LMFDB - Embedded newform 1881.2.a.i.1.2

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			1.2
Root			$0.239123$ 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)

$f(q)$	$=$	\(q+0.760877 q^{2} -1.42107 q^{4} -1.94282 q^{5} -5.18194 q^{7} -2.60301 q^{8} -1.47825 q^{10} -1.00000 q^{11} -4.23912 q^{13} -3.94282 q^{14} +0.861564 q^{16} +2.28263 q^{17} -1.00000 q^{19} +2.76088 q^{20} -0.760877 q^{22} +5.00000 q^{23} -1.22545 q^{25} -3.22545 q^{26} +7.36389 q^{28} +5.13844 q^{29} +6.66019 q^{31} +5.86156 q^{32} +1.73680 q^{34} +10.0676 q^{35} -9.72777 q^{37} -0.760877 q^{38} +5.05718 q^{40} +0.239123 q^{41} -0.578933 q^{43} +1.42107 q^{44} +3.80438 q^{46} -3.12476 q^{47} +19.8525 q^{49} -0.932417 q^{50} +6.02408 q^{52} +8.82846 q^{53} +1.94282 q^{55} +13.4887 q^{56} +3.90972 q^{58} -9.54583 q^{59} -9.46457 q^{61} +5.06758 q^{62} +2.73680 q^{64} +8.23585 q^{65} -10.9669 q^{67} -3.24377 q^{68} +7.66019 q^{70} +7.23912 q^{71} -5.12476 q^{73} -7.40164 q^{74} +1.42107 q^{76} +5.18194 q^{77} +7.03775 q^{79} -1.67386 q^{80} +0.181943 q^{82} -13.3502 q^{83} -4.43474 q^{85} -0.440497 q^{86} +2.60301 q^{88} +6.66019 q^{89} +21.9669 q^{91} -7.10533 q^{92} -2.37756 q^{94} +1.94282 q^{95} -8.74720 q^{97} +15.1053 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.760877	0.538021	0.269011	−	0.963137i	$-0.413303\pi$
$2$	0.760877	0.538021	0.269011	+	0.963137i	$0.413303\pi$
$3$	0	0
$3$	0	0
$4$	−1.42107	−0.710533
$5$	−1.94282	−0.868856	−0.434428	−	0.900707i	$-0.643049\pi$
$5$	−1.94282	−0.868856	−0.434428	+	0.900707i	$0.643049\pi$
$6$	0	0
$7$	−5.18194	−1.95859	−0.979295	−	0.202437i	$-0.935114\pi$
$7$	−5.18194	−1.95859	−0.979295	+	0.202437i	$0.935114\pi$
$8$	−2.60301	−0.920303
$9$	0	0
$10$	−1.47825	−0.467463
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.23912	−1.17572	−0.587861	−	0.808962i	$-0.700030\pi$
$13$	−4.23912	−1.17572	−0.587861	+	0.808962i	$0.700030\pi$
$14$	−3.94282	−1.05376
$15$	0	0
$16$	0.861564	0.215391
$17$	2.28263	0.553619	0.276810	−	0.960925i	$-0.410723\pi$
$17$	2.28263	0.553619	0.276810	+	0.960925i	$0.410723\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.76088	0.617351
$21$	0	0
$22$	−0.760877	−0.162219
$23$	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
$25$	−1.22545	−0.245090
$26$	−3.22545	−0.632563
$27$	0	0
$28$	7.36389	1.39164
$29$	5.13844	0.954184	0.477092	−	0.878853i	$-0.341691\pi$
$29$	5.13844	0.954184	0.477092	+	0.878853i	$0.341691\pi$
$30$	0	0
$31$	6.66019	1.19621	0.598103	−	0.801419i	$-0.295921\pi$
$31$	6.66019	1.19621	0.598103	+	0.801419i	$0.295921\pi$
$32$	5.86156	1.03619
$33$	0	0
$34$	1.73680	0.297859
$35$	10.0676	1.70173
$36$	0	0
$37$	−9.72777	−1.59924	−0.799618	−	0.600509i	$-0.794965\pi$
$37$	−9.72777	−1.59924	−0.799618	+	0.600509i	$0.794965\pi$
$38$	−0.760877	−0.123431
$39$	0	0
$40$	5.05718	0.799610
$41$	0.239123	0.0373448	0.0186724	−	0.999826i	$-0.494056\pi$
$41$	0.239123	0.0373448	0.0186724	+	0.999826i	$0.494056\pi$
$42$	0	0
$43$	−0.578933	−0.0882865	−0.0441433	−	0.999025i	$-0.514056\pi$
$43$	−0.578933	−0.0882865	−0.0441433	+	0.999025i	$0.514056\pi$
$44$	1.42107	0.214234
$45$	0	0
$46$	3.80438	0.560926
$47$	−3.12476	−0.455794	−0.227897	−	0.973685i	$-0.573185\pi$
$47$	−3.12476	−0.455794	−0.227897	+	0.973685i	$0.573185\pi$
$48$	0	0
$49$	19.8525	2.83608
$50$	−0.932417	−0.131864
$51$	0	0
$52$	6.02408	0.835389
$53$	8.82846	1.21268	0.606341	−	0.795205i	$-0.292637\pi$
$53$	8.82846	1.21268	0.606341	+	0.795205i	$0.292637\pi$
$54$	0	0
$55$	1.94282	0.261970
$56$	13.4887	1.80250
$57$	0	0
$58$	3.90972	0.513371
$59$	−9.54583	−1.24276	−0.621381	−	0.783509i	$-0.713428\pi$
$59$	−9.54583	−1.24276	−0.621381	+	0.783509i	$0.713428\pi$
$60$	0	0
$61$	−9.46457	−1.21181	−0.605907	−	0.795535i	$-0.707190\pi$
$61$	−9.46457	−1.21181	−0.605907	+	0.795535i	$0.707190\pi$
$62$	5.06758	0.643584
$63$	0	0
$64$	2.73680	0.342100
$65$	8.23585	1.02153
$66$	0	0
$67$	−10.9669	−1.33982	−0.669910	−	0.742442i	$-0.733667\pi$
$67$	−10.9669	−1.33982	−0.669910	+	0.742442i	$0.733667\pi$
$68$	−3.24377	−0.393365
$69$	0	0
$70$	7.66019	0.915568
$71$	7.23912	0.859126	0.429563	−	0.903037i	$-0.358668\pi$
$71$	7.23912	0.859126	0.429563	+	0.903037i	$0.358668\pi$
$72$	0	0
$73$	−5.12476	−0.599808	−0.299904	−	0.953969i	$-0.596955\pi$
$73$	−5.12476	−0.599808	−0.299904	+	0.953969i	$0.596955\pi$
$74$	−7.40164	−0.860423
$75$	0	0
$76$	1.42107	0.163008
$77$	5.18194	0.590537
$78$	0	0
$79$	7.03775	0.791809	0.395904	−	0.918292i	$-0.370431\pi$
$79$	7.03775	0.791809	0.395904	+	0.918292i	$0.370431\pi$
$80$	−1.67386	−0.187144
$81$	0	0
$82$	0.181943	0.0200923
$83$	−13.3502	−1.46538	−0.732688	−	0.680565i	$-0.761735\pi$
$83$	−13.3502	−1.46538	−0.732688	+	0.680565i	$0.761735\pi$
$84$	0	0
$85$	−4.43474	−0.481015
$86$	−0.440497	−0.0475000
$87$	0	0
$88$	2.60301	0.277482
