LMFDB - Embedded newform 77.4.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [77,4,Mod(1,77)] 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("77.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(77, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	77.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$4.54314707044$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.22767$ of defining polynomial
Character	$\chi$	$=$	77.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.22767 q^{2} -7.89221 q^{3} -6.49284 q^{4} -2.21191 q^{5} +9.68899 q^{6} +7.00000 q^{7} +17.7924 q^{8} +35.2869 q^{9} +2.71549 q^{10} +11.0000 q^{11} +51.2428 q^{12} +12.8361 q^{13} -8.59366 q^{14} +17.4569 q^{15} +30.0996 q^{16} -45.5444 q^{17} -43.3205 q^{18} +11.0493 q^{19} +14.3616 q^{20} -55.2454 q^{21} -13.5043 q^{22} +177.525 q^{23} -140.421 q^{24} -120.107 q^{25} -15.7584 q^{26} -65.4021 q^{27} -45.4499 q^{28} +58.5230 q^{29} -21.4312 q^{30} +175.188 q^{31} -179.291 q^{32} -86.8143 q^{33} +55.9132 q^{34} -15.4834 q^{35} -229.112 q^{36} +221.135 q^{37} -13.5648 q^{38} -101.305 q^{39} -39.3551 q^{40} -307.706 q^{41} +67.8229 q^{42} +462.781 q^{43} -71.4212 q^{44} -78.0515 q^{45} -217.942 q^{46} -293.151 q^{47} -237.552 q^{48} +49.0000 q^{49} +147.452 q^{50} +359.445 q^{51} -83.3427 q^{52} +400.608 q^{53} +80.2919 q^{54} -24.3310 q^{55} +124.547 q^{56} -87.2032 q^{57} -71.8467 q^{58} +16.3417 q^{59} -113.344 q^{60} +509.546 q^{61} -215.072 q^{62} +247.008 q^{63} -20.6874 q^{64} -28.3923 q^{65} +106.579 q^{66} +483.585 q^{67} +295.712 q^{68} -1401.07 q^{69} +19.0084 q^{70} +202.883 q^{71} +627.838 q^{72} +885.910 q^{73} -271.479 q^{74} +947.913 q^{75} -71.7412 q^{76} +77.0000 q^{77} +124.369 q^{78} +289.526 q^{79} -66.5777 q^{80} -436.580 q^{81} +377.760 q^{82} +106.577 q^{83} +358.700 q^{84} +100.740 q^{85} -568.140 q^{86} -461.876 q^{87} +195.716 q^{88} -1586.10 q^{89} +95.8211 q^{90} +89.8527 q^{91} -1152.64 q^{92} -1382.62 q^{93} +359.891 q^{94} -24.4400 q^{95} +1415.00 q^{96} -990.599 q^{97} -60.1556 q^{98} +388.156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9} - 10 q^{10} + 55 q^{11} + 24 q^{12} - 50 q^{13} + 7 q^{14} - 146 q^{15} + 433 q^{16} + 222 q^{17} + 245 q^{18} + 160 q^{19}+ \cdots + 693 q^{99}+O(q^{100}) $ 5 * q + q^2 + 2 * q^3 + 45 * q^4 - 24 * q^5 + 4 * q^6 + 35 * q^7 + 57 * q^8 + 63 * q^9 - 10 * q^10 + 55 * q^11 + 24 * q^12 - 50 * q^13 + 7 * q^14 - 146 * q^15 + 433 * q^16 + 222 * q^17 + 245 * q^18 + 160 * q^19 - 430 * q^20 + 14 * q^21 + 11 * q^22 + 54 * q^23 - 220 * q^24 + 125 * q^25 - 1026 * q^26 + 110 * q^27 + 315 * q^28 + 14 * q^29 - 1080 * q^30 - 34 * q^31 - 583 * q^32 + 22 * q^33 - 750 * q^34 - 168 * q^35 - 411 * q^36 + 1044 * q^37 - 156 * q^38 - 124 * q^39 + 1158 * q^40 - 114 * q^41 + 28 * q^42 + 672 * q^43 + 495 * q^44 - 530 * q^45 - 1224 * q^46 - 292 * q^47 - 1652 * q^48 + 245 * q^49 + 1143 * q^50 + 768 * q^51 - 914 * q^52 - 710 * q^53 + 1608 * q^54 - 264 * q^55 + 399 * q^56 + 2012 * q^57 - 810 * q^58 + 270 * q^59 - 3068 * q^60 + 138 * q^61 - 480 * q^62 + 441 * q^63 + 3001 * q^64 - 196 * q^65 + 44 * q^66 + 1942 * q^67 + 3130 * q^68 - 1306 * q^69 - 70 * q^70 - 278 * q^71 + 3565 * q^72 - 338 * q^73 + 462 * q^74 + 3960 * q^75 - 1332 * q^76 + 385 * q^77 + 280 * q^78 + 576 * q^79 - 1602 * q^80 - 1439 * q^81 + 1386 * q^82 + 1644 * q^83 + 168 * q^84 + 360 * q^85 + 2020 * q^86 - 1956 * q^87 + 627 * q^88 - 3656 * q^89 - 6162 * q^90 - 350 * q^91 - 748 * q^92 - 1442 * q^93 + 976 * q^94 - 1276 * q^95 - 4508 * q^96 + 692 * q^97 + 49 * q^98 + 693 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.22767	−0.434045	−0.217023	−	0.976167i	$-0.569635\pi$
$2$	−1.22767	−0.434045	−0.217023	+	0.976167i	$0.569635\pi$
$3$	−7.89221	−1.51886	−0.759428	−	0.650591i	$-0.774521\pi$
$3$	−7.89221	−1.51886	−0.759428	+	0.650591i	$0.774521\pi$
$4$	−6.49284	−0.811605
$5$	−2.21191	−0.197839	−0.0989196	−	0.995095i	$-0.531539\pi$
$5$	−2.21191	−0.197839	−0.0989196	+	0.995095i	$0.531539\pi$
$6$	9.68899	0.659252
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	17.7924	0.786319
$9$	35.2869	1.30692
$10$	2.71549	0.0858712
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	51.2428	1.23271
$13$	12.8361	0.273853	0.136927	−	0.990581i	$-0.456278\pi$
$13$	12.8361	0.273853	0.136927	+	0.990581i	$0.456278\pi$
$14$	−8.59366	−0.164054
$15$	17.4569	0.300489
$16$	30.0996	0.470307
$17$	−45.5444	−0.649772	−0.324886	−	0.945753i	$-0.605326\pi$
$17$	−45.5444	−0.649772	−0.324886	+	0.945753i	$0.605326\pi$
$18$	−43.3205	−0.567264
$19$	11.0493	0.133415	0.0667074	−	0.997773i	$-0.478751\pi$
$19$	11.0493	0.133415	0.0667074	+	0.997773i	$0.478751\pi$
$20$	14.3616	0.160567
$21$	−55.2454	−0.574074
$22$	−13.5043	−0.130870
$23$	177.525	1.60942	0.804708	−	0.593671i	$-0.202322\pi$
$23$	177.525	1.60942	0.804708	+	0.593671i	$0.202322\pi$
$24$	−140.421	−1.19430
$25$	−120.107	−0.960860
$26$	−15.7584	−0.118865
$27$	−65.4021	−0.466172
$28$	−45.4499	−0.306758
$29$	58.5230	0.374740	0.187370	−	0.982289i	$-0.440004\pi$
$29$	58.5230	0.374740	0.187370	+	0.982289i	$0.440004\pi$
$30$	−21.4312	−0.130426
$31$	175.188	1.01499	0.507495	−	0.861654i	$-0.330571\pi$
$31$	175.188	1.01499	0.507495	+	0.861654i	$0.330571\pi$
$32$	−179.291	−0.990453
$33$	−86.8143	−0.457952
$34$	55.9132	0.282031
$35$	−15.4834	−0.0747762
$36$	−229.112	−1.06070
$37$	221.135	0.982549	0.491274	−	0.871005i	$-0.336531\pi$
$37$	221.135	0.982549	0.491274	+	0.871005i	$0.336531\pi$
$38$	−13.5648	−0.0579080
$39$	−101.305	−0.415944
$40$	−39.3551	−0.155565
$41$	−307.706	−1.17209	−0.586043	−	0.810280i	$-0.699315\pi$
$41$	−307.706	−1.17209	−0.586043	+	0.810280i	$0.699315\pi$
$42$	67.8229	0.249174
$43$	462.781	1.64124	0.820621	−	0.571473i	$-0.193628\pi$
$43$	462.781	1.64124	0.820621	+	0.571473i	$0.193628\pi$
$44$	−71.4212	−0.244708
$45$	−78.0515	−0.258561
$46$	−217.942	−0.698560
$47$	−293.151	−0.909796	−0.454898	−	0.890544i	$-0.650324\pi$
