LMFDB - Embedded newform 931.2.a.l.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:	$1$
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	$-0.735016\pi$
$3$	−2.33152	−1.34610	−0.673050	+	0.739597i	$0.735016\pi$
$4$	−1.43596	−0.717981
$5$	−4.26543	−1.90756	−0.953779	−	0.300509i	$-0.902844\pi$
$5$	−4.26543	−1.90756	−0.953779	+	0.300509i	$0.902844\pi$
$6$	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	$-0.401918\pi$
$13$	2.18699	0.606561	0.303281	+	0.952901i	$0.401918\pi$
$14$	0	0
$15$	9.94491	2.56777
$16$	0.933914	0.233479
$17$	−0.590042	−0.143106	−0.0715531	−	0.997437i	$-0.522796\pi$
$17$	−0.590042	−0.143106	−0.0715531	+	0.997437i	$0.522796\pi$
$18$	−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	$-0.632998\pi$
$31$	−4.51850	−0.811547	−0.405773	+	0.913974i	$0.632998\pi$
$32$	−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	$-0.445614\pi$
$41$	2.17744	0.340058	0.170029	+	0.985439i	$0.445614\pi$
$42$	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	$-0.168563\pi$
$47$	11.8333	1.72606	0.863032	+	0.505150i	$0.168563\pi$
$48$	−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	$-0.745781\pi$
$59$	−10.7178	−1.39534	−0.697672	+	0.716417i	$0.745781\pi$
$60$	−14.2805	−1.84361
$61$	2.43051	0.311195	0.155597	−	0.987821i	$-0.450270\pi$
$61$	2.43051	0.311195	0.155597	+	0.987821i	$0.450270\pi$
$62$	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	$-0.285111\pi$
$73$	10.6795	1.24994	0.624970	+	0.780649i	$0.285111\pi$
$74$	−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	$-0.424916\pi$
$83$	4.25826	0.467404	0.233702	+	0.972308i	$0.424916\pi$
$84$	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	$-0.564396\pi$
$89$	−3.79110	−0.401856	−0.200928	+	0.979606i	$0.564396\pi$
$90$	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	$-0.534167\pi$
$97$	−2.11026	−0.214265	−0.107132	+	0.994245i	$0.534167\pi$
$98$	0	0
$99$	9.98499	1.00353
$100$	−18.9459	−1.89459
$101$	−15.0606	−1.49859	−0.749294	−	0.662237i	$-0.769607\pi$
$101$	−15.0606	−1.49859	−0.749294	+	0.662237i	$0.769607\pi$
$102$	−1.03318	−0.102300
$103$	−4.89383	−0.482204	−0.241102	−	0.970500i	$-0.577509\pi$
$103$	−4.89383	−0.482204	−0.241102	+	0.970500i	$0.577509\pi$
$104$	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	$-0.551778\pi$
$131$	−3.70721	−0.323900	−0.161950	+	0.986799i	$0.551778\pi$
$132$	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	$-0.442363\pi$
$139$	4.24633	0.360169	0.180084	+	0.983651i	$0.442363\pi$
$140$	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	$-0.556680\pi$
$157$	−4.43877	−0.354252	−0.177126	+	0.984188i	$0.556680\pi$
$158$	−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	$-0.510909\pi$
$167$	−0.885641	−0.0685329	−0.0342665	+	0.999413i	$0.510909\pi$
$168$	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	$-0.873751\pi$
$173$	−24.2638	−1.84474	−0.922371	+	0.386305i	$0.873751\pi$
$174$	−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	$-0.743002\pi$
$181$	−18.6034	−1.38278	−0.691391	+	0.722481i	$0.743002\pi$
$182$	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	$-0.255498\pi$
$199$	19.6024	1.38958	0.694789	+	0.719214i	$0.255498\pi$
$200$	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	$-0.282133\pi$
$223$	18.8829	1.26449	0.632247	+	0.774767i	$0.282133\pi$
$224$	0	0
$225$	32.1398	2.14265
$226$	9.70030	0.645255
$227$	20.5405	1.36332	0.681660	−	0.731669i	$-0.261258\pi$
$227$	20.5405	1.36332	0.681660	+	0.731669i	$0.261258\pi$
$228$	3.34797	0.221725
$229$	6.98499	0.461581	0.230791	−	0.973003i	$-0.425869\pi$
$229$	6.98499	0.461581	0.230791	+	0.973003i	$0.425869\pi$
$230$	−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	$-0.336843\pi$
$241$	15.2268	0.980844	0.490422	+	0.871485i	$0.336843\pi$
$242$	−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	$-0.432149\pi$
$251$	6.70311	0.423097	0.211548	+	0.977368i	$0.432149\pi$
$252$	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	$-0.417150\pi$
$257$	8.25135	0.514705	0.257353	+	0.966318i	$0.417150\pi$
$258$	−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	$-0.360015\pi$
$269$	13.9652	0.851473	0.425736	+	0.904847i	$0.360015\pi$
