LMFDB - Embedded newform 900.4.d.d.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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 180)
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.d.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{7} +O(q^{10})$	$q+2.00000i q^{7} -30.0000 q^{11} +4.00000i q^{13} -90.0000i q^{17} +28.0000 q^{19} +120.000i q^{23} +210.000 q^{29} -4.00000 q^{31} +200.000i q^{37} -240.000 q^{41} +136.000i q^{43} +120.000i q^{47} +339.000 q^{49} -30.0000i q^{53} -450.000 q^{59} -166.000 q^{61} +908.000i q^{67} +1020.00 q^{71} +250.000i q^{73} -60.0000i q^{77} +916.000 q^{79} -1140.00i q^{83} -420.000 q^{89} -8.00000 q^{91} +1538.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 60 q^{11} + 56 q^{19} + 420 q^{29} - 8 q^{31} - 480 q^{41} + 678 q^{49} - 900 q^{59} - 332 q^{61} + 2040 q^{71} + 1832 q^{79} - 840 q^{89} - 16 q^{91}+O(q^{100}) $ 2 * q - 60 * q^11 + 56 * q^19 + 420 * q^29 - 8 * q^31 - 480 * q^41 + 678 * q^49 - 900 * q^59 - 332 * q^61 + 2040 * q^71 + 1832 * q^79 - 840 * q^89 - 16 * 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
$7$	2.00000i	0.107990i	0.998541	+	0.0539949i	$0.0171955\pi$
$7$	2.00000i	0.107990i	−0.998541	+	0.0539949i	$0.982805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−30.0000	−0.822304	−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000	−0.822304	−0.411152	+	0.911567i	$0.634873\pi$
$12$	0	0
$13$	4.00000i	0.0853385i	0.999089	+	0.0426692i	$0.0135862\pi$
$13$	4.00000i	0.0853385i	−0.999089	+	0.0426692i	$0.986414\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 90.0000i	− 1.28401i	−0.766700	−	0.642006i	$-0.778102\pi$
$17$	− 90.0000i	− 1.28401i	0.766700	−	0.642006i	$-0.221898\pi$
$18$	0	0
$19$	28.0000	0.338086	0.169043	−	0.985609i	$-0.445932\pi$
$19$	28.0000	0.338086	0.169043	+	0.985609i	$0.445932\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	120.000i	1.08790i	0.839117	+	0.543951i	$0.183072\pi$
$23$	120.000i	1.08790i	−0.839117	+	0.543951i	$0.816928\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	210.000	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$29$	210.000	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$30$	0	0
$31$	−4.00000	−0.0231749	−0.0115874	−	0.999933i	$-0.503688\pi$
$31$	−4.00000	−0.0231749	−0.0115874	+	0.999933i	$0.503688\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	200.000i	0.888643i	0.895867	+	0.444322i	$0.146555\pi$
$37$	200.000i	0.888643i	−0.895867	+	0.444322i	$0.853445\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−240.000	−0.914188	−0.457094	−	0.889418i	$-0.651110\pi$
$41$	−240.000	−0.914188	−0.457094	+	0.889418i	$0.651110\pi$
$42$	0	0
$43$	136.000i	0.482321i	0.970485	+	0.241161i	$0.0775280\pi$
$43$	136.000i	0.482321i	−0.970485	+	0.241161i	$0.922472\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	120.000i	0.372421i	0.982510	+	0.186211i	$0.0596207\pi$
$47$	120.000i	0.372421i	−0.982510	+	0.186211i	$0.940379\pi$
$48$	0	0
$49$	339.000	0.988338
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 30.0000i	− 0.0777513i	−0.999244	−	0.0388756i	$-0.987622\pi$
$53$	− 30.0000i	− 0.0777513i	0.999244	−	0.0388756i	$-0.0123776\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−450.000	−0.992966	−0.496483	−	0.868046i	$-0.665376\pi$
$59$	−450.000	−0.992966	−0.496483	+	0.868046i	$0.665376\pi$
$60$	0	0
$61$	−166.000	−0.348428	−0.174214	−	0.984708i	$-0.555738\pi$
$61$	−166.000	−0.348428	−0.174214	+	0.984708i	$0.555738\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	908.000i	1.65567i	0.560972	+	0.827835i	$0.310428\pi$
$67$	908.000i	1.65567i	−0.560972	+	0.827835i	$0.689572\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1020.00	1.70495	0.852477	−	0.522765i	$-0.175099\pi$
$71$	1020.00	1.70495	0.852477	+	0.522765i	$0.175099\pi$
$72$	0	0
$73$	250.000i	0.400826i	0.979712	+	0.200413i	$0.0642284\pi$
$73$	250.000i	0.400826i	−0.979712	+	0.200413i	$0.935772\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 60.0000i	− 0.0888004i
$78$	0	0
$79$	916.000	1.30453	0.652266	−	0.757990i	$-0.273818\pi$
$79$	916.000	1.30453	0.652266	+	0.757990i	$0.273818\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1140.00i	− 1.50761i	−0.657101	−	0.753803i	$-0.728217\pi$
