LMFDB - Embedded newform 3264.2.c.q.577.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3264,2,Mod(577,3264)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3264, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3264.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3264 = 2^{6} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3264.c (of order $2$, degree $1$, not 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:	$26.0631712197$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 12x^{8} + 38x^{6} + 44x^{4} + 17x^{2} + 1 $ x^10 + 12x^8 + 38x^6 + 44x^4 + 17x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 1632)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.5
Root			$-0.266708i$ of defining polynomial
Character	$\chi$	$=$	3264.577
Dual form			3264.2.c.q.577.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.82600i q^{5} +2.57427i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.82600i q^{5} +2.57427i q^{7} -1.00000 q^{9} +4.35941i q^{11} +5.29258 q^{13} +3.82600 q^{15} +(-2.67284 + 3.13942i) q^{17} +1.25173 q^{19} +2.57427 q^{21} +8.06399i q^{23} -9.63825 q^{25} +1.00000i q^{27} -0.771405i q^{29} +5.07773i q^{31} +4.35941 q^{33} -9.84913 q^{35} -9.91994i q^{37} -5.29258i q^{39} -4.18490i q^{41} +0.121425 q^{43} -3.82600i q^{45} +6.21536 q^{47} +0.373156 q^{49} +(3.13942 + 2.67284i) q^{51} -0.869694 q^{53} -16.6791 q^{55} -1.25173i q^{57} +5.37316 q^{59} -11.3566i q^{61} -2.57427i q^{63} +20.2494i q^{65} +12.0645 q^{67} +8.06399 q^{69} -1.73206i q^{71} -4.08170i q^{73} +9.63825i q^{75} -11.2223 q^{77} +3.64110i q^{79} +1.00000 q^{81} -16.1462 q^{83} +(-12.0114 - 10.2263i) q^{85} -0.771405 q^{87} -18.4343 q^{89} +13.6245i q^{91} +5.07773 q^{93} +4.78912i q^{95} -9.43663i q^{97} -4.35941i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^9	$ 10 q - 10 q^{9} + 8 q^{13} - 4 q^{15} - 6 q^{17} - 8 q^{19} + 4 q^{21} - 10 q^{25} + 4 q^{33} - 16 q^{35} - 16 q^{43} + 24 q^{47} - 34 q^{49} + 8 q^{51} - 12 q^{53} - 56 q^{55} + 16 q^{59} + 16 q^{69} + 8 q^{77} + 10 q^{81} + 8 q^{83} + 4 q^{85} + 12 q^{87} - 12 q^{89} - 12 q^{93}+O(q^{100}) $ 10 * q - 10 * q^9 + 8 * q^13 - 4 * q^15 - 6 * q^17 - 8 * q^19 + 4 * q^21 - 10 * q^25 + 4 * q^33 - 16 * q^35 - 16 * q^43 + 24 * q^47 - 34 * q^49 + 8 * q^51 - 12 * q^53 - 56 * q^55 + 16 * q^59 + 16 * q^69 + 8 * q^77 + 10 * q^81 + 8 * q^83 + 4 * q^85 + 12 * q^87 - 12 * q^89 - 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times$.

$n$	$511$	$2177$	$2245$	$2689$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$			3.82600i			1.71104i	0.517772	+	0.855519i	$0.326762\pi$
$5$			3.82600i			1.71104i	−0.517772	+	0.855519i	$0.673238\pi$
$6$		0			0
$7$			2.57427i			0.972981i	0.873686	+	0.486491i	$0.161723\pi$
$7$			2.57427i			0.972981i	−0.873686	+	0.486491i	$0.838277\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			4.35941i			1.31441i	0.753711	+	0.657206i	$0.228262\pi$
$11$			4.35941i			1.31441i	−0.753711	+	0.657206i	$0.771738\pi$
$12$		0			0
$13$	5.29258			1.46790			0.733949	−	0.679205i	$-0.237675\pi$
$13$	5.29258			1.46790			0.733949	+	0.679205i	$0.237675\pi$
$14$		0			0
$15$	3.82600			0.987868
$16$		0			0
$17$	−2.67284	+	3.13942i	−0.648258	+	0.761421i
$17$	−2.67284	+	3.13942i	−0.648258	+	0.761421i
$18$		0			0
$19$	1.25173			0.287167			0.143583	−	0.989638i	$-0.454137\pi$
$19$	1.25173			0.287167			0.143583	+	0.989638i	$0.454137\pi$
$20$		0			0
$21$	2.57427			0.561751
$22$		0			0
$23$			8.06399i			1.68146i	0.541457	+	0.840729i	$0.317873\pi$
$23$			8.06399i			1.68146i	−0.541457	+	0.840729i	$0.682127\pi$
$24$		0			0
$25$	−9.63825			−1.92765
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	0.771405i		−	0.143246i	−0.997432	−	0.0716231i	$-0.977182\pi$
$29$		−	0.771405i		−	0.143246i	0.997432	−	0.0716231i	$-0.0228179\pi$
$30$		0			0
$31$			5.07773i			0.911987i	0.889983	+	0.455993i	$0.150716\pi$
$31$			5.07773i			0.911987i	−0.889983	+	0.455993i	$0.849284\pi$
$32$		0			0
$33$	4.35941			0.758876
$34$		0			0
$35$	−9.84913			−1.66481
$36$		0			0
$37$		−	9.91994i		−	1.63083i	−0.578879	−	0.815414i	$-0.696509\pi$
$37$		−	9.91994i		−	1.63083i	0.578879	−	0.815414i	$-0.303491\pi$
$38$		0			0
$39$		−	5.29258i		−	0.847491i
$40$		0			0
$41$		−	4.18490i		−	0.653571i	−0.945098	−	0.326786i	$-0.894034\pi$
$41$		−	4.18490i		−	0.653571i	0.945098	−	0.326786i	$-0.105966\pi$
$42$		0			0
$43$	0.121425			0.0185171			0.00925854	−	0.999957i	$-0.497053\pi$
$43$	0.121425			0.0185171			0.00925854	+	0.999957i	$0.497053\pi$
$44$		0			0
$45$		−	3.82600i		−	0.570346i
$46$		0			0
$47$	6.21536			0.906604			0.453302	−	0.891357i	$-0.350246\pi$
$47$	6.21536			0.906604			0.453302	+	0.891357i	$0.350246\pi$
$48$		0			0
$49$	0.373156			0.0533079
$50$		0			0
$51$	3.13942	+	2.67284i	0.439607	+	0.374272i
$52$		0			0
$53$	−0.869694			−0.119462			−0.0597308	−	0.998215i	$-0.519024\pi$
$53$	−0.869694			−0.119462			−0.0597308	+	0.998215i	$0.519024\pi$
$54$		0			0
$55$	−16.6791			−2.24901
$56$		0			0
$57$		−	1.25173i		−	0.165796i
$58$		0			0
$59$	5.37316			0.699525			0.349763	−	0.936838i	$-0.386262\pi$
$59$	5.37316			0.699525			0.349763	+	0.936838i	$0.386262\pi$
$60$		0			0
$61$		−	11.3566i		−	1.45406i	−0.686606	−	0.727030i	$-0.740900\pi$
$61$		−	11.3566i		−	1.45406i	0.686606	−	0.727030i	$-0.259100\pi$
$62$		0			0
$63$		−	2.57427i		−	0.324327i
$64$		0			0
$65$			20.2494i			2.51163i
$66$		0			0
