LMFDB - Embedded newform 900.4.d.h.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(649,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("900.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.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:	$53.1017190052$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	900.649
Dual form			900.4.d.h.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.0000i q^{7} +O(q^{10})$	$q+28.0000i q^{7} +24.0000 q^{11} -70.0000i q^{13} +102.000i q^{17} -20.0000 q^{19} +72.0000i q^{23} +306.000 q^{29} -136.000 q^{31} +214.000i q^{37} +150.000 q^{41} -292.000i q^{43} -72.0000i q^{47} -441.000 q^{49} +414.000i q^{53} -744.000 q^{59} -418.000 q^{61} -188.000i q^{67} -480.000 q^{71} +434.000i q^{73} +672.000i q^{77} -1352.00 q^{79} +612.000i q^{83} -30.0000 q^{89} +1960.00 q^{91} +286.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 48 q^{11} - 40 q^{19} + 612 q^{29} - 272 q^{31} + 300 q^{41} - 882 q^{49} - 1488 q^{59} - 836 q^{61} - 960 q^{71} - 2704 q^{79} - 60 q^{89} + 3920 q^{91}+O(q^{100}) $ 2 * q + 48 * q^11 - 40 * q^19 + 612 * q^29 - 272 * q^31 + 300 * q^41 - 882 * q^49 - 1488 * q^59 - 836 * q^61 - 960 * q^71 - 2704 * q^79 - 60 * q^89 + 3920 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	28.0000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	28.0000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	24.0000	0.657843	0.328921	−	0.944357i	$-0.393315\pi$
$11$	24.0000	0.657843	0.328921	+	0.944357i	$0.393315\pi$
$12$	0	0
$13$	− 70.0000i	− 1.49342i	−0.665148	−	0.746712i	$-0.731631\pi$
$13$	− 70.0000i	− 1.49342i	0.665148	−	0.746712i	$-0.268369\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	102.000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	102.000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	72.0000i	0.652741i	0.945242	+	0.326370i	$0.105826\pi$
$23$	72.0000i	0.652741i	−0.945242	+	0.326370i	$0.894174\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	306.000	1.95941	0.979703	−	0.200455i	$-0.0642419\pi$
$29$	306.000	1.95941	0.979703	+	0.200455i	$0.0642419\pi$
$30$	0	0
$31$	−136.000	−0.787946	−0.393973	−	0.919122i	$-0.628900\pi$
$31$	−136.000	−0.787946	−0.393973	+	0.919122i	$0.628900\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	214.000i	0.950848i	0.879757	+	0.475424i	$0.157705\pi$
$37$	214.000i	0.950848i	−0.879757	+	0.475424i	$0.842295\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	150.000	0.571367	0.285684	−	0.958324i	$-0.407779\pi$
$41$	150.000	0.571367	0.285684	+	0.958324i	$0.407779\pi$
$42$	0	0
$43$	− 292.000i	− 1.03557i	−0.855510	−	0.517786i	$-0.826756\pi$
$43$	− 292.000i	− 1.03557i	0.855510	−	0.517786i	$-0.173244\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 72.0000i	− 0.223453i	−0.993739	−	0.111726i	$-0.964362\pi$
$47$	− 72.0000i	− 0.223453i	0.993739	−	0.111726i	$-0.0356380\pi$
$48$	0	0
$49$	−441.000	−1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	414.000i	1.07297i	0.843911	+	0.536484i	$0.180248\pi$
$53$	414.000i	1.07297i	−0.843911	+	0.536484i	$0.819752\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−744.000	−1.64170	−0.820852	−	0.571141i	$-0.806501\pi$
$59$	−744.000	−1.64170	−0.820852	+	0.571141i	$0.806501\pi$
$60$	0	0
$61$	−418.000	−0.877367	−0.438684	−	0.898642i	$-0.644555\pi$
$61$	−418.000	−0.877367	−0.438684	+	0.898642i	$0.644555\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 188.000i	− 0.342804i	−0.985201	−	0.171402i	$-0.945170\pi$
$67$	− 188.000i	− 0.342804i	0.985201	−	0.171402i	$-0.0548297\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−480.000	−0.802331	−0.401166	−	0.916006i	$-0.631395\pi$
$71$	−480.000	−0.802331	−0.401166	+	0.916006i	$0.631395\pi$
$72$	0	0
$73$	434.000i	0.695834i	0.937525	+	0.347917i	$0.113111\pi$
$73$	434.000i	0.695834i	−0.937525	+	0.347917i	$0.886889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	672.000i	0.994565i
$78$	0	0
$79$	−1352.00	−1.92547	−0.962733	−	0.270452i	$-0.912827\pi$
$79$	−1352.00	−1.92547	−0.962733	+	0.270452i	$0.912827\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	612.000i	0.809346i	0.914461	+	0.404673i	$0.132615\pi$
$83$	612.000i	0.809346i	−0.914461	+	0.404673i	$0.867385\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−30.0000	−0.0357303	−0.0178651	−	0.999840i	$-0.505687\pi$
