LMFDB - Embedded newform 1089.3.b.j.485.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 921x^{12} + 8986x^{10} + 46812x^{8} + 125072x^{6} + 152129x^{4} + 65614x^{2} + 5041 $ x^16 + 48x^14 + 921x^12 + 8986x^10 + 46812x^8 + 125072x^6 + 152129x^4 + 65614*x^2 + 5041
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.12
Root			$1.96467i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.j.485.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.96467i q^{2} +0.140085 q^{4} +9.62261i q^{5} +5.23056 q^{7} +8.13389i q^{8} +O(q^{10})$	$q+1.96467i q^{2} +0.140085 q^{4} +9.62261i q^{5} +5.23056 q^{7} +8.13389i q^{8} -18.9052 q^{10} -17.2247 q^{13} +10.2763i q^{14} -15.4200 q^{16} -7.71825i q^{17} -14.0290 q^{19} +1.34798i q^{20} +5.68512i q^{23} -67.5947 q^{25} -33.8407i q^{26} +0.732721 q^{28} -24.0845i q^{29} +15.0159 q^{31} +2.24031i q^{32} +15.1638 q^{34} +50.3317i q^{35} -7.65533 q^{37} -27.5624i q^{38} -78.2692 q^{40} -18.1953i q^{41} +53.6955 q^{43} -11.1694 q^{46} +38.8113i q^{47} -21.6412 q^{49} -132.801i q^{50} -2.41291 q^{52} +81.6140i q^{53} +42.5448i q^{56} +47.3181 q^{58} +0.000434171i q^{59} -4.77983 q^{61} +29.5012i q^{62} -66.0816 q^{64} -165.746i q^{65} +111.796 q^{67} -1.08121i q^{68} -98.8850 q^{70} +99.6336i q^{71} +34.2957 q^{73} -15.0402i q^{74} -1.96525 q^{76} +61.7290 q^{79} -148.381i q^{80} +35.7478 q^{82} -65.2314i q^{83} +74.2697 q^{85} +105.494i q^{86} +92.5976i q^{89} -90.0947 q^{91} +0.796398i q^{92} -76.2513 q^{94} -134.996i q^{95} -52.4236 q^{97} -42.5178i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 8 q^{7}+O(q^{10}) $ 16 * q - 32 * q^4 + 8 * q^7	$ 16 q - 32 q^{4} + 8 q^{7} - 24 q^{10} - 4 q^{13} + 28 q^{16} + 20 q^{19} - 44 q^{25} - 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} - 224 q^{40} - 272 q^{43} - 208 q^{46} + 348 q^{49} - 520 q^{52} - 44 q^{58} - 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} + 4 q^{73} - 1052 q^{76} - 216 q^{79} + 348 q^{82} - 416 q^{85} - 168 q^{91} - 1140 q^{94} - 44 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 8 * q^7 - 24 * q^10 - 4 * q^13 + 28 * q^16 + 20 * q^19 - 44 * q^25 - 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 - 224 * q^40 - 272 * q^43 - 208 * q^46 + 348 * q^49 - 520 * q^52 - 44 * q^58 - 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 + 4 * q^73 - 1052 * q^76 - 216 * q^79 + 348 * q^82 - 416 * q^85 - 168 * q^91 - 1140 * q^94 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.96467i	0.982333i	0.871066	+	0.491167i	$0.163429\pi$
$2$	1.96467i	0.982333i	−0.871066	+	0.491167i	$0.836571\pi$
$3$	0	0
$3$	0	0
$4$	0.140085	0.0350211
$5$	9.62261i	1.92452i	0.272127	+	0.962261i	$0.412273\pi$
$5$	9.62261i	1.92452i	−0.272127	+	0.962261i	$0.587727\pi$
$6$	0	0
$7$	5.23056	0.747223	0.373612	−	0.927585i	$-0.378119\pi$
$7$	5.23056	0.747223	0.373612	+	0.927585i	$0.378119\pi$
$8$	8.13389i	1.01674i
$9$	0	0
$10$	−18.9052	−1.89052
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−17.2247	−1.32497	−0.662487	−	0.749073i	$-0.730499\pi$
$13$	−17.2247	−1.32497	−0.662487	+	0.749073i	$0.730499\pi$
$14$	10.2763i	0.734022i
$15$	0	0
$16$	−15.4200	−0.963752
$17$	− 7.71825i	− 0.454015i	−0.973893	−	0.227007i	$-0.927106\pi$
$17$	− 7.71825i	− 0.454015i	0.973893	−	0.227007i	$-0.0728941\pi$
$18$	0	0
$19$	−14.0290	−0.738370	−0.369185	−	0.929356i	$-0.620363\pi$
$19$	−14.0290	−0.738370	−0.369185	+	0.929356i	$0.620363\pi$
$20$	1.34798i	0.0673989i
$21$	0	0
$22$	0	0
$23$	5.68512i	0.247179i	0.992333	+	0.123590i	$0.0394407\pi$
$23$	5.68512i	0.247179i	−0.992333	+	0.123590i	$0.960559\pi$
$24$	0	0
$25$	−67.5947	−2.70379
$26$	− 33.8407i	− 1.30157i
$27$	0	0
$28$	0.732721	0.0261686
$29$	− 24.0845i	− 0.830501i	−0.909707	−	0.415250i	$-0.863694\pi$
$29$	− 24.0845i	− 0.830501i	0.909707	−	0.415250i	$-0.136306\pi$
$30$	0	0
$31$	15.0159	0.484383	0.242192	−	0.970228i	$-0.422134\pi$
$31$	15.0159	0.484383	0.242192	+	0.970228i	$0.422134\pi$
$32$	2.24031i	0.0700097i
$33$	0	0
$34$	15.1638	0.445994
$35$	50.3317i	1.43805i
$36$	0	0
$37$	−7.65533	−0.206901	−0.103450	−	0.994635i	$-0.532988\pi$
$37$	−7.65533	−0.206901	−0.103450	+	0.994635i	$0.532988\pi$
$38$	− 27.5624i	− 0.725325i
$39$	0	0
$40$	−78.2692	−1.95673
$41$	− 18.1953i	− 0.443789i	−0.975071	−	0.221894i	$-0.928776\pi$
$41$	− 18.1953i	− 0.443789i	0.975071	−	0.221894i	$-0.0712240\pi$
$42$	0	0
$43$	53.6955	1.24873	0.624366	−	0.781132i	$-0.285358\pi$
$43$	53.6955	1.24873	0.624366	+	0.781132i	$0.285358\pi$
$44$	0	0
$45$	0	0
$46$	−11.1694	−0.242812
$47$	38.8113i	0.825772i	0.910783	+	0.412886i	$0.135479\pi$
$47$	38.8113i	0.825772i	−0.910783	+	0.412886i	$0.864521\pi$
$48$	0	0
$49$	−21.6412	−0.441658
$50$	− 132.801i	− 2.65602i
$51$	0	0
$52$	−2.41291	−0.0464021
$53$	81.6140i	1.53989i	0.638112	+	0.769944i	$0.279716\pi$
$53$	81.6140i	1.53989i	−0.638112	+	0.769944i	$0.720284\pi$
$54$	0	0
$55$	0	0
$56$	42.5448i	0.759728i
$57$	0	0
$58$	47.3181	0.815829
$59$	0 0.000434171i	0 7.35883e-6i	1.00000		3.67942e-6i	$1.17119e-6\pi$
$59$	0 0.000434171i	0 7.35883e-6i	−1.00000		3.67942e-6i	$0.999999\pi$
$60$	0	0
$61$	−4.77983	−0.0783579	−0.0391789	−	0.999232i	$-0.512474\pi$
