LMFDB - Embedded newform 525.4.d.f.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.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:	$30.9760027530$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.4.d.f.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +8.00000 q^{4} +7.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +8.00000 q^{4} +7.00000i q^{7} -9.00000 q^{9} +42.0000 q^{11} +24.0000i q^{12} -20.0000i q^{13} +64.0000 q^{16} +66.0000i q^{17} -38.0000 q^{19} -21.0000 q^{21} -12.0000i q^{23} -27.0000i q^{27} +56.0000i q^{28} +258.000 q^{29} +146.000 q^{31} +126.000i q^{33} -72.0000 q^{36} +434.000i q^{37} +60.0000 q^{39} -282.000 q^{41} -20.0000i q^{43} +336.000 q^{44} -72.0000i q^{47} +192.000i q^{48} -49.0000 q^{49} -198.000 q^{51} -160.000i q^{52} -336.000i q^{53} -114.000i q^{57} +360.000 q^{59} -682.000 q^{61} -63.0000i q^{63} +512.000 q^{64} +812.000i q^{67} +528.000i q^{68} +36.0000 q^{69} +810.000 q^{71} +124.000i q^{73} -304.000 q^{76} +294.000i q^{77} -1136.00 q^{79} +81.0000 q^{81} -156.000i q^{83} -168.000 q^{84} +774.000i q^{87} +1038.00 q^{89} +140.000 q^{91} -96.0000i q^{92} +438.000i q^{93} +1208.00i q^{97} -378.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{4} - 18 q^{9}+O(q^{10}) $ 2 * q + 16 * q^4 - 18 * q^9	$ 2 q + 16 q^{4} - 18 q^{9} + 84 q^{11} + 128 q^{16} - 76 q^{19} - 42 q^{21} + 516 q^{29} + 292 q^{31} - 144 q^{36} + 120 q^{39} - 564 q^{41} + 672 q^{44} - 98 q^{49} - 396 q^{51} + 720 q^{59} - 1364 q^{61} + 1024 q^{64} + 72 q^{69} + 1620 q^{71} - 608 q^{76} - 2272 q^{79} + 162 q^{81} - 336 q^{84} + 2076 q^{89} + 280 q^{91} - 756 q^{99}+O(q^{100}) $ 2 * q + 16 * q^4 - 18 * q^9 + 84 * q^11 + 128 * q^16 - 76 * q^19 - 42 * q^21 + 516 * q^29 + 292 * q^31 - 144 * q^36 + 120 * q^39 - 564 * q^41 + 672 * q^44 - 98 * q^49 - 396 * q^51 + 720 * q^59 - 1364 * q^61 + 1024 * q^64 + 72 * q^69 + 1620 * q^71 - 608 * q^76 - 2272 * q^79 + 162 * q^81 - 336 * q^84 + 2076 * q^89 + 280 * q^91 - 756 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	8.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	42.0000	1.15123	0.575613	−	0.817723i	$-0.304764\pi$
$11$	42.0000	1.15123	0.575613	+	0.817723i	$0.304764\pi$
$12$	24.0000i	0.577350i
$13$	− 20.0000i	− 0.426692i	−0.976977	−	0.213346i	$-0.931564\pi$
$13$	− 20.0000i	− 0.426692i	0.976977	−	0.213346i	$-0.0684362\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	66.0000i	0.941609i	0.882238	+	0.470804i	$0.156036\pi$
$17$	66.0000i	0.941609i	−0.882238	+	0.470804i	$0.843964\pi$
$18$	0	0
$19$	−38.0000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−38.0000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	− 12.0000i	− 0.108790i	−0.998519	−	0.0543951i	$-0.982677\pi$
$23$	− 12.0000i	− 0.108790i	0.998519	−	0.0543951i	$-0.0173230\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	56.0000i	0.377964i
$29$	258.000	1.65205	0.826024	−	0.563635i	$-0.190597\pi$
$29$	258.000	1.65205	0.826024	+	0.563635i	$0.190597\pi$
$30$	0	0
$31$	146.000	0.845883	0.422942	−	0.906157i	$-0.360998\pi$
$31$	146.000	0.845883	0.422942	+	0.906157i	$0.360998\pi$
$32$	0	0
$33$	126.000i	0.664660i
$34$	0	0
$35$	0	0
$36$	−72.0000	−0.333333
$37$	434.000i	1.92836i	0.265257	+	0.964178i	$0.414543\pi$
$37$	434.000i	1.92836i	−0.265257	+	0.964178i	$0.585457\pi$
$38$	0	0
$39$	60.0000	0.246351
$40$	0	0
$41$	−282.000	−1.07417	−0.537085	−	0.843528i	$-0.680475\pi$
$41$	−282.000	−1.07417	−0.537085	+	0.843528i	$0.680475\pi$
$42$	0	0
$43$	− 20.0000i	− 0.0709296i	−0.999371	−	0.0354648i	$-0.988709\pi$
$43$	− 20.0000i	− 0.0709296i	0.999371	−	0.0354648i	$-0.0112912\pi$
$44$	336.000	1.15123
$45$	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	$-0.0356380\pi$
$48$	192.000i	0.577350i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−198.000	−0.543638
$52$	− 160.000i	− 0.426692i
$53$	− 336.000i	− 0.870814i	−0.900234	−	0.435407i	$-0.856604\pi$
$53$	− 336.000i	− 0.870814i	0.900234	−	0.435407i	$-0.143396\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 114.000i	− 0.264906i
$58$	0	0
$59$	360.000	0.794373	0.397187	−	0.917738i	$-0.369987\pi$
$59$	360.000	0.794373	0.397187	+	0.917738i	$0.369987\pi$
$60$	0	0
$61$	−682.000	−1.43149	−0.715747	−	0.698360i	$-0.753914\pi$
$61$	−682.000	−1.43149	−0.715747	+	0.698360i	$0.753914\pi$
$62$	0	0
$63$	− 63.0000i	− 0.125988i
$64$	512.000	1.00000
$65$	0	0
$66$	0	0
$67$	812.000i	1.48062i	0.672265	+	0.740310i	$0.265321\pi$
$67$	812.000i	1.48062i	−0.672265	+	0.740310i	$0.734679\pi$
$68$	528.000i	0.941609i
$69$	36.0000	0.0628100
$70$	0	0
$71$	810.000	1.35393	0.676967	−	0.736013i	$-0.263294\pi$
$71$	810.000	1.35393	0.676967	+	0.736013i	$0.263294\pi$
$72$	0	0
$73$	124.000i	0.198810i	0.995047	+	0.0994048i	$0.0316939\pi$
$73$	124.000i	0.198810i	−0.995047	+	0.0994048i	$0.968306\pi$
$74$	0	0
$75$	0	0
$76$	−304.000	−0.458831
$77$	294.000i	0.435122i
$78$	0	0
$79$	−1136.00	−1.61785	−0.808924	−	0.587913i	$-0.799950\pi$