$89$	−30.0000	−0.0357303	−0.0178651	+	0.999840i	$0.505687\pi$
$90$	0	0
$91$	1960.00	2.25784
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	286.000i	0.299370i	0.988734	+	0.149685i	$0.0478260\pi$
$97$	286.000i	0.299370i	−0.988734	+	0.149685i	$0.952174\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1542.00	1.51916	0.759578	−	0.650416i	$-0.225406\pi$
$101$	1542.00	1.51916	0.759578	+	0.650416i	$0.225406\pi$
$102$	0	0
$103$	1172.00i	1.12117i	0.828097	+	0.560585i	$0.189424\pi$
$103$	1172.00i	1.12117i	−0.828097	+	0.560585i	$0.810576\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1956.00i	1.76723i	0.468214	+	0.883615i	$0.344898\pi$
$107$	1956.00i	1.76723i	−0.468214	+	0.883615i	$0.655102\pi$
$108$	0	0
$109$	1858.00	1.63270	0.816349	−	0.577559i	$-0.195995\pi$
$109$	1858.00	1.63270	0.816349	+	0.577559i	$0.195995\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 174.000i	− 0.144854i	−0.997374	−	0.0724272i	$-0.976926\pi$
$113$	− 174.000i	− 0.144854i	0.997374	−	0.0724272i	$-0.0230745\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2856.00	−2.20008
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2068.00i	1.44492i	0.691411	+	0.722462i	$0.256990\pi$
$127$	2068.00i	1.44492i	−0.691411	+	0.722462i	$0.743010\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−312.000	−0.208088	−0.104044	−	0.994573i	$-0.533178\pi$
$131$	−312.000	−0.208088	−0.104044	+	0.994573i	$0.533178\pi$
$132$	0	0
$133$	− 560.000i	− 0.365099i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2646.00i	1.65010i	0.565063	+	0.825048i	$0.308852\pi$
$137$	2646.00i	1.65010i	−0.565063	+	0.825048i	$0.691148\pi$
$138$	0	0
$139$	1276.00	0.778625	0.389313	−	0.921106i	$-0.372713\pi$
$139$	1276.00	0.778625	0.389313	+	0.921106i	$0.372713\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1680.00i	− 0.982438i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3198.00	−1.75832	−0.879162	−	0.476522i	$-0.841897\pi$
$149$	−3198.00	−1.75832	−0.879162	+	0.476522i	$0.841897\pi$
$150$	0	0
$151$	−760.000	−0.409589	−0.204794	−	0.978805i	$-0.565653\pi$
$151$	−760.000	−0.409589	−0.204794	+	0.978805i	$0.565653\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	166.000i	0.0843837i	0.999110	+	0.0421919i	$0.0134341\pi$
$157$	166.000i	0.0843837i	−0.999110	+	0.0421919i	$0.986566\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2016.00	−0.986851
$162$	0	0
$163$	3020.00i	1.45119i	0.688120	+	0.725597i	$0.258436\pi$
$163$	3020.00i	1.45119i	−0.688120	+	0.725597i	$0.741564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 984.000i	− 0.455953i	−0.973667	−	0.227977i	$-0.926789\pi$
$167$	− 984.000i	− 0.455953i	0.973667	−	0.227977i	$-0.0732110\pi$
$168$	0	0
$169$	−2703.00	−1.23031
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1962.00i	− 0.862243i	−0.902294	−	0.431122i	$-0.858118\pi$
$173$	− 1962.00i	− 0.862243i	0.902294	−	0.431122i	$-0.141882\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	576.000	0.240515	0.120258	−	0.992743i	$-0.461628\pi$
$179$	576.000	0.240515	0.120258	+	0.992743i	$0.461628\pi$
$180$	0	0
$181$	−1210.00	−0.496898	−0.248449	−	0.968645i	$-0.579921\pi$
$181$	−1210.00	−0.496898	−0.248449	+	0.968645i	$0.579921\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2448.00i	0.957302i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3384.00	−1.28198	−0.640989	−	0.767550i	$-0.721475\pi$
$191$	−3384.00	−1.28198	−0.640989	+	0.767550i	$0.721475\pi$
$192$	0	0
$193$	− 2038.00i	− 0.760096i	−0.924967	−	0.380048i	$-0.875908\pi$
$193$	− 2038.00i	− 0.760096i	0.924967	−	0.380048i	$-0.124092\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4098.00i	1.48208i	0.671459	+	0.741042i	$0.265668\pi$
$197$	4098.00i	1.48208i	−0.671459	+	0.741042i	$0.734332\pi$
$198$	0	0
$199$	2248.00	0.800786	0.400393	−	0.916343i	$-0.368874\pi$
$199$	2248.00	0.800786	0.400393	+	0.916343i	$0.368874\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8568.00i	2.96234i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−480.000	−0.158863
$210$	0	0
$211$	3260.00	1.06364	0.531819	−	0.846858i	$-0.321509\pi$
$211$	3260.00	1.06364	0.531819	+	0.846858i	$0.321509\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3808.00i	− 1.19126i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7140.00	2.17325
$222$	0	0
$223$	− 2980.00i	− 0.894868i	−0.894317	−	0.447434i	$-0.852338\pi$
