−	0.996312i	$-0.527347\pi$
$173$	−2.25725	−0.171616	−0.0858079	+	0.996312i	$0.527347\pi$
$174$	−1.17075	−0.0887540
$175$	13.3743	1.01100
$176$	−3.19669	−0.240960
$177$	−8.86173	−0.666089
$178$	11.8101	0.885205
$179$	21.4655	1.60441	0.802203	−	0.597052i	$-0.203661\pi$
$179$	21.4655	1.60441	0.802203	+	0.597052i	$0.203661\pi$
$180$	−0.399742	−0.0297950
$181$	−2.29750	−0.170772	−0.0853861	−	0.996348i	$-0.527212\pi$
$181$	−2.29750	−0.170772	−0.0853861	+	0.996348i	$0.527212\pi$
$182$	−9.08226	−0.673222
$183$	1.81865	0.134439
$184$	4.99371	0.368141
$185$	1.65844	0.121931
$186$	−0.743340	−0.0545043
$187$	16.1557	1.18142
$188$	−9.20442	−0.671301
$189$	−9.09853	−0.661820
$190$	0.483903	0.0351060
$191$	20.8096	1.50573	0.752865	−	0.658174i	$-0.228671\pi$
$191$	20.8096	1.50573	0.752865	+	0.658174i	$0.228671\pi$
$192$	−0.600384	−0.0433290
$193$	−4.07240	−0.293138	−0.146569	−	0.989200i	$-0.546823\pi$
$193$	−4.07240	−0.293138	−0.146569	+	0.989200i	$0.546823\pi$
$194$	−0.970934	−0.0697090
$195$	0.307309	0.0220069
$196$	0.220992	0.0157851
$197$	−11.4726	−0.817391	−0.408696	−	0.912671i	$-0.634016\pi$
$197$	−11.4726	−0.817391	−0.408696	+	0.912671i	$0.634016\pi$
$198$	−8.43780	−0.599648
$199$	3.47150	0.246088	0.123044	−	0.992401i	$-0.460734\pi$
$199$	3.47150	0.246088	0.123044	+	0.992401i	$0.460734\pi$
$200$	4.97706	0.351932
$201$	−4.61609	−0.325594
$202$	5.66965	0.398915
$203$	5.24001	0.367777
$204$	3.03427	0.212441
$205$	−0.474675	−0.0331528
$206$	−0.648425	−0.0451779
$207$	13.1811	0.916149
$208$	−3.37983	−0.234349
$209$	10.2143	0.706536
$210$	−0.244331	−0.0168605
$211$	−19.5994	−1.34928	−0.674641	−	0.738146i	$-0.735701\pi$
$211$	−19.5994	−1.34928	−0.674641	+	0.738146i	$0.735701\pi$
$212$	−7.20538	−0.494867
$213$	−9.33458	−0.639595
$214$	4.46203	0.305018
$215$	−0.952349	−0.0649497
$216$	−3.38589	−0.230380
$217$	3.32703	0.225854
$218$	−8.59001	−0.581789
$219$	−2.91958	−0.197287
$220$	−0.484119	−0.0326393
$221$	17.0813	1.14901
$222$	6.57473	0.441267
$223$	−20.1695	−1.35065	−0.675323	−	0.737522i	$-0.735996\pi$
$223$	−20.1695	−1.35065	−0.675323	+	0.737522i	$0.735996\pi$
$224$	2.68719	0.179545
$225$	13.1372	0.875811
$226$	4.09107	0.272134
$227$	−21.0485	−1.39704	−0.698520	−	0.715591i	$-0.746158\pi$
$227$	−21.0485	−1.39704	−0.698520	+	0.715591i	$0.746158\pi$
$228$	1.91838	0.127048
$229$	−0.918463	−0.0606938	−0.0303469	−	0.999539i	$-0.509661\pi$
$229$	−0.918463	−0.0606938	−0.0303469	+	0.999539i	$0.509661\pi$
$230$	0.756266	0.0498667
$231$	−5.15737	−0.339330
$232$	1.95000	0.128023
$233$	−11.5142	−0.754318	−0.377159	−	0.926148i	$-0.623099\pi$
$233$	−11.5142	−0.754318	−0.377159	+	0.926148i	$0.623099\pi$
$234$	−8.92120	−0.583197
$235$	−1.39395	−0.0909314
$236$	14.7601	0.960801
$237$	5.84117	0.379424
$238$	−13.5807	−0.880308
$239$	−26.2413	−1.69741	−0.848703	−	0.528869i	$-0.822616\pi$
$239$	−26.2413	−1.69741	−0.848703	+	0.528869i	$0.822616\pi$
$240$	−0.0909244	−0.00586915
$241$	−9.40894	−0.606083	−0.303041	−	0.952977i	$-0.598002\pi$
$241$	−9.40894	−0.606083	−0.303041	+	0.952977i	$0.598002\pi$
$242$	0.781157	0.0502147
$243$	−13.6914	−0.878303
$244$	−3.02915	−0.193922
$245$	0.0334679	0.00213819
$246$	−1.88180	−0.119979
$247$	10.7995	0.687153
$248$	1.23811	0.0786199
$249$	−2.05816	−0.130431
$250$	1.51097	0.0955618
$251$	−1.76932	−0.111678	−0.0558392	−	0.998440i	$-0.517783\pi$
$251$	−1.76932	−0.111678	−0.0558392	+	0.998440i	$0.517783\pi$
$252$	7.09294	0.446814
$253$	15.9633	1.00361
$254$	8.74345	0.548613
$255$	0.459521	0.0287763
$256$	1.00000	0.0625000
$257$	14.3609	0.895807	0.447904	−	0.894082i	$-0.352171\pi$
$257$	14.3609	0.895807	0.447904	+	0.894082i	$0.352171\pi$
$258$	−3.77549	−0.235052
$259$	−29.4271	−1.82851
$260$	−0.511855	−0.0317439
$261$	5.14709	0.318597
$262$	−10.4005	−0.642545
$263$	−10.6773	−0.658392	−0.329196	−	0.944262i	$-0.606778\pi$
$263$	−10.6773	−0.658392	−0.329196	+	0.944262i	$0.606778\pi$
$264$	−1.91924	−0.118121
$265$	−1.09121	−0.0670325
$266$	−8.58628	−0.526459
$267$	7.09060	0.433937
$268$	7.68856	0.469653
$269$	−17.0604	−1.04019	−0.520095	−	0.854108i	$-0.674103\pi$
$269$	−17.0604	−1.04019	−0.520095	+	0.854108i	$0.674103\pi$
$270$	−0.512772	−0.0312063
$271$	15.6864	0.952881	0.476440	−	0.879207i	$-0.341927\pi$
$271$	15.6864	0.952881	0.476440	+	0.879207i	$0.341927\pi$
$272$	−5.05388	−0.306436
$273$	−5.45284	−0.330021
$274$	13.5036	0.815785
$275$	15.9101	0.959418
$276$	2.99814	0.180467
$277$	−16.4529	−0.988560	−0.494280	−	0.869303i	$-0.664568\pi$
$277$	−16.4529	−0.988560	−0.494280	+	0.869303i	$0.664568\pi$
$278$	9.47559	0.568308
$279$	3.26803	0.195652
$280$	0.406958	0.0243204
$281$	−6.45173	−0.384878	−0.192439	−	0.981309i	$-0.561640\pi$
$281$	−6.45173	−0.384878	−0.192439	+	0.981309i	$0.561640\pi$
