LMFDB - Embedded newform 453.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [453,2,Mod(1,453)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(453, 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("453.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 453 = 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	453.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:	$3.61722321156$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	453.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.763932 q^{11} -1.61803 q^{12} -1.00000 q^{13} -1.85410 q^{14} -1.61803 q^{15} +1.85410 q^{16} -2.23607 q^{17} +0.618034 q^{18} -4.00000 q^{19} +2.61803 q^{20} -3.00000 q^{21} -0.472136 q^{22} -7.47214 q^{23} -2.23607 q^{24} -2.38197 q^{25} -0.618034 q^{26} +1.00000 q^{27} +4.85410 q^{28} +7.85410 q^{29} -1.00000 q^{30} +3.09017 q^{31} +5.61803 q^{32} -0.763932 q^{33} -1.38197 q^{34} +4.85410 q^{35} -1.61803 q^{36} -4.85410 q^{37} -2.47214 q^{38} -1.00000 q^{39} +3.61803 q^{40} +1.14590 q^{41} -1.85410 q^{42} -0.527864 q^{43} +1.23607 q^{44} -1.61803 q^{45} -4.61803 q^{46} +4.47214 q^{47} +1.85410 q^{48} +2.00000 q^{49} -1.47214 q^{50} -2.23607 q^{51} +1.61803 q^{52} +7.94427 q^{53} +0.618034 q^{54} +1.23607 q^{55} +6.70820 q^{56} -4.00000 q^{57} +4.85410 q^{58} -3.61803 q^{59} +2.61803 q^{60} +12.5623 q^{61} +1.90983 q^{62} -3.00000 q^{63} -0.236068 q^{64} +1.61803 q^{65} -0.472136 q^{66} -10.8541 q^{67} +3.61803 q^{68} -7.47214 q^{69} +3.00000 q^{70} +12.0902 q^{71} -2.23607 q^{72} -14.7082 q^{73} -3.00000 q^{74} -2.38197 q^{75} +6.47214 q^{76} +2.29180 q^{77} -0.618034 q^{78} -8.79837 q^{79} -3.00000 q^{80} +1.00000 q^{81} +0.708204 q^{82} -6.38197 q^{83} +4.85410 q^{84} +3.61803 q^{85} -0.326238 q^{86} +7.85410 q^{87} +1.70820 q^{88} -3.00000 q^{89} -1.00000 q^{90} +3.00000 q^{91} +12.0902 q^{92} +3.09017 q^{93} +2.76393 q^{94} +6.47214 q^{95} +5.61803 q^{96} -13.4164 q^{97} +1.23607 q^{98} -0.763932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^5 - q^6 - 6 * q^7 + 2 * q^9	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{5} - q^{6} - 6 q^{7} + 2 q^{9} - 2 q^{10} - 6 q^{11} - q^{12} - 2 q^{13} + 3 q^{14} - q^{15} - 3 q^{16} - q^{18} - 8 q^{19} + 3 q^{20} - 6 q^{21} + 8 q^{22} - 6 q^{23} - 7 q^{25} + q^{26} + 2 q^{27} + 3 q^{28} + 9 q^{29} - 2 q^{30} - 5 q^{31} + 9 q^{32} - 6 q^{33} - 5 q^{34} + 3 q^{35} - q^{36} - 3 q^{37} + 4 q^{38} - 2 q^{39} + 5 q^{40} + 9 q^{41} + 3 q^{42} - 10 q^{43} - 2 q^{44} - q^{45} - 7 q^{46} - 3 q^{48} + 4 q^{49} + 6 q^{50} + q^{52} - 2 q^{53} - q^{54} - 2 q^{55} - 8 q^{57} + 3 q^{58} - 5 q^{59} + 3 q^{60} + 5 q^{61} + 15 q^{62} - 6 q^{63} + 4 q^{64} + q^{65} + 8 q^{66} - 15 q^{67} + 5 q^{68} - 6 q^{69} + 6 q^{70} + 13 q^{71} - 16 q^{73} - 6 q^{74} - 7 q^{75} + 4 q^{76} + 18 q^{77} + q^{78} + 7 q^{79} - 6 q^{80} + 2 q^{81} - 12 q^{82} - 15 q^{83} + 3 q^{84} + 5 q^{85} + 15 q^{86} + 9 q^{87} - 10 q^{88} - 6 q^{89} - 2 q^{90} + 6 q^{91} + 13 q^{92} - 5 q^{93} + 10 q^{94} + 4 q^{95} + 9 q^{96} - 2 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^5 - q^6 - 6 * q^7 + 2 * q^9 - 2 * q^10 - 6 * q^11 - q^12 - 2 * q^13 + 3 * q^14 - q^15 - 3 * q^16 - q^18 - 8 * q^19 + 3 * q^20 - 6 * q^21 + 8 * q^22 - 6 * q^23 - 7 * q^25 + q^26 + 2 * q^27 + 3 * q^28 + 9 * q^29 - 2 * q^30 - 5 * q^31 + 9 * q^32 - 6 * q^33 - 5 * q^34 + 3 * q^35 - q^36 - 3 * q^37 + 4 * q^38 - 2 * q^39 + 5 * q^40 + 9 * q^41 + 3 * q^42 - 10 * q^43 - 2 * q^44 - q^45 - 7 * q^46 - 3 * q^48 + 4 * q^49 + 6 * q^50 + q^52 - 2 * q^53 - q^54 - 2 * q^55 - 8 * q^57 + 3 * q^58 - 5 * q^59 + 3 * q^60 + 5 * q^61 + 15 * q^62 - 6 * q^63 + 4 * q^64 + q^65 + 8 * q^66 - 15 * q^67 + 5 * q^68 - 6 * q^69 + 6 * q^70 + 13 * q^71 - 16 * q^73 - 6 * q^74 - 7 * q^75 + 4 * q^76 + 18 * q^77 + q^78 + 7 * q^79 - 6 * q^80 + 2 * q^81 - 12 * q^82 - 15 * q^83 + 3 * q^84 + 5 * q^85 + 15 * q^86 + 9 * q^87 - 10 * q^88 - 6 * q^89 - 2 * q^90 + 6 * q^91 + 13 * q^92 - 5 * q^93 + 10 * q^94 + 4 * q^95 + 9 * q^96 - 2 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0.618034	0.252311
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	−1.61803	−0.467086
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.85410	−0.495530
$15$	−1.61803	−0.417775
$16$	1.85410	0.463525
$17$	−2.23607	−0.542326	−0.271163	−	0.962533i	$-0.587408\pi$
$17$	−2.23607	−0.542326	−0.271163	+	0.962533i	$0.587408\pi$
$18$	0.618034	0.145672
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	2.61803	0.585410
$21$	−3.00000	−0.654654
$22$	−0.472136	−0.100660
$23$	−7.47214	−1.55805	−0.779024	−	0.626994i	$-0.784285\pi$
$23$	−7.47214	−1.55805	−0.779024	+	0.626994i	$0.784285\pi$
$24$	−2.23607	−0.456435
$25$	−2.38197	−0.476393
$26$	−0.618034	−0.121206
$27$	1.00000	0.192450
$28$	4.85410	0.917339
$29$	7.85410	1.45847	0.729235	−	0.684263i	$-0.239876\pi$
$29$	7.85410	1.45847	0.729235	+	0.684263i	$0.239876\pi$
$30$	−1.00000	−0.182574
$31$	3.09017	0.555011	0.277505	−	0.960724i	$-0.410492\pi$
$31$	3.09017	0.555011	0.277505	+	0.960724i	$0.410492\pi$
$32$	5.61803	0.993137
$33$	−0.763932	−0.132983
$34$	−1.38197	−0.237005
$35$	4.85410	0.820493
$36$	−1.61803	−0.269672
$37$	−4.85410	−0.798009	−0.399005	−	0.916949i	$-0.630644\pi$
$37$	−4.85410	−0.798009	−0.399005	+	0.916949i	$0.630644\pi$
$38$	−2.47214	−0.401033
$39$	−1.00000	−0.160128
$40$	3.61803	0.572061
$41$	1.14590	0.178959	0.0894796	−	0.995989i	$-0.471480\pi$
$41$	1.14590	0.178959	0.0894796	+	0.995989i	$0.471480\pi$
$42$	−1.85410	−0.286094
$43$	−0.527864	−0.0804985	−0.0402493	−	0.999190i	$-0.512815\pi$
$43$	−0.527864	−0.0804985	−0.0402493	+	0.999190i	$0.512815\pi$
$44$	1.23607	0.186344
$45$	−1.61803	−0.241202
$46$	−4.61803	−0.680892
$47$	4.47214	0.652328	0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214	0.652328	0.326164	+	0.945313i	$0.394244\pi$
$48$	1.85410	0.267617
$49$	2.00000	0.285714
$50$	−1.47214	−0.208191
$51$	−2.23607	−0.313112
