LMFDB - Embedded newform 720.6.o.d.719.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(719,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("720.719");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.o (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:	$115.476350265$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			719.24
Character	$\chi$	$=$	720.719
Dual form			720.6.o.d.719.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(21.5882 + 51.5650i) q^{5} +168.759 q^{7} +O(q^{10})$	$q+(21.5882 + 51.5650i) q^{5} +168.759 q^{7} +307.480 q^{11} +191.336i q^{13} -1131.52 q^{17} +1274.09i q^{19} +2347.48i q^{23} +(-2192.90 + 2226.39i) q^{25} -1057.25i q^{29} +3223.99i q^{31} +(3643.20 + 8702.04i) q^{35} +11966.3i q^{37} -10448.9i q^{41} -4148.22 q^{43} -2969.37i q^{47} +11672.5 q^{49} +8220.41 q^{53} +(6637.95 + 15855.2i) q^{55} +17357.9 q^{59} +18178.8 q^{61} +(-9866.22 + 4130.60i) q^{65} -44404.4 q^{67} -26272.3 q^{71} -47234.8i q^{73} +51889.9 q^{77} +379.315i q^{79} -30659.1i q^{83} +(-24427.5 - 58346.9i) q^{85} +112227. i q^{89} +32289.6i q^{91} +(-65698.6 + 27505.4i) q^{95} +84324.9i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 10648 q^{25} + 36248 q^{49} - 133136 q^{61} - 336816 q^{85}+O(q^{100}) $ 40 * q + 10648 * q^25 + 36248 * q^49 - 133136 * q^61 - 336816 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	21.5882	+	51.5650i	0.386182	+	0.922423i
$5$	21.5882	+	51.5650i	0.386182	+	0.922423i
$6$		0			0
$7$	168.759			1.30173			0.650865	−	0.759193i	$-0.274406\pi$
$7$	168.759			1.30173			0.650865	+	0.759193i	$0.274406\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	307.480			0.766188			0.383094	−	0.923709i	$-0.374859\pi$
$11$	307.480			0.766188			0.383094	+	0.923709i	$0.374859\pi$
$12$		0			0
$13$			191.336i			0.314006i	0.987598	+	0.157003i	$0.0501832\pi$
$13$			191.336i			0.314006i	−0.987598	+	0.157003i	$0.949817\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1131.52			−0.949599			−0.474800	−	0.880094i	$-0.657479\pi$
$17$	−1131.52			−0.949599			−0.474800	+	0.880094i	$0.657479\pi$
$18$		0			0
$19$			1274.09i			0.809686i	0.914386	+	0.404843i	$0.132674\pi$
$19$			1274.09i			0.809686i	−0.914386	+	0.404843i	$0.867326\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2347.48i			0.925300i	0.886541	+	0.462650i	$0.153101\pi$
$23$			2347.48i			0.925300i	−0.886541	+	0.462650i	$0.846899\pi$
$24$		0			0
$25$	−2192.90	+	2226.39i	−0.701727	+	0.712446i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1057.25i		−	0.233443i	−0.993165	−	0.116721i	$-0.962762\pi$
$29$		−	1057.25i		−	0.233443i	0.993165	−	0.116721i	$-0.0372385\pi$
$30$		0			0
$31$			3223.99i			0.602546i	0.953538	+	0.301273i	$0.0974115\pi$
$31$			3223.99i			0.602546i	−0.953538	+	0.301273i	$0.902588\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3643.20	+	8702.04i	0.502705	+	1.20075i
$36$		0			0
$37$			11966.3i			1.43700i	0.695527	+	0.718500i	$0.255171\pi$
$37$			11966.3i			1.43700i	−0.695527	+	0.718500i	$0.744829\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10448.9i		−	0.970756i	−0.874304	−	0.485378i	$-0.838682\pi$
$41$		−	10448.9i		−	0.970756i	0.874304	−	0.485378i	$-0.161318\pi$
$42$		0			0
$43$	−4148.22			−0.342129			−0.171065	−	0.985260i	$-0.554721\pi$
$43$	−4148.22			−0.342129			−0.171065	+	0.985260i	$0.554721\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2969.37i		−	0.196074i	−0.995183	−	0.0980369i	$-0.968744\pi$
$47$		−	2969.37i		−	0.196074i	0.995183	−	0.0980369i	$-0.0312563\pi$
$48$		0			0
$49$	11672.5			0.694502
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8220.41			0.401979			0.200990	−	0.979593i	$-0.435584\pi$
$53$	8220.41			0.401979			0.200990	+	0.979593i	$0.435584\pi$
$54$		0			0
$55$	6637.95	+	15855.2i	0.295888	+	0.706749i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	17357.9			0.649182			0.324591	−	0.945854i	$-0.394773\pi$
$59$	17357.9			0.649182			0.324591	+	0.945854i	$0.394773\pi$
$60$		0			0
$61$	18178.8			0.625520			0.312760	−	0.949832i	$-0.398746\pi$
$61$	18178.8			0.625520			0.312760	+	0.949832i	$0.398746\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9866.22	+	4130.60i	−0.289646	+	0.121263i
$66$		0			0
$67$	−44404.4			−1.20848			−0.604239	−	0.796803i	$-0.706523\pi$
$67$	−44404.4			−1.20848			−0.604239	+	0.796803i	$0.706523\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−26272.3			−0.618518			−0.309259	−	0.950978i	$-0.600081\pi$
$71$	−26272.3			−0.618518			−0.309259	+	0.950978i	$0.600081\pi$
$72$		0			0
$73$		−	47234.8i		−	1.03742i	−0.854950	−	0.518711i	$-0.826412\pi$
$73$		−	47234.8i		−	1.03742i	0.854950	−	0.518711i	$-0.173588\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	51889.9			0.997370
$78$		0			0
$79$			379.315i			0.00683804i	0.999994	+	0.00341902i	$0.00108831\pi$
$79$			379.315i			0.00683804i	−0.999994	+	0.00341902i	$0.998912\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	30659.1i		−	0.488499i	−0.969712	−	0.244250i	$-0.921458\pi$
$83$		−	30659.1i		−	0.488499i	0.969712	−	0.244250i	$-0.0785416\pi$
$84$		0			0