$89$	6.66019	0.705979	0.352989	−	0.935627i	$-0.385165\pi$
$89$	6.66019	0.705979	0.352989	+	0.935627i	$0.385165\pi$
$90$	0	0
$91$	21.9669	2.30276
$92$	−7.10533	−0.740782
$93$	0	0
$94$	−2.37756	−0.245227
$95$	1.94282	0.199329
$96$	0	0
$97$	−8.74720	−0.888144	−0.444072	−	0.895991i	$-0.646467\pi$
$97$	−8.74720	−0.888144	−0.444072	+	0.895991i	$0.646467\pi$
$98$	15.1053	1.52587
$99$	0	0
$100$	1.74145	0.174145
$101$	1.94282	0.193318	0.0966589	−	0.995318i	$-0.469184\pi$
$101$	1.94282	0.193318	0.0966589	+	0.995318i	$0.469184\pi$
$102$	0	0
$103$	17.7954	1.75343	0.876714	−	0.481011i	$-0.159730\pi$
$103$	17.7954	1.75343	0.876714	+	0.481011i	$0.159730\pi$
$104$	11.0345	1.08202
$105$	0	0
$106$	6.71737	0.652449
$107$	5.29630	0.512013	0.256006	−	0.966675i	$-0.417593\pi$
$107$	5.29630	0.512013	0.256006	+	0.966675i	$0.417593\pi$
$108$	0	0
$109$	2.54583	0.243846	0.121923	−	0.992540i	$-0.461094\pi$
$109$	2.54583	0.243846	0.121923	+	0.992540i	$0.461094\pi$
$110$	1.47825	0.140945
$111$	0	0
$112$	−4.46457	−0.421863
$113$	−16.1923	−1.52325	−0.761624	−	0.648019i	$-0.775598\pi$
$113$	−16.1923	−1.52325	−0.761624	+	0.648019i	$0.775598\pi$
$114$	0	0
$115$	−9.71410	−0.905845
$116$	−7.30206	−0.677979
$117$	0	0
$118$	−7.26320	−0.668632
$119$	−11.8285	−1.08431
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.20137	−0.651982
$123$	0	0
$124$	−9.46457	−0.849944
$125$	12.0949	1.08180
$126$	0	0
$127$	12.1111	1.07469	0.537343	−	0.843364i	$-0.319428\pi$
$127$	12.1111	1.07469	0.537343	+	0.843364i	$0.319428\pi$
$128$	−9.64076	−0.852131
$129$	0	0
$130$	6.26647	0.549606
$131$	−12.8148	−1.11963	−0.559817	−	0.828617i	$-0.689128\pi$
$131$	−12.8148	−1.11963	−0.559817	+	0.828617i	$0.689128\pi$
$132$	0	0
$133$	5.18194	0.449331
$134$	−8.34446	−0.720851
$135$	0	0
$136$	−5.94171	−0.509497
$137$	19.1923	1.63971	0.819856	−	0.572569i	$-0.194053\pi$
$137$	19.1923	1.63971	0.819856	+	0.572569i	$0.194053\pi$
$138$	0	0
$139$	9.01040	0.764252	0.382126	−	0.924110i	$-0.375192\pi$
$139$	9.01040	0.764252	0.382126	+	0.924110i	$0.375192\pi$
$140$	−14.3067	−1.20914
$141$	0	0
$142$	5.50808	0.462228
$143$	4.23912	0.354493
$144$	0	0
$145$	−9.98306	−0.829048
$146$	−3.89931	−0.322709
$147$	0	0
$148$	13.8238	1.13631
$149$	−10.2391	−0.838822	−0.419411	−	0.907797i	$-0.637763\pi$
$149$	−10.2391	−0.838822	−0.419411	+	0.907797i	$0.637763\pi$
$150$	0	0
$151$	−10.0345	−0.816594	−0.408297	−	0.912849i	$-0.633877\pi$
$151$	−10.0345	−0.816594	−0.408297	+	0.912849i	$0.633877\pi$
$152$	2.60301	0.211132
$153$	0	0
$154$	3.94282	0.317721
$155$	−12.9396	−1.03933
$156$	0	0
$157$	3.53216	0.281897	0.140948	−	0.990017i	$-0.454985\pi$
$157$	3.53216	0.281897	0.140948	+	0.990017i	$0.454985\pi$
$158$	5.35486	0.426010
$159$	0	0
$160$	−11.3880	−0.900298
$161$	−25.9097	−2.04197
$162$	0	0
$163$	11.7472	0.920112	0.460056	−	0.887890i	$-0.347829\pi$
$163$	11.7472	0.920112	0.460056	+	0.887890i	$0.347829\pi$
$164$	−0.339810	−0.0265347
$165$	0	0
$166$	−10.1579	−0.788403
$167$	3.94282	0.305105	0.152552	−	0.988295i	$-0.451251\pi$
$167$	3.94282	0.305105	0.152552	+	0.988295i	$0.451251\pi$
$168$	0	0
$169$	4.97017	0.382320
$170$	−3.37429	−0.258796
$171$	0	0
$172$	0.822703	0.0627305
$173$	1.17154	0.0890705	0.0445353	−	0.999008i	$-0.485819\pi$
$173$	1.17154	0.0890705	0.0445353	+	0.999008i	$0.485819\pi$
$174$	0	0
$175$	6.35021	0.480031
$176$	−0.861564	−0.0649428
$177$	0	0
$178$	5.06758	0.379831
$179$	−3.93242	−0.293923	−0.146961	−	0.989142i	$-0.546949\pi$
$179$	−3.93242	−0.293923	−0.146961	+	0.989142i	$0.546949\pi$
$180$	0	0
$181$	13.5322	1.00584	0.502919	−	0.864334i	$-0.332260\pi$
$181$	13.5322	1.00584	0.502919	+	0.864334i	$0.332260\pi$
$182$	16.7141	1.23893
$183$	0	0
$184$	−13.0150	−0.959482
$185$	18.8993	1.38951
$186$	0	0
$187$	−2.28263	−0.166922
$188$	4.44050	0.323857
$189$	0	0
$190$	1.47825	0.107243
$191$	15.2255	1.10167	0.550837	−	0.834613i	$-0.314308\pi$
$191$	15.2255	1.10167	0.550837	+	0.834613i	$0.314308\pi$
$192$	0	0
$193$	18.2826	1.31601	0.658006	−	0.753012i	$-0.271400\pi$
$193$	18.2826	1.31601	0.658006	+	0.753012i	$0.271400\pi$
$194$	−6.65554	−0.477840
$195$	0	0
$196$	−28.2118	−2.01513
$197$	−14.1625	−1.00904	−0.504519	−	0.863401i	$-0.668330\pi$
$197$	−14.1625	−1.00904	−0.504519	+	0.863401i	$0.668330\pi$
$198$	0	0
$199$	17.7850	1.26074	0.630371	−	0.776294i	$-0.282903\pi$
$199$	17.7850	1.26074	0.630371	+	0.776294i	$0.282903\pi$
$200$	3.18986	0.225557
$201$	0	0
$202$	1.47825	0.104009
$203$	−26.6271	−1.86886
$204$	0	0
$205$	−0.464574	−0.0324472
$206$	13.5401	0.943382
$207$	0	0
$208$	−3.65227	−0.253240
$209$	1.00000	0.0691714
$210$	0	0
$211$	−18.5562	−1.27746	−0.638732	−	0.769429i	$-0.720541\pi$
$211$	−18.5562	−1.27746	−0.638732	+	0.769429i	$0.720541\pi$
$212$	−12.5458	−0.861651
$213$	0	0
$214$	4.02983	0.275474
$215$	1.12476	0.0767082
$216$	0	0
$217$	−34.5127	−2.34288
