LMFDB - Embedded newform 1089.3.b.a.485.2

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:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.a.485.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +O(q^{10})$	$q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +8.00000 q^{10} -16.0000 q^{13} -1.41421i q^{14} -4.00000 q^{16} -11.3137i q^{17} -33.0000 q^{19} -11.3137i q^{20} -24.0416i q^{23} -7.00000 q^{25} -22.6274i q^{26} -2.00000 q^{28} +12.7279i q^{29} +31.0000 q^{31} +28.2843i q^{32} +16.0000 q^{34} +5.65685i q^{35} -57.0000 q^{37} -46.6690i q^{38} +48.0000 q^{40} -15.5563i q^{41} +34.0000 q^{46} +60.8112i q^{47} -48.0000 q^{49} -9.89949i q^{50} -32.0000 q^{52} -89.0955i q^{53} -8.48528i q^{56} -18.0000 q^{58} +69.2965i q^{59} -105.000 q^{61} +43.8406i q^{62} -56.0000 q^{64} +90.5097i q^{65} -103.000 q^{67} -22.6274i q^{68} -8.00000 q^{70} -118.794i q^{71} +47.0000 q^{73} -80.6102i q^{74} -66.0000 q^{76} -23.0000 q^{79} +22.6274i q^{80} +22.0000 q^{82} -106.066i q^{83} -64.0000 q^{85} +50.9117i q^{89} +16.0000 q^{91} -48.0833i q^{92} -86.0000 q^{94} +186.676i q^{95} -25.0000 q^{97} -67.8823i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 2 q^{7}+O(q^{10}) $ 2 * q + 4 * q^4 - 2 * q^7	$ 2 q + 4 q^{4} - 2 q^{7} + 16 q^{10} - 32 q^{13} - 8 q^{16} - 66 q^{19} - 14 q^{25} - 4 q^{28} + 62 q^{31} + 32 q^{34} - 114 q^{37} + 96 q^{40} + 68 q^{46} - 96 q^{49} - 64 q^{52} - 36 q^{58} - 210 q^{61} - 112 q^{64} - 206 q^{67} - 16 q^{70} + 94 q^{73} - 132 q^{76} - 46 q^{79} + 44 q^{82} - 128 q^{85} + 32 q^{91} - 172 q^{94} - 50 q^{97}+O(q^{100}) $ 2 * q + 4 * q^4 - 2 * q^7 + 16 * q^10 - 32 * q^13 - 8 * q^16 - 66 * q^19 - 14 * q^25 - 4 * q^28 + 62 * q^31 + 32 * q^34 - 114 * q^37 + 96 * q^40 + 68 * q^46 - 96 * q^49 - 64 * q^52 - 36 * q^58 - 210 * q^61 - 112 * q^64 - 206 * q^67 - 16 * q^70 + 94 * q^73 - 132 * q^76 - 46 * q^79 + 44 * q^82 - 128 * q^85 + 32 * q^91 - 172 * q^94 - 50 * 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 $

$2$	1.41421i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.41421i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	− 5.65685i	− 1.13137i	−0.824621	−	0.565685i	$-0.808612\pi$
$5$	− 5.65685i	− 1.13137i	0.824621	−	0.565685i	$-0.191388\pi$
$6$	0	0
$7$	−1.00000	−0.142857	−0.0714286	−	0.997446i	$-0.522756\pi$
$7$	−1.00000	−0.142857	−0.0714286	+	0.997446i	$0.522756\pi$
$8$	8.48528i	1.06066i
$9$	0	0
$10$	8.00000	0.800000
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−16.0000	−1.23077	−0.615385	−	0.788227i	$-0.710999\pi$
$13$	−16.0000	−1.23077	−0.615385	+	0.788227i	$0.710999\pi$
$14$	− 1.41421i	− 0.101015i
$15$	0	0
$16$	−4.00000	−0.250000
$17$	− 11.3137i	− 0.665512i	−0.943013	−	0.332756i	$-0.892021\pi$
$17$	− 11.3137i	− 0.665512i	0.943013	−	0.332756i	$-0.107979\pi$
$18$	0	0
$19$	−33.0000	−1.73684	−0.868421	−	0.495827i	$-0.834865\pi$
$19$	−33.0000	−1.73684	−0.868421	+	0.495827i	$0.834865\pi$
$20$	− 11.3137i	− 0.565685i
$21$	0	0
$22$	0	0
$23$	− 24.0416i	− 1.04529i	−0.852551	−	0.522644i	$-0.824946\pi$
$23$	− 24.0416i	− 1.04529i	0.852551	−	0.522644i	$-0.175054\pi$
$24$	0	0
$25$	−7.00000	−0.280000
$26$	− 22.6274i	− 0.870285i
$27$	0	0
$28$	−2.00000	−0.0714286
$29$	12.7279i	0.438894i	0.975624	+	0.219447i	$0.0704253\pi$
$29$	12.7279i	0.438894i	−0.975624	+	0.219447i	$0.929575\pi$
$30$	0	0
$31$	31.0000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$31$	31.0000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$32$	28.2843i	0.883883i
$33$	0	0
$34$	16.0000	0.470588
$35$	5.65685i	0.161624i
$36$	0	0
$37$	−57.0000	−1.54054	−0.770270	−	0.637718i	$-0.779879\pi$
$37$	−57.0000	−1.54054	−0.770270	+	0.637718i	$0.779879\pi$
$38$	− 46.6690i	− 1.22813i
$39$	0	0
$40$	48.0000	1.20000
$41$	− 15.5563i	− 0.379423i	−0.981840	−	0.189712i	$-0.939245\pi$
$41$	− 15.5563i	− 0.379423i	0.981840	−	0.189712i	$-0.0607553\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	34.0000	0.739130
$47$	60.8112i	1.29385i	0.762552	+	0.646927i	$0.223946\pi$
$47$	60.8112i	1.29385i	−0.762552	+	0.646927i	$0.776054\pi$
$48$	0	0
$49$	−48.0000	−0.979592
$50$	− 9.89949i	− 0.197990i
$51$	0	0
$52$	−32.0000	−0.615385
$53$	− 89.0955i	− 1.68105i	−0.541776	−	0.840523i	$-0.682248\pi$
$53$	− 89.0955i	− 1.68105i	0.541776	−	0.840523i	$-0.317752\pi$
$54$	0	0
$55$	0	0
$56$	− 8.48528i	− 0.151523i
$57$	0	0
$58$	−18.0000	−0.310345
$59$	69.2965i	1.17452i	0.809400	+	0.587258i	$0.199793\pi$
$59$	69.2965i	1.17452i	−0.809400	+	0.587258i	$0.800207\pi$
$60$	0	0
$61$	−105.000	−1.72131	−0.860656	−	0.509187i	$-0.829946\pi$
$61$	−105.000	−1.72131	−0.860656	+	0.509187i	$0.829946\pi$
$62$	43.8406i	0.707107i
$63$	0	0
$64$	−56.0000	−0.875000
$65$	90.5097i	1.39246i
$66$	0	0
$67$	−103.000	−1.53731	−0.768657	−	0.639662i	$-0.779075\pi$
$67$	−103.000	−1.53731	−0.768657	+	0.639662i	$0.779075\pi$
$68$	− 22.6274i	− 0.332756i
$69$	0	0
$70$	−8.00000	−0.114286
$71$	− 118.794i	− 1.67315i	−0.547849	−	0.836577i	$-0.684553\pi$
$71$	− 118.794i	− 1.67315i	0.547849	−	0.836577i	$-0.315447\pi$
