LMFDB - Embedded newform 4560.2.d.i.2431.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.d (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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} + \cdots + 64 $ x^12 - 4x^11 + 8x^10 + 4x^9 + 36x^8 - 128x^7 + 232x^6 + 104x^5 + 324x^4 - 784x^3 + 800x^2 - 320*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.10
Root			$1.62658 - 1.62658i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.i.2431.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +2.44391i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +2.44391i q^{7} +1.00000 q^{9} -5.73545i q^{11} -3.19074i q^{13} +1.00000 q^{15} -1.96162 q^{17} +(0.679626 + 4.30559i) q^{19} -2.44391i q^{21} -7.49018i q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.19459i q^{29} -4.75564 q^{31} +5.73545i q^{33} -2.44391i q^{35} +6.54876i q^{37} +3.19074i q^{39} -8.05124i q^{41} +3.08465i q^{43} -1.00000 q^{45} -7.96162i q^{47} +1.02732 q^{49} +1.96162 q^{51} +6.48025i q^{53} +5.73545i q^{55} +(-0.679626 - 4.30559i) q^{57} +13.4990 q^{59} +6.82010 q^{61} +2.44391i q^{63} +3.19074i q^{65} -15.2508 q^{67} +7.49018i q^{69} -12.8462 q^{71} +1.35802 q^{73} -1.00000 q^{75} +14.0169 q^{77} +0.478237 q^{79} +1.00000 q^{81} +12.5719i q^{83} +1.96162 q^{85} -6.19459i q^{87} +16.2218i q^{89} +7.79787 q^{91} +4.75564 q^{93} +(-0.679626 - 4.30559i) q^{95} -4.92214i q^{97} -5.73545i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9	$ 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9} + 12 q^{15} - 8 q^{17} - 8 q^{19} + 12 q^{25} - 12 q^{27} - 12 q^{45} - 20 q^{49} + 8 q^{51} + 8 q^{57} - 8 q^{59} + 8 q^{67} - 32 q^{71} - 24 q^{73} - 12 q^{75} + 56 q^{77} - 16 q^{79} + 12 q^{81} + 8 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) $ 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9 + 12 * q^15 - 8 * q^17 - 8 * q^19 + 12 * q^25 - 12 * q^27 - 12 * q^45 - 20 * q^49 + 8 * q^51 + 8 * q^57 - 8 * q^59 + 8 * q^67 - 32 * q^71 - 24 * q^73 - 12 * q^75 + 56 * q^77 - 16 * q^79 + 12 * q^81 + 8 * q^85 + 16 * q^91 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			2.44391i			0.923710i	0.886955	+	0.461855i	$0.152816\pi$
$7$			2.44391i			0.923710i	−0.886955	+	0.461855i	$0.847184\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	5.73545i		−	1.72930i	−0.502372	−	0.864652i	$-0.667539\pi$
$11$		−	5.73545i		−	1.72930i	0.502372	−	0.864652i	$-0.332461\pi$
$12$		0			0
$13$		−	3.19074i		−	0.884952i	−0.896780	−	0.442476i	$-0.854100\pi$
$13$		−	3.19074i		−	0.884952i	0.896780	−	0.442476i	$-0.145900\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	−1.96162			−0.475763			−0.237882	−	0.971294i	$-0.576453\pi$
$17$	−1.96162			−0.475763			−0.237882	+	0.971294i	$0.576453\pi$
$18$		0			0
$19$	0.679626	+	4.30559i	0.155917	+	0.987770i
$19$	0.679626	+	4.30559i	0.155917	+	0.987770i
$20$		0			0
$21$		−	2.44391i		−	0.533304i
$22$		0			0
$23$		−	7.49018i		−	1.56181i	−0.624649	−	0.780906i	$-0.714758\pi$
$23$		−	7.49018i		−	1.56181i	0.624649	−	0.780906i	$-0.285242\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			6.19459i			1.15031i	0.818045	+	0.575154i	$0.195058\pi$
$29$			6.19459i			1.15031i	−0.818045	+	0.575154i	$0.804942\pi$
$30$		0			0
$31$	−4.75564			−0.854139			−0.427070	−	0.904219i	$-0.640454\pi$
$31$	−4.75564			−0.854139			−0.427070	+	0.904219i	$0.640454\pi$
$32$		0			0
$33$			5.73545i			0.998414i
$34$		0			0
$35$		−	2.44391i		−	0.413096i
$36$		0			0
$37$			6.54876i			1.07661i	0.842750	+	0.538304i	$0.180935\pi$
$37$			6.54876i			1.07661i	−0.842750	+	0.538304i	$0.819065\pi$
$38$		0			0
$39$			3.19074i			0.510928i
$40$		0			0
$41$		−	8.05124i		−	1.25739i	−0.777651	−	0.628696i	$-0.783589\pi$
$41$		−	8.05124i		−	1.25739i	0.777651	−	0.628696i	$-0.216411\pi$
$42$		0			0
$43$			3.08465i			0.470405i	0.971946	+	0.235203i	$0.0755754\pi$
$43$			3.08465i			0.470405i	−0.971946	+	0.235203i	$0.924425\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$		−	7.96162i		−	1.16132i	−0.814145	−	0.580661i	$-0.802794\pi$
$47$		−	7.96162i		−	1.16132i	0.814145	−	0.580661i	$-0.197206\pi$
$48$		0			0
$49$	1.02732			0.146760
$50$		0			0
$51$	1.96162			0.274682
$52$		0			0
$53$			6.48025i			0.890131i	0.895498	+	0.445065i	$0.146820\pi$
$53$			6.48025i			0.890131i	−0.895498	+	0.445065i	$0.853180\pi$
$54$		0			0
$55$			5.73545i			0.773368i
$56$		0			0
$57$	−0.679626	−	4.30559i	−0.0900187	−	0.570289i
$58$		0			0
$59$	13.4990			1.75742			0.878710	−	0.477357i	$-0.158405\pi$
$59$	13.4990			1.75742			0.878710	+	0.477357i	$0.158405\pi$
$60$		0			0
$61$	6.82010			0.873225			0.436612	−	0.899650i	$-0.356178\pi$
$61$	6.82010			0.873225			0.436612	+	0.899650i	$0.356178\pi$
$62$		0			0
$63$			2.44391i			0.307903i
$64$		0			0
$65$			3.19074i			0.395763i
$66$		0			0
$67$	−15.2508			−1.86318			−0.931590	−	0.363510i	$-0.881578\pi$
$67$	−15.2508			−1.86318			−0.931590	+	0.363510i	$0.881578\pi$
$68$		0			0
$69$			7.49018i			0.901712i
$70$		0			0
$71$	−12.8462			−1.52456			−0.762281	−	0.647246i	$-0.775921\pi$
$71$	−12.8462			−1.52456			−0.762281	+	0.647246i	$0.775921\pi$
$72$		0			0
$73$	1.35802			0.158944			0.0794718	−	0.996837i	$-0.474677\pi$
$73$	1.35802			0.158944			0.0794718	+	0.996837i	$0.474677\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	14.0169			1.59737
