LMFDB - Embedded newform 225.3.c.c.26.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(26,225)]

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, 0]))

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

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.c (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:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.3
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.3.c.c.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.821854i q^{2} +3.32456 q^{4} -0.837722 q^{7} +6.01972i q^{8} +O(q^{10})$	$q+0.821854i q^{2} +3.32456 q^{4} -0.837722 q^{7} +6.01972i q^{8} +14.3716i q^{11} +21.8114 q^{13} -0.688486i q^{14} +8.35089 q^{16} -23.5454i q^{17} -6.32456 q^{19} -11.8114 q^{22} +38.8723i q^{23} +17.9258i q^{26} -2.78505 q^{28} -0.266737i q^{29} +30.2719 q^{31} +30.9421i q^{32} +19.3509 q^{34} -9.53950 q^{37} -5.19786i q^{38} -19.3028i q^{41} -19.6228 q^{43} +47.7793i q^{44} -31.9473 q^{46} -22.1684i q^{47} -48.2982 q^{49} +72.5132 q^{52} -49.0012i q^{53} -5.04285i q^{56} +0.219219 q^{58} -73.2351i q^{59} -48.3246 q^{61} +24.8791i q^{62} +7.97367 q^{64} -77.2982 q^{67} -78.2780i q^{68} -104.044i q^{71} -47.6754 q^{73} -7.84008i q^{74} -21.0263 q^{76} -12.0394i q^{77} +68.2192 q^{79} +15.8641 q^{82} -28.2098i q^{83} -16.1271i q^{86} -86.5132 q^{88} -53.7774i q^{89} -18.2719 q^{91} +129.233i q^{92} +18.2192 q^{94} +114.921 q^{97} -39.6941i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4} - 16 q^{7}+O(q^{10}) $ 4 * q - 12 * q^4 - 16 * q^7	$ 4 q - 12 q^{4} - 16 q^{7} + 24 q^{13} + 84 q^{16} + 16 q^{22} + 128 q^{28} - 56 q^{31} + 128 q^{34} - 152 q^{37} + 48 q^{43} + 24 q^{46} - 92 q^{49} + 328 q^{52} - 328 q^{58} - 168 q^{61} - 44 q^{64} - 208 q^{67} - 216 q^{73} - 160 q^{76} - 56 q^{79} + 152 q^{82} - 384 q^{88} + 104 q^{91} - 256 q^{94} + 232 q^{97}+O(q^{100}) $ 4 * q - 12 * q^4 - 16 * q^7 + 24 * q^13 + 84 * q^16 + 16 * q^22 + 128 * q^28 - 56 * q^31 + 128 * q^34 - 152 * q^37 + 48 * q^43 + 24 * q^46 - 92 * q^49 + 328 * q^52 - 328 * q^58 - 168 * q^61 - 44 * q^64 - 208 * q^67 - 216 * q^73 - 160 * q^76 - 56 * q^79 + 152 * q^82 - 384 * q^88 + 104 * q^91 - 256 * q^94 + 232 * q^97

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 $

$2$	0.821854i	0.410927i	0.978665	+	0.205464i	$0.0658702\pi$
$2$	0.821854i	0.410927i	−0.978665	+	0.205464i	$0.934130\pi$
$3$	0	0
$3$	0	0
$4$	3.32456	0.831139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.837722	−0.119675	−0.0598373	−	0.998208i	$-0.519058\pi$
$7$	−0.837722	−0.119675	−0.0598373	+	0.998208i	$0.519058\pi$
$8$	6.01972i	0.752465i
$9$	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
$13$	21.8114	1.67780	0.838900	−	0.544286i	$-0.183199\pi$
$13$	21.8114	1.67780	0.838900	+	0.544286i	$0.183199\pi$
$14$	− 0.688486i	− 0.0491776i
$15$	0	0
$16$	8.35089	0.521931
$17$	− 23.5454i	− 1.38502i	−0.721407	−	0.692512i	$-0.756504\pi$
$17$	− 23.5454i	− 1.38502i	0.721407	−	0.692512i	$-0.243496\pi$
$18$	0	0
$19$	−6.32456	−0.332871	−0.166436	−	0.986052i	$-0.553226\pi$
$19$	−6.32456	−0.332871	−0.166436	+	0.986052i	$0.553226\pi$
$20$	0	0
$21$	0	0
$22$	−11.8114	−0.536881
$23$	38.8723i	1.69010i	0.534689	+	0.845049i	$0.320429\pi$
$23$	38.8723i	1.69010i	−0.534689	+	0.845049i	$0.679571\pi$
$24$	0	0
$25$	0	0
$26$	17.9258i	0.689453i
$27$	0	0
$28$	−2.78505	−0.0994662
$29$	− 0.266737i	− 0.00919784i	−0.999989	−	0.00459892i	$-0.998536\pi$
$29$	− 0.266737i	− 0.00919784i	0.999989	−	0.00459892i	$-0.00146389\pi$
$30$	0	0
$31$	30.2719	0.976512	0.488256	−	0.872700i	$-0.337633\pi$
$31$	30.2719	0.976512	0.488256	+	0.872700i	$0.337633\pi$
$32$	30.9421i	0.966940i
$33$	0	0
$34$	19.3509	0.569144
$35$	0	0
$36$	0	0
$37$	−9.53950	−0.257824	−0.128912	−	0.991656i	$-0.541149\pi$
$37$	−9.53950	−0.257824	−0.128912	+	0.991656i	$0.541149\pi$
$38$	− 5.19786i	− 0.136786i
$39$	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
$43$	−19.6228	−0.456344	−0.228172	−	0.973621i	$-0.573275\pi$
$43$	−19.6228	−0.456344	−0.228172	+	0.973621i	$0.573275\pi$
$44$	47.7793i	1.08589i
$45$	0	0
$46$	−31.9473	−0.694507
$47$	− 22.1684i	− 0.471669i	−0.971793	−	0.235834i	$-0.924218\pi$
$47$	− 22.1684i	− 0.471669i	0.971793	−	0.235834i	$-0.0757823\pi$
$48$	0	0
$49$	−48.2982	−0.985678
$50$	0	0
$51$	0	0
$52$	72.5132	1.39448
$53$	− 49.0012i	− 0.924552i	−0.886736	−	0.462276i	$-0.847033\pi$
$53$	− 49.0012i	− 0.924552i	0.886736	−	0.462276i	$-0.152967\pi$
$54$	0	0
$55$	0	0
$56$	− 5.04285i	− 0.0900509i
$57$	0	0
$58$	0.219219	0.00377964
$59$	− 73.2351i	− 1.24127i	−0.784098	−	0.620637i	$-0.786874\pi$
$59$	− 73.2351i	− 1.24127i	0.784098	−	0.620637i	$-0.213126\pi$
$60$	0	0
$61$	−48.3246	−0.792206	−0.396103	−	0.918206i	$-0.629638\pi$
$61$	−48.3246	−0.792206	−0.396103	+	0.918206i	$0.629638\pi$
