LMFDB - Embedded newform 931.4.a.c.1.1

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			1.1
Root			$-3.20905$ 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)

$f(q)$	$=$	\(q-3.96257 q^{2} -6.71610 q^{3} +7.70200 q^{4} -18.1342 q^{5} +26.6130 q^{6} +1.18085 q^{8} +18.1060 q^{9} +71.8581 q^{10} -8.13420 q^{11} -51.7274 q^{12} +4.56640 q^{13} +121.791 q^{15} -66.2952 q^{16} +62.5850 q^{17} -71.7464 q^{18} +19.0000 q^{19} -139.670 q^{20} +32.2324 q^{22} -52.7502 q^{23} -7.93070 q^{24} +203.849 q^{25} -18.0947 q^{26} +59.7330 q^{27} +171.620 q^{29} -482.606 q^{30} -168.749 q^{31} +253.253 q^{32} +54.6301 q^{33} -247.998 q^{34} +139.452 q^{36} -147.534 q^{37} -75.2889 q^{38} -30.6684 q^{39} -21.4138 q^{40} -308.774 q^{41} -448.950 q^{43} -62.6496 q^{44} -328.338 q^{45} +209.027 q^{46} -113.335 q^{47} +445.245 q^{48} -807.768 q^{50} -420.327 q^{51} +35.1704 q^{52} +155.402 q^{53} -236.696 q^{54} +147.507 q^{55} -127.606 q^{57} -680.059 q^{58} -182.347 q^{59} +938.035 q^{60} -404.080 q^{61} +668.681 q^{62} -473.172 q^{64} -82.8080 q^{65} -216.476 q^{66} -106.400 q^{67} +482.030 q^{68} +354.276 q^{69} +472.079 q^{71} +21.3805 q^{72} -843.821 q^{73} +584.616 q^{74} -1369.07 q^{75} +146.338 q^{76} +121.526 q^{78} -591.036 q^{79} +1202.21 q^{80} -890.035 q^{81} +1223.54 q^{82} -290.388 q^{83} -1134.93 q^{85} +1779.00 q^{86} -1152.62 q^{87} -9.60526 q^{88} +964.896 q^{89} +1301.06 q^{90} -406.282 q^{92} +1133.34 q^{93} +449.099 q^{94} -344.550 q^{95} -1700.87 q^{96} +219.495 q^{97} -147.278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.96257	−1.40098	−0.700491	−	0.713661i	$-0.747036\pi$
$2$	−3.96257	−1.40098	−0.700491	+	0.713661i	$0.747036\pi$
$3$	−6.71610	−1.29251	−0.646257	−	0.763120i	$-0.723667\pi$
$3$	−6.71610	−1.29251	−0.646257	+	0.763120i	$0.723667\pi$
$4$	7.70200	0.962750
$5$	−18.1342	−1.62197	−0.810986	−	0.585065i	$-0.801069\pi$
$5$	−18.1342	−1.62197	−0.810986	+	0.585065i	$0.801069\pi$
$6$	26.6130	1.81079
$7$	0	0
$7$	0	0
$8$	1.18085	0.0521866
$9$	18.1060	0.670593
$10$	71.8581	2.27235
$11$	−8.13420	−0.222959	−0.111480	−	0.993767i	$-0.535559\pi$
$11$	−8.13420	−0.222959	−0.111480	+	0.993767i	$0.535559\pi$
$12$	−51.7274	−1.24437
$13$	4.56640	0.0974224	0.0487112	−	0.998813i	$-0.484489\pi$
$13$	4.56640	0.0974224	0.0487112	+	0.998813i	$0.484489\pi$
$14$	0	0
$15$	121.791	2.09642
$16$	−66.2952	−1.03586
$17$	62.5850	0.892888	0.446444	−	0.894812i	$-0.352690\pi$
$17$	62.5850	0.892888	0.446444	+	0.894812i	$0.352690\pi$
$18$	−71.7464	−0.939488
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−139.670	−1.56155
$21$	0	0
$22$	32.2324	0.312362
$23$	−52.7502	−0.478225	−0.239113	−	0.970992i	$-0.576856\pi$
$23$	−52.7502	−0.478225	−0.239113	+	0.970992i	$0.576856\pi$
$24$	−7.93070	−0.0674520
$25$	203.849	1.63079
$26$	−18.0947	−0.136487
$27$	59.7330	0.425764
$28$	0	0
$29$	171.620	1.09894	0.549468	−	0.835515i	$-0.314831\pi$
$29$	171.620	1.09894	0.549468	+	0.835515i	$0.314831\pi$
$30$	−482.606	−2.93705
$31$	−168.749	−0.977685	−0.488842	−	0.872372i	$-0.662581\pi$
$31$	−168.749	−0.977685	−0.488842	+	0.872372i	$0.662581\pi$
$32$	253.253	1.39904
$33$	54.6301	0.288178
$34$	−247.998	−1.25092
$35$	0	0
$36$	139.452	0.645613
$37$	−147.534	−0.655528	−0.327764	−	0.944760i	$-0.606295\pi$
$37$	−147.534	−0.655528	−0.327764	+	0.944760i	$0.606295\pi$
$38$	−75.2889	−0.321407
$39$	−30.6684	−0.125920
$40$	−21.4138	−0.0846453
$41$	−308.774	−1.17616	−0.588078	−	0.808804i	$-0.700115\pi$
$41$	−308.774	−1.17616	−0.588078	+	0.808804i	$0.700115\pi$
$42$	0	0
$43$	−448.950	−1.59219	−0.796096	−	0.605170i	$-0.793105\pi$
$43$	−448.950	−1.59219	−0.796096	+	0.605170i	$0.793105\pi$
$44$	−62.6496	−0.214654
$45$	−328.338	−1.08768
$46$	209.027	0.669985
$47$	−113.335	−0.351737	−0.175868	−	0.984414i	$-0.556273\pi$
$47$	−113.335	−0.351737	−0.175868	+	0.984414i	$0.556273\pi$
$48$	445.245	1.33887
$49$	0	0
$50$	−807.768	−2.28471
$51$	−420.327	−1.15407
$52$	35.1704	0.0937934
$53$	155.402	0.402758	0.201379	−	0.979513i	$-0.435458\pi$
$53$	155.402	0.402758	0.201379	+	0.979513i	$0.435458\pi$
$54$	−236.696	−0.596487
$55$	147.507	0.361634
$56$	0	0
$57$	−127.606	−0.296523
$58$	−680.059	−1.53959
$59$	−182.347	−0.402365	−0.201183	−	0.979554i	$-0.564478\pi$
$59$	−182.347	−0.402365	−0.201183	+	0.979554i	$0.564478\pi$
$60$	938.035	2.01833
$61$	−404.080	−0.848149	−0.424075	−	0.905627i	$-0.639401\pi$
$61$	−404.080	−0.848149	−0.424075	+	0.905627i	$0.639401\pi$
$62$	668.681	1.36972
$63$	0	0
$64$	−473.172	−0.924164
$65$	−82.8080	−0.158016
$66$	−216.476	−0.403732
$67$	−106.400	−0.194013	−0.0970064	−	0.995284i	$-0.530927\pi$
$67$	−106.400	−0.194013	−0.0970064	+	0.995284i	$0.530927\pi$
$68$	482.030	0.859628
$69$	354.276	0.618113
$70$	0	0
$71$	472.079	0.789091	0.394546	−	0.918876i	$-0.370902\pi$
$71$	472.079	0.789091	0.394546	+	0.918876i	$0.370902\pi$
$72$	21.3805	0.0349960
$73$	−843.821	−1.35290	−0.676451	−	0.736488i	$-0.736483\pi$
$73$	−843.821	−1.35290	−0.676451	+	0.736488i	$0.736483\pi$
$74$	584.616	0.918382
$75$	−1369.07	−2.10782
$76$	146.338	0.220870
$77$	0	0
$78$	121.526	0.176411
$79$	−591.036	−0.841731	−0.420866	−	0.907123i	$-0.638274\pi$
$79$	−591.036	−0.841731	−0.420866	+	0.907123i	$0.638274\pi$
$80$	1202.21	1.68014
$81$	−890.035	−1.22090
$82$	1223.54	1.64777
$83$	−290.388	−0.384027	−0.192013	−	0.981392i	$-0.561502\pi$
