LMFDB - Embedded newform 1089.3.c.i.604.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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([0, 1]))

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$8$
Coefficient field:	8.0.79010463744.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} - 44x^{5} + 108x^{3} + 538x^{2} + 360x + 825 $ x^8 - 6x^6 - 44x^5 + 108x^3 + 538x^2 + 360*x + 825
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.7
Root			$-1.67344 + 1.81863i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.i.604.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.11962i q^{2} -5.73205 q^{4} -6.02665 q^{5} +7.07107i q^{7} -5.40335i q^{8} +O(q^{10})$	$q+3.11962i q^{2} -5.73205 q^{4} -6.02665 q^{5} +7.07107i q^{7} -5.40335i q^{8} -18.8009i q^{10} +16.7303i q^{13} -22.0591 q^{14} -6.07180 q^{16} +30.9723i q^{17} +32.9802i q^{19} +34.5451 q^{20} -3.54643 q^{23} +11.3205 q^{25} -52.1923 q^{26} -40.5317i q^{28} +4.17950i q^{29} +34.0000 q^{31} -40.5551i q^{32} -96.6218 q^{34} -42.6148i q^{35} +28.6603 q^{37} -102.886 q^{38} +32.5641i q^{40} -81.0502i q^{41} -36.7423i q^{43} -11.0635i q^{46} -52.0764 q^{47} -1.00000 q^{49} +35.3157i q^{50} -95.8991i q^{52} -32.1808 q^{53} +38.2074 q^{56} -13.0385 q^{58} -5.91071 q^{59} +35.0235i q^{61} +106.067i q^{62} +102.229 q^{64} -100.828i q^{65} +90.9808 q^{67} -177.535i q^{68} +132.942 q^{70} -98.4739 q^{71} +47.0951i q^{73} +89.4092i q^{74} -189.044i q^{76} +27.1203i q^{79} +36.5926 q^{80} +252.846 q^{82} +11.2547i q^{83} -186.659i q^{85} +114.622 q^{86} +62.4300 q^{89} -118.301 q^{91} +20.3283 q^{92} -162.459i q^{94} -198.760i q^{95} -110.622 q^{97} -3.11962i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4}+O(q^{10}) $ 8 * q - 32 * q^4	$ 8 q - 32 q^{4} - 104 q^{16} - 48 q^{25} + 272 q^{31} - 288 q^{34} + 160 q^{37} - 8 q^{49} - 520 q^{58} + 416 q^{64} + 520 q^{67} + 440 q^{70} + 360 q^{82} - 600 q^{91} - 400 q^{97}+O(q^{100}) $ 8 * q - 32 * q^4 - 104 * q^16 - 48 * q^25 + 272 * q^31 - 288 * q^34 + 160 * q^37 - 8 * q^49 - 520 * q^58 + 416 * q^64 + 520 * q^67 + 440 * q^70 + 360 * q^82 - 600 * q^91 - 400 * 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 $

$2$	3.11962i	1.55981i	0.625897	+	0.779906i	$0.284733\pi$
$2$	3.11962i	1.55981i	−0.625897	+	0.779906i	$0.715267\pi$
$3$	0	0
$3$	0	0
$4$	−5.73205	−1.43301
$5$	−6.02665	−1.20533	−0.602665	−	0.797994i	$-0.705894\pi$
$5$	−6.02665	−1.20533	−0.602665	+	0.797994i	$0.705894\pi$
$6$	0	0
$7$	7.07107i	1.01015i	0.863075	+	0.505076i	$0.168536\pi$
$7$	7.07107i	1.01015i	−0.863075	+	0.505076i	$0.831464\pi$
$8$	− 5.40335i	− 0.675418i
$9$	0	0
$10$	− 18.8009i	− 1.88009i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	16.7303i	1.28695i	0.765468	+	0.643474i	$0.222508\pi$
$13$	16.7303i	1.28695i	−0.765468	+	0.643474i	$0.777492\pi$
$14$	−22.0591	−1.57565
$15$	0	0
$16$	−6.07180	−0.379487
$17$	30.9723i	1.82190i	0.412520	+	0.910949i	$0.364649\pi$
$17$	30.9723i	1.82190i	−0.412520	+	0.910949i	$0.635351\pi$
$18$	0	0
$19$	32.9802i	1.73580i	0.496740	+	0.867899i	$0.334530\pi$
$19$	32.9802i	1.73580i	−0.496740	+	0.867899i	$0.665470\pi$
$20$	34.5451	1.72725
$21$	0	0
$22$	0	0
$23$	−3.54643	−0.154192	−0.0770962	−	0.997024i	$-0.524565\pi$
$23$	−3.54643	−0.154192	−0.0770962	+	0.997024i	$0.524565\pi$
$24$	0	0
$25$	11.3205	0.452820
$26$	−52.1923	−2.00740
$27$	0	0
$28$	− 40.5317i	− 1.44756i
$29$	4.17950i	0.144121i	0.997400	+	0.0720604i	$0.0229574\pi$
$29$	4.17950i	0.144121i	−0.997400	+	0.0720604i	$0.977043\pi$
$30$	0	0
$31$	34.0000	1.09677	0.548387	−	0.836225i	$-0.315242\pi$
$31$	34.0000	1.09677	0.548387	+	0.836225i	$0.315242\pi$
$32$	− 40.5551i	− 1.26735i
$33$	0	0
$34$	−96.6218	−2.84182
$35$	− 42.6148i	− 1.21757i
$36$	0	0
$37$	28.6603	0.774601	0.387301	−	0.921953i	$-0.373408\pi$
$37$	28.6603	0.774601	0.387301	+	0.921953i	$0.373408\pi$
$38$	−102.886	−2.70752
$39$	0	0
$40$	32.5641i	0.814102i
$41$	− 81.0502i	− 1.97683i	−0.151762	−	0.988417i	$-0.548495\pi$
$41$	− 81.0502i	− 1.97683i	0.151762	−	0.988417i	$-0.451505\pi$
$42$	0	0
$43$	− 36.7423i	− 0.854473i	−0.904140	−	0.427237i	$-0.859487\pi$
$43$	− 36.7423i	− 0.854473i	0.904140	−	0.427237i	$-0.140513\pi$
$44$	0	0
$45$	0	0
$46$	− 11.0635i	− 0.240511i
$47$	−52.0764	−1.10801	−0.554004	−	0.832514i	$-0.686901\pi$
$47$	−52.0764	−1.10801	−0.554004	+	0.832514i	$0.686901\pi$
$48$	0	0
$49$	−1.00000	−0.0204082
$50$	35.3157i	0.706314i
$51$	0	0
$52$	− 95.8991i	− 1.84421i
$53$	−32.1808	−0.607185	−0.303592	−	0.952802i	$-0.598186\pi$
$53$	−32.1808	−0.607185	−0.303592	+	0.952802i	$0.598186\pi$
$54$	0	0
$55$	0	0
$56$	38.2074	0.682276
$57$	0	0
$58$	−13.0385	−0.224801
$59$	−5.91071	−0.100182	−0.0500908	−	0.998745i	$-0.515951\pi$
$59$	−5.91071	−0.100182	−0.0500908	+	0.998745i	$0.515951\pi$
$60$	0	0
$61$	35.0235i	0.574156i	0.957907	+	0.287078i	$0.0926839\pi$
$61$	35.0235i	0.574156i	−0.957907	+	0.287078i	$0.907316\pi$
$62$	106.067i	1.71076i
$63$	0	0
$64$	102.229	1.59734
$65$	− 100.828i	− 1.55120i
$66$	0	0
