LMFDB - Embedded newform 927.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(1,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, 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("927.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	927.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:	$7.40213226737$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 13x^{6} + 11x^{5} + 52x^{4} - 35x^{3} - 59x^{2} + 27x + 1 $ x^8 - x^7 - 13x^6 + 11x^5 + 52x^4 - 35x^3 - 59x^2 + 27x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 309)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.27306$ of defining polynomial
Character	$\chi$	$=$	927.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27306 q^{2} -0.379326 q^{4} -1.25442 q^{5} +2.55696 q^{7} +3.02902 q^{8} +O(q^{10})$	\(q-1.27306 q^{2} -0.379326 q^{4} -1.25442 q^{5} +2.55696 q^{7} +3.02902 q^{8} +1.59695 q^{10} +0.263412 q^{11} +2.56998 q^{13} -3.25516 q^{14} -3.09746 q^{16} -0.293552 q^{17} -5.31376 q^{19} +0.475836 q^{20} -0.335338 q^{22} -3.68477 q^{23} -3.42642 q^{25} -3.27173 q^{26} -0.969922 q^{28} +8.22506 q^{29} +3.61056 q^{31} -2.11479 q^{32} +0.373708 q^{34} -3.20752 q^{35} +0.269739 q^{37} +6.76472 q^{38} -3.79968 q^{40} +10.3358 q^{41} +3.52503 q^{43} -0.0999190 q^{44} +4.69092 q^{46} +11.4483 q^{47} -0.461936 q^{49} +4.36203 q^{50} -0.974859 q^{52} -4.35871 q^{53} -0.330431 q^{55} +7.74509 q^{56} -10.4710 q^{58} +9.99270 q^{59} +12.7035 q^{61} -4.59645 q^{62} +8.88717 q^{64} -3.22384 q^{65} +3.51633 q^{67} +0.111352 q^{68} +4.08335 q^{70} +0.0409897 q^{71} -0.813252 q^{73} -0.343393 q^{74} +2.01565 q^{76} +0.673535 q^{77} +0.783502 q^{79} +3.88553 q^{80} -13.1581 q^{82} -8.45280 q^{83} +0.368239 q^{85} -4.48756 q^{86} +0.797879 q^{88} +2.01361 q^{89} +6.57134 q^{91} +1.39773 q^{92} -14.5744 q^{94} +6.66571 q^{95} +11.7859 q^{97} +0.588071 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 11 q^{4} + q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) $ 8 * q + q^2 + 11 * q^4 + q^5 + 6 * q^7 + 3 * q^8	$ 8 q + q^{2} + 11 q^{4} + q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{10} - 6 q^{11} + 9 q^{13} + 6 q^{14} + 9 q^{16} + 4 q^{17} + 16 q^{19} + 23 q^{20} - 4 q^{22} + 11 q^{23} + 15 q^{25} + 14 q^{26} - 5 q^{28} + 17 q^{31} + 12 q^{32} - 8 q^{34} - 4 q^{35} - 6 q^{37} + 3 q^{38} + 3 q^{40} - 12 q^{41} + 9 q^{43} - 8 q^{44} - 30 q^{46} + 6 q^{47} + 18 q^{49} + 36 q^{50} + 23 q^{52} + 16 q^{53} - 10 q^{55} + 13 q^{56} - 22 q^{58} - 11 q^{59} + 5 q^{61} + 25 q^{62} - 35 q^{64} + 41 q^{65} + 5 q^{67} + 19 q^{68} - 48 q^{70} + 10 q^{71} + 14 q^{73} - 4 q^{74} - 12 q^{76} + 40 q^{77} + 14 q^{79} + 19 q^{80} - 13 q^{82} + 23 q^{83} - 4 q^{85} + 3 q^{86} - 30 q^{88} + 14 q^{89} - 6 q^{91} + 21 q^{92} + 22 q^{94} - 6 q^{95} + 3 q^{97} + 18 q^{98}+O(q^{100}) $ 8 * q + q^2 + 11 * q^4 + q^5 + 6 * q^7 + 3 * q^8 + 3 * q^10 - 6 * q^11 + 9 * q^13 + 6 * q^14 + 9 * q^16 + 4 * q^17 + 16 * q^19 + 23 * q^20 - 4 * q^22 + 11 * q^23 + 15 * q^25 + 14 * q^26 - 5 * q^28 + 17 * q^31 + 12 * q^32 - 8 * q^34 - 4 * q^35 - 6 * q^37 + 3 * q^38 + 3 * q^40 - 12 * q^41 + 9 * q^43 - 8 * q^44 - 30 * q^46 + 6 * q^47 + 18 * q^49 + 36 * q^50 + 23 * q^52 + 16 * q^53 - 10 * q^55 + 13 * q^56 - 22 * q^58 - 11 * q^59 + 5 * q^61 + 25 * q^62 - 35 * q^64 + 41 * q^65 + 5 * q^67 + 19 * q^68 - 48 * q^70 + 10 * q^71 + 14 * q^73 - 4 * q^74 - 12 * q^76 + 40 * q^77 + 14 * q^79 + 19 * q^80 - 13 * q^82 + 23 * q^83 - 4 * q^85 + 3 * q^86 - 30 * q^88 + 14 * q^89 - 6 * q^91 + 21 * q^92 + 22 * q^94 - 6 * q^95 + 3 * q^97 + 18 * 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 $

$2$	−1.27306	−0.900187	−0.450094	−	0.892981i	$-0.648609\pi$
$2$	−1.27306	−0.900187	−0.450094	+	0.892981i	$0.648609\pi$
$3$	0	0
$3$	0	0
$4$	−0.379326	−0.189663
$5$	−1.25442	−0.560996	−0.280498	−	0.959855i	$-0.590500\pi$
$5$	−1.25442	−0.560996	−0.280498	+	0.959855i	$0.590500\pi$
$6$	0	0
$7$	2.55696	0.966441	0.483221	−	0.875499i	$-0.339467\pi$
$7$	2.55696	0.966441	0.483221	+	0.875499i	$0.339467\pi$
$8$	3.02902	1.07092
$9$	0	0
$10$	1.59695	0.505001
$11$	0.263412	0.0794217	0.0397108	−	0.999211i	$-0.487356\pi$
$11$	0.263412	0.0794217	0.0397108	+	0.999211i	$0.487356\pi$
$12$	0	0
$13$	2.56998	0.712784	0.356392	−	0.934337i	$-0.384007\pi$
$13$	2.56998	0.712784	0.356392	+	0.934337i	$0.384007\pi$
$14$	−3.25516	−0.869978
$15$	0	0
$16$	−3.09746	−0.774365
$17$	−0.293552	−0.0711968	−0.0355984	−	0.999366i	$-0.511334\pi$
$17$	−0.293552	−0.0711968	−0.0355984	+	0.999366i	$0.511334\pi$
$18$	0	0
$19$	−5.31376	−1.21906	−0.609530	−	0.792763i	$-0.708642\pi$
$19$	−5.31376	−1.21906	−0.609530	+	0.792763i	$0.708642\pi$
$20$	0.475836	0.106400
$21$	0	0
$22$	−0.335338	−0.0714944
$23$	−3.68477	−0.768328	−0.384164	−	0.923265i	$-0.625510\pi$
$23$	−3.68477	−0.768328	−0.384164	+	0.923265i	$0.625510\pi$
$24$	0	0
$25$	−3.42642	−0.685284
$26$	−3.27173	−0.641639
$27$	0	0
$28$	−0.969922	−0.183298
$29$	8.22506	1.52736	0.763678	−	0.645598i	$-0.223392\pi$
$29$	8.22506	1.52736	0.763678	+	0.645598i	$0.223392\pi$
$30$	0	0
$31$	3.61056	0.648477	0.324238	−	0.945975i	$-0.394892\pi$
$31$	3.61056	0.648477	0.324238	+	0.945975i	$0.394892\pi$
$32$	−2.11479	−0.373846
$33$	0	0
$34$	0.373708	0.0640904
$35$	−3.20752	−0.542170
$36$	0	0
$37$	0.269739	0.0443449	0.0221724	−	0.999754i	$-0.492942\pi$
