LMFDB - Embedded newform 720.4.f.l.289.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,4,Mod(289,720)]

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, 0, 1]))

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

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

chi := DirichletCharacter("720.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	720.f (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:	$42.4813752041$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{31})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 15x^{2} + 64 $ x^4 - 15*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.3
Root			$-2.78388 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.4.f.l.289.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.1355 - 1.00000i) q^{5} -22.2711i q^{7} +O(q^{10})$	$q+(11.1355 - 1.00000i) q^{5} -22.2711i q^{7} -22.2711 q^{11} +66.8132i q^{13} +62.0000i q^{17} +84.0000 q^{19} +140.000i q^{23} +(123.000 - 22.2711i) q^{25} +200.440 q^{29} -16.0000 q^{31} +(-22.2711 - 248.000i) q^{35} -244.982i q^{37} +222.711 q^{41} +356.337i q^{43} -100.000i q^{47} -153.000 q^{49} -738.000i q^{53} +(-248.000 + 22.2711i) q^{55} +645.861 q^{59} -358.000 q^{61} +(66.8132 + 744.000i) q^{65} -846.300i q^{67} +935.384 q^{71} +445.421i q^{73} +496.000i q^{77} -936.000 q^{79} +1304.00i q^{83} +(62.0000 + 690.403i) q^{85} +712.674 q^{89} +1488.00 q^{91} +(935.384 - 84.0000i) q^{95} -757.216i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 336 q^{19} + 492 q^{25} - 64 q^{31} - 612 q^{49} - 992 q^{55} - 1432 q^{61} - 3744 q^{79} + 248 q^{85} + 5952 q^{91}+O(q^{100}) $ 4 * q + 336 * q^19 + 492 * q^25 - 64 * q^31 - 612 * q^49 - 992 * q^55 - 1432 * q^61 - 3744 * q^79 + 248 * q^85 + 5952 * q^91

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
$5$	11.1355	−	1.00000i	0.995992	−	0.0894427i
$5$	11.1355	−	1.00000i	0.995992	−	0.0894427i
$6$		0			0
$7$		−	22.2711i		−	1.20252i	−0.799052	−	0.601262i	$-0.794665\pi$
$7$		−	22.2711i		−	1.20252i	0.799052	−	0.601262i	$-0.205335\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−22.2711			−0.610452			−0.305226	−	0.952280i	$-0.598732\pi$
$11$	−22.2711			−0.610452			−0.305226	+	0.952280i	$0.598732\pi$
$12$		0			0
$13$			66.8132i			1.42543i	0.701452	+	0.712717i	$0.252536\pi$
$13$			66.8132i			1.42543i	−0.701452	+	0.712717i	$0.747464\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			62.0000i			0.884542i	0.896882	+	0.442271i	$0.145827\pi$
$17$			62.0000i			0.884542i	−0.896882	+	0.442271i	$0.854173\pi$
$18$		0			0
$19$	84.0000			1.01426			0.507130	−	0.861870i	$-0.330707\pi$
$19$	84.0000			1.01426			0.507130	+	0.861870i	$0.330707\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			140.000i			1.26922i	0.772833	+	0.634609i	$0.218839\pi$
$23$			140.000i			1.26922i	−0.772833	+	0.634609i	$0.781161\pi$
$24$		0			0
$25$	123.000	−	22.2711i	0.984000	−	0.178168i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	200.440			1.28347			0.641736	−	0.766926i	$-0.278215\pi$
$29$	200.440			1.28347			0.641736	+	0.766926i	$0.278215\pi$
$30$		0			0
$31$	−16.0000			−0.0926995			−0.0463498	−	0.998925i	$-0.514759\pi$
$31$	−16.0000			−0.0926995			−0.0463498	+	0.998925i	$0.514759\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−22.2711	−	248.000i	−0.107557	−	1.19770i
$36$		0			0
$37$		−	244.982i		−	1.08851i	−0.838921	−	0.544253i	$-0.816813\pi$
$37$		−	244.982i		−	1.08851i	0.838921	−	0.544253i	$-0.183187\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	222.711			0.848330			0.424165	−	0.905585i	$-0.360568\pi$
$41$	222.711			0.848330			0.424165	+	0.905585i	$0.360568\pi$
$42$		0			0
$43$			356.337i			1.26374i	0.775074	+	0.631871i	$0.217713\pi$
$43$			356.337i			1.26374i	−0.775074	+	0.631871i	$0.782287\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	100.000i		−	0.310351i	−0.987887	−	0.155176i	$-0.950406\pi$
$47$		−	100.000i		−	0.310351i	0.987887	−	0.155176i	$-0.0495943\pi$
$48$		0			0
$49$	−153.000			−0.446064
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	738.000i		−	1.91268i	−0.292255	−	0.956341i	$-0.594405\pi$
$53$		−	738.000i		−	1.91268i	0.292255	−	0.956341i	$-0.405595\pi$
$54$		0			0
$55$	−248.000	+	22.2711i	−0.608006	+	0.0546005i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	645.861			1.42515			0.712575	−	0.701596i	$-0.247529\pi$
$59$	645.861			1.42515			0.712575	+	0.701596i	$0.247529\pi$
$60$		0			0
$61$	−358.000			−0.751430			−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000			−0.751430			−0.375715	+	0.926735i	$0.622603\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	66.8132	+	744.000i	0.127495	+	1.41972i
$66$		0			0
$67$		−	846.300i		−	1.54316i	−0.636130	−	0.771582i	$-0.719466\pi$
$67$		−	846.300i		−	1.54316i	0.636130	−	0.771582i	$-0.280534\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	935.384			1.56352			0.781758	−	0.623581i	$-0.214323\pi$
$71$	935.384			1.56352			0.781758	+	0.623581i	$0.214323\pi$
$72$		0			0
$73$			445.421i			0.714145i	0.934077	+	0.357073i	$0.116225\pi$
$73$			445.421i			0.714145i	−0.934077	+	0.357073i	$0.883775\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			496.000i			0.734084i