$67$	90.9808	1.35792	0.678961	−	0.734174i	$-0.262431\pi$
$67$	90.9808	1.35792	0.678961	+	0.734174i	$0.262431\pi$
$68$	− 177.535i	− 2.61080i
$69$	0	0
$70$	132.942	1.89918
$71$	−98.4739	−1.38696	−0.693478	−	0.720477i	$-0.743923\pi$
$71$	−98.4739	−1.38696	−0.693478	+	0.720477i	$0.743923\pi$
$72$	0	0
$73$	47.0951i	0.645138i	0.946546	+	0.322569i	$0.104547\pi$
$73$	47.0951i	0.645138i	−0.946546	+	0.322569i	$0.895453\pi$
$74$	89.4092i	1.20823i
$75$	0	0
$76$	− 189.044i	− 2.48742i
$77$	0	0
$78$	0	0
$79$	27.1203i	0.343294i	0.985158	+	0.171647i	$0.0549089\pi$
$79$	27.1203i	0.343294i	−0.985158	+	0.171647i	$0.945091\pi$
$80$	36.5926	0.457407
$81$	0	0
$82$	252.846	3.08349
$83$	11.2547i	0.135598i	0.997699	+	0.0677991i	$0.0215977\pi$
$83$	11.2547i	0.135598i	−0.997699	+	0.0677991i	$0.978402\pi$
$84$	0	0
$85$	− 186.659i	− 2.19599i
$86$	114.622	1.33282
$87$	0	0
$88$	0	0
$89$	62.4300	0.701460	0.350730	−	0.936477i	$-0.385933\pi$
$89$	62.4300	0.701460	0.350730	+	0.936477i	$0.385933\pi$
$90$	0	0
$91$	−118.301	−1.30001
$92$	20.3283	0.220960
$93$	0	0
$94$	− 162.459i	− 1.72828i
$95$	− 198.760i	− 2.09221i
$96$	0	0
$97$	−110.622	−1.14043	−0.570215	−	0.821495i	$-0.693140\pi$
$97$	−110.622	−1.14043	−0.570215	+	0.821495i	$0.693140\pi$
$98$	− 3.11962i	− 0.0318329i
$99$	0	0
$100$	−64.8897	−0.648897
$101$	96.6483i	0.956914i	0.878111	+	0.478457i	$0.158804\pi$
$101$	96.6483i	0.956914i	−0.878111	+	0.478457i	$0.841196\pi$
$102$	0	0
$103$	57.5833	0.559061	0.279531	−	0.960137i	$-0.409821\pi$
$103$	57.5833	0.559061	0.279531	+	0.960137i	$0.409821\pi$
$104$	90.3997	0.869228
$105$	0	0
$106$	− 100.392i	− 0.947094i
$107$	− 120.829i	− 1.12925i	−0.825349	−	0.564623i	$-0.809021\pi$
$107$	− 120.829i	− 1.12925i	0.825349	−	0.564623i	$-0.190979\pi$
$108$	0	0
$109$	22.8949i	0.210045i	0.994470	+	0.105022i	$0.0334914\pi$
$109$	22.8949i	0.210045i	−0.994470	+	0.105022i	$0.966509\pi$
$110$	0	0
$111$	0	0
$112$	− 42.9341i	− 0.383340i
$113$	54.5566	0.482802	0.241401	−	0.970425i	$-0.422393\pi$
$113$	54.5566	0.482802	0.241401	+	0.970425i	$0.422393\pi$
$114$	0	0
$115$	21.3731	0.185853
$116$	− 23.9571i	− 0.206527i
$117$	0	0
$118$	− 18.4392i	− 0.156264i
$119$	−219.007	−1.84039
$120$	0	0
$121$	0	0
$122$	−109.260	−0.895575
$123$	0	0
$124$	−194.890	−1.57169
$125$	82.4415	0.659532
$126$	0	0
$127$	− 156.950i	− 1.23583i	−0.786245	−	0.617915i	$-0.787977\pi$
$127$	− 156.950i	− 1.23583i	0.786245	−	0.617915i	$-0.212023\pi$
$128$	156.697i	1.22420i
$129$	0	0
$130$	314.545	2.41958
$131$	167.400i	1.27786i	0.769264	+	0.638930i	$0.220623\pi$
$131$	167.400i	1.27786i	−0.769264	+	0.638930i	$0.779377\pi$
$132$	0	0
$133$	−233.205	−1.75342
$134$	283.826i	2.11810i
$135$	0	0
$136$	167.354	1.23054
$137$	132.818	0.969476	0.484738	−	0.874659i	$-0.338915\pi$
$137$	132.818	0.969476	0.484738	+	0.874659i	$0.338915\pi$
$138$	0	0
$139$	52.1228i	0.374984i	0.982266	+	0.187492i	$0.0600359\pi$
$139$	52.1228i	0.374984i	−0.982266	+	0.187492i	$0.939964\pi$
$140$	244.270i	1.74479i
$141$	0	0
$142$	− 307.202i	− 2.16339i
$143$	0	0
$144$	0	0
$145$	− 25.1884i	− 0.173713i
$146$	−146.919	−1.00629
$147$	0	0
$148$	−164.282	−1.11001
$149$	− 75.7508i	− 0.508395i	−0.967152	−	0.254197i	$-0.918189\pi$
$149$	− 75.7508i	− 0.508395i	0.967152	−	0.254197i	$-0.0818113\pi$
$150$	0	0
$151$	90.7127i	0.600747i	0.953822	+	0.300373i	$0.0971113\pi$
$151$	90.7127i	0.600747i	−0.953822	+	0.300373i	$0.902889\pi$
$152$	178.203	1.17239
$153$	0	0
$154$	0	0
$155$	−204.906	−1.32197
$156$	0	0
$157$	140.526	0.895067	0.447534	−	0.894267i	$-0.352302\pi$
$157$	140.526	0.895067	0.447534	+	0.894267i	$0.352302\pi$
$158$	−84.6050	−0.535475
$159$	0	0
$160$	244.411i	1.52757i
$161$	− 25.0770i	− 0.155758i
$162$	0	0
$163$	−34.9038	−0.214134	−0.107067	−	0.994252i	$-0.534146\pi$
$163$	−34.9038	−0.214134	−0.107067	+	0.994252i	$0.534146\pi$
$164$	464.584i	2.83283i
$165$	0	0
$166$	−35.1103	−0.211508
$167$	220.089i	1.31790i	0.752187	+	0.658950i	$0.228999\pi$
$167$	220.089i	1.31790i	−0.752187	+	0.658950i	$0.771001\pi$
$168$	0	0
$169$	−110.904	−0.656236
$170$	582.306	3.42533
$171$	0	0
$172$	210.609i	1.22447i
$173$	− 116.262i	− 0.672035i	−0.941856	−	0.336017i	$-0.890920\pi$
$173$	− 116.262i	− 0.672035i	0.941856	−	0.336017i	$-0.109080\pi$
$174$	0	0
$175$	80.0481i	0.457418i
$176$	0	0
$177$	0	0
$178$	194.758i	1.09415i
$179$	336.797	1.88155	0.940773	−	0.339037i	$-0.110101\pi$
$179$	336.797	1.88155	0.940773	+	0.339037i	$0.110101\pi$
$180$	0	0
$181$	−108.660	−0.600333	−0.300166	−	0.953887i	$-0.597042\pi$
$181$	−108.660	−0.600333	−0.300166	+	0.953887i	$0.597042\pi$
$182$	− 369.055i	− 2.02778i
$183$	0	0
$184$	19.1626i	0.104144i
$185$	−172.725	−0.933650
$186$	0	0
$187$	0	0
$188$	298.504	1.58779
$189$	0	0
$190$	620.056	3.26345
$191$	120.533	0.631063	0.315531	−	0.948915i	$-0.397817\pi$
$191$	120.533	0.631063	0.315531	+	0.948915i	$0.397817\pi$
$192$	0	0
$193$	− 215.971i	− 1.11902i	−0.828823	−	0.559511i	$-0.810989\pi$
$193$	− 215.971i	− 1.11902i	0.828823	−	0.559511i	$-0.189011\pi$
$194$	− 345.098i	− 1.77886i
$195$	0	0