$78$		0			0
$79$	0.478237			0.0538058			0.0269029	−	0.999638i	$-0.491435\pi$
$79$	0.478237			0.0538058			0.0269029	+	0.999638i	$0.491435\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.5719i			1.37995i	0.723836	+	0.689973i	$0.242377\pi$
$83$			12.5719i			1.37995i	−0.723836	+	0.689973i	$0.757623\pi$
$84$		0			0
$85$	1.96162			0.212768
$86$		0			0
$87$		−	6.19459i		−	0.664130i
$88$		0			0
$89$			16.2218i			1.71951i	0.510711	+	0.859753i	$0.329382\pi$
$89$			16.2218i			1.71951i	−0.510711	+	0.859753i	$0.670618\pi$
$90$		0			0
$91$	7.79787			0.817439
$92$		0			0
$93$	4.75564			0.493137
$94$		0			0
$95$	−0.679626	−	4.30559i	−0.0697282	−	0.441744i
$96$		0			0
$97$		−	4.92214i		−	0.499768i	−0.968276	−	0.249884i	$-0.919608\pi$
$97$		−	4.92214i		−	0.499768i	0.968276	−	0.249884i	$-0.0803925\pi$
$98$		0			0
$99$		−	5.73545i		−	0.576434i
$100$		0			0
$101$	1.66298			0.165473			0.0827363	−	0.996571i	$-0.473634\pi$
$101$	1.66298			0.165473			0.0827363	+	0.996571i	$0.473634\pi$
$102$		0			0
$103$	−12.5348			−1.23509			−0.617543	−	0.786537i	$-0.711872\pi$
$103$	−12.5348			−1.23509			−0.617543	+	0.786537i	$0.711872\pi$
$104$		0			0
$105$			2.44391i			0.238501i
$106$		0			0
$107$	−4.94548			−0.478097			−0.239049	−	0.971008i	$-0.576836\pi$
$107$	−4.94548			−0.478097			−0.239049	+	0.971008i	$0.576836\pi$
$108$		0			0
$109$			9.39414i			0.899796i	0.893080	+	0.449898i	$0.148540\pi$
$109$			9.39414i			0.899796i	−0.893080	+	0.449898i	$0.851460\pi$
$110$		0			0
$111$		−	6.54876i		−	0.621580i
$112$		0			0
$113$		−	8.93849i		−	0.840862i	−0.907324	−	0.420431i	$-0.861879\pi$
$113$		−	8.93849i		−	0.840862i	0.907324	−	0.420431i	$-0.138121\pi$
$114$		0			0
$115$			7.49018i			0.698463i
$116$		0			0
$117$		−	3.19074i		−	0.294984i
$118$		0			0
$119$		−	4.79402i		−	0.439467i
$120$		0			0
$121$	−21.8954			−1.99049
$122$		0			0
$123$			8.05124i			0.725955i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	5.54176			0.491752			0.245876	−	0.969301i	$-0.420924\pi$
$127$	5.54176			0.491752			0.245876	+	0.969301i	$0.420924\pi$
$128$		0			0
$129$		−	3.08465i		−	0.271589i
$130$		0			0
$131$		−	12.9414i		−	1.13070i	−0.824852	−	0.565349i	$-0.808742\pi$
$131$		−	12.9414i		−	1.13070i	0.824852	−	0.565349i	$-0.191258\pi$
$132$		0			0
$133$	−10.5225	+	1.66094i	−0.912413	+	0.144022i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	0.393778			0.0336427			0.0168213	−	0.999859i	$-0.494645\pi$
$137$	0.393778			0.0336427			0.0168213	+	0.999859i	$0.494645\pi$
$138$		0			0
$139$		−	5.34696i		−	0.453523i	−0.973950	−	0.226761i	$-0.927186\pi$
$139$		−	5.34696i		−	0.453523i	0.973950	−	0.226761i	$-0.0728138\pi$
$140$		0			0
$141$			7.96162i			0.670490i
$142$		0			0
$143$	−18.3003			−1.53035
$144$		0			0
$145$		−	6.19459i		−	0.514433i
$146$		0			0
$147$	−1.02732			−0.0847319
$148$		0			0
$149$	−16.2286			−1.32950			−0.664748	−	0.747068i	$-0.731461\pi$
$149$	−16.2286			−1.32950			−0.664748	+	0.747068i	$0.731461\pi$
$150$		0			0
$151$	−10.1917			−0.829385			−0.414692	−	0.909962i	$-0.636111\pi$
$151$	−10.1917			−0.829385			−0.414692	+	0.909962i	$0.636111\pi$
$152$		0			0
$153$	−1.96162			−0.158588
$154$		0			0
$155$	4.75564			0.381983
$156$		0			0
$157$	−1.65316			−0.131936			−0.0659681	−	0.997822i	$-0.521014\pi$
$157$	−1.65316			−0.131936			−0.0659681	+	0.997822i	$0.521014\pi$
$158$		0			0
$159$		−	6.48025i		−	0.513917i
$160$		0			0
$161$	18.3053			1.44266
$162$		0			0
$163$		−	10.0314i		−	0.785717i	−0.919599	−	0.392859i	$-0.871486\pi$
$163$		−	10.0314i		−	0.785717i	0.919599	−	0.392859i	$-0.128514\pi$
$164$		0			0
$165$		−	5.73545i		−	0.446504i
$166$		0			0
$167$	−25.0164			−1.93582			−0.967912	−	0.251288i	$-0.919146\pi$
$167$	−25.0164			−1.93582			−0.967912	+	0.251288i	$0.919146\pi$
$168$		0			0
$169$	2.81917			0.216859
$170$		0			0
$171$	0.679626	+	4.30559i	0.0519723	+	0.329257i
$172$		0			0
$173$		−	2.66930i		−	0.202943i	−0.994838	−	0.101472i	$-0.967645\pi$
$173$		−	2.66930i		−	0.202943i	0.994838	−	0.101472i	$-0.0323551\pi$
$174$		0			0
$175$			2.44391i			0.184742i
$176$		0			0
$177$	−13.4990			−1.01465
$178$		0			0
$179$	10.2924			0.769292			0.384646	−	0.923064i	$-0.374323\pi$
$179$	10.2924			0.769292			0.384646	+	0.923064i	$0.374323\pi$
$180$		0			0
$181$			21.8326i			1.62281i	0.584486	+	0.811404i	$0.301296\pi$
$181$			21.8326i			1.62281i	−0.584486	+	0.811404i	$0.698704\pi$
$182$		0			0
$183$	−6.82010			−0.504157
$184$		0			0
$185$		−	6.54876i		−	0.481474i
$186$		0			0
$187$			11.2508i			0.822739i
$188$		0			0
$189$		−	2.44391i		−	0.177768i
$190$		0			0
$191$			0.368258i			0.0266462i	0.999911	+	0.0133231i	$0.00424100\pi$
$191$			0.368258i			0.0266462i	−0.999911	+	0.0133231i	$0.995759\pi$
$192$		0			0
$193$			4.32078i			0.311017i	0.987835	+	0.155508i	$0.0497016\pi$
$193$			4.32078i			0.311017i	−0.987835	+	0.155508i	$0.950298\pi$
$194$		0			0
$195$		−	3.19074i		−	0.228494i
$196$		0			0
$197$	−3.43892			−0.245013			−0.122507	−	0.992468i	$-0.539093\pi$
$197$	−3.43892			−0.245013			−0.122507	+	0.992468i	$0.539093\pi$
$198$		0			0
$199$		−	8.82743i		−	0.625760i	−0.949793	−	0.312880i	$-0.898706\pi$
$199$		−	8.82743i		−	0.625760i	0.949793	−	0.312880i	$-0.101294\pi$
$200$		0			0
$201$	15.2508			1.07571
$202$		0			0
