LMFDB - Embedded newform 2300.3.d.c.1149.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1149,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1149");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$62.6704608029$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.15
Character	$\chi$	$=$	2300.1149
Dual form			2300.3.d.c.1149.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.10848i q^{3} +1.98337 q^{7} -0.662652 q^{9} +O(q^{10})$	$q-3.10848i q^{3} +1.98337 q^{7} -0.662652 q^{9} -10.6844i q^{11} +1.75531i q^{13} -21.2683 q^{17} +4.57847i q^{19} -6.16528i q^{21} +(6.72368 + 21.9953i) q^{23} -25.9165i q^{27} -9.21257 q^{29} -42.5991 q^{31} -33.2121 q^{33} +63.8869 q^{37} +5.45635 q^{39} -22.0962 q^{41} -61.7003 q^{43} -20.7975i q^{47} -45.0662 q^{49} +66.1121i q^{51} +55.7539 q^{53} +14.2321 q^{57} -34.0834 q^{59} -76.1879i q^{61} -1.31429 q^{63} -64.4207 q^{67} +(68.3719 - 20.9004i) q^{69} -126.165 q^{71} +103.394i q^{73} -21.1911i q^{77} -39.5132i q^{79} -86.5248 q^{81} +117.708 q^{83} +28.6371i q^{87} +78.6012i q^{89} +3.48144i q^{91} +132.419i q^{93} +74.1276 q^{97} +7.08001i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 128 q^{9}+O(q^{10}) $ 32 * q - 128 * q^9	$ 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) $ 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$		−	3.10848i		−	1.03616i	−0.855332	−	0.518080i	$-0.826647\pi$
$3$		−	3.10848i		−	1.03616i	0.855332	−	0.518080i	$-0.173353\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.98337			0.283339			0.141670	−	0.989914i	$-0.454753\pi$
$7$	1.98337			0.283339			0.141670	+	0.989914i	$0.454753\pi$
$8$		0			0
$9$	−0.662652			−0.0736280
$10$		0			0
$11$		−	10.6844i		−	0.971305i	−0.874152	−	0.485652i	$-0.838582\pi$
$11$		−	10.6844i		−	0.971305i	0.874152	−	0.485652i	$-0.161418\pi$
$12$		0			0
$13$			1.75531i			0.135024i	0.997718	+	0.0675120i	$0.0215061\pi$
$13$			1.75531i			0.135024i	−0.997718	+	0.0675120i	$0.978494\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21.2683			−1.25108			−0.625538	−	0.780193i	$-0.715121\pi$
$17$	−21.2683			−1.25108			−0.625538	+	0.780193i	$0.715121\pi$
$18$		0			0
$19$			4.57847i			0.240972i	0.992715	+	0.120486i	$0.0384453\pi$
$19$			4.57847i			0.240972i	−0.992715	+	0.120486i	$0.961555\pi$
$20$		0			0
$21$		−	6.16528i		−	0.293585i
$22$		0			0
$23$	6.72368	+	21.9953i	0.292334	+	0.956316i
$23$	6.72368	+	21.9953i	0.292334	+	0.956316i
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	25.9165i		−	0.959870i
$28$		0			0
$29$	−9.21257			−0.317675			−0.158837	−	0.987305i	$-0.550775\pi$
$29$	−9.21257			−0.317675			−0.158837	+	0.987305i	$0.550775\pi$
$30$		0			0
$31$	−42.5991			−1.37417			−0.687083	−	0.726579i	$-0.741109\pi$
$31$	−42.5991			−1.37417			−0.687083	+	0.726579i	$0.741109\pi$
$32$		0			0
$33$	−33.2121			−1.00643
$34$		0			0
$35$		0			0
$36$		0			0
$37$	63.8869			1.72667			0.863337	−	0.504628i	$-0.168370\pi$
$37$	63.8869			1.72667			0.863337	+	0.504628i	$0.168370\pi$
$38$		0			0
$39$	5.45635			0.139907
$40$		0			0
$41$	−22.0962			−0.538932			−0.269466	−	0.963010i	$-0.586847\pi$
$41$	−22.0962			−0.538932			−0.269466	+	0.963010i	$0.586847\pi$
$42$		0			0
$43$	−61.7003			−1.43489			−0.717445	−	0.696615i	$-0.754689\pi$
$43$	−61.7003			−1.43489			−0.717445	+	0.696615i	$0.754689\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	20.7975i		−	0.442500i	−0.975217	−	0.221250i	$-0.928986\pi$
$47$		−	20.7975i		−	0.442500i	0.975217	−	0.221250i	$-0.0710136\pi$
$48$		0			0
$49$	−45.0662			−0.919719
$50$		0			0
$51$			66.1121i			1.29632i
$52$		0			0
$53$	55.7539			1.05196			0.525980	−	0.850497i	$-0.323699\pi$
$53$	55.7539			1.05196			0.525980	+	0.850497i	$0.323699\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	14.2321			0.249686
$58$		0			0
$59$	−34.0834			−0.577685			−0.288843	−	0.957377i	$-0.593270\pi$
$59$	−34.0834			−0.577685			−0.288843	+	0.957377i	$0.593270\pi$
$60$		0			0
$61$		−	76.1879i		−	1.24898i	−0.781032	−	0.624491i	$-0.785307\pi$
$61$		−	76.1879i		−	1.24898i	0.781032	−	0.624491i	$-0.214693\pi$
$62$		0			0
$63$	−1.31429			−0.0208617
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−64.4207			−0.961503			−0.480751	−	0.876857i	$-0.659636\pi$
$67$	−64.4207			−0.961503			−0.480751	+	0.876857i	$0.659636\pi$
$68$		0			0
$69$	68.3719	−	20.9004i	0.990897	−	0.302905i
$70$		0			0
$71$	−126.165			−1.77697			−0.888483	−	0.458910i	$-0.848240\pi$
$71$	−126.165			−1.77697			−0.888483	+	0.458910i	$0.848240\pi$
$72$		0			0
$73$			103.394i			1.41635i	0.706036	+	0.708176i	$0.250482\pi$
$73$			103.394i			1.41635i	−0.706036	+	0.708176i	$0.749518\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	21.1911i		−	0.275209i
$78$		0			0
$79$		−	39.5132i		−	0.500166i	−0.968224	−	0.250083i	$-0.919542\pi$
$79$		−	39.5132i		−	0.500166i	0.968224	−	0.250083i	$-0.0804580\pi$
$80$		0			0
$81$	−86.5248			−1.06821
$82$		0			0
$83$	117.708			1.41817			0.709083	−	0.705125i	$-0.249109\pi$
$83$	117.708			1.41817			0.709083	+	0.705125i	$0.249109\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			28.6371i			0.329162i