$196$	5.73205	0.0292452
$197$	− 243.315i	− 1.23510i	−0.786532	−	0.617550i	$-0.788125\pi$
$197$	− 243.315i	− 1.23510i	0.786532	−	0.617550i	$-0.211875\pi$
$198$	0	0
$199$	2.02439	0.0101728	0.00508641	−	0.999987i	$-0.498381\pi$
$199$	2.02439	0.0101728	0.00508641	+	0.999987i	$0.498381\pi$
$200$	− 61.1686i	− 0.305843i
$201$	0	0
$202$	−301.506	−1.49261
$203$	−29.5535	−0.145584
$204$	0	0
$205$	488.461i	2.38274i
$206$	179.638i	0.872030i
$207$	0	0
$208$	− 101.583i	− 0.488380i
$209$	0	0
$210$	0	0
$211$	247.636i	1.17363i	0.809721	+	0.586815i	$0.199619\pi$
$211$	247.636i	1.17363i	−0.809721	+	0.586815i	$0.800381\pi$
$212$	184.462	0.870103
$213$	0	0
$214$	376.942	1.76141
$215$	221.433i	1.02992i
$216$	0	0
$217$	240.416i	1.10791i
$218$	−71.4233	−0.327630
$219$	0	0
$220$	0	0
$221$	−518.176	−2.34469
$222$	0	0
$223$	−263.205	−1.18029	−0.590146	−	0.807297i	$-0.700930\pi$
$223$	−263.205	−1.18029	−0.590146	+	0.807297i	$0.700930\pi$
$224$	286.768	1.28021
$225$	0	0
$226$	170.196i	0.753080i
$227$	− 40.6151i	− 0.178921i	−0.995990	−	0.0894606i	$-0.971486\pi$
$227$	− 40.6151i	− 0.178921i	0.995990	−	0.0894606i	$-0.0285143\pi$
$228$	0	0
$229$	71.0141	0.310105	0.155053	−	0.987906i	$-0.450445\pi$
$229$	71.0141	0.310105	0.155053	+	0.987906i	$0.450445\pi$
$230$	66.6759i	0.289895i
$231$	0	0
$232$	22.5833	0.0973418
$233$	292.125i	1.25375i	0.779118	+	0.626877i	$0.215667\pi$
$233$	292.125i	1.25375i	−0.779118	+	0.626877i	$0.784333\pi$
$234$	0	0
$235$	313.846	1.33552
$236$	33.8805	0.143561
$237$	0	0
$238$	− 683.219i	− 2.87067i
$239$	343.159i	1.43581i	0.696141	+	0.717905i	$0.254899\pi$
$239$	343.159i	1.43581i	−0.696141	+	0.717905i	$0.745101\pi$
$240$	0	0
$241$	305.294i	1.26678i	0.773832	+	0.633391i	$0.218337\pi$
$241$	305.294i	1.26678i	−0.773832	+	0.633391i	$0.781663\pi$
$242$	0	0
$243$	0	0
$244$	− 200.757i	− 0.822773i
$245$	6.02665	0.0245986
$246$	0	0
$247$	−551.769	−2.23388
$248$	− 183.714i	− 0.740781i
$249$	0	0
$250$	257.186i	1.02875i
$251$	−448.251	−1.78586	−0.892931	−	0.450193i	$-0.851355\pi$
$251$	−448.251	−1.78586	−0.892931	+	0.450193i	$0.851355\pi$
$252$	0	0
$253$	0	0
$254$	489.626	1.92766
$255$	0	0
$256$	−79.9179	−0.312179
$257$	240.865	0.937219	0.468609	−	0.883406i	$-0.344755\pi$
$257$	240.865	0.937219	0.468609	+	0.883406i	$0.344755\pi$
$258$	0	0
$259$	202.659i	0.782466i
$260$	577.950i	2.22289i
$261$	0	0
$262$	−522.224	−1.99322
$263$	− 49.7939i	− 0.189331i	−0.995509	−	0.0946653i	$-0.969822\pi$
$263$	− 49.7939i	− 0.189331i	0.995509	−	0.0946653i	$-0.0301781\pi$
$264$	0	0
$265$	193.942	0.731858
$266$	− 727.512i	− 2.73501i
$267$	0	0
$268$	−521.506	−1.94592
$269$	−241.646	−0.898311	−0.449156	−	0.893454i	$-0.648275\pi$
$269$	−241.646	−0.898311	−0.449156	+	0.893454i	$0.648275\pi$
$270$	0	0
$271$	− 312.338i	− 1.15254i	−0.817260	−	0.576270i	$-0.804508\pi$
$271$	− 312.338i	− 1.15254i	0.817260	−	0.576270i	$-0.195492\pi$
$272$	− 188.057i	− 0.691387i
$273$	0	0
$274$	414.343i	1.51220i
$275$	0	0
$276$	0	0
$277$	− 278.088i	− 1.00393i	−0.864889	−	0.501963i	$-0.832611\pi$
$277$	− 278.088i	− 1.00393i	0.864889	−	0.501963i	$-0.167389\pi$
$278$	−162.604	−0.584905
$279$	0	0
$280$	−230.263	−0.822367
$281$	− 181.878i	− 0.647253i	−0.946185	−	0.323626i	$-0.895098\pi$
$281$	− 181.878i	− 0.647253i	0.946185	−	0.323626i	$-0.104902\pi$
$282$	0	0
$283$	− 254.422i	− 0.899019i	−0.893276	−	0.449510i	$-0.851599\pi$
$283$	− 254.422i	− 0.899019i	0.893276	−	0.449510i	$-0.148401\pi$
$284$	564.458	1.98753
$285$	0	0
$286$	0	0
$287$	573.111	1.99690
$288$	0	0
$289$	−670.281	−2.31931
$290$	78.5783	0.270960
$291$	0	0
$292$	− 269.952i	− 0.924492i
$293$	− 230.672i	− 0.787277i	−0.919265	−	0.393638i	$-0.871216\pi$
$293$	− 230.672i	− 0.787277i	0.919265	−	0.393638i	$-0.128784\pi$
$294$	0	0
$295$	35.6218	0.120752
$296$	− 154.861i	− 0.523180i
$297$	0	0
$298$	236.314	0.793000
$299$	− 59.3329i	− 0.198438i
$300$	0	0
$301$	259.808	0.863148
$302$	−282.990	−0.937052
$303$	0	0
$304$	− 200.249i	− 0.658714i
$305$	− 211.075i	− 0.692048i
$306$	0	0
$307$	− 3.92541i	− 0.0127863i	−0.999980	−	0.00639317i	$-0.997965\pi$
$307$	− 3.92541i	− 0.0127863i	0.999980	−	0.00639317i	$-0.00203502\pi$
$308$	0	0
$309$	0	0
$310$	− 639.230i	− 2.06203i
$311$	−79.5824	−0.255892	−0.127946	−	0.991781i	$-0.540838\pi$
$311$	−79.5824	−0.255892	−0.127946	+	0.991781i	$0.540838\pi$
$312$	0	0
$313$	315.000	1.00639	0.503195	−	0.864173i	$-0.332158\pi$
$313$	315.000	1.00639	0.503195	+	0.864173i	$0.332158\pi$
$314$	438.387i	1.39614i
$315$	0	0
$316$	− 155.455i	− 0.491945i
$317$	194.584	0.613828	0.306914	−	0.951737i	$-0.400704\pi$
$317$	194.584	0.613828	0.306914	+	0.951737i	$0.400704\pi$
$318$	0	0
$319$	0	0
$320$	−616.101	−1.92532
$321$	0	0
$322$	78.2309	0.242953
$323$	−1021.47	−3.16245
$324$	0	0
$325$	189.396i	0.582756i
$326$	− 108.887i	− 0.334008i
$327$	0	0
$328$	−437.942	−1.33519
$329$	− 368.236i	− 1.11926i
$330$	0	0
$331$	514.315	1.55382	0.776911	−	0.629610i	$-0.216785\pi$
$331$	514.315	1.55382	0.776911	+	0.629610i	$0.216785\pi$
$332$	− 64.5122i	− 0.194314i
$333$	0	0
