LMFDB - Embedded newform 3234.2.a.bd.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3234,2,Mod(1,3234)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3234.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,2,2,-4,2,0,2,2,-4,-2,2,-8,0,-4,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.8236200137$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.58579 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -6.24264 q^{19} -2.00000 q^{20} -1.00000 q^{22} +0.828427 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.58579 q^{26} +1.00000 q^{27} -1.65685 q^{29} -2.00000 q^{30} +2.24264 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -4.82843 q^{37} -6.24264 q^{38} -2.58579 q^{39} -2.00000 q^{40} -0.343146 q^{41} +0.828427 q^{43} -1.00000 q^{44} -2.00000 q^{45} +0.828427 q^{46} -11.8995 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -2.58579 q^{52} +6.48528 q^{53} +1.00000 q^{54} +2.00000 q^{55} -6.24264 q^{57} -1.65685 q^{58} -1.17157 q^{59} -2.00000 q^{60} -5.41421 q^{61} +2.24264 q^{62} +1.00000 q^{64} +5.17157 q^{65} -1.00000 q^{66} -6.82843 q^{67} -2.00000 q^{68} +0.828427 q^{69} -5.65685 q^{71} +1.00000 q^{72} -0.343146 q^{73} -4.82843 q^{74} -1.00000 q^{75} -6.24264 q^{76} -2.58579 q^{78} +0.485281 q^{79} -2.00000 q^{80} +1.00000 q^{81} -0.343146 q^{82} +9.07107 q^{83} +4.00000 q^{85} +0.828427 q^{86} -1.65685 q^{87} -1.00000 q^{88} -12.2426 q^{89} -2.00000 q^{90} +0.828427 q^{92} +2.24264 q^{93} -11.8995 q^{94} +12.4853 q^{95} +1.00000 q^{96} -9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 8 q^{13} - 4 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^10 - 2 * q^11 + 2 * q^12 - 8 * q^13 - 4 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 - 4 * q^19 - 4 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^24 - 2 * q^25 - 8 * q^26 + 2 * q^27 + 8 * q^29 - 4 * q^30 - 4 * q^31 + 2 * q^32 - 2 * q^33 - 4 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 8 * q^39 - 4 * q^40 - 12 * q^41 - 4 * q^43 - 2 * q^44 - 4 * q^45 - 4 * q^46 - 4 * q^47 + 2 * q^48 - 2 * q^50 - 4 * q^51 - 8 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^65 - 2 * q^66 - 8 * q^67 - 4 * q^68 - 4 * q^69 + 2 * q^72 - 12 * q^73 - 4 * q^74 - 2 * q^75 - 4 * q^76 - 8 * q^78 - 16 * q^79 - 4 * q^80 + 2 * q^81 - 12 * q^82 + 4 * q^83 + 8 * q^85 - 4 * q^86 + 8 * q^87 - 2 * q^88 - 16 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 4 * q^94 + 8 * q^95 + 2 * q^96 - 2 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−6.24264	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	0.828427	0.172739	0.0863695	−	0.996263i	$-0.472473\pi$
$23$	0.828427	0.172739	0.0863695	+	0.996263i	$0.472473\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	−2.58579	−0.507114
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.65685	−0.307670	−0.153835	−	0.988097i	$-0.549162\pi$
$29$	−1.65685	−0.307670	−0.153835	+	0.988097i	$0.549162\pi$
$30$	−2.00000	−0.365148
$31$	2.24264	0.402790	0.201395	−	0.979510i	$-0.435452\pi$
$31$	2.24264	0.402790	0.201395	+	0.979510i	$0.435452\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.82843	−0.793789	−0.396894	−	0.917864i	$-0.629912\pi$
$37$	−4.82843	−0.793789	−0.396894	+	0.917864i	$0.629912\pi$
$38$	−6.24264	−1.01269
$39$	−2.58579	−0.414057
$40$	−2.00000	−0.316228
$41$	−0.343146	−0.0535904	−0.0267952	−	0.999641i	$-0.508530\pi$
$41$	−0.343146	−0.0535904	−0.0267952	+	0.999641i	$0.508530\pi$
$42$	0	0
$43$	0.828427	0.126334	0.0631670	−	0.998003i	$-0.479880\pi$
$43$	0.828427	0.126334	0.0631670	+	0.998003i	$0.479880\pi$
$44$	−1.00000	−0.150756
$45$	−2.00000	−0.298142
$46$	0.828427	0.122145
$47$	−11.8995	−1.73572	−0.867860	−	0.496809i	$-0.834505\pi$
$47$	−11.8995	−1.73572	−0.867860	+	0.496809i	$0.834505\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	−2.58579	−0.358584
$53$	6.48528	0.890822	0.445411	−	0.895326i	$-0.353058\pi$
$53$	6.48528	0.890822	0.445411	+	0.895326i	$0.353058\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	0	0
$57$	−6.24264	−0.826858
$58$	−1.65685	−0.217556
$59$	−1.17157	−0.152526	−0.0762629	−	0.997088i	$-0.524299\pi$
$59$	−1.17157	−0.152526	−0.0762629	+	0.997088i	$0.524299\pi$
$60$	−2.00000	−0.258199
$61$	−5.41421	−0.693219	−0.346610	−	0.938010i	$-0.612667\pi$
$61$	−5.41421	−0.693219	−0.346610	+	0.938010i	$0.612667\pi$
$62$	2.24264	0.284816
$63$	0	0
$64$	1.00000	0.125000
$65$	5.17157	0.641455
$66$	−1.00000	−0.123091
$67$	−6.82843	−0.834225	−0.417113	−	0.908855i	$-0.636958\pi$
$67$	−6.82843	−0.834225	−0.417113	+	0.908855i	$0.636958\pi$
$68$	−2.00000	−0.242536
$69$	0.828427	0.0997309
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	1.00000	0.117851
$73$	−0.343146	−0.0401622	−0.0200811	−	0.999798i	$-0.506392\pi$
$73$	−0.343146	−0.0401622	−0.0200811	+	0.999798i	$0.506392\pi$
$74$	−4.82843	−0.561293
$75$	−1.00000	−0.115470
$76$	−6.24264	−0.716080
$77$	0	0
$78$	−2.58579	−0.292783
$79$	0.485281	0.0545984	0.0272992	−	0.999627i	$-0.491309\pi$
$79$	0.485281	0.0545984	0.0272992	+	0.999627i	$0.491309\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−0.343146	−0.0378941
$83$	9.07107	0.995679	0.497840	−	0.867269i	$-0.334127\pi$
