LMFDB - Embedded newform 4114.2.a.bg.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4114,2,Mod(1,4114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4114, 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("4114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4114 = 2 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4114.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.8504553916$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 28x^{5} + 51x^{4} - 80x^{3} - 92x^{2} + 67x + 59 $ x^8 - 3x^7 - 12x^6 + 28x^5 + 51x^4 - 80x^3 - 92x^2 + 67*x + 59
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 374)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.22721$ of defining polynomial
Character	$\chi$	$=$	4114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.227215 q^{3} +1.00000 q^{4} -3.76434 q^{5} -0.227215 q^{6} -2.36764 q^{7} -1.00000 q^{8} -2.94837 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.227215 q^{3} +1.00000 q^{4} -3.76434 q^{5} -0.227215 q^{6} -2.36764 q^{7} -1.00000 q^{8} -2.94837 q^{9} +3.76434 q^{10} +0.227215 q^{12} +2.52727 q^{13} +2.36764 q^{14} -0.855313 q^{15} +1.00000 q^{16} +1.00000 q^{17} +2.94837 q^{18} +7.29357 q^{19} -3.76434 q^{20} -0.537963 q^{21} -1.09577 q^{23} -0.227215 q^{24} +9.17024 q^{25} -2.52727 q^{26} -1.35156 q^{27} -2.36764 q^{28} -10.6985 q^{29} +0.855313 q^{30} +9.45428 q^{31} -1.00000 q^{32} -1.00000 q^{34} +8.91260 q^{35} -2.94837 q^{36} +5.97864 q^{37} -7.29357 q^{38} +0.574233 q^{39} +3.76434 q^{40} -2.21430 q^{41} +0.537963 q^{42} -4.57948 q^{43} +11.0987 q^{45} +1.09577 q^{46} -2.17774 q^{47} +0.227215 q^{48} -1.39428 q^{49} -9.17024 q^{50} +0.227215 q^{51} +2.52727 q^{52} +3.87457 q^{53} +1.35156 q^{54} +2.36764 q^{56} +1.65721 q^{57} +10.6985 q^{58} +11.1526 q^{59} -0.855313 q^{60} -1.21781 q^{61} -9.45428 q^{62} +6.98069 q^{63} +1.00000 q^{64} -9.51350 q^{65} +5.55004 q^{67} +1.00000 q^{68} -0.248974 q^{69} -8.91260 q^{70} +7.65607 q^{71} +2.94837 q^{72} +10.8252 q^{73} -5.97864 q^{74} +2.08361 q^{75} +7.29357 q^{76} -0.574233 q^{78} -2.91706 q^{79} -3.76434 q^{80} +8.53803 q^{81} +2.21430 q^{82} -7.45828 q^{83} -0.537963 q^{84} -3.76434 q^{85} +4.57948 q^{86} -2.43086 q^{87} -6.13086 q^{89} -11.0987 q^{90} -5.98367 q^{91} -1.09577 q^{92} +2.14815 q^{93} +2.17774 q^{94} -27.4555 q^{95} -0.227215 q^{96} -19.5462 q^{97} +1.39428 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.227215	0.131182	0.0655912	−	0.997847i	$-0.479107\pi$
$3$	0.227215	0.131182	0.0655912	+	0.997847i	$0.479107\pi$
$4$	1.00000	0.500000
$5$	−3.76434	−1.68346	−0.841732	−	0.539896i	$-0.818463\pi$
$5$	−3.76434	−1.68346	−0.841732	+	0.539896i	$0.818463\pi$
$6$	−0.227215	−0.0927600
$7$	−2.36764	−0.894884	−0.447442	−	0.894313i	$-0.647665\pi$
$7$	−2.36764	−0.894884	−0.447442	+	0.894313i	$0.647665\pi$
$8$	−1.00000	−0.353553
$9$	−2.94837	−0.982791
$10$	3.76434	1.19039
$11$	0	0
$11$	0	0
$12$	0.227215	0.0655912
$13$	2.52727	0.700939	0.350469	−	0.936574i	$-0.386022\pi$
$13$	2.52727	0.700939	0.350469	+	0.936574i	$0.386022\pi$
$14$	2.36764	0.632779
$15$	−0.855313	−0.220841
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.94837	0.694938
$19$	7.29357	1.67326	0.836630	−	0.547769i	$-0.184523\pi$
$19$	7.29357	1.67326	0.836630	+	0.547769i	$0.184523\pi$
$20$	−3.76434	−0.841732
$21$	−0.537963	−0.117393
$22$	0	0
$23$	−1.09577	−0.228483	−0.114241	−	0.993453i	$-0.536444\pi$
$23$	−1.09577	−0.228483	−0.114241	+	0.993453i	$0.536444\pi$
$24$	−0.227215	−0.0463800
$25$	9.17024	1.83405
$26$	−2.52727	−0.495639
$27$	−1.35156	−0.260107
$28$	−2.36764	−0.447442
$29$	−10.6985	−1.98666	−0.993332	−	0.115291i	$-0.963220\pi$
$29$	−10.6985	−1.98666	−0.993332	+	0.115291i	$0.963220\pi$
$30$	0.855313	0.156158
$31$	9.45428	1.69804	0.849020	−	0.528361i	$-0.177193\pi$
$31$	9.45428	1.69804	0.849020	+	0.528361i	$0.177193\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	8.91260	1.50650
$36$	−2.94837	−0.491396
$37$	5.97864	0.982882	0.491441	−	0.870911i	$-0.336470\pi$
$37$	5.97864	0.982882	0.491441	+	0.870911i	$0.336470\pi$
$38$	−7.29357	−1.18317
$39$	0.574233	0.0919509
$40$	3.76434	0.595194
$41$	−2.21430	−0.345816	−0.172908	−	0.984938i	$-0.555316\pi$
$41$	−2.21430	−0.345816	−0.172908	+	0.984938i	$0.555316\pi$
$42$	0.537963	0.0830095
$43$	−4.57948	−0.698364	−0.349182	−	0.937055i	$-0.613541\pi$
$43$	−4.57948	−0.698364	−0.349182	+	0.937055i	$0.613541\pi$
$44$	0	0
$45$	11.0987	1.65449
$46$	1.09577	0.161562
$47$	−2.17774	−0.317656	−0.158828	−	0.987306i	$-0.550772\pi$
$47$	−2.17774	−0.317656	−0.158828	+	0.987306i	$0.550772\pi$
$48$	0.227215	0.0327956
$49$	−1.39428	−0.199182
$50$	−9.17024	−1.29687
$51$	0.227215	0.0318164
$52$	2.52727	0.350469
$53$	3.87457	0.532213	0.266107	−	0.963944i	$-0.414263\pi$
$53$	3.87457	0.532213	0.266107	+	0.963944i	$0.414263\pi$
$54$	1.35156	0.183924
$55$	0	0
$56$	2.36764	0.316389
$57$	1.65721	0.219502
$58$	10.6985	1.40478
$59$	11.1526	1.45194	0.725972	−	0.687724i	$-0.241390\pi$
$59$	11.1526	1.45194	0.725972	+	0.687724i	$0.241390\pi$
$60$	−0.855313	−0.110420
$61$	−1.21781	−0.155925	−0.0779623	−	0.996956i	$-0.524841\pi$
$61$	−1.21781	−0.155925	−0.0779623	+	0.996956i	$0.524841\pi$
$62$	−9.45428	−1.20070
$63$	6.98069	0.879484
$64$	1.00000	0.125000
$65$	−9.51350	−1.18000
$66$	0	0
$67$	5.55004	0.678045	0.339022	−	0.940778i	$-0.389904\pi$
$67$	5.55004	0.678045	0.339022	+	0.940778i	$0.389904\pi$
$68$	1.00000	0.121268
$69$	−0.248974	−0.0299730
$70$	−8.91260	−1.06526
