LMFDB - Embedded newform 931.2.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [931,2,Mod(1,931)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("931.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(931, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 931 = 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	931.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,2,2,8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.43407242818$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.751024$ of defining polynomial
Character	$\chi$	$=$	931.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.751024 q^{2} +2.33152 q^{3} -1.43596 q^{4} +4.26543 q^{5} -1.75102 q^{6} +2.58049 q^{8} +2.43596 q^{9} -3.20344 q^{10} +4.09899 q^{11} -3.34797 q^{12} -2.18699 q^{13} +9.94491 q^{15} +0.933914 q^{16} +0.590042 q^{17} -1.82947 q^{18} -1.00000 q^{19} -6.12500 q^{20} -3.07844 q^{22} -4.36988 q^{23} +6.01645 q^{24} +13.1939 q^{25} +1.64248 q^{26} -1.31506 q^{27} -7.36442 q^{29} -7.46887 q^{30} +4.51850 q^{31} -5.86237 q^{32} +9.55686 q^{33} -0.443136 q^{34} -3.49795 q^{36} +8.95446 q^{37} +0.751024 q^{38} -5.09899 q^{39} +11.0069 q^{40} -2.17744 q^{41} -8.69594 q^{43} -5.88600 q^{44} +10.3904 q^{45} +3.28188 q^{46} -11.8333 q^{47} +2.17744 q^{48} -9.90893 q^{50} +1.37569 q^{51} +3.14043 q^{52} +4.40996 q^{53} +0.987643 q^{54} +17.4840 q^{55} -2.33152 q^{57} +5.53086 q^{58} +10.7178 q^{59} -14.2805 q^{60} -2.43051 q^{61} -3.39350 q^{62} +2.53496 q^{64} -9.32844 q^{65} -7.17744 q^{66} -5.65348 q^{67} -0.847279 q^{68} -10.1884 q^{69} +8.32434 q^{71} +6.28598 q^{72} -10.6795 q^{73} -6.72502 q^{74} +30.7618 q^{75} +1.43596 q^{76} +3.82947 q^{78} +8.56377 q^{79} +3.98355 q^{80} -10.3740 q^{81} +1.63531 q^{82} -4.25826 q^{83} +2.51678 q^{85} +6.53086 q^{86} -17.1703 q^{87} +10.5774 q^{88} +3.79110 q^{89} -7.80346 q^{90} +6.27498 q^{92} +10.5350 q^{93} +8.88709 q^{94} -4.26543 q^{95} -13.6682 q^{96} +2.11026 q^{97} +9.98499 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 2 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{8} + 2 q^{9} + 10 q^{10} - 6 q^{11} + 6 q^{12} + 2 q^{13} + 4 q^{15} + 2 q^{16} + 8 q^{17} - 6 q^{18} - 4 q^{19} - 14 q^{22} - 8 q^{23} + 12 q^{24} + 20 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 2 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^8 + 2 * q^9 + 10 * q^10 - 6 * q^11 + 6 * q^12 + 2 * q^13 + 4 * q^15 + 2 * q^16 + 8 * q^17 - 6 * q^18 - 4 * q^19 - 14 * q^22 - 8 * q^23 + 12 * q^24 + 20 * q^25 + 16 * q^26 - 10 * q^27 + 2 * q^29 + 2 * q^30 + 2 * q^32 + 18 * q^33 - 22 * q^34 - 20 * q^36 + 10 * q^37 + 2 * q^39 + 22 * q^40 + 12 * q^41 + 4 * q^43 - 14 * q^44 + 8 * q^45 - 8 * q^46 + 16 * q^47 - 12 * q^48 + 12 * q^50 - 10 * q^51 + 28 * q^52 + 12 * q^53 + 4 * q^54 - 8 * q^55 - 2 * q^57 + 4 * q^58 + 14 * q^59 - 2 * q^60 + 20 * q^61 - 20 * q^62 - 20 * q^64 + 10 * q^65 - 8 * q^66 + 2 * q^67 - 24 * q^68 + 14 * q^69 - 2 * q^71 - 8 * q^72 - 16 * q^73 - 26 * q^74 + 36 * q^75 - 2 * q^76 + 14 * q^78 - 8 * q^79 + 28 * q^80 - 20 * q^81 - 12 * q^82 + 20 * q^83 - 14 * q^85 + 8 * q^86 - 20 * q^87 - 2 * q^88 + 16 * q^89 - 32 * q^90 + 26 * q^92 + 12 * q^93 + 10 * q^94 - 8 * q^95 - 12 * q^96 + 2 * q^97 + 8 * 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 $

$2$	−0.751024	−0.531054	−0.265527	−	0.964103i	$-0.585546\pi$
$2$	−0.751024	−0.531054	−0.265527	+	0.964103i	$0.585546\pi$
$3$	2.33152	1.34610	0.673050	−	0.739597i	$-0.264984\pi$
$3$	2.33152	1.34610	0.673050	+	0.739597i	$0.264984\pi$
$4$	−1.43596	−0.717981
$5$	4.26543	1.90756	0.953779	−	0.300509i	$-0.0971565\pi$
$5$	4.26543	1.90756	0.953779	+	0.300509i	$0.0971565\pi$
$6$	−1.75102	−0.714853
$7$	0	0
$7$	0	0
$8$	2.58049	0.912341
$9$	2.43596	0.811988
$10$	−3.20344	−1.01302
$11$	4.09899	1.23589	0.617946	−	0.786220i	$-0.287965\pi$
$11$	4.09899	1.23589	0.617946	+	0.786220i	$0.287965\pi$
$12$	−3.34797	−0.966475
$13$	−2.18699	−0.606561	−0.303281	−	0.952901i	$-0.598082\pi$
$13$	−2.18699	−0.606561	−0.303281	+	0.952901i	$0.598082\pi$
$14$	0	0
$15$	9.94491	2.56777
$16$	0.933914	0.233479
$17$	0.590042	0.143106	0.0715531	−	0.997437i	$-0.477204\pi$
$17$	0.590042	0.143106	0.0715531	+	0.997437i	$0.477204\pi$
$18$	−1.82947	−0.431209
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−6.12500	−1.36959
$21$	0	0
$22$	−3.07844	−0.656326
$23$	−4.36988	−0.911182	−0.455591	−	0.890189i	$-0.650572\pi$
$23$	−4.36988	−0.911182	−0.455591	+	0.890189i	$0.650572\pi$
$24$	6.01645	1.22810
$25$	13.1939	2.63878
$26$	1.64248	0.322117
$27$	−1.31506	−0.253084
$28$	0	0
$29$	−7.36442	−1.36754	−0.683769	−	0.729698i	$-0.739661\pi$
$29$	−7.36442	−1.36754	−0.683769	+	0.729698i	$0.739661\pi$
$30$	−7.46887	−1.36362
$31$	4.51850	0.811547	0.405773	−	0.913974i	$-0.367002\pi$
$31$	4.51850	0.811547	0.405773	+	0.913974i	$0.367002\pi$
$32$	−5.86237	−1.03633
$33$	9.55686	1.66364
$34$	−0.443136	−0.0759972
$35$	0	0
$36$	−3.49795	−0.582992
$37$	8.95446	1.47210	0.736052	−	0.676924i	$-0.236688\pi$
$37$	8.95446	1.47210	0.736052	+	0.676924i	$0.236688\pi$
$38$	0.751024	0.121832
$39$	−5.09899	−0.816492
$40$	11.0069	1.74034
$41$	−2.17744	−0.340058	−0.170029	−	0.985439i	$-0.554386\pi$
$41$	−2.17744	−0.340058	−0.170029	+	0.985439i	$0.554386\pi$
$42$	0	0
$43$	−8.69594	−1.32612	−0.663059	−	0.748567i	$-0.730742\pi$
$43$	−8.69594	−1.32612	−0.663059	+	0.748567i	$0.730742\pi$
$44$	−5.88600	−0.887348
$45$	10.3904	1.54891
$46$	3.28188	0.483887
$47$	−11.8333	−1.72606	−0.863032	−	0.505150i	$-0.831437\pi$
$47$	−11.8333	−1.72606	−0.863032	+	0.505150i	$0.831437\pi$
$48$	2.17744	0.314286
$49$	0	0
$50$	−9.90893	−1.40133
$51$	1.37569	0.192635