$88$		0			0
$89$			78.6012i			0.883160i	0.897222	+	0.441580i	$0.145582\pi$
$89$			78.6012i			0.883160i	−0.897222	+	0.441580i	$0.854418\pi$
$90$		0			0
$91$			3.48144i			0.0382576i
$92$		0			0
$93$			132.419i			1.42386i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	74.1276			0.764202			0.382101	−	0.924121i	$-0.375201\pi$
$97$	74.1276			0.764202			0.382101	+	0.924121i	$0.375201\pi$
$98$		0			0
$99$			7.08001i			0.0715153i
$100$		0			0
$101$	−37.8585			−0.374836			−0.187418	−	0.982280i	$-0.560012\pi$
$101$	−37.8585			−0.374836			−0.187418	+	0.982280i	$0.560012\pi$
$102$		0			0
$103$	−29.7009			−0.288358			−0.144179	−	0.989552i	$-0.546054\pi$
$103$	−29.7009			−0.288358			−0.144179	+	0.989552i	$0.546054\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−41.4366			−0.387258			−0.193629	−	0.981075i	$-0.562026\pi$
$107$	−41.4366			−0.387258			−0.193629	+	0.981075i	$0.562026\pi$
$108$		0			0
$109$		−	207.967i		−	1.90795i	−0.299885	−	0.953975i	$-0.596948\pi$
$109$		−	207.967i		−	1.90795i	0.299885	−	0.953975i	$-0.403052\pi$
$110$		0			0
$111$		−	198.591i		−	1.78911i
$112$		0			0
$113$	−171.262			−1.51559			−0.757796	−	0.652491i	$-0.773724\pi$
$113$	−171.262			−1.51559			−0.757796	+	0.652491i	$0.773724\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	1.16316i		−	0.00994155i
$118$		0			0
$119$	−42.1830			−0.354479
$120$		0			0
$121$	6.84460			0.0565669
$122$		0			0
$123$			68.6857i			0.558420i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	13.5576i		−	0.106753i	−0.998574	−	0.0533765i	$-0.983002\pi$
$127$		−	13.5576i		−	0.106753i	0.998574	−	0.0533765i	$-0.0169983\pi$
$128$		0			0
$129$			191.794i			1.48678i
$130$		0			0
$131$	−111.249			−0.849228			−0.424614	−	0.905374i	$-0.639590\pi$
$131$	−111.249			−0.849228			−0.424614	+	0.905374i	$0.639590\pi$
$132$		0			0
$133$			9.08082i			0.0682769i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−40.8903			−0.298470			−0.149235	−	0.988802i	$-0.547681\pi$
$137$	−40.8903			−0.298470			−0.149235	+	0.988802i	$0.547681\pi$
$138$		0			0
$139$	116.972			0.841524			0.420762	−	0.907171i	$-0.361763\pi$
$139$	116.972			0.841524			0.420762	+	0.907171i	$0.361763\pi$
$140$		0			0
$141$	−64.6486			−0.458500
$142$		0			0
$143$	18.7544			0.131149
$144$		0			0
$145$		0			0
$146$		0			0
$147$			140.087i			0.952976i
$148$		0			0
$149$			190.028i			1.27536i	0.770303	+	0.637679i	$0.220105\pi$
$149$			190.028i			1.27536i	−0.770303	+	0.637679i	$0.779895\pi$
$150$		0			0
$151$	−219.572			−1.45412			−0.727060	−	0.686574i	$-0.759114\pi$
$151$	−219.572			−1.45412			−0.727060	+	0.686574i	$0.759114\pi$
$152$		0			0
$153$	14.0935			0.0921143
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−181.926			−1.15877			−0.579383	−	0.815055i	$-0.696706\pi$
$157$	−181.926			−1.15877			−0.579383	+	0.815055i	$0.696706\pi$
$158$		0			0
$159$		−	173.310i		−	1.09000i
$160$		0			0
$161$	13.3356	+	43.6249i	0.0828296	+	0.270962i
$162$		0			0
$163$		−	148.985i		−	0.914019i	−0.889462	−	0.457009i	$-0.848921\pi$
$163$		−	148.985i		−	0.914019i	0.889462	−	0.457009i	$-0.151079\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			104.424i			0.625295i	0.949869	+	0.312648i	$0.101216\pi$
$167$			104.424i			0.625295i	−0.949869	+	0.312648i	$0.898784\pi$
$168$		0			0
$169$	165.919			0.981769
$170$		0			0
$171$		−	3.03393i		−	0.0177423i
$172$		0			0
$173$			149.922i			0.866599i	0.901250	+	0.433299i	$0.142651\pi$
$173$			149.922i			0.866599i	−0.901250	+	0.433299i	$0.857349\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			105.948i			0.598574i
$178$		0			0
$179$	38.2719			0.213810			0.106905	−	0.994269i	$-0.465906\pi$
$179$	38.2719			0.213810			0.106905	+	0.994269i	$0.465906\pi$
$180$		0			0
$181$		−	98.1669i		−	0.542359i	−0.962529	−	0.271179i	$-0.912586\pi$
$181$		−	98.1669i		−	0.542359i	0.962529	−	0.271179i	$-0.0874137\pi$
$182$		0			0
$183$	−236.829			−1.29415
$184$		0			0
$185$		0			0
$186$		0			0
$187$			227.238i			1.21518i
$188$		0			0
$189$		−	51.4021i		−	0.271969i
$190$		0			0
$191$		−	193.804i		−	1.01468i	−0.861746	−	0.507340i	$-0.830629\pi$
$191$		−	193.804i		−	1.01468i	0.861746	−	0.507340i	$-0.169371\pi$
$192$		0			0
$193$			80.0520i			0.414777i	0.978259	+	0.207389i	$0.0664964\pi$
$193$			80.0520i			0.414777i	−0.978259	+	0.207389i	$0.933504\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			291.126i			1.47780i	0.673817	+	0.738898i	$0.264654\pi$
$197$			291.126i			1.47780i	−0.673817	+	0.738898i	$0.735346\pi$
$198$		0			0
$199$			374.264i			1.88072i	0.340177	+	0.940361i	$0.389513\pi$
$199$			374.264i			1.88072i	−0.340177	+	0.940361i	$0.610487\pi$
$200$		0			0
$201$			200.250i			0.996271i
$202$		0			0
$203$	−18.2720			−0.0900097
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−4.45546	−	14.5752i	−0.0215240	−	0.0704117i
$208$		0			0
$209$	48.9180			0.234058
$210$		0			0
$211$	−246.226			−1.16695			−0.583474	−	0.812132i	$-0.698307\pi$
$211$	−246.226			−1.16695			−0.583474	+	0.812132i	$0.698307\pi$
