LMFDB - Embedded newform 960.3.j.e.319.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(319,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("960.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.j (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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.389136420864.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 $ x^8 + 5x^6 + 24x^4 + 80*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.2
Root			$1.52274 + 1.29664i$ of defining polynomial
Character	$\chi$	$=$	960.319
Dual form			960.3.j.e.319.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +(-4.27492 + 2.59328i) q^{5} +0.837253 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +(-4.27492 + 2.59328i) q^{5} +0.837253 q^{7} +3.00000 q^{9} +15.7955i q^{11} -5.18655i q^{13} +(7.40437 - 4.49169i) q^{15} +27.3586i q^{17} -17.9667i q^{19} -1.45017 q^{21} -19.1101 q^{23} +(11.5498 - 22.1721i) q^{25} -5.19615 q^{27} +45.6495 q^{29} +13.6243i q^{31} -27.3586i q^{33} +(-3.57919 + 2.17123i) q^{35} +15.5597i q^{37} +8.98337i q^{39} +13.2990 q^{41} -27.9430 q^{43} +(-12.8248 + 7.77983i) q^{45} -55.6558 q^{47} -48.2990 q^{49} -47.3865i q^{51} +15.5597i q^{53} +(-40.9621 - 67.5245i) q^{55} +31.1193i q^{57} -87.6625i q^{59} -38.0000 q^{61} +2.51176 q^{63} +(13.4502 + 22.1721i) q^{65} -92.2015 q^{67} +33.0997 q^{69} -130.707i q^{71} +54.7173i q^{73} +(-20.0049 + 38.4032i) q^{75} +13.2249i q^{77} -13.6243i q^{79} +9.00000 q^{81} -59.0048 q^{83} +(-70.9485 - 116.956i) q^{85} -79.0673 q^{87} +39.8007 q^{89} -4.34246i q^{91} -23.5980i q^{93} +(46.5927 + 76.8064i) q^{95} -168.821i q^{97} +47.3865i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} + 24 q^{9}+O(q^{10}) $ 8 * q - 4 * q^5 + 24 * q^9	$ 8 q - 4 q^{5} + 24 q^{9} - 72 q^{21} + 32 q^{25} + 184 q^{29} - 256 q^{41} - 12 q^{45} - 24 q^{49} - 304 q^{61} + 168 q^{65} + 144 q^{69} + 72 q^{81} - 24 q^{85} + 560 q^{89}+O(q^{100}) $ 8 * q - 4 * q^5 + 24 * q^9 - 72 * q^21 + 32 * q^25 + 184 * q^29 - 256 * q^41 - 12 * q^45 - 24 * q^49 - 304 * q^61 + 168 * q^65 + 144 * q^69 + 72 * q^81 - 24 * q^85 + 560 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$		0			0
$5$	−4.27492	+	2.59328i	−0.854983	+	0.518655i
$5$	−4.27492	+	2.59328i	−0.854983	+	0.518655i
$6$		0			0
$7$	0.837253			0.119608			0.0598038	−	0.998210i	$-0.480952\pi$
$7$	0.837253			0.119608			0.0598038	+	0.998210i	$0.480952\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$			15.7955i			1.43596i	0.696066	+	0.717978i	$0.254932\pi$
$11$			15.7955i			1.43596i	−0.696066	+	0.717978i	$0.745068\pi$
$12$		0			0
$13$		−	5.18655i		−	0.398966i	−0.979901	−	0.199483i	$-0.936074\pi$
$13$		−	5.18655i		−	0.398966i	0.979901	−	0.199483i	$-0.0639262\pi$
$14$		0			0
$15$	7.40437	−	4.49169i	0.493625	−	0.299446i
$16$		0			0
$17$			27.3586i			1.60933i	0.593728	+	0.804666i	$0.297656\pi$
$17$			27.3586i			1.60933i	−0.593728	+	0.804666i	$0.702344\pi$
$18$		0			0
$19$		−	17.9667i		−	0.945618i	−0.881165	−	0.472809i	$-0.843240\pi$
$19$		−	17.9667i		−	0.945618i	0.881165	−	0.472809i	$-0.156760\pi$
$20$		0			0
$21$	−1.45017			−0.0690555
$22$		0			0
$23$	−19.1101			−0.830874			−0.415437	−	0.909622i	$-0.636371\pi$
$23$	−19.1101			−0.830874			−0.415437	+	0.909622i	$0.636371\pi$
$24$		0			0
$25$	11.5498	−	22.1721i	0.461993	−	0.886883i
$26$		0			0
$27$	−5.19615			−0.192450
$28$		0			0
$29$	45.6495			1.57412			0.787060	−	0.616876i	$-0.211602\pi$
$29$	45.6495			1.57412			0.787060	+	0.616876i	$0.211602\pi$
$30$		0			0
$31$			13.6243i			0.439493i	0.975557	+	0.219747i	$0.0705230\pi$
$31$			13.6243i			0.439493i	−0.975557	+	0.219747i	$0.929477\pi$
$32$		0			0
$33$		−	27.3586i		−	0.829050i
$34$		0			0
$35$	−3.57919	+	2.17123i	−0.102263	+	0.0620351i
$36$		0			0
$37$			15.5597i			0.420531i	0.977644	+	0.210266i	$0.0674329\pi$
$37$			15.5597i			0.420531i	−0.977644	+	0.210266i	$0.932567\pi$
$38$		0			0
$39$			8.98337i			0.230343i
$40$		0			0
$41$	13.2990			0.324366			0.162183	−	0.986761i	$-0.448147\pi$
$41$	13.2990			0.324366			0.162183	+	0.986761i	$0.448147\pi$
$42$		0			0
$43$	−27.9430			−0.649837			−0.324918	−	0.945742i	$-0.605337\pi$
$43$	−27.9430			−0.649837			−0.324918	+	0.945742i	$0.605337\pi$
$44$		0			0
$45$	−12.8248	+	7.77983i	−0.284994	+	0.172885i
$46$		0			0
$47$	−55.6558			−1.18417			−0.592083	−	0.805877i	$-0.701694\pi$
$47$	−55.6558			−1.18417			−0.592083	+	0.805877i	$0.701694\pi$
$48$		0			0
$49$	−48.2990			−0.985694
$50$		0			0
$51$		−	47.3865i		−	0.929148i
$52$		0			0
$53$			15.5597i			0.293578i	0.989168	+	0.146789i	$0.0468939\pi$
$53$			15.5597i			0.293578i	−0.989168	+	0.146789i	$0.953106\pi$
$54$		0			0
$55$	−40.9621	−	67.5245i	−0.744766	−	1.22772i
$56$		0			0
$57$			31.1193i			0.545953i
$58$		0			0
$59$		−	87.6625i		−	1.48581i	−0.669400	−	0.742903i	$-0.733449\pi$
$59$		−	87.6625i		−	1.48581i	0.669400	−	0.742903i	$-0.266551\pi$
$60$		0			0
$61$	−38.0000			−0.622951			−0.311475	−	0.950254i	$-0.600823\pi$
$61$	−38.0000			−0.622951			−0.311475	+	0.950254i	$0.600823\pi$
$62$		0			0
$63$	2.51176			0.0398692
$64$		0			0
$65$	13.4502	+	22.1721i	0.206926	+	0.341109i
$66$		0			0
$67$	−92.2015			−1.37614			−0.688071	−	0.725643i	$-0.741542\pi$
$67$	−92.2015			−1.37614			−0.688071	+	0.725643i	$0.741542\pi$
$68$		0			0
$69$	33.0997			0.479705
$70$		0			0
$71$		−	130.707i		−	1.84094i	−0.390816	−	0.920469i	$-0.627807\pi$
$71$		−	130.707i		−	1.84094i	0.390816	−	0.920469i	$-0.372193\pi$
$72$		0			0
$73$			54.7173i			0.749552i	0.927115	+	0.374776i	$0.122280\pi$
$73$			54.7173i			0.749552i	−0.927115	+	0.374776i	$0.877720\pi$
$74$		0			0
$75$	−20.0049	+	38.4032i	−0.266732	+	0.512042i
