LMFDB - Embedded newform 225.3.d.b.224.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(224,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.224");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.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:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			224.6
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.3.d.b.224.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.821854 q^{2} -3.32456 q^{4} +0.837722i q^{7} -6.01972 q^{8} +O(q^{10})$	$q+0.821854 q^{2} -3.32456 q^{4} +0.837722i q^{7} -6.01972 q^{8} +14.3716i q^{11} +21.8114i q^{13} +0.688486i q^{14} +8.35089 q^{16} -23.5454 q^{17} +6.32456 q^{19} +11.8114i q^{22} -38.8723 q^{23} +17.9258i q^{26} -2.78505i q^{28} +0.266737i q^{29} +30.2719 q^{31} +30.9421 q^{32} -19.3509 q^{34} +9.53950i q^{37} +5.19786 q^{38} -19.3028i q^{41} -19.6228i q^{43} -47.7793i q^{44} -31.9473 q^{46} -22.1684 q^{47} +48.2982 q^{49} -72.5132i q^{52} +49.0012 q^{53} -5.04285i q^{56} +0.219219i q^{58} +73.2351i q^{59} -48.3246 q^{61} +24.8791 q^{62} -7.97367 q^{64} +77.2982i q^{67} +78.2780 q^{68} -104.044i q^{71} -47.6754i q^{73} +7.84008i q^{74} -21.0263 q^{76} -12.0394 q^{77} -68.2192 q^{79} -15.8641i q^{82} +28.2098 q^{83} -16.1271i q^{86} -86.5132i q^{88} +53.7774i q^{89} -18.2719 q^{91} +129.233 q^{92} -18.2192 q^{94} -114.921i q^{97} +39.6941 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{4}+O(q^{10}) $ 8 * q + 24 * q^4	$ 8 q + 24 q^{4} + 168 q^{16} - 112 q^{31} - 256 q^{34} + 48 q^{46} + 184 q^{49} - 336 q^{61} + 88 q^{64} - 320 q^{76} + 112 q^{79} + 208 q^{91} + 512 q^{94}+O(q^{100}) $ 8 * q + 24 * q^4 + 168 * q^16 - 112 * q^31 - 256 * q^34 + 48 * q^46 + 184 * q^49 - 336 * q^61 + 88 * q^64 - 320 * q^76 + 112 * q^79 + 208 * q^91 + 512 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.821854	0.410927	0.205464	−	0.978665i	$-0.434130\pi$
$2$	0.821854	0.410927	0.205464	+	0.978665i	$0.434130\pi$
$3$	0	0
$3$	0	0
$4$	−3.32456	−0.831139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.837722i	0.119675i	0.998208	+	0.0598373i	$0.0190582\pi$
$7$	0.837722i	0.119675i	−0.998208	+	0.0598373i	$0.980942\pi$
$8$	−6.01972	−0.752465
$9$	0	0
$10$	0	0
$11$	14.3716i	1.30651i	0.757137	+	0.653256i	$0.226597\pi$
$11$	14.3716i	1.30651i	−0.757137	+	0.653256i	$0.773403\pi$
$12$	0	0
$13$	21.8114i	1.67780i	0.544286	+	0.838900i	$0.316801\pi$
$13$	21.8114i	1.67780i	−0.544286	+	0.838900i	$0.683199\pi$
$14$	0.688486i	0.0491776i
$15$	0	0
$16$	8.35089	0.521931
$17$	−23.5454	−1.38502	−0.692512	−	0.721407i	$-0.743496\pi$
$17$	−23.5454	−1.38502	−0.692512	+	0.721407i	$0.743496\pi$
$18$	0	0
$19$	6.32456	0.332871	0.166436	−	0.986052i	$-0.446774\pi$
$19$	6.32456	0.332871	0.166436	+	0.986052i	$0.446774\pi$
$20$	0	0
$21$	0	0
$22$	11.8114i	0.536881i
$23$	−38.8723	−1.69010	−0.845049	−	0.534689i	$-0.820429\pi$
$23$	−38.8723	−1.69010	−0.845049	+	0.534689i	$0.820429\pi$
$24$	0	0
$25$	0	0
$26$	17.9258i	0.689453i
$27$	0	0
$28$	− 2.78505i	− 0.0994662i
$29$	0.266737i	0.00919784i	0.999989	+	0.00459892i	$0.00146389\pi$
$29$	0.266737i	0.00919784i	−0.999989	+	0.00459892i	$0.998536\pi$
$30$	0	0
$31$	30.2719	0.976512	0.488256	−	0.872700i	$-0.337633\pi$
$31$	30.2719	0.976512	0.488256	+	0.872700i	$0.337633\pi$
$32$	30.9421	0.966940
$33$	0	0
$34$	−19.3509	−0.569144
$35$	0	0
$36$	0	0
$37$	9.53950i	0.257824i	0.991656	+	0.128912i	$0.0411485\pi$
$37$	9.53950i	0.257824i	−0.991656	+	0.128912i	$0.958851\pi$
$38$	5.19786	0.136786
$39$	0	0
$40$	0	0
$41$	− 19.3028i	− 0.470799i	−0.971899	−	0.235399i	$-0.924360\pi$
$41$	− 19.3028i	− 0.470799i	0.971899	−	0.235399i	$-0.0756398\pi$
$42$	0	0
$43$	− 19.6228i	− 0.456344i	−0.973621	−	0.228172i	$-0.926725\pi$
$43$	− 19.6228i	− 0.456344i	0.973621	−	0.228172i	$-0.0732748\pi$
$44$	− 47.7793i	− 1.08589i
$45$	0	0
$46$	−31.9473	−0.694507
$47$	−22.1684	−0.471669	−0.235834	−	0.971793i	$-0.575782\pi$
$47$	−22.1684	−0.471669	−0.235834	+	0.971793i	$0.575782\pi$
$48$	0	0
$49$	48.2982	0.985678
$50$	0	0
$51$	0	0
$52$	− 72.5132i	− 1.39448i
$53$	49.0012	0.924552	0.462276	−	0.886736i	$-0.347033\pi$
$53$	49.0012	0.924552	0.462276	+	0.886736i	$0.347033\pi$
$54$	0	0
$55$	0	0
$56$	− 5.04285i	− 0.0900509i
$57$	0	0
$58$	0.219219i	0.00377964i
$59$	73.2351i	1.24127i	0.784098	+	0.620637i	$0.213126\pi$
$59$	73.2351i	1.24127i	−0.784098	+	0.620637i	$0.786874\pi$
$60$	0	0
$61$	−48.3246	−0.792206	−0.396103	−	0.918206i	$-0.629638\pi$
$61$	−48.3246	−0.792206	−0.396103	+	0.918206i	$0.629638\pi$
$62$	24.8791	0.401276
$63$	0	0
$64$	−7.97367	−0.124589
$65$	0	0
$66$	0	0
$67$	77.2982i	1.15370i	0.816848	+	0.576852i	$0.195719\pi$
$67$	77.2982i	1.15370i	−0.816848	+	0.576852i	$0.804281\pi$
$68$	78.2780	1.15115
$69$	0	0
$70$	0	0
$71$	− 104.044i	− 1.46541i	−0.680548	−	0.732703i	$-0.738258\pi$
$71$	− 104.044i	− 1.46541i	0.680548	−	0.732703i	$-0.261742\pi$
$72$	0	0
$73$	− 47.6754i	− 0.653088i	−0.945182	−	0.326544i	$-0.894116\pi$
$73$	− 47.6754i	− 0.653088i	0.945182	−	0.326544i	$-0.105884\pi$