$79$	−1136.00	−1.61785	−0.808924	+	0.587913i	$0.799950\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 156.000i	− 0.206304i	−0.994666	−	0.103152i	$-0.967107\pi$
$83$	− 156.000i	− 0.206304i	0.994666	−	0.103152i	$-0.0328928\pi$
$84$	−168.000	−0.218218
$85$	0	0
$86$	0	0
$87$	774.000i	0.953810i
$88$	0	0
$89$	1038.00	1.23627	0.618134	−	0.786073i	$-0.287889\pi$
$89$	1038.00	1.23627	0.618134	+	0.786073i	$0.287889\pi$
$90$	0	0
$91$	140.000	0.161275
$92$	− 96.0000i	− 0.108790i
$93$	438.000i	0.488371i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1208.00i	1.26447i	0.774776	+	0.632236i	$0.217863\pi$
$97$	1208.00i	1.26447i	−0.774776	+	0.632236i	$0.782137\pi$
$98$	0	0
$99$	−378.000	−0.383742
$100$	0	0
$101$	546.000	0.537911	0.268956	−	0.963153i	$-0.413322\pi$
$101$	546.000	0.537911	0.268956	+	0.963153i	$0.413322\pi$
$102$	0	0
$103$	520.000i	0.497448i	0.968574	+	0.248724i	$0.0800112\pi$
$103$	520.000i	0.497448i	−0.968574	+	0.248724i	$0.919989\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1212.00i	1.09503i	0.836795	+	0.547516i	$0.184427\pi$
$107$	1212.00i	1.09503i	−0.836795	+	0.547516i	$0.815573\pi$
$108$	− 216.000i	− 0.192450i
$109$	1078.00	0.947281	0.473641	−	0.880718i	$-0.342940\pi$
$109$	1078.00	0.947281	0.473641	+	0.880718i	$0.342940\pi$
$110$	0	0
$111$	−1302.00	−1.11334
$112$	448.000i	0.377964i
$113$	1452.00i	1.20878i	0.796687	+	0.604392i	$0.206584\pi$
$113$	1452.00i	1.20878i	−0.796687	+	0.604392i	$0.793416\pi$
$114$	0	0
$115$	0	0
$116$	2064.00	1.65205
$117$	180.000i	0.142231i
$118$	0	0
$119$	−462.000	−0.355895
$120$	0	0
$121$	433.000	0.325319
$122$	0	0
$123$	− 846.000i	− 0.620173i
$124$	1168.00	0.845883
$125$	0	0
$126$	0	0
$127$	− 1312.00i	− 0.916702i	−0.888771	−	0.458351i	$-0.848440\pi$
$127$	− 1312.00i	− 0.916702i	0.888771	−	0.458351i	$-0.151560\pi$
$128$	0	0
$129$	60.0000	0.0409512
$130$	0	0
$131$	−1356.00	−0.904384	−0.452192	−	0.891921i	$-0.649358\pi$
$131$	−1356.00	−0.904384	−0.452192	+	0.891921i	$0.649358\pi$
$132$	1008.00i	0.664660i
$133$	− 266.000i	− 0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 984.000i	− 0.613641i	−0.951767	−	0.306820i	$-0.900735\pi$
$137$	− 984.000i	− 0.613641i	0.951767	−	0.306820i	$-0.0992651\pi$
$138$	0	0
$139$	394.000	0.240422	0.120211	−	0.992748i	$-0.461643\pi$
$139$	394.000	0.240422	0.120211	+	0.992748i	$0.461643\pi$
$140$	0	0
$141$	216.000	0.129011
$142$	0	0
$143$	− 840.000i	− 0.491219i
$144$	−576.000	−0.333333
$145$	0	0
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	3472.00i	1.92836i
$149$	1014.00	0.557518	0.278759	−	0.960361i	$-0.410077\pi$
$149$	1014.00	0.557518	0.278759	+	0.960361i	$0.410077\pi$
$150$	0	0
$151$	−1996.00	−1.07571	−0.537855	−	0.843037i	$-0.680765\pi$
$151$	−1996.00	−1.07571	−0.537855	+	0.843037i	$0.680765\pi$
$152$	0	0
$153$	− 594.000i	− 0.313870i
$154$	0	0
$155$	0	0
$156$	480.000	0.246351
$157$	− 2392.00i	− 1.21594i	−0.793960	−	0.607969i	$-0.791984\pi$
$157$	− 2392.00i	− 1.21594i	0.793960	−	0.607969i	$-0.208016\pi$
$158$	0	0
$159$	1008.00	0.502765
$160$	0	0
$161$	84.0000	0.0411188
$162$	0	0
$163$	− 2036.00i	− 0.978355i	−0.872184	−	0.489177i	$-0.837297\pi$
$163$	− 2036.00i	− 0.978355i	0.872184	−	0.489177i	$-0.162703\pi$
$164$	−2256.00	−1.07417
$165$	0	0
$166$	0	0
$167$	− 3936.00i	− 1.82381i	−0.410398	−	0.911907i	$-0.634610\pi$
$167$	− 3936.00i	− 1.82381i	0.410398	−	0.911907i	$-0.365390\pi$
$168$	0	0
$169$	1797.00	0.817934
$170$	0	0
$171$	342.000	0.152944
$172$	− 160.000i	− 0.0709296i
$173$	− 378.000i	− 0.166120i	−0.996545	−	0.0830601i	$-0.973531\pi$
$173$	− 378.000i	− 0.166120i	0.996545	−	0.0830601i	$-0.0264694\pi$
$174$	0	0
$175$	0	0
$176$	2688.00	1.15123
$177$	1080.00i	0.458631i
$178$	0	0
$179$	222.000	0.0926987	0.0463493	−	0.998925i	$-0.485241\pi$
$179$	222.000	0.0926987	0.0463493	+	0.998925i	$0.485241\pi$
$180$	0	0
$181$	−2590.00	−1.06361	−0.531804	−	0.846867i	$-0.678486\pi$
$181$	−2590.00	−1.06361	−0.531804	+	0.846867i	$0.678486\pi$
$182$	0	0
$183$	− 2046.00i	− 0.826474i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2772.00i	1.08400i
$188$	− 576.000i	− 0.223453i
$189$	189.000	0.0727393
$190$	0	0
$191$	2214.00	0.838740	0.419370	−	0.907815i	$-0.362251\pi$
$191$	2214.00	0.838740	0.419370	+	0.907815i	$0.362251\pi$
$192$	1536.00i	0.577350i
$193$	− 4178.00i	− 1.55823i	−0.626879	−	0.779117i	$-0.715668\pi$
$193$	− 4178.00i	− 1.55823i	0.626879	−	0.779117i	$-0.284332\pi$
$194$	0	0
$195$	0	0
$196$	−392.000	−0.142857
$197$	− 3060.00i	− 1.10668i	−0.832955	−	0.553340i	$-0.813353\pi$
$197$	− 3060.00i	− 1.10668i	0.832955	−	0.553340i	$-0.186647\pi$
$198$	0	0
$199$	−2666.00	−0.949687	−0.474844	−	0.880070i	$-0.657495\pi$
$199$	−2666.00	−0.949687	−0.474844	+	0.880070i	$0.657495\pi$
$200$	0	0
$201$	−2436.00	−0.854837
$202$	0	0
$203$	1806.00i	0.624416i
$204$	−1584.00	−0.543638
$205$	0	0
$206$	0	0
$207$	108.000i	0.0362634i
$208$	− 1280.00i	− 0.426692i
$209$	−1596.00	−0.528218
$210$	0	0