$270$	4.21272	0.256378
$271$	−8.32570	−0.505750	−0.252875	−	0.967499i	$-0.581376\pi$
$271$	−8.32570	−0.505750	−0.252875	+	0.967499i	$0.581376\pi$
$272$	−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	$-0.559872\pi$
$283$	−6.29116	−0.373971	−0.186985	+	0.982363i	$0.559872\pi$
$284$	−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	$-0.513792\pi$
$293$	−1.48286	−0.0866294	−0.0433147	+	0.999061i	$0.513792\pi$
$294$	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	$-0.173544\pi$
$307$	29.9624	1.71004	0.855022	+	0.518592i	$0.173544\pi$
$308$	0	0
$309$	11.4100	0.649095
$310$	−14.4748	−0.822111
$311$	−4.38115	−0.248432	−0.124216	−	0.992255i	$-0.539642\pi$
$311$	−4.38115	−0.248432	−0.124216	+	0.992255i	$0.539642\pi$
$312$	−13.1579	−0.744920
$313$	−20.4536	−1.15611	−0.578053	−	0.815999i	$-0.696187\pi$
$313$	−20.4536	−1.15611	−0.578053	+	0.815999i	$0.696187\pi$
$314$	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	$-0.821783\pi$
$349$	−31.6583	−1.69463	−0.847316	+	0.531090i	$0.821783\pi$
$350$	0	0
$351$	2.87602	0.153511
$352$	−24.0298	−1.28079
$353$	−8.90381	−0.473902	−0.236951	−	0.971522i	$-0.576148\pi$
$353$	−8.90381	−0.473902	−0.236951	+	0.971522i	$0.576148\pi$
$354$	−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	$-0.608224\pi$
$367$	−12.7772	−0.666964	−0.333482	+	0.942757i	$0.608224\pi$
$368$	−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	$-0.460410\pi$
$383$	4.85563	0.248111	0.124056	+	0.992275i	$0.460410\pi$
$384$	−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	$-0.603678\pi$
$397$	−12.7514	−0.639974	−0.319987	+	0.947422i	$0.603678\pi$
$398$	−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	$-0.524734\pi$
$409$	−3.13980	−0.155253	−0.0776266	+	0.996983i	$0.524734\pi$
$410$	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	$-0.598671\pi$
$419$	−12.4881	−0.610085	−0.305043	+	0.952339i	$0.598671\pi$
$420$	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	$-0.535402\pi$
$433$	−4.61912	−0.221981	−0.110990	+	0.993821i	$0.535402\pi$
$434$	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	$-0.649454\pi$
$439$	−18.9603	−0.904925	−0.452462	+	0.891783i	$0.649454\pi$
$440$	−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	$-0.707035\pi$
$461$	−26.0021	−1.21104	−0.605519	+	0.795831i	$0.707035\pi$
$462$	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	$-0.613842\pi$
$467$	−15.1302	−0.700141	−0.350071	+	0.936723i	$0.613842\pi$
$468$	−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	$-0.720237\pi$
$479$	−27.9265	−1.27599	−0.637997	+	0.770039i	$0.720237\pi$
$480$	−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	$-0.754267\pi$
$503$	−32.1398	−1.43304	−0.716522	+	0.697565i	$0.754267\pi$
$504$	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	$-0.799520\pi$
$509$	−36.4645	−1.61626	−0.808130	+	0.589004i	$0.799520\pi$
$510$	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	$-0.783820\pi$
$521$	−35.5212	−1.55621	−0.778107	+	0.628132i	$0.783820\pi$
$522$	13.4730	0.589696
$523$	−14.1623	−0.619273	−0.309636	−	0.950855i	$-0.600207\pi$
$523$	−14.1623	−0.619273	−0.309636	+	0.950855i	$0.600207\pi$
$524$	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	$-0.476784\pi$
$563$	3.45814	0.145743	0.0728717	+	0.997341i	$0.476784\pi$
$564$	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	$-0.661107\pi$
$577$	−23.2905	−0.969596	−0.484798	+	0.874626i	$0.661107\pi$
$578$	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	$-0.283234\pi$
$587$	30.5061	1.25912	0.629562	+	0.776951i	$0.283234\pi$
$588$	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	$-0.265810\pi$
$593$	32.6861	1.34226	0.671129	+	0.741341i	$0.265810\pi$
$594$	−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	$-0.581313\pi$
$601$	−12.3891	−0.505363	−0.252682	+	0.967549i	$0.581313\pi$
$602$	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	$-0.263376\pi$
$607$	33.3480	1.35355	0.676776	+	0.736189i	$0.263376\pi$
$608$	−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	$-0.584147\pi$
$619$	−13.0014	−0.522572	−0.261286	+	0.965261i	$0.584147\pi$
$620$	−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	$-0.340462\pi$