$61$	−4.77983	−0.0783579	−0.0391789	+	0.999232i	$0.512474\pi$
$62$	29.5012i	0.475826i
$63$	0	0
$64$	−66.0816	−1.03253
$65$	− 165.746i	− 2.54994i
$66$	0	0
$67$	111.796	1.66860	0.834298	−	0.551313i	$-0.185873\pi$
$67$	111.796	1.66860	0.834298	+	0.551313i	$0.185873\pi$
$68$	− 1.08121i	− 0.0159001i
$69$	0	0
$70$	−98.8850	−1.41264
$71$	99.6336i	1.40329i	0.712526	+	0.701645i	$0.247551\pi$
$71$	99.6336i	1.40329i	−0.712526	+	0.701645i	$0.752449\pi$
$72$	0	0
$73$	34.2957	0.469804	0.234902	−	0.972019i	$-0.424523\pi$
$73$	34.2957	0.469804	0.234902	+	0.972019i	$0.424523\pi$
$74$	− 15.0402i	− 0.203246i
$75$	0	0
$76$	−1.96525	−0.0258585
$77$	0	0
$78$	0	0
$79$	61.7290	0.781380	0.390690	−	0.920522i	$-0.372236\pi$
$79$	61.7290	0.781380	0.390690	+	0.920522i	$0.372236\pi$
$80$	− 148.381i	− 1.85476i
$81$	0	0
$82$	35.7478	0.435948
$83$	− 65.2314i	− 0.785921i	−0.919555	−	0.392961i	$-0.871451\pi$
$83$	− 65.2314i	− 0.785921i	0.919555	−	0.392961i	$-0.128549\pi$
$84$	0	0
$85$	74.2697	0.873761
$86$	105.494i	1.22667i
$87$	0	0
$88$	0	0
$89$	92.5976i	1.04042i	0.854038	+	0.520211i	$0.174147\pi$
$89$	92.5976i	1.04042i	−0.854038	+	0.520211i	$0.825853\pi$
$90$	0	0
$91$	−90.0947	−0.990051
$92$	0.796398i	0.00865650i
$93$	0	0
$94$	−76.2513	−0.811184
$95$	− 134.996i	− 1.42101i
$96$	0	0
$97$	−52.4236	−0.540450	−0.270225	−	0.962797i	$-0.587098\pi$
$97$	−52.4236	−0.540450	−0.270225	+	0.962797i	$0.587098\pi$
$98$	− 42.5178i	− 0.433855i
$99$	0	0
$100$	−9.46897	−0.0946897
$101$	− 58.9222i	− 0.583389i	−0.956512	−	0.291694i	$-0.905781\pi$
$101$	− 58.9222i	− 0.583389i	0.956512	−	0.291694i	$-0.0942190\pi$
$102$	0	0
$103$	−96.7152	−0.938982	−0.469491	−	0.882937i	$-0.655563\pi$
$103$	−96.7152	−0.938982	−0.469491	+	0.882937i	$0.655563\pi$
$104$	− 140.103i	− 1.34715i
$105$	0	0
$106$	−160.344	−1.51268
$107$	− 18.3916i	− 0.171884i	−0.996300	−	0.0859422i	$-0.972610\pi$
$107$	− 18.3916i	− 0.171884i	0.996300	−	0.0859422i	$-0.0273900\pi$
$108$	0	0
$109$	−50.6495	−0.464674	−0.232337	−	0.972635i	$-0.574637\pi$
$109$	−50.6495	−0.464674	−0.232337	+	0.972635i	$0.574637\pi$
$110$	0	0
$111$	0	0
$112$	−80.6555	−0.720138
$113$	9.83566i	0.0870412i	0.999053	+	0.0435206i	$0.0138574\pi$
$113$	9.83566i	0.0870412i	−0.999053	+	0.0435206i	$0.986143\pi$
$114$	0	0
$115$	−54.7058	−0.475702
$116$	− 3.37387i	− 0.0290851i
$117$	0	0
$118$	−0.000853002 0	−7.22883e−6 0
$119$	− 40.3708i	− 0.339250i
$120$	0	0
$121$	0	0
$122$	− 9.39077i	− 0.0769736i
$123$	0	0
$124$	2.10349	0.0169637
$125$	− 409.872i	− 3.27898i
$126$	0	0
$127$	−215.481	−1.69670	−0.848349	−	0.529437i	$-0.822403\pi$
$127$	−215.481	−1.69670	−0.848349	+	0.529437i	$0.822403\pi$
$128$	− 120.867i	− 0.944274i
$129$	0	0
$130$	325.636	2.50489
$131$	166.588i	1.27166i	0.771827	+	0.635832i	$0.219343\pi$
$131$	166.588i	1.27166i	−0.771827	+	0.635832i	$0.780657\pi$
$132$	0	0
$133$	−73.3797	−0.551727
$134$	219.642i	1.63912i
$135$	0	0
$136$	62.7794	0.461613
$137$	− 43.7479i	− 0.319328i	−0.987171	−	0.159664i	$-0.948959\pi$
$137$	− 43.7479i	− 0.319328i	0.987171	−	0.159664i	$-0.0510411\pi$
$138$	0	0
$139$	103.074	0.741539	0.370769	−	0.928725i	$-0.379094\pi$
$139$	103.074	0.741539	0.370769	+	0.928725i	$0.379094\pi$
$140$	7.05069i	0.0503620i
$141$	0	0
$142$	−195.747	−1.37850
$143$	0	0
$144$	0	0
$145$	231.756	1.59832
$146$	67.3796i	0.461504i
$147$	0	0
$148$	−1.07239	−0.00724590
$149$	− 90.4963i	− 0.607358i	−0.952774	−	0.303679i	$-0.901785\pi$
$149$	− 90.4963i	− 0.607358i	0.952774	−	0.303679i	$-0.0982150\pi$
$150$	0	0
$151$	−240.354	−1.59175	−0.795873	−	0.605464i	$-0.792988\pi$
$151$	−240.354	−1.59175	−0.795873	+	0.605464i	$0.792988\pi$
$152$	− 114.110i	− 0.750727i
$153$	0	0
$154$	0	0
$155$	144.492i	0.932207i
$156$	0	0
$157$	−29.6036	−0.188558	−0.0942789	−	0.995546i	$-0.530055\pi$
$157$	−29.6036	−0.188558	−0.0942789	+	0.995546i	$0.530055\pi$
$158$	121.277i	0.767576i
$159$	0	0
$160$	−21.5576	−0.134735
$161$	29.7364i	0.184698i
$162$	0	0
$163$	210.084	1.28886	0.644430	−	0.764663i	$-0.277095\pi$
$163$	210.084	1.28886	0.644430	+	0.764663i	$0.277095\pi$
$164$	− 2.54888i	− 0.0155420i
$165$	0	0
$166$	128.158	0.772036
$167$	− 15.2464i	− 0.0912955i	−0.998958	−	0.0456478i	$-0.985465\pi$
$167$	− 15.2464i	− 0.0912955i	0.998958	−	0.0456478i	$-0.0145352\pi$
$168$	0	0
$169$	127.689	0.755557
$170$	145.915i	0.858325i
$171$	0	0
$172$	7.52190	0.0437320
$173$	214.396i	1.23928i	0.784885	+	0.619642i	$0.212722\pi$
$173$	214.396i	1.23928i	−0.784885	+	0.619642i	$0.787278\pi$
$174$	0	0
$175$	−353.558	−2.02033
$176$	0	0
$177$	0	0
$178$	−181.923	−1.02204
$179$	− 204.595i	− 1.14299i	−0.820605	−	0.571495i	$-0.806364\pi$
$179$	− 204.595i	− 1.14299i	0.820605	−	0.571495i	$-0.193636\pi$
$180$	0	0
$181$	136.759	0.755572	0.377786	−	0.925893i	$-0.376685\pi$
$181$	136.759	0.755572	0.377786	+	0.925893i	$0.376685\pi$
$182$	− 177.006i	− 0.972561i
$183$	0	0
$184$	−46.2422	−0.251316
$185$	− 73.6643i	− 0.398185i
$186$	0	0
$187$	0	0
$188$	5.43686i	0.0289195i
$189$	0	0
$190$	265.222	1.39590
$191$	244.731i	1.28132i	0.767827	+	0.640658i	$0.221338\pi$
$191$	244.731i	1.28132i	−0.767827	+	0.640658i	$0.778662\pi$
$192$	0	0
$193$	15.7364	0.0815358	0.0407679	−	0.999169i	$-0.487020\pi$