$211$	−1348.00	−0.439811	−0.219906	−	0.975521i	$-0.570575\pi$
$211$	−1348.00	−0.439811	−0.219906	+	0.975521i	$0.570575\pi$
$212$	− 2688.00i	− 0.870814i
$213$	2430.00i	0.781694i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1022.00i	0.319714i
$218$	0	0
$219$	−372.000	−0.114783
$220$	0	0
$221$	1320.00	0.401777
$222$	0	0
$223$	− 3188.00i	− 0.957329i	−0.877998	−	0.478664i	$-0.841121\pi$
$223$	− 3188.00i	− 0.957329i	0.877998	−	0.478664i	$-0.158879\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3396.00i	− 0.992953i	−0.868050	−	0.496477i	$-0.834627\pi$
$227$	− 3396.00i	− 0.992953i	0.868050	−	0.496477i	$-0.165373\pi$
$228$	− 912.000i	− 0.264906i
$229$	−5294.00	−1.52767	−0.763837	−	0.645409i	$-0.776687\pi$
$229$	−5294.00	−1.52767	−0.763837	+	0.645409i	$0.776687\pi$
$230$	0	0
$231$	−882.000	−0.251218
$232$	0	0
$233$	− 852.000i	− 0.239555i	−0.992801	−	0.119778i	$-0.961782\pi$
$233$	− 852.000i	− 0.239555i	0.992801	−	0.119778i	$-0.0382182\pi$
$234$	0	0
$235$	0	0
$236$	2880.00	0.794373
$237$	− 3408.00i	− 0.934065i
$238$	0	0
$239$	−4866.00	−1.31697	−0.658484	−	0.752595i	$-0.728802\pi$
$239$	−4866.00	−1.31697	−0.658484	+	0.752595i	$0.728802\pi$
$240$	0	0
$241$	−2050.00	−0.547934	−0.273967	−	0.961739i	$-0.588336\pi$
$241$	−2050.00	−0.547934	−0.273967	+	0.961739i	$0.588336\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	−5456.00	−1.43149
$245$	0	0
$246$	0	0
$247$	760.000i	0.195780i
$248$	0	0
$249$	468.000	0.119110
$250$	0	0
$251$	−1152.00	−0.289696	−0.144848	−	0.989454i	$-0.546269\pi$
$251$	−1152.00	−0.289696	−0.144848	+	0.989454i	$0.546269\pi$
$252$	− 504.000i	− 0.125988i
$253$	− 504.000i	− 0.125242i
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	− 6450.00i	− 1.56553i	−0.622321	−	0.782763i	$-0.713810\pi$
$257$	− 6450.00i	− 1.56553i	0.622321	−	0.782763i	$-0.286190\pi$
$258$	0	0
$259$	−3038.00	−0.728850
$260$	0	0
$261$	−2322.00	−0.550683
$262$	0	0
$263$	− 1968.00i	− 0.461415i	−0.973023	−	0.230707i	$-0.925896\pi$
$263$	− 1968.00i	− 0.461415i	0.973023	−	0.230707i	$-0.0741040\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3114.00i	0.713759i
$268$	6496.00i	1.48062i
$269$	3894.00	0.882607	0.441304	−	0.897358i	$-0.354516\pi$
$269$	3894.00	0.882607	0.441304	+	0.897358i	$0.354516\pi$
$270$	0	0
$271$	7094.00	1.59015	0.795073	−	0.606513i	$-0.207432\pi$
$271$	7094.00	1.59015	0.795073	+	0.606513i	$0.207432\pi$
$272$	4224.00i	0.941609i
$273$	420.000i	0.0931119i
$274$	0	0
$275$	0	0
$276$	288.000	0.0628100
$277$	− 3310.00i	− 0.717973i	−0.933343	−	0.358987i	$-0.883122\pi$
$277$	− 3310.00i	− 0.717973i	0.933343	−	0.358987i	$-0.116878\pi$
$278$	0	0
$279$	−1314.00	−0.281961
$280$	0	0
$281$	7158.00	1.51961	0.759805	−	0.650151i	$-0.225294\pi$
$281$	7158.00	1.51961	0.759805	+	0.650151i	$0.225294\pi$
$282$	0	0
$283$	5164.00i	1.08469i	0.840155	+	0.542346i	$0.182464\pi$
$283$	5164.00i	1.08469i	−0.840155	+	0.542346i	$0.817536\pi$
$284$	6480.00	1.35393
$285$	0	0
$286$	0	0
$287$	− 1974.00i	− 0.405998i
$288$	0	0
$289$	557.000	0.113373
$290$	0	0
$291$	−3624.00	−0.730043
$292$	992.000i	0.198810i
$293$	8598.00i	1.71434i	0.515037	+	0.857168i	$0.327778\pi$
$293$	8598.00i	1.71434i	−0.515037	+	0.857168i	$0.672222\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1134.00i	− 0.221553i
$298$	0	0
$299$	−240.000	−0.0464199
$300$	0	0
$301$	140.000	0.0268089
$302$	0	0
$303$	1638.00i	0.310563i
$304$	−2432.00	−0.458831
$305$	0	0
$306$	0	0
$307$	− 448.000i	− 0.0832857i	−0.999133	−	0.0416429i	$-0.986741\pi$
$307$	− 448.000i	− 0.0832857i	0.999133	−	0.0416429i	$-0.0132592\pi$
$308$	2352.00i	0.435122i
$309$	−1560.00	−0.287202
$310$	0	0
$311$	−5832.00	−1.06335	−0.531676	−	0.846948i	$-0.678438\pi$
$311$	−5832.00	−1.06335	−0.531676	+	0.846948i	$0.678438\pi$
$312$	0	0
$313$	− 9848.00i	− 1.77841i	−0.457510	−	0.889204i	$-0.651259\pi$
$313$	− 9848.00i	− 1.77841i	0.457510	−	0.889204i	$-0.348741\pi$
$314$	0	0
$315$	0	0
$316$	−9088.00	−1.61785
$317$	− 5616.00i	− 0.995035i	−0.867454	−	0.497517i	$-0.834245\pi$
$317$	− 5616.00i	− 0.995035i	0.867454	−	0.497517i	$-0.165755\pi$
$318$	0	0
$319$	10836.0	1.90188
$320$	0	0
$321$	−3636.00	−0.632217
$322$	0	0
$323$	− 2508.00i	− 0.432040i
$324$	648.000	0.111111
$325$	0	0
$326$	0	0
$327$	3234.00i	0.546913i
$328$	0	0
$329$	504.000	0.0844572
$330$	0	0
$331$	452.000	0.0750579	0.0375290	−	0.999296i	$-0.488051\pi$
$331$	452.000	0.0750579	0.0375290	+	0.999296i	$0.488051\pi$
$332$	− 1248.00i	− 0.206304i
$333$	− 3906.00i	− 0.642785i
$334$	0	0
$335$	0	0
$336$	−1344.00	−0.218218
$337$	− 2302.00i	− 0.372101i	−0.982540	−	0.186050i	$-0.940431\pi$
$337$	− 2302.00i	− 0.372101i	0.982540	−	0.186050i	$-0.0595688\pi$
$338$	0	0
$339$	−4356.00	−0.697892
$340$	0	0
$341$	6132.00	0.973802
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1584.00i	0.245054i	0.992465	+	0.122527i	$0.0390997\pi$
$347$	1584.00i	0.245054i	−0.992465	+	0.122527i	$0.960900\pi$
$348$	6192.00i	0.953810i
