LMFDB - Embedded newform 3330.2.h.m.2071.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(2071,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3330.2071");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.279290944.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 26x^{4} + 169x^{2} + 16 $ x^6 + 26x^4 + 169x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2071.5
Root			$0.309984i$ of defining polynomial
Character	$\chi$	$=$	3330.2071
Dual form			3330.2.h.m.2071.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.30998 q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.30998 q^{7} -1.00000i q^{8} -1.00000 q^{10} -0.690016 q^{11} +2.69002i q^{13} -1.30998i q^{14} +1.00000 q^{16} +6.90391i q^{17} -2.69002i q^{19} -1.00000i q^{20} -0.690016i q^{22} +8.90391i q^{23} -1.00000 q^{25} -2.69002 q^{26} +1.30998 q^{28} -2.00000i q^{29} -9.59393i q^{31} +1.00000i q^{32} -6.90391 q^{34} -1.30998i q^{35} +(4.10695 - 4.48698i) q^{37} +2.69002 q^{38} +1.00000 q^{40} +2.00000 q^{41} +8.21389i q^{43} +0.690016 q^{44} -8.90391 q^{46} +4.61997 q^{47} -5.28394 q^{49} -1.00000i q^{50} -2.69002i q^{52} -0.690016 q^{53} -0.690016i q^{55} +1.30998i q^{56} +2.00000 q^{58} -6.21389i q^{59} +2.00000i q^{61} +9.59393 q^{62} -1.00000 q^{64} -2.69002 q^{65} -1.23993 q^{67} -6.90391i q^{68} +1.30998 q^{70} -13.8078 q^{71} -12.2839 q^{73} +(4.48698 + 4.10695i) q^{74} +2.69002i q^{76} +0.903910 q^{77} +1.59393i q^{79} +1.00000i q^{80} +2.00000i q^{82} -10.2839 q^{83} -6.90391 q^{85} -8.21389 q^{86} +0.690016i q^{88} +5.92995i q^{89} -3.52388i q^{91} -8.90391i q^{92} +4.61997i q^{94} +2.69002 q^{95} -3.38003i q^{97} -5.28394i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 6 q^{7}+O(q^{10}) $ 6 * q - 6 * q^4 - 6 * q^7	$ 6 q - 6 q^{4} - 6 q^{7} - 6 q^{10} - 6 q^{11} + 6 q^{16} - 6 q^{25} - 18 q^{26} + 6 q^{28} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 6 q^{40} + 12 q^{41} + 6 q^{44} - 2 q^{46} + 24 q^{47} + 16 q^{49} - 6 q^{53} + 12 q^{58} + 8 q^{62} - 6 q^{64} - 18 q^{65} + 6 q^{70} + 20 q^{71} - 26 q^{73} + 4 q^{74} - 46 q^{77} - 14 q^{83} + 10 q^{85} + 4 q^{86} + 18 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 - 6 * q^7 - 6 * q^10 - 6 * q^11 + 6 * q^16 - 6 * q^25 - 18 * q^26 + 6 * q^28 + 10 * q^34 - 2 * q^37 + 18 * q^38 + 6 * q^40 + 12 * q^41 + 6 * q^44 - 2 * q^46 + 24 * q^47 + 16 * q^49 - 6 * q^53 + 12 * q^58 + 8 * q^62 - 6 * q^64 - 18 * q^65 + 6 * q^70 + 20 * q^71 - 26 * q^73 + 4 * q^74 - 46 * q^77 - 14 * q^83 + 10 * q^85 + 4 * q^86 + 18 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−1.30998			−0.495127			−0.247564	−	0.968872i	$-0.579630\pi$
$7$	−1.30998			−0.495127			−0.247564	+	0.968872i	$0.579630\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.00000			−0.316228
$11$	−0.690016			−0.208048			−0.104024	−	0.994575i	$-0.533172\pi$
$11$	−0.690016			−0.208048			−0.104024	+	0.994575i	$0.533172\pi$
$12$		0			0
$13$			2.69002i			0.746076i	0.927816	+	0.373038i	$0.121684\pi$
$13$			2.69002i			0.746076i	−0.927816	+	0.373038i	$0.878316\pi$
$14$		−	1.30998i		−	0.350108i
$15$		0			0
$16$	1.00000			0.250000
$17$			6.90391i			1.67444i	0.546863	+	0.837222i	$0.315822\pi$
$17$			6.90391i			1.67444i	−0.546863	+	0.837222i	$0.684178\pi$
$18$		0			0
$19$		−	2.69002i		−	0.617132i	−0.951203	−	0.308566i	$-0.900151\pi$
$19$		−	2.69002i		−	0.617132i	0.951203	−	0.308566i	$-0.0998491\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$		−	0.690016i		−	0.147112i
$23$			8.90391i			1.85659i	0.371840	+	0.928297i	$0.378727\pi$
$23$			8.90391i			1.85659i	−0.371840	+	0.928297i	$0.621273\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−2.69002			−0.527556
$27$		0			0
$28$	1.30998			0.247564
$29$		−	2.00000i		−	0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$		−	2.00000i		−	0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$		0			0
$31$		−	9.59393i		−	1.72312i	−0.507656	−	0.861560i	$-0.669488\pi$
$31$		−	9.59393i		−	1.72312i	0.507656	−	0.861560i	$-0.330512\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−6.90391			−1.18401
$35$		−	1.30998i		−	0.221428i
$36$		0			0
$37$	4.10695	−	4.48698i	0.675178	−	0.737655i
$37$	4.10695	−	4.48698i	0.675178	−	0.737655i
$38$	2.69002			0.436378
$39$		0			0
$40$	1.00000			0.158114
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$			8.21389i			1.25261i	0.779579	+	0.626304i	$0.215433\pi$
$43$			8.21389i			1.25261i	−0.779579	+	0.626304i	$0.784567\pi$
$44$	0.690016			0.104024
$45$		0			0
$46$	−8.90391			−1.31281
$47$	4.61997			0.673891			0.336946	−	0.941524i	$-0.390606\pi$
$47$	4.61997			0.673891			0.336946	+	0.941524i	$0.390606\pi$
$48$		0			0
$49$	−5.28394			−0.754849
$50$		−	1.00000i		−	0.141421i
$51$		0			0
$52$		−	2.69002i		−	0.373038i
$53$	−0.690016			−0.0947810			−0.0473905	−	0.998876i	$-0.515091\pi$
$53$	−0.690016			−0.0947810			−0.0473905	+	0.998876i	$0.515091\pi$
$54$		0			0
$55$		−	0.690016i		−	0.0930418i
$56$			1.30998i			0.175054i
$57$		0			0
$58$	2.00000			0.262613
$59$		−	6.21389i		−	0.808980i	−0.914542	−	0.404490i	$-0.867449\pi$
$59$		−	6.21389i		−	0.808980i	0.914542	−	0.404490i	$-0.132551\pi$
$60$		0			0
$61$			2.00000i			0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$			2.00000i			0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$	9.59393			1.21843
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−2.69002			−0.333655
$66$		0			0
$67$	−1.23993			−0.151482			−0.0757410	−	0.997128i	$-0.524132\pi$
$67$	−1.23993			−0.151482			−0.0757410	+	0.997128i	$0.524132\pi$
$68$		−	6.90391i		−	0.837222i
$69$		0			0
$70$	1.30998			0.156573
$71$	−13.8078			−1.63869			−0.819343	−	0.573303i	$-0.805662\pi$
$71$	−13.8078			−1.63869			−0.819343	+	0.573303i	$0.805662\pi$
