LMFDB - Embedded newform 3234.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$25.8236200137$
Analytic rank:	$0$
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)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} +1.41421 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +5.41421 q^{19} -1.00000 q^{22} +7.65685 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.41421 q^{26} -1.00000 q^{27} -5.65685 q^{29} -3.07107 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +3.65685 q^{37} -5.41421 q^{38} -1.41421 q^{39} +9.65685 q^{41} -10.0000 q^{43} +1.00000 q^{44} -7.65685 q^{46} -1.41421 q^{47} -1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} +1.41421 q^{52} -3.65685 q^{53} +1.00000 q^{54} -5.41421 q^{57} +5.65685 q^{58} +6.82843 q^{59} -1.41421 q^{61} +3.07107 q^{62} +1.00000 q^{64} +1.00000 q^{66} -11.3137 q^{67} +4.00000 q^{68} -7.65685 q^{69} +13.6569 q^{71} -1.00000 q^{72} +4.00000 q^{73} -3.65685 q^{74} +5.00000 q^{75} +5.41421 q^{76} +1.41421 q^{78} +1.65685 q^{79} +1.00000 q^{81} -9.65685 q^{82} -15.0711 q^{83} +10.0000 q^{86} +5.65685 q^{87} -1.00000 q^{88} +2.58579 q^{89} +7.65685 q^{92} +3.07107 q^{93} +1.41421 q^{94} +1.00000 q^{96} +12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 8 q^{19} - 2 q^{22} + 4 q^{23} + 2 q^{24} - 10 q^{25} - 2 q^{27} + 8 q^{31} - 2 q^{32}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 8 * q^19 - 2 * q^22 + 4 * q^23 + 2 * q^24 - 10 * q^25 - 2 * q^27 + 8 * q^31 - 2 * q^32 - 2 * q^33 - 8 * q^34 + 2 * q^36 - 4 * q^37 - 8 * q^38 + 8 * q^41 - 20 * q^43 + 2 * q^44 - 4 * q^46 - 2 * q^48 + 10 * q^50 - 8 * q^51 + 4 * q^53 + 2 * q^54 - 8 * q^57 + 8 * q^59 - 8 * q^62 + 2 * q^64 + 2 * q^66 + 8 * q^68 - 4 * q^69 + 16 * q^71 - 2 * q^72 + 8 * q^73 + 4 * q^74 + 10 * q^75 + 8 * q^76 - 8 * q^79 + 2 * q^81 - 8 * q^82 - 16 * q^83 + 20 * q^86 - 2 * q^88 + 8 * q^89 + 4 * q^92 - 8 * q^93 + 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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	5.41421	1.24211	0.621053	−	0.783769i	$-0.286705\pi$
$19$	5.41421	1.24211	0.621053	+	0.783769i	$0.286705\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	7.65685	1.59656	0.798282	−	0.602284i	$-0.205742\pi$
$23$	7.65685	1.59656	0.798282	+	0.602284i	$0.205742\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.41421	−0.277350
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.65685	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$30$	0	0
$31$	−3.07107	−0.551580	−0.275790	−	0.961218i	$-0.588939\pi$
$31$	−3.07107	−0.551580	−0.275790	+	0.961218i	$0.588939\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	3.65685	0.601183	0.300592	−	0.953753i	$-0.402816\pi$
$37$	3.65685	0.601183	0.300592	+	0.953753i	$0.402816\pi$
$38$	−5.41421	−0.878301
$39$	−1.41421	−0.226455
$40$	0	0
$41$	9.65685	1.50815	0.754074	−	0.656790i	$-0.228086\pi$
$41$	9.65685	1.50815	0.754074	+	0.656790i	$0.228086\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−7.65685	−1.12894
$47$	−1.41421	−0.206284	−0.103142	−	0.994667i	$-0.532890\pi$
$47$	−1.41421	−0.206284	−0.103142	+	0.994667i	$0.532890\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	5.00000	0.707107
$51$	−4.00000	−0.560112
$52$	1.41421	0.196116
$53$	−3.65685	−0.502308	−0.251154	−	0.967947i	$-0.580810\pi$
$53$	−3.65685	−0.502308	−0.251154	+	0.967947i	$0.580810\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−5.41421	−0.717130
$58$	5.65685	0.742781
$59$	6.82843	0.888985	0.444493	−	0.895782i	$-0.353384\pi$
$59$	6.82843	0.888985	0.444493	+	0.895782i	$0.353384\pi$
$60$	0	0
$61$	−1.41421	−0.181071	−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	−1.41421	−0.181071	−0.0905357	+	0.995893i	$0.528858\pi$
$62$	3.07107	0.390026
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	4.00000	0.485071
$69$	−7.65685	−0.921777
$70$	0	0
$71$	13.6569	1.62077	0.810385	−	0.585897i	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−3.65685	−0.425101
$75$	5.00000	0.577350
$76$	5.41421	0.621053
$77$	0	0
$78$	1.41421	0.160128
$79$	1.65685	0.186411	0.0932053	−	0.995647i	$-0.470289\pi$
$79$	1.65685	0.186411	0.0932053	+	0.995647i	$0.470289\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.65685	−1.06642
$83$	−15.0711	−1.65426	−0.827132	−	0.562007i	$-0.810029\pi$
$83$	−15.0711	−1.65426	−0.827132	+	0.562007i	$0.810029\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	5.65685	0.606478
$88$	−1.00000	−0.106600
$89$	2.58579	0.274093	0.137046	−	0.990565i	$-0.456239\pi$
$89$	2.58579	0.274093	0.137046	+	0.990565i	$0.456239\pi$
$90$	0	0
$91$	0	0
$92$	7.65685	0.798282
$93$	3.07107	0.318455
$94$	1.41421	0.145865
$95$	0	0
$96$	1.00000	0.102062
$97$	12.7279	1.29232	0.646162	−	0.763200i	$-0.276373\pi$
$97$	12.7279	1.29232	0.646162	+	0.763200i	$0.276373\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−5.00000	−0.500000