$218$	1.93706	0.131194
$219$	0	0
$220$	−2.76088	−0.186138
$221$	−9.67635	−0.650902
$222$	0	0
$223$	−7.72777	−0.517490	−0.258745	−	0.965946i	$-0.583309\pi$
$223$	−7.72777	−0.517490	−0.258745	+	0.965946i	$0.583309\pi$
$224$	−30.3743	−2.02947
$225$	0	0
$226$	−12.3204	−0.819539
$227$	−0.545830	−0.0362280	−0.0181140	−	0.999836i	$-0.505766\pi$
$227$	−0.545830	−0.0362280	−0.0181140	+	0.999836i	$0.505766\pi$
$228$	0	0
$229$	7.16251	0.473312	0.236656	−	0.971593i	$-0.423949\pi$
$229$	7.16251	0.473312	0.236656	+	0.971593i	$0.423949\pi$
$230$	−7.39123	−0.487363
$231$	0	0
$232$	−13.3754	−0.878138
$233$	−3.25855	−0.213475	−0.106737	−	0.994287i	$-0.534040\pi$
$233$	−3.25855	−0.213475	−0.106737	+	0.994287i	$0.534040\pi$
$234$	0	0
$235$	6.07085	0.396019
$236$	13.5653	0.883023
$237$	0	0
$238$	−9.00000	−0.583383
$239$	25.3171	1.63763	0.818814	−	0.574059i	$-0.194632\pi$
$239$	25.3171	1.63763	0.818814	+	0.574059i	$0.194632\pi$
$240$	0	0
$241$	−10.5836	−0.681748	−0.340874	−	0.940109i	$-0.610723\pi$
$241$	−10.5836	−0.681748	−0.340874	+	0.940109i	$0.610723\pi$
$242$	0.760877	0.0489110
$243$	0	0
$244$	13.4498	0.861035
$245$	−38.5699	−2.46414
$246$	0	0
$247$	4.23912	0.269729
$248$	−17.3365	−1.10087
$249$	0	0
$250$	9.20275	0.582033
$251$	30.7187	1.93895	0.969475	−	0.245190i	$-0.0788503\pi$
$251$	30.7187	1.93895	0.969475	+	0.245190i	$0.0788503\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	9.21505	0.578203
$255$	0	0
$256$	−12.8090	−0.800564
$257$	−25.7817	−1.60822	−0.804109	−	0.594482i	$-0.797357\pi$
$257$	−25.7817	−1.60822	−0.804109	+	0.594482i	$0.797357\pi$
$258$	0	0
$259$	50.4088	3.13225
$260$	−11.7037	−0.725832
$261$	0	0
$262$	−9.75047	−0.602386
$263$	25.5893	1.57791	0.788953	−	0.614453i	$-0.210623\pi$
$263$	25.5893	1.57791	0.788953	+	0.614453i	$0.210623\pi$
$264$	0	0
$265$	−17.1521	−1.05365
$266$	3.94282	0.241750
$267$	0	0
$268$	15.5847	0.951986
$269$	26.4555	1.61302	0.806512	−	0.591218i	$-0.201353\pi$
$269$	26.4555	1.61302	0.806512	+	0.591218i	$0.201353\pi$
$270$	0	0
$271$	9.44514	0.573752	0.286876	−	0.957968i	$-0.407383\pi$
$271$	9.44514	0.573752	0.286876	+	0.957968i	$0.407383\pi$
$272$	1.96663	0.119245
$273$	0	0
$274$	14.6030	0.882200
$275$	1.22545	0.0738974
$276$	0	0
$277$	−12.3398	−0.741427	−0.370714	−	0.928747i	$-0.620887\pi$
$277$	−12.3398	−0.741427	−0.370714	+	0.928747i	$0.620887\pi$
$278$	6.85581	0.411184
$279$	0	0
$280$	−26.2060	−1.56611
$281$	0.899313	0.0536485	0.0268243	−	0.999640i	$-0.491461\pi$
$281$	0.899313	0.0536485	0.0268243	+	0.999640i	$0.491461\pi$
$282$	0	0
$283$	−1.37756	−0.0818874	−0.0409437	−	0.999161i	$-0.513036\pi$
$283$	−1.37756	−0.0818874	−0.0409437	+	0.999161i	$0.513036\pi$
$284$	−10.2873	−0.610438
$285$	0	0
$286$	3.22545	0.190725
$287$	−1.23912	−0.0731431
$288$	0	0
$289$	−11.7896	−0.693506
$290$	−7.59588	−0.446045
$291$	0	0
$292$	7.28263	0.426184
$293$	28.5322	1.66687	0.833433	−	0.552620i	$-0.186372\pi$
$293$	28.5322	1.66687	0.833433	+	0.552620i	$0.186372\pi$
$294$	0	0
$295$	18.5458	1.07978
$296$	25.3215	1.47178
$297$	0	0
$298$	−7.79071	−0.451304
$299$	−21.1956	−1.22577
$300$	0	0
$301$	3.00000	0.172917
$302$	−7.63500	−0.439345
$303$	0	0
$304$	−0.861564	−0.0494141
$305$	18.3880	1.05289
$306$	0	0
$307$	3.42107	0.195251	0.0976253	−	0.995223i	$-0.468875\pi$
$307$	3.42107	0.195251	0.0976253	+	0.995223i	$0.468875\pi$
$308$	−7.36389	−0.419596
$309$	0	0
$310$	−9.84540	−0.559181
$311$	16.8525	0.955620	0.477810	−	0.878463i	$-0.341431\pi$
$311$	16.8525	0.955620	0.477810	+	0.878463i	$0.341431\pi$
$312$	0	0
$313$	−1.07661	−0.0608536	−0.0304268	−	0.999537i	$-0.509687\pi$
$313$	−1.07661	−0.0608536	−0.0304268	+	0.999537i	$0.509687\pi$
$314$	2.68754	0.151666
$315$	0	0
$316$	−10.0011	−0.562606
$317$	−18.2197	−1.02332	−0.511660	−	0.859188i	$-0.670969\pi$
$317$	−18.2197	−1.02332	−0.511660	+	0.859188i	$0.670969\pi$
$318$	0	0
$319$	−5.13844	−0.287697
$320$	−5.31711	−0.297235
$321$	0	0
$322$	−19.7141	−1.09862
$323$	−2.28263	−0.127009
$324$	0	0
$325$	5.19483	0.288158
$326$	8.93817	0.495040
$327$	0	0
$328$	−0.622440	−0.0343685
$329$	16.1923	0.892713
$330$	0	0
$331$	−8.82846	−0.485256	−0.242628	−	0.970119i	$-0.578009\pi$
$331$	−8.82846	−0.485256	−0.242628	+	0.970119i	$0.578009\pi$
$332$	18.9715	1.04120
$333$	0	0
$334$	3.00000	0.164153
$335$	21.3067	1.16411
$336$	0	0
$337$	−3.85254	−0.209861	−0.104931	−	0.994480i	$-0.533462\pi$
$337$	−3.85254	−0.209861	−0.104931	+	0.994480i	$0.533462\pi$
$338$	3.78168	0.205696
$339$	0	0
$340$	6.30206	0.341777
$341$	−6.66019	−0.360670
$342$	0	0
$343$	−66.6011	−3.59612
$344$	1.50697	0.0812503
$345$	0	0
$346$	0.891397	0.0479218
$347$	19.4120	1.04209	0.521046	−	0.853528i	$-0.325542\pi$
$347$	19.4120	1.04209	0.521046	+	0.853528i	$0.325542\pi$
$348$	0	0
$349$	−32.9669	−1.76468	−0.882339	−	0.470615i	$-0.844032\pi$
