LMFDB - Embedded newform 261.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,2,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("261.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	261.a (trivial)

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:	yes
Analytic conductor:	$2.08409549276$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
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:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -2.82843 q^{7} +4.41421 q^{8} +O(q^{10})$	\(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -2.82843 q^{7} +4.41421 q^{8} +2.41421 q^{10} +0.414214 q^{11} -3.82843 q^{13} -6.82843 q^{14} +3.00000 q^{16} -0.828427 q^{17} +6.00000 q^{19} +3.82843 q^{20} +1.00000 q^{22} -3.65685 q^{23} -4.00000 q^{25} -9.24264 q^{26} -10.8284 q^{28} -1.00000 q^{29} +10.0711 q^{31} -1.58579 q^{32} -2.00000 q^{34} -2.82843 q^{35} -4.00000 q^{37} +14.4853 q^{38} +4.41421 q^{40} +4.48528 q^{41} +3.58579 q^{43} +1.58579 q^{44} -8.82843 q^{46} +3.24264 q^{47} +1.00000 q^{49} -9.65685 q^{50} -14.6569 q^{52} -9.48528 q^{53} +0.414214 q^{55} -12.4853 q^{56} -2.41421 q^{58} +3.65685 q^{59} -4.82843 q^{61} +24.3137 q^{62} -9.82843 q^{64} -3.82843 q^{65} +5.65685 q^{67} -3.17157 q^{68} -6.82843 q^{70} +8.82843 q^{71} +4.00000 q^{73} -9.65685 q^{74} +22.9706 q^{76} -1.17157 q^{77} -2.41421 q^{79} +3.00000 q^{80} +10.8284 q^{82} -7.65685 q^{83} -0.828427 q^{85} +8.65685 q^{86} +1.82843 q^{88} +12.4853 q^{89} +10.8284 q^{91} -14.0000 q^{92} +7.82843 q^{94} +6.00000 q^{95} +4.48528 q^{97} +2.41421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 8 q^{14} + 6 q^{16} + 4 q^{17} + 12 q^{19} + 2 q^{20} + 2 q^{22} + 4 q^{23} - 8 q^{25} - 10 q^{26} - 16 q^{28} - 2 q^{29} + 6 q^{31} - 6 q^{32} - 4 q^{34} - 8 q^{37} + 12 q^{38} + 6 q^{40} - 8 q^{41} + 10 q^{43} + 6 q^{44} - 12 q^{46} - 2 q^{47} + 2 q^{49} - 8 q^{50} - 18 q^{52} - 2 q^{53} - 2 q^{55} - 8 q^{56} - 2 q^{58} - 4 q^{59} - 4 q^{61} + 26 q^{62} - 14 q^{64} - 2 q^{65} - 12 q^{68} - 8 q^{70} + 12 q^{71} + 8 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{77} - 2 q^{79} + 6 q^{80} + 16 q^{82} - 4 q^{83} + 4 q^{85} + 6 q^{86} - 2 q^{88} + 8 q^{89} + 16 q^{91} - 28 q^{92} + 10 q^{94} + 12 q^{95} - 8 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 2 * q^10 - 2 * q^11 - 2 * q^13 - 8 * q^14 + 6 * q^16 + 4 * q^17 + 12 * q^19 + 2 * q^20 + 2 * q^22 + 4 * q^23 - 8 * q^25 - 10 * q^26 - 16 * q^28 - 2 * q^29 + 6 * q^31 - 6 * q^32 - 4 * q^34 - 8 * q^37 + 12 * q^38 + 6 * q^40 - 8 * q^41 + 10 * q^43 + 6 * q^44 - 12 * q^46 - 2 * q^47 + 2 * q^49 - 8 * q^50 - 18 * q^52 - 2 * q^53 - 2 * q^55 - 8 * q^56 - 2 * q^58 - 4 * q^59 - 4 * q^61 + 26 * q^62 - 14 * q^64 - 2 * q^65 - 12 * q^68 - 8 * q^70 + 12 * q^71 + 8 * q^73 - 8 * q^74 + 12 * q^76 - 8 * q^77 - 2 * q^79 + 6 * q^80 + 16 * q^82 - 4 * q^83 + 4 * q^85 + 6 * q^86 - 2 * q^88 + 8 * q^89 + 16 * q^91 - 28 * q^92 + 10 * q^94 + 12 * q^95 - 8 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	2.41421	0.763441
$11$	0.414214	0.124890	0.0624450	−	0.998048i	$-0.480110\pi$
$11$	0.414214	0.124890	0.0624450	+	0.998048i	$0.480110\pi$
$12$	0	0
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	−6.82843	−1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	3.82843	0.856062
$21$	0	0
$22$	1.00000	0.213201
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−9.24264	−1.81263
$27$	0	0
$28$	−10.8284	−2.04638
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	10.0711	1.80882	0.904409	−	0.426667i	$-0.140313\pi$
$31$	10.0711	1.80882	0.904409	+	0.426667i	$0.140313\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	14.4853	2.34982
$39$	0	0
$40$	4.41421	0.697948
$41$	4.48528	0.700483	0.350242	−	0.936659i	$-0.386099\pi$
$41$	4.48528	0.700483	0.350242	+	0.936659i	$0.386099\pi$
$42$	0	0
$43$	3.58579	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$43$	3.58579	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$44$	1.58579	0.239066
$45$	0	0
$46$	−8.82843	−1.30168
$47$	3.24264	0.472988	0.236494	−	0.971633i	$-0.424002\pi$
$47$	3.24264	0.472988	0.236494	+	0.971633i	$0.424002\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.65685	−1.36569
$51$	0	0
$52$	−14.6569	−2.03254
$53$	−9.48528	−1.30290	−0.651452	−	0.758690i	$-0.725840\pi$
$53$	−9.48528	−1.30290	−0.651452	+	0.758690i	$0.725840\pi$
$54$	0	0
$55$	0.414214	0.0558525
$56$	−12.4853	−1.66842
$57$	0	0
$58$	−2.41421	−0.317002
$59$	3.65685	0.476082	0.238041	−	0.971255i	$-0.423495\pi$
$59$	3.65685	0.476082	0.238041	+	0.971255i	$0.423495\pi$
$60$	0	0
$61$	−4.82843	−0.618217	−0.309108	−	0.951027i	$-0.600031\pi$
$61$	−4.82843	−0.618217	−0.309108	+	0.951027i	$0.600031\pi$
$62$	24.3137	3.08784
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−3.82843	−0.474858
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	−3.17157	−0.384610
$69$	0	0
$70$	−6.82843	−0.816153
$71$	8.82843	1.04774	0.523871	−	0.851798i	$-0.324487\pi$
$71$	8.82843	1.04774	0.523871	+	0.851798i	$0.324487\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−9.65685	−1.12259
$75$	0	0
$76$	22.9706	2.63490
$77$	−1.17157	−0.133513
$78$	0	0
$79$	−2.41421	−0.271620	−0.135810	−	0.990735i	$-0.543364\pi$
