LMFDB - Embedded newform 63.4.c.b.62.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [63,4,Mod(62,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.62"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	63.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.71712033036$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{111})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 112x^{2} + 3025 $ x^4 + 112*x^2 + 3025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			62.1
Root			$8.15694i$ of defining polynomial
Character	$\chi$	$=$	63.62
Dual form			63.4.c.b.62.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.82843i q^{2} -21.0713 q^{5} +(-11.0000 - 14.8997i) q^{7} -22.6274i q^{8} +59.5987i q^{10} -15.5563i q^{11} +29.7993i q^{13} +(-42.1426 + 31.1127i) q^{14} -64.0000 q^{16} +63.2139 q^{17} -89.3980i q^{19} -44.0000 q^{22} +77.7817i q^{23} +319.000 q^{25} +84.2852 q^{26} -125.865i q^{29} -238.395i q^{31} -178.796i q^{34} +(231.784 + 313.955i) q^{35} -184.000 q^{37} -252.856 q^{38} +476.789i q^{40} +105.357 q^{41} -190.000 q^{43} +220.000 q^{46} -42.1426 q^{47} +(-101.000 + 327.793i) q^{49} -902.268i q^{50} +357.796i q^{53} +327.793i q^{55} +(-337.141 + 248.902i) q^{56} -356.000 q^{58} -84.2852 q^{59} -655.585i q^{61} -674.282 q^{62} -512.000 q^{64} -627.911i q^{65} +296.000 q^{67} +(888.000 - 655.585i) q^{70} -329.512i q^{71} +804.582i q^{73} +520.431i q^{74} +(-231.784 + 171.120i) q^{77} +836.000 q^{79} +1348.56 q^{80} -297.993i q^{82} +1222.14 q^{83} -1332.00 q^{85} +537.401i q^{86} -352.000 q^{88} -695.353 q^{89} +(444.000 - 327.793i) q^{91} +119.197i q^{94} +1883.73i q^{95} -566.187i q^{97} +(927.138 + 285.671i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 44 q^{7} - 256 q^{16} - 176 q^{22} + 1276 q^{25} - 736 q^{37} - 760 q^{43} + 880 q^{46} - 404 q^{49} - 1424 q^{58} - 2048 q^{64} + 1184 q^{67} + 3552 q^{70} + 3344 q^{79} - 5328 q^{85} - 1408 q^{88}+ \cdots + 1776 q^{91}+O(q^{100}) $ 4 * q - 44 * q^7 - 256 * q^16 - 176 * q^22 + 1276 * q^25 - 736 * q^37 - 760 * q^43 + 880 * q^46 - 404 * q^49 - 1424 * q^58 - 2048 * q^64 + 1184 * q^67 + 3552 * q^70 + 3344 * q^79 - 5328 * q^85 - 1408 * q^88 + 1776 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.82843i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$		−	2.82843i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−21.0713			−1.88468			−0.942338	−	0.334664i	$-0.891377\pi$
$5$	−21.0713			−1.88468			−0.942338	+	0.334664i	$0.891377\pi$
$6$		0			0
$7$	−11.0000	−	14.8997i	−0.593944	−	0.804506i
$7$	−11.0000	−	14.8997i	−0.593944	−	0.804506i
$8$		−	22.6274i		−	1.00000i
$9$		0			0
$10$			59.5987i			1.88468i
$11$		−	15.5563i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	15.5563i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$			29.7993i			0.635757i	0.948131	+	0.317879i	$0.102970\pi$
$13$			29.7993i			0.635757i	−0.948131	+	0.317879i	$0.897030\pi$
$14$	−42.1426	+	31.1127i	−0.804506	+	0.593944i
$15$		0			0
$16$	−64.0000			−1.00000
$17$	63.2139			0.901860			0.450930	−	0.892559i	$-0.351092\pi$
$17$	63.2139			0.901860			0.450930	+	0.892559i	$0.351092\pi$
$18$		0			0
$19$		−	89.3980i		−	1.07944i	−0.841846	−	0.539719i	$-0.818531\pi$
$19$		−	89.3980i		−	1.07944i	0.841846	−	0.539719i	$-0.181469\pi$
$20$		0			0
$21$		0			0
$22$	−44.0000			−0.426401
$23$			77.7817i			0.705157i	0.935782	+	0.352579i	$0.114695\pi$
$23$			77.7817i			0.705157i	−0.935782	+	0.352579i	$0.885305\pi$
$24$		0			0
$25$	319.000			2.55200
$26$	84.2852			0.635757
$27$		0			0
$28$		0			0
$29$		−	125.865i		−	0.805950i	−0.915211	−	0.402975i	$-0.867976\pi$
$29$		−	125.865i		−	0.805950i	0.915211	−	0.402975i	$-0.132024\pi$
$30$		0			0
$31$		−	238.395i		−	1.38119i	−0.723241	−	0.690596i	$-0.757348\pi$
$31$		−	238.395i		−	1.38119i	0.723241	−	0.690596i	$-0.242652\pi$
$32$		0			0
$33$		0			0
$34$		−	178.796i		−	0.901860i
$35$	231.784	+	313.955i	1.11939	+	1.51623i
$36$		0			0
$37$	−184.000			−0.817552			−0.408776	−	0.912635i	$-0.634044\pi$
$37$	−184.000			−0.817552			−0.408776	+	0.912635i	$0.634044\pi$
$38$	−252.856			−1.07944
$39$		0			0
$40$			476.789i			1.88468i
$41$	105.357			0.401315			0.200658	−	0.979661i	$-0.435692\pi$
$41$	105.357			0.401315			0.200658	+	0.979661i	$0.435692\pi$
$42$		0			0
$43$	−190.000			−0.673831			−0.336915	−	0.941535i	$-0.609384\pi$
$43$	−190.000			−0.673831			−0.336915	+	0.941535i	$0.609384\pi$
$44$		0			0
$45$		0			0
$46$	220.000			0.705157
$47$	−42.1426			−0.130790			−0.0653950	−	0.997859i	$-0.520831\pi$
$47$	−42.1426			−0.130790			−0.0653950	+	0.997859i	$0.520831\pi$
$48$		0			0
$49$	−101.000	+	327.793i	−0.294461	+	0.955664i
$50$		−	902.268i		−	2.55200i
$51$		0			0
$52$		0			0
$53$			357.796i			0.927303i	0.886018	+	0.463652i	$0.153461\pi$
$53$			357.796i			0.927303i	−0.886018	+	0.463652i	$0.846539\pi$
$54$		0			0
$55$			327.793i			0.803628i
$56$	−337.141	+	248.902i	−0.804506	+	0.593944i
$57$		0			0
$58$	−356.000			−0.805950
$59$	−84.2852			−0.185983			−0.0929915	−	0.995667i	$-0.529643\pi$
$59$	−84.2852			−0.185983			−0.0929915	+	0.995667i	$0.529643\pi$
$60$		0			0
$61$		−	655.585i		−	1.37605i	−0.725687	−	0.688025i	$-0.758478\pi$
$61$		−	655.585i		−	1.37605i	0.725687	−	0.688025i	$-0.241522\pi$
$62$	−674.282			−1.38119
$63$		0			0
$64$	−512.000			−1.00000
$65$		−	627.911i		−	1.19820i
$66$		0			0
$67$	296.000			0.539734			0.269867	−	0.962898i	$-0.413020\pi$
$67$	296.000			0.539734			0.269867	+	0.962898i	$0.413020\pi$
$68$		0			0
$69$		0			0
$70$	888.000	−	655.585i	1.51623	−	1.11939i
$71$		−	329.512i		−	0.550787i	−0.961332	−	0.275393i	$-0.911192\pi$
$71$		−	329.512i		−	0.550787i	0.961332	−	0.275393i	$-0.0888081\pi$
