LMFDB - Embedded newform 2300.3.f.d.1701.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1701,2300)]

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, 0, 1]))

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

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

chi := DirichletCharacter("2300.1701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.f (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:	$62.6704608029$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 487 x^{14} + 91703 x^{12} + 8599549 x^{10} + 437649516 x^{8} + 12136718132 x^{6} + \cdots + 1845424439296 $ x^16 + 487x^14 + 91703x^12 + 8599549x^10 + 437649516x^8 + 12136718132x^6 + 169813416064x^4 + 973986747392*x^2 + 1845424439296
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1701.12
Root			$1.98337i$ of defining polynomial
Character	$\chi$	$=$	2300.1701
Dual form			2300.3.f.d.1701.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.10848 q^{3} +1.98337i q^{7} +0.662652 q^{9} +O(q^{10})$	$q+3.10848 q^{3} +1.98337i q^{7} +0.662652 q^{9} +10.6844i q^{11} -1.75531 q^{13} -21.2683i q^{17} +4.57847i q^{19} +6.16528i q^{21} +(-21.9953 - 6.72368i) q^{23} -25.9165 q^{27} +9.21257 q^{29} -42.5991 q^{31} +33.2121i q^{33} +63.8869i q^{37} -5.45635 q^{39} -22.0962 q^{41} +61.7003i q^{43} -20.7975 q^{47} +45.0662 q^{49} -66.1121i q^{51} -55.7539i q^{53} +14.2321i q^{57} +34.0834 q^{59} +76.1879i q^{61} +1.31429i q^{63} -64.4207i q^{67} +(-68.3719 - 20.9004i) q^{69} -126.165 q^{71} -103.394 q^{73} -21.1911 q^{77} -39.5132i q^{79} -86.5248 q^{81} -117.708i q^{83} +28.6371 q^{87} +78.6012i q^{89} -3.48144i q^{91} -132.419 q^{93} +74.1276i q^{97} +7.08001i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} + 6 q^{13} - 8 q^{23} + 66 q^{27} - 6 q^{29} + 28 q^{31} + 74 q^{39} - 90 q^{41} + 40 q^{47} - 190 q^{49} - 174 q^{59} + 50 q^{69} + 116 q^{71} + 110 q^{73} + 198 q^{77} + 56 q^{81} - 362 q^{87} - 50 q^{93}+O(q^{100}) $ 16 * q + 64 * q^9 + 6 * q^13 - 8 * q^23 + 66 * q^27 - 6 * q^29 + 28 * q^31 + 74 * q^39 - 90 * q^41 + 40 * q^47 - 190 * q^49 - 174 * q^59 + 50 * q^69 + 116 * q^71 + 110 * q^73 + 198 * q^77 + 56 * q^81 - 362 * q^87 - 50 * q^93

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
$3$	3.10848			1.03616			0.518080	−	0.855332i	$-0.326647\pi$
$3$	3.10848			1.03616			0.518080	+	0.855332i	$0.326647\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			1.98337i			0.283339i	0.989914	+	0.141670i	$0.0452471\pi$
$7$			1.98337i			0.283339i	−0.989914	+	0.141670i	$0.954753\pi$
$8$		0			0
$9$	0.662652			0.0736280
$10$		0			0
$11$			10.6844i			0.971305i	0.874152	+	0.485652i	$0.161418\pi$
$11$			10.6844i			0.971305i	−0.874152	+	0.485652i	$0.838582\pi$
$12$		0			0
$13$	−1.75531			−0.135024			−0.0675120	−	0.997718i	$-0.521506\pi$
$13$	−1.75531			−0.135024			−0.0675120	+	0.997718i	$0.521506\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	21.2683i		−	1.25108i	−0.780193	−	0.625538i	$-0.784879\pi$
$17$		−	21.2683i		−	1.25108i	0.780193	−	0.625538i	$-0.215121\pi$
$18$		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
$23$	−21.9953	−	6.72368i	−0.956316	−	0.292334i
$23$	−21.9953	−	6.72368i	−0.956316	−	0.292334i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−25.9165			−0.959870
$28$		0			0
$29$	9.21257			0.317675			0.158837	−	0.987305i	$-0.449225\pi$
$29$	9.21257			0.317675			0.158837	+	0.987305i	$0.449225\pi$
$30$		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
$33$			33.2121i			1.00643i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			63.8869i			1.72667i	0.504628	+	0.863337i	$0.331630\pi$
$37$			63.8869i			1.72667i	−0.504628	+	0.863337i	$0.668370\pi$
$38$		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
$43$			61.7003i			1.43489i	0.696615	+	0.717445i	$0.254689\pi$
$43$			61.7003i			1.43489i	−0.696615	+	0.717445i	$0.745311\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−20.7975			−0.442500			−0.221250	−	0.975217i	$-0.571014\pi$
$47$	−20.7975			−0.442500			−0.221250	+	0.975217i	$0.571014\pi$
$48$		0			0
$49$	45.0662			0.919719
$50$		0			0
$51$		−	66.1121i		−	1.29632i
$52$		0			0
$53$		−	55.7539i		−	1.05196i	−0.850497	−	0.525980i	$-0.823699\pi$
$53$		−	55.7539i		−	1.05196i	0.850497	−	0.525980i	$-0.176301\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			14.2321i			0.249686i
$58$		0			0
$59$	34.0834			0.577685			0.288843	−	0.957377i	$-0.406730\pi$
$59$	34.0834			0.577685			0.288843	+	0.957377i	$0.406730\pi$
$60$		0			0
$61$			76.1879i			1.24898i	0.781032	+	0.624491i	$0.214693\pi$
