LMFDB - Embedded newform 3234.2.a.bf.1.3

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:	$3$
Coefficient field:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.41883$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.41883 q^{10} -1.00000 q^{11} -1.00000 q^{12} -4.41883 q^{15} +1.00000 q^{16} +2.37683 q^{17} +1.00000 q^{18} -3.68842 q^{19} +4.41883 q^{20} -1.00000 q^{22} +4.73042 q^{23} -1.00000 q^{24} +14.5261 q^{25} -1.00000 q^{27} -6.52608 q^{29} -4.41883 q^{30} -3.37683 q^{31} +1.00000 q^{32} +1.00000 q^{33} +2.37683 q^{34} +1.00000 q^{36} +7.68842 q^{37} -3.68842 q^{38} +4.41883 q^{40} +12.1492 q^{41} +3.06525 q^{43} -1.00000 q^{44} +4.41883 q^{45} +4.73042 q^{46} -4.10725 q^{47} -1.00000 q^{48} +14.5261 q^{50} -2.37683 q^{51} +8.00000 q^{53} -1.00000 q^{54} -4.41883 q^{55} +3.68842 q^{57} -6.52608 q^{58} -12.5261 q^{59} -4.41883 q^{60} +3.79567 q^{61} -3.37683 q^{62} +1.00000 q^{64} +1.00000 q^{66} +10.1492 q^{67} +2.37683 q^{68} -4.73042 q^{69} +0.934749 q^{71} +1.00000 q^{72} -7.46083 q^{73} +7.68842 q^{74} -14.5261 q^{75} -3.68842 q^{76} -0.418833 q^{79} +4.41883 q^{80} +1.00000 q^{81} +12.1492 q^{82} -2.14925 q^{83} +10.5028 q^{85} +3.06525 q^{86} +6.52608 q^{87} -1.00000 q^{88} -12.8377 q^{89} +4.41883 q^{90} +4.73042 q^{92} +3.37683 q^{93} -4.10725 q^{94} -16.2985 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 3 q^{12} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 15 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} + 3 q^{33} + 3 q^{34} + 3 q^{36} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} - 3 q^{54} + 9 q^{57} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} + 3 q^{66} + 6 q^{67} + 3 q^{68} - 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} - 15 q^{75} - 9 q^{76} + 12 q^{79} + 3 q^{81} + 12 q^{82} + 18 q^{83} + 3 q^{86} - 9 q^{87} - 3 q^{88} - 12 q^{89} + 3 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{96} + 21 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 3 * q^11 - 3 * q^12 + 3 * q^16 + 3 * q^17 + 3 * q^18 - 9 * q^19 - 3 * q^22 + 3 * q^23 - 3 * q^24 + 15 * q^25 - 3 * q^27 + 9 * q^29 - 6 * q^31 + 3 * q^32 + 3 * q^33 + 3 * q^34 + 3 * q^36 + 21 * q^37 - 9 * q^38 + 12 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 3 * q^47 - 3 * q^48 + 15 * q^50 - 3 * q^51 + 24 * q^53 - 3 * q^54 + 9 * q^57 + 9 * q^58 - 9 * q^59 - 6 * q^61 - 6 * q^62 + 3 * q^64 + 3 * q^66 + 6 * q^67 + 3 * q^68 - 3 * q^69 + 9 * q^71 + 3 * q^72 + 21 * q^74 - 15 * q^75 - 9 * q^76 + 12 * q^79 + 3 * q^81 + 12 * q^82 + 18 * q^83 + 3 * q^86 - 9 * q^87 - 3 * q^88 - 12 * q^89 + 3 * q^92 + 6 * q^93 + 3 * q^94 - 3 * q^96 + 21 * q^97 - 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	4.41883	1.97616	0.988081	−	0.153935i	$-0.0491945\pi$
$5$	4.41883	1.97616	0.988081	+	0.153935i	$0.0491945\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	4.41883	1.39736
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−4.41883	−1.14094
$16$	1.00000	0.250000
$17$	2.37683	0.576467	0.288233	−	0.957560i	$-0.406932\pi$
$17$	2.37683	0.576467	0.288233	+	0.957560i	$0.406932\pi$
$18$	1.00000	0.235702
$19$	−3.68842	−0.846181	−0.423090	−	0.906087i	$-0.639055\pi$
$19$	−3.68842	−0.846181	−0.423090	+	0.906087i	$0.639055\pi$
$20$	4.41883	0.988081
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.73042	0.986360	0.493180	−	0.869927i	$-0.335834\pi$
$23$	4.73042	0.986360	0.493180	+	0.869927i	$0.335834\pi$
$24$	−1.00000	−0.204124
$25$	14.5261	2.90522
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.52608	−1.21186	−0.605932	−	0.795517i	$-0.707199\pi$
$29$	−6.52608	−1.21186	−0.605932	+	0.795517i	$0.707199\pi$
$30$	−4.41883	−0.806765
$31$	−3.37683	−0.606497	−0.303249	−	0.952911i	$-0.598071\pi$
$31$	−3.37683	−0.606497	−0.303249	+	0.952911i	$0.598071\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	2.37683	0.407624
$35$	0	0
$36$	1.00000	0.166667
$37$	7.68842	1.26397	0.631984	−	0.774981i	$-0.282241\pi$
$37$	7.68842	1.26397	0.631984	+	0.774981i	$0.282241\pi$
$38$	−3.68842	−0.598340
$39$	0	0
$40$	4.41883	0.698679
$41$	12.1492	1.89739	0.948697	−	0.316187i	$-0.102403\pi$
$41$	12.1492	1.89739	0.948697	+	0.316187i	$0.102403\pi$
$42$	0	0
$43$	3.06525	0.467446	0.233723	−	0.972303i	$-0.424909\pi$
$43$	3.06525	0.467446	0.233723	+	0.972303i	$0.424909\pi$
$44$	−1.00000	−0.150756
$45$	4.41883	0.658721
$46$	4.73042	0.697462
$47$	−4.10725	−0.599104	−0.299552	−	0.954080i	$-0.596837\pi$
$47$	−4.10725	−0.599104	−0.299552	+	0.954080i	$0.596837\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	14.5261	2.05430
$51$	−2.37683	−0.332823
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	−1.00000	−0.136083
$55$	−4.41883	−0.595835
$56$	0	0
$57$	3.68842	0.488543
$58$	−6.52608	−0.856917
$59$	−12.5261	−1.63076	−0.815379	−	0.578928i	$-0.803471\pi$
$59$	−12.5261	−1.63076	−0.815379	+	0.578928i	$0.803471\pi$
$60$	−4.41883	−0.570469
$61$	3.79567	0.485985	0.242993	−	0.970028i	$-0.421871\pi$
$61$	3.79567	0.485985	0.242993	+	0.970028i	$0.421871\pi$
$62$	−3.37683	−0.428858
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	10.1492	1.23993	0.619964	−	0.784630i	$-0.287147\pi$
$67$	10.1492	1.23993	0.619964	+	0.784630i	$0.287147\pi$
$68$	2.37683	0.288233
$69$	−4.73042	−0.569475
$70$	0	0
$71$	0.934749	0.110934	0.0554672	−	0.998461i	$-0.482335\pi$