$71$	7.65607	0.908608	0.454304	−	0.890847i	$-0.349888\pi$
$71$	7.65607	0.908608	0.454304	+	0.890847i	$0.349888\pi$
$72$	2.94837	0.347469
$73$	10.8252	1.26700	0.633498	−	0.773744i	$-0.281619\pi$
$73$	10.8252	1.26700	0.633498	+	0.773744i	$0.281619\pi$
$74$	−5.97864	−0.695003
$75$	2.08361	0.240595
$76$	7.29357	0.836630
$77$	0	0
$78$	−0.574233	−0.0650191
$79$	−2.91706	−0.328195	−0.164098	−	0.986444i	$-0.552471\pi$
$79$	−2.91706	−0.328195	−0.164098	+	0.986444i	$0.552471\pi$
$80$	−3.76434	−0.420866
$81$	8.53803	0.948670
$82$	2.21430	0.244529
$83$	−7.45828	−0.818653	−0.409326	−	0.912388i	$-0.634236\pi$
$83$	−7.45828	−0.818653	−0.409326	+	0.912388i	$0.634236\pi$
$84$	−0.537963	−0.0586966
$85$	−3.76434	−0.408300
$86$	4.57948	0.493818
$87$	−2.43086	−0.260615
$88$	0	0
$89$	−6.13086	−0.649869	−0.324935	−	0.945736i	$-0.605342\pi$
$89$	−6.13086	−0.649869	−0.324935	+	0.945736i	$0.605342\pi$
$90$	−11.0987	−1.16990
$91$	−5.98367	−0.627259
$92$	−1.09577	−0.114241
$93$	2.14815	0.222753
$94$	2.17774	0.224617
$95$	−27.4555	−2.81687
$96$	−0.227215	−0.0231900
$97$	−19.5462	−1.98462	−0.992309	−	0.123787i	$-0.960496\pi$
$97$	−19.5462	−1.98462	−0.992309	+	0.123787i	$0.960496\pi$
$98$	1.39428	0.140843
$99$	0	0
$100$	9.17024	0.917024
$101$	−12.8408	−1.27770	−0.638851	−	0.769330i	$-0.720590\pi$
$101$	−12.8408	−1.27770	−0.638851	+	0.769330i	$0.720590\pi$
$102$	−0.227215	−0.0224976
$103$	−7.39712	−0.728860	−0.364430	−	0.931231i	$-0.618736\pi$
$103$	−7.39712	−0.728860	−0.364430	+	0.931231i	$0.618736\pi$
$104$	−2.52727	−0.247819
$105$	2.02507	0.197627
$106$	−3.87457	−0.376332
$107$	18.2495	1.76425	0.882124	−	0.471017i	$-0.156113\pi$
$107$	18.2495	1.76425	0.882124	+	0.471017i	$0.156113\pi$
$108$	−1.35156	−0.130054
$109$	−2.62526	−0.251454	−0.125727	−	0.992065i	$-0.540126\pi$
$109$	−2.62526	−0.251454	−0.125727	+	0.992065i	$0.540126\pi$
$110$	0	0
$111$	1.35844	0.128937
$112$	−2.36764	−0.223721
$113$	13.2038	1.24211	0.621057	−	0.783766i	$-0.286704\pi$
$113$	13.2038	1.24211	0.621057	+	0.783766i	$0.286704\pi$
$114$	−1.65721	−0.155212
$115$	4.12483	0.384643
$116$	−10.6985	−0.993332
$117$	−7.45134	−0.688876
$118$	−11.1526	−1.02668
$119$	−2.36764	−0.217041
$120$	0.855313	0.0780790
$121$	0	0
$122$	1.21781	0.110255
$123$	−0.503122	−0.0453650
$124$	9.45428	0.849020
$125$	−15.6982	−1.40409
$126$	−6.98069	−0.621889
$127$	8.88945	0.788811	0.394406	−	0.918936i	$-0.370950\pi$
$127$	8.88945	0.788811	0.394406	+	0.918936i	$0.370950\pi$
$128$	−1.00000	−0.0883883
$129$	−1.04053	−0.0916131
$130$	9.51350	0.834389
$131$	−10.2778	−0.897978	−0.448989	−	0.893537i	$-0.648216\pi$
$131$	−10.2778	−0.897978	−0.448989	+	0.893537i	$0.648216\pi$
$132$	0	0
$133$	−17.2686	−1.49737
$134$	−5.55004	−0.479450
$135$	5.08772	0.437881
$136$	−1.00000	−0.0857493
$137$	−0.998749	−0.0853288	−0.0426644	−	0.999089i	$-0.513585\pi$
$137$	−0.998749	−0.0853288	−0.0426644	+	0.999089i	$0.513585\pi$
$138$	0.248974	0.0211941
$139$	1.58592	0.134516	0.0672578	−	0.997736i	$-0.478575\pi$
$139$	1.58592	0.134516	0.0672578	+	0.997736i	$0.478575\pi$
$140$	8.91260	0.753252
$141$	−0.494815	−0.0416709
$142$	−7.65607	−0.642483
$143$	0	0
$144$	−2.94837	−0.245698
$145$	40.2728	3.34448
$146$	−10.8252	−0.895901
$147$	−0.316800	−0.0261292
$148$	5.97864	0.491441
$149$	−18.3858	−1.50622	−0.753112	−	0.657892i	$-0.771448\pi$
$149$	−18.3858	−1.50622	−0.753112	+	0.657892i	$0.771448\pi$
$150$	−2.08361	−0.170126
$151$	−12.0778	−0.982876	−0.491438	−	0.870913i	$-0.663529\pi$
$151$	−12.0778	−0.982876	−0.491438	+	0.870913i	$0.663529\pi$
$152$	−7.29357	−0.591587
$153$	−2.94837	−0.238362
$154$	0	0
$155$	−35.5891	−2.85859
$156$	0.574233	0.0459754
$157$	−7.61868	−0.608037	−0.304018	−	0.952666i	$-0.598328\pi$
$157$	−7.61868	−0.608037	−0.304018	+	0.952666i	$0.598328\pi$
$158$	2.91706	0.232069
$159$	0.880359	0.0698170
$160$	3.76434	0.297597
$161$	2.59438	0.204466
$162$	−8.53803	−0.670811
$163$	0.0528140	0.00413671	0.00206835	−	0.999998i	$-0.499342\pi$
$163$	0.0528140	0.00413671	0.00206835	+	0.999998i	$0.499342\pi$
$164$	−2.21430	−0.172908
$165$	0	0
$166$	7.45828	0.578875
$167$	−19.3960	−1.50091	−0.750455	−	0.660921i	$-0.770166\pi$
$167$	−19.3960	−1.50091	−0.750455	+	0.660921i	$0.770166\pi$
$168$	0.537963	0.0415047
$169$	−6.61290	−0.508685
$170$	3.76434	0.288712
$171$	−21.5042	−1.64446
$172$	−4.57948	−0.349182
$173$	0.493499	0.0375200	0.0187600	−	0.999824i	$-0.494028\pi$
$173$	0.493499	0.0375200	0.0187600	+	0.999824i	$0.494028\pi$
$174$	2.43086	0.184283
$175$	−21.7118	−1.64126
$176$	0	0
$177$	2.53403	0.190470
$178$	6.13086	0.459527
$179$	6.59389	0.492851	0.246425	−	0.969162i	$-0.420744\pi$
$179$	6.59389	0.492851	0.246425	+	0.969162i	$0.420744\pi$
$180$	11.0987	0.827246
$181$	−13.5189	−1.00485	−0.502427	−	0.864619i	$-0.667560\pi$
$181$	−13.5189	−1.00485	−0.502427	+	0.864619i	$0.667560\pi$
$182$	5.98367	0.443539
$183$	−0.276704	−0.0204546
$184$	1.09577	0.0807809
$185$	−22.5056	−1.65465
$186$	−2.14815	−0.157510
$187$	0	0
$188$	−2.17774	−0.158828
$189$	3.20000	0.232766
$190$	27.4555	1.99183
$191$	7.00508	0.506870	0.253435	−	0.967352i	$-0.418440\pi$
$191$	7.00508	0.506870	0.253435	+	0.967352i	$0.418440\pi$
$192$	0.227215	0.0163978
$193$	−12.6669	−0.911786	−0.455893	−	0.890035i	$-0.650680\pi$
$193$	−12.6669	−0.911786	−0.455893	+	0.890035i	$0.650680\pi$
$194$	19.5462	1.40334
$195$	−2.16161	−0.154796
$196$	−1.39428	−0.0995911
$197$	−4.74061	−0.337755	−0.168877	−	0.985637i	$-0.554014\pi$
$197$	−4.74061	−0.337755	−0.168877	+	0.985637i	$0.554014\pi$
$198$	0	0
