LMFDB - Embedded newform 161.4.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [161,4,Mod(1,161)] 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("161.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(161, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 161 = 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	161.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.49930751092$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 44x^{6} - 23x^{5} + 587x^{4} + 594x^{3} - 2430x^{2} - 3403x + 110 $ x^8 - 44x^6 - 23x^5 + 587x^4 + 594x^3 - 2430x^2 - 3403x + 110
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.91250$ of defining polynomial
Character	$\chi$	$=$	161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.91250 q^{2} +3.10924 q^{3} +0.482647 q^{4} +2.69280 q^{5} -9.05565 q^{6} -7.00000 q^{7} +21.8943 q^{8} -17.3326 q^{9} -7.84277 q^{10} -6.61362 q^{11} +1.50066 q^{12} +21.3608 q^{13} +20.3875 q^{14} +8.37256 q^{15} -67.6282 q^{16} -117.295 q^{17} +50.4813 q^{18} +92.4022 q^{19} +1.29967 q^{20} -21.7647 q^{21} +19.2621 q^{22} +23.0000 q^{23} +68.0746 q^{24} -117.749 q^{25} -62.2133 q^{26} -137.841 q^{27} -3.37853 q^{28} -137.861 q^{29} -24.3851 q^{30} -245.826 q^{31} +21.8129 q^{32} -20.5633 q^{33} +341.621 q^{34} -18.8496 q^{35} -8.36554 q^{36} -78.4997 q^{37} -269.121 q^{38} +66.4158 q^{39} +58.9569 q^{40} -226.452 q^{41} +63.3896 q^{42} -557.321 q^{43} -3.19204 q^{44} -46.6733 q^{45} -66.9875 q^{46} +278.354 q^{47} -210.272 q^{48} +49.0000 q^{49} +342.943 q^{50} -364.698 q^{51} +10.3097 q^{52} +114.979 q^{53} +401.461 q^{54} -17.8091 q^{55} -153.260 q^{56} +287.301 q^{57} +401.521 q^{58} +311.388 q^{59} +4.04099 q^{60} +449.956 q^{61} +715.967 q^{62} +121.328 q^{63} +477.496 q^{64} +57.5203 q^{65} +59.8906 q^{66} -881.349 q^{67} -56.6120 q^{68} +71.5125 q^{69} +54.8994 q^{70} +1040.80 q^{71} -379.485 q^{72} +83.8276 q^{73} +228.630 q^{74} -366.109 q^{75} +44.5976 q^{76} +46.2953 q^{77} -193.436 q^{78} -622.942 q^{79} -182.109 q^{80} +39.4012 q^{81} +659.540 q^{82} +1410.12 q^{83} -10.5046 q^{84} -315.852 q^{85} +1623.20 q^{86} -428.644 q^{87} -144.800 q^{88} +1240.47 q^{89} +135.936 q^{90} -149.526 q^{91} +11.1009 q^{92} -764.331 q^{93} -810.705 q^{94} +248.821 q^{95} +67.8214 q^{96} +727.667 q^{97} -142.712 q^{98} +114.631 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{3} + 24 q^{4} - 24 q^{5} - 41 q^{6} - 56 q^{7} - 69 q^{8} + 95 q^{9} - 30 q^{10} - 98 q^{11} - 131 q^{12} - 145 q^{13} - 232 q^{15} - 76 q^{16} - 96 q^{17} - 69 q^{18} - 226 q^{19} - 22 q^{20}+ \cdots - 1676 q^{99}+O(q^{100}) $ 8 * q - 3 * q^3 + 24 * q^4 - 24 * q^5 - 41 * q^6 - 56 * q^7 - 69 * q^8 + 95 * q^9 - 30 * q^10 - 98 * q^11 - 131 * q^12 - 145 * q^13 - 232 * q^15 - 76 * q^16 - 96 * q^17 - 69 * q^18 - 226 * q^19 - 22 * q^20 + 21 * q^21 - 98 * q^22 + 184 * q^23 - 216 * q^24 + 60 * q^25 - 185 * q^26 - 531 * q^27 - 168 * q^28 + 73 * q^29 - 416 * q^30 - 987 * q^31 + 118 * q^32 + 16 * q^33 - 846 * q^34 + 168 * q^35 - 69 * q^36 - 2 * q^37 - 454 * q^38 - 675 * q^39 - 640 * q^40 - 907 * q^41 + 287 * q^42 + 636 * q^43 + 126 * q^44 - 126 * q^45 + 421 * q^47 + 2165 * q^48 + 392 * q^49 + 2308 * q^50 - 408 * q^51 - 931 * q^52 + 138 * q^53 + 1717 * q^54 - 90 * q^55 + 483 * q^56 + 984 * q^57 + 1329 * q^58 - 924 * q^59 + 2642 * q^60 - 890 * q^61 + 2175 * q^62 - 665 * q^63 + 303 * q^64 - 392 * q^65 + 408 * q^66 - 1392 * q^67 + 2592 * q^68 - 69 * q^69 + 210 * q^70 + 259 * q^71 + 141 * q^72 - 297 * q^73 + 2314 * q^74 - 851 * q^75 - 428 * q^76 + 686 * q^77 + 6789 * q^78 - 1788 * q^79 - 2104 * q^80 + 2300 * q^81 - 219 * q^82 + 2114 * q^83 + 917 * q^84 + 528 * q^85 - 2360 * q^86 - 2441 * q^87 + 2144 * q^88 - 1040 * q^89 + 4872 * q^90 + 1015 * q^91 + 552 * q^92 - 621 * q^93 - 1009 * q^94 + 1268 * q^95 - 1163 * q^96 - 1452 * q^97 - 1676 * 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 $

$2$	−2.91250	−1.02972	−0.514862	−	0.857273i	$-0.672157\pi$
$2$	−2.91250	−1.02972	−0.514862	+	0.857273i	$0.672157\pi$
$3$	3.10924	0.598373	0.299187	−	0.954195i	$-0.403285\pi$
$3$	3.10924	0.598373	0.299187	+	0.954195i	$0.403285\pi$
$4$	0.482647	0.0603308
$5$	2.69280	0.240851	0.120426	−	0.992722i	$-0.461574\pi$
$5$	2.69280	0.240851	0.120426	+	0.992722i	$0.461574\pi$
$6$	−9.05565	−0.616159
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	21.8943	0.967600
$9$	−17.3326	−0.641949
$10$	−7.84277	−0.248010
$11$	−6.61362	−0.181280	−0.0906400	−	0.995884i	$-0.528891\pi$
$11$	−6.61362	−0.181280	−0.0906400	+	0.995884i	$0.528891\pi$
$12$	1.50066	0.0361004
$13$	21.3608	0.455724	0.227862	−	0.973693i	$-0.426826\pi$
$13$	21.3608	0.455724	0.227862	+	0.973693i	$0.426826\pi$
$14$	20.3875	0.389199
$15$	8.37256	0.144119
$16$	−67.6282	−1.05669
$17$	−117.295	−1.67342	−0.836712	−	0.547643i	$-0.815525\pi$
$17$	−117.295	−1.67342	−0.836712	+	0.547643i	$0.815525\pi$
$18$	50.4813	0.661030
$19$	92.4022	1.11571	0.557856	−	0.829938i	$-0.311624\pi$
$19$	92.4022	1.11571	0.557856	+	0.829938i	$0.311624\pi$
$20$	1.29967	0.0145308
$21$	−21.7647	−0.226164
$22$	19.2621	0.186668
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	68.0746	0.578986
$25$	−117.749	−0.941991
$26$	−62.2133	−0.469270
$27$	−137.841	−0.982499
$28$	−3.37853	−0.0228029
$29$	−137.861	−0.882766	−0.441383	−	0.897319i	$-0.645512\pi$
$29$	−137.861	−0.882766	−0.441383	+	0.897319i	$0.645512\pi$
$30$	−24.3851	−0.148403
$31$	−245.826	−1.42425	−0.712123	−	0.702055i	$-0.752266\pi$
