LMFDB - Embedded newform 6845.2.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6845,2,Mod(1,6845)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6845 = 5 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6845.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,6,-5,2,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.6576001836$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.09027$ of defining polynomial
Character	$\chi$	$=$	6845.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.46516 q^{2} +0.744131 q^{3} +0.146703 q^{4} -1.00000 q^{5} -1.09027 q^{6} +3.94357 q^{7} +2.71538 q^{8} -2.44627 q^{9} +1.46516 q^{10} -4.52210 q^{11} +0.109166 q^{12} -1.36924 q^{13} -5.77797 q^{14} -0.744131 q^{15} -4.27188 q^{16} -2.29957 q^{17} +3.58418 q^{18} -4.84765 q^{19} -0.146703 q^{20} +2.93453 q^{21} +6.62562 q^{22} +4.41859 q^{23} +2.02060 q^{24} +1.00000 q^{25} +2.00616 q^{26} -4.05274 q^{27} +0.578534 q^{28} +9.55595 q^{29} +1.09027 q^{30} +4.75908 q^{31} +0.828243 q^{32} -3.36504 q^{33} +3.36924 q^{34} -3.94357 q^{35} -0.358875 q^{36} +7.10259 q^{38} -1.01889 q^{39} -2.71538 q^{40} +5.21439 q^{41} -4.29957 q^{42} +1.19485 q^{43} -0.663407 q^{44} +2.44627 q^{45} -6.47395 q^{46} +5.34785 q^{47} -3.17884 q^{48} +8.55174 q^{49} -1.46516 q^{50} -1.71118 q^{51} -0.200872 q^{52} +1.03384 q^{53} +5.93792 q^{54} +4.52210 q^{55} +10.7083 q^{56} -3.60728 q^{57} -14.0010 q^{58} -14.6197 q^{59} -0.109166 q^{60} -1.30772 q^{61} -6.97283 q^{62} -9.64703 q^{63} +7.33026 q^{64} +1.36924 q^{65} +4.93033 q^{66} +7.23975 q^{67} -0.337354 q^{68} +3.28801 q^{69} +5.77797 q^{70} +15.1787 q^{71} -6.64256 q^{72} -4.81551 q^{73} +0.744131 q^{75} -0.711165 q^{76} -17.8332 q^{77} +1.49285 q^{78} -4.64520 q^{79} +4.27188 q^{80} +4.32304 q^{81} -7.63993 q^{82} -5.89279 q^{83} +0.430505 q^{84} +2.29957 q^{85} -1.75066 q^{86} +7.11087 q^{87} -12.2792 q^{88} +10.3435 q^{89} -3.58418 q^{90} -5.39969 q^{91} +0.648221 q^{92} +3.54138 q^{93} -7.83547 q^{94} +4.84765 q^{95} +0.616321 q^{96} -6.37000 q^{97} -12.5297 q^{98} +11.0623 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} + 7 q^{7} + 6 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{13} - 4 q^{14} + q^{15} + 8 q^{16} + 8 q^{17} + 6 q^{18} - 14 q^{19} - 6 q^{20} - 9 q^{21} - 2 q^{22} - 2 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) $ 5 * q - q^3 + 6 * q^4 - 5 * q^5 + 2 * q^6 + 7 * q^7 + 6 * q^8 + 2 * q^9 + 7 * q^11 - 2 * q^13 - 4 * q^14 + q^15 + 8 * q^16 + 8 * q^17 + 6 * q^18 - 14 * q^19 - 6 * q^20 - 9 * q^21 - 2 * q^22 - 2 * q^23 - 12 * q^24 + 5 * q^25 - 20 * q^26 - 7 * q^27 - 14 * q^28 - 2 * q^29 - 2 * q^30 - 8 * q^31 + 22 * q^32 - 21 * q^33 + 12 * q^34 - 7 * q^35 - 36 * q^36 - 32 * q^38 - 12 * q^39 - 6 * q^40 - 9 * q^41 - 2 * q^42 - 14 * q^43 + 18 * q^44 - 2 * q^45 - 28 * q^46 - 5 * q^47 - 30 * q^48 + 2 * q^49 - 10 * q^51 - 20 * q^52 - 15 * q^53 + 12 * q^54 - 7 * q^55 + 14 * q^56 - 4 * q^57 - 12 * q^59 - 12 * q^61 + 10 * q^62 + 24 * q^63 + 12 * q^64 + 2 * q^65 + 10 * q^66 - 2 * q^67 + 4 * q^68 + 30 * q^69 + 4 * q^70 + 13 * q^71 - 26 * q^72 - 5 * q^73 - q^75 - 18 * q^76 - 7 * q^77 + 10 * q^78 - 36 * q^79 - 8 * q^80 + 21 * q^81 + 2 * q^82 - 9 * q^83 + 30 * q^84 - 8 * q^85 - 8 * q^86 + 6 * q^87 - 8 * q^88 + 16 * q^89 - 6 * q^90 + 4 * q^91 - 12 * q^92 + 16 * q^93 - 20 * q^94 + 14 * q^95 - 30 * q^98 + 18 * q^99

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 $

$2$	−1.46516	−1.03603	−0.518013	−	0.855372i	$-0.673328\pi$
$2$	−1.46516	−1.03603	−0.518013	+	0.855372i	$0.673328\pi$
$3$	0.744131	0.429624	0.214812	−	0.976655i	$-0.431086\pi$
$3$	0.744131	0.429624	0.214812	+	0.976655i	$0.431086\pi$
$4$	0.146703	0.0733516
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.09027	−0.445102
$7$	3.94357	1.49053	0.745265	−	0.666769i	$-0.232323\pi$
$7$	3.94357	1.49053	0.745265	+	0.666769i	$0.232323\pi$
$8$	2.71538	0.960033
$9$	−2.44627	−0.815423
$10$	1.46516	0.463325
$11$	−4.52210	−1.36347	−0.681733	−	0.731601i	$-0.738773\pi$
$11$	−4.52210	−1.36347	−0.681733	+	0.731601i	$0.738773\pi$
$12$	0.109166	0.0315136
$13$	−1.36924	−0.379759	−0.189879	−	0.981807i	$-0.560810\pi$
$13$	−1.36924	−0.379759	−0.189879	+	0.981807i	$0.560810\pi$
$14$	−5.77797	−1.54423
$15$	−0.744131	−0.192134
$16$	−4.27188	−1.06797
$17$	−2.29957	−0.557727	−0.278863	−	0.960331i	$-0.589958\pi$
$17$	−2.29957	−0.557727	−0.278863	+	0.960331i	$0.589958\pi$
$18$	3.58418	0.844800
$19$	−4.84765	−1.11213	−0.556063	−	0.831140i	$-0.687689\pi$
$19$	−4.84765	−1.11213	−0.556063	+	0.831140i	$0.687689\pi$
$20$	−0.146703	−0.0328038
$21$	2.93453	0.640367
$22$	6.62562	1.41259
$23$	4.41859	0.921339	0.460670	−	0.887572i	$-0.347609\pi$
$23$	4.41859	0.921339	0.460670	+	0.887572i	$0.347609\pi$
$24$	2.02060	0.412453
$25$	1.00000	0.200000
$26$	2.00616	0.393440
$27$	−4.05274	−0.779949
$28$	0.578534	0.109333
$29$	9.55595	1.77449	0.887247	−	0.461294i	$-0.152615\pi$
$29$	9.55595	1.77449	0.887247	+	0.461294i	$0.152615\pi$
$30$	1.09027	0.199056
$31$	4.75908	0.854756	0.427378	−	0.904073i	$-0.359437\pi$
$31$	4.75908	0.854756	0.427378	+	0.904073i	$0.359437\pi$
$32$	0.828243	0.146414
$33$	−3.36504	−0.585778
$34$	3.36924	0.577820
$35$	−3.94357	−0.666585
$36$	−0.358875	−0.0598126
$37$	0	0
$37$	0	0
$38$	7.10259	1.15219
$39$	−1.01889	−0.163154
$40$	−2.71538	−0.429340
$41$	5.21439	0.814351	0.407175	−	0.913350i	$-0.366514\pi$
$41$	5.21439	0.814351	0.407175	+	0.913350i	$0.366514\pi$
$42$	−4.29957	−0.663438
$43$	1.19485	0.182214	0.0911068	−	0.995841i	$-0.470960\pi$
$43$	1.19485	0.182214	0.0911068	+	0.995841i	$0.470960\pi$
$44$	−0.663407	−0.100012
$45$	2.44627	0.364668
$46$	−6.47395	−0.954532
$47$	5.34785	0.780064	0.390032	−	0.920801i	$-0.372464\pi$
$47$	5.34785	0.780064	0.390032	+	0.920801i	$0.372464\pi$
$48$	−3.17884	−0.458826
$49$	8.55174	1.22168
$50$	−1.46516	−0.207205
$51$	−1.71118	−0.239613
$52$	−0.200872	−0.0278559
$53$	1.03384	0.142009	0.0710046	−	0.997476i	$-0.477380\pi$