$47$	−293.151	−0.909796	−0.454898	+	0.890544i	$0.650324\pi$
$48$	−237.552	−0.714328
$49$	49.0000	0.142857
$50$	147.452	0.417057
$51$	359.445	0.986910
$52$	−83.3427	−0.222261
$53$	400.608	1.03826	0.519130	−	0.854696i	$-0.326256\pi$
$53$	400.608	1.03826	0.519130	+	0.854696i	$0.326256\pi$
$54$	80.2919	0.202340
$55$	−24.3310	−0.0596508
$56$	124.547	0.297201
$57$	−87.2032	−0.202638
$58$	−71.8467	−0.162654
$59$	16.3417	0.0360595	0.0180298	−	0.999837i	$-0.494261\pi$
$59$	16.3417	0.0360595	0.0180298	+	0.999837i	$0.494261\pi$
$60$	−113.344	−0.243879
$61$	509.546	1.06952	0.534760	−	0.845004i	$-0.320402\pi$
$61$	509.546	1.06952	0.534760	+	0.845004i	$0.320402\pi$
$62$	−215.072	−0.440552
$63$	247.008	0.493970
$64$	−20.6874	−0.0404051
$65$	−28.3923	−0.0541790
$66$	106.579	0.198772
$67$	483.585	0.881781	0.440891	−	0.897561i	$-0.354663\pi$
$67$	483.585	0.881781	0.440891	+	0.897561i	$0.354663\pi$
$68$	295.712	0.527358
$69$	−1401.07	−2.44447
$70$	19.0084	0.0324563
$71$	202.883	0.339124	0.169562	−	0.985519i	$-0.445765\pi$
$71$	202.883	0.339124	0.169562	+	0.985519i	$0.445765\pi$
$72$	627.838	1.02766
$73$	885.910	1.42038	0.710191	−	0.704009i	$-0.248608\pi$
$73$	885.910	1.42038	0.710191	+	0.704009i	$0.248608\pi$
$74$	−271.479	−0.426471
$75$	947.913	1.45941
$76$	−71.7412	−0.108280
$77$	77.0000	0.113961
$78$	124.369	0.180539
$79$	289.526	0.412331	0.206166	−	0.978517i	$-0.433901\pi$
$79$	289.526	0.412331	0.206166	+	0.978517i	$0.433901\pi$
$80$	−66.5777	−0.0930451
$81$	−436.580	−0.598875
$82$	377.760	0.508739
$83$	106.577	0.140944	0.0704722	−	0.997514i	$-0.477549\pi$
$83$	106.577	0.140944	0.0704722	+	0.997514i	$0.477549\pi$
$84$	358.700	0.465921
$85$	100.740	0.128550
$86$	−568.140	−0.712373
$87$	−461.876	−0.569175
$88$	195.716	0.237084
$89$	−1586.10	−1.88906	−0.944528	−	0.328432i	$-0.893480\pi$
$89$	−1586.10	−1.88906	−0.944528	+	0.328432i	$0.893480\pi$
$90$	95.8211	0.112227
$91$	89.8527	0.103507
$92$	−1152.64	−1.30621
$93$	−1382.62	−1.54162
$94$	359.891	0.394893
$95$	−24.4400	−0.0263947
$96$	1415.00	1.50436
$97$	−990.599	−1.03691	−0.518454	−	0.855105i	$-0.673492\pi$
$97$	−990.599	−1.03691	−0.518454	+	0.855105i	$0.673492\pi$
$98$	−60.1556	−0.0620065
$99$	388.156	0.394052
$100$	779.838	0.779838
$101$	−496.401	−0.489047	−0.244523	−	0.969643i	$-0.578631\pi$
$101$	−496.401	−0.489047	−0.244523	+	0.969643i	$0.578631\pi$
$102$	−441.279	−0.428364
$103$	287.602	0.275128	0.137564	−	0.990493i	$-0.456073\pi$
$103$	287.602	0.275128	0.137564	+	0.990493i	$0.456073\pi$
$104$	228.385	0.215336
$105$	122.198	0.113574
$106$	−491.813	−0.450652
$107$	−1310.33	−1.18388	−0.591938	−	0.805984i	$-0.701637\pi$
$107$	−1310.33	−1.18388	−0.591938	+	0.805984i	$0.701637\pi$
$108$	424.645	0.378347
$109$	−1226.79	−1.07803	−0.539014	−	0.842297i	$-0.681203\pi$
$109$	−1226.79	−1.07803	−0.539014	+	0.842297i	$0.681203\pi$
$110$	29.8703	0.0258911
$111$	−1745.24	−1.49235
$112$	210.697	0.177759
$113$	1717.53	1.42984	0.714921	−	0.699206i	$-0.246463\pi$
$113$	1717.53	1.42984	0.714921	+	0.699206i	$0.246463\pi$
$114$	107.056	0.0879540
$115$	−392.670	−0.318406
$116$	−379.980	−0.304140
$117$	452.947	0.357905
$118$	−20.0622	−0.0156515
$119$	−318.810	−0.245591
$120$	310.599	0.236280
$121$	121.000	0.0909091
$122$	−625.552	−0.464220
$123$	2428.48	1.78023
$124$	−1137.47	−0.823771
$125$	542.156	0.387935
$126$	−303.244	−0.214406
$127$	2764.27	1.93141	0.965707	−	0.259633i	$-0.0836015\pi$
$127$	2764.27	1.93141	0.965707	+	0.259633i	$0.0836015\pi$
$128$	1459.73	1.00799
$129$	−3652.36	−2.49281
$130$	34.8563	0.0235161
$131$	−781.116	−0.520965	−0.260482	−	0.965479i	$-0.583882\pi$
$131$	−781.116	−0.520965	−0.260482	+	0.965479i	$0.583882\pi$
$132$	563.671	0.371676
$133$	77.3450	0.0504260
$134$	−593.681	−0.382733
$135$	144.664	0.0922271
$136$	−810.341	−0.510928
$137$	−1562.95	−0.974685	−0.487343	−	0.873211i	$-0.662034\pi$
$137$	−1562.95	−0.974685	−0.487343	+	0.873211i	$0.662034\pi$
$138$	1720.04	1.06101
$139$	1707.58	1.04198	0.520991	−	0.853562i	$-0.325563\pi$
$139$	1707.58	1.04198	0.520991	+	0.853562i	$0.325563\pi$
$140$	100.531	0.0606887
$141$	2313.60	1.38185
$142$	−249.073	−0.147195
$143$	141.197	0.0825699
$144$	1062.12	0.614655
$145$	−129.448	−0.0741382
$146$	−1087.60	−0.616511
$147$	−386.718	−0.216979
$148$	−1435.79	−0.797441
$149$	1825.21	1.00353	0.501767	−	0.865003i	$-0.332683\pi$
$149$	1825.21	1.00353	0.501767	+	0.865003i	$0.332683\pi$
$150$	−1163.72	−0.633449
$151$	−1300.18	−0.700708	−0.350354	−	0.936617i	$-0.613939\pi$
$151$	−1300.18	−0.700708	−0.350354	+	0.936617i	$0.613939\pi$
$152$	196.593	0.104906
$153$	−1607.12	−0.849202
$154$	−94.5303	−0.0494641
$155$	−387.500	−0.200805
$156$	657.758	0.337582
$157$	2068.27	1.05138	0.525688	−	0.850677i	$-0.323808\pi$
$157$	2068.27	1.05138	0.525688	+	0.850677i	$0.323808\pi$
$158$	−355.441	−0.178971
$159$	−3161.68	−1.57697
$160$	396.576	0.195950
$161$	1242.68	0.608302
$162$	535.974	0.259939
$163$	1463.67	0.703336	0.351668	−	0.936125i	$-0.385615\pi$
$163$	1463.67	0.703336	0.351668	+	0.936125i	$0.385615\pi$
$164$	1997.88	0.951271
$165$	192.025	0.0906009
$166$	−130.841	−0.0611763
$167$	3276.22	1.51809	0.759045	−	0.651038i	$-0.225666\pi$
$167$	3276.22	1.51809	0.759045	+	0.651038i	$0.225666\pi$
$168$	−982.947	−0.451405
$169$	−2032.23	−0.925004
$170$	−123.675	−0.0557967
$171$	389.895	0.174363
$172$	−3004.76	−1.33204
$173$	−795.944	−0.349795	−0.174897	−	0.984587i	$-0.555959\pi$
$173$	−795.944	−0.349795	−0.174897	+	0.984587i	$0.555959\pi$
$174$	567.029	0.247048
$175$	−840.752	−0.363171
$176$	331.096	0.141803
$177$	−128.972	−0.0547692
$178$	1947.20	0.819936
$179$	1636.50	0.683340	0.341670	−	0.939820i	$-0.389008\pi$
$179$	1636.50	0.683340	0.341670	+	0.939820i	$0.389008\pi$
$180$	506.776	0.209849
$181$	3631.26	1.49121	0.745606	−	0.666387i	$-0.232160\pi$
$181$	3631.26	1.49121	0.745606	+	0.666387i	$0.232160\pi$
$182$	−110.309	−0.0449267
$183$	−4021.44	−1.62445
$184$	3158.59	1.26551
$185$	−489.130	−0.194387
$186$	1697.40	0.669135
$187$	−500.988	−0.195914
