LMFDB - Embedded newform 931.4.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [931,4,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, 4); 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, 4, names="a")

Level:	$ N $	$=$	$ 931 = 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	931.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,-1,21,-14] 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:	$54.9307782153$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.526440$ 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)

$f(q)$	$=$	\(q+5.07177 q^{2} +8.66998 q^{3} +17.7229 q^{4} +2.61710 q^{5} +43.9722 q^{6} +49.3121 q^{8} +48.1686 q^{9} +13.2733 q^{10} +12.6171 q^{11} +153.657 q^{12} -46.9571 q^{13} +22.6902 q^{15} +108.317 q^{16} +28.9745 q^{17} +244.300 q^{18} +19.0000 q^{19} +46.3825 q^{20} +63.9911 q^{22} -112.166 q^{23} +427.535 q^{24} -118.151 q^{25} -238.155 q^{26} +183.531 q^{27} +295.107 q^{29} +115.080 q^{30} +57.6979 q^{31} +154.862 q^{32} +109.390 q^{33} +146.952 q^{34} +853.685 q^{36} -341.167 q^{37} +96.3636 q^{38} -407.117 q^{39} +129.055 q^{40} -274.056 q^{41} +327.536 q^{43} +223.611 q^{44} +126.062 q^{45} -568.879 q^{46} -139.140 q^{47} +939.106 q^{48} -599.234 q^{50} +251.209 q^{51} -832.214 q^{52} +296.715 q^{53} +930.828 q^{54} +33.0202 q^{55} +164.730 q^{57} +1496.72 q^{58} -459.383 q^{59} +402.136 q^{60} +232.911 q^{61} +292.631 q^{62} -81.1127 q^{64} -122.891 q^{65} +554.801 q^{66} -320.784 q^{67} +513.511 q^{68} -972.475 q^{69} -9.54518 q^{71} +2375.30 q^{72} -320.868 q^{73} -1730.32 q^{74} -1024.37 q^{75} +336.734 q^{76} -2064.80 q^{78} -89.2323 q^{79} +283.476 q^{80} +290.661 q^{81} -1389.95 q^{82} +439.455 q^{83} +75.8293 q^{85} +1661.19 q^{86} +2558.58 q^{87} +622.176 q^{88} -883.164 q^{89} +639.358 q^{90} -1987.90 q^{92} +500.240 q^{93} -705.688 q^{94} +49.7249 q^{95} +1342.65 q^{96} +1705.87 q^{97} +607.748 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 21 q^{4} - 14 q^{5} + 65 q^{6} + 27 q^{8} + 48 q^{9} + 88 q^{10} + 16 q^{11} + 115 q^{12} - 65 q^{13} + 140 q^{15} + 33 q^{16} - 29 q^{17} + 138 q^{18} + 57 q^{19} - 100 q^{20}+ \cdots + 250 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 21 * q^4 - 14 * q^5 + 65 * q^6 + 27 * q^8 + 48 * q^9 + 88 * q^10 + 16 * q^11 + 115 * q^12 - 65 * q^13 + 140 * q^15 + 33 * q^16 - 29 * q^17 + 138 * q^18 + 57 * q^19 - 100 * q^20 + 118 * q^22 - 101 * q^23 + 489 * q^24 - 37 * q^25 - 299 * q^26 + 377 * q^27 + 377 * q^29 - 376 * q^30 + 140 * q^31 + 579 * q^32 + 130 * q^33 - 329 * q^34 + 1074 * q^36 - 290 * q^37 + 57 * q^38 - 371 * q^39 + 72 * q^40 - 956 * q^41 - 570 * q^43 + 110 * q^44 - 230 * q^45 - 239 * q^46 - 66 * q^47 + 1411 * q^48 - 1639 * q^50 + 187 * q^51 - 697 * q^52 + 817 * q^53 + 947 * q^54 + 198 * q^55 - 19 * q^57 + 647 * q^58 - 265 * q^59 + 1360 * q^60 - 988 * q^61 + 1436 * q^62 - 159 * q^64 - 240 * q^65 + 274 * q^66 - 207 * q^67 + 1529 * q^68 - 807 * q^69 + 846 * q^71 + 2826 * q^72 - 627 * q^73 - 770 * q^74 - 2031 * q^75 + 399 * q^76 - 1817 * q^78 + 382 * q^79 + 1472 * q^80 - 501 * q^81 - 872 * q^82 + 766 * q^83 - 1242 * q^85 + 2592 * q^86 + 1671 * q^87 + 342 * q^88 + 172 * q^89 + 1888 * q^90 - 2677 * q^92 + 892 * q^93 + 96 * q^94 - 266 * q^95 - 863 * q^96 + 2450 * q^97 + 250 * 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 $

$2$	5.07177	1.79314	0.896571	−	0.442900i	$-0.146050\pi$
$2$	5.07177	1.79314	0.896571	+	0.442900i	$0.146050\pi$
$3$	8.66998	1.66854	0.834269	−	0.551357i	$-0.185890\pi$
$3$	8.66998	1.66854	0.834269	+	0.551357i	$0.185890\pi$
$4$	17.7229	2.21536
$5$	2.61710	0.234081	0.117040	−	0.993127i	$-0.462659\pi$
$5$	2.61710	0.234081	0.117040	+	0.993127i	$0.462659\pi$
$6$	43.9722	2.99193
$7$	0	0
$7$	0	0
$8$	49.3121	2.17931
$9$	48.1686	1.78402
$10$	13.2733	0.419740
$11$	12.6171	0.345836	0.172918	−	0.984936i	$-0.444680\pi$
$11$	12.6171	0.345836	0.172918	+	0.984936i	$0.444680\pi$
$12$	153.657	3.69641
$13$	−46.9571	−1.00181	−0.500906	−	0.865502i	$-0.667000\pi$
$13$	−46.9571	−1.00181	−0.500906	+	0.865502i	$0.667000\pi$
$14$	0	0
$15$	22.6902	0.390573
$16$	108.317	1.69245
$17$	28.9745	0.413374	0.206687	−	0.978407i	$-0.433732\pi$
$17$	28.9745	0.413374	0.206687	+	0.978407i	$0.433732\pi$
$18$	244.300	3.19900
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	46.3825	0.518573
$21$	0	0
$22$	63.9911	0.620134
$23$	−112.166	−1.01688	−0.508439	−	0.861098i	$-0.669777\pi$
$23$	−112.166	−1.01688	−0.508439	+	0.861098i	$0.669777\pi$
$24$	427.535	3.63626
$25$	−118.151	−0.945206
$26$	−238.155	−1.79639
$27$	183.531	1.30817
$28$	0	0
$29$	295.107	1.88966	0.944828	−	0.327565i	$-0.106228\pi$
$29$	295.107	1.88966	0.944828	+	0.327565i	$0.106228\pi$
$30$	115.080	0.700352
$31$	57.6979	0.334285	0.167143	−	0.985933i	$-0.446546\pi$
$31$	57.6979	0.334285	0.167143	+	0.985933i	$0.446546\pi$
$32$	154.862	0.855498
$33$	109.390	0.577041
$34$	146.952	0.741238
$35$	0	0
$36$	853.685	3.95225
$37$	−341.167	−1.51588	−0.757940	−	0.652324i	$-0.773794\pi$
$37$	−341.167	−1.51588	−0.757940	+	0.652324i	$0.773794\pi$
$38$	96.3636	0.411375
$39$	−407.117	−1.67156
$40$	129.055	0.510134
$41$	−274.056	−1.04391	−0.521955	−	0.852973i	$-0.674797\pi$
$41$	−274.056	−1.04391	−0.521955	+	0.852973i	$0.674797\pi$
$42$	0	0
$43$	327.536	1.16160	0.580800	−	0.814046i	$-0.302740\pi$
$43$	327.536	1.16160	0.580800	+	0.814046i	$0.302740\pi$
$44$	223.611	0.766151
$45$	126.062	0.417605
$46$	−568.879	−1.82341
$47$	−139.140	−0.431823	−0.215912	−	0.976413i	$-0.569272\pi$
$47$	−139.140	−0.431823	−0.215912	+	0.976413i	$0.569272\pi$
$48$	939.106	2.82392
$49$	0	0
$50$	−599.234	−1.69489
$51$	251.209	0.689730