$203$	−15.1390			−1.06255
$204$		0			0
$205$			8.05124i			0.562323i
$206$		0			0
$207$		−	7.49018i		−	0.520604i
$208$		0			0
$209$	24.6945	−	3.89796i	1.70815	−	0.269628i
$210$		0			0
$211$	−7.35008			−0.506000			−0.253000	−	0.967466i	$-0.581417\pi$
$211$	−7.35008			−0.506000			−0.253000	+	0.967466i	$0.581417\pi$
$212$		0			0
$213$	12.8462			0.880206
$214$		0			0
$215$		−	3.08465i		−	0.210372i
$216$		0			0
$217$		−	11.6224i		−	0.788977i
$218$		0			0
$219$	−1.35802			−0.0917662
$220$		0			0
$221$			6.25903i			0.421028i
$222$		0			0
$223$	−28.4797			−1.90714			−0.953571	−	0.301169i	$-0.902623\pi$
$223$	−28.4797			−1.90714			−0.953571	+	0.301169i	$0.902623\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−23.2568			−1.54361			−0.771804	−	0.635860i	$-0.780646\pi$
$227$	−23.2568			−1.54361			−0.771804	+	0.635860i	$0.780646\pi$
$228$		0			0
$229$	−19.4527			−1.28547			−0.642735	−	0.766089i	$-0.722200\pi$
$229$	−19.4527			−1.28547			−0.642735	+	0.766089i	$0.722200\pi$
$230$		0			0
$231$	−14.0169			−0.922245
$232$		0			0
$233$	4.55170			0.298192			0.149096	−	0.988823i	$-0.452364\pi$
$233$	4.55170			0.298192			0.149096	+	0.988823i	$0.452364\pi$
$234$		0			0
$235$			7.96162i			0.519359i
$236$		0			0
$237$	−0.478237			−0.0310648
$238$		0			0
$239$			21.2912i			1.37721i	0.725135	+	0.688607i	$0.241777\pi$
$239$			21.2912i			1.37721i	−0.725135	+	0.688607i	$0.758223\pi$
$240$		0			0
$241$		−	5.91004i		−	0.380699i	−0.981716	−	0.190350i	$-0.939038\pi$
$241$		−	5.91004i		−	0.380699i	0.981716	−	0.190350i	$-0.0609622\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−1.02732			−0.0656330
$246$		0			0
$247$	13.7380	−	2.16851i	0.874130	−	0.137979i
$248$		0			0
$249$		−	12.5719i		−	0.796712i
$250$		0			0
$251$			30.5593i			1.92888i	0.264296	+	0.964442i	$0.414861\pi$
$251$			30.5593i			1.92888i	−0.264296	+	0.964442i	$0.585139\pi$
$252$		0			0
$253$	−42.9596			−2.70085
$254$		0			0
$255$	−1.96162			−0.122842
$256$		0			0
$257$			6.53791i			0.407824i	0.978989	+	0.203912i	$0.0653656\pi$
$257$			6.53791i			0.407824i	−0.978989	+	0.203912i	$0.934634\pi$
$258$		0			0
$259$	−16.0046			−0.994474
$260$		0			0
$261$			6.19459i			0.383436i
$262$		0			0
$263$			1.65949i			0.102329i	0.998690	+	0.0511644i	$0.0162932\pi$
$263$			1.65949i			0.102329i	−0.998690	+	0.0511644i	$0.983707\pi$
$264$		0			0
$265$		−	6.48025i		−	0.398079i
$266$		0			0
$267$		−	16.2218i		−	0.992757i
$268$		0			0
$269$			1.41866i			0.0864974i	0.999064	+	0.0432487i	$0.0137708\pi$
$269$			1.41866i			0.0864974i	−0.999064	+	0.0432487i	$0.986229\pi$
$270$		0			0
$271$		−	2.61183i		−	0.158658i	−0.996849	−	0.0793288i	$-0.974722\pi$
$271$		−	2.61183i		−	0.158658i	0.996849	−	0.0793288i	$-0.0252777\pi$
$272$		0			0
$273$	−7.79787			−0.471949
$274$		0			0
$275$		−	5.73545i		−	0.345861i
$276$		0			0
$277$	20.9415			1.25825			0.629125	−	0.777304i	$-0.283413\pi$
$277$	20.9415			1.25825			0.629125	+	0.777304i	$0.283413\pi$
$278$		0			0
$279$	−4.75564			−0.284713
$280$		0			0
$281$			13.0328i			0.777474i	0.921349	+	0.388737i	$0.127089\pi$
$281$			13.0328i			0.777474i	−0.921349	+	0.388737i	$0.872911\pi$
$282$		0			0
$283$		−	15.3330i		−	0.911450i	−0.890121	−	0.455725i	$-0.849380\pi$
$283$		−	15.3330i		−	0.911450i	0.890121	−	0.455725i	$-0.150620\pi$
$284$		0			0
$285$	0.679626	+	4.30559i	0.0402576	+	0.255041i
$286$		0			0
$287$	19.6765			1.16147
$288$		0			0
$289$	−13.1520			−0.773649
$290$		0			0
$291$			4.92214i			0.288541i
$292$		0			0
$293$			2.59008i			0.151314i	0.997134	+	0.0756569i	$0.0241054\pi$
$293$			2.59008i			0.151314i	−0.997134	+	0.0756569i	$0.975895\pi$
$294$		0			0
$295$	−13.4990			−0.785942
$296$		0			0
$297$			5.73545i			0.332805i
$298$		0			0
$299$	−23.8992			−1.38213
$300$		0			0
$301$	−7.53861			−0.434518
$302$		0			0
$303$	−1.66298			−0.0955356
$304$		0			0
$305$	−6.82010			−0.390518
$306$		0			0
$307$	−30.1207			−1.71908			−0.859541	−	0.511067i	$-0.829250\pi$
$307$	−30.1207			−1.71908			−0.859541	+	0.511067i	$0.829250\pi$
$308$		0			0
$309$	12.5348			0.713077
$310$		0			0
$311$		−	26.1504i		−	1.48285i	−0.671035	−	0.741425i	$-0.734150\pi$
$311$		−	26.1504i		−	1.48285i	0.671035	−	0.741425i	$-0.265850\pi$
$312$		0			0
$313$	−5.73179			−0.323980			−0.161990	−	0.986792i	$-0.551791\pi$
$313$	−5.73179			−0.323980			−0.161990	+	0.986792i	$0.551791\pi$
$314$		0			0
$315$		−	2.44391i		−	0.137699i
$316$		0			0
$317$			10.4035i			0.584318i	0.956370	+	0.292159i	$0.0943737\pi$
$317$			10.4035i			0.584318i	−0.956370	+	0.292159i	$0.905626\pi$
$318$		0			0
$319$	35.5288			1.98923
$320$		0			0
$321$	4.94548			0.276030
$322$		0			0
$323$	−1.33317	−	8.44594i	−0.0741795	−	0.469945i
$324$		0			0
$325$		−	3.19074i		−	0.176990i
$326$		0			0
$327$		−	9.39414i		−	0.519497i
$328$		0			0
$329$	19.4575			1.07272
$330$		0			0
$331$	16.9269			0.930384			0.465192	−	0.885210i	$-0.345985\pi$
$331$	16.9269			0.930384			0.465192	+	0.885210i	$0.345985\pi$
$332$		0			0
$333$			6.54876i			0.358870i
$334$		0			0
$335$	15.2508			0.833240
$336$		0			0
$337$			11.8359i			0.644743i	0.946613	+	0.322372i	$0.104480\pi$
$337$			11.8359i			0.644743i	−0.946613	+	0.322372i	$0.895520\pi$
$338$		0			0
$339$			8.93849i			0.485472i
$340$		0			0
$341$			27.2758i			1.47707i
$342$		0			0
$343$			19.6180i			1.05927i
