LMFDB - Embedded newform 1334.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1334,2,Mod(1,1334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1334, 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("1334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1334 = 2 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1334.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:	$10.6520436296$
Analytic rank:	$0$
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} - 22x^{6} + 151x^{4} - 332x^{2} + 6 $ x^8 - 22x^6 + 151x^4 - 332*x^2 + 6
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			1.5
Root			$0.134992$ of defining polynomial
Character	$\chi$	$=$	1334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.77365 q^{3} +1.00000 q^{4} -0.134992 q^{5} +1.77365 q^{6} +4.95530 q^{7} +1.00000 q^{8} +0.145840 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.77365 q^{3} +1.00000 q^{4} -0.134992 q^{5} +1.77365 q^{6} +4.95530 q^{7} +1.00000 q^{8} +0.145840 q^{9} -0.134992 q^{10} +2.17939 q^{11} +1.77365 q^{12} +2.10699 q^{13} +4.95530 q^{14} -0.239429 q^{15} +1.00000 q^{16} -5.44530 q^{17} +0.145840 q^{18} -1.70247 q^{19} -0.134992 q^{20} +8.78898 q^{21} +2.17939 q^{22} -1.00000 q^{23} +1.77365 q^{24} -4.98178 q^{25} +2.10699 q^{26} -5.06229 q^{27} +4.95530 q^{28} -1.00000 q^{29} -0.239429 q^{30} -2.56495 q^{31} +1.00000 q^{32} +3.86547 q^{33} -5.44530 q^{34} -0.668927 q^{35} +0.145840 q^{36} -2.44965 q^{37} -1.70247 q^{38} +3.73706 q^{39} -0.134992 q^{40} +6.83522 q^{41} +8.78898 q^{42} -7.63729 q^{43} +2.17939 q^{44} -0.0196873 q^{45} -1.00000 q^{46} +0.902069 q^{47} +1.77365 q^{48} +17.5550 q^{49} -4.98178 q^{50} -9.65807 q^{51} +2.10699 q^{52} -2.72943 q^{53} -5.06229 q^{54} -0.294201 q^{55} +4.95530 q^{56} -3.01960 q^{57} -1.00000 q^{58} -5.59801 q^{59} -0.239429 q^{60} +8.08971 q^{61} -2.56495 q^{62} +0.722681 q^{63} +1.00000 q^{64} -0.284427 q^{65} +3.86547 q^{66} +8.63159 q^{67} -5.44530 q^{68} -1.77365 q^{69} -0.668927 q^{70} +3.37742 q^{71} +0.145840 q^{72} -2.22792 q^{73} -2.44965 q^{74} -8.83594 q^{75} -1.70247 q^{76} +10.7995 q^{77} +3.73706 q^{78} +9.54644 q^{79} -0.134992 q^{80} -9.41625 q^{81} +6.83522 q^{82} -15.4624 q^{83} +8.78898 q^{84} +0.735074 q^{85} -7.63729 q^{86} -1.77365 q^{87} +2.17939 q^{88} +5.04760 q^{89} -0.0196873 q^{90} +10.4407 q^{91} -1.00000 q^{92} -4.54934 q^{93} +0.902069 q^{94} +0.229821 q^{95} +1.77365 q^{96} +7.70718 q^{97} +17.5550 q^{98} +0.317842 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{14} + 8 q^{16} + 8 q^{17} + 12 q^{18} + 16 q^{19} - 4 q^{21} + 8 q^{22} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 8 q^{28} - 8 q^{29} + 8 q^{31} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{35} + 12 q^{36} + 8 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} - 4 q^{42} + 32 q^{43} + 8 q^{44} - 8 q^{46} + 16 q^{47} + 4 q^{48} + 4 q^{49} + 4 q^{50} + 12 q^{51} + 4 q^{52} + 4 q^{54} + 8 q^{56} + 8 q^{57} - 8 q^{58} - 4 q^{59} + 8 q^{61} + 8 q^{62} + 24 q^{63} + 8 q^{64} - 4 q^{65} + 12 q^{66} + 20 q^{67} + 8 q^{68} - 4 q^{69} + 4 q^{70} - 24 q^{71} + 12 q^{72} - 4 q^{73} + 8 q^{74} - 16 q^{75} + 16 q^{76} - 16 q^{77} - 16 q^{78} + 4 q^{79} - 8 q^{81} + 8 q^{82} - 4 q^{83} - 4 q^{84} - 44 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} + 20 q^{89} - 8 q^{92} - 4 q^{93} + 16 q^{94} - 8 q^{95} + 4 q^{96} + 4 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^12 + 4 * q^13 + 8 * q^14 + 8 * q^16 + 8 * q^17 + 12 * q^18 + 16 * q^19 - 4 * q^21 + 8 * q^22 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 8 * q^28 - 8 * q^29 + 8 * q^31 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^35 + 12 * q^36 + 8 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 - 4 * q^42 + 32 * q^43 + 8 * q^44 - 8 * q^46 + 16 * q^47 + 4 * q^48 + 4 * q^49 + 4 * q^50 + 12 * q^51 + 4 * q^52 + 4 * q^54 + 8 * q^56 + 8 * q^57 - 8 * q^58 - 4 * q^59 + 8 * q^61 + 8 * q^62 + 24 * q^63 + 8 * q^64 - 4 * q^65 + 12 * q^66 + 20 * q^67 + 8 * q^68 - 4 * q^69 + 4 * q^70 - 24 * q^71 + 12 * q^72 - 4 * q^73 + 8 * q^74 - 16 * q^75 + 16 * q^76 - 16 * q^77 - 16 * q^78 + 4 * q^79 - 8 * q^81 + 8 * q^82 - 4 * q^83 - 4 * q^84 - 44 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 + 20 * q^89 - 8 * q^92 - 4 * q^93 + 16 * q^94 - 8 * q^95 + 4 * q^96 + 4 * q^97 + 4 * 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
$3$	1.77365	1.02402	0.512009	−	0.858980i	$-0.328901\pi$
$3$	1.77365	1.02402	0.512009	+	0.858980i	$0.328901\pi$
$4$	1.00000	0.500000
$5$	−0.134992	−0.0603704	−0.0301852	−	0.999544i	$-0.509610\pi$
$5$	−0.134992	−0.0603704	−0.0301852	+	0.999544i	$0.509610\pi$
$6$	1.77365	0.724090
$7$	4.95530	1.87293	0.936464	−	0.350764i	$-0.114078\pi$
$7$	4.95530	1.87293	0.936464	+	0.350764i	$0.114078\pi$
$8$	1.00000	0.353553
$9$	0.145840	0.0486134
$10$	−0.134992	−0.0426883
$11$	2.17939	0.657110	0.328555	−	0.944485i	$-0.393438\pi$
$11$	2.17939	0.657110	0.328555	+	0.944485i	$0.393438\pi$
$12$	1.77365	0.512009
$13$	2.10699	0.584373	0.292186	−	0.956361i	$-0.405617\pi$
$13$	2.10699	0.584373	0.292186	+	0.956361i	$0.405617\pi$
$14$	4.95530	1.32436
$15$	−0.239429	−0.0618204
$16$	1.00000	0.250000
$17$	−5.44530	−1.32068	−0.660340	−	0.750967i	$-0.729588\pi$
$17$	−5.44530	−1.32068	−0.660340	+	0.750967i	$0.729588\pi$
$18$	0.145840	0.0343748
$19$	−1.70247	−0.390575	−0.195287	−	0.980746i	$-0.562564\pi$
$19$	−1.70247	−0.390575	−0.195287	+	0.980746i	$0.562564\pi$
$20$	−0.134992	−0.0301852
$21$	8.78898	1.91791
$22$	2.17939	0.464647
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.77365	0.362045
$25$	−4.98178	−0.996355
$26$	2.10699	0.413214
$27$	−5.06229	−0.974237
$28$	4.95530	0.936464
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−0.239429	−0.0437136
$31$	−2.56495	−0.460679	−0.230340	−	0.973110i	$-0.573984\pi$
$31$	−2.56495	−0.460679	−0.230340	+	0.973110i	$0.573984\pi$
$32$	1.00000	0.176777
$33$	3.86547	0.672893
$34$	−5.44530	−0.933862
$35$	−0.668927	−0.113069
$36$	0.145840	0.0243067
$37$	−2.44965	−0.402720	−0.201360	−	0.979517i	$-0.564536\pi$
$37$	−2.44965	−0.402720	−0.201360	+	0.979517i	$0.564536\pi$