$52$	−832.214	−2.21937
$53$	296.715	0.769000	0.384500	−	0.923125i	$-0.374374\pi$
$53$	296.715	0.769000	0.384500	+	0.923125i	$0.374374\pi$
$54$	930.828	2.34574
$55$	33.0202	0.0809536
$56$	0	0
$57$	164.730	0.382789
$58$	1496.72	3.38842
$59$	−459.383	−1.01367	−0.506836	−	0.862043i	$-0.669185\pi$
$59$	−459.383	−1.01367	−0.506836	+	0.862043i	$0.669185\pi$
$60$	402.136	0.865258
$61$	232.911	0.488873	0.244436	−	0.969665i	$-0.421397\pi$
$61$	232.911	0.488873	0.244436	+	0.969665i	$0.421397\pi$
$62$	292.631	0.599421
$63$	0	0
$64$	−81.1127	−0.158423
$65$	−122.891	−0.234505
$66$	554.801	1.03472
$67$	−320.784	−0.584926	−0.292463	−	0.956277i	$-0.594475\pi$
$67$	−320.784	−0.584926	−0.292463	+	0.956277i	$0.594475\pi$
$68$	513.511	0.915771
$69$	−972.475	−1.69670
$70$	0	0
$71$	−9.54518	−0.0159550	−0.00797749	−	0.999968i	$-0.502539\pi$
$71$	−9.54518	−0.0159550	−0.00797749	+	0.999968i	$0.502539\pi$
$72$	2375.30	3.88793
$73$	−320.868	−0.514448	−0.257224	−	0.966352i	$-0.582808\pi$
$73$	−320.868	−0.514448	−0.257224	+	0.966352i	$0.582808\pi$
$74$	−1730.32	−2.71819
$75$	−1024.37	−1.57711
$76$	336.734	0.508238
$77$	0	0
$78$	−2064.80	−2.99735
$79$	−89.2323	−0.127081	−0.0635406	−	0.997979i	$-0.520239\pi$
$79$	−89.2323	−0.127081	−0.0635406	+	0.997979i	$0.520239\pi$
$80$	283.476	0.396170
$81$	290.661	0.398712
$82$	−1389.95	−1.87188
$83$	439.455	0.581163	0.290581	−	0.956850i	$-0.406151\pi$
$83$	439.455	0.581163	0.290581	+	0.956850i	$0.406151\pi$
$84$	0	0
$85$	75.8293	0.0967628
$86$	1661.19	2.08292
$87$	2558.58	3.15297
$88$	622.176	0.753684
$89$	−883.164	−1.05186	−0.525928	−	0.850529i	$-0.676282\pi$
$89$	−883.164	−1.05186	−0.525928	+	0.850529i	$0.676282\pi$
$90$	639.358	0.748825
$91$	0	0
$92$	−1987.90	−2.25275
$93$	500.240	0.557768
$94$	−705.688	−0.774321
$95$	49.7249	0.0537018
$96$	1342.65	1.42743
$97$	1705.87	1.78562	0.892808	−	0.450437i	$-0.148732\pi$
$97$	1705.87	1.78562	0.892808	+	0.450437i	$0.148732\pi$
$98$	0	0
$99$	607.748	0.616979
$100$	−2093.97	−2.09397
$101$	961.422	0.947178	0.473589	−	0.880746i	$-0.342958\pi$
$101$	961.422	0.947178	0.473589	+	0.880746i	$0.342958\pi$
$102$	1274.07	1.23678
$103$	−173.142	−0.165633	−0.0828166	−	0.996565i	$-0.526392\pi$
$103$	−173.142	−0.165633	−0.0828166	+	0.996565i	$0.526392\pi$
$104$	−2315.55	−2.18326
$105$	0	0
$106$	1504.87	1.37893
$107$	921.087	0.832194	0.416097	−	0.909320i	$-0.363398\pi$
$107$	921.087	0.832194	0.416097	+	0.909320i	$0.363398\pi$
$108$	3252.70	2.89807
$109$	552.454	0.485463	0.242731	−	0.970094i	$-0.421957\pi$
$109$	552.454	0.485463	0.242731	+	0.970094i	$0.421957\pi$
$110$	167.471	0.145161
$111$	−2957.91	−2.52930
$112$	0	0
$113$	−1395.26	−1.16155	−0.580774	−	0.814064i	$-0.697250\pi$
$113$	−1395.26	−1.16155	−0.580774	+	0.814064i	$0.697250\pi$
$114$	835.471	0.686395
$115$	−293.549	−0.238031
$116$	5230.15	4.18627
$117$	−2261.86	−1.78725
$118$	−2329.89	−1.81766
$119$	0	0
$120$	1118.90	0.851179
$121$	−1171.81	−0.880397
$122$	1181.27	0.876618
$123$	−2376.06	−1.74180
$124$	1022.57	0.740562
$125$	−636.350	−0.455335
$126$	0	0
$127$	793.013	0.554083	0.277041	−	0.960858i	$-0.410646\pi$
$127$	793.013	0.554083	0.277041	+	0.960858i	$0.410646\pi$
$128$	−1650.28	−1.13957
$129$	2839.73	1.93818
$130$	−623.277	−0.420500
$131$	−25.0544	−0.0167100	−0.00835501	−	0.999965i	$-0.502660\pi$
$131$	−25.0544	−0.0167100	−0.00835501	+	0.999965i	$0.502660\pi$
$132$	1938.70	1.27835
$133$	0	0
$134$	−1626.95	−1.04886
$135$	480.320	0.306218
$136$	1428.80	0.900869
$137$	−716.056	−0.446546	−0.223273	−	0.974756i	$-0.571674\pi$
$137$	−716.056	−0.446546	−0.223273	+	0.974756i	$0.571674\pi$
$138$	−4932.17	−3.04242
$139$	−666.566	−0.406744	−0.203372	−	0.979102i	$-0.565190\pi$
$139$	−666.566	−0.406744	−0.203372	+	0.979102i	$0.565190\pi$
$140$	0	0
$141$	−1206.34	−0.720514
$142$	−48.4109	−0.0286095
$143$	−592.462	−0.346463
$144$	5217.47	3.01937
$145$	772.326	0.442332
$146$	−1627.37	−0.922479
$147$	0	0
$148$	−6046.46	−3.35822
$149$	−268.063	−0.147386	−0.0736931	−	0.997281i	$-0.523479\pi$
$149$	−268.063	−0.147386	−0.0736931	+	0.997281i	$0.523479\pi$
$150$	−5195.35	−2.82799
$151$	2809.46	1.51411	0.757055	−	0.653352i	$-0.226638\pi$
$151$	2809.46	1.51411	0.757055	+	0.653352i	$0.226638\pi$
$152$	936.930	0.499968
$153$	1395.66	0.737468
$154$	0	0
$155$	151.001	0.0782498
$156$	−7215.28	−3.70311
$157$	1999.77	1.01656	0.508278	−	0.861193i	$-0.330282\pi$
$157$	1999.77	1.01656	0.508278	+	0.861193i	$0.330282\pi$
$158$	−452.566	−0.227875
$159$	2572.52	1.28311
$160$	405.289	0.200256
$161$	0	0
$162$	1474.16	0.714946
$163$	1642.08	0.789064	0.394532	−	0.918882i	$-0.370907\pi$
$163$	1642.08	0.789064	0.394532	+	0.918882i	$0.370907\pi$
$164$	−4857.05	−2.31263
$165$	286.285	0.135074
$166$	2228.82	1.04211
$167$	−2965.92	−1.37431	−0.687155	−	0.726511i	$-0.741141\pi$
$167$	−2965.92	−1.37431	−0.687155	+	0.726511i	$0.741141\pi$
$168$	0	0
$169$	7.96603	0.00362587
$170$	384.589	0.173509
$171$	915.203	0.409283
$172$	5804.88	2.57336
$173$	−92.7872	−0.0407773	−0.0203887	−	0.999792i	$-0.506490\pi$
$173$	−92.7872	−0.0407773	−0.0203887	+	0.999792i	$0.506490\pi$
$174$	12976.5	5.65372
$175$	0	0
$176$	1366.65	0.585311
$177$	−3982.84	−1.69135
$178$	−4479.21	−1.88613
$179$	−3294.07	−1.37548	−0.687738	−	0.725959i	$-0.741396\pi$
$179$	−3294.07	−1.37548	−0.687738	+	0.725959i	$0.741396\pi$
$180$	2234.18	0.925145
$181$	−3590.17	−1.47434	−0.737168	−	0.675709i	$-0.763838\pi$
$181$	−3590.17	−1.47434	−0.737168	+	0.675709i	$0.763838\pi$
$182$	0	0
$183$	2019.34	0.815703
$184$	−5531.13	−2.21609
$185$	−892.870	−0.354838
$186$	2537.10	1.00016
$187$	365.574	0.142960
$188$	−2465.96	−0.956643
$189$	0	0
