LMFDB - Embedded newform 961.2.a.j.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [961,2,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,2,3,8,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.67362363425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.2051578125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 $ x^8 - 2x^7 - 9x^6 + 19x^5 + 14x^4 - 28x^3 - 11x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.431370$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.26660 q^{2} -1.48431 q^{3} -0.395721 q^{4} -3.80032 q^{5} -1.88004 q^{6} +2.18899 q^{7} -3.03442 q^{8} -0.796809 q^{9} -4.81349 q^{10} +0.950596 q^{11} +0.587374 q^{12} -0.168142 q^{13} +2.77258 q^{14} +5.64087 q^{15} -3.05196 q^{16} +6.57666 q^{17} -1.00924 q^{18} -1.15284 q^{19} +1.50387 q^{20} -3.24915 q^{21} +1.20403 q^{22} +4.62850 q^{23} +4.50404 q^{24} +9.44244 q^{25} -0.212969 q^{26} +5.63566 q^{27} -0.866228 q^{28} -1.33672 q^{29} +7.14474 q^{30} +2.20322 q^{32} -1.41098 q^{33} +8.33000 q^{34} -8.31886 q^{35} +0.315314 q^{36} +3.87165 q^{37} -1.46019 q^{38} +0.249576 q^{39} +11.5318 q^{40} +0.328203 q^{41} -4.11538 q^{42} -9.63057 q^{43} -0.376170 q^{44} +3.02813 q^{45} +5.86247 q^{46} +5.63858 q^{47} +4.53008 q^{48} -2.20832 q^{49} +11.9598 q^{50} -9.76183 q^{51} +0.0665374 q^{52} +7.33294 q^{53} +7.13814 q^{54} -3.61257 q^{55} -6.64232 q^{56} +1.71118 q^{57} -1.69309 q^{58} -2.65312 q^{59} -2.23221 q^{60} +1.74967 q^{61} -1.74421 q^{63} +8.89454 q^{64} +0.638995 q^{65} -1.78715 q^{66} +0.552007 q^{67} -2.60252 q^{68} -6.87015 q^{69} -10.5367 q^{70} +1.13699 q^{71} +2.41786 q^{72} +7.92260 q^{73} +4.90384 q^{74} -14.0156 q^{75} +0.456203 q^{76} +2.08084 q^{77} +0.316114 q^{78} -4.54228 q^{79} +11.5984 q^{80} -5.97467 q^{81} +0.415702 q^{82} -0.326396 q^{83} +1.28576 q^{84} -24.9934 q^{85} -12.1981 q^{86} +1.98411 q^{87} -2.88451 q^{88} +14.7102 q^{89} +3.83543 q^{90} -0.368062 q^{91} -1.83159 q^{92} +7.14183 q^{94} +4.38117 q^{95} -3.27028 q^{96} +15.5192 q^{97} -2.79707 q^{98} -0.757443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 3 q^{5} + 11 q^{6} - 2 q^{7} - 9 q^{8} + 5 q^{9} - 13 q^{10} + 18 q^{11} + 8 q^{13} - 9 q^{14} + 18 q^{15} + 4 q^{16} + 14 q^{17} + 23 q^{18} - 6 q^{19} - 7 q^{20} - q^{21}+ \cdots + 6 q^{99}+O(q^{100}) $ 8 * q + 2 * q^2 + 3 * q^3 + 8 * q^4 + 3 * q^5 + 11 * q^6 - 2 * q^7 - 9 * q^8 + 5 * q^9 - 13 * q^10 + 18 * q^11 + 8 * q^13 - 9 * q^14 + 18 * q^15 + 4 * q^16 + 14 * q^17 + 23 * q^18 - 6 * q^19 - 7 * q^20 - q^21 + 4 * q^22 + 22 * q^23 + 30 * q^24 + 13 * q^25 + 9 * q^26 + 18 * q^27 - 5 * q^28 + 12 * q^29 + 11 * q^30 - 21 * q^32 + 6 * q^33 - 7 * q^34 - 12 * q^35 - q^36 - 8 * q^37 + 7 * q^38 + q^39 + 11 * q^40 - 22 * q^41 - 6 * q^42 - 2 * q^43 + 4 * q^44 + 18 * q^46 - 18 * q^47 - q^48 - 2 * q^49 + 27 * q^50 + 2 * q^51 - q^52 + 6 * q^53 - 7 * q^54 + 28 * q^55 + 30 * q^56 - 17 * q^57 - 5 * q^58 - 4 * q^59 - 40 * q^60 + 30 * q^61 - 23 * q^63 + 9 * q^64 + 3 * q^65 + 10 * q^66 - 13 * q^67 + 30 * q^68 + 22 * q^69 - 39 * q^70 - q^71 + 3 * q^72 + 2 * q^73 + 8 * q^74 + 23 * q^75 - 33 * q^76 - 24 * q^77 + 8 * q^79 + 24 * q^80 - 28 * q^81 - 19 * q^82 + 39 * q^83 + 8 * q^84 - 26 * q^85 - 16 * q^86 - 15 * q^87 - 17 * q^88 + 27 * q^89 + 8 * q^90 + 16 * q^91 + 32 * q^92 + 22 * q^94 - 6 * q^95 - 12 * q^96 + 34 * q^97 + 10 * q^98 + 6 * q^99

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 $

$2$	1.26660	0.895623	0.447811	−	0.894128i	$-0.352204\pi$
$2$	1.26660	0.895623	0.447811	+	0.894128i	$0.352204\pi$
$3$	−1.48431	−0.856970	−0.428485	−	0.903549i	$-0.640952\pi$
$3$	−1.48431	−0.856970	−0.428485	+	0.903549i	$0.640952\pi$
$4$	−0.395721	−0.197860
$5$	−3.80032	−1.69956	−0.849778	−	0.527141i	$-0.823264\pi$
$5$	−3.80032	−1.69956	−0.849778	+	0.527141i	$0.823264\pi$
$6$	−1.88004	−0.767521
$7$	2.18899	0.827360	0.413680	−	0.910422i	$-0.364243\pi$
$7$	2.18899	0.827360	0.413680	+	0.910422i	$0.364243\pi$
$8$	−3.03442	−1.07283
$9$	−0.796809	−0.265603
$10$	−4.81349	−1.52216
$11$	0.950596	0.286615	0.143308	−	0.989678i	$-0.454226\pi$
$11$	0.950596	0.286615	0.143308	+	0.989678i	$0.454226\pi$
$12$	0.587374	0.169560
$13$	−0.168142	−0.0466343	−0.0233172	−	0.999728i	$-0.507423\pi$
$13$	−0.168142	−0.0466343	−0.0233172	+	0.999728i	$0.507423\pi$
$14$	2.77258	0.741003
$15$	5.64087	1.45647
$16$	−3.05196	−0.762991
$17$	6.57666	1.59507	0.797537	−	0.603271i	$-0.206136\pi$
$17$	6.57666	1.59507	0.797537	+	0.603271i	$0.206136\pi$
$18$	−1.00924	−0.237880
$19$	−1.15284	−0.264480	−0.132240	−	0.991218i	$-0.542217\pi$
$19$	−1.15284	−0.264480	−0.132240	+	0.991218i	$0.542217\pi$
$20$	1.50387	0.336274
$21$	−3.24915	−0.709023
$22$	1.20403	0.256699
$23$	4.62850	0.965109	0.482554	−	0.875866i	$-0.339709\pi$
$23$	4.62850	0.965109	0.482554	+	0.875866i	$0.339709\pi$
$24$	4.50404	0.919383
$25$	9.44244	1.88849
$26$	−0.212969	−0.0417667
$27$	5.63566	1.08458
$28$	−0.866228	−0.163702
$29$	−1.33672	−0.248222	−0.124111	−	0.992268i	$-0.539608\pi$
$29$	−1.33672	−0.248222	−0.124111	+	0.992268i	$0.539608\pi$
$30$	7.14474	1.30444
$31$	0	0
$31$	0	0
$32$	2.20322	0.389479
$33$	−1.41098	−0.245621
$34$	8.33000	1.42858
$35$	−8.31886	−1.40614
$36$	0.315314	0.0525523
$37$	3.87165	0.636495	0.318248	−	0.948008i	$-0.396906\pi$
$37$	3.87165	0.636495	0.318248	+	0.948008i	$0.396906\pi$
$38$	−1.46019	−0.236874
$39$	0.249576	0.0399642
$40$	11.5318	1.82333
$41$	0.328203	0.0512566	0.0256283	−	0.999672i	$-0.491841\pi$
$41$	0.328203	0.0512566	0.0256283	+	0.999672i	$0.491841\pi$
$42$	−4.11538	−0.635017
$43$	−9.63057	−1.46865	−0.734324	−	0.678799i	$-0.762501\pi$
$43$	−9.63057	−1.46865	−0.734324	+	0.678799i	$0.762501\pi$
$44$	−0.376170	−0.0567098
$45$	3.02813	0.451407
$46$	5.86247	0.864373
$47$	5.63858	0.822471	0.411235	−	0.911529i	$-0.365097\pi$
$47$	5.63858	0.822471	0.411235	+	0.911529i	$0.365097\pi$
$48$	4.53008	0.653860
$49$	−2.20832	−0.315475
$50$	11.9598	1.69137
$51$	−9.76183	−1.36693
$52$	0.0665374	0.00922708
$53$	7.33294	1.00726	0.503629	−	0.863920i	$-0.331998\pi$
$53$	7.33294	1.00726	0.503629	+	0.863920i	$0.331998\pi$
$54$	7.13814	0.971377
$55$	−3.61257	−0.487119
$56$	−6.64232	−0.887617
$57$	1.71118	0.226651
$58$	−1.69309	−0.222313