$344$		0			0
$345$		−	7.49018i		−	0.403258i
$346$		0			0
$347$		−	5.09952i		−	0.273757i	−0.990588	−	0.136878i	$-0.956293\pi$
$347$		−	5.09952i		−	0.273757i	0.990588	−	0.136878i	$-0.0437070\pi$
$348$		0			0
$349$	−2.28343			−0.122229			−0.0611146	−	0.998131i	$-0.519466\pi$
$349$	−2.28343			−0.122229			−0.0611146	+	0.998131i	$0.519466\pi$
$350$		0			0
$351$			3.19074i			0.170309i
$352$		0			0
$353$	−27.1325			−1.44412			−0.722059	−	0.691832i	$-0.756804\pi$
$353$	−27.1325			−1.44412			−0.722059	+	0.691832i	$0.756804\pi$
$354$		0			0
$355$	12.8462			0.681805
$356$		0			0
$357$			4.79402i			0.253727i
$358$		0			0
$359$			8.44280i			0.445594i	0.974865	+	0.222797i	$0.0715187\pi$
$359$			8.44280i			0.445594i	−0.974865	+	0.222797i	$0.928481\pi$
$360$		0			0
$361$	−18.0762	+	5.85238i	−0.951380	+	0.308020i
$362$		0			0
$363$	21.8954			1.14921
$364$		0			0
$365$	−1.35802			−0.0710818
$366$		0			0
$367$			30.6978i			1.60241i	0.598387	+	0.801207i	$0.295808\pi$
$367$			30.6978i			1.60241i	−0.598387	+	0.801207i	$0.704192\pi$
$368$		0			0
$369$		−	8.05124i		−	0.419131i
$370$		0			0
$371$	−15.8371			−0.822223
$372$		0			0
$373$			29.6596i			1.53572i	0.640619	+	0.767859i	$0.278678\pi$
$373$			29.6596i			1.53572i	−0.640619	+	0.767859i	$0.721322\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	19.7653			1.01797
$378$		0			0
$379$	−2.72936			−0.140198			−0.0700989	−	0.997540i	$-0.522331\pi$
$379$	−2.72936			−0.140198			−0.0700989	+	0.997540i	$0.522331\pi$
$380$		0			0
$381$	−5.54176			−0.283913
$382$		0			0
$383$	−0.449061			−0.0229459			−0.0114730	−	0.999934i	$-0.503652\pi$
$383$	−0.449061			−0.0229459			−0.0114730	+	0.999934i	$0.503652\pi$
$384$		0			0
$385$	−14.0169			−0.714368
$386$		0			0
$387$			3.08465i			0.156802i
$388$		0			0
$389$	−14.8151			−0.751157			−0.375578	−	0.926791i	$-0.622556\pi$
$389$	−14.8151			−0.751157			−0.375578	+	0.926791i	$0.622556\pi$
$390$		0			0
$391$			14.6929i			0.743053i
$392$		0			0
$393$			12.9414i			0.652809i
$394$		0			0
$395$	−0.478237			−0.0240627
$396$		0			0
$397$	−36.7942			−1.84665			−0.923325	−	0.384019i	$-0.874540\pi$
$397$	−36.7942			−1.84665			−0.923325	+	0.384019i	$0.874540\pi$
$398$		0			0
$399$	10.5225	−	1.66094i	0.526782	−	0.0831511i
$400$		0			0
$401$			11.2221i			0.560404i	0.959941	+	0.280202i	$0.0904015\pi$
$401$			11.2221i			0.560404i	−0.959941	+	0.280202i	$0.909599\pi$
$402$		0			0
$403$			15.1740i			0.755872i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	37.5601			1.86178
$408$		0			0
$409$		−	15.2396i		−	0.753551i	−0.926305	−	0.376776i	$-0.877033\pi$
$409$		−	15.2396i		−	0.753551i	0.926305	−	0.376776i	$-0.122967\pi$
$410$		0			0
$411$	−0.393778			−0.0194236
$412$		0			0
$413$			32.9903i			1.62335i
$414$		0			0
$415$		−	12.5719i		−	0.617130i
$416$		0			0
$417$			5.34696i			0.261842i
$418$		0			0
$419$			35.7713i			1.74754i	0.486335	+	0.873772i	$0.338333\pi$
$419$			35.7713i			1.74754i	−0.486335	+	0.873772i	$0.661667\pi$
$420$		0			0
$421$			3.33366i			0.162473i	0.996695	+	0.0812363i	$0.0258869\pi$
$421$			3.33366i			0.162473i	−0.996695	+	0.0812363i	$0.974113\pi$
$422$		0			0
$423$		−	7.96162i		−	0.387107i
$424$		0			0
$425$	−1.96162			−0.0951527
$426$		0			0
$427$			16.6677i			0.806607i
$428$		0			0
$429$	18.3003			0.883549
$430$		0			0
$431$	23.7914			1.14599			0.572996	−	0.819558i	$-0.305781\pi$
$431$	23.7914			1.14599			0.572996	+	0.819558i	$0.305781\pi$
$432$		0			0
$433$			37.7284i			1.81311i	0.422084	+	0.906557i	$0.361299\pi$
$433$			37.7284i			1.81311i	−0.422084	+	0.906557i	$0.638701\pi$
$434$		0			0
$435$			6.19459i			0.297008i
$436$		0			0
$437$	32.2497	−	5.09052i	1.54271	−	0.243513i
$438$		0			0
$439$	20.6203			0.984154			0.492077	−	0.870552i	$-0.336238\pi$
$439$	20.6203			0.984154			0.492077	+	0.870552i	$0.336238\pi$
$440$		0			0
$441$	1.02732			0.0489200
$442$		0			0
$443$		−	12.2773i		−	0.583314i	−0.956523	−	0.291657i	$-0.905793\pi$
$443$		−	12.2773i		−	0.583314i	0.956523	−	0.291657i	$-0.0942066\pi$
$444$		0			0
$445$		−	16.2218i		−	0.768986i
$446$		0			0
$447$	16.2286			0.767584
$448$		0			0
$449$		−	24.7774i		−	1.16932i	−0.811278	−	0.584660i	$-0.801228\pi$
$449$		−	24.7774i		−	1.16932i	0.811278	−	0.584660i	$-0.198772\pi$
$450$		0			0
$451$	−46.1775			−2.17441
$452$		0			0
$453$	10.1917			0.478846
$454$		0			0
$455$	−7.79787			−0.365570
$456$		0			0
$457$	13.0635			0.611083			0.305542	−	0.952179i	$-0.401163\pi$
$457$	13.0635			0.611083			0.305542	+	0.952179i	$0.401163\pi$
$458$		0			0
$459$	1.96162			0.0915607
$460$		0			0
$461$	−2.74186			−0.127701			−0.0638505	−	0.997959i	$-0.520338\pi$
$461$	−2.74186			−0.127701			−0.0638505	+	0.997959i	$0.520338\pi$
$462$		0			0
$463$		−	24.5972i		−	1.14313i	−0.820558	−	0.571564i	$-0.806337\pi$
$463$		−	24.5972i		−	1.14313i	0.820558	−	0.571564i	$-0.193663\pi$
$464$		0			0
$465$	−4.75564			−0.220538
$466$		0			0
$467$			2.96619i			0.137259i	0.997642	+	0.0686294i	$0.0218626\pi$
$467$			2.96619i			0.137259i	−0.997642	+	0.0686294i	$0.978137\pi$
$468$		0			0
$469$		−	37.2715i		−	1.72104i
$470$		0			0
$471$	1.65316			0.0761735
$472$		0			0
$473$	17.6919			0.813474
$474$		0			0
$475$	0.679626	+	4.30559i	0.0311834	+	0.197554i
$476$		0			0
$477$			6.48025i			0.296710i
$478$		0			0
