LMFDB - Embedded newform 2645.2.a.v.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2645,2,Mod(1,2645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2645, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2645 = 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2645.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:	$21.1204313346$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 88 x^{13} + 93 x^{12} - 728 x^{11} + 58 x^{10} + 2760 x^{9} - 1764 x^{8} + \cdots - 18 $ x^16 - 4x^15 - 18x^14 + 88x^13 + 93x^12 - 728x^11 + 58x^10 + 2760x^9 - 1764x^8 - 4568x^7 + 4414x^6 + 2440x^5 - 2759x^4 - 684x^3 + 558x^2 + 108*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.23086$ of defining polynomial
Character	$\chi$	$=$	2645.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23086 q^{2} -0.203445 q^{3} -0.484991 q^{4} -1.00000 q^{5} -0.250412 q^{6} +4.57348 q^{7} -3.05867 q^{8} -2.95861 q^{9} +O(q^{10})$	\(q+1.23086 q^{2} -0.203445 q^{3} -0.484991 q^{4} -1.00000 q^{5} -0.250412 q^{6} +4.57348 q^{7} -3.05867 q^{8} -2.95861 q^{9} -1.23086 q^{10} -2.09396 q^{11} +0.0986692 q^{12} +1.79111 q^{13} +5.62930 q^{14} +0.203445 q^{15} -2.79480 q^{16} +3.59694 q^{17} -3.64163 q^{18} -6.49064 q^{19} +0.484991 q^{20} -0.930454 q^{21} -2.57736 q^{22} +0.622272 q^{24} +1.00000 q^{25} +2.20460 q^{26} +1.21225 q^{27} -2.21810 q^{28} +7.60349 q^{29} +0.250412 q^{30} -3.15873 q^{31} +2.67734 q^{32} +0.426006 q^{33} +4.42732 q^{34} -4.57348 q^{35} +1.43490 q^{36} -11.4307 q^{37} -7.98905 q^{38} -0.364392 q^{39} +3.05867 q^{40} -10.2446 q^{41} -1.14526 q^{42} -9.57259 q^{43} +1.01555 q^{44} +2.95861 q^{45} -2.89196 q^{47} +0.568590 q^{48} +13.9168 q^{49} +1.23086 q^{50} -0.731781 q^{51} -0.868671 q^{52} -2.18675 q^{53} +1.49211 q^{54} +2.09396 q^{55} -13.9888 q^{56} +1.32049 q^{57} +9.35881 q^{58} -1.57089 q^{59} -0.0986692 q^{60} +1.93354 q^{61} -3.88795 q^{62} -13.5312 q^{63} +8.88502 q^{64} -1.79111 q^{65} +0.524352 q^{66} -10.5701 q^{67} -1.74448 q^{68} -5.62930 q^{70} -7.11617 q^{71} +9.04941 q^{72} +2.95109 q^{73} -14.0696 q^{74} -0.203445 q^{75} +3.14790 q^{76} -9.57668 q^{77} -0.448515 q^{78} -5.52879 q^{79} +2.79480 q^{80} +8.62920 q^{81} -12.6097 q^{82} -10.9551 q^{83} +0.451262 q^{84} -3.59694 q^{85} -11.7825 q^{86} -1.54689 q^{87} +6.40472 q^{88} +12.3360 q^{89} +3.64163 q^{90} +8.19160 q^{91} +0.642630 q^{93} -3.55959 q^{94} +6.49064 q^{95} -0.544692 q^{96} -5.90357 q^{97} +17.1295 q^{98} +6.19520 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9}+O(q^{10}) $ 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9	\( 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9} - 4 q^{10} - 16 q^{11} - 4 q^{12} + 8 q^{13} - 8 q^{14} - 4 q^{15} + 28 q^{16} + 8 q^{17} + 4 q^{18} - 32 q^{19} - 20 q^{20} - 16 q^{21} - 44 q^{22} - 28 q^{24} + 16 q^{25} - 20 q^{26} - 8 q^{27} - 52 q^{28} - 12 q^{29} + 12 q^{30} - 12 q^{31} + 4 q^{32} - 20 q^{33} - 24 q^{34} + 12 q^{35} - 4 q^{36} - 28 q^{37} + 20 q^{38} - 8 q^{39} - 4 q^{41} + 8 q^{42} - 48 q^{43} - 32 q^{44} - 8 q^{45} + 48 q^{47} + 24 q^{48} + 12 q^{49} + 4 q^{50} - 24 q^{51} + 12 q^{52} + 4 q^{53} - 44 q^{54} + 16 q^{55} - 64 q^{56} - 52 q^{57} - 28 q^{58} - 40 q^{59} + 4 q^{60} - 16 q^{61} - 4 q^{63} - 16 q^{64} - 8 q^{65} - 8 q^{66} - 68 q^{67} + 4 q^{68} + 8 q^{70} - 24 q^{71} - 12 q^{72} + 52 q^{73} - 40 q^{74} + 4 q^{75} - 24 q^{76} + 44 q^{77} + 12 q^{78} - 72 q^{79} - 28 q^{80} + 20 q^{81} - 20 q^{82} - 12 q^{83} - 32 q^{84} - 8 q^{85} + 56 q^{86} - 104 q^{88} - 48 q^{89} - 4 q^{90} - 48 q^{91} - 48 q^{93} - 60 q^{94} + 32 q^{95} - 108 q^{96} - 4 q^{97} + 8 q^{98} - 24 q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9 - 4 * q^10 - 16 * q^11 - 4 * q^12 + 8 * q^13 - 8 * q^14 - 4 * q^15 + 28 * q^16 + 8 * q^17 + 4 * q^18 - 32 * q^19 - 20 * q^20 - 16 * q^21 - 44 * q^22 - 28 * q^24 + 16 * q^25 - 20 * q^26 - 8 * q^27 - 52 * q^28 - 12 * q^29 + 12 * q^30 - 12 * q^31 + 4 * q^32 - 20 * q^33 - 24 * q^34 + 12 * q^35 - 4 * q^36 - 28 * q^37 + 20 * q^38 - 8 * q^39 - 4 * q^41 + 8 * q^42 - 48 * q^43 - 32 * q^44 - 8 * q^45 + 48 * q^47 + 24 * q^48 + 12 * q^49 + 4 * q^50 - 24 * q^51 + 12 * q^52 + 4 * q^53 - 44 * q^54 + 16 * q^55 - 64 * q^56 - 52 * q^57 - 28 * q^58 - 40 * q^59 + 4 * q^60 - 16 * q^61 - 4 * q^63 - 16 * q^64 - 8 * q^65 - 8 * q^66 - 68 * q^67 + 4 * q^68 + 8 * q^70 - 24 * q^71 - 12 * q^72 + 52 * q^73 - 40 * q^74 + 4 * q^75 - 24 * q^76 + 44 * q^77 + 12 * q^78 - 72 * q^79 - 28 * q^80 + 20 * q^81 - 20 * q^82 - 12 * q^83 - 32 * q^84 - 8 * q^85 + 56 * q^86 - 104 * q^88 - 48 * q^89 - 4 * q^90 - 48 * q^91 - 48 * q^93 - 60 * q^94 + 32 * q^95 - 108 * q^96 - 4 * q^97 + 8 * q^98 - 24 * 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 $

$2$	1.23086	0.870347	0.435174	−	0.900347i	$-0.356687\pi$
$2$	1.23086	0.870347	0.435174	+	0.900347i	$0.356687\pi$
$3$	−0.203445	−0.117459	−0.0587296	−	0.998274i	$-0.518705\pi$
$3$	−0.203445	−0.117459	−0.0587296	+	0.998274i	$0.518705\pi$
$4$	−0.484991	−0.242495
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.250412	−0.102230
$7$	4.57348	1.72861	0.864307	−	0.502964i	$-0.167757\pi$
$7$	4.57348	1.72861	0.864307	+	0.502964i	$0.167757\pi$
$8$	−3.05867	−1.08140
$9$	−2.95861	−0.986203
$10$	−1.23086	−0.389231
$11$	−2.09396	−0.631352	−0.315676	−	0.948867i	$-0.602231\pi$
$11$	−2.09396	−0.631352	−0.315676	+	0.948867i	$0.602231\pi$
$12$	0.0986692	0.0284833
$13$	1.79111	0.496764	0.248382	−	0.968662i	$-0.420101\pi$
$13$	1.79111	0.496764	0.248382	+	0.968662i	$0.420101\pi$
$14$	5.62930	1.50450
$15$	0.203445	0.0525294
$16$	−2.79480	−0.698700
$17$	3.59694	0.872387	0.436193	−	0.899853i	$-0.356326\pi$
$17$	3.59694	0.872387	0.436193	+	0.899853i	$0.356326\pi$
$18$	−3.64163	−0.858339
$19$	−6.49064	−1.48905	−0.744527	−	0.667592i	$-0.767325\pi$
$19$	−6.49064	−1.48905	−0.744527	+	0.667592i	$0.767325\pi$
$20$	0.484991	0.108447
$21$	−0.930454	−0.203042
$22$	−2.57736	−0.549495
$23$	0	0
$23$	0	0
$24$	0.622272	0.127021
$25$	1.00000	0.200000
$26$	2.20460	0.432357
$27$	1.21225	0.233298
$28$	−2.21810	−0.419181
$29$	7.60349	1.41193	0.705966	−	0.708246i	$-0.250513\pi$