$72$	0	0
$73$	47.0000	0.643836	0.321918	−	0.946768i	$-0.395673\pi$
$73$	47.0000	0.643836	0.321918	+	0.946768i	$0.395673\pi$
$74$	− 80.6102i	− 1.08933i
$75$	0	0
$76$	−66.0000	−0.868421
$77$	0	0
$78$	0	0
$79$	−23.0000	−0.291139	−0.145570	−	0.989348i	$-0.546501\pi$
$79$	−23.0000	−0.291139	−0.145570	+	0.989348i	$0.546501\pi$
$80$	22.6274i	0.282843i
$81$	0	0
$82$	22.0000	0.268293
$83$	− 106.066i	− 1.27790i	−0.769247	−	0.638952i	$-0.779368\pi$
$83$	− 106.066i	− 1.27790i	0.769247	−	0.638952i	$-0.220632\pi$
$84$	0	0
$85$	−64.0000	−0.752941
$86$	0	0
$87$	0	0
$88$	0	0
$89$	50.9117i	0.572041i	0.958223	+	0.286021i	$0.0923326\pi$
$89$	50.9117i	0.572041i	−0.958223	+	0.286021i	$0.907667\pi$
$90$	0	0
$91$	16.0000	0.175824
$92$	− 48.0833i	− 0.522644i
$93$	0	0
$94$	−86.0000	−0.914894
$95$	186.676i	1.96501i
$96$	0	0
$97$	−25.0000	−0.257732	−0.128866	−	0.991662i	$-0.541134\pi$
$97$	−25.0000	−0.257732	−0.128866	+	0.991662i	$0.541134\pi$
$98$	− 67.8823i	− 0.692676i
$99$	0	0
$100$	−14.0000	−0.140000
$101$	151.321i	1.49823i	0.662442	+	0.749113i	$0.269520\pi$
$101$	151.321i	1.49823i	−0.662442	+	0.749113i	$0.730480\pi$
$102$	0	0
$103$	−25.0000	−0.242718	−0.121359	−	0.992609i	$-0.538725\pi$
$103$	−25.0000	−0.242718	−0.121359	+	0.992609i	$0.538725\pi$
$104$	− 135.765i	− 1.30543i
$105$	0	0
$106$	126.000	1.18868
$107$	− 131.522i	− 1.22918i	−0.788848	−	0.614588i	$-0.789322\pi$
$107$	− 131.522i	− 1.22918i	0.788848	−	0.614588i	$-0.210678\pi$
$108$	0	0
$109$	119.000	1.09174	0.545872	−	0.837869i	$-0.316199\pi$
$109$	119.000	1.09174	0.545872	+	0.837869i	$0.316199\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.0357143
$113$	12.7279i	0.112636i	0.998413	+	0.0563182i	$0.0179361\pi$
$113$	12.7279i	0.112636i	−0.998413	+	0.0563182i	$0.982064\pi$
$114$	0	0
$115$	−136.000	−1.18261
$116$	25.4558i	0.219447i
$117$	0	0
$118$	−98.0000	−0.830508
$119$	11.3137i	0.0950732i
$120$	0	0
$121$	0	0
$122$	− 148.492i	− 1.21715i
$123$	0	0
$124$	62.0000	0.500000
$125$	− 101.823i	− 0.814587i
$126$	0	0
$127$	79.0000	0.622047	0.311024	−	0.950402i	$-0.399328\pi$
$127$	79.0000	0.622047	0.311024	+	0.950402i	$0.399328\pi$
$128$	33.9411i	0.265165i
$129$	0	0
$130$	−128.000	−0.984615
$131$	114.551i	0.874437i	0.899355	+	0.437219i	$0.144036\pi$
$131$	114.551i	0.874437i	−0.899355	+	0.437219i	$0.855964\pi$
$132$	0	0
$133$	33.0000	0.248120
$134$	− 145.664i	− 1.08704i
$135$	0	0
$136$	96.0000	0.705882
$137$	90.5097i	0.660655i	0.943866	+	0.330327i	$0.107159\pi$
$137$	90.5097i	0.660655i	−0.943866	+	0.330327i	$0.892841\pi$
$138$	0	0
$139$	−58.0000	−0.417266	−0.208633	−	0.977994i	$-0.566901\pi$
$139$	−58.0000	−0.417266	−0.208633	+	0.977994i	$0.566901\pi$
$140$	11.3137i	0.0808122i
$141$	0	0
$142$	168.000	1.18310
$143$	0	0
$144$	0	0
$145$	72.0000	0.496552
$146$	66.4680i	0.455261i
$147$	0	0
$148$	−114.000	−0.770270
$149$	− 260.215i	− 1.74641i	−0.487352	−	0.873206i	$-0.662037\pi$
$149$	− 260.215i	− 1.74641i	0.487352	−	0.873206i	$-0.337963\pi$
$150$	0	0
$151$	64.0000	0.423841	0.211921	−	0.977287i	$-0.432028\pi$
$151$	64.0000	0.423841	0.211921	+	0.977287i	$0.432028\pi$
$152$	− 280.014i	− 1.84220i
$153$	0	0
$154$	0	0
$155$	− 175.362i	− 1.13137i
$156$	0	0
$157$	257.000	1.63694	0.818471	−	0.574547i	$-0.194822\pi$
$157$	257.000	1.63694	0.818471	+	0.574547i	$0.194822\pi$
$158$	− 32.5269i	− 0.205867i
$159$	0	0
$160$	160.000	1.00000
$161$	24.0416i	0.149327i
$162$	0	0
$163$	79.0000	0.484663	0.242331	−	0.970194i	$-0.422088\pi$
$163$	79.0000	0.484663	0.242331	+	0.970194i	$0.422088\pi$
$164$	− 31.1127i	− 0.189712i
$165$	0	0
$166$	150.000	0.903614
$167$	29.6985i	0.177835i	0.996039	+	0.0889176i	$0.0283408\pi$
$167$	29.6985i	0.177835i	−0.996039	+	0.0889176i	$0.971659\pi$
$168$	0	0
$169$	87.0000	0.514793
$170$	− 90.5097i	− 0.532410i
$171$	0	0
$172$	0	0
$173$	− 67.8823i	− 0.392383i	−0.980566	−	0.196191i	$-0.937143\pi$
$173$	− 67.8823i	− 0.392383i	0.980566	−	0.196191i	$-0.0628574\pi$
$174$	0	0
$175$	7.00000	0.0400000
$176$	0	0
$177$	0	0
$178$	−72.0000	−0.404494
$179$	− 199.404i	− 1.11399i	−0.830516	−	0.556995i	$-0.811954\pi$
$179$	− 199.404i	− 1.11399i	0.830516	−	0.556995i	$-0.188046\pi$
$180$	0	0
$181$	−287.000	−1.58564	−0.792818	−	0.609459i	$-0.791387\pi$
$181$	−287.000	−1.58564	−0.792818	+	0.609459i	$0.791387\pi$
$182$	22.6274i	0.124326i
$183$	0	0
$184$	204.000	1.10870
$185$	322.441i	1.74292i
$186$	0	0
$187$	0	0
$188$	121.622i	0.646927i
$189$	0	0
$190$	−264.000	−1.38947
$191$	− 164.049i	− 0.858894i	−0.903092	−	0.429447i	$-0.858708\pi$
$191$	− 164.049i	− 0.858894i	0.903092	−	0.429447i	$-0.141292\pi$
$192$	0	0
$193$	−145.000	−0.751295	−0.375648	−	0.926763i	$-0.622580\pi$
$193$	−145.000	−0.751295	−0.375648	+	0.926763i	$0.622580\pi$
$194$	− 35.3553i	− 0.182244i
$195$	0	0
$196$	−96.0000	−0.489796
$197$	− 45.2548i	− 0.229720i	−0.993382	−	0.114860i	$-0.963358\pi$
$197$	− 45.2548i	− 0.229720i	0.993382	−	0.114860i	$-0.0366419\pi$
$198$	0	0
$199$	−57.0000	−0.286432	−0.143216	−	0.989691i	$-0.545744\pi$