$72$		0			0
$73$	−12.2839			−1.43773			−0.718863	−	0.695151i	$-0.755337\pi$
$73$	−12.2839			−1.43773			−0.718863	+	0.695151i	$0.755337\pi$
$74$	4.48698	+	4.10695i	0.521601	+	0.477423i
$75$		0			0
$76$			2.69002i			0.308566i
$77$	0.903910			0.103010
$78$		0			0
$79$			1.59393i			0.179331i	0.995972	+	0.0896654i	$0.0285798\pi$
$79$			1.59393i			0.179331i	−0.995972	+	0.0896654i	$0.971420\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$			2.00000i			0.220863i
$83$	−10.2839			−1.12881			−0.564405	−	0.825498i	$-0.690894\pi$
$83$	−10.2839			−1.12881			−0.564405	+	0.825498i	$0.690894\pi$
$84$		0			0
$85$	−6.90391			−0.748834
$86$	−8.21389			−0.885727
$87$		0			0
$88$			0.690016i			0.0735560i
$89$			5.92995i			0.628574i	0.949328	+	0.314287i	$0.101765\pi$
$89$			5.92995i			0.628574i	−0.949328	+	0.314287i	$0.898235\pi$
$90$		0			0
$91$		−	3.52388i		−	0.369403i
$92$		−	8.90391i		−	0.928297i
$93$		0			0
$94$			4.61997i			0.476513i
$95$	2.69002			0.275990
$96$		0			0
$97$		−	3.38003i		−	0.343190i	−0.985168	−	0.171595i	$-0.945108\pi$
$97$		−	3.38003i		−	0.343190i	0.985168	−	0.171595i	$-0.0548921\pi$
$98$		−	5.28394i		−	0.533759i
$99$		0			0
$100$	1.00000			0.100000
$101$	−8.97396			−0.892942			−0.446471	−	0.894798i	$-0.647319\pi$
$101$	−8.97396			−0.892942			−0.446471	+	0.894798i	$0.647319\pi$
$102$		0			0
$103$		−	19.1879i		−	1.89064i	−0.326151	−	0.945318i	$-0.605752\pi$
$103$		−	19.1879i		−	1.89064i	0.326151	−	0.945318i	$-0.394248\pi$
$104$	2.69002			0.263778
$105$		0			0
$106$		−	0.690016i		−	0.0670203i
$107$	−18.1438			−1.75403			−0.877016	−	0.480461i	$-0.840469\pi$
$107$	−18.1438			−1.75403			−0.877016	+	0.480461i	$0.840469\pi$
$108$		0			0
$109$			17.5239i			1.67848i	0.543759	+	0.839242i	$0.317001\pi$
$109$			17.5239i			1.67848i	−0.543759	+	0.839242i	$0.682999\pi$
$110$	0.690016			0.0657905
$111$		0			0
$112$	−1.30998			−0.123782
$113$			2.00000i			0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$			2.00000i			0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$		0			0
$115$	−8.90391			−0.830294
$116$			2.00000i			0.185695i
$117$		0			0
$118$	6.21389			0.572035
$119$		−	9.04401i		−	0.829063i
$120$		0			0
$121$	−10.5239			−0.956716
$122$	−2.00000			−0.181071
$123$		0			0
$124$			9.59393i			0.861560i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−10.6900			−0.948586			−0.474293	−	0.880367i	$-0.657296\pi$
$127$	−10.6900			−0.948586			−0.474293	+	0.880367i	$0.657296\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		−	2.69002i		−	0.235930i
$131$		−	8.97396i		−	0.784058i	−0.919953	−	0.392029i	$-0.871773\pi$
$131$		−	8.97396i		−	0.784058i	0.919953	−	0.392029i	$-0.128227\pi$
$132$		0			0
$133$			3.52388i			0.305559i
$134$		−	1.23993i		−	0.107114i
$135$		0			0
$136$	6.90391			0.592005
$137$	11.5939			0.990536			0.495268	−	0.868740i	$-0.335070\pi$
$137$	11.5939			0.990536			0.495268	+	0.868740i	$0.335070\pi$
$138$		0			0
$139$	−6.76007			−0.573381			−0.286691	−	0.958023i	$-0.592555\pi$
$139$	−6.76007			−0.573381			−0.286691	+	0.958023i	$0.592555\pi$
$140$			1.30998i			0.110714i
$141$		0			0
$142$		−	13.8078i		−	1.15873i
$143$		−	1.85616i		−	0.155220i
$144$		0			0
$145$	2.00000			0.166091
$146$		−	12.2839i		−	1.01663i
$147$		0			0
$148$	−4.10695	+	4.48698i	−0.337589	+	0.368827i
$149$	−10.2139			−0.836755			−0.418377	−	0.908273i	$-0.637401\pi$
$149$	−10.2139			−0.836755			−0.418377	+	0.908273i	$0.637401\pi$
$150$		0			0
$151$	22.2839			1.81344			0.906721	−	0.421731i	$-0.138577\pi$
$151$	22.2839			1.81344			0.906721	+	0.421731i	$0.138577\pi$
$152$	−2.69002			−0.218189
$153$		0			0
$154$			0.903910i			0.0728392i
$155$	9.59393			0.770603
$156$		0			0
$157$	12.2139			0.974775			0.487387	−	0.873186i	$-0.337950\pi$
$157$	12.2139			0.974775			0.487387	+	0.873186i	$0.337950\pi$
$158$	−1.59393			−0.126806
$159$		0			0
$160$	−1.00000			−0.0790569
$161$		−	11.6640i		−	0.919250i
$162$		0			0
$163$		−	6.07005i		−	0.475443i	−0.971333	−	0.237721i	$-0.923599\pi$
$163$		−	6.07005i		−	0.475443i	0.971333	−	0.237721i	$-0.0764006\pi$
$164$	−2.00000			−0.156174
$165$		0			0
$166$		−	10.2839i		−	0.798189i
$167$		−	8.90391i		−	0.689005i	−0.938785	−	0.344503i	$-0.888048\pi$
$167$		−	8.90391i		−	0.689005i	0.938785	−	0.344503i	$-0.111952\pi$
$168$		0			0
$169$	5.76381			0.443370
$170$		−	6.90391i		−	0.529506i
$171$		0			0
$172$		−	8.21389i		−	0.626304i
$173$	−19.8779			−1.51129			−0.755643	−	0.654983i	$-0.772676\pi$
$173$	−19.8779			−1.51129			−0.755643	+	0.654983i	$0.772676\pi$
$174$		0			0
$175$	1.30998			0.0990254
$176$	−0.690016			−0.0520119
$177$		0			0
$178$	−5.92995			−0.444469
$179$		−	19.5939i		−	1.46452i	−0.681026	−	0.732259i	$-0.738466\pi$
$179$		−	19.5939i		−	1.46452i	0.681026	−	0.732259i	$-0.261534\pi$
$180$		0			0
$181$	−9.18785			−0.682928			−0.341464	−	0.939895i	$-0.610923\pi$
$181$	−9.18785			−0.682928			−0.341464	+	0.939895i	$0.610923\pi$
$182$	3.52388			0.261207
$183$		0			0
$184$	8.90391			0.656405
$185$	4.48698	+	4.10695i	0.329889	+	0.301949i
$186$		0			0
$187$		−	4.76381i		−	0.348364i
$188$	−4.61997			−0.336946
$189$		0			0
$190$			2.69002i			0.195154i
$191$			20.9039i			1.51255i	0.654252	+	0.756277i	$0.272984\pi$
$191$			20.9039i			1.51255i	−0.654252	+	0.756277i	$0.727016\pi$
$192$		0			0
$193$			8.61997i			0.620479i	0.950658	+	0.310239i	$0.100409\pi$
$193$			8.61997i			0.620479i	−0.950658	+	0.310239i	$0.899591\pi$
$194$	3.38003			0.242672
$195$		0			0
$196$	5.28394			0.377425
$197$	12.6900			0.904126			0.452063	−	0.891986i	$-0.350688\pi$
$197$	12.6900			0.904126			0.452063	+	0.891986i	$0.350688\pi$
$198$		0			0
$199$			22.8339i			1.61865i	0.587361	+	0.809325i	$0.300167\pi$