$59$	−2.65312	−0.345407	−0.172703	−	0.984974i	$-0.555250\pi$
$59$	−2.65312	−0.345407	−0.172703	+	0.984974i	$0.555250\pi$
$60$	−2.23221	−0.288177
$61$	1.74967	0.224023	0.112011	−	0.993707i	$-0.464271\pi$
$61$	1.74967	0.224023	0.112011	+	0.993707i	$0.464271\pi$
$62$	0	0
$63$	−1.74421	−0.219749
$64$	8.89454	1.11182
$65$	0.638995	0.0792576
$66$	−1.78715	−0.219983
$67$	0.552007	0.0674383	0.0337192	−	0.999431i	$-0.489265\pi$
$67$	0.552007	0.0674383	0.0337192	+	0.999431i	$0.489265\pi$
$68$	−2.60252	−0.315602
$69$	−6.87015	−0.827069
$70$	−10.5367	−1.25937
$71$	1.13699	0.134936	0.0674679	−	0.997721i	$-0.478508\pi$
$71$	1.13699	0.134936	0.0674679	+	0.997721i	$0.478508\pi$
$72$	2.41786	0.284947
$73$	7.92260	0.927270	0.463635	−	0.886026i	$-0.346545\pi$
$73$	7.92260	0.927270	0.463635	+	0.886026i	$0.346545\pi$
$74$	4.90384	0.570059
$75$	−14.0156	−1.61838
$76$	0.456203	0.0523301
$77$	2.08084	0.237134
$78$	0.316114	0.0357928
$79$	−4.54228	−0.511046	−0.255523	−	0.966803i	$-0.582248\pi$
$79$	−4.54228	−0.511046	−0.255523	+	0.966803i	$0.582248\pi$
$80$	11.5984	1.29675
$81$	−5.97467	−0.663852
$82$	0.415702	0.0459066
$83$	−0.326396	−0.0358266	−0.0179133	−	0.999840i	$-0.505702\pi$
$83$	−0.326396	−0.0358266	−0.0179133	+	0.999840i	$0.505702\pi$
$84$	1.28576	0.140287
$85$	−24.9934	−2.71091
$86$	−12.1981	−1.31535
$87$	1.98411	0.212719
$88$	−2.88451	−0.307490
$89$	14.7102	1.55927	0.779637	−	0.626232i	$-0.215404\pi$
$89$	14.7102	1.55927	0.779637	+	0.626232i	$0.215404\pi$
$90$	3.83543	0.404290
$91$	−0.368062	−0.0385834
$92$	−1.83159	−0.190957
$93$	0	0
$94$	7.14183	0.736623
$95$	4.38117	0.449498
$96$	−3.27028	−0.333771
$97$	15.5192	1.57573	0.787867	−	0.615845i	$-0.211185\pi$
$97$	15.5192	1.57573	0.787867	+	0.615845i	$0.211185\pi$
$98$	−2.79707	−0.282546
$99$	−0.757443	−0.0761259
$100$	−3.73657	−0.373657
$101$	4.87360	0.484941	0.242471	−	0.970159i	$-0.422042\pi$
$101$	4.87360	0.484941	0.242471	+	0.970159i	$0.422042\pi$
$102$	−12.3643	−1.22425
$103$	1.93062	0.190230	0.0951148	−	0.995466i	$-0.469678\pi$
$103$	1.93062	0.190230	0.0951148	+	0.995466i	$0.469678\pi$
$104$	0.510215	0.0500307
$105$	12.3478	1.20502
$106$	9.28792	0.902122
$107$	12.5275	1.21108	0.605540	−	0.795815i	$-0.292957\pi$
$107$	12.5275	1.21108	0.605540	+	0.795815i	$0.292957\pi$
$108$	−2.23015	−0.214596
$109$	−10.2625	−0.982968	−0.491484	−	0.870887i	$-0.663545\pi$
$109$	−10.2625	−0.982968	−0.491484	+	0.870887i	$0.663545\pi$
$110$	−4.57568	−0.436274
$111$	−5.74675	−0.545457
$112$	−6.68072	−0.631268
$113$	−3.84294	−0.361514	−0.180757	−	0.983528i	$-0.557855\pi$
$113$	−3.84294	−0.361514	−0.180757	+	0.983528i	$0.557855\pi$
$114$	2.16738	0.202994
$115$	−17.5898	−1.64026
$116$	0.528966	0.0491132
$117$	0.133977	0.0123862
$118$	−3.36045	−0.309354
$119$	14.3962	1.31970
$120$	−17.1168	−1.56254
$121$	−10.0964	−0.917852
$122$	2.21614	0.200640
$123$	−0.487156	−0.0439254
$124$	0	0
$125$	−16.8827	−1.51003
$126$	−2.20922	−0.196813
$127$	−7.34389	−0.651665	−0.325832	−	0.945428i	$-0.605644\pi$
$127$	−7.34389	−0.651665	−0.325832	+	0.945428i	$0.605644\pi$
$128$	6.85938	0.606290
$129$	14.2948	1.25859
$130$	0.809352	0.0709849
$131$	13.0337	1.13876	0.569379	−	0.822075i	$-0.307184\pi$
$131$	13.0337	1.13876	0.569379	+	0.822075i	$0.307184\pi$
$132$	0.558355	0.0485986
$133$	−2.52356	−0.218820
$134$	0.699172	0.0603993
$135$	−21.4173	−1.84331
$136$	−19.9564	−1.71124
$137$	0.134299	0.0114740	0.00573698	−	0.999984i	$-0.498174\pi$
$137$	0.134299	0.0114740	0.00573698	+	0.999984i	$0.498174\pi$
$138$	−8.70174	−0.740742
$139$	−11.5275	−0.977749	−0.488874	−	0.872354i	$-0.662592\pi$
$139$	−11.5275	−0.977749	−0.488874	+	0.872354i	$0.662592\pi$
$140$	3.29195	0.278220
$141$	−8.36942	−0.704832
$142$	1.44011	0.120851
$143$	−0.159835	−0.0133661
$144$	2.43183	0.202653
$145$	5.07995	0.421867
$146$	10.0348	0.830484
$147$	3.27785	0.270352
$148$	−1.53209	−0.125937
$149$	−5.44108	−0.445751	−0.222875	−	0.974847i	$-0.571544\pi$
$149$	−5.44108	−0.445751	−0.222875	+	0.974847i	$0.571544\pi$
$150$	−17.7521	−1.44945
$151$	−13.6892	−1.11401	−0.557005	−	0.830509i	$-0.688050\pi$
$151$	−13.6892	−1.11401	−0.557005	+	0.830509i	$0.688050\pi$
$152$	3.49821	0.283742
$153$	−5.24034	−0.423656
$154$	2.63560	0.212383
$155$	0	0
$156$	−0.0987625	−0.00790733
$157$	−15.0553	−1.20155	−0.600774	−	0.799419i	$-0.705141\pi$
$157$	−15.0553	−1.20155	−0.600774	+	0.799419i	$0.705141\pi$
$158$	−5.75326	−0.457705
$159$	−10.8844	−0.863189
$160$	−8.37296	−0.661941
$161$	10.1317	0.798493
$162$	−7.56752	−0.594561
$163$	17.0381	1.33453	0.667263	−	0.744822i	$-0.267466\pi$
$163$	17.0381	1.33453	0.667263	+	0.744822i	$0.267466\pi$
$164$	−0.129877	−0.0101417
$165$	5.36219	0.417446
$166$	−0.413414	−0.0320871
$167$	16.1982	1.25346	0.626729	−	0.779238i	$-0.284393\pi$
$167$	16.1982	1.25346	0.626729	+	0.779238i	$0.284393\pi$
$168$	9.85930	0.760661
$169$	−12.9717	−0.997825
$170$	−31.6567	−2.42796
$171$	0.918595	0.0702467
$172$	3.81102	0.290587
$173$	−2.33122	−0.177239	−0.0886196	−	0.996066i	$-0.528246\pi$
$173$	−2.33122	−0.177239	−0.0886196	+	0.996066i	$0.528246\pi$
$174$	2.51307	0.190515
$175$	20.6694	1.56246
$176$	−2.90118	−0.218685
$177$	3.93806	0.296003
$178$	18.6319	1.39652
$179$	16.9706	1.26844	0.634222	−	0.773151i	$-0.281321\pi$
$179$	16.9706	1.26844	0.634222	+	0.773151i	$0.281321\pi$
$180$	−1.19829	−0.0893155
$181$	7.33576	0.545263	0.272631	−	0.962119i	$-0.412106\pi$
$181$	7.33576	0.545263	0.272631	+	0.962119i	$0.412106\pi$
$182$	−0.466188	−0.0345561
$183$	−2.59707	−0.191981
$184$	−14.0448	−1.03540
$185$	−14.7135	−1.08176
$186$	0	0
$187$	6.25174	0.457173
$188$	−2.23130	−0.162734
$189$	12.3364	0.897341
$190$	5.54919	0.402581
$191$	7.82276	0.566035	0.283018	−	0.959115i	$-0.408665\pi$
$191$	7.82276	0.566035	0.283018	+	0.959115i	$0.408665\pi$
$192$	−13.2023	−0.952793
$193$	−4.63151	−0.333383	−0.166692	−	0.986009i	$-0.553308\pi$
$193$	−4.63151	−0.333383	−0.166692	+	0.986009i	$0.553308\pi$
$194$	19.6566	1.41126
$195$	−0.948470	−0.0679213
$196$	0.873880	0.0624200
$197$	22.3844	1.59482	0.797411	−	0.603437i	$-0.206203\pi$
$197$	22.3844	1.59482	0.797411	+	0.603437i	$0.206203\pi$
$198$	−0.959379	−0.0681801
$199$	26.6038	1.88589	0.942945	−	0.332948i	$-0.108043\pi$