$282$	−5.52618	−0.329079
$283$	0.0698754	0.00415366	0.00207683	−	0.999998i	$-0.499339\pi$
$283$	0.0698754	0.00415366	0.00207683	+	0.999998i	$0.499339\pi$
$284$	15.5477	0.922586
$285$	0.290527	0.0172094
$286$	−10.8043	−0.638871
$287$	8.42255	0.497167
$288$	2.63954	0.155536
$289$	8.54169	0.502452
$290$	0.295315	0.0173415
$291$	−0.582933	−0.0341721
$292$	4.86285	0.284577
$293$	1.13330	0.0662080	0.0331040	−	0.999452i	$-0.489461\pi$
$293$	1.13330	0.0662080	0.0331040	+	0.999452i	$0.489461\pi$
$294$	0.132680	0.00773806
$295$	2.23533	0.130146
$296$	−10.9509	−0.636507
$297$	−10.8236	−0.628051
$298$	7.31620	0.423816
$299$	16.8779	0.976074
$300$	2.98815	0.172521
$301$	16.8983	0.974002
$302$	1.15435	0.0664251
$303$	3.40396	0.195553
$304$	−3.19526	−0.183261
$305$	−0.458747	−0.0262678
$306$	−13.3399	−0.762592
$307$	30.7512	1.75506	0.877531	−	0.479520i	$-0.159189\pi$
$307$	30.7512	1.75506	0.877531	+	0.479520i	$0.159189\pi$
$308$	8.59012	0.489468
$309$	−0.389304	−0.0221467
$310$	0.187504	0.0106495
$311$	−7.94991	−0.450798	−0.225399	−	0.974267i	$-0.572369\pi$
$311$	−7.94991	−0.450798	−0.225399	+	0.974267i	$0.572369\pi$
$312$	−2.02920	−0.114881
$313$	8.36491	0.472813	0.236406	−	0.971654i	$-0.424030\pi$
$313$	8.36491	0.472813	0.236406	+	0.971654i	$0.424030\pi$
$314$	−18.1408	−1.02374
$315$	1.07418	0.0605234
$316$	−9.72905	−0.547302
$317$	5.75648	0.323316	0.161658	−	0.986847i	$-0.448316\pi$
$317$	5.75648	0.323316	0.161658	+	0.986847i	$0.448316\pi$
$318$	−4.32599	−0.242590
$319$	6.23354	0.349011
$320$	0.151444	0.00846597
$321$	2.67893	0.149523
$322$	−13.4190	−0.747814
$323$	16.1485	0.898525
$324$	5.88579	0.326988
$325$	16.8216	0.933097
$326$	−1.24873	−0.0691606
$327$	−5.15730	−0.285199
$328$	3.13433	0.173064
$329$	24.7340	1.36363
$330$	−0.290657	−0.0160002
$331$	16.5938	0.912080	0.456040	−	0.889959i	$-0.349267\pi$
$331$	16.5938	0.912080	0.456040	+	0.889959i	$0.349267\pi$
$332$	3.42807	0.188140
$333$	−28.9053	−1.58400
$334$	22.6773	1.24085
$335$	1.16439	0.0636172
$336$	1.61335	0.0880152
$337$	22.5547	1.22863	0.614317	−	0.789059i	$-0.289432\pi$
$337$	22.5547	1.22863	0.614317	+	0.789059i	$0.289432\pi$
$338$	1.57673	0.0857627
$339$	2.45621	0.133403
$340$	−0.765379	−0.0415085
$341$	3.95785	0.214330
$342$	−8.43402	−0.456060
$343$	18.2165	0.983598
$344$	6.28846	0.339051
$345$	0.454050	0.0244452
$346$	2.25725	0.121351
$347$	13.3649	0.717464	0.358732	−	0.933441i	$-0.383209\pi$
$347$	13.3649	0.717464	0.358732	+	0.933441i	$0.383209\pi$
$348$	1.17075	0.0627586
$349$	23.4001	1.25258	0.626290	−	0.779590i	$-0.284573\pi$
$349$	23.4001	1.25258	0.626290	+	0.779590i	$0.284573\pi$
$350$	−13.3743	−0.714888
$351$	−11.4437	−0.610821
$352$	3.19669	0.170384
$353$	−4.00016	−0.212907	−0.106454	−	0.994318i	$-0.533950\pi$
$353$	−4.00016	−0.212907	−0.106454	+	0.994318i	$0.533950\pi$
$354$	8.86173	0.470996
$355$	2.35460	0.124969
$356$	−11.8101	−0.625934
$357$	−8.15365	−0.431537
$358$	−21.4655	−1.13449
$359$	37.0523	1.95555	0.977773	−	0.209664i	$-0.0672372\pi$
$359$	37.0523	1.95555	0.977773	+	0.209664i	$0.0672372\pi$
$360$	0.399742	0.0210683
$361$	−8.79030	−0.462647
$362$	2.29750	0.120754
$363$	0.468994	0.0246158
$364$	9.08226	0.476040
$365$	0.736449	0.0385475
$366$	−1.81865	−0.0950626
$367$	−16.5337	−0.863052	−0.431526	−	0.902101i	$-0.642025\pi$
$367$	−16.5337	−0.863052	−0.431526	+	0.902101i	$0.642025\pi$
$368$	−4.99371	−0.260315
$369$	8.27319	0.430685
$370$	−1.65844	−0.0862184
$371$	19.3622	1.00524
$372$	0.743340	0.0385404
$373$	−23.1867	−1.20056	−0.600282	−	0.799789i	$-0.704945\pi$
$373$	−23.1867	−1.20056	−0.600282	+	0.799789i	$0.704945\pi$
$374$	−16.1557	−0.835391
$375$	0.907159	0.0468455
$376$	9.20442	0.474682
$377$	6.59066	0.339436
$378$	9.09853	0.467978
$379$	27.6659	1.42110	0.710552	−	0.703645i	$-0.248445\pi$
$379$	27.6659	1.42110	0.710552	+	0.703645i	$0.248445\pi$
$380$	−0.483903	−0.0248237
$381$	5.24943	0.268936
$382$	−20.8096	−1.06471
$383$	17.5015	0.894286	0.447143	−	0.894463i	$-0.352442\pi$
$383$	17.5015	0.894286	0.447143	+	0.894463i	$0.352442\pi$
$384$	0.600384	0.0306382
$385$	1.30092	0.0663011
$386$	4.07240	0.207280
$387$	16.5986	0.843757
$388$	0.970934	0.0492917
$389$	−9.65839	−0.489700	−0.244850	−	0.969561i	$-0.578739\pi$
$389$	−9.65839	−0.489700	−0.244850	+	0.969561i	$0.578739\pi$
$390$	−0.307309	−0.0155612
$391$	25.2376	1.27632
$392$	−0.220992	−0.0111618
$393$	−6.24429	−0.314983
$394$	11.4726	0.577983
$395$	−1.47341	−0.0741351
$396$	8.43780	0.424015
$397$	−19.0689	−0.957041	−0.478521	−	0.878076i	$-0.658827\pi$
$397$	−19.0689	−0.957041	−0.478521	+	0.878076i	$0.658827\pi$
$398$	−3.47150	−0.174011
$399$	−5.15506	−0.258076
$400$	−4.97706	−0.248853