$85$	−24427.5	−	58346.9i	−0.366718	−	0.875932i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			112227.i			1.50184i	0.660395	+	0.750918i	$0.270389\pi$
$89$			112227.i			1.50184i	−0.660395	+	0.750918i	$0.729611\pi$
$90$		0			0
$91$			32289.6i			0.408751i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−65698.6	+	27505.4i	−0.746873	+	0.312686i
$96$		0			0
$97$			84324.9i			0.909969i	0.890499	+	0.454984i	$0.150355\pi$
$97$			84324.9i			0.909969i	−0.890499	+	0.454984i	$0.849645\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			26316.7i			0.256702i	0.991729	+	0.128351i	$0.0409684\pi$
$101$			26316.7i			0.256702i	−0.991729	+	0.128351i	$0.959032\pi$
$102$		0			0
$103$	23099.7			0.214543			0.107271	−	0.994230i	$-0.465789\pi$
$103$	23099.7			0.214543			0.107271	+	0.994230i	$0.465789\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	210729.i		−	1.77937i	−0.456580	−	0.889683i	$-0.650926\pi$
$107$		−	210729.i		−	1.77937i	0.456580	−	0.889683i	$-0.349074\pi$
$108$		0			0
$109$	151866.			1.22432			0.612160	−	0.790734i	$-0.290301\pi$
$109$	151866.			1.22432			0.612160	+	0.790734i	$0.290301\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−229446.			−1.69038			−0.845188	−	0.534468i	$-0.820512\pi$
$113$	−229446.			−1.69038			−0.845188	+	0.534468i	$0.820512\pi$
$114$		0			0
$115$	−121048.	+	50678.0i	−0.853518	+	0.357334i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−190954.			−1.23612
$120$		0			0
$121$	−66507.0			−0.412956
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−162145.	−	65012.7i	−0.928171	−	0.372155i
$126$		0			0
$127$	260300.			1.43207			0.716035	−	0.698064i	$-0.245955\pi$
$127$	260300.			1.43207			0.716035	+	0.698064i	$0.245955\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−207042.			−1.05410			−0.527048	−	0.849835i	$-0.676701\pi$
$131$	−207042.			−1.05410			−0.527048	+	0.849835i	$0.676701\pi$
$132$		0			0
$133$			215014.i			1.05399i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	222643.			1.01346			0.506730	−	0.862105i	$-0.330854\pi$
$137$	222643.			1.01346			0.506730	+	0.862105i	$0.330854\pi$
$138$		0			0
$139$			204205.i			0.896458i	0.893919	+	0.448229i	$0.147945\pi$
$139$			204205.i			0.896458i	−0.893919	+	0.448229i	$0.852055\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			58831.9i			0.240587i
$144$		0			0
$145$	54516.8	−	22824.1i	0.215333	−	0.0901515i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	60363.5i		−	0.222745i	−0.993779	−	0.111373i	$-0.964475\pi$
$149$		−	60363.5i		−	0.222745i	0.993779	−	0.111373i	$-0.0355247\pi$
$150$		0			0
$151$			503254.i			1.79616i	0.439832	+	0.898080i	$0.355038\pi$
$151$			503254.i			1.79616i	−0.439832	+	0.898080i	$0.644962\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−166245.	+	69600.3i	−0.555802	+	0.232692i
$156$		0			0
$157$			364962.i			1.18168i	0.806790	+	0.590838i	$0.201203\pi$
$157$			364962.i			1.18168i	−0.806790	+	0.590838i	$0.798797\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			396158.i			1.20449i
$162$		0			0
$163$	−441953.			−1.30289			−0.651444	−	0.758696i	$-0.725837\pi$
$163$	−441953.			−1.30289			−0.651444	+	0.758696i	$0.725837\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17833.9i		−	0.0494828i	−0.999694	−	0.0247414i	$-0.992124\pi$
$167$		−	17833.9i		−	0.0494828i	0.999694	−	0.0247414i	$-0.00787623\pi$
$168$		0			0
$169$	334684.			0.901400
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−51132.3			−0.129891			−0.0649457	−	0.997889i	$-0.520687\pi$
$173$	−51132.3			−0.129891			−0.0649457	+	0.997889i	$0.520687\pi$
$174$		0			0
$175$	−370070.	+	375723.i	−0.913459	+	0.927413i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−26668.9			−0.0622118			−0.0311059	−	0.999516i	$-0.509903\pi$
$179$	−26668.9			−0.0622118			−0.0311059	+	0.999516i	$0.509903\pi$
$180$		0			0
$181$	550930.			1.24997			0.624986	−	0.780636i	$-0.285105\pi$
$181$	550930.			1.24997			0.624986	+	0.780636i	$0.285105\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−617044.	+	258332.i	−1.32552	+	0.554944i
$186$		0			0
$187$	−347920.			−0.727572
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−57189.3			−0.113431			−0.0567155	−	0.998390i	$-0.518063\pi$
$191$	−57189.3			−0.113431			−0.0567155	+	0.998390i	$0.518063\pi$
$192$		0			0
$193$		−	554830.i		−	1.07218i	−0.844162	−	0.536089i	$-0.819901\pi$
$193$		−	554830.i		−	1.07218i	0.844162	−	0.536089i	$-0.180099\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.01603e6			1.86526			0.932630	−	0.360834i	$-0.117508\pi$
$197$	1.01603e6			1.86526			0.932630	+	0.360834i	$0.117508\pi$
$198$		0			0
$199$			1.02132e6i			1.82823i	0.405455	+	0.914115i	$0.367113\pi$
$199$			1.02132e6i			1.82823i	−0.405455	+	0.914115i	$0.632887\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	178419.i		−	0.303880i
$204$		0			0
$205$	538797.	−	225573.i	0.895448	−	0.374889i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			391758.i			0.620372i
$210$		0			0
$211$			777698.i			1.20256i	0.799040	+	0.601278i	$0.205341\pi$
$211$			777698.i			1.20256i	−0.799040	+	0.601278i	$0.794659\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−89552.6	−	213903.i	−0.132124	−	0.315588i
$216$		0			0