$212$		0			0
$213$			392.180i			1.84122i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−84.4900			−0.389355
$218$		0			0
$219$	321.397			1.46757
$220$		0			0
$221$		−	37.3325i		−	0.168925i
$222$		0			0
$223$		−	216.987i		−	0.973037i	−0.873670	−	0.486519i	$-0.838267\pi$
$223$		−	216.987i		−	0.973037i	0.873670	−	0.486519i	$-0.161733\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	106.791			0.470445			0.235222	−	0.971942i	$-0.424418\pi$
$227$	106.791			0.470445			0.235222	+	0.971942i	$0.424418\pi$
$228$		0			0
$229$			6.34304i			0.0276989i	0.999904	+	0.0138494i	$0.00440855\pi$
$229$			6.34304i			0.0276989i	−0.999904	+	0.0138494i	$0.995591\pi$
$230$		0			0
$231$	−65.8720			−0.285160
$232$		0			0
$233$			101.825i			0.437017i	0.975835	+	0.218508i	$0.0701191\pi$
$233$			101.825i			0.437017i	−0.975835	+	0.218508i	$0.929881\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−122.826			−0.518253
$238$		0			0
$239$	−11.6391			−0.0486992			−0.0243496	−	0.999704i	$-0.507751\pi$
$239$	−11.6391			−0.0486992			−0.0243496	+	0.999704i	$0.507751\pi$
$240$		0			0
$241$			71.1303i			0.295146i	0.989051	+	0.147573i	$0.0471462\pi$
$241$			71.1303i			0.295146i	−0.989051	+	0.147573i	$0.952854\pi$
$242$		0			0
$243$			35.7122i			0.146964i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.03665			−0.0325370
$248$		0			0
$249$		−	365.892i		−	1.46945i
$250$		0			0
$251$			281.356i			1.12094i	0.828174	+	0.560471i	$0.189380\pi$
$251$			281.356i			1.12094i	−0.828174	+	0.560471i	$0.810620\pi$
$252$		0			0
$253$	235.005	−	71.8381i	0.928875	−	0.283945i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			419.878i			1.63377i	0.576804	+	0.816883i	$0.304300\pi$
$257$			419.878i			1.63377i	−0.576804	+	0.816883i	$0.695700\pi$
$258$		0			0
$259$	126.712			0.489234
$260$		0			0
$261$	6.10473			0.0233898
$262$		0			0
$263$	412.742			1.56936			0.784681	−	0.619899i	$-0.212827\pi$
$263$	412.742			1.56936			0.784681	+	0.619899i	$0.212827\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	244.330			0.915095
$268$		0			0
$269$	70.6401			0.262602			0.131301	−	0.991343i	$-0.458084\pi$
$269$	70.6401			0.262602			0.131301	+	0.991343i	$0.458084\pi$
$270$		0			0
$271$	−108.563			−0.400600			−0.200300	−	0.979735i	$-0.564192\pi$
$271$	−108.563			−0.400600			−0.200300	+	0.979735i	$0.564192\pi$
$272$		0			0
$273$	10.8220			0.0396410
$274$		0			0
$275$		0			0
$276$		0			0
$277$			198.446i			0.716411i	0.933643	+	0.358206i	$0.116611\pi$
$277$			198.446i			0.716411i	−0.933643	+	0.358206i	$0.883389\pi$
$278$		0			0
$279$	28.2284			0.101177
$280$		0			0
$281$		−	329.525i		−	1.17269i	−0.810063	−	0.586343i	$-0.800567\pi$
$281$		−	329.525i		−	1.17269i	0.810063	−	0.586343i	$-0.199433\pi$
$282$		0			0
$283$	168.334			0.594821			0.297411	−	0.954750i	$-0.403877\pi$
$283$	168.334			0.594821			0.297411	+	0.954750i	$0.403877\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−43.8251			−0.152701
$288$		0			0
$289$	163.341			0.565192
$290$		0			0
$291$		−	230.424i		−	0.791836i
$292$		0			0
$293$	−453.853			−1.54899			−0.774493	−	0.632583i	$-0.781995\pi$
$293$	−453.853			−1.54899			−0.774493	+	0.632583i	$0.781995\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−276.901			−0.932326
$298$		0			0
$299$	−38.6086	+	11.8022i	−0.129126	+	0.0394721i
$300$		0			0
$301$	−122.375			−0.406561
$302$		0			0
$303$			117.682i			0.388390i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			364.912i			1.18864i	0.804229	+	0.594319i	$0.202578\pi$
$307$			364.912i			1.18864i	−0.804229	+	0.594319i	$0.797422\pi$
$308$		0			0
$309$			92.3245i			0.298785i
$310$		0			0
$311$	−60.5174			−0.194590			−0.0972949	−	0.995256i	$-0.531019\pi$
$311$	−60.5174			−0.194590			−0.0972949	+	0.995256i	$0.531019\pi$
$312$		0			0
$313$	−426.545			−1.36276			−0.681381	−	0.731929i	$-0.738620\pi$
$313$	−426.545			−1.36276			−0.681381	+	0.731929i	$0.738620\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	30.7769i		−	0.0970880i	−0.998821	−	0.0485440i	$-0.984542\pi$
$317$		−	30.7769i		−	0.0970880i	0.998821	−	0.0485440i	$-0.0154581\pi$
$318$		0			0
$319$			98.4304i			0.308559i
$320$		0			0
$321$			128.805i			0.401261i
$322$		0			0
$323$		−	97.3763i		−	0.301475i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−646.460			−1.97694
$328$		0			0
$329$		−	41.2492i		−	0.125377i
$330$		0			0
$331$	−278.125			−0.840257			−0.420129	−	0.907465i	$-0.638015\pi$
$331$	−278.125			−0.840257			−0.420129	+	0.907465i	$0.638015\pi$
$332$		0			0
$333$	−42.3348			−0.127132
$334$		0			0
$335$		0			0
$336$		0			0
$337$	419.816			1.24574			0.622872	−	0.782324i	$-0.285966\pi$
$337$	419.816			1.24574			0.622872	+	0.782324i	$0.285966\pi$
$338$		0			0
$339$			532.365i			1.57040i
$340$		0			0
$341$			455.144i			1.33473i
$342$		0			0
$343$	−186.569			−0.543932
$344$		0			0
$345$		0			0
$346$		0			0
$347$			286.347i			0.825208i	0.910911	+	0.412604i	$0.135381\pi$
$347$			286.347i			0.825208i	−0.910911	+	0.412604i	$0.864619\pi$
$348$		0			0