$349$	−8174.00	−1.25371	−0.626854	−	0.779137i	$-0.715658\pi$
$349$	−8174.00	−1.25371	−0.626854	+	0.779137i	$0.715658\pi$
$350$	0	0
$351$	−540.000	−0.0821170
$352$	0	0
$353$	− 8610.00i	− 1.29820i	−0.760704	−	0.649099i	$-0.775146\pi$
$353$	− 8610.00i	− 1.29820i	0.760704	−	0.649099i	$-0.224854\pi$
$354$	0	0
$355$	0	0
$356$	8304.00	1.23627
$357$	− 1386.00i	− 0.205476i
$358$	0	0
$359$	2154.00	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	2154.00	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−5415.00	−0.789474
$362$	0	0
$363$	1299.00i	0.187823i
$364$	1120.00	0.161275
$365$	0	0
$366$	0	0
$367$	6644.00i	0.944997i	0.881331	+	0.472499i	$0.156648\pi$
$367$	6644.00i	0.944997i	−0.881331	+	0.472499i	$0.843352\pi$
$368$	− 768.000i	− 0.108790i
$369$	2538.00	0.358057
$370$	0	0
$371$	2352.00	0.329137
$372$	3504.00i	0.488371i
$373$	− 7958.00i	− 1.10469i	−0.833615	−	0.552345i	$-0.813733\pi$
$373$	− 7958.00i	− 1.10469i	0.833615	−	0.552345i	$-0.186267\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5160.00i	− 0.704917i
$378$	0	0
$379$	−3440.00	−0.466229	−0.233115	−	0.972449i	$-0.574892\pi$
$379$	−3440.00	−0.466229	−0.233115	+	0.972449i	$0.574892\pi$
$380$	0	0
$381$	3936.00	0.529258
$382$	0	0
$383$	− 12936.0i	− 1.72585i	−0.505336	−	0.862923i	$-0.668631\pi$
$383$	− 12936.0i	− 1.72585i	0.505336	−	0.862923i	$-0.331369\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	180.000i	0.0236432i
$388$	9664.00i	1.26447i
$389$	14862.0	1.93710	0.968552	−	0.248812i	$-0.0800401\pi$
$389$	14862.0	1.93710	0.968552	+	0.248812i	$0.0800401\pi$
$390$	0	0
$391$	792.000	0.102438
$392$	0	0
$393$	− 4068.00i	− 0.522146i
$394$	0	0
$395$	0	0
$396$	−3024.00	−0.383742
$397$	10460.0i	1.32235i	0.750232	+	0.661174i	$0.229942\pi$
$397$	10460.0i	1.32235i	−0.750232	+	0.661174i	$0.770058\pi$
$398$	0	0
$399$	798.000	0.100125
$400$	0	0
$401$	−9150.00	−1.13947	−0.569737	−	0.821827i	$-0.692955\pi$
$401$	−9150.00	−1.13947	−0.569737	+	0.821827i	$0.692955\pi$
$402$	0	0
$403$	− 2920.00i	− 0.360932i
$404$	4368.00	0.537911
$405$	0	0
$406$	0	0
$407$	18228.0i	2.21997i
$408$	0	0
$409$	4894.00	0.591669	0.295835	−	0.955239i	$-0.404402\pi$
$409$	4894.00	0.591669	0.295835	+	0.955239i	$0.404402\pi$
$410$	0	0
$411$	2952.00	0.354286
$412$	4160.00i	0.497448i
$413$	2520.00i	0.300245i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1182.00i	0.138808i
$418$	0	0
$419$	1668.00	0.194480	0.0972400	−	0.995261i	$-0.468999\pi$
$419$	1668.00	0.194480	0.0972400	+	0.995261i	$0.468999\pi$
$420$	0	0
$421$	−12418.0	−1.43757	−0.718784	−	0.695233i	$-0.755301\pi$
$421$	−12418.0	−1.43757	−0.718784	+	0.695233i	$0.755301\pi$
$422$	0	0
$423$	648.000i	0.0744843i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4774.00i	− 0.541054i
$428$	9696.00i	1.09503i
$429$	2520.00	0.283605
$430$	0	0
$431$	15186.0	1.69718	0.848589	−	0.529052i	$-0.177452\pi$
$431$	15186.0	1.69718	0.848589	+	0.529052i	$0.177452\pi$
$432$	− 1728.00i	− 0.192450i
$433$	5704.00i	0.633064i	0.948582	+	0.316532i	$0.102518\pi$
$433$	5704.00i	0.633064i	−0.948582	+	0.316532i	$0.897482\pi$
$434$	0	0
$435$	0	0
$436$	8624.00	0.947281
$437$	456.000i	0.0499163i
$438$	0	0
$439$	17206.0	1.87061	0.935305	−	0.353843i	$-0.115125\pi$
$439$	17206.0	1.87061	0.935305	+	0.353843i	$0.115125\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	− 3456.00i	− 0.370654i	−0.982677	−	0.185327i	$-0.940666\pi$
$443$	− 3456.00i	− 0.370654i	0.982677	−	0.185327i	$-0.0593344\pi$
$444$	−10416.0	−1.11334
$445$	0	0
$446$	0	0
$447$	3042.00i	0.321883i
$448$	3584.00i	0.377964i
$449$	−16074.0	−1.68949	−0.844743	−	0.535173i	$-0.820247\pi$
$449$	−16074.0	−1.68949	−0.844743	+	0.535173i	$0.820247\pi$
$450$	0	0
$451$	−11844.0	−1.23661
$452$	11616.0i	1.20878i
$453$	− 5988.00i	− 0.621061i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7526.00i	0.770353i	0.922843	+	0.385177i	$0.125859\pi$
$457$	7526.00i	0.770353i	−0.922843	+	0.385177i	$0.874141\pi$
$458$	0	0
$459$	1782.00	0.181213
$460$	0	0
$461$	−2274.00	−0.229741	−0.114871	−	0.993380i	$-0.536645\pi$
$461$	−2274.00	−0.229741	−0.114871	+	0.993380i	$0.536645\pi$
$462$	0	0
$463$	10024.0i	1.00617i	0.864238	+	0.503083i	$0.167801\pi$
$463$	10024.0i	1.00617i	−0.864238	+	0.503083i	$0.832199\pi$
$464$	16512.0	1.65205
$465$	0	0
$466$	0	0
$467$	− 2460.00i	− 0.243759i	−0.992545	−	0.121879i	$-0.961108\pi$
$467$	− 2460.00i	− 0.243759i	0.992545	−	0.121879i	$-0.0388921\pi$
$468$	1440.00i	0.142231i
$469$	−5684.00	−0.559622
$470$	0	0
$471$	7176.00	0.702023
$472$	0	0
$473$	− 840.000i	− 0.0816559i
$474$	0	0
$475$	0	0
$476$	−3696.00	−0.355895
$477$	3024.00i	0.290271i
$478$	0	0
$479$	−19320.0	−1.84291	−0.921454	−	0.388486i	$-0.872998\pi$
$479$	−19320.0	−1.84291	−0.921454	+	0.388486i	$0.872998\pi$
$480$	0	0
$481$	8680.00	0.822815
$482$	0	0
$483$	252.000i	0.0237400i
$484$	3464.00	0.325319
$485$	0	0
$486$	0	0
$487$	− 12544.0i	− 1.16719i	−0.812044	−	0.583596i	$-0.801645\pi$
$487$	− 12544.0i	− 1.16719i	0.812044	−	0.583596i	$-0.198355\pi$