$190$	252.194	0.0962950
$191$	480.480	0.182023	0.0910114	−	0.995850i	$-0.470990\pi$
$191$	480.480	0.182023	0.0910114	+	0.995850i	$0.470990\pi$
$192$	−703.246	−0.264335
$193$	3504.07	1.30688	0.653442	−	0.756977i	$-0.273325\pi$
$193$	3504.07	1.30688	0.653442	+	0.756977i	$0.273325\pi$
$194$	8651.78	3.20186
$195$	−1065.47	−0.391280
$196$	0	0
$197$	603.790	0.218367	0.109183	−	0.994022i	$-0.465176\pi$
$197$	603.790	0.218367	0.109183	+	0.994022i	$0.465176\pi$
$198$	3082.36	1.10633
$199$	−3063.80	−1.09139	−0.545695	−	0.837984i	$-0.683734\pi$
$199$	−3063.80	−1.09139	−0.545695	+	0.837984i	$0.683734\pi$
$200$	−5826.27	−2.05990
$201$	−2781.20	−0.975972
$202$	4876.11	1.69843
$203$	0	0
$204$	4452.13	1.52800
$205$	−717.232	−0.244359
$206$	−878.139	−0.297004
$207$	−5402.87	−1.81413
$208$	−5086.24	−1.69552
$209$	239.725	0.0793403
$210$	0	0
$211$	−2772.16	−0.904470	−0.452235	−	0.891899i	$-0.649373\pi$
$211$	−2772.16	−0.904470	−0.452235	+	0.891899i	$0.649373\pi$
$212$	5258.65	1.70361
$213$	−82.7565	−0.0266215
$214$	4671.54	1.49224
$215$	857.196	0.271908
$216$	9050.32	2.85091
$217$	0	0
$218$	2801.92	0.870504
$219$	−2781.92	−0.858377
$220$	585.213	0.179341
$221$	−1360.56	−0.414122
$222$	−15001.9	−4.53540
$223$	5653.47	1.69769	0.848844	−	0.528644i	$-0.177299\pi$
$223$	5653.47	1.69769	0.848844	+	0.528644i	$0.177299\pi$
$224$	0	0
$225$	−5691.16	−1.68627
$226$	−7076.44	−2.08282
$227$	5083.00	1.48621	0.743106	−	0.669173i	$-0.233352\pi$
$227$	5083.00	1.48621	0.743106	+	0.669173i	$0.233352\pi$
$228$	2919.48	0.848015
$229$	−501.966	−0.144851	−0.0724254	−	0.997374i	$-0.523074\pi$
$229$	−501.966	−0.144851	−0.0724254	+	0.997374i	$0.523074\pi$
$230$	−1488.82	−0.426824
$231$	0	0
$232$	14552.4	4.11815
$233$	2809.38	0.789909	0.394955	−	0.918701i	$-0.370760\pi$
$233$	2809.38	0.789909	0.394955	+	0.918701i	$0.370760\pi$
$234$	−11471.6	−3.20480
$235$	−364.144	−0.101082
$236$	−8141.58	−2.24564
$237$	−773.642	−0.212040
$238$	0	0
$239$	2239.56	0.606130	0.303065	−	0.952970i	$-0.401990\pi$
$239$	2239.56	0.606130	0.303065	+	0.952970i	$0.401990\pi$
$240$	2457.74	0.661026
$241$	7214.25	1.92826	0.964130	−	0.265431i	$-0.0855142\pi$
$241$	7214.25	1.92826	0.964130	+	0.265431i	$0.0855142\pi$
$242$	−5943.15	−1.57868
$243$	−2435.32	−0.642905
$244$	4127.85	1.08303
$245$	0	0
$246$	−12050.8	−3.12330
$247$	−892.184	−0.229831
$248$	2845.21	0.728511
$249$	3810.07	0.969693
$250$	−3227.42	−0.816481
$251$	−6212.82	−1.56235	−0.781174	−	0.624313i	$-0.785379\pi$
$251$	−6212.82	−1.56235	−0.781174	+	0.624313i	$0.785379\pi$
$252$	0	0
$253$	−1415.21	−0.351673
$254$	4021.98	0.993549
$255$	657.438	0.161453
$256$	−7720.93	−1.88499
$257$	2796.37	0.678727	0.339364	−	0.940655i	$-0.389788\pi$
$257$	2796.37	0.678727	0.339364	+	0.940655i	$0.389788\pi$
$258$	14402.5	3.47542
$259$	0	0
$260$	−2177.99	−0.519512
$261$	14214.9	3.37119
$262$	−127.070	−0.0299634
$263$	2976.00	0.697748	0.348874	−	0.937170i	$-0.386564\pi$
$263$	2976.00	0.697748	0.348874	+	0.937170i	$0.386564\pi$
$264$	5394.26	1.25755
$265$	776.534	0.180008
$266$	0	0
$267$	−7657.02	−1.75506
$268$	−5685.22	−1.29582
$269$	25.4734	0.00577376	0.00288688	−	0.999996i	$-0.499081\pi$
$269$	25.4734	0.00577376	0.00288688	+	0.999996i	$0.499081\pi$
$270$	2436.07	0.549091
$271$	4338.19	0.972421	0.486211	−	0.873842i	$-0.338379\pi$
$271$	4338.19	0.972421	0.486211	+	0.873842i	$0.338379\pi$
$272$	3138.43	0.699615
$273$	0	0
$274$	−3631.67	−0.800720
$275$	−1490.72	−0.326887
$276$	−17235.0	−3.75880
$277$	4276.02	0.927514	0.463757	−	0.885962i	$-0.346501\pi$
$277$	4276.02	0.927514	0.463757	+	0.885962i	$0.346501\pi$
$278$	−3380.67	−0.729349
$279$	2779.23	0.596373
$280$	0	0
$281$	−3716.43	−0.788980	−0.394490	−	0.918900i	$-0.629079\pi$
$281$	−3716.43	−0.788980	−0.394490	+	0.918900i	$0.629079\pi$
$282$	−6118.30	−1.29198
$283$	−3748.01	−0.787265	−0.393633	−	0.919268i	$-0.628782\pi$
$283$	−3748.01	−0.787265	−0.393633	+	0.919268i	$0.628782\pi$
$284$	−169.168	−0.0353460
$285$	431.114	0.0896035
$286$	−3004.83	−0.621257
$287$	0	0
$288$	7459.46	1.52623
$289$	−4073.48	−0.829122
$290$	3917.06	0.793165
$291$	14789.9	2.97937
$292$	−5686.69	−1.13969
$293$	−8210.10	−1.63699	−0.818497	−	0.574510i	$-0.805192\pi$
$293$	−8210.10	−1.63699	−0.818497	+	0.574510i	$0.805192\pi$
$294$	0	0
$295$	−1202.25	−0.237281
$296$	−16823.7	−3.30357
$297$	2315.63	0.452413
$298$	−1359.55	−0.264284
$299$	5266.98	1.01872
$300$	−18154.7	−3.49387
$301$	0	0
$302$	14248.9	2.71501
$303$	8335.51	1.58040
$304$	2058.02	0.388275
$305$	609.553	0.114436
$306$	7078.48	1.32238
$307$	2814.79	0.523285	0.261642	−	0.965165i	$-0.415736\pi$
$307$	2814.79	0.523285	0.261642	+	0.965165i	$0.415736\pi$
$308$	0	0
$309$	−1501.14	−0.276366
$310$	765.844	0.140313
$311$	8650.75	1.57730	0.788648	−	0.614845i	$-0.210782\pi$
$311$	8650.75	1.57730	0.788648	+	0.614845i	$0.210782\pi$
$312$	−20075.8	−3.64285
$313$	−1623.48	−0.293178	−0.146589	−	0.989197i	$-0.546829\pi$
$313$	−1623.48	−0.293178	−0.146589	+	0.989197i	$0.546829\pi$
$314$	10142.4	1.82283
$315$	0	0
$316$	−1581.45	−0.281530
$317$	9372.31	1.66057	0.830286	−	0.557337i	$-0.188177\pi$
$317$	9372.31	1.66057	0.830286	+	0.557337i	$0.188177\pi$
$318$	13047.2	2.30079
$319$	3723.40	0.653512
$320$	−212.280	−0.0370838
$321$	7985.80	1.38855
$322$	0	0
$323$	550.516	0.0948344
$324$	5151.34	0.883289
$325$	5548.01	0.946918
$326$	8328.24	1.41490
$327$	4789.76	0.810014
$328$	−13514.3	−2.27500
$329$	0	0
$330$	1451.97	0.242207
$331$	−1765.55	−0.293182	−0.146591	−	0.989197i	$-0.546830\pi$
$331$	−1765.55	−0.293182	−0.146591	+	0.989197i	$0.546830\pi$
$332$	7788.41	1.28748
$333$	−16433.5	−2.70436
$334$	−15042.5	−2.46433
$335$	−839.526	−0.136920