$193$	15.7364	0.0815358	0.0407679	+	0.999169i	$0.487020\pi$
$194$	− 102.995i	− 0.530902i
$195$	0	0
$196$	−3.03160	−0.0154674
$197$	95.4172i	0.484351i	0.970232	+	0.242176i	$0.0778610\pi$
$197$	95.4172i	0.484351i	−0.970232	+	0.242176i	$0.922139\pi$
$198$	0	0
$199$	−65.2214	−0.327746	−0.163873	−	0.986481i	$-0.552399\pi$
$199$	−65.2214	−0.327746	−0.163873	+	0.986481i	$0.552399\pi$
$200$	− 549.807i	− 2.74904i
$201$	0	0
$202$	115.763	0.573082
$203$	− 125.976i	− 0.620569i
$204$	0	0
$205$	175.087	0.854081
$206$	− 190.013i	− 0.922394i
$207$	0	0
$208$	265.605	1.27695
$209$	0	0
$210$	0	0
$211$	275.123	1.30390	0.651951	−	0.758261i	$-0.273951\pi$
$211$	275.123	1.30390	0.651951	+	0.758261i	$0.273951\pi$
$212$	11.4329i	0.0539286i
$213$	0	0
$214$	36.1334	0.168848
$215$	516.691i	2.40321i
$216$	0	0
$217$	78.5415	0.361942
$218$	− 99.5094i	− 0.456465i
$219$	0	0
$220$	0	0
$221$	132.944i	0.601558i
$222$	0	0
$223$	182.909	0.820218	0.410109	−	0.912037i	$-0.365491\pi$
$223$	182.909	0.820218	0.410109	+	0.912037i	$0.365491\pi$
$224$	11.7181i	0.0523128i
$225$	0	0
$226$	−19.3238	−0.0855035
$227$	250.411i	1.10313i	0.834132	+	0.551565i	$0.185969\pi$
$227$	250.411i	1.10313i	−0.834132	+	0.551565i	$0.814031\pi$
$228$	0	0
$229$	−350.485	−1.53050	−0.765252	−	0.643731i	$-0.777386\pi$
$229$	−350.485	−1.53050	−0.765252	+	0.643731i	$0.777386\pi$
$230$	− 107.479i	− 0.467298i
$231$	0	0
$232$	195.901	0.844400
$233$	364.712i	1.56529i	0.622471	+	0.782643i	$0.286129\pi$
$233$	364.712i	1.56529i	−0.622471	+	0.782643i	$0.713871\pi$
$234$	0	0
$235$	−373.466	−1.58922
$236$	0 6.08206e-5i	0 2.57715e-7i
$237$	0	0
$238$	79.3151	0.333257
$239$	− 242.816i	− 1.01597i	−0.861367	−	0.507983i	$-0.830391\pi$
$239$	− 242.816i	− 1.01597i	0.861367	−	0.507983i	$-0.169609\pi$
$240$	0	0
$241$	418.345	1.73587	0.867935	−	0.496677i	$-0.165447\pi$
$241$	418.345	1.73587	0.867935	+	0.496677i	$0.165447\pi$
$242$	0	0
$243$	0	0
$244$	−0.669580	−0.00274418
$245$	− 208.245i	− 0.849980i
$246$	0	0
$247$	241.645	0.978321
$248$	122.138i	0.492490i
$249$	0	0
$250$	805.262	3.22105
$251$	276.186i	1.10034i	0.835051	+	0.550172i	$0.185438\pi$
$251$	276.186i	1.10034i	−0.835051	+	0.550172i	$0.814562\pi$
$252$	0	0
$253$	0	0
$254$	− 423.348i	− 1.66672i
$255$	0	0
$256$	−26.8629	−0.104933
$257$	− 161.198i	− 0.627231i	−0.949550	−	0.313615i	$-0.898460\pi$
$257$	− 161.198i	− 0.627231i	0.949550	−	0.313615i	$-0.101540\pi$
$258$	0	0
$259$	−40.0417	−0.154601
$260$	− 23.2185i	− 0.0893019i
$261$	0	0
$262$	−327.290	−1.24920
$263$	− 55.4784i	− 0.210944i	−0.994422	−	0.105472i	$-0.966365\pi$
$263$	− 55.4784i	− 0.210944i	0.994422	−	0.105472i	$-0.0336354\pi$
$264$	0	0
$265$	−785.340	−2.96355
$266$	− 144.167i	− 0.541980i
$267$	0	0
$268$	15.6609	0.0584361
$269$	− 58.2016i	− 0.216363i	−0.994131	−	0.108181i	$-0.965497\pi$
$269$	− 58.2016i	− 0.216363i	0.994131	−	0.108181i	$-0.0345027\pi$
$270$	0	0
$271$	−425.905	−1.57161	−0.785803	−	0.618477i	$-0.787750\pi$
$271$	−425.905	−1.57161	−0.785803	+	0.618477i	$0.787750\pi$
$272$	119.016i	0.437558i
$273$	0	0
$274$	85.9501	0.313686
$275$	0	0
$276$	0	0
$277$	127.884	0.461677	0.230838	−	0.972992i	$-0.425853\pi$
$277$	127.884	0.461677	0.230838	+	0.972992i	$0.425853\pi$
$278$	202.506i	0.728438i
$279$	0	0
$280$	−409.392	−1.46211
$281$	74.3983i	0.264763i	0.991199	+	0.132381i	$0.0422624\pi$
$281$	74.3983i	0.264763i	−0.991199	+	0.132381i	$0.957738\pi$
$282$	0	0
$283$	−494.336	−1.74677	−0.873386	−	0.487029i	$-0.838081\pi$
$283$	−494.336	−1.74677	−0.873386	+	0.487029i	$0.838081\pi$
$284$	13.9571i	0.0491448i
$285$	0	0
$286$	0	0
$287$	− 95.1718i	− 0.331609i
$288$	0	0
$289$	229.429	0.793871
$290$	455.323i	1.57008i
$291$	0	0
$292$	4.80429	0.0164531
$293$	13.6119i	0.0464568i	0.999730	+	0.0232284i	$0.00739450\pi$
$293$	13.6119i	0.0464568i	−0.999730	+	0.0232284i	$0.992606\pi$
$294$	0	0
$295$	−0.00417786	−1.41622e−5 0
$296$	− 62.2676i	− 0.210363i
$297$	0	0
$298$	177.795	0.596628
$299$	− 97.9244i	− 0.327506i
$300$	0	0
$301$	280.857	0.933081
$302$	− 472.215i	− 1.56362i
$303$	0	0
$304$	216.328	0.711605
$305$	− 45.9945i	− 0.150801i
$306$	0	0
$307$	215.268	0.701198	0.350599	−	0.936526i	$-0.385978\pi$
$307$	215.268	0.701198	0.350599	+	0.936526i	$0.385978\pi$
$308$	0	0
$309$	0	0
$310$	−283.879	−0.915738
$311$	90.0470i	0.289540i	0.989465	+	0.144770i	$0.0462442\pi$
$311$	90.0470i	0.289540i	−0.989465	+	0.144770i	$0.953756\pi$
$312$	0	0
$313$	258.266	0.825130	0.412565	−	0.910928i	$-0.364633\pi$
$313$	258.266	0.825130	0.412565	+	0.910928i	$0.364633\pi$
$314$	− 58.1611i	− 0.185227i
$315$	0	0
$316$	8.64728	0.0273648
$317$	105.042i	0.331363i	0.986179	+	0.165681i	$0.0529824\pi$
$317$	105.042i	0.331363i	−0.986179	+	0.165681i	$0.947018\pi$
$318$	0	0
$319$	0	0
$320$	− 635.878i	− 1.98712i
$321$	0	0
$322$	−58.4221	−0.181435
$323$	108.279i	0.335231i
$324$	0	0
$325$	1164.30	3.58245
$326$	412.746i	1.26609i
$327$	0	0
$328$	147.999	0.451216
$329$	203.005i	0.617036i
$330$	0	0
$331$	258.731	0.781664	0.390832	−	0.920462i	$-0.372187\pi$
$331$	258.731	0.781664	0.390832	+	0.920462i	$0.372187\pi$
$332$	− 9.13791i	− 0.0275238i
$333$	0	0
$334$	29.9540	0.0896826
$335$	1075.77i	3.21125i
$336$	0	0
