LMFDB - Embedded newform 3600.3.e.ba.3151.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(3151,3600)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3600.3151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.e (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:	$98.0928951697$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3151.2
Root			$-0.309017 - 0.535233i$ of defining polynomial
Character	$\chi$	$=$	3600.3151
Dual form			3600.3.e.ba.3151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.74597i q^{7} +O(q^{10})$	$q-7.74597i q^{7} +17.3205i q^{11} -4.00000 q^{13} +13.4164 q^{17} -30.9839i q^{19} +34.6410i q^{23} -40.2492 q^{29} +15.4919i q^{31} +16.0000 q^{37} -53.6656 q^{41} -61.9677i q^{43} -34.6410i q^{47} -11.0000 q^{49} +93.9149 q^{53} +17.3205i q^{59} -58.0000 q^{61} +46.4758i q^{67} -34.6410i q^{71} -94.0000 q^{73} +134.164 q^{77} -15.4919i q^{79} -103.923i q^{83} -26.8328 q^{89} +30.9839i q^{91} +14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{13} + 64 q^{37} - 44 q^{49} - 232 q^{61} - 376 q^{73} + 56 q^{97}+O(q^{100}) $ 4 * q - 16 * q^13 + 64 * q^37 - 44 * q^49 - 232 * q^61 - 376 * q^73 + 56 * q^97

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
$7$	− 7.74597i	− 1.10657i	−0.832993	−	0.553283i	$-0.813375\pi$
$7$	− 7.74597i	− 1.10657i	0.832993	−	0.553283i	$-0.186625\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	17.3205i	1.57459i	0.616575	+	0.787296i	$0.288520\pi$
$11$	17.3205i	1.57459i	−0.616575	+	0.787296i	$0.711480\pi$
$12$	0	0
$13$	−4.00000	−0.307692	−0.153846	−	0.988095i	$-0.549166\pi$
$13$	−4.00000	−0.307692	−0.153846	+	0.988095i	$0.549166\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	13.4164	0.789200	0.394600	−	0.918853i	$-0.370883\pi$
$17$	13.4164	0.789200	0.394600	+	0.918853i	$0.370883\pi$
$18$	0	0
$19$	− 30.9839i	− 1.63073i	−0.578947	−	0.815365i	$-0.696536\pi$
$19$	− 30.9839i	− 1.63073i	0.578947	−	0.815365i	$-0.303464\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	34.6410i	1.50613i	0.657945	+	0.753066i	$0.271426\pi$
$23$	34.6410i	1.50613i	−0.657945	+	0.753066i	$0.728574\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−40.2492	−1.38790	−0.693952	−	0.720021i	$-0.744132\pi$
$29$	−40.2492	−1.38790	−0.693952	+	0.720021i	$0.744132\pi$
$30$	0	0
$31$	15.4919i	0.499740i	0.968279	+	0.249870i	$0.0803879\pi$
$31$	15.4919i	0.499740i	−0.968279	+	0.249870i	$0.919612\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	16.0000	0.432432	0.216216	−	0.976346i	$-0.430628\pi$
$37$	16.0000	0.432432	0.216216	+	0.976346i	$0.430628\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−53.6656	−1.30892	−0.654459	−	0.756098i	$-0.727104\pi$
$41$	−53.6656	−1.30892	−0.654459	+	0.756098i	$0.727104\pi$
$42$	0	0
$43$	− 61.9677i	− 1.44111i	−0.693398	−	0.720555i	$-0.743887\pi$
$43$	− 61.9677i	− 1.44111i	0.693398	−	0.720555i	$-0.256113\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 34.6410i	− 0.737043i	−0.929619	−	0.368521i	$-0.879864\pi$
$47$	− 34.6410i	− 0.737043i	0.929619	−	0.368521i	$-0.120136\pi$
$48$	0	0
$49$	−11.0000	−0.224490
$50$	0	0
$51$	0	0
$52$	0	0
$53$	93.9149	1.77198	0.885989	−	0.463706i	$-0.153481\pi$
$53$	93.9149	1.77198	0.885989	+	0.463706i	$0.153481\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	17.3205i	0.293568i	0.989169	+	0.146784i	$0.0468922\pi$
$59$	17.3205i	0.293568i	−0.989169	+	0.146784i	$0.953108\pi$
$60$	0	0
$61$	−58.0000	−0.950820	−0.475410	−	0.879764i	$-0.657700\pi$
$61$	−58.0000	−0.950820	−0.475410	+	0.879764i	$0.657700\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	46.4758i	0.693669i	0.937926	+	0.346834i	$0.112743\pi$
$67$	46.4758i	0.693669i	−0.937926	+	0.346834i	$0.887257\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 34.6410i	− 0.487902i	−0.969788	−	0.243951i	$-0.921556\pi$
$71$	− 34.6410i	− 0.487902i	0.969788	−	0.243951i	$-0.0784435\pi$
$72$	0	0
$73$	−94.0000	−1.28767	−0.643836	−	0.765164i	$-0.722658\pi$
$73$	−94.0000	−1.28767	−0.643836	+	0.765164i	$0.722658\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	134.164	1.74239
$78$	0	0
$79$	− 15.4919i	− 0.196100i	−0.995181	−	0.0980502i	$-0.968739\pi$
$79$	− 15.4919i	− 0.196100i	0.995181	−	0.0980502i	$-0.0312606\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 103.923i	− 1.25208i	−0.779789	−	0.626042i	$-0.784674\pi$