$52$	3.14043	0.435500
$53$	4.40996	0.605754	0.302877	−	0.953030i	$-0.402053\pi$
$53$	4.40996	0.605754	0.302877	+	0.953030i	$0.402053\pi$
$54$	0.987643	0.134401
$55$	17.4840	2.35754
$56$	0	0
$57$	−2.33152	−0.308817
$58$	5.53086	0.726237
$59$	10.7178	1.39534	0.697672	−	0.716417i	$-0.254219\pi$
$59$	10.7178	1.39534	0.697672	+	0.716417i	$0.254219\pi$
$60$	−14.2805	−1.84361
$61$	−2.43051	−0.311195	−0.155597	−	0.987821i	$-0.549730\pi$
$61$	−2.43051	−0.311195	−0.155597	+	0.987821i	$0.549730\pi$
$62$	−3.39350	−0.430975
$63$	0	0
$64$	2.53496	0.316869
$65$	−9.32844	−1.15705
$66$	−7.17744	−0.883481
$67$	−5.65348	−0.690682	−0.345341	−	0.938477i	$-0.612237\pi$
$67$	−5.65348	−0.690682	−0.345341	+	0.938477i	$0.612237\pi$
$68$	−0.847279	−0.102748
$69$	−10.1884	−1.22654
$70$	0	0
$71$	8.32434	0.987918	0.493959	−	0.869485i	$-0.335549\pi$
$71$	8.32434	0.987918	0.493959	+	0.869485i	$0.335549\pi$
$72$	6.28598	0.740810
$73$	−10.6795	−1.24994	−0.624970	−	0.780649i	$-0.714889\pi$
$73$	−10.6795	−1.24994	−0.624970	+	0.780649i	$0.714889\pi$
$74$	−6.72502	−0.781768
$75$	30.7618	3.55206
$76$	1.43596	0.164716
$77$	0	0
$78$	3.82947	0.433602
$79$	8.56377	0.963499	0.481750	−	0.876309i	$-0.340002\pi$
$79$	8.56377	0.963499	0.481750	+	0.876309i	$0.340002\pi$
$80$	3.98355	0.445374
$81$	−10.3740	−1.15266
$82$	1.63531	0.180589
$83$	−4.25826	−0.467404	−0.233702	−	0.972308i	$-0.575084\pi$
$83$	−4.25826	−0.467404	−0.233702	+	0.972308i	$0.575084\pi$
$84$	0	0
$85$	2.51678	0.272983
$86$	6.53086	0.704241
$87$	−17.1703	−1.84085
$88$	10.5774	1.12756
$89$	3.79110	0.401856	0.200928	−	0.979606i	$-0.435604\pi$
$89$	3.79110	0.401856	0.200928	+	0.979606i	$0.435604\pi$
$90$	−7.80346	−0.822557
$91$	0	0
$92$	6.27498	0.654212
$93$	10.5350	1.09242
$94$	8.88709	0.916633
$95$	−4.26543	−0.437624
$96$	−13.6682	−1.39501
$97$	2.11026	0.214265	0.107132	−	0.994245i	$-0.465833\pi$
$97$	2.11026	0.214265	0.107132	+	0.994245i	$0.465833\pi$
$98$	0	0
$99$	9.98499	1.00353
$100$	−18.9459	−1.89459
$101$	15.0606	1.49859	0.749294	−	0.662237i	$-0.230393\pi$
$101$	15.0606	1.49859	0.749294	+	0.662237i	$0.230393\pi$
$102$	−1.03318	−0.102300
$103$	4.89383	0.482204	0.241102	−	0.970500i	$-0.422491\pi$
$103$	4.89383	0.482204	0.241102	+	0.970500i	$0.422491\pi$
$104$	−5.64350	−0.553391
$105$	0	0
$106$	−3.31198	−0.321688
$107$	0.888650	0.0859090	0.0429545	−	0.999077i	$-0.486323\pi$
$107$	0.888650	0.0859090	0.0429545	+	0.999077i	$0.486323\pi$
$108$	1.88838	0.181709
$109$	−1.11135	−0.106448	−0.0532240	−	0.998583i	$-0.516950\pi$
$109$	−1.11135	−0.106448	−0.0532240	+	0.998583i	$0.516950\pi$
$110$	−13.1309	−1.25198
$111$	20.8775	1.98160
$112$	0	0
$113$	−12.9161	−1.21504	−0.607522	−	0.794303i	$-0.707836\pi$
$113$	−12.9161	−1.21504	−0.607522	+	0.794303i	$0.707836\pi$
$114$	1.75102	0.163998
$115$	−18.6394	−1.73813
$116$	10.5750	0.981867
$117$	−5.32742	−0.492520
$118$	−8.04936	−0.741004
$119$	0	0
$120$	25.6628	2.34268
$121$	5.80174	0.527431
$122$	1.82537	0.165261
$123$	−5.07672	−0.457753
$124$	−6.48840	−0.582676
$125$	34.9505	3.12606
$126$	0	0
$127$	−3.11373	−0.276299	−0.138149	−	0.990411i	$-0.544115\pi$
$127$	−3.11373	−0.276299	−0.138149	+	0.990411i	$0.544115\pi$
$128$	9.82094	0.868056
$129$	−20.2747	−1.78509
$130$	7.00588	0.614457
$131$	3.70721	0.323900	0.161950	−	0.986799i	$-0.448222\pi$
$131$	3.70721	0.323900	0.161950	+	0.986799i	$0.448222\pi$
$132$	−13.7233	−1.19446
$133$	0	0
$134$	4.24590	0.366790
$135$	−5.60930	−0.482772
$136$	1.52260	0.130562
$137$	1.16098	0.0991894	0.0495947	−	0.998769i	$-0.484207\pi$
$137$	1.16098	0.0991894	0.0495947	+	0.998769i	$0.484207\pi$
$138$	7.65176	0.651361
$139$	−4.24633	−0.360169	−0.180084	−	0.983651i	$-0.557637\pi$
$139$	−4.24633	−0.360169	−0.180084	+	0.983651i	$0.557637\pi$
$140$	0	0
$141$	−27.5895	−2.32346
$142$	−6.25178	−0.524638
$143$	−8.96444	−0.749644
$144$	2.27498	0.189582
$145$	−31.4124	−2.60866
$146$	8.02055	0.663785
$147$	0	0
$148$	−12.8583	−1.05694
$149$	−23.9932	−1.96560	−0.982799	−	0.184677i	$-0.940876\pi$
$149$	−23.9932	−1.96560	−0.982799	+	0.184677i	$0.940876\pi$
$150$	−23.1028	−1.88634
$151$	8.50069	0.691776	0.345888	−	0.938276i	$-0.387578\pi$
$151$	8.50069	0.691776	0.345888	+	0.938276i	$0.387578\pi$
$152$	−2.58049	−0.209305
$153$	1.43732	0.116201
$154$	0	0
$155$	19.2734	1.54807
$156$	7.32196	0.586226
$157$	4.43877	0.354252	0.177126	−	0.984188i	$-0.443320\pi$
$157$	4.43877	0.354252	0.177126	+	0.984188i	$0.443320\pi$
$158$	−6.43160	−0.511670
$159$	10.2819	0.815406
$160$	−25.0055	−1.97686
$161$	0	0
$162$	7.79110	0.612127
$163$	−0.146906	−0.0115066	−0.00575329	−	0.999983i	$-0.501831\pi$
$163$	−0.146906	−0.0115066	−0.00575329	+	0.999983i	$0.501831\pi$
$164$	3.12672	0.244156
$165$	40.7641	3.17348
$166$	3.19805	0.248217
$167$	0.885641	0.0685329	0.0342665	−	0.999413i	$-0.489091\pi$
$167$	0.885641	0.0685329	0.0342665	+	0.999413i	$0.489091\pi$
$168$	0	0
$169$	−8.21709	−0.632084
$170$	−1.89017	−0.144969
$171$	−2.43596	−0.186283
$172$	12.4870	0.952128
$173$	24.2638	1.84474	0.922371	−	0.386305i	$-0.126249\pi$
$173$	24.2638	1.84474	0.922371	+	0.386305i	$0.126249\pi$
$174$	12.8953	0.977589
$175$	0	0
$176$	3.82811	0.288555
$177$	24.9888	1.87827
$178$	−2.84721	−0.213408
$179$	−12.9466	−0.967677	−0.483838	−	0.875157i	$-0.660758\pi$
$179$	−12.9466	−0.967677	−0.483838	+	0.875157i	$0.660758\pi$
$180$	−14.9203	−1.11209
$181$	18.6034	1.38278	0.691391	−	0.722481i	$-0.256998\pi$
$181$	18.6034	1.38278	0.691391	+	0.722481i	$0.256998\pi$
$182$	0	0
$183$	−5.66677	−0.418899
$184$	−11.2764	−0.831309
$185$	38.1946	2.80813
$186$	−7.91201	−0.580136
$187$	2.41858	0.176864
$188$	16.9922	1.23928
$189$	0	0