$76$		0			0
$77$			13.2249i			0.171751i
$78$		0			0
$79$		−	13.6243i		−	0.172459i	−0.996275	−	0.0862297i	$-0.972518\pi$
$79$		−	13.6243i		−	0.172459i	0.996275	−	0.0862297i	$-0.0274819\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−59.0048			−0.710901			−0.355451	−	0.934695i	$-0.615673\pi$
$83$	−59.0048			−0.710901			−0.355451	+	0.934695i	$0.615673\pi$
$84$		0			0
$85$	−70.9485	−	116.956i	−0.834688	−	1.37595i
$86$		0			0
$87$	−79.0673			−0.908819
$88$		0			0
$89$	39.8007			0.447198			0.223599	−	0.974681i	$-0.428219\pi$
$89$	39.8007			0.447198			0.223599	+	0.974681i	$0.428219\pi$
$90$		0			0
$91$		−	4.34246i		−	0.0477193i
$92$		0			0
$93$		−	23.5980i		−	0.253741i
$94$		0			0
$95$	46.5927	+	76.8064i	0.490450	+	0.808488i
$96$		0			0
$97$		−	168.821i		−	1.74043i	−0.492675	−	0.870214i	$-0.663981\pi$
$97$		−	168.821i		−	1.74043i	0.492675	−	0.870214i	$-0.336019\pi$
$98$		0			0
$99$			47.3865i			0.478652i
$100$		0			0
$101$	−44.5498			−0.441087			−0.220544	−	0.975377i	$-0.570783\pi$
$101$	−44.5498			−0.441087			−0.220544	+	0.975377i	$0.570783\pi$
$102$		0			0
$103$	−126.466			−1.22782			−0.613911	−	0.789375i	$-0.710405\pi$
$103$	−126.466			−1.22782			−0.613911	+	0.789375i	$0.710405\pi$
$104$		0			0
$105$	6.19934	−	3.76068i	0.0590413	−	0.0358160i
$106$		0			0
$107$	−104.383			−0.975546			−0.487773	−	0.872971i	$-0.662191\pi$
$107$	−104.383			−0.975546			−0.487773	+	0.872971i	$0.662191\pi$
$108$		0			0
$109$	0.501656			0.00460235			0.00230117	−	0.999997i	$-0.499268\pi$
$109$	0.501656			0.00460235			0.00230117	+	0.999997i	$0.499268\pi$
$110$		0			0
$111$		−	26.9501i		−	0.242794i
$112$		0			0
$113$		−	16.9855i		−	0.150314i	−0.997172	−	0.0751572i	$-0.976054\pi$
$113$		−	16.9855i		−	0.150314i	0.997172	−	0.0751572i	$-0.0239459\pi$
$114$		0			0
$115$	81.6941	−	49.5578i	0.710384	−	0.430937i
$116$		0			0
$117$		−	15.5597i		−	0.132989i
$118$		0			0
$119$			22.9061i			0.192488i
$120$		0			0
$121$	−128.498			−1.06197
$122$		0			0
$123$	−23.0346			−0.187273
$124$		0			0
$125$	8.12376	+	124.736i	0.0649901	+	0.997886i
$126$		0			0
$127$	8.45598			0.0665825			0.0332913	−	0.999446i	$-0.489401\pi$
$127$	8.45598			0.0665825			0.0332913	+	0.999446i	$0.489401\pi$
$128$		0			0
$129$	48.3987			0.375184
$130$		0			0
$131$		−	51.7290i		−	0.394878i	−0.980315	−	0.197439i	$-0.936738\pi$
$131$		−	51.7290i		−	0.394878i	0.980315	−	0.197439i	$-0.0632624\pi$
$132$		0			0
$133$		−	15.0427i		−	0.113103i
$134$		0			0
$135$	22.2131	−	13.4751i	0.164542	−	0.0998153i
$136$		0			0
$137$		−	53.8083i		−	0.392762i	−0.980528	−	0.196381i	$-0.937081\pi$
$137$		−	53.8083i		−	0.392762i	0.980528	−	0.196381i	$-0.0629189\pi$
$138$		0			0
$139$		−	13.6243i		−	0.0980165i	−0.998798	−	0.0490082i	$-0.984394\pi$
$139$		−	13.6243i		−	0.0980165i	0.998798	−	0.0490082i	$-0.0156061\pi$
$140$		0			0
$141$	96.3987			0.683679
$142$		0			0
$143$	81.9243			0.572897
$144$		0			0
$145$	−195.148	+	118.382i	−1.34585	+	0.816426i
$146$		0			0
$147$	83.6563			0.569091
$148$		0			0
$149$	33.6495			0.225836			0.112918	−	0.993604i	$-0.463980\pi$
$149$	33.6495			0.225836			0.112918	+	0.993604i	$0.463980\pi$
$150$		0			0
$151$			139.988i			0.927076i	0.886077	+	0.463538i	$0.153420\pi$
$151$			139.988i			0.927076i	−0.886077	+	0.463538i	$0.846580\pi$
$152$		0			0
$153$			82.0759i			0.536444i
$154$		0			0
$155$	−35.3315	−	58.2427i	−0.227945	−	0.375759i
$156$		0			0
$157$		−	21.2631i		−	0.135434i	−0.997705	−	0.0677170i	$-0.978428\pi$
$157$		−	21.2631i		−	0.135434i	0.997705	−	0.0677170i	$-0.0215715\pi$
$158$		0			0
$159$		−	26.9501i		−	0.169498i
$160$		0			0
$161$	−16.0000			−0.0993789
$162$		0			0
$163$	210.211			1.28964			0.644819	−	0.764335i	$-0.276933\pi$
$163$	210.211			1.28964			0.644819	+	0.764335i	$0.276933\pi$
$164$		0			0
$165$	70.9485	+	116.956i	0.429991	+	0.708824i
$166$		0			0
$167$	−238.384			−1.42745			−0.713725	−	0.700426i	$-0.752994\pi$
$167$	−238.384			−1.42745			−0.713725	+	0.700426i	$0.752994\pi$
$168$		0			0
$169$	142.100			0.840826
$170$		0			0
$171$		−	53.9002i		−	0.315206i
$172$		0			0
$173$			2.33481i			0.0134960i	0.999977	+	0.00674800i	$0.00214797\pi$
$173$			2.33481i			0.0134960i	−0.999977	+	0.00674800i	$0.997852\pi$
$174$		0			0
$175$	9.67014	−	18.5637i	0.0552579	−	0.106078i
$176$		0			0
$177$			151.836i			0.857830i
$178$		0			0
$179$			227.054i			1.26846i	0.773145	+	0.634229i	$0.218682\pi$
$179$			227.054i			1.26846i	−0.773145	+	0.634229i	$0.781318\pi$
$180$		0			0
$181$	−114.096			−0.630367			−0.315183	−	0.949031i	$-0.602066\pi$
$181$	−114.096			−0.630367			−0.315183	+	0.949031i	$0.602066\pi$
$182$		0			0
$183$	65.8179			0.359661
$184$		0			0
$185$	−40.3505	−	66.5163i	−0.218111	−	0.359547i
$186$		0			0
$187$	−432.144			−2.31093
$188$		0			0
$189$	−4.35050			−0.0230185
$190$		0			0
$191$			139.392i			0.729798i	0.931047	+	0.364899i	$0.118897\pi$
$191$			139.392i			0.729798i	−0.931047	+	0.364899i	$0.881103\pi$
$192$		0			0
$193$		−	182.046i		−	0.943245i	−0.881801	−	0.471623i	$-0.843669\pi$
$193$		−	182.046i		−	0.943245i	0.881801	−	0.471623i	$-0.156331\pi$
$194$		0			0
$195$	−23.2964	−	38.4032i	−0.119469	−	0.196939i
$196$		0			0
$197$		−	258.027i		−	1.30978i	−0.755724	−	0.654890i	$-0.772715\pi$
$197$		−	258.027i		−	1.30978i	0.755724	−	0.654890i	$-0.227285\pi$
$198$		0			0
$199$		−	256.474i		−	1.28881i	−0.764683	−	0.644407i	$-0.777104\pi$
$199$		−	256.474i		−	1.28881i	0.764683	−	0.644407i	$-0.222896\pi$
$200$		0			0
$201$	159.698			0.794516
$202$		0			0
$203$	38.2202			0.188277
$204$		0			0
$205$	−56.8522	+	34.4880i	−0.277328	+	0.168234i
$206$		0			0
$207$	−57.3303			−0.276958