$29$	7.60349	1.41193	0.705966	+	0.708246i	$0.250513\pi$
$30$	0.250412	0.0457188
$31$	−3.15873	−0.567325	−0.283663	−	0.958924i	$-0.591550\pi$
$31$	−3.15873	−0.567325	−0.283663	+	0.958924i	$0.591550\pi$
$32$	2.67734	0.473291
$33$	0.426006	0.0741581
$34$	4.42732	0.759279
$35$	−4.57348	−0.773060
$36$	1.43490	0.239150
$37$	−11.4307	−1.87920	−0.939598	−	0.342280i	$-0.888801\pi$
$37$	−11.4307	−1.87920	−0.939598	+	0.342280i	$0.888801\pi$
$38$	−7.98905	−1.29599
$39$	−0.364392	−0.0583495
$40$	3.05867	0.483618
$41$	−10.2446	−1.59994	−0.799971	−	0.600039i	$-0.795152\pi$
$41$	−10.2446	−1.59994	−0.799971	+	0.600039i	$0.795152\pi$
$42$	−1.14526	−0.176717
$43$	−9.57259	−1.45981	−0.729903	−	0.683550i	$-0.760435\pi$
$43$	−9.57259	−1.45981	−0.729903	+	0.683550i	$0.760435\pi$
$44$	1.01555	0.153100
$45$	2.95861	0.441044
$46$	0	0
$47$	−2.89196	−0.421836	−0.210918	−	0.977504i	$-0.567645\pi$
$47$	−2.89196	−0.421836	−0.210918	+	0.977504i	$0.567645\pi$
$48$	0.568590	0.0820688
$49$	13.9168	1.98811
$50$	1.23086	0.174069
$51$	−0.731781	−0.102470
$52$	−0.868671	−0.120463
$53$	−2.18675	−0.300373	−0.150187	−	0.988658i	$-0.547987\pi$
$53$	−2.18675	−0.300373	−0.150187	+	0.988658i	$0.547987\pi$
$54$	1.49211	0.203050
$55$	2.09396	0.282349
$56$	−13.9888	−1.86933
$57$	1.32049	0.174903
$58$	9.35881	1.22887
$59$	−1.57089	−0.204512	−0.102256	−	0.994758i	$-0.532606\pi$
$59$	−1.57089	−0.204512	−0.102256	+	0.994758i	$0.532606\pi$
$60$	−0.0986692	−0.0127381
$61$	1.93354	0.247565	0.123782	−	0.992309i	$-0.460498\pi$
$61$	1.93354	0.247565	0.123782	+	0.992309i	$0.460498\pi$
$62$	−3.88795	−0.493770
$63$	−13.5312	−1.70477
$64$	8.88502	1.11063
$65$	−1.79111	−0.222159
$66$	0.524352	0.0645433
$67$	−10.5701	−1.29135	−0.645673	−	0.763614i	$-0.723423\pi$
$67$	−10.5701	−1.29135	−0.645673	+	0.763614i	$0.723423\pi$
$68$	−1.74448	−0.211550
$69$	0	0
$70$	−5.62930	−0.672831
$71$	−7.11617	−0.844534	−0.422267	−	0.906471i	$-0.638766\pi$
$71$	−7.11617	−0.844534	−0.422267	+	0.906471i	$0.638766\pi$
$72$	9.04941	1.06648
$73$	2.95109	0.345399	0.172700	−	0.984975i	$-0.444751\pi$
$73$	2.95109	0.345399	0.172700	+	0.984975i	$0.444751\pi$
$74$	−14.0696	−1.63555
$75$	−0.203445	−0.0234919
$76$	3.14790	0.361089
$77$	−9.57668	−1.09136
$78$	−0.448515	−0.0507843
$79$	−5.52879	−0.622038	−0.311019	−	0.950404i	$-0.600670\pi$
$79$	−5.52879	−0.622038	−0.311019	+	0.950404i	$0.600670\pi$
$80$	2.79480	0.312468
$81$	8.62920	0.958800
$82$	−12.6097	−1.39250
$83$	−10.9551	−1.20248	−0.601239	−	0.799070i	$-0.705326\pi$
$83$	−10.9551	−1.20248	−0.601239	+	0.799070i	$0.705326\pi$
$84$	0.451262	0.0492367
$85$	−3.59694	−0.390143
$86$	−11.7825	−1.27054
$87$	−1.54689	−0.165845
$88$	6.40472	0.682745
$89$	12.3360	1.30762	0.653808	−	0.756660i	$-0.273170\pi$
$89$	12.3360	1.30762	0.653808	+	0.756660i	$0.273170\pi$
$90$	3.64163	0.383861
$91$	8.19160	0.858713
$92$	0	0
$93$	0.642630	0.0666376
$94$	−3.55959	−0.367144
$95$	6.49064	0.665925
$96$	−0.544692	−0.0555924
$97$	−5.90357	−0.599417	−0.299709	−	0.954031i	$-0.596889\pi$
$97$	−5.90357	−0.599417	−0.299709	+	0.954031i	$0.596889\pi$
$98$	17.1295	1.73034
$99$	6.19520	0.622641
$100$	−0.484991	−0.0484991
$101$	−2.97647	−0.296169	−0.148085	−	0.988975i	$-0.547311\pi$
$101$	−2.97647	−0.296169	−0.148085	+	0.988975i	$0.547311\pi$
$102$	−0.900718	−0.0891844
$103$	−0.463769	−0.0456965	−0.0228483	−	0.999739i	$-0.507273\pi$
$103$	−0.463769	−0.0456965	−0.0228483	+	0.999739i	$0.507273\pi$
$104$	−5.47840	−0.537202
$105$	0.930454	0.0908030
$106$	−2.69158	−0.261429
$107$	0.0133367	0.00128930	0.000644652	−	1.00000i	$-0.499795\pi$
$107$	0.0133367	0.00128930	0.000644652		1.00000i	$0.499795\pi$
$108$	−0.587931	−0.0565737
$109$	12.7996	1.22598	0.612990	−	0.790091i	$-0.289967\pi$
$109$	12.7996	1.22598	0.612990	+	0.790091i	$0.289967\pi$
$110$	2.57736	0.245742
$111$	2.32552	0.220729
$112$	−12.7820	−1.20778
$113$	−15.8401	−1.49011	−0.745055	−	0.667003i	$-0.767577\pi$
$113$	−15.8401	−1.49011	−0.745055	+	0.667003i	$0.767577\pi$
$114$	1.62533	0.152227
$115$	0	0
$116$	−3.68762	−0.342387
$117$	−5.29919	−0.489910
$118$	−1.93354	−0.177997
$119$	16.4506	1.50802
$120$	−0.622272	−0.0568054
$121$	−6.61534	−0.601395
$122$	2.37991	0.215467
$123$	2.08422	0.187928
$124$	1.53196	0.137574
$125$	−1.00000	−0.0894427
$126$	−16.6549	−1.48374
$127$	12.6019	1.11824	0.559118	−	0.829088i	$-0.311140\pi$
$127$	12.6019	1.11824	0.559118	+	0.829088i	$0.311140\pi$
$128$	5.58152	0.493341
$129$	1.94750	0.171468
$130$	−2.20460	−0.193356
$131$	−10.5361	−0.920547	−0.460273	−	0.887777i	$-0.652249\pi$
$131$	−10.5361	−0.920547	−0.460273	+	0.887777i	$0.652249\pi$
$132$	−0.206609	−0.0179830
$133$	−29.6848	−2.57400
$134$	−13.0103	−1.12392
$135$	−1.21225	−0.104334
$136$	−11.0019	−0.943401
$137$	1.42748	0.121958	0.0609789	−	0.998139i	$-0.480578\pi$
$137$	1.42748	0.121958	0.0609789	+	0.998139i	$0.480578\pi$
$138$	0	0
$139$	0.00961267	0.000815336 0	0.000407668	−	1.00000i	$-0.499870\pi$
$139$	0.00961267	0.000815336 0	0.000407668		1.00000i	$0.499870\pi$
$140$	2.21810	0.187464
$141$	0.588356	0.0495485
$142$	−8.75899	−0.735038
$143$	−3.75050	−0.313633
$144$	8.26873	0.689061
$145$	−7.60349	−0.631435
$146$	3.63237	0.300617
$147$	−2.83130	−0.233522
$148$	5.54379	0.455697
$149$	−9.86086	−0.807833	−0.403916	−	0.914796i	$-0.632351\pi$
$149$	−9.86086	−0.807833	−0.403916	+	0.914796i	$0.632351\pi$
$150$	−0.250412	−0.0204461
$151$	−3.94804	−0.321287	−0.160643	−	0.987013i	$-0.551357\pi$
$151$	−3.94804	−0.321287	−0.160643	+	0.987013i	$0.551357\pi$
$152$	19.8527	1.61027
$153$	−10.6419	−0.860350
$154$	−11.7875	−0.949866
$155$	3.15873	0.253716
$156$	0.176727	0.0141495
$157$	1.12433	0.0897316	0.0448658	−	0.998993i	$-0.485714\pi$
$157$	1.12433	0.0897316	0.0448658	+	0.998993i	$0.485714\pi$
$158$	−6.80515	−0.541389
$159$	0.444884	0.0352816
$160$	−2.67734	−0.211662
$161$	0	0
$162$	10.6213	0.834489
$163$	12.2611	0.960361	0.480180	−	0.877170i	$-0.340571\pi$
$163$	12.2611	0.960361	0.480180	+	0.877170i	$0.340571\pi$
$164$	4.96855	0.387979
$165$	−0.426006	−0.0331645
$166$	−13.4841	−1.04657
$167$	9.28982	0.718868	0.359434	−	0.933171i	$-0.382970\pi$
$167$	9.28982	0.718868	0.359434	+	0.933171i	$0.382970\pi$