$37$	0.269739	0.0443449	0.0221724	+	0.999754i	$0.492942\pi$
$38$	6.76472	1.09738
$39$	0	0
$40$	−3.79968	−0.600781
$41$	10.3358	1.61418	0.807092	−	0.590425i	$-0.201040\pi$
$41$	10.3358	1.61418	0.807092	+	0.590425i	$0.201040\pi$
$42$	0	0
$43$	3.52503	0.537562	0.268781	−	0.963201i	$-0.413379\pi$
$43$	3.52503	0.537562	0.268781	+	0.963201i	$0.413379\pi$
$44$	−0.0999190	−0.0150633
$45$	0	0
$46$	4.69092	0.691639
$47$	11.4483	1.66991	0.834956	−	0.550317i	$-0.185493\pi$
$47$	11.4483	1.66991	0.834956	+	0.550317i	$0.185493\pi$
$48$	0	0
$49$	−0.461936	−0.0659909
$50$	4.36203	0.616884
$51$	0	0
$52$	−0.974859	−0.135189
$53$	−4.35871	−0.598715	−0.299357	−	0.954141i	$-0.596772\pi$
$53$	−4.35871	−0.598715	−0.299357	+	0.954141i	$0.596772\pi$
$54$	0	0
$55$	−0.330431	−0.0445552
$56$	7.74509	1.03498
$57$	0	0
$58$	−10.4710	−1.37491
$59$	9.99270	1.30094	0.650469	−	0.759533i	$-0.274572\pi$
$59$	9.99270	1.30094	0.650469	+	0.759533i	$0.274572\pi$
$60$	0	0
$61$	12.7035	1.62651	0.813256	−	0.581906i	$-0.197693\pi$
$61$	12.7035	1.62651	0.813256	+	0.581906i	$0.197693\pi$
$62$	−4.59645	−0.583750
$63$	0	0
$64$	8.88717	1.11090
$65$	−3.22384	−0.399869
$66$	0	0
$67$	3.51633	0.429588	0.214794	−	0.976659i	$-0.431092\pi$
$67$	3.51633	0.429588	0.214794	+	0.976659i	$0.431092\pi$
$68$	0.111352	0.0135034
$69$	0	0
$70$	4.08335	0.488054
$71$	0.0409897	0.00486458	0.00243229	−	0.999997i	$-0.499226\pi$
$71$	0.0409897	0.00486458	0.00243229	+	0.999997i	$0.499226\pi$
$72$	0	0
$73$	−0.813252	−0.0951839	−0.0475919	−	0.998867i	$-0.515155\pi$
$73$	−0.813252	−0.0951839	−0.0475919	+	0.998867i	$0.515155\pi$
$74$	−0.343393	−0.0399187
$75$	0	0
$76$	2.01565	0.231211
$77$	0.673535	0.0767564
$78$	0	0
$79$	0.783502	0.0881508	0.0440754	−	0.999028i	$-0.485966\pi$
$79$	0.783502	0.0881508	0.0440754	+	0.999028i	$0.485966\pi$
$80$	3.88553	0.434416
$81$	0	0
$82$	−13.1581	−1.45307
$83$	−8.45280	−0.927816	−0.463908	−	0.885884i	$-0.653553\pi$
$83$	−8.45280	−0.927816	−0.463908	+	0.885884i	$0.653553\pi$
$84$	0	0
$85$	0.368239	0.0399411
$86$	−4.48756	−0.483906
$87$	0	0
$88$	0.797879	0.0850542
$89$	2.01361	0.213442	0.106721	−	0.994289i	$-0.465965\pi$
$89$	2.01361	0.213442	0.106721	+	0.994289i	$0.465965\pi$
$90$	0	0
$91$	6.57134	0.688864
$92$	1.39773	0.145723
$93$	0	0
$94$	−14.5744	−1.50323
$95$	6.66571	0.683888
$96$	0	0
$97$	11.7859	1.19668	0.598340	−	0.801242i	$-0.295827\pi$
$97$	11.7859	1.19668	0.598340	+	0.801242i	$0.295827\pi$
$98$	0.588071	0.0594042
$99$	0	0
$100$	1.29973	0.129973
$101$	−2.74529	−0.273167	−0.136583	−	0.990629i	$-0.543612\pi$
$101$	−2.74529	−0.273167	−0.136583	+	0.990629i	$0.543612\pi$
$102$	0	0
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	7.78451	0.763334
$105$	0	0
$106$	5.54889	0.538956
$107$	−8.98639	−0.868747	−0.434374	−	0.900733i	$-0.643030\pi$
$107$	−8.98639	−0.868747	−0.434374	+	0.900733i	$0.643030\pi$
$108$	0	0
$109$	−8.83401	−0.846145	−0.423073	−	0.906096i	$-0.639048\pi$
$109$	−8.83401	−0.846145	−0.423073	+	0.906096i	$0.639048\pi$
$110$	0.420657	0.0401081
$111$	0	0
$112$	−7.92009	−0.748379
$113$	4.95002	0.465659	0.232830	−	0.972518i	$-0.425202\pi$
$113$	4.95002	0.465659	0.232830	+	0.972518i	$0.425202\pi$
$114$	0	0
$115$	4.62227	0.431029
$116$	−3.11998	−0.289683
$117$	0	0
$118$	−12.7213	−1.17109
$119$	−0.750601	−0.0688075
$120$	0	0
$121$	−10.9306	−0.993692
$122$	−16.1722	−1.46417
$123$	0	0
$124$	−1.36958	−0.122992
$125$	10.5703	0.945437
$126$	0	0
$127$	20.3717	1.80770	0.903849	−	0.427852i	$-0.140729\pi$
$127$	20.3717	1.80770	0.903849	+	0.427852i	$0.140729\pi$
$128$	−7.08429	−0.626169
$129$	0	0
$130$	4.10414	0.359957
$131$	−2.06488	−0.180410	−0.0902049	−	0.995923i	$-0.528752\pi$
$131$	−2.06488	−0.180410	−0.0902049	+	0.995923i	$0.528752\pi$
$132$	0	0
$133$	−13.5871	−1.17815
$134$	−4.47648	−0.386709
$135$	0	0
$136$	−0.889174	−0.0762460
$137$	−2.22059	−0.189718	−0.0948588	−	0.995491i	$-0.530240\pi$
$137$	−2.22059	−0.189718	−0.0948588	+	0.995491i	$0.530240\pi$
$138$	0	0
$139$	12.1026	1.02653	0.513264	−	0.858231i	$-0.328436\pi$
$139$	12.1026	1.02653	0.513264	+	0.858231i	$0.328436\pi$
$140$	1.21669	0.102829
$141$	0	0
$142$	−0.0521822	−0.00437903
$143$	0.676963	0.0566105
$144$	0	0
$145$	−10.3177	−0.856840
$146$	1.03532	0.0856833
$147$	0	0
$148$	−0.102319	−0.00841057
$149$	3.39846	0.278413	0.139207	−	0.990263i	$-0.455545\pi$
$149$	3.39846	0.278413	0.139207	+	0.990263i	$0.455545\pi$
$150$	0	0
$151$	23.7639	1.93388	0.966941	−	0.255002i	$-0.0820760\pi$
$151$	23.7639	1.93388	0.966941	+	0.255002i	$0.0820760\pi$
$152$	−16.0955	−1.30552
$153$	0	0
$154$	−0.857448	−0.0690951
$155$	−4.52918	−0.363793
$156$	0	0
$157$	19.3703	1.54592	0.772960	−	0.634455i	$-0.218775\pi$
$157$	19.3703	1.54592	0.772960	+	0.634455i	$0.218775\pi$
$158$	−0.997443	−0.0793523
$159$	0	0
$160$	2.65285	0.209726
$161$	−9.42182	−0.742544
$162$	0	0
$163$	6.13272	0.480352	0.240176	−	0.970729i	$-0.422795\pi$
$163$	6.13272	0.480352	0.240176	+	0.970729i	$0.422795\pi$
$164$	−3.92065	−0.306151
$165$	0	0
$166$	10.7609	0.835208
$167$	−4.77970	−0.369865	−0.184932	−	0.982751i	$-0.559207\pi$
$167$	−4.77970	−0.369865	−0.184932	+	0.982751i	$0.559207\pi$
$168$	0	0
$169$	−6.39521	−0.491939
$170$	−0.468789	−0.0359545
$171$	0	0
$172$	−1.33713	−0.101956
$173$	2.71132	0.206138	0.103069	−	0.994674i	$-0.467134\pi$
$173$	2.71132	0.206138	0.103069	+	0.994674i	$0.467134\pi$
$174$	0	0
$175$	−8.76123	−0.662286