$643$	24.3676	0.960964	0.480482	+	0.877005i	$0.340462\pi$
$644$	0	0
$645$	−86.4803	−3.40516
$646$	0.443136	0.0174350
$647$	−38.2402	−1.50338	−0.751688	−	0.659519i	$-0.770760\pi$
$647$	−38.2402	−1.50338	−0.751688	+	0.659519i	$0.770760\pi$
$648$	−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	$-0.766508\pi$
$661$	−38.1953	−1.48562	−0.742812	+	0.669500i	$0.766508\pi$
$662$	−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	$-0.532084\pi$
$677$	−5.23635	−0.201249	−0.100625	+	0.994924i	$0.532084\pi$
$678$	−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	$-0.0828012\pi$
$691$	50.8050	1.93271	0.966357	+	0.257204i	$0.0828012\pi$
$692$	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	$-0.612549\pi$
$719$	−18.5694	−0.692521	−0.346260	+	0.938138i	$0.612549\pi$
$720$	−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	$-0.296320\pi$
$727$	32.1991	1.19420	0.597099	+	0.802168i	$0.296320\pi$
$728$	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	$-0.556472\pi$
$733$	−9.55623	−0.352968	−0.176484	+	0.984304i	$0.556472\pi$
$734$	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	$-0.608748\pi$
$761$	−18.4847	−0.670070	−0.335035	+	0.942206i	$0.608748\pi$
$762$	−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	$-0.475063\pi$
$769$	4.34052	0.156523	0.0782617	+	0.996933i	$0.475063\pi$
$770$	0	0
$771$	−19.2382	−0.692845
$772$	−4.57847	−0.164783
$773$	−33.5513	−1.20676	−0.603379	−	0.797455i	$-0.706179\pi$
$773$	−33.5513	−1.20676	−0.603379	+	0.797455i	$0.706179\pi$
$774$	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	$-0.230252\pi$
$787$	42.0571	1.49917	0.749587	+	0.661906i	$0.230252\pi$
$788$	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	$-0.515289\pi$
$797$	−2.71101	−0.0960289	−0.0480145	+	0.998847i	$0.515289\pi$
$798$	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	$-0.414410\pi$
$811$	15.1310	0.531320	0.265660	+	0.964067i	$0.414410\pi$
$812$	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	$-0.845965\pi$
$829$	−50.9728	−1.77036	−0.885180	+	0.465249i	$0.845965\pi$
$830$	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	$-0.467840\pi$
$839$	5.84311	0.201727	0.100863	+	0.994900i	$0.467840\pi$
$840$	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	$-0.281579\pi$
$853$	37.0096	1.26719	0.633593	+	0.773667i	$0.281579\pi$
$854$	0	0
$855$	−10.3904	−0.355345
$856$	2.29315	0.0783784
$857$	−40.1269	−1.37071	−0.685355	−	0.728209i	$-0.740353\pi$
$857$	−40.1269	−1.37071	−0.685355	+	0.728209i	$0.740353\pi$
$858$	15.6970	0.535885
$859$	14.4388	0.492644	0.246322	−	0.969188i	$-0.420778\pi$
$859$	14.4388	0.492644	0.246322	+	0.969188i	$0.420778\pi$
$860$	−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	$-0.262858\pi$
$881$	40.2468	1.35595	0.677975	+	0.735085i	$0.262858\pi$
$882$	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	$-0.296204\pi$
$887$	35.5836	1.19478	0.597390	+	0.801951i	$0.296204\pi$
$888$	−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	$-0.398477\pi$
$929$	19.1145	0.627127	0.313563	+	0.949567i	$0.398477\pi$
$930$	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	$-0.758105\pi$
$937$	−44.3779	−1.44976	−0.724881	+	0.688874i	$0.758105\pi$
$938$	0	0
$939$	47.6879	1.55623
$940$	72.4789	2.36400
$941$	37.6172	1.22628	0.613142	−	0.789973i	$-0.289905\pi$
$941$	37.6172	1.22628	0.613142	+	0.789973i	$0.289905\pi$
$942$	−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	$-0.516348\pi$
$971$	−3.19928	−0.102670	−0.0513348	+	0.998681i	$0.516348\pi$
$972$	−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	$-0.472261\pi$
$983$	5.45760	0.174070	0.0870352	+	0.996205i	$0.472261\pi$
$984$	−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	$-0.619818\pi$
$997$	−23.2138	−0.735188	−0.367594	+	0.929986i	$0.619818\pi$
$998$	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.l.1.2	✓	4
3.2	odd	2		8379.2.a.bv.1.3		4
7.2	even	3		931.2.f.o.704.3		8
7.3	odd	6		931.2.f.n.324.3		8
7.4	even	3		931.2.f.o.324.3		8
7.5	odd	6		931.2.f.n.704.3		8
7.6	odd	2		931.2.a.m.1.2	yes	4
21.20	even	2		8379.2.a.bu.1.3		4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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