LMFDB - Embedded newform 4560.2.d.l.2431.9

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} + 35x^{10} + 202x^{8} + 362x^{6} + 245x^{4} + 63x^{2} + 4 $ x^12 + 35x^10 + 202x^8 + 362x^6 + 245x^4 + 63*x^2 + 4
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			2431.9
Root			$-2.03924i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.l.2431.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +2.79053i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +2.79053i q^{7} +1.00000 q^{9} +5.01148i q^{11} +3.47756i q^{13} +1.00000 q^{15} +2.25824 q^{17} +(4.19273 + 1.19207i) q^{19} +2.79053i q^{21} +0.453944i q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.54737i q^{29} -5.12801 q^{31} +5.01148i q^{33} +2.79053i q^{35} -4.77172i q^{37} +3.47756i q^{39} -2.33658i q^{41} -1.78713i q^{43} +1.00000 q^{45} +8.48904i q^{47} -0.787044 q^{49} +2.25824 q^{51} -3.01016i q^{53} +5.01148i q^{55} +(4.19273 + 1.19207i) q^{57} -1.34096 q^{59} -6.21296 q^{61} +2.79053i q^{63} +3.47756i q^{65} -3.58856 q^{67} +0.453944i q^{69} -5.38625 q^{71} -11.7717 q^{73} +1.00000 q^{75} -13.9847 q^{77} +2.18919 q^{79} +1.00000 q^{81} -9.49244i q^{83} +2.25824 q^{85} -1.54737i q^{87} +3.89792i q^{89} -9.70424 q^{91} -5.12801 q^{93} +(4.19273 + 1.19207i) q^{95} -1.67697i q^{97} +5.01148i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{17} + 12 q^{25} + 12 q^{27} - 12 q^{31} + 12 q^{45} - 28 q^{49} + 4 q^{51} + 52 q^{59} - 56 q^{61} - 32 q^{67} + 8 q^{71} + 32 q^{73} + 12 q^{75} + 24 q^{77} - 28 q^{79} + 12 q^{81} + 4 q^{85} + 32 q^{91} - 12 q^{93}+O(q^{100}) $ 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9 + 12 * q^15 + 4 * q^17 + 12 * q^25 + 12 * q^27 - 12 * q^31 + 12 * q^45 - 28 * q^49 + 4 * q^51 + 52 * q^59 - 56 * q^61 - 32 * q^67 + 8 * q^71 + 32 * q^73 + 12 * q^75 + 24 * q^77 - 28 * q^79 + 12 * q^81 + 4 * q^85 + 32 * q^91 - 12 * q^93

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
$7$			2.79053i			1.05472i	0.849642	+	0.527360i	$0.176818\pi$
$7$			2.79053i			1.05472i	−0.849642	+	0.527360i	$0.823182\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			5.01148i			1.51102i	0.655139	+	0.755508i	$0.272610\pi$
$11$			5.01148i			1.51102i	−0.655139	+	0.755508i	$0.727390\pi$
$12$		0			0
$13$			3.47756i			0.964503i	0.876033	+	0.482251i	$0.160181\pi$
$13$			3.47756i			0.964503i	−0.876033	+	0.482251i	$0.839819\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	2.25824			0.547703			0.273852	−	0.961772i	$-0.411702\pi$
$17$	2.25824			0.547703			0.273852	+	0.961772i	$0.411702\pi$
$18$		0			0
$19$	4.19273	+	1.19207i	0.961878	+	0.273479i
$19$	4.19273	+	1.19207i	0.961878	+	0.273479i
$20$		0			0
$21$			2.79053i			0.608943i
$22$		0			0
$23$			0.453944i			0.0946539i	0.998879	+	0.0473269i	$0.0150703\pi$
$23$			0.453944i			0.0946539i	−0.998879	+	0.0473269i	$0.984930\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	1.54737i		−	0.287340i	−0.989626	−	0.143670i	$-0.954110\pi$
$29$		−	1.54737i		−	0.287340i	0.989626	−	0.143670i	$-0.0458904\pi$
$30$		0			0
$31$	−5.12801			−0.921018			−0.460509	−	0.887655i	$-0.652333\pi$
$31$	−5.12801			−0.921018			−0.460509	+	0.887655i	$0.652333\pi$
$32$		0			0
$33$			5.01148i			0.872386i
$34$		0			0
$35$			2.79053i			0.471685i
$36$		0			0
$37$		−	4.77172i		−	0.784466i	−0.919866	−	0.392233i	$-0.871703\pi$
$37$		−	4.77172i		−	0.784466i	0.919866	−	0.392233i	$-0.128297\pi$
$38$		0			0
$39$			3.47756i			0.556856i
$40$		0			0
$41$		−	2.33658i		−	0.364913i	−0.983214	−	0.182457i	$-0.941595\pi$
$41$		−	2.33658i		−	0.364913i	0.983214	−	0.182457i	$-0.0584049\pi$
$42$		0			0
$43$		−	1.78713i		−	0.272534i	−0.990672	−	0.136267i	$-0.956489\pi$
$43$		−	1.78713i		−	0.272534i	0.990672	−	0.136267i	$-0.0435106\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$			8.48904i			1.23825i	0.785291	+	0.619127i	$0.212513\pi$
$47$			8.48904i			1.23825i	−0.785291	+	0.619127i	$0.787487\pi$
$48$		0			0
$49$	−0.787044			−0.112435
$50$		0			0
$51$	2.25824			0.316217
$52$		0			0
$53$		−	3.01016i		−	0.413477i	−0.978396	−	0.206738i	$-0.933715\pi$
$53$		−	3.01016i		−	0.413477i	0.978396	−	0.206738i	$-0.0662849\pi$
$54$		0			0
$55$			5.01148i			0.675747i
$56$		0			0
$57$	4.19273	+	1.19207i	0.555341	+	0.157893i
$58$		0			0
$59$	−1.34096			−0.174579			−0.0872894	−	0.996183i	$-0.527820\pi$
$59$	−1.34096			−0.174579			−0.0872894	+	0.996183i	$0.527820\pi$
$60$		0			0
$61$	−6.21296			−0.795488			−0.397744	−	0.917497i	$-0.630207\pi$
$61$	−6.21296			−0.795488			−0.397744	+	0.917497i	$0.630207\pi$
$62$		0			0
$63$			2.79053i			0.351573i
$64$		0			0
$65$			3.47756i			0.431339i
$66$		0			0
$67$	−3.58856			−0.438412			−0.219206	−	0.975679i	$-0.570347\pi$
$67$	−3.58856			−0.438412			−0.219206	+	0.975679i	$0.570347\pi$
$68$		0			0
$69$			0.453944i			0.0546484i
$70$		0			0
$71$	−5.38625			−0.639230			−0.319615	−	0.947547i	$-0.603554\pi$
$71$	−5.38625			−0.639230			−0.319615	+	0.947547i	$0.603554\pi$
$72$		0			0
$73$	−11.7717			−1.37777			−0.688887	−	0.724869i	$-0.741900\pi$
$73$	−11.7717			−1.37777			−0.688887	+	0.724869i	$0.741900\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	−13.9847			−1.59370
$78$		0			0
$79$	2.18919			0.246303			0.123152	−	0.992388i	$-0.460700\pi$