$71$	0.934749	0.110934	0.0554672	+	0.998461i	$0.482335\pi$
$72$	1.00000	0.117851
$73$	−7.46083	−0.873224	−0.436612	−	0.899650i	$-0.643822\pi$
$73$	−7.46083	−0.873224	−0.436612	+	0.899650i	$0.643822\pi$
$74$	7.68842	0.893760
$75$	−14.5261	−1.67733
$76$	−3.68842	−0.423090
$77$	0	0
$78$	0	0
$79$	−0.418833	−0.0471224	−0.0235612	−	0.999722i	$-0.507500\pi$
$79$	−0.418833	−0.0471224	−0.0235612	+	0.999722i	$0.507500\pi$
$80$	4.41883	0.494041
$81$	1.00000	0.111111
$82$	12.1492	1.34166
$83$	−2.14925	−0.235911	−0.117955	−	0.993019i	$-0.537634\pi$
$83$	−2.14925	−0.235911	−0.117955	+	0.993019i	$0.537634\pi$
$84$	0	0
$85$	10.5028	1.13919
$86$	3.06525	0.330534
$87$	6.52608	0.699669
$88$	−1.00000	−0.106600
$89$	−12.8377	−1.36079	−0.680395	−	0.732846i	$-0.738192\pi$
$89$	−12.8377	−1.36079	−0.680395	+	0.732846i	$0.738192\pi$
$90$	4.41883	0.465786
$91$	0	0
$92$	4.73042	0.493180
$93$	3.37683	0.350161
$94$	−4.10725	−0.423630
$95$	−16.2985	−1.67219
$96$	−1.00000	−0.102062
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	14.5261	1.45261
$101$	−15.3637	−1.52875	−0.764375	−	0.644772i	$-0.776952\pi$
$101$	−15.3637	−1.52875	−0.764375	+	0.644772i	$0.776952\pi$
$102$	−2.37683	−0.235342
$103$	−2.62317	−0.258468	−0.129234	−	0.991614i	$-0.541252\pi$
$103$	−2.62317	−0.258468	−0.129234	+	0.991614i	$0.541252\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	10.9029	1.05402	0.527012	−	0.849858i	$-0.323312\pi$
$107$	10.9029	1.05402	0.527012	+	0.849858i	$0.323312\pi$
$108$	−1.00000	−0.0962250
$109$	−11.7957	−1.12982	−0.564910	−	0.825153i	$-0.691089\pi$
$109$	−11.7957	−1.12982	−0.564910	+	0.825153i	$0.691089\pi$
$110$	−4.41883	−0.421319
$111$	−7.68842	−0.729752
$112$	0	0
$113$	5.37683	0.505810	0.252905	−	0.967491i	$-0.418614\pi$
$113$	5.37683	0.505810	0.252905	+	0.967491i	$0.418614\pi$
$114$	3.68842	0.345452
$115$	20.9029	1.94921
$116$	−6.52608	−0.605932
$117$	0	0
$118$	−12.5261	−1.15312
$119$	0	0
$120$	−4.41883	−0.403382
$121$	1.00000	0.0909091
$122$	3.79567	0.343643
$123$	−12.1492	−1.09546
$124$	−3.37683	−0.303249
$125$	42.0942	3.76502
$126$	0	0
$127$	0.730416	0.0648139	0.0324070	−	0.999475i	$-0.489683\pi$
$127$	0.730416	0.0648139	0.0324070	+	0.999475i	$0.489683\pi$
$128$	1.00000	0.0883883
$129$	−3.06525	−0.269880
$130$	0	0
$131$	7.37683	0.644517	0.322258	−	0.946652i	$-0.395558\pi$
$131$	7.37683	0.644517	0.322258	+	0.946652i	$0.395558\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	10.1492	0.876762
$135$	−4.41883	−0.380313
$136$	2.37683	0.203812
$137$	−8.83767	−0.755053	−0.377526	−	0.925999i	$-0.623225\pi$
$137$	−8.83767	−0.755053	−0.377526	+	0.925999i	$0.623225\pi$
$138$	−4.73042	−0.402680
$139$	1.47392	0.125016	0.0625080	−	0.998044i	$-0.480090\pi$
$139$	1.47392	0.125016	0.0625080	+	0.998044i	$0.480090\pi$
$140$	0	0
$141$	4.10725	0.345893
$142$	0.934749	0.0784424
$143$	0	0
$144$	1.00000	0.0833333
$145$	−28.8377	−2.39484
$146$	−7.46083	−0.617463
$147$	0	0
$148$	7.68842	0.631984
$149$	−9.36375	−0.767108	−0.383554	−	0.923518i	$-0.625300\pi$
$149$	−9.36375	−0.767108	−0.383554	+	0.923518i	$0.625300\pi$
$150$	−14.5261	−1.18605
$151$	−13.5681	−1.10415	−0.552077	−	0.833793i	$-0.686165\pi$
$151$	−13.5681	−1.10415	−0.552077	+	0.833793i	$0.686165\pi$
$152$	−3.68842	−0.299170
$153$	2.37683	0.192156
$154$	0	0
$155$	−14.9217	−1.19854
$156$	0	0
$157$	−18.7406	−1.49566	−0.747831	−	0.663890i	$-0.768904\pi$
$157$	−18.7406	−1.49566	−0.747831	+	0.663890i	$0.768904\pi$
$158$	−0.418833	−0.0333205
$159$	−8.00000	−0.634441
$160$	4.41883	0.349339
$161$	0	0
$162$	1.00000	0.0785674
$163$	18.7724	1.47037	0.735185	−	0.677867i	$-0.237096\pi$
$163$	18.7724	1.47037	0.735185	+	0.677867i	$0.237096\pi$
$164$	12.1492	0.948697
$165$	4.41883	0.344006
$166$	−2.14925	−0.166814
$167$	−23.8898	−1.84865	−0.924325	−	0.381606i	$-0.875371\pi$
$167$	−23.8898	−1.84865	−0.924325	+	0.381606i	$0.875371\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	10.5028	0.805530
$171$	−3.68842	−0.282060
$172$	3.06525	0.233723
$173$	11.3768	0.864965	0.432482	−	0.901642i	$-0.357638\pi$
$173$	11.3768	0.864965	0.432482	+	0.901642i	$0.357638\pi$
$174$	6.52608	0.494741
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	12.5261	0.941518
$178$	−12.8377	−0.962224
$179$	24.6101	1.83944	0.919722	−	0.392571i	$-0.128414\pi$
$179$	24.6101	1.83944	0.919722	+	0.392571i	$0.128414\pi$
$180$	4.41883	0.329360
$181$	17.4608	1.29785	0.648927	−	0.760851i	$-0.275218\pi$
$181$	17.4608	1.29785	0.648927	+	0.760851i	$0.275218\pi$
$182$	0	0
$183$	−3.79567	−0.280584
$184$	4.73042	0.348731
$185$	33.9738	2.49781
$186$	3.37683	0.247601
$187$	−2.37683	−0.173811
$188$	−4.10725	−0.299552
$189$	0	0
$190$	−16.2985	−1.18242
$191$	−16.8377	−1.21833	−0.609165	−	0.793043i	$-0.708495\pi$
$191$	−16.8377	−1.21833	−0.609165	+	0.793043i	$0.708495\pi$
$192$	−1.00000	−0.0721688
$193$	5.46083	0.393079	0.196540	−	0.980496i	$-0.437030\pi$
$193$	5.46083	0.393079	0.196540	+	0.980496i	$0.437030\pi$
$194$	7.00000	0.502571
$195$	0	0
$196$	0	0
$197$	−9.68842	−0.690271	−0.345136	−	0.938553i	$-0.612167\pi$
$197$	−9.68842	−0.690271	−0.345136	+	0.938553i	$0.612167\pi$
$198$	−1.00000	−0.0710669
$199$	20.2985	1.43892	0.719461	−	0.694533i	$-0.244389\pi$
$199$	20.2985	1.43892	0.719461	+	0.694533i	$0.244389\pi$
$200$	14.5261	1.02715
$201$	−10.1492	−0.715873
$202$	−15.3637	−1.08099
$203$	0	0
$204$	−2.37683	−0.166412
$205$	53.6855	3.74956