$101$	−0.242641	−0.0241437	−0.0120718	−	0.999927i	$-0.503843\pi$
$101$	−0.242641	−0.0241437	−0.0120718	+	0.999927i	$0.503843\pi$
$102$	4.00000	0.396059
$103$	13.8995	1.36956	0.684779	−	0.728751i	$-0.259899\pi$
$103$	13.8995	1.36956	0.684779	+	0.728751i	$0.259899\pi$
$104$	−1.41421	−0.138675
$105$	0	0
$106$	3.65685	0.355185
$107$	17.3137	1.67378	0.836890	−	0.547372i	$-0.184372\pi$
$107$	17.3137	1.67378	0.836890	+	0.547372i	$0.184372\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−3.65685	−0.347093
$112$	0	0
$113$	−5.65685	−0.532152	−0.266076	−	0.963952i	$-0.585727\pi$
$113$	−5.65685	−0.532152	−0.266076	+	0.963952i	$0.585727\pi$
$114$	5.41421	0.507088
$115$	0	0
$116$	−5.65685	−0.525226
$117$	1.41421	0.130744
$118$	−6.82843	−0.628608
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.41421	0.128037
$123$	−9.65685	−0.870729
$124$	−3.07107	−0.275790
$125$	0	0
$126$	0	0
$127$	−15.3137	−1.35887	−0.679436	−	0.733735i	$-0.737775\pi$
$127$	−15.3137	−1.35887	−0.679436	+	0.733735i	$0.737775\pi$
$128$	−1.00000	−0.0883883
$129$	10.0000	0.880451
$130$	0	0
$131$	3.75736	0.328282	0.164141	−	0.986437i	$-0.447515\pi$
$131$	3.75736	0.328282	0.164141	+	0.986437i	$0.447515\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	11.3137	0.977356
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−9.31371	−0.795724	−0.397862	−	0.917445i	$-0.630248\pi$
$137$	−9.31371	−0.795724	−0.397862	+	0.917445i	$0.630248\pi$
$138$	7.65685	0.651795
$139$	2.10051	0.178163	0.0890813	−	0.996024i	$-0.471607\pi$
$139$	2.10051	0.178163	0.0890813	+	0.996024i	$0.471607\pi$
$140$	0	0
$141$	1.41421	0.119098
$142$	−13.6569	−1.14606
$143$	1.41421	0.118262
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	3.65685	0.300592
$149$	12.9706	1.06259	0.531295	−	0.847187i	$-0.321706\pi$
$149$	12.9706	1.06259	0.531295	+	0.847187i	$0.321706\pi$
$150$	−5.00000	−0.408248
$151$	−1.65685	−0.134833	−0.0674164	−	0.997725i	$-0.521476\pi$
$151$	−1.65685	−0.134833	−0.0674164	+	0.997725i	$0.521476\pi$
$152$	−5.41421	−0.439151
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	−1.41421	−0.113228
$157$	−2.82843	−0.225733	−0.112867	−	0.993610i	$-0.536003\pi$
$157$	−2.82843	−0.225733	−0.112867	+	0.993610i	$0.536003\pi$
$158$	−1.65685	−0.131812
$159$	3.65685	0.290007
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	5.65685	0.443079	0.221540	−	0.975151i	$-0.428892\pi$
$163$	5.65685	0.443079	0.221540	+	0.975151i	$0.428892\pi$
$164$	9.65685	0.754074
$165$	0	0
$166$	15.0711	1.16974
$167$	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
$169$	−11.0000	−0.846154
$170$	0	0
$171$	5.41421	0.414035
$172$	−10.0000	−0.762493
$173$	4.92893	0.374740	0.187370	−	0.982289i	$-0.440004\pi$
$173$	4.92893	0.374740	0.187370	+	0.982289i	$0.440004\pi$
$174$	−5.65685	−0.428845
$175$	0	0
$176$	1.00000	0.0753778
$177$	−6.82843	−0.513256
$178$	−2.58579	−0.193813
$179$	2.34315	0.175135	0.0875675	−	0.996159i	$-0.472091\pi$
$179$	2.34315	0.175135	0.0875675	+	0.996159i	$0.472091\pi$
$180$	0	0
$181$	−21.6569	−1.60974	−0.804871	−	0.593450i	$-0.797765\pi$
$181$	−21.6569	−1.60974	−0.804871	+	0.593450i	$0.797765\pi$
$182$	0	0
$183$	1.41421	0.104542
$184$	−7.65685	−0.564471
$185$	0	0
$186$	−3.07107	−0.225182
$187$	4.00000	0.292509
$188$	−1.41421	−0.103142
$189$	0	0
$190$	0	0
$191$	−10.9706	−0.793802	−0.396901	−	0.917861i	$-0.629914\pi$
$191$	−10.9706	−0.793802	−0.396901	+	0.917861i	$0.629914\pi$
$192$	−1.00000	−0.0721688
$193$	7.65685	0.551152	0.275576	−	0.961279i	$-0.411131\pi$
$193$	7.65685	0.551152	0.275576	+	0.961279i	$0.411131\pi$
$194$	−12.7279	−0.913812
$195$	0	0
$196$	0	0
$197$	21.3137	1.51854	0.759269	−	0.650776i	$-0.225556\pi$
$197$	21.3137	1.51854	0.759269	+	0.650776i	$0.225556\pi$
$198$	−1.00000	−0.0710669
$199$	10.5858	0.750407	0.375203	−	0.926943i	$-0.377573\pi$
$199$	10.5858	0.750407	0.375203	+	0.926943i	$0.377573\pi$
$200$	5.00000	0.353553
$201$	11.3137	0.798007
$202$	0.242641	0.0170721
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−13.8995	−0.968424
$207$	7.65685	0.532188
$208$	1.41421	0.0980581
$209$	5.41421	0.374509
$210$	0	0
$211$	17.3137	1.19192	0.595962	−	0.803012i	$-0.296771\pi$
$211$	17.3137	1.19192	0.595962	+	0.803012i	$0.296771\pi$
$212$	−3.65685	−0.251154
$213$	−13.6569	−0.935752
$214$	−17.3137	−1.18354
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	−4.00000	−0.270295
$220$	0	0
$221$	5.65685	0.380521
$222$	3.65685	0.245432
$223$	19.5563	1.30959	0.654795	−	0.755807i	$-0.272755\pi$
$223$	19.5563	1.30959	0.654795	+	0.755807i	$0.272755\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	5.65685	0.376288
$227$	23.5563	1.56349	0.781745	−	0.623598i	$-0.214330\pi$
$227$	23.5563	1.56349	0.781745	+	0.623598i	$0.214330\pi$
$228$	−5.41421	−0.358565