$72$		0			0
$73$			804.582i			1.28999i	0.764187	+	0.644994i	$0.223140\pi$
$73$			804.582i			1.28999i	−0.764187	+	0.644994i	$0.776860\pi$
$74$			520.431i			0.817552i
$75$		0			0
$76$		0			0
$77$	−231.784	+	171.120i	−0.343043	+	0.253259i
$78$		0			0
$79$	836.000			1.19060			0.595300	−	0.803504i	$-0.297033\pi$
$79$	836.000			1.19060			0.595300	+	0.803504i	$0.297033\pi$
$80$	1348.56			1.88468
$81$		0			0
$82$		−	297.993i		−	0.401315i
$83$	1222.14			1.61623			0.808113	−	0.589027i	$-0.200489\pi$
$83$	1222.14			1.61623			0.808113	+	0.589027i	$0.200489\pi$
$84$		0			0
$85$	−1332.00			−1.69971
$86$			537.401i			0.673831i
$87$		0			0
$88$	−352.000			−0.426401
$89$	−695.353			−0.828172			−0.414086	−	0.910238i	$-0.635899\pi$
$89$	−695.353			−0.828172			−0.414086	+	0.910238i	$0.635899\pi$
$90$		0			0
$91$	444.000	−	327.793i	0.511471	−	0.377604i
$92$		0			0
$93$		0			0
$94$			119.197i			0.130790i
$95$			1883.73i			2.03439i
$96$		0			0
$97$		−	566.187i		−	0.592656i	−0.955086	−	0.296328i	$-0.904238\pi$
$97$		−	566.187i		−	0.592656i	0.955086	−	0.296328i	$-0.0957621\pi$
$98$	927.138	+	285.671i	0.955664	+	0.294461i
$99$		0			0
$100$		0			0
$101$	−737.496			−0.726570			−0.363285	−	0.931678i	$-0.618345\pi$
$101$	−737.496			−0.726570			−0.363285	+	0.931678i	$0.618345\pi$
$102$		0			0
$103$			655.585i			0.627153i	0.949563	+	0.313576i	$0.101527\pi$
$103$			655.585i			0.627153i	−0.949563	+	0.313576i	$0.898473\pi$
$104$	674.282			0.635757
$105$		0			0
$106$	1012.00			0.927303
$107$		−	1984.14i		−	1.79266i	−0.443391	−	0.896328i	$-0.646225\pi$
$107$		−	1984.14i		−	1.79266i	0.443391	−	0.896328i	$-0.353775\pi$
$108$		0			0
$109$	−844.000			−0.741656			−0.370828	−	0.928702i	$-0.620926\pi$
$109$	−844.000			−0.741656			−0.370828	+	0.928702i	$0.620926\pi$
$110$	927.138			0.803628
$111$		0			0
$112$	704.000	+	953.579i	0.593944	+	0.804506i
$113$		−	575.585i		−	0.479172i	−0.970875	−	0.239586i	$-0.922988\pi$
$113$		−	575.585i		−	0.479172i	0.970875	−	0.239586i	$-0.0770118\pi$
$114$		0			0
$115$		−	1638.96i		−	1.32899i
$116$		0			0
$117$		0			0
$118$			238.395i			0.185983i
$119$	−695.353	−	941.866i	−0.535655	−	0.725552i
$120$		0			0
$121$	1089.00			0.818182
$122$	−1854.28			−1.37605
$123$		0			0
$124$		0			0
$125$	−4087.83			−2.92502
$126$		0			0
$127$	−220.000			−0.153715			−0.0768577	−	0.997042i	$-0.524489\pi$
$127$	−220.000			−0.153715			−0.0768577	+	0.997042i	$0.524489\pi$
$128$			1448.15i			1.00000i
$129$		0			0
$130$	−1776.00			−1.19820
$131$	1306.42			0.871317			0.435659	−	0.900112i	$-0.356516\pi$
$131$	1306.42			0.871317			0.435659	+	0.900112i	$0.356516\pi$
$132$		0			0
$133$	−1332.00	+	983.378i	−0.868414	+	0.641125i
$134$		−	837.214i		−	0.539734i
$135$		0			0
$136$		−	1430.37i		−	0.901860i
$137$		−	1568.36i		−	0.978060i	−0.872267	−	0.489030i	$-0.837351\pi$
$137$		−	1568.36i		−	0.978060i	0.872267	−	0.489030i	$-0.162649\pi$
$138$		0			0
$139$			655.585i			0.400043i	0.979791	+	0.200022i	$0.0641012\pi$
$139$			655.585i			0.400043i	−0.979791	+	0.200022i	$0.935899\pi$
$140$		0			0
$141$		0			0
$142$	−932.000			−0.550787
$143$	463.569			0.271088
$144$		0			0
$145$			2652.14i			1.51895i
$146$	2275.70			1.28999
$147$		0			0
$148$		0			0
$149$		−	1466.54i		−	0.806333i	−0.915127	−	0.403166i	$-0.867910\pi$
$149$		−	1466.54i		−	0.806333i	0.915127	−	0.403166i	$-0.132090\pi$
$150$		0			0
$151$	1970.00			1.06170			0.530849	−	0.847467i	$-0.321873\pi$
$151$	1970.00			1.06170			0.530849	+	0.847467i	$0.321873\pi$
$152$	−2022.85			−1.07944
$153$		0			0
$154$	484.000	+	655.585i	0.253259	+	0.343043i
$155$			5023.29i			2.60310i
$156$		0			0
$157$			2622.34i			1.33303i	0.745492	+	0.666515i	$0.232215\pi$
$157$			2622.34i			1.33303i	−0.745492	+	0.666515i	$0.767785\pi$
$158$		−	2364.57i		−	1.19060i
$159$		0			0
$160$		0			0
$161$	1158.92	−	855.599i	0.567303	−	0.418824i
$162$		0			0
$163$	−1336.00			−0.641985			−0.320993	−	0.947082i	$-0.604016\pi$
$163$	−1336.00			−0.641985			−0.320993	+	0.947082i	$0.604016\pi$
$164$		0			0
$165$		0			0
$166$		−	3456.72i		−	1.61623i
$167$	2317.84			1.07401			0.537006	−	0.843578i	$-0.319555\pi$
$167$	2317.84			1.07401			0.537006	+	0.843578i	$0.319555\pi$
$168$		0			0
$169$	1309.00			0.595812
$170$			3767.46i			1.69971i
$171$		0			0
$172$		0			0
$173$	−231.784			−0.101863			−0.0509313	−	0.998702i	$-0.516219\pi$
$173$	−231.784			−0.101863			−0.0509313	+	0.998702i	$0.516219\pi$
$174$		0			0
$175$	−3509.00	−	4752.99i	−1.51575	−	2.05310i
$176$			995.606i			0.426401i
$177$		0			0
$178$			1966.76i			0.828172i
$179$		−	108.894i		−	0.0454701i	−0.999742	−	0.0227351i	$-0.992763\pi$
$179$		−	108.894i		−	0.0454701i	0.999742	−	0.0227351i	$-0.00723742\pi$
$180$		0			0
$181$		−	2592.54i		−	1.06465i	−0.846539	−	0.532326i	$-0.821318\pi$
$181$		−	2592.54i		−	1.06465i	0.846539	−	0.532326i	$-0.178682\pi$
$182$	−927.138	−	1255.82i	−0.377604	−	0.511471i
$183$		0			0
$184$	1760.00			0.705157
$185$	3877.12			1.54082
$186$		0			0
$187$		−	983.378i		−	0.384555i
$188$		0			0
$189$		0			0
$190$	5328.00			2.03439
$191$			2224.56i			0.842740i	0.906889	+	0.421370i	$0.138451\pi$
$191$			2224.56i			0.842740i	−0.906889	+	0.421370i	$0.861549\pi$
$192$		0			0
$193$	3740.00			1.39488			0.697438	−	0.716645i	$-0.254323\pi$
$193$	3740.00			1.39488			0.697438	+	0.716645i	$0.254323\pi$
$194$	−1601.42			−0.592656
$195$		0			0
$196$		0			0
$197$			1197.84i			0.433211i	0.976259	+	0.216605i	$0.0694985\pi$
$197$			1197.84i			0.433211i	−0.976259	+	0.216605i	$0.930502\pi$
$198$		0			0
$199$		−	804.582i		−	0.286610i	−0.989679	−	0.143305i	$-0.954227\pi$
$199$		−	804.582i		−	0.286610i	0.989679	−	0.143305i	$-0.0457729\pi$
$200$		−	7218.15i		−	2.55200i
$201$		0			0