$206$	−2.62317	−0.182765
$207$	4.73042	0.328787
$208$	0	0
$209$	3.68842	0.255133
$210$	0	0
$211$	22.5130	1.54986	0.774929	−	0.632048i	$-0.217785\pi$
$211$	22.5130	1.54986	0.774929	+	0.632048i	$0.217785\pi$
$212$	8.00000	0.549442
$213$	−0.934749	−0.0640480
$214$	10.9029	0.745308
$215$	13.5448	0.923750
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−11.7957	−0.798903
$219$	7.46083	0.504156
$220$	−4.41883	−0.297918
$221$	0	0
$222$	−7.68842	−0.516013
$223$	−7.46083	−0.499614	−0.249807	−	0.968296i	$-0.580367\pi$
$223$	−7.46083	−0.499614	−0.249807	+	0.968296i	$0.580367\pi$
$224$	0	0
$225$	14.5261	0.968405
$226$	5.37683	0.357662
$227$	−2.77241	−0.184012	−0.0920058	−	0.995758i	$-0.529328\pi$
$227$	−2.77241	−0.184012	−0.0920058	+	0.995758i	$0.529328\pi$
$228$	3.68842	0.244271
$229$	−2.53917	−0.167793	−0.0838965	−	0.996474i	$-0.526737\pi$
$229$	−2.53917	−0.167793	−0.0838965	+	0.996474i	$0.526737\pi$
$230$	20.9029	1.37830
$231$	0	0
$232$	−6.52608	−0.428458
$233$	−24.0522	−1.57571	−0.787855	−	0.615861i	$-0.788808\pi$
$233$	−24.0522	−1.57571	−0.787855	+	0.615861i	$0.788808\pi$
$234$	0	0
$235$	−18.1492	−1.18393
$236$	−12.5261	−0.815379
$237$	0.418833	0.0272061
$238$	0	0
$239$	7.59133	0.491042	0.245521	−	0.969391i	$-0.421041\pi$
$239$	7.59133	0.491042	0.245521	+	0.969391i	$0.421041\pi$
$240$	−4.41883	−0.285234
$241$	16.2985	1.04988	0.524939	−	0.851140i	$-0.324088\pi$
$241$	16.2985	1.04988	0.524939	+	0.851140i	$0.324088\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	3.79567	0.242993
$245$	0	0
$246$	−12.1492	−0.774608
$247$	0	0
$248$	−3.37683	−0.214429
$249$	2.14925	0.136203
$250$	42.0942	2.66227
$251$	−8.61008	−0.543463	−0.271732	−	0.962373i	$-0.587596\pi$
$251$	−8.61008	−0.543463	−0.271732	+	0.962373i	$0.587596\pi$
$252$	0	0
$253$	−4.73042	−0.297399
$254$	0.730416	0.0458304
$255$	−10.5028	−0.657713
$256$	1.00000	0.0625000
$257$	0.837665	0.0522521	0.0261261	−	0.999659i	$-0.491683\pi$
$257$	0.837665	0.0522521	0.0261261	+	0.999659i	$0.491683\pi$
$258$	−3.06525	−0.190834
$259$	0	0
$260$	0	0
$261$	−6.52608	−0.403954
$262$	7.37683	0.455742
$263$	10.8377	0.668279	0.334140	−	0.942524i	$-0.391554\pi$
$263$	10.8377	0.668279	0.334140	+	0.942524i	$0.391554\pi$
$264$	1.00000	0.0615457
$265$	35.3507	2.17157
$266$	0	0
$267$	12.8377	0.785652
$268$	10.1492	0.619964
$269$	17.8797	1.09014	0.545071	−	0.838390i	$-0.316503\pi$
$269$	17.8797	1.09014	0.545071	+	0.838390i	$0.316503\pi$
$270$	−4.41883	−0.268922
$271$	−8.83767	−0.536850	−0.268425	−	0.963301i	$-0.586503\pi$
$271$	−8.83767	−0.536850	−0.268425	+	0.963301i	$0.586503\pi$
$272$	2.37683	0.144117
$273$	0	0
$274$	−8.83767	−0.533903
$275$	−14.5261	−0.875956
$276$	−4.73042	−0.284738
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	1.47392	0.0883997
$279$	−3.37683	−0.202166
$280$	0	0
$281$	−15.9217	−0.949807	−0.474903	−	0.880038i	$-0.657517\pi$
$281$	−15.9217	−0.949807	−0.474903	+	0.880038i	$0.657517\pi$
$282$	4.10725	0.244583
$283$	−18.0840	−1.07498	−0.537491	−	0.843269i	$-0.680628\pi$
$283$	−18.0840	−1.07498	−0.537491	+	0.843269i	$0.680628\pi$
$284$	0.934749	0.0554672
$285$	16.2985	0.965440
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−11.3507	−0.667686
$290$	−28.8377	−1.69341
$291$	−7.00000	−0.410347
$292$	−7.46083	−0.436612
$293$	25.1492	1.46923	0.734617	−	0.678482i	$-0.237362\pi$
$293$	25.1492	1.46923	0.734617	+	0.678482i	$0.237362\pi$
$294$	0	0
$295$	−55.3507	−3.22264
$296$	7.68842	0.446880
$297$	1.00000	0.0580259
$298$	−9.36375	−0.542427
$299$	0	0
$300$	−14.5261	−0.838664
$301$	0	0
$302$	−13.5681	−0.780755
$303$	15.3637	0.882624
$304$	−3.68842	−0.211545
$305$	16.7724	0.960386
$306$	2.37683	0.135875
$307$	−3.86950	−0.220844	−0.110422	−	0.993885i	$-0.535220\pi$
$307$	−3.86950	−0.220844	−0.110422	+	0.993885i	$0.535220\pi$
$308$	0	0
$309$	2.62317	0.149227
$310$	−14.9217	−0.847494
$311$	−13.5681	−0.769375	−0.384688	−	0.923047i	$-0.625691\pi$
$311$	−13.5681	−0.769375	−0.384688	+	0.923047i	$0.625691\pi$
$312$	0	0
$313$	2.52608	0.142783	0.0713913	−	0.997448i	$-0.477256\pi$
$313$	2.52608	0.142783	0.0713913	+	0.997448i	$0.477256\pi$
$314$	−18.7406	−1.05759
$315$	0	0
$316$	−0.418833	−0.0235612
$317$	−20.0102	−1.12388	−0.561941	−	0.827177i	$-0.689945\pi$
$317$	−20.0102	−1.12388	−0.561941	+	0.827177i	$0.689945\pi$
$318$	−8.00000	−0.448618
$319$	6.52608	0.365390
$320$	4.41883	0.247020
$321$	−10.9029	−0.608541
$322$	0	0
$323$	−8.76675	−0.487795
$324$	1.00000	0.0555556
$325$	0	0
$326$	18.7724	1.03971
$327$	11.7957	0.652302
$328$	12.1492	0.670830
$329$	0	0
$330$	4.41883	0.243249
$331$	−16.1492	−0.887643	−0.443821	−	0.896115i	$-0.646378\pi$
$331$	−16.1492	−0.887643	−0.443821	+	0.896115i	$0.646378\pi$
$332$	−2.14925	−0.117955
$333$	7.68842	0.421323
$334$	−23.8898	−1.30719
$335$	44.8478	2.45030
$336$	0	0
$337$	23.1362	1.26031	0.630154	−	0.776471i	$-0.282992\pi$
$337$	23.1362	1.26031	0.630154	+	0.776471i	$0.282992\pi$
$338$	−13.0000	−0.707107
$339$	−5.37683	−0.292030
$340$	10.5028	0.569596
$341$	3.37683	0.182866
$342$	−3.68842	−0.199447
$343$	0	0
$344$	3.06525	0.165267
$345$	−20.9029	−1.12538
$346$	11.3768	0.611622
$347$	15.6566	0.840489	0.420245	−	0.907411i	$-0.361944\pi$
$347$	15.6566	0.840489	0.420245	+	0.907411i	$0.361944\pi$
$348$	6.52608	0.349835
$349$	−20.0102	−1.07112	−0.535560	−	0.844497i	$-0.679899\pi$