$83$	−290.388	−0.384027	−0.192013	+	0.981392i	$0.561502\pi$
$84$	0	0
$85$	−1134.93	−1.44824
$86$	1779.00	2.23063
$87$	−1152.62	−1.42039
$88$	−9.60526	−0.0116355
$89$	964.896	1.14920	0.574600	−	0.818435i	$-0.305158\pi$
$89$	964.896	1.14920	0.574600	+	0.818435i	$0.305158\pi$
$90$	1301.06	1.52382
$91$	0	0
$92$	−406.282	−0.460411
$93$	1133.34	1.26367
$94$	449.099	0.492777
$95$	−344.550	−0.372106
$96$	−1700.87	−1.80828
$97$	219.495	0.229756	0.114878	−	0.993380i	$-0.463352\pi$
$97$	219.495	0.229756	0.114878	+	0.993380i	$0.463352\pi$
$98$	0	0
$99$	−147.278	−0.149515
$100$	1570.05	1.57005
$101$	−1447.94	−1.42649	−0.713247	−	0.700913i	$-0.752776\pi$
$101$	−1447.94	−1.42649	−0.713247	+	0.700913i	$0.752776\pi$
$102$	1665.58	1.61683
$103$	−883.567	−0.845247	−0.422623	−	0.906305i	$-0.638891\pi$
$103$	−883.567	−0.845247	−0.422623	+	0.906305i	$0.638891\pi$
$104$	5.39223	0.00508415
$105$	0	0
$106$	−615.793	−0.564256
$107$	−1307.82	−1.18160	−0.590801	−	0.806817i	$-0.701188\pi$
$107$	−1307.82	−1.18160	−0.590801	+	0.806817i	$0.701188\pi$
$108$	460.064	0.409904
$109$	870.507	0.764949	0.382475	−	0.923966i	$-0.375072\pi$
$109$	870.507	0.764949	0.382475	+	0.923966i	$0.375072\pi$
$110$	−584.508	−0.506643
$111$	990.856	0.847279
$112$	0	0
$113$	−1181.41	−0.983521	−0.491761	−	0.870730i	$-0.663646\pi$
$113$	−1181.41	−0.983521	−0.491761	+	0.870730i	$0.663646\pi$
$114$	505.648	0.415423
$115$	956.583	0.775668
$116$	1321.82	1.05800
$117$	82.6792	0.0653307
$118$	722.564	0.563707
$119$	0	0
$120$	143.817	0.109405
$121$	−1264.83	−0.950289
$122$	1601.20	1.18824
$123$	2073.76	1.52020
$124$	−1299.71	−0.941266
$125$	−1429.87	−1.02313
$126$	0	0
$127$	887.509	0.620108	0.310054	−	0.950719i	$-0.399653\pi$
$127$	887.509	0.620108	0.310054	+	0.950719i	$0.399653\pi$
$128$	−151.044	−0.104301
$129$	3015.19	2.05793
$130$	328.133	0.221378
$131$	2344.76	1.56384	0.781920	−	0.623379i	$-0.214241\pi$
$131$	2344.76	1.56384	0.781920	+	0.623379i	$0.214241\pi$
$132$	420.761	0.277444
$133$	0	0
$134$	421.619	0.271809
$135$	−1083.21	−0.690577
$136$	73.9034	0.0465968
$137$	2244.82	1.39991	0.699956	−	0.714186i	$-0.253203\pi$
$137$	2244.82	1.39991	0.699956	+	0.714186i	$0.253203\pi$
$138$	−1403.84	−0.865964
$139$	296.146	0.180711	0.0903554	−	0.995910i	$-0.471200\pi$
$139$	296.146	0.180711	0.0903554	+	0.995910i	$0.471200\pi$
$140$	0	0
$141$	761.170	0.454625
$142$	−1870.65	−1.10550
$143$	−37.1440	−0.0217212
$144$	−1200.34	−0.694642
$145$	−3112.20	−1.78244
$146$	3343.70	1.89539
$147$	0	0
$148$	−1136.31	−0.631109
$149$	1791.09	0.984780	0.492390	−	0.870375i	$-0.336123\pi$
$149$	1791.09	0.984780	0.492390	+	0.870375i	$0.336123\pi$
$150$	5425.05	2.95302
$151$	−2352.65	−1.26792	−0.633960	−	0.773366i	$-0.718571\pi$
$151$	−2352.65	−1.26792	−0.633960	+	0.773366i	$0.718571\pi$
$152$	22.4361	0.0119724
$153$	1133.16	0.598764
$154$	0	0
$155$	3060.13	1.58578
$156$	−236.208	−0.121229
$157$	1438.26	0.731118	0.365559	−	0.930788i	$-0.380878\pi$
$157$	1438.26	0.731118	0.365559	+	0.930788i	$0.380878\pi$
$158$	2342.02	1.17925
$159$	−1043.70	−0.520570
$160$	−4592.54	−2.26920
$161$	0	0
$162$	3526.83	1.71046
$163$	127.493	0.0612640	0.0306320	−	0.999531i	$-0.490248\pi$
$163$	127.493	0.0612640	0.0306320	+	0.999531i	$0.490248\pi$
$164$	−2378.18	−1.13234
$165$	−990.673	−0.467417
$166$	1150.68	0.538014
$167$	−3419.05	−1.58428	−0.792139	−	0.610341i	$-0.791033\pi$
$167$	−3419.05	−1.58428	−0.792139	+	0.610341i	$0.791033\pi$
$168$	0	0
$169$	−2176.15	−0.990509
$170$	4497.24	2.02896
$171$	344.014	0.153844
$172$	−3457.81	−1.53288
$173$	362.598	0.159352	0.0796758	−	0.996821i	$-0.474611\pi$
$173$	362.598	0.159352	0.0796758	+	0.996821i	$0.474611\pi$
$174$	4567.34	1.98994
$175$	0	0
$176$	539.258	0.230955
$177$	1224.66	0.520063
$178$	−3823.47	−1.61001
$179$	2417.89	1.00962	0.504809	−	0.863231i	$-0.331563\pi$
$179$	2417.89	1.00962	0.504809	+	0.863231i	$0.331563\pi$
$180$	−2528.86	−1.04717
$181$	2444.64	1.00391	0.501957	−	0.864892i	$-0.332613\pi$
$181$	2444.64	1.00391	0.501957	+	0.864892i	$0.332613\pi$
$182$	0	0
$183$	2713.84	1.09624
$184$	−62.2900	−0.0249570
$185$	2675.42	1.06325
$186$	−4490.93	−1.77038
$187$	−509.079	−0.199078
$188$	−872.908	−0.338635
$189$	0	0
$190$	1365.30	0.521314
$191$	−1387.66	−0.525693	−0.262846	−	0.964838i	$-0.584661\pi$
$191$	−1387.66	−0.525693	−0.262846	+	0.964838i	$0.584661\pi$
$192$	3177.87	1.19449
$193$	−3208.03	−1.19647	−0.598237	−	0.801319i	$-0.704132\pi$
$193$	−3208.03	−1.19647	−0.598237	+	0.801319i	$0.704132\pi$
$194$	−869.764	−0.321884
$195$	556.147	0.204238
$196$	0	0
$197$	−3445.36	−1.24605	−0.623025	−	0.782202i	$-0.714097\pi$
$197$	−3445.36	−1.24605	−0.623025	+	0.782202i	$0.714097\pi$
$198$	583.599	0.209468
$199$	−2025.71	−0.721602	−0.360801	−	0.932643i	$-0.617497\pi$
$199$	−2025.71	−0.721602	−0.360801	+	0.932643i	$0.617497\pi$
$200$	240.715	0.0851056
$201$	714.595	0.250764
$202$	5737.59	1.99849
$203$	0	0
$204$	−3237.36	−1.11108
$205$	5599.37	1.90769
$206$	3501.20	1.18418
$207$	−955.095	−0.320694
$208$	−302.730	−0.100916
$209$	−154.550	−0.0511504
$210$	0	0
$211$	4309.54	1.40607	0.703036	−	0.711155i	$-0.251827\pi$
$211$	4309.54	1.40607	0.703036	+	0.711155i	$0.251827\pi$
$212$	1196.91	0.387755
$213$	−3170.53	−1.01991
$214$	5182.33	1.65540
$215$	8141.35	2.58249
$216$	70.5357	0.0222192