$199$	26.6038	1.88589	0.942945	+	0.332948i	$0.108043\pi$
$200$	−28.6524	−2.02603
$201$	−0.819352	−0.0577926
$202$	6.17291	0.434324
$203$	−2.92606	−0.205369
$204$	3.86296	0.270461
$205$	−1.24728	−0.0871135
$206$	2.44533	0.170374
$207$	−3.68803	−0.256336
$208$	0.513165	0.0355816
$209$	−1.09589	−0.0758040
$210$	15.6398	1.07925
$211$	1.32614	0.0912951	0.0456476	−	0.998958i	$-0.485465\pi$
$211$	1.32614	0.0912951	0.0456476	+	0.998958i	$0.485465\pi$
$212$	−2.90180	−0.199296
$213$	−1.68765	−0.115636
$214$	15.8674	1.08467
$215$	36.5993	2.49605
$216$	−17.1010	−1.16357
$217$	0	0
$218$	−12.9985	−0.880368
$219$	−11.7596	−0.794642
$220$	1.42957	0.0963814
$221$	−1.10581	−0.0743851
$222$	−7.27884	−0.488524
$223$	12.1199	0.811612	0.405806	−	0.913959i	$-0.366991\pi$
$223$	12.1199	0.811612	0.405806	+	0.913959i	$0.366991\pi$
$224$	4.82284	0.322239
$225$	−7.52382	−0.501588
$226$	−4.86748	−0.323780
$227$	−15.9022	−1.05546	−0.527732	−	0.849411i	$-0.676957\pi$
$227$	−15.9022	−1.05546	−0.527732	+	0.849411i	$0.676957\pi$
$228$	−0.677149	−0.0448453
$229$	16.2986	1.07704	0.538521	−	0.842612i	$-0.318983\pi$
$229$	16.2986	1.07704	0.538521	+	0.842612i	$0.318983\pi$
$230$	−22.2792	−1.46905
$231$	−3.08863	−0.203217
$232$	4.05616	0.266300
$233$	−5.12296	−0.335616	−0.167808	−	0.985820i	$-0.553669\pi$
$233$	−5.12296	−0.335616	−0.167808	+	0.985820i	$0.553669\pi$
$234$	0.169696	0.0110934
$235$	−21.4284	−1.39783
$236$	1.04989	0.0683423
$237$	6.74217	0.437951
$238$	18.2343	1.18195
$239$	−8.56800	−0.554218	−0.277109	−	0.960839i	$-0.589376\pi$
$239$	−8.56800	−0.554218	−0.277109	+	0.960839i	$0.589376\pi$
$240$	−17.2157	−1.11127
$241$	6.84676	0.441039	0.220519	−	0.975383i	$-0.429225\pi$
$241$	6.84676	0.441039	0.220519	+	0.975383i	$0.429225\pi$
$242$	−12.7881	−0.822049
$243$	−8.03869	−0.515682
$244$	−0.692382	−0.0443252
$245$	8.39234	0.536167
$246$	−0.617033	−0.0393406
$247$	0.193842	0.0123338
$248$	0	0
$249$	0.484474	0.0307023
$250$	−21.3836	−1.35242
$251$	−22.6661	−1.43067	−0.715336	−	0.698781i	$-0.753726\pi$
$251$	−22.6661	−1.43067	−0.715336	+	0.698781i	$0.753726\pi$
$252$	0.690219	0.0434797
$253$	4.39983	0.276615
$254$	−9.30178	−0.583645
$255$	37.0981	2.32317
$256$	−9.10097	−0.568810
$257$	−16.6584	−1.03912	−0.519560	−	0.854434i	$-0.673904\pi$
$257$	−16.6584	−1.03912	−0.519560	+	0.854434i	$0.673904\pi$
$258$	18.1058	1.12722
$259$	8.47500	0.526611
$260$	−0.252864	−0.0156819
$261$	1.06511	0.0659285
$262$	16.5085	1.01990
$263$	24.5864	1.51606	0.758030	−	0.652220i	$-0.226162\pi$
$263$	24.5864	1.51606	0.758030	+	0.652220i	$0.226162\pi$
$264$	4.28152	0.263509
$265$	−27.8675	−1.71189
$266$	−3.19634	−0.195980
$267$	−21.8345	−1.33625
$268$	−0.218440	−0.0133434
$269$	−12.5141	−0.763001	−0.381501	−	0.924369i	$-0.624593\pi$
$269$	−12.5141	−0.763001	−0.381501	+	0.924369i	$0.624593\pi$
$270$	−27.1272	−1.65091
$271$	27.1673	1.65030	0.825148	−	0.564917i	$-0.191092\pi$
$271$	27.1673	1.65030	0.825148	+	0.564917i	$0.191092\pi$
$272$	−20.0717	−1.21703
$273$	0.546320	0.0330648
$274$	0.170104	0.0102763
$275$	8.97594	0.541270
$276$	2.71866	0.163644
$277$	15.1573	0.910716	0.455358	−	0.890308i	$-0.349511\pi$
$277$	15.1573	0.910716	0.455358	+	0.890308i	$0.349511\pi$
$278$	−14.6007	−0.875694
$279$	0	0
$280$	25.2430	1.50855
$281$	30.2477	1.80443	0.902214	−	0.431289i	$-0.141941\pi$
$281$	30.2477	1.80443	0.902214	+	0.431289i	$0.141941\pi$
$282$	−10.6007	−0.631264
$283$	3.14532	0.186970	0.0934850	−	0.995621i	$-0.470199\pi$
$283$	3.14532	0.186970	0.0934850	+	0.995621i	$0.470199\pi$
$284$	−0.449930	−0.0266984
$285$	−6.50303	−0.385206
$286$	−0.202448	−0.0119710
$287$	0.718432	0.0424077
$288$	−1.75555	−0.103447
$289$	26.2524	1.54426
$290$	6.43427	0.377833
$291$	−23.0354	−1.35036
$292$	−3.13514	−0.183470
$293$	1.90008	0.111004	0.0555020	−	0.998459i	$-0.482324\pi$
$293$	1.90008	0.111004	0.0555020	+	0.998459i	$0.482324\pi$
$294$	4.15173	0.242134
$295$	10.0827	0.587038
$296$	−11.7482	−0.682851
$297$	5.35723	0.310858
$298$	−6.89168	−0.399224
$299$	−0.778247	−0.0450072
$300$	5.54624	0.320212
$301$	−21.0812	−1.21510
$302$	−17.3387	−0.997733
$303$	−7.23395	−0.415580
$304$	3.51843	0.201796
$305$	−6.64932	−0.380739
$306$	−6.63742	−0.379436
$307$	22.7281	1.29716	0.648580	−	0.761146i	$-0.275363\pi$
$307$	22.7281	1.29716	0.648580	+	0.761146i	$0.275363\pi$
$308$	−0.823433	−0.0469194
$309$	−2.86565	−0.163021
$310$	0	0
$311$	15.8754	0.900213	0.450106	−	0.892975i	$-0.351386\pi$
$311$	15.8754	0.900213	0.450106	+	0.892975i	$0.351386\pi$
$312$	−0.757320	−0.0428748
$313$	−6.34935	−0.358886	−0.179443	−	0.983768i	$-0.557430\pi$
$313$	−6.34935	−0.358886	−0.179443	+	0.983768i	$0.557430\pi$
$314$	−19.0691	−1.07613
$315$	6.62855	0.373476
$316$	1.79747	0.101116
$317$	−15.3010	−0.859390	−0.429695	−	0.902974i	$-0.641379\pi$
$317$	−15.3010	−0.859390	−0.429695	+	0.902974i	$0.641379\pi$
$318$	−13.7862	−0.773092
$319$	−1.27068	−0.0711442
$320$	−33.8021	−1.88959
$321$	−18.5948	−1.03786
$322$	12.8329	0.715148
$323$	−7.58184	−0.421865
$324$	2.36430	0.131350
$325$	−1.58767	−0.0880683
$326$	21.5805	1.19523
$327$	15.2328	0.842374
$328$	−0.995906	−0.0549897
$329$	12.3428	0.680480
$330$	6.79176	0.373874
$331$	8.96795	0.492923	0.246462	−	0.969153i	$-0.420732\pi$
$331$	8.96795	0.492923	0.246462	+	0.969153i	$0.420732\pi$
$332$	0.129162	0.00708866
$333$	−3.08497	−0.169055
$334$	20.5167	1.12262
$335$	−2.09780	−0.114615
$336$	9.91629	0.540978
$337$	4.36877	0.237982	0.118991	−	0.992895i	$-0.462034\pi$
$337$	4.36877	0.237982	0.118991	+	0.992895i	$0.462034\pi$
$338$	−16.4300	−0.893675
$339$	5.70414	0.309806
$340$	9.89040	0.536382
$341$	0	0
$342$	1.16349	0.0629145
$343$	−20.1569	−1.08837
$344$	29.2232	1.57561
$345$	26.1088	1.40565
$346$	−2.95272	−0.158739
$347$	−24.4053	−1.31015	−0.655073	−	0.755566i	$-0.727362\pi$
$347$	−24.4053	−1.31015	−0.655073	+	0.755566i	$0.727362\pi$
$348$	−0.785152	−0.0420885
$349$	13.0439	0.698224	0.349112	−	0.937081i	$-0.386483\pi$
$349$	13.0439	0.698224	0.349112	+	0.937081i	$0.386483\pi$
$350$	26.1799	1.39937
$351$	−0.947594	−0.0505788
$352$	2.09438	0.111631
$353$	−10.8248	−0.576146	−0.288073	−	0.957608i	$-0.593015\pi$
$353$	−10.8248	−0.576146	−0.288073	+	0.957608i	$0.593015\pi$
