LMFDB - Embedded newform 1881.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.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:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 627)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.46050 q^{2} +0.133074 q^{4} +4.05408 q^{5} -1.40642 q^{7} +2.72665 q^{8} -5.92101 q^{10} -1.00000 q^{11} -6.46050 q^{13} +2.05408 q^{14} -4.24844 q^{16} -4.38151 q^{17} -1.00000 q^{19} +0.539495 q^{20} +1.46050 q^{22} +5.00000 q^{23} +11.4356 q^{25} +9.43560 q^{26} -0.187159 q^{28} +10.2484 q^{29} +7.32743 q^{31} +0.751560 q^{32} +6.39922 q^{34} -5.70175 q^{35} +5.37432 q^{37} +1.46050 q^{38} +11.0541 q^{40} +2.46050 q^{41} -2.13307 q^{43} -0.133074 q^{44} -7.30252 q^{46} +6.64766 q^{47} -5.02198 q^{49} -16.7017 q^{50} -0.859728 q^{52} -9.16225 q^{53} -4.05408 q^{55} -3.83482 q^{56} -14.9679 q^{58} +1.78074 q^{59} +0.975094 q^{61} -10.7017 q^{62} +7.39922 q^{64} -26.1914 q^{65} +1.91381 q^{67} -0.583068 q^{68} +8.32743 q^{70} +9.46050 q^{71} +4.64766 q^{73} -7.84922 q^{74} -0.133074 q^{76} +1.40642 q^{77} +15.0364 q^{79} -17.2235 q^{80} -3.59358 q^{82} +9.08326 q^{83} -17.7630 q^{85} +3.11537 q^{86} -2.72665 q^{88} +7.32743 q^{89} +9.08619 q^{91} +0.665372 q^{92} -9.70895 q^{94} -4.05408 q^{95} +8.35661 q^{97} +7.33463 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 7 q^{7} + 9 q^{8} - 5 q^{10} - 3 q^{11} - 13 q^{13} - 3 q^{14} + 10 q^{16} + 6 q^{17} - 3 q^{19} + 8 q^{20} - 2 q^{22} + 15 q^{23} + 6 q^{25} + 5 q^{28} + 8 q^{29} + 12 q^{31}+ \cdots + 4 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 7 * q^7 + 9 * q^8 - 5 * q^10 - 3 * q^11 - 13 * q^13 - 3 * q^14 + 10 * q^16 + 6 * q^17 - 3 * q^19 + 8 * q^20 - 2 * q^22 + 15 * q^23 + 6 * q^25 + 5 * q^28 + 8 * q^29 + 12 * q^31 + 25 * q^32 + 30 * q^34 + 4 * q^35 + 5 * q^37 - 2 * q^38 + 24 * q^40 + q^41 - 10 * q^43 - 4 * q^44 + 10 * q^46 + 8 * q^47 + 8 * q^49 - 29 * q^50 - 7 * q^52 - 3 * q^55 + 6 * q^56 - 31 * q^58 - 3 * q^59 - 19 * q^61 - 11 * q^62 + 33 * q^64 - 20 * q^65 + q^67 + 39 * q^68 + 15 * q^70 + 22 * q^71 + 2 * q^73 + 10 * q^74 - 4 * q^76 + 7 * q^77 + 6 * q^79 - 7 * q^80 - 8 * q^82 - 13 * q^83 - 15 * q^85 - 17 * q^86 - 9 * q^88 + 12 * q^89 + 32 * q^91 + 20 * q^92 - 3 * q^95 - 16 * q^97 + 4 * q^98

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 $

$2$	−1.46050	−1.03273	−0.516366	−	0.856368i	$-0.672716\pi$
$2$	−1.46050	−1.03273	−0.516366	+	0.856368i	$0.672716\pi$
$3$	0	0
$3$	0	0
$4$	0.133074	0.0665372
$5$	4.05408	1.81304	0.906521	−	0.422161i	$-0.138728\pi$
$5$	4.05408	1.81304	0.906521	+	0.422161i	$0.138728\pi$
$6$	0	0
$7$	−1.40642	−0.531577	−0.265789	−	0.964031i	$-0.585632\pi$
$7$	−1.40642	−0.531577	−0.265789	+	0.964031i	$0.585632\pi$
$8$	2.72665	0.964018
$9$	0	0
$10$	−5.92101	−1.87239
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.46050	−1.79182	−0.895911	−	0.444234i	$-0.853476\pi$
$13$	−6.46050	−1.79182	−0.895911	+	0.444234i	$0.853476\pi$
$14$	2.05408	0.548977
$15$	0	0
$16$	−4.24844	−1.06211
$17$	−4.38151	−1.06267	−0.531337	−	0.847161i	$-0.678310\pi$
$17$	−4.38151	−1.06267	−0.531337	+	0.847161i	$0.678310\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.539495	0.120635
$21$	0	0
$22$	1.46050	0.311381
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	11.4356	2.28712
$26$	9.43560	1.85047
$27$	0	0
$28$	−0.187159	−0.0353697
$29$	10.2484	1.90309	0.951544	−	0.307513i	$-0.0994969\pi$
$29$	10.2484	1.90309	0.951544	+	0.307513i	$0.0994969\pi$
$30$	0	0
$31$	7.32743	1.31605	0.658023	−	0.752998i	$-0.271393\pi$
$31$	7.32743	1.31605	0.658023	+	0.752998i	$0.271393\pi$
$32$	0.751560	0.132858
$33$	0	0
$34$	6.39922	1.09746
$35$	−5.70175	−0.963771
$36$	0	0
$37$	5.37432	0.883532	0.441766	−	0.897130i	$-0.354352\pi$
$37$	5.37432	0.883532	0.441766	+	0.897130i	$0.354352\pi$
$38$	1.46050	0.236925
$39$	0	0
$40$	11.0541	1.74780
$41$	2.46050	0.384266	0.192133	−	0.981369i	$-0.438459\pi$
$41$	2.46050	0.384266	0.192133	+	0.981369i	$0.438459\pi$
$42$	0	0
$43$	−2.13307	−0.325291	−0.162645	−	0.986685i	$-0.552003\pi$
$43$	−2.13307	−0.325291	−0.162645	+	0.986685i	$0.552003\pi$
$44$	−0.133074	−0.0200617
$45$	0	0
$46$	−7.30252	−1.07670
$47$	6.64766	0.969661	0.484831	−	0.874608i	$-0.338881\pi$
$47$	6.64766	0.969661	0.484831	+	0.874608i	$0.338881\pi$
$48$	0	0
$49$	−5.02198	−0.717426
$50$	−16.7017	−2.36198
$51$	0	0
$52$	−0.859728	−0.119223
$53$	−9.16225	−1.25853	−0.629266	−	0.777190i	$-0.716644\pi$
$53$	−9.16225	−1.25853	−0.629266	+	0.777190i	$0.716644\pi$
$54$	0	0
$55$	−4.05408	−0.546653
$56$	−3.83482	−0.512450
$57$	0	0
$58$	−14.9679	−1.96538
$59$	1.78074	0.231832	0.115916	−	0.993259i	$-0.463020\pi$
$59$	1.78074	0.231832	0.115916	+	0.993259i	$0.463020\pi$
$60$	0	0
$61$	0.975094	0.124848	0.0624240	−	0.998050i	$-0.480117\pi$
$61$	0.975094	0.124848	0.0624240	+	0.998050i	$0.480117\pi$
$62$	−10.7017	−1.35912
$63$	0	0
$64$	7.39922	0.924903
$65$	−26.1914	−3.24865
$66$	0	0
$67$	1.91381	0.233809	0.116905	−	0.993143i	$-0.462703\pi$
$67$	1.91381	0.233809	0.116905	+	0.993143i	$0.462703\pi$