$337$	367.989	1.09196	0.545978	−	0.837799i	$-0.316158\pi$
$337$	367.989	1.09196	0.545978	+	0.837799i	$0.316158\pi$
$338$	250.867i	0.742209i
$339$	0	0
$340$	10.4040	0.0306001
$341$	0	0
$342$	0	0
$343$	−369.493	−1.07724
$344$	436.753i	1.26963i
$345$	0	0
$346$	−421.217	−1.21739
$347$	417.073i	1.20194i	0.799272	+	0.600969i	$0.205219\pi$
$347$	417.073i	1.20194i	−0.799272	+	0.600969i	$0.794781\pi$
$348$	0	0
$349$	−279.831	−0.801807	−0.400903	−	0.916120i	$-0.631304\pi$
$349$	−279.831	−0.801807	−0.400903	+	0.916120i	$0.631304\pi$
$350$	− 694.624i	− 1.98464i
$351$	0	0
$352$	0	0
$353$	333.271i	0.944111i	0.881569	+	0.472056i	$0.156488\pi$
$353$	333.271i	0.944111i	−0.881569	+	0.472056i	$0.843512\pi$
$354$	0	0
$355$	−958.736	−2.70066
$356$	12.9715i	0.0364368i
$357$	0	0
$358$	401.962	1.12280
$359$	496.763i	1.38374i	0.722022	+	0.691871i	$0.243213\pi$
$359$	496.763i	1.38374i	−0.722022	+	0.691871i	$0.756787\pi$
$360$	0	0
$361$	−164.187	−0.454810
$362$	268.685i	0.742224i
$363$	0	0
$364$	−12.6209	−0.0346727
$365$	330.014i	0.904148i
$366$	0	0
$367$	165.189	0.450106	0.225053	−	0.974346i	$-0.427744\pi$
$367$	165.189	0.450106	0.225053	+	0.974346i	$0.427744\pi$
$368$	− 87.6648i	− 0.238220i
$369$	0	0
$370$	144.726	0.391151
$371$	426.887i	1.15064i
$372$	0	0
$373$	−393.501	−1.05496	−0.527481	−	0.849567i	$-0.676864\pi$
$373$	−393.501	−1.05496	−0.527481	+	0.849567i	$0.676864\pi$
$374$	0	0
$375$	0	0
$376$	−315.687	−0.839592
$377$	414.848i	1.10039i
$378$	0	0
$379$	189.140	0.499049	0.249525	−	0.968368i	$-0.419726\pi$
$379$	189.140	0.499049	0.249525	+	0.968368i	$0.419726\pi$
$380$	− 18.9108i	− 0.0497653i
$381$	0	0
$382$	−480.815	−1.25868
$383$	− 468.923i	− 1.22434i	−0.790725	−	0.612171i	$-0.790296\pi$
$383$	− 468.923i	− 1.22434i	0.790725	−	0.612171i	$-0.209704\pi$
$384$	0	0
$385$	0	0
$386$	30.9168i	0.0800954i
$387$	0	0
$388$	−7.34374	−0.0189272
$389$	454.980i	1.16961i	0.811173	+	0.584807i	$0.198830\pi$
$389$	454.980i	1.16961i	−0.811173	+	0.584807i	$0.801170\pi$
$390$	0	0
$391$	43.8792	0.112223
$392$	− 176.027i	− 0.449049i
$393$	0	0
$394$	−187.463	−0.475794
$395$	593.994i	1.50378i
$396$	0	0
$397$	168.638	0.424782	0.212391	−	0.977185i	$-0.431875\pi$
$397$	168.638	0.424782	0.212391	+	0.977185i	$0.431875\pi$
$398$	− 128.138i	− 0.321956i
$399$	0	0
$400$	1042.31	2.60578
$401$	69.7963i	0.174056i	0.996206	+	0.0870278i	$0.0277369\pi$
$401$	69.7963i	0.174056i	−0.996206	+	0.0870278i	$0.972263\pi$
$402$	0	0
$403$	−258.644	−0.641796
$404$	− 8.25409i	− 0.0204309i
$405$	0	0
$406$	247.500	0.609606
$407$	0	0
$408$	0	0
$409$	−553.915	−1.35431	−0.677157	−	0.735838i	$-0.736788\pi$
$409$	−553.915	−1.35431	−0.677157	+	0.735838i	$0.736788\pi$
$410$	343.987i	0.838992i
$411$	0	0
$412$	−13.5483	−0.0328842
$413$	0.00227096i	0 5.49869e-6i
$414$	0	0
$415$	627.697	1.51252
$416$	− 38.5886i	− 0.0927610i
$417$	0	0
$418$	0	0
$419$	− 307.485i	− 0.733855i	−0.930250	−	0.366927i	$-0.880410\pi$
$419$	− 307.485i	− 0.733855i	0.930250	−	0.366927i	$-0.119590\pi$
$420$	0	0
$421$	−3.54659	−0.00842420	−0.00421210	−	0.999991i	$-0.501341\pi$
$421$	−3.54659	−0.00842420	−0.00421210	+	0.999991i	$0.501341\pi$
$422$	540.526i	1.28087i
$423$	0	0
$424$	−663.839	−1.56566
$425$	521.713i	1.22756i
$426$	0	0
$427$	−25.0012	−0.0585508
$428$	− 2.57638i	− 0.00601959i
$429$	0	0
$430$	−1015.12	−2.36076
$431$	118.926i	0.275930i	0.990437	+	0.137965i	$0.0440562\pi$
$431$	118.926i	0.275930i	−0.990437	+	0.137965i	$0.955944\pi$
$432$	0	0
$433$	199.452	0.460628	0.230314	−	0.973116i	$-0.426025\pi$
$433$	199.452	0.460628	0.230314	+	0.973116i	$0.426025\pi$
$434$	154.308i	0.355548i
$435$	0	0
$436$	−7.09521	−0.0162734
$437$	− 79.7567i	− 0.182510i
$438$	0	0
$439$	7.94474	0.0180973	0.00904867	−	0.999959i	$-0.497120\pi$
$439$	7.94474	0.0180973	0.00904867	+	0.999959i	$0.497120\pi$
$440$	0	0
$441$	0	0
$442$	−261.191	−0.590930
$443$	469.310i	1.05939i	0.848188	+	0.529696i	$0.177694\pi$
$443$	469.310i	1.05939i	−0.848188	+	0.529696i	$0.822306\pi$
$444$	0	0
$445$	−891.031	−2.00232
$446$	359.354i	0.805728i
$447$	0	0
$448$	−345.644	−0.771527
$449$	738.513i	1.64479i	0.568913	+	0.822397i	$0.307364\pi$
$449$	738.513i	1.64479i	−0.568913	+	0.822397i	$0.692636\pi$
$450$	0	0
$451$	0	0
$452$	1.37782i	0.00304828i
$453$	0	0
$454$	−491.974	−1.08364
$455$	− 866.946i	− 1.90538i
$456$	0	0
$457$	181.188	0.396473	0.198236	−	0.980154i	$-0.436479\pi$
$457$	181.188	0.396473	0.198236	+	0.980154i	$0.436479\pi$
$458$	− 688.587i	− 1.50347i
$459$	0	0
$460$	−7.66343	−0.0166596
$461$	− 561.409i	− 1.21781i	−0.793244	−	0.608903i	$-0.791610\pi$
$461$	− 561.409i	− 1.21781i	0.793244	−	0.608903i	$-0.208390\pi$
$462$	0	0
$463$	379.155	0.818909	0.409454	−	0.912331i	$-0.365719\pi$
$463$	379.155	0.818909	0.409454	+	0.912331i	$0.365719\pi$
$464$	371.384i	0.800397i
$465$	0	0
$466$	−716.537	−1.53763
$467$	− 456.305i	− 0.977098i	−0.872536	−	0.488549i	$-0.837526\pi$
$467$	− 456.305i	− 0.977098i	0.872536	−	0.488549i	$-0.162474\pi$
$468$	0	0
$469$	584.756	1.24681
$470$	− 733.737i	− 1.56114i
$471$	0	0
$472$	−0.00353150	−7.48199e−6 0
$473$	0	0
$474$	0	0
$475$	948.287	1.99639
$476$	− 5.65532i	− 0.0118809i
$477$	0	0
$478$	477.052	0.998017