$62$	24.8791i	0.401276i
$63$	0	0
$64$	7.97367	0.124589
$65$	0	0
$66$	0	0
$67$	−77.2982	−1.15370	−0.576852	−	0.816848i	$-0.695719\pi$
$67$	−77.2982	−1.15370	−0.576852	+	0.816848i	$0.695719\pi$
$68$	− 78.2780i	− 1.15115i
$69$	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
$73$	−47.6754	−0.653088	−0.326544	−	0.945182i	$-0.605884\pi$
$73$	−47.6754	−0.653088	−0.326544	+	0.945182i	$0.605884\pi$
$74$	− 7.84008i	− 0.105947i
$75$	0	0
$76$	−21.0263	−0.276662
$77$	− 12.0394i	− 0.156356i
$78$	0	0
$79$	68.2192	0.863534	0.431767	−	0.901985i	$-0.357890\pi$
$79$	68.2192	0.863534	0.431767	+	0.901985i	$0.357890\pi$
$80$	0	0
$81$	0	0
$82$	15.8641	0.193464
$83$	− 28.2098i	− 0.339877i	−0.985455	−	0.169938i	$-0.945643\pi$
$83$	− 28.2098i	− 0.339877i	0.985455	−	0.169938i	$-0.0543569\pi$
$84$	0	0
$85$	0	0
$86$	− 16.1271i	− 0.187524i
$87$	0	0
$88$	−86.5132	−0.983104
$89$	− 53.7774i	− 0.604240i	−0.953270	−	0.302120i	$-0.902306\pi$
$89$	− 53.7774i	− 0.604240i	0.953270	−	0.302120i	$-0.0976943\pi$
$90$	0	0
$91$	−18.2719	−0.200790
$92$	129.233i	1.40471i
$93$	0	0
$94$	18.2192	0.193821
$95$	0	0
$96$	0	0
$97$	114.921	1.18475	0.592376	−	0.805661i	$-0.298190\pi$
$97$	114.921	1.18475	0.592376	+	0.805661i	$0.298190\pi$
$98$	− 39.6941i	− 0.405042i
$99$	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
$103$	71.5395	0.694558	0.347279	−	0.937762i	$-0.387106\pi$
$103$	71.5395	0.694558	0.347279	+	0.937762i	$0.387106\pi$
$104$	131.298i	1.26248i
$105$	0	0
$106$	40.2719	0.379923
$107$	− 76.3675i	− 0.713715i	−0.934159	−	0.356858i	$-0.883848\pi$
$107$	− 76.3675i	− 0.713715i	0.934159	−	0.356858i	$-0.116152\pi$
$108$	0	0
$109$	−126.921	−1.16441	−0.582206	−	0.813041i	$-0.697810\pi$
$109$	−126.921	−1.16441	−0.582206	+	0.813041i	$0.697810\pi$
$110$	0	0
$111$	0	0
$112$	−6.99573	−0.0624618
$113$	− 15.0601i	− 0.133275i	−0.997777	−	0.0666377i	$-0.978773\pi$
$113$	− 15.0601i	− 0.133275i	0.997777	−	0.0666377i	$-0.0212272\pi$
$114$	0	0
$115$	0	0
$116$	− 0.886783i	− 0.00764468i
$117$	0	0
$118$	60.1886	0.510073
$119$	19.7245i	0.165752i
$120$	0	0
$121$	−85.5438	−0.706973
$122$	− 39.7157i	− 0.325539i
$123$	0	0
$124$	100.641	0.811617
$125$	0	0
$126$	0	0
$127$	−158.031	−1.24434	−0.622168	−	0.782884i	$-0.713748\pi$
$127$	−158.031	−1.24434	−0.622168	+	0.782884i	$0.713748\pi$
$128$	130.322i	1.01814i
$129$	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
$133$	5.29822	0.0398363
$134$	− 63.5279i	− 0.474089i
$135$	0	0
$136$	141.737	1.04218
$137$	− 69.2592i	− 0.505542i	−0.967526	−	0.252771i	$-0.918658\pi$
$137$	− 69.2592i	− 0.505542i	0.967526	−	0.252771i	$-0.0813419\pi$
$138$	0	0
$139$	−159.842	−1.14994	−0.574971	−	0.818174i	$-0.694987\pi$
$139$	−159.842	−1.14994	−0.574971	+	0.818174i	$0.694987\pi$
$140$	0	0
$141$	0	0
$142$	85.5089	0.602175
$143$	313.465i	2.19206i
$144$	0	0
$145$	0	0
$146$	− 39.1823i	− 0.268372i
$147$	0	0
$148$	−31.7146	−0.214288
$149$	9.81897i	0.0658991i	0.999457	+	0.0329496i	$0.0104901\pi$
$149$	9.81897i	0.0658991i	−0.999457	+	0.0329496i	$0.989510\pi$
$150$	0	0
$151$	210.649	1.39503	0.697514	−	0.716572i	$-0.254290\pi$
$151$	210.649	1.39503	0.697514	+	0.716572i	$0.254290\pi$
$152$	− 38.0720i	− 0.250474i
$153$	0	0
$154$	9.89466	0.0642511
$155$	0	0
$156$	0	0
$157$	−211.276	−1.34571	−0.672854	−	0.739775i	$-0.734932\pi$
$157$	−211.276	−1.34571	−0.672854	+	0.739775i	$0.734932\pi$
$158$	56.0663i	0.354850i
$159$	0	0
$160$	0	0
$161$	− 32.5642i	− 0.202262i
$162$	0	0
$163$	222.763	1.36664	0.683322	−	0.730117i	$-0.260535\pi$
$163$	222.763	1.36664	0.683322	+	0.730117i	$0.260535\pi$
$164$	− 64.1731i	− 0.391299i
$165$	0	0
$166$	23.1843	0.139665
$167$	− 33.3644i	− 0.199787i	−0.994998	−	0.0998933i	$-0.968150\pi$
$167$	− 33.3644i	− 0.199787i	0.994998	−	0.0998933i	$-0.0318501\pi$
$168$	0	0
$169$	306.737	1.81501
$170$	0	0
$171$	0	0
$172$	−65.2370	−0.379285
$173$	− 29.8102i	− 0.172313i	−0.996282	−	0.0861567i	$-0.972541\pi$
$173$	− 29.8102i	− 0.172313i	0.996282	−	0.0861567i	$-0.0274586\pi$
$174$	0	0
$175$	0	0
$176$	120.016i	0.681909i
$177$	0	0
$178$	44.1972	0.248299
$179$	111.841i	0.624808i	0.949949	+	0.312404i	$0.101134\pi$
$179$	111.841i	0.624808i	−0.949949	+	0.312404i	$0.898866\pi$
$180$	0	0
$181$	−49.0790	−0.271155	−0.135577	−	0.990767i	$-0.543289\pi$
$181$	−49.0790	−0.271155	−0.135577	+	0.990767i	$0.543289\pi$
$182$	− 15.0168i	− 0.0825101i
$183$	0	0
$184$	−234.000	−1.27174
$185$	0	0
$186$	0	0
$187$	338.386	1.80955
$188$	− 73.7002i	− 0.392022i
$189$	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
$193$	−89.8947	−0.465775	−0.232888	−	0.972504i	$-0.574817\pi$
$193$	−89.8947	−0.465775	−0.232888	+	0.972504i	$0.574817\pi$
$194$	94.4483i	0.486847i
$195$	0	0
$196$	−160.570	−0.819235
$197$	212.709i	1.07974i	0.841748	+	0.539870i	$0.181527\pi$