$79$	2.18919			0.246303			0.123152	+	0.992388i	$0.460700\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	9.49244i		−	1.04193i	−0.853578	−	0.520965i	$-0.825572\pi$
$83$		−	9.49244i		−	1.04193i	0.853578	−	0.520965i	$-0.174428\pi$
$84$		0			0
$85$	2.25824			0.244940
$86$		0			0
$87$		−	1.54737i		−	0.165896i
$88$		0			0
$89$			3.89792i			0.413179i	0.978428	+	0.206589i	$0.0662364\pi$
$89$			3.89792i			0.413179i	−0.978428	+	0.206589i	$0.933764\pi$
$90$		0			0
$91$	−9.70424			−1.01728
$92$		0			0
$93$	−5.12801			−0.531750
$94$		0			0
$95$	4.19273	+	1.19207i	0.430165	+	0.122303i
$96$		0			0
$97$		−	1.67697i		−	0.170271i	−0.996369	−	0.0851354i	$-0.972868\pi$
$97$		−	1.67697i		−	0.170271i	0.996369	−	0.0851354i	$-0.0271323\pi$
$98$		0			0
$99$			5.01148i			0.503672i
$100$		0			0
$101$	−5.10425			−0.507891			−0.253946	−	0.967218i	$-0.581728\pi$
$101$	−5.10425			−0.507891			−0.253946	+	0.967218i	$0.581728\pi$
$102$		0			0
$103$	3.58856			0.353591			0.176796	−	0.984248i	$-0.443427\pi$
$103$	3.58856			0.353591			0.176796	+	0.984248i	$0.443427\pi$
$104$		0			0
$105$			2.79053i			0.272328i
$106$		0			0
$107$	9.79769			0.947179			0.473589	−	0.880746i	$-0.342958\pi$
$107$	9.79769			0.947179			0.473589	+	0.880746i	$0.342958\pi$
$108$		0			0
$109$		−	0.0765684i		−	0.00733392i	−0.999993	−	0.00366696i	$-0.998833\pi$
$109$		−	0.0765684i		−	0.00733392i	0.999993	−	0.00366696i	$-0.00116723\pi$
$110$		0			0
$111$		−	4.77172i		−	0.452912i
$112$		0			0
$113$			17.2670i			1.62434i	0.583418	+	0.812172i	$0.301715\pi$
$113$			17.2670i			1.62434i	−0.583418	+	0.812172i	$0.698285\pi$
$114$		0			0
$115$			0.453944i			0.0423305i
$116$		0			0
$117$			3.47756i			0.321501i
$118$		0			0
$119$			6.30168i			0.577674i
$120$		0			0
$121$	−14.1149			−1.28317
$122$		0			0
$123$		−	2.33658i		−	0.210683i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−10.1831			−0.903608			−0.451804	−	0.892117i	$-0.649219\pi$
$127$	−10.1831			−0.903608			−0.451804	+	0.892117i	$0.649219\pi$
$128$		0			0
$129$		−	1.78713i		−	0.157348i
$130$		0			0
$131$			16.8540i			1.47254i	0.676689	+	0.736269i	$0.263414\pi$
$131$			16.8540i			1.47254i	−0.676689	+	0.736269i	$0.736586\pi$
$132$		0			0
$133$	−3.32650	+	11.6999i	−0.288444	+	1.01451i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	8.58172			0.733186			0.366593	−	0.930381i	$-0.380524\pi$
$137$	8.58172			0.733186			0.366593	+	0.930381i	$0.380524\pi$
$138$		0			0
$139$			12.1130i			1.02741i	0.857967	+	0.513705i	$0.171727\pi$
$139$			12.1130i			1.02741i	−0.857967	+	0.513705i	$0.828273\pi$
$140$		0			0
$141$			8.48904i			0.714906i
$142$		0			0
$143$	−17.4277			−1.45738
$144$		0			0
$145$		−	1.54737i		−	0.128502i
$146$		0			0
$147$	−0.787044			−0.0649143
$148$		0			0
$149$	−17.6423			−1.44531			−0.722655	−	0.691209i	$-0.757079\pi$
$149$	−17.6423			−1.44531			−0.722655	+	0.691209i	$0.757079\pi$
$150$		0			0
$151$	22.4559			1.82743			0.913716	−	0.406354i	$-0.133200\pi$
$151$	22.4559			1.82743			0.913716	+	0.406354i	$0.133200\pi$
$152$		0			0
$153$	2.25824			0.182568
$154$		0			0
$155$	−5.12801			−0.411892
$156$		0			0
$157$	4.75320			0.379346			0.189673	−	0.981847i	$-0.439257\pi$
$157$	4.75320			0.379346			0.189673	+	0.981847i	$0.439257\pi$
$158$		0			0
$159$		−	3.01016i		−	0.238721i
$160$		0			0
$161$	−1.26674			−0.0998334
$162$		0			0
$163$		−	1.33369i		−	0.104462i	−0.998635	−	0.0522312i	$-0.983367\pi$
$163$		−	1.33369i		−	0.104462i	0.998635	−	0.0522312i	$-0.0166333\pi$
$164$		0			0
$165$			5.01148i			0.390143i
$166$		0			0
$167$	11.1717			0.864493			0.432247	−	0.901755i	$-0.357721\pi$
$167$	11.1717			0.864493			0.432247	+	0.901755i	$0.357721\pi$
$168$		0			0
$169$	0.906548			0.0697344
$170$		0			0
$171$	4.19273	+	1.19207i	0.320626	+	0.0911596i
$172$		0			0
$173$		−	9.69811i		−	0.737333i	−0.929562	−	0.368667i	$-0.879814\pi$
$173$		−	9.69811i		−	0.737333i	0.929562	−	0.368667i	$-0.120186\pi$
$174$		0			0
$175$			2.79053i			0.210944i
$176$		0			0
$177$	−1.34096			−0.100793
$178$		0			0
$179$	1.67272			0.125025			0.0625123	−	0.998044i	$-0.480089\pi$
$179$	1.67272			0.125025			0.0625123	+	0.998044i	$0.480089\pi$
$180$		0			0
$181$		−	10.1587i		−	0.755091i	−0.925991	−	0.377546i	$-0.876768\pi$
$181$		−	10.1587i		−	0.755091i	0.925991	−	0.377546i	$-0.123232\pi$
$182$		0			0
$183$	−6.21296			−0.459275
$184$		0			0
$185$		−	4.77172i		−	0.350824i
$186$		0			0
$187$			11.3171i			0.827589i
$188$		0			0
$189$			2.79053i			0.202981i
$190$		0			0
$191$		−	21.2959i		−	1.54091i	−0.637492	−	0.770457i	$-0.720028\pi$
$191$		−	21.2959i		−	1.54091i	0.637492	−	0.770457i	$-0.279972\pi$
$192$		0			0
$193$		−	11.7269i		−	0.844117i	−0.906569	−	0.422059i	$-0.861308\pi$
$193$		−	11.7269i		−	0.844117i	0.906569	−	0.422059i	$-0.138692\pi$
$194$		0			0
$195$			3.47756i			0.249034i
$196$		0			0
$197$	12.1885			0.868398			0.434199	−	0.900817i	$-0.357031\pi$
$197$	12.1885			0.868398			0.434199	+	0.900817i	$0.357031\pi$
$198$		0			0
$199$			15.7331i			1.11529i	0.830079	+	0.557645i	$0.188295\pi$
$199$			15.7331i			1.11529i	−0.830079	+	0.557645i	$0.811705\pi$
$200$		0			0
$201$	−3.58856			−0.253117
$202$		0			0
$203$	4.31799			0.303064
$204$		0			0
$205$		−	2.33658i		−	0.163194i
$206$		0			0
$207$			0.453944i			0.0315513i
$208$		0			0
$209$	−5.97401	+	21.0118i	−0.413231	+	1.45341i