$78$		0			0
$79$	−936.000			−1.33302			−0.666508	−	0.745498i	$-0.732212\pi$
$79$	−936.000			−1.33302			−0.666508	+	0.745498i	$0.732212\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1304.00i			1.72449i	0.506492	+	0.862245i	$0.330942\pi$
$83$			1304.00i			1.72449i	−0.506492	+	0.862245i	$0.669058\pi$
$84$		0			0
$85$	62.0000	+	690.403i	0.0791158	+	0.880996i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	712.674			0.848801			0.424400	−	0.905475i	$-0.360485\pi$
$89$	712.674			0.848801			0.424400	+	0.905475i	$0.360485\pi$
$90$		0			0
$91$	1488.00			1.71412
$92$		0			0
$93$		0			0
$94$		0			0
$95$	935.384	−	84.0000i	1.01019	−	0.0907181i
$96$		0			0
$97$		−	757.216i		−	0.792615i	−0.918118	−	0.396307i	$-0.870291\pi$
$97$		−	757.216i		−	0.792615i	0.918118	−	0.396307i	$-0.129709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	645.861			0.636292			0.318146	−	0.948042i	$-0.396940\pi$
$101$	645.861			0.636292			0.318146	+	0.948042i	$0.396940\pi$
$102$		0			0
$103$		−	690.403i		−	0.660460i	−0.943900	−	0.330230i	$-0.892874\pi$
$103$		−	690.403i		−	0.660460i	0.943900	−	0.330230i	$-0.107126\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	696.000i		−	0.628830i	−0.949286	−	0.314415i	$-0.898192\pi$
$107$		−	696.000i		−	0.628830i	0.949286	−	0.314415i	$-0.101808\pi$
$108$		0			0
$109$	1142.00			1.00352			0.501760	−	0.865007i	$-0.332686\pi$
$109$	1142.00			1.00352			0.501760	+	0.865007i	$0.332686\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			78.0000i			0.0649347i	0.999473	+	0.0324674i	$0.0103365\pi$
$113$			78.0000i			0.0649347i	−0.999473	+	0.0324674i	$0.989664\pi$
$114$		0			0
$115$	140.000	+	1558.97i	0.113522	+	1.26413i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1380.81			1.06368
$120$		0			0
$121$	−835.000			−0.627348
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1347.40	−	371.000i	0.964120	−	0.265466i
$126$		0			0
$127$			1313.99i			0.918094i	0.888412	+	0.459047i	$0.151809\pi$
$127$			1313.99i			0.918094i	−0.888412	+	0.459047i	$0.848191\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	467.692			0.311927			0.155964	−	0.987763i	$-0.450152\pi$
$131$	467.692			0.311927			0.155964	+	0.987763i	$0.450152\pi$
$132$		0			0
$133$		−	1870.77i		−	1.21967i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1062.00i		−	0.662283i	−0.943581	−	0.331142i	$-0.892566\pi$
$137$		−	1062.00i		−	0.662283i	0.943581	−	0.331142i	$-0.107434\pi$
$138$		0			0
$139$	−860.000			−0.524779			−0.262389	−	0.964962i	$-0.584510\pi$
$139$	−860.000			−0.524779			−0.262389	+	0.964962i	$0.584510\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	1488.00i		−	0.870160i
$144$		0			0
$145$	2232.00	−	200.440i	1.27833	−	0.114797i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1358.53			0.746950			0.373475	−	0.927640i	$-0.378166\pi$
$149$	1358.53			0.746950			0.373475	+	0.927640i	$0.378166\pi$
$150$		0			0
$151$	1376.00			0.741571			0.370786	−	0.928718i	$-0.379088\pi$
$151$	1376.00			0.741571			0.370786	+	0.928718i	$0.379088\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−178.168	+	16.0000i	−0.0923280	+	0.00829130i
$156$		0			0
$157$			1848.50i			0.939657i	0.882758	+	0.469829i	$0.155684\pi$
$157$			1848.50i			0.939657i	−0.882758	+	0.469829i	$0.844316\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3117.95			1.52627
$162$		0			0
$163$		−	222.711i		−	0.107019i	−0.998567	−	0.0535093i	$-0.982959\pi$
$163$		−	222.711i		−	0.107019i	0.998567	−	0.0535093i	$-0.0170407\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2172.00i			1.00643i	0.864160	+	0.503217i	$0.167850\pi$
$167$			2172.00i			1.00643i	−0.864160	+	0.503217i	$0.832150\pi$
$168$		0			0
$169$	−2267.00			−1.03186
$170$		0			0
$171$		0			0
$172$		0			0
$173$			3434.00i			1.50915i	0.656216	+	0.754573i	$0.272156\pi$
$173$			3434.00i			1.50915i	−0.656216	+	0.754573i	$0.727844\pi$
$174$		0			0
$175$	−496.000	−	2739.34i	−0.214252	−	1.18328i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3362.93			−1.40423			−0.702115	−	0.712064i	$-0.747761\pi$
$179$	−3362.93			−1.40423			−0.702115	+	0.712064i	$0.747761\pi$
$180$		0			0
$181$	−1678.00			−0.689087			−0.344544	−	0.938770i	$-0.611966\pi$
$181$	−1678.00			−0.689087			−0.344544	+	0.938770i	$0.611966\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−244.982	−	2728.00i	−0.0973590	−	1.08414i
$186$		0			0
$187$		−	1380.81i		−	0.539971i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1737.14			0.658090			0.329045	−	0.944314i	$-0.393273\pi$
$191$	1737.14			0.658090			0.329045	+	0.944314i	$0.393273\pi$
$192$		0			0
$193$			2449.82i			0.913687i	0.889547	+	0.456844i	$0.151020\pi$
$193$			2449.82i			0.913687i	−0.889547	+	0.456844i	$0.848980\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			526.000i			0.190233i	0.995466	+	0.0951166i	$0.0303224\pi$
$197$			526.000i			0.190233i	−0.995466	+	0.0951166i	$0.969678\pi$
$198$		0			0
$199$	−744.000			−0.265029			−0.132514	−	0.991181i	$-0.542305\pi$
$199$	−744.000			−0.265029			−0.132514	+	0.991181i	$0.542305\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	4464.00i		−	1.54341i
$204$		0			0