$197$	212.709i	1.07974i	−0.841748	+	0.539870i	$0.818473\pi$
$198$	0	0
$199$	96.4911	0.484880	0.242440	−	0.970166i	$-0.422052\pi$
$199$	96.4911	0.484880	0.242440	+	0.970166i	$0.422052\pi$
$200$	0	0
$201$	0	0
$202$	−14.4213	−0.0713928
$203$	0.223452i	0.00110075i
$204$	0	0
$205$	0	0
$206$	58.7951i	0.285413i
$207$	0	0
$208$	182.144	0.875695
$209$	− 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	$0.549827\pi$
$212$	− 162.907i	− 0.768431i
$213$	0	0
$214$	62.7630	0.293285
$215$	0	0
$216$	0	0
$217$	−25.3594	−0.116864
$218$	− 104.311i	− 0.478489i
$219$	0	0
$220$	0	0
$221$	− 513.558i	− 2.32379i
$222$	0	0
$223$	−102.302	−0.458756	−0.229378	−	0.973337i	$-0.573669\pi$
$223$	−102.302	−0.458756	−0.229378	+	0.973337i	$0.573669\pi$
$224$	− 25.9209i	− 0.115718i
$225$	0	0
$226$	12.3772	0.0547665
$227$	12.5296i	0.0551966i	0.999619	+	0.0275983i	$0.00878593\pi$
$227$	12.5296i	0.0551966i	−0.999619	+	0.0275983i	$0.991214\pi$
$228$	0	0
$229$	23.2982	0.101739	0.0508695	−	0.998705i	$-0.483801\pi$
$229$	23.2982	0.101739	0.0508695	+	0.998705i	$0.483801\pi$
$230$	0	0
$231$	0	0
$232$	1.60568	0.00692105
$233$	− 356.382i	− 1.52954i	−0.644306	−	0.764768i	$-0.722854\pi$
$233$	− 356.382i	− 1.52954i	0.644306	−	0.764768i	$-0.277146\pi$
$234$	0	0
$235$	0	0
$236$	− 243.474i	− 1.03167i
$237$	0	0
$238$	−16.2107	−0.0681121
$239$	− 175.524i	− 0.734408i	−0.930140	−	0.367204i	$-0.880315\pi$
$239$	− 175.524i	− 0.734408i	0.930140	−	0.367204i	$-0.119685\pi$
$240$	0	0
$241$	104.438	0.433355	0.216677	−	0.976243i	$-0.430478\pi$
$241$	104.438	0.433355	0.216677	+	0.976243i	$0.430478\pi$
$242$	− 70.3045i	− 0.290515i
$243$	0	0
$244$	−160.658	−0.658433
$245$	0	0
$246$	0	0
$247$	−137.947	−0.558491
$248$	182.228i	0.734791i
$249$	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
$253$	−558.658	−2.20813
$254$	− 129.878i	− 0.511331i
$255$	0	0
$256$	−75.2107	−0.293792
$257$	425.641i	1.65619i	0.560587	+	0.828095i	$0.310575\pi$
$257$	425.641i	1.65619i	−0.560587	+	0.828095i	$0.689425\pi$
$258$	0	0
$259$	7.99145	0.0308550
$260$	0	0
$261$	0	0
$262$	−173.592	−0.662566
$263$	− 74.5004i	− 0.283271i	−0.989919	−	0.141636i	$-0.954764\pi$
$263$	− 74.5004i	− 0.283271i	0.989919	−	0.141636i	$-0.0452362\pi$
$264$	0	0
$265$	0	0
$266$	4.35437i	0.0163698i
$267$	0	0
$268$	−256.982	−0.958889
$269$	205.067i	0.762331i	0.924507	+	0.381165i	$0.124477\pi$
$269$	205.067i	0.762331i	−0.924507	+	0.381165i	$0.875523\pi$
$270$	0	0
$271$	−233.351	−0.861073	−0.430537	−	0.902573i	$-0.641676\pi$
$271$	−233.351	−0.861073	−0.430537	+	0.902573i	$0.641676\pi$
$272$	− 196.625i	− 0.722886i
$273$	0	0
$274$	56.9210	0.207741
$275$	0	0
$276$	0	0
$277$	423.715	1.52966	0.764828	−	0.644235i	$-0.222824\pi$
$277$	423.715	1.52966	0.764828	+	0.644235i	$0.222824\pi$
$278$	− 131.367i	− 0.472543i
$279$	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
$283$	272.333	0.962308	0.481154	−	0.876636i	$-0.340218\pi$
$283$	272.333	0.962308	0.481154	+	0.876636i	$0.340218\pi$
$284$	− 345.900i	− 1.21796i
$285$	0	0
$286$	−257.623	−0.900779
$287$	16.1704i	0.0563427i
$288$	0	0
$289$	−265.386	−0.918290
$290$	0	0
$291$	0	0
$292$	−158.500	−0.542807
$293$	− 443.188i	− 1.51259i	−0.654232	−	0.756294i	$-0.727008\pi$
$293$	− 443.188i	− 1.51259i	0.654232	−	0.756294i	$-0.272992\pi$
$294$	0	0
$295$	0	0
$296$	− 57.4251i	− 0.194004i
$297$	0	0
$298$	−8.06976	−0.0270797
$299$	847.858i	2.83564i
$300$	0	0
$301$	16.4384	0.0546128
$302$	173.123i	0.573255i
$303$	0	0
$304$	−52.8157	−0.173736
$305$	0	0
$306$	0	0
$307$	390.824	1.27304	0.636522	−	0.771259i	$-0.280373\pi$
$307$	390.824	1.27304	0.636522	+	0.771259i	$0.280373\pi$
$308$	− 40.0258i	− 0.129954i
$309$	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
$313$	−130.105	−0.415672	−0.207836	−	0.978164i	$-0.566642\pi$
$313$	−130.105	−0.415672	−0.207836	+	0.978164i	$0.566642\pi$
$314$	− 173.638i	− 0.552988i
$315$	0	0
$316$	226.799	0.717717
$317$	131.677i	0.415384i	0.978194	+	0.207692i	$0.0665953\pi$
$317$	131.677i	0.415384i	−0.978194	+	0.207692i	$0.933405\pi$
$318$	0	0
$319$	3.83345	0.0120171
$320$	0	0
$321$	0	0
$322$	26.7630	0.0831149
$323$	148.914i	0.461035i
$324$	0	0
$325$	0	0
$326$	183.079i	0.561591i
$327$	0	0
$328$	116.197	0.354260
$329$	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	$0.422059\pi$
$332$	− 93.7850i	− 0.282485i
$333$	0	0
$334$	27.4207	0.0820978
$335$	0	0
$336$	0	0
$337$	128.114	0.380160	0.190080	−	0.981769i	$-0.439125\pi$
$337$	128.114	0.380160	0.190080	+	0.981769i	$0.439125\pi$
$338$	252.093i	0.745837i
$339$	0	0
$340$	0	0
$341$	435.056i	1.27583i
$342$	0	0
$343$	81.5089	0.237635
$344$	− 118.124i	− 0.343383i
$345$	0	0
$346$	24.4997	0.0708082
$347$	− 219.637i	− 0.632959i	−0.948599	−	0.316480i	$-0.897499\pi$