$168$	2.84595	0.219570
$169$	−9.79194	−0.753226
$170$	−4.42732	−0.339560
$171$	19.2033	1.46851
$172$	4.64262	0.353997
$173$	−16.5597	−1.25901	−0.629506	−	0.776996i	$-0.716743\pi$
$173$	−16.5597	−1.25901	−0.629506	+	0.776996i	$0.716743\pi$
$174$	−1.90401	−0.144342
$175$	4.57348	0.345723
$176$	5.85219	0.441126
$177$	0.319590	0.0240219
$178$	15.1839	1.13808
$179$	13.1205	0.980671	0.490335	−	0.871534i	$-0.336874\pi$
$179$	13.1205	0.980671	0.490335	+	0.871534i	$0.336874\pi$
$180$	−1.43490	−0.106951
$181$	−16.1850	−1.20302	−0.601511	−	0.798864i	$-0.705434\pi$
$181$	−16.1850	−1.20302	−0.601511	+	0.798864i	$0.705434\pi$
$182$	10.0827	0.747378
$183$	−0.393370	−0.0290788
$184$	0	0
$185$	11.4307	0.840402
$186$	0.790985	0.0579979
$187$	−7.53184	−0.550783
$188$	1.40257	0.102293
$189$	5.54421	0.403282
$190$	7.98905	0.579586
$191$	27.1036	1.96115	0.980573	−	0.196153i	$-0.0628451\pi$
$191$	27.1036	1.96115	0.980573	+	0.196153i	$0.0628451\pi$
$192$	−1.80762	−0.130454
$193$	20.5951	1.48246	0.741232	−	0.671248i	$-0.234242\pi$
$193$	20.5951	1.48246	0.741232	+	0.671248i	$0.234242\pi$
$194$	−7.26646	−0.521701
$195$	0.364392	0.0260947
$196$	−6.74950	−0.482107
$197$	−3.21537	−0.229086	−0.114543	−	0.993418i	$-0.536540\pi$
$197$	−3.21537	−0.229086	−0.114543	+	0.993418i	$0.536540\pi$
$198$	7.62541	0.541914
$199$	−26.4919	−1.87796	−0.938982	−	0.343966i	$-0.888229\pi$
$199$	−26.4919	−1.87796	−0.938982	+	0.343966i	$0.888229\pi$
$200$	−3.05867	−0.216281
$201$	2.15044	0.151681
$202$	−3.66360	−0.257770
$203$	34.7744	2.44069
$204$	0.354907	0.0248485
$205$	10.2446	0.715516
$206$	−0.570834	−0.0397719
$207$	0	0
$208$	−5.00579	−0.347089
$209$	13.5911	0.940117
$210$	1.14526	0.0790302
$211$	−15.3557	−1.05713	−0.528564	−	0.848893i	$-0.677269\pi$
$211$	−15.3557	−1.05713	−0.528564	+	0.848893i	$0.677269\pi$
$212$	1.06055	0.0728392
$213$	1.44775	0.0991983
$214$	0.0164155	0.00112214
$215$	9.57259	0.652845
$216$	−3.70788	−0.252289
$217$	−14.4464	−0.980687
$218$	15.7545	1.06703
$219$	−0.600386	−0.0405703
$220$	−1.01555	−0.0684684
$221$	6.44251	0.433370
$222$	2.86239	0.192111
$223$	−18.2500	−1.22211	−0.611055	−	0.791588i	$-0.709255\pi$
$223$	−18.2500	−1.22211	−0.611055	+	0.791588i	$0.709255\pi$
$224$	12.2448	0.818137
$225$	−2.95861	−0.197241
$226$	−19.4969	−1.29691
$227$	−0.998024	−0.0662411	−0.0331206	−	0.999451i	$-0.510545\pi$
$227$	−0.998024	−0.0662411	−0.0331206	+	0.999451i	$0.510545\pi$
$228$	−0.640426	−0.0424132
$229$	11.4983	0.759827	0.379913	−	0.925022i	$-0.375954\pi$
$229$	11.4983	0.759827	0.379913	+	0.925022i	$0.375954\pi$
$230$	0	0
$231$	1.94833	0.128191
$232$	−23.2566	−1.52687
$233$	−3.85730	−0.252700	−0.126350	−	0.991986i	$-0.540326\pi$
$233$	−3.85730	−0.252700	−0.126350	+	0.991986i	$0.540326\pi$
$234$	−6.52254	−0.426392
$235$	2.89196	0.188651
$236$	0.761867	0.0495933
$237$	1.12481	0.0730641
$238$	20.2483	1.31250
$239$	6.46079	0.417914	0.208957	−	0.977925i	$-0.432993\pi$
$239$	6.46079	0.417914	0.208957	+	0.977925i	$0.432993\pi$
$240$	−0.568590	−0.0367023
$241$	−1.15962	−0.0746980	−0.0373490	−	0.999302i	$-0.511891\pi$
$241$	−1.15962	−0.0746980	−0.0373490	+	0.999302i	$0.511891\pi$
$242$	−8.14254	−0.523423
$243$	−5.39233	−0.345918
$244$	−0.937750	−0.0600333
$245$	−13.9168	−0.889109
$246$	2.56538	0.163563
$247$	−11.6254	−0.739708
$248$	9.66152	0.613507
$249$	2.22876	0.141242
$250$	−1.23086	−0.0778462
$251$	−11.9425	−0.753802	−0.376901	−	0.926254i	$-0.623010\pi$
$251$	−11.9425	−0.753802	−0.376901	+	0.926254i	$0.623010\pi$
$252$	6.56249	0.413398
$253$	0	0
$254$	15.5111	0.973254
$255$	0.731781	0.0458259
$256$	−10.9000	−0.681249
$257$	8.26404	0.515496	0.257748	−	0.966212i	$-0.417019\pi$
$257$	8.26404	0.515496	0.257748	+	0.966212i	$0.417019\pi$
$258$	2.39709	0.149237
$259$	−52.2781	−3.24841
$260$	0.868671	0.0538727
$261$	−22.4958	−1.39245
$262$	−12.9685	−0.801196
$263$	6.94515	0.428256	0.214128	−	0.976806i	$-0.431309\pi$
$263$	6.94515	0.428256	0.214128	+	0.976806i	$0.431309\pi$
$264$	−1.30301	−0.0801948
$265$	2.18675	0.134331
$266$	−36.5378	−2.24027
$267$	−2.50971	−0.153592
$268$	5.12642	0.313146
$269$	15.5928	0.950710	0.475355	−	0.879794i	$-0.342320\pi$
$269$	15.5928	0.950710	0.475355	+	0.879794i	$0.342320\pi$
$270$	−1.49211	−0.0908068
$271$	16.4734	1.00069	0.500344	−	0.865827i	$-0.333207\pi$
$271$	16.4734	1.00069	0.500344	+	0.865827i	$0.333207\pi$
$272$	−10.0527	−0.609537
$273$	−1.66654	−0.100864
$274$	1.75702	0.106146
$275$	−2.09396	−0.126270
$276$	0	0
$277$	3.75737	0.225759	0.112879	−	0.993609i	$-0.463993\pi$
$277$	3.75737	0.225759	0.112879	+	0.993609i	$0.463993\pi$
$278$	0.0118318	0.000709625 0
$279$	9.34546	0.559498
$280$	13.9888	0.835989
$281$	16.6302	0.992077	0.496038	−	0.868301i	$-0.334788\pi$
$281$	16.6302	0.992077	0.496038	+	0.868301i	$0.334788\pi$
$282$	0.724182	0.0431244
$283$	−7.87230	−0.467960	−0.233980	−	0.972241i	$-0.575175\pi$
$283$	−7.87230	−0.467960	−0.233980	+	0.972241i	$0.575175\pi$
$284$	3.45128	0.204796
$285$	−1.32049	−0.0782191
$286$	−4.61633	−0.272969
$287$	−46.8536	−2.76568
$288$	−7.92119	−0.466761
$289$	−4.06201	−0.238942
$290$	−9.35881	−0.549568
$291$	1.20105	0.0704071
$292$	−1.43125	−0.0837577
$293$	−13.1869	−0.770388	−0.385194	−	0.922836i	$-0.625865\pi$
$293$	−13.1869	−0.770388	−0.385194	+	0.922836i	$0.625865\pi$
$294$	−3.48493	−0.203245
$295$	1.57089	0.0914607
$296$	34.9627	2.03217
$297$	−2.53840	−0.147293
$298$	−12.1373	−0.703095
$299$	0	0
$300$	0.0986692	0.00569667
$301$	−43.7801	−2.52344
$302$	−4.85947	−0.279631
$303$	0.605548	0.0347878
$304$	18.1400	1.04040
$305$	−1.93354	−0.110714
$306$	−13.0987	−0.748804
$307$	23.3860	1.33471	0.667355	−	0.744740i	$-0.267426\pi$
$307$	23.3860	1.33471	0.667355	+	0.744740i	$0.267426\pi$
$308$	4.64460	0.264651
$309$	0.0943517	0.00536748
$310$	3.88795	0.220821
$311$	27.1272	1.53824	0.769122	−	0.639102i	$-0.220694\pi$
$311$	27.1272	1.53824	0.769122	+	0.639102i	$0.220694\pi$
$312$	1.11456	0.0630993
$313$	3.72850	0.210748	0.105374	−	0.994433i	$-0.466396\pi$
$313$	3.72850	0.210748	0.105374	+	0.994433i	$0.466396\pi$
$314$	1.38389	0.0780977
$315$	13.5312	0.762394
$316$	2.68141	0.150841