$202$			2085.95i			0.726570i
$203$	−1875.35	+	1384.52i	−0.648392	+	0.478689i
$204$		0			0
$205$	−2220.00			−0.756349
$206$	1854.28			0.627153
$207$		0			0
$208$		−	1907.16i		−	0.635757i
$209$	−1390.71			−0.460274
$210$		0			0
$211$	3590.00			1.17131			0.585654	−	0.810561i	$-0.300838\pi$
$211$	3590.00			1.17131			0.585654	+	0.810561i	$0.300838\pi$
$212$		0			0
$213$		0			0
$214$	−5612.00			−1.79266
$215$	4003.55			1.26995
$216$		0			0
$217$	−3552.00	+	2622.34i	−1.11118	+	0.820351i
$218$			2387.19i			0.741656i
$219$		0			0
$220$		0			0
$221$			1883.73i			0.573365i
$222$		0			0
$223$			3009.73i			0.903796i	0.892070	+	0.451898i	$0.149253\pi$
$223$			3009.73i			0.903796i	−0.892070	+	0.451898i	$0.850747\pi$
$224$		0			0
$225$		0			0
$226$	−1628.00			−0.479172
$227$	−3708.55			−1.08434			−0.542170	−	0.840269i	$-0.682397\pi$
$227$	−3708.55			−1.08434			−0.542170	+	0.840269i	$0.682397\pi$
$228$		0			0
$229$		−	327.793i		−	0.0945902i	−0.998881	−	0.0472951i	$-0.984940\pi$
$229$		−	327.793i		−	0.0945902i	0.998881	−	0.0472951i	$-0.0150601\pi$
$230$	−4635.69			−1.32899
$231$		0			0
$232$	−2848.00			−0.805950
$233$		−	643.467i		−	0.180922i	−0.995900	−	0.0904612i	$-0.971166\pi$
$233$		−	643.467i		−	0.180922i	0.995900	−	0.0904612i	$-0.0288341\pi$
$234$		0			0
$235$	888.000			0.246497
$236$		0			0
$237$		0			0
$238$	−2664.00	+	1966.76i	−0.725552	+	0.535655i
$239$			646.296i			0.174918i	0.996168	+	0.0874590i	$0.0278747\pi$
$239$			646.296i			0.174918i	−0.996168	+	0.0874590i	$0.972125\pi$
$240$		0			0
$241$		−	387.391i		−	0.103544i	−0.998659	−	0.0517719i	$-0.983513\pi$
$241$		−	387.391i		−	0.103544i	0.998659	−	0.0517719i	$-0.0164869\pi$
$242$		−	3080.16i		−	0.818182i
$243$		0			0
$244$		0			0
$245$	2128.20	−	6907.02i	0.554963	−	1.80112i
$246$		0			0
$247$	2664.00			0.686260
$248$	−5394.25			−1.38119
$249$		0			0
$250$			11562.1i			2.92502i
$251$	6194.96			1.55786			0.778930	−	0.627111i	$-0.215763\pi$
$251$	6194.96			1.55786			0.778930	+	0.627111i	$0.215763\pi$
$252$		0			0
$253$	1210.00			0.300680
$254$			622.254i			0.153715i
$255$		0			0
$256$		0			0
$257$	−3561.05			−0.864328			−0.432164	−	0.901795i	$-0.642250\pi$
$257$	−3561.05			−0.864328			−0.432164	+	0.901795i	$0.642250\pi$
$258$		0			0
$259$	2024.00	+	2741.54i	0.485580	+	0.657725i
$260$		0			0
$261$		0			0
$262$		−	3695.12i		−	0.871317i
$263$		−	601.041i		−	0.140919i	−0.997515	−	0.0704596i	$-0.977553\pi$
$263$		−	601.041i		−	0.140919i	0.997515	−	0.0704596i	$-0.0224466\pi$
$264$		0			0
$265$		−	7539.23i		−	1.74767i
$266$	2781.41	+	3767.46i	0.641125	+	0.868414i
$267$		0			0
$268$		0			0
$269$	−4867.47			−1.10325			−0.551626	−	0.834091i	$-0.685993\pi$
$269$	−4867.47			−1.10325			−0.551626	+	0.834091i	$0.685993\pi$
$270$		0			0
$271$		−	1787.96i		−	0.400778i	−0.979716	−	0.200389i	$-0.935779\pi$
$271$		−	1787.96i		−	0.400778i	0.979716	−	0.200389i	$-0.0642206\pi$
$272$	−4045.69			−0.901860
$273$		0			0
$274$	−4436.00			−0.978060
$275$		−	4962.48i		−	1.08818i
$276$		0			0
$277$	−1126.00			−0.244241			−0.122121	−	0.992515i	$-0.538969\pi$
$277$	−1126.00			−0.244241			−0.122121	+	0.992515i	$0.538969\pi$
$278$	1854.28			0.400043
$279$		0			0
$280$	7104.00	−	5244.68i	1.51623	−	1.11939i
$281$			5075.61i			1.07753i	0.842456	+	0.538765i	$0.181109\pi$
$281$			5075.61i			1.07753i	−0.842456	+	0.538765i	$0.818891\pi$
$282$		0			0
$283$		−	7122.04i		−	1.49598i	−0.663712	−	0.747988i	$-0.731020\pi$
$283$		−	7122.04i		−	1.49598i	0.663712	−	0.747988i	$-0.268980\pi$
$284$		0			0
$285$		0			0
$286$		−	1311.17i		−	0.271088i
$287$	−1158.92	−	1569.78i	−0.238359	−	0.322861i
$288$		0			0
$289$	−917.000			−0.186648
$290$	7501.39			1.51895
$291$		0			0
$292$		0			0
$293$	231.784			0.0462150			0.0231075	−	0.999733i	$-0.492644\pi$
$293$	231.784			0.0462150			0.0231075	+	0.999733i	$0.492644\pi$
$294$		0			0
$295$	1776.00			0.350518
$296$			4163.44i			0.817552i
$297$		0			0
$298$	−4148.00			−0.806333
$299$	−2317.84			−0.448309
$300$		0			0
$301$	2090.00	+	2830.94i	0.400218	+	0.542101i
$302$		−	5572.00i		−	1.06170i
$303$		0			0
$304$			5721.47i			1.07944i
$305$			13814.0i			2.59341i
$306$		0			0
$307$			8850.40i			1.64534i	0.568520	+	0.822669i	$0.307516\pi$
$307$			8850.40i			1.64534i	−0.568520	+	0.822669i	$0.692484\pi$
$308$		0			0
$309$		0			0
$310$	14208.0			2.60310
$311$	6489.96			1.18332			0.591659	−	0.806188i	$-0.298473\pi$
$311$	6489.96			1.18332			0.591659	+	0.806188i	$0.298473\pi$
$312$		0			0
$313$		−	10847.0i		−	1.95881i	−0.201916	−	0.979403i	$-0.564717\pi$
$313$		−	10847.0i		−	1.95881i	0.201916	−	0.979403i	$-0.435283\pi$
$314$	7417.10			1.33303
$315$		0			0
$316$		0			0
$317$		−	1780.49i		−	0.315465i	−0.987482	−	0.157733i	$-0.949582\pi$
$317$		−	1780.49i		−	0.315465i	0.987482	−	0.157733i	$-0.0504184\pi$
$318$		0			0
$319$	−1958.00			−0.343658
$320$	10788.5			1.88468
$321$		0			0
$322$	−2420.00	−	3277.93i	−0.418824	−	0.567303i
$323$		−	5651.20i		−	0.973502i
$324$		0			0
$325$			9505.99i			1.62245i
$326$			3778.78i			0.641985i
$327$		0			0
$328$		−	2383.95i		−	0.401315i
$329$	463.569	+	627.911i	0.0776820	+	0.105221i
$330$		0			0
$331$	−9526.00			−1.58186			−0.790931	−	0.611905i	$-0.790403\pi$
$331$	−9526.00			−1.58186			−0.790931	+	0.611905i	$0.790403\pi$
$332$		0			0
$333$		0			0
$334$		−	6555.85i		−	1.07401i
$335$	−6237.11			−1.01722
$336$		0			0
$337$	−8272.00			−1.33711			−0.668553	−	0.743665i	$-0.733086\pi$
$337$	−8272.00			−1.33711			−0.668553	+	0.743665i	$0.733086\pi$
$338$		−	3702.41i		−	0.595812i
$339$		0			0
$340$		0			0
$341$	−3708.55			−0.588942
$342$		0			0
$343$	5995.00	−	2100.85i	0.943731	−	0.330715i
$344$			4299.21i			0.673831i
$345$		0			0
$346$			655.585i			0.101863i