$349$	−32.9669	−1.76468	−0.882339	+	0.470615i	$0.844032\pi$
$350$	4.83173	0.258267
$351$	0	0
$352$	−5.86156	−0.312422
$353$	−23.1546	−1.23239	−0.616197	−	0.787592i	$-0.711328\pi$
$353$	−23.1546	−1.23239	−0.616197	+	0.787592i	$0.711328\pi$
$354$	0	0
$355$	−14.0643	−0.746456
$356$	−9.46457	−0.501621
$357$	0	0
$358$	−2.99208	−0.158137
$359$	35.3685	1.86668	0.933340	−	0.358994i	$-0.116880\pi$
$359$	35.3685	1.86668	0.933340	+	0.358994i	$0.116880\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	10.2963	0.541162
$363$	0	0
$364$	−31.2164	−1.63619
$365$	9.95649	0.521147
$366$	0	0
$367$	−28.4224	−1.48364	−0.741820	−	0.670599i	$-0.766037\pi$
$367$	−28.4224	−1.48364	−0.741820	+	0.670599i	$0.766037\pi$
$368$	4.30782	0.224561
$369$	0	0
$370$	14.3800	0.747583
$371$	−45.7486	−2.37515
$372$	0	0
$373$	17.5997	0.911280	0.455640	−	0.890164i	$-0.349410\pi$
$373$	17.5997	0.911280	0.455640	+	0.890164i	$0.349410\pi$
$374$	−1.73680	−0.0898078
$375$	0	0
$376$	8.13379	0.419468
$377$	−21.7825	−1.12185
$378$	0	0
$379$	12.9669	0.666065	0.333032	−	0.942915i	$-0.391928\pi$
$379$	12.9669	0.666065	0.333032	+	0.942915i	$0.391928\pi$
$380$	−2.76088	−0.141630
$381$	0	0
$382$	11.5847	0.592724
$383$	−8.91299	−0.455432	−0.227716	−	0.973728i	$-0.573126\pi$
$383$	−8.91299	−0.455432	−0.227716	+	0.973728i	$0.573126\pi$
$384$	0	0
$385$	−10.0676	−0.513092
$386$	13.9108	0.708042
$387$	0	0
$388$	12.4304	0.631056
$389$	17.2118	0.872672	0.436336	−	0.899784i	$-0.356276\pi$
$389$	17.2118	0.872672	0.436336	+	0.899784i	$0.356276\pi$
$390$	0	0
$391$	11.4132	0.577188
$392$	−51.6764	−2.61005
$393$	0	0
$394$	−10.7759	−0.542883
$395$	−13.6731	−0.687967
$396$	0	0
$397$	−11.0733	−0.555755	−0.277878	−	0.960617i	$-0.589631\pi$
$397$	−11.0733	−0.555755	−0.277878	+	0.960617i	$0.589631\pi$
$398$	13.5322	0.678306
$399$	0	0
$400$	−1.05580	−0.0527902
$401$	15.9759	0.797800	0.398900	−	0.916995i	$-0.369392\pi$
$401$	15.9759	0.797800	0.398900	+	0.916995i	$0.369392\pi$
$402$	0	0
$403$	−28.2334	−1.40640
$404$	−2.76088	−0.137359
$405$	0	0
$406$	−20.2599	−1.00548
$407$	9.72777	0.482188
$408$	0	0
$409$	0.336541	0.0166409	0.00832043	−	0.999965i	$-0.497351\pi$
$409$	0.336541	0.0166409	0.00832043	+	0.999965i	$0.497351\pi$
$410$	−0.353483	−0.0174573
$411$	0	0
$412$	−25.2884	−1.24587
$413$	49.4660	2.43406
$414$	0	0
$415$	25.9371	1.27320
$416$	−24.8479	−1.21827
$417$	0	0
$418$	0.760877	0.0372157
$419$	33.4088	1.63213	0.816063	−	0.577964i	$-0.196152\pi$
$419$	33.4088	1.63213	0.816063	+	0.577964i	$0.196152\pi$
$420$	0	0
$421$	−8.09742	−0.394644	−0.197322	−	0.980339i	$-0.563224\pi$
$421$	−8.09742	−0.394644	−0.197322	+	0.980339i	$0.563224\pi$
$422$	−14.1190	−0.687302
$423$	0	0
$424$	−22.9806	−1.11604
$425$	−2.79725	−0.135687
$426$	0	0
$427$	49.0449	2.37345
$428$	−7.52640	−0.363802
$429$	0	0
$430$	0.855806	0.0412706
$431$	−37.4660	−1.80467	−0.902336	−	0.431034i	$-0.858149\pi$
$431$	−37.4660	−1.80467	−0.902336	+	0.431034i	$0.858149\pi$
$432$	0	0
$433$	2.39372	0.115035	0.0575174	−	0.998345i	$-0.481682\pi$
$433$	2.39372	0.115035	0.0575174	+	0.998345i	$0.481682\pi$
$434$	−26.2599	−1.26052
$435$	0	0
$436$	−3.61779	−0.173261
$437$	−5.00000	−0.239182
$438$	0	0
$439$	12.4016	0.591898	0.295949	−	0.955204i	$-0.404364\pi$
$439$	12.4016	0.591898	0.295949	+	0.955204i	$0.404364\pi$
$440$	−5.05718	−0.241092
$441$	0	0
$442$	−7.36251	−0.350199
$443$	−1.73818	−0.0825833	−0.0412916	−	0.999147i	$-0.513147\pi$
$443$	−1.73818	−0.0825833	−0.0412916	+	0.999147i	$0.513147\pi$
$444$	0	0
$445$	−12.9396	−0.613394
$446$	−5.87988	−0.278421
$447$	0	0
$448$	−14.1819	−0.670034
$449$	0.185213	0.00874074	0.00437037	−	0.999990i	$-0.498609\pi$
$449$	0.185213	0.00874074	0.00437037	+	0.999990i	$0.498609\pi$
$450$	0	0
$451$	−0.239123	−0.0112599
$452$	23.0104	1.08232
$453$	0	0
$454$	−0.415309	−0.0194914
$455$	−42.6777	−2.00076
$456$	0	0
$457$	18.3034	0.856199	0.428099	−	0.903732i	$-0.359183\pi$
$457$	18.3034	0.856199	0.428099	+	0.903732i	$0.359183\pi$
$458$	5.44979	0.254652
$459$	0	0
$460$	13.8044	0.643633
$461$	−30.1488	−1.40417	−0.702086	−	0.712092i	$-0.747748\pi$
$461$	−30.1488	−1.40417	−0.702086	+	0.712092i	$0.747748\pi$
$462$	0	0
$463$	−4.46922	−0.207702	−0.103851	−	0.994593i	$-0.533117\pi$
$463$	−4.46922	−0.207702	−0.103851	+	0.994593i	$0.533117\pi$
$464$	4.42709	0.205522
$465$	0	0
$466$	−2.47936	−0.114854
$467$	6.63611	0.307083	0.153541	−	0.988142i	$-0.450932\pi$
$467$	6.63611	0.307083	0.153541	+	0.988142i	$0.450932\pi$
$468$	0	0
$469$	56.8298	2.62416
$470$	4.61917	0.213066
$471$	0	0
$472$	24.8479	1.14372
$473$	0.578933	0.0266194
$474$	0	0
$475$	1.22545	0.0562275
$476$	16.8090	0.770441
$477$	0	0
$478$	19.2632	0.881078
$479$	−24.5368	−1.12112	−0.560558	−	0.828115i	$-0.689413\pi$
$479$	−24.5368	−1.12112	−0.560558	+	0.828115i	$0.689413\pi$
$480$	0	0
$481$	41.2372	1.88026
$482$	−8.05280	−0.366795