$52$	1.61803	0.224381
$53$	7.94427	1.09123	0.545615	−	0.838036i	$-0.316296\pi$
$53$	7.94427	1.09123	0.545615	+	0.838036i	$0.316296\pi$
$54$	0.618034	0.0841038
$55$	1.23607	0.166671
$56$	6.70820	0.896421
$57$	−4.00000	−0.529813
$58$	4.85410	0.637375
$59$	−3.61803	−0.471028	−0.235514	−	0.971871i	$-0.575677\pi$
$59$	−3.61803	−0.471028	−0.235514	+	0.971871i	$0.575677\pi$
$60$	2.61803	0.337987
$61$	12.5623	1.60844	0.804219	−	0.594333i	$-0.202584\pi$
$61$	12.5623	1.60844	0.804219	+	0.594333i	$0.202584\pi$
$62$	1.90983	0.242549
$63$	−3.00000	−0.377964
$64$	−0.236068	−0.0295085
$65$	1.61803	0.200692
$66$	−0.472136	−0.0581159
$67$	−10.8541	−1.32604	−0.663020	−	0.748602i	$-0.730725\pi$
$67$	−10.8541	−1.32604	−0.663020	+	0.748602i	$0.730725\pi$
$68$	3.61803	0.438751
$69$	−7.47214	−0.899539
$70$	3.00000	0.358569
$71$	12.0902	1.43484	0.717420	−	0.696641i	$-0.245323\pi$
$71$	12.0902	1.43484	0.717420	+	0.696641i	$0.245323\pi$
$72$	−2.23607	−0.263523
$73$	−14.7082	−1.72147	−0.860733	−	0.509057i	$-0.829994\pi$
$73$	−14.7082	−1.72147	−0.860733	+	0.509057i	$0.829994\pi$
$74$	−3.00000	−0.348743
$75$	−2.38197	−0.275046
$76$	6.47214	0.742405
$77$	2.29180	0.261174
$78$	−0.618034	−0.0699786
$79$	−8.79837	−0.989894	−0.494947	−	0.868923i	$-0.664813\pi$
$79$	−8.79837	−0.989894	−0.494947	+	0.868923i	$0.664813\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	0.708204	0.0782080
$83$	−6.38197	−0.700512	−0.350256	−	0.936654i	$-0.613905\pi$
$83$	−6.38197	−0.700512	−0.350256	+	0.936654i	$0.613905\pi$
$84$	4.85410	0.529626
$85$	3.61803	0.392431
$86$	−0.326238	−0.0351791
$87$	7.85410	0.842048
$88$	1.70820	0.182095
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	−1.00000	−0.105409
$91$	3.00000	0.314485
$92$	12.0902	1.26049
$93$	3.09017	0.320436
$94$	2.76393	0.285078
$95$	6.47214	0.664027
$96$	5.61803	0.573388
$97$	−13.4164	−1.36223	−0.681115	−	0.732177i	$-0.738505\pi$
$97$	−13.4164	−1.36223	−0.681115	+	0.732177i	$0.738505\pi$
$98$	1.23607	0.124862
$99$	−0.763932	−0.0767781
$100$	3.85410	0.385410
$101$	1.90983	0.190035	0.0950176	−	0.995476i	$-0.469709\pi$
$101$	1.90983	0.190035	0.0950176	+	0.995476i	$0.469709\pi$
$102$	−1.38197	−0.136835
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	2.23607	0.219265
$105$	4.85410	0.473712
$106$	4.90983	0.476885
$107$	−9.94427	−0.961349	−0.480675	−	0.876899i	$-0.659608\pi$
$107$	−9.94427	−0.961349	−0.480675	+	0.876899i	$0.659608\pi$
$108$	−1.61803	−0.155695
$109$	−12.0344	−1.15269	−0.576345	−	0.817206i	$-0.695522\pi$
$109$	−12.0344	−1.15269	−0.576345	+	0.817206i	$0.695522\pi$
$110$	0.763932	0.0728381
$111$	−4.85410	−0.460731
$112$	−5.56231	−0.525589
$113$	1.23607	0.116279	0.0581397	−	0.998308i	$-0.481483\pi$
$113$	1.23607	0.116279	0.0581397	+	0.998308i	$0.481483\pi$
$114$	−2.47214	−0.231537
$115$	12.0902	1.12741
$116$	−12.7082	−1.17993
$117$	−1.00000	−0.0924500
$118$	−2.23607	−0.205847
$119$	6.70820	0.614940
$120$	3.61803	0.330280
$121$	−10.4164	−0.946946
$122$	7.76393	0.702913
$123$	1.14590	0.103322
$124$	−5.00000	−0.449013
$125$	11.9443	1.06833
$126$	−1.85410	−0.165177
$127$	−2.85410	−0.253261	−0.126630	−	0.991950i	$-0.540416\pi$
$127$	−2.85410	−0.253261	−0.126630	+	0.991950i	$0.540416\pi$
$128$	−11.3820	−1.00603
$129$	−0.527864	−0.0464758
$130$	1.00000	0.0877058
$131$	3.90983	0.341603	0.170802	−	0.985305i	$-0.445364\pi$
$131$	3.90983	0.341603	0.170802	+	0.985305i	$0.445364\pi$
$132$	1.23607	0.107586
$133$	12.0000	1.04053
$134$	−6.70820	−0.579501
$135$	−1.61803	−0.139258
$136$	5.00000	0.428746
$137$	15.1803	1.29694	0.648472	−	0.761239i	$-0.275408\pi$
$137$	15.1803	1.29694	0.648472	+	0.761239i	$0.275408\pi$
$138$	−4.61803	−0.393113
$139$	3.14590	0.266832	0.133416	−	0.991060i	$-0.457405\pi$
$139$	3.14590	0.266832	0.133416	+	0.991060i	$0.457405\pi$
$140$	−7.85410	−0.663793
$141$	4.47214	0.376622
$142$	7.47214	0.627048
$143$	0.763932	0.0638832
$144$	1.85410	0.154508
$145$	−12.7082	−1.05536
$146$	−9.09017	−0.752308
$147$	2.00000	0.164957
$148$	7.85410	0.645603
$149$	2.32624	0.190573	0.0952864	−	0.995450i	$-0.469623\pi$
$149$	2.32624	0.190573	0.0952864	+	0.995450i	$0.469623\pi$
$150$	−1.47214	−0.120199
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	8.94427	0.725476
$153$	−2.23607	−0.180775
$154$	1.41641	0.114137
$155$	−5.00000	−0.401610
$156$	1.61803	0.129546
$157$	3.56231	0.284303	0.142151	−	0.989845i	$-0.454598\pi$
$157$	3.56231	0.284303	0.142151	+	0.989845i	$0.454598\pi$
$158$	−5.43769	−0.432600
$159$	7.94427	0.630022
$160$	−9.09017	−0.718641
$161$	22.4164	1.76666
$162$	0.618034	0.0485573
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	−1.85410	−0.144781
$165$	1.23607	0.0962278
$166$	−3.94427	−0.306135
$167$	−6.38197	−0.493851	−0.246926	−	0.969034i	$-0.579420\pi$
$167$	−6.38197	−0.493851	−0.246926	+	0.969034i	$0.579420\pi$
$168$	6.70820	0.517549
$169$	−12.0000	−0.923077
$170$	2.23607	0.171499
$171$	−4.00000	−0.305888
$172$	0.854102	0.0651247
$173$	13.2361	1.00632	0.503160	−	0.864193i	$-0.332171\pi$
$173$	13.2361	1.00632	0.503160	+	0.864193i	$0.332171\pi$
$174$	4.85410	0.367989
$175$	7.14590	0.540179
$176$	−1.41641	−0.106766
$177$	−3.61803	−0.271948
$178$	−1.85410	−0.138971
$179$	−4.41641	−0.330098	−0.165049	−	0.986285i	$-0.552778\pi$
$179$	−4.41641	−0.330098	−0.165049	+	0.986285i	$0.552778\pi$
$180$	2.61803	0.195137
$181$	−9.29180	−0.690654	−0.345327	−	0.938482i	$-0.612232\pi$
$181$	−9.29180	−0.690654	−0.345327	+	0.938482i	$0.612232\pi$
$182$	1.85410	0.137435
$183$	12.5623	0.928632
$184$	16.7082	1.23175
$185$	7.85410	0.577445
$186$	1.90983	0.140036
$187$	1.70820	0.124916
$188$	−7.23607	−0.527744
$189$	−3.00000	−0.218218
$190$	4.00000	0.290191
$191$	−19.3607	−1.40089	−0.700445	−	0.713707i	$-0.747015\pi$
$191$	−19.3607	−1.40089	−0.700445	+	0.713707i	$0.747015\pi$
$192$	−0.236068	−0.0170367
$193$	25.8885	1.86350	0.931749	−	0.363103i	$-0.118283\pi$
$193$	25.8885	1.86350	0.931749	+	0.363103i	$0.118283\pi$