$83$	− 1140.00i	− 1.50761i	0.657101	−	0.753803i	$-0.271783\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−420.000	−0.500224	−0.250112	−	0.968217i	$-0.580467\pi$
$89$	−420.000	−0.500224	−0.250112	+	0.968217i	$0.580467\pi$
$90$	0	0
$91$	−8.00000	−0.00921569
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1538.00i	1.60990i	0.593343	+	0.804950i	$0.297808\pi$
$97$	1538.00i	1.60990i	−0.593343	+	0.804950i	$0.702192\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−450.000	−0.443333	−0.221667	−	0.975122i	$-0.571150\pi$
$101$	−450.000	−0.443333	−0.221667	+	0.975122i	$0.571150\pi$
$102$	0	0
$103$	1150.00i	1.10012i	0.835124	+	0.550062i	$0.185396\pi$
$103$	1150.00i	1.10012i	−0.835124	+	0.550062i	$0.814604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1620.00i	1.46366i	0.681489	+	0.731829i	$0.261333\pi$
$107$	1620.00i	1.46366i	−0.681489	+	0.731829i	$0.738667\pi$
$108$	0	0
$109$	1702.00	1.49561	0.747807	−	0.663916i	$-0.231107\pi$
$109$	1702.00	1.49561	0.747807	+	0.663916i	$0.231107\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1350.00i	1.12387i	0.827181	+	0.561935i	$0.189943\pi$
$113$	1350.00i	1.12387i	−0.827181	+	0.561935i	$0.810057\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	180.000	0.138660
$120$	0	0
$121$	−431.000	−0.323817
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2450.00i	1.71183i	0.517117	+	0.855915i	$0.327005\pi$
$127$	2450.00i	1.71183i	−0.517117	+	0.855915i	$0.672995\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−690.000	−0.460195	−0.230098	−	0.973168i	$-0.573905\pi$
$131$	−690.000	−0.460195	−0.230098	+	0.973168i	$0.573905\pi$
$132$	0	0
$133$	56.0000i	0.0365099i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2070.00i	− 1.29089i	−0.763806	−	0.645445i	$-0.776672\pi$
$137$	− 2070.00i	− 1.29089i	0.763806	−	0.645445i	$-0.223328\pi$
$138$	0	0
$139$	1924.00	1.17404	0.587020	−	0.809572i	$-0.300301\pi$
$139$	1924.00	1.17404	0.587020	+	0.809572i	$0.300301\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 120.000i	− 0.0701742i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2910.00	1.59998	0.799988	−	0.600016i	$-0.204839\pi$
$149$	2910.00	1.59998	0.799988	+	0.600016i	$0.204839\pi$
$150$	0	0
$151$	176.000	0.0948522	0.0474261	−	0.998875i	$-0.484898\pi$
$151$	176.000	0.0948522	0.0474261	+	0.998875i	$0.484898\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2348.00i	1.19357i	0.802400	+	0.596786i	$0.203556\pi$
$157$	2348.00i	1.19357i	−0.802400	+	0.596786i	$0.796444\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−240.000	−0.117482
$162$	0	0
$163$	1996.00i	0.959134i	0.877505	+	0.479567i	$0.159206\pi$
$163$	1996.00i	0.959134i	−0.877505	+	0.479567i	$0.840794\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3120.00i	− 1.44571i	−0.691002	−	0.722853i	$-0.742830\pi$
$167$	− 3120.00i	− 1.44571i	0.691002	−	0.722853i	$-0.257170\pi$
$168$	0	0
$169$	2181.00	0.992717
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1770.00i	0.777865i	0.921266	+	0.388932i	$0.127156\pi$
$173$	1770.00i	0.777865i	−0.921266	+	0.388932i	$0.872844\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2130.00	−0.889406	−0.444703	−	0.895678i	$-0.646691\pi$
$179$	−2130.00	−0.889406	−0.444703	+	0.895678i	$0.646691\pi$
$180$	0	0
$181$	−1654.00	−0.679231	−0.339616	−	0.940564i	$-0.610297\pi$
$181$	−1654.00	−0.679231	−0.339616	+	0.940564i	$0.610297\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2700.00i	1.05585i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1740.00	−0.659173	−0.329586	−	0.944125i	$-0.606909\pi$
$191$	−1740.00	−0.659173	−0.329586	+	0.944125i	$0.606909\pi$
$192$	0	0
$193$	− 86.0000i	− 0.0320747i	−0.999871	−	0.0160373i	$-0.994895\pi$
$193$	− 86.0000i	− 0.0320747i	0.999871	−	0.0160373i	$-0.00510506\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2490.00i	0.900534i	0.892894	+	0.450267i	$0.148671\pi$
$197$	2490.00i	0.900534i	−0.892894	+	0.450267i	$0.851329\pi$
$198$	0	0
$199$	832.000	0.296376	0.148188	−	0.988959i	$-0.452656\pi$
$199$	832.000	0.296376	0.148188	+	0.988959i	$0.452656\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	420.000i	0.145213i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−840.000	−0.278010
$210$	0	0
$211$	2084.00	0.679945	0.339973	−	0.940435i	$-0.389582\pi$
$211$	2084.00	0.679945	0.339973	+	0.940435i	$0.389582\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.00250265i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	360.000	0.109576