$199$	−57.0000	−0.286432	−0.143216	+	0.989691i	$0.545744\pi$
$200$	− 59.3970i	− 0.296985i
$201$	0	0
$202$	−214.000	−1.05941
$203$	− 12.7279i	− 0.0626991i
$204$	0	0
$205$	−88.0000	−0.429268
$206$	− 35.3553i	− 0.171628i
$207$	0	0
$208$	64.0000	0.307692
$209$	0	0
$210$	0	0
$211$	105.000	0.497630	0.248815	−	0.968551i	$-0.419959\pi$
$211$	105.000	0.497630	0.248815	+	0.968551i	$0.419959\pi$
$212$	− 178.191i	− 0.840523i
$213$	0	0
$214$	186.000	0.869159
$215$	0	0
$216$	0	0
$217$	−31.0000	−0.142857
$218$	168.291i	0.771979i
$219$	0	0
$220$	0	0
$221$	181.019i	0.819092i
$222$	0	0
$223$	199.000	0.892377	0.446188	−	0.894939i	$-0.352781\pi$
$223$	199.000	0.892377	0.446188	+	0.894939i	$0.352781\pi$
$224$	− 28.2843i	− 0.126269i
$225$	0	0
$226$	−18.0000	−0.0796460
$227$	− 186.676i	− 0.822362i	−0.911554	−	0.411181i	$-0.865116\pi$
$227$	− 186.676i	− 0.822362i	0.911554	−	0.411181i	$-0.134884\pi$
$228$	0	0
$229$	−72.0000	−0.314410	−0.157205	−	0.987566i	$-0.550248\pi$
$229$	−72.0000	−0.314410	−0.157205	+	0.987566i	$0.550248\pi$
$230$	− 192.333i	− 0.836231i
$231$	0	0
$232$	−108.000	−0.465517
$233$	− 131.522i	− 0.564472i	−0.959345	−	0.282236i	$-0.908924\pi$
$233$	− 131.522i	− 0.564472i	0.959345	−	0.282236i	$-0.0910760\pi$
$234$	0	0
$235$	344.000	1.46383
$236$	138.593i	0.587258i
$237$	0	0
$238$	−16.0000	−0.0672269
$239$	− 4.24264i	− 0.0177516i	−0.999961	−	0.00887582i	$-0.997175\pi$
$239$	− 4.24264i	− 0.0177516i	0.999961	−	0.00887582i	$-0.00282530\pi$
$240$	0	0
$241$	−160.000	−0.663900	−0.331950	−	0.943297i	$-0.607707\pi$
$241$	−160.000	−0.663900	−0.331950	+	0.943297i	$0.607707\pi$
$242$	0	0
$243$	0	0
$244$	−210.000	−0.860656
$245$	271.529i	1.10828i
$246$	0	0
$247$	528.000	2.13765
$248$	263.044i	1.06066i
$249$	0	0
$250$	144.000	0.576000
$251$	108.894i	0.433842i	0.976189	+	0.216921i	$0.0696015\pi$
$251$	108.894i	0.433842i	−0.976189	+	0.216921i	$0.930399\pi$
$252$	0	0
$253$	0	0
$254$	111.723i	0.439854i
$255$	0	0
$256$	−272.000	−1.06250
$257$	261.630i	1.01801i	0.860762	+	0.509007i	$0.169987\pi$
$257$	261.630i	1.01801i	−0.860762	+	0.509007i	$0.830013\pi$
$258$	0	0
$259$	57.0000	0.220077
$260$	181.019i	0.696228i
$261$	0	0
$262$	−162.000	−0.618321
$263$	439.820i	1.67232i	0.548485	+	0.836160i	$0.315205\pi$
$263$	439.820i	1.67232i	−0.548485	+	0.836160i	$0.684795\pi$
$264$	0	0
$265$	−504.000	−1.90189
$266$	46.6690i	0.175448i
$267$	0	0
$268$	−206.000	−0.768657
$269$	244.659i	0.909513i	0.890616	+	0.454756i	$0.150274\pi$
$269$	244.659i	0.909513i	−0.890616	+	0.454756i	$0.849726\pi$
$270$	0	0
$271$	−344.000	−1.26937	−0.634686	−	0.772770i	$-0.718871\pi$
$271$	−344.000	−1.26937	−0.634686	+	0.772770i	$0.718871\pi$
$272$	45.2548i	0.166378i
$273$	0	0
$274$	−128.000	−0.467153
$275$	0	0
$276$	0	0
$277$	95.0000	0.342960	0.171480	−	0.985188i	$-0.445145\pi$
$277$	95.0000	0.342960	0.171480	+	0.985188i	$0.445145\pi$
$278$	− 82.0244i	− 0.295052i
$279$	0	0
$280$	−48.0000	−0.171429
$281$	390.323i	1.38905i	0.719469	+	0.694525i	$0.244385\pi$
$281$	390.323i	1.38905i	−0.719469	+	0.694525i	$0.755615\pi$
$282$	0	0
$283$	335.000	1.18375	0.591873	−	0.806031i	$-0.298389\pi$
$283$	335.000	1.18375	0.591873	+	0.806031i	$0.298389\pi$
$284$	− 237.588i	− 0.836577i
$285$	0	0
$286$	0	0
$287$	15.5563i	0.0542033i
$288$	0	0
$289$	161.000	0.557093
$290$	101.823i	0.351115i
$291$	0	0
$292$	94.0000	0.321918
$293$	− 560.029i	− 1.91136i	−0.294407	−	0.955680i	$-0.595122\pi$
$293$	− 560.029i	− 1.91136i	0.294407	−	0.955680i	$-0.404878\pi$
$294$	0	0
$295$	392.000	1.32881
$296$	− 483.661i	− 1.63399i
$297$	0	0
$298$	368.000	1.23490
$299$	384.666i	1.28651i
$300$	0	0
$301$	0	0
$302$	90.5097i	0.299701i
$303$	0	0
$304$	132.000	0.434211
$305$	593.970i	1.94744i
$306$	0	0
$307$	−311.000	−1.01303	−0.506515	−	0.862231i	$-0.669066\pi$
$307$	−311.000	−1.01303	−0.506515	+	0.862231i	$0.669066\pi$
$308$	0	0
$309$	0	0
$310$	248.000	0.800000
$311$	16.9706i	0.0545677i	0.999628	+	0.0272839i	$0.00868580\pi$
$311$	16.9706i	0.0545677i	−0.999628	+	0.0272839i	$0.991314\pi$
$312$	0	0
$313$	200.000	0.638978	0.319489	−	0.947590i	$-0.396489\pi$
$313$	200.000	0.638978	0.319489	+	0.947590i	$0.396489\pi$
$314$	363.453i	1.15749i
$315$	0	0
$316$	−46.0000	−0.145570
$317$	282.843i	0.892248i	0.894971	+	0.446124i	$0.147196\pi$
$317$	282.843i	0.892248i	−0.894971	+	0.446124i	$0.852804\pi$
$318$	0	0
$319$	0	0
$320$	316.784i	0.989949i
$321$	0	0
$322$	−34.0000	−0.105590
$323$	373.352i	1.15589i
$324$	0	0
$325$	112.000	0.344615
$326$	111.723i	0.342708i
$327$	0	0
$328$	132.000	0.402439
$329$	− 60.8112i	− 0.184836i
$330$	0	0
$331$	577.000	1.74320	0.871601	−	0.490216i	$-0.163082\pi$
$331$	577.000	1.74320	0.871601	+	0.490216i	$0.163082\pi$
$332$	− 212.132i	− 0.638952i
$333$	0	0
$334$	−42.0000	−0.125749
$335$	582.656i	1.73927i
$336$	0	0
$337$	289.000	0.857567	0.428783	−	0.903407i	$-0.358942\pi$
$337$	289.000	0.857567	0.428783	+	0.903407i	$0.358942\pi$
$338$	123.037i	0.364014i
$339$	0	0
$340$	−128.000	−0.376471
$341$	0	0
$342$	0	0