$190$	3.20344	0.232402
$191$	−23.2888	−1.68512	−0.842559	−	0.538605i	$-0.818952\pi$
$191$	−23.2888	−1.68512	−0.842559	+	0.538605i	$0.818952\pi$
$192$	5.91029	0.426538
$193$	3.18843	0.229509	0.114754	−	0.993394i	$-0.463392\pi$
$193$	3.18843	0.229509	0.114754	+	0.993394i	$0.463392\pi$
$194$	−1.58486	−0.113786
$195$	−21.7494	−1.55751
$196$	0	0
$197$	−8.25178	−0.587915	−0.293958	−	0.955818i	$-0.594972\pi$
$197$	−8.25178	−0.587915	−0.293958	+	0.955818i	$0.594972\pi$
$198$	−7.49897	−0.532929
$199$	−19.6024	−1.38958	−0.694789	−	0.719214i	$-0.744502\pi$
$199$	−19.6024	−1.38958	−0.694789	+	0.719214i	$0.744502\pi$
$200$	34.0467	2.40747
$201$	−13.1812	−0.929728
$202$	−11.3109	−0.795832
$203$	0	0
$204$	−1.97544	−0.138309
$205$	−9.28770	−0.648681
$206$	−3.67539	−0.256076
$207$	−10.6449	−0.739869
$208$	−2.04246	−0.141619
$209$	−4.09899	−0.283533
$210$	0	0
$211$	9.49078	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$211$	9.49078	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$212$	−6.33254	−0.434920
$213$	19.4083	1.32984
$214$	−0.667398	−0.0456224
$215$	−37.0919	−2.52965
$216$	−3.39350	−0.230899
$217$	0	0
$218$	0.834651	0.0565297
$219$	−24.8994	−1.68254
$220$	−25.1063	−1.69267
$221$	−1.29041	−0.0868027
$222$	−15.6795	−1.05234
$223$	−18.8829	−1.26449	−0.632247	−	0.774767i	$-0.717867\pi$
$223$	−18.8829	−1.26449	−0.632247	+	0.774767i	$0.717867\pi$
$224$	0	0
$225$	32.1398	2.14265
$226$	9.70030	0.645255
$227$	−20.5405	−1.36332	−0.681660	−	0.731669i	$-0.738742\pi$
$227$	−20.5405	−1.36332	−0.681660	+	0.731669i	$0.738742\pi$
$228$	3.34797	0.221725
$229$	−6.98499	−0.461581	−0.230791	−	0.973003i	$-0.574131\pi$
$229$	−6.98499	−0.461581	−0.230791	+	0.973003i	$0.574131\pi$
$230$	13.9986	0.923043
$231$	0	0
$232$	−19.0038	−1.24766
$233$	19.5265	1.27922	0.639612	−	0.768698i	$-0.279095\pi$
$233$	19.5265	1.27922	0.639612	+	0.768698i	$0.279095\pi$
$234$	4.00102	0.261555
$235$	−50.4741	−3.29257
$236$	−15.3904	−1.00183
$237$	19.9666	1.29697
$238$	0	0
$239$	−17.8056	−1.15175	−0.575873	−	0.817539i	$-0.695338\pi$
$239$	−17.8056	−1.15175	−0.575873	+	0.817539i	$0.695338\pi$
$240$	9.28770	0.599518
$241$	−15.2268	−0.980844	−0.490422	−	0.871485i	$-0.663157\pi$
$241$	−15.2268	−0.980844	−0.490422	+	0.871485i	$0.663157\pi$
$242$	−4.35725	−0.280095
$243$	−20.2419	−1.29852
$244$	3.49012	0.223432
$245$	0	0
$246$	3.81274	0.243092
$247$	2.18699	0.139155
$248$	11.6600	0.740408
$249$	−9.92819	−0.629173
$250$	−26.2486	−1.66011
$251$	−6.70311	−0.423097	−0.211548	−	0.977368i	$-0.567851\pi$
$251$	−6.70311	−0.423097	−0.211548	+	0.977368i	$0.567851\pi$
$252$	0	0
$253$	−17.9121	−1.12612
$254$	2.33848	0.146730
$255$	5.86792	0.367463
$256$	−12.4457	−0.777854
$257$	−8.25135	−0.514705	−0.257353	−	0.966318i	$-0.582850\pi$
$257$	−8.25135	−0.514705	−0.257353	+	0.966318i	$0.582850\pi$
$258$	15.2268	0.947979
$259$	0	0
$260$	13.3953	0.830741
$261$	−17.9395	−1.11042
$262$	−2.78420	−0.172009
$263$	−0.700305	−0.0431827	−0.0215913	−	0.999767i	$-0.506873\pi$
$263$	−0.700305	−0.0431827	−0.0215913	+	0.999767i	$0.506873\pi$
$264$	24.6614	1.51780
$265$	18.8104	1.15551
$266$	0	0
$267$	8.83902	0.540939
$268$	8.11819	0.495897
$269$	−13.9652	−0.851473	−0.425736	−	0.904847i	$-0.639985\pi$
$269$	−13.9652	−0.851473	−0.425736	+	0.904847i	$0.639985\pi$
$270$	4.21272	0.256378
$271$	8.32570	0.505750	0.252875	−	0.967499i	$-0.418624\pi$
$271$	8.32570	0.505750	0.252875	+	0.967499i	$0.418624\pi$
$272$	0.551049	0.0334122
$273$	0	0
$274$	−0.871925	−0.0526749
$275$	54.0817	3.26125
$276$	14.6302	0.880635
$277$	−15.1501	−0.910280	−0.455140	−	0.890420i	$-0.650411\pi$
$277$	−15.1501	−0.910280	−0.455140	+	0.890420i	$0.650411\pi$
$278$	3.18909	0.191269
$279$	11.0069	0.658966
$280$	0	0
$281$	−10.2038	−0.608708	−0.304354	−	0.952559i	$-0.598441\pi$
$281$	−10.2038	−0.608708	−0.304354	+	0.952559i	$0.598441\pi$
$282$	20.7204	1.23388
$283$	6.29116	0.373971	0.186985	−	0.982363i	$-0.440128\pi$
$283$	6.29116	0.373971	0.186985	+	0.982363i	$0.440128\pi$
$284$	−11.9534	−0.709306
$285$	−9.94491	−0.589086
$286$	6.73251	0.398102
$287$	0	0
$288$	−14.2805	−0.841488
$289$	−16.6519	−0.979521
$290$	23.5915	1.38534
$291$	4.92011	0.288422
$292$	15.3353	0.897433
$293$	1.48286	0.0866294	0.0433147	−	0.999061i	$-0.486208\pi$
$293$	1.48286	0.0866294	0.0433147	+	0.999061i	$0.486208\pi$
$294$	0	0
$295$	45.7162	2.66170
$296$	23.1069	1.34306
$297$	−5.39043	−0.312784
$298$	18.0195	1.04384
$299$	9.55686	0.552688
$300$	−44.1727	−2.55031
$301$	0	0
$302$	−6.38422	−0.367371
$303$	35.1141	2.01725
$304$	−0.933914	−0.0535637
$305$	−10.3672	−0.593622
$306$	−1.07946	−0.0617088
$307$	−29.9624	−1.71004	−0.855022	−	0.518592i	$-0.826456\pi$
$307$	−29.9624	−1.71004	−0.855022	+	0.518592i	$0.826456\pi$
$308$	0	0
$309$	11.4100	0.649095
$310$	−14.4748	−0.822111
$311$	4.38115	0.248432	0.124216	−	0.992255i	$-0.460358\pi$
$311$	4.38115	0.248432	0.124216	+	0.992255i	$0.460358\pi$
$312$	−13.1579	−0.744920
$313$	20.4536	1.15611	0.578053	−	0.815999i	$-0.303813\pi$
$313$	20.4536	1.15611	0.578053	+	0.815999i	$0.303813\pi$
$314$	−3.33362	−0.188127
$315$	0	0
$316$	−12.2972	−0.691774
$317$	−8.81648	−0.495183	−0.247591	−	0.968865i	$-0.579639\pi$
$317$	−8.81648	−0.495183	−0.247591	+	0.968865i	$0.579639\pi$
$318$	−7.72194	−0.433025
$319$	−30.1867	−1.69013
$320$	10.8127	0.604447
$321$	2.07190	0.115642
$322$	0	0
$323$	−0.590042	−0.0328308
$324$	14.8966	0.827591
$325$	−28.8549	−1.60058
$326$	0.110330	0.00611062
$327$	−2.59113	−0.143290
$328$	−5.61885	−0.310249
$329$	0	0
$330$	−30.6148	−1.68529
$331$	12.0580	0.662767	0.331383	−	0.943496i	$-0.392485\pi$
$331$	12.0580	0.662767	0.331383	+	0.943496i	$0.392485\pi$
