LMFDB - Embedded newform 3600.2.h.g.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(1151,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3600.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$28.7461447277$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3600.1151
Dual form			3600.2.h.g.1151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{7} +O(q^{10})$	$q-4.24264i q^{7} -2.44949 q^{11} -2.44949 q^{13} -6.92820i q^{17} +6.92820i q^{19} +6.00000 q^{23} -2.82843i q^{29} -3.46410i q^{31} -2.44949 q^{37} -7.07107i q^{41} +8.48528i q^{43} -11.0000 q^{49} -12.2474 q^{59} -2.00000 q^{61} -9.79796 q^{71} -4.89898 q^{73} +10.3923i q^{77} +10.3923i q^{79} -12.0000 q^{83} -7.07107i q^{89} +10.3923i q^{91} +14.6969 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{23} - 44 q^{49} - 8 q^{61} - 48 q^{83}+O(q^{100}) $ 4 * q + 24 * q^23 - 44 * q^49 - 8 * q^61 - 48 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$7$	− 4.24264i	− 1.60357i	−0.597614	−	0.801784i	$-0.703885\pi$
$7$	− 4.24264i	− 1.60357i	0.597614	−	0.801784i	$-0.296115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	−2.44949	−0.679366	−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949	−0.679366	−0.339683	+	0.940540i	$0.610320\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.92820i	− 1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	− 6.92820i	− 1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$18$	0	0
$19$	6.92820i	1.58944i	0.606977	+	0.794719i	$0.292382\pi$
$19$	6.92820i	1.58944i	−0.606977	+	0.794719i	$0.707618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.82843i	− 0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$	− 2.82843i	− 0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$	0	0
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.44949	−0.402694	−0.201347	−	0.979520i	$-0.564532\pi$
$37$	−2.44949	−0.402694	−0.201347	+	0.979520i	$0.564532\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.07107i	− 1.10432i	−0.833740	−	0.552158i	$-0.813805\pi$
$41$	− 7.07107i	− 1.10432i	0.833740	−	0.552158i	$-0.186195\pi$
$42$	0	0
$43$	8.48528i	1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$	8.48528i	1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−11.0000	−1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.2474	−1.59448	−0.797241	−	0.603661i	$-0.793708\pi$
$59$	−12.2474	−1.59448	−0.797241	+	0.603661i	$0.793708\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.79796	−1.16280	−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796	−1.16280	−0.581402	+	0.813617i	$0.697496\pi$
$72$	0	0
$73$	−4.89898	−0.573382	−0.286691	−	0.958023i	$-0.592555\pi$
$73$	−4.89898	−0.573382	−0.286691	+	0.958023i	$0.592555\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3923i	1.18431i
$78$	0	0
$79$	10.3923i	1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$	10.3923i	1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7.07107i	− 0.749532i	−0.927119	−	0.374766i	$-0.877723\pi$
$89$	− 7.07107i	− 0.749532i	0.927119	−	0.374766i	$-0.122277\pi$
$90$	0	0
$91$	10.3923i	1.08941i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.6969	1.49225	0.746124	−	0.665807i	$-0.231913\pi$
$97$	14.6969	1.49225	0.746124	+	0.665807i	$0.231913\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.3137i	1.12576i	0.826540	+	0.562878i	$0.190306\pi$
$101$	11.3137i	1.12576i	−0.826540	+	0.562878i	$0.809694\pi$
$102$	0	0
$103$	− 12.7279i	− 1.25412i	−0.778971	−	0.627060i	$-0.784258\pi$
$103$	− 12.7279i	− 1.25412i	0.778971	−	0.627060i	$-0.215742\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.92820i	0.651751i	0.945413	+	0.325875i	$0.105659\pi$
$113$	6.92820i	0.651751i	−0.945413	+	0.325875i	$0.894341\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−29.3939	−2.69453