$199$	−9.53658	−0.676030	−0.338015	−	0.941141i	$-0.609755\pi$
$199$	−9.53658	−0.676030	−0.338015	+	0.941141i	$0.609755\pi$
$200$	−9.17024	−0.648434
$201$	1.26105	0.0889476
$202$	12.8408	0.903472
$203$	25.3302	1.77783
$204$	0.227215	0.0159082
$205$	8.33538	0.582168
$206$	7.39712	0.515382
$207$	3.23073	0.224551
$208$	2.52727	0.175235
$209$	0	0
$210$	−2.02507	−0.139743
$211$	−23.6865	−1.63065	−0.815324	−	0.579005i	$-0.803441\pi$
$211$	−23.6865	−1.63065	−0.815324	+	0.579005i	$0.803441\pi$
$212$	3.87457	0.266107
$213$	1.73957	0.119193
$214$	−18.2495	−1.24751
$215$	17.2387	1.17567
$216$	1.35156	0.0919619
$217$	−22.3844	−1.51955
$218$	2.62526	0.177805
$219$	2.45965	0.166208
$220$	0	0
$221$	2.52727	0.170003
$222$	−1.35844	−0.0911722
$223$	21.9765	1.47166	0.735828	−	0.677168i	$-0.236793\pi$
$223$	21.9765	1.47166	0.735828	+	0.677168i	$0.236793\pi$
$224$	2.36764	0.158195
$225$	−27.0373	−1.80249
$226$	−13.2038	−0.878307
$227$	−13.8802	−0.921259	−0.460629	−	0.887593i	$-0.652376\pi$
$227$	−13.8802	−0.921259	−0.460629	+	0.887593i	$0.652376\pi$
$228$	1.65721	0.109751
$229$	1.15349	0.0762249	0.0381124	−	0.999273i	$-0.487865\pi$
$229$	1.15349	0.0762249	0.0381124	+	0.999273i	$0.487865\pi$
$230$	−4.12483	−0.271983
$231$	0	0
$232$	10.6985	0.702392
$233$	−12.5295	−0.820835	−0.410418	−	0.911898i	$-0.634617\pi$
$233$	−12.5295	−0.820835	−0.410418	+	0.911898i	$0.634617\pi$
$234$	7.45134	0.487109
$235$	8.19775	0.534762
$236$	11.1526	0.725972
$237$	−0.662800	−0.0430535
$238$	2.36764	0.153471
$239$	18.4473	1.19326	0.596628	−	0.802518i	$-0.296507\pi$
$239$	18.4473	1.19326	0.596628	+	0.802518i	$0.296507\pi$
$240$	−0.855313	−0.0552102
$241$	−5.79907	−0.373551	−0.186776	−	0.982403i	$-0.559804\pi$
$241$	−5.79907	−0.373551	−0.186776	+	0.982403i	$0.559804\pi$
$242$	0	0
$243$	5.99464	0.384556
$244$	−1.21781	−0.0779623
$245$	5.24852	0.335316
$246$	0.503122	0.0320779
$247$	18.4328	1.17285
$248$	−9.45428	−0.600348
$249$	−1.69463	−0.107393
$250$	15.6982	0.992841
$251$	6.40289	0.404147	0.202073	−	0.979370i	$-0.435232\pi$
$251$	6.40289	0.404147	0.202073	+	0.979370i	$0.435232\pi$
$252$	6.98069	0.439742
$253$	0	0
$254$	−8.88945	−0.557774
$255$	−0.855313	−0.0535618
$256$	1.00000	0.0625000
$257$	20.8977	1.30356	0.651781	−	0.758407i	$-0.274022\pi$
$257$	20.8977	1.30356	0.651781	+	0.758407i	$0.274022\pi$
$258$	1.04053	0.0647803
$259$	−14.1553	−0.879566
$260$	−9.51350	−0.590002
$261$	31.5432	1.95248
$262$	10.2778	0.634966
$263$	−10.8761	−0.670650	−0.335325	−	0.942102i	$-0.608846\pi$
$263$	−10.8761	−0.670650	−0.335325	+	0.942102i	$0.608846\pi$
$264$	0	0
$265$	−14.5852	−0.895961
$266$	17.2686	1.05880
$267$	−1.39302	−0.0852515
$268$	5.55004	0.339022
$269$	−18.4843	−1.12701	−0.563503	−	0.826114i	$-0.690547\pi$
$269$	−18.4843	−1.12701	−0.563503	+	0.826114i	$0.690547\pi$
$270$	−5.08772	−0.309629
$271$	−4.10367	−0.249280	−0.124640	−	0.992202i	$-0.539778\pi$
$271$	−4.10367	−0.249280	−0.124640	+	0.992202i	$0.539778\pi$
$272$	1.00000	0.0606339
$273$	−1.35958	−0.0822854
$274$	0.998749	0.0603366
$275$	0	0
$276$	−0.248974	−0.0149865
$277$	6.83580	0.410724	0.205362	−	0.978686i	$-0.434163\pi$
$277$	6.83580	0.410724	0.205362	+	0.978686i	$0.434163\pi$
$278$	−1.58592	−0.0951169
$279$	−27.8748	−1.66882
$280$	−8.91260	−0.532630
$281$	28.9642	1.72786	0.863929	−	0.503614i	$-0.167997\pi$
$281$	28.9642	1.72786	0.863929	+	0.503614i	$0.167997\pi$
$282$	0.494815	0.0294658
$283$	−14.2797	−0.848842	−0.424421	−	0.905465i	$-0.639522\pi$
$283$	−14.2797	−0.848842	−0.424421	+	0.905465i	$0.639522\pi$
$284$	7.65607	0.454304
$285$	−6.23828	−0.369524
$286$	0	0
$287$	5.24267	0.309465
$288$	2.94837	0.173735
$289$	1.00000	0.0588235
$290$	−40.2728	−2.36490
$291$	−4.44119	−0.260347
$292$	10.8252	0.633498
$293$	16.7036	0.975832	0.487916	−	0.872891i	$-0.337757\pi$
$293$	16.7036	0.975832	0.487916	+	0.872891i	$0.337757\pi$
$294$	0.316800	0.0184761
$295$	−41.9821	−2.44430
$296$	−5.97864	−0.347501
$297$	0	0
$298$	18.3858	1.06506
$299$	−2.76930	−0.160153
$300$	2.08361	0.120297
$301$	10.8426	0.624955
$302$	12.0778	0.694998
$303$	−2.91761	−0.167612
$304$	7.29357	0.418315
$305$	4.58425	0.262493
$306$	2.94837	0.168547
$307$	−1.96030	−0.111880	−0.0559402	−	0.998434i	$-0.517816\pi$
$307$	−1.96030	−0.111880	−0.0559402	+	0.998434i	$0.517816\pi$
$308$	0	0
$309$	−1.68073	−0.0956137
$310$	35.5891	2.02133
$311$	18.3723	1.04180	0.520898	−	0.853619i	$-0.325597\pi$
$311$	18.3723	1.04180	0.520898	+	0.853619i	$0.325597\pi$
$312$	−0.574233	−0.0325095
$313$	−22.8404	−1.29102	−0.645508	−	0.763753i	$-0.723354\pi$
$313$	−22.8404	−1.29102	−0.645508	+	0.763753i	$0.723354\pi$
$314$	7.61868	0.429947
$315$	−26.2777	−1.48058
$316$	−2.91706	−0.164098
$317$	−25.2240	−1.41672	−0.708360	−	0.705851i	$-0.750565\pi$
$317$	−25.2240	−1.41672	−0.708360	+	0.705851i	$0.750565\pi$
$318$	−0.880359	−0.0493681
$319$	0	0
$320$	−3.76434	−0.210433
$321$	4.14656	0.231439
$322$	−2.59438	−0.144579
$323$	7.29357	0.405825
$324$	8.53803	0.474335
$325$	23.1757	1.28556
$326$	−0.0528140	−0.00292509
$327$	−0.596497	−0.0329864
$328$	2.21430	0.122264
$329$	5.15611	0.284265
$330$	0	0
$331$	25.3774	1.39487	0.697435	−	0.716648i	$-0.254324\pi$
$331$	25.3774	1.39487	0.697435	+	0.716648i	$0.254324\pi$
$332$	−7.45828	−0.409326
$333$	−17.6273	−0.965968
$334$	19.3960	1.06130
$335$	−20.8922	−1.14146
$336$	−0.537963	−0.0293483
$337$	22.3776	1.21898	0.609492	−	0.792792i	$-0.291373\pi$
$337$	22.3776	1.21898	0.609492	+	0.792792i	$0.291373\pi$
$338$	6.61290	0.359695
$339$	3.00011	0.162943