$205$	2480.00	−	222.711i	0.844930	−	0.0758770i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1870.77			−0.619157
$210$		0			0
$211$	−1476.00			−0.481574			−0.240787	−	0.970578i	$-0.577405\pi$
$211$	−1476.00			−0.481574			−0.240787	+	0.970578i	$0.577405\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	356.337	+	3968.00i	0.113032	+	1.25868i
$216$		0			0
$217$			356.337i			0.111473i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4142.42			−1.26086
$222$		0			0
$223$			824.029i			0.247449i	0.992317	+	0.123724i	$0.0394839\pi$
$223$			824.029i			0.247449i	−0.992317	+	0.123724i	$0.960516\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2200.00i			0.643256i	0.946866	+	0.321628i	$0.104230\pi$
$227$			2200.00i			0.643256i	−0.946866	+	0.321628i	$0.895770\pi$
$228$		0			0
$229$	−5246.00			−1.51382			−0.756911	−	0.653517i	$-0.773293\pi$
$229$	−5246.00			−1.51382			−0.756911	+	0.653517i	$0.773293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			842.000i			0.236744i	0.992969	+	0.118372i	$0.0377675\pi$
$233$			842.000i			0.236744i	−0.992969	+	0.118372i	$0.962233\pi$
$234$		0			0
$235$	−100.000	−	1113.55i	−0.0277586	−	0.309107i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5835.02			1.57923			0.789615	−	0.613603i	$-0.210280\pi$
$239$	5835.02			1.57923			0.789615	+	0.613603i	$0.210280\pi$
$240$		0			0
$241$	2530.00			0.676231			0.338115	−	0.941105i	$-0.390211\pi$
$241$	2530.00			0.676231			0.338115	+	0.941105i	$0.390211\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1703.74	+	153.000i	−0.444276	+	0.0398972i
$246$		0			0
$247$			5612.31i			1.44576i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6748.13			−1.69696			−0.848482	−	0.529223i	$-0.822483\pi$
$251$	−6748.13			−1.69696			−0.848482	+	0.529223i	$0.822483\pi$
$252$		0			0
$253$		−	3117.95i		−	0.774797i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			258.000i			0.0626210i	0.999510	+	0.0313105i	$0.00996807\pi$
$257$			258.000i			0.0626210i	−0.999510	+	0.0313105i	$0.990032\pi$
$258$		0			0
$259$	−5456.00			−1.30895
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	772.000i		−	0.181002i	−0.995896	−	0.0905011i	$-0.971153\pi$
$263$		−	772.000i		−	0.181002i	0.995896	−	0.0905011i	$-0.0288469\pi$
$264$		0			0
$265$	−738.000	−	8218.02i	−0.171075	−	1.90501i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2605.71			−0.590607			−0.295303	−	0.955404i	$-0.595421\pi$
$269$	−2605.71			−0.590607			−0.295303	+	0.955404i	$0.595421\pi$
$270$		0			0
$271$	6392.00			1.43279			0.716395	−	0.697694i	$-0.245791\pi$
$271$	6392.00			1.43279			0.716395	+	0.697694i	$0.245791\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2739.34	+	496.000i	−0.600685	+	0.108763i
$276$		0			0
$277$			1759.41i			0.381635i	0.981626	+	0.190818i	$0.0611139\pi$
$277$			1759.41i			0.381635i	−0.981626	+	0.190818i	$0.938886\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6414.06			−1.36168			−0.680838	−	0.732434i	$-0.738384\pi$
$281$	−6414.06			−1.36168			−0.680838	+	0.732434i	$0.738384\pi$
$282$		0			0
$283$			1380.81i			0.290037i	0.989429	+	0.145018i	$0.0463241\pi$
$283$			1380.81i			0.290037i	−0.989429	+	0.145018i	$0.953676\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4960.00i		−	1.02014i
$288$		0			0
$289$	1069.00			0.217586
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1374.00i		−	0.273959i	−0.990574	−	0.136979i	$-0.956261\pi$
$293$		−	1374.00i		−	0.273959i	0.990574	−	0.136979i	$-0.0437394\pi$
$294$		0			0
$295$	7192.00	−	645.861i	1.41944	−	0.127469i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9353.84			−1.80919
$300$		0			0
$301$	7936.00			1.51968
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3986.52	+	358.000i	−0.748418	+	0.0672099i
$306$		0			0
$307$		−	267.253i		−	0.0496838i	−0.999691	−	0.0248419i	$-0.992092\pi$
$307$		−	267.253i		−	0.0496838i	0.999691	−	0.0248419i	$-0.00790823\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3607.91			0.657832			0.328916	−	0.944359i	$-0.393317\pi$
$311$	3607.91			0.657832			0.328916	+	0.944359i	$0.393317\pi$
$312$		0			0
$313$			7794.87i			1.40764i	0.710377	+	0.703821i	$0.248524\pi$
$313$			7794.87i			1.40764i	−0.710377	+	0.703821i	$0.751476\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9334.00i		−	1.65378i	−0.562360	−	0.826892i	$-0.690107\pi$
$317$		−	9334.00i		−	1.65378i	0.562360	−	0.826892i	$-0.309893\pi$
$318$		0			0
$319$	−4464.00			−0.783498
$320$		0			0
$321$		0			0
$322$		0			0
$323$			5208.00i			0.897154i
$324$		0			0
$325$	1488.00	+	8218.02i	0.253967	+	1.40263i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2227.11			−0.373205
$330$		0			0
$331$	−1636.00			−0.271670			−0.135835	−	0.990731i	$-0.543372\pi$
$331$	−1636.00			−0.271670			−0.135835	+	0.990731i	$0.543372\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−846.300	−	9424.00i	−0.138025	−	1.53698i
$336$		0			0
$337$		−	178.168i		−	0.0287996i	−0.999896	−	0.0143998i	$-0.995416\pi$
$337$		−	178.168i		−	0.0287996i	0.999896	−	0.0143998i	$-0.00458375\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	356.337			0.0565886
$342$		0			0
