LMFDB - Embedded newform 2252.3.d.b.1125.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2252,3,Mod(1125,2252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2252.1125"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2252 = 2^{2} \cdot 563 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2252.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$61.3625555339$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1125.14
Character	$\chi$	$=$	2252.1125
Dual form			2252.3.d.b.1125.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.03034 q^{3} -4.25476i q^{5} -6.68184 q^{7} +7.24360 q^{9} -2.45745 q^{11} +15.0750 q^{13} +17.1481i q^{15} -14.0894 q^{17} -16.7283 q^{19} +26.9301 q^{21} -9.19372 q^{23} +6.89699 q^{25} +7.07886 q^{27} -14.2032i q^{29} +23.2203i q^{31} +9.90433 q^{33} +28.4297i q^{35} -39.2466i q^{37} -60.7572 q^{39} +78.3003i q^{41} +40.5013i q^{43} -30.8198i q^{45} -28.9200 q^{47} -4.35296 q^{49} +56.7850 q^{51} +96.6004i q^{53} +10.4558i q^{55} +67.4205 q^{57} -47.3123 q^{59} +92.6486 q^{61} -48.4006 q^{63} -64.1404i q^{65} +17.9997 q^{67} +37.0538 q^{69} -22.2559 q^{71} +7.30071i q^{73} -27.7972 q^{75} +16.4203 q^{77} +15.6568i q^{79} -93.7226 q^{81} +3.48742i q^{83} +59.9471i q^{85} +57.2438i q^{87} -97.5347i q^{89} -100.729 q^{91} -93.5855i q^{93} +71.1748i q^{95} -27.8558i q^{97} -17.8008 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q + 4 q^{3} - 8 q^{7} + 128 q^{9} + 2 q^{11} - 6 q^{13} + 22 q^{17} + 12 q^{19} - 6 q^{21} + 24 q^{23} - 912 q^{25} + 22 q^{27} - 52 q^{33} - 70 q^{39} - 28 q^{47} - 340 q^{49} + 314 q^{51} - 98 q^{57}+ \cdots + 348 q^{99}+O(q^{100}) $ 76 * q + 4 * q^3 - 8 * q^7 + 128 * q^9 + 2 * q^11 - 6 * q^13 + 22 * q^17 + 12 * q^19 - 6 * q^21 + 24 * q^23 - 912 * q^25 + 22 * q^27 - 52 * q^33 - 70 * q^39 - 28 * q^47 - 340 * q^49 + 314 * q^51 - 98 * q^57 + 18 * q^59 - 162 * q^61 - 246 * q^63 - 178 * q^67 + 2 * q^69 + 72 * q^71 + 14 * q^75 - 498 * q^77 - 468 * q^81 + 132 * q^91 + 348 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2252\mathbb{Z}\right)^\times$.

$n$	$565$	$1127$
$\chi(n)$	$-1$	$1$

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
$3$	−4.03034			−1.34345			−0.671723	−	0.740803i	$-0.734445\pi$
$3$	−4.03034			−1.34345			−0.671723	+	0.740803i	$0.734445\pi$
$4$		0			0
$5$		−	4.25476i		−	0.850953i	−0.904970	−	0.425476i	$-0.860107\pi$
$5$		−	4.25476i		−	0.850953i	0.904970	−	0.425476i	$-0.139893\pi$
$6$		0			0
$7$	−6.68184			−0.954549			−0.477275	−	0.878754i	$-0.658375\pi$
$7$	−6.68184			−0.954549			−0.477275	+	0.878754i	$0.658375\pi$
$8$		0			0
$9$	7.24360			0.804845
$10$		0			0
$11$	−2.45745			−0.223404			−0.111702	−	0.993742i	$-0.535630\pi$
$11$	−2.45745			−0.223404			−0.111702	+	0.993742i	$0.535630\pi$
$12$		0			0
$13$	15.0750			1.15961			0.579807	−	0.814754i	$-0.303128\pi$
$13$	15.0750			1.15961			0.579807	+	0.814754i	$0.303128\pi$
$14$		0			0
$15$			17.1481i			1.14321i
$16$		0			0
$17$	−14.0894			−0.828788			−0.414394	−	0.910097i	$-0.636007\pi$
$17$	−14.0894			−0.828788			−0.414394	+	0.910097i	$0.636007\pi$
$18$		0			0
$19$	−16.7283			−0.880435			−0.440218	−	0.897891i	$-0.645099\pi$
$19$	−16.7283			−0.880435			−0.440218	+	0.897891i	$0.645099\pi$
$20$		0			0
$21$	26.9301			1.28238
$22$		0			0
$23$	−9.19372			−0.399727			−0.199863	−	0.979824i	$-0.564050\pi$
$23$	−9.19372			−0.399727			−0.199863	+	0.979824i	$0.564050\pi$
$24$		0			0
$25$	6.89699			0.275880
$26$		0			0
$27$	7.07886			0.262180
$28$		0			0
$29$		−	14.2032i		−	0.489766i	−0.969553	−	0.244883i	$-0.921250\pi$
$29$		−	14.2032i		−	0.489766i	0.969553	−	0.244883i	$-0.0787496\pi$
$30$		0			0
$31$			23.2203i			0.749041i	0.927219	+	0.374520i	$0.122193\pi$
$31$			23.2203i			0.749041i	−0.927219	+	0.374520i	$0.877807\pi$
$32$		0			0
$33$	9.90433			0.300131
$34$		0			0
$35$			28.4297i			0.812276i
$36$		0			0
$37$		−	39.2466i		−	1.06072i	−0.847773	−	0.530359i	$-0.822057\pi$
$37$		−	39.2466i		−	1.06072i	0.847773	−	0.530359i	$-0.177943\pi$
$38$		0			0
$39$	−60.7572			−1.55788
$40$		0			0
$41$			78.3003i			1.90976i	0.296985	+	0.954882i	$0.404019\pi$
$41$			78.3003i			1.90976i	−0.296985	+	0.954882i	$0.595981\pi$
$42$		0			0
$43$			40.5013i			0.941890i	0.882163	+	0.470945i	$0.156087\pi$
$43$			40.5013i			0.941890i	−0.882163	+	0.470945i	$0.843913\pi$
$44$		0			0
$45$		−	30.8198i		−	0.684885i
$46$		0			0
$47$	−28.9200			−0.615319			−0.307660	−	0.951496i	$-0.599546\pi$
$47$	−28.9200			−0.615319			−0.307660	+	0.951496i	$0.599546\pi$
$48$		0			0
$49$	−4.35296			−0.0888359
$50$		0			0
$51$	56.7850			1.11343
$52$		0			0
$53$			96.6004i			1.82265i	0.411688	+	0.911325i	$0.364939\pi$
$53$			96.6004i			1.82265i	−0.411688	+	0.911325i	$0.635061\pi$
$54$		0			0
$55$			10.4558i			0.190106i
$56$		0			0
$57$	67.4205			1.18282
$58$		0			0
$59$	−47.3123			−0.801904			−0.400952	−	0.916099i	$-0.631321\pi$
$59$	−47.3123			−0.801904			−0.400952	+	0.916099i	$0.631321\pi$
$60$		0			0
$61$	92.6486			1.51883			0.759415	−	0.650606i	$-0.225485\pi$
$61$	92.6486			1.51883			0.759415	+	0.650606i	$0.225485\pi$
$62$		0			0
$63$	−48.4006			−0.768264
$64$		0			0
$65$		−	64.1404i		−	0.986776i
$66$		0			0
$67$	17.9997			0.268652			0.134326	−	0.990937i	$-0.457113\pi$