$194$	−8.29180	−0.595316
$195$	1.61803	0.115870
$196$	−3.23607	−0.231148
$197$	−20.0344	−1.42739	−0.713697	−	0.700454i	$-0.752981\pi$
$197$	−20.0344	−1.42739	−0.713697	+	0.700454i	$0.752981\pi$
$198$	−0.472136	−0.0335532
$199$	0.236068	0.0167344	0.00836721	−	0.999965i	$-0.497337\pi$
$199$	0.236068	0.0167344	0.00836721	+	0.999965i	$0.497337\pi$
$200$	5.32624	0.376622
$201$	−10.8541	−0.765589
$202$	1.18034	0.0830484
$203$	−23.5623	−1.65375
$204$	3.61803	0.253313
$205$	−1.85410	−0.129496
$206$	4.32624	0.301423
$207$	−7.47214	−0.519349
$208$	−1.85410	−0.128559
$209$	3.05573	0.211369
$210$	3.00000	0.207020
$211$	14.5623	1.00251	0.501255	−	0.865299i	$-0.332872\pi$
$211$	14.5623	1.00251	0.501255	+	0.865299i	$0.332872\pi$
$212$	−12.8541	−0.882823
$213$	12.0902	0.828405
$214$	−6.14590	−0.420125
$215$	0.854102	0.0582493
$216$	−2.23607	−0.152145
$217$	−9.27051	−0.629323
$218$	−7.43769	−0.503744
$219$	−14.7082	−0.993888
$220$	−2.00000	−0.134840
$221$	2.23607	0.150414
$222$	−3.00000	−0.201347
$223$	0.673762	0.0451184	0.0225592	−	0.999746i	$-0.492819\pi$
$223$	0.673762	0.0451184	0.0225592	+	0.999746i	$0.492819\pi$
$224$	−16.8541	−1.12611
$225$	−2.38197	−0.158798
$226$	0.763932	0.0508160
$227$	−13.9098	−0.923228	−0.461614	−	0.887081i	$-0.652729\pi$
$227$	−13.9098	−0.923228	−0.461614	+	0.887081i	$0.652729\pi$
$228$	6.47214	0.428628
$229$	−22.2361	−1.46940	−0.734700	−	0.678392i	$-0.762677\pi$
$229$	−22.2361	−1.46940	−0.734700	+	0.678392i	$0.762677\pi$
$230$	7.47214	0.492698
$231$	2.29180	0.150789
$232$	−17.5623	−1.15302
$233$	−2.23607	−0.146490	−0.0732448	−	0.997314i	$-0.523335\pi$
$233$	−2.23607	−0.146490	−0.0732448	+	0.997314i	$0.523335\pi$
$234$	−0.618034	−0.0404021
$235$	−7.23607	−0.472029
$236$	5.85410	0.381070
$237$	−8.79837	−0.571516
$238$	4.14590	0.268739
$239$	2.94427	0.190449	0.0952246	−	0.995456i	$-0.469643\pi$
$239$	2.94427	0.190449	0.0952246	+	0.995456i	$0.469643\pi$
$240$	−3.00000	−0.193649
$241$	9.94427	0.640567	0.320283	−	0.947322i	$-0.396222\pi$
$241$	9.94427	0.640567	0.320283	+	0.947322i	$0.396222\pi$
$242$	−6.43769	−0.413831
$243$	1.00000	0.0641500
$244$	−20.3262	−1.30125
$245$	−3.23607	−0.206745
$246$	0.708204	0.0451534
$247$	4.00000	0.254514
$248$	−6.90983	−0.438775
$249$	−6.38197	−0.404441
$250$	7.38197	0.466877
$251$	−16.5279	−1.04323	−0.521615	−	0.853181i	$-0.674670\pi$
$251$	−16.5279	−1.04323	−0.521615	+	0.853181i	$0.674670\pi$
$252$	4.85410	0.305780
$253$	5.70820	0.358872
$254$	−1.76393	−0.110679
$255$	3.61803	0.226570
$256$	−6.56231	−0.410144
$257$	−10.4164	−0.649758	−0.324879	−	0.945756i	$-0.605324\pi$
$257$	−10.4164	−0.649758	−0.324879	+	0.945756i	$0.605324\pi$
$258$	−0.326238	−0.0203107
$259$	14.5623	0.904858
$260$	−2.61803	−0.162364
$261$	7.85410	0.486157
$262$	2.41641	0.149286
$263$	10.0344	0.618750	0.309375	−	0.950940i	$-0.399880\pi$
$263$	10.0344	0.618750	0.309375	+	0.950940i	$0.399880\pi$
$264$	1.70820	0.105133
$265$	−12.8541	−0.789621
$266$	7.41641	0.454729
$267$	−3.00000	−0.183597
$268$	17.5623	1.07279
$269$	−9.03444	−0.550840	−0.275420	−	0.961324i	$-0.588817\pi$
$269$	−9.03444	−0.550840	−0.275420	+	0.961324i	$0.588817\pi$
$270$	−1.00000	−0.0608581
$271$	−16.9443	−1.02929	−0.514646	−	0.857403i	$-0.672076\pi$
$271$	−16.9443	−1.02929	−0.514646	+	0.857403i	$0.672076\pi$
$272$	−4.14590	−0.251382
$273$	3.00000	0.181568
$274$	9.38197	0.566785
$275$	1.81966	0.109730
$276$	12.0902	0.727743
$277$	17.7639	1.06733	0.533666	−	0.845696i	$-0.320814\pi$
$277$	17.7639	1.06733	0.533666	+	0.845696i	$0.320814\pi$
$278$	1.94427	0.116610
$279$	3.09017	0.185004
$280$	−10.8541	−0.648657
$281$	20.1803	1.20386	0.601929	−	0.798550i	$-0.294399\pi$
$281$	20.1803	1.20386	0.601929	+	0.798550i	$0.294399\pi$
$282$	2.76393	0.164590
$283$	−21.9443	−1.30445	−0.652226	−	0.758025i	$-0.726165\pi$
$283$	−21.9443	−1.30445	−0.652226	+	0.758025i	$0.726165\pi$
$284$	−19.5623	−1.16081
$285$	6.47214	0.383376
$286$	0.472136	0.0279180
$287$	−3.43769	−0.202921
$288$	5.61803	0.331046
$289$	−12.0000	−0.705882
$290$	−7.85410	−0.461209
$291$	−13.4164	−0.786484
$292$	23.7984	1.39269
$293$	−11.6525	−0.680745	−0.340372	−	0.940291i	$-0.610553\pi$
$293$	−11.6525	−0.680745	−0.340372	+	0.940291i	$0.610553\pi$
$294$	1.23607	0.0720889
$295$	5.85410	0.340839
$296$	10.8541	0.630882
$297$	−0.763932	−0.0443278
$298$	1.43769	0.0832834
$299$	7.47214	0.432125
$300$	3.85410	0.222517
$301$	1.58359	0.0912767
$302$	0.618034	0.0355639
$303$	1.90983	0.109717
$304$	−7.41641	−0.425360
$305$	−20.3262	−1.16388
$306$	−1.38197	−0.0790017
$307$	17.0344	0.972207	0.486103	−	0.873901i	$-0.338418\pi$
$307$	17.0344	0.972207	0.486103	+	0.873901i	$0.338418\pi$
$308$	−3.70820	−0.211295
$309$	7.00000	0.398216
$310$	−3.09017	−0.175510
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	2.23607	0.126592
$313$	13.5066	0.763437	0.381718	−	0.924279i	$-0.375332\pi$
$313$	13.5066	0.763437	0.381718	+	0.924279i	$0.375332\pi$
$314$	2.20163	0.124245
$315$	4.85410	0.273498
$316$	14.2361	0.800841
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	4.90983	0.275330
$319$	−6.00000	−0.335936
$320$	0.381966	0.0213525
$321$	−9.94427	−0.555035
$322$	13.8541	0.772059
$323$	8.94427	0.497673
$324$	−1.61803	−0.0898908
$325$	2.38197	0.132128
$326$	−8.65248	−0.479216
$327$	−12.0344	−0.665506
$328$	−2.56231	−0.141480
$329$	−13.4164	−0.739671
$330$	0.763932	0.0420531
$331$	19.1246	1.05118	0.525592	−	0.850737i	$-0.323844\pi$
$331$	19.1246	1.05118	0.525592	+	0.850737i	$0.323844\pi$
$332$	10.3262	0.566726
$333$	−4.85410	−0.266003
$334$	−3.94427	−0.215821
$335$	17.5623	0.959531
$336$	−5.56231	−0.303449
$337$	17.7984	0.969539	0.484770	−	0.874642i	$-0.338903\pi$
$337$	17.7984	0.969539	0.484770	+	0.874642i	$0.338903\pi$
$338$	−7.41641	−0.403399
$339$	1.23607	0.0671340
$340$	−5.85410	−0.317483
$341$	−2.36068	−0.127838
$342$	−2.47214	−0.133678
$343$	15.0000	0.809924
$344$	1.18034	0.0636397
$345$	12.0902	0.650913