$222$	0	0
$223$	1174.00i	0.352542i	0.984342	+	0.176271i	$0.0564035\pi$
$223$	1174.00i	0.352542i	−0.984342	+	0.176271i	$0.943597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3120.00i	0.912254i	0.889915	+	0.456127i	$0.150764\pi$
$227$	3120.00i	0.912254i	−0.889915	+	0.456127i	$0.849236\pi$
$228$	0	0
$229$	58.0000	0.0167369	0.00836845	−	0.999965i	$-0.497336\pi$
$229$	58.0000	0.0167369	0.00836845	+	0.999965i	$0.497336\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5910.00i	− 1.66170i	−0.556494	−	0.830852i	$-0.687854\pi$
$233$	− 5910.00i	− 1.66170i	0.556494	−	0.830852i	$-0.312146\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3300.00	0.893135	0.446567	−	0.894750i	$-0.352646\pi$
$239$	3300.00	0.893135	0.446567	+	0.894750i	$0.352646\pi$
$240$	0	0
$241$	−2986.00	−0.798113	−0.399056	−	0.916926i	$-0.630662\pi$
$241$	−2986.00	−0.798113	−0.399056	+	0.916926i	$0.630662\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	112.000i	0.0288518i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6630.00	1.66726	0.833629	−	0.552324i	$-0.186259\pi$
$251$	6630.00	1.66726	0.833629	+	0.552324i	$0.186259\pi$
$252$	0	0
$253$	− 3600.00i	− 0.894585i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1530.00i	0.371357i	0.982611	+	0.185679i	$0.0594483\pi$
$257$	1530.00i	0.371357i	−0.982611	+	0.185679i	$0.940552\pi$
$258$	0	0
$259$	−400.000	−0.0959644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2640.00i	0.618971i	0.950904	+	0.309486i	$0.100157\pi$
$263$	2640.00i	0.618971i	−0.950904	+	0.309486i	$0.899843\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7350.00	−1.66594	−0.832969	−	0.553319i	$-0.813361\pi$
$269$	−7350.00	−1.66594	−0.832969	+	0.553319i	$0.813361\pi$
$270$	0	0
$271$	3512.00	0.787228	0.393614	−	0.919276i	$-0.371225\pi$
$271$	3512.00	0.787228	0.393614	+	0.919276i	$0.371225\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5368.00i	− 1.16437i	−0.813055	−	0.582187i	$-0.802197\pi$
$277$	− 5368.00i	− 1.16437i	0.813055	−	0.582187i	$-0.197803\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3060.00	0.649624	0.324812	−	0.945779i	$-0.394699\pi$
$281$	3060.00	0.649624	0.324812	+	0.945779i	$0.394699\pi$
$282$	0	0
$283$	5044.00i	1.05949i	0.848158	+	0.529743i	$0.177712\pi$
$283$	5044.00i	1.05949i	−0.848158	+	0.529743i	$0.822288\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 480.000i	− 0.0987230i
$288$	0	0
$289$	−3187.00	−0.648687
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2010.00i	− 0.400769i	−0.979717	−	0.200385i	$-0.935781\pi$
$293$	− 2010.00i	− 0.400769i	0.979717	−	0.200385i	$-0.0642192\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−480.000	−0.0928399
$300$	0	0
$301$	−272.000	−0.0520858
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2752.00i	− 0.511612i	−0.966728	−	0.255806i	$-0.917659\pi$
$307$	− 2752.00i	− 0.511612i	0.966728	−	0.255806i	$-0.0823409\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9540.00	−1.73943	−0.869717	−	0.493551i	$-0.835699\pi$
$311$	−9540.00	−1.73943	−0.869717	+	0.493551i	$0.835699\pi$
$312$	0	0
$313$	− 9254.00i	− 1.67114i	−0.549384	−	0.835570i	$-0.685137\pi$
$313$	− 9254.00i	− 1.67114i	0.549384	−	0.835570i	$-0.314863\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 150.000i	− 0.0265768i	−0.999912	−	0.0132884i	$-0.995770\pi$
$317$	− 150.000i	− 0.0265768i	0.999912	−	0.0132884i	$-0.00422995\pi$
$318$	0	0
$319$	−6300.00	−1.10574
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2520.00i	− 0.434107i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−240.000	−0.0402177
$330$	0	0
$331$	1892.00	0.314180	0.157090	−	0.987584i	$-0.449789\pi$
$331$	1892.00	0.314180	0.157090	+	0.987584i	$0.449789\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7378.00i	− 1.19260i	−0.802763	−	0.596299i	$-0.796637\pi$
$337$	− 7378.00i	− 1.19260i	0.802763	−	0.596299i	$-0.203363\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	120.000	0.0190568
$342$	0	0
$343$	1364.00i	0.214720i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6720.00i	− 1.03962i	−0.854282	−	0.519811i	$-0.826003\pi$
$347$	− 6720.00i	− 1.03962i	0.854282	−	0.519811i	$-0.173997\pi$
$348$	0	0
$349$	−5186.00	−0.795416	−0.397708	−	0.917512i	$-0.630194\pi$