$67$	17.9997			0.268652			0.134326	+	0.990937i	$0.457113\pi$
$68$		0			0
$69$	37.0538			0.537011
$70$		0			0
$71$	−22.2559			−0.313464			−0.156732	−	0.987641i	$-0.550096\pi$
$71$	−22.2559			−0.313464			−0.156732	+	0.987641i	$0.550096\pi$
$72$		0			0
$73$			7.30071i			0.100010i	0.998749	+	0.0500049i	$0.0159237\pi$
$73$			7.30071i			0.100010i	−0.998749	+	0.0500049i	$0.984076\pi$
$74$		0			0
$75$	−27.7972			−0.370629
$76$		0			0
$77$	16.4203			0.213250
$78$		0			0
$79$			15.6568i			0.198188i	0.995078	+	0.0990939i	$0.0315944\pi$
$79$			15.6568i			0.198188i	−0.995078	+	0.0990939i	$0.968406\pi$
$80$		0			0
$81$	−93.7226			−1.15707
$82$		0			0
$83$			3.48742i			0.0420171i	0.999779	+	0.0210086i	$0.00668773\pi$
$83$			3.48742i			0.0420171i	−0.999779	+	0.0210086i	$0.993312\pi$
$84$		0			0
$85$			59.9471i			0.705260i
$86$		0			0
$87$			57.2438i			0.657974i
$88$		0			0
$89$		−	97.5347i		−	1.09590i	−0.836512	−	0.547948i	$-0.815409\pi$
$89$		−	97.5347i		−	1.09590i	0.836512	−	0.547948i	$-0.184591\pi$
$90$		0			0
$91$	−100.729			−1.10691
$92$		0			0
$93$		−	93.5855i		−	1.00630i
$94$		0			0
$95$			71.1748i			0.749209i
$96$		0			0
$97$		−	27.8558i		−	0.287174i	−0.989638	−	0.143587i	$-0.954136\pi$
$97$		−	27.8558i		−	0.287174i	0.989638	−	0.143587i	$-0.0458636\pi$
$98$		0			0
$99$	−17.8008			−0.179806
$100$		0			0
$101$	−49.9764			−0.494816			−0.247408	−	0.968911i	$-0.579579\pi$
$101$	−49.9764			−0.494816			−0.247408	+	0.968911i	$0.579579\pi$
$102$		0			0
$103$	−39.3526			−0.382064			−0.191032	−	0.981584i	$-0.561183\pi$
$103$	−39.3526			−0.382064			−0.191032	+	0.981584i	$0.561183\pi$
$104$		0			0
$105$		−	114.581i		−	1.09125i
$106$		0			0
$107$	10.3256			0.0965011			0.0482505	−	0.998835i	$-0.484635\pi$
$107$	10.3256			0.0965011			0.0482505	+	0.998835i	$0.484635\pi$
$108$		0			0
$109$		−	74.1329i		−	0.680118i	−0.940404	−	0.340059i	$-0.889553\pi$
$109$		−	74.1329i		−	0.680118i	0.940404	−	0.340059i	$-0.110447\pi$
$110$		0			0
$111$			158.177i			1.42502i
$112$		0			0
$113$	88.8153			0.785976			0.392988	−	0.919544i	$-0.371441\pi$
$113$	88.8153			0.785976			0.392988	+	0.919544i	$0.371441\pi$
$114$		0			0
$115$			39.1171i			0.340149i
$116$		0			0
$117$	109.197			0.933309
$118$		0			0
$119$	94.1432			0.791119
$120$		0			0
$121$	−114.961			−0.950091
$122$		0			0
$123$		−	315.577i		−	2.56566i
$124$		0			0
$125$		−	135.714i		−	1.08571i
$126$		0			0
$127$	14.6041			0.114993			0.0574966	−	0.998346i	$-0.481688\pi$
$127$	14.6041			0.114993			0.0574966	+	0.998346i	$0.481688\pi$
$128$		0			0
$129$		−	163.234i		−	1.26538i
$130$		0			0
$131$			115.146i			0.878977i	0.898248	+	0.439488i	$0.144840\pi$
$131$			115.146i			0.878977i	−0.898248	+	0.439488i	$0.855160\pi$
$132$		0			0
$133$	111.776			0.840419
$134$		0			0
$135$		−	30.1189i		−	0.223103i
$136$		0			0
$137$	87.0151			0.635147			0.317573	−	0.948234i	$-0.397132\pi$
$137$	87.0151			0.635147			0.317573	+	0.948234i	$0.397132\pi$
$138$		0			0
$139$		−	135.554i		−	0.975210i	−0.873064	−	0.487605i	$-0.837871\pi$
$139$		−	135.554i		−	0.975210i	0.873064	−	0.487605i	$-0.162129\pi$
$140$		0			0
$141$	116.557			0.826648
$142$		0			0
$143$	−37.0459			−0.259062
$144$		0			0
$145$	−60.4314			−0.416768
$146$		0			0
$147$	17.5439			0.119346
$148$		0			0
$149$	219.999			1.47650			0.738252	−	0.674525i	$-0.235651\pi$
$149$	219.999			1.47650			0.738252	+	0.674525i	$0.235651\pi$
$150$		0			0
$151$		−	137.429i		−	0.910123i	−0.890460	−	0.455062i	$-0.849617\pi$
$151$		−	137.429i		−	0.910123i	0.890460	−	0.455062i	$-0.150383\pi$
$152$		0			0
$153$	−102.058			−0.667046
$154$		0			0
$155$	98.7967			0.637398
$156$		0			0
$157$		−	90.6307i		−	0.577266i	−0.957440	−	0.288633i	$-0.906799\pi$
$157$		−	90.6307i		−	0.577266i	0.957440	−	0.288633i	$-0.0932007\pi$
$158$		0			0
$159$		−	389.332i		−	2.44863i
$160$		0			0
$161$	61.4310			0.381559
$162$		0			0
$163$			43.8455i			0.268991i	0.990914	+	0.134495i	$0.0429414\pi$
$163$			43.8455i			0.268991i	−0.990914	+	0.134495i	$0.957059\pi$
$164$		0			0
$165$		−	42.1406i		−	0.255397i
$166$		0			0
$167$			137.139i			0.821193i	0.911817	+	0.410597i	$0.134680\pi$
$167$			137.139i			0.821193i	−0.911817	+	0.410597i	$0.865320\pi$
$168$		0			0
$169$	58.2548			0.344703
$170$		0			0
$171$	−121.173			−0.708614
$172$		0			0
$173$		−	34.4048i		−	0.198872i	−0.995044	−	0.0994359i	$-0.968296\pi$
$173$		−	34.4048i		−	0.198872i	0.995044	−	0.0994359i	$-0.0317038\pi$
$174$		0			0
$175$	−46.0846			−0.263341
$176$		0			0
$177$	190.685			1.07731
$178$		0			0
$179$	152.256			0.850591			0.425295	−	0.905055i	$-0.360170\pi$
$179$	152.256			0.850591			0.425295	+	0.905055i	$0.360170\pi$
$180$		0			0
$181$	329.821			1.82221			0.911106	−	0.412171i	$-0.135229\pi$
$181$	329.821			1.82221			0.911106	+	0.412171i	$0.135229\pi$
$182$		0			0
$183$	−373.405			−2.04047
$184$		0			0
$185$	−166.985			−0.902621
$186$		0			0
$187$	34.6239			0.185155
$188$		0			0
$189$	−47.2999			−0.250264
$190$		0			0
$191$	363.082			1.90095			0.950476	−	0.310798i	$-0.100596\pi$
$191$	363.082			1.90095			0.950476	+	0.310798i	$0.100596\pi$
$192$		0			0
$193$	−3.72383			−0.0192944			−0.00964722	−	0.999953i	$-0.503071\pi$
$193$	−3.72383			−0.0192944			−0.00964722	+	0.999953i	$0.503071\pi$
$194$		0			0
$195$			258.507i			1.32568i