$61$			76.1879i			1.24898i	−0.781032	+	0.624491i	$0.785307\pi$
$62$		0			0
$63$			1.31429i			0.0208617i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	64.4207i		−	0.961503i	−0.876857	−	0.480751i	$-0.840364\pi$
$67$		−	64.4207i		−	0.961503i	0.876857	−	0.480751i	$-0.159636\pi$
$68$		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
$73$	−103.394			−1.41635			−0.708176	−	0.706036i	$-0.750482\pi$
$73$	−103.394			−1.41635			−0.708176	+	0.706036i	$0.750482\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−21.1911			−0.275209
$78$		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
$83$		−	117.708i		−	1.41817i	−0.705125	−	0.709083i	$-0.749109\pi$
$83$		−	117.708i		−	1.41817i	0.705125	−	0.709083i	$-0.250891\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	28.6371			0.329162
$88$		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
$93$	−132.419			−1.42386
$94$		0			0
$95$		0			0
$96$		0			0
$97$			74.1276i			0.764202i	0.924121	+	0.382101i	$0.124799\pi$
$97$			74.1276i			0.764202i	−0.924121	+	0.382101i	$0.875201\pi$
$98$		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
$103$			29.7009i			0.288358i	0.989552	+	0.144179i	$0.0460541\pi$
$103$			29.7009i			0.288358i	−0.989552	+	0.144179i	$0.953946\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	41.4366i		−	0.387258i	−0.981075	−	0.193629i	$-0.937974\pi$
$107$		−	41.4366i		−	0.387258i	0.981075	−	0.193629i	$-0.0620258\pi$
$108$		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
$113$			171.262i			1.51559i	0.652491	+	0.757796i	$0.273724\pi$
$113$			171.262i			1.51559i	−0.652491	+	0.757796i	$0.726276\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.16316			−0.00994155
$118$		0			0
$119$	42.1830			0.354479
$120$		0			0
$121$	6.84460			0.0565669
$122$		0			0
$123$	−68.6857			−0.558420
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−13.5576			−0.106753			−0.0533765	−	0.998574i	$-0.516998\pi$
$127$	−13.5576			−0.106753			−0.0533765	+	0.998574i	$0.516998\pi$
$128$		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
$133$	−9.08082			−0.0682769
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	40.8903i		−	0.298470i	−0.988802	−	0.149235i	$-0.952319\pi$
$137$		−	40.8903i		−	0.298470i	0.988802	−	0.149235i	$-0.0476810\pi$
$138$		0			0
$139$	−116.972			−0.841524			−0.420762	−	0.907171i	$-0.638237\pi$
$139$	−116.972			−0.841524			−0.420762	+	0.907171i	$0.638237\pi$
$140$		0			0
$141$	−64.6486			−0.458500
$142$		0			0
$143$		−	18.7544i		−	0.131149i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	140.087			0.952976
$148$		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
$153$		−	14.0935i		−	0.0921143i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	181.926i		−	1.15877i	−0.815055	−	0.579383i	$-0.803294\pi$
$157$		−	181.926i		−	1.15877i	0.815055	−	0.579383i	$-0.196706\pi$
$158$		0			0
$159$		−	173.310i		−	1.09000i
$160$		0			0
$161$	13.3356	−	43.6249i	0.0828296	−	0.270962i
$162$		0			0
$163$	148.985			0.914019			0.457009	−	0.889462i	$-0.348921\pi$
$163$	148.985			0.914019			0.457009	+	0.889462i	$0.348921\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	104.424			0.625295			0.312648	−	0.949869i	$-0.398784\pi$
$167$	104.424			0.625295			0.312648	+	0.949869i	$0.398784\pi$
$168$		0			0
$169$	−165.919			−0.981769
$170$		0			0
$171$			3.03393i			0.0177423i
$172$		0			0
$173$	−149.922			−0.866599			−0.433299	−	0.901250i	$-0.642651\pi$
$173$	−149.922			−0.866599			−0.433299	+	0.901250i	$0.642651\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	105.948			0.598574
$178$		0			0
$179$	−38.2719			−0.213810			−0.106905	−	0.994269i	$-0.534094\pi$
$179$	−38.2719			−0.213810			−0.106905	+	0.994269i	$0.534094\pi$
$180$		0			0
$181$			98.1669i			0.542359i	0.962529	+	0.271179i	$0.0874137\pi$
$181$			98.1669i			0.542359i	−0.962529	+	0.271179i	$0.912586\pi$
$182$		0			0
$183$			236.829i			1.29415i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	227.238			1.21518
$188$		0			0
$189$		−	51.4021i		−	0.271969i
$190$		0			0
$191$			193.804i			1.01468i	0.861746	+	0.507340i	$0.169371\pi$
$191$			193.804i			1.01468i	−0.861746	+	0.507340i	$0.830629\pi$
$192$		0			0
$193$	−80.0520			−0.414777			−0.207389	−	0.978259i	$-0.566496\pi$
$193$	−80.0520			−0.414777			−0.207389	+	0.978259i	$0.566496\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	291.126			1.47780			0.738898	−	0.673817i	$-0.235346\pi$
$197$	291.126			1.47780			0.738898	+	0.673817i	$0.235346\pi$
