LMFDB - Embedded newform 7200.2.f.h.6049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(6049,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7200.6049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6049.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	7200.6049
Dual form			7200.2.f.h.6049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{7} +O(q^{10})$	$q+2.00000i q^{7} -2.00000 q^{11} +2.00000i q^{17} -4.00000 q^{19} +2.00000 q^{29} +8.00000 q^{31} -4.00000i q^{37} +8.00000 q^{41} -8.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} +10.0000i q^{53} -6.00000 q^{59} +2.00000 q^{61} +12.0000i q^{67} -12.0000 q^{71} +2.00000i q^{73} -4.00000i q^{77} -8.00000 q^{79} -4.00000i q^{83} -12.0000 q^{89} +10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 4 q^{11} - 8 q^{19} + 4 q^{29} + 16 q^{31} + 16 q^{41} + 6 q^{49} - 12 q^{59} + 4 q^{61} - 24 q^{71} - 16 q^{79} - 24 q^{89}+O(q^{100}) $ 2 * q - 4 * q^11 - 8 * q^19 + 4 * q^29 + 16 * q^31 + 16 * q^41 + 6 * q^49 - 12 * q^59 + 4 * q^61 - 24 * q^71 - 16 * q^79 - 24 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	− 10.0000i	− 0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$	− 10.0000i	− 0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−22.0000	−1.92215	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000	−1.92215	−0.961074	+	0.276289i	$0.910895\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.00000i	− 0.638470i	−0.947676	−	0.319235i	$-0.896574\pi$
$157$	− 8.00000i	− 0.638470i	0.947676	−	0.319235i	$-0.103426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 26.0000i	− 1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	− 26.0000i	− 1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−22.0000	−1.38863	−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000	−1.38863	−0.694314	+	0.719672i	$0.744292\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 10.0000i	− 0.623783i	−0.950118	−	0.311891i	$-0.899037\pi$
$257$	− 10.0000i	− 0.623783i	0.950118	−	0.311891i	$-0.100963\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.0000i	0.944450i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.0000i	1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$	26.0000i	1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6.00000i	− 0.313197i	−0.987662	−	0.156599i	$-0.949947\pi$
$367$	− 6.00000i	− 0.313197i	0.987662	−	0.156599i	$-0.0500529\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.0000	−1.03835
$372$	0	0
$373$	− 16.0000i	− 0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$	− 16.0000i	− 0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.00000i	0.396545i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	0	0
$493$	4.00000i	0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000i	0.696971i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	− 20.0000i	− 0.828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.0000i	1.54237i	0.636610	+	0.771186i	$0.280336\pi$
$607$	38.0000i	1.54237i	−0.636610	+	0.771186i	$0.719664\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 44.0000i	− 1.77714i	−0.458738	−	0.888572i	$-0.651698\pi$
$613$	− 44.0000i	− 1.77714i	0.458738	−	0.888572i	$-0.348302\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 24.0000i	− 0.961540i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.0000i	0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$	26.0000i	0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	16.0000i	0.603451i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.0000i	0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$	14.0000i	0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	36.0000i	1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$	36.0000i	1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 44.0000i	− 1.59921i	−0.600528	−	0.799604i	$-0.705043\pi$
