LMFDB - Embedded newform 720.3.c.c.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,3,Mod(449,720)]

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

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

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

chi := DirichletCharacter("720.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$720 = 2^{4} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	720.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.449
Dual form			720.3.c.c.449.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.53553 - 3.53553i) q^{5} +4.00000i q^{7} +11.3137i q^{11} -18.0000i q^{13} -1.41421 q^{17} +24.0000 q^{19} -39.5980 q^{23} +25.0000i q^{25} +38.1838i q^{29} -4.00000 q^{31} +(14.1421 - 14.1421i) q^{35} +56.0000i q^{37} -24.0416i q^{41} +80.0000i q^{43} +28.2843 q^{47} +33.0000 q^{49} -4.24264 q^{53} +(40.0000 - 40.0000i) q^{55} +62.2254i q^{59} +110.000 q^{61} +(-63.6396 + 63.6396i) q^{65} +32.0000i q^{67} +50.9117i q^{71} +46.0000i q^{73} -45.2548 q^{77} -36.0000 q^{79} -5.65685 q^{83} +(5.00000 + 5.00000i) q^{85} +57.9828i q^{89} +72.0000 q^{91} +(-84.8528 - 84.8528i) q^{95} -14.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 96 q^{19} - 16 q^{31} + 132 q^{49} + 160 q^{55} + 440 q^{61} - 144 q^{79} + 20 q^{85} + 288 q^{91}+O(q^{100})$ 4 * q + 96 * q^19 - 16 * q^31 + 132 * q^49 + 160 * q^55 + 440 * q^61 - 144 * q^79 + 20 * q^85 + 288 * q^91

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
$5$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$5$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$6$		0			0
$7$			4.00000i			0.571429i	0.958315	+	0.285714i	$0.0922308\pi$
$7$			4.00000i			0.571429i	−0.958315	+	0.285714i	$0.907769\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			11.3137i			1.02852i	0.857635	+	0.514259i	$0.171933\pi$
$11$			11.3137i			1.02852i	−0.857635	+	0.514259i	$0.828067\pi$
$12$		0			0
$13$		−	18.0000i		−	1.38462i	−0.721602	−	0.692308i	$-0.756594\pi$
$13$		−	18.0000i		−	1.38462i	0.721602	−	0.692308i	$-0.243406\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.41421			−0.0831890			−0.0415945	−	0.999135i	$-0.513244\pi$
$17$	−1.41421			−0.0831890			−0.0415945	+	0.999135i	$0.513244\pi$
$18$		0			0
$19$	24.0000			1.26316			0.631579	−	0.775312i	$-0.282407\pi$
$19$	24.0000			1.26316			0.631579	+	0.775312i	$0.282407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−39.5980			−1.72165			−0.860826	−	0.508900i	$-0.830052\pi$
$23$	−39.5980			−1.72165			−0.860826	+	0.508900i	$0.830052\pi$
$24$		0			0
$25$			25.0000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			38.1838i			1.31668i	0.752720	+	0.658341i	$0.228741\pi$
$29$			38.1838i			1.31668i	−0.752720	+	0.658341i	$0.771259\pi$
$30$		0			0
$31$	−4.00000			−0.129032			−0.0645161	−	0.997917i	$-0.520550\pi$
$31$	−4.00000			−0.129032			−0.0645161	+	0.997917i	$0.520550\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	14.1421	−	14.1421i	0.404061	−	0.404061i
$36$		0			0
$37$			56.0000i			1.51351i	0.653697	+	0.756757i	$0.273217\pi$
$37$			56.0000i			1.51351i	−0.653697	+	0.756757i	$0.726783\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	24.0416i		−	0.586381i	−0.956054	−	0.293191i	$-0.905283\pi$
$41$		−	24.0416i		−	0.586381i	0.956054	−	0.293191i	$-0.0947171\pi$
$42$		0			0
$43$			80.0000i			1.86047i	0.366971	+	0.930233i	$0.380395\pi$
$43$			80.0000i			1.86047i	−0.366971	+	0.930233i	$0.619605\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	28.2843			0.601793			0.300897	−	0.953657i	$-0.402714\pi$
$47$	28.2843			0.601793			0.300897	+	0.953657i	$0.402714\pi$
$48$		0			0
$49$	33.0000			0.673469
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.24264			−0.0800498			−0.0400249	−	0.999199i	$-0.512744\pi$
$53$	−4.24264			−0.0800498			−0.0400249	+	0.999199i	$0.512744\pi$
$54$		0			0
$55$	40.0000	−	40.0000i	0.727273	−	0.727273i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			62.2254i			1.05467i	0.849658	+	0.527334i	$0.176808\pi$
$59$			62.2254i			1.05467i	−0.849658	+	0.527334i	$0.823192\pi$
$60$		0			0
$61$	110.000			1.80328			0.901639	−	0.432489i	$-0.142364\pi$
$61$	110.000			1.80328			0.901639	+	0.432489i	$0.142364\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−63.6396	+	63.6396i	−0.979071	+	0.979071i
$66$		0			0
$67$			32.0000i			0.477612i	0.971067	+	0.238806i	$0.0767560\pi$
$67$			32.0000i			0.477612i	−0.971067	+	0.238806i	$0.923244\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			50.9117i			0.717066i	0.933517	+	0.358533i	$0.116723\pi$
$71$			50.9117i			0.717066i	−0.933517	+	0.358533i	$0.883277\pi$
$72$		0			0
$73$			46.0000i			0.630137i	0.949069	+	0.315068i	$0.102027\pi$
$73$			46.0000i			0.630137i	−0.949069	+	0.315068i	$0.897973\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−45.2548			−0.587725
$78$		0			0
$79$	−36.0000			−0.455696			−0.227848	−	0.973697i	$-0.573169\pi$
$79$	−36.0000			−0.455696			−0.227848	+	0.973697i	$0.573169\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.65685			−0.0681549			−0.0340774	−	0.999419i	$-0.510849\pi$
$83$	−5.65685			−0.0681549			−0.0340774	+	0.999419i	$0.510849\pi$