$340$	−3.76434	−0.204150
$341$	0	0
$342$	21.5042	1.16281
$343$	19.8746	1.07313
$344$	4.57948	0.246909
$345$	0.937223	0.0504584
$346$	−0.493499	−0.0265307
$347$	2.45988	0.132053	0.0660267	−	0.997818i	$-0.478968\pi$
$347$	2.45988	0.132053	0.0660267	+	0.997818i	$0.478968\pi$
$348$	−2.43086	−0.130308
$349$	−23.0054	−1.23145	−0.615724	−	0.787962i	$-0.711136\pi$
$349$	−23.0054	−1.23145	−0.615724	+	0.787962i	$0.711136\pi$
$350$	21.7118	1.16055
$351$	−3.41575	−0.182319
$352$	0	0
$353$	−9.15891	−0.487480	−0.243740	−	0.969841i	$-0.578374\pi$
$353$	−9.15891	−0.487480	−0.243740	+	0.969841i	$0.578374\pi$
$354$	−2.53403	−0.134682
$355$	−28.8200	−1.52961
$356$	−6.13086	−0.324935
$357$	−0.537963	−0.0284720
$358$	−6.59389	−0.348498
$359$	−3.65091	−0.192688	−0.0963439	−	0.995348i	$-0.530715\pi$
$359$	−3.65091	−0.192688	−0.0963439	+	0.995348i	$0.530715\pi$
$360$	−11.0987	−0.584952
$361$	34.1961	1.79980
$362$	13.5189	0.710540
$363$	0	0
$364$	−5.98367	−0.313630
$365$	−40.7498	−2.13294
$366$	0.276704	0.0144636
$367$	−20.4126	−1.06553	−0.532765	−	0.846263i	$-0.678847\pi$
$367$	−20.4126	−1.06553	−0.532765	+	0.846263i	$0.678847\pi$
$368$	−1.09577	−0.0571207
$369$	6.52859	0.339865
$370$	22.5056	1.17001
$371$	−9.17359	−0.476269
$372$	2.14815	0.111377
$373$	−5.72182	−0.296265	−0.148132	−	0.988968i	$-0.547326\pi$
$373$	−5.72182	−0.296265	−0.148132	+	0.988968i	$0.547326\pi$
$374$	0	0
$375$	−3.56686	−0.184192
$376$	2.17774	0.112308
$377$	−27.0380	−1.39253
$378$	−3.20000	−0.164590
$379$	−22.8617	−1.17432	−0.587162	−	0.809469i	$-0.699755\pi$
$379$	−22.8617	−1.17432	−0.587162	+	0.809469i	$0.699755\pi$
$380$	−27.4555	−1.40844
$381$	2.01981	0.103478
$382$	−7.00508	−0.358411
$383$	−2.25134	−0.115038	−0.0575191	−	0.998344i	$-0.518319\pi$
$383$	−2.25134	−0.115038	−0.0575191	+	0.998344i	$0.518319\pi$
$384$	−0.227215	−0.0115950
$385$	0	0
$386$	12.6669	0.644730
$387$	13.5020	0.686346
$388$	−19.5462	−0.992309
$389$	18.9128	0.958917	0.479458	−	0.877565i	$-0.340833\pi$
$389$	18.9128	0.958917	0.479458	+	0.877565i	$0.340833\pi$
$390$	2.16161	0.109457
$391$	−1.09577	−0.0554153
$392$	1.39428	0.0704215
$393$	−2.33527	−0.117799
$394$	4.74061	0.238829
$395$	10.9808	0.552504
$396$	0	0
$397$	30.8598	1.54881	0.774404	−	0.632691i	$-0.218050\pi$
$397$	30.8598	1.54881	0.774404	+	0.632691i	$0.218050\pi$
$398$	9.53658	0.478026
$399$	−3.92367	−0.196429
$400$	9.17024	0.458512
$401$	−28.4338	−1.41992	−0.709958	−	0.704244i	$-0.751286\pi$
$401$	−28.4338	−1.41992	−0.709958	+	0.704244i	$0.751286\pi$
$402$	−1.26105	−0.0628955
$403$	23.8935	1.19022
$404$	−12.8408	−0.638851
$405$	−32.1400	−1.59705
$406$	−25.3302	−1.25712
$407$	0	0
$408$	−0.227215	−0.0112488
$409$	−2.86421	−0.141626	−0.0708129	−	0.997490i	$-0.522559\pi$
$409$	−2.86421	−0.141626	−0.0708129	+	0.997490i	$0.522559\pi$
$410$	−8.33538	−0.411655
$411$	−0.226930	−0.0111936
$412$	−7.39712	−0.364430
$413$	−26.4054	−1.29932
$414$	−3.23073	−0.158782
$415$	28.0755	1.37817
$416$	−2.52727	−0.123910
$417$	0.360343	0.0176461
$418$	0	0
$419$	12.7407	0.622424	0.311212	−	0.950341i	$-0.399265\pi$
$419$	12.7407	0.622424	0.311212	+	0.950341i	$0.399265\pi$
$420$	2.02507	0.0988135
$421$	6.94861	0.338655	0.169327	−	0.985560i	$-0.445840\pi$
$421$	6.94861	0.338655	0.169327	+	0.985560i	$0.445840\pi$
$422$	23.6865	1.15304
$423$	6.42079	0.312189
$424$	−3.87457	−0.188166
$425$	9.17024	0.444822
$426$	−1.73957	−0.0842825
$427$	2.88334	0.139535
$428$	18.2495	0.882124
$429$	0	0
$430$	−17.2387	−0.831324
$431$	1.56216	0.0752467	0.0376234	−	0.999292i	$-0.488021\pi$
$431$	1.56216	0.0752467	0.0376234	+	0.999292i	$0.488021\pi$
$432$	−1.35156	−0.0650269
$433$	25.0206	1.20241	0.601206	−	0.799094i	$-0.294687\pi$
$433$	25.0206	1.20241	0.601206	+	0.799094i	$0.294687\pi$
$434$	22.3844	1.07448
$435$	9.15058	0.438737
$436$	−2.62526	−0.125727
$437$	−7.99204	−0.382311
$438$	−2.45965	−0.117527
$439$	0.210595	0.0100512	0.00502558	−	0.999987i	$-0.498400\pi$
$439$	0.210595	0.0100512	0.00502558	+	0.999987i	$0.498400\pi$
$440$	0	0
$441$	4.11084	0.195755
$442$	−2.52727	−0.120210
$443$	22.2250	1.05594	0.527971	−	0.849263i	$-0.322953\pi$
$443$	22.2250	1.05594	0.527971	+	0.849263i	$0.322953\pi$
$444$	1.35844	0.0644685
$445$	23.0786	1.09403
$446$	−21.9765	−1.04062
$447$	−4.17753	−0.197590
$448$	−2.36764	−0.111861
$449$	−32.4238	−1.53017	−0.765087	−	0.643927i	$-0.777304\pi$
$449$	−32.4238	−1.53017	−0.765087	+	0.643927i	$0.777304\pi$
$450$	27.0373	1.27455
$451$	0	0
$452$	13.2038	0.621057
$453$	−2.74425	−0.128936
$454$	13.8802	0.651428
$455$	22.5246	1.05597
$456$	−1.65721	−0.0776058
$457$	10.6218	0.496867	0.248433	−	0.968649i	$-0.420084\pi$
$457$	10.6218	0.496867	0.248433	+	0.968649i	$0.420084\pi$
$458$	−1.15349	−0.0538991
$459$	−1.35156	−0.0630853
$460$	4.12483	0.192321
$461$	−20.4870	−0.954174	−0.477087	−	0.878856i	$-0.658307\pi$
$461$	−20.4870	−0.954174	−0.477087	+	0.878856i	$0.658307\pi$
$462$	0	0
$463$	−32.9066	−1.52930	−0.764649	−	0.644447i	$-0.777088\pi$
$463$	−32.9066	−1.52930	−0.764649	+	0.644447i	$0.777088\pi$
$464$	−10.6985	−0.496666
$465$	−8.08637	−0.374997
$466$	12.5295	0.580418
$467$	21.0846	0.975681	0.487841	−	0.872933i	$-0.337785\pi$
$467$	21.0846	0.975681	0.487841	+	0.872933i	$0.337785\pi$
$468$	−7.45134	−0.344438
$469$	−13.1405	−0.606772
$470$	−8.19775	−0.378134
$471$	−1.73108	−0.0797638
$472$	−11.1526	−0.513340
$473$	0	0
$474$	0.662800	0.0304434
$475$	66.8838	3.06884
$476$	−2.36764	−0.108521
$477$	−11.4237	−0.523054
$478$	−18.4473	−0.843760