$196$		0			0
$197$	−132.379			−0.671975			−0.335988	−	0.941866i	$-0.609070\pi$
$197$	−132.379			−0.671975			−0.335988	+	0.941866i	$0.609070\pi$
$198$		0			0
$199$		−	307.793i		−	1.54670i	−0.633981	−	0.773349i	$-0.718580\pi$
$199$		−	307.793i		−	1.54670i	0.633981	−	0.773349i	$-0.281420\pi$
$200$		0			0
$201$	−72.5448			−0.360919
$202$		0			0
$203$			94.9038i			0.467506i
$204$		0			0
$205$	333.149			1.62512
$206$		0			0
$207$	−66.5956			−0.321718
$208$		0			0
$209$	41.1088			0.196693
$210$		0			0
$211$	−150.577			−0.713634			−0.356817	−	0.934174i	$-0.616138\pi$
$211$	−150.577			−0.713634			−0.356817	+	0.934174i	$0.616138\pi$
$212$		0			0
$213$	89.6989			0.421121
$214$		0			0
$215$	172.323			0.801504
$216$		0			0
$217$		−	155.154i		−	0.714996i
$218$		0			0
$219$		−	29.4243i		−	0.134358i
$220$		0			0
$221$	−212.397			−0.961074
$222$		0			0
$223$	−103.866			−0.465766			−0.232883	−	0.972505i	$-0.574816\pi$
$223$	−103.866			−0.465766			−0.232883	+	0.972505i	$0.574816\pi$
$224$		0			0
$225$	49.9591			0.222040
$226$		0			0
$227$			23.4658i			0.103374i	0.998663	+	0.0516868i	$0.0164598\pi$
$227$			23.4658i			0.103374i	−0.998663	+	0.0516868i	$0.983540\pi$
$228$		0			0
$229$		−	334.059i		−	1.45877i	−0.684101	−	0.729387i	$-0.739805\pi$
$229$		−	334.059i		−	1.45877i	0.684101	−	0.729387i	$-0.260195\pi$
$230$		0			0
$231$	−66.1792			−0.286490
$232$		0			0
$233$		−	92.7672i		−	0.398143i	−0.979985	−	0.199071i	$-0.936207\pi$
$233$		−	92.7672i		−	0.398143i	0.979985	−	0.199071i	$-0.0637925\pi$
$234$		0			0
$235$			123.048i			0.523608i
$236$		0			0
$237$		−	63.1023i		−	0.266254i
$238$		0			0
$239$		−	280.914i		−	1.17537i	−0.809089	−	0.587686i	$-0.800039\pi$
$239$		−	280.914i		−	1.17537i	0.809089	−	0.587686i	$-0.199961\pi$
$240$		0			0
$241$	94.4147			0.391762			0.195881	−	0.980628i	$-0.437243\pi$
$241$	94.4147			0.391762			0.195881	+	0.980628i	$0.437243\pi$
$242$		0			0
$243$	314.024			1.29228
$244$		0			0
$245$			18.5208i			0.0755952i
$246$		0			0
$247$	−252.178			−1.02096
$248$		0			0
$249$		−	14.0555i		−	0.0564477i
$250$		0			0
$251$	−82.4747			−0.328585			−0.164292	−	0.986412i	$-0.552534\pi$
$251$	−82.4747			−0.328585			−0.164292	+	0.986412i	$0.552534\pi$
$252$		0			0
$253$	22.5931			0.0893006
$254$		0			0
$255$		−	241.607i		−	0.947478i
$256$		0			0
$257$	−3.35802			−0.0130662			−0.00653311	−	0.999979i	$-0.502080\pi$
$257$	−3.35802			−0.0130662			−0.00653311	+	0.999979i	$0.502080\pi$
$258$		0			0
$259$			262.240i			1.01251i
$260$		0			0
$261$		−	102.883i		−	0.394186i
$262$		0			0
$263$			131.107i			0.498505i	0.968439	+	0.249252i	$0.0801849\pi$
$263$			131.107i			0.498505i	−0.968439	+	0.249252i	$0.919815\pi$
$264$		0			0
$265$	411.012			1.55099
$266$		0			0
$267$			393.098i			1.47228i
$268$		0			0
$269$	172.580			0.641562			0.320781	−	0.947153i	$-0.396055\pi$
$269$	172.580			0.641562			0.320781	+	0.947153i	$0.396055\pi$
$270$		0			0
$271$	261.014			0.963152			0.481576	−	0.876404i	$-0.340065\pi$
$271$	261.014			0.963152			0.481576	+	0.876404i	$0.340065\pi$
$272$		0			0
$273$	405.970			1.48707
$274$		0			0
$275$	−16.9490			−0.0616326
$276$		0			0
$277$	−283.277			−1.02266			−0.511330	−	0.859384i	$-0.670847\pi$
$277$	−283.277			−1.02266			−0.511330	+	0.859384i	$0.670847\pi$
$278$		0			0
$279$			168.198i			0.602862i
$280$		0			0
$281$	36.3551			0.129378			0.0646888	−	0.997905i	$-0.479395\pi$
$281$	36.3551			0.129378			0.0646888	+	0.997905i	$0.479395\pi$
$282$		0			0
$283$		−	301.611i		−	1.06576i	−0.846190	−	0.532881i	$-0.821109\pi$
$283$		−	301.611i		−	1.06576i	0.846190	−	0.532881i	$-0.178891\pi$
$284$		0			0
$285$		−	286.858i		−	1.00652i
$286$		0			0
$287$		−	523.191i		−	1.82296i
$288$		0			0
$289$	−90.4887			−0.313110
$290$		0			0
$291$			112.268i			0.385802i
$292$		0			0
$293$			56.6144i			0.193223i	0.995322	+	0.0966116i	$0.0308005\pi$
$293$			56.6144i			0.193223i	−0.995322	+	0.0966116i	$0.969200\pi$
$294$		0			0
$295$			201.303i			0.682382i
$296$		0			0
$297$	−17.3959			−0.0585721
$298$		0			0
$299$	−138.595			−0.463528
$300$		0			0
$301$		−	270.623i		−	0.899080i
$302$		0			0
$303$	201.422			0.664758
$304$		0			0
$305$		−	394.198i		−	1.29245i
$306$		0			0
$307$		−	110.756i		−	0.360768i	−0.983596	−	0.180384i	$-0.942266\pi$
$307$		−	110.756i		−	0.360768i	0.983596	−	0.180384i	$-0.0577340\pi$
$308$		0			0
$309$	158.604			0.513282
$310$		0			0
$311$		−	234.379i		−	0.753629i	−0.926289	−	0.376814i	$-0.877019\pi$
$311$		−	234.379i		−	0.753629i	0.926289	−	0.376814i	$-0.122981\pi$
$312$		0			0
$313$			38.6992i			0.123640i	0.998087	+	0.0618199i	$0.0196904\pi$
$313$			38.6992i			0.123640i	−0.998087	+	0.0618199i	$0.980310\pi$
$314$		0			0
$315$			205.933i			0.653756i
$316$		0			0
$317$			46.6350i			0.147114i	0.997291	+	0.0735568i	$0.0234350\pi$
$317$			46.6350i			0.147114i	−0.997291	+	0.0735568i	$0.976565\pi$
$318$		0			0
$319$			34.9037i			0.109416i
$320$		0			0
$321$	−41.6157			−0.129644
$322$		0			0
$323$	235.691			0.729694
$324$		0			0
$325$	103.972			0.319914
$326$		0			0
$327$			298.780i			0.913701i
$328$		0			0
$329$	193.239			0.587352
$330$		0			0
$331$		−	83.4597i		−	0.252144i	−0.992021	−	0.126072i	$-0.959763\pi$
$331$		−	83.4597i		−	0.252144i	0.992021	−	0.126072i	$-0.0402371\pi$