$334$	−686.596	−2.05568
$335$	−548.309	−1.63674
$336$	0	0
$337$	115.961i	0.344098i	0.985088	+	0.172049i	$0.0550387\pi$
$337$	115.961i	0.344098i	−0.985088	+	0.172049i	$0.944961\pi$
$338$	− 345.978i	− 1.02360i
$339$	0	0
$340$	1069.94i	3.14688i
$341$	0	0
$342$	0	0
$343$	339.411i	0.989537i
$344$	−198.532	−0.577127
$345$	0	0
$346$	362.694	1.04825
$347$	135.144i	0.389463i	0.980857	+	0.194732i	$0.0623836\pi$
$347$	135.144i	0.389463i	−0.980857	+	0.194732i	$0.937616\pi$
$348$	0	0
$349$	− 144.902i	− 0.415191i	−0.978215	−	0.207595i	$-0.933436\pi$
$349$	− 144.902i	− 0.415191i	0.978215	−	0.207595i	$-0.0665637\pi$
$350$	−249.720	−0.713485
$351$	0	0
$352$	0	0
$353$	−579.562	−1.64182	−0.820910	−	0.571058i	$-0.806533\pi$
$353$	−579.562	−1.64182	−0.820910	+	0.571058i	$0.806533\pi$
$354$	0	0
$355$	593.468	1.67174
$356$	−357.852	−1.00520
$357$	0	0
$358$	1050.68i	2.93486i
$359$	215.314i	0.599761i	0.953977	+	0.299880i	$0.0969467\pi$
$359$	215.314i	0.599761i	−0.953977	+	0.299880i	$0.903053\pi$
$360$	0	0
$361$	−726.692	−2.01300
$362$	− 338.979i	− 0.936406i
$363$	0	0
$364$	678.109	1.86294
$365$	− 283.826i	− 0.777605i
$366$	0	0
$367$	363.468	0.990376	0.495188	−	0.868786i	$-0.335099\pi$
$367$	363.468	0.990376	0.495188	+	0.868786i	$0.335099\pi$
$368$	21.5332	0.0585141
$369$	0	0
$370$	− 538.838i	− 1.45632i
$371$	− 227.552i	− 0.613349i
$372$	0	0
$373$	159.861i	0.428581i	0.976770	+	0.214290i	$0.0687438\pi$
$373$	159.861i	0.428581i	−0.976770	+	0.214290i	$0.931256\pi$
$374$	0	0
$375$	0	0
$376$	281.387i	0.748369i
$377$	−69.9244	−0.185476
$378$	0	0
$379$	−260.711	−0.687893	−0.343946	−	0.938989i	$-0.611764\pi$
$379$	−260.711	−0.687893	−0.343946	+	0.938989i	$0.611764\pi$
$380$	1139.30i	2.99816i
$381$	0	0
$382$	376.018i	0.984339i
$383$	275.874	0.720298	0.360149	−	0.932895i	$-0.382726\pi$
$383$	275.874	0.720298	0.360149	+	0.932895i	$0.382726\pi$
$384$	0	0
$385$	0	0
$386$	673.749	1.74546
$387$	0	0
$388$	634.090	1.63425
$389$	127.023	0.326538	0.163269	−	0.986582i	$-0.447796\pi$
$389$	127.023	0.326538	0.163269	+	0.986582i	$0.447796\pi$
$390$	0	0
$391$	− 109.841i	− 0.280923i
$392$	5.40335i	0.0137840i
$393$	0	0
$394$	759.050	1.92652
$395$	− 163.444i	− 0.413783i
$396$	0	0
$397$	278.731	0.702092	0.351046	−	0.936358i	$-0.385826\pi$
$397$	278.731	0.702092	0.351046	+	0.936358i	$0.385826\pi$
$398$	6.31534i	0.0158677i
$399$	0	0
$400$	−68.7358	−0.171840
$401$	−529.997	−1.32169	−0.660845	−	0.750523i	$-0.729802\pi$
$401$	−529.997	−1.32169	−0.660845	+	0.750523i	$0.729802\pi$
$402$	0	0
$403$	568.831i	1.41149i
$404$	− 553.993i	− 1.37127i
$405$	0	0
$406$	− 92.1959i	− 0.227084i
$407$	0	0
$408$	0	0
$409$	614.685i	1.50290i	0.659791	+	0.751449i	$0.270645\pi$
$409$	614.685i	1.50290i	−0.659791	+	0.751449i	$0.729355\pi$
$410$	−1523.81	−3.71662
$411$	0	0
$412$	−330.070	−0.801142
$413$	− 41.7950i	− 0.101199i
$414$	0	0
$415$	− 67.8278i	− 0.163441i
$416$	678.500	1.63101
$417$	0	0
$418$	0	0
$419$	166.235	0.396742	0.198371	−	0.980127i	$-0.436435\pi$
$419$	166.235	0.396742	0.198371	+	0.980127i	$0.436435\pi$
$420$	0	0
$421$	−318.808	−0.757263	−0.378631	−	0.925548i	$-0.623605\pi$
$421$	−318.808	−0.757263	−0.378631	+	0.925548i	$0.623605\pi$
$422$	−772.531	−1.83064
$423$	0	0
$424$	173.884i	0.410104i
$425$	350.622i	0.824992i
$426$	0	0
$427$	−247.654	−0.579985
$428$	692.600i	1.61823i
$429$	0	0
$430$	−690.788	−1.60648
$431$	317.262i	0.736106i	0.929805	+	0.368053i	$0.119976\pi$
$431$	317.262i	0.736106i	−0.929805	+	0.368053i	$0.880024\pi$
$432$	0	0
$433$	−260.429	−0.601454	−0.300727	−	0.953710i	$-0.597229\pi$
$433$	−260.429	−0.601454	−0.300727	+	0.953710i	$0.597229\pi$
$434$	−750.008	−1.72813
$435$	0	0
$436$	− 131.235i	− 0.300997i
$437$	− 116.962i	− 0.267647i
$438$	0	0
$439$	579.867i	1.32088i	0.750878	+	0.660441i	$0.229631\pi$
$439$	579.867i	1.32088i	−0.750878	+	0.660441i	$0.770369\pi$
$440$	0	0
$441$	0	0
$442$	− 1616.51i	− 3.65727i
$443$	115.550	0.260835	0.130417	−	0.991459i	$-0.458368\pi$
$443$	115.550	0.260835	0.130417	+	0.991459i	$0.458368\pi$
$444$	0	0
$445$	−376.244	−0.845491
$446$	− 821.101i	− 1.84103i
$447$	0	0
$448$	722.872i	1.61355i
$449$	659.609	1.46906	0.734531	−	0.678575i	$-0.237402\pi$
$449$	659.609	1.46906	0.734531	+	0.678575i	$0.237402\pi$
$450$	0	0
$451$	0	0
$452$	−312.721	−0.691861
$453$	0	0
$454$	126.704	0.279083
$455$	712.960	1.56695
$456$	0	0
$457$	− 305.121i	− 0.667661i	−0.942633	−	0.333830i	$-0.891659\pi$
$457$	− 305.121i	− 0.667661i	0.942633	−	0.333830i	$-0.108341\pi$
$458$	221.537i	0.483706i
$459$	0	0
$460$	−122.512	−0.266329
$461$	43.7347i	0.0948693i	0.998874	+	0.0474346i	$0.0151046\pi$
$461$	43.7347i	0.0948693i	−0.998874	+	0.0474346i	$0.984895\pi$
$462$	0	0
$463$	−602.032	−1.30028	−0.650142	−	0.759812i	$-0.725291\pi$
$463$	−602.032	−1.30028	−0.650142	+	0.759812i	$0.725291\pi$
$464$	− 25.3771i	− 0.0546920i
$465$	0	0
$466$	−911.319	−1.95562
$467$	461.278	0.987747	0.493873	−	0.869534i	$-0.335581\pi$
$467$	461.278	0.987747	0.493873	+	0.869534i	$0.335581\pi$
$468$	0	0
$469$	643.331i	1.37171i