$84$		0			0
$85$	5.00000	+	5.00000i	0.0588235	+	0.0588235i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			57.9828i			0.651492i	0.945457	+	0.325746i	$0.105615\pi$
$89$			57.9828i			0.651492i	−0.945457	+	0.325746i	$0.894385\pi$
$90$		0			0
$91$	72.0000			0.791209
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−84.8528	−	84.8528i	−0.893188	−	0.893188i
$96$		0			0
$97$		−	14.0000i		−	0.144330i	−0.997393	−	0.0721649i	$-0.977009\pi$
$97$		−	14.0000i		−	0.144330i	0.997393	−	0.0721649i	$-0.0229908\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	26.8701i		−	0.266040i	−0.991113	−	0.133020i	$-0.957533\pi$
$101$		−	26.8701i		−	0.266040i	0.991113	−	0.133020i	$-0.0424675\pi$
$102$		0			0
$103$		−	68.0000i		−	0.660194i	−0.943947	−	0.330097i	$-0.892919\pi$
$103$		−	68.0000i		−	0.660194i	0.943947	−	0.330097i	$-0.107081\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−169.706			−1.58603			−0.793017	−	0.609200i	$-0.791491\pi$
$107$	−169.706			−1.58603			−0.793017	+	0.609200i	$0.791491\pi$
$108$		0			0
$109$	−136.000			−1.24771			−0.623853	−	0.781542i	$-0.714434\pi$
$109$	−136.000			−1.24771			−0.623853	+	0.781542i	$0.714434\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.7279			0.112636			0.0563182	−	0.998413i	$-0.482064\pi$
$113$	12.7279			0.112636			0.0563182	+	0.998413i	$0.482064\pi$
$114$		0			0
$115$	140.000	+	140.000i	1.21739	+	1.21739i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.65685i		−	0.0475366i
$120$		0			0
$121$	−7.00000			−0.0578512
$122$		0			0
$123$		0			0
$124$		0			0
$125$	88.3883	−	88.3883i	0.707107	−	0.707107i
$126$		0			0
$127$			28.0000i			0.220472i	0.993905	+	0.110236i	$0.0351607\pi$
$127$			28.0000i			0.220472i	−0.993905	+	0.110236i	$0.964839\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			84.8528i			0.647731i	0.946103	+	0.323866i	$0.104983\pi$
$131$			84.8528i			0.647731i	−0.946103	+	0.323866i	$0.895017\pi$
$132$		0			0
$133$			96.0000i			0.721805i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.7279			0.0929045			0.0464523	−	0.998921i	$-0.485208\pi$
$137$	12.7279			0.0929045			0.0464523	+	0.998921i	$0.485208\pi$
$138$		0			0
$139$	−8.00000			−0.0575540			−0.0287770	−	0.999586i	$-0.509161\pi$
$139$	−8.00000			−0.0575540			−0.0287770	+	0.999586i	$0.509161\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	203.647			1.42410
$144$		0			0
$145$	135.000	−	135.000i	0.931034	−	0.931034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			120.208i			0.806766i	0.915031	+	0.403383i	$0.132166\pi$
$149$			120.208i			0.806766i	−0.915031	+	0.403383i	$0.867834\pi$
$150$		0			0
$151$	−172.000			−1.13907			−0.569536	−	0.821966i	$-0.692877\pi$
$151$	−172.000			−1.13907			−0.569536	+	0.821966i	$0.692877\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.1421	+	14.1421i	0.0912396	+	0.0912396i
$156$		0			0
$157$		−	152.000i		−	0.968153i	−0.875026	−	0.484076i	$-0.839156\pi$
$157$		−	152.000i		−	0.968153i	0.875026	−	0.484076i	$-0.160844\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	158.392i		−	0.983801i
$162$		0			0
$163$		−	104.000i		−	0.638037i	−0.947749	−	0.319018i	$-0.896647\pi$
$163$		−	104.000i		−	0.638037i	0.947749	−	0.319018i	$-0.103353\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	203.647			1.21944			0.609721	−	0.792616i	$-0.291282\pi$
$167$	203.647			1.21944			0.609721	+	0.792616i	$0.291282\pi$
$168$		0			0
$169$	−155.000			−0.917160
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−230.517			−1.33247			−0.666234	−	0.745743i	$-0.732095\pi$
$173$	−230.517			−1.33247			−0.666234	+	0.745743i	$0.732095\pi$
$174$		0			0
$175$	−100.000			−0.571429
$176$		0			0
$177$		0			0
$178$		0			0
$179$			164.049i			0.916474i	0.888830	+	0.458237i	$0.151519\pi$
$179$			164.049i			0.916474i	−0.888830	+	0.458237i	$0.848481\pi$
$180$		0			0
$181$	8.00000			0.0441989			0.0220994	−	0.999756i	$-0.492965\pi$
$181$	8.00000			0.0441989			0.0220994	+	0.999756i	$0.492965\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	197.990	−	197.990i	1.07022	−	1.07022i
$186$		0			0
$187$		−	16.0000i		−	0.0855615i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	305.470i		−	1.59932i	−0.600453	−	0.799660i	$-0.705013\pi$
$191$		−	305.470i		−	1.59932i	0.600453	−	0.799660i	$-0.294987\pi$
$192$		0			0
$193$			64.0000i			0.331606i	0.986159	+	0.165803i	$0.0530216\pi$
$193$			64.0000i			0.331606i	−0.986159	+	0.165803i	$0.946978\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	230.517			1.17014			0.585068	−	0.810984i	$-0.301068\pi$
$197$	230.517			1.17014			0.585068	+	0.810984i	$0.301068\pi$
$198$		0			0
$199$	−300.000			−1.50754			−0.753769	−	0.657140i	$-0.771766\pi$
$199$	−300.000			−1.50754			−0.753769	+	0.657140i	$0.771766\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−152.735			−0.752389
$204$		0			0
$205$	−85.0000	+	85.0000i	−0.414634	+	0.414634i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			271.529i			1.29918i
$210$		0			0
$211$	264.000			1.25118			0.625592	−	0.780150i	$-0.284857\pi$
$211$	264.000			1.25118			0.625592	+	0.780150i	$0.284857\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	282.843	−	282.843i	1.31555	−	1.31555i