$401$	−25.6358	−1.28019	−0.640096	−	0.768295i	$-0.721105\pi$
$401$	−25.6358	−1.28019	−0.640096	+	0.768295i	$0.721105\pi$
$402$	4.61609	0.230229
$403$	4.18460	0.208450
$404$	−5.66965	−0.282075
$405$	0.891366	0.0442923
$406$	−5.24001	−0.260057
$407$	−35.0066	−1.73521
$408$	−3.03427	−0.150219
$409$	13.3681	0.661012	0.330506	−	0.943804i	$-0.392781\pi$
$409$	13.3681	0.661012	0.330506	+	0.943804i	$0.392781\pi$
$410$	0.474675	0.0234425
$411$	8.10736	0.399907
$412$	0.648425	0.0319456
$413$	−39.6632	−1.95170
$414$	−13.1811	−0.647815
$415$	0.519160	0.0254846
$416$	3.37983	0.165710
$417$	5.68899	0.278591
$418$	−10.2143	−0.499596
$419$	29.6516	1.44858	0.724289	−	0.689497i	$-0.242168\pi$
$419$	29.6516	1.44858	0.724289	+	0.689497i	$0.242168\pi$
$420$	0.244331	0.0119221
$421$	−40.1919	−1.95884	−0.979418	−	0.201845i	$-0.935306\pi$
$421$	−40.1919	−1.95884	−0.979418	+	0.201845i	$0.935306\pi$
$422$	19.5994	0.954086
$423$	24.2954	1.18128
$424$	7.20538	0.349924
$425$	25.1535	1.22012
$426$	9.33458	0.452262
$427$	8.13991	0.393918
$428$	−4.46203	−0.215680
$429$	−6.48672	−0.313182
$430$	0.952349	0.0459264
$431$	−10.1661	−0.489683	−0.244842	−	0.969563i	$-0.578736\pi$
$431$	−10.1661	−0.489683	−0.244842	+	0.969563i	$0.578736\pi$
$432$	3.38589	0.162904
$433$	4.72305	0.226975	0.113488	−	0.993539i	$-0.463798\pi$
$433$	4.72305	0.226975	0.113488	+	0.993539i	$0.463798\pi$
$434$	−3.32703	−0.159703
$435$	0.177302	0.00850099
$436$	8.59001	0.411387
$437$	15.9562	0.763289
$438$	2.91958	0.139503
$439$	23.5759	1.12522	0.562608	−	0.826724i	$-0.309798\pi$
$439$	23.5759	1.12522	0.562608	+	0.826724i	$0.309798\pi$
$440$	0.484119	0.0230795
$441$	−0.583317	−0.0277770
$442$	−17.0813	−0.812473
$443$	−37.3555	−1.77481	−0.887407	−	0.460987i	$-0.847496\pi$
$443$	−37.3555	−1.77481	−0.887407	+	0.460987i	$0.847496\pi$
$444$	−6.57473	−0.312023
$445$	−1.78857	−0.0847863
$446$	20.1695	0.955052
$447$	4.39253	0.207759
$448$	−2.68719	−0.126958
$449$	−18.5098	−0.873533	−0.436767	−	0.899575i	$-0.643876\pi$
$449$	−18.5098	−0.873533	−0.436767	+	0.899575i	$0.643876\pi$
$450$	−13.1372	−0.619292
$451$	10.0195	0.471800
$452$	−4.09107	−0.192428
$453$	0.693050	0.0325623
$454$	21.0485	0.987856
$455$	1.37545	0.0644822
$456$	−1.91838	−0.0898366
$457$	16.9839	0.794473	0.397237	−	0.917716i	$-0.369969\pi$
$457$	16.9839	0.794473	0.397237	+	0.917716i	$0.369969\pi$
$458$	0.918463	0.0429170
$459$	−17.1119	−0.798714
$460$	−0.756266	−0.0352611
$461$	8.02306	0.373671	0.186836	−	0.982391i	$-0.440177\pi$
$461$	8.02306	0.373671	0.186836	+	0.982391i	$0.440177\pi$
$462$	5.15737	0.239943
$463$	−1.16113	−0.0539622	−0.0269811	−	0.999636i	$-0.508589\pi$
$463$	−1.16113	−0.0539622	−0.0269811	+	0.999636i	$0.508589\pi$
$464$	−1.95000	−0.0905263
$465$	0.112574	0.00522051
$466$	11.5142	0.533384
$467$	−25.4181	−1.17621	−0.588105	−	0.808784i	$-0.700126\pi$
$467$	−25.4181	−1.17621	−0.588105	+	0.808784i	$0.700126\pi$
$468$	8.92120	0.412383
$469$	−20.6606	−0.954019
$470$	1.39395	0.0642982
$471$	−10.8914	−0.501850
$472$	−14.7601	−0.679389
$473$	20.1023	0.924304
$474$	−5.84117	−0.268294
$475$	15.9030	0.729681
$476$	13.5807	0.622472
$477$	19.0189	0.870815
$478$	26.2413	1.20025
$479$	21.0953	0.963871	0.481935	−	0.876207i	$-0.339934\pi$
$479$	21.0953	0.963871	0.481935	+	0.876207i	$0.339934\pi$
$480$	0.0909244	0.00415011
$481$	−37.0122	−1.68761
$482$	9.40894	0.428565
$483$	−8.05657	−0.366587
$484$	−0.781157	−0.0355072
$485$	0.147042	0.00667683
$486$	13.6914	0.621054
$487$	33.1388	1.50166	0.750832	−	0.660494i	$-0.229653\pi$
$487$	33.1388	1.50166	0.750832	+	0.660494i	$0.229653\pi$
$488$	3.02915	0.137123
$489$	−0.749716	−0.0339033
$490$	−0.0334679	−0.00151193
$491$	20.5934	0.929366	0.464683	−	0.885477i	$-0.346168\pi$
$491$	20.5934	0.929366	0.464683	+	0.885477i	$0.346168\pi$
$492$	1.88180	0.0848382
$493$	9.85504	0.443849
$494$	−10.7995	−0.485890
$495$	1.27785	0.0574352
$496$	−1.23811	−0.0555927
$497$	−41.7796	−1.87407
$498$	2.05816	0.0922283
$499$	24.0252	1.07552	0.537758	−	0.843099i	$-0.319272\pi$
$499$	24.0252	1.07552	0.537758	+	0.843099i	$0.319272\pi$
$500$	−1.51097	−0.0675724
$501$	13.6151	0.608278
$502$	1.76932	0.0789685
$503$	−29.1140	−1.29813	−0.649064	−	0.760733i	$-0.724839\pi$
$503$	−29.1140	−1.29813	−0.649064	+	0.760733i	$0.724839\pi$
$504$	−7.09294	−0.315945
$505$	−0.858633	−0.0382087
$506$	−15.9633	−0.709657
$507$	0.946642	0.0420418
$508$	−8.74345	−0.387928
$509$	−6.81856	−0.302228	−0.151114	−	0.988516i	$-0.548286\pi$
$509$	−6.81856	−0.302228	−0.151114	+	0.988516i	$0.548286\pi$
$510$	−0.459521	−0.0203479
$511$	−13.0674	−0.578068
$512$	−1.00000	−0.0441942
$513$	−10.8188	−0.477662
$514$	−14.3609	−0.633431
$515$	0.0982000	0.00432721
$516$	3.77549	0.166207
$517$	29.4237	1.29405