$83$	9.07107	0.995679	0.497840	+	0.867269i	$0.334127\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0.828427	0.0893316
$87$	−1.65685	−0.177633
$88$	−1.00000	−0.106600
$89$	−12.2426	−1.29772	−0.648859	−	0.760909i	$-0.724753\pi$
$89$	−12.2426	−1.29772	−0.648859	+	0.760909i	$0.724753\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0.828427	0.0863695
$93$	2.24264	0.232551
$94$	−11.8995	−1.22734
$95$	12.4853	1.28096
$96$	1.00000	0.102062
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	0.928932	0.0924322	0.0462161	−	0.998931i	$-0.485284\pi$
$101$	0.928932	0.0924322	0.0462161	+	0.998931i	$0.485284\pi$
$102$	−2.00000	−0.198030
$103$	2.24264	0.220974	0.110487	−	0.993878i	$-0.464759\pi$
$103$	2.24264	0.220974	0.110487	+	0.993878i	$0.464759\pi$
$104$	−2.58579	−0.253557
$105$	0	0
$106$	6.48528	0.629906
$107$	7.17157	0.693302	0.346651	−	0.937994i	$-0.387319\pi$
$107$	7.17157	0.693302	0.346651	+	0.937994i	$0.387319\pi$
$108$	1.00000	0.0962250
$109$	15.6569	1.49965	0.749827	−	0.661634i	$-0.230137\pi$
$109$	15.6569	1.49965	0.749827	+	0.661634i	$0.230137\pi$
$110$	2.00000	0.190693
$111$	−4.82843	−0.458294
$112$	0	0
$113$	−5.65685	−0.532152	−0.266076	−	0.963952i	$-0.585727\pi$
$113$	−5.65685	−0.532152	−0.266076	+	0.963952i	$0.585727\pi$
$114$	−6.24264	−0.584677
$115$	−1.65685	−0.154502
$116$	−1.65685	−0.153835
$117$	−2.58579	−0.239056
$118$	−1.17157	−0.107852
$119$	0	0
$120$	−2.00000	−0.182574
$121$	1.00000	0.0909091
$122$	−5.41421	−0.490180
$123$	−0.343146	−0.0309404
$124$	2.24264	0.201395
$125$	12.0000	1.07331
$126$	0	0
$127$	−14.1421	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$128$	1.00000	0.0883883
$129$	0.828427	0.0729389
$130$	5.17157	0.453577
$131$	13.5563	1.18442	0.592212	−	0.805782i	$-0.298255\pi$
$131$	13.5563	1.18442	0.592212	+	0.805782i	$0.298255\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−6.82843	−0.589886
$135$	−2.00000	−0.172133
$136$	−2.00000	−0.171499
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	0.828427	0.0705204
$139$	−19.8995	−1.68785	−0.843927	−	0.536459i	$-0.819762\pi$
$139$	−19.8995	−1.68785	−0.843927	+	0.536459i	$0.819762\pi$
$140$	0	0
$141$	−11.8995	−1.00212
$142$	−5.65685	−0.474713
$143$	2.58579	0.216234
$144$	1.00000	0.0833333
$145$	3.31371	0.275189
$146$	−0.343146	−0.0283989
$147$	0	0
$148$	−4.82843	−0.396894
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−1.00000	−0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−6.24264	−0.506345
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−4.48528	−0.360266
$156$	−2.58579	−0.207029
$157$	−7.17157	−0.572354	−0.286177	−	0.958177i	$-0.592385\pi$
$157$	−7.17157	−0.572354	−0.286177	+	0.958177i	$0.592385\pi$
$158$	0.485281	0.0386069
$159$	6.48528	0.514316
$160$	−2.00000	−0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	−13.6569	−1.06969	−0.534844	−	0.844951i	$-0.679630\pi$
$163$	−13.6569	−1.06969	−0.534844	+	0.844951i	$0.679630\pi$
$164$	−0.343146	−0.0267952
$165$	2.00000	0.155700
$166$	9.07107	0.704051
$167$	10.8284	0.837929	0.418964	−	0.908003i	$-0.362393\pi$
$167$	10.8284	0.837929	0.418964	+	0.908003i	$0.362393\pi$
$168$	0	0
$169$	−6.31371	−0.485670
$170$	4.00000	0.306786
$171$	−6.24264	−0.477387
$172$	0.828427	0.0631670
$173$	−13.8995	−1.05676	−0.528380	−	0.849008i	$-0.677200\pi$
$173$	−13.8995	−1.05676	−0.528380	+	0.849008i	$0.677200\pi$
$174$	−1.65685	−0.125606
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−1.17157	−0.0880608
$178$	−12.2426	−0.917625
$179$	−3.51472	−0.262702	−0.131351	−	0.991336i	$-0.541932\pi$
$179$	−3.51472	−0.262702	−0.131351	+	0.991336i	$0.541932\pi$
$180$	−2.00000	−0.149071
$181$	−13.3137	−0.989600	−0.494800	−	0.869007i	$-0.664759\pi$
$181$	−13.3137	−0.989600	−0.494800	+	0.869007i	$0.664759\pi$
$182$	0	0
$183$	−5.41421	−0.400230
$184$	0.828427	0.0610725
$185$	9.65685	0.709986
$186$	2.24264	0.164438
$187$	2.00000	0.146254
$188$	−11.8995	−0.867860
$189$	0	0
$190$	12.4853	0.905778
$191$	−17.7990	−1.28789	−0.643945	−	0.765072i	$-0.722703\pi$
$191$	−17.7990	−1.28789	−0.643945	+	0.765072i	$0.722703\pi$
$192$	1.00000	0.0721688
$193$	18.9706	1.36553	0.682765	−	0.730638i	$-0.260777\pi$
$193$	18.9706	1.36553	0.682765	+	0.730638i	$0.260777\pi$
$194$	−9.89949	−0.710742
$195$	5.17157	0.370344
$196$	0	0
$197$	4.34315	0.309436	0.154718	−	0.987959i	$-0.450553\pi$
$197$	4.34315	0.309436	0.154718	+	0.987959i	$0.450553\pi$
$198$	−1.00000	−0.0710669
$199$	27.2132	1.92909	0.964546	−	0.263913i	$-0.0850133\pi$
$199$	27.2132	1.92909	0.964546	+	0.263913i	$0.0850133\pi$
$200$	−1.00000	−0.0707107
$201$	−6.82843	−0.481640
$202$	0.928932	0.0653594
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0.686292	0.0479327
$206$	2.24264	0.156252
$207$	0.828427	0.0575797
$208$	−2.58579	−0.179292
$209$	6.24264	0.431812
$210$	0	0
$211$	−28.8284	−1.98463	−0.992315	−	0.123734i	$-0.960513\pi$
$211$	−28.8284	−1.98463	−0.992315	+	0.123734i	$0.960513\pi$
$212$	6.48528	0.445411
$213$	−5.65685	−0.387601
$214$	7.17157	0.490239
$215$	−1.65685	−0.112997
$216$	1.00000	0.0680414
$217$	0	0
$218$	15.6569	1.06042
$219$	−0.343146	−0.0231876
$220$	2.00000	0.134840
$221$	5.17157	0.347878
$222$	−4.82843	−0.324063
$223$	2.24264	0.150178	0.0750892	−	0.997177i	$-0.476076\pi$