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.7279i	1.12942i	0.825289	+	0.564710i	$0.191012\pi$
$127$	12.7279i	1.12942i	−0.825289	+	0.564710i	$0.808988\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.44949	0.214013	0.107006	−	0.994258i	$-0.465873\pi$
$131$	2.44949	0.214013	0.107006	+	0.994258i	$0.465873\pi$
$132$	0	0
$133$	29.3939	2.54877
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.8564i	1.18383i	0.805999	+	0.591916i	$0.201628\pi$
$137$	13.8564i	1.18383i	−0.805999	+	0.591916i	$0.798372\pi$
$138$	0	0
$139$	3.46410i	0.293821i	0.989150	+	0.146911i	$0.0469330\pi$
$139$	3.46410i	0.293821i	−0.989150	+	0.146911i	$0.953067\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 22.6274i	− 1.85371i	−0.375419	−	0.926855i	$-0.622501\pi$
$149$	− 22.6274i	− 1.85371i	0.375419	−	0.926855i	$-0.377499\pi$
$150$	0	0
$151$	17.3205i	1.40952i	0.709444	+	0.704761i	$0.248946\pi$
$151$	17.3205i	1.40952i	−0.709444	+	0.704761i	$0.751054\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.2474	−0.977453	−0.488726	−	0.872437i	$-0.662538\pi$
$157$	−12.2474	−0.977453	−0.488726	+	0.872437i	$0.662538\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 25.4558i	− 2.00620i
$162$	0	0
$163$	− 8.48528i	− 0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$	− 8.48528i	− 0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.8564i	− 1.05348i	−0.850026	−	0.526742i	$-0.823414\pi$
$173$	− 13.8564i	− 1.05348i	0.850026	−	0.526742i	$-0.176586\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.2474	−0.915417	−0.457709	−	0.889102i	$-0.651330\pi$
$179$	−12.2474	−0.915417	−0.457709	+	0.889102i	$0.651330\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.9706i	1.24101i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.89898	0.354478	0.177239	−	0.984168i	$-0.443283\pi$
$191$	4.89898	0.354478	0.177239	+	0.984168i	$0.443283\pi$
$192$	0	0
$193$	−19.5959	−1.41055	−0.705273	−	0.708936i	$-0.749175\pi$
$193$	−19.5959	−1.41055	−0.705273	+	0.708936i	$0.749175\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.46410i	− 0.246807i	−0.992357	−	0.123404i	$-0.960619\pi$
$197$	− 3.46410i	− 0.246807i	0.992357	−	0.123404i	$-0.0393809\pi$
$198$	0	0
$199$	10.3923i	0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$	10.3923i	0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 16.9706i	− 1.17388i
$210$	0	0
$211$	− 3.46410i	− 0.238479i	−0.992866	−	0.119239i	$-0.961954\pi$
$211$	− 3.46410i	− 0.238479i	0.992866	−	0.119239i	$-0.0380456\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.6969	−0.997693
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.9706i	1.14156i
$222$	0	0
$223$	4.24264i	0.284108i	0.989859	+	0.142054i	$0.0453707\pi$
$223$	4.24264i	0.284108i	−0.989859	+	0.142054i	$0.954629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.92820i	0.453882i	0.973909	+	0.226941i	$0.0728724\pi$
$233$	6.92820i	0.453882i	−0.973909	+	0.226941i	$0.927128\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.6969	−0.950666	−0.475333	−	0.879806i	$-0.657672\pi$
$239$	−14.6969	−0.950666	−0.475333	+	0.879806i	$0.657672\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 16.9706i	− 1.07981i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−22.0454	−1.39149	−0.695747	−	0.718287i	$-0.744926\pi$
$251$	−22.0454	−1.39149	−0.695747	+	0.718287i	$0.744926\pi$
$252$	0	0
$253$	−14.6969	−0.923989
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 27.7128i	− 1.72868i	−0.502910	−	0.864339i	$-0.667737\pi$
$257$	− 27.7128i	− 1.72868i	0.502910	−	0.864339i	$-0.332263\pi$
$258$	0	0
$259$	10.3923i	0.645746i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.7990i	1.20717i	0.797300	+	0.603583i	$0.206261\pi$
$269$	19.7990i	1.20717i	−0.797300	+	0.603583i	$0.793739\pi$
$270$	0	0
$271$	10.3923i	0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$	10.3923i	0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.34847	−0.441527	−0.220763	−	0.975327i	$-0.570855\pi$