$349$	−20.0102	−1.07112	−0.535560	+	0.844497i	$0.679899\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	7.46083	0.397100	0.198550	−	0.980091i	$-0.436377\pi$
$353$	7.46083	0.397100	0.198550	+	0.980091i	$0.436377\pi$
$354$	12.5261	0.665754
$355$	4.13050	0.219224
$356$	−12.8377	−0.680395
$357$	0	0
$358$	24.6101	1.30068
$359$	−11.2463	−0.593559	−0.296779	−	0.954946i	$-0.595913\pi$
$359$	−11.2463	−0.593559	−0.296779	+	0.954946i	$0.595913\pi$
$360$	4.41883	0.232893
$361$	−5.39558	−0.283978
$362$	17.4608	0.917721
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−32.9682	−1.72563
$366$	−3.79567	−0.198403
$367$	−5.91600	−0.308813	−0.154406	−	0.988007i	$-0.549347\pi$
$367$	−5.91600	−0.308813	−0.154406	+	0.988007i	$0.549347\pi$
$368$	4.73042	0.246590
$369$	12.1492	0.632465
$370$	33.9738	1.76622
$371$	0	0
$372$	3.37683	0.175081
$373$	−23.1260	−1.19742	−0.598709	−	0.800966i	$-0.704320\pi$
$373$	−23.1260	−1.19742	−0.598709	+	0.800966i	$0.704320\pi$
$374$	−2.37683	−0.122903
$375$	−42.0942	−2.17373
$376$	−4.10725	−0.211815
$377$	0	0
$378$	0	0
$379$	−6.77241	−0.347876	−0.173938	−	0.984757i	$-0.555649\pi$
$379$	−6.77241	−0.347876	−0.173938	+	0.984757i	$0.555649\pi$
$380$	−16.2985	−0.836095
$381$	−0.730416	−0.0374203
$382$	−16.8377	−0.861490
$383$	25.3637	1.29603	0.648013	−	0.761629i	$-0.275600\pi$
$383$	25.3637	1.29603	0.648013	+	0.761629i	$0.275600\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	5.46083	0.277949
$387$	3.06525	0.155815
$388$	7.00000	0.355371
$389$	25.2565	1.28056	0.640278	−	0.768144i	$-0.278819\pi$
$389$	25.2565	1.28056	0.640278	+	0.768144i	$0.278819\pi$
$390$	0	0
$391$	11.2434	0.568604
$392$	0	0
$393$	−7.37683	−0.372112
$394$	−9.68842	−0.488095
$395$	−1.85075	−0.0931214
$396$	−1.00000	−0.0502519
$397$	−13.4739	−0.676237	−0.338118	−	0.941104i	$-0.609790\pi$
$397$	−13.4739	−0.676237	−0.338118	+	0.941104i	$0.609790\pi$
$398$	20.2985	1.01747
$399$	0	0
$400$	14.5261	0.726304
$401$	33.8898	1.69238	0.846189	−	0.532883i	$-0.178892\pi$
$401$	33.8898	1.69238	0.846189	+	0.532883i	$0.178892\pi$
$402$	−10.1492	−0.506199
$403$	0	0
$404$	−15.3637	−0.764375
$405$	4.41883	0.219574
$406$	0	0
$407$	−7.68842	−0.381101
$408$	−2.37683	−0.117671
$409$	−4.21450	−0.208394	−0.104197	−	0.994557i	$-0.533227\pi$
$409$	−4.21450	−0.208394	−0.104197	+	0.994557i	$0.533227\pi$
$410$	53.6855	2.65134
$411$	8.83767	0.435930
$412$	−2.62317	−0.129234
$413$	0	0
$414$	4.73042	0.232487
$415$	−9.49717	−0.466198
$416$	0	0
$417$	−1.47392	−0.0721781
$418$	3.68842	0.180406
$419$	−22.3116	−1.08999	−0.544996	−	0.838439i	$-0.683469\pi$
$419$	−22.3116	−1.08999	−0.544996	+	0.838439i	$0.683469\pi$
$420$	0	0
$421$	18.3116	0.892452	0.446226	−	0.894920i	$-0.352768\pi$
$421$	18.3116	0.892452	0.446226	+	0.894920i	$0.352768\pi$
$422$	22.5130	1.09592
$423$	−4.10725	−0.199701
$424$	8.00000	0.388514
$425$	34.5261	1.67476
$426$	−0.934749	−0.0452888
$427$	0	0
$428$	10.9029	0.527012
$429$	0	0
$430$	13.5448	0.653190
$431$	−28.8377	−1.38906	−0.694531	−	0.719463i	$-0.744388\pi$
$431$	−28.8377	−1.38906	−0.694531	+	0.719463i	$0.744388\pi$
$432$	−1.00000	−0.0481125
$433$	7.62317	0.366346	0.183173	−	0.983081i	$-0.441363\pi$
$433$	7.62317	0.366346	0.183173	+	0.983081i	$0.441363\pi$
$434$	0	0
$435$	28.8377	1.38266
$436$	−11.7957	−0.564910
$437$	−17.4477	−0.834639
$438$	7.46083	0.356492
$439$	−33.3739	−1.59285	−0.796425	−	0.604737i	$-0.793278\pi$
$439$	−33.3739	−1.59285	−0.796425	+	0.604737i	$0.793278\pi$
$440$	−4.41883	−0.210660
$441$	0	0
$442$	0	0
$443$	−4.85075	−0.230466	−0.115233	−	0.993338i	$-0.536761\pi$
$443$	−4.85075	−0.230466	−0.115233	+	0.993338i	$0.536761\pi$
$444$	−7.68842	−0.364876
$445$	−56.7275	−2.68914
$446$	−7.46083	−0.353281
$447$	9.36375	0.442890
$448$	0	0
$449$	14.3450	0.676982	0.338491	−	0.940970i	$-0.390083\pi$
$449$	14.3450	0.676982	0.338491	+	0.940970i	$0.390083\pi$
$450$	14.5261	0.684766
$451$	−12.1492	−0.572086
$452$	5.37683	0.252905
$453$	13.5681	0.637484
$454$	−2.77241	−0.130116
$455$	0	0
$456$	3.68842	0.172726
$457$	−25.1827	−1.17800	−0.588998	−	0.808135i	$-0.700477\pi$
$457$	−25.1827	−1.17800	−0.588998	+	0.808135i	$0.700477\pi$
$458$	−2.53917	−0.118648
$459$	−2.37683	−0.110941
$460$	20.9029	0.974603
$461$	13.0187	0.606344	0.303172	−	0.952936i	$-0.401954\pi$
$461$	13.0187	0.606344	0.303172	+	0.952936i	$0.401954\pi$
$462$	0	0
$463$	−14.2145	−0.660604	−0.330302	−	0.943875i	$-0.607151\pi$
$463$	−14.2145	−0.660604	−0.330302	+	0.943875i	$0.607151\pi$
$464$	−6.52608	−0.302966
$465$	14.9217	0.691976
$466$	−24.0522	−1.11420
$467$	−2.39558	−0.110854	−0.0554271	−	0.998463i	$-0.517652\pi$
$467$	−2.39558	−0.110854	−0.0554271	+	0.998463i	$0.517652\pi$
$468$	0	0
$469$	0	0
$470$	−18.1492	−0.837162
$471$	18.7406	0.863520
$472$	−12.5261	−0.576560
$473$	−3.06525	−0.140940
$474$	0.418833	0.0192376
$475$	−53.5782	−2.45834
$476$	0	0
$477$	8.00000	0.366295
$478$	7.59133	0.347219
$479$	10.8377	0.495186	0.247593	−	0.968864i	$-0.420360\pi$
$479$	10.8377	0.495186	0.247593	+	0.968864i	$0.420360\pi$
$480$	−4.41883	−0.201691
$481$	0	0
$482$	16.2985	0.742376
$483$	0	0
$484$	1.00000	0.0454545
$485$	30.9318	1.40454
$486$	−1.00000	−0.0453609
$487$	−34.6435	−1.56985	−0.784923	−	0.619593i	$-0.787298\pi$
$487$	−34.6435	−1.56985	−0.784923	+	0.619593i	$0.787298\pi$
$488$	3.79567	0.171822
$489$	−18.7724	−0.848918
$490$	0	0
$491$	38.1492	1.72165	0.860826	−	0.508900i	$-0.169948\pi$