$346$	8.18034	0.439778
$347$	−29.7639	−1.59781	−0.798906	−	0.601456i	$-0.794587\pi$
$347$	−29.7639	−1.59781	−0.798906	+	0.601456i	$0.794587\pi$
$348$	−12.7082	−0.681231
$349$	30.1803	1.61552	0.807758	−	0.589514i	$-0.200681\pi$
$349$	30.1803	1.61552	0.807758	+	0.589514i	$0.200681\pi$
$350$	4.41641	0.236067
$351$	−1.00000	−0.0533761
$352$	−4.29180	−0.228753
$353$	20.3607	1.08369	0.541845	−	0.840479i	$-0.317726\pi$
$353$	20.3607	1.08369	0.541845	+	0.840479i	$0.317726\pi$
$354$	−2.23607	−0.118846
$355$	−19.5623	−1.03826
$356$	4.85410	0.257267
$357$	6.70820	0.355036
$358$	−2.72949	−0.144258
$359$	−18.0344	−0.951821	−0.475911	−	0.879494i	$-0.657881\pi$
$359$	−18.0344	−0.951821	−0.475911	+	0.879494i	$0.657881\pi$
$360$	3.61803	0.190687
$361$	−3.00000	−0.157895
$362$	−5.74265	−0.301827
$363$	−10.4164	−0.546720
$364$	−4.85410	−0.254424
$365$	23.7984	1.24566
$366$	7.76393	0.405827
$367$	−4.32624	−0.225828	−0.112914	−	0.993605i	$-0.536018\pi$
$367$	−4.32624	−0.225828	−0.112914	+	0.993605i	$0.536018\pi$
$368$	−13.8541	−0.722195
$369$	1.14590	0.0596531
$370$	4.85410	0.252353
$371$	−23.8328	−1.23734
$372$	−5.00000	−0.259238
$373$	17.1459	0.887781	0.443890	−	0.896081i	$-0.353598\pi$
$373$	17.1459	0.887781	0.443890	+	0.896081i	$0.353598\pi$
$374$	1.05573	0.0545904
$375$	11.9443	0.616800
$376$	−10.0000	−0.515711
$377$	−7.85410	−0.404507
$378$	−1.85410	−0.0953647
$379$	−34.1246	−1.75286	−0.876432	−	0.481526i	$-0.840083\pi$
$379$	−34.1246	−1.75286	−0.876432	+	0.481526i	$0.840083\pi$
$380$	−10.4721	−0.537209
$381$	−2.85410	−0.146220
$382$	−11.9656	−0.612211
$383$	−3.76393	−0.192328	−0.0961640	−	0.995366i	$-0.530657\pi$
$383$	−3.76393	−0.192328	−0.0961640	+	0.995366i	$0.530657\pi$
$384$	−11.3820	−0.580834
$385$	−3.70820	−0.188988
$386$	16.0000	0.814379
$387$	−0.527864	−0.0268328
$388$	21.7082	1.10207
$389$	15.9787	0.810153	0.405076	−	0.914283i	$-0.367245\pi$
$389$	15.9787	0.810153	0.405076	+	0.914283i	$0.367245\pi$
$390$	1.00000	0.0506370
$391$	16.7082	0.844970
$392$	−4.47214	−0.225877
$393$	3.90983	0.197225
$394$	−12.3820	−0.623794
$395$	14.2361	0.716294
$396$	1.23607	0.0621148
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	0.145898	0.00731321
$399$	12.0000	0.600751
$400$	−4.41641	−0.220820
$401$	−1.47214	−0.0735150	−0.0367575	−	0.999324i	$-0.511703\pi$
$401$	−1.47214	−0.0735150	−0.0367575	+	0.999324i	$0.511703\pi$
$402$	−6.70820	−0.334575
$403$	−3.09017	−0.153932
$404$	−3.09017	−0.153742
$405$	−1.61803	−0.0804008
$406$	−14.5623	−0.722715
$407$	3.70820	0.183809
$408$	5.00000	0.247537
$409$	24.4164	1.20731	0.603657	−	0.797244i	$-0.293710\pi$
$409$	24.4164	1.20731	0.603657	+	0.797244i	$0.293710\pi$
$410$	−1.14590	−0.0565919
$411$	15.1803	0.748791
$412$	−11.3262	−0.558004
$413$	10.8541	0.534095
$414$	−4.61803	−0.226964
$415$	10.3262	0.506895
$416$	−5.61803	−0.275447
$417$	3.14590	0.154055
$418$	1.88854	0.0923717
$419$	24.5279	1.19826	0.599132	−	0.800650i	$-0.295512\pi$
$419$	24.5279	1.19826	0.599132	+	0.800650i	$0.295512\pi$
$420$	−7.85410	−0.383241
$421$	−6.27051	−0.305606	−0.152803	−	0.988257i	$-0.548830\pi$
$421$	−6.27051	−0.305606	−0.152803	+	0.988257i	$0.548830\pi$
$422$	9.00000	0.438113
$423$	4.47214	0.217443
$424$	−17.7639	−0.862693
$425$	5.32624	0.258360
$426$	7.47214	0.362026
$427$	−37.6869	−1.82380
$428$	16.0902	0.777748
$429$	0.763932	0.0368830
$430$	0.527864	0.0254559
$431$	−30.2148	−1.45539	−0.727697	−	0.685898i	$-0.759409\pi$
$431$	−30.2148	−1.45539	−0.727697	+	0.685898i	$0.759409\pi$
$432$	1.85410	0.0892055
$433$	35.3951	1.70098	0.850490	−	0.525990i	$-0.176305\pi$
$433$	35.3951	1.70098	0.850490	+	0.525990i	$0.176305\pi$
$434$	−5.72949	−0.275024
$435$	−12.7082	−0.609312
$436$	19.4721	0.932546
$437$	29.8885	1.42976
$438$	−9.09017	−0.434345
$439$	23.4721	1.12026	0.560132	−	0.828403i	$-0.310750\pi$
$439$	23.4721	1.12026	0.560132	+	0.828403i	$0.310750\pi$
$440$	−2.76393	−0.131765
$441$	2.00000	0.0952381
$442$	1.38197	0.0657334
$443$	−26.7639	−1.27159	−0.635796	−	0.771857i	$-0.719328\pi$
$443$	−26.7639	−1.27159	−0.635796	+	0.771857i	$0.719328\pi$
$444$	7.85410	0.372739
$445$	4.85410	0.230107
$446$	0.416408	0.0197175
$447$	2.32624	0.110027
$448$	0.708204	0.0334595
$449$	31.9443	1.50754	0.753772	−	0.657136i	$-0.228232\pi$
$449$	31.9443	1.50754	0.753772	+	0.657136i	$0.228232\pi$
$450$	−1.47214	−0.0693972
$451$	−0.875388	−0.0412204
$452$	−2.00000	−0.0940721
$453$	1.00000	0.0469841
$454$	−8.59675	−0.403465
$455$	−4.85410	−0.227564
$456$	8.94427	0.418854
$457$	−19.4164	−0.908261	−0.454131	−	0.890935i	$-0.650050\pi$
$457$	−19.4164	−0.908261	−0.454131	+	0.890935i	$0.650050\pi$
$458$	−13.7426	−0.642152
$459$	−2.23607	−0.104371
$460$	−19.5623	−0.912097
$461$	42.0344	1.95774	0.978870	−	0.204486i	$-0.0655522\pi$
$461$	42.0344	1.95774	0.978870	+	0.204486i	$0.0655522\pi$
$462$	1.41641	0.0658973
$463$	4.65248	0.216219	0.108109	−	0.994139i	$-0.465520\pi$
$463$	4.65248	0.216219	0.108109	+	0.994139i	$0.465520\pi$
$464$	14.5623	0.676038
$465$	−5.00000	−0.231869
$466$	−1.38197	−0.0640183
$467$	41.8328	1.93579	0.967896	−	0.251351i	$-0.0808749\pi$
$467$	41.8328	1.93579	0.967896	+	0.251351i	$0.0808749\pi$
$468$	1.61803	0.0747936
$469$	32.5623	1.50359
$470$	−4.47214	−0.206284
$471$	3.56231	0.164142
$472$	8.09017	0.372380
$473$	0.403252	0.0185416
$474$	−5.43769	−0.249762
$475$	9.52786	0.437168
$476$	−10.8541	−0.497497
$477$	7.94427	0.363743
$478$	1.81966	0.0832293
$479$	−9.43769	−0.431219	−0.215610	−	0.976480i	$-0.569174\pi$
$479$	−9.43769	−0.431219	−0.215610	+	0.976480i	$0.569174\pi$
$480$	−9.09017	−0.414908
$481$	4.85410	0.221328
$482$	6.14590	0.279938
$483$	22.4164	1.01998
$484$	16.8541	0.766096
$485$	21.7082	0.985719
$486$	0.618034	0.0280346
$487$	−10.8197	−0.490286	−0.245143	−	0.969487i	$-0.578835\pi$
$487$	−10.8197	−0.490286	−0.245143	+	0.969487i	$0.578835\pi$
$488$	−28.0902	−1.27158
$489$	−14.0000	−0.633102
$490$	−2.00000	−0.0903508