$479$	538.316i	1.12383i	0.827194	+	0.561917i	$0.189936\pi$
$479$	538.316i	1.12383i	−0.827194	+	0.561917i	$0.810064\pi$
$480$	0	0
$481$	131.861	0.274138
$482$	821.908i	1.70520i
$483$	0	0
$484$	0	0
$485$	− 504.452i	− 1.04011i
$486$	0	0
$487$	194.690	0.399775	0.199887	−	0.979819i	$-0.435942\pi$
$487$	194.690	0.399775	0.199887	+	0.979819i	$0.435942\pi$
$488$	− 38.8786i	− 0.0796693i
$489$	0	0
$490$	409.132	0.834964
$491$	203.856i	0.415185i	0.978215	+	0.207593i	$0.0665628\pi$
$491$	203.856i	0.415185i	−0.978215	+	0.207593i	$0.933437\pi$
$492$	0	0
$493$	−185.890	−0.377060
$494$	474.752i	0.961037i
$495$	0	0
$496$	−231.546	−0.466826
$497$	521.140i	1.04857i
$498$	0	0
$499$	−855.119	−1.71367	−0.856833	−	0.515595i	$-0.827571\pi$
$499$	−855.119	−1.71367	−0.856833	+	0.515595i	$0.827571\pi$
$500$	− 57.4167i	− 0.114833i
$501$	0	0
$502$	−542.614	−1.08090
$503$	− 272.283i	− 0.541319i	−0.962675	−	0.270659i	$-0.912758\pi$
$503$	− 272.283i	− 0.541319i	0.962675	−	0.270659i	$-0.0872417\pi$
$504$	0	0
$505$	566.986	1.12274
$506$	0	0
$507$	0	0
$508$	−30.1855	−0.0594203
$509$	− 943.531i	− 1.85370i	−0.375437	−	0.926848i	$-0.622507\pi$
$509$	− 943.531i	− 1.85370i	0.375437	−	0.926848i	$-0.377493\pi$
$510$	0	0
$511$	179.386	0.351048
$512$	− 536.245i	− 1.04735i
$513$	0	0
$514$	316.701	0.616150
$515$	− 930.653i	− 1.80709i
$516$	0	0
$517$	0	0
$518$	− 78.6685i	− 0.151870i
$519$	0	0
$520$	1348.16	2.59262
$521$	324.370i	0.622591i	0.950313	+	0.311295i	$0.100763\pi$
$521$	324.370i	0.622591i	−0.950313	+	0.311295i	$0.899237\pi$
$522$	0	0
$523$	−121.137	−0.231620	−0.115810	−	0.993271i	$-0.536946\pi$
$523$	−121.137	−0.231620	−0.115810	+	0.993271i	$0.536946\pi$
$524$	23.3364i	0.0445351i
$525$	0	0
$526$	108.997	0.207218
$527$	− 115.896i	− 0.219917i
$528$	0	0
$529$	496.679	0.938902
$530$	− 1542.93i	− 2.91119i
$531$	0	0
$532$	−10.2794	−0.0193221
$533$	313.408i	0.588008i
$534$	0	0
$535$	176.976	0.330795
$536$	909.336i	1.69652i
$537$	0	0
$538$	114.347	0.212540
$539$	0	0
$540$	0	0
$541$	596.978	1.10347	0.551736	−	0.834019i	$-0.313966\pi$
$541$	596.978	1.10347	0.551736	+	0.834019i	$0.313966\pi$
$542$	− 836.762i	− 1.54384i
$543$	0	0
$544$	17.2913	0.0317854
$545$	− 487.381i	− 0.894277i
$546$	0	0
$547$	338.385	0.618620	0.309310	−	0.950961i	$-0.399902\pi$
$547$	338.385	0.618620	0.309310	+	0.950961i	$0.399902\pi$
$548$	− 6.12841i	− 0.0111832i
$549$	0	0
$550$	0	0
$551$	337.882i	0.613217i
$552$	0	0
$553$	322.877	0.583865
$554$	251.250i	0.453520i
$555$	0	0
$556$	14.4391	0.0259695
$557$	− 477.662i	− 0.857562i	−0.903408	−	0.428781i	$-0.858943\pi$
$557$	− 477.662i	− 0.857562i	0.903408	−	0.428781i	$-0.141057\pi$
$558$	0	0
$559$	−924.886	−1.65454
$560$	− 776.116i	− 1.38592i
$561$	0	0
$562$	−146.168	−0.260085
$563$	− 471.956i	− 0.838288i	−0.907920	−	0.419144i	$-0.862330\pi$
$563$	− 471.956i	− 0.838288i	0.907920	−	0.419144i	$-0.137670\pi$
$564$	0	0
$565$	−94.6447	−0.167513
$566$	− 971.206i	− 1.71591i
$567$	0	0
$568$	−810.409	−1.42678
$569$	128.554i	0.225929i	0.993599	+	0.112965i	$0.0360346\pi$
$569$	128.554i	0.225929i	−0.993599	+	0.112965i	$0.963965\pi$
$570$	0	0
$571$	−266.234	−0.466259	−0.233130	−	0.972446i	$-0.574897\pi$
$571$	−266.234	−0.466259	−0.233130	+	0.972446i	$0.574897\pi$
$572$	0	0
$573$	0	0
$574$	186.981	0.325751
$575$	− 384.284i	− 0.668320i
$576$	0	0
$577$	720.993	1.24955	0.624777	−	0.780803i	$-0.285190\pi$
$577$	720.993	1.24955	0.624777	+	0.780803i	$0.285190\pi$
$578$	450.751i	0.779846i
$579$	0	0
$580$	32.4654	0.0559749
$581$	− 341.197i	− 0.587258i
$582$	0	0
$583$	0	0
$584$	278.957i	0.477667i
$585$	0	0
$586$	−26.7428	−0.0456361
$587$	− 411.867i	− 0.701648i	−0.936441	−	0.350824i	$-0.885902\pi$
$587$	− 411.867i	− 0.701648i	0.936441	−	0.350824i	$-0.114098\pi$
$588$	0	0
$589$	−210.658	−0.357654
$590$	− 0.00820810i	0 1.39120e-5i
$591$	0	0
$592$	118.045	0.199401
$593$	− 750.336i	− 1.26532i	−0.774429	−	0.632661i	$-0.781963\pi$
$593$	− 750.336i	− 1.26532i	0.774429	−	0.632661i	$-0.218037\pi$
$594$	0	0
$595$	388.472	0.652895
$596$	− 12.6771i	− 0.0212704i
$597$	0	0
$598$	192.389	0.321720
$599$	543.614i	0.907535i	0.891120	+	0.453768i	$0.149920\pi$
$599$	543.614i	0.907535i	−0.891120	+	0.453768i	$0.850080\pi$
$600$	0	0
$601$	736.195	1.22495	0.612475	−	0.790490i	$-0.290174\pi$
$601$	736.195	1.22495	0.612475	+	0.790490i	$0.290174\pi$
$602$	551.791i	0.916596i
$603$	0	0
$604$	−33.6698	−0.0557447
$605$	0	0
$606$	0	0
$607$	−559.641	−0.921979	−0.460989	−	0.887406i	$-0.652505\pi$
$607$	−559.641	−0.921979	−0.460989	+	0.887406i	$0.652505\pi$
$608$	− 31.4293i	− 0.0516930i
$609$	0	0
$610$	90.3638	0.148137
$611$	− 668.512i	− 1.09413i
$612$	0	0
$613$	109.915	0.179306	0.0896530	−	0.995973i	$-0.471424\pi$
$613$	109.915	0.179306	0.0896530	+	0.995973i	$0.471424\pi$
$614$	422.929i	0.688810i
$615$	0	0
$616$	0	0
$617$	61.1594i	0.0991238i	0.998771	+	0.0495619i	$0.0157825\pi$
$617$	61.1594i	0.0991238i	−0.998771	+	0.0495619i	$0.984217\pi$
$618$	0	0
$619$	81.8184	0.132178	0.0660892	−	0.997814i	$-0.478948\pi$
$619$	81.8184	0.132178	0.0660892	+	0.997814i	$0.478948\pi$
$620$	20.2411i	0.0326469i
$621$	0	0
$622$	−176.912	−0.284425
$623$	484.337i	0.777427i
$624$	0	0
$625$	2254.17	3.60668