$199$			22.8339i			1.61865i	−0.587361	+	0.809325i	$0.699833\pi$
$200$			1.00000i			0.0707107i
$201$		0			0
$202$		−	8.97396i		−	0.631406i
$203$			2.61997i			0.183886i
$204$		0			0
$205$			2.00000i			0.139686i
$206$	19.1879			1.33688
$207$		0			0
$208$			2.69002i			0.186519i
$209$			1.85616i			0.128393i
$210$		0			0
$211$	−5.38003			−0.370377			−0.185188	−	0.982703i	$-0.559290\pi$
$211$	−5.38003			−0.370377			−0.185188	+	0.982703i	$0.559290\pi$
$212$	0.690016			0.0473905
$213$		0			0
$214$		−	18.1438i		−	1.24029i
$215$	−8.21389			−0.560183
$216$		0			0
$217$			12.5679i			0.853164i
$218$	−17.5239			−1.18687
$219$		0			0
$220$			0.690016i			0.0465209i
$221$	−18.5716			−1.24926
$222$		0			0
$223$	−0.833861			−0.0558395			−0.0279197	−	0.999610i	$-0.508888\pi$
$223$	−0.833861			−0.0558395			−0.0279197	+	0.999610i	$0.508888\pi$
$224$		−	1.30998i		−	0.0875270i
$225$		0			0
$226$	−2.00000			−0.133038
$227$			15.1879i			1.00805i	0.863688	+	0.504027i	$0.168149\pi$
$227$			15.1879i			1.00805i	−0.863688	+	0.504027i	$0.831851\pi$
$228$		0			0
$229$	−7.80782			−0.515955			−0.257978	−	0.966151i	$-0.583056\pi$
$229$	−7.80782			−0.515955			−0.257978	+	0.966151i	$0.583056\pi$
$230$		−	8.90391i		−	0.587106i
$231$		0			0
$232$	−2.00000			−0.131306
$233$	−3.59393			−0.235446			−0.117723	−	0.993046i	$-0.537559\pi$
$233$	−3.59393			−0.235446			−0.117723	+	0.993046i	$0.537559\pi$
$234$		0			0
$235$			4.61997i			0.301373i
$236$			6.21389i			0.404490i
$237$		0			0
$238$	9.04401			0.586236
$239$		−	7.18785i		−	0.464944i	−0.972603	−	0.232472i	$-0.925319\pi$
$239$		−	7.18785i		−	0.464944i	0.972603	−	0.232472i	$-0.0746813\pi$
$240$		0			0
$241$			3.18785i			0.205348i	0.994715	+	0.102674i	$0.0327398\pi$
$241$			3.18785i			0.205348i	−0.994715	+	0.102674i	$0.967260\pi$
$242$		−	10.5239i		−	0.676500i
$243$		0			0
$244$		−	2.00000i		−	0.128037i
$245$		−	5.28394i		−	0.337579i
$246$		0			0
$247$	7.23619			0.460428
$248$	−9.59393			−0.609215
$249$		0			0
$250$	1.00000			0.0632456
$251$		−	11.5939i		−	0.731802i	−0.930654	−	0.365901i	$-0.880761\pi$
$251$		−	11.5939i		−	0.731802i	0.930654	−	0.365901i	$-0.119239\pi$
$252$		0			0
$253$		−	6.14384i		−	0.386260i
$254$		−	10.6900i		−	0.670751i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	25.5239i		−	1.59214i	−0.605207	−	0.796068i	$-0.706910\pi$
$257$		−	25.5239i		−	1.59214i	0.605207	−	0.796068i	$-0.293090\pi$
$258$		0			0
$259$	−5.38003	+	5.87787i	−0.334299	+	0.365233i
$260$	2.69002			0.166828
$261$		0			0
$262$	8.97396			0.554413
$263$	−18.5679			−1.14494			−0.572472	−	0.819924i	$-0.694016\pi$
$263$	−18.5679			−1.14494			−0.572472	+	0.819924i	$0.694016\pi$
$264$		0			0
$265$		−	0.690016i		−	0.0423874i
$266$	−3.52388			−0.216063
$267$		0			0
$268$	1.23993			0.0757410
$269$	13.3100			0.811524			0.405762	−	0.913979i	$-0.367006\pi$
$269$	13.3100			0.811524			0.405762	+	0.913979i	$0.367006\pi$
$270$		0			0
$271$	17.9479			1.09026			0.545129	−	0.838352i	$-0.316481\pi$
$271$	17.9479			1.09026			0.545129	+	0.838352i	$0.316481\pi$
$272$			6.90391i			0.418611i
$273$		0			0
$274$			11.5939i			0.700415i
$275$	0.690016			0.0416096
$276$		0			0
$277$		−	13.7378i		−	0.825423i	−0.910862	−	0.412711i	$-0.864582\pi$
$277$		−	13.7378i		−	0.825423i	0.910862	−	0.412711i	$-0.135418\pi$
$278$		−	6.76007i		−	0.405442i
$279$		0			0
$280$	−1.30998			−0.0782865
$281$			4.54992i			0.271425i	0.990748	+	0.135713i	$0.0433324\pi$
$281$			4.54992i			0.271425i	−0.990748	+	0.135713i	$0.956668\pi$
$282$		0			0
$283$		−	27.7378i		−	1.64884i	−0.565979	−	0.824420i	$-0.691502\pi$
$283$		−	27.7378i		−	1.64884i	0.565979	−	0.824420i	$-0.308498\pi$
$284$	13.8078			0.819343
$285$		0			0
$286$	1.85616			0.109757
$287$	−2.61997			−0.154652
$288$		0			0
$289$	−30.6640			−1.80376
$290$			2.00000i			0.117444i
$291$		0			0
$292$	12.2839			0.718863
$293$	3.30998			0.193371			0.0966857	−	0.995315i	$-0.469176\pi$
$293$	3.30998			0.193371			0.0966857	+	0.995315i	$0.469176\pi$
$294$		0			0
$295$	6.21389			0.361787
$296$	−4.48698	−	4.10695i	−0.260800	−	0.238711i
$297$		0			0
$298$		−	10.2139i		−	0.591675i
$299$	−23.9517			−1.38516
$300$		0			0
$301$		−	10.7601i		−	0.620200i
$302$			22.2839i			1.28230i
$303$		0			0
$304$		−	2.69002i		−	0.154283i
$305$	−2.00000			−0.114520
$306$		0			0
$307$	21.9479			1.25263			0.626317	−	0.779569i	$-0.284562\pi$
$307$	21.9479			1.25263			0.626317	+	0.779569i	$0.284562\pi$
$308$	−0.903910			−0.0515051
$309$		0			0
$310$			9.59393i			0.544898i
$311$		−	18.7601i		−	1.06379i	−0.846812	−	0.531893i	$-0.821481\pi$
$311$		−	18.7601i		−	1.06379i	0.846812	−	0.531893i	$-0.178519\pi$
$312$		0			0
$313$			21.1879i			1.19761i	0.800896	+	0.598804i	$0.204357\pi$
$313$			21.1879i			1.19761i	−0.800896	+	0.598804i	$0.795643\pi$
$314$			12.2139i			0.689270i
$315$		0			0
$316$		−	1.59393i		−	0.0896654i
$317$	−23.5418			−1.32224			−0.661121	−	0.750279i	$-0.729919\pi$
$317$	−23.5418			−1.32224			−0.661121	+	0.750279i	$0.729919\pi$
$318$		0			0
$319$			1.38003i			0.0772670i
$320$		−	1.00000i		−	0.0559017i
$321$		0			0
$322$	11.6640			0.650008
$323$	18.5716			1.03335
$324$		0			0
$325$		−	2.69002i		−	0.149215i
$326$	6.07005			0.336189
$327$		0			0
$328$		−	2.00000i		−	0.110432i
$329$	−6.05208			−0.333662
$330$		0			0
$331$			16.4061i			0.901759i	0.892585	+	0.450880i	$0.148890\pi$
$331$			16.4061i			0.901759i	−0.892585	+	0.450880i	$0.851110\pi$
$332$	10.2839			0.564405
$333$		0			0
$334$	8.90391			0.487200
$335$		−	1.23993i		−	0.0677448i
$336$		0			0
$337$	23.3317			1.27096			0.635479	−	0.772118i	$-0.280803\pi$
$337$	23.3317			1.27096			0.635479	+	0.772118i	$0.280803\pi$
$338$			5.76381i			0.313510i
$339$		0			0
$340$	6.90391			0.374417