$68$	−0.583068	−0.0707074
$69$	0	0
$70$	8.32743	0.995318
$71$	9.46050	1.12276	0.561378	−	0.827560i	$-0.310272\pi$
$71$	9.46050	1.12276	0.561378	+	0.827560i	$0.310272\pi$
$72$	0	0
$73$	4.64766	0.543968	0.271984	−	0.962302i	$-0.412320\pi$
$73$	4.64766	0.543968	0.271984	+	0.962302i	$0.412320\pi$
$74$	−7.84922	−0.912453
$75$	0	0
$76$	−0.133074	−0.0152647
$77$	1.40642	0.160277
$78$	0	0
$79$	15.0364	1.69172	0.845862	−	0.533401i	$-0.179086\pi$
$79$	15.0364	1.69172	0.845862	+	0.533401i	$0.179086\pi$
$80$	−17.2235	−1.92565
$81$	0	0
$82$	−3.59358	−0.396844
$83$	9.08326	0.997018	0.498509	−	0.866885i	$-0.333881\pi$
$83$	9.08326	0.997018	0.498509	+	0.866885i	$0.333881\pi$
$84$	0	0
$85$	−17.7630	−1.92667
$86$	3.11537	0.335939
$87$	0	0
$88$	−2.72665	−0.290662
$89$	7.32743	0.776706	0.388353	−	0.921511i	$-0.373044\pi$
$89$	7.32743	0.776706	0.388353	+	0.921511i	$0.373044\pi$
$90$	0	0
$91$	9.08619	0.952491
$92$	0.665372	0.0693699
$93$	0	0
$94$	−9.70895	−1.00140
$95$	−4.05408	−0.415940
$96$	0	0
$97$	8.35661	0.848485	0.424243	−	0.905549i	$-0.360540\pi$
$97$	8.35661	0.848485	0.424243	+	0.905549i	$0.360540\pi$
$98$	7.33463	0.740909
$99$	0	0
$100$	1.52179	0.152179
$101$	−4.05408	−0.403396	−0.201698	−	0.979448i	$-0.564646\pi$
$101$	−4.05408	−0.403396	−0.201698	+	0.979448i	$0.564646\pi$
$102$	0	0
$103$	−13.0761	−1.28842	−0.644211	−	0.764847i	$-0.722814\pi$
$103$	−13.0761	−1.28842	−0.644211	+	0.764847i	$0.722814\pi$
$104$	−17.6156	−1.72735
$105$	0	0
$106$	13.3815	1.29973
$107$	13.5146	1.30650	0.653252	−	0.757140i	$-0.273404\pi$
$107$	13.5146	1.30650	0.653252	+	0.757140i	$0.273404\pi$
$108$	0	0
$109$	−8.78074	−0.841042	−0.420521	−	0.907283i	$-0.638153\pi$
$109$	−8.78074	−0.841042	−0.420521	+	0.907283i	$0.638153\pi$
$110$	5.92101	0.564546
$111$	0	0
$112$	5.97509	0.564593
$113$	9.34941	0.879519	0.439759	−	0.898116i	$-0.355064\pi$
$113$	9.34941	0.879519	0.439759	+	0.898116i	$0.355064\pi$
$114$	0	0
$115$	20.2704	1.89023
$116$	1.36381	0.126626
$117$	0	0
$118$	−2.60078	−0.239421
$119$	6.16225	0.564893
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.42413	−0.128935
$123$	0	0
$124$	0.975094	0.0875660
$125$	26.0905	2.33360
$126$	0	0
$127$	−12.5438	−1.11308	−0.556540	−	0.830821i	$-0.687871\pi$
$127$	−12.5438	−1.11308	−0.556540	+	0.830821i	$0.687871\pi$
$128$	−12.3097	−1.08804
$129$	0	0
$130$	38.2527	3.35498
$131$	20.0584	1.75251	0.876253	−	0.481851i	$-0.160035\pi$
$131$	20.0584	1.75251	0.876253	+	0.481851i	$0.160035\pi$
$132$	0	0
$133$	1.40642	0.121952
$134$	−2.79513	−0.241463
$135$	0	0
$136$	−11.9469	−1.02444
$137$	−6.34941	−0.542467	−0.271233	−	0.962514i	$-0.587432\pi$
$137$	−6.34941	−0.542467	−0.271233	+	0.962514i	$0.587432\pi$
$138$	0	0
$139$	−12.7558	−1.08194	−0.540968	−	0.841043i	$-0.681942\pi$
$139$	−12.7558	−1.08194	−0.540968	+	0.841043i	$0.681942\pi$
$140$	−0.758757	−0.0641267
$141$	0	0
$142$	−13.8171	−1.15951
$143$	6.46050	0.540255
$144$	0	0
$145$	41.5480	3.45038
$146$	−6.78794	−0.561774
$147$	0	0
$148$	0.715184	0.0587878
$149$	−12.4605	−1.02080	−0.510402	−	0.859936i	$-0.670503\pi$
$149$	−12.4605	−1.02080	−0.510402	+	0.859936i	$0.670503\pi$
$150$	0	0
$151$	18.6156	1.51491	0.757456	−	0.652886i	$-0.226442\pi$
$151$	18.6156	1.51491	0.757456	+	0.652886i	$0.226442\pi$
$152$	−2.72665	−0.221161
$153$	0	0
$154$	−2.05408	−0.165523
$155$	29.7060	2.38604
$156$	0	0
$157$	−22.6768	−1.80981	−0.904904	−	0.425615i	$-0.860058\pi$
$157$	−22.6768	−1.80981	−0.904904	+	0.425615i	$0.860058\pi$
$158$	−21.9607	−1.74710
$159$	0	0
$160$	3.04689	0.240878
$161$	−7.03210	−0.554207
$162$	0	0
$163$	−5.35661	−0.419562	−0.209781	−	0.977748i	$-0.567275\pi$
$163$	−5.35661	−0.419562	−0.209781	+	0.977748i	$0.567275\pi$
$164$	0.327430	0.0255680
$165$	0	0
$166$	−13.2661	−1.02965
$167$	−2.05408	−0.158950	−0.0794749	−	0.996837i	$-0.525324\pi$
$167$	−2.05408	−0.158950	−0.0794749	+	0.996837i	$0.525324\pi$
$168$	0	0
$169$	28.7381	2.21062
$170$	25.9430	1.98974
$171$	0	0
$172$	−0.283858	−0.0216440
$173$	19.1623	1.45688	0.728440	−	0.685110i	$-0.240246\pi$
$173$	19.1623	1.45688	0.728440	+	0.685110i	$0.240246\pi$
$174$	0	0
$175$	−16.0833	−1.21578
$176$	4.24844	0.320238
$177$	0	0
$178$	−10.7017	−0.802130
$179$	−19.7017	−1.47258	−0.736289	−	0.676667i	$-0.763424\pi$
$179$	−19.7017	−1.47258	−0.736289	+	0.676667i	$0.763424\pi$
$180$	0	0
$181$	−12.6768	−0.942262	−0.471131	−	0.882063i	$-0.656154\pi$
$181$	−12.6768	−0.942262	−0.471131	+	0.882063i	$0.656154\pi$
$182$	−13.2704	−0.983669
$183$	0	0
$184$	13.6333	1.00506
$185$	21.7879	1.60188
$186$	0	0
$187$	4.38151	0.320408
$188$	0.884634	0.0645186
$189$	0	0
$190$	5.92101	0.429555
$191$	2.56440	0.185554	0.0927768	−	0.995687i	$-0.470426\pi$
$191$	2.56440	0.185554	0.0927768	+	0.995687i	$0.470426\pi$
$192$	0	0
$193$	11.6185	0.836317	0.418158	−	0.908374i	$-0.362676\pi$
$193$	11.6185	0.836317	0.418158	+	0.908374i	$0.362676\pi$
$194$	−12.2049	−0.876258
$195$	0	0
$196$	−0.668297	−0.0477355
$197$	−12.3887	−0.882659	−0.441330	−	0.897345i	$-0.645493\pi$