$349$	176.419			0.505498			0.252749	−	0.967532i	$-0.418665\pi$
$349$	176.419			0.505498			0.252749	+	0.967532i	$0.418665\pi$
$350$		0			0
$351$	45.4915			0.129605
$352$		0			0
$353$			100.063i			0.283465i	0.989905	+	0.141732i	$0.0452672\pi$
$353$			100.063i			0.283465i	−0.989905	+	0.141732i	$0.954733\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			131.125i			0.367297i
$358$		0			0
$359$			318.742i			0.887862i	0.896061	+	0.443931i	$0.146416\pi$
$359$			318.742i			0.887862i	−0.896061	+	0.443931i	$0.853584\pi$
$360$		0			0
$361$	340.038			0.941932
$362$		0			0
$363$		−	21.2763i		−	0.0586124i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−46.2696			−0.126075			−0.0630376	−	0.998011i	$-0.520079\pi$
$367$	−46.2696			−0.126075			−0.0630376	+	0.998011i	$0.520079\pi$
$368$		0			0
$369$	14.6421			0.0396805
$370$		0			0
$371$	110.581			0.298061
$372$		0			0
$373$	−101.366			−0.271757			−0.135879	−	0.990725i	$-0.543386\pi$
$373$	−101.366			−0.271757			−0.135879	+	0.990725i	$0.543386\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	16.1709i		−	0.0428937i
$378$		0			0
$379$		−	565.306i		−	1.49157i	−0.666185	−	0.745786i	$-0.732074\pi$
$379$		−	565.306i		−	1.49157i	0.666185	−	0.745786i	$-0.267926\pi$
$380$		0			0
$381$	−42.1436			−0.110613
$382$		0			0
$383$	242.620			0.633473			0.316737	−	0.948514i	$-0.397413\pi$
$383$	242.620			0.633473			0.316737	+	0.948514i	$0.397413\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	40.8858			0.105648
$388$		0			0
$389$			77.8285i			0.200073i	0.994984	+	0.100037i	$0.0318960\pi$
$389$			77.8285i			0.200073i	−0.994984	+	0.100037i	$0.968104\pi$
$390$		0			0
$391$	−143.001	−	467.802i	−0.365732	−	1.19642i
$392$		0			0
$393$			345.815i			0.879936i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	304.863i		−	0.767917i	−0.923350	−	0.383958i	$-0.874561\pi$
$397$		−	304.863i		−	0.767917i	0.923350	−	0.383958i	$-0.125439\pi$
$398$		0			0
$399$	28.2276			0.0707458
$400$		0			0
$401$		−	703.077i		−	1.75331i	−0.481121	−	0.876654i	$-0.659770\pi$
$401$		−	703.077i		−	1.75331i	0.481121	−	0.876654i	$-0.340230\pi$
$402$		0			0
$403$		−	74.7748i		−	0.185545i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	682.590i		−	1.67713i
$408$		0			0
$409$	−416.790			−1.01905			−0.509523	−	0.860457i	$-0.670178\pi$
$409$	−416.790			−1.01905			−0.509523	+	0.860457i	$0.670178\pi$
$410$		0			0
$411$			127.107i			0.309262i
$412$		0			0
$413$	−67.6002			−0.163681
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	363.605i		−	0.871954i
$418$		0			0
$419$		−	558.416i		−	1.33273i	−0.745624	−	0.666367i	$-0.767848\pi$
$419$		−	558.416i		−	1.33273i	0.745624	−	0.666367i	$-0.232152\pi$
$420$		0			0
$421$		−	101.013i		−	0.239936i	−0.992778	−	0.119968i	$-0.961721\pi$
$421$		−	101.013i		−	0.239936i	0.992778	−	0.119968i	$-0.0382791\pi$
$422$		0			0
$423$			13.7815i			0.0325804i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	151.109i		−	0.353885i
$428$		0			0
$429$		−	58.2976i		−	0.135892i
$430$		0			0
$431$		−	489.410i		−	1.13552i	−0.823194	−	0.567760i	$-0.807810\pi$
$431$		−	489.410i		−	1.13552i	0.823194	−	0.567760i	$-0.192190\pi$
$432$		0			0
$433$	471.464			1.08883			0.544415	−	0.838816i	$-0.316752\pi$
$433$	471.464			1.08883			0.544415	+	0.838816i	$0.316752\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−100.705	+	30.7842i	−0.230446	+	0.0704443i
$438$		0			0
$439$	123.984			0.282425			0.141212	−	0.989979i	$-0.454900\pi$
$439$	123.984			0.282425			0.141212	+	0.989979i	$0.454900\pi$
$440$		0			0
$441$	29.8632			0.0677171
$442$		0			0
$443$		−	859.972i		−	1.94125i	−0.240606	−	0.970623i	$-0.577346\pi$
$443$		−	859.972i		−	1.94125i	0.240606	−	0.970623i	$-0.422654\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	590.699			1.32147
$448$		0			0
$449$	−359.990			−0.801759			−0.400880	−	0.916131i	$-0.631295\pi$
$449$	−359.990			−0.801759			−0.400880	+	0.916131i	$0.631295\pi$
$450$		0			0
$451$			236.084i			0.523468i
$452$		0			0
$453$			682.535i			1.50670i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	120.754			0.264232			0.132116	−	0.991234i	$-0.457823\pi$
$457$	120.754			0.264232			0.132116	+	0.991234i	$0.457823\pi$
$458$		0			0
$459$			551.200i			1.20087i
$460$		0			0
$461$	−14.2261			−0.0308592			−0.0154296	−	0.999881i	$-0.504912\pi$
$461$	−14.2261			−0.0308592			−0.0154296	+	0.999881i	$0.504912\pi$
$462$		0			0
$463$		−	365.620i		−	0.789676i	−0.918751	−	0.394838i	$-0.870801\pi$
$463$		−	365.620i		−	0.789676i	0.918751	−	0.394838i	$-0.129199\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−189.176			−0.405089			−0.202544	−	0.979273i	$-0.564921\pi$
$467$	−189.176			−0.405089			−0.202544	+	0.979273i	$0.564921\pi$
$468$		0			0
$469$	−127.770			−0.272431
$470$		0			0
$471$			565.514i			1.20067i
$472$		0			0
$473$			659.228i			1.39372i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−36.9454			−0.0774537
$478$		0			0
$479$		−	896.832i		−	1.87230i	−0.351600	−	0.936150i	$-0.614362\pi$
$479$		−	896.832i		−	1.87230i	0.351600	−	0.936150i	$-0.385638\pi$