$332$	6.11470	0.335588
$333$	21.8127	1.19533
$334$	−0.665138	−0.0363947
$335$	−24.1145	−1.31752
$336$	0	0
$337$	27.5871	1.50277	0.751383	−	0.659866i	$-0.229387\pi$
$337$	27.5871	1.50277	0.751383	+	0.659866i	$0.229387\pi$
$338$	6.17123	0.335671
$339$	−30.1141	−1.63557
$340$	−3.61401	−0.195997
$341$	18.5213	1.00299
$342$	1.82947	0.0989262
$343$	0	0
$344$	−22.4398	−1.20987
$345$	−43.4581	−2.33970
$346$	−18.2227	−0.979658
$347$	0.991536	0.0532284	0.0266142	−	0.999646i	$-0.491527\pi$
$347$	0.991536	0.0532284	0.0266142	+	0.999646i	$0.491527\pi$
$348$	24.6559	1.32169
$349$	31.6583	1.69463	0.847316	−	0.531090i	$-0.178217\pi$
$349$	31.6583	1.69463	0.847316	+	0.531090i	$0.178217\pi$
$350$	0	0
$351$	2.87602	0.153511
$352$	−24.0298	−1.28079
$353$	8.90381	0.473902	0.236951	−	0.971522i	$-0.423852\pi$
$353$	8.90381	0.473902	0.236951	+	0.971522i	$0.423852\pi$
$354$	−18.7672	−0.997466
$355$	35.5069	1.88451
$356$	−5.44389	−0.288525
$357$	0	0
$358$	9.72323	0.513889
$359$	−21.6998	−1.14527	−0.572635	−	0.819811i	$-0.694079\pi$
$359$	−21.6998	−1.14527	−0.572635	+	0.819811i	$0.694079\pi$
$360$	26.8124	1.41314
$361$	1.00000	0.0526316
$362$	−13.9716	−0.734332
$363$	13.5269	0.709976
$364$	0	0
$365$	−45.5526	−2.38433
$366$	4.25588	0.222458
$367$	12.7772	0.666964	0.333482	−	0.942757i	$-0.391776\pi$
$367$	12.7772	0.666964	0.333482	+	0.942757i	$0.391776\pi$
$368$	−4.08109	−0.212742
$369$	−5.30415	−0.276123
$370$	−28.6851	−1.49127
$371$	0	0
$372$	−15.1278	−0.784340
$373$	−13.2182	−0.684415	−0.342207	−	0.939624i	$-0.611175\pi$
$373$	−13.2182	−0.684415	−0.342207	+	0.939624i	$0.611175\pi$
$374$	−1.81641	−0.0939244
$375$	81.4875	4.20800
$376$	−30.5357	−1.57476
$377$	16.1059	0.829496
$378$	0	0
$379$	−1.72767	−0.0887443	−0.0443722	−	0.999015i	$-0.514129\pi$
$379$	−1.72767	−0.0887443	−0.0443722	+	0.999015i	$0.514129\pi$
$380$	6.12500	0.314206
$381$	−7.25970	−0.371926
$382$	17.4904	0.894889
$383$	−4.85563	−0.248111	−0.124056	−	0.992275i	$-0.539590\pi$
$383$	−4.85563	−0.248111	−0.124056	+	0.992275i	$0.539590\pi$
$384$	22.8977	1.16849
$385$	0	0
$386$	−2.39459	−0.121881
$387$	−21.1830	−1.07679
$388$	−3.03026	−0.153838
$389$	10.6299	0.538955	0.269477	−	0.963007i	$-0.413149\pi$
$389$	10.6299	0.538955	0.269477	+	0.963007i	$0.413149\pi$
$390$	16.3343	0.827120
$391$	−2.57841	−0.130396
$392$	0	0
$393$	8.64341	0.436002
$394$	6.19729	0.312215
$395$	36.5281	1.83793
$396$	−14.3381	−0.720516
$397$	12.7514	0.639974	0.319987	−	0.947422i	$-0.396322\pi$
$397$	12.7514	0.639974	0.319987	+	0.947422i	$0.396322\pi$
$398$	14.7219	0.737941
$399$	0	0
$400$	12.3220	0.616098
$401$	19.2514	0.961367	0.480683	−	0.876894i	$-0.340389\pi$
$401$	19.2514	0.961367	0.480683	+	0.876894i	$0.340389\pi$
$402$	9.89938	0.493736
$403$	−9.88190	−0.492253
$404$	−21.6265	−1.07596
$405$	−44.2495	−2.19877
$406$	0	0
$407$	36.7043	1.81936
$408$	3.54996	0.175749
$409$	3.13980	0.155253	0.0776266	−	0.996983i	$-0.475266\pi$
$409$	3.13980	0.155253	0.0776266	+	0.996983i	$0.475266\pi$
$410$	6.97529	0.344485
$411$	2.70685	0.133519
$412$	−7.02736	−0.346213
$413$	0	0
$414$	7.99455	0.392910
$415$	−18.1633	−0.891601
$416$	12.8209	0.628598
$417$	−9.90038	−0.484823
$418$	3.07844	0.150572
$419$	12.4881	0.610085	0.305043	−	0.952339i	$-0.401329\pi$
$419$	12.4881	0.610085	0.305043	+	0.952339i	$0.401329\pi$
$420$	0	0
$421$	−14.5141	−0.707376	−0.353688	−	0.935364i	$-0.615072\pi$
$421$	−14.5141	−0.707376	−0.353688	+	0.935364i	$0.615072\pi$
$422$	−7.12780	−0.346976
$423$	−28.8255	−1.40154
$424$	11.3799	0.552655
$425$	7.78495	0.377626
$426$	−14.5761	−0.706215
$427$	0	0
$428$	−1.27607	−0.0616811
$429$	−20.9007	−1.00910
$430$	27.8569	1.34338
$431$	23.6671	1.14001	0.570003	−	0.821643i	$-0.306942\pi$
$431$	23.6671	1.14001	0.570003	+	0.821643i	$0.306942\pi$
$432$	−1.22815	−0.0590896
$433$	4.61912	0.221981	0.110990	−	0.993821i	$-0.464598\pi$
$433$	4.61912	0.221981	0.110990	+	0.993821i	$0.464598\pi$
$434$	0	0
$435$	−73.2385	−3.51152
$436$	1.59586	0.0764277
$437$	4.36988	0.209040
$438$	18.7000	0.893522
$439$	18.9603	0.904925	0.452462	−	0.891783i	$-0.350546\pi$
$439$	18.9603	0.904925	0.452462	+	0.891783i	$0.350546\pi$
$440$	45.1172	2.15088
$441$	0	0
$442$	0.969132	0.0460969
$443$	−22.4689	−1.06753	−0.533764	−	0.845633i	$-0.679223\pi$
$443$	−22.4689	−1.06753	−0.533764	+	0.845633i	$0.679223\pi$
$444$	−29.9793	−1.42275
$445$	16.1707	0.766564
$446$	14.1815	0.671515
$447$	−55.9405	−2.64589
$448$	0	0
$449$	15.7360	0.742629	0.371314	−	0.928507i	$-0.378907\pi$
$449$	15.7360	0.742629	0.371314	+	0.928507i	$0.378907\pi$
$450$	−24.1378	−1.13787
$451$	−8.92529	−0.420276
$452$	18.5470	0.872379
$453$	19.8195	0.931201
$454$	15.4264	0.723997
$455$	0	0
$456$	−6.01645	−0.281746
$457$	22.8566	1.06919	0.534594	−	0.845109i	$-0.320464\pi$
$457$	22.8566	1.06919	0.534594	+	0.845109i	$0.320464\pi$
$458$	5.24590	0.245125
$459$	−0.775942	−0.0362179
$460$	26.7655	1.24795
$461$	26.0021	1.21104	0.605519	−	0.795831i	$-0.292965\pi$
$461$	26.0021	1.21104	0.605519	+	0.795831i	$0.292965\pi$
$462$	0	0
$463$	20.3481	0.945654	0.472827	−	0.881155i	$-0.343233\pi$
$463$	20.3481	0.945654	0.472827	+	0.881155i	$0.343233\pi$
$464$	−6.87774	−0.319291
$465$	44.9361	2.08386
$466$	−14.6649	−0.679337
$467$	15.1302	0.700141	0.350071	−	0.936723i	$-0.386158\pi$
$467$	15.1302	0.700141	0.350071	+	0.936723i	$0.386158\pi$
$468$	7.64997	0.353620
$469$	0	0
$470$	37.9073	1.74853
$471$	10.3491	0.476859
$472$	27.6573	1.27303
$473$	−35.6446	−1.63894
$474$	−14.9954	−0.688760
$475$	−13.1939	−0.605377
$476$	0	0
$477$	10.7425	0.491865
$478$	13.3724	0.611640
$479$	27.9265	1.27599	0.637997	−	0.770039i	$-0.279763\pi$