$347$			9784.94i			1.51378i	0.653540	+	0.756892i	$0.273283\pi$
$347$			9784.94i			1.51378i	−0.653540	+	0.756892i	$0.726717\pi$
$348$		0			0
$349$			6317.46i			0.968956i	0.874803	+	0.484478i	$0.160990\pi$
$349$			6317.46i			0.968956i	−0.874803	+	0.484478i	$0.839010\pi$
$350$	−13443.5	+	9924.95i	−2.05310	+	1.51575i
$351$		0			0
$352$		0			0
$353$	5457.47			0.822866			0.411433	−	0.911440i	$-0.365028\pi$
$353$	5457.47			0.822866			0.411433	+	0.911440i	$0.365028\pi$
$354$		0			0
$355$			6943.24i			1.03805i
$356$		0			0
$357$		0			0
$358$	−308.000			−0.0454701
$359$		−	10842.8i		−	1.59404i	−0.603954	−	0.797019i	$-0.706409\pi$
$359$		−	10842.8i		−	1.59404i	0.603954	−	0.797019i	$-0.293591\pi$
$360$		0			0
$361$	−1133.00			−0.165184
$362$	−7332.82			−1.06465
$363$		0			0
$364$		0			0
$365$		−	16953.6i		−	2.43121i
$366$		0			0
$367$			3724.92i			0.529807i	0.964275	+	0.264903i	$0.0853400\pi$
$367$			3724.92i			0.529807i	−0.964275	+	0.264903i	$0.914660\pi$
$368$		−	4978.03i		−	0.705157i
$369$		0			0
$370$		−	10966.2i		−	1.54082i
$371$	5331.04	−	3935.76i	0.746021	−	0.550766i
$372$		0			0
$373$	4202.00			0.583301			0.291651	−	0.956525i	$-0.405796\pi$
$373$	4202.00			0.583301			0.291651	+	0.956525i	$0.405796\pi$
$374$	−2781.41			−0.384555
$375$		0			0
$376$			953.579i			0.130790i
$377$	3750.69			0.512389
$378$		0			0
$379$	−2506.00			−0.339643			−0.169821	−	0.985475i	$-0.554319\pi$
$379$	−2506.00			−0.339643			−0.169821	+	0.985475i	$0.554319\pi$
$380$		0			0
$381$		0			0
$382$	6292.00			0.842740
$383$	−13106.4			−1.74857			−0.874286	−	0.485410i	$-0.838670\pi$
$383$	−13106.4			−1.74857			−0.874286	+	0.485410i	$0.838670\pi$
$384$		0			0
$385$	4884.00	−	3605.72i	0.646524	−	0.477310i
$386$		−	10578.3i		−	1.39488i
$387$		0			0
$388$		0			0
$389$		−	4631.55i		−	0.603673i	−0.953360	−	0.301837i	$-0.902400\pi$
$389$		−	4631.55i		−	0.603673i	0.953360	−	0.301837i	$-0.0975997\pi$
$390$		0			0
$391$			4916.89i			0.635953i
$392$	7417.10	+	2285.37i	0.955664	+	0.294461i
$393$		0			0
$394$	3388.00			0.433211
$395$	−17615.6			−2.24389
$396$		0			0
$397$			8045.82i			1.01715i	0.861018	+	0.508574i	$0.169827\pi$
$397$			8045.82i			1.01715i	−0.861018	+	0.508574i	$0.830173\pi$
$398$	−2275.70			−0.286610
$399$		0			0
$400$	−20416.0			−2.55200
$401$			13552.4i			1.68772i	0.536565	+	0.843859i	$0.319722\pi$
$401$			13552.4i			1.68772i	−0.536565	+	0.843859i	$0.680278\pi$
$402$		0			0
$403$	7104.00			0.878103
$404$		0			0
$405$		0			0
$406$	3916.00	+	5304.28i	0.478689	+	0.648392i
$407$			2862.37i			0.348605i
$408$		0			0
$409$		−	10161.6i		−	1.22850i	−0.789111	−	0.614251i	$-0.789458\pi$
$409$		−	10161.6i		−	1.22850i	0.789111	−	0.614251i	$-0.210542\pi$
$410$			6279.11i			0.756349i
$411$		0			0
$412$		0			0
$413$	927.138	+	1255.82i	0.110464	+	0.149625i
$414$		0			0
$415$	−25752.0			−3.04606
$416$		0			0
$417$		0			0
$418$			3933.51i			0.460274i
$419$	−8934.23			−1.04168			−0.520842	−	0.853653i	$-0.674382\pi$
$419$	−8934.23			−1.04168			−0.520842	+	0.853653i	$0.674382\pi$
$420$		0			0
$421$	5606.00			0.648978			0.324489	−	0.945889i	$-0.394808\pi$
$421$	5606.00			0.648978			0.324489	+	0.945889i	$0.394808\pi$
$422$		−	10154.1i		−	1.17131i
$423$		0			0
$424$	8096.00			0.927303
$425$	20165.2			2.30155
$426$		0			0
$427$	−9768.00	+	7211.44i	−1.10704	+	0.817297i
$428$		0			0
$429$		0			0
$430$		−	11323.7i		−	1.26995i
$431$			883.883i			0.0987823i	0.998780	+	0.0493911i	$0.0157281\pi$
$431$			883.883i			0.0987823i	−0.998780	+	0.0493911i	$0.984272\pi$
$432$		0			0
$433$		−	1966.76i		−	0.218282i	−0.994026	−	0.109141i	$-0.965190\pi$
$433$		−	1966.76i		−	0.218282i	0.994026	−	0.109141i	$-0.0348101\pi$
$434$	7417.10	+	10046.6i	0.820351	+	1.11118i
$435$		0			0
$436$		0			0
$437$	6953.53			0.761173
$438$		0			0
$439$		−	387.391i		−	0.0421166i	−0.999778	−	0.0210583i	$-0.993296\pi$
$439$		−	387.391i		−	0.0421166i	0.999778	−	0.0210583i	$-0.00670356\pi$
$440$	7417.10			0.803628
$441$		0			0
$442$	5328.00			0.573365
$443$			2623.37i			0.281354i	0.990056	+	0.140677i	$0.0449279\pi$
$443$			2623.37i			0.281354i	−0.990056	+	0.140677i	$0.955072\pi$
$444$		0			0
$445$	14652.0			1.56083
$446$	8512.81			0.903796
$447$		0			0
$448$	5632.00	+	7628.63i	0.593944	+	0.804506i
$449$		−	5140.67i		−	0.540319i	−0.962816	−	0.270159i	$-0.912924\pi$
$449$		−	5140.67i		−	0.540319i	0.962816	−	0.270159i	$-0.0870764\pi$
$450$		0			0
$451$		−	1638.96i		−	0.171121i
$452$		0			0
$453$		0			0
$454$			10489.4i			1.08434i
$455$	−9355.66	+	6907.02i	−0.963956	+	0.711662i
$456$		0			0
$457$	3608.00			0.369311			0.184655	−	0.982803i	$-0.440883\pi$
$457$	3608.00			0.369311			0.184655	+	0.982803i	$0.440883\pi$
$458$	−927.138			−0.0945902
$459$		0			0
$460$		0			0
$461$	1538.21			0.155404			0.0777021	−	0.996977i	$-0.475242\pi$
$461$	1538.21			0.155404			0.0777021	+	0.996977i	$0.475242\pi$
$462$		0			0
$463$	1772.00			0.177866			0.0889329	−	0.996038i	$-0.471654\pi$
$463$	1772.00			0.177866			0.0889329	+	0.996038i	$0.471654\pi$
$464$			8055.36i			0.805950i
$465$		0			0
$466$	−1820.00			−0.180922
$467$	12769.2			1.26529			0.632643	−	0.774443i	$-0.281970\pi$
$467$	12769.2			1.26529			0.632643	+	0.774443i	$0.281970\pi$
$468$		0			0
$469$	−3256.00	−	4410.30i	−0.320572	−	0.434219i
$470$		−	2511.64i		−	0.246497i
$471$		0			0
$472$			1907.16i			0.185983i
$473$			2955.71i			0.287322i
$474$		0			0
$475$		−	28518.0i		−	2.75472i
$476$		0			0
$477$		0			0
$478$	1828.00			0.174918
$479$	4635.69			0.442192			0.221096	−	0.975252i	$-0.429037\pi$
$479$	4635.69			0.442192			0.221096	+	0.975252i	$0.429037\pi$
$480$		0			0
$481$		−	5483.08i		−	0.519765i
$482$	−1095.71			−0.103544
$483$		0			0
$484$		0			0