$488$	0	0
$489$	6108.00	0.564853
$490$	0	0
$491$	−15510.0	−1.42557	−0.712787	−	0.701381i	$-0.752567\pi$
$491$	−15510.0	−1.42557	−0.712787	+	0.701381i	$0.752567\pi$
$492$	− 6768.00i	− 0.620173i
$493$	17028.0i	1.55558i
$494$	0	0
$495$	0	0
$496$	9344.00	0.845883
$497$	5670.00i	0.511739i
$498$	0	0
$499$	14344.0	1.28682	0.643412	−	0.765520i	$-0.277518\pi$
$499$	14344.0	1.28682	0.643412	+	0.765520i	$0.277518\pi$
$500$	0	0
$501$	11808.0	1.05298
$502$	0	0
$503$	21384.0i	1.89556i	0.318929	+	0.947779i	$0.396677\pi$
$503$	21384.0i	1.89556i	−0.318929	+	0.947779i	$0.603323\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5391.00i	0.472234i
$508$	− 10496.0i	− 0.916702i
$509$	7134.00	0.621236	0.310618	−	0.950535i	$-0.399464\pi$
$509$	7134.00	0.621236	0.310618	+	0.950535i	$0.399464\pi$
$510$	0	0
$511$	−868.000	−0.0751430
$512$	0	0
$513$	1026.00i	0.0883022i
$514$	0	0
$515$	0	0
$516$	480.000	0.0409512
$517$	− 3024.00i	− 0.257244i
$518$	0	0
$519$	1134.00	0.0959096
$520$	0	0
$521$	−19122.0	−1.60797	−0.803983	−	0.594653i	$-0.797290\pi$
$521$	−19122.0	−1.60797	−0.803983	+	0.594653i	$0.797290\pi$
$522$	0	0
$523$	15640.0i	1.30763i	0.756655	+	0.653814i	$0.226832\pi$
$523$	15640.0i	1.30763i	−0.756655	+	0.653814i	$0.773168\pi$
$524$	−10848.0	−0.904384
$525$	0	0
$526$	0	0
$527$	9636.00i	0.796491i
$528$	8064.00i	0.664660i
$529$	12023.0	0.988165
$530$	0	0
$531$	−3240.00	−0.264791
$532$	− 2128.00i	− 0.173422i
$533$	5640.00i	0.458341i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	666.000i	0.0535196i
$538$	0	0
$539$	−2058.00	−0.164461
$540$	0	0
$541$	2846.00	0.226172	0.113086	−	0.993585i	$-0.463926\pi$
$541$	2846.00	0.226172	0.113086	+	0.993585i	$0.463926\pi$
$542$	0	0
$543$	− 7770.00i	− 0.614075i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4444.00i	− 0.347371i	−0.984801	−	0.173685i	$-0.944432\pi$
$547$	− 4444.00i	− 0.347371i	0.984801	−	0.173685i	$-0.0555676\pi$
$548$	− 7872.00i	− 0.613641i
$549$	6138.00	0.477165
$550$	0	0
$551$	−9804.00	−0.758012
$552$	0	0
$553$	− 7952.00i	− 0.611489i
$554$	0	0
$555$	0	0
$556$	3152.00	0.240422
$557$	18552.0i	1.41126i	0.708579	+	0.705631i	$0.249337\pi$
$557$	18552.0i	1.41126i	−0.708579	+	0.705631i	$0.750663\pi$
$558$	0	0
$559$	−400.000	−0.0302651
$560$	0	0
$561$	−8316.00	−0.625850
$562$	0	0
$563$	16452.0i	1.23156i	0.787918	+	0.615781i	$0.211159\pi$
$563$	16452.0i	1.23156i	−0.787918	+	0.615781i	$0.788841\pi$
$564$	1728.00	0.129011
$565$	0	0
$566$	0	0
$567$	567.000i	0.0419961i
$568$	0	0
$569$	−7722.00	−0.568933	−0.284467	−	0.958686i	$-0.591817\pi$
$569$	−7722.00	−0.568933	−0.284467	+	0.958686i	$0.591817\pi$
$570$	0	0
$571$	2576.00	0.188796	0.0943978	−	0.995535i	$-0.469907\pi$
$571$	2576.00	0.188796	0.0943978	+	0.995535i	$0.469907\pi$
$572$	− 6720.00i	− 0.491219i
$573$	6642.00i	0.484247i
$574$	0	0
$575$	0	0
$576$	−4608.00	−0.333333
$577$	− 2464.00i	− 0.177778i	−0.996042	−	0.0888888i	$-0.971668\pi$
$577$	− 2464.00i	− 0.177778i	0.996042	−	0.0888888i	$-0.0283316\pi$
$578$	0	0
$579$	12534.0	0.899646
$580$	0	0
$581$	1092.00	0.0779755
$582$	0	0
$583$	− 14112.0i	− 1.00250i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1452.00i	0.102096i	0.998696	+	0.0510481i	$0.0162562\pi$
$587$	1452.00i	0.102096i	−0.998696	+	0.0510481i	$0.983744\pi$
$588$	− 1176.00i	− 0.0824786i
$589$	−5548.00	−0.388118
$590$	0	0
$591$	9180.00	0.638942
$592$	27776.0i	1.92836i
$593$	− 10698.0i	− 0.740833i	−0.928866	−	0.370417i	$-0.879215\pi$
$593$	− 10698.0i	− 0.740833i	0.928866	−	0.370417i	$-0.120785\pi$
$594$	0	0
$595$	0	0
$596$	8112.00	0.557518
$597$	− 7998.00i	− 0.548302i
$598$	0	0
$599$	8730.00	0.595489	0.297745	−	0.954646i	$-0.403766\pi$
$599$	8730.00	0.595489	0.297745	+	0.954646i	$0.403766\pi$
$600$	0	0
$601$	1910.00	0.129635	0.0648174	−	0.997897i	$-0.479354\pi$
$601$	1910.00	0.129635	0.0648174	+	0.997897i	$0.479354\pi$
$602$	0	0
$603$	− 7308.00i	− 0.493540i
$604$	−15968.0	−1.07571
$605$	0	0
$606$	0	0
$607$	− 5596.00i	− 0.374192i	−0.982342	−	0.187096i	$-0.940092\pi$
$607$	− 5596.00i	− 0.374192i	0.982342	−	0.187096i	$-0.0599076\pi$
$608$	0	0
$609$	−5418.00	−0.360506
$610$	0	0
$611$	−1440.00	−0.0953456
$612$	− 4752.00i	− 0.313870i
$613$	− 28586.0i	− 1.88349i	−0.336332	−	0.941744i	$-0.609186\pi$
$613$	− 28586.0i	− 1.88349i	0.336332	−	0.941744i	$-0.390814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19236.0i	− 1.25513i	−0.778566	−	0.627563i	$-0.784053\pi$
$617$	− 19236.0i	− 1.25513i	0.778566	−	0.627563i	$-0.215947\pi$
$618$	0	0
$619$	−6734.00	−0.437257	−0.218629	−	0.975808i	$-0.570158\pi$
$619$	−6734.00	−0.437257	−0.218629	+	0.975808i	$0.570158\pi$
$620$	0	0
$621$	−324.000	−0.0209367
$622$	0	0
$623$	7266.00i	0.467265i
$624$	3840.00	0.246351
$625$	0	0
$626$	0	0
$627$	− 4788.00i	− 0.304967i
$628$	− 19136.0i	− 1.21594i
$629$	−28644.0	−1.81576
$630$	0	0
$631$	7184.00	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$631$	7184.00	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$632$	0	0