$176$	−0.815908	−0.0615014
$177$	0	0
$178$	−2.56344	−0.192138
$179$	−6.79008	−0.507514	−0.253757	−	0.967268i	$-0.581666\pi$
$179$	−6.79008	−0.507514	−0.253757	+	0.967268i	$0.581666\pi$
$180$	0	0
$181$	−3.15097	−0.234209	−0.117105	−	0.993120i	$-0.537361\pi$
$181$	−3.15097	−0.234209	−0.117105	+	0.993120i	$0.537361\pi$
$182$	−8.36569	−0.620106
$183$	0	0
$184$	−11.1612	−0.822817
$185$	−0.338368	−0.0248773
$186$	0	0
$187$	−0.0773251	−0.00565457
$188$	−4.34265	−0.316720
$189$	0	0
$190$	−8.48583	−0.615627
$191$	9.68027	0.700440	0.350220	−	0.936668i	$-0.386107\pi$
$191$	9.68027	0.700440	0.350220	+	0.936668i	$0.386107\pi$
$192$	0	0
$193$	7.90112	0.568735	0.284367	−	0.958715i	$-0.408216\pi$
$193$	7.90112	0.568735	0.284367	+	0.958715i	$0.408216\pi$
$194$	−15.0042	−1.07724
$195$	0	0
$196$	0.175224	0.0125160
$197$	6.63176	0.472494	0.236247	−	0.971693i	$-0.424083\pi$
$197$	6.63176	0.472494	0.236247	+	0.971693i	$0.424083\pi$
$198$	0	0
$199$	−16.3709	−1.16050	−0.580251	−	0.814438i	$-0.697045\pi$
$199$	−16.3709	−1.16050	−0.580251	+	0.814438i	$0.697045\pi$
$200$	−10.3787	−0.733883
$201$	0	0
$202$	3.49492	0.245901
$203$	21.0312	1.47610
$204$	0	0
$205$	−12.9655	−0.905551
$206$	1.27306	0.0886981
$207$	0	0
$208$	−7.96041	−0.551955
$209$	−1.39971	−0.0968198
$210$	0	0
$211$	−6.73471	−0.463636	−0.231818	−	0.972759i	$-0.574467\pi$
$211$	−6.73471	−0.463636	−0.231818	+	0.972759i	$0.574467\pi$
$212$	1.65337	0.113554
$213$	0	0
$214$	11.4402	0.782035
$215$	−4.42188	−0.301570
$216$	0	0
$217$	9.23208	0.626715
$218$	11.2462	0.761689
$219$	0	0
$220$	0.125341	0.00845048
$221$	−0.754422	−0.0507479
$222$	0	0
$223$	9.87916	0.661558	0.330779	−	0.943708i	$-0.392689\pi$
$223$	9.87916	0.661558	0.330779	+	0.943708i	$0.392689\pi$
$224$	−5.40745	−0.361300
$225$	0	0
$226$	−6.30166	−0.419180
$227$	−26.4553	−1.75590	−0.877951	−	0.478751i	$-0.841090\pi$
$227$	−26.4553	−1.75590	−0.877951	+	0.478751i	$0.841090\pi$
$228$	0	0
$229$	−6.59664	−0.435919	−0.217959	−	0.975958i	$-0.569940\pi$
$229$	−6.59664	−0.435919	−0.217959	+	0.975958i	$0.569940\pi$
$230$	−5.88441	−0.388006
$231$	0	0
$232$	24.9139	1.63567
$233$	2.80917	0.184035	0.0920174	−	0.995757i	$-0.470668\pi$
$233$	2.80917	0.184035	0.0920174	+	0.995757i	$0.470668\pi$
$234$	0	0
$235$	−14.3611	−0.936814
$236$	−3.79049	−0.246740
$237$	0	0
$238$	0.955558	0.0619397
$239$	−5.51054	−0.356447	−0.178223	−	0.983990i	$-0.557035\pi$
$239$	−5.51054	−0.356447	−0.178223	+	0.983990i	$0.557035\pi$
$240$	0	0
$241$	−3.17113	−0.204270	−0.102135	−	0.994771i	$-0.532567\pi$
$241$	−3.17113	−0.204270	−0.102135	+	0.994771i	$0.532567\pi$
$242$	13.9153	0.894509
$243$	0	0
$244$	−4.81875	−0.308489
$245$	0.579464	0.0370206
$246$	0	0
$247$	−13.6562	−0.868926
$248$	10.9365	0.694466
$249$	0	0
$250$	−13.4566	−0.851071
$251$	−30.2544	−1.90964	−0.954820	−	0.297186i	$-0.903952\pi$
$251$	−30.2544	−1.90964	−0.954820	+	0.297186i	$0.903952\pi$
$252$	0	0
$253$	−0.970612	−0.0610219
$254$	−25.9344	−1.62727
$255$	0	0
$256$	−8.75563	−0.547227
$257$	4.05791	0.253125	0.126563	−	0.991959i	$-0.459606\pi$
$257$	4.05791	0.253125	0.126563	+	0.991959i	$0.459606\pi$
$258$	0	0
$259$	0.689713	0.0428567
$260$	1.22289	0.0758403
$261$	0	0
$262$	2.62872	0.162403
$263$	−6.35259	−0.391717	−0.195859	−	0.980632i	$-0.562749\pi$
$263$	−6.35259	−0.391717	−0.195859	+	0.980632i	$0.562749\pi$
$264$	0	0
$265$	5.46768	0.335877
$266$	17.2971	1.06056
$267$	0	0
$268$	−1.33383	−0.0814768
$269$	1.98537	0.121050	0.0605250	−	0.998167i	$-0.480723\pi$
$269$	1.98537	0.121050	0.0605250	+	0.998167i	$0.480723\pi$
$270$	0	0
$271$	−25.9354	−1.57546	−0.787731	−	0.616019i	$-0.788744\pi$
$271$	−25.9354	−1.57546	−0.787731	+	0.616019i	$0.788744\pi$
$272$	0.909265	0.0551323
$273$	0	0
$274$	2.82693	0.170781
$275$	−0.902559	−0.0544264
$276$	0	0
$277$	−27.8754	−1.67487	−0.837437	−	0.546535i	$-0.815947\pi$
$277$	−27.8754	−1.67487	−0.837437	+	0.546535i	$0.815947\pi$
$278$	−15.4073	−0.924067
$279$	0	0
$280$	−9.71563	−0.580620
$281$	9.95610	0.593931	0.296966	−	0.954888i	$-0.404025\pi$
$281$	9.95610	0.593931	0.296966	+	0.954888i	$0.404025\pi$
$282$	0	0
$283$	−17.5111	−1.04093	−0.520463	−	0.853884i	$-0.674241\pi$
$283$	−17.5111	−1.04093	−0.520463	+	0.853884i	$0.674241\pi$
$284$	−0.0155484	−0.000922631 0
$285$	0	0
$286$	−0.861812	−0.0509600
$287$	26.4283	1.56001
$288$	0	0
$289$	−16.9138	−0.994931
$290$	13.1350	0.771317
$291$	0	0
$292$	0.308487	0.0180529
$293$	5.46064	0.319014	0.159507	−	0.987197i	$-0.449010\pi$
$293$	5.46064	0.319014	0.159507	+	0.987197i	$0.449010\pi$
$294$	0	0
$295$	−12.5351	−0.729821
$296$	0.817045	0.0474898
$297$	0	0
$298$	−4.32644	−0.250624
$299$	−9.46978	−0.547651
$300$	0	0
$301$	9.01337	0.519522
$302$	−30.2528	−1.74086
$303$	0	0
$304$	16.4592	0.943998
$305$	−15.9355	−0.912467
$306$	0	0
$307$	−16.8318	−0.960641	−0.480321	−	0.877093i	$-0.659480\pi$
$307$	−16.8318	−0.960641	−0.480321	+	0.877093i	$0.659480\pi$
$308$	−0.255489	−0.0145578
$309$	0	0
$310$	5.76591	0.327482
$311$	14.9534	0.847932	0.423966	−	0.905678i	$-0.360638\pi$
$311$	14.9534	0.847932	0.423966	+	0.905678i	$0.360638\pi$
$312$	0	0
$313$	6.59657	0.372860	0.186430	−	0.982468i	$-0.440308\pi$
$313$	6.59657	0.372860	0.186430	+	0.982468i	$0.440308\pi$
$314$	−24.6595	−1.39162
$315$	0	0
$316$	−0.297202	−0.0167189
$317$	34.8832	1.95924	0.979619	−	0.200865i	$-0.0643753\pi$
$317$	34.8832	1.95924	0.979619	+	0.200865i	$0.0643753\pi$