$518$	29.4271	1.29295
$519$	1.35522	0.0594875
$520$	0.511855	0.0224463
$521$	−21.6472	−0.948383	−0.474192	−	0.880422i	$-0.657260\pi$
$521$	−21.6472	−0.948383	−0.474192	+	0.880422i	$0.657260\pi$
$522$	−5.14709	−0.225282
$523$	9.69869	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$523$	9.69869	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$524$	10.4005	0.454348
$525$	−8.02972	−0.350446
$526$	10.6773	0.465553
$527$	6.25725	0.272570
$528$	1.91924	0.0835243
$529$	1.93709	0.0842215
$530$	1.09121	0.0473992
$531$	−38.9599	−1.69072
$532$	8.58628	0.372262
$533$	10.5935	0.458856
$534$	−7.09060	−0.306840
$535$	−0.675746	−0.0292151
$536$	−7.68856	−0.332095
$537$	−12.8875	−0.556138
$538$	17.0604	0.735525
$539$	−0.706444	−0.0304287
$540$	0.512772	0.0220662
$541$	25.7986	1.10917	0.554584	−	0.832128i	$-0.312877\pi$
$541$	25.7986	1.10917	0.554584	+	0.832128i	$0.312877\pi$
$542$	−15.6864	−0.673788
$543$	1.37938	0.0591951
$544$	5.05388	0.216683
$545$	1.30090	0.0557246
$546$	5.45284	0.233360
$547$	6.11718	0.261552	0.130776	−	0.991412i	$-0.458253\pi$
$547$	6.11718	0.261552	0.130776	+	0.991412i	$0.458253\pi$
$548$	−13.5036	−0.576847
$549$	7.99557	0.341243
$550$	−15.9101	−0.678411
$551$	6.23075	0.265439
$552$	−2.99814	−0.127609
$553$	26.1438	1.11175
$554$	16.4529	0.699018
$555$	−0.995703	−0.0422652
$556$	−9.47559	−0.401855
$557$	28.2842	1.19844	0.599219	−	0.800585i	$-0.295478\pi$
$557$	28.2842	1.19844	0.599219	+	0.800585i	$0.295478\pi$
$558$	−3.26803	−0.138347
$559$	21.2540	0.898947
$560$	−0.406958	−0.0171971
$561$	−9.69962	−0.409518
$562$	6.45173	0.272150
$563$	−37.3315	−1.57334	−0.786668	−	0.617376i	$-0.788196\pi$
$563$	−37.3315	−1.57334	−0.786668	+	0.617376i	$0.788196\pi$
$564$	5.52618	0.232694
$565$	−0.619567	−0.0260654
$566$	−0.0698754	−0.00293708
$567$	−15.8162	−0.664219
$568$	−15.5477	−0.652367
$569$	7.75204	0.324982	0.162491	−	0.986710i	$-0.448047\pi$
$569$	7.75204	0.324982	0.162491	+	0.986710i	$0.448047\pi$
$570$	−0.290527	−0.0121689
$571$	3.39533	0.142090	0.0710450	−	0.997473i	$-0.477367\pi$
$571$	3.39533	0.142090	0.0710450	+	0.997473i	$0.477367\pi$
$572$	10.8043	0.451750
$573$	−12.4938	−0.521934
$574$	−8.42255	−0.351550
$575$	24.8540	1.03648
$576$	−2.63954	−0.109981
$577$	9.35469	0.389441	0.194720	−	0.980859i	$-0.437620\pi$
$577$	9.35469	0.389441	0.194720	+	0.980859i	$0.437620\pi$
$578$	−8.54169	−0.355288
$579$	2.44500	0.101611
$580$	−0.295315	−0.0122623
$581$	−9.21188	−0.382173
$582$	0.582933	0.0241633
$583$	23.0334	0.953945
$584$	−4.86285	−0.201226
$585$	1.35106	0.0558595
$586$	−1.13330	−0.0468161
$587$	−10.2299	−0.422234	−0.211117	−	0.977461i	$-0.567710\pi$
$587$	−10.2299	−0.422234	−0.211117	+	0.977461i	$0.567710\pi$
$588$	−0.132680	−0.00547163
$589$	3.95608	0.163007
$590$	−2.23533	−0.0920270
$591$	6.88798	0.283334
$592$	10.9509	0.450079
$593$	−35.4236	−1.45467	−0.727336	−	0.686281i	$-0.759242\pi$
$593$	−35.4236	−1.45467	−0.727336	+	0.686281i	$0.759242\pi$
$594$	10.8236	0.444099
$595$	2.05672	0.0843173
$596$	−7.31620	−0.299683
$597$	−2.08423	−0.0853019
$598$	−16.8779	−0.690188
$599$	−24.8691	−1.01612	−0.508062	−	0.861320i	$-0.669638\pi$
$599$	−24.8691	−1.01612	−0.508062	+	0.861320i	$0.669638\pi$
$600$	−2.98815	−0.121991
$601$	4.32278	0.176330	0.0881649	−	0.996106i	$-0.471900\pi$
$601$	4.32278	0.176330	0.0881649	+	0.996106i	$0.471900\pi$
$602$	−16.8983	−0.688723
$603$	−20.2943	−0.826446
$604$	−1.15435	−0.0469697
$605$	−0.118301	−0.00480964
$606$	−3.40396	−0.138277
$607$	−19.1967	−0.779171	−0.389585	−	0.920990i	$-0.627382\pi$
$607$	−19.1967	−0.779171	−0.389585	+	0.920990i	$0.627382\pi$
$608$	3.19526	0.129585
$609$	−3.14602	−0.127483
$610$	0.458747	0.0185741
$611$	31.1094	1.25855
$612$	13.3399	0.539234
$613$	16.1860	0.653748	0.326874	−	0.945068i	$-0.394005\pi$
$613$	16.1860	0.653748	0.326874	+	0.945068i	$0.394005\pi$
$614$	−30.7512	−1.24102
$615$	0.284987	0.0114918
$616$	−8.59012	−0.346106
$617$	−17.2203	−0.693263	−0.346631	−	0.938001i	$-0.612675\pi$
$617$	−17.2203	−0.693263	−0.346631	+	0.938001i	$0.612675\pi$
$618$	0.389304	0.0156601
$619$	46.7912	1.88070	0.940348	−	0.340215i	$-0.110500\pi$
$619$	46.7912	1.88070	0.940348	+	0.340215i	$0.110500\pi$
$620$	−0.187504	−0.00753034
$621$	−16.9081	−0.678500
$622$	7.94991	0.318763
$623$	31.7360	1.27148
$624$	2.02920	0.0812329
$625$	24.6565	0.986260
$626$	−8.36491	−0.334329
$627$	−6.13248	−0.244908
$628$	18.1408	0.723896
$629$	−55.3444	−2.20673
$630$	−1.07418	−0.0427965
$631$	11.4934	0.457543	0.228772	−	0.973480i	$-0.426529\pi$
$631$	11.4934	0.457543	0.228772	+	0.973480i	$0.426529\pi$
$632$	9.72905	0.387001
$633$	11.7672	0.467704
$634$	−5.75648	−0.228619
$635$	−1.32414	−0.0525470
$636$	4.32599	0.171537
$637$	−0.746916	−0.0295939