$470$	979.082i	2.08315i
$471$	0	0
$472$	31.9376i	0.0676644i
$473$	0	0
$474$	0	0
$475$	373.352i	0.786005i
$476$	1255.36	2.63731
$477$	0	0
$478$	−1070.53	−2.23959
$479$	522.797i	1.09143i	0.837970	+	0.545717i	$0.183743\pi$
$479$	522.797i	1.09143i	−0.837970	+	0.545717i	$0.816257\pi$
$480$	0	0
$481$	479.495i	0.996872i
$482$	−952.403	−1.97594
$483$	0	0
$484$	0	0
$485$	666.679	1.37460
$486$	0	0
$487$	222.750	0.457392	0.228696	−	0.973498i	$-0.426554\pi$
$487$	222.750	0.457392	0.228696	+	0.973498i	$0.426554\pi$
$488$	189.244	0.387796
$489$	0	0
$490$	18.8009i	0.0383691i
$491$	− 394.952i	− 0.804383i	−0.915555	−	0.402192i	$-0.868249\pi$
$491$	− 394.952i	− 0.804383i	0.915555	−	0.402192i	$-0.131751\pi$
$492$	0	0
$493$	−129.449	−0.262573
$494$	− 1721.31i	− 3.48444i
$495$	0	0
$496$	−206.441	−0.416212
$497$	− 696.316i	− 1.40104i
$498$	0	0
$499$	−502.942	−1.00790	−0.503950	−	0.863733i	$-0.668120\pi$
$499$	−502.942	−1.00790	−0.503950	+	0.863733i	$0.668120\pi$
$500$	−472.559	−0.945118
$501$	0	0
$502$	− 1398.38i	− 2.78561i
$503$	− 51.3339i	− 0.102056i	−0.998697	−	0.0510278i	$-0.983750\pi$
$503$	− 51.3339i	− 0.102056i	0.998697	−	0.0510278i	$-0.0162497\pi$
$504$	0	0
$505$	− 582.466i	− 1.15340i
$506$	0	0
$507$	0	0
$508$	899.648i	1.77096i
$509$	−450.995	−0.886041	−0.443020	−	0.896512i	$-0.646093\pi$
$509$	−450.995	−0.886041	−0.443020	+	0.896512i	$0.646093\pi$
$510$	0	0
$511$	−333.013	−0.651688
$512$	377.474i	0.737255i
$513$	0	0
$514$	751.409i	1.46188i
$515$	−347.034	−0.673853
$516$	0	0
$517$	0	0
$518$	−632.219	−1.22050
$519$	0	0
$520$	−544.808	−1.04771
$521$	10.6620	0.0204645	0.0102323	−	0.999948i	$-0.496743\pi$
$521$	10.6620	0.0204645	0.0102323	+	0.999948i	$0.496743\pi$
$522$	0	0
$523$	− 829.309i	− 1.58568i	−0.609432	−	0.792839i	$-0.708602\pi$
$523$	− 829.309i	− 1.58568i	0.609432	−	0.792839i	$-0.291398\pi$
$524$	− 959.544i	− 1.83119i
$525$	0	0
$526$	155.338	0.295320
$527$	1053.06i	1.99821i
$528$	0	0
$529$	−516.423	−0.976225
$530$	605.027i	1.14156i
$531$	0	0
$532$	1336.74	2.51268
$533$	1356.00	2.54408
$534$	0	0
$535$	728.197i	1.36112i
$536$	− 491.601i	− 0.917165i
$537$	0	0
$538$	− 753.844i	− 1.40120i
$539$	0	0
$540$	0	0
$541$	− 357.043i	− 0.659969i	−0.943986	−	0.329985i	$-0.892956\pi$
$541$	− 357.043i	− 0.659969i	0.943986	−	0.329985i	$-0.107044\pi$
$542$	974.377	1.79774
$543$	0	0
$544$	1256.08	2.30898
$545$	− 137.979i	− 0.253173i
$546$	0	0
$547$	− 331.461i	− 0.605961i	−0.952997	−	0.302981i	$-0.902018\pi$
$547$	− 331.461i	− 0.605961i	0.952997	−	0.302981i	$-0.0979818\pi$
$548$	−761.321	−1.38927
$549$	0	0
$550$	0	0
$551$	−137.841	−0.250165
$552$	0	0
$553$	−191.769	−0.346780
$554$	867.529	1.56594
$555$	0	0
$556$	− 298.771i	− 0.537357i
$557$	499.064i	0.895985i	0.894037	+	0.447993i	$0.147861\pi$
$557$	499.064i	0.895985i	−0.894037	+	0.447993i	$0.852139\pi$
$558$	0	0
$559$	614.711	1.09966
$560$	258.749i	0.462051i
$561$	0	0
$562$	567.391	1.00959
$563$	− 902.615i	− 1.60322i	−0.597845	−	0.801612i	$-0.703976\pi$
$563$	− 902.615i	− 1.60322i	0.597845	−	0.801612i	$-0.296024\pi$
$564$	0	0
$565$	−328.794	−0.581935
$566$	793.702	1.40230
$567$	0	0
$568$	532.089i	0.936776i
$569$	616.386i	1.08328i	0.840611	+	0.541639i	$0.182196\pi$
$569$	616.386i	1.08328i	−0.840611	+	0.541639i	$0.817804\pi$
$570$	0	0
$571$	174.706i	0.305965i	0.988229	+	0.152983i	$0.0488879\pi$
$571$	174.706i	0.305965i	−0.988229	+	0.152983i	$0.951112\pi$
$572$	0	0
$573$	0	0
$574$	1787.89i	3.11479i
$575$	−40.1473	−0.0698215
$576$	0	0
$577$	419.519	0.727069	0.363535	−	0.931581i	$-0.381570\pi$
$577$	419.519	0.727069	0.363535	+	0.931581i	$0.381570\pi$
$578$	− 2091.02i	− 3.61769i
$579$	0	0
$580$	144.381i	0.248933i
$581$	−79.5824	−0.136975
$582$	0	0
$583$	0	0
$584$	254.471	0.435738
$585$	0	0
$586$	719.610	1.22800
$587$	−460.452	−0.784415	−0.392208	−	0.919877i	$-0.628289\pi$
$587$	−460.452	−0.784415	−0.392208	+	0.919877i	$0.628289\pi$
$588$	0	0
$589$	1121.33i	1.90378i
$590$	111.127i	0.188350i
$591$	0	0
$592$	−174.019	−0.293951
$593$	329.636i	0.555879i	0.960599	+	0.277940i	$0.0896515\pi$
$593$	329.636i	0.555879i	−0.960599	+	0.277940i	$0.910349\pi$
$594$	0	0
$595$	1319.88	2.21828
$596$	434.207i	0.728536i
$597$	0	0
$598$	185.096	0.309525
$599$	61.4259	0.102547	0.0512737	−	0.998685i	$-0.483672\pi$
$599$	61.4259	0.102547	0.0512737	+	0.998685i	$0.483672\pi$
$600$	0	0
$601$	− 98.7003i	− 0.164227i	−0.996623	−	0.0821134i	$-0.973833\pi$
$601$	− 98.7003i	− 0.164227i	0.996623	−	0.0821134i	$-0.0261670\pi$
$602$	810.502i	1.34635i
$603$	0	0
$604$	− 519.970i	− 0.860878i
$605$	0	0
$606$	0	0
$607$	− 945.864i	− 1.55826i	−0.626862	−	0.779130i	$-0.715661\pi$
$607$	− 945.864i	− 1.55826i	0.626862	−	0.779130i	$-0.284339\pi$
$608$	1337.51	2.19986
$609$	0	0
$610$	658.473	1.07946
$611$	− 871.255i	− 1.42595i
$612$	0	0
$613$	− 140.320i	− 0.228907i	−0.993429	−	0.114453i	$-0.963488\pi$
$613$	− 140.320i	− 0.228907i	0.993429	−	0.114453i	$-0.0365116\pi$
$614$	12.2458	0.0199443
$615$	0	0
$616$	0	0
$617$	564.302	0.914590	0.457295	−	0.889315i	$-0.348818\pi$