$277$	−7.34847	−0.441527	−0.220763	+	0.975327i	$0.570855\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7.07107i	− 0.421825i	−0.977505	−	0.210912i	$-0.932357\pi$
$281$	− 7.07107i	− 0.421825i	0.977505	−	0.210912i	$-0.0676434\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−30.0000	−1.77084
$288$	0	0
$289$	−31.0000	−1.82353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 17.3205i	− 1.01187i	−0.862570	−	0.505937i	$-0.831147\pi$
$293$	− 17.3205i	− 1.01187i	0.862570	−	0.505937i	$-0.168853\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.6969	−0.849946
$300$	0	0
$301$	36.0000	2.07501
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 33.9411i	− 1.93712i	−0.248776	−	0.968561i	$-0.580028\pi$
$307$	− 33.9411i	− 1.93712i	0.248776	−	0.968561i	$-0.419972\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.89898	−0.277796	−0.138898	−	0.990307i	$-0.544356\pi$
$311$	−4.89898	−0.277796	−0.138898	+	0.990307i	$0.544356\pi$
$312$	0	0
$313$	19.5959	1.10763	0.553813	−	0.832641i	$-0.313172\pi$
$313$	19.5959	1.10763	0.553813	+	0.832641i	$0.313172\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.2487i	− 1.36194i	−0.732310	−	0.680972i	$-0.761558\pi$
$317$	− 24.2487i	− 1.36194i	0.732310	−	0.680972i	$-0.238442\pi$
$318$	0	0
$319$	6.92820i	0.387905i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	48.0000	2.67079
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 13.8564i	− 0.761617i	−0.924654	−	0.380808i	$-0.875646\pi$
$331$	− 13.8564i	− 0.761617i	0.924654	−	0.380808i	$-0.124354\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.6969	0.800593	0.400297	−	0.916386i	$-0.368907\pi$
$337$	14.6969	0.800593	0.400297	+	0.916386i	$0.368907\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.48528i	0.459504i
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	34.2929	1.80991	0.904954	−	0.425510i	$-0.139905\pi$
$359$	34.2929	1.80991	0.904954	+	0.425510i	$0.139905\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.24264i	− 0.221464i	−0.993850	−	0.110732i	$-0.964680\pi$
$367$	− 4.24264i	− 0.221464i	0.993850	−	0.110732i	$-0.0353195\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.0454	−1.14147	−0.570734	−	0.821135i	$-0.693341\pi$
$373$	−22.0454	−1.14147	−0.570734	+	0.821135i	$0.693341\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.92820i	0.356821i
$378$	0	0
$379$	10.3923i	0.533817i	0.963722	+	0.266908i	$0.0860021\pi$
$379$	10.3923i	0.533817i	−0.963722	+	0.266908i	$0.913998\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.82843i	0.143407i	0.997426	+	0.0717035i	$0.0228435\pi$
$389$	2.82843i	0.143407i	−0.997426	+	0.0717035i	$0.977156\pi$
$390$	0	0
$391$	− 41.5692i	− 2.10225i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.8434	1.59817	0.799086	−	0.601216i	$-0.205317\pi$
$397$	31.8434	1.59817	0.799086	+	0.601216i	$0.205317\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 35.3553i	− 1.76556i	−0.469785	−	0.882781i	$-0.655669\pi$
$401$	− 35.3553i	− 1.76556i	0.469785	−	0.882781i	$-0.344331\pi$
$402$	0	0
$403$	8.48528i	0.422682i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	51.9615i	2.55686i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.34847	0.358996	0.179498	−	0.983758i	$-0.442553\pi$
$419$	7.34847	0.358996	0.179498	+	0.983758i	$0.442553\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.48528i	0.410632i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.89898	−0.235976	−0.117988	−	0.993015i	$-0.537644\pi$
$431$	−4.89898	−0.235976	−0.117988	+	0.993015i	$0.537644\pi$
$432$	0	0
$433$	4.89898	0.235430	0.117715	−	0.993047i	$-0.462443\pi$
$433$	4.89898	0.235430	0.117715	+	0.993047i	$0.462443\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.5692i	1.98853i
$438$	0	0
