LMFDB - Embedded newform 960.4.d.a.289.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(289,960)]

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([0, 1, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("960.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.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:	$56.6418336055$
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} - 6 x^{11} + 18 x^{10} + 300 x^{9} + 14598 x^{8} - 60928 x^{7} + 147804 x^{6} + 1450180 x^{5} + \cdots + 2737382400 $ x^12 - 6x^11 + 18x^10 + 300x^9 + 14598x^8 - 60928x^7 + 147804x^6 + 1450180x^5 + 13518601x^4 - 29643466x^3 + 68047778x^2 + 610365120*x + 2737382400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.8
Root			$-2.21647 - 2.21647i$ of defining polynomial
Character	$\chi$	$=$	960.289
Dual form			960.4.d.a.289.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +(4.70786 + 10.1408i) q^{5} -21.3630i q^{7} +9.00000 q^{9} +62.3850i q^{11} +61.9262 q^{13} +(-14.1236 - 30.4224i) q^{15} -97.6721i q^{17} +77.4831i q^{19} +64.0890i q^{21} +127.834i q^{23} +(-80.6721 + 95.4831i) q^{25} -27.0000 q^{27} -294.782i q^{29} -45.2846 q^{31} -187.155i q^{33} +(216.638 - 100.574i) q^{35} -50.3207 q^{37} -185.779 q^{39} +326.114 q^{41} +112.378 q^{43} +(42.3708 + 91.2673i) q^{45} +61.9512i q^{47} -113.378 q^{49} +293.016i q^{51} +287.558 q^{53} +(-632.635 + 293.700i) q^{55} -232.449i q^{57} -387.661i q^{59} +472.891i q^{61} -192.267i q^{63} +(291.540 + 627.982i) q^{65} -1050.93 q^{67} -383.503i q^{69} +115.039 q^{71} +756.343i q^{73} +(242.016 - 286.449i) q^{75} +1332.73 q^{77} +579.824 q^{79} +81.0000 q^{81} +604.520 q^{83} +(990.474 - 459.827i) q^{85} +884.345i q^{87} -560.602 q^{89} -1322.93i q^{91} +135.854 q^{93} +(-785.741 + 364.780i) q^{95} -1183.96i q^{97} +561.465i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{3} + 108 q^{9} - 44 q^{25} - 324 q^{27} - 296 q^{35} + 792 q^{41} + 1472 q^{43} - 1484 q^{49} + 952 q^{65} - 1152 q^{67} + 132 q^{75} + 972 q^{81} + 1216 q^{83} - 3112 q^{89}+O(q^{100}) $ 12 * q - 36 * q^3 + 108 * q^9 - 44 * q^25 - 324 * q^27 - 296 * q^35 + 792 * q^41 + 1472 * q^43 - 1484 * q^49 + 952 * q^65 - 1152 * q^67 + 132 * q^75 + 972 * q^81 + 1216 * q^83 - 3112 * q^89

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
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$		0			0
$5$	4.70786	+	10.1408i	0.421084	+	0.907022i
$5$	4.70786	+	10.1408i	0.421084	+	0.907022i
$6$		0			0
$7$		−	21.3630i		−	1.15349i	−0.816923	−	0.576747i	$-0.804322\pi$
$7$		−	21.3630i		−	1.15349i	0.816923	−	0.576747i	$-0.195678\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			62.3850i			1.70998i	0.518643	+	0.854991i	$0.326437\pi$
$11$			62.3850i			1.70998i	−0.518643	+	0.854991i	$0.673563\pi$
$12$		0			0
$13$	61.9262			1.32117			0.660587	−	0.750750i	$-0.270308\pi$
$13$	61.9262			1.32117			0.660587	+	0.750750i	$0.270308\pi$
$14$		0			0
$15$	−14.1236	−	30.4224i	−0.243113	−	0.523669i
$16$		0			0
$17$		−	97.6721i		−	1.39347i	−0.717329	−	0.696734i	$-0.754636\pi$
$17$		−	97.6721i		−	1.39347i	0.717329	−	0.696734i	$-0.245364\pi$
$18$		0			0
$19$			77.4831i			0.935570i	0.883842	+	0.467785i	$0.154948\pi$
$19$			77.4831i			0.935570i	−0.883842	+	0.467785i	$0.845052\pi$
$20$		0			0
$21$			64.0890i			0.665970i
$22$		0			0
$23$			127.834i			1.15893i	0.814999	+	0.579463i	$0.196738\pi$
$23$			127.834i			1.15893i	−0.814999	+	0.579463i	$0.803262\pi$
$24$		0			0
$25$	−80.6721	+	95.4831i	−0.645377	+	0.763864i
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$		−	294.782i		−	1.88757i	−0.330559	−	0.943785i	$-0.607237\pi$
$29$		−	294.782i		−	1.88757i	0.330559	−	0.943785i	$-0.392763\pi$
$30$		0			0
$31$	−45.2846			−0.262366			−0.131183	−	0.991358i	$-0.541878\pi$
$31$	−45.2846			−0.262366			−0.131183	+	0.991358i	$0.541878\pi$
$32$		0			0
$33$		−	187.155i		−	0.987258i
$34$		0			0
$35$	216.638	−	100.574i	1.04624	−	0.485718i
$36$		0			0
$37$	−50.3207			−0.223586			−0.111793	−	0.993732i	$-0.535659\pi$
$37$	−50.3207			−0.223586			−0.111793	+	0.993732i	$0.535659\pi$
$38$		0			0
$39$	−185.779			−0.762780
$40$		0			0
$41$	326.114			1.24221			0.621104	−	0.783729i	$-0.286685\pi$
$41$	326.114			1.24221			0.621104	+	0.783729i	$0.286685\pi$
$42$		0			0
$43$	112.378			0.398546			0.199273	−	0.979944i	$-0.436142\pi$
$43$	112.378			0.398546			0.199273	+	0.979944i	$0.436142\pi$
$44$		0			0
$45$	42.3708	+	91.2673i	0.140361	+	0.302341i
$46$		0			0
$47$			61.9512i			0.192266i	0.995368	+	0.0961332i	$0.0306475\pi$
$47$			61.9512i			0.192266i	−0.995368	+	0.0961332i	$0.969353\pi$
$48$		0			0
$49$	−113.378			−0.330548
$50$		0			0
$51$			293.016i			0.804519i
$52$		0			0
$53$	287.558			0.745266			0.372633	−	0.927979i	$-0.378455\pi$
$53$	287.558			0.745266			0.372633	+	0.927979i	$0.378455\pi$
$54$		0			0
$55$	−632.635	+	293.700i	−1.55099	+	0.720046i
$56$		0			0
$57$		−	232.449i		−	0.540152i
$58$		0			0
$59$		−	387.661i		−	0.855411i	−0.903918	−	0.427705i	$-0.859322\pi$
$59$		−	387.661i		−	0.855411i	0.903918	−	0.427705i	$-0.140678\pi$
$60$		0			0
$61$			472.891i			0.992581i	0.868157	+	0.496291i	$0.165305\pi$
$61$			472.891i			0.992581i	−0.868157	+	0.496291i	$0.834695\pi$
$62$		0			0
$63$		−	192.267i		−	0.384498i
$64$		0			0
$65$	291.540	+	627.982i	0.556325	+	1.19833i
$66$		0			0
$67$	−1050.93			−1.91629			−0.958147	−	0.286278i	$-0.907582\pi$
$67$	−1050.93			−1.91629			−0.958147	+	0.286278i	$0.907582\pi$
$68$		0			0
$69$		−	383.503i		−	0.669106i
$70$		0			0