$31$	−245.826	−1.42425	−0.712123	+	0.702055i	$0.752266\pi$
$32$	21.8129	0.120500
$33$	−20.5633	−0.108473
$34$	341.621	1.72316
$35$	−18.8496	−0.0910332
$36$	−8.36554	−0.0387293
$37$	−78.4997	−0.348791	−0.174396	−	0.984676i	$-0.555797\pi$
$37$	−78.4997	−0.348791	−0.174396	+	0.984676i	$0.555797\pi$
$38$	−269.121	−1.14887
$39$	66.4158	0.272693
$40$	58.9569	0.233048
$41$	−226.452	−0.862581	−0.431290	−	0.902213i	$-0.641942\pi$
$41$	−226.452	−0.862581	−0.431290	+	0.902213i	$0.641942\pi$
$42$	63.3896	0.232886
$43$	−557.321	−1.97653	−0.988263	−	0.152762i	$-0.951183\pi$
$43$	−557.321	−1.97653	−0.988263	+	0.152762i	$0.951183\pi$
$44$	−3.19204	−0.0109368
$45$	−46.6733	−0.154614
$46$	−66.9875	−0.214712
$47$	278.354	0.863874	0.431937	−	0.901904i	$-0.357830\pi$
$47$	278.354	0.863874	0.431937	+	0.901904i	$0.357830\pi$
$48$	−210.272	−0.632296
$49$	49.0000	0.142857
$50$	342.943	0.969990
$51$	−364.698	−1.00133
$52$	10.3097	0.0274942
$53$	114.979	0.297993	0.148997	−	0.988838i	$-0.452396\pi$
$53$	114.979	0.297993	0.148997	+	0.988838i	$0.452396\pi$
$54$	401.461	1.01170
$55$	−17.8091	−0.0436615
$56$	−153.260	−0.365718
$57$	287.301	0.667612
$58$	401.521	0.909005
$59$	311.388	0.687106	0.343553	−	0.939133i	$-0.388370\pi$
$59$	311.388	0.687106	0.343553	+	0.939133i	$0.388370\pi$
$60$	4.04099	0.00869482
$61$	449.956	0.944442	0.472221	−	0.881480i	$-0.343452\pi$
$61$	449.956	0.944442	0.472221	+	0.881480i	$0.343452\pi$
$62$	715.967	1.46658
$63$	121.328	0.242634
$64$	477.496	0.932609
$65$	57.5203	0.109762
$66$	59.8906	0.111697
$67$	−881.349	−1.60707	−0.803537	−	0.595255i	$-0.797051\pi$
$67$	−881.349	−1.60707	−0.803537	+	0.595255i	$0.797051\pi$
$68$	−56.6120	−0.100959
$69$	71.5125	0.124769
$70$	54.8994	0.0937391
$71$	1040.80	1.73971	0.869857	−	0.493303i	$-0.164211\pi$
$71$	1040.80	1.73971	0.869857	+	0.493303i	$0.164211\pi$
$72$	−379.485	−0.621150
$73$	83.8276	0.134401	0.0672006	−	0.997739i	$-0.478593\pi$
$73$	83.8276	0.134401	0.0672006	+	0.997739i	$0.478593\pi$
$74$	228.630	0.359159
$75$	−366.109	−0.563662
$76$	44.5976	0.0673118
$77$	46.2953	0.0685174
$78$	−193.436	−0.280799
$79$	−622.942	−0.887170	−0.443585	−	0.896232i	$-0.646294\pi$
$79$	−622.942	−0.887170	−0.443585	+	0.896232i	$0.646294\pi$
$80$	−182.109	−0.254505
$81$	39.4012	0.0540483
$82$	659.540	0.888220
$83$	1410.12	1.86483	0.932417	−	0.361385i	$-0.117696\pi$
$83$	1410.12	1.86483	0.932417	+	0.361385i	$0.117696\pi$
$84$	−10.5046	−0.0136447
$85$	−315.852	−0.403046
$86$	1623.20	2.03528
$87$	−428.644	−0.528224
$88$	−144.800	−0.175406
$89$	1240.47	1.47742	0.738708	−	0.674025i	$-0.235436\pi$
$89$	1240.47	1.47742	0.738708	+	0.674025i	$0.235436\pi$
$90$	135.936	0.159210
$91$	−149.526	−0.172248
$92$	11.1009	0.0125798
$93$	−764.331	−0.852231
$94$	−810.705	−0.889551
$95$	248.821	0.268721
$96$	67.8214	0.0721041
$97$	727.667	0.761684	0.380842	−	0.924640i	$-0.375634\pi$
$97$	727.667	0.761684	0.380842	+	0.924640i	$0.375634\pi$
$98$	−142.712	−0.147103
$99$	114.631	0.116373
$100$	−56.8311	−0.0568311
$101$	−1658.97	−1.63440	−0.817198	−	0.576357i	$-0.804474\pi$
$101$	−1658.97	−1.63440	−0.817198	+	0.576357i	$0.804474\pi$
$102$	1062.18	1.03110
$103$	−1045.68	−1.00033	−0.500163	−	0.865931i	$-0.666727\pi$
$103$	−1045.68	−1.00033	−0.500163	+	0.865931i	$0.666727\pi$
$104$	467.679	0.440959
$105$	−58.6079	−0.0544719
$106$	−334.877	−0.306851
$107$	−1074.22	−0.970552	−0.485276	−	0.874361i	$-0.661281\pi$
$107$	−1074.22	−0.970552	−0.485276	+	0.874361i	$0.661281\pi$
$108$	−66.5284	−0.0592750
$109$	−1081.53	−0.950382	−0.475191	−	0.879883i	$-0.657621\pi$
$109$	−1081.53	−0.950382	−0.475191	+	0.879883i	$0.657621\pi$
$110$	51.8691	0.0449593
$111$	−244.074	−0.208707
$112$	473.398	0.399392
$113$	237.825	0.197989	0.0989944	−	0.995088i	$-0.468437\pi$
$113$	237.825	0.197989	0.0989944	+	0.995088i	$0.468437\pi$
$114$	−836.763	−0.687456
$115$	61.9344	0.0502210
$116$	−66.5383	−0.0532580
$117$	−370.239	−0.292552
$118$	−906.917	−0.707529
$119$	821.065	0.632495
$120$	183.311	0.139449
$121$	−1287.26	−0.967138
$122$	−1310.50	−0.972515
$123$	−704.093	−0.516145
$124$	−118.647	−0.0859260
$125$	−653.674	−0.467731
$126$	−353.369	−0.249846
$127$	2634.80	1.84095	0.920476	−	0.390798i	$-0.127801\pi$
$127$	2634.80	1.84095	0.920476	+	0.390798i	$0.127801\pi$
$128$	−1565.21	−1.08083
$129$	−1732.84	−1.18270
$130$	−167.528	−0.113024
$131$	−607.236	−0.404996	−0.202498	−	0.979283i	$-0.564906\pi$
$131$	−607.236	−0.404996	−0.202498	+	0.979283i	$0.564906\pi$
$132$	−9.92481	−0.00654427
$133$	−646.816	−0.421699
$134$	2566.93	1.65484
$135$	−371.178	−0.236636
$136$	−2568.09	−1.61920
$137$	−353.025	−0.220153	−0.110076	−	0.993923i	$-0.535110\pi$
$137$	−353.025	−0.220153	−0.110076	+	0.993923i	$0.535110\pi$
$138$	−208.280	−0.128478
$139$	−1536.42	−0.937537	−0.468768	−	0.883321i	$-0.655302\pi$
$139$	−1536.42	−0.937537	−0.468768	+	0.883321i	$0.655302\pi$
$140$	−9.09770	−0.00549211
$141$	865.469	0.516919
$142$	−3031.32	−1.79143
$143$	−141.272	−0.0826137
$144$	1172.18	0.678342
$145$	−371.233	−0.212615
$146$	−244.148	−0.138396
$147$	152.353	0.0854819
$148$	−37.8876	−0.0210429
$149$	2973.83	1.63507	0.817537	−	0.575877i	$-0.195339\pi$
$149$	2973.83	1.63507	0.817537	+	0.575877i	$0.195339\pi$
$150$	1066.29	0.580416
$151$	2846.85	1.53426	0.767130	−	0.641491i	$-0.221684\pi$
$151$	2846.85	1.53426	0.767130	+	0.641491i	$0.221684\pi$
$152$	2023.08	1.07956
$153$	2033.03	1.07425
$154$	−134.835	−0.0705540
$155$	−661.960	−0.343032
$156$	32.0554	0.0164518