$638$	−6.23354	−0.246788
$639$	−41.0387	−1.62347
$640$	−0.151444	−0.00598634
$641$	23.2519	0.918393	0.459197	−	0.888335i	$-0.348137\pi$
$641$	23.2519	0.918393	0.459197	+	0.888335i	$0.348137\pi$
$642$	−2.67893	−0.105729
$643$	2.62372	0.103469	0.0517347	−	0.998661i	$-0.483525\pi$
$643$	2.62372	0.103469	0.0517347	+	0.998661i	$0.483525\pi$
$644$	13.4190	0.528784
$645$	0.571775	0.0225136
$646$	−16.1485	−0.635353
$647$	−13.4249	−0.527789	−0.263894	−	0.964552i	$-0.585007\pi$
$647$	−13.4249	−0.527789	−0.263894	+	0.964552i	$0.585007\pi$
$648$	−5.88579	−0.231216
$649$	−47.1835	−1.85212
$650$	−16.8216	−0.659799
$651$	−1.99750	−0.0782880
$652$	1.24873	0.0489039
$653$	−12.4928	−0.488881	−0.244441	−	0.969664i	$-0.578604\pi$
$653$	−12.4928	−0.488881	−0.244441	+	0.969664i	$0.578604\pi$
$654$	5.15730	0.201666
$655$	1.57509	0.0615439
$656$	−3.13433	−0.122375
$657$	−12.8357	−0.500768
$658$	−24.7340	−0.964232
$659$	−19.5776	−0.762636	−0.381318	−	0.924444i	$-0.624530\pi$
$659$	−19.5776	−0.762636	−0.381318	+	0.924444i	$0.624530\pi$
$660$	0.290657	0.0113138
$661$	−26.1340	−1.01649	−0.508247	−	0.861211i	$-0.669706\pi$
$661$	−26.1340	−1.01649	−0.508247	+	0.861211i	$0.669706\pi$
$662$	−16.5938	−0.644938
$663$	−10.2553	−0.398283
$664$	−3.42807	−0.133035
$665$	1.30034	0.0504250
$666$	28.9053	1.12006
$667$	9.73770	0.377045
$668$	−22.6773	−0.877412
$669$	12.1094	0.468177
$670$	−1.16439	−0.0449841
$671$	9.68327	0.373819
$672$	−1.61335	−0.0622362
$673$	−24.5360	−0.945793	−0.472896	−	0.881118i	$-0.656791\pi$
$673$	−24.5360	−0.945793	−0.472896	+	0.881118i	$0.656791\pi$
$674$	−22.5547	−0.868776
$675$	−16.8518	−0.648625
$676$	−1.57673	−0.0606434
$677$	−48.4213	−1.86098	−0.930491	−	0.366314i	$-0.880620\pi$
$677$	−48.4213	−1.86098	−0.930491	+	0.366314i	$0.880620\pi$
$678$	−2.45621	−0.0943302
$679$	−2.60908	−0.100127
$680$	0.765379	0.0293509
$681$	12.6372	0.484258
$682$	−3.95785	−0.151554
$683$	27.7580	1.06213	0.531064	−	0.847332i	$-0.321792\pi$
$683$	27.7580	1.06213	0.531064	+	0.847332i	$0.321792\pi$
$684$	8.43402	0.322483
$685$	−2.04504	−0.0781371
$686$	−18.2165	−0.695509
$687$	0.551431	0.0210384
$688$	−6.28846	−0.239745
$689$	24.3530	0.927775
$690$	−0.454050	−0.0172854
$691$	29.1512	1.10896	0.554481	−	0.832196i	$-0.312917\pi$
$691$	29.1512	1.10896	0.554481	+	0.832196i	$0.312917\pi$
$692$	−2.25725	−0.0858079
$693$	−22.6740	−0.861313
$694$	−13.3649	−0.507324
$695$	−1.43502	−0.0544334
$696$	−1.17075	−0.0443770
$697$	15.8405	0.600003
$698$	−23.4001	−0.885708
$699$	6.91292	0.261471
$700$	13.3743	0.505502
$701$	18.5695	0.701358	0.350679	−	0.936496i	$-0.385951\pi$
$701$	18.5695	0.701358	0.350679	+	0.936496i	$0.385951\pi$
$702$	11.4437	0.431916
$703$	−34.9909	−1.31971
$704$	−3.19669	−0.120480
$705$	0.836906	0.0315197
$706$	4.00016	0.150548
$707$	15.2354	0.572987
$708$	−8.86173	−0.333044
$709$	−20.5180	−0.770570	−0.385285	−	0.922798i	$-0.625897\pi$
$709$	−20.5180	−0.770570	−0.385285	+	0.922798i	$0.625897\pi$
$710$	−2.35460	−0.0883666
$711$	25.6802	0.963083
$712$	11.8101	0.442602
$713$	6.18275	0.231546
$714$	8.15365	0.305143
$715$	1.63624	0.0611920
$716$	21.4655	0.802203
$717$	15.7548	0.588375
$718$	−37.0523	−1.38278
$719$	−30.3443	−1.13165	−0.565825	−	0.824525i	$-0.691442\pi$
$719$	−30.3443	−1.13165	−0.565825	+	0.824525i	$0.691442\pi$
$720$	−0.399742	−0.0148975
$721$	−1.74244	−0.0648919
$722$	8.79030	0.327141
$723$	5.64897	0.210088
$724$	−2.29750	−0.0853861
$725$	9.70525	0.360444
$726$	−0.468994	−0.0174060
$727$	−6.12223	−0.227061	−0.113530	−	0.993535i	$-0.536216\pi$
$727$	−6.12223	−0.227061	−0.113530	+	0.993535i	$0.536216\pi$
$728$	−9.08226	−0.336611
$729$	−9.43727	−0.349528
$730$	−0.736449	−0.0272572
$731$	31.7811	1.17547
$732$	1.81865	0.0672194
$733$	−2.89308	−0.106858	−0.0534291	−	0.998572i	$-0.517015\pi$
$733$	−2.89308	−0.106858	−0.0534291	+	0.998572i	$0.517015\pi$
$734$	16.5337	0.610270
$735$	−0.0200936	−0.000741163 0
$736$	4.99371	0.184070
$737$	−24.5780	−0.905341
$738$	−8.27319	−0.304540
$739$	−4.05532	−0.149177	−0.0745886	−	0.997214i	$-0.523764\pi$
$739$	−4.05532	−0.149177	−0.0745886	+	0.997214i	$0.523764\pi$
$740$	1.65844	0.0609656
$741$	−6.48382	−0.238189
$742$	−19.3622	−0.710810
$743$	−15.9544	−0.585309	−0.292655	−	0.956218i	$-0.594539\pi$
$743$	−15.9544	−0.585309	−0.292655	+	0.956218i	$0.594539\pi$
$744$	−0.743340	−0.0272522
$745$	−1.10799	−0.0405937
$746$	23.1867	0.848926
$747$	−9.04853	−0.331068
$748$	16.1557	0.590711
$749$	11.9903	0.438116
$750$	−0.907159	−0.0331248
$751$	−15.4057	−0.562164	−0.281082	−	0.959684i	$-0.590693\pi$
$751$	−15.4057	−0.562164	−0.281082	+	0.959684i	$0.590693\pi$
$752$	−9.20442	−0.335651
$753$	1.06227	0.0387112
$754$	−6.59066	−0.240018
$755$	−0.174819	−0.00636230