$223$	2.24264	0.150178	0.0750892	+	0.997177i	$0.476076\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−5.65685	−0.376288
$227$	−7.89949	−0.524308	−0.262154	−	0.965026i	$-0.584433\pi$
$227$	−7.89949	−0.524308	−0.262154	+	0.965026i	$0.584433\pi$
$228$	−6.24264	−0.413429
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−1.65685	−0.109250
$231$	0	0
$232$	−1.65685	−0.108778
$233$	3.65685	0.239568	0.119784	−	0.992800i	$-0.461780\pi$
$233$	3.65685	0.239568	0.119784	+	0.992800i	$0.461780\pi$
$234$	−2.58579	−0.169038
$235$	23.7990	1.55247
$236$	−1.17157	−0.0762629
$237$	0.485281	0.0315224
$238$	0	0
$239$	23.3137	1.50804	0.754019	−	0.656852i	$-0.228113\pi$
$239$	23.3137	1.50804	0.754019	+	0.656852i	$0.228113\pi$
$240$	−2.00000	−0.129099
$241$	−3.65685	−0.235559	−0.117779	−	0.993040i	$-0.537578\pi$
$241$	−3.65685	−0.235559	−0.117779	+	0.993040i	$0.537578\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−5.41421	−0.346610
$245$	0	0
$246$	−0.343146	−0.0218782
$247$	16.1421	1.02710
$248$	2.24264	0.142408
$249$	9.07107	0.574856
$250$	12.0000	0.758947
$251$	8.48528	0.535586	0.267793	−	0.963476i	$-0.413706\pi$
$251$	8.48528	0.535586	0.267793	+	0.963476i	$0.413706\pi$
$252$	0	0
$253$	−0.828427	−0.0520828
$254$	−14.1421	−0.887357
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−13.2132	−0.824217	−0.412108	−	0.911135i	$-0.635208\pi$
$257$	−13.2132	−0.824217	−0.412108	+	0.911135i	$0.635208\pi$
$258$	0.828427	0.0515756
$259$	0	0
$260$	5.17157	0.320727
$261$	−1.65685	−0.102557
$262$	13.5563	0.837514
$263$	14.8284	0.914360	0.457180	−	0.889374i	$-0.348860\pi$
$263$	14.8284	0.914360	0.457180	+	0.889374i	$0.348860\pi$
$264$	−1.00000	−0.0615457
$265$	−12.9706	−0.796775
$266$	0	0
$267$	−12.2426	−0.749237
$268$	−6.82843	−0.417113
$269$	17.3137	1.05564	0.527818	−	0.849358i	$-0.323010\pi$
$269$	17.3137	1.05564	0.527818	+	0.849358i	$0.323010\pi$
$270$	−2.00000	−0.121716
$271$	6.14214	0.373108	0.186554	−	0.982445i	$-0.440268\pi$
$271$	6.14214	0.373108	0.186554	+	0.982445i	$0.440268\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	17.3137	1.04596
$275$	1.00000	0.0603023
$276$	0.828427	0.0498655
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	−19.8995	−1.19349
$279$	2.24264	0.134263
$280$	0	0
$281$	21.3137	1.27147	0.635735	−	0.771908i	$-0.280697\pi$
$281$	21.3137	1.27147	0.635735	+	0.771908i	$0.280697\pi$
$282$	−11.8995	−0.708605
$283$	−6.24264	−0.371086	−0.185543	−	0.982636i	$-0.559404\pi$
$283$	−6.24264	−0.371086	−0.185543	+	0.982636i	$0.559404\pi$
$284$	−5.65685	−0.335673
$285$	12.4853	0.739564
$286$	2.58579	0.152901
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	3.31371	0.194588
$291$	−9.89949	−0.580319
$292$	−0.343146	−0.0200811
$293$	19.0711	1.11414	0.557072	−	0.830464i	$-0.311925\pi$
$293$	19.0711	1.11414	0.557072	+	0.830464i	$0.311925\pi$
$294$	0	0
$295$	2.34315	0.136423
$296$	−4.82843	−0.280647
$297$	−1.00000	−0.0580259
$298$	4.00000	0.231714
$299$	−2.14214	−0.123883
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	4.00000	0.230174
$303$	0.928932	0.0533658
$304$	−6.24264	−0.358040
$305$	10.8284	0.620034
$306$	−2.00000	−0.114332
$307$	10.9289	0.623747	0.311874	−	0.950124i	$-0.399043\pi$
$307$	10.9289	0.623747	0.311874	+	0.950124i	$0.399043\pi$
$308$	0	0
$309$	2.24264	0.127579
$310$	−4.48528	−0.254747
$311$	−27.8995	−1.58204	−0.791018	−	0.611793i	$-0.790448\pi$
$311$	−27.8995	−1.58204	−0.791018	+	0.611793i	$0.790448\pi$
$312$	−2.58579	−0.146391
$313$	−0.242641	−0.0137149	−0.00685743	−	0.999976i	$-0.502183\pi$
$313$	−0.242641	−0.0137149	−0.00685743	+	0.999976i	$0.502183\pi$
$314$	−7.17157	−0.404715
$315$	0	0
$316$	0.485281	0.0272992
$317$	7.17157	0.402796	0.201398	−	0.979510i	$-0.435452\pi$
$317$	7.17157	0.402796	0.201398	+	0.979510i	$0.435452\pi$
$318$	6.48528	0.363677
$319$	1.65685	0.0927660
$320$	−2.00000	−0.111803
$321$	7.17157	0.400278
$322$	0	0
$323$	12.4853	0.694700
$324$	1.00000	0.0555556
$325$	2.58579	0.143434
$326$	−13.6569	−0.756383
$327$	15.6569	0.865826
$328$	−0.343146	−0.0189471
$329$	0	0
$330$	2.00000	0.110096
$331$	11.7990	0.648531	0.324266	−	0.945966i	$-0.394883\pi$
$331$	11.7990	0.648531	0.324266	+	0.945966i	$0.394883\pi$
$332$	9.07107	0.497840
$333$	−4.82843	−0.264596
$334$	10.8284	0.592505
$335$	13.6569	0.746154
$336$	0	0
$337$	7.17157	0.390660	0.195330	−	0.980738i	$-0.437422\pi$
$337$	7.17157	0.390660	0.195330	+	0.980738i	$0.437422\pi$
$338$	−6.31371	−0.343420
$339$	−5.65685	−0.307238
$340$	4.00000	0.216930
$341$	−2.24264	−0.121446
$342$	−6.24264	−0.337563
$343$	0	0
$344$	0.828427	0.0446658
$345$	−1.65685	−0.0892020
$346$	−13.8995	−0.747241
$347$	−31.4558	−1.68864	−0.844319	−	0.535841i	$-0.819995\pi$
$347$	−31.4558	−1.68864	−0.844319	+	0.535841i	$0.819995\pi$
$348$	−1.65685	−0.0888167
$349$	14.5858	0.780759	0.390380	−	0.920654i	$-0.372344\pi$
$349$	14.5858	0.780759	0.390380	+	0.920654i	$0.372344\pi$
$350$	0	0
$351$	−2.58579	−0.138019
$352$	−1.00000	−0.0533002
$353$	21.2132	1.12906	0.564532	−	0.825411i	$-0.309057\pi$
$353$	21.2132	1.12906	0.564532	+	0.825411i	$0.309057\pi$
$354$	−1.17157	−0.0622684
$355$	11.3137	0.600469
$356$	−12.2426	−0.648859
$357$	0	0
$358$	−3.51472	−0.185759