$188$	1903.38	0.738394
$189$	−457.815	−0.176196
$190$	30.0042	0.0114565
$191$	208.684	0.0790568	0.0395284	−	0.999218i	$-0.487414\pi$
$191$	208.684	0.0790568	0.0395284	+	0.999218i	$0.487414\pi$
$192$	163.269	0.0613695
$193$	808.914	0.301694	0.150847	−	0.988557i	$-0.451800\pi$
$193$	808.914	0.301694	0.150847	+	0.988557i	$0.451800\pi$
$194$	1216.12	0.450065
$195$	224.078	0.0822900
$196$	−318.149	−0.115944
$197$	4224.83	1.52795	0.763976	−	0.645245i	$-0.223245\pi$
$197$	4224.83	1.52795	0.763976	+	0.645245i	$0.223245\pi$
$198$	−476.526	−0.171036
$199$	344.463	0.122705	0.0613526	−	0.998116i	$-0.480459\pi$
$199$	344.463	0.122705	0.0613526	+	0.998116i	$0.480459\pi$
$200$	−2136.99	−0.755542
$201$	−3816.55	−1.33930
$202$	609.414	0.212268
$203$	409.661	0.141638
$204$	−2333.82	−0.800981
$205$	680.617	0.231885
$206$	−353.079	−0.119418
$207$	6264.32	2.10338
$208$	386.362	0.128795
$209$	121.542	0.0402261
$210$	−150.018	−0.0492964
$211$	−5991.08	−1.95470	−0.977352	−	0.211619i	$-0.932126\pi$
$211$	−5991.08	−1.95470	−0.977352	+	0.211619i	$0.932126\pi$
$212$	−2601.08	−0.842656
$213$	−1601.20	−0.515081
$214$	1608.65	0.513856
$215$	−1023.63	−0.324702
$216$	−1163.66	−0.366560
$217$	1226.32	0.383631
$218$	1506.09	0.467913
$219$	−6991.79	−2.15736
$220$	157.977	0.0484128
$221$	−584.612	−0.177942
$222$	2142.57	0.647747
$223$	−2148.98	−0.645320	−0.322660	−	0.946515i	$-0.604577\pi$
$223$	−2148.98	−0.645320	−0.322660	+	0.946515i	$0.604577\pi$
$224$	−1255.04	−0.374356
$225$	−4238.22	−1.25577
$226$	−2108.56	−0.620616
$227$	−2067.49	−0.604513	−0.302256	−	0.953227i	$-0.597740\pi$
$227$	−2067.49	−0.604513	−0.302256	+	0.953227i	$0.597740\pi$
$228$	566.196	0.164462
$229$	−6394.72	−1.84530	−0.922652	−	0.385633i	$-0.873983\pi$
$229$	−6394.72	−1.84530	−0.922652	+	0.385633i	$0.873983\pi$
$230$	482.067	0.138203
$231$	−607.700	−0.173090
$232$	1041.26	0.294665
$233$	−5341.34	−1.50181	−0.750907	−	0.660408i	$-0.770383\pi$
$233$	−5341.34	−1.50181	−0.750907	+	0.660408i	$0.770383\pi$
$234$	−556.067	−0.155347
$235$	648.423	0.179993
$236$	−106.104	−0.0292661
$237$	−2285.00	−0.626272
$238$	391.393	0.106598
$239$	−4325.48	−1.17068	−0.585339	−	0.810789i	$-0.699038\pi$
$239$	−4325.48	−1.17068	−0.585339	+	0.810789i	$0.699038\pi$
$240$	525.445	0.141322
$241$	6620.57	1.76958	0.884790	−	0.465991i	$-0.154302\pi$
$241$	6620.57	1.76958	0.884790	+	0.465991i	$0.154302\pi$
$242$	−148.548	−0.0394587
$243$	5211.44	1.37578
$244$	−3308.40	−0.868027
$245$	−108.384	−0.0282628
$246$	−2981.36	−0.772701
$247$	141.830	0.0365361
$248$	3117.01	0.798106
$249$	−841.130	−0.214074
$250$	−665.586	−0.168381
$251$	4621.48	1.16217	0.581086	−	0.813842i	$-0.302628\pi$
$251$	4621.48	1.16217	0.581086	+	0.813842i	$0.302628\pi$
$252$	−1603.79	−0.400909
$253$	1952.78	0.485257
$254$	−3393.61	−0.838322
$255$	−795.061	−0.195250
$256$	−1626.56	−0.397109
$257$	6233.98	1.51309	0.756547	−	0.653939i	$-0.226885\pi$
$257$	6233.98	1.51309	0.756547	+	0.653939i	$0.226885\pi$
$258$	4483.88	1.08199
$259$	1547.94	0.371368
$260$	184.347	0.0439719
$261$	2065.10	0.489756
$262$	958.949	0.226122
$263$	1289.39	0.302308	0.151154	−	0.988510i	$-0.451701\pi$
$263$	1289.39	0.302308	0.151154	+	0.988510i	$0.451701\pi$
$264$	−1544.63	−0.360096
$265$	−886.109	−0.205408
$266$	−94.9538	−0.0218872
$267$	12517.8	2.86920
$268$	−3139.84	−0.715658
$269$	2310.16	0.523616	0.261808	−	0.965120i	$-0.415681\pi$
$269$	2310.16	0.523616	0.261808	+	0.965120i	$0.415681\pi$
$270$	−177.598	−0.0400307
$271$	7584.99	1.70020	0.850102	−	0.526619i	$-0.176540\pi$
$271$	7584.99	1.70020	0.850102	+	0.526619i	$0.176540\pi$
$272$	−1370.87	−0.305592
$273$	−709.136	−0.157212
$274$	1918.78	0.423058
$275$	−1321.18	−0.289710
$276$	9096.89	1.98394
$277$	−339.509	−0.0736431	−0.0368215	−	0.999322i	$-0.511723\pi$
$277$	−339.509	−0.0736431	−0.0368215	+	0.999322i	$0.511723\pi$
$278$	−2096.34	−0.452267
$279$	6181.85	1.32651
$280$	−275.486	−0.0587979
$281$	−4829.29	−1.02524	−0.512618	−	0.858617i	$-0.671324\pi$
$281$	−4829.29	−1.02524	−0.512618	+	0.858617i	$0.671324\pi$
$282$	−2840.33	−0.599785
$283$	−2858.99	−0.600527	−0.300264	−	0.953856i	$-0.597075\pi$
$283$	−2858.99	−0.600527	−0.300264	+	0.953856i	$0.597075\pi$
$284$	−1317.29	−0.275235
$285$	192.886	0.0400897
$286$	−173.343	−0.0358391
$287$	−2153.94	−0.443007
$288$	−6326.63	−1.29445
$289$	−2838.71	−0.577796
$290$	158.918	0.0321793
$291$	7818.01	1.57491
$292$	−5752.07	−1.15279
$293$	8749.66	1.74458	0.872288	−	0.488993i	$-0.162636\pi$
$293$	8749.66	1.74458	0.872288	+	0.488993i	$0.162636\pi$
$294$	474.761	0.0941789
$295$	−36.1465	−0.00713399
$296$	3934.51	0.772596
$297$	−719.423	−0.140556
$298$	−2240.74	−0.435580
$299$	2278.73	0.440744
$300$	−6154.64	−1.18446
$301$	3239.46	0.620331
$302$	1596.18	0.304139
$303$	3917.70	0.742791
$304$	332.579	0.0627458
$305$	−1127.07	−0.211593
$306$	1973.01	0.368592
$307$	−3970.00	−0.738045	−0.369023	−	0.929420i	$-0.620308\pi$
$307$	−3970.00	−0.738045	−0.369023	+	0.929420i	$0.620308\pi$
$308$	−499.948	−0.0924909
$309$	−2269.81	−0.417880
$310$	475.721	0.0871585
$311$	2412.36	0.439847	0.219923	−	0.975517i	$-0.429419\pi$
$311$	2412.36	0.439847	0.219923	+	0.975517i	$0.429419\pi$
$312$	−1802.46	−0.327064
$313$	−6809.58	−1.22971	−0.614857	−	0.788639i	$-0.710786\pi$
$313$	−6809.58	−1.22971	−0.614857	+	0.788639i	$0.710786\pi$
$314$	−2539.15	−0.456345
$315$	−546.360	−0.0977267
$316$	−1879.84	−0.334650
$317$	−10686.8	−1.89347	−0.946733	−	0.322021i	$-0.895638\pi$
$317$	−10686.8	−1.89347	−0.946733	+	0.322021i	$0.895638\pi$
$318$	3881.49	0.684475
$319$	643.753	0.112988
$320$	45.7587	0.00799371
$321$	10341.4	1.79814
$322$	−1525.59	−0.264031
$323$	−503.232	−0.0866892
$324$	2834.64	0.486050
$325$	−1541.71	−0.263135
$326$	−1796.90	−0.305280
$327$	9682.06	1.63737
$328$	−5474.81	−0.921634
$329$	−2052.05	−0.343870
$330$	−235.743	−0.0393249
$331$	3924.27	0.651654	0.325827	−	0.945429i	$-0.394357\pi$
$331$	3924.27	0.651654	0.325827	+	0.945429i	$0.394357\pi$