$343$	97.0000	0.282799
$344$	0	0
$345$	0	0
$346$	96.0000	0.277457
$347$	− 541.644i	− 1.56093i	−0.625198	−	0.780467i	$-0.714982\pi$
$347$	− 541.644i	− 1.56093i	0.625198	−	0.780467i	$-0.285018\pi$
$348$	0	0
$349$	−513.000	−1.46991	−0.734957	−	0.678114i	$-0.762798\pi$
$349$	−513.000	−1.46991	−0.734957	+	0.678114i	$0.762798\pi$
$350$	9.89949i	0.0282843i
$351$	0	0
$352$	0	0
$353$	− 254.558i	− 0.721129i	−0.932734	−	0.360564i	$-0.882584\pi$
$353$	− 254.558i	− 0.721129i	0.932734	−	0.360564i	$-0.117416\pi$
$354$	0	0
$355$	−672.000	−1.89296
$356$	101.823i	0.286021i
$357$	0	0
$358$	282.000	0.787709
$359$	− 452.548i	− 1.26058i	−0.776360	−	0.630290i	$-0.782936\pi$
$359$	− 452.548i	− 1.26058i	0.776360	−	0.630290i	$-0.217064\pi$
$360$	0	0
$361$	728.000	2.01662
$362$	− 405.879i	− 1.12121i
$363$	0	0
$364$	32.0000	0.0879121
$365$	− 265.872i	− 0.728417i
$366$	0	0
$367$	160.000	0.435967	0.217984	−	0.975952i	$-0.430052\pi$
$367$	160.000	0.435967	0.217984	+	0.975952i	$0.430052\pi$
$368$	96.1665i	0.261322i
$369$	0	0
$370$	−456.000	−1.23243
$371$	89.0955i	0.240149i
$372$	0	0
$373$	−127.000	−0.340483	−0.170241	−	0.985402i	$-0.554455\pi$
$373$	−127.000	−0.340483	−0.170241	+	0.985402i	$0.554455\pi$
$374$	0	0
$375$	0	0
$376$	−516.000	−1.37234
$377$	− 203.647i	− 0.540177i
$378$	0	0
$379$	454.000	1.19789	0.598945	−	0.800790i	$-0.295587\pi$
$379$	454.000	1.19789	0.598945	+	0.800790i	$0.295587\pi$
$380$	373.352i	0.982506i
$381$	0	0
$382$	232.000	0.607330
$383$	− 74.9533i	− 0.195701i	−0.995201	−	0.0978503i	$-0.968803\pi$
$383$	− 74.9533i	− 0.195701i	0.995201	−	0.0978503i	$-0.0311966\pi$
$384$	0	0
$385$	0	0
$386$	− 205.061i	− 0.531246i
$387$	0	0
$388$	−50.0000	−0.128866
$389$	4.24264i	0.0109065i	0.999985	+	0.00545327i	$0.00173584\pi$
$389$	4.24264i	0.0109065i	−0.999985	+	0.00545327i	$0.998264\pi$
$390$	0	0
$391$	−272.000	−0.695652
$392$	− 407.294i	− 1.03901i
$393$	0	0
$394$	64.0000	0.162437
$395$	130.108i	0.329386i
$396$	0	0
$397$	481.000	1.21159	0.605793	−	0.795622i	$-0.292856\pi$
$397$	481.000	1.21159	0.605793	+	0.795622i	$0.292856\pi$
$398$	− 80.6102i	− 0.202538i
$399$	0	0
$400$	28.0000	0.0700000
$401$	429.921i	1.07212i	0.844179	+	0.536061i	$0.180088\pi$
$401$	429.921i	1.07212i	−0.844179	+	0.536061i	$0.819912\pi$
$402$	0	0
$403$	−496.000	−1.23077
$404$	302.642i	0.749113i
$405$	0	0
$406$	18.0000	0.0443350
$407$	0	0
$408$	0	0
$409$	−177.000	−0.432763	−0.216381	−	0.976309i	$-0.569425\pi$
$409$	−177.000	−0.432763	−0.216381	+	0.976309i	$0.569425\pi$
$410$	− 124.451i	− 0.303539i
$411$	0	0
$412$	−50.0000	−0.121359
$413$	− 69.2965i	− 0.167788i
$414$	0	0
$415$	−600.000	−1.44578
$416$	− 452.548i	− 1.08786i
$417$	0	0
$418$	0	0
$419$	306.884i	0.732421i	0.930532	+	0.366210i	$0.119345\pi$
$419$	306.884i	0.732421i	−0.930532	+	0.366210i	$0.880655\pi$
$420$	0	0
$421$	160.000	0.380048	0.190024	−	0.981779i	$-0.439143\pi$
$421$	160.000	0.380048	0.190024	+	0.981779i	$0.439143\pi$
$422$	148.492i	0.351878i
$423$	0	0
$424$	756.000	1.78302
$425$	79.1960i	0.186343i
$426$	0	0
$427$	105.000	0.245902
$428$	− 263.044i	− 0.614588i
$429$	0	0
$430$	0	0
$431$	691.550i	1.60453i	0.596971	+	0.802263i	$0.296371\pi$
$431$	691.550i	1.60453i	−0.596971	+	0.802263i	$0.703629\pi$
$432$	0	0
$433$	−143.000	−0.330254	−0.165127	−	0.986272i	$-0.552803\pi$
$433$	−143.000	−0.330254	−0.165127	+	0.986272i	$0.552803\pi$
$434$	− 43.8406i	− 0.101015i
$435$	0	0
$436$	238.000	0.545872
$437$	793.374i	1.81550i
$438$	0	0
$439$	335.000	0.763098	0.381549	−	0.924349i	$-0.375391\pi$
$439$	335.000	0.763098	0.381549	+	0.924349i	$0.375391\pi$
$440$	0	0
$441$	0	0
$442$	−256.000	−0.579186
$443$	130.108i	0.293697i	0.989159	+	0.146848i	$0.0469129\pi$
$443$	130.108i	0.293697i	−0.989159	+	0.146848i	$0.953087\pi$
$444$	0	0
$445$	288.000	0.647191
$446$	281.428i	0.631006i
$447$	0	0
$448$	56.0000	0.125000
$449$	854.185i	1.90242i	0.308550	+	0.951208i	$0.400156\pi$
$449$	854.185i	1.90242i	−0.308550	+	0.951208i	$0.599844\pi$
$450$	0	0
$451$	0	0
$452$	25.4558i	0.0563182i
$453$	0	0
$454$	264.000	0.581498
$455$	− 90.5097i	− 0.198922i
$456$	0	0
$457$	−264.000	−0.577681	−0.288840	−	0.957377i	$-0.593270\pi$
$457$	−264.000	−0.577681	−0.288840	+	0.957377i	$0.593270\pi$
$458$	− 101.823i	− 0.222322i
$459$	0	0
$460$	−272.000	−0.591304
$461$	123.037i	0.266891i	0.991056	+	0.133445i	$0.0426041\pi$
$461$	123.037i	0.266891i	−0.991056	+	0.133445i	$0.957396\pi$
$462$	0	0
$463$	−608.000	−1.31317	−0.656587	−	0.754250i	$-0.728001\pi$
$463$	−608.000	−1.31317	−0.656587	+	0.754250i	$0.728001\pi$
$464$	− 50.9117i	− 0.109723i
$465$	0	0
$466$	186.000	0.399142
$467$	− 424.264i	− 0.908488i	−0.890877	−	0.454244i	$-0.849909\pi$
$467$	− 424.264i	− 0.908488i	0.890877	−	0.454244i	$-0.150091\pi$
$468$	0	0
$469$	103.000	0.219616
$470$	486.489i	1.03508i
$471$	0	0
$472$	−588.000	−1.24576
$473$	0	0
$474$	0	0
$475$	231.000	0.486316
$476$	22.6274i	0.0475366i
$477$	0	0
$478$	6.00000	0.0125523
$479$	830.143i	1.73308i	0.499111	+	0.866538i	$0.333660\pi$