$349$	−5186.00	−0.795416	−0.397708	+	0.917512i	$0.630194\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3330.00i	0.502091i	0.967975	+	0.251045i	$0.0807743\pi$
$353$	3330.00i	0.502091i	−0.967975	+	0.251045i	$0.919226\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9000.00	1.32312	0.661562	−	0.749890i	$-0.269894\pi$
$359$	9000.00	1.32312	0.661562	+	0.749890i	$0.269894\pi$
$360$	0	0
$361$	−6075.00	−0.885698
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8758.00i	− 1.24568i	−0.782350	−	0.622839i	$-0.785979\pi$
$367$	− 8758.00i	− 1.24568i	0.782350	−	0.622839i	$-0.214021\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	60.0000	0.00839635
$372$	0	0
$373$	− 4724.00i	− 0.655763i	−0.944719	−	0.327881i	$-0.893665\pi$
$373$	− 4724.00i	− 0.655763i	0.944719	−	0.327881i	$-0.106335\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	840.000i	0.114754i
$378$	0	0
$379$	−7292.00	−0.988298	−0.494149	−	0.869377i	$-0.664520\pi$
$379$	−7292.00	−0.988298	−0.494149	+	0.869377i	$0.664520\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14520.0i	1.93717i	0.248676	+	0.968587i	$0.420004\pi$
$383$	14520.0i	1.93717i	−0.248676	+	0.968587i	$0.579996\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7110.00	0.926713	0.463356	−	0.886172i	$-0.346645\pi$
$389$	7110.00	0.926713	0.463356	+	0.886172i	$0.346645\pi$
$390$	0	0
$391$	10800.0	1.39688
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 11488.0i	− 1.45231i	−0.687532	−	0.726154i	$-0.741306\pi$
$397$	− 11488.0i	− 1.45231i	0.687532	−	0.726154i	$-0.258694\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−780.000	−0.0971355	−0.0485678	−	0.998820i	$-0.515466\pi$
$401$	−780.000	−0.0971355	−0.0485678	+	0.998820i	$0.515466\pi$
$402$	0	0
$403$	− 16.0000i	− 0.00197771i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6000.00i	− 0.730735i
$408$	0	0
$409$	−5402.00	−0.653085	−0.326542	−	0.945183i	$-0.605884\pi$
$409$	−5402.00	−0.653085	−0.326542	+	0.945183i	$0.605884\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 900.000i	− 0.107230i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2190.00	−0.255342	−0.127671	−	0.991817i	$-0.540750\pi$
$419$	−2190.00	−0.255342	−0.127671	+	0.991817i	$0.540750\pi$
$420$	0	0
$421$	−7162.00	−0.829108	−0.414554	−	0.910025i	$-0.636062\pi$
$421$	−7162.00	−0.829108	−0.414554	+	0.910025i	$0.636062\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 332.000i	− 0.0376267i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9360.00	−1.04607	−0.523034	−	0.852312i	$-0.675200\pi$
$431$	−9360.00	−1.04607	−0.523034	+	0.852312i	$0.675200\pi$
$432$	0	0
$433$	− 12806.0i	− 1.42129i	−0.703552	−	0.710643i	$-0.748404\pi$
$433$	− 12806.0i	− 1.42129i	0.703552	−	0.710643i	$-0.251596\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3360.00i	0.367805i
$438$	0	0
$439$	−11288.0	−1.22721	−0.613607	−	0.789612i	$-0.710282\pi$
$439$	−11288.0	−1.22721	−0.613607	+	0.789612i	$0.710282\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8520.00i	− 0.913764i	−0.889527	−	0.456882i	$-0.848966\pi$
$443$	− 8520.00i	− 0.913764i	0.889527	−	0.456882i	$-0.151034\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1260.00	−0.132434	−0.0662172	−	0.997805i	$-0.521093\pi$
$449$	−1260.00	−0.132434	−0.0662172	+	0.997805i	$0.521093\pi$
$450$	0	0
$451$	7200.00	0.751740
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 13750.0i	− 1.40744i	−0.710480	−	0.703718i	$-0.751522\pi$
$457$	− 13750.0i	− 1.40744i	0.710480	−	0.703718i	$-0.248478\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3210.00	0.324305	0.162152	−	0.986766i	$-0.448156\pi$
$461$	3210.00	0.324305	0.162152	+	0.986766i	$0.448156\pi$
$462$	0	0
$463$	12850.0i	1.28983i	0.764255	+	0.644914i	$0.223107\pi$
$463$	12850.0i	1.28983i	−0.764255	+	0.644914i	$0.776893\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8220.00i	0.814510i	0.913314	+	0.407255i	$0.133514\pi$
$467$	8220.00i	0.814510i	−0.913314	+	0.407255i	$0.866486\pi$
$468$	0	0
$469$	−1816.00	−0.178795
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4080.00i	− 0.396614i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7020.00	−0.669628	−0.334814	−	0.942284i	$-0.608674\pi$
$479$	−7020.00	−0.669628	−0.334814	+	0.942284i	$0.608674\pi$
$480$	0	0
$481$	−800.000	−0.0758355