$479$		−	6.28624i		−	0.287226i	−0.989634	−	0.143613i	$-0.954128\pi$
$479$		−	6.28624i		−	0.287226i	0.989634	−	0.143613i	$-0.0458720\pi$
$480$		0			0
$481$	20.8954			0.952748
$482$		0			0
$483$	−18.3053			−0.832921
$484$		0			0
$485$			4.92214i			0.223503i
$486$		0			0
$487$	−2.25844			−0.102340			−0.0511700	−	0.998690i	$-0.516295\pi$
$487$	−2.25844			−0.102340			−0.0511700	+	0.998690i	$0.516295\pi$
$488$		0			0
$489$			10.0314i			0.453634i
$490$		0			0
$491$		−	39.5287i		−	1.78391i	−0.452129	−	0.891953i	$-0.649335\pi$
$491$		−	39.5287i		−	1.78391i	0.452129	−	0.891953i	$-0.350665\pi$
$492$		0			0
$493$		−	12.1515i		−	0.547274i
$494$		0			0
$495$			5.73545i			0.257789i
$496$		0			0
$497$		−	31.3949i		−	1.40825i
$498$		0			0
$499$		−	26.0920i		−	1.16804i	−0.811740	−	0.584019i	$-0.801479\pi$
$499$		−	26.0920i		−	1.16804i	0.811740	−	0.584019i	$-0.198521\pi$
$500$		0			0
$501$	25.0164			1.11765
$502$		0			0
$503$			23.7366i			1.05836i	0.848509	+	0.529181i	$0.177501\pi$
$503$			23.7366i			1.05836i	−0.848509	+	0.529181i	$0.822499\pi$
$504$		0			0
$505$	−1.66298			−0.0740016
$506$		0			0
$507$	−2.81917			−0.125204
$508$		0			0
$509$		−	12.2064i		−	0.541038i	−0.962715	−	0.270519i	$-0.912805\pi$
$509$		−	12.2064i		−	0.541038i	0.962715	−	0.270519i	$-0.0871953\pi$
$510$		0			0
$511$			3.31886i			0.146818i
$512$		0			0
$513$	−0.679626	−	4.30559i	−0.0300062	−	0.190096i
$514$		0			0
$515$	12.5348			0.552347
$516$		0			0
$517$	−45.6635			−2.00828
$518$		0			0
$519$			2.66930i			0.117169i
$520$		0			0
$521$		−	40.6929i		−	1.78279i	−0.453230	−	0.891394i	$-0.649728\pi$
$521$		−	40.6929i		−	1.78279i	0.453230	−	0.891394i	$-0.350272\pi$
$522$		0			0
$523$	13.9451			0.609778			0.304889	−	0.952388i	$-0.401381\pi$
$523$	13.9451			0.609778			0.304889	+	0.952388i	$0.401381\pi$
$524$		0			0
$525$		−	2.44391i		−	0.106661i
$526$		0			0
$527$	9.32878			0.406368
$528$		0			0
$529$	−33.1029			−1.43925
$530$		0			0
$531$	13.4990			0.585806
$532$		0			0
$533$	−25.6894			−1.11273
$534$		0			0
$535$	4.94548			0.213812
$536$		0			0
$537$	−10.2924			−0.444151
$538$		0			0
$539$		−	5.89214i		−	0.253792i
$540$		0			0
$541$	26.8527			1.15449			0.577243	−	0.816572i	$-0.304129\pi$
$541$	26.8527			1.15449			0.577243	+	0.816572i	$0.304129\pi$
$542$		0			0
$543$		−	21.8326i		−	0.936928i
$544$		0			0
$545$		−	9.39414i		−	0.402401i
$546$		0			0
$547$	−25.1287			−1.07443			−0.537213	−	0.843446i	$-0.680523\pi$
$547$	−25.1287			−1.07443			−0.537213	+	0.843446i	$0.680523\pi$
$548$		0			0
$549$	6.82010			0.291075
$550$		0			0
$551$	−26.6714	+	4.21001i	−1.13624	+	0.179352i
$552$		0			0
$553$			1.16877i			0.0497010i
$554$		0			0
$555$			6.54876i			0.277979i
$556$		0			0
$557$	21.7320			0.920812			0.460406	−	0.887708i	$-0.347704\pi$
$557$	21.7320			0.920812			0.460406	+	0.887708i	$0.347704\pi$
$558$		0			0
$559$	9.84234			0.416286
$560$		0			0
$561$		−	11.2508i		−	0.475009i
$562$		0			0
$563$	−17.7430			−0.747780			−0.373890	−	0.927473i	$-0.621976\pi$
$563$	−17.7430			−0.747780			−0.373890	+	0.927473i	$0.621976\pi$
$564$		0			0
$565$			8.93849i			0.376045i
$566$		0			0
$567$			2.44391i			0.102634i
$568$		0			0
$569$		−	32.7750i		−	1.37400i	−0.726658	−	0.686999i	$-0.758928\pi$
$569$		−	32.7750i		−	1.37400i	0.726658	−	0.686999i	$-0.241072\pi$
$570$		0			0
$571$		−	17.8977i		−	0.748997i	−0.927227	−	0.374499i	$-0.877815\pi$
$571$		−	17.8977i		−	0.748997i	0.927227	−	0.374499i	$-0.122185\pi$
$572$		0			0
$573$		−	0.368258i		−	0.0153842i
$574$		0			0
$575$		−	7.49018i		−	0.312362i
$576$		0			0
$577$	−8.02435			−0.334058			−0.167029	−	0.985952i	$-0.553417\pi$
$577$	−8.02435			−0.334058			−0.167029	+	0.985952i	$0.553417\pi$
$578$		0			0
$579$		−	4.32078i		−	0.179566i
$580$		0			0
$581$	−30.7245			−1.27467
$582$		0			0
$583$	37.1671			1.53931
$584$		0			0
$585$			3.19074i			0.131921i
$586$		0			0
$587$		−	2.21786i		−	0.0915411i	−0.998952	−	0.0457705i	$-0.985426\pi$
$587$		−	2.21786i		−	0.0915411i	0.998952	−	0.0457705i	$-0.0145743\pi$
$588$		0			0
$589$	−3.23206	−	20.4759i	−0.133175	−	0.843693i
$590$		0			0
$591$	3.43892			0.141458
$592$		0			0
$593$	−27.4464			−1.12709			−0.563544	−	0.826086i	$-0.690563\pi$
$593$	−27.4464			−1.12709			−0.563544	+	0.826086i	$0.690563\pi$
$594$		0			0
$595$			4.79402i			0.196536i
$596$		0			0
$597$			8.82743i			0.361282i
$598$		0			0
$599$	25.6554			1.04825			0.524126	−	0.851641i	$-0.324392\pi$
$599$	25.6554			1.04825			0.524126	+	0.851641i	$0.324392\pi$
$600$		0			0
$601$		−	30.1469i		−	1.22972i	−0.788637	−	0.614859i	$-0.789213\pi$
$601$		−	30.1469i		−	1.22972i	0.788637	−	0.614859i	$-0.210787\pi$
$602$		0			0
$603$	−15.2508			−0.621060
$604$		0			0
$605$	21.8954			0.890174
$606$		0			0
$607$	−26.3118			−1.06796			−0.533981	−	0.845496i	$-0.679305\pi$
$607$	−26.3118			−1.06796			−0.533981	+	0.845496i	$0.679305\pi$
$608$		0			0
$609$	15.1390			0.613464
$610$		0			0
$611$	−25.4035			−1.02771
$612$		0			0
$613$	−12.9119			−0.521506			−0.260753	−	0.965406i	$-0.583971\pi$
$613$	−12.9119			−0.521506			−0.260753	+	0.965406i	$0.583971\pi$
$614$		0			0
$615$		−	8.05124i		−	0.324657i
$616$		0			0
$617$	4.79364			0.192985			0.0964923	−	0.995334i	$-0.469238\pi$
$617$	4.79364			0.192985			0.0964923	+	0.995334i	$0.469238\pi$
$618$		0			0