$439$	− 3.46410i	− 0.165333i	−0.996577	−	0.0826663i	$-0.973656\pi$
$439$	− 3.46410i	− 0.165333i	0.996577	−	0.0826663i	$-0.0263436\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 9.89949i	− 0.467186i	−0.972334	−	0.233593i	$-0.924952\pi$
$449$	− 9.89949i	− 0.467186i	0.972334	−	0.233593i	$-0.0750483\pi$
$450$	0	0
$451$	17.3205i	0.815591i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.89898	−0.229165	−0.114582	−	0.993414i	$-0.536553\pi$
$457$	−4.89898	−0.229165	−0.114582	+	0.993414i	$0.536553\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.3137i	0.526932i	0.964669	+	0.263466i	$0.0848657\pi$
$461$	11.3137i	0.526932i	−0.964669	+	0.263466i	$0.915134\pi$
$462$	0	0
$463$	21.2132i	0.985861i	0.870069	+	0.492931i	$0.164074\pi$
$463$	21.2132i	0.985861i	−0.870069	+	0.492931i	$0.835926\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 20.7846i	− 0.955677i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.5959	0.895360	0.447680	−	0.894194i	$-0.352250\pi$
$479$	19.5959	0.895360	0.447680	+	0.894194i	$0.352250\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	21.2132i	0.961262i	0.876923	+	0.480631i	$0.159592\pi$
$487$	21.2132i	0.961262i	−0.876923	+	0.480631i	$0.840408\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.1464	0.773807	0.386904	−	0.922120i	$-0.373545\pi$
$491$	17.1464	0.773807	0.386904	+	0.922120i	$0.373545\pi$
$492$	0	0
$493$	−19.5959	−0.882556
$494$	0	0
$495$	0	0
$496$	0	0
$497$	41.5692i	1.86463i
$498$	0	0
$499$	− 13.8564i	− 0.620298i	−0.950688	−	0.310149i	$-0.899621\pi$
$499$	− 13.8564i	− 0.620298i	0.950688	−	0.310149i	$-0.100379\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.5980i	1.75515i	0.479440	+	0.877575i	$0.340840\pi$
$509$	39.5980i	1.75515i	−0.479440	+	0.877575i	$0.659160\pi$
$510$	0	0
$511$	20.7846i	0.919457i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.07107i	0.309789i	0.987931	+	0.154895i	$0.0495038\pi$
$521$	7.07107i	0.309789i	−0.987931	+	0.154895i	$0.950496\pi$
$522$	0	0
$523$	− 25.4558i	− 1.11311i	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	− 25.4558i	− 1.11311i	0.830812	−	0.556553i	$-0.187876\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.3205i	0.750234i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	26.9444	1.16058
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.48528i	0.362804i	0.983409	+	0.181402i	$0.0580636\pi$
$547$	8.48528i	0.362804i	−0.983409	+	0.181402i	$0.941936\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.5959	0.834814
$552$	0	0
$553$	44.0908	1.87493
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.7128i	− 1.17423i	−0.809504	−	0.587115i	$-0.800264\pi$
$557$	− 27.7128i	− 1.17423i	0.809504	−	0.587115i	$-0.199736\pi$
$558$	0	0
$559$	− 20.7846i	− 0.879095i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.89949i	− 0.415008i	−0.978234	−	0.207504i	$-0.933466\pi$
$569$	− 9.89949i	− 0.415008i	0.978234	−	0.207504i	$-0.0665341\pi$
$570$	0	0
$571$	− 27.7128i	− 1.15975i	−0.814707	−	0.579873i	$-0.803102\pi$
$571$	− 27.7128i	− 1.15975i	0.814707	−	0.579873i	$-0.196898\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.89898	−0.203947	−0.101974	−	0.994787i	$-0.532516\pi$
$577$	−4.89898	−0.203947	−0.101974	+	0.994787i	$0.532516\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	50.9117i	2.11217i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 27.7128i	− 1.13803i	−0.822328	−	0.569014i	$-0.807325\pi$
$593$	− 27.7128i	− 1.13803i	0.822328	−	0.569014i	$-0.192675\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.89898	−0.200167	−0.100083	−	0.994979i	$-0.531911\pi$
$599$	−4.89898	−0.200167	−0.100083	+	0.994979i	$0.531911\pi$
$600$	0	0
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.2132i	0.861017i	0.902586	+	0.430509i	$0.141666\pi$