$71$	115.039			0.192290			0.0961452	−	0.995367i	$-0.469349\pi$
$71$	115.039			0.192290			0.0961452	+	0.995367i	$0.469349\pi$
$72$		0			0
$73$			756.343i			1.21265i	0.795218	+	0.606323i	$0.207356\pi$
$73$			756.343i			1.21265i	−0.795218	+	0.606323i	$0.792644\pi$
$74$		0			0
$75$	242.016	−	286.449i	0.372608	−	0.441017i
$76$		0			0
$77$	1332.73			1.97245
$78$		0			0
$79$	579.824			0.825764			0.412882	−	0.910785i	$-0.364522\pi$
$79$	579.824			0.825764			0.412882	+	0.910785i	$0.364522\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	604.520			0.799455			0.399727	−	0.916634i	$-0.369105\pi$
$83$	604.520			0.799455			0.399727	+	0.916634i	$0.369105\pi$
$84$		0			0
$85$	990.474	−	459.827i	1.26391	−	0.586767i
$86$		0			0
$87$			884.345i			1.08979i
$88$		0			0
$89$	−560.602			−0.667682			−0.333841	−	0.942629i	$-0.608345\pi$
$89$	−560.602			−0.667682			−0.333841	+	0.942629i	$0.608345\pi$
$90$		0			0
$91$		−	1322.93i		−	1.52397i
$92$		0			0
$93$	135.854			0.151477
$94$		0			0
$95$	−785.741	+	364.780i	−0.848582	+	0.393954i
$96$		0			0
$97$		−	1183.96i		−	1.23931i	−0.784873	−	0.619657i	$-0.787272\pi$
$97$		−	1183.96i		−	1.23931i	0.784873	−	0.619657i	$-0.212728\pi$
$98$		0			0
$99$			561.465i			0.569994i
$100$		0			0
$101$			1772.90i			1.74664i	0.487149	+	0.873319i	$0.338037\pi$
$101$			1772.90i			1.74664i	−0.487149	+	0.873319i	$0.661963\pi$
$102$		0			0
$103$			1092.73i			1.04534i	0.852535	+	0.522669i	$0.175064\pi$
$103$			1092.73i			1.04534i	−0.852535	+	0.522669i	$0.824936\pi$
$104$		0			0
$105$	−649.915	+	301.722i	−0.604049	+	0.280429i
$106$		0			0
$107$	561.932			0.507701			0.253851	−	0.967243i	$-0.418303\pi$
$107$	561.932			0.507701			0.253851	+	0.967243i	$0.418303\pi$
$108$		0			0
$109$			1443.62i			1.26856i	0.773102	+	0.634282i	$0.218704\pi$
$109$			1443.62i			1.26856i	−0.773102	+	0.634282i	$0.781296\pi$
$110$		0			0
$111$	150.962			0.129087
$112$		0			0
$113$			8.15990i			0.00679308i	0.999994	+	0.00339654i	$0.00108116\pi$
$113$			8.15990i			0.00679308i	−0.999994	+	0.00339654i	$0.998919\pi$
$114$		0			0
$115$	−1296.34	+	601.826i	−1.05117	+	0.488005i
$116$		0			0
$117$	557.336			0.440391
$118$		0			0
$119$	−2086.57			−1.60736
$120$		0			0
$121$	−2560.89			−1.92404
$122$		0			0
$123$	−978.343			−0.717189
$124$		0			0
$125$	−1348.07	−	368.559i	−0.964599	−	0.263720i
$126$		0			0
$127$			2522.43i			1.76244i	0.472711	+	0.881218i	$0.343276\pi$
$127$			2522.43i			1.76244i	−0.472711	+	0.881218i	$0.656724\pi$
$128$		0			0
$129$	−337.134			−0.230101
$130$		0			0
$131$			71.1832i			0.0474756i	0.999718	+	0.0237378i	$0.00755668\pi$
$131$			71.1832i			0.0474756i	−0.999718	+	0.0237378i	$0.992443\pi$
$132$		0			0
$133$	1655.27			1.07917
$134$		0			0
$135$	−127.112	−	273.802i	−0.0810376	−	0.174556i
$136$		0			0
$137$			705.637i			0.440048i	0.975494	+	0.220024i	$0.0706136\pi$
$137$			705.637i			0.440048i	−0.975494	+	0.220024i	$0.929386\pi$
$138$		0			0
$139$			2738.07i			1.67079i	0.549649	+	0.835395i	$0.314761\pi$
$139$			2738.07i			1.67079i	−0.549649	+	0.835395i	$0.685239\pi$
$140$		0			0
$141$		−	185.854i		−	0.111005i
$142$		0			0
$143$			3863.27i			2.25918i
$144$		0			0
$145$	2989.32	−	1387.79i	1.71207	−	0.794826i
$146$		0			0
$147$	340.134			0.190842
$148$		0			0
$149$			1330.57i			0.731571i	0.930699	+	0.365786i	$0.119200\pi$
$149$			1330.57i			0.731571i	−0.930699	+	0.365786i	$0.880800\pi$
$150$		0			0
$151$	2504.12			1.34955			0.674775	−	0.738023i	$-0.264241\pi$
$151$	2504.12			1.34955			0.674775	+	0.738023i	$0.264241\pi$
$152$		0			0
$153$		−	879.049i		−	0.464489i
$154$		0			0
$155$	−213.194	−	459.223i	−0.110478	−	0.237972i
$156$		0			0
$157$	−444.260			−0.225833			−0.112917	−	0.993604i	$-0.536019\pi$
$157$	−444.260			−0.225833			−0.112917	+	0.993604i	$0.536019\pi$
$158$		0			0
$159$	−862.673			−0.430279
$160$		0			0
$161$	2730.92			1.33681
$162$		0			0
$163$	2351.99			1.13020			0.565099	−	0.825023i	$-0.308838\pi$
$163$	2351.99			1.13020			0.565099	+	0.825023i	$0.308838\pi$
$164$		0			0
$165$	1897.90	−	881.100i	0.895465	−	0.415719i
$166$		0			0
$167$		−	1293.63i		−	0.599426i	−0.954029	−	0.299713i	$-0.903109\pi$
$167$		−	1293.63i		−	0.599426i	0.954029	−	0.299713i	$-0.0968909\pi$
$168$		0			0
$169$	1637.86			0.745499
$170$		0			0
$171$			697.348i			0.311857i
$172$		0			0
$173$	3599.40			1.58183			0.790917	−	0.611923i	$-0.209604\pi$
$173$	3599.40			1.58183			0.790917	+	0.611923i	$0.209604\pi$
$174$		0			0
$175$	2039.81	+	1723.40i	0.881113	+	0.744438i
$176$		0			0
$177$			1162.98i			0.493872i
$178$		0			0
$179$			177.522i			0.0741263i	0.999313	+	0.0370632i	$0.0118003\pi$
$179$			177.522i			0.0741263i	−0.999313	+	0.0370632i	$0.988200\pi$
$180$		0			0
$181$			4532.03i			1.86112i	0.366136	+	0.930562i	$0.380681\pi$
$181$			4532.03i			1.86112i	−0.366136	+	0.930562i	$0.619319\pi$
$182$		0			0
$183$		−	1418.67i		−	0.573067i
$184$		0			0
$185$	−236.903	−	510.293i	−0.0941484	−	0.202797i
$186$		0			0
$187$	6093.28			2.38281
$188$		0			0
$189$			576.801i			0.221990i
$190$		0			0
$191$	−1060.25			−0.401661			−0.200830	−	0.979626i	$-0.564364\pi$
$191$	−1060.25			−0.401661			−0.200830	+	0.979626i	$0.564364\pi$
$192$		0			0
$193$			38.7168i			0.0144399i	0.999974	+	0.00721994i	$0.00229820\pi$
$193$			38.7168i			0.0144399i	−0.999974	+	0.00721994i	$0.997702\pi$
$194$		0			0
$195$	−874.621	−	1883.95i	−0.321194	−	0.691858i
$196$		0			0
$197$	3836.30			1.38744			0.693718	−	0.720247i	$-0.255972\pi$
$197$	3836.30			1.38744			0.693718	+	0.720247i	$0.255972\pi$
$198$		0			0
$199$	−3402.71			−1.21212			−0.606059	−	0.795419i	$-0.707251\pi$