$332$		0			0
$333$		−	284.287i		−	0.853714i
$334$		0			0
$335$		−	76.5844i		−	0.228610i
$336$		0			0
$337$	16.1289			0.0478601			0.0239300	−	0.999714i	$-0.492382\pi$
$337$	16.1289			0.0478601			0.0239300	+	0.999714i	$0.492382\pi$
$338$		0			0
$339$	−357.955			−1.05592
$340$		0			0
$341$		−	57.0625i		−	0.167339i
$342$		0			0
$343$	356.496			1.03935
$344$		0			0
$345$		−	157.655i		−	0.456971i
$346$		0			0
$347$	326.930			0.942161			0.471081	−	0.882090i	$-0.343864\pi$
$347$	326.930			0.942161			0.471081	+	0.882090i	$0.343864\pi$
$348$		0			0
$349$	664.022			1.90264			0.951320	−	0.308204i	$-0.0997278\pi$
$349$	664.022			1.90264			0.951320	+	0.308204i	$0.0997278\pi$
$350$		0			0
$351$	106.714			0.304028
$352$		0			0
$353$			467.667i			1.32484i	0.749134	+	0.662418i	$0.230470\pi$
$353$			467.667i			1.32484i	−0.749134	+	0.662418i	$0.769530\pi$
$354$		0			0
$355$			94.6937i			0.266743i
$356$		0			0
$357$	−379.429			−1.06283
$358$		0			0
$359$		−	406.278i		−	1.13169i	−0.824511	−	0.565846i	$-0.808550\pi$
$359$		−	406.278i		−	1.13169i	0.824511	−	0.565846i	$-0.191450\pi$
$360$		0			0
$361$	−81.1651			−0.224834
$362$		0			0
$363$	463.331			1.27639
$364$		0			0
$365$	31.0628			0.0851036
$366$		0			0
$367$			63.4160i			0.172796i	0.996261	+	0.0863978i	$0.0275356\pi$
$367$			63.4160i			0.172796i	−0.996261	+	0.0863978i	$0.972464\pi$
$368$		0			0
$369$			567.177i			1.53706i
$370$		0			0
$371$		−	645.469i		−	1.73981i
$372$		0			0
$373$		−	538.512i		−	1.44373i	−0.692033	−	0.721866i	$-0.743285\pi$
$373$		−	538.512i		−	1.44373i	0.692033	−	0.721866i	$-0.256715\pi$
$374$		0			0
$375$			546.974i			1.45860i
$376$		0			0
$377$		−	214.113i		−	0.567940i
$378$		0			0
$379$	−257.670			−0.679869			−0.339935	−	0.940449i	$-0.610405\pi$
$379$	−257.670			−0.679869			−0.339935	+	0.940449i	$0.610405\pi$
$380$		0			0
$381$	−58.8596			−0.154487
$382$		0			0
$383$	291.461			0.760994			0.380497	−	0.924782i	$-0.375753\pi$
$383$	291.461			0.760994			0.380497	+	0.924782i	$0.375753\pi$
$384$		0			0
$385$		−	69.8643i		−	0.181466i
$386$		0			0
$387$			293.375i			0.758075i
$388$		0			0
$389$		−	147.113i		−	0.378183i	−0.981959	−	0.189092i	$-0.939446\pi$
$389$		−	147.113i		−	0.378183i	0.981959	−	0.189092i	$-0.0605543\pi$
$390$		0			0
$391$	129.534			0.331289
$392$		0			0
$393$		−	464.077i		−	1.18086i
$394$		0			0
$395$	66.6161			0.168648
$396$		0			0
$397$			281.789i			0.709797i	0.934905	+	0.354898i	$0.115485\pi$
$397$			281.789i			0.709797i	−0.934905	+	0.354898i	$0.884515\pi$
$398$		0			0
$399$	−450.493			−1.12906
$400$		0			0
$401$	311.189			0.776031			0.388016	−	0.921653i	$-0.373161\pi$
$401$	311.189			0.776031			0.388016	+	0.921653i	$0.373161\pi$
$402$		0			0
$403$			350.045i			0.868598i
$404$		0			0
$405$			398.768i			0.984611i
$406$		0			0
$407$			96.4463i			0.236969i
$408$		0			0
$409$	40.7348			0.0995961			0.0497980	−	0.998759i	$-0.484142\pi$
$409$	40.7348			0.0995961			0.0497980	+	0.998759i	$0.484142\pi$
$410$		0			0
$411$	−350.700			−0.853285
$412$		0			0
$413$	316.134			0.765456
$414$		0			0
$415$	14.8382			0.0357546
$416$		0			0
$417$			546.329i			1.31014i
$418$		0			0
$419$			534.408i			1.27544i	0.770270	+	0.637718i	$0.220121\pi$
$419$			534.408i			1.27544i	−0.770270	+	0.637718i	$0.779879\pi$
$420$		0			0
$421$	−462.655			−1.09894			−0.549472	−	0.835512i	$-0.685171\pi$
$421$	−462.655			−1.09894			−0.549472	+	0.835512i	$0.685171\pi$
$422$		0			0
$423$	−209.485			−0.495237
$424$		0			0
$425$	−97.1745			−0.228646
$426$		0			0
$427$	−619.064			−1.44980
$428$		0			0
$429$	149.307			0.348036
$430$		0			0
$431$			176.518i			0.409556i	0.978808	+	0.204778i	$0.0656472\pi$
$431$			176.518i			0.409556i	−0.978808	+	0.204778i	$0.934353\pi$
$432$		0			0
$433$			141.667i			0.327176i	0.986529	+	0.163588i	$0.0523068\pi$
$433$			141.667i			0.327176i	−0.986529	+	0.163588i	$0.947693\pi$
$434$		0			0
$435$	243.559			0.559905
$436$		0			0
$437$	153.795			0.351933
$438$		0			0
$439$	39.1999			0.0892935			0.0446468	−	0.999003i	$-0.485784\pi$
$439$	39.1999			0.0892935			0.0446468	+	0.999003i	$0.485784\pi$
$440$		0			0
$441$	−31.5311			−0.0714991
$442$		0			0
$443$		−	830.230i		−	1.87411i	−0.349184	−	0.937054i	$-0.613541\pi$
$443$		−	830.230i		−	1.87411i	0.349184	−	0.937054i	$-0.386459\pi$
$444$		0			0
$445$	−414.987			−0.932556
$446$		0			0
$447$	−886.670			−1.98360
$448$		0			0
$449$	361.534			0.805197			0.402599	−	0.915377i	$-0.368107\pi$
$449$	361.534			0.805197			0.402599	+	0.915377i	$0.368107\pi$
$450$		0			0
$451$		−	192.419i		−	0.426649i
$452$		0			0
$453$			553.883i			1.22270i
$454$		0			0
$455$			428.576i			0.941926i
$456$		0			0
$457$		−	723.336i		−	1.58279i	−0.611303	−	0.791396i	$-0.709354\pi$
$457$		−	723.336i		−	1.58279i	0.611303	−	0.791396i	$-0.290646\pi$
$458$		0			0
$459$	−99.7370			−0.217292
$460$		0			0
$461$	−428.479			−0.929456			−0.464728	−	0.885454i	$-0.653848\pi$
$461$	−428.479			−0.929456			−0.464728	+	0.885454i	$0.653848\pi$
$462$		0			0
$463$		−	251.468i		−	0.543128i	−0.962420	−	0.271564i	$-0.912459\pi$
$463$		−	251.468i		−	0.543128i	0.962420	−	0.271564i	$-0.0875408\pi$
$464$		0			0
$465$	−398.184			−0.856310
$466$		0			0
$467$	−723.554			−1.54937			−0.774683	−	0.632350i	$-0.782090\pi$