$229$	5.65685	0.373815	0.186908	−	0.982377i	$-0.440153\pi$
$229$	5.65685	0.373815	0.186908	+	0.982377i	$0.440153\pi$
$230$	0	0
$231$	0	0
$232$	5.65685	0.371391
$233$	12.3431	0.808626	0.404313	−	0.914621i	$-0.367511\pi$
$233$	12.3431	0.808626	0.404313	+	0.914621i	$0.367511\pi$
$234$	−1.41421	−0.0924500
$235$	0	0
$236$	6.82843	0.444493
$237$	−1.65685	−0.107624
$238$	0	0
$239$	14.3431	0.927781	0.463890	−	0.885893i	$-0.346453\pi$
$239$	14.3431	0.927781	0.463890	+	0.885893i	$0.346453\pi$
$240$	0	0
$241$	−7.31371	−0.471117	−0.235559	−	0.971860i	$-0.575692\pi$
$241$	−7.31371	−0.471117	−0.235559	+	0.971860i	$0.575692\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−1.41421	−0.0905357
$245$	0	0
$246$	9.65685	0.615699
$247$	7.65685	0.487194
$248$	3.07107	0.195013
$249$	15.0711	0.955090
$250$	0	0
$251$	14.8284	0.935962	0.467981	−	0.883739i	$-0.344982\pi$
$251$	14.8284	0.935962	0.467981	+	0.883739i	$0.344982\pi$
$252$	0	0
$253$	7.65685	0.481382
$254$	15.3137	0.960868
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.7279	−1.04346	−0.521730	−	0.853111i	$-0.674713\pi$
$257$	−16.7279	−1.04346	−0.521730	+	0.853111i	$0.674713\pi$
$258$	−10.0000	−0.622573
$259$	0	0
$260$	0	0
$261$	−5.65685	−0.350150
$262$	−3.75736	−0.232130
$263$	13.6569	0.842118	0.421059	−	0.907033i	$-0.361659\pi$
$263$	13.6569	0.842118	0.421059	+	0.907033i	$0.361659\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	0	0
$267$	−2.58579	−0.158248
$268$	−11.3137	−0.691095
$269$	22.6274	1.37962	0.689809	−	0.723991i	$-0.257694\pi$
$269$	22.6274	1.37962	0.689809	+	0.723991i	$0.257694\pi$
$270$	0	0
$271$	21.1716	1.28608	0.643041	−	0.765832i	$-0.277673\pi$
$271$	21.1716	1.28608	0.643041	+	0.765832i	$0.277673\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	9.31371	0.562662
$275$	−5.00000	−0.301511
$276$	−7.65685	−0.460888
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	−2.10051	−0.125980
$279$	−3.07107	−0.183860
$280$	0	0
$281$	−18.9706	−1.13169	−0.565844	−	0.824512i	$-0.691450\pi$
$281$	−18.9706	−1.13169	−0.565844	+	0.824512i	$0.691450\pi$
$282$	−1.41421	−0.0842152
$283$	15.7574	0.936678	0.468339	−	0.883549i	$-0.344853\pi$
$283$	15.7574	0.936678	0.468339	+	0.883549i	$0.344853\pi$
$284$	13.6569	0.810385
$285$	0	0
$286$	−1.41421	−0.0836242
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−12.7279	−0.746124
$292$	4.00000	0.234082
$293$	−3.07107	−0.179414	−0.0897068	−	0.995968i	$-0.528593\pi$
$293$	−3.07107	−0.179414	−0.0897068	+	0.995968i	$0.528593\pi$
$294$	0	0
$295$	0	0
$296$	−3.65685	−0.212550
$297$	−1.00000	−0.0580259
$298$	−12.9706	−0.751365
$299$	10.8284	0.626224
$300$	5.00000	0.288675
$301$	0	0
$302$	1.65685	0.0953412
$303$	0.242641	0.0139393
$304$	5.41421	0.310526
$305$	0	0
$306$	−4.00000	−0.228665
$307$	−0.727922	−0.0415447	−0.0207724	−	0.999784i	$-0.506613\pi$
$307$	−0.727922	−0.0415447	−0.0207724	+	0.999784i	$0.506613\pi$
$308$	0	0
$309$	−13.8995	−0.790715
$310$	0	0
$311$	20.2426	1.14785	0.573927	−	0.818906i	$-0.305419\pi$
$311$	20.2426	1.14785	0.573927	+	0.818906i	$0.305419\pi$
$312$	1.41421	0.0800641
$313$	−12.2426	−0.691995	−0.345997	−	0.938235i	$-0.612459\pi$
$313$	−12.2426	−0.691995	−0.345997	+	0.938235i	$0.612459\pi$
$314$	2.82843	0.159617
$315$	0	0
$316$	1.65685	0.0932053
$317$	−5.31371	−0.298448	−0.149224	−	0.988803i	$-0.547677\pi$
$317$	−5.31371	−0.298448	−0.149224	+	0.988803i	$0.547677\pi$
$318$	−3.65685	−0.205066
$319$	−5.65685	−0.316723
$320$	0	0
$321$	−17.3137	−0.966357
$322$	0	0
$323$	21.6569	1.20502
$324$	1.00000	0.0555556
$325$	−7.07107	−0.392232
$326$	−5.65685	−0.313304
$327$	10.0000	0.553001
$328$	−9.65685	−0.533211
$329$	0	0
$330$	0	0
$331$	28.9706	1.59237	0.796183	−	0.605056i	$-0.206849\pi$
$331$	28.9706	1.59237	0.796183	+	0.605056i	$0.206849\pi$
$332$	−15.0711	−0.827132
$333$	3.65685	0.200394
$334$	−10.8284	−0.592505
$335$	0	0
$336$	0	0
$337$	−17.3137	−0.943138	−0.471569	−	0.881829i	$-0.656312\pi$
$337$	−17.3137	−0.943138	−0.471569	+	0.881829i	$0.656312\pi$
$338$	11.0000	0.598321
$339$	5.65685	0.307238
$340$	0	0
$341$	−3.07107	−0.166308
$342$	−5.41421	−0.292767
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	−4.92893	−0.264981
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	5.65685	0.303239
$349$	22.5858	1.20899	0.604495	−	0.796609i	$-0.293375\pi$
$349$	22.5858	1.20899	0.604495	+	0.796609i	$0.293375\pi$
$350$	0	0
$351$	−1.41421	−0.0754851
$352$	−1.00000	−0.0533002
$353$	−13.8995	−0.739795	−0.369898	−	0.929072i	$-0.620607\pi$
$353$	−13.8995	−0.739795	−0.369898	+	0.929072i	$0.620607\pi$
$354$	6.82843	0.362927
$355$	0	0
$356$	2.58579	0.137046
$357$	0	0
$358$	−2.34315	−0.123839
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	10.3137	0.542827
$362$	21.6569	1.13826
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	−1.41421	−0.0739221
$367$	−30.8701	−1.61140	−0.805702	−	0.592321i	$-0.798212\pi$