$491$	19.0902	0.861527	0.430764	−	0.902465i	$-0.358244\pi$
$491$	19.0902	0.861527	0.430764	+	0.902465i	$0.358244\pi$
$492$	−1.85410	−0.0835894
$493$	−17.5623	−0.790966
$494$	2.47214	0.111227
$495$	1.23607	0.0555571
$496$	5.72949	0.257262
$497$	−36.2705	−1.62695
$498$	−3.94427	−0.176747
$499$	−38.1246	−1.70669	−0.853346	−	0.521345i	$-0.825430\pi$
$499$	−38.1246	−1.70669	−0.853346	+	0.521345i	$0.825430\pi$
$500$	−19.3262	−0.864296
$501$	−6.38197	−0.285125
$502$	−10.2148	−0.455908
$503$	−5.67376	−0.252981	−0.126490	−	0.991968i	$-0.540371\pi$
$503$	−5.67376	−0.252981	−0.126490	+	0.991968i	$0.540371\pi$
$504$	6.70820	0.298807
$505$	−3.09017	−0.137511
$506$	3.52786	0.156833
$507$	−12.0000	−0.532939
$508$	4.61803	0.204892
$509$	−22.3607	−0.991120	−0.495560	−	0.868574i	$-0.665037\pi$
$509$	−22.3607	−0.991120	−0.495560	+	0.868574i	$0.665037\pi$
$510$	2.23607	0.0990148
$511$	44.1246	1.95196
$512$	18.7082	0.826794
$513$	−4.00000	−0.176604
$514$	−6.43769	−0.283955
$515$	−11.3262	−0.499094
$516$	0.854102	0.0375997
$517$	−3.41641	−0.150253
$518$	9.00000	0.395437
$519$	13.2361	0.580999
$520$	−3.61803	−0.158661
$521$	19.0344	0.833914	0.416957	−	0.908926i	$-0.363097\pi$
$521$	19.0344	0.833914	0.416957	+	0.908926i	$0.363097\pi$
$522$	4.85410	0.212458
$523$	0.798374	0.0349105	0.0174552	−	0.999848i	$-0.494444\pi$
$523$	0.798374	0.0349105	0.0174552	+	0.999848i	$0.494444\pi$
$524$	−6.32624	−0.276363
$525$	7.14590	0.311873
$526$	6.20163	0.270404
$527$	−6.90983	−0.300997
$528$	−1.41641	−0.0616412
$529$	32.8328	1.42751
$530$	−7.94427	−0.345077
$531$	−3.61803	−0.157009
$532$	−19.4164	−0.841808
$533$	−1.14590	−0.0496344
$534$	−1.85410	−0.0802348
$535$	16.0902	0.695639
$536$	24.2705	1.04833
$537$	−4.41641	−0.190582
$538$	−5.58359	−0.240726
$539$	−1.52786	−0.0658098
$540$	2.61803	0.112662
$541$	13.8328	0.594719	0.297360	−	0.954766i	$-0.403894\pi$
$541$	13.8328	0.594719	0.297360	+	0.954766i	$0.403894\pi$
$542$	−10.4721	−0.449817
$543$	−9.29180	−0.398749
$544$	−12.5623	−0.538604
$545$	19.4721	0.834095
$546$	1.85410	0.0793482
$547$	−8.12461	−0.347383	−0.173692	−	0.984800i	$-0.555570\pi$
$547$	−8.12461	−0.347383	−0.173692	+	0.984800i	$0.555570\pi$
$548$	−24.5623	−1.04925
$549$	12.5623	0.536146
$550$	1.12461	0.0479536
$551$	−31.4164	−1.33838
$552$	16.7082	0.711148
$553$	26.3951	1.12243
$554$	10.9787	0.466441
$555$	7.85410	0.333388
$556$	−5.09017	−0.215871
$557$	−46.0902	−1.95290	−0.976452	−	0.215737i	$-0.930785\pi$
$557$	−46.0902	−1.95290	−0.976452	+	0.215737i	$0.930785\pi$
$558$	1.90983	0.0808496
$559$	0.527864	0.0223263
$560$	9.00000	0.380319
$561$	1.70820	0.0721204
$562$	12.4721	0.526105
$563$	−33.6525	−1.41828	−0.709141	−	0.705066i	$-0.750917\pi$
$563$	−33.6525	−1.41828	−0.709141	+	0.705066i	$0.750917\pi$
$564$	−7.23607	−0.304693
$565$	−2.00000	−0.0841406
$566$	−13.5623	−0.570066
$567$	−3.00000	−0.125988
$568$	−27.0344	−1.13434
$569$	1.34752	0.0564912	0.0282456	−	0.999601i	$-0.491008\pi$
$569$	1.34752	0.0564912	0.0282456	+	0.999601i	$0.491008\pi$
$570$	4.00000	0.167542
$571$	20.3607	0.852068	0.426034	−	0.904707i	$-0.359910\pi$
$571$	20.3607	0.852068	0.426034	+	0.904707i	$0.359910\pi$
$572$	−1.23607	−0.0516826
$573$	−19.3607	−0.808804
$574$	−2.12461	−0.0886796
$575$	17.7984	0.742243
$576$	−0.236068	−0.00983617
$577$	−9.90983	−0.412552	−0.206276	−	0.978494i	$-0.566134\pi$
$577$	−9.90983	−0.412552	−0.206276	+	0.978494i	$0.566134\pi$
$578$	−7.41641	−0.308482
$579$	25.8885	1.07589
$580$	20.5623	0.853803
$581$	19.1459	0.794306
$582$	−8.29180	−0.343706
$583$	−6.06888	−0.251347
$584$	32.8885	1.36094
$585$	1.61803	0.0668975
$586$	−7.20163	−0.297496
$587$	25.7426	1.06251	0.531256	−	0.847211i	$-0.321720\pi$
$587$	25.7426	1.06251	0.531256	+	0.847211i	$0.321720\pi$
$588$	−3.23607	−0.133453
$589$	−12.3607	−0.509313
$590$	3.61803	0.148952
$591$	−20.0344	−0.824107
$592$	−9.00000	−0.369898
$593$	30.9787	1.27214	0.636072	−	0.771630i	$-0.280558\pi$
$593$	30.9787	1.27214	0.636072	+	0.771630i	$0.280558\pi$
$594$	−0.472136	−0.0193720
$595$	−10.8541	−0.444975
$596$	−3.76393	−0.154177
$597$	0.236068	0.00966162
$598$	4.61803	0.188845
$599$	−12.8197	−0.523797	−0.261899	−	0.965095i	$-0.584349\pi$
$599$	−12.8197	−0.523797	−0.261899	+	0.965095i	$0.584349\pi$
$600$	5.32624	0.217443
$601$	−35.2148	−1.43644	−0.718220	−	0.695816i	$-0.755043\pi$
$601$	−35.2148	−1.43644	−0.718220	+	0.695816i	$0.755043\pi$
$602$	0.978714	0.0398894
$603$	−10.8541	−0.442013
$604$	−1.61803	−0.0658369
$605$	16.8541	0.685217
$606$	1.18034	0.0479480
$607$	−34.2705	−1.39100	−0.695499	−	0.718528i	$-0.744816\pi$
$607$	−34.2705	−1.39100	−0.695499	+	0.718528i	$0.744816\pi$
$608$	−22.4721	−0.911365
$609$	−23.5623	−0.954793
$610$	−12.5623	−0.508633
$611$	−4.47214	−0.180923
$612$	3.61803	0.146250
$613$	−2.29180	−0.0925648	−0.0462824	−	0.998928i	$-0.514737\pi$
$613$	−2.29180	−0.0925648	−0.0462824	+	0.998928i	$0.514737\pi$
$614$	10.5279	0.424870
$615$	−1.85410	−0.0747646
$616$	−5.12461	−0.206476
$617$	20.2918	0.816917	0.408458	−	0.912777i	$-0.366066\pi$
$617$	20.2918	0.816917	0.408458	+	0.912777i	$0.366066\pi$
$618$	4.32624	0.174027
$619$	10.1803	0.409182	0.204591	−	0.978848i	$-0.434414\pi$
$619$	10.1803	0.409182	0.204591	+	0.978848i	$0.434414\pi$
$620$	8.09017	0.324909
$621$	−7.47214	−0.299846
$622$	1.85410	0.0743427
$623$	9.00000	0.360577
$624$	−1.85410	−0.0742235
$625$	−7.41641	−0.296656
$626$	8.34752	0.333634
$627$	3.05573	0.122034
$628$	−5.76393	−0.230006
$629$	10.8541	0.432781
$630$	3.00000	0.119523
$631$	−46.9787	−1.87019	−0.935097	−	0.354393i	$-0.884687\pi$
$631$	−46.9787	−1.87019	−0.935097	+	0.354393i	$0.884687\pi$
$632$	19.6738	0.782580
$633$	14.5623	0.578800
$634$	7.41641	0.294543
$635$	4.61803	0.183261
$636$	−12.8541	−0.509698
$637$	−2.00000	−0.0792429
$638$	−3.70820	−0.146809
$639$	12.0902	0.478280
$640$	18.4164	0.727972
$641$	−23.5066	−0.928454	−0.464227	−	0.885716i	$-0.653668\pi$