$197$	−12.3887	−0.882659	−0.441330	+	0.897345i	$0.645493\pi$
$198$	0	0
$199$	8.67977	0.615292	0.307646	−	0.951501i	$-0.400459\pi$
$199$	8.67977	0.615292	0.307646	+	0.951501i	$0.400459\pi$
$200$	31.1809	2.20482
$201$	0	0
$202$	5.92101	0.416601
$203$	−14.4136	−1.01164
$204$	0	0
$205$	9.97509	0.696691
$206$	19.0977	1.33060
$207$	0	0
$208$	27.4471	1.90311
$209$	1.00000	0.0691714
$210$	0	0
$211$	14.5366	1.00074	0.500369	−	0.865812i	$-0.333198\pi$
$211$	14.5366	1.00074	0.500369	+	0.865812i	$0.333198\pi$
$212$	−1.21926	−0.0837393
$213$	0	0
$214$	−19.7381	−1.34927
$215$	−8.64766	−0.589766
$216$	0	0
$217$	−10.3054	−0.699579
$218$	12.8243	0.868572
$219$	0	0
$220$	−0.539495	−0.0363728
$221$	28.3068	1.90412
$222$	0	0
$223$	7.37432	0.493821	0.246910	−	0.969038i	$-0.420585\pi$
$223$	7.37432	0.493821	0.246910	+	0.969038i	$0.420585\pi$
$224$	−1.05701	−0.0706244
$225$	0	0
$226$	−13.6549	−0.908308
$227$	10.7807	0.715543	0.357771	−	0.933809i	$-0.383537\pi$
$227$	10.7807	0.715543	0.357771	+	0.933809i	$0.383537\pi$
$228$	0	0
$229$	5.38871	0.356096	0.178048	−	0.984022i	$-0.443022\pi$
$229$	5.38871	0.356096	0.178048	+	0.984022i	$0.443022\pi$
$230$	−29.6050	−1.95210
$231$	0	0
$232$	27.9439	1.83461
$233$	−3.47821	−0.227865	−0.113933	−	0.993488i	$-0.536345\pi$
$233$	−3.47821	−0.227865	−0.113933	+	0.993488i	$0.536345\pi$
$234$	0	0
$235$	26.9502	1.75804
$236$	0.236971	0.0154255
$237$	0	0
$238$	−9.00000	−0.583383
$239$	−9.99707	−0.646657	−0.323329	−	0.946287i	$-0.604802\pi$
$239$	−9.99707	−0.646657	−0.323329	+	0.946287i	$0.604802\pi$
$240$	0	0
$241$	−7.25564	−0.467377	−0.233688	−	0.972312i	$-0.575080\pi$
$241$	−7.25564	−0.467377	−0.233688	+	0.972312i	$0.575080\pi$
$242$	−1.46050	−0.0938848
$243$	0	0
$244$	0.129760	0.00830704
$245$	−20.3595	−1.30072
$246$	0	0
$247$	6.46050	0.411072
$248$	19.9794	1.26869
$249$	0	0
$250$	−38.1052	−2.40999
$251$	−4.14786	−0.261810	−0.130905	−	0.991395i	$-0.541788\pi$
$251$	−4.14786	−0.261810	−0.130905	+	0.991395i	$0.541788\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	18.3202	1.14951
$255$	0	0
$256$	3.17996	0.198748
$257$	19.9722	1.24583	0.622915	−	0.782290i	$-0.285948\pi$
$257$	19.9722	1.24583	0.622915	+	0.782290i	$0.285948\pi$
$258$	0	0
$259$	−7.55855	−0.469666
$260$	−3.48541	−0.216156
$261$	0	0
$262$	−29.2953	−1.80987
$263$	5.37724	0.331575	0.165787	−	0.986162i	$-0.446983\pi$
$263$	5.37724	0.331575	0.165787	+	0.986162i	$0.446983\pi$
$264$	0	0
$265$	−37.1445	−2.28177
$266$	−2.05408	−0.125944
$267$	0	0
$268$	0.254680	0.0155570
$269$	−3.74863	−0.228558	−0.114279	−	0.993449i	$-0.536456\pi$
$269$	−3.74863	−0.228558	−0.114279	+	0.993449i	$0.536456\pi$
$270$	0	0
$271$	1.00720	0.0611829	0.0305914	−	0.999532i	$-0.490261\pi$
$271$	1.00720	0.0611829	0.0305914	+	0.999532i	$0.490261\pi$
$272$	18.6146	1.12868
$273$	0	0
$274$	9.27335	0.560223
$275$	−11.4356	−0.689593
$276$	0	0
$277$	−11.6726	−0.701337	−0.350668	−	0.936500i	$-0.614046\pi$
$277$	−11.6726	−0.701337	−0.350668	+	0.936500i	$0.614046\pi$
$278$	18.6300	1.11735
$279$	0	0
$280$	−15.5467	−0.929093
$281$	3.78794	0.225969	0.112985	−	0.993597i	$-0.463959\pi$
$281$	3.78794	0.225969	0.112985	+	0.993597i	$0.463959\pi$
$282$	0	0
$283$	−8.70895	−0.517693	−0.258847	−	0.965918i	$-0.583342\pi$
$283$	−8.70895	−0.517693	−0.258847	+	0.965918i	$0.583342\pi$
$284$	1.25895	0.0747050
$285$	0	0
$286$	−9.43560	−0.557939
$287$	−3.46050	−0.204267
$288$	0	0
$289$	2.19767	0.129275
$290$	−60.6811	−3.56332
$291$	0	0
$292$	0.618485	0.0361941
$293$	2.32316	0.135720	0.0678602	−	0.997695i	$-0.478383\pi$
$293$	2.32316	0.135720	0.0678602	+	0.997695i	$0.478383\pi$
$294$	0	0
$295$	7.21926	0.420322
$296$	14.6539	0.851741
$297$	0	0
$298$	18.1986	1.05422
$299$	−32.3025	−1.86810
$300$	0	0
$301$	3.00000	0.172917
$302$	−27.1881	−1.56450
$303$	0	0
$304$	4.24844	0.243665
$305$	3.95311	0.226355
$306$	0	0
$307$	1.86693	0.106551	0.0532755	−	0.998580i	$-0.483034\pi$
$307$	1.86693	0.106551	0.0532755	+	0.998580i	$0.483034\pi$
$308$	0.187159	0.0106644
$309$	0	0
$310$	−43.3858	−2.46415
$311$	−8.02198	−0.454885	−0.227442	−	0.973792i	$-0.573036\pi$
$311$	−8.02198	−0.454885	−0.227442	+	0.973792i	$0.573036\pi$
$312$	0	0
$313$	−5.07179	−0.286675	−0.143337	−	0.989674i	$-0.545783\pi$
$313$	−5.07179	−0.286675	−0.143337	+	0.989674i	$0.545783\pi$
$314$	33.1196	1.86905
$315$	0	0
$316$	2.00096	0.112563
$317$	−22.4428	−1.26051	−0.630257	−	0.776387i	$-0.717050\pi$
$317$	−22.4428	−1.26051	−0.630257	+	0.776387i	$0.717050\pi$
$318$	0	0
$319$	−10.2484	−0.573802
$320$	29.9971	1.67689
$321$	0	0
$322$	10.2704	0.572348
$323$	4.38151	0.243794
$324$	0	0
$325$	−73.8797	−4.09811
$326$	7.82335	0.433295
$327$	0	0
$328$	6.70895	0.370440
$329$	−9.34941	−0.515450
$330$	0	0
$331$	9.16225	0.503603	0.251801	−	0.967779i	$-0.418977\pi$
$331$	9.16225	0.503603	0.251801	+	0.967779i	$0.418977\pi$
$332$	1.20875	0.0663388
$333$	0	0
$334$	3.00000	0.164153
$335$	7.75876	0.423906
$336$	0	0
$337$	21.0220	1.14514	0.572570	−	0.819856i	$-0.305946\pi$
$337$	21.0220	1.14514	0.572570	+	0.819856i	$0.305946\pi$