$38$	−1.70247	−0.276178
$39$	3.73706	0.598408
$40$	−0.134992	−0.0213442
$41$	6.83522	1.06748	0.533741	−	0.845648i	$-0.320786\pi$
$41$	6.83522	1.06748	0.533741	+	0.845648i	$0.320786\pi$
$42$	8.78898	1.35617
$43$	−7.63729	−1.16468	−0.582338	−	0.812947i	$-0.697862\pi$
$43$	−7.63729	−1.16468	−0.582338	+	0.812947i	$0.697862\pi$
$44$	2.17939	0.328555
$45$	−0.0196873	−0.00293481
$46$	−1.00000	−0.147442
$47$	0.902069	0.131580	0.0657901	−	0.997833i	$-0.479043\pi$
$47$	0.902069	0.131580	0.0657901	+	0.997833i	$0.479043\pi$
$48$	1.77365	0.256005
$49$	17.5550	2.50786
$50$	−4.98178	−0.704530
$51$	−9.65807	−1.35240
$52$	2.10699	0.292186
$53$	−2.72943	−0.374917	−0.187458	−	0.982273i	$-0.560025\pi$
$53$	−2.72943	−0.374917	−0.187458	+	0.982273i	$0.560025\pi$
$54$	−5.06229	−0.688890
$55$	−0.294201	−0.0396700
$56$	4.95530	0.662180
$57$	−3.01960	−0.399955
$58$	−1.00000	−0.131306
$59$	−5.59801	−0.728799	−0.364399	−	0.931243i	$-0.618726\pi$
$59$	−5.59801	−0.728799	−0.364399	+	0.931243i	$0.618726\pi$
$60$	−0.239429	−0.0309102
$61$	8.08971	1.03578	0.517891	−	0.855447i	$-0.326717\pi$
$61$	8.08971	1.03578	0.517891	+	0.855447i	$0.326717\pi$
$62$	−2.56495	−0.325750
$63$	0.722681	0.0910493
$64$	1.00000	0.125000
$65$	−0.284427	−0.0352788
$66$	3.86547	0.475807
$67$	8.63159	1.05452	0.527258	−	0.849705i	$-0.323220\pi$
$67$	8.63159	1.05452	0.527258	+	0.849705i	$0.323220\pi$
$68$	−5.44530	−0.660340
$69$	−1.77365	−0.213523
$70$	−0.668927	−0.0799521
$71$	3.37742	0.400827	0.200413	−	0.979711i	$-0.435772\pi$
$71$	3.37742	0.400827	0.200413	+	0.979711i	$0.435772\pi$
$72$	0.145840	0.0171874
$73$	−2.22792	−0.260758	−0.130379	−	0.991464i	$-0.541619\pi$
$73$	−2.22792	−0.260758	−0.130379	+	0.991464i	$0.541619\pi$
$74$	−2.44965	−0.284766
$75$	−8.83594	−1.02029
$76$	−1.70247	−0.195287
$77$	10.7995	1.23072
$78$	3.73706	0.423138
$79$	9.54644	1.07406	0.537029	−	0.843563i	$-0.319546\pi$
$79$	9.54644	1.07406	0.537029	+	0.843563i	$0.319546\pi$
$80$	−0.134992	−0.0150926
$81$	−9.41625	−1.04625
$82$	6.83522	0.754823
$83$	−15.4624	−1.69722	−0.848609	−	0.529020i	$-0.822560\pi$
$83$	−15.4624	−1.69722	−0.848609	+	0.529020i	$0.822560\pi$
$84$	8.78898	0.958956
$85$	0.735074	0.0797300
$86$	−7.63729	−0.823550
$87$	−1.77365	−0.190155
$88$	2.17939	0.232323
$89$	5.04760	0.535045	0.267522	−	0.963552i	$-0.413795\pi$
$89$	5.04760	0.535045	0.267522	+	0.963552i	$0.413795\pi$
$90$	−0.0196873	−0.00207522
$91$	10.4407	1.09449
$92$	−1.00000	−0.104257
$93$	−4.54934	−0.471744
$94$	0.902069	0.0930413
$95$	0.229821	0.0235791
$96$	1.77365	0.181023
$97$	7.70718	0.782546	0.391273	−	0.920275i	$-0.372035\pi$
$97$	7.70718	0.782546	0.391273	+	0.920275i	$0.372035\pi$
$98$	17.5550	1.77332
$99$	0.317842	0.0319443
$100$	−4.98178	−0.498178
$101$	1.31545	0.130893	0.0654463	−	0.997856i	$-0.479153\pi$
$101$	1.31545	0.130893	0.0654463	+	0.997856i	$0.479153\pi$
$102$	−9.65807	−0.956292
$103$	−4.64515	−0.457700	−0.228850	−	0.973462i	$-0.573497\pi$
$103$	−4.64515	−0.457700	−0.228850	+	0.973462i	$0.573497\pi$
$104$	2.10699	0.206607
$105$	−1.18644	−0.115785
$106$	−2.72943	−0.265106
$107$	−4.64518	−0.449066	−0.224533	−	0.974466i	$-0.572086\pi$
$107$	−4.64518	−0.449066	−0.224533	+	0.974466i	$0.572086\pi$
$108$	−5.06229	−0.487119
$109$	10.5094	1.00662	0.503309	−	0.864106i	$-0.332116\pi$
$109$	10.5094	1.00662	0.503309	+	0.864106i	$0.332116\pi$
$110$	−0.294201	−0.0280509
$111$	−4.34482	−0.412393
$112$	4.95530	0.468232
$113$	−7.57071	−0.712193	−0.356096	−	0.934449i	$-0.615893\pi$
$113$	−7.57071	−0.712193	−0.356096	+	0.934449i	$0.615893\pi$
$114$	−3.01960	−0.282811
$115$	0.134992	0.0125881
$116$	−1.00000	−0.0928477
$117$	0.307283	0.0284083
$118$	−5.59801	−0.515339
$119$	−26.9831	−2.47354
$120$	−0.239429	−0.0218568
$121$	−6.25027	−0.568206
$122$	8.08971	0.732408
$123$	12.1233	1.09312
$124$	−2.56495	−0.230340
$125$	1.34746	0.120521
$126$	0.722681	0.0643816
$127$	5.79052	0.513825	0.256913	−	0.966435i	$-0.417295\pi$
$127$	5.79052	0.513825	0.256913	+	0.966435i	$0.417295\pi$
$128$	1.00000	0.0883883
$129$	−13.5459	−1.19265
$130$	−0.284427	−0.0249459
$131$	2.59145	0.226416	0.113208	−	0.993571i	$-0.463887\pi$
$131$	2.59145	0.226416	0.113208	+	0.993571i	$0.463887\pi$
$132$	3.86547	0.336446
$133$	−8.43627	−0.731518
$134$	8.63159	0.745655
$135$	0.683370	0.0588151
$136$	−5.44530	−0.466931
$137$	−10.6882	−0.913156	−0.456578	−	0.889683i	$-0.650925\pi$
$137$	−10.6882	−0.913156	−0.456578	+	0.889683i	$0.650925\pi$
$138$	−1.77365	−0.150983
$139$	−13.3500	−1.13233	−0.566167	−	0.824290i	$-0.691574\pi$
$139$	−13.3500	−1.13233	−0.566167	+	0.824290i	$0.691574\pi$
$140$	−0.668927	−0.0565347
$141$	1.59996	0.134741
$142$	3.37742	0.283427
$143$	4.59194	0.383997
$144$	0.145840	0.0121533
$145$	0.134992	0.0112105
$146$	−2.22792	−0.184384
$147$	31.1365	2.56809
$148$	−2.44965	−0.201360
$149$	2.38806	0.195638	0.0978189	−	0.995204i	$-0.468813\pi$
$149$	2.38806	0.195638	0.0978189	+	0.995204i	$0.468813\pi$
$150$	−8.83594	−0.721451
$151$	0.144628	0.0117697	0.00588485	−	0.999983i	$-0.498127\pi$
$151$	0.144628	0.0117697	0.00588485	+	0.999983i	$0.498127\pi$
$152$	−1.70247	−0.138089
$153$	−0.794144	−0.0642027
$154$	10.7995	0.870250
$155$	0.346249	0.0278114
$156$	3.73706	0.299204
$157$	12.1438	0.969180	0.484590	−	0.874741i	$-0.338969\pi$
$157$	12.1438	0.969180	0.484590	+	0.874741i	$0.338969\pi$
$158$	9.54644	0.759474
$159$	−4.84106	−0.383921
$160$	−0.134992	−0.0106721
$161$	−4.95530	−0.390532
$162$	−9.41625	−0.739811
$163$	6.56116	0.513909	0.256955	−	0.966423i	$-0.417281\pi$
$163$	6.56116	0.513909	0.256955	+	0.966423i	$0.417281\pi$
$164$	6.83522	0.533741
$165$	−0.521809	−0.0406228
$166$	−15.4624	−1.20011
$167$	−11.2113	−0.867559	−0.433780	−	0.901019i	$-0.642820\pi$
$167$	−11.2113	−0.867559	−0.433780	+	0.901019i	$0.642820\pi$
$168$	8.78898	0.678084
$169$	−8.56061	−0.658509
$170$	0.735074	0.0563776
$171$	−0.248289	−0.0189871
$172$	−7.63729	−0.582338
$173$	−6.12763	−0.465875	−0.232938	−	0.972492i	$-0.574834\pi$
$173$	−6.12763	−0.465875	−0.232938	+	0.972492i	$0.574834\pi$
$174$	−1.77365	−0.134460
$175$	−24.6862	−1.86610
$176$	2.17939	0.164278
$177$	−9.92892	−0.746303