$53$	1.03384	0.142009	0.0710046	+	0.997476i	$0.477380\pi$
$54$	5.93792	0.808048
$55$	4.52210	0.609760
$56$	10.7083	1.43096
$57$	−3.60728	−0.477796
$58$	−14.0010	−1.83842
$59$	−14.6197	−1.90333	−0.951664	−	0.307143i	$-0.900627\pi$
$59$	−14.6197	−1.90333	−0.951664	+	0.307143i	$0.900627\pi$
$60$	−0.109166	−0.0140933
$61$	−1.30772	−0.167436	−0.0837179	−	0.996489i	$-0.526679\pi$
$61$	−1.30772	−0.167436	−0.0837179	+	0.996489i	$0.526679\pi$
$62$	−6.97283	−0.885550
$63$	−9.64703	−1.21541
$64$	7.33026	0.916282
$65$	1.36924	0.169833
$66$	4.93033	0.606881
$67$	7.23975	0.884476	0.442238	−	0.896898i	$-0.354185\pi$
$67$	7.23975	0.884476	0.442238	+	0.896898i	$0.354185\pi$
$68$	−0.337354	−0.0409101
$69$	3.28801	0.395829
$70$	5.77797	0.690600
$71$	15.1787	1.80138	0.900692	−	0.434458i	$-0.143060\pi$
$71$	15.1787	1.80138	0.900692	+	0.434458i	$0.143060\pi$
$72$	−6.64256	−0.782833
$73$	−4.81551	−0.563613	−0.281806	−	0.959471i	$-0.590934\pi$
$73$	−4.81551	−0.563613	−0.281806	+	0.959471i	$0.590934\pi$
$74$	0	0
$75$	0.744131	0.0859248
$76$	−0.711165	−0.0815762
$77$	−17.8332	−2.03229
$78$	1.49285	0.169031
$79$	−4.64520	−0.522626	−0.261313	−	0.965254i	$-0.584155\pi$
$79$	−4.64520	−0.522626	−0.261313	+	0.965254i	$0.584155\pi$
$80$	4.27188	0.477611
$81$	4.32304	0.480338
$82$	−7.63993	−0.843689
$83$	−5.89279	−0.646818	−0.323409	−	0.946259i	$-0.604829\pi$
$83$	−5.89279	−0.646818	−0.323409	+	0.946259i	$0.604829\pi$
$84$	0.430505	0.0469719
$85$	2.29957	0.249423
$86$	−1.75066	−0.188778
$87$	7.11087	0.762365
$88$	−12.2792	−1.30897
$89$	10.3435	1.09641	0.548205	−	0.836344i	$-0.315311\pi$
$89$	10.3435	1.09641	0.548205	+	0.836344i	$0.315311\pi$
$90$	−3.58418	−0.377806
$91$	−5.39969	−0.566042
$92$	0.648221	0.0675817
$93$	3.54138	0.367224
$94$	−7.83547	−0.808167
$95$	4.84765	0.497358
$96$	0.616321	0.0629030
$97$	−6.37000	−0.646776	−0.323388	−	0.946267i	$-0.604822\pi$
$97$	−6.37000	−0.646776	−0.323388	+	0.946267i	$0.604822\pi$
$98$	−12.5297	−1.26569
$99$	11.0623	1.11180
$100$	0.146703	0.0146703
$101$	−4.62142	−0.459848	−0.229924	−	0.973209i	$-0.573848\pi$
$101$	−4.62142	−0.459848	−0.229924	+	0.973209i	$0.573848\pi$
$102$	2.50715	0.248245
$103$	−18.3734	−1.81039	−0.905193	−	0.425001i	$-0.860274\pi$
$103$	−18.3734	−1.81039	−0.905193	+	0.425001i	$0.860274\pi$
$104$	−3.71801	−0.364581
$105$	−2.93453	−0.286381
$106$	−1.51475	−0.147125
$107$	−17.8700	−1.72755	−0.863777	−	0.503875i	$-0.831907\pi$
$107$	−17.8700	−1.72755	−0.863777	+	0.503875i	$0.831907\pi$
$108$	−0.594549	−0.0572105
$109$	3.03189	0.290402	0.145201	−	0.989402i	$-0.453617\pi$
$109$	3.03189	0.290402	0.145201	+	0.989402i	$0.453617\pi$
$110$	−6.62562	−0.631728
$111$	0	0
$112$	−16.8465	−1.59184
$113$	20.1901	1.89933	0.949663	−	0.313273i	$-0.101425\pi$
$113$	20.1901	1.89933	0.949663	+	0.313273i	$0.101425\pi$
$114$	5.28526	0.495010
$115$	−4.41859	−0.412035
$116$	1.40189	0.130162
$117$	3.34953	0.309664
$118$	21.4203	1.97190
$119$	−9.06850	−0.831308
$120$	−2.02060	−0.184455
$121$	9.44942	0.859038
$122$	1.91602	0.173468
$123$	3.88019	0.349865
$124$	0.698172	0.0626977
$125$	−1.00000	−0.0894427
$126$	14.1345	1.25920
$127$	−13.5915	−1.20605	−0.603025	−	0.797722i	$-0.706038\pi$
$127$	−13.5915	−1.20605	−0.603025	+	0.797722i	$0.706038\pi$
$128$	−12.3965	−1.09571
$129$	0.889128	0.0782834
$130$	−2.00616	−0.175952
$131$	−21.1799	−1.85049	−0.925246	−	0.379367i	$-0.876142\pi$
$131$	−21.1799	−1.85049	−0.925246	+	0.379367i	$0.876142\pi$
$132$	−0.493661	−0.0429677
$133$	−19.1170	−1.65766
$134$	−10.6074	−0.916341
$135$	4.05274	0.348804
$136$	−6.24420	−0.535436
$137$	7.17856	0.613305	0.306653	−	0.951821i	$-0.400791\pi$
$137$	7.17856	0.613305	0.306653	+	0.951821i	$0.400791\pi$
$138$	−4.81747	−0.410090
$139$	−12.9602	−1.09927	−0.549636	−	0.835404i	$-0.685234\pi$
$139$	−12.9602	−1.09927	−0.549636	+	0.835404i	$0.685234\pi$
$140$	−0.578534	−0.0488950
$141$	3.97950	0.335134
$142$	−22.2393	−1.86628
$143$	6.19185	0.517788
$144$	10.4502	0.870848
$145$	−9.55595	−0.793578
$146$	7.05551	0.583918
$147$	6.36361	0.524862
$148$	0	0
$149$	−15.8451	−1.29809	−0.649043	−	0.760752i	$-0.724830\pi$
$149$	−15.8451	−1.29809	−0.649043	+	0.760752i	$0.724830\pi$
$150$	−1.09027	−0.0890204
$151$	14.2944	1.16326	0.581632	−	0.813452i	$-0.302414\pi$
$151$	14.2944	1.16326	0.581632	+	0.813452i	$0.302414\pi$
$152$	−13.1632	−1.06768
$153$	5.62536	0.454783
$154$	26.1286	2.10550
$155$	−4.75908	−0.382258
$156$	−0.149475	−0.0119676
$157$	−1.95398	−0.155945	−0.0779723	−	0.996956i	$-0.524845\pi$
$157$	−1.95398	−0.155945	−0.0779723	+	0.996956i	$0.524845\pi$
$158$	6.80597	0.541454
$159$	0.769314	0.0610105
$160$	−0.828243	−0.0654784
$161$	17.4250	1.37328
$162$	−6.33397	−0.497643
$163$	−3.01431	−0.236099	−0.118049	−	0.993008i	$-0.537664\pi$
$163$	−3.01431	−0.236099	−0.118049	+	0.993008i	$0.537664\pi$
$164$	0.764967	0.0597339
$165$	3.36504	0.261968
$166$	8.63390	0.670120
$167$	−4.16008	−0.321916	−0.160958	−	0.986961i	$-0.551458\pi$
$167$	−4.16008	−0.321916	−0.160958	+	0.986961i	$0.551458\pi$
$168$	7.96837	0.614773
$169$	−11.1252	−0.855783
$170$	−3.36924	−0.258409
$171$	11.8587	0.906854
$172$	0.175289	0.0133657
$173$	−3.79279	−0.288361	−0.144180	−	0.989551i	$-0.546055\pi$
$173$	−3.79279	−0.288361	−0.144180	+	0.989551i	$0.546055\pi$
$174$	−10.4186	−0.789831
$175$	3.94357	0.298106
$176$	19.3179	1.45614
$177$	−10.8790	−0.817715
$178$	−15.1549	−1.13591
$179$	13.0017	0.971793	0.485897	−	0.874016i	$-0.338493\pi$
$179$	13.0017	0.971793	0.485897	+	0.874016i	$0.338493\pi$
$180$	0.358875	0.0267490
$181$	−16.6739	−1.23936	−0.619681	−	0.784854i	$-0.712738\pi$
$181$	−16.6739	−1.23936	−0.619681	+	0.784854i	$0.712738\pi$
$182$	7.91143	0.586434
$183$	−0.973111	−0.0719345
$184$	11.9982	0.884516
$185$	0	0
$186$	−5.18869	−0.380454
$187$	10.3989	0.760441
$188$	0.784546	0.0572189
$189$	−15.9822	−1.16254
$190$	−7.10259	−0.515276
$191$	−8.73937	−0.632359	−0.316179	−	0.948699i	$-0.602400\pi$