$338$	−41.9722	−2.28299
$339$	0	0
$340$	−2.36381	−0.128195
$341$	−7.32743	−0.396803
$342$	0	0
$343$	16.9080	0.912944
$344$	−5.81616	−0.313586
$345$	0	0
$346$	−27.9866	−1.50457
$347$	−1.90662	−0.102352	−0.0511762	−	0.998690i	$-0.516297\pi$
$347$	−1.90662	−0.102352	−0.0511762	+	0.998690i	$0.516297\pi$
$348$	0	0
$349$	−20.0862	−1.07519	−0.537594	−	0.843204i	$-0.680667\pi$
$349$	−20.0862	−1.07519	−0.537594	+	0.843204i	$0.680667\pi$
$350$	23.4897	1.25558
$351$	0	0
$352$	−0.751560	−0.0400583
$353$	10.3858	0.552780	0.276390	−	0.961046i	$-0.410862\pi$
$353$	10.3858	0.552780	0.276390	+	0.961046i	$0.410862\pi$
$354$	0	0
$355$	38.3537	2.03560
$356$	0.975094	0.0516799
$357$	0	0
$358$	28.7745	1.52078
$359$	22.9354	1.21048	0.605242	−	0.796041i	$-0.293076\pi$
$359$	22.9354	1.21048	0.605242	+	0.796041i	$0.293076\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.5146	0.973105
$363$	0	0
$364$	1.20914	0.0633761
$365$	18.8420	0.986236
$366$	0	0
$367$	14.6624	0.765374	0.382687	−	0.923878i	$-0.374999\pi$
$367$	14.6624	0.765374	0.382687	+	0.923878i	$0.374999\pi$
$368$	−21.2422	−1.10733
$369$	0	0
$370$	−31.8214	−1.65432
$371$	12.8860	0.669007
$372$	0	0
$373$	−24.3786	−1.26228	−0.631138	−	0.775671i	$-0.717412\pi$
$373$	−24.3786	−1.26228	−0.631138	+	0.775671i	$0.717412\pi$
$374$	−6.39922	−0.330896
$375$	0	0
$376$	18.1259	0.934771
$377$	−66.2101	−3.40999
$378$	0	0
$379$	0.0861875	0.00442715	0.00221358	−	0.999998i	$-0.499295\pi$
$379$	0.0861875	0.00442715	0.00221358	+	0.999998i	$0.499295\pi$
$380$	−0.539495	−0.0276755
$381$	0	0
$382$	−3.74532	−0.191627
$383$	−26.6840	−1.36349	−0.681745	−	0.731590i	$-0.738779\pi$
$383$	−26.6840	−1.36349	−0.681745	+	0.731590i	$0.738779\pi$
$384$	0	0
$385$	5.70175	0.290588
$386$	−16.9689	−0.863692
$387$	0	0
$388$	1.11205	0.0564559
$389$	−10.3317	−0.523838	−0.261919	−	0.965090i	$-0.584355\pi$
$389$	−10.3317	−0.523838	−0.261919	+	0.965090i	$0.584355\pi$
$390$	0	0
$391$	−21.9076	−1.10791
$392$	−13.6932	−0.691611
$393$	0	0
$394$	18.0938	0.911551
$395$	60.9587	3.06717
$396$	0	0
$397$	21.5801	1.08308	0.541538	−	0.840676i	$-0.317842\pi$
$397$	21.5801	1.08308	0.541538	+	0.840676i	$0.317842\pi$
$398$	−12.6768	−0.635433
$399$	0	0
$400$	−48.5835	−2.42917
$401$	22.8597	1.14156	0.570780	−	0.821103i	$-0.306641\pi$
$401$	22.8597	1.14156	0.570780	+	0.821103i	$0.306641\pi$
$402$	0	0
$403$	−47.3389	−2.35812
$404$	−0.539495	−0.0268409
$405$	0	0
$406$	21.0512	1.04475
$407$	−5.37432	−0.266395
$408$	0	0
$409$	−36.9794	−1.82851	−0.914256	−	0.405137i	$-0.867224\pi$
$409$	−36.9794	−1.82851	−0.914256	+	0.405137i	$0.867224\pi$
$410$	−14.5687	−0.719495
$411$	0	0
$412$	−1.74009	−0.0857281
$413$	−2.50447	−0.123237
$414$	0	0
$415$	36.8243	1.80763
$416$	−4.85546	−0.238058
$417$	0	0
$418$	−1.46050	−0.0714356
$419$	−24.5586	−1.19976	−0.599882	−	0.800089i	$-0.704786\pi$
$419$	−24.5586	−1.19976	−0.599882	+	0.800089i	$0.704786\pi$
$420$	0	0
$421$	31.4399	1.53229	0.766143	−	0.642670i	$-0.222174\pi$
$421$	31.4399	1.53229	0.766143	+	0.642670i	$0.222174\pi$
$422$	−21.2307	−1.03350
$423$	0	0
$424$	−24.9823	−1.21325
$425$	−50.1052	−2.43046
$426$	0	0
$427$	−1.37139	−0.0663663
$428$	1.79845	0.0869312
$429$	0	0
$430$	12.6300	0.609071
$431$	14.5045	0.698656	0.349328	−	0.937001i	$-0.386410\pi$
$431$	14.5045	0.698656	0.349328	+	0.937001i	$0.386410\pi$
$432$	0	0
$433$	−28.9253	−1.39006	−0.695030	−	0.718981i	$-0.744609\pi$
$433$	−28.9253	−1.39006	−0.695030	+	0.718981i	$0.744609\pi$
$434$	15.0512	0.722479
$435$	0	0
$436$	−1.16849	−0.0559606
$437$	−5.00000	−0.239182
$438$	0	0
$439$	12.8492	0.613260	0.306630	−	0.951829i	$-0.400799\pi$
$439$	12.8492	0.613260	0.306630	+	0.951829i	$0.400799\pi$
$440$	−11.0541	−0.526983
$441$	0	0
$442$	−41.3422	−1.96645
$443$	35.1301	1.66908	0.834542	−	0.550945i	$-0.185732\pi$
$443$	35.1301	1.66908	0.834542	+	0.550945i	$0.185732\pi$
$444$	0	0
$445$	29.7060	1.40820
$446$	−10.7702	−0.509985
$447$	0	0
$448$	−10.4064	−0.491657
$449$	33.0584	1.56012	0.780060	−	0.625705i	$-0.215188\pi$
$449$	33.0584	1.56012	0.780060	+	0.625705i	$0.215188\pi$
$450$	0	0
$451$	−2.46050	−0.115861
$452$	1.24417	0.0585207
$453$	0	0
$454$	−15.7453	−0.738965
$455$	36.8362	1.72691
$456$	0	0
$457$	−31.8932	−1.49190	−0.745950	−	0.666002i	$-0.768004\pi$
$457$	−31.8932	−1.49190	−0.745950	+	0.666002i	$0.768004\pi$
$458$	−7.87024	−0.367752
$459$	0	0
$460$	2.69748	0.125770
$461$	−13.4926	−0.628413	−0.314207	−	0.949355i	$-0.601738\pi$
$461$	−13.4926	−0.628413	−0.314207	+	0.949355i	$0.601738\pi$
$462$	0	0
$463$	10.8525	0.504360	0.252180	−	0.967680i	$-0.418853\pi$
$463$	10.8525	0.504360	0.252180	+	0.967680i	$0.418853\pi$
$464$	−43.5399	−2.02129
$465$	0	0
$466$	5.07995	0.235324
$467$	14.1872	0.656503	0.328252	−	0.944590i	$-0.393541\pi$
$467$	14.1872	0.656503	0.328252	+	0.944590i	$0.393541\pi$
$468$	0	0
$469$	−2.69163	−0.124288
$470$	−39.3609	−1.81558
$471$	0	0
$472$	4.85546	0.223490
$473$	2.13307	0.0980789
$474$	0	0
$475$	−11.4356	−0.524701
$476$	0.820039	0.0375864
$477$	0	0
$478$	14.6008	0.667824