$210$		0			0
$211$	24.9810			1.71976			0.859881	−	0.510495i	$-0.170538\pi$
$211$	24.9810			1.71976			0.859881	+	0.510495i	$0.170538\pi$
$212$		0			0
$213$	−5.38625			−0.369060
$214$		0			0
$215$		−	1.78713i		−	0.121881i
$216$		0			0
$217$		−	14.3098i		−	0.971416i
$218$		0			0
$219$	−11.7717			−0.795458
$220$		0			0
$221$			7.85317i			0.528261i
$222$		0			0
$223$	10.0781			0.674880			0.337440	−	0.941347i	$-0.390439\pi$
$223$	10.0781			0.674880			0.337440	+	0.941347i	$0.390439\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	21.9442			1.45649			0.728244	−	0.685318i	$-0.240337\pi$
$227$	21.9442			1.45649			0.728244	+	0.685318i	$0.240337\pi$
$228$		0			0
$229$	−3.91648			−0.258809			−0.129404	−	0.991592i	$-0.541307\pi$
$229$	−3.91648			−0.258809			−0.129404	+	0.991592i	$0.541307\pi$
$230$		0			0
$231$	−13.9847			−0.920123
$232$		0			0
$233$	9.98688			0.654262			0.327131	−	0.944979i	$-0.393918\pi$
$233$	9.98688			0.654262			0.327131	+	0.944979i	$0.393918\pi$
$234$		0			0
$235$			8.48904i			0.553764i
$236$		0			0
$237$	2.18919			0.142203
$238$		0			0
$239$			8.43260i			0.545459i	0.962091	+	0.272730i	$0.0879264\pi$
$239$			8.43260i			0.545459i	−0.962091	+	0.272730i	$0.912074\pi$
$240$		0			0
$241$		−	30.1280i		−	1.94072i	−0.241668	−	0.970359i	$-0.577694\pi$
$241$		−	30.1280i		−	1.94072i	0.241668	−	0.970359i	$-0.422306\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−0.787044			−0.0502824
$246$		0			0
$247$	−4.14549	+	14.5805i	−0.263771	+	0.927734i
$248$		0			0
$249$		−	9.49244i		−	0.601559i
$250$		0			0
$251$			19.8949i			1.25575i	0.778313	+	0.627876i	$0.216076\pi$
$251$			19.8949i			1.25575i	−0.778313	+	0.627876i	$0.783924\pi$
$252$		0			0
$253$	−2.27493			−0.143024
$254$		0			0
$255$	2.25824			0.141416
$256$		0			0
$257$			1.93019i			0.120402i	0.998186	+	0.0602010i	$0.0191742\pi$
$257$			1.93019i			0.120402i	−0.998186	+	0.0602010i	$0.980826\pi$
$258$		0			0
$259$	13.3156			0.827393
$260$		0			0
$261$		−	1.54737i		−	0.0957801i
$262$		0			0
$263$			5.66147i			0.349101i	0.984648	+	0.174551i	$0.0558473\pi$
$263$			5.66147i			0.349101i	−0.984648	+	0.174551i	$0.944153\pi$
$264$		0			0
$265$		−	3.01016i		−	0.184913i
$266$		0			0
$267$			3.89792i			0.238549i
$268$		0			0
$269$		−	3.92150i		−	0.239098i	−0.992828	−	0.119549i	$-0.961855\pi$
$269$		−	3.92150i		−	0.239098i	0.992828	−	0.119549i	$-0.0381449\pi$
$270$		0			0
$271$		−	12.6114i		−	0.766090i	−0.923730	−	0.383045i	$-0.874875\pi$
$271$		−	12.6114i		−	0.766090i	0.923730	−	0.383045i	$-0.125125\pi$
$272$		0			0
$273$	−9.70424			−0.587327
$274$		0			0
$275$			5.01148i			0.302203i
$276$		0			0
$277$	3.39936			0.204248			0.102124	−	0.994772i	$-0.467436\pi$
$277$	3.39936			0.204248			0.102124	+	0.994772i	$0.467436\pi$
$278$		0			0
$279$	−5.12801			−0.307006
$280$		0			0
$281$		−	7.44525i		−	0.444147i	−0.975030	−	0.222073i	$-0.928718\pi$
$281$		−	7.44525i		−	0.444147i	0.975030	−	0.222073i	$-0.0712824\pi$
$282$		0			0
$283$		−	5.08809i		−	0.302455i	−0.988499	−	0.151228i	$-0.951677\pi$
$283$		−	5.08809i		−	0.302455i	0.988499	−	0.151228i	$-0.0483226\pi$
$284$		0			0
$285$	4.19273	+	1.19207i	0.248356	+	0.0706119i
$286$		0			0
$287$	6.52030			0.384881
$288$		0			0
$289$	−11.9004			−0.700021
$290$		0			0
$291$		−	1.67697i		−	0.0983059i
$292$		0			0
$293$			22.8707i			1.33612i	0.744108	+	0.668059i	$0.232875\pi$
$293$			22.8707i			1.33612i	−0.744108	+	0.668059i	$0.767125\pi$
$294$		0			0
$295$	−1.34096			−0.0780740
$296$		0			0
$297$			5.01148i			0.290795i
$298$		0			0
$299$	−1.57862			−0.0912939
$300$		0			0
$301$	4.98703			0.287448
$302$		0			0
$303$	−5.10425			−0.293231
$304$		0			0
$305$	−6.21296			−0.355753
$306$		0			0
$307$	−33.7089			−1.92387			−0.961933	−	0.273285i	$-0.911890\pi$
$307$	−33.7089			−1.92387			−0.961933	+	0.273285i	$0.911890\pi$
$308$		0			0
$309$	3.58856			0.204146
$310$		0			0
$311$			4.53197i			0.256984i	0.991711	+	0.128492i	$0.0410137\pi$
$311$			4.53197i			0.256984i	−0.991711	+	0.128492i	$0.958986\pi$
$312$		0			0
$313$	21.9549			1.24096			0.620481	−	0.784222i	$-0.286937\pi$
$313$	21.9549			1.24096			0.620481	+	0.784222i	$0.286937\pi$
$314$		0			0
$315$			2.79053i			0.157228i
$316$		0			0
$317$			9.73402i			0.546717i	0.961912	+	0.273358i	$0.0881345\pi$
$317$			9.73402i			0.546717i	−0.961912	+	0.273358i	$0.911866\pi$
$318$		0			0
$319$	7.75463			0.434176
$320$		0			0
$321$	9.79769			0.546854
$322$		0			0
$323$	9.46818	+	2.69197i	0.526824	+	0.149785i
$324$		0			0
$325$			3.47756i			0.192901i
$326$		0			0
$327$		−	0.0765684i		−	0.00423424i
$328$		0			0
$329$	−23.6889			−1.30601
$330$		0			0
$331$	19.7097			1.08334			0.541672	−	0.840590i	$-0.317791\pi$
$331$	19.7097			1.08334			0.541672	+	0.840590i	$0.317791\pi$
$332$		0			0
$333$		−	4.77172i		−	0.261489i
$334$		0			0
$335$	−3.58856			−0.196064
$336$		0			0
$337$		−	26.9177i		−	1.46630i	−0.680068	−	0.733149i	$-0.738050\pi$
$337$		−	26.9177i		−	1.46630i	0.680068	−	0.733149i	$-0.261950\pi$
$338$		0			0
$339$			17.2670i			0.937816i
$340$		0			0
$341$		−	25.6989i		−	1.39167i
$342$		0			0
$343$			17.3374i			0.936133i
$344$		0			0
$345$			0.453944i			0.0244395i
$346$		0			0
$347$		−	0.345111i		−	0.0185265i	−0.999957	−	0.00926325i	$-0.997051\pi$