$199$	−3402.71			−1.21212			−0.606059	+	0.795419i	$0.707251\pi$
$200$		0			0
$201$	3152.79			1.10637
$202$		0			0
$203$	−6297.42			−2.17730
$204$		0			0
$205$	1535.30	+	3307.06i	0.523073	+	1.12671i
$206$		0			0
$207$			1150.51i			0.386308i
$208$		0			0
$209$	−4833.78			−1.59981
$210$		0			0
$211$		−	2878.08i		−	0.939028i	−0.882925	−	0.469514i	$-0.844429\pi$
$211$		−	2878.08i		−	0.939028i	0.882925	−	0.469514i	$-0.155571\pi$
$212$		0			0
$213$	−345.117			−0.111019
$214$		0			0
$215$	529.060	+	1139.60i	0.167821	+	0.361490i
$216$		0			0
$217$			967.416i			0.302638i
$218$		0			0
$219$		−	2269.03i		−	0.700122i
$220$		0			0
$221$		−	6048.47i		−	1.84101i
$222$		0			0
$223$		−	430.971i		−	0.129417i	−0.997904	−	0.0647085i	$-0.979388\pi$
$223$		−	430.971i		−	0.129417i	0.997904	−	0.0647085i	$-0.0206117\pi$
$224$		0			0
$225$	−726.049	+	859.348i	−0.215126	+	0.254621i
$226$		0			0
$227$	3613.48			1.05654			0.528271	−	0.849076i	$-0.322841\pi$
$227$	3613.48			1.05654			0.528271	+	0.849076i	$0.322841\pi$
$228$		0			0
$229$			1668.14i			0.481369i	0.970603	+	0.240684i	$0.0773719\pi$
$229$			1668.14i			0.481369i	−0.970603	+	0.240684i	$0.922628\pi$
$230$		0			0
$231$	−3998.20			−1.13880
$232$		0			0
$233$		−	1373.79i		−	0.386265i	−0.981173	−	0.193132i	$-0.938135\pi$
$233$		−	1373.79i		−	0.386265i	0.981173	−	0.193132i	$-0.0618647\pi$
$234$		0			0
$235$	−628.236	+	291.658i	−0.174390	+	0.0809603i
$236$		0			0
$237$	−1739.47			−0.476755
$238$		0			0
$239$	1969.40			0.533011			0.266506	−	0.963833i	$-0.414131\pi$
$239$	1969.40			0.533011			0.266506	+	0.963833i	$0.414131\pi$
$240$		0			0
$241$	−491.738			−0.131434			−0.0657171	−	0.997838i	$-0.520934\pi$
$241$	−491.738			−0.131434			−0.0657171	+	0.997838i	$0.520934\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$	−533.768	−	1149.75i	−0.139189	−	0.299814i
$246$		0			0
$247$			4798.24i			1.23605i
$248$		0			0
$249$	−1813.56			−0.461565
$250$		0			0
$251$			3517.30i			0.884502i	0.896891	+	0.442251i	$0.145820\pi$
$251$			3517.30i			0.884502i	−0.896891	+	0.442251i	$0.854180\pi$
$252$		0			0
$253$	−7974.94			−1.98174
$254$		0			0
$255$	−2971.42	+	1379.48i	−0.729716	+	0.338770i
$256$		0			0
$257$		−	3461.61i		−	0.840191i	−0.907480	−	0.420096i	$-0.861997\pi$
$257$		−	3461.61i		−	0.840191i	0.907480	−	0.420096i	$-0.138003\pi$
$258$		0			0
$259$			1075.00i			0.257905i
$260$		0			0
$261$		−	2653.03i		−	0.629190i
$262$		0			0
$263$			2684.50i			0.629405i	0.949190	+	0.314703i	$0.101905\pi$
$263$			2684.50i			0.629405i	−0.949190	+	0.314703i	$0.898095\pi$
$264$		0			0
$265$	1353.78	+	2916.07i	0.313819	+	0.675972i
$266$		0			0
$267$	1681.81			0.385486
$268$		0			0
$269$		−	2944.72i		−	0.667445i	−0.942671	−	0.333722i	$-0.891695\pi$
$269$		−	2944.72i		−	0.667445i	0.942671	−	0.333722i	$-0.108305\pi$
$270$		0			0
$271$	−1668.02			−0.373894			−0.186947	−	0.982370i	$-0.559859\pi$
$271$	−1668.02			−0.373894			−0.186947	+	0.982370i	$0.559859\pi$
$272$		0			0
$273$			3968.79i			0.879862i
$274$		0			0
$275$	−5956.72	−	5032.73i	−1.30619	−	1.10358i
$276$		0			0
$277$	3299.29			0.715651			0.357826	−	0.933788i	$-0.383518\pi$
$277$	3299.29			0.715651			0.357826	+	0.933788i	$0.383518\pi$
$278$		0			0
$279$	−407.562			−0.0874555
$280$		0			0
$281$	22.9341			0.00486881			0.00243440	−	0.999997i	$-0.499225\pi$
$281$	22.9341			0.00486881			0.00243440	+	0.999997i	$0.499225\pi$
$282$		0			0
$283$	−8073.88			−1.69591			−0.847955	−	0.530069i	$-0.822166\pi$
$283$	−8073.88			−1.69591			−0.847955	+	0.530069i	$0.822166\pi$
$284$		0			0
$285$	2357.22	−	1094.34i	0.489929	−	0.227449i
$286$		0			0
$287$		−	6966.78i		−	1.43288i
$288$		0			0
$289$	−4626.84			−0.941754
$290$		0			0
$291$			3551.89i			0.715518i
$292$		0			0
$293$	7144.38			1.42450			0.712251	−	0.701925i	$-0.247676\pi$
$293$	7144.38			1.42450			0.712251	+	0.701925i	$0.247676\pi$
$294$		0			0
$295$	3931.20	−	1825.06i	0.775876	−	0.360200i
$296$		0			0
$297$		−	1684.40i		−	0.329086i
$298$		0			0
$299$			7916.29i			1.53114i
$300$		0			0
$301$		−	2400.73i		−	0.459721i
$302$		0			0
$303$		−	5318.71i		−	1.00842i
$304$		0			0
$305$	−4795.49	+	2226.30i	−0.900293	+	0.417960i
$306$		0			0
$307$	−2285.79			−0.424940			−0.212470	−	0.977168i	$-0.568151\pi$
$307$	−2285.79			−0.424940			−0.212470	+	0.977168i	$0.568151\pi$
$308$		0			0
$309$		−	3278.19i		−	0.603527i
$310$		0			0
$311$	−9333.00			−1.70169			−0.850845	−	0.525416i	$-0.823910\pi$
$311$	−9333.00			−1.70169			−0.850845	+	0.525416i	$0.823910\pi$
$312$		0			0
$313$			2557.79i			0.461900i	0.972966	+	0.230950i	$0.0741834\pi$
$313$			2557.79i			0.461900i	−0.972966	+	0.230950i	$0.925817\pi$
$314$		0			0
$315$	1949.74	−	905.167i	0.348748	−	0.161906i
$316$		0			0
$317$	−8465.99			−1.49999			−0.749995	−	0.661443i	$-0.769944\pi$
$317$	−8465.99			−1.49999			−0.749995	+	0.661443i	$0.769944\pi$
$318$		0			0
$319$	18390.0			3.22771
$320$		0			0
$321$	−1685.80			−0.293121
$322$		0			0
$323$	7567.93			1.30369
$324$		0			0
$325$	−4995.72	+	5912.91i	−0.852654	+	1.00920i
$326$		0			0
$327$		−	4330.85i		−	0.732405i
$328$		0			0
$329$	1323.46			0.221778
$330$		0			0
$331$		−	7456.35i		−	1.23818i	−0.785319	−	0.619091i	$-0.787501\pi$
$331$		−	7456.35i		−	1.23818i	0.785319	−	0.619091i	$-0.212499\pi$
$332$		0			0
$333$	−452.886			−0.0745286
$334$		0			0
$335$	−4947.64	−	10657.3i	−0.806920	−	1.73812i
$336$		0			0
$337$		−	8621.94i		−	1.39367i	−0.717231	−	0.696835i	$-0.754591\pi$
$337$		−	8621.94i		−	1.39367i	0.717231	−	0.696835i	$-0.245409\pi$