$479$	830.143i	1.73308i	−0.499111	+	0.866538i	$0.666340\pi$
$480$	0	0
$481$	912.000	1.89605
$482$	− 226.274i	− 0.469448i
$483$	0	0
$484$	0	0
$485$	141.421i	0.291590i
$486$	0	0
$487$	56.0000	0.114990	0.0574949	−	0.998346i	$-0.481689\pi$
$487$	56.0000	0.114990	0.0574949	+	0.998346i	$0.481689\pi$
$488$	− 890.955i	− 1.82573i
$489$	0	0
$490$	−384.000	−0.783673
$491$	− 278.600i	− 0.567414i	−0.958911	−	0.283707i	$-0.908436\pi$
$491$	− 278.600i	− 0.567414i	0.958911	−	0.283707i	$-0.0915642\pi$
$492$	0	0
$493$	144.000	0.292089
$494$	746.705i	1.51155i
$495$	0	0
$496$	−124.000	−0.250000
$497$	118.794i	0.239022i
$498$	0	0
$499$	−439.000	−0.879760	−0.439880	−	0.898057i	$-0.644979\pi$
$499$	−439.000	−0.879760	−0.439880	+	0.898057i	$0.644979\pi$
$500$	− 203.647i	− 0.407294i
$501$	0	0
$502$	−154.000	−0.306773
$503$	− 605.283i	− 1.20335i	−0.798742	−	0.601673i	$-0.794501\pi$
$503$	− 605.283i	− 1.20335i	0.798742	−	0.601673i	$-0.205499\pi$
$504$	0	0
$505$	856.000	1.69505
$506$	0	0
$507$	0	0
$508$	158.000	0.311024
$509$	352.139i	0.691825i	0.938267	+	0.345913i	$0.112431\pi$
$509$	352.139i	0.691825i	−0.938267	+	0.345913i	$0.887569\pi$
$510$	0	0
$511$	−47.0000	−0.0919765
$512$	− 248.902i	− 0.486136i
$513$	0	0
$514$	−370.000	−0.719844
$515$	141.421i	0.274605i
$516$	0	0
$517$	0	0
$518$	80.6102i	0.155618i
$519$	0	0
$520$	−768.000	−1.47692
$521$	581.242i	1.11563i	0.829966	+	0.557814i	$0.188360\pi$
$521$	581.242i	1.11563i	−0.829966	+	0.557814i	$0.811640\pi$
$522$	0	0
$523$	385.000	0.736138	0.368069	−	0.929799i	$-0.380019\pi$
$523$	385.000	0.736138	0.368069	+	0.929799i	$0.380019\pi$
$524$	229.103i	0.437219i
$525$	0	0
$526$	−622.000	−1.18251
$527$	− 350.725i	− 0.665512i
$528$	0	0
$529$	−49.0000	−0.0926276
$530$	− 712.764i	− 1.34484i
$531$	0	0
$532$	66.0000	0.124060
$533$	248.902i	0.466982i
$534$	0	0
$535$	−744.000	−1.39065
$536$	− 873.984i	− 1.63057i
$537$	0	0
$538$	−346.000	−0.643123
$539$	0	0
$540$	0	0
$541$	−88.0000	−0.162662	−0.0813309	−	0.996687i	$-0.525917\pi$
$541$	−88.0000	−0.162662	−0.0813309	+	0.996687i	$0.525917\pi$
$542$	− 486.489i	− 0.897582i
$543$	0	0
$544$	320.000	0.588235
$545$	− 673.166i	− 1.23517i
$546$	0	0
$547$	424.000	0.775137	0.387569	−	0.921841i	$-0.373315\pi$
$547$	424.000	0.775137	0.387569	+	0.921841i	$0.373315\pi$
$548$	181.019i	0.330327i
$549$	0	0
$550$	0	0
$551$	− 420.021i	− 0.762289i
$552$	0	0
$553$	23.0000	0.0415913
$554$	134.350i	0.242510i
$555$	0	0
$556$	−116.000	−0.208633
$557$	209.304i	0.375769i	0.982191	+	0.187885i	$0.0601631\pi$
$557$	209.304i	0.375769i	−0.982191	+	0.187885i	$0.939837\pi$
$558$	0	0
$559$	0	0
$560$	− 22.6274i	− 0.0404061i
$561$	0	0
$562$	−552.000	−0.982206
$563$	− 469.519i	− 0.833959i	−0.908916	−	0.416979i	$-0.863089\pi$
$563$	− 469.519i	− 0.833959i	0.908916	−	0.416979i	$-0.136911\pi$
$564$	0	0
$565$	72.0000	0.127434
$566$	473.762i	0.837035i
$567$	0	0
$568$	1008.00	1.77465
$569$	− 831.558i	− 1.46144i	−0.682679	−	0.730718i	$-0.739185\pi$
$569$	− 831.558i	− 1.46144i	0.682679	−	0.730718i	$-0.260815\pi$
$570$	0	0
$571$	−503.000	−0.880911	−0.440455	−	0.897775i	$-0.645183\pi$
$571$	−503.000	−0.880911	−0.440455	+	0.897775i	$0.645183\pi$
$572$	0	0
$573$	0	0
$574$	−22.0000	−0.0383275
$575$	168.291i	0.292681i
$576$	0	0
$577$	−615.000	−1.06586	−0.532929	−	0.846160i	$-0.678909\pi$
$577$	−615.000	−1.06586	−0.532929	+	0.846160i	$0.678909\pi$
$578$	227.688i	0.393925i
$579$	0	0
$580$	144.000	0.248276
$581$	106.066i	0.182558i
$582$	0	0
$583$	0	0
$584$	398.808i	0.682891i
$585$	0	0
$586$	792.000	1.35154
$587$	− 5.65685i	− 0.00963689i	−0.999988	−	0.00481844i	$-0.998466\pi$
$587$	− 5.65685i	− 0.00963689i	0.999988	−	0.00481844i	$-0.00153376\pi$
$588$	0	0
$589$	−1023.00	−1.73684
$590$	554.372i	0.939613i
$591$	0	0
$592$	228.000	0.385135
$593$	287.085i	0.484124i	0.970261	+	0.242062i	$0.0778237\pi$
$593$	287.085i	0.484124i	−0.970261	+	0.242062i	$0.922176\pi$
$594$	0	0
$595$	64.0000	0.107563
$596$	− 520.431i	− 0.873206i
$597$	0	0
$598$	−544.000	−0.909699
$599$	− 956.008i	− 1.59601i	−0.602653	−	0.798004i	$-0.705890\pi$
$599$	− 956.008i	− 1.59601i	0.602653	−	0.798004i	$-0.294110\pi$
$600$	0	0
$601$	601.000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$601$	601.000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$602$	0	0
$603$	0	0
$604$	128.000	0.211921
$605$	0	0
$606$	0	0
$607$	1024.00	1.68699	0.843493	−	0.537141i	$-0.180496\pi$
$607$	1024.00	1.68699	0.843493	+	0.537141i	$0.180496\pi$
$608$	− 933.381i	− 1.53517i
$609$	0	0
$610$	−840.000	−1.37705
$611$	− 972.979i	− 1.59244i
$612$	0	0
$613$	721.000	1.17618	0.588091	−	0.808795i	$-0.299880\pi$
$613$	721.000	1.17618	0.588091	+	0.808795i	$0.299880\pi$
$614$	− 439.820i	− 0.716320i
$615$	0	0
$616$	0	0
$617$	50.9117i	0.0825149i	0.999149	+	0.0412574i	$0.0131364\pi$
$617$	50.9117i	0.0825149i	−0.999149	+	0.0412574i	$0.986864\pi$
$618$	0	0
$619$	152.000	0.245557	0.122779	−	0.992434i	$-0.460819\pi$
$619$	152.000	0.245557	0.122779	+	0.992434i	$0.460819\pi$
$620$	− 350.725i	− 0.565685i
$621$	0	0