$347$		−	0.345111i		−	0.0185265i	0.999957	−	0.00926325i	$-0.00294863\pi$
$348$		0			0
$349$	−33.9623			−1.81796			−0.908979	−	0.416842i	$-0.863137\pi$
$349$	−33.9623			−1.81796			−0.908979	+	0.416842i	$0.863137\pi$
$350$		0			0
$351$			3.47756i			0.185619i
$352$		0			0
$353$	25.5090			1.35771			0.678854	−	0.734274i	$-0.262477\pi$
$353$	25.5090			1.35771			0.678854	+	0.734274i	$0.262477\pi$
$354$		0			0
$355$	−5.38625			−0.285872
$356$		0			0
$357$			6.30168i			0.333520i
$358$		0			0
$359$			7.48070i			0.394816i	0.980321	+	0.197408i	$0.0632524\pi$
$359$			7.48070i			0.394816i	−0.980321	+	0.197408i	$0.936748\pi$
$360$		0			0
$361$	16.1580	+	9.99603i	0.850419	+	0.526107i
$362$		0			0
$363$	−14.1149			−0.740840
$364$		0			0
$365$	−11.7717			−0.616159
$366$		0			0
$367$		−	2.36401i		−	0.123400i	−0.998095	−	0.0617002i	$-0.980348\pi$
$367$		−	2.36401i		−	0.123400i	0.998095	−	0.0617002i	$-0.0196523\pi$
$368$		0			0
$369$		−	2.33658i		−	0.121638i
$370$		0			0
$371$	8.39993			0.436102
$372$		0			0
$373$			2.58300i			0.133743i	0.997762	+	0.0668713i	$0.0213017\pi$
$373$			2.58300i			0.133743i	−0.997762	+	0.0668713i	$0.978698\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	5.38109			0.277140
$378$		0			0
$379$	26.3961			1.35588			0.677938	−	0.735119i	$-0.262874\pi$
$379$	26.3961			1.35588			0.677938	+	0.735119i	$0.262874\pi$
$380$		0			0
$381$	−10.1831			−0.521699
$382$		0			0
$383$	−23.9156			−1.22203			−0.611015	−	0.791619i	$-0.709239\pi$
$383$	−23.9156			−1.22203			−0.611015	+	0.791619i	$0.709239\pi$
$384$		0			0
$385$	−13.9847			−0.712724
$386$		0			0
$387$		−	1.78713i		−	0.0908448i
$388$		0			0
$389$	−10.2544			−0.519921			−0.259960	−	0.965619i	$-0.583709\pi$
$389$	−10.2544			−0.519921			−0.259960	+	0.965619i	$0.583709\pi$
$390$		0			0
$391$			1.02511i			0.0518423i
$392$		0			0
$393$			16.8540i			0.850170i
$394$		0			0
$395$	2.18919			0.110150
$396$		0			0
$397$	−11.5809			−0.581230			−0.290615	−	0.956840i	$-0.593860\pi$
$397$	−11.5809			−0.581230			−0.290615	+	0.956840i	$0.593860\pi$
$398$		0			0
$399$	−3.32650	+	11.6999i	−0.166533	+	0.585729i
$400$		0			0
$401$			13.4206i			0.670191i	0.942184	+	0.335095i	$0.108769\pi$
$401$			13.4206i			0.670191i	−0.942184	+	0.335095i	$0.891231\pi$
$402$		0			0
$403$		−	17.8330i		−	0.888324i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	23.9134			1.18534
$408$		0			0
$409$		−	18.4595i		−	0.912764i	−0.889784	−	0.456382i	$-0.849145\pi$
$409$		−	18.4595i		−	0.912764i	0.889784	−	0.456382i	$-0.150855\pi$
$410$		0			0
$411$	8.58172			0.423305
$412$		0			0
$413$		−	3.74200i		−	0.184132i
$414$		0			0
$415$		−	9.49244i		−	0.465966i
$416$		0			0
$417$			12.1130i			0.593175i
$418$		0			0
$419$			22.8974i			1.11861i	0.828961	+	0.559307i	$0.188933\pi$
$419$			22.8974i			1.11861i	−0.828961	+	0.559307i	$0.811067\pi$
$420$		0			0
$421$			3.89670i			0.189914i	0.995481	+	0.0949568i	$0.0302713\pi$
$421$			3.89670i			0.189914i	−0.995481	+	0.0949568i	$0.969729\pi$
$422$		0			0
$423$			8.48904i			0.412751i
$424$		0			0
$425$	2.25824			0.109541
$426$		0			0
$427$		−	17.3374i		−	0.839017i
$428$		0			0
$429$	−17.4277			−0.841419
$430$		0			0
$431$	−41.2101			−1.98502			−0.992510	−	0.122164i	$-0.961017\pi$
$431$	−41.2101			−1.98502			−0.992510	+	0.122164i	$0.961017\pi$
$432$		0			0
$433$		−	27.3990i		−	1.31671i	−0.752706	−	0.658357i	$-0.771252\pi$
$433$		−	27.3990i		−	1.31671i	0.752706	−	0.658357i	$-0.228748\pi$
$434$		0			0
$435$		−	1.54737i		−	0.0741909i
$436$		0			0
$437$	−0.541132	+	1.90326i	−0.0258858	+	0.0910455i
$438$		0			0
$439$	27.6076			1.31764			0.658820	−	0.752301i	$-0.271056\pi$
$439$	27.6076			1.31764			0.658820	+	0.752301i	$0.271056\pi$
$440$		0			0
$441$	−0.787044			−0.0374783
$442$		0			0
$443$		−	9.94502i		−	0.472502i	−0.971692	−	0.236251i	$-0.924081\pi$
$443$		−	9.94502i		−	0.472502i	0.971692	−	0.236251i	$-0.0759188\pi$
$444$		0			0
$445$			3.89792i			0.184779i
$446$		0			0
$447$	−17.6423			−0.834450
$448$		0			0
$449$			25.5454i			1.20556i	0.797906	+	0.602781i	$0.205941\pi$
$449$			25.5454i			1.20556i	−0.797906	+	0.602781i	$0.794059\pi$
$450$		0			0
$451$	11.7097			0.551390
$452$		0			0
$453$	22.4559			1.05507
$454$		0			0
$455$	−9.70424			−0.454942
$456$		0			0
$457$	−12.3655			−0.578433			−0.289217	−	0.957264i	$-0.593395\pi$
$457$	−12.3655			−0.578433			−0.289217	+	0.957264i	$0.593395\pi$
$458$		0			0
$459$	2.25824			0.105406
$460$		0			0
$461$	−9.94867			−0.463356			−0.231678	−	0.972793i	$-0.574422\pi$
$461$	−9.94867			−0.463356			−0.231678	+	0.972793i	$0.574422\pi$
$462$		0			0
$463$		−	18.4744i		−	0.858580i	−0.903167	−	0.429290i	$-0.858764\pi$
$463$		−	18.4744i		−	0.858580i	0.903167	−	0.429290i	$-0.141236\pi$
$464$		0			0
$465$	−5.12801			−0.237806
$466$		0			0
$467$		−	9.09632i		−	0.420928i	−0.977602	−	0.210464i	$-0.932503\pi$
$467$		−	9.09632i		−	0.420928i	0.977602	−	0.210464i	$-0.0674974\pi$
$468$		0			0
$469$		−	10.0140i		−	0.462402i
$470$		0			0
$471$	4.75320			0.219016
$472$		0			0
$473$	8.95615			0.411804
$474$		0			0
$475$	4.19273	+	1.19207i	0.192376	+	0.0546958i
$476$		0			0
$477$		−	3.01016i		−	0.137826i
$478$		0			0
$479$			20.5135i			0.937284i	0.883388	+	0.468642i	$0.155257\pi$
$479$			20.5135i			0.937284i	−0.883388	+	0.468642i	$0.844743\pi$