$479$	6.55428	0.299473	0.149736	−	0.988726i	$-0.452158\pi$
$479$	6.55428	0.299473	0.149736	+	0.988726i	$0.452158\pi$
$480$	0	0
$481$	−34.7208	−1.58313
$482$	10.5969	0.482675
$483$	0	0
$484$	0.133074	0.00604884
$485$	33.8784	1.53834
$486$	0	0
$487$	39.0833	1.77103	0.885516	−	0.464609i	$-0.153805\pi$
$487$	39.0833	1.77103	0.885516	+	0.464609i	$0.153805\pi$
$488$	2.65874	0.120356
$489$	0	0
$490$	29.7352	1.34330
$491$	16.2163	0.731833	0.365917	−	0.930648i	$-0.380756\pi$
$491$	16.2163	0.731833	0.365917	+	0.930648i	$0.380756\pi$
$492$	0	0
$493$	−44.9037	−2.02236
$494$	−9.43560	−0.424528
$495$	0	0
$496$	−31.1301	−1.39778
$497$	−13.3054	−0.596831
$498$	0	0
$499$	30.9866	1.38715	0.693575	−	0.720385i	$-0.256035\pi$
$499$	30.9866	1.38715	0.693575	+	0.720385i	$0.256035\pi$
$500$	3.47197	0.155271
$501$	0	0
$502$	6.05797	0.270380
$503$	−2.57918	−0.115000	−0.0575001	−	0.998346i	$-0.518313\pi$
$503$	−2.57918	−0.115000	−0.0575001	+	0.998346i	$0.518313\pi$
$504$	0	0
$505$	−16.4356	−0.731375
$506$	7.30252	0.324637
$507$	0	0
$508$	−1.66926	−0.0740612
$509$	−20.3959	−0.904033	−0.452016	−	0.892010i	$-0.649295\pi$
$509$	−20.3959	−0.904033	−0.452016	+	0.892010i	$0.649295\pi$
$510$	0	0
$511$	−6.53657	−0.289161
$512$	19.9751	0.882783
$513$	0	0
$514$	−29.1694	−1.28661
$515$	−53.0115	−2.33596
$516$	0	0
$517$	−6.64766	−0.292364
$518$	11.0393	0.485039
$519$	0	0
$520$	−71.4150	−3.13175
$521$	−31.8607	−1.39584	−0.697921	−	0.716175i	$-0.745891\pi$
$521$	−31.8607	−1.39584	−0.697921	+	0.716175i	$0.745891\pi$
$522$	0	0
$523$	5.01478	0.219281	0.109641	−	0.993971i	$-0.465030\pi$
$523$	5.01478	0.219281	0.109641	+	0.993971i	$0.465030\pi$
$524$	2.66926	0.116607
$525$	0	0
$526$	−7.85349	−0.342428
$527$	−32.1052	−1.39853
$528$	0	0
$529$	2.00000	0.0869565
$530$	54.2498	2.35646
$531$	0	0
$532$	0.187159	0.00811436
$533$	−15.8961	−0.688537
$534$	0	0
$535$	54.7893	2.36875
$536$	5.21830	0.225396
$537$	0	0
$538$	5.47490	0.236040
$539$	5.02198	0.216312
$540$	0	0
$541$	5.80857	0.249730	0.124865	−	0.992174i	$-0.460150\pi$
$541$	5.80857	0.249730	0.124865	+	0.992174i	$0.460150\pi$
$542$	−1.47102	−0.0631856
$543$	0	0
$544$	−3.29297	−0.141185
$545$	−35.5979	−1.52484
$546$	0	0
$547$	−31.1019	−1.32982	−0.664911	−	0.746922i	$-0.731531\pi$
$547$	−31.1019	−1.32982	−0.664911	+	0.746922i	$0.731531\pi$
$548$	−0.844945	−0.0360942
$549$	0	0
$550$	16.7017	0.712165
$551$	−10.2484	−0.436598
$552$	0	0
$553$	−21.1475	−0.899282
$554$	17.0478	0.724294
$555$	0	0
$556$	−1.69748	−0.0719890
$557$	8.32743	0.352845	0.176422	−	0.984315i	$-0.443548\pi$
$557$	8.32743	0.352845	0.176422	+	0.984315i	$0.443548\pi$
$558$	0	0
$559$	13.7807	0.582863
$560$	24.2235	1.02363
$561$	0	0
$562$	−5.53230	−0.233366
$563$	−28.9502	−1.22010	−0.610052	−	0.792361i	$-0.708852\pi$
$563$	−28.9502	−1.22010	−0.610052	+	0.792361i	$0.708852\pi$
$564$	0	0
$565$	37.9033	1.59460
$566$	12.7195	0.534639
$567$	0	0
$568$	25.7955	1.08236
$569$	21.6693	0.908422	0.454211	−	0.890894i	$-0.349921\pi$
$569$	21.6693	0.908422	0.454211	+	0.890894i	$0.349921\pi$
$570$	0	0
$571$	10.8391	0.453602	0.226801	−	0.973941i	$-0.427173\pi$
$571$	10.8391	0.453602	0.226801	+	0.973941i	$0.427173\pi$
$572$	0.859728	0.0359470
$573$	0	0
$574$	5.05408	0.210953
$575$	57.1780	2.38449
$576$	0	0
$577$	−28.9282	−1.20430	−0.602149	−	0.798384i	$-0.705688\pi$
$577$	−28.9282	−1.20430	−0.602149	+	0.798384i	$0.705688\pi$
$578$	−3.20971	−0.133506
$579$	0	0
$580$	5.52898	0.229579
$581$	−12.7749	−0.529992
$582$	0	0
$583$	9.16225	0.379462
$584$	12.6726	0.524395
$585$	0	0
$586$	−3.39298	−0.140163
$587$	−38.3360	−1.58230	−0.791148	−	0.611625i	$-0.790516\pi$
$587$	−38.3360	−1.58230	−0.791148	+	0.611625i	$0.790516\pi$
$588$	0	0
$589$	−7.32743	−0.301922
$590$	−10.5438	−0.434080
$591$	0	0
$592$	−22.8325	−0.938409
$593$	10.8918	0.447274	0.223637	−	0.974673i	$-0.428207\pi$
$593$	10.8918	0.447274	0.223637	+	0.974673i	$0.428207\pi$
$594$	0	0
$595$	24.9823	1.02417
$596$	−1.65818	−0.0679215
$597$	0	0
$598$	47.1780	1.92925
$599$	−27.5831	−1.12701	−0.563507	−	0.826111i	$-0.690548\pi$
$599$	−27.5831	−1.12701	−0.563507	+	0.826111i	$0.690548\pi$
$600$	0	0
$601$	23.6663	0.965370	0.482685	−	0.875794i	$-0.339662\pi$
$601$	23.6663	0.965370	0.482685	+	0.875794i	$0.339662\pi$
$602$	−4.38151	−0.178577
$603$	0	0
$604$	2.47726	0.100798
$605$	4.05408	0.164822
$606$	0	0
$607$	−30.0364	−1.21914	−0.609569	−	0.792733i	$-0.708658\pi$
$607$	−30.0364	−1.21914	−0.609569	+	0.792733i	$0.708658\pi$
$608$	−0.751560	−0.0304798
$609$	0	0
$610$	−5.77354	−0.233764
$611$	−42.9473	−1.73746
$612$	0	0
$613$	21.6477	0.874341	0.437170	−	0.899379i	$-0.355981\pi$
$613$	21.6477	0.874341	0.437170	+	0.899379i	$0.355981\pi$
$614$	−2.72665	−0.110039
$615$	0	0
$616$	3.83482	0.154509
$617$	−43.3402	−1.74481	−0.872406	−	0.488781i	$-0.837442\pi$
$617$	−43.3402	−1.74481	−0.872406	+	0.488781i	$0.837442\pi$
$618$	0	0
$619$	19.9387	0.801405	0.400702	−	0.916208i	$-0.368766\pi$
$619$	19.9387	0.801405	0.400702	+	0.916208i	$0.368766\pi$
$620$	3.95311	0.158761
$621$	0	0
$622$	11.7161	0.469775