$354$	4.98796	0.265107
$355$	−4.32092	−0.229331
$356$	−5.82111	−0.308518
$357$	−21.3685	−1.13094
$358$	21.4950	1.13605
$359$	33.7834	1.78302	0.891510	−	0.453001i	$-0.149647\pi$
$359$	33.7834	1.78302	0.891510	+	0.453001i	$0.149647\pi$
$360$	−9.18863	−0.484283
$361$	−17.6710	−0.930050
$362$	9.29149	0.488350
$363$	14.9862	0.786571
$364$	0.145650	0.00763412
$365$	−30.1084	−1.57595
$366$	−3.28945	−0.171942
$367$	−22.6313	−1.18135	−0.590673	−	0.806911i	$-0.701138\pi$
$367$	−22.6313	−1.18135	−0.590673	+	0.806911i	$0.701138\pi$
$368$	−14.1260	−0.736369
$369$	−0.261515	−0.0136139
$370$	−18.6362	−0.968847
$371$	16.0517	0.833365
$372$	0	0
$373$	−32.9720	−1.70723	−0.853613	−	0.520908i	$-0.825593\pi$
$373$	−32.9720	−1.70723	−0.853613	+	0.520908i	$0.825593\pi$
$374$	7.91846	0.409454
$375$	25.0592	1.29405
$376$	−17.1098	−0.882372
$377$	0.224759	0.0115757
$378$	15.6253	0.803679
$379$	−32.4919	−1.66900	−0.834498	−	0.551012i	$-0.814242\pi$
$379$	−32.4919	−1.66900	−0.834498	+	0.551012i	$0.814242\pi$
$380$	−1.73372	−0.0889379
$381$	10.9006	0.558457
$382$	9.90832	0.506954
$383$	26.7804	1.36842	0.684208	−	0.729287i	$-0.260148\pi$
$383$	26.7804	1.36842	0.684208	+	0.729287i	$0.260148\pi$
$384$	−10.1815	−0.519572
$385$	−7.90787	−0.403023
$386$	−5.86627	−0.298585
$387$	7.67373	0.390078
$388$	−6.14126	−0.311775
$389$	17.7375	0.899327	0.449663	−	0.893198i	$-0.351544\pi$
$389$	17.7375	0.899327	0.449663	+	0.893198i	$0.351544\pi$
$390$	−1.20133	−0.0608319
$391$	30.4400	1.53942
$392$	6.70099	0.338451
$393$	−19.3461	−0.975880
$394$	28.3521	1.42836
$395$	17.2621	0.868552
$396$	0.299736	0.0150623
$397$	9.94953	0.499352	0.249676	−	0.968329i	$-0.419676\pi$
$397$	9.94953	0.499352	0.249676	+	0.968329i	$0.419676\pi$
$398$	33.6964	1.68905
$399$	3.74576	0.187522
$400$	−28.8180	−1.44090
$401$	−19.8956	−0.993538	−0.496769	−	0.867883i	$-0.665480\pi$
$401$	−19.8956	−0.993538	−0.496769	+	0.867883i	$0.665480\pi$
$402$	−1.03779	−0.0517604
$403$	0	0
$404$	−1.92858	−0.0959506
$405$	22.7057	1.12825
$406$	−3.70615	−0.183933
$407$	3.68037	0.182429
$408$	29.6215	1.46648
$409$	24.3255	1.20282	0.601410	−	0.798941i	$-0.294606\pi$
$409$	24.3255	1.20282	0.601410	+	0.798941i	$0.294606\pi$
$410$	−1.57980	−0.0780208
$411$	−0.199342	−0.00983284
$412$	−0.763986	−0.0376389
$413$	−5.80765	−0.285776
$414$	−4.67127	−0.229580
$415$	1.24041	0.0608893
$416$	−0.370456	−0.0181631
$417$	17.1104	0.837901
$418$	−1.38805	−0.0678918
$419$	−4.40675	−0.215284	−0.107642	−	0.994190i	$-0.534330\pi$
$419$	−4.40675	−0.215284	−0.107642	+	0.994190i	$0.534330\pi$
$420$	−4.88628	−0.238426
$421$	−12.4237	−0.605493	−0.302746	−	0.953071i	$-0.597904\pi$
$421$	−12.4237	−0.605493	−0.302746	+	0.953071i	$0.597904\pi$
$422$	1.67969	0.0817660
$423$	−4.49287	−0.218451
$424$	−22.2513	−1.08062
$425$	62.0997	3.01228
$426$	−2.13758	−0.103566
$427$	3.83002	0.185348
$428$	−4.95739	−0.239625
$429$	0.237246	0.0114544
$430$	46.3567	2.23552
$431$	−18.5653	−0.894259	−0.447130	−	0.894469i	$-0.647554\pi$
$431$	−18.5653	−0.894259	−0.447130	+	0.894469i	$0.647554\pi$
$432$	−17.1998	−0.827527
$433$	36.1204	1.73584	0.867918	−	0.496708i	$-0.165458\pi$
$433$	36.1204	1.73584	0.867918	+	0.496708i	$0.165458\pi$
$434$	0	0
$435$	−7.54024	−0.361527
$436$	4.06108	0.194490
$437$	−5.33593	−0.255252
$438$	−14.8948	−0.711699
$439$	−19.0094	−0.907268	−0.453634	−	0.891188i	$-0.649873\pi$
$439$	−19.0094	−0.907268	−0.453634	+	0.891188i	$0.649873\pi$
$440$	10.9621	0.522596
$441$	1.75961	0.0837911
$442$	−1.40063	−0.0666210
$443$	12.4540	0.591709	0.295854	−	0.955233i	$-0.404396\pi$
$443$	12.4540	0.591709	0.295854	+	0.955233i	$0.404396\pi$
$444$	2.27411	0.107924
$445$	−55.9033	−2.65007
$446$	15.3511	0.726898
$447$	8.07627	0.381995
$448$	19.4700	0.919873
$449$	−25.4366	−1.20043	−0.600214	−	0.799840i	$-0.704918\pi$
$449$	−25.4366	−1.20043	−0.600214	+	0.799840i	$0.704918\pi$
$450$	−9.52968	−0.449234
$451$	0.311988	0.0146909
$452$	1.52073	0.0715292
$453$	20.3191	0.954673
$454$	−20.1417	−0.945297
$455$	1.39875	0.0655746
$456$	−5.19244	−0.243159
$457$	−31.9740	−1.49568	−0.747839	−	0.663880i	$-0.768909\pi$
$457$	−31.9740	−1.49568	−0.747839	+	0.663880i	$0.768909\pi$
$458$	20.6438	0.964623
$459$	37.0638	1.72999
$460$	6.96064	0.324542
$461$	−19.3501	−0.901223	−0.450612	−	0.892720i	$-0.648794\pi$
$461$	−19.3501	−0.901223	−0.450612	+	0.892720i	$0.648794\pi$
$462$	−3.91206	−0.182006
$463$	−0.0414625	−0.00192693	−0.000963463	−	1.00000i	$-0.500307\pi$
$463$	−0.0414625	−0.00192693	−0.000963463		1.00000i	$0.500307\pi$
$464$	4.07961	0.189391
$465$	0	0
$466$	−6.48875	−0.300586
$467$	25.3083	1.17113	0.585564	−	0.810626i	$-0.300873\pi$
$467$	25.3083	1.17113	0.585564	+	0.810626i	$0.300873\pi$
$468$	−0.0530176	−0.00245074
$469$	1.20834	0.0557958
$470$	−27.1412	−1.25193
$471$	22.3469	1.02969
$472$	8.05069	0.370563
$473$	−9.15478	−0.420937
$474$	8.53965	0.392239
$475$	−10.8856	−0.499467
$476$	−5.69688	−0.261116
$477$	−5.84296	−0.267531
$478$	−10.8522	−0.496370
$479$	−21.2411	−0.970533	−0.485266	−	0.874366i	$-0.661277\pi$
$479$	−21.2411	−0.970533	−0.485266	+	0.874366i	$0.661277\pi$
$480$	12.4281	0.567263
$481$	−0.650988	−0.0296825
$482$	8.67212	0.395004
$483$	−15.0387	−0.684284
$484$	3.99534	0.181606
$485$	−58.9779	−2.67805
$486$	−10.1818	−0.461857
$487$	−26.7620	−1.21270	−0.606350	−	0.795198i	$-0.707367\pi$
$487$	−26.7620	−1.21270	−0.606350	+	0.795198i	$0.707367\pi$
$488$	−5.30925	−0.240338
$489$	−25.2899	−1.14365
$490$	10.6298	0.480203
$491$	12.5664	0.567113	0.283557	−	0.958955i	$-0.408486\pi$
$491$	12.5664	0.567113	0.283557	+	0.958955i	$0.408486\pi$
$492$	0.192778	0.00869109
$493$	−8.79111	−0.395932
$494$	0.245520	0.0110465
$495$	2.87853	0.129380
$496$	0	0
$497$	2.48886	0.111640
$498$	0.613636	0.0274977
$499$	23.0304	1.03098	0.515490	−	0.856895i	$-0.327610\pi$
$499$	23.0304	1.03098	0.515490	+	0.856895i	$0.327610\pi$
$500$	6.68083	0.298776
$501$	−24.0433	−1.07417
$502$	−28.7089	−1.28134
$503$	0.661578	0.0294983	0.0147492	−	0.999891i	$-0.495305\pi$
$503$	0.661578	0.0294983	0.0147492	+	0.999891i	$0.495305\pi$
$504$	5.29266	0.235754
$505$	−18.5212	−0.824184
$506$	5.57283	0.247743
$507$	19.2541	0.855106
$508$	2.90613	0.128939