$338$		0			0
$339$		−	24.4797i		−	0.00392199i
$340$		0			0
$341$		−	2825.08i		−	0.448642i
$342$		0			0
$343$		−	4905.42i		−	0.772209i
$344$		0			0
$345$	3889.03	−	1805.48i	0.606893	−	0.281750i
$346$		0			0
$347$	−2049.21			−0.317025			−0.158512	−	0.987357i	$-0.550670\pi$
$347$	−2049.21			−0.317025			−0.158512	+	0.987357i	$0.550670\pi$
$348$		0			0
$349$			3697.81i			0.567161i	0.958948	+	0.283580i	$0.0915222\pi$
$349$			3697.81i			0.567161i	−0.958948	+	0.283580i	$0.908478\pi$
$350$		0			0
$351$	−1672.01			−0.254260
$352$		0			0
$353$			7804.32i			1.17672i	0.808599	+	0.588360i	$0.200226\pi$
$353$			7804.32i			1.17672i	−0.808599	+	0.588360i	$0.799774\pi$
$354$		0			0
$355$	541.588	+	1166.59i	0.0809704	+	0.174412i
$356$		0			0
$357$	6259.71			0.928008
$358$		0			0
$359$	−3008.97			−0.442360			−0.221180	−	0.975233i	$-0.570991\pi$
$359$	−3008.97			−0.442360			−0.221180	+	0.975233i	$0.570991\pi$
$360$		0			0
$361$	855.375			0.124708
$362$		0			0
$363$	7682.68			1.11084
$364$		0			0
$365$	−7669.93	+	3560.76i	−1.09990	+	0.510626i
$366$		0			0
$367$			1103.47i			0.156951i	0.996916	+	0.0784753i	$0.0250052\pi$
$367$			1103.47i			0.156951i	−0.996916	+	0.0784753i	$0.974995\pi$
$368$		0			0
$369$	2935.03			0.414069
$370$		0			0
$371$		−	6143.10i		−	0.859660i
$372$		0			0
$373$	−2803.77			−0.389206			−0.194603	−	0.980882i	$-0.562342\pi$
$373$	−2803.77			−0.389206			−0.194603	+	0.980882i	$0.562342\pi$
$374$		0			0
$375$	4044.21	+	1105.68i	0.556912	+	0.152259i
$376$		0			0
$377$		−	18254.7i		−	2.49381i
$378$		0			0
$379$		−	3063.52i		−	0.415204i	−0.978213	−	0.207602i	$-0.933434\pi$
$379$		−	3063.52i		−	0.415204i	0.978213	−	0.207602i	$-0.0665659\pi$
$380$		0			0
$381$		−	7567.28i		−	1.01754i
$382$		0			0
$383$		−	679.079i		−	0.0905987i	−0.998973	−	0.0452993i	$-0.985576\pi$
$383$		−	679.079i		−	0.0905987i	0.998973	−	0.0452993i	$-0.0144242\pi$
$384$		0			0
$385$	6274.32	+	13515.0i	0.830568	+	1.78906i
$386$		0			0
$387$	1011.40			0.132849
$388$		0			0
$389$		−	7916.81i		−	1.03187i	−0.856627	−	0.515936i	$-0.827444\pi$
$389$		−	7916.81i		−	1.03187i	0.856627	−	0.515936i	$-0.172556\pi$
$390$		0			0
$391$	12485.8			1.61493
$392$		0			0
$393$		−	213.549i		−	0.0274100i
$394$		0			0
$395$	2729.73	+	5879.89i	0.347716	+	0.748986i
$396$		0			0
$397$	−13539.1			−1.71161			−0.855803	−	0.517302i	$-0.826937\pi$
$397$	−13539.1			−1.71161			−0.855803	+	0.517302i	$0.826937\pi$
$398$		0			0
$399$	−4965.81			−0.623062
$400$		0			0
$401$	243.973			0.0303826			0.0151913	−	0.999885i	$-0.495164\pi$
$401$	243.973			0.0303826			0.0151913	+	0.999885i	$0.495164\pi$
$402$		0			0
$403$	−2804.31			−0.346632
$404$		0			0
$405$	381.337	+	821.406i	0.0467871	+	0.100780i
$406$		0			0
$407$		−	3139.26i		−	0.382328i
$408$		0			0
$409$	−244.569			−0.0295676			−0.0147838	−	0.999891i	$-0.504706\pi$
$409$	−244.569			−0.0295676			−0.0147838	+	0.999891i	$0.504706\pi$
$410$		0			0
$411$		−	2116.91i		−	0.254062i
$412$		0			0
$413$	−8281.61			−0.986711
$414$		0			0
$415$	2846.00	+	6130.33i	0.336637	+	0.725123i
$416$		0			0
$417$		−	8214.21i		−	0.964632i
$418$		0			0
$419$			835.764i			0.0974457i	0.998812	+	0.0487229i	$0.0155151\pi$
$419$			835.764i			0.0974457i	−0.998812	+	0.0487229i	$0.984485\pi$
$420$		0			0
$421$			6457.84i			0.747591i	0.927511	+	0.373796i	$0.121944\pi$
$421$			6457.84i			0.747591i	−0.927511	+	0.373796i	$0.878056\pi$
$422$		0			0
$423$			557.561i			0.0640888i
$424$		0			0
$425$	9326.03	+	7879.41i	1.06442	+	0.899312i
$426$		0			0
$427$	10102.4			1.14494
$428$		0			0
$429$		−	11589.8i		−	1.30434i
$430$		0			0
$431$	913.052			0.102042			0.0510211	−	0.998698i	$-0.483752\pi$
$431$	913.052			0.102042			0.0510211	+	0.998698i	$0.483752\pi$
$432$		0			0
$433$		−	193.572i		−	0.0214837i	−0.999942	−	0.0107419i	$-0.996581\pi$
$433$		−	193.572i		−	0.0214837i	0.999942	−	0.0107419i	$-0.00341931\pi$
$434$		0			0
$435$	−8967.97	+	4163.37i	−0.988463	+	0.458893i
$436$		0			0
$437$	−9904.99			−1.08426
$438$		0			0
$439$	8320.59			0.904602			0.452301	−	0.891865i	$-0.350603\pi$
$439$	8320.59			0.904602			0.452301	+	0.891865i	$0.350603\pi$
$440$		0			0
$441$	−1020.40			−0.110183
$442$		0			0
$443$	555.054			0.0595292			0.0297646	−	0.999557i	$-0.490524\pi$
$443$	555.054			0.0595292			0.0297646	+	0.999557i	$0.490524\pi$
$444$		0			0
$445$	−2639.24	−	5684.96i	−0.281150	−	0.605602i
$446$		0			0
$447$		−	3991.70i		−	0.422373i
$448$		0			0
$449$	12300.2			1.29283			0.646416	−	0.762985i	$-0.276267\pi$
$449$	12300.2			1.29283			0.646416	+	0.762985i	$0.276267\pi$
$450$		0			0
$451$			20344.7i			2.12415i
$452$		0			0
$453$	−7512.35			−0.779163
$454$		0			0
$455$	13415.6	−	6228.18i	1.38227	−	0.641717i
$456$		0			0
$457$		−	6858.10i		−	0.701988i	−0.936378	−	0.350994i	$-0.885844\pi$
$457$		−	6858.10i		−	0.701988i	0.936378	−	0.350994i	$-0.114156\pi$
$458$		0			0
$459$			2637.15i			0.268173i
$460$		0			0
$461$		−	2076.92i		−	0.209831i	−0.994481	−	0.104915i	$-0.966543\pi$
$461$		−	2076.92i		−	0.209831i	0.994481	−	0.104915i	$-0.0334571\pi$
$462$		0			0
$463$		−	1184.82i		−	0.118927i	−0.998230	−	0.0594636i	$-0.981061\pi$
$463$		−	1184.82i		−	0.118927i	0.998230	−	0.0594636i	$-0.0189390\pi$
$464$		0			0
$465$	639.581	+	1377.67i	0.0637847	+	0.137393i
$466$		0			0
$467$	16649.7			1.64980			0.824901	−	0.565277i	$-0.191231\pi$
$467$	16649.7			1.64980			0.824901	+	0.565277i	$0.191231\pi$
$468$		0			0
$469$			22451.0i			2.21043i
$470$		0			0
$471$	1332.78			0.130385
$472$		0			0
$473$			7010.71i			0.681507i
$474$		0			0