$341$			6.61997i			0.358491i
$342$		0			0
$343$	16.0918			0.868874
$344$	8.21389			0.442863
$345$		0			0
$346$		−	19.8779i		−	1.06864i
$347$			5.38003i			0.288815i	0.989518	+	0.144408i	$0.0461277\pi$
$347$			5.38003i			0.288815i	−0.989518	+	0.144408i	$0.953872\pi$
$348$		0			0
$349$	−29.1879			−1.56239			−0.781195	−	0.624287i	$-0.785390\pi$
$349$	−29.1879			−1.56239			−0.781195	+	0.624287i	$0.785390\pi$
$350$			1.30998i			0.0700216i
$351$		0			0
$352$		−	0.690016i		−	0.0367780i
$353$			22.4278i			1.19371i	0.802349	+	0.596855i	$0.203583\pi$
$353$			22.4278i			1.19371i	−0.802349	+	0.596855i	$0.796417\pi$
$354$		0			0
$355$		−	13.8078i		−	0.732843i
$356$		−	5.92995i		−	0.314287i
$357$		0			0
$358$	19.5939			1.03557
$359$	−20.5679			−1.08553			−0.542766	−	0.839884i	$-0.682623\pi$
$359$	−20.5679			−1.08553			−0.542766	+	0.839884i	$0.682623\pi$
$360$		0			0
$361$	11.7638			0.619148
$362$		−	9.18785i		−	0.482903i
$363$		0			0
$364$			3.52388i			0.184701i
$365$		−	12.2839i		−	0.642971i
$366$		0			0
$367$	17.8779			0.933217			0.466609	−	0.884464i	$-0.345476\pi$
$367$	17.8779			0.933217			0.466609	+	0.884464i	$0.345476\pi$
$368$			8.90391i			0.464148i
$369$		0			0
$370$	−4.10695	+	4.48698i	−0.213510	+	0.233267i
$371$	0.903910			0.0469287
$372$		0			0
$373$	−34.5896			−1.79098			−0.895491	−	0.445080i	$-0.853175\pi$
$373$	−34.5896			−1.79098			−0.895491	+	0.445080i	$0.853175\pi$
$374$	4.76381			0.246331
$375$		0			0
$376$		−	4.61997i		−	0.238257i
$377$	5.38003			0.277086
$378$		0			0
$379$	15.8599			0.814668			0.407334	−	0.913279i	$-0.366458\pi$
$379$	15.8599			0.814668			0.407334	+	0.913279i	$0.366458\pi$
$380$	−2.69002			−0.137995
$381$		0			0
$382$	−20.9039			−1.06954
$383$			26.7117i			1.36491i	0.730930	+	0.682453i	$0.239087\pi$
$383$			26.7117i			1.36491i	−0.730930	+	0.682453i	$0.760913\pi$
$384$		0			0
$385$			0.903910i			0.0460675i
$386$	−8.61997			−0.438745
$387$		0			0
$388$			3.38003i			0.171595i
$389$		−	4.19218i		−	0.212552i	−0.994337	−	0.106276i	$-0.966107\pi$
$389$		−	4.19218i		−	0.212552i	0.994337	−	0.106276i	$-0.0338927\pi$
$390$		0			0
$391$	−61.4718			−3.10876
$392$			5.28394i			0.266879i
$393$		0			0
$394$			12.6900i			0.639314i
$395$	−1.59393			−0.0801992
$396$		0			0
$397$	5.45383			0.273720			0.136860	−	0.990590i	$-0.456299\pi$
$397$	5.45383			0.273720			0.136860	+	0.990590i	$0.456299\pi$
$398$	−22.8339			−1.14456
$399$		0			0
$400$	−1.00000			−0.0500000
$401$		−	34.4978i		−	1.72274i	−0.507978	−	0.861370i	$-0.669607\pi$
$401$		−	34.4978i		−	1.72274i	0.507978	−	0.861370i	$-0.330393\pi$
$402$		0			0
$403$	25.8078			1.28558
$404$	8.97396			0.446471
$405$		0			0
$406$	−2.61997			−0.130027
$407$	−2.83386	+	3.09609i	−0.140469	+	0.153467i
$408$		0			0
$409$			30.9077i			1.52829i	0.645047	+	0.764143i	$0.276838\pi$
$409$			30.9077i			1.52829i	−0.645047	+	0.764143i	$0.723162\pi$
$410$	−2.00000			−0.0987730
$411$		0			0
$412$			19.1879i			0.945318i
$413$			8.14010i			0.400548i
$414$		0			0
$415$		−	10.2839i		−	0.504819i
$416$	−2.69002			−0.131889
$417$		0			0
$418$	−1.85616			−0.0907875
$419$	21.6857			1.05942			0.529708	−	0.848180i	$-0.322302\pi$
$419$	21.6857			1.05942			0.529708	+	0.848180i	$0.322302\pi$
$420$		0			0
$421$			36.3757i			1.77284i	0.462879	+	0.886422i	$0.346817\pi$
$421$			36.3757i			1.77284i	−0.462879	+	0.886422i	$0.653183\pi$
$422$		−	5.38003i		−	0.261896i
$423$		0			0
$424$			0.690016i			0.0335102i
$425$		−	6.90391i		−	0.334889i
$426$		0			0
$427$		−	2.61997i		−	0.126789i
$428$	18.1438			0.877016
$429$		0			0
$430$		−	8.21389i		−	0.396109i
$431$			4.47612i			0.215607i	0.994172	+	0.107804i	$0.0343818\pi$
$431$			4.47612i			0.215607i	−0.994172	+	0.107804i	$0.965618\pi$
$432$		0			0
$433$	−13.5239			−0.649916			−0.324958	−	0.945728i	$-0.605350\pi$
$433$	−13.5239			−0.649916			−0.324958	+	0.945728i	$0.605350\pi$
$434$	−12.5679			−0.603278
$435$		0			0
$436$		−	17.5239i		−	0.839242i
$437$	23.9517			1.14576
$438$		0			0
$439$			1.59393i			0.0760740i	0.999276	+	0.0380370i	$0.0121105\pi$
$439$			1.59393i			0.0760740i	−0.999276	+	0.0380370i	$0.987890\pi$
$440$	−0.690016			−0.0328952
$441$		0			0
$442$		−	18.5716i		−	0.883362i
$443$	36.8556			1.75106			0.875531	−	0.483163i	$-0.160512\pi$
$443$	36.8556			1.75106			0.875531	+	0.483163i	$0.160512\pi$
$444$		0			0
$445$	−5.92995			−0.281107
$446$		−	0.833861i		−	0.0394845i
$447$		0			0
$448$	1.30998			0.0618909
$449$		−	12.0738i		−	0.569798i	−0.958558	−	0.284899i	$-0.908040\pi$
$449$		−	12.0738i		−	0.569798i	0.958558	−	0.284899i	$-0.0919600\pi$
$450$		0			0
$451$	−1.38003			−0.0649832
$452$		−	2.00000i		−	0.0940721i
$453$		0			0
$454$	−15.1879			−0.712801
$455$	3.52388			0.165202
$456$		0			0
$457$			9.18785i			0.429790i	0.976637	+	0.214895i	$0.0689409\pi$
$457$			9.18785i			0.429790i	−0.976637	+	0.214895i	$0.931059\pi$
$458$		−	7.80782i		−	0.364835i
$459$		0			0
$460$	8.90391			0.415147
$461$			6.14010i			0.285973i	0.989725	+	0.142986i	$0.0456705\pi$
$461$			6.14010i			0.285973i	−0.989725	+	0.142986i	$0.954329\pi$
$462$		0			0
$463$			1.09984i			0.0511137i	0.999673	+	0.0255568i	$0.00813588\pi$
$463$			1.09984i			0.0511137i	−0.999673	+	0.0255568i	$0.991864\pi$
$464$		−	2.00000i		−	0.0928477i
$465$		0			0
$466$		−	3.59393i		−	0.166485i
$467$		−	3.18785i		−	0.147516i	−0.997276	−	0.0737581i	$-0.976501\pi$
$467$		−	3.18785i		−	0.147516i	0.997276	−	0.0737581i	$-0.0234993\pi$
$468$		0			0
$469$	1.62429			0.0750029
$470$	−4.61997			−0.213103
$471$		0			0
$472$	−6.21389			−0.286018
$473$		−	5.66772i		−	0.260602i
$474$		0			0
$475$			2.69002i			0.123426i
$476$			9.04401i			0.414531i
$477$		0			0
$478$	7.18785			0.328765
$479$			41.1916i			1.88209i	0.338278	+	0.941046i	$0.390155\pi$