$83$	− 103.923i	− 1.25208i	0.779789	−	0.626042i	$-0.215326\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−26.8328	−0.301492	−0.150746	−	0.988573i	$-0.548168\pi$
$89$	−26.8328	−0.301492	−0.150746	+	0.988573i	$0.548168\pi$
$90$	0	0
$91$	30.9839i	0.340482i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000	0.144330	0.0721649	−	0.997393i	$-0.477009\pi$
$97$	14.0000	0.144330	0.0721649	+	0.997393i	$0.477009\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−93.9149	−0.929850	−0.464925	−	0.885350i	$-0.653919\pi$
$101$	−93.9149	−0.929850	−0.464925	+	0.885350i	$0.653919\pi$
$102$	0	0
$103$	54.2218i	0.526425i	0.964738	+	0.263212i	$0.0847820\pi$
$103$	54.2218i	0.526425i	−0.964738	+	0.263212i	$0.915218\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	173.205i	1.61874i	0.587300	+	0.809370i	$0.300191\pi$
$107$	173.205i	1.61874i	−0.587300	+	0.809370i	$0.699809\pi$
$108$	0	0
$109$	22.0000	0.201835	0.100917	−	0.994895i	$-0.467822\pi$
$109$	22.0000	0.201835	0.100917	+	0.994895i	$0.467822\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.4164	−0.118729	−0.0593646	−	0.998236i	$-0.518907\pi$
$113$	−13.4164	−0.118729	−0.0593646	+	0.998236i	$0.518907\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 103.923i	− 0.873303i
$120$	0	0
$121$	−179.000	−1.47934
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 162.665i	− 1.28083i	−0.768029	−	0.640415i	$-0.778763\pi$
$127$	− 162.665i	− 1.28083i	0.768029	−	0.640415i	$-0.221237\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 86.6025i	− 0.661088i	−0.943791	−	0.330544i	$-0.892768\pi$
$131$	− 86.6025i	− 0.661088i	0.943791	−	0.330544i	$-0.107232\pi$
$132$	0	0
$133$	−240.000	−1.80451
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−174.413	−1.27309	−0.636545	−	0.771240i	$-0.719637\pi$
$137$	−174.413	−1.27309	−0.636545	+	0.771240i	$0.719637\pi$
$138$	0	0
$139$	61.9677i	0.445811i	0.974840	+	0.222906i	$0.0715541\pi$
$139$	61.9677i	0.445811i	−0.974840	+	0.222906i	$0.928446\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 69.2820i	− 0.484490i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−120.748	−0.810387	−0.405194	−	0.914231i	$-0.632796\pi$
$149$	−120.748	−0.810387	−0.405194	+	0.914231i	$0.632796\pi$
$150$	0	0
$151$	− 216.887i	− 1.43634i	−0.695869	−	0.718169i	$-0.744980\pi$
$151$	− 216.887i	− 1.43634i	0.695869	−	0.718169i	$-0.255020\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	196.000	1.24841	0.624204	−	0.781262i	$-0.285423\pi$
$157$	196.000	1.24841	0.624204	+	0.781262i	$0.285423\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	268.328	1.66663
$162$	0	0
$163$	232.379i	1.42564i	0.701348	+	0.712819i	$0.252582\pi$
$163$	232.379i	1.42564i	−0.701348	+	0.712819i	$0.747418\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 242.487i	− 1.45202i	−0.687685	−	0.726009i	$-0.741373\pi$
$167$	− 242.487i	− 1.45202i	0.687685	−	0.726009i	$-0.258627\pi$
$168$	0	0
$169$	−153.000	−0.905325
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−120.748	−0.697963	−0.348982	−	0.937130i	$-0.613472\pi$
$173$	−120.748	−0.697963	−0.348982	+	0.937130i	$0.613472\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	225.167i	1.25791i	0.777440	+	0.628957i	$0.216518\pi$
$179$	225.167i	1.25791i	−0.777440	+	0.628957i	$0.783482\pi$
$180$	0	0
$181$	−242.000	−1.33702	−0.668508	−	0.743705i	$-0.733067\pi$
$181$	−242.000	−1.33702	−0.668508	+	0.743705i	$0.733067\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	232.379i	1.24267i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	103.923i	0.544100i	0.962283	+	0.272050i	$0.0877016\pi$
$191$	103.923i	0.544100i	−0.962283	+	0.272050i	$0.912298\pi$
$192$	0	0
$193$	−326.000	−1.68912	−0.844560	−	0.535462i	$-0.820138\pi$
$193$	−326.000	−1.68912	−0.844560	+	0.535462i	$0.820138\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−40.2492	−0.204311	−0.102155	−	0.994768i	$-0.532574\pi$
$197$	−40.2492	−0.204311	−0.102155	+	0.994768i	$0.532574\pi$
$198$	0	0
$199$	371.806i	1.86837i	0.356784	+	0.934187i	$0.383873\pi$
$199$	371.806i	1.86837i	−0.356784	+	0.934187i	$0.616127\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	311.769i	1.53581i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	536.656	2.56773
$210$	0	0
$211$	278.855i	1.32159i	0.750568	+	0.660793i	$0.229780\pi$