$622$	−24.0000	−0.0385852
$623$	− 50.9117i	− 0.0817202i
$624$	0	0
$625$	−751.000	−1.20160
$626$	282.843i	0.451825i
$627$	0	0
$628$	514.000	0.818471
$629$	644.881i	1.02525i
$630$	0	0
$631$	544.000	0.862124	0.431062	−	0.902322i	$-0.358139\pi$
$631$	544.000	0.862124	0.431062	+	0.902322i	$0.358139\pi$
$632$	− 195.161i	− 0.308800i
$633$	0	0
$634$	−400.000	−0.630915
$635$	− 446.891i	− 0.703766i
$636$	0	0
$637$	768.000	1.20565
$638$	0	0
$639$	0	0
$640$	192.000	0.300000
$641$	814.587i	1.27081i	0.772180	+	0.635403i	$0.219166\pi$
$641$	814.587i	1.27081i	−0.772180	+	0.635403i	$0.780834\pi$
$642$	0	0
$643$	−1231.00	−1.91446	−0.957232	−	0.289322i	$-0.906570\pi$
$643$	−1231.00	−1.91446	−0.957232	+	0.289322i	$0.906570\pi$
$644$	48.0833i	0.0746634i
$645$	0	0
$646$	−528.000	−0.817337
$647$	158.392i	0.244810i	0.992480	+	0.122405i	$0.0390606\pi$
$647$	158.392i	0.244810i	−0.992480	+	0.122405i	$0.960939\pi$
$648$	0	0
$649$	0	0
$650$	158.392i	0.243680i
$651$	0	0
$652$	158.000	0.242331
$653$	− 264.458i	− 0.404989i	−0.979283	−	0.202495i	$-0.935095\pi$
$653$	− 264.458i	− 0.404989i	0.979283	−	0.202495i	$-0.0649049\pi$
$654$	0	0
$655$	648.000	0.989313
$656$	62.2254i	0.0948558i
$657$	0	0
$658$	86.0000	0.130699
$659$	− 490.732i	− 0.744662i	−0.928100	−	0.372331i	$-0.878559\pi$
$659$	− 490.732i	− 0.744662i	0.928100	−	0.372331i	$-0.121441\pi$
$660$	0	0
$661$	−521.000	−0.788200	−0.394100	−	0.919068i	$-0.628944\pi$
$661$	−521.000	−0.788200	−0.394100	+	0.919068i	$0.628944\pi$
$662$	816.001i	1.23263i
$663$	0	0
$664$	900.000	1.35542
$665$	− 186.676i	− 0.280716i
$666$	0	0
$667$	306.000	0.458771
$668$	59.3970i	0.0889176i
$669$	0	0
$670$	−824.000	−1.22985
$671$	0	0
$672$	0	0
$673$	−239.000	−0.355126	−0.177563	−	0.984109i	$-0.556821\pi$
$673$	−239.000	−0.355126	−0.177563	+	0.984109i	$0.556821\pi$
$674$	408.708i	0.606391i
$675$	0	0
$676$	174.000	0.257396
$677$	− 305.470i	− 0.451211i	−0.974219	−	0.225606i	$-0.927564\pi$
$677$	− 305.470i	− 0.451211i	0.974219	−	0.225606i	$-0.0724361\pi$
$678$	0	0
$679$	25.0000	0.0368189
$680$	− 543.058i	− 0.798615i
$681$	0	0
$682$	0	0
$683$	− 1101.67i	− 1.61299i	−0.591241	−	0.806495i	$-0.701362\pi$
$683$	− 1101.67i	− 1.61299i	0.591241	−	0.806495i	$-0.298638\pi$
$684$	0	0
$685$	512.000	0.747445
$686$	137.179i	0.199969i
$687$	0	0
$688$	0	0
$689$	1425.53i	2.06898i
$690$	0	0
$691$	−919.000	−1.32996	−0.664978	−	0.746863i	$-0.731559\pi$
$691$	−919.000	−1.32996	−0.664978	+	0.746863i	$0.731559\pi$
$692$	− 135.765i	− 0.196191i
$693$	0	0
$694$	766.000	1.10375
$695$	328.098i	0.472083i
$696$	0	0
$697$	−176.000	−0.252511
$698$	− 725.492i	− 1.03939i
$699$	0	0
$700$	14.0000	0.0200000
$701$	− 646.296i	− 0.921962i	−0.887410	−	0.460981i	$-0.847498\pi$
$701$	− 646.296i	− 0.921962i	0.887410	−	0.460981i	$-0.152502\pi$
$702$	0	0
$703$	1881.00	2.67568
$704$	0	0
$705$	0	0
$706$	360.000	0.509915
$707$	− 151.321i	− 0.214032i
$708$	0	0
$709$	518.000	0.730606	0.365303	−	0.930889i	$-0.380965\pi$
$709$	518.000	0.730606	0.365303	+	0.930889i	$0.380965\pi$
$710$	− 950.352i	− 1.33852i
$711$	0	0
$712$	−432.000	−0.606742
$713$	− 745.291i	− 1.04529i
$714$	0	0
$715$	0	0
$716$	− 398.808i	− 0.556995i
$717$	0	0
$718$	640.000	0.891365
$719$	− 182.434i	− 0.253732i	−0.991920	−	0.126866i	$-0.959508\pi$
$719$	− 182.434i	− 0.253732i	0.991920	−	0.126866i	$-0.0404919\pi$
$720$	0	0
$721$	25.0000	0.0346741
$722$	1029.55i	1.42597i
$723$	0	0
$724$	−574.000	−0.792818
$725$	− 89.0955i	− 0.122890i
$726$	0	0
$727$	−114.000	−0.156809	−0.0784044	−	0.996922i	$-0.524983\pi$
$727$	−114.000	−0.156809	−0.0784044	+	0.996922i	$0.524983\pi$
$728$	135.765i	0.186490i
$729$	0	0
$730$	376.000	0.515068
$731$	0	0
$732$	0	0
$733$	234.000	0.319236	0.159618	−	0.987179i	$-0.448974\pi$
$733$	234.000	0.319236	0.159618	+	0.987179i	$0.448974\pi$
$734$	226.274i	0.308275i
$735$	0	0
$736$	680.000	0.923913
$737$	0	0
$738$	0	0
$739$	−1241.00	−1.67930	−0.839648	−	0.543131i	$-0.817239\pi$
$739$	−1241.00	−1.67930	−0.839648	+	0.543131i	$0.817239\pi$
$740$	644.881i	0.871461i
$741$	0	0
$742$	−126.000	−0.169811
$743$	141.421i	0.190338i	0.995461	+	0.0951691i	$0.0303392\pi$
$743$	141.421i	0.190338i	−0.995461	+	0.0951691i	$0.969661\pi$
$744$	0	0
$745$	−1472.00	−1.97584
$746$	− 179.605i	− 0.240758i
$747$	0	0
$748$	0	0
$749$	131.522i	0.175597i
$750$	0	0
$751$	−113.000	−0.150466	−0.0752330	−	0.997166i	$-0.523970\pi$
$751$	−113.000	−0.150466	−0.0752330	+	0.997166i	$0.523970\pi$
$752$	− 243.245i	− 0.323464i
$753$	0	0
$754$	288.000	0.381963
$755$	− 362.039i	− 0.479521i
$756$	0	0
$757$	−55.0000	−0.0726552	−0.0363276	−	0.999340i	$-0.511566\pi$
$757$	−55.0000	−0.0726552	−0.0363276	+	0.999340i	$0.511566\pi$
$758$	642.053i	0.847036i
$759$	0	0
$760$	−1584.00	−2.08421
$761$	− 882.469i	− 1.15962i	−0.814752	−	0.579809i	$-0.803127\pi$
$761$	− 882.469i	− 1.15962i	0.814752	−	0.579809i	$-0.196873\pi$
$762$	0	0
$763$	−119.000	−0.155963
$764$	− 328.098i	− 0.429447i
$765$	0	0
$766$	106.000	0.138381
$767$	− 1108.74i	− 1.44556i
$768$	0	0