$191$	−8.73937	−0.632359	−0.316179	+	0.948699i	$0.602400\pi$
$192$	5.45467	0.393657
$193$	−21.6457	−1.55809	−0.779047	−	0.626966i	$-0.784297\pi$
$193$	−21.6457	−1.55809	−0.779047	+	0.626966i	$0.784297\pi$
$194$	9.33309	0.670077
$195$	1.01889	0.0729645
$196$	1.25457	0.0896119
$197$	−2.60609	−0.185676	−0.0928380	−	0.995681i	$-0.529594\pi$
$197$	−2.60609	−0.185676	−0.0928380	+	0.995681i	$0.529594\pi$
$198$	−16.2081	−1.15186
$199$	6.57573	0.466141	0.233071	−	0.972460i	$-0.425123\pi$
$199$	6.57573	0.466141	0.233071	+	0.972460i	$0.425123\pi$
$200$	2.71538	0.192007
$201$	5.38732	0.379992
$202$	6.77113	0.476415
$203$	37.6845	2.64494
$204$	−0.251035	−0.0175760
$205$	−5.21439	−0.364189
$206$	26.9200	1.87561
$207$	−10.8091	−0.751281
$208$	5.84924	0.405572
$209$	21.9216	1.51635
$210$	4.29957	0.296698
$211$	−2.00196	−0.137820	−0.0689102	−	0.997623i	$-0.521952\pi$
$211$	−2.00196	−0.137820	−0.0689102	+	0.997623i	$0.521952\pi$
$212$	0.151668	0.0104166
$213$	11.2950	0.773918
$214$	26.1824	1.78979
$215$	−1.19485	−0.0814884
$216$	−11.0047	−0.748777
$217$	18.7678	1.27404
$218$	−4.44221	−0.300864
$219$	−3.58337	−0.242142
$220$	0.663407	0.0447269
$221$	3.14866	0.211802
$222$	0	0
$223$	−7.43402	−0.497819	−0.248910	−	0.968527i	$-0.580072\pi$
$223$	−7.43402	−0.497819	−0.248910	+	0.968527i	$0.580072\pi$
$224$	3.26623	0.218234
$225$	−2.44627	−0.163085
$226$	−29.5818	−1.96775
$227$	0.633009	0.0420143	0.0210071	−	0.999779i	$-0.493313\pi$
$227$	0.633009	0.0420143	0.0210071	+	0.999779i	$0.493313\pi$
$228$	−0.529200	−0.0350471
$229$	15.9122	1.05151	0.525754	−	0.850636i	$-0.323783\pi$
$229$	15.9122	1.05151	0.525754	+	0.850636i	$0.323783\pi$
$230$	6.47395	0.426880
$231$	−13.2703	−0.873119
$232$	25.9480	1.70357
$233$	−8.89467	−0.582709	−0.291355	−	0.956615i	$-0.594106\pi$
$233$	−8.89467	−0.582709	−0.291355	+	0.956615i	$0.594106\pi$
$234$	−4.90761	−0.320820
$235$	−5.34785	−0.348855
$236$	−2.14476	−0.139612
$237$	−3.45663	−0.224533
$238$	13.2868	0.861257
$239$	0.0826797	0.00534810	0.00267405	−	0.999996i	$-0.499149\pi$
$239$	0.0826797	0.00534810	0.00267405	+	0.999996i	$0.499149\pi$
$240$	3.17884	0.205193
$241$	19.6320	1.26461	0.632305	−	0.774719i	$-0.282109\pi$
$241$	19.6320	1.26461	0.632305	+	0.774719i	$0.282109\pi$
$242$	−13.8449	−0.889987
$243$	15.3751	0.986314
$244$	−0.191846	−0.0122817
$245$	−8.55174	−0.546351
$246$	−5.68511	−0.362469
$247$	6.63759	0.422340
$248$	12.9227	0.820594
$249$	−4.38501	−0.277888
$250$	1.46516	0.0926651
$251$	3.16254	0.199618	0.0998089	−	0.995007i	$-0.468177\pi$
$251$	3.16254	0.199618	0.0998089	+	0.995007i	$0.468177\pi$
$252$	−1.41525	−0.0891524
$253$	−19.9813	−1.25621
$254$	19.9138	1.24950
$255$	1.71118	0.107158
$256$	3.50239	0.218900
$257$	11.7279	0.731569	0.365784	−	0.930700i	$-0.380801\pi$
$257$	11.7279	0.731569	0.365784	+	0.930700i	$0.380801\pi$
$258$	−1.30272	−0.0811037
$259$	0	0
$260$	0.200872	0.0124575
$261$	−23.3764	−1.44696
$262$	31.0320	1.91716
$263$	0.755431	0.0465819	0.0232909	−	0.999729i	$-0.492586\pi$
$263$	0.755431	0.0465819	0.0232909	+	0.999729i	$0.492586\pi$
$264$	−9.13736	−0.562366
$265$	−1.03384	−0.0635084
$266$	28.0096	1.71738
$267$	7.69693	0.471044
$268$	1.06209	0.0648777
$269$	13.4542	0.820320	0.410160	−	0.912014i	$-0.365473\pi$
$269$	13.4542	0.820320	0.410160	+	0.912014i	$0.365473\pi$
$270$	−5.93792	−0.361370
$271$	7.94268	0.482483	0.241242	−	0.970465i	$-0.422445\pi$
$271$	7.94268	0.482483	0.241242	+	0.970465i	$0.422445\pi$
$272$	9.82348	0.595636
$273$	−4.01808	−0.243185
$274$	−10.5178	−0.635401
$275$	−4.52210	−0.272693
$276$	0.482361	0.0290347
$277$	−6.88638	−0.413762	−0.206881	−	0.978366i	$-0.566331\pi$
$277$	−6.88638	−0.413762	−0.206881	+	0.978366i	$0.566331\pi$
$278$	18.9888	1.13888
$279$	−11.6420	−0.696988
$280$	−10.7083	−0.639943
$281$	−16.0874	−0.959693	−0.479847	−	0.877352i	$-0.659308\pi$
$281$	−16.0874	−0.959693	−0.479847	+	0.877352i	$0.659308\pi$
$282$	−5.83061	−0.347208
$283$	−24.9263	−1.48172	−0.740859	−	0.671661i	$-0.765581\pi$
$283$	−24.9263	−1.48172	−0.740859	+	0.671661i	$0.765581\pi$
$284$	2.22677	0.132134
$285$	3.60728	0.213677
$286$	−9.07206	−0.536442
$287$	20.5633	1.21381
$288$	−2.02611	−0.119389
$289$	−11.7120	−0.688941
$290$	14.0010	0.822168
$291$	−4.74011	−0.277870
$292$	−0.706450	−0.0413419
$293$	26.0732	1.52321	0.761607	−	0.648039i	$-0.224411\pi$
$293$	26.0732	1.52321	0.761607	+	0.648039i	$0.224411\pi$
$294$	−9.32373	−0.543771
$295$	14.6197	0.851194
$296$	0	0
$297$	18.3269	1.06343
$298$	23.2157	1.34485
$299$	−6.05011	−0.349887
$300$	0.109166	0.00630272
$301$	4.71199	0.271595
$302$	−20.9437	−1.20517
$303$	−3.43894	−0.197562
$304$	20.7086	1.18772
$305$	1.30772	0.0748796
$306$	−8.24207	−0.471168
$307$	−18.2979	−1.04432	−0.522158	−	0.852849i	$-0.674873\pi$
$307$	−18.2979	−1.04432	−0.522158	+	0.852849i	$0.674873\pi$
$308$	−2.61619	−0.149071
$309$	−13.6722	−0.777785
$310$	6.97283	0.396030
$311$	1.44118	0.0817216	0.0408608	−	0.999165i	$-0.486990\pi$
$311$	1.44118	0.0817216	0.0408608	+	0.999165i	$0.486990\pi$
$312$	−2.76669	−0.156633
$313$	18.5015	1.04576	0.522882	−	0.852405i	$-0.324857\pi$
$313$	18.5015	1.04576	0.522882	+	0.852405i	$0.324857\pi$
$314$	2.86290	0.161563
$315$	9.64703	0.543549
$316$	−0.681465	−0.0383354
$317$	−24.1703	−1.35754	−0.678769	−	0.734352i	$-0.737486\pi$
$317$	−24.1703	−1.35754	−0.678769	+	0.734352i	$0.737486\pi$
$318$	−1.12717	−0.0632086
$319$	−43.2130	−2.41946
$320$	−7.33026	−0.409774
$321$	−13.2976	−0.742198
$322$	−25.5305	−1.42276
$323$	11.1475	0.620263
$324$	0.634204	0.0352336
$325$	−1.36924	−0.0759518
$326$	4.41646	0.244605
$327$	2.25612	0.124764
$328$	14.1591	0.781803
$329$	21.0896	1.16271
$330$	−4.93033	−0.271406
$331$	0.308401	0.0169513	0.00847563	−	0.999964i	$-0.497302\pi$
$331$	0.308401	0.0169513	0.00847563	+	0.999964i	$0.497302\pi$
$332$	−0.864491	−0.0474451
$333$	0	0
$334$	6.09519	0.333514
$335$	−7.23975	−0.395550
$336$	−12.5360	−0.683894