$317$	20.8141	1.16904	0.584518	−	0.811381i	$-0.301284\pi$
$317$	20.8141	1.16904	0.584518	+	0.811381i	$0.301284\pi$
$318$	0.547589	0.0307073
$319$	−15.9214	−0.891426
$320$	−8.88502	−0.496688
$321$	−0.00271328	−0.000151441 0
$322$	0	0
$323$	−23.3464	−1.29903
$324$	−4.18509	−0.232505
$325$	1.79111	0.0993527
$326$	15.0916	0.835848
$327$	−2.60402	−0.144003
$328$	31.3349	1.73018
$329$	−13.2263	−0.729192
$330$	−0.524352	−0.0288646
$331$	19.8169	1.08924	0.544618	−	0.838684i	$-0.316675\pi$
$331$	19.8169	1.08924	0.544618	+	0.838684i	$0.316675\pi$
$332$	5.31312	0.291595
$333$	33.8190	1.85327
$334$	11.4344	0.625665
$335$	10.5701	0.577508
$336$	2.60044	0.141865
$337$	1.83687	0.100061	0.0500304	−	0.998748i	$-0.484068\pi$
$337$	1.83687	0.100061	0.0500304	+	0.998748i	$0.484068\pi$
$338$	−12.0525	−0.655568
$339$	3.22259	0.175027
$340$	1.74448	0.0946079
$341$	6.61425	0.358182
$342$	23.6365	1.27811
$343$	31.6337	1.70806
$344$	29.2794	1.57864
$345$	0	0
$346$	−20.3826	−1.09578
$347$	22.2572	1.19483	0.597415	−	0.801932i	$-0.296195\pi$
$347$	22.2572	1.19483	0.597415	+	0.801932i	$0.296195\pi$
$348$	0.750230	0.0402166
$349$	9.45858	0.506306	0.253153	−	0.967426i	$-0.418532\pi$
$349$	9.45858	0.506306	0.253153	+	0.967426i	$0.418532\pi$
$350$	5.62930	0.300899
$351$	2.17127	0.115894
$352$	−5.60623	−0.298813
$353$	23.3935	1.24511	0.622556	−	0.782576i	$-0.286094\pi$
$353$	23.3935	1.24511	0.622556	+	0.782576i	$0.286094\pi$
$354$	0.393370	0.0209074
$355$	7.11617	0.377687
$356$	−5.98286	−0.317091
$357$	−3.34679	−0.177131
$358$	16.1494	0.853524
$359$	−27.3266	−1.44224	−0.721122	−	0.692808i	$-0.756373\pi$
$359$	−27.3266	−1.44224	−0.721122	+	0.692808i	$0.756373\pi$
$360$	−9.04941	−0.476946
$361$	23.1284	1.21728
$362$	−19.9214	−1.04705
$363$	1.34586	0.0706394
$364$	−3.97285	−0.208234
$365$	−2.95109	−0.154467
$366$	−0.484182	−0.0253086
$367$	2.20386	0.115041	0.0575203	−	0.998344i	$-0.481681\pi$
$367$	2.20386	0.115041	0.0575203	+	0.998344i	$0.481681\pi$
$368$	0	0
$369$	30.3098	1.57787
$370$	14.0696	0.731442
$371$	−10.0011	−0.519230
$372$	−0.311670	−0.0161593
$373$	−20.3814	−1.05531	−0.527655	−	0.849459i	$-0.676928\pi$
$373$	−20.3814	−1.05531	−0.527655	+	0.849459i	$0.676928\pi$
$374$	−9.27062	−0.479372
$375$	0.203445	0.0105059
$376$	8.84555	0.456174
$377$	13.6187	0.701397
$378$	6.82414	0.350996
$379$	−6.17437	−0.317156	−0.158578	−	0.987346i	$-0.550691\pi$
$379$	−6.17437	−0.317156	−0.158578	+	0.987346i	$0.550691\pi$
$380$	−3.14790	−0.161484
$381$	−2.56380	−0.131347
$382$	33.3606	1.70688
$383$	−1.93132	−0.0986858	−0.0493429	−	0.998782i	$-0.515713\pi$
$383$	−1.93132	−0.0986858	−0.0493429	+	0.998782i	$0.515713\pi$
$384$	−1.13553	−0.0579475
$385$	9.57668	0.488073
$386$	25.3496	1.29026
$387$	28.3216	1.43967
$388$	2.86318	0.145356
$389$	9.82584	0.498190	0.249095	−	0.968479i	$-0.419867\pi$
$389$	9.82584	0.498190	0.249095	+	0.968479i	$0.419867\pi$
$390$	0.448515	0.0227114
$391$	0	0
$392$	−42.5667	−2.14995
$393$	2.14353	0.108127
$394$	−3.95766	−0.199384
$395$	5.52879	0.278184
$396$	−3.00462	−0.150988
$397$	−31.6630	−1.58912	−0.794561	−	0.607185i	$-0.792299\pi$
$397$	−31.6630	−1.58912	−0.794561	+	0.607185i	$0.792299\pi$
$398$	−32.6078	−1.63448
$399$	6.03924	0.302340
$400$	−2.79480	−0.139740
$401$	9.59388	0.479095	0.239548	−	0.970885i	$-0.423001\pi$
$401$	9.59388	0.479095	0.239548	+	0.970885i	$0.423001\pi$
$402$	2.64689	0.132015
$403$	−5.65763	−0.281827
$404$	1.44356	0.0718197
$405$	−8.62920	−0.428789
$406$	42.8024	2.12425
$407$	23.9354	1.18643
$408$	2.23828	0.110811
$409$	−24.3656	−1.20480	−0.602402	−	0.798193i	$-0.705789\pi$
$409$	−24.3656	−1.20480	−0.602402	+	0.798193i	$0.705789\pi$
$410$	12.6097	0.622747
$411$	−0.290414	−0.0143251
$412$	0.224924	0.0110812
$413$	−7.18443	−0.353523
$414$	0	0
$415$	10.9551	0.537764
$416$	4.79539	0.235114
$417$	−0.00195565	−9.57688e−5 0
$418$	16.7287	0.818228
$419$	27.1952	1.32857	0.664287	−	0.747478i	$-0.268735\pi$
$419$	27.1952	1.32857	0.664287	+	0.747478i	$0.268735\pi$
$420$	−0.451262	−0.0220193
$421$	1.68301	0.0820249	0.0410125	−	0.999159i	$-0.486942\pi$
$421$	1.68301	0.0820249	0.0410125	+	0.999159i	$0.486942\pi$
$422$	−18.9006	−0.920069
$423$	8.55618	0.416016
$424$	6.68854	0.324824
$425$	3.59694	0.174477
$426$	1.78198	0.0863370
$427$	8.84302	0.427944
$428$	−0.00646817	−0.000312651 0
$429$	0.763022	0.0368391
$430$	11.7825	0.568202
$431$	1.22272	0.0588964	0.0294482	−	0.999566i	$-0.490625\pi$
$431$	1.22272	0.0588964	0.0294482	+	0.999566i	$0.490625\pi$
$432$	−3.38800	−0.163005
$433$	22.5403	1.08322	0.541608	−	0.840631i	$-0.317816\pi$
$433$	22.5403	1.08322	0.541608	+	0.840631i	$0.317816\pi$
$434$	−17.7815	−0.853538
$435$	1.54689	0.0741679
$436$	−6.20769	−0.297295
$437$	0	0
$438$	−0.738989	−0.0353103
$439$	21.4696	1.02469	0.512344	−	0.858780i	$-0.328777\pi$
$439$	21.4696	1.02469	0.512344	+	0.858780i	$0.328777\pi$
$440$	−6.40472	−0.305333
$441$	−41.1743	−1.96068
$442$	7.92981	0.377182
$443$	24.6179	1.16963	0.584816	−	0.811166i	$-0.301167\pi$
$443$	24.6179	1.16963	0.584816	+	0.811166i	$0.301167\pi$
$444$	−1.12786	−0.0535258
$445$	−12.3360	−0.584784
$446$	−22.4631	−1.06366
$447$	2.00615	0.0948875
$448$	40.6355	1.91985
$449$	33.2738	1.57029	0.785144	−	0.619313i	$-0.212589\pi$
$449$	33.2738	1.57029	0.785144	+	0.619313i	$0.212589\pi$
$450$	−3.64163	−0.171668
$451$	21.4518	1.01013
$452$	7.68230	0.361345
$453$	0.803210	0.0377381
$454$	−1.22842	−0.0576528
$455$	−8.19160	−0.384028
$456$	−4.03894	−0.189141
$457$	21.9113	1.02497	0.512483	−	0.858698i	$-0.328726\pi$
$457$	21.9113	1.02497	0.512483	+	0.858698i	$0.328726\pi$
$458$	14.1527	0.661313
$459$	4.36040	0.203526
$460$	0	0
$461$	32.1173	1.49585	0.747925	−	0.663783i	$-0.231050\pi$
$461$	32.1173	1.49585	0.747925	+	0.663783i	$0.231050\pi$
$462$	2.39812	0.111571
$463$	−32.6712	−1.51836	−0.759180	−	0.650881i	$-0.774400\pi$
$463$	−32.6712	−1.51836	−0.759180	+	0.650881i	$0.774400\pi$
$464$	−21.2502	−0.986518
$465$	−0.642630	−0.0298012
$466$	−4.74778	−0.219937
$467$	35.7918	1.65625	0.828124	−	0.560545i	$-0.189408\pi$
$467$	35.7918	1.65625	0.828124	+	0.560545i	$0.189408\pi$
$468$	2.57006	0.118801
$469$	−48.3423	−2.23224
$470$	3.55959	0.164192