$157$	1548.18	0.786994	0.393497	−	0.919326i	$-0.371265\pi$
$157$	1548.18	0.786994	0.393497	+	0.919326i	$0.371265\pi$
$158$	1814.32	0.913540
$159$	357.499	0.178311
$160$	58.7377	0.0290226
$161$	−161.000	−0.0788110
$162$	−114.756	−0.0556548
$163$	310.390	0.149151	0.0745755	−	0.997215i	$-0.476240\pi$
$163$	310.390	0.149151	0.0745755	+	0.997215i	$0.476240\pi$
$164$	−109.296	−0.0520402
$165$	−55.3729	−0.0261259
$166$	−4106.98	−1.92026
$167$	−1510.10	−0.699733	−0.349866	−	0.936800i	$-0.613773\pi$
$167$	−1510.10	−0.699733	−0.349866	+	0.936800i	$0.613773\pi$
$168$	−476.522	−0.218836
$169$	−1740.72	−0.792315
$170$	919.918	0.415026
$171$	−1601.57	−0.716230
$172$	−268.989	−0.119245
$173$	482.353	0.211981	0.105990	−	0.994367i	$-0.466199\pi$
$173$	482.353	0.211981	0.105990	+	0.994367i	$0.466199\pi$
$174$	1248.43	0.543924
$175$	824.242	0.356039
$176$	447.267	0.191557
$177$	968.179	0.411146
$178$	−3612.88	−1.52133
$179$	−511.329	−0.213511	−0.106756	−	0.994285i	$-0.534046\pi$
$179$	−511.329	−0.213511	−0.106756	+	0.994285i	$0.534046\pi$
$180$	−22.5267	−0.00932801
$181$	1042.16	0.427975	0.213987	−	0.976836i	$-0.431355\pi$
$181$	1042.16	0.427975	0.213987	+	0.976836i	$0.431355\pi$
$182$	435.493	0.177367
$183$	1399.02	0.565129
$184$	503.568	0.201758
$185$	−211.384	−0.0840068
$186$	2226.11	0.877562
$187$	775.744	0.303358
$188$	134.347	0.0521182
$189$	964.885	0.371350
$190$	−724.690	−0.276708
$191$	−2127.59	−0.806006	−0.403003	−	0.915199i	$-0.632034\pi$
$191$	−2127.59	−0.806006	−0.403003	+	0.915199i	$0.632034\pi$
$192$	1484.65	0.558048
$193$	4077.76	1.52085	0.760424	−	0.649427i	$-0.224991\pi$
$193$	4077.76	1.52085	0.760424	+	0.649427i	$0.224991\pi$
$194$	−2119.33	−0.784324
$195$	178.844	0.0656786
$196$	23.6497	0.00861869
$197$	369.376	0.133589	0.0667943	−	0.997767i	$-0.478723\pi$
$197$	369.376	0.133589	0.0667943	+	0.997767i	$0.478723\pi$
$198$	−333.864	−0.119832
$199$	−2522.81	−0.898680	−0.449340	−	0.893361i	$-0.648341\pi$
$199$	−2522.81	−0.898680	−0.449340	+	0.893361i	$0.648341\pi$
$200$	−2578.03	−0.911470
$201$	−2740.33	−0.961630
$202$	4831.76	1.68298
$203$	965.030	0.333654
$204$	−176.020	−0.0604112
$205$	−609.789	−0.207754
$206$	3045.53	1.03006
$207$	−398.651	−0.133856
$208$	−1444.59	−0.481560
$209$	−611.113	−0.202256
$210$	170.695	0.0560910
$211$	−2591.27	−0.845454	−0.422727	−	0.906257i	$-0.638927\pi$
$211$	−2591.27	−0.845454	−0.422727	+	0.906257i	$0.638927\pi$
$212$	55.4944	0.0179782
$213$	3236.08	1.04100
$214$	3128.67	0.999401
$215$	−1500.75	−0.476049
$216$	−3017.92	−0.950665
$217$	1720.78	0.538314
$218$	3149.95	0.978631
$219$	260.640	0.0804221
$220$	−8.59552	−0.00263414
$221$	−2505.51	−0.762620
$222$	710.866	0.214911
$223$	−4704.56	−1.41274	−0.706369	−	0.707844i	$-0.749668\pi$
$223$	−4704.56	−1.41274	−0.706369	+	0.707844i	$0.749668\pi$
$224$	−152.690	−0.0455448
$225$	2040.90	0.604710
$226$	−692.666	−0.203874
$227$	4721.94	1.38065	0.690323	−	0.723502i	$-0.257469\pi$
$227$	4721.94	1.38065	0.690323	+	0.723502i	$0.257469\pi$
$228$	138.665	0.0402776
$229$	−1926.09	−0.555806	−0.277903	−	0.960609i	$-0.589639\pi$
$229$	−1926.09	−0.555806	−0.277903	+	0.960609i	$0.589639\pi$
$230$	−180.384	−0.0517137
$231$	143.943	0.0409990
$232$	−3018.38	−0.854164
$233$	4781.29	1.34435	0.672173	−	0.740394i	$-0.265361\pi$
$233$	4781.29	1.34435	0.672173	+	0.740394i	$0.265361\pi$
$234$	1078.32	0.301248
$235$	749.551	0.208065
$236$	150.290	0.0414537
$237$	−1936.87	−0.530859
$238$	−2391.35	−0.651295
$239$	1729.86	0.468182	0.234091	−	0.972215i	$-0.424789\pi$
$239$	1729.86	0.468182	0.234091	+	0.972215i	$0.424789\pi$
$240$	−566.221	−0.152289
$241$	−3219.05	−0.860404	−0.430202	−	0.902733i	$-0.641558\pi$
$241$	−3219.05	−0.860404	−0.430202	+	0.902733i	$0.641558\pi$
$242$	3749.14	0.995884
$243$	3844.21	1.01484
$244$	217.170	0.0569790
$245$	131.947	0.0344073
$246$	2050.67	0.531487
$247$	1973.78	0.508457
$248$	−5382.18	−1.37810
$249$	4384.41	1.11587
$250$	1903.82	0.481634
$251$	−5176.51	−1.30175	−0.650873	−	0.759187i	$-0.725597\pi$
$251$	−5176.51	−1.30175	−0.650873	+	0.759187i	$0.725597\pi$
$252$	58.5588	0.0146383
$253$	−152.113	−0.0377995
$254$	−7673.86	−1.89567
$255$	−982.059	−0.241172
$256$	738.701	0.180347
$257$	−4945.64	−1.20039	−0.600196	−	0.799853i	$-0.704911\pi$
$257$	−4945.64	−1.20039	−0.600196	+	0.799853i	$0.704911\pi$
$258$	5046.90	1.21785
$259$	549.498	0.131831
$260$	27.7620	0.00662202
$261$	2389.50	0.566691
$262$	1768.57	0.417034
$263$	−891.240	−0.208959	−0.104479	−	0.994527i	$-0.533318\pi$
$263$	−891.240	−0.208959	−0.104479	+	0.994527i	$0.533318\pi$
$264$	−450.219	−0.104959
$265$	309.617	0.0717720
$266$	1883.85	0.434234
$267$	3856.93	0.884047
$268$	−425.380	−0.0969561
$269$	−1444.07	−0.327311	−0.163656	−	0.986518i	$-0.552329\pi$
$269$	−1444.07	−0.327311	−0.163656	+	0.986518i	$0.552329\pi$
$270$	1081.05	0.243670
$271$	−6168.63	−1.38272	−0.691361	−	0.722510i	$-0.742989\pi$
$271$	−6168.63	−1.38272	−0.691361	+	0.722510i	$0.742989\pi$
$272$	7932.45	1.76829
$273$	−464.911	−0.103068
$274$	1028.18	0.226697
$275$	778.746	0.170764
$276$	34.5153	0.00752745
$277$	−3884.61	−0.842612	−0.421306	−	0.906918i	$-0.638428\pi$
$277$	−3884.61	−0.842612	−0.421306	+	0.906918i	$0.638428\pi$
$278$	4474.83	0.965404
$279$	4260.81	0.914294
$280$	−412.698	−0.0880837
$281$	2170.19	0.460722	0.230361	−	0.973105i	$-0.426009\pi$
$281$	2170.19	0.460722	0.230361	+	0.973105i	$0.426009\pi$
$282$	−2520.68	−0.532284
$283$	−5305.24	−1.11436	−0.557180	−	0.830392i	$-0.688117\pi$