$332$	−691.989	−0.114391
$333$	7803.16	1.28412
$334$	−4022.10	−0.658920
$335$	−1069.65	−0.174451
$336$	−1662.87	−0.269991
$337$	−4292.08	−0.693783	−0.346891	−	0.937905i	$-0.612763\pi$
$337$	−4292.08	−0.693783	−0.346891	+	0.937905i	$0.612763\pi$
$338$	2494.90	0.401494
$339$	−13555.1	−2.17172
$340$	−654.088	−0.104332
$341$	1927.07	0.306031
$342$	−478.661	−0.0756814
$343$	343.000	0.0539949
$344$	8233.96	1.29054
$345$	3099.03	0.483612
$346$	977.153	0.151827
$347$	−1481.08	−0.229131	−0.114565	−	0.993416i	$-0.536548\pi$
$347$	−1481.08	−0.229131	−0.114565	+	0.993416i	$0.536548\pi$
$348$	2998.88	0.461945
$349$	8449.08	1.29590	0.647949	−	0.761683i	$-0.275627\pi$
$349$	8449.08	1.29590	0.647949	+	0.761683i	$0.275627\pi$
$350$	1032.16	0.157633
$351$	−839.508	−0.127663
$352$	−1972.20	−0.298633
$353$	−4430.47	−0.668017	−0.334008	−	0.942570i	$-0.608401\pi$
$353$	−4430.47	−0.668017	−0.334008	+	0.942570i	$0.608401\pi$
$354$	158.335	0.0237723
$355$	−448.760	−0.0670921
$356$	10298.3	1.53317
$357$	2516.12	0.373017
$358$	−2009.08	−0.296601
$359$	1340.69	0.197101	0.0985503	−	0.995132i	$-0.468579\pi$
$359$	1340.69	0.197101	0.0985503	+	0.995132i	$0.468579\pi$
$360$	−1388.72	−0.203311
$361$	−6736.91	−0.982201
$362$	−4457.97	−0.647254
$363$	−954.957	−0.138078
$364$	−583.399	−0.0840067
$365$	−1959.55	−0.281007
$366$	4936.99	0.705083
$367$	−10543.8	−1.49967	−0.749837	−	0.661623i	$-0.769868\pi$
$367$	−10543.8	−1.49967	−0.749837	+	0.661623i	$0.769868\pi$
$368$	5343.44	0.756919
$369$	−10858.0	−1.53183
$370$	600.488	0.0843726
$371$	2804.26	0.392425
$372$	8977.13	1.25119
$373$	2702.46	0.375142	0.187571	−	0.982251i	$-0.439938\pi$
$373$	2702.46	0.375142	0.187571	+	0.982251i	$0.439938\pi$
$374$	615.046	0.0850354
$375$	−4278.80	−0.589217
$376$	−5215.84	−0.715389
$377$	751.207	0.102624
$378$	562.043	0.0764772
$379$	2784.94	0.377447	0.188724	−	0.982030i	$-0.439565\pi$
$379$	2784.94	0.377447	0.188724	+	0.982030i	$0.439565\pi$
$380$	158.685	0.0214220
$381$	−21816.2	−2.93354
$382$	−256.194	−0.0343142
$383$	−9176.46	−1.22427	−0.612135	−	0.790753i	$-0.709689\pi$
$383$	−9176.46	−1.22427	−0.612135	+	0.790753i	$0.709689\pi$
$384$	−11520.5	−1.53099
$385$	−170.317	−0.0225459
$386$	−993.077	−0.130949
$387$	16330.1	2.14498
$388$	6431.80	0.841560
$389$	−3142.13	−0.409544	−0.204772	−	0.978810i	$-0.565645\pi$
$389$	−3142.13	−0.409544	−0.204772	+	0.978810i	$0.565645\pi$
$390$	−275.093	−0.0357176
$391$	−8085.27	−1.04575
$392$	871.826	0.112331
$393$	6164.73	0.791271
$394$	−5186.67	−0.663200
$395$	−640.405	−0.0815753
$396$	−2520.23	−0.319815
$397$	−11192.0	−1.41489	−0.707444	−	0.706769i	$-0.750152\pi$
$397$	−11192.0	−1.41489	−0.707444	+	0.706769i	$0.750152\pi$
$398$	−422.885	−0.0532596
$399$	−610.423	−0.0765899
$400$	−3615.19	−0.451899
$401$	−3429.07	−0.427031	−0.213515	−	0.976940i	$-0.568491\pi$
$401$	−3429.07	−0.427031	−0.213515	+	0.976940i	$0.568491\pi$
$402$	4685.45	0.581316
$403$	2248.73	0.277959
$404$	3223.05	0.396912
$405$	965.676	0.118481
$406$	−502.927	−0.0614774
$407$	2432.48	0.296250
$408$	6395.38	0.776026
$409$	12313.7	1.48869	0.744347	−	0.667793i	$-0.232761\pi$
$409$	12313.7	1.48869	0.744347	+	0.667793i	$0.232761\pi$
$410$	−835.571	−0.100649
$411$	12335.1	1.48041
$412$	−1867.35	−0.223295
$413$	114.392	0.0136292
$414$	−7690.49	−0.912964
$415$	−235.740	−0.0278843
$416$	−2301.40	−0.271239
$417$	−13476.6	−1.58262
$418$	−149.213	−0.0174599
$419$	−11674.8	−1.36122	−0.680609	−	0.732647i	$-0.738285\pi$
$419$	−11674.8	−1.36122	−0.680609	+	0.732647i	$0.738285\pi$
$420$	−793.411	−0.0921774
$421$	6294.54	0.728687	0.364344	−	0.931265i	$-0.381293\pi$
$421$	6294.54	0.728687	0.364344	+	0.931265i	$0.381293\pi$
$422$	7355.04	0.848430
$423$	−10344.4	−1.18903
$424$	7127.76	0.816403
$425$	5470.22	0.624340
$426$	1965.74	0.223569
$427$	3566.82	0.404240
$428$	8507.78	0.960838
$429$	−1114.36	−0.125412
$430$	1256.67	0.140935
$431$	−5110.96	−0.571198	−0.285599	−	0.958349i	$-0.592193\pi$
$431$	−5110.96	−0.571198	−0.285599	+	0.958349i	$0.592193\pi$
$432$	−1968.58	−0.219244
$433$	−6754.23	−0.749624	−0.374812	−	0.927101i	$-0.622293\pi$
$433$	−6754.23	−0.749624	−0.374812	+	0.927101i	$0.622293\pi$
$434$	−1505.51	−0.166513
$435$	1021.63	0.112605
$436$	7965.33	0.874932
$437$	1961.53	0.214720
$438$	8583.58	0.936391
$439$	−6529.02	−0.709825	−0.354912	−	0.934900i	$-0.615489\pi$
$439$	−6529.02	−0.709825	−0.354912	+	0.934900i	$0.615489\pi$
$440$	−432.906	−0.0469045
$441$	1729.06	0.186703
$442$	717.708	0.0772351
$443$	−1529.54	−0.164042	−0.0820211	−	0.996631i	$-0.526137\pi$
$443$	−1529.54	−0.164042	−0.0820211	+	0.996631i	$0.526137\pi$
$444$	11331.6	1.21120
$445$	3508.30	0.373729
$446$	2638.23	0.280098
$447$	−14404.9	−1.52422
$448$	−144.812	−0.0152717
$449$	9778.96	1.02783	0.513917	−	0.857840i	$-0.328194\pi$
$449$	9778.96	1.02783	0.513917	+	0.857840i	$0.328194\pi$
$450$	5203.12	0.545061
$451$	−3384.76	−0.353397
$452$	−11151.7	−1.16047
$453$	10261.3	1.06427
$454$	2538.19	0.262386
$455$	−198.746	−0.0204777
$456$	−1551.55	−0.159338
$457$	10614.0	1.08643	0.543217	−	0.839592i	$-0.317206\pi$
$457$	10614.0	1.08643	0.543217	+	0.839592i	$0.317206\pi$
$458$	7850.58	0.800946
$459$	2978.70	0.302905
$460$	2549.54	0.258420
$461$	7536.27	0.761386	0.380693	−	0.924701i	$-0.375685\pi$
$461$	7536.27	0.761386	0.380693	+	0.924701i	$0.375685\pi$
$462$	746.052	0.0751288
$463$	−3945.44	−0.396026	−0.198013	−	0.980199i	$-0.563449\pi$
$463$	−3945.44	−0.396026	−0.198013	+	0.980199i	$0.563449\pi$
$464$	1761.52	0.176242
$465$	3058.23	0.304994
$466$	6557.37	0.651855
$467$	9456.18	0.937002	0.468501	−	0.883463i	$-0.344794\pi$
$467$	9456.18	0.937002	0.468501	+	0.883463i	$0.344794\pi$
$468$	−2940.91	−0.290478
$469$	3385.10	0.333282
$470$	−796.046	−0.0781253
$471$	−16323.2	−1.59689
$472$	290.758	0.0283543
$473$	5090.59	0.494853
$474$	2805.21	0.271830
$475$	−1327.10	−0.128193
$476$	2069.98	0.199323
$477$	14136.2	1.35692
$478$	5310.24	0.508127
$479$	−2775.63	−0.264764	−0.132382	−	0.991199i	$-0.542263\pi$