$337$	−17.7374	−0.966219	−0.483110	−	0.875560i	$-0.660493\pi$
$337$	−17.7374	−0.966219	−0.483110	+	0.875560i	$0.660493\pi$
$338$	16.3002	0.886614
$339$	15.0241	0.815996
$340$	0.337354	0.0182956
$341$	−21.5211	−1.16543
$342$	−17.3749	−0.939525
$343$	6.11940	0.330417
$344$	3.24449	0.174931
$345$	−3.28801	−0.177020
$346$	5.55706	0.298749
$347$	−13.4103	−0.719902	−0.359951	−	0.932971i	$-0.617207\pi$
$347$	−13.4103	−0.719902	−0.359951	+	0.932971i	$0.617207\pi$
$348$	1.04319	0.0559207
$349$	−18.0187	−0.964521	−0.482261	−	0.876028i	$-0.660184\pi$
$349$	−18.0187	−0.964521	−0.482261	+	0.876028i	$0.660184\pi$
$350$	−5.77797	−0.308846
$351$	5.54917	0.296193
$352$	−3.74540	−0.199631
$353$	16.5967	0.883355	0.441678	−	0.897174i	$-0.354383\pi$
$353$	16.5967	0.883355	0.441678	+	0.897174i	$0.354383\pi$
$354$	15.9395	0.847175
$355$	−15.1787	−0.805604
$356$	1.51743	0.0804234
$357$	−6.74815	−0.357150
$358$	−19.0496	−1.00680
$359$	0.294342	0.0155348	0.00776739	−	0.999970i	$-0.497528\pi$
$359$	0.294342	0.0155348	0.00776739	+	0.999970i	$0.497528\pi$
$360$	6.64256	0.350094
$361$	4.49968	0.236825
$362$	24.4300	1.28401
$363$	7.03160	0.369063
$364$	−0.792152	−0.0415200
$365$	4.81551	0.252055
$366$	1.42577	0.0745260
$367$	23.2638	1.21436	0.607181	−	0.794563i	$-0.292300\pi$
$367$	23.2638	1.21436	0.607181	+	0.794563i	$0.292300\pi$
$368$	−18.8757	−0.983964
$369$	−12.7558	−0.664040
$370$	0	0
$371$	4.07703	0.211669
$372$	0.519531	0.0269364
$373$	−37.2311	−1.92776	−0.963878	−	0.266345i	$-0.914184\pi$
$373$	−37.2311	−1.92776	−0.963878	+	0.266345i	$0.914184\pi$
$374$	−15.2361	−0.787838
$375$	−0.744131	−0.0384267
$376$	14.5215	0.748887
$377$	−13.0844	−0.673880
$378$	23.4166	1.20442
$379$	−23.0305	−1.18300	−0.591498	−	0.806306i	$-0.701463\pi$
$379$	−23.0305	−1.18300	−0.591498	+	0.806306i	$0.701463\pi$
$380$	0.711165	0.0364820
$381$	−10.1138	−0.518148
$382$	12.8046	0.655140
$383$	12.8637	0.657302	0.328651	−	0.944451i	$-0.393406\pi$
$383$	12.8637	0.657302	0.328651	+	0.944451i	$0.393406\pi$
$384$	−9.22462	−0.470742
$385$	17.8332	0.908866
$386$	31.7145	1.61423
$387$	−2.92294	−0.148581
$388$	−0.934499	−0.0474420
$389$	16.1194	0.817287	0.408644	−	0.912694i	$-0.366002\pi$
$389$	16.1194	0.817287	0.408644	+	0.912694i	$0.366002\pi$
$390$	−1.49285	−0.0755932
$391$	−10.1608	−0.513856
$392$	23.2212	1.17285
$393$	−15.7606	−0.795016
$394$	3.81834	0.192365
$395$	4.64520	0.233725
$396$	1.62287	0.0815524
$397$	−12.9491	−0.649897	−0.324948	−	0.945732i	$-0.605347\pi$
$397$	−12.9491	−0.649897	−0.324948	+	0.945732i	$0.605347\pi$
$398$	−9.63452	−0.482935
$399$	−14.2256	−0.712169
$400$	−4.27188	−0.213594
$401$	17.2452	0.861185	0.430593	−	0.902546i	$-0.358305\pi$
$401$	17.2452	0.861185	0.430593	+	0.902546i	$0.358305\pi$
$402$	−7.89330	−0.393682
$403$	−6.51632	−0.324601
$404$	−0.677976	−0.0337306
$405$	−4.32304	−0.214814
$406$	−55.2140	−2.74022
$407$	0	0
$408$	−4.64650	−0.230036
$409$	−31.2920	−1.54729	−0.773644	−	0.633621i	$-0.781568\pi$
$409$	−31.2920	−1.54729	−0.773644	+	0.633621i	$0.781568\pi$
$410$	7.63993	0.377309
$411$	5.34178	0.263491
$412$	−2.69544	−0.132795
$413$	−57.6539	−2.83696
$414$	15.8370	0.778348
$415$	5.89279	0.289266
$416$	−1.13406	−0.0556020
$417$	−9.64410	−0.472274
$418$	−32.1187	−1.57098
$419$	−11.8203	−0.577459	−0.288730	−	0.957411i	$-0.593233\pi$
$419$	−11.8203	−0.577459	−0.288730	+	0.957411i	$0.593233\pi$
$420$	−0.430505	−0.0210065
$421$	−19.1673	−0.934155	−0.467077	−	0.884216i	$-0.654693\pi$
$421$	−19.1673	−0.934155	−0.467077	+	0.884216i	$0.654693\pi$
$422$	2.93319	0.142786
$423$	−13.0823	−0.636082
$424$	2.80728	0.136333
$425$	−2.29957	−0.111545
$426$	−16.5490	−0.801800
$427$	−5.15707	−0.249568
$428$	−2.62158	−0.126719
$429$	4.60754	0.222454
$430$	1.75066	0.0844242
$431$	−9.78469	−0.471312	−0.235656	−	0.971837i	$-0.575724\pi$
$431$	−9.78469	−0.471312	−0.235656	+	0.971837i	$0.575724\pi$
$432$	17.3128	0.832963
$433$	35.2661	1.69478	0.847391	−	0.530969i	$-0.178172\pi$
$433$	35.2661	1.69478	0.847391	+	0.530969i	$0.178172\pi$
$434$	−27.4978	−1.31994
$435$	−7.11087	−0.340940
$436$	0.444787	0.0213014
$437$	−21.4198	−1.02465
$438$	5.25022	0.250865
$439$	−24.4934	−1.16901	−0.584503	−	0.811392i	$-0.698710\pi$
$439$	−24.4934	−1.16901	−0.584503	+	0.811392i	$0.698710\pi$
$440$	12.2792	0.585390
$441$	−20.9199	−0.996184
$442$	−4.61330	−0.219432
$443$	−26.1850	−1.24409	−0.622044	−	0.782982i	$-0.713698\pi$
$443$	−26.1850	−1.24409	−0.622044	+	0.782982i	$0.713698\pi$
$444$	0	0
$445$	−10.3435	−0.490330
$446$	10.8921	0.515754
$447$	−11.7909	−0.557689
$448$	28.9074	1.36575
$449$	−39.6969	−1.87341	−0.936705	−	0.350119i	$-0.886141\pi$
$449$	−39.6969	−1.87341	−0.936705	+	0.350119i	$0.886141\pi$
$450$	3.58418	0.168960
$451$	−23.5800	−1.11034
$452$	2.96195	0.139319
$453$	10.6369	0.499766
$454$	−0.927461	−0.0435279
$455$	5.39969	0.253142
$456$	−9.79515	−0.458700
$457$	−20.6802	−0.967380	−0.483690	−	0.875239i	$-0.660704\pi$
$457$	−20.6802	−0.967380	−0.483690	+	0.875239i	$0.660704\pi$
$458$	−23.3140	−1.08939
$459$	9.31954	0.434999
$460$	−0.648221	−0.0302234
$461$	5.51764	0.256982	0.128491	−	0.991711i	$-0.458987\pi$
$461$	5.51764	0.256982	0.128491	+	0.991711i	$0.458987\pi$
$462$	19.4431	0.904574
$463$	−20.9783	−0.974944	−0.487472	−	0.873139i	$-0.662081\pi$
$463$	−20.9783	−0.974944	−0.487472	+	0.873139i	$0.662081\pi$
$464$	−40.8219	−1.89511
$465$	−3.54138	−0.164227
$466$	13.0321	0.603702
$467$	7.62261	0.352732	0.176366	−	0.984325i	$-0.443566\pi$
$467$	7.62261	0.352732	0.176366	+	0.984325i	$0.443566\pi$
$468$	0.491387	0.0227144
$469$	28.5504	1.31834
$470$	7.83547	0.361423
$471$	−1.45402	−0.0669976
$472$	−39.6982	−1.82726
$473$	−5.40326	−0.248442
$474$	5.06453	0.232622
$475$	−4.84765	−0.222425
$476$	−1.33038	−0.0609777
$477$	−2.52906	−0.115798
$478$	−0.121139	−0.00554078
$479$	−32.7445	−1.49613	−0.748067	−	0.663623i	$-0.769018\pi$
$479$	−32.7445	−1.49613	−0.748067	+	0.663623i	$0.769018\pi$