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8122.00i	− 0.755735i	−0.925860	−	0.377868i	$-0.876657\pi$
$487$	− 8122.00i	− 0.755735i	0.925860	−	0.377868i	$-0.123343\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13470.0	1.23807	0.619035	−	0.785363i	$-0.287524\pi$
$491$	13470.0	1.23807	0.619035	+	0.785363i	$0.287524\pi$
$492$	0	0
$493$	− 18900.0i	− 1.72660i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2040.00i	0.184118i
$498$	0	0
$499$	−2468.00	−0.221409	−0.110704	−	0.993853i	$-0.535311\pi$
$499$	−2468.00	−0.221409	−0.110704	+	0.993853i	$0.535311\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 4440.00i	− 0.393578i	−0.980446	−	0.196789i	$-0.936949\pi$
$503$	− 4440.00i	− 0.393578i	0.980446	−	0.196789i	$-0.0630514\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11190.0	−0.974436	−0.487218	−	0.873280i	$-0.661988\pi$
$509$	−11190.0	−0.974436	−0.487218	+	0.873280i	$0.661988\pi$
$510$	0	0
$511$	−500.000	−0.0432851
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3600.00i	− 0.306243i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4020.00	0.338041	0.169021	−	0.985613i	$-0.445940\pi$
$521$	4020.00	0.338041	0.169021	+	0.985613i	$0.445940\pi$
$522$	0	0
$523$	9076.00i	0.758826i	0.925228	+	0.379413i	$0.123874\pi$
$523$	9076.00i	0.758826i	−0.925228	+	0.379413i	$0.876126\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	360.000i	0.0297568i
$528$	0	0
$529$	−2233.00	−0.183529
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 960.000i	− 0.0780154i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10170.0	−0.812714
$540$	0	0
$541$	−7486.00	−0.594914	−0.297457	−	0.954735i	$-0.596138\pi$
$541$	−7486.00	−0.594914	−0.297457	+	0.954735i	$0.596138\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7400.00i	0.578430i	0.957264	+	0.289215i	$0.0933942\pi$
$547$	7400.00i	0.578430i	−0.957264	+	0.289215i	$0.906606\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5880.00	0.454621
$552$	0	0
$553$	1832.00i	0.140876i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11490.0i	− 0.874052i	−0.899449	−	0.437026i	$-0.856032\pi$
$557$	− 11490.0i	− 0.874052i	0.899449	−	0.437026i	$-0.143968\pi$
$558$	0	0
$559$	−544.000	−0.0411606
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 19320.0i	− 1.44625i	−0.690715	−	0.723127i	$-0.742704\pi$
$563$	− 19320.0i	− 1.44625i	0.690715	−	0.723127i	$-0.257296\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8340.00	0.614466	0.307233	−	0.951634i	$-0.400597\pi$
$569$	8340.00	0.614466	0.307233	+	0.951634i	$0.400597\pi$
$570$	0	0
$571$	21044.0	1.54232	0.771159	−	0.636642i	$-0.219677\pi$
$571$	21044.0	1.54232	0.771159	+	0.636642i	$0.219677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1418.00i	0.102309i	0.998691	+	0.0511543i	$0.0162900\pi$
$577$	1418.00i	0.102309i	−0.998691	+	0.0511543i	$0.983710\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2280.00	0.162806
$582$	0	0
$583$	900.000i	0.0639351i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22020.0i	1.54832i	0.632991	+	0.774159i	$0.281827\pi$
$587$	22020.0i	1.54832i	−0.632991	+	0.774159i	$0.718173\pi$
$588$	0	0
$589$	−112.000	−0.00783511
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25230.0i	1.74717i	0.486671	+	0.873585i	$0.338211\pi$
$593$	25230.0i	1.74717i	−0.486671	+	0.873585i	$0.661789\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8280.00	0.564794	0.282397	−	0.959298i	$-0.408870\pi$
$599$	8280.00	0.564794	0.282397	+	0.959298i	$0.408870\pi$
$600$	0	0
$601$	−18874.0	−1.28101	−0.640505	−	0.767954i	$-0.721275\pi$
$601$	−18874.0	−1.28101	−0.640505	+	0.767954i	$0.721275\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10550.0i	0.705455i	0.935726	+	0.352728i	$0.114746\pi$
$607$	10550.0i	0.705455i	−0.935726	+	0.352728i	$0.885254\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−480.000	−0.0317819
$612$	0	0
$613$	− 11000.0i	− 0.724773i	−0.932028	−	0.362386i	$-0.881962\pi$
$613$	− 11000.0i	− 0.724773i	0.932028	−	0.362386i	$-0.118038\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11310.0i	0.737963i	0.929437	+	0.368982i	$0.120294\pi$
$617$	11310.0i	0.737963i	−0.929437	+	0.368982i	$0.879706\pi$
$618$	0	0
$619$	17572.0	1.14100	0.570499	−	0.821298i	$-0.306750\pi$
$619$	17572.0	1.14100	0.570499	+	0.821298i	$0.306750\pi$
$620$	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