$491$	38.1492	1.72165	0.860826	+	0.508900i	$0.169948\pi$
$492$	−12.1492	−0.547730
$493$	−15.5114	−0.698599
$494$	0	0
$495$	−4.41883	−0.198612
$496$	−3.37683	−0.151624
$497$	0	0
$498$	2.14925	0.0963101
$499$	−0.623166	−0.0278968	−0.0139484	−	0.999903i	$-0.504440\pi$
$499$	−0.623166	−0.0278968	−0.0139484	+	0.999903i	$0.504440\pi$
$500$	42.0942	1.88251
$501$	23.8898	1.06732
$502$	−8.61008	−0.384287
$503$	31.1362	1.38829	0.694146	−	0.719834i	$-0.255782\pi$
$503$	31.1362	1.38829	0.694146	+	0.719834i	$0.255782\pi$
$504$	0	0
$505$	−67.8898	−3.02106
$506$	−4.73042	−0.210293
$507$	13.0000	0.577350
$508$	0.730416	0.0324070
$509$	−42.1043	−1.86624	−0.933121	−	0.359563i	$-0.882926\pi$
$509$	−42.1043	−1.86624	−0.933121	+	0.359563i	$0.882926\pi$
$510$	−10.5028	−0.465073
$511$	0	0
$512$	1.00000	0.0441942
$513$	3.68842	0.162848
$514$	0.837665	0.0369478
$515$	−11.5913	−0.510775
$516$	−3.06525	−0.134940
$517$	4.10725	0.180637
$518$	0	0
$519$	−11.3768	−0.499388
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−6.52608	−0.285639
$523$	−25.2667	−1.10483	−0.552417	−	0.833568i	$-0.686294\pi$
$523$	−25.2667	−1.10483	−0.552417	+	0.833568i	$0.686294\pi$
$524$	7.37683	0.322258
$525$	0	0
$526$	10.8377	0.472545
$527$	−8.02617	−0.349626
$528$	1.00000	0.0435194
$529$	−0.623166	−0.0270942
$530$	35.3507	1.53553
$531$	−12.5261	−0.543586
$532$	0	0
$533$	0	0
$534$	12.8377	0.555540
$535$	48.1782	2.08292
$536$	10.1492	0.438381
$537$	−24.6101	−1.06200
$538$	17.8797	0.770847
$539$	0	0
$540$	−4.41883	−0.190156
$541$	−4.41883	−0.189980	−0.0949902	−	0.995478i	$-0.530282\pi$
$541$	−4.41883	−0.189980	−0.0949902	+	0.995478i	$0.530282\pi$
$542$	−8.83767	−0.379610
$543$	−17.4608	−0.749316
$544$	2.37683	0.101906
$545$	−52.1231	−2.23271
$546$	0	0
$547$	−27.6622	−1.18275	−0.591376	−	0.806396i	$-0.701415\pi$
$547$	−27.6622	−1.18275	−0.591376	+	0.806396i	$0.701415\pi$
$548$	−8.83767	−0.377526
$549$	3.79567	0.161995
$550$	−14.5261	−0.619394
$551$	24.0709	1.02546
$552$	−4.73042	−0.201340
$553$	0	0
$554$	−16.0000	−0.679775
$555$	−33.9738	−1.44211
$556$	1.47392	0.0625080
$557$	−7.98691	−0.338416	−0.169208	−	0.985580i	$-0.554121\pi$
$557$	−7.98691	−0.338416	−0.169208	+	0.985580i	$0.554121\pi$
$558$	−3.37683	−0.142953
$559$	0	0
$560$	0	0
$561$	2.37683	0.100350
$562$	−15.9217	−0.671615
$563$	−27.3507	−1.15269	−0.576346	−	0.817205i	$-0.695522\pi$
$563$	−27.3507	−1.15269	−0.576346	+	0.817205i	$0.695522\pi$
$564$	4.10725	0.172946
$565$	23.7593	0.999562
$566$	−18.0840	−0.760127
$567$	0	0
$568$	0.934749	0.0392212
$569$	−29.4477	−1.23451	−0.617257	−	0.786762i	$-0.711756\pi$
$569$	−29.4477	−1.23451	−0.617257	+	0.786762i	$0.711756\pi$
$570$	16.2985	0.682669
$571$	−14.5261	−0.607898	−0.303949	−	0.952688i	$-0.598305\pi$
$571$	−14.5261	−0.607898	−0.303949	+	0.952688i	$0.598305\pi$
$572$	0	0
$573$	16.8377	0.703404
$574$	0	0
$575$	68.7144	2.86559
$576$	1.00000	0.0416667
$577$	13.3956	0.557665	0.278833	−	0.960340i	$-0.410053\pi$
$577$	13.3956	0.557665	0.278833	+	0.960340i	$0.410053\pi$
$578$	−11.3507	−0.472125
$579$	−5.46083	−0.226944
$580$	−28.8377	−1.19742
$581$	0	0
$582$	−7.00000	−0.290159
$583$	−8.00000	−0.331326
$584$	−7.46083	−0.308731
$585$	0	0
$586$	25.1492	1.03891
$587$	−11.2928	−0.466105	−0.233053	−	0.972464i	$-0.574871\pi$
$587$	−11.2928	−0.466105	−0.233053	+	0.972464i	$0.574871\pi$
$588$	0	0
$589$	12.4552	0.513206
$590$	−55.3507	−2.27875
$591$	9.68842	0.398528
$592$	7.68842	0.315992
$593$	−19.5782	−0.803982	−0.401991	−	0.915644i	$-0.631682\pi$
$593$	−19.5782	−0.803982	−0.401991	+	0.915644i	$0.631682\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−9.36375	−0.383554
$597$	−20.2985	−0.830762
$598$	0	0
$599$	10.9318	0.446662	0.223331	−	0.974743i	$-0.428307\pi$
$599$	10.9318	0.446662	0.223331	+	0.974743i	$0.428307\pi$
$600$	−14.5261	−0.593025
$601$	−41.2202	−1.68141	−0.840703	−	0.541497i	$-0.817858\pi$
$601$	−41.2202	−1.68141	−0.840703	+	0.541497i	$0.817858\pi$
$602$	0	0
$603$	10.1492	0.413309
$604$	−13.5681	−0.552077
$605$	4.41883	0.179651
$606$	15.3637	0.624110
$607$	−31.3405	−1.27207	−0.636036	−	0.771660i	$-0.719427\pi$
$607$	−31.3405	−1.27207	−0.636036	+	0.771660i	$0.719427\pi$
$608$	−3.68842	−0.149585
$609$	0	0
$610$	16.7724	0.679095
$611$	0	0
$612$	2.37683	0.0960778
$613$	17.0885	0.690198	0.345099	−	0.938566i	$-0.387845\pi$
$613$	17.0885	0.690198	0.345099	+	0.938566i	$0.387845\pi$
$614$	−3.86950	−0.156160
$615$	−53.6855	−2.16481
$616$	0	0
$617$	−6.96817	−0.280528	−0.140264	−	0.990114i	$-0.544795\pi$
$617$	−6.96817	−0.280528	−0.140264	+	0.990114i	$0.544795\pi$
$618$	2.62317	0.105519
$619$	−0.0187473	−0.000753517 0	−0.000376759	−	1.00000i	$-0.500120\pi$
$619$	−0.0187473	−0.000753517 0	−0.000376759		1.00000i	$0.500120\pi$
$620$	−14.9217	−0.599268
$621$	−4.73042	−0.189825
$622$	−13.5681	−0.544030
$623$	0	0
$624$	0	0
$625$	113.377	4.53507
$626$	2.52608	0.100963
$627$	−3.68842	−0.147301
$628$	−18.7406	−0.747831
$629$	18.2741	0.728636
$630$	0	0
$631$	−5.78550	−0.230317	−0.115159	−	0.993347i	$-0.536738\pi$
$631$	−5.78550	−0.230317	−0.115159	+	0.993347i	$0.536738\pi$
$632$	−0.418833	−0.0166603
$633$	−22.5130	−0.894811
$634$	−20.0102	−0.794705
$635$	3.22759	0.128083
$636$	−8.00000	−0.317221
$637$	0	0
$638$	6.52608	0.258370
$639$	0.934749	0.0369781
$640$	4.41883	0.174670