$217$			544077.i			0.784352i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	216500.i		−	0.298180i
$222$		0			0
$223$	−1.13326e6			−1.52604			−0.763020	−	0.646375i	$-0.776284\pi$
$223$	−1.13326e6			−1.52604			−0.763020	+	0.646375i	$0.776284\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	739890.i		−	0.953022i	−0.879169	−	0.476511i	$-0.841901\pi$
$227$		−	739890.i		−	0.953022i	0.879169	−	0.476511i	$-0.158099\pi$
$228$		0			0
$229$	67413.6			0.0849491			0.0424746	−	0.999098i	$-0.486476\pi$
$229$	67413.6			0.0849491			0.0424746	+	0.999098i	$0.486476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	261927.			0.316075			0.158038	−	0.987433i	$-0.449483\pi$
$233$	261927.			0.316075			0.158038	+	0.987433i	$0.449483\pi$
$234$		0			0
$235$	153115.	−	64103.4i	0.180863	−	0.0757202i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.21975e6			−1.38127			−0.690634	−	0.723205i	$-0.742668\pi$
$239$	−1.21975e6			−1.38127			−0.690634	+	0.723205i	$0.742668\pi$
$240$		0			0
$241$	1.16735e6			1.29467			0.647334	−	0.762206i	$-0.275884\pi$
$241$	1.16735e6			1.29467			0.647334	+	0.762206i	$0.275884\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	251989.	+	601892.i	0.268204	+	0.640625i
$246$		0			0
$247$	−243779.			−0.254246
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.80064e6			−1.80403			−0.902015	−	0.431705i	$-0.857912\pi$
$251$	−1.80064e6			−1.80403			−0.902015	+	0.431705i	$0.857912\pi$
$252$		0			0
$253$			721804.i			0.708954i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.82960e6			−1.72792			−0.863960	−	0.503561i	$-0.832023\pi$
$257$	−1.82960e6			−1.72792			−0.863960	+	0.503561i	$0.832023\pi$
$258$		0			0
$259$			2.01942e6i			1.87059i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			109784.i			0.0978699i	0.998802	+	0.0489349i	$0.0155827\pi$
$263$			109784.i			0.0978699i	−0.998802	+	0.0489349i	$0.984417\pi$
$264$		0			0
$265$	177464.	+	423885.i	0.155237	+	0.370795i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			551.216i			0.000464452i	1.00000		0.000232226i	$7.39198e-5\pi$
$269$			551.216i			0.000464452i	−1.00000		0.000232226i	$0.999926\pi$
$270$		0			0
$271$		−	306185.i		−	0.253257i	−0.991950	−	0.126628i	$-0.959584\pi$
$271$		−	306185.i		−	0.253257i	0.991950	−	0.126628i	$-0.0404156\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−674272.	+	684572.i	−0.537655	+	0.545868i
$276$		0			0
$277$			245795.i			0.192475i	0.995358	+	0.0962374i	$0.0306808\pi$
$277$			245795.i			0.192475i	−0.995358	+	0.0962374i	$0.969319\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.08142e6i			0.817012i	0.912756	+	0.408506i	$0.133950\pi$
$281$			1.08142e6i			0.817012i	−0.912756	+	0.408506i	$0.866050\pi$
$282$		0			0
$283$	2.03706e6			1.51195			0.755975	−	0.654600i	$-0.227163\pi$
$283$	2.03706e6			1.51195			0.755975	+	0.654600i	$0.227163\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.76334e6i		−	1.26366i
$288$		0			0
$289$	−139517.			−0.0982610
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.86589e6			−1.26975			−0.634873	−	0.772617i	$-0.718947\pi$
$293$	−1.86589e6			−1.26975			−0.634873	+	0.772617i	$0.718947\pi$
$294$		0			0
$295$	374726.	+	895058.i	0.250702	+	0.598820i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−449157.			−0.290550
$300$		0			0
$301$	−700047.			−0.445360
$302$		0			0
$303$		0			0
$304$		0			0
$305$	392449.	+	937391.i	0.241565	+	0.576994i
$306$		0			0
$307$	−977552.			−0.591962			−0.295981	−	0.955194i	$-0.595647\pi$
$307$	−977552.			−0.591962			−0.295981	+	0.955194i	$0.595647\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−591279.			−0.346650			−0.173325	−	0.984865i	$-0.555451\pi$
$311$	−591279.			−0.346650			−0.173325	+	0.984865i	$0.555451\pi$
$312$		0			0
$313$			1.57319e6i			0.907652i	0.891090	+	0.453826i	$0.149941\pi$
$313$			1.57319e6i			0.907652i	−0.891090	+	0.453826i	$0.850059\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.42391e6			−0.795853			−0.397927	−	0.917417i	$-0.630270\pi$
$317$	−1.42391e6			−0.795853			−0.397927	+	0.917417i	$0.630270\pi$
$318$		0			0
$319$		−	325082.i		−	0.178861i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.44166e6i		−	0.768878i
$324$		0			0
$325$	−425989.	−	419579.i	−0.223712	−	0.220346i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	501107.i		−	0.255235i
$330$		0			0
$331$			2.65384e6i			1.33139i	0.746225	+	0.665694i	$0.231864\pi$
$331$			2.65384e6i			1.33139i	−0.746225	+	0.665694i	$0.768136\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−958612.	−	2.28971e6i	−0.466692	−	1.11473i
$336$		0			0
$337$		−	1.30398e6i		−	0.625456i	−0.949843	−	0.312728i	$-0.898757\pi$
$337$		−	1.30398e6i		−	0.625456i	0.949843	−	0.312728i	$-0.101243\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			991314.i			0.461663i
$342$		0			0
$343$	−866492.			−0.397676
$344$		0			0
$345$		0			0
$346$		0			0
$347$			4.19503e6i			1.87030i	0.354251	+	0.935150i	$0.384736\pi$
$347$			4.19503e6i			1.87030i	−0.354251	+	0.935150i	$0.615264\pi$
$348$		0			0
$349$	1.11653e6			0.490692			0.245346	−	0.969436i	$-0.421099\pi$