$216$		0			0
$217$		−	16.0000i		−	0.0737327i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			25.4558i			0.115185i
$222$		0			0
$223$		−	316.000i		−	1.41704i	−0.705691	−	0.708520i	$-0.749363\pi$
$223$		−	316.000i		−	1.41704i	0.705691	−	0.708520i	$-0.250637\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−96.1665			−0.423641			−0.211821	−	0.977309i	$-0.567939\pi$
$227$	−96.1665			−0.423641			−0.211821	+	0.977309i	$0.567939\pi$
$228$		0			0
$229$	−8.00000			−0.0349345			−0.0174672	−	0.999847i	$-0.505560\pi$
$229$	−8.00000			−0.0349345			−0.0174672	+	0.999847i	$0.505560\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	227.688			0.977203			0.488602	−	0.872507i	$-0.337507\pi$
$233$	227.688			0.977203			0.488602	+	0.872507i	$0.337507\pi$
$234$		0			0
$235$	−100.000	−	100.000i	−0.425532	−	0.425532i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			192.333i			0.804741i	0.915477	+	0.402370i	$0.131814\pi$
$239$			192.333i			0.804741i	−0.915477	+	0.402370i	$0.868186\pi$
$240$		0			0
$241$	−128.000			−0.531120			−0.265560	−	0.964094i	$-0.585557\pi$
$241$	−128.000			−0.531120			−0.265560	+	0.964094i	$0.585557\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−116.673	−	116.673i	−0.476215	−	0.476215i
$246$		0			0
$247$		−	432.000i		−	1.74899i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	475.176i		−	1.89313i	−0.322512	−	0.946565i	$-0.604527\pi$
$251$		−	475.176i		−	1.89313i	0.322512	−	0.946565i	$-0.395473\pi$
$252$		0			0
$253$		−	448.000i		−	1.77075i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−326.683			−1.27114			−0.635571	−	0.772043i	$-0.719235\pi$
$257$	−326.683			−1.27114			−0.635571	+	0.772043i	$0.719235\pi$
$258$		0			0
$259$	−224.000			−0.864865
$260$		0			0
$261$		0			0
$262$		0			0
$263$	79.1960			0.301125			0.150563	−	0.988600i	$-0.451891\pi$
$263$	79.1960			0.301125			0.150563	+	0.988600i	$0.451891\pi$
$264$		0			0
$265$	15.0000	+	15.0000i	0.0566038	+	0.0566038i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	165.463i		−	0.615104i	−0.951531	−	0.307552i	$-0.900490\pi$
$269$		−	165.463i		−	0.615104i	0.951531	−	0.307552i	$-0.0995098\pi$
$270$		0			0
$271$	−220.000			−0.811808			−0.405904	−	0.913916i	$-0.633043\pi$
$271$	−220.000			−0.811808			−0.405904	+	0.913916i	$0.633043\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−282.843			−1.02852
$276$		0			0
$277$			2.00000i			0.00722022i	0.999993	+	0.00361011i	$0.00114914\pi$
$277$			2.00000i			0.00722022i	−0.999993	+	0.00361011i	$0.998851\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	12.7279i		−	0.0452951i	−0.999744	−	0.0226475i	$-0.992790\pi$
$281$		−	12.7279i		−	0.0452951i	0.999744	−	0.0226475i	$-0.00720955\pi$
$282$		0			0
$283$			472.000i			1.66784i	0.551882	+	0.833922i	$0.313910\pi$
$283$			472.000i			1.66784i	−0.551882	+	0.833922i	$0.686090\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	96.1665			0.335075
$288$		0			0
$289$	−287.000			−0.993080
$290$		0			0
$291$		0			0
$292$		0			0
$293$	411.536			1.40456			0.702280	−	0.711901i	$-0.252165\pi$
$293$	411.536			1.40456			0.702280	+	0.711901i	$0.252165\pi$
$294$		0			0
$295$	220.000	−	220.000i	0.745763	−	0.745763i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			712.764i			2.38382i
$300$		0			0
$301$	−320.000			−1.06312
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−388.909	−	388.909i	−1.27511	−	1.27511i
$306$		0			0
$307$			360.000i			1.17264i	0.810080	+	0.586319i	$0.199424\pi$
$307$			360.000i			1.17264i	−0.810080	+	0.586319i	$0.800576\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			33.9411i			0.109135i	0.998510	+	0.0545677i	$0.0173781\pi$
$311$			33.9411i			0.109135i	−0.998510	+	0.0545677i	$0.982622\pi$
$312$		0			0
$313$		−	416.000i		−	1.32907i	−0.747256	−	0.664537i	$-0.768629\pi$
$313$		−	416.000i		−	1.32907i	0.747256	−	0.664537i	$-0.231371\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	74.9533			0.236446			0.118223	−	0.992987i	$-0.462280\pi$
$317$	74.9533			0.236446			0.118223	+	0.992987i	$0.462280\pi$
$318$		0			0
$319$	−432.000			−1.35423
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−33.9411			−0.105081
$324$		0			0
$325$	450.000			1.38462
$326$		0			0
$327$		0			0
$328$		0			0
$329$			113.137i			0.343882i
$330$		0			0
$331$	464.000			1.40181			0.700906	−	0.713253i	$-0.252779\pi$
$331$	464.000			1.40181			0.700906	+	0.713253i	$0.252779\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	113.137	−	113.137i	0.337723	−	0.337723i
$336$		0			0
$337$			62.0000i			0.183976i	0.995760	+	0.0919881i	$0.0293222\pi$
$337$			62.0000i			0.183976i	−0.995760	+	0.0919881i	$0.970678\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	45.2548i		−	0.132712i
$342$		0			0
$343$			328.000i			0.956268i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	458.205			1.32048			0.660238	−	0.751056i	$-0.270455\pi$
$347$	458.205			1.32048			0.660238	+	0.751056i	$0.270455\pi$
$348$		0			0
$349$	−142.000			−0.406877			−0.203438	−	0.979088i	$-0.565212\pi$