$67$	12.0645			1.47391			0.736956	−	0.675940i	$-0.236262\pi$
$67$	12.0645			1.47391			0.736956	+	0.675940i	$0.236262\pi$
$68$		0			0
$69$	8.06399			0.970790
$70$		0			0
$71$		−	1.73206i		−	0.205557i	−0.994704	−	0.102779i	$-0.967227\pi$
$71$		−	1.73206i		−	0.205557i	0.994704	−	0.102779i	$-0.0327733\pi$
$72$		0			0
$73$		−	4.08170i		−	0.477727i	−0.971053	−	0.238863i	$-0.923225\pi$
$73$		−	4.08170i		−	0.477727i	0.971053	−	0.238863i	$-0.0767748\pi$
$74$		0			0
$75$			9.63825i			1.11293i
$76$		0			0
$77$	−11.2223			−1.27890
$78$		0			0
$79$			3.64110i			0.409656i	0.978798	+	0.204828i	$0.0656634\pi$
$79$			3.64110i			0.409656i	−0.978798	+	0.204828i	$0.934337\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−16.1462			−1.77228			−0.886138	−	0.463422i	$-0.846621\pi$
$83$	−16.1462			−1.77228			−0.886138	+	0.463422i	$0.846621\pi$
$84$		0			0
$85$	−12.0114	−	10.2263i	−1.30282	−	1.10919i
$86$		0			0
$87$	−0.771405			−0.0827033
$88$		0			0
$89$	−18.4343			−1.95403			−0.977016	−	0.213168i	$-0.931622\pi$
$89$	−18.4343			−1.95403			−0.977016	+	0.213168i	$0.931622\pi$
$90$		0			0
$91$			13.6245i			1.42824i
$92$		0			0
$93$	5.07773			0.526536
$94$		0			0
$95$			4.78912i			0.491353i
$96$		0			0
$97$		−	9.43663i		−	0.958145i	−0.877775	−	0.479072i	$-0.840973\pi$
$97$		−	9.43663i		−	0.958145i	0.877775	−	0.479072i	$-0.159027\pi$
$98$		0			0
$99$		−	4.35941i		−	0.438138i
$100$		0			0
$101$	9.34567			0.929929			0.464964	−	0.885329i	$-0.346067\pi$
$101$	9.34567			0.929929			0.464964	+	0.885329i	$0.346067\pi$
$102$		0			0
$103$	−2.17564			−0.214372			−0.107186	−	0.994239i	$-0.534184\pi$
$103$	−2.17564			−0.214372			−0.107186	+	0.994239i	$0.534184\pi$
$104$		0			0
$105$			9.84913i			0.961177i
$106$		0			0
$107$		−	10.2257i		−	0.988560i	−0.869303	−	0.494280i	$-0.835432\pi$
$107$		−	10.2257i		−	0.988560i	0.869303	−	0.494280i	$-0.164568\pi$
$108$		0			0
$109$			10.8806i			1.04217i	0.853504	+	0.521086i	$0.174473\pi$
$109$			10.8806i			1.04217i	−0.853504	+	0.521086i	$0.825527\pi$
$110$		0			0
$111$	−9.91994			−0.941559
$112$		0			0
$113$			1.33343i			0.125439i	0.998031	+	0.0627193i	$0.0199773\pi$
$113$			1.33343i			0.125439i	−0.998031	+	0.0627193i	$0.980023\pi$
$114$		0			0
$115$	−30.8528			−2.87704
$116$		0			0
$117$	−5.29258			−0.489299
$118$		0			0
$119$	−8.08170	−	6.88059i	−0.740848	−	0.630743i
$120$		0			0
$121$	−8.00448			−0.727680
$122$		0			0
$123$	−4.18490			−0.377340
$124$		0			0
$125$		−	17.7459i		−	1.58724i
$126$		0			0
$127$	−2.01224			−0.178558			−0.0892788	−	0.996007i	$-0.528456\pi$
$127$	−2.01224			−0.178558			−0.0892788	+	0.996007i	$0.528456\pi$
$128$		0			0
$129$		−	0.121425i		−	0.0106908i
$130$		0			0
$131$		−	10.9446i		−	0.956232i	−0.878297	−	0.478116i	$-0.841320\pi$
$131$		−	10.9446i		−	0.956232i	0.878297	−	0.478116i	$-0.158680\pi$
$132$		0			0
$133$			3.22229i			0.279408i
$134$		0			0
$135$	−3.82600			−0.329289
$136$		0			0
$137$	−6.08170			−0.519595			−0.259797	−	0.965663i	$-0.583656\pi$
$137$	−6.08170			−0.519595			−0.259797	+	0.965663i	$0.583656\pi$
$138$		0			0
$139$			18.3433i			1.55586i	0.628350	+	0.777931i	$0.283731\pi$
$139$			18.3433i			1.55586i	−0.628350	+	0.777931i	$0.716269\pi$
$140$		0			0
$141$		−	6.21536i		−	0.523428i
$142$		0			0
$143$			23.0725i			1.92942i
$144$		0			0
$145$	2.95139			0.245100
$146$		0			0
$147$		−	0.373156i		−	0.0307774i
$148$		0			0
$149$	18.1462			1.48659			0.743297	−	0.668961i	$-0.233261\pi$
$149$	18.1462			1.48659			0.743297	+	0.668961i	$0.233261\pi$
$150$		0			0
$151$	−21.5646			−1.75490			−0.877451	−	0.479666i	$-0.840758\pi$
$151$	−21.5646			−1.75490			−0.877451	+	0.479666i	$0.840758\pi$
$152$		0			0
$153$	2.67284	−	3.13942i	0.216086	−	0.253807i
$154$		0			0
$155$	−19.4274			−1.56044
$156$		0			0
$157$	16.5966			1.32455			0.662275	−	0.749261i	$-0.269591\pi$
$157$	16.5966			1.32455			0.662275	+	0.749261i	$0.269591\pi$
$158$		0			0
$159$			0.869694i			0.0689712i
$160$		0			0
$161$	−20.7588			−1.63603
$162$		0			0
$163$		−	4.71883i		−	0.369607i	−0.982775	−	0.184803i	$-0.940835\pi$
$163$		−	4.71883i		−	0.369607i	0.982775	−	0.184803i	$-0.0591649\pi$
$164$		0			0
$165$			16.6791i			1.29847i
$166$		0			0
$167$			4.49369i			0.347732i	0.984769	+	0.173866i	$0.0556260\pi$
$167$			4.49369i			0.347732i	−0.984769	+	0.173866i	$0.944374\pi$
$168$		0			0
$169$	15.0114			1.15472
$170$		0			0
$171$	−1.25173			−0.0957223
$172$		0			0
$173$			25.8213i			1.96316i	0.191054	+	0.981579i	$0.438809\pi$
$173$			25.8213i			1.96316i	−0.191054	+	0.981579i	$0.561191\pi$
$174$		0			0
$175$		−	24.8114i		−	1.87557i
$176$		0			0
$177$		−	5.37316i		−	0.403871i
$178$		0			0
$179$	16.8005			1.25573			0.627865	−	0.778322i	$-0.283929\pi$
$179$	16.8005			1.25573			0.627865	+	0.778322i	$0.283929\pi$
$180$		0			0
$181$			9.44396i			0.701964i	0.936382	+	0.350982i	$0.114152\pi$
$181$			9.44396i			0.701964i	−0.936382	+	0.350982i	$0.885848\pi$
$182$		0			0
$183$	−11.3566			−0.839501
$184$		0			0
$185$	37.9536			2.79041
$186$		0			0
$187$	−13.6860	−	11.6520i	−1.00082	−	0.852078i
$188$		0			0
$189$	−2.57427			−0.187250
$190$		0			0
$191$	10.9794			0.794444			0.397222	−	0.917722i	$-0.369974\pi$
$191$	10.9794			0.794444			0.397222	+	0.917722i	$0.369974\pi$
$192$		0			0
$193$			5.35493i			0.385456i	0.981252	+	0.192728i	$0.0617336\pi$
$193$			5.35493i			0.385456i	−0.981252	+	0.192728i	$0.938266\pi$
$194$		0			0
$195$	20.2494			1.45009
$196$		0			0