$347$	− 219.637i	− 0.632959i	0.948599	−	0.316480i	$-0.102501\pi$
$348$	0	0
$349$	403.465	1.15606	0.578030	−	0.816016i	$-0.303822\pi$
$349$	403.465	1.15606	0.578030	+	0.816016i	$0.303822\pi$
$350$	0	0
$351$	0	0
$352$	−444.688	−1.26332
$353$	54.8192i	0.155295i	0.996981	+	0.0776475i	$0.0247409\pi$
$353$	54.8192i	0.155295i	−0.996981	+	0.0776475i	$0.975259\pi$
$354$	0	0
$355$	0	0
$356$	− 178.786i	− 0.502207i
$357$	0	0
$358$	−91.9167	−0.256751
$359$	− 480.460i	− 1.33833i	−0.743114	−	0.669165i	$-0.766652\pi$
$359$	− 480.460i	− 1.33833i	0.743114	−	0.669165i	$-0.233348\pi$
$360$	0	0
$361$	−321.000	−0.889197
$362$	− 40.3358i	− 0.111425i
$363$	0	0
$364$	−60.7459	−0.166884
$365$	0	0
$366$	0	0
$367$	−522.364	−1.42333	−0.711667	−	0.702517i	$-0.752060\pi$
$367$	−522.364	−1.42333	−0.711667	+	0.702517i	$0.752060\pi$
$368$	324.618i	0.882114i
$369$	0	0
$370$	0	0
$371$	41.0494i	0.110645i
$372$	0	0
$373$	233.285	0.625428	0.312714	−	0.949847i	$-0.398762\pi$
$373$	233.285	0.625428	0.312714	+	0.949847i	$0.398762\pi$
$374$	278.104i	0.743593i
$375$	0	0
$376$	133.448	0.354914
$377$	− 5.81791i	− 0.0154321i
$378$	0	0
$379$	248.596	0.655927	0.327964	−	0.944690i	$-0.393638\pi$
$379$	248.596	0.655927	0.327964	+	0.944690i	$0.393638\pi$
$380$	0	0
$381$	0	0
$382$	−229.254	−0.600142
$383$	468.291i	1.22269i	0.791364	+	0.611346i	$0.209372\pi$
$383$	468.291i	1.22269i	−0.791364	+	0.611346i	$0.790628\pi$
$384$	0	0
$385$	0	0
$386$	− 73.8803i	− 0.191400i
$387$	0	0
$388$	382.061	0.984694
$389$	− 484.238i	− 1.24483i	−0.782688	−	0.622414i	$-0.786152\pi$
$389$	− 484.238i	− 1.24483i	0.782688	−	0.622414i	$-0.213848\pi$
$390$	0	0
$391$	915.263	2.34083
$392$	− 290.742i	− 0.741688i
$393$	0	0
$394$	−174.816	−0.443695
$395$	0	0
$396$	0	0
$397$	298.943	0.753005	0.376503	−	0.926416i	$-0.377127\pi$
$397$	298.943	0.753005	0.376503	+	0.926416i	$0.377127\pi$
$398$	79.3016i	0.199250i
$399$	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
$403$	660.272	1.63839
$404$	58.3370i	0.144399i
$405$	0	0
$406$	−0.183645	−0.000452327 0
$407$	− 137.098i	− 0.336851i
$408$	0	0
$409$	−184.158	−0.450264	−0.225132	−	0.974328i	$-0.572281\pi$
$409$	−184.158	−0.450264	−0.225132	+	0.974328i	$0.572281\pi$
$410$	0	0
$411$	0	0
$412$	237.837	0.577274
$413$	61.3507i	0.148549i
$414$	0	0
$415$	0	0
$416$	674.890i	1.62233i
$417$	0	0
$418$	74.7018	0.178712
$419$	429.840i	1.02587i	0.858427	+	0.512936i	$0.171442\pi$
$419$	429.840i	1.02587i	−0.858427	+	0.512936i	$0.828558\pi$
$420$	0	0
$421$	−305.035	−0.724548	−0.362274	−	0.932072i	$-0.618000\pi$
$421$	−305.035	−0.724548	−0.362274	+	0.932072i	$0.618000\pi$
$422$	− 54.0692i	− 0.128126i
$423$	0	0
$424$	294.974	0.695693
$425$	0	0
$426$	0	0
$427$	40.4826	0.0948069
$428$	− 253.888i	− 0.593196i
$429$	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
$433$	−243.886	−0.563247	−0.281624	−	0.959525i	$-0.590873\pi$
$433$	−243.886	−0.563247	−0.281624	+	0.959525i	$0.590873\pi$
$434$	− 20.8418i	− 0.0480225i
$435$	0	0
$436$	−421.956	−0.967789
$437$	− 245.850i	− 0.562585i
$438$	0	0
$439$	−259.614	−0.591376	−0.295688	−	0.955285i	$-0.595549\pi$
$439$	−259.614	−0.591376	−0.295688	+	0.955285i	$0.595549\pi$
$440$	0	0
$441$	0	0
$442$	422.070	0.954909
$443$	541.011i	1.22124i	0.791923	+	0.610622i	$0.209080\pi$
$443$	541.011i	1.22124i	−0.791923	+	0.610622i	$0.790920\pi$
$444$	0	0
$445$	0	0
$446$	− 84.0778i	− 0.188515i
$447$	0	0
$448$	−6.67972	−0.0149101
$449$	791.947i	1.76380i	0.471436	+	0.881900i	$0.343736\pi$
$449$	791.947i	1.76380i	−0.471436	+	0.881900i	$0.656264\pi$
$450$	0	0
$451$	277.412	0.615104
$452$	− 50.0682i	− 0.110770i
$453$	0	0
$454$	−10.2975	−0.0226818
$455$	0	0
$456$	0	0
$457$	−611.359	−1.33777	−0.668883	−	0.743367i	$-0.733227\pi$
$457$	−611.359	−1.33777	−0.668883	+	0.743367i	$0.733227\pi$
$458$	19.1477i	0.0418073i
$459$	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
$463$	−195.285	−0.421781	−0.210891	−	0.977510i	$-0.567636\pi$
$463$	−195.285	−0.421781	−0.210891	+	0.977510i	$0.567636\pi$
$464$	− 2.22749i	− 0.00480063i
$465$	0	0
$466$	292.894	0.628528
$467$	753.763i	1.61405i	0.590515	+	0.807027i	$0.298925\pi$
$467$	753.763i	1.61405i	−0.590515	+	0.807027i	$0.701075\pi$
$468$	0	0
$469$	64.7544	0.138069
$470$	0	0
$471$	0	0
$472$	440.855	0.934014
$473$	− 282.011i	− 0.596218i
$474$	0	0
$475$	0	0
$476$	65.5752i	0.137763i
$477$	0	0
$478$	144.255	0.301788
$479$	− 614.848i	− 1.28361i	−0.766869	−	0.641803i	$-0.778186\pi$
$479$	− 614.848i	− 1.28361i	0.766869	−	0.641803i	$-0.221814\pi$
$480$	0	0
$481$	−208.070	−0.432577
$482$	85.8332i	0.178077i
$483$	0	0
$484$	−284.395	−0.587593
$485$	0	0
$486$	0	0
$487$	−478.197	−0.981924	−0.490962	−	0.871181i	$-0.663355\pi$
$487$	−478.197	−0.981924	−0.490962	+	0.871181i	$0.663355\pi$
$488$	− 290.900i	− 0.596107i