$283$	−5305.24	−1.11436	−0.557180	+	0.830392i	$0.688117\pi$
$284$	502.337	0.104958
$285$	773.643	0.160795
$286$	411.455	0.0850693
$287$	1585.16	0.326025
$288$	−378.074	−0.0773550
$289$	8845.11	1.80035
$290$	1081.22	0.218935
$291$	2262.49	0.455772
$292$	40.4591	0.00810853
$293$	1724.96	0.343936	0.171968	−	0.985103i	$-0.444987\pi$
$293$	1724.96	0.343936	0.171968	+	0.985103i	$0.444987\pi$
$294$	−443.727	−0.0880227
$295$	838.505	0.165490
$296$	−1718.70	−0.337490
$297$	911.626	0.178107
$298$	−8661.28	−1.68367
$299$	491.298	0.0950251
$300$	−176.701	−0.0340062
$301$	3901.25	0.747057
$302$	−8291.44	−1.57986
$303$	−5158.15	−0.977979
$304$	−6249.00	−1.17896
$305$	1211.64	0.227470
$306$	−5921.20	−1.10618
$307$	6971.91	1.29612	0.648058	−	0.761591i	$-0.275581\pi$
$307$	6971.91	1.29612	0.648058	+	0.761591i	$0.275581\pi$
$308$	22.3443	0.00413371
$309$	−3251.26	−0.598568
$310$	1927.96	0.353228
$311$	7105.27	1.29551	0.647754	−	0.761850i	$-0.275709\pi$
$311$	7105.27	1.29551	0.647754	+	0.761850i	$0.275709\pi$
$312$	1454.13	0.263858
$313$	−4465.70	−0.806443	−0.403221	−	0.915102i	$-0.632110\pi$
$313$	−4465.70	−0.806443	−0.403221	+	0.915102i	$0.632110\pi$
$314$	−4509.06	−0.810386
$315$	326.713	0.0584387
$316$	−300.661	−0.0535237
$317$	7756.50	1.37428	0.687142	−	0.726523i	$-0.258865\pi$
$317$	7756.50	1.37428	0.687142	+	0.726523i	$0.258865\pi$
$318$	−1041.21	−0.183611
$319$	911.762	0.160028
$320$	1285.80	0.224620
$321$	−3340.02	−0.580753
$322$	468.912	0.0811536
$323$	−10838.3	−1.86706
$324$	19.0168	0.00326078
$325$	−2515.21	−0.429288
$326$	−904.010	−0.153584
$327$	−3362.73	−0.568683
$328$	−4958.00	−0.834633
$329$	−1948.48	−0.326514
$330$	161.273	0.0269025
$331$	−3157.24	−0.524283	−0.262142	−	0.965029i	$-0.584429\pi$
$331$	−3157.24	−0.524283	−0.262142	+	0.965029i	$0.584429\pi$
$332$	680.592	0.112507
$333$	1360.61	0.223906
$334$	4398.17	0.720531
$335$	−2373.30	−0.387066
$336$	1471.91	0.238985
$337$	11196.9	1.80990	0.904950	−	0.425518i	$-0.139908\pi$
$337$	11196.9	1.80990	0.904950	+	0.425518i	$0.139908\pi$
$338$	5069.83	0.815866
$339$	739.456	0.118471
$340$	−152.445	−0.0243161
$341$	1625.80	0.258187
$342$	4664.58	0.737519
$343$	−343.000	−0.0539949
$344$	−12202.1	−1.91249
$345$	192.569	0.0300509
$346$	−1404.85	−0.218281
$347$	1661.59	0.257058	0.128529	−	0.991706i	$-0.458975\pi$
$347$	1661.59	0.257058	0.128529	+	0.991706i	$0.458975\pi$
$348$	−206.884	−0.0318682
$349$	−5996.92	−0.919793	−0.459896	−	0.887973i	$-0.652113\pi$
$349$	−5996.92	−0.919793	−0.459896	+	0.887973i	$0.652113\pi$
$350$	−2400.60	−0.366622
$351$	−2944.39	−0.447749
$352$	−144.262	−0.0218443
$353$	6627.56	0.999291	0.499645	−	0.866230i	$-0.333464\pi$
$353$	6627.56	0.999291	0.499645	+	0.866230i	$0.333464\pi$
$354$	−2819.82	−0.423367
$355$	2802.65	0.419013
$356$	598.711	0.0891338
$357$	2552.89	0.378468
$358$	1489.24	0.219858
$359$	−5522.59	−0.811897	−0.405949	−	0.913896i	$-0.633059\pi$
$359$	−5522.59	−0.811897	−0.405949	+	0.913896i	$0.633059\pi$
$360$	−1021.88	−0.149605
$361$	1679.17	0.244813
$362$	−3035.30	−0.440696
$363$	−4002.40	−0.578709
$364$	−72.1680	−0.0103918
$365$	225.731	0.0323707
$366$	−4074.65	−0.581927
$367$	−7253.13	−1.03164	−0.515818	−	0.856698i	$-0.672512\pi$
$367$	−7253.13	−1.03164	−0.515818	+	0.856698i	$0.672512\pi$
$368$	−1555.45	−0.220335
$369$	3925.00	0.553733
$370$	615.656	0.0865038
$371$	−804.856	−0.112631
$372$	−368.902	−0.0514158
$373$	4090.05	0.567760	0.283880	−	0.958860i	$-0.408378\pi$
$373$	4090.05	0.567760	0.283880	+	0.958860i	$0.408378\pi$
$374$	−2259.35	−0.312375
$375$	−2032.43	−0.279878
$376$	6094.36	0.835884
$377$	−2944.83	−0.402298
$378$	−2810.23	−0.382387
$379$	−6261.48	−0.848630	−0.424315	−	0.905515i	$-0.639485\pi$
$379$	−6261.48	−0.848630	−0.424315	+	0.905515i	$0.639485\pi$
$380$	120.092	0.0162121
$381$	8192.23	1.10158
$382$	6196.61	0.829964
$383$	−8653.13	−1.15445	−0.577225	−	0.816585i	$-0.695864\pi$
$383$	−8653.13	−1.15445	−0.577225	+	0.816585i	$0.695864\pi$
$384$	−4866.61	−0.646740
$385$	124.664	0.0165025
$386$	−11876.5	−1.56605
$387$	9659.83	1.26883
$388$	351.206	0.0459531
$389$	−9670.90	−1.26050	−0.630250	−	0.776393i	$-0.717047\pi$
$389$	−9670.90	−1.26050	−0.630250	+	0.776393i	$0.717047\pi$
$390$	−520.884	−0.0676308
$391$	−2697.78	−0.348933
$392$	1072.82	0.138229
$393$	−1888.04	−0.242339
$394$	−1075.81	−0.137559
$395$	−1677.46	−0.213676
$396$	55.3264	0.00702086
$397$	−1547.24	−0.195602	−0.0978009	−	0.995206i	$-0.531181\pi$
$397$	−1547.24	−0.195602	−0.0978009	+	0.995206i	$0.531181\pi$
$398$	7347.68	0.925392
$399$	−2011.10	−0.252334
$400$	7963.14	0.995393
$401$	−3779.82	−0.470712	−0.235356	−	0.971909i	$-0.575626\pi$
$401$	−3779.82	−0.470712	−0.235356	+	0.971909i	$0.575626\pi$
$402$	7981.20	0.990214
$403$	−5251.04	−0.649064
$404$	−800.698	−0.0986045
$405$	106.099	0.0130176
$406$	−2810.65	−0.343572
$407$	519.167	0.0632289
$408$	−7984.80	−0.968889
$409$	13562.5	1.63967	0.819833	−	0.572603i	$-0.194066\pi$
$409$	13562.5	1.63967	0.819833	+	0.572603i	$0.194066\pi$
$410$	1776.01	0.213929
$411$	−1097.64	−0.131734
$412$	−504.692	−0.0603505
$413$	−2179.72	−0.259702
$414$	1161.07	0.137834
$415$	3797.18	0.449148
$416$	465.940	0.0549149
$417$	−4777.10	−0.560997
$418$	1779.86	0.208268
$419$	230.091	0.0268274	0.0134137	−	0.999910i	$-0.495730\pi$
$419$	230.091	0.0268274	0.0134137	+	0.999910i	$0.495730\pi$
$420$	−28.2869	−0.00328633
$421$	−1459.37	−0.168943	−0.0844716	−	0.996426i	$-0.526920\pi$