$475$	−7398.32	−	6250.72i	−0.714649	−	0.603795i
$476$		0			0
$477$	2588.02			0.248422
$478$		0			0
$479$	−7169.47			−0.683886			−0.341943	−	0.939721i	$-0.611085\pi$
$479$	−7169.47			−0.683886			−0.341943	+	0.939721i	$0.611085\pi$
$480$		0			0
$481$	−3116.17			−0.295395
$482$		0			0
$483$	−8192.77			−0.771809
$484$		0			0
$485$	12006.4	−	5573.94i	1.12408	−	0.521855i
$486$		0			0
$487$		−	15806.7i		−	1.47078i	−0.677645	−	0.735389i	$-0.736999\pi$
$487$		−	15806.7i		−	1.47078i	0.677645	−	0.735389i	$-0.263001\pi$
$488$		0			0
$489$	−7055.98			−0.652521
$490$		0			0
$491$		−	19581.5i		−	1.79979i	−0.436103	−	0.899897i	$-0.643642\pi$
$491$		−	19581.5i		−	1.79979i	0.436103	−	0.899897i	$-0.356358\pi$
$492$		0			0
$493$	−28791.9			−2.63027
$494$		0			0
$495$	−5693.71	+	2643.30i	−0.516997	+	0.240015i
$496$		0			0
$497$		−	2457.58i		−	0.221806i
$498$		0			0
$499$			156.561i			0.0140454i	0.999975	+	0.00702270i	$0.00223541\pi$
$499$			156.561i			0.0140454i	−0.999975	+	0.00702270i	$0.997765\pi$
$500$		0			0
$501$			3880.89i			0.346079i
$502$		0			0
$503$		−	19995.3i		−	1.77246i	−0.463244	−	0.886231i	$-0.653315\pi$
$503$		−	19995.3i		−	1.77246i	0.463244	−	0.886231i	$-0.346685\pi$
$504$		0			0
$505$	−17978.7	+	8346.58i	−1.58424	+	0.735481i
$506$		0			0
$507$	−4913.58			−0.430414
$508$		0			0
$509$		−	7200.81i		−	0.627054i	−0.949579	−	0.313527i	$-0.898489\pi$
$509$		−	7200.81i		−	0.627054i	0.949579	−	0.313527i	$-0.101511\pi$
$510$		0			0
$511$	16157.8			1.39878
$512$		0			0
$513$		−	2092.04i		−	0.180051i
$514$		0			0
$515$	−11081.2	+	5144.42i	−0.948145	+	0.440175i
$516$		0			0
$517$	−3864.83			−0.328772
$518$		0			0
$519$	−10798.2			−0.913273
$520$		0			0
$521$	4435.42			0.372974			0.186487	−	0.982457i	$-0.440290\pi$
$521$	4435.42			0.372974			0.186487	+	0.982457i	$0.440290\pi$
$522$		0			0
$523$	−6398.95			−0.535003			−0.267502	−	0.963557i	$-0.586198\pi$
$523$	−6398.95			−0.535003			−0.267502	+	0.963557i	$0.586198\pi$
$524$		0			0
$525$	−6119.42	−	5170.19i	−0.508711	−	0.429802i
$526$		0			0
$527$			4423.04i			0.365599i
$528$		0			0
$529$	−4174.59			−0.343108
$530$		0			0
$531$		−	3488.95i		−	0.285137i
$532$		0			0
$533$	20195.0			1.64117
$534$		0			0
$535$	2645.50	+	5698.45i	0.213785	+	0.460496i
$536$		0			0
$537$		−	532.566i		−	0.0427969i
$538$		0			0
$539$		−	7073.09i		−	0.565231i
$540$		0			0
$541$			13296.1i			1.05664i	0.849044	+	0.528321i	$0.177178\pi$
$541$			13296.1i			1.05664i	−0.849044	+	0.528321i	$0.822822\pi$
$542$		0			0
$543$		−	13596.1i		−	1.07452i
$544$		0			0
$545$	−14639.4	+	6796.35i	−1.15061	+	0.534171i
$546$		0			0
$547$	16352.8			1.27824			0.639120	−	0.769107i	$-0.279299\pi$
$547$	16352.8			1.27824			0.639120	+	0.769107i	$0.279299\pi$
$548$		0			0
$549$			4256.02i			0.330860i
$550$		0			0
$551$	22840.6			1.76596
$552$		0			0
$553$		−	12386.8i		−	0.952514i
$554$		0			0
$555$	710.709	+	1530.88i	0.0543566	+	0.117085i
$556$		0			0
$557$	−6507.85			−0.495057			−0.247528	−	0.968881i	$-0.579618\pi$
$557$	−6507.85			−0.495057			−0.247528	+	0.968881i	$0.579618\pi$
$558$		0			0
$559$	6959.15			0.526549
$560$		0			0
$561$	−18279.8			−1.37571
$562$		0			0
$563$	−17211.3			−1.28840			−0.644201	−	0.764856i	$-0.722810\pi$
$563$	−17211.3			−1.28840			−0.644201	+	0.764856i	$0.722810\pi$
$564$		0			0
$565$	−82.7480	+	38.4157i	−0.00616148	+	0.00286046i
$566$		0			0
$567$		−	1730.40i		−	0.128166i
$568$		0			0
$569$	−20548.4			−1.51394			−0.756972	−	0.653447i	$-0.773322\pi$
$569$	−20548.4			−1.51394			−0.756972	+	0.653447i	$0.773322\pi$
$570$		0			0
$571$			23344.0i			1.71089i	0.517896	+	0.855443i	$0.326715\pi$
$571$			23344.0i			1.71089i	−0.517896	+	0.855443i	$0.673285\pi$
$572$		0			0
$573$	3180.76			0.231899
$574$		0			0
$575$	−12206.0	−	10312.7i	−0.885262	−	0.747943i
$576$		0			0
$577$		−	15538.9i		−	1.12113i	−0.828111	−	0.560565i	$-0.810584\pi$
$577$		−	15538.9i		−	1.12113i	0.828111	−	0.560565i	$-0.189416\pi$
$578$		0			0
$579$		−	116.151i		−	0.00833687i
$580$		0			0
$581$		−	12914.4i		−	0.922166i
$582$		0			0
$583$			17939.3i			1.27439i
$584$		0			0
$585$	2623.86	+	5651.84i	0.185442	+	0.399444i
$586$		0			0
$587$	19057.9			1.34004			0.670019	−	0.742344i	$-0.266286\pi$
$587$	19057.9			1.34004			0.670019	+	0.742344i	$0.266286\pi$
$588$		0			0
$589$		−	3508.79i		−	0.245462i
$590$		0			0
$591$	−11508.9			−0.801036
$592$		0			0
$593$			2882.70i			0.199626i	0.995006	+	0.0998130i	$0.0318244\pi$
$593$			2882.70i			0.199626i	−0.995006	+	0.0998130i	$0.968176\pi$
$594$		0			0
$595$	−9823.28	−	21159.5i	−0.676832	−	1.45791i
$596$		0			0
$597$	10208.1			0.699817
$598$		0			0
$599$	−2793.66			−0.190560			−0.0952802	−	0.995450i	$-0.530375\pi$
$599$	−2793.66			−0.190560			−0.0952802	+	0.995450i	$0.530375\pi$
$600$		0			0
$601$	16728.9			1.13542			0.567708	−	0.823230i	$-0.307830\pi$
$601$	16728.9			1.13542			0.567708	+	0.823230i	$0.307830\pi$
$602$		0			0
$603$	−9458.38			−0.638764
$604$		0			0
$605$	−12056.3	−	25969.5i	−0.810181	−	1.74514i
$606$		0			0
$607$		−	4085.51i		−	0.273189i	−0.990627	−	0.136595i	$-0.956384\pi$
$607$		−	4085.51i		−	0.273189i	0.990627	−	0.136595i	$-0.0436158\pi$
$608$		0			0
$609$	18892.3			1.25707
$610$		0			0
$611$			3836.41i			0.254017i
$612$		0			0
$613$	21091.0			1.38965			0.694827	−	0.719177i	$-0.255481\pi$
$613$	21091.0			1.38965			0.694827	+	0.719177i	$0.255481\pi$
$614$		0			0
$615$	−4605.90	−	9921.19i	−0.301997	−	0.650506i
$616$		0			0
$617$			21775.6i			1.42083i	0.703784	+	0.710414i	$0.251492\pi$
$617$			21775.6i			1.42083i	−0.703784	+	0.710414i	$0.748508\pi$