$198$		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
$203$			18.2720i			0.0900097i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−14.5752	−	4.45546i	−0.0704117	−	0.0215240i
$208$		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
$213$	−392.180			−1.84122
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	84.4900i		−	0.389355i
$218$		0			0
$219$	−321.397			−1.46757
$220$		0			0
$221$			37.3325i			0.168925i
$222$		0			0
$223$	216.987			0.973037			0.486519	−	0.873670i	$-0.338267\pi$
$223$	216.987			0.973037			0.486519	+	0.873670i	$0.338267\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			106.791i			0.470445i	0.971942	+	0.235222i	$0.0755819\pi$
$227$			106.791i			0.470445i	−0.971942	+	0.235222i	$0.924418\pi$
$228$		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
$233$	−101.825			−0.437017			−0.218508	−	0.975835i	$-0.570119\pi$
$233$	−101.825			−0.437017			−0.218508	+	0.975835i	$0.570119\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	122.826i		−	0.518253i
$238$		0			0
$239$	11.6391			0.0486992			0.0243496	−	0.999704i	$-0.492249\pi$
$239$	11.6391			0.0486992			0.0243496	+	0.999704i	$0.492249\pi$
$240$		0			0
$241$		−	71.1303i		−	0.295146i	−0.989051	−	0.147573i	$-0.952854\pi$
$241$		−	71.1303i		−	0.295146i	0.989051	−	0.147573i	$-0.0471462\pi$
$242$		0			0
$243$	−35.7122			−0.146964
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	8.03665i		−	0.0325370i
$248$		0			0
$249$		−	365.892i		−	1.46945i
$250$		0			0
$251$		−	281.356i		−	1.12094i	−0.828174	−	0.560471i	$-0.810620\pi$
$251$		−	281.356i		−	1.12094i	0.828174	−	0.560471i	$-0.189380\pi$
$252$		0			0
$253$	71.8381	−	235.005i	0.283945	−	0.928875i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	419.878			1.63377			0.816883	−	0.576804i	$-0.195700\pi$
$257$	419.878			1.63377			0.816883	+	0.576804i	$0.195700\pi$
$258$		0			0
$259$	−126.712			−0.489234
$260$		0			0
$261$	6.10473			0.0233898
$262$		0			0
$263$		−	412.742i		−	1.56936i	−0.619899	−	0.784681i	$-0.712827\pi$
$263$		−	412.742i		−	1.56936i	0.619899	−	0.784681i	$-0.287173\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			244.330i			0.915095i
$268$		0			0
$269$	−70.6401			−0.262602			−0.131301	−	0.991343i	$-0.541916\pi$
$269$	−70.6401			−0.262602			−0.131301	+	0.991343i	$0.541916\pi$
$270$		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
$273$		−	10.8220i		−	0.0396410i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	198.446			0.716411			0.358206	−	0.933643i	$-0.383389\pi$
$277$	198.446			0.716411			0.358206	+	0.933643i	$0.383389\pi$
$278$		0			0
$279$	−28.2284			−0.101177
$280$		0			0
$281$			329.525i			1.17269i	0.810063	+	0.586343i	$0.199433\pi$
$281$			329.525i			1.17269i	−0.810063	+	0.586343i	$0.800567\pi$
$282$		0			0
$283$		−	168.334i		−	0.594821i	−0.954750	−	0.297411i	$-0.903877\pi$
$283$		−	168.334i		−	0.594821i	0.954750	−	0.297411i	$-0.0961231\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	43.8251i		−	0.152701i
$288$		0			0
$289$	−163.341			−0.565192
$290$		0			0
$291$			230.424i			0.791836i
$292$		0			0
$293$			453.853i			1.54899i	0.632583	+	0.774493i	$0.281995\pi$
$293$			453.853i			1.54899i	−0.632583	+	0.774493i	$0.718005\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	276.901i		−	0.932326i
$298$		0			0
$299$	38.6086	+	11.8022i	0.129126	+	0.0394721i
$300$		0			0
$301$	−122.375			−0.406561
$302$		0			0
$303$	−117.682			−0.388390
$304$		0			0
$305$		0			0
$306$		0			0
$307$	364.912			1.18864			0.594319	−	0.804229i	$-0.297422\pi$
$307$	364.912			1.18864			0.594319	+	0.804229i	$0.297422\pi$
$308$		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
$313$			426.545i			1.36276i	0.731929	+	0.681381i	$0.238620\pi$
$313$			426.545i			1.36276i	−0.731929	+	0.681381i	$0.761380\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−30.7769			−0.0970880			−0.0485440	−	0.998821i	$-0.515458\pi$
$317$	−30.7769			−0.0970880			−0.0485440	+	0.998821i	$0.515458\pi$
$318$		0			0
$319$			98.4304i			0.308559i
$320$		0			0
$321$		−	128.805i		−	0.401261i
$322$		0			0
$323$	97.3763			0.301475
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	646.460i		−	1.97694i
$328$		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
$333$			42.3348i			0.127132i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			419.816i			1.24574i	0.782324	+	0.622872i	$0.214034\pi$
$337$			419.816i			1.24574i	−0.782324	+	0.622872i	$0.785966\pi$
$338$		0			0
$339$			532.365i			1.57040i
$340$		0			0
$341$		−	455.144i		−	1.33473i
$342$		0			0