$483$	0	0
$484$	−1.42107	−0.0645939
$485$	16.9942	0.771669
$486$	0	0
$487$	16.6498	0.754474	0.377237	−	0.926117i	$-0.376874\pi$
$487$	16.6498	0.754474	0.377237	+	0.926117i	$0.376874\pi$
$488$	24.6364	1.11524
$489$	0	0
$490$	−29.3469	−1.32576
$491$	−7.77128	−0.350713	−0.175356	−	0.984505i	$-0.556108\pi$
$491$	−7.77128	−0.350713	−0.175356	+	0.984505i	$0.556108\pi$
$492$	0	0
$493$	11.7292	0.528254
$494$	3.22545	0.145120
$495$	0	0
$496$	5.73818	0.257652
$497$	−37.5127	−1.68268
$498$	0	0
$499$	2.10860	0.0943940	0.0471970	−	0.998886i	$-0.484971\pi$
$499$	2.10860	0.0943940	0.0471970	+	0.998886i	$0.484971\pi$
$500$	−17.1877	−0.768657
$501$	0	0
$502$	23.3732	1.04320
$503$	18.0722	0.805801	0.402900	−	0.915244i	$-0.368002\pi$
$503$	18.0722	0.805801	0.402900	+	0.915244i	$0.368002\pi$
$504$	0	0
$505$	−3.77455	−0.167965
$506$	−3.80438	−0.169125
$507$	0	0
$508$	−17.2107	−0.763600
$509$	−30.6077	−1.35666	−0.678330	−	0.734757i	$-0.737296\pi$
$509$	−30.6077	−1.35666	−0.678330	+	0.734757i	$0.737296\pi$
$510$	0	0
$511$	26.5562	1.17478
$512$	9.53543	0.421410
$513$	0	0
$514$	−19.6167	−0.865255
$515$	−34.5732	−1.52348
$516$	0	0
$517$	3.12476	0.137427
$518$	38.3549	1.68522
$519$	0	0
$520$	−21.4380	−0.940119
$521$	−12.9748	−0.568437	−0.284218	−	0.958760i	$-0.591734\pi$
$521$	−12.9748	−0.568437	−0.284218	+	0.958760i	$0.591734\pi$
$522$	0	0
$523$	−28.2977	−1.23737	−0.618686	−	0.785639i	$-0.712335\pi$
$523$	−28.2977	−1.23737	−0.618686	+	0.785639i	$0.712335\pi$
$524$	18.2107	0.795537
$525$	0	0
$526$	19.4703	0.848947
$527$	15.2028	0.662242
$528$	0	0
$529$	2.00000	0.0869565
$530$	−13.0506	−0.566884
$531$	0	0
$532$	−7.36389	−0.319265
$533$	−1.01367	−0.0439071
$534$	0	0
$535$	−10.2898	−0.444865
$536$	28.5469	1.23304
$537$	0	0
$538$	20.1294	0.867840
$539$	−19.8525	−0.855109
$540$	0	0
$541$	40.2359	1.72987	0.864937	−	0.501880i	$-0.167358\pi$
$541$	40.2359	1.72987	0.864937	+	0.501880i	$0.167358\pi$
$542$	7.18659	0.308690
$543$	0	0
$544$	13.3798	0.573653
$545$	−4.94609	−0.211867
$546$	0	0
$547$	1.33189	0.0569477	0.0284738	−	0.999595i	$-0.490935\pi$
$547$	1.33189	0.0569477	0.0284738	+	0.999595i	$0.490935\pi$
$548$	−27.2736	−1.16507
$549$	0	0
$550$	0.932417	0.0397584
$551$	−5.13844	−0.218905
$552$	0	0
$553$	−36.4692	−1.55083
$554$	−9.38907	−0.398904
$555$	0	0
$556$	−12.8044	−0.543027
$557$	7.66019	0.324573	0.162286	−	0.986744i	$-0.448113\pi$
$557$	7.66019	0.324573	0.162286	+	0.986744i	$0.448113\pi$
$558$	0	0
$559$	2.45417	0.103800
$560$	8.67386	0.366538
$561$	0	0
$562$	0.684266	0.0288640
$563$	−8.07085	−0.340146	−0.170073	−	0.985431i	$-0.554400\pi$
$563$	−8.07085	−0.340146	−0.170073	+	0.985431i	$0.554400\pi$
$564$	0	0
$565$	31.4588	1.32348
$566$	−1.04815	−0.0440572
$567$	0	0
$568$	−18.8435	−0.790656
$569$	37.2107	1.55995	0.779976	−	0.625809i	$-0.215231\pi$
$569$	37.2107	1.55995	0.779976	+	0.625809i	$0.215231\pi$
$570$	0	0
$571$	−33.3606	−1.39610	−0.698049	−	0.716050i	$-0.745948\pi$
$571$	−33.3606	−1.39610	−0.698049	+	0.716050i	$0.745948\pi$
$572$	−6.02408	−0.251879
$573$	0	0
$574$	−0.942820	−0.0393525
$575$	−6.12725	−0.255524
$576$	0	0
$577$	−32.9234	−1.37062	−0.685309	−	0.728252i	$-0.740333\pi$
$577$	−32.9234	−1.37062	−0.685309	+	0.728252i	$0.740333\pi$
$578$	−8.97043	−0.373121
$579$	0	0
$580$	14.1866	0.589066
$581$	69.1801	2.87007
$582$	0	0
$583$	−8.82846	−0.365637
$584$	13.3398	0.552005
$585$	0	0
$586$	21.7095	0.896809
$587$	16.0837	0.663847	0.331924	−	0.943306i	$-0.392302\pi$
$587$	16.0837	0.663847	0.331924	+	0.943306i	$0.392302\pi$
$588$	0	0
$589$	−6.66019	−0.274428
$590$	14.1111	0.580944
$591$	0	0
$592$	−8.38109	−0.344461
$593$	22.8856	0.939801	0.469900	−	0.882719i	$-0.344290\pi$
$593$	22.8856	0.939801	0.469900	+	0.882719i	$0.344290\pi$
$594$	0	0
$595$	22.9806	0.942112
$596$	14.5505	0.596011
$597$	0	0
$598$	−16.1273	−0.659492
$599$	−30.2438	−1.23573	−0.617863	−	0.786285i	$-0.712002\pi$
$599$	−30.2438	−1.23573	−0.617863	+	0.786285i	$0.712002\pi$
$600$	0	0
$601$	3.89356	0.158821	0.0794107	−	0.996842i	$-0.474696\pi$
$601$	3.89356	0.158821	0.0794107	+	0.996842i	$0.474696\pi$
$602$	2.28263	0.0930331
$603$	0	0
$604$	14.2597	0.580218
$605$	−1.94282	−0.0789869
$606$	0	0
$607$	−22.0377	−0.894485	−0.447242	−	0.894413i	$-0.647594\pi$
$607$	−22.0377	−0.894485	−0.447242	+	0.894413i	$0.647594\pi$
$608$	−5.86156	−0.237718
$609$	0	0
$610$	13.9910	0.566478
$611$	13.2463	0.535886
$612$	0	0
$613$	11.8752	0.479636	0.239818	−	0.970818i	$-0.422912\pi$
$613$	11.8752	0.479636	0.239818	+	0.970818i	$0.422912\pi$
$614$	2.60301	0.105049
$615$	0	0
$616$	−13.4887	−0.543473
$617$	37.9557	1.52804	0.764020	−	0.645193i	$-0.223223\pi$
$617$	37.9557	1.52804	0.764020	+	0.645193i	$0.223223\pi$
$618$	0	0
$619$	17.4977	0.703291	0.351646	−	0.936133i	$-0.385622\pi$
$619$	17.4977	0.703291	0.351646	+	0.936133i	$0.385622\pi$