$626$	507.406i	0.810553i
$627$	0	0
$628$	−4.14700	−0.00660351
$629$	59.0857i	0.0939360i
$630$	0	0
$631$	9.29210	0.0147260	0.00736300	−	0.999973i	$-0.497656\pi$
$631$	9.29210	0.0147260	0.00736300	+	0.999973i	$0.497656\pi$
$632$	502.097i	0.794457i
$633$	0	0
$634$	−206.373	−0.325509
$635$	− 2073.49i	− 3.26533i
$636$	0	0
$637$	372.763	0.585185
$638$	0	0
$639$	0	0
$640$	1163.06	1.81728
$641$	− 349.041i	− 0.544525i	−0.962223	−	0.272263i	$-0.912228\pi$
$641$	− 349.041i	− 0.544525i	0.962223	−	0.272263i	$-0.0877719\pi$
$642$	0	0
$643$	906.166	1.40928	0.704639	−	0.709566i	$-0.251109\pi$
$643$	906.166	1.40928	0.704639	+	0.709566i	$0.251109\pi$
$644$	4.16561i	0.00646833i
$645$	0	0
$646$	−212.733	−0.329308
$647$	− 56.5419i	− 0.0873909i	−0.999045	−	0.0436954i	$-0.986087\pi$
$647$	− 56.5419i	− 0.0873909i	0.999045	−	0.0436954i	$-0.0139131\pi$
$648$	0	0
$649$	0	0
$650$	2287.45i	3.51916i
$651$	0	0
$652$	29.4296	0.0451373
$653$	− 568.547i	− 0.870669i	−0.900269	−	0.435334i	$-0.856630\pi$
$653$	− 568.547i	− 0.870669i	0.900269	−	0.435334i	$-0.143370\pi$
$654$	0	0
$655$	−1603.01	−2.44735
$656$	280.573i	0.427702i
$657$	0	0
$658$	−398.837	−0.606135
$659$	− 380.801i	− 0.577847i	−0.957352	−	0.288923i	$-0.906703\pi$
$659$	− 380.801i	− 0.577847i	0.957352	−	0.288923i	$-0.0932973\pi$
$660$	0	0
$661$	350.773	0.530671	0.265335	−	0.964156i	$-0.414517\pi$
$661$	350.773	0.530671	0.265335	+	0.964156i	$0.414517\pi$
$662$	508.319i	0.767854i
$663$	0	0
$664$	530.585	0.799074
$665$	− 706.104i	− 1.06181i
$666$	0	0
$667$	136.924	0.205283
$668$	− 2.13578i	− 0.00319727i
$669$	0	0
$670$	−2113.53	−3.15452
$671$	0	0
$672$	0	0
$673$	482.896	0.717527	0.358763	−	0.933429i	$-0.383198\pi$
$673$	482.896	0.717527	0.358763	+	0.933429i	$0.383198\pi$
$674$	722.976i	1.07267i
$675$	0	0
$676$	17.8873	0.0264605
$677$	849.679i	1.25507i	0.778590	+	0.627533i	$0.215935\pi$
$677$	849.679i	1.25507i	−0.778590	+	0.627533i	$0.784065\pi$
$678$	0	0
$679$	−274.205	−0.403836
$680$	604.101i	0.888384i
$681$	0	0
$682$	0	0
$683$	− 831.602i	− 1.21757i	−0.793334	−	0.608786i	$-0.791657\pi$
$683$	− 831.602i	− 1.21757i	0.793334	−	0.608786i	$-0.208343\pi$
$684$	0	0
$685$	420.969	0.614554
$686$	− 725.931i	− 1.05821i
$687$	0	0
$688$	−827.986	−1.20347
$689$	− 1405.77i	− 2.04031i
$690$	0	0
$691$	910.760	1.31803	0.659016	−	0.752129i	$-0.270973\pi$
$691$	910.760	1.31803	0.659016	+	0.752129i	$0.270973\pi$
$692$	30.0336i	0.0434011i
$693$	0	0
$694$	−819.409	−1.18070
$695$	991.840i	1.42711i
$696$	0	0
$697$	−140.436	−0.201486
$698$	− 549.774i	− 0.787642i
$699$	0	0
$700$	−49.5280	−0.0707543
$701$	900.411i	1.28447i	0.766509	+	0.642234i	$0.221992\pi$
$701$	900.411i	1.28447i	−0.766509	+	0.642234i	$0.778008\pi$
$702$	0	0
$703$	107.397	0.152769
$704$	0	0
$705$	0	0
$706$	−654.767	−0.927432
$707$	− 308.196i	− 0.435921i
$708$	0	0
$709$	−922.297	−1.30084	−0.650421	−	0.759574i	$-0.725408\pi$
$709$	−922.297	−1.30084	−0.650421	+	0.759574i	$0.725408\pi$
$710$	− 1883.60i	− 2.65295i
$711$	0	0
$712$	−753.178	−1.05783
$713$	85.3672i	0.119730i
$714$	0	0
$715$	0	0
$716$	− 28.6606i	− 0.0400288i
$717$	0	0
$718$	−975.974	−1.35930
$719$	− 10.3358i	− 0.0143753i	−0.999974	−	0.00718763i	$-0.997712\pi$
$719$	− 10.3358i	− 0.0143753i	0.999974	−	0.00718763i	$-0.00228791\pi$
$720$	0	0
$721$	−505.875	−0.701629
$722$	− 322.572i	− 0.446775i
$723$	0	0
$724$	19.1578	0.0264610
$725$	1627.99i	2.24550i
$726$	0	0
$727$	119.809	0.164799	0.0823995	−	0.996599i	$-0.473742\pi$
$727$	119.809	0.164799	0.0823995	+	0.996599i	$0.473742\pi$
$728$	− 732.820i	− 1.00662i
$729$	0	0
$730$	−648.368	−0.888175
$731$	− 414.435i	− 0.566942i
$732$	0	0
$733$	−1272.26	−1.73570	−0.867848	−	0.496830i	$-0.834497\pi$
$733$	−1272.26	−1.73570	−0.867848	+	0.496830i	$0.834497\pi$
$734$	324.541i	0.442154i
$735$	0	0
$736$	−12.7364	−0.0173049
$737$	0	0
$738$	0	0
$739$	168.978	0.228658	0.114329	−	0.993443i	$-0.463528\pi$
$739$	168.978	0.228658	0.114329	+	0.993443i	$0.463528\pi$
$740$	− 10.3192i	− 0.0139449i
$741$	0	0
$742$	−838.691	−1.13031
$743$	− 963.291i	− 1.29649i	−0.761432	−	0.648245i	$-0.775503\pi$
$743$	− 963.291i	− 1.29649i	0.761432	−	0.648245i	$-0.224497\pi$
$744$	0	0
$745$	870.811	1.16887
$746$	− 773.098i	− 1.03632i
$747$	0	0
$748$	0	0
$749$	− 96.1986i	− 0.128436i
$750$	0	0
$751$	−412.835	−0.549713	−0.274857	−	0.961485i	$-0.588630\pi$
$751$	−412.835	−0.549713	−0.274857	+	0.961485i	$0.588630\pi$
$752$	− 598.472i	− 0.795840i
$753$	0	0
$754$	−815.038	−1.08095
$755$	− 2312.83i	− 3.06335i
$756$	0	0
$757$	902.073	1.19164	0.595821	−	0.803117i	$-0.296827\pi$
$757$	902.073	1.19164	0.595821	+	0.803117i	$0.296827\pi$
$758$	371.596i	0.490233i
$759$	0	0
$760$	1098.04	1.44479
$761$	972.116i	1.27742i	0.769448	+	0.638709i	$0.220531\pi$
$761$	972.116i	1.27742i	−0.769448	+	0.638709i	$0.779469\pi$
$762$	0	0
$763$	−264.925	−0.347215
$764$	34.2830i	0.0448731i
$765$	0	0
$766$	921.278	1.20271
$767$	− 0.00747845i	0 9.75027e-6i
$768$	0	0
$769$	−1404.29	−1.82613	−0.913064	−	0.407817i	$-0.866290\pi$
$769$	−1404.29	−1.82613	−0.913064	+	0.407817i	$0.866290\pi$
$770$	0	0
$771$	0	0
$772$	2.20443	0.00285548
$773$	145.730i	0.188525i	0.995547	+	0.0942627i	$0.0300494\pi$