$359$	10.8284	0.571503	0.285751	−	0.958304i	$-0.407757\pi$
$359$	10.8284	0.571503	0.285751	+	0.958304i	$0.407757\pi$
$360$	−2.00000	−0.105409
$361$	19.9706	1.05108
$362$	−13.3137	−0.699753
$363$	1.00000	0.0524864
$364$	0	0
$365$	0.686292	0.0359221
$366$	−5.41421	−0.283005
$367$	35.4142	1.84861	0.924303	−	0.381658i	$-0.124647\pi$
$367$	35.4142	1.84861	0.924303	+	0.381658i	$0.124647\pi$
$368$	0.828427	0.0431847
$369$	−0.343146	−0.0178635
$370$	9.65685	0.502036
$371$	0	0
$372$	2.24264	0.116276
$373$	−31.9411	−1.65385	−0.826924	−	0.562313i	$-0.809912\pi$
$373$	−31.9411	−1.65385	−0.826924	+	0.562313i	$0.809912\pi$
$374$	2.00000	0.103418
$375$	12.0000	0.619677
$376$	−11.8995	−0.613670
$377$	4.28427	0.220651
$378$	0	0
$379$	−24.2843	−1.24740	−0.623700	−	0.781664i	$-0.714371\pi$
$379$	−24.2843	−1.24740	−0.623700	+	0.781664i	$0.714371\pi$
$380$	12.4853	0.640481
$381$	−14.1421	−0.724524
$382$	−17.7990	−0.910676
$383$	−2.72792	−0.139390	−0.0696952	−	0.997568i	$-0.522203\pi$
$383$	−2.72792	−0.139390	−0.0696952	+	0.997568i	$0.522203\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	18.9706	0.965576
$387$	0.828427	0.0421113
$388$	−9.89949	−0.502571
$389$	22.2843	1.12986	0.564929	−	0.825140i	$-0.308904\pi$
$389$	22.2843	1.12986	0.564929	+	0.825140i	$0.308904\pi$
$390$	5.17157	0.261873
$391$	−1.65685	−0.0837907
$392$	0	0
$393$	13.5563	0.683827
$394$	4.34315	0.218805
$395$	−0.970563	−0.0488343
$396$	−1.00000	−0.0502519
$397$	16.8284	0.844595	0.422297	−	0.906457i	$-0.361224\pi$
$397$	16.8284	0.844595	0.422297	+	0.906457i	$0.361224\pi$
$398$	27.2132	1.36407
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	33.3137	1.66361	0.831804	−	0.555070i	$-0.187309\pi$
$401$	33.3137	1.66361	0.831804	+	0.555070i	$0.187309\pi$
$402$	−6.82843	−0.340571
$403$	−5.79899	−0.288868
$404$	0.928932	0.0462161
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	4.82843	0.239336
$408$	−2.00000	−0.0990148
$409$	18.4853	0.914038	0.457019	−	0.889457i	$-0.348917\pi$
$409$	18.4853	0.914038	0.457019	+	0.889457i	$0.348917\pi$
$410$	0.686292	0.0338935
$411$	17.3137	0.854022
$412$	2.24264	0.110487
$413$	0	0
$414$	0.828427	0.0407150
$415$	−18.1421	−0.890562
$416$	−2.58579	−0.126779
$417$	−19.8995	−0.974482
$418$	6.24264	0.305338
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	−16.3431	−0.796516	−0.398258	−	0.917273i	$-0.630385\pi$
$421$	−16.3431	−0.796516	−0.398258	+	0.917273i	$0.630385\pi$
$422$	−28.8284	−1.40335
$423$	−11.8995	−0.578573
$424$	6.48528	0.314953
$425$	2.00000	0.0970143
$426$	−5.65685	−0.274075
$427$	0	0
$428$	7.17157	0.346651
$429$	2.58579	0.124843
$430$	−1.65685	−0.0799006
$431$	−26.1421	−1.25922	−0.629611	−	0.776910i	$-0.716786\pi$
$431$	−26.1421	−1.25922	−0.629611	+	0.776910i	$0.716786\pi$
$432$	1.00000	0.0481125
$433$	−2.38478	−0.114605	−0.0573025	−	0.998357i	$-0.518250\pi$
$433$	−2.38478	−0.114605	−0.0573025	+	0.998357i	$0.518250\pi$
$434$	0	0
$435$	3.31371	0.158880
$436$	15.6569	0.749827
$437$	−5.17157	−0.247390
$438$	−0.343146	−0.0163961
$439$	34.8284	1.66227	0.831135	−	0.556071i	$-0.187692\pi$
$439$	34.8284	1.66227	0.831135	+	0.556071i	$0.187692\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	5.17157	0.245987
$443$	−4.48528	−0.213102	−0.106551	−	0.994307i	$-0.533981\pi$
$443$	−4.48528	−0.213102	−0.106551	+	0.994307i	$0.533981\pi$
$444$	−4.82843	−0.229147
$445$	24.4853	1.16071
$446$	2.24264	0.106192
$447$	4.00000	0.189194
$448$	0	0
$449$	−16.9706	−0.800890	−0.400445	−	0.916321i	$-0.631145\pi$
$449$	−16.9706	−0.800890	−0.400445	+	0.916321i	$0.631145\pi$
$450$	−1.00000	−0.0471405
$451$	0.343146	0.0161581
$452$	−5.65685	−0.266076
$453$	4.00000	0.187936
$454$	−7.89949	−0.370742
$455$	0	0
$456$	−6.24264	−0.292338
$457$	−6.68629	−0.312772	−0.156386	−	0.987696i	$-0.549984\pi$
$457$	−6.68629	−0.312772	−0.156386	+	0.987696i	$0.549984\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	−1.65685	−0.0772512
$461$	33.4142	1.55626	0.778128	−	0.628106i	$-0.216170\pi$
$461$	33.4142	1.55626	0.778128	+	0.628106i	$0.216170\pi$
$462$	0	0
$463$	19.3137	0.897584	0.448792	−	0.893636i	$-0.351854\pi$
$463$	19.3137	0.897584	0.448792	+	0.893636i	$0.351854\pi$
$464$	−1.65685	−0.0769175
$465$	−4.48528	−0.208000
$466$	3.65685	0.169401
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.58579	−0.119528
$469$	0	0
$470$	23.7990	1.09777
$471$	−7.17157	−0.330449
$472$	−1.17157	−0.0539260
$473$	−0.828427	−0.0380911
$474$	0.485281	0.0222897
$475$	6.24264	0.286432
$476$	0	0
$477$	6.48528	0.296941
$478$	23.3137	1.06634
$479$	15.3137	0.699701	0.349851	−	0.936806i	$-0.386232\pi$
$479$	15.3137	0.699701	0.349851	+	0.936806i	$0.386232\pi$
$480$	−2.00000	−0.0912871
$481$	12.4853	0.569280
$482$	−3.65685	−0.166565
$483$	0	0
$484$	1.00000	0.0454545
$485$	19.7990	0.899026
$486$	1.00000	0.0453609
$487$	−28.1421	−1.27524	−0.637621	−	0.770350i	$-0.720081\pi$
$487$	−28.1421	−1.27524	−0.637621	+	0.770350i	$0.720081\pi$
$488$	−5.41421	−0.245090
$489$	−13.6569	−0.617584
$490$	0	0
$491$	2.62742	0.118574	0.0592868	−	0.998241i	$-0.481117\pi$
$491$	2.62742	0.118574	0.0592868	+	0.998241i	$0.481117\pi$
$492$	−0.343146	−0.0154702
$493$	3.31371	0.149242
$494$	16.1421	0.726269
$495$	2.00000	0.0898933
$496$	2.24264	0.100698