$79$	−2.41421	−0.271620	−0.135810	+	0.990735i	$0.543364\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	10.8284	1.19580
$83$	−7.65685	−0.840449	−0.420224	−	0.907420i	$-0.638049\pi$
$83$	−7.65685	−0.840449	−0.420224	+	0.907420i	$0.638049\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	8.65685	0.933493
$87$	0	0
$88$	1.82843	0.194911
$89$	12.4853	1.32344	0.661719	−	0.749752i	$-0.269827\pi$
$89$	12.4853	1.32344	0.661719	+	0.749752i	$0.269827\pi$
$90$	0	0
$91$	10.8284	1.13513
$92$	−14.0000	−1.45960
$93$	0	0
$94$	7.82843	0.807441
$95$	6.00000	0.615587
$96$	0	0
$97$	4.48528	0.455411	0.227706	−	0.973730i	$-0.426878\pi$
$97$	4.48528	0.455411	0.227706	+	0.973730i	$0.426878\pi$
$98$	2.41421	0.243872
$99$	0	0
$100$	−15.3137	−1.53137
$101$	2.34315	0.233152	0.116576	−	0.993182i	$-0.462808\pi$
$101$	2.34315	0.233152	0.116576	+	0.993182i	$0.462808\pi$
$102$	0	0
$103$	−4.82843	−0.475759	−0.237880	−	0.971295i	$-0.576452\pi$
$103$	−4.82843	−0.475759	−0.237880	+	0.971295i	$0.576452\pi$
$104$	−16.8995	−1.65713
$105$	0	0
$106$	−22.8995	−2.22420
$107$	14.8284	1.43352	0.716759	−	0.697321i	$-0.245625\pi$
$107$	14.8284	1.43352	0.716759	+	0.697321i	$0.245625\pi$
$108$	0	0
$109$	12.6569	1.21231	0.606153	−	0.795348i	$-0.292712\pi$
$109$	12.6569	1.21231	0.606153	+	0.795348i	$0.292712\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−8.48528	−0.801784
$113$	13.3137	1.25245	0.626224	−	0.779643i	$-0.284599\pi$
$113$	13.3137	1.25245	0.626224	+	0.779643i	$0.284599\pi$
$114$	0	0
$115$	−3.65685	−0.341003
$116$	−3.82843	−0.355461
$117$	0	0
$118$	8.82843	0.812723
$119$	2.34315	0.214796
$120$	0	0
$121$	−10.8284	−0.984402
$122$	−11.6569	−1.05536
$123$	0	0
$124$	38.5563	3.46246
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	−9.24264	−0.810633
$131$	−21.3137	−1.86219	−0.931094	−	0.364780i	$-0.881144\pi$
$131$	−21.3137	−1.86219	−0.931094	+	0.364780i	$0.881144\pi$
$132$	0	0
$133$	−16.9706	−1.47153
$134$	13.6569	1.17977
$135$	0	0
$136$	−3.65685	−0.313573
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−10.8284	−0.915169
$141$	0	0
$142$	21.3137	1.78861
$143$	−1.58579	−0.132610
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	9.65685	0.799207
$147$	0	0
$148$	−15.3137	−1.25878
$149$	7.82843	0.641330	0.320665	−	0.947193i	$-0.396094\pi$
$149$	7.82843	0.641330	0.320665	+	0.947193i	$0.396094\pi$
$150$	0	0
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	26.4853	2.14824
$153$	0	0
$154$	−2.82843	−0.227921
$155$	10.0711	0.808928
$156$	0	0
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	−5.82843	−0.463685
$159$	0	0
$160$	−1.58579	−0.125367
$161$	10.3431	0.815154
$162$	0	0
$163$	3.92893	0.307738	0.153869	−	0.988091i	$-0.450827\pi$
$163$	3.92893	0.307738	0.153869	+	0.988091i	$0.450827\pi$
$164$	17.1716	1.34087
$165$	0	0
$166$	−18.4853	−1.43474
$167$	3.17157	0.245424	0.122712	−	0.992442i	$-0.460841\pi$
$167$	3.17157	0.245424	0.122712	+	0.992442i	$0.460841\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	−2.00000	−0.153393
$171$	0	0
$172$	13.7279	1.04674
$173$	−12.3431	−0.938432	−0.469216	−	0.883083i	$-0.655463\pi$
$173$	−12.3431	−0.938432	−0.469216	+	0.883083i	$0.655463\pi$
$174$	0	0
$175$	11.3137	0.855236
$176$	1.24264	0.0936676
$177$	0	0
$178$	30.1421	2.25925
$179$	6.48528	0.484733	0.242366	−	0.970185i	$-0.422076\pi$
$179$	6.48528	0.484733	0.242366	+	0.970185i	$0.422076\pi$
$180$	0	0
$181$	8.31371	0.617953	0.308977	−	0.951070i	$-0.400014\pi$
$181$	8.31371	0.617953	0.308977	+	0.951070i	$0.400014\pi$
$182$	26.1421	1.93778
$183$	0	0
$184$	−16.1421	−1.19001
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−0.343146	−0.0250933
$188$	12.4142	0.905400
$189$	0	0
$190$	14.4853	1.05087
$191$	−25.3137	−1.83164	−0.915818	−	0.401594i	$-0.868456\pi$
$191$	−25.3137	−1.83164	−0.915818	+	0.401594i	$0.868456\pi$
$192$	0	0
$193$	−5.17157	−0.372258	−0.186129	−	0.982525i	$-0.559594\pi$
$193$	−5.17157	−0.372258	−0.186129	+	0.982525i	$0.559594\pi$
$194$	10.8284	0.777436
$195$	0	0
$196$	3.82843	0.273459
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−0.485281	−0.0344007	−0.0172003	−	0.999852i	$-0.505475\pi$
$199$	−0.485281	−0.0344007	−0.0172003	+	0.999852i	$0.505475\pi$
$200$	−17.6569	−1.24853
$201$	0	0
$202$	5.65685	0.398015
$203$	2.82843	0.198517
$204$	0	0
$205$	4.48528	0.313266
$206$	−11.6569	−0.812172
$207$	0	0
$208$	−11.4853	−0.796361
$209$	2.48528	0.171911
$210$	0	0
$211$	−19.3848	−1.33450	−0.667252	−	0.744832i	$-0.732529\pi$
$211$	−19.3848	−1.33450	−0.667252	+	0.744832i	$0.732529\pi$
$212$	−36.3137	−2.49404
$213$	0	0
$214$	35.7990	2.44717
$215$	3.58579	0.244549
$216$	0	0
$217$	−28.4853	−1.93371
$218$	30.5563	2.06954
$219$	0	0
$220$	1.58579	0.106914
$221$	3.17157	0.213343
$222$	0	0
$223$	−3.17157	−0.212384	−0.106192	−	0.994346i	$-0.533866\pi$
$223$	−3.17157	−0.212384	−0.106192	+	0.994346i	$0.533866\pi$
$224$	4.48528	0.299685
$225$	0	0
$226$	32.1421	2.13806
$227$	8.14214	0.540413	0.270206	−	0.962802i	$-0.412908\pi$
$227$	8.14214	0.540413	0.270206	+	0.962802i	$0.412908\pi$
$228$	0	0
$229$	−3.51472	−0.232259	−0.116130	−	0.993234i	$-0.537049\pi$
$229$	−3.51472	−0.232259	−0.116130	+	0.993234i	$0.537049\pi$
$230$	−8.82843	−0.582129
$231$	0	0
$232$	−4.41421	−0.289807
$233$	−18.3137	−1.19977	−0.599885	−	0.800086i	$-0.704787\pi$