$223$	− 2980.00i	− 0.894868i	0.894317	−	0.447434i	$-0.147662\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3180.00i	− 0.929797i	−0.885364	−	0.464899i	$-0.846091\pi$
$227$	− 3180.00i	− 0.929797i	0.885364	−	0.464899i	$-0.153909\pi$
$228$	0	0
$229$	−3374.00	−0.973625	−0.486813	−	0.873506i	$-0.661841\pi$
$229$	−3374.00	−0.973625	−0.486813	+	0.873506i	$0.661841\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1950.00i	− 0.548278i	−0.961690	−	0.274139i	$-0.911607\pi$
$233$	− 1950.00i	− 0.548278i	0.961690	−	0.274139i	$-0.0883928\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2232.00	0.604084	0.302042	−	0.953295i	$-0.402332\pi$
$239$	2232.00	0.604084	0.302042	+	0.953295i	$0.402332\pi$
$240$	0	0
$241$	−1822.00	−0.486993	−0.243497	−	0.969902i	$-0.578294\pi$
$241$	−1822.00	−0.486993	−0.243497	+	0.969902i	$0.578294\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1400.00i	0.360647i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1488.00	−0.374190	−0.187095	−	0.982342i	$-0.559907\pi$
$251$	−1488.00	−0.374190	−0.187095	+	0.982342i	$0.559907\pi$
$252$	0	0
$253$	1728.00i	0.429401i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2994.00i	− 0.726695i	−0.931654	−	0.363347i	$-0.881634\pi$
$257$	− 2994.00i	− 0.726695i	0.931654	−	0.363347i	$-0.118366\pi$
$258$	0	0
$259$	−5992.00	−1.43755
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2472.00i	0.579582i	0.957090	+	0.289791i	$0.0935858\pi$
$263$	2472.00i	0.579582i	−0.957090	+	0.289791i	$0.906414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3954.00	0.896207	0.448103	−	0.893982i	$-0.352100\pi$
$269$	3954.00	0.896207	0.448103	+	0.893982i	$0.352100\pi$
$270$	0	0
$271$	−2176.00	−0.487759	−0.243879	−	0.969806i	$-0.578420\pi$
$271$	−2176.00	−0.487759	−0.243879	+	0.969806i	$0.578420\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1034.00i	− 0.224285i	−0.993692	−	0.112143i	$-0.964229\pi$
$277$	− 1034.00i	− 0.224285i	0.993692	−	0.112143i	$-0.0357714\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6654.00	1.41261	0.706307	−	0.707906i	$-0.250360\pi$
$281$	6654.00	1.41261	0.706307	+	0.707906i	$0.250360\pi$
$282$	0	0
$283$	− 1756.00i	− 0.368846i	−0.982847	−	0.184423i	$-0.940958\pi$
$283$	− 1756.00i	− 0.368846i	0.982847	−	0.184423i	$-0.0590416\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4200.00i	0.863826i
$288$	0	0
$289$	−5491.00	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3234.00i	− 0.644820i	−0.946600	−	0.322410i	$-0.895507\pi$
$293$	− 3234.00i	− 0.644820i	0.946600	−	0.322410i	$-0.104493\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5040.00	0.974818
$300$	0	0
$301$	8176.00	1.56564
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2036.00i	− 0.378504i	−0.981929	−	0.189252i	$-0.939394\pi$
$307$	− 2036.00i	− 0.378504i	0.981929	−	0.189252i	$-0.0606063\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	96.0000	0.0175037	0.00875187	−	0.999962i	$-0.497214\pi$
$311$	96.0000	0.0175037	0.00875187	+	0.999962i	$0.497214\pi$
$312$	0	0
$313$	1202.00i	0.217064i	0.994093	+	0.108532i	$0.0346150\pi$
$313$	1202.00i	0.217064i	−0.994093	+	0.108532i	$0.965385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3798.00i	− 0.672924i	−0.941697	−	0.336462i	$-0.890770\pi$
$317$	− 3798.00i	− 0.672924i	0.941697	−	0.336462i	$-0.109230\pi$
$318$	0	0
$319$	7344.00	1.28898
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2040.00i	− 0.351420i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2016.00	0.337829
$330$	0	0
$331$	−5668.00	−0.941213	−0.470606	−	0.882343i	$-0.655965\pi$
$331$	−5668.00	−0.941213	−0.470606	+	0.882343i	$0.655965\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	454.000i	0.0733856i	0.999327	+	0.0366928i	$0.0116823\pi$
$337$	454.000i	0.0733856i	−0.999327	+	0.0366928i	$0.988318\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3264.00	−0.518345
$342$	0	0
$343$	− 2744.00i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5604.00i	− 0.866970i	−0.901161	−	0.433485i	$-0.857284\pi$
$347$	− 5604.00i	− 0.866970i	0.901161	−	0.433485i	$-0.142716\pi$
$348$	0	0
$349$	11266.0	1.72795	0.863976	−	0.503533i	$-0.167967\pi$
$349$	11266.0	1.72795	0.863976	+	0.503533i	$0.167967\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6426.00i	0.968899i	0.874819	+	0.484450i	$0.160980\pi$