$497$	0	0
$498$	9.07107	0.406484
$499$	22.1421	0.991218	0.495609	−	0.868546i	$-0.334945\pi$
$499$	22.1421	0.991218	0.495609	+	0.868546i	$0.334945\pi$
$500$	12.0000	0.536656
$501$	10.8284	0.483778
$502$	8.48528	0.378717
$503$	−8.48528	−0.378340	−0.189170	−	0.981944i	$-0.560580\pi$
$503$	−8.48528	−0.378340	−0.189170	+	0.981944i	$0.560580\pi$
$504$	0	0
$505$	−1.85786	−0.0826739
$506$	−0.828427	−0.0368281
$507$	−6.31371	−0.280402
$508$	−14.1421	−0.627456
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	−6.24264	−0.275619
$514$	−13.2132	−0.582809
$515$	−4.48528	−0.197645
$516$	0.828427	0.0364695
$517$	11.8995	0.523339
$518$	0	0
$519$	−13.8995	−0.610120
$520$	5.17157	0.226788
$521$	21.2132	0.929367	0.464684	−	0.885477i	$-0.346168\pi$
$521$	21.2132	0.929367	0.464684	+	0.885477i	$0.346168\pi$
$522$	−1.65685	−0.0725185
$523$	−5.07107	−0.221742	−0.110871	−	0.993835i	$-0.535364\pi$
$523$	−5.07107	−0.221742	−0.110871	+	0.993835i	$0.535364\pi$
$524$	13.5563	0.592212
$525$	0	0
$526$	14.8284	0.646550
$527$	−4.48528	−0.195382
$528$	−1.00000	−0.0435194
$529$	−22.3137	−0.970161
$530$	−12.9706	−0.563405
$531$	−1.17157	−0.0508419
$532$	0	0
$533$	0.887302	0.0384333
$534$	−12.2426	−0.529791
$535$	−14.3431	−0.620108
$536$	−6.82843	−0.294943
$537$	−3.51472	−0.151671
$538$	17.3137	0.746447
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−22.6863	−0.975360	−0.487680	−	0.873023i	$-0.662157\pi$
$541$	−22.6863	−0.975360	−0.487680	+	0.873023i	$0.662157\pi$
$542$	6.14214	0.263827
$543$	−13.3137	−0.571346
$544$	−2.00000	−0.0857493
$545$	−31.3137	−1.34133
$546$	0	0
$547$	2.62742	0.112340	0.0561701	−	0.998421i	$-0.482111\pi$
$547$	2.62742	0.112340	0.0561701	+	0.998421i	$0.482111\pi$
$548$	17.3137	0.739605
$549$	−5.41421	−0.231073
$550$	1.00000	0.0426401
$551$	10.3431	0.440633
$552$	0.828427	0.0352602
$553$	0	0
$554$	−16.0000	−0.679775
$555$	9.65685	0.409911
$556$	−19.8995	−0.843927
$557$	8.34315	0.353510	0.176755	−	0.984255i	$-0.443440\pi$
$557$	8.34315	0.353510	0.176755	+	0.984255i	$0.443440\pi$
$558$	2.24264	0.0949386
$559$	−2.14214	−0.0906027
$560$	0	0
$561$	2.00000	0.0844401
$562$	21.3137	0.899065
$563$	6.92893	0.292020	0.146010	−	0.989283i	$-0.453357\pi$
$563$	6.92893	0.292020	0.146010	+	0.989283i	$0.453357\pi$
$564$	−11.8995	−0.501059
$565$	11.3137	0.475971
$566$	−6.24264	−0.262398
$567$	0	0
$568$	−5.65685	−0.237356
$569$	−33.3137	−1.39658	−0.698292	−	0.715813i	$-0.746056\pi$
$569$	−33.3137	−1.39658	−0.698292	+	0.715813i	$0.746056\pi$
$570$	12.4853	0.522951
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	2.58579	0.108117
$573$	−17.7990	−0.743563
$574$	0	0
$575$	−0.828427	−0.0345478
$576$	1.00000	0.0416667
$577$	35.5563	1.48023	0.740115	−	0.672480i	$-0.234771\pi$
$577$	35.5563	1.48023	0.740115	+	0.672480i	$0.234771\pi$
$578$	−13.0000	−0.540729
$579$	18.9706	0.788390
$580$	3.31371	0.137594
$581$	0	0
$582$	−9.89949	−0.410347
$583$	−6.48528	−0.268593
$584$	−0.343146	−0.0141995
$585$	5.17157	0.213818
$586$	19.0711	0.787819
$587$	27.7990	1.14739	0.573694	−	0.819070i	$-0.305510\pi$
$587$	27.7990	1.14739	0.573694	+	0.819070i	$0.305510\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	2.34315	0.0964658
$591$	4.34315	0.178653
$592$	−4.82843	−0.198447
$593$	14.6863	0.603094	0.301547	−	0.953451i	$-0.402497\pi$
$593$	14.6863	0.603094	0.301547	+	0.953451i	$0.402497\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	4.00000	0.163846
$597$	27.2132	1.11376
$598$	−2.14214	−0.0875984
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	−1.00000	−0.0408248
$601$	−20.8284	−0.849609	−0.424805	−	0.905285i	$-0.639657\pi$
$601$	−20.8284	−0.849609	−0.424805	+	0.905285i	$0.639657\pi$
$602$	0	0
$603$	−6.82843	−0.278075
$604$	4.00000	0.162758
$605$	−2.00000	−0.0813116
$606$	0.928932	0.0377353
$607$	12.6863	0.514921	0.257460	−	0.966289i	$-0.417114\pi$
$607$	12.6863	0.514921	0.257460	+	0.966289i	$0.417114\pi$
$608$	−6.24264	−0.253173
$609$	0	0
$610$	10.8284	0.438430
$611$	30.7696	1.24480
$612$	−2.00000	−0.0808452
$613$	−41.6569	−1.68250	−0.841252	−	0.540643i	$-0.818181\pi$
$613$	−41.6569	−1.68250	−0.841252	+	0.540643i	$0.818181\pi$
$614$	10.9289	0.441056
$615$	0.686292	0.0276739
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	2.24264	0.0902122
$619$	10.3431	0.415726	0.207863	−	0.978158i	$-0.433349\pi$
$619$	10.3431	0.415726	0.207863	+	0.978158i	$0.433349\pi$
$620$	−4.48528	−0.180133
$621$	0.828427	0.0332436
$622$	−27.8995	−1.11867
$623$	0	0
$624$	−2.58579	−0.103514
$625$	−19.0000	−0.760000
$626$	−0.242641	−0.00969787
$627$	6.24264	0.249307
$628$	−7.17157	−0.286177
$629$	9.65685	0.385044
$630$	0	0
$631$	7.85786	0.312817	0.156408	−	0.987692i	$-0.450008\pi$
$631$	7.85786	0.312817	0.156408	+	0.987692i	$0.450008\pi$
$632$	0.485281	0.0193035
$633$	−28.8284	−1.14583
$634$	7.17157	0.284820
$635$	28.2843	1.12243
$636$	6.48528	0.257158
$637$	0	0
$638$	1.65685	0.0655955
$639$	−5.65685	−0.223782
$640$	−2.00000	−0.0790569
$641$	−12.6863	−0.501078	−0.250539	−	0.968106i	$-0.580608\pi$
$641$	−12.6863	−0.501078	−0.250539	+	0.968106i	$0.580608\pi$
$642$	7.17157	0.283039
$643$	13.4558	0.530647	0.265323	−	0.964159i	$-0.414521\pi$
$643$	13.4558	0.530647	0.265323	+	0.964159i	$0.414521\pi$