$479$	27.9265	1.27599	0.637997	+	0.770039i	$0.279763\pi$
$480$	−58.3008	−2.66106
$481$	−19.5833	−0.892921
$482$	11.4357	0.520881
$483$	0	0
$484$	−8.33109	−0.378686
$485$	9.00118	0.408722
$486$	15.2021	0.689584
$487$	0.660652	0.0299370	0.0149685	−	0.999888i	$-0.495235\pi$
$487$	0.660652	0.0299370	0.0149685	+	0.999888i	$0.495235\pi$
$488$	−6.27190	−0.283916
$489$	−0.342514	−0.0154890
$490$	0	0
$491$	−5.09463	−0.229917	−0.114959	−	0.993370i	$-0.536674\pi$
$491$	−5.09463	−0.229917	−0.114959	+	0.993370i	$0.536674\pi$
$492$	7.28999	0.328658
$493$	−4.34532	−0.195703
$494$	−1.64248	−0.0738987
$495$	42.5903	1.91429
$496$	4.21989	0.189479
$497$	0	0
$498$	7.45631	0.334125
$499$	−12.7990	−0.572963	−0.286482	−	0.958086i	$-0.592486\pi$
$499$	−12.7990	−0.572963	−0.286482	+	0.958086i	$0.592486\pi$
$500$	−50.1876	−2.24446
$501$	2.06488	0.0922522
$502$	5.03420	0.224687
$503$	32.1398	1.43304	0.716522	−	0.697565i	$-0.245733\pi$
$503$	32.1398	1.43304	0.716522	+	0.697565i	$0.245733\pi$
$504$	0	0
$505$	64.2401	2.85865
$506$	13.4524	0.598033
$507$	−19.1583	−0.850848
$508$	4.47120	0.198377
$509$	36.4645	1.61626	0.808130	−	0.589004i	$-0.200480\pi$
$509$	36.4645	1.61626	0.808130	+	0.589004i	$0.200480\pi$
$510$	−4.40695	−0.195143
$511$	0	0
$512$	−10.2949	−0.454973
$513$	1.31506	0.0580614
$514$	6.19697	0.273336
$515$	20.8743	0.919832
$516$	29.1137	1.28166
$517$	−48.5046	−2.13323
$518$	0	0
$519$	56.5714	2.48321
$520$	−24.0720	−1.05562
$521$	35.5212	1.55621	0.778107	−	0.628132i	$-0.216180\pi$
$521$	35.5212	1.55621	0.778107	+	0.628132i	$0.216180\pi$
$522$	13.4730	0.589696
$523$	14.1623	0.619273	0.309636	−	0.950855i	$-0.399793\pi$
$523$	14.1623	0.619273	0.309636	+	0.950855i	$0.399793\pi$
$524$	−5.32341	−0.232554
$525$	0	0
$526$	0.525946	0.0229323
$527$	2.66611	0.116137
$528$	8.92529	0.388424
$529$	−3.90417	−0.169747
$530$	−14.1270	−0.613639
$531$	26.1083	1.13300
$532$	0	0
$533$	4.76202	0.206266
$534$	−6.63832	−0.287268
$535$	3.79047	0.163876
$536$	−14.5888	−0.630138
$537$	−30.1853	−1.30259
$538$	10.4882	0.452178
$539$	0	0
$540$	8.05475	0.346621
$541$	−16.7454	−0.719940	−0.359970	−	0.932964i	$-0.617213\pi$
$541$	−16.7454	−0.719940	−0.359970	+	0.932964i	$0.617213\pi$
$542$	−6.25280	−0.268581
$543$	43.3742	1.86136
$544$	−3.45905	−0.148305
$545$	−4.74039	−0.203056
$546$	0	0
$547$	−41.3888	−1.76966	−0.884829	−	0.465917i	$-0.845725\pi$
$547$	−41.3888	−1.76966	−0.884829	+	0.465917i	$0.845725\pi$
$548$	−1.66713	−0.0712161
$549$	−5.92063	−0.252686
$550$	−40.6166	−1.73190
$551$	7.36442	0.313735
$552$	−26.2912	−1.11903
$553$	0	0
$554$	11.3781	0.483408
$555$	89.0514	3.78002
$556$	6.09757	0.258594
$557$	37.6750	1.59634	0.798172	−	0.602430i	$-0.205801\pi$
$557$	37.6750	1.59634	0.798172	+	0.602430i	$0.205801\pi$
$558$	−8.26645	−0.349947
$559$	19.0179	0.804372
$560$	0	0
$561$	5.63895	0.238077
$562$	7.66330	0.323257
$563$	−3.45814	−0.145743	−0.0728717	−	0.997341i	$-0.523216\pi$
$563$	−3.45814	−0.145743	−0.0728717	+	0.997341i	$0.523216\pi$
$564$	39.6175	1.66820
$565$	−55.0927	−2.31777
$566$	−4.72482	−0.198599
$567$	0	0
$568$	21.4809	0.901318
$569$	7.69204	0.322467	0.161234	−	0.986916i	$-0.448453\pi$
$569$	7.69204	0.322467	0.161234	+	0.986916i	$0.448453\pi$
$570$	7.46887	0.312837
$571$	−0.417131	−0.0174564	−0.00872819	−	0.999962i	$-0.502778\pi$
$571$	−0.417131	−0.0174564	−0.00872819	+	0.999962i	$0.502778\pi$
$572$	12.8726	0.538231
$573$	−54.2982	−2.26834
$574$	0	0
$575$	−57.6557	−2.40441
$576$	6.17506	0.257294
$577$	23.2905	0.969596	0.484798	−	0.874626i	$-0.338893\pi$
$577$	23.2905	0.969596	0.484798	+	0.874626i	$0.338893\pi$
$578$	12.5059	0.520179
$579$	7.43388	0.308942
$580$	45.1071	1.87297
$581$	0	0
$582$	−3.69512	−0.153168
$583$	18.0764	0.748647
$584$	−27.5583	−1.14037
$585$	−22.7237	−0.939511
$586$	−1.11366	−0.0460049
$587$	−30.5061	−1.25912	−0.629562	−	0.776951i	$-0.716766\pi$
$587$	−30.5061	−1.25912	−0.629562	+	0.776951i	$0.716766\pi$
$588$	0	0
$589$	−4.51850	−0.186182
$590$	−34.3340	−1.41351
$591$	−19.2392	−0.791393
$592$	8.36270	0.343705
$593$	−32.6861	−1.34226	−0.671129	−	0.741341i	$-0.734190\pi$
$593$	−32.6861	−1.34226	−0.671129	+	0.741341i	$0.734190\pi$
$594$	4.04834	0.166105
$595$	0	0
$596$	34.4533	1.41126
$597$	−45.7033	−1.87051
$598$	−7.17744	−0.293507
$599$	−20.3738	−0.832451	−0.416225	−	0.909262i	$-0.636647\pi$
$599$	−20.3738	−0.832451	−0.416225	+	0.909262i	$0.636647\pi$
$600$	79.3804	3.24069
$601$	12.3891	0.505363	0.252682	−	0.967549i	$-0.418687\pi$
$601$	12.3891	0.505363	0.252682	+	0.967549i	$0.418687\pi$
$602$	0	0
$603$	−13.7717	−0.560826
$604$	−12.2067	−0.496683
$605$	24.7469	1.00611
$606$	−26.3715	−1.07127
$607$	−33.3480	−1.35355	−0.676776	−	0.736189i	$-0.736624\pi$
$607$	−33.3480	−1.35355	−0.676776	+	0.736189i	$0.736624\pi$
$608$	5.86237	0.237751
$609$	0	0
$610$	7.78599	0.315245
$611$	25.8793	1.04696
$612$	−2.06394	−0.0834298
$613$	−38.7985	−1.56706	−0.783528	−	0.621356i	$-0.786582\pi$
$613$	−38.7985	−1.56706	−0.783528	+	0.621356i	$0.786582\pi$
$614$	22.5025	0.908126
$615$	−21.6544	−0.873190
$616$	0	0
$617$	1.98936	0.0800887	0.0400443	−	0.999198i	$-0.487250\pi$
$617$	1.98936	0.0800887	0.0400443	+	0.999198i	$0.487250\pi$
$618$	−8.56922	−0.344705
$619$	13.0014	0.522572	0.261286	−	0.965261i	$-0.415853\pi$
$619$	13.0014	0.522572	0.261286	+	0.965261i	$0.415853\pi$
$620$	−27.6758	−1.11149
$621$	5.74666	0.230605
$622$	−3.29035	−0.131931
$623$	0	0
$624$	−4.76202	−0.190633
$625$	83.1093	3.32437
$626$	−15.3611	−0.613955
$627$	−9.55686	−0.381664
$628$	−6.37391	−0.254347
$629$	5.28351	0.210667
$630$	0	0
$631$	−14.9918	−0.596814	−0.298407	−	0.954439i	$-0.596455\pi$