$353$	6426.00i	0.968899i	−0.874819	+	0.484450i	$0.839020\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6936.00	1.01969	0.509844	−	0.860267i	$-0.329703\pi$
$359$	6936.00	1.01969	0.509844	+	0.860267i	$0.329703\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	388.000i	0.0551865i	0.999619	+	0.0275932i	$0.00878431\pi$
$367$	388.000i	0.0551865i	−0.999619	+	0.0275932i	$0.991216\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11592.0	−1.62217
$372$	0	0
$373$	− 8062.00i	− 1.11913i	−0.828787	−	0.559564i	$-0.810969\pi$
$373$	− 8062.00i	− 1.11913i	0.828787	−	0.559564i	$-0.189031\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 21420.0i	− 2.92622i
$378$	0	0
$379$	3388.00	0.459182	0.229591	−	0.973287i	$-0.426261\pi$
$379$	3388.00	0.459182	0.229591	+	0.973287i	$0.426261\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 6984.00i	− 0.931764i	−0.884847	−	0.465882i	$-0.845737\pi$
$383$	− 6984.00i	− 0.931764i	0.884847	−	0.465882i	$-0.154263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2526.00	−0.329237	−0.164619	−	0.986357i	$-0.552639\pi$
$389$	−2526.00	−0.329237	−0.164619	+	0.986357i	$0.552639\pi$
$390$	0	0
$391$	−7344.00	−0.949877
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6146.00i	− 0.776975i	−0.921454	−	0.388487i	$-0.872998\pi$
$397$	− 6146.00i	− 0.776975i	0.921454	−	0.388487i	$-0.127002\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9786.00	−1.21868	−0.609339	−	0.792910i	$-0.708565\pi$
$401$	−9786.00	−1.21868	−0.609339	+	0.792910i	$0.708565\pi$
$402$	0	0
$403$	9520.00i	1.17674i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5136.00i	0.625509i
$408$	0	0
$409$	886.000	0.107115	0.0535573	−	0.998565i	$-0.482944\pi$
$409$	886.000	0.107115	0.0535573	+	0.998565i	$0.482944\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 20832.0i	− 2.48202i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11352.0	−1.32358	−0.661792	−	0.749688i	$-0.730204\pi$
$419$	−11352.0	−1.32358	−0.661792	+	0.749688i	$0.730204\pi$
$420$	0	0
$421$	10190.0	1.17964	0.589822	−	0.807533i	$-0.299198\pi$
$421$	10190.0	1.17964	0.589822	+	0.807533i	$0.299198\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11704.0i	− 1.32645i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2448.00	−0.273587	−0.136794	−	0.990600i	$-0.543680\pi$
$431$	−2448.00	−0.273587	−0.136794	+	0.990600i	$0.543680\pi$
$432$	0	0
$433$	− 7078.00i	− 0.785559i	−0.919633	−	0.392779i	$-0.871514\pi$
$433$	− 7078.00i	− 0.785559i	0.919633	−	0.392779i	$-0.128486\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1440.00i	− 0.157631i
$438$	0	0
$439$	18088.0	1.96650	0.983250	−	0.182264i	$-0.0583426\pi$
$439$	18088.0	1.96650	0.983250	+	0.182264i	$0.0583426\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3852.00i	− 0.413124i	−0.978433	−	0.206562i	$-0.933772\pi$
$443$	− 3852.00i	− 0.413124i	0.978433	−	0.206562i	$-0.0662276\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6522.00	0.685506	0.342753	−	0.939426i	$-0.388641\pi$
$449$	6522.00	0.685506	0.342753	+	0.939426i	$0.388641\pi$
$450$	0	0
$451$	3600.00	0.375870
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2090.00i	− 0.213930i	−0.994263	−	0.106965i	$-0.965887\pi$
$457$	− 2090.00i	− 0.213930i	0.994263	−	0.106965i	$-0.0341133\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9894.00	0.999587	0.499793	−	0.866145i	$-0.333409\pi$
$461$	9894.00	0.999587	0.499793	+	0.866145i	$0.333409\pi$
$462$	0	0
$463$	3044.00i	0.305544i	0.988261	+	0.152772i	$0.0488199\pi$
$463$	3044.00i	0.305544i	−0.988261	+	0.152772i	$0.951180\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10236.0i	1.01427i	0.861866	+	0.507137i	$0.169296\pi$
$467$	10236.0i	1.01427i	−0.861866	+	0.507137i	$0.830704\pi$
$468$	0	0
$469$	5264.00	0.518271
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 7008.00i	− 0.681244i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11496.0	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	11496.0	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	14980.0	1.42002
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15316.0i	1.42512i	0.701610	+	0.712561i	$0.252465\pi$
$487$	15316.0i	1.42512i	−0.701610	+	0.712561i	$0.747535\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11616.0	1.06766	0.533832	−	0.845591i	$-0.320752\pi$
$491$	11616.0	1.06766	0.533832	+	0.845591i	$0.320752\pi$