$197$			3.58315i			0.255289i	0.991820	+	0.127644i	$0.0407416\pi$
$197$			3.58315i			0.255289i	−0.991820	+	0.127644i	$0.959258\pi$
$198$		0			0
$199$		−	9.15943i		−	0.649295i	−0.945835	−	0.324647i	$-0.894754\pi$
$199$		−	9.15943i		−	0.649295i	0.945835	−	0.324647i	$-0.105246\pi$
$200$		0			0
$201$		−	12.0645i		−	0.850964i
$202$		0			0
$203$	1.98580			0.139376
$204$		0			0
$205$	16.0114			1.11829
$206$		0			0
$207$		−	8.06399i		−	0.560486i
$208$		0			0
$209$			5.45681i			0.377456i
$210$		0			0
$211$			7.73369i			0.532409i	0.963917	+	0.266205i	$0.0857697\pi$
$211$			7.73369i			0.532409i	−0.963917	+	0.266205i	$0.914230\pi$
$212$		0			0
$213$	−1.73206			−0.118679
$214$		0			0
$215$			0.464570i			0.0316834i
$216$		0			0
$217$	−13.0714			−0.887346
$218$		0			0
$219$	−4.08170			−0.275816
$220$		0			0
$221$	−14.1462	+	16.6156i	−0.951576	+	1.11769i
$222$		0			0
$223$	14.8128			0.991936			0.495968	−	0.868341i	$-0.334813\pi$
$223$	14.8128			0.991936			0.495968	+	0.868341i	$0.334813\pi$
$224$		0			0
$225$	9.63825			0.642550
$226$		0			0
$227$			1.23837i			0.0821933i	0.999155	+	0.0410966i	$0.0130852\pi$
$227$			1.23837i			0.0821933i	−0.999155	+	0.0410966i	$0.986915\pi$
$228$		0			0
$229$	6.61265			0.436976			0.218488	−	0.975840i	$-0.429888\pi$
$229$	6.61265			0.436976			0.218488	+	0.975840i	$0.429888\pi$
$230$		0			0
$231$			11.2223i			0.738372i
$232$		0			0
$233$		−	8.96363i		−	0.587227i	−0.955924	−	0.293614i	$-0.905142\pi$
$233$		−	8.96363i		−	0.587227i	0.955924	−	0.293614i	$-0.0948579\pi$
$234$		0			0
$235$			23.7800i			1.55123i
$236$		0			0
$237$	3.64110			0.236515
$238$		0			0
$239$	−21.8891			−1.41589			−0.707946	−	0.706267i	$-0.750378\pi$
$239$	−21.8891			−1.41589			−0.707946	+	0.706267i	$0.750378\pi$
$240$		0			0
$241$			12.6726i			0.816311i	0.912912	+	0.408156i	$0.133828\pi$
$241$			12.6726i			0.816311i	−0.912912	+	0.408156i	$0.866172\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			1.42769i			0.0912119i
$246$		0			0
$247$	6.62489			0.421531
$248$		0			0
$249$			16.1462i			1.02322i
$250$		0			0
$251$	−23.4919			−1.48279			−0.741397	−	0.671067i	$-0.765836\pi$
$251$	−23.4919			−1.48279			−0.741397	+	0.671067i	$0.765836\pi$
$252$		0			0
$253$	−35.1542			−2.21013
$254$		0			0
$255$	−10.2263	+	12.0114i	−0.640393	+	0.752183i
$256$		0			0
$257$	−10.2428			−0.638931			−0.319466	−	0.947598i	$-0.603503\pi$
$257$	−10.2428			−0.638931			−0.319466	+	0.947598i	$0.603503\pi$
$258$		0			0
$259$	25.5366			1.58676
$260$		0			0
$261$			0.771405i			0.0477488i
$262$		0			0
$263$	−1.06581			−0.0657206			−0.0328603	−	0.999460i	$-0.510462\pi$
$263$	−1.06581			−0.0657206			−0.0328603	+	0.999460i	$0.510462\pi$
$264$		0			0
$265$		−	3.32745i		−	0.204403i
$266$		0			0
$267$			18.4343i			1.12816i
$268$		0			0
$269$		−	24.1151i		−	1.47032i	−0.677891	−	0.735162i	$-0.737106\pi$
$269$		−	24.1151i		−	1.47032i	0.677891	−	0.735162i	$-0.262894\pi$
$270$		0			0
$271$	23.7952			1.44546			0.722728	−	0.691133i	$-0.242888\pi$
$271$	23.7952			1.44546			0.722728	+	0.691133i	$0.242888\pi$
$272$		0			0
$273$	13.6245			0.824593
$274$		0			0
$275$		−	42.0171i		−	2.53373i
$276$		0			0
$277$		−	24.3635i		−	1.46386i	−0.681380	−	0.731930i	$-0.738620\pi$
$277$		−	24.3635i		−	1.46386i	0.681380	−	0.731930i	$-0.261380\pi$
$278$		0			0
$279$		−	5.07773i		−	0.303996i
$280$		0			0
$281$	26.5977			1.58669			0.793343	−	0.608775i	$-0.208339\pi$
$281$	26.5977			1.58669			0.793343	+	0.608775i	$0.208339\pi$
$282$		0			0
$283$		−	4.02179i		−	0.239071i	−0.992830	−	0.119535i	$-0.961860\pi$
$283$		−	4.02179i		−	0.239071i	0.992830	−	0.119535i	$-0.0381405\pi$
$284$		0			0
$285$	4.78912			0.283683
$286$		0			0
$287$	10.7730			0.635912
$288$		0			0
$289$	−2.71190	−	16.7823i	−0.159524	−	0.987194i
$290$		0			0
$291$	−9.43663			−0.553185
$292$		0			0
$293$	3.80388			0.222225			0.111113	−	0.993808i	$-0.464559\pi$
$293$	3.80388			0.222225			0.111113	+	0.993808i	$0.464559\pi$
$294$		0			0
$295$			20.5577i			1.19691i
$296$		0			0
$297$	−4.35941			−0.252959
$298$		0			0
$299$			42.6793i			2.46821i
$300$		0			0
$301$			0.312579i			0.0180168i
$302$		0			0
$303$		−	9.34567i		−	0.536895i
$304$		0			0
$305$	43.4502			2.48795
$306$		0			0
$307$	−5.30866			−0.302981			−0.151491	−	0.988459i	$-0.548407\pi$
$307$	−5.30866			−0.302981			−0.151491	+	0.988459i	$0.548407\pi$
$308$		0			0
$309$			2.17564i			0.123768i
$310$		0			0
$311$			26.0244i			1.47571i	0.674959	+	0.737856i	$0.264161\pi$
$311$			26.0244i			1.47571i	−0.674959	+	0.737856i	$0.735839\pi$
$312$		0			0
$313$		−	21.2547i		−	1.20139i	−0.799479	−	0.600694i	$-0.794891\pi$
$313$		−	21.2547i		−	1.20139i	0.799479	−	0.600694i	$-0.205109\pi$
$314$		0			0
$315$	9.84913			0.554936
$316$		0			0
$317$			26.9269i			1.51236i	0.654362	+	0.756182i	$0.272937\pi$
$317$			26.9269i			1.51236i	−0.654362	+	0.756182i	$0.727063\pi$
$318$		0			0
$319$	3.36287			0.188285
$320$		0			0
$321$	−10.2257			−0.570746
$322$		0			0
$323$	−3.34567	+	3.92971i	−0.186158	+	0.218655i
$324$		0			0
$325$	−51.0112			−2.82959
$326$		0			0
$327$	10.8806			0.601698
$328$		0			0
$329$			16.0000i			0.882109i
$330$		0			0
$331$	2.62489			0.144277			0.0721384	−	0.997395i	$-0.477018\pi$
$331$	2.62489			0.144277			0.0721384	+	0.997395i	$0.477018\pi$
$332$		0			0
$333$			9.91994i			0.543609i
$334$		0			0
$335$			46.1587i			2.52192i
$336$		0			0
$337$			2.02748i			0.110444i	0.998474	+	0.0552221i	$0.0175867\pi$