$623$	−10.3054	−0.412879
$624$	0	0
$625$	48.5949	1.94380
$626$	7.40738	0.296058
$627$	0	0
$628$	−3.01771	−0.120420
$629$	−23.5477	−0.938906
$630$	0	0
$631$	−17.8712	−0.711441	−0.355721	−	0.934592i	$-0.615765\pi$
$631$	−17.8712	−0.711441	−0.355721	+	0.934592i	$0.615765\pi$
$632$	40.9990	1.63085
$633$	0	0
$634$	32.7778	1.30177
$635$	−50.8535	−2.01806
$636$	0	0
$637$	32.4445	1.28550
$638$	14.9679	0.592585
$639$	0	0
$640$	−49.9046	−1.97265
$641$	−22.6372	−0.894114	−0.447057	−	0.894506i	$-0.647528\pi$
$641$	−22.6372	−0.894114	−0.447057	+	0.894506i	$0.647528\pi$
$642$	0	0
$643$	−6.80272	−0.268273	−0.134137	−	0.990963i	$-0.542826\pi$
$643$	−6.80272	−0.268273	−0.134137	+	0.990963i	$0.542826\pi$
$644$	−0.935793	−0.0368754
$645$	0	0
$646$	−6.39922	−0.251774
$647$	−38.0010	−1.49397	−0.746986	−	0.664840i	$-0.768500\pi$
$647$	−38.0010	−1.49397	−0.746986	+	0.664840i	$0.768500\pi$
$648$	0	0
$649$	−1.78074	−0.0699001
$650$	107.902	4.23225
$651$	0	0
$652$	−0.712828	−0.0279165
$653$	27.5395	1.07770	0.538852	−	0.842401i	$-0.318858\pi$
$653$	27.5395	1.07770	0.538852	+	0.842401i	$0.318858\pi$
$654$	0	0
$655$	81.3183	3.17737
$656$	−10.4533	−0.408133
$657$	0	0
$658$	13.6549	0.532322
$659$	23.9473	0.932853	0.466426	−	0.884560i	$-0.345541\pi$
$659$	23.9473	0.932853	0.466426	+	0.884560i	$0.345541\pi$
$660$	0	0
$661$	16.3202	0.634783	0.317392	−	0.948295i	$-0.397193\pi$
$661$	16.3202	0.634783	0.317392	+	0.948295i	$0.397193\pi$
$662$	−13.3815	−0.520087
$663$	0	0
$664$	24.7669	0.961143
$665$	5.70175	0.221104
$666$	0	0
$667$	51.2422	1.98411
$668$	−0.273346	−0.0105761
$669$	0	0
$670$	−11.3317	−0.437782
$671$	−0.975094	−0.0376431
$672$	0	0
$673$	−11.8306	−0.456034	−0.228017	−	0.973657i	$-0.573224\pi$
$673$	−11.8306	−0.456034	−0.228017	+	0.973657i	$0.573224\pi$
$674$	−30.7027	−1.18262
$675$	0	0
$676$	3.82431	0.147089
$677$	13.6874	0.526048	0.263024	−	0.964789i	$-0.415280\pi$
$677$	13.6874	0.526048	0.263024	+	0.964789i	$0.415280\pi$
$678$	0	0
$679$	−11.7529	−0.451035
$680$	−48.4336	−1.85734
$681$	0	0
$682$	10.7017	0.409791
$683$	18.5582	0.710108	0.355054	−	0.934846i	$-0.384462\pi$
$683$	18.5582	0.710108	0.355054	+	0.934846i	$0.384462\pi$
$684$	0	0
$685$	−25.7410	−0.983515
$686$	−24.6942	−0.942827
$687$	0	0
$688$	9.06224	0.345495
$689$	59.1928	2.25507
$690$	0	0
$691$	1.23990	0.0471679	0.0235839	−	0.999722i	$-0.492492\pi$
$691$	1.23990	0.0471679	0.0235839	+	0.999722i	$0.492492\pi$
$692$	2.55001	0.0969367
$693$	0	0
$694$	2.78462	0.105703
$695$	−51.7132	−1.96159
$696$	0	0
$697$	−10.7807	−0.408350
$698$	29.3360	1.11038
$699$	0	0
$700$	−2.14027	−0.0808947
$701$	−2.47821	−0.0936008	−0.0468004	−	0.998904i	$-0.514902\pi$
$701$	−2.47821	−0.0936008	−0.0468004	+	0.998904i	$0.514902\pi$
$702$	0	0
$703$	−5.37432	−0.202696
$704$	−7.39922	−0.278869
$705$	0	0
$706$	−15.1685	−0.570874
$707$	5.70175	0.214436
$708$	0	0
$709$	−33.7745	−1.26843	−0.634214	−	0.773158i	$-0.718676\pi$
$709$	−33.7745	−1.26843	−0.634214	+	0.773158i	$0.718676\pi$
$710$	−56.0157	−2.10223
$711$	0	0
$712$	19.9794	0.748758
$713$	36.6372	1.37207
$714$	0	0
$715$	26.1914	0.979504
$716$	−2.62180	−0.0979813
$717$	0	0
$718$	−33.4973	−1.25011
$719$	−16.7601	−0.625046	−0.312523	−	0.949910i	$-0.601174\pi$
$719$	−16.7601	−0.625046	−0.312523	+	0.949910i	$0.601174\pi$
$720$	0	0
$721$	18.3904	0.684896
$722$	−1.46050	−0.0543544
$723$	0	0
$724$	−1.68696	−0.0626955
$725$	117.197	4.35259
$726$	0	0
$727$	24.5586	0.910826	0.455413	−	0.890280i	$-0.349492\pi$
$727$	24.5586	0.910826	0.455413	+	0.890280i	$0.349492\pi$
$728$	24.7749	0.918218
$729$	0	0
$730$	−27.5189	−1.01852
$731$	9.34610	0.345678
$732$	0	0
$733$	−24.8056	−0.916217	−0.458109	−	0.888896i	$-0.651473\pi$
$733$	−24.8056	−0.916217	−0.458109	+	0.888896i	$0.651473\pi$
$734$	−21.4146	−0.790426
$735$	0	0
$736$	3.75780	0.138514
$737$	−1.91381	−0.0704962
$738$	0	0
$739$	19.4792	0.716553	0.358276	−	0.933616i	$-0.383365\pi$
$739$	19.4792	0.716553	0.358276	+	0.933616i	$0.383365\pi$
$740$	2.89942	0.106585
$741$	0	0
$742$	−18.8200	−0.690905
$743$	−27.3858	−1.00469	−0.502344	−	0.864668i	$-0.667529\pi$
$743$	−27.3858	−1.00469	−0.502344	+	0.864668i	$0.667529\pi$
$744$	0	0
$745$	−50.5159	−1.85076
$746$	35.6050	1.30359
$747$	0	0
$748$	0.583068	0.0213191
$749$	−19.0072	−0.694508
$750$	0	0
$751$	−25.2924	−0.922933	−0.461466	−	0.887158i	$-0.652677\pi$
$751$	−25.2924	−0.922933	−0.461466	+	0.887158i	$0.652677\pi$
$752$	−28.2422	−1.02989
$753$	0	0
$754$	96.7002	3.52161
$755$	75.4690	2.74660
$756$	0	0
$757$	−41.9646	−1.52523	−0.762614	−	0.646853i	$-0.776085\pi$
$757$	−41.9646	−1.52523	−0.762614	+	0.646853i	$0.776085\pi$
$758$	−0.125877	−0.00457207
$759$	0	0
$760$	−11.0541	−0.400974
$761$	33.9866	1.23201	0.616006	−	0.787741i	$-0.288750\pi$
$761$	33.9866	1.23201	0.616006	+	0.787741i	$0.288750\pi$
$762$	0	0
$763$	12.3494	0.447079
$764$	0.341256	0.0123462
$765$	0	0
$766$	38.9722	1.40812
$767$	−11.5045	−0.415402
$768$	0	0
$769$	−20.1039	−0.724965	−0.362483	−	0.931991i	$-0.618071\pi$
$769$	−20.1039	−0.724965	−0.362483	+	0.931991i	$0.618071\pi$