$492$	0	0
$493$	31212.0i	2.85135i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13440.0i	− 1.21301i
$498$	0	0
$499$	−14996.0	−1.34532	−0.672658	−	0.739953i	$-0.734848\pi$
$499$	−14996.0	−1.34532	−0.672658	+	0.739953i	$0.734848\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21648.0i	1.91896i	0.281778	+	0.959480i	$0.409076\pi$
$503$	21648.0i	1.91896i	−0.281778	+	0.959480i	$0.590924\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3378.00	0.294160	0.147080	−	0.989125i	$-0.453013\pi$
$509$	3378.00	0.294160	0.147080	+	0.989125i	$0.453013\pi$
$510$	0	0
$511$	−12152.0	−1.05200
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1728.00i	− 0.146997i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16158.0	1.35872	0.679362	−	0.733804i	$-0.262257\pi$
$521$	16158.0	1.35872	0.679362	+	0.733804i	$0.262257\pi$
$522$	0	0
$523$	− 76.0000i	− 0.00635420i	−0.999995	−	0.00317710i	$-0.998989\pi$
$523$	− 76.0000i	− 0.00635420i	0.999995	−	0.00317710i	$-0.00101130\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 13872.0i	− 1.14663i
$528$	0	0
$529$	6983.00	0.573929
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 10500.0i	− 0.853294i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10584.0	−0.845798
$540$	0	0
$541$	9278.00	0.737324	0.368662	−	0.929563i	$-0.379816\pi$
$541$	9278.00	0.737324	0.368662	+	0.929563i	$0.379816\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14564.0i	− 1.13841i	−0.822195	−	0.569206i	$-0.807251\pi$
$547$	− 14564.0i	− 1.13841i	0.822195	−	0.569206i	$-0.192749\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6120.00	−0.473177
$552$	0	0
$553$	− 37856.0i	− 2.91103i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2154.00i	0.163856i	0.996638	+	0.0819281i	$0.0261078\pi$
$557$	2154.00i	0.163856i	−0.996638	+	0.0819281i	$0.973892\pi$
$558$	0	0
$559$	−20440.0	−1.54655
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8700.00i	0.651263i	0.945497	+	0.325632i	$0.105577\pi$
$563$	8700.00i	0.651263i	−0.945497	+	0.325632i	$0.894423\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4194.00	0.309001	0.154501	−	0.987993i	$-0.450623\pi$
$569$	4194.00	0.309001	0.154501	+	0.987993i	$0.450623\pi$
$570$	0	0
$571$	−8020.00	−0.587787	−0.293894	−	0.955838i	$-0.594951\pi$
$571$	−8020.00	−0.587787	−0.293894	+	0.955838i	$0.594951\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2686.00i	0.193795i	0.995294	+	0.0968974i	$0.0308919\pi$
$577$	2686.00i	0.193795i	−0.995294	+	0.0968974i	$0.969108\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17136.0	−1.22362
$582$	0	0
$583$	9936.00i	0.705844i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3012.00i	0.211786i	0.994378	+	0.105893i	$0.0337701\pi$
$587$	3012.00i	0.211786i	−0.994378	+	0.105893i	$0.966230\pi$
$588$	0	0
$589$	2720.00	0.190281
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15522.0i	1.07489i	0.843298	+	0.537447i	$0.180611\pi$
$593$	15522.0i	1.07489i	−0.843298	+	0.537447i	$0.819389\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19224.0	1.31130	0.655652	−	0.755063i	$-0.272394\pi$
$599$	19224.0	1.31130	0.655652	+	0.755063i	$0.272394\pi$
$600$	0	0
$601$	−6502.00	−0.441301	−0.220651	−	0.975353i	$-0.570818\pi$
$601$	−6502.00	−0.441301	−0.220651	+	0.975353i	$0.570818\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 29396.0i	− 1.96565i	−0.184552	−	0.982823i	$-0.559083\pi$
$607$	− 29396.0i	− 1.96565i	0.184552	−	0.982823i	$-0.440917\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5040.00	−0.333710
$612$	0	0
$613$	− 10006.0i	− 0.659280i	−0.944107	−	0.329640i	$-0.893073\pi$
$613$	− 10006.0i	− 0.659280i	0.944107	−	0.329640i	$-0.106927\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23118.0i	1.50842i	0.656633	+	0.754210i	$0.271980\pi$
$617$	23118.0i	1.50842i	−0.656633	+	0.754210i	$0.728020\pi$
$618$	0	0
$619$	−14036.0	−0.911397	−0.455698	−	0.890134i	$-0.650610\pi$
$619$	−14036.0	−0.911397	−0.455698	+	0.890134i	$0.650610\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 840.000i	− 0.0540191i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−21828.0	−1.38369
$630$	0	0
$631$	−4288.00	−0.270527	−0.135264	−	0.990810i	$-0.543188\pi$
$631$	−4288.00	−0.270527	−0.135264	+	0.990810i	$0.543188\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	30870.0i	1.92012i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1314.00	−0.0809671	−0.0404835	−	0.999180i	$-0.512890\pi$