$644$	0	0
$645$	−1.65685	−0.0652386
$646$	12.4853	0.491227
$647$	13.0711	0.513877	0.256938	−	0.966428i	$-0.417286\pi$
$647$	13.0711	0.513877	0.256938	+	0.966428i	$0.417286\pi$
$648$	1.00000	0.0392837
$649$	1.17157	0.0459883
$650$	2.58579	0.101423
$651$	0	0
$652$	−13.6569	−0.534844
$653$	−34.2843	−1.34165	−0.670824	−	0.741617i	$-0.734059\pi$
$653$	−34.2843	−1.34165	−0.670824	+	0.741617i	$0.734059\pi$
$654$	15.6569	0.612231
$655$	−27.1127	−1.05938
$656$	−0.343146	−0.0133976
$657$	−0.343146	−0.0133874
$658$	0	0
$659$	13.1127	0.510798	0.255399	−	0.966836i	$-0.417793\pi$
$659$	13.1127	0.510798	0.255399	+	0.966836i	$0.417793\pi$
$660$	2.00000	0.0778499
$661$	−27.9411	−1.08678	−0.543392	−	0.839479i	$-0.682860\pi$
$661$	−27.9411	−1.08678	−0.543392	+	0.839479i	$0.682860\pi$
$662$	11.7990	0.458581
$663$	5.17157	0.200847
$664$	9.07107	0.352026
$665$	0	0
$666$	−4.82843	−0.187098
$667$	−1.37258	−0.0531466
$668$	10.8284	0.418964
$669$	2.24264	0.0867055
$670$	13.6569	0.527610
$671$	5.41421	0.209013
$672$	0	0
$673$	−34.4853	−1.32931	−0.664655	−	0.747150i	$-0.731421\pi$
$673$	−34.4853	−1.32931	−0.664655	+	0.747150i	$0.731421\pi$
$674$	7.17157	0.276239
$675$	−1.00000	−0.0384900
$676$	−6.31371	−0.242835
$677$	−12.0416	−0.462797	−0.231399	−	0.972859i	$-0.574330\pi$
$677$	−12.0416	−0.462797	−0.231399	+	0.972859i	$0.574330\pi$
$678$	−5.65685	−0.217250
$679$	0	0
$680$	4.00000	0.153393
$681$	−7.89949	−0.302709
$682$	−2.24264	−0.0858752
$683$	−44.7696	−1.71306	−0.856530	−	0.516098i	$-0.827384\pi$
$683$	−44.7696	−1.71306	−0.856530	+	0.516098i	$0.827384\pi$
$684$	−6.24264	−0.238693
$685$	−34.6274	−1.32305
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0.828427	0.0315835
$689$	−16.7696	−0.638869
$690$	−1.65685	−0.0630754
$691$	−14.1421	−0.537992	−0.268996	−	0.963141i	$-0.586692\pi$
$691$	−14.1421	−0.537992	−0.268996	+	0.963141i	$0.586692\pi$
$692$	−13.8995	−0.528380
$693$	0	0
$694$	−31.4558	−1.19405
$695$	39.7990	1.50966
$696$	−1.65685	−0.0628029
$697$	0.686292	0.0259951
$698$	14.5858	0.552080
$699$	3.65685	0.138315
$700$	0	0
$701$	−13.3137	−0.502852	−0.251426	−	0.967877i	$-0.580899\pi$
$701$	−13.3137	−0.502852	−0.251426	+	0.967877i	$0.580899\pi$
$702$	−2.58579	−0.0975942
$703$	30.1421	1.13683
$704$	−1.00000	−0.0376889
$705$	23.7990	0.896322
$706$	21.2132	0.798369
$707$	0	0
$708$	−1.17157	−0.0440304
$709$	38.7696	1.45602	0.728011	−	0.685566i	$-0.240445\pi$
$709$	38.7696	1.45602	0.728011	+	0.685566i	$0.240445\pi$
$710$	11.3137	0.424596
$711$	0.485281	0.0181995
$712$	−12.2426	−0.458812
$713$	1.85786	0.0695776
$714$	0	0
$715$	−5.17157	−0.193406
$716$	−3.51472	−0.131351
$717$	23.3137	0.870666
$718$	10.8284	0.404113
$719$	−32.1838	−1.20025	−0.600126	−	0.799906i	$-0.704883\pi$
$719$	−32.1838	−1.20025	−0.600126	+	0.799906i	$0.704883\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	19.9706	0.743227
$723$	−3.65685	−0.136000
$724$	−13.3137	−0.494800
$725$	1.65685	0.0615340
$726$	1.00000	0.0371135
$727$	11.2132	0.415875	0.207937	−	0.978142i	$-0.433325\pi$
$727$	11.2132	0.415875	0.207937	+	0.978142i	$0.433325\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.686292	0.0254008
$731$	−1.65685	−0.0612810
$732$	−5.41421	−0.200115
$733$	11.5563	0.426843	0.213422	−	0.976960i	$-0.431539\pi$
$733$	11.5563	0.426843	0.213422	+	0.976960i	$0.431539\pi$
$734$	35.4142	1.30716
$735$	0	0
$736$	0.828427	0.0305362
$737$	6.82843	0.251528
$738$	−0.343146	−0.0126314
$739$	−42.6274	−1.56807	−0.784037	−	0.620714i	$-0.786843\pi$
$739$	−42.6274	−1.56807	−0.784037	+	0.620714i	$0.786843\pi$
$740$	9.65685	0.354993
$741$	16.1421	0.592996
$742$	0	0
$743$	26.6274	0.976865	0.488433	−	0.872602i	$-0.337569\pi$
$743$	26.6274	0.976865	0.488433	+	0.872602i	$0.337569\pi$
$744$	2.24264	0.0822192
$745$	−8.00000	−0.293097
$746$	−31.9411	−1.16945
$747$	9.07107	0.331893
$748$	2.00000	0.0731272
$749$	0	0
$750$	12.0000	0.438178
$751$	16.1421	0.589035	0.294517	−	0.955646i	$-0.404841\pi$
$751$	16.1421	0.589035	0.294517	+	0.955646i	$0.404841\pi$
$752$	−11.8995	−0.433930
$753$	8.48528	0.309221
$754$	4.28427	0.156024
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−19.8579	−0.721746	−0.360873	−	0.932615i	$-0.617521\pi$
$757$	−19.8579	−0.721746	−0.360873	+	0.932615i	$0.617521\pi$
$758$	−24.2843	−0.882044
$759$	−0.828427	−0.0300700
$760$	12.4853	0.452889
$761$	18.4853	0.670091	0.335045	−	0.942202i	$-0.391248\pi$
$761$	18.4853	0.670091	0.335045	+	0.942202i	$0.391248\pi$
$762$	−14.1421	−0.512316
$763$	0	0
$764$	−17.7990	−0.643945
$765$	4.00000	0.144620
$766$	−2.72792	−0.0985638
$767$	3.02944	0.109387
$768$	1.00000	0.0360844
$769$	44.1421	1.59181	0.795903	−	0.605424i	$-0.206996\pi$
$769$	44.1421	1.59181	0.795903	+	0.605424i	$0.206996\pi$
$770$	0	0
$771$	−13.2132	−0.475862
$772$	18.9706	0.682765
$773$	−22.4853	−0.808739	−0.404370	−	0.914596i	$-0.632509\pi$
$773$	−22.4853	−0.808739	−0.404370	+	0.914596i	$0.632509\pi$
$774$	0.828427	0.0297772
$775$	−2.24264	−0.0805580
$776$	−9.89949	−0.355371
$777$	0	0
$778$	22.2843	0.798930
$779$	2.14214	0.0767500
$780$	5.17157	0.185172
$781$	5.65685	0.202418
$782$	−1.65685	−0.0592490
$783$	−1.65685	−0.0592111
$784$	0	0
$785$	14.3431	0.511929