$773$	145.730i	0.188525i	−0.995547	+	0.0942627i	$0.969951\pi$
$774$	0	0
$775$	−1014.99	−1.30967
$776$	− 426.408i	− 0.549494i
$777$	0	0
$778$	−893.883	−1.14895
$779$	255.263i	0.327680i
$780$	0	0
$781$	0	0
$782$	86.2080i	0.110240i
$783$	0	0
$784$	333.709	0.425649
$785$	− 284.864i	− 0.362884i
$786$	0	0
$787$	914.791	1.16238	0.581189	−	0.813769i	$-0.302588\pi$
$787$	914.791	1.16238	0.581189	+	0.813769i	$0.302588\pi$
$788$	13.3665i	0.0169625i
$789$	0	0
$790$	−1167.00	−1.47722
$791$	51.4460i	0.0650392i
$792$	0	0
$793$	82.3310	0.103822
$794$	331.318i	0.417277i
$795$	0	0
$796$	−9.13651	−0.0114780
$797$	373.062i	0.468082i	0.972227	+	0.234041i	$0.0751951\pi$
$797$	373.062i	0.468082i	−0.972227	+	0.234041i	$0.924805\pi$
$798$	0	0
$799$	299.555	0.374913
$800$	− 151.433i	− 0.189291i
$801$	0	0
$802$	−137.126	−0.170981
$803$	0	0
$804$	0	0
$805$	−286.142	−0.355456
$806$	− 508.149i	− 0.630457i
$807$	0	0
$808$	479.267	0.593152
$809$	1578.26i	1.95088i	0.220263	+	0.975440i	$0.429308\pi$
$809$	1578.26i	1.95088i	−0.220263	+	0.975440i	$0.570692\pi$
$810$	0	0
$811$	−720.185	−0.888021	−0.444010	−	0.896022i	$-0.646445\pi$
$811$	−720.185	−0.888021	−0.444010	+	0.896022i	$0.646445\pi$
$812$	− 17.6472i	− 0.0217330i
$813$	0	0
$814$	0	0
$815$	2021.56i	2.48044i
$816$	0	0
$817$	−753.295	−0.922025
$818$	− 1088.26i	− 1.33039i
$819$	0	0
$820$	24.5269	0.0299109
$821$	932.683i	1.13603i	0.823017	+	0.568017i	$0.192289\pi$
$821$	932.683i	1.13603i	−0.823017	+	0.568017i	$0.807711\pi$
$822$	0	0
$823$	499.235	0.606604	0.303302	−	0.952895i	$-0.401911\pi$
$823$	499.235	0.606604	0.303302	+	0.952895i	$0.401911\pi$
$824$	− 786.670i	− 0.954697i
$825$	0	0
$826$	−0.00446168	−5.40155e−6 0
$827$	− 729.473i	− 0.882072i	−0.897490	−	0.441036i	$-0.854611\pi$
$827$	− 729.473i	− 0.882072i	0.897490	−	0.441036i	$-0.145389\pi$
$828$	0	0
$829$	−1480.29	−1.78564	−0.892818	−	0.450418i	$-0.851275\pi$
$829$	−1480.29	−1.78564	−0.892818	+	0.450418i	$0.851275\pi$
$830$	1233.22i	1.48580i
$831$	0	0
$832$	1138.23	1.36807
$833$	167.032i	0.200519i
$834$	0	0
$835$	146.710	0.175700
$836$	0	0
$837$	0	0
$838$	604.106	0.720890
$839$	− 980.006i	− 1.16806i	−0.811730	−	0.584032i	$-0.801474\pi$
$839$	− 980.006i	− 1.16806i	0.811730	−	0.584032i	$-0.198526\pi$
$840$	0	0
$841$	260.936	0.310268
$842$	− 6.96787i	− 0.00827538i
$843$	0	0
$844$	38.5405	0.0456641
$845$	1228.70i	1.45409i
$846$	0	0
$847$	0	0
$848$	− 1258.49i	− 1.48407i
$849$	0	0
$850$	−1024.99	−1.20587
$851$	− 43.5215i	− 0.0511416i
$852$	0	0
$853$	1390.45	1.63007	0.815037	−	0.579408i	$-0.196716\pi$
$853$	1390.45	1.63007	0.815037	+	0.579408i	$0.196716\pi$
$854$	− 49.1190i	− 0.0575164i
$855$	0	0
$856$	149.595	0.174761
$857$	− 1044.64i	− 1.21895i	−0.792804	−	0.609477i	$-0.791379\pi$
$857$	− 1044.64i	− 1.21895i	0.792804	−	0.609477i	$-0.208621\pi$
$858$	0	0
$859$	480.571	0.559454	0.279727	−	0.960080i	$-0.409756\pi$
$859$	480.571	0.559454	0.279727	+	0.960080i	$0.409756\pi$
$860$	72.3803i	0.0841632i
$861$	0	0
$862$	−233.650	−0.271056
$863$	652.084i	0.755602i	0.925887	+	0.377801i	$0.123320\pi$
$863$	652.084i	0.755602i	−0.925887	+	0.377801i	$0.876680\pi$
$864$	0	0
$865$	−2063.05	−2.38503
$866$	391.856i	0.452490i
$867$	0	0
$868$	11.0024	0.0126756
$869$	0	0
$870$	0	0
$871$	−1925.65	−2.21085
$872$	− 411.977i	− 0.472451i
$873$	0	0
$874$	156.695	0.179285
$875$	− 2143.86i	− 2.45013i
$876$	0	0
$877$	348.132	0.396958	0.198479	−	0.980105i	$-0.436400\pi$
$877$	348.132	0.396958	0.198479	+	0.980105i	$0.436400\pi$
$878$	15.6088i	0.0177776i
$879$	0	0
$880$	0	0
$881$	− 81.0204i	− 0.0919641i	−0.998942	−	0.0459821i	$-0.985358\pi$
$881$	− 81.0204i	− 0.0919641i	0.998942	−	0.0459821i	$-0.0146417\pi$
$882$	0	0
$883$	975.621	1.10489	0.552447	−	0.833548i	$-0.313694\pi$
$883$	975.621	1.10489	0.552447	+	0.833548i	$0.313694\pi$
$884$	18.6234i	0.0210672i
$885$	0	0
$886$	−922.039	−1.04068
$887$	1534.16i	1.72960i	0.502113	+	0.864802i	$0.332556\pi$
$887$	1534.16i	1.72960i	−0.502113	+	0.864802i	$0.667444\pi$
$888$	0	0
$889$	−1127.08	−1.26781
$890$	− 1750.58i	− 1.96694i
$891$	0	0
$892$	25.6227	0.0287250
$893$	− 544.485i	− 0.609725i
$894$	0	0
$895$	1968.74	2.19971
$896$	− 632.203i	− 0.705584i
$897$	0	0
$898$	−1450.93	−1.61574
$899$	− 361.650i	− 0.402281i
$900$	0	0
$901$	629.917	0.699131
$902$	0	0
$903$	0	0
$904$	−80.0021	−0.0884979
$905$	1315.97i	1.45412i
$906$	0	0
$907$	1040.31	1.14698	0.573490	−	0.819213i	$-0.305589\pi$
$907$	1040.31	1.14698	0.573490	+	0.819213i	$0.305589\pi$
$908$	35.0787i	0.0386329i
$909$	0	0
$910$	1703.26	1.87171
$911$	373.647i	0.410150i	0.978746	+	0.205075i	$0.0657439\pi$
$911$	373.647i	0.410150i	−0.978746	+	0.205075i	$0.934256\pi$
$912$	0	0
$913$	0	0
$914$	355.974i	0.389468i
$915$	0	0
$916$	−49.0976	−0.0536000
$917$	871.349i	0.950217i
$918$	0	0
$919$	−4.74930	−0.00516790	−0.00258395	−	0.999997i	$-0.500822\pi$
$919$	−4.74930	−0.00516790	−0.00258395	+	0.999997i	$0.500822\pi$
$920$	− 444.970i	− 0.483663i
$921$	0	0
$922$	1102.98	1.19629
$923$	− 1716.16i	− 1.85932i
$924$	0	0
$925$	517.460	0.559416
$926$	744.913i	0.804442i
$927$	0	0
$928$	53.9568	0.0581431
$929$	− 493.733i	− 0.531467i	−0.964047	−	0.265733i	$-0.914386\pi$