$480$		0			0
$481$			112.141i			0.233142i
$482$		0			0
$483$	135.607	−	41.4534i	0.280760	−	0.0858247i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	330.861i		−	0.679385i	−0.940537	−	0.339693i	$-0.889677\pi$
$487$		−	330.861i		−	0.679385i	0.940537	−	0.339693i	$-0.110323\pi$
$488$		0			0
$489$	−463.117			−0.947070
$490$		0			0
$491$	−801.670			−1.63273			−0.816365	−	0.577536i	$-0.804014\pi$
$491$	−801.670			−1.63273			−0.816365	+	0.577536i	$0.804014\pi$
$492$		0			0
$493$	195.936			0.397436
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−250.232			−0.503484
$498$		0			0
$499$	−40.8056			−0.0817748			−0.0408874	−	0.999164i	$-0.513018\pi$
$499$	−40.8056			−0.0817748			−0.0408874	+	0.999164i	$0.513018\pi$
$500$		0			0
$501$	324.601			0.647906
$502$		0			0
$503$	−44.0719			−0.0876180			−0.0438090	−	0.999040i	$-0.513949\pi$
$503$	−44.0719			−0.0876180			−0.0438090	+	0.999040i	$0.513949\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	515.756i		−	1.01727i
$508$		0			0
$509$	−234.886			−0.461466			−0.230733	−	0.973017i	$-0.574112\pi$
$509$	−234.886			−0.461466			−0.230733	+	0.973017i	$0.574112\pi$
$510$		0			0
$511$			205.068i			0.401308i
$512$		0			0
$513$	118.658			0.231302
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−222.208			−0.429802
$518$		0			0
$519$	466.028			0.897935
$520$		0			0
$521$			944.850i			1.81353i	0.421635	+	0.906765i	$0.361456\pi$
$521$			944.850i			1.81353i	−0.421635	+	0.906765i	$0.638544\pi$
$522$		0			0
$523$	355.863			0.680427			0.340213	−	0.940348i	$-0.389501\pi$
$523$	355.863			0.680427			0.340213	+	0.940348i	$0.389501\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	906.011			1.71919
$528$		0			0
$529$	−438.584	+	295.778i	−0.829082	+	0.559127i
$530$		0			0
$531$	22.5855			0.0425338
$532$		0			0
$533$		−	38.7858i		−	0.0727688i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	118.968i		−	0.221541i
$538$		0			0
$539$			481.503i			0.893327i
$540$		0			0
$541$	−559.206			−1.03365			−0.516826	−	0.856091i	$-0.672887\pi$
$541$	−559.206			−1.03365			−0.516826	+	0.856091i	$0.672887\pi$
$542$		0			0
$543$	−305.150			−0.561971
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	428.257i		−	0.782920i	−0.920195	−	0.391460i	$-0.871970\pi$
$547$		−	428.257i		−	0.782920i	0.920195	−	0.391460i	$-0.128030\pi$
$548$		0			0
$549$			50.4861i			0.0919600i
$550$		0			0
$551$		−	42.1795i		−	0.0765508i
$552$		0			0
$553$		−	78.3694i		−	0.141717i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	410.029			0.736138			0.368069	−	0.929798i	$-0.380019\pi$
$557$	410.029			0.736138			0.368069	+	0.929798i	$0.380019\pi$
$558$		0			0
$559$		−	108.303i		−	0.193745i
$560$		0			0
$561$	706.365			1.25912
$562$		0			0
$563$	−727.785			−1.29269			−0.646345	−	0.763045i	$-0.723703\pi$
$563$	−727.785			−1.29269			−0.646345	+	0.763045i	$0.723703\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−171.611			−0.302665
$568$		0			0
$569$		−	812.848i		−	1.42856i	−0.699862	−	0.714278i	$-0.746755\pi$
$569$		−	812.848i		−	1.42856i	0.699862	−	0.714278i	$-0.253245\pi$
$570$		0			0
$571$		−	594.402i		−	1.04098i	−0.853866	−	0.520492i	$-0.825748\pi$
$571$		−	594.402i		−	1.04098i	0.853866	−	0.520492i	$-0.174252\pi$
$572$		0			0
$573$	−602.435			−1.05137
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	73.6344i		−	0.127616i	−0.997962	−	0.0638079i	$-0.979675\pi$
$577$		−	73.6344i		−	0.127616i	0.997962	−	0.0638079i	$-0.0203245\pi$
$578$		0			0
$579$	248.840			0.429776
$580$		0			0
$581$	233.459			0.401822
$582$		0			0
$583$		−	595.694i		−	1.02177i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			229.976i			0.391782i	0.980626	+	0.195891i	$0.0627598\pi$
$587$			229.976i			0.391782i	−0.980626	+	0.195891i	$0.937240\pi$
$588$		0			0
$589$		−	195.039i		−	0.331136i
$590$		0			0
$591$	904.959			1.53123
$592$		0			0
$593$		−	745.294i		−	1.25682i	−0.777882	−	0.628410i	$-0.783706\pi$
$593$		−	745.294i		−	1.25682i	0.777882	−	0.628410i	$-0.216294\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1163.39			1.94873
$598$		0			0
$599$	−447.095			−0.746403			−0.373201	−	0.927750i	$-0.621740\pi$
$599$	−447.095			−0.746403			−0.373201	+	0.927750i	$0.621740\pi$
$600$		0			0
$601$	598.459			0.995771			0.497886	−	0.867243i	$-0.334110\pi$
$601$	598.459			0.995771			0.497886	+	0.867243i	$0.334110\pi$
$602$		0			0
$603$	42.6885			0.0707935
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	998.935i		−	1.64569i	−0.568264	−	0.822846i	$-0.692385\pi$
$607$		−	998.935i		−	1.64569i	0.568264	−	0.822846i	$-0.307615\pi$
$608$		0			0
$609$			56.7981i			0.0932645i
$610$		0			0
$611$	36.5061			0.0597481
$612$		0			0
$613$	732.465			1.19489			0.597443	−	0.801912i	$-0.296184\pi$
$613$	732.465			1.19489			0.597443	+	0.801912i	$0.296184\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−450.432			−0.730036			−0.365018	−	0.931001i	$-0.618937\pi$
$617$	−450.432			−0.730036			−0.365018	+	0.931001i	$0.618937\pi$
$618$		0			0
$619$			574.519i			0.928141i	0.885798	+	0.464071i	$0.153612\pi$