$208$		0			0
$209$	283.794			1.35787
$210$		0			0
$211$			211.855i			1.00405i	0.864852	+	0.502027i	$0.167412\pi$
$211$			211.855i			1.00405i	−0.864852	+	0.502027i	$0.832588\pi$
$212$		0			0
$213$			226.390i			1.06287i
$214$		0			0
$215$	119.454	−	72.4639i	0.555600	−	0.337041i
$216$		0			0
$217$			11.4070i			0.0525667i
$218$		0			0
$219$		−	94.7731i		−	0.432754i
$220$		0			0
$221$	141.897			0.642068
$222$		0			0
$223$	349.843			1.56880			0.784401	−	0.620255i	$-0.212971\pi$
$223$	349.843			1.56880			0.784401	+	0.620255i	$0.212971\pi$
$224$		0			0
$225$	34.6495	−	66.5163i	0.153998	−	0.295628i
$226$		0			0
$227$	185.554			0.817418			0.408709	−	0.912665i	$-0.365979\pi$
$227$	185.554			0.817418			0.408709	+	0.912665i	$0.365979\pi$
$228$		0			0
$229$	−263.897			−1.15239			−0.576194	−	0.817313i	$-0.695463\pi$
$229$	−263.897			−1.15239			−0.576194	+	0.817313i	$0.695463\pi$
$230$		0			0
$231$		−	22.9061i		−	0.0991607i
$232$		0			0
$233$			58.4780i			0.250978i	0.992095	+	0.125489i	$0.0400500\pi$
$233$			58.4780i			0.250978i	−0.992095	+	0.125489i	$0.959950\pi$
$234$		0			0
$235$	237.924	−	144.331i	1.01244	−	0.614174i
$236$		0			0
$237$			23.5980i			0.0995694i
$238$		0			0
$239$		−	113.337i		−	0.474212i	−0.971484	−	0.237106i	$-0.923801\pi$
$239$		−	113.337i		−	0.474212i	0.971484	−	0.237106i	$-0.0761989\pi$
$240$		0			0
$241$	−77.7940			−0.322797			−0.161398	−	0.986889i	$-0.551600\pi$
$241$	−77.7940			−0.322797			−0.161398	+	0.986889i	$0.551600\pi$
$242$		0			0
$243$	−15.5885			−0.0641500
$244$		0			0
$245$	206.474	−	125.253i	0.842752	−	0.511235i
$246$		0			0
$247$	−93.1855			−0.377269
$248$		0			0
$249$	102.199			0.410439
$250$		0			0
$251$		−	106.226i		−	0.423212i	−0.977355	−	0.211606i	$-0.932131\pi$
$251$		−	106.226i		−	0.423212i	0.977355	−	0.211606i	$-0.0678693\pi$
$252$		0			0
$253$		−	301.854i		−	1.19310i
$254$		0			0
$255$	122.886	+	202.574i	0.481908	+	0.794406i
$256$		0			0
$257$			381.078i			1.48279i	0.671067	+	0.741397i	$0.265836\pi$
$257$			381.078i			1.48279i	−0.671067	+	0.741397i	$0.734164\pi$
$258$		0			0
$259$			13.0274i			0.0502988i
$260$		0			0
$261$	136.949			0.524707
$262$		0			0
$263$	−11.4914			−0.0436934			−0.0218467	−	0.999761i	$-0.506955\pi$
$263$	−11.4914			−0.0436934			−0.0218467	+	0.999761i	$0.506955\pi$
$264$		0			0
$265$	−40.3505	−	66.5163i	−0.152266	−	0.251005i
$266$		0			0
$267$	−68.9368			−0.258190
$268$		0			0
$269$	77.9518			0.289784			0.144892	−	0.989447i	$-0.453717\pi$
$269$	77.9518			0.289784			0.144892	+	0.989447i	$0.453717\pi$
$270$		0			0
$271$		−	86.6851i		−	0.319871i	−0.987127	−	0.159936i	$-0.948871\pi$
$271$		−	86.6851i		−	0.319871i	0.987127	−	0.159936i	$-0.0511287\pi$
$272$		0			0
$273$			7.52136i			0.0275508i
$274$		0			0
$275$	350.220	+	182.436i	1.27353	+	0.663402i
$276$		0			0
$277$			287.328i			1.03729i	0.854991	+	0.518643i	$0.173563\pi$
$277$			287.328i			1.03729i	−0.854991	+	0.518643i	$0.826437\pi$
$278$		0			0
$279$			40.8729i			0.146498i
$280$		0			0
$281$	−224.598			−0.799281			−0.399641	−	0.916672i	$-0.630865\pi$
$281$	−224.598			−0.799281			−0.399641	+	0.916672i	$0.630865\pi$
$282$		0			0
$283$	84.1224			0.297252			0.148626	−	0.988893i	$-0.452515\pi$
$283$	84.1224			0.297252			0.148626	+	0.988893i	$0.452515\pi$
$284$		0			0
$285$	−80.7010	−	133.033i	−0.283161	−	0.466781i
$286$		0			0
$287$	11.1346			0.0387967
$288$		0			0
$289$	−459.495			−1.58995
$290$		0			0
$291$			292.407i			1.00484i
$292$		0			0
$293$			246.620i			0.841706i	0.907129	+	0.420853i	$0.138269\pi$
$293$			246.620i			0.841706i	−0.907129	+	0.420853i	$0.861731\pi$
$294$		0			0
$295$	227.333	+	374.750i	0.770621	+	1.27034i
$296$		0			0
$297$		−	82.0759i		−	0.276350i
$298$		0			0
$299$			99.1156i			0.331490i
$300$		0			0
$301$	−23.3954			−0.0777255
$302$		0			0
$303$	77.1626			0.254662
$304$		0			0
$305$	162.447	−	98.5445i	0.532613	−	0.323097i
$306$		0			0
$307$	115.811			0.377236			0.188618	−	0.982051i	$-0.439599\pi$
$307$	115.811			0.377236			0.188618	+	0.982051i	$0.439599\pi$
$308$		0			0
$309$	219.045			0.708883
$310$		0			0
$311$			203.767i			0.655201i	0.944816	+	0.327600i	$0.106240\pi$
$311$			203.767i			0.655201i	−0.944816	+	0.327600i	$0.893760\pi$
$312$		0			0
$313$		−	99.0614i		−	0.316490i	−0.987400	−	0.158245i	$-0.949416\pi$
$313$		−	99.0614i		−	0.316490i	0.987400	−	0.158245i	$-0.0505836\pi$
$314$		0			0
$315$	−10.7376	+	6.51369i	−0.0340875	+	0.0206784i
$316$		0			0
$317$			471.192i			1.48641i	0.669063	+	0.743206i	$0.266696\pi$
$317$			471.192i			1.48641i	−0.669063	+	0.743206i	$0.733304\pi$
$318$		0			0
$319$			721.057i			2.26037i
$320$		0			0
$321$	180.797			0.563232
$322$		0			0
$323$	491.546			1.52181
$324$		0			0
$325$	−114.997	−	59.9038i	−0.353836	−	0.184319i
$326$		0			0
$327$	−0.868893			−0.00265717
$328$		0			0
$329$	−46.5980			−0.141635
$330$		0			0
$331$		−	270.695i		−	0.817810i	−0.912577	−	0.408905i	$-0.865911\pi$
$331$		−	270.695i		−	0.817810i	0.912577	−	0.408905i	$-0.134089\pi$
$332$		0			0
$333$			46.6790i			0.140177i
$334$		0			0
$335$	394.154	−	239.104i	1.17658	−	0.713743i
$336$		0			0
$337$		−	377.317i		−	1.11964i	−0.828615	−	0.559818i	$-0.810871\pi$
$337$		−	377.317i		−	1.11964i	0.828615	−	0.559818i	$-0.189129\pi$
$338$		0			0
$339$			29.4198i			0.0867841i
$340$		0			0
$341$	−215.203			−0.631093
$342$		0			0
$343$	−81.4639			−0.237504
$344$		0			0
$345$	−141.498	+	85.8366i	−0.410140	+	0.248802i
$346$		0			0
$347$	−462.222			−1.33205			−0.666025	−	0.745929i	$-0.732006\pi$
$347$	−462.222			−1.33205			−0.666025	+	0.745929i	$0.732006\pi$
$348$		0			0
$349$	−200.598			−0.574779			−0.287390	−	0.957814i	$-0.592787\pi$