$489$	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
$493$	−6.28043	−0.0127392
$494$	− 113.373i	− 0.229499i
$495$	0	0
$496$	252.797	0.509672
$497$	87.1599i	0.175372i
$498$	0	0
$499$	783.096	1.56933	0.784665	−	0.619919i	$-0.212835\pi$
$499$	783.096	1.56933	0.784665	+	0.619919i	$0.212835\pi$
$500$	0	0
$501$	0	0
$502$	−107.618	−0.214378
$503$	369.395i	0.734383i	0.930145	+	0.367191i	$0.119681\pi$
$503$	369.395i	0.734383i	−0.930145	+	0.367191i	$0.880319\pi$
$504$	0	0
$505$	0	0
$506$	− 459.135i	− 0.907382i
$507$	0	0
$508$	−525.381	−1.03422
$509$	− 225.635i	− 0.443291i	−0.975127	−	0.221645i	$-0.928857\pi$
$509$	− 225.635i	− 0.443291i	0.975127	−	0.221645i	$-0.0711427\pi$
$510$	0	0
$511$	39.9388	0.0781581
$512$	459.474i	0.897410i
$513$	0	0
$514$	−349.815	−0.680574
$515$	0	0
$516$	0	0
$517$	318.596	0.616241
$518$	6.56781i	0.0126792i
$519$	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
$523$	−935.517	−1.78875	−0.894376	−	0.447316i	$-0.852380\pi$
$523$	−935.517	−1.78875	−0.894376	+	0.447316i	$0.852380\pi$
$524$	702.213i	1.34010i
$525$	0	0
$526$	61.2285	0.116404
$527$	− 712.764i	− 1.35249i
$528$	0	0
$529$	−982.052	−1.85643
$530$	0	0
$531$	0	0
$532$	17.6142	0.0331095
$533$	− 421.020i	− 0.789906i
$534$	0	0
$535$	0	0
$536$	− 465.314i	− 0.868122i
$537$	0	0
$538$	−168.535	−0.313263
$539$	− 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	$0.620265\pi$
$542$	− 191.780i	− 0.353838i
$543$	0	0
$544$	728.544	1.33923
$545$	0	0
$546$	0	0
$547$	−480.833	−0.879036	−0.439518	−	0.898234i	$-0.644851\pi$
$547$	−480.833	−0.879036	−0.439518	+	0.898234i	$0.644851\pi$
$548$	− 230.256i	− 0.420175i
$549$	0	0
$550$	0	0
$551$	1.68699i	0.00306170i
$552$	0	0
$553$	−57.1488	−0.103343
$554$	348.232i	0.628577i
$555$	0	0
$556$	−531.404	−0.955762
$557$	− 751.542i	− 1.34927i	−0.738152	−	0.674634i	$-0.764301\pi$
$557$	− 751.542i	− 1.34927i	0.738152	−	0.674634i	$-0.235699\pi$
$558$	0	0
$559$	−428.000	−0.765653
$560$	0	0
$561$	0	0
$562$	330.882	0.588758
$563$	− 670.820i	− 1.19151i	−0.803166	−	0.595755i	$-0.796853\pi$
$563$	− 670.820i	− 1.19151i	0.803166	−	0.595755i	$-0.203147\pi$
$564$	0	0
$565$	0	0
$566$	223.818i	0.395438i
$567$	0	0
$568$	626.315	1.10267
$569$	− 275.325i	− 0.483875i	−0.970292	−	0.241937i	$-0.922217\pi$
$569$	− 275.325i	− 0.483875i	0.970292	−	0.241937i	$-0.0777829\pi$
$570$	0	0
$571$	−900.289	−1.57669	−0.788344	−	0.615235i	$-0.789061\pi$
$571$	−900.289	−1.57669	−0.788344	+	0.615235i	$0.789061\pi$
$572$	1042.13i	1.82191i
$573$	0	0
$574$	−13.2897	−0.0231527
$575$	0	0
$576$	0	0
$577$	596.236	1.03334	0.516669	−	0.856185i	$-0.327172\pi$
$577$	596.236	1.03334	0.516669	+	0.856185i	$0.327172\pi$
$578$	− 218.108i	− 0.377350i
$579$	0	0
$580$	0	0
$581$	23.6320i	0.0406746i
$582$	0	0
$583$	704.228	1.20794
$584$	− 286.993i	− 0.491426i
$585$	0	0
$586$	364.236	0.621564
$587$	497.431i	0.847412i	0.905800	+	0.423706i	$0.139271\pi$
$587$	497.431i	0.847412i	−0.905800	+	0.423706i	$0.860729\pi$
$588$	0	0
$589$	−191.456	−0.325053
$590$	0	0
$591$	0	0
$592$	−79.6633	−0.134566
$593$	− 898.856i	− 1.51578i	−0.652384	−	0.757889i	$-0.726231\pi$
$593$	− 898.856i	− 1.51578i	0.652384	−	0.757889i	$-0.273769\pi$
$594$	0	0
$595$	0	0
$596$	32.6437i	0.0547713i
$597$	0	0
$598$	−696.816	−1.16524
$599$	− 28.5701i	− 0.0476964i	−0.999716	−	0.0238482i	$-0.992408\pi$
$599$	− 28.5701i	− 0.0476964i	0.999716	−	0.0238482i	$-0.00759183\pi$
$600$	0	0
$601$	−20.2107	−0.0336284	−0.0168142	−	0.999859i	$-0.505352\pi$
$601$	−20.2107	−0.0336284	−0.0168142	+	0.999859i	$0.505352\pi$
$602$	13.5100i	0.0224419i
$603$	0	0
$604$	700.315	1.15946
$605$	0	0
$606$	0	0
$607$	713.626	1.17566	0.587831	−	0.808984i	$-0.299982\pi$
$607$	713.626	1.17566	0.587831	+	0.808984i	$0.299982\pi$
$608$	− 195.695i	− 0.321867i
$609$	0	0
$610$	0	0
$611$	− 483.524i	− 0.791365i
$612$	0	0
$613$	76.1530	0.124230	0.0621150	−	0.998069i	$-0.480215\pi$
$613$	76.1530	0.124230	0.0621150	+	0.998069i	$0.480215\pi$
$614$	321.201i	0.523128i
$615$	0	0
$616$	72.4740	0.117653
$617$	201.599i	0.326741i	0.986565	+	0.163371i	$0.0522367\pi$
$617$	201.599i	0.326741i	−0.986565	+	0.163371i	$0.947763\pi$
$618$	0	0
$619$	−204.263	−0.329989	−0.164995	−	0.986294i	$-0.552761\pi$
$619$	−204.263	−0.329989	−0.164995	+	0.986294i	$0.552761\pi$
$620$	0	0
$621$	0	0
$622$	−2.44699	−0.00393406
$623$	45.0505i	0.0723122i
$624$	0	0
$625$	0	0
$626$	− 106.928i	− 0.170811i
$627$	0	0
$628$	−702.399	−1.11847
$629$	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	$0.669266\pi$
$632$	410.660i	0.649779i
$633$	0	0
$634$	−108.219	−0.170693
$635$	0	0
$636$	0	0
$637$	−1053.45	−1.65377
$638$	3.15054i	0.00493815i
$639$	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
$643$	−453.693	−0.705587	−0.352794	−	0.935701i	$-0.614768\pi$