$74$	7.84008i	0.105947i
$75$	0	0
$76$	−21.0263	−0.276662
$77$	−12.0394	−0.156356
$78$	0	0
$79$	−68.2192	−0.863534	−0.431767	−	0.901985i	$-0.642110\pi$
$79$	−68.2192	−0.863534	−0.431767	+	0.901985i	$0.642110\pi$
$80$	0	0
$81$	0	0
$82$	− 15.8641i	− 0.193464i
$83$	28.2098	0.339877	0.169938	−	0.985455i	$-0.445643\pi$
$83$	28.2098	0.339877	0.169938	+	0.985455i	$0.445643\pi$
$84$	0	0
$85$	0	0
$86$	− 16.1271i	− 0.187524i
$87$	0	0
$88$	− 86.5132i	− 0.983104i
$89$	53.7774i	0.604240i	0.953270	+	0.302120i	$0.0976943\pi$
$89$	53.7774i	0.604240i	−0.953270	+	0.302120i	$0.902306\pi$
$90$	0	0
$91$	−18.2719	−0.200790
$92$	129.233	1.40471
$93$	0	0
$94$	−18.2192	−0.193821
$95$	0	0
$96$	0	0
$97$	− 114.921i	− 1.18475i	−0.805661	−	0.592376i	$-0.798190\pi$
$97$	− 114.921i	− 1.18475i	0.805661	−	0.592376i	$-0.201810\pi$
$98$	39.6941	0.405042
$99$	0	0
$100$	0	0
$101$	17.5473i	0.173736i	0.996220	+	0.0868679i	$0.0276858\pi$
$101$	17.5473i	0.173736i	−0.996220	+	0.0868679i	$0.972314\pi$
$102$	0	0
$103$	71.5395i	0.694558i	0.937762	+	0.347279i	$0.112894\pi$
$103$	71.5395i	0.694558i	−0.937762	+	0.347279i	$0.887106\pi$
$104$	− 131.298i	− 1.26248i
$105$	0	0
$106$	40.2719	0.379923
$107$	−76.3675	−0.713715	−0.356858	−	0.934159i	$-0.616152\pi$
$107$	−76.3675	−0.713715	−0.356858	+	0.934159i	$0.616152\pi$
$108$	0	0
$109$	126.921	1.16441	0.582206	−	0.813041i	$-0.302190\pi$
$109$	126.921	1.16441	0.582206	+	0.813041i	$0.302190\pi$
$110$	0	0
$111$	0	0
$112$	6.99573i	0.0624618i
$113$	15.0601	0.133275	0.0666377	−	0.997777i	$-0.478773\pi$
$113$	15.0601	0.133275	0.0666377	+	0.997777i	$0.478773\pi$
$114$	0	0
$115$	0	0
$116$	− 0.886783i	− 0.00764468i
$117$	0	0
$118$	60.1886i	0.510073i
$119$	− 19.7245i	− 0.165752i
$120$	0	0
$121$	−85.5438	−0.706973
$122$	−39.7157	−0.325539
$123$	0	0
$124$	−100.641	−0.811617
$125$	0	0
$126$	0	0
$127$	158.031i	1.24434i	0.782884	+	0.622168i	$0.213748\pi$
$127$	158.031i	1.24434i	−0.782884	+	0.622168i	$0.786252\pi$
$128$	−130.322	−1.01814
$129$	0	0
$130$	0	0
$131$	211.220i	1.61237i	0.591665	+	0.806184i	$0.298471\pi$
$131$	211.220i	1.61237i	−0.591665	+	0.806184i	$0.701529\pi$
$132$	0	0
$133$	5.29822i	0.0398363i
$134$	63.5279i	0.474089i
$135$	0	0
$136$	141.737	1.04218
$137$	−69.2592	−0.505542	−0.252771	−	0.967526i	$-0.581342\pi$
$137$	−69.2592	−0.505542	−0.252771	+	0.967526i	$0.581342\pi$
$138$	0	0
$139$	159.842	1.14994	0.574971	−	0.818174i	$-0.305013\pi$
$139$	159.842	1.14994	0.574971	+	0.818174i	$0.305013\pi$
$140$	0	0
$141$	0	0
$142$	− 85.5089i	− 0.602175i
$143$	−313.465	−2.19206
$144$	0	0
$145$	0	0
$146$	− 39.1823i	− 0.268372i
$147$	0	0
$148$	− 31.7146i	− 0.214288i
$149$	− 9.81897i	− 0.0658991i	−0.999457	−	0.0329496i	$-0.989510\pi$
$149$	− 9.81897i	− 0.0658991i	0.999457	−	0.0329496i	$-0.0104901\pi$
$150$	0	0
$151$	210.649	1.39503	0.697514	−	0.716572i	$-0.254290\pi$
$151$	210.649	1.39503	0.697514	+	0.716572i	$0.254290\pi$
$152$	−38.0720	−0.250474
$153$	0	0
$154$	−9.89466	−0.0642511
$155$	0	0
$156$	0	0
$157$	211.276i	1.34571i	0.739775	+	0.672854i	$0.234932\pi$
$157$	211.276i	1.34571i	−0.739775	+	0.672854i	$0.765068\pi$
$158$	−56.0663	−0.354850
$159$	0	0
$160$	0	0
$161$	− 32.5642i	− 0.202262i
$162$	0	0
$163$	222.763i	1.36664i	0.730117	+	0.683322i	$0.239465\pi$
$163$	222.763i	1.36664i	−0.730117	+	0.683322i	$0.760535\pi$
$164$	64.1731i	0.391299i
$165$	0	0
$166$	23.1843	0.139665
$167$	−33.3644	−0.199787	−0.0998933	−	0.994998i	$-0.531850\pi$
$167$	−33.3644	−0.199787	−0.0998933	+	0.994998i	$0.531850\pi$
$168$	0	0
$169$	−306.737	−1.81501
$170$	0	0
$171$	0	0
$172$	65.2370i	0.379285i
$173$	29.8102	0.172313	0.0861567	−	0.996282i	$-0.472541\pi$
$173$	29.8102	0.172313	0.0861567	+	0.996282i	$0.472541\pi$
$174$	0	0
$175$	0	0
$176$	120.016i	0.681909i
$177$	0	0
$178$	44.1972i	0.248299i
$179$	− 111.841i	− 0.624808i	−0.949949	−	0.312404i	$-0.898866\pi$
$179$	− 111.841i	− 0.624808i	0.949949	−	0.312404i	$-0.101134\pi$
$180$	0	0
$181$	−49.0790	−0.271155	−0.135577	−	0.990767i	$-0.543289\pi$
$181$	−49.0790	−0.271155	−0.135577	+	0.990767i	$0.543289\pi$
$182$	−15.0168	−0.0825101
$183$	0	0
$184$	234.000	1.27174
$185$	0	0
$186$	0	0
$187$	− 338.386i	− 1.80955i
$188$	73.7002	0.392022
$189$	0	0
$190$	0	0
$191$	278.947i	1.46046i	0.683203	+	0.730229i	$0.260586\pi$
$191$	278.947i	1.46046i	−0.683203	+	0.730229i	$0.739414\pi$
$192$	0	0
$193$	− 89.8947i	− 0.465775i	−0.972504	−	0.232888i	$-0.925183\pi$
$193$	− 89.8947i	− 0.465775i	0.972504	−	0.232888i	$-0.0748175\pi$
$194$	− 94.4483i	− 0.486847i
$195$	0	0
$196$	−160.570	−0.819235
$197$	212.709	1.07974	0.539870	−	0.841748i	$-0.318473\pi$
$197$	212.709	1.07974	0.539870	+	0.841748i	$0.318473\pi$
$198$	0	0
$199$	−96.4911	−0.484880	−0.242440	−	0.970166i	$-0.577948\pi$
$199$	−96.4911	−0.484880	−0.242440	+	0.970166i	$0.577948\pi$
$200$	0	0
$201$	0	0
$202$	14.4213i	0.0713928i
$203$	−0.223452	−0.00110075
$204$	0	0
$205$	0	0
$206$	58.7951i	0.285413i
$207$	0	0
$208$	182.144i	0.875695i
$209$	90.8942i	0.434900i