$421$	−1459.37	−0.168943	−0.0844716	+	0.996426i	$0.526920\pi$
$422$	7547.08	0.870584
$423$	−4824.60	−0.554563
$424$	2517.39	0.288338
$425$	13811.3	1.57635
$426$	−9425.09	−1.07194
$427$	−3149.69	−0.356966
$428$	−518.470	−0.0585542
$429$	−439.249	−0.0494339
$430$	4370.94	0.490199
$431$	−12449.7	−1.39138	−0.695688	−	0.718344i	$-0.744900\pi$
$431$	−12449.7	−1.39138	−0.695688	+	0.718344i	$0.744900\pi$
$432$	9321.93	1.03820
$433$	10062.9	1.11684	0.558420	−	0.829559i	$-0.311408\pi$
$433$	10062.9	1.11684	0.558420	+	0.829559i	$0.311408\pi$
$434$	−5011.77	−0.554315
$435$	−1154.25	−0.127223
$436$	−521.996	−0.0573373
$437$	2125.25	0.232642
$438$	−759.114	−0.0828125
$439$	1422.15	0.154614	0.0773071	−	0.997007i	$-0.475368\pi$
$439$	1422.15	0.154614	0.0773071	+	0.997007i	$0.475368\pi$
$440$	−389.918	−0.0422469
$441$	−849.299	−0.0917070
$442$	7297.31	0.785288
$443$	472.036	0.0506255	0.0253128	−	0.999680i	$-0.491942\pi$
$443$	472.036	0.0506255	0.0253128	+	0.999680i	$0.491942\pi$
$444$	−117.802	−0.0125915
$445$	3340.35	0.355838
$446$	13702.0	1.45473
$447$	9246.36	0.978384
$448$	−3342.47	−0.352493
$449$	15094.7	1.58655	0.793277	−	0.608860i	$-0.208373\pi$
$449$	15094.7	1.58655	0.793277	+	0.608860i	$0.208373\pi$
$450$	−5944.11	−0.622684
$451$	1497.66	0.156369
$452$	114.786	0.0119448
$453$	8851.54	0.918061
$454$	−13752.7	−1.42168
$455$	−402.642	−0.0414861
$456$	6290.24	0.645981
$457$	519.824	0.0532086	0.0266043	−	0.999646i	$-0.491531\pi$
$457$	519.824	0.0532086	0.0266043	+	0.999646i	$0.491531\pi$
$458$	5609.74	0.572327
$459$	16168.0	1.64414
$460$	29.8924	0.00302987
$461$	2.96323	0.000299374 0	0.000149687	−	1.00000i	$-0.499952\pi$
$461$	2.96323	0.000299374 0	0.000149687		1.00000i	$0.499952\pi$
$462$	−419.234	−0.0422176
$463$	−4341.22	−0.435753	−0.217876	−	0.975976i	$-0.569913\pi$
$463$	−4341.22	−0.435753	−0.217876	+	0.975976i	$0.569913\pi$
$464$	9323.32	0.932811
$465$	−2058.19	−0.205261
$466$	−13925.5	−1.38431
$467$	−8282.26	−0.820679	−0.410340	−	0.911933i	$-0.634590\pi$
$467$	−8282.26	−0.820679	−0.410340	+	0.911933i	$0.634590\pi$
$468$	−178.695	−0.0176499
$469$	6169.45	0.607417
$470$	−2183.07	−0.214250
$471$	4813.65	0.470916
$472$	6817.61	0.664843
$473$	3685.91	0.358305
$474$	5641.14	0.546638
$475$	−10880.3	−1.05099
$476$	396.284	0.0381589
$477$	−1992.90	−0.191297
$478$	−5038.22	−0.482098
$479$	1242.40	0.118511	0.0592555	−	0.998243i	$-0.481127\pi$
$479$	1242.40	0.118511	0.0592555	+	0.998243i	$0.481127\pi$
$480$	182.629	0.0173664
$481$	−1676.82	−0.158953
$482$	9375.48	0.885978
$483$	−500.588	−0.0471584
$484$	−621.292	−0.0583482
$485$	1959.46	0.183453
$486$	−11196.2	−1.04500
$487$	−7727.82	−0.719057	−0.359529	−	0.933134i	$-0.617063\pi$
$487$	−7727.82	−0.719057	−0.359529	+	0.933134i	$0.617063\pi$
$488$	9851.46	0.913842
$489$	965.076	0.0892479
$490$	−384.296	−0.0354300
$491$	−495.763	−0.0455671	−0.0227836	−	0.999740i	$-0.507253\pi$
$491$	−495.763	−0.0455671	−0.0227836	+	0.999740i	$0.507253\pi$
$492$	−339.828	−0.0311395
$493$	16170.4	1.47724
$494$	−5748.64	−0.523570
$495$	308.679	0.0280285
$496$	16624.8	1.50499
$497$	−7285.57	−0.657550
$498$	−12769.6	−1.14903
$499$	−11405.3	−1.02319	−0.511596	−	0.859226i	$-0.670945\pi$
$499$	−11405.3	−1.02319	−0.511596	+	0.859226i	$0.670945\pi$
$500$	−315.494	−0.0282186
$501$	−4695.27	−0.418701
$502$	15076.6	1.34044
$503$	−1742.19	−0.154434	−0.0772171	−	0.997014i	$-0.524603\pi$
$503$	−1742.19	−0.154434	−0.0772171	+	0.997014i	$0.524603\pi$
$504$	2656.40	0.234773
$505$	−4467.28	−0.393647
$506$	443.029	0.0389230
$507$	−5412.30	−0.474100
$508$	1271.68	0.111066
$509$	−5183.27	−0.451364	−0.225682	−	0.974201i	$-0.572461\pi$
$509$	−5183.27	−0.451364	−0.225682	+	0.974201i	$0.572461\pi$
$510$	2860.25	0.248341
$511$	−586.794	−0.0507989
$512$	10370.2	0.895122
$513$	−12736.8	−1.09619
$514$	14404.2	1.23607
$515$	−2815.80	−0.240930
$516$	−836.351	−0.0713533
$517$	−1840.93	−0.156603
$518$	−1600.41	−0.135749
$519$	1499.75	0.126844
$520$	1259.37	0.106206
$521$	11495.7	0.966672	0.483336	−	0.875435i	$-0.339425\pi$
$521$	11495.7	0.966672	0.483336	+	0.875435i	$0.339425\pi$
$522$	−6959.42	−0.583535
$523$	17005.7	1.42181	0.710907	−	0.703286i	$-0.248285\pi$
$523$	17005.7	1.42181	0.710907	+	0.703286i	$0.248285\pi$
$524$	−293.080	−0.0244337
$525$	2562.76	0.213044
$526$	2595.73	0.215170
$527$	28834.1	2.38337
$528$	1390.66	0.114623
$529$	529.000	0.0434783
$530$	−901.758	−0.0739054
$531$	−5397.17	−0.441087
$532$	−312.183	−0.0254415
$533$	−4837.19	−0.393099
$534$	−11233.3	−0.910324
$535$	−2892.67	−0.233759
$536$	−19296.5	−1.55500
$537$	−1589.84	−0.127759
$538$	4205.86	0.337040
$539$	−324.067	−0.0258971
$540$	−179.148	−0.0142765
$541$	6975.36	0.554333	0.277167	−	0.960822i	$-0.410605\pi$
$541$	6975.36	0.554333	0.277167	+	0.960822i	$0.410605\pi$
$542$	17966.1	1.42382
$543$	3240.34	0.256089
$544$	−2558.54	−0.201648
$545$	−2912.34	−0.228901
$546$	1354.05	0.106132
$547$	7492.37	0.585650	0.292825	−	0.956166i	$-0.405405\pi$
$547$	7492.37	0.585650	0.292825	+	0.956166i	$0.405405\pi$
$548$	−170.386	−0.0132820
$549$	−7798.92	−0.606284
$550$	−2268.10	−0.175840
$551$	−12738.7	−0.984913
$552$	1565.71	0.120727
$553$	4360.59	0.335319
$554$	11313.9	0.867658
$555$	−657.244	−0.0502675
$556$	−741.549	−0.0565624
$557$	15894.4	1.20909	0.604547	−	0.796569i	$-0.293354\pi$
$557$	15894.4	1.20909	0.604547	+	0.796569i	$0.293354\pi$
$558$	−12409.6	−0.941470
$559$	−11904.8	−0.900751
$560$	1274.76	0.0961940
$561$	2411.97	0.181522
$562$	−6320.69	−0.474417