$479$			41.1916i			1.88209i	−0.338278	+	0.941046i	$0.609845\pi$
$480$		0			0
$481$	12.0700	+	11.0478i	0.550347	+	0.503734i
$482$	−3.18785			−0.145203
$483$		0			0
$484$	10.5239			0.478358
$485$	3.38003			0.153479
$486$		0			0
$487$		−	24.4278i		−	1.10693i	−0.832873	−	0.553464i	$-0.813306\pi$
$487$		−	24.4278i		−	1.10693i	0.832873	−	0.553464i	$-0.186694\pi$
$488$	2.00000			0.0905357
$489$		0			0
$490$	5.28394			0.238704
$491$	0.690016			0.0311400			0.0155700	−	0.999879i	$-0.495044\pi$
$491$	0.690016			0.0311400			0.0155700	+	0.999879i	$0.495044\pi$
$492$		0			0
$493$	13.8078			0.621873
$494$			7.23619i			0.325572i
$495$		0			0
$496$		−	9.59393i		−	0.430780i
$497$	18.0880			0.811358
$498$		0			0
$499$		−	8.63794i		−	0.386687i	−0.981131	−	0.193344i	$-0.938067\pi$
$499$		−	8.63794i		−	0.386687i	0.981131	−	0.193344i	$-0.0619332\pi$
$500$			1.00000i			0.0447214i
$501$		0			0
$502$	11.5939			0.517462
$503$			30.3757i			1.35439i	0.735806	+	0.677193i	$0.236804\pi$
$503$			30.3757i			1.35439i	−0.735806	+	0.677193i	$0.763196\pi$
$504$		0			0
$505$		−	8.97396i		−	0.399336i
$506$	6.14384			0.273127
$507$		0			0
$508$	10.6900			0.474293
$509$	36.9256			1.63670			0.818350	−	0.574721i	$-0.194889\pi$
$509$	36.9256			1.63670			0.818350	+	0.574721i	$0.194889\pi$
$510$		0			0
$511$	16.0918			0.711858
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	25.5239			1.12581
$515$	19.1879			0.845518
$516$		0			0
$517$	−3.18785			−0.140202
$518$	−5.87787	−	5.38003i	−0.258259	−	0.236385i
$519$		0			0
$520$			2.69002i			0.117965i
$521$	−10.8121			−0.473689			−0.236844	−	0.971548i	$-0.576113\pi$
$521$	−10.8121			−0.473689			−0.236844	+	0.971548i	$0.576113\pi$
$522$		0			0
$523$			20.9219i			0.914850i	0.889248	+	0.457425i	$0.151228\pi$
$523$			20.9219i			0.914850i	−0.889248	+	0.457425i	$0.848772\pi$
$524$			8.97396i			0.392029i
$525$		0			0
$526$		−	18.5679i		−	0.809598i
$527$	66.2356			2.88527
$528$		0			0
$529$	−56.2796			−2.44694
$530$	0.690016			0.0299724
$531$		0			0
$532$		−	3.52388i		−	0.152779i
$533$			5.38003i			0.233035i
$534$		0			0
$535$		−	18.1438i		−	0.784427i
$536$			1.23993i			0.0535570i
$537$		0			0
$538$			13.3100i			0.573834i
$539$	3.64601			0.157045
$540$		0			0
$541$		−	25.5239i		−	1.09736i	−0.836033	−	0.548679i	$-0.815131\pi$
$541$		−	25.5239i		−	1.09736i	0.836033	−	0.548679i	$-0.184869\pi$
$542$			17.9479i			0.770929i
$543$		0			0
$544$	−6.90391			−0.296003
$545$	−17.5239			−0.750640
$546$		0			0
$547$		−	6.49784i		−	0.277827i	−0.990304	−	0.138914i	$-0.955639\pi$
$547$		−	6.49784i		−	0.277827i	0.990304	−	0.138914i	$-0.0443611\pi$
$548$	−11.5939			−0.495268
$549$		0			0
$550$			0.690016i			0.0294224i
$551$	−5.38003			−0.229197
$552$		0			0
$553$		−	2.08802i		−	0.0887915i
$554$	13.7378			0.583662
$555$		0			0
$556$	6.76007			0.286691
$557$			0.760066i			0.0322050i	0.999870	+	0.0161025i	$0.00512581\pi$
$557$			0.760066i			0.0322050i	−0.999870	+	0.0161025i	$0.994874\pi$
$558$		0			0
$559$	−22.0955			−0.934540
$560$		−	1.30998i		−	0.0553569i
$561$		0			0
$562$	−4.54992			−0.191927
$563$			18.0880i			0.762319i	0.924509	+	0.381160i	$0.124475\pi$
$563$			18.0880i			0.762319i	−0.924509	+	0.381160i	$0.875525\pi$
$564$		0			0
$565$	−2.00000			−0.0841406
$566$	27.7378			1.16591
$567$		0			0
$568$			13.8078i			0.579363i
$569$		−	2.74210i		−	0.114955i	−0.998347	−	0.0574774i	$-0.981694\pi$
$569$		−	2.74210i		−	0.114955i	0.998347	−	0.0574774i	$-0.0183057\pi$
$570$		0			0
$571$	−25.2399			−1.05626			−0.528129	−	0.849164i	$-0.677106\pi$
$571$	−25.2399			−1.05626			−0.528129	+	0.849164i	$0.677106\pi$
$572$			1.85616i			0.0776098i
$573$		0			0
$574$		−	2.61997i		−	0.109355i
$575$		−	8.90391i		−	0.371319i
$576$		0			0
$577$			38.5679i			1.60560i	0.596247	+	0.802801i	$0.296658\pi$
$577$			38.5679i			1.60560i	−0.596247	+	0.802801i	$0.703342\pi$
$578$		−	30.6640i		−	1.27545i
$579$		0			0
$580$	−2.00000			−0.0830455
$581$	13.4718			0.558904
$582$		0			0
$583$	0.476123			0.0197190
$584$			12.2839i			0.508313i
$585$		0			0
$586$			3.30998i			0.136734i
$587$			3.32795i			0.137359i	0.997639	+	0.0686796i	$0.0218786\pi$
$587$			3.32795i			0.137359i	−0.997639	+	0.0686796i	$0.978121\pi$
$588$		0			0
$589$	−25.8078			−1.06339
$590$			6.21389i			0.255822i
$591$		0			0
$592$	4.10695	−	4.48698i	0.168794	−	0.184414i
$593$	24.5896			1.00977			0.504887	−	0.863185i	$-0.331534\pi$
$593$	24.5896			1.00977			0.504887	+	0.863185i	$0.331534\pi$
$594$		0			0
$595$	9.04401			0.370768
$596$	10.2139			0.418377
$597$		0			0
$598$		−	23.9517i		−	0.979456i
$599$	−7.85990			−0.321147			−0.160573	−	0.987024i	$-0.551334\pi$
$599$	−7.85990			−0.321147			−0.160573	+	0.987024i	$0.551334\pi$
$600$		0			0
$601$	0.143844			0.00586754			0.00293377	−	0.999996i	$-0.499066\pi$
$601$	0.143844			0.00586754			0.00293377	+	0.999996i	$0.499066\pi$
$602$	10.7601			0.438548
$603$		0			0
$604$	−22.2839			−0.906721
$605$		−	10.5239i		−	0.427856i
$606$		0			0
$607$		−	27.1879i		−	1.10352i	−0.834003	−	0.551760i	$-0.813956\pi$
$607$		−	27.1879i		−	1.10352i	0.834003	−	0.551760i	$-0.186044\pi$
$608$	2.69002			0.109095
$609$		0			0
$610$		−	2.00000i		−	0.0809776i
$611$			12.4278i			0.502774i
$612$		0			0
$613$	−10.9740			−0.443234			−0.221617	−	0.975134i	$-0.571133\pi$
$613$	−10.9740			−0.443234			−0.221617	+	0.975134i	$0.571133\pi$
$614$			21.9479i			0.885746i
$615$		0			0
$616$		−	0.903910i		−	0.0364196i
$617$	−8.97396			−0.361278			−0.180639	−	0.983549i	$-0.557817\pi$
$617$	−8.97396			−0.361278			−0.180639	+	0.983549i	$0.557817\pi$
$618$		0			0
$619$	−1.52013			−0.0610992			−0.0305496	−	0.999533i	$-0.509726\pi$
$619$	−1.52013			−0.0610992			−0.0305496	+	0.999533i	$0.509726\pi$