$233$	−18.3137	−1.19977	−0.599885	+	0.800086i	$0.704787\pi$
$234$	0	0
$235$	3.24264	0.211527
$236$	14.0000	0.911322
$237$	0	0
$238$	5.65685	0.366679
$239$	19.6569	1.27150	0.635748	−	0.771897i	$-0.280692\pi$
$239$	19.6569	1.27150	0.635748	+	0.771897i	$0.280692\pi$
$240$	0	0
$241$	−18.3137	−1.17969	−0.589845	−	0.807517i	$-0.700811\pi$
$241$	−18.3137	−1.17969	−0.589845	+	0.807517i	$0.700811\pi$
$242$	−26.1421	−1.68048
$243$	0	0
$244$	−18.4853	−1.18340
$245$	1.00000	0.0638877
$246$	0	0
$247$	−22.9706	−1.46158
$248$	44.4558	2.82295
$249$	0	0
$250$	−21.7279	−1.37419
$251$	−20.0711	−1.26687	−0.633437	−	0.773794i	$-0.718357\pi$
$251$	−20.0711	−1.26687	−0.633437	+	0.773794i	$0.718357\pi$
$252$	0	0
$253$	−1.51472	−0.0952295
$254$	−10.4853	−0.657905
$255$	0	0
$256$	−29.9706	−1.87316
$257$	18.1716	1.13351	0.566756	−	0.823886i	$-0.308198\pi$
$257$	18.1716	1.13351	0.566756	+	0.823886i	$0.308198\pi$
$258$	0	0
$259$	11.3137	0.703000
$260$	−14.6569	−0.908980
$261$	0	0
$262$	−51.4558	−3.17895
$263$	−2.75736	−0.170026	−0.0850130	−	0.996380i	$-0.527093\pi$
$263$	−2.75736	−0.170026	−0.0850130	+	0.996380i	$0.527093\pi$
$264$	0	0
$265$	−9.48528	−0.582676
$266$	−40.9706	−2.51207
$267$	0	0
$268$	21.6569	1.32290
$269$	−31.4558	−1.91790	−0.958948	−	0.283581i	$-0.908478\pi$
$269$	−31.4558	−1.91790	−0.958948	+	0.283581i	$0.908478\pi$
$270$	0	0
$271$	16.5563	1.00573	0.502863	−	0.864366i	$-0.332280\pi$
$271$	16.5563	1.00573	0.502863	+	0.864366i	$0.332280\pi$
$272$	−2.48528	−0.150692
$273$	0	0
$274$	−28.9706	−1.75018
$275$	−1.65685	−0.0999121
$276$	0	0
$277$	−17.3137	−1.04028	−0.520140	−	0.854081i	$-0.674120\pi$
$277$	−17.3137	−1.04028	−0.520140	+	0.854081i	$0.674120\pi$
$278$	33.7990	2.02713
$279$	0	0
$280$	−12.4853	−0.746138
$281$	−31.9706	−1.90720	−0.953602	−	0.301070i	$-0.902656\pi$
$281$	−31.9706	−1.90720	−0.953602	+	0.301070i	$0.902656\pi$
$282$	0	0
$283$	11.6569	0.692928	0.346464	−	0.938063i	$-0.387382\pi$
$283$	11.6569	0.692928	0.346464	+	0.938063i	$0.387382\pi$
$284$	33.7990	2.00560
$285$	0	0
$286$	−3.82843	−0.226380
$287$	−12.6863	−0.748848
$288$	0	0
$289$	−16.3137	−0.959630
$290$	−2.41421	−0.141768
$291$	0	0
$292$	15.3137	0.896167
$293$	−7.65685	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$293$	−7.65685	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$294$	0	0
$295$	3.65685	0.212910
$296$	−17.6569	−1.02628
$297$	0	0
$298$	18.8995	1.09482
$299$	14.0000	0.809641
$300$	0	0
$301$	−10.1421	−0.584583
$302$	−34.1421	−1.96466
$303$	0	0
$304$	18.0000	1.03237
$305$	−4.82843	−0.276475
$306$	0	0
$307$	2.89949	0.165483	0.0827415	−	0.996571i	$-0.473632\pi$
$307$	2.89949	0.165483	0.0827415	+	0.996571i	$0.473632\pi$
$308$	−4.48528	−0.255573
$309$	0	0
$310$	24.3137	1.38093
$311$	−2.68629	−0.152326	−0.0761628	−	0.997095i	$-0.524267\pi$
$311$	−2.68629	−0.152326	−0.0761628	+	0.997095i	$0.524267\pi$
$312$	0	0
$313$	9.82843	0.555536	0.277768	−	0.960648i	$-0.410405\pi$
$313$	9.82843	0.555536	0.277768	+	0.960648i	$0.410405\pi$
$314$	20.4853	1.15605
$315$	0	0
$316$	−9.24264	−0.519939
$317$	31.4558	1.76674	0.883368	−	0.468680i	$-0.155270\pi$
$317$	31.4558	1.76674	0.883368	+	0.468680i	$0.155270\pi$
$318$	0	0
$319$	−0.414214	−0.0231915
$320$	−9.82843	−0.549426
$321$	0	0
$322$	24.9706	1.39156
$323$	−4.97056	−0.276570
$324$	0	0
$325$	15.3137	0.849452
$326$	9.48528	0.525341
$327$	0	0
$328$	19.7990	1.09322
$329$	−9.17157	−0.505645
$330$	0	0
$331$	−2.41421	−0.132697	−0.0663486	−	0.997797i	$-0.521135\pi$
$331$	−2.41421	−0.132697	−0.0663486	+	0.997797i	$0.521135\pi$
$332$	−29.3137	−1.60880
$333$	0	0
$334$	7.65685	0.418964
$335$	5.65685	0.309067
$336$	0	0
$337$	21.7990	1.18747	0.593733	−	0.804662i	$-0.297653\pi$
$337$	21.7990	1.18747	0.593733	+	0.804662i	$0.297653\pi$
$338$	4.00000	0.217571
$339$	0	0
$340$	−3.17157	−0.172003
$341$	4.17157	0.225903
$342$	0	0
$343$	16.9706	0.916324
$344$	15.8284	0.853412
$345$	0	0
$346$	−29.7990	−1.60200
$347$	−2.48528	−0.133417	−0.0667084	−	0.997773i	$-0.521250\pi$
$347$	−2.48528	−0.133417	−0.0667084	+	0.997773i	$0.521250\pi$
$348$	0	0
$349$	−5.14214	−0.275252	−0.137626	−	0.990484i	$-0.543947\pi$
$349$	−5.14214	−0.275252	−0.137626	+	0.990484i	$0.543947\pi$
$350$	27.3137	1.45998
$351$	0	0
$352$	−0.656854	−0.0350104
$353$	−26.9706	−1.43550	−0.717749	−	0.696302i	$-0.754828\pi$
$353$	−26.9706	−1.43550	−0.717749	+	0.696302i	$0.754828\pi$
$354$	0	0
$355$	8.82843	0.468564
$356$	47.7990	2.53334
$357$	0	0
$358$	15.6569	0.827490
$359$	−3.92893	−0.207361	−0.103681	−	0.994611i	$-0.533062\pi$
$359$	−3.92893	−0.207361	−0.103681	+	0.994611i	$0.533062\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	20.0711	1.05491
$363$	0	0
$364$	41.4558	2.17288
$365$	4.00000	0.209370
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−10.9706	−0.571880
$369$	0	0
$370$	−9.65685	−0.502036
$371$	26.8284	1.39286
$372$	0	0
$373$	−26.3137	−1.36247	−0.681236	−	0.732064i	$-0.738557\pi$
$373$	−26.3137	−1.36247	−0.681236	+	0.732064i	$0.738557\pi$
$374$	−0.828427	−0.0428369
$375$	0	0
$376$	14.3137	0.738173
$377$	3.82843	0.197174
$378$	0	0
$379$	−6.97056	−0.358054	−0.179027	−	0.983844i	$-0.557295\pi$
$379$	−6.97056	−0.358054	−0.179027	+	0.983844i	$0.557295\pi$
$380$	22.9706	1.17837