$770$	−8.32743	−0.300100
$771$	0	0
$772$	1.54612	0.0556462
$773$	44.3753	1.59607	0.798034	−	0.602613i	$-0.205874\pi$
$773$	44.3753	1.59607	0.798034	+	0.602613i	$0.205874\pi$
$774$	0	0
$775$	83.7936	3.00995
$776$	22.7856	0.817955
$777$	0	0
$778$	15.0895	0.540985
$779$	−2.46050	−0.0881567
$780$	0	0
$781$	−9.46050	−0.338523
$782$	31.9961	1.14418
$783$	0	0
$784$	21.3356	0.761985
$785$	−91.9338	−3.28126
$786$	0	0
$787$	−28.7103	−1.02341	−0.511706	−	0.859161i	$-0.670986\pi$
$787$	−28.7103	−1.02341	−0.511706	+	0.859161i	$0.670986\pi$
$788$	−1.64862	−0.0587297
$789$	0	0
$790$	−89.0305	−3.16756
$791$	−13.1492	−0.467532
$792$	0	0
$793$	−6.29960	−0.223705
$794$	−31.5179	−1.11853
$795$	0	0
$796$	1.15506	0.0409399
$797$	9.82004	0.347844	0.173922	−	0.984759i	$-0.444356\pi$
$797$	9.82004	0.347844	0.173922	+	0.984759i	$0.444356\pi$
$798$	0	0
$799$	−29.1268	−1.03043
$800$	8.59454	0.303863
$801$	0	0
$802$	−33.3867	−1.17893
$803$	−4.64766	−0.164012
$804$	0	0
$805$	−28.5087	−1.00480
$806$	69.1387	2.43531
$807$	0	0
$808$	−11.0541	−0.388881
$809$	−55.9368	−1.96663	−0.983316	−	0.181907i	$-0.941773\pi$
$809$	−55.9368	−1.96663	−0.983316	+	0.181907i	$0.941773\pi$
$810$	0	0
$811$	4.70175	0.165101	0.0825503	−	0.996587i	$-0.473693\pi$
$811$	4.70175	0.165101	0.0825503	+	0.996587i	$0.473693\pi$
$812$	−1.91808	−0.0673116
$813$	0	0
$814$	7.84922	0.275115
$815$	−21.7161	−0.760683
$816$	0	0
$817$	2.13307	0.0746268
$818$	54.0085	1.88836
$819$	0	0
$820$	1.32743	0.0463559
$821$	−14.3274	−0.500031	−0.250015	−	0.968242i	$-0.580436\pi$
$821$	−14.3274	−0.500031	−0.250015	+	0.968242i	$0.580436\pi$
$822$	0	0
$823$	−5.83482	−0.203389	−0.101695	−	0.994816i	$-0.532426\pi$
$823$	−5.83482	−0.203389	−0.101695	+	0.994816i	$0.532426\pi$
$824$	−35.6539	−1.24206
$825$	0	0
$826$	3.65779	0.127271
$827$	−36.8784	−1.28239	−0.641194	−	0.767379i	$-0.721560\pi$
$827$	−36.8784	−1.28239	−0.641194	+	0.767379i	$0.721560\pi$
$828$	0	0
$829$	−8.00720	−0.278101	−0.139051	−	0.990285i	$-0.544405\pi$
$829$	−8.00720	−0.278101	−0.139051	+	0.990285i	$0.544405\pi$
$830$	−53.7821	−1.86680
$831$	0	0
$832$	−47.8027	−1.65726
$833$	22.0039	0.762389
$834$	0	0
$835$	−8.32743	−0.288183
$836$	0.133074	0.00460248
$837$	0	0
$838$	35.8679	1.23904
$839$	20.0029	0.690578	0.345289	−	0.938496i	$-0.387781\pi$
$839$	20.0029	0.690578	0.345289	+	0.938496i	$0.387781\pi$
$840$	0	0
$841$	76.0305	2.62174
$842$	−45.9181	−1.58244
$843$	0	0
$844$	1.93445	0.0665864
$845$	116.507	4.00795
$846$	0	0
$847$	−1.40642	−0.0483252
$848$	38.9253	1.33670
$849$	0	0
$850$	73.1790	2.51002
$851$	26.8716	0.921146
$852$	0	0
$853$	−38.5083	−1.31850	−0.659250	−	0.751923i	$-0.729126\pi$
$853$	−38.5083	−1.31850	−0.659250	+	0.751923i	$0.729126\pi$
$854$	2.00293	0.0685387
$855$	0	0
$856$	36.8496	1.25949
$857$	−6.19863	−0.211741	−0.105871	−	0.994380i	$-0.533763\pi$
$857$	−6.19863	−0.211741	−0.105871	+	0.994380i	$0.533763\pi$
$858$	0	0
$859$	37.6883	1.28591	0.642954	−	0.765905i	$-0.277709\pi$
$859$	37.6883	1.28591	0.642954	+	0.765905i	$0.277709\pi$
$860$	−1.15078	−0.0392414
$861$	0	0
$862$	−21.1838	−0.721525
$863$	−48.0085	−1.63423	−0.817115	−	0.576475i	$-0.804428\pi$
$863$	−48.0085	−1.63423	−0.817115	+	0.576475i	$0.804428\pi$
$864$	0	0
$865$	77.6854	2.64138
$866$	42.2455	1.43556
$867$	0	0
$868$	−1.37139	−0.0465481
$869$	−15.0364	−0.510074
$870$	0	0
$871$	−12.3642	−0.418945
$872$	−23.9420	−0.810780
$873$	0	0
$874$	7.30252	0.247012
$875$	−36.6942	−1.24049
$876$	0	0
$877$	35.1838	1.18807	0.594037	−	0.804438i	$-0.297533\pi$
$877$	35.1838	1.18807	0.594037	+	0.804438i	$0.297533\pi$
$878$	−18.7663	−0.633333
$879$	0	0
$880$	17.2235	0.580605
$881$	−24.6621	−0.830886	−0.415443	−	0.909619i	$-0.636373\pi$
$881$	−24.6621	−0.830886	−0.415443	+	0.909619i	$0.636373\pi$
$882$	0	0
$883$	−19.2193	−0.646780	−0.323390	−	0.946266i	$-0.604823\pi$
$883$	−19.2193	−0.646780	−0.323390	+	0.946266i	$0.604823\pi$
$884$	3.76691	0.126695
$885$	0	0
$886$	−51.3078	−1.72372
$887$	22.2498	0.747075	0.373537	−	0.927615i	$-0.378145\pi$
$887$	22.2498	0.747075	0.373537	+	0.927615i	$0.378145\pi$
$888$	0	0
$889$	17.6418	0.591687
$890$	−43.3858	−1.45429
$891$	0	0
$892$	0.981333	0.0328575
$893$	−6.64766	−0.222456
$894$	0	0
$895$	−79.8725	−2.66984
$896$	17.3126	0.578375
$897$	0	0
$898$	−48.2819	−1.61119
$899$	75.0947	2.50455
$900$	0	0
$901$	40.1445	1.33741
$902$	3.59358	0.119653
$903$	0	0
$904$	25.4926	0.847872
$905$	−51.3930	−1.70836
$906$	0	0
$907$	−38.5480	−1.27997	−0.639983	−	0.768389i	$-0.721059\pi$
$907$	−38.5480	−1.27997	−0.639983	+	0.768389i	$0.721059\pi$
$908$	1.43464	0.0476102
$909$	0	0
$910$	−53.7994	−1.78343
$911$	−4.28093	−0.141834	−0.0709168	−	0.997482i	$-0.522592\pi$
$911$	−4.28093	−0.141834	−0.0709168	+	0.997482i	$0.522592\pi$
$912$	0	0
$913$	−9.08326	−0.300612
$914$	46.5801	1.54073
$915$	0	0
$916$	0.717100	0.0236937
$917$	−28.2105	−0.931592
$918$	0	0
$919$	−7.52313	−0.248165	−0.124083	−	0.992272i	$-0.539599\pi$
$919$	−7.52313	−0.248165	−0.124083	+	0.992272i	$0.539599\pi$