$349$	1.11653e6			0.490692			0.245346	+	0.969436i	$0.421099\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.50537e6			1.07013			0.535063	−	0.844812i	$-0.320288\pi$
$353$	2.50537e6			1.07013			0.535063	+	0.844812i	$0.320288\pi$
$354$		0			0
$355$	−567173.	−	1.35473e6i	−0.238861	−	0.570535i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.52264e6			0.623533			0.311767	−	0.950159i	$-0.399079\pi$
$359$	1.52264e6			0.623533			0.311767	+	0.950159i	$0.399079\pi$
$360$		0			0
$361$	852788.			0.344408
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.43566e6	−	1.01972e6i	0.956941	−	0.400634i
$366$		0			0
$367$	−2.94842e6			−1.14268			−0.571339	−	0.820714i	$-0.693576\pi$
$367$	−2.94842e6			−1.14268			−0.571339	+	0.820714i	$0.693576\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.38727e6			0.523269
$372$		0			0
$373$		−	1.76054e6i		−	0.655201i	−0.944816	−	0.327600i	$-0.893760\pi$
$373$		−	1.76054e6i		−	0.655201i	0.944816	−	0.327600i	$-0.106240\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	202289.			0.0733024
$378$		0			0
$379$		−	458988.i		−	0.164136i	−0.996627	−	0.0820679i	$-0.973848\pi$
$379$		−	458988.i		−	0.164136i	0.996627	−	0.0820679i	$-0.0261524\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			7672.63i			0.00267268i	0.999999	+	0.00133634i	$0.000425371\pi$
$383$			7672.63i			0.00267268i	−0.999999	+	0.00133634i	$0.999575\pi$
$384$		0			0
$385$	1.12021e6	+	2.67570e6i	0.385166	+	0.919997i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	1.09208e6i		−	0.365914i	−0.983121	−	0.182957i	$-0.941433\pi$
$389$		−	1.09208e6i		−	0.365914i	0.983121	−	0.182957i	$-0.0585669\pi$
$390$		0			0
$391$		−	2.65623e6i		−	0.878665i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−19559.3	+	8188.73i	−0.00630756	+	0.00264073i
$396$		0			0
$397$		−	4.78165e6i		−	1.52265i	−0.648368	−	0.761327i	$-0.724548\pi$
$397$		−	4.78165e6i		−	1.52265i	0.648368	−	0.761327i	$-0.275452\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	997900.i		−	0.309903i	−0.987922	−	0.154952i	$-0.950478\pi$
$401$		−	997900.i		−	0.309903i	0.987922	−	0.154952i	$-0.0495222\pi$
$402$		0			0
$403$	−616865.			−0.189203
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.67941e6i			1.10101i
$408$		0			0
$409$	5.63443e6			1.66549			0.832745	−	0.553657i	$-0.186768\pi$
$409$	5.63443e6			1.66549			0.832745	+	0.553657i	$0.186768\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.92929e6			0.845060
$414$		0			0
$415$	1.58094e6	−	661875.i	0.450603	−	0.188650i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.99209e6			1.66741			0.833707	−	0.552208i	$-0.186214\pi$
$419$	5.99209e6			1.66741			0.833707	+	0.552208i	$0.186214\pi$
$420$		0			0
$421$	−4.09629e6			−1.12638			−0.563191	−	0.826327i	$-0.690426\pi$
$421$	−4.09629e6			−1.12638			−0.563191	+	0.826327i	$0.690426\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.48131e6	−	2.51921e6i	0.666359	−	0.676538i
$426$		0			0
$427$	3.06784e6			0.814259
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.20971e6			0.572983			0.286491	−	0.958083i	$-0.407511\pi$
$431$	2.20971e6			0.572983			0.286491	+	0.958083i	$0.407511\pi$
$432$		0			0
$433$			307349.i			0.0787792i	0.999224	+	0.0393896i	$0.0125413\pi$
$433$			307349.i			0.0787792i	−0.999224	+	0.0393896i	$0.987459\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.99091e6			−0.749203
$438$		0			0
$439$			2.26018e6i			0.559733i	0.960039	+	0.279866i	$0.0902902\pi$
$439$			2.26018e6i			0.559733i	−0.960039	+	0.279866i	$0.909710\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	2.49779e6i		−	0.604708i	−0.953196	−	0.302354i	$-0.902227\pi$
$443$		−	2.49779e6i		−	0.604708i	0.953196	−	0.302354i	$-0.0977725\pi$
$444$		0			0
$445$	−5.78699e6	+	2.42279e6i	−1.38533	+	0.579983i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			288514.i			0.0675385i	0.999430	+	0.0337692i	$0.0107511\pi$
$449$			288514.i			0.0675385i	−0.999430	+	0.0337692i	$0.989249\pi$
$450$		0			0
$451$		−	3.21282e6i		−	0.743782i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.66501e6	+	697075.i	−0.377041	+	0.157852i
$456$		0			0
$457$			212928.i			0.0476915i	0.999716	+	0.0238458i	$0.00759106\pi$
$457$			212928.i			0.0476915i	−0.999716	+	0.0238458i	$0.992409\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	2.17600e6i		−	0.476878i	−0.971157	−	0.238439i	$-0.923364\pi$
$461$		−	2.17600e6i		−	0.476878i	0.971157	−	0.238439i	$-0.0766357\pi$
$462$		0			0
$463$	129289.			0.0280291			0.0140146	−	0.999902i	$-0.495539\pi$
$463$	129289.			0.0280291			0.0140146	+	0.999902i	$0.495539\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			1.20715e6i			0.256134i	0.991765	+	0.128067i	$0.0408773\pi$
$467$			1.20715e6i			0.256134i	−0.991765	+	0.128067i	$0.959123\pi$
$468$		0			0
$469$	−7.49362e6			−1.57311
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.27549e6			−0.262135
$474$		0			0
$475$	−2.83663e6	−	2.79395e6i	−0.576858	−	0.568179i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.95921e6			1.38587			0.692933	−	0.721002i	$-0.256318\pi$
$479$	6.95921e6			1.38587			0.692933	+	0.721002i	$0.256318\pi$
$480$		0			0
$481$	−2.28959e6			−0.451226