$367$	−30.8701	−1.61140	−0.805702	+	0.592321i	$0.798212\pi$
$368$	7.65685	0.399141
$369$	9.65685	0.502716
$370$	0	0
$371$	0	0
$372$	3.07107	0.159227
$373$	−36.6274	−1.89650	−0.948248	−	0.317531i	$-0.897146\pi$
$373$	−36.6274	−1.89650	−0.948248	+	0.317531i	$0.897146\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	1.41421	0.0729325
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−17.6569	−0.906972	−0.453486	−	0.891263i	$-0.649820\pi$
$379$	−17.6569	−0.906972	−0.453486	+	0.891263i	$0.649820\pi$
$380$	0	0
$381$	15.3137	0.784545
$382$	10.9706	0.561303
$383$	−7.55635	−0.386111	−0.193056	−	0.981188i	$-0.561840\pi$
$383$	−7.55635	−0.386111	−0.193056	+	0.981188i	$0.561840\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−7.65685	−0.389724
$387$	−10.0000	−0.508329
$388$	12.7279	0.646162
$389$	12.6274	0.640235	0.320118	−	0.947378i	$-0.396278\pi$
$389$	12.6274	0.640235	0.320118	+	0.947378i	$0.396278\pi$
$390$	0	0
$391$	30.6274	1.54890
$392$	0	0
$393$	−3.75736	−0.189534
$394$	−21.3137	−1.07377
$395$	0	0
$396$	1.00000	0.0502519
$397$	11.7990	0.592174	0.296087	−	0.955161i	$-0.404318\pi$
$397$	11.7990	0.592174	0.296087	+	0.955161i	$0.404318\pi$
$398$	−10.5858	−0.530618
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−11.3137	−0.564276
$403$	−4.34315	−0.216347
$404$	−0.242641	−0.0120718
$405$	0	0
$406$	0	0
$407$	3.65685	0.181264
$408$	4.00000	0.198030
$409$	−12.4853	−0.617357	−0.308679	−	0.951166i	$-0.599887\pi$
$409$	−12.4853	−0.617357	−0.308679	+	0.951166i	$0.599887\pi$
$410$	0	0
$411$	9.31371	0.459411
$412$	13.8995	0.684779
$413$	0	0
$414$	−7.65685	−0.376314
$415$	0	0
$416$	−1.41421	−0.0693375
$417$	−2.10051	−0.102862
$418$	−5.41421	−0.264818
$419$	16.2843	0.795539	0.397769	−	0.917485i	$-0.369784\pi$
$419$	16.2843	0.795539	0.397769	+	0.917485i	$0.369784\pi$
$420$	0	0
$421$	−14.9706	−0.729621	−0.364810	−	0.931082i	$-0.618866\pi$
$421$	−14.9706	−0.729621	−0.364810	+	0.931082i	$0.618866\pi$
$422$	−17.3137	−0.842818
$423$	−1.41421	−0.0687614
$424$	3.65685	0.177593
$425$	−20.0000	−0.970143
$426$	13.6569	0.661677
$427$	0	0
$428$	17.3137	0.836890
$429$	−1.41421	−0.0682789
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	37.2132	1.78835	0.894176	−	0.447715i	$-0.147762\pi$
$433$	37.2132	1.78835	0.894176	+	0.447715i	$0.147762\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	41.4558	1.98310
$438$	4.00000	0.191127
$439$	35.7990	1.70859	0.854296	−	0.519786i	$-0.173988\pi$
$439$	35.7990	1.70859	0.854296	+	0.519786i	$0.173988\pi$
$440$	0	0
$441$	0	0
$442$	−5.65685	−0.269069
$443$	28.2843	1.34383	0.671913	−	0.740630i	$-0.265473\pi$
$443$	28.2843	1.34383	0.671913	+	0.740630i	$0.265473\pi$
$444$	−3.65685	−0.173547
$445$	0	0
$446$	−19.5563	−0.926020
$447$	−12.9706	−0.613487
$448$	0	0
$449$	10.3431	0.488123	0.244062	−	0.969760i	$-0.421520\pi$
$449$	10.3431	0.488123	0.244062	+	0.969760i	$0.421520\pi$
$450$	5.00000	0.235702
$451$	9.65685	0.454724
$452$	−5.65685	−0.266076
$453$	1.65685	0.0778458
$454$	−23.5563	−1.10555
$455$	0	0
$456$	5.41421	0.253544
$457$	19.6569	0.919509	0.459754	−	0.888046i	$-0.347937\pi$
$457$	19.6569	0.919509	0.459754	+	0.888046i	$0.347937\pi$
$458$	−5.65685	−0.264327
$459$	−4.00000	−0.186704
$460$	0	0
$461$	37.8995	1.76516	0.882578	−	0.470167i	$-0.155806\pi$
$461$	37.8995	1.76516	0.882578	+	0.470167i	$0.155806\pi$
$462$	0	0
$463$	−27.3137	−1.26938	−0.634688	−	0.772769i	$-0.718871\pi$
$463$	−27.3137	−1.26938	−0.634688	+	0.772769i	$0.718871\pi$
$464$	−5.65685	−0.262613
$465$	0	0
$466$	−12.3431	−0.571785
$467$	−37.9411	−1.75571	−0.877853	−	0.478930i	$-0.841025\pi$
$467$	−37.9411	−1.75571	−0.877853	+	0.478930i	$0.841025\pi$
$468$	1.41421	0.0653720
$469$	0	0
$470$	0	0
$471$	2.82843	0.130327
$472$	−6.82843	−0.314304
$473$	−10.0000	−0.459800
$474$	1.65685	0.0761018
$475$	−27.0711	−1.24211
$476$	0	0
$477$	−3.65685	−0.167436
$478$	−14.3431	−0.656040
$479$	32.9706	1.50646	0.753232	−	0.657755i	$-0.228494\pi$
$479$	32.9706	1.50646	0.753232	+	0.657755i	$0.228494\pi$
$480$	0	0
$481$	5.17157	0.235803
$482$	7.31371	0.333130
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−0.343146	−0.0155494	−0.00777471	−	0.999970i	$-0.502475\pi$
$487$	−0.343146	−0.0155494	−0.00777471	+	0.999970i	$0.502475\pi$
$488$	1.41421	0.0640184
$489$	−5.65685	−0.255812
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−9.65685	−0.435365
$493$	−22.6274	−1.01909
$494$	−7.65685	−0.344498
$495$	0	0
$496$	−3.07107	−0.137895
$497$	0	0
$498$	−15.0711	−0.675351
$499$	−10.6274	−0.475749	−0.237874	−	0.971296i	$-0.576451\pi$
$499$	−10.6274	−0.475749	−0.237874	+	0.971296i	$0.576451\pi$
$500$	0	0
$501$	−10.8284	−0.483778
$502$	−14.8284	−0.661825
$503$	−0.485281	−0.0216376	−0.0108188	−	0.999941i	$-0.503444\pi$
$503$	−0.485281	−0.0216376	−0.0108188	+	0.999941i	$0.503444\pi$