$480$	−0.616321	−0.0281311
$481$	0	0
$482$	−28.7641	−1.31017
$483$	12.9665	0.589995
$484$	1.38626	0.0630118
$485$	6.37000	0.289247
$486$	−22.5271	−1.02185
$487$	24.3487	1.10335	0.551673	−	0.834061i	$-0.313990\pi$
$487$	24.3487	1.10335	0.551673	+	0.834061i	$0.313990\pi$
$488$	−3.55095	−0.160744
$489$	−2.24304	−0.101434
$490$	12.5297	0.566034
$491$	9.88214	0.445975	0.222987	−	0.974821i	$-0.428419\pi$
$491$	9.88214	0.445975	0.222987	+	0.974821i	$0.428419\pi$
$492$	0.569235	0.0256631
$493$	−21.9745	−0.989683
$494$	−9.72516	−0.437555
$495$	−11.0623	−0.497213
$496$	−20.3302	−0.912855
$497$	59.8584	2.68502
$498$	6.42475	0.287900
$499$	−38.4155	−1.71971	−0.859857	−	0.510535i	$-0.829447\pi$
$499$	−38.4155	−1.71971	−0.859857	+	0.510535i	$0.829447\pi$
$500$	−0.146703	−0.00656076
$501$	−3.09564	−0.138303
$502$	−4.63364	−0.206809
$503$	14.2482	0.635297	0.317649	−	0.948209i	$-0.397107\pi$
$503$	14.2482	0.635297	0.317649	+	0.948209i	$0.397107\pi$
$504$	−26.1954	−1.16684
$505$	4.62142	0.205650
$506$	29.2759	1.30147
$507$	−8.27859	−0.367665
$508$	−1.99391	−0.0884657
$509$	28.0410	1.24290	0.621448	−	0.783455i	$-0.286545\pi$
$509$	28.0410	1.24290	0.621448	+	0.783455i	$0.286545\pi$
$510$	−2.50715	−0.111019
$511$	−18.9903	−0.840081
$512$	19.6614	0.868921
$513$	19.6462	0.867402
$514$	−17.1833	−0.757925
$515$	18.3734	0.809629
$516$	0.130438	0.00574221
$517$	−24.1835	−1.06359
$518$	0	0
$519$	−2.82233	−0.123887
$520$	3.71801	0.163046
$521$	31.3715	1.37441	0.687204	−	0.726464i	$-0.258838\pi$
$521$	31.3715	1.37441	0.687204	+	0.726464i	$0.258838\pi$
$522$	34.2503	1.49909
$523$	37.1439	1.62419	0.812095	−	0.583525i	$-0.198327\pi$
$523$	37.1439	1.62419	0.812095	+	0.583525i	$0.198327\pi$
$524$	−3.10715	−0.135737
$525$	2.93453	0.128073
$526$	−1.10683	−0.0482601
$527$	−10.9438	−0.476720
$528$	14.3750	0.625593
$529$	−3.47608	−0.151134
$530$	1.51475	0.0657964
$531$	35.7638	1.55202
$532$	−2.80453	−0.121592
$533$	−7.13975	−0.309257
$534$	−11.2773	−0.488014
$535$	17.8700	0.772585
$536$	19.6587	0.849126
$537$	9.67497	0.417506
$538$	−19.7127	−0.849873
$539$	−38.6719	−1.66571
$540$	0.594549	0.0255853
$541$	4.01732	0.172718	0.0863590	−	0.996264i	$-0.472477\pi$
$541$	4.01732	0.172718	0.0863590	+	0.996264i	$0.472477\pi$
$542$	−11.6373	−0.499866
$543$	−12.4076	−0.532460
$544$	−1.90460	−0.0816591
$545$	−3.03189	−0.129872
$546$	5.88714	0.251946
$547$	40.4991	1.73162	0.865809	−	0.500375i	$-0.166805\pi$
$547$	40.4991	1.73162	0.865809	+	0.500375i	$0.166805\pi$
$548$	1.05312	0.0449869
$549$	3.19903	0.136531
$550$	6.62562	0.282517
$551$	−46.3238	−1.97346
$552$	8.92819	0.380009
$553$	−18.3187	−0.778989
$554$	10.0897	0.428669
$555$	0	0
$556$	−1.90131	−0.0806333
$557$	−41.5078	−1.75874	−0.879370	−	0.476139i	$-0.842036\pi$
$557$	−41.5078	−1.75874	−0.879370	+	0.476139i	$0.842036\pi$
$558$	17.0574	0.722098
$559$	−1.63604	−0.0691973
$560$	16.8465	0.711893
$561$	7.73812	0.326704
$562$	23.5707	0.994268
$563$	−8.82377	−0.371877	−0.185939	−	0.982561i	$-0.559533\pi$
$563$	−8.82377	−0.371877	−0.185939	+	0.982561i	$0.559533\pi$
$564$	0.583805	0.0245826
$565$	−20.1901	−0.849405
$566$	36.5212	1.53510
$567$	17.0482	0.715958
$568$	41.2161	1.72939
$569$	1.06707	0.0447338	0.0223669	−	0.999750i	$-0.492880\pi$
$569$	1.06707	0.0447338	0.0223669	+	0.999750i	$0.492880\pi$
$570$	−5.28526	−0.221375
$571$	3.43435	0.143723	0.0718616	−	0.997415i	$-0.477106\pi$
$571$	3.43435	0.143723	0.0718616	+	0.997415i	$0.477106\pi$
$572$	0.908363	0.0379806
$573$	−6.50323	−0.271676
$574$	−30.1286	−1.25754
$575$	4.41859	0.184268
$576$	−17.9318	−0.747158
$577$	−30.2801	−1.26058	−0.630289	−	0.776361i	$-0.717064\pi$
$577$	−30.2801	−1.26058	−0.630289	+	0.776361i	$0.717064\pi$
$578$	17.1600	0.713761
$579$	−16.1073	−0.669395
$580$	−1.40189	−0.0582102
$581$	−23.2386	−0.964101
$582$	6.94504	0.287881
$583$	−4.67514	−0.193625
$584$	−13.0760	−0.541087
$585$	−3.34953	−0.138486
$586$	−38.2015	−1.57809
$587$	10.4265	0.430347	0.215173	−	0.976576i	$-0.430968\pi$
$587$	10.4265	0.430347	0.215173	+	0.976576i	$0.430968\pi$
$588$	0.933562	0.0384994
$589$	−23.0703	−0.950597
$590$	−21.4203	−0.881860
$591$	−1.93927	−0.0797709
$592$	0	0
$593$	19.5382	0.802337	0.401169	−	0.916004i	$-0.368604\pi$
$593$	19.5382	0.802337	0.401169	+	0.916004i	$0.368604\pi$
$594$	−26.8519	−1.10175
$595$	9.06850	0.371772
$596$	−2.32453	−0.0952166
$597$	4.89320	0.200265
$598$	8.86439	0.362492
$599$	−9.95917	−0.406921	−0.203460	−	0.979083i	$-0.565219\pi$
$599$	−9.95917	−0.406921	−0.203460	+	0.979083i	$0.565219\pi$
$600$	2.02060	0.0824906
$601$	0.201892	0.00823534	0.00411767	−	0.999992i	$-0.498689\pi$
$601$	0.201892	0.00823534	0.00411767	+	0.999992i	$0.498689\pi$
$602$	−6.90384	−0.281380
$603$	−17.7104	−0.721222
$604$	2.09704	0.0853272
$605$	−9.44942	−0.384174
$606$	5.03860	0.204679
$607$	−33.5993	−1.36375	−0.681876	−	0.731468i	$-0.738836\pi$
$607$	−33.5993	−1.36375	−0.681876	+	0.731468i	$0.738836\pi$
$608$	−4.01503	−0.162831
$609$	28.0422	1.13633
$610$	−1.91602	−0.0775773
$611$	−7.32249	−0.296236
$612$	0.825258	0.0333591
$613$	8.63736	0.348859	0.174430	−	0.984670i	$-0.444192\pi$
$613$	8.63736	0.348859	0.174430	+	0.984670i	$0.444192\pi$
$614$	26.8094	1.08194
$615$	−3.88019	−0.156464
$616$	−48.4240	−1.95106
$617$	8.37257	0.337067	0.168533	−	0.985696i	$-0.446097\pi$
$617$	8.37257	0.337067	0.168533	+	0.985696i	$0.446097\pi$
$618$	20.0320	0.805806
$619$	−0.670360	−0.0269441	−0.0134720	−	0.999909i	$-0.504288\pi$
$619$	−0.670360	−0.0269441	−0.0134720	+	0.999909i	$0.504288\pi$
$620$	−0.698172	−0.0280393
$621$	−17.9074	−0.718598
$622$	−2.11156	−0.0846657
$623$	40.7904	1.63423
$624$	4.35260	0.174243
$625$	1.00000	0.0400000
$626$	−27.1077	−1.08344
$627$	16.3125	0.651459
$628$	−0.286655	−0.0114388
$629$	0	0
$630$	−14.1345	−0.563131
$631$	−15.0416	−0.598798	−0.299399	−	0.954128i	$-0.596786\pi$
$631$	−15.0416	−0.598798	−0.299399	+	0.954128i	$0.596786\pi$
$632$	−12.6135	−0.501738
$633$	−1.48972	−0.0592109