$210$	0	0
$211$	−65.7893	−0.311798	−0.155899	−	0.987773i	$-0.549827\pi$
$211$	−65.7893	−0.311798	−0.155899	+	0.987773i	$0.549827\pi$
$212$	−162.907	−0.768431
$213$	0	0
$214$	−62.7630	−0.293285
$215$	0	0
$216$	0	0
$217$	25.3594i	0.116864i
$218$	104.311	0.478489
$219$	0	0
$220$	0	0
$221$	− 513.558i	− 2.32379i
$222$	0	0
$223$	− 102.302i	− 0.458756i	−0.973337	−	0.229378i	$-0.926331\pi$
$223$	− 102.302i	− 0.458756i	0.973337	−	0.229378i	$-0.0736691\pi$
$224$	25.9209i	0.115718i
$225$	0	0
$226$	12.3772	0.0547665
$227$	12.5296	0.0551966	0.0275983	−	0.999619i	$-0.491214\pi$
$227$	12.5296	0.0551966	0.0275983	+	0.999619i	$0.491214\pi$
$228$	0	0
$229$	−23.2982	−0.101739	−0.0508695	−	0.998705i	$-0.516199\pi$
$229$	−23.2982	−0.101739	−0.0508695	+	0.998705i	$0.516199\pi$
$230$	0	0
$231$	0	0
$232$	− 1.60568i	− 0.00692105i
$233$	356.382	1.52954	0.764768	−	0.644306i	$-0.222854\pi$
$233$	356.382	1.52954	0.764768	+	0.644306i	$0.222854\pi$
$234$	0	0
$235$	0	0
$236$	− 243.474i	− 1.03167i
$237$	0	0
$238$	− 16.2107i	− 0.0681121i
$239$	175.524i	0.734408i	0.930140	+	0.367204i	$0.119685\pi$
$239$	175.524i	0.734408i	−0.930140	+	0.367204i	$0.880315\pi$
$240$	0	0
$241$	104.438	0.433355	0.216677	−	0.976243i	$-0.430478\pi$
$241$	104.438	0.433355	0.216677	+	0.976243i	$0.430478\pi$
$242$	−70.3045	−0.290515
$243$	0	0
$244$	160.658	0.658433
$245$	0	0
$246$	0	0
$247$	137.947i	0.558491i
$248$	−182.228	−0.734791
$249$	0	0
$250$	0	0
$251$	130.945i	0.521694i	0.965380	+	0.260847i	$0.0840018\pi$
$251$	130.945i	0.521694i	−0.965380	+	0.260847i	$0.915998\pi$
$252$	0	0
$253$	− 558.658i	− 2.20813i
$254$	129.878i	0.511331i
$255$	0	0
$256$	−75.2107	−0.293792
$257$	425.641	1.65619	0.828095	−	0.560587i	$-0.189425\pi$
$257$	425.641	1.65619	0.828095	+	0.560587i	$0.189425\pi$
$258$	0	0
$259$	−7.99145	−0.0308550
$260$	0	0
$261$	0	0
$262$	173.592i	0.662566i
$263$	74.5004	0.283271	0.141636	−	0.989919i	$-0.454764\pi$
$263$	74.5004	0.283271	0.141636	+	0.989919i	$0.454764\pi$
$264$	0	0
$265$	0	0
$266$	4.35437i	0.0163698i
$267$	0	0
$268$	− 256.982i	− 0.958889i
$269$	− 205.067i	− 0.762331i	−0.924507	−	0.381165i	$-0.875523\pi$
$269$	− 205.067i	− 0.762331i	0.924507	−	0.381165i	$-0.124477\pi$
$270$	0	0
$271$	−233.351	−0.861073	−0.430537	−	0.902573i	$-0.641676\pi$
$271$	−233.351	−0.861073	−0.430537	+	0.902573i	$0.641676\pi$
$272$	−196.625	−0.722886
$273$	0	0
$274$	−56.9210	−0.207741
$275$	0	0
$276$	0	0
$277$	− 423.715i	− 1.52966i	−0.644235	−	0.764828i	$-0.722824\pi$
$277$	− 423.715i	− 1.52966i	0.644235	−	0.764828i	$-0.277176\pi$
$278$	131.367	0.472543
$279$	0	0
$280$	0	0
$281$	− 402.604i	− 1.43275i	−0.697713	−	0.716377i	$-0.745799\pi$
$281$	− 402.604i	− 1.43275i	0.697713	−	0.716377i	$-0.254201\pi$
$282$	0	0
$283$	272.333i	0.962308i	0.876636	+	0.481154i	$0.159782\pi$
$283$	272.333i	0.962308i	−0.876636	+	0.481154i	$0.840218\pi$
$284$	345.900i	1.21796i
$285$	0	0
$286$	−257.623	−0.900779
$287$	16.1704	0.0563427
$288$	0	0
$289$	265.386	0.918290
$290$	0	0
$291$	0	0
$292$	158.500i	0.542807i
$293$	443.188	1.51259	0.756294	−	0.654232i	$-0.227008\pi$
$293$	443.188	1.51259	0.756294	+	0.654232i	$0.227008\pi$
$294$	0	0
$295$	0	0
$296$	− 57.4251i	− 0.194004i
$297$	0	0
$298$	− 8.06976i	− 0.0270797i
$299$	− 847.858i	− 2.83564i
$300$	0	0
$301$	16.4384	0.0546128
$302$	173.123	0.573255
$303$	0	0
$304$	52.8157	0.173736
$305$	0	0
$306$	0	0
$307$	− 390.824i	− 1.27304i	−0.771259	−	0.636522i	$-0.780373\pi$
$307$	− 390.824i	− 1.27304i	0.771259	−	0.636522i	$-0.219627\pi$
$308$	40.0258	0.129954
$309$	0	0
$310$	0	0
$311$	2.97739i	0.00957362i	0.999989	+	0.00478681i	$0.00152369\pi$
$311$	2.97739i	0.00957362i	−0.999989	+	0.00478681i	$0.998476\pi$
$312$	0	0
$313$	− 130.105i	− 0.415672i	−0.978164	−	0.207836i	$-0.933358\pi$
$313$	− 130.105i	− 0.415672i	0.978164	−	0.207836i	$-0.0666420\pi$
$314$	173.638i	0.552988i
$315$	0	0
$316$	226.799	0.717717
$317$	131.677	0.415384	0.207692	−	0.978194i	$-0.433405\pi$
$317$	131.677	0.415384	0.207692	+	0.978194i	$0.433405\pi$
$318$	0	0
$319$	−3.83345	−0.0120171
$320$	0	0
$321$	0	0
$322$	− 26.7630i	− 0.0831149i
$323$	−148.914	−0.461035
$324$	0	0
$325$	0	0
$326$	183.079i	0.561591i
$327$	0	0
$328$	116.197i	0.354260i
$329$	− 18.5710i	− 0.0564468i
$330$	0	0
$331$	160.483	0.484842	0.242421	−	0.970171i	$-0.422059\pi$
$331$	160.483	0.484842	0.242421	+	0.970171i	$0.422059\pi$
$332$	−93.7850	−0.282485
$333$	0	0
$334$	−27.4207	−0.0820978
$335$	0	0
$336$	0	0
$337$	− 128.114i	− 0.380160i	−0.981769	−	0.190080i	$-0.939125\pi$
$337$	− 128.114i	− 0.380160i	0.981769	−	0.190080i	$-0.0608747\pi$
$338$	−252.093	−0.745837
$339$	0	0
$340$	0	0
$341$	435.056i	1.27583i
$342$	0	0
$343$	81.5089i	0.237635i
$344$	118.124i	0.343383i
$345$	0	0
$346$	24.4997	0.0708082
$347$	−219.637	−0.632959	−0.316480	−	0.948599i	$-0.602501\pi$
$347$	−219.637	−0.632959	−0.316480	+	0.948599i	$0.602501\pi$
$348$	0	0
$349$	−403.465	−1.15606	−0.578030	−	0.816016i	$-0.696178\pi$
$349$	−403.465	−1.15606	−0.578030	+	0.816016i	$0.696178\pi$