$641$	−1314.00	−0.0809671	−0.0404835	+	0.999180i	$0.512890\pi$
$642$	0	0
$643$	− 628.000i	− 0.0385162i	−0.999815	−	0.0192581i	$-0.993870\pi$
$643$	− 628.000i	− 0.0385162i	0.999815	−	0.0192581i	$-0.00613042\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10944.0i	− 0.664997i	−0.943104	−	0.332498i	$-0.892108\pi$
$647$	− 10944.0i	− 0.664997i	0.943104	−	0.332498i	$-0.107892\pi$
$648$	0	0
$649$	−17856.0	−1.07998
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1098.00i	− 0.0658010i	−0.999459	−	0.0329005i	$-0.989526\pi$
$653$	− 1098.00i	− 0.0658010i	0.999459	−	0.0329005i	$-0.0104744\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	312.000	0.0184428	0.00922139	−	0.999957i	$-0.497065\pi$
$659$	312.000	0.0184428	0.00922139	+	0.999957i	$0.497065\pi$
$660$	0	0
$661$	8678.00	0.510643	0.255322	−	0.966856i	$-0.417819\pi$
$661$	8678.00	0.510643	0.255322	+	0.966856i	$0.417819\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22032.0i	1.27898i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10032.0	−0.577170
$672$	0	0
$673$	− 14470.0i	− 0.828793i	−0.910097	−	0.414396i	$-0.863993\pi$
$673$	− 14470.0i	− 0.828793i	0.910097	−	0.414396i	$-0.136007\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 11838.0i	− 0.672040i	−0.941855	−	0.336020i	$-0.890919\pi$
$677$	− 11838.0i	− 0.672040i	0.941855	−	0.336020i	$-0.109081\pi$
$678$	0	0
$679$	−8008.00	−0.452605
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25548.0i	1.43128i	0.698467	+	0.715642i	$0.253866\pi$
$683$	25548.0i	1.43128i	−0.698467	+	0.715642i	$0.746134\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	28980.0	1.60239
$690$	0	0
$691$	−18412.0	−1.01364	−0.506820	−	0.862052i	$-0.669179\pi$
$691$	−18412.0	−1.01364	−0.506820	+	0.862052i	$0.669179\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15300.0i	0.831462i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8814.00	0.474893	0.237447	−	0.971401i	$-0.423690\pi$
$701$	8814.00	0.474893	0.237447	+	0.971401i	$0.423690\pi$
$702$	0	0
$703$	− 4280.00i	− 0.229621i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43176.0i	2.29675i
$708$	0	0
$709$	17314.0	0.917124	0.458562	−	0.888662i	$-0.348365\pi$
$709$	17314.0	0.917124	0.458562	+	0.888662i	$0.348365\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 9792.00i	− 0.514324i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−768.000	−0.0398353	−0.0199176	−	0.999802i	$-0.506340\pi$
$719$	−768.000	−0.0398353	−0.0199176	+	0.999802i	$0.506340\pi$
$720$	0	0
$721$	−32816.0	−1.69505
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18196.0i	0.928270i	0.885764	+	0.464135i	$0.153635\pi$
$727$	18196.0i	0.928270i	−0.885764	+	0.464135i	$0.846365\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29784.0	1.50698
$732$	0	0
$733$	− 18142.0i	− 0.914175i	−0.889422	−	0.457087i	$-0.848893\pi$
$733$	− 18142.0i	− 0.914175i	0.889422	−	0.457087i	$-0.151107\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4512.00i	− 0.225511i
$738$	0	0
$739$	13660.0	0.679961	0.339981	−	0.940432i	$-0.389580\pi$
$739$	13660.0	0.679961	0.339981	+	0.940432i	$0.389580\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12768.0i	0.630434i	0.949020	+	0.315217i	$0.102077\pi$
$743$	12768.0i	0.630434i	−0.949020	+	0.315217i	$0.897923\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−54768.0	−2.67180
$750$	0	0
$751$	22952.0	1.11522	0.557610	−	0.830103i	$-0.311718\pi$
$751$	22952.0	1.11522	0.557610	+	0.830103i	$0.311718\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 15818.0i	− 0.759465i	−0.925096	−	0.379732i	$-0.876016\pi$
$757$	− 15818.0i	− 0.759465i	0.925096	−	0.379732i	$-0.123984\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18558.0	0.884004	0.442002	−	0.897014i	$-0.354268\pi$
$761$	18558.0	0.884004	0.442002	+	0.897014i	$0.354268\pi$
$762$	0	0
$763$	52024.0i	2.46841i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	52080.0i	2.45176i
$768$	0	0
$769$	−14978.0	−0.702367	−0.351184	−	0.936307i	$-0.614221\pi$
$769$	−14978.0	−0.702367	−0.351184	+	0.936307i	$0.614221\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 8946.00i	− 0.416255i	−0.978102	−	0.208128i	$-0.933263\pi$
$773$	− 8946.00i	− 0.416255i	0.978102	−	0.208128i	$-0.0667369\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3000.00	−0.137980