$217$	0	0
$218$	−3449.45	−1.07168
$219$	5667.19	1.74864
$220$	1136.10	0.348163
$221$	285.788	0.0869873
$222$	−3926.34	−1.18702
$223$	−825.648	−0.247935	−0.123968	−	0.992286i	$-0.539562\pi$
$223$	−825.648	−0.247935	−0.123968	+	0.992286i	$0.539562\pi$
$224$	0	0
$225$	3690.89	1.09360
$226$	4681.43	1.37790
$227$	1501.19	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$227$	1501.19	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$228$	−982.821	−0.285478
$229$	5250.40	1.51509	0.757547	−	0.652781i	$-0.226398\pi$
$229$	5250.40	1.51509	0.757547	+	0.652781i	$0.226398\pi$
$230$	−3790.53	−1.08670
$231$	0	0
$232$	202.658	0.0573497
$233$	2139.06	0.601435	0.300717	−	0.953713i	$-0.402774\pi$
$233$	2139.06	0.601435	0.300717	+	0.953713i	$0.402774\pi$
$234$	−327.623	−0.0915272
$235$	2055.24	0.570508
$236$	−1404.44	−0.387377
$237$	3969.46	1.08795
$238$	0	0
$239$	3772.70	1.02107	0.510534	−	0.859857i	$-0.329448\pi$
$239$	3772.70	1.02107	0.510534	+	0.859857i	$0.329448\pi$
$240$	−8074.17	−2.17160
$241$	−6415.39	−1.71474	−0.857369	−	0.514702i	$-0.827903\pi$
$241$	−6415.39	−1.71474	−0.857369	+	0.514702i	$0.827903\pi$
$242$	5012.00	1.33134
$243$	4364.77	1.15226
$244$	−3112.22	−0.816555
$245$	0	0
$246$	−8217.41	−2.12977
$247$	86.7616	0.0223502
$248$	−199.267	−0.0510221
$249$	1950.27	0.496360
$250$	5665.95	1.43339
$251$	6277.31	1.57857	0.789283	−	0.614029i	$-0.210452\pi$
$251$	6277.31	1.57857	0.789283	+	0.614029i	$0.210452\pi$
$252$	0	0
$253$	429.081	0.106625
$254$	−3516.82	−0.868760
$255$	7622.30	1.87187
$256$	4383.90	1.07029
$257$	3183.98	0.772807	0.386404	−	0.922330i	$-0.373717\pi$
$257$	3183.98	0.772807	0.386404	+	0.922330i	$0.373717\pi$
$258$	−11947.9	−2.88312
$259$	0	0
$260$	−637.787	−0.152130
$261$	3107.36	0.736938
$262$	−9291.31	−2.19091
$263$	−2624.18	−0.615261	−0.307630	−	0.951506i	$-0.599536\pi$
$263$	−2624.18	−0.615261	−0.307630	+	0.951506i	$0.599536\pi$
$264$	64.5099	0.0150391
$265$	−2818.10	−0.653261
$266$	0	0
$267$	−6480.34	−1.48536
$268$	−819.495	−0.186786
$269$	−7444.76	−1.68742	−0.843708	−	0.536803i	$-0.819632\pi$
$269$	−7444.76	−1.68742	−0.843708	+	0.536803i	$0.819632\pi$
$270$	4292.30	0.967486
$271$	4004.49	0.897621	0.448810	−	0.893627i	$-0.351848\pi$
$271$	4004.49	0.897621	0.448810	+	0.893627i	$0.351848\pi$
$272$	−4149.08	−0.924909
$273$	0	0
$274$	−8895.27	−1.96125
$275$	−1658.15	−0.363601
$276$	2728.63	0.595088
$277$	−5830.66	−1.26473	−0.632365	−	0.774671i	$-0.717916\pi$
$277$	−5830.66	−1.26473	−0.632365	+	0.774671i	$0.717916\pi$
$278$	−1173.50	−0.253172
$279$	−3055.37	−0.655628
$280$	0	0
$281$	−7504.37	−1.59314	−0.796572	−	0.604544i	$-0.793355\pi$
$281$	−7504.37	−1.59314	−0.796572	+	0.604544i	$0.793355\pi$
$282$	−3016.19	−0.636921
$283$	−5910.87	−1.24157	−0.620785	−	0.783980i	$-0.713186\pi$
$283$	−5910.87	−1.24157	−0.620785	+	0.783980i	$0.713186\pi$
$284$	3635.95	0.759697
$285$	2314.03	0.480952
$286$	147.186	0.0304311
$287$	0	0
$288$	4585.40	0.938184
$289$	−996.118	−0.202752
$290$	12332.3	2.49717
$291$	−1474.15	−0.296962
$292$	−6499.11	−1.30251
$293$	−3245.59	−0.647131	−0.323566	−	0.946206i	$-0.604882\pi$
$293$	−3245.59	−0.647131	−0.323566	+	0.946206i	$0.604882\pi$
$294$	0	0
$295$	3306.72	0.652625
$296$	−174.216	−0.0342098
$297$	−485.880	−0.0949280
$298$	−7097.35	−1.37966
$299$	−240.878	−0.0465898
$300$	−10544.6	−2.02931
$301$	0	0
$302$	9322.54	1.77633
$303$	9724.54	1.84376
$304$	−1259.61	−0.237643
$305$	7327.66	1.37567
$306$	−4490.25	−0.838857
$307$	−7489.14	−1.39227	−0.696137	−	0.717909i	$-0.745099\pi$
$307$	−7489.14	−1.39227	−0.696137	+	0.717909i	$0.745099\pi$
$308$	0	0
$309$	5934.12	1.09249
$310$	−12126.0	−2.22165
$311$	−2136.71	−0.389588	−0.194794	−	0.980844i	$-0.562404\pi$
$311$	−2136.71	−0.389588	−0.194794	+	0.980844i	$0.562404\pi$
$312$	−36.2147	−0.00657133
$313$	−2212.15	−0.399483	−0.199742	−	0.979849i	$-0.564010\pi$
$313$	−2212.15	−0.399483	−0.199742	+	0.979849i	$0.564010\pi$
$314$	−5699.21	−1.02428
$315$	0	0
$316$	−4552.16	−0.810377
$317$	429.326	0.0760674	0.0380337	−	0.999276i	$-0.487891\pi$
$317$	429.326	0.0760674	0.0380337	+	0.999276i	$0.487891\pi$
$318$	4135.73	0.729309
$319$	−1396.00	−0.245018
$320$	8580.60	1.49897
$321$	8783.43	1.52724
$322$	0	0
$323$	1189.11	0.204842
$324$	−6855.05	−1.17542
$325$	930.857	0.158876
$326$	−505.201	−0.0858297
$327$	−5846.42	−0.988708
$328$	−364.615	−0.0613796
$329$	0	0
$330$	3925.62	0.654843
$331$	−765.454	−0.127109	−0.0635546	−	0.997978i	$-0.520244\pi$
$331$	−765.454	−0.127109	−0.0635546	+	0.997978i	$0.520244\pi$
$332$	−2236.57	−0.369722
$333$	−2671.26	−0.439592
$334$	13548.3	2.21954
$335$	1929.48	0.314684
$336$	0	0
$337$	−3049.81	−0.492978	−0.246489	−	0.969146i	$-0.579277\pi$
$337$	−3049.81	−0.492978	−0.246489	+	0.969146i	$0.579277\pi$
$338$	8623.15	1.38768
$339$	7934.48	1.27121
$340$	−8741.22	−1.39429
$341$	1372.64	0.217984
$342$	−1363.18	−0.215533
$343$	0	0
$344$	−530.142	−0.0830912
$345$	−6424.50	−1.00256
$346$	−1436.82	−0.223249
$347$	−5907.00	−0.913845	−0.456922	−	0.889507i	$-0.651048\pi$
$347$	−5907.00	−0.913845	−0.456922	+	0.889507i	$0.651048\pi$
$348$	−8877.48	−1.36748
$349$	12107.4	1.85700	0.928502	−	0.371327i	$-0.121097\pi$
$349$	12107.4	1.85700	0.928502	+	0.371327i	$0.121097\pi$
$350$	0	0
$351$	272.765	0.0414789
$352$	−2060.01	−0.311929
$353$	2420.40	0.364943	0.182471	−	0.983211i	$-0.441590\pi$