$381$	0	0
$382$	−61.1127	−3.12680
$383$	3.51472	0.179594	0.0897969	−	0.995960i	$-0.471378\pi$
$383$	3.51472	0.179594	0.0897969	+	0.995960i	$0.471378\pi$
$384$	0	0
$385$	−1.17157	−0.0597089
$386$	−12.4853	−0.635484
$387$	0	0
$388$	17.1716	0.871755
$389$	−3.02944	−0.153599	−0.0767993	−	0.997047i	$-0.524470\pi$
$389$	−3.02944	−0.153599	−0.0767993	+	0.997047i	$0.524470\pi$
$390$	0	0
$391$	3.02944	0.153205
$392$	4.41421	0.222951
$393$	0	0
$394$	−4.82843	−0.243253
$395$	−2.41421	−0.121472
$396$	0	0
$397$	19.3431	0.970805	0.485402	−	0.874291i	$-0.338673\pi$
$397$	19.3431	0.970805	0.485402	+	0.874291i	$0.338673\pi$
$398$	−1.17157	−0.0587256
$399$	0	0
$400$	−12.0000	−0.600000
$401$	18.6569	0.931679	0.465839	−	0.884869i	$-0.345752\pi$
$401$	18.6569	0.931679	0.465839	+	0.884869i	$0.345752\pi$
$402$	0	0
$403$	−38.5563	−1.92063
$404$	8.97056	0.446302
$405$	0	0
$406$	6.82843	0.338889
$407$	−1.65685	−0.0821272
$408$	0	0
$409$	−18.9706	−0.938034	−0.469017	−	0.883189i	$-0.655392\pi$
$409$	−18.9706	−0.938034	−0.469017	+	0.883189i	$0.655392\pi$
$410$	10.8284	0.534778
$411$	0	0
$412$	−18.4853	−0.910704
$413$	−10.3431	−0.508953
$414$	0	0
$415$	−7.65685	−0.375860
$416$	6.07107	0.297659
$417$	0	0
$418$	6.00000	0.293470
$419$	9.51472	0.464824	0.232412	−	0.972617i	$-0.425338\pi$
$419$	9.51472	0.464824	0.232412	+	0.972617i	$0.425338\pi$
$420$	0	0
$421$	37.1127	1.80876	0.904381	−	0.426726i	$-0.140333\pi$
$421$	37.1127	1.80876	0.904381	+	0.426726i	$0.140333\pi$
$422$	−46.7990	−2.27814
$423$	0	0
$424$	−41.8701	−2.03339
$425$	3.31371	0.160738
$426$	0	0
$427$	13.6569	0.660901
$428$	56.7696	2.74406
$429$	0	0
$430$	8.65685	0.417471
$431$	−19.6569	−0.946837	−0.473419	−	0.880838i	$-0.656980\pi$
$431$	−19.6569	−0.946837	−0.473419	+	0.880838i	$0.656980\pi$
$432$	0	0
$433$	30.6274	1.47186	0.735930	−	0.677058i	$-0.236745\pi$
$433$	30.6274	1.47186	0.735930	+	0.677058i	$0.236745\pi$
$434$	−68.7696	−3.30104
$435$	0	0
$436$	48.4558	2.32061
$437$	−21.9411	−1.04959
$438$	0	0
$439$	−0.343146	−0.0163775	−0.00818873	−	0.999966i	$-0.502607\pi$
$439$	−0.343146	−0.0163775	−0.00818873	+	0.999966i	$0.502607\pi$
$440$	1.82843	0.0871668
$441$	0	0
$442$	7.65685	0.364199
$443$	24.3431	1.15658	0.578289	−	0.815832i	$-0.303721\pi$
$443$	24.3431	1.15658	0.578289	+	0.815832i	$0.303721\pi$
$444$	0	0
$445$	12.4853	0.591859
$446$	−7.65685	−0.362563
$447$	0	0
$448$	27.7990	1.31338
$449$	34.9706	1.65036	0.825181	−	0.564868i	$-0.191073\pi$
$449$	34.9706	1.65036	0.825181	+	0.564868i	$0.191073\pi$
$450$	0	0
$451$	1.85786	0.0874834
$452$	50.9706	2.39745
$453$	0	0
$454$	19.6569	0.922542
$455$	10.8284	0.507644
$456$	0	0
$457$	1.02944	0.0481550	0.0240775	−	0.999710i	$-0.492335\pi$
$457$	1.02944	0.0481550	0.0240775	+	0.999710i	$0.492335\pi$
$458$	−8.48528	−0.396491
$459$	0	0
$460$	−14.0000	−0.652753
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−44.2132	−2.04814
$467$	38.3553	1.77487	0.887437	−	0.460930i	$-0.152484\pi$
$467$	38.3553	1.77487	0.887437	+	0.460930i	$0.152484\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	7.82843	0.361098
$471$	0	0
$472$	16.1421	0.743002
$473$	1.48528	0.0682933
$474$	0	0
$475$	−24.0000	−1.10120
$476$	8.97056	0.411165
$477$	0	0
$478$	47.4558	2.17058
$479$	−6.89949	−0.315246	−0.157623	−	0.987499i	$-0.550383\pi$
$479$	−6.89949	−0.315246	−0.157623	+	0.987499i	$0.550383\pi$
$480$	0	0
$481$	15.3137	0.698245
$482$	−44.2132	−2.01386
$483$	0	0
$484$	−41.4558	−1.88436
$485$	4.48528	0.203666
$486$	0	0
$487$	−11.5147	−0.521782	−0.260891	−	0.965368i	$-0.584016\pi$
$487$	−11.5147	−0.521782	−0.260891	+	0.965368i	$0.584016\pi$
$488$	−21.3137	−0.964826
$489$	0	0
$490$	2.41421	0.109063
$491$	21.2426	0.958667	0.479333	−	0.877633i	$-0.340878\pi$
$491$	21.2426	0.958667	0.479333	+	0.877633i	$0.340878\pi$
$492$	0	0
$493$	0.828427	0.0373105
$494$	−55.4558	−2.49508
$495$	0	0
$496$	30.2132	1.35661
$497$	−24.9706	−1.12008
$498$	0	0
$499$	18.9706	0.849239	0.424620	−	0.905372i	$-0.360408\pi$
$499$	18.9706	0.849239	0.424620	+	0.905372i	$0.360408\pi$
$500$	−34.4558	−1.54091
$501$	0	0
$502$	−48.4558	−2.16269
$503$	−0.272078	−0.0121314	−0.00606568	−	0.999982i	$-0.501931\pi$
$503$	−0.272078	−0.0121314	−0.00606568	+	0.999982i	$0.501931\pi$
$504$	0	0
$505$	2.34315	0.104269
$506$	−3.65685	−0.162567
$507$	0	0
$508$	−16.6274	−0.737722
$509$	10.5147	0.466057	0.233028	−	0.972470i	$-0.425137\pi$
$509$	10.5147	0.466057	0.233028	+	0.972470i	$0.425137\pi$
$510$	0	0
$511$	−11.3137	−0.500489
$512$	−31.2426	−1.38074
$513$	0	0
$514$	43.8701	1.93503
$515$	−4.82843	−0.212766
$516$	0	0
$517$	1.34315	0.0590715
$518$	27.3137	1.20010
$519$	0	0
$520$	−16.8995	−0.741092
$521$	29.1421	1.27674	0.638370	−	0.769730i	$-0.279609\pi$
$521$	29.1421	1.27674	0.638370	+	0.769730i	$0.279609\pi$
$522$	0	0
$523$	4.68629	0.204917	0.102459	−	0.994737i	$-0.467329\pi$
$523$	4.68629	0.204917	0.102459	+	0.994737i	$0.467329\pi$
$524$	−81.5980	−3.56462
$525$	0	0
$526$	−6.65685	−0.290253
$527$	−8.34315	−0.363433
$528$	0	0
$529$	−9.62742	−0.418583
$530$	−22.8995	−0.994690
$531$	0	0
$532$	−64.9706	−2.81683
$533$	−17.1716	−0.743783
$534$	0	0
$535$	14.8284	0.641089
$536$	24.9706	1.07856
$537$	0	0
$538$	−75.9411	−3.27405
$539$	0.414214	0.0178414
$540$	0	0