$617$	564.302	0.914590	0.457295	+	0.889315i	$0.348818\pi$
$618$	0	0
$619$	−375.814	−0.607131	−0.303566	−	0.952811i	$-0.598177\pi$
$619$	−375.814	−0.607131	−0.303566	+	0.952811i	$0.598177\pi$
$620$	1174.53	1.89441
$621$	0	0
$622$	− 248.267i	− 0.399143i
$623$	441.447i	0.708582i
$624$	0	0
$625$	−779.859	−1.24777
$626$	982.681i	1.56978i
$627$	0	0
$628$	−805.500	−1.28264
$629$	887.673i	1.41124i
$630$	0	0
$631$	190.922	0.302570	0.151285	−	0.988490i	$-0.451659\pi$
$631$	190.922	0.302570	0.151285	+	0.988490i	$0.451659\pi$
$632$	146.540	0.231867
$633$	0	0
$634$	607.027i	0.957457i
$635$	945.886i	1.48958i
$636$	0	0
$637$	− 16.7303i	− 0.0262642i
$638$	0	0
$639$	0	0
$640$	− 944.358i	− 1.47556i
$641$	−335.793	−0.523858	−0.261929	−	0.965087i	$-0.584359\pi$
$641$	−335.793	−0.523858	−0.261929	+	0.965087i	$0.584359\pi$
$642$	0	0
$643$	109.808	0.170774	0.0853870	−	0.996348i	$-0.472787\pi$
$643$	109.808	0.170774	0.0853870	+	0.996348i	$0.472787\pi$
$644$	143.743i	0.223203i
$645$	0	0
$646$	− 3186.60i	− 4.93282i
$647$	393.200	0.607728	0.303864	−	0.952715i	$-0.401723\pi$
$647$	393.200	0.607728	0.303864	+	0.952715i	$0.401723\pi$
$648$	0	0
$649$	0	0
$650$	−590.844	−0.908990
$651$	0	0
$652$	200.070	0.306856
$653$	187.135	0.286577	0.143288	−	0.989681i	$-0.454232\pi$
$653$	187.135	0.286577	0.143288	+	0.989681i	$0.454232\pi$
$654$	0	0
$655$	− 1008.86i	− 1.54024i
$656$	492.120i	0.750183i
$657$	0	0
$658$	1148.76	1.74583
$659$	353.157i	0.535899i	0.963433	+	0.267949i	$0.0863460\pi$
$659$	353.157i	0.535899i	−0.963433	+	0.267949i	$0.913654\pi$
$660$	0	0
$661$	−580.885	−0.878797	−0.439398	−	0.898292i	$-0.644808\pi$
$661$	−580.885	−0.878797	−0.439398	+	0.898292i	$0.644808\pi$
$662$	1604.47i	2.42367i
$663$	0	0
$664$	60.8128	0.0915855
$665$	1405.45	2.11345
$666$	0	0
$667$	− 14.8223i	− 0.0222223i
$668$	− 1261.56i	− 1.88857i
$669$	0	0
$670$	− 1710.52i	− 2.55301i
$671$	0	0
$672$	0	0
$673$	541.526i	0.804644i	0.915498	+	0.402322i	$0.131797\pi$
$673$	541.526i	0.804644i	−0.915498	+	0.402322i	$0.868203\pi$
$674$	−361.754	−0.536727
$675$	0	0
$676$	635.706	0.940394
$677$	− 160.817i	− 0.237543i	−0.992922	−	0.118771i	$-0.962104\pi$
$677$	− 160.817i	− 0.237543i	0.992922	−	0.118771i	$-0.0378956\pi$
$678$	0	0
$679$	− 782.214i	− 1.15201i
$680$	−1008.58	−1.48321
$681$	0	0
$682$	0	0
$683$	33.6486	0.0492659	0.0246329	−	0.999697i	$-0.492158\pi$
$683$	33.6486	0.0492659	0.0246329	+	0.999697i	$0.492158\pi$
$684$	0	0
$685$	−800.449	−1.16854
$686$	−1058.84	−1.54349
$687$	0	0
$688$	223.092i	0.324262i
$689$	− 538.395i	− 0.781415i
$690$	0	0
$691$	1000.13	1.44737	0.723686	−	0.690129i	$-0.242446\pi$
$691$	1000.13	1.44737	0.723686	+	0.690129i	$0.242446\pi$
$692$	666.420i	0.963034i
$693$	0	0
$694$	−421.597	−0.607489
$695$	− 314.126i	− 0.451980i
$696$	0	0
$697$	2510.31	3.60159
$698$	452.038	0.647619
$699$	0	0
$700$	− 458.840i	− 0.655485i
$701$	− 161.581i	− 0.230500i	−0.993337	−	0.115250i	$-0.963233\pi$
$701$	− 161.581i	− 0.230500i	0.993337	−	0.115250i	$-0.0367669\pi$
$702$	0	0
$703$	945.220i	1.34455i
$704$	0	0
$705$	0	0
$706$	− 1808.02i	− 2.56093i
$707$	−683.407	−0.966629
$708$	0	0
$709$	−646.820	−0.912299	−0.456150	−	0.889903i	$-0.650772\pi$
$709$	−646.820	−0.912299	−0.456150	+	0.889903i	$0.650772\pi$
$710$	1851.40i	2.60760i
$711$	0	0
$712$	− 337.331i	− 0.473779i
$713$	−120.578	−0.169114
$714$	0	0
$715$	0	0
$716$	−1930.54	−2.69628
$717$	0	0
$718$	−671.699	−0.935514
$719$	1230.56	1.71148	0.855742	−	0.517403i	$-0.173101\pi$
$719$	1230.56	1.71148	0.855742	+	0.517403i	$0.173101\pi$
$720$	0	0
$721$	407.175i	0.564737i
$722$	− 2267.01i	− 3.13990i
$723$	0	0
$724$	622.846	0.860285
$725$	47.3141i	0.0652608i
$726$	0	0
$727$	178.494	0.245521	0.122760	−	0.992436i	$-0.460825\pi$
$727$	178.494	0.245521	0.122760	+	0.992436i	$0.460825\pi$
$728$	639.223i	0.878053i
$729$	0	0
$730$	885.429	1.21292
$731$	1137.99	1.55676
$732$	0	0
$733$	− 569.996i	− 0.777620i	−0.921318	−	0.388810i	$-0.872886\pi$
$733$	− 569.996i	− 0.777620i	0.921318	−	0.388810i	$-0.127114\pi$
$734$	1133.88i	1.54480i
$735$	0	0
$736$	143.826i	0.195415i
$737$	0	0
$738$	0	0
$739$	36.9182i	0.0499570i	0.999688	+	0.0249785i	$0.00795173\pi$
$739$	36.9182i	0.0499570i	−0.999688	+	0.0249785i	$0.992048\pi$
$740$	990.070	1.33793
$741$	0	0
$742$	709.878	0.956709
$743$	− 935.363i	− 1.25890i	−0.777041	−	0.629450i	$-0.783280\pi$
$743$	− 935.363i	− 1.25890i	0.777041	−	0.629450i	$-0.216720\pi$
$744$	0	0
$745$	456.524i	0.612783i
$746$	−498.705	−0.668505
$747$	0	0
$748$	0	0
$749$	854.393	1.14071
$750$	0	0
$751$	−944.053	−1.25706	−0.628530	−	0.777785i	$-0.716343\pi$
$751$	−944.053	−1.25706	−0.628530	+	0.777785i	$0.716343\pi$
$752$	316.197	0.420475
$753$	0	0
$754$	− 218.138i	− 0.289308i
$755$	− 546.694i	− 0.724098i
$756$	0	0
$757$	−228.205	−0.301460	−0.150730	−	0.988575i	$-0.548162\pi$
$757$	−228.205	−0.301460	−0.150730	+	0.988575i	$0.548162\pi$
$758$	− 813.322i	− 1.07298i
$759$	0	0
$760$	−1073.97	−1.41312
$761$	438.687i	0.576461i	0.957561	+	0.288231i	$0.0930670\pi$