$607$	21.2132i	0.861017i	−0.902586	+	0.430509i	$0.858334\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	22.0454	0.890406	0.445203	−	0.895430i	$-0.353132\pi$
$613$	22.0454	0.890406	0.445203	+	0.895430i	$0.353132\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13.8564i	− 0.557838i	−0.960315	−	0.278919i	$-0.910024\pi$
$617$	− 13.8564i	− 0.557838i	0.960315	−	0.278919i	$-0.0899761\pi$
$618$	0	0
$619$	− 31.1769i	− 1.25311i	−0.779379	−	0.626553i	$-0.784465\pi$
$619$	− 31.1769i	− 1.25311i	0.779379	−	0.626553i	$-0.215535\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−30.0000	−1.20192
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.9706i	0.676661i
$630$	0	0
$631$	17.3205i	0.689519i	0.938691	+	0.344759i	$0.112039\pi$
$631$	17.3205i	0.689519i	−0.938691	+	0.344759i	$0.887961\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.9444	1.06758
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 15.5563i	− 0.614439i	−0.951639	−	0.307219i	$-0.900601\pi$
$641$	− 15.5563i	− 0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$	0	0
$643$	− 25.4558i	− 1.00388i	−0.864902	−	0.501940i	$-0.832620\pi$
$643$	− 25.4558i	− 1.00388i	0.864902	−	0.501940i	$-0.167380\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 10.3923i	− 0.406682i	−0.979108	−	0.203341i	$-0.934820\pi$
$653$	− 10.3923i	− 0.406682i	0.979108	−	0.203341i	$-0.0651801\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.7423	1.43128	0.715639	−	0.698470i	$-0.246135\pi$
$659$	36.7423	1.43128	0.715639	+	0.698470i	$0.246135\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.9706i	− 0.657103i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.89898	0.189123
$672$	0	0
$673$	−34.2929	−1.32189	−0.660946	−	0.750433i	$-0.729845\pi$
$673$	−34.2929	−1.32189	−0.660946	+	0.750433i	$0.729845\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10.3923i	− 0.399409i	−0.979856	−	0.199704i	$-0.936002\pi$
$677$	− 10.3923i	− 0.399409i	0.979856	−	0.199704i	$-0.0639982\pi$
$678$	0	0
$679$	− 62.3538i	− 2.39292i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	20.7846i	0.790684i	0.918534	+	0.395342i	$0.129374\pi$
$691$	20.7846i	0.790684i	−0.918534	+	0.395342i	$0.870626\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−48.9898	−1.85562
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.65685i	0.213656i	0.994277	+	0.106828i	$0.0340695\pi$
$701$	5.65685i	0.213656i	−0.994277	+	0.106828i	$0.965931\pi$
$702$	0	0
$703$	− 16.9706i	− 0.640057i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	48.0000	1.80523
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 20.7846i	− 0.778390i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.89898	−0.182701	−0.0913506	−	0.995819i	$-0.529118\pi$
$719$	−4.89898	−0.182701	−0.0913506	+	0.995819i	$0.529118\pi$
$720$	0	0
$721$	−54.0000	−2.01107
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 21.2132i	− 0.786754i	−0.919377	−	0.393377i	$-0.871307\pi$
$727$	− 21.2132i	− 0.786754i	0.919377	−	0.393377i	$-0.128693\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	58.7878	2.17434
$732$	0	0
$733$	7.34847	0.271422	0.135711	−	0.990748i	$-0.456668\pi$
$733$	7.34847	0.271422	0.135711	+	0.990748i	$0.456668\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 27.7128i	− 1.01943i	−0.860343	−	0.509716i	$-0.829750\pi$
$739$	− 27.7128i	− 1.01943i	0.860343	−	0.509716i	$-0.170250\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	50.9117i	1.86027i
$750$	0	0
$751$	− 51.9615i	− 1.89610i	−0.318117	−	0.948051i	$-0.603050\pi$
$751$	− 51.9615i	− 1.89610i	0.318117	−	0.948051i	$-0.396950\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.7423	−1.33542	−0.667712	−	0.744420i	$-0.732726\pi$
$757$	−36.7423	−1.33542	−0.667712	+	0.744420i	$0.732726\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.41421i	− 0.0512652i	−0.999671	−	0.0256326i	$-0.991840\pi$
$761$	− 1.41421i	− 0.0512652i	0.999671	−	0.0256326i	$-0.00816000\pi$