$504$	0	0
$505$	0	0
$506$	−7.65685	−0.340389
$507$	11.0000	0.488527
$508$	−15.3137	−0.679436
$509$	−41.9411	−1.85901	−0.929504	−	0.368812i	$-0.879764\pi$
$509$	−41.9411	−1.85901	−0.929504	+	0.368812i	$0.879764\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−5.41421	−0.239043
$514$	16.7279	0.737837
$515$	0	0
$516$	10.0000	0.440225
$517$	−1.41421	−0.0621970
$518$	0	0
$519$	−4.92893	−0.216356
$520$	0	0
$521$	−7.27208	−0.318596	−0.159298	−	0.987231i	$-0.550923\pi$
$521$	−7.27208	−0.318596	−0.159298	+	0.987231i	$0.550923\pi$
$522$	5.65685	0.247594
$523$	−31.7574	−1.38865	−0.694326	−	0.719660i	$-0.744297\pi$
$523$	−31.7574	−1.38865	−0.694326	+	0.719660i	$0.744297\pi$
$524$	3.75736	0.164141
$525$	0	0
$526$	−13.6569	−0.595467
$527$	−12.2843	−0.535111
$528$	−1.00000	−0.0435194
$529$	35.6274	1.54902
$530$	0	0
$531$	6.82843	0.296328
$532$	0	0
$533$	13.6569	0.591544
$534$	2.58579	0.111898
$535$	0	0
$536$	11.3137	0.488678
$537$	−2.34315	−0.101114
$538$	−22.6274	−0.975537
$539$	0	0
$540$	0	0
$541$	−8.62742	−0.370922	−0.185461	−	0.982652i	$-0.559378\pi$
$541$	−8.62742	−0.370922	−0.185461	+	0.982652i	$0.559378\pi$
$542$	−21.1716	−0.909397
$543$	21.6569	0.929385
$544$	−4.00000	−0.171499
$545$	0	0
$546$	0	0
$547$	−1.65685	−0.0708420	−0.0354210	−	0.999372i	$-0.511277\pi$
$547$	−1.65685	−0.0708420	−0.0354210	+	0.999372i	$0.511277\pi$
$548$	−9.31371	−0.397862
$549$	−1.41421	−0.0603572
$550$	5.00000	0.213201
$551$	−30.6274	−1.30477
$552$	7.65685	0.325897
$553$	0	0
$554$	−4.00000	−0.169944
$555$	0	0
$556$	2.10051	0.0890813
$557$	31.9411	1.35339	0.676694	−	0.736264i	$-0.263412\pi$
$557$	31.9411	1.35339	0.676694	+	0.736264i	$0.263412\pi$
$558$	3.07107	0.130009
$559$	−14.1421	−0.598149
$560$	0	0
$561$	−4.00000	−0.168880
$562$	18.9706	0.800225
$563$	−22.5858	−0.951877	−0.475939	−	0.879478i	$-0.657892\pi$
$563$	−22.5858	−0.951877	−0.475939	+	0.879478i	$0.657892\pi$
$564$	1.41421	0.0595491
$565$	0	0
$566$	−15.7574	−0.662331
$567$	0	0
$568$	−13.6569	−0.573029
$569$	−6.68629	−0.280304	−0.140152	−	0.990130i	$-0.544759\pi$
$569$	−6.68629	−0.280304	−0.140152	+	0.990130i	$0.544759\pi$
$570$	0	0
$571$	40.2843	1.68584	0.842922	−	0.538036i	$-0.180833\pi$
$571$	40.2843	1.68584	0.842922	+	0.538036i	$0.180833\pi$
$572$	1.41421	0.0591312
$573$	10.9706	0.458302
$574$	0	0
$575$	−38.2843	−1.59656
$576$	1.00000	0.0416667
$577$	−27.7574	−1.15555	−0.577777	−	0.816195i	$-0.696080\pi$
$577$	−27.7574	−1.15555	−0.577777	+	0.816195i	$0.696080\pi$
$578$	1.00000	0.0415945
$579$	−7.65685	−0.318208
$580$	0	0
$581$	0	0
$582$	12.7279	0.527589
$583$	−3.65685	−0.151451
$584$	−4.00000	−0.165521
$585$	0	0
$586$	3.07107	0.126865
$587$	3.51472	0.145068	0.0725340	−	0.997366i	$-0.476891\pi$
$587$	3.51472	0.145068	0.0725340	+	0.997366i	$0.476891\pi$
$588$	0	0
$589$	−16.6274	−0.685121
$590$	0	0
$591$	−21.3137	−0.876729
$592$	3.65685	0.150296
$593$	−23.3137	−0.957379	−0.478690	−	0.877984i	$-0.658888\pi$
$593$	−23.3137	−0.957379	−0.478690	+	0.877984i	$0.658888\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	12.9706	0.531295
$597$	−10.5858	−0.433247
$598$	−10.8284	−0.442807
$599$	−32.9706	−1.34714	−0.673570	−	0.739123i	$-0.735240\pi$
$599$	−32.9706	−1.34714	−0.673570	+	0.739123i	$0.735240\pi$
$600$	−5.00000	−0.204124
$601$	13.4558	0.548875	0.274438	−	0.961605i	$-0.411508\pi$
$601$	13.4558	0.548875	0.274438	+	0.961605i	$0.411508\pi$
$602$	0	0
$603$	−11.3137	−0.460730
$604$	−1.65685	−0.0674164
$605$	0	0
$606$	−0.242641	−0.00985660
$607$	3.31371	0.134499	0.0672496	−	0.997736i	$-0.478578\pi$
$607$	3.31371	0.134499	0.0672496	+	0.997736i	$0.478578\pi$
$608$	−5.41421	−0.219575
$609$	0	0
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	4.00000	0.161690
$613$	12.0000	0.484675	0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000	0.484675	0.242338	+	0.970192i	$0.422086\pi$
$614$	0.727922	0.0293765
$615$	0	0
$616$	0	0
$617$	14.3431	0.577433	0.288717	−	0.957415i	$-0.406771\pi$
$617$	14.3431	0.577433	0.288717	+	0.957415i	$0.406771\pi$
$618$	13.8995	0.559120
$619$	−7.31371	−0.293963	−0.146981	−	0.989139i	$-0.546956\pi$
$619$	−7.31371	−0.293963	−0.146981	+	0.989139i	$0.546956\pi$
$620$	0	0
$621$	−7.65685	−0.307259
$622$	−20.2426	−0.811656
$623$	0	0
$624$	−1.41421	−0.0566139
$625$	25.0000	1.00000
$626$	12.2426	0.489314
$627$	−5.41421	−0.216223
$628$	−2.82843	−0.112867
$629$	14.6274	0.583233
$630$	0	0
$631$	37.5980	1.49675	0.748376	−	0.663275i	$-0.230834\pi$
$631$	37.5980	1.49675	0.748376	+	0.663275i	$0.230834\pi$
$632$	−1.65685	−0.0659061
$633$	−17.3137	−0.688158
$634$	5.31371	0.211034
$635$	0	0
$636$	3.65685	0.145004
$637$	0	0
$638$	5.65685	0.223957
$639$	13.6569	0.540257
$640$	0	0
$641$	8.28427	0.327209	0.163605	−	0.986526i	$-0.447688\pi$
$641$	8.28427	0.327209	0.163605	+	0.986526i	$0.447688\pi$