$479$	−2775.63	−0.264764	−0.132382	+	0.991199i	$0.542263\pi$
$480$	−3129.86	−0.297621
$481$	2838.51	0.269074
$482$	−8127.85	−0.768078
$483$	−9807.46	−0.923923
$484$	−785.633	−0.0737822
$485$	2191.12	0.205141
$486$	−6397.90	−0.597150
$487$	20902.4	1.94492	0.972461	−	0.233066i	$-0.0748759\pi$
$487$	20902.4	1.94492	0.972461	+	0.233066i	$0.0748759\pi$
$488$	9066.02	0.840983
$489$	−11551.6	−1.06827
$490$	133.059	0.0122673
$491$	−7832.44	−0.719904	−0.359952	−	0.932971i	$-0.617207\pi$
$491$	−7832.44	−0.719904	−0.359952	+	0.932971i	$0.617207\pi$
$492$	−15767.7	−1.44484
$493$	−2665.39	−0.243495
$494$	−174.120	−0.0158583
$495$	−858.566	−0.0779590
$496$	5273.10	0.477357
$497$	1420.18	0.128177
$498$	1032.63	0.0929179
$499$	−11585.6	−1.03936	−0.519681	−	0.854360i	$-0.673949\pi$
$499$	−11585.6	−1.03936	−0.519681	+	0.854360i	$0.673949\pi$
$500$	−3520.13	−0.314850
$501$	−25856.6	−2.30576
$502$	−5673.63	−0.504435
$503$	−168.525	−0.0149387	−0.00746934	−	0.999972i	$-0.502378\pi$
$503$	−168.525	−0.0149387	−0.00746934	+	0.999972i	$0.502378\pi$
$504$	4394.86	0.388418
$505$	1097.99	0.0967526
$506$	−2397.36	−0.210624
$507$	16038.8	1.40495
$508$	−17948.0	−1.56755
$509$	−5404.59	−0.470637	−0.235319	−	0.971918i	$-0.575613\pi$
$509$	−5404.59	−0.470637	−0.235319	+	0.971918i	$0.575613\pi$
$510$	976.069	0.0847472
$511$	6201.37	0.536854
$512$	−9680.94	−0.835628
$513$	−722.646	−0.0621942
$514$	−7653.25	−0.656752
$515$	−636.149	−0.0544312
$516$	23714.2	2.02318
$517$	−3224.66	−0.274314
$518$	−1900.36	−0.161191
$519$	6281.75	0.531288
$520$	−505.166	−0.0426019
$521$	5372.28	0.451754	0.225877	−	0.974156i	$-0.427475\pi$
$521$	5372.28	0.451754	0.225877	+	0.974156i	$0.427475\pi$
$522$	−2535.25	−0.212576
$523$	18418.3	1.53991	0.769957	−	0.638096i	$-0.220278\pi$
$523$	18418.3	1.53991	0.769957	+	0.638096i	$0.220278\pi$
$524$	5071.66	0.422818
$525$	6635.39	0.551604
$526$	−1582.94	−0.131215
$527$	−7978.83	−0.659513
$528$	−2613.08	−0.215378
$529$	19348.2	1.59022
$530$	1087.85	0.0891566
$531$	576.650	0.0471270
$532$	−502.188	−0.0409260
$533$	−3949.74	−0.320980
$534$	−15367.7	−1.24536
$535$	2898.34	0.234217
$536$	8604.12	0.693361
$537$	−12915.6	−1.03789
$538$	−2836.10	−0.227273
$539$	539.000	0.0430730
$540$	−939.277	−0.0748519
$541$	2953.93	0.234749	0.117375	−	0.993088i	$-0.462552\pi$
$541$	2953.93	0.234749	0.117375	+	0.993088i	$0.462552\pi$
$542$	−9311.83	−0.737965
$543$	−28658.7	−2.26494
$544$	8165.70	0.643569
$545$	2713.54	0.213276
$546$	870.582	0.0682372
$547$	13476.6	1.05341	0.526706	−	0.850048i	$-0.323427\pi$
$547$	13476.6	1.05341	0.526706	+	0.850048i	$0.323427\pi$
$548$	10148.0	0.791059
$549$	17980.3	1.39778
$550$	1621.97	0.125747
$551$	646.637	0.0499958
$552$	−24928.3	−1.92213
$553$	2026.68	0.155847
$554$	416.804	0.0319644
$555$	3860.31	0.295245
$556$	−11087.1	−0.845677
$557$	8327.77	0.633499	0.316749	−	0.948509i	$-0.397409\pi$
$557$	8327.77	0.633499	0.316749	+	0.948509i	$0.397409\pi$
$558$	−7589.25	−0.575768
$559$	5940.30	0.449460
$560$	−466.044	−0.0351677
$561$	3953.90	0.297565
$562$	5928.75	0.444999
$563$	4910.23	0.367569	0.183785	−	0.982967i	$-0.441165\pi$
$563$	4910.23	0.367569	0.183785	+	0.982967i	$0.441165\pi$
$564$	−15021.9	−1.12151
$565$	−3799.03	−0.282879
$566$	3509.88	0.260656
$567$	−3056.06	−0.226354
$568$	3609.77	0.266660
$569$	−9451.76	−0.696377	−0.348188	−	0.937425i	$-0.613203\pi$
$569$	−9451.76	−0.696377	−0.348188	+	0.937425i	$0.613203\pi$
$570$	−236.799	−0.0174007
$571$	−9705.19	−0.711295	−0.355648	−	0.934620i	$-0.615740\pi$
$571$	−9705.19	−0.711295	−0.355648	+	0.934620i	$0.615740\pi$
$572$	−916.770	−0.0670141
$573$	−1646.98	−0.120076
$574$	2644.32	0.192285
$575$	−21322.1	−1.54642
$576$	−729.994	−0.0528063
$577$	2933.73	0.211669	0.105834	−	0.994384i	$-0.466249\pi$
$577$	2933.73	0.211669	0.105834	+	0.994384i	$0.466249\pi$
$578$	3484.99	0.250790
$579$	−6384.12	−0.458230
$580$	840.482	0.0601709
$581$	746.041	0.0532720
$582$	−9597.91	−0.683584
$583$	4406.69	0.313047
$584$	15762.4	1.11687
$585$	−1001.88	−0.0708077
$586$	−10741.7	−0.757225
$587$	−23344.2	−1.64143	−0.820714	−	0.571339i	$-0.806424\pi$
$587$	−23344.2	−1.64143	−0.820714	+	0.571339i	$0.806424\pi$
$588$	2510.90	0.176101
$589$	1935.70	0.135415
$590$	44.3758	0.00309648
$591$	−33343.2	−2.32074
$592$	6656.07	0.462099
$593$	26704.7	1.84929	0.924645	−	0.380830i	$-0.124362\pi$
$593$	26704.7	1.84929	0.924645	+	0.380830i	$0.124362\pi$
$594$	883.211	0.0610077
$595$	705.180	0.0485875
$596$	−11850.8	−0.814473
$597$	−2718.57	−0.186372
$598$	−2797.52	−0.191303
$599$	−1767.91	−0.120593	−0.0602963	−	0.998181i	$-0.519205\pi$
$599$	−1767.91	−0.120593	−0.0602963	+	0.998181i	$0.519205\pi$
$600$	16865.6	1.14756
$601$	−3390.95	−0.230149	−0.115075	−	0.993357i	$-0.536711\pi$
$601$	−3390.95	−0.230149	−0.115075	+	0.993357i	$0.536711\pi$
$602$	−3976.98	−0.269252
$603$	17064.2	1.15242
$604$	8441.83	0.568697
$605$	−267.641	−0.0179854
$606$	−4809.62	−0.322405
$607$	−12925.9	−0.864323	−0.432162	−	0.901796i	$-0.642249\pi$
$607$	−12925.9	−0.864323	−0.432162	+	0.901796i	$0.642249\pi$
$608$	−1981.04	−0.132141
$609$	−3233.13	−0.215128
$610$	1383.66	0.0918409
$611$	−3762.91	−0.249151
$612$	10434.8	0.689216
$613$	12339.8	0.813049	0.406525	−	0.913640i	$-0.366741\pi$
$613$	12339.8	0.813049	0.406525	+	0.913640i	$0.366741\pi$
$614$	4873.83	0.320345
$615$	−5371.57	−0.352200
$616$	1370.01	0.0896093
$617$	−6235.47	−0.406857	−0.203428	−	0.979090i	$-0.565208\pi$
$617$	−6235.47	−0.406857	−0.203428	+	0.979090i	$0.565208\pi$
$618$	2786.57	0.181379
$619$	9830.39	0.638314	0.319157	−	0.947702i	$-0.396600\pi$
$619$	9830.39	0.638314	0.319157	+	0.947702i	$0.396600\pi$
$620$	2515.98	0.162974
$621$	−11610.5	−0.750264
$622$	−2961.57	−0.190913
$623$	−11102.7	−0.713996
$624$	−3049.25	−0.195621
$625$	13814.2	0.884111
$626$	8359.89	0.533752
$627$	−959.236	−0.0610976
$628$	−13428.9	−0.853302
$629$	−10071.4	−0.638433
$630$	670.748	0.0424178
$631$	−13040.9	−0.822742	−0.411371	−	0.911468i	$-0.634950\pi$