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.34821e6	+	1.82043e6i	−0.839376	+	0.351414i
$486$		0			0
$487$	7.58622e6			1.44945			0.724725	−	0.689039i	$-0.241967\pi$
$487$	7.58622e6			1.44945			0.724725	+	0.689039i	$0.241967\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.51148e6			−0.470138			−0.235069	−	0.971979i	$-0.575532\pi$
$491$	−2.51148e6			−0.470138			−0.235069	+	0.971979i	$0.575532\pi$
$492$		0			0
$493$			1.19630e6i			0.221677i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.43368e6			−0.805144
$498$		0			0
$499$		−	9.81681e6i		−	1.76490i	−0.470410	−	0.882448i	$-0.655894\pi$
$499$		−	9.81681e6i		−	1.76490i	0.470410	−	0.882448i	$-0.344106\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.08728e7i			1.91611i	0.286584	+	0.958055i	$0.407480\pi$
$503$			1.08728e7i			1.91611i	−0.286584	+	0.958055i	$0.592520\pi$
$504$		0			0
$505$	−1.35702e6	+	568132.i	−0.236787	+	0.0991336i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	5.21618e6i		−	0.892396i	−0.894934	−	0.446198i	$-0.852778\pi$
$509$		−	5.21618e6i		−	0.892396i	0.894934	−	0.446198i	$-0.147222\pi$
$510$		0			0
$511$		−	7.97129e6i		−	1.35044i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	498682.	+	1.19114e6i	0.0828526	+	0.197899i
$516$		0			0
$517$		−	913022.i		−	0.150229i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	2.03638e6i		−	0.328673i	−0.986404	−	0.164337i	$-0.947452\pi$
$521$		−	2.03638e6i		−	0.328673i	0.986404	−	0.164337i	$-0.0525483\pi$
$522$		0			0
$523$	8.49706e6			1.35836			0.679180	−	0.733972i	$-0.262336\pi$
$523$	8.49706e6			1.35836			0.679180	+	0.733972i	$0.262336\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.64802e6i		−	0.572177i
$528$		0			0
$529$	925669.			0.143819
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.99924e6			0.304823
$534$		0			0
$535$	1.08662e7	−	4.54927e6i	1.64133	−	0.687159i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.58906e6			0.532119
$540$		0			0
$541$	−6.33145e6			−0.930059			−0.465029	−	0.885295i	$-0.653956\pi$
$541$	−6.33145e6			−0.930059			−0.465029	+	0.885295i	$0.653956\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.27853e6	+	7.83099e6i	0.472811	+	1.12934i
$546$		0			0
$547$	−4.90337e6			−0.700691			−0.350345	−	0.936621i	$-0.613936\pi$
$547$	−4.90337e6			−0.700691			−0.350345	+	0.936621i	$0.613936\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.34703e6			0.189016
$552$		0			0
$553$			64012.6i			0.00890129i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.90371e6			0.942855			0.471427	−	0.881905i	$-0.343739\pi$
$557$	6.90371e6			0.942855			0.471427	+	0.881905i	$0.343739\pi$
$558$		0			0
$559$		−	793702.i		−	0.107431i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.19446e7i		−	1.58818i	−0.607799	−	0.794091i	$-0.707948\pi$
$563$		−	1.19446e7i		−	1.58818i	0.607799	−	0.794091i	$-0.292052\pi$
$564$		0			0
$565$	−4.95332e6	−	1.18314e7i	−0.652793	−	1.55924i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	7.60046e6i		−	0.984145i	−0.870554	−	0.492072i	$-0.836240\pi$
$569$		−	7.60046e6i		−	0.984145i	0.870554	−	0.492072i	$-0.163760\pi$
$570$		0			0
$571$		−	1.03204e7i		−	1.32466i	−0.749211	−	0.662331i	$-0.769567\pi$
$571$		−	1.03204e7i		−	1.32466i	0.749211	−	0.662331i	$-0.230433\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.22642e6	−	5.14779e6i	−0.659227	−	0.649308i
$576$		0			0
$577$			2.93053e6i			0.366444i	0.983072	+	0.183222i	$0.0586527\pi$
$577$			2.93053e6i			0.366444i	−0.983072	+	0.183222i	$0.941347\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	5.17399e6i		−	0.635894i
$582$		0			0
$583$	2.52761e6			0.307992
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.00950e6i		−	0.120924i	−0.998171	−	0.0604619i	$-0.980743\pi$
$587$		−	1.00950e6i		−	0.120924i	0.998171	−	0.0604619i	$-0.0192574\pi$
$588$		0			0
$589$	−4.10767e6			−0.487873
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.24195e6			−0.962484			−0.481242	−	0.876588i	$-0.659814\pi$
$593$	−8.24195e6			−0.962484			−0.481242	+	0.876588i	$0.659814\pi$
$594$		0			0
$595$	−4.12236e6	−	9.84654e6i	−0.477368	−	1.14023i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.41100e7			1.60680			0.803399	−	0.595441i	$-0.203023\pi$
$599$	1.41100e7			1.60680			0.803399	+	0.595441i	$0.203023\pi$
$600$		0			0
$601$	6.91600e6			0.781032			0.390516	−	0.920596i	$-0.372297\pi$
$601$	6.91600e6			0.781032			0.390516	+	0.920596i	$0.372297\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.43577e6	−	3.42943e6i	−0.159476	−	0.380920i
$606$		0			0
$607$	507880.			0.0559486			0.0279743	−	0.999609i	$-0.491094\pi$
$607$	507880.			0.0559486			0.0279743	+	0.999609i	$0.491094\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	568146.			0.0615683
$612$		0			0
$613$		−	1.53663e7i		−	1.65165i	−0.563929	−	0.825824i	$-0.690711\pi$
$613$		−	1.53663e7i		−	1.65165i	0.563929	−	0.825824i	$-0.309289\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.74410e7			1.84442			0.922209	−	0.386693i	$-0.126383\pi$
$617$	1.74410e7			1.84442			0.922209	+	0.386693i	$0.126383\pi$
$618$		0			0
$619$			7.15644e6i			0.750707i	0.926882	+	0.375354i	$0.122479\pi$