$761$	438.687i	0.576461i	−0.957561	+	0.288231i	$0.906933\pi$
$762$	0	0
$763$	−161.891	−0.212177
$764$	−690.901	−0.904321
$765$	0	0
$766$	860.623i	1.12353i
$767$	− 98.8881i	− 0.128928i
$768$	0	0
$769$	113.000i	0.146945i	0.997297	+	0.0734723i	$0.0234080\pi$
$769$	113.000i	0.146945i	−0.997297	+	0.0734723i	$0.976592\pi$
$770$	0	0
$771$	0	0
$772$	1237.96i	1.60357i
$773$	1020.56	1.32026	0.660129	−	0.751152i	$-0.270502\pi$
$773$	1020.56	1.32026	0.660129	+	0.751152i	$0.270502\pi$
$774$	0	0
$775$	384.897	0.496642
$776$	597.728i	0.770268i
$777$	0	0
$778$	396.265i	0.509338i
$779$	2673.05	3.43139
$780$	0	0
$781$	0	0
$782$	342.662	0.438187
$783$	0	0
$784$	6.07180	0.00774464
$785$	−846.899	−1.07885
$786$	0	0
$787$	− 776.158i	− 0.986224i	−0.869966	−	0.493112i	$-0.835859\pi$
$787$	− 776.158i	− 0.986224i	0.869966	−	0.493112i	$-0.164141\pi$
$788$	1394.69i	1.76991i
$789$	0	0
$790$	509.885	0.645424
$791$	385.773i	0.487703i
$792$	0	0
$793$	−585.955	−0.738909
$794$	869.535i	1.09513i
$795$	0	0
$796$	−11.6039	−0.0145778
$797$	−1228.97	−1.54200	−0.770999	−	0.636836i	$-0.780243\pi$
$797$	−1228.97	−1.54200	−0.770999	+	0.636836i	$0.780243\pi$
$798$	0	0
$799$	− 1612.92i	− 2.01868i
$800$	− 459.104i	− 0.573880i
$801$	0	0
$802$	− 1653.39i	− 2.06159i
$803$	0	0
$804$	0	0
$805$	151.130i	0.187740i
$806$	−1774.54	−2.20166
$807$	0	0
$808$	522.224	0.646317
$809$	223.073i	0.275739i	0.990450	+	0.137870i	$0.0440255\pi$
$809$	223.073i	0.275739i	−0.990450	+	0.137870i	$0.955975\pi$
$810$	0	0
$811$	− 197.930i	− 0.244057i	−0.992527	−	0.122028i	$-0.961060\pi$
$811$	− 197.930i	− 0.244057i	0.992527	−	0.122028i	$-0.0389399\pi$
$812$	169.402	0.208624
$813$	0	0
$814$	0	0
$815$	210.353	0.258102
$816$	0	0
$817$	1211.77	1.48319
$818$	−1917.59	−2.34424
$819$	0	0
$820$	− 2799.88i	− 3.41449i
$821$	1165.08i	1.41910i	0.704656	+	0.709549i	$0.251101\pi$
$821$	1165.08i	1.41910i	−0.704656	+	0.709549i	$0.748899\pi$
$822$	0	0
$823$	−253.468	−0.307980	−0.153990	−	0.988072i	$-0.549212\pi$
$823$	−253.468	−0.307980	−0.153990	+	0.988072i	$0.549212\pi$
$824$	− 311.143i	− 0.377600i
$825$	0	0
$826$	130.385	0.157851
$827$	− 816.785i	− 0.987648i	−0.869562	−	0.493824i	$-0.835599\pi$
$827$	− 816.785i	− 0.987648i	0.869562	−	0.493824i	$-0.164401\pi$
$828$	0	0
$829$	522.454	0.630222	0.315111	−	0.949055i	$-0.397958\pi$
$829$	522.454	0.630222	0.315111	+	0.949055i	$0.397958\pi$
$830$	211.597	0.254937
$831$	0	0
$832$	1710.33i	2.05569i
$833$	− 30.9723i	− 0.0371816i
$834$	0	0
$835$	− 1326.40i	− 1.58850i
$836$	0	0
$837$	0	0
$838$	518.590i	0.618843i
$839$	−1008.38	−1.20189	−0.600943	−	0.799292i	$-0.705208\pi$
$839$	−1008.38	−1.20189	−0.600943	+	0.799292i	$0.705208\pi$
$840$	0	0
$841$	823.532	0.979229
$842$	− 994.560i	− 1.18119i
$843$	0	0
$844$	− 1419.46i	− 1.68183i
$845$	668.378	0.790980
$846$	0	0
$847$	0	0
$848$	195.395	0.230419
$849$	0	0
$850$	−1093.81	−1.28683
$851$	−101.641	−0.119438
$852$	0	0
$853$	1379.14i	1.61682i	0.588623	+	0.808408i	$0.299670\pi$
$853$	1379.14i	1.61682i	−0.588623	+	0.808408i	$0.700330\pi$
$854$	− 772.586i	− 0.904668i
$855$	0	0
$856$	−652.883	−0.762714
$857$	992.620i	1.15825i	0.815239	+	0.579125i	$0.196606\pi$
$857$	992.620i	1.15825i	−0.815239	+	0.579125i	$0.803394\pi$
$858$	0	0
$859$	−768.301	−0.894414	−0.447207	−	0.894431i	$-0.647581\pi$
$859$	−768.301	−0.894414	−0.447207	+	0.894431i	$0.647581\pi$
$860$	− 1269.27i	− 1.47589i
$861$	0	0
$862$	−989.737	−1.14819
$863$	−758.006	−0.878338	−0.439169	−	0.898404i	$-0.644727\pi$
$863$	−758.006	−0.878338	−0.439169	+	0.898404i	$0.644727\pi$
$864$	0	0
$865$	700.670i	0.810023i
$866$	− 812.442i	− 0.938154i
$867$	0	0
$868$	− 1378.08i	− 1.58765i
$869$	0	0
$870$	0	0
$871$	1522.14i	1.74757i
$872$	123.709	0.141868
$873$	0	0
$874$	364.877	0.417479
$875$	582.949i	0.666228i
$876$	0	0
$877$	1483.59i	1.69166i	0.533451	+	0.845831i	$0.320895\pi$
$877$	1483.59i	1.69166i	−0.533451	+	0.845831i	$0.679105\pi$
$878$	−1808.97	−2.06033
$879$	0	0
$880$	0	0
$881$	277.110	0.314540	0.157270	−	0.987556i	$-0.449731\pi$
$881$	277.110	0.314540	0.157270	+	0.987556i	$0.449731\pi$
$882$	0	0
$883$	1747.60	1.97916	0.989582	−	0.143969i	$-0.0459865\pi$
$883$	1747.60	1.97916	0.989582	+	0.143969i	$0.0459865\pi$
$884$	2970.21	3.35997
$885$	0	0
$886$	360.472i	0.406853i
$887$	1682.19i	1.89650i	0.317529	+	0.948249i	$0.397147\pi$
$887$	1682.19i	1.89650i	−0.317529	+	0.948249i	$0.602853\pi$
$888$	0	0
$889$	1109.81	1.24838
$890$	− 1173.74i	− 1.31881i
$891$	0	0
$892$	1508.70	1.69137
$893$	− 1717.49i	− 1.92328i
$894$	0	0
$895$	−2029.76	−2.26788
$896$	−1108.02	−1.23662
$897$	0	0
$898$	2057.73i	2.29146i
$899$	142.103i	0.158068i
$900$	0	0
$901$	− 996.711i	− 1.10623i
$902$	0	0
$903$	0	0
$904$	− 294.788i	− 0.326093i
$905$	654.857	0.723599
$906$	0	0
$907$	−1243.80	−1.37134	−0.685668	−	0.727915i	$-0.740490\pi$
$907$	−1243.80	−1.37134	−0.685668	+	0.727915i	$0.740490\pi$
$908$	232.808i	0.256396i
$909$	0	0
$910$	2224.17i	2.44414i
$911$	−1164.80	−1.27860	−0.639300	−	0.768958i	$-0.720776\pi$
$911$	−1164.80	−1.27860	−0.639300	+	0.768958i	$0.720776\pi$