$619$			574.519i			0.928141i	−0.885798	+	0.464071i	$0.846388\pi$
$620$		0			0
$621$	570.040	−	174.254i	0.917939	−	0.280602i
$622$		0			0
$623$			155.896i			0.250234i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	152.061i		−	0.242521i
$628$		0			0
$629$	−1358.77			−2.16020
$630$		0			0
$631$			601.263i			0.952873i	0.879209	+	0.476436i	$0.158072\pi$
$631$			601.263i			0.952873i	−0.879209	+	0.476436i	$0.841928\pi$
$632$		0			0
$633$			765.389i			1.20914i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	79.1053i		−	0.124184i
$638$		0			0
$639$	83.6032			0.130834
$640$		0			0
$641$			738.437i			1.15201i	0.817447	+	0.576004i	$0.195389\pi$
$641$			738.437i			1.15201i	−0.817447	+	0.576004i	$0.804611\pi$
$642$		0			0
$643$	135.341			0.210484			0.105242	−	0.994447i	$-0.466438\pi$
$643$	135.341			0.210484			0.105242	+	0.994447i	$0.466438\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	978.141i		−	1.51181i	−0.654682	−	0.755905i	$-0.727197\pi$
$647$		−	978.141i		−	1.51181i	0.654682	−	0.755905i	$-0.272803\pi$
$648$		0			0
$649$			364.159i			0.561108i
$650$		0			0
$651$			262.636i			0.403434i
$652$		0			0
$653$		−	412.686i		−	0.631984i	−0.948762	−	0.315992i	$-0.897663\pi$
$653$		−	412.686i		−	0.631984i	0.948762	−	0.315992i	$-0.102337\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	68.5140i		−	0.104283i
$658$		0			0
$659$		−	638.479i		−	0.968860i	−0.874830	−	0.484430i	$-0.839027\pi$
$659$		−	638.479i		−	0.968860i	0.874830	−	0.484430i	$-0.160973\pi$
$660$		0			0
$661$			49.7030i			0.0751937i	0.999293	+	0.0375968i	$0.0119703\pi$
$661$			49.7030i			0.0751937i	−0.999293	+	0.0375968i	$0.988030\pi$
$662$		0			0
$663$	−116.047			−0.175034
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−61.9423	−	202.633i	−0.0928671	−	0.303798i
$668$		0			0
$669$	−674.501			−1.00822
$670$		0			0
$671$	−814.018			−1.21314
$672$		0			0
$673$		−	645.714i		−	0.959456i	−0.877417	−	0.479728i	$-0.840735\pi$
$673$		−	645.714i		−	0.959456i	0.877417	−	0.479728i	$-0.159265\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−839.079			−1.23941			−0.619704	−	0.784836i	$-0.712747\pi$
$677$	−839.079			−1.23941			−0.619704	+	0.784836i	$0.712747\pi$
$678$		0			0
$679$	147.023			0.216528
$680$		0			0
$681$		−	331.958i		−	0.487456i
$682$		0			0
$683$		−	468.404i		−	0.685804i	−0.939371	−	0.342902i	$-0.888590\pi$
$683$		−	468.404i		−	0.685804i	0.939371	−	0.342902i	$-0.111410\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	19.7172			0.0287005
$688$		0			0
$689$			97.8654i			0.142040i
$690$		0			0
$691$	247.952			0.358831			0.179415	−	0.983773i	$-0.442579\pi$
$691$	247.952			0.358831			0.179415	+	0.983773i	$0.442579\pi$
$692$		0			0
$693$			14.0423i			0.0202631i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	469.949			0.674246
$698$		0			0
$699$	316.521			0.452819
$700$		0			0
$701$		−	952.474i		−	1.35874i	−0.733798	−	0.679368i	$-0.762254\pi$
$701$		−	952.474i		−	1.35874i	0.733798	−	0.679368i	$-0.237746\pi$
$702$		0			0
$703$			292.504i			0.416080i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−75.0875			−0.106206
$708$		0			0
$709$			773.110i			1.09042i	0.838299	+	0.545212i	$0.183551\pi$
$709$			773.110i			1.09042i	−0.838299	+	0.545212i	$0.816449\pi$
$710$		0			0
$711$			26.1835i			0.0368263i
$712$		0			0
$713$	−286.423	−	936.980i	−0.401715	−	1.31414i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			36.1799i			0.0504602i
$718$		0			0
$719$	389.128			0.541207			0.270603	−	0.962691i	$-0.412777\pi$
$719$	389.128			0.541207			0.270603	+	0.962691i	$0.412777\pi$
$720$		0			0
$721$	−58.9079			−0.0817031
$722$		0			0
$723$	221.107			0.305819
$724$		0			0
$725$		0			0
$726$		0			0
$727$	85.6314			0.117787			0.0588937	−	0.998264i	$-0.481243\pi$
$727$	85.6314			0.117787			0.0588937	+	0.998264i	$0.481243\pi$
$728$		0			0
$729$	−667.712			−0.915929
$730$		0			0
$731$	1312.26			1.79516
$732$		0			0
$733$	−885.478			−1.20802			−0.604009	−	0.796977i	$-0.706431\pi$
$733$	−885.478			−1.20802			−0.604009	+	0.796977i	$0.706431\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			688.293i			0.933912i
$738$		0			0
$739$	807.772			1.09306			0.546530	−	0.837439i	$-0.315948\pi$
$739$	807.772			1.09306			0.546530	+	0.837439i	$0.315948\pi$
$740$		0			0
$741$			24.9818i			0.0337136i
$742$		0			0
$743$	612.748			0.824694			0.412347	−	0.911027i	$-0.364709\pi$
$743$	612.748			0.824694			0.412347	+	0.911027i	$0.364709\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−77.9993			−0.104417
$748$		0			0
$749$	−82.1842			−0.109725
$750$		0			0
$751$		−	1210.29i		−	1.61157i	−0.592210	−	0.805784i	$-0.701744\pi$
$751$		−	1210.29i		−	1.61157i	0.592210	−	0.805784i	$-0.298256\pi$
$752$		0			0
$753$	874.591			1.16148
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−643.582			−0.850174			−0.425087	−	0.905153i	$-0.639756\pi$
$757$	−643.582			−0.850174			−0.425087	+	0.905153i	$0.639756\pi$
$758$		0			0
$759$	−223.307	−	730.509i	−0.294213	−	0.962463i
$760$		0			0