$343$		−	4231.50i		−	0.666121i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3840.00i		−	0.594069i	−0.954867	−	0.297035i	$-0.904002\pi$
$347$		−	3840.00i		−	0.594069i	0.954867	−	0.297035i	$-0.0959977\pi$
$348$		0			0
$349$	−6682.00			−1.02487			−0.512434	−	0.858726i	$-0.671256\pi$
$349$	−6682.00			−1.02487			−0.512434	+	0.858726i	$0.671256\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2110.00i		−	0.318142i	−0.987267	−	0.159071i	$-0.949150\pi$
$353$		−	2110.00i		−	0.318142i	0.987267	−	0.159071i	$-0.0508498\pi$
$354$		0			0
$355$	10416.0	−	935.384i	1.55725	−	0.139845i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1870.77			−0.275029			−0.137514	−	0.990500i	$-0.543911\pi$
$359$	−1870.77			−0.275029			−0.137514	+	0.990500i	$0.543911\pi$
$360$		0			0
$361$	197.000			0.0287214
$362$		0			0
$363$		0			0
$364$		0			0
$365$	445.421	+	4960.00i	0.0638751	+	0.711283i
$366$		0			0
$367$		−	9643.37i		−	1.37161i	−0.727787	−	0.685803i	$-0.759451\pi$
$367$		−	9643.37i		−	1.37161i	0.727787	−	0.685803i	$-0.240549\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−16436.0			−2.30005
$372$		0			0
$373$		−	11558.7i		−	1.60452i	−0.596975	−	0.802260i	$-0.703631\pi$
$373$		−	11558.7i		−	1.60452i	0.596975	−	0.802260i	$-0.296369\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			13392.0i			1.82950i
$378$		0			0
$379$	−6220.00			−0.843008			−0.421504	−	0.906827i	$-0.638498\pi$
$379$	−6220.00			−0.843008			−0.421504	+	0.906827i	$0.638498\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			13900.0i			1.85446i	0.374497	+	0.927228i	$0.377815\pi$
$383$			13900.0i			1.85446i	−0.374497	+	0.927228i	$0.622185\pi$
$384$		0			0
$385$	496.000	+	5523.22i	0.0656584	+	0.731141i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3630.18			−0.473156			−0.236578	−	0.971613i	$-0.576026\pi$
$389$	−3630.18			−0.473156			−0.236578	+	0.971613i	$0.576026\pi$
$390$		0			0
$391$	−8680.00			−1.12268
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10422.9	+	936.000i	−1.32767	+	0.119229i
$396$		0			0
$397$		−	512.234i		−	0.0647564i	−0.999476	−	0.0323782i	$-0.989692\pi$
$397$		−	512.234i		−	0.0647564i	0.999476	−	0.0323782i	$-0.0103081\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2984.32			0.371646			0.185823	−	0.982583i	$-0.440505\pi$
$401$	2984.32			0.371646			0.185823	+	0.982583i	$0.440505\pi$
$402$		0			0
$403$		−	1069.01i		−	0.132137i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5456.00i			0.664481i
$408$		0			0
$409$	−14822.0			−1.79193			−0.895967	−	0.444121i	$-0.853516\pi$
$409$	−14822.0			−1.79193			−0.895967	+	0.444121i	$0.853516\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	14384.0i		−	1.71378i
$414$		0			0
$415$	1304.00	+	14520.7i	0.154243	+	1.71758i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8663.44			−1.01011			−0.505056	−	0.863087i	$-0.668528\pi$
$419$	−8663.44			−1.01011			−0.505056	+	0.863087i	$0.668528\pi$
$420$		0			0
$421$	4894.00			0.566553			0.283277	−	0.959038i	$-0.408579\pi$
$421$	4894.00			0.566553			0.283277	+	0.959038i	$0.408579\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1380.81	+	7626.00i	0.157597	+	0.870389i
$426$		0			0
$427$			7973.04i			0.903612i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9799.27			1.09516			0.547580	−	0.836753i	$-0.315549\pi$
$431$	9799.27			1.09516			0.547580	+	0.836753i	$0.315549\pi$
$432$		0			0
$433$		−	3697.00i		−	0.410315i	−0.978729	−	0.205157i	$-0.934229\pi$
$433$		−	3697.00i		−	0.410315i	0.978729	−	0.205157i	$-0.0657706\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			11760.0i			1.28732i
$438$		0			0
$439$	−9288.00			−1.00978			−0.504888	−	0.863185i	$-0.668466\pi$
$439$	−9288.00			−1.00978			−0.504888	+	0.863185i	$0.668466\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	7368.00i		−	0.790213i	−0.918635	−	0.395106i	$-0.870708\pi$
$443$		−	7368.00i		−	0.790213i	0.918635	−	0.395106i	$-0.129292\pi$
$444$		0			0
$445$	7936.00	−	712.674i	0.845399	−	0.0759191i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	400.879			0.0421351			0.0210675	−	0.999778i	$-0.493293\pi$
$449$	400.879			0.0421351			0.0210675	+	0.999778i	$0.493293\pi$
$450$		0			0
$451$	−4960.00			−0.517865
$452$		0			0
$453$		0			0
$454$		0			0
$455$	16569.7	−	1488.00i	1.70725	−	0.153315i
$456$		0			0
$457$		−	14743.4i		−	1.50912i	−0.656230	−	0.754561i	$-0.727850\pi$
$457$		−	14743.4i		−	1.50912i	0.656230	−	0.754561i	$-0.272150\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4743.74			−0.479258			−0.239629	−	0.970865i	$-0.577026\pi$
$461$	−4743.74			−0.479258			−0.239629	+	0.970865i	$0.577026\pi$
$462$		0			0
$463$			7282.64i			0.731000i	0.930811	+	0.365500i	$0.119102\pi$
$463$			7282.64i			0.731000i	−0.930811	+	0.365500i	$0.880898\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	5664.00i		−	0.561239i	−0.959819	−	0.280620i	$-0.909460\pi$
$467$		−	5664.00i		−	0.561239i	0.959819	−	0.280620i	$-0.0905399\pi$
$468$		0			0
$469$	−18848.0			−1.85569
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	7936.00i		−	0.771454i
$474$		0			0
$475$	10332.0	−	1870.77i	0.998031	−	0.180709i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10111.1			0.964480			0.482240	−	0.876039i	$-0.339823\pi$