$336$	0	0
$337$	10189.8	1.64711	0.823555	−	0.567237i	$-0.191988\pi$
$337$	10189.8	1.64711	0.823555	+	0.567237i	$0.191988\pi$
$338$	40.4019	0.00650169
$339$	−12096.9	−1.93809
$340$	1343.91	0.214364
$341$	727.980	0.115608
$342$	4641.70	0.733902
$343$	0	0
$344$	16151.5	2.53149
$345$	−2545.07	−0.397165
$346$	−470.595	−0.0731195
$347$	−11350.2	−1.75594	−0.877971	−	0.478714i	$-0.841103\pi$
$347$	−11350.2	−1.75594	−0.877971	+	0.478714i	$0.841103\pi$
$348$	45345.3	6.98495
$349$	−511.619	−0.0784709	−0.0392354	−	0.999230i	$-0.512492\pi$
$349$	−511.619	−0.0784709	−0.0392354	+	0.999230i	$0.512492\pi$
$350$	0	0
$351$	−8618.09	−1.31054
$352$	1953.90	0.295862
$353$	−816.097	−0.123049	−0.0615247	−	0.998106i	$-0.519596\pi$
$353$	−816.097	−0.123049	−0.0615247	+	0.998106i	$0.519596\pi$
$354$	−20200.1	−3.03283
$355$	−24.9807	−0.00373475
$356$	−15652.2	−2.33024
$357$	0	0
$358$	−16706.8	−2.46642
$359$	3998.35	0.587813	0.293906	−	0.955834i	$-0.405045\pi$
$359$	3998.35	0.587813	0.293906	+	0.955834i	$0.405045\pi$
$360$	6216.39	0.910090
$361$	361.000	0.0526316
$362$	−18208.5	−2.64369
$363$	−10159.6	−1.46898
$364$	0	0
$365$	−839.744	−0.120422
$366$	10241.6	1.46267
$367$	3123.19	0.444221	0.222111	−	0.975021i	$-0.428705\pi$
$367$	3123.19	0.444221	0.222111	+	0.975021i	$0.428705\pi$
$368$	−12149.5	−1.72102
$369$	−13200.9	−1.86236
$370$	−4528.43	−0.636276
$371$	0	0
$372$	8865.68	1.23566
$373$	−7026.28	−0.975354	−0.487677	−	0.873024i	$-0.662156\pi$
$373$	−7026.28	−0.975354	−0.487677	+	0.873024i	$0.662156\pi$
$374$	1854.11	0.256347
$375$	−5517.15	−0.759745
$376$	−6861.30	−0.941076
$377$	−13857.4	−1.89308
$378$	0	0
$379$	−11595.9	−1.57162	−0.785808	−	0.618470i	$-0.787753\pi$
$379$	−11595.9	−1.57162	−0.785808	+	0.618470i	$0.787753\pi$
$380$	881.268	0.118969
$381$	6875.41	0.924508
$382$	2436.89	0.326393
$383$	10962.4	1.46254	0.731268	−	0.682091i	$-0.238929\pi$
$383$	10962.4	1.46254	0.731268	+	0.682091i	$0.238929\pi$
$384$	−14307.9	−1.90142
$385$	0	0
$386$	17771.8	2.34343
$387$	15777.0	2.07232
$388$	30232.9	3.95578
$389$	7126.21	0.928825	0.464413	−	0.885619i	$-0.346265\pi$
$389$	7126.21	0.928825	0.464413	+	0.885619i	$0.346265\pi$
$390$	−5403.80	−0.701621
$391$	−3249.95	−0.420350
$392$	0	0
$393$	−217.221	−0.0278813
$394$	3062.29	0.391563
$395$	−233.530	−0.0297473
$396$	10771.0	1.36683
$397$	−4895.59	−0.618899	−0.309449	−	0.950916i	$-0.600145\pi$
$397$	−4895.59	−0.618899	−0.309449	+	0.950916i	$0.600145\pi$
$398$	−15538.9	−1.95702
$399$	0	0
$400$	−12797.7	−1.59972
$401$	−6148.05	−0.765634	−0.382817	−	0.923824i	$-0.625046\pi$
$401$	−6148.05	−0.765634	−0.382817	+	0.923824i	$0.625046\pi$
$402$	−14105.6	−1.75006
$403$	−2709.32	−0.334891
$404$	17039.1	2.09834
$405$	760.689	0.0933307
$406$	0	0
$407$	−4304.54	−0.524246
$408$	12387.6	1.50313
$409$	−2968.47	−0.358878	−0.179439	−	0.983769i	$-0.557428\pi$
$409$	−2968.47	−0.358878	−0.179439	+	0.983769i	$0.557428\pi$
$410$	−3637.64	−0.438171
$411$	−6208.19	−0.745079
$412$	−3068.58	−0.366937
$413$	0	0
$414$	−27402.1	−3.25300
$415$	1150.10	0.136039
$416$	−7271.85	−0.857047
$417$	−5779.12	−0.678668
$418$	1215.83	0.142268
$419$	11101.1	1.29432	0.647162	−	0.762352i	$-0.275956\pi$
$419$	11101.1	1.29432	0.647162	+	0.762352i	$0.275956\pi$
$420$	0	0
$421$	3241.73	0.375278	0.187639	−	0.982238i	$-0.439916\pi$
$421$	3241.73	0.375278	0.187639	+	0.982238i	$0.439916\pi$
$422$	−14059.7	−1.62184
$423$	−6702.19	−0.770382
$424$	14631.7	1.67589
$425$	−3423.36	−0.390723
$426$	−419.722	−0.0477361
$427$	0	0
$428$	16324.3	1.84361
$429$	−5136.64	−0.578086
$430$	4347.50	0.487570
$431$	290.271	0.0324405	0.0162202	−	0.999868i	$-0.494837\pi$
$431$	290.271	0.0324405	0.0162202	+	0.999868i	$0.494837\pi$
$432$	19879.5	2.21402
$433$	−2104.16	−0.233533	−0.116766	−	0.993159i	$-0.537253\pi$
$433$	−2104.16	−0.233533	−0.116766	+	0.993159i	$0.537253\pi$
$434$	0	0
$435$	6696.05	0.738049
$436$	9791.06	1.07547
$437$	−2131.15	−0.233288
$438$	−14109.2	−1.53919
$439$	−4013.82	−0.436376	−0.218188	−	0.975907i	$-0.570015\pi$
$439$	−4013.82	−0.436376	−0.218188	+	0.975907i	$0.570015\pi$
$440$	1628.30	0.176423
$441$	0	0
$442$	−6900.44	−0.742580
$443$	12013.8	1.28847	0.644236	−	0.764827i	$-0.277176\pi$
$443$	12013.8	1.28847	0.644236	+	0.764827i	$0.277176\pi$
$444$	−52422.7	−5.60331
$445$	−2311.33	−0.246219
$446$	28673.1	3.04419
$447$	−2324.10	−0.245920
$448$	0	0
$449$	4281.55	0.450020	0.225010	−	0.974356i	$-0.427759\pi$
$449$	4281.55	0.450020	0.225010	+	0.974356i	$0.427759\pi$
$450$	−28864.2	−3.02372
$451$	−3457.79	−0.361022
$452$	−24728.0	−2.57325
$453$	24358.0	2.52635
$454$	25779.8	2.66499
$455$	0	0
$456$	8123.17	0.834215
$457$	293.330	0.0300249	0.0150125	−	0.999887i	$-0.495221\pi$
$457$	293.330	0.0300249	0.0150125	+	0.999887i	$0.495221\pi$
$458$	−2545.85	−0.259738
$459$	5317.73	0.540763
$460$	−5202.53	−0.527325
$461$	5267.87	0.532210	0.266105	−	0.963944i	$-0.414263\pi$
$461$	5267.87	0.532210	0.266105	+	0.963944i	$0.414263\pi$
$462$	0	0
$463$	8076.98	0.810733	0.405366	−	0.914154i	$-0.367144\pi$
$463$	8076.98	0.810733	0.405366	+	0.914154i	$0.367144\pi$
$464$	31965.1	3.19815
$465$	1309.18	0.130563
$466$	14248.5	1.41642
$467$	3160.43	0.313163	0.156582	−	0.987665i	$-0.449953\pi$
$467$	3160.43	0.313163	0.156582	+	0.987665i	$0.449953\pi$
$468$	−40086.5	−3.95940
$469$	0	0
$470$	−1846.86	−0.181254
$471$	17338.0	1.69616
$472$	−22653.2	−2.20910
$473$	4132.56	0.401724
$474$	−3923.74	−0.380218
$475$	−2244.86	−0.216845
$476$	0	0
$477$	14292.4	1.37191
$478$	11358.5	1.08688
$479$	249.277	0.0237782	0.0118891	−	0.999929i	$-0.496215\pi$
$479$	249.277	0.0237782	0.0118891	+	0.999929i	$0.496215\pi$
$480$	3513.85	0.334134