$633$	− 4044.00i	− 0.253925i
$634$	0	0
$635$	0	0
$636$	8064.00	0.502765
$637$	980.000i	0.0609561i
$638$	0	0
$639$	−7290.00	−0.451311
$640$	0	0
$641$	510.000	0.0314256	0.0157128	−	0.999877i	$-0.494998\pi$
$641$	510.000	0.0314256	0.0157128	+	0.999877i	$0.494998\pi$
$642$	0	0
$643$	20752.0i	1.27275i	0.771379	+	0.636376i	$0.219567\pi$
$643$	20752.0i	1.27275i	−0.771379	+	0.636376i	$0.780433\pi$
$644$	672.000	0.0411188
$645$	0	0
$646$	0	0
$647$	21072.0i	1.28041i	0.768204	+	0.640205i	$0.221151\pi$
$647$	21072.0i	1.28041i	−0.768204	+	0.640205i	$0.778849\pi$
$648$	0	0
$649$	15120.0	0.914502
$650$	0	0
$651$	−3066.00	−0.184587
$652$	− 16288.0i	− 0.978355i
$653$	− 2892.00i	− 0.173312i	−0.996238	−	0.0866560i	$-0.972382\pi$
$653$	− 2892.00i	− 0.173312i	0.996238	−	0.0866560i	$-0.0276181\pi$
$654$	0	0
$655$	0	0
$656$	−18048.0	−1.07417
$657$	− 1116.00i	− 0.0662699i
$658$	0	0
$659$	−750.000	−0.0443336	−0.0221668	−	0.999754i	$-0.507056\pi$
$659$	−750.000	−0.0443336	−0.0221668	+	0.999754i	$0.507056\pi$
$660$	0	0
$661$	30062.0	1.76895	0.884475	−	0.466587i	$-0.154517\pi$
$661$	30062.0	1.76895	0.884475	+	0.466587i	$0.154517\pi$
$662$	0	0
$663$	3960.00i	0.231966i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3096.00i	− 0.179727i
$668$	− 31488.0i	− 1.82381i
$669$	9564.00	0.552714
$670$	0	0
$671$	−28644.0	−1.64797
$672$	0	0
$673$	− 15446.0i	− 0.884695i	−0.896844	−	0.442347i	$-0.854146\pi$
$673$	− 15446.0i	− 0.884695i	0.896844	−	0.442347i	$-0.145854\pi$
$674$	0	0
$675$	0	0
$676$	14376.0	0.817934
$677$	− 25110.0i	− 1.42549i	−0.701424	−	0.712744i	$-0.747452\pi$
$677$	− 25110.0i	− 1.42549i	0.701424	−	0.712744i	$-0.252548\pi$
$678$	0	0
$679$	−8456.00	−0.477926
$680$	0	0
$681$	10188.0	0.573282
$682$	0	0
$683$	− 7968.00i	− 0.446394i	−0.974773	−	0.223197i	$-0.928351\pi$
$683$	− 7968.00i	− 0.446394i	0.974773	−	0.223197i	$-0.0716493\pi$
$684$	2736.00	0.152944
$685$	0	0
$686$	0	0
$687$	− 15882.0i	− 0.882003i
$688$	− 1280.00i	− 0.0709296i
$689$	−6720.00	−0.371570
$690$	0	0
$691$	−14398.0	−0.792657	−0.396328	−	0.918109i	$-0.629716\pi$
$691$	−14398.0	−0.792657	−0.396328	+	0.918109i	$0.629716\pi$
$692$	− 3024.00i	− 0.166120i
$693$	− 2646.00i	− 0.145041i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18612.0i	− 1.01145i
$698$	0	0
$699$	2556.00	0.138307
$700$	0	0
$701$	−9234.00	−0.497523	−0.248761	−	0.968565i	$-0.580023\pi$
$701$	−9234.00	−0.497523	−0.248761	+	0.968565i	$0.580023\pi$
$702$	0	0
$703$	− 16492.0i	− 0.884790i
$704$	21504.0	1.15123
$705$	0	0
$706$	0	0
$707$	3822.00i	0.203311i
$708$	8640.00i	0.458631i
$709$	−8030.00	−0.425350	−0.212675	−	0.977123i	$-0.568218\pi$
$709$	−8030.00	−0.425350	−0.212675	+	0.977123i	$0.568218\pi$
$710$	0	0
$711$	10224.0	0.539283
$712$	0	0
$713$	− 1752.00i	− 0.0920237i
$714$	0	0
$715$	0	0
$716$	1776.00	0.0926987
$717$	− 14598.0i	− 0.760352i
$718$	0	0
$719$	−27060.0	−1.40357	−0.701786	−	0.712388i	$-0.747614\pi$
$719$	−27060.0	−1.40357	−0.701786	+	0.712388i	$0.747614\pi$
$720$	0	0
$721$	−3640.00	−0.188018
$722$	0	0
$723$	− 6150.00i	− 0.316350i
$724$	−20720.0	−1.06361
$725$	0	0
$726$	0	0
$727$	− 3724.00i	− 0.189980i	−0.995478	−	0.0949900i	$-0.969718\pi$
$727$	− 3724.00i	− 0.189980i	0.995478	−	0.0949900i	$-0.0302819\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	1320.00	0.0667879
$732$	− 16368.0i	− 0.826474i
$733$	5668.00i	0.285610i	0.989751	+	0.142805i	$0.0456122\pi$
$733$	5668.00i	0.285610i	−0.989751	+	0.142805i	$0.954388\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34104.0i	1.70453i
$738$	0	0
$739$	16072.0	0.800024	0.400012	−	0.916510i	$-0.369006\pi$
$739$	16072.0	0.800024	0.400012	+	0.916510i	$0.369006\pi$
$740$	0	0
$741$	−2280.00	−0.113034
$742$	0	0
$743$	8256.00i	0.407649i	0.979007	+	0.203825i	$0.0653372\pi$
$743$	8256.00i	0.407649i	−0.979007	+	0.203825i	$0.934663\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1404.00i	0.0687680i
$748$	22176.0i	1.08400i
$749$	−8484.00	−0.413883
$750$	0	0
$751$	−6352.00	−0.308639	−0.154319	−	0.988021i	$-0.549318\pi$
$751$	−6352.00	−0.308639	−0.154319	+	0.988021i	$0.549318\pi$
$752$	− 4608.00i	− 0.223453i
$753$	− 3456.00i	− 0.167256i
$754$	0	0
$755$	0	0
$756$	1512.00	0.0727393
$757$	11558.0i	0.554931i	0.960736	+	0.277465i	$0.0894944\pi$
$757$	11558.0i	0.554931i	−0.960736	+	0.277465i	$0.910506\pi$
$758$	0	0
$759$	1512.00	0.0723085
$760$	0	0
$761$	7770.00	0.370121	0.185061	−	0.982727i	$-0.440752\pi$
$761$	7770.00	0.370121	0.185061	+	0.982727i	$0.440752\pi$
$762$	0	0
$763$	7546.00i	0.358039i
$764$	17712.0	0.838740
$765$	0	0
$766$	0	0
$767$	− 7200.00i	− 0.338953i
$768$	12288.0i	0.577350i
$769$	−22646.0	−1.06194	−0.530972	−	0.847389i	$-0.678173\pi$
$769$	−22646.0	−1.06194	−0.530972	+	0.847389i	$0.678173\pi$
$770$	0	0
$771$	19350.0	0.903856
$772$	− 33424.0i	− 1.55823i
$773$	35502.0i	1.65190i	0.563744	+	0.825950i	$0.309361\pi$