$641$	15.3768	0.607348	0.303674	−	0.952776i	$-0.401787\pi$
$641$	15.3768	0.607348	0.303674	+	0.952776i	$0.401787\pi$
$642$	−10.9029	−0.430304
$643$	−35.8058	−1.41204	−0.706022	−	0.708190i	$-0.749512\pi$
$643$	−35.8058	−1.41204	−0.706022	+	0.708190i	$0.749512\pi$
$644$	0	0
$645$	−13.5448	−0.533327
$646$	−8.76675	−0.344923
$647$	9.25650	0.363910	0.181955	−	0.983307i	$-0.441757\pi$
$647$	9.25650	0.363910	0.181955	+	0.983307i	$0.441757\pi$
$648$	1.00000	0.0392837
$649$	12.5261	0.491692
$650$	0	0
$651$	0	0
$652$	18.7724	0.735185
$653$	24.8478	0.972371	0.486185	−	0.873856i	$-0.338388\pi$
$653$	24.8478	0.972371	0.486185	+	0.873856i	$0.338388\pi$
$654$	11.7957	0.461247
$655$	32.5970	1.27367
$656$	12.1492	0.474348
$657$	−7.46083	−0.291075
$658$	0	0
$659$	−6.60442	−0.257272	−0.128636	−	0.991692i	$-0.541060\pi$
$659$	−6.60442	−0.257272	−0.128636	+	0.991692i	$0.541060\pi$
$660$	4.41883	0.172003
$661$	−13.2332	−0.514714	−0.257357	−	0.966316i	$-0.582852\pi$
$661$	−13.2332	−0.514714	−0.257357	+	0.966316i	$0.582852\pi$
$662$	−16.1492	−0.627658
$663$	0	0
$664$	−2.14925	−0.0834070
$665$	0	0
$666$	7.68842	0.297920
$667$	−30.8711	−1.19533
$668$	−23.8898	−0.924325
$669$	7.46083	0.288452
$670$	44.8478	1.73262
$671$	−3.79567	−0.146530
$672$	0	0
$673$	0.130501	0.00503045	0.00251523	−	0.999997i	$-0.499199\pi$
$673$	0.130501	0.00503045	0.00251523	+	0.999997i	$0.499199\pi$
$674$	23.1362	0.891172
$675$	−14.5261	−0.559109
$676$	−13.0000	−0.500000
$677$	28.1174	1.08064	0.540320	−	0.841460i	$-0.318303\pi$
$677$	28.1174	1.08064	0.540320	+	0.841460i	$0.318303\pi$
$678$	−5.37683	−0.206496
$679$	0	0
$680$	10.5028	0.402765
$681$	2.77241	0.106239
$682$	3.37683	0.129306
$683$	15.9029	0.608508	0.304254	−	0.952591i	$-0.401593\pi$
$683$	15.9029	0.608508	0.304254	+	0.952591i	$0.401593\pi$
$684$	−3.68842	−0.141030
$685$	−39.0522	−1.49211
$686$	0	0
$687$	2.53917	0.0968753
$688$	3.06525	0.116862
$689$	0	0
$690$	−20.9029	−0.795760
$691$	−15.9813	−0.607956	−0.303978	−	0.952679i	$-0.598315\pi$
$691$	−15.9813	−0.607956	−0.303978	+	0.952679i	$0.598315\pi$
$692$	11.3768	0.432482
$693$	0	0
$694$	15.6566	0.594316
$695$	6.51300	0.247052
$696$	6.52608	0.247371
$697$	28.8767	1.09378
$698$	−20.0102	−0.757396
$699$	24.0522	0.909736
$700$	0	0
$701$	13.9869	0.528278	0.264139	−	0.964485i	$-0.414912\pi$
$701$	13.9869	0.528278	0.264139	+	0.964485i	$0.414912\pi$
$702$	0	0
$703$	−28.3581	−1.06955
$704$	−1.00000	−0.0376889
$705$	18.1492	0.683540
$706$	7.46083	0.280792
$707$	0	0
$708$	12.5261	0.470759
$709$	7.64191	0.286998	0.143499	−	0.989650i	$-0.454165\pi$
$709$	7.64191	0.286998	0.143499	+	0.989650i	$0.454165\pi$
$710$	4.13050	0.155015
$711$	−0.418833	−0.0157075
$712$	−12.8377	−0.481112
$713$	−15.9738	−0.598225
$714$	0	0
$715$	0	0
$716$	24.6101	0.919722
$717$	−7.59133	−0.283504
$718$	−11.2463	−0.419709
$719$	−19.4376	−0.724899	−0.362450	−	0.932003i	$-0.618060\pi$
$719$	−19.4376	−0.724899	−0.362450	+	0.932003i	$0.618060\pi$
$720$	4.41883	0.164680
$721$	0	0
$722$	−5.39558	−0.200803
$723$	−16.2985	−0.606148
$724$	17.4608	0.648927
$725$	−94.7984	−3.52072
$726$	−1.00000	−0.0371135
$727$	−44.9217	−1.66605	−0.833026	−	0.553234i	$-0.813394\pi$
$727$	−44.9217	−1.66605	−0.833026	+	0.553234i	$0.813394\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−32.9682	−1.22021
$731$	7.28559	0.269467
$732$	−3.79567	−0.140292
$733$	25.0420	0.924947	0.462474	−	0.886633i	$-0.346962\pi$
$733$	25.0420	0.924947	0.462474	+	0.886633i	$0.346962\pi$
$734$	−5.91600	−0.218364
$735$	0	0
$736$	4.73042	0.174365
$737$	−10.1492	−0.373852
$738$	12.1492	0.447220
$739$	−6.62317	−0.243637	−0.121819	−	0.992552i	$-0.538873\pi$
$739$	−6.62317	−0.243637	−0.121819	+	0.992552i	$0.538873\pi$
$740$	33.9738	1.24890
$741$	0	0
$742$	0	0
$743$	−39.3972	−1.44534	−0.722671	−	0.691192i	$-0.757086\pi$
$743$	−39.3972	−1.44534	−0.722671	+	0.691192i	$0.757086\pi$
$744$	3.37683	0.123801
$745$	−41.3768	−1.51593
$746$	−23.1260	−0.846703
$747$	−2.14925	−0.0786369
$748$	−2.37683	−0.0869056
$749$	0	0
$750$	−42.0942	−1.53706
$751$	−10.7072	−0.390710	−0.195355	−	0.980733i	$-0.562586\pi$
$751$	−10.7072	−0.390710	−0.195355	+	0.980733i	$0.562586\pi$
$752$	−4.10725	−0.149776
$753$	8.61008	0.313769
$754$	0	0
$755$	−59.9551	−2.18199
$756$	0	0
$757$	−17.2797	−0.628043	−0.314022	−	0.949416i	$-0.601676\pi$
$757$	−17.2797	−0.628043	−0.314022	+	0.949416i	$0.601676\pi$
$758$	−6.77241	−0.245985
$759$	4.73042	0.171703
$760$	−16.2985	−0.591209
$761$	−32.7087	−1.18569	−0.592846	−	0.805316i	$-0.701996\pi$
$761$	−32.7087	−1.18569	−0.592846	+	0.805316i	$0.701996\pi$
$762$	−0.730416	−0.0264602
$763$	0	0
$764$	−16.8377	−0.609165
$765$	10.5028	0.379731
$766$	25.3637	0.916429
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	12.2145	0.440466	0.220233	−	0.975447i	$-0.429318\pi$
$769$	12.2145	0.440466	0.220233	+	0.975447i	$0.429318\pi$
$770$	0	0
$771$	−0.837665	−0.0301678
$772$	5.46083	0.196540
$773$	0.204334	0.00734937	0.00367468	−	0.999993i	$-0.498830\pi$
$773$	0.204334	0.00734937	0.00367468	+	0.999993i	$0.498830\pi$
$774$	3.06525	0.110178
$775$	−49.0522	−1.76201
$776$	7.00000	0.251285
$777$	0	0
$778$	25.2565	0.905489
$779$	−44.8115	−1.60554
$780$	0	0
$781$	−0.934749	−0.0334480
$782$	11.2434	0.402064
$783$	6.52608	0.233223
$784$	0	0
$785$	−82.8115	−2.95567
$786$	−7.37683	−0.263123
$787$	25.5579	0.911041	0.455521	−	0.890225i	$-0.349453\pi$