$480$		0			0
$481$	16.5940			0.756620
$482$		0			0
$483$	−1.26674			−0.0576388
$484$		0			0
$485$		−	1.67697i		−	0.0761474i
$486$		0			0
$487$	26.2253			1.18838			0.594192	−	0.804323i	$-0.297472\pi$
$487$	26.2253			1.18838			0.594192	+	0.804323i	$0.297472\pi$
$488$		0			0
$489$		−	1.33369i		−	0.0603114i
$490$		0			0
$491$		−	10.1642i		−	0.458702i	−0.973344	−	0.229351i	$-0.926340\pi$
$491$		−	10.1642i		−	0.458702i	0.973344	−	0.229351i	$-0.0736603\pi$
$492$		0			0
$493$		−	3.49434i		−	0.157377i
$494$		0			0
$495$			5.01148i			0.225249i
$496$		0			0
$497$		−	15.0305i		−	0.674209i
$498$		0			0
$499$			4.94452i			0.221347i	0.993857	+	0.110674i	$0.0353008\pi$
$499$			4.94452i			0.221347i	−0.993857	+	0.110674i	$0.964699\pi$
$500$		0			0
$501$	11.1717			0.499115
$502$		0			0
$503$			20.1453i			0.898233i	0.893473	+	0.449116i	$0.148261\pi$
$503$			20.1453i			0.898233i	−0.893473	+	0.449116i	$0.851739\pi$
$504$		0			0
$505$	−5.10425			−0.227136
$506$		0			0
$507$	0.906548			0.0402612
$508$		0			0
$509$			10.7845i			0.478013i	0.971018	+	0.239006i	$0.0768217\pi$
$509$			10.7845i			0.478013i	−0.971018	+	0.239006i	$0.923178\pi$
$510$		0			0
$511$		−	32.8493i		−	1.45317i
$512$		0			0
$513$	4.19273	+	1.19207i	0.185114	+	0.0526310i
$514$		0			0
$515$	3.58856			0.158131
$516$		0			0
$517$	−42.5426			−1.87102
$518$		0			0
$519$		−	9.69811i		−	0.425700i
$520$		0			0
$521$		−	22.2501i		−	0.974794i	−0.873181	−	0.487397i	$-0.837947\pi$
$521$		−	22.2501i		−	0.974794i	0.873181	−	0.487397i	$-0.162053\pi$
$522$		0			0
$523$	−17.5616			−0.767917			−0.383958	−	0.923350i	$-0.625439\pi$
$523$	−17.5616			−0.767917			−0.383958	+	0.923350i	$0.625439\pi$
$524$		0			0
$525$			2.79053i			0.121789i
$526$		0			0
$527$	−11.5803			−0.504444
$528$		0			0
$529$	22.7939			0.991041
$530$		0			0
$531$	−1.34096			−0.0581929
$532$		0			0
$533$	8.12562			0.351960
$534$		0			0
$535$	9.79769			0.423591
$536$		0			0
$537$	1.67272			0.0721830
$538$		0			0
$539$		−	3.94425i		−	0.169891i
$540$		0			0
$541$	−18.4645			−0.793851			−0.396926	−	0.917851i	$-0.629923\pi$
$541$	−18.4645			−0.793851			−0.396926	+	0.917851i	$0.629923\pi$
$542$		0			0
$543$		−	10.1587i		−	0.435952i
$544$		0			0
$545$		−	0.0765684i		−	0.00327983i
$546$		0			0
$547$	16.3273			0.698105			0.349053	−	0.937103i	$-0.386503\pi$
$547$	16.3273			0.698105			0.349053	+	0.937103i	$0.386503\pi$
$548$		0			0
$549$	−6.21296			−0.265163
$550$		0			0
$551$	1.84457	−	6.48772i	0.0785815	−	0.276386i
$552$		0			0
$553$			6.10900i			0.259781i
$554$		0			0
$555$		−	4.77172i		−	0.202548i
$556$		0			0
$557$	−34.3987			−1.45752			−0.728760	−	0.684769i	$-0.759903\pi$
$557$	−34.3987			−1.45752			−0.728760	+	0.684769i	$0.759903\pi$
$558$		0			0
$559$	6.21485			0.262860
$560$		0			0
$561$			11.3171i			0.477809i
$562$		0			0
$563$	−21.1471			−0.891242			−0.445621	−	0.895222i	$-0.647017\pi$
$563$	−21.1471			−0.891242			−0.445621	+	0.895222i	$0.647017\pi$
$564$		0			0
$565$			17.2670i			0.726429i
$566$		0			0
$567$			2.79053i			0.117191i
$568$		0			0
$569$			26.5895i			1.11469i	0.830281	+	0.557344i	$0.188180\pi$
$569$			26.5895i			1.11469i	−0.830281	+	0.557344i	$0.811820\pi$
$570$		0			0
$571$			27.6017i			1.15509i	0.816358	+	0.577547i	$0.195990\pi$
$571$			27.6017i			1.15509i	−0.816358	+	0.577547i	$0.804010\pi$
$572$		0			0
$573$		−	21.2959i		−	0.889647i
$574$		0			0
$575$			0.453944i			0.0189308i
$576$		0			0
$577$	23.2290			0.967036			0.483518	−	0.875334i	$-0.339359\pi$
$577$	23.2290			0.967036			0.483518	+	0.875334i	$0.339359\pi$
$578$		0			0
$579$		−	11.7269i		−	0.487351i
$580$		0			0
$581$	26.4889			1.09895
$582$		0			0
$583$	15.0853			0.624771
$584$		0			0
$585$			3.47756i			0.143780i
$586$		0			0
$587$		−	24.5441i		−	1.01305i	−0.862227	−	0.506523i	$-0.830931\pi$
$587$		−	24.5441i		−	1.01305i	0.862227	−	0.506523i	$-0.169069\pi$
$588$		0			0
$589$	−21.5004	−	6.11293i	−0.885906	−	0.251879i
$590$		0			0
$591$	12.1885			0.501370
$592$		0			0
$593$	30.1832			1.23947			0.619737	−	0.784809i	$-0.287239\pi$
$593$	30.1832			1.23947			0.619737	+	0.784809i	$0.287239\pi$
$594$		0			0
$595$			6.30168i			0.258344i
$596$		0			0
$597$			15.7331i			0.643913i
$598$		0			0
$599$	−26.6559			−1.08913			−0.544566	−	0.838718i	$-0.683306\pi$
$599$	−26.6559			−1.08913			−0.544566	+	0.838718i	$0.683306\pi$
$600$		0			0
$601$			6.28750i			0.256472i	0.991744	+	0.128236i	$0.0409315\pi$
$601$			6.28750i			0.256472i	−0.991744	+	0.128236i	$0.959068\pi$
$602$		0			0
$603$	−3.58856			−0.146137
$604$		0			0
$605$	−14.1149			−0.573852
$606$		0			0
$607$	18.4629			0.749387			0.374694	−	0.927149i	$-0.377748\pi$
$607$	18.4629			0.749387			0.374694	+	0.927149i	$0.377748\pi$
$608$		0			0
$609$	4.31799			0.174974
$610$		0			0
$611$	−29.5212			−1.19430
$612$		0			0
$613$	23.2890			0.940634			0.470317	−	0.882498i	$-0.344140\pi$
$613$	23.2890			0.940634			0.470317	+	0.882498i	$0.344140\pi$
$614$		0			0
$615$		−	2.33658i		−	0.0942201i
$616$		0			0
$617$	30.2276			1.21692			0.608458	−	0.793586i	$-0.291788\pi$
$617$	30.2276			1.21692			0.608458	+	0.793586i	$0.291788\pi$
$618$		0			0
$619$		−	10.9782i		−	0.441249i	−0.975359	−	0.220625i	$-0.929190\pi$
$619$		−	10.9782i		−	0.441249i	0.975359	−	0.220625i	$-0.0708096\pi$
$620$		0			0