$479$	3.87404	0.177009	0.0885046	−	0.996076i	$-0.471791\pi$
$479$	3.87404	0.177009	0.0885046	+	0.996076i	$0.471791\pi$
$480$	0.855313	0.0390395
$481$	15.1096	0.688940
$482$	5.79907	0.264140
$483$	0.589481	0.0268223
$484$	0	0
$485$	73.5786	3.34103
$486$	−5.99464	−0.271922
$487$	−28.1654	−1.27630	−0.638148	−	0.769914i	$-0.720299\pi$
$487$	−28.1654	−1.27630	−0.638148	+	0.769914i	$0.720299\pi$
$488$	1.21781	0.0551277
$489$	0.0120001	0.000542664 0
$490$	−5.24852	−0.237104
$491$	5.56723	0.251246	0.125623	−	0.992078i	$-0.459907\pi$
$491$	5.56723	0.251246	0.125623	+	0.992078i	$0.459907\pi$
$492$	−0.503122	−0.0226825
$493$	−10.6985	−0.481837
$494$	−18.4328	−0.829332
$495$	0	0
$496$	9.45428	0.424510
$497$	−18.1268	−0.813099
$498$	1.69463	0.0759382
$499$	−9.48411	−0.424567	−0.212284	−	0.977208i	$-0.568090\pi$
$499$	−9.48411	−0.424567	−0.212284	+	0.977208i	$0.568090\pi$
$500$	−15.6982	−0.702045
$501$	−4.40707	−0.196893
$502$	−6.40289	−0.285775
$503$	−23.2426	−1.03634	−0.518169	−	0.855278i	$-0.673386\pi$
$503$	−23.2426	−1.03634	−0.518169	+	0.855278i	$0.673386\pi$
$504$	−6.98069	−0.310945
$505$	48.3369	2.15097
$506$	0	0
$507$	−1.50255	−0.0667305
$508$	8.88945	0.394406
$509$	−39.9407	−1.77034	−0.885170	−	0.465269i	$-0.845958\pi$
$509$	−39.9407	−1.77034	−0.885170	+	0.465269i	$0.845958\pi$
$510$	0.855313	0.0378739
$511$	−25.6302	−1.13381
$512$	−1.00000	−0.0441942
$513$	−9.85768	−0.435227
$514$	−20.8977	−0.921758
$515$	27.8453	1.22701
$516$	−1.04053	−0.0458066
$517$	0	0
$518$	14.1553	0.621947
$519$	0.112130	0.00492197
$520$	9.51350	0.417195
$521$	−9.53250	−0.417626	−0.208813	−	0.977956i	$-0.566960\pi$
$521$	−9.53250	−0.417626	−0.208813	+	0.977956i	$0.566960\pi$
$522$	−31.5432	−1.38061
$523$	32.7606	1.43252	0.716260	−	0.697834i	$-0.245853\pi$
$523$	32.7606	1.43252	0.716260	+	0.697834i	$0.245853\pi$
$524$	−10.2778	−0.448989
$525$	−4.93325	−0.215305
$526$	10.8761	0.474221
$527$	9.45428	0.411835
$528$	0	0
$529$	−21.7993	−0.947796
$530$	14.5852	0.633540
$531$	−32.8820	−1.42696
$532$	−17.2686	−0.748687
$533$	−5.59614	−0.242396
$534$	1.39302	0.0602819
$535$	−68.6974	−2.97005
$536$	−5.55004	−0.239725
$537$	1.49823	0.0646534
$538$	18.4843	0.796914
$539$	0	0
$540$	5.08772	0.218941
$541$	−9.55871	−0.410961	−0.205480	−	0.978661i	$-0.565876\pi$
$541$	−9.55871	−0.410961	−0.205480	+	0.978661i	$0.565876\pi$
$542$	4.10367	0.176268
$543$	−3.07170	−0.131819
$544$	−1.00000	−0.0428746
$545$	9.88235	0.423314
$546$	1.35958	0.0581846
$547$	−4.24720	−0.181597	−0.0907985	−	0.995869i	$-0.528942\pi$
$547$	−4.24720	−0.181597	−0.0907985	+	0.995869i	$0.528942\pi$
$548$	−0.998749	−0.0426644
$549$	3.59056	0.153241
$550$	0	0
$551$	−78.0303	−3.32420
$552$	0.248974	0.0105970
$553$	6.90656	0.293697
$554$	−6.83580	−0.290425
$555$	−5.11361	−0.217061
$556$	1.58592	0.0672578
$557$	2.95348	0.125143	0.0625714	−	0.998040i	$-0.480070\pi$
$557$	2.95348	0.125143	0.0625714	+	0.998040i	$0.480070\pi$
$558$	27.8748	1.18003
$559$	−11.5736	−0.489510
$560$	8.91260	0.376626
$561$	0	0
$562$	−28.9642	−1.22178
$563$	16.6464	0.701564	0.350782	−	0.936457i	$-0.385916\pi$
$563$	16.6464	0.701564	0.350782	+	0.936457i	$0.385916\pi$
$564$	−0.494815	−0.0208354
$565$	−49.7037	−2.09105
$566$	14.2797	0.600222
$567$	−20.2150	−0.848949
$568$	−7.65607	−0.321241
$569$	−13.4136	−0.562327	−0.281164	−	0.959660i	$-0.590720\pi$
$569$	−13.4136	−0.562327	−0.281164	+	0.959660i	$0.590720\pi$
$570$	6.23828	0.261293
$571$	19.9613	0.835355	0.417678	−	0.908595i	$-0.362844\pi$
$571$	19.9613	0.835355	0.417678	+	0.908595i	$0.362844\pi$
$572$	0	0
$573$	1.59166	0.0664925
$574$	−5.24267	−0.218825
$575$	−10.0484	−0.419049
$576$	−2.94837	−0.122849
$577$	−1.88354	−0.0784129	−0.0392064	−	0.999231i	$-0.512483\pi$
$577$	−1.88354	−0.0784129	−0.0392064	+	0.999231i	$0.512483\pi$
$578$	−1.00000	−0.0415945
$579$	−2.87811	−0.119610
$580$	40.2728	1.67224
$581$	17.6585	0.732599
$582$	4.44119	0.184093
$583$	0	0
$584$	−10.8252	−0.447951
$585$	28.0494	1.15970
$586$	−16.7036	−0.690017
$587$	0.243685	0.0100580	0.00502898	−	0.999987i	$-0.498399\pi$
$587$	0.243685	0.0100580	0.00502898	+	0.999987i	$0.498399\pi$
$588$	−0.316800	−0.0130646
$589$	68.9555	2.84126
$590$	41.9821	1.72838
$591$	−1.07714	−0.0443075
$592$	5.97864	0.245721
$593$	−5.48695	−0.225322	−0.112661	−	0.993633i	$-0.535937\pi$
$593$	−5.48695	−0.225322	−0.112661	+	0.993633i	$0.535937\pi$
$594$	0	0
$595$	8.91260	0.365381
$596$	−18.3858	−0.753112
$597$	−2.16685	−0.0886833
$598$	2.76930	0.113245
$599$	−12.0206	−0.491148	−0.245574	−	0.969378i	$-0.578976\pi$
$599$	−12.0206	−0.491148	−0.245574	+	0.969378i	$0.578976\pi$
$600$	−2.08361	−0.0850632
$601$	32.8015	1.33800	0.669000	−	0.743262i	$-0.266722\pi$
$601$	32.8015	1.33800	0.669000	+	0.743262i	$0.266722\pi$
$602$	−10.8426	−0.441910
$603$	−16.3636	−0.666376
$604$	−12.0778	−0.491438
$605$	0	0
$606$	2.91761	0.118520
$607$	16.0345	0.650819	0.325409	−	0.945573i	$-0.394498\pi$
$607$	16.0345	0.650819	0.325409	+	0.945573i	$0.394498\pi$
$608$	−7.29357	−0.295793
$609$	5.75540	0.233221
$610$	−4.58425	−0.185611
$611$	−5.50374	−0.222657
$612$	−2.94837	−0.119181
$613$	−8.57736	−0.346436	−0.173218	−	0.984883i	$-0.555417\pi$
$613$	−8.57736	−0.346436	−0.173218	+	0.984883i	$0.555417\pi$
$614$	1.96030	0.0791114
$615$	1.89392	0.0763703
$616$	0	0
$617$	−12.1471	−0.489023	−0.244512	−	0.969646i	$-0.578628\pi$
$617$	−12.1471	−0.489023	−0.244512	+	0.969646i	$0.578628\pi$
$618$	1.68073	0.0676091
$619$	11.3304	0.455407	0.227703	−	0.973731i	$-0.426878\pi$