$349$	−200.598			−0.574779			−0.287390	+	0.957814i	$0.592787\pi$
$350$		0			0
$351$			26.9501i			0.0767810i
$352$		0			0
$353$			250.897i			0.710757i	0.934722	+	0.355379i	$0.115648\pi$
$353$			250.897i			0.710757i	−0.934722	+	0.355379i	$0.884352\pi$
$354$		0			0
$355$	338.958	+	558.760i	0.954812	+	1.57397i
$356$		0			0
$357$		−	39.6746i		−	0.111133i
$358$		0			0
$359$			215.601i			0.600560i	0.953851	+	0.300280i	$0.0970801\pi$
$359$			215.601i			0.600560i	−0.953851	+	0.300280i	$0.902920\pi$
$360$		0			0
$361$	38.1960			0.105806
$362$		0			0
$363$	222.566			0.613129
$364$		0			0
$365$	−141.897	−	233.912i	−0.388759	−	0.640854i
$366$		0			0
$367$	−67.0637			−0.182735			−0.0913675	−	0.995817i	$-0.529124\pi$
$367$	−67.0637			−0.182735			−0.0913675	+	0.995817i	$0.529124\pi$
$368$		0			0
$369$	39.8970			0.108122
$370$		0			0
$371$			13.0274i			0.0351142i
$372$		0			0
$373$		−	567.402i		−	1.52119i	−0.649230	−	0.760593i	$-0.724909\pi$
$373$		−	567.402i		−	1.52119i	0.649230	−	0.760593i	$-0.275091\pi$
$374$		0			0
$375$	−14.0708	−	216.049i	−0.0375220	−	0.576130i
$376$		0			0
$377$		−	236.764i		−	0.628020i
$378$		0			0
$379$			240.298i			0.634031i	0.948420	+	0.317016i	$0.102681\pi$
$379$			240.298i			0.634031i	−0.948420	+	0.317016i	$0.897319\pi$
$380$		0			0
$381$	−14.6462			−0.0384414
$382$		0			0
$383$	670.068			1.74952			0.874762	−	0.484553i	$-0.161018\pi$
$383$	670.068			1.74952			0.874762	+	0.484553i	$0.161018\pi$
$384$		0			0
$385$	−34.2957	−	56.5351i	−0.0890797	−	0.146845i
$386$		0			0
$387$	−83.8290			−0.216612
$388$		0			0
$389$	474.640			1.22015			0.610077	−	0.792342i	$-0.291139\pi$
$389$	474.640			1.22015			0.610077	+	0.792342i	$0.291139\pi$
$390$		0			0
$391$		−	522.826i		−	1.33715i
$392$		0			0
$393$			89.5973i			0.227983i
$394$		0			0
$395$	35.3315	+	58.2427i	0.0894469	+	0.147450i
$396$		0			0
$397$		−	499.460i		−	1.25809i	−0.777371	−	0.629043i	$-0.783447\pi$
$397$		−	499.460i		−	1.25809i	0.777371	−	0.629043i	$-0.216553\pi$
$398$		0			0
$399$			26.0548i			0.0653001i
$400$		0			0
$401$	344.694			0.859587			0.429793	−	0.902927i	$-0.358586\pi$
$401$	344.694			0.859587			0.429793	+	0.902927i	$0.358586\pi$
$402$		0			0
$403$	70.6631			0.175343
$404$		0			0
$405$	−38.4743	+	23.3395i	−0.0949982	+	0.0576284i
$406$		0			0
$407$	−245.773			−0.603864
$408$		0			0
$409$	−501.890			−1.22712			−0.613558	−	0.789650i	$-0.710262\pi$
$409$	−501.890			−1.22712			−0.613558	+	0.789650i	$0.710262\pi$
$410$		0			0
$411$			93.1988i			0.226761i
$412$		0			0
$413$		−	73.3957i		−	0.177714i
$414$		0			0
$415$	252.241	−	153.016i	0.607809	−	0.368713i
$416$		0			0
$417$			23.5980i			0.0565898i
$418$		0			0
$419$		−	218.369i		−	0.521167i	−0.965451	−	0.260584i	$-0.916085\pi$
$419$		−	218.369i		−	0.521167i	0.965451	−	0.260584i	$-0.0839150\pi$
$420$		0			0
$421$	−281.698			−0.669116			−0.334558	−	0.942375i	$-0.608587\pi$
$421$	−281.698			−0.669116			−0.334558	+	0.942375i	$0.608587\pi$
$422$		0			0
$423$	−166.967			−0.394722
$424$		0			0
$425$	606.598	+	315.988i	1.42729	+	0.743501i
$426$		0			0
$427$	−31.8156			−0.0745097
$428$		0			0
$429$	−141.897			−0.330762
$430$		0			0
$431$			441.081i			1.02339i	0.859167	+	0.511694i	$0.170982\pi$
$431$			441.081i			1.02339i	−0.859167	+	0.511694i	$0.829018\pi$
$432$		0			0
$433$		−	123.443i		−	0.285089i	−0.989788	−	0.142544i	$-0.954472\pi$
$433$		−	123.443i		−	0.285089i	0.989788	−	0.142544i	$-0.0455283\pi$
$434$		0			0
$435$	338.006	−	205.043i	0.777025	−	0.471364i
$436$		0			0
$437$			343.346i			0.785690i
$438$		0			0
$439$		−	330.728i		−	0.753368i	−0.926342	−	0.376684i	$-0.877064\pi$
$439$		−	330.728i		−	0.753368i	0.926342	−	0.376684i	$-0.122936\pi$
$440$		0			0
$441$	−144.897			−0.328565
$442$		0			0
$443$	−154.952			−0.349780			−0.174890	−	0.984588i	$-0.555957\pi$
$443$	−154.952			−0.349780			−0.174890	+	0.984588i	$0.555957\pi$
$444$		0			0
$445$	−170.145	+	103.214i	−0.382347	+	0.231942i
$446$		0			0
$447$	−58.2826			−0.130386
$448$		0			0
$449$	−95.8970			−0.213579			−0.106790	−	0.994282i	$-0.534057\pi$
$449$	−95.8970			−0.213579			−0.106790	+	0.994282i	$0.534057\pi$
$450$		0			0
$451$			210.065i			0.465775i
$452$		0			0
$453$		−	242.467i		−	0.535247i
$454$		0			0
$455$	11.2612	+	18.5637i	0.0247499	+	0.0407992i
$456$		0			0
$457$			485.718i			1.06284i	0.847108	+	0.531420i	$0.178341\pi$
$457$			485.718i			1.06284i	−0.847108	+	0.531420i	$0.821659\pi$
$458$		0			0
$459$		−	142.160i		−	0.309716i
$460$		0			0
$461$	−353.650			−0.767136			−0.383568	−	0.923513i	$-0.625305\pi$
$461$	−353.650			−0.767136			−0.383568	+	0.923513i	$0.625305\pi$
$462$		0			0
$463$	421.720			0.910842			0.455421	−	0.890276i	$-0.349489\pi$
$463$	421.720			0.910842			0.455421	+	0.890276i	$0.349489\pi$
$464$		0			0
$465$	61.1960	+	100.879i	0.131604	+	0.216945i
$466$		0			0
$467$	−640.974			−1.37254			−0.686268	−	0.727349i	$-0.740752\pi$
$467$	−640.974			−1.37254			−0.686268	+	0.727349i	$0.740752\pi$
$468$		0			0
$469$	−77.1960			−0.164597
$470$		0			0
$471$			36.8289i			0.0781929i
$472$		0			0
$473$		−	441.374i		−	0.933137i
$474$		0			0
$475$	−398.360	−	207.513i	−0.838653	−	0.436869i
$476$		0			0
$477$			46.6790i			0.0978595i
$478$		0			0
$479$			221.137i			0.461664i	0.972994	+	0.230832i	$0.0741448\pi$
$479$			221.137i			0.461664i	−0.972994	+	0.230832i	$0.925855\pi$
$480$		0			0
$481$	80.7010			0.167778
$482$		0			0
$483$	27.7128			0.0573764
$484$		0			0
$485$	437.801	+	721.698i	0.902682	+	1.48804i
$486$		0			0
$487$	−889.949			−1.82741			−0.913705	−	0.406377i	$-0.866792\pi$
$487$	−889.949			−1.82741			−0.913705	+	0.406377i	$0.866792\pi$
$488$		0			0