$786$	13.5563	0.483539
$787$	0.585786	0.0208810	0.0104405	−	0.999945i	$-0.496677\pi$
$787$	0.585786	0.0208810	0.0104405	+	0.999945i	$0.496677\pi$
$788$	4.34315	0.154718
$789$	14.8284	0.527906
$790$	−0.970563	−0.0345311
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	14.0000	0.497155
$794$	16.8284	0.597219
$795$	−12.9706	−0.460018
$796$	27.2132	0.964546
$797$	−34.4853	−1.22153	−0.610766	−	0.791811i	$-0.709138\pi$
$797$	−34.4853	−1.22153	−0.610766	+	0.791811i	$0.709138\pi$
$798$	0	0
$799$	23.7990	0.841948
$800$	−1.00000	−0.0353553
$801$	−12.2426	−0.432572
$802$	33.3137	1.17635
$803$	0.343146	0.0121094
$804$	−6.82843	−0.240820
$805$	0	0
$806$	−5.79899	−0.204261
$807$	17.3137	0.609471
$808$	0.928932	0.0326797
$809$	−46.7696	−1.64433	−0.822165	−	0.569249i	$-0.807234\pi$
$809$	−46.7696	−1.64433	−0.822165	+	0.569249i	$0.807234\pi$
$810$	−2.00000	−0.0702728
$811$	0.384776	0.0135113	0.00675566	−	0.999977i	$-0.497850\pi$
$811$	0.384776	0.0135113	0.00675566	+	0.999977i	$0.497850\pi$
$812$	0	0
$813$	6.14214	0.215414
$814$	4.82843	0.169236
$815$	27.3137	0.956757
$816$	−2.00000	−0.0700140
$817$	−5.17157	−0.180930
$818$	18.4853	0.646323
$819$	0	0
$820$	0.686292	0.0239663
$821$	24.6863	0.861558	0.430779	−	0.902458i	$-0.358239\pi$
$821$	24.6863	0.861558	0.430779	+	0.902458i	$0.358239\pi$
$822$	17.3137	0.603885
$823$	0.142136	0.00495454	0.00247727	−	0.999997i	$-0.499211\pi$
$823$	0.142136	0.00495454	0.00247727	+	0.999997i	$0.499211\pi$
$824$	2.24264	0.0781261
$825$	1.00000	0.0348155
$826$	0	0
$827$	16.6863	0.580239	0.290120	−	0.956990i	$-0.406305\pi$
$827$	16.6863	0.580239	0.290120	+	0.956990i	$0.406305\pi$
$828$	0.828427	0.0287898
$829$	6.28427	0.218262	0.109131	−	0.994027i	$-0.465193\pi$
$829$	6.28427	0.218262	0.109131	+	0.994027i	$0.465193\pi$
$830$	−18.1421	−0.629723
$831$	−16.0000	−0.555034
$832$	−2.58579	−0.0896460
$833$	0	0
$834$	−19.8995	−0.689063
$835$	−21.6569	−0.749466
$836$	6.24264	0.215906
$837$	2.24264	0.0775170
$838$	−16.0000	−0.552711
$839$	2.92893	0.101118	0.0505590	−	0.998721i	$-0.483900\pi$
$839$	2.92893	0.101118	0.0505590	+	0.998721i	$0.483900\pi$
$840$	0	0
$841$	−26.2548	−0.905339
$842$	−16.3431	−0.563222
$843$	21.3137	0.734083
$844$	−28.8284	−0.992315
$845$	12.6274	0.434396
$846$	−11.8995	−0.409113
$847$	0	0
$848$	6.48528	0.222705
$849$	−6.24264	−0.214247
$850$	2.00000	0.0685994
$851$	−4.00000	−0.137118
$852$	−5.65685	−0.193801
$853$	−4.44365	−0.152148	−0.0760739	−	0.997102i	$-0.524238\pi$
$853$	−4.44365	−0.152148	−0.0760739	+	0.997102i	$0.524238\pi$
$854$	0	0
$855$	12.4853	0.426988
$856$	7.17157	0.245119
$857$	28.1421	0.961317	0.480659	−	0.876908i	$-0.340398\pi$
$857$	28.1421	0.961317	0.480659	+	0.876908i	$0.340398\pi$
$858$	2.58579	0.0882773
$859$	−54.4264	−1.85701	−0.928503	−	0.371326i	$-0.878903\pi$
$859$	−54.4264	−1.85701	−0.928503	+	0.371326i	$0.878903\pi$
$860$	−1.65685	−0.0564983
$861$	0	0
$862$	−26.1421	−0.890405
$863$	34.7696	1.18357	0.591785	−	0.806096i	$-0.298424\pi$
$863$	34.7696	1.18357	0.591785	+	0.806096i	$0.298424\pi$
$864$	1.00000	0.0340207
$865$	27.7990	0.945194
$866$	−2.38478	−0.0810380
$867$	−13.0000	−0.441503
$868$	0	0
$869$	−0.485281	−0.0164620
$870$	3.31371	0.112345
$871$	17.6569	0.598280
$872$	15.6569	0.530208
$873$	−9.89949	−0.335047
$874$	−5.17157	−0.174931
$875$	0	0
$876$	−0.343146	−0.0115938
$877$	−29.3137	−0.989854	−0.494927	−	0.868935i	$-0.664805\pi$
$877$	−29.3137	−0.989854	−0.494927	+	0.868935i	$0.664805\pi$
$878$	34.8284	1.17540
$879$	19.0711	0.643251
$880$	2.00000	0.0674200
$881$	−18.1005	−0.609822	−0.304911	−	0.952381i	$-0.598627\pi$
$881$	−18.1005	−0.609822	−0.304911	+	0.952381i	$0.598627\pi$
$882$	0	0
$883$	34.6274	1.16531	0.582653	−	0.812721i	$-0.302015\pi$
$883$	34.6274	1.16531	0.582653	+	0.812721i	$0.302015\pi$
$884$	5.17157	0.173939
$885$	2.34315	0.0787640
$886$	−4.48528	−0.150686
$887$	52.2843	1.75553	0.877767	−	0.479088i	$-0.159032\pi$
$887$	52.2843	1.75553	0.877767	+	0.479088i	$0.159032\pi$
$888$	−4.82843	−0.162031
$889$	0	0
$890$	24.4853	0.820748
$891$	−1.00000	−0.0335013
$892$	2.24264	0.0750892
$893$	74.2843	2.48583
$894$	4.00000	0.133780
$895$	7.02944	0.234968
$896$	0	0
$897$	−2.14214	−0.0715238
$898$	−16.9706	−0.566315
$899$	−3.71573	−0.123926
$900$	−1.00000	−0.0333333
$901$	−12.9706	−0.432112
$902$	0.343146	0.0114255
$903$	0	0
$904$	−5.65685	−0.188144
$905$	26.6274	0.885125
$906$	4.00000	0.132891
$907$	−47.7990	−1.58714	−0.793570	−	0.608479i	$-0.791780\pi$
$907$	−47.7990	−1.58714	−0.793570	+	0.608479i	$0.791780\pi$
$908$	−7.89949	−0.262154
$909$	0.928932	0.0308107
$910$	0	0
$911$	39.4558	1.30723	0.653615	−	0.756827i	$-0.273251\pi$
$911$	39.4558	1.30723	0.653615	+	0.756827i	$0.273251\pi$
$912$	−6.24264	−0.206714
$913$	−9.07107	−0.300209
$914$	−6.68629	−0.221163
$915$	10.8284	0.357977
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	32.9706	1.08760	0.543799	−	0.839215i	$-0.316985\pi$
$919$	32.9706	1.08760	0.543799	+	0.839215i	$0.316985\pi$
$920$	−1.65685	−0.0546249
$921$	10.9289	0.360121
$922$	33.4142	1.10044
$923$	14.6274	0.481467
$924$	0	0
$925$	4.82843	0.158758
$926$	19.3137	0.634688
$927$	2.24264	0.0736580
$928$	−1.65685	−0.0543889
$929$	−53.2132	−1.74587	−0.872934	−	0.487838i	$-0.837786\pi$