$343$			186.569i			0.543932i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	286.347			0.825208			0.412604	−	0.910911i	$-0.364619\pi$
$347$	286.347			0.825208			0.412604	+	0.910911i	$0.364619\pi$
$348$		0			0
$349$	−176.419			−0.505498			−0.252749	−	0.967532i	$-0.581335\pi$
$349$	−176.419			−0.505498			−0.252749	+	0.967532i	$0.581335\pi$
$350$		0			0
$351$	45.4915			0.129605
$352$		0			0
$353$	−100.063			−0.283465			−0.141732	−	0.989905i	$-0.545267\pi$
$353$	−100.063			−0.283465			−0.141732	+	0.989905i	$0.545267\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	131.125			0.367297
$358$		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
$363$	21.2763			0.0586124
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	46.2696i		−	0.126075i	−0.998011	−	0.0630376i	$-0.979921\pi$
$367$		−	46.2696i		−	0.126075i	0.998011	−	0.0630376i	$-0.0200788\pi$
$368$		0			0
$369$	−14.6421			−0.0396805
$370$		0			0
$371$	110.581			0.298061
$372$		0			0
$373$			101.366i			0.271757i	0.990725	+	0.135879i	$0.0433857\pi$
$373$			101.366i			0.271757i	−0.990725	+	0.135879i	$0.956614\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−16.1709			−0.0428937
$378$		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
$383$		−	242.620i		−	0.633473i	−0.948514	−	0.316737i	$-0.897413\pi$
$383$		−	242.620i		−	0.633473i	0.948514	−	0.316737i	$-0.102587\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			40.8858i			0.105648i
$388$		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
$393$	−345.815			−0.879936
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−304.863			−0.767917			−0.383958	−	0.923350i	$-0.625439\pi$
$397$	−304.863			−0.767917			−0.383958	+	0.923350i	$0.625439\pi$
$398$		0			0
$399$	−28.2276			−0.0707458
$400$		0			0
$401$			703.077i			1.75331i	0.481121	+	0.876654i	$0.340230\pi$
$401$			703.077i			1.75331i	−0.481121	+	0.876654i	$0.659770\pi$
$402$		0			0
$403$	74.7748			0.185545
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−682.590			−1.67713
$408$		0			0
$409$	416.790			1.01905			0.509523	−	0.860457i	$-0.329822\pi$
$409$	416.790			1.01905			0.509523	+	0.860457i	$0.329822\pi$
$410$		0			0
$411$		−	127.107i		−	0.309262i
$412$		0			0
$413$			67.6002i			0.163681i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−363.605			−0.871954
$418$		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	$0.0382791\pi$
$421$			101.013i			0.239936i	−0.992778	+	0.119968i	$0.961721\pi$
$422$		0			0
$423$	−13.7815			−0.0325804
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−151.109			−0.353885
$428$		0			0
$429$		−	58.2976i		−	0.135892i
$430$		0			0
$431$			489.410i			1.13552i	0.823194	+	0.567760i	$0.192190\pi$
$431$			489.410i			1.13552i	−0.823194	+	0.567760i	$0.807810\pi$
$432$		0			0
$433$		−	471.464i		−	1.08883i	−0.838816	−	0.544415i	$-0.816752\pi$
$433$		−	471.464i		−	1.08883i	0.838816	−	0.544415i	$-0.183248\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	30.7842	−	100.705i	0.0704443	−	0.230446i
$438$		0			0
$439$	−123.984			−0.282425			−0.141212	−	0.989979i	$-0.545100\pi$
$439$	−123.984			−0.282425			−0.141212	+	0.989979i	$0.545100\pi$
$440$		0			0
$441$	29.8632			0.0677171
$442$		0			0
$443$	859.972			1.94125			0.970623	−	0.240606i	$-0.0773462\pi$
$443$	859.972			1.94125			0.970623	+	0.240606i	$0.0773462\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			590.699i			1.32147i
$448$		0			0
$449$	359.990			0.801759			0.400880	−	0.916131i	$-0.368705\pi$
$449$	359.990			0.801759			0.400880	+	0.916131i	$0.368705\pi$
$450$		0			0
$451$		−	236.084i		−	0.523468i
$452$		0			0
$453$	−682.535			−1.50670
$454$		0			0
$455$		0			0
$456$		0			0
$457$			120.754i			0.264232i	0.991234	+	0.132116i	$0.0421772\pi$
$457$			120.754i			0.264232i	−0.991234	+	0.132116i	$0.957823\pi$
$458$		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
$463$	365.620			0.789676			0.394838	−	0.918751i	$-0.370801\pi$
$463$	365.620			0.789676			0.394838	+	0.918751i	$0.370801\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	189.176i		−	0.405089i	−0.979273	−	0.202544i	$-0.935079\pi$
$467$		−	189.176i		−	0.405089i	0.979273	−	0.202544i	$-0.0649210\pi$
$468$		0			0
$469$	127.770			0.272431
$470$		0			0
$471$		−	565.514i		−	1.20067i
$472$		0			0
$473$	−659.228			−1.39372
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	36.9454i		−	0.0774537i
$478$		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
$483$	41.4534	−	135.607i	0.0858247	−	0.280760i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−330.861			−0.679385			−0.339693	−	0.940537i	$-0.610323\pi$