$211$	278.855i	1.32159i	−0.750568	+	0.660793i	$0.770220\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	120.000	0.552995
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−53.6656	−0.242831
$222$	0	0
$223$	− 131.681i	− 0.590500i	−0.955420	−	0.295250i	$-0.904597\pi$
$223$	− 131.681i	− 0.590500i	0.955420	−	0.295250i	$-0.0954029\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 69.2820i	− 0.305207i	−0.988287	−	0.152604i	$-0.951234\pi$
$227$	− 69.2820i	− 0.305207i	0.988287	−	0.152604i	$-0.0487658\pi$
$228$	0	0
$229$	82.0000	0.358079	0.179039	−	0.983842i	$-0.442701\pi$
$229$	82.0000	0.358079	0.179039	+	0.983842i	$0.442701\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−362.243	−1.55469	−0.777346	−	0.629074i	$-0.783434\pi$
$233$	−362.243	−1.55469	−0.777346	+	0.629074i	$0.783434\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 173.205i	− 0.724707i	−0.932041	−	0.362354i	$-0.881973\pi$
$239$	− 173.205i	− 0.724707i	0.932041	−	0.362354i	$-0.118027\pi$
$240$	0	0
$241$	−362.000	−1.50207	−0.751037	−	0.660260i	$-0.770446\pi$
$241$	−362.000	−1.50207	−0.751037	+	0.660260i	$0.770446\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	123.935i	0.501763i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	329.090i	1.31111i	0.755146	+	0.655557i	$0.227566\pi$
$251$	329.090i	1.31111i	−0.755146	+	0.655557i	$0.772434\pi$
$252$	0	0
$253$	−600.000	−2.37154
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−228.079	−0.887467	−0.443733	−	0.896159i	$-0.646346\pi$
$257$	−228.079	−0.887467	−0.443733	+	0.896159i	$0.646346\pi$
$258$	0	0
$259$	− 123.935i	− 0.478515i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	242.487i	0.922004i	0.887399	+	0.461002i	$0.152510\pi$
$263$	242.487i	0.922004i	−0.887399	+	0.461002i	$0.847490\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	120.748	0.448876	0.224438	−	0.974488i	$-0.427945\pi$
$269$	120.748	0.448876	0.224438	+	0.974488i	$0.427945\pi$
$270$	0	0
$271$	− 340.823i	− 1.25765i	−0.777548	−	0.628824i	$-0.783537\pi$
$271$	− 340.823i	− 1.25765i	0.777548	−	0.628824i	$-0.216463\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	224.000	0.808664	0.404332	−	0.914612i	$-0.367504\pi$
$277$	224.000	0.808664	0.404332	+	0.914612i	$0.367504\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	456.158	1.62334	0.811669	−	0.584118i	$-0.198559\pi$
$281$	456.158	1.62334	0.811669	+	0.584118i	$0.198559\pi$
$282$	0	0
$283$	− 77.4597i	− 0.273709i	−0.990591	−	0.136855i	$-0.956301\pi$
$283$	− 77.4597i	− 0.273709i	0.990591	−	0.136855i	$-0.0436993\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	415.692i	1.44840i
$288$	0	0
$289$	−109.000	−0.377163
$290$	0	0
$291$	0	0
$292$	0	0
$293$	389.076	1.32790	0.663952	−	0.747775i	$-0.268878\pi$
$293$	389.076	1.32790	0.663952	+	0.747775i	$0.268878\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 138.564i	− 0.463425i
$300$	0	0
$301$	−480.000	−1.59468
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 402.790i	− 1.31202i	−0.754752	−	0.656010i	$-0.772243\pi$
$307$	− 402.790i	− 1.31202i	0.754752	−	0.656010i	$-0.227757\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 34.6410i	− 0.111386i	−0.998448	−	0.0556930i	$-0.982263\pi$
$311$	− 34.6410i	− 0.111386i	0.998448	−	0.0556930i	$-0.0177368\pi$
$312$	0	0
$313$	−446.000	−1.42492	−0.712460	−	0.701713i	$-0.752419\pi$
$313$	−446.000	−1.42492	−0.712460	+	0.701713i	$0.752419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−201.246	−0.634846	−0.317423	−	0.948284i	$-0.602817\pi$
$317$	−201.246	−0.634846	−0.317423	+	0.948284i	$0.602817\pi$
$318$	0	0
$319$	− 697.137i	− 2.18538i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 415.692i	− 1.28697i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−268.328	−0.815587
$330$	0	0
$331$	340.823i	1.02968i	0.857288	+	0.514838i	$0.172148\pi$
$331$	340.823i	1.02968i	−0.857288	+	0.514838i	$0.827852\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−46.0000	−0.136499	−0.0682493	−	0.997668i	$-0.521741\pi$
$337$	−46.0000	−0.136499	−0.0682493	+	0.997668i	$0.521741\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−268.328	−0.786886
$342$	0	0
$343$	− 294.347i	− 0.858154i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 415.692i	− 1.19796i	−0.800764	−	0.598980i	$-0.795573\pi$
$347$	− 415.692i	− 1.19796i	0.800764	−	0.598980i	$-0.204427\pi$
$348$	0	0
$349$	−82.0000	−0.234957	−0.117479	−	0.993075i	$-0.537481\pi$