$350$	0	0
$351$	0	0
$352$	444.688i	1.26332i
$353$	−54.8192	−0.155295	−0.0776475	−	0.996981i	$-0.524741\pi$
$353$	−54.8192	−0.155295	−0.0776475	+	0.996981i	$0.524741\pi$
$354$	0	0
$355$	0	0
$356$	− 178.786i	− 0.502207i
$357$	0	0
$358$	− 91.9167i	− 0.256751i
$359$	480.460i	1.33833i	0.743114	+	0.669165i	$0.233348\pi$
$359$	480.460i	1.33833i	−0.743114	+	0.669165i	$0.766652\pi$
$360$	0	0
$361$	−321.000	−0.889197
$362$	−40.3358	−0.111425
$363$	0	0
$364$	60.7459	0.166884
$365$	0	0
$366$	0	0
$367$	522.364i	1.42333i	0.702517	+	0.711667i	$0.252060\pi$
$367$	522.364i	1.42333i	−0.702517	+	0.711667i	$0.747940\pi$
$368$	−324.618	−0.882114
$369$	0	0
$370$	0	0
$371$	41.0494i	0.110645i
$372$	0	0
$373$	233.285i	0.625428i	0.949847	+	0.312714i	$0.101238\pi$
$373$	233.285i	0.625428i	−0.949847	+	0.312714i	$0.898762\pi$
$374$	− 278.104i	− 0.743593i
$375$	0	0
$376$	133.448	0.354914
$377$	−5.81791	−0.0154321
$378$	0	0
$379$	−248.596	−0.655927	−0.327964	−	0.944690i	$-0.606362\pi$
$379$	−248.596	−0.655927	−0.327964	+	0.944690i	$0.606362\pi$
$380$	0	0
$381$	0	0
$382$	229.254i	0.600142i
$383$	−468.291	−1.22269	−0.611346	−	0.791364i	$-0.709372\pi$
$383$	−468.291	−1.22269	−0.611346	+	0.791364i	$0.709372\pi$
$384$	0	0
$385$	0	0
$386$	− 73.8803i	− 0.191400i
$387$	0	0
$388$	382.061i	0.984694i
$389$	484.238i	1.24483i	0.782688	+	0.622414i	$0.213848\pi$
$389$	484.238i	1.24483i	−0.782688	+	0.622414i	$0.786152\pi$
$390$	0	0
$391$	915.263	2.34083
$392$	−290.742	−0.741688
$393$	0	0
$394$	174.816	0.443695
$395$	0	0
$396$	0	0
$397$	− 298.943i	− 0.753005i	−0.926416	−	0.376503i	$-0.877127\pi$
$397$	− 298.943i	− 0.753005i	0.926416	−	0.376503i	$-0.122873\pi$
$398$	−79.3016	−0.199250
$399$	0	0
$400$	0	0
$401$	467.509i	1.16586i	0.812523	+	0.582929i	$0.198093\pi$
$401$	467.509i	1.16586i	−0.812523	+	0.582929i	$0.801907\pi$
$402$	0	0
$403$	660.272i	1.63839i
$404$	− 58.3370i	− 0.144399i
$405$	0	0
$406$	−0.183645	−0.000452327 0
$407$	−137.098	−0.336851
$408$	0	0
$409$	184.158	0.450264	0.225132	−	0.974328i	$-0.427719\pi$
$409$	184.158	0.450264	0.225132	+	0.974328i	$0.427719\pi$
$410$	0	0
$411$	0	0
$412$	− 237.837i	− 0.577274i
$413$	−61.3507	−0.148549
$414$	0	0
$415$	0	0
$416$	674.890i	1.62233i
$417$	0	0
$418$	74.7018i	0.178712i
$419$	− 429.840i	− 1.02587i	−0.858427	−	0.512936i	$-0.828558\pi$
$419$	− 429.840i	− 1.02587i	0.858427	−	0.512936i	$-0.171442\pi$
$420$	0	0
$421$	−305.035	−0.724548	−0.362274	−	0.932072i	$-0.618000\pi$
$421$	−305.035	−0.724548	−0.362274	+	0.932072i	$0.618000\pi$
$422$	−54.0692	−0.128126
$423$	0	0
$424$	−294.974	−0.695693
$425$	0	0
$426$	0	0
$427$	− 40.4826i	− 0.0948069i
$428$	253.888	0.593196
$429$	0	0
$430$	0	0
$431$	128.880i	0.299025i	0.988760	+	0.149512i	$0.0477704\pi$
$431$	128.880i	0.299025i	−0.988760	+	0.149512i	$0.952230\pi$
$432$	0	0
$433$	− 243.886i	− 0.563247i	−0.959525	−	0.281624i	$-0.909127\pi$
$433$	− 243.886i	− 0.563247i	0.959525	−	0.281624i	$-0.0908730\pi$
$434$	20.8418i	0.0480225i
$435$	0	0
$436$	−421.956	−0.967789
$437$	−245.850	−0.562585
$438$	0	0
$439$	259.614	0.591376	0.295688	−	0.955285i	$-0.404451\pi$
$439$	259.614	0.591376	0.295688	+	0.955285i	$0.404451\pi$
$440$	0	0
$441$	0	0
$442$	− 422.070i	− 0.954909i
$443$	−541.011	−1.22124	−0.610622	−	0.791923i	$-0.709080\pi$
$443$	−541.011	−1.22124	−0.610622	+	0.791923i	$0.709080\pi$
$444$	0	0
$445$	0	0
$446$	− 84.0778i	− 0.188515i
$447$	0	0
$448$	− 6.67972i	− 0.0149101i
$449$	− 791.947i	− 1.76380i	−0.471436	−	0.881900i	$-0.656264\pi$
$449$	− 791.947i	− 1.76380i	0.471436	−	0.881900i	$-0.343736\pi$
$450$	0	0
$451$	277.412	0.615104
$452$	−50.0682	−0.110770
$453$	0	0
$454$	10.2975	0.0226818
$455$	0	0
$456$	0	0
$457$	611.359i	1.33777i	0.743367	+	0.668883i	$0.233227\pi$
$457$	611.359i	1.33777i	−0.743367	+	0.668883i	$0.766773\pi$
$458$	−19.1477	−0.0418073
$459$	0	0
$460$	0	0
$461$	− 586.991i	− 1.27330i	−0.771153	−	0.636650i	$-0.780320\pi$
$461$	− 586.991i	− 1.27330i	0.771153	−	0.636650i	$-0.219680\pi$
$462$	0	0
$463$	− 195.285i	− 0.421781i	−0.977510	−	0.210891i	$-0.932364\pi$
$463$	− 195.285i	− 0.421781i	0.977510	−	0.210891i	$-0.0676364\pi$
$464$	2.22749i	0.00480063i
$465$	0	0
$466$	292.894	0.628528
$467$	753.763	1.61405	0.807027	−	0.590515i	$-0.201075\pi$
$467$	753.763	1.61405	0.807027	+	0.590515i	$0.201075\pi$
$468$	0	0
$469$	−64.7544	−0.138069
$470$	0	0
$471$	0	0
$472$	− 440.855i	− 0.934014i
$473$	282.011	0.596218
$474$	0	0
$475$	0	0
$476$	65.5752i	0.137763i
$477$	0	0
$478$	144.255i	0.301788i
$479$	614.848i	1.28361i	0.766869	+	0.641803i	$0.221814\pi$
$479$	614.848i	1.28361i	−0.766869	+	0.641803i	$0.778186\pi$
$480$	0	0
$481$	−208.070	−0.432577
$482$	85.8332	0.178077
$483$	0	0
$484$	284.395	0.587593
$485$	0	0
$486$	0	0
$487$	478.197i	0.981924i	0.871181	+	0.490962i	$0.163355\pi$
$487$	478.197i	0.981924i	−0.871181	+	0.490962i	$0.836645\pi$
$488$	290.900	0.596107
$489$	0	0
$490$	0	0
$491$	617.223i	1.25707i	0.777780	+	0.628537i	$0.216346\pi$
$491$	617.223i	1.25707i	−0.777780	+	0.628537i	$0.783654\pi$