$621$			0.453944i			0.0182161i
$622$		0			0
$623$	−10.8773			−0.435788
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−5.97401	+	21.0118i	−0.238579	+	0.839129i
$628$		0			0
$629$		−	10.7757i		−	0.429655i
$630$		0			0
$631$		−	0.188288i		−	0.00749561i	−0.999993	−	0.00374780i	$-0.998807\pi$
$631$		−	0.188288i		−	0.00749561i	0.999993	−	0.00374780i	$-0.00119297\pi$
$632$		0			0
$633$	24.9810			0.992905
$634$		0			0
$635$	−10.1831			−0.404106
$636$		0			0
$637$		−	2.73700i		−	0.108444i
$638$		0			0
$639$	−5.38625			−0.213077
$640$		0			0
$641$			19.5155i			0.770817i	0.922746	+	0.385409i	$0.125939\pi$
$641$			19.5155i			0.770817i	−0.922746	+	0.385409i	$0.874061\pi$
$642$		0			0
$643$		−	21.6799i		−	0.854972i	−0.904022	−	0.427486i	$-0.859399\pi$
$643$		−	21.6799i		−	0.854972i	0.904022	−	0.427486i	$-0.140601\pi$
$644$		0			0
$645$		−	1.78713i		−	0.0703681i
$646$		0			0
$647$		−	7.19675i		−	0.282933i	−0.989943	−	0.141467i	$-0.954818\pi$
$647$		−	7.19675i		−	0.282933i	0.989943	−	0.141467i	$-0.0451818\pi$
$648$		0			0
$649$		−	6.72021i		−	0.263791i
$650$		0			0
$651$		−	14.3098i		−	0.560847i
$652$		0			0
$653$	−8.95947			−0.350611			−0.175305	−	0.984514i	$-0.556091\pi$
$653$	−8.95947			−0.350611			−0.175305	+	0.984514i	$0.556091\pi$
$654$		0			0
$655$			16.8540i			0.658539i
$656$		0			0
$657$	−11.7717			−0.459258
$658$		0			0
$659$	4.61309			0.179700			0.0898502	−	0.995955i	$-0.471361\pi$
$659$	4.61309			0.179700			0.0898502	+	0.995955i	$0.471361\pi$
$660$		0			0
$661$		−	25.6833i		−	0.998964i	−0.866324	−	0.499482i	$-0.833524\pi$
$661$		−	25.6833i		−	0.998964i	0.866324	−	0.499482i	$-0.166476\pi$
$662$		0			0
$663$			7.85317i			0.304992i
$664$		0			0
$665$	−3.32650	+	11.6999i	−0.128996	+	0.453704i
$666$		0			0
$667$	0.702422			0.0271979
$668$		0			0
$669$	10.0781			0.389642
$670$		0			0
$671$		−	31.1361i		−	1.20200i
$672$		0			0
$673$		−	24.0542i		−	0.927221i	−0.886039	−	0.463610i	$-0.846554\pi$
$673$		−	24.0542i		−	0.927221i	0.886039	−	0.463610i	$-0.153446\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$			16.5914i			0.637659i	0.947812	+	0.318830i	$0.103290\pi$
$677$			16.5914i			0.637659i	−0.947812	+	0.318830i	$0.896710\pi$
$678$		0			0
$679$	4.67964			0.179588
$680$		0			0
$681$	21.9442			0.840904
$682$		0			0
$683$	5.79325			0.221672			0.110836	−	0.993839i	$-0.464647\pi$
$683$	5.79325			0.221672			0.110836	+	0.993839i	$0.464647\pi$
$684$		0			0
$685$	8.58172			0.327891
$686$		0			0
$687$	−3.91648			−0.149423
$688$		0			0
$689$	10.4680			0.398800
$690$		0			0
$691$		−	38.9878i		−	1.48317i	−0.670861	−	0.741583i	$-0.734075\pi$
$691$		−	38.9878i		−	1.48317i	0.670861	−	0.741583i	$-0.265925\pi$
$692$		0			0
$693$	−13.9847			−0.531233
$694$		0			0
$695$			12.1130i			0.459472i
$696$		0			0
$697$		−	5.27656i		−	0.199864i
$698$		0			0
$699$	9.98688			0.377739
$700$		0			0
$701$	30.3226			1.14527			0.572635	−	0.819811i	$-0.305921\pi$
$701$	30.3226			1.14527			0.572635	+	0.819811i	$0.305921\pi$
$702$		0			0
$703$	5.68821	−	20.0065i	0.214535	−	0.754561i
$704$		0			0
$705$			8.48904i			0.319716i
$706$		0			0
$707$		−	14.2435i		−	0.535683i
$708$		0			0
$709$	−4.21138			−0.158162			−0.0790808	−	0.996868i	$-0.525199\pi$
$709$	−4.21138			−0.158162			−0.0790808	+	0.996868i	$0.525199\pi$
$710$		0			0
$711$	2.18919			0.0821011
$712$		0			0
$713$		−	2.32783i		−	0.0871779i
$714$		0			0
$715$	−17.4277			−0.651760
$716$		0			0
$717$			8.43260i			0.314921i
$718$		0			0
$719$			45.7427i			1.70591i	0.521980	+	0.852957i	$0.325193\pi$
$719$			45.7427i			1.70591i	−0.521980	+	0.852957i	$0.674807\pi$
$720$		0			0
$721$			10.0140i			0.372940i
$722$		0			0
$723$		−	30.1280i		−	1.12047i
$724$		0			0
$725$		−	1.54737i		−	0.0574680i
$726$		0			0
$727$			13.6129i			0.504876i	0.967613	+	0.252438i	$0.0812324\pi$
$727$			13.6129i			0.504876i	−0.967613	+	0.252438i	$0.918768\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	4.03576i		−	0.149268i
$732$		0			0
$733$	0.680329			0.0251285			0.0125643	−	0.999921i	$-0.496001\pi$
$733$	0.680329			0.0251285			0.0125643	+	0.999921i	$0.496001\pi$
$734$		0			0
$735$	−0.787044			−0.0290305
$736$		0			0
$737$		−	17.9840i		−	0.662448i
$738$		0			0
$739$			30.8656i			1.13541i	0.823232	+	0.567704i	$0.192168\pi$
$739$			30.8656i			1.13541i	−0.823232	+	0.567704i	$0.807832\pi$
$740$		0			0
$741$	−4.14549	+	14.5805i	−0.152288	+	0.535627i
$742$		0			0
$743$	−30.9470			−1.13534			−0.567668	−	0.823258i	$-0.692154\pi$
$743$	−30.9470			−1.13534			−0.567668	+	0.823258i	$0.692154\pi$
$744$		0			0
$745$	−17.6423			−0.646363
$746$		0			0
$747$		−	9.49244i		−	0.347310i
$748$		0			0
$749$			27.3407i			0.999008i
$750$		0			0
$751$	34.0255			1.24161			0.620804	−	0.783966i	$-0.286806\pi$
$751$	34.0255			1.24161			0.620804	+	0.783966i	$0.286806\pi$
$752$		0			0
$753$			19.8949i			0.725009i
$754$		0			0
$755$	22.4559			0.817252
$756$		0			0
$757$	−34.9975			−1.27200			−0.636002	−	0.771687i	$-0.719413\pi$
$757$	−34.9975			−1.27200			−0.636002	+	0.771687i	$0.719413\pi$
$758$		0			0
$759$	−2.27493			−0.0825747
$760$		0			0
$761$	3.12865			0.113414			0.0567068	−	0.998391i	$-0.481940\pi$
$761$	3.12865			0.113414			0.0567068	+	0.998391i	$0.481940\pi$
$762$		0			0
$763$	0.213666			0.00773524
$764$		0			0