$634$	35.4134	1.40645
$635$	13.5915	0.539362
$636$	0.112861	0.00447522
$637$	−11.7094	−0.463943
$638$	63.3141	2.50663
$639$	−37.1313	−1.46889
$640$	12.3965	0.490015
$641$	18.9440	0.748241	0.374121	−	0.927380i	$-0.377945\pi$
$641$	18.9440	0.748241	0.374121	+	0.927380i	$0.377945\pi$
$642$	19.4831	0.768937
$643$	27.9869	1.10369	0.551847	−	0.833945i	$-0.313923\pi$
$643$	27.9869	1.10369	0.551847	+	0.833945i	$0.313923\pi$
$644$	2.55630	0.100732
$645$	−0.889128	−0.0350094
$646$	−16.3329	−0.642609
$647$	−20.3334	−0.799387	−0.399694	−	0.916649i	$-0.630883\pi$
$647$	−20.3334	−0.799387	−0.399694	+	0.916649i	$0.630883\pi$
$648$	11.7387	0.461140
$649$	66.1119	2.59512
$650$	2.00616	0.0786881
$651$	13.9657	0.547358
$652$	−0.442209	−0.0173182
$653$	12.5212	0.489991	0.244996	−	0.969524i	$-0.421213\pi$
$653$	12.5212	0.489991	0.244996	+	0.969524i	$0.421213\pi$
$654$	−3.30558	−0.129259
$655$	21.1799	0.827566
$656$	−22.2753	−0.869703
$657$	11.7800	0.459583
$658$	−30.8997	−1.20460
$659$	−10.5140	−0.409565	−0.204783	−	0.978807i	$-0.565649\pi$
$659$	−10.5140	−0.409565	−0.204783	+	0.978807i	$0.565649\pi$
$660$	0.493661	0.0192157
$661$	−12.9804	−0.504880	−0.252440	−	0.967613i	$-0.581233\pi$
$661$	−12.9804	−0.504880	−0.252440	+	0.967613i	$0.581233\pi$
$662$	−0.451858	−0.0175620
$663$	2.34301	0.0909951
$664$	−16.0012	−0.620966
$665$	19.1170	0.741327
$666$	0	0
$667$	42.2238	1.63491
$668$	−0.610296	−0.0236131
$669$	−5.53188	−0.213875
$670$	10.6074	0.409800
$671$	5.91363	0.228293
$672$	2.43050	0.0937588
$673$	2.47979	0.0955890	0.0477945	−	0.998857i	$-0.484781\pi$
$673$	2.47979	0.0955890	0.0477945	+	0.998857i	$0.484781\pi$
$674$	25.9882	1.00103
$675$	−4.05274	−0.155990
$676$	−1.63210	−0.0627730
$677$	−23.8883	−0.918103	−0.459052	−	0.888410i	$-0.651811\pi$
$677$	−23.8883	−0.918103	−0.459052	+	0.888410i	$0.651811\pi$
$678$	−22.0127	−0.845394
$679$	−25.1205	−0.964038
$680$	6.24420	0.239454
$681$	0.471041	0.0180503
$682$	31.5319	1.20742
$683$	36.3338	1.39028	0.695138	−	0.718876i	$-0.255343\pi$
$683$	36.3338	1.39028	0.695138	+	0.718876i	$0.255343\pi$
$684$	1.73970	0.0665191
$685$	−7.17856	−0.274279
$686$	−8.96592	−0.342320
$687$	11.8408	0.451753
$688$	−5.10428	−0.194599
$689$	−1.41558	−0.0539292
$690$	4.81747	0.183398
$691$	2.36561	0.0899922	0.0449961	−	0.998987i	$-0.485672\pi$
$691$	2.36561	0.0899922	0.0449961	+	0.998987i	$0.485672\pi$
$692$	−0.556414	−0.0211517
$693$	43.6249	1.65717
$694$	19.6483	0.745838
$695$	12.9602	0.491609
$696$	19.3087	0.731896
$697$	−11.9908	−0.454185
$698$	26.4004	0.999270
$699$	−6.61880	−0.250346
$700$	0.578534	0.0218665
$701$	36.1238	1.36438	0.682189	−	0.731176i	$-0.261028\pi$
$701$	36.1238	1.36438	0.682189	+	0.731176i	$0.261028\pi$
$702$	−8.13044	−0.306864
$703$	0	0
$704$	−33.1482	−1.24932
$705$	−3.97950	−0.149877
$706$	−24.3169	−0.915180
$707$	−18.2249	−0.685417
$708$	−1.59598	−0.0599807
$709$	−24.4167	−0.916989	−0.458495	−	0.888697i	$-0.651611\pi$
$709$	−24.4167	−0.916989	−0.458495	+	0.888697i	$0.651611\pi$
$710$	22.2393	0.834627
$711$	11.3634	0.426161
$712$	28.0866	1.05259
$713$	21.0284	0.787520
$714$	9.88714	0.370017
$715$	−6.19185	−0.231562
$716$	1.90739	0.0712825
$717$	0.0615245	0.00229767
$718$	−0.431259	−0.0160944
$719$	−26.7224	−0.996576	−0.498288	−	0.867012i	$-0.666038\pi$
$719$	−26.7224	−0.996576	−0.498288	+	0.867012i	$0.666038\pi$
$720$	−10.4502	−0.389455
$721$	−72.4568	−2.69843
$722$	−6.59276	−0.245357
$723$	14.6088	0.543307
$724$	−2.44612	−0.0909092
$725$	9.55595	0.354899
$726$	−10.3024	−0.382360
$727$	−2.85449	−0.105867	−0.0529336	−	0.998598i	$-0.516857\pi$
$727$	−2.85449	−0.105867	−0.0529336	+	0.998598i	$0.516857\pi$
$728$	−14.6622	−0.543419
$729$	−1.52804	−0.0565940
$730$	−7.05551	−0.261136
$731$	−2.74765	−0.101625
$732$	−0.142758	−0.00527650
$733$	−23.9471	−0.884507	−0.442253	−	0.896890i	$-0.645821\pi$
$733$	−23.9471	−0.884507	−0.442253	+	0.896890i	$0.645821\pi$
$734$	−34.0853	−1.25811
$735$	−6.36361	−0.234725
$736$	3.65966	0.134897
$737$	−32.7389	−1.20595
$738$	18.6893	0.687964
$739$	27.0438	0.994823	0.497411	−	0.867515i	$-0.334284\pi$
$739$	27.0438	0.994823	0.497411	+	0.867515i	$0.334284\pi$
$740$	0	0
$741$	4.93924	0.181447
$742$	−5.97351	−0.219295
$743$	38.5442	1.41405	0.707026	−	0.707188i	$-0.250037\pi$
$743$	38.5442	1.41405	0.707026	+	0.707188i	$0.250037\pi$
$744$	9.61619	0.352547
$745$	15.8451	0.580521
$746$	54.5497	1.99721
$747$	14.4154	0.527430
$748$	1.52555	0.0557796
$749$	−70.4714	−2.57497
$750$	1.09027	0.0398111
$751$	−42.9011	−1.56548	−0.782741	−	0.622348i	$-0.786179\pi$
$751$	−42.9011	−1.56548	−0.782741	+	0.622348i	$0.786179\pi$
$752$	−22.8454	−0.833086
$753$	2.35334	0.0857606
$754$	19.1708	0.698158
$755$	−14.2944	−0.520227
$756$	−2.34465	−0.0852739
$757$	−17.3376	−0.630146	−0.315073	−	0.949067i	$-0.602029\pi$
$757$	−17.3376	−0.630146	−0.315073	+	0.949067i	$0.602029\pi$
$758$	33.7434	1.22562
$759$	−14.8687	−0.539700
$760$	13.1632	0.477480
$761$	41.5752	1.50710	0.753550	−	0.657390i	$-0.228340\pi$
$761$	41.5752	1.50710	0.753550	+	0.657390i	$0.228340\pi$
$762$	14.8184	0.536815
$763$	11.9565	0.432853
$764$	−1.28209	−0.0463845
$765$	−5.62536	−0.203385
$766$	−18.8474	−0.680983
$767$	20.0179	0.722805
$768$	2.60624	0.0940445
$769$	10.1940	0.367603	0.183802	−	0.982963i	$-0.441160\pi$
$769$	10.1940	0.367603	0.183802	+	0.982963i	$0.441160\pi$
$770$	−26.1286	−0.941609
$771$	8.72712	0.314300
$772$	−3.17550	−0.114289
$773$	−1.79833	−0.0646816	−0.0323408	−	0.999477i	$-0.510296\pi$
$773$	−1.79833	−0.0646816	−0.0323408	+	0.999477i	$0.510296\pi$
$774$	4.28258	0.153934
$775$	4.75908	0.170951
$776$	−17.2970	−0.620926
$777$	0	0
$778$	−23.6176	−0.846732
$779$	−25.2775	−0.905661
$780$	0.149475	0.00535206
$781$	−68.6398	−2.45613
$782$	14.8873	0.532368
$783$	−38.7277	−1.38402
$784$	−36.5321	−1.30472
$785$	1.95398	0.0697406
$786$	23.0918	0.823658
$787$	−44.7062	−1.59360	−0.796802	−	0.604241i	$-0.793477\pi$