$353$	2420.40	0.364943	0.182471	+	0.983211i	$0.441590\pi$
$354$	−4852.81	−0.728599
$355$	−8560.77	−1.27988
$356$	7431.63	1.10639
$357$	0	0
$358$	−9581.07	−1.41446
$359$	−1455.80	−0.214023	−0.107011	−	0.994258i	$-0.534128\pi$
$359$	−1455.80	−0.214023	−0.107011	+	0.994258i	$0.534128\pi$
$360$	−387.717	−0.0567625
$361$	361.000	0.0526316
$362$	−9687.06	−1.40647
$363$	8494.76	1.22826
$364$	0	0
$365$	15302.0	2.19437
$366$	−10753.8	−1.53582
$367$	−8783.80	−1.24935	−0.624674	−	0.780886i	$-0.714768\pi$
$367$	−8783.80	−1.24935	−0.624674	+	0.780886i	$0.714768\pi$
$368$	3497.08	0.495375
$369$	−5590.66	−0.788721
$370$	−10601.6	−1.48959
$371$	0	0
$372$	8728.95	1.21660
$373$	−9199.84	−1.27708	−0.638538	−	0.769590i	$-0.720461\pi$
$373$	−9199.84	−1.27708	−0.638538	+	0.769590i	$0.720461\pi$
$374$	2017.26	0.278904
$375$	9603.13	1.32241
$376$	−133.832	−0.0183560
$377$	783.688	0.107061
$378$	0	0
$379$	−6161.38	−0.835063	−0.417531	−	0.908662i	$-0.637105\pi$
$379$	−6161.38	−0.835063	−0.417531	+	0.908662i	$0.637105\pi$
$380$	−2653.72	−0.358245
$381$	−5960.60	−0.801498
$382$	5498.70	0.736486
$383$	−2630.79	−0.350985	−0.175492	−	0.984481i	$-0.556152\pi$
$383$	−2630.79	−0.350985	−0.175492	+	0.984481i	$0.556152\pi$
$384$	1014.43	0.134810
$385$	0	0
$386$	12712.1	1.67624
$387$	−8128.69	−1.06771
$388$	1690.55	0.221197
$389$	−5866.48	−0.764633	−0.382317	−	0.924031i	$-0.624874\pi$
$389$	−5866.48	−0.764633	−0.382317	+	0.924031i	$0.624874\pi$
$390$	−2203.77	−0.286134
$391$	−3301.37	−0.427001
$392$	0	0
$393$	−15747.7	−2.02129
$394$	13652.5	1.74569
$395$	10718.0	1.36526
$396$	−1134.33	−0.143945
$397$	14254.0	1.80199	0.900993	−	0.433833i	$-0.142839\pi$
$397$	14254.0	1.80199	0.900993	+	0.433833i	$0.142839\pi$
$398$	8027.03	1.01095
$399$	0	0
$400$	−13514.2	−1.68928
$401$	9909.27	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$401$	9909.27	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$402$	−2831.64	−0.351316
$403$	−770.576	−0.0952484
$404$	−11152.1	−1.37336
$405$	16140.1	1.98026
$406$	0	0
$407$	1200.07	0.146156
$408$	−496.343	−0.0602270
$409$	−5805.87	−0.701912	−0.350956	−	0.936392i	$-0.614143\pi$
$409$	−5805.87	−0.701912	−0.350956	+	0.936392i	$0.614143\pi$
$410$	−22187.9	−2.67264
$411$	−15076.4	−1.80941
$412$	−6805.23	−0.813761
$413$	0	0
$414$	3784.64	0.449287
$415$	5265.95	0.622881
$416$	1156.45	0.136298
$417$	−1988.95	−0.233571
$418$	612.415	0.0716608
$419$	−12260.9	−1.42955	−0.714777	−	0.699353i	$-0.753472\pi$
$419$	−12260.9	−1.42955	−0.714777	+	0.699353i	$0.753472\pi$
$420$	0	0
$421$	5837.85	0.675818	0.337909	−	0.941179i	$-0.390280\pi$
$421$	5837.85	0.675818	0.337909	+	0.941179i	$0.390280\pi$
$422$	−17076.9	−1.96988
$423$	−2052.05	−0.235872
$424$	183.507	0.0210186
$425$	12757.9	1.45612
$426$	12563.5	1.42888
$427$	0	0
$428$	−10072.8	−1.13759
$429$	249.463	0.0280750
$430$	−32260.7	−3.61802
$431$	−2770.16	−0.309591	−0.154796	−	0.987946i	$-0.549472\pi$
$431$	−2770.16	−0.309591	−0.154796	+	0.987946i	$0.549472\pi$
$432$	−3960.01	−0.441033
$433$	−5663.00	−0.628513	−0.314257	−	0.949338i	$-0.601755\pi$
$433$	−5663.00	−0.628513	−0.314257	+	0.949338i	$0.601755\pi$
$434$	0	0
$435$	20901.8	2.30383
$436$	6704.65	0.736455
$437$	−1002.25	−0.109712
$438$	−22456.6	−2.44982
$439$	8399.20	0.913148	0.456574	−	0.889685i	$-0.349076\pi$
$439$	8399.20	0.913148	0.456574	+	0.889685i	$0.349076\pi$
$440$	174.184	0.0188725
$441$	0	0
$442$	−1132.46	−0.121868
$443$	6154.68	0.660085	0.330043	−	0.943966i	$-0.392937\pi$
$443$	6154.68	0.660085	0.330043	+	0.943966i	$0.392937\pi$
$444$	7631.58	0.815717
$445$	−17497.6	−1.86397
$446$	3271.69	0.347352
$447$	−12029.2	−1.27284
$448$	0	0
$449$	3445.03	0.362095	0.181048	−	0.983474i	$-0.442051\pi$
$449$	3445.03	0.362095	0.181048	+	0.983474i	$0.442051\pi$
$450$	−14625.4	−1.53211
$451$	2511.63	0.262235
$452$	−9099.24	−0.946885
$453$	15800.6	1.63880
$454$	−5948.57	−0.614935
$455$	0	0
$456$	−150.683	−0.0154745
$457$	−502.346	−0.0514196	−0.0257098	−	0.999669i	$-0.508185\pi$
$457$	−502.346	−0.0514196	−0.0257098	+	0.999669i	$0.508185\pi$
$458$	−20805.1	−2.12262
$459$	3738.39	0.380159
$460$	7367.60	0.746774
$461$	−546.259	−0.0551883	−0.0275942	−	0.999619i	$-0.508785\pi$
$461$	−546.259	−0.0551883	−0.0275942	+	0.999619i	$0.508785\pi$
$462$	0	0
$463$	18540.2	1.86098	0.930490	−	0.366316i	$-0.119381\pi$
$463$	18540.2	1.86098	0.930490	+	0.366316i	$0.119381\pi$
$464$	−11377.6	−1.13835
$465$	−20552.1	−2.04964
$466$	−8476.18	−0.842599
$467$	−12475.1	−1.23614	−0.618070	−	0.786123i	$-0.712085\pi$
$467$	−12475.1	−1.23614	−0.618070	+	0.786123i	$0.712085\pi$
$468$	636.795	0.0628972
$469$	0	0
$470$	−8144.05	−0.799271
$471$	−9659.49	−0.944981
$472$	−215.324	−0.0209981
$473$	3651.85	0.354994
$474$	−15729.3	−1.52420
$475$	3873.13	0.374130
$476$	0	0
$477$	2813.71	0.270086
$478$	−14949.6	−1.43050
$479$	10569.2	1.00818	0.504091	−	0.863651i	$-0.331828\pi$
$479$	10569.2	1.00818	0.504091	+	0.863651i	$0.331828\pi$
$480$	30843.9	2.93297
$481$	−673.701	−0.0638631
$482$	25421.5	2.40232
$483$	0	0
$484$	−9741.76	−0.914891
$485$	−3980.36	−0.372657
$486$	−17295.7	−1.61430
$487$	−11227.9	−1.04473	−0.522366	−	0.852721i	$-0.674951\pi$
$487$	−11227.9	−1.04473	−0.522366	+	0.852721i	$0.674951\pi$
$488$	−477.157	−0.0442621
$489$	−856.256	−0.0791846
$490$	0	0