$618$		0			0
$619$		−	14992.4i		−	0.973497i	−0.873542	−	0.486748i	$-0.838183\pi$
$619$		−	14992.4i		−	0.973497i	0.873542	−	0.486748i	$-0.161817\pi$
$620$		0			0
$621$		−	3451.52i		−	0.223035i
$622$		0			0
$623$			11976.1i			0.770167i
$624$		0			0
$625$	−2609.03	−	15405.6i	−0.166978	−	0.985961i
$626$		0			0
$627$	14501.4			0.923650
$628$		0			0
$629$			4914.93i			0.311560i
$630$		0			0
$631$	5547.82			0.350008			0.175004	−	0.984568i	$-0.444006\pi$
$631$	5547.82			0.350008			0.175004	+	0.984568i	$0.444006\pi$
$632$		0			0
$633$			8634.23i			0.542148i
$634$		0			0
$635$	−25579.5	+	11875.2i	−1.59857	+	0.742133i
$636$		0			0
$637$	−7021.08			−0.436711
$638$		0			0
$639$	1035.35			0.0640968
$640$		0			0
$641$	9758.42			0.601302			0.300651	−	0.953734i	$-0.402796\pi$
$641$	9758.42			0.601302			0.300651	+	0.953734i	$0.402796\pi$
$642$		0			0
$643$	−8281.64			−0.507925			−0.253963	−	0.967214i	$-0.581734\pi$
$643$	−8281.64			−0.507925			−0.253963	+	0.967214i	$0.581734\pi$
$644$		0			0
$645$	−1587.18	−	3418.81i	−0.0968918	−	0.208706i
$646$		0			0
$647$			25179.4i			1.52999i	0.644035	+	0.764996i	$0.277259\pi$
$647$			25179.4i			1.52999i	−0.644035	+	0.764996i	$0.722741\pi$
$648$		0			0
$649$	24184.3			1.46274
$650$		0			0
$651$		−	2902.25i		−	0.174728i
$652$		0			0
$653$	−16406.4			−0.983203			−0.491601	−	0.870820i	$-0.663588\pi$
$653$	−16406.4			−0.983203			−0.491601	+	0.870820i	$0.663588\pi$
$654$		0			0
$655$	−721.855	+	335.120i	−0.0430614	+	0.0199912i
$656$		0			0
$657$			6807.09i			0.404216i
$658$		0			0
$659$		−	17109.3i		−	1.01136i	−0.862722	−	0.505678i	$-0.831242\pi$
$659$		−	17109.3i		−	1.01136i	0.862722	−	0.505678i	$-0.168758\pi$
$660$		0			0
$661$			16545.4i			0.973588i	0.873517	+	0.486794i	$0.161834\pi$
$661$			16545.4i			0.973588i	−0.873517	+	0.486794i	$0.838166\pi$
$662$		0			0
$663$			18145.4i			1.06291i
$664$		0			0
$665$	7792.79	+	16785.8i	0.454423	+	0.978835i
$666$		0			0
$667$	37683.2			2.18755
$668$		0			0
$669$			1292.91i			0.0747189i
$670$		0			0
$671$	−29501.3			−1.69730
$672$		0			0
$673$			30667.2i			1.75651i	0.478189	+	0.878257i	$0.341293\pi$
$673$			30667.2i			1.75651i	−0.478189	+	0.878257i	$0.658707\pi$
$674$		0			0
$675$	2178.15	−	2578.04i	0.124203	−	0.147006i
$676$		0			0
$677$	15668.9			0.889522			0.444761	−	0.895649i	$-0.353289\pi$
$677$	15668.9			0.889522			0.444761	+	0.895649i	$0.353289\pi$
$678$		0			0
$679$	−25293.0			−1.42954
$680$		0			0
$681$	−10840.4			−0.609994
$682$		0			0
$683$	2979.28			0.166909			0.0834547	−	0.996512i	$-0.473405\pi$
$683$	2979.28			0.166909			0.0834547	+	0.996512i	$0.473405\pi$
$684$		0			0
$685$	−7155.73	+	3322.04i	−0.399133	+	0.185297i
$686$		0			0
$687$		−	5004.41i		−	0.277918i
$688$		0			0
$689$	17807.4			0.984625
$690$		0			0
$691$			10017.1i			0.551475i	0.961233	+	0.275737i	$0.0889220\pi$
$691$			10017.1i			0.551475i	−0.961233	+	0.275737i	$0.911078\pi$
$692$		0			0
$693$	11994.6			0.657485
$694$		0			0
$695$	−27766.2	+	12890.4i	−1.51544	+	0.703543i
$696$		0			0
$697$		−	31852.3i		−	1.73098i
$698$		0			0
$699$			4121.36i			0.223010i
$700$		0			0
$701$			19134.6i			1.03096i	0.856901	+	0.515481i	$0.172387\pi$
$701$			19134.6i			1.03096i	−0.856901	+	0.515481i	$0.827613\pi$
$702$		0			0
$703$		−	3899.00i		−	0.209180i
$704$		0			0
$705$	1884.71	−	874.974i	0.100684	−	0.0467424i
$706$		0			0
$707$	37874.5			2.01474
$708$		0			0
$709$		−	17862.1i		−	0.946159i	−0.881020	−	0.473079i	$-0.843142\pi$
$709$		−	17862.1i		−	0.946159i	0.881020	−	0.473079i	$-0.156858\pi$
$710$		0			0
$711$	5218.42			0.275255
$712$		0			0
$713$		−	5788.93i		−	0.304063i
$714$		0			0
$715$	−39176.7	+	18187.7i	−2.04913	+	0.951305i
$716$		0			0
$717$	−5908.19			−0.307734
$718$		0			0
$719$	−2464.40			−0.127826			−0.0639129	−	0.997955i	$-0.520358\pi$
$719$	−2464.40			−0.127826			−0.0639129	+	0.997955i	$0.520358\pi$
$720$		0			0
$721$	23344.0			1.20579
$722$		0			0
$723$	1475.22			0.0758836
$724$		0			0
$725$	28146.6	+	23780.6i	1.44185	+	1.21819i
$726$		0			0
$727$			17483.6i			0.891929i	0.895051	+	0.445964i	$0.147139\pi$
$727$			17483.6i			0.891929i	−0.895051	+	0.445964i	$0.852861\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$		−	10976.2i		−	0.555362i
$732$		0			0
$733$	21187.2			1.06762			0.533811	−	0.845604i	$-0.320759\pi$
$733$	21187.2			1.06762			0.533811	+	0.845604i	$0.320759\pi$
$734$		0			0
$735$	1601.30	+	3449.24i	0.0803605	+	0.173098i
$736$		0			0
$737$		−	65562.4i		−	3.27683i
$738$		0			0
$739$		−	15519.8i		−	0.772535i	−0.922387	−	0.386267i	$-0.873764\pi$
$739$		−	15519.8i		−	0.772535i	0.922387	−	0.386267i	$-0.126236\pi$
$740$		0			0
$741$		−	14394.7i		−	0.713634i
$742$		0			0
$743$		−	14837.1i		−	0.732599i	−0.930497	−	0.366300i	$-0.880625\pi$
$743$		−	14837.1i		−	0.732599i	0.930497	−	0.366300i	$-0.119375\pi$
$744$		0			0
$745$	−13493.0	+	6264.12i	−0.663551	+	0.308053i
$746$		0			0
$747$	5440.68			0.266485
$748$		0			0
$749$		−	12004.6i		−	0.585630i
$750$		0			0
$751$	13643.9			0.662945			0.331473	−	0.943465i	$-0.392455\pi$
$751$	13643.9			0.662945			0.331473	+	0.943465i	$0.392455\pi$
$752$		0			0
$753$		−	10551.9i		−	0.510667i
$754$		0			0
$755$	11789.0	+	25393.8i	0.568274	+	1.22407i
$756$		0			0
$757$	34745.6			1.66823			0.834114	−	0.551592i	$-0.185979\pi$
$757$	34745.6			1.66823			0.834114	+	0.551592i	$0.185979\pi$
$758$		0			0
$759$	23924.8			1.14416
$760$		0			0
$761$	−17980.2			−0.856481			−0.428241	−	0.903665i	$-0.640866\pi$
$761$	−17980.2			−0.856481			−0.428241	+	0.903665i	$0.640866\pi$