$349$	−142.000			−0.406877			−0.203438	+	0.979088i	$0.565212\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	100.409			0.284445			0.142223	−	0.989835i	$-0.454575\pi$
$353$	100.409			0.284445			0.142223	+	0.989835i	$0.454575\pi$
$354$		0			0
$355$	180.000	−	180.000i	0.507042	−	0.507042i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	390.323i		−	1.08725i	−0.839328	−	0.543625i	$-0.817051\pi$
$359$		−	390.323i		−	1.08725i	0.839328	−	0.543625i	$-0.182949\pi$
$360$		0			0
$361$	215.000			0.595568
$362$		0			0
$363$		0			0
$364$		0			0
$365$	162.635	−	162.635i	0.445574	−	0.445574i
$366$		0			0
$367$		−	284.000i		−	0.773842i	−0.922113	−	0.386921i	$-0.873539\pi$
$367$		−	284.000i		−	0.773842i	0.922113	−	0.386921i	$-0.126461\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	16.9706i		−	0.0457428i
$372$		0			0
$373$			360.000i			0.965147i	0.875855	+	0.482574i	$0.160298\pi$
$373$			360.000i			0.965147i	−0.875855	+	0.482574i	$0.839702\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	687.308			1.82310
$378$		0			0
$379$	320.000			0.844327			0.422164	−	0.906520i	$-0.361271\pi$
$379$	320.000			0.844327			0.422164	+	0.906520i	$0.361271\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−181.019			−0.472635			−0.236318	−	0.971676i	$-0.575941\pi$
$383$	−181.019			−0.472635			−0.236318	+	0.971676i	$0.575941\pi$
$384$		0			0
$385$	160.000	+	160.000i	0.415584	+	0.415584i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			685.894i			1.76322i	0.471976	+	0.881611i	$0.343541\pi$
$389$			685.894i			1.76322i	−0.471976	+	0.881611i	$0.656459\pi$
$390$		0			0
$391$	56.0000			0.143223
$392$		0			0
$393$		0			0
$394$		0			0
$395$	127.279	+	127.279i	0.322226	+	0.322226i
$396$		0			0
$397$			296.000i			0.745592i	0.927913	+	0.372796i	$0.121601\pi$
$397$			296.000i			0.745592i	−0.927913	+	0.372796i	$0.878399\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			439.820i			1.09681i	0.836213	+	0.548405i	$0.184765\pi$
$401$			439.820i			1.09681i	−0.836213	+	0.548405i	$0.815235\pi$
$402$		0			0
$403$			72.0000i			0.178660i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−633.568			−1.55668
$408$		0			0
$409$	−368.000			−0.899756			−0.449878	−	0.893090i	$-0.648532\pi$
$409$	−368.000			−0.899756			−0.449878	+	0.893090i	$0.648532\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−248.902			−0.602667
$414$		0			0
$415$	20.0000	+	20.0000i	0.0481928	+	0.0481928i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			192.333i			0.459029i	0.973305	+	0.229514i	$0.0737138\pi$
$419$			192.333i			0.459029i	−0.973305	+	0.229514i	$0.926286\pi$
$420$		0			0
$421$	280.000			0.665083			0.332542	−	0.943089i	$-0.392094\pi$
$421$	280.000			0.665083			0.332542	+	0.943089i	$0.392094\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	35.3553i		−	0.0831890i
$426$		0			0
$427$			440.000i			1.03044i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			96.1665i			0.223124i	0.993757	+	0.111562i	$0.0355854\pi$
$431$			96.1665i			0.223124i	−0.993757	+	0.111562i	$0.964415\pi$
$432$		0			0
$433$			688.000i			1.58891i	0.607320	+	0.794457i	$0.292245\pi$
$433$			688.000i			1.58891i	−0.607320	+	0.794457i	$0.707755\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−950.352			−2.17472
$438$		0			0
$439$	−324.000			−0.738041			−0.369021	−	0.929421i	$-0.620307\pi$
$439$	−324.000			−0.738041			−0.369021	+	0.929421i	$0.620307\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−322.441			−0.727857			−0.363929	−	0.931427i	$-0.618565\pi$
$443$	−322.441			−0.727857			−0.363929	+	0.931427i	$0.618565\pi$
$444$		0			0
$445$	205.000	−	205.000i	0.460674	−	0.460674i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			123.037i			0.274024i	0.990569	+	0.137012i	$0.0437498\pi$
$449$			123.037i			0.274024i	−0.990569	+	0.137012i	$0.956250\pi$
$450$		0			0
$451$	272.000			0.603104
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−254.558	−	254.558i	−0.559469	−	0.559469i
$456$		0			0
$457$		−	130.000i		−	0.284464i	−0.989833	−	0.142232i	$-0.954572\pi$
$457$		−	130.000i		−	0.284464i	0.989833	−	0.142232i	$-0.0454279\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	470.933i		−	1.02155i	−0.859715	−	0.510773i	$-0.829359\pi$
$461$		−	470.933i		−	1.02155i	0.859715	−	0.510773i	$-0.170641\pi$
$462$		0			0
$463$		−	444.000i		−	0.958963i	−0.877552	−	0.479482i	$-0.840825\pi$
$463$		−	444.000i		−	0.958963i	0.877552	−	0.479482i	$-0.159175\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	186.676			0.399735			0.199867	−	0.979823i	$-0.435949\pi$
$467$	186.676			0.399735			0.199867	+	0.979823i	$0.435949\pi$
$468$		0			0
$469$	−128.000			−0.272921
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−905.097			−1.91352
$474$		0			0
$475$			600.000i			1.26316i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			582.656i			1.21640i	0.793784	+	0.608200i	$0.208108\pi$
$479$			582.656i			1.21640i	−0.793784	+	0.608200i	$0.791892\pi$
$480$		0			0
$481$	1008.00			2.09563
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−49.4975	+	49.4975i	−0.102057	+	0.102057i
$486$		0			0
$487$			84.0000i			0.172485i	0.996274	+	0.0862423i	$0.0274859\pi$