$467$	−723.554			−1.54937			−0.774683	+	0.632350i	$0.782090\pi$
$468$		0			0
$469$	−120.271			−0.256442
$470$		0			0
$471$			365.272i			0.775525i
$472$		0			0
$473$		−	99.5297i		−	0.210422i
$474$		0			0
$475$	−115.375			−0.242894
$476$		0			0
$477$			699.735i			1.46695i
$478$		0			0
$479$			445.530i			0.930126i	0.885278	+	0.465063i	$0.153968\pi$
$479$			445.530i			0.930126i	−0.885278	+	0.465063i	$0.846032\pi$
$480$		0			0
$481$		−	591.641i		−	1.23002i
$482$		0			0
$483$	−247.587			−0.512603
$484$		0			0
$485$	−118.520			−0.244371
$486$		0			0
$487$			288.547i			0.592500i	0.955110	+	0.296250i	$0.0957361\pi$
$487$			288.547i			0.592500i	−0.955110	+	0.296250i	$0.904264\pi$
$488$		0			0
$489$		−	176.712i		−	0.361374i
$490$		0			0
$491$	273.941			0.557924			0.278962	−	0.960302i	$-0.410010\pi$
$491$	273.941			0.557924			0.278962	+	0.960302i	$0.410010\pi$
$492$		0			0
$493$			200.115i			0.405913i
$494$		0			0
$495$			75.7380i			0.153006i
$496$		0			0
$497$	148.711			0.299217
$498$		0			0
$499$			774.221i			1.55155i	0.631012	+	0.775773i	$0.282640\pi$
$499$			774.221i			1.55155i	−0.631012	+	0.775773i	$0.717360\pi$
$500$		0			0
$501$		−	552.717i		−	1.10323i
$502$		0			0
$503$	−414.999			−0.825048			−0.412524	−	0.910947i	$-0.635353\pi$
$503$	−414.999			−0.825048			−0.412524	+	0.910947i	$0.635353\pi$
$504$		0			0
$505$			212.638i			0.421065i
$506$		0			0
$507$	−234.786			−0.463089
$508$		0			0
$509$	138.696			0.272487			0.136243	−	0.990675i	$-0.456497\pi$
$509$	138.696			0.272487			0.136243	+	0.990675i	$0.456497\pi$
$510$		0			0
$511$		−	48.7822i		−	0.0954642i
$512$		0			0
$513$	−118.417			−0.230833
$514$		0			0
$515$			167.436i			0.325119i
$516$		0			0
$517$	71.0693			0.137465
$518$		0			0
$519$			138.663i			0.267173i
$520$		0			0
$521$	879.921			1.68891			0.844454	−	0.535629i	$-0.179925\pi$
$521$	879.921			1.68891			0.844454	+	0.535629i	$0.179925\pi$
$522$		0			0
$523$			284.281i			0.543559i	0.962360	+	0.271779i	$0.0876121\pi$
$523$			284.281i			0.543559i	−0.962360	+	0.271779i	$0.912388\pi$
$524$		0			0
$525$	185.736			0.353784
$526$		0			0
$527$		−	327.160i		−	0.620796i
$528$		0			0
$529$	−444.476			−0.840219
$530$		0			0
$531$	−342.712			−0.645408
$532$		0			0
$533$			1180.38i			2.21459i
$534$		0			0
$535$		−	43.9330i		−	0.0821178i
$536$		0			0
$537$	−613.642			−1.14272
$538$		0			0
$539$	10.6972			0.0198463
$540$		0			0
$541$	−185.988			−0.343786			−0.171893	−	0.985116i	$-0.554988\pi$
$541$	−185.988			−0.343786			−0.171893	+	0.985116i	$0.554988\pi$
$542$		0			0
$543$	−1329.29			−2.44804
$544$		0			0
$545$	−315.418			−0.578748
$546$		0			0
$547$		−	637.414i		−	1.16529i	−0.812727	−	0.582645i	$-0.802018\pi$
$547$		−	637.414i		−	1.16529i	0.812727	−	0.582645i	$-0.197982\pi$
$548$		0			0
$549$	671.110			1.22242
$550$		0			0
$551$			237.595i			0.431208i
$552$		0			0
$553$		−	104.617i		−	0.189180i
$554$		0			0
$555$	673.005			1.21262
$556$		0			0
$557$	258.749			0.464541			0.232271	−	0.972651i	$-0.425385\pi$
$557$	258.749			0.464541			0.232271	+	0.972651i	$0.425385\pi$
$558$		0			0
$559$			610.556i			1.09223i
$560$		0			0
$561$	−139.546			−0.248745
$562$		0			0
$563$	−358.152	−	434.392i	−0.636149	−	0.771566i
$563$	−358.152	−	434.392i	−0.636149	−	0.771566i
$564$		0			0
$565$		−	377.888i		−	0.668828i
$566$		0			0
$567$	626.240			1.10448
$568$		0			0
$569$		−	415.359i		−	0.729981i	−0.931011	−	0.364991i	$-0.881072\pi$
$569$		−	415.359i		−	0.729981i	0.931011	−	0.364991i	$-0.118928\pi$
$570$		0			0
$571$			348.772i			0.610810i	0.952223	+	0.305405i	$0.0987918\pi$
$571$			348.772i			0.610810i	−0.952223	+	0.305405i	$0.901208\pi$
$572$		0			0
$573$	−1463.34			−2.55382
$574$		0			0
$575$	−63.4090			−0.110276
$576$		0			0
$577$		−	464.581i		−	0.805166i	−0.915384	−	0.402583i	$-0.868113\pi$
$577$		−	464.581i		−	0.805166i	0.915384	−	0.402583i	$-0.131887\pi$
$578$		0			0
$579$	15.0083			0.0259210
$580$		0			0
$581$		−	23.3024i		−	0.0401074i
$582$		0			0
$583$		−	237.390i		−	0.407187i
$584$		0			0
$585$		−	464.608i		−	0.794202i
$586$		0			0
$587$			1012.69i			1.72519i	0.505893	+	0.862596i	$0.331163\pi$
$587$			1012.69i			1.72519i	−0.505893	+	0.862596i	$0.668837\pi$
$588$		0			0
$589$		−	388.435i		−	0.659482i
$590$		0			0
$591$	533.532			0.902762
$592$		0			0
$593$	−1119.79			−1.88836			−0.944178	−	0.329436i	$-0.893141\pi$
$593$	−1119.79			−1.88836			−0.944178	+	0.329436i	$0.893141\pi$
$594$		0			0
$595$		−	400.557i		−	0.673205i
$596$		0			0
$597$			1240.51i			2.07790i
$598$		0			0
$599$	1041.44			1.73863			0.869316	−	0.494257i	$-0.164560\pi$
$599$	1041.44			1.73863			0.869316	+	0.494257i	$0.164560\pi$
$600$		0			0
$601$		−	908.444i		−	1.51155i	−0.654829	−	0.755777i	$-0.727259\pi$
$601$		−	908.444i		−	1.51155i	0.654829	−	0.755777i	$-0.272741\pi$
$602$		0			0
$603$	130.383			0.216223
$604$		0			0
$605$			489.132i			0.808482i
$606$		0			0
$607$	−699.315			−1.15208			−0.576042	−	0.817420i	$-0.695404\pi$
$607$	−699.315			−1.15208			−0.576042	+	0.817420i	$0.695404\pi$
$608$		0			0
$609$		−	382.494i		−	0.628069i
$610$		0			0
$611$	−435.968			−0.713532
$612$		0			0
$613$			368.757i			0.601562i	0.953693	+	0.300781i	$0.0972473\pi$
$613$			368.757i			0.601562i	−0.953693	+	0.300781i	$0.902753\pi$
$614$		0			0