$631$	−14.9918	−0.596814	−0.298407	+	0.954439i	$0.596455\pi$
$632$	22.0987	0.879040
$633$	22.1279	0.879505
$634$	6.62139	0.262969
$635$	−13.2814	−0.527056
$636$	−14.7644	−0.585447
$637$	0	0
$638$	22.6710	0.897552
$639$	20.2778	0.802177
$640$	41.8905	1.65587
$641$	24.0547	0.950105	0.475053	−	0.879957i	$-0.342429\pi$
$641$	24.0547	0.950105	0.475053	+	0.879957i	$0.342429\pi$
$642$	−1.55605	−0.0614123
$643$	−24.3676	−0.960964	−0.480482	−	0.877005i	$-0.659538\pi$
$643$	−24.3676	−0.960964	−0.480482	+	0.877005i	$0.659538\pi$
$644$	0	0
$645$	−86.4803	−3.40516
$646$	0.443136	0.0174350
$647$	38.2402	1.50338	0.751688	−	0.659519i	$-0.229240\pi$
$647$	38.2402	1.50338	0.751688	+	0.659519i	$0.229240\pi$
$648$	−26.7699	−1.05162
$649$	43.9324	1.72450
$650$	21.6707	0.849995
$651$	0	0
$652$	0.210952	0.00826151
$653$	21.1624	0.828147	0.414074	−	0.910243i	$-0.364106\pi$
$653$	21.1624	0.828147	0.414074	+	0.910243i	$0.364106\pi$
$654$	1.94600	0.0760947
$655$	15.8128	0.617858
$656$	−2.03354	−0.0793963
$657$	−26.0148	−1.01493
$658$	0	0
$659$	24.6069	0.958548	0.479274	−	0.877665i	$-0.340900\pi$
$659$	24.6069	0.958548	0.479274	+	0.877665i	$0.340900\pi$
$660$	−58.5358	−2.27850
$661$	38.1953	1.48562	0.742812	−	0.669500i	$-0.233492\pi$
$661$	38.1953	1.48562	0.742812	+	0.669500i	$0.233492\pi$
$662$	−9.05584	−0.351965
$663$	−3.00862	−0.116845
$664$	−10.9884	−0.426432
$665$	0	0
$666$	−16.3819	−0.634786
$667$	32.1816	1.24608
$668$	−1.27175	−0.0492054
$669$	−44.0258	−1.70214
$670$	18.1106	0.699673
$671$	−9.96264	−0.384603
$672$	0	0
$673$	−28.0333	−1.08060	−0.540302	−	0.841471i	$-0.681690\pi$
$673$	−28.0333	−1.08060	−0.540302	+	0.841471i	$0.681690\pi$
$674$	−20.7186	−0.798050
$675$	−17.3508	−0.667832
$676$	11.7994	0.453824
$677$	5.23635	0.201249	0.100625	−	0.994924i	$-0.467916\pi$
$677$	5.23635	0.201249	0.100625	+	0.994924i	$0.467916\pi$
$678$	22.6164	0.868578
$679$	0	0
$680$	6.49454	0.249054
$681$	−47.8905	−1.83517
$682$	−13.9099	−0.532639
$683$	−1.25171	−0.0478955	−0.0239478	−	0.999713i	$-0.507624\pi$
$683$	−1.25171	−0.0478955	−0.0239478	+	0.999713i	$0.507624\pi$
$684$	3.49795	0.133748
$685$	4.95209	0.189210
$686$	0	0
$687$	−16.2856	−0.621335
$688$	−8.12126	−0.309620
$689$	−9.64452	−0.367427
$690$	32.6380	1.24251
$691$	−50.8050	−1.93271	−0.966357	−	0.257204i	$-0.917199\pi$
$691$	−50.8050	−1.93271	−0.966357	+	0.257204i	$0.917199\pi$
$692$	−34.8419	−1.32449
$693$	0	0
$694$	−0.744668	−0.0282672
$695$	−18.1124	−0.687043
$696$	−44.3077	−1.67948
$697$	−1.28478	−0.0486645
$698$	−23.7762	−0.899941
$699$	45.5263	1.72196
$700$	0	0
$701$	0.998799	0.0377241	0.0188621	−	0.999822i	$-0.493996\pi$
$701$	0.998799	0.0377241	0.0188621	+	0.999822i	$0.493996\pi$
$702$	−2.15996	−0.0815225
$703$	−8.95446	−0.337724
$704$	10.3908	0.391617
$705$	−117.681	−4.43213
$706$	−6.68698	−0.251668
$707$	0	0
$708$	−35.8830	−1.34857
$709$	−13.9625	−0.524371	−0.262185	−	0.965017i	$-0.584443\pi$
$709$	−13.9625	−0.524371	−0.262185	+	0.965017i	$0.584443\pi$
$710$	−26.6665	−1.00078
$711$	20.8610	0.782349
$712$	9.78291	0.366630
$713$	−19.7453	−0.739467
$714$	0	0
$715$	−38.2372	−1.42999
$716$	18.5909	0.694774
$717$	−41.5140	−1.55037
$718$	16.2970	0.608200
$719$	18.5694	0.692521	0.346260	−	0.938138i	$-0.387451\pi$
$719$	18.5694	0.692521	0.346260	+	0.938138i	$0.387451\pi$
$720$	9.70377	0.361638
$721$	0	0
$722$	−0.751024	−0.0279502
$723$	−35.5015	−1.32032
$724$	−26.7138	−0.992811
$725$	−97.1654	−3.60863
$726$	−10.1590	−0.377036
$727$	−32.1991	−1.19420	−0.597099	−	0.802168i	$-0.703680\pi$
$727$	−32.1991	−1.19420	−0.597099	+	0.802168i	$0.703680\pi$
$728$	0	0
$729$	−16.0724	−0.595272
$730$	34.2111	1.26621
$731$	−5.13097	−0.189776
$732$	8.13727	0.300762
$733$	9.55623	0.352968	0.176484	−	0.984304i	$-0.443528\pi$
$733$	9.55623	0.352968	0.176484	+	0.984304i	$0.443528\pi$
$734$	−9.59598	−0.354194
$735$	0	0
$736$	25.6179	0.944287
$737$	−23.1736	−0.853609
$738$	3.98355	0.146636
$739$	−20.2850	−0.746194	−0.373097	−	0.927792i	$-0.621704\pi$
$739$	−20.2850	−0.746194	−0.373097	+	0.927792i	$0.621704\pi$
$740$	−54.8461	−2.01618
$741$	5.09899	0.187316
$742$	0	0
$743$	0.965374	0.0354161	0.0177081	−	0.999843i	$-0.494363\pi$
$743$	0.965374	0.0354161	0.0177081	+	0.999843i	$0.494363\pi$
$744$	27.1854	0.996664
$745$	−102.341	−3.74949
$746$	9.92722	0.363461
$747$	−10.3730	−0.379527
$748$	−3.47299	−0.126985
$749$	0	0
$750$	−61.1991	−2.23468
$751$	−0.657779	−0.0240027	−0.0120013	−	0.999928i	$-0.503820\pi$
$751$	−0.657779	−0.0240027	−0.0120013	+	0.999928i	$0.503820\pi$
$752$	−11.0513	−0.402999
$753$	−15.6284	−0.569531
$754$	−12.0959	−0.440507
$755$	36.2591	1.31960
$756$	0	0
$757$	−1.47679	−0.0536749	−0.0268375	−	0.999640i	$-0.508544\pi$
$757$	−1.47679	−0.0536749	−0.0268375	+	0.999640i	$0.508544\pi$
$758$	1.29752	0.0471281
$759$	−41.7623	−1.51588
$760$	−11.0069	−0.399262
$761$	18.4847	0.670070	0.335035	−	0.942206i	$-0.391252\pi$
$761$	18.4847	0.670070	0.335035	+	0.942206i	$0.391252\pi$
$762$	5.45221	0.197513
$763$	0	0
$764$	33.4418	1.20988
$765$	6.13079	0.221659
$766$	3.64669	0.131760
$767$	−23.4398	−0.846362
$768$	−29.0173	−1.04707
$769$	−4.34052	−0.156523	−0.0782617	−	0.996933i	$-0.524937\pi$
$769$	−4.34052	−0.156523	−0.0782617	+	0.996933i	$0.524937\pi$
$770$	0	0
$771$	−19.2382	−0.692845
$772$	−4.57847	−0.164783
$773$	33.5513	1.20676	0.603379	−	0.797455i	$-0.293821\pi$
$773$	33.5513	1.20676	0.603379	+	0.797455i	$0.293821\pi$
$774$	15.9089	0.571835
$775$	59.6166	2.14149
$776$	5.44551	0.195483
$777$	0	0
$778$	−7.98328	−0.286214
$779$	2.17744	0.0780147
$780$	31.2313	1.11826
$781$	34.1214	1.22096
$782$	1.93645	0.0692473
$783$	9.68467	0.346102
$784$	0	0