$787$	−44.7062	−1.59360	−0.796802	+	0.604241i	$0.793477\pi$
$788$	−0.382321	−0.0136196
$789$	0.562139	0.0200127
$790$	−6.80597	−0.242146
$791$	79.6211	2.83100
$792$	30.0383	1.06737
$793$	1.79058	0.0635852
$794$	18.9725	0.673310
$795$	−0.769314	−0.0272847
$796$	0.964680	0.0341922
$797$	−1.95203	−0.0691444	−0.0345722	−	0.999402i	$-0.511007\pi$
$797$	−1.95203	−0.0691444	−0.0345722	+	0.999402i	$0.511007\pi$
$798$	20.8428	0.737826
$799$	−12.2977	−0.435062
$800$	0.828243	0.0292828
$801$	−25.3030	−0.894038
$802$	−25.2671	−0.892211
$803$	21.7762	0.768467
$804$	0.790336	0.0278730
$805$	−17.4250	−0.614151
$806$	9.54748	0.336296
$807$	10.0117	0.352429
$808$	−12.5489	−0.441469
$809$	−32.1732	−1.13115	−0.565574	−	0.824698i	$-0.691345\pi$
$809$	−32.1732	−1.13115	−0.565574	+	0.824698i	$0.691345\pi$
$810$	6.33397	0.222553
$811$	43.5729	1.53005	0.765025	−	0.644000i	$-0.222726\pi$
$811$	43.5729	1.53005	0.765025	+	0.644000i	$0.222726\pi$
$812$	5.52844	0.194010
$813$	5.91039	0.207286
$814$	0	0
$815$	3.01431	0.105587
$816$	7.30995	0.255900
$817$	−5.79223	−0.202645
$818$	45.8478	1.60303
$819$	13.2091	0.461564
$820$	−0.764967	−0.0267138
$821$	−42.7990	−1.49370	−0.746848	−	0.664995i	$-0.768434\pi$
$821$	−42.7990	−1.49370	−0.746848	+	0.664995i	$0.768434\pi$
$822$	−7.82658	−0.272983
$823$	52.9850	1.84694	0.923470	−	0.383670i	$-0.125340\pi$
$823$	52.9850	1.84694	0.923470	+	0.383670i	$0.125340\pi$
$824$	−49.8908	−1.73803
$825$	−3.36504	−0.117156
$826$	84.4724	2.93917
$827$	−23.3945	−0.813508	−0.406754	−	0.913538i	$-0.633339\pi$
$827$	−23.3945	−0.813508	−0.406754	+	0.913538i	$0.633339\pi$
$828$	−1.58572	−0.0551077
$829$	−0.375803	−0.0130522	−0.00652608	−	0.999979i	$-0.502077\pi$
$829$	−0.375803	−0.0130522	−0.00652608	+	0.999979i	$0.502077\pi$
$830$	−8.63390	−0.299687
$831$	−5.12437	−0.177762
$832$	−10.0369	−0.347966
$833$	−19.6653	−0.681362
$834$	14.1302	0.489288
$835$	4.16008	0.143965
$836$	3.21596	0.111226
$837$	−19.2873	−0.666666
$838$	17.3187	0.598263
$839$	15.8320	0.546583	0.273291	−	0.961931i	$-0.411888\pi$
$839$	15.8320	0.546583	0.273291	+	0.961931i	$0.411888\pi$
$840$	−7.96837	−0.274935
$841$	62.3161	2.14883
$842$	28.0832	0.967810
$843$	−11.9711	−0.412307
$844$	−0.293693	−0.0101093
$845$	11.1252	0.382718
$846$	19.1677	0.658998
$847$	37.2644	1.28042
$848$	−4.41646	−0.151662
$849$	−18.5485	−0.636581
$850$	3.36924	0.115564
$851$	0	0
$852$	1.65701	0.0567681
$853$	9.32579	0.319309	0.159655	−	0.987173i	$-0.448962\pi$
$853$	9.32579	0.319309	0.159655	+	0.987173i	$0.448962\pi$
$854$	7.55595	0.258559
$855$	−11.8587	−0.405557
$856$	−48.5238	−1.65851
$857$	5.99313	0.204721	0.102361	−	0.994747i	$-0.467360\pi$
$857$	5.99313	0.204721	0.102361	+	0.994747i	$0.467360\pi$
$858$	−6.75080	−0.230469
$859$	44.6541	1.52358	0.761788	−	0.647826i	$-0.224322\pi$
$859$	44.6541	1.52358	0.761788	+	0.647826i	$0.224322\pi$
$860$	−0.175289	−0.00597730
$861$	15.3018	0.521483
$862$	14.3362	0.488292
$863$	−21.8515	−0.743832	−0.371916	−	0.928266i	$-0.621299\pi$
$863$	−21.8515	−0.743832	−0.371916	+	0.928266i	$0.621299\pi$
$864$	−3.35665	−0.114196
$865$	3.79279	0.128959
$866$	−51.6706	−1.75584
$867$	−8.71525	−0.295986
$868$	2.75329	0.0934527
$869$	21.0061	0.712582
$870$	10.4186	0.353223
$871$	−9.91295	−0.335888
$872$	8.23273	0.278795
$873$	15.5827	0.527396
$874$	31.3834	1.06156
$875$	−3.94357	−0.133317
$876$	−0.525691	−0.0177615
$877$	−11.3951	−0.384786	−0.192393	−	0.981318i	$-0.561625\pi$
$877$	−11.3951	−0.384786	−0.192393	+	0.981318i	$0.561625\pi$
$878$	35.8868	1.21112
$879$	19.4019	0.654409
$880$	−19.3179	−0.651206
$881$	−48.2087	−1.62419	−0.812096	−	0.583523i	$-0.801674\pi$
$881$	−48.2087	−1.62419	−0.812096	+	0.583523i	$0.801674\pi$
$882$	30.6510	1.03207
$883$	36.2461	1.21978	0.609890	−	0.792486i	$-0.291214\pi$
$883$	36.2461	1.21978	0.609890	+	0.792486i	$0.291214\pi$
$884$	0.461918	0.0155360
$885$	10.8790	0.365693
$886$	38.3653	1.28891
$887$	−23.8002	−0.799134	−0.399567	−	0.916704i	$-0.630840\pi$
$887$	−23.8002	−0.799134	−0.399567	+	0.916704i	$0.630840\pi$
$888$	0	0
$889$	−53.5990	−1.79765
$890$	15.1549	0.507995
$891$	−19.5493	−0.654925
$892$	−1.09059	−0.0365158
$893$	−25.9245	−0.867530
$894$	17.2755	0.577780
$895$	−13.0017	−0.434599
$896$	−48.8865	−1.63318
$897$	−4.50207	−0.150320
$898$	58.1624	1.94090
$899$	45.4775	1.51676
$900$	−0.358875	−0.0119625
$901$	−2.37739	−0.0792023
$902$	34.5486	1.15034
$903$	3.50634	0.116684
$904$	54.8239	1.82342
$905$	16.6739	0.554260
$906$	−15.5848	−0.517771
$907$	9.82566	0.326256	0.163128	−	0.986605i	$-0.447842\pi$
$907$	9.82566	0.326256	0.163128	+	0.986605i	$0.447842\pi$
$908$	0.0928643	0.00308181
$909$	11.3052	0.374971
$910$	−7.91143	−0.262261
$911$	−31.2125	−1.03412	−0.517058	−	0.855950i	$-0.672973\pi$
$911$	−31.2125	−1.03412	−0.517058	+	0.855950i	$0.672973\pi$
$912$	15.4099	0.510273
$913$	26.6478	0.881914
$914$	30.2999	1.00223
$915$	0.973111	0.0321701
$916$	2.33437	0.0771298
$917$	−83.5243	−2.75821
$918$	−13.6546	−0.450670
$919$	−28.5950	−0.943263	−0.471631	−	0.881796i	$-0.656335\pi$
$919$	−28.5950	−0.943263	−0.471631	+	0.881796i	$0.656335\pi$
$920$	−11.9982	−0.395567
$921$	−13.6160	−0.448663
$922$	−8.08424	−0.266240
$923$	−20.7833	−0.684092
$924$	−1.94679	−0.0640446
$925$	0	0
$926$	30.7366	1.01007
$927$	44.9463	1.47623
$928$	7.91465	0.259811
$929$	−45.3832	−1.48898	−0.744488	−	0.667636i	$-0.767306\pi$
$929$	−45.3832	−1.48898	−0.744488	+	0.667636i	$0.767306\pi$
$930$	5.18869	0.170144
$931$	−41.4558	−1.35866
$932$	−1.30488	−0.0427426
$933$	1.07242	0.0351095
$934$	−11.1684	−0.365440
$935$	−10.3989	−0.340080
$936$	9.09526	0.297288
$937$	−27.0457	−0.883543	−0.441772	−	0.897128i	$-0.645650\pi$
$937$	−27.0457	−0.883543	−0.441772	+	0.897128i	$0.645650\pi$
$938$	−41.8311	−1.36583
$939$	13.7675	0.449285
$940$	−0.784546	−0.0255891
$941$	31.3367	1.02155	0.510773	−	0.859716i	$-0.329359\pi$
$941$	31.3367	1.02155	0.510773	+	0.859716i	$0.329359\pi$
$942$	2.13037	0.0694113