$471$	−0.228741	−0.0105398
$472$	4.80483	0.221160
$473$	20.0446	0.921651
$474$	1.38448	0.0635911
$475$	−6.49064	−0.297811
$476$	−7.97837	−0.365688
$477$	6.46974	0.296229
$478$	7.95231	0.363730
$479$	−12.3975	−0.566456	−0.283228	−	0.959053i	$-0.591405\pi$
$479$	−12.3975	−0.566456	−0.283228	+	0.959053i	$0.591405\pi$
$480$	0.544692	0.0248617
$481$	−20.4736	−0.933516
$482$	−1.42733	−0.0650132
$483$	0	0
$484$	3.20838	0.145836
$485$	5.90357	0.268067
$486$	−6.63718	−0.301069
$487$	0.308921	0.0139986	0.00699928	−	0.999976i	$-0.497772\pi$
$487$	0.308921	0.0139986	0.00699928	+	0.999976i	$0.497772\pi$
$488$	−5.91406	−0.267717
$489$	−2.49446	−0.112803
$490$	−17.1295	−0.773834
$491$	−22.5651	−1.01835	−0.509174	−	0.860664i	$-0.670049\pi$
$491$	−22.5651	−1.01835	−0.509174	+	0.860664i	$0.670049\pi$
$492$	−1.01083	−0.0455717
$493$	27.3493	1.23175
$494$	−14.3092	−0.643803
$495$	−6.19520	−0.278454
$496$	8.82803	0.396390
$497$	−32.5457	−1.45987
$498$	2.74329	0.122930
$499$	23.0578	1.03221	0.516103	−	0.856526i	$-0.327382\pi$
$499$	23.0578	1.03221	0.516103	+	0.856526i	$0.327382\pi$
$500$	0.484991	0.0216895
$501$	−1.88997	−0.0844377
$502$	−14.6995	−0.656069
$503$	0.956205	0.0426351	0.0213175	−	0.999773i	$-0.493214\pi$
$503$	0.956205	0.0426351	0.0213175	+	0.999773i	$0.493214\pi$
$504$	41.3873	1.84354
$505$	2.97647	0.132451
$506$	0	0
$507$	1.99212	0.0884734
$508$	−6.11180	−0.271167
$509$	−40.8664	−1.81137	−0.905686	−	0.423950i	$-0.860643\pi$
$509$	−40.8664	−1.81137	−0.905686	+	0.423950i	$0.860643\pi$
$510$	0.900718	0.0398845
$511$	13.4968	0.597062
$512$	−24.5794	−1.08626
$513$	−7.86829	−0.347393
$514$	10.1718	0.448661
$515$	0.463769	0.0204361
$516$	−0.944520	−0.0415802
$517$	6.05564	0.266327
$518$	−64.3469	−2.82724
$519$	3.36900	0.147883
$520$	5.47840	0.240244
$521$	42.5169	1.86270	0.931349	−	0.364128i	$-0.118633\pi$
$521$	42.5169	1.86270	0.931349	+	0.364128i	$0.118633\pi$
$522$	−27.6891	−1.21192
$523$	14.1539	0.618905	0.309452	−	0.950915i	$-0.399854\pi$
$523$	14.1539	0.618905	0.309452	+	0.950915i	$0.399854\pi$
$524$	5.10993	0.223228
$525$	−0.930454	−0.0406084
$526$	8.54849	0.372732
$527$	−11.3618	−0.494927
$528$	−1.19060	−0.0518143
$529$	0	0
$530$	2.69158	0.116915
$531$	4.64765	0.201691
$532$	14.3969	0.624184
$533$	−18.3492	−0.794793
$534$	−3.08909	−0.133678
$535$	−0.0133367	−0.000576595 0
$536$	32.3305	1.39647
$537$	−2.66930	−0.115189
$538$	19.1925	0.827448
$539$	−29.1411	−1.25520
$540$	0.587931	0.0253005
$541$	−20.6627	−0.888359	−0.444180	−	0.895938i	$-0.646505\pi$
$541$	−20.6627	−0.888359	−0.444180	+	0.895938i	$0.646505\pi$
$542$	20.2764	0.870947
$543$	3.29277	0.141306
$544$	9.63022	0.412892
$545$	−12.7996	−0.548275
$546$	−2.05128	−0.0877865
$547$	−22.2896	−0.953033	−0.476516	−	0.879166i	$-0.658101\pi$
$547$	−22.2896	−0.953033	−0.476516	+	0.879166i	$0.658101\pi$
$548$	−0.692315	−0.0295742
$549$	−5.72060	−0.244149
$550$	−2.57736	−0.109899
$551$	−49.3515	−2.10244
$552$	0	0
$553$	−25.2858	−1.07526
$554$	4.62479	0.196488
$555$	−2.32552	−0.0987130
$556$	−0.00466206	−0.000197715 0
$557$	14.7364	0.624403	0.312201	−	0.950016i	$-0.398934\pi$
$557$	14.7364	0.624403	0.312201	+	0.950016i	$0.398934\pi$
$558$	11.5029	0.486958
$559$	−17.1455	−0.725179
$560$	12.7820	0.540137
$561$	1.53232	0.0646945
$562$	20.4694	0.863451
$563$	−0.929704	−0.0391824	−0.0195912	−	0.999808i	$-0.506236\pi$
$563$	−0.929704	−0.0391824	−0.0195912	+	0.999808i	$0.506236\pi$
$564$	−0.285347	−0.0120153
$565$	15.8401	0.666397
$566$	−9.68968	−0.407287
$567$	39.4655	1.65740
$568$	21.7660	0.913281
$569$	−9.93224	−0.416381	−0.208191	−	0.978088i	$-0.566757\pi$
$569$	−9.93224	−0.416381	−0.208191	+	0.978088i	$0.566757\pi$
$570$	−1.62533	−0.0680778
$571$	−44.0179	−1.84209	−0.921045	−	0.389456i	$-0.872663\pi$
$571$	−44.0179	−1.84209	−0.921045	+	0.389456i	$0.872663\pi$
$572$	1.81896	0.0760545
$573$	−5.51410	−0.230355
$574$	−57.6701	−2.40710
$575$	0	0
$576$	−26.2873	−1.09530
$577$	−37.6414	−1.56703	−0.783517	−	0.621371i	$-0.786576\pi$
$577$	−37.6414	−1.56703	−0.783517	+	0.621371i	$0.786576\pi$
$578$	−4.99975	−0.207962
$579$	−4.18997	−0.174129
$580$	3.68762	0.153120
$581$	−50.1029	−2.07862
$582$	1.47833	0.0612786
$583$	4.57896	0.189641
$584$	−9.02641	−0.373516
$585$	5.29919	0.219094
$586$	−16.2312	−0.670505
$587$	−8.24345	−0.340243	−0.170122	−	0.985423i	$-0.554416\pi$
$587$	−8.24345	−0.340243	−0.170122	+	0.985423i	$0.554416\pi$
$588$	1.37316	0.0566280
$589$	20.5022	0.844778
$590$	1.93354	0.0796026
$591$	0.654152	0.0269082
$592$	31.9466	1.31300
$593$	−25.3215	−1.03983	−0.519915	−	0.854218i	$-0.674037\pi$
$593$	−25.3215	−1.03983	−0.519915	+	0.854218i	$0.674037\pi$
$594$	−3.12441	−0.128196
$595$	−16.4506	−0.674407
$596$	4.78243	0.195896
$597$	5.38966	0.220584
$598$	0	0
$599$	−8.10753	−0.331265	−0.165632	−	0.986188i	$-0.552966\pi$
$599$	−8.10753	−0.331265	−0.165632	+	0.986188i	$0.552966\pi$
$600$	0.622272	0.0254042
$601$	4.23039	0.172561	0.0862805	−	0.996271i	$-0.472502\pi$
$601$	4.23039	0.172561	0.0862805	+	0.996271i	$0.472502\pi$
$602$	−53.8870	−2.19627
$603$	31.2729	1.27353
$604$	1.91476	0.0779106
$605$	6.61534	0.268952
$606$	0.745343	0.0302775
$607$	−20.0849	−0.815220	−0.407610	−	0.913156i	$-0.633638\pi$
$607$	−20.0849	−0.815220	−0.407610	+	0.913156i	$0.633638\pi$
$608$	−17.3776	−0.704755
$609$	−7.07470	−0.286681
$610$	−2.37991	−0.0963599
$611$	−5.17981	−0.209553
$612$	5.16125	0.208631
$613$	−12.9615	−0.523509	−0.261754	−	0.965134i	$-0.584301\pi$
$613$	−12.9615	−0.523509	−0.261754	+	0.965134i	$0.584301\pi$
$614$	28.7848	1.16166
$615$	−2.08422	−0.0840439
$616$	29.2919	1.18020
$617$	−27.5402	−1.10873	−0.554364	−	0.832274i	$-0.687038\pi$
$617$	−27.5402	−1.10873	−0.554364	+	0.832274i	$0.687038\pi$
$618$	0.116133	0.00467157
$619$	−36.6272	−1.47217	−0.736085	−	0.676889i	$-0.763328\pi$
$619$	−36.6272	−1.47217	−0.736085	+	0.676889i	$0.763328\pi$
$620$	−1.53196	−0.0615249
$621$	0	0
$622$	33.3897	1.33881
$623$	56.4186	2.26036
$624$	1.01840	0.0407688
$625$	1.00000	0.0400000
$626$	4.58926	0.183424
$627$	−2.76505	−0.110425
$628$	−0.545292	−0.0217595
$629$	−41.1156	−1.63939
$630$	16.6549	0.663548