$643$	−453.693	−0.705587	−0.352794	+	0.935701i	$0.614768\pi$
$644$	− 108.261i	− 0.168108i
$645$	0	0
$646$	−122.386	−0.189452
$647$	− 983.536i	− 1.52015i	−0.649837	−	0.760074i	$-0.725163\pi$
$647$	− 983.536i	− 1.52015i	0.649837	−	0.760074i	$-0.274837\pi$
$648$	0	0
$649$	1052.51	1.62174
$650$	0	0
$651$	0	0
$652$	740.588	1.13587
$653$	544.478i	0.833811i	0.908950	+	0.416905i	$0.136885\pi$
$653$	544.478i	0.833811i	−0.908950	+	0.416905i	$0.863115\pi$
$654$	0	0
$655$	0	0
$656$	− 161.195i	− 0.245724i
$657$	0	0
$658$	−15.2626	−0.0231955
$659$	107.933i	0.163783i	0.996641	+	0.0818916i	$0.0260961\pi$
$659$	107.933i	0.163783i	−0.996641	+	0.0818916i	$0.973904\pi$
$660$	0	0
$661$	345.728	0.523038	0.261519	−	0.965198i	$-0.415777\pi$
$661$	345.728	0.523038	0.261519	+	0.965198i	$0.415777\pi$
$662$	131.893i	0.199235i
$663$	0	0
$664$	169.815	0.255745
$665$	0	0
$666$	0	0
$667$	10.3687	0.0155452
$668$	− 110.922i	− 0.166050i
$669$	0	0
$670$	0	0
$671$	− 694.503i	− 1.03503i
$672$	0	0
$673$	1204.78	1.79016	0.895082	−	0.445902i	$-0.147117\pi$
$673$	1204.78	1.79016	0.895082	+	0.445902i	$0.147117\pi$
$674$	105.291i	0.156218i
$675$	0	0
$676$	1019.76	1.50853
$677$	574.598i	0.848742i	0.905488	+	0.424371i	$0.139505\pi$
$677$	574.598i	0.848742i	−0.905488	+	0.424371i	$0.860495\pi$
$678$	0	0
$679$	−96.2719	−0.141785
$680$	0	0
$681$	0	0
$682$	−357.553	−0.524271
$683$	334.387i	0.489585i	0.969575	+	0.244792i	$0.0787198\pi$
$683$	334.387i	0.489585i	−0.969575	+	0.244792i	$0.921280\pi$
$684$	0	0
$685$	0	0
$686$	66.9884i	0.0976508i
$687$	0	0
$688$	−163.868	−0.238180
$689$	− 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	$0.273904\pi$
$692$	− 99.1057i	− 0.143216i
$693$	0	0
$694$	180.510	0.260100
$695$	0	0
$696$	0	0
$697$	−454.491	−0.652068
$698$	331.589i	0.475056i
$699$	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
$703$	60.3331	0.0858223
$704$	114.595i	0.162776i
$705$	0	0
$706$	−45.0534	−0.0638150
$707$	− 14.6998i	− 0.0207918i
$708$	0	0
$709$	−1026.35	−1.44760	−0.723801	−	0.690008i	$-0.757607\pi$
$709$	−1026.35	−1.44760	−0.723801	+	0.690008i	$0.757607\pi$
$710$	0	0
$711$	0	0
$712$	323.725	0.454669
$713$	1176.74i	1.65040i
$714$	0	0
$715$	0	0
$716$	371.820i	0.519302i
$717$	0	0
$718$	394.868	0.549956
$719$	740.080i	1.02932i	0.857395	+	0.514659i	$0.172081\pi$
$719$	740.080i	1.02932i	−0.857395	+	0.514659i	$0.827919\pi$
$720$	0	0
$721$	−59.9302	−0.0831210
$722$	− 263.815i	− 0.365395i
$723$	0	0
$724$	−163.166	−0.225367
$725$	0	0
$726$	0	0
$727$	−1063.75	−1.46321	−0.731603	−	0.681731i	$-0.761227\pi$
$727$	−1063.75	−1.46321	−0.731603	+	0.681731i	$0.761227\pi$
$728$	− 109.992i	− 0.151087i
$729$	0	0
$730$	0	0
$731$	462.026i	0.632047i
$732$	0	0
$733$	−134.749	−0.183833	−0.0919164	−	0.995767i	$-0.529299\pi$
$733$	−134.749	−0.183833	−0.0919164	+	0.995767i	$0.529299\pi$
$734$	− 429.307i	− 0.584887i
$735$	0	0
$736$	−1202.79	−1.63422
$737$	− 1110.90i	− 1.50733i
$738$	0	0
$739$	−711.429	−0.962692	−0.481346	−	0.876531i	$-0.659852\pi$
$739$	−711.429	−0.962692	−0.481346	+	0.876531i	$0.659852\pi$
$740$	0	0
$741$	0	0
$742$	−33.7367	−0.0454672
$743$	466.330i	0.627631i	0.949484	+	0.313816i	$0.101607\pi$
$743$	466.330i	0.627631i	−0.949484	+	0.313816i	$0.898393\pi$
$744$	0	0
$745$	0	0
$746$	191.726i	0.257005i
$747$	0	0
$748$	1124.98	1.50399
$749$	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	$0.451631\pi$
$752$	− 185.126i	− 0.246178i
$753$	0	0
$754$	4.78147	0.00634148
$755$	0	0
$756$	0	0
$757$	552.258	0.729535	0.364768	−	0.931099i	$-0.381148\pi$
$757$	552.258	0.729535	0.364768	+	0.931099i	$0.381148\pi$
$758$	204.310i	0.269538i
$759$	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
$763$	106.325	0.139351
$764$	927.376i	1.21384i
$765$	0	0
$766$	−384.867	−0.502437
$767$	− 1597.36i	− 2.08261i
$768$	0	0
$769$	739.684	0.961878	0.480939	−	0.876754i	$-0.340296\pi$
$769$	739.684	0.961878	0.480939	+	0.876754i	$0.340296\pi$
$770$	0	0
$771$	0	0
$772$	−298.860	−0.387124
$773$	603.479i	0.780697i	0.920667	+	0.390348i	$0.127645\pi$
$773$	603.479i	0.780697i	−0.920667	+	0.390348i	$0.872355\pi$
$774$	0	0
$775$	0	0
$776$	691.792i	0.891485i
$777$	0	0
$778$	397.973	0.511533
$779$	122.081i	0.156715i
$780$	0	0
$781$	1495.28	1.91457
$782$	752.213i	0.961909i
$783$	0	0
$784$	−403.333	−0.514455
$785$	0	0
$786$	0	0
$787$	660.483	0.839241	0.419620	−	0.907700i	$-0.362163\pi$
$787$	660.483	0.839241	0.419620	+	0.907700i	$0.362163\pi$
$788$	707.162i	0.897414i
$789$	0	0
$790$	0	0
$791$	12.6162i	0.0159497i
$792$	0	0
$793$	−1054.03	−1.32916
$794$	245.688i	0.309430i
$795$	0	0
$796$	320.790	0.403003
$797$	634.969i	0.796699i	0.917234	+	0.398349i	$0.130417\pi$
$797$	634.969i	0.796699i	−0.917234	+	0.398349i	$0.869583\pi$
$798$	0	0
$799$	−521.964	−0.653272
$800$	0	0
$801$	0	0