$337$			2.02748i			0.110444i	−0.998474	+	0.0552221i	$0.982413\pi$
$338$		0			0
$339$	1.33343			0.0724220
$340$		0			0
$341$	−22.1359			−1.19873
$342$		0			0
$343$			18.9805i			1.02485i
$344$		0			0
$345$			30.8528i			1.66106i
$346$		0			0
$347$		−	29.6011i		−	1.58907i	−0.607220	−	0.794534i	$-0.707715\pi$
$347$		−	29.6011i		−	1.58907i	0.607220	−	0.794534i	$-0.292285\pi$
$348$		0			0
$349$	−33.7097			−1.80444			−0.902219	−	0.431279i	$-0.858063\pi$
$349$	−33.7097			−1.80444			−0.902219	+	0.431279i	$0.858063\pi$
$350$		0			0
$351$			5.29258i			0.282497i
$352$		0			0
$353$	11.0301			0.587072			0.293536	−	0.955948i	$-0.405168\pi$
$353$	11.0301			0.587072			0.293536	+	0.955948i	$0.405168\pi$
$354$		0			0
$355$	6.62684			0.351716
$356$		0			0
$357$	−6.88059	+	8.08170i	−0.364159	+	0.427729i
$358$		0			0
$359$	28.8005			1.52003			0.760017	−	0.649903i	$-0.225191\pi$
$359$	28.8005			1.52003			0.760017	+	0.649903i	$0.225191\pi$
$360$		0			0
$361$	−17.4332			−0.917535
$362$		0			0
$363$			8.00448i			0.420126i
$364$		0			0
$365$	15.6166			0.817408
$366$		0			0
$367$			5.79553i			0.302524i	0.988494	+	0.151262i	$0.0483337\pi$
$367$			5.79553i			0.302524i	−0.988494	+	0.151262i	$0.951666\pi$
$368$		0			0
$369$			4.18490i			0.217857i
$370$		0			0
$371$		−	2.23882i		−	0.116234i
$372$		0			0
$373$	1.44232			0.0746807			0.0373404	−	0.999303i	$-0.488111\pi$
$373$	1.44232			0.0746807			0.0373404	+	0.999303i	$0.488111\pi$
$374$		0			0
$375$	−17.7459			−0.916396
$376$		0			0
$377$		−	4.08272i		−	0.210271i
$378$		0			0
$379$		−	26.8280i		−	1.37806i	−0.724732	−	0.689031i	$-0.758036\pi$
$379$		−	26.8280i		−	1.37806i	0.724732	−	0.689031i	$-0.241964\pi$
$380$		0			0
$381$			2.01224i			0.103090i
$382$		0			0
$383$	13.2765			0.678398			0.339199	−	0.940715i	$-0.389844\pi$
$383$	13.2765			0.678398			0.339199	+	0.940715i	$0.389844\pi$
$384$		0			0
$385$		−	42.9364i		−	2.18824i
$386$		0			0
$387$	−0.121425			−0.00617236
$388$		0			0
$389$	30.6222			1.55261			0.776303	−	0.630360i	$-0.217093\pi$
$389$	30.6222			1.55261			0.776303	+	0.630360i	$0.217093\pi$
$390$		0			0
$391$	−25.3162	−	21.5537i	−1.28030	−	1.09002i
$392$		0			0
$393$	−10.9446			−0.552081
$394$		0			0
$395$	−13.9308			−0.700936
$396$		0			0
$397$			7.43929i			0.373367i	0.982420	+	0.186684i	$0.0597739\pi$
$397$			7.43929i			0.373367i	−0.982420	+	0.186684i	$0.940226\pi$
$398$		0			0
$399$	3.22229			0.161316
$400$		0			0
$401$		−	14.4820i		−	0.723195i	−0.932334	−	0.361597i	$-0.882232\pi$
$401$		−	14.4820i		−	0.723195i	0.932334	−	0.361597i	$-0.117768\pi$
$402$		0			0
$403$			26.8743i			1.33870i
$404$		0			0
$405$			3.82600i			0.190115i
$406$		0			0
$407$	43.2451			2.14358
$408$		0			0
$409$	0.368673			0.0182297			0.00911485	−	0.999958i	$-0.497099\pi$
$409$	0.368673			0.0182297			0.00911485	+	0.999958i	$0.497099\pi$
$410$		0			0
$411$			6.08170i			0.299988i
$412$		0			0
$413$			13.8319i			0.680625i
$414$		0			0
$415$		−	61.7753i		−	3.03243i
$416$		0			0
$417$	18.3433			0.898277
$418$		0			0
$419$			30.1772i			1.47425i	0.675754	+	0.737127i	$0.263818\pi$
$419$			30.1772i			1.47425i	−0.675754	+	0.737127i	$0.736182\pi$
$420$		0			0
$421$	−3.65937			−0.178347			−0.0891735	−	0.996016i	$-0.528423\pi$
$421$	−3.65937			−0.178347			−0.0891735	+	0.996016i	$0.528423\pi$
$422$		0			0
$423$	−6.21536			−0.302201
$424$		0			0
$425$	25.7615	−	30.2585i	1.24961	−	1.46775i
$426$		0			0
$427$	29.2348			1.41477
$428$		0			0
$429$	23.0725			1.11395
$430$		0			0
$431$		−	23.2781i		−	1.12127i	−0.828064	−	0.560634i	$-0.810557\pi$
$431$		−	23.2781i		−	1.12127i	0.828064	−	0.560634i	$-0.189443\pi$
$432$		0			0
$433$	−10.8084			−0.519417			−0.259708	−	0.965687i	$-0.583626\pi$
$433$	−10.8084			−0.519417			−0.259708	+	0.965687i	$0.583626\pi$
$434$		0			0
$435$		−	2.95139i		−	0.141508i
$436$		0			0
$437$			10.0939i			0.482859i
$438$		0			0
$439$			25.6364i			1.22356i	0.791028	+	0.611780i	$0.209546\pi$
$439$			25.6364i			1.22356i	−0.791028	+	0.611780i	$0.790454\pi$
$440$		0			0
$441$	−0.373156			−0.0177693
$442$		0			0
$443$	−28.3926			−1.34897			−0.674487	−	0.738287i	$-0.735635\pi$
$443$	−28.3926			−1.34897			−0.674487	+	0.738287i	$0.735635\pi$
$444$		0			0
$445$		−	70.5295i		−	3.34342i
$446$		0			0
$447$		−	18.1462i		−	0.858286i
$448$		0			0
$449$			5.58282i			0.263470i	0.991285	+	0.131735i	$0.0420548\pi$
$449$			5.58282i			0.263470i	−0.991285	+	0.131735i	$0.957945\pi$
$450$		0			0
$451$	18.2437			0.859062
$452$		0			0
$453$			21.5646i			1.01319i
$454$		0			0
$455$	−52.1273			−2.44377
$456$		0			0
$457$	4.63825			0.216968			0.108484	−	0.994098i	$-0.465400\pi$
$457$	4.63825			0.216968			0.108484	+	0.994098i	$0.465400\pi$
$458$		0			0
$459$	−3.13942	−	2.67284i	−0.146536	−	0.124757i
$460$		0			0
$461$	−23.2120			−1.08109			−0.540545	−	0.841315i	$-0.681782\pi$
$461$	−23.2120			−1.08109			−0.540545	+	0.841315i	$0.681782\pi$
$462$		0			0
$463$	−40.3109			−1.87341			−0.936703	−	0.350124i	$-0.886140\pi$
$463$	−40.3109			−1.87341			−0.936703	+	0.350124i	$0.886140\pi$
$464$		0			0
$465$			19.4274i			0.900923i
$466$		0			0
$467$	31.1428			1.44112			0.720559	−	0.693393i	$-0.243885\pi$
$467$	31.1428			1.44112			0.720559	+	0.693393i	$0.243885\pi$
$468$		0			0
$469$			31.0572i			1.43409i
$470$		0			0
$471$		−	16.5966i		−	0.764729i
$472$		0			0
$473$			0.529340i			0.0243391i
$474$		0			0
$475$	−12.0645			−0.553557
$476$		0			0
$477$	0.869694			0.0398205
$478$		0			0