$318$	0	0
$319$	2.16658	0.121305
$320$	−11.1483	−0.623208
$321$	0	0
$322$	11.9945	0.668428
$323$	1.55986	0.0867932
$324$	0	0
$325$	−8.80582	−0.488459
$326$	−7.80730	−0.432407
$327$	0	0
$328$	31.3074	1.72866
$329$	29.2730	1.61387
$330$	0	0
$331$	29.0160	1.59487	0.797433	−	0.603408i	$-0.206191\pi$
$331$	29.0160	1.59487	0.797433	+	0.603408i	$0.206191\pi$
$332$	3.20637	0.175972
$333$	0	0
$334$	6.08484	0.332947
$335$	−4.41097	−0.240997
$336$	0	0
$337$	17.6065	0.959086	0.479543	−	0.877518i	$-0.340802\pi$
$337$	17.6065	0.959086	0.479543	+	0.877518i	$0.340802\pi$
$338$	8.14147	0.442838
$339$	0	0
$340$	−0.139682	−0.00757535
$341$	0.951066	0.0515031
$342$	0	0
$343$	−19.0799	−1.03022
$344$	10.6774	0.575685
$345$	0	0
$346$	−3.45167	−0.185563
$347$	12.1889	0.654334	0.327167	−	0.944966i	$-0.393906\pi$
$347$	12.1889	0.654334	0.327167	+	0.944966i	$0.393906\pi$
$348$	0	0
$349$	−30.8182	−1.64966	−0.824831	−	0.565379i	$-0.808730\pi$
$349$	−30.8182	−1.64966	−0.824831	+	0.565379i	$0.808730\pi$
$350$	11.1535	0.596182
$351$	0	0
$352$	−0.557061	−0.0296915
$353$	13.3102	0.708428	0.354214	−	0.935164i	$-0.384748\pi$
$353$	13.3102	0.708428	0.354214	+	0.935164i	$0.384748\pi$
$354$	0	0
$355$	−0.0514185	−0.00272901
$356$	−0.763814	−0.0404821
$357$	0	0
$358$	8.64416	0.456858
$359$	7.37338	0.389152	0.194576	−	0.980887i	$-0.437667\pi$
$359$	7.37338	0.389152	0.194576	+	0.980887i	$0.437667\pi$
$360$	0	0
$361$	9.23605	0.486108
$362$	4.01136	0.210832
$363$	0	0
$364$	−2.49268	−0.130652
$365$	1.02016	0.0533978
$366$	0	0
$367$	11.4010	0.595126	0.297563	−	0.954702i	$-0.403826\pi$
$367$	11.4010	0.595126	0.297563	+	0.954702i	$0.403826\pi$
$368$	11.4134	0.594966
$369$	0	0
$370$	0.430761	0.0223942
$371$	−11.1451	−0.578623
$372$	0	0
$373$	−27.7872	−1.43876	−0.719382	−	0.694614i	$-0.755575\pi$
$373$	−27.7872	−1.43876	−0.719382	+	0.694614i	$0.755575\pi$
$374$	0.0984392	0.00509017
$375$	0	0
$376$	34.6772	1.78834
$377$	21.1382	1.08867
$378$	0	0
$379$	3.71150	0.190647	0.0953235	−	0.995446i	$-0.469611\pi$
$379$	3.71150	0.190647	0.0953235	+	0.995446i	$0.469611\pi$
$380$	−2.52848	−0.129708
$381$	0	0
$382$	−12.3235	−0.630527
$383$	−8.15093	−0.416493	−0.208247	−	0.978076i	$-0.566776\pi$
$383$	−8.15093	−0.416493	−0.208247	+	0.978076i	$0.566776\pi$
$384$	0	0
$385$	−0.844899	−0.0430600
$386$	−10.0586	−0.511968
$387$	0	0
$388$	−4.47071	−0.226966
$389$	−13.9731	−0.708465	−0.354232	−	0.935157i	$-0.615258\pi$
$389$	−13.9731	−0.708465	−0.354232	+	0.935157i	$0.615258\pi$
$390$	0	0
$391$	1.08167	0.0547025
$392$	−1.39921	−0.0706709
$393$	0	0
$394$	−8.44261	−0.425333
$395$	−0.982844	−0.0494523
$396$	0	0
$397$	36.7059	1.84222	0.921109	−	0.389305i	$-0.127285\pi$
$397$	36.7059	1.84222	0.921109	+	0.389305i	$0.127285\pi$
$398$	20.8411	1.04467
$399$	0	0
$400$	10.6132	0.530660
$401$	36.1907	1.80728	0.903638	−	0.428298i	$-0.140887\pi$
$401$	36.1907	1.80728	0.903638	+	0.428298i	$0.140887\pi$
$402$	0	0
$403$	9.27907	0.462224
$404$	1.04136	0.0518096
$405$	0	0
$406$	−26.7739	−1.32877
$407$	0.0710525	0.00352194
$408$	0	0
$409$	−1.40554	−0.0694993	−0.0347496	−	0.999396i	$-0.511063\pi$
$409$	−1.40554	−0.0694993	−0.0347496	+	0.999396i	$0.511063\pi$
$410$	16.5058	0.815165
$411$	0	0
$412$	0.379326	0.0186880
$413$	25.5510	1.25728
$414$	0	0
$415$	10.6034	0.520501
$416$	−5.43497	−0.266471
$417$	0	0
$418$	1.78191	0.0871560
$419$	−24.8083	−1.21196	−0.605982	−	0.795478i	$-0.707220\pi$
$419$	−24.8083	−1.21196	−0.605982	+	0.795478i	$0.707220\pi$
$420$	0	0
$421$	26.9710	1.31449	0.657243	−	0.753679i	$-0.271723\pi$
$421$	26.9710	1.31449	0.657243	+	0.753679i	$0.271723\pi$
$422$	8.57367	0.417360
$423$	0	0
$424$	−13.2026	−0.641175
$425$	1.00583	0.0487900
$426$	0	0
$427$	32.4823	1.57193
$428$	3.40877	0.164769
$429$	0	0
$430$	5.62931	0.271469
$431$	−19.8881	−0.957974	−0.478987	−	0.877822i	$-0.658996\pi$
$431$	−19.8881	−0.957974	−0.478987	+	0.877822i	$0.658996\pi$
$432$	0	0
$433$	18.6204	0.894837	0.447419	−	0.894325i	$-0.352343\pi$
$433$	18.6204	0.894837	0.447419	+	0.894325i	$0.352343\pi$
$434$	−11.7530	−0.564161
$435$	0	0
$436$	3.35097	0.160482
$437$	19.5800	0.936638
$438$	0	0
$439$	28.3051	1.35093	0.675465	−	0.737392i	$-0.263943\pi$
$439$	28.3051	1.35093	0.675465	+	0.737392i	$0.263943\pi$
$440$	−1.00088	−0.0477151
$441$	0	0
$442$	0.960422	0.0456826
$443$	14.8788	0.706913	0.353456	−	0.935451i	$-0.385006\pi$
$443$	14.8788	0.706913	0.353456	+	0.935451i	$0.385006\pi$
$444$	0	0
$445$	−2.52592	−0.119740
$446$	−12.5767	−0.595526
$447$	0	0
$448$	22.7242	1.07362
$449$	14.7581	0.696477	0.348238	−	0.937406i	$-0.386780\pi$
$449$	14.7581	0.696477	0.348238	+	0.937406i	$0.386780\pi$
$450$	0	0
$451$	2.72258	0.128201
$452$	−1.87767	−0.0883183
$453$	0	0
$454$	33.6791	1.58064
$455$	−8.24325	−0.386450
$456$	0	0
$457$	6.00491	0.280898	0.140449	−	0.990088i	$-0.455145\pi$
$457$	6.00491	0.280898	0.140449	+	0.990088i	$0.455145\pi$
$458$	8.39790	0.392408
$459$	0	0
$460$	−1.75335	−0.0817501
$461$	−40.1228	−1.86871	−0.934353	−	0.356348i	$-0.884022\pi$
$461$	−40.1228	−1.86871	−0.934353	+	0.356348i	$0.884022\pi$
$462$	0	0
$463$	−35.2182	−1.63673	−0.818365	−	0.574699i	$-0.805119\pi$
$463$	−35.2182	−1.63673	−0.818365	+	0.574699i	$0.805119\pi$
$464$	−25.4768	−1.18273
$465$	0	0
$466$	−3.57623	−0.165666
$467$	−38.1236	−1.76415	−0.882074	−	0.471110i	$-0.843854\pi$
$467$	−38.1236	−1.76415	−0.882074	+	0.471110i	$0.843854\pi$