$563$	−9735.19	−0.728755	−0.364378	−	0.931251i	$-0.618718\pi$
$563$	−9735.19	−0.728755	−0.364378	+	0.931251i	$0.618718\pi$
$564$	417.716	0.0311862
$565$	640.416	0.0476858
$566$	15451.5	1.14748
$567$	−275.808	−0.0204283
$568$	22787.5	1.68335
$569$	−14929.5	−1.09996	−0.549981	−	0.835177i	$-0.685365\pi$
$569$	−14929.5	−1.09996	−0.549981	+	0.835177i	$0.685365\pi$
$570$	−2253.23	−0.165575
$571$	2644.32	0.193802	0.0969012	−	0.995294i	$-0.469107\pi$
$571$	2644.32	0.193802	0.0969012	+	0.995294i	$0.469107\pi$
$572$	−68.1845	−0.00498416
$573$	−6615.20	−0.482293
$574$	−4616.78	−0.335716
$575$	−2708.22	−0.196419
$576$	−8276.26	−0.598688
$577$	3788.86	0.273366	0.136683	−	0.990615i	$-0.456356\pi$
$577$	3788.86	0.273366	0.136683	+	0.990615i	$0.456356\pi$
$578$	−25761.4	−1.85386
$579$	12678.7	0.910035
$580$	−179.174	−0.0128273
$581$	−9870.87	−0.704841
$582$	−6589.50	−0.469319
$583$	−760.430	−0.0540202
$584$	1835.35	0.130046
$585$	−996.979	−0.0704615
$586$	−5023.94	−0.354159
$587$	−21854.2	−1.53666	−0.768332	−	0.640052i	$-0.778913\pi$
$587$	−21854.2	−1.53666	−0.768332	+	0.640052i	$0.778913\pi$
$588$	73.5325	0.00515719
$589$	−22714.9	−1.58905
$590$	−2442.15	−0.170409
$591$	1148.48	0.0799359
$592$	5308.80	0.368565
$593$	−8515.06	−0.589665	−0.294833	−	0.955549i	$-0.595264\pi$
$593$	−8515.06	−0.589665	−0.294833	+	0.955549i	$0.595264\pi$
$594$	−2655.11	−0.183401
$595$	2210.96	0.152337
$596$	1435.31	0.0986453
$597$	−7844.02	−0.537746
$598$	−1430.91	−0.0978496
$599$	−1484.88	−0.101286	−0.0506431	−	0.998717i	$-0.516127\pi$
$599$	−1484.88	−0.101286	−0.0506431	+	0.998717i	$0.516127\pi$
$600$	−8015.70	−0.545399
$601$	−13766.2	−0.934332	−0.467166	−	0.884170i	$-0.654725\pi$
$601$	−13766.2	−0.934332	−0.467166	+	0.884170i	$0.654725\pi$
$602$	−11362.4	−0.769262
$603$	15276.1	1.03166
$604$	1374.02	0.0925632
$605$	−3466.33	−0.232936
$606$	15023.1	1.00705
$607$	−1255.44	−0.0839484	−0.0419742	−	0.999119i	$-0.513365\pi$
$607$	−1255.44	−0.0839484	−0.0419742	+	0.999119i	$0.513365\pi$
$608$	2015.56	0.134443
$609$	3000.51	0.199650
$610$	−3528.90	−0.234231
$611$	5945.86	0.393688
$612$	981.236	0.0648106
$613$	−21086.5	−1.38936	−0.694678	−	0.719321i	$-0.744453\pi$
$613$	−21086.5	−1.38936	−0.694678	+	0.719321i	$0.744453\pi$
$614$	−20305.7	−1.33464
$615$	−1895.98	−0.124314
$616$	1013.60	0.0662974
$617$	−17617.6	−1.14953	−0.574765	−	0.818319i	$-0.694906\pi$
$617$	−17617.6	−1.14953	−0.574765	+	0.818319i	$0.694906\pi$
$618$	9469.29	0.616360
$619$	−4192.59	−0.272236	−0.136118	−	0.990693i	$-0.543463\pi$
$619$	−4192.59	−0.272236	−0.136118	+	0.990693i	$0.543463\pi$
$620$	−319.493	−0.0206954
$621$	−3170.34	−0.204865
$622$	−20694.1	−1.33402
$623$	−8683.32	−0.558411
$624$	−4491.58	−0.288153
$625$	12958.4	0.829337
$626$	13006.4	0.830413
$627$	−1900.10	−0.121025
$628$	747.223	0.0474800
$629$	9207.63	0.583676
$630$	−951.551	−0.0601757
$631$	−2700.44	−0.170369	−0.0851844	−	0.996365i	$-0.527148\pi$
$631$	−2700.44	−0.170369	−0.0851844	+	0.996365i	$0.527148\pi$
$632$	−13638.9	−0.858425
$633$	−8056.89	−0.505897
$634$	−22590.8	−1.41513
$635$	7095.00	0.443396
$636$	172.545	0.0107577
$637$	1046.68	0.0651035
$638$	−2655.51	−0.164784
$639$	−18039.7	−1.11681
$640$	−4214.79	−0.260319
$641$	−5002.08	−0.308222	−0.154111	−	0.988054i	$-0.549251\pi$
$641$	−5002.08	−0.308222	−0.154111	+	0.988054i	$0.549251\pi$
$642$	9727.79	0.598015
$643$	−5426.44	−0.332811	−0.166406	−	0.986057i	$-0.553216\pi$
$643$	−5426.44	−0.332811	−0.166406	+	0.986057i	$0.553216\pi$
$644$	−77.7061	−0.00475474
$645$	−4666.20	−0.284855
$646$	31566.6	1.92255
$647$	23639.0	1.43639	0.718194	−	0.695843i	$-0.244969\pi$
$647$	23639.0	1.43639	0.718194	+	0.695843i	$0.244969\pi$
$648$	862.660	0.0522971
$649$	−2059.40	−0.124559
$650$	7325.54	0.442048
$651$	5350.32	0.322113
$652$	149.809	0.00899840
$653$	18706.7	1.12105	0.560527	−	0.828136i	$-0.310599\pi$
$653$	18706.7	1.12105	0.560527	+	0.828136i	$0.310599\pi$
$654$	9793.94	0.585586
$655$	−1635.16	−0.0975437
$656$	15314.5	0.911481
$657$	−1452.95	−0.0862787
$658$	5674.93	0.336219
$659$	−30984.1	−1.83152	−0.915759	−	0.401727i	$-0.868410\pi$
$659$	−30984.1	−1.83152	−0.915759	+	0.401727i	$0.868410\pi$
$660$	−26.7255	−0.00157620
$661$	−30744.2	−1.80910	−0.904548	−	0.426372i	$-0.859792\pi$
$661$	−30744.2	−1.80910	−0.904548	+	0.426372i	$0.859792\pi$
$662$	9195.46	0.539867
$663$	−7790.24	−0.456332
$664$	30873.6	1.80441
$665$	−1741.74	−0.101567
$666$	−3962.77	−0.230562
$667$	−3170.81	−0.184069
$668$	−728.846	−0.0422154
$669$	−14627.6	−0.845344
$670$	6912.23	0.398571
$671$	−2975.84	−0.171209
$672$	−474.750	−0.0272528
$673$	16123.7	0.923513	0.461756	−	0.887007i	$-0.347219\pi$
$673$	16123.7	0.923513	0.461756	+	0.887007i	$0.347219\pi$
$674$	−32611.1	−1.86370
$675$	16230.6	0.925505
$676$	−840.151	−0.0478010
$677$	5015.44	0.284725	0.142363	−	0.989815i	$-0.454530\pi$
$677$	5015.44	0.284725	0.142363	+	0.989815i	$0.454530\pi$
$678$	−2153.66	−0.121993
$679$	−5093.67	−0.287890
$680$	−6915.35	−0.389988
$681$	14681.7	0.826141
$682$	−4735.13	−0.265862
$683$	30484.3	1.70783	0.853915	−	0.520413i	$-0.174222\pi$
$683$	30484.3	1.70783	0.853915	+	0.520413i	$0.174222\pi$
$684$	−772.994	−0.0432108
$685$	−950.625	−0.0530241
$686$	998.987	0.0555999
$687$	−5988.68	−0.332580
$688$	37690.6	2.08858
$689$	2456.05	0.135803
$690$	−560.856	−0.0309441
$691$	−24595.1	−1.35404	−0.677021	−	0.735964i	$-0.736729\pi$
$691$	−24595.1	−1.35404	−0.677021	+	0.735964i	$0.736729\pi$
$692$	232.806	0.0127890
$693$	−802.420	−0.0439847