$785$	18.9333	0.675757
$786$	−6.49141	−0.231541
$787$	−42.0571	−1.49917	−0.749587	−	0.661906i	$-0.769748\pi$
$787$	−42.0571	−1.49917	−0.749587	+	0.661906i	$0.769748\pi$
$788$	11.8493	0.422112
$789$	−1.63277	−0.0581282
$790$	−27.4335	−0.976041
$791$	0	0
$792$	25.7662	0.915562
$793$	5.31549	0.188759
$794$	−9.57660	−0.339861
$795$	43.8566	1.55544
$796$	28.1483	0.997691
$797$	2.71101	0.0960289	0.0480145	−	0.998847i	$-0.484711\pi$
$797$	2.71101	0.0960289	0.0480145	+	0.998847i	$0.484711\pi$
$798$	0	0
$799$	−6.98214	−0.247010
$800$	−77.3475	−2.73465
$801$	9.23499	0.326302
$802$	−14.4582	−0.510538
$803$	−43.7751	−1.54479
$804$	18.9277	0.667528
$805$	0	0
$806$	7.42155	0.261413
$807$	−32.5601	−1.14617
$808$	38.8638	1.36722
$809$	13.3158	0.468159	0.234079	−	0.972217i	$-0.424792\pi$
$809$	13.3158	0.468159	0.234079	+	0.972217i	$0.424792\pi$
$810$	33.2324	1.16767
$811$	−15.1310	−0.531320	−0.265660	−	0.964067i	$-0.585590\pi$
$811$	−15.1310	−0.531320	−0.265660	+	0.964067i	$0.585590\pi$
$812$	0	0
$813$	19.4115	0.680791
$814$	−27.5658	−0.966181
$815$	−0.626618	−0.0219495
$816$	1.28478	0.0449763
$817$	8.69594	0.304232
$818$	−2.35807	−0.0824478
$819$	0	0
$820$	13.3368	0.465741
$821$	−24.3734	−0.850639	−0.425319	−	0.905043i	$-0.639838\pi$
$821$	−24.3734	−0.850639	−0.425319	+	0.905043i	$0.639838\pi$
$822$	−2.03291	−0.0709058
$823$	11.7877	0.410892	0.205446	−	0.978668i	$-0.434135\pi$
$823$	11.7877	0.410892	0.205446	+	0.978668i	$0.434135\pi$
$824$	12.6285	0.439934
$825$	126.092	4.38997
$826$	0	0
$827$	17.5429	0.610028	0.305014	−	0.952348i	$-0.401339\pi$
$827$	17.5429	0.610028	0.305014	+	0.952348i	$0.401339\pi$
$828$	15.2856	0.531212
$829$	50.9728	1.77036	0.885180	−	0.465249i	$-0.154035\pi$
$829$	50.9728	1.77036	0.885180	+	0.465249i	$0.154035\pi$
$830$	13.6411	0.473488
$831$	−35.3226	−1.22533
$832$	−5.54391	−0.192201
$833$	0	0
$834$	7.43542	0.257468
$835$	3.77764	0.130731
$836$	5.88600	0.203572
$837$	−5.94211	−0.205389
$838$	−9.37889	−0.323988
$839$	−5.84311	−0.201727	−0.100863	−	0.994900i	$-0.532160\pi$
$839$	−5.84311	−0.201727	−0.100863	+	0.994900i	$0.532160\pi$
$840$	0	0
$841$	25.2347	0.870163
$842$	10.9005	0.375655
$843$	−23.7903	−0.819382
$844$	−13.6284	−0.469109
$845$	−35.0494	−1.20574
$846$	21.6486	0.744295
$847$	0	0
$848$	4.11852	0.141431
$849$	14.6679	0.503403
$850$	−5.84669	−0.200540
$851$	−39.1299	−1.34136
$852$	−27.8696	−0.954798
$853$	−37.0096	−1.26719	−0.633593	−	0.773667i	$-0.718421\pi$
$853$	−37.0096	−1.26719	−0.633593	+	0.773667i	$0.718421\pi$
$854$	0	0
$855$	−10.3904	−0.355345
$856$	2.29315	0.0783784
$857$	40.1269	1.37071	0.685355	−	0.728209i	$-0.259647\pi$
$857$	40.1269	1.37071	0.685355	+	0.728209i	$0.259647\pi$
$858$	15.6970	0.535885
$859$	−14.4388	−0.492644	−0.246322	−	0.969188i	$-0.579222\pi$
$859$	−14.4388	−0.492644	−0.246322	+	0.969188i	$0.579222\pi$
$860$	53.2626	1.81624
$861$	0	0
$862$	−17.7746	−0.605405
$863$	30.6183	1.04226	0.521130	−	0.853477i	$-0.325511\pi$
$863$	30.6183	1.04226	0.521130	+	0.853477i	$0.325511\pi$
$864$	7.70938	0.262279
$865$	103.496	3.51895
$866$	−3.46907	−0.117884
$867$	−38.8240	−1.31853
$868$	0	0
$869$	35.1028	1.19078
$870$	55.0039	1.86481
$871$	12.3641	0.418941
$872$	−2.86783	−0.0971169
$873$	5.14052	0.173980
$874$	−3.28188	−0.111011
$875$	0	0
$876$	35.7546	1.20804
$877$	−16.0685	−0.542596	−0.271298	−	0.962495i	$-0.587453\pi$
$877$	−16.0685	−0.542596	−0.271298	+	0.962495i	$0.587453\pi$
$878$	−14.2396	−0.480564
$879$	3.45730	0.116612
$880$	16.3285	0.550435
$881$	−40.2468	−1.35595	−0.677975	−	0.735085i	$-0.737142\pi$
$881$	−40.2468	−1.35595	−0.677975	+	0.735085i	$0.737142\pi$
$882$	0	0
$883$	−0.564426	−0.0189944	−0.00949722	−	0.999955i	$-0.503023\pi$
$883$	−0.564426	−0.0189944	−0.00949722	+	0.999955i	$0.503023\pi$
$884$	1.85299	0.0623227
$885$	106.588	3.58292
$886$	16.8747	0.566915
$887$	−35.5836	−1.19478	−0.597390	−	0.801951i	$-0.703796\pi$
$887$	−35.5836	−1.19478	−0.597390	+	0.801951i	$0.703796\pi$
$888$	53.8741	1.80790
$889$	0	0
$890$	−12.1446	−0.407087
$891$	−42.5228	−1.42457
$892$	27.1152	0.907883
$893$	11.8333	0.395986
$894$	42.0127	1.40511
$895$	−55.2229	−1.84590
$896$	0	0
$897$	22.2820	0.743973
$898$	−11.8181	−0.394376
$899$	−33.2762	−1.10982
$900$	−46.1516	−1.53839
$901$	2.60206	0.0866872
$902$	6.70311	0.223189
$903$	0	0
$904$	−33.3299	−1.10854
$905$	79.3516	2.63774
$906$	−14.8849	−0.494518
$907$	30.3682	1.00836	0.504179	−	0.863599i	$-0.331795\pi$
$907$	30.3682	1.00836	0.504179	+	0.863599i	$0.331795\pi$
$908$	29.4954	0.978839
$909$	36.6871	1.21684
$910$	0	0
$911$	17.9947	0.596191	0.298096	−	0.954536i	$-0.403649\pi$
$911$	17.9947	0.596191	0.298096	+	0.954536i	$0.403649\pi$
$912$	−2.17744	−0.0721021
$913$	−17.4546	−0.577662
$914$	−17.1659	−0.567797
$915$	−24.1712	−0.799075
$916$	10.0302	0.331407
$917$	0	0
$918$	0.582751	0.0192336
$919$	41.9568	1.38403	0.692014	−	0.721884i	$-0.256724\pi$
$919$	41.9568	1.38403	0.692014	+	0.721884i	$0.256724\pi$
$920$	−48.0988	−1.58577
$921$	−69.8578	−2.30189
$922$	−19.5282	−0.643127
$923$	−18.2052	−0.599232
$924$	0	0
$925$	118.144	3.88456
$926$	−15.2819	−0.502194
$927$	11.9212	0.391543
$928$	43.1730	1.41722
$929$	−19.1145	−0.627127	−0.313563	−	0.949567i	$-0.601523\pi$
$929$	−19.1145	−0.627127	−0.313563	+	0.949567i	$0.601523\pi$
$930$	−33.7481	−1.10664
$931$	0	0
$932$	−28.0393	−0.918458
$933$	10.2147	0.334415
$934$	−11.3631	−0.371813
$935$	10.3163	0.337378
$936$	−13.7474	−0.449346
$937$	44.3779	1.44976	0.724881	−	0.688874i	$-0.241895\pi$
$937$	44.3779	1.44976	0.724881	+	0.688874i	$0.241895\pi$
$938$	0	0
$939$	47.6879	1.55623
$940$	72.4789	2.36400
$941$	−37.6172	−1.22628	−0.613142	−	0.789973i	$-0.710095\pi$