$468$	0	0
$469$	8.99112	0.415171
$470$	18.2825	0.843308
$471$	0	0
$472$	30.2681	1.39320
$473$	0.928534	0.0426941
$474$	0	0
$475$	18.2072	0.835402
$476$	0.284723	0.0130502
$477$	0	0
$478$	7.01523	0.320869
$479$	2.53398	0.115781	0.0578904	−	0.998323i	$-0.481563\pi$
$479$	2.53398	0.115781	0.0578904	+	0.998323i	$0.481563\pi$
$480$	0	0
$481$	0.693224	0.0316083
$482$	4.03703	0.183882
$483$	0	0
$484$	4.14626	0.188467
$485$	−14.7846	−0.671333
$486$	0	0
$487$	−27.4844	−1.24544	−0.622718	−	0.782446i	$-0.713972\pi$
$487$	−27.4844	−1.24544	−0.622718	+	0.782446i	$0.713972\pi$
$488$	38.4790	1.74186
$489$	0	0
$490$	−0.737691	−0.0333255
$491$	−23.4158	−1.05674	−0.528369	−	0.849015i	$-0.677196\pi$
$491$	−23.4158	−1.05674	−0.528369	+	0.849015i	$0.677196\pi$
$492$	0	0
$493$	−2.41448	−0.108743
$494$	17.3852	0.782196
$495$	0	0
$496$	−11.1836	−0.502158
$497$	0.104809	0.00470133
$498$	0	0
$499$	−29.8193	−1.33490	−0.667449	−	0.744656i	$-0.732614\pi$
$499$	−29.8193	−1.33490	−0.667449	+	0.744656i	$0.732614\pi$
$500$	−4.00959	−0.179314
$501$	0	0
$502$	38.5155	1.71903
$503$	−25.2475	−1.12573	−0.562865	−	0.826549i	$-0.690301\pi$
$503$	−25.2475	−1.12573	−0.562865	+	0.826549i	$0.690301\pi$
$504$	0	0
$505$	3.44376	0.153246
$506$	1.23564	0.0549311
$507$	0	0
$508$	−7.72752	−0.342853
$509$	−31.3946	−1.39154	−0.695772	−	0.718263i	$-0.744937\pi$
$509$	−31.3946	−1.39154	−0.695772	+	0.718263i	$0.744937\pi$
$510$	0	0
$511$	−2.07945	−0.0919897
$512$	25.3150	1.11878
$513$	0	0
$514$	−5.16594	−0.227860
$515$	1.25442	0.0552766
$516$	0	0
$517$	3.01563	0.132627
$518$	−0.878044	−0.0385791
$519$	0	0
$520$	−9.76508	−0.428227
$521$	−10.5563	−0.462479	−0.231240	−	0.972897i	$-0.574278\pi$
$521$	−10.5563	−0.462479	−0.231240	+	0.972897i	$0.574278\pi$
$522$	0	0
$523$	20.6556	0.903205	0.451603	−	0.892219i	$-0.350852\pi$
$523$	20.6556	0.903205	0.451603	+	0.892219i	$0.350852\pi$
$524$	0.783264	0.0342170
$525$	0	0
$526$	8.08720	0.352619
$527$	−1.05989	−0.0461694
$528$	0	0
$529$	−9.42247	−0.409673
$530$	−6.96066	−0.302352
$531$	0	0
$532$	5.15394	0.223451
$533$	26.5628	1.15056
$534$	0	0
$535$	11.2728	0.487364
$536$	10.6510	0.460054
$537$	0	0
$538$	−2.52748	−0.108968
$539$	−0.121680	−0.00524111
$540$	0	0
$541$	−6.49326	−0.279167	−0.139584	−	0.990210i	$-0.544576\pi$
$541$	−6.49326	−0.279167	−0.139584	+	0.990210i	$0.544576\pi$
$542$	33.0172	1.41821
$543$	0	0
$544$	0.620801	0.0266166
$545$	11.0816	0.474684
$546$	0	0
$547$	−12.2400	−0.523344	−0.261672	−	0.965157i	$-0.584274\pi$
$547$	−12.2400	−0.523344	−0.261672	+	0.965157i	$0.584274\pi$
$548$	0.842326	0.0359824
$549$	0	0
$550$	1.14901	0.0489939
$551$	−43.7060	−1.86194
$552$	0	0
$553$	2.00339	0.0851926
$554$	35.4870	1.50770
$555$	0	0
$556$	−4.59082	−0.194694
$557$	31.3177	1.32697	0.663486	−	0.748189i	$-0.269076\pi$
$557$	31.3177	1.32697	0.663486	+	0.748189i	$0.269076\pi$
$558$	0	0
$559$	9.05925	0.383165
$560$	9.93516	0.419837
$561$	0	0
$562$	−12.6747	−0.534649
$563$	38.7773	1.63427	0.817135	−	0.576446i	$-0.195561\pi$
$563$	38.7773	1.63427	0.817135	+	0.576446i	$0.195561\pi$
$564$	0	0
$565$	−6.20943	−0.261233
$566$	22.2926	0.937028
$567$	0	0
$568$	0.124158	0.00520957
$569$	−21.8182	−0.914668	−0.457334	−	0.889295i	$-0.651196\pi$
$569$	−21.8182	−0.914668	−0.457334	+	0.889295i	$0.651196\pi$
$570$	0	0
$571$	−11.5086	−0.481619	−0.240810	−	0.970572i	$-0.577413\pi$
$571$	−11.5086	−0.481619	−0.240810	+	0.970572i	$0.577413\pi$
$572$	−0.256790	−0.0107369
$573$	0	0
$574$	−33.6448	−1.40431
$575$	12.6256	0.526522
$576$	0	0
$577$	20.8550	0.868204	0.434102	−	0.900864i	$-0.357066\pi$
$577$	20.8550	0.868204	0.434102	+	0.900864i	$0.357066\pi$
$578$	21.5323	0.895624
$579$	0	0
$580$	3.91378	0.162511
$581$	−21.6135	−0.896679
$582$	0	0
$583$	−1.14814	−0.0475510
$584$	−2.46335	−0.101934
$585$	0	0
$586$	−6.95171	−0.287173
$587$	−44.1721	−1.82318	−0.911588	−	0.411105i	$-0.865143\pi$
$587$	−44.1721	−1.82318	−0.911588	+	0.411105i	$0.865143\pi$
$588$	0	0
$589$	−19.1857	−0.790532
$590$	15.9579	0.656976
$591$	0	0
$592$	−0.835506	−0.0343391
$593$	−48.4720	−1.99051	−0.995254	−	0.0973121i	$-0.968976\pi$
$593$	−48.4720	−1.99051	−0.995254	+	0.0973121i	$0.968976\pi$
$594$	0	0
$595$	0.941573	0.0386007
$596$	−1.28913	−0.0528046
$597$	0	0
$598$	12.0556	0.492989
$599$	−7.96629	−0.325494	−0.162747	−	0.986668i	$-0.552035\pi$
$599$	−7.96629	−0.325494	−0.162747	+	0.986668i	$0.552035\pi$
$600$	0	0
$601$	37.1883	1.51694	0.758471	−	0.651706i	$-0.225947\pi$
$601$	37.1883	1.51694	0.758471	+	0.651706i	$0.225947\pi$
$602$	−11.4745	−0.467667
$603$	0	0
$604$	−9.01427	−0.366786
$605$	13.7116	0.557457
$606$	0	0
$607$	−47.7514	−1.93817	−0.969085	−	0.246726i	$-0.920645\pi$
$607$	−47.7514	−1.93817	−0.969085	+	0.246726i	$0.920645\pi$
$608$	11.2375	0.455741
$609$	0	0
$610$	20.2869	0.821391
$611$	29.4220	1.19029
$612$	0	0
$613$	34.4227	1.39032	0.695161	−	0.718854i	$-0.255333\pi$
$613$	34.4227	1.39032	0.695161	+	0.718854i	$0.255333\pi$
$614$	21.4278	0.864757
$615$	0	0
$616$	2.04015	0.0821999
$617$	24.1875	0.973751	0.486875	−	0.873471i	$-0.338137\pi$
$617$	24.1875	0.973751	0.486875	+	0.873471i	$0.338137\pi$
$618$	0	0
$619$	−19.7687	−0.794569	−0.397285	−	0.917695i	$-0.630047\pi$
$619$	−19.7687	−0.794569	−0.397285	+	0.917695i	$0.630047\pi$
$620$	1.71804	0.0689980
$621$	0	0
$622$	−19.0366	−0.763297
$623$	5.14873	0.206279
$624$	0	0