$487$	−330.861			−0.679385			−0.339693	+	0.940537i	$0.610323\pi$
$488$		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
$493$		−	195.936i		−	0.397436i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	250.232i		−	0.503484i
$498$		0			0
$499$	40.8056			0.0817748			0.0408874	−	0.999164i	$-0.486982\pi$
$499$	40.8056			0.0817748			0.0408874	+	0.999164i	$0.486982\pi$
$500$		0			0
$501$	324.601			0.647906
$502$		0			0
$503$			44.0719i			0.0876180i	0.999040	+	0.0438090i	$0.0139493\pi$
$503$			44.0719i			0.0876180i	−0.999040	+	0.0438090i	$0.986051\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−515.756			−1.01727
$508$		0			0
$509$	234.886			0.461466			0.230733	−	0.973017i	$-0.425888\pi$
$509$	234.886			0.461466			0.230733	+	0.973017i	$0.425888\pi$
$510$		0			0
$511$		−	205.068i		−	0.401308i
$512$		0			0
$513$		−	118.658i		−	0.231302i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	222.208i		−	0.429802i
$518$		0			0
$519$	−466.028			−0.897935
$520$		0			0
$521$		−	944.850i		−	1.81353i	−0.421635	−	0.906765i	$-0.638544\pi$
$521$		−	944.850i		−	1.81353i	0.421635	−	0.906765i	$-0.361456\pi$
$522$		0			0
$523$		−	355.863i		−	0.680427i	−0.940348	−	0.340213i	$-0.889501\pi$
$523$		−	355.863i		−	0.680427i	0.940348	−	0.340213i	$-0.110499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			906.011i			1.71919i
$528$		0			0
$529$	438.584	+	295.778i	0.829082	+	0.559127i
$530$		0			0
$531$	22.5855			0.0425338
$532$		0			0
$533$	38.7858			0.0727688
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−118.968			−0.221541
$538$		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
$543$			305.150i			0.561971i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−428.257			−0.782920			−0.391460	−	0.920195i	$-0.628030\pi$
$547$	−428.257			−0.782920			−0.391460	+	0.920195i	$0.628030\pi$
$548$		0			0
$549$			50.4861i			0.0919600i
$550$		0			0
$551$			42.1795i			0.0765508i
$552$		0			0
$553$	78.3694			0.141717
$554$		0			0
$555$		0			0
$556$		0			0
$557$			410.029i			0.736138i	0.929798	+	0.368069i	$0.119981\pi$
$557$			410.029i			0.736138i	−0.929798	+	0.368069i	$0.880019\pi$
$558$		0			0
$559$		−	108.303i		−	0.193745i
$560$		0			0
$561$	706.365			1.25912
$562$		0			0
$563$			727.785i			1.29269i	0.763045	+	0.646345i	$0.223703\pi$
$563$			727.785i			1.29269i	−0.763045	+	0.646345i	$0.776297\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	171.611i		−	0.302665i
$568$		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	$0.174252\pi$
$571$			594.402i			1.04098i	−0.853866	+	0.520492i	$0.825748\pi$
$572$		0			0
$573$			602.435i			1.05137i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−73.6344			−0.127616			−0.0638079	−	0.997962i	$-0.520325\pi$
$577$	−73.6344			−0.127616			−0.0638079	+	0.997962i	$0.520325\pi$
$578$		0			0
$579$	−248.840			−0.429776
$580$		0			0
$581$	233.459			0.401822
$582$		0			0
$583$	595.694			1.02177
$584$		0			0
$585$		0			0
$586$		0			0
$587$	229.976			0.391782			0.195891	−	0.980626i	$-0.437240\pi$
$587$	229.976			0.391782			0.195891	+	0.980626i	$0.437240\pi$
$588$		0			0
$589$		−	195.039i		−	0.331136i
$590$		0			0
$591$	904.959			1.53123
$592$		0			0
$593$	745.294			1.25682			0.628410	−	0.777882i	$-0.283706\pi$
$593$	745.294			1.25682			0.628410	+	0.777882i	$0.283706\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			1163.39i			1.94873i
$598$		0			0
$599$	447.095			0.746403			0.373201	−	0.927750i	$-0.378260\pi$
$599$	447.095			0.746403			0.373201	+	0.927750i	$0.378260\pi$
$600$		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
$603$		−	42.6885i		−	0.0707935i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−998.935			−1.64569			−0.822846	−	0.568264i	$-0.807615\pi$
$607$	−998.935			−1.64569			−0.822846	+	0.568264i	$0.807615\pi$
$608$		0			0
$609$			56.7981i			0.0932645i
$610$		0			0
$611$	36.5061			0.0597481
$612$		0			0
$613$		−	732.465i		−	1.19489i	−0.801912	−	0.597443i	$-0.796184\pi$
$613$		−	732.465i		−	1.19489i	0.801912	−	0.597443i	$-0.203816\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	450.432i		−	0.730036i	−0.931001	−	0.365018i	$-0.881063\pi$
$617$		−	450.432i		−	0.730036i	0.931001	−	0.365018i	$-0.118937\pi$
$618$		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
$623$	−155.896			−0.250234
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−152.061			−0.242521
$628$		0			0
$629$	1358.77			2.16020
$630$		0			0
$631$		−	601.263i		−	0.952873i	−0.879209	−	0.476436i	$-0.841928\pi$