$642$	17.3137	0.683318
$643$	−46.8284	−1.84673	−0.923366	−	0.383920i	$-0.874574\pi$
$643$	−46.8284	−1.84673	−0.923366	+	0.383920i	$0.874574\pi$
$644$	0	0
$645$	0	0
$646$	−21.6569	−0.852078
$647$	17.8995	0.703702	0.351851	−	0.936056i	$-0.385552\pi$
$647$	17.8995	0.703702	0.351851	+	0.936056i	$0.385552\pi$
$648$	−1.00000	−0.0392837
$649$	6.82843	0.268039
$650$	7.07107	0.277350
$651$	0	0
$652$	5.65685	0.221540
$653$	−32.6274	−1.27681	−0.638405	−	0.769701i	$-0.720405\pi$
$653$	−32.6274	−1.27681	−0.638405	+	0.769701i	$0.720405\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	9.65685	0.377037
$657$	4.00000	0.156055
$658$	0	0
$659$	−25.3137	−0.986082	−0.493041	−	0.870006i	$-0.664115\pi$
$659$	−25.3137	−0.986082	−0.493041	+	0.870006i	$0.664115\pi$
$660$	0	0
$661$	−24.9706	−0.971242	−0.485621	−	0.874169i	$-0.661407\pi$
$661$	−24.9706	−0.971242	−0.485621	+	0.874169i	$0.661407\pi$
$662$	−28.9706	−1.12597
$663$	−5.65685	−0.219694
$664$	15.0711	0.584871
$665$	0	0
$666$	−3.65685	−0.141700
$667$	−43.3137	−1.67711
$668$	10.8284	0.418964
$669$	−19.5563	−0.756092
$670$	0	0
$671$	−1.41421	−0.0545951
$672$	0	0
$673$	19.6569	0.757716	0.378858	−	0.925455i	$-0.376317\pi$
$673$	19.6569	0.757716	0.378858	+	0.925455i	$0.376317\pi$
$674$	17.3137	0.666899
$675$	5.00000	0.192450
$676$	−11.0000	−0.423077
$677$	−18.5858	−0.714310	−0.357155	−	0.934045i	$-0.616253\pi$
$677$	−18.5858	−0.714310	−0.357155	+	0.934045i	$0.616253\pi$
$678$	−5.65685	−0.217250
$679$	0	0
$680$	0	0
$681$	−23.5563	−0.902681
$682$	3.07107	0.117597
$683$	−48.2843	−1.84755	−0.923773	−	0.382940i	$-0.874912\pi$
$683$	−48.2843	−1.84755	−0.923773	+	0.382940i	$0.874912\pi$
$684$	5.41421	0.207018
$685$	0	0
$686$	0	0
$687$	−5.65685	−0.215822
$688$	−10.0000	−0.381246
$689$	−5.17157	−0.197021
$690$	0	0
$691$	4.48528	0.170628	0.0853141	−	0.996354i	$-0.472811\pi$
$691$	4.48528	0.170628	0.0853141	+	0.996354i	$0.472811\pi$
$692$	4.92893	0.187370
$693$	0	0
$694$	2.00000	0.0759190
$695$	0	0
$696$	−5.65685	−0.214423
$697$	38.6274	1.46312
$698$	−22.5858	−0.854885
$699$	−12.3431	−0.466861
$700$	0	0
$701$	−41.3137	−1.56040	−0.780199	−	0.625532i	$-0.784882\pi$
$701$	−41.3137	−1.56040	−0.780199	+	0.625532i	$0.784882\pi$
$702$	1.41421	0.0533761
$703$	19.7990	0.746733
$704$	1.00000	0.0376889
$705$	0	0
$706$	13.8995	0.523114
$707$	0	0
$708$	−6.82843	−0.256628
$709$	0.627417	0.0235631	0.0117816	−	0.999931i	$-0.496250\pi$
$709$	0.627417	0.0235631	0.0117816	+	0.999931i	$0.496250\pi$
$710$	0	0
$711$	1.65685	0.0621369
$712$	−2.58579	−0.0969064
$713$	−23.5147	−0.880633
$714$	0	0
$715$	0	0
$716$	2.34315	0.0875675
$717$	−14.3431	−0.535655
$718$	4.00000	0.149279
$719$	14.5858	0.543958	0.271979	−	0.962303i	$-0.412322\pi$
$719$	14.5858	0.543958	0.271979	+	0.962303i	$0.412322\pi$
$720$	0	0
$721$	0	0
$722$	−10.3137	−0.383836
$723$	7.31371	0.272000
$724$	−21.6569	−0.804871
$725$	28.2843	1.05045
$726$	1.00000	0.0371135
$727$	−39.3553	−1.45961	−0.729804	−	0.683656i	$-0.760389\pi$
$727$	−39.3553	−1.45961	−0.729804	+	0.683656i	$0.760389\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	1.41421	0.0522708
$733$	−35.7574	−1.32073	−0.660364	−	0.750946i	$-0.729598\pi$
$733$	−35.7574	−1.32073	−0.660364	+	0.750946i	$0.729598\pi$
$734$	30.8701	1.13943
$735$	0	0
$736$	−7.65685	−0.282235
$737$	−11.3137	−0.416746
$738$	−9.65685	−0.355474
$739$	6.34315	0.233336	0.116668	−	0.993171i	$-0.462779\pi$
$739$	6.34315	0.233336	0.116668	+	0.993171i	$0.462779\pi$
$740$	0	0
$741$	−7.65685	−0.281282
$742$	0	0
$743$	−4.97056	−0.182352	−0.0911761	−	0.995835i	$-0.529063\pi$
$743$	−4.97056	−0.182352	−0.0911761	+	0.995835i	$0.529063\pi$
$744$	−3.07107	−0.112591
$745$	0	0
$746$	36.6274	1.34103
$747$	−15.0711	−0.551422
$748$	4.00000	0.146254
$749$	0	0
$750$	0	0
$751$	26.9706	0.984170	0.492085	−	0.870547i	$-0.336235\pi$
$751$	26.9706	0.984170	0.492085	+	0.870547i	$0.336235\pi$
$752$	−1.41421	−0.0515711
$753$	−14.8284	−0.540378
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	−6.68629	−0.243017	−0.121509	−	0.992590i	$-0.538773\pi$
$757$	−6.68629	−0.243017	−0.121509	+	0.992590i	$0.538773\pi$
$758$	17.6569	0.641326
$759$	−7.65685	−0.277926
$760$	0	0
$761$	−13.8579	−0.502347	−0.251174	−	0.967942i	$-0.580817\pi$
$761$	−13.8579	−0.502347	−0.251174	+	0.967942i	$0.580817\pi$
$762$	−15.3137	−0.554757
$763$	0	0
$764$	−10.9706	−0.396901
$765$	0	0
$766$	7.55635	0.273022
$767$	9.65685	0.348689
$768$	−1.00000	−0.0360844
$769$	39.7990	1.43519	0.717594	−	0.696462i	$-0.245243\pi$
$769$	39.7990	1.43519	0.717594	+	0.696462i	$0.245243\pi$
$770$	0	0
$771$	16.7279	0.602441
$772$	7.65685	0.275576
$773$	−14.1421	−0.508657	−0.254329	−	0.967118i	$-0.581854\pi$
$773$	−14.1421	−0.508657	−0.254329	+	0.967118i	$0.581854\pi$
$774$	10.0000	0.359443
$775$	15.3553	0.551580
$776$	−12.7279	−0.456906
$777$	0	0