$349$	−82.0000	−0.234957	−0.117479	+	0.993075i	$0.537481\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−469.574	−1.33024	−0.665119	−	0.746737i	$-0.731619\pi$
$353$	−469.574	−1.33024	−0.665119	+	0.746737i	$0.731619\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	554.256i	1.54389i	0.635690	+	0.771945i	$0.280716\pi$
$359$	554.256i	1.54389i	−0.635690	+	0.771945i	$0.719284\pi$
$360$	0	0
$361$	−599.000	−1.65928
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 255.617i	− 0.696504i	−0.937401	−	0.348252i	$-0.886775\pi$
$367$	− 255.617i	− 0.696504i	0.937401	−	0.348252i	$-0.113225\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 727.461i	− 1.96081i
$372$	0	0
$373$	−596.000	−1.59786	−0.798928	−	0.601427i	$-0.794599\pi$
$373$	−596.000	−1.59786	−0.798928	+	0.601427i	$0.794599\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	160.997	0.427047
$378$	0	0
$379$	− 371.806i	− 0.981020i	−0.871436	−	0.490510i	$-0.836811\pi$
$379$	− 371.806i	− 0.981020i	0.871436	−	0.490510i	$-0.163189\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 658.179i	− 1.71848i	−0.511569	−	0.859242i	$-0.670936\pi$
$383$	− 658.179i	− 1.71848i	0.511569	−	0.859242i	$-0.329064\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−281.745	−0.724279	−0.362140	−	0.932124i	$-0.617954\pi$
$389$	−281.745	−0.724279	−0.362140	+	0.932124i	$0.617954\pi$
$390$	0	0
$391$	464.758i	1.18864i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	736.000	1.85390	0.926952	−	0.375180i	$-0.122419\pi$
$397$	736.000	1.85390	0.926952	+	0.375180i	$0.122419\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−348.827	−0.869892	−0.434946	−	0.900457i	$-0.643233\pi$
$401$	−348.827	−0.869892	−0.434946	+	0.900457i	$0.643233\pi$
$402$	0	0
$403$	− 61.9677i	− 0.153766i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	277.128i	0.680904i
$408$	0	0
$409$	−158.000	−0.386308	−0.193154	−	0.981168i	$-0.561872\pi$
$409$	−158.000	−0.386308	−0.193154	+	0.981168i	$0.561872\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	134.164	0.324852
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 779.423i	− 1.86020i	−0.367309	−	0.930099i	$-0.619721\pi$
$419$	− 779.423i	− 1.86020i	0.367309	−	0.930099i	$-0.380279\pi$
$420$	0	0
$421$	−302.000	−0.717340	−0.358670	−	0.933464i	$-0.616770\pi$
$421$	−302.000	−0.717340	−0.358670	+	0.933464i	$0.616770\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	449.266i	1.05215i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 484.974i	− 1.12523i	−0.826719	−	0.562615i	$-0.809795\pi$
$431$	− 484.974i	− 1.12523i	0.826719	−	0.562615i	$-0.190205\pi$
$432$	0	0
$433$	−326.000	−0.752887	−0.376443	−	0.926440i	$-0.622853\pi$
$433$	−326.000	−0.752887	−0.376443	+	0.926440i	$0.622853\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1073.31	2.45609
$438$	0	0
$439$	340.823i	0.776361i	0.921583	+	0.388181i	$0.126896\pi$
$439$	340.823i	0.776361i	−0.921583	+	0.388181i	$0.873104\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	138.564i	0.312786i	0.987695	+	0.156393i	$0.0499866\pi$
$443$	138.564i	0.312786i	−0.987695	+	0.156393i	$0.950013\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.8328	−0.0597613	−0.0298806	−	0.999553i	$-0.509513\pi$
$449$	−26.8328	−0.0597613	−0.0298806	+	0.999553i	$0.509513\pi$
$450$	0	0
$451$	− 929.516i	− 2.06101i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−466.000	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$457$	−466.000	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	93.9149	0.203720	0.101860	−	0.994799i	$-0.467521\pi$
$461$	93.9149	0.203720	0.101860	+	0.994799i	$0.467521\pi$
$462$	0	0
$463$	100.698i	0.217489i	0.994070	+	0.108745i	$0.0346831\pi$
$463$	100.698i	0.217489i	−0.994070	+	0.108745i	$0.965317\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.6410i	0.0741778i	0.999312	+	0.0370889i	$0.0118085\pi$
$467$	34.6410i	0.0741778i	−0.999312	+	0.0370889i	$0.988192\pi$
$468$	0	0
$469$	360.000	0.767591
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1073.31	2.26916
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	519.615i	1.08479i	0.840123	+	0.542396i	$0.182483\pi$
$479$	519.615i	1.08479i	−0.840123	+	0.542396i	$0.817517\pi$
$480$	0	0
$481$	−64.0000	−0.133056
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	348.569i	0.715746i	0.933770	+	0.357873i	$0.116498\pi$