$479$	10111.1			0.964480			0.482240	+	0.876039i	$0.339823\pi$
$480$		0			0
$481$	16368.0			1.55159
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−757.216	−	8432.00i	−0.0708936	−	0.789438i
$486$		0			0
$487$		−	11558.7i		−	1.07551i	−0.843101	−	0.537755i	$-0.819272\pi$
$487$		−	11558.7i		−	1.07551i	0.843101	−	0.537755i	$-0.180728\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	467.692			0.0429871			0.0214935	−	0.999769i	$-0.493158\pi$
$491$	467.692			0.0429871			0.0214935	+	0.999769i	$0.493158\pi$
$492$		0			0
$493$			12427.3i			1.13528i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	20832.0i		−	1.88017i
$498$		0			0
$499$	9204.00			0.825707			0.412853	−	0.910798i	$-0.364532\pi$
$499$	9204.00			0.825707			0.412853	+	0.910798i	$0.364532\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	1260.00i		−	0.111691i	−0.998439	−	0.0558455i	$-0.982215\pi$
$503$		−	1260.00i		−	0.111691i	0.998439	−	0.0558455i	$-0.0177854\pi$
$504$		0			0
$505$	7192.00	−	645.861i	0.633742	−	0.0569117i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2071.21			0.180363			0.0901814	−	0.995925i	$-0.471255\pi$
$509$	2071.21			0.180363			0.0901814	+	0.995925i	$0.471255\pi$
$510$		0			0
$511$	9920.00			0.858777
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−690.403	−	7688.00i	−0.0590734	−	0.657813i
$516$		0			0
$517$			2227.11i			0.189455i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5300.51			−0.445719			−0.222860	−	0.974851i	$-0.571539\pi$
$521$	−5300.51			−0.445719			−0.222860	+	0.974851i	$0.571539\pi$
$522$		0			0
$523$		−	9309.30i		−	0.778331i	−0.921168	−	0.389166i	$-0.872763\pi$
$523$		−	9309.30i		−	0.778331i	0.921168	−	0.389166i	$-0.127237\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	992.000i		−	0.0819966i
$528$		0			0
$529$	−7433.00			−0.610915
$530$		0			0
$531$		0			0
$532$		0			0
$533$			14880.0i			1.20924i
$534$		0			0
$535$	−696.000	−	7750.33i	−0.0562443	−	0.626310i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3407.47			0.272301
$540$		0			0
$541$	9658.00			0.767523			0.383761	−	0.923432i	$-0.374629\pi$
$541$	9658.00			0.767523			0.383761	+	0.923432i	$0.374629\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12716.8	−	1142.00i	0.999499	−	0.0897576i
$546$		0			0
$547$		−	890.842i		−	0.0696338i	−0.999394	−	0.0348169i	$-0.988915\pi$
$547$		−	890.842i		−	0.0696338i	0.999394	−	0.0348169i	$-0.0110848\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	16836.9			1.30177
$552$		0			0
$553$			20845.7i			1.60298i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			9066.00i			0.689657i	0.938666	+	0.344828i	$0.112063\pi$
$557$			9066.00i			0.689657i	−0.938666	+	0.344828i	$0.887937\pi$
$558$		0			0
$559$	−23808.0			−1.80138
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1568.00i		−	0.117377i	−0.998276	−	0.0586886i	$-0.981308\pi$
$563$		−	1568.00i		−	0.117377i	0.998276	−	0.0586886i	$-0.0186919\pi$
$564$		0			0
$565$	78.0000	+	868.571i	0.00580794	+	0.0646745i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2093.48			−0.154241			−0.0771206	−	0.997022i	$-0.524573\pi$
$569$	−2093.48			−0.154241			−0.0771206	+	0.997022i	$0.524573\pi$
$570$		0			0
$571$	13916.0			1.01991			0.509953	−	0.860202i	$-0.329663\pi$
$571$	13916.0			1.01991			0.509953	+	0.860202i	$0.329663\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3117.95	+	17220.0i	0.226135	+	1.24891i
$576$		0			0
$577$			8106.66i			0.584896i	0.956281	+	0.292448i	$0.0944698\pi$
$577$			8106.66i			0.584896i	−0.956281	+	0.292448i	$0.905530\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	29041.5			2.07374
$582$		0			0
$583$			16436.0i			1.16760i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			20808.0i			1.46310i	0.681789	+	0.731549i	$0.261202\pi$
$587$			20808.0i			1.46310i	−0.681789	+	0.731549i	$0.738798\pi$
$588$		0			0
$589$	−1344.00			−0.0940213
$590$		0			0
$591$		0			0
$592$		0			0
$593$			21486.0i			1.48790i	0.668236	+	0.743950i	$0.267050\pi$
$593$			21486.0i			1.48790i	−0.668236	+	0.743950i	$0.732950\pi$
$594$		0			0
$595$	15376.0	−	1380.81i	1.05942	−	0.0951387i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25923.5			1.76829			0.884145	−	0.467212i	$-0.154742\pi$
$599$	25923.5			1.76829			0.884145	+	0.467212i	$0.154742\pi$
$600$		0			0
$601$	−250.000			−0.0169679			−0.00848395	−	0.999964i	$-0.502701\pi$
$601$	−250.000			−0.0169679			−0.00848395	+	0.999964i	$0.502701\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9298.17	+	835.000i	−0.624833	+	0.0561117i
$606$		0			0
$607$		−	13340.4i		−	0.892041i	−0.895023	−	0.446020i	$-0.852841\pi$
$607$		−	13340.4i		−	0.892041i	0.895023	−	0.446020i	$-0.147159\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6681.32			0.442385
$612$		0			0
$613$		−	14721.2i		−	0.969955i	−0.874527	−	0.484977i	$-0.838828\pi$
$613$		−	14721.2i		−	0.969955i	0.874527	−	0.484977i	$-0.161172\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	14410.0i		−	0.940235i	−0.882604	−	0.470117i	$-0.844212\pi$
$617$		−	14410.0i		−	0.940235i	0.882604	−	0.470117i	$-0.155788\pi$
$618$		0			0