$929$	−53.2132	−1.74587	−0.872934	+	0.487838i	$0.837786\pi$
$930$	−4.48528	−0.147078
$931$	0	0
$932$	3.65685	0.119784
$933$	−27.8995	−0.913388
$934$	−12.0000	−0.392652
$935$	−4.00000	−0.130814
$936$	−2.58579	−0.0845191
$937$	−44.9117	−1.46720	−0.733600	−	0.679581i	$-0.762162\pi$
$937$	−44.9117	−1.46720	−0.733600	+	0.679581i	$0.762162\pi$
$938$	0	0
$939$	−0.242641	−0.00791828
$940$	23.7990	0.776237
$941$	3.55635	0.115934	0.0579668	−	0.998319i	$-0.481538\pi$
$941$	3.55635	0.115934	0.0579668	+	0.998319i	$0.481538\pi$
$942$	−7.17157	−0.233662
$943$	−0.284271	−0.00925715
$944$	−1.17157	−0.0381314
$945$	0	0
$946$	−0.828427	−0.0269345
$947$	60.5685	1.96821	0.984107	−	0.177579i	$-0.0568265\pi$
$947$	60.5685	1.96821	0.984107	+	0.177579i	$0.0568265\pi$
$948$	0.485281	0.0157612
$949$	0.887302	0.0288030
$950$	6.24264	0.202538
$951$	7.17157	0.232554
$952$	0	0
$953$	26.4853	0.857942	0.428971	−	0.903318i	$-0.358876\pi$
$953$	26.4853	0.857942	0.428971	+	0.903318i	$0.358876\pi$
$954$	6.48528	0.209969
$955$	35.5980	1.15192
$956$	23.3137	0.754019
$957$	1.65685	0.0535585
$958$	15.3137	0.494763
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−25.9706	−0.837760
$962$	12.4853	0.402542
$963$	7.17157	0.231101
$964$	−3.65685	−0.117779
$965$	−37.9411	−1.22137
$966$	0	0
$967$	57.9411	1.86326	0.931630	−	0.363407i	$-0.118387\pi$
$967$	57.9411	1.86326	0.931630	+	0.363407i	$0.118387\pi$
$968$	1.00000	0.0321412
$969$	12.4853	0.401085
$970$	19.7990	0.635707
$971$	−27.3137	−0.876539	−0.438269	−	0.898844i	$-0.644408\pi$
$971$	−27.3137	−0.876539	−0.438269	+	0.898844i	$0.644408\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−28.1421	−0.901732
$975$	2.58579	0.0828114
$976$	−5.41421	−0.173305
$977$	−18.6274	−0.595944	−0.297972	−	0.954575i	$-0.596310\pi$
$977$	−18.6274	−0.595944	−0.297972	+	0.954575i	$0.596310\pi$
$978$	−13.6569	−0.436698
$979$	12.2426	0.391276
$980$	0	0
$981$	15.6569	0.499885
$982$	2.62742	0.0838442
$983$	52.8701	1.68629	0.843146	−	0.537684i	$-0.180701\pi$
$983$	52.8701	1.68629	0.843146	+	0.537684i	$0.180701\pi$
$984$	−0.343146	−0.0109391
$985$	−8.68629	−0.276768
$986$	3.31371	0.105530
$987$	0	0
$988$	16.1421	0.513550
$989$	0.686292	0.0218228
$990$	2.00000	0.0635642
$991$	−50.9117	−1.61726	−0.808632	−	0.588315i	$-0.799791\pi$
$991$	−50.9117	−1.61726	−0.808632	+	0.588315i	$0.799791\pi$
$992$	2.24264	0.0712039
$993$	11.7990	0.374430
$994$	0	0
$995$	−54.4264	−1.72543
$996$	9.07107	0.287428
$997$	−33.4142	−1.05824	−0.529119	−	0.848547i	$-0.677478\pi$
$997$	−33.4142	−1.05824	−0.529119	+	0.848547i	$0.677478\pi$
$998$	22.1421	0.700897
$999$	−4.82843	−0.152765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bd.1.2	yes	2
3.2	odd	2		9702.2.a.cy.1.2		2
7.6	odd	2		3234.2.a.bc.1.1	✓	2
21.20	even	2		9702.2.a.ci.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bc.1.1	✓	2	7.6	odd	2
3234.2.a.bd.1.2	yes	2	1.1	even	1	trivial
9702.2.a.ci.1.1		2	21.20	even	2
9702.2.a.cy.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.58579 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -6.24264 q^{19} -2.00000 q^{20} -1.00000 q^{22} +0.828427 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.58579 q^{26} +1.00000 q^{27} -1.65685 q^{29} -2.00000 q^{30} +2.24264 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -4.82843 q^{37} -6.24264 q^{38} -2.58579 q^{39} -2.00000 q^{40} -0.343146 q^{41} +0.828427 q^{43} -1.00000 q^{44} -2.00000 q^{45} +0.828427 q^{46} -11.8995 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -2.58579 q^{52} +6.48528 q^{53} +1.00000 q^{54} +2.00000 q^{55} -6.24264 q^{57} -1.65685 q^{58} -1.17157 q^{59} -2.00000 q^{60} -5.41421 q^{61} +2.24264 q^{62} +1.00000 q^{64} +5.17157 q^{65} -1.00000 q^{66} -6.82843 q^{67} -2.00000 q^{68} +0.828427 q^{69} -5.65685 q^{71} +1.00000 q^{72} -0.343146 q^{73} -4.82843 q^{74} -1.00000 q^{75} -6.24264 q^{76} -2.58579 q^{78} +0.485281 q^{79} -2.00000 q^{80} +1.00000 q^{81} -0.343146 q^{82} +9.07107 q^{83} +4.00000 q^{85} +0.828427 q^{86} -1.65685 q^{87} -1.00000 q^{88} -12.2426 q^{89} -2.00000 q^{90} +0.828427 q^{92} +2.24264 q^{93} -11.8995 q^{94} +12.4853 q^{95} +1.00000 q^{96} -9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 8 q^{13} - 4 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^10 - 2 * q^11 + 2 * q^12 - 8 * q^13 - 4 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 - 4 * q^19 - 4 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^24 - 2 * q^25 - 8 * q^26 + 2 * q^27 + 8 * q^29 - 4 * q^30 - 4 * q^31 + 2 * q^32 - 2 * q^33 - 4 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 8 * q^39 - 4 * q^40 - 12 * q^41 - 4 * q^43 - 2 * q^44 - 4 * q^45 - 4 * q^46 - 4 * q^47 + 2 * q^48 - 2 * q^50 - 4 * q^51 - 8 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^65 - 2 * q^66 - 8 * q^67 - 4 * q^68 - 4 * q^69 + 2 * q^72 - 12 * q^73 - 4 * q^74 - 2 * q^75 - 4 * q^76 - 8 * q^78 - 16 * q^79 - 4 * q^80 + 2 * q^81 - 12 * q^82 + 4 * q^83 + 8 * q^85 - 4 * q^86 + 8 * q^87 - 2 * q^88 - 16 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 4 * q^94 + 8 * q^95 + 2 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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