$487$	348.569i	0.715746i	−0.933770	+	0.357873i	$0.883502\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	675.500i	1.37576i	0.725823	+	0.687882i	$0.241459\pi$
$491$	675.500i	1.37576i	−0.725823	+	0.687882i	$0.758541\pi$
$492$	0	0
$493$	−540.000	−1.09533
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−268.328	−0.539896
$498$	0	0
$499$	774.597i	1.55230i	0.630550	+	0.776149i	$0.282830\pi$
$499$	774.597i	1.55230i	−0.630550	+	0.776149i	$0.717170\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	519.615i	1.03303i	0.856277	+	0.516516i	$0.172771\pi$
$503$	519.615i	1.03303i	−0.856277	+	0.516516i	$0.827229\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−630.571	−1.23884	−0.619422	−	0.785059i	$-0.712633\pi$
$509$	−630.571	−1.23884	−0.619422	+	0.785059i	$0.712633\pi$
$510$	0	0
$511$	728.121i	1.42489i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	600.000	1.16054
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−724.486	−1.39057	−0.695284	−	0.718735i	$-0.744721\pi$
$521$	−724.486	−1.39057	−0.695284	+	0.718735i	$0.744721\pi$
$522$	0	0
$523$	− 573.202i	− 1.09599i	−0.836482	−	0.547994i	$-0.815392\pi$
$523$	− 573.202i	− 1.09599i	0.836482	−	0.547994i	$-0.184608\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	207.846i	0.394395i
$528$	0	0
$529$	−671.000	−1.26843
$530$	0	0
$531$	0	0
$532$	0	0
$533$	214.663	0.402744
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 190.526i	− 0.353480i
$540$	0	0
$541$	−2.00000	−0.00369686	−0.00184843	−	0.999998i	$-0.500588\pi$
$541$	−2.00000	−0.00369686	−0.00184843	+	0.999998i	$0.500588\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 61.9677i	− 0.113287i	−0.998394	−	0.0566433i	$-0.981960\pi$
$547$	− 61.9677i	− 0.113287i	0.998394	−	0.0566433i	$-0.0180398\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1247.08i	2.26330i
$552$	0	0
$553$	−120.000	−0.216998
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−281.745	−0.505825	−0.252913	−	0.967489i	$-0.581388\pi$
$557$	−281.745	−0.505825	−0.252913	+	0.967489i	$0.581388\pi$
$558$	0	0
$559$	247.871i	0.443418i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	484.974i	0.861411i	0.902493	+	0.430705i	$0.141735\pi$
$563$	484.974i	0.861411i	−0.902493	+	0.430705i	$0.858265\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.8328	0.0471578	0.0235789	−	0.999722i	$-0.492494\pi$
$569$	26.8328	0.0471578	0.0235789	+	0.999722i	$0.492494\pi$
$570$	0	0
$571$	− 402.790i	− 0.705412i	−0.935734	−	0.352706i	$-0.885262\pi$
$571$	− 402.790i	− 0.705412i	0.935734	−	0.352706i	$-0.114738\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−74.0000	−0.128250	−0.0641248	−	0.997942i	$-0.520426\pi$
$577$	−74.0000	−0.128250	−0.0641248	+	0.997942i	$0.520426\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−804.984	−1.38552
$582$	0	0
$583$	1626.65i	2.79014i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 519.615i	− 0.885205i	−0.896718	−	0.442602i	$-0.854055\pi$
$587$	− 519.615i	− 0.885205i	0.896718	−	0.442602i	$-0.145945\pi$
$588$	0	0
$589$	480.000	0.814941
$590$	0	0
$591$	0	0
$592$	0	0
$593$	147.580	0.248871	0.124435	−	0.992228i	$-0.460288\pi$
$593$	147.580	0.248871	0.124435	+	0.992228i	$0.460288\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−418.000	−0.695507	−0.347754	−	0.937586i	$-0.613055\pi$
$601$	−418.000	−0.695507	−0.347754	+	0.937586i	$0.613055\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.74597i	0.0127611i	0.999980	+	0.00638053i	$0.00203100\pi$
$607$	7.74597i	0.0127611i	−0.999980	+	0.00638053i	$0.997969\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	138.564i	0.226782i
$612$	0	0
$613$	−56.0000	−0.0913540	−0.0456770	−	0.998956i	$-0.514545\pi$
$613$	−56.0000	−0.0913540	−0.0456770	+	0.998956i	$0.514545\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	979.398	1.58735	0.793677	−	0.608339i	$-0.208164\pi$
$617$	979.398	1.58735	0.793677	+	0.608339i	$0.208164\pi$
$618$	0	0
$619$	− 123.935i	− 0.200219i	−0.994976	−	0.100109i	$-0.968081\pi$
$619$	− 123.935i	− 0.200219i	0.994976	−	0.100109i	$-0.0319193\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	207.846i	0.333621i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	214.663	0.341276
$630$	0	0
$631$	− 418.282i	− 0.662888i	−0.943475	−	0.331444i	$-0.892464\pi$