$641$	−23.5066	−0.928454	−0.464227	+	0.885716i	$0.653668\pi$
$642$	−6.14590	−0.242559
$643$	10.4508	0.412141	0.206071	−	0.978537i	$-0.433932\pi$
$643$	10.4508	0.412141	0.206071	+	0.978537i	$0.433932\pi$
$644$	−36.2705	−1.42926
$645$	0.854102	0.0336302
$646$	5.52786	0.217491
$647$	−2.88854	−0.113560	−0.0567802	−	0.998387i	$-0.518083\pi$
$647$	−2.88854	−0.113560	−0.0567802	+	0.998387i	$0.518083\pi$
$648$	−2.23607	−0.0878410
$649$	2.76393	0.108494
$650$	1.47214	0.0577419
$651$	−9.27051	−0.363340
$652$	22.6525	0.887139
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−7.43769	−0.290837
$655$	−6.32624	−0.247187
$656$	2.12461	0.0829522
$657$	−14.7082	−0.573822
$658$	−8.29180	−0.323248
$659$	23.6180	0.920028	0.460014	−	0.887912i	$-0.347844\pi$
$659$	23.6180	0.920028	0.460014	+	0.887912i	$0.347844\pi$
$660$	−2.00000	−0.0778499
$661$	21.7984	0.847858	0.423929	−	0.905695i	$-0.360651\pi$
$661$	21.7984	0.847858	0.423929	+	0.905695i	$0.360651\pi$
$662$	11.8197	0.459384
$663$	2.23607	0.0868417
$664$	14.2705	0.553803
$665$	−19.4164	−0.752936
$666$	−3.00000	−0.116248
$667$	−58.6869	−2.27237
$668$	10.3262	0.399534
$669$	0.673762	0.0260491
$670$	10.8541	0.419331
$671$	−9.59675	−0.370478
$672$	−16.8541	−0.650161
$673$	−18.9443	−0.730248	−0.365124	−	0.930959i	$-0.618973\pi$
$673$	−18.9443	−0.730248	−0.365124	+	0.930959i	$0.618973\pi$
$674$	11.0000	0.423704
$675$	−2.38197	−0.0916819
$676$	19.4164	0.746785
$677$	−2.40325	−0.0923645	−0.0461822	−	0.998933i	$-0.514705\pi$
$677$	−2.40325	−0.0923645	−0.0461822	+	0.998933i	$0.514705\pi$
$678$	0.763932	0.0293386
$679$	40.2492	1.54462
$680$	−8.09017	−0.310244
$681$	−13.9098	−0.533026
$682$	−1.45898	−0.0558672
$683$	10.0557	0.384772	0.192386	−	0.981319i	$-0.438377\pi$
$683$	10.0557	0.384772	0.192386	+	0.981319i	$0.438377\pi$
$684$	6.47214	0.247468
$685$	−24.5623	−0.938477
$686$	9.27051	0.353950
$687$	−22.2361	−0.848359
$688$	−0.978714	−0.0373131
$689$	−7.94427	−0.302653
$690$	7.47214	0.284459
$691$	−37.0689	−1.41017	−0.705083	−	0.709124i	$-0.749090\pi$
$691$	−37.0689	−1.41017	−0.705083	+	0.709124i	$0.749090\pi$
$692$	−21.4164	−0.814130
$693$	2.29180	0.0870581
$694$	−18.3951	−0.698269
$695$	−5.09017	−0.193081
$696$	−17.5623	−0.665697
$697$	−2.56231	−0.0970543
$698$	18.6525	0.706007
$699$	−2.23607	−0.0845759
$700$	−11.5623	−0.437014
$701$	41.1803	1.55536	0.777680	−	0.628660i	$-0.216396\pi$
$701$	41.1803	1.55536	0.777680	+	0.628660i	$0.216396\pi$
$702$	−0.618034	−0.0233262
$703$	19.4164	0.732304
$704$	0.180340	0.00679682
$705$	−7.23607	−0.272526
$706$	12.5836	0.473590
$707$	−5.72949	−0.215480
$708$	5.85410	0.220011
$709$	−35.6525	−1.33896	−0.669478	−	0.742832i	$-0.733482\pi$
$709$	−35.6525	−1.33896	−0.669478	+	0.742832i	$0.733482\pi$
$710$	−12.0902	−0.453736
$711$	−8.79837	−0.329965
$712$	6.70820	0.251401
$713$	−23.0902	−0.864734
$714$	4.14590	0.155156
$715$	−1.23607	−0.0462263
$716$	7.14590	0.267055
$717$	2.94427	0.109956
$718$	−11.1459	−0.415961
$719$	19.7426	0.736276	0.368138	−	0.929771i	$-0.379995\pi$
$719$	19.7426	0.736276	0.368138	+	0.929771i	$0.379995\pi$
$720$	−3.00000	−0.111803
$721$	−21.0000	−0.782081
$722$	−1.85410	−0.0690025
$723$	9.94427	0.369831
$724$	15.0344	0.558751
$725$	−18.7082	−0.694805
$726$	−6.43769	−0.238925
$727$	2.29180	0.0849980	0.0424990	−	0.999097i	$-0.486468\pi$
$727$	2.29180	0.0849980	0.0424990	+	0.999097i	$0.486468\pi$
$728$	−6.70820	−0.248623
$729$	1.00000	0.0370370
$730$	14.7082	0.544375
$731$	1.18034	0.0436564
$732$	−20.3262	−0.751279
$733$	−34.8673	−1.28785	−0.643926	−	0.765088i	$-0.722695\pi$
$733$	−34.8673	−1.28785	−0.643926	+	0.765088i	$0.722695\pi$
$734$	−2.67376	−0.0986904
$735$	−3.23607	−0.119364
$736$	−41.9787	−1.54736
$737$	8.29180	0.305432
$738$	0.708204	0.0260693
$739$	−22.6869	−0.834552	−0.417276	−	0.908780i	$-0.637015\pi$
$739$	−22.6869	−0.834552	−0.417276	+	0.908780i	$0.637015\pi$
$740$	−12.7082	−0.467163
$741$	4.00000	0.146944
$742$	−14.7295	−0.540737
$743$	−29.3607	−1.07714	−0.538569	−	0.842581i	$-0.681035\pi$
$743$	−29.3607	−1.07714	−0.538569	+	0.842581i	$0.681035\pi$
$744$	−6.90983	−0.253327
$745$	−3.76393	−0.137900
$746$	10.5967	0.387975
$747$	−6.38197	−0.233504
$748$	−2.76393	−0.101059
$749$	29.8328	1.09007
$750$	7.38197	0.269551
$751$	42.7082	1.55844	0.779222	−	0.626748i	$-0.215614\pi$
$751$	42.7082	1.55844	0.779222	+	0.626748i	$0.215614\pi$
$752$	8.29180	0.302371
$753$	−16.5279	−0.602309
$754$	−4.85410	−0.176776
$755$	−1.61803	−0.0588863
$756$	4.85410	0.176542
$757$	−46.9574	−1.70670	−0.853348	−	0.521341i	$-0.825432\pi$
$757$	−46.9574	−1.70670	−0.853348	+	0.521341i	$0.825432\pi$
$758$	−21.0902	−0.766029
$759$	5.70820	0.207195
$760$	−14.4721	−0.524960
$761$	38.6525	1.40115	0.700576	−	0.713578i	$-0.252927\pi$
$761$	38.6525	1.40115	0.700576	+	0.713578i	$0.252927\pi$
$762$	−1.76393	−0.0639005
$763$	36.1033	1.30703
$764$	31.3262	1.13334
$765$	3.61803	0.130810
$766$	−2.32624	−0.0840504
$767$	3.61803	0.130640
$768$	−6.56231	−0.236797
$769$	15.6869	0.565685	0.282842	−	0.959166i	$-0.408723\pi$
$769$	15.6869	0.565685	0.282842	+	0.959166i	$0.408723\pi$
$770$	−2.29180	−0.0825906
$771$	−10.4164	−0.375138
$772$	−41.8885	−1.50760
$773$	−50.4853	−1.81583	−0.907915	−	0.419155i	$-0.862327\pi$
$773$	−50.4853	−1.81583	−0.907915	+	0.419155i	$0.862327\pi$
$774$	−0.326238	−0.0117264
$775$	−7.36068	−0.264403
$776$	30.0000	1.07694
$777$	14.5623	0.522420
$778$	9.87539	0.354050
$779$	−4.58359	−0.164224
$780$	−2.61803	−0.0937407
$781$	−9.23607	−0.330492
$782$	10.3262	0.369266
$783$	7.85410	0.280683
$784$	3.70820	0.132436
$785$	−5.76393	−0.205724
$786$	2.41641	0.0861904
$787$	−26.7082	−0.952045	−0.476022	−	0.879433i	$-0.657922\pi$
$787$	−26.7082	−0.952045	−0.476022	+	0.879433i	$0.657922\pi$
$788$	32.4164	1.15479
$789$	10.0344	0.357236
$790$	8.79837	0.313032
$791$	−3.70820	−0.131849
$792$	1.70820	0.0606984
$793$	−12.5623	−0.446101
$794$	−1.85410	−0.0657996
$795$	−12.8541	−0.455888
$796$	−0.381966	−0.0135384