$491$	−536.840	−0.0493427	−0.0246713	−	0.999696i	$-0.507854\pi$
$491$	−536.840	−0.0493427	−0.0246713	+	0.999696i	$0.507854\pi$
$492$	15972.1	1.46357
$493$	10740.9	0.981226
$494$	−343.799	−0.0313123
$495$	2670.77	0.242509
$496$	11187.3	1.01275
$497$	0	0
$498$	−7728.11	−0.695391
$499$	1319.91	0.118412	0.0592058	−	0.998246i	$-0.481143\pi$
$499$	1319.91	0.118412	0.0592058	+	0.998246i	$0.481143\pi$
$500$	−11012.8	−0.985018
$501$	22962.7	2.04770
$502$	−24874.3	−2.21154
$503$	−1749.27	−0.155062	−0.0775310	−	0.996990i	$-0.524704\pi$
$503$	−1749.27	−0.155062	−0.0775310	+	0.996990i	$0.524704\pi$
$504$	0	0
$505$	26257.3	2.31373
$506$	−1700.26	−0.149379
$507$	14615.2	1.28025
$508$	6835.59	0.597009
$509$	−1882.19	−0.163903	−0.0819516	−	0.996636i	$-0.526115\pi$
$509$	−1882.19	−0.163903	−0.0819516	+	0.996636i	$0.526115\pi$
$510$	−30203.9	−2.62245
$511$	0	0
$512$	−16163.2	−1.39515
$513$	1134.93	0.0976769
$514$	−12616.8	−1.08269
$515$	16022.8	1.37097
$516$	23223.0	1.98127
$517$	921.891	0.0784231
$518$	0	0
$519$	−2435.25	−0.205964
$520$	−97.7837	−0.00824635
$521$	3238.50	0.272325	0.136163	−	0.990686i	$-0.456523\pi$
$521$	3238.50	0.272325	0.136163	+	0.990686i	$0.456523\pi$
$522$	−12313.1	−1.03244
$523$	−99.0144	−0.00827839	−0.00413919	−	0.999991i	$-0.501318\pi$
$523$	−99.0144	−0.00827839	−0.00413919	+	0.999991i	$0.501318\pi$
$524$	18059.4	1.50559
$525$	0	0
$526$	10398.5	0.861969
$527$	−10561.2	−0.872963
$528$	−3621.71	−0.298513
$529$	−9384.42	−0.771301
$530$	11166.9	0.915207
$531$	−3301.57	−0.269823
$532$	0	0
$533$	−1409.98	−0.114584
$534$	25678.8	2.08096
$535$	23716.2	1.91653
$536$	−125.643	−0.0101249
$537$	−16238.8	−1.30495
$538$	29500.4	2.36404
$539$	0	0
$540$	−8342.88	−0.664853
$541$	17183.7	1.36559	0.682794	−	0.730611i	$-0.260765\pi$
$541$	17183.7	1.36559	0.682794	+	0.730611i	$0.260765\pi$
$542$	−15868.1	−1.25755
$543$	−16418.4	−1.29757
$544$	15849.8	1.24918
$545$	−15786.0	−1.24073
$546$	0	0
$547$	−1965.86	−0.153664	−0.0768319	−	0.997044i	$-0.524480\pi$
$547$	−1965.86	−0.153664	−0.0768319	+	0.997044i	$0.524480\pi$
$548$	17289.6	1.34777
$549$	−7316.27	−0.568762
$550$	6570.54	0.509398
$551$	3260.79	0.252113
$552$	418.346	0.0322572
$553$	0	0
$554$	23104.4	1.77186
$555$	−17968.4	−1.37426
$556$	2280.92	0.173979
$557$	6039.93	0.459461	0.229731	−	0.973254i	$-0.426215\pi$
$557$	6039.93	0.459461	0.229731	+	0.973254i	$0.426215\pi$
$558$	12107.1	0.918523
$559$	−2050.09	−0.155115
$560$	0	0
$561$	3419.02	0.257311
$562$	29736.6	2.23197
$563$	−5260.06	−0.393757	−0.196878	−	0.980428i	$-0.563080\pi$
$563$	−5260.06	−0.393757	−0.196878	+	0.980428i	$0.563080\pi$
$564$	5862.53	0.437690
$565$	21424.0	1.59524
$566$	23422.3	1.73942
$567$	0	0
$568$	557.454	0.0411800
$569$	20567.4	1.51534	0.757672	−	0.652635i	$-0.226337\pi$
$569$	20567.4	1.51534	0.757672	+	0.652635i	$0.226337\pi$
$570$	−9169.52	−0.673805
$571$	11462.4	0.840080	0.420040	−	0.907506i	$-0.362016\pi$
$571$	11462.4	0.840080	0.420040	+	0.907506i	$0.362016\pi$
$572$	−286.083	−0.0209121
$573$	9319.64	0.679466
$574$	0	0
$575$	−10753.1	−0.779886
$576$	−8567.25	−0.619737
$577$	27029.6	1.95019	0.975094	−	0.221790i	$-0.0711899\pi$
$577$	27029.6	1.95019	0.975094	+	0.221790i	$0.0711899\pi$
$578$	3947.19	0.284051
$579$	21545.5	1.54646
$580$	−23970.2	−1.71605
$581$	0	0
$582$	5841.42	0.416039
$583$	−1264.07	−0.0897986
$584$	−996.425	−0.0706034
$585$	−1499.32	−0.105965
$586$	12860.9	0.906619
$587$	−15200.4	−1.06881	−0.534403	−	0.845230i	$-0.679464\pi$
$587$	−15200.4	−1.06881	−0.534403	+	0.845230i	$0.679464\pi$
$588$	0	0
$589$	−3206.23	−0.224296
$590$	−13103.1	−0.914316
$591$	23139.4	1.61054
$592$	9780.83	0.679036
$593$	19026.7	1.31759	0.658796	−	0.752322i	$-0.271066\pi$
$593$	19026.7	1.31759	0.658796	+	0.752322i	$0.271066\pi$
$594$	1925.34	0.132992
$595$	0	0
$596$	13795.0	0.948096
$597$	13604.9	0.932681
$598$	954.499	0.0652715
$599$	3927.31	0.267889	0.133945	−	0.990989i	$-0.457236\pi$
$599$	3927.31	0.267889	0.133945	+	0.990989i	$0.457236\pi$
$600$	−1616.67	−0.110000
$601$	−13718.1	−0.931069	−0.465534	−	0.885030i	$-0.654138\pi$
$601$	−13718.1	−0.931069	−0.465534	+	0.885030i	$0.654138\pi$
$602$	0	0
$603$	−1926.48	−0.130104
$604$	−18120.1	−1.22069
$605$	22936.8	1.54134
$606$	−38534.2	−2.58308
$607$	−26461.5	−1.76942	−0.884712	−	0.466138i	$-0.845645\pi$
$607$	−26461.5	−1.76942	−0.884712	+	0.466138i	$0.845645\pi$
$608$	4811.81	0.320961
$609$	0	0
$610$	−29036.4	−1.92729
$611$	−517.534	−0.0342671
$612$	8727.63	0.576460
$613$	−233.384	−0.0153773	−0.00768865	−	0.999970i	$-0.502447\pi$
$613$	−233.384	−0.0153773	−0.00768865	+	0.999970i	$0.502447\pi$
$614$	29676.3	1.95055
$615$	−37605.9	−2.46572
$616$	0	0
$617$	4202.77	0.274225	0.137113	−	0.990555i	$-0.456218\pi$
$617$	4202.77	0.274225	0.137113	+	0.990555i	$0.456218\pi$
$618$	−23514.4	−1.53056
$619$	23009.4	1.49407	0.747033	−	0.664787i	$-0.231478\pi$
$619$	23009.4	1.49407	0.747033	+	0.664787i	$0.231478\pi$
$620$	23569.1	1.52671
$621$	−3150.93	−0.203611
$622$	8466.89	0.545806
$623$	0	0
$624$	2033.17	0.130436
$625$	448.344	0.0286940
$626$	8765.82	0.559668
$627$	1037.97	0.0661126
$628$	11077.5	0.703884
$629$	−9233.45	−0.585313
$630$	0	0
$631$	18819.3	1.18730	0.593650	−	0.804723i	$-0.297686\pi$
$631$	18819.3	1.18730	0.593650	+	0.804723i	$0.297686\pi$
$632$	−697.924	−0.0439271
$633$	−28943.3	−1.81737