$631$	− 418.282i	− 0.662888i	0.943475	−	0.331444i	$-0.107536\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	44.0000	0.0690738
$638$	0	0
$639$	0	0
$640$	0	0
$641$	804.984	1.25583	0.627913	−	0.778284i	$-0.283909\pi$
$641$	804.984	1.25583	0.627913	+	0.778284i	$0.283909\pi$
$642$	0	0
$643$	511.234i	0.795076i	0.917586	+	0.397538i	$0.130135\pi$
$643$	511.234i	0.795076i	−0.917586	+	0.397538i	$0.869865\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 658.179i	− 1.01728i	−0.860980	−	0.508639i	$-0.830149\pi$
$647$	− 658.179i	− 1.01728i	0.860980	−	0.508639i	$-0.169851\pi$
$648$	0	0
$649$	−300.000	−0.462250
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−523.240	−0.801286	−0.400643	−	0.916234i	$-0.631213\pi$
$653$	−523.240	−0.801286	−0.400643	+	0.916234i	$0.631213\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	675.500i	1.02504i	0.858676	+	0.512519i	$0.171288\pi$
$659$	675.500i	1.02504i	−0.858676	+	0.512519i	$0.828712\pi$
$660$	0	0
$661$	62.0000	0.0937973	0.0468986	−	0.998900i	$-0.485066\pi$
$661$	62.0000	0.0937973	0.0468986	+	0.998900i	$0.485066\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1394.27i	− 2.09037i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1004.59i	− 1.49715i
$672$	0	0
$673$	994.000	1.47697	0.738484	−	0.674271i	$-0.235542\pi$
$673$	994.000	1.47697	0.738484	+	0.674271i	$0.235542\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−201.246	−0.297262	−0.148631	−	0.988893i	$-0.547487\pi$
$677$	−201.246	−0.297262	−0.148631	+	0.988893i	$0.547487\pi$
$678$	0	0
$679$	− 108.444i	− 0.159711i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	381.051i	0.557908i	0.960304	+	0.278954i	$0.0899877\pi$
$683$	381.051i	0.557908i	−0.960304	+	0.278954i	$0.910012\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−375.659	−0.545224
$690$	0	0
$691$	− 371.806i	− 0.538070i	−0.963130	−	0.269035i	$-0.913295\pi$
$691$	− 371.806i	− 0.538070i	0.963130	−	0.269035i	$-0.0867047\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−720.000	−1.03300
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1355.06	1.93303	0.966517	−	0.256602i	$-0.0826028\pi$
$701$	1355.06	1.93303	0.966517	+	0.256602i	$0.0826028\pi$
$702$	0	0
$703$	− 495.742i	− 0.705180i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	727.461i	1.02894i
$708$	0	0
$709$	562.000	0.792666	0.396333	−	0.918107i	$-0.370283\pi$
$709$	562.000	0.792666	0.396333	+	0.918107i	$0.370283\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−536.656	−0.752674
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1316.36i	− 1.83082i	−0.402524	−	0.915409i	$-0.631867\pi$
$719$	− 1316.36i	− 1.83082i	0.402524	−	0.915409i	$-0.368133\pi$
$720$	0	0
$721$	420.000	0.582524
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	472.504i	0.649937i	0.945725	+	0.324968i	$0.105354\pi$
$727$	472.504i	0.649937i	−0.945725	+	0.324968i	$0.894646\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 831.384i	− 1.13732i
$732$	0	0
$733$	−184.000	−0.251023	−0.125512	−	0.992092i	$-0.540057\pi$
$733$	−184.000	−0.251023	−0.125512	+	0.992092i	$0.540057\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−804.984	−1.09224
$738$	0	0
$739$	743.613i	1.00624i	0.864216	+	0.503121i	$0.167815\pi$
$739$	743.613i	1.00624i	−0.864216	+	0.503121i	$0.832185\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 588.897i	− 0.792594i	−0.918122	−	0.396297i	$-0.870295\pi$
$743$	− 588.897i	− 0.792594i	0.918122	−	0.396297i	$-0.129705\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1341.64	1.79124
$750$	0	0
$751$	728.121i	0.969535i	0.874643	+	0.484768i	$0.161096\pi$
$751$	728.121i	0.969535i	−0.874643	+	0.484768i	$0.838904\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	704.000	0.929987	0.464993	−	0.885314i	$-0.346057\pi$
$757$	704.000	0.929987	0.464993	+	0.885314i	$0.346057\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−992.814	−1.30462	−0.652309	−	0.757953i	$-0.726200\pi$
$761$	−992.814	−1.30462	−0.652309	+	0.757953i	$0.726200\pi$
$762$	0	0
$763$	− 170.411i	− 0.223344i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 69.2820i	− 0.0903286i
$768$	0	0
$769$	−802.000	−1.04291	−0.521456	−	0.853278i	$-0.674611\pi$
$769$	−802.000	−1.04291	−0.521456	+	0.853278i	$0.674611\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−630.571	−0.815745	−0.407873	−	0.913039i	$-0.633729\pi$