$620$	18.3880	0.738478
$621$	0	0
$622$	12.8227	0.514144
$623$	−34.5127	−1.38272
$624$	0	0
$625$	−17.3710	−0.694841
$626$	−0.819168	−0.0327405
$627$	0	0
$628$	−5.01943	−0.200297
$629$	−22.2049	−0.885368
$630$	0	0
$631$	7.45090	0.296616	0.148308	−	0.988941i	$-0.452617\pi$
$631$	7.45090	0.296616	0.148308	+	0.988941i	$0.452617\pi$
$632$	−18.3193	−0.728704
$633$	0	0
$634$	−13.8629	−0.550568
$635$	−23.5297	−0.933746
$636$	0	0
$637$	−84.1574	−3.33444
$638$	−3.90972	−0.154787
$639$	0	0
$640$	18.7303	0.740379
$641$	−19.3009	−0.762342	−0.381171	−	0.924505i	$-0.624479\pi$
$641$	−19.3009	−0.762342	−0.381171	+	0.924505i	$0.624479\pi$
$642$	0	0
$643$	29.3984	1.15936	0.579679	−	0.814845i	$-0.303178\pi$
$643$	29.3984	1.15936	0.579679	+	0.814845i	$0.303178\pi$
$644$	36.8194	1.45089
$645$	0	0
$646$	−1.73680	−0.0683335
$647$	−25.9989	−1.02212	−0.511061	−	0.859545i	$-0.670747\pi$
$647$	−25.9989	−1.02212	−0.511061	+	0.859545i	$0.670747\pi$
$648$	0	0
$649$	9.54583	0.374707
$650$	3.95263	0.155035
$651$	0	0
$652$	−16.6936	−0.653770
$653$	29.7609	1.16463	0.582317	−	0.812962i	$-0.302146\pi$
$653$	29.7609	1.16463	0.582317	+	0.812962i	$0.302146\pi$
$654$	0	0
$655$	24.8968	0.972799
$656$	0.206020	0.00804373
$657$	0	0
$658$	12.3204	0.480298
$659$	−32.2463	−1.25614	−0.628068	−	0.778159i	$-0.716154\pi$
$659$	−32.2463	−1.25614	−0.628068	+	0.778159i	$0.716154\pi$
$660$	0	0
$661$	7.21505	0.280633	0.140316	−	0.990107i	$-0.455188\pi$
$661$	7.21505	0.280633	0.140316	+	0.990107i	$0.455188\pi$
$662$	−6.71737	−0.261078
$663$	0	0
$664$	34.7507	1.34859
$665$	−10.0676	−0.390404
$666$	0	0
$667$	25.6922	0.994805
$668$	−5.60301	−0.216787
$669$	0	0
$670$	16.2118	0.626316
$671$	9.46457	0.365376
$672$	0	0
$673$	−21.3833	−0.824266	−0.412133	−	0.911124i	$-0.635216\pi$
$673$	−21.3833	−0.824266	−0.412133	+	0.911124i	$0.635216\pi$
$674$	−2.93131	−0.112910
$675$	0	0
$676$	−7.06294	−0.271651
$677$	−18.9579	−0.728610	−0.364305	−	0.931280i	$-0.618693\pi$
$677$	−18.9579	−0.728610	−0.364305	+	0.931280i	$0.618693\pi$
$678$	0	0
$679$	45.3275	1.73951
$680$	11.5437	0.442680
$681$	0	0
$682$	−5.06758	−0.194048
$683$	10.7792	0.412454	0.206227	−	0.978504i	$-0.433881\pi$
$683$	10.7792	0.412454	0.206227	+	0.978504i	$0.433881\pi$
$684$	0	0
$685$	−37.2873	−1.42467
$686$	−50.6752	−1.93479
$687$	0	0
$688$	−0.498788	−0.0190161
$689$	−37.4249	−1.42578
$690$	0	0
$691$	49.8824	1.89761	0.948807	−	0.315855i	$-0.102291\pi$
$691$	49.8824	1.89761	0.948807	+	0.315855i	$0.102291\pi$
$692$	−1.66484	−0.0632876
$693$	0	0
$694$	14.7702	0.560668
$695$	−17.5056	−0.664025
$696$	0	0
$697$	0.545830	0.0206748
$698$	−25.0837	−0.949434
$699$	0	0
$700$	−9.02408	−0.341078
$701$	−2.25855	−0.0853044	−0.0426522	−	0.999090i	$-0.513581\pi$
$701$	−2.25855	−0.0853044	−0.0426522	+	0.999090i	$0.513581\pi$
$702$	0	0
$703$	9.72777	0.366890
$704$	−2.73680	−0.103147
$705$	0	0
$706$	−17.6178	−0.663054
$707$	−10.0676	−0.378630
$708$	0	0
$709$	−2.00792	−0.0754089	−0.0377044	−	0.999289i	$-0.512005\pi$
$709$	−2.00792	−0.0754089	−0.0377044	+	0.999289i	$0.512005\pi$
$710$	−10.7012	−0.401609
$711$	0	0
$712$	−17.3365	−0.649714
$713$	33.3009	1.24713
$714$	0	0
$715$	−8.23585	−0.308003
$716$	5.58823	0.208842
$717$	0	0
$718$	26.9111	1.00431
$719$	31.8824	1.18901	0.594506	−	0.804091i	$-0.297348\pi$
$719$	31.8824	1.18901	0.594506	+	0.804091i	$0.297348\pi$
$720$	0	0
$721$	−92.2145	−3.43425
$722$	0.760877	0.0283169
$723$	0	0
$724$	−19.2301	−0.714681
$725$	−6.29690	−0.233861
$726$	0	0
$727$	−33.4088	−1.23906	−0.619531	−	0.784972i	$-0.712677\pi$
$727$	−33.4088	−1.23906	−0.619531	+	0.784972i	$0.712677\pi$
$728$	−57.1801	−2.11923
$729$	0	0
$730$	7.57566	0.280388
$731$	−1.32149	−0.0488771
$732$	0	0
$733$	−23.9187	−0.883459	−0.441729	−	0.897148i	$-0.645635\pi$
$733$	−23.9187	−0.883459	−0.441729	+	0.897148i	$0.645635\pi$
$734$	−21.6260	−0.798229
$735$	0	0
$736$	29.3078	1.08030
$737$	10.9669	0.403971
$738$	0	0
$739$	7.25744	0.266969	0.133485	−	0.991051i	$-0.457383\pi$
$739$	7.25744	0.266969	0.133485	+	0.991051i	$0.457383\pi$
$740$	−26.8572	−0.987290
$741$	0	0
$742$	−34.8090	−1.27788
$743$	6.15460	0.225790	0.112895	−	0.993607i	$-0.463988\pi$
$743$	6.15460	0.225790	0.112895	+	0.993607i	$0.463988\pi$
$744$	0	0
$745$	19.8928	0.728815
$746$	13.3912	0.490288
$747$	0	0
$748$	3.24377	0.118604
$749$	−27.4451	−1.00282
$750$	0	0
$751$	29.5666	1.07890	0.539451	−	0.842017i	$-0.318632\pi$
$751$	29.5666	1.07890	0.539451	+	0.842017i	$0.318632\pi$
$752$	−2.69218	−0.0981738
$753$	0	0
$754$	−16.5738	−0.603581
$755$	19.4952	0.709503
$756$	0	0
$757$	−37.9611	−1.37972	−0.689861	−	0.723942i	$-0.742328\pi$
$757$	−37.9611	−1.37972	−0.689861	+	0.723942i	$0.742328\pi$
$758$	9.86621	0.358357
$759$	0	0
$760$	−5.05718	−0.183443
$761$	5.10860	0.185187	0.0925934	−	0.995704i	$-0.470484\pi$