$631$		−	601.263i		−	0.952873i	0.879209	−	0.476436i	$-0.158072\pi$
$632$		0			0
$633$	−765.389			−1.20914
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−79.1053			−0.124184
$638$		0			0
$639$	−83.6032			−0.130834
$640$		0			0
$641$		−	738.437i		−	1.15201i	−0.817447	−	0.576004i	$-0.804611\pi$
$641$		−	738.437i		−	1.15201i	0.817447	−	0.576004i	$-0.195389\pi$
$642$		0			0
$643$		−	135.341i		−	0.210484i	−0.994447	−	0.105242i	$-0.966438\pi$
$643$		−	135.341i		−	0.210484i	0.994447	−	0.105242i	$-0.0335617\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−978.141			−1.51181			−0.755905	−	0.654682i	$-0.772803\pi$
$647$	−978.141			−1.51181			−0.755905	+	0.654682i	$0.772803\pi$
$648$		0			0
$649$			364.159i			0.561108i
$650$		0			0
$651$		−	262.636i		−	0.403434i
$652$		0			0
$653$	412.686			0.631984			0.315992	−	0.948762i	$-0.397663\pi$
$653$	412.686			0.631984			0.315992	+	0.948762i	$0.397663\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−68.5140			−0.104283
$658$		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	$-0.988030\pi$
$661$		−	49.7030i		−	0.0751937i	0.999293	−	0.0375968i	$-0.0119703\pi$
$662$		0			0
$663$			116.047i			0.175034i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−202.633	−	61.9423i	−0.303798	−	0.0928671i
$668$		0			0
$669$	674.501			1.00822
$670$		0			0
$671$	−814.018			−1.21314
$672$		0			0
$673$	645.714			0.959456			0.479728	−	0.877417i	$-0.340735\pi$
$673$	645.714			0.959456			0.479728	+	0.877417i	$0.340735\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	839.079i		−	1.23941i	−0.784836	−	0.619704i	$-0.787253\pi$
$677$		−	839.079i		−	1.23941i	0.784836	−	0.619704i	$-0.212747\pi$
$678$		0			0
$679$	−147.023			−0.216528
$680$		0			0
$681$			331.958i			0.487456i
$682$		0			0
$683$	468.404			0.685804			0.342902	−	0.939371i	$-0.388590\pi$
$683$	468.404			0.685804			0.342902	+	0.939371i	$0.388590\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			19.7172i			0.0287005i
$688$		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
$693$	−14.0423			−0.0202631
$694$		0			0
$695$		0			0
$696$		0			0
$697$			469.949i			0.674246i
$698$		0			0
$699$	−316.521			−0.452819
$700$		0			0
$701$			952.474i			1.35874i	0.733798	+	0.679368i	$0.237746\pi$
$701$			952.474i			1.35874i	−0.733798	+	0.679368i	$0.762254\pi$
$702$		0			0
$703$	−292.504			−0.416080
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	75.0875i		−	0.106206i
$708$		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
$713$	936.980	+	286.423i	1.31414	+	0.401715i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	36.1799			0.0504602
$718$		0			0
$719$	−389.128			−0.541207			−0.270603	−	0.962691i	$-0.587223\pi$
$719$	−389.128			−0.541207			−0.270603	+	0.962691i	$0.587223\pi$
$720$		0			0
$721$	−58.9079			−0.0817031
$722$		0			0
$723$		−	221.107i		−	0.305819i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			85.6314i			0.117787i	0.998264	+	0.0588937i	$0.0187573\pi$
$727$			85.6314i			0.117787i	−0.998264	+	0.0588937i	$0.981243\pi$
$728$		0			0
$729$	667.712			0.915929
$730$		0			0
$731$	1312.26			1.79516
$732$		0			0
$733$			885.478i			1.20802i	0.796977	+	0.604009i	$0.206431\pi$
$733$			885.478i			1.20802i	−0.796977	+	0.604009i	$0.793569\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	688.293			0.933912
$738$		0			0
$739$	−807.772			−1.09306			−0.546530	−	0.837439i	$-0.684052\pi$
$739$	−807.772			−1.09306			−0.546530	+	0.837439i	$0.684052\pi$
$740$		0			0
$741$		−	24.9818i		−	0.0337136i
$742$		0			0
$743$		−	612.748i		−	0.824694i	−0.911027	−	0.412347i	$-0.864709\pi$
$743$		−	612.748i		−	0.824694i	0.911027	−	0.412347i	$-0.135291\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	77.9993i		−	0.104417i
$748$		0			0
$749$	82.1842			0.109725
$750$		0			0
$751$			1210.29i			1.61157i	0.592210	+	0.805784i	$0.298256\pi$
$751$			1210.29i			1.61157i	−0.592210	+	0.805784i	$0.701744\pi$
$752$		0			0
$753$		−	874.591i		−	1.16148i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	643.582i		−	0.850174i	−0.905153	−	0.425087i	$-0.860244\pi$
$757$		−	643.582i		−	0.850174i	0.905153	−	0.425087i	$-0.139756\pi$
$758$		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
$763$	412.476			0.540597
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−59.8271			−0.0780014
$768$		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
$773$		−	23.1809i		−	0.0299882i	−0.999888	−	0.0149941i	$-0.995227\pi$
$773$		−	23.1809i		−	0.0299882i	0.999888	−	0.0149941i	$-0.00477295\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−393.881			−0.506925