$489$	−364.096			−0.744573
$490$		0			0
$491$			552.843i			1.12595i	0.826473	+	0.562977i	$0.190344\pi$
$491$			552.843i			1.12595i	−0.826473	+	0.562977i	$0.809656\pi$
$492$		0			0
$493$			1248.91i			2.53328i
$494$		0			0
$495$	−122.886	−	202.574i	−0.248255	−	0.409240i
$496$		0			0
$497$		−	109.435i		−	0.220190i
$498$		0			0
$499$		−	533.302i		−	1.06874i	−0.845250	−	0.534371i	$-0.820549\pi$
$499$		−	533.302i		−	1.06874i	0.845250	−	0.534371i	$-0.179451\pi$
$500$		0			0
$501$	412.894			0.824139
$502$		0			0
$503$	−574.914			−1.14297			−0.571485	−	0.820612i	$-0.693633\pi$
$503$	−574.914			−1.14297			−0.571485	+	0.820612i	$0.693633\pi$
$504$		0			0
$505$	190.447	−	115.530i	0.377122	−	0.228772i
$506$		0			0
$507$	−246.124			−0.485451
$508$		0			0
$509$	−207.547			−0.407753			−0.203877	−	0.978997i	$-0.565354\pi$
$509$	−207.547			−0.407753			−0.203877	+	0.978997i	$0.565354\pi$
$510$		0			0
$511$			45.8122i			0.0896521i
$512$		0			0
$513$			93.3580i			0.181984i
$514$		0			0
$515$	540.630	−	327.960i	1.04977	−	0.636816i
$516$		0			0
$517$		−	879.112i		−	1.70041i
$518$		0			0
$519$		−	4.04401i		−	0.00779192i
$520$		0			0
$521$	−712.900			−1.36833			−0.684165	−	0.729327i	$-0.739833\pi$
$521$	−712.900			−1.36833			−0.684165	+	0.729327i	$0.739833\pi$
$522$		0			0
$523$	139.548			0.266822			0.133411	−	0.991061i	$-0.457407\pi$
$523$	139.548			0.266822			0.133411	+	0.991061i	$0.457407\pi$
$524$		0			0
$525$	−16.7492	+	32.1532i	−0.0319032	+	0.0612442i
$526$		0			0
$527$	−372.742			−0.707290
$528$		0			0
$529$	−163.804			−0.309648
$530$		0			0
$531$		−	262.988i		−	0.495268i
$532$		0			0
$533$		−	68.9760i		−	0.129411i
$534$		0			0
$535$	446.230	−	270.695i	0.834075	−	0.505972i
$536$		0			0
$537$		−	393.269i		−	0.732345i
$538$		0			0
$539$		−	762.908i		−	1.41541i
$540$		0			0
$541$	946.688			1.74988			0.874942	−	0.484227i	$-0.160899\pi$
$541$	946.688			1.74988			0.874942	+	0.484227i	$0.160899\pi$
$542$		0			0
$543$	197.621			0.363942
$544$		0			0
$545$	−2.14454	+	1.30093i	−0.00393493	+	0.00238703i
$546$		0			0
$547$	50.3388			0.0920271			0.0460136	−	0.998941i	$-0.485348\pi$
$547$	50.3388			0.0920271			0.0460136	+	0.998941i	$0.485348\pi$
$548$		0			0
$549$	−114.000			−0.207650
$550$		0			0
$551$		−	820.173i		−	1.48852i
$552$		0			0
$553$		−	11.4070i		−	0.0206275i
$554$		0			0
$555$	69.8891	+	115.210i	0.125926	+	0.207585i
$556$		0			0
$557$		−	790.157i		−	1.41859i	−0.704910	−	0.709297i	$-0.749013\pi$
$557$		−	790.157i		−	1.41859i	0.704910	−	0.709297i	$-0.250987\pi$
$558$		0			0
$559$			144.928i			0.259263i
$560$		0			0
$561$	748.495			1.33422
$562$		0			0
$563$	−354.133			−0.629010			−0.314505	−	0.949256i	$-0.601838\pi$
$563$	−354.133			−0.629010			−0.314505	+	0.949256i	$0.601838\pi$
$564$		0			0
$565$	44.0482	+	72.6117i	0.0779614	+	0.128516i
$566$		0			0
$567$	7.53528			0.0132897
$568$		0			0
$569$	55.4983			0.0975366			0.0487683	−	0.998810i	$-0.484470\pi$
$569$	55.4983			0.0975366			0.0487683	+	0.998810i	$0.484470\pi$
$570$		0			0
$571$		−	791.134i		−	1.38552i	−0.721167	−	0.692762i	$-0.756394\pi$
$571$		−	791.134i		−	1.38552i	0.721167	−	0.692762i	$-0.243606\pi$
$572$		0			0
$573$		−	241.433i		−	0.421349i
$574$		0			0
$575$	−220.719	+	423.711i	−0.383858	+	0.736888i
$576$		0			0
$577$			201.759i			0.349668i	0.984598	+	0.174834i	$0.0559389\pi$
$577$			201.759i			0.349668i	−0.984598	+	0.174834i	$0.944061\pi$
$578$		0			0
$579$			315.313i			0.544583i
$580$		0			0
$581$	−49.4020			−0.0850292
$582$		0			0
$583$	−245.773			−0.421566
$584$		0			0
$585$	40.3505	+	66.5163i	0.0689752	+	0.113703i
$586$		0			0
$587$	−444.556			−0.757335			−0.378668	−	0.925533i	$-0.623618\pi$
$587$	−444.556			−0.757335			−0.378668	+	0.925533i	$0.623618\pi$
$588$		0			0
$589$	244.784			0.415593
$590$		0			0
$591$			446.915i			0.756202i
$592$		0			0
$593$			563.908i			0.950942i	0.879731	+	0.475471i	$0.157722\pi$
$593$			563.908i			0.950942i	−0.879731	+	0.475471i	$0.842278\pi$
$594$		0			0
$595$	−59.4019	−	97.9217i	−0.0998351	−	0.164574i
$596$		0			0
$597$			444.226i			0.744097i
$598$		0			0
$599$		−	845.034i		−	1.41074i	−0.708839	−	0.705371i	$-0.750781\pi$
$599$		−	845.034i		−	1.41074i	0.708839	−	0.705371i	$-0.249219\pi$
$600$		0			0
$601$	672.296			1.11863			0.559314	−	0.828956i	$-0.311065\pi$
$601$	672.296			1.11863			0.559314	+	0.828956i	$0.311065\pi$
$602$		0			0
$603$	−276.604			−0.458714
$604$		0			0
$605$	549.320	−	333.232i	0.907967	−	0.550796i
$606$		0			0
$607$	−882.664			−1.45414			−0.727071	−	0.686562i	$-0.759119\pi$
$607$	−882.664			−1.45414			−0.727071	+	0.686562i	$0.759119\pi$
$608$		0			0
$609$	−66.1993			−0.108702
$610$		0			0
$611$			288.662i			0.472442i
$612$		0			0
$613$			469.374i			0.765701i	0.923810	+	0.382850i	$0.125057\pi$
$613$			469.374i			0.765701i	−0.923810	+	0.382850i	$0.874943\pi$
$614$		0			0
$615$	98.4708	−	59.7350i	0.160115	−	0.0971300i
$616$		0			0
$617$			218.994i			0.354934i	0.984127	+	0.177467i	$0.0567902\pi$
$617$			218.994i			0.354934i	−0.984127	+	0.177467i	$0.943210\pi$
$618$		0			0
$619$			879.610i			1.42102i	0.703689	+	0.710509i	$0.251535\pi$
$619$			879.610i			1.42102i	−0.703689	+	0.710509i	$0.748465\pi$
$620$		0			0
$621$	99.2990			0.159902
$622$		0			0
$623$	33.3232			0.0534884
$624$		0			0
$625$	−358.203	−	512.168i	−0.573124	−	0.819468i
$626$		0			0
$627$	−491.546			−0.783964
$628$		0			0
$629$	−425.691			−0.676774
$630$		0			0
$631$			635.566i			1.00724i	0.863926	+	0.503618i	$0.167998\pi$
$631$			635.566i			1.00724i	−0.863926	+	0.503618i	$0.832002\pi$
$632$		0			0
$633$		−	366.944i		−	0.579691i
$634$		0			0
$635$	−36.1486	+	21.9287i	−0.0569270	+	0.0345334i