$492$	0	0
$493$	− 6.28043i	− 0.0127392i
$494$	113.373i	0.229499i
$495$	0	0
$496$	252.797	0.509672
$497$	87.1599	0.175372
$498$	0	0
$499$	−783.096	−1.56933	−0.784665	−	0.619919i	$-0.787165\pi$
$499$	−783.096	−1.56933	−0.784665	+	0.619919i	$0.787165\pi$
$500$	0	0
$501$	0	0
$502$	107.618i	0.214378i
$503$	−369.395	−0.734383	−0.367191	−	0.930145i	$-0.619681\pi$
$503$	−369.395	−0.734383	−0.367191	+	0.930145i	$0.619681\pi$
$504$	0	0
$505$	0	0
$506$	− 459.135i	− 0.907382i
$507$	0	0
$508$	− 525.381i	− 1.03422i
$509$	225.635i	0.443291i	0.975127	+	0.221645i	$0.0711427\pi$
$509$	225.635i	0.443291i	−0.975127	+	0.221645i	$0.928857\pi$
$510$	0	0
$511$	39.9388	0.0781581
$512$	459.474	0.897410
$513$	0	0
$514$	349.815	0.680574
$515$	0	0
$516$	0	0
$517$	− 318.596i	− 0.616241i
$518$	−6.56781	−0.0126792
$519$	0	0
$520$	0	0
$521$	− 1007.73i	− 1.93422i	−0.254367	−	0.967108i	$-0.581867\pi$
$521$	− 1007.73i	− 1.93422i	0.254367	−	0.967108i	$-0.418133\pi$
$522$	0	0
$523$	− 935.517i	− 1.78875i	−0.447316	−	0.894376i	$-0.647620\pi$
$523$	− 935.517i	− 1.78875i	0.447316	−	0.894376i	$-0.352380\pi$
$524$	− 702.213i	− 1.34010i
$525$	0	0
$526$	61.2285	0.116404
$527$	−712.764	−1.35249
$528$	0	0
$529$	982.052	1.85643
$530$	0	0
$531$	0	0
$532$	− 17.6142i	− 0.0331095i
$533$	421.020	0.789906
$534$	0	0
$535$	0	0
$536$	− 465.314i	− 0.868122i
$537$	0	0
$538$	− 168.535i	− 0.313263i
$539$	694.124i	1.28780i
$540$	0	0
$541$	−399.149	−0.737798	−0.368899	−	0.929469i	$-0.620265\pi$
$541$	−399.149	−0.737798	−0.368899	+	0.929469i	$0.620265\pi$
$542$	−191.780	−0.353838
$543$	0	0
$544$	−728.544	−1.33923
$545$	0	0
$546$	0	0
$547$	480.833i	0.879036i	0.898234	+	0.439518i	$0.144851\pi$
$547$	480.833i	0.879036i	−0.898234	+	0.439518i	$0.855149\pi$
$548$	230.256	0.420175
$549$	0	0
$550$	0	0
$551$	1.68699i	0.00306170i
$552$	0	0
$553$	− 57.1488i	− 0.103343i
$554$	− 348.232i	− 0.628577i
$555$	0	0
$556$	−531.404	−0.955762
$557$	−751.542	−1.34927	−0.674634	−	0.738152i	$-0.735699\pi$
$557$	−751.542	−1.34927	−0.674634	+	0.738152i	$0.735699\pi$
$558$	0	0
$559$	428.000	0.765653
$560$	0	0
$561$	0	0
$562$	− 330.882i	− 0.588758i
$563$	670.820	1.19151	0.595755	−	0.803166i	$-0.296853\pi$
$563$	670.820	1.19151	0.595755	+	0.803166i	$0.296853\pi$
$564$	0	0
$565$	0	0
$566$	223.818i	0.395438i
$567$	0	0
$568$	626.315i	1.10267i
$569$	275.325i	0.483875i	0.970292	+	0.241937i	$0.0777829\pi$
$569$	275.325i	0.483875i	−0.970292	+	0.241937i	$0.922217\pi$
$570$	0	0
$571$	−900.289	−1.57669	−0.788344	−	0.615235i	$-0.789061\pi$
$571$	−900.289	−1.57669	−0.788344	+	0.615235i	$0.789061\pi$
$572$	1042.13	1.82191
$573$	0	0
$574$	13.2897	0.0231527
$575$	0	0
$576$	0	0
$577$	− 596.236i	− 1.03334i	−0.856185	−	0.516669i	$-0.827172\pi$
$577$	− 596.236i	− 1.03334i	0.856185	−	0.516669i	$-0.172828\pi$
$578$	218.108	0.377350
$579$	0	0
$580$	0	0
$581$	23.6320i	0.0406746i
$582$	0	0
$583$	704.228i	1.20794i
$584$	286.993i	0.491426i
$585$	0	0
$586$	364.236	0.621564
$587$	497.431	0.847412	0.423706	−	0.905800i	$-0.360729\pi$
$587$	497.431	0.847412	0.423706	+	0.905800i	$0.360729\pi$
$588$	0	0
$589$	191.456	0.325053
$590$	0	0
$591$	0	0
$592$	79.6633i	0.134566i
$593$	898.856	1.51578	0.757889	−	0.652384i	$-0.226231\pi$
$593$	898.856	1.51578	0.757889	+	0.652384i	$0.226231\pi$
$594$	0	0
$595$	0	0
$596$	32.6437i	0.0547713i
$597$	0	0
$598$	− 696.816i	− 1.16524i
$599$	28.5701i	0.0476964i	0.999716	+	0.0238482i	$0.00759183\pi$
$599$	28.5701i	0.0476964i	−0.999716	+	0.0238482i	$0.992408\pi$
$600$	0	0
$601$	−20.2107	−0.0336284	−0.0168142	−	0.999859i	$-0.505352\pi$
$601$	−20.2107	−0.0336284	−0.0168142	+	0.999859i	$0.505352\pi$
$602$	13.5100	0.0224419
$603$	0	0
$604$	−700.315	−1.15946
$605$	0	0
$606$	0	0
$607$	− 713.626i	− 1.17566i	−0.808984	−	0.587831i	$-0.799982\pi$
$607$	− 713.626i	− 1.17566i	0.808984	−	0.587831i	$-0.200018\pi$
$608$	195.695	0.321867
$609$	0	0
$610$	0	0
$611$	− 483.524i	− 0.791365i
$612$	0	0
$613$	76.1530i	0.124230i	0.998069	+	0.0621150i	$0.0197846\pi$
$613$	76.1530i	0.124230i	−0.998069	+	0.0621150i	$0.980215\pi$
$614$	− 321.201i	− 0.523128i
$615$	0	0
$616$	72.4740	0.117653
$617$	201.599	0.326741	0.163371	−	0.986565i	$-0.447763\pi$
$617$	201.599	0.326741	0.163371	+	0.986565i	$0.447763\pi$
$618$	0	0
$619$	204.263	0.329989	0.164995	−	0.986294i	$-0.447239\pi$
$619$	204.263	0.329989	0.164995	+	0.986294i	$0.447239\pi$
$620$	0	0
$621$	0	0
$622$	2.44699i	0.00393406i
$623$	−45.0505	−0.0723122
$624$	0	0
$625$	0	0
$626$	− 106.928i	− 0.170811i
$627$	0	0
$628$	− 702.399i	− 1.11847i
$629$	− 224.611i	− 0.357093i
$630$	0	0
$631$	−639.903	−1.01411	−0.507055	−	0.861914i	$-0.669266\pi$
$631$	−639.903	−1.01411	−0.507055	+	0.861914i	$0.669266\pi$
$632$	410.660	0.649779
$633$	0	0
$634$	108.219	0.170693
$635$	0	0
$636$	0	0
$637$	1053.45i	1.65377i
$638$	−3.15054	−0.00493815
$639$	0	0
$640$	0	0
$641$	570.709i	0.890342i	0.895446	+	0.445171i	$0.146857\pi$
$641$	570.709i	0.890342i	−0.895446	+	0.445171i	$0.853143\pi$