$761$	5.10860	0.185187	0.0925934	+	0.995704i	$0.470484\pi$
$762$	0	0
$763$	−13.1923	−0.477595
$764$	−21.6364	−0.782777
$765$	0	0
$766$	−6.78168	−0.245032
$767$	40.4660	1.46114
$768$	0	0
$769$	−34.9863	−1.26164	−0.630820	−	0.775930i	$-0.717281\pi$
$769$	−34.9863	−1.26164	−0.630820	+	0.775930i	$0.717281\pi$
$770$	−7.66019	−0.276054
$771$	0	0
$772$	−25.9808	−0.935071
$773$	17.2711	0.621199	0.310599	−	0.950541i	$-0.399470\pi$
$773$	17.2711	0.621199	0.310599	+	0.950541i	$0.399470\pi$
$774$	0	0
$775$	−8.16173	−0.293178
$776$	22.7691	0.817362
$777$	0	0
$778$	13.0960	0.469516
$779$	−0.239123	−0.00856748
$780$	0	0
$781$	−7.23912	−0.259036
$782$	8.68400	0.310539
$783$	0	0
$784$	17.1042	0.610865
$785$	−6.86235	−0.244928
$786$	0	0
$787$	40.8115	1.45477	0.727387	−	0.686228i	$-0.240735\pi$
$787$	40.8115	1.45477	0.727387	+	0.686228i	$0.240735\pi$
$788$	20.1259	0.716955
$789$	0	0
$790$	−10.4035	−0.370141
$791$	83.9078	2.98342
$792$	0	0
$793$	40.1215	1.42476
$794$	−8.42545	−0.299008
$795$	0	0
$796$	−25.2736	−0.895799
$797$	25.8090	0.914203	0.457101	−	0.889415i	$-0.348888\pi$
$797$	25.8090	0.914203	0.457101	+	0.889415i	$0.348888\pi$
$798$	0	0
$799$	−7.13268	−0.252336
$800$	−7.18305	−0.253959
$801$	0	0
$802$	12.1557	0.429233
$803$	5.12476	0.180849
$804$	0	0
$805$	50.3379	1.77418
$806$	−21.4821	−0.756675
$807$	0	0
$808$	−5.05718	−0.177911
$809$	−6.17946	−0.217258	−0.108629	−	0.994082i	$-0.534646\pi$
$809$	−6.17946	−0.217258	−0.108629	+	0.994082i	$0.534646\pi$
$810$	0	0
$811$	−11.0676	−0.388635	−0.194318	−	0.980939i	$-0.562249\pi$
$811$	−11.0676	−0.388635	−0.194318	+	0.980939i	$0.562249\pi$
$812$	37.8389	1.32788
$813$	0	0
$814$	7.40164	0.259427
$815$	−22.8227	−0.799444
$816$	0	0
$817$	0.578933	0.0202543
$818$	0.256066	0.00895313
$819$	0	0
$820$	0.660190	0.0230548
$821$	−13.6602	−0.476744	−0.238372	−	0.971174i	$-0.576614\pi$
$821$	−13.6602	−0.476744	−0.238372	+	0.971174i	$0.576614\pi$
$822$	0	0
$823$	11.4887	0.400469	0.200235	−	0.979748i	$-0.435830\pi$
$823$	11.4887	0.400469	0.200235	+	0.979748i	$0.435830\pi$
$824$	−46.3215	−1.61369
$825$	0	0
$826$	37.6375	1.30958
$827$	−19.9942	−0.695268	−0.347634	−	0.937630i	$-0.613015\pi$
$827$	−19.9942	−0.695268	−0.347634	+	0.937630i	$0.613015\pi$
$828$	0	0
$829$	−16.4451	−0.571163	−0.285582	−	0.958354i	$-0.592187\pi$
$829$	−16.4451	−0.571163	−0.285582	+	0.958354i	$0.592187\pi$
$830$	19.7349	0.685009
$831$	0	0
$832$	−11.6016	−0.402214
$833$	45.3160	1.57011
$834$	0	0
$835$	−7.66019	−0.265092
$836$	−1.42107	−0.0491486
$837$	0	0
$838$	25.4200	0.878118
$839$	55.3171	1.90976	0.954879	−	0.296994i	$-0.0959841\pi$
$839$	55.3171	1.90976	0.954879	+	0.296994i	$0.0959841\pi$
$840$	0	0
$841$	−2.59647	−0.0895335
$842$	−6.16114	−0.212327
$843$	0	0
$844$	26.3696	0.907681
$845$	−9.65614	−0.332181
$846$	0	0
$847$	−5.18194	−0.178054
$848$	7.60628	0.261201
$849$	0	0
$850$	−2.12836	−0.0730022
$851$	−48.6389	−1.66732
$852$	0	0
$853$	−9.85005	−0.337259	−0.168630	−	0.985679i	$-0.553934\pi$
$853$	−9.85005	−0.337259	−0.168630	+	0.985679i	$0.553934\pi$
$854$	37.3171	1.27697
$855$	0	0
$856$	−13.7863	−0.471207
$857$	19.7907	0.676038	0.338019	−	0.941139i	$-0.390243\pi$
$857$	19.7907	0.676038	0.338019	+	0.941139i	$0.390243\pi$
$858$	0	0
$859$	−6.95898	−0.237437	−0.118719	−	0.992928i	$-0.537879\pi$
$859$	−6.95898	−0.237437	−0.118719	+	0.992928i	$0.537879\pi$
$860$	−1.59836	−0.0545038
$861$	0	0
$862$	−28.5070	−0.970951
$863$	5.74393	0.195526	0.0977629	−	0.995210i	$-0.468831\pi$
$863$	5.74393	0.195526	0.0977629	+	0.995210i	$0.468831\pi$
$864$	0	0
$865$	−2.27609	−0.0773894
$866$	1.82133	0.0618912
$867$	0	0
$868$	49.0449	1.66469
$869$	−7.03775	−0.238739
$870$	0	0
$871$	46.4900	1.57525
$872$	−6.62682	−0.224412
$873$	0	0
$874$	−3.80438	−0.128685
$875$	−62.6752	−2.11881
$876$	0	0
$877$	42.5070	1.43536	0.717679	−	0.696374i	$-0.245204\pi$
$877$	42.5070	1.43536	0.717679	+	0.696374i	$0.245204\pi$
$878$	9.43612	0.318454
$879$	0	0
$880$	1.67386	0.0564259
$881$	−31.7655	−1.07021	−0.535104	−	0.844786i	$-0.679728\pi$
$881$	−31.7655	−1.07021	−0.535104	+	0.844786i	$0.679728\pi$
$882$	0	0
$883$	−30.5458	−1.02795	−0.513975	−	0.857805i	$-0.671827\pi$
$883$	−30.5458	−1.02795	−0.513975	+	0.857805i	$0.671827\pi$
$884$	13.7507	0.462487
$885$	0	0
$886$	−1.32254	−0.0444315
$887$	−45.0506	−1.51265	−0.756326	−	0.654195i	$-0.773008\pi$
$887$	−45.0506	−1.51265	−0.756326	+	0.654195i	$0.773008\pi$
$888$	0	0
$889$	−62.7590	−2.10487
$890$	−9.84540	−0.330019
$891$	0	0
$892$	10.9817	0.367694
$893$	3.12476	0.104566
$894$	0	0
$895$	7.63998	0.255376
$896$	49.9579	1.66898
$897$	0	0
$898$	0.140924	0.00470270
$899$	34.2230	1.14140
$900$	0	0
$901$	20.1521	0.671364
$902$	−0.181943	−0.00605805
$903$	0	0
$904$	42.1488	1.40185
$905$	−26.2905	−0.873927
$906$	0	0
$907$	12.9831	0.431095	0.215548	−	0.976493i	$-0.430846\pi$