$631$	−13040.9	−0.822742	−0.411371	+	0.911468i	$0.634950\pi$
$632$	5151.34	0.324224
$633$	47282.8	2.96891
$634$	13119.8	0.821850
$635$	−6114.33	−0.382110
$636$	20528.3	1.27987
$637$	628.969	0.0391219
$638$	−790.313	−0.0490420
$639$	7159.13	0.443210
$640$	−3228.78	−0.199420
$641$	−8274.75	−0.509880	−0.254940	−	0.966957i	$-0.582056\pi$
$641$	−8274.75	−0.509880	−0.254940	+	0.966957i	$0.582056\pi$
$642$	−12695.8	−0.780472
$643$	−2407.55	−0.147659	−0.0738294	−	0.997271i	$-0.523522\pi$
$643$	−2407.55	−0.147659	−0.0738294	+	0.997271i	$0.523522\pi$
$644$	−8068.50	−0.493701
$645$	8078.69	0.493175
$646$	617.801	0.0376270
$647$	−24597.2	−1.49462	−0.747309	−	0.664477i	$-0.768654\pi$
$647$	−24597.2	−1.49462	−0.747309	+	0.664477i	$0.768654\pi$
$648$	−7767.79	−0.470907
$649$	179.759	0.0108724
$650$	1892.71	0.114212
$651$	−9678.35	−0.582679
$652$	−9503.39	−0.570831
$653$	6681.60	0.400415	0.200208	−	0.979753i	$-0.435838\pi$
$653$	6681.60	0.400415	0.200208	+	0.979753i	$0.435838\pi$
$654$	−11886.3	−0.710692
$655$	1727.76	0.103067
$656$	−9261.83	−0.551240
$657$	31261.0	1.85633
$658$	2519.24	0.149255
$659$	−8185.27	−0.483844	−0.241922	−	0.970296i	$-0.577778\pi$
$659$	−8185.27	−0.483844	−0.241922	+	0.970296i	$0.577778\pi$
$660$	−1246.79	−0.0735321
$661$	−5645.12	−0.332178	−0.166089	−	0.986111i	$-0.553114\pi$
$661$	−5645.12	−0.332178	−0.166089	+	0.986111i	$0.553114\pi$
$662$	−4817.70	−0.282848
$663$	4613.88	0.270269
$664$	1896.26	0.110827
$665$	−171.080	−0.00997625
$666$	−9579.67	−0.557364
$667$	10389.3	0.603112
$668$	−21271.9	−1.23209
$669$	16960.2	0.980147
$670$	1313.17	0.0757196
$671$	5605.00	0.322472
$672$	9905.02	0.568593
$673$	−5422.21	−0.310566	−0.155283	−	0.987870i	$-0.549629\pi$
$673$	−5422.21	−0.310566	−0.155283	+	0.987870i	$0.549629\pi$
$674$	5269.24	0.301133
$675$	7855.28	0.447926
$676$	13195.0	0.750738
$677$	−21717.0	−1.23287	−0.616435	−	0.787406i	$-0.711424\pi$
$677$	−21717.0	−1.23287	−0.616435	+	0.787406i	$0.711424\pi$
$678$	16641.2	0.942626
$679$	−6934.19	−0.391915
$680$	1792.40	0.101082
$681$	16317.1	0.918168
$682$	−2365.80	−0.132831
$683$	5011.91	0.280784	0.140392	−	0.990096i	$-0.455164\pi$
$683$	5011.91	0.280784	0.140392	+	0.990096i	$0.455164\pi$
$684$	−2531.53	−0.141514
$685$	3457.11	0.192831
$686$	−421.089	−0.0234362
$687$	50468.4	2.80275
$688$	13929.5	0.771887
$689$	5142.25	0.284331
$690$	−3804.57	−0.209910
$691$	8199.88	0.451430	0.225715	−	0.974193i	$-0.427528\pi$
$691$	8199.88	0.451430	0.225715	+	0.974193i	$0.427528\pi$
$692$	5167.93	0.283895
$693$	2717.09	0.148938
$694$	1818.27	0.0994532
$695$	−3777.02	−0.206145
$696$	−8217.86	−0.447553
$697$	14014.3	0.761589
$698$	−10372.6	−0.562479
$699$	42154.9	2.28104
$700$	5458.87	0.294751
$701$	33113.7	1.78415	0.892073	−	0.451891i	$-0.149251\pi$
$701$	33113.7	1.78415	0.892073	+	0.451891i	$0.149251\pi$
$702$	1030.64	0.0554114
$703$	2443.38	0.131086
$704$	−227.561	−0.0121826
$705$	−5117.49	−0.273384
$706$	5439.13	0.289950
$707$	−3474.80	−0.184842
$708$	837.396	0.0444510
$709$	−8338.06	−0.441668	−0.220834	−	0.975311i	$-0.570878\pi$
$709$	−8338.06	−0.441668	−0.220834	+	0.975311i	$0.570878\pi$
$710$	550.927	0.0291210
$711$	10216.5	0.538885
$712$	−28220.4	−1.48540
$713$	31100.3	1.63354
$714$	−3088.95	−0.161906
$715$	−312.315	−0.0163356
$716$	−10625.5	−0.554602
$717$	34137.6	1.77809
$718$	−1645.92	−0.0855506
$719$	13316.7	0.690724	0.345362	−	0.938470i	$-0.387756\pi$
$719$	13316.7	0.690724	0.345362	+	0.938470i	$0.387756\pi$
$720$	−2349.32	−0.121603
$721$	2013.21	0.103989
$722$	8270.68	0.426320
$723$	−52250.9	−2.68774
$724$	−23577.2	−1.21027
$725$	−7029.05	−0.360072
$726$	1172.37	0.0599320
$727$	−14439.2	−0.736616	−0.368308	−	0.929704i	$-0.620063\pi$
$727$	−14439.2	−0.736616	−0.368308	+	0.929704i	$0.620063\pi$
$728$	1598.69	0.0813894
$729$	−29342.1	−1.49073
$730$	2405.68	0.121970
$731$	−21077.0	−1.06643
$732$	26110.6	1.31841
$733$	−26816.8	−1.35130	−0.675650	−	0.737223i	$-0.736137\pi$
$733$	−26816.8	−1.35130	−0.675650	+	0.737223i	$0.736137\pi$
$734$	12944.2	0.650926
$735$	855.386	0.0429270
$736$	−31828.7	−1.59405
$737$	5319.44	0.265867
$738$	13330.0	0.664882
$739$	15313.3	0.762257	0.381129	−	0.924522i	$-0.375536\pi$
$739$	15313.3	0.762257	0.381129	+	0.924522i	$0.375536\pi$
$740$	3175.84	0.157765
$741$	−1119.35	−0.0554930
$742$	−3442.69	−0.170330
$743$	4586.65	0.226471	0.113235	−	0.993568i	$-0.463879\pi$
$743$	4586.65	0.226471	0.113235	+	0.993568i	$0.463879\pi$
$744$	−24600.1	−1.21221
$745$	−4037.19	−0.198539
$746$	−3317.72	−0.162829
$747$	3760.79	0.184203
$748$	3252.83	0.159004
$749$	−9172.33	−0.447463
$750$	5252.94	0.255747
$751$	20980.9	1.01945	0.509724	−	0.860338i	$-0.329748\pi$
$751$	20980.9	1.01945	0.509724	+	0.860338i	$0.329748\pi$
$752$	−8823.72	−0.427883
$753$	−36473.6	−1.76517
$754$	−922.231	−0.0445434
$755$	2875.87	0.138627
$756$	2972.52	0.143002
$757$	2320.42	0.111410	0.0557049	−	0.998447i	$-0.482259\pi$
$757$	2320.42	0.111410	0.0557049	+	0.998447i	$0.482259\pi$
$758$	−3418.97	−0.163829
$759$	−15411.7	−0.737036
$760$	−434.846	−0.0207546
$761$	15977.5	0.761084	0.380542	−	0.924764i	$-0.375737\pi$
$761$	15977.5	0.761084	0.380542	+	0.924764i	$0.375737\pi$
$762$	26783.0	1.27329
$763$	−8587.52	−0.407456
$764$	−1354.95	−0.0641629
$765$	3554.80	0.168006
$766$	11265.6	0.531389
$767$	209.764	0.00987503
$768$	12837.1	0.603151
$769$	−5354.46	−0.251088	−0.125544	−	0.992088i	$-0.540068\pi$
$769$	−5354.46	−0.251088	−0.125544	+	0.992088i	$0.540068\pi$
$770$	209.092	0.00978593
$771$	−49199.9	−2.29817
$772$	−5252.15	−0.244856
$773$	−36773.3	−1.71105	−0.855527	−	0.517758i	$-0.826767\pi$
$773$	−36773.3	−1.71105	−0.855527	+	0.517758i	$0.826767\pi$
$774$	−20047.9	−0.931017
$775$	−21041.4	−0.975264
$776$	−17625.1	−0.815340
$777$	−12216.7	−0.564055
$778$	3857.49	0.177761
$779$	−3399.93	−0.156374
$780$	−1454.90	−0.0667870
$781$	2231.72	0.102250
$782$	9926.01	0.453905
$783$	−3827.53	−0.174693
$784$	1474.88	0.0671867
$785$	−4574.83	−0.208003