$802$	−384.224	−0.479083
$803$	− 685.174i	− 0.853268i
$804$	0	0
$805$	0	0
$806$	542.647i	0.673260i
$807$	0	0
$808$	−105.630	−0.130730
$809$	− 405.048i	− 0.500677i	−0.968158	−	0.250339i	$-0.919458\pi$
$809$	− 405.048i	− 0.500677i	0.968158	−	0.250339i	$-0.0805419\pi$
$810$	0	0
$811$	406.034	0.500659	0.250329	−	0.968161i	$-0.419461\pi$
$811$	406.034	0.500659	0.250329	+	0.968161i	$0.419461\pi$
$812$	0.742878i	0 0.000914874i
$813$	0	0
$814$	112.675	0.138421
$815$	0	0
$816$	0	0
$817$	124.105	0.151904
$818$	− 151.351i	− 0.185026i
$819$	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
$823$	1472.51	1.78920	0.894600	−	0.446868i	$-0.147460\pi$
$823$	1472.51	1.78920	0.894600	+	0.446868i	$0.147460\pi$
$824$	430.648i	0.522631i
$825$	0	0
$826$	−50.4213	−0.0610428
$827$	1510.47i	1.82644i	0.407466	+	0.913220i	$0.366413\pi$
$827$	1510.47i	1.82644i	−0.407466	+	0.913220i	$0.633587\pi$
$828$	0	0
$829$	712.692	0.859701	0.429850	−	0.902900i	$-0.358566\pi$
$829$	712.692	0.859701	0.429850	+	0.902900i	$0.358566\pi$
$830$	0	0
$831$	0	0
$832$	173.917	0.209035
$833$	1137.20i	1.36519i
$834$	0	0
$835$	0	0
$836$	− 302.183i	− 0.361463i
$837$	0	0
$838$	−353.266	−0.421559
$839$	− 530.579i	− 0.632394i	−0.948694	−	0.316197i	$-0.897594\pi$
$839$	− 530.579i	− 0.632394i	0.948694	−	0.316197i	$-0.102406\pi$
$840$	0	0
$841$	840.929	0.999915
$842$	− 250.694i	− 0.297737i
$843$	0	0
$844$	−218.720	−0.259147
$845$	0	0
$846$	0	0
$847$	71.6619	0.0846068
$848$	− 409.204i	− 0.482552i
$849$	0	0
$850$	0	0
$851$	− 370.822i	− 0.435748i
$852$	0	0
$853$	−215.232	−0.252324	−0.126162	−	0.992010i	$-0.540266\pi$
$853$	−215.232	−0.252324	−0.126162	+	0.992010i	$0.540266\pi$
$854$	33.2708i	0.0389587i
$855$	0	0
$856$	459.711	0.537046
$857$	− 4.97441i	− 0.00580445i	−0.999996	−	0.00290222i	$-0.999076\pi$
$857$	− 4.97441i	− 0.00580445i	0.999996	−	0.00290222i	$-0.000923808\pi$
$858$	0	0
$859$	1652.28	1.92349	0.961746	−	0.273941i	$-0.0883273\pi$
$859$	1652.28	1.92349	0.961746	+	0.273941i	$0.0883273\pi$
$860$	0	0
$861$	0	0
$862$	−105.920	−0.122877
$863$	− 379.077i	− 0.439255i	−0.975584	−	0.219627i	$-0.929516\pi$
$863$	− 379.077i	− 0.439255i	0.975584	−	0.219627i	$-0.0704841\pi$
$864$	0	0
$865$	0	0
$866$	− 200.439i	− 0.231454i
$867$	0	0
$868$	−84.3088	−0.0971300
$869$	980.421i	1.12822i
$870$	0	0
$871$	−1685.98	−1.93568
$872$	− 764.029i	− 0.876180i
$873$	0	0
$874$	202.053	0.231182
$875$	0	0
$876$	0	0
$877$	−1101.60	−1.25610	−0.628051	−	0.778173i	$-0.716147\pi$
$877$	−1101.60	−1.25610	−0.628051	+	0.778173i	$0.716147\pi$
$878$	− 213.365i	− 0.243013i
$879$	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
$883$	−978.236	−1.10786	−0.553928	−	0.832565i	$-0.686872\pi$
$883$	−978.236	−1.10786	−0.553928	+	0.832565i	$0.686872\pi$
$884$	− 1707.35i	− 1.93139i
$885$	0	0
$886$	−444.632	−0.501842
$887$	667.937i	0.753029i	0.926411	+	0.376514i	$0.122877\pi$
$887$	667.937i	0.753029i	−0.926411	+	0.376514i	$0.877123\pi$
$888$	0	0
$889$	132.386	0.148915
$890$	0	0
$891$	0	0
$892$	−340.110	−0.381290
$893$	140.205i	0.157005i
$894$	0	0
$895$	0	0
$896$	− 109.173i	− 0.121845i
$897$	0	0
$898$	−650.865	−0.724794
$899$	− 8.07464i	− 0.00898180i
$900$	0	0
$901$	−1153.75	−1.28053
$902$	227.992i	0.252763i
$903$	0	0
$904$	90.6577	0.100285
$905$	0	0
$906$	0	0
$907$	832.622	0.917996	0.458998	−	0.888437i	$-0.348209\pi$
$907$	832.622	0.917996	0.458998	+	0.888437i	$0.348209\pi$
$908$	41.6554i	0.0458760i
$909$	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
$913$	405.421	0.444053
$914$	− 502.448i	− 0.549725i
$915$	0	0
$916$	77.4562	0.0845592
$917$	− 176.944i	− 0.192959i
$918$	0	0
$919$	−1185.75	−1.29026	−0.645128	−	0.764075i	$-0.723196\pi$
$919$	−1185.75	−1.29026	−0.645128	+	0.764075i	$0.723196\pi$
$920$	0	0
$921$	0	0
$922$	482.421	0.523234
$923$	− 2269.34i	− 2.45866i
$924$	0	0
$925$	0	0
$926$	− 160.496i	− 0.173321i
$927$	0	0
$928$	8.25341	0.00889376
$929$	942.995i	1.01506i	0.861633	+	0.507532i	$0.169442\pi$
$929$	942.995i	1.01506i	−0.861633	+	0.507532i	$0.830558\pi$
$930$	0	0
$931$	305.465	0.328104
$932$	− 1184.81i	− 1.27126i
$933$	0	0
$934$	−619.483	−0.663258
$935$	0	0
$936$	0	0
$937$	531.281	0.567002	0.283501	−	0.958972i	$-0.408504\pi$
$937$	531.281	0.567002	0.283501	+	0.958972i	$0.408504\pi$
$938$	53.2187i	0.0567364i
$939$	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
$943$	750.342	0.795696
$944$	− 611.578i	− 0.647859i
$945$	0	0
$946$	231.772	0.245002
$947$	782.246i	0.826026i	0.910725	+	0.413013i	$0.135524\pi$
$947$	782.246i	0.826026i	−0.910725	+	0.413013i	$0.864476\pi$
$948$	0	0
$949$	−1039.87	−1.09575
$950$	0	0
$951$	0	0
$952$	−118.736	−0.124723
$953$	− 426.708i	− 0.447752i	−0.974618	−	0.223876i	$-0.928129\pi$
$953$	− 426.708i	− 0.447752i	0.974618	−	0.223876i	$-0.0718711\pi$
$954$	0	0