$642$	0	0
$643$	− 453.693i	− 0.705587i	−0.935701	−	0.352794i	$-0.885232\pi$
$643$	− 453.693i	− 0.705587i	0.935701	−	0.352794i	$-0.114768\pi$
$644$	108.261i	0.168108i
$645$	0	0
$646$	−122.386	−0.189452
$647$	−983.536	−1.52015	−0.760074	−	0.649837i	$-0.774837\pi$
$647$	−983.536	−1.52015	−0.760074	+	0.649837i	$0.774837\pi$
$648$	0	0
$649$	−1052.51	−1.62174
$650$	0	0
$651$	0	0
$652$	− 740.588i	− 1.13587i
$653$	−544.478	−0.833811	−0.416905	−	0.908950i	$-0.636885\pi$
$653$	−544.478	−0.833811	−0.416905	+	0.908950i	$0.636885\pi$
$654$	0	0
$655$	0	0
$656$	− 161.195i	− 0.245724i
$657$	0	0
$658$	− 15.2626i	− 0.0231955i
$659$	− 107.933i	− 0.163783i	−0.996641	−	0.0818916i	$-0.973904\pi$
$659$	− 107.933i	− 0.163783i	0.996641	−	0.0818916i	$-0.0260961\pi$
$660$	0	0
$661$	345.728	0.523038	0.261519	−	0.965198i	$-0.415777\pi$
$661$	345.728	0.523038	0.261519	+	0.965198i	$0.415777\pi$
$662$	131.893	0.199235
$663$	0	0
$664$	−169.815	−0.255745
$665$	0	0
$666$	0	0
$667$	− 10.3687i	− 0.0155452i
$668$	110.922	0.166050
$669$	0	0
$670$	0	0
$671$	− 694.503i	− 1.03503i
$672$	0	0
$673$	1204.78i	1.79016i	0.445902	+	0.895082i	$0.352883\pi$
$673$	1204.78i	1.79016i	−0.445902	+	0.895082i	$0.647117\pi$
$674$	− 105.291i	− 0.156218i
$675$	0	0
$676$	1019.76	1.50853
$677$	574.598	0.848742	0.424371	−	0.905488i	$-0.360495\pi$
$677$	574.598	0.848742	0.424371	+	0.905488i	$0.360495\pi$
$678$	0	0
$679$	96.2719	0.141785
$680$	0	0
$681$	0	0
$682$	357.553i	0.524271i
$683$	−334.387	−0.489585	−0.244792	−	0.969575i	$-0.578720\pi$
$683$	−334.387	−0.489585	−0.244792	+	0.969575i	$0.578720\pi$
$684$	0	0
$685$	0	0
$686$	66.9884i	0.0976508i
$687$	0	0
$688$	− 163.868i	− 0.238180i
$689$	1068.79i	1.55121i
$690$	0	0
$691$	901.149	1.30412	0.652061	−	0.758166i	$-0.273904\pi$
$691$	901.149	1.30412	0.652061	+	0.758166i	$0.273904\pi$
$692$	−99.1057	−0.143216
$693$	0	0
$694$	−180.510	−0.260100
$695$	0	0
$696$	0	0
$697$	454.491i	0.652068i
$698$	−331.589	−0.475056
$699$	0	0
$700$	0	0
$701$	887.934i	1.26667i	0.773879	+	0.633334i	$0.218314\pi$
$701$	887.934i	1.26667i	−0.773879	+	0.633334i	$0.781686\pi$
$702$	0	0
$703$	60.3331i	0.0858223i
$704$	− 114.595i	− 0.162776i
$705$	0	0
$706$	−45.0534	−0.0638150
$707$	−14.6998	−0.0207918
$708$	0	0
$709$	1026.35	1.44760	0.723801	−	0.690008i	$-0.242393\pi$
$709$	1026.35	1.44760	0.723801	+	0.690008i	$0.242393\pi$
$710$	0	0
$711$	0	0
$712$	− 323.725i	− 0.454669i
$713$	−1176.74	−1.65040
$714$	0	0
$715$	0	0
$716$	371.820i	0.519302i
$717$	0	0
$718$	394.868i	0.549956i
$719$	− 740.080i	− 1.02932i	−0.857395	−	0.514659i	$-0.827919\pi$
$719$	− 740.080i	− 1.02932i	0.857395	−	0.514659i	$-0.172081\pi$
$720$	0	0
$721$	−59.9302	−0.0831210
$722$	−263.815	−0.365395
$723$	0	0
$724$	163.166	0.225367
$725$	0	0
$726$	0	0
$727$	1063.75i	1.46321i	0.681731	+	0.731603i	$0.261227\pi$
$727$	1063.75i	1.46321i	−0.681731	+	0.731603i	$0.738773\pi$
$728$	109.992	0.151087
$729$	0	0
$730$	0	0
$731$	462.026i	0.632047i
$732$	0	0
$733$	− 134.749i	− 0.183833i	−0.995767	−	0.0919164i	$-0.970701\pi$
$733$	− 134.749i	− 0.183833i	0.995767	−	0.0919164i	$-0.0292993\pi$
$734$	429.307i	0.584887i
$735$	0	0
$736$	−1202.79	−1.63422
$737$	−1110.90	−1.50733
$738$	0	0
$739$	711.429	0.962692	0.481346	−	0.876531i	$-0.340148\pi$
$739$	711.429	0.962692	0.481346	+	0.876531i	$0.340148\pi$
$740$	0	0
$741$	0	0
$742$	33.7367i	0.0454672i
$743$	−466.330	−0.627631	−0.313816	−	0.949484i	$-0.601607\pi$
$743$	−466.330	−0.627631	−0.313816	+	0.949484i	$0.601607\pi$
$744$	0	0
$745$	0	0
$746$	191.726i	0.257005i
$747$	0	0
$748$	1124.98i	1.50399i
$749$	− 63.9748i	− 0.0854136i
$750$	0	0
$751$	227.359	0.302742	0.151371	−	0.988477i	$-0.451631\pi$
$751$	227.359	0.302742	0.151371	+	0.988477i	$0.451631\pi$
$752$	−185.126	−0.246178
$753$	0	0
$754$	−4.78147	−0.00634148
$755$	0	0
$756$	0	0
$757$	− 552.258i	− 0.729535i	−0.931099	−	0.364768i	$-0.881148\pi$
$757$	− 552.258i	− 0.729535i	0.931099	−	0.364768i	$-0.118852\pi$
$758$	−204.310	−0.269538
$759$	0	0
$760$	0	0
$761$	303.051i	0.398228i	0.979976	+	0.199114i	$0.0638064\pi$
$761$	303.051i	0.398228i	−0.979976	+	0.199114i	$0.936194\pi$
$762$	0	0
$763$	106.325i	0.139351i
$764$	− 927.376i	− 1.21384i
$765$	0	0
$766$	−384.867	−0.502437
$767$	−1597.36	−2.08261
$768$	0	0
$769$	−739.684	−0.961878	−0.480939	−	0.876754i	$-0.659704\pi$
$769$	−739.684	−0.961878	−0.480939	+	0.876754i	$0.659704\pi$
$770$	0	0
$771$	0	0
$772$	298.860i	0.387124i
$773$	−603.479	−0.780697	−0.390348	−	0.920667i	$-0.627645\pi$
$773$	−603.479	−0.780697	−0.390348	+	0.920667i	$0.627645\pi$
$774$	0	0
$775$	0	0
$776$	691.792i	0.891485i
$777$	0	0
$778$	397.973i	0.511533i
$779$	− 122.081i	− 0.156715i
$780$	0	0
$781$	1495.28	1.91457
$782$	752.213	0.961909
$783$	0	0
$784$	403.333	0.514455
$785$	0	0
$786$	0	0
$787$	− 660.483i	− 0.839241i	−0.907700	−	0.419620i	$-0.862163\pi$
$787$	− 660.483i	− 0.839241i	0.907700	−	0.419620i	$-0.137837\pi$
$788$	−707.162	−0.897414
$789$	0	0
$790$	0	0
$791$	12.6162i	0.0159497i
$792$	0	0