$778$	−12.6274	−0.452715
$779$	52.2843	1.87328
$780$	0	0
$781$	13.6569	0.488681
$782$	−30.6274	−1.09523
$783$	5.65685	0.202159
$784$	0	0
$785$	0	0
$786$	3.75736	0.134021
$787$	−38.3848	−1.36827	−0.684135	−	0.729356i	$-0.739820\pi$
$787$	−38.3848	−1.36827	−0.684135	+	0.729356i	$0.739820\pi$
$788$	21.3137	0.759269
$789$	−13.6569	−0.486197
$790$	0	0
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−2.00000	−0.0710221
$794$	−11.7990	−0.418730
$795$	0	0
$796$	10.5858	0.375203
$797$	−50.8284	−1.80044	−0.900218	−	0.435440i	$-0.856593\pi$
$797$	−50.8284	−1.80044	−0.900218	+	0.435440i	$0.856593\pi$
$798$	0	0
$799$	−5.65685	−0.200125
$800$	5.00000	0.176777
$801$	2.58579	0.0913643
$802$	6.00000	0.211867
$803$	4.00000	0.141157
$804$	11.3137	0.399004
$805$	0	0
$806$	4.34315	0.152981
$807$	−22.6274	−0.796523
$808$	0.242641	0.00853607
$809$	18.9706	0.666969	0.333485	−	0.942755i	$-0.391775\pi$
$809$	18.9706	0.666969	0.333485	+	0.942755i	$0.391775\pi$
$810$	0	0
$811$	1.12994	0.0396776	0.0198388	−	0.999803i	$-0.493685\pi$
$811$	1.12994	0.0396776	0.0198388	+	0.999803i	$0.493685\pi$
$812$	0	0
$813$	−21.1716	−0.742519
$814$	−3.65685	−0.128173
$815$	0	0
$816$	−4.00000	−0.140028
$817$	−54.1421	−1.89419
$818$	12.4853	0.436538
$819$	0	0
$820$	0	0
$821$	3.02944	0.105728	0.0528640	−	0.998602i	$-0.483165\pi$
$821$	3.02944	0.105728	0.0528640	+	0.998602i	$0.483165\pi$
$822$	−9.31371	−0.324853
$823$	−53.5980	−1.86831	−0.934154	−	0.356870i	$-0.883844\pi$
$823$	−53.5980	−1.86831	−0.934154	+	0.356870i	$0.883844\pi$
$824$	−13.8995	−0.484212
$825$	5.00000	0.174078
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	7.65685	0.266094
$829$	19.3137	0.670793	0.335396	−	0.942077i	$-0.391130\pi$
$829$	19.3137	0.670793	0.335396	+	0.942077i	$0.391130\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	1.41421	0.0490290
$833$	0	0
$834$	2.10051	0.0727345
$835$	0	0
$836$	5.41421	0.187254
$837$	3.07107	0.106152
$838$	−16.2843	−0.562531
$839$	12.7279	0.439417	0.219708	−	0.975566i	$-0.429489\pi$
$839$	12.7279	0.439417	0.219708	+	0.975566i	$0.429489\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	14.9706	0.515920
$843$	18.9706	0.653381
$844$	17.3137	0.595962
$845$	0	0
$846$	1.41421	0.0486217
$847$	0	0
$848$	−3.65685	−0.125577
$849$	−15.7574	−0.540791
$850$	20.0000	0.685994
$851$	28.0000	0.959828
$852$	−13.6569	−0.467876
$853$	−52.7279	−1.80537	−0.902685	−	0.430302i	$-0.858407\pi$
$853$	−52.7279	−1.80537	−0.902685	+	0.430302i	$0.858407\pi$
$854$	0	0
$855$	0	0
$856$	−17.3137	−0.591770
$857$	−14.8284	−0.506529	−0.253265	−	0.967397i	$-0.581504\pi$
$857$	−14.8284	−0.506529	−0.253265	+	0.967397i	$0.581504\pi$
$858$	1.41421	0.0482805
$859$	46.4264	1.58405	0.792024	−	0.610490i	$-0.209027\pi$
$859$	46.4264	1.58405	0.792024	+	0.610490i	$0.209027\pi$
$860$	0	0
$861$	0	0
$862$	−16.0000	−0.544962
$863$	49.5980	1.68833	0.844167	−	0.536080i	$-0.180095\pi$
$863$	49.5980	1.68833	0.844167	+	0.536080i	$0.180095\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−37.2132	−1.26456
$867$	1.00000	0.0339618
$868$	0	0
$869$	1.65685	0.0562049
$870$	0	0
$871$	−16.0000	−0.542139
$872$	10.0000	0.338643
$873$	12.7279	0.430775
$874$	−41.4558	−1.40226
$875$	0	0
$876$	−4.00000	−0.135147
$877$	53.3137	1.80028	0.900138	−	0.435605i	$-0.143465\pi$
$877$	53.3137	1.80028	0.900138	+	0.435605i	$0.143465\pi$
$878$	−35.7990	−1.20816
$879$	3.07107	0.103585
$880$	0	0
$881$	−21.8995	−0.737813	−0.368906	−	0.929467i	$-0.620268\pi$
$881$	−21.8995	−0.737813	−0.368906	+	0.929467i	$0.620268\pi$
$882$	0	0
$883$	−26.6274	−0.896084	−0.448042	−	0.894013i	$-0.647878\pi$
$883$	−26.6274	−0.896084	−0.448042	+	0.894013i	$0.647878\pi$
$884$	5.65685	0.190261
$885$	0	0
$886$	−28.2843	−0.950229
$887$	−4.28427	−0.143852	−0.0719259	−	0.997410i	$-0.522915\pi$
$887$	−4.28427	−0.143852	−0.0719259	+	0.997410i	$0.522915\pi$
$888$	3.65685	0.122716
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	19.5563	0.654795
$893$	−7.65685	−0.256227
$894$	12.9706	0.433801
$895$	0	0
$896$	0	0
$897$	−10.8284	−0.361551
$898$	−10.3431	−0.345155
$899$	17.3726	0.579408
$900$	−5.00000	−0.166667
$901$	−14.6274	−0.487310
$902$	−9.65685	−0.321538
$903$	0	0
$904$	5.65685	0.188144
$905$	0	0
$906$	−1.65685	−0.0550453
$907$	−33.9411	−1.12700	−0.563498	−	0.826117i	$-0.690545\pi$
$907$	−33.9411	−1.12700	−0.563498	+	0.826117i	$0.690545\pi$
$908$	23.5563	0.781745
$909$	−0.242641	−0.00804788
$910$	0	0
$911$	41.5980	1.37820	0.689101	−	0.724665i	$-0.258006\pi$
$911$	41.5980	1.37820	0.689101	+	0.724665i	$0.258006\pi$
$912$	−5.41421	−0.179283
$913$	−15.0711	−0.498780
$914$	−19.6569	−0.650191
$915$	0	0
$916$	5.65685	0.186908
$917$	0	0
$918$	4.00000	0.132020
$919$	14.6274	0.482514	0.241257	−	0.970461i	$-0.422440\pi$
$919$	14.6274	0.482514	0.241257	+	0.970461i	$0.422440\pi$
$920$	0	0