$631$	−20.2005	−0.804169	−0.402084	−	0.915603i	$-0.631714\pi$
$631$	−20.2005	−0.804169	−0.402084	+	0.915603i	$0.631714\pi$
$632$	16.9107	0.672673
$633$	3.12404	0.124169
$634$	25.6192	1.01747
$635$	−12.6019	−0.500091
$636$	−0.215765	−0.00855563
$637$	24.9264	0.987620
$638$	−19.5969	−0.775850
$639$	21.0540	0.832882
$640$	−5.58152	−0.220629
$641$	21.1800	0.836560	0.418280	−	0.908318i	$-0.362633\pi$
$641$	21.1800	0.836560	0.418280	+	0.908318i	$0.362633\pi$
$642$	−0.00333967	−0.000131806 0
$643$	−29.0613	−1.14606	−0.573032	−	0.819533i	$-0.694233\pi$
$643$	−29.0613	−1.14606	−0.573032	+	0.819533i	$0.694233\pi$
$644$	0	0
$645$	−1.94750	−0.0766827
$646$	−28.7361	−1.13061
$647$	−32.0881	−1.26151	−0.630757	−	0.775980i	$-0.717256\pi$
$647$	−32.0881	−1.26151	−0.630757	+	0.775980i	$0.717256\pi$
$648$	−26.3939	−1.03685
$649$	3.28937	0.129119
$650$	2.20460	0.0864714
$651$	2.93906	0.115191
$652$	−5.94651	−0.232883
$653$	−21.1860	−0.829073	−0.414537	−	0.910033i	$-0.636056\pi$
$653$	−21.1860	−0.829073	−0.414537	+	0.910033i	$0.636056\pi$
$654$	−3.20518	−0.125332
$655$	10.5361	0.411681
$656$	28.6317	1.11788
$657$	−8.73113	−0.340634
$658$	−16.2797	−0.634650
$659$	−28.0287	−1.09184	−0.545922	−	0.837836i	$-0.683821\pi$
$659$	−28.0287	−1.09184	−0.545922	+	0.837836i	$0.683821\pi$
$660$	0.206609	0.00804225
$661$	−43.5169	−1.69261	−0.846306	−	0.532697i	$-0.821179\pi$
$661$	−43.5169	−1.69261	−0.846306	+	0.532697i	$0.821179\pi$
$662$	24.3918	0.948014
$663$	−1.31070	−0.0509033
$664$	33.5080	1.30036
$665$	29.6848	1.15113
$666$	41.6263	1.61299
$667$	0	0
$668$	−4.50548	−0.174322
$669$	3.71288	0.143548
$670$	13.0103	0.502632
$671$	−4.04875	−0.156300
$672$	−2.49114	−0.0960978
$673$	13.3566	0.514858	0.257429	−	0.966297i	$-0.417125\pi$
$673$	13.3566	0.514858	0.257429	+	0.966297i	$0.417125\pi$
$674$	2.26093	0.0870877
$675$	1.21225	0.0466596
$676$	4.74900	0.182654
$677$	2.82051	0.108401	0.0542004	−	0.998530i	$-0.482739\pi$
$677$	2.82051	0.108401	0.0542004	+	0.998530i	$0.482739\pi$
$678$	3.96655	0.152334
$679$	−26.9999	−1.03616
$680$	11.0019	0.421902
$681$	0.203043	0.00778064
$682$	8.14120	0.311743
$683$	19.3823	0.741642	0.370821	−	0.928704i	$-0.379076\pi$
$683$	19.3823	0.741642	0.370821	+	0.928704i	$0.379076\pi$
$684$	−9.31341	−0.356107
$685$	−1.42748	−0.0545412
$686$	38.9365	1.48660
$687$	−2.33927	−0.0892487
$688$	26.7535	1.01997
$689$	−3.91670	−0.149215
$690$	0	0
$691$	8.55512	0.325452	0.162726	−	0.986671i	$-0.447971\pi$
$691$	8.55512	0.325452	0.162726	+	0.986671i	$0.447971\pi$
$692$	8.03131	0.305305
$693$	28.3337	1.07631
$694$	27.3955	1.03992
$695$	−0.00961267	−0.000364629 0
$696$	4.73144	0.179345
$697$	−36.8493	−1.39577
$698$	11.6422	0.440662
$699$	0.784749	0.0296819
$700$	−2.21810	−0.0838362
$701$	−2.52479	−0.0953601	−0.0476800	−	0.998863i	$-0.515183\pi$
$701$	−2.52479	−0.0953601	−0.0476800	+	0.998863i	$0.515183\pi$
$702$	2.67253	0.100868
$703$	74.1925	2.79822
$704$	−18.6049	−0.701197
$705$	−0.588356	−0.0221588
$706$	28.7941	1.08368
$707$	−13.6128	−0.511963
$708$	−0.154998	−0.00582519
$709$	10.6238	0.398985	0.199493	−	0.979899i	$-0.436071\pi$
$709$	10.6238	0.398985	0.199493	+	0.979899i	$0.436071\pi$
$710$	8.75899	0.328719
$711$	16.3575	0.613456
$712$	−37.7318	−1.41406
$713$	0	0
$714$	−4.11942	−0.154165
$715$	3.75050	0.140261
$716$	−6.36331	−0.237808
$717$	−1.31442	−0.0490878
$718$	−33.6352	−1.25525
$719$	0.356375	0.0132905	0.00664527	−	0.999978i	$-0.497885\pi$
$719$	0.356375	0.0132905	0.00664527	+	0.999978i	$0.497885\pi$
$720$	−8.26873	−0.308157
$721$	−2.12104	−0.0789917
$722$	28.4677	1.05946
$723$	0.235920	0.00877397
$724$	7.84958	0.291728
$725$	7.60349	0.282386
$726$	1.65656	0.0614808
$727$	−29.6000	−1.09780	−0.548902	−	0.835887i	$-0.684954\pi$
$727$	−29.6000	−1.09780	−0.548902	+	0.835887i	$0.684954\pi$
$728$	−25.0554	−0.928614
$729$	−24.7906	−0.918169
$730$	−3.63237	−0.134440
$731$	−34.4321	−1.27352
$732$	0.190781	0.00705147
$733$	−31.7442	−1.17250	−0.586250	−	0.810130i	$-0.699396\pi$
$733$	−31.7442	−1.17250	−0.586250	+	0.810130i	$0.699396\pi$
$734$	2.71264	0.100125
$735$	2.83130	0.104434
$736$	0	0
$737$	22.1334	0.815294
$738$	37.3071	1.37329
$739$	41.2958	1.51909	0.759545	−	0.650455i	$-0.225422\pi$
$739$	41.2958	1.51909	0.759545	+	0.650455i	$0.225422\pi$
$740$	−5.54379	−0.203794
$741$	2.36514	0.0868856
$742$	−12.3099	−0.451910
$743$	−52.4946	−1.92584	−0.962919	−	0.269789i	$-0.913046\pi$
$743$	−52.4946	−1.92584	−0.962919	+	0.269789i	$0.913046\pi$
$744$	−1.96559	−0.0720621
$745$	9.86086	0.361274
$746$	−25.0866	−0.918486
$747$	32.4118	1.18589
$748$	3.65287	0.133562
$749$	0.0609951	0.00222871
$750$	0.250412	0.00914376
$751$	10.4277	0.380510	0.190255	−	0.981735i	$-0.439068\pi$
$751$	10.4277	0.380510	0.190255	+	0.981735i	$0.439068\pi$
$752$	8.08246	0.294737
$753$	2.42964	0.0885410
$754$	16.7626	0.610459
$755$	3.94804	0.143684
$756$	−2.68889	−0.0977941
$757$	28.3935	1.03198	0.515989	−	0.856595i	$-0.327424\pi$
$757$	28.3935	1.03198	0.515989	+	0.856595i	$0.327424\pi$
$758$	−7.59977	−0.276036
$759$	0	0
$760$	−19.8527	−0.720133
$761$	10.5399	0.382070	0.191035	−	0.981583i	$-0.438816\pi$
$761$	10.5399	0.382070	0.191035	+	0.981583i	$0.438816\pi$
$762$	−3.15567	−0.114318
$763$	58.5388	2.11925
$764$	−13.1450	−0.475569
$765$	10.6419	0.384760
$766$	−2.37718	−0.0858910
$767$	−2.81363	−0.101594
$768$	2.21755	0.0800191
$769$	53.8169	1.94069	0.970344	−	0.241729i	$-0.0777145\pi$
$769$	53.8169	1.94069	0.970344	+	0.241729i	$0.0777145\pi$
$770$	11.7875	0.424793
$771$	−1.68128	−0.0605498
$772$	−9.98842	−0.359491
$773$	−22.9804	−0.826547	−0.413273	−	0.910607i	$-0.635615\pi$
$773$	−22.9804	−0.826547	−0.413273	+	0.910607i	$0.635615\pi$
$774$	34.8598	1.25301
$775$	−3.15873	−0.113465
$776$	18.0571	0.648211
$777$	10.6357	0.381555
$778$	12.0942	0.433598
$779$	66.4941	2.38240
$780$	−0.176727	−0.00632784
$781$	14.9010	0.533198
$782$	0	0
$783$	9.21734	0.329401
$784$	−38.8946	−1.38909
$785$	−1.12433	−0.0401292
$786$	2.63838	0.0941078
$787$	−29.1543	−1.03924	−0.519619	−	0.854398i	$-0.673926\pi$
$787$	−29.1543	−1.03924	−0.519619	+	0.854398i	$0.673926\pi$
$788$	1.55943	0.0555522
$789$	−1.41296	−0.0503027
$790$	6.80515	0.242117