$694$	−4839.39	−0.264698
$695$	−4137.28	−0.225807
$696$	−9384.85	−0.511109
$697$	26561.7	1.44346
$698$	17466.0	0.947133
$699$	14866.2	0.804421
$700$	397.818	0.0214801
$701$	−11700.0	−0.630390	−0.315195	−	0.949027i	$-0.602070\pi$
$701$	−11700.0	−0.630390	−0.315195	+	0.949027i	$0.602070\pi$
$702$	8575.52	0.461057
$703$	−7253.55	−0.389151
$704$	−3157.97	−0.169063
$705$	2330.53	0.124501
$706$	−19302.8	−1.02899
$707$	11612.8	0.617744
$708$	467.289	0.0248048
$709$	390.131	0.0206652	0.0103326	−	0.999947i	$-0.496711\pi$
$709$	390.131	0.0206652	0.0103326	+	0.999947i	$0.496711\pi$
$710$	−8162.73	−0.431467
$711$	10797.2	0.569518
$712$	27159.3	1.42955
$713$	−5654.00	−0.296976
$714$	−7435.28	−0.389718
$715$	−380.417	−0.0198976
$716$	−246.791	−0.0128813
$717$	5378.56	0.280148
$718$	16084.5	0.836030
$719$	−32849.9	−1.70389	−0.851944	−	0.523633i	$-0.824576\pi$
$719$	−32849.9	−1.70389	−0.851944	+	0.523633i	$0.824576\pi$
$720$	3156.43	0.163380
$721$	7319.74	0.378088
$722$	−4890.58	−0.252089
$723$	−10008.8	−0.514843
$724$	502.997	0.0258201
$725$	16233.0	0.831557
$726$	11657.0	0.595911
$727$	−24574.4	−1.25367	−0.626833	−	0.779153i	$-0.715649\pi$
$727$	−24574.4	−1.25367	−0.626833	+	0.779153i	$0.715649\pi$
$728$	−3273.75	−0.166667
$729$	10888.7	0.553205
$730$	−657.441	−0.0333329
$731$	65370.9	3.30757
$732$	675.233	0.0340947
$733$	−23806.8	−1.19963	−0.599813	−	0.800140i	$-0.704758\pi$
$733$	−23806.8	−1.19963	−0.599813	+	0.800140i	$0.704758\pi$
$734$	21124.7	1.06230
$735$	410.255	0.0205884
$736$	501.696	0.0251260
$737$	5828.91	0.291330
$738$	−11431.6	−0.570192
$739$	14089.3	0.701333	0.350666	−	0.936500i	$-0.385955\pi$
$739$	14089.3	0.701333	0.350666	+	0.936500i	$0.385955\pi$
$740$	−102.024	−0.00506820
$741$	6136.97	0.304247
$742$	2344.14	0.115979
$743$	−12735.1	−0.628809	−0.314405	−	0.949289i	$-0.601805\pi$
$743$	−12735.1	−0.628809	−0.314405	+	0.949289i	$0.601805\pi$
$744$	−16734.5	−0.824618
$745$	8007.94	0.393810
$746$	−11912.3	−0.584636
$747$	−24441.2	−1.19713
$748$	374.410	0.0183019
$749$	7519.56	0.366834
$750$	5919.45	0.288197
$751$	25958.0	1.26128	0.630640	−	0.776076i	$-0.282793\pi$
$751$	25958.0	1.26128	0.630640	+	0.776076i	$0.282793\pi$
$752$	−18824.6	−0.912848
$753$	−16095.0	−0.778930
$754$	8576.81	0.414256
$755$	7666.00	0.369529
$756$	465.699	0.0224038
$757$	33354.8	1.60146	0.800728	−	0.599028i	$-0.204446\pi$
$757$	33354.8	1.60146	0.800728	+	0.599028i	$0.204446\pi$
$758$	18236.6	0.873854
$759$	−472.956	−0.0226182
$760$	5447.75	0.260014
$761$	16836.5	0.802002	0.401001	−	0.916078i	$-0.368662\pi$
$761$	16836.5	0.802002	0.401001	+	0.916078i	$0.368662\pi$
$762$	−23859.9	−1.13432
$763$	7570.70	0.359211
$764$	−1026.88	−0.0486270
$765$	5474.54	0.258735
$766$	25202.2	1.18876
$767$	6651.49	0.313131
$768$	2296.80	0.107915
$769$	−6471.65	−0.303477	−0.151738	−	0.988421i	$-0.548487\pi$
$769$	−6471.65	−0.303477	−0.151738	+	0.988421i	$0.548487\pi$
$770$	−363.084	−0.0169930
$771$	−15377.2	−0.718283
$772$	1968.12	0.0917540
$773$	1188.93	0.0553207	0.0276603	−	0.999617i	$-0.491194\pi$
$773$	1188.93	0.0553207	0.0276603	+	0.999617i	$0.491194\pi$
$774$	−28134.3	−1.30654
$775$	28945.7	1.34163
$776$	15931.7	0.737006
$777$	1708.52	0.0788840
$778$	28166.5	1.29797
$779$	−20924.6	−0.962392
$780$	86.3187	0.00396244
$781$	−6883.42	−0.315376
$782$	7857.29	0.359305
$783$	19002.9	0.867317
$784$	−3313.78	−0.150956
$785$	4168.93	0.189548
$786$	5498.92	0.249542
$787$	−13521.2	−0.612424	−0.306212	−	0.951963i	$-0.599062\pi$
$787$	−13521.2	−0.612424	−0.306212	+	0.951963i	$0.599062\pi$
$788$	178.278	0.00805951
$789$	−2771.08	−0.125035
$790$	4885.59	0.220027
$791$	−1664.78	−0.0748327
$792$	2509.77	0.112602
$793$	9611.42	0.430405
$794$	4506.34	0.201416
$795$	962.672	0.0429465
$796$	−1217.63	−0.0542181
$797$	−12158.2	−0.540358	−0.270179	−	0.962810i	$-0.587083\pi$
$797$	−12158.2	−0.540358	−0.270179	+	0.962810i	$0.587083\pi$
$798$	5857.34	0.259834
$799$	−32649.5	−1.44563
$800$	−2568.44	−0.113510
$801$	−21500.7	−0.948426
$802$	11008.7	0.484703
$803$	−554.404	−0.0243642
$804$	−1322.61	−0.0580160
$805$	−433.541	−0.0189817
$806$	15293.6	0.668356
$807$	−4489.97	−0.195854
$808$	−36322.0	−1.58144
$809$	13478.2	0.585745	0.292873	−	0.956151i	$-0.405389\pi$
$809$	13478.2	0.585745	0.292873	+	0.956151i	$0.405389\pi$
$810$	−309.015	−0.0134045
$811$	38395.2	1.66244	0.831219	−	0.555944i	$-0.187643\pi$
$811$	38395.2	1.66244	0.831219	+	0.555944i	$0.187643\pi$
$812$	465.768	0.0201296
$813$	−19179.7	−0.827384
$814$	−1512.07	−0.0651083
$815$	835.817	0.0359232
$816$	24663.9	1.05810
$817$	−51497.7	−2.20523
$818$	−39500.8	−1.68840
$819$	2591.67	0.110574
$820$	−294.313	−0.0125340
$821$	31613.8	1.34388	0.671942	−	0.740604i	$-0.265461\pi$
$821$	31613.8	1.34388	0.671942	+	0.740604i	$0.265461\pi$
$822$	3196.87	0.135649
$823$	15618.7	0.661524	0.330762	−	0.943714i	$-0.392694\pi$
$823$	15618.7	0.661524	0.330762	+	0.943714i	$0.392694\pi$
$824$	−22894.3	−0.967915
$825$	2421.31	0.102181
$826$	6348.42	0.267421
$827$	−13646.1	−0.573785	−0.286892	−	0.957963i	$-0.592622\pi$
$827$	−13646.1	−0.573785	−0.286892	+	0.957963i	$0.592622\pi$
$828$	−192.407	−0.00807563
$829$	−14216.6	−0.595614	−0.297807	−	0.954626i	$-0.596255\pi$
$829$	−14216.6	−0.595614	−0.297807	+	0.954626i	$0.596255\pi$
$830$	−11059.3	−0.462498
$831$	−12078.2	−0.504197
$832$	10199.7	0.425013
$833$	−5747.45	−0.239061
$834$	13913.3	0.577672
$835$	−4066.41	−0.168531
$836$	−294.952	−0.0122023
$837$	33884.8	1.39932