$487$			84.0000i			0.172485i	−0.996274	+	0.0862423i	$0.972514\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	322.441i		−	0.656702i	−0.944556	−	0.328351i	$-0.893507\pi$
$491$		−	322.441i		−	0.656702i	0.944556	−	0.328351i	$-0.106493\pi$
$492$		0			0
$493$		−	54.0000i		−	0.109533i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−203.647			−0.409752
$498$		0			0
$499$	−456.000			−0.913828			−0.456914	−	0.889511i	$-0.651045\pi$
$499$	−456.000			−0.913828			−0.456914	+	0.889511i	$0.651045\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	492.146			0.978422			0.489211	−	0.872165i	$-0.337285\pi$
$503$	492.146			0.978422			0.489211	+	0.872165i	$0.337285\pi$
$504$		0			0
$505$	−95.0000	+	95.0000i	−0.188119	+	0.188119i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			581.242i			1.14193i	0.820975	+	0.570964i	$0.193431\pi$
$509$			581.242i			1.14193i	−0.820975	+	0.570964i	$0.806569\pi$
$510$		0			0
$511$	−184.000			−0.360078
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−240.416	+	240.416i	−0.466828	+	0.466828i
$516$		0			0
$517$			320.000i			0.618956i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	408.708i		−	0.784468i	−0.919866	−	0.392234i	$-0.871702\pi$
$521$		−	408.708i		−	0.784468i	0.919866	−	0.392234i	$-0.128298\pi$
$522$		0			0
$523$		−	656.000i		−	1.25430i	−0.778898	−	0.627151i	$-0.784221\pi$
$523$		−	656.000i		−	1.25430i	0.778898	−	0.627151i	$-0.215779\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.65685			0.0107341
$528$		0			0
$529$	1039.00			1.96408
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−432.749			−0.811912
$534$		0			0
$535$	600.000	+	600.000i	1.12150	+	1.12150i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			373.352i			0.692676i
$540$		0			0
$541$	−248.000			−0.458410			−0.229205	−	0.973378i	$-0.573613\pi$
$541$	−248.000			−0.458410			−0.229205	+	0.973378i	$0.573613\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	480.833	+	480.833i	0.882262	+	0.882262i
$546$		0			0
$547$			304.000i			0.555759i	0.960616	+	0.277879i	$0.0896315\pi$
$547$			304.000i			0.555759i	−0.960616	+	0.277879i	$0.910368\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			916.410i			1.66318i
$552$		0			0
$553$		−	144.000i		−	0.260398i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	886.712			1.59194			0.795971	−	0.605335i	$-0.206961\pi$
$557$	886.712			1.59194			0.795971	+	0.605335i	$0.206961\pi$
$558$		0			0
$559$	1440.00			2.57603
$560$		0			0
$561$		0			0
$562$		0			0
$563$	39.5980			0.0703339			0.0351669	−	0.999381i	$-0.488804\pi$
$563$	39.5980			0.0703339			0.0351669	+	0.999381i	$0.488804\pi$
$564$		0			0
$565$	−45.0000	−	45.0000i	−0.0796460	−	0.0796460i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	43.8406i		−	0.0770485i	−0.999258	−	0.0385243i	$-0.987734\pi$
$569$		−	43.8406i		−	0.0770485i	0.999258	−	0.0385243i	$-0.0122657\pi$
$570$		0			0
$571$	392.000			0.686515			0.343257	−	0.939241i	$-0.388470\pi$
$571$	392.000			0.686515			0.343257	+	0.939241i	$0.388470\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	989.949i		−	1.72165i
$576$		0			0
$577$		−	112.000i		−	0.194107i	−0.995279	−	0.0970537i	$-0.969058\pi$
$577$		−	112.000i		−	0.194107i	0.995279	−	0.0970537i	$-0.0309419\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	22.6274i		−	0.0389456i
$582$		0			0
$583$		−	48.0000i		−	0.0823328i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1154.00			−1.96593			−0.982963	−	0.183805i	$-0.941159\pi$
$587$	−1154.00			−1.96593			−0.982963	+	0.183805i	$0.941159\pi$
$588$		0			0
$589$	−96.0000			−0.162988
$590$		0			0
$591$		0			0
$592$		0			0
$593$	258.801			0.436427			0.218213	−	0.975901i	$-0.429977\pi$
$593$	258.801			0.436427			0.218213	+	0.975901i	$0.429977\pi$
$594$		0			0
$595$	−20.0000	+	20.0000i	−0.0336134	+	0.0336134i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			288.500i			0.481635i	0.970570	+	0.240818i	$0.0774156\pi$
$599$			288.500i			0.481635i	−0.970570	+	0.240818i	$0.922584\pi$
$600$		0			0
$601$	−862.000			−1.43428			−0.717138	−	0.696931i	$-0.754548\pi$
$601$	−862.000			−1.43428			−0.717138	+	0.696931i	$0.754548\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.7487	+	24.7487i	0.0409070	+	0.0409070i
$606$		0			0
$607$			788.000i			1.29819i	0.760708	+	0.649094i	$0.224852\pi$
$607$			788.000i			1.29819i	−0.760708	+	0.649094i	$0.775148\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	509.117i		−	0.833252i
$612$		0			0
$613$			952.000i			1.55302i	0.630106	+	0.776509i	$0.283011\pi$
$613$			952.000i			1.55302i	−0.630106	+	0.776509i	$0.716989\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−383.252			−0.621154			−0.310577	−	0.950548i	$-0.600522\pi$
$617$	−383.252			−0.621154			−0.310577	+	0.950548i	$0.600522\pi$
$618$		0			0
$619$	−304.000			−0.491115			−0.245557	−	0.969382i	$-0.578971\pi$
$619$	−304.000			−0.491115			−0.245557	+	0.969382i	$0.578971\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−231.931			−0.372281
$624$		0			0
$625$	−625.000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	79.1960i		−	0.125908i
$630$		0			0
$631$	−860.000			−1.36292			−0.681458	−	0.731857i	$-0.738654\pi$