$955$	0	0
$956$	− 583.538i	− 0.610395i
$957$	0	0
$958$	505.315	0.527469
$959$	58.0200i	0.0605005i
$960$	0	0
$961$	−44.6128	−0.0464234
$962$	− 171.003i	− 0.177758i
$963$	0	0
$964$	347.211	0.360178
$965$	0	0
$966$	0	0
$967$	1210.91	1.25223	0.626116	−	0.779730i	$-0.284644\pi$
$967$	1210.91	1.25223	0.626116	+	0.779730i	$0.284644\pi$
$968$	− 514.949i	− 0.531973i
$969$	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
$973$	133.903	0.137619
$974$	− 393.008i	− 0.403499i
$975$	0	0
$976$	−403.553	−0.413476
$977$	− 909.475i	− 0.930886i	−0.885078	−	0.465443i	$-0.845895\pi$
$977$	− 909.475i	− 0.930886i	0.885078	−	0.465443i	$-0.154105\pi$
$978$	0	0
$979$	772.868	0.789447
$980$	0	0
$981$	0	0
$982$	−507.268	−0.516566
$983$	− 1560.88i	− 1.58788i	−0.607999	−	0.793938i	$-0.708028\pi$
$983$	− 1560.88i	− 1.58788i	0.607999	−	0.793938i	$-0.291972\pi$
$984$	0	0
$985$	0	0
$986$	− 5.16160i	− 0.00523489i
$987$	0	0
$988$	−458.614	−0.464184
$989$	− 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	$0.825831\pi$
$992$	936.675i	0.944229i
$993$	0	0
$994$	−71.6327	−0.0720651
$995$	0	0
$996$	0	0
$997$	−16.5744	−0.0166243	−0.00831213	−	0.999965i	$-0.502646\pi$
$997$	−16.5744	−0.0166243	−0.00831213	+	0.999965i	$0.502646\pi$
$998$	643.591i	0.644881i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.c.c.26.3		4
3.2	odd	2	inner	225.3.c.c.26.2		4
4.3	odd	2		3600.3.l.v.1601.1		4
5.2	odd	4		225.3.d.b.224.3		8
5.3	odd	4		225.3.d.b.224.6		8
5.4	even	2		45.3.c.a.26.2	✓	4
12.11	even	2		3600.3.l.v.1601.2		4
15.2	even	4		225.3.d.b.224.5		8
15.8	even	4		225.3.d.b.224.4		8
15.14	odd	2		45.3.c.a.26.3	yes	4
20.3	even	4		3600.3.c.i.449.3		8
20.7	even	4		3600.3.c.i.449.5		8
20.19	odd	2		720.3.l.a.161.2		4
40.19	odd	2		2880.3.l.c.1601.4		4
40.29	even	2		2880.3.l.g.1601.3		4
45.4	even	6		405.3.i.d.26.3		8
45.14	odd	6		405.3.i.d.26.2		8
45.29	odd	6		405.3.i.d.296.3		8
45.34	even	6		405.3.i.d.296.2		8
60.23	odd	4		3600.3.c.i.449.4		8
60.47	odd	4		3600.3.c.i.449.6		8
60.59	even	2		720.3.l.a.161.4		4
120.29	odd	2		2880.3.l.g.1601.1		4
120.59	even	2		2880.3.l.c.1601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.2	✓	4	5.4	even	2
45.3.c.a.26.3	yes	4	15.14	odd	2
225.3.c.c.26.2		4	3.2	odd	2	inner
225.3.c.c.26.3		4	1.1	even	1	trivial
225.3.d.b.224.3		8	5.2	odd	4
225.3.d.b.224.4		8	15.8	even	4
225.3.d.b.224.5		8	15.2	even	4
225.3.d.b.224.6		8	5.3	odd	4
405.3.i.d.26.2		8	45.14	odd	6
405.3.i.d.26.3		8	45.4	even	6
405.3.i.d.296.2		8	45.34	even	6
405.3.i.d.296.3		8	45.29	odd	6
720.3.l.a.161.2		4	20.19	odd	2
720.3.l.a.161.4		4	60.59	even	2
2880.3.l.c.1601.2		4	120.59	even	2
2880.3.l.c.1601.4		4	40.19	odd	2
2880.3.l.g.1601.1		4	120.29	odd	2
2880.3.l.g.1601.3		4	40.29	even	2
3600.3.c.i.449.3		8	20.3	even	4
3600.3.c.i.449.4		8	60.23	odd	4
3600.3.c.i.449.5		8	20.7	even	4
3600.3.c.i.449.6		8	60.47	odd	4
3600.3.l.v.1601.1		4	4.3	odd	2
3600.3.l.v.1601.2		4	12.11	even	2

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.821854i q^{2} +3.32456 q^{4} -0.837722 q^{7} +6.01972i q^{8} +O(q^{10})\)	\(q+0.821854i q^{2} +3.32456 q^{4} -0.837722 q^{7} +6.01972i q^{8} +14.3716i q^{11} +21.8114 q^{13} -0.688486i q^{14} +8.35089 q^{16} -23.5454i q^{17} -6.32456 q^{19} -11.8114 q^{22} +38.8723i q^{23} +17.9258i q^{26} -2.78505 q^{28} -0.266737i q^{29} +30.2719 q^{31} +30.9421i q^{32} +19.3509 q^{34} -9.53950 q^{37} -5.19786i q^{38} -19.3028i q^{41} -19.6228 q^{43} +47.7793i q^{44} -31.9473 q^{46} -22.1684i q^{47} -48.2982 q^{49} +72.5132 q^{52} -49.0012i q^{53} -5.04285i q^{56} +0.219219 q^{58} -73.2351i q^{59} -48.3246 q^{61} +24.8791i q^{62} +7.97367 q^{64} -77.2982 q^{67} -78.2780i q^{68} -104.044i q^{71} -47.6754 q^{73} -7.84008i q^{74} -21.0263 q^{76} -12.0394i q^{77} +68.2192 q^{79} +15.8641 q^{82} -28.2098i q^{83} -16.1271i q^{86} -86.5132 q^{88} -53.7774i q^{89} -18.2719 q^{91} +129.233i q^{92} +18.2192 q^{94} +114.921 q^{97} -39.6941i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4} - 16 q^{7}+O(q^{10}) \) 4 * q - 12 * q^4 - 16 * q^7	\( 4 q - 12 q^{4} - 16 q^{7} + 24 q^{13} + 84 q^{16} + 16 q^{22} + 128 q^{28} - 56 q^{31} + 128 q^{34} - 152 q^{37} + 48 q^{43} + 24 q^{46} - 92 q^{49} + 328 q^{52} - 328 q^{58} - 168 q^{61} - 44 q^{64} - 208 q^{67} - 216 q^{73} - 160 q^{76} - 56 q^{79} + 152 q^{82} - 384 q^{88} + 104 q^{91} - 256 q^{94} + 232 q^{97}+O(q^{100}) \) 4 * q - 12 * q^4 - 16 * q^7 + 24 * q^13 + 84 * q^16 + 16 * q^22 + 128 * q^28 - 56 * q^31 + 128 * q^34 - 152 * q^37 + 48 * q^43 + 24 * q^46 - 92 * q^49 + 328 * q^52 - 328 * q^58 - 168 * q^61 - 44 * q^64 - 208 * q^67 - 216 * q^73 - 160 * q^76 - 56 * q^79 + 152 * q^82 - 384 * q^88 + 104 * q^91 - 256 * q^94 + 232 * q^97

Introduction

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