$629$	18000.0	1.14103
$630$	0	0
$631$	1604.00	0.101195	0.0505976	−	0.998719i	$-0.483887\pi$
$631$	1604.00	0.101195	0.0505976	+	0.998719i	$0.483887\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1356.00i	0.0843433i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−31320.0	−1.92990	−0.964950	−	0.262435i	$-0.915475\pi$
$641$	−31320.0	−1.92990	−0.964950	+	0.262435i	$0.915475\pi$
$642$	0	0
$643$	31300.0i	1.91968i	0.280555	+	0.959838i	$0.409481\pi$
$643$	31300.0i	1.91968i	−0.280555	+	0.959838i	$0.590519\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10920.0i	− 0.663539i	−0.943361	−	0.331769i	$-0.892354\pi$
$647$	− 10920.0i	− 0.663539i	0.943361	−	0.331769i	$-0.107646\pi$
$648$	0	0
$649$	13500.0	0.816520
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3210.00i	− 0.192369i	−0.995364	−	0.0961845i	$-0.969336\pi$
$653$	− 3210.00i	− 0.192369i	0.995364	−	0.0961845i	$-0.0306639\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11910.0	−0.704018	−0.352009	−	0.935997i	$-0.614501\pi$
$659$	−11910.0	−0.704018	−0.352009	+	0.935997i	$0.614501\pi$
$660$	0	0
$661$	−3382.00	−0.199008	−0.0995042	−	0.995037i	$-0.531726\pi$
$661$	−3382.00	−0.199008	−0.0995042	+	0.995037i	$0.531726\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25200.0i	1.46289i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4980.00	0.286514
$672$	0	0
$673$	− 15950.0i	− 0.913562i	−0.889579	−	0.456781i	$-0.849002\pi$
$673$	− 15950.0i	− 0.913562i	0.889579	−	0.456781i	$-0.150998\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 32190.0i	− 1.82742i	−0.406369	−	0.913709i	$-0.633205\pi$
$677$	− 32190.0i	− 1.82742i	0.406369	−	0.913709i	$-0.366795\pi$
$678$	0	0
$679$	−3076.00	−0.173853
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22140.0i	1.24036i	0.784461	+	0.620178i	$0.212940\pi$
$683$	22140.0i	1.24036i	−0.784461	+	0.620178i	$0.787060\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	120.000	0.00663518
$690$	0	0
$691$	−6172.00	−0.339789	−0.169894	−	0.985462i	$-0.554343\pi$
$691$	−6172.00	−0.339789	−0.169894	+	0.985462i	$0.554343\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21600.0i	1.17383i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19170.0	−1.03287	−0.516434	−	0.856327i	$-0.672741\pi$
$701$	−19170.0	−1.03287	−0.516434	+	0.856327i	$0.672741\pi$
$702$	0	0
$703$	5600.00i	0.300438i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 900.000i	− 0.0478755i
$708$	0	0
$709$	21898.0	1.15994	0.579969	−	0.814638i	$-0.303064\pi$
$709$	21898.0	1.15994	0.579969	+	0.814638i	$0.303064\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 480.000i	− 0.0252120i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16680.0	0.865173	0.432586	−	0.901593i	$-0.357601\pi$
$719$	16680.0	0.865173	0.432586	+	0.901593i	$0.357601\pi$
$720$	0	0
$721$	−2300.00	−0.118802
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6518.00i	0.332516i	0.986082	+	0.166258i	$0.0531685\pi$
$727$	6518.00i	0.332516i	−0.986082	+	0.166258i	$0.946832\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12240.0	0.619306
$732$	0	0
$733$	23200.0i	1.16905i	0.811377	+	0.584524i	$0.198719\pi$
$733$	23200.0i	1.16905i	−0.811377	+	0.584524i	$0.801281\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 27240.0i	− 1.36146i
$738$	0	0
$739$	16324.0	0.812568	0.406284	−	0.913747i	$-0.366824\pi$
$739$	16324.0	0.812568	0.406284	+	0.913747i	$0.366824\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 120.000i	− 0.00592513i	−0.999996	−	0.00296257i	$-0.999057\pi$
$743$	− 120.000i	− 0.00592513i	0.999996	−	0.00296257i	$-0.000943015\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3240.00	−0.158060
$750$	0	0
$751$	30548.0	1.48430	0.742152	−	0.670232i	$-0.233805\pi$
$751$	30548.0	1.48430	0.742152	+	0.670232i	$0.233805\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16952.0i	0.813911i	0.913448	+	0.406956i	$0.133410\pi$
$757$	16952.0i	0.813911i	−0.913448	+	0.406956i	$0.866590\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20220.0	0.963173	0.481586	−	0.876399i	$-0.340061\pi$
$761$	20220.0	0.963173	0.481586	+	0.876399i	$0.340061\pi$
$762$	0	0
$763$	3404.00i	0.161511i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1800.00i	− 0.0847382i
$768$	0	0
$769$	20722.0	0.971722	0.485861	−	0.874036i	$-0.338506\pi$