$619$	11.3304	0.455407	0.227703	+	0.973731i	$0.426878\pi$
$620$	−35.5891	−1.42929
$621$	1.48099	0.0594301
$622$	−18.3723	−0.736661
$623$	14.5157	0.581558
$624$	0.574233	0.0229877
$625$	13.2421	0.529685
$626$	22.8404	0.912886
$627$	0	0
$628$	−7.61868	−0.304018
$629$	5.97864	0.238384
$630$	26.2777	1.04693
$631$	−5.56788	−0.221654	−0.110827	−	0.993840i	$-0.535350\pi$
$631$	−5.56788	−0.221654	−0.110827	+	0.993840i	$0.535350\pi$
$632$	2.91706	0.116035
$633$	−5.38193	−0.213912
$634$	25.2240	1.00177
$635$	−33.4629	−1.32793
$636$	0.880359	0.0349085
$637$	−3.52371	−0.139615
$638$	0	0
$639$	−22.5729	−0.892972
$640$	3.76434	0.148799
$641$	50.2989	1.98669	0.993344	−	0.115185i	$-0.0367462\pi$
$641$	50.2989	1.98669	0.993344	+	0.115185i	$0.0367462\pi$
$642$	−4.14656	−0.163652
$643$	11.5905	0.457083	0.228542	−	0.973534i	$-0.426604\pi$
$643$	11.5905	0.457083	0.228542	+	0.973534i	$0.426604\pi$
$644$	2.59438	0.102233
$645$	3.91689	0.154227
$646$	−7.29357	−0.286962
$647$	−17.6881	−0.695389	−0.347695	−	0.937608i	$-0.613035\pi$
$647$	−17.6881	−0.695389	−0.347695	+	0.937608i	$0.613035\pi$
$648$	−8.53803	−0.335405
$649$	0	0
$650$	−23.1757	−0.909025
$651$	−5.08605	−0.199338
$652$	0.0528140	0.00206835
$653$	−8.03429	−0.314406	−0.157203	−	0.987566i	$-0.550248\pi$
$653$	−8.03429	−0.314406	−0.157203	+	0.987566i	$0.550248\pi$
$654$	0.596497	0.0233249
$655$	38.6892	1.51171
$656$	−2.21430	−0.0864540
$657$	−31.9168	−1.24519
$658$	−5.15611	−0.201006
$659$	−20.9890	−0.817617	−0.408808	−	0.912620i	$-0.634056\pi$
$659$	−20.9890	−0.817617	−0.408808	+	0.912620i	$0.634056\pi$
$660$	0	0
$661$	−6.00797	−0.233683	−0.116842	−	0.993151i	$-0.537277\pi$
$661$	−6.00797	−0.233683	−0.116842	+	0.993151i	$0.537277\pi$
$662$	−25.3774	−0.986323
$663$	0.574233	0.0223014
$664$	7.45828	0.289437
$665$	65.0047	2.52077
$666$	17.6273	0.683043
$667$	11.7231	0.453919
$668$	−19.3960	−0.750455
$669$	4.99339	0.193056
$670$	20.8922	0.807137
$671$	0	0
$672$	0.537963	0.0207524
$673$	−45.0592	−1.73691	−0.868453	−	0.495772i	$-0.834885\pi$
$673$	−45.0592	−1.73691	−0.868453	+	0.495772i	$0.834885\pi$
$674$	−22.3776	−0.861952
$675$	−12.3941	−0.477050
$676$	−6.61290	−0.254342
$677$	−36.5092	−1.40316	−0.701582	−	0.712589i	$-0.747523\pi$
$677$	−36.5092	−1.40316	−0.701582	+	0.712589i	$0.747523\pi$
$678$	−3.00011	−0.115218
$679$	46.2784	1.77600
$680$	3.76434	0.144356
$681$	−3.15378	−0.120853
$682$	0	0
$683$	32.6374	1.24883	0.624417	−	0.781091i	$-0.285336\pi$
$683$	32.6374	1.24883	0.624417	+	0.781091i	$0.285336\pi$
$684$	−21.5042	−0.822232
$685$	3.75963	0.143648
$686$	−19.8746	−0.758817
$687$	0.262090	0.00999937
$688$	−4.57948	−0.174591
$689$	9.79209	0.373049
$690$	−0.937223	−0.0356795
$691$	26.8622	1.02189	0.510944	−	0.859614i	$-0.329296\pi$
$691$	26.8622	1.02189	0.510944	+	0.859614i	$0.329296\pi$
$692$	0.493499	0.0187600
$693$	0	0
$694$	−2.45988	−0.0933758
$695$	−5.96992	−0.226452
$696$	2.43086	0.0921415
$697$	−2.21430	−0.0838727
$698$	23.0054	0.870766
$699$	−2.84689	−0.107679
$700$	−21.7118	−0.820630
$701$	−11.4552	−0.432657	−0.216329	−	0.976321i	$-0.569408\pi$
$701$	−11.4552	−0.432657	−0.216329	+	0.976321i	$0.569408\pi$
$702$	3.41575	0.128919
$703$	43.6056	1.64462
$704$	0	0
$705$	1.86265	0.0701514
$706$	9.15891	0.344700
$707$	30.4023	1.14340
$708$	2.53403	0.0952348
$709$	9.60627	0.360771	0.180386	−	0.983596i	$-0.442265\pi$
$709$	9.60627	0.360771	0.180386	+	0.983596i	$0.442265\pi$
$710$	28.8200	1.08160
$711$	8.60059	0.322547
$712$	6.13086	0.229764
$713$	−10.3597	−0.387973
$714$	0.537963	0.0201328
$715$	0	0
$716$	6.59389	0.246425
$717$	4.19150	0.156534
$718$	3.65091	0.136251
$719$	−47.2671	−1.76277	−0.881383	−	0.472402i	$-0.843387\pi$
$719$	−47.2671	−1.76277	−0.881383	+	0.472402i	$0.843387\pi$
$720$	11.0987	0.413623
$721$	17.5137	0.652245
$722$	−34.1961	−1.27265
$723$	−1.31763	−0.0490033
$724$	−13.5189	−0.502427
$725$	−98.1079	−3.64364
$726$	0	0
$727$	19.6655	0.729354	0.364677	−	0.931134i	$-0.381179\pi$
$727$	19.6655	0.729354	0.364677	+	0.931134i	$0.381179\pi$
$728$	5.98367	0.221770
$729$	−24.2520	−0.898223
$730$	40.7498	1.50822
$731$	−4.57948	−0.169378
$732$	−0.276704	−0.0102273
$733$	2.19780	0.0811778	0.0405889	−	0.999176i	$-0.487077\pi$
$733$	2.19780	0.0811778	0.0405889	+	0.999176i	$0.487077\pi$
$734$	20.4126	0.753444
$735$	1.19254	0.0439876
$736$	1.09577	0.0403905
$737$	0	0
$738$	−6.52859	−0.240321
$739$	−42.1742	−1.55140	−0.775702	−	0.631099i	$-0.782604\pi$
$739$	−42.1742	−1.55140	−0.775702	+	0.631099i	$0.782604\pi$
$740$	−22.5056	−0.827323
$741$	4.18821	0.153858
$742$	9.17359	0.336773
$743$	−17.8557	−0.655061	−0.327530	−	0.944841i	$-0.606216\pi$
$743$	−17.8557	−0.655061	−0.327530	+	0.944841i	$0.606216\pi$
$744$	−2.14815	−0.0787551
$745$	69.2104	2.53567
$746$	5.72182	0.209491
$747$	21.9898	0.804565
$748$	0	0
$749$	−43.2083	−1.57880
$750$	3.56686	0.130243
$751$	−8.06573	−0.294323	−0.147161	−	0.989113i	$-0.547014\pi$
$751$	−8.06573	−0.294323	−0.147161	+	0.989113i	$0.547014\pi$
$752$	−2.17774	−0.0794140
$753$	1.45483	0.0530170
$754$	27.0380	0.984667
$755$	45.4648	1.65464
$756$	3.20000	0.116383
$757$	42.6063	1.54855	0.774276	−	0.632849i	$-0.218114\pi$
$757$	42.6063	1.54855	0.774276	+	0.632849i	$0.218114\pi$
$758$	22.8617	0.830373
$759$	0	0
$760$	27.4555	0.995914
$761$	−15.8084	−0.573056	−0.286528	−	0.958072i	$-0.592501\pi$
$761$	−15.8084	−0.573056	−0.286528	+	0.958072i	$0.592501\pi$
$762$	−2.01981	−0.0731702
$763$	6.21567	0.225022
$764$	7.00508	0.253435
$765$	11.0987	0.401273
$766$	2.25134	0.0813442
$767$	28.1856	1.01772