$793$	− 1054.03i	− 1.32916i
$794$	− 245.688i	− 0.309430i
$795$	0	0
$796$	320.790	0.403003
$797$	634.969	0.796699	0.398349	−	0.917234i	$-0.369583\pi$
$797$	634.969	0.796699	0.398349	+	0.917234i	$0.369583\pi$
$798$	0	0
$799$	521.964	0.653272
$800$	0	0
$801$	0	0
$802$	384.224i	0.479083i
$803$	685.174	0.853268
$804$	0	0
$805$	0	0
$806$	542.647i	0.673260i
$807$	0	0
$808$	− 105.630i	− 0.130730i
$809$	405.048i	0.500677i	0.968158	+	0.250339i	$0.0805419\pi$
$809$	405.048i	0.500677i	−0.968158	+	0.250339i	$0.919458\pi$
$810$	0	0
$811$	406.034	0.500659	0.250329	−	0.968161i	$-0.419461\pi$
$811$	406.034	0.500659	0.250329	+	0.968161i	$0.419461\pi$
$812$	0.742878	0.000914874 0
$813$	0	0
$814$	−112.675	−0.138421
$815$	0	0
$816$	0	0
$817$	− 124.105i	− 0.151904i
$818$	151.351	0.185026
$819$	0	0
$820$	0	0
$821$	− 550.073i	− 0.670003i	−0.942218	−	0.335002i	$-0.891263\pi$
$821$	− 550.073i	− 0.670003i	0.942218	−	0.335002i	$-0.108737\pi$
$822$	0	0
$823$	1472.51i	1.78920i	0.446868	+	0.894600i	$0.352540\pi$
$823$	1472.51i	1.78920i	−0.446868	+	0.894600i	$0.647460\pi$
$824$	− 430.648i	− 0.522631i
$825$	0	0
$826$	−50.4213	−0.0610428
$827$	1510.47	1.82644	0.913220	−	0.407466i	$-0.133587\pi$
$827$	1510.47	1.82644	0.913220	+	0.407466i	$0.133587\pi$
$828$	0	0
$829$	−712.692	−0.859701	−0.429850	−	0.902900i	$-0.641434\pi$
$829$	−712.692	−0.859701	−0.429850	+	0.902900i	$0.641434\pi$
$830$	0	0
$831$	0	0
$832$	− 173.917i	− 0.209035i
$833$	−1137.20	−1.36519
$834$	0	0
$835$	0	0
$836$	− 302.183i	− 0.361463i
$837$	0	0
$838$	− 353.266i	− 0.421559i
$839$	530.579i	0.632394i	0.948694	+	0.316197i	$0.102406\pi$
$839$	530.579i	0.632394i	−0.948694	+	0.316197i	$0.897594\pi$
$840$	0	0
$841$	840.929	0.999915
$842$	−250.694	−0.297737
$843$	0	0
$844$	218.720	0.259147
$845$	0	0
$846$	0	0
$847$	− 71.6619i	− 0.0846068i
$848$	409.204	0.482552
$849$	0	0
$850$	0	0
$851$	− 370.822i	− 0.435748i
$852$	0	0
$853$	− 215.232i	− 0.252324i	−0.992010	−	0.126162i	$-0.959734\pi$
$853$	− 215.232i	− 0.252324i	0.992010	−	0.126162i	$-0.0402659\pi$
$854$	− 33.2708i	− 0.0389587i
$855$	0	0
$856$	459.711	0.537046
$857$	−4.97441	−0.00580445	−0.00290222	−	0.999996i	$-0.500924\pi$
$857$	−4.97441	−0.00580445	−0.00290222	+	0.999996i	$0.500924\pi$
$858$	0	0
$859$	−1652.28	−1.92349	−0.961746	−	0.273941i	$-0.911673\pi$
$859$	−1652.28	−1.92349	−0.961746	+	0.273941i	$0.911673\pi$
$860$	0	0
$861$	0	0
$862$	105.920i	0.122877i
$863$	379.077	0.439255	0.219627	−	0.975584i	$-0.429516\pi$
$863$	379.077	0.439255	0.219627	+	0.975584i	$0.429516\pi$
$864$	0	0
$865$	0	0
$866$	− 200.439i	− 0.231454i
$867$	0	0
$868$	− 84.3088i	− 0.0971300i
$869$	− 980.421i	− 1.12822i
$870$	0	0
$871$	−1685.98	−1.93568
$872$	−764.029	−0.876180
$873$	0	0
$874$	−202.053	−0.231182
$875$	0	0
$876$	0	0
$877$	1101.60i	1.25610i	0.778173	+	0.628051i	$0.216147\pi$
$877$	1101.60i	1.25610i	−0.778173	+	0.628051i	$0.783853\pi$
$878$	213.365	0.243013
$879$	0	0
$880$	0	0
$881$	184.877i	0.209850i	0.994480	+	0.104925i	$0.0334602\pi$
$881$	184.877i	0.209850i	−0.994480	+	0.104925i	$0.966540\pi$
$882$	0	0
$883$	− 978.236i	− 1.10786i	−0.832565	−	0.553928i	$-0.813128\pi$
$883$	− 978.236i	− 1.10786i	0.832565	−	0.553928i	$-0.186872\pi$
$884$	1707.35i	1.93139i
$885$	0	0
$886$	−444.632	−0.501842
$887$	667.937	0.753029	0.376514	−	0.926411i	$-0.377123\pi$
$887$	667.937	0.753029	0.376514	+	0.926411i	$0.377123\pi$
$888$	0	0
$889$	−132.386	−0.148915
$890$	0	0
$891$	0	0
$892$	340.110i	0.381290i
$893$	−140.205	−0.157005
$894$	0	0
$895$	0	0
$896$	− 109.173i	− 0.121845i
$897$	0	0
$898$	− 650.865i	− 0.724794i
$899$	8.07464i	0.00898180i
$900$	0	0
$901$	−1153.75	−1.28053
$902$	227.992	0.252763
$903$	0	0
$904$	−90.6577	−0.100285
$905$	0	0
$906$	0	0
$907$	− 832.622i	− 0.917996i	−0.888437	−	0.458998i	$-0.848209\pi$
$907$	− 832.622i	− 0.917996i	0.888437	−	0.458998i	$-0.151791\pi$
$908$	−41.6554	−0.0458760
$909$	0	0
$910$	0	0
$911$	565.263i	0.620486i	0.950657	+	0.310243i	$0.100410\pi$
$911$	565.263i	0.620486i	−0.950657	+	0.310243i	$0.899590\pi$
$912$	0	0
$913$	405.421i	0.444053i
$914$	502.448i	0.549725i
$915$	0	0
$916$	77.4562	0.0845592
$917$	−176.944	−0.192959
$918$	0	0
$919$	1185.75	1.29026	0.645128	−	0.764075i	$-0.276804\pi$
$919$	1185.75	1.29026	0.645128	+	0.764075i	$0.276804\pi$
$920$	0	0
$921$	0	0
$922$	− 482.421i	− 0.523234i
$923$	2269.34	2.45866
$924$	0	0
$925$	0	0
$926$	− 160.496i	− 0.173321i
$927$	0	0
$928$	8.25341i	0.00889376i
$929$	− 942.995i	− 1.01506i	−0.861633	−	0.507532i	$-0.830558\pi$
$929$	− 942.995i	− 1.01506i	0.861633	−	0.507532i	$-0.169442\pi$
$930$	0	0
$931$	305.465	0.328104
$932$	−1184.81	−1.27126
$933$	0	0
$934$	619.483	0.663258
$935$	0	0
$936$	0	0
$937$	− 531.281i	− 0.567002i	−0.958972	−	0.283501i	$-0.908504\pi$
$937$	− 531.281i	− 0.567002i	0.958972	−	0.283501i	$-0.0914960\pi$
$938$	−53.2187	−0.0567364
$939$	0	0
$940$	0	0
$941$	− 595.650i	− 0.632996i	−0.948593	−	0.316498i	$-0.897493\pi$
$941$	− 595.650i	− 0.632996i	0.948593	−	0.316498i	$-0.102507\pi$
$942$	0	0