$941$	−37.6172	−1.22628	−0.613142	+	0.789973i	$0.710095\pi$
$942$	−7.77239	−0.253238
$943$	9.51513	0.309855
$944$	10.0096	0.325783
$945$	0	0
$946$	26.7699	0.870366
$947$	29.1696	0.947883	0.473942	−	0.880556i	$-0.342831\pi$
$947$	29.1696	0.947883	0.473942	+	0.880556i	$0.342831\pi$
$948$	−28.6712	−0.931198
$949$	23.3559	0.758164
$950$	9.90893	0.321488
$951$	−20.5558	−0.666566
$952$	0	0
$953$	25.0234	0.810586	0.405293	−	0.914187i	$-0.367170\pi$
$953$	25.0234	0.810586	0.405293	+	0.914187i	$0.367170\pi$
$954$	−8.06787	−0.261207
$955$	−99.3367	−3.21446
$956$	25.5681	0.826933
$957$	−70.3808	−2.27509
$958$	−20.9735	−0.677622
$959$	0	0
$960$	25.2099	0.813647
$961$	−10.5831	−0.341392
$962$	14.7075	0.474190
$963$	2.16472	0.0697571
$964$	21.8651	0.704228
$965$	13.6000	0.437801
$966$	0	0
$967$	−28.8469	−0.927655	−0.463828	−	0.885926i	$-0.653524\pi$
$967$	−28.8469	−0.927655	−0.463828	+	0.885926i	$0.653524\pi$
$968$	14.9713	0.481197
$969$	−1.37569	−0.0441936
$970$	−6.76010	−0.217054
$971$	3.19928	0.102670	0.0513348	−	0.998681i	$-0.483652\pi$
$971$	3.19928	0.102670	0.0513348	+	0.998681i	$0.483652\pi$
$972$	29.0666	0.932312
$973$	0	0
$974$	−0.496166	−0.0158982
$975$	−67.2755	−2.15454
$976$	−2.26989	−0.0726573
$977$	−54.4969	−1.74351	−0.871755	−	0.489943i	$-0.837018\pi$
$977$	−54.4969	−1.74351	−0.871755	+	0.489943i	$0.837018\pi$
$978$	0.257236	0.00822551
$979$	15.5397	0.496651
$980$	0	0
$981$	−2.70721	−0.0864345
$982$	3.82619	0.122099
$983$	−5.45760	−0.174070	−0.0870352	−	0.996205i	$-0.527739\pi$
$983$	−5.45760	−0.174070	−0.0870352	+	0.996205i	$0.527739\pi$
$984$	−13.1004	−0.417627
$985$	−35.1974	−1.12148
$986$	3.26344	0.103929
$987$	0	0
$988$	−3.14043	−0.0999104
$989$	38.0002	1.20834
$990$	−31.9863	−1.01659
$991$	34.2911	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$991$	34.2911	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$992$	−26.4891	−0.841031
$993$	28.1134	0.892151
$994$	0	0
$995$	−83.6127	−2.65070
$996$	14.2565	0.451735
$997$	23.2138	0.735188	0.367594	−	0.929986i	$-0.380182\pi$
$997$	23.2138	0.735188	0.367594	+	0.929986i	$0.380182\pi$
$998$	9.61238	0.304275
$999$	−11.7757	−0.372566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	931.2.a.m.1.2	yes	4
3.2	odd	2		8379.2.a.bu.1.3		4
7.2	even	3		931.2.f.n.704.3		8
7.3	odd	6		931.2.f.o.324.3		8
7.4	even	3		931.2.f.n.324.3		8
7.5	odd	6		931.2.f.o.704.3		8
7.6	odd	2		931.2.a.l.1.2	✓	4
21.20	even	2		8379.2.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
931.2.a.l.1.2	✓	4	7.6	odd	2
931.2.a.m.1.2	yes	4	1.1	even	1	trivial
931.2.f.n.324.3		8	7.4	even	3
931.2.f.n.704.3		8	7.2	even	3
931.2.f.o.324.3		8	7.3	odd	6
931.2.f.o.704.3		8	7.5	odd	6
8379.2.a.bu.1.3		4	3.2	odd	2
8379.2.a.bv.1.3		4	21.20	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:	\( N \)	\(=\)	\( 931 = 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	931.a (trivial)

\(f(q)\)	\(=\)	\(q-0.751024 q^{2} +2.33152 q^{3} -1.43596 q^{4} +4.26543 q^{5} -1.75102 q^{6} +2.58049 q^{8} +2.43596 q^{9} -3.20344 q^{10} +4.09899 q^{11} -3.34797 q^{12} -2.18699 q^{13} +9.94491 q^{15} +0.933914 q^{16} +0.590042 q^{17} -1.82947 q^{18} -1.00000 q^{19} -6.12500 q^{20} -3.07844 q^{22} -4.36988 q^{23} +6.01645 q^{24} +13.1939 q^{25} +1.64248 q^{26} -1.31506 q^{27} -7.36442 q^{29} -7.46887 q^{30} +4.51850 q^{31} -5.86237 q^{32} +9.55686 q^{33} -0.443136 q^{34} -3.49795 q^{36} +8.95446 q^{37} +0.751024 q^{38} -5.09899 q^{39} +11.0069 q^{40} -2.17744 q^{41} -8.69594 q^{43} -5.88600 q^{44} +10.3904 q^{45} +3.28188 q^{46} -11.8333 q^{47} +2.17744 q^{48} -9.90893 q^{50} +1.37569 q^{51} +3.14043 q^{52} +4.40996 q^{53} +0.987643 q^{54} +17.4840 q^{55} -2.33152 q^{57} +5.53086 q^{58} +10.7178 q^{59} -14.2805 q^{60} -2.43051 q^{61} -3.39350 q^{62} +2.53496 q^{64} -9.32844 q^{65} -7.17744 q^{66} -5.65348 q^{67} -0.847279 q^{68} -10.1884 q^{69} +8.32434 q^{71} +6.28598 q^{72} -10.6795 q^{73} -6.72502 q^{74} +30.7618 q^{75} +1.43596 q^{76} +3.82947 q^{78} +8.56377 q^{79} +3.98355 q^{80} -10.3740 q^{81} +1.63531 q^{82} -4.25826 q^{83} +2.51678 q^{85} +6.53086 q^{86} -17.1703 q^{87} +10.5774 q^{88} +3.79110 q^{89} -7.80346 q^{90} +6.27498 q^{92} +10.5350 q^{93} +8.88709 q^{94} -4.26543 q^{95} -13.6682 q^{96} +2.11026 q^{97} +9.98499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 2 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{8} + 2 q^{9} + 10 q^{10} - 6 q^{11} + 6 q^{12} + 2 q^{13} + 4 q^{15} + 2 q^{16} + 8 q^{17} - 6 q^{18} - 4 q^{19} - 14 q^{22} - 8 q^{23} + 12 q^{24} + 20 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 2 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^8 + 2 * q^9 + 10 * q^10 - 6 * q^11 + 6 * q^12 + 2 * q^13 + 4 * q^15 + 2 * q^16 + 8 * q^17 - 6 * q^18 - 4 * q^19 - 14 * q^22 - 8 * q^23 + 12 * q^24 + 20 * q^25 + 16 * q^26 - 10 * q^27 + 2 * q^29 + 2 * q^30 + 2 * q^32 + 18 * q^33 - 22 * q^34 - 20 * q^36 + 10 * q^37 + 2 * q^39 + 22 * q^40 + 12 * q^41 + 4 * q^43 - 14 * q^44 + 8 * q^45 - 8 * q^46 + 16 * q^47 - 12 * q^48 + 12 * q^50 - 10 * q^51 + 28 * q^52 + 12 * q^53 + 4 * q^54 - 8 * q^55 - 2 * q^57 + 4 * q^58 + 14 * q^59 - 2 * q^60 + 20 * q^61 - 20 * q^62 - 20 * q^64 + 10 * q^65 - 8 * q^66 + 2 * q^67 - 24 * q^68 + 14 * q^69 - 2 * q^71 - 8 * q^72 - 16 * q^73 - 26 * q^74 + 36 * q^75 - 2 * q^76 + 14 * q^78 - 8 * q^79 + 28 * q^80 - 20 * q^81 - 12 * q^82 + 20 * q^83 - 14 * q^85 + 8 * q^86 - 20 * q^87 - 2 * q^88 + 16 * q^89 - 32 * q^90 + 26 * q^92 + 12 * q^93 + 10 * q^94 - 8 * q^95 - 12 * q^96 + 2 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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