$481$	16020.2	1.51863
$482$	36589.0	3.45764
$483$	0	0
$484$	−20767.8	−1.95039
$485$	4464.44	0.417979
$486$	−12351.4	−1.15282
$487$	7265.71	0.676059	0.338029	−	0.941136i	$-0.390240\pi$
$487$	7265.71	0.676059	0.338029	+	0.941136i	$0.390240\pi$
$488$	11485.4	1.06540
$489$	14236.8	1.31658
$490$	0	0
$491$	−8456.47	−0.777261	−0.388630	−	0.921394i	$-0.627052\pi$
$491$	−8456.47	−0.777261	−0.388630	+	0.921394i	$0.627052\pi$
$492$	−42110.5	−3.85872
$493$	8550.59	0.781134
$494$	−4524.95	−0.412120
$495$	1590.54	0.144423
$496$	6249.66	0.565762
$497$	0	0
$498$	19323.8	1.73880
$499$	1905.15	0.170914	0.0854571	−	0.996342i	$-0.472765\pi$
$499$	1905.15	0.170914	0.0854571	+	0.996342i	$0.472765\pi$
$500$	−11278.0	−1.00873
$501$	−25714.5	−2.29309
$502$	−31510.0	−2.80151
$503$	−6082.63	−0.539187	−0.269593	−	0.962974i	$-0.586889\pi$
$503$	−6082.63	−0.539187	−0.269593	+	0.962974i	$0.586889\pi$
$504$	0	0
$505$	2516.14	0.221716
$506$	−7177.61	−0.630600
$507$	69.0653	0.00604990
$508$	14054.5	1.22749
$509$	−13858.9	−1.20684	−0.603422	−	0.797422i	$-0.706197\pi$
$509$	−13858.9	−1.20684	−0.603422	+	0.797422i	$0.706197\pi$
$510$	3334.38	0.289507
$511$	0	0
$512$	−25956.6	−2.24049
$513$	3487.09	0.300115
$514$	14182.6	1.21705
$515$	−453.131	−0.0387716
$516$	50328.2	4.29375
$517$	−1755.55	−0.149340
$518$	0	0
$519$	−804.463	−0.0680385
$520$	−6060.04	−0.511058
$521$	−4086.72	−0.343651	−0.171826	−	0.985127i	$-0.554967\pi$
$521$	−4086.72	−0.343651	−0.171826	+	0.985127i	$0.554967\pi$
$522$	72094.7	6.04502
$523$	−20188.4	−1.68791	−0.843957	−	0.536411i	$-0.819780\pi$
$523$	−20188.4	−1.68791	−0.843957	+	0.536411i	$0.819780\pi$
$524$	−444.036	−0.0370187
$525$	0	0
$526$	15093.6	1.25116
$527$	1671.77	0.138185
$528$	11848.8	0.976615
$529$	414.164	0.0340399
$530$	3938.40	0.322780
$531$	−22127.8	−1.80841
$532$	0	0
$533$	12868.9	1.04580
$534$	−38834.7	−3.14708
$535$	2410.58	0.194801
$536$	−15818.6	−1.27473
$537$	−28559.5	−2.29503
$538$	129.195	0.0103532
$539$	0	0
$540$	8512.64	0.678381
$541$	16356.6	1.29986	0.649932	−	0.759992i	$-0.274797\pi$
$541$	16356.6	1.29986	0.649932	+	0.759992i	$0.274797\pi$
$542$	22002.3	1.74369
$543$	−31126.7	−2.45999
$544$	4487.04	0.353640
$545$	1445.83	0.113638
$546$	0	0
$547$	−12751.0	−0.996697	−0.498349	−	0.866977i	$-0.666060\pi$
$547$	−12751.0	−0.996697	−0.498349	+	0.866977i	$0.666060\pi$
$548$	−12690.6	−0.989259
$549$	11219.0	0.872159
$550$	−7560.59	−0.586154
$551$	5607.04	0.433517
$552$	−47954.8	−3.69763
$553$	0	0
$554$	21687.0	1.66316
$555$	−7741.17	−0.592062
$556$	−11813.5	−0.901083
$557$	−7353.77	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$557$	−7353.77	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$558$	14095.6	1.06938
$559$	−15380.1	−1.16370
$560$	0	0
$561$	3169.52	0.238534
$562$	−18848.9	−1.41475
$563$	5597.74	0.419035	0.209517	−	0.977805i	$-0.432811\pi$
$563$	5597.74	0.419035	0.209517	+	0.977805i	$0.432811\pi$
$564$	−21379.9	−1.59620
$565$	−3651.54	−0.271896
$566$	−19009.0	−1.41168
$567$	0	0
$568$	−470.693	−0.0347708
$569$	−19640.2	−1.44703	−0.723515	−	0.690309i	$-0.757475\pi$
$569$	−19640.2	−1.44703	−0.723515	+	0.690309i	$0.757475\pi$
$570$	2186.51	0.160672
$571$	−9749.71	−0.714558	−0.357279	−	0.933998i	$-0.616295\pi$
$571$	−9749.71	−0.714558	−0.357279	+	0.933998i	$0.616295\pi$
$572$	−10500.1	−0.767539
$573$	4165.76	0.303712
$574$	0	0
$575$	13252.5	0.961159
$576$	−3907.09	−0.282631
$577$	−11197.9	−0.807927	−0.403964	−	0.914775i	$-0.632368\pi$
$577$	−11197.9	−0.807927	−0.403964	+	0.914775i	$0.632368\pi$
$578$	−20659.7	−1.48673
$579$	30380.2	2.18059
$580$	13687.8	0.979924
$581$	0	0
$582$	75010.8	5.34243
$583$	3743.69	0.265948
$584$	−15822.7	−1.12114
$585$	−5919.51	−0.418362
$586$	−41639.8	−2.93536
$587$	18654.4	1.31167	0.655833	−	0.754906i	$-0.272318\pi$
$587$	18654.4	1.31167	0.655833	+	0.754906i	$0.272318\pi$
$588$	0	0
$589$	1096.26	0.0766903
$590$	−6097.55	−0.425478
$591$	5234.85	0.364354
$592$	−36954.2	−2.56555
$593$	−21513.4	−1.48980	−0.744900	−	0.667176i	$-0.767503\pi$
$593$	−21513.4	−1.48980	−0.744900	+	0.667176i	$0.767503\pi$
$594$	11744.4	0.811240
$595$	0	0
$596$	−4750.84	−0.326513
$597$	−26563.1	−1.82103
$598$	26712.9	1.82671
$599$	22762.8	1.55269	0.776346	−	0.630308i	$-0.217071\pi$
$599$	22762.8	1.55269	0.776346	+	0.630308i	$0.217071\pi$
$600$	−50513.6	−3.43702
$601$	13941.0	0.946195	0.473098	−	0.881010i	$-0.343136\pi$
$601$	13941.0	0.946195	0.473098	+	0.881010i	$0.343136\pi$
$602$	0	0
$603$	−15451.7	−1.04352
$604$	49791.6	3.35429
$605$	−3066.74	−0.206084
$606$	42275.8	2.83389
$607$	−8031.64	−0.537058	−0.268529	−	0.963272i	$-0.586538\pi$
$607$	−8031.64	−0.537058	−0.268529	+	0.963272i	$0.586538\pi$
$608$	2942.37	0.196265
$609$	0	0
$610$	3091.51	0.205199
$611$	6533.62	0.432606
$612$	24735.1	1.63375
$613$	−19429.7	−1.28019	−0.640095	−	0.768296i	$-0.721105\pi$
$613$	−19429.7	−1.28019	−0.640095	+	0.768296i	$0.721105\pi$
$614$	14276.0	0.938324
$615$	−6218.39	−0.407723
$616$	0	0
$617$	−11264.0	−0.734959	−0.367480	−	0.930032i	$-0.619779\pi$
$617$	−11264.0	−0.734959	−0.367480	+	0.930032i	$0.619779\pi$
$618$	−7613.45	−0.495563
$619$	22183.9	1.44046	0.720231	−	0.693734i	$-0.244036\pi$
$619$	22183.9	1.44046	0.720231	+	0.693734i	$0.244036\pi$
$620$	2676.18	0.173351
$621$	−20585.9	−1.33025
$622$	43874.6	2.82831
$623$	0	0
$624$	−44097.6	−2.82904
$625$	13103.5	0.838621
$626$	−8233.93	−0.525710
$627$	2078.41	0.132382
$628$	35441.7	2.25204
$629$	−9885.16	−0.626625
$630$	0	0
$631$	−23139.3	−1.45984	−0.729920	−	0.683532i	$-0.760443\pi$
$631$	−23139.3	−1.45984	−0.729920	+	0.683532i	$0.760443\pi$
$632$	−4400.23	−0.276949
$633$	−24034.5	−1.50914
$634$	47534.2	2.97764