$509$	11.6674	0.517149	0.258574	−	0.965991i	$-0.416747\pi$
$509$	11.6674	0.517149	0.258574	+	0.965991i	$0.416747\pi$
$510$	46.9885	2.08068
$511$	17.3425	0.767186
$512$	−25.2461	−1.11573
$513$	−6.49702	−0.286851
$514$	−21.0995	−0.930659
$515$	−7.33697	−0.323306
$516$	−5.65675	−0.249024
$517$	5.36001	0.235733
$518$	10.7344	0.471644
$519$	3.46026	0.151889
$520$	−1.93898	−0.0850300
$521$	−31.9184	−1.39837	−0.699186	−	0.714940i	$-0.746454\pi$
$521$	−31.9184	−1.39837	−0.699186	+	0.714940i	$0.746454\pi$
$522$	1.34907	0.0590470
$523$	0.00415040	0.000181484 0	9.07421e−5	−	1.00000i	$-0.499971\pi$
$523$	0.00415040	0.000181484 0	9.07421e−5		1.00000i	$0.499971\pi$
$524$	−5.15769	−0.225315
$525$	−30.6799	−1.33898
$526$	31.1411	1.35782
$527$	0	0
$528$	4.30627	0.187406
$529$	−1.57699	−0.0685646
$530$	−35.2971	−1.53321
$531$	2.11403	0.0917411
$532$	0.998624	0.0432958
$533$	−0.0551848	−0.00239032
$534$	−27.6556	−1.19678
$535$	−47.6086	−2.05830
$536$	−1.67502	−0.0723499
$537$	−25.1898	−1.08702
$538$	−15.8504	−0.683361
$539$	−2.09922	−0.0904200
$540$	8.47527	0.364718
$541$	29.2750	1.25863	0.629316	−	0.777150i	$-0.283335\pi$
$541$	29.2750	1.25863	0.629316	+	0.777150i	$0.283335\pi$
$542$	34.4101	1.47804
$543$	−10.8886	−0.467274
$544$	14.4898	0.621247
$545$	39.0007	1.67061
$546$	0.691970	0.0296136
$547$	41.1653	1.76010	0.880050	−	0.474880i	$-0.157509\pi$
$547$	41.1653	1.76010	0.880050	+	0.474880i	$0.157509\pi$
$548$	−0.0531450	−0.00227024
$549$	−1.39416	−0.0595011
$550$	11.3689	0.484773
$551$	1.54102	0.0656497
$552$	20.8469	0.887305
$553$	−9.94300	−0.422820
$554$	19.1983	0.815658
$555$	21.8395	0.927034
$556$	4.56167	0.193458
$557$	5.73810	0.243131	0.121566	−	0.992583i	$-0.461209\pi$
$557$	5.73810	0.243131	0.121566	+	0.992583i	$0.461209\pi$
$558$	0	0
$559$	1.61931	0.0684894
$560$	25.3889	1.07288
$561$	−9.27955	−0.391783
$562$	38.3118	1.61609
$563$	−14.2942	−0.602428	−0.301214	−	0.953557i	$-0.597392\pi$
$563$	−14.2942	−0.602428	−0.301214	+	0.953557i	$0.597392\pi$
$564$	3.31195	0.139458
$565$	14.6044	0.614412
$566$	3.98387	0.167455
$567$	−13.0785	−0.549245
$568$	−3.45011	−0.144763
$569$	−14.3618	−0.602078	−0.301039	−	0.953612i	$-0.597333\pi$
$569$	−14.3618	−0.602078	−0.301039	+	0.953612i	$0.597333\pi$
$570$	−8.23675	−0.345000
$571$	−33.3790	−1.39687	−0.698434	−	0.715674i	$-0.746119\pi$
$571$	−33.3790	−1.39687	−0.698434	+	0.715674i	$0.746119\pi$
$572$	0.0632502	0.00264462
$573$	−11.6114	−0.485075
$574$	0.909967	0.0379813
$575$	43.7043	1.82260
$576$	−7.08725	−0.295302
$577$	−22.4796	−0.935838	−0.467919	−	0.883771i	$-0.654996\pi$
$577$	−22.4796	−0.935838	−0.467919	+	0.883771i	$0.654996\pi$
$578$	33.2513	1.38307
$579$	6.87461	0.285699
$580$	−2.01024	−0.0834706
$581$	−0.714477	−0.0296415
$582$	−29.1766	−1.20941
$583$	6.97067	0.288695
$584$	−24.0405	−0.994804
$585$	−0.509157	−0.0210511
$586$	2.40665	0.0994176
$587$	6.41617	0.264824	0.132412	−	0.991195i	$-0.457728\pi$
$587$	6.41617	0.264824	0.132412	+	0.991195i	$0.457728\pi$
$588$	−1.29711	−0.0534920
$589$	0	0
$590$	12.7708	0.525764
$591$	−33.2255	−1.36671
$592$	−11.8161	−0.485640
$593$	−20.8601	−0.856623	−0.428312	−	0.903631i	$-0.640891\pi$
$593$	−20.8601	−0.856623	−0.428312	+	0.903631i	$0.640891\pi$
$594$	6.78548	0.278412
$595$	−54.7103	−2.24290
$596$	2.15315	0.0881963
$597$	−39.4884	−1.61615
$598$	−0.985729	−0.0403095
$599$	−11.5584	−0.472263	−0.236131	−	0.971721i	$-0.575880\pi$
$599$	−11.5584	−0.472263	−0.236131	+	0.971721i	$0.575880\pi$
$600$	42.5291	1.73624
$601$	−4.31742	−0.176111	−0.0880555	−	0.996116i	$-0.528065\pi$
$601$	−4.31742	−0.176111	−0.0880555	+	0.996116i	$0.528065\pi$
$602$	−26.7015	−1.08827
$603$	−0.439844	−0.0179118
$604$	5.41709	0.220418
$605$	38.3694	1.55994
$606$	−9.16254	−0.372203
$607$	44.9092	1.82281	0.911405	−	0.411510i	$-0.134999\pi$
$607$	44.9092	1.82281	0.911405	+	0.411510i	$0.134999\pi$
$608$	−2.53997	−0.103009
$609$	4.34319	0.175995
$610$	−8.42204	−0.340998
$611$	−0.948084	−0.0383554
$612$	2.07371	0.0838248
$613$	2.60516	0.105222	0.0526108	−	0.998615i	$-0.483246\pi$
$613$	2.60516	0.105222	0.0526108	+	0.998615i	$0.483246\pi$
$614$	28.7874	1.16177
$615$	1.85135	0.0746536
$616$	−6.31416	−0.254405
$617$	−24.9868	−1.00593	−0.502965	−	0.864307i	$-0.667758\pi$
$617$	−24.9868	−1.00593	−0.502965	+	0.864307i	$0.667758\pi$
$618$	−3.62963	−0.146005
$619$	−31.9083	−1.28250	−0.641252	−	0.767330i	$-0.721585\pi$
$619$	−31.9083	−1.28250	−0.641252	+	0.767330i	$0.721585\pi$
$620$	0	0
$621$	26.0847	1.04674
$622$	20.1078	0.806251
$623$	32.2004	1.29008
$624$	−0.761698	−0.0304923
$625$	16.9474	0.677898
$626$	−8.04209	−0.321427
$627$	1.62664	0.0649618
$628$	5.95771	0.237739
$629$	25.4625	1.01526
$630$	8.39573	0.334494
$631$	0.359751	0.0143215	0.00716073	−	0.999974i	$-0.497721\pi$
$631$	0.359751	0.0143215	0.00716073	+	0.999974i	$0.497721\pi$
$632$	13.7832	0.548266
$633$	−1.96841	−0.0782372
$634$	−19.3803	−0.769689
$635$	27.9091	1.10754
$636$	4.30718	0.170791
$637$	0.371313	0.0147120
$638$	−1.60944	−0.0637183
$639$	−0.905963	−0.0358394
$640$	−26.0679	−1.03042
$641$	30.8069	1.21680	0.608400	−	0.793630i	$-0.291812\pi$
$641$	30.8069	1.21680	0.608400	+	0.793630i	$0.291812\pi$
$642$	−23.5522	−0.929530
$643$	1.75184	0.0690856	0.0345428	−	0.999403i	$-0.489002\pi$
$643$	1.75184	0.0690856	0.0345428	+	0.999403i	$0.489002\pi$
$644$	−4.00934	−0.157990
$645$	−54.3248	−2.13904
$646$	−9.60317	−0.377832
$647$	24.6504	0.969106	0.484553	−	0.874762i	$-0.338982\pi$
$647$	24.6504	0.969106	0.484553	+	0.874762i	$0.338982\pi$
$648$	18.1297	0.712201
$649$	−2.52204	−0.0989989
$650$	−2.01095	−0.0788760
$651$	0	0
$652$	−6.74233	−0.264050
$653$	−0.285664	−0.0111789	−0.00558946	−	0.999984i	$-0.501779\pi$
$653$	−0.285664	−0.0111789	−0.00558946	+	0.999984i	$0.501779\pi$
$654$	19.2938	0.754449
$655$	−49.5321	−1.93538
$656$	−1.00166	−0.0391083
$657$	−6.31280	−0.246286
$658$	15.6334	0.609453
$659$	37.4247	1.45786	0.728929	−	0.684589i	$-0.240018\pi$
$659$	37.4247	1.45786	0.728929	+	0.684589i	$0.240018\pi$
$660$	−2.12193	−0.0825960
$661$	4.19157	0.163033	0.0815167	−	0.996672i	$-0.474024\pi$
$661$	4.19157	0.163033	0.0815167	+	0.996672i	$0.474024\pi$
$662$	11.3588	0.441473
$663$	1.64138	0.0637458
$664$	0.990423	0.0384359