$634$	−1701.24	−0.106569
$635$	−16094.3	−1.00580
$636$	−8038.56	−0.501178
$637$	0	0
$638$	5531.74	0.343266
$639$	8547.46	0.529159
$640$	2739.06	0.169173
$641$	25253.7	1.55610	0.778052	−	0.628200i	$-0.216208\pi$
$641$	25253.7	1.55610	0.778052	+	0.628200i	$0.216208\pi$
$642$	−34805.0	−2.13963
$643$	−11712.1	−0.718324	−0.359162	−	0.933275i	$-0.616937\pi$
$643$	−11712.1	−0.718324	−0.359162	+	0.933275i	$0.616937\pi$
$644$	0	0
$645$	−54678.1	−3.33791
$646$	−4711.96	−0.286981
$647$	26533.3	1.61226	0.806131	−	0.591737i	$-0.201558\pi$
$647$	26533.3	1.61226	0.806131	+	0.591737i	$0.201558\pi$
$648$	−1051.00	−0.0637146
$649$	1483.25	0.0897111
$650$	−3688.59	−0.222582
$651$	0	0
$652$	981.952	0.0589819
$653$	27898.9	1.67193	0.835964	−	0.548785i	$-0.184909\pi$
$653$	27898.9	1.67193	0.835964	+	0.548785i	$0.184909\pi$
$654$	23166.9	1.38516
$655$	−42520.4	−2.53650
$656$	20470.2	1.21834
$657$	−15278.2	−0.907245
$658$	0	0
$659$	−1274.66	−0.0753468	−0.0376734	−	0.999290i	$-0.511995\pi$
$659$	−1274.66	−0.0753468	−0.0376734	+	0.999290i	$0.511995\pi$
$660$	−7630.16	−0.450006
$661$	5049.52	0.297131	0.148565	−	0.988903i	$-0.452534\pi$
$661$	5049.52	0.297131	0.148565	+	0.988903i	$0.452534\pi$
$662$	3033.17	0.178078
$663$	−1919.38	−0.112432
$664$	−342.904	−0.0200411
$665$	0	0
$666$	10585.1	0.615860
$667$	−9053.01	−0.525538
$668$	−26333.6	−1.52526
$669$	5545.14	0.320460
$670$	−7645.73	−0.440866
$671$	3286.86	0.189103
$672$	0	0
$673$	8398.64	0.481045	0.240523	−	0.970644i	$-0.422681\pi$
$673$	8398.64	0.481045	0.240523	+	0.970644i	$0.422681\pi$
$674$	12085.1	0.690653
$675$	12176.5	0.694333
$676$	−16760.7	−0.953612
$677$	−9875.31	−0.560619	−0.280309	−	0.959910i	$-0.590437\pi$
$677$	−9875.31	−0.560619	−0.280309	+	0.959910i	$0.590437\pi$
$678$	−31441.0	−1.78095
$679$	0	0
$680$	−1340.18	−0.0755787
$681$	−10082.1	−0.567325
$682$	−5439.18	−0.305392
$683$	−8653.78	−0.484814	−0.242407	−	0.970175i	$-0.577937\pi$
$683$	−8653.78	−0.484814	−0.242407	+	0.970175i	$0.577937\pi$
$684$	2649.60	0.148114
$685$	−40708.0	−2.27062
$686$	0	0
$687$	−35262.2	−1.95828
$688$	29763.2	1.64929
$689$	709.629	0.0392376
$690$	25457.6	1.40457
$691$	−2916.50	−0.160563	−0.0802813	−	0.996772i	$-0.525582\pi$
$691$	−2916.50	−0.160563	−0.0802813	+	0.996772i	$0.525582\pi$
$692$	2792.73	0.153416
$693$	0	0
$694$	23406.9	1.28028
$695$	−5370.37	−0.293108
$696$	−1361.07	−0.0741254
$697$	−19324.6	−1.05017
$698$	−47976.5	−2.60163
$699$	−14366.1	−0.777363
$700$	0	0
$701$	−9070.78	−0.488729	−0.244364	−	0.969683i	$-0.578579\pi$
$701$	−9070.78	−0.488729	−0.244364	+	0.969683i	$0.578579\pi$
$702$	−1080.85	−0.0581112
$703$	−2803.16	−0.150388
$704$	3848.88	0.206051
$705$	−13803.2	−0.737389
$706$	−9591.00	−0.511278
$707$	0	0
$708$	9432.34	0.500690
$709$	−5957.11	−0.315549	−0.157774	−	0.987475i	$-0.550432\pi$
$709$	−5957.11	−0.315549	−0.157774	+	0.987475i	$0.550432\pi$
$710$	33922.7	1.79309
$711$	−10701.3	−0.564459
$712$	1139.40	0.0599729
$713$	8901.55	0.467553
$714$	0	0
$715$	673.577	0.0352312
$716$	18622.6	0.972010
$717$	−25337.8	−1.31975
$718$	5768.71	0.299842
$719$	31140.8	1.61524	0.807620	−	0.589703i	$-0.200755\pi$
$719$	31140.8	1.61524	0.807620	+	0.589703i	$0.200755\pi$
$720$	21767.2	1.12669
$721$	0	0
$722$	−1430.49	−0.0737359
$723$	43086.4	2.21632
$724$	18828.6	0.966519
$725$	34984.7	1.79214
$726$	−33661.1	−1.72077
$727$	−14969.7	−0.763682	−0.381841	−	0.924228i	$-0.624710\pi$
$727$	−14969.7	−0.763682	−0.381841	+	0.924228i	$0.624710\pi$
$728$	0	0
$729$	−5283.30	−0.268420
$730$	−60635.4	−3.07427
$731$	−28097.5	−1.42165
$732$	20902.0	1.05541
$733$	12414.1	0.625545	0.312772	−	0.949828i	$-0.398742\pi$
$733$	12414.1	0.625545	0.312772	+	0.949828i	$0.398742\pi$
$734$	34806.5	1.75031
$735$	0	0
$736$	−13359.1	−0.669055
$737$	865.481	0.0432570
$738$	22153.4	1.10498
$739$	1324.11	0.0659111	0.0329555	−	0.999457i	$-0.489508\pi$
$739$	1324.11	0.0659111	0.0329555	+	0.999457i	$0.489508\pi$
$740$	20606.1	1.02364
$741$	−582.700	−0.0288880
$742$	0	0
$743$	−4391.55	−0.216838	−0.108419	−	0.994105i	$-0.534579\pi$
$743$	−4391.55	−0.216838	−0.108419	+	0.994105i	$0.534579\pi$
$744$	1338.30	0.0659468
$745$	−32480.1	−1.59729
$746$	36455.0	1.78916
$747$	−5257.76	−0.257525
$748$	−3920.92	−0.191662
$749$	0	0
$750$	−38053.1	−1.85267
$751$	−31947.5	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$751$	−31947.5	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$752$	7513.58	0.364351
$753$	−42159.0	−2.04032
$754$	−3105.42	−0.149990
$755$	42663.4	2.05653
$756$	0	0
$757$	18569.8	0.891585	0.445793	−	0.895136i	$-0.352922\pi$
$757$	18569.8	0.891585	0.445793	+	0.895136i	$0.352922\pi$
$758$	24414.9	1.16991
$759$	−2881.75	−0.137814
$760$	−406.861	−0.0194190
$761$	−5507.32	−0.262339	−0.131170	−	0.991360i	$-0.541873\pi$
$761$	−5507.32	−0.262339	−0.131170	+	0.991360i	$0.541873\pi$
$762$	23619.3	1.12288
$763$	0	0
$764$	−10687.7	−0.506111
$765$	−20549.0	−0.971178
$766$	10424.7	0.491723
$767$	−832.669	−0.0391994
$768$	−29442.7	−1.38336
$769$	14977.9	0.702362	0.351181	−	0.936308i	$-0.385780\pi$
$769$	14977.9	0.702362	0.351181	+	0.936308i	$0.385780\pi$
$770$	0	0
$771$	−21384.0	−0.998864
$772$	−24708.3	−1.15190
$773$	−19545.6	−0.909450	−0.454725	−	0.890632i	$-0.650262\pi$
$773$	−19545.6	−0.909450	−0.454725	+	0.890632i	$0.650262\pi$
$774$	32210.5	1.49585