$786$	−7568.23	−0.343447
$787$	38254.9	1.73271	0.866353	−	0.499432i	$-0.166458\pi$
$787$	38254.9	1.73271	0.866353	+	0.499432i	$0.166458\pi$
$788$	−27431.1	−1.24009
$789$	−10176.1	−0.459162
$790$	786.203	0.0354074
$791$	12022.7	0.540429
$792$	6906.21	0.309851
$793$	6540.58	0.292892
$794$	13740.0	0.614126
$795$	6993.35	0.311986
$796$	−2236.54	−0.0995881
$797$	−26969.6	−1.19864	−0.599318	−	0.800511i	$-0.704562\pi$
$797$	−26969.6	−1.19864	−0.599318	+	0.800511i	$0.704562\pi$
$798$	749.395	0.0332435
$799$	13351.4	0.591160
$800$	21534.2	0.951686
$801$	−55968.5	−2.46885
$802$	4209.75	0.185351
$803$	9745.01	0.428262
$804$	24780.3	1.08698
$805$	−2748.69	−0.120346
$806$	−2760.69	−0.120647
$807$	−18232.2	−0.795297
$808$	−8832.14	−0.384546
$809$	7996.42	0.347514	0.173757	−	0.984789i	$-0.444409\pi$
$809$	7996.42	0.347514	0.173757	+	0.984789i	$0.444409\pi$
$810$	−1185.53	−0.0514261
$811$	2150.99	0.0931337	0.0465669	−	0.998915i	$-0.485172\pi$
$811$	2150.99	0.0931337	0.0465669	+	0.998915i	$0.485172\pi$
$812$	−2659.86	−0.114954
$813$	−59862.3	−2.58236
$814$	−2986.27	−0.128586
$815$	−3237.51	−0.139147
$816$	10819.2	0.464150
$817$	5113.39	0.218966
$818$	−15117.2	−0.646160
$819$	3170.63	0.135276
$820$	−4419.14	−0.188199
$821$	−27882.7	−1.18528	−0.592638	−	0.805469i	$-0.701914\pi$
$821$	−27882.7	−1.18528	−0.592638	+	0.805469i	$0.701914\pi$
$822$	−15143.4	−0.642564
$823$	18462.7	0.781978	0.390989	−	0.920395i	$-0.372133\pi$
$823$	18462.7	0.781978	0.390989	+	0.920395i	$0.372133\pi$
$824$	5117.11	0.216339
$825$	10427.0	0.440028
$826$	−140.435	−0.00591570
$827$	28769.2	1.20968	0.604838	−	0.796348i	$-0.293238\pi$
$827$	28769.2	1.20968	0.604838	+	0.796348i	$0.293238\pi$
$828$	−40673.2	−1.70712
$829$	31112.2	1.30346	0.651732	−	0.758449i	$-0.274043\pi$
$829$	31112.2	1.30346	0.651732	+	0.758449i	$0.274043\pi$
$830$	289.409	0.0121031
$831$	2679.48	0.111853
$832$	−265.546	−0.0110651
$833$	−2231.67	−0.0928246
$834$	16544.8	0.686929
$835$	−7246.69	−0.300338
$836$	−789.153	−0.0326476
$837$	−11457.7	−0.473160
$838$	14332.7	0.590830
$839$	13494.6	0.555287	0.277643	−	0.960684i	$-0.410447\pi$
$839$	13494.6	0.555287	0.277643	+	0.960684i	$0.410447\pi$
$840$	2174.19	0.0893056
$841$	−20964.1	−0.859570
$842$	−7727.60	−0.316283
$843$	38113.7	1.55719
$844$	38899.1	1.58645
$845$	4495.12	0.183002
$846$	12699.4	0.516094
$847$	847.000	0.0343604
$848$	12058.2	0.488300
$849$	22563.7	0.912114
$850$	−6715.60	−0.270992
$851$	39257.0	1.58133
$852$	10396.3	0.418042
$853$	850.969	0.0341578	0.0170789	−	0.999854i	$-0.494563\pi$
$853$	850.969	0.0341578	0.0170789	+	0.999854i	$0.494563\pi$
$854$	−4378.86	−0.175459
$855$	−862.413	−0.0344958
$856$	−23313.9	−0.930903
$857$	47221.3	1.88220	0.941102	−	0.338122i	$-0.109792\pi$
$857$	47221.3	1.88220	0.941102	+	0.338122i	$0.109792\pi$
$858$	1368.06	0.0544344
$859$	32876.5	1.30586	0.652929	−	0.757419i	$-0.273540\pi$
$859$	32876.5	1.30586	0.652929	+	0.757419i	$0.273540\pi$
$860$	6646.26	0.263530
$861$	16999.3	0.672864
$862$	6274.55	0.247926
$863$	−33117.4	−1.30629	−0.653145	−	0.757232i	$-0.726551\pi$
$863$	−33117.4	−1.30629	−0.653145	+	0.757232i	$0.726551\pi$
$864$	11726.0	0.461721
$865$	1760.56	0.0692031
$866$	8291.93	0.325371
$867$	22403.7	0.877589
$868$	−7962.28	−0.311356
$869$	3184.78	0.124323
$870$	−1254.22	−0.0488758
$871$	6207.35	0.241479
$872$	−21827.5	−0.847673
$873$	−34955.2	−1.35516
$874$	−2408.10	−0.0931981
$875$	3795.09	0.146626
$876$	45396.5	1.75092
$877$	−16817.3	−0.647526	−0.323763	−	0.946138i	$-0.604948\pi$
$877$	−16817.3	−0.647526	−0.323763	+	0.946138i	$0.604948\pi$
$878$	8015.45	0.308096
$879$	−69054.1	−2.64976
$880$	−732.354	−0.0280542
$881$	−23372.2	−0.893792	−0.446896	−	0.894586i	$-0.647471\pi$
$881$	−23372.2	−0.893792	−0.446896	+	0.894586i	$0.647471\pi$
$882$	−2122.71	−0.0810377
$883$	−46228.1	−1.76183	−0.880916	−	0.473272i	$-0.843073\pi$
$883$	−46228.1	−1.76183	−0.880916	+	0.473272i	$0.843073\pi$
$884$	3795.79	0.144419
$885$	285.275	0.0108355
$886$	1877.76	0.0712017
$887$	36945.6	1.39855	0.699275	−	0.714853i	$-0.253506\pi$
$887$	36945.6	1.39855	0.699275	+	0.714853i	$0.253506\pi$
$888$	−31051.9	−1.17346
$889$	19349.9	0.730006
$890$	−4307.02	−0.162215
$891$	−4802.38	−0.180568
$892$	13953.0	0.523744
$893$	−3239.10	−0.121380
$894$	17684.4	0.661583
$895$	−3619.79	−0.135191
$896$	10218.1	0.380985
$897$	−17984.2	−0.669427
$898$	−12005.3	−0.446127
$899$	10252.5	0.380357
$900$	27518.1	1.01919
$901$	−18245.4	−0.674632
$902$	4155.36	0.153391
$903$	−25566.5	−0.942193
$904$	30559.0	1.12431
$905$	−8032.02	−0.295020
$906$	−12597.4	−0.461943
$907$	16331.0	0.597864	0.298932	−	0.954274i	$-0.403370\pi$
$907$	16331.0	0.597864	0.298932	+	0.954274i	$0.403370\pi$
$908$	13423.9	0.490625
$909$	−17516.4	−0.639146
$910$	243.994	0.00888826
$911$	−20518.3	−0.746213	−0.373107	−	0.927788i	$-0.621707\pi$
$911$	−20518.3	−0.746213	−0.373107	+	0.927788i	$0.621707\pi$
$912$	−2624.78	−0.0953019
$913$	1172.35	0.0424963
$914$	−13030.4	−0.471562
$915$	8895.07	0.321379
$916$	41519.9	1.49766
$917$	−5467.81	−0.196906
$918$	−3656.84	−0.131475
$919$	47280.4	1.69710	0.848550	−	0.529115i	$-0.177476\pi$
$919$	47280.4	1.69710	0.848550	+	0.529115i	$0.177476\pi$
$920$	−6986.52	−0.250368
$921$	31332.1	1.12098
$922$	−9252.02	−0.330476
$923$	2604.23	0.0928704
$924$	3945.70	0.140480
$925$	−26559.9	−0.944091
$926$	4843.68	0.171893
$927$	10148.6	0.359572
$928$	−10492.7	−0.371162
$929$	−35214.0	−1.24363	−0.621816	−	0.783163i	$-0.713605\pi$
$929$	−35214.0	−1.24363	−0.621816	+	0.783163i	$0.713605\pi$
$930$	−3754.49	−0.132381
$931$	541.415	0.0190592
$932$	34680.4	1.21888
$933$	−19038.8	−0.668064
$934$	−11609.0	−0.406701
$935$	1108.14	0.0387594
$936$	8058.99	0.281428
$937$	−53330.3	−1.85937	−0.929683	−	0.368360i	$-0.879919\pi$
$937$	−53330.3	−1.85937	−0.929683	+	0.368360i	$0.879919\pi$
$938$	−4155.77	−0.144659
$939$	53742.6	1.86776
$940$	−4210.10	−0.146083
$941$	−20304.7	−0.703416	−0.351708	−	0.936110i	$-0.614399\pi$