$778$		0			0
$779$		−	101.167i		−	0.129868i
$780$		0			0
$781$		−	1347.99i		−	1.72598i
$782$		0			0
$783$	−238.757			−0.304927
$784$		0			0
$785$		0			0
$786$		0			0
$787$			961.062i			1.22117i	0.791950	+	0.610585i	$0.209066\pi$
$787$			961.062i			1.22117i	−0.791950	+	0.610585i	$0.790934\pi$
$788$		0			0
$789$		−	1283.00i		−	1.62611i
$790$		0			0
$791$	−339.677			−0.429427
$792$		0			0
$793$		−	133.734i		−	0.168643i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	288.472i		−	0.361947i	−0.983488	−	0.180974i	$-0.942075\pi$
$797$		−	288.472i		−	0.361947i	0.983488	−	0.180974i	$-0.0579249\pi$
$798$		0			0
$799$			442.327i			0.553601i
$800$		0			0
$801$			52.0853i			0.0650253i
$802$		0			0
$803$		−	1104.69i		−	1.37571i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−219.583			−0.272098
$808$		0			0
$809$	354.079			0.437675			0.218837	−	0.975761i	$-0.429774\pi$
$809$	354.079			0.437675			0.218837	+	0.975761i	$0.429774\pi$
$810$		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
$813$	−337.465			−0.415086
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−282.493			−0.345769
$818$		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
$823$	959.645			1.16603			0.583017	−	0.812460i	$-0.301872\pi$
$823$	959.645			1.16603			0.583017	+	0.812460i	$0.301872\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	526.528i		−	0.636672i	−0.947978	−	0.318336i	$-0.896876\pi$
$827$		−	526.528i		−	0.636672i	0.947978	−	0.318336i	$-0.103124\pi$
$828$		0			0
$829$	−508.750			−0.613692			−0.306846	−	0.951759i	$-0.599274\pi$
$829$	−508.750			−0.613692			−0.306846	+	0.951759i	$0.599274\pi$
$830$		0			0
$831$	616.865			0.742317
$832$		0			0
$833$		−	958.482i		−	1.15064i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1104.02			1.31902
$838$		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
$843$			1024.32i			1.21509i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			13.5754i			0.0160276i
$848$		0			0
$849$		−	523.264i		−	0.616330i
$850$		0			0
$851$	429.555	−	1405.21i	0.504765	−	1.65125i
$852$		0			0
$853$	1147.14			1.34482			0.672412	−	0.740177i	$-0.265258\pi$
$853$	1147.14			1.34482			0.672412	+	0.740177i	$0.265258\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−97.7654			−0.114079			−0.0570393	−	0.998372i	$-0.518166\pi$
$857$	−97.7654			−0.114079			−0.0570393	+	0.998372i	$0.518166\pi$
$858$		0			0
$859$	−1503.31			−1.75007			−0.875033	−	0.484063i	$-0.839161\pi$
$859$	−1503.31			−1.75007			−0.875033	+	0.484063i	$0.839161\pi$
$860$		0			0
$861$		−	136.229i		−	0.158222i
$862$		0			0
$863$	−655.859			−0.759976			−0.379988	−	0.924992i	$-0.624072\pi$
$863$	−655.859			−0.759976			−0.379988	+	0.924992i	$0.624072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−507.741			−0.585630
$868$		0			0
$869$	422.172			0.485814
$870$		0			0
$871$			113.078i			0.129826i
$872$		0			0
$873$			49.1208i			0.0562667i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	251.321			0.286568			0.143284	−	0.989682i	$-0.454234\pi$
$877$	251.321			0.286568			0.143284	+	0.989682i	$0.454234\pi$
$878$		0			0
$879$			1410.79i			1.60500i
$880$		0			0
$881$			96.6276i			0.109679i	0.998495	+	0.0548397i	$0.0174648\pi$
$881$			96.6276i			0.109679i	−0.998495	+	0.0548397i	$0.982535\pi$
$882$		0			0
$883$	1075.24			1.21772			0.608859	−	0.793279i	$-0.291628\pi$
$883$	1075.24			1.21772			0.608859	+	0.793279i	$0.291628\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	281.233			0.317061			0.158530	−	0.987354i	$-0.449324\pi$
$887$	281.233			0.317061			0.158530	+	0.987354i	$0.449324\pi$
$888$		0			0
$889$		−	26.8898i		−	0.0302473i
$890$		0			0
$891$		−	924.461i		−	1.03755i
$892$		0			0
$893$		−	95.2207i		−	0.106630i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	120.014	+	36.6868i	0.133795	+	0.0408994i
$898$		0			0
$899$	−392.448			−0.436538
$900$		0			0
$901$	−1185.79			−1.31608
$902$		0			0
$903$	−380.400			−0.421262
$904$		0			0
$905$		0			0
$906$		0			0
$907$			509.375i			0.561604i	0.959766	+	0.280802i	$0.0906004\pi$
$907$			509.375i			0.561604i	−0.959766	+	0.280802i	$0.909400\pi$
$908$		0			0
$909$	−25.0870			−0.0275984
$910$		0			0
$911$		−	1258.48i		−	1.38143i	−0.723127	−	0.690715i	$-0.757296\pi$
$911$		−	1258.48i		−	1.38143i	0.723127	−	0.690715i	$-0.242704\pi$
$912$		0			0
$913$	1257.63			1.37747
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	220.648i		−	0.240620i
$918$		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
$923$	221.458			0.239933