$773$	35502.0i	1.65190i	−0.563744	+	0.825950i	$0.690639\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 9114.00i	− 0.420802i
$778$	0	0
$779$	10716.0	0.492863
$780$	0	0
$781$	34020.0	1.55868
$782$	0	0
$783$	− 6966.00i	− 0.317937i
$784$	−3136.00	−0.142857
$785$	0	0
$786$	0	0
$787$	− 17080.0i	− 0.773617i	−0.922160	−	0.386808i	$-0.873578\pi$
$787$	− 17080.0i	− 0.773617i	0.922160	−	0.386808i	$-0.126422\pi$
$788$	− 24480.0i	− 1.10668i
$789$	5904.00	0.266398
$790$	0	0
$791$	−10164.0	−0.456878
$792$	0	0
$793$	13640.0i	0.610808i
$794$	0	0
$795$	0	0
$796$	−21328.0	−0.949687
$797$	5730.00i	0.254664i	0.991860	+	0.127332i	$0.0406413\pi$
$797$	5730.00i	0.254664i	−0.991860	+	0.127332i	$0.959359\pi$
$798$	0	0
$799$	4752.00	0.210405
$800$	0	0
$801$	−9342.00	−0.412089
$802$	0	0
$803$	5208.00i	0.228875i
$804$	−19488.0	−0.854837
$805$	0	0
$806$	0	0
$807$	11682.0i	0.509574i
$808$	0	0
$809$	−2550.00	−0.110820	−0.0554099	−	0.998464i	$-0.517647\pi$
$809$	−2550.00	−0.110820	−0.0554099	+	0.998464i	$0.517647\pi$
$810$	0	0
$811$	−27538.0	−1.19234	−0.596171	−	0.802857i	$-0.703312\pi$
$811$	−27538.0	−1.19234	−0.596171	+	0.802857i	$0.703312\pi$
$812$	14448.0i	0.624416i
$813$	21282.0i	0.918072i
$814$	0	0
$815$	0	0
$816$	−12672.0	−0.543638
$817$	760.000i	0.0325447i
$818$	0	0
$819$	−1260.00	−0.0537582
$820$	0	0
$821$	−19242.0	−0.817966	−0.408983	−	0.912542i	$-0.634117\pi$
$821$	−19242.0	−0.817966	−0.408983	+	0.912542i	$0.634117\pi$
$822$	0	0
$823$	11752.0i	0.497751i	0.968536	+	0.248875i	$0.0800610\pi$
$823$	11752.0i	0.497751i	−0.968536	+	0.248875i	$0.919939\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28692.0i	− 1.20643i	−0.797578	−	0.603216i	$-0.793886\pi$
$827$	− 28692.0i	− 1.20643i	0.797578	−	0.603216i	$-0.206114\pi$
$828$	864.000i	0.0362634i
$829$	−28442.0	−1.19159	−0.595797	−	0.803135i	$-0.703164\pi$
$829$	−28442.0	−1.19159	−0.595797	+	0.803135i	$0.703164\pi$
$830$	0	0
$831$	9930.00	0.414522
$832$	− 10240.0i	− 0.426692i
$833$	− 3234.00i	− 0.134516i
$834$	0	0
$835$	0	0
$836$	−12768.0	−0.528218
$837$	− 3942.00i	− 0.162790i
$838$	0	0
$839$	−20172.0	−0.830053	−0.415027	−	0.909809i	$-0.636228\pi$
$839$	−20172.0	−0.830053	−0.415027	+	0.909809i	$0.636228\pi$
$840$	0	0
$841$	42175.0	1.72926
$842$	0	0
$843$	21474.0i	0.877347i
$844$	−10784.0	−0.439811
$845$	0	0
$846$	0	0
$847$	3031.00i	0.122959i
$848$	− 21504.0i	− 0.870814i
$849$	−15492.0	−0.626247
$850$	0	0
$851$	5208.00	0.209786
$852$	19440.0i	0.781694i
$853$	− 19820.0i	− 0.795573i	−0.917478	−	0.397787i	$-0.869778\pi$
$853$	− 19820.0i	− 0.795573i	0.917478	−	0.397787i	$-0.130222\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 10290.0i	− 0.410151i	−0.978746	−	0.205076i	$-0.934256\pi$
$857$	− 10290.0i	− 0.410151i	0.978746	−	0.205076i	$-0.0657440\pi$
$858$	0	0
$859$	31606.0	1.25539	0.627697	−	0.778458i	$-0.283998\pi$
$859$	31606.0	1.25539	0.627697	+	0.778458i	$0.283998\pi$
$860$	0	0
$861$	5922.00	0.234403
$862$	0	0
$863$	− 23172.0i	− 0.914002i	−0.889466	−	0.457001i	$-0.848924\pi$
$863$	− 23172.0i	− 0.914002i	0.889466	−	0.457001i	$-0.151076\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1671.00i	0.0654558i
$868$	8176.00i	0.319714i
$869$	−47712.0	−1.86251
$870$	0	0
$871$	16240.0	0.631770
$872$	0	0
$873$	− 10872.0i	− 0.421491i
$874$	0	0
$875$	0	0
$876$	−2976.00	−0.114783
$877$	− 15550.0i	− 0.598730i	−0.954139	−	0.299365i	$-0.903225\pi$
$877$	− 15550.0i	− 0.598730i	0.954139	−	0.299365i	$-0.0967748\pi$
$878$	0	0
$879$	−25794.0	−0.989772
$880$	0	0
$881$	28530.0	1.09103	0.545517	−	0.838100i	$-0.316334\pi$
$881$	28530.0	1.09103	0.545517	+	0.838100i	$0.316334\pi$
$882$	0	0
$883$	28780.0i	1.09686i	0.836198	+	0.548428i	$0.184774\pi$
$883$	28780.0i	1.09686i	−0.836198	+	0.548428i	$0.815226\pi$
$884$	10560.0	0.401777
$885$	0	0
$886$	0	0
$887$	22872.0i	0.865802i	0.901441	+	0.432901i	$0.142510\pi$
$887$	22872.0i	0.865802i	−0.901441	+	0.432901i	$0.857490\pi$
$888$	0	0
$889$	9184.00	0.346481
$890$	0	0
$891$	3402.00	0.127914
$892$	− 25504.0i	− 0.957329i
$893$	2736.00i	0.102527i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 720.000i	− 0.0268006i
$898$	0	0
$899$	37668.0	1.39744
$900$	0	0
$901$	22176.0	0.819966
$902$	0	0
$903$	420.000i	0.0154781i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 10708.0i	− 0.392010i	−0.980603	−	0.196005i	$-0.937203\pi$
$907$	− 10708.0i	− 0.392010i	0.980603	−	0.196005i	$-0.0627969\pi$
$908$	− 27168.0i	− 0.992953i
$909$	−4914.00	−0.179304
$910$	0	0
$911$	1326.00	0.0482243	0.0241122	−	0.999709i	$-0.492324\pi$
$911$	1326.00	0.0482243	0.0241122	+	0.999709i	$0.492324\pi$
$912$	− 7296.00i	− 0.264906i
$913$	− 6552.00i	− 0.237502i
$914$	0	0
$915$	0	0
$916$	−42352.0	−1.52767
$917$	− 9492.00i	− 0.341825i
$918$	0	0
$919$	13696.0	0.491610	0.245805	−	0.969319i	$-0.420948\pi$
$919$	13696.0	0.491610	0.245805	+	0.969319i	$0.420948\pi$
$920$	0	0
$921$	1344.00	0.0480850
$922$	0	0
$923$	− 16200.0i	− 0.577713i
$924$	−7056.00	−0.251218
$925$	0	0
$926$	0	0
$927$	− 4680.00i	− 0.165816i