$479$		−	17.0306i		−	0.778148i	−0.921207	−	0.389074i	$-0.872795\pi$
$479$		−	17.0306i		−	0.778148i	0.921207	−	0.389074i	$-0.127205\pi$
$480$		0			0
$481$		−	52.5021i		−	2.39389i
$482$		0			0
$483$			20.7588i			0.944560i
$484$		0			0
$485$	36.1045			1.63942
$486$		0			0
$487$			23.0001i			1.04223i	0.853486	+	0.521116i	$0.174484\pi$
$487$			23.0001i			1.04223i	−0.853486	+	0.521116i	$0.825516\pi$
$488$		0			0
$489$	−4.71883			−0.213393
$490$		0			0
$491$	24.9067			1.12402			0.562012	−	0.827129i	$-0.310027\pi$
$491$	24.9067			1.12402			0.562012	+	0.827129i	$0.310027\pi$
$492$		0			0
$493$	2.42176	+	2.06184i	0.109071	+	0.0928605i
$494$		0			0
$495$	16.6791			0.749670
$496$		0			0
$497$	4.45878			0.200003
$498$		0			0
$499$		−	13.1958i		−	0.590726i	−0.955385	−	0.295363i	$-0.904559\pi$
$499$		−	13.1958i		−	0.590726i	0.955385	−	0.295363i	$-0.0954406\pi$
$500$		0			0
$501$	4.49369			0.200763
$502$		0			0
$503$			11.1632i			0.497744i	0.968536	+	0.248872i	$0.0800599\pi$
$503$			11.1632i			0.497744i	−0.968536	+	0.248872i	$0.919940\pi$
$504$		0			0
$505$			35.7565i			1.59114i
$506$		0			0
$507$		−	15.0114i		−	0.666680i
$508$		0			0
$509$	−5.73995			−0.254419			−0.127209	−	0.991876i	$-0.540602\pi$
$509$	−5.73995			−0.254419			−0.127209	+	0.991876i	$0.540602\pi$
$510$		0			0
$511$	10.5074			0.464819
$512$		0			0
$513$			1.25173i			0.0552653i
$514$		0			0
$515$		−	8.32399i		−	0.366799i
$516$		0			0
$517$			27.0953i			1.19165i
$518$		0			0
$519$	25.8213			1.13343
$520$		0			0
$521$		−	35.4889i		−	1.55480i	−0.629009	−	0.777398i	$-0.716539\pi$
$521$		−	35.4889i		−	1.55480i	0.629009	−	0.777398i	$-0.283461\pi$
$522$		0			0
$523$	24.4952			1.07110			0.535551	−	0.844503i	$-0.320104\pi$
$523$	24.4952			1.07110			0.535551	+	0.844503i	$0.320104\pi$
$524$		0			0
$525$	−24.8114			−1.08286
$526$		0			0
$527$	−15.9411	−	13.5719i	−0.694406	−	0.591203i
$528$		0			0
$529$	−42.0279			−1.82730
$530$		0			0
$531$	−5.37316			−0.233175
$532$		0			0
$533$		−	22.1489i		−	0.959376i
$534$		0			0
$535$	39.1237			1.69146
$536$		0			0
$537$		−	16.8005i		−	0.724996i
$538$		0			0
$539$			1.62674i			0.0700686i
$540$		0			0
$541$			9.81478i			0.421970i	0.977489	+	0.210985i	$0.0676672\pi$
$541$			9.81478i			0.421970i	−0.977489	+	0.210985i	$0.932333\pi$
$542$		0			0
$543$	9.44396			0.405279
$544$		0			0
$545$	−41.6291			−1.78319
$546$		0			0
$547$		−	4.61758i		−	0.197434i	−0.995116	−	0.0987168i	$-0.968526\pi$
$547$		−	4.61758i		−	0.197434i	0.995116	−	0.0987168i	$-0.0314738\pi$
$548$		0			0
$549$			11.3566i			0.484686i
$550$		0			0
$551$		−	0.965591i		−	0.0411356i
$552$		0			0
$553$	−9.37316			−0.398587
$554$		0			0
$555$		−	37.9536i		−	1.61104i
$556$		0			0
$557$	4.33875			0.183839			0.0919193	−	0.995766i	$-0.470700\pi$
$557$	4.33875			0.183839			0.0919193	+	0.995766i	$0.470700\pi$
$558$		0			0
$559$	0.642650			0.0271812
$560$		0			0
$561$	−11.6520	+	13.6860i	−0.491948	+	0.577824i
$562$		0			0
$563$	5.29370			0.223103			0.111552	−	0.993759i	$-0.464418\pi$
$563$	5.29370			0.223103			0.111552	+	0.993759i	$0.464418\pi$
$564$		0			0
$565$	−5.10170			−0.214630
$566$		0			0
$567$			2.57427i			0.108109i
$568$		0			0
$569$	10.3490			0.433854			0.216927	−	0.976188i	$-0.430397\pi$
$569$	10.3490			0.433854			0.216927	+	0.976188i	$0.430397\pi$
$570$		0			0
$571$		−	23.9776i		−	1.00343i	−0.865033	−	0.501715i	$-0.832703\pi$
$571$		−	23.9776i		−	1.00343i	0.865033	−	0.501715i	$-0.167297\pi$
$572$		0			0
$573$		−	10.9794i		−	0.458673i
$574$		0			0
$575$		−	77.7227i		−	3.24126i
$576$		0			0
$577$	−10.9634			−0.456411			−0.228205	−	0.973613i	$-0.573286\pi$
$577$	−10.9634			−0.456411			−0.228205	+	0.973613i	$0.573286\pi$
$578$		0			0
$579$	5.35493			0.222543
$580$		0			0
$581$		−	41.5646i		−	1.72439i
$582$		0			0
$583$		−	3.79135i		−	0.157022i
$584$		0			0
$585$		−	20.2494i		−	0.837209i
$586$		0			0
$587$	−25.9953			−1.07294			−0.536471	−	0.843919i	$-0.680243\pi$
$587$	−25.9953			−1.07294			−0.536471	+	0.843919i	$0.680243\pi$
$588$		0			0
$589$			6.35595i			0.261892i
$590$		0			0
$591$	3.58315			0.147391
$592$		0			0
$593$	22.9275			0.941518			0.470759	−	0.882262i	$-0.343980\pi$
$593$	22.9275			0.941518			0.470759	+	0.882262i	$0.343980\pi$
$594$		0			0
$595$	26.3251	−	30.9206i	1.07922	−	1.26762i
$596$		0			0
$597$	−9.15943			−0.374870
$598$		0			0
$599$	−2.36679			−0.0967045			−0.0483523	−	0.998830i	$-0.515397\pi$
$599$	−2.36679			−0.0967045			−0.0483523	+	0.998830i	$0.515397\pi$
$600$		0			0
$601$		−	6.36512i		−	0.259639i	−0.991538	−	0.129819i	$-0.958560\pi$
$601$		−	6.36512i		−	0.259639i	0.991538	−	0.129819i	$-0.0414398\pi$
$602$		0			0
$603$	−12.0645			−0.491304
$604$		0			0
$605$		−	30.6251i		−	1.24509i
$606$		0			0
$607$			5.91041i			0.239896i	0.992780	+	0.119948i	$0.0382728\pi$
$607$			5.91041i			0.239896i	−0.992780	+	0.119948i	$0.961727\pi$
$608$		0			0
$609$		−	1.98580i		−	0.0804687i
$610$		0			0
$611$	32.8953			1.33080
$612$		0			0
$613$	−33.1335			−1.33825			−0.669124	−	0.743151i	$-0.733331\pi$
$613$	−33.1335			−1.33825			−0.669124	+	0.743151i	$0.733331\pi$
$614$		0			0
$615$		−	16.0114i		−	0.645642i
$616$		0			0
$617$			25.0469i			1.00835i	0.863601	+	0.504176i	$0.168204\pi$
$617$			25.0469i			1.00835i	−0.863601	+	0.504176i	$0.831796\pi$
$618$		0			0
$619$		−	18.8183i		−	0.756371i	−0.925730	−	0.378185i	$-0.876548\pi$
$619$		−	18.8183i		−	0.756371i	0.925730	−	0.378185i	$-0.123452\pi$
$620$		0			0