$929$	− 493.733i	− 0.531467i	0.964047	−	0.265733i	$-0.0856141\pi$
$930$	0	0
$931$	303.605	0.326107
$932$	51.0904i	0.0548181i
$933$	0	0
$934$	896.487	0.959836
$935$	0	0
$936$	0	0
$937$	587.140	0.626617	0.313308	−	0.949651i	$-0.398563\pi$
$937$	587.140	0.626617	0.313308	+	0.949651i	$0.398563\pi$
$938$	1148.85i	1.22479i
$939$	0	0
$940$	−52.3168	−0.0556562
$941$	− 1066.36i	− 1.13322i	−0.823986	−	0.566610i	$-0.808255\pi$
$941$	− 1066.36i	− 1.13322i	0.823986	−	0.566610i	$-0.191745\pi$
$942$	0	0
$943$	103.443	0.109695
$944$	− 0.00669494i	0 7.09209e-6i
$945$	0	0
$946$	0	0
$947$	198.618i	0.209734i	0.994486	+	0.104867i	$0.0334416\pi$
$947$	198.618i	0.209734i	−0.994486	+	0.104867i	$0.966558\pi$
$948$	0	0
$949$	−590.732	−0.622478
$950$	1863.07i	1.96112i
$951$	0	0
$952$	328.371	0.344928
$953$	1114.25i	1.16920i	0.811321	+	0.584601i	$0.198749\pi$
$953$	1114.25i	1.16920i	−0.811321	+	0.584601i	$0.801251\pi$
$954$	0	0
$955$	−2354.95	−2.46592
$956$	− 34.0147i	− 0.0355802i
$957$	0	0
$958$	−1057.61	−1.10398
$959$	− 228.826i	− 0.238609i
$960$	0	0
$961$	−735.523	−0.765373
$962$	259.062i	0.269295i
$963$	0	0
$964$	58.6036	0.0607921
$965$	151.425i	0.156918i
$966$	0	0
$967$	983.820	1.01739	0.508697	−	0.860945i	$-0.330127\pi$
$967$	983.820	1.01739	0.508697	+	0.860945i	$0.330127\pi$
$968$	0	0
$969$	0	0
$970$	991.080	1.02173
$971$	− 115.916i	− 0.119378i	−0.998217	−	0.0596890i	$-0.980989\pi$
$971$	− 115.916i	− 0.119378i	0.998217	−	0.0596890i	$-0.0190109\pi$
$972$	0	0
$973$	539.134	0.554095
$974$	382.502i	0.392712i
$975$	0	0
$976$	73.7052	0.0755176
$977$	114.282i	0.116973i	0.998288	+	0.0584863i	$0.0186274\pi$
$977$	114.282i	0.116973i	−0.998288	+	0.0584863i	$0.981373\pi$
$978$	0	0
$979$	0	0
$980$	− 29.1719i	− 0.0297673i
$981$	0	0
$982$	−400.509	−0.407850
$983$	− 1020.67i	− 1.03832i	−0.854677	−	0.519160i	$-0.826245\pi$
$983$	− 1020.67i	− 1.03832i	0.854677	−	0.519160i	$-0.173755\pi$
$984$	0	0
$985$	−918.163	−0.932145
$986$	− 365.213i	− 0.370398i
$987$	0	0
$988$	33.8508	0.0342619
$989$	305.265i	0.308661i
$990$	0	0
$991$	544.638	0.549584	0.274792	−	0.961504i	$-0.411391\pi$
$991$	544.638	0.549584	0.274792	+	0.961504i	$0.411391\pi$
$992$	33.6402i	0.0339115i
$993$	0	0
$994$	−1023.87	−1.03005
$995$	− 627.600i	− 0.630754i
$996$	0	0
$997$	−1249.59	−1.25335	−0.626677	−	0.779279i	$-0.715585\pi$
$997$	−1249.59	−1.25335	−0.626677	+	0.779279i	$0.715585\pi$
$998$	− 1680.02i	− 1.68339i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.j.485.12		16
3.2	odd	2	inner	1089.3.b.j.485.5		16
11.2	odd	10		99.3.l.a.26.3	✓	32
11.6	odd	10		99.3.l.a.80.6	yes	32
11.10	odd	2		1089.3.b.i.485.5		16
33.2	even	10		99.3.l.a.26.6	yes	32
33.17	even	10		99.3.l.a.80.3	yes	32
33.32	even	2		1089.3.b.i.485.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.26.3	✓	32	11.2	odd	10
99.3.l.a.26.6	yes	32	33.2	even	10
99.3.l.a.80.3	yes	32	33.17	even	10
99.3.l.a.80.6	yes	32	11.6	odd	10
1089.3.b.i.485.5		16	11.10	odd	2
1089.3.b.i.485.12		16	33.32	even	2
1089.3.b.j.485.5		16	3.2	odd	2	inner
1089.3.b.j.485.12		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.96467i q^{2} +0.140085 q^{4} +9.62261i q^{5} +5.23056 q^{7} +8.13389i q^{8} +O(q^{10})\)	\(q+1.96467i q^{2} +0.140085 q^{4} +9.62261i q^{5} +5.23056 q^{7} +8.13389i q^{8} -18.9052 q^{10} -17.2247 q^{13} +10.2763i q^{14} -15.4200 q^{16} -7.71825i q^{17} -14.0290 q^{19} +1.34798i q^{20} +5.68512i q^{23} -67.5947 q^{25} -33.8407i q^{26} +0.732721 q^{28} -24.0845i q^{29} +15.0159 q^{31} +2.24031i q^{32} +15.1638 q^{34} +50.3317i q^{35} -7.65533 q^{37} -27.5624i q^{38} -78.2692 q^{40} -18.1953i q^{41} +53.6955 q^{43} -11.1694 q^{46} +38.8113i q^{47} -21.6412 q^{49} -132.801i q^{50} -2.41291 q^{52} +81.6140i q^{53} +42.5448i q^{56} +47.3181 q^{58} +0.000434171i q^{59} -4.77983 q^{61} +29.5012i q^{62} -66.0816 q^{64} -165.746i q^{65} +111.796 q^{67} -1.08121i q^{68} -98.8850 q^{70} +99.6336i q^{71} +34.2957 q^{73} -15.0402i q^{74} -1.96525 q^{76} +61.7290 q^{79} -148.381i q^{80} +35.7478 q^{82} -65.2314i q^{83} +74.2697 q^{85} +105.494i q^{86} +92.5976i q^{89} -90.0947 q^{91} +0.796398i q^{92} -76.2513 q^{94} -134.996i q^{95} -52.4236 q^{97} -42.5178i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 8 q^{7}+O(q^{10}) \) 16 * q - 32 * q^4 + 8 * q^7	\( 16 q - 32 q^{4} + 8 q^{7} - 24 q^{10} - 4 q^{13} + 28 q^{16} + 20 q^{19} - 44 q^{25} - 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} - 224 q^{40} - 272 q^{43} - 208 q^{46} + 348 q^{49} - 520 q^{52} - 44 q^{58} - 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} + 4 q^{73} - 1052 q^{76} - 216 q^{79} + 348 q^{82} - 416 q^{85} - 168 q^{91} - 1140 q^{94} - 44 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 8 * q^7 - 24 * q^10 - 4 * q^13 + 28 * q^16 + 20 * q^19 - 44 * q^25 - 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 - 224 * q^40 - 272 * q^43 - 208 * q^46 + 348 * q^49 - 520 * q^52 - 44 * q^58 - 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 + 4 * q^73 - 1052 * q^76 - 216 * q^79 + 348 * q^82 - 416 * q^85 - 168 * q^91 - 1140 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

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