$625$	3.87243	0.154897
$626$	−8.39781	−0.335644
$627$	0	0
$628$	−7.34766	−0.293204
$629$	−0.0791824	−0.00315721
$630$	0	0
$631$	1.03744	0.0412998	0.0206499	−	0.999787i	$-0.493426\pi$
$631$	1.03744	0.0412998	0.0206499	+	0.999787i	$0.493426\pi$
$632$	2.37324	0.0944024
$633$	0	0
$634$	−44.4083	−1.76368
$635$	−25.5548	−1.01411
$636$	0	0
$637$	−1.18717	−0.0470372
$638$	−2.75818	−0.109197
$639$	0	0
$640$	8.88671	0.351278
$641$	34.4425	1.36040	0.680199	−	0.733028i	$-0.261894\pi$
$641$	34.4425	1.36040	0.680199	+	0.733028i	$0.261894\pi$
$642$	0	0
$643$	−26.9604	−1.06321	−0.531606	−	0.846991i	$-0.678411\pi$
$643$	−26.9604	−1.06321	−0.531606	+	0.846991i	$0.678411\pi$
$644$	3.57394	0.140833
$645$	0	0
$646$	−1.98580	−0.0781301
$647$	28.2016	1.10872	0.554359	−	0.832278i	$-0.312964\pi$
$647$	28.2016	1.10872	0.554359	+	0.832278i	$0.312964\pi$
$648$	0	0
$649$	2.63220	0.103323
$650$	11.2103	0.439705
$651$	0	0
$652$	−2.32630	−0.0911049
$653$	25.4710	0.996757	0.498379	−	0.866959i	$-0.333929\pi$
$653$	25.4710	0.996757	0.498379	+	0.866959i	$0.333929\pi$
$654$	0	0
$655$	2.59024	0.101209
$656$	−32.0148	−1.24997
$657$	0	0
$658$	−37.2662	−1.45279
$659$	−46.2987	−1.80354	−0.901771	−	0.432213i	$-0.857733\pi$
$659$	−46.2987	−1.80354	−0.901771	+	0.432213i	$0.857733\pi$
$660$	0	0
$661$	−33.0137	−1.28408	−0.642041	−	0.766670i	$-0.721912\pi$
$661$	−33.0137	−1.28408	−0.642041	+	0.766670i	$0.721912\pi$
$662$	−36.9391	−1.43568
$663$	0	0
$664$	−25.6037	−0.993616
$665$	17.0440	0.660938
$666$	0	0
$667$	−30.3075	−1.17351
$668$	1.81307	0.0701496
$669$	0	0
$670$	5.61541	0.216942
$671$	3.34624	0.129180
$672$	0	0
$673$	26.9204	1.03771	0.518853	−	0.854863i	$-0.326359\pi$
$673$	26.9204	1.03771	0.518853	+	0.854863i	$0.326359\pi$
$674$	−22.4141	−0.863357
$675$	0	0
$676$	2.42587	0.0933027
$677$	19.4416	0.747203	0.373601	−	0.927589i	$-0.378123\pi$
$677$	19.4416	0.747203	0.373601	+	0.927589i	$0.378123\pi$
$678$	0	0
$679$	30.1362	1.15652
$680$	1.11540	0.0427737
$681$	0	0
$682$	−1.21076	−0.0463624
$683$	0.863761	0.0330509	0.0165254	−	0.999863i	$-0.494740\pi$
$683$	0.863761	0.0330509	0.0165254	+	0.999863i	$0.494740\pi$
$684$	0	0
$685$	2.78556	0.106431
$686$	24.2898	0.927389
$687$	0	0
$688$	−10.9186	−0.416269
$689$	−11.2018	−0.426754
$690$	0	0
$691$	21.1991	0.806450	0.403225	−	0.915101i	$-0.367889\pi$
$691$	21.1991	0.806450	0.403225	+	0.915101i	$0.367889\pi$
$692$	−1.02848	−0.0390968
$693$	0	0
$694$	−15.5172	−0.589023
$695$	−15.1818	−0.575878
$696$	0	0
$697$	−3.03410	−0.114925
$698$	39.2334	1.48500
$699$	0	0
$700$	3.32336	0.125611
$701$	1.56912	0.0592647	0.0296323	−	0.999561i	$-0.490566\pi$
$701$	1.56912	0.0592647	0.0296323	+	0.999561i	$0.490566\pi$
$702$	0	0
$703$	−1.43333	−0.0540591
$704$	2.34099	0.0882293
$705$	0	0
$706$	−16.9446	−0.637718
$707$	−7.01962	−0.264000
$708$	0	0
$709$	−26.9726	−1.01298	−0.506489	−	0.862246i	$-0.669057\pi$
$709$	−26.9726	−1.01298	−0.506489	+	0.862246i	$0.669057\pi$
$710$	0.0654587	0.00245662
$711$	0	0
$712$	6.09926	0.228579
$713$	−13.3041	−0.498242
$714$	0	0
$715$	−0.849199	−0.0317582
$716$	2.57565	0.0962566
$717$	0	0
$718$	−9.38673	−0.350310
$719$	18.7596	0.699614	0.349807	−	0.936822i	$-0.386247\pi$
$719$	18.7596	0.699614	0.349807	+	0.936822i	$0.386247\pi$
$720$	0	0
$721$	−2.55696	−0.0952263
$722$	−11.7580	−0.437588
$723$	0	0
$724$	1.19524	0.0444208
$725$	−28.1825	−1.04667
$726$	0	0
$727$	−21.5057	−0.797601	−0.398800	−	0.917038i	$-0.630573\pi$
$727$	−21.5057	−0.797601	−0.398800	+	0.917038i	$0.630573\pi$
$728$	19.9047	0.737718
$729$	0	0
$730$	−1.29873	−0.0480680
$731$	−1.03478	−0.0382727
$732$	0	0
$733$	−46.0387	−1.70048	−0.850240	−	0.526396i	$-0.823543\pi$
$733$	−46.0387	−1.70048	−0.850240	+	0.526396i	$0.823543\pi$
$734$	−14.5141	−0.535725
$735$	0	0
$736$	7.79252	0.287236
$737$	0.926242	0.0341186
$738$	0	0
$739$	43.6526	1.60579	0.802893	−	0.596123i	$-0.203293\pi$
$739$	43.6526	1.60579	0.802893	+	0.596123i	$0.203293\pi$
$740$	0.128352	0.00471830
$741$	0	0
$742$	14.1883	0.520869
$743$	−37.8923	−1.39013	−0.695066	−	0.718946i	$-0.744625\pi$
$743$	−37.8923	−1.39013	−0.695066	+	0.718946i	$0.744625\pi$
$744$	0	0
$745$	−4.26312	−0.156189
$746$	35.3746	1.29516
$747$	0	0
$748$	0.0293314	0.00107246
$749$	−22.9779	−0.839594
$750$	0	0
$751$	5.41830	0.197716	0.0988582	−	0.995102i	$-0.468481\pi$
$751$	5.41830	0.197716	0.0988582	+	0.995102i	$0.468481\pi$
$752$	−35.4608	−1.29312
$753$	0	0
$754$	−26.9102	−0.980010
$755$	−29.8101	−1.08490
$756$	0	0
$757$	22.4762	0.816912	0.408456	−	0.912778i	$-0.366067\pi$
$757$	22.4762	0.816912	0.408456	+	0.912778i	$0.366067\pi$
$758$	−4.72495	−0.171618
$759$	0	0
$760$	20.1906	0.732389
$761$	29.3780	1.06495	0.532476	−	0.846445i	$-0.321262\pi$
$761$	29.3780	1.06495	0.532476	+	0.846445i	$0.321262\pi$
$762$	0	0
$763$	−22.5883	−0.817750
$764$	−3.67198	−0.132847
$765$	0	0
$766$	10.3766	0.374922
$767$	25.6810	0.927288
$768$	0	0
$769$	34.0082	1.22637	0.613184	−	0.789940i	$-0.289889\pi$
$769$	34.0082	1.22637	0.613184	+	0.789940i	$0.289889\pi$
$770$	1.07560	0.0387621
$771$	0	0
$772$	−2.99710	−0.107868
$773$	−39.3389	−1.41492	−0.707461	−	0.706753i	$-0.750159\pi$
$773$	−39.3389	−1.41492	−0.707461	+	0.706753i	$0.750159\pi$
$774$	0	0
$775$	−12.3713	−0.444390
$776$	35.6998	1.28155
$777$	0	0
$778$	17.7886	0.637751
$779$	−54.9221	−1.96779
$780$	0	0
$781$	0.0107972	0.000386353 0
$782$	−1.37703	−0.0492425