$765$	2.25824			0.0816468
$766$		0			0
$767$		−	4.66329i		−	0.168382i
$768$		0			0
$769$	−13.4129			−0.483682			−0.241841	−	0.970316i	$-0.577751\pi$
$769$	−13.4129			−0.483682			−0.241841	+	0.970316i	$0.577751\pi$
$770$		0			0
$771$			1.93019i			0.0695141i
$772$		0			0
$773$			16.3402i			0.587716i	0.955849	+	0.293858i	$0.0949392\pi$
$773$			16.3402i			0.587716i	−0.955849	+	0.293858i	$0.905061\pi$
$774$		0			0
$775$	−5.12801			−0.184204
$776$		0			0
$777$	13.3156			0.477695
$778$		0			0
$779$	2.78536	−	9.79666i	0.0997960	−	0.351002i
$780$		0			0
$781$		−	26.9931i		−	0.965887i
$782$		0			0
$783$		−	1.54737i		−	0.0552987i
$784$		0			0
$785$	4.75320			0.169649
$786$		0			0
$787$	−38.2407			−1.36313			−0.681566	−	0.731757i	$-0.738701\pi$
$787$	−38.2407			−1.36313			−0.681566	+	0.731757i	$0.738701\pi$
$788$		0			0
$789$			5.66147i			0.201554i
$790$		0			0
$791$	−48.1841			−1.71323
$792$		0			0
$793$		−	21.6060i		−	0.767250i
$794$		0			0
$795$		−	3.01016i		−	0.106759i
$796$		0			0
$797$			9.62113i			0.340798i	0.985375	+	0.170399i	$0.0545056\pi$
$797$			9.62113i			0.340798i	−0.985375	+	0.170399i	$0.945494\pi$
$798$		0			0
$799$			19.1703i			0.678196i
$800$		0			0
$801$			3.89792i			0.137726i
$802$		0			0
$803$		−	58.9936i		−	2.08184i
$804$		0			0
$805$	−1.26674			−0.0446468
$806$		0			0
$807$		−	3.92150i		−	0.138043i
$808$		0			0
$809$	27.3647			0.962092			0.481046	−	0.876695i	$-0.340257\pi$
$809$	27.3647			0.962092			0.481046	+	0.876695i	$0.340257\pi$
$810$		0			0
$811$	−52.6092			−1.84736			−0.923680	−	0.383165i	$-0.874834\pi$
$811$	−52.6092			−1.84736			−0.923680	+	0.383165i	$0.874834\pi$
$812$		0			0
$813$		−	12.6114i		−	0.442302i
$814$		0			0
$815$		−	1.33369i		−	0.0467170i
$816$		0			0
$817$	2.13038	−	7.49294i	0.0745324	−	0.262145i
$818$		0			0
$819$	−9.70424			−0.339094
$820$		0			0
$821$	4.27371			0.149154			0.0745768	−	0.997215i	$-0.476239\pi$
$821$	4.27371			0.149154			0.0745768	+	0.997215i	$0.476239\pi$
$822$		0			0
$823$		−	30.5714i		−	1.06565i	−0.846225	−	0.532826i	$-0.821130\pi$
$823$		−	30.5714i		−	1.06565i	0.846225	−	0.532826i	$-0.178870\pi$
$824$		0			0
$825$			5.01148i			0.174477i
$826$		0			0
$827$	−56.4650			−1.96348			−0.981740	−	0.190229i	$-0.939077\pi$
$827$	−56.4650			−1.96348			−0.981740	+	0.190229i	$0.939077\pi$
$828$		0			0
$829$			18.7202i			0.650179i	0.945683	+	0.325089i	$0.105394\pi$
$829$			18.7202i			0.650179i	−0.945683	+	0.325089i	$0.894606\pi$
$830$		0			0
$831$	3.39936			0.117923
$832$		0			0
$833$	−1.77733			−0.0615809
$834$		0			0
$835$	11.1717			0.386613
$836$		0			0
$837$	−5.12801			−0.177250
$838$		0			0
$839$	−17.5811			−0.606966			−0.303483	−	0.952837i	$-0.598150\pi$
$839$	−17.5811			−0.606966			−0.303483	+	0.952837i	$0.598150\pi$
$840$		0			0
$841$	26.6056			0.917436
$842$		0			0
$843$		−	7.44525i		−	0.256428i
$844$		0			0
$845$	0.906548			0.0311862
$846$		0			0
$847$		−	39.3880i		−	1.35339i
$848$		0			0
$849$		−	5.08809i		−	0.174623i
$850$		0			0
$851$	2.16610			0.0742528
$852$		0			0
$853$	3.39936			0.116392			0.0581960	−	0.998305i	$-0.481465\pi$
$853$	3.39936			0.116392			0.0581960	+	0.998305i	$0.481465\pi$
$854$		0			0
$855$	4.19273	+	1.19207i	0.143388	+	0.0407678i
$856$		0			0
$857$		−	35.9868i		−	1.22929i	−0.788806	−	0.614643i	$-0.789300\pi$
$857$		−	35.9868i		−	1.22929i	0.788806	−	0.614643i	$-0.210700\pi$
$858$		0			0
$859$			19.7234i			0.672954i	0.941692	+	0.336477i	$0.109236\pi$
$859$			19.7234i			0.672954i	−0.941692	+	0.336477i	$0.890764\pi$
$860$		0			0
$861$	6.52030			0.222211
$862$		0			0
$863$	−2.30483			−0.0784573			−0.0392287	−	0.999230i	$-0.512490\pi$
$863$	−2.30483			−0.0784573			−0.0392287	+	0.999230i	$0.512490\pi$
$864$		0			0
$865$		−	9.69811i		−	0.329745i
$866$		0			0
$867$	−11.9004			−0.404157
$868$		0			0
$869$			10.9711i			0.372169i
$870$		0			0
$871$		−	12.4794i		−	0.422850i
$872$		0			0
$873$		−	1.67697i		−	0.0567570i
$874$		0			0
$875$			2.79053i			0.0943370i
$876$		0			0
$877$		−	0.248047i		−	0.00837594i	−0.999991	−	0.00418797i	$-0.998667\pi$
$877$		−	0.248047i		−	0.00837594i	0.999991	−	0.00418797i	$-0.00133308\pi$
$878$		0			0
$879$			22.8707i			0.771408i
$880$		0			0
$881$	10.9135			0.367684			0.183842	−	0.982956i	$-0.441147\pi$
$881$	10.9135			0.367684			0.183842	+	0.982956i	$0.441147\pi$
$882$		0			0
$883$		−	45.6930i		−	1.53769i	−0.639434	−	0.768846i	$-0.720831\pi$
$883$		−	45.6930i		−	1.53769i	0.639434	−	0.768846i	$-0.279169\pi$
$884$		0			0
$885$	−1.34096			−0.0450760
$886$		0			0
$887$	26.4723			0.888852			0.444426	−	0.895816i	$-0.353408\pi$
$887$	26.4723			0.888852			0.444426	+	0.895816i	$0.353408\pi$
$888$		0			0
$889$		−	28.4164i		−	0.953054i
$890$		0			0
$891$			5.01148i			0.167891i
$892$		0			0
$893$	−10.1195	+	35.5922i	−0.338636	+	1.19105i
$894$		0			0
$895$	1.67272			0.0559127
$896$		0			0
$897$	−1.57862			−0.0527086
$898$		0			0
$899$			7.93495i			0.264645i
$900$		0			0
$901$		−	6.79765i		−	0.226463i
$902$		0			0
$903$	4.98703			0.165958
$904$		0			0
$905$		−	10.1587i		−	0.337687i
$906$		0			0
$907$	3.42828			0.113834			0.0569171	−	0.998379i	$-0.481873\pi$
$907$	3.42828			0.113834			0.0569171	+	0.998379i	$0.481873\pi$
$908$		0			0
$909$	−5.10425			−0.169297
$910$		0			0
$911$	−1.27414			−0.0422142			−0.0211071	−	0.999777i	$-0.506719\pi$