$485$			11930.3i			1.11696i
$486$		0			0
$487$	1958.00			0.182188			0.0910939	−	0.995842i	$-0.470964\pi$
$487$	1958.00			0.182188			0.0910939	+	0.995842i	$0.470964\pi$
$488$	−14834.2			−1.37605
$489$		0			0
$490$	−19536.0	−	6019.46i	−1.80112	−	0.554963i
$491$		−	7126.22i		−	0.654994i	−0.944852	−	0.327497i	$-0.893795\pi$
$491$		−	7126.22i		−	0.654994i	0.944852	−	0.327497i	$-0.106205\pi$
$492$		0			0
$493$		−	7956.42i		−	0.726854i
$494$		−	7534.93i		−	0.686260i
$495$		0			0
$496$			15257.3i			1.38119i
$497$	−4909.61	+	3624.63i	−0.443111	+	0.327137i
$498$		0			0
$499$	10310.0			0.924928			0.462464	−	0.886638i	$-0.346965\pi$
$499$	10310.0			0.924928			0.462464	+	0.886638i	$0.346965\pi$
$500$		0			0
$501$		0			0
$502$		−	17522.0i		−	1.55786i
$503$	126.428			0.0112070			0.00560352	−	0.999984i	$-0.498216\pi$
$503$	126.428			0.0112070			0.00560352	+	0.999984i	$0.498216\pi$
$504$		0			0
$505$	15540.0			1.36935
$506$		−	3422.40i		−	0.300680i
$507$		0			0
$508$		0			0
$509$	−7101.03			−0.618365			−0.309182	−	0.951003i	$-0.600055\pi$
$509$	−7101.03			−0.618365			−0.309182	+	0.951003i	$0.600055\pi$
$510$		0			0
$511$	11988.0	−	8850.40i	1.03780	−	0.766181i
$512$			11585.2i			1.00000i
$513$		0			0
$514$			10072.2i			0.864328i
$515$		−	13814.0i		−	1.18198i
$516$		0			0
$517$			655.585i			0.0557691i
$518$	7754.24	−	5724.74i	0.657725	−	0.485580i
$519$		0			0
$520$	−14208.0			−1.19820
$521$	4488.19			0.377411			0.188705	−	0.982034i	$-0.439571\pi$
$521$	4488.19			0.377411			0.188705	+	0.982034i	$0.439571\pi$
$522$		0			0
$523$			9297.39i			0.777336i	0.921378	+	0.388668i	$0.127065\pi$
$523$			9297.39i			0.777336i	−0.921378	+	0.388668i	$0.872935\pi$
$524$		0			0
$525$		0			0
$526$	−1700.00			−0.140919
$527$		−	15069.9i		−	1.24564i
$528$		0			0
$529$	6117.00			0.502753
$530$	−21324.2			−1.74767
$531$		0			0
$532$		0			0
$533$			3139.55i			0.255139i
$534$		0			0
$535$			41808.5i			3.37857i
$536$		−	6697.72i		−	0.539734i
$537$		0			0
$538$			13767.3i			1.10325i
$539$	5099.26	+	1571.19i	0.407496	+	0.125558i
$540$		0			0
$541$	−15646.0			−1.24339			−0.621695	−	0.783259i	$-0.713556\pi$
$541$	−15646.0			−1.24339			−0.621695	+	0.783259i	$0.713556\pi$
$542$	−5057.11			−0.400778
$543$		0			0
$544$		0			0
$545$	17784.2			1.39778
$546$		0			0
$547$	1880.00			0.146952			0.0734762	−	0.997297i	$-0.476591\pi$
$547$	1880.00			0.146952			0.0734762	+	0.997297i	$0.476591\pi$
$548$		0			0
$549$		0			0
$550$	−14036.0			−1.08818
$551$	−11252.1			−0.869972
$552$		0			0
$553$	−9196.00	−	12456.1i	−0.707150	−	0.957845i
$554$			3184.81i			0.244241i
$555$		0			0
$556$		0			0
$557$			7001.77i			0.532629i	0.963886	+	0.266315i	$0.0858060\pi$
$557$			7001.77i			0.532629i	−0.963886	+	0.266315i	$0.914194\pi$
$558$		0			0
$559$		−	5661.87i		−	0.428393i
$560$	−14834.2	−	20093.1i	−1.11939	−	1.51623i
$561$		0			0
$562$	14356.0			1.07753
$563$	8386.38			0.627786			0.313893	−	0.949458i	$-0.398367\pi$
$563$	8386.38			0.627786			0.313893	+	0.949458i	$0.398367\pi$
$564$		0			0
$565$			12128.3i			0.903084i
$566$	−20144.2			−1.49598
$567$		0			0
$568$	−7456.00			−0.550787
$569$			17905.4i			1.31921i	0.751612	+	0.659606i	$0.229277\pi$
$569$			17905.4i			1.31921i	−0.751612	+	0.659606i	$0.770723\pi$
$570$		0			0
$571$	−24736.0			−1.81291			−0.906453	−	0.422307i	$-0.861221\pi$
$571$	−24736.0			−1.81291			−0.906453	+	0.422307i	$0.861221\pi$
$572$		0			0
$573$		0			0
$574$	−4440.00	+	3277.93i	−0.322861	+	0.238359i
$575$			24812.4i			1.79956i
$576$		0			0
$577$		−	17343.2i		−	1.25131i	−0.780099	−	0.625656i	$-0.784831\pi$
$577$		−	17343.2i		−	1.25131i	0.780099	−	0.625656i	$-0.215169\pi$
$578$			2593.67i			0.186648i
$579$		0			0
$580$		0			0
$581$	−13443.5	−	18209.4i	−0.959949	−	1.30026i
$582$		0			0
$583$	5566.00			0.395403
$584$	18205.6			1.28999
$585$		0			0
$586$		−	655.585i		−	0.0462150i
$587$	−3329.27			−0.234095			−0.117047	−	0.993126i	$-0.537343\pi$
$587$	−3329.27			−0.234095			−0.117047	+	0.993126i	$0.537343\pi$
$588$		0			0
$589$	−21312.0			−1.49091
$590$		−	5023.29i		−	0.350518i
$591$		0			0
$592$	11776.0			0.817552
$593$	22567.4			1.56278			0.781392	−	0.624041i	$-0.214510\pi$
$593$	22567.4			1.56278			0.781392	+	0.624041i	$0.214510\pi$
$594$		0			0
$595$	14652.0	+	19846.4i	1.00954	+	1.36743i
$596$		0			0
$597$		0			0
$598$			6555.85i			0.448309i
$599$		−	25216.8i		−	1.72009i	−0.510221	−	0.860044i	$-0.670436\pi$
$599$		−	25216.8i		−	1.72009i	0.510221	−	0.860044i	$-0.329564\pi$
$600$		0			0
$601$			13290.5i			0.902048i	0.892512	+	0.451024i	$0.148941\pi$
$601$			13290.5i			0.902048i	−0.892512	+	0.451024i	$0.851059\pi$
$602$	8007.10	−	5911.41i	0.542101	−	0.400218i
$603$		0			0
$604$		0			0
$605$	−22946.7			−1.54201
$606$		0			0
$607$		−	18386.2i		−	1.22944i	−0.788744	−	0.614722i	$-0.789268\pi$
$607$		−	18386.2i		−	1.22944i	0.788744	−	0.614722i	$-0.210732\pi$
$608$		0			0
$609$		0			0
$610$	39072.0			2.59341
$611$		−	1255.82i		−	0.0831507i
$612$		0			0
$613$	−26554.0			−1.74960			−0.874801	−	0.484483i	$-0.839008\pi$
$613$	−26554.0			−1.74960			−0.874801	+	0.484483i	$0.839008\pi$
$614$	25032.7			1.64534
$615$		0			0
$616$	3872.00	+	5244.68i	0.253259	+	0.343043i
$617$			26475.5i			1.72749i	0.503927	+	0.863746i	$0.331888\pi$
$617$			26475.5i			1.72749i	−0.503927	+	0.863746i	$0.668112\pi$
$618$		0			0
$619$		−	16389.6i		−	1.06422i	−0.846674	−	0.532112i	$-0.821398\pi$
$619$		−	16389.6i		−	1.06422i	0.846674	−	0.532112i	$-0.178602\pi$
$620$		0			0
$621$		0			0
$622$		−	18356.4i		−	1.18332i
$623$	7648.88	+	10360.5i	0.491888	+	0.666269i
$624$		0			0
$625$	46261.0			2.96070
$626$	−30679.8			−1.95881
$627$		0			0
$628$		0			0
$629$	−11631.4			−0.737318
$630$		0			0