$541$	−10.3431	−0.444687	−0.222343	−	0.974968i	$-0.571371\pi$
$541$	−10.3431	−0.444687	−0.222343	+	0.974968i	$0.571371\pi$
$542$	39.9706	1.71688
$543$	0	0
$544$	1.31371	0.0563248
$545$	12.6569	0.542160
$546$	0	0
$547$	35.7990	1.53065	0.765327	−	0.643641i	$-0.222577\pi$
$547$	35.7990	1.53065	0.765327	+	0.643641i	$0.222577\pi$
$548$	−45.9411	−1.96251
$549$	0	0
$550$	−4.00000	−0.170561
$551$	−6.00000	−0.255609
$552$	0	0
$553$	6.82843	0.290374
$554$	−41.7990	−1.77587
$555$	0	0
$556$	53.5980	2.27306
$557$	17.3137	0.733605	0.366803	−	0.930299i	$-0.380452\pi$
$557$	17.3137	0.733605	0.366803	+	0.930299i	$0.380452\pi$
$558$	0	0
$559$	−13.7279	−0.580629
$560$	−8.48528	−0.358569
$561$	0	0
$562$	−77.1838	−3.25580
$563$	0.757359	0.0319189	0.0159594	−	0.999873i	$-0.494920\pi$
$563$	0.757359	0.0319189	0.0159594	+	0.999873i	$0.494920\pi$
$564$	0	0
$565$	13.3137	0.560112
$566$	28.1421	1.18290
$567$	0	0
$568$	38.9706	1.63517
$569$	39.6569	1.66250	0.831251	−	0.555897i	$-0.187625\pi$
$569$	39.6569	1.66250	0.831251	+	0.555897i	$0.187625\pi$
$570$	0	0
$571$	14.6274	0.612138	0.306069	−	0.952009i	$-0.400986\pi$
$571$	14.6274	0.612138	0.306069	+	0.952009i	$0.400986\pi$
$572$	−6.07107	−0.253844
$573$	0	0
$574$	−30.6274	−1.27836
$575$	14.6274	0.610005
$576$	0	0
$577$	−29.7990	−1.24055	−0.620274	−	0.784385i	$-0.712979\pi$
$577$	−29.7990	−1.24055	−0.620274	+	0.784385i	$0.712979\pi$
$578$	−39.3848	−1.63819
$579$	0	0
$580$	−3.82843	−0.158967
$581$	21.6569	0.898478
$582$	0	0
$583$	−3.92893	−0.162720
$584$	17.6569	0.730646
$585$	0	0
$586$	−18.4853	−0.763620
$587$	−7.65685	−0.316032	−0.158016	−	0.987437i	$-0.550510\pi$
$587$	−7.65685	−0.316032	−0.158016	+	0.987437i	$0.550510\pi$
$588$	0	0
$589$	60.4264	2.48983
$590$	8.82843	0.363461
$591$	0	0
$592$	−12.0000	−0.493197
$593$	19.4853	0.800165	0.400082	−	0.916479i	$-0.368982\pi$
$593$	19.4853	0.800165	0.400082	+	0.916479i	$0.368982\pi$
$594$	0	0
$595$	2.34315	0.0960596
$596$	29.9706	1.22764
$597$	0	0
$598$	33.7990	1.38214
$599$	−9.87006	−0.403280	−0.201640	−	0.979460i	$-0.564627\pi$
$599$	−9.87006	−0.403280	−0.201640	+	0.979460i	$0.564627\pi$
$600$	0	0
$601$	−17.1716	−0.700443	−0.350222	−	0.936667i	$-0.613894\pi$
$601$	−17.1716	−0.700443	−0.350222	+	0.936667i	$0.613894\pi$
$602$	−24.4853	−0.997946
$603$	0	0
$604$	−54.1421	−2.20301
$605$	−10.8284	−0.440238
$606$	0	0
$607$	−7.72792	−0.313667	−0.156833	−	0.987625i	$-0.550129\pi$
$607$	−7.72792	−0.313667	−0.156833	+	0.987625i	$0.550129\pi$
$608$	−9.51472	−0.385873
$609$	0	0
$610$	−11.6569	−0.471972
$611$	−12.4142	−0.502225
$612$	0	0
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	−5.17157	−0.208369
$617$	−0.686292	−0.0276291	−0.0138145	−	0.999905i	$-0.504397\pi$
$617$	−0.686292	−0.0276291	−0.0138145	+	0.999905i	$0.504397\pi$
$618$	0	0
$619$	33.5858	1.34993	0.674963	−	0.737851i	$-0.264159\pi$
$619$	33.5858	1.34993	0.674963	+	0.737851i	$0.264159\pi$
$620$	38.5563	1.54846
$621$	0	0
$622$	−6.48528	−0.260036
$623$	−35.3137	−1.41481
$624$	0	0
$625$	11.0000	0.440000
$626$	23.7279	0.948358
$627$	0	0
$628$	32.4853	1.29630
$629$	3.31371	0.132126
$630$	0	0
$631$	−36.8284	−1.46612	−0.733058	−	0.680166i	$-0.761908\pi$
$631$	−36.8284	−1.46612	−0.733058	+	0.680166i	$0.761908\pi$
$632$	−10.6569	−0.423907
$633$	0	0
$634$	75.9411	3.01601
$635$	−4.34315	−0.172352
$636$	0	0
$637$	−3.82843	−0.151688
$638$	−1.00000	−0.0395904
$639$	0	0
$640$	−20.5563	−0.812561
$641$	−17.7990	−0.703018	−0.351509	−	0.936185i	$-0.614331\pi$
$641$	−17.7990	−0.703018	−0.351509	+	0.936185i	$0.614331\pi$
$642$	0	0
$643$	32.4853	1.28109	0.640547	−	0.767919i	$-0.278708\pi$
$643$	32.4853	1.28109	0.640547	+	0.767919i	$0.278708\pi$
$644$	39.5980	1.56038
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−39.6569	−1.55907	−0.779536	−	0.626358i	$-0.784545\pi$
$647$	−39.6569	−1.55907	−0.779536	+	0.626358i	$0.784545\pi$
$648$	0	0
$649$	1.51472	0.0594579
$650$	36.9706	1.45010
$651$	0	0
$652$	15.0416	0.589076
$653$	30.1421	1.17955	0.589776	−	0.807567i	$-0.299216\pi$
$653$	30.1421	1.17955	0.589776	+	0.807567i	$0.299216\pi$
$654$	0	0
$655$	−21.3137	−0.832796
$656$	13.4558	0.525362
$657$	0	0
$658$	−22.1421	−0.863190
$659$	−14.4142	−0.561498	−0.280749	−	0.959781i	$-0.590583\pi$
$659$	−14.4142	−0.561498	−0.280749	+	0.959781i	$0.590583\pi$
$660$	0	0
$661$	33.3137	1.29575	0.647877	−	0.761745i	$-0.275657\pi$
$661$	33.3137	1.29575	0.647877	+	0.761745i	$0.275657\pi$
$662$	−5.82843	−0.226528
$663$	0	0
$664$	−33.7990	−1.31166
$665$	−16.9706	−0.658090
$666$	0	0
$667$	3.65685	0.141594
$668$	12.1421	0.469793
$669$	0	0
$670$	13.6569	0.527610
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−21.6274	−0.833676	−0.416838	−	0.908981i	$-0.636862\pi$
$673$	−21.6274	−0.833676	−0.416838	+	0.908981i	$0.636862\pi$
$674$	52.6274	2.02713
$675$	0	0
$676$	6.34315	0.243967
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−12.6863	−0.486855
$680$	−3.65685	−0.140234
$681$	0	0
$682$	10.0711	0.385641
$683$	−20.9706	−0.802416	−0.401208	−	0.915987i	$-0.631410\pi$
$683$	−20.9706	−0.802416	−0.401208	+	0.915987i	$0.631410\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	40.9706	1.56426
$687$	0	0
$688$	10.7574	0.410120
$689$	36.3137	1.38344
$690$	0	0