$911$	−1.27414			−0.0422142			−0.0211071	+	0.999777i	$0.506719\pi$
$912$		0			0
$913$	47.5711			1.57437
$914$		0			0
$915$	−6.21296			−0.205394
$916$		0			0
$917$	−47.0314			−1.55312
$918$		0			0
$919$			4.73219i			0.156101i	0.996949	+	0.0780504i	$0.0248695\pi$
$919$			4.73219i			0.156101i	−0.996949	+	0.0780504i	$0.975131\pi$
$920$		0			0
$921$	−33.7089			−1.11074
$922$		0			0
$923$		−	18.7310i		−	0.616539i
$924$		0			0
$925$		−	4.77172i		−	0.156893i
$926$		0			0
$927$	3.58856			0.117864
$928$		0			0
$929$	17.7772			0.583251			0.291625	−	0.956533i	$-0.405804\pi$
$929$	17.7772			0.583251			0.291625	+	0.956533i	$0.405804\pi$
$930$		0			0
$931$	−3.29986	−	0.938209i	−0.108149	−	0.0307486i
$932$		0			0
$933$			4.53197i			0.148370i
$934$		0			0
$935$			11.3171i			0.370109i
$936$		0			0
$937$	48.1049			1.57152			0.785760	−	0.618531i	$-0.212272\pi$
$937$	48.1049			1.57152			0.785760	+	0.618531i	$0.212272\pi$
$938$		0			0
$939$	21.9549			0.716470
$940$		0			0
$941$			49.1215i			1.60132i	0.599121	+	0.800658i	$0.295517\pi$
$941$			49.1215i			1.60132i	−0.599121	+	0.800658i	$0.704483\pi$
$942$		0			0
$943$	1.06068			0.0345404
$944$		0			0
$945$			2.79053i			0.0907759i
$946$		0			0
$947$		−	32.8252i		−	1.06668i	−0.845902	−	0.533338i	$-0.820938\pi$
$947$		−	32.8252i		−	1.06668i	0.845902	−	0.533338i	$-0.179062\pi$
$948$		0			0
$949$		−	40.9369i		−	1.32887i
$950$		0			0
$951$			9.73402i			0.315647i
$952$		0			0
$953$			61.1052i			1.97939i	0.143190	+	0.989695i	$0.454264\pi$
$953$			61.1052i			1.97939i	−0.143190	+	0.989695i	$0.545736\pi$
$954$		0			0
$955$		−	21.2959i		−	0.689118i
$956$		0			0
$957$	7.75463			0.250672
$958$		0			0
$959$			23.9475i			0.773306i
$960$		0			0
$961$	−4.70353			−0.151727
$962$		0			0
$963$	9.79769			0.315726
$964$		0			0
$965$		−	11.7269i		−	0.377501i
$966$		0			0
$967$		−	48.6878i		−	1.56569i	−0.622215	−	0.782846i	$-0.713767\pi$
$967$		−	48.6878i		−	1.56569i	0.622215	−	0.782846i	$-0.286233\pi$
$968$		0			0
$969$	9.46818	+	2.69197i	0.304162	+	0.0864786i
$970$		0			0
$971$	−9.64648			−0.309570			−0.154785	−	0.987948i	$-0.549469\pi$
$971$	−9.64648			−0.309570			−0.154785	+	0.987948i	$0.549469\pi$
$972$		0			0
$973$	−33.8016			−1.08363
$974$		0			0
$975$			3.47756i			0.111371i
$976$		0			0
$977$			10.3497i			0.331116i	0.986200	+	0.165558i	$0.0529425\pi$
$977$			10.3497i			0.331116i	−0.986200	+	0.165558i	$0.947057\pi$
$978$		0			0
$979$	−19.5343			−0.624320
$980$		0			0
$981$		−	0.0765684i		−	0.00244464i
$982$		0			0
$983$	55.7912			1.77946			0.889731	−	0.456485i	$-0.150892\pi$
$983$	55.7912			1.77946			0.889731	+	0.456485i	$0.150892\pi$
$984$		0			0
$985$	12.1885			0.388359
$986$		0			0
$987$	−23.6889			−0.754026
$988$		0			0
$989$	0.811256			0.0257964
$990$		0			0
$991$	−22.0505			−0.700458			−0.350229	−	0.936664i	$-0.613896\pi$
$991$	−22.0505			−0.700458			−0.350229	+	0.936664i	$0.613896\pi$
$992$		0			0
$993$	19.7097			0.625469
$994$		0			0
$995$			15.7331i			0.498773i
$996$		0			0
$997$	49.3087			1.56162			0.780811	−	0.624767i	$-0.214806\pi$
$997$	49.3087			1.56162			0.780811	+	0.624767i	$0.214806\pi$
$998$		0			0
$999$		−	4.77172i		−	0.150971i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.l.2431.9	yes	12
4.3	odd	2		4560.2.d.j.2431.4	✓	12
19.18	odd	2		4560.2.d.j.2431.9	yes	12
76.75	even	2	inner	4560.2.d.l.2431.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.j.2431.4	✓	12	4.3	odd	2
4560.2.d.j.2431.9	yes	12	19.18	odd	2
4560.2.d.l.2431.4	yes	12	76.75	even	2	inner
4560.2.d.l.2431.9	yes	12	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.79053i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.79053i q^{7} +1.00000 q^{9} +5.01148i q^{11} +3.47756i q^{13} +1.00000 q^{15} +2.25824 q^{17} +(4.19273 + 1.19207i) q^{19} +2.79053i q^{21} +0.453944i q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.54737i q^{29} -5.12801 q^{31} +5.01148i q^{33} +2.79053i q^{35} -4.77172i q^{37} +3.47756i q^{39} -2.33658i q^{41} -1.78713i q^{43} +1.00000 q^{45} +8.48904i q^{47} -0.787044 q^{49} +2.25824 q^{51} -3.01016i q^{53} +5.01148i q^{55} +(4.19273 + 1.19207i) q^{57} -1.34096 q^{59} -6.21296 q^{61} +2.79053i q^{63} +3.47756i q^{65} -3.58856 q^{67} +0.453944i q^{69} -5.38625 q^{71} -11.7717 q^{73} +1.00000 q^{75} -13.9847 q^{77} +2.18919 q^{79} +1.00000 q^{81} -9.49244i q^{83} +2.25824 q^{85} -1.54737i q^{87} +3.89792i q^{89} -9.70424 q^{91} -5.12801 q^{93} +(4.19273 + 1.19207i) q^{95} -1.67697i q^{97} +5.01148i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9	\( 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9} + 12 q^{15} + 4 q^{17} + 12 q^{25} + 12 q^{27} - 12 q^{31} + 12 q^{45} - 28 q^{49} + 4 q^{51} + 52 q^{59} - 56 q^{61} - 32 q^{67} + 8 q^{71} + 32 q^{73} + 12 q^{75} + 24 q^{77} - 28 q^{79} + 12 q^{81} + 4 q^{85} + 32 q^{91} - 12 q^{93}+O(q^{100}) \) 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9 + 12 * q^15 + 4 * q^17 + 12 * q^25 + 12 * q^27 - 12 * q^31 + 12 * q^45 - 28 * q^49 + 4 * q^51 + 52 * q^59 - 56 * q^61 - 32 * q^67 + 8 * q^71 + 32 * q^73 + 12 * q^75 + 24 * q^77 - 28 * q^79 + 12 * q^81 + 4 * q^85 + 32 * q^91 - 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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