$783$	0	0
$784$	1.43083	0.0511010
$785$	−24.2986	−0.867255
$786$	0	0
$787$	5.45784	0.194551	0.0972755	−	0.995257i	$-0.468987\pi$
$787$	5.45784	0.194551	0.0972755	+	0.995257i	$0.468987\pi$
$788$	−2.51560	−0.0896145
$789$	0	0
$790$	1.25122	0.0445163
$791$	12.6570	0.450032
$792$	0	0
$793$	32.6476	1.15935
$794$	−46.7287	−1.65834
$795$	0	0
$796$	6.20990	0.220104
$797$	−42.8590	−1.51814	−0.759071	−	0.651007i	$-0.774347\pi$
$797$	−42.8590	−1.51814	−0.759071	+	0.651007i	$0.774347\pi$
$798$	0	0
$799$	−3.36068	−0.118892
$800$	7.24616	0.256190
$801$	0	0
$802$	−46.0728	−1.62689
$803$	−0.214220	−0.00755967
$804$	0	0
$805$	11.8190	0.416564
$806$	−11.8128	−0.416088
$807$	0	0
$808$	−8.31554	−0.292540
$809$	16.9790	0.596950	0.298475	−	0.954417i	$-0.403522\pi$
$809$	16.9790	0.596950	0.298475	+	0.954417i	$0.403522\pi$
$810$	0	0
$811$	39.2430	1.37801	0.689003	−	0.724758i	$-0.258049\pi$
$811$	39.2430	1.37801	0.689003	+	0.724758i	$0.258049\pi$
$812$	−7.97767	−0.279961
$813$	0	0
$814$	−0.0904539	−0.00317041
$815$	−7.69304	−0.269475
$816$	0	0
$817$	−18.7312	−0.655320
$818$	1.78933	0.0625624
$819$	0	0
$820$	4.91816	0.171749
$821$	−41.3371	−1.44267	−0.721337	−	0.692584i	$-0.756472\pi$
$821$	−41.3371	−1.44267	−0.721337	+	0.692584i	$0.756472\pi$
$822$	0	0
$823$	−6.66487	−0.232323	−0.116161	−	0.993230i	$-0.537059\pi$
$823$	−6.66487	−0.232323	−0.116161	+	0.993230i	$0.537059\pi$
$824$	−3.02902	−0.105521
$825$	0	0
$826$	−32.5278	−1.13179
$827$	36.3819	1.26512	0.632561	−	0.774511i	$-0.282004\pi$
$827$	36.3819	1.26512	0.632561	+	0.774511i	$0.282004\pi$
$828$	0	0
$829$	−41.8998	−1.45524	−0.727620	−	0.685981i	$-0.759373\pi$
$829$	−41.8998	−1.45524	−0.727620	+	0.685981i	$0.759373\pi$
$830$	−13.4987	−0.468548
$831$	0	0
$832$	22.8398	0.791829
$833$	0.135602	0.00469834
$834$	0	0
$835$	5.99578	0.207493
$836$	0.530945	0.0183631
$837$	0	0
$838$	31.5824	1.09099
$839$	−22.3434	−0.771379	−0.385689	−	0.922629i	$-0.626036\pi$
$839$	−22.3434	−0.771379	−0.385689	+	0.922629i	$0.626036\pi$
$840$	0	0
$841$	38.6516	1.33281
$842$	−34.3356	−1.18328
$843$	0	0
$844$	2.55465	0.0879346
$845$	8.02231	0.275976
$846$	0	0
$847$	−27.9492	−0.960345
$848$	13.5009	0.463624
$849$	0	0
$850$	−1.28048	−0.0439201
$851$	−0.993927	−0.0340714
$852$	0	0
$853$	14.7622	0.505447	0.252724	−	0.967538i	$-0.418674\pi$
$853$	14.7622	0.505447	0.252724	+	0.967538i	$0.418674\pi$
$854$	−41.3518	−1.41503
$855$	0	0
$856$	−27.2199	−0.930358
$857$	44.8558	1.53224	0.766122	−	0.642695i	$-0.222184\pi$
$857$	44.8558	1.53224	0.766122	+	0.642695i	$0.222184\pi$
$858$	0	0
$859$	−35.4111	−1.20821	−0.604106	−	0.796904i	$-0.706470\pi$
$859$	−35.4111	−1.20821	−0.604106	+	0.796904i	$0.706470\pi$
$860$	1.67733	0.0571966
$861$	0	0
$862$	25.3186	0.862356
$863$	38.7935	1.32055	0.660273	−	0.751026i	$-0.270441\pi$
$863$	38.7935	1.32055	0.660273	+	0.751026i	$0.270441\pi$
$864$	0	0
$865$	−3.40115	−0.115643
$866$	−23.7048	−0.805521
$867$	0	0
$868$	−3.50197	−0.118865
$869$	0.206384	0.00700109
$870$	0	0
$871$	9.03688	0.306203
$872$	−26.7584	−0.906153
$873$	0	0
$874$	−24.9264	−0.843149
$875$	27.0279	0.913710
$876$	0	0
$877$	25.7890	0.870833	0.435416	−	0.900229i	$-0.356601\pi$
$877$	25.7890	0.870833	0.435416	+	0.900229i	$0.356601\pi$
$878$	−36.0340	−1.21609
$879$	0	0
$880$	1.02350	0.0345020
$881$	3.96753	0.133669	0.0668347	−	0.997764i	$-0.478710\pi$
$881$	3.96753	0.133669	0.0668347	+	0.997764i	$0.478710\pi$
$882$	0	0
$883$	−32.2023	−1.08369	−0.541847	−	0.840477i	$-0.682275\pi$
$883$	−32.2023	−1.08369	−0.541847	+	0.840477i	$0.682275\pi$
$884$	0.286172	0.00962500
$885$	0	0
$886$	−18.9416	−0.636354
$887$	9.09894	0.305512	0.152756	−	0.988264i	$-0.451185\pi$
$887$	9.09894	0.305512	0.152756	+	0.988264i	$0.451185\pi$
$888$	0	0
$889$	52.0898	1.74703
$890$	3.21564	0.107789
$891$	0	0
$892$	−3.74742	−0.125473
$893$	−60.8337	−2.03572
$894$	0	0
$895$	8.51764	0.284713
$896$	−18.1143	−0.605156
$897$	0	0
$898$	−18.7879	−0.626960
$899$	29.6971	0.990454
$900$	0	0
$901$	1.27951	0.0426266
$902$	−3.46600	−0.115405
$903$	0	0
$904$	14.9937	0.498683
$905$	3.95265	0.131391
$906$	0	0
$907$	7.41765	0.246299	0.123150	−	0.992388i	$-0.460701\pi$
$907$	7.41765	0.246299	0.123150	+	0.992388i	$0.460701\pi$
$908$	10.0352	0.333029
$909$	0	0
$910$	10.4941	0.347877
$911$	56.1794	1.86131	0.930653	−	0.365902i	$-0.119239\pi$
$911$	56.1794	1.86131	0.930653	+	0.365902i	$0.119239\pi$
$912$	0	0
$913$	−2.22657	−0.0736887
$914$	−7.64459	−0.252861
$915$	0	0
$916$	2.50228	0.0826776
$917$	−5.27983	−0.174356
$918$	0	0
$919$	−11.4294	−0.377022	−0.188511	−	0.982071i	$-0.560366\pi$
$919$	−11.4294	−0.377022	−0.188511	+	0.982071i	$0.560366\pi$
$920$	14.0009	0.461597
$921$	0	0
$922$	51.0787	1.68219
$923$	0.105343	0.00346739
$924$	0	0
$925$	−0.924239	−0.0303888
$926$	44.8348	1.47336
$927$	0	0
$928$	−17.3943	−0.570995
$929$	−46.5203	−1.52628	−0.763140	−	0.646233i	$-0.776344\pi$
$929$	−46.5203	−1.52628	−0.763140	+	0.646233i	$0.776344\pi$
$930$	0	0
$931$	2.45462	0.0804469
$932$	−1.06559	−0.0349046
$933$	0	0
$934$	48.5335	1.58806
$935$	0.0969985	0.00317219
$936$	0	0
$937$	−28.6109	−0.934676	−0.467338	−	0.884079i	$-0.654787\pi$
$937$	−28.6109	−0.934676	−0.467338	+	0.884079i	$0.654787\pi$
$938$	−11.4462	−0.373732
$939$	0	0
$940$	5.44753	0.177679
$941$	−3.88733	−0.126723	−0.0633617	−	0.997991i	$-0.520182\pi$
$941$	−3.88733	−0.126723	−0.0633617	+	0.997991i	$0.520182\pi$