$615$	−1342.70			−2.18326
$616$		0			0
$617$			724.386i			1.17404i	0.809571	+	0.587022i	$0.199700\pi$
$617$			724.386i			1.17404i	−0.809571	+	0.587022i	$0.800300\pi$
$618$		0			0
$619$			226.575i			0.366034i	0.983110	+	0.183017i	$0.0585863\pi$
$619$			226.575i			0.366034i	−0.983110	+	0.183017i	$0.941414\pi$
$620$		0			0
$621$	−65.0811			−0.104800
$622$		0			0
$623$			651.712i			1.04609i
$624$		0			0
$625$	−405.007			−0.648011
$626$		0			0
$627$	−165.682			−0.264246
$628$		0			0
$629$			552.961i			0.879111i
$630$		0			0
$631$	489.789			0.776210			0.388105	−	0.921615i	$-0.373130\pi$
$631$	489.789			0.776210			0.388105	+	0.921615i	$0.373130\pi$
$632$		0			0
$633$	606.875			0.958728
$634$		0			0
$635$		−	62.1372i		−	0.0978538i
$636$		0			0
$637$	−65.6207			−0.103015
$638$		0			0
$639$	−161.213			−0.252290
$640$		0			0
$641$		−	330.916i		−	0.516249i	−0.966112	−	0.258125i	$-0.916895\pi$
$641$		−	330.916i		−	0.516249i	0.966112	−	0.258125i	$-0.0831045\pi$
$642$		0			0
$643$			1005.42i			1.56364i	0.623507	+	0.781818i	$0.285707\pi$
$643$			1005.42i			1.56364i	−0.623507	+	0.781818i	$0.714293\pi$
$644$		0			0
$645$	−694.521			−1.07678
$646$		0			0
$647$	151.293			0.233837			0.116919	−	0.993141i	$-0.462698\pi$
$647$	151.293			0.233837			0.116919	+	0.993141i	$0.462698\pi$
$648$		0			0
$649$	116.267			0.179149
$650$		0			0
$651$			625.323i			0.960558i
$652$		0			0
$653$	−1274.74			−1.95213			−0.976066	−	0.217476i	$-0.930218\pi$
$653$	−1274.74			−1.95213			−0.976066	+	0.217476i	$0.930218\pi$
$654$		0			0
$655$	489.919			0.747968
$656$		0			0
$657$			52.8835i			0.0804923i
$658$		0			0
$659$		−	332.781i		−	0.504979i	−0.967600	−	0.252490i	$-0.918751\pi$
$659$		−	332.781i		−	0.504979i	0.967600	−	0.252490i	$-0.0812494\pi$
$660$		0			0
$661$			538.739i			0.815036i	0.913197	+	0.407518i	$0.133606\pi$
$661$			538.739i			0.815036i	−0.913197	+	0.407518i	$0.866394\pi$
$662$		0			0
$663$	856.033			1.29115
$664$		0			0
$665$		−	475.579i		−	0.715156i
$666$		0			0
$667$			130.580i			0.195773i
$668$		0			0
$669$	418.614			0.625731
$670$		0			0
$671$	−227.679			−0.339313
$672$		0			0
$673$	113.060			0.167994			0.0839970	−	0.996466i	$-0.473231\pi$
$673$	113.060			0.167994			0.0839970	+	0.996466i	$0.473231\pi$
$674$		0			0
$675$	48.8229			0.0723301
$676$		0			0
$677$			375.558i			0.554738i	0.960763	+	0.277369i	$0.0894625\pi$
$677$			375.558i			0.554738i	−0.960763	+	0.277369i	$0.910537\pi$
$678$		0			0
$679$			186.128i			0.274121i
$680$		0			0
$681$		−	94.5751i		−	0.138877i
$682$		0			0
$683$	−250.675			−0.367021			−0.183511	−	0.983018i	$-0.558746\pi$
$683$	−250.675			−0.367021			−0.183511	+	0.983018i	$0.558746\pi$
$684$		0			0
$685$		−	370.229i		−	0.540480i
$686$		0			0
$687$			1346.37i			1.95978i
$688$		0			0
$689$			1456.25i			2.11357i
$690$		0			0
$691$			898.966i			1.30096i	0.759522	+	0.650482i	$0.225433\pi$
$691$			898.966i			1.30096i	−0.759522	+	0.650482i	$0.774567\pi$
$692$		0			0
$693$	118.942			0.171633
$694$		0			0
$695$	−576.751			−0.829857
$696$		0			0
$697$		−	1103.20i		−	1.58279i
$698$		0			0
$699$			373.883i			0.534883i
$700$		0			0
$701$			637.027i			0.908741i	0.890813	+	0.454370i	$0.150136\pi$
$701$			637.027i			0.908741i	−0.890813	+	0.454370i	$0.849864\pi$
$702$		0			0
$703$			656.527i			0.933894i
$704$		0			0
$705$		−	495.924i		−	0.703438i
$706$		0			0
$707$	333.935			0.472326
$708$		0			0
$709$	520.466			0.734085			0.367042	−	0.930204i	$-0.380370\pi$
$709$	520.466			0.734085			0.367042	+	0.930204i	$0.380370\pi$
$710$		0			0
$711$			113.412i			0.159510i
$712$		0			0
$713$		−	213.481i		−	0.299412i
$714$		0			0
$715$			157.622i			0.220450i
$716$		0			0
$717$			1132.18i			1.57905i
$718$		0			0
$719$	1002.60			1.39443			0.697216	−	0.716861i	$-0.254422\pi$
$719$	1002.60			1.39443			0.697216	+	0.716861i	$0.254422\pi$
$720$		0			0
$721$	262.948			0.364699
$722$		0			0
$723$	−380.523			−0.526311
$724$		0			0
$725$		−	97.9595i		−	0.135117i
$726$		0			0
$727$			832.460i			1.14506i	0.819883	+	0.572531i	$0.194038\pi$
$727$			832.460i			1.14506i	−0.819883	+	0.572531i	$0.805962\pi$
$728$		0			0
$729$	−422.118			−0.579037
$730$		0			0
$731$		−	570.639i		−	0.780628i
$732$		0			0
$733$	−377.212			−0.514614			−0.257307	−	0.966330i	$-0.582835\pi$
$733$	−377.212			−0.514614			−0.257307	+	0.966330i	$0.582835\pi$
$734$		0			0
$735$		−	74.6451i		−	0.101558i
$736$		0			0
$737$	−44.2332			−0.0600180
$738$		0			0
$739$	−905.019			−1.22465			−0.612327	−	0.790605i	$-0.709766\pi$
$739$	−905.019			−1.22465			−0.612327	+	0.790605i	$0.709766\pi$
$740$		0			0
$741$	1016.36			1.37161
$742$		0			0
$743$			314.877i			0.423791i	0.977292	+	0.211896i	$0.0679637\pi$
$743$			314.877i			0.423791i	−0.977292	+	0.211896i	$0.932036\pi$
$744$		0			0
$745$		−	936.044i		−	1.25644i
$746$		0			0
$747$			25.2615i			0.0338173i
$748$		0			0
$749$	−68.9941			−0.0921150
$750$		0			0
$751$	−452.327			−0.602299			−0.301149	−	0.953577i	$-0.597370\pi$
$751$	−452.327			−0.602299			−0.301149	+	0.953577i	$0.597370\pi$
$752$		0			0
$753$	332.401			0.441435
$754$		0			0
$755$	−584.726			−0.774472
$756$		0			0
$757$	596.036			0.787366			0.393683	−	0.919246i	$-0.371201\pi$
$757$	596.036			0.787366			0.393683	+	0.919246i	$0.371201\pi$