$775$	−34399.4	−1.59440
$776$	259.190	0.0119902
$777$	0	0
$778$	23246.4	1.07124
$779$	−5866.70	−0.269829
$780$	4283.44	0.196631
$781$	−3839.98	−0.175935
$782$	13081.9	0.598221
$783$	10251.4	0.467887
$784$	0	0
$785$	−26081.7	−1.18585
$786$	62401.3	2.83178
$787$	4274.62	0.193613	0.0968067	−	0.995303i	$-0.469137\pi$
$787$	4274.62	0.193613	0.0968067	+	0.995303i	$0.469137\pi$
$788$	−26536.2	−1.19963
$789$	17624.2	0.795233
$790$	−42470.7	−1.91271
$791$	0	0
$792$	−173.913	−0.00780268
$793$	−1845.19	−0.0826287
$794$	−56482.6	−2.52455
$795$	18926.6	0.844350
$796$	−15602.0	−0.694722
$797$	25450.6	1.13112	0.565562	−	0.824706i	$-0.308659\pi$
$797$	25450.6	1.13112	0.565562	+	0.824706i	$0.308659\pi$
$798$	0	0
$799$	−7093.08	−0.314062
$800$	51625.4	2.28154
$801$	17470.4	0.770645
$802$	−39266.2	−1.72885
$803$	6863.81	0.301642
$804$	5503.81	0.241423
$805$	0	0
$806$	3053.46	0.133441
$807$	49999.7	2.18101
$808$	−1709.80	−0.0744439
$809$	4002.04	0.173924	0.0869619	−	0.996212i	$-0.472284\pi$
$809$	4002.04	0.173924	0.0869619	+	0.996212i	$0.472284\pi$
$810$	−63956.2	−2.77431
$811$	37915.1	1.64165	0.820826	−	0.571179i	$-0.193514\pi$
$811$	37915.1	1.64165	0.820826	+	0.571179i	$0.193514\pi$
$812$	0	0
$813$	−26894.5	−1.16019
$814$	−4755.39	−0.204762
$815$	−2311.99	−0.0993685
$816$	27865.7	1.19546
$817$	−8530.05	−0.365274
$818$	23006.2	0.983366
$819$	0	0
$820$	43126.3	1.83663
$821$	4739.43	0.201470	0.100735	−	0.994913i	$-0.467881\pi$
$821$	4739.43	0.201470	0.100735	+	0.994913i	$0.467881\pi$
$822$	59741.5	2.53494
$823$	20752.2	0.878952	0.439476	−	0.898254i	$-0.355164\pi$
$823$	20752.2	0.878952	0.439476	+	0.898254i	$0.355164\pi$
$824$	−1043.36	−0.0441106
$825$	11136.3	0.469959
$826$	0	0
$827$	34264.8	1.44075	0.720377	−	0.693583i	$-0.243969\pi$
$827$	34264.8	1.44075	0.720377	+	0.693583i	$0.243969\pi$
$828$	−7356.14	−0.308748
$829$	39707.5	1.66357	0.831784	−	0.555100i	$-0.187320\pi$
$829$	39707.5	1.66357	0.831784	+	0.555100i	$0.187320\pi$
$830$	−20866.7	−0.872645
$831$	39159.3	1.63468
$832$	−2160.69	−0.0900343
$833$	0	0
$834$	7881.35	0.327229
$835$	62001.8	2.56965
$836$	−1190.34	−0.0492450
$837$	−10079.9	−0.416263
$838$	48584.6	2.00278
$839$	4524.04	0.186159	0.0930794	−	0.995659i	$-0.470329\pi$
$839$	4524.04	0.186159	0.0930794	+	0.995659i	$0.470329\pi$
$840$	0	0
$841$	5064.59	0.207659
$842$	−23132.9	−0.946809
$843$	50400.1	2.05916
$844$	33192.1	1.35370
$845$	39462.7	1.60658
$846$	8131.39	0.330453
$847$	0	0
$848$	−10302.4	−0.417201
$849$	39698.0	1.60475
$850$	−50554.1	−2.03999
$851$	7782.47	0.313490
$852$	−24419.4	−0.981920
$853$	7595.54	0.304884	0.152442	−	0.988312i	$-0.451286\pi$
$853$	7595.54	0.304884	0.152442	+	0.988312i	$0.451286\pi$
$854$	0	0
$855$	−6238.42	−0.249531
$856$	−1544.34	−0.0616639
$857$	19528.9	0.778405	0.389203	−	0.921152i	$-0.372751\pi$
$857$	19528.9	0.778405	0.389203	+	0.921152i	$0.372751\pi$
$858$	−988.515	−0.0393326
$859$	−25980.8	−1.03196	−0.515979	−	0.856601i	$-0.672572\pi$
$859$	−25980.8	−1.03196	−0.515979	+	0.856601i	$0.672572\pi$
$860$	62704.7	2.48629
$861$	0	0
$862$	10977.0	0.433732
$863$	−48294.6	−1.90494	−0.952472	−	0.304625i	$-0.901469\pi$
$863$	−48294.6	−1.90494	−0.952472	+	0.304625i	$0.901469\pi$
$864$	15127.6	0.595660
$865$	−6575.43	−0.258464
$866$	22440.1	0.880536
$867$	6690.03	0.262059
$868$	0	0
$869$	4807.61	0.187672
$870$	−82825.1	−3.22763
$871$	−485.866	−0.0189012
$872$	1027.94	0.0399201
$873$	3974.17	0.154072
$874$	3971.51	0.153705
$875$	0	0
$876$	43648.7	1.68351
$877$	44377.5	1.70869	0.854346	−	0.519705i	$-0.173958\pi$
$877$	44377.5	1.70869	0.854346	+	0.519705i	$0.173958\pi$
$878$	−33282.5	−1.27930
$879$	21797.7	0.836426
$880$	−9779.02	−0.374603
$881$	12139.4	0.464231	0.232116	−	0.972688i	$-0.425435\pi$
$881$	12139.4	0.464231	0.232116	+	0.972688i	$0.425435\pi$
$882$	0	0
$883$	5048.07	0.192391	0.0961954	−	0.995362i	$-0.469333\pi$
$883$	5048.07	0.192391	0.0961954	+	0.995362i	$0.469333\pi$
$884$	2201.14	0.0837470
$885$	−22208.2	−0.843527
$886$	−24388.4	−0.924767
$887$	−20373.4	−0.771221	−0.385610	−	0.922662i	$-0.626009\pi$
$887$	−20373.4	−0.771221	−0.385610	+	0.922662i	$0.626009\pi$
$888$	1170.05	0.0442166
$889$	0	0
$890$	69335.6	2.61139
$891$	7239.72	0.272211
$892$	−6359.14	−0.238699
$893$	−2153.37	−0.0806940
$894$	47666.5	1.78323
$895$	−43846.5	−1.63757
$896$	0	0
$897$	1617.76	0.0602180
$898$	−13651.2	−0.507289
$899$	−28960.8	−1.07441
$900$	28427.3	1.05286
$901$	9725.85	0.359617
$902$	−9952.52	−0.367386
$903$	0	0
$904$	−1395.07	−0.0513267
$905$	−44331.6	−1.62832
$906$	−62611.1	−2.29593
$907$	−7456.13	−0.272962	−0.136481	−	0.990643i	$-0.543579\pi$
$907$	−7456.13	−0.272962	−0.136481	+	0.990643i	$0.543579\pi$
$908$	11562.2	0.422581
$909$	−26216.5	−0.956596
$910$	0	0
$911$	−10653.2	−0.387440	−0.193720	−	0.981057i	$-0.562055\pi$
$911$	−10653.2	−0.387440	−0.193720	+	0.981057i	$0.562055\pi$
$912$	8459.66	0.307157
$913$	2362.07	0.0856224
$914$	1990.59	0.0720380
$915$	−49213.3	−1.77808
$916$	40438.6	1.45866
$917$	0	0
$918$	−14813.6	−0.532596
$919$	−12569.7	−0.451183	−0.225591	−	0.974222i	$-0.572431\pi$
$919$	−12569.7	−0.451183	−0.225591	+	0.974222i	$0.572431\pi$
$920$	1129.58	0.0404795
$921$	50297.8	1.79953
$922$	2164.59	0.0773179
$923$	2155.70	0.0768752
$924$	0	0
$925$	−30074.8	−1.06903
$926$	−73466.8	−2.60720
$927$	−15997.9	−0.566816