$631$	24860.0			1.56840			0.784200	−	0.620508i	$-0.213073\pi$
$631$	24860.0			1.56840			0.784200	+	0.620508i	$0.213073\pi$
$632$		−	18916.5i		−	1.19060i
$633$		0			0
$634$	−5036.00			−0.315465
$635$	4635.69			0.289703
$636$		0			0
$637$	−9768.00	−	3009.73i	−0.607570	−	0.187206i
$638$			5538.06i			0.343658i
$639$		0			0
$640$		−	30514.5i		−	1.88468i
$641$			5279.26i			0.325301i	0.986684	+	0.162651i	$0.0520044\pi$
$641$			5279.26i			0.325301i	−0.986684	+	0.162651i	$0.947996\pi$
$642$		0			0
$643$		−	2652.14i		−	0.162660i	−0.996687	−	0.0813299i	$-0.974083\pi$
$643$		−	2652.14i		−	0.162660i	0.996687	−	0.0813299i	$-0.0259167\pi$
$644$		0			0
$645$		0			0
$646$	−15984.0			−0.973502
$647$	23009.9			1.39816			0.699081	−	0.715042i	$-0.253593\pi$
$647$	23009.9			1.39816			0.699081	+	0.715042i	$0.253593\pi$
$648$		0			0
$649$			1311.17i			0.0793035i
$650$	26887.0			1.62245
$651$		0			0
$652$		0			0
$653$			5864.74i			0.351463i	0.984438	+	0.175731i	$0.0562290\pi$
$653$			5864.74i			0.351463i	−0.984438	+	0.175731i	$0.943771\pi$
$654$		0			0
$655$	−27528.0			−1.64215
$656$	−6742.82			−0.401315
$657$		0			0
$658$	1776.00	−	1311.17i	0.105221	−	0.0776820i
$659$			2759.13i			0.163096i	0.996669	+	0.0815482i	$0.0259864\pi$
$659$			2759.13i			0.163096i	−0.996669	+	0.0815482i	$0.974014\pi$
$660$		0			0
$661$		−	20978.7i		−	1.23446i	−0.786783	−	0.617230i	$-0.788255\pi$
$661$		−	20978.7i		−	1.23446i	0.786783	−	0.617230i	$-0.211745\pi$
$662$			26943.6i			1.58186i
$663$		0			0
$664$		−	27653.8i		−	1.61623i
$665$	28067.0	−	20721.1i	1.63668	−	1.20831i
$666$		0			0
$667$	9790.00			0.568321
$668$		0			0
$669$		0			0
$670$			17641.2i			1.01722i
$671$	−10198.5			−0.586750
$672$		0			0
$673$	−13636.0			−0.781024			−0.390512	−	0.920598i	$-0.627702\pi$
$673$	−13636.0			−0.781024			−0.390512	+	0.920598i	$0.627702\pi$
$674$			23396.7i			1.33711i
$675$		0			0
$676$		0			0
$677$	−9966.73			−0.565809			−0.282904	−	0.959148i	$-0.591298\pi$
$677$	−9966.73			−0.565809			−0.282904	+	0.959148i	$0.591298\pi$
$678$		0			0
$679$	−8436.00	+	6228.06i	−0.476795	+	0.352004i
$680$			30139.7i			1.69971i
$681$		0			0
$682$			10489.4i			0.588942i
$683$		−	202.233i		−	0.0113297i	−0.999984	−	0.00566487i	$-0.998197\pi$
$683$		−	202.233i		−	0.0113297i	0.999984	−	0.00566487i	$-0.00180319\pi$
$684$		0			0
$685$			33047.5i			1.84333i
$686$	−5942.11	−	16956.4i	−0.330715	−	0.943731i
$687$		0			0
$688$	12160.0			0.673831
$689$	−10662.1			−0.589540
$690$		0			0
$691$			20084.7i			1.10573i	0.833271	+	0.552865i	$0.186466\pi$
$691$			20084.7i			1.10573i	−0.833271	+	0.552865i	$0.813534\pi$
$692$		0			0
$693$		0			0
$694$	27676.0			1.51378
$695$		−	13814.0i		−	0.753952i
$696$		0			0
$697$	6660.00			0.361930
$698$	17868.5			0.968956
$699$		0			0
$700$		0			0
$701$		−	22883.4i		−	1.23294i	−0.787377	−	0.616472i	$-0.788561\pi$
$701$		−	22883.4i		−	1.23294i	0.787377	−	0.616472i	$-0.211439\pi$
$702$		0			0
$703$			16449.2i			0.882496i
$704$			7964.85i			0.426401i
$705$		0			0
$706$		−	15436.1i		−	0.822866i
$707$	8112.45	+	10988.4i	0.431542	+	0.584530i
$708$		0			0
$709$	12212.0			0.646871			0.323435	−	0.946250i	$-0.395162\pi$
$709$	12212.0			0.646871			0.323435	+	0.946250i	$0.395162\pi$
$710$	19638.5			1.03805
$711$		0			0
$712$			15734.0i			0.828172i
$713$	18542.8			0.973957
$714$		0			0
$715$	−9768.00			−0.510913
$716$		0			0
$717$		0			0
$718$	−30668.0			−1.59404
$719$	−28825.5			−1.49515			−0.747574	−	0.664178i	$-0.768782\pi$
$719$	−28825.5			−1.49515			−0.747574	+	0.664178i	$0.768782\pi$
$720$		0			0
$721$	9768.00	−	7211.44i	0.504548	−	0.372494i
$722$			3204.61i			0.165184i
$723$		0			0
$724$		0			0
$725$		−	40150.9i		−	2.05678i
$726$		0			0
$727$			3277.93i			0.167224i	0.996498	+	0.0836118i	$0.0266456\pi$
$727$			3277.93i			0.167224i	−0.996498	+	0.0836118i	$0.973354\pi$
$728$	−7417.10	−	10046.6i	−0.377604	−	0.511471i
$729$		0			0
$730$	−47952.0			−2.43121
$731$	−12010.6			−0.607701
$732$		0			0
$733$		−	12128.3i		−	0.611146i	−0.952169	−	0.305573i	$-0.901152\pi$
$733$		−	12128.3i		−	0.611146i	0.952169	−	0.305573i	$-0.0988480\pi$
$734$	10535.7			0.529807
$735$		0			0
$736$		0			0
$737$		−	4604.68i		−	0.230143i
$738$		0			0
$739$	1760.00			0.0876085			0.0438042	−	0.999040i	$-0.486052\pi$
$739$	1760.00			0.0876085			0.0438042	+	0.999040i	$0.486052\pi$
$740$		0			0
$741$		0			0
$742$	−11132.0	−	15078.5i	−0.550766	−	0.746021i
$743$			22436.5i			1.10783i	0.832574	+	0.553913i	$0.186866\pi$
$743$			22436.5i			1.10783i	−0.832574	+	0.553913i	$0.813134\pi$
$744$		0			0
$745$			30901.9i			1.51968i
$746$		−	11885.1i		−	0.583301i
$747$		0			0
$748$		0			0
$749$	−29563.0	+	21825.6i	−1.44220	+	1.06474i
$750$		0			0
$751$	−5122.00			−0.248874			−0.124437	−	0.992228i	$-0.539712\pi$
$751$	−5122.00			−0.248874			−0.124437	+	0.992228i	$0.539712\pi$
$752$	2697.13			0.130790
$753$		0			0
$754$		−	10608.6i		−	0.512389i
$755$	−41510.5			−2.00095
$756$		0			0
$757$	−18772.0			−0.901295			−0.450647	−	0.892702i	$-0.648807\pi$
$757$	−18772.0			−0.901295			−0.450647	+	0.892702i	$0.648807\pi$
$758$			7088.04i			0.339643i
$759$		0			0
$760$	42624.0			2.03439
$761$	−28973.0			−1.38012			−0.690061	−	0.723752i	$-0.742416\pi$
$761$	−28973.0			−1.38012			−0.690061	+	0.723752i	$0.742416\pi$
$762$		0			0
$763$	9284.00	+	12575.3i	0.440502	+	0.596667i
$764$		0			0
$765$		0			0
$766$			37070.4i			1.74857i
$767$		−	2511.64i		−	0.118240i
$768$		0			0
$769$			11740.9i			0.550571i	0.961363	+	0.275285i	$0.0887724\pi$
$769$			11740.9i			0.550571i	−0.961363	+	0.275285i	$0.911228\pi$
$770$	−10198.5	−	13814.0i	−0.477310	−	0.646524i
$771$		0			0
$772$		0			0
$773$	−12706.0			−0.591207			−0.295603	−	0.955311i	$-0.595521\pi$