$636$		0			0
$637$			250.505i			0.393258i
$638$		0			0
$639$		−	392.120i		−	0.613646i
$640$		0			0
$641$	−296.309			−0.462260			−0.231130	−	0.972923i	$-0.574242\pi$
$641$	−296.309			−0.462260			−0.231130	+	0.972923i	$0.574242\pi$
$642$		0			0
$643$	−591.032			−0.919179			−0.459590	−	0.888131i	$-0.652003\pi$
$643$	−591.032			−0.919179			−0.459590	+	0.888131i	$0.652003\pi$
$644$		0			0
$645$	−206.900	+	125.511i	−0.320776	+	0.194591i
$646$		0			0
$647$	−166.507			−0.257352			−0.128676	−	0.991687i	$-0.541073\pi$
$647$	−166.507			−0.257352			−0.128676	+	0.991687i	$0.541073\pi$
$648$		0			0
$649$	1384.67			2.13355
$650$		0			0
$651$		−	19.7575i		−	0.0303494i
$652$		0			0
$653$		−	621.335i		−	0.951509i	−0.879578	−	0.475754i	$-0.842175\pi$
$653$		−	621.335i		−	0.951509i	0.879578	−	0.475754i	$-0.157825\pi$
$654$		0			0
$655$	134.148	+	221.137i	0.204806	+	0.337614i
$656$		0			0
$657$			164.152i			0.249851i
$658$		0			0
$659$			702.113i			1.06542i	0.846297	+	0.532711i	$0.178827\pi$
$659$			702.113i			1.06542i	−0.846297	+	0.532711i	$0.821173\pi$
$660$		0			0
$661$	−358.193			−0.541895			−0.270948	−	0.962594i	$-0.587337\pi$
$661$	−358.193			−0.541895			−0.270948	+	0.962594i	$0.587337\pi$
$662$		0			0
$663$	−245.773			−0.370698
$664$		0			0
$665$	39.0099	+	64.3064i	0.0586616	+	0.0967013i
$666$		0			0
$667$	−872.367			−1.30790
$668$		0			0
$669$	−605.945			−0.905748
$670$		0			0
$671$		−	600.230i		−	0.894530i
$672$		0			0
$673$			714.176i			1.06118i	0.847628	+	0.530592i	$0.178030\pi$
$673$			714.176i			1.06118i	−0.847628	+	0.530592i	$0.821970\pi$
$674$		0			0
$675$	−60.0147	+	115.210i	−0.0889107	+	0.170681i
$676$		0			0
$677$			509.833i			0.753077i	0.926401	+	0.376538i	$0.122886\pi$
$677$			509.833i			0.753077i	−0.926401	+	0.376538i	$0.877114\pi$
$678$		0			0
$679$		−	141.346i		−	0.208168i
$680$		0			0
$681$	−321.389			−0.471936
$682$		0			0
$683$	1263.93			1.85055			0.925275	−	0.379298i	$-0.123834\pi$
$683$	1263.93			1.85055			0.925275	+	0.379298i	$0.123834\pi$
$684$		0			0
$685$	139.540	+	230.026i	0.203708	+	0.335805i
$686$		0			0
$687$	457.083			0.665332
$688$		0			0
$689$	80.7010			0.117128
$690$		0			0
$691$		−	512.351i		−	0.741463i	−0.928740	−	0.370731i	$-0.879107\pi$
$691$		−	512.351i		−	0.741463i	0.928740	−	0.370731i	$-0.120893\pi$
$692$		0			0
$693$			39.6746i			0.0572504i
$694$		0			0
$695$	35.3315	+	58.2427i	0.0508368	+	0.0838024i
$696$		0			0
$697$			363.843i			0.522012i
$698$		0			0
$699$		−	101.287i		−	0.144902i
$700$		0			0
$701$	−1092.03			−1.55781			−0.778907	−	0.627139i	$-0.784226\pi$
$701$	−1092.03			−1.55781			−0.778907	+	0.627139i	$0.784226\pi$
$702$		0			0
$703$	279.556			0.397662
$704$		0			0
$705$	−412.096	+	249.988i	−0.584534	+	0.354593i
$706$		0			0
$707$	−37.2995			−0.0527574
$708$		0			0
$709$	−416.887			−0.587993			−0.293997	−	0.955806i	$-0.594985\pi$
$709$	−416.887			−0.587993			−0.293997	+	0.955806i	$0.594985\pi$
$710$		0			0
$711$		−	40.8729i		−	0.0574864i
$712$		0			0
$713$		−	260.362i		−	0.365163i
$714$		0			0
$715$	−350.220	+	212.452i	−0.489818	+	0.297136i
$716$		0			0
$717$			196.305i			0.273787i
$718$		0			0
$719$		−	395.268i		−	0.549747i	−0.961480	−	0.274874i	$-0.911364\pi$
$719$		−	395.268i		−	0.549747i	0.961480	−	0.274874i	$-0.0886361\pi$
$720$		0			0
$721$	−105.884			−0.146857
$722$		0			0
$723$	134.743			0.186367
$724$		0			0
$725$	527.244	−	1012.14i	0.727233	−	1.39606i
$726$		0			0
$727$	−597.583			−0.821985			−0.410993	−	0.911639i	$-0.634818\pi$
$727$	−597.583			−0.821985			−0.410993	+	0.911639i	$0.634818\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	764.482i		−	1.04580i
$732$		0			0
$733$			23.8650i			0.0325580i	0.999867	+	0.0162790i	$0.00518200\pi$
$733$			23.8650i			0.0325580i	−0.999867	+	0.0162790i	$0.994818\pi$
$734$		0			0
$735$	−357.624	+	216.944i	−0.486563	+	0.295162i
$736$		0			0
$737$		−	1456.37i		−	1.97608i
$738$		0			0
$739$		−	125.767i		−	0.170186i	−0.996373	−	0.0850928i	$-0.972881\pi$
$739$		−	125.767i		−	0.170186i	0.996373	−	0.0850928i	$-0.0271187\pi$
$740$		0			0
$741$	161.402			0.217816
$742$		0			0
$743$	−148.841			−0.200325			−0.100162	−	0.994971i	$-0.531936\pi$
$743$	−148.841			−0.200325			−0.100162	+	0.994971i	$0.531936\pi$
$744$		0			0
$745$	−143.849	+	87.2625i	−0.193086	+	0.117131i
$746$		0			0
$747$	−177.014			−0.236967
$748$		0			0
$749$	−87.3954			−0.116683
$750$		0			0
$751$		−	463.390i		−	0.617030i	−0.951219	−	0.308515i	$-0.900168\pi$
$751$		−	463.390i		−	0.617030i	0.951219	−	0.308515i	$-0.0998321\pi$
$752$		0			0
$753$			183.989i			0.244341i
$754$		0			0
$755$	−363.029	−	598.439i	−0.480833	−	0.792634i
$756$		0			0
$757$			719.363i			0.950281i	0.879910	+	0.475141i	$0.157603\pi$
$757$			719.363i			0.950281i	−0.879910	+	0.475141i	$0.842397\pi$
$758$		0			0
$759$			522.826i			0.688836i
$760$		0			0
$761$	−1107.49			−1.45530			−0.727651	−	0.685947i	$-0.759388\pi$
$761$	−1107.49			−1.45530			−0.727651	+	0.685947i	$0.759388\pi$
$762$		0			0
$763$	0.420013			0.000550476
$764$		0			0
$765$	−212.846	−	350.868i	−0.278229	−	0.458651i
$766$		0			0
$767$	−454.666			−0.592785
$768$		0			0
$769$	−231.691			−0.301289			−0.150644	−	0.988588i	$-0.548135\pi$
$769$	−231.691			−0.301289			−0.150644	+	0.988588i	$0.548135\pi$
$770$		0			0
$771$		−	660.047i		−	0.856092i
$772$		0			0
$773$		−	519.956i		−	0.672647i	−0.941746	−	0.336324i	$-0.890816\pi$
$773$		−	519.956i		−	0.672647i	0.941746	−	0.336324i	$-0.109184\pi$
$774$		0			0
$775$	302.079	+	157.358i	0.389779	+	0.203043i
$776$		0			0
$777$		−	22.5641i		−	0.0290400i
$778$		0			0
$779$		−	238.940i		−	0.306726i
$780$		0			0