$631$	−860.000			−1.36292			−0.681458	+	0.731857i	$0.738654\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	98.9949	−	98.9949i	0.155898	−	0.155898i
$636$		0			0
$637$		−	594.000i		−	0.932496i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			442.649i			0.690560i	0.938500	+	0.345280i	$0.112216\pi$
$641$			442.649i			0.690560i	−0.938500	+	0.345280i	$0.887784\pi$
$642$		0			0
$643$			648.000i			1.00778i	0.863769	+	0.503888i	$0.168098\pi$
$643$			648.000i			1.00778i	−0.863769	+	0.503888i	$0.831902\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	390.323			0.603281			0.301641	−	0.953422i	$-0.402466\pi$
$647$	390.323			0.603281			0.301641	+	0.953422i	$0.402466\pi$
$648$		0			0
$649$	−704.000			−1.08475
$650$		0			0
$651$		0			0
$652$		0			0
$653$	909.339			1.39256			0.696278	−	0.717772i	$-0.254838\pi$
$653$	909.339			1.39256			0.696278	+	0.717772i	$0.254838\pi$
$654$		0			0
$655$	300.000	−	300.000i	0.458015	−	0.458015i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	345.068i		−	0.523624i	−0.965119	−	0.261812i	$-0.915680\pi$
$659$		−	345.068i		−	0.523624i	0.965119	−	0.261812i	$-0.0843200\pi$
$660$		0			0
$661$	962.000			1.45537			0.727685	−	0.685911i	$-0.240596\pi$
$661$	962.000			1.45537			0.727685	+	0.685911i	$0.240596\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	339.411	−	339.411i	0.510393	−	0.510393i
$666$		0			0
$667$		−	1512.00i		−	2.26687i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1244.51i			1.85471i
$672$		0			0
$673$			240.000i			0.356612i	0.983975	+	0.178306i	$0.0570617\pi$
$673$			240.000i			0.356612i	−0.983975	+	0.178306i	$0.942938\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−332.340			−0.490901			−0.245451	−	0.969409i	$-0.578936\pi$
$677$	−332.340			−0.490901			−0.245451	+	0.969409i	$0.578936\pi$
$678$		0			0
$679$	56.0000			0.0824742
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1012.58			1.48254			0.741272	−	0.671205i	$-0.234223\pi$
$683$	1012.58			1.48254			0.741272	+	0.671205i	$0.234223\pi$
$684$		0			0
$685$	−45.0000	−	45.0000i	−0.0656934	−	0.0656934i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			76.3675i			0.110838i
$690$		0			0
$691$	80.0000			0.115774			0.0578871	−	0.998323i	$-0.481564\pi$
$691$	80.0000			0.115774			0.0578871	+	0.998323i	$0.481564\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.2843	+	28.2843i	0.0406968	+	0.0406968i
$696$		0			0
$697$			34.0000i			0.0487805i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	558.614i		−	0.796882i	−0.917194	−	0.398441i	$-0.869551\pi$
$701$		−	558.614i		−	0.796882i	0.917194	−	0.398441i	$-0.130449\pi$
$702$		0			0
$703$			1344.00i			1.91181i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	107.480			0.152023
$708$		0			0
$709$	312.000			0.440056			0.220028	−	0.975494i	$-0.429385\pi$
$709$	312.000			0.440056			0.220028	+	0.975494i	$0.429385\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	158.392			0.222149
$714$		0			0
$715$	−720.000	−	720.000i	−1.00699	−	1.00699i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1148.34i		−	1.59714i	−0.601904	−	0.798568i	$-0.705591\pi$
$719$		−	1148.34i		−	1.59714i	0.601904	−	0.798568i	$-0.294409\pi$
$720$		0			0
$721$	272.000			0.377254
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−954.594			−1.31668
$726$		0			0
$727$			468.000i			0.643741i	0.946784	+	0.321871i	$0.104312\pi$
$727$			468.000i			0.643741i	−0.946784	+	0.321871i	$0.895688\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	113.137i		−	0.154770i
$732$		0			0
$733$		−	526.000i		−	0.717599i	−0.933415	−	0.358799i	$-0.883186\pi$
$733$		−	526.000i		−	0.717599i	0.933415	−	0.358799i	$-0.116814\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−362.039			−0.491233
$738$		0			0
$739$	−344.000			−0.465494			−0.232747	−	0.972537i	$-0.574771\pi$
$739$	−344.000			−0.465494			−0.232747	+	0.972537i	$0.574771\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−678.823			−0.913624			−0.456812	−	0.889563i	$-0.651009\pi$
$743$	−678.823			−0.913624			−0.456812	+	0.889563i	$0.651009\pi$
$744$		0			0
$745$	425.000	−	425.000i	0.570470	−	0.570470i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	678.823i		−	0.906305i
$750$		0			0
$751$	−308.000			−0.410120			−0.205060	−	0.978749i	$-0.565739\pi$
$751$	−308.000			−0.410120			−0.205060	+	0.978749i	$0.565739\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	608.112	+	608.112i	0.805446	+	0.805446i
$756$		0			0
$757$		−	610.000i		−	0.805812i	−0.915241	−	0.402906i	$-0.868000\pi$
$757$		−	610.000i		−	0.805812i	0.915241	−	0.402906i	$-0.132000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1175.21i		−	1.54430i	−0.635441	−	0.772149i	$-0.719182\pi$
$761$		−	1175.21i		−	1.54430i	0.635441	−	0.772149i	$-0.280818\pi$
$762$		0			0
$763$		−	544.000i		−	0.712975i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1120.06			1.46031
$768$		0			0
$769$	−270.000			−0.351105			−0.175553	−	0.984470i	$-0.556171\pi$
$769$	−270.000			−0.351105			−0.175553	+	0.984470i	$0.556171\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	267.286			0.345778			0.172889	−	0.984941i	$-0.444690\pi$