$762$	0	0
$763$	− 8.48528i	− 0.307188i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.0000	1.08324
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.1051i	− 1.37055i	−0.728286	−	0.685273i	$-0.759683\pi$
$773$	− 38.1051i	− 1.37055i	0.728286	−	0.685273i	$-0.240317\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48.9898	1.75524
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.4558i	0.907403i	0.891154	+	0.453701i	$0.149897\pi$
$787$	25.4558i	0.907403i	−0.891154	+	0.453701i	$0.850103\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	29.3939	1.04513
$792$	0	0
$793$	4.89898	0.173968
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13.8564i	− 0.490819i	−0.969419	−	0.245410i	$-0.921078\pi$
$797$	− 13.8564i	− 0.490819i	0.969419	−	0.245410i	$-0.0789224\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.3848i	0.646374i	0.946335	+	0.323187i	$0.104754\pi$
$809$	18.3848i	0.646374i	−0.946335	+	0.323187i	$0.895246\pi$
$810$	0	0
$811$	38.1051i	1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$811$	38.1051i	1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−58.7878	−2.05672
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.82843i	− 0.0987128i	−0.998781	−	0.0493564i	$-0.984283\pi$
$821$	− 2.82843i	− 0.0987128i	0.998781	−	0.0493564i	$-0.0157170\pi$
$822$	0	0
$823$	− 21.2132i	− 0.739446i	−0.929142	−	0.369723i	$-0.879453\pi$
$823$	− 21.2132i	− 0.739446i	0.929142	−	0.369723i	$-0.120547\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	76.2102i	2.64053i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34.2929	1.18392	0.591960	−	0.805967i	$-0.298354\pi$
$839$	34.2929	1.18392	0.591960	+	0.805967i	$0.298354\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21.2132i	0.728894i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.6969	−0.503805
$852$	0	0
$853$	−36.7423	−1.25803	−0.629017	−	0.777392i	$-0.716542\pi$
$853$	−36.7423	−1.25803	−0.629017	+	0.777392i	$0.716542\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 13.8564i	− 0.473326i	−0.971592	−	0.236663i	$-0.923946\pi$
$857$	− 13.8564i	− 0.473326i	0.971592	−	0.236663i	$-0.0760537\pi$
$858$	0	0
$859$	− 45.0333i	− 1.53652i	−0.640140	−	0.768259i	$-0.721124\pi$
$859$	− 45.0333i	− 1.53652i	0.640140	−	0.768259i	$-0.278876\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 25.4558i	− 0.863530i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.6413	1.40613	0.703064	−	0.711127i	$-0.251815\pi$
$877$	41.6413	1.40613	0.703064	+	0.711127i	$0.251815\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.89949i	0.333522i	0.985997	+	0.166761i	$0.0533309\pi$
$881$	9.89949i	0.333522i	−0.985997	+	0.166761i	$0.946669\pi$
$882$	0	0
$883$	42.4264i	1.42776i	0.700267	+	0.713881i	$0.253064\pi$
$883$	42.4264i	1.42776i	−0.700267	+	0.713881i	$0.746936\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	0	0
$889$	54.0000	1.81110
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.79796	−0.326780
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.48528i	0.281749i	0.990027	+	0.140875i	$0.0449914\pi$
$907$	8.48528i	0.281749i	−0.990027	+	0.140875i	$0.955009\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.9898	−1.62310	−0.811552	−	0.584280i	$-0.801377\pi$
$911$	−48.9898	−1.62310	−0.811552	+	0.584280i	$0.801377\pi$
$912$	0	0
$913$	29.3939	0.972795
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.3923i	− 0.343184i
$918$	0	0
$919$	45.0333i	1.48551i	0.669562	+	0.742756i	$0.266482\pi$
$919$	45.0333i	1.48551i	−0.669562	+	0.742756i	$0.733518\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 9.89949i	− 0.324792i	−0.986726	−	0.162396i	$-0.948078\pi$
$929$	− 9.89949i	− 0.324792i	0.986726	−	0.162396i	$-0.0519222\pi$
$930$	0	0
$931$	− 76.2102i	− 2.49769i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48.9898	1.60043	0.800213	−	0.599715i	$-0.204720\pi$