$768$	0.227215	0.00819891
$769$	−52.0723	−1.87777	−0.938887	−	0.344226i	$-0.888141\pi$
$769$	−52.0723	−1.87777	−0.938887	+	0.344226i	$0.888141\pi$
$770$	0	0
$771$	4.74827	0.171005
$772$	−12.6669	−0.455893
$773$	16.5507	0.595286	0.297643	−	0.954677i	$-0.403800\pi$
$773$	16.5507	0.595286	0.297643	+	0.954677i	$0.403800\pi$
$774$	−13.5020	−0.485320
$775$	86.6981	3.11429
$776$	19.5462	0.701668
$777$	−3.21629	−0.115384
$778$	−18.9128	−0.678057
$779$	−16.1502	−0.578640
$780$	−2.16161	−0.0773980
$781$	0	0
$782$	1.09577	0.0391845
$783$	14.4597	0.516746
$784$	−1.39428	−0.0497955
$785$	28.6793	1.02361
$786$	2.33527	0.0832965
$787$	−40.9521	−1.45978	−0.729892	−	0.683562i	$-0.760430\pi$
$787$	−40.9521	−1.45978	−0.729892	+	0.683562i	$0.760430\pi$
$788$	−4.74061	−0.168877
$789$	−2.47122	−0.0879776
$790$	−10.9808	−0.390680
$791$	−31.2620	−1.11155
$792$	0	0
$793$	−3.07774	−0.109294
$794$	−30.8598	−1.09517
$795$	−3.31397	−0.117534
$796$	−9.53658	−0.338015
$797$	−15.8384	−0.561026	−0.280513	−	0.959850i	$-0.590505\pi$
$797$	−15.8384	−0.561026	−0.280513	+	0.959850i	$0.590505\pi$
$798$	3.92367	0.138896
$799$	−2.17774	−0.0770429
$800$	−9.17024	−0.324217
$801$	18.0761	0.638686
$802$	28.4338	1.00403
$803$	0	0
$804$	1.26105	0.0444738
$805$	−9.76613	−0.344211
$806$	−23.8935	−0.841614
$807$	−4.19990	−0.147843
$808$	12.8408	0.451736
$809$	−1.33928	−0.0470864	−0.0235432	−	0.999723i	$-0.507495\pi$
$809$	−1.33928	−0.0470864	−0.0235432	+	0.999723i	$0.507495\pi$
$810$	32.1400	1.12929
$811$	−14.1643	−0.497376	−0.248688	−	0.968584i	$-0.579999\pi$
$811$	−14.1643	−0.497376	−0.248688	+	0.968584i	$0.579999\pi$
$812$	25.3302	0.888917
$813$	−0.932414	−0.0327012
$814$	0	0
$815$	−0.198810	−0.00696400
$816$	0.227215	0.00795411
$817$	−33.4007	−1.16854
$818$	2.86421	0.100145
$819$	17.6421	0.616465
$820$	8.33538	0.291084
$821$	−33.3635	−1.16439	−0.582196	−	0.813048i	$-0.697806\pi$
$821$	−33.3635	−1.16439	−0.582196	+	0.813048i	$0.697806\pi$
$822$	0.226930	0.00791511
$823$	−6.70420	−0.233694	−0.116847	−	0.993150i	$-0.537279\pi$
$823$	−6.70420	−0.233694	−0.116847	+	0.993150i	$0.537279\pi$
$824$	7.39712	0.257691
$825$	0	0
$826$	26.4054	0.918760
$827$	−6.29189	−0.218790	−0.109395	−	0.993998i	$-0.534891\pi$
$827$	−6.29189	−0.218790	−0.109395	+	0.993998i	$0.534891\pi$
$828$	3.23073	0.112276
$829$	7.45481	0.258916	0.129458	−	0.991585i	$-0.458676\pi$
$829$	7.45481	0.258916	0.129458	+	0.991585i	$0.458676\pi$
$830$	−28.0755	−0.974514
$831$	1.55319	0.0538797
$832$	2.52727	0.0876173
$833$	−1.39428	−0.0483088
$834$	−0.360343	−0.0124777
$835$	73.0132	2.52673
$836$	0	0
$837$	−12.7780	−0.441673
$838$	−12.7407	−0.440120
$839$	−33.4653	−1.15535	−0.577675	−	0.816267i	$-0.696040\pi$
$839$	−33.4653	−1.15535	−0.577675	+	0.816267i	$0.696040\pi$
$840$	−2.02507	−0.0698717
$841$	85.4581	2.94683
$842$	−6.94861	−0.239465
$843$	6.58109	0.226665
$844$	−23.6865	−0.815324
$845$	24.8932	0.856352
$846$	−6.42079	−0.220751
$847$	0	0
$848$	3.87457	0.133053
$849$	−3.24456	−0.111353
$850$	−9.17024	−0.314537
$851$	−6.55119	−0.224572
$852$	1.73957	0.0595967
$853$	39.9664	1.36842	0.684212	−	0.729283i	$-0.260146\pi$
$853$	39.9664	1.36842	0.684212	+	0.729283i	$0.260146\pi$
$854$	−2.88334	−0.0986658
$855$	80.9489	2.76840
$856$	−18.2495	−0.623756
$857$	16.1973	0.553288	0.276644	−	0.960972i	$-0.410778\pi$
$857$	16.1973	0.553288	0.276644	+	0.960972i	$0.410778\pi$
$858$	0	0
$859$	−0.468663	−0.0159906	−0.00799530	−	0.999968i	$-0.502545\pi$
$859$	−0.468663	−0.0159906	−0.00799530	+	0.999968i	$0.502545\pi$
$860$	17.2387	0.587835
$861$	1.19121	0.0405964
$862$	−1.56216	−0.0532075
$863$	44.2543	1.50643	0.753216	−	0.657773i	$-0.228501\pi$
$863$	44.2543	1.50643	0.753216	+	0.657773i	$0.228501\pi$
$864$	1.35156	0.0459809
$865$	−1.85770	−0.0631636
$866$	−25.0206	−0.850233
$867$	0.227215	0.00771662
$868$	−22.3844	−0.759774
$869$	0	0
$870$	−9.15058	−0.310234
$871$	14.0264	0.475268
$872$	2.62526	0.0889024
$873$	57.6295	1.95046
$874$	7.99204	0.270335
$875$	37.1677	1.25650
$876$	2.45965	0.0831038
$877$	8.82749	0.298083	0.149042	−	0.988831i	$-0.452381\pi$
$877$	8.82749	0.298083	0.149042	+	0.988831i	$0.452381\pi$
$878$	−0.210595	−0.00710725
$879$	3.79529	0.128012
$880$	0	0
$881$	−3.60582	−0.121483	−0.0607415	−	0.998154i	$-0.519347\pi$
$881$	−3.60582	−0.121483	−0.0607415	+	0.998154i	$0.519347\pi$
$882$	−4.11084	−0.138419
$883$	−45.3727	−1.52691	−0.763457	−	0.645859i	$-0.776499\pi$
$883$	−45.3727	−1.52691	−0.763457	+	0.645859i	$0.776499\pi$
$884$	2.52727	0.0850013
$885$	−9.53896	−0.320649
$886$	−22.2250	−0.746663
$887$	30.4970	1.02399	0.511995	−	0.858988i	$-0.328907\pi$
$887$	30.4970	1.02399	0.511995	+	0.858988i	$0.328907\pi$
$888$	−1.35844	−0.0455861
$889$	−21.0470	−0.705895
$890$	−23.0786	−0.773597
$891$	0	0
$892$	21.9765	0.735828
$893$	−15.8835	−0.531521
$894$	4.17753	0.139717
$895$	−24.8216	−0.829696
$896$	2.36764	0.0790973
$897$	−0.629225	−0.0210092
$898$	32.4238	1.08200
$899$	−101.147	−3.37343
$900$	−27.0373	−0.901243
$901$	3.87457	0.129081
$902$	0	0
$903$	2.46359	0.0819832
$904$	−13.2038	−0.439153
$905$	50.8899	1.69164
$906$	2.74425	0.0911716
$907$	−52.0011	−1.72667	−0.863334	−	0.504632i	$-0.831628\pi$
$907$	−52.0011	−1.72667	−0.863334	+	0.504632i	$0.831628\pi$
$908$	−13.8802	−0.460629
$909$	37.8593	1.25572
$910$	−22.5246	−0.746682
$911$	40.2107	1.33224	0.666121	−	0.745844i	$-0.267954\pi$
$911$	40.2107	1.33224	0.666121	+	0.745844i	$0.267954\pi$
$912$	1.65721	0.0548756
$913$	0	0
$914$	−10.6218	−0.351338