$773$	267.286			0.345778			0.172889	+	0.984941i	$0.444690\pi$
$774$		0			0
$775$		−	100.000i		−	0.129032i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	576.999i		−	0.740692i
$780$		0			0
$781$	−576.000			−0.737516
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−537.401	+	537.401i	−0.684587	+	0.684587i
$786$		0			0
$787$		−	1520.00i		−	1.93139i	−0.259688	−	0.965693i	$-0.583620\pi$
$787$		−	1520.00i		−	1.93139i	0.259688	−	0.965693i	$-0.416380\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			50.9117i			0.0643637i
$792$		0			0
$793$		−	1980.00i		−	2.49685i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	388.909			0.487966			0.243983	−	0.969780i	$-0.421546\pi$
$797$	388.909			0.487966			0.243983	+	0.969780i	$0.421546\pi$
$798$		0			0
$799$	−40.0000			−0.0500626
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−520.431			−0.648108
$804$		0			0
$805$	−560.000	+	560.000i	−0.695652	+	0.695652i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			46.6690i			0.0576873i	0.999584	+	0.0288437i	$0.00918250\pi$
$809$			46.6690i			0.0576873i	−0.999584	+	0.0288437i	$0.990818\pi$
$810$		0			0
$811$	−1120.00			−1.38101			−0.690506	−	0.723327i	$-0.742612\pi$
$811$	−1120.00			−1.38101			−0.690506	+	0.723327i	$0.742612\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−367.696	+	367.696i	−0.451160	+	0.451160i
$816$		0			0
$817$			1920.00i			2.35006i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1271.38i			1.54857i	0.632836	+	0.774286i	$0.281891\pi$
$821$			1271.38i			1.54857i	−0.632836	+	0.774286i	$0.718109\pi$
$822$		0			0
$823$		−	828.000i		−	1.00608i	−0.864264	−	0.503038i	$-0.832216\pi$
$823$		−	828.000i		−	1.00608i	0.864264	−	0.503038i	$-0.167784\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	158.392			0.191526			0.0957629	−	0.995404i	$-0.469471\pi$
$827$	158.392			0.191526			0.0957629	+	0.995404i	$0.469471\pi$
$828$		0			0
$829$	−552.000			−0.665862			−0.332931	−	0.942951i	$-0.608038\pi$
$829$	−552.000			−0.665862			−0.332931	+	0.942951i	$0.608038\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−46.6690			−0.0560253
$834$		0			0
$835$	−720.000	−	720.000i	−0.862275	−	0.862275i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	854.185i		−	1.01810i	−0.860737	−	0.509049i	$-0.829997\pi$
$839$		−	854.185i		−	1.01810i	0.860737	−	0.509049i	$-0.170003\pi$
$840$		0			0
$841$	−617.000			−0.733650
$842$		0			0
$843$		0			0
$844$		0			0
$845$	548.008	+	548.008i	0.648530	+	0.648530i
$846$		0			0
$847$		−	28.0000i		−	0.0330579i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2217.49i		−	2.60574i
$852$		0			0
$853$		−	856.000i		−	1.00352i	−0.865008	−	0.501758i	$-0.832687\pi$
$853$		−	856.000i		−	1.00352i	0.865008	−	0.501758i	$-0.167313\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	555.786			0.648525			0.324263	−	0.945967i	$-0.394884\pi$
$857$	555.786			0.648525			0.324263	+	0.945967i	$0.394884\pi$
$858$		0			0
$859$	−472.000			−0.549476			−0.274738	−	0.961519i	$-0.588591\pi$
$859$	−472.000			−0.549476			−0.274738	+	0.961519i	$0.588591\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−441.235			−0.511280			−0.255640	−	0.966772i	$-0.582286\pi$
$863$	−441.235			−0.511280			−0.255640	+	0.966772i	$0.582286\pi$
$864$		0			0
$865$	815.000	+	815.000i	0.942197	+	0.942197i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	407.294i		−	0.468692i
$870$		0			0
$871$	576.000			0.661309
$872$		0			0
$873$		0			0
$874$		0			0
$875$	353.553	+	353.553i	0.404061	+	0.404061i
$876$		0			0
$877$		−	680.000i		−	0.775371i	−0.921792	−	0.387685i	$-0.873275\pi$
$877$		−	680.000i		−	0.775371i	0.921792	−	0.387685i	$-0.126725\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	292.742i		−	0.332284i	−0.986102	−	0.166142i	$-0.946869\pi$
$881$		−	292.742i		−	0.332284i	0.986102	−	0.166142i	$-0.0531310\pi$
$882$		0			0
$883$		−	696.000i		−	0.788222i	−0.919063	−	0.394111i	$-0.871053\pi$
$883$		−	696.000i		−	0.788222i	0.919063	−	0.394111i	$-0.128947\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−882.469			−0.994892			−0.497446	−	0.867495i	$-0.665729\pi$
$887$	−882.469			−0.994892			−0.497446	+	0.867495i	$0.665729\pi$
$888$		0			0
$889$	−112.000			−0.125984
$890$		0			0
$891$		0			0
$892$		0			0
$893$	678.823			0.760160
$894$		0			0
$895$	580.000	−	580.000i	0.648045	−	0.648045i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	152.735i		−	0.169894i
$900$		0			0
$901$	6.00000			0.00665927
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−28.2843	−	28.2843i	−0.0312533	−	0.0312533i
$906$		0			0
$907$			528.000i			0.582139i	0.956702	+	0.291069i	$0.0940111\pi$
$907$			528.000i			0.582139i	−0.956702	+	0.291069i	$0.905989\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1595.23i			1.75108i	0.483147	+	0.875539i	$0.339494\pi$
$911$			1595.23i			1.75108i	−0.483147	+	0.875539i	$0.660506\pi$
$912$		0			0
$913$		−	64.0000i		−	0.0700986i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−339.411			−0.370132
$918$		0			0
$919$	660.000			0.718172			0.359086	−	0.933304i	$-0.383088\pi$