$907$	12.9831	0.431095	0.215548	+	0.976493i	$0.430846\pi$
$908$	0.775661	0.0257412
$909$	0	0
$910$	−32.4725	−1.07645
$911$	32.1398	1.06484	0.532420	−	0.846480i	$-0.321283\pi$
$911$	32.1398	1.06484	0.532420	+	0.846480i	$0.321283\pi$
$912$	0	0
$913$	13.3502	0.441828
$914$	13.9267	0.460653
$915$	0	0
$916$	−10.1784	−0.336304
$917$	66.4055	2.19290
$918$	0	0
$919$	54.4476	1.79606	0.898031	−	0.439933i	$-0.144998\pi$
$919$	54.4476	1.79606	0.898031	+	0.439933i	$0.144998\pi$
$920$	25.2859	0.833651
$921$	0	0
$922$	−22.9396	−0.755474
$923$	−30.6875	−1.01009
$924$	0	0
$925$	11.9209	0.391957
$926$	−3.40053	−0.111748
$927$	0	0
$928$	30.1193	0.988714
$929$	11.3502	0.372388	0.186194	−	0.982513i	$-0.440385\pi$
$929$	11.3502	0.372388	0.186194	+	0.982513i	$0.440385\pi$
$930$	0	0
$931$	−19.8525	−0.650641
$932$	4.63062	0.151681
$933$	0	0
$934$	5.04926	0.165217
$935$	4.43474	0.145031
$936$	0	0
$937$	−5.70697	−0.186438	−0.0932192	−	0.995646i	$-0.529716\pi$
$937$	−5.70697	−0.186438	−0.0932192	+	0.995646i	$0.529716\pi$
$938$	43.2405	1.41185
$939$	0	0
$940$	−8.62709	−0.281385
$941$	17.4646	0.569329	0.284664	−	0.958627i	$-0.408118\pi$
$941$	17.4646	0.569329	0.284664	+	0.958627i	$0.408118\pi$
$942$	0	0
$943$	1.19562	0.0389346
$944$	−8.22434	−0.267679
$945$	0	0
$946$	0.440497	0.0143218
$947$	42.0449	1.36628	0.683138	−	0.730290i	$-0.260615\pi$
$947$	42.0449	1.36628	0.683138	+	0.730290i	$0.260615\pi$
$948$	0	0
$949$	21.7245	0.705207
$950$	0.932417	0.0302516
$951$	0	0
$952$	30.7896	0.997897
$953$	21.1704	0.685777	0.342889	−	0.939376i	$-0.388595\pi$
$953$	21.1704	0.685777	0.342889	+	0.939376i	$0.388595\pi$
$954$	0	0
$955$	−29.5803	−0.957196
$956$	−35.9773	−1.16359
$957$	0	0
$958$	−18.6695	−0.603184
$959$	−99.4537	−3.21153
$960$	0	0
$961$	13.3581	0.430907
$962$	31.3764	1.01162
$963$	0	0
$964$	15.0400	0.484405
$965$	−35.5199	−1.14342
$966$	0	0
$967$	36.9787	1.18915	0.594577	−	0.804039i	$-0.297320\pi$
$967$	36.9787	1.18915	0.594577	+	0.804039i	$0.297320\pi$
$968$	−2.60301	−0.0836639
$969$	0	0
$970$	12.9305	0.415174
$971$	34.0917	1.09405	0.547027	−	0.837115i	$-0.315760\pi$
$971$	34.0917	1.09405	0.547027	+	0.837115i	$0.315760\pi$
$972$	0	0
$973$	−46.6914	−1.49686
$974$	12.6684	0.405923
$975$	0	0
$976$	−8.15433	−0.261014
$977$	1.30422	0.0417257	0.0208628	−	0.999782i	$-0.493359\pi$
$977$	1.30422	0.0417257	0.0208628	+	0.999782i	$0.493359\pi$
$978$	0	0
$979$	−6.66019	−0.212861
$980$	54.8104	1.75085
$981$	0	0
$982$	−5.91299	−0.188691
$983$	17.3387	0.553019	0.276509	−	0.961011i	$-0.410822\pi$
$983$	17.3387	0.553019	0.276509	+	0.961011i	$0.410822\pi$
$984$	0	0
$985$	27.5152	0.876708
$986$	8.92444	0.284212
$987$	0	0
$988$	−6.02408	−0.191651
$989$	−2.89467	−0.0920451
$990$	0	0
$991$	−19.9475	−0.633652	−0.316826	−	0.948484i	$-0.602617\pi$
$991$	−19.9475	−0.633652	−0.316826	+	0.948484i	$0.602617\pi$
$992$	39.0391	1.23949
$993$	0	0
$994$	−28.5426	−0.905315
$995$	−34.5530	−1.09540
$996$	0	0
$997$	−6.80438	−0.215497	−0.107748	−	0.994178i	$-0.534364\pi$
$997$	−6.80438	−0.215497	−0.107748	+	0.994178i	$0.534364\pi$
$998$	1.60439	0.0507860
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.i.1.2		3
3.2	odd	2		627.2.a.e.1.2	✓	3
33.32	even	2		6897.2.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.a.e.1.2	✓	3	3.2	odd	2
1881.2.a.i.1.2		3	1.1	even	1	trivial
6897.2.a.p.1.2		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)

\(f(q)\)	\(=\)	\(q+0.760877 q^{2} -1.42107 q^{4} -1.94282 q^{5} -5.18194 q^{7} -2.60301 q^{8} -1.47825 q^{10} -1.00000 q^{11} -4.23912 q^{13} -3.94282 q^{14} +0.861564 q^{16} +2.28263 q^{17} -1.00000 q^{19} +2.76088 q^{20} -0.760877 q^{22} +5.00000 q^{23} -1.22545 q^{25} -3.22545 q^{26} +7.36389 q^{28} +5.13844 q^{29} +6.66019 q^{31} +5.86156 q^{32} +1.73680 q^{34} +10.0676 q^{35} -9.72777 q^{37} -0.760877 q^{38} +5.05718 q^{40} +0.239123 q^{41} -0.578933 q^{43} +1.42107 q^{44} +3.80438 q^{46} -3.12476 q^{47} +19.8525 q^{49} -0.932417 q^{50} +6.02408 q^{52} +8.82846 q^{53} +1.94282 q^{55} +13.4887 q^{56} +3.90972 q^{58} -9.54583 q^{59} -9.46457 q^{61} +5.06758 q^{62} +2.73680 q^{64} +8.23585 q^{65} -10.9669 q^{67} -3.24377 q^{68} +7.66019 q^{70} +7.23912 q^{71} -5.12476 q^{73} -7.40164 q^{74} +1.42107 q^{76} +5.18194 q^{77} +7.03775 q^{79} -1.67386 q^{80} +0.181943 q^{82} -13.3502 q^{83} -4.43474 q^{85} -0.440497 q^{86} +2.60301 q^{88} +6.66019 q^{89} +21.9669 q^{91} -7.10533 q^{92} -2.37756 q^{94} +1.94282 q^{95} -8.74720 q^{97} +15.1053 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 7 q^{7} + 9 q^{8} - 5 q^{10} - 3 q^{11} - 13 q^{13} - 3 q^{14} + 10 q^{16} + 6 q^{17} - 3 q^{19} + 8 q^{20} - 2 q^{22} + 15 q^{23} + 6 q^{25} + 5 q^{28} + 8 q^{29} + 12 q^{31}+ \cdots + 4 q^{98}+O(q^{100}) \) 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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