$769$	20722.0	0.971722	0.485861	+	0.874036i	$0.338506\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4350.00i	0.202404i	0.994866	+	0.101202i	$0.0322689\pi$
$773$	4350.00i	0.202404i	−0.994866	+	0.101202i	$0.967731\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6720.00	−0.309074
$780$	0	0
$781$	−30600.0	−1.40199
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41972.0i	1.90107i	0.310621	+	0.950534i	$0.399463\pi$
$787$	41972.0i	1.90107i	−0.310621	+	0.950534i	$0.600537\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2700.00	−0.121367
$792$	0	0
$793$	− 664.000i	− 0.0297343i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39510.0i	1.75598i	0.478679	+	0.877990i	$0.341116\pi$
$797$	39510.0i	1.75598i	−0.478679	+	0.877990i	$0.658884\pi$
$798$	0	0
$799$	10800.0	0.478193
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 7500.00i	− 0.329601i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16680.0	0.724892	0.362446	−	0.932005i	$-0.381942\pi$
$809$	16680.0	0.724892	0.362446	+	0.932005i	$0.381942\pi$
$810$	0	0
$811$	−15484.0	−0.670428	−0.335214	−	0.942142i	$-0.608809\pi$
$811$	−15484.0	−0.670428	−0.335214	+	0.942142i	$0.608809\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3808.00i	0.163066i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4170.00	−0.177264	−0.0886322	−	0.996064i	$-0.528250\pi$
$821$	−4170.00	−0.177264	−0.0886322	+	0.996064i	$0.528250\pi$
$822$	0	0
$823$	30226.0i	1.28021i	0.768288	+	0.640105i	$0.221109\pi$
$823$	30226.0i	1.28021i	−0.768288	+	0.640105i	$0.778891\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14760.0i	0.620623i	0.950635	+	0.310312i	$0.100433\pi$
$827$	14760.0i	0.620623i	−0.950635	+	0.310312i	$0.899567\pi$
$828$	0	0
$829$	9934.00	0.416191	0.208095	−	0.978109i	$-0.433274\pi$
$829$	9934.00	0.416191	0.208095	+	0.978109i	$0.433274\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 30510.0i	− 1.26904i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23520.0	−0.967820	−0.483910	−	0.875118i	$-0.660784\pi$
$839$	−23520.0	−0.967820	−0.483910	+	0.875118i	$0.660784\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 862.000i	− 0.0349689i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24000.0	−0.966756
$852$	0	0
$853$	− 29816.0i	− 1.19681i	−0.801193	−	0.598406i	$-0.795801\pi$
$853$	− 29816.0i	− 1.19681i	0.801193	−	0.598406i	$-0.204199\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35430.0i	1.41221i	0.708106	+	0.706106i	$0.249550\pi$
$857$	35430.0i	1.41221i	−0.708106	+	0.706106i	$0.750450\pi$
$858$	0	0
$859$	36196.0	1.43771	0.718854	−	0.695161i	$-0.244667\pi$
$859$	36196.0	1.43771	0.718854	+	0.695161i	$0.244667\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	480.000i	0.0189332i	0.999955	+	0.00946662i	$0.00301336\pi$
$863$	480.000i	0.0189332i	−0.999955	+	0.00946662i	$0.996987\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−27480.0	−1.07272
$870$	0	0
$871$	−3632.00	−0.141292
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28532.0i	1.09858i	0.835631	+	0.549291i	$0.185102\pi$
$877$	28532.0i	1.09858i	−0.835631	+	0.549291i	$0.814898\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20340.0	0.777834	0.388917	−	0.921273i	$-0.372849\pi$
$881$	20340.0	0.777834	0.388917	+	0.921273i	$0.372849\pi$
$882$	0	0
$883$	10756.0i	0.409930i	0.978769	+	0.204965i	$0.0657081\pi$
$883$	10756.0i	0.409930i	−0.978769	+	0.204965i	$0.934292\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 600.000i	− 0.0227125i	−0.999936	−	0.0113563i	$-0.996385\pi$
$887$	− 600.000i	− 0.0227125i	0.999936	−	0.0113563i	$-0.00361489\pi$
$888$	0	0
$889$	−4900.00	−0.184860
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3360.00i	0.125911i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−840.000	−0.0311630
$900$	0	0
$901$	−2700.00	−0.0998336
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25400.0i	0.929871i	0.885345	+	0.464936i	$0.153923\pi$
$907$	25400.0i	0.929871i	−0.885345	+	0.464936i	$0.846077\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36240.0	1.31799	0.658993	−	0.752149i	$-0.270983\pi$
$911$	36240.0	1.31799	0.658993	+	0.752149i	$0.270983\pi$
$912$	0	0
$913$	34200.0i	1.23971i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1380.00i	− 0.0496964i
$918$	0	0
$919$	−6572.00	−0.235898	−0.117949	−	0.993020i	$-0.537632\pi$