$635$	2075.40	0.129700
$636$	45592.4	2.84254
$637$	0	0
$638$	18884.2	1.17184
$639$	−459.778	−0.0284640
$640$	−4318.95	−0.266752
$641$	−4988.45	−0.307382	−0.153691	−	0.988119i	$-0.549116\pi$
$641$	−4988.45	−0.307382	−0.153691	+	0.988119i	$0.549116\pi$
$642$	40502.2	2.48986
$643$	11115.2	0.681712	0.340856	−	0.940115i	$-0.389283\pi$
$643$	11115.2	0.681712	0.340856	+	0.940115i	$0.389283\pi$
$644$	0	0
$645$	7431.88	0.453690
$646$	2792.09	0.170052
$647$	−11916.2	−0.724071	−0.362036	−	0.932164i	$-0.617918\pi$
$647$	−11916.2	−0.724071	−0.362036	+	0.932164i	$0.617918\pi$
$648$	14333.1	0.868915
$649$	−5796.09	−0.350564
$650$	28138.3	1.69796
$651$	0	0
$652$	29102.3	1.74806
$653$	−18100.5	−1.08473	−0.542363	−	0.840144i	$-0.682470\pi$
$653$	−18100.5	−1.08473	−0.542363	+	0.840144i	$0.682470\pi$
$654$	24292.6	1.45247
$655$	−65.5699	−0.00391149
$656$	−29684.9	−1.76677
$657$	−15455.7	−0.917787
$658$	0	0
$659$	331.740	0.0196096	0.00980481	−	0.999952i	$-0.496879\pi$
$659$	331.740	0.0196096	0.00980481	+	0.999952i	$0.496879\pi$
$660$	5073.79	0.299238
$661$	30555.8	1.79801	0.899004	−	0.437939i	$-0.144292\pi$
$661$	30555.8	1.79801	0.899004	+	0.437939i	$0.144292\pi$
$662$	−8954.46	−0.525717
$663$	−11796.0	−0.690979
$664$	21670.5	1.26653
$665$	0	0
$666$	−83347.2	−4.84931
$667$	−33100.9	−1.92155
$668$	−52564.6	−3.04459
$669$	49015.5	2.83266
$670$	−4257.88	−0.245517
$671$	2938.67	0.169070
$672$	0	0
$673$	3261.14	0.186787	0.0933936	−	0.995629i	$-0.470228\pi$
$673$	3261.14	0.186787	0.0933936	+	0.995629i	$0.470228\pi$
$674$	51680.5	2.95350
$675$	−21684.4	−1.23649
$676$	141.181	0.00803259
$677$	−11556.6	−0.656063	−0.328031	−	0.944667i	$-0.606385\pi$
$677$	−11556.6	−0.656063	−0.328031	+	0.944667i	$0.606385\pi$
$678$	−61352.6	−3.47527
$679$	0	0
$680$	3739.30	0.210876
$681$	44069.5	2.47980
$682$	3692.15	0.207302
$683$	19704.1	1.10389	0.551944	−	0.833881i	$-0.313886\pi$
$683$	19704.1	1.10389	0.551944	+	0.833881i	$0.313886\pi$
$684$	16220.0	0.906707
$685$	−1873.99	−0.104528
$686$	0	0
$687$	−4352.03	−0.241689
$688$	35477.7	1.96595
$689$	−13932.9	−0.770393
$690$	−12908.0	−0.712173
$691$	2956.51	0.162765	0.0813827	−	0.996683i	$-0.474066\pi$
$691$	2956.51	0.162765	0.0813827	+	0.996683i	$0.474066\pi$
$692$	−1644.45	−0.0903364
$693$	0	0
$694$	−57565.7	−3.14865
$695$	−1744.47	−0.0952109
$696$	126169.	6.87129
$697$	−7940.63	−0.431525
$698$	−2594.81	−0.140709
$699$	24357.3	1.31799
$700$	0	0
$701$	29022.1	1.56370	0.781848	−	0.623469i	$-0.214277\pi$
$701$	29022.1	1.56370	0.781848	+	0.623469i	$0.214277\pi$
$702$	−43709.0	−2.34998
$703$	−6482.18	−0.347767
$704$	−1023.41	−0.0547885
$705$	−3157.13	−0.168658
$706$	−4139.05	−0.220645
$707$	0	0
$708$	−70587.4	−3.74694
$709$	−5110.64	−0.270711	−0.135356	−	0.990797i	$-0.543218\pi$
$709$	−5110.64	−0.270711	−0.135356	+	0.990797i	$0.543218\pi$
$710$	−126.696	−0.00669694
$711$	−4298.19	−0.226716
$712$	−43550.7	−2.29232
$713$	−6471.73	−0.339927
$714$	0	0
$715$	−1550.53	−0.0811003
$716$	−58380.3	−3.04717
$717$	19416.9	1.01135
$718$	20278.7	1.05403
$719$	25225.9	1.30844	0.654219	−	0.756305i	$-0.272998\pi$
$719$	25225.9	1.30844	0.654219	+	0.756305i	$0.272998\pi$
$720$	13654.7	0.706777
$721$	0	0
$722$	1830.91	0.0943759
$723$	62547.4	3.21738
$724$	−63628.0	−3.26618
$725$	−34867.2	−1.78612
$726$	−51527.0	−2.63408
$727$	12817.7	0.653893	0.326947	−	0.945043i	$-0.393980\pi$
$727$	12817.7	0.653893	0.326947	+	0.945043i	$0.393980\pi$
$728$	0	0
$729$	−28962.0	−1.47142
$730$	−4258.99	−0.215935
$731$	9490.21	0.480175
$732$	35788.4	1.80707
$733$	15307.5	0.771342	0.385671	−	0.922636i	$-0.373970\pi$
$733$	15307.5	0.771342	0.385671	+	0.922636i	$0.373970\pi$
$734$	15840.1	0.796551
$735$	0	0
$736$	−17370.2	−0.869936
$737$	−4047.37	−0.202289
$738$	−66951.8	−3.33947
$739$	34340.1	1.70936	0.854682	−	0.519152i	$-0.173752\pi$
$739$	34340.1	1.70936	0.854682	+	0.519152i	$0.173752\pi$
$740$	−15824.2	−0.786094
$741$	−7735.22	−0.383482
$742$	0	0
$743$	−1883.40	−0.0929948	−0.0464974	−	0.998918i	$-0.514806\pi$
$743$	−1883.40	−0.0929948	−0.0464974	+	0.998918i	$0.514806\pi$
$744$	24667.9	1.21555
$745$	−701.547	−0.0345003
$746$	−35635.7	−1.74895
$747$	21167.9	1.03681
$748$	6479.03	0.316707
$749$	0	0
$750$	−27981.7	−1.36233
$751$	−9431.98	−0.458292	−0.229146	−	0.973392i	$-0.573593\pi$
$751$	−9431.98	−0.458292	−0.229146	+	0.973392i	$0.573593\pi$
$752$	−15071.2	−0.730840
$753$	−53865.0	−2.60684
$754$	−70281.4	−3.39456
$755$	7352.64	0.354424
$756$	0	0
$757$	−12355.0	−0.593196	−0.296598	−	0.955002i	$-0.595852\pi$
$757$	−12355.0	−0.593196	−0.296598	+	0.955002i	$0.595852\pi$
$758$	−58811.9	−2.81813
$759$	−12269.8	−0.586780
$760$	2452.04	0.117033
$761$	27257.6	1.29841	0.649204	−	0.760614i	$-0.275102\pi$
$761$	27257.6	1.29841	0.649204	+	0.760614i	$0.275102\pi$
$762$	34870.5	1.65777
$763$	0	0
$764$	8515.49	0.403246
$765$	3652.59	0.172627
$766$	55598.6	2.62253
$767$	21571.3	1.01551
$768$	−66940.3	−3.14518
$769$	1191.72	0.0558835	0.0279418	−	0.999610i	$-0.491105\pi$
$769$	1191.72	0.0558835	0.0279418	+	0.999610i	$0.491105\pi$
$770$	0	0
$771$	24244.5	1.13248
$772$	62102.1	2.89521
$773$	7481.03	0.348091	0.174045	−	0.984738i	$-0.444316\pi$
$773$	7481.03	0.348091	0.174045	+	0.984738i	$0.444316\pi$
$774$	80017.2	3.71597
$775$	−6817.05	−0.315969
$776$	84120.1	3.89141
$777$	0	0
$778$	36142.5	1.66552
$779$	−5207.06	−0.239489
$780$	−18883.1	−0.866826
$781$	−120.432	−0.00551781
$782$	−16483.0	−0.753748
$783$	54161.4	2.47199
$784$	0	0
$785$	5233.61	0.237956
$786$	−1101.70	−0.0499952
$787$	−27403.6	−1.24121	−0.620604	−	0.784124i	$-0.713113\pi$
$787$	−27403.6	−1.24121	−0.620604	+	0.784124i	$0.713113\pi$