$937$	48.9898	1.60043	0.800213	+	0.599715i	$0.204720\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.7401i	1.75188i	0.482422	+	0.875939i	$0.339757\pi$
$941$	53.7401i	1.75188i	−0.482422	+	0.875939i	$0.660243\pi$
$942$	0	0
$943$	− 42.4264i	− 1.38159i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 27.7128i	− 0.897706i	−0.893606	−	0.448853i	$-0.851833\pi$
$953$	− 27.7128i	− 0.897706i	0.893606	−	0.448853i	$-0.148167\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	58.7878	1.89836
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 46.6690i	− 1.50078i	−0.660998	−	0.750388i	$-0.729867\pi$
$967$	− 46.6690i	− 1.50078i	0.660998	−	0.750388i	$-0.270133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.2474	0.393039	0.196520	−	0.980500i	$-0.437036\pi$
$971$	12.2474	0.393039	0.196520	+	0.980500i	$0.437036\pi$
$972$	0	0
$973$	14.6969	0.471162
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	17.3205i	0.553566i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	50.9117i	1.61890i
$990$	0	0
$991$	17.3205i	0.550204i	0.961415	+	0.275102i	$0.0887116\pi$
$991$	17.3205i	0.550204i	−0.961415	+	0.275102i	$0.911288\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.44949	−0.0775761	−0.0387881	−	0.999247i	$-0.512350\pi$
$997$	−2.44949	−0.0775761	−0.0387881	+	0.999247i	$0.512350\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.h.g.1151.1		4
3.2	odd	2		3600.2.h.f.1151.2		4
4.3	odd	2		3600.2.h.f.1151.4		4
5.2	odd	4		720.2.o.b.719.2	yes	8
5.3	odd	4		720.2.o.b.719.5	yes	8
5.4	even	2		3600.2.h.f.1151.3		4
12.11	even	2	inner	3600.2.h.g.1151.3		4
15.2	even	4		720.2.o.b.719.8	yes	8
15.8	even	4		720.2.o.b.719.3	yes	8
15.14	odd	2	inner	3600.2.h.g.1151.4		4
20.3	even	4		720.2.o.b.719.6	yes	8
20.7	even	4		720.2.o.b.719.1	✓	8
20.19	odd	2	inner	3600.2.h.g.1151.2		4
40.3	even	4		2880.2.o.d.2879.4		8
40.13	odd	4		2880.2.o.d.2879.3		8
40.27	even	4		2880.2.o.d.2879.7		8
40.37	odd	4		2880.2.o.d.2879.8		8
60.23	odd	4		720.2.o.b.719.4	yes	8
60.47	odd	4		720.2.o.b.719.7	yes	8
60.59	even	2		3600.2.h.f.1151.1		4
120.53	even	4		2880.2.o.d.2879.5		8
120.77	even	4		2880.2.o.d.2879.2		8
120.83	odd	4		2880.2.o.d.2879.6		8
120.107	odd	4		2880.2.o.d.2879.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.o.b.719.1	✓	8	20.7	even	4
720.2.o.b.719.2	yes	8	5.2	odd	4
720.2.o.b.719.3	yes	8	15.8	even	4
720.2.o.b.719.4	yes	8	60.23	odd	4
720.2.o.b.719.5	yes	8	5.3	odd	4
720.2.o.b.719.6	yes	8	20.3	even	4
720.2.o.b.719.7	yes	8	60.47	odd	4
720.2.o.b.719.8	yes	8	15.2	even	4
2880.2.o.d.2879.1		8	120.107	odd	4
2880.2.o.d.2879.2		8	120.77	even	4
2880.2.o.d.2879.3		8	40.13	odd	4
2880.2.o.d.2879.4		8	40.3	even	4
2880.2.o.d.2879.5		8	120.53	even	4
2880.2.o.d.2879.6		8	120.83	odd	4
2880.2.o.d.2879.7		8	40.27	even	4
2880.2.o.d.2879.8		8	40.37	odd	4
3600.2.h.f.1151.1		4	60.59	even	2
3600.2.h.f.1151.2		4	3.2	odd	2
3600.2.h.f.1151.3		4	5.4	even	2
3600.2.h.f.1151.4		4	4.3	odd	2
3600.2.h.g.1151.1		4	1.1	even	1	trivial
3600.2.h.g.1151.2		4	20.19	odd	2	inner
3600.2.h.g.1151.3		4	12.11	even	2	inner
3600.2.h.g.1151.4		4	15.14	odd	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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{7} +O(q^{10})\)	\(q-4.24264i q^{7} -2.44949 q^{11} -2.44949 q^{13} -6.92820i q^{17} +6.92820i q^{19} +6.00000 q^{23} -2.82843i q^{29} -3.46410i q^{31} -2.44949 q^{37} -7.07107i q^{41} +8.48528i q^{43} -11.0000 q^{49} -12.2474 q^{59} -2.00000 q^{61} -9.79796 q^{71} -4.89898 q^{73} +10.3923i q^{77} +10.3923i q^{79} -12.0000 q^{83} -7.07107i q^{89} +10.3923i q^{91} +14.6969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{23} - 44 q^{49} - 8 q^{61} - 48 q^{83}+O(q^{100}) \) 4 * q + 24 * q^23 - 44 * q^49 - 8 * q^61 - 48 * q^83

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