$915$	1.04161	0.0344345
$916$	1.15349	0.0381124
$917$	24.3342	0.803586
$918$	1.35156	0.0446081
$919$	−1.63171	−0.0538250	−0.0269125	−	0.999638i	$-0.508568\pi$
$919$	−1.63171	−0.0538250	−0.0269125	+	0.999638i	$0.508568\pi$
$920$	−4.12483	−0.135992
$921$	−0.445410	−0.0146767
$922$	20.4870	0.674703
$923$	19.3490	0.636878
$924$	0	0
$925$	54.8256	1.80265
$926$	32.9066	1.08138
$927$	21.8095	0.716317
$928$	10.6985	0.351196
$929$	40.8795	1.34121	0.670606	−	0.741814i	$-0.266034\pi$
$929$	40.8795	1.34121	0.670606	+	0.741814i	$0.266034\pi$
$930$	8.08637	0.265163
$931$	−10.1692	−0.333283
$932$	−12.5295	−0.410418
$933$	4.17445	0.136665
$934$	−21.0846	−0.689911
$935$	0	0
$936$	7.45134	0.243555
$937$	−29.0669	−0.949575	−0.474787	−	0.880101i	$-0.657475\pi$
$937$	−29.0669	−0.949575	−0.474787	+	0.880101i	$0.657475\pi$
$938$	13.1405	0.429052
$939$	−5.18968	−0.169359
$940$	8.19775	0.267381
$941$	27.7947	0.906081	0.453041	−	0.891490i	$-0.350339\pi$
$941$	27.7947	0.906081	0.453041	+	0.891490i	$0.350339\pi$
$942$	1.73108	0.0564015
$943$	2.42636	0.0790131
$944$	11.1526	0.362986
$945$	−12.0459	−0.391853
$946$	0	0
$947$	−20.5649	−0.668268	−0.334134	−	0.942526i	$-0.608444\pi$
$947$	−20.5649	−0.668268	−0.334134	+	0.942526i	$0.608444\pi$
$948$	−0.662800	−0.0215267
$949$	27.3583	0.888087
$950$	−66.8838	−2.17000
$951$	−5.73126	−0.185849
$952$	2.36764	0.0767357
$953$	−26.6731	−0.864025	−0.432012	−	0.901868i	$-0.642196\pi$
$953$	−26.6731	−0.864025	−0.432012	+	0.901868i	$0.642196\pi$
$954$	11.4237	0.369855
$955$	−26.3695	−0.853297
$956$	18.4473	0.596628
$957$	0	0
$958$	−3.87404	−0.125164
$959$	2.36468	0.0763594
$960$	−0.855313	−0.0276051
$961$	58.3835	1.88334
$962$	−15.1096	−0.487154
$963$	−53.8064	−1.73389
$964$	−5.79907	−0.186776
$965$	47.6826	1.53496
$966$	−0.589481	−0.0189663
$967$	−19.3791	−0.623188	−0.311594	−	0.950215i	$-0.600863\pi$
$967$	−19.3791	−0.623188	−0.311594	+	0.950215i	$0.600863\pi$
$968$	0	0
$969$	1.65721	0.0532371
$970$	−73.5786	−2.36247
$971$	−20.0049	−0.641987	−0.320993	−	0.947081i	$-0.604017\pi$
$971$	−20.0049	−0.641987	−0.320993	+	0.947081i	$0.604017\pi$
$972$	5.99464	0.192278
$973$	−3.75488	−0.120376
$974$	28.1654	0.902477
$975$	5.26586	0.168642
$976$	−1.21781	−0.0389812
$977$	27.1004	0.867019	0.433509	−	0.901149i	$-0.357275\pi$
$977$	27.1004	0.867019	0.433509	+	0.901149i	$0.357275\pi$
$978$	−0.0120001	−0.000383721 0
$979$	0	0
$980$	5.24852	0.167658
$981$	7.74024	0.247127
$982$	−5.56723	−0.177657
$983$	42.3022	1.34923	0.674616	−	0.738169i	$-0.264309\pi$
$983$	42.3022	1.34923	0.674616	+	0.738169i	$0.264309\pi$
$984$	0.503122	0.0160389
$985$	17.8453	0.568597
$986$	10.6985	0.340710
$987$	1.17154	0.0372906
$988$	18.4328	0.586426
$989$	5.01804	0.159564
$990$	0	0
$991$	−32.3009	−1.02607	−0.513035	−	0.858367i	$-0.671479\pi$
$991$	−32.3009	−1.02607	−0.513035	+	0.858367i	$0.671479\pi$
$992$	−9.45428	−0.300174
$993$	5.76613	0.182983
$994$	18.1268	0.574948
$995$	35.8989	1.13807
$996$	−1.69463	−0.0536964
$997$	0.857939	0.0271712	0.0135856	−	0.999908i	$-0.495675\pi$
$997$	0.857939	0.0271712	0.0135856	+	0.999908i	$0.495675\pi$
$998$	9.48411	0.300214
$999$	−8.08048	−0.255655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4114.2.a.bg.1.5		8
11.7	odd	10		374.2.g.f.137.2	✓	16
11.8	odd	10		374.2.g.f.273.2	yes	16
11.10	odd	2		4114.2.a.bi.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.g.f.137.2	✓	16	11.7	odd	10
374.2.g.f.273.2	yes	16	11.8	odd	10
4114.2.a.bg.1.5		8	1.1	even	1	trivial
4114.2.a.bi.1.5		8	11.10	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:	\( N \)	\(=\)	\( 4114 = 2 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4114.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.227215 q^{3} +1.00000 q^{4} -3.76434 q^{5} -0.227215 q^{6} -2.36764 q^{7} -1.00000 q^{8} -2.94837 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.227215 q^{3} +1.00000 q^{4} -3.76434 q^{5} -0.227215 q^{6} -2.36764 q^{7} -1.00000 q^{8} -2.94837 q^{9} +3.76434 q^{10} +0.227215 q^{12} +2.52727 q^{13} +2.36764 q^{14} -0.855313 q^{15} +1.00000 q^{16} +1.00000 q^{17} +2.94837 q^{18} +7.29357 q^{19} -3.76434 q^{20} -0.537963 q^{21} -1.09577 q^{23} -0.227215 q^{24} +9.17024 q^{25} -2.52727 q^{26} -1.35156 q^{27} -2.36764 q^{28} -10.6985 q^{29} +0.855313 q^{30} +9.45428 q^{31} -1.00000 q^{32} -1.00000 q^{34} +8.91260 q^{35} -2.94837 q^{36} +5.97864 q^{37} -7.29357 q^{38} +0.574233 q^{39} +3.76434 q^{40} -2.21430 q^{41} +0.537963 q^{42} -4.57948 q^{43} +11.0987 q^{45} +1.09577 q^{46} -2.17774 q^{47} +0.227215 q^{48} -1.39428 q^{49} -9.17024 q^{50} +0.227215 q^{51} +2.52727 q^{52} +3.87457 q^{53} +1.35156 q^{54} +2.36764 q^{56} +1.65721 q^{57} +10.6985 q^{58} +11.1526 q^{59} -0.855313 q^{60} -1.21781 q^{61} -9.45428 q^{62} +6.98069 q^{63} +1.00000 q^{64} -9.51350 q^{65} +5.55004 q^{67} +1.00000 q^{68} -0.248974 q^{69} -8.91260 q^{70} +7.65607 q^{71} +2.94837 q^{72} +10.8252 q^{73} -5.97864 q^{74} +2.08361 q^{75} +7.29357 q^{76} -0.574233 q^{78} -2.91706 q^{79} -3.76434 q^{80} +8.53803 q^{81} +2.21430 q^{82} -7.45828 q^{83} -0.537963 q^{84} -3.76434 q^{85} +4.57948 q^{86} -2.43086 q^{87} -6.13086 q^{89} -11.0987 q^{90} -5.98367 q^{91} -1.09577 q^{92} +2.14815 q^{93} +2.17774 q^{94} -27.4555 q^{95} -0.227215 q^{96} -19.5462 q^{97} +1.39428 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - q^98

Introduction

L-functions

Modular forms

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Database

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