$762$		0			0
$763$	30840.0			1.46328
$764$		0			0
$765$	8914.27	−	4138.44i	0.421302	−	0.195589i
$766$		0			0
$767$		−	24006.4i		−	1.13015i
$768$		0			0
$769$	−21791.9			−1.02189			−0.510947	−	0.859612i	$-0.670705\pi$
$769$	−21791.9			−1.02189			−0.510947	+	0.859612i	$0.670705\pi$
$770$		0			0
$771$			10384.8i			0.485085i
$772$		0			0
$773$	−5416.23			−0.252016			−0.126008	−	0.992029i	$-0.540216\pi$
$773$	−5416.23			−0.252016			−0.126008	+	0.992029i	$0.540216\pi$
$774$		0			0
$775$	3653.21	−	4323.92i	0.169325	−	0.200412i
$776$		0			0
$777$		−	3225.00i		−	0.148901i
$778$		0			0
$779$			25268.3i			1.16217i
$780$		0			0
$781$			7176.72i			0.328813i
$782$		0			0
$783$			7959.10i			0.363263i
$784$		0			0
$785$	−2091.51	−	4505.16i	−0.0950947	−	0.204836i
$786$		0			0
$787$	−9727.86			−0.440611			−0.220305	−	0.975431i	$-0.570705\pi$
$787$	−9727.86			−0.440611			−0.220305	+	0.975431i	$0.570705\pi$
$788$		0			0
$789$		−	8053.51i		−	0.363387i
$790$		0			0
$791$	174.320			0.00783578
$792$		0			0
$793$			29284.3i			1.31137i
$794$		0			0
$795$	−4061.35	−	8748.21i	−0.181184	−	0.390273i
$796$		0			0
$797$	36030.9			1.60136			0.800678	−	0.599095i	$-0.204473\pi$
$797$	36030.9			1.60136			0.800678	+	0.599095i	$0.204473\pi$
$798$		0			0
$799$	6050.91			0.267917
$800$		0			0
$801$	−5045.42			−0.222561
$802$		0			0
$803$	−47184.5			−2.07360
$804$		0			0
$805$	12856.8	+	27693.8i	0.562911	+	1.21252i
$806$		0			0
$807$			8834.16i			0.385349i
$808$		0			0
$809$	−6298.55			−0.273727			−0.136864	−	0.990590i	$-0.543702\pi$
$809$	−6298.55			−0.273727			−0.136864	+	0.990590i	$0.543702\pi$
$810$		0			0
$811$		−	10654.2i		−	0.461306i	−0.973036	−	0.230653i	$-0.925914\pi$
$811$		−	10654.2i		−	0.461306i	0.973036	−	0.230653i	$-0.0740863\pi$
$812$		0			0
$813$	5004.07			0.215868
$814$		0			0
$815$	11072.9	+	23851.1i	0.475909	+	1.02511i
$816$		0			0
$817$			8707.40i			0.372868i
$818$		0			0
$819$		−	11906.4i		−	0.507988i
$820$		0			0
$821$			25369.3i			1.07844i	0.842166	+	0.539218i	$0.181280\pi$
$821$			25369.3i			1.07844i	−0.842166	+	0.539218i	$0.818720\pi$
$822$		0			0
$823$			17794.8i			0.753692i	0.926276	+	0.376846i	$0.122991\pi$
$823$			17794.8i			0.753692i	−0.926276	+	0.376846i	$0.877009\pi$
$824$		0			0
$825$	17870.1	+	15098.2i	0.754132	+	0.637154i
$826$		0			0
$827$	−14614.5			−0.614506			−0.307253	−	0.951628i	$-0.599410\pi$
$827$	−14614.5			−0.614506			−0.307253	+	0.951628i	$0.599410\pi$
$828$		0			0
$829$		−	19898.4i		−	0.833654i	−0.908986	−	0.416827i	$-0.863142\pi$
$829$		−	19898.4i		−	0.833654i	0.908986	−	0.416827i	$-0.136858\pi$
$830$		0			0
$831$	−9897.88			−0.413181
$832$		0			0
$833$			11073.9i			0.460608i
$834$		0			0
$835$	13118.5	−	6090.23i	0.543692	−	0.252408i
$836$		0			0
$837$	1222.69			0.0504924
$838$		0			0
$839$	−1164.17			−0.0479043			−0.0239522	−	0.999713i	$-0.507625\pi$
$839$	−1164.17			−0.0479043			−0.0239522	+	0.999713i	$0.507625\pi$
$840$		0			0
$841$	−62507.2			−2.56292
$842$		0			0
$843$	−68.8023			−0.00281101
$844$		0			0
$845$	7710.82	+	16609.2i	0.313917	+	0.676183i
$846$		0			0
$847$			54708.4i			2.21937i
$848$		0			0
$849$	24221.6			0.979134
$850$		0			0
$851$		−	6432.71i		−	0.259119i
$852$		0			0
$853$	−3513.03			−0.141013			−0.0705063	−	0.997511i	$-0.522461\pi$
$853$	−3513.03			−0.141013			−0.0705063	+	0.997511i	$0.522461\pi$
$854$		0			0
$855$	−7071.67	+	3283.02i	−0.282861	+	0.131318i
$856$		0			0
$857$		−	45814.0i		−	1.82611i	−0.407837	−	0.913055i	$-0.633717\pi$
$857$		−	45814.0i		−	1.82611i	0.407837	−	0.913055i	$-0.366283\pi$
$858$		0			0
$859$			15334.5i			0.609087i	0.952499	+	0.304543i	$0.0985038\pi$
$859$			15334.5i			0.609087i	−0.952499	+	0.304543i	$0.901496\pi$
$860$		0			0
$861$			20900.3i			0.827273i
$862$		0			0
$863$			5461.91i			0.215441i	0.994181	+	0.107721i	$0.0343552\pi$
$863$			5461.91i			0.215441i	−0.994181	+	0.107721i	$0.965645\pi$
$864$		0			0
$865$	16945.5	+	36500.8i	0.666085	+	1.43476i
$866$		0			0
$867$	13880.5			0.543722
$868$		0			0
$869$			36172.4i			1.41204i
$870$		0			0
$871$	−65080.2			−2.53176
$872$		0			0
$873$		−	10655.7i		−	0.413104i
$874$		0			0
$875$	−7873.54	+	28798.8i	−0.304199	+	1.11266i
$876$		0			0
$877$	−27282.4			−1.05047			−0.525235	−	0.850957i	$-0.676022\pi$
$877$	−27282.4			−1.05047			−0.525235	+	0.850957i	$0.676022\pi$
$878$		0			0
$879$	−21433.1			−0.822437
$880$		0			0
$881$	10368.4			0.396505			0.198252	−	0.980151i	$-0.436473\pi$
$881$	10368.4			0.396505			0.198252	+	0.980151i	$0.436473\pi$
$882$		0			0
$883$	−15902.8			−0.606084			−0.303042	−	0.952977i	$-0.598002\pi$
$883$	−15902.8			−0.606084			−0.303042	+	0.952977i	$0.598002\pi$
$884$		0			0
$885$	−11793.6	+	5475.17i	−0.447952	+	0.207961i
$886$		0			0
$887$			9899.94i			0.374755i	0.982288	+	0.187377i	$0.0599987\pi$
$887$			9899.94i			0.374755i	−0.982288	+	0.187377i	$0.940001\pi$
$888$		0			0
$889$	53886.7			2.03296
$890$		0			0
$891$			5053.19i			0.189998i
$892$		0			0
$893$	−4800.17			−0.179879
$894$		0			0
$895$	−1800.22	+	835.749i	−0.0672342	+	0.0312134i
$896$		0			0
$897$		−	23748.9i		−	0.884005i
$898$		0			0
$899$			13349.1i			0.495235i
$900$		0			0
$901$		−	28086.4i		−	1.03850i
$902$		0			0
$903$			7202.20i			0.265420i
$904$		0			0
$905$	−45958.5	+	21336.2i	−1.68808	+	0.783689i
$906$		0			0
$907$	−7186.49			−0.263091			−0.131545	−	0.991310i	$-0.541994\pi$
$907$	−7186.49			−0.263091			−0.131545	+	0.991310i	$0.541994\pi$
$908$		0			0
$909$			15956.1i			0.582212i
$910$		0			0
$911$	−17888.6			−0.650578			−0.325289	−	0.945615i	$-0.605462\pi$