$691$	48.0000	1.82601	0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000	1.82601	0.913003	+	0.407953i	$0.133757\pi$
$692$	−47.2548	−1.79636
$693$	0	0
$694$	−6.00000	−0.227757
$695$	14.0000	0.531050
$696$	0	0
$697$	−3.71573	−0.140743
$698$	−12.4142	−0.469885
$699$	0	0
$700$	43.3137	1.63710
$701$	40.1127	1.51504	0.757518	−	0.652814i	$-0.226412\pi$
$701$	40.1127	1.51504	0.757518	+	0.652814i	$0.226412\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	−4.07107	−0.153434
$705$	0	0
$706$	−65.1127	−2.45055
$707$	−6.62742	−0.249250
$708$	0	0
$709$	29.1421	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$709$	29.1421	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$710$	21.3137	0.799889
$711$	0	0
$712$	55.1127	2.06544
$713$	−36.8284	−1.37924
$714$	0	0
$715$	−1.58579	−0.0593051
$716$	24.8284	0.927882
$717$	0	0
$718$	−9.48528	−0.353988
$719$	20.1421	0.751175	0.375587	−	0.926787i	$-0.377441\pi$
$719$	20.1421	0.751175	0.375587	+	0.926787i	$0.377441\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	41.0416	1.52741
$723$	0	0
$724$	31.8284	1.18289
$725$	4.00000	0.148556
$726$	0	0
$727$	1.31371	0.0487228	0.0243614	−	0.999703i	$-0.492245\pi$
$727$	1.31371	0.0487228	0.0243614	+	0.999703i	$0.492245\pi$
$728$	47.7990	1.77155
$729$	0	0
$730$	9.65685	0.357416
$731$	−2.97056	−0.109870
$732$	0	0
$733$	−41.2548	−1.52378	−0.761891	−	0.647705i	$-0.775729\pi$
$733$	−41.2548	−1.52378	−0.761891	+	0.647705i	$0.775729\pi$
$734$	43.4558	1.60398
$735$	0	0
$736$	5.79899	0.213754
$737$	2.34315	0.0863109
$738$	0	0
$739$	4.07107	0.149757	0.0748783	−	0.997193i	$-0.476143\pi$
$739$	4.07107	0.149757	0.0748783	+	0.997193i	$0.476143\pi$
$740$	−15.3137	−0.562943
$741$	0	0
$742$	64.7696	2.37777
$743$	−23.6569	−0.867886	−0.433943	−	0.900940i	$-0.642878\pi$
$743$	−23.6569	−0.867886	−0.433943	+	0.900940i	$0.642878\pi$
$744$	0	0
$745$	7.82843	0.286811
$746$	−63.5269	−2.32589
$747$	0	0
$748$	−1.31371	−0.0480339
$749$	−41.9411	−1.53250
$750$	0	0
$751$	25.3137	0.923710	0.461855	−	0.886955i	$-0.347184\pi$
$751$	25.3137	0.923710	0.461855	+	0.886955i	$0.347184\pi$
$752$	9.72792	0.354741
$753$	0	0
$754$	9.24264	0.336597
$755$	−14.1421	−0.514685
$756$	0	0
$757$	25.5147	0.927348	0.463674	−	0.886006i	$-0.346531\pi$
$757$	25.5147	0.927348	0.463674	+	0.886006i	$0.346531\pi$
$758$	−16.8284	−0.611236
$759$	0	0
$760$	26.4853	0.960722
$761$	−45.5980	−1.65293	−0.826463	−	0.562991i	$-0.809650\pi$
$761$	−45.5980	−1.65293	−0.826463	+	0.562991i	$0.809650\pi$
$762$	0	0
$763$	−35.7990	−1.29601
$764$	−96.9117	−3.50614
$765$	0	0
$766$	8.48528	0.306586
$767$	−14.0000	−0.505511
$768$	0	0
$769$	−49.1127	−1.77105	−0.885525	−	0.464592i	$-0.846201\pi$
$769$	−49.1127	−1.77105	−0.885525	+	0.464592i	$0.846201\pi$
$770$	−2.82843	−0.101929
$771$	0	0
$772$	−19.7990	−0.712581
$773$	19.5147	0.701896	0.350948	−	0.936395i	$-0.385859\pi$
$773$	19.5147	0.701896	0.350948	+	0.936395i	$0.385859\pi$
$774$	0	0
$775$	−40.2843	−1.44705
$776$	19.7990	0.710742
$777$	0	0
$778$	−7.31371	−0.262209
$779$	26.9117	0.964211
$780$	0	0
$781$	3.65685	0.130853
$782$	7.31371	0.261538
$783$	0	0
$784$	3.00000	0.107143
$785$	8.48528	0.302853
$786$	0	0
$787$	−54.0833	−1.92786	−0.963930	−	0.266156i	$-0.914246\pi$
$787$	−54.0833	−1.92786	−0.963930	+	0.266156i	$0.914246\pi$
$788$	−7.65685	−0.272764
$789$	0	0
$790$	−5.82843	−0.207366
$791$	−37.6569	−1.33892
$792$	0	0
$793$	18.4853	0.656432
$794$	46.6985	1.65727
$795$	0	0
$796$	−1.85786	−0.0658503
$797$	−51.7401	−1.83273	−0.916364	−	0.400345i	$-0.868890\pi$
$797$	−51.7401	−1.83273	−0.916364	+	0.400345i	$0.868890\pi$
$798$	0	0
$799$	−2.68629	−0.0950342
$800$	6.34315	0.224264
$801$	0	0
$802$	45.0416	1.59048
$803$	1.65685	0.0584691
$804$	0	0
$805$	10.3431	0.364548
$806$	−93.0833	−3.27872
$807$	0	0
$808$	10.3431	0.363871
$809$	−36.2843	−1.27569	−0.637844	−	0.770166i	$-0.720173\pi$
$809$	−36.2843	−1.27569	−0.637844	+	0.770166i	$0.720173\pi$
$810$	0	0
$811$	10.8284	0.380238	0.190119	−	0.981761i	$-0.439113\pi$
$811$	10.8284	0.380238	0.190119	+	0.981761i	$0.439113\pi$
$812$	10.8284	0.380003
$813$	0	0
$814$	−4.00000	−0.140200
$815$	3.92893	0.137624
$816$	0	0
$817$	21.5147	0.752705
$818$	−45.7990	−1.60132
$819$	0	0
$820$	17.1716	0.599657
$821$	1.48528	0.0518367	0.0259183	−	0.999664i	$-0.491749\pi$
$821$	1.48528	0.0518367	0.0259183	+	0.999664i	$0.491749\pi$
$822$	0	0
$823$	−54.2843	−1.89223	−0.946115	−	0.323830i	$-0.895029\pi$
$823$	−54.2843	−1.89223	−0.946115	+	0.323830i	$0.895029\pi$
$824$	−21.3137	−0.742498
$825$	0	0
$826$	−24.9706	−0.868837
$827$	−32.8995	−1.14403	−0.572014	−	0.820244i	$-0.693838\pi$
$827$	−32.8995	−1.14403	−0.572014	+	0.820244i	$0.693838\pi$
$828$	0	0
$829$	−29.7990	−1.03496	−0.517481	−	0.855695i	$-0.673130\pi$
$829$	−29.7990	−1.03496	−0.517481	+	0.855695i	$0.673130\pi$
$830$	−18.4853	−0.641633
$831$	0	0
$832$	37.6274	1.30450
$833$	−0.828427	−0.0287033
$834$	0	0
$835$	3.17157	0.109757
$836$	9.51472	0.329073
$837$	0	0
$838$	22.9706	0.793505
$839$	7.92893	0.273737	0.136869	−	0.990589i	$-0.456296\pi$
$839$	7.92893	0.273737	0.136869	+	0.990589i	$0.456296\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	89.5980	3.08775
$843$	0	0
$844$	−74.2132	−2.55452
$845$	1.65685	0.0569975
$846$	0	0
$847$	30.6274	1.05237
$848$	−28.4558	−0.977178