$178$	5.04760	0.378334
$179$	4.31108	0.322225	0.161113	−	0.986936i	$-0.448492\pi$
$179$	4.31108	0.322225	0.161113	+	0.986936i	$0.448492\pi$
$180$	−0.0196873	−0.00146740
$181$	19.3720	1.43991	0.719956	−	0.694020i	$-0.244162\pi$
$181$	19.3720	1.43991	0.719956	+	0.694020i	$0.244162\pi$
$182$	10.4407	0.773919
$183$	14.3483	1.06066
$184$	−1.00000	−0.0737210
$185$	0.330684	0.0243124
$186$	−4.54934	−0.333573
$187$	−11.8674	−0.867832
$188$	0.902069	0.0657901
$189$	−25.0851	−1.82468
$190$	0.229821	0.0166730
$191$	−1.72801	−0.125034	−0.0625171	−	0.998044i	$-0.519913\pi$
$191$	−1.72801	−0.125034	−0.0625171	+	0.998044i	$0.519913\pi$
$192$	1.77365	0.128002
$193$	−2.14013	−0.154050	−0.0770249	−	0.997029i	$-0.524542\pi$
$193$	−2.14013	−0.154050	−0.0770249	+	0.997029i	$0.524542\pi$
$194$	7.70718	0.553344
$195$	−0.504474	−0.0361261
$196$	17.5550	1.25393
$197$	−18.4461	−1.31423	−0.657114	−	0.753791i	$-0.728223\pi$
$197$	−18.4461	−1.31423	−0.657114	+	0.753791i	$0.728223\pi$
$198$	0.317842	0.0225881
$199$	0.205915	0.0145969	0.00729847	−	0.999973i	$-0.497677\pi$
$199$	0.205915	0.0145969	0.00729847	+	0.999973i	$0.497677\pi$
$200$	−4.98178	−0.352265
$201$	15.3094	1.07984
$202$	1.31545	0.0925550
$203$	−4.95530	−0.347794
$204$	−9.65807	−0.676200
$205$	−0.922702	−0.0644443
$206$	−4.64515	−0.323643
$207$	−0.145840	−0.0101366
$208$	2.10699	0.146093
$209$	−3.71035	−0.256650
$210$	−1.18644	−0.0818724
$211$	−7.35825	−0.506563	−0.253282	−	0.967393i	$-0.581510\pi$
$211$	−7.35825	−0.506563	−0.253282	+	0.967393i	$0.581510\pi$
$212$	−2.72943	−0.187458
$213$	5.99037	0.410454
$214$	−4.64518	−0.317538
$215$	1.03098	0.0703119
$216$	−5.06229	−0.344445
$217$	−12.7101	−0.862819
$218$	10.5094	0.711787
$219$	−3.95156	−0.267021
$220$	−0.294201	−0.0198350
$221$	−11.4732	−0.771769
$222$	−4.34482	−0.291606
$223$	−16.9128	−1.13257	−0.566283	−	0.824211i	$-0.691619\pi$
$223$	−16.9128	−1.13257	−0.566283	+	0.824211i	$0.691619\pi$
$224$	4.95530	0.331090
$225$	−0.726543	−0.0484362
$226$	−7.57071	−0.503596
$227$	−1.57932	−0.104823	−0.0524115	−	0.998626i	$-0.516691\pi$
$227$	−1.57932	−0.104823	−0.0524115	+	0.998626i	$0.516691\pi$
$228$	−3.01960	−0.199978
$229$	19.0826	1.26101	0.630506	−	0.776184i	$-0.282847\pi$
$229$	19.0826	1.26101	0.630506	+	0.776184i	$0.282847\pi$
$230$	0.134992	0.00890113
$231$	19.1546	1.26028
$232$	−1.00000	−0.0656532
$233$	−7.42009	−0.486106	−0.243053	−	0.970013i	$-0.578149\pi$
$233$	−7.42009	−0.486106	−0.243053	+	0.970013i	$0.578149\pi$
$234$	0.307283	0.0200877
$235$	−0.121772	−0.00794355
$236$	−5.59801	−0.364399
$237$	16.9321	1.09986
$238$	−26.9831	−1.74906
$239$	−1.36914	−0.0885624	−0.0442812	−	0.999019i	$-0.514100\pi$
$239$	−1.36914	−0.0885624	−0.0442812	+	0.999019i	$0.514100\pi$
$240$	−0.239429	−0.0154551
$241$	13.2520	0.853635	0.426817	−	0.904338i	$-0.359635\pi$
$241$	13.2520	0.853635	0.426817	+	0.904338i	$0.359635\pi$
$242$	−6.25027	−0.401783
$243$	−1.51429	−0.0971419
$244$	8.08971	0.517891
$245$	−2.36979	−0.151400
$246$	12.1233	0.772953
$247$	−3.58709	−0.228241
$248$	−2.56495	−0.162875
$249$	−27.4249	−1.73798
$250$	1.34746	0.0852210
$251$	22.5558	1.42371	0.711855	−	0.702327i	$-0.247855\pi$
$251$	22.5558	1.42371	0.711855	+	0.702327i	$0.247855\pi$
$252$	0.722681	0.0455246
$253$	−2.17939	−0.137017
$254$	5.79052	0.363329
$255$	1.30377	0.0816450
$256$	1.00000	0.0625000
$257$	−20.6081	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$257$	−20.6081	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$258$	−13.5459	−0.843331
$259$	−12.1387	−0.754265
$260$	−0.284427	−0.0176394
$261$	−0.145840	−0.00902727
$262$	2.59145	0.160100
$263$	9.47769	0.584420	0.292210	−	0.956354i	$-0.405609\pi$
$263$	9.47769	0.584420	0.292210	+	0.956354i	$0.405609\pi$
$264$	3.86547	0.237903
$265$	0.368453	0.0226339
$266$	−8.43627	−0.517261
$267$	8.95269	0.547896
$268$	8.63159	0.527258
$269$	−12.5482	−0.765080	−0.382540	−	0.923939i	$-0.624951\pi$
$269$	−12.5482	−0.765080	−0.382540	+	0.923939i	$0.624951\pi$
$270$	0.683370	0.0415885
$271$	5.44356	0.330672	0.165336	−	0.986237i	$-0.447129\pi$
$271$	5.44356	0.330672	0.165336	+	0.986237i	$0.447129\pi$
$272$	−5.44530	−0.330170
$273$	18.5182	1.12078
$274$	−10.6882	−0.645699
$275$	−10.8572	−0.654715
$276$	−1.77365	−0.106761
$277$	24.5864	1.47726	0.738628	−	0.674113i	$-0.235474\pi$
$277$	24.5864	1.47726	0.738628	+	0.674113i	$0.235474\pi$
$278$	−13.3500	−0.800682
$279$	−0.374073	−0.0223952
$280$	−0.668927	−0.0399761
$281$	−13.5983	−0.811205	−0.405603	−	0.914050i	$-0.632938\pi$
$281$	−13.5983	−0.811205	−0.405603	+	0.914050i	$0.632938\pi$
$282$	1.59996	0.0952760
$283$	30.7884	1.83018	0.915091	−	0.403247i	$-0.132118\pi$
$283$	30.7884	1.83018	0.915091	+	0.403247i	$0.132118\pi$
$284$	3.37742	0.200413
$285$	0.407622	0.0241455
$286$	4.59194	0.271527
$287$	33.8705	1.99932
$288$	0.145840	0.00859371
$289$	12.6513	0.744196
$290$	0.134992	0.00792702
$291$	13.6699	0.801341
$292$	−2.22792	−0.130379
$293$	17.8019	1.04000	0.520000	−	0.854166i	$-0.325932\pi$
$293$	17.8019	1.04000	0.520000	+	0.854166i	$0.325932\pi$
$294$	31.1365	1.81591
$295$	0.755688	0.0439979
$296$	−2.44965	−0.142383
$297$	−11.0327	−0.640181
$298$	2.38806	0.138337
$299$	−2.10699	−0.121850
$300$	−8.83594	−0.510143
$301$	−37.8451	−2.18135
$302$	0.144628	0.00832243
$303$	2.33316	0.134036
$304$	−1.70247	−0.0976436
$305$	−1.09205	−0.0625305
$306$	−0.794144	−0.0453982
$307$	24.4778	1.39702	0.698510	−	0.715600i	$-0.253847\pi$
$307$	24.4778	1.39702	0.698510	+	0.715600i	$0.253847\pi$
$308$	10.7995	0.615360
$309$	−8.23887	−0.468693
$310$	0.346249	0.0196656
$311$	20.4737	1.16096	0.580478	−	0.814276i	$-0.302866\pi$
$311$	20.4737	1.16096	0.580478	+	0.814276i	$0.302866\pi$
$312$	3.73706	0.211569
$313$	21.7868	1.23146	0.615730	−	0.787957i	$-0.288861\pi$
$313$	21.7868	1.23146	0.615730	+	0.787957i	$0.288861\pi$
$314$	12.1438	0.685314
$315$	−0.0975564	−0.00549668
$316$	9.54644	0.537029
$317$	−17.8529	−1.00272	−0.501360	−	0.865239i	$-0.667167\pi$
$317$	−17.8529	−1.00272	−0.501360	+	0.865239i	$0.667167\pi$
$318$	−4.84106	−0.271473
$319$	−2.17939	−0.122022
$320$	−0.134992	−0.00754630
$321$	−8.23892	−0.459852