$621$	−8.06399			−0.323597
$622$		0			0
$623$		−	47.4548i		−	1.90124i
$624$		0			0
$625$	19.7046			0.788185
$626$		0			0
$627$	5.45681			0.217924
$628$		0			0
$629$	31.1428	+	26.5144i	1.24175	+	1.05720i
$630$		0			0
$631$	−40.1710			−1.59918			−0.799590	−	0.600546i	$-0.794950\pi$
$631$	−40.1710			−1.59918			−0.799590	+	0.600546i	$0.794950\pi$
$632$		0			0
$633$	7.73369			0.307387
$634$		0			0
$635$		−	7.69883i		−	0.305519i
$636$		0			0
$637$	1.97496			0.0782506
$638$		0			0
$639$			1.73206i			0.0685191i
$640$		0			0
$641$		−	6.31856i		−	0.249568i	−0.992184	−	0.124784i	$-0.960176\pi$
$641$		−	6.31856i		−	0.249568i	0.992184	−	0.124784i	$-0.0398238\pi$
$642$		0			0
$643$		−	40.9275i		−	1.61402i	−0.590536	−	0.807011i	$-0.701084\pi$
$643$		−	40.9275i		−	1.61402i	0.590536	−	0.807011i	$-0.298916\pi$
$644$		0			0
$645$	0.464570			0.0182924
$646$		0			0
$647$	14.1359			0.555740			0.277870	−	0.960619i	$-0.410372\pi$
$647$	14.1359			0.555740			0.277870	+	0.960619i	$0.410372\pi$
$648$		0			0
$649$			23.4238i			0.919465i
$650$		0			0
$651$			13.0714i			0.512309i
$652$		0			0
$653$		−	4.81440i		−	0.188402i	−0.995553	−	0.0942010i	$-0.969970\pi$
$653$		−	4.81440i		−	0.188402i	0.995553	−	0.0942010i	$-0.0300296\pi$
$654$		0			0
$655$	41.8739			1.63615
$656$		0			0
$657$			4.08170i			0.159242i
$658$		0			0
$659$	−10.2739			−0.400214			−0.200107	−	0.979774i	$-0.564129\pi$
$659$	−10.2739			−0.400214			−0.200107	+	0.979774i	$0.564129\pi$
$660$		0			0
$661$	−8.84109			−0.343878			−0.171939	−	0.985108i	$-0.555003\pi$
$661$	−8.84109			−0.343878			−0.171939	+	0.985108i	$0.555003\pi$
$662$		0			0
$663$	16.6156	+	14.1462i	0.645297	+	0.549393i
$664$		0			0
$665$	−12.3285			−0.478077
$666$		0			0
$667$	6.22060			0.240862
$668$		0			0
$669$		−	14.8128i		−	0.572695i
$670$		0			0
$671$	49.5080			1.91123
$672$		0			0
$673$			16.3708i			0.631049i	0.948917	+	0.315524i	$0.102180\pi$
$673$			16.3708i			0.631049i	−0.948917	+	0.315524i	$0.897820\pi$
$674$		0			0
$675$		−	9.63825i		−	0.370976i
$676$		0			0
$677$		−	14.5723i		−	0.560059i	−0.959991	−	0.280030i	$-0.909656\pi$
$677$		−	14.5723i		−	0.560059i	0.959991	−	0.280030i	$-0.0903443\pi$
$678$		0			0
$679$	24.2924			0.932256
$680$		0			0
$681$	1.23837			0.0474543
$682$		0			0
$683$			4.34475i			0.166247i	0.996539	+	0.0831237i	$0.0264896\pi$
$683$			4.34475i			0.166247i	−0.996539	+	0.0831237i	$0.973510\pi$
$684$		0			0
$685$		−	23.2686i		−	0.889046i
$686$		0			0
$687$		−	6.61265i		−	0.252288i
$688$		0			0
$689$	−4.60292			−0.175357
$690$		0			0
$691$			13.2035i			0.502285i	0.967950	+	0.251142i	$0.0808063\pi$
$691$			13.2035i			0.502285i	−0.967950	+	0.251142i	$0.919194\pi$
$692$		0			0
$693$	11.2223			0.426299
$694$		0			0
$695$	−70.1815			−2.66214
$696$		0			0
$697$	13.1381	+	11.1855i	0.497643	+	0.423683i
$698$		0			0
$699$	−8.96363			−0.339036
$700$		0			0
$701$	−41.6930			−1.57472			−0.787362	−	0.616491i	$-0.788554\pi$
$701$	−41.6930			−1.57472			−0.787362	+	0.616491i	$0.788554\pi$
$702$		0			0
$703$		−	12.4171i		−	0.468319i
$704$		0			0
$705$	23.7800			0.895605
$706$		0			0
$707$			24.0582i			0.904803i
$708$		0			0
$709$		−	37.2969i		−	1.40072i	−0.713792	−	0.700358i	$-0.753024\pi$
$709$		−	37.2969i		−	1.40072i	0.713792	−	0.700358i	$-0.246976\pi$
$710$		0			0
$711$		−	3.64110i		−	0.136552i
$712$		0			0
$713$	−40.9467			−1.53347
$714$		0			0
$715$	−88.2755			−3.30132
$716$		0			0
$717$			21.8891i			0.817465i
$718$		0			0
$719$			13.1074i			0.488822i	0.969672	+	0.244411i	$0.0785946\pi$
$719$			13.1074i			0.488822i	−0.969672	+	0.244411i	$0.921405\pi$
$720$		0			0
$721$		−	5.60067i		−	0.208580i
$722$		0			0
$723$	12.6726			0.471297
$724$		0			0
$725$			7.43499i			0.276129i
$726$		0			0
$727$	−42.7714			−1.58630			−0.793151	−	0.609025i	$-0.791561\pi$
$727$	−42.7714			−1.58630			−0.793151	+	0.609025i	$0.791561\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−0.324548	+	0.381203i	−0.0120038	+	0.0140993i
$732$		0			0
$733$	45.0387			1.66354			0.831771	−	0.555119i	$-0.187327\pi$
$733$	45.0387			1.66354			0.831771	+	0.555119i	$0.187327\pi$
$734$		0			0
$735$	1.42769			0.0526612
$736$		0			0
$737$			52.5941i			1.93733i
$738$		0			0
$739$	16.1065			0.592486			0.296243	−	0.955113i	$-0.404266\pi$
$739$	16.1065			0.592486			0.296243	+	0.955113i	$0.404266\pi$
$740$		0			0
$741$		−	6.62489i		−	0.243371i
$742$		0			0
$743$			42.4737i			1.55821i	0.626894	+	0.779104i	$0.284326\pi$
$743$			42.4737i			1.55821i	−0.626894	+	0.779104i	$0.715674\pi$
$744$		0			0
$745$			69.4273i			2.54362i
$746$		0			0
$747$	16.1462			0.590758
$748$		0			0
$749$	26.3238			0.961850
$750$		0			0
$751$			4.39433i			0.160351i	0.996781	+	0.0801757i	$0.0255481\pi$
$751$			4.39433i			0.160351i	−0.996781	+	0.0801757i	$0.974452\pi$
$752$		0			0
$753$			23.4919i			0.856091i
$754$		0			0
$755$		−	82.5061i		−	3.00270i
$756$		0			0
$757$	29.4216			1.06935			0.534673	−	0.845059i	$-0.320435\pi$
$757$	29.4216			1.06935			0.534673	+	0.845059i	$0.320435\pi$
$758$		0			0
$759$			35.1542i			1.27602i
$760$		0			0
$761$	−0.691341			−0.0250611			−0.0125306	−	0.999921i	$-0.503989\pi$
$761$	−0.691341			−0.0250611			−0.0125306	+	0.999921i	$0.503989\pi$
$762$		0			0
$763$	−28.0095			−1.01401
$764$		0			0
$765$	12.0114	+	10.2263i	0.434273	+	0.369731i
$766$		0			0
$767$	28.4379			1.02683
$768$		0			0
$769$	8.65154			0.311982			0.155991	−	0.987758i	$-0.450143\pi$