$769$	−41.0000	−0.0533160	−0.0266580	−	0.999645i	$-0.508487\pi$
$769$	−41.0000	−0.0533160	−0.0266580	+	0.999645i	$0.508487\pi$
$770$	0	0
$771$	0	0
$772$	−290.000	−0.375648
$773$	339.411i	0.439083i	0.975603	+	0.219542i	$0.0704562\pi$
$773$	339.411i	0.439083i	−0.975603	+	0.219542i	$0.929544\pi$
$774$	0	0
$775$	−217.000	−0.280000
$776$	− 212.132i	− 0.273366i
$777$	0	0
$778$	−6.00000	−0.00771208
$779$	513.360i	0.658998i
$780$	0	0
$781$	0	0
$782$	− 384.666i	− 0.491900i
$783$	0	0
$784$	192.000	0.244898
$785$	− 1453.81i	− 1.85199i
$786$	0	0
$787$	−1024.00	−1.30114	−0.650572	−	0.759445i	$-0.725471\pi$
$787$	−1024.00	−1.30114	−0.650572	+	0.759445i	$0.725471\pi$
$788$	− 90.5097i	− 0.114860i
$789$	0	0
$790$	−184.000	−0.232911
$791$	− 12.7279i	− 0.0160909i
$792$	0	0
$793$	1680.00	2.11854
$794$	680.237i	0.856721i
$795$	0	0
$796$	−114.000	−0.143216
$797$	− 7.07107i	− 0.00887211i	−0.999990	−	0.00443605i	$-0.998588\pi$
$797$	− 7.07107i	− 0.00887211i	0.999990	−	0.00443605i	$-0.00141204\pi$
$798$	0	0
$799$	688.000	0.861076
$800$	− 197.990i	− 0.247487i
$801$	0	0
$802$	−608.000	−0.758105
$803$	0	0
$804$	0	0
$805$	136.000	0.168944
$806$	− 701.450i	− 0.870285i
$807$	0	0
$808$	−1284.00	−1.58911
$809$	− 933.381i	− 1.15375i	−0.816834	−	0.576873i	$-0.804273\pi$
$809$	− 933.381i	− 1.15375i	0.816834	−	0.576873i	$-0.195727\pi$
$810$	0	0
$811$	−1513.00	−1.86560	−0.932799	−	0.360397i	$-0.882641\pi$
$811$	−1513.00	−1.86560	−0.932799	+	0.360397i	$0.882641\pi$
$812$	− 25.4558i	− 0.0313496i
$813$	0	0
$814$	0	0
$815$	− 446.891i	− 0.548333i
$816$	0	0
$817$	0	0
$818$	− 250.316i	− 0.306010i
$819$	0	0
$820$	−176.000	−0.214634
$821$	926.310i	1.12827i	0.825682	+	0.564135i	$0.190790\pi$
$821$	926.310i	1.12827i	−0.825682	+	0.564135i	$0.809210\pi$
$822$	0	0
$823$	615.000	0.747266	0.373633	−	0.927577i	$-0.378112\pi$
$823$	615.000	0.747266	0.373633	+	0.927577i	$0.378112\pi$
$824$	− 212.132i	− 0.257442i
$825$	0	0
$826$	98.0000	0.118644
$827$	1084.70i	1.31161i	0.754930	+	0.655805i	$0.227671\pi$
$827$	1084.70i	1.31161i	−0.754930	+	0.655805i	$0.772329\pi$
$828$	0	0
$829$	97.0000	0.117008	0.0585042	−	0.998287i	$-0.481367\pi$
$829$	97.0000	0.117008	0.0585042	+	0.998287i	$0.481367\pi$
$830$	− 848.528i	− 1.02232i
$831$	0	0
$832$	896.000	1.07692
$833$	543.058i	0.651930i
$834$	0	0
$835$	168.000	0.201198
$836$	0	0
$837$	0	0
$838$	−434.000	−0.517900
$839$	− 1363.30i	− 1.62491i	−0.583022	−	0.812456i	$-0.698130\pi$
$839$	− 1363.30i	− 1.62491i	0.583022	−	0.812456i	$-0.301870\pi$
$840$	0	0
$841$	679.000	0.807372
$842$	226.274i	0.268734i
$843$	0	0
$844$	210.000	0.248815
$845$	− 492.146i	− 0.582422i
$846$	0	0
$847$	0	0
$848$	356.382i	0.420262i
$849$	0	0
$850$	−112.000	−0.131765
$851$	1370.37i	1.61031i
$852$	0	0
$853$	−41.0000	−0.0480657	−0.0240328	−	0.999711i	$-0.507651\pi$
$853$	−41.0000	−0.0480657	−0.0240328	+	0.999711i	$0.507651\pi$
$854$	148.492i	0.173879i
$855$	0	0
$856$	1116.00	1.30374
$857$	401.637i	0.468654i	0.972158	+	0.234327i	$0.0752887\pi$
$857$	401.637i	0.468654i	−0.972158	+	0.234327i	$0.924711\pi$
$858$	0	0
$859$	−1425.00	−1.65891	−0.829453	−	0.558577i	$-0.811348\pi$
$859$	−1425.00	−1.65891	−0.829453	+	0.558577i	$0.811348\pi$
$860$	0	0
$861$	0	0
$862$	−978.000	−1.13457
$863$	− 1107.33i	− 1.28312i	−0.767074	−	0.641558i	$-0.778288\pi$
$863$	− 1107.33i	− 1.28312i	0.767074	−	0.641558i	$-0.221712\pi$
$864$	0	0
$865$	−384.000	−0.443931
$866$	− 202.233i	− 0.233525i
$867$	0	0
$868$	−62.0000	−0.0714286
$869$	0	0
$870$	0	0
$871$	1648.00	1.89208
$872$	1009.75i	1.15797i
$873$	0	0
$874$	−1122.00	−1.28375
$875$	101.823i	0.116370i
$876$	0	0
$877$	1593.00	1.81642	0.908210	−	0.418515i	$-0.137449\pi$
$877$	1593.00	1.81642	0.908210	+	0.418515i	$0.137449\pi$
$878$	473.762i	0.539592i
$879$	0	0
$880$	0	0
$881$	646.296i	0.733593i	0.930301	+	0.366797i	$0.119546\pi$
$881$	646.296i	0.733593i	−0.930301	+	0.366797i	$0.880454\pi$
$882$	0	0
$883$	−1145.00	−1.29672	−0.648358	−	0.761336i	$-0.724544\pi$
$883$	−1145.00	−1.29672	−0.648358	+	0.761336i	$0.724544\pi$
$884$	362.039i	0.409546i
$885$	0	0
$886$	−184.000	−0.207675
$887$	439.820i	0.495852i	0.968779	+	0.247926i	$0.0797489\pi$
$887$	439.820i	0.495852i	−0.968779	+	0.247926i	$0.920251\pi$
$888$	0	0
$889$	−79.0000	−0.0888639
$890$	407.294i	0.457633i
$891$	0	0
$892$	398.000	0.446188
$893$	− 2006.77i	− 2.24722i
$894$	0	0
$895$	−1128.00	−1.26034
$896$	− 33.9411i	− 0.0378807i
$897$	0	0
$898$	−1208.00	−1.34521
$899$	394.566i	0.438894i
$900$	0	0
$901$	−1008.00	−1.11876
$902$	0	0
$903$	0	0
$904$	−108.000	−0.119469
$905$	1623.52i	1.79394i
$906$	0	0
$907$	−87.0000	−0.0959206	−0.0479603	−	0.998849i	$-0.515272\pi$
$907$	−87.0000	−0.0959206	−0.0479603	+	0.998849i	$0.515272\pi$
$908$	− 373.352i	− 0.411181i
$909$	0	0
$910$	128.000	0.140659
$911$	− 1493.41i	− 1.63931i	−0.572859	−	0.819654i	$-0.694166\pi$
$911$	− 1493.41i	− 1.63931i	0.572859	−	0.819654i	$-0.305834\pi$
$912$	0	0
$913$	0	0
$914$	− 373.352i	− 0.408482i
$915$	0	0
$916$	−144.000	−0.157205
$917$	− 114.551i	− 0.124920i