$921$	0.727922	0.0239858
$922$	−37.8995	−1.24815
$923$	19.3137	0.635718
$924$	0	0
$925$	−18.2843	−0.601183
$926$	27.3137	0.897584
$927$	13.8995	0.456519
$928$	5.65685	0.185695
$929$	−35.0711	−1.15064	−0.575322	−	0.817927i	$-0.695123\pi$
$929$	−35.0711	−1.15064	−0.575322	+	0.817927i	$0.695123\pi$
$930$	0	0
$931$	0	0
$932$	12.3431	0.404313
$933$	−20.2426	−0.662714
$934$	37.9411	1.24147
$935$	0	0
$936$	−1.41421	−0.0462250
$937$	−50.6274	−1.65393	−0.826963	−	0.562257i	$-0.809933\pi$
$937$	−50.6274	−1.65393	−0.826963	+	0.562257i	$0.809933\pi$
$938$	0	0
$939$	12.2426	0.399523
$940$	0	0
$941$	1.69848	0.0553690	0.0276845	−	0.999617i	$-0.491187\pi$
$941$	1.69848	0.0553690	0.0276845	+	0.999617i	$0.491187\pi$
$942$	−2.82843	−0.0921551
$943$	73.9411	2.40785
$944$	6.82843	0.222246
$945$	0	0
$946$	10.0000	0.325128
$947$	−24.6863	−0.802197	−0.401098	−	0.916035i	$-0.631371\pi$
$947$	−24.6863	−0.802197	−0.401098	+	0.916035i	$0.631371\pi$
$948$	−1.65685	−0.0538121
$949$	5.65685	0.183629
$950$	27.0711	0.878301
$951$	5.31371	0.172309
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	3.65685	0.118395
$955$	0	0
$956$	14.3431	0.463890
$957$	5.65685	0.182860
$958$	−32.9706	−1.06523
$959$	0	0
$960$	0	0
$961$	−21.5685	−0.695759
$962$	−5.17157	−0.166738
$963$	17.3137	0.557926
$964$	−7.31371	−0.235559
$965$	0	0
$966$	0	0
$967$	44.2843	1.42409	0.712043	−	0.702136i	$-0.247770\pi$
$967$	44.2843	1.42409	0.712043	+	0.702136i	$0.247770\pi$
$968$	−1.00000	−0.0321412
$969$	−21.6569	−0.695718
$970$	0	0
$971$	46.9117	1.50547	0.752734	−	0.658325i	$-0.228735\pi$
$971$	46.9117	1.50547	0.752734	+	0.658325i	$0.228735\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	0.343146	0.0109951
$975$	7.07107	0.226455
$976$	−1.41421	−0.0452679
$977$	26.3431	0.842792	0.421396	−	0.906877i	$-0.361540\pi$
$977$	26.3431	0.842792	0.421396	+	0.906877i	$0.361540\pi$
$978$	5.65685	0.180886
$979$	2.58579	0.0826421
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−20.0000	−0.638226
$983$	41.0122	1.30809	0.654043	−	0.756457i	$-0.273072\pi$
$983$	41.0122	1.30809	0.654043	+	0.756457i	$0.273072\pi$
$984$	9.65685	0.307849
$985$	0	0
$986$	22.6274	0.720604
$987$	0	0
$988$	7.65685	0.243597
$989$	−76.5685	−2.43474
$990$	0	0
$991$	43.3137	1.37591	0.687953	−	0.725756i	$-0.258510\pi$
$991$	43.3137	1.37591	0.687953	+	0.725756i	$0.258510\pi$
$992$	3.07107	0.0975065
$993$	−28.9706	−0.919353
$994$	0	0
$995$	0	0
$996$	15.0711	0.477545
$997$	9.89949	0.313520	0.156760	−	0.987637i	$-0.449895\pi$
$997$	9.89949	0.313520	0.156760	+	0.987637i	$0.449895\pi$
$998$	10.6274	0.336405
$999$	−3.65685	−0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.y.1.2	✓	2
3.2	odd	2		9702.2.a.de.1.2		2
7.6	odd	2		3234.2.a.z.1.1	yes	2
21.20	even	2		9702.2.a.dl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.y.1.2	✓	2	1.1	even	1	trivial
3234.2.a.z.1.1	yes	2	7.6	odd	2
9702.2.a.de.1.2		2	3.2	odd	2
9702.2.a.dl.1.1		2	21.20	even	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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} +1.41421 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +5.41421 q^{19} -1.00000 q^{22} +7.65685 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.41421 q^{26} -1.00000 q^{27} -5.65685 q^{29} -3.07107 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +3.65685 q^{37} -5.41421 q^{38} -1.41421 q^{39} +9.65685 q^{41} -10.0000 q^{43} +1.00000 q^{44} -7.65685 q^{46} -1.41421 q^{47} -1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} +1.41421 q^{52} -3.65685 q^{53} +1.00000 q^{54} -5.41421 q^{57} +5.65685 q^{58} +6.82843 q^{59} -1.41421 q^{61} +3.07107 q^{62} +1.00000 q^{64} +1.00000 q^{66} -11.3137 q^{67} +4.00000 q^{68} -7.65685 q^{69} +13.6569 q^{71} -1.00000 q^{72} +4.00000 q^{73} -3.65685 q^{74} +5.00000 q^{75} +5.41421 q^{76} +1.41421 q^{78} +1.65685 q^{79} +1.00000 q^{81} -9.65685 q^{82} -15.0711 q^{83} +10.0000 q^{86} +5.65685 q^{87} -1.00000 q^{88} +2.58579 q^{89} +7.65685 q^{92} +3.07107 q^{93} +1.41421 q^{94} +1.00000 q^{96} +12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 8 q^{19} - 2 q^{22} + 4 q^{23} + 2 q^{24} - 10 q^{25} - 2 q^{27} + 8 q^{31} - 2 q^{32}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 8 * q^19 - 2 * q^22 + 4 * q^23 + 2 * q^24 - 10 * q^25 - 2 * q^27 + 8 * q^31 - 2 * q^32 - 2 * q^33 - 8 * q^34 + 2 * q^36 - 4 * q^37 - 8 * q^38 + 8 * q^41 - 20 * q^43 + 2 * q^44 - 4 * q^46 - 2 * q^48 + 10 * q^50 - 8 * q^51 + 4 * q^53 + 2 * q^54 - 8 * q^57 + 8 * q^59 - 8 * q^62 + 2 * q^64 + 2 * q^66 + 8 * q^68 - 4 * q^69 + 16 * q^71 - 2 * q^72 + 8 * q^73 + 4 * q^74 + 10 * q^75 + 8 * q^76 - 8 * q^79 + 2 * q^81 - 8 * q^82 - 16 * q^83 + 20 * q^86 - 2 * q^88 + 8 * q^89 + 4 * q^92 - 8 * q^93 + 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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