$769$	8.65154			0.311982			0.155991	+	0.987758i	$0.450143\pi$
$770$		0			0
$771$			10.2428i			0.368887i
$772$		0			0
$773$	−18.0264			−0.648364			−0.324182	−	0.945995i	$-0.605089\pi$
$773$	−18.0264			−0.648364			−0.324182	+	0.945995i	$0.605089\pi$
$774$		0			0
$775$		−	48.9404i		−	1.75799i
$776$		0			0
$777$		−	25.5366i		−	0.916119i
$778$		0			0
$779$		−	5.23837i		−	0.187684i
$780$		0			0
$781$	7.55075			0.270187
$782$		0			0
$783$	0.771405			0.0275678
$784$		0			0
$785$			63.4984i			2.26636i
$786$		0			0
$787$			38.4419i			1.37031i	0.728400	+	0.685153i	$0.240264\pi$
$787$			38.4419i			1.37031i	−0.728400	+	0.685153i	$0.759736\pi$
$788$		0			0
$789$			1.06581i			0.0379438i
$790$		0			0
$791$	−3.43260			−0.122049
$792$		0			0
$793$		−	60.1055i		−	2.13441i
$794$		0			0
$795$	−3.32745			−0.118012
$796$		0			0
$797$	22.6979			0.804001			0.402001	−	0.915639i	$-0.368315\pi$
$797$	22.6979			0.804001			0.402001	+	0.915639i	$0.368315\pi$
$798$		0			0
$799$	−16.6126	+	19.5126i	−0.587713	+	0.690307i
$800$		0			0
$801$	18.4343			0.651344
$802$		0			0
$803$	17.7938			0.627930
$804$		0			0
$805$		−	79.4233i		−	2.79930i
$806$		0			0
$807$	−24.1151			−0.848892
$808$		0			0
$809$			38.3681i			1.34895i	0.738297	+	0.674475i	$0.235630\pi$
$809$			38.3681i			1.34895i	−0.738297	+	0.674475i	$0.764370\pi$
$810$		0			0
$811$			32.6943i			1.14805i	0.818836	+	0.574027i	$0.194620\pi$
$811$			32.6943i			1.14805i	−0.818836	+	0.574027i	$0.805380\pi$
$812$		0			0
$813$		−	23.7952i		−	0.834534i
$814$		0			0
$815$	18.0542			0.632412
$816$		0			0
$817$	0.151991			0.00531749
$818$		0			0
$819$		−	13.6245i		−	0.476079i
$820$		0			0
$821$			12.4604i			0.434872i	0.976075	+	0.217436i	$0.0697694\pi$
$821$			12.4604i			0.434872i	−0.976075	+	0.217436i	$0.930231\pi$
$822$		0			0
$823$			16.3050i			0.568355i	0.958772	+	0.284178i	$0.0917205\pi$
$823$			16.3050i			0.568355i	−0.958772	+	0.284178i	$0.908279\pi$
$824$		0			0
$825$	−42.0171			−1.46285
$826$		0			0
$827$			29.2416i			1.01683i	0.861112	+	0.508416i	$0.169769\pi$
$827$			29.2416i			1.01683i	−0.861112	+	0.508416i	$0.830231\pi$
$828$		0			0
$829$	43.1154			1.49746			0.748729	−	0.662876i	$-0.230664\pi$
$829$	43.1154			1.49746			0.748729	+	0.662876i	$0.230664\pi$
$830$		0			0
$831$	−24.3635			−0.845160
$832$		0			0
$833$	−0.997383	+	1.17149i	−0.0345573	+	0.0405898i
$834$		0			0
$835$	−17.1928			−0.594983
$836$		0			0
$837$	−5.07773			−0.175512
$838$		0			0
$839$			20.9429i			0.723031i	0.932366	+	0.361515i	$0.117740\pi$
$839$			20.9429i			0.723031i	−0.932366	+	0.361515i	$0.882260\pi$
$840$		0			0
$841$	28.4049			0.979481
$842$		0			0
$843$		−	26.5977i		−	0.916073i
$844$		0			0
$845$			57.4336i			1.97578i
$846$		0			0
$847$		−	20.6057i		−	0.708019i
$848$		0			0
$849$	−4.02179			−0.138027
$850$		0			0
$851$	79.9942			2.74217
$852$		0			0
$853$		−	22.4863i		−	0.769917i	−0.922934	−	0.384958i	$-0.874216\pi$
$853$		−	22.4863i		−	0.769917i	0.922934	−	0.384958i	$-0.125784\pi$
$854$		0			0
$855$		−	4.78912i		−	0.163784i
$856$		0			0
$857$		−	21.3269i		−	0.728512i	−0.931299	−	0.364256i	$-0.881323\pi$
$857$		−	21.3269i		−	0.728512i	0.931299	−	0.364256i	$-0.118677\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		−	10.7730i		−	0.367144i
$862$		0			0
$863$	11.3492			0.386332			0.193166	−	0.981166i	$-0.438124\pi$
$863$	11.3492			0.386332			0.193166	+	0.981166i	$0.438124\pi$
$864$		0			0
$865$	−98.7923			−3.35904
$866$		0			0
$867$	−16.7823	+	2.71190i	−0.569957	+	0.0921010i
$868$		0			0
$869$	−15.8731			−0.538456
$870$		0			0
$871$	63.8523			2.16355
$872$		0			0
$873$			9.43663i			0.319382i
$874$		0			0
$875$	45.6827			1.54436
$876$		0			0
$877$			28.6783i			0.968398i	0.874958	+	0.484199i	$0.160889\pi$
$877$			28.6783i			0.968398i	−0.874958	+	0.484199i	$0.839111\pi$
$878$		0			0
$879$		−	3.80388i		−	0.128302i
$880$		0			0
$881$		−	46.4661i		−	1.56548i	−0.622348	−	0.782741i	$-0.713821\pi$
$881$		−	46.4661i		−	1.56548i	0.622348	−	0.782741i	$-0.286179\pi$
$882$		0			0
$883$	38.6649			1.30118			0.650589	−	0.759430i	$-0.274522\pi$
$883$	38.6649			1.30118			0.650589	+	0.759430i	$0.274522\pi$
$884$		0			0
$885$	20.5577			0.691039
$886$		0			0
$887$		−	17.5006i		−	0.587613i	−0.955865	−	0.293807i	$-0.905078\pi$
$887$		−	17.5006i		−	0.587613i	0.955865	−	0.293807i	$-0.0949222\pi$
$888$		0			0
$889$		−	5.18004i		−	0.173733i
$890$		0			0
$891$			4.35941i			0.146046i
$892$		0			0
$893$	7.77996			0.260347
$894$		0			0
$895$			64.2788i			2.14860i
$896$		0			0
$897$	42.6793			1.42502
$898$		0			0
$899$	3.91698			0.130639
$900$		0			0
$901$	2.32455	−	2.73033i	0.0774419	−	0.0909606i
$902$		0			0
$903$	0.312579			0.0104020
$904$		0			0
$905$	−36.1326			−1.20109
$906$		0			0
$907$			29.8221i			0.990227i	0.868828	+	0.495113i	$0.164874\pi$
$907$			29.8221i			0.990227i	−0.868828	+	0.495113i	$0.835126\pi$
$908$		0			0
$909$	−9.34567			−0.309976
$910$		0			0
$911$		−	31.6011i		−	1.04699i	−0.852028	−	0.523496i	$-0.824628\pi$
$911$		−	31.6011i		−	1.04699i	0.852028	−	0.523496i	$-0.175372\pi$
$912$		0			0
$913$		−	70.3879i		−	2.32950i
$914$		0			0
$915$		−	43.4502i		−	1.43642i
$916$		0			0
$917$	28.1742			0.930395
$918$		0			0
$919$	44.9584			1.48304			0.741520	−	0.670930i	$-0.234105\pi$
$919$	44.9584			1.48304			0.741520	+	0.670930i	$0.234105\pi$
$920$		0			0
$921$			5.30866i			0.174926i
$922$		0			0
$923$		−	9.16705i		−	0.301737i
$924$		0			0
$925$			95.6108i			3.14366i
$926$		0			0