$781$	2064.58			2.64351
$782$		0			0
$783$	−237.202			−0.302940
$784$		0			0
$785$	55.1412	+	90.8982i	0.0702436	+	0.115794i
$786$		0			0
$787$	46.0288			0.0584864			0.0292432	−	0.999572i	$-0.490690\pi$
$787$	46.0288			0.0584864			0.0292432	+	0.999572i	$0.490690\pi$
$788$		0			0
$789$	19.9036			0.0252264
$790$		0			0
$791$		−	14.2212i		−	0.0179788i
$792$		0			0
$793$			197.089i			0.248536i
$794$		0			0
$795$	69.8891	+	115.210i	0.0879108	+	0.144918i
$796$		0			0
$797$		−	15.3098i		−	0.0192093i	−0.999954	−	0.00960463i	$-0.996943\pi$
$797$		−	15.3098i		−	0.0192093i	0.999954	−	0.00960463i	$-0.00305730\pi$
$798$		0			0
$799$		−	1522.67i		−	1.90572i
$800$		0			0
$801$	119.402			0.149066
$802$		0			0
$803$	−864.288			−1.07632
$804$		0			0
$805$	68.3987	−	41.4924i	0.0849673	−	0.0515434i
$806$		0			0
$807$	−135.017			−0.167307
$808$		0			0
$809$	−313.093			−0.387012			−0.193506	−	0.981099i	$-0.561986\pi$
$809$	−313.093			−0.387012			−0.193506	+	0.981099i	$0.561986\pi$
$810$		0			0
$811$			1056.89i			1.30319i	0.758566	+	0.651596i	$0.225900\pi$
$811$			1056.89i			1.30319i	−0.758566	+	0.651596i	$0.774100\pi$
$812$		0			0
$813$			150.143i			0.184678i
$814$		0			0
$815$	−898.635	+	545.136i	−1.10262	+	0.668878i
$816$		0			0
$817$			502.045i			0.614498i
$818$		0			0
$819$		−	13.0274i		−	0.0159064i
$820$		0			0
$821$	−308.659			−0.375955			−0.187978	−	0.982173i	$-0.560193\pi$
$821$	−308.659			−0.375955			−0.187978	+	0.982173i	$0.560193\pi$
$822$		0			0
$823$	−109.680			−0.133269			−0.0666344	−	0.997777i	$-0.521226\pi$
$823$	−109.680			−0.133269			−0.0666344	+	0.997777i	$0.521226\pi$
$824$		0			0
$825$	−606.598	−	315.988i	−0.735270	−	0.383015i
$826$		0			0
$827$	−711.971			−0.860908			−0.430454	−	0.902613i	$-0.641646\pi$
$827$	−711.971			−0.860908			−0.430454	+	0.902613i	$0.641646\pi$
$828$		0			0
$829$	118.688			0.143170			0.0715849	−	0.997435i	$-0.477194\pi$
$829$	118.688			0.143170			0.0715849	+	0.997435i	$0.477194\pi$
$830$		0			0
$831$		−	497.667i		−	0.598877i
$832$		0			0
$833$		−	1321.40i		−	1.58631i
$834$		0			0
$835$	1019.07	−	618.196i	1.22045	−	0.740355i
$836$		0			0
$837$		−	70.7939i		−	0.0845805i
$838$		0			0
$839$			1413.67i			1.68495i	0.538736	+	0.842475i	$0.318902\pi$
$839$			1413.67i			1.68495i	−0.538736	+	0.842475i	$0.681098\pi$
$840$		0			0
$841$	1242.88			1.47786
$842$		0			0
$843$	389.015			0.461465
$844$		0			0
$845$	−607.464	+	368.504i	−0.718893	+	0.436099i
$846$		0			0
$847$	−107.586			−0.127020
$848$		0			0
$849$	−145.704			−0.171619
$850$		0			0
$851$		−	297.347i		−	0.349409i
$852$		0			0
$853$		−	1308.03i		−	1.53344i	−0.641979	−	0.766722i	$-0.721886\pi$
$853$		−	1308.03i		−	1.53344i	0.641979	−	0.766722i	$-0.278114\pi$
$854$		0			0
$855$	139.778	+	230.419i	0.163483	+	0.269496i
$856$		0			0
$857$			719.755i			0.839854i	0.907558	+	0.419927i	$0.137944\pi$
$857$			719.755i			0.839854i	−0.907558	+	0.419927i	$0.862056\pi$
$858$		0			0
$859$		−	1402.44i		−	1.63264i	−0.577601	−	0.816319i	$-0.696011\pi$
$859$		−	1402.44i		−	1.63264i	0.577601	−	0.816319i	$-0.303989\pi$
$860$		0			0
$861$	−19.2858			−0.0223993
$862$		0			0
$863$	−72.4412			−0.0839411			−0.0419706	−	0.999119i	$-0.513364\pi$
$863$	−72.4412			−0.0839411			−0.0419706	+	0.999119i	$0.513364\pi$
$864$		0			0
$865$	−6.05480	−	9.98111i	−0.00699977	−	0.0115389i
$866$		0			0
$867$	795.869			0.917957
$868$		0			0
$869$	215.203			0.247644
$870$		0			0
$871$			478.208i			0.549033i
$872$		0			0
$873$		−	506.464i		−	0.580142i
$874$		0			0
$875$	6.80164	+	104.435i	0.00777331	+	0.119355i
$876$		0			0
$877$			1382.21i			1.57606i	0.615635	+	0.788032i	$0.288900\pi$
$877$			1382.21i			1.57606i	−0.615635	+	0.788032i	$0.711100\pi$
$878$		0			0
$879$		−	427.158i		−	0.485959i
$880$		0			0
$881$	−1131.38			−1.28419			−0.642097	−	0.766623i	$-0.721936\pi$
$881$	−1131.38			−1.28419			−0.642097	+	0.766623i	$0.721936\pi$
$882$		0			0
$883$	−1077.39			−1.22014			−0.610072	−	0.792346i	$-0.708860\pi$
$883$	−1077.39			−1.22014			−0.610072	+	0.792346i	$0.708860\pi$
$884$		0			0
$885$	−393.752	−	649.086i	−0.444918	−	0.733430i
$886$		0			0
$887$	766.896			0.864595			0.432297	−	0.901731i	$-0.357703\pi$
$887$	766.896			0.864595			0.432297	+	0.901731i	$0.357703\pi$
$888$		0			0
$889$	7.07980			0.00796378
$890$		0			0
$891$			142.160i			0.159551i
$892$		0			0
$893$			999.954i			1.11977i
$894$		0			0
$895$	−588.814	−	970.637i	−0.657893	−	1.08451i
$896$		0			0
$897$		−	171.673i		−	0.191386i
$898$		0			0
$899$			621.942i			0.691815i
$900$		0			0
$901$	−425.691			−0.472465
$902$		0			0
$903$	40.5220			0.0448748
$904$		0			0
$905$	487.752	−	295.883i	0.538953	−	0.326943i
$906$		0			0
$907$	1086.43			1.19783			0.598913	−	0.800814i	$-0.295600\pi$
$907$	1086.43			1.19783			0.598913	+	0.800814i	$0.295600\pi$
$908$		0			0
$909$	−133.650			−0.147029
$910$		0			0
$911$			237.746i			0.260973i	0.991450	+	0.130486i	$0.0416539\pi$
$911$			237.746i			0.260973i	−0.991450	+	0.130486i	$0.958346\pi$
$912$		0			0
$913$		−	932.012i		−	1.02082i
$914$		0			0
$915$	−281.366	+	170.684i	−0.307504	+	0.186540i
$916$		0			0
$917$		−	43.3103i		−	0.0472304i
$918$		0			0
$919$			344.517i			0.374882i	0.982276	+	0.187441i	$0.0600194\pi$
$919$			344.517i			0.374882i	−0.982276	+	0.187441i	$0.939981\pi$
$920$		0			0
$921$	−200.591			−0.217797
$922$		0			0
$923$	−677.917			−0.734471
$924$		0			0
$925$	344.990	+	179.711i	0.372962	+	0.194283i
$926$		0			0
$927$	−379.397			−0.409274
$928$		0			0
$929$	1478.10			1.59106			0.795531	−	0.605913i	$-0.207192\pi$
$929$	1478.10			1.59106			0.795531	+	0.605913i	$0.207192\pi$
$930$		0			0
$931$			867.776i			0.932090i
$932$		0			0