$619$			7.15644e6i			0.750707i	−0.926882	+	0.375354i	$0.877521\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1.89393e7i			1.95499i
$624$		0			0
$625$	−148038.	−	9.76450e6i	−0.0151590	−	0.999885i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.35402e7i		−	1.36457i
$630$		0			0
$631$		−	8.45652e6i		−	0.845509i	−0.906244	−	0.422755i	$-0.861063\pi$
$631$		−	8.45652e6i		−	0.845509i	0.906244	−	0.422755i	$-0.138937\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.61941e6	+	1.34224e7i	0.553040	+	1.32097i
$636$		0			0
$637$			2.23337e6i			0.218078i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.88522e7i			1.81225i	0.423014	+	0.906123i	$0.360972\pi$
$641$			1.88522e7i			1.81225i	−0.423014	+	0.906123i	$0.639028\pi$
$642$		0			0
$643$	−3.95867e6			−0.377591			−0.188796	−	0.982016i	$-0.560458\pi$
$643$	−3.95867e6			−0.377591			−0.188796	+	0.982016i	$0.560458\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.10407e7i		−	1.03690i	−0.855109	−	0.518449i	$-0.826510\pi$
$647$		−	1.10407e7i		−	1.03690i	0.855109	−	0.518449i	$-0.173490\pi$
$648$		0			0
$649$	5.33720e6			0.497395
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.25634e6			0.849487			0.424743	−	0.905314i	$-0.360364\pi$
$653$	9.25634e6			0.849487			0.424743	+	0.905314i	$0.360364\pi$
$654$		0			0
$655$	−4.46967e6	−	1.06761e7i	−0.407073	−	0.972322i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.53010e7			1.37248			0.686242	−	0.727373i	$-0.259259\pi$
$659$	1.53010e7			1.37248			0.686242	+	0.727373i	$0.259259\pi$
$660$		0			0
$661$	−1.39464e7			−1.24153			−0.620767	−	0.783995i	$-0.713179\pi$
$661$	−1.39464e7			−1.24153			−0.620767	+	0.783995i	$0.713179\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.10872e7	+	4.64178e6i	−0.972227	+	0.407033i
$666$		0			0
$667$	2.48186e6			0.216005
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.58963e6			0.479266
$672$		0			0
$673$			1.12897e7i			0.960827i	0.877042	+	0.480414i	$0.159513\pi$
$673$			1.12897e7i			0.960827i	−0.877042	+	0.480414i	$0.840487\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.58998e6			0.720312			0.360156	−	0.932892i	$-0.382724\pi$
$677$	8.58998e6			0.720312			0.360156	+	0.932892i	$0.382724\pi$
$678$		0			0
$679$			1.42306e7i			1.18453i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.21190e6i			0.345482i	0.984967	+	0.172741i	$0.0552624\pi$
$683$			4.21190e6i			0.345482i	−0.984967	+	0.172741i	$0.944738\pi$
$684$		0			0
$685$	4.80646e6	+	1.14806e7i	0.391380	+	0.934839i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1.57286e6i			0.126224i
$690$		0			0
$691$			476000.i			0.0379238i	0.999820	+	0.0189619i	$0.00603613\pi$
$691$			476000.i			0.0379238i	−0.999820	+	0.0189619i	$0.993964\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.05298e7	+	4.40843e6i	−0.826913	+	0.346196i
$696$		0			0
$697$			1.18231e7i			0.921830i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.19544e7i		−	0.918828i	−0.888222	−	0.459414i	$-0.848059\pi$
$701$		−	1.19544e7i		−	0.918828i	0.888222	−	0.459414i	$-0.151941\pi$
$702$		0			0
$703$	−1.52462e7			−1.16352
$704$		0			0
$705$		0			0
$706$		0			0
$707$			4.44118e6i			0.334156i
$708$		0			0
$709$	−1.00045e7			−0.747443			−0.373721	−	0.927541i	$-0.621918\pi$
$709$	−1.00045e7			−0.747443			−0.373721	+	0.927541i	$0.621918\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.56827e6			−0.557536
$714$		0			0
$715$	−3.03367e6	+	1.27008e6i	−0.221923	+	0.0929106i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.07965e7			0.778860			0.389430	−	0.921056i	$-0.372672\pi$
$719$	1.07965e7			0.778860			0.389430	+	0.921056i	$0.372672\pi$
$720$		0			0
$721$	3.89828e6			0.279277
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.35384e6	+	2.31843e6i	0.166316	+	0.163813i
$726$		0			0
$727$	1.66845e7			1.17078			0.585391	−	0.810751i	$-0.300941\pi$
$727$	1.66845e7			1.17078			0.585391	+	0.810751i	$0.300941\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.69379e6			0.324886
$732$		0			0
$733$		−	8.44284e6i		−	0.580402i	−0.956966	−	0.290201i	$-0.906278\pi$
$733$		−	8.44284e6i		−	0.580402i	0.956966	−	0.290201i	$-0.0937221\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.36535e7			−0.925921
$738$		0			0
$739$			1.82932e7i			1.23219i	0.787671	+	0.616096i	$0.211287\pi$
$739$			1.82932e7i			1.23219i	−0.787671	+	0.616096i	$0.788713\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.85490e6i		−	0.521998i	−0.965339	−	0.260999i	$-0.915948\pi$
$743$		−	7.85490e6i		−	0.521998i	0.965339	−	0.260999i	$-0.0840519\pi$
$744$		0			0
$745$	3.11264e6	−	1.30314e6i	0.205465	−	0.0860202i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	3.55624e7i		−	2.31625i
$750$		0			0
$751$			1.12755e7i			0.729517i	0.931102	+	0.364759i	$0.118848\pi$
$751$			1.12755e7i			0.729517i	−0.931102	+	0.364759i	$0.881152\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.59503e7	+	1.08644e7i	−1.65682	+	0.693645i
$756$		0			0
$757$			1.24488e7i			0.789564i	0.918775	+	0.394782i	$0.129180\pi$
$757$			1.24488e7i			0.789564i	−0.918775	+	0.394782i	$0.870820\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1.23480e7i			0.772922i	0.922306	+	0.386461i	$0.126303\pi$