$322$	−4.95530	−0.276148
$323$	9.27049	0.515824
$324$	−9.41625	−0.523125
$325$	−10.4965	−0.582243
$326$	6.56116	0.363389
$327$	18.6400	1.03080
$328$	6.83522	0.377412
$329$	4.47002	0.246440
$330$	−0.521809	−0.0287247
$331$	−13.2751	−0.729664	−0.364832	−	0.931073i	$-0.618874\pi$
$331$	−13.2751	−0.729664	−0.364832	+	0.931073i	$0.618874\pi$
$332$	−15.4624	−0.848609
$333$	−0.357257	−0.0195776
$334$	−11.2113	−0.613457
$335$	−1.16520	−0.0636615
$336$	8.78898	0.479478
$337$	24.8811	1.35536	0.677681	−	0.735356i	$-0.262985\pi$
$337$	24.8811	1.35536	0.677681	+	0.735356i	$0.262985\pi$
$338$	−8.56061	−0.465636
$339$	−13.4278	−0.729298
$340$	0.735074	0.0398650
$341$	−5.59003	−0.302717
$342$	−0.248289	−0.0134259
$343$	52.3032	2.82411
$344$	−7.63729	−0.411775
$345$	0.239429	0.0128904
$346$	−6.12763	−0.329423
$347$	−6.57340	−0.352879	−0.176439	−	0.984312i	$-0.556458\pi$
$347$	−6.57340	−0.352879	−0.176439	+	0.984312i	$0.556458\pi$
$348$	−1.77365	−0.0950777
$349$	−21.1063	−1.12980	−0.564898	−	0.825160i	$-0.691085\pi$
$349$	−21.1063	−1.12980	−0.564898	+	0.825160i	$0.691085\pi$
$350$	−24.6862	−1.31953
$351$	−10.6662	−0.569318
$352$	2.17939	0.116162
$353$	16.4039	0.873093	0.436547	−	0.899682i	$-0.356201\pi$
$353$	16.4039	0.873093	0.436547	+	0.899682i	$0.356201\pi$
$354$	−9.92892	−0.527716
$355$	−0.455926	−0.0241981
$356$	5.04760	0.267522
$357$	−47.8586	−2.53295
$358$	4.31108	0.227848
$359$	−18.5796	−0.980593	−0.490297	−	0.871556i	$-0.663112\pi$
$359$	−18.5796	−0.980593	−0.490297	+	0.871556i	$0.663112\pi$
$360$	−0.0196873	−0.00103761
$361$	−16.1016	−0.847452
$362$	19.3720	1.01817
$363$	−11.0858	−0.581854
$364$	10.4407	0.547244
$365$	0.300752	0.0157421
$366$	14.3483	0.749999
$367$	−1.19082	−0.0621605	−0.0310802	−	0.999517i	$-0.509895\pi$
$367$	−1.19082	−0.0621605	−0.0310802	+	0.999517i	$0.509895\pi$
$368$	−1.00000	−0.0521286
$369$	0.996849	0.0518939
$370$	0.330684	0.0171914
$371$	−13.5252	−0.702191
$372$	−4.54934	−0.235872
$373$	−33.1221	−1.71500	−0.857499	−	0.514485i	$-0.827983\pi$
$373$	−33.1221	−1.71500	−0.857499	+	0.514485i	$0.827983\pi$
$374$	−11.8674	−0.613650
$375$	2.38993	0.123415
$376$	0.902069	0.0465207
$377$	−2.10699	−0.108515
$378$	−25.0851	−1.29024
$379$	−11.2907	−0.579966	−0.289983	−	0.957032i	$-0.593650\pi$
$379$	−11.2907	−0.579966	−0.289983	+	0.957032i	$0.593650\pi$
$380$	0.229821	0.0117896
$381$	10.2704	0.526166
$382$	−1.72801	−0.0884125
$383$	−26.8436	−1.37164	−0.685822	−	0.727769i	$-0.740557\pi$
$383$	−26.8436	−1.37164	−0.685822	+	0.727769i	$0.740557\pi$
$384$	1.77365	0.0905113
$385$	−1.45785	−0.0742990
$386$	−2.14013	−0.108930
$387$	−1.11382	−0.0566188
$388$	7.70718	0.391273
$389$	6.56507	0.332862	0.166431	−	0.986053i	$-0.446776\pi$
$389$	6.56507	0.332862	0.166431	+	0.986053i	$0.446776\pi$
$390$	−0.504474	−0.0255450
$391$	5.44530	0.275381
$392$	17.5550	0.886661
$393$	4.59632	0.231854
$394$	−18.4461	−0.929299
$395$	−1.28870	−0.0648414
$396$	0.317842	0.0159722
$397$	−36.5300	−1.83339	−0.916694	−	0.399590i	$-0.869153\pi$
$397$	−36.5300	−1.83339	−0.916694	+	0.399590i	$0.869153\pi$
$398$	0.205915	0.0103216
$399$	−14.9630	−0.749087
$400$	−4.98178	−0.249089
$401$	35.8895	1.79224	0.896118	−	0.443816i	$-0.146376\pi$
$401$	35.8895	1.79224	0.896118	+	0.443816i	$0.146376\pi$
$402$	15.3094	0.763565
$403$	−5.40432	−0.269208
$404$	1.31545	0.0654463
$405$	1.27112	0.0631625
$406$	−4.95530	−0.245927
$407$	−5.33874	−0.264631
$408$	−9.65807	−0.478146
$409$	−39.1844	−1.93754	−0.968772	−	0.247955i	$-0.920241\pi$
$409$	−39.1844	−1.93754	−0.968772	+	0.247955i	$0.920241\pi$
$410$	−0.922702	−0.0455690
$411$	−18.9572	−0.935089
$412$	−4.64515	−0.228850
$413$	−27.7398	−1.36499
$414$	−0.145840	−0.00716765
$415$	2.08730	0.102462
$416$	2.10699	0.103303
$417$	−23.6783	−1.15953
$418$	−3.71035	−0.181479
$419$	−26.1186	−1.27598	−0.637988	−	0.770046i	$-0.720233\pi$
$419$	−26.1186	−1.27598	−0.637988	+	0.770046i	$0.720233\pi$
$420$	−1.18644	−0.0578925
$421$	−19.4423	−0.947562	−0.473781	−	0.880643i	$-0.657111\pi$
$421$	−19.4423	−0.947562	−0.473781	+	0.880643i	$0.657111\pi$
$422$	−7.35825	−0.358194
$423$	0.131558	0.00639656
$424$	−2.72943	−0.132553
$425$	27.1273	1.31587
$426$	5.99037	0.290235
$427$	40.0869	1.93994
$428$	−4.64518	−0.224533
$429$	8.14450	0.393220
$430$	1.03098	0.0497181
$431$	−8.63079	−0.415731	−0.207865	−	0.978157i	$-0.566652\pi$
$431$	−8.63079	−0.415731	−0.207865	+	0.978157i	$0.566652\pi$
$432$	−5.06229	−0.243559
$433$	12.6636	0.608574	0.304287	−	0.952580i	$-0.401582\pi$
$433$	12.6636	0.608574	0.304287	+	0.952580i	$0.401582\pi$
$434$	−12.7101	−0.610105
$435$	0.239429	0.0114798
$436$	10.5094	0.503309
$437$	1.70247	0.0814404
$438$	−3.95156	−0.188813
$439$	−32.1780	−1.53577	−0.767886	−	0.640586i	$-0.778691\pi$
$439$	−32.1780	−1.53577	−0.767886	+	0.640586i	$0.778691\pi$
$440$	−0.294201	−0.0140255
$441$	2.56022	0.121915
$442$	−11.4732	−0.545723
$443$	30.1222	1.43115	0.715576	−	0.698535i	$-0.246165\pi$
$443$	30.1222	1.43115	0.715576	+	0.698535i	$0.246165\pi$
$444$	−4.34482	−0.206196
$445$	−0.681388	−0.0323009
$446$	−16.9128	−0.800846
$447$	4.23559	0.200337
$448$	4.95530	0.234116
$449$	38.5294	1.81832	0.909158	−	0.416451i	$-0.136726\pi$
$449$	38.5294	1.81832	0.909158	+	0.416451i	$0.136726\pi$
$450$	−0.726543	−0.0342496
$451$	14.8966	0.701453
$452$	−7.57071	−0.356096
$453$	0.256520	0.0120524
$454$	−1.57932	−0.0741211
$455$	−1.40942	−0.0660746
$456$	−3.01960	−0.141406
$457$	−2.00798	−0.0939292	−0.0469646	−	0.998897i	$-0.514955\pi$
$457$	−2.00798	−0.0939292	−0.0469646	+	0.998897i	$0.514955\pi$
$458$	19.0826	0.891670
$459$	27.5657	1.28666
$460$	0.134992	0.00629405
$461$	−1.12655	−0.0524687	−0.0262343	−	0.999656i	$-0.508352\pi$
$461$	−1.12655	−0.0524687	−0.0262343	+	0.999656i	$0.508352\pi$
$462$	19.1546	0.891152
$463$	1.46771	0.0682103	0.0341051	−	0.999418i	$-0.489142\pi$
$463$	1.46771	0.0682103	0.0341051	+	0.999418i	$0.489142\pi$
$464$	−1.00000	−0.0464238
$465$	0.614125	0.0284794
$466$	−7.42009	−0.343729
$467$	−4.51222	−0.208800	−0.104400	−	0.994535i	$-0.533292\pi$
$467$	−4.51222	−0.208800	−0.104400	+	0.994535i	$0.533292\pi$
$468$	0.307283	0.0142042
$469$	42.7721	1.97503