$849$	0	0
$850$	8.00000	0.274398
$851$	14.6274	0.501421
$852$	0	0
$853$	−22.9706	−0.786497	−0.393249	−	0.919432i	$-0.628649\pi$
$853$	−22.9706	−0.786497	−0.393249	+	0.919432i	$0.628649\pi$
$854$	32.9706	1.12823
$855$	0	0
$856$	65.4558	2.23723
$857$	6.17157	0.210817	0.105408	−	0.994429i	$-0.466385\pi$
$857$	6.17157	0.210817	0.105408	+	0.994429i	$0.466385\pi$
$858$	0	0
$859$	19.7279	0.673108	0.336554	−	0.941664i	$-0.390739\pi$
$859$	19.7279	0.673108	0.336554	+	0.941664i	$0.390739\pi$
$860$	13.7279	0.468118
$861$	0	0
$862$	−47.4558	−1.61635
$863$	−17.1127	−0.582523	−0.291262	−	0.956643i	$-0.594075\pi$
$863$	−17.1127	−0.582523	−0.291262	+	0.956643i	$0.594075\pi$
$864$	0	0
$865$	−12.3431	−0.419680
$866$	73.9411	2.51262
$867$	0	0
$868$	−109.054	−3.70153
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	−21.6569	−0.733815
$872$	55.8701	1.89200
$873$	0	0
$874$	−52.9706	−1.79176
$875$	25.4558	0.860565
$876$	0	0
$877$	−37.1421	−1.25420	−0.627100	−	0.778938i	$-0.715758\pi$
$877$	−37.1421	−1.25420	−0.627100	+	0.778938i	$0.715758\pi$
$878$	−0.828427	−0.0279581
$879$	0	0
$880$	1.24264	0.0418894
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	38.4264	1.29315	0.646576	−	0.762850i	$-0.276200\pi$
$883$	38.4264	1.29315	0.646576	+	0.762850i	$0.276200\pi$
$884$	12.1421	0.408384
$885$	0	0
$886$	58.7696	1.97440
$887$	−17.1005	−0.574179	−0.287089	−	0.957904i	$-0.592688\pi$
$887$	−17.1005	−0.574179	−0.287089	+	0.957904i	$0.592688\pi$
$888$	0	0
$889$	12.2843	0.412001
$890$	30.1421	1.01037
$891$	0	0
$892$	−12.1421	−0.406549
$893$	19.4558	0.651065
$894$	0	0
$895$	6.48528	0.216779
$896$	58.1421	1.94239
$897$	0	0
$898$	84.4264	2.81735
$899$	−10.0711	−0.335889
$900$	0	0
$901$	7.85786	0.261783
$902$	4.48528	0.149344
$903$	0	0
$904$	58.7696	1.95465
$905$	8.31371	0.276357
$906$	0	0
$907$	22.2843	0.739937	0.369969	−	0.929044i	$-0.379368\pi$
$907$	22.2843	0.739937	0.369969	+	0.929044i	$0.379368\pi$
$908$	31.1716	1.03446
$909$	0	0
$910$	26.1421	0.866603
$911$	15.4437	0.511671	0.255835	−	0.966720i	$-0.417649\pi$
$911$	15.4437	0.511671	0.255835	+	0.966720i	$0.417649\pi$
$912$	0	0
$913$	−3.17157	−0.104964
$914$	2.48528	0.0822058
$915$	0	0
$916$	−13.4558	−0.444594
$917$	60.2843	1.99076
$918$	0	0
$919$	8.14214	0.268584	0.134292	−	0.990942i	$-0.457124\pi$
$919$	8.14214	0.268584	0.134292	+	0.990942i	$0.457124\pi$
$920$	−16.1421	−0.532190
$921$	0	0
$922$	−33.7990	−1.11311
$923$	−33.7990	−1.11251
$924$	0	0
$925$	16.0000	0.526077
$926$	−62.7696	−2.06274
$927$	0	0
$928$	1.58579	0.0520560
$929$	−18.6863	−0.613077	−0.306539	−	0.951858i	$-0.599171\pi$
$929$	−18.6863	−0.613077	−0.306539	+	0.951858i	$0.599171\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−70.1127	−2.29662
$933$	0	0
$934$	92.5980	3.02990
$935$	−0.343146	−0.0112221
$936$	0	0
$937$	−16.6274	−0.543194	−0.271597	−	0.962411i	$-0.587552\pi$
$937$	−16.6274	−0.543194	−0.271597	+	0.962411i	$0.587552\pi$
$938$	−38.6274	−1.26123
$939$	0	0
$940$	12.4142	0.404907
$941$	56.5980	1.84504	0.922521	−	0.385948i	$-0.126125\pi$
$941$	56.5980	1.84504	0.922521	+	0.385948i	$0.126125\pi$
$942$	0	0
$943$	−16.4020	−0.534123
$944$	10.9706	0.357061
$945$	0	0
$946$	3.58579	0.116584
$947$	−2.61522	−0.0849834	−0.0424917	−	0.999097i	$-0.513530\pi$
$947$	−2.61522	−0.0849834	−0.0424917	+	0.999097i	$0.513530\pi$
$948$	0	0
$949$	−15.3137	−0.497104
$950$	−57.9411	−1.87986
$951$	0	0
$952$	10.3431	0.335223
$953$	35.6274	1.15409	0.577043	−	0.816714i	$-0.304207\pi$
$953$	35.6274	1.15409	0.577043	+	0.816714i	$0.304207\pi$
$954$	0	0
$955$	−25.3137	−0.819132
$956$	75.2548	2.43392
$957$	0	0
$958$	−16.6569	−0.538159
$959$	33.9411	1.09602
$960$	0	0
$961$	70.4264	2.27182
$962$	36.9706	1.19198
$963$	0	0
$964$	−70.1127	−2.25818
$965$	−5.17157	−0.166479
$966$	0	0
$967$	−35.2426	−1.13333	−0.566663	−	0.823949i	$-0.691766\pi$
$967$	−35.2426	−1.13333	−0.566663	+	0.823949i	$0.691766\pi$
$968$	−47.7990	−1.53632
$969$	0	0
$970$	10.8284	0.347680
$971$	15.6569	0.502452	0.251226	−	0.967928i	$-0.419166\pi$
$971$	15.6569	0.502452	0.251226	+	0.967928i	$0.419166\pi$
$972$	0	0
$973$	−39.5980	−1.26945
$974$	−27.7990	−0.890737
$975$	0	0
$976$	−14.4853	−0.463663
$977$	−36.1716	−1.15723	−0.578616	−	0.815600i	$-0.696407\pi$
$977$	−36.1716	−1.15723	−0.578616	+	0.815600i	$0.696407\pi$
$978$	0	0
$979$	5.17157	0.165284
$980$	3.82843	0.122295
$981$	0	0
$982$	51.2843	1.63655
$983$	21.8701	0.697547	0.348773	−	0.937207i	$-0.386598\pi$
$983$	21.8701	0.697547	0.348773	+	0.937207i	$0.386598\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	2.00000	0.0636930
$987$	0	0
$988$	−87.9411	−2.79778
$989$	−13.1127	−0.416960
$990$	0	0
$991$	−12.8284	−0.407508	−0.203754	−	0.979022i	$-0.565314\pi$
$991$	−12.8284	−0.407508	−0.203754	+	0.979022i	$0.565314\pi$
$992$	−15.9706	−0.507066
$993$	0	0
$994$	−60.2843	−1.91210
$995$	−0.485281	−0.0153845
$996$	0	0
$997$	28.2843	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$997$	28.2843	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$998$	45.7990	1.44974
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.2.a.d.1.2		2
3.2	odd	2		29.2.a.a.1.1	✓	2
4.3	odd	2		4176.2.a.bq.1.2		2
5.4	even	2		6525.2.a.o.1.1		2
12.11	even	2		464.2.a.h.1.1		2
15.2	even	4		725.2.b.b.349.1		4