$761$			1.23480e7i			0.772922i	−0.922306	+	0.386461i	$0.873697\pi$
$762$		0			0
$763$	2.56288e7			1.59374
$764$		0			0
$765$		0			0
$766$		0			0
$767$			3.32118e6i			0.203847i
$768$		0			0
$769$	2.37874e7			1.45054			0.725272	−	0.688462i	$-0.241714\pi$
$769$	2.37874e7			1.45054			0.725272	+	0.688462i	$0.241714\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.99739e6			−0.240618			−0.120309	−	0.992737i	$-0.538389\pi$
$773$	−3.99739e6			−0.240618			−0.120309	+	0.992737i	$0.538389\pi$
$774$		0			0
$775$	−7.17788e6	−	7.06988e6i	−0.429281	−	0.422822i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.33128e7			0.786008
$780$		0			0
$781$	−8.07822e6			−0.473901
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.88193e7	+	7.87889e6i	−1.09000	+	0.456342i
$786$		0			0
$787$	7.39354e6			0.425516			0.212758	−	0.977105i	$-0.431755\pi$
$787$	7.39354e6			0.425516			0.212758	+	0.977105i	$0.431755\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.87209e7			−2.20042
$792$		0			0
$793$			3.47826e6i			0.196417i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.15520e7			1.20183			0.600913	−	0.799315i	$-0.294804\pi$
$797$	2.15520e7			1.20183			0.600913	+	0.799315i	$0.294804\pi$
$798$		0			0
$799$			3.35990e6i			0.186192i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.45238e7i		−	0.794860i
$804$		0			0
$805$	−2.04279e7	+	8.55235e6i	−1.11105	+	0.465153i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.54764e7i			0.831377i	0.909507	+	0.415689i	$0.136459\pi$
$809$			1.54764e7i			0.831377i	−0.909507	+	0.415689i	$0.863541\pi$
$810$		0			0
$811$		−	1.99618e6i		−	0.106573i	−0.998579	−	0.0532864i	$-0.983030\pi$
$811$		−	1.99618e6i		−	0.106573i	0.998579	−	0.0532864i	$-0.0169696\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.54099e6	−	2.27893e7i	−0.503152	−	1.20181i
$816$		0			0
$817$		−	5.28521e6i		−	0.277017i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			5.70697e6i			0.295494i	0.989025	+	0.147747i	$0.0472021\pi$
$821$			5.70697e6i			0.295494i	−0.989025	+	0.147747i	$0.952798\pi$
$822$		0			0
$823$	−3.37790e7			−1.73839			−0.869195	−	0.494469i	$-0.835362\pi$
$823$	−3.37790e7			−1.73839			−0.869195	+	0.494469i	$0.835362\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.19876e6i			0.162637i	0.996688	+	0.0813183i	$0.0259130\pi$
$827$			3.19876e6i			0.162637i	−0.996688	+	0.0813183i	$0.974087\pi$
$828$		0			0
$829$	1.92356e7			0.972118			0.486059	−	0.873926i	$-0.338434\pi$
$829$	1.92356e7			0.972118			0.486059	+	0.873926i	$0.338434\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.32077e7			−0.659499
$834$		0			0
$835$	919603.	−	385001.i	0.0456440	−	0.0191094i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.29251e7			0.633912			0.316956	−	0.948440i	$-0.397339\pi$
$839$	1.29251e7			0.633912			0.316956	+	0.948440i	$0.397339\pi$
$840$		0			0
$841$	1.93934e7			0.945504
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.22523e6	+	1.72580e7i	0.348105	+	0.831472i
$846$		0			0
$847$	−1.12236e7			−0.537558
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.80908e7			−1.32966
$852$		0			0
$853$			2.63935e6i			0.124201i	0.998070	+	0.0621003i	$0.0197799\pi$
$853$			2.63935e6i			0.124201i	−0.998070	+	0.0621003i	$0.980220\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	411706.			0.0191485			0.00957425	−	0.999954i	$-0.496952\pi$
$857$	411706.			0.0191485			0.00957425	+	0.999954i	$0.496952\pi$
$858$		0			0
$859$			4.12735e7i			1.90848i	0.299039	+	0.954241i	$0.403334\pi$
$859$			4.12735e7i			1.90848i	−0.299039	+	0.954241i	$0.596666\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.73896e7i		−	0.794808i	−0.917644	−	0.397404i	$-0.869911\pi$
$863$		−	1.73896e7i		−	0.794808i	0.917644	−	0.397404i	$-0.130089\pi$
$864$		0			0
$865$	−1.10386e6	−	2.63664e6i	−0.0501617	−	0.119815i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			116632.i			0.00523922i
$870$		0			0
$871$		−	8.49614e6i		−	0.379469i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.73633e7	−	1.09715e7i	−1.20823	−	0.484445i
$876$		0			0
$877$		−	2.05555e7i		−	0.902462i	−0.892407	−	0.451231i	$-0.850985\pi$
$877$		−	2.05555e7i		−	0.902462i	0.892407	−	0.451231i	$-0.149015\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			3.61850e7i			1.57069i	0.619061	+	0.785343i	$0.287513\pi$
$881$			3.61850e7i			1.57069i	−0.619061	+	0.785343i	$0.712487\pi$
$882$		0			0
$883$	1.64125e6			0.0708389			0.0354195	−	0.999373i	$-0.488723\pi$
$883$	1.64125e6			0.0708389			0.0354195	+	0.999373i	$0.488723\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.10707e7i		−	0.472462i	−0.971697	−	0.236231i	$-0.924088\pi$
$887$		−	1.10707e7i		−	0.472462i	0.971697	−	0.236231i	$-0.0759122\pi$
$888$		0			0
$889$	4.39279e7			1.86417
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.78325e6			0.158758
$894$		0			0
$895$	−575734.	−	1.37518e6i	−0.0240251	−	0.0573855i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.40855e6			0.140660
$900$		0			0
$901$	−9.30157e6			−0.381719
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.18936e7	+	2.84087e7i	0.482717	+	1.15300i
$906$		0			0