$924$		0			0
$925$		0			0
$926$		0			0
$927$			19.6813i			0.0212312i
$928$		0			0
$929$	1160.51			1.24920			0.624599	−	0.780945i	$-0.285262\pi$
$929$	1160.51			1.24920			0.624599	+	0.780945i	$0.285262\pi$
$930$		0			0
$931$			206.334i			0.221627i
$932$		0			0
$933$	−188.117			−0.201626
$934$		0			0
$935$		0			0
$936$		0			0
$937$			267.963i			0.285980i	0.989724	+	0.142990i	$0.0456717\pi$
$937$			267.963i			0.285980i	−0.989724	+	0.142990i	$0.954328\pi$
$938$		0			0
$939$			1325.91i			1.41204i
$940$		0			0
$941$			251.727i			0.267510i	0.991014	+	0.133755i	$0.0427035\pi$
$941$			251.727i			0.267510i	−0.991014	+	0.133755i	$0.957297\pi$
$942$		0			0
$943$	486.013	+	148.568i	0.515390	+	0.157548i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1302.05			1.37492			0.687462	−	0.726220i	$-0.258725\pi$
$947$	1302.05			1.37492			0.687462	+	0.726220i	$0.258725\pi$
$948$		0			0
$949$	181.488			0.191241
$950$		0			0
$951$	−95.6694			−0.100599
$952$		0			0
$953$			1156.60i			1.21364i	0.794839	+	0.606821i	$0.207555\pi$
$953$			1156.60i			1.21364i	−0.794839	+	0.606821i	$0.792445\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			305.969i			0.319717i
$958$		0			0
$959$	81.1008			0.0845681
$960$		0			0
$961$	853.687			0.888332
$962$		0			0
$963$		−	27.4580i		−	0.0285130i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1048.39			−1.08417			−0.542086	−	0.840323i	$-0.682365\pi$
$967$	−1048.39			−1.08417			−0.542086	+	0.840323i	$0.682365\pi$
$968$		0			0
$969$	302.692			0.312376
$970$		0			0
$971$		−	257.911i		−	0.265614i	−0.991142	−	0.132807i	$-0.957601\pi$
$971$		−	257.911i		−	0.265614i	0.991142	−	0.132807i	$-0.0423990\pi$
$972$		0			0
$973$		−	231.999i		−	0.238437i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			72.9096i			0.0746260i	0.999304	+	0.0373130i	$0.0118799\pi$
$977$			72.9096i			0.0746260i	−0.999304	+	0.0373130i	$0.988120\pi$
$978$		0			0
$979$	−839.803			−0.857817
$980$		0			0
$981$		−	137.810i		−	0.140479i
$982$		0			0
$983$			393.989i			0.400803i	0.979714	+	0.200401i	$0.0642247\pi$
$983$			393.989i			0.400803i	−0.979714	+	0.200401i	$0.935775\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	128.222i		−	0.129911i
$988$		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
$993$	−864.547			−0.870641
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1240.46			1.24420			0.622098	−	0.782939i	$-0.286281\pi$
$997$	1240.46			1.24420			0.622098	+	0.782939i	$0.286281\pi$
$998$		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.f.d.1701.12	yes	16
5.2	odd	4		2300.3.d.c.1149.17		32
5.3	odd	4		2300.3.d.c.1149.16		32
5.4	even	2		2300.3.f.c.1701.5	✓	16
23.22	odd	2	inner	2300.3.f.d.1701.11	yes	16
115.22	even	4		2300.3.d.c.1149.15		32
115.68	even	4		2300.3.d.c.1149.18		32
115.114	odd	2		2300.3.f.c.1701.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.15		32	115.22	even	4
2300.3.d.c.1149.16		32	5.3	odd	4
2300.3.d.c.1149.17		32	5.2	odd	4
2300.3.d.c.1149.18		32	115.68	even	4
2300.3.f.c.1701.5	✓	16	5.4	even	2
2300.3.f.c.1701.6	yes	16	115.114	odd	2
2300.3.f.d.1701.11	yes	16	23.22	odd	2	inner
2300.3.f.d.1701.12	yes	16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.10848 q^{3} +1.98337i q^{7} +0.662652 q^{9} +O(q^{10})\)	\(q+3.10848 q^{3} +1.98337i q^{7} +0.662652 q^{9} +10.6844i q^{11} -1.75531 q^{13} -21.2683i q^{17} +4.57847i q^{19} +6.16528i q^{21} +(-21.9953 - 6.72368i) q^{23} -25.9165 q^{27} +9.21257 q^{29} -42.5991 q^{31} +33.2121i q^{33} +63.8869i q^{37} -5.45635 q^{39} -22.0962 q^{41} +61.7003i q^{43} -20.7975 q^{47} +45.0662 q^{49} -66.1121i q^{51} -55.7539i q^{53} +14.2321i q^{57} +34.0834 q^{59} +76.1879i q^{61} +1.31429i q^{63} -64.4207i q^{67} +(-68.3719 - 20.9004i) q^{69} -126.165 q^{71} -103.394 q^{73} -21.1911 q^{77} -39.5132i q^{79} -86.5248 q^{81} -117.708i q^{83} +28.6371 q^{87} +78.6012i q^{89} -3.48144i q^{91} -132.419 q^{93} +74.1276i q^{97} +7.08001i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} + 6 q^{13} - 8 q^{23} + 66 q^{27} - 6 q^{29} + 28 q^{31} + 74 q^{39} - 90 q^{41} + 40 q^{47} - 190 q^{49} - 174 q^{59} + 50 q^{69} + 116 q^{71} + 110 q^{73} + 198 q^{77} + 56 q^{81} - 362 q^{87} - 50 q^{93}+O(q^{100}) \) 16 * q + 64 * q^9 + 6 * q^13 - 8 * q^23 + 66 * q^27 - 6 * q^29 + 28 * q^31 + 74 * q^39 - 90 * q^41 + 40 * q^47 - 190 * q^49 - 174 * q^59 + 50 * q^69 + 116 * q^71 + 110 * q^73 + 198 * q^77 + 56 * q^81 - 362 * q^87 - 50 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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