$787$	25.5579	0.911041	0.455521	+	0.890225i	$0.349453\pi$
$788$	−9.68842	−0.345136
$789$	−10.8377	−0.385831
$790$	−1.85075	−0.0658468
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	−13.4739	−0.478171
$795$	−35.3507	−1.25376
$796$	20.2985	0.719461
$797$	30.0477	1.06434	0.532171	−	0.846637i	$-0.321376\pi$
$797$	30.0477	1.06434	0.532171	+	0.846637i	$0.321376\pi$
$798$	0	0
$799$	−9.76225	−0.345364
$800$	14.5261	0.513575
$801$	−12.8377	−0.453597
$802$	33.8898	1.19669
$803$	7.46083	0.263287
$804$	−10.1492	−0.357936
$805$	0	0
$806$	0	0
$807$	−17.8797	−0.629394
$808$	−15.3637	−0.540495
$809$	2.34342	0.0823901	0.0411951	−	0.999151i	$-0.486883\pi$
$809$	2.34342	0.0823901	0.0411951	+	0.999151i	$0.486883\pi$
$810$	4.41883	0.155262
$811$	−11.8695	−0.416794	−0.208397	−	0.978044i	$-0.566825\pi$
$811$	−11.8695	−0.416794	−0.208397	+	0.978044i	$0.566825\pi$
$812$	0	0
$813$	8.83767	0.309950
$814$	−7.68842	−0.269479
$815$	82.9522	2.90569
$816$	−2.37683	−0.0832058
$817$	−11.3059	−0.395544
$818$	−4.21450	−0.147357
$819$	0	0
$820$	53.6855	1.87478
$821$	−37.4812	−1.30810	−0.654051	−	0.756451i	$-0.726932\pi$
$821$	−37.4812	−1.30810	−0.654051	+	0.756451i	$0.726932\pi$
$822$	8.83767	0.308249
$823$	1.29284	0.0450654	0.0225327	−	0.999746i	$-0.492827\pi$
$823$	1.29284	0.0450654	0.0225327	+	0.999746i	$0.492827\pi$
$824$	−2.62317	−0.0913823
$825$	14.5261	0.505733
$826$	0	0
$827$	43.3319	1.50680	0.753399	−	0.657563i	$-0.228413\pi$
$827$	43.3319	1.50680	0.753399	+	0.657563i	$0.228413\pi$
$828$	4.73042	0.164393
$829$	40.1174	1.39334	0.696668	−	0.717394i	$-0.254665\pi$
$829$	40.1174	1.39334	0.696668	+	0.717394i	$0.254665\pi$
$830$	−9.49717	−0.329652
$831$	16.0000	0.555034
$832$	0	0
$833$	0	0
$834$	−1.47392	−0.0510376
$835$	−105.565	−3.65323
$836$	3.68842	0.127567
$837$	3.37683	0.116720
$838$	−22.3116	−0.770741
$839$	54.6912	1.88815	0.944074	−	0.329733i	$-0.106959\pi$
$839$	54.6912	1.88815	0.944074	+	0.329733i	$0.106959\pi$
$840$	0	0
$841$	13.5897	0.468612
$842$	18.3116	0.631059
$843$	15.9217	0.548371
$844$	22.5130	0.774929
$845$	−57.4448	−1.97616
$846$	−4.10725	−0.141210
$847$	0	0
$848$	8.00000	0.274721
$849$	18.0840	0.620641
$850$	34.5261	1.18423
$851$	36.3694	1.24673
$852$	−0.934749	−0.0320240
$853$	−20.8275	−0.713120	−0.356560	−	0.934272i	$-0.616050\pi$
$853$	−20.8275	−0.713120	−0.356560	+	0.934272i	$0.616050\pi$
$854$	0	0
$855$	−16.2985	−0.557397
$856$	10.9029	0.372654
$857$	−36.8433	−1.25854	−0.629272	−	0.777185i	$-0.716647\pi$
$857$	−36.8433	−1.25854	−0.629272	+	0.777185i	$0.716647\pi$
$858$	0	0
$859$	38.5782	1.31627	0.658136	−	0.752899i	$-0.271345\pi$
$859$	38.5782	1.31627	0.658136	+	0.752899i	$0.271345\pi$
$860$	13.5448	0.461875
$861$	0	0
$862$	−28.8377	−0.982215
$863$	−17.2565	−0.587418	−0.293709	−	0.955895i	$-0.594890\pi$
$863$	−17.2565	−0.587418	−0.293709	+	0.955895i	$0.594890\pi$
$864$	−1.00000	−0.0340207
$865$	50.2723	1.70931
$866$	7.62317	0.259046
$867$	11.3507	0.385489
$868$	0	0
$869$	0.418833	0.0142079
$870$	28.8377	0.977688
$871$	0	0
$872$	−11.7957	−0.399452
$873$	7.00000	0.236914
$874$	−17.4477	−0.590179
$875$	0	0
$876$	7.46083	0.252078
$877$	0.250837	0.00847016	0.00423508	−	0.999991i	$-0.498652\pi$
$877$	0.250837	0.00847016	0.00423508	+	0.999991i	$0.498652\pi$
$878$	−33.3739	−1.12632
$879$	−25.1492	−0.848263
$880$	−4.41883	−0.148959
$881$	16.2985	0.549110	0.274555	−	0.961571i	$-0.411469\pi$
$881$	16.2985	0.549110	0.274555	+	0.961571i	$0.411469\pi$
$882$	0	0
$883$	−33.9551	−1.14268	−0.571340	−	0.820714i	$-0.693576\pi$
$883$	−33.9551	−1.14268	−0.571340	+	0.820714i	$0.693576\pi$
$884$	0	0
$885$	55.3507	1.86059
$886$	−4.85075	−0.162964
$887$	19.4608	0.653431	0.326715	−	0.945123i	$-0.394058\pi$
$887$	19.4608	0.653431	0.326715	+	0.945123i	$0.394058\pi$
$888$	−7.68842	−0.258006
$889$	0	0
$890$	−56.7275	−1.90151
$891$	−1.00000	−0.0335013
$892$	−7.46083	−0.249807
$893$	15.1492	0.506950
$894$	9.36375	0.313171
$895$	108.748	3.63504
$896$	0	0
$897$	0	0
$898$	14.3450	0.478699
$899$	22.0375	0.734992
$900$	14.5261	0.484203
$901$	19.0147	0.633470
$902$	−12.1492	−0.404526
$903$	0	0
$904$	5.37683	0.178831
$905$	77.1565	2.56477
$906$	13.5681	0.450769
$907$	0.772415	0.0256476	0.0128238	−	0.999918i	$-0.495918\pi$
$907$	0.772415	0.0256476	0.0128238	+	0.999918i	$0.495918\pi$
$908$	−2.77241	−0.0920058
$909$	−15.3637	−0.509583
$910$	0	0
$911$	−8.94491	−0.296358	−0.148179	−	0.988961i	$-0.547341\pi$
$911$	−8.94491	−0.296358	−0.148179	+	0.988961i	$0.547341\pi$
$912$	3.68842	0.122136
$913$	2.14925	0.0711297
$914$	−25.1827	−0.832969
$915$	−16.7724	−0.554479
$916$	−2.53917	−0.0838965
$917$	0	0
$918$	−2.37683	−0.0784472
$919$	58.8812	1.94231	0.971157	−	0.238443i	$-0.0766369\pi$
$919$	58.8812	1.94231	0.971157	+	0.238443i	$0.0766369\pi$
$920$	20.9029	0.689149
$921$	3.86950	0.127504
$922$	13.0187	0.428750
$923$	0	0
$924$	0	0
$925$	111.683	3.67210
$926$	−14.2145	−0.467117
$927$	−2.62317	−0.0861561
$928$	−6.52608	−0.214229
$929$	−22.5392	−0.739486	−0.369743	−	0.929134i	$-0.620554\pi$
$929$	−22.5392	−0.739486	−0.369743	+	0.929134i	$0.620554\pi$
$930$	14.9217	0.489301
$931$	0	0
$932$	−24.0522	−0.787855
$933$	13.5681	0.444199
$934$	−2.39558	−0.0783858
$935$	−10.5028	−0.343479
$936$	0	0
$937$	−12.9478	−0.422987	−0.211494	−	0.977379i	$-0.567833\pi$
$937$	−12.9478	−0.422987	−0.211494	+	0.977379i	$0.567833\pi$
$938$	0	0