$470$	−0.121772	−0.00561694
$471$	21.5389	0.992458
$472$	−5.59801	−0.257669
$473$	−16.6446	−0.765320
$474$	16.9321	0.777716
$475$	8.48135	0.389151
$476$	−26.9831	−1.23677
$477$	−0.398061	−0.0182260
$478$	−1.36914	−0.0626231
$479$	25.4637	1.16346	0.581732	−	0.813380i	$-0.302375\pi$
$479$	25.4637	1.16346	0.581732	+	0.813380i	$0.302375\pi$
$480$	−0.239429	−0.0109284
$481$	−5.16138	−0.235338
$482$	13.2520	0.603611
$483$	−8.78898	−0.399912
$484$	−6.25027	−0.284103
$485$	−1.04041	−0.0472426
$486$	−1.51429	−0.0686897
$487$	11.8527	0.537097	0.268549	−	0.963266i	$-0.413456\pi$
$487$	11.8527	0.537097	0.268549	+	0.963266i	$0.413456\pi$
$488$	8.08971	0.366204
$489$	11.6372	0.526253
$490$	−2.36979	−0.107056
$491$	29.7685	1.34343	0.671716	−	0.740808i	$-0.265557\pi$
$491$	29.7685	1.34343	0.671716	+	0.740808i	$0.265557\pi$
$492$	12.1233	0.546560
$493$	5.44530	0.245244
$494$	−3.58709	−0.161391
$495$	−0.0429062	−0.00192849
$496$	−2.56495	−0.115170
$497$	16.7362	0.750719
$498$	−27.4249	−1.22894
$499$	1.27421	0.0570417	0.0285208	−	0.999593i	$-0.490920\pi$
$499$	1.27421	0.0570417	0.0285208	+	0.999593i	$0.490920\pi$
$500$	1.34746	0.0602604
$501$	−19.8850	−0.888396
$502$	22.5558	1.00671
$503$	9.28651	0.414065	0.207032	−	0.978334i	$-0.433619\pi$
$503$	9.28651	0.414065	0.207032	+	0.978334i	$0.433619\pi$
$504$	0.722681	0.0321908
$505$	−0.177576	−0.00790203
$506$	−2.17939	−0.0968856
$507$	−15.1835	−0.674325
$508$	5.79052	0.256913
$509$	−9.26011	−0.410447	−0.205224	−	0.978715i	$-0.565792\pi$
$509$	−9.26011	−0.410447	−0.205224	+	0.978715i	$0.565792\pi$
$510$	1.30377	0.0577317
$511$	−11.0400	−0.488382
$512$	1.00000	0.0441942
$513$	8.61841	0.380512
$514$	−20.6081	−0.908984
$515$	0.627059	0.0276315
$516$	−13.5459	−0.596325
$517$	1.96596	0.0864627
$518$	−12.1387	−0.533346
$519$	−10.8683	−0.477065
$520$	−0.284427	−0.0124729
$521$	14.3471	0.628559	0.314279	−	0.949331i	$-0.398237\pi$
$521$	14.3471	0.628559	0.314279	+	0.949331i	$0.398237\pi$
$522$	−0.145840	−0.00638325
$523$	−29.0416	−1.26990	−0.634949	−	0.772554i	$-0.718979\pi$
$523$	−29.0416	−1.26990	−0.634949	+	0.772554i	$0.718979\pi$
$524$	2.59145	0.113208
$525$	−43.7847	−1.91092
$526$	9.47769	0.413247
$527$	13.9670	0.608410
$528$	3.86547	0.168223
$529$	1.00000	0.0434783
$530$	0.368453	0.0160046
$531$	−0.816414	−0.0354294
$532$	−8.43627	−0.365759
$533$	14.4017	0.623807
$534$	8.95269	0.387421
$535$	0.627063	0.0271103
$536$	8.63159	0.372828
$537$	7.64636	0.329965
$538$	−12.5482	−0.540993
$539$	38.2591	1.64794
$540$	0.683370	0.0294075
$541$	−18.9698	−0.815575	−0.407788	−	0.913077i	$-0.633700\pi$
$541$	−18.9698	−0.815575	−0.407788	+	0.913077i	$0.633700\pi$
$542$	5.44356	0.233821
$543$	34.3592	1.47450
$544$	−5.44530	−0.233465
$545$	−1.41869	−0.0607700
$546$	18.5182	0.792508
$547$	19.0963	0.816498	0.408249	−	0.912871i	$-0.366140\pi$
$547$	19.0963	0.816498	0.408249	+	0.912871i	$0.366140\pi$
$548$	−10.6882	−0.456578
$549$	1.17980	0.0503528
$550$	−10.8572	−0.462954
$551$	1.70247	0.0725279
$552$	−1.77365	−0.0754916
$553$	47.3055	2.01163
$554$	24.5864	1.04458
$555$	0.586518	0.0248963
$556$	−13.3500	−0.566167
$557$	−5.51734	−0.233777	−0.116889	−	0.993145i	$-0.537292\pi$
$557$	−5.51734	−0.233777	−0.116889	+	0.993145i	$0.537292\pi$
$558$	−0.374073	−0.0158358
$559$	−16.0917	−0.680605
$560$	−0.668927	−0.0282673
$561$	−21.0487	−0.888676
$562$	−13.5983	−0.573609
$563$	−25.6072	−1.07922	−0.539608	−	0.841916i	$-0.681428\pi$
$563$	−25.6072	−1.07922	−0.539608	+	0.841916i	$0.681428\pi$
$564$	1.59996	0.0673703
$565$	1.02199	0.0429954
$566$	30.7884	1.29413
$567$	−46.6603	−1.95955
$568$	3.37742	0.141714
$569$	6.77820	0.284157	0.142078	−	0.989855i	$-0.454621\pi$
$569$	6.77820	0.284157	0.142078	+	0.989855i	$0.454621\pi$
$570$	0.407622	0.0170734
$571$	10.0555	0.420809	0.210405	−	0.977614i	$-0.432522\pi$
$571$	10.0555	0.420809	0.210405	+	0.977614i	$0.432522\pi$
$572$	4.59194	0.191999
$573$	−3.06488	−0.128037
$574$	33.8705	1.41373
$575$	4.98178	0.207754
$576$	0.145840	0.00607667
$577$	36.4024	1.51545	0.757727	−	0.652572i	$-0.226310\pi$
$577$	36.4024	1.51545	0.757727	+	0.652572i	$0.226310\pi$
$578$	12.6513	0.526226
$579$	−3.79584	−0.157750
$580$	0.134992	0.00560525
$581$	−76.6208	−3.17877
$582$	13.6699	0.566634
$583$	−5.94849	−0.246361
$584$	−2.22792	−0.0921920
$585$	−0.0414808	−0.00171502
$586$	17.8019	0.735391
$587$	21.1664	0.873630	0.436815	−	0.899551i	$-0.356106\pi$
$587$	21.1664	0.873630	0.436815	+	0.899551i	$0.356106\pi$
$588$	31.1365	1.28405
$589$	4.36677	0.179930
$590$	0.755688	0.0311112
$591$	−32.7169	−1.34579
$592$	−2.44965	−0.100680
$593$	19.4872	0.800242	0.400121	−	0.916462i	$-0.368968\pi$
$593$	19.4872	0.800242	0.400121	+	0.916462i	$0.368968\pi$
$594$	−11.0327	−0.452676
$595$	3.64251	0.149328
$596$	2.38806	0.0978189
$597$	0.365222	0.0149475
$598$	−2.10699	−0.0861610
$599$	−5.99383	−0.244901	−0.122451	−	0.992475i	$-0.539075\pi$
$599$	−5.99383	−0.244901	−0.122451	+	0.992475i	$0.539075\pi$
$600$	−8.83594	−0.360726
$601$	8.02261	0.327249	0.163625	−	0.986523i	$-0.447681\pi$
$601$	8.02261	0.327249	0.163625	+	0.986523i	$0.447681\pi$
$602$	−37.8451	−1.54245
$603$	1.25883	0.0512636
$604$	0.144628	0.00588485
$605$	0.843738	0.0343028
$606$	2.33316	0.0947780
$607$	36.7362	1.49108	0.745538	−	0.666463i	$-0.232193\pi$
$607$	36.7362	1.49108	0.745538	+	0.666463i	$0.232193\pi$
$608$	−1.70247	−0.0690445
$609$	−8.78898	−0.356147
$610$	−1.09205	−0.0442158
$611$	1.90065	0.0768919
$612$	−0.794144	−0.0321014
$613$	−47.4815	−1.91776	−0.958879	−	0.283815i	$-0.908400\pi$
$613$	−47.4815	−1.91776	−0.958879	+	0.283815i	$0.908400\pi$
$614$	24.4778	0.987843
$615$	−1.63655	−0.0659921
$616$	10.7995	0.435125
$617$	6.96049	0.280219	0.140109	−	0.990136i	$-0.455255\pi$
$617$	6.96049	0.280219	0.140109	+	0.990136i	$0.455255\pi$
$618$	−8.23887	−0.331416
$619$	21.4801	0.863358	0.431679	−	0.902027i	$-0.357921\pi$
$619$	21.4801	0.863358	0.431679	+	0.902027i	$0.357921\pi$
$620$	0.346249	0.0139057
$621$	5.06229	0.203143
$622$	20.4737	0.820919
$623$	25.0124	1.00210
$624$	3.73706	0.149602
$625$	24.7270	0.989080
$626$	21.7868	0.870774
$627$	−6.58087	−0.262815