$788$	10700.9	0.483761
$789$	25801.8	1.16422
$790$	−1184.41	−0.0533411
$791$	0	0
$792$	29969.3	1.34459
$793$	−10936.8	−0.489758
$794$	−24829.3	−1.10977
$795$	6732.54	0.300350
$796$	−54299.2	−2.41782
$797$	−30558.5	−1.35814	−0.679070	−	0.734073i	$-0.737617\pi$
$797$	−30558.5	−1.35814	−0.679070	+	0.734073i	$0.737617\pi$
$798$	0	0
$799$	−4031.52	−0.178504
$800$	−18297.0	−0.808622
$801$	−42540.8	−1.87654
$802$	−31181.5	−1.37289
$803$	−4048.42	−0.177915
$804$	−49290.7	−2.16213
$805$	0	0
$806$	−13741.1	−0.600507
$807$	220.854	0.00963374
$808$	47409.7	2.06419
$809$	−28018.7	−1.21766	−0.608829	−	0.793302i	$-0.708360\pi$
$809$	−28018.7	−1.21766	−0.608829	+	0.793302i	$0.708360\pi$
$810$	3858.04	0.167355
$811$	−2520.45	−0.109131	−0.0545654	−	0.998510i	$-0.517377\pi$
$811$	−2520.45	−0.109131	−0.0545654	+	0.998510i	$0.517377\pi$
$812$	0	0
$813$	37612.0	1.62252
$814$	−21831.7	−0.940048
$815$	4297.49	0.184705
$816$	27210.1	1.16733
$817$	6223.19	0.266490
$818$	−15055.4	−0.643520
$819$	0	0
$820$	−12711.4	−0.541343
$821$	21887.7	0.930434	0.465217	−	0.885197i	$-0.345976\pi$
$821$	21887.7	0.930434	0.465217	+	0.885197i	$0.345976\pi$
$822$	−31486.5	−1.33603
$823$	8149.95	0.345188	0.172594	−	0.984993i	$-0.444785\pi$
$823$	8149.95	0.345188	0.172594	+	0.984993i	$0.444785\pi$
$824$	−8538.02	−0.360966
$825$	−12924.5	−0.545423
$826$	0	0
$827$	22007.0	0.925345	0.462672	−	0.886529i	$-0.346891\pi$
$827$	22007.0	0.925345	0.462672	+	0.886529i	$0.346891\pi$
$828$	−95754.3	−4.01895
$829$	−8082.69	−0.338629	−0.169314	−	0.985562i	$-0.554155\pi$
$829$	−8082.69	−0.338629	−0.169314	+	0.985562i	$0.554155\pi$
$830$	5833.04	0.243937
$831$	37073.0	1.54759
$832$	3808.82	0.158710
$833$	0	0
$834$	−29310.4	−1.21695
$835$	−7762.12	−0.321700
$836$	4248.61	0.175767
$837$	10589.4	0.437302
$838$	56302.0	2.32091
$839$	1144.88	0.0471105	0.0235552	−	0.999723i	$-0.492501\pi$
$839$	1144.88	0.0471105	0.0235552	+	0.999723i	$0.492501\pi$
$840$	0	0
$841$	62699.3	2.57080
$842$	16441.3	0.672927
$843$	−32221.3	−1.31644
$844$	−49130.5	−2.00372
$845$	20.8479	0.000848746 0
$846$	−33992.0	−1.38140
$847$	0	0
$848$	32139.3	1.30150
$849$	−32495.2	−1.31358
$850$	−17362.5	−0.700622
$851$	38267.3	1.54146
$852$	−1466.68	−0.0589762
$853$	12853.7	0.515948	0.257974	−	0.966152i	$-0.416945\pi$
$853$	12853.7	0.515948	0.257974	+	0.966152i	$0.416945\pi$
$854$	0	0
$855$	2395.18	0.0958052
$856$	45420.7	1.81361
$857$	−24528.1	−0.977669	−0.488834	−	0.872377i	$-0.662578\pi$
$857$	−24528.1	−0.977669	−0.488834	+	0.872377i	$0.662578\pi$
$858$	−26051.8	−1.03659
$859$	37536.3	1.49095	0.745473	−	0.666536i	$-0.232224\pi$
$859$	37536.3	1.49095	0.745473	+	0.666536i	$0.232224\pi$
$860$	15192.0	0.602374
$861$	0	0
$862$	1472.19	0.0581704
$863$	−2098.30	−0.0827659	−0.0413830	−	0.999143i	$-0.513176\pi$
$863$	−2098.30	−0.0827659	−0.0413830	+	0.999143i	$0.513176\pi$
$864$	28421.9	1.11914
$865$	−242.833	−0.00954519
$866$	−10671.8	−0.418757
$867$	−35317.0	−1.38342
$868$	0	0
$869$	−1125.85	−0.0439493
$870$	33960.8	1.32343
$871$	15063.1	0.585986
$872$	27242.7	1.05797
$873$	82169.3	3.18558
$874$	−10808.7	−0.418318
$875$	0	0
$876$	−49303.5	−1.90161
$877$	−9857.12	−0.379534	−0.189767	−	0.981829i	$-0.560773\pi$
$877$	−9857.12	−0.379534	−0.189767	+	0.981829i	$0.560773\pi$
$878$	−20357.2	−0.782484
$879$	−71181.5	−2.73139
$880$	3576.65	0.137010
$881$	−2301.91	−0.0880289	−0.0440144	−	0.999031i	$-0.514015\pi$
$881$	−2301.91	−0.0880289	−0.0440144	+	0.999031i	$0.514015\pi$
$882$	0	0
$883$	25401.4	0.968093	0.484047	−	0.875042i	$-0.339166\pi$
$883$	25401.4	0.968093	0.484047	+	0.875042i	$0.339166\pi$
$884$	−24113.0	−0.917429
$885$	−10423.5	−0.395912
$886$	60931.2	2.31041
$887$	11451.6	0.433493	0.216746	−	0.976228i	$-0.430456\pi$
$887$	11451.6	0.433493	0.216746	+	0.976228i	$0.430456\pi$
$888$	−145861.	−5.51214
$889$	0	0
$890$	−11722.5	−0.441506
$891$	3667.30	0.137889
$892$	100196.	3.76098
$893$	−2643.67	−0.0990671
$894$	−11787.3	−0.440969
$895$	−8620.91	−0.321972
$896$	0	0
$897$	45664.6	1.69977
$898$	21715.0	0.806949
$899$	17027.1	0.631685
$900$	−100864.	−3.73569
$901$	8597.18	0.317884
$902$	−17537.1	−0.647364
$903$	0	0
$904$	−68803.2	−2.53137
$905$	−9395.83	−0.345114
$906$	123538.	4.53010
$907$	53074.7	1.94302	0.971508	−	0.237005i	$-0.0761659\pi$
$907$	53074.7	1.94302	0.971508	+	0.237005i	$0.0761659\pi$
$908$	90085.2	3.29249
$909$	46310.3	1.68979
$910$	0	0
$911$	−20284.7	−0.737718	−0.368859	−	0.929485i	$-0.620251\pi$
$911$	−20284.7	−0.737718	−0.368859	+	0.929485i	$0.620251\pi$
$912$	17843.0	0.647852
$913$	5544.65	0.200987
$914$	1487.70	0.0538389
$915$	5284.81	0.190940
$916$	−8896.27	−0.320896
$917$	0	0
$918$	26970.3	0.969665
$919$	34782.0	1.24848	0.624240	−	0.781233i	$-0.285409\pi$
$919$	34782.0	1.24848	0.624240	+	0.781233i	$0.285409\pi$
$920$	−14475.5	−0.518744
$921$	24404.2	0.873121
$922$	26717.4	0.954329
$923$	448.213	0.0159839
$924$	0	0
$925$	40309.2	1.43282
$926$	40964.6	1.45376
$927$	−8340.02	−0.295493
$928$	45700.8	1.61660
$929$	15586.2	0.550448	0.275224	−	0.961380i	$-0.411248\pi$
$929$	15586.2	0.550448	0.275224	+	0.961380i	$0.411248\pi$
$930$	6639.85	0.234118
$931$	0	0
$932$	49790.3	1.74993
$933$	75001.8	2.63178
$934$	16029.0	0.561546
$935$	956.746	0.0334641
$936$	−111537.	−3.89498
$937$	15194.9	0.529770	0.264885	−	0.964280i	$-0.414666\pi$
$937$	15194.9	0.529770	0.264885	+	0.964280i	$0.414666\pi$
$938$	0	0
$939$	−14075.6	−0.489179
$940$	−6453.68	−0.223932
$941$	48650.1	1.68538	0.842692	−	0.538396i	$-0.180969\pi$
$941$	48650.1	1.68538	0.842692	+	0.538396i	$0.180969\pi$
$942$	87934.4	3.04146
$943$	30739.7	1.06153
$944$	−49759.0	−1.71559
$945$	0	0
$946$	20959.4	0.720348