$939$	−2.52608	−0.0824356
$940$	−18.1492	−0.591963
$941$	48.8246	1.59164	0.795818	−	0.605536i	$-0.207041\pi$
$941$	48.8246	1.59164	0.795818	+	0.605536i	$0.207041\pi$
$942$	18.7406	0.610601
$943$	57.4710	1.87151
$944$	−12.5261	−0.407689
$945$	0	0
$946$	−3.06525	−0.0996599
$947$	−31.1231	−1.01136	−0.505682	−	0.862720i	$-0.668759\pi$
$947$	−31.1231	−1.01136	−0.505682	+	0.862720i	$0.668759\pi$
$948$	0.418833	0.0136031
$949$	0	0
$950$	−53.5782	−1.73831
$951$	20.0102	0.648874
$952$	0	0
$953$	16.2797	0.527353	0.263676	−	0.964611i	$-0.415065\pi$
$953$	16.2797	0.527353	0.263676	+	0.964611i	$0.415065\pi$
$954$	8.00000	0.259010
$955$	−74.4028	−2.40762
$956$	7.59133	0.245521
$957$	−6.52608	−0.210958
$958$	10.8377	0.350149
$959$	0	0
$960$	−4.41883	−0.142617
$961$	−19.5970	−0.632161
$962$	0	0
$963$	10.9029	0.351342
$964$	16.2985	0.524939
$965$	24.1305	0.776788
$966$	0	0
$967$	3.91308	0.125836	0.0629181	−	0.998019i	$-0.479959\pi$
$967$	3.91308	0.125836	0.0629181	+	0.998019i	$0.479959\pi$
$968$	1.00000	0.0321412
$969$	8.76675	0.281629
$970$	30.9318	0.993161
$971$	39.3132	1.26162	0.630810	−	0.775938i	$-0.282723\pi$
$971$	39.3132	1.26162	0.630810	+	0.775938i	$0.282723\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−34.6435	−1.11005
$975$	0	0
$976$	3.79567	0.121496
$977$	−23.3507	−0.747054	−0.373527	−	0.927619i	$-0.621852\pi$
$977$	−23.3507	−0.747054	−0.373527	+	0.927619i	$0.621852\pi$
$978$	−18.7724	−0.600276
$979$	12.8377	0.410294
$980$	0	0
$981$	−11.7957	−0.376607
$982$	38.1492	1.21739
$983$	−4.51592	−0.144035	−0.0720177	−	0.997403i	$-0.522944\pi$
$983$	−4.51592	−0.144035	−0.0720177	+	0.997403i	$0.522944\pi$
$984$	−12.1492	−0.387304
$985$	−42.8115	−1.36409
$986$	−15.5114	−0.493984
$987$	0	0
$988$	0	0
$989$	14.4999	0.461070
$990$	−4.41883	−0.140440
$991$	59.9273	1.90365	0.951827	−	0.306635i	$-0.0992032\pi$
$991$	59.9273	1.90365	0.951827	+	0.306635i	$0.0992032\pi$
$992$	−3.37683	−0.107215
$993$	16.1492	0.512481
$994$	0	0
$995$	89.6957	2.84354
$996$	2.14925	0.0681015
$997$	−21.5073	−0.681144	−0.340572	−	0.940218i	$-0.610621\pi$
$997$	−21.5073	−0.681144	−0.340572	+	0.940218i	$0.610621\pi$
$998$	−0.623166	−0.0197260
$999$	−7.68842	−0.243251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bf.1.3		3
3.2	odd	2		9702.2.a.dv.1.1		3
7.2	even	3		462.2.i.g.67.1	✓	6
7.4	even	3		462.2.i.g.331.1	yes	6
7.6	odd	2		3234.2.a.bh.1.1		3
21.2	odd	6		1386.2.k.v.991.3		6
21.11	odd	6		1386.2.k.v.793.3		6
21.20	even	2		9702.2.a.dw.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.g.67.1	✓	6	7.2	even	3
462.2.i.g.331.1	yes	6	7.4	even	3
1386.2.k.v.793.3		6	21.11	odd	6
1386.2.k.v.991.3		6	21.2	odd	6
3234.2.a.bf.1.3		3	1.1	even	1	trivial
3234.2.a.bh.1.1		3	7.6	odd	2
9702.2.a.dv.1.1		3	3.2	odd	2
9702.2.a.dw.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.41883 q^{10} -1.00000 q^{11} -1.00000 q^{12} -4.41883 q^{15} +1.00000 q^{16} +2.37683 q^{17} +1.00000 q^{18} -3.68842 q^{19} +4.41883 q^{20} -1.00000 q^{22} +4.73042 q^{23} -1.00000 q^{24} +14.5261 q^{25} -1.00000 q^{27} -6.52608 q^{29} -4.41883 q^{30} -3.37683 q^{31} +1.00000 q^{32} +1.00000 q^{33} +2.37683 q^{34} +1.00000 q^{36} +7.68842 q^{37} -3.68842 q^{38} +4.41883 q^{40} +12.1492 q^{41} +3.06525 q^{43} -1.00000 q^{44} +4.41883 q^{45} +4.73042 q^{46} -4.10725 q^{47} -1.00000 q^{48} +14.5261 q^{50} -2.37683 q^{51} +8.00000 q^{53} -1.00000 q^{54} -4.41883 q^{55} +3.68842 q^{57} -6.52608 q^{58} -12.5261 q^{59} -4.41883 q^{60} +3.79567 q^{61} -3.37683 q^{62} +1.00000 q^{64} +1.00000 q^{66} +10.1492 q^{67} +2.37683 q^{68} -4.73042 q^{69} +0.934749 q^{71} +1.00000 q^{72} -7.46083 q^{73} +7.68842 q^{74} -14.5261 q^{75} -3.68842 q^{76} -0.418833 q^{79} +4.41883 q^{80} +1.00000 q^{81} +12.1492 q^{82} -2.14925 q^{83} +10.5028 q^{85} +3.06525 q^{86} +6.52608 q^{87} -1.00000 q^{88} -12.8377 q^{89} +4.41883 q^{90} +4.73042 q^{92} +3.37683 q^{93} -4.10725 q^{94} -16.2985 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 3 q^{12} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 15 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} + 3 q^{33} + 3 q^{34} + 3 q^{36} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} - 3 q^{54} + 9 q^{57} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} + 3 q^{66} + 6 q^{67} + 3 q^{68} - 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} - 15 q^{75} - 9 q^{76} + 12 q^{79} + 3 q^{81} + 12 q^{82} + 18 q^{83} + 3 q^{86} - 9 q^{87} - 3 q^{88} - 12 q^{89} + 3 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{96} + 21 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 3 * q^11 - 3 * q^12 + 3 * q^16 + 3 * q^17 + 3 * q^18 - 9 * q^19 - 3 * q^22 + 3 * q^23 - 3 * q^24 + 15 * q^25 - 3 * q^27 + 9 * q^29 - 6 * q^31 + 3 * q^32 + 3 * q^33 + 3 * q^34 + 3 * q^36 + 21 * q^37 - 9 * q^38 + 12 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 3 * q^47 - 3 * q^48 + 15 * q^50 - 3 * q^51 + 24 * q^53 - 3 * q^54 + 9 * q^57 + 9 * q^58 - 9 * q^59 - 6 * q^61 - 6 * q^62 + 3 * q^64 + 3 * q^66 + 6 * q^67 + 3 * q^68 - 3 * q^69 + 9 * q^71 + 3 * q^72 + 21 * q^74 - 15 * q^75 - 9 * q^76 + 12 * q^79 + 3 * q^81 + 12 * q^82 + 18 * q^83 + 3 * q^86 - 9 * q^87 - 3 * q^88 - 12 * q^89 + 3 * q^92 + 6 * q^93 + 3 * q^94 - 3 * q^96 + 21 * q^97 - 3 * q^99

Introduction

L-functions

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