$665$	9.59033	0.371897
$666$	−3.90742	−0.151410
$667$	−6.18699	−0.239561
$668$	−6.40998	−0.248009
$669$	−17.9898	−0.695526
$670$	−2.65708	−0.102652
$671$	1.66323	0.0642084
$672$	−7.15861	−0.276149
$673$	−9.97267	−0.384418	−0.192209	−	0.981354i	$-0.561565\pi$
$673$	−9.97267	−0.384418	−0.192209	+	0.981354i	$0.561565\pi$
$674$	5.53349	0.213142
$675$	53.2144	2.04822
$676$	5.13318	0.197430
$677$	47.7577	1.83548	0.917738	−	0.397186i	$-0.130013\pi$
$677$	47.7577	1.83548	0.917738	+	0.397186i	$0.130013\pi$
$678$	7.22487	0.277469
$679$	33.9713	1.30370
$680$	75.8406	2.90835
$681$	23.6038	0.904500
$682$	0	0
$683$	27.7600	1.06221	0.531104	−	0.847307i	$-0.321777\pi$
$683$	27.7600	1.06221	0.531104	+	0.847307i	$0.321777\pi$
$684$	−0.363507	−0.0138990
$685$	−0.510380	−0.0195006
$686$	−25.5308	−0.974770
$687$	−24.1923	−0.922993
$688$	29.3922	1.12057
$689$	−1.23298	−0.0469728
$690$	33.0694	1.25893
$691$	38.0310	1.44677	0.723383	−	0.690447i	$-0.242586\pi$
$691$	38.0310	1.44677	0.723383	+	0.690447i	$0.242586\pi$
$692$	0.922511	0.0350686
$693$	−1.65804	−0.0629836
$694$	−30.9118	−1.17340
$695$	43.8082	1.66174
$696$	−6.02062	−0.228211
$697$	2.15848	0.0817581
$698$	16.5214	0.625346
$699$	7.60409	0.287613
$700$	−8.17931	−0.309149
$701$	41.0562	1.55067	0.775336	−	0.631548i	$-0.217580\pi$
$701$	41.0562	1.55067	0.775336	+	0.631548i	$0.217580\pi$
$702$	−1.20022	−0.0452995
$703$	−4.46340	−0.168340
$704$	8.45511	0.318664
$705$	31.8065	1.19790
$706$	−13.7107	−0.516010
$707$	10.6683	0.401221
$708$	−1.55837	−0.0585673
$709$	−42.0657	−1.57981	−0.789906	−	0.613229i	$-0.789870\pi$
$709$	−42.0657	−1.57981	−0.789906	+	0.613229i	$0.789870\pi$
$710$	−5.47289	−0.205394
$711$	3.61933	0.135736
$712$	−44.6368	−1.67284
$713$	0	0
$714$	−27.0654	−1.01290
$715$	0.607426	0.0227164
$716$	−6.71563	−0.250975
$717$	12.7176	0.474948
$718$	42.7901	1.59691
$719$	33.2690	1.24072	0.620362	−	0.784316i	$-0.286986\pi$
$719$	33.2690	1.24072	0.620362	+	0.784316i	$0.286986\pi$
$720$	−9.24175	−0.344420
$721$	4.22611	0.157388
$722$	−22.3821	−0.832974
$723$	−10.1628	−0.377957
$724$	−2.90291	−0.107886
$725$	−12.6218	−0.468764
$726$	18.9815	0.704471
$727$	−32.7229	−1.21363	−0.606813	−	0.794845i	$-0.707552\pi$
$727$	−32.7229	−1.21363	−0.606813	+	0.794845i	$0.707552\pi$
$728$	1.11686	0.0413934
$729$	29.8560	1.10578
$730$	−38.1354	−1.41145
$731$	−63.3370	−2.34260
$732$	1.02771	0.0379854
$733$	−9.96391	−0.368025	−0.184013	−	0.982924i	$-0.558909\pi$
$733$	−9.96391	−0.368025	−0.184013	+	0.982924i	$0.558909\pi$
$734$	−28.6649	−1.05804
$735$	−12.4569	−0.459479
$736$	10.1976	0.375889
$737$	0.524735	0.0193289
$738$	−0.331235	−0.0121929
$739$	−30.9584	−1.13882	−0.569412	−	0.822052i	$-0.692829\pi$
$739$	−30.9584	−1.13882	−0.569412	+	0.822052i	$0.692829\pi$
$740$	5.82244	0.214037
$741$	−0.287722	−0.0105697
$742$	20.3312	0.746380
$743$	−1.11003	−0.0407231	−0.0203615	−	0.999793i	$-0.506482\pi$
$743$	−1.11003	−0.0407231	−0.0203615	+	0.999793i	$0.506482\pi$
$744$	0	0
$745$	20.6778	0.757578
$746$	−41.7624	−1.52903
$747$	0.260075	0.00951566
$748$	−2.47394	−0.0904563
$749$	27.4226	1.00200
$750$	31.7401	1.15898
$751$	28.1553	1.02740	0.513701	−	0.857969i	$-0.328274\pi$
$751$	28.1553	1.02740	0.513701	+	0.857969i	$0.328274\pi$
$752$	−17.2087	−0.627538
$753$	33.6436	1.22604
$754$	0.284679	0.0103674
$755$	52.0233	1.89332
$756$	−4.88177	−0.177548
$757$	−38.7223	−1.40739	−0.703693	−	0.710504i	$-0.748467\pi$
$757$	−38.7223	−1.40739	−0.703693	+	0.710504i	$0.748467\pi$
$758$	−41.1542	−1.49479
$759$	−6.53074	−0.237051
$760$	−13.2943	−0.482236
$761$	−43.4369	−1.57459	−0.787293	−	0.616580i	$-0.788518\pi$
$761$	−43.4369	−1.57459	−0.787293	+	0.616580i	$0.788518\pi$
$762$	13.8068	0.500166
$763$	−22.4645	−0.813269
$764$	−3.09563	−0.111996
$765$	19.9150	0.720027
$766$	33.9201	1.22558
$767$	0.446102	0.0161078
$768$	13.5087	0.487453
$769$	−3.11019	−0.112156	−0.0560781	−	0.998426i	$-0.517860\pi$
$769$	−3.11019	−0.112156	−0.0560781	+	0.998426i	$0.517860\pi$
$770$	−10.0161	−0.360956
$771$	24.7263	0.890494
$772$	1.83278	0.0659633
$773$	−21.6402	−0.778343	−0.389172	−	0.921165i	$-0.627239\pi$
$773$	−21.6402	−0.778343	−0.389172	+	0.921165i	$0.627239\pi$
$774$	9.71956	0.349362
$775$	0	0
$776$	−47.0918	−1.69050
$777$	−12.5796	−0.451289
$778$	22.4663	0.805457
$779$	−0.378366	−0.0135564
$780$	0.375329	0.0134389
$781$	1.08082	0.0386747
$782$	38.5554	1.37874
$783$	−7.53327	−0.269217
$784$	6.73973	0.240705
$785$	57.2151	2.04210
$786$	−24.5038	−0.874020
$787$	7.87653	0.280768	0.140384	−	0.990097i	$-0.455166\pi$
$787$	7.87653	0.280768	0.140384	+	0.990097i	$0.455166\pi$
$788$	−8.85796	−0.315552
$789$	−36.4939	−1.29922
$790$	21.8642	0.777894
$791$	−8.41216	−0.299102
$792$	2.29840	0.0816702
$793$	−0.294194	−0.0104471
$794$	12.6021	0.447231
$795$	41.3642	1.46704
$796$	−10.5277	−0.373143
$797$	−29.5182	−1.04559	−0.522793	−	0.852459i	$-0.675110\pi$
$797$	−29.5182	−1.04559	−0.522793	+	0.852459i	$0.675110\pi$
$798$	4.74438	0.167949
$799$	37.0830	1.31190
$800$	20.8038	0.735526
$801$	−11.7212	−0.414148
$802$	−25.1998	−0.889835
$803$	7.53119	0.265770
$804$	0.324234	0.0114349
$805$	−38.5039	−1.35708
$806$	0	0
$807$	18.5749	0.653869
$808$	−14.7886	−0.520260
$809$	2.12753	0.0747998	0.0373999	−	0.999300i	$-0.488092\pi$
$809$	2.12753	0.0747998	0.0373999	+	0.999300i	$0.488092\pi$
$810$	28.7590	1.01049
$811$	7.20504	0.253003	0.126502	−	0.991966i	$-0.459625\pi$
$811$	7.20504	0.253003	0.126502	+	0.991966i	$0.459625\pi$
$812$	1.15790	0.0406343
$813$	−40.3248	−1.41425
$814$	4.66157	0.163388
$815$	−64.7502	−2.26810
$816$	29.7927	1.04295
$817$	11.1025	0.388428
$818$	30.8107	1.07727
$819$	0.293275	0.0102479
$820$	0.493572	0.0172363
$821$	−42.0597	−1.46789	−0.733947	−	0.679206i	$-0.762324\pi$
$821$	−42.0597	−1.46789	−0.733947	+	0.679206i	$0.762324\pi$
$822$	−0.252487	−0.00880651
$823$	30.1730	1.05176	0.525882	−	0.850558i	$-0.323735\pi$
$823$	30.1730	1.05176	0.525882	+	0.850558i	$0.323735\pi$
$824$	−5.85832	−0.204084
$825$	−13.3231	−0.463852
$826$	−7.35598	−0.255947
$827$	3.87046	0.134589	0.0672945	−	0.997733i	$-0.478563\pi$
$827$	3.87046	0.134589	0.0672945	+	0.997733i	$0.478563\pi$
$828$	1.45943	0.0507187