$919$	660.000			0.718172			0.359086	+	0.933304i	$0.383088\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	916.410			0.992861
$924$		0			0
$925$	−1400.00			−1.51351
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1005.51i		−	1.08235i	−0.840909	−	0.541176i	$-0.817979\pi$
$929$		−	1005.51i		−	1.08235i	0.840909	−	0.541176i	$-0.182021\pi$
$930$		0			0
$931$	792.000			0.850698
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−56.5685	+	56.5685i	−0.0605011	+	0.0605011i
$936$		0			0
$937$		−	1568.00i		−	1.67343i	−0.547642	−	0.836713i	$-0.684474\pi$
$937$		−	1568.00i		−	1.67343i	0.547642	−	0.836713i	$-0.315526\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			637.810i			0.677801i	0.940822	+	0.338900i	$0.110055\pi$
$941$			637.810i			0.677801i	−0.940822	+	0.338900i	$0.889945\pi$
$942$		0			0
$943$			952.000i			1.00954i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1629.17			1.72035			0.860176	−	0.509997i	$-0.170353\pi$
$947$	1629.17			1.72035			0.860176	+	0.509997i	$0.170353\pi$
$948$		0			0
$949$	828.000			0.872497
$950$		0			0
$951$		0			0
$952$		0			0
$953$	914.996			0.960122			0.480061	−	0.877235i	$-0.340615\pi$
$953$	914.996			0.960122			0.480061	+	0.877235i	$0.340615\pi$
$954$		0			0
$955$	−1080.00	+	1080.00i	−1.13089	+	1.13089i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			50.9117i			0.0530883i
$960$		0			0
$961$	−945.000			−0.983351
$962$		0			0
$963$		0			0
$964$		0			0
$965$	226.274	−	226.274i	0.234481	−	0.234481i
$966$		0			0
$967$			500.000i			0.517063i	0.966003	+	0.258532i	$0.0832386\pi$
$967$			500.000i			0.517063i	−0.966003	+	0.258532i	$0.916761\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1221.88i		−	1.25837i	−0.777254	−	0.629187i	$-0.783388\pi$
$971$		−	1221.88i		−	1.25837i	0.777254	−	0.629187i	$-0.216612\pi$
$972$		0			0
$973$		−	32.0000i		−	0.0328880i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1743.73			1.78478			0.892388	−	0.451270i	$-0.149029\pi$
$977$	1743.73			1.78478			0.892388	+	0.451270i	$0.149029\pi$
$978$		0			0
$979$	−656.000			−0.670072
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−79.1960			−0.0805656			−0.0402828	−	0.999188i	$-0.512826\pi$
$983$	−79.1960			−0.0805656			−0.0402828	+	0.999188i	$0.512826\pi$
$984$		0			0
$985$	−815.000	−	815.000i	−0.827411	−	0.827411i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	3167.84i		−	3.20307i
$990$		0			0
$991$	−748.000			−0.754793			−0.377397	−	0.926052i	$-0.623181\pi$
$991$	−748.000			−0.754793			−0.377397	+	0.926052i	$0.623181\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1060.66	+	1060.66i	1.06599	+	1.06599i
$996$		0			0
$997$			1464.00i			1.46841i	0.678930	+	0.734203i	$0.262444\pi$
$997$			1464.00i			1.46841i	−0.678930	+	0.734203i	$0.737556\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.c.c.449.1		4
3.2	odd	2	inner	720.3.c.c.449.4		4
4.3	odd	2		90.3.b.a.89.1	✓	4
5.2	odd	4		3600.3.l.c.1601.2		2
5.3	odd	4		3600.3.l.i.1601.2		2
5.4	even	2	inner	720.3.c.c.449.3		4
8.3	odd	2		2880.3.c.h.449.4		4
8.5	even	2		2880.3.c.a.449.4		4
12.11	even	2		90.3.b.a.89.4	yes	4
15.2	even	4		3600.3.l.c.1601.1		2
15.8	even	4		3600.3.l.i.1601.1		2
15.14	odd	2	inner	720.3.c.c.449.2		4
20.3	even	4		450.3.d.b.251.2		2
20.7	even	4		450.3.d.e.251.1		2
20.19	odd	2		90.3.b.a.89.3	yes	4
24.5	odd	2		2880.3.c.a.449.1		4
24.11	even	2		2880.3.c.h.449.1		4
36.7	odd	6		810.3.j.d.539.4		8
36.11	even	6		810.3.j.d.539.1		8
36.23	even	6		810.3.j.d.269.2		8
36.31	odd	6		810.3.j.d.269.3		8
40.19	odd	2		2880.3.c.h.449.2		4
40.29	even	2		2880.3.c.a.449.2		4
60.23	odd	4		450.3.d.b.251.1		2
60.47	odd	4		450.3.d.e.251.2		2
60.59	even	2		90.3.b.a.89.2	yes	4
120.29	odd	2		2880.3.c.a.449.3		4
120.59	even	2		2880.3.c.h.449.3		4
180.59	even	6		810.3.j.d.269.4		8
180.79	odd	6		810.3.j.d.539.2		8
180.119	even	6		810.3.j.d.539.3		8
180.139	odd	6		810.3.j.d.269.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.b.a.89.1	✓	4	4.3	odd	2
90.3.b.a.89.2	yes	4	60.59	even	2
90.3.b.a.89.3	yes	4	20.19	odd	2
90.3.b.a.89.4	yes	4	12.11	even	2
450.3.d.b.251.1		2	60.23	odd	4
450.3.d.b.251.2		2	20.3	even	4
450.3.d.e.251.1		2	20.7	even	4
450.3.d.e.251.2		2	60.47	odd	4
720.3.c.c.449.1		4	1.1	even	1	trivial
720.3.c.c.449.2		4	15.14	odd	2	inner
720.3.c.c.449.3		4	5.4	even	2	inner
720.3.c.c.449.4		4	3.2	odd	2	inner
810.3.j.d.269.1		8	180.139	odd	6
810.3.j.d.269.2		8	36.23	even	6
810.3.j.d.269.3		8	36.31	odd	6
810.3.j.d.269.4		8	180.59	even	6
810.3.j.d.539.1		8	36.11	even	6
810.3.j.d.539.2		8	180.79	odd	6
810.3.j.d.539.3		8	180.119	even	6
810.3.j.d.539.4		8	36.7	odd	6
2880.3.c.a.449.1		4	24.5	odd	2
2880.3.c.a.449.2		4	40.29	even	2
2880.3.c.a.449.3		4	120.29	odd	2
2880.3.c.a.449.4		4	8.5	even	2
2880.3.c.h.449.1		4	24.11	even	2
2880.3.c.h.449.2		4	40.19	odd	2
2880.3.c.h.449.3		4	120.59	even	2
2880.3.c.h.449.4		4	8.3	odd	2
3600.3.l.c.1601.1		2	15.2	even	4
3600.3.l.c.1601.2		2	5.2	odd	4
3600.3.l.i.1601.1		2	15.8	even	4
3600.3.l.i.1601.2		2	5.3	odd	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	720.3.c.c.449.1

Level	$720$
Weight	$3$
Character	720.449
Analytic conductor	$19.619$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$