$907$	−3.70573e7			−1.49574			−0.747870	−	0.663845i	$-0.768923\pi$
$907$	−3.70573e7			−1.49574			−0.747870	+	0.663845i	$0.768923\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.29115e7			−0.515443			−0.257722	−	0.966219i	$-0.582972\pi$
$911$	−1.29115e7			−0.515443			−0.257722	+	0.966219i	$0.582972\pi$
$912$		0			0
$913$		−	9.42706e6i		−	0.374282i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.49401e7			−1.37215
$918$		0			0
$919$			4.12584e7i			1.61148i	0.592272	+	0.805738i	$0.298231\pi$
$919$			4.12584e7i			1.61148i	−0.592272	+	0.805738i	$0.701769\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	5.02683e6i		−	0.194218i
$924$		0			0
$925$	−2.66418e7	−	2.62409e7i	−1.02379	−	1.00838i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.15725e7i			0.439936i	0.975507	+	0.219968i	$0.0705952\pi$
$929$			1.15725e7i			0.439936i	−0.975507	+	0.219968i	$0.929405\pi$
$930$		0			0
$931$			1.48718e7i			0.562329i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7.51098e6	−	1.79405e7i	−0.280975	−	0.671128i
$936$		0			0
$937$		−	7.07224e6i		−	0.263153i	−0.991306	−	0.131576i	$-0.957996\pi$
$937$		−	7.07224e6i		−	0.263153i	0.991306	−	0.131576i	$-0.0420039\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	2.69230e7i		−	0.991174i	−0.868558	−	0.495587i	$-0.834953\pi$
$941$		−	2.69230e7i		−	0.991174i	0.868558	−	0.495587i	$-0.165047\pi$
$942$		0			0
$943$	2.45286e7			0.898241
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.65593e7i		−	0.600024i	−0.953935	−	0.300012i	$-0.903009\pi$
$947$		−	1.65593e7i		−	0.600024i	0.953935	−	0.300012i	$-0.0969906\pi$
$948$		0			0
$949$	9.03771e6			0.325756
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.32915e7			0.474070			0.237035	−	0.971501i	$-0.423824\pi$
$953$	1.32915e7			0.474070			0.237035	+	0.971501i	$0.423824\pi$
$954$		0			0
$955$	−1.23462e6	−	2.94897e6i	−0.0438050	−	0.104631i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.75729e7			1.31925
$960$		0			0
$961$	1.82350e7			0.636939
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.86098e7	−	1.19778e7i	0.989001	−	0.414056i
$966$		0			0
$967$	−2.62458e7			−0.902597			−0.451298	−	0.892373i	$-0.649039\pi$
$967$	−2.62458e7			−0.902597			−0.451298	+	0.892373i	$0.649039\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.99034e7			−1.69856			−0.849282	−	0.527939i	$-0.822965\pi$
$971$	−4.99034e7			−1.69856			−0.849282	+	0.527939i	$0.822965\pi$
$972$		0			0
$973$			3.44614e7i			1.16695i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.58429e7			−1.53651			−0.768256	−	0.640143i	$-0.778875\pi$
$977$	−4.58429e7			−1.53651			−0.768256	+	0.640143i	$0.778875\pi$
$978$		0			0
$979$			3.45076e7i			1.15069i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.68063e7i			0.884815i	0.896814	+	0.442408i	$0.145876\pi$
$983$			2.68063e7i			0.884815i	−0.896814	+	0.442408i	$0.854124\pi$
$984$		0			0
$985$	2.19342e7	+	5.23914e7i	0.720330	+	1.72056i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	9.73786e6i		−	0.316572i
$990$		0			0
$991$			4.99141e7i			1.61451i	0.590206	+	0.807253i	$0.299046\pi$
$991$			4.99141e7i			1.61451i	−0.590206	+	0.807253i	$0.700954\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.26646e7	+	2.20486e7i	−1.68640	+	0.706030i
$996$		0			0
$997$		−	3.45397e7i		−	1.10048i	−0.835007	−	0.550239i	$-0.814537\pi$
$997$		−	3.45397e7i		−	1.10048i	0.835007	−	0.550239i	$-0.185463\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.o.d.719.24	yes	40
3.2	odd	2	inner	720.6.o.d.719.18	yes	40
4.3	odd	2	inner	720.6.o.d.719.23	yes	40
5.4	even	2	inner	720.6.o.d.719.19	yes	40
12.11	even	2	inner	720.6.o.d.719.17	✓	40
15.14	odd	2	inner	720.6.o.d.719.21	yes	40
20.19	odd	2	inner	720.6.o.d.719.20	yes	40
60.59	even	2	inner	720.6.o.d.719.22	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.6.o.d.719.17	✓	40	12.11	even	2	inner
720.6.o.d.719.18	yes	40	3.2	odd	2	inner
720.6.o.d.719.19	yes	40	5.4	even	2	inner
720.6.o.d.719.20	yes	40	20.19	odd	2	inner
720.6.o.d.719.21	yes	40	15.14	odd	2	inner
720.6.o.d.719.22	yes	40	60.59	even	2	inner
720.6.o.d.719.23	yes	40	4.3	odd	2	inner
720.6.o.d.719.24	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(21.5882 + 51.5650i) q^{5} +168.759 q^{7} +O(q^{10})\)	\(q+(21.5882 + 51.5650i) q^{5} +168.759 q^{7} +307.480 q^{11} +191.336i q^{13} -1131.52 q^{17} +1274.09i q^{19} +2347.48i q^{23} +(-2192.90 + 2226.39i) q^{25} -1057.25i q^{29} +3223.99i q^{31} +(3643.20 + 8702.04i) q^{35} +11966.3i q^{37} -10448.9i q^{41} -4148.22 q^{43} -2969.37i q^{47} +11672.5 q^{49} +8220.41 q^{53} +(6637.95 + 15855.2i) q^{55} +17357.9 q^{59} +18178.8 q^{61} +(-9866.22 + 4130.60i) q^{65} -44404.4 q^{67} -26272.3 q^{71} -47234.8i q^{73} +51889.9 q^{77} +379.315i q^{79} -30659.1i q^{83} +(-24427.5 - 58346.9i) q^{85} +112227. i q^{89} +32289.6i q^{91} +(-65698.6 + 27505.4i) q^{95} +84324.9i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 10648 q^{25} + 36248 q^{49} - 133136 q^{61} - 336816 q^{85}+O(q^{100}) \) 40 * q + 10648 * q^25 + 36248 * q^49 - 133136 * q^61 - 336816 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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