$943$	23.0402	0.750293
$944$	62.4538	2.03270
$945$	15.9822	0.519903
$946$	7.91665	0.257393
$947$	27.4446	0.891830	0.445915	−	0.895075i	$-0.352878\pi$
$947$	27.4446	0.891830	0.445915	+	0.895075i	$0.352878\pi$
$948$	−0.507099	−0.0164698
$949$	6.59359	0.214037
$950$	7.10259	0.230439
$951$	−17.9858	−0.583231
$952$	−24.6244	−0.798083
$953$	−35.3324	−1.14453	−0.572265	−	0.820069i	$-0.693935\pi$
$953$	−35.3324	−1.14453	−0.572265	+	0.820069i	$0.693935\pi$
$954$	3.70548	0.119969
$955$	8.73937	0.282799
$956$	0.0121294	0.000392292 0
$957$	−32.1561	−1.03946
$958$	47.9760	1.55004
$959$	28.3091	0.914150
$960$	−5.45467	−0.176049
$961$	−8.35116	−0.269392
$962$	0	0
$963$	43.7147	1.40869
$964$	2.88008	0.0927612
$965$	21.6457	0.696801
$966$	−18.9980	−0.611251
$967$	−31.4137	−1.01020	−0.505098	−	0.863062i	$-0.668544\pi$
$967$	−31.4137	−1.01020	−0.505098	+	0.863062i	$0.668544\pi$
$968$	25.6588	0.824705
$969$	8.29519	0.266480
$970$	−9.33309	−0.299667
$971$	6.60982	0.212119	0.106060	−	0.994360i	$-0.466177\pi$
$971$	6.60982	0.212119	0.106060	+	0.994360i	$0.466177\pi$
$972$	2.25558	0.0723477
$973$	−51.1095	−1.63850
$974$	−35.6748	−1.14310
$975$	−1.01889	−0.0326307
$976$	5.58641	0.178817
$977$	0.318781	0.0101987	0.00509935	−	0.999987i	$-0.498377\pi$
$977$	0.318781	0.0101987	0.00509935	+	0.999987i	$0.498377\pi$
$978$	3.28642	0.105088
$979$	−46.7744	−1.49492
$980$	−1.25457	−0.0400757
$981$	−7.41681	−0.236801
$982$	−14.4789	−0.462042
$983$	−10.7688	−0.343473	−0.171736	−	0.985143i	$-0.554938\pi$
$983$	−10.7688	−0.343473	−0.171736	+	0.985143i	$0.554938\pi$
$984$	10.5362	0.335881
$985$	2.60609	0.0830368
$986$	32.1963	1.02534
$987$	15.6934	0.499527
$988$	0.973756	0.0309793
$989$	5.27957	0.167881
$990$	16.2081	0.515126
$991$	23.6638	0.751706	0.375853	−	0.926679i	$-0.377350\pi$
$991$	23.6638	0.751706	0.375853	+	0.926679i	$0.377350\pi$
$992$	3.94167	0.125148
$993$	0.229491	0.00728267
$994$	−87.7023	−2.78175
$995$	−6.57573	−0.208465
$996$	−0.643294	−0.0203835
$997$	4.48006	0.141885	0.0709425	−	0.997480i	$-0.477399\pi$
$997$	4.48006	0.141885	0.0709425	+	0.997480i	$0.477399\pi$
$998$	56.2850	1.78167
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6845.2.a.g.1.2		5
37.36	even	2		185.2.a.d.1.4	✓	5
111.110	odd	2		1665.2.a.q.1.2		5
148.147	odd	2		2960.2.a.ba.1.2		5
185.73	odd	4		925.2.b.g.149.3		10
185.147	odd	4		925.2.b.g.149.8		10
185.184	even	2		925.2.a.h.1.2		5
259.258	odd	2		9065.2.a.j.1.4		5
555.554	odd	2		8325.2.a.cc.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.d.1.4	✓	5	37.36	even	2
925.2.a.h.1.2		5	185.184	even	2
925.2.b.g.149.3		10	185.73	odd	4
925.2.b.g.149.8		10	185.147	odd	4
1665.2.a.q.1.2		5	111.110	odd	2
2960.2.a.ba.1.2		5	148.147	odd	2
6845.2.a.g.1.2		5	1.1	even	1	trivial
8325.2.a.cc.1.4		5	555.554	odd	2
9065.2.a.j.1.4		5	259.258	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6845 = 5 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6845.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46516 q^{2} +0.744131 q^{3} +0.146703 q^{4} -1.00000 q^{5} -1.09027 q^{6} +3.94357 q^{7} +2.71538 q^{8} -2.44627 q^{9} +1.46516 q^{10} -4.52210 q^{11} +0.109166 q^{12} -1.36924 q^{13} -5.77797 q^{14} -0.744131 q^{15} -4.27188 q^{16} -2.29957 q^{17} +3.58418 q^{18} -4.84765 q^{19} -0.146703 q^{20} +2.93453 q^{21} +6.62562 q^{22} +4.41859 q^{23} +2.02060 q^{24} +1.00000 q^{25} +2.00616 q^{26} -4.05274 q^{27} +0.578534 q^{28} +9.55595 q^{29} +1.09027 q^{30} +4.75908 q^{31} +0.828243 q^{32} -3.36504 q^{33} +3.36924 q^{34} -3.94357 q^{35} -0.358875 q^{36} +7.10259 q^{38} -1.01889 q^{39} -2.71538 q^{40} +5.21439 q^{41} -4.29957 q^{42} +1.19485 q^{43} -0.663407 q^{44} +2.44627 q^{45} -6.47395 q^{46} +5.34785 q^{47} -3.17884 q^{48} +8.55174 q^{49} -1.46516 q^{50} -1.71118 q^{51} -0.200872 q^{52} +1.03384 q^{53} +5.93792 q^{54} +4.52210 q^{55} +10.7083 q^{56} -3.60728 q^{57} -14.0010 q^{58} -14.6197 q^{59} -0.109166 q^{60} -1.30772 q^{61} -6.97283 q^{62} -9.64703 q^{63} +7.33026 q^{64} +1.36924 q^{65} +4.93033 q^{66} +7.23975 q^{67} -0.337354 q^{68} +3.28801 q^{69} +5.77797 q^{70} +15.1787 q^{71} -6.64256 q^{72} -4.81551 q^{73} +0.744131 q^{75} -0.711165 q^{76} -17.8332 q^{77} +1.49285 q^{78} -4.64520 q^{79} +4.27188 q^{80} +4.32304 q^{81} -7.63993 q^{82} -5.89279 q^{83} +0.430505 q^{84} +2.29957 q^{85} -1.75066 q^{86} +7.11087 q^{87} -12.2792 q^{88} +10.3435 q^{89} -3.58418 q^{90} -5.39969 q^{91} +0.648221 q^{92} +3.54138 q^{93} -7.83547 q^{94} +4.84765 q^{95} +0.616321 q^{96} -6.37000 q^{97} -12.5297 q^{98} +11.0623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} + 7 q^{7} + 6 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{13} - 4 q^{14} + q^{15} + 8 q^{16} + 8 q^{17} + 6 q^{18} - 14 q^{19} - 6 q^{20} - 9 q^{21} - 2 q^{22} - 2 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) \) 5 * q - q^3 + 6 * q^4 - 5 * q^5 + 2 * q^6 + 7 * q^7 + 6 * q^8 + 2 * q^9 + 7 * q^11 - 2 * q^13 - 4 * q^14 + q^15 + 8 * q^16 + 8 * q^17 + 6 * q^18 - 14 * q^19 - 6 * q^20 - 9 * q^21 - 2 * q^22 - 2 * q^23 - 12 * q^24 + 5 * q^25 - 20 * q^26 - 7 * q^27 - 14 * q^28 - 2 * q^29 - 2 * q^30 - 8 * q^31 + 22 * q^32 - 21 * q^33 + 12 * q^34 - 7 * q^35 - 36 * q^36 - 32 * q^38 - 12 * q^39 - 6 * q^40 - 9 * q^41 - 2 * q^42 - 14 * q^43 + 18 * q^44 - 2 * q^45 - 28 * q^46 - 5 * q^47 - 30 * q^48 + 2 * q^49 - 10 * q^51 - 20 * q^52 - 15 * q^53 + 12 * q^54 - 7 * q^55 + 14 * q^56 - 4 * q^57 - 12 * q^59 - 12 * q^61 + 10 * q^62 + 24 * q^63 + 12 * q^64 + 2 * q^65 + 10 * q^66 - 2 * q^67 + 4 * q^68 + 30 * q^69 + 4 * q^70 + 13 * q^71 - 26 * q^72 - 5 * q^73 - q^75 - 18 * q^76 - 7 * q^77 + 10 * q^78 - 36 * q^79 - 8 * q^80 + 21 * q^81 + 2 * q^82 - 9 * q^83 + 30 * q^84 - 8 * q^85 - 8 * q^86 + 6 * q^87 - 8 * q^88 + 16 * q^89 - 6 * q^90 + 4 * q^91 - 12 * q^92 + 16 * q^93 - 20 * q^94 + 14 * q^95 - 30 * q^98 + 18 * q^99

Introduction

L-functions

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