$933$		−	352.935i		−	0.378280i
$934$		0			0
$935$	1847.38	−	1120.67i	1.97581	−	1.19858i
$936$		0			0
$937$			246.887i			0.263486i	0.991284	+	0.131743i	$0.0420574\pi$
$937$			246.887i			0.263486i	−0.991284	+	0.131743i	$0.957943\pi$
$938$		0			0
$939$			171.579i			0.182726i
$940$		0			0
$941$	−1658.64			−1.76264			−0.881318	−	0.472525i	$-0.843343\pi$
$941$	−1658.64			−1.76264			−0.881318	+	0.472525i	$0.843343\pi$
$942$		0			0
$943$	−254.145			−0.269507
$944$		0			0
$945$	18.5980	−	11.2820i	0.0196804	−	0.0119387i
$946$		0			0
$947$	−6.00750			−0.00634372			−0.00317186	−	0.999995i	$-0.501010\pi$
$947$	−6.00750			−0.00634372			−0.00317186	+	0.999995i	$0.501010\pi$
$948$		0			0
$949$	283.794			0.299045
$950$		0			0
$951$		−	816.129i		−	0.858180i
$952$		0			0
$953$			1089.55i			1.14329i	0.820503	+	0.571643i	$0.193694\pi$
$953$			1089.55i			1.14329i	−0.820503	+	0.571643i	$0.806306\pi$
$954$		0			0
$955$	−361.481	−	595.887i	−0.378514	−	0.623966i
$956$		0			0
$957$		−	1248.91i		−	1.30502i
$958$		0			0
$959$		−	45.0512i		−	0.0469773i
$960$		0			0
$961$	775.379			0.806846
$962$		0			0
$963$	−313.150			−0.325182
$964$		0			0
$965$	472.096	+	778.233i	0.489219	+	0.806459i
$966$		0			0
$967$	1699.27			1.75726			0.878630	−	0.477502i	$-0.158458\pi$
$967$	1699.27			1.75726			0.878630	+	0.477502i	$0.158458\pi$
$968$		0			0
$969$	−851.382			−0.878619
$970$		0			0
$971$		−	197.851i		−	0.203760i	−0.994797	−	0.101880i	$-0.967514\pi$
$971$		−	197.851i		−	0.203760i	0.994797	−	0.101880i	$-0.0324857\pi$
$972$		0			0
$973$		−	11.4070i		−	0.0117235i
$974$		0			0
$975$	199.180	+	103.756i	0.204287	+	0.106417i
$976$		0			0
$977$			847.868i			0.867828i	0.900954	+	0.433914i	$0.142868\pi$
$977$			847.868i			0.867828i	−0.900954	+	0.433914i	$0.857132\pi$
$978$		0			0
$979$			628.672i			0.642157i
$980$		0			0
$981$	1.50497			0.00153412
$982$		0			0
$983$	96.0512			0.0977123			0.0488562	−	0.998806i	$-0.484442\pi$
$983$	96.0512			0.0977123			0.0488562	+	0.998806i	$0.484442\pi$
$984$		0			0
$985$	669.135	+	1103.04i	0.679324	+	1.11984i
$986$		0			0
$987$	80.7101			0.0817732
$988$		0			0
$989$	533.993			0.539933
$990$		0			0
$991$			1184.45i			1.19520i	0.801793	+	0.597602i	$0.203880\pi$
$991$			1184.45i			1.19520i	−0.801793	+	0.597602i	$0.796120\pi$
$992$		0			0
$993$			468.858i			0.472163i
$994$		0			0
$995$	665.108	+	1096.40i	0.668450	+	1.10191i
$996$		0			0
$997$		−	1887.35i		−	1.89303i	−0.322655	−	0.946517i	$-0.604575\pi$
$997$		−	1887.35i		−	1.89303i	0.322655	−	0.946517i	$-0.395425\pi$
$998$		0			0
$999$		−	80.8504i		−	0.0809313i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.j.e.319.2		8
4.3	odd	2	inner	960.3.j.e.319.6		8
5.4	even	2	inner	960.3.j.e.319.5		8
8.3	odd	2		60.3.f.b.19.7	yes	8
8.5	even	2		60.3.f.b.19.1	✓	8
20.19	odd	2	inner	960.3.j.e.319.1		8
24.5	odd	2		180.3.f.h.19.8		8
24.11	even	2		180.3.f.h.19.2		8
40.3	even	4		300.3.c.f.151.3		8
40.13	odd	4		300.3.c.f.151.4		8
40.19	odd	2		60.3.f.b.19.2	yes	8
40.27	even	4		300.3.c.f.151.6		8
40.29	even	2		60.3.f.b.19.8	yes	8
40.37	odd	4		300.3.c.f.151.5		8
120.29	odd	2		180.3.f.h.19.1		8
120.53	even	4		900.3.c.r.451.5		8
120.59	even	2		180.3.f.h.19.7		8
120.77	even	4		900.3.c.r.451.4		8
120.83	odd	4		900.3.c.r.451.6		8
120.107	odd	4		900.3.c.r.451.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.f.b.19.1	✓	8	8.5	even	2
60.3.f.b.19.2	yes	8	40.19	odd	2
60.3.f.b.19.7	yes	8	8.3	odd	2
60.3.f.b.19.8	yes	8	40.29	even	2
180.3.f.h.19.1		8	120.29	odd	2
180.3.f.h.19.2		8	24.11	even	2
180.3.f.h.19.7		8	120.59	even	2
180.3.f.h.19.8		8	24.5	odd	2
300.3.c.f.151.3		8	40.3	even	4
300.3.c.f.151.4		8	40.13	odd	4
300.3.c.f.151.5		8	40.37	odd	4
300.3.c.f.151.6		8	40.27	even	4
900.3.c.r.451.3		8	120.107	odd	4
900.3.c.r.451.4		8	120.77	even	4
900.3.c.r.451.5		8	120.53	even	4
900.3.c.r.451.6		8	120.83	odd	4
960.3.j.e.319.1		8	20.19	odd	2	inner
960.3.j.e.319.2		8	1.1	even	1	trivial
960.3.j.e.319.5		8	5.4	even	2	inner
960.3.j.e.319.6		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(-4.27492 + 2.59328i) q^{5} +0.837253 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +(-4.27492 + 2.59328i) q^{5} +0.837253 q^{7} +3.00000 q^{9} +15.7955i q^{11} -5.18655i q^{13} +(7.40437 - 4.49169i) q^{15} +27.3586i q^{17} -17.9667i q^{19} -1.45017 q^{21} -19.1101 q^{23} +(11.5498 - 22.1721i) q^{25} -5.19615 q^{27} +45.6495 q^{29} +13.6243i q^{31} -27.3586i q^{33} +(-3.57919 + 2.17123i) q^{35} +15.5597i q^{37} +8.98337i q^{39} +13.2990 q^{41} -27.9430 q^{43} +(-12.8248 + 7.77983i) q^{45} -55.6558 q^{47} -48.2990 q^{49} -47.3865i q^{51} +15.5597i q^{53} +(-40.9621 - 67.5245i) q^{55} +31.1193i q^{57} -87.6625i q^{59} -38.0000 q^{61} +2.51176 q^{63} +(13.4502 + 22.1721i) q^{65} -92.2015 q^{67} +33.0997 q^{69} -130.707i q^{71} +54.7173i q^{73} +(-20.0049 + 38.4032i) q^{75} +13.2249i q^{77} -13.6243i q^{79} +9.00000 q^{81} -59.0048 q^{83} +(-70.9485 - 116.956i) q^{85} -79.0673 q^{87} +39.8007 q^{89} -4.34246i q^{91} -23.5980i q^{93} +(46.5927 + 76.8064i) q^{95} -168.821i q^{97} +47.3865i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} + 24 q^{9}+O(q^{10}) \) 8 * q - 4 * q^5 + 24 * q^9	\( 8 q - 4 q^{5} + 24 q^{9} - 72 q^{21} + 32 q^{25} + 184 q^{29} - 256 q^{41} - 12 q^{45} - 24 q^{49} - 304 q^{61} + 168 q^{65} + 144 q^{69} + 72 q^{81} - 24 q^{85} + 560 q^{89}+O(q^{100}) \) 8 * q - 4 * q^5 + 24 * q^9 - 72 * q^21 + 32 * q^25 + 184 * q^29 - 256 * q^41 - 12 * q^45 - 24 * q^49 - 304 * q^61 + 168 * q^65 + 144 * q^69 + 72 * q^81 - 24 * q^85 + 560 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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