$947$	3258.29	0.111806	0.0559030	−	0.998436i	$-0.482196\pi$
$947$	3258.29	0.111806	0.0559030	+	0.998436i	$0.482196\pi$
$948$	−13711.2	−0.469744
$949$	15067.0	0.515380
$950$	−11385.4	−0.388834
$951$	81257.8	2.77073
$952$	0	0
$953$	−44488.8	−1.51221	−0.756104	−	0.654451i	$-0.772900\pi$
$953$	−44488.8	−1.51221	−0.756104	+	0.654451i	$0.772900\pi$
$954$	72487.6	2.46003
$955$	1257.47	0.0426080
$956$	39691.4	1.34279
$957$	32281.8	1.09041
$958$	1264.27	0.0426376
$959$	0	0
$960$	−1840.47	−0.0618758
$961$	−26462.0	−0.888253
$962$	81250.9	2.72311
$963$	44367.4	1.48465
$964$	127857.	4.27179
$965$	9170.51	0.305916
$966$	0	0
$967$	−5791.53	−0.192599	−0.0962994	−	0.995352i	$-0.530701\pi$
$967$	−5791.53	−0.192599	−0.0962994	+	0.995352i	$0.530701\pi$
$968$	−57784.4	−1.91866
$969$	4772.96	0.158235
$970$	22642.6	0.749495
$971$	17829.5	0.589265	0.294632	−	0.955611i	$-0.404803\pi$
$971$	17829.5	0.589265	0.294632	+	0.955611i	$0.404803\pi$
$972$	−43160.8	−1.42426
$973$	0	0
$974$	36850.0	1.21227
$975$	48101.2	1.57997
$976$	25228.2	0.827394
$977$	−2645.66	−0.0866349	−0.0433174	−	0.999061i	$-0.513793\pi$
$977$	−2645.66	−0.0866349	−0.0433174	+	0.999061i	$0.513793\pi$
$978$	72205.7	2.36082
$979$	−11143.0	−0.363770
$980$	0	0
$981$	26610.9	0.866076
$982$	−42889.3	−1.39374
$983$	−3880.50	−0.125909	−0.0629545	−	0.998016i	$-0.520052\pi$
$983$	−3880.50	−0.125909	−0.0629545	+	0.998016i	$0.520052\pi$
$984$	−117168.	−3.79593
$985$	1580.18	0.0511155
$986$	43366.6	1.40068
$987$	0	0
$988$	−15812.1	−0.509158
$989$	−36738.4	−1.18121
$990$	8066.85	0.258971
$991$	57787.1	1.85234	0.926170	−	0.377106i	$-0.123081\pi$
$991$	57787.1	1.85234	0.926170	+	0.377106i	$0.123081\pi$
$992$	8935.19	0.285980
$993$	−15307.3	−0.489186
$994$	0	0
$995$	−8018.27	−0.255474
$996$	67525.4	2.14822
$997$	−30649.0	−0.973585	−0.486793	−	0.873518i	$-0.661833\pi$
$997$	−30649.0	−0.973585	−0.486793	+	0.873518i	$0.661833\pi$
$998$	9662.48	0.306473
$999$	−62614.9	−1.98303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	931.4.a.c.1.3		3
7.6	odd	2		19.4.a.b.1.3	✓	3
21.20	even	2		171.4.a.f.1.1		3
28.27	even	2		304.4.a.i.1.3		3
35.13	even	4		475.4.b.f.324.1		6
35.27	even	4		475.4.b.f.324.6		6
35.34	odd	2		475.4.a.f.1.1		3
56.13	odd	2		1216.4.a.s.1.3		3
56.27	even	2		1216.4.a.u.1.1		3
77.76	even	2		2299.4.a.h.1.1		3
133.132	even	2		361.4.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.3	✓	3	7.6	odd	2
171.4.a.f.1.1		3	21.20	even	2
304.4.a.i.1.3		3	28.27	even	2
361.4.a.i.1.1		3	133.132	even	2
475.4.a.f.1.1		3	35.34	odd	2
475.4.b.f.324.1		6	35.13	even	4
475.4.b.f.324.6		6	35.27	even	4
931.4.a.c.1.3		3	1.1	even	1	trivial
1216.4.a.s.1.3		3	56.13	odd	2
1216.4.a.u.1.1		3	56.27	even	2
2299.4.a.h.1.1		3	77.76	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	931.a (trivial)

\(f(q)\)	\(=\)	\(q+5.07177 q^{2} +8.66998 q^{3} +17.7229 q^{4} +2.61710 q^{5} +43.9722 q^{6} +49.3121 q^{8} +48.1686 q^{9} +13.2733 q^{10} +12.6171 q^{11} +153.657 q^{12} -46.9571 q^{13} +22.6902 q^{15} +108.317 q^{16} +28.9745 q^{17} +244.300 q^{18} +19.0000 q^{19} +46.3825 q^{20} +63.9911 q^{22} -112.166 q^{23} +427.535 q^{24} -118.151 q^{25} -238.155 q^{26} +183.531 q^{27} +295.107 q^{29} +115.080 q^{30} +57.6979 q^{31} +154.862 q^{32} +109.390 q^{33} +146.952 q^{34} +853.685 q^{36} -341.167 q^{37} +96.3636 q^{38} -407.117 q^{39} +129.055 q^{40} -274.056 q^{41} +327.536 q^{43} +223.611 q^{44} +126.062 q^{45} -568.879 q^{46} -139.140 q^{47} +939.106 q^{48} -599.234 q^{50} +251.209 q^{51} -832.214 q^{52} +296.715 q^{53} +930.828 q^{54} +33.0202 q^{55} +164.730 q^{57} +1496.72 q^{58} -459.383 q^{59} +402.136 q^{60} +232.911 q^{61} +292.631 q^{62} -81.1127 q^{64} -122.891 q^{65} +554.801 q^{66} -320.784 q^{67} +513.511 q^{68} -972.475 q^{69} -9.54518 q^{71} +2375.30 q^{72} -320.868 q^{73} -1730.32 q^{74} -1024.37 q^{75} +336.734 q^{76} -2064.80 q^{78} -89.2323 q^{79} +283.476 q^{80} +290.661 q^{81} -1389.95 q^{82} +439.455 q^{83} +75.8293 q^{85} +1661.19 q^{86} +2558.58 q^{87} +622.176 q^{88} -883.164 q^{89} +639.358 q^{90} -1987.90 q^{92} +500.240 q^{93} -705.688 q^{94} +49.7249 q^{95} +1342.65 q^{96} +1705.87 q^{97} +607.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 21 q^{4} - 14 q^{5} + 65 q^{6} + 27 q^{8} + 48 q^{9} + 88 q^{10} + 16 q^{11} + 115 q^{12} - 65 q^{13} + 140 q^{15} + 33 q^{16} - 29 q^{17} + 138 q^{18} + 57 q^{19} - 100 q^{20}+ \cdots + 250 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 21 * q^4 - 14 * q^5 + 65 * q^6 + 27 * q^8 + 48 * q^9 + 88 * q^10 + 16 * q^11 + 115 * q^12 - 65 * q^13 + 140 * q^15 + 33 * q^16 - 29 * q^17 + 138 * q^18 + 57 * q^19 - 100 * q^20 + 118 * q^22 - 101 * q^23 + 489 * q^24 - 37 * q^25 - 299 * q^26 + 377 * q^27 + 377 * q^29 - 376 * q^30 + 140 * q^31 + 579 * q^32 + 130 * q^33 - 329 * q^34 + 1074 * q^36 - 290 * q^37 + 57 * q^38 - 371 * q^39 + 72 * q^40 - 956 * q^41 - 570 * q^43 + 110 * q^44 - 230 * q^45 - 239 * q^46 - 66 * q^47 + 1411 * q^48 - 1639 * q^50 + 187 * q^51 - 697 * q^52 + 817 * q^53 + 947 * q^54 + 198 * q^55 - 19 * q^57 + 647 * q^58 - 265 * q^59 + 1360 * q^60 - 988 * q^61 + 1436 * q^62 - 159 * q^64 - 240 * q^65 + 274 * q^66 - 207 * q^67 + 1529 * q^68 - 807 * q^69 + 846 * q^71 + 2826 * q^72 - 627 * q^73 - 770 * q^74 - 2031 * q^75 + 399 * q^76 - 1817 * q^78 + 382 * q^79 + 1472 * q^80 - 501 * q^81 - 872 * q^82 + 766 * q^83 - 1242 * q^85 + 2592 * q^86 + 1671 * q^87 + 342 * q^88 + 172 * q^89 + 1888 * q^90 - 2677 * q^92 + 892 * q^93 + 96 * q^94 - 266 * q^95 - 863 * q^96 + 2450 * q^97 + 250 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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