$773$	−12706.0			−0.591207			−0.295603	+	0.955311i	$0.595521\pi$
$774$		0			0
$775$		−	76047.9i		−	3.52480i
$776$	−12811.4			−0.592656
$777$		0			0
$778$	−13100.0			−0.603673
$779$		−	9418.66i		−	0.433195i
$780$		0			0
$781$	−5126.00			−0.234856
$782$	13907.1			0.635953
$783$		0			0
$784$	6464.00	−	20978.7i	0.294461	−	0.955664i
$785$		−	55256.2i		−	2.51233i
$786$		0			0
$787$		−	26342.6i		−	1.19315i	−0.802556	−	0.596577i	$-0.796527\pi$
$787$		−	26342.6i		−	1.19315i	0.802556	−	0.596577i	$-0.203473\pi$
$788$		0			0
$789$		0			0
$790$			49824.5i			2.24389i
$791$	−8576.02	+	6331.43i	−0.385497	+	0.284602i
$792$		0			0
$793$	19536.0			0.874834
$794$	22757.0			1.01715
$795$		0			0
$796$		0			0
$797$	19448.8			0.864382			0.432191	−	0.901782i	$-0.357741\pi$
$797$	19448.8			0.864382			0.432191	+	0.901782i	$0.357741\pi$
$798$		0			0
$799$	−2664.00			−0.117954
$800$		0			0
$801$		0			0
$802$	38332.0			1.68772
$803$	12516.4			0.550053
$804$		0			0
$805$	−24420.0	+	18028.6i	−1.06918	+	0.789347i
$806$		−	20093.1i		−	0.878103i
$807$		0			0
$808$			16687.6i			0.726570i
$809$		−	34593.1i		−	1.50337i	−0.659521	−	0.751686i	$-0.729241\pi$
$809$		−	34593.1i		−	1.50337i	0.659521	−	0.751686i	$-0.270759\pi$
$810$		0			0
$811$		−	28607.4i		−	1.23864i	−0.785137	−	0.619322i	$-0.787408\pi$
$811$		−	28607.4i		−	1.23864i	0.785137	−	0.619322i	$-0.212592\pi$
$812$		0			0
$813$		0			0
$814$	8096.00			0.348605
$815$	28151.3			1.20993
$816$		0			0
$817$			16985.6i			0.727358i
$818$	−28741.3			−1.22850
$819$		0			0
$820$		0			0
$821$			28469.5i			1.21022i	0.796141	+	0.605112i	$0.206872\pi$
$821$			28469.5i			1.21022i	−0.796141	+	0.605112i	$0.793128\pi$
$822$		0			0
$823$	23468.0			0.993977			0.496988	−	0.867757i	$-0.334439\pi$
$823$	23468.0			0.993977			0.496988	+	0.867757i	$0.334439\pi$
$824$	14834.2			0.627153
$825$		0			0
$826$	3552.00	−	2622.34i	0.149625	−	0.110464i
$827$			37447.0i			1.57456i	0.616598	+	0.787278i	$0.288511\pi$
$827$			37447.0i			1.57456i	−0.616598	+	0.787278i	$0.711489\pi$
$828$		0			0
$829$			10817.2i			0.453191i	0.973989	+	0.226596i	$0.0727595\pi$
$829$			10817.2i			0.453191i	−0.973989	+	0.226596i	$0.927240\pi$
$830$			72837.7i			3.04606i
$831$		0			0
$832$		−	15257.3i		−	0.635757i
$833$	−6384.61	+	20721.1i	−0.265562	+	0.861875i
$834$		0			0
$835$	−48840.0			−2.02417
$836$		0			0
$837$		0			0
$838$			25269.8i			1.04168i
$839$	42690.5			1.75666			0.878331	−	0.478054i	$-0.158658\pi$
$839$	42690.5			1.75666			0.878331	+	0.478054i	$0.158658\pi$
$840$		0			0
$841$	8547.00			0.350445
$842$		−	15856.2i		−	0.648978i
$843$		0			0
$844$		0			0
$845$	−27582.3			−1.12291
$846$		0			0
$847$	−11979.0	−	16225.7i	−0.485954	−	0.658232i
$848$		−	22898.9i		−	0.927303i
$849$		0			0
$850$		−	57035.9i		−	2.30155i
$851$		−	14311.8i		−	0.576502i
$852$		0			0
$853$		−	4589.10i		−	0.184206i	−0.995749	−	0.0921030i	$-0.970641\pi$
$853$		−	4589.10i		−	0.184206i	0.995749	−	0.0921030i	$-0.0293589\pi$
$854$	20397.0	+	27628.1i	0.817297	+	1.10704i
$855$		0			0
$856$	−44896.0			−1.79266
$857$	−4783.19			−0.190654			−0.0953270	−	0.995446i	$-0.530390\pi$
$857$	−4783.19			−0.190654			−0.0953270	+	0.995446i	$0.530390\pi$
$858$		0			0
$859$			37040.6i			1.47125i	0.677386	+	0.735627i	$0.263113\pi$
$859$			37040.6i			1.47125i	−0.677386	+	0.735627i	$0.736887\pi$
$860$		0			0
$861$		0			0
$862$	2500.00			0.0987823
$863$		−	1195.01i		−	0.0471363i	−0.999722	−	0.0235682i	$-0.992497\pi$
$863$		−	1195.01i		−	0.0471363i	0.999722	−	0.0235682i	$-0.00750267\pi$
$864$		0			0
$865$	4884.00			0.191978
$866$	−5562.83			−0.218282
$867$		0			0
$868$		0			0
$869$		−	13005.1i		−	0.507673i
$870$		0			0
$871$			8820.60i			0.343140i
$872$			19097.5i			0.741656i
$873$		0			0
$874$		−	19667.6i		−	0.761173i
$875$	44966.2	+	60907.3i	1.73730	+	2.35319i
$876$		0			0
$877$	34298.0			1.32059			0.660297	−	0.751004i	$-0.270430\pi$
$877$	34298.0			1.32059			0.660297	+	0.751004i	$0.270430\pi$
$878$	−1095.71			−0.0421166
$879$		0			0
$880$		−	20978.7i		−	0.803628i
$881$	21682.4			0.829169			0.414584	−	0.910011i	$-0.363927\pi$
$881$	21682.4			0.829169			0.414584	+	0.910011i	$0.363927\pi$
$882$		0			0
$883$	16034.0			0.611084			0.305542	−	0.952179i	$-0.401162\pi$
$883$	16034.0			0.611084			0.305542	+	0.952179i	$0.401162\pi$
$884$		0			0
$885$		0			0
$886$	7420.00			0.281354
$887$	46820.4			1.77235			0.886176	−	0.463348i	$-0.153352\pi$
$887$	46820.4			1.77235			0.886176	+	0.463348i	$0.153352\pi$
$888$		0			0
$889$	2420.00	+	3277.93i	0.0912983	+	0.123665i
$890$		−	41442.1i		−	1.56083i
$891$		0			0
$892$		0			0
$893$			3767.46i			0.141180i
$894$		0			0
$895$			2294.55i			0.0856964i
$896$	21577.0	−	15929.7i	0.804506	−	0.593944i
$897$		0			0
$898$	−14540.0			−0.540319
$899$	−30005.5			−1.11317
$900$		0			0
$901$			22617.7i			0.836298i
$902$	−4635.69			−0.171121
$903$		0			0
$904$	−13024.0			−0.479172
$905$			54628.2i			2.00652i
$906$		0			0
$907$	51722.0			1.89350			0.946748	−	0.321976i	$-0.104347\pi$
$907$	51722.0			1.89350			0.946748	+	0.321976i	$0.104347\pi$
$908$		0			0
$909$		0			0
$910$	19536.0	+	26461.8i	0.711662	+	0.963956i
$911$		−	38564.2i		−	1.40251i	−0.712909	−	0.701256i	$-0.752623\pi$
$911$		−	38564.2i		−	1.40251i	0.712909	−	0.701256i	$-0.247377\pi$
$912$		0			0
$913$		−	19012.0i		−	0.689161i
$914$		−	10205.0i		−	0.369311i
$915$		0			0
$916$		0			0
$917$	−14370.6	−	19465.2i	−0.517514	−	0.700980i
$918$		0			0
$919$	30470.0			1.09370			0.546851	−	0.837230i	$-0.315826\pi$
$919$	30470.0			1.09370			0.546851	+	0.837230i	$0.315826\pi$
$920$	−37085.5			−1.32899
$921$		0			0
$922$		−	4350.70i		−	0.155404i
$923$	9819.23			0.350167
$924$		0			0
$925$	−58696.0			−2.08639