$619$		−	34.4346i		−	1.38405i	−0.721876	−	0.692023i	$-0.756720\pi$
$619$		−	34.4346i		−	1.38405i	0.721876	−	0.692023i	$-0.243280\pi$
$620$		0			0
$621$			7.49018i			0.300571i
$622$		0			0
$623$	−39.6445			−1.58832
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−24.6945	+	3.89796i	−0.986203	+	0.155670i
$628$		0			0
$629$		−	12.8462i		−	0.512211i
$630$		0			0
$631$			6.96950i			0.277451i	0.990331	+	0.138726i	$0.0443006\pi$
$631$			6.96950i			0.277451i	−0.990331	+	0.138726i	$0.955699\pi$
$632$		0			0
$633$	7.35008			0.292139
$634$		0			0
$635$	−5.54176			−0.219918
$636$		0			0
$637$		−	3.27791i		−	0.129876i
$638$		0			0
$639$	−12.8462			−0.508187
$640$		0			0
$641$			14.1990i			0.560827i	0.959879	+	0.280414i	$0.0904716\pi$
$641$			14.1990i			0.560827i	−0.959879	+	0.280414i	$0.909528\pi$
$642$		0			0
$643$			34.6851i			1.36785i	0.729554	+	0.683923i	$0.239728\pi$
$643$			34.6851i			1.36785i	−0.729554	+	0.683923i	$0.760272\pi$
$644$		0			0
$645$			3.08465i			0.121458i
$646$		0			0
$647$		−	10.5928i		−	0.416446i	−0.978081	−	0.208223i	$-0.933232\pi$
$647$		−	10.5928i		−	0.416446i	0.978081	−	0.208223i	$-0.0667680\pi$
$648$		0			0
$649$		−	77.4228i		−	3.03911i
$650$		0			0
$651$			11.6224i			0.455516i
$652$		0			0
$653$	29.5914			1.15800			0.579000	−	0.815328i	$-0.303443\pi$
$653$	29.5914			1.15800			0.579000	+	0.815328i	$0.303443\pi$
$654$		0			0
$655$			12.9414i			0.505663i
$656$		0			0
$657$	1.35802			0.0529812
$658$		0			0
$659$	42.7963			1.66711			0.833554	−	0.552439i	$-0.186303\pi$
$659$	42.7963			1.66711			0.833554	+	0.552439i	$0.186303\pi$
$660$		0			0
$661$		−	21.2870i		−	0.827967i	−0.910284	−	0.413983i	$-0.864137\pi$
$661$		−	21.2870i		−	0.827967i	0.910284	−	0.413983i	$-0.135863\pi$
$662$		0			0
$663$		−	6.25903i		−	0.243081i
$664$		0			0
$665$	10.5225	−	1.66094i	0.408044	−	0.0644086i
$666$		0			0
$667$	46.3986			1.79656
$668$		0			0
$669$	28.4797			1.10109
$670$		0			0
$671$		−	39.1164i		−	1.51007i
$672$		0			0
$673$		−	10.7449i		−	0.414185i	−0.978321	−	0.207093i	$-0.933600\pi$
$673$		−	10.7449i		−	0.414185i	0.978321	−	0.207093i	$-0.0664001\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$			3.77782i			0.145193i	0.997361	+	0.0725966i	$0.0231286\pi$
$677$			3.77782i			0.145193i	−0.997361	+	0.0725966i	$0.976871\pi$
$678$		0			0
$679$	12.0293			0.461641
$680$		0			0
$681$	23.2568			0.891203
$682$		0			0
$683$	14.0003			0.535708			0.267854	−	0.963459i	$-0.413685\pi$
$683$	14.0003			0.535708			0.267854	+	0.963459i	$0.413685\pi$
$684$		0			0
$685$	−0.393778			−0.0150455
$686$		0			0
$687$	19.4527			0.742166
$688$		0			0
$689$	20.6768			0.787723
$690$		0			0
$691$			33.4711i			1.27330i	0.771152	+	0.636651i	$0.219681\pi$
$691$			33.4711i			1.27330i	−0.771152	+	0.636651i	$0.780319\pi$
$692$		0			0
$693$	14.0169			0.532458
$694$		0			0
$695$			5.34696i			0.202822i
$696$		0			0
$697$			15.7935i			0.598221i
$698$		0			0
$699$	−4.55170			−0.172161
$700$		0			0
$701$	24.9890			0.943823			0.471912	−	0.881646i	$-0.343564\pi$
$701$	24.9890			0.943823			0.471912	+	0.881646i	$0.343564\pi$
$702$		0			0
$703$	−28.1963	+	4.45071i	−1.06344	+	0.167862i
$704$		0			0
$705$		−	7.96162i		−	0.299852i
$706$		0			0
$707$			4.06416i			0.152849i
$708$		0			0
$709$	32.4102			1.21719			0.608594	−	0.793481i	$-0.291734\pi$
$709$	32.4102			1.21719			0.608594	+	0.793481i	$0.291734\pi$
$710$		0			0
$711$	0.478237			0.0179353
$712$		0			0
$713$			35.6207i			1.33400i
$714$		0			0
$715$	18.3003			0.684394
$716$		0			0
$717$		−	21.2912i		−	0.795134i
$718$		0			0
$719$			31.0429i			1.15771i	0.815432	+	0.578853i	$0.196499\pi$
$719$			31.0429i			1.15771i	−0.815432	+	0.578853i	$0.803501\pi$
$720$		0			0
$721$		−	30.6338i		−	1.14086i
$722$		0			0
$723$			5.91004i			0.219797i
$724$		0			0
$725$			6.19459i			0.230061i
$726$		0			0
$727$			1.31123i			0.0486310i	0.999704	+	0.0243155i	$0.00774063\pi$
$727$			1.31123i			0.0486310i	−0.999704	+	0.0243155i	$0.992259\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	6.05093i		−	0.223802i
$732$		0			0
$733$	−42.5762			−1.57259			−0.786294	−	0.617852i	$-0.788003\pi$
$733$	−42.5762			−1.57259			−0.786294	+	0.617852i	$0.788003\pi$
$734$		0			0
$735$	1.02732			0.0378932
$736$		0			0
$737$			87.4701i			3.22200i
$738$		0			0
$739$			1.45016i			0.0533450i	0.999644	+	0.0266725i	$0.00849113\pi$
$739$			1.45016i			0.0533450i	−0.999644	+	0.0266725i	$0.991509\pi$
$740$		0			0
$741$	−13.7380	+	2.16851i	−0.504679	+	0.0796622i
$742$		0			0
$743$	4.18251			0.153441			0.0767207	−	0.997053i	$-0.475555\pi$
$743$	4.18251			0.153441			0.0767207	+	0.997053i	$0.475555\pi$
$744$		0			0
$745$	16.2286			0.594568
$746$		0			0
$747$			12.5719i			0.459982i
$748$		0			0
$749$		−	12.0863i		−	0.441623i
$750$		0			0
$751$	−44.7112			−1.63154			−0.815768	−	0.578379i	$-0.803685\pi$
$751$	−44.7112			−1.63154			−0.815768	+	0.578379i	$0.803685\pi$
$752$		0			0
$753$		−	30.5593i		−	1.11364i
$754$		0			0
$755$	10.1917			0.370912
$756$		0			0
$757$	12.6836			0.460994			0.230497	−	0.973073i	$-0.425965\pi$
$757$	12.6836			0.460994			0.230497	+	0.973073i	$0.425965\pi$
$758$		0			0
$759$	42.9596			1.55933
$760$		0			0
$761$	−7.59451			−0.275301			−0.137650	−	0.990481i	$-0.543955\pi$
$761$	−7.59451			−0.275301			−0.137650	+	0.990481i	$0.543955\pi$
$762$		0			0