$619$	−21580.0			−1.40125			−0.700625	−	0.713530i	$-0.747095\pi$
$619$	−21580.0			−1.40125			−0.700625	+	0.713530i	$0.747095\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	15872.0i		−	1.02070i
$624$		0			0
$625$	14633.0	−	5478.68i	0.936512	−	0.350636i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15188.9			0.962829
$630$		0			0
$631$	15160.0			0.956434			0.478217	−	0.878242i	$-0.341283\pi$
$631$	15160.0			0.956434			0.478217	+	0.878242i	$0.341283\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1313.99	+	14632.0i	0.0821168	+	0.914415i
$636$		0			0
$637$		−	10222.4i		−	0.635835i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6102.27			0.376014			0.188007	−	0.982168i	$-0.439797\pi$
$641$	6102.27			0.376014			0.188007	+	0.982168i	$0.439797\pi$
$642$		0			0
$643$			14298.0i			0.876919i	0.898751	+	0.438459i	$0.144476\pi$
$643$			14298.0i			0.876919i	−0.898751	+	0.438459i	$0.855524\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	11916.0i		−	0.724059i	−0.932167	−	0.362030i	$-0.882084\pi$
$647$		−	11916.0i		−	0.724059i	0.932167	−	0.362030i	$-0.117916\pi$
$648$		0			0
$649$	−14384.0			−0.869987
$650$		0			0
$651$		0			0
$652$		0			0
$653$			24646.0i			1.47699i	0.674261	+	0.738493i	$0.264462\pi$
$653$			24646.0i			1.47699i	−0.674261	+	0.738493i	$0.735538\pi$
$654$		0			0
$655$	5208.00	−	467.692i	0.310677	−	0.0278996i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17928.2			−1.05976			−0.529881	−	0.848072i	$-0.677764\pi$
$659$	−17928.2			−1.05976			−0.529881	+	0.848072i	$0.677764\pi$
$660$		0			0
$661$	−14638.0			−0.861350			−0.430675	−	0.902507i	$-0.641724\pi$
$661$	−14638.0			−0.861350			−0.430675	+	0.902507i	$0.641724\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1870.77	−	20832.0i	−0.109091	−	1.21478i
$666$		0			0
$667$			28061.5i			1.62901i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7973.04			0.458712
$672$		0			0
$673$		−	31134.9i		−	1.78330i	−0.452721	−	0.891652i	$-0.649547\pi$
$673$		−	31134.9i		−	1.78330i	0.452721	−	0.891652i	$-0.350453\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	6382.00i		−	0.362305i	−0.983455	−	0.181152i	$-0.942017\pi$
$677$		−	6382.00i		−	0.362305i	0.983455	−	0.181152i	$-0.0579827\pi$
$678$		0			0
$679$	−16864.0			−0.953138
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4544.00i			0.254570i	0.991866	+	0.127285i	$0.0406263\pi$
$683$			4544.00i			0.254570i	−0.991866	+	0.127285i	$0.959374\pi$
$684$		0			0
$685$	−1062.00	−	11825.9i	−0.0592364	−	0.659629i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	49308.1			2.72640
$690$		0			0
$691$	−13564.0			−0.746742			−0.373371	−	0.927682i	$-0.621798\pi$
$691$	−13564.0			−0.746742			−0.373371	+	0.927682i	$0.621798\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9576.55	+	860.000i	−0.522675	+	0.0469376i
$696$		0			0
$697$			13808.1i			0.750384i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1492.16			0.0803968			0.0401984	−	0.999192i	$-0.487201\pi$
$701$	1492.16			0.0803968			0.0401984	+	0.999192i	$0.487201\pi$
$702$		0			0
$703$		−	20578.5i		−	1.10403i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	14384.0i		−	0.765157i
$708$		0			0
$709$	17826.0			0.944245			0.472122	−	0.881533i	$-0.343488\pi$
$709$	17826.0			0.944245			0.472122	+	0.881533i	$0.343488\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2240.00i		−	0.117656i
$714$		0			0
$715$	−1488.00	−	16569.7i	−0.0778294	−	0.866672i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18796.8			0.974967			0.487484	−	0.873132i	$-0.337915\pi$
$719$	18796.8			0.974967			0.487484	+	0.873132i	$0.337915\pi$
$720$		0			0
$721$	−15376.0			−0.794219
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24654.1	−	4464.00i	1.26294	−	0.228674i
$726$		0			0
$727$			33072.5i			1.68720i	0.536975	+	0.843598i	$0.319567\pi$
$727$			33072.5i			1.68720i	−0.536975	+	0.843598i	$0.680433\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−22092.9			−1.11783
$732$		0			0
$733$			34765.1i			1.75181i	0.482481	+	0.875907i	$0.339736\pi$
$733$			34765.1i			1.75181i	−0.482481	+	0.875907i	$0.660264\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			18848.0i			0.942028i
$738$		0			0
$739$	−22940.0			−1.14190			−0.570948	−	0.820986i	$-0.693424\pi$
$739$	−22940.0			−1.14190			−0.570948	+	0.820986i	$0.693424\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	26460.0i		−	1.30649i	−0.757146	−	0.653246i	$-0.773407\pi$
$743$		−	26460.0i		−	1.30649i	0.757146	−	0.653246i	$-0.226593\pi$
$744$		0			0
$745$	15128.0	−	1358.53i	0.743956	−	0.0668092i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15500.7			−0.756184
$750$		0			0
$751$	30160.0			1.46545			0.732726	−	0.680524i	$-0.238248\pi$
$751$	30160.0			1.46545			0.732726	+	0.680524i	$0.238248\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15322.5	−	1376.00i	0.738599	−	0.0663282i
$756$		0			0
$757$			18596.3i			0.892860i	0.894818	+	0.446430i	$0.147305\pi$
$757$			18596.3i			0.892860i	−0.894818	+	0.446430i	$0.852695\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5478.68			0.260975			0.130488	−	0.991450i	$-0.458346\pi$
$761$	5478.68			0.260975			0.130488	+	0.991450i	$0.458346\pi$