$628$	12.1438	0.484590
$629$	13.3391	0.531864
$630$	−0.0975564	−0.00388674
$631$	40.0234	1.59331	0.796653	−	0.604436i	$-0.206602\pi$
$631$	40.0234	1.59331	0.796653	+	0.604436i	$0.206602\pi$
$632$	9.54644	0.379737
$633$	−13.0510	−0.518730
$634$	−17.8529	−0.709030
$635$	−0.781675	−0.0310198
$636$	−4.84106	−0.191961
$637$	36.9881	1.46552
$638$	−2.17939	−0.0862828
$639$	0.492564	0.0194855
$640$	−0.134992	−0.00533604
$641$	36.5759	1.44466	0.722331	−	0.691547i	$-0.243071\pi$
$641$	36.5759	1.44466	0.722331	+	0.691547i	$0.243071\pi$
$642$	−8.23892	−0.325164
$643$	35.4742	1.39897	0.699484	−	0.714649i	$-0.253413\pi$
$643$	35.4742	1.39897	0.699484	+	0.714649i	$0.253413\pi$
$644$	−4.95530	−0.195266
$645$	1.82859	0.0720007
$646$	9.27049	0.364743
$647$	42.3164	1.66363	0.831814	−	0.555054i	$-0.187302\pi$
$647$	42.3164	1.66363	0.831814	+	0.555054i	$0.187302\pi$
$648$	−9.41625	−0.369905
$649$	−12.2002	−0.478901
$650$	−10.4965	−0.411708
$651$	−22.5433	−0.883543
$652$	6.56116	0.256955
$653$	−37.2056	−1.45597	−0.727983	−	0.685595i	$-0.759542\pi$
$653$	−37.2056	−1.45597	−0.727983	+	0.685595i	$0.759542\pi$
$654$	18.6400	0.728883
$655$	−0.349825	−0.0136688
$656$	6.83522	0.266870
$657$	−0.324920	−0.0126763
$658$	4.47002	0.174260
$659$	−44.5149	−1.73405	−0.867026	−	0.498262i	$-0.833972\pi$
$659$	−44.5149	−1.73405	−0.867026	+	0.498262i	$0.833972\pi$
$660$	−0.521809	−0.0203114
$661$	−37.7685	−1.46903	−0.734513	−	0.678595i	$-0.762589\pi$
$661$	−37.7685	−1.46903	−0.734513	+	0.678595i	$0.762589\pi$
$662$	−13.2751	−0.515950
$663$	−20.3494	−0.790306
$664$	−15.4624	−0.600057
$665$	1.13883	0.0441620
$666$	−0.357257	−0.0138434
$667$	1.00000	0.0387202
$668$	−11.2113	−0.433780
$669$	−29.9975	−1.15977
$670$	−1.16520	−0.0450155
$671$	17.6306	0.680622
$672$	8.78898	0.339042
$673$	22.4916	0.866989	0.433495	−	0.901156i	$-0.357280\pi$
$673$	22.4916	0.866989	0.433495	+	0.901156i	$0.357280\pi$
$674$	24.8811	0.958386
$675$	25.2192	0.970687
$676$	−8.56061	−0.329254
$677$	−17.8955	−0.687779	−0.343889	−	0.939010i	$-0.611745\pi$
$677$	−17.8955	−0.687779	−0.343889	+	0.939010i	$0.611745\pi$
$678$	−13.4278	−0.515692
$679$	38.1914	1.46565
$680$	0.735074	0.0281888
$681$	−2.80116	−0.107341
$682$	−5.59003	−0.214053
$683$	−23.7865	−0.910163	−0.455082	−	0.890450i	$-0.650390\pi$
$683$	−23.7865	−0.910163	−0.455082	+	0.890450i	$0.650390\pi$
$684$	−0.248289	−0.00949357
$685$	1.44283	0.0551276
$686$	52.3032	1.99695
$687$	33.8458	1.29130
$688$	−7.63729	−0.291169
$689$	−5.75088	−0.219091
$690$	0.239429	0.00911492
$691$	50.9397	1.93784	0.968918	−	0.247381i	$-0.0795700\pi$
$691$	50.9397	1.93784	0.968918	+	0.247381i	$0.0795700\pi$
$692$	−6.12763	−0.232938
$693$	1.57500	0.0598294
$694$	−6.57340	−0.249523
$695$	1.80215	0.0683595
$696$	−1.77365	−0.0672301
$697$	−37.2198	−1.40980
$698$	−21.1063	−0.798887
$699$	−13.1607	−0.497782
$700$	−24.6862	−0.933051
$701$	29.9685	1.13189	0.565947	−	0.824441i	$-0.308511\pi$
$701$	29.9685	1.13189	0.565947	+	0.824441i	$0.308511\pi$
$702$	−10.6662	−0.402568
$703$	4.17047	0.157292
$704$	2.17939	0.0821388
$705$	−0.215982	−0.00813434
$706$	16.4039	0.617370
$707$	6.51847	0.245152
$708$	−9.92892	−0.373152
$709$	50.2701	1.88793	0.943966	−	0.330043i	$-0.107063\pi$
$709$	50.2701	1.88793	0.943966	+	0.330043i	$0.107063\pi$
$710$	−0.455926	−0.0171106
$711$	1.39225	0.0522136
$712$	5.04760	0.189167
$713$	2.56495	0.0960583
$714$	−47.8586	−1.79106
$715$	−0.619876	−0.0231821
$716$	4.31108	0.161113
$717$	−2.42838	−0.0906895
$718$	−18.5796	−0.693384
$719$	−42.7981	−1.59610	−0.798049	−	0.602592i	$-0.794135\pi$
$719$	−42.7981	−1.59610	−0.798049	+	0.602592i	$0.794135\pi$
$720$	−0.0196873	−0.000733702 0
$721$	−23.0181	−0.857239
$722$	−16.1016	−0.599239
$723$	23.5044	0.874138
$724$	19.3720	0.719956
$725$	4.98178	0.185019
$726$	−11.0858	−0.411433
$727$	0.543743	0.0201663	0.0100832	−	0.999949i	$-0.496790\pi$
$727$	0.543743	0.0201663	0.0100832	+	0.999949i	$0.496790\pi$
$728$	10.4407	0.386960
$729$	25.5629	0.946775
$730$	0.300752	0.0111313
$731$	41.5874	1.53816
$732$	14.3483	0.530329
$733$	−31.8986	−1.17820	−0.589101	−	0.808059i	$-0.700518\pi$
$733$	−31.8986	−1.17820	−0.589101	+	0.808059i	$0.700518\pi$
$734$	−1.19082	−0.0439541
$735$	−4.20318	−0.155037
$736$	−1.00000	−0.0368605
$737$	18.8116	0.692933
$738$	0.996849	0.0366945
$739$	34.2988	1.26170	0.630851	−	0.775904i	$-0.282706\pi$
$739$	34.2988	1.26170	0.630851	+	0.775904i	$0.282706\pi$
$740$	0.330684	0.0121562
$741$	−6.36225	−0.233723
$742$	−13.5252	−0.496524
$743$	−35.2435	−1.29296	−0.646478	−	0.762932i	$-0.723759\pi$
$743$	−35.2435	−1.29296	−0.646478	+	0.762932i	$0.723759\pi$
$744$	−4.54934	−0.166787
$745$	−0.322370	−0.0118107
$746$	−33.1221	−1.21269
$747$	−2.25504	−0.0825075
$748$	−11.8674	−0.433916
$749$	−23.0182	−0.841068
$750$	2.38993	0.0872679
$751$	−13.1638	−0.480352	−0.240176	−	0.970729i	$-0.577205\pi$
$751$	−13.1638	−0.480352	−0.240176	+	0.970729i	$0.577205\pi$
$752$	0.902069	0.0328951
$753$	40.0061	1.45790
$754$	−2.10699	−0.0767319
$755$	−0.0195237	−0.000710541 0
$756$	−25.0851	−0.912338
$757$	46.3122	1.68324	0.841622	−	0.540067i	$-0.181601\pi$
$757$	46.3122	1.68324	0.841622	+	0.540067i	$0.181601\pi$
$758$	−11.2907	−0.410098
$759$	−3.86547	−0.140308
$760$	0.229821	0.00833648
$761$	8.39699	0.304391	0.152195	−	0.988350i	$-0.451366\pi$
$761$	8.39699	0.304391	0.152195	+	0.988350i	$0.451366\pi$
$762$	10.2704	0.372056
$763$	52.0773	1.88532
$764$	−1.72801	−0.0625171
$765$	0.107203	0.00387594
$766$	−26.8436	−0.969899
$767$	−11.7949	−0.425890
$768$	1.77365	0.0640011
$769$	−19.0187	−0.685832	−0.342916	−	0.939366i	$-0.611415\pi$
$769$	−19.0187	−0.685832	−0.342916	+	0.939366i	$0.611415\pi$
$770$	−1.45785	−0.0525373
$771$	−36.5516	−1.31637
$772$	−2.14013	−0.0770249
$773$	−19.7986	−0.712108	−0.356054	−	0.934465i	$-0.615878\pi$
$773$	−19.7986	−0.712108	−0.356054	+	0.934465i	$0.615878\pi$
$774$	−1.11382	−0.0400355
$775$	12.7780	0.459000
$776$	7.70718	0.276672
$777$	−21.5299	−0.772381
$778$	6.56507	0.235369
$779$	−11.6368	−0.416931
$780$	−0.504474	−0.0180631
$781$	7.36072	0.263387
$782$	5.44530	0.194724
$783$	5.06229	0.180911
$784$	17.5550	0.626964