$912$	0	0
$913$	0	0
$914$	951.863	1.04143
$915$	0	0
$916$	−407.056	−0.444385
$917$	−1183.70	−1.29083
$918$	0	0
$919$	− 853.559i	− 0.928791i	−0.885628	−	0.464396i	$-0.846272\pi$
$919$	− 853.559i	− 0.928791i	0.885628	−	0.464396i	$-0.153728\pi$
$920$	− 115.486i	− 0.125528i
$921$	0	0
$922$	−136.436	−0.147978
$923$	− 1647.50i	− 1.78494i
$924$	0	0
$925$	324.449	0.350755
$926$	− 1878.11i	− 2.02820i
$927$	0	0
$928$	169.500	0.182651
$929$	−1411.51	−1.51939	−0.759694	−	0.650281i	$-0.774651\pi$
$929$	−1411.51	−1.51939	−0.759694	+	0.650281i	$0.774651\pi$
$930$	0	0
$931$	− 32.9802i	− 0.0354245i
$932$	− 1674.47i	− 1.79665i
$933$	0	0
$934$	1439.01i	1.54070i
$935$	0	0
$936$	0	0
$937$	− 1181.80i	− 1.26126i	−0.776085	−	0.630628i	$-0.782797\pi$
$937$	− 1181.80i	− 1.26126i	0.776085	−	0.630628i	$-0.217203\pi$
$938$	−2006.95	−2.13961
$939$	0	0
$940$	−1798.98	−1.91381
$941$	1345.84i	1.43022i	0.699012	+	0.715110i	$0.253623\pi$
$941$	1345.84i	1.43022i	−0.699012	+	0.715110i	$0.746377\pi$
$942$	0	0
$943$	287.439i	0.304813i
$944$	35.8886	0.0380176
$945$	0	0
$946$	0	0
$947$	−1244.85	−1.31452	−0.657260	−	0.753664i	$-0.728285\pi$
$947$	−1244.85	−1.31452	−0.657260	+	0.753664i	$0.728285\pi$
$948$	0	0
$949$	−787.917	−0.830260
$950$	−1164.72	−1.22602
$951$	0	0
$952$	1183.37i	1.24304i
$953$	425.793i	0.446792i	0.974728	+	0.223396i	$0.0717143\pi$
$953$	425.793i	0.446792i	−0.974728	+	0.223396i	$0.928286\pi$
$954$	0	0
$955$	−726.410	−0.760639
$956$	− 1967.00i	− 2.05753i
$957$	0	0
$958$	−1630.93	−1.70243
$959$	939.166i	0.979318i
$960$	0	0
$961$	195.000	0.202914
$962$	−1495.85	−1.55493
$963$	0	0
$964$	− 1749.96i	− 1.81531i
$965$	1301.58i	1.34879i
$966$	0	0
$967$	− 1096.76i	− 1.13419i	−0.823653	−	0.567093i	$-0.808068\pi$
$967$	− 1096.76i	− 1.13419i	0.823653	−	0.567093i	$-0.191932\pi$
$968$	0	0
$969$	0	0
$970$	2079.79i	2.14411i
$971$	947.691	0.975995	0.487998	−	0.872845i	$-0.337727\pi$
$971$	947.691	0.975995	0.487998	+	0.872845i	$0.337727\pi$
$972$	0	0
$973$	−368.564	−0.378791
$974$	694.896i	0.713445i
$975$	0	0
$976$	− 212.656i	− 0.217885i
$977$	228.201	0.233573	0.116787	−	0.993157i	$-0.462741\pi$
$977$	228.201	0.233573	0.116787	+	0.993157i	$0.462741\pi$
$978$	0	0
$979$	0	0
$980$	−34.5451	−0.0352501
$981$	0	0
$982$	1232.10	1.25469
$983$	−1677.50	−1.70651	−0.853253	−	0.521497i	$-0.825374\pi$
$983$	−1677.50	−1.70651	−0.853253	+	0.521497i	$0.825374\pi$
$984$	0	0
$985$	1466.37i	1.48870i
$986$	− 403.831i	− 0.409565i
$987$	0	0
$988$	3162.77	3.20118
$989$	130.304i	0.131753i
$990$	0	0
$991$	−122.603	−0.123716	−0.0618580	−	0.998085i	$-0.519703\pi$
$991$	−122.603	−0.123716	−0.0618580	+	0.998085i	$0.519703\pi$
$992$	− 1378.87i	− 1.38999i
$993$	0	0
$994$	2172.24	2.18536
$995$	−12.2003	−0.0122616
$996$	0	0
$997$	1052.25i	1.05542i	0.849425	+	0.527709i	$0.176949\pi$
$997$	1052.25i	1.05542i	−0.849425	+	0.527709i	$0.823051\pi$
$998$	− 1568.99i	− 1.57213i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.i.604.7	yes	8
3.2	odd	2	inner	1089.3.c.i.604.2	yes	8
11.10	odd	2	inner	1089.3.c.i.604.1	✓	8
33.32	even	2	inner	1089.3.c.i.604.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.c.i.604.1	✓	8	11.10	odd	2	inner
1089.3.c.i.604.2	yes	8	3.2	odd	2	inner
1089.3.c.i.604.7	yes	8	1.1	even	1	trivial
1089.3.c.i.604.8	yes	8	33.32	even	2	inner

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.11962i q^{2} -5.73205 q^{4} -6.02665 q^{5} +7.07107i q^{7} -5.40335i q^{8} +O(q^{10})\)	\(q+3.11962i q^{2} -5.73205 q^{4} -6.02665 q^{5} +7.07107i q^{7} -5.40335i q^{8} -18.8009i q^{10} +16.7303i q^{13} -22.0591 q^{14} -6.07180 q^{16} +30.9723i q^{17} +32.9802i q^{19} +34.5451 q^{20} -3.54643 q^{23} +11.3205 q^{25} -52.1923 q^{26} -40.5317i q^{28} +4.17950i q^{29} +34.0000 q^{31} -40.5551i q^{32} -96.6218 q^{34} -42.6148i q^{35} +28.6603 q^{37} -102.886 q^{38} +32.5641i q^{40} -81.0502i q^{41} -36.7423i q^{43} -11.0635i q^{46} -52.0764 q^{47} -1.00000 q^{49} +35.3157i q^{50} -95.8991i q^{52} -32.1808 q^{53} +38.2074 q^{56} -13.0385 q^{58} -5.91071 q^{59} +35.0235i q^{61} +106.067i q^{62} +102.229 q^{64} -100.828i q^{65} +90.9808 q^{67} -177.535i q^{68} +132.942 q^{70} -98.4739 q^{71} +47.0951i q^{73} +89.4092i q^{74} -189.044i q^{76} +27.1203i q^{79} +36.5926 q^{80} +252.846 q^{82} +11.2547i q^{83} -186.659i q^{85} +114.622 q^{86} +62.4300 q^{89} -118.301 q^{91} +20.3283 q^{92} -162.459i q^{94} -198.760i q^{95} -110.622 q^{97} -3.11962i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4}+O(q^{10}) \) 8 * q - 32 * q^4	\( 8 q - 32 q^{4} - 104 q^{16} - 48 q^{25} + 272 q^{31} - 288 q^{34} + 160 q^{37} - 8 q^{49} - 520 q^{58} + 416 q^{64} + 520 q^{67} + 440 q^{70} + 360 q^{82} - 600 q^{91} - 400 q^{97}+O(q^{100}) \) 8 * q - 32 * q^4 - 104 * q^16 - 48 * q^25 + 272 * q^31 - 288 * q^34 + 160 * q^37 - 8 * q^49 - 520 * q^58 + 416 * q^64 + 520 * q^67 + 440 * q^70 + 360 * q^82 - 600 * q^91 - 400 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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