$941$	−20304.7	−0.703416	−0.351708	+	0.936110i	$0.614399\pi$
$942$	20039.5	0.693122
$943$	−54625.5	−1.88638
$944$	491.880	0.0169590
$945$	1012.64	0.0348586
$946$	−6249.54	−0.214789
$947$	−20902.4	−0.717252	−0.358626	−	0.933481i	$-0.616755\pi$
$947$	−20902.4	−0.717252	−0.358626	+	0.933481i	$0.616755\pi$
$948$	14836.1	0.508285
$949$	11371.6	0.388977
$950$	1629.24	0.0556415
$951$	84342.1	2.87590
$952$	−5672.39	−0.193113
$953$	44662.5	1.51811	0.759055	−	0.651027i	$-0.225661\pi$
$953$	44662.5	1.51811	0.759055	+	0.651027i	$0.225661\pi$
$954$	−17354.6	−0.588967
$955$	−461.590	−0.0156405
$956$	28084.6	0.950127
$957$	−5080.63	−0.171613
$958$	3407.55	0.114920
$959$	−10940.7	−0.368396
$960$	−361.137	−0.0121413
$961$	899.884	0.0302066
$962$	−3484.74	−0.116790
$963$	−46237.6	−1.54723
$964$	−42986.3	−1.43620
$965$	−1789.25	−0.0596869
$966$	12040.3	0.401025
$967$	−40483.6	−1.34629	−0.673146	−	0.739509i	$-0.735058\pi$
$967$	−40483.6	−1.34629	−0.673146	+	0.739509i	$0.735058\pi$
$968$	2152.88	0.0714835
$969$	3971.61	0.131668
$970$	−2689.96	−0.0890406
$971$	55314.7	1.82815	0.914075	−	0.405544i	$-0.132918\pi$
$971$	55314.7	1.82815	0.914075	+	0.405544i	$0.132918\pi$
$972$	−33837.0	−1.11659
$973$	11953.1	0.393832
$974$	−25661.1	−0.844184
$975$	12167.5	0.399664
$976$	15337.1	0.503002
$977$	37534.5	1.22910	0.614552	−	0.788876i	$-0.289337\pi$
$977$	37534.5	1.22910	0.614552	+	0.788876i	$0.289337\pi$
$978$	14181.5	0.463676
$979$	−17447.1	−0.569572
$980$	703.717	0.0229382
$981$	−43289.6	−1.40890
$982$	9615.61	0.312471
$983$	985.287	0.0319692	0.0159846	−	0.999872i	$-0.494912\pi$
$983$	985.287	0.0319692	0.0159846	+	0.999872i	$0.494912\pi$
$984$	43208.3	1.39983
$985$	−9344.93	−0.302289
$986$	3272.21	0.105688
$987$	16195.2	0.522290
$988$	−920.878	−0.0296529
$989$	82155.2	2.64144
$990$	1054.03	0.0338377
$991$	−9167.83	−0.293870	−0.146935	−	0.989146i	$-0.546941\pi$
$991$	−9167.83	−0.293870	−0.146935	+	0.989146i	$0.546941\pi$
$992$	−31409.7	−1.00530
$993$	−30971.2	−0.989769
$994$	−1743.51	−0.0556346
$995$	−761.921	−0.0242759
$996$	5461.32	0.173744
$997$	−21433.7	−0.680854	−0.340427	−	0.940271i	$-0.610572\pi$
$997$	−21433.7	−0.680854	−0.340427	+	0.940271i	$0.610572\pi$
$998$	14223.2	0.451131
$999$	−14462.7	−0.458036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.a.e.1.3	✓	5
3.2	odd	2		693.4.a.o.1.3		5
4.3	odd	2		1232.4.a.y.1.5		5
5.4	even	2		1925.4.a.r.1.3		5
7.6	odd	2		539.4.a.h.1.3		5
11.10	odd	2		847.4.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.3	✓	5	1.1	even	1	trivial
539.4.a.h.1.3		5	7.6	odd	2
693.4.a.o.1.3		5	3.2	odd	2
847.4.a.f.1.3		5	11.10	odd	2
1232.4.a.y.1.5		5	4.3	odd	2
1925.4.a.r.1.3		5	5.4	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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)

\(f(q)\)	\(=\)	\(q-1.22767 q^{2} -7.89221 q^{3} -6.49284 q^{4} -2.21191 q^{5} +9.68899 q^{6} +7.00000 q^{7} +17.7924 q^{8} +35.2869 q^{9} +2.71549 q^{10} +11.0000 q^{11} +51.2428 q^{12} +12.8361 q^{13} -8.59366 q^{14} +17.4569 q^{15} +30.0996 q^{16} -45.5444 q^{17} -43.3205 q^{18} +11.0493 q^{19} +14.3616 q^{20} -55.2454 q^{21} -13.5043 q^{22} +177.525 q^{23} -140.421 q^{24} -120.107 q^{25} -15.7584 q^{26} -65.4021 q^{27} -45.4499 q^{28} +58.5230 q^{29} -21.4312 q^{30} +175.188 q^{31} -179.291 q^{32} -86.8143 q^{33} +55.9132 q^{34} -15.4834 q^{35} -229.112 q^{36} +221.135 q^{37} -13.5648 q^{38} -101.305 q^{39} -39.3551 q^{40} -307.706 q^{41} +67.8229 q^{42} +462.781 q^{43} -71.4212 q^{44} -78.0515 q^{45} -217.942 q^{46} -293.151 q^{47} -237.552 q^{48} +49.0000 q^{49} +147.452 q^{50} +359.445 q^{51} -83.3427 q^{52} +400.608 q^{53} +80.2919 q^{54} -24.3310 q^{55} +124.547 q^{56} -87.2032 q^{57} -71.8467 q^{58} +16.3417 q^{59} -113.344 q^{60} +509.546 q^{61} -215.072 q^{62} +247.008 q^{63} -20.6874 q^{64} -28.3923 q^{65} +106.579 q^{66} +483.585 q^{67} +295.712 q^{68} -1401.07 q^{69} +19.0084 q^{70} +202.883 q^{71} +627.838 q^{72} +885.910 q^{73} -271.479 q^{74} +947.913 q^{75} -71.7412 q^{76} +77.0000 q^{77} +124.369 q^{78} +289.526 q^{79} -66.5777 q^{80} -436.580 q^{81} +377.760 q^{82} +106.577 q^{83} +358.700 q^{84} +100.740 q^{85} -568.140 q^{86} -461.876 q^{87} +195.716 q^{88} -1586.10 q^{89} +95.8211 q^{90} +89.8527 q^{91} -1152.64 q^{92} -1382.62 q^{93} +359.891 q^{94} -24.4400 q^{95} +1415.00 q^{96} -990.599 q^{97} -60.1556 q^{98} +388.156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9} - 10 q^{10} + 55 q^{11} + 24 q^{12} - 50 q^{13} + 7 q^{14} - 146 q^{15} + 433 q^{16} + 222 q^{17} + 245 q^{18} + 160 q^{19}+ \cdots + 693 q^{99}+O(q^{100}) \) 5 * q + q^2 + 2 * q^3 + 45 * q^4 - 24 * q^5 + 4 * q^6 + 35 * q^7 + 57 * q^8 + 63 * q^9 - 10 * q^10 + 55 * q^11 + 24 * q^12 - 50 * q^13 + 7 * q^14 - 146 * q^15 + 433 * q^16 + 222 * q^17 + 245 * q^18 + 160 * q^19 - 430 * q^20 + 14 * q^21 + 11 * q^22 + 54 * q^23 - 220 * q^24 + 125 * q^25 - 1026 * q^26 + 110 * q^27 + 315 * q^28 + 14 * q^29 - 1080 * q^30 - 34 * q^31 - 583 * q^32 + 22 * q^33 - 750 * q^34 - 168 * q^35 - 411 * q^36 + 1044 * q^37 - 156 * q^38 - 124 * q^39 + 1158 * q^40 - 114 * q^41 + 28 * q^42 + 672 * q^43 + 495 * q^44 - 530 * q^45 - 1224 * q^46 - 292 * q^47 - 1652 * q^48 + 245 * q^49 + 1143 * q^50 + 768 * q^51 - 914 * q^52 - 710 * q^53 + 1608 * q^54 - 264 * q^55 + 399 * q^56 + 2012 * q^57 - 810 * q^58 + 270 * q^59 - 3068 * q^60 + 138 * q^61 - 480 * q^62 + 441 * q^63 + 3001 * q^64 - 196 * q^65 + 44 * q^66 + 1942 * q^67 + 3130 * q^68 - 1306 * q^69 - 70 * q^70 - 278 * q^71 + 3565 * q^72 - 338 * q^73 + 462 * q^74 + 3960 * q^75 - 1332 * q^76 + 385 * q^77 + 280 * q^78 + 576 * q^79 - 1602 * q^80 - 1439 * q^81 + 1386 * q^82 + 1644 * q^83 + 168 * q^84 + 360 * q^85 + 2020 * q^86 - 1956 * q^87 + 627 * q^88 - 3656 * q^89 - 6162 * q^90 - 350 * q^91 - 748 * q^92 - 1442 * q^93 + 976 * q^94 - 1276 * q^95 - 4508 * q^96 + 692 * q^97 + 49 * q^98 + 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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