$942$	0	0
$943$	−38.0851	−1.24022
$944$	−30.9520	−1.00740
$945$	0	0
$946$	−1.18208	−0.0384326
$947$	16.1937	0.526226	0.263113	−	0.964765i	$-0.415251\pi$
$947$	16.1937	0.526226	0.263113	+	0.964765i	$0.415251\pi$
$948$	0	0
$949$	−2.09004	−0.0678455
$950$	−23.1788	−0.752018
$951$	0	0
$952$	−2.27358	−0.0736873
$953$	4.46552	0.144652	0.0723262	−	0.997381i	$-0.476958\pi$
$953$	4.46552	0.144652	0.0723262	+	0.997381i	$0.476958\pi$
$954$	0	0
$955$	−12.1432	−0.392944
$956$	2.09029	0.0676048
$957$	0	0
$958$	−3.22591	−0.104224
$959$	−5.67796	−0.183351
$960$	0	0
$961$	−17.9638	−0.579478
$962$	−0.882514	−0.0284534
$963$	0	0
$964$	1.20289	0.0387425
$965$	−9.91136	−0.319058
$966$	0	0
$967$	−33.7167	−1.08426	−0.542128	−	0.840296i	$-0.682381\pi$
$967$	−33.7167	−1.08426	−0.542128	+	0.840296i	$0.682381\pi$
$968$	−33.1090	−1.06416
$969$	0	0
$970$	18.8216	0.604325
$971$	−6.32560	−0.202998	−0.101499	−	0.994836i	$-0.532364\pi$
$971$	−6.32560	−0.202998	−0.101499	+	0.994836i	$0.532364\pi$
$972$	0	0
$973$	30.9459	0.992079
$974$	34.9892	1.12113
$975$	0	0
$976$	−39.3485	−1.25951
$977$	38.0919	1.21867	0.609333	−	0.792914i	$-0.291437\pi$
$977$	38.0919	1.21867	0.609333	+	0.792914i	$0.291437\pi$
$978$	0	0
$979$	0.530409	0.0169519
$980$	−0.219806	−0.00702144
$981$	0	0
$982$	29.8096	0.951262
$983$	35.7252	1.13946	0.569729	−	0.821833i	$-0.307048\pi$
$983$	35.7252	1.13946	0.569729	+	0.821833i	$0.307048\pi$
$984$	0	0
$985$	−8.31905	−0.265067
$986$	3.07377	0.0978889
$987$	0	0
$988$	5.18017	0.164803
$989$	−12.9889	−0.413024
$990$	0	0
$991$	4.29051	0.136292	0.0681462	−	0.997675i	$-0.478292\pi$
$991$	4.29051	0.136292	0.0681462	+	0.997675i	$0.478292\pi$
$992$	−7.63559	−0.242430
$993$	0	0
$994$	−0.133428	−0.00423208
$995$	20.5361	0.651037
$996$	0	0
$997$	25.9878	0.823042	0.411521	−	0.911400i	$-0.364998\pi$
$997$	25.9878	0.823042	0.411521	+	0.911400i	$0.364998\pi$
$998$	37.9617	1.20166
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.a.g.1.3		8
3.2	odd	2		309.2.a.d.1.6	✓	8
12.11	even	2		4944.2.a.bf.1.6		8
15.14	odd	2		7725.2.a.z.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.d.1.6	✓	8	3.2	odd	2
927.2.a.g.1.3		8	1.1	even	1	trivial
4944.2.a.bf.1.6		8	12.11	even	2
7725.2.a.z.1.3		8	15.14	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:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27306 q^{2} -0.379326 q^{4} -1.25442 q^{5} +2.55696 q^{7} +3.02902 q^{8} +O(q^{10})\)	\(q-1.27306 q^{2} -0.379326 q^{4} -1.25442 q^{5} +2.55696 q^{7} +3.02902 q^{8} +1.59695 q^{10} +0.263412 q^{11} +2.56998 q^{13} -3.25516 q^{14} -3.09746 q^{16} -0.293552 q^{17} -5.31376 q^{19} +0.475836 q^{20} -0.335338 q^{22} -3.68477 q^{23} -3.42642 q^{25} -3.27173 q^{26} -0.969922 q^{28} +8.22506 q^{29} +3.61056 q^{31} -2.11479 q^{32} +0.373708 q^{34} -3.20752 q^{35} +0.269739 q^{37} +6.76472 q^{38} -3.79968 q^{40} +10.3358 q^{41} +3.52503 q^{43} -0.0999190 q^{44} +4.69092 q^{46} +11.4483 q^{47} -0.461936 q^{49} +4.36203 q^{50} -0.974859 q^{52} -4.35871 q^{53} -0.330431 q^{55} +7.74509 q^{56} -10.4710 q^{58} +9.99270 q^{59} +12.7035 q^{61} -4.59645 q^{62} +8.88717 q^{64} -3.22384 q^{65} +3.51633 q^{67} +0.111352 q^{68} +4.08335 q^{70} +0.0409897 q^{71} -0.813252 q^{73} -0.343393 q^{74} +2.01565 q^{76} +0.673535 q^{77} +0.783502 q^{79} +3.88553 q^{80} -13.1581 q^{82} -8.45280 q^{83} +0.368239 q^{85} -4.48756 q^{86} +0.797879 q^{88} +2.01361 q^{89} +6.57134 q^{91} +1.39773 q^{92} -14.5744 q^{94} +6.66571 q^{95} +11.7859 q^{97} +0.588071 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 11 q^{4} + q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) \) 8 * q + q^2 + 11 * q^4 + q^5 + 6 * q^7 + 3 * q^8	\( 8 q + q^{2} + 11 q^{4} + q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{10} - 6 q^{11} + 9 q^{13} + 6 q^{14} + 9 q^{16} + 4 q^{17} + 16 q^{19} + 23 q^{20} - 4 q^{22} + 11 q^{23} + 15 q^{25} + 14 q^{26} - 5 q^{28} + 17 q^{31} + 12 q^{32} - 8 q^{34} - 4 q^{35} - 6 q^{37} + 3 q^{38} + 3 q^{40} - 12 q^{41} + 9 q^{43} - 8 q^{44} - 30 q^{46} + 6 q^{47} + 18 q^{49} + 36 q^{50} + 23 q^{52} + 16 q^{53} - 10 q^{55} + 13 q^{56} - 22 q^{58} - 11 q^{59} + 5 q^{61} + 25 q^{62} - 35 q^{64} + 41 q^{65} + 5 q^{67} + 19 q^{68} - 48 q^{70} + 10 q^{71} + 14 q^{73} - 4 q^{74} - 12 q^{76} + 40 q^{77} + 14 q^{79} + 19 q^{80} - 13 q^{82} + 23 q^{83} - 4 q^{85} + 3 q^{86} - 30 q^{88} + 14 q^{89} - 6 q^{91} + 21 q^{92} + 22 q^{94} - 6 q^{95} + 3 q^{97} + 18 q^{98}+O(q^{100}) \) 8 * q + q^2 + 11 * q^4 + q^5 + 6 * q^7 + 3 * q^8 + 3 * q^10 - 6 * q^11 + 9 * q^13 + 6 * q^14 + 9 * q^16 + 4 * q^17 + 16 * q^19 + 23 * q^20 - 4 * q^22 + 11 * q^23 + 15 * q^25 + 14 * q^26 - 5 * q^28 + 17 * q^31 + 12 * q^32 - 8 * q^34 - 4 * q^35 - 6 * q^37 + 3 * q^38 + 3 * q^40 - 12 * q^41 + 9 * q^43 - 8 * q^44 - 30 * q^46 + 6 * q^47 + 18 * q^49 + 36 * q^50 + 23 * q^52 + 16 * q^53 - 10 * q^55 + 13 * q^56 - 22 * q^58 - 11 * q^59 + 5 * q^61 + 25 * q^62 - 35 * q^64 + 41 * q^65 + 5 * q^67 + 19 * q^68 - 48 * q^70 + 10 * q^71 + 14 * q^73 - 4 * q^74 - 12 * q^76 + 40 * q^77 + 14 * q^79 + 19 * q^80 - 13 * q^82 + 23 * q^83 - 4 * q^85 + 3 * q^86 - 30 * q^88 + 14 * q^89 - 6 * q^91 + 21 * q^92 + 22 * q^94 - 6 * q^95 + 3 * q^97 + 18 * q^98

Introduction

L-functions

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Database

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