$791$	−72.4443	−2.57582
$792$	−18.9491	−0.673326
$793$	3.46318	0.122981
$794$	−38.9726	−1.38309
$795$	−0.444884	−0.0157784
$796$	12.8484	0.455398
$797$	16.8257	0.595995	0.297998	−	0.954567i	$-0.403681\pi$
$797$	16.8257	0.595995	0.297998	+	0.954567i	$0.403681\pi$
$798$	7.43344	0.263141
$799$	−10.4022	−0.368004
$800$	2.67734	0.0946581
$801$	−36.4975	−1.28958
$802$	11.8087	0.416979
$803$	−6.17946	−0.218068
$804$	−1.04295	−0.0367819
$805$	0	0
$806$	−6.96373	−0.245287
$807$	−3.17228	−0.111670
$808$	9.10402	0.320278
$809$	8.61770	0.302982	0.151491	−	0.988459i	$-0.451592\pi$
$809$	8.61770	0.302982	0.151491	+	0.988459i	$0.451592\pi$
$810$	−10.6213	−0.373195
$811$	−35.1372	−1.23383	−0.616917	−	0.787028i	$-0.711619\pi$
$811$	−35.1372	−1.23383	−0.616917	+	0.787028i	$0.711619\pi$
$812$	−16.8653	−0.591856
$813$	−3.35144	−0.117540
$814$	29.4611	1.03261
$815$	−12.2611	−0.429486
$816$	2.04518	0.0715957
$817$	62.1322	2.17373
$818$	−29.9906	−1.04860
$819$	−24.2357	−0.846865
$820$	−4.96855	−0.173509
$821$	−35.4744	−1.23807	−0.619033	−	0.785365i	$-0.712475\pi$
$821$	−35.4744	−1.23807	−0.619033	+	0.785365i	$0.712475\pi$
$822$	−0.357458	−0.0124678
$823$	−11.8854	−0.414300	−0.207150	−	0.978309i	$-0.566419\pi$
$823$	−11.8854	−0.414300	−0.207150	+	0.978309i	$0.566419\pi$
$824$	1.41852	0.0494164
$825$	0.426006	0.0148316
$826$	−8.84301	−0.307688
$827$	30.6075	1.06433	0.532163	−	0.846642i	$-0.321379\pi$
$827$	30.6075	1.06433	0.532163	+	0.846642i	$0.321379\pi$
$828$	0	0
$829$	−46.4296	−1.61257	−0.806283	−	0.591529i	$-0.798524\pi$
$829$	−46.4296	−1.61257	−0.806283	+	0.591529i	$0.798524\pi$
$830$	13.4841	0.468042
$831$	−0.764420	−0.0265174
$832$	15.9140	0.551719
$833$	50.0578	1.73440
$834$	−0.00240713	−8.33521e−5 0
$835$	−9.28982	−0.321487
$836$	−6.59157	−0.227974
$837$	−3.82918	−0.132356
$838$	33.4734	1.15632
$839$	15.2467	0.526375	0.263188	−	0.964745i	$-0.415226\pi$
$839$	15.2467	0.526375	0.263188	+	0.964745i	$0.415226\pi$
$840$	−2.84595	−0.0981947
$841$	28.8130	0.993553
$842$	2.07155	0.0713902
$843$	−3.38335	−0.116529
$844$	7.44736	0.256349
$845$	9.79194	0.336853
$846$	10.5314	0.362078
$847$	−30.2552	−1.03958
$848$	6.11153	0.209871
$849$	1.60158	0.0549662
$850$	4.42732	0.151856
$851$	0	0
$852$	−0.702147	−0.0240552
$853$	−16.7582	−0.573789	−0.286895	−	0.957962i	$-0.592623\pi$
$853$	−16.7582	−0.573789	−0.286895	+	0.957962i	$0.592623\pi$
$854$	10.8845	0.372460
$855$	−19.2033	−0.656738
$856$	−0.0407925	−0.00139426
$857$	33.7430	1.15264	0.576319	−	0.817225i	$-0.304489\pi$
$857$	33.7430	1.15264	0.576319	+	0.817225i	$0.304489\pi$
$858$	0.939171	0.0320628
$859$	15.6490	0.533937	0.266969	−	0.963705i	$-0.413978\pi$
$859$	15.6490	0.533937	0.266969	+	0.963705i	$0.413978\pi$
$860$	−4.64262	−0.158312
$861$	9.53216	0.324855
$862$	1.50499	0.0512603
$863$	36.3586	1.23766	0.618831	−	0.785524i	$-0.287607\pi$
$863$	36.3586	1.23766	0.618831	+	0.785524i	$0.287607\pi$
$864$	3.24561	0.110418
$865$	16.5597	0.563047
$866$	27.7438	0.942774
$867$	0.826397	0.0280659
$868$	7.00638	0.237812
$869$	11.5771	0.392725
$870$	1.90401	0.0645519
$871$	−18.9322	−0.641494
$872$	−39.1497	−1.32578
$873$	17.4664	0.591147
$874$	0	0
$875$	−4.57348	−0.154612
$876$	0.291182	0.00983812
$877$	−11.5949	−0.391531	−0.195765	−	0.980651i	$-0.562719\pi$
$877$	−11.5949	−0.391531	−0.195765	+	0.980651i	$0.562719\pi$
$878$	26.4260	0.891835
$879$	2.68282	0.0904892
$880$	−5.85219	−0.197277
$881$	13.2564	0.446618	0.223309	−	0.974748i	$-0.428314\pi$
$881$	13.2564	0.446618	0.223309	+	0.974748i	$0.428314\pi$
$882$	−50.6796	−1.70647
$883$	−17.9625	−0.604487	−0.302244	−	0.953231i	$-0.597736\pi$
$883$	−17.9625	−0.604487	−0.302244	+	0.953231i	$0.597736\pi$
$884$	−3.12456	−0.105090
$885$	−0.319590	−0.0107429
$886$	30.3011	1.01799
$887$	−44.0435	−1.47884	−0.739418	−	0.673247i	$-0.764899\pi$
$887$	−44.0435	−1.47884	−0.739418	+	0.673247i	$0.764899\pi$
$888$	−7.11301	−0.238697
$889$	57.6345	1.93300
$890$	−15.1839	−0.508965
$891$	−18.0692	−0.605340
$892$	8.85109	0.296356
$893$	18.7707	0.628136
$894$	2.46928	0.0825850
$895$	−13.1205	−0.438569
$896$	25.5270	0.852797
$897$	0	0
$898$	40.9553	1.36670
$899$	−24.0174	−0.801025
$900$	1.43490	0.0478300
$901$	−7.86561	−0.262042
$902$	26.4041	0.879160
$903$	8.90686	0.296402
$904$	48.4496	1.61141
$905$	16.1850	0.538008
$906$	0.988637	0.0328453
$907$	−24.6042	−0.816971	−0.408485	−	0.912765i	$-0.633943\pi$
$907$	−24.6042	−0.816971	−0.408485	+	0.912765i	$0.633943\pi$
$908$	0.484032	0.0160632
$909$	8.80620	0.292083
$910$	−10.0827	−0.334238
$911$	−25.5981	−0.848101	−0.424051	−	0.905638i	$-0.639392\pi$
$911$	−25.5981	−0.848101	−0.424051	+	0.905638i	$0.639392\pi$
$912$	−3.69051	−0.122205
$913$	22.9395	0.759186
$914$	26.9696	0.892076
$915$	0.393370	0.0130044
$916$	−5.57656	−0.184255
$917$	−48.1869	−1.59127
$918$	5.36703	0.177138
$919$	−0.198711	−0.00655486	−0.00327743	−	0.999995i	$-0.501043\pi$
$919$	−0.198711	−0.00655486	−0.00327743	+	0.999995i	$0.501043\pi$
$920$	0	0
$921$	−4.75778	−0.156774
$922$	39.5318	1.30191
$923$	−12.7458	−0.419534
$924$	−0.944923	−0.0310857
$925$	−11.4307	−0.375839
$926$	−40.2136	−1.32150
$927$	1.37211	0.0450661
$928$	20.3571	0.668254
$929$	24.4935	0.803606	0.401803	−	0.915726i	$-0.368384\pi$
$929$	24.4935	0.803606	0.401803	+	0.915726i	$0.368384\pi$
$930$	−0.790985	−0.0259374
$931$	−90.3286	−2.96040
$932$	1.87075	0.0612786
$933$	−5.51891	−0.180681
$934$	44.0546	1.44151
$935$	7.53184	0.246318
$936$	16.2085	0.529790
$937$	−6.91953	−0.226051	−0.113026	−	0.993592i	$-0.536054\pi$
$937$	−6.91953	−0.226051	−0.113026	+	0.993592i	$0.536054\pi$
$938$	−59.5025	−1.94282
$939$	−0.758547	−0.0247542
$940$	−1.40257	−0.0457469
$941$	21.2436	0.692521	0.346260	−	0.938138i	$-0.387451\pi$
$941$	21.2436	0.692521	0.346260	+	0.938138i	$0.387451\pi$
$942$	−0.281547	−0.00917330
$943$	0	0
$944$	4.39032	0.142893
$945$	−5.54421	−0.180353
$946$	24.6720	0.802157
$947$	18.9509	0.615822	0.307911	−	0.951415i	$-0.400370\pi$
$947$	18.9509	0.615822	0.307911	+	0.951415i	$0.400370\pi$
$948$	−0.545522	−0.0177177
$949$	5.28572	0.171582
$950$	−7.98905	−0.259199
$951$	−4.23453	−0.137314
$952$	−50.3168	−1.63078