$943$	750.342i	0.795696i
$944$	611.578i	0.647859i
$945$	0	0
$946$	231.772	0.245002
$947$	782.246	0.826026	0.413013	−	0.910725i	$-0.364476\pi$
$947$	782.246	0.826026	0.413013	+	0.910725i	$0.364476\pi$
$948$	0	0
$949$	1039.87	1.09575
$950$	0	0
$951$	0	0
$952$	118.736i	0.124723i
$953$	426.708	0.447752	0.223876	−	0.974618i	$-0.428129\pi$
$953$	426.708	0.447752	0.223876	+	0.974618i	$0.428129\pi$
$954$	0	0
$955$	0	0
$956$	− 583.538i	− 0.610395i
$957$	0	0
$958$	505.315i	0.527469i
$959$	− 58.0200i	− 0.0605005i
$960$	0	0
$961$	−44.6128	−0.0464234
$962$	−171.003	−0.177758
$963$	0	0
$964$	−347.211	−0.360178
$965$	0	0
$966$	0	0
$967$	− 1210.91i	− 1.25223i	−0.779730	−	0.626116i	$-0.784644\pi$
$967$	− 1210.91i	− 1.25223i	0.779730	−	0.626116i	$-0.215356\pi$
$968$	514.949	0.531973
$969$	0	0
$970$	0	0
$971$	− 1193.69i	− 1.22934i	−0.788784	−	0.614670i	$-0.789289\pi$
$971$	− 1193.69i	− 1.22934i	0.788784	−	0.614670i	$-0.210711\pi$
$972$	0	0
$973$	133.903i	0.137619i
$974$	393.008i	0.403499i
$975$	0	0
$976$	−403.553	−0.413476
$977$	−909.475	−0.930886	−0.465443	−	0.885078i	$-0.654105\pi$
$977$	−909.475	−0.930886	−0.465443	+	0.885078i	$0.654105\pi$
$978$	0	0
$979$	−772.868	−0.789447
$980$	0	0
$981$	0	0
$982$	507.268i	0.516566i
$983$	1560.88	1.58788	0.793938	−	0.607999i	$-0.208028\pi$
$983$	1560.88	1.58788	0.793938	+	0.607999i	$0.208028\pi$
$984$	0	0
$985$	0	0
$986$	− 5.16160i	− 0.00523489i
$987$	0	0
$988$	− 458.614i	− 0.464184i
$989$	762.782i	0.771265i
$990$	0	0
$991$	−1692.63	−1.70800	−0.854001	−	0.520271i	$-0.825831\pi$
$991$	−1692.63	−1.70800	−0.854001	+	0.520271i	$0.825831\pi$
$992$	936.675	0.944229
$993$	0	0
$994$	71.6327	0.0720651
$995$	0	0
$996$	0	0
$997$	16.5744i	0.0166243i	0.999965	+	0.00831213i	$0.00264586\pi$
$997$	16.5744i	0.0166243i	−0.999965	+	0.00831213i	$0.997354\pi$
$998$	−643.591	−0.644881
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.d.b.224.6		8
3.2	odd	2	inner	225.3.d.b.224.4		8
4.3	odd	2		3600.3.c.i.449.3		8
5.2	odd	4		225.3.c.c.26.3		4
5.3	odd	4		45.3.c.a.26.2	✓	4
5.4	even	2	inner	225.3.d.b.224.3		8
12.11	even	2		3600.3.c.i.449.4		8
15.2	even	4		225.3.c.c.26.2		4
15.8	even	4		45.3.c.a.26.3	yes	4
15.14	odd	2	inner	225.3.d.b.224.5		8
20.3	even	4		720.3.l.a.161.2		4
20.7	even	4		3600.3.l.v.1601.1		4
20.19	odd	2		3600.3.c.i.449.5		8
40.3	even	4		2880.3.l.c.1601.4		4
40.13	odd	4		2880.3.l.g.1601.3		4
45.13	odd	12		405.3.i.d.26.3		8
45.23	even	12		405.3.i.d.26.2		8
45.38	even	12		405.3.i.d.296.3		8
45.43	odd	12		405.3.i.d.296.2		8
60.23	odd	4		720.3.l.a.161.4		4
60.47	odd	4		3600.3.l.v.1601.2		4
60.59	even	2		3600.3.c.i.449.6		8
120.53	even	4		2880.3.l.g.1601.1		4
120.83	odd	4		2880.3.l.c.1601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.2	✓	4	5.3	odd	4
45.3.c.a.26.3	yes	4	15.8	even	4
225.3.c.c.26.2		4	15.2	even	4
225.3.c.c.26.3		4	5.2	odd	4
225.3.d.b.224.3		8	5.4	even	2	inner
225.3.d.b.224.4		8	3.2	odd	2	inner
225.3.d.b.224.5		8	15.14	odd	2	inner
225.3.d.b.224.6		8	1.1	even	1	trivial
405.3.i.d.26.2		8	45.23	even	12
405.3.i.d.26.3		8	45.13	odd	12
405.3.i.d.296.2		8	45.43	odd	12
405.3.i.d.296.3		8	45.38	even	12
720.3.l.a.161.2		4	20.3	even	4
720.3.l.a.161.4		4	60.23	odd	4
2880.3.l.c.1601.2		4	120.83	odd	4
2880.3.l.c.1601.4		4	40.3	even	4
2880.3.l.g.1601.1		4	120.53	even	4
2880.3.l.g.1601.3		4	40.13	odd	4
3600.3.c.i.449.3		8	4.3	odd	2
3600.3.c.i.449.4		8	12.11	even	2
3600.3.c.i.449.5		8	20.19	odd	2
3600.3.c.i.449.6		8	60.59	even	2
3600.3.l.v.1601.1		4	20.7	even	4
3600.3.l.v.1601.2		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.821854 q^{2} -3.32456 q^{4} +0.837722i q^{7} -6.01972 q^{8} +O(q^{10})\)	\(q+0.821854 q^{2} -3.32456 q^{4} +0.837722i q^{7} -6.01972 q^{8} +14.3716i q^{11} +21.8114i q^{13} +0.688486i q^{14} +8.35089 q^{16} -23.5454 q^{17} +6.32456 q^{19} +11.8114i q^{22} -38.8723 q^{23} +17.9258i q^{26} -2.78505i q^{28} +0.266737i q^{29} +30.2719 q^{31} +30.9421 q^{32} -19.3509 q^{34} +9.53950i q^{37} +5.19786 q^{38} -19.3028i q^{41} -19.6228i q^{43} -47.7793i q^{44} -31.9473 q^{46} -22.1684 q^{47} +48.2982 q^{49} -72.5132i q^{52} +49.0012 q^{53} -5.04285i q^{56} +0.219219i q^{58} +73.2351i q^{59} -48.3246 q^{61} +24.8791 q^{62} -7.97367 q^{64} +77.2982i q^{67} +78.2780 q^{68} -104.044i q^{71} -47.6754i q^{73} +7.84008i q^{74} -21.0263 q^{76} -12.0394 q^{77} -68.2192 q^{79} -15.8641i q^{82} +28.2098 q^{83} -16.1271i q^{86} -86.5132i q^{88} +53.7774i q^{89} -18.2719 q^{91} +129.233 q^{92} -18.2192 q^{94} -114.921i q^{97} +39.6941 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{4}+O(q^{10}) \) 8 * q + 24 * q^4	\( 8 q + 24 q^{4} + 168 q^{16} - 112 q^{31} - 256 q^{34} + 48 q^{46} + 184 q^{49} - 336 q^{61} + 88 q^{64} - 320 q^{76} + 112 q^{79} + 208 q^{91} + 512 q^{94}+O(q^{100}) \) 8 * q + 24 * q^4 + 168 * q^16 - 112 * q^31 - 256 * q^34 + 48 * q^46 + 184 * q^49 - 336 * q^61 + 88 * q^64 - 320 * q^76 + 112 * q^79 + 208 * q^91 + 512 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more