$620$	−9.59393			−0.385301
$621$		0			0
$622$	18.7601			0.752210
$623$		−	7.76814i		−	0.311224i
$624$		0			0
$625$	1.00000			0.0400000
$626$	−21.1879			−0.846837
$627$		0			0
$628$	−12.2139			−0.487387
$629$	30.9777	+	28.3540i	1.23516	+	1.13055i
$630$		0			0
$631$		−	12.2139i		−	0.486227i	−0.969998	−	0.243114i	$-0.921831\pi$
$631$		−	12.2139i		−	0.486227i	0.969998	−	0.243114i	$-0.0781688\pi$
$632$	1.59393			0.0634030
$633$		0			0
$634$		−	23.5418i		−	0.934966i
$635$		−	10.6900i		−	0.424220i
$636$		0			0
$637$		−	14.2139i		−	0.563175i
$638$	−1.38003			−0.0546360
$639$		0			0
$640$	1.00000			0.0395285
$641$	1.43211			0.0565651			0.0282825	−	0.999600i	$-0.490996\pi$
$641$	1.43211			0.0565651			0.0282825	+	0.999600i	$0.490996\pi$
$642$		0			0
$643$			10.0700i			0.397124i	0.980088	+	0.198562i	$0.0636271\pi$
$643$			10.0700i			0.397124i	−0.980088	+	0.198562i	$0.936373\pi$
$644$			11.6640i			0.459625i
$645$		0			0
$646$			18.5716i			0.730691i
$647$		−	6.14384i		−	0.241539i	−0.992681	−	0.120770i	$-0.961464\pi$
$647$		−	6.14384i		−	0.241539i	0.992681	−	0.120770i	$-0.0385363\pi$
$648$		0			0
$649$			4.28769i			0.168307i
$650$	2.69002			0.105511
$651$		0			0
$652$			6.07005i			0.237721i
$653$			45.1879i			1.76834i	0.467168	+	0.884169i	$0.345274\pi$
$653$			45.1879i			1.76834i	−0.467168	+	0.884169i	$0.654726\pi$
$654$		0			0
$655$	8.97396			0.350642
$656$	2.00000			0.0780869
$657$		0			0
$658$		−	6.05208i		−	0.235935i
$659$	−19.4017			−0.755785			−0.377892	−	0.925850i	$-0.623351\pi$
$659$	−19.4017			−0.755785			−0.377892	+	0.925850i	$0.623351\pi$
$660$		0			0
$661$		−	0.424042i		−	0.0164933i	−0.999966	−	0.00824666i	$-0.997375\pi$
$661$		−	0.424042i		−	0.0164933i	0.999966	−	0.00824666i	$-0.00262502\pi$
$662$	−16.4061			−0.637640
$663$		0			0
$664$			10.2839i			0.399094i
$665$	−3.52388			−0.136650
$666$		0			0
$667$	17.8078			0.689522
$668$			8.90391i			0.344503i
$669$		0			0
$670$	1.23993			0.0479028
$671$		−	1.38003i		−	0.0532756i
$672$		0			0
$673$	2.76381			0.106537			0.0532686	−	0.998580i	$-0.483036\pi$
$673$	2.76381			0.106537			0.0532686	+	0.998580i	$0.483036\pi$
$674$			23.3317i			0.898703i
$675$		0			0
$676$	−5.76381			−0.221685
$677$	−23.5977			−0.906932			−0.453466	−	0.891274i	$-0.649813\pi$
$677$	−23.5977			−0.906932			−0.453466	+	0.891274i	$0.649813\pi$
$678$		0			0
$679$			4.42779i			0.169923i
$680$			6.90391i			0.264753i
$681$		0			0
$682$	−6.61997			−0.253492
$683$			17.9479i			0.686758i	0.939197	+	0.343379i	$0.111572\pi$
$683$			17.9479i			0.686758i	−0.939197	+	0.343379i	$0.888428\pi$
$684$		0			0
$685$			11.5939i			0.442981i
$686$			16.0918i			0.614386i
$687$		0			0
$688$			8.21389i			0.313152i
$689$		−	1.85616i		−	0.0707139i
$690$		0			0
$691$	46.2356			1.75889			0.879443	−	0.476005i	$-0.157915\pi$
$691$	46.2356			1.75889			0.879443	+	0.476005i	$0.157915\pi$
$692$	19.8779			0.755643
$693$		0			0
$694$	−5.38003			−0.204223
$695$		−	6.76007i		−	0.256424i
$696$		0			0
$697$			13.8078i			0.523008i
$698$		−	29.1879i		−	1.10478i
$699$		0			0
$700$	−1.30998			−0.0495127
$701$		−	0.192180i		−	0.00725852i	−0.999993	−	0.00362926i	$-0.998845\pi$
$701$		−	0.192180i		−	0.00725852i	0.999993	−	0.00362926i	$-0.00115523\pi$
$702$		0			0
$703$	−12.0700	−	11.0478i	−0.455231	−	0.416674i
$704$	0.690016			0.0260060
$705$		0			0
$706$	−22.4278			−0.844081
$707$	11.7557			0.442120
$708$		0			0
$709$		−	3.85616i		−	0.144821i	−0.997375	−	0.0724105i	$-0.976931\pi$
$709$		−	3.85616i		−	0.144821i	0.997375	−	0.0724105i	$-0.0230692\pi$
$710$	13.8078			0.518198
$711$		0			0
$712$	5.92995			0.222234
$713$	85.4235			3.19913
$714$		0			0
$715$	1.85616			0.0694163
$716$			19.5939i			0.732259i
$717$		0			0
$718$		−	20.5679i		−	0.767587i
$719$	−15.6156			−0.582365			−0.291183	−	0.956668i	$-0.594049\pi$
$719$	−15.6156			−0.582365			−0.291183	+	0.956668i	$0.594049\pi$
$720$		0			0
$721$			25.1358i			0.936105i
$722$			11.7638i			0.437804i
$723$		0			0
$724$	9.18785			0.341464
$725$			2.00000i			0.0742781i
$726$		0			0
$727$			31.0478i			1.15150i	0.817627	+	0.575749i	$0.195289\pi$
$727$			31.0478i			1.15150i	−0.817627	+	0.575749i	$0.804711\pi$
$728$	−3.52388			−0.130604
$729$		0			0
$730$	12.2839			0.454649
$731$	−56.7080			−2.09742
$732$		0			0
$733$	−36.7818			−1.35857			−0.679283	−	0.733876i	$-0.737709\pi$
$733$	−36.7818			−1.35857			−0.679283	+	0.733876i	$0.737709\pi$
$734$			17.8779i			0.659884i
$735$		0			0
$736$	−8.90391			−0.328202
$737$	0.855575			0.0315155
$738$		0			0
$739$	31.8958			1.17331			0.586654	−	0.809838i	$-0.300445\pi$
$739$	31.8958			1.17331			0.586654	+	0.809838i	$0.300445\pi$
$740$	−4.48698	−	4.10695i	−0.164945	−	0.150974i
$741$		0			0
$742$			0.903910i			0.0331836i
$743$	−21.1879			−0.777307			−0.388653	−	0.921384i	$-0.627060\pi$
$743$	−21.1879			−0.777307			−0.388653	+	0.921384i	$0.627060\pi$
$744$		0			0
$745$		−	10.2139i		−	0.374208i
$746$		−	34.5896i		−	1.26642i
$747$		0			0
$748$			4.76381i			0.174182i
$749$	23.7681			0.868469
$750$		0			0
$751$	16.0000			0.583848			0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000			0.583848			0.291924	+	0.956441i	$0.405705\pi$
$752$	4.61997			0.168473
$753$		0			0
$754$			5.38003i			0.195929i
$755$			22.2839i			0.810996i
$756$		0			0
$757$			24.7780i			0.900573i	0.892884	+	0.450286i	$0.148678\pi$
$757$			24.7780i			0.900573i	−0.892884	+	0.450286i	$0.851322\pi$
$758$			15.8599i			0.576058i
$759$		0			0
$760$		−	2.69002i		−	0.0975772i
$761$	10.8121			0.391940			0.195970	−	0.980610i	$-0.437214\pi$
$761$	10.8121			0.391940			0.195970	+	0.980610i	$0.437214\pi$
$762$		0			0
$763$		−	22.9560i		−	0.831063i
$764$		−	20.9039i		−	0.756277i
$765$		0			0
$766$	−26.7117			−0.965134