$785$	−1.63932	−0.0585098
$786$	4.59632	0.163945
$787$	14.1260	0.503537	0.251769	−	0.967787i	$-0.418988\pi$
$787$	14.1260	0.503537	0.251769	+	0.967787i	$0.418988\pi$
$788$	−18.4461	−0.657114
$789$	16.8101	0.598456
$790$	−1.28870	−0.0458498
$791$	−37.5151	−1.33389
$792$	0.317842	0.0112940
$793$	17.0449	0.605282
$794$	−36.5300	−1.29640
$795$	0.653506	0.0231775
$796$	0.205915	0.00729847
$797$	46.3093	1.64036	0.820179	−	0.572107i	$-0.193874\pi$
$797$	46.3093	1.64036	0.820179	+	0.572107i	$0.193874\pi$
$798$	−14.9630	−0.529685
$799$	−4.91204	−0.173775
$800$	−4.98178	−0.176132
$801$	0.736143	0.0260103
$802$	35.8895	1.26730
$803$	−4.85550	−0.171347
$804$	15.3094	0.539922
$805$	0.668927	0.0235766
$806$	−5.40432	−0.190359
$807$	−22.2562	−0.783456
$808$	1.31545	0.0462775
$809$	5.52012	0.194077	0.0970385	−	0.995281i	$-0.469063\pi$
$809$	5.52012	0.194077	0.0970385	+	0.995281i	$0.469063\pi$
$810$	1.27112	0.0446627
$811$	−12.6374	−0.443758	−0.221879	−	0.975074i	$-0.571219\pi$
$811$	−12.6374	−0.443758	−0.221879	+	0.975074i	$0.571219\pi$
$812$	−4.95530	−0.173897
$813$	9.65497	0.338615
$814$	−5.33874	−0.187123
$815$	−0.885706	−0.0310249
$816$	−9.65807	−0.338100
$817$	13.0023	0.454893
$818$	−39.1844	−1.37005
$819$	1.52268	0.0532067
$820$	−0.922702	−0.0322221
$821$	−0.512545	−0.0178880	−0.00894398	−	0.999960i	$-0.502847\pi$
$821$	−0.512545	−0.0178880	−0.00894398	+	0.999960i	$0.502847\pi$
$822$	−18.9572	−0.661208
$823$	−33.4753	−1.16687	−0.583437	−	0.812158i	$-0.698293\pi$
$823$	−33.4753	−1.16687	−0.583437	+	0.812158i	$0.698293\pi$
$824$	−4.64515	−0.161821
$825$	−19.2569	−0.670440
$826$	−27.7398	−0.965192
$827$	50.7138	1.76349	0.881746	−	0.471725i	$-0.156368\pi$
$827$	50.7138	1.76349	0.881746	+	0.471725i	$0.156368\pi$
$828$	−0.145840	−0.00506829
$829$	35.5468	1.23459	0.617295	−	0.786731i	$-0.288228\pi$
$829$	35.5468	1.23459	0.617295	+	0.786731i	$0.288228\pi$
$830$	2.08730	0.0724514
$831$	43.6078	1.51274
$832$	2.10699	0.0730466
$833$	−95.5923	−3.31208
$834$	−23.6783	−0.819913
$835$	1.51344	0.0523749
$836$	−3.71035	−0.128325
$837$	12.9845	0.448811
$838$	−26.1186	−0.902252
$839$	19.8140	0.684055	0.342027	−	0.939690i	$-0.388886\pi$
$839$	19.8140	0.684055	0.342027	+	0.939690i	$0.388886\pi$
$840$	−1.18644	−0.0409362
$841$	1.00000	0.0344828
$842$	−19.4423	−0.670028
$843$	−24.1186	−0.830689
$844$	−7.35825	−0.253282
$845$	1.15562	0.0397544
$846$	0.131558	0.00452305
$847$	−30.9720	−1.06421
$848$	−2.72943	−0.0937291
$849$	54.6080	1.87414
$850$	27.1273	0.930458
$851$	2.44965	0.0839729
$852$	5.99037	0.205227
$853$	5.42362	0.185701	0.0928506	−	0.995680i	$-0.470402\pi$
$853$	5.42362	0.185701	0.0928506	+	0.995680i	$0.470402\pi$
$854$	40.0869	1.37175
$855$	0.0335171	0.00114626
$856$	−4.64518	−0.158769
$857$	−15.3085	−0.522929	−0.261465	−	0.965213i	$-0.584206\pi$
$857$	−15.3085	−0.522929	−0.261465	+	0.965213i	$0.584206\pi$
$858$	8.14450	0.278049
$859$	−49.1676	−1.67758	−0.838788	−	0.544458i	$-0.816735\pi$
$859$	−49.1676	−1.67758	−0.838788	+	0.544458i	$0.816735\pi$
$860$	1.03098	0.0351560
$861$	60.0746	2.04734
$862$	−8.63079	−0.293966
$863$	50.6146	1.72294	0.861470	−	0.507809i	$-0.169544\pi$
$863$	50.6146	1.72294	0.861470	+	0.507809i	$0.169544\pi$
$864$	−5.06229	−0.172222
$865$	0.827183	0.0281251
$866$	12.6636	0.430327
$867$	22.4391	0.762071
$868$	−12.7101	−0.431410
$869$	20.8054	0.705775
$870$	0.239429	0.00811741
$871$	18.1866	0.616230
$872$	10.5094	0.355893
$873$	1.12402	0.0380422
$874$	1.70247	0.0575871
$875$	6.67708	0.225727
$876$	−3.95156	−0.133511
$877$	−19.8575	−0.670540	−0.335270	−	0.942122i	$-0.608828\pi$
$877$	−19.8575	−0.670540	−0.335270	+	0.942122i	$0.608828\pi$
$878$	−32.1780	−1.08596
$879$	31.5744	1.06498
$880$	−0.294201	−0.00991750
$881$	−0.512245	−0.0172580	−0.00862898	−	0.999963i	$-0.502747\pi$
$881$	−0.512245	−0.0172580	−0.00862898	+	0.999963i	$0.502747\pi$
$882$	2.56022	0.0862072
$883$	−17.5793	−0.591591	−0.295795	−	0.955251i	$-0.595585\pi$
$883$	−17.5793	−0.591591	−0.295795	+	0.955251i	$0.595585\pi$
$884$	−11.4732	−0.385885
$885$	1.34033	0.0450546
$886$	30.1222	1.01198
$887$	38.9015	1.30619	0.653093	−	0.757278i	$-0.273471\pi$
$887$	38.9015	1.30619	0.653093	+	0.757278i	$0.273471\pi$
$888$	−4.34482	−0.145803
$889$	28.6937	0.962357
$890$	−0.681388	−0.0228402
$891$	−20.5217	−0.687501
$892$	−16.9128	−0.566283
$893$	−1.53575	−0.0513919
$894$	4.23559	0.141659
$895$	−0.581963	−0.0194529
$896$	4.95530	0.165545
$897$	−3.73706	−0.124777
$898$	38.5294	1.28574
$899$	2.56495	0.0855460
$900$	−0.726543	−0.0242181
$901$	14.8626	0.495145
$902$	14.8966	0.496002
$903$	−67.1240	−2.23375
$904$	−7.57071	−0.251798
$905$	−2.61507	−0.0869280
$906$	0.256520	0.00852232
$907$	54.0435	1.79448	0.897242	−	0.441539i	$-0.145567\pi$
$907$	54.0435	1.79448	0.897242	+	0.441539i	$0.145567\pi$
$908$	−1.57932	−0.0524115
$909$	0.191846	0.00636313
$910$	−1.40942	−0.0467218
$911$	39.8057	1.31882	0.659412	−	0.751782i	$-0.270806\pi$
$911$	39.8057	1.31882	0.659412	+	0.751782i	$0.270806\pi$
$912$	−3.01960	−0.0999889
$913$	−33.6986	−1.11526
$914$	−2.00798	−0.0664180
$915$	−1.93691	−0.0640324
$916$	19.0826	0.630506
$917$	12.8414	0.424060
$918$	27.5657	0.909803
$919$	36.4836	1.20348	0.601741	−	0.798691i	$-0.294474\pi$
$919$	36.4836	1.20348	0.601741	+	0.798691i	$0.294474\pi$
$920$	0.134992	0.00445056
$921$	43.4151	1.43057
$922$	−1.12655	−0.0371010
$923$	7.11618	0.234232
$924$	19.1546	0.630140
$925$	12.2036	0.401252
$926$	1.46771	0.0482319
$927$	−0.677449	−0.0222503
$928$	−1.00000	−0.0328266
$929$	−40.7304	−1.33632	−0.668161	−	0.744017i	$-0.732918\pi$
$929$	−40.7304	−1.33632	−0.668161	+	0.744017i	$0.732918\pi$
$930$	0.614125	0.0201380
$931$	−29.8869	−0.979505
$932$	−7.42009	−0.243053
$933$	36.3132	1.18884
$934$	−4.51222	−0.147644
$935$	1.60201	0.0523914
$936$	0.307283	0.0100439
$937$	7.93359	0.259179	0.129589	−	0.991568i	$-0.458634\pi$
$937$	7.93359	0.259179	0.129589	+	0.991568i	$0.458634\pi$
$938$	42.7721	1.39656
$939$	38.6421	1.26104
$940$	−0.121772	−0.00397178
$941$	8.01629	0.261324	0.130662	−	0.991427i	$-0.458290\pi$
$941$	8.01629	0.261324	0.130662	+	0.991427i	$0.458290\pi$