$756$	−9.09853	−0.330910
$757$	15.5502	0.565180	0.282590	−	0.959241i	$-0.408806\pi$
$757$	15.5502	0.565180	0.282590	+	0.959241i	$0.408806\pi$
$758$	−27.6659	−1.00487
$759$	−9.58413	−0.347882
$760$	0.483903	0.0175530
$761$	−29.5537	−1.07132	−0.535660	−	0.844434i	$-0.679937\pi$
$761$	−29.5537	−1.07132	−0.535660	+	0.844434i	$0.679937\pi$
$762$	−5.24943	−0.190167
$763$	−23.0830	−0.835660
$764$	20.8096	0.752865
$765$	2.02025	0.0730422
$766$	−17.5015	−0.632356
$767$	−49.8867	−1.80131
$768$	−0.600384	−0.0216645
$769$	−18.7106	−0.674723	−0.337361	−	0.941375i	$-0.609534\pi$
$769$	−18.7106	−0.674723	−0.337361	+	0.941375i	$0.609534\pi$
$770$	−1.30092	−0.0468819
$771$	−8.62204	−0.310515
$772$	−4.07240	−0.146569
$773$	−4.66585	−0.167819	−0.0839095	−	0.996473i	$-0.526741\pi$
$773$	−4.66585	−0.167819	−0.0839095	+	0.996473i	$0.526741\pi$
$774$	−16.5986	−0.596626
$775$	6.16214	0.221351
$776$	−0.970934	−0.0348545
$777$	17.6676	0.633820
$778$	9.65839	0.346270
$779$	10.0150	0.358825
$780$	0.307309	0.0110034
$781$	−49.7012	−1.77845
$782$	−25.2376	−0.902494
$783$	−6.60247	−0.235953
$784$	0.220992	0.00789257
$785$	2.74731	0.0980556
$786$	6.24429	0.222727
$787$	−27.6472	−0.985518	−0.492759	−	0.870166i	$-0.664012\pi$
$787$	−27.6472	−0.985518	−0.492759	+	0.870166i	$0.664012\pi$
$788$	−11.4726	−0.408696
$789$	6.41049	0.228220
$790$	1.47341	0.0524214
$791$	10.9935	0.390883
$792$	−8.43780	−0.299824
$793$	10.2380	0.363563
$794$	19.0689	0.676730
$795$	0.655145	0.0232356
$796$	3.47150	0.123044
$797$	2.63512	0.0933409	0.0466704	−	0.998910i	$-0.485139\pi$
$797$	2.63512	0.0933409	0.0466704	+	0.998910i	$0.485139\pi$
$798$	5.15506	0.182487
$799$	46.5180	1.64569
$800$	4.97706	0.175966
$801$	31.1732	1.10145
$802$	25.6358	0.905232
$803$	−15.5450	−0.548572
$804$	−4.61609	−0.162797
$805$	2.03223	0.0716267
$806$	−4.18460	−0.147396
$807$	10.2428	0.360563
$808$	5.66965	0.199457
$809$	45.1158	1.58619	0.793094	−	0.609100i	$-0.208469\pi$
$809$	45.1158	1.58619	0.793094	+	0.609100i	$0.208469\pi$
$810$	−0.891366	−0.0313194
$811$	−40.3714	−1.41763	−0.708816	−	0.705394i	$-0.750770\pi$
$811$	−40.3714	−1.41763	−0.708816	+	0.705394i	$0.750770\pi$
$812$	5.24001	0.183888
$813$	−9.41786	−0.330299
$814$	35.0066	1.22698
$815$	0.189112	0.00662431
$816$	3.03427	0.106221
$817$	20.0933	0.702975
$818$	−13.3681	−0.467406
$819$	−23.9730	−0.837683
$820$	−0.474675	−0.0165764
$821$	−4.82419	−0.168365	−0.0841826	−	0.996450i	$-0.526828\pi$
$821$	−4.82419	−0.168365	−0.0841826	+	0.996450i	$0.526828\pi$
$822$	−8.10736	−0.282777
$823$	−22.5877	−0.787357	−0.393679	−	0.919248i	$-0.628798\pi$
$823$	−22.5877	−0.787357	−0.393679	+	0.919248i	$0.628798\pi$
$824$	−0.648425	−0.0225890
$825$	−9.55219	−0.332565
$826$	39.6632	1.38006
$827$	−41.8709	−1.45600	−0.727998	−	0.685579i	$-0.759549\pi$
$827$	−41.8709	−1.45600	−0.727998	+	0.685579i	$0.759549\pi$
$828$	13.1811	0.458074
$829$	14.4426	0.501611	0.250806	−	0.968037i	$-0.419305\pi$
$829$	14.4426	0.501611	0.250806	+	0.968037i	$0.419305\pi$
$830$	−0.519160	−0.0180203
$831$	9.87807	0.342666
$832$	−3.37983	−0.117175
$833$	−1.11687	−0.0386972
$834$	−5.68899	−0.196994
$835$	−3.43434	−0.118850
$836$	10.2143	0.353268
$837$	−4.19209	−0.144900
$838$	−29.6516	−1.02430
$839$	13.9252	0.480750	0.240375	−	0.970680i	$-0.422730\pi$
$839$	13.9252	0.480750	0.240375	+	0.970680i	$0.422730\pi$
$840$	−0.244331	−0.00843023
$841$	−25.1975	−0.868880
$842$	40.1919	1.38511
$843$	3.87351	0.133411
$844$	−19.5994	−0.674641
$845$	−0.238786	−0.00821448
$846$	−24.2954	−0.835294
$847$	2.09912	0.0721266
$848$	−7.20538	−0.247434
$849$	−0.0419521	−0.00143979
$850$	−25.1535	−0.862757
$851$	−54.6855	−1.87459
$852$	−9.33458	−0.319797
$853$	3.49095	0.119528	0.0597639	−	0.998213i	$-0.480965\pi$
$853$	3.49095	0.119528	0.0597639	+	0.998213i	$0.480965\pi$
$854$	−8.13991	−0.278542
$855$	1.27728	0.0436821
$856$	4.46203	0.152509
$857$	53.9116	1.84158	0.920792	−	0.390054i	$-0.127543\pi$
$857$	53.9116	1.84158	0.920792	+	0.390054i	$0.127543\pi$
$858$	6.48672	0.221453
$859$	15.1979	0.518544	0.259272	−	0.965804i	$-0.416517\pi$
$859$	15.1979	0.518544	0.259272	+	0.965804i	$0.416517\pi$
$860$	−0.952349	−0.0324748
$861$	−5.05676	−0.172334
$862$	10.1661	0.346259
$863$	57.7249	1.96498	0.982490	−	0.186316i	$-0.0596549\pi$
$863$	57.7249	1.96498	0.982490	+	0.186316i	$0.0596549\pi$
$864$	−3.38589	−0.115190
$865$	−0.341847	−0.0116231
$866$	−4.72305	−0.160496
$867$	−5.12829	−0.174166
$868$	3.32703	0.112927
$869$	31.1008	1.05502
$870$	−0.177302	−0.00601111
$871$	−25.9860	−0.880504
$872$	−8.59001	−0.290894
$873$	−2.56282	−0.0867382
$874$	−15.9562	−0.539727
$875$	4.06025	0.137262
$876$	−2.91958	−0.0986434
$877$	0.0555540	0.00187592	0.000937962	−	1.00000i	$-0.499701\pi$