$919$	−6572.00	−0.235898	−0.117949	+	0.993020i	$0.537632\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4080.00i	0.145498i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2340.00	0.0826404	0.0413202	−	0.999146i	$-0.486844\pi$
$929$	2340.00	0.0826404	0.0413202	+	0.999146i	$0.486844\pi$
$930$	0	0
$931$	9492.00	0.334144
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2522.00i	0.0879297i	0.999033	+	0.0439649i	$0.0139990\pi$
$937$	2522.00i	0.0879297i	−0.999033	+	0.0439649i	$0.986001\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52770.0	1.82811	0.914056	−	0.405589i	$-0.132933\pi$
$941$	52770.0	1.82811	0.914056	+	0.405589i	$0.132933\pi$
$942$	0	0
$943$	− 28800.0i	− 0.994546i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28200.0i	− 0.967663i	−0.875161	−	0.483832i	$-0.839245\pi$
$947$	− 28200.0i	− 0.967663i	0.875161	−	0.483832i	$-0.160755\pi$
$948$	0	0
$949$	−1000.00	−0.0342059
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 15570.0i	− 0.529236i	−0.964353	−	0.264618i	$-0.914754\pi$
$953$	− 15570.0i	− 0.529236i	0.964353	−	0.264618i	$-0.0852458\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4140.00	0.139403
$960$	0	0
$961$	−29775.0	−0.999463
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8350.00i	− 0.277681i	−0.990315	−	0.138841i	$-0.955662\pi$
$967$	− 8350.00i	− 0.277681i	0.990315	−	0.138841i	$-0.0443376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−43650.0	−1.44263	−0.721316	−	0.692606i	$-0.756462\pi$
$971$	−43650.0	−1.44263	−0.721316	+	0.692606i	$0.756462\pi$
$972$	0	0
$973$	3848.00i	0.126784i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18810.0i	− 0.615952i	−0.951394	−	0.307976i	$-0.900348\pi$
$977$	− 18810.0i	− 0.615952i	0.951394	−	0.307976i	$-0.0996517\pi$
$978$	0	0
$979$	12600.0	0.411336
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 25320.0i	− 0.821549i	−0.911737	−	0.410774i	$-0.865258\pi$
$983$	− 25320.0i	− 0.821549i	0.911737	−	0.410774i	$-0.134742\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16320.0	−0.524718
$990$	0	0
$991$	−6736.00	−0.215919	−0.107960	−	0.994155i	$-0.534432\pi$
$991$	−6736.00	−0.215919	−0.107960	+	0.994155i	$0.534432\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 20500.0i	− 0.651195i	−0.945508	−	0.325598i	$-0.894435\pi$
$997$	− 20500.0i	− 0.651195i	0.945508	−	0.325598i	$-0.105565\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.d.d.649.2		2
3.2	odd	2		900.4.d.i.649.2		2
5.2	odd	4		900.4.a.i.1.1		1
5.3	odd	4		180.4.a.b.1.1	✓	1
5.4	even	2	inner	900.4.d.d.649.1		2
15.2	even	4		900.4.a.j.1.1		1
15.8	even	4		180.4.a.e.1.1	yes	1
15.14	odd	2		900.4.d.i.649.1		2
20.3	even	4		720.4.a.h.1.1		1
45.13	odd	12		1620.4.i.i.541.1		2
45.23	even	12		1620.4.i.c.541.1		2
45.38	even	12		1620.4.i.c.1081.1		2
45.43	odd	12		1620.4.i.i.1081.1		2
60.23	odd	4		720.4.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.a.b.1.1	✓	1	5.3	odd	4
180.4.a.e.1.1	yes	1	15.8	even	4
720.4.a.h.1.1		1	20.3	even	4
720.4.a.w.1.1		1	60.23	odd	4
900.4.a.i.1.1		1	5.2	odd	4
900.4.a.j.1.1		1	15.2	even	4
900.4.d.d.649.1		2	5.4	even	2	inner
900.4.d.d.649.2		2	1.1	even	1	trivial
900.4.d.i.649.1		2	15.14	odd	2
900.4.d.i.649.2		2	3.2	odd	2
1620.4.i.c.541.1		2	45.23	even	12
1620.4.i.c.1081.1		2	45.38	even	12
1620.4.i.i.541.1		2	45.13	odd	12
1620.4.i.i.1081.1		2	45.43	odd	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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{7} +O(q^{10})\)	\(q+2.00000i q^{7} -30.0000 q^{11} +4.00000i q^{13} -90.0000i q^{17} +28.0000 q^{19} +120.000i q^{23} +210.000 q^{29} -4.00000 q^{31} +200.000i q^{37} -240.000 q^{41} +136.000i q^{43} +120.000i q^{47} +339.000 q^{49} -30.0000i q^{53} -450.000 q^{59} -166.000 q^{61} +908.000i q^{67} +1020.00 q^{71} +250.000i q^{73} -60.0000i q^{77} +916.000 q^{79} -1140.00i q^{83} -420.000 q^{89} -8.00000 q^{91} +1538.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 60 q^{11} + 56 q^{19} + 420 q^{29} - 8 q^{31} - 480 q^{41} + 678 q^{49} - 900 q^{59} - 332 q^{61} + 2040 q^{71} + 1832 q^{79} - 840 q^{89} - 16 q^{91}+O(q^{100}) \) 2 * q - 60 * q^11 + 56 * q^19 + 420 * q^29 - 8 * q^31 - 480 * q^41 + 678 * q^49 - 900 * q^59 - 332 * q^61 + 2040 * q^71 + 1832 * q^79 - 840 * q^89 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more