$927$	2.17564			0.0714573
$928$		0			0
$929$			41.0350i			1.34631i	0.739500	+	0.673157i	$0.235062\pi$
$929$			41.0350i			1.34631i	−0.739500	+	0.673157i	$0.764938\pi$
$930$		0			0
$931$	0.467090			0.0153083
$932$		0			0
$933$	26.0244			0.852002
$934$		0			0
$935$	44.5805	−	52.3627i	1.45794	−	1.71244i
$936$		0			0
$937$	23.2838			0.760648			0.380324	−	0.924853i	$-0.375812\pi$
$937$	23.2838			0.760648			0.380324	+	0.924853i	$0.375812\pi$
$938$		0			0
$939$	−21.2547			−0.693621
$940$		0			0
$941$		−	9.46742i		−	0.308629i	−0.988022	−	0.154315i	$-0.950683\pi$
$941$		−	9.46742i		−	0.308629i	0.988022	−	0.154315i	$-0.0493169\pi$
$942$		0			0
$943$	33.7470			1.09895
$944$		0			0
$945$		−	9.84913i		−	0.320392i
$946$		0			0
$947$		−	9.27753i		−	0.301479i	−0.988574	−	0.150740i	$-0.951835\pi$
$947$		−	9.27753i		−	0.301479i	0.988574	−	0.150740i	$-0.0481655\pi$
$948$		0			0
$949$		−	21.6027i		−	0.701254i
$950$		0			0
$951$	26.9269			0.873163
$952$		0			0
$953$	14.2976			0.463145			0.231573	−	0.972818i	$-0.425613\pi$
$953$	14.2976			0.463145			0.231573	+	0.972818i	$0.425613\pi$
$954$		0			0
$955$			42.0073i			1.35932i
$956$		0			0
$957$		−	3.36287i		−	0.108706i
$958$		0			0
$959$		−	15.6559i		−	0.505556i
$960$		0			0
$961$	5.21668			0.168280
$962$		0			0
$963$			10.2257i			0.329520i
$964$		0			0
$965$	−20.4879			−0.659530
$966$		0			0
$967$	18.1392			0.583317			0.291658	−	0.956523i	$-0.405793\pi$
$967$	18.1392			0.583317			0.291658	+	0.956523i	$0.405793\pi$
$968$		0			0
$969$	3.92971	+	3.34567i	0.126240	+	0.107478i
$970$		0			0
$971$	44.5997			1.43127			0.715637	−	0.698473i	$-0.246137\pi$
$971$	44.5997			1.43127			0.715637	+	0.698473i	$0.246137\pi$
$972$		0			0
$973$	−47.2206			−1.51382
$974$		0			0
$975$			51.0112i			1.63367i
$976$		0			0
$977$	6.73735			0.215547			0.107773	−	0.994175i	$-0.465628\pi$
$977$	6.73735			0.215547			0.107773	+	0.994175i	$0.465628\pi$
$978$		0			0
$979$		−	80.3627i		−	2.56840i
$980$		0			0
$981$		−	10.8806i		−	0.347390i
$982$		0			0
$983$			6.62838i			0.211412i	0.994397	+	0.105706i	$0.0337103\pi$
$983$			6.62838i			0.211412i	−0.994397	+	0.105706i	$0.966290\pi$
$984$		0			0
$985$	−13.7091			−0.436809
$986$		0			0
$987$	16.0000			0.509286
$988$		0			0
$989$			0.979167i			0.0311357i
$990$		0			0
$991$		−	43.0998i		−	1.36911i	−0.728961	−	0.684555i	$-0.759997\pi$
$991$		−	43.0998i		−	1.36911i	0.728961	−	0.684555i	$-0.240003\pi$
$992$		0			0
$993$		−	2.62489i		−	0.0832983i
$994$		0			0
$995$	35.0439			1.11097
$996$		0			0
$997$		−	42.6039i		−	1.34928i	−0.738147	−	0.674640i	$-0.764299\pi$
$997$		−	42.6039i		−	1.34928i	0.738147	−	0.674640i	$-0.235701\pi$
$998$		0			0
$999$	9.91994			0.313853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3264.2.c.q.577.5		10
4.3	odd	2		3264.2.c.r.577.10		10
8.3	odd	2		1632.2.c.f.577.1	yes	10
8.5	even	2		1632.2.c.e.577.6	yes	10
17.16	even	2	inner	3264.2.c.q.577.6		10
24.5	odd	2		4896.2.c.s.577.10		10
24.11	even	2		4896.2.c.r.577.10		10
68.67	odd	2		3264.2.c.r.577.1		10
136.67	odd	2		1632.2.c.f.577.10	yes	10
136.101	even	2		1632.2.c.e.577.5	✓	10
408.101	odd	2		4896.2.c.s.577.1		10
408.203	even	2		4896.2.c.r.577.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1632.2.c.e.577.5	✓	10	136.101	even	2
1632.2.c.e.577.6	yes	10	8.5	even	2
1632.2.c.f.577.1	yes	10	8.3	odd	2
1632.2.c.f.577.10	yes	10	136.67	odd	2
3264.2.c.q.577.5		10	1.1	even	1	trivial
3264.2.c.q.577.6		10	17.16	even	2	inner
3264.2.c.r.577.1		10	68.67	odd	2
3264.2.c.r.577.10		10	4.3	odd	2
4896.2.c.r.577.1		10	408.203	even	2
4896.2.c.r.577.10		10	24.11	even	2
4896.2.c.s.577.1		10	408.101	odd	2
4896.2.c.s.577.10		10	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3264.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.82600i q^{5} +2.57427i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.82600i q^{5} +2.57427i q^{7} -1.00000 q^{9} +4.35941i q^{11} +5.29258 q^{13} +3.82600 q^{15} +(-2.67284 + 3.13942i) q^{17} +1.25173 q^{19} +2.57427 q^{21} +8.06399i q^{23} -9.63825 q^{25} +1.00000i q^{27} -0.771405i q^{29} +5.07773i q^{31} +4.35941 q^{33} -9.84913 q^{35} -9.91994i q^{37} -5.29258i q^{39} -4.18490i q^{41} +0.121425 q^{43} -3.82600i q^{45} +6.21536 q^{47} +0.373156 q^{49} +(3.13942 + 2.67284i) q^{51} -0.869694 q^{53} -16.6791 q^{55} -1.25173i q^{57} +5.37316 q^{59} -11.3566i q^{61} -2.57427i q^{63} +20.2494i q^{65} +12.0645 q^{67} +8.06399 q^{69} -1.73206i q^{71} -4.08170i q^{73} +9.63825i q^{75} -11.2223 q^{77} +3.64110i q^{79} +1.00000 q^{81} -16.1462 q^{83} +(-12.0114 - 10.2263i) q^{85} -0.771405 q^{87} -18.4343 q^{89} +13.6245i q^{91} +5.07773 q^{93} +4.78912i q^{95} -9.43663i q^{97} -4.35941i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^9	\( 10 q - 10 q^{9} + 8 q^{13} - 4 q^{15} - 6 q^{17} - 8 q^{19} + 4 q^{21} - 10 q^{25} + 4 q^{33} - 16 q^{35} - 16 q^{43} + 24 q^{47} - 34 q^{49} + 8 q^{51} - 12 q^{53} - 56 q^{55} + 16 q^{59} + 16 q^{69} + 8 q^{77} + 10 q^{81} + 8 q^{83} + 4 q^{85} + 12 q^{87} - 12 q^{89} - 12 q^{93}+O(q^{100}) \) 10 * q - 10 * q^9 + 8 * q^13 - 4 * q^15 - 6 * q^17 - 8 * q^19 + 4 * q^21 - 10 * q^25 + 4 * q^33 - 16 * q^35 - 16 * q^43 + 24 * q^47 - 34 * q^49 + 8 * q^51 - 12 * q^53 - 56 * q^55 + 16 * q^59 + 16 * q^69 + 8 * q^77 + 10 * q^81 + 8 * q^83 + 4 * q^85 + 12 * q^87 - 12 * q^89 - 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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