$928$	43463.4	1.53745
$929$	−4920.06	−0.173759	−0.0868795	−	0.996219i	$-0.527689\pi$
$929$	−4920.06	−0.173759	−0.0868795	+	0.996219i	$0.527689\pi$
$930$	81439.4	2.87151
$931$	0	0
$932$	16475.0	0.579031
$933$	14350.4	0.503548
$934$	49433.4	1.73181
$935$	9231.74	0.322898
$936$	97.6317	0.00340939
$937$	−1991.87	−0.0694465	−0.0347233	−	0.999397i	$-0.511055\pi$
$937$	−1991.87	−0.0694465	−0.0347233	+	0.999397i	$0.511055\pi$
$938$	0	0
$939$	14857.0	0.516337
$940$	15829.5	0.549256
$941$	7640.33	0.264684	0.132342	−	0.991204i	$-0.457750\pi$
$941$	7640.33	0.264684	0.132342	+	0.991204i	$0.457750\pi$
$942$	38276.5	1.32390
$943$	16287.9	0.562467
$944$	12088.7	0.416795
$945$	0	0
$946$	−14470.7	−0.497340
$947$	−6521.15	−0.223769	−0.111884	−	0.993721i	$-0.535689\pi$
$947$	−6521.15	−0.223769	−0.111884	+	0.993721i	$0.535689\pi$
$948$	30572.8	1.04742
$949$	−3853.22	−0.131803
$950$	−15347.6	−0.524149
$951$	−2883.40	−0.0983182
$952$	0	0
$953$	−35757.9	−1.21544	−0.607719	−	0.794152i	$-0.707915\pi$
$953$	−35757.9	−1.21544	−0.607719	+	0.794152i	$0.707915\pi$
$954$	−11149.6	−0.378386
$955$	25164.1	0.852659
$956$	29057.3	0.983034
$957$	9375.64	0.316689
$958$	−41881.3	−1.41244
$959$	0	0
$960$	−57628.1	−1.93744
$961$	−1314.75	−0.0441323
$962$	2669.59	0.0894710
$963$	−23679.3	−0.792374
$964$	−49411.4	−1.65086
$965$	58175.1	1.94065
$966$	0	0
$967$	−6342.30	−0.210915	−0.105457	−	0.994424i	$-0.533631\pi$
$967$	−6342.30	−0.210915	−0.105457	+	0.994424i	$0.533631\pi$
$968$	−1493.58	−0.0495924
$969$	−7986.21	−0.264762
$970$	15772.5	0.522086
$971$	−30351.2	−1.00311	−0.501553	−	0.865127i	$-0.667238\pi$
$971$	−30351.2	−1.00311	−0.501553	+	0.865127i	$0.667238\pi$
$972$	33617.5	1.10934
$973$	0	0
$974$	44491.4	1.46365
$975$	−6251.73	−0.205349
$976$	26788.5	0.878566
$977$	39843.6	1.30472	0.652359	−	0.757910i	$-0.273780\pi$
$977$	39843.6	1.30472	0.652359	+	0.757910i	$0.273780\pi$
$978$	3392.98	0.110936
$979$	−7848.66	−0.256225
$980$	0	0
$981$	15761.4	0.512969
$982$	2127.27	0.0691282
$983$	−24068.7	−0.780949	−0.390475	−	0.920614i	$-0.627689\pi$
$983$	−24068.7	−0.780949	−0.390475	+	0.920614i	$0.627689\pi$
$984$	2448.79	0.0793340
$985$	62478.9	2.02106
$986$	−42561.5	−1.37468
$987$	0	0
$988$	668.238	0.0215177
$989$	23682.2	0.761426
$990$	−10583.1	−0.339751
$991$	3235.83	0.103723	0.0518615	−	0.998654i	$-0.483485\pi$
$991$	3235.83	0.103723	0.0518615	+	0.998654i	$0.483485\pi$
$992$	−42736.2	−1.36782
$993$	5140.86	0.164290
$994$	0	0
$995$	36734.6	1.17042
$996$	15021.0	0.477871
$997$	−19444.5	−0.617665	−0.308833	−	0.951116i	$-0.599938\pi$
$997$	−19444.5	−0.617665	−0.308833	+	0.951116i	$0.599938\pi$
$998$	−5230.25	−0.165892
$999$	−8812.68	−0.279100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	931.4.a.c.1.1		3
7.6	odd	2		19.4.a.b.1.1	✓	3
21.20	even	2		171.4.a.f.1.3		3
28.27	even	2		304.4.a.i.1.1		3
35.13	even	4		475.4.b.f.324.5		6
35.27	even	4		475.4.b.f.324.2		6
35.34	odd	2		475.4.a.f.1.3		3
56.13	odd	2		1216.4.a.s.1.1		3
56.27	even	2		1216.4.a.u.1.3		3
77.76	even	2		2299.4.a.h.1.3		3
133.132	even	2		361.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.1	✓	3	7.6	odd	2
171.4.a.f.1.3		3	21.20	even	2
304.4.a.i.1.1		3	28.27	even	2
361.4.a.i.1.3		3	133.132	even	2
475.4.a.f.1.3		3	35.34	odd	2
475.4.b.f.324.2		6	35.27	even	4
475.4.b.f.324.5		6	35.13	even	4
931.4.a.c.1.1		3	1.1	even	1	trivial
1216.4.a.s.1.1		3	56.13	odd	2
1216.4.a.u.1.3		3	56.27	even	2
2299.4.a.h.1.3		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)

\(f(q)\)	\(=\)	\(q-3.96257 q^{2} -6.71610 q^{3} +7.70200 q^{4} -18.1342 q^{5} +26.6130 q^{6} +1.18085 q^{8} +18.1060 q^{9} +71.8581 q^{10} -8.13420 q^{11} -51.7274 q^{12} +4.56640 q^{13} +121.791 q^{15} -66.2952 q^{16} +62.5850 q^{17} -71.7464 q^{18} +19.0000 q^{19} -139.670 q^{20} +32.2324 q^{22} -52.7502 q^{23} -7.93070 q^{24} +203.849 q^{25} -18.0947 q^{26} +59.7330 q^{27} +171.620 q^{29} -482.606 q^{30} -168.749 q^{31} +253.253 q^{32} +54.6301 q^{33} -247.998 q^{34} +139.452 q^{36} -147.534 q^{37} -75.2889 q^{38} -30.6684 q^{39} -21.4138 q^{40} -308.774 q^{41} -448.950 q^{43} -62.6496 q^{44} -328.338 q^{45} +209.027 q^{46} -113.335 q^{47} +445.245 q^{48} -807.768 q^{50} -420.327 q^{51} +35.1704 q^{52} +155.402 q^{53} -236.696 q^{54} +147.507 q^{55} -127.606 q^{57} -680.059 q^{58} -182.347 q^{59} +938.035 q^{60} -404.080 q^{61} +668.681 q^{62} -473.172 q^{64} -82.8080 q^{65} -216.476 q^{66} -106.400 q^{67} +482.030 q^{68} +354.276 q^{69} +472.079 q^{71} +21.3805 q^{72} -843.821 q^{73} +584.616 q^{74} -1369.07 q^{75} +146.338 q^{76} +121.526 q^{78} -591.036 q^{79} +1202.21 q^{80} -890.035 q^{81} +1223.54 q^{82} -290.388 q^{83} -1134.93 q^{85} +1779.00 q^{86} -1152.62 q^{87} -9.60526 q^{88} +964.896 q^{89} +1301.06 q^{90} -406.282 q^{92} +1133.34 q^{93} +449.099 q^{94} -344.550 q^{95} -1700.87 q^{96} +219.495 q^{97} -147.278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 21 q^{4} - 14 q^{5} + 65 q^{6} + 27 q^{8} + 48 q^{9} + 88 q^{10} + 16 q^{11} + 115 q^{12} - 65 q^{13} + 140 q^{15} + 33 q^{16} - 29 q^{17} + 138 q^{18} + 57 q^{19} - 100 q^{20}+ \cdots + 250 q^{99}+O(q^{100}) \) 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

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Database

Properties

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