$761$	910.469			1.19641			0.598206	−	0.801342i	$-0.295881\pi$
$761$	910.469			1.19641			0.598206	+	0.801342i	$0.295881\pi$
$762$		0			0
$763$		−	412.476i		−	0.540597i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	59.8271i		−	0.0780014i
$768$		0			0
$769$		−	365.775i		−	0.475650i	−0.971308	−	0.237825i	$-0.923565\pi$
$769$		−	365.775i		−	0.475650i	0.971308	−	0.237825i	$-0.0764345\pi$
$770$		0			0
$771$	1305.18			1.69284
$772$		0			0
$773$	23.1809			0.0299882			0.0149941	−	0.999888i	$-0.495227\pi$
$773$	23.1809			0.0299882			0.0149941	+	0.999888i	$0.495227\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	393.881i		−	0.506925i
$778$		0			0
$779$		−	101.167i		−	0.129868i
$780$		0			0
$781$			1347.99i			1.72598i
$782$		0			0
$783$			238.757i			0.304927i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	961.062			1.22117			0.610585	−	0.791950i	$-0.290934\pi$
$787$	961.062			1.22117			0.610585	+	0.791950i	$0.290934\pi$
$788$		0			0
$789$		−	1283.00i		−	1.62611i
$790$		0			0
$791$	−339.677			−0.429427
$792$		0			0
$793$	133.734			0.168643
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−288.472			−0.361947			−0.180974	−	0.983488i	$-0.557925\pi$
$797$	−288.472			−0.361947			−0.180974	+	0.983488i	$0.557925\pi$
$798$		0			0
$799$			442.327i			0.553601i
$800$		0			0
$801$		−	52.0853i		−	0.0650253i
$802$		0			0
$803$	1104.69			1.37571
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	219.583i		−	0.272098i
$808$		0			0
$809$	−354.079			−0.437675			−0.218837	−	0.975761i	$-0.570226\pi$
$809$	−354.079			−0.437675			−0.218837	+	0.975761i	$0.570226\pi$
$810$		0			0
$811$	1398.07			1.72389			0.861943	−	0.507005i	$-0.169247\pi$
$811$	1398.07			1.72389			0.861943	+	0.507005i	$0.169247\pi$
$812$		0			0
$813$			337.465i			0.415086i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	282.493i		−	0.345769i
$818$		0			0
$819$		−	2.30698i		−	0.00281683i
$820$		0			0
$821$	1571.81			1.91451			0.957253	−	0.289252i	$-0.0934065\pi$
$821$	1571.81			1.91451			0.957253	+	0.289252i	$0.0934065\pi$
$822$		0			0
$823$		−	959.645i		−	1.16603i	−0.812460	−	0.583017i	$-0.801872\pi$
$823$		−	959.645i		−	1.16603i	0.812460	−	0.583017i	$-0.198128\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−526.528			−0.636672			−0.318336	−	0.947978i	$-0.603124\pi$
$827$	−526.528			−0.636672			−0.318336	+	0.947978i	$0.603124\pi$
$828$		0			0
$829$	508.750			0.613692			0.306846	−	0.951759i	$-0.400726\pi$
$829$	508.750			0.613692			0.306846	+	0.951759i	$0.400726\pi$
$830$		0			0
$831$	616.865			0.742317
$832$		0			0
$833$	958.482			1.15064
$834$		0			0
$835$		0			0
$836$		0			0
$837$			1104.02i			1.31902i
$838$		0			0
$839$			522.306i			0.622534i	0.950322	+	0.311267i	$0.100753\pi$
$839$			522.306i			0.622534i	−0.950322	+	0.311267i	$0.899247\pi$
$840$		0			0
$841$	−756.129			−0.899083
$842$		0			0
$843$	−1024.32			−1.21509
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.5754			0.0160276
$848$		0			0
$849$		−	523.264i		−	0.616330i
$850$		0			0
$851$	429.555	+	1405.21i	0.504765	+	1.65125i
$852$		0			0
$853$		−	1147.14i		−	1.34482i	−0.740177	−	0.672412i	$-0.765258\pi$
$853$		−	1147.14i		−	1.34482i	0.740177	−	0.672412i	$-0.234742\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	97.7654i		−	0.114079i	−0.998372	−	0.0570393i	$-0.981834\pi$
$857$		−	97.7654i		−	0.114079i	0.998372	−	0.0570393i	$-0.0181660\pi$
$858$		0			0
$859$	1503.31			1.75007			0.875033	−	0.484063i	$-0.160839\pi$
$859$	1503.31			1.75007			0.875033	+	0.484063i	$0.160839\pi$
$860$		0			0
$861$			136.229i			0.158222i
$862$		0			0
$863$			655.859i			0.759976i	0.924992	+	0.379988i	$0.124072\pi$
$863$			655.859i			0.759976i	−0.924992	+	0.379988i	$0.875928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	507.741i		−	0.585630i
$868$		0			0
$869$	−422.172			−0.485814
$870$		0			0
$871$		−	113.078i		−	0.129826i
$872$		0			0
$873$	−49.1208			−0.0562667
$874$		0			0
$875$		0			0
$876$		0			0
$877$			251.321i			0.286568i	0.989682	+	0.143284i	$0.0457663\pi$
$877$			251.321i			0.286568i	−0.989682	+	0.143284i	$0.954234\pi$
$878$		0			0
$879$			1410.79i			1.60500i
$880$		0			0
$881$		−	96.6276i		−	0.109679i	−0.998495	−	0.0548397i	$-0.982535\pi$
$881$		−	96.6276i		−	0.109679i	0.998495	−	0.0548397i	$-0.0174648\pi$
$882$		0			0
$883$		−	1075.24i		−	1.21772i	−0.793279	−	0.608859i	$-0.791628\pi$
$883$		−	1075.24i		−	1.21772i	0.793279	−	0.608859i	$-0.208372\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			281.233i			0.317061i	0.987354	+	0.158530i	$0.0506756\pi$
$887$			281.233i			0.317061i	−0.987354	+	0.158530i	$0.949324\pi$
$888$		0			0
$889$		−	26.8898i		−	0.0302473i
$890$		0			0
$891$			924.461i			1.03755i
$892$		0			0
$893$	95.2207			0.106630
$894$		0			0
$895$		0			0
$896$		0			0
$897$	36.6868	+	120.014i	0.0408994	+	0.133795i
$898$		0			0
$899$	392.448			0.436538
$900$		0			0
$901$	−1185.79			−1.31608
$902$		0			0
$903$			380.400i			0.421262i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	509.375			0.561604			0.280802	−	0.959766i	$-0.409400\pi$