$877$	0.0555540	0.00187592	0.000937962		1.00000i	$0.499701\pi$
$878$	−23.5759	−0.795648
$879$	−0.680414	−0.0229498
$880$	−0.484119	−0.0163197
$881$	12.9528	0.436392	0.218196	−	0.975905i	$-0.429983\pi$
$881$	12.9528	0.436392	0.218196	+	0.975905i	$0.429983\pi$
$882$	0.583317	0.0196413
$883$	−30.1111	−1.01332	−0.506659	−	0.862146i	$-0.669120\pi$
$883$	−30.1111	−1.01332	−0.506659	+	0.862146i	$0.669120\pi$
$884$	17.0813	0.574505
$885$	−1.34205	−0.0451127
$886$	37.3555	1.25498
$887$	−43.1946	−1.45033	−0.725166	−	0.688575i	$-0.758237\pi$
$887$	−43.1946	−1.45033	−0.725166	+	0.688575i	$0.758237\pi$
$888$	6.57473	0.220634
$889$	23.4953	0.788008
$890$	1.78857	0.0599529
$891$	−18.8150	−0.630328
$892$	−20.1695	−0.675323
$893$	29.4105	0.984186
$894$	−4.39253	−0.146908
$895$	3.25081	0.108663
$896$	2.68719	0.0897727
$897$	−10.1332	−0.338338
$898$	18.5098	0.617681
$899$	2.41431	0.0805216
$900$	13.1372	0.437905
$901$	36.4151	1.21316
$902$	−10.0195	−0.333613
$903$	−10.1455	−0.337620
$904$	4.09107	0.136067
$905$	−0.347943	−0.0115660
$906$	−0.693050	−0.0230251
$907$	−26.7176	−0.887144	−0.443572	−	0.896239i	$-0.646289\pi$
$907$	−26.7176	−0.887144	−0.443572	+	0.896239i	$0.646289\pi$
$908$	−21.0485	−0.698520
$909$	14.9653	0.496366
$910$	−1.37545	−0.0455958
$911$	−52.6617	−1.74476	−0.872380	−	0.488829i	$-0.837424\pi$
$911$	−52.6617	−1.74476	−0.872380	+	0.488829i	$0.837424\pi$
$912$	1.91838	0.0635240
$913$	−10.9585	−0.362673
$914$	−16.9839	−0.561777
$915$	0.275424	0.00910524
$916$	−0.918463	−0.0303469
$917$	−27.9481	−0.922929
$918$	17.1119	0.564776
$919$	−5.03869	−0.166211	−0.0831055	−	0.996541i	$-0.526484\pi$
$919$	−5.03869	−0.166211	−0.0831055	+	0.996541i	$0.526484\pi$
$920$	0.756266	0.0249334
$921$	−18.4625	−0.608360
$922$	−8.02306	−0.264225
$923$	−52.5486	−1.72966
$924$	−5.15737	−0.169665
$925$	−54.5033	−1.79206
$926$	1.16113	0.0381571
$927$	−1.71154	−0.0562145
$928$	1.95000	0.0640117
$929$	−27.5642	−0.904351	−0.452175	−	0.891929i	$-0.649352\pi$
$929$	−27.5642	−0.904351	−0.452175	+	0.891929i	$0.649352\pi$
$930$	−0.112574	−0.00369146
$931$	−0.706128	−0.0231424
$932$	−11.5142	−0.377159
$933$	4.77300	0.156261
$934$	25.4181	0.831707
$935$	2.44668	0.0800150
$936$	−8.92120	−0.291599
$937$	−18.4721	−0.603457	−0.301728	−	0.953394i	$-0.597564\pi$
$937$	−18.4721	−0.603457	−0.301728	+	0.953394i	$0.597564\pi$
$938$	20.6606	0.674593
$939$	−5.02216	−0.163892
$940$	−1.39395	−0.0454657
$941$	−14.4465	−0.470943	−0.235471	−	0.971881i	$-0.575663\pi$
$941$	−14.4465	−0.470943	−0.235471	+	0.971881i	$0.575663\pi$
$942$	10.8914	0.354862
$943$	15.6519	0.509697
$944$	14.7601	0.480401
$945$	−1.37792	−0.0448236
$946$	−20.1023	−0.653582
$947$	33.6961	1.09498	0.547488	−	0.836813i	$-0.315584\pi$
$947$	33.6961	1.09498	0.547488	+	0.836813i	$0.315584\pi$
$948$	5.84117	0.189712
$949$	−16.4356	−0.533523
$950$	−15.9030	−0.515962
$951$	−3.45610	−0.112072
$952$	−13.5807	−0.440154
$953$	−13.2505	−0.429224	−0.214612	−	0.976699i	$-0.568849\pi$
$953$	−13.2505	−0.429224	−0.214612	+	0.976699i	$0.568849\pi$
$954$	−19.0189	−0.615759
$955$	3.15149	0.101980
$956$	−26.2413	−0.848703
$957$	−3.74251	−0.120978
$958$	−21.0953	−0.681560
$959$	36.2868	1.17176
$960$	−0.0909244	−0.00293457
$961$	−29.4671	−0.950551
$962$	37.0122	1.19332
$963$	11.7777	0.379531
$964$	−9.40894	−0.303041
$965$	−0.616740	−0.0198536
$966$	8.05657	0.259216
$967$	46.8008	1.50501	0.752507	−	0.658584i	$-0.228844\pi$
$967$	46.8008	1.50501	0.752507	+	0.658584i	$0.228844\pi$
$968$	0.781157	0.0251073
$969$	−9.69528	−0.311457
$970$	−0.147042	−0.00472123
$971$	−6.86171	−0.220203	−0.110101	−	0.993920i	$-0.535118\pi$
$971$	−6.86171	−0.220203	−0.110101	+	0.993920i	$0.535118\pi$
$972$	−13.6914	−0.439152
$973$	25.4627	0.816297
$974$	−33.1388	−1.06184
$975$	−10.0994	−0.323441
$976$	−3.02915	−0.0969609
$977$	−45.8332	−1.46634	−0.733168	−	0.680048i	$-0.761959\pi$
$977$	−45.8332	−1.46634	−0.733168	+	0.680048i	$0.761959\pi$
$978$	0.749716	0.0239733
$979$	37.7533	1.20660
$980$	0.0334679	0.00106909
$981$	−22.6737	−0.723914
$982$	−20.5934	−0.657161
$983$	35.0591	1.11821	0.559106	−	0.829096i	$-0.311144\pi$
$983$	35.0591	1.11821	0.559106	+	0.829096i	$0.311144\pi$
$984$	−1.88180	−0.0599896
$985$	−1.73746	−0.0553601
$986$	−9.85504	−0.313848
$987$	−14.8499	−0.472678
$988$	10.7995	0.343576
$989$	31.4027	0.998549
$990$	−1.27785	−0.0406128
$991$	32.7274	1.03962	0.519809	−	0.854282i	$-0.326003\pi$
$991$	32.7274	1.03962	0.519809	+	0.854282i	$0.326003\pi$
$992$	1.23811	0.0393100
$993$	−9.96267	−0.316156
$994$	41.7796	1.32517
$995$	0.525737	0.0166670
$996$	−2.05816	−0.0652153
$997$	−23.7757	−0.752985	−0.376493	−	0.926420i	$-0.622870\pi$
$997$	−23.7757	−0.752985	−0.376493	+	0.926420i	$0.622870\pi$