$926$		−	5011.97i		−	0.177866i
$927$		0			0
$928$		0			0
$929$	26444.5			0.933924			0.466962	−	0.884277i	$-0.345348\pi$
$929$	26444.5			0.933924			0.466962	+	0.884277i	$0.345348\pi$
$930$		0			0
$931$	29304.0	+	9029.20i	1.03158	+	0.317852i
$932$		0			0
$933$		0			0
$934$		−	36116.8i		−	1.26529i
$935$			20721.1i			0.724760i
$936$		0			0
$937$		−	19667.6i		−	0.685711i	−0.939388	−	0.342855i	$-0.888606\pi$
$937$		−	19667.6i		−	0.685711i	0.939388	−	0.342855i	$-0.111394\pi$
$938$	−12474.2	+	9209.36i	−0.434219	+	0.320572i
$939$		0			0
$940$		0			0
$941$	−3940.33			−0.136505			−0.0682525	−	0.997668i	$-0.521742\pi$
$941$	−3940.33			−0.136505			−0.0682525	+	0.997668i	$0.521742\pi$
$942$		0			0
$943$			8194.82i			0.282990i
$944$	5394.25			0.185983
$945$		0			0
$946$	8360.00			0.287322
$947$		−	52081.2i		−	1.78713i	−0.448933	−	0.893565i	$-0.648196\pi$
$947$		−	52081.2i		−	1.78713i	0.448933	−	0.893565i	$-0.351804\pi$
$948$		0			0
$949$	−23976.0			−0.820120
$950$	−80661.0			−2.75472
$951$		0			0
$952$	−21312.0	+	15734.0i	−0.725552	+	0.535655i
$953$		−	15993.3i		−	0.543626i	−0.962350	−	0.271813i	$-0.912377\pi$
$953$		−	15993.3i		−	0.543626i	0.962350	−	0.271813i	$-0.0876231\pi$
$954$		0			0
$955$		−	46874.3i		−	1.58829i
$956$		0			0
$957$		0			0
$958$		−	13111.7i		−	0.442192i
$959$	−23368.1	+	17252.0i	−0.786856	+	0.580913i
$960$		0			0
$961$	−27041.0			−0.907690
$962$	−15508.5			−0.519765
$963$		0			0
$964$		0			0
$965$	−78806.7			−2.62889
$966$		0			0
$967$	−24772.0			−0.823799			−0.411900	−	0.911229i	$-0.635135\pi$
$967$	−24772.0			−0.823799			−0.411900	+	0.911229i	$0.635135\pi$
$968$		−	24641.3i		−	0.818182i
$969$		0			0
$970$	33744.0			1.11696
$971$	55122.5			1.82180			0.910899	−	0.412629i	$-0.135389\pi$
$971$	55122.5			1.82180			0.910899	+	0.412629i	$0.135389\pi$
$972$		0			0
$973$	9768.00	−	7211.44i	0.321837	−	0.237603i
$974$		−	5538.06i		−	0.182188i
$975$		0			0
$976$			41957.5i			1.37605i
$977$			45160.1i			1.47881i	0.673260	+	0.739406i	$0.264893\pi$
$977$			45160.1i			1.47881i	−0.673260	+	0.739406i	$0.735107\pi$
$978$		0			0
$979$			10817.2i			0.353134i
$980$		0			0
$981$		0			0
$982$	−20156.0			−0.654994
$983$	33840.5			1.09801			0.549006	−	0.835819i	$-0.315007\pi$
$983$	33840.5			1.09801			0.549006	+	0.835819i	$0.315007\pi$
$984$		0			0
$985$		−	25240.0i		−	0.816461i
$986$	−22504.2			−0.726854
$987$		0			0
$988$		0			0
$989$		−	14778.5i		−	0.475157i
$990$		0			0
$991$	−18382.0			−0.589227			−0.294613	−	0.955617i	$-0.595191\pi$
$991$	−18382.0			−0.589227			−0.294613	+	0.955617i	$0.595191\pi$
$992$		0			0
$993$		0			0
$994$	10252.0	+	13886.5i	0.327137	+	0.443111i
$995$			16953.6i			0.540166i
$996$		0			0
$997$			38679.5i			1.22868i	0.789042	+	0.614340i	$0.210577\pi$
$997$			38679.5i			1.22868i	−0.789042	+	0.614340i	$0.789423\pi$
$998$		−	29161.1i		−	0.924928i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.c.b.62.1	✓	4
3.2	odd	2	inner	63.4.c.b.62.4	yes	4
4.3	odd	2		1008.4.k.b.881.2		4
7.2	even	3		441.4.p.a.80.2		8
7.3	odd	6		441.4.p.a.215.3		8
7.4	even	3		441.4.p.a.215.4		8
7.5	odd	6		441.4.p.a.80.1		8
7.6	odd	2	inner	63.4.c.b.62.2	yes	4
12.11	even	2		1008.4.k.b.881.4		4
21.2	odd	6		441.4.p.a.80.3		8
21.5	even	6		441.4.p.a.80.4		8
21.11	odd	6		441.4.p.a.215.1		8
21.17	even	6		441.4.p.a.215.2		8
21.20	even	2	inner	63.4.c.b.62.3	yes	4
28.27	even	2		1008.4.k.b.881.3		4
84.83	odd	2		1008.4.k.b.881.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.c.b.62.1	✓	4	1.1	even	1	trivial
63.4.c.b.62.2	yes	4	7.6	odd	2	inner
63.4.c.b.62.3	yes	4	21.20	even	2	inner
63.4.c.b.62.4	yes	4	3.2	odd	2	inner
441.4.p.a.80.1		8	7.5	odd	6
441.4.p.a.80.2		8	7.2	even	3
441.4.p.a.80.3		8	21.2	odd	6
441.4.p.a.80.4		8	21.5	even	6
441.4.p.a.215.1		8	21.11	odd	6
441.4.p.a.215.2		8	21.17	even	6
441.4.p.a.215.3		8	7.3	odd	6
441.4.p.a.215.4		8	7.4	even	3
1008.4.k.b.881.1		4	84.83	odd	2
1008.4.k.b.881.2		4	4.3	odd	2
1008.4.k.b.881.3		4	28.27	even	2
1008.4.k.b.881.4		4	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -21.0713 q^{5} +(-11.0000 - 14.8997i) q^{7} -22.6274i q^{8} +59.5987i q^{10} -15.5563i q^{11} +29.7993i q^{13} +(-42.1426 + 31.1127i) q^{14} -64.0000 q^{16} +63.2139 q^{17} -89.3980i q^{19} -44.0000 q^{22} +77.7817i q^{23} +319.000 q^{25} +84.2852 q^{26} -125.865i q^{29} -238.395i q^{31} -178.796i q^{34} +(231.784 + 313.955i) q^{35} -184.000 q^{37} -252.856 q^{38} +476.789i q^{40} +105.357 q^{41} -190.000 q^{43} +220.000 q^{46} -42.1426 q^{47} +(-101.000 + 327.793i) q^{49} -902.268i q^{50} +357.796i q^{53} +327.793i q^{55} +(-337.141 + 248.902i) q^{56} -356.000 q^{58} -84.2852 q^{59} -655.585i q^{61} -674.282 q^{62} -512.000 q^{64} -627.911i q^{65} +296.000 q^{67} +(888.000 - 655.585i) q^{70} -329.512i q^{71} +804.582i q^{73} +520.431i q^{74} +(-231.784 + 171.120i) q^{77} +836.000 q^{79} +1348.56 q^{80} -297.993i q^{82} +1222.14 q^{83} -1332.00 q^{85} +537.401i q^{86} -352.000 q^{88} -695.353 q^{89} +(444.000 - 327.793i) q^{91} +119.197i q^{94} +1883.73i q^{95} -566.187i q^{97} +(927.138 + 285.671i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 44 q^{7} - 256 q^{16} - 176 q^{22} + 1276 q^{25} - 736 q^{37} - 760 q^{43} + 880 q^{46} - 404 q^{49} - 1424 q^{58} - 2048 q^{64} + 1184 q^{67} + 3552 q^{70} + 3344 q^{79} - 5328 q^{85} - 1408 q^{88}+ \cdots + 1776 q^{91}+O(q^{100}) \) 4 * q - 44 * q^7 - 256 * q^16 - 176 * q^22 + 1276 * q^25 - 736 * q^37 - 760 * q^43 + 880 * q^46 - 404 * q^49 - 1424 * q^58 - 2048 * q^64 + 1184 * q^67 + 3552 * q^70 + 3344 * q^79 - 5328 * q^85 - 1408 * q^88 + 1776 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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