$797$	15.2016	0.538469	0.269235	−	0.963075i	$-0.413229\pi$
$797$	15.2016	0.538469	0.269235	+	0.963075i	$0.413229\pi$
$798$	7.41641	0.262538
$799$	−10.0000	−0.353775
$800$	−13.3820	−0.473124
$801$	−3.00000	−0.106000
$802$	−0.909830	−0.0321272
$803$	11.2361	0.396512
$804$	17.5623	0.619375
$805$	−36.2705	−1.27837
$806$	−1.90983	−0.0672709
$807$	−9.03444	−0.318027
$808$	−4.27051	−0.150236
$809$	−1.20163	−0.0422469	−0.0211235	−	0.999777i	$-0.506724\pi$
$809$	−1.20163	−0.0422469	−0.0211235	+	0.999777i	$0.506724\pi$
$810$	−1.00000	−0.0351364
$811$	12.1246	0.425753	0.212876	−	0.977079i	$-0.431717\pi$
$811$	12.1246	0.425753	0.212876	+	0.977079i	$0.431717\pi$
$812$	38.1246	1.33791
$813$	−16.9443	−0.594262
$814$	2.29180	0.0803274
$815$	22.6525	0.793482
$816$	−4.14590	−0.145135
$817$	2.11146	0.0738705
$818$	15.0902	0.527615
$819$	3.00000	0.104828
$820$	3.00000	0.104765
$821$	−1.47214	−0.0513779	−0.0256889	−	0.999670i	$-0.508178\pi$
$821$	−1.47214	−0.0513779	−0.0256889	+	0.999670i	$0.508178\pi$
$822$	9.38197	0.327234
$823$	−32.0344	−1.11665	−0.558325	−	0.829622i	$-0.688556\pi$
$823$	−32.0344	−1.11665	−0.558325	+	0.829622i	$0.688556\pi$
$824$	−15.6525	−0.545280
$825$	1.81966	0.0633524
$826$	6.70820	0.233408
$827$	−10.9098	−0.379372	−0.189686	−	0.981845i	$-0.560747\pi$
$827$	−10.9098	−0.379372	−0.189686	+	0.981845i	$0.560747\pi$
$828$	12.0902	0.420162
$829$	24.8328	0.862479	0.431240	−	0.902237i	$-0.358076\pi$
$829$	24.8328	0.862479	0.431240	+	0.902237i	$0.358076\pi$
$830$	6.38197	0.221521
$831$	17.7639	0.616224
$832$	0.236068	0.00818418
$833$	−4.47214	−0.154950
$834$	1.94427	0.0673246
$835$	10.3262	0.357354
$836$	−4.94427	−0.171001
$837$	3.09017	0.106812
$838$	15.1591	0.523661
$839$	3.96556	0.136906	0.0684531	−	0.997654i	$-0.478194\pi$
$839$	3.96556	0.136906	0.0684531	+	0.997654i	$0.478194\pi$
$840$	−10.8541	−0.374502
$841$	32.6869	1.12714
$842$	−3.87539	−0.133555
$843$	20.1803	0.695048
$844$	−23.5623	−0.811048
$845$	19.4164	0.667945
$846$	2.76393	0.0950259
$847$	31.2492	1.07374
$848$	14.7295	0.505813
$849$	−21.9443	−0.753125
$850$	3.29180	0.112908
$851$	36.2705	1.24334
$852$	−19.5623	−0.670194
$853$	−41.5279	−1.42189	−0.710943	−	0.703249i	$-0.751732\pi$
$853$	−41.5279	−1.42189	−0.710943	+	0.703249i	$0.751732\pi$
$854$	−23.2918	−0.797029
$855$	6.47214	0.221342
$856$	22.2361	0.760013
$857$	7.58359	0.259051	0.129525	−	0.991576i	$-0.458655\pi$
$857$	7.58359	0.259051	0.129525	+	0.991576i	$0.458655\pi$
$858$	0.472136	0.0161185
$859$	13.9787	0.476948	0.238474	−	0.971149i	$-0.423353\pi$
$859$	13.9787	0.476948	0.238474	+	0.971149i	$0.423353\pi$
$860$	−1.38197	−0.0471246
$861$	−3.43769	−0.117156
$862$	−18.6738	−0.636031
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	5.61803	0.191129
$865$	−21.4164	−0.728180
$866$	21.8754	0.743356
$867$	−12.0000	−0.407541
$868$	15.0000	0.509133
$869$	6.72136	0.228007
$870$	−7.85410	−0.266279
$871$	10.8541	0.367777
$872$	26.9098	0.911282
$873$	−13.4164	−0.454077
$874$	18.4721	0.624829
$875$	−35.8328	−1.21137
$876$	23.7984	0.804073
$877$	15.8885	0.536518	0.268259	−	0.963347i	$-0.413552\pi$
$877$	15.8885	0.536518	0.268259	+	0.963347i	$0.413552\pi$
$878$	14.5066	0.489573
$879$	−11.6525	−0.393028
$880$	2.29180	0.0772564
$881$	51.1033	1.72171	0.860857	−	0.508846i	$-0.169928\pi$
$881$	51.1033	1.72171	0.860857	+	0.508846i	$0.169928\pi$
$882$	1.23607	0.0416206
$883$	−18.5410	−0.623955	−0.311977	−	0.950089i	$-0.600991\pi$
$883$	−18.5410	−0.623955	−0.311977	+	0.950089i	$0.600991\pi$
$884$	−3.61803	−0.121688
$885$	5.85410	0.196783
$886$	−16.5410	−0.555706
$887$	−14.8885	−0.499908	−0.249954	−	0.968258i	$-0.580416\pi$
$887$	−14.8885	−0.499908	−0.249954	+	0.968258i	$0.580416\pi$
$888$	10.8541	0.364240
$889$	8.56231	0.287171
$890$	3.00000	0.100560
$891$	−0.763932	−0.0255927
$892$	−1.09017	−0.0365016
$893$	−17.8885	−0.598617
$894$	1.43769	0.0480837
$895$	7.14590	0.238861
$896$	34.1459	1.14073
$897$	7.47214	0.249487
$898$	19.7426	0.658821
$899$	24.2705	0.809467
$900$	3.85410	0.128470
$901$	−17.7639	−0.591802
$902$	−0.541020	−0.0180140
$903$	1.58359	0.0526986
$904$	−2.76393	−0.0919270
$905$	15.0344	0.499762
$906$	0.618034	0.0205328
$907$	17.5410	0.582440	0.291220	−	0.956656i	$-0.405939\pi$
$907$	17.5410	0.582440	0.291220	+	0.956656i	$0.405939\pi$
$908$	22.5066	0.746907
$909$	1.90983	0.0633451
$910$	−3.00000	−0.0994490
$911$	56.5967	1.87513	0.937567	−	0.347805i	$-0.113073\pi$
$911$	56.5967	1.87513	0.937567	+	0.347805i	$0.113073\pi$
$912$	−7.41641	−0.245582
$913$	4.87539	0.161352
$914$	−12.0000	−0.396925
$915$	−20.3262	−0.671965
$916$	35.9787	1.18877
$917$	−11.7295	−0.387342
$918$	−1.38197	−0.0456117
$919$	21.4377	0.707164	0.353582	−	0.935403i	$-0.384963\pi$
$919$	21.4377	0.707164	0.353582	+	0.935403i	$0.384963\pi$
$920$	−27.0344	−0.891299
$921$	17.0344	0.561304
$922$	25.9787	0.855563
$923$	−12.0902	−0.397953
$924$	−3.70820	−0.121991
$925$	11.5623	0.380166
$926$	2.87539	0.0944911
$927$	7.00000	0.229910
$928$	44.1246	1.44846
$929$	−24.7984	−0.813608	−0.406804	−	0.913515i	$-0.633357\pi$
$929$	−24.7984	−0.813608	−0.406804	+	0.913515i	$0.633357\pi$
$930$	−3.09017	−0.101331
$931$	−8.00000	−0.262189
$932$	3.61803	0.118513
$933$	3.00000	0.0982156
$934$	25.8541	0.845972
$935$	−2.76393	−0.0903902
$936$	2.23607	0.0730882
$937$	29.0000	0.947389	0.473694	−	0.880689i	$-0.342920\pi$
$937$	29.0000	0.947389	0.473694	+	0.880689i	$0.342920\pi$
$938$	20.1246	0.657092
$939$	13.5066	0.440771
$940$	11.7082	0.381880
$941$	−21.4721	−0.699972	−0.349986	−	0.936755i	$-0.613814\pi$
$941$	−21.4721	−0.699972	−0.349986	+	0.936755i	$0.613814\pi$
$942$	2.20163	0.0717329
$943$	−8.56231	−0.278827
$944$	−6.70820	−0.218333
$945$	4.85410	0.157904
$946$	0.249224	0.00810296
$947$	−54.7426	−1.77890	−0.889448	−	0.457035i	$-0.848911\pi$
$947$	−54.7426	−1.77890	−0.889448	+	0.457035i	$0.848911\pi$
$948$	14.2361	0.462366
$949$	14.7082	0.477449
$950$	5.88854	0.191050