$907$	509.375			0.561604			0.280802	+	0.959766i	$0.409400\pi$
$908$		0			0
$909$	25.0870			0.0275984
$910$		0			0
$911$			1258.48i			1.38143i	0.723127	+	0.690715i	$0.242704\pi$
$911$			1258.48i			1.38143i	−0.723127	+	0.690715i	$0.757296\pi$
$912$		0			0
$913$		−	1257.63i		−	1.37747i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−220.648			−0.240620
$918$		0			0
$919$			287.355i			0.312682i	0.987703	+	0.156341i	$0.0499699\pi$
$919$			287.355i			0.312682i	−0.987703	+	0.156341i	$0.950030\pi$
$920$		0			0
$921$	1134.32			1.23162
$922$		0			0
$923$		−	221.458i		−	0.239933i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	19.6813			0.0212312
$928$		0			0
$929$	−1160.51			−1.24920			−0.624599	−	0.780945i	$-0.714738\pi$
$929$	−1160.51			−1.24920			−0.624599	+	0.780945i	$0.714738\pi$
$930$		0			0
$931$		−	206.334i		−	0.221627i
$932$		0			0
$933$			188.117i			0.201626i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	267.963			0.285980			0.142990	−	0.989724i	$-0.454328\pi$
$937$	267.963			0.285980			0.142990	+	0.989724i	$0.454328\pi$
$938$		0			0
$939$			1325.91i			1.41204i
$940$		0			0
$941$		−	251.727i		−	0.267510i	−0.991014	−	0.133755i	$-0.957297\pi$
$941$		−	251.727i		−	0.267510i	0.991014	−	0.133755i	$-0.0427035\pi$
$942$		0			0
$943$	−148.568	−	486.013i	−0.157548	−	0.515390i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1302.05i			1.37492i	0.726220	+	0.687462i	$0.241275\pi$
$947$			1302.05i			1.37492i	−0.726220	+	0.687462i	$0.758725\pi$
$948$		0			0
$949$	−181.488			−0.191241
$950$		0			0
$951$	−95.6694			−0.100599
$952$		0			0
$953$	−1156.60			−1.21364			−0.606821	−	0.794839i	$-0.707555\pi$
$953$	−1156.60			−1.21364			−0.606821	+	0.794839i	$0.707555\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	305.969			0.319717
$958$		0			0
$959$	−81.1008			−0.0845681
$960$		0			0
$961$	853.687			0.888332
$962$		0			0
$963$	27.4580			0.0285130
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1048.39i		−	1.08417i	−0.840323	−	0.542086i	$-0.817635\pi$
$967$		−	1048.39i		−	1.08417i	0.840323	−	0.542086i	$-0.182365\pi$
$968$		0			0
$969$	−302.692			−0.312376
$970$		0			0
$971$			257.911i			0.265614i	0.991142	+	0.132807i	$0.0423990\pi$
$971$			257.911i			0.265614i	−0.991142	+	0.132807i	$0.957601\pi$
$972$		0			0
$973$	231.999			0.238437
$974$		0			0
$975$		0			0
$976$		0			0
$977$	72.9096			0.0746260			0.0373130	−	0.999304i	$-0.488120\pi$
$977$	72.9096			0.0746260			0.0373130	+	0.999304i	$0.488120\pi$
$978$		0			0
$979$	839.803			0.857817
$980$		0			0
$981$			137.810i			0.140479i
$982$		0			0
$983$	−393.989			−0.400803			−0.200401	−	0.979714i	$-0.564225\pi$
$983$	−393.989			−0.400803			−0.200401	+	0.979714i	$0.564225\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−128.222			−0.129911
$988$		0			0
$989$	−414.853	−	1357.11i	−0.419467	−	1.37221i
$990$		0			0
$991$	−1913.19			−1.93057			−0.965283	−	0.261206i	$-0.915880\pi$
$991$	−1913.19			−1.93057			−0.965283	+	0.261206i	$0.915880\pi$
$992$		0			0
$993$			864.547i			0.870641i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1240.46i			1.24420i	0.782939	+	0.622098i	$0.213719\pi$
$997$			1240.46i			1.24420i	−0.782939	+	0.622098i	$0.786281\pi$
$998$		0			0
$999$		−	1655.72i		−	1.65738i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.3.d.c.1149.15		32
5.2	odd	4		2300.3.f.c.1701.6	yes	16
5.3	odd	4		2300.3.f.d.1701.11	yes	16
5.4	even	2	inner	2300.3.d.c.1149.18		32
23.22	odd	2	inner	2300.3.d.c.1149.17		32
115.22	even	4		2300.3.f.c.1701.5	✓	16
115.68	even	4		2300.3.f.d.1701.12	yes	16
115.114	odd	2	inner	2300.3.d.c.1149.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.15		32	1.1	even	1	trivial
2300.3.d.c.1149.16		32	115.114	odd	2	inner
2300.3.d.c.1149.17		32	23.22	odd	2	inner
2300.3.d.c.1149.18		32	5.4	even	2	inner
2300.3.f.c.1701.5	✓	16	115.22	even	4
2300.3.f.c.1701.6	yes	16	5.2	odd	4
2300.3.f.d.1701.11	yes	16	5.3	odd	4
2300.3.f.d.1701.12	yes	16	115.68	even	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.10848i q^{3} +1.98337 q^{7} -0.662652 q^{9} +O(q^{10})\)	\(q-3.10848i q^{3} +1.98337 q^{7} -0.662652 q^{9} -10.6844i q^{11} +1.75531i q^{13} -21.2683 q^{17} +4.57847i q^{19} -6.16528i q^{21} +(6.72368 + 21.9953i) q^{23} -25.9165i q^{27} -9.21257 q^{29} -42.5991 q^{31} -33.2121 q^{33} +63.8869 q^{37} +5.45635 q^{39} -22.0962 q^{41} -61.7003 q^{43} -20.7975i q^{47} -45.0662 q^{49} +66.1121i q^{51} +55.7539 q^{53} +14.2321 q^{57} -34.0834 q^{59} -76.1879i q^{61} -1.31429 q^{63} -64.4207 q^{67} +(68.3719 - 20.9004i) q^{69} -126.165 q^{71} +103.394i q^{73} -21.1911i q^{77} -39.5132i q^{79} -86.5248 q^{81} +117.708 q^{83} +28.6371i q^{87} +78.6012i q^{89} +3.48144i q^{91} +132.419i q^{93} +74.1276 q^{97} +7.08001i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 128 q^{9}+O(q^{10}) \) 32 * q - 128 * q^9	\( 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) \) 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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