$757$	− 44.0000i	− 1.59921i	0.600528	−	0.799604i	$-0.294957\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	− 20.0000i	− 0.724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.0000i	0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$	20.0000i	0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.00000i	0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$	6.00000i	0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	32.0000i	1.11954i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	− 54.0000i	− 1.88232i	−0.337959	−	0.941161i	$-0.609737\pi$
$823$	− 54.0000i	− 1.88232i	0.337959	−	0.941161i	$-0.390263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.00000i	− 0.278187i	−0.990279	−	0.139094i	$-0.955581\pi$
$827$	− 8.00000i	− 0.278187i	0.990279	−	0.139094i	$-0.0444189\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.0000i	− 0.481046i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 28.0000i	− 0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$	− 28.0000i	− 0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 30.0000i	− 1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$	− 30.0000i	− 1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.00000	−0.134763	−0.0673817	−	0.997727i	$-0.521465\pi$
$881$	−4.00000	−0.134763	−0.0673817	+	0.997727i	$0.521465\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 40.0000i	− 1.34307i	−0.740973	−	0.671534i	$-0.765636\pi$
$887$	− 40.0000i	− 1.34307i	0.740973	−	0.671534i	$-0.234364\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 32.0000i	− 1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.0000	0.533630
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.0000i	1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$	40.0000i	1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	8.00000i	0.264761i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 44.0000i	− 1.45301i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28.0000	−0.904167
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 24.0000i	− 0.760088i	−0.924968	−	0.380044i	$-0.875909\pi$
$997$	− 24.0000i	− 0.760088i	0.924968	−	0.380044i	$-0.124091\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.f.h.6049.2		2
3.2	odd	2		7200.2.f.u.6049.2		2
4.3	odd	2		7200.2.f.v.6049.1		2
5.2	odd	4		7200.2.a.m.1.1		1
5.3	odd	4		1440.2.a.e.1.1	yes	1
5.4	even	2	inner	7200.2.f.h.6049.1		2
12.11	even	2		7200.2.f.i.6049.1		2
15.2	even	4		7200.2.a.n.1.1		1
15.8	even	4		1440.2.a.m.1.1	yes	1
15.14	odd	2		7200.2.f.u.6049.1		2
20.3	even	4		1440.2.a.b.1.1	✓	1
20.7	even	4		7200.2.a.bo.1.1		1
20.19	odd	2		7200.2.f.v.6049.2		2
40.3	even	4		2880.2.a.v.1.1		1
40.13	odd	4		2880.2.a.be.1.1		1
60.23	odd	4		1440.2.a.h.1.1	yes	1
60.47	odd	4		7200.2.a.bn.1.1		1
60.59	even	2		7200.2.f.i.6049.2		2
120.53	even	4		2880.2.a.l.1.1		1
120.83	odd	4		2880.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.a.b.1.1	✓	1	20.3	even	4
1440.2.a.e.1.1	yes	1	5.3	odd	4
1440.2.a.h.1.1	yes	1	60.23	odd	4
1440.2.a.m.1.1	yes	1	15.8	even	4
2880.2.a.g.1.1		1	120.83	odd	4
2880.2.a.l.1.1		1	120.53	even	4
2880.2.a.v.1.1		1	40.3	even	4
2880.2.a.be.1.1		1	40.13	odd	4
7200.2.a.m.1.1		1	5.2	odd	4
7200.2.a.n.1.1		1	15.2	even	4
7200.2.a.bn.1.1		1	60.47	odd	4
7200.2.a.bo.1.1		1	20.7	even	4
7200.2.f.h.6049.1		2	5.4	even	2	inner
7200.2.f.h.6049.2		2	1.1	even	1	trivial
7200.2.f.i.6049.1		2	12.11	even	2
7200.2.f.i.6049.2		2	60.59	even	2
7200.2.f.u.6049.1		2	15.14	odd	2
7200.2.f.u.6049.2		2	3.2	odd	2
7200.2.f.v.6049.1		2	4.3	odd	2
7200.2.f.v.6049.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{7} +O(q^{10})\)	\(q+2.00000i q^{7} -2.00000 q^{11} +2.00000i q^{17} -4.00000 q^{19} +2.00000 q^{29} +8.00000 q^{31} -4.00000i q^{37} +8.00000 q^{41} -8.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} +10.0000i q^{53} -6.00000 q^{59} +2.00000 q^{61} +12.0000i q^{67} -12.0000 q^{71} +2.00000i q^{73} -4.00000i q^{77} -8.00000 q^{79} -4.00000i q^{83} -12.0000 q^{89} +10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 4 q^{11} - 8 q^{19} + 4 q^{29} + 16 q^{31} + 16 q^{41} + 6 q^{49} - 12 q^{59} + 4 q^{61} - 24 q^{71} - 16 q^{79} - 24 q^{89}+O(q^{100}) \) 2 * q - 4 * q^11 - 8 * q^19 + 4 * q^29 + 16 * q^31 + 16 * q^41 + 6 * q^49 - 12 * q^59 + 4 * q^61 - 24 * q^71 - 16 * q^79 - 24 * q^89

Introduction

L-functions

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