$773$	−630.571	−0.815745	−0.407873	+	0.913039i	$0.633729\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1662.77i	2.13449i
$780$	0	0
$781$	600.000	0.768246
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1006.98i	1.27951i	0.768578	+	0.639756i	$0.220965\pi$
$787$	1006.98i	1.27951i	−0.768578	+	0.639756i	$0.779035\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	103.923i	0.131382i
$792$	0	0
$793$	232.000	0.292560
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1301.39	−1.63286	−0.816431	−	0.577443i	$-0.804051\pi$
$797$	−1301.39	−1.63286	−0.816431	+	0.577443i	$0.804051\pi$
$798$	0	0
$799$	− 464.758i	− 0.581675i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1628.13i	− 2.02756i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−375.659	−0.464350	−0.232175	−	0.972674i	$-0.574584\pi$
$809$	−375.659	−0.464350	−0.232175	+	0.972674i	$0.574584\pi$
$810$	0	0
$811$	309.839i	0.382045i	0.981586	+	0.191023i	$0.0611804\pi$
$811$	309.839i	0.382045i	−0.981586	+	0.191023i	$0.938820\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1920.00	−2.35006
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1140.39	1.38903	0.694516	−	0.719478i	$-0.255619\pi$
$821$	1140.39	1.38903	0.694516	+	0.719478i	$0.255619\pi$
$822$	0	0
$823$	− 1417.51i	− 1.72237i	−0.508290	−	0.861186i	$-0.669722\pi$
$823$	− 1417.51i	− 1.72237i	0.508290	−	0.861186i	$-0.330278\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	346.410i	0.418876i	0.977822	+	0.209438i	$0.0671634\pi$
$827$	346.410i	0.418876i	−0.977822	+	0.209438i	$0.932837\pi$
$828$	0	0
$829$	1582.00	1.90832	0.954162	−	0.299292i	$-0.0967504\pi$
$829$	1582.00	1.90832	0.954162	+	0.299292i	$0.0967504\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−147.580	−0.177167
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1316.36i	− 1.56896i	−0.620153	−	0.784481i	$-0.712930\pi$
$839$	− 1316.36i	− 1.56896i	0.620153	−	0.784481i	$-0.287070\pi$
$840$	0	0
$841$	779.000	0.926278
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1386.53i	1.63699i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	554.256i	0.651300i
$852$	0	0
$853$	1384.00	1.62251	0.811254	−	0.584693i	$-0.198785\pi$
$853$	1384.00	1.62251	0.811254	+	0.584693i	$0.198785\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	67.0820	0.0782754	0.0391377	−	0.999234i	$-0.487539\pi$
$857$	67.0820	0.0782754	0.0391377	+	0.999234i	$0.487539\pi$
$858$	0	0
$859$	− 681.645i	− 0.793533i	−0.917920	−	0.396767i	$-0.870132\pi$
$859$	− 681.645i	− 0.793533i	0.917920	−	0.396767i	$-0.129868\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 519.615i	− 0.602103i	−0.953608	−	0.301052i	$-0.902662\pi$
$863$	− 519.615i	− 0.602103i	0.953608	−	0.301052i	$-0.0973377\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	268.328	0.308778
$870$	0	0
$871$	− 185.903i	− 0.213437i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1036.00	1.18130	0.590650	−	0.806928i	$-0.298871\pi$
$877$	1036.00	1.18130	0.590650	+	0.806928i	$0.298871\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	402.492	0.456858	0.228429	−	0.973561i	$-0.426641\pi$
$881$	402.492	0.456858	0.228429	+	0.973561i	$0.426641\pi$
$882$	0	0
$883$	759.105i	0.859688i	0.902903	+	0.429844i	$0.141431\pi$
$883$	759.105i	0.859688i	−0.902903	+	0.429844i	$0.858569\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	658.179i	0.742029i	0.928627	+	0.371014i	$0.120990\pi$
$887$	658.179i	0.742029i	−0.928627	+	0.371014i	$0.879010\pi$
$888$	0	0
$889$	−1260.00	−1.41732
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1073.31	−1.20192
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 623.538i	− 0.693591i
$900$	0	0
$901$	1260.00	1.39845
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1146.40i	− 1.26395i	−0.774989	−	0.631975i	$-0.782244\pi$
$907$	− 1146.40i	− 1.26395i	0.774989	−	0.631975i	$-0.217756\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 69.2820i	− 0.0760505i	−0.999277	−	0.0380253i	$-0.987893\pi$
$911$	− 69.2820i	− 0.0760505i	0.999277	−	0.0380253i	$-0.0121067\pi$
$912$	0	0
$913$	1800.00	1.97152
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−670.820	−0.731538
$918$	0	0
$919$	− 480.250i	− 0.522579i	−0.965261	−	0.261289i	$-0.915852\pi$
$919$	− 480.250i	− 0.522579i	0.965261	−	0.261289i	$-0.0841477\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	138.564i	0.150124i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1046.48	1.12646	0.563229	−	0.826301i	$-0.309559\pi$