$762$		0			0
$763$		−	25433.5i		−	1.20676i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			43152.0i			2.03146i
$768$		0			0
$769$	−29406.0			−1.37894			−0.689472	−	0.724313i	$-0.742157\pi$
$769$	−29406.0			−1.37894			−0.689472	+	0.724313i	$0.742157\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	31122.0i		−	1.44810i	−0.689748	−	0.724050i	$-0.742279\pi$
$773$		−	31122.0i		−	1.44810i	0.689748	−	0.724050i	$-0.257721\pi$
$774$		0			0
$775$	−1968.00	+	356.337i	−0.0912163	+	0.0165161i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18707.7			0.860427
$780$		0			0
$781$	−20832.0			−0.954453
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1848.50	+	20584.0i	0.0840455	+	0.935891i
$786$		0			0
$787$		−	34431.1i		−	1.55951i	−0.626085	−	0.779755i	$-0.715344\pi$
$787$		−	34431.1i		−	1.55951i	0.626085	−	0.779755i	$-0.284656\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1737.14			0.0780856
$792$		0			0
$793$		−	23919.1i		−	1.07111i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13066.0i		−	0.580704i	−0.956920	−	0.290352i	$-0.906228\pi$
$797$		−	13066.0i		−	0.580704i	0.956920	−	0.290352i	$-0.0937725\pi$
$798$		0			0
$799$	6200.00			0.274518
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	9920.00i		−	0.435952i
$804$		0			0
$805$	34720.0	−	3117.95i	1.52015	−	0.136513i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39954.3			−1.73636			−0.868181	−	0.496247i	$-0.834711\pi$
$809$	−39954.3			−1.73636			−0.868181	+	0.496247i	$0.834711\pi$
$810$		0			0
$811$	−14428.0			−0.624705			−0.312352	−	0.949966i	$-0.601117\pi$
$811$	−14428.0			−0.624705			−0.312352	+	0.949966i	$0.601117\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−222.711	−	2480.00i	−0.00957204	−	0.106590i
$816$		0			0
$817$			29932.3i			1.28176i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16591.9			−0.705314			−0.352657	−	0.935753i	$-0.614722\pi$
$821$	−16591.9			−0.705314			−0.352657	+	0.935753i	$0.614722\pi$
$822$		0			0
$823$		−	25322.2i		−	1.07251i	−0.844056	−	0.536255i	$-0.819838\pi$
$823$		−	25322.2i		−	1.07251i	0.844056	−	0.536255i	$-0.180162\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	16592.0i		−	0.697655i	−0.937187	−	0.348827i	$-0.886580\pi$
$827$		−	16592.0i		−	0.697655i	0.937187	−	0.348827i	$-0.113420\pi$
$828$		0			0
$829$	−18234.0			−0.763924			−0.381962	−	0.924178i	$-0.624751\pi$
$829$	−18234.0			−0.763924			−0.381962	+	0.924178i	$0.624751\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	9486.00i		−	0.394562i
$834$		0			0
$835$	2172.00	+	24186.4i	0.0900182	+	1.00240i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−31357.6			−1.29033			−0.645165	−	0.764044i	$-0.723211\pi$
$839$	−31357.6			−1.29033			−0.645165	+	0.764044i	$0.723211\pi$
$840$		0			0
$841$	15787.0			0.647300
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−25244.2	+	2267.00i	−1.02773	+	0.0922925i
$846$		0			0
$847$			18596.3i			0.754401i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	34297.4			1.38155
$852$		0			0
$853$		−	2249.38i		−	0.0902898i	−0.998980	−	0.0451449i	$-0.985625\pi$
$853$		−	2249.38i		−	0.0902898i	0.998980	−	0.0451449i	$-0.0143750\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			25206.0i			1.00469i	0.864667	+	0.502346i	$0.167530\pi$
$857$			25206.0i			1.00469i	−0.864667	+	0.502346i	$0.832470\pi$
$858$		0			0
$859$	17540.0			0.696690			0.348345	−	0.937366i	$-0.386744\pi$
$859$	17540.0			0.696690			0.348345	+	0.937366i	$0.386744\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			33108.0i			1.30592i	0.757392	+	0.652960i	$0.226473\pi$
$863$			33108.0i			1.30592i	−0.757392	+	0.652960i	$0.773527\pi$
$864$		0			0
$865$	3434.00	+	38239.4i	0.134982	+	1.50310i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20845.7			0.813743
$870$		0			0
$871$	56544.0			2.19968
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8262.56	−	30008.0i	−0.319229	−	1.15938i
$876$		0			0
$877$		−	29152.8i		−	1.12249i	−0.827651	−	0.561243i	$-0.810323\pi$
$877$		−	29152.8i		−	1.12249i	0.827651	−	0.561243i	$-0.189677\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8819.34			−0.337266			−0.168633	−	0.985679i	$-0.553935\pi$
$881$	−8819.34			−0.337266			−0.168633	+	0.985679i	$0.553935\pi$
$882$		0			0
$883$			19643.1i			0.748632i	0.927301	+	0.374316i	$0.122122\pi$
$883$			19643.1i			0.748632i	−0.927301	+	0.374316i	$0.877878\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	8004.00i		−	0.302985i	−0.988458	−	0.151493i	$-0.951592\pi$
$887$		−	8004.00i		−	0.302985i	0.988458	−	0.151493i	$-0.0484080\pi$
$888$		0			0
$889$	29264.0			1.10403
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8400.00i		−	0.314776i
$894$		0			0
$895$	−37448.0	+	3362.93i	−1.39860	+	0.125598i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3207.03			−0.118977
$900$		0			0
$901$	45756.0			1.69185
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−18685.4	+	1678.00i	−0.686325	+	0.0616338i
$906$		0			0
$907$		−	1336.26i		−	0.0489194i	−0.999701	−	0.0244597i	$-0.992213\pi$
$907$		−	1336.26i		−	0.0489194i	0.999701	−	0.0244597i	$-0.00778654\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32070.3			−1.16634			−0.583171	−	0.812350i	$-0.698188\pi$