$953$	2.93340	0.0950221	0.0475110	−	0.998871i	$-0.484871\pi$
$953$	2.93340	0.0950221	0.0475110	+	0.998871i	$0.484871\pi$
$954$	7.96333	0.257822
$955$	−27.1036	−0.877051
$956$	−3.13342	−0.101342
$957$	3.23913	0.104706
$958$	−15.2596	−0.493014
$959$	6.52856	0.210818
$960$	1.80762	0.0583406
$961$	−21.0224	−0.678142
$962$	−25.2001	−0.812483
$963$	−0.0394580	−0.00127152
$964$	0.562407	0.0181139
$965$	−20.5951	−0.662978
$966$	0	0
$967$	−32.0254	−1.02987	−0.514934	−	0.857230i	$-0.672184\pi$
$967$	−32.0254	−1.02987	−0.514934	+	0.857230i	$0.672184\pi$
$968$	20.2341	0.650350
$969$	4.74973	0.152583
$970$	7.26646	0.233312
$971$	−35.7888	−1.14852	−0.574258	−	0.818674i	$-0.694709\pi$
$971$	−35.7888	−1.14852	−0.574258	+	0.818674i	$0.694709\pi$
$972$	2.61523	0.0838835
$973$	0.0439634	0.00140940
$974$	0.380238	0.0121836
$975$	−0.364392	−0.0116699
$976$	−5.40387	−0.172974
$977$	−45.1891	−1.44573	−0.722863	−	0.690991i	$-0.757174\pi$
$977$	−45.1891	−1.44573	−0.722863	+	0.690991i	$0.757174\pi$
$978$	−3.07032	−0.0981780
$979$	−25.8311	−0.825566
$980$	6.74950	0.215605
$981$	−37.8690	−1.20907
$982$	−27.7744	−0.886316
$983$	49.1348	1.56716	0.783578	−	0.621293i	$-0.213392\pi$
$983$	49.1348	1.56716	0.783578	+	0.621293i	$0.213392\pi$
$984$	−6.37494	−0.203226
$985$	3.21537	0.102450
$986$	33.6631	1.07205
$987$	2.69084	0.0856503
$988$	5.63823	0.179376
$989$	0	0
$990$	−7.62541	−0.242351
$991$	14.4763	0.459853	0.229927	−	0.973208i	$-0.426151\pi$
$991$	14.4763	0.459853	0.229927	+	0.973208i	$0.426151\pi$
$992$	−8.45699	−0.268510
$993$	−4.03166	−0.127941
$994$	−40.0591	−1.27060
$995$	26.4919	0.839851
$996$	−1.08093	−0.0342506
$997$	48.0660	1.52227	0.761133	−	0.648596i	$-0.224643\pi$
$997$	48.0660	1.52227	0.761133	+	0.648596i	$0.224643\pi$
$998$	28.3808	0.898378
$999$	−13.8569	−0.438413

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2645.2.a.v.1.10	✓	16
23.22	odd	2		2645.2.a.w.1.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2645.2.a.v.1.10	✓	16	1.1	even	1	trivial
2645.2.a.w.1.10	yes	16	23.22	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2645 = 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2645.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23086 q^{2} -0.203445 q^{3} -0.484991 q^{4} -1.00000 q^{5} -0.250412 q^{6} +4.57348 q^{7} -3.05867 q^{8} -2.95861 q^{9} +O(q^{10})\)	\(q+1.23086 q^{2} -0.203445 q^{3} -0.484991 q^{4} -1.00000 q^{5} -0.250412 q^{6} +4.57348 q^{7} -3.05867 q^{8} -2.95861 q^{9} -1.23086 q^{10} -2.09396 q^{11} +0.0986692 q^{12} +1.79111 q^{13} +5.62930 q^{14} +0.203445 q^{15} -2.79480 q^{16} +3.59694 q^{17} -3.64163 q^{18} -6.49064 q^{19} +0.484991 q^{20} -0.930454 q^{21} -2.57736 q^{22} +0.622272 q^{24} +1.00000 q^{25} +2.20460 q^{26} +1.21225 q^{27} -2.21810 q^{28} +7.60349 q^{29} +0.250412 q^{30} -3.15873 q^{31} +2.67734 q^{32} +0.426006 q^{33} +4.42732 q^{34} -4.57348 q^{35} +1.43490 q^{36} -11.4307 q^{37} -7.98905 q^{38} -0.364392 q^{39} +3.05867 q^{40} -10.2446 q^{41} -1.14526 q^{42} -9.57259 q^{43} +1.01555 q^{44} +2.95861 q^{45} -2.89196 q^{47} +0.568590 q^{48} +13.9168 q^{49} +1.23086 q^{50} -0.731781 q^{51} -0.868671 q^{52} -2.18675 q^{53} +1.49211 q^{54} +2.09396 q^{55} -13.9888 q^{56} +1.32049 q^{57} +9.35881 q^{58} -1.57089 q^{59} -0.0986692 q^{60} +1.93354 q^{61} -3.88795 q^{62} -13.5312 q^{63} +8.88502 q^{64} -1.79111 q^{65} +0.524352 q^{66} -10.5701 q^{67} -1.74448 q^{68} -5.62930 q^{70} -7.11617 q^{71} +9.04941 q^{72} +2.95109 q^{73} -14.0696 q^{74} -0.203445 q^{75} +3.14790 q^{76} -9.57668 q^{77} -0.448515 q^{78} -5.52879 q^{79} +2.79480 q^{80} +8.62920 q^{81} -12.6097 q^{82} -10.9551 q^{83} +0.451262 q^{84} -3.59694 q^{85} -11.7825 q^{86} -1.54689 q^{87} +6.40472 q^{88} +12.3360 q^{89} +3.64163 q^{90} +8.19160 q^{91} +0.642630 q^{93} -3.55959 q^{94} +6.49064 q^{95} -0.544692 q^{96} -5.90357 q^{97} +17.1295 q^{98} +6.19520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9}+O(q^{10}) \) 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9	\( 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9} - 4 q^{10} - 16 q^{11} - 4 q^{12} + 8 q^{13} - 8 q^{14} - 4 q^{15} + 28 q^{16} + 8 q^{17} + 4 q^{18} - 32 q^{19} - 20 q^{20} - 16 q^{21} - 44 q^{22} - 28 q^{24} + 16 q^{25} - 20 q^{26} - 8 q^{27} - 52 q^{28} - 12 q^{29} + 12 q^{30} - 12 q^{31} + 4 q^{32} - 20 q^{33} - 24 q^{34} + 12 q^{35} - 4 q^{36} - 28 q^{37} + 20 q^{38} - 8 q^{39} - 4 q^{41} + 8 q^{42} - 48 q^{43} - 32 q^{44} - 8 q^{45} + 48 q^{47} + 24 q^{48} + 12 q^{49} + 4 q^{50} - 24 q^{51} + 12 q^{52} + 4 q^{53} - 44 q^{54} + 16 q^{55} - 64 q^{56} - 52 q^{57} - 28 q^{58} - 40 q^{59} + 4 q^{60} - 16 q^{61} - 4 q^{63} - 16 q^{64} - 8 q^{65} - 8 q^{66} - 68 q^{67} + 4 q^{68} + 8 q^{70} - 24 q^{71} - 12 q^{72} + 52 q^{73} - 40 q^{74} + 4 q^{75} - 24 q^{76} + 44 q^{77} + 12 q^{78} - 72 q^{79} - 28 q^{80} + 20 q^{81} - 20 q^{82} - 12 q^{83} - 32 q^{84} - 8 q^{85} + 56 q^{86} - 104 q^{88} - 48 q^{89} - 4 q^{90} - 48 q^{91} - 48 q^{93} - 60 q^{94} + 32 q^{95} - 108 q^{96} - 4 q^{97} + 8 q^{98} - 24 q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9 - 4 * q^10 - 16 * q^11 - 4 * q^12 + 8 * q^13 - 8 * q^14 - 4 * q^15 + 28 * q^16 + 8 * q^17 + 4 * q^18 - 32 * q^19 - 20 * q^20 - 16 * q^21 - 44 * q^22 - 28 * q^24 + 16 * q^25 - 20 * q^26 - 8 * q^27 - 52 * q^28 - 12 * q^29 + 12 * q^30 - 12 * q^31 + 4 * q^32 - 20 * q^33 - 24 * q^34 + 12 * q^35 - 4 * q^36 - 28 * q^37 + 20 * q^38 - 8 * q^39 - 4 * q^41 + 8 * q^42 - 48 * q^43 - 32 * q^44 - 8 * q^45 + 48 * q^47 + 24 * q^48 + 12 * q^49 + 4 * q^50 - 24 * q^51 + 12 * q^52 + 4 * q^53 - 44 * q^54 + 16 * q^55 - 64 * q^56 - 52 * q^57 - 28 * q^58 - 40 * q^59 + 4 * q^60 - 16 * q^61 - 4 * q^63 - 16 * q^64 - 8 * q^65 - 8 * q^66 - 68 * q^67 + 4 * q^68 + 8 * q^70 - 24 * q^71 - 12 * q^72 + 52 * q^73 - 40 * q^74 + 4 * q^75 - 24 * q^76 + 44 * q^77 + 12 * q^78 - 72 * q^79 - 28 * q^80 + 20 * q^81 - 20 * q^82 - 12 * q^83 - 32 * q^84 - 8 * q^85 + 56 * q^86 - 104 * q^88 - 48 * q^89 - 4 * q^90 - 48 * q^91 - 48 * q^93 - 60 * q^94 + 32 * q^95 - 108 * q^96 - 4 * q^97 + 8 * q^98 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more