$920$	55.2704	1.82221
$921$	0	0
$922$	19.7060	0.648983
$923$	−61.1196	−2.01178
$924$	0	0
$925$	61.4585	2.02074
$926$	−15.8502	−0.520869
$927$	0	0
$928$	7.70232	0.252841
$929$	−11.0833	−0.363630	−0.181815	−	0.983333i	$-0.558197\pi$
$929$	−11.0833	−0.363630	−0.181815	+	0.983333i	$0.558197\pi$
$930$	0	0
$931$	5.02198	0.164589
$932$	−0.462861	−0.0151615
$933$	0	0
$934$	−20.7204	−0.677993
$935$	17.7630	0.580913
$936$	0	0
$937$	−34.1373	−1.11522	−0.557609	−	0.830104i	$-0.688281\pi$
$937$	−34.1373	−1.11522	−0.557609	+	0.830104i	$0.688281\pi$
$938$	3.93113	0.128356
$939$	0	0
$940$	3.58638	0.116975
$941$	7.02491	0.229005	0.114503	−	0.993423i	$-0.463473\pi$
$941$	7.02491	0.229005	0.114503	+	0.993423i	$0.463473\pi$
$942$	0	0
$943$	12.3025	0.400625
$944$	−7.56536	−0.246231
$945$	0	0
$946$	−3.11537	−0.101289
$947$	−8.37139	−0.272034	−0.136017	−	0.990707i	$-0.543430\pi$
$947$	−8.37139	−0.272034	−0.136017	+	0.990707i	$0.543430\pi$
$948$	0	0
$949$	−30.0263	−0.974693
$950$	16.7017	0.541876
$951$	0	0
$952$	16.8023	0.544567
$953$	51.1632	1.65734	0.828669	−	0.559738i	$-0.189098\pi$
$953$	51.1632	1.65734	0.828669	+	0.559738i	$0.189098\pi$
$954$	0	0
$955$	10.3963	0.336416
$956$	−1.33036	−0.0430268
$957$	0	0
$958$	−9.57256	−0.309275
$959$	8.92994	0.288363
$960$	0	0
$961$	22.6912	0.731975
$962$	50.7099	1.63495
$963$	0	0
$964$	−0.965540	−0.0310980
$965$	47.1023	1.51628
$966$	0	0
$967$	−39.1990	−1.26056	−0.630278	−	0.776370i	$-0.717059\pi$
$967$	−39.1990	−1.26056	−0.630278	+	0.776370i	$0.717059\pi$
$968$	2.72665	0.0876380
$969$	0	0
$970$	−49.4796	−1.58869
$971$	11.4385	0.367080	0.183540	−	0.983012i	$-0.441244\pi$
$971$	11.4385	0.367080	0.183540	+	0.983012i	$0.441244\pi$
$972$	0	0
$973$	17.9401	0.575132
$974$	−57.0813	−1.82900
$975$	0	0
$976$	−4.14263	−0.132602
$977$	41.2891	1.32095	0.660477	−	0.750846i	$-0.270354\pi$
$977$	41.2891	1.32095	0.660477	+	0.750846i	$0.270354\pi$
$978$	0	0
$979$	−7.32743	−0.234186
$980$	−2.70933	−0.0865465
$981$	0	0
$982$	−23.6840	−0.755788
$983$	28.6735	0.914543	0.457272	−	0.889327i	$-0.348827\pi$
$983$	28.6735	0.914543	0.457272	+	0.889327i	$0.348827\pi$
$984$	0	0
$985$	−50.2249	−1.60030
$986$	65.5821	2.08856
$987$	0	0
$988$	0.859728	0.0273516
$989$	−10.6654	−0.339139
$990$	0	0
$991$	−9.06848	−0.288070	−0.144035	−	0.989573i	$-0.546008\pi$
$991$	−9.06848	−0.288070	−0.144035	+	0.989573i	$0.546008\pi$
$992$	5.50700	0.174848
$993$	0	0
$994$	19.4327	0.616367
$995$	35.1885	1.11555
$996$	0	0
$997$	4.30252	0.136262	0.0681312	−	0.997676i	$-0.478296\pi$
$997$	4.30252	0.136262	0.0681312	+	0.997676i	$0.478296\pi$
$998$	−45.2560	−1.43255
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.i.1.1		3
3.2	odd	2		627.2.a.e.1.3	✓	3
33.32	even	2		6897.2.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.a.e.1.3	✓	3	3.2	odd	2
1881.2.a.i.1.1		3	1.1	even	1	trivial
6897.2.a.p.1.1		3	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46050 q^{2} +0.133074 q^{4} +4.05408 q^{5} -1.40642 q^{7} +2.72665 q^{8} -5.92101 q^{10} -1.00000 q^{11} -6.46050 q^{13} +2.05408 q^{14} -4.24844 q^{16} -4.38151 q^{17} -1.00000 q^{19} +0.539495 q^{20} +1.46050 q^{22} +5.00000 q^{23} +11.4356 q^{25} +9.43560 q^{26} -0.187159 q^{28} +10.2484 q^{29} +7.32743 q^{31} +0.751560 q^{32} +6.39922 q^{34} -5.70175 q^{35} +5.37432 q^{37} +1.46050 q^{38} +11.0541 q^{40} +2.46050 q^{41} -2.13307 q^{43} -0.133074 q^{44} -7.30252 q^{46} +6.64766 q^{47} -5.02198 q^{49} -16.7017 q^{50} -0.859728 q^{52} -9.16225 q^{53} -4.05408 q^{55} -3.83482 q^{56} -14.9679 q^{58} +1.78074 q^{59} +0.975094 q^{61} -10.7017 q^{62} +7.39922 q^{64} -26.1914 q^{65} +1.91381 q^{67} -0.583068 q^{68} +8.32743 q^{70} +9.46050 q^{71} +4.64766 q^{73} -7.84922 q^{74} -0.133074 q^{76} +1.40642 q^{77} +15.0364 q^{79} -17.2235 q^{80} -3.59358 q^{82} +9.08326 q^{83} -17.7630 q^{85} +3.11537 q^{86} -2.72665 q^{88} +7.32743 q^{89} +9.08619 q^{91} +0.665372 q^{92} -9.70895 q^{94} -4.05408 q^{95} +8.35661 q^{97} +7.33463 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 7 q^{7} + 9 q^{8} - 5 q^{10} - 3 q^{11} - 13 q^{13} - 3 q^{14} + 10 q^{16} + 6 q^{17} - 3 q^{19} + 8 q^{20} - 2 q^{22} + 15 q^{23} + 6 q^{25} + 5 q^{28} + 8 q^{29} + 12 q^{31}+ \cdots + 4 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 7 * q^7 + 9 * q^8 - 5 * q^10 - 3 * q^11 - 13 * q^13 - 3 * q^14 + 10 * q^16 + 6 * q^17 - 3 * q^19 + 8 * q^20 - 2 * q^22 + 15 * q^23 + 6 * q^25 + 5 * q^28 + 8 * q^29 + 12 * q^31 + 25 * q^32 + 30 * q^34 + 4 * q^35 + 5 * q^37 - 2 * q^38 + 24 * q^40 + q^41 - 10 * q^43 - 4 * q^44 + 10 * q^46 + 8 * q^47 + 8 * q^49 - 29 * q^50 - 7 * q^52 - 3 * q^55 + 6 * q^56 - 31 * q^58 - 3 * q^59 - 19 * q^61 - 11 * q^62 + 33 * q^64 - 20 * q^65 + q^67 + 39 * q^68 + 15 * q^70 + 22 * q^71 + 2 * q^73 + 10 * q^74 - 4 * q^76 + 7 * q^77 + 6 * q^79 - 7 * q^80 - 8 * q^82 - 13 * q^83 - 15 * q^85 - 17 * q^86 - 9 * q^88 + 12 * q^89 + 32 * q^91 + 20 * q^92 - 3 * q^95 - 16 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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