$758$		0			0
$759$	−91.0576			−0.119970
$760$		0			0
$761$		−	205.721i		−	0.270330i	−0.990823	−	0.135165i	$-0.956844\pi$
$761$		−	205.721i		−	0.270330i	0.990823	−	0.135165i	$-0.0431564\pi$
$762$		0			0
$763$			495.344i			0.649206i
$764$		0			0
$765$			434.233i			0.567625i
$766$		0			0
$767$	−713.232			−0.929898
$768$		0			0
$769$		−	906.018i		−	1.17818i	−0.808069	−	0.589088i	$-0.799487\pi$
$769$		−	906.018i		−	1.17818i	0.808069	−	0.589088i	$-0.200513\pi$
$770$		0			0
$771$	13.5339			0.0175537
$772$		0			0
$773$	992.037			1.28336			0.641679	−	0.766973i	$-0.278238\pi$
$773$	992.037			1.28336			0.641679	+	0.766973i	$0.278238\pi$
$774$		0			0
$775$			160.150i			0.206645i
$776$		0			0
$777$		−	1056.91i		−	1.36025i
$778$		0			0
$779$		−	1309.83i		−	1.68142i
$780$		0			0
$781$	54.6927			0.0700291
$782$		0			0
$783$		−	100.543i		−	0.128407i
$784$		0			0
$785$	−385.612			−0.491226
$786$		0			0
$787$			617.318i			0.784394i	0.919881	+	0.392197i	$0.128285\pi$
$787$			617.318i			0.784394i	−0.919881	+	0.392197i	$0.871715\pi$
$788$		0			0
$789$		−	528.404i		−	0.669714i
$790$		0			0
$791$	−593.450			−0.750253
$792$		0			0
$793$	1396.68			1.76126
$794$		0			0
$795$	−1656.52			−2.08367
$796$		0			0
$797$		−	789.555i		−	0.990658i	−0.868705	−	0.495329i	$-0.835047\pi$
$797$		−	789.555i		−	0.990658i	0.868705	−	0.495329i	$-0.164953\pi$
$798$		0			0
$799$	407.466			0.509969
$800$		0			0
$801$		−	706.503i		−	0.882026i
$802$		0			0
$803$		−	17.9411i		−	0.0223426i
$804$		0			0
$805$		−	261.374i		−	0.324689i
$806$		0			0
$807$	−695.556			−0.861904
$808$		0			0
$809$	−802.650			−0.992150			−0.496075	−	0.868280i	$-0.665226\pi$
$809$	−802.650			−0.992150			−0.496075	+	0.868280i	$0.665226\pi$
$810$		0			0
$811$	699.255			0.862214			0.431107	−	0.902301i	$-0.358123\pi$
$811$	699.255			0.862214			0.431107	+	0.902301i	$0.358123\pi$
$812$		0			0
$813$	−1051.97			−1.29394
$814$		0			0
$815$	186.552			0.228898
$816$		0			0
$817$		−	677.516i		−	0.829273i
$818$		0			0
$819$	−729.638			−0.890889
$820$		0			0
$821$	363.539			0.442800			0.221400	−	0.975183i	$-0.428937\pi$
$821$	363.539			0.442800			0.221400	+	0.975183i	$0.428937\pi$
$822$		0			0
$823$		−	1054.92i		−	1.28180i	−0.767625	−	0.640900i	$-0.778561\pi$
$823$		−	1054.92i		−	1.28180i	0.767625	−	0.640900i	$-0.221439\pi$
$824$		0			0
$825$	68.3100			0.0828000
$826$		0			0
$827$		−	261.052i		−	0.315661i	−0.987466	−	0.157830i	$-0.949550\pi$
$827$		−	261.052i		−	0.315661i	0.987466	−	0.157830i	$-0.0504500\pi$
$828$		0			0
$829$		−	1185.82i		−	1.43043i	−0.698907	−	0.715213i	$-0.746330\pi$
$829$		−	1185.82i		−	1.43043i	0.698907	−	0.715213i	$-0.253670\pi$
$830$		0			0
$831$	1141.70			1.37389
$832$		0			0
$833$	61.3306			0.0736262
$834$		0			0
$835$	583.495			0.698797
$836$		0			0
$837$			164.373i			0.196384i
$838$		0			0
$839$	110.824			0.132091			0.0660456	−	0.997817i	$-0.478962\pi$
$839$	110.824			0.132091			0.0660456	+	0.997817i	$0.478962\pi$
$840$		0			0
$841$	639.268			0.760129
$842$		0			0
$843$	−146.523			−0.173812
$844$		0			0
$845$		−	247.860i		−	0.293326i
$846$		0			0
$847$	768.151			0.906908
$848$		0			0
$849$			1215.59i			1.43179i
$850$		0			0
$851$			360.822i			0.423998i
$852$		0			0
$853$			558.243i			0.654446i	0.944947	+	0.327223i	$0.106113\pi$
$853$			558.243i			0.654446i	−0.944947	+	0.327223i	$0.893887\pi$
$854$		0			0
$855$			515.562i			0.602997i
$856$		0			0
$857$			234.119i			0.273185i	0.990627	+	0.136592i	$0.0436151\pi$
$857$			234.119i			0.273185i	−0.990627	+	0.136592i	$0.956385\pi$
$858$		0			0
$859$	920.390			1.07147			0.535733	−	0.844387i	$-0.320035\pi$
$859$	920.390			1.07147			0.535733	+	0.844387i	$0.320035\pi$
$860$		0			0
$861$			2108.63i			2.44905i
$862$		0			0
$863$	639.808			0.741376			0.370688	−	0.928757i	$-0.379122\pi$
$863$	639.808			0.741376			0.370688	+	0.928757i	$0.379122\pi$
$864$		0			0
$865$	−146.384			−0.169231
$866$		0			0
$867$	364.700			0.420646
$868$		0			0
$869$		−	38.4758i		−	0.0442760i
$870$		0			0
$871$	271.345			0.311532
$872$		0			0
$873$		−	201.777i		−	0.231130i
$874$		0			0
$875$			906.821i			1.03637i
$876$		0			0
$877$	973.200			1.10969			0.554846	−	0.831953i	$-0.312777\pi$
$877$	973.200			1.10969			0.554846	+	0.831953i	$0.312777\pi$
$878$		0			0
$879$		−	228.175i		−	0.259585i
$880$		0			0
$881$	−133.233			−0.151229			−0.0756147	−	0.997137i	$-0.524092\pi$
$881$	−133.233			−0.151229			−0.0756147	+	0.997137i	$0.524092\pi$
$882$		0			0
$883$		−	1380.72i		−	1.56367i	−0.623485	−	0.781835i	$-0.714284\pi$
$883$		−	1380.72i		−	1.56367i	0.623485	−	0.781835i	$-0.285716\pi$
$884$		0			0
$885$		−	811.317i		−	0.916743i
$886$		0			0
$887$	1074.34			1.21121			0.605603	−	0.795767i	$-0.292932\pi$
$887$	1074.34			1.21121			0.605603	+	0.795767i	$0.292932\pi$
$888$		0			0
$889$	−97.5826			−0.109767
$890$		0			0
$891$	230.318			0.258494
$892$		0			0
$893$	483.782			0.541749
$894$		0			0
$895$		−	647.812i		−	0.723812i
$896$		0			0
$897$	558.584			0.622725
$898$		0			0
$899$	329.803			0.366855
$900$		0			0
$901$		−	1361.04i		−	1.51059i
$902$		0			0
$903$			1090.70i			1.20787i
$904$		0			0
$905$		−	1403.31i		−	1.55062i
$906$		0			0
$907$	540.676			0.596114			0.298057	−	0.954548i	$-0.403661\pi$
$907$	540.676			0.596114			0.298057	+	0.954548i	$0.403661\pi$