$998$	−24.0252	−0.760504
$999$	37.0785	1.17311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4034.2.a.c.1.19	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4034.2.a.c.1.19	✓	49	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:	\( N \)	\(=\)	\( 4034 = 2 \cdot 2017 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.600384 q^{3} +1.00000 q^{4} +0.151444 q^{5} +0.600384 q^{6} -2.68719 q^{7} -1.00000 q^{8} -2.63954 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.600384 q^{3} +1.00000 q^{4} +0.151444 q^{5} +0.600384 q^{6} -2.68719 q^{7} -1.00000 q^{8} -2.63954 q^{9} -0.151444 q^{10} -3.19669 q^{11} -0.600384 q^{12} -3.37983 q^{13} +2.68719 q^{14} -0.0909244 q^{15} +1.00000 q^{16} -5.05388 q^{17} +2.63954 q^{18} -3.19526 q^{19} +0.151444 q^{20} +1.61335 q^{21} +3.19669 q^{22} -4.99371 q^{23} +0.600384 q^{24} -4.97706 q^{25} +3.37983 q^{26} +3.38589 q^{27} -2.68719 q^{28} -1.95000 q^{29} +0.0909244 q^{30} -1.23811 q^{31} -1.00000 q^{32} +1.91924 q^{33} +5.05388 q^{34} -0.406958 q^{35} -2.63954 q^{36} +10.9509 q^{37} +3.19526 q^{38} +2.02920 q^{39} -0.151444 q^{40} -3.13433 q^{41} -1.61335 q^{42} -6.28846 q^{43} -3.19669 q^{44} -0.399742 q^{45} +4.99371 q^{46} -9.20442 q^{47} -0.600384 q^{48} +0.220992 q^{49} +4.97706 q^{50} +3.03427 q^{51} -3.37983 q^{52} -7.20538 q^{53} -3.38589 q^{54} -0.484119 q^{55} +2.68719 q^{56} +1.91838 q^{57} +1.95000 q^{58} +14.7601 q^{59} -0.0909244 q^{60} -3.02915 q^{61} +1.23811 q^{62} +7.09294 q^{63} +1.00000 q^{64} -0.511855 q^{65} -1.91924 q^{66} +7.68856 q^{67} -5.05388 q^{68} +2.99814 q^{69} +0.406958 q^{70} +15.5477 q^{71} +2.63954 q^{72} +4.86285 q^{73} -10.9509 q^{74} +2.98815 q^{75} -3.19526 q^{76} +8.59012 q^{77} -2.02920 q^{78} -9.72905 q^{79} +0.151444 q^{80} +5.88579 q^{81} +3.13433 q^{82} +3.42807 q^{83} +1.61335 q^{84} -0.765379 q^{85} +6.28846 q^{86} +1.17075 q^{87} +3.19669 q^{88} -11.8101 q^{89} +0.399742 q^{90} +9.08226 q^{91} -4.99371 q^{92} +0.743340 q^{93} +9.20442 q^{94} -0.483903 q^{95} +0.600384 q^{96} +0.970934 q^{97} -0.220992 q^{98} +8.43780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 49 q^{2} + 8 q^{3} + 49 q^{4} - 8 q^{5} - 8 q^{6} + 18 q^{7} - 49 q^{8} + 59 q^{9}+O(q^{10}) \) 49 * q - 49 * q^2 + 8 * q^3 + 49 * q^4 - 8 * q^5 - 8 * q^6 + 18 * q^7 - 49 * q^8 + 59 * q^9	\( 49 q - 49 q^{2} + 8 q^{3} + 49 q^{4} - 8 q^{5} - 8 q^{6} + 18 q^{7} - 49 q^{8} + 59 q^{9} + 8 q^{10} + q^{11} + 8 q^{12} + 9 q^{13} - 18 q^{14} + 15 q^{15} + 49 q^{16} - 27 q^{17} - 59 q^{18} + 27 q^{19} - 8 q^{20} + 13 q^{21} - q^{22} + 16 q^{23} - 8 q^{24} + 71 q^{25} - 9 q^{26} + 29 q^{27} + 18 q^{28} - 7 q^{29} - 15 q^{30} + 75 q^{31} - 49 q^{32} - 3 q^{33} + 27 q^{34} - 16 q^{35} + 59 q^{36} + 36 q^{37} - 27 q^{38} + 24 q^{39} + 8 q^{40} - 12 q^{41} - 13 q^{42} + 22 q^{43} + q^{44} + 5 q^{45} - 16 q^{46} + 26 q^{47} + 8 q^{48} + 107 q^{49} - 71 q^{50} + 35 q^{51} + 9 q^{52} - 10 q^{53} - 29 q^{54} + 76 q^{55} - 18 q^{56} - 10 q^{57} + 7 q^{58} + 9 q^{59} + 15 q^{60} + 87 q^{61} - 75 q^{62} + 68 q^{63} + 49 q^{64} - 6 q^{65} + 3 q^{66} + 46 q^{67} - 27 q^{68} + 70 q^{69} + 16 q^{70} + 40 q^{71} - 59 q^{72} + 6 q^{73} - 36 q^{74} + 69 q^{75} + 27 q^{76} - 12 q^{77} - 24 q^{78} + 76 q^{79} - 8 q^{80} + 77 q^{81} + 12 q^{82} - 32 q^{83} + 13 q^{84} + 19 q^{85} - 22 q^{86} + 36 q^{87} - q^{88} + 34 q^{89} - 5 q^{90} + 119 q^{91} + 16 q^{92} - 5 q^{93} - 26 q^{94} - 2 q^{95} - 8 q^{96} + 52 q^{97} - 107 q^{98} + 26 q^{99}+O(q^{100}) \) 49 * q - 49 * q^2 + 8 * q^3 + 49 * q^4 - 8 * q^5 - 8 * q^6 + 18 * q^7 - 49 * q^8 + 59 * q^9 + 8 * q^10 + q^11 + 8 * q^12 + 9 * q^13 - 18 * q^14 + 15 * q^15 + 49 * q^16 - 27 * q^17 - 59 * q^18 + 27 * q^19 - 8 * q^20 + 13 * q^21 - q^22 + 16 * q^23 - 8 * q^24 + 71 * q^25 - 9 * q^26 + 29 * q^27 + 18 * q^28 - 7 * q^29 - 15 * q^30 + 75 * q^31 - 49 * q^32 - 3 * q^33 + 27 * q^34 - 16 * q^35 + 59 * q^36 + 36 * q^37 - 27 * q^38 + 24 * q^39 + 8 * q^40 - 12 * q^41 - 13 * q^42 + 22 * q^43 + q^44 + 5 * q^45 - 16 * q^46 + 26 * q^47 + 8 * q^48 + 107 * q^49 - 71 * q^50 + 35 * q^51 + 9 * q^52 - 10 * q^53 - 29 * q^54 + 76 * q^55 - 18 * q^56 - 10 * q^57 + 7 * q^58 + 9 * q^59 + 15 * q^60 + 87 * q^61 - 75 * q^62 + 68 * q^63 + 49 * q^64 - 6 * q^65 + 3 * q^66 + 46 * q^67 - 27 * q^68 + 70 * q^69 + 16 * q^70 + 40 * q^71 - 59 * q^72 + 6 * q^73 - 36 * q^74 + 69 * q^75 + 27 * q^76 - 12 * q^77 - 24 * q^78 + 76 * q^79 - 8 * q^80 + 77 * q^81 + 12 * q^82 - 32 * q^83 + 13 * q^84 + 19 * q^85 - 22 * q^86 + 36 * q^87 - q^88 + 34 * q^89 - 5 * q^90 + 119 * q^91 + 16 * q^92 - 5 * q^93 - 26 * q^94 - 2 * q^95 - 8 * q^96 + 52 * q^97 - 107 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more