$767$	16.7155			0.603561
$768$		0			0
$769$			15.0478i			0.542636i	0.962490	+	0.271318i	$0.0874595\pi$
$769$			15.0478i			0.542636i	−0.962490	+	0.271318i	$0.912541\pi$
$770$	−0.903910			−0.0325747
$771$		0			0
$772$		−	8.61997i		−	0.310239i
$773$	−34.0700			−1.22541			−0.612707	−	0.790310i	$-0.709920\pi$
$773$	−34.0700			−1.22541			−0.612707	+	0.790310i	$0.709920\pi$
$774$		0			0
$775$			9.59393i			0.344624i
$776$	−3.38003			−0.121336
$777$		0			0
$778$	4.19218			0.150297
$779$		−	5.38003i		−	0.192760i
$780$		0			0
$781$	9.52762			0.340925
$782$		−	61.4718i		−	2.19823i
$783$		0			0
$784$	−5.28394			−0.188712
$785$			12.2139i			0.435933i
$786$		0			0
$787$	11.3280			0.403798			0.201899	−	0.979406i	$-0.435289\pi$
$787$	11.3280			0.403798			0.201899	+	0.979406i	$0.435289\pi$
$788$	−12.6900			−0.452063
$789$		0			0
$790$		−	1.59393i		−	0.0567094i
$791$		−	2.61997i		−	0.0931553i
$792$		0			0
$793$	−5.38003			−0.191051
$794$			5.45383i			0.193549i
$795$		0			0
$796$		−	22.8339i		−	0.809325i
$797$			14.2877i			0.506096i	0.967454	+	0.253048i	$0.0814330\pi$
$797$			14.2877i			0.506096i	−0.967454	+	0.253048i	$0.918567\pi$
$798$		0			0
$799$			31.8958i			1.12839i
$800$		−	1.00000i		−	0.0353553i
$801$		0			0
$802$	34.4978			1.21816
$803$	8.47612			0.299116
$804$		0			0
$805$	11.6640			0.411101
$806$			25.8078i			0.909042i
$807$		0			0
$808$			8.97396i			0.315703i
$809$			31.1699i			1.09587i	0.836519	+	0.547937i	$0.184587\pi$
$809$			31.1699i			1.09587i	−0.836519	+	0.547937i	$0.815413\pi$
$810$		0			0
$811$	7.85990			0.275998			0.137999	−	0.990432i	$-0.455933\pi$
$811$	7.85990			0.275998			0.137999	+	0.990432i	$0.455933\pi$
$812$		−	2.61997i		−	0.0919428i
$813$		0			0
$814$	−3.09609	−	2.83386i	−0.108518	−	0.0993268i
$815$	6.07005			0.212625
$816$		0			0
$817$	22.0955			0.773024
$818$	−30.9077			−1.08066
$819$		0			0
$820$		−	2.00000i		−	0.0698430i
$821$	−20.9256			−0.730309			−0.365155	−	0.930947i	$-0.618984\pi$
$821$	−20.9256			−0.730309			−0.365155	+	0.930947i	$0.618984\pi$
$822$		0			0
$823$	−52.2536			−1.82145			−0.910723	−	0.413019i	$-0.864474\pi$
$823$	−52.2536			−1.82145			−0.910723	+	0.413019i	$0.864474\pi$
$824$	−19.1879			−0.668441
$825$		0			0
$826$	−8.14010			−0.283230
$827$		−	13.8078i		−	0.480145i	−0.970755	−	0.240072i	$-0.922829\pi$
$827$		−	13.8078i		−	0.480145i	0.970755	−	0.240072i	$-0.0771712\pi$
$828$		0			0
$829$		−	17.5239i		−	0.608629i	−0.952572	−	0.304315i	$-0.901573\pi$
$829$		−	17.5239i		−	0.608629i	0.952572	−	0.304315i	$-0.0984274\pi$
$830$	10.2839			0.356961
$831$		0			0
$832$		−	2.69002i		−	0.0932595i
$833$		−	36.4799i		−	1.26395i
$834$		0			0
$835$	8.90391			0.308133
$836$		−	1.85616i		−	0.0641965i
$837$		0			0
$838$			21.6857i			0.749120i
$839$	−45.9479			−1.58630			−0.793149	−	0.609027i	$-0.791560\pi$
$839$	−45.9479			−1.58630			−0.793149	+	0.609027i	$0.791560\pi$
$840$		0			0
$841$	25.0000			0.862069
$842$	−36.3757			−1.25359
$843$		0			0
$844$	5.38003			0.185188
$845$			5.76381i			0.198281i
$846$		0			0
$847$	13.7861			0.473696
$848$	−0.690016			−0.0236953
$849$		0			0
$850$	6.90391			0.236802
$851$	39.9517	+	36.5679i	1.36953	+	1.25353i
$852$		0			0
$853$			44.4978i			1.52358i	0.647826	+	0.761788i	$0.275678\pi$
$853$			44.4978i			1.52358i	−0.647826	+	0.761788i	$0.724322\pi$
$854$	2.61997			0.0896534
$855$		0			0
$856$			18.1438i			0.620144i
$857$		−	20.8518i		−	0.712285i	−0.934432	−	0.356142i	$-0.884092\pi$
$857$		−	20.8518i		−	0.712285i	0.934432	−	0.356142i	$-0.115908\pi$
$858$		0			0
$859$		−	14.9777i		−	0.511033i	−0.966805	−	0.255516i	$-0.917755\pi$
$859$		−	14.9777i		−	0.511033i	0.966805	−	0.255516i	$-0.0822455\pi$
$860$	8.21389			0.280091
$861$		0			0
$862$	−4.47612			−0.152457
$863$	22.1401			0.753658			0.376829	−	0.926283i	$-0.377014\pi$
$863$	22.1401			0.753658			0.376829	+	0.926283i	$0.377014\pi$
$864$		0			0
$865$		−	19.8779i		−	0.675868i
$866$		−	13.5239i		−	0.459560i
$867$		0			0
$868$		−	12.5679i		−	0.426582i
$869$		−	1.09984i		−	0.0373094i
$870$		0			0
$871$		−	3.33544i		−	0.113017i
$872$	17.5239			0.593433
$873$		0			0
$874$			23.9517i			0.810177i
$875$			1.30998i			0.0442855i
$876$		0			0
$877$	23.4017			0.790221			0.395110	−	0.918634i	$-0.370706\pi$
$877$	23.4017			0.790221			0.395110	+	0.918634i	$0.370706\pi$
$878$	−1.59393			−0.0537924
$879$		0			0
$880$		−	0.690016i		−	0.0232604i
$881$	29.8958			1.00722			0.503608	−	0.863932i	$-0.332006\pi$
$881$	29.8958			1.00722			0.503608	+	0.863932i	$0.332006\pi$
$882$		0			0
$883$			32.8376i			1.10507i	0.833489	+	0.552537i	$0.186340\pi$
$883$			32.8376i			1.10507i	−0.833489	+	0.552537i	$0.813660\pi$
$884$	18.5716			0.624632
$885$		0			0
$886$			36.8556i			1.23819i
$887$	−43.4235			−1.45802			−0.729009	−	0.684505i	$-0.760019\pi$
$887$	−43.4235			−1.45802			−0.729009	+	0.684505i	$0.760019\pi$
$888$		0			0
$889$	14.0037			0.469671
$890$		−	5.92995i		−	0.198772i
$891$		0			0
$892$	0.833861			0.0279197
$893$		−	12.4278i		−	0.415880i
$894$		0			0
$895$	19.5939			0.654953
$896$			1.30998i			0.0437635i
$897$		0			0
$898$	12.0738			0.402908
$899$	−19.1879			−0.639951
$900$		0			0
$901$		−	4.76381i		−	0.158706i
$902$		−	1.38003i		−	0.0459501i
$903$		0			0
$904$	2.00000			0.0665190
$905$		−	9.18785i		−	0.305415i
$906$		0			0
$907$		−	25.5456i		−	0.848227i	−0.905609	−	0.424114i	$-0.860586\pi$
$907$		−	25.5456i		−	0.848227i	0.905609	−	0.424114i	$-0.139414\pi$
$908$		−	15.1879i		−	0.504027i
$909$		0			0
$910$			3.52388i			0.116815i
$911$			24.0000i			0.795155i	0.917568	+	0.397578i	$0.130149\pi$
$911$			24.0000i			0.795155i	−0.917568	+	0.397578i	$0.869851\pi$
$912$		0			0
$913$	7.09609			0.234846
$914$	−9.18785			−0.303907
$915$		0			0
$916$	7.80782			0.257978