$908$		0			0
$909$	−362.009			−0.398250
$910$		0			0
$911$		−	1474.26i		−	1.61828i	−0.587614	−	0.809141i	$-0.699933\pi$
$911$		−	1474.26i		−	1.61828i	0.587614	−	0.809141i	$-0.300067\pi$
$912$		0			0
$913$		−	8.57015i		−	0.00938680i
$914$		0			0
$915$			1588.75i			1.73634i
$916$		0			0
$917$		−	769.387i		−	0.839027i
$918$		0			0
$919$		−	1540.17i		−	1.67591i	−0.545736	−	0.837957i	$-0.683750\pi$
$919$		−	1540.17i		−	1.67591i	0.545736	−	0.837957i	$-0.316250\pi$
$920$		0			0
$921$			446.382i			0.484671i
$922$		0			0
$923$	−335.507			−0.363497
$924$		0			0
$925$		−	270.683i		−	0.292631i
$926$		0			0
$927$	−285.055			−0.307502
$928$		0			0
$929$			1165.91i			1.25501i	0.778611	+	0.627507i	$0.215925\pi$
$929$			1165.91i			1.25501i	−0.778611	+	0.627507i	$0.784075\pi$
$930$		0			0
$931$	72.8175			0.0782143
$932$		0			0
$933$			944.624i			1.01246i
$934$		0			0
$935$		−	147.317i		−	0.157558i
$936$		0			0
$937$			1406.47i			1.50104i	0.660848	+	0.750520i	$0.270197\pi$
$937$			1406.47i			1.50104i	−0.660848	+	0.750520i	$0.729803\pi$
$938$		0			0
$939$		−	155.971i		−	0.166103i
$940$		0			0
$941$		−	832.947i		−	0.885172i	−0.896726	−	0.442586i	$-0.854061\pi$
$941$		−	832.947i		−	0.885172i	0.896726	−	0.442586i	$-0.145939\pi$
$942$		0			0
$943$		−	719.871i		−	0.763384i
$944$		0			0
$945$			201.250i			0.212963i
$946$		0			0
$947$		−	23.1923i		−	0.0244903i	−0.999925	−	0.0122452i	$-0.996102\pi$
$947$		−	23.1923i		−	0.0244903i	0.999925	−	0.0122452i	$-0.00389785\pi$
$948$		0			0
$949$			110.058i			0.115973i
$950$		0			0
$951$		−	187.955i		−	0.197639i
$952$		0			0
$953$	1517.73			1.59258			0.796290	−	0.604914i	$-0.206793\pi$
$953$	1517.73			1.59258			0.796290	+	0.604914i	$0.206793\pi$
$954$		0			0
$955$		−	1544.83i		−	1.61762i
$956$		0			0
$957$		−	140.673i		−	0.146994i
$958$		0			0
$959$	−581.421			−0.606279
$960$		0			0
$961$	421.819			0.438938
$962$		0			0
$963$	74.7947			0.0776684
$964$		0			0
$965$			15.8440i			0.0164187i
$966$		0			0
$967$	933.308			0.965158			0.482579	−	0.875852i	$-0.339700\pi$
$967$	933.308			0.965158			0.482579	+	0.875852i	$0.339700\pi$
$968$		0			0
$969$	−949.915			−0.980304
$970$		0			0
$971$			949.908i			0.978278i	0.872206	+	0.489139i	$0.162689\pi$
$971$			949.908i			0.978278i	−0.872206	+	0.489139i	$0.837311\pi$
$972$		0			0
$973$			905.752i			0.930886i
$974$		0			0
$975$	−419.042			−0.429786
$976$		0			0
$977$			1319.54i			1.35061i	0.737540	+	0.675304i	$0.235988\pi$
$977$			1319.54i			1.35061i	−0.737540	+	0.675304i	$0.764012\pi$
$978$		0			0
$979$			239.686i			0.244828i
$980$		0			0
$981$		−	536.989i		−	0.547390i
$982$		0			0
$983$		−	648.594i		−	0.659811i	−0.944014	−	0.329906i	$-0.892983\pi$
$983$		−	648.594i		−	0.659811i	0.944014	−	0.329906i	$-0.107017\pi$
$984$		0			0
$985$			563.242i			0.571819i
$986$		0			0
$987$	−778.818			−0.789076
$988$		0			0
$989$		−	372.357i		−	0.376499i
$990$		0			0
$991$	330.317			0.333317			0.166658	−	0.986015i	$-0.446702\pi$
$991$	330.317			0.333317			0.166658	+	0.986015i	$0.446702\pi$
$992$		0			0
$993$			336.371i			0.338742i
$994$		0			0
$995$	−1309.59			−1.31617
$996$		0			0
$997$	250.592			0.251346			0.125673	−	0.992072i	$-0.459891\pi$
$997$	250.592			0.251346			0.125673	+	0.992072i	$0.459891\pi$
$998$		0			0
$999$		−	277.821i		−	0.278099i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2252.3.d.b.1125.14	yes	76
563.562	odd	2	inner	2252.3.d.b.1125.13	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2252.3.d.b.1125.13	✓	76	563.562	odd	2	inner
2252.3.d.b.1125.14	yes	76	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2252 = 2^{2} \cdot 563 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.03034 q^{3} -4.25476i q^{5} -6.68184 q^{7} +7.24360 q^{9} -2.45745 q^{11} +15.0750 q^{13} +17.1481i q^{15} -14.0894 q^{17} -16.7283 q^{19} +26.9301 q^{21} -9.19372 q^{23} +6.89699 q^{25} +7.07886 q^{27} -14.2032i q^{29} +23.2203i q^{31} +9.90433 q^{33} +28.4297i q^{35} -39.2466i q^{37} -60.7572 q^{39} +78.3003i q^{41} +40.5013i q^{43} -30.8198i q^{45} -28.9200 q^{47} -4.35296 q^{49} +56.7850 q^{51} +96.6004i q^{53} +10.4558i q^{55} +67.4205 q^{57} -47.3123 q^{59} +92.6486 q^{61} -48.4006 q^{63} -64.1404i q^{65} +17.9997 q^{67} +37.0538 q^{69} -22.2559 q^{71} +7.30071i q^{73} -27.7972 q^{75} +16.4203 q^{77} +15.6568i q^{79} -93.7226 q^{81} +3.48742i q^{83} +59.9471i q^{85} +57.2438i q^{87} -97.5347i q^{89} -100.729 q^{91} -93.5855i q^{93} +71.1748i q^{95} -27.8558i q^{97} -17.8008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 4 q^{3} - 8 q^{7} + 128 q^{9} + 2 q^{11} - 6 q^{13} + 22 q^{17} + 12 q^{19} - 6 q^{21} + 24 q^{23} - 912 q^{25} + 22 q^{27} - 52 q^{33} - 70 q^{39} - 28 q^{47} - 340 q^{49} + 314 q^{51} - 98 q^{57}+ \cdots + 348 q^{99}+O(q^{100}) \) 76 * q + 4 * q^3 - 8 * q^7 + 128 * q^9 + 2 * q^11 - 6 * q^13 + 22 * q^17 + 12 * q^19 - 6 * q^21 + 24 * q^23 - 912 * q^25 + 22 * q^27 - 52 * q^33 - 70 * q^39 - 28 * q^47 - 340 * q^49 + 314 * q^51 - 98 * q^57 + 18 * q^59 - 162 * q^61 - 246 * q^63 - 178 * q^67 + 2 * q^69 + 72 * q^71 + 14 * q^75 - 498 * q^77 - 468 * q^81 + 132 * q^91 + 348 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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