$780$	0	0
$781$	−11520.0	−0.527808
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18436.0i	0.835035i	0.908669	+	0.417517i	$0.137100\pi$
$787$	18436.0i	0.835035i	−0.908669	+	0.417517i	$0.862900\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4872.00	0.218999
$792$	0	0
$793$	29260.0i	1.31028i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16314.0i	0.725058i	0.931972	+	0.362529i	$0.118087\pi$
$797$	16314.0i	0.725058i	−0.931972	+	0.362529i	$0.881913\pi$
$798$	0	0
$799$	7344.00	0.325172
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10416.0i	0.457749i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−25446.0	−1.10585	−0.552926	−	0.833231i	$-0.686489\pi$
$809$	−25446.0	−1.10585	−0.552926	+	0.833231i	$0.686489\pi$
$810$	0	0
$811$	42740.0	1.85056	0.925280	−	0.379284i	$-0.123830\pi$
$811$	42740.0	1.85056	0.925280	+	0.379284i	$0.123830\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5840.00i	0.250080i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29946.0	−1.27299	−0.636494	−	0.771282i	$-0.719616\pi$
$821$	−29946.0	−1.27299	−0.636494	+	0.771282i	$0.719616\pi$
$822$	0	0
$823$	− 32596.0i	− 1.38059i	−0.723528	−	0.690295i	$-0.757481\pi$
$823$	− 32596.0i	− 1.38059i	0.723528	−	0.690295i	$-0.242519\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3804.00i	0.159949i	0.996797	+	0.0799746i	$0.0254839\pi$
$827$	3804.00i	0.159949i	−0.996797	+	0.0799746i	$0.974516\pi$
$828$	0	0
$829$	−3278.00	−0.137334	−0.0686669	−	0.997640i	$-0.521875\pi$
$829$	−3278.00	−0.137334	−0.0686669	+	0.997640i	$0.521875\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 44982.0i	− 1.87099i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5784.00	−0.238005	−0.119002	−	0.992894i	$-0.537970\pi$
$839$	−5784.00	−0.238005	−0.119002	+	0.992894i	$0.537970\pi$
$840$	0	0
$841$	69247.0	2.83927
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 21140.0i	− 0.857590i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15408.0	−0.620657
$852$	0	0
$853$	17306.0i	0.694661i	0.937743	+	0.347331i	$0.112912\pi$
$853$	17306.0i	0.694661i	−0.937743	+	0.347331i	$0.887088\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31134.0i	1.24098i	0.784216	+	0.620488i	$0.213066\pi$
$857$	31134.0i	1.24098i	−0.784216	+	0.620488i	$0.786934\pi$
$858$	0	0
$859$	10780.0	0.428183	0.214091	−	0.976814i	$-0.431321\pi$
$859$	10780.0	0.428183	0.214091	+	0.976814i	$0.431321\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3456.00i	− 0.136319i	−0.997674	−	0.0681597i	$-0.978287\pi$
$863$	− 3456.00i	− 0.136319i	0.997674	−	0.0681597i	$-0.0217127\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−32448.0	−1.26665
$870$	0	0
$871$	−13160.0	−0.511951
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2618.00i	− 0.100802i	−0.998729	−	0.0504011i	$-0.983950\pi$
$877$	− 2618.00i	− 0.100802i	0.998729	−	0.0504011i	$-0.0160500\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26550.0	1.01531	0.507657	−	0.861559i	$-0.330512\pi$
$881$	26550.0	1.01531	0.507657	+	0.861559i	$0.330512\pi$
$882$	0	0
$883$	27596.0i	1.05173i	0.850567	+	0.525866i	$0.176259\pi$
$883$	27596.0i	1.05173i	−0.850567	+	0.525866i	$0.823741\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37848.0i	− 1.43271i	−0.697737	−	0.716354i	$-0.745810\pi$
$887$	− 37848.0i	− 1.43271i	0.697737	−	0.716354i	$-0.254190\pi$
$888$	0	0
$889$	−57904.0	−2.18452
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1440.00i	0.0539617i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41616.0	−1.54391
$900$	0	0
$901$	−42228.0	−1.56140
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4804.00i	0.175870i	0.996126	+	0.0879351i	$0.0280268\pi$
$907$	4804.00i	0.175870i	−0.996126	+	0.0879351i	$0.971973\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28608.0	−1.04042	−0.520211	−	0.854037i	$-0.674147\pi$
$911$	−28608.0	−1.04042	−0.520211	+	0.854037i	$0.674147\pi$
$912$	0	0
$913$	14688.0i	0.532423i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8736.00i	− 0.314600i
$918$	0	0
$919$	40768.0	1.46334	0.731672	−	0.681657i	$-0.238741\pi$
$919$	40768.0	1.46334	0.731672	+	0.681657i	$0.238741\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	33600.0i	1.19822i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27642.0	0.976216	0.488108	−	0.872783i	$-0.337687\pi$