$917$			11.7557i			0.388209i
$918$		0			0
$919$		−	6.69376i		−	0.220807i	−0.993887	−	0.110403i	$-0.964786\pi$
$919$		−	6.69376i		−	0.220807i	0.993887	−	0.110403i	$-0.0352143\pi$
$920$			8.90391i			0.293553i
$921$		0			0
$922$	−6.14010			−0.202213
$923$		−	37.1433i		−	1.22259i
$924$		0			0
$925$	−4.10695	+	4.48698i	−0.135036	+	0.147531i
$926$	−1.09984			−0.0361428
$927$		0			0
$928$	2.00000			0.0656532
$929$	−35.8078			−1.17482			−0.587408	−	0.809291i	$-0.699852\pi$
$929$	−35.8078			−1.17482			−0.587408	+	0.809291i	$0.699852\pi$
$930$		0			0
$931$			14.2139i			0.465842i
$932$	3.59393			0.117723
$933$		0			0
$934$	3.18785			0.104310
$935$	4.76381			0.155793
$936$		0			0
$937$	−36.7601			−1.20090			−0.600450	−	0.799663i	$-0.705012\pi$
$937$	−36.7601			−1.20090			−0.600450	+	0.799663i	$0.705012\pi$
$938$			1.62429i			0.0530351i
$939$		0			0
$940$		−	4.61997i		−	0.150687i
$941$	−33.6819			−1.09800			−0.549000	−	0.835822i	$-0.684991\pi$
$941$	−33.6819			−1.09800			−0.549000	+	0.835822i	$0.684991\pi$
$942$		0			0
$943$			17.8078i			0.579902i
$944$		−	6.21389i		−	0.202245i
$945$		0			0
$946$	5.66772			0.184274
$947$		−	13.5201i		−	0.439345i	−0.975574	−	0.219673i	$-0.929501\pi$
$947$		−	13.5201i		−	0.439345i	0.975574	−	0.219673i	$-0.0704989\pi$
$948$		0			0
$949$		−	33.0440i		−	1.07265i
$950$	−2.69002			−0.0872757
$951$		0			0
$952$	−9.04401			−0.293118
$953$	13.2616			0.429587			0.214793	−	0.976659i	$-0.431092\pi$
$953$	13.2616			0.429587			0.214793	+	0.976659i	$0.431092\pi$
$954$		0			0
$955$	−20.9039			−0.676435
$956$			7.18785i			0.232472i
$957$		0			0
$958$	−41.1916			−1.33084
$959$	−15.1879			−0.490441
$960$		0			0
$961$	−61.0434			−1.96914
$962$	−11.0478	+	12.0700i	−0.356194	+	0.389154i
$963$		0			0
$964$		−	3.18785i		−	0.102674i
$965$	−8.61997			−0.277487
$966$		0			0
$967$		−	48.8556i		−	1.57109i	−0.618805	−	0.785545i	$-0.712383\pi$
$967$		−	48.8556i		−	1.57109i	0.618805	−	0.785545i	$-0.287617\pi$
$968$			10.5239i			0.338250i
$969$		0			0
$970$			3.38003i			0.108526i
$971$	39.7861			1.27680			0.638398	−	0.769706i	$-0.279597\pi$
$971$	39.7861			1.27680			0.638398	+	0.769706i	$0.279597\pi$
$972$		0			0
$973$	8.85558			0.283897
$974$	24.4278			0.782717
$975$		0			0
$976$			2.00000i			0.0640184i
$977$			49.5239i			1.58441i	0.610256	+	0.792205i	$0.291067\pi$
$977$			49.5239i			1.58441i	−0.610256	+	0.792205i	$0.708933\pi$
$978$		0			0
$979$		−	4.09176i		−	0.130773i
$980$			5.28394i			0.168789i
$981$		0			0
$982$			0.690016i			0.0220193i
$983$	40.2356			1.28332			0.641658	−	0.766991i	$-0.278247\pi$
$983$	40.2356			1.28332			0.641658	+	0.766991i	$0.278247\pi$
$984$		0			0
$985$			12.6900i			0.404338i
$986$			13.8078i			0.439731i
$987$		0			0
$988$	−7.23619			−0.230214
$989$	−73.1358			−2.32558
$990$		0			0
$991$			54.4495i			1.72965i	0.502077	+	0.864823i	$0.332569\pi$
$991$			54.4495i			1.72965i	−0.502077	+	0.864823i	$0.667431\pi$
$992$	9.59393			0.304607
$993$		0			0
$994$			18.0880i			0.573717i
$995$	−22.8339			−0.723882
$996$		0			0
$997$			51.2579i			1.62335i	0.584106	+	0.811677i	$0.301445\pi$
$997$			51.2579i			1.62335i	−0.584106	+	0.811677i	$0.698555\pi$
$998$	8.63794			0.273429
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.h.m.2071.5		6
3.2	odd	2		1110.2.h.f.961.2	✓	6
37.36	even	2	inner	3330.2.h.m.2071.2		6
111.110	odd	2		1110.2.h.f.961.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.h.f.961.2	✓	6	3.2	odd	2
1110.2.h.f.961.5	yes	6	111.110	odd	2
3330.2.h.m.2071.2		6	37.36	even	2	inner
3330.2.h.m.2071.5		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.30998 q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.30998 q^{7} -1.00000i q^{8} -1.00000 q^{10} -0.690016 q^{11} +2.69002i q^{13} -1.30998i q^{14} +1.00000 q^{16} +6.90391i q^{17} -2.69002i q^{19} -1.00000i q^{20} -0.690016i q^{22} +8.90391i q^{23} -1.00000 q^{25} -2.69002 q^{26} +1.30998 q^{28} -2.00000i q^{29} -9.59393i q^{31} +1.00000i q^{32} -6.90391 q^{34} -1.30998i q^{35} +(4.10695 - 4.48698i) q^{37} +2.69002 q^{38} +1.00000 q^{40} +2.00000 q^{41} +8.21389i q^{43} +0.690016 q^{44} -8.90391 q^{46} +4.61997 q^{47} -5.28394 q^{49} -1.00000i q^{50} -2.69002i q^{52} -0.690016 q^{53} -0.690016i q^{55} +1.30998i q^{56} +2.00000 q^{58} -6.21389i q^{59} +2.00000i q^{61} +9.59393 q^{62} -1.00000 q^{64} -2.69002 q^{65} -1.23993 q^{67} -6.90391i q^{68} +1.30998 q^{70} -13.8078 q^{71} -12.2839 q^{73} +(4.48698 + 4.10695i) q^{74} +2.69002i q^{76} +0.903910 q^{77} +1.59393i q^{79} +1.00000i q^{80} +2.00000i q^{82} -10.2839 q^{83} -6.90391 q^{85} -8.21389 q^{86} +0.690016i q^{88} +5.92995i q^{89} -3.52388i q^{91} -8.90391i q^{92} +4.61997i q^{94} +2.69002 q^{95} -3.38003i q^{97} -5.28394i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 6 q^{7}+O(q^{10}) \) 6 * q - 6 * q^4 - 6 * q^7	\( 6 q - 6 q^{4} - 6 q^{7} - 6 q^{10} - 6 q^{11} + 6 q^{16} - 6 q^{25} - 18 q^{26} + 6 q^{28} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 6 q^{40} + 12 q^{41} + 6 q^{44} - 2 q^{46} + 24 q^{47} + 16 q^{49} - 6 q^{53} + 12 q^{58} + 8 q^{62} - 6 q^{64} - 18 q^{65} + 6 q^{70} + 20 q^{71} - 26 q^{73} + 4 q^{74} - 46 q^{77} - 14 q^{83} + 10 q^{85} + 4 q^{86} + 18 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 - 6 * q^7 - 6 * q^10 - 6 * q^11 + 6 * q^16 - 6 * q^25 - 18 * q^26 + 6 * q^28 + 10 * q^34 - 2 * q^37 + 18 * q^38 + 6 * q^40 + 12 * q^41 + 6 * q^44 - 2 * q^46 + 24 * q^47 + 16 * q^49 - 6 * q^53 + 12 * q^58 + 8 * q^62 - 6 * q^64 - 18 * q^65 + 6 * q^70 + 20 * q^71 - 26 * q^73 + 4 * q^74 - 46 * q^77 - 14 * q^83 + 10 * q^85 + 4 * q^86 + 18 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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