$928$	0	0
$929$	−42354.0	−1.49579	−0.747895	−	0.663817i	$-0.768936\pi$
$929$	−42354.0	−1.49579	−0.747895	+	0.663817i	$0.768936\pi$
$930$	0	0
$931$	1862.00	0.0655474
$932$	− 6816.00i	− 0.239555i
$933$	− 17496.0i	− 0.613926i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6644.00i	0.231644i	0.993270	+	0.115822i	$0.0369501\pi$
$937$	6644.00i	0.231644i	−0.993270	+	0.115822i	$0.963050\pi$
$938$	0	0
$939$	29544.0	1.02676
$940$	0	0
$941$	1350.00	0.0467681	0.0233840	−	0.999727i	$-0.492556\pi$
$941$	1350.00	0.0467681	0.0233840	+	0.999727i	$0.492556\pi$
$942$	0	0
$943$	3384.00i	0.116859i
$944$	23040.0	0.794373
$945$	0	0
$946$	0	0
$947$	49320.0i	1.69238i	0.532881	+	0.846190i	$0.321109\pi$
$947$	49320.0i	1.69238i	−0.532881	+	0.846190i	$0.678891\pi$
$948$	− 27264.0i	− 0.934065i
$949$	2480.00	0.0848306
$950$	0	0
$951$	16848.0	0.574484
$952$	0	0
$953$	− 5940.00i	− 0.201905i	−0.994891	−	0.100953i	$-0.967811\pi$
$953$	− 5940.00i	− 0.201905i	0.994891	−	0.100953i	$-0.0321890\pi$
$954$	0	0
$955$	0	0
$956$	−38928.0	−1.31697
$957$	32508.0i	1.09805i
$958$	0	0
$959$	6888.00	0.231934
$960$	0	0
$961$	−8475.00	−0.284482
$962$	0	0
$963$	− 10908.0i	− 0.365011i
$964$	−16400.0	−0.547934
$965$	0	0
$966$	0	0
$967$	47216.0i	1.57018i	0.619382	+	0.785090i	$0.287383\pi$
$967$	47216.0i	1.57018i	−0.619382	+	0.785090i	$0.712617\pi$
$968$	0	0
$969$	7524.00	0.249438
$970$	0	0
$971$	12552.0	0.414843	0.207422	−	0.978252i	$-0.433493\pi$
$971$	12552.0	0.414843	0.207422	+	0.978252i	$0.433493\pi$
$972$	1944.00i	0.0641500i
$973$	2758.00i	0.0908709i
$974$	0	0
$975$	0	0
$976$	−43648.0	−1.43149
$977$	46908.0i	1.53605i	0.640420	+	0.768025i	$0.278760\pi$
$977$	46908.0i	1.53605i	−0.640420	+	0.768025i	$0.721240\pi$
$978$	0	0
$979$	43596.0	1.42322
$980$	0	0
$981$	−9702.00	−0.315760
$982$	0	0
$983$	46128.0i	1.49670i	0.663305	+	0.748349i	$0.269153\pi$
$983$	46128.0i	1.49670i	−0.663305	+	0.748349i	$0.730847\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1512.00i	0.0487614i
$988$	6080.00i	0.195780i
$989$	−240.000	−0.00771644
$990$	0	0
$991$	−12184.0	−0.390552	−0.195276	−	0.980748i	$-0.562560\pi$
$991$	−12184.0	−0.390552	−0.195276	+	0.980748i	$0.562560\pi$
$992$	0	0
$993$	1356.00i	0.0433347i
$994$	0	0
$995$	0	0
$996$	3744.00	0.119110
$997$	− 5164.00i	− 0.164038i	−0.996631	−	0.0820188i	$-0.973863\pi$
$997$	− 5164.00i	− 0.164038i	0.996631	−	0.0820188i	$-0.0261368\pi$
$998$	0	0
$999$	11718.0	0.371112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.d.f.274.2		2
5.2	odd	4		525.4.a.e.1.1		1
5.3	odd	4		105.4.a.a.1.1	✓	1
5.4	even	2	inner	525.4.d.f.274.1		2
15.2	even	4		1575.4.a.f.1.1		1
15.8	even	4		315.4.a.d.1.1		1
20.3	even	4		1680.4.a.s.1.1		1
35.13	even	4		735.4.a.c.1.1		1
105.83	odd	4		2205.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.a.1.1	✓	1	5.3	odd	4
315.4.a.d.1.1		1	15.8	even	4
525.4.a.e.1.1		1	5.2	odd	4
525.4.d.f.274.1		2	5.4	even	2	inner
525.4.d.f.274.2		2	1.1	even	1	trivial
735.4.a.c.1.1		1	35.13	even	4
1575.4.a.f.1.1		1	15.2	even	4
1680.4.a.s.1.1		1	20.3	even	4
2205.4.a.o.1.1		1	105.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +8.00000 q^{4} +7.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +8.00000 q^{4} +7.00000i q^{7} -9.00000 q^{9} +42.0000 q^{11} +24.0000i q^{12} -20.0000i q^{13} +64.0000 q^{16} +66.0000i q^{17} -38.0000 q^{19} -21.0000 q^{21} -12.0000i q^{23} -27.0000i q^{27} +56.0000i q^{28} +258.000 q^{29} +146.000 q^{31} +126.000i q^{33} -72.0000 q^{36} +434.000i q^{37} +60.0000 q^{39} -282.000 q^{41} -20.0000i q^{43} +336.000 q^{44} -72.0000i q^{47} +192.000i q^{48} -49.0000 q^{49} -198.000 q^{51} -160.000i q^{52} -336.000i q^{53} -114.000i q^{57} +360.000 q^{59} -682.000 q^{61} -63.0000i q^{63} +512.000 q^{64} +812.000i q^{67} +528.000i q^{68} +36.0000 q^{69} +810.000 q^{71} +124.000i q^{73} -304.000 q^{76} +294.000i q^{77} -1136.00 q^{79} +81.0000 q^{81} -156.000i q^{83} -168.000 q^{84} +774.000i q^{87} +1038.00 q^{89} +140.000 q^{91} -96.0000i q^{92} +438.000i q^{93} +1208.00i q^{97} -378.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{4} - 18 q^{9}+O(q^{10}) \) 2 * q + 16 * q^4 - 18 * q^9	\( 2 q + 16 q^{4} - 18 q^{9} + 84 q^{11} + 128 q^{16} - 76 q^{19} - 42 q^{21} + 516 q^{29} + 292 q^{31} - 144 q^{36} + 120 q^{39} - 564 q^{41} + 672 q^{44} - 98 q^{49} - 396 q^{51} + 720 q^{59} - 1364 q^{61} + 1024 q^{64} + 72 q^{69} + 1620 q^{71} - 608 q^{76} - 2272 q^{79} + 162 q^{81} - 336 q^{84} + 2076 q^{89} + 280 q^{91} - 756 q^{99}+O(q^{100}) \) 2 * q + 16 * q^4 - 18 * q^9 + 84 * q^11 + 128 * q^16 - 76 * q^19 - 42 * q^21 + 516 * q^29 + 292 * q^31 - 144 * q^36 + 120 * q^39 - 564 * q^41 + 672 * q^44 - 98 * q^49 - 396 * q^51 + 720 * q^59 - 1364 * q^61 + 1024 * q^64 + 72 * q^69 + 1620 * q^71 - 608 * q^76 - 2272 * q^79 + 162 * q^81 - 336 * q^84 + 2076 * q^89 + 280 * q^91 - 756 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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