$911$	−32070.3			−1.16634			−0.583171	+	0.812350i	$0.698188\pi$
$912$		0			0
$913$		−	29041.5i		−	1.05272i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	10416.0i		−	0.375100i
$918$		0			0
$919$	2832.00			0.101653			0.0508265	−	0.998707i	$-0.483814\pi$
$919$	2832.00			0.101653			0.0508265	+	0.998707i	$0.483814\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			62496.0i			2.22869i
$924$		0			0
$925$	−5456.00	−	30132.7i	−0.193937	−	1.07109i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−31936.7			−1.12789			−0.563945	−	0.825813i	$-0.690717\pi$
$929$	−31936.7			−1.12789			−0.563945	+	0.825813i	$0.690717\pi$
$930$		0			0
$931$	−12852.0			−0.452425
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1380.81	−	15376.0i	−0.0482964	−	0.537806i
$936$		0			0
$937$		−	25745.3i		−	0.897613i	−0.893629	−	0.448807i	$-0.851849\pi$
$937$		−	25745.3i		−	0.897613i	0.893629	−	0.448807i	$-0.148151\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7059.93			−0.244577			−0.122289	−	0.992495i	$-0.539023\pi$
$941$	−7059.93			−0.244577			−0.122289	+	0.992495i	$0.539023\pi$
$942$		0			0
$943$			31179.5i			1.07672i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1752.00i		−	0.0601186i	−0.999548	−	0.0300593i	$-0.990430\pi$
$947$		−	1752.00i		−	0.0601186i	0.999548	−	0.0300593i	$-0.00956962\pi$
$948$		0			0
$949$	−29760.0			−1.01797
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	3850.00i		−	0.130864i	−0.997857	−	0.0654322i	$-0.979157\pi$
$953$		−	3850.00i		−	0.130864i	0.997857	−	0.0654322i	$-0.0208426\pi$
$954$		0			0
$955$	19344.0	−	1737.14i	0.655453	−	0.0588614i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−23651.9			−0.796411
$960$		0			0
$961$	−29535.0			−0.991407
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2449.82	+	27280.0i	0.0817227	+	0.910025i
$966$		0			0
$967$			24475.9i			0.813952i	0.913439	+	0.406976i	$0.133417\pi$
$967$			24475.9i			0.813952i	−0.913439	+	0.406976i	$0.866583\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2204.83			−0.0728697			−0.0364349	−	0.999336i	$-0.511600\pi$
$971$	−2204.83			−0.0728697			−0.0364349	+	0.999336i	$0.511600\pi$
$972$		0			0
$973$			19153.1i			0.631059i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			29982.0i			0.981790i	0.871219	+	0.490895i	$0.163330\pi$
$977$			29982.0i			0.981790i	−0.871219	+	0.490895i	$0.836670\pi$
$978$		0			0
$979$	−15872.0			−0.518153
$980$		0			0
$981$		0			0
$982$		0			0
$983$			35284.0i			1.14485i	0.819958	+	0.572424i	$0.193997\pi$
$983$			35284.0i			1.14485i	−0.819958	+	0.572424i	$0.806003\pi$
$984$		0			0
$985$	526.000	+	5857.29i	0.0170150	+	0.189471i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−49887.2			−1.60396
$990$		0			0
$991$	11528.0			0.369525			0.184762	−	0.982783i	$-0.440848\pi$
$991$	11528.0			0.369525			0.184762	+	0.982783i	$0.440848\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8284.83	+	744.000i	−0.263967	+	0.0237049i
$996$		0			0
$997$			9955.16i			0.316232i	0.987421	+	0.158116i	$0.0505420\pi$
$997$			9955.16i			0.316232i	−0.987421	+	0.158116i	$0.949458\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.f.l.289.3		4
3.2	odd	2	inner	720.4.f.l.289.2		4
4.3	odd	2		90.4.c.c.19.4	yes	4
5.4	even	2	inner	720.4.f.l.289.4		4
12.11	even	2		90.4.c.c.19.1	✓	4
15.14	odd	2	inner	720.4.f.l.289.1		4
20.3	even	4		450.4.a.v.1.2		2
20.7	even	4		450.4.a.u.1.1		2
20.19	odd	2		90.4.c.c.19.2	yes	4
60.23	odd	4		450.4.a.u.1.2		2
60.47	odd	4		450.4.a.v.1.1		2
60.59	even	2		90.4.c.c.19.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.4.c.c.19.1	✓	4	12.11	even	2
90.4.c.c.19.2	yes	4	20.19	odd	2
90.4.c.c.19.3	yes	4	60.59	even	2
90.4.c.c.19.4	yes	4	4.3	odd	2
450.4.a.u.1.1		2	20.7	even	4
450.4.a.u.1.2		2	60.23	odd	4
450.4.a.v.1.1		2	60.47	odd	4
450.4.a.v.1.2		2	20.3	even	4
720.4.f.l.289.1		4	15.14	odd	2	inner
720.4.f.l.289.2		4	3.2	odd	2	inner
720.4.f.l.289.3		4	1.1	even	1	trivial
720.4.f.l.289.4		4	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(11.1355 - 1.00000i) q^{5} -22.2711i q^{7} +O(q^{10})\)	\(q+(11.1355 - 1.00000i) q^{5} -22.2711i q^{7} -22.2711 q^{11} +66.8132i q^{13} +62.0000i q^{17} +84.0000 q^{19} +140.000i q^{23} +(123.000 - 22.2711i) q^{25} +200.440 q^{29} -16.0000 q^{31} +(-22.2711 - 248.000i) q^{35} -244.982i q^{37} +222.711 q^{41} +356.337i q^{43} -100.000i q^{47} -153.000 q^{49} -738.000i q^{53} +(-248.000 + 22.2711i) q^{55} +645.861 q^{59} -358.000 q^{61} +(66.8132 + 744.000i) q^{65} -846.300i q^{67} +935.384 q^{71} +445.421i q^{73} +496.000i q^{77} -936.000 q^{79} +1304.00i q^{83} +(62.0000 + 690.403i) q^{85} +712.674 q^{89} +1488.00 q^{91} +(935.384 - 84.0000i) q^{95} -757.216i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 336 q^{19} + 492 q^{25} - 64 q^{31} - 612 q^{49} - 992 q^{55} - 1432 q^{61} - 3744 q^{79} + 248 q^{85} + 5952 q^{91}+O(q^{100}) \) 4 * q + 336 * q^19 + 492 * q^25 - 64 * q^31 - 612 * q^49 - 992 * q^55 - 1432 * q^61 - 3744 * q^79 + 248 * q^85 + 5952 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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