$763$	−22.9584			−0.831150
$764$		0			0
$765$	1.96162			0.0709226
$766$		0			0
$767$		−	43.0718i		−	1.55523i
$768$		0			0
$769$	23.2810			0.839535			0.419768	−	0.907632i	$-0.362112\pi$
$769$	23.2810			0.839535			0.419768	+	0.907632i	$0.362112\pi$
$770$		0			0
$771$		−	6.53791i		−	0.235457i
$772$		0			0
$773$		−	27.4716i		−	0.988084i	−0.869438	−	0.494042i	$-0.835519\pi$
$773$		−	27.4716i		−	0.988084i	0.869438	−	0.494042i	$-0.164481\pi$
$774$		0			0
$775$	−4.75564			−0.170828
$776$		0			0
$777$	16.0046			0.574160
$778$		0			0
$779$	34.6653	−	5.47183i	1.24201	−	0.196049i
$780$		0			0
$781$			73.6787i			2.63643i
$782$		0			0
$783$		−	6.19459i		−	0.221377i
$784$		0			0
$785$	1.65316			0.0590037
$786$		0			0
$787$	30.1820			1.07587			0.537936	−	0.842985i	$-0.319204\pi$
$787$	30.1820			1.07587			0.537936	+	0.842985i	$0.319204\pi$
$788$		0			0
$789$		−	1.65949i		−	0.0590795i
$790$		0			0
$791$	21.8448			0.776713
$792$		0			0
$793$		−	21.7612i		−	0.772763i
$794$		0			0
$795$			6.48025i			0.229831i
$796$		0			0
$797$			33.7169i			1.19431i	0.802125	+	0.597156i	$0.203703\pi$
$797$			33.7169i			1.19431i	−0.802125	+	0.597156i	$0.796297\pi$
$798$		0			0
$799$			15.6177i			0.552514i
$800$		0			0
$801$			16.2218i			0.573168i
$802$		0			0
$803$		−	7.78883i		−	0.274862i
$804$		0			0
$805$	−18.3053			−0.645177
$806$		0			0
$807$		−	1.41866i		−	0.0499393i
$808$		0			0
$809$	−52.3089			−1.83908			−0.919541	−	0.392994i	$-0.871439\pi$
$809$	−52.3089			−1.83908			−0.919541	+	0.392994i	$0.871439\pi$
$810$		0			0
$811$	23.0757			0.810299			0.405150	−	0.914250i	$-0.367219\pi$
$811$	23.0757			0.810299			0.405150	+	0.914250i	$0.367219\pi$
$812$		0			0
$813$			2.61183i			0.0916010i
$814$		0			0
$815$			10.0314i			0.351383i
$816$		0			0
$817$	−13.2813	+	2.09641i	−0.464652	+	0.0733442i
$818$		0			0
$819$	7.79787			0.272480
$820$		0			0
$821$	−41.6537			−1.45373			−0.726863	−	0.686783i	$-0.759022\pi$
$821$	−41.6537			−1.45373			−0.726863	+	0.686783i	$0.759022\pi$
$822$		0			0
$823$			38.6993i			1.34897i	0.738288	+	0.674486i	$0.235635\pi$
$823$			38.6993i			1.34897i	−0.738288	+	0.674486i	$0.764365\pi$
$824$		0			0
$825$			5.73545i			0.199683i
$826$		0			0
$827$	49.0464			1.70551			0.852755	−	0.522310i	$-0.174930\pi$
$827$	49.0464			1.70551			0.852755	+	0.522310i	$0.174930\pi$
$828$		0			0
$829$			49.2855i			1.71176i	0.517178	+	0.855878i	$0.326983\pi$
$829$			49.2855i			1.71176i	−0.517178	+	0.855878i	$0.673017\pi$
$830$		0			0
$831$	−20.9415			−0.726452
$832$		0			0
$833$	−2.01521			−0.0698230
$834$		0			0
$835$	25.0164			0.865727
$836$		0			0
$837$	4.75564			0.164379
$838$		0			0
$839$	−1.23827			−0.0427499			−0.0213749	−	0.999772i	$-0.506804\pi$
$839$	−1.23827			−0.0427499			−0.0213749	+	0.999772i	$0.506804\pi$
$840$		0			0
$841$	−9.37299			−0.323206
$842$		0			0
$843$		−	13.0328i		−	0.448875i
$844$		0			0
$845$	−2.81917			−0.0969824
$846$		0			0
$847$		−	53.5103i		−	1.83864i
$848$		0			0
$849$			15.3330i			0.526226i
$850$		0			0
$851$	49.0514			1.68146
$852$		0			0
$853$	8.87688			0.303939			0.151969	−	0.988385i	$-0.451439\pi$
$853$	8.87688			0.303939			0.151969	+	0.988385i	$0.451439\pi$
$854$		0			0
$855$	−0.679626	−	4.30559i	−0.0232427	−	0.147248i
$856$		0			0
$857$		−	26.0250i		−	0.888997i	−0.895780	−	0.444499i	$-0.853382\pi$
$857$		−	26.0250i		−	0.888997i	0.895780	−	0.444499i	$-0.146618\pi$
$858$		0			0
$859$			31.1563i			1.06304i	0.847046	+	0.531519i	$0.178379\pi$
$859$			31.1563i			1.06304i	−0.847046	+	0.531519i	$0.821621\pi$
$860$		0			0
$861$	−19.6765			−0.670572
$862$		0			0
$863$	−41.4646			−1.41147			−0.705735	−	0.708476i	$-0.749383\pi$
$863$	−41.4646			−1.41147			−0.705735	+	0.708476i	$0.749383\pi$
$864$		0			0
$865$			2.66930i			0.0907591i
$866$		0			0
$867$	13.1520			0.446667
$868$		0			0
$869$		−	2.74290i		−	0.0930466i
$870$		0			0
$871$			48.6613i			1.64883i
$872$		0			0
$873$		−	4.92214i		−	0.166589i
$874$		0			0
$875$		−	2.44391i		−	0.0826191i
$876$		0			0
$877$			20.1592i			0.680729i	0.940294	+	0.340364i	$0.110550\pi$
$877$			20.1592i			0.680729i	−0.940294	+	0.340364i	$0.889450\pi$
$878$		0			0
$879$		−	2.59008i		−	0.0873611i
$880$		0			0
$881$	5.52147			0.186023			0.0930115	−	0.995665i	$-0.470351\pi$
$881$	5.52147			0.186023			0.0930115	+	0.995665i	$0.470351\pi$
$882$		0			0
$883$		−	38.9790i		−	1.31175i	−0.754870	−	0.655874i	$-0.772300\pi$
$883$		−	38.9790i		−	1.31175i	0.754870	−	0.655874i	$-0.227700\pi$
$884$		0			0
$885$	13.4990			0.453764
$886$		0			0
$887$	−14.4297			−0.484502			−0.242251	−	0.970214i	$-0.577886\pi$
$887$	−14.4297			−0.484502			−0.242251	+	0.970214i	$0.577886\pi$
$888$		0			0
$889$			13.5436i			0.454236i
$890$		0			0
$891$		−	5.73545i		−	0.192145i
$892$		0			0
$893$	34.2795	−	5.41093i	1.14712	−	0.181070i
$894$		0			0
$895$	−10.2924			−0.344038
$896$		0			0
$897$	23.8992			0.797972
$898$		0			0
$899$		−	29.4593i		−	0.982522i
$900$		0			0
$901$		−	12.7118i		−	0.423491i
$902$		0			0
$903$	7.53861			0.250869
$904$		0			0
$905$		−	21.8326i		−	0.725741i
$906$		0			0
$907$	54.9837			1.82571			0.912853	−	0.408289i	$-0.133875\pi$
$907$	54.9837			1.82571			0.912853	+	0.408289i	$0.133875\pi$
$908$		0			0
$909$	1.66298			0.0551575
$910$		0			0
$911$	35.5991			1.17945			0.589725	−	0.807604i	$-0.299236\pi$