$929$	1046.48	1.12646	0.563229	+	0.826301i	$0.309559\pi$
$930$	0	0
$931$	340.823i	0.366082i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−434.000	−0.463180	−0.231590	−	0.972813i	$-0.574393\pi$
$937$	−434.000	−0.463180	−0.231590	+	0.972813i	$0.574393\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	576.906	0.613077	0.306539	−	0.951858i	$-0.400829\pi$
$941$	576.906	0.613077	0.306539	+	0.951858i	$0.400829\pi$
$942$	0	0
$943$	− 1859.03i	− 1.97140i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1247.08i	1.31687i	0.752637	+	0.658435i	$0.228781\pi$
$947$	1247.08i	1.31687i	−0.752637	+	0.658435i	$0.771219\pi$
$948$	0	0
$949$	376.000	0.396207
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1059.90	1.11217	0.556084	−	0.831126i	$-0.312303\pi$
$953$	1059.90	1.11217	0.556084	+	0.831126i	$0.312303\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1351.00i	1.40876i
$960$	0	0
$961$	721.000	0.750260
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1866.78i	− 1.93048i	−0.261358	−	0.965242i	$-0.584170\pi$
$967$	− 1866.78i	− 1.93048i	0.261358	−	0.965242i	$-0.415830\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 710.141i	− 0.731350i	−0.930743	−	0.365675i	$-0.880838\pi$
$971$	− 710.141i	− 0.731350i	0.930743	−	0.365675i	$-0.119162\pi$
$972$	0	0
$973$	480.000	0.493320
$974$	0	0
$975$	0	0
$976$	0	0
$977$	764.735	0.782738	0.391369	−	0.920234i	$-0.372002\pi$
$977$	764.735	0.782738	0.391369	+	0.920234i	$0.372002\pi$
$978$	0	0
$979$	− 464.758i	− 0.474727i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1212.44i	1.23340i	0.787197	+	0.616702i	$0.211532\pi$
$983$	1212.44i	1.23340i	−0.787197	+	0.616702i	$0.788468\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2146.63	2.17050
$990$	0	0
$991$	495.742i	0.500244i	0.968214	+	0.250122i	$0.0804707\pi$
$991$	495.742i	0.500244i	−0.968214	+	0.250122i	$0.919529\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−164.000	−0.164493	−0.0822467	−	0.996612i	$-0.526210\pi$
$997$	−164.000	−0.164493	−0.0822467	+	0.996612i	$0.526210\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.e.ba.3151.2		4
3.2	odd	2	inner	3600.3.e.ba.3151.1		4
4.3	odd	2	inner	3600.3.e.ba.3151.3		4
5.2	odd	4		3600.3.j.o.1999.8		8
5.3	odd	4		3600.3.j.o.1999.3		8
5.4	even	2		720.3.e.d.271.2	yes	4
12.11	even	2	inner	3600.3.e.ba.3151.4		4
15.2	even	4		3600.3.j.o.1999.6		8
15.8	even	4		3600.3.j.o.1999.1		8
15.14	odd	2		720.3.e.d.271.4	yes	4
20.3	even	4		3600.3.j.o.1999.5		8
20.7	even	4		3600.3.j.o.1999.2		8
20.19	odd	2		720.3.e.d.271.1	✓	4
40.19	odd	2		2880.3.e.c.2431.3		4
40.29	even	2		2880.3.e.c.2431.4		4
60.23	odd	4		3600.3.j.o.1999.7		8
60.47	odd	4		3600.3.j.o.1999.4		8
60.59	even	2		720.3.e.d.271.3	yes	4
120.29	odd	2		2880.3.e.c.2431.2		4
120.59	even	2		2880.3.e.c.2431.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.3.e.d.271.1	✓	4	20.19	odd	2
720.3.e.d.271.2	yes	4	5.4	even	2
720.3.e.d.271.3	yes	4	60.59	even	2
720.3.e.d.271.4	yes	4	15.14	odd	2
2880.3.e.c.2431.1		4	120.59	even	2
2880.3.e.c.2431.2		4	120.29	odd	2
2880.3.e.c.2431.3		4	40.19	odd	2
2880.3.e.c.2431.4		4	40.29	even	2
3600.3.e.ba.3151.1		4	3.2	odd	2	inner
3600.3.e.ba.3151.2		4	1.1	even	1	trivial
3600.3.e.ba.3151.3		4	4.3	odd	2	inner
3600.3.e.ba.3151.4		4	12.11	even	2	inner
3600.3.j.o.1999.1		8	15.8	even	4
3600.3.j.o.1999.2		8	20.7	even	4
3600.3.j.o.1999.3		8	5.3	odd	4
3600.3.j.o.1999.4		8	60.47	odd	4
3600.3.j.o.1999.5		8	20.3	even	4
3600.3.j.o.1999.6		8	15.2	even	4
3600.3.j.o.1999.7		8	60.23	odd	4
3600.3.j.o.1999.8		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-7.74597i q^{7} +O(q^{10})\)	\(q-7.74597i q^{7} +17.3205i q^{11} -4.00000 q^{13} +13.4164 q^{17} -30.9839i q^{19} +34.6410i q^{23} -40.2492 q^{29} +15.4919i q^{31} +16.0000 q^{37} -53.6656 q^{41} -61.9677i q^{43} -34.6410i q^{47} -11.0000 q^{49} +93.9149 q^{53} +17.3205i q^{59} -58.0000 q^{61} +46.4758i q^{67} -34.6410i q^{71} -94.0000 q^{73} +134.164 q^{77} -15.4919i q^{79} -103.923i q^{83} -26.8328 q^{89} +30.9839i q^{91} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{13} + 64 q^{37} - 44 q^{49} - 232 q^{61} - 376 q^{73} + 56 q^{97}+O(q^{100}) \) 4 * q - 16 * q^13 + 64 * q^37 - 44 * q^49 - 232 * q^61 - 376 * q^73 + 56 * q^97

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