$829$	−13.1716	−0.457469	−0.228735	−	0.973489i	$-0.573459\pi$
$829$	−13.1716	−0.457469	−0.228735	+	0.973489i	$0.573459\pi$
$830$	1.57110	0.0545338
$831$	−22.4983	−0.780456
$832$	−1.49555	−0.0518488
$833$	−14.5234	−0.503206
$834$	21.6721	0.750443
$835$	−61.5585	−2.13032
$836$	0.433665	0.0149986
$837$	0	0
$838$	−5.58159	−0.192813
$839$	−34.2490	−1.18241	−0.591204	−	0.806522i	$-0.701347\pi$
$839$	−34.2490	−1.18241	−0.591204	+	0.806522i	$0.701347\pi$
$840$	−37.4685	−1.29279
$841$	−27.2132	−0.938386
$842$	−15.7358	−0.542293
$843$	−44.8971	−1.54634
$844$	−0.524780	−0.0180637
$845$	49.2967	1.69586
$846$	−5.69067	−0.195649
$847$	−22.1008	−0.759394
$848$	−22.3799	−0.768528
$849$	−4.66865	−0.160228
$850$	78.6555	2.69786
$851$	17.9199	0.614287
$852$	0.667838	0.0228797
$853$	−14.4479	−0.494686	−0.247343	−	0.968928i	$-0.579557\pi$
$853$	−14.4479	−0.494686	−0.247343	+	0.968928i	$0.579557\pi$
$854$	4.85111	0.166001
$855$	−3.49096	−0.119388
$856$	−38.0138	−1.29928
$857$	27.8331	0.950759	0.475379	−	0.879781i	$-0.342311\pi$
$857$	27.8331	0.950759	0.475379	+	0.879781i	$0.342311\pi$
$858$	0.300496	0.0102588
$859$	47.3202	1.61454	0.807272	−	0.590180i	$-0.200943\pi$
$859$	47.3202	1.61454	0.807272	+	0.590180i	$0.200943\pi$
$860$	−14.4831	−0.493869
$861$	−1.06638	−0.0363421
$862$	−23.5148	−0.800919
$863$	−26.0534	−0.886867	−0.443434	−	0.896307i	$-0.646240\pi$
$863$	−26.0534	−0.886867	−0.443434	+	0.896307i	$0.646240\pi$
$864$	12.4166	0.422422
$865$	8.85937	0.301228
$866$	45.7502	1.55465
$867$	−38.9668	−1.32338
$868$	0	0
$869$	−4.31787	−0.146474
$870$	−9.55048	−0.323792
$871$	−0.0928157	−0.00314494
$872$	31.1407	1.05456
$873$	−12.3658	−0.418520
$874$	−6.75849	−0.228609
$875$	−36.9560	−1.24934
$876$	4.65353	0.157228
$877$	−34.3215	−1.15895	−0.579477	−	0.814989i	$-0.696743\pi$
$877$	−34.3215	−1.15895	−0.579477	+	0.814989i	$0.696743\pi$
$878$	−24.0773	−0.812570
$879$	−2.82032	−0.0951270
$880$	11.0254	0.371667
$881$	3.63467	0.122455	0.0612276	−	0.998124i	$-0.480498\pi$
$881$	3.63467	0.122455	0.0612276	+	0.998124i	$0.480498\pi$
$882$	2.22873	0.0750452
$883$	−39.4574	−1.32785	−0.663923	−	0.747801i	$-0.731110\pi$
$883$	−39.4574	−1.32785	−0.663923	+	0.747801i	$0.731110\pi$
$884$	0.437594	0.0147179
$885$	−14.9659	−0.503074
$886$	15.7743	0.529948
$887$	−42.3027	−1.42039	−0.710194	−	0.704007i	$-0.751393\pi$
$887$	−42.3027	−1.42039	−0.710194	+	0.704007i	$0.751393\pi$
$888$	17.4381	0.585183
$889$	−16.0757	−0.539161
$890$	−70.8072	−2.37346
$891$	−5.67949	−0.190270
$892$	−4.79611	−0.160586
$893$	−6.50038	−0.217527
$894$	10.2294	0.342123
$895$	−64.4938	−2.15579
$896$	15.0151	0.501620
$897$	1.15516	0.0385698
$898$	−32.2180	−1.07513
$899$	0	0
$900$	2.97733	0.0992444
$901$	48.2262	1.60665
$902$	0.395164	0.0131575
$903$	31.2912	1.04131
$904$	11.6611	0.387843
$905$	−27.8782	−0.926704
$906$	25.7362	0.855027
$907$	−32.3680	−1.07476	−0.537381	−	0.843339i	$-0.680586\pi$
$907$	−32.3680	−1.07476	−0.537381	+	0.843339i	$0.680586\pi$
$908$	6.29281	0.208834
$909$	−3.88333	−0.128802
$910$	1.77166	0.0587301
$911$	27.0607	0.896563	0.448281	−	0.893893i	$-0.352036\pi$
$911$	27.0607	0.896563	0.448281	+	0.893893i	$0.352036\pi$
$912$	−5.22246	−0.172933
$913$	−0.310271	−0.0102685
$914$	−40.4983	−1.33956
$915$	9.86969	0.326282
$916$	−6.44969	−0.213104
$917$	28.5306	0.942162
$918$	46.9451	1.54942
$919$	20.7649	0.684972	0.342486	−	0.939523i	$-0.388731\pi$
$919$	20.7649	0.684972	0.342486	+	0.939523i	$0.388731\pi$
$920$	53.3749	1.75972
$921$	−33.7356	−1.11163
$922$	−24.5088	−0.807156
$923$	−0.191176	−0.00629264
$924$	1.22223	0.0402085
$925$	36.5578	1.20201
$926$	−0.0525165	−0.00172580
$927$	−1.53834	−0.0505256
$928$	−2.94508	−0.0966771
$929$	20.6589	0.677798	0.338899	−	0.940823i	$-0.389945\pi$
$929$	20.6589	0.677798	0.338899	+	0.940823i	$0.389945\pi$
$930$	0	0
$931$	2.54585	0.0834368
$932$	2.02726	0.0664052
$933$	−23.5641	−0.771455
$934$	32.0555	1.04889
$935$	−23.7586	−0.776990
$936$	−0.406544	−0.0132883
$937$	2.92060	0.0954119	0.0477059	−	0.998861i	$-0.484809\pi$
$937$	2.92060	0.0954119	0.0477059	+	0.998861i	$0.484809\pi$
$938$	1.53048	0.0499720
$939$	9.42443	0.307555
$940$	8.47966	0.276576
$941$	−32.8501	−1.07088	−0.535441	−	0.844573i	$-0.679855\pi$
$941$	−32.8501	−1.07088	−0.535441	+	0.844573i	$0.679855\pi$
$942$	28.3046	0.922213
$943$	1.51909	0.0494682
$944$	8.09722	0.263542
$945$	−46.8823	−1.52508
$946$	−11.5955	−0.377001
$947$	−39.1247	−1.27138	−0.635692	−	0.771943i	$-0.719285\pi$
$947$	−39.1247	−1.27138	−0.635692	+	0.771943i	$0.719285\pi$
$948$	−2.66802	−0.0866532
$949$	−1.33212	−0.0432426
$950$	−13.7878	−0.447334
$951$	22.7115	0.736471
$952$	−43.6843	−1.41581
$953$	−6.01975	−0.194999	−0.0974994	−	0.995236i	$-0.531084\pi$
$953$	−6.01975	−0.194999	−0.0974994	+	0.995236i	$0.531084\pi$
$954$	−7.40070	−0.239607
$955$	−29.7290	−0.962008
$956$	3.39053	0.109658
$957$	1.88608	0.0609684
$958$	−26.9041	−0.869231
$959$	0.293980	0.00949310
$960$	50.1729	1.61932
$961$	0	0
$962$	−0.824543	−0.0265843
$963$	−9.98204	−0.321667
$964$	−2.70941	−0.0872641
$965$	17.6012	0.566603
$966$	−19.0480	−0.612860
$967$	−26.3103	−0.846083	−0.423042	−	0.906110i	$-0.639038\pi$
$967$	−26.3103	−0.846083	−0.423042	+	0.906110i	$0.639038\pi$
$968$	30.6367	0.984699
$969$	11.2538	0.361525
$970$	−74.7015	−2.39852
$971$	−4.73587	−0.151981	−0.0759906	−	0.997109i	$-0.524212\pi$
$971$	−4.73587	−0.151981	−0.0759906	+	0.997109i	$0.524212\pi$
$972$	3.18108	0.102033
$973$	−25.2336	−0.808951
$974$	−33.8967	−1.08612
$975$	2.35661	0.0754719
$976$	−5.33994	−0.170927
$977$	−17.2800	−0.552837	−0.276419	−	0.961037i	$-0.589148\pi$
$977$	−17.2800	−0.552837	−0.276419	+	0.961037i	$0.589148\pi$
$978$	−32.0322	−1.02428
$979$	13.9834	0.446912
$980$	−3.32102	−0.106086
$981$	8.17725	0.261079
$982$	15.9166	0.507920
$983$	40.9665	1.30663	0.653315	−	0.757087i	$-0.273378\pi$
$983$	40.9665	1.30663	0.653315	+	0.757087i	$0.273378\pi$
$984$	1.47824	0.0471245
$985$	−85.0678	−2.71049
$986$	−11.1348	−0.354605
$987$	−18.3206	−0.583150
$988$	−0.0767071	−0.00244038
$989$	−44.5751	−1.41741
$990$	3.64595	0.115876
$991$	−14.1338	−0.448975	−0.224487	−	0.974477i	$-0.572071\pi$
$991$	−14.1338	−0.448975	−0.224487	+	0.974477i	$0.572071\pi$
$992$	0	0