$911$	−17888.6			−0.650578			−0.325289	+	0.945615i	$0.605462\pi$
$912$		0			0
$913$			37713.0i			1.36705i
$914$		0			0
$915$	14386.5	−	6678.91i	0.519784	−	0.241309i
$916$		0			0
$917$	1520.69			0.0547628
$918$		0			0
$919$	10224.4			0.366997			0.183499	−	0.983020i	$-0.441258\pi$
$919$	10224.4			0.366997			0.183499	+	0.983020i	$0.441258\pi$
$920$		0			0
$921$	6857.36			0.245339
$922$		0			0
$923$	7123.94			0.254049
$924$		0			0
$925$	4059.48	−	4804.77i	0.144297	−	0.170789i
$926$		0			0
$927$			9834.57i			0.348446i
$928$		0			0
$929$	−22646.4			−0.799790			−0.399895	−	0.916561i	$-0.630953\pi$
$929$	−22646.4			−0.799790			−0.399895	+	0.916561i	$0.630953\pi$
$930$		0			0
$931$		−	8784.88i		−	0.309251i
$932$		0			0
$933$	27999.0			0.982471
$934$		0			0
$935$	28686.3	+	61790.8i	1.00336	+	2.16126i
$936$		0			0
$937$		−	32230.8i		−	1.12373i	−0.827229	−	0.561865i	$-0.810084\pi$
$937$		−	32230.8i		−	1.12373i	0.827229	−	0.561865i	$-0.189916\pi$
$938$		0			0
$939$		−	7673.36i		−	0.266678i
$940$		0			0
$941$		−	46578.3i		−	1.61361i	−0.590817	−	0.806805i	$-0.701195\pi$
$941$		−	46578.3i		−	1.61361i	0.590817	−	0.806805i	$-0.298805\pi$
$942$		0			0
$943$			41688.6i			1.43963i
$944$		0			0
$945$	−5849.23	+	2715.50i	−0.201350	+	0.0934764i
$946$		0			0
$947$	−13542.9			−0.464715			−0.232358	−	0.972630i	$-0.574644\pi$
$947$	−13542.9			−0.464715			−0.232358	+	0.972630i	$0.574644\pi$
$948$		0			0
$949$			46837.5i			1.60212i
$950$		0			0
$951$	25398.0			0.866020
$952$		0			0
$953$		−	7754.57i		−	0.263583i	−0.991277	−	0.131792i	$-0.957927\pi$
$953$		−	7754.57i		−	0.263583i	0.991277	−	0.131792i	$-0.0420730\pi$
$954$		0			0
$955$	−4991.53	−	10751.8i	−0.169133	−	0.364315i
$956$		0			0
$957$	−55169.9			−1.86352
$958$		0			0
$959$	15074.5			0.507593
$960$		0			0
$961$	−27740.3			−0.931164
$962$		0			0
$963$	5057.39			0.169234
$964$		0			0
$965$	−392.620	+	182.273i	−0.0130973	+	0.00608041i
$966$		0			0
$967$		−	30390.3i		−	1.01064i	−0.862933	−	0.505318i	$-0.831375\pi$
$967$		−	30390.3i		−	1.01064i	0.862933	−	0.505318i	$-0.168625\pi$
$968$		0			0
$969$	−22703.8			−0.752684
$970$		0			0
$971$		−	34151.1i		−	1.12869i	−0.825538	−	0.564347i	$-0.809128\pi$
$971$		−	34151.1i		−	1.12869i	0.825538	−	0.564347i	$-0.190872\pi$
$972$		0			0
$973$	58493.4			1.92725
$974$		0			0
$975$	14987.2	−	17738.7i	0.492280	−	0.582660i
$976$		0			0
$977$			19542.1i			0.639926i	0.947430	+	0.319963i	$0.103670\pi$
$977$			19542.1i			0.639926i	−0.947430	+	0.319963i	$0.896330\pi$
$978$		0			0
$979$		−	34973.2i		−	1.14172i
$980$		0			0
$981$			12992.5i			0.422854i
$982$		0			0
$983$			25296.5i			0.820787i	0.911909	+	0.410394i	$0.134609\pi$
$983$			25296.5i			0.820787i	−0.911909	+	0.410394i	$0.865391\pi$
$984$		0			0
$985$	18060.7	+	38903.1i	0.584227	+	1.25843i
$986$		0			0
$987$	−3970.39			−0.128044
$988$		0			0
$989$			14365.8i			0.461885i
$990$		0			0
$991$	21175.5			0.678772			0.339386	−	0.940647i	$-0.389781\pi$
$991$	21175.5			0.678772			0.339386	+	0.940647i	$0.389781\pi$
$992$		0			0
$993$			22369.1i			0.714865i
$994$		0			0
$995$	−16019.5	−	34506.2i	−0.510404	−	1.09942i
$996$		0			0
$997$	53310.9			1.69345			0.846727	−	0.532028i	$-0.178570\pi$
$997$	53310.9			1.69345			0.846727	+	0.532028i	$0.178570\pi$
$998$		0			0
$999$	1358.66			0.0430291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.d.a.289.8	yes	12
4.3	odd	2		960.4.d.b.289.8	yes	12
5.4	even	2		960.4.d.b.289.6	yes	12
8.3	odd	2	inner	960.4.d.a.289.5	✓	12
8.5	even	2		960.4.d.b.289.5	yes	12
20.19	odd	2	inner	960.4.d.a.289.6	yes	12
40.19	odd	2		960.4.d.b.289.7	yes	12
40.29	even	2	inner	960.4.d.a.289.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.4.d.a.289.5	✓	12	8.3	odd	2	inner
960.4.d.a.289.6	yes	12	20.19	odd	2	inner
960.4.d.a.289.7	yes	12	40.29	even	2	inner
960.4.d.a.289.8	yes	12	1.1	even	1	trivial
960.4.d.b.289.5	yes	12	8.5	even	2
960.4.d.b.289.6	yes	12	5.4	even	2
960.4.d.b.289.7	yes	12	40.19	odd	2
960.4.d.b.289.8	yes	12	4.3	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +(4.70786 + 10.1408i) q^{5} -21.3630i q^{7} +9.00000 q^{9} +62.3850i q^{11} +61.9262 q^{13} +(-14.1236 - 30.4224i) q^{15} -97.6721i q^{17} +77.4831i q^{19} +64.0890i q^{21} +127.834i q^{23} +(-80.6721 + 95.4831i) q^{25} -27.0000 q^{27} -294.782i q^{29} -45.2846 q^{31} -187.155i q^{33} +(216.638 - 100.574i) q^{35} -50.3207 q^{37} -185.779 q^{39} +326.114 q^{41} +112.378 q^{43} +(42.3708 + 91.2673i) q^{45} +61.9512i q^{47} -113.378 q^{49} +293.016i q^{51} +287.558 q^{53} +(-632.635 + 293.700i) q^{55} -232.449i q^{57} -387.661i q^{59} +472.891i q^{61} -192.267i q^{63} +(291.540 + 627.982i) q^{65} -1050.93 q^{67} -383.503i q^{69} +115.039 q^{71} +756.343i q^{73} +(242.016 - 286.449i) q^{75} +1332.73 q^{77} +579.824 q^{79} +81.0000 q^{81} +604.520 q^{83} +(990.474 - 459.827i) q^{85} +884.345i q^{87} -560.602 q^{89} -1322.93i q^{91} +135.854 q^{93} +(-785.741 + 364.780i) q^{95} -1183.96i q^{97} +561.465i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{3} + 108 q^{9} - 44 q^{25} - 324 q^{27} - 296 q^{35} + 792 q^{41} + 1472 q^{43} - 1484 q^{49} + 952 q^{65} - 1152 q^{67} + 132 q^{75} + 972 q^{81} + 1216 q^{83} - 3112 q^{89}+O(q^{100}) \) 12 * q - 36 * q^3 + 108 * q^9 - 44 * q^25 - 324 * q^27 - 296 * q^35 + 792 * q^41 + 1472 * q^43 - 1484 * q^49 + 952 * q^65 - 1152 * q^67 + 132 * q^75 + 972 * q^81 + 1216 * q^83 - 3112 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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