$929$	27642.0	0.976216	0.488108	+	0.872783i	$0.337687\pi$
$930$	0	0
$931$	8820.00	0.310487
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28106.0i	− 0.979918i	−0.871746	−	0.489959i	$-0.837012\pi$
$937$	− 28106.0i	− 0.979918i	0.871746	−	0.489959i	$-0.162988\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14730.0	−0.510291	−0.255146	−	0.966903i	$-0.582123\pi$
$941$	−14730.0	−0.510291	−0.255146	+	0.966903i	$0.582123\pi$
$942$	0	0
$943$	10800.0i	0.372955i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9564.00i	− 0.328182i	−0.986445	−	0.164091i	$-0.947531\pi$
$947$	− 9564.00i	− 0.328182i	0.986445	−	0.164091i	$-0.0524691\pi$
$948$	0	0
$949$	30380.0	1.03917
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53898.0i	1.83203i	0.401141	+	0.916017i	$0.368614\pi$
$953$	53898.0i	1.83203i	−0.401141	+	0.916017i	$0.631386\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−74088.0	−2.49471
$960$	0	0
$961$	−11295.0	−0.379141
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 15140.0i	− 0.503485i	−0.967794	−	0.251742i	$-0.918996\pi$
$967$	− 15140.0i	− 0.503485i	0.967794	−	0.251742i	$-0.0810035\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23808.0	−0.786854	−0.393427	−	0.919356i	$-0.628711\pi$
$971$	−23808.0	−0.786854	−0.393427	+	0.919356i	$0.628711\pi$
$972$	0	0
$973$	35728.0i	1.17717i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23094.0i	0.756236i	0.925757	+	0.378118i	$0.123429\pi$
$977$	23094.0i	0.756236i	−0.925757	+	0.378118i	$0.876571\pi$
$978$	0	0
$979$	−720.000	−0.0235049
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 7584.00i	− 0.246075i	−0.992402	−	0.123038i	$-0.960736\pi$
$983$	− 7584.00i	− 0.246075i	0.992402	−	0.123038i	$-0.0392636\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21024.0	0.675960
$990$	0	0
$991$	−26752.0	−0.857523	−0.428761	−	0.903418i	$-0.641050\pi$
$991$	−26752.0	−0.857523	−0.428761	+	0.903418i	$0.641050\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 7778.00i	− 0.247073i	−0.992340	−	0.123536i	$-0.960576\pi$
$997$	− 7778.00i	− 0.247073i	0.992340	−	0.123536i	$-0.0394236\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.d.h.649.2		2
3.2	odd	2		300.4.d.b.49.1		2
5.2	odd	4		180.4.a.d.1.1		1
5.3	odd	4		900.4.a.q.1.1		1
5.4	even	2	inner	900.4.d.h.649.1		2
12.11	even	2		1200.4.f.n.49.2		2
15.2	even	4		60.4.a.a.1.1	✓	1
15.8	even	4		300.4.a.i.1.1		1
15.14	odd	2		300.4.d.b.49.2		2
20.7	even	4		720.4.a.bb.1.1		1
45.2	even	12		1620.4.i.l.1081.1		2
45.7	odd	12		1620.4.i.f.1081.1		2
45.22	odd	12		1620.4.i.f.541.1		2
45.32	even	12		1620.4.i.l.541.1		2
60.23	odd	4		1200.4.a.a.1.1		1
60.47	odd	4		240.4.a.i.1.1		1
60.59	even	2		1200.4.f.n.49.1		2
120.77	even	4		960.4.a.bc.1.1		1
120.107	odd	4		960.4.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.a.a.1.1	✓	1	15.2	even	4
180.4.a.d.1.1		1	5.2	odd	4
240.4.a.i.1.1		1	60.47	odd	4
300.4.a.i.1.1		1	15.8	even	4
300.4.d.b.49.1		2	3.2	odd	2
300.4.d.b.49.2		2	15.14	odd	2
720.4.a.bb.1.1		1	20.7	even	4
900.4.a.q.1.1		1	5.3	odd	4
900.4.d.h.649.1		2	5.4	even	2	inner
900.4.d.h.649.2		2	1.1	even	1	trivial
960.4.a.r.1.1		1	120.107	odd	4
960.4.a.bc.1.1		1	120.77	even	4
1200.4.a.a.1.1		1	60.23	odd	4
1200.4.f.n.49.1		2	60.59	even	2
1200.4.f.n.49.2		2	12.11	even	2
1620.4.i.f.541.1		2	45.22	odd	12
1620.4.i.f.1081.1		2	45.7	odd	12
1620.4.i.l.541.1		2	45.32	even	12
1620.4.i.l.1081.1		2	45.2	even	12

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+28.0000i q^{7} +O(q^{10})\)	\(q+28.0000i q^{7} +24.0000 q^{11} -70.0000i q^{13} +102.000i q^{17} -20.0000 q^{19} +72.0000i q^{23} +306.000 q^{29} -136.000 q^{31} +214.000i q^{37} +150.000 q^{41} -292.000i q^{43} -72.0000i q^{47} -441.000 q^{49} +414.000i q^{53} -744.000 q^{59} -418.000 q^{61} -188.000i q^{67} -480.000 q^{71} +434.000i q^{73} +672.000i q^{77} -1352.00 q^{79} +612.000i q^{83} -30.0000 q^{89} +1960.00 q^{91} +286.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 48 q^{11} - 40 q^{19} + 612 q^{29} - 272 q^{31} + 300 q^{41} - 882 q^{49} - 1488 q^{59} - 836 q^{61} - 960 q^{71} - 2704 q^{79} - 60 q^{89} + 3920 q^{91}+O(q^{100}) \) 2 * q + 48 * q^11 - 40 * q^19 + 612 * q^29 - 272 * q^31 + 300 * q^41 - 882 * q^49 - 1488 * q^59 - 836 * q^61 - 960 * q^71 - 2704 * q^79 - 60 * q^89 + 3920 * q^91

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