$993$	−13.3113	−0.422420
$994$	3.15239	0.0999877
$995$	−101.103	−3.20517
$996$	−0.191716	−0.00607477
$997$	15.4538	0.489428	0.244714	−	0.969595i	$-0.421306\pi$
$997$	15.4538	0.489428	0.244714	+	0.969595i	$0.421306\pi$
$998$	29.1703	0.923370
$999$	21.8193	0.690332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.2.a.j.1.6		8
3.2	odd	2		8649.2.a.be.1.3		8
31.2	even	5		961.2.d.q.531.3		16
31.3	odd	30		961.2.g.t.846.2		16
31.4	even	5		961.2.d.n.388.2		16
31.5	even	3		961.2.c.i.521.6		16
31.6	odd	6		961.2.c.j.439.6		16
31.7	even	15		961.2.g.j.235.1		16
31.8	even	5		961.2.d.n.374.2		16
31.9	even	15		961.2.g.j.732.1		16
31.10	even	15		961.2.g.n.844.2		16
31.11	odd	30		31.2.g.a.28.1	yes	16
31.12	odd	30		961.2.g.s.547.2		16
31.13	odd	30		961.2.g.s.448.2		16
31.14	even	15		961.2.g.l.816.1		16
31.15	odd	10		961.2.d.p.628.3		16
31.16	even	5		961.2.d.q.628.3		16
31.17	odd	30		31.2.g.a.10.1	✓	16
31.18	even	15		961.2.g.m.448.2		16
31.19	even	15		961.2.g.m.547.2		16
31.20	even	15		961.2.g.l.338.1		16
31.21	odd	30		961.2.g.t.844.2		16
31.22	odd	30		961.2.g.k.732.1		16
31.23	odd	10		961.2.d.o.374.2		16
31.24	odd	30		961.2.g.k.235.1		16
31.25	even	3		961.2.c.i.439.6		16
31.26	odd	6		961.2.c.j.521.6		16
31.27	odd	10		961.2.d.o.388.2		16
31.28	even	15		961.2.g.n.846.2		16
31.29	odd	10		961.2.d.p.531.3		16
31.30	odd	2		961.2.a.i.1.6		8
93.11	even	30		279.2.y.c.28.2		16
93.17	even	30		279.2.y.c.10.2		16
93.92	even	2		8649.2.a.bf.1.3		8
124.11	even	30		496.2.bg.c.369.2		16
124.79	even	30		496.2.bg.c.289.2		16
155.17	even	60		775.2.ck.a.599.4		32
155.42	even	60		775.2.ck.a.524.1		32
155.48	even	60		775.2.ck.a.599.1		32
155.73	even	60		775.2.ck.a.524.4		32
155.79	odd	30		775.2.bl.a.351.2		16
155.104	odd	30		775.2.bl.a.276.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.10.1	✓	16	31.17	odd	30
31.2.g.a.28.1	yes	16	31.11	odd	30
279.2.y.c.10.2		16	93.17	even	30
279.2.y.c.28.2		16	93.11	even	30
496.2.bg.c.289.2		16	124.79	even	30
496.2.bg.c.369.2		16	124.11	even	30
775.2.bl.a.276.2		16	155.104	odd	30
775.2.bl.a.351.2		16	155.79	odd	30
775.2.ck.a.524.1		32	155.42	even	60
775.2.ck.a.524.4		32	155.73	even	60
775.2.ck.a.599.1		32	155.48	even	60
775.2.ck.a.599.4		32	155.17	even	60
961.2.a.i.1.6		8	31.30	odd	2
961.2.a.j.1.6		8	1.1	even	1	trivial
961.2.c.i.439.6		16	31.25	even	3
961.2.c.i.521.6		16	31.5	even	3
961.2.c.j.439.6		16	31.6	odd	6
961.2.c.j.521.6		16	31.26	odd	6
961.2.d.n.374.2		16	31.8	even	5
961.2.d.n.388.2		16	31.4	even	5
961.2.d.o.374.2		16	31.23	odd	10
961.2.d.o.388.2		16	31.27	odd	10
961.2.d.p.531.3		16	31.29	odd	10
961.2.d.p.628.3		16	31.15	odd	10
961.2.d.q.531.3		16	31.2	even	5
961.2.d.q.628.3		16	31.16	even	5
961.2.g.j.235.1		16	31.7	even	15
961.2.g.j.732.1		16	31.9	even	15
961.2.g.k.235.1		16	31.24	odd	30
961.2.g.k.732.1		16	31.22	odd	30
961.2.g.l.338.1		16	31.20	even	15
961.2.g.l.816.1		16	31.14	even	15
961.2.g.m.448.2		16	31.18	even	15
961.2.g.m.547.2		16	31.19	even	15
961.2.g.n.844.2		16	31.10	even	15
961.2.g.n.846.2		16	31.28	even	15
961.2.g.s.448.2		16	31.13	odd	30
961.2.g.s.547.2		16	31.12	odd	30
961.2.g.t.844.2		16	31.21	odd	30
961.2.g.t.846.2		16	31.3	odd	30
8649.2.a.be.1.3		8	3.2	odd	2
8649.2.a.bf.1.3		8	93.92	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26660 q^{2} -1.48431 q^{3} -0.395721 q^{4} -3.80032 q^{5} -1.88004 q^{6} +2.18899 q^{7} -3.03442 q^{8} -0.796809 q^{9} -4.81349 q^{10} +0.950596 q^{11} +0.587374 q^{12} -0.168142 q^{13} +2.77258 q^{14} +5.64087 q^{15} -3.05196 q^{16} +6.57666 q^{17} -1.00924 q^{18} -1.15284 q^{19} +1.50387 q^{20} -3.24915 q^{21} +1.20403 q^{22} +4.62850 q^{23} +4.50404 q^{24} +9.44244 q^{25} -0.212969 q^{26} +5.63566 q^{27} -0.866228 q^{28} -1.33672 q^{29} +7.14474 q^{30} +2.20322 q^{32} -1.41098 q^{33} +8.33000 q^{34} -8.31886 q^{35} +0.315314 q^{36} +3.87165 q^{37} -1.46019 q^{38} +0.249576 q^{39} +11.5318 q^{40} +0.328203 q^{41} -4.11538 q^{42} -9.63057 q^{43} -0.376170 q^{44} +3.02813 q^{45} +5.86247 q^{46} +5.63858 q^{47} +4.53008 q^{48} -2.20832 q^{49} +11.9598 q^{50} -9.76183 q^{51} +0.0665374 q^{52} +7.33294 q^{53} +7.13814 q^{54} -3.61257 q^{55} -6.64232 q^{56} +1.71118 q^{57} -1.69309 q^{58} -2.65312 q^{59} -2.23221 q^{60} +1.74967 q^{61} -1.74421 q^{63} +8.89454 q^{64} +0.638995 q^{65} -1.78715 q^{66} +0.552007 q^{67} -2.60252 q^{68} -6.87015 q^{69} -10.5367 q^{70} +1.13699 q^{71} +2.41786 q^{72} +7.92260 q^{73} +4.90384 q^{74} -14.0156 q^{75} +0.456203 q^{76} +2.08084 q^{77} +0.316114 q^{78} -4.54228 q^{79} +11.5984 q^{80} -5.97467 q^{81} +0.415702 q^{82} -0.326396 q^{83} +1.28576 q^{84} -24.9934 q^{85} -12.1981 q^{86} +1.98411 q^{87} -2.88451 q^{88} +14.7102 q^{89} +3.83543 q^{90} -0.368062 q^{91} -1.83159 q^{92} +7.14183 q^{94} +4.38117 q^{95} -3.27028 q^{96} +15.5192 q^{97} -2.79707 q^{98} -0.757443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 3 q^{5} + 11 q^{6} - 2 q^{7} - 9 q^{8} + 5 q^{9} - 13 q^{10} + 18 q^{11} + 8 q^{13} - 9 q^{14} + 18 q^{15} + 4 q^{16} + 14 q^{17} + 23 q^{18} - 6 q^{19} - 7 q^{20} - q^{21}+ \cdots + 6 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 + 3 * q^3 + 8 * q^4 + 3 * q^5 + 11 * q^6 - 2 * q^7 - 9 * q^8 + 5 * q^9 - 13 * q^10 + 18 * q^11 + 8 * q^13 - 9 * q^14 + 18 * q^15 + 4 * q^16 + 14 * q^17 + 23 * q^18 - 6 * q^19 - 7 * q^20 - q^21 + 4 * q^22 + 22 * q^23 + 30 * q^24 + 13 * q^25 + 9 * q^26 + 18 * q^27 - 5 * q^28 + 12 * q^29 + 11 * q^30 - 21 * q^32 + 6 * q^33 - 7 * q^34 - 12 * q^35 - q^36 - 8 * q^37 + 7 * q^38 + q^39 + 11 * q^40 - 22 * q^41 - 6 * q^42 - 2 * q^43 + 4 * q^44 + 18 * q^46 - 18 * q^47 - q^48 - 2 * q^49 + 27 * q^50 + 2 * q^51 - q^52 + 6 * q^53 - 7 * q^54 + 28 * q^55 + 30 * q^56 - 17 * q^57 - 5 * q^58 - 4 * q^59 - 40 * q^60 + 30 * q^61 - 23 * q^63 + 9 * q^64 + 3 * q^65 + 10 * q^66 - 13 * q^67 + 30 * q^68 + 22 * q^69 - 39 * q^70 - q^71 + 3 * q^72 + 2 * q^73 + 8 * q^74 + 23 * q^75 - 33 * q^76 - 24 * q^77 + 8 * q^79 + 24 * q^80 - 28 * q^81 - 19 * q^82 + 39 * q^83 + 8 * q^84 - 26 * q^85 - 16 * q^86 - 15 * q^87 - 17 * q^88 + 27 * q^89 + 8 * q^90 + 16 * q^91 + 32 * q^92 + 22 * q^94 - 6 * q^95 - 12 * q^96 + 34 * q^97 + 10 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more