$951$	12.0000	0.389127
$952$	−15.0000	−0.486153
$953$	−42.1033	−1.36386	−0.681930	−	0.731417i	$-0.738859\pi$
$953$	−42.1033	−1.36386	−0.681930	+	0.731417i	$0.738859\pi$
$954$	4.90983	0.158962
$955$	31.3262	1.01369
$956$	−4.76393	−0.154077
$957$	−6.00000	−0.193952
$958$	−5.83282	−0.188450
$959$	−45.5410	−1.47060
$960$	0.381966	0.0123279
$961$	−21.4508	−0.691963
$962$	3.00000	0.0967239
$963$	−9.94427	−0.320450
$964$	−16.0902	−0.518229
$965$	−41.8885	−1.34844
$966$	13.8541	0.445748
$967$	23.7082	0.762404	0.381202	−	0.924492i	$-0.375510\pi$
$967$	23.7082	0.762404	0.381202	+	0.924492i	$0.375510\pi$
$968$	23.2918	0.748627
$969$	8.94427	0.287331
$970$	13.4164	0.430775
$971$	−20.2361	−0.649406	−0.324703	−	0.945816i	$-0.605264\pi$
$971$	−20.2361	−0.649406	−0.324703	+	0.945816i	$0.605264\pi$
$972$	−1.61803	−0.0518985
$973$	−9.43769	−0.302559
$974$	−6.68692	−0.214263
$975$	2.38197	0.0762840
$976$	23.2918	0.745552
$977$	−23.5967	−0.754927	−0.377463	−	0.926024i	$-0.623204\pi$
$977$	−23.5967	−0.754927	−0.377463	+	0.926024i	$0.623204\pi$
$978$	−8.65248	−0.276676
$979$	2.29180	0.0732461
$980$	5.23607	0.167260
$981$	−12.0344	−0.384230
$982$	11.7984	0.376501
$983$	38.5967	1.23105	0.615523	−	0.788119i	$-0.288945\pi$
$983$	38.5967	1.23105	0.615523	+	0.788119i	$0.288945\pi$
$984$	−2.56231	−0.0816833
$985$	32.4164	1.03287
$986$	−10.8541	−0.345665
$987$	−13.4164	−0.427049
$988$	−6.47214	−0.205906
$989$	3.94427	0.125421
$990$	0.763932	0.0242794
$991$	13.7082	0.435455	0.217728	−	0.976010i	$-0.430136\pi$
$991$	13.7082	0.435455	0.217728	+	0.976010i	$0.430136\pi$
$992$	17.3607	0.551202
$993$	19.1246	0.606901
$994$	−22.4164	−0.711005
$995$	−0.381966	−0.0121091
$996$	10.3262	0.327199
$997$	44.6869	1.41525	0.707624	−	0.706589i	$-0.249767\pi$
$997$	44.6869	1.41525	0.707624	+	0.706589i	$0.249767\pi$
$998$	−23.5623	−0.745852
$999$	−4.85410	−0.153577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	453.2.a.b.1.2	✓	2
3.2	odd	2		1359.2.a.c.1.1		2
4.3	odd	2		7248.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
453.2.a.b.1.2	✓	2	1.1	even	1	trivial
1359.2.a.c.1.1		2	3.2	odd	2
7248.2.a.o.1.1		2	4.3	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:	\( N \)	\(=\)	\( 453 = 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	453.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -1.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.763932 q^{11} -1.61803 q^{12} -1.00000 q^{13} -1.85410 q^{14} -1.61803 q^{15} +1.85410 q^{16} -2.23607 q^{17} +0.618034 q^{18} -4.00000 q^{19} +2.61803 q^{20} -3.00000 q^{21} -0.472136 q^{22} -7.47214 q^{23} -2.23607 q^{24} -2.38197 q^{25} -0.618034 q^{26} +1.00000 q^{27} +4.85410 q^{28} +7.85410 q^{29} -1.00000 q^{30} +3.09017 q^{31} +5.61803 q^{32} -0.763932 q^{33} -1.38197 q^{34} +4.85410 q^{35} -1.61803 q^{36} -4.85410 q^{37} -2.47214 q^{38} -1.00000 q^{39} +3.61803 q^{40} +1.14590 q^{41} -1.85410 q^{42} -0.527864 q^{43} +1.23607 q^{44} -1.61803 q^{45} -4.61803 q^{46} +4.47214 q^{47} +1.85410 q^{48} +2.00000 q^{49} -1.47214 q^{50} -2.23607 q^{51} +1.61803 q^{52} +7.94427 q^{53} +0.618034 q^{54} +1.23607 q^{55} +6.70820 q^{56} -4.00000 q^{57} +4.85410 q^{58} -3.61803 q^{59} +2.61803 q^{60} +12.5623 q^{61} +1.90983 q^{62} -3.00000 q^{63} -0.236068 q^{64} +1.61803 q^{65} -0.472136 q^{66} -10.8541 q^{67} +3.61803 q^{68} -7.47214 q^{69} +3.00000 q^{70} +12.0902 q^{71} -2.23607 q^{72} -14.7082 q^{73} -3.00000 q^{74} -2.38197 q^{75} +6.47214 q^{76} +2.29180 q^{77} -0.618034 q^{78} -8.79837 q^{79} -3.00000 q^{80} +1.00000 q^{81} +0.708204 q^{82} -6.38197 q^{83} +4.85410 q^{84} +3.61803 q^{85} -0.326238 q^{86} +7.85410 q^{87} +1.70820 q^{88} -3.00000 q^{89} -1.00000 q^{90} +3.00000 q^{91} +12.0902 q^{92} +3.09017 q^{93} +2.76393 q^{94} +6.47214 q^{95} +5.61803 q^{96} -13.4164 q^{97} +1.23607 q^{98} -0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^5 - q^6 - 6 * q^7 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{5} - q^{6} - 6 q^{7} + 2 q^{9} - 2 q^{10} - 6 q^{11} - q^{12} - 2 q^{13} + 3 q^{14} - q^{15} - 3 q^{16} - q^{18} - 8 q^{19} + 3 q^{20} - 6 q^{21} + 8 q^{22} - 6 q^{23} - 7 q^{25} + q^{26} + 2 q^{27} + 3 q^{28} + 9 q^{29} - 2 q^{30} - 5 q^{31} + 9 q^{32} - 6 q^{33} - 5 q^{34} + 3 q^{35} - q^{36} - 3 q^{37} + 4 q^{38} - 2 q^{39} + 5 q^{40} + 9 q^{41} + 3 q^{42} - 10 q^{43} - 2 q^{44} - q^{45} - 7 q^{46} - 3 q^{48} + 4 q^{49} + 6 q^{50} + q^{52} - 2 q^{53} - q^{54} - 2 q^{55} - 8 q^{57} + 3 q^{58} - 5 q^{59} + 3 q^{60} + 5 q^{61} + 15 q^{62} - 6 q^{63} + 4 q^{64} + q^{65} + 8 q^{66} - 15 q^{67} + 5 q^{68} - 6 q^{69} + 6 q^{70} + 13 q^{71} - 16 q^{73} - 6 q^{74} - 7 q^{75} + 4 q^{76} + 18 q^{77} + q^{78} + 7 q^{79} - 6 q^{80} + 2 q^{81} - 12 q^{82} - 15 q^{83} + 3 q^{84} + 5 q^{85} + 15 q^{86} + 9 q^{87} - 10 q^{88} - 6 q^{89} - 2 q^{90} + 6 q^{91} + 13 q^{92} - 5 q^{93} + 10 q^{94} + 4 q^{95} + 9 q^{96} - 2 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^5 - q^6 - 6 * q^7 + 2 * q^9 - 2 * q^10 - 6 * q^11 - q^12 - 2 * q^13 + 3 * q^14 - q^15 - 3 * q^16 - q^18 - 8 * q^19 + 3 * q^20 - 6 * q^21 + 8 * q^22 - 6 * q^23 - 7 * q^25 + q^26 + 2 * q^27 + 3 * q^28 + 9 * q^29 - 2 * q^30 - 5 * q^31 + 9 * q^32 - 6 * q^33 - 5 * q^34 + 3 * q^35 - q^36 - 3 * q^37 + 4 * q^38 - 2 * q^39 + 5 * q^40 + 9 * q^41 + 3 * q^42 - 10 * q^43 - 2 * q^44 - q^45 - 7 * q^46 - 3 * q^48 + 4 * q^49 + 6 * q^50 + q^52 - 2 * q^53 - q^54 - 2 * q^55 - 8 * q^57 + 3 * q^58 - 5 * q^59 + 3 * q^60 + 5 * q^61 + 15 * q^62 - 6 * q^63 + 4 * q^64 + q^65 + 8 * q^66 - 15 * q^67 + 5 * q^68 - 6 * q^69 + 6 * q^70 + 13 * q^71 - 16 * q^73 - 6 * q^74 - 7 * q^75 + 4 * q^76 + 18 * q^77 + q^78 + 7 * q^79 - 6 * q^80 + 2 * q^81 - 12 * q^82 - 15 * q^83 + 3 * q^84 + 5 * q^85 + 15 * q^86 + 9 * q^87 - 10 * q^88 - 6 * q^89 - 2 * q^90 + 6 * q^91 + 13 * q^92 - 5 * q^93 + 10 * q^94 + 4 * q^95 + 9 * q^96 - 2 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more