$918$	0	0
$919$	−865.000	−0.941240	−0.470620	−	0.882336i	$-0.655970\pi$
$919$	−865.000	−0.941240	−0.470620	+	0.882336i	$0.655970\pi$
$920$	− 1154.00i	− 1.25435i
$921$	0	0
$922$	−174.000	−0.188720
$923$	1900.70i	2.05927i
$924$	0	0
$925$	399.000	0.431351
$926$	− 859.842i	− 0.928555i
$927$	0	0
$928$	−360.000	−0.387931
$929$	− 1192.18i	− 1.28330i	−0.766999	−	0.641648i	$-0.778251\pi$
$929$	− 1192.18i	− 1.28330i	0.766999	−	0.641648i	$-0.221749\pi$
$930$	0	0
$931$	1584.00	1.70140
$932$	− 263.044i	− 0.282236i
$933$	0	0
$934$	600.000	0.642398
$935$	0	0
$936$	0	0
$937$	−473.000	−0.504803	−0.252401	−	0.967623i	$-0.581220\pi$
$937$	−473.000	−0.504803	−0.252401	+	0.967623i	$0.581220\pi$
$938$	145.664i	0.155292i
$939$	0	0
$940$	688.000	0.731915
$941$	114.551i	0.121734i	0.998146	+	0.0608668i	$0.0193865\pi$
$941$	114.551i	0.121734i	−0.998146	+	0.0608668i	$0.980614\pi$
$942$	0	0
$943$	−374.000	−0.396607
$944$	− 277.186i	− 0.293629i
$945$	0	0
$946$	0	0
$947$	− 818.830i	− 0.864656i	−0.901716	−	0.432328i	$-0.857692\pi$
$947$	− 818.830i	− 0.864656i	0.901716	−	0.432328i	$-0.142308\pi$
$948$	0	0
$949$	−752.000	−0.792413
$950$	326.683i	0.343877i
$951$	0	0
$952$	−96.0000	−0.100840
$953$	− 1042.28i	− 1.09368i	−0.837238	−	0.546839i	$-0.815831\pi$
$953$	− 1042.28i	− 1.09368i	0.837238	−	0.546839i	$-0.184169\pi$
$954$	0	0
$955$	−928.000	−0.971728
$956$	− 8.48528i	− 0.00887582i
$957$	0	0
$958$	−1174.00	−1.22547
$959$	− 90.5097i	− 0.0943792i
$960$	0	0
$961$	0	0
$962$	1289.76i	1.34071i
$963$	0	0
$964$	−320.000	−0.331950
$965$	820.244i	0.849994i
$966$	0	0
$967$	−937.000	−0.968976	−0.484488	−	0.874798i	$-0.660994\pi$
$967$	−937.000	−0.968976	−0.484488	+	0.874798i	$0.660994\pi$
$968$	0	0
$969$	0	0
$970$	−200.000	−0.206186
$971$	− 694.379i	− 0.715117i	−0.933891	−	0.357559i	$-0.883609\pi$
$971$	− 694.379i	− 0.715117i	0.933891	−	0.357559i	$-0.116391\pi$
$972$	0	0
$973$	58.0000	0.0596095
$974$	79.1960i	0.0813100i
$975$	0	0
$976$	420.000	0.430328
$977$	− 708.521i	− 0.725201i	−0.931945	−	0.362600i	$-0.881889\pi$
$977$	− 708.521i	− 0.725201i	0.931945	−	0.362600i	$-0.118111\pi$
$978$	0	0
$979$	0	0
$980$	543.058i	0.554141i
$981$	0	0
$982$	394.000	0.401222
$983$	547.301i	0.556766i	0.960470	+	0.278383i	$0.0897984\pi$
$983$	547.301i	0.556766i	−0.960470	+	0.278383i	$0.910202\pi$
$984$	0	0
$985$	−256.000	−0.259898
$986$	203.647i	0.206538i
$987$	0	0
$988$	1056.00	1.06883
$989$	0	0
$990$	0	0
$991$	286.000	0.288597	0.144299	−	0.989534i	$-0.453907\pi$
$991$	286.000	0.288597	0.144299	+	0.989534i	$0.453907\pi$
$992$	876.812i	0.883883i
$993$	0	0
$994$	−168.000	−0.169014
$995$	322.441i	0.324061i
$996$	0	0
$997$	265.000	0.265797	0.132899	−	0.991130i	$-0.457572\pi$
$997$	265.000	0.265797	0.132899	+	0.991130i	$0.457572\pi$
$998$	− 620.840i	− 0.622084i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.a.485.2	yes	2
3.2	odd	2	inner	1089.3.b.a.485.1	✓	2
11.10	odd	2		1089.3.b.b.485.1	yes	2
33.32	even	2		1089.3.b.b.485.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.b.a.485.1	✓	2	3.2	odd	2	inner
1089.3.b.a.485.2	yes	2	1.1	even	1	trivial
1089.3.b.b.485.1	yes	2	11.10	odd	2
1089.3.b.b.485.2	yes	2	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +8.00000 q^{10} -16.0000 q^{13} -1.41421i q^{14} -4.00000 q^{16} -11.3137i q^{17} -33.0000 q^{19} -11.3137i q^{20} -24.0416i q^{23} -7.00000 q^{25} -22.6274i q^{26} -2.00000 q^{28} +12.7279i q^{29} +31.0000 q^{31} +28.2843i q^{32} +16.0000 q^{34} +5.65685i q^{35} -57.0000 q^{37} -46.6690i q^{38} +48.0000 q^{40} -15.5563i q^{41} +34.0000 q^{46} +60.8112i q^{47} -48.0000 q^{49} -9.89949i q^{50} -32.0000 q^{52} -89.0955i q^{53} -8.48528i q^{56} -18.0000 q^{58} +69.2965i q^{59} -105.000 q^{61} +43.8406i q^{62} -56.0000 q^{64} +90.5097i q^{65} -103.000 q^{67} -22.6274i q^{68} -8.00000 q^{70} -118.794i q^{71} +47.0000 q^{73} -80.6102i q^{74} -66.0000 q^{76} -23.0000 q^{79} +22.6274i q^{80} +22.0000 q^{82} -106.066i q^{83} -64.0000 q^{85} +50.9117i q^{89} +16.0000 q^{91} -48.0833i q^{92} -86.0000 q^{94} +186.676i q^{95} -25.0000 q^{97} -67.8823i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 2 q^{7}+O(q^{10}) \) 2 * q + 4 * q^4 - 2 * q^7	\( 2 q + 4 q^{4} - 2 q^{7} + 16 q^{10} - 32 q^{13} - 8 q^{16} - 66 q^{19} - 14 q^{25} - 4 q^{28} + 62 q^{31} + 32 q^{34} - 114 q^{37} + 96 q^{40} + 68 q^{46} - 96 q^{49} - 64 q^{52} - 36 q^{58} - 210 q^{61} - 112 q^{64} - 206 q^{67} - 16 q^{70} + 94 q^{73} - 132 q^{76} - 46 q^{79} + 44 q^{82} - 128 q^{85} + 32 q^{91} - 172 q^{94} - 50 q^{97}+O(q^{100}) \) 2 * q + 4 * q^4 - 2 * q^7 + 16 * q^10 - 32 * q^13 - 8 * q^16 - 66 * q^19 - 14 * q^25 - 4 * q^28 + 62 * q^31 + 32 * q^34 - 114 * q^37 + 96 * q^40 + 68 * q^46 - 96 * q^49 - 64 * q^52 - 36 * q^58 - 210 * q^61 - 112 * q^64 - 206 * q^67 - 16 * q^70 + 94 * q^73 - 132 * q^76 - 46 * q^79 + 44 * q^82 - 128 * q^85 + 32 * q^91 - 172 * q^94 - 50 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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