$911$	35.5991			1.17945			0.589725	+	0.807604i	$0.299236\pi$
$912$		0			0
$913$	72.1055			2.38634
$914$		0			0
$915$	6.82010			0.225466
$916$		0			0
$917$	31.6276			1.04444
$918$		0			0
$919$			50.8517i			1.67744i	0.544561	+	0.838721i	$0.316696\pi$
$919$			50.8517i			1.67744i	−0.544561	+	0.838721i	$0.683304\pi$
$920$		0			0
$921$	30.1207			0.992512
$922$		0			0
$923$			40.9889i			1.34916i
$924$		0			0
$925$			6.54876i			0.215322i
$926$		0			0
$927$	−12.5348			−0.411695
$928$		0			0
$929$	−14.8524			−0.487292			−0.243646	−	0.969864i	$-0.578344\pi$
$929$	−14.8524			−0.487292			−0.243646	+	0.969864i	$0.578344\pi$
$930$		0			0
$931$	0.698193	+	4.42322i	0.0228823	+	0.144965i
$932$		0			0
$933$			26.1504i			0.856124i
$934$		0			0
$935$		−	11.2508i		−	0.367940i
$936$		0			0
$937$	−47.8733			−1.56395			−0.781977	−	0.623307i	$-0.785789\pi$
$937$	−47.8733			−1.56395			−0.781977	+	0.623307i	$0.785789\pi$
$938$		0			0
$939$	5.73179			0.187050
$940$		0			0
$941$		−	9.50773i		−	0.309943i	−0.987919	−	0.154972i	$-0.950471\pi$
$941$		−	9.50773i		−	0.309943i	0.987919	−	0.154972i	$-0.0495286\pi$
$942$		0			0
$943$	−60.3052			−1.96381
$944$		0			0
$945$			2.44391i			0.0795003i
$946$		0			0
$947$		−	19.2206i		−	0.624586i	−0.949986	−	0.312293i	$-0.898903\pi$
$947$		−	19.2206i		−	0.624586i	0.949986	−	0.312293i	$-0.101097\pi$
$948$		0			0
$949$		−	4.33307i		−	0.140658i
$950$		0			0
$951$		−	10.4035i		−	0.337356i
$952$		0			0
$953$		−	48.0760i		−	1.55734i	−0.627436	−	0.778668i	$-0.715896\pi$
$953$		−	48.0760i		−	1.55734i	0.627436	−	0.778668i	$-0.284104\pi$
$954$		0			0
$955$		−	0.368258i		−	0.0119166i
$956$		0			0
$957$	−35.5288			−1.14848
$958$		0			0
$959$			0.962356i			0.0310761i
$960$		0			0
$961$	−8.38384			−0.270447
$962$		0			0
$963$	−4.94548			−0.159366
$964$		0			0
$965$		−	4.32078i		−	0.139091i
$966$		0			0
$967$			36.8103i			1.18374i	0.806033	+	0.591870i	$0.201610\pi$
$967$			36.8103i			1.18374i	−0.806033	+	0.591870i	$0.798390\pi$
$968$		0			0
$969$	1.33317	+	8.44594i	0.0428276	+	0.271323i
$970$		0			0
$971$	−21.7480			−0.697927			−0.348963	−	0.937136i	$-0.613466\pi$
$971$	−21.7480			−0.697927			−0.348963	+	0.937136i	$0.613466\pi$
$972$		0			0
$973$	13.0675			0.418924
$974$		0			0
$975$			3.19074i			0.102186i
$976$		0			0
$977$		−	27.4236i		−	0.877358i	−0.898644	−	0.438679i	$-0.855447\pi$
$977$		−	27.4236i		−	0.877358i	0.898644	−	0.438679i	$-0.144553\pi$
$978$		0			0
$979$	93.0392			2.97355
$980$		0			0
$981$			9.39414i			0.299932i
$982$		0			0
$983$	38.8706			1.23978			0.619889	−	0.784689i	$-0.287178\pi$
$983$	38.8706			1.23978			0.619889	+	0.784689i	$0.287178\pi$
$984$		0			0
$985$	3.43892			0.109573
$986$		0			0
$987$	−19.4575			−0.619338
$988$		0			0
$989$	23.1046			0.734685
$990$		0			0
$991$	16.8164			0.534191			0.267096	−	0.963670i	$-0.413936\pi$
$991$	16.8164			0.534191			0.267096	+	0.963670i	$0.413936\pi$
$992$		0			0
$993$	−16.9269			−0.537158
$994$		0			0
$995$			8.82743i			0.279848i
$996$		0			0
$997$	−36.2290			−1.14738			−0.573691	−	0.819072i	$-0.694489\pi$
$997$	−36.2290			−1.14738			−0.573691	+	0.819072i	$0.694489\pi$
$998$		0			0
$999$		−	6.54876i		−	0.207193i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.i.2431.10	yes	12
4.3	odd	2		4560.2.d.k.2431.3	yes	12
19.18	odd	2		4560.2.d.k.2431.10	yes	12
76.75	even	2	inner	4560.2.d.i.2431.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.i.2431.3	✓	12	76.75	even	2	inner
4560.2.d.i.2431.10	yes	12	1.1	even	1	trivial
4560.2.d.k.2431.3	yes	12	4.3	odd	2
4560.2.d.k.2431.10	yes	12	19.18	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.44391i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.44391i q^{7} +1.00000 q^{9} -5.73545i q^{11} -3.19074i q^{13} +1.00000 q^{15} -1.96162 q^{17} +(0.679626 + 4.30559i) q^{19} -2.44391i q^{21} -7.49018i q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.19459i q^{29} -4.75564 q^{31} +5.73545i q^{33} -2.44391i q^{35} +6.54876i q^{37} +3.19074i q^{39} -8.05124i q^{41} +3.08465i q^{43} -1.00000 q^{45} -7.96162i q^{47} +1.02732 q^{49} +1.96162 q^{51} +6.48025i q^{53} +5.73545i q^{55} +(-0.679626 - 4.30559i) q^{57} +13.4990 q^{59} +6.82010 q^{61} +2.44391i q^{63} +3.19074i q^{65} -15.2508 q^{67} +7.49018i q^{69} -12.8462 q^{71} +1.35802 q^{73} -1.00000 q^{75} +14.0169 q^{77} +0.478237 q^{79} +1.00000 q^{81} +12.5719i q^{83} +1.96162 q^{85} -6.19459i q^{87} +16.2218i q^{89} +7.79787 q^{91} +4.75564 q^{93} +(-0.679626 - 4.30559i) q^{95} -4.92214i q^{97} -5.73545i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9	\( 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9} + 12 q^{15} - 8 q^{17} - 8 q^{19} + 12 q^{25} - 12 q^{27} - 12 q^{45} - 20 q^{49} + 8 q^{51} + 8 q^{57} - 8 q^{59} + 8 q^{67} - 32 q^{71} - 24 q^{73} - 12 q^{75} + 56 q^{77} - 16 q^{79} + 12 q^{81} + 8 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) \) 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9 + 12 * q^15 - 8 * q^17 - 8 * q^19 + 12 * q^25 - 12 * q^27 - 12 * q^45 - 20 * q^49 + 8 * q^51 + 8 * q^57 - 8 * q^59 + 8 * q^67 - 32 * q^71 - 24 * q^73 - 12 * q^75 + 56 * q^77 - 16 * q^79 + 12 * q^81 + 8 * q^85 + 16 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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