$942$	21.5389	0.701774
$943$	−6.83522	−0.222585
$944$	−5.59801	−0.182200
$945$	3.38630	0.110156
$946$	−16.6446	−0.541163
$947$	−35.3847	−1.14985	−0.574924	−	0.818207i	$-0.694968\pi$
$947$	−35.3847	−1.14985	−0.574924	+	0.818207i	$0.694968\pi$
$948$	16.9321	0.549928
$949$	−4.69420	−0.152380
$950$	8.48135	0.275171
$951$	−31.6649	−1.02680
$952$	−26.9831	−0.874528
$953$	−56.9240	−1.84395	−0.921975	−	0.387250i	$-0.873425\pi$
$953$	−56.9240	−1.84395	−0.921975	+	0.387250i	$0.873425\pi$
$954$	−0.398061	−0.0128877
$955$	0.233268	0.00754836
$956$	−1.36914	−0.0442812
$957$	−3.86547	−0.124953
$958$	25.4637	0.822694
$959$	−52.9633	−1.71028
$960$	−0.239429	−0.00772755
$961$	−24.4210	−0.787774
$962$	−5.16138	−0.166409
$963$	−0.677453	−0.0218306
$964$	13.2520	0.426817
$965$	0.288901	0.00930004
$966$	−8.78898	−0.282781
$967$	−25.3311	−0.814592	−0.407296	−	0.913296i	$-0.633528\pi$
$967$	−25.3311	−0.814592	−0.407296	+	0.913296i	$0.633528\pi$
$968$	−6.25027	−0.200891
$969$	16.4426	0.528213
$970$	−1.04041	−0.0334056
$971$	−34.4408	−1.10526	−0.552629	−	0.833428i	$-0.686375\pi$
$971$	−34.4408	−1.10526	−0.552629	+	0.833428i	$0.686375\pi$
$972$	−1.51429	−0.0485710
$973$	−66.1534	−2.12078
$974$	11.8527	0.379785
$975$	−18.6172	−0.596227
$976$	8.08971	0.258945
$977$	−57.6849	−1.84550	−0.922752	−	0.385394i	$-0.874066\pi$
$977$	−57.6849	−1.84550	−0.922752	+	0.385394i	$0.874066\pi$
$978$	11.6372	0.372117
$979$	11.0007	0.351583
$980$	−2.36979	−0.0757002
$981$	1.53269	0.0489351
$982$	29.7685	0.949951
$983$	−16.5269	−0.527127	−0.263564	−	0.964642i	$-0.584898\pi$
$983$	−16.5269	−0.527127	−0.263564	+	0.964642i	$0.584898\pi$
$984$	12.1233	0.386476
$985$	2.49008	0.0793405
$986$	5.44530	0.173414
$987$	7.92826	0.252359
$988$	−3.58709	−0.114121
$989$	7.63729	0.242852
$990$	−0.0429062	−0.00136365
$991$	−16.3115	−0.518153	−0.259077	−	0.965857i	$-0.583418\pi$
$991$	−16.3115	−0.518153	−0.259077	+	0.965857i	$0.583418\pi$
$992$	−2.56495	−0.0814374
$993$	−23.5454	−0.747189
$994$	16.7362	0.530838
$995$	−0.0277970	−0.000881223 0
$996$	−27.4249	−0.868991
$997$	−27.0005	−0.855115	−0.427558	−	0.903988i	$-0.640626\pi$
$997$	−27.0005	−0.855115	−0.427558	+	0.903988i	$0.640626\pi$
$998$	1.27421	0.0403346
$999$	12.4008	0.392345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1334.2.a.i.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1334.2.a.i.1.5	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1334 = 2 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.77365 q^{3} +1.00000 q^{4} -0.134992 q^{5} +1.77365 q^{6} +4.95530 q^{7} +1.00000 q^{8} +0.145840 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.77365 q^{3} +1.00000 q^{4} -0.134992 q^{5} +1.77365 q^{6} +4.95530 q^{7} +1.00000 q^{8} +0.145840 q^{9} -0.134992 q^{10} +2.17939 q^{11} +1.77365 q^{12} +2.10699 q^{13} +4.95530 q^{14} -0.239429 q^{15} +1.00000 q^{16} -5.44530 q^{17} +0.145840 q^{18} -1.70247 q^{19} -0.134992 q^{20} +8.78898 q^{21} +2.17939 q^{22} -1.00000 q^{23} +1.77365 q^{24} -4.98178 q^{25} +2.10699 q^{26} -5.06229 q^{27} +4.95530 q^{28} -1.00000 q^{29} -0.239429 q^{30} -2.56495 q^{31} +1.00000 q^{32} +3.86547 q^{33} -5.44530 q^{34} -0.668927 q^{35} +0.145840 q^{36} -2.44965 q^{37} -1.70247 q^{38} +3.73706 q^{39} -0.134992 q^{40} +6.83522 q^{41} +8.78898 q^{42} -7.63729 q^{43} +2.17939 q^{44} -0.0196873 q^{45} -1.00000 q^{46} +0.902069 q^{47} +1.77365 q^{48} +17.5550 q^{49} -4.98178 q^{50} -9.65807 q^{51} +2.10699 q^{52} -2.72943 q^{53} -5.06229 q^{54} -0.294201 q^{55} +4.95530 q^{56} -3.01960 q^{57} -1.00000 q^{58} -5.59801 q^{59} -0.239429 q^{60} +8.08971 q^{61} -2.56495 q^{62} +0.722681 q^{63} +1.00000 q^{64} -0.284427 q^{65} +3.86547 q^{66} +8.63159 q^{67} -5.44530 q^{68} -1.77365 q^{69} -0.668927 q^{70} +3.37742 q^{71} +0.145840 q^{72} -2.22792 q^{73} -2.44965 q^{74} -8.83594 q^{75} -1.70247 q^{76} +10.7995 q^{77} +3.73706 q^{78} +9.54644 q^{79} -0.134992 q^{80} -9.41625 q^{81} +6.83522 q^{82} -15.4624 q^{83} +8.78898 q^{84} +0.735074 q^{85} -7.63729 q^{86} -1.77365 q^{87} +2.17939 q^{88} +5.04760 q^{89} -0.0196873 q^{90} +10.4407 q^{91} -1.00000 q^{92} -4.54934 q^{93} +0.902069 q^{94} +0.229821 q^{95} +1.77365 q^{96} +7.70718 q^{97} +17.5550 q^{98} +0.317842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{14} + 8 q^{16} + 8 q^{17} + 12 q^{18} + 16 q^{19} - 4 q^{21} + 8 q^{22} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 8 q^{28} - 8 q^{29} + 8 q^{31} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{35} + 12 q^{36} + 8 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} - 4 q^{42} + 32 q^{43} + 8 q^{44} - 8 q^{46} + 16 q^{47} + 4 q^{48} + 4 q^{49} + 4 q^{50} + 12 q^{51} + 4 q^{52} + 4 q^{54} + 8 q^{56} + 8 q^{57} - 8 q^{58} - 4 q^{59} + 8 q^{61} + 8 q^{62} + 24 q^{63} + 8 q^{64} - 4 q^{65} + 12 q^{66} + 20 q^{67} + 8 q^{68} - 4 q^{69} + 4 q^{70} - 24 q^{71} + 12 q^{72} - 4 q^{73} + 8 q^{74} - 16 q^{75} + 16 q^{76} - 16 q^{77} - 16 q^{78} + 4 q^{79} - 8 q^{81} + 8 q^{82} - 4 q^{83} - 4 q^{84} - 44 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} + 20 q^{89} - 8 q^{92} - 4 q^{93} + 16 q^{94} - 8 q^{95} + 4 q^{96} + 4 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^12 + 4 * q^13 + 8 * q^14 + 8 * q^16 + 8 * q^17 + 12 * q^18 + 16 * q^19 - 4 * q^21 + 8 * q^22 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 8 * q^28 - 8 * q^29 + 8 * q^31 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^35 + 12 * q^36 + 8 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 - 4 * q^42 + 32 * q^43 + 8 * q^44 - 8 * q^46 + 16 * q^47 + 4 * q^48 + 4 * q^49 + 4 * q^50 + 12 * q^51 + 4 * q^52 + 4 * q^54 + 8 * q^56 + 8 * q^57 - 8 * q^58 - 4 * q^59 + 8 * q^61 + 8 * q^62 + 24 * q^63 + 8 * q^64 - 4 * q^65 + 12 * q^66 + 20 * q^67 + 8 * q^68 - 4 * q^69 + 4 * q^70 - 24 * q^71 + 12 * q^72 - 4 * q^73 + 8 * q^74 - 16 * q^75 + 16 * q^76 - 16 * q^77 - 16 * q^78 + 4 * q^79 - 8 * q^81 + 8 * q^82 - 4 * q^83 - 4 * q^84 - 44 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 + 20 * q^89 - 8 * q^92 - 4 * q^93 + 16 * q^94 - 8 * q^95 + 4 * q^96 + 4 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more