15.8	even	4		725.2.b.b.349.4		4
15.14	odd	2		725.2.a.b.1.2		2
21.20	even	2		1421.2.a.j.1.1		2
24.5	odd	2		1856.2.a.r.1.1		2
24.11	even	2		1856.2.a.w.1.2		2
29.28	even	2		7569.2.a.c.1.1		2
33.32	even	2		3509.2.a.j.1.2		2
39.38	odd	2		4901.2.a.g.1.2		2
51.50	odd	2		8381.2.a.e.1.1		2
87.2	even	28		841.2.e.k.236.1		24
87.5	odd	14		841.2.d.f.605.1		12
87.8	even	28		841.2.e.k.267.1		24
87.11	even	28		841.2.e.k.63.1		24
87.14	even	28		841.2.e.k.196.4		24
87.17	even	4		841.2.b.a.840.4		4
87.20	odd	14		841.2.d.j.574.1		12
87.23	odd	14		841.2.d.j.645.2		12
87.26	even	28		841.2.e.k.270.4		24
87.32	even	28		841.2.e.k.270.1		24
87.35	odd	14		841.2.d.f.645.1		12
87.38	odd	14		841.2.d.f.574.2		12
87.41	even	4		841.2.b.a.840.1		4
87.44	even	28		841.2.e.k.196.1		24
87.47	even	28		841.2.e.k.63.4		24
87.50	even	28		841.2.e.k.267.4		24
87.53	odd	14		841.2.d.j.605.2		12
87.56	even	28		841.2.e.k.236.4		24
87.62	odd	14		841.2.d.f.190.1		12
87.65	odd	14		841.2.d.j.571.2		12
87.68	even	28		841.2.e.k.651.1		24
87.71	odd	14		841.2.d.f.778.2		12
87.74	odd	14		841.2.d.j.778.1		12
87.77	even	28		841.2.e.k.651.4		24
87.80	odd	14		841.2.d.f.571.1		12
87.83	odd	14		841.2.d.j.190.2		12
87.86	odd	2		841.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.a.a.1.1	✓	2	3.2	odd	2
261.2.a.d.1.2		2	1.1	even	1	trivial
464.2.a.h.1.1		2	12.11	even	2
725.2.a.b.1.2		2	15.14	odd	2
725.2.b.b.349.1		4	15.2	even	4
725.2.b.b.349.4		4	15.8	even	4
841.2.a.d.1.2		2	87.86	odd	2
841.2.b.a.840.1		4	87.41	even	4
841.2.b.a.840.4		4	87.17	even	4
841.2.d.f.190.1		12	87.62	odd	14
841.2.d.f.571.1		12	87.80	odd	14
841.2.d.f.574.2		12	87.38	odd	14
841.2.d.f.605.1		12	87.5	odd	14
841.2.d.f.645.1		12	87.35	odd	14
841.2.d.f.778.2		12	87.71	odd	14
841.2.d.j.190.2		12	87.83	odd	14
841.2.d.j.571.2		12	87.65	odd	14
841.2.d.j.574.1		12	87.20	odd	14
841.2.d.j.605.2		12	87.53	odd	14
841.2.d.j.645.2		12	87.23	odd	14
841.2.d.j.778.1		12	87.74	odd	14
841.2.e.k.63.1		24	87.11	even	28
841.2.e.k.63.4		24	87.47	even	28
841.2.e.k.196.1		24	87.44	even	28
841.2.e.k.196.4		24	87.14	even	28
841.2.e.k.236.1		24	87.2	even	28
841.2.e.k.236.4		24	87.56	even	28
841.2.e.k.267.1		24	87.8	even	28
841.2.e.k.267.4		24	87.50	even	28
841.2.e.k.270.1		24	87.32	even	28
841.2.e.k.270.4		24	87.26	even	28
841.2.e.k.651.1		24	87.68	even	28
841.2.e.k.651.4		24	87.77	even	28
1421.2.a.j.1.1		2	21.20	even	2
1856.2.a.r.1.1		2	24.5	odd	2
1856.2.a.w.1.2		2	24.11	even	2
3509.2.a.j.1.2		2	33.32	even	2
4176.2.a.bq.1.2		2	4.3	odd	2
4901.2.a.g.1.2		2	39.38	odd	2
6525.2.a.o.1.1		2	5.4	even	2
7569.2.a.c.1.1		2	29.28	even	2
8381.2.a.e.1.1		2	51.50	odd	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:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -2.82843 q^{7} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -2.82843 q^{7} +4.41421 q^{8} +2.41421 q^{10} +0.414214 q^{11} -3.82843 q^{13} -6.82843 q^{14} +3.00000 q^{16} -0.828427 q^{17} +6.00000 q^{19} +3.82843 q^{20} +1.00000 q^{22} -3.65685 q^{23} -4.00000 q^{25} -9.24264 q^{26} -10.8284 q^{28} -1.00000 q^{29} +10.0711 q^{31} -1.58579 q^{32} -2.00000 q^{34} -2.82843 q^{35} -4.00000 q^{37} +14.4853 q^{38} +4.41421 q^{40} +4.48528 q^{41} +3.58579 q^{43} +1.58579 q^{44} -8.82843 q^{46} +3.24264 q^{47} +1.00000 q^{49} -9.65685 q^{50} -14.6569 q^{52} -9.48528 q^{53} +0.414214 q^{55} -12.4853 q^{56} -2.41421 q^{58} +3.65685 q^{59} -4.82843 q^{61} +24.3137 q^{62} -9.82843 q^{64} -3.82843 q^{65} +5.65685 q^{67} -3.17157 q^{68} -6.82843 q^{70} +8.82843 q^{71} +4.00000 q^{73} -9.65685 q^{74} +22.9706 q^{76} -1.17157 q^{77} -2.41421 q^{79} +3.00000 q^{80} +10.8284 q^{82} -7.65685 q^{83} -0.828427 q^{85} +8.65685 q^{86} +1.82843 q^{88} +12.4853 q^{89} +10.8284 q^{91} -14.0000 q^{92} +7.82843 q^{94} +6.00000 q^{95} +4.48528 q^{97} +2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 8 q^{14} + 6 q^{16} + 4 q^{17} + 12 q^{19} + 2 q^{20} + 2 q^{22} + 4 q^{23} - 8 q^{25} - 10 q^{26} - 16 q^{28} - 2 q^{29} + 6 q^{31} - 6 q^{32} - 4 q^{34} - 8 q^{37} + 12 q^{38} + 6 q^{40} - 8 q^{41} + 10 q^{43} + 6 q^{44} - 12 q^{46} - 2 q^{47} + 2 q^{49} - 8 q^{50} - 18 q^{52} - 2 q^{53} - 2 q^{55} - 8 q^{56} - 2 q^{58} - 4 q^{59} - 4 q^{61} + 26 q^{62} - 14 q^{64} - 2 q^{65} - 12 q^{68} - 8 q^{70} + 12 q^{71} + 8 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{77} - 2 q^{79} + 6 q^{80} + 16 q^{82} - 4 q^{83} + 4 q^{85} + 6 q^{86} - 2 q^{88} + 8 q^{89} + 16 q^{91} - 28 q^{92} + 10 q^{94} + 12 q^{95} - 8 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 2 * q^10 - 2 * q^11 - 2 * q^13 - 8 * q^14 + 6 * q^16 + 4 * q^17 + 12 * q^19 + 2 * q^20 + 2 * q^22 + 4 * q^23 - 8 * q^25 - 10 * q^26 - 16 * q^28 - 2 * q^29 + 6 * q^31 - 6 * q^32 - 4 * q^34 - 8 * q^37 + 12 * q^38 + 6 * q^40 - 8 * q^41 + 10 * q^43 + 6 * q^44 - 12 * q^46 - 2 * q^47 + 2 * q^49 - 8 * q^50 - 18 * q^52 - 2 * q^53 - 2 * q^55 - 8 * q^56 - 2 * q^58 - 4 * q^59 - 4 * q^61 + 26 * q^62 - 14 * q^64 - 2 * q^65 - 12 * q^68 - 8 * q^70 + 12 * q^71 + 8 * q^73 - 8 * q^74 + 12 * q^76 - 8 * q^77 - 2 * q^79 + 6 * q^80 + 16 * q^82 - 4 * q^83 + 4 * q^85 + 6 * q^86 - 2 * q^88 + 8 * q^89 + 16 * q^91 - 28 * q^92 + 10 * q^94 + 12 * q^95 - 8 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more