$838$	−670.139	−0.0276248
$839$	6041.07	0.248583	0.124291	−	0.992246i	$-0.460334\pi$
$839$	6041.07	0.248583	0.124291	+	0.992246i	$0.460334\pi$
$840$	−1283.18	−0.0527070
$841$	−5383.24	−0.220724
$842$	4250.40	0.173965
$843$	6747.65	0.275684
$844$	−1250.67	−0.0510069
$845$	−4687.40	−0.190830
$846$	14051.7	0.571047
$847$	9010.82	0.365544
$848$	−7775.85	−0.314887
$849$	−16495.3	−0.666803
$850$	−40225.5	−1.62320
$851$	−1805.49	−0.0727280
$852$	1561.88	0.0628043
$853$	1124.39	0.0451331	0.0225665	−	0.999745i	$-0.492816\pi$
$853$	1124.39	0.0451331	0.0225665	+	0.999745i	$0.492816\pi$
$854$	9173.48	0.367576
$855$	−4312.72	−0.172505
$856$	−23519.3	−0.939106
$857$	−29228.0	−1.16501	−0.582503	−	0.812828i	$-0.697927\pi$
$857$	−29228.0	−1.16501	−0.582503	+	0.812828i	$0.697927\pi$
$858$	1279.31	0.0509032
$859$	19510.5	0.774960	0.387480	−	0.921878i	$-0.373346\pi$
$859$	19510.5	0.774960	0.387480	+	0.921878i	$0.373346\pi$
$860$	−724.333	−0.0287204
$861$	4928.65	0.195085
$862$	36259.8	1.43273
$863$	−40855.4	−1.61151	−0.805756	−	0.592247i	$-0.798241\pi$
$863$	−40855.4	−1.61151	−0.805756	+	0.592247i	$0.798241\pi$
$864$	−3006.70	−0.118391
$865$	1298.88	0.0510558
$866$	−29308.1	−1.15004
$867$	27501.6	1.07728
$868$	830.529	0.0324770
$869$	4119.90	0.160826
$870$	3361.76	0.131005
$871$	−18826.3	−0.732383
$872$	−23679.3	−0.919589
$873$	−12612.4	−0.488963
$874$	−6189.79	−0.239557
$875$	4575.72	0.176786
$876$	125.797	0.00485193
$877$	2696.93	0.103841	0.0519207	−	0.998651i	$-0.483466\pi$
$877$	2696.93	0.103841	0.0519207	+	0.998651i	$0.483466\pi$
$878$	−4142.02	−0.159210
$879$	5363.31	0.205802
$880$	1204.40	0.0461367
$881$	−20196.6	−0.772351	−0.386176	−	0.922425i	$-0.626204\pi$
$881$	−20196.6	−0.772351	−0.386176	+	0.922425i	$0.626204\pi$
$882$	2473.58	0.0944329
$883$	37775.3	1.43968	0.719841	−	0.694139i	$-0.244215\pi$
$883$	37775.3	1.43968	0.719841	+	0.694139i	$0.244215\pi$
$884$	−1209.28	−0.0460095
$885$	2607.11	0.0990250
$886$	−1374.80	−0.0521303
$887$	41023.6	1.55292	0.776459	−	0.630168i	$-0.217014\pi$
$887$	41023.6	1.55292	0.776459	+	0.630168i	$0.217014\pi$
$888$	−5343.83	−0.201945
$889$	−18443.6	−0.695815
$890$	−9728.76	−0.366414
$891$	−260.584	−0.00979787
$892$	−2270.64	−0.0852316
$893$	25720.5	0.963834
$894$	−26930.0	−1.00747
$895$	−1376.91	−0.0514245
$896$	10956.5	0.408515
$897$	1527.56	0.0568605
$898$	−43963.3	−1.63371
$899$	33889.9	1.25728
$900$	985.032	0.0364827
$901$	−13486.5	−0.498669
$902$	−4361.95	−0.161017
$903$	12129.9	0.447019
$904$	5207.01	0.191574
$905$	2806.34	0.103078
$906$	−25780.1	−0.945349
$907$	18445.5	0.675275	0.337637	−	0.941276i	$-0.390372\pi$
$907$	18445.5	0.675275	0.337637	+	0.941276i	$0.390372\pi$
$908$	2279.03	0.0832955
$909$	28754.4	1.04920
$910$	1172.70	0.0427192
$911$	−19734.9	−0.717725	−0.358862	−	0.933390i	$-0.616835\pi$
$911$	−19734.9	−0.717725	−0.358862	+	0.933390i	$0.616835\pi$
$912$	−19429.6	−0.705460
$913$	−9326.02	−0.338057
$914$	−1513.99	−0.0547901
$915$	3767.28	0.136112
$916$	−929.621	−0.0335323
$917$	4250.65	0.153074
$918$	−47089.4	−1.69301
$919$	7846.45	0.281644	0.140822	−	0.990035i	$-0.455026\pi$
$919$	7846.45	0.281644	0.140822	+	0.990035i	$0.455026\pi$
$920$	1356.01	0.0485938
$921$	21677.3	0.775562
$922$	−8.63042	−0.000308273 0
$923$	22232.2	0.792831
$924$	69.4737	0.00247350
$925$	9243.25	0.328558
$926$	12643.8	0.448705
$927$	18124.3	0.642159
$928$	−3007.15	−0.106373
$929$	−27574.5	−0.973831	−0.486916	−	0.873449i	$-0.661878\pi$
$929$	−27574.5	−0.973831	−0.486916	+	0.873449i	$0.661878\pi$
$930$	5994.48	0.211362
$931$	4527.71	0.159387
$932$	2307.67	0.0811055
$933$	22092.0	0.775197
$934$	24122.1	0.845073
$935$	2088.92	0.0730643
$936$	−8106.11	−0.283073
$937$	9195.66	0.320607	0.160304	−	0.987068i	$-0.448753\pi$
$937$	9195.66	0.320607	0.160304	+	0.987068i	$0.448753\pi$
$938$	−17968.5	−0.625472
$939$	−13884.9	−0.482554
$940$	361.768	0.0125527
$941$	5669.90	0.196422	0.0982112	−	0.995166i	$-0.468688\pi$
$941$	5669.90	0.196422	0.0982112	+	0.995166i	$0.468688\pi$
$942$	−14019.8	−0.484913
$943$	−5208.39	−0.179861
$944$	−21058.6	−0.726059
$945$	2598.24	0.0894400
$946$	−10735.2	−0.368955
$947$	25608.2	0.878726	0.439363	−	0.898310i	$-0.355204\pi$
$947$	25608.2	0.878726	0.439363	+	0.898310i	$0.355204\pi$
$948$	−934.826	−0.0320272
$949$	1790.62	0.0612499
$950$	31688.7	1.08223
$951$	24116.8	0.822335
$952$	17976.6	0.612002
$953$	−9059.61	−0.307943	−0.153971	−	0.988075i	$-0.549206\pi$
$953$	−9059.61	−0.307943	−0.153971	+	0.988075i	$0.549206\pi$
$954$	5804.31	0.196983
$955$	−5729.18	−0.194128
$956$	834.912	0.0282458
$957$	2834.89	0.0957564
$958$	−3618.49	−0.122034
$959$	2471.17	0.0832099
$960$	3997.86	0.134407
$961$	30639.4	1.02848
$962$	4883.73	0.163677
$963$	18619.1	0.623045
$964$	−1553.66	−0.0519089
$965$	10980.6	0.366298
$966$	1457.96	0.0485601
$967$	−11377.9	−0.378376	−0.189188	−	0.981941i	$-0.560586\pi$
$967$	−11377.9	−0.378376	−0.189188	+	0.981941i	$0.560586\pi$
$968$	−28183.6	−0.935802
$969$	−33698.9	−1.11720
$970$	−5706.93	−0.188906
$971$	−41723.8	−1.37897	−0.689485	−	0.724300i	$-0.742163\pi$
$971$	−41723.8	−1.37897	−0.689485	+	0.724300i	$0.742163\pi$
$972$	1855.39	0.0612261
$973$	10755.0	0.354355
$974$	22507.3	0.740430
$975$	−7820.38	−0.256875
$976$	−30429.7	−0.997984
$977$	−33302.6	−1.09053	−0.545264	−	0.838265i	$-0.683570\pi$
$977$	−33302.6	−1.09053	−0.545264	+	0.838265i	$0.683570\pi$
$978$	−2810.78	−0.0919007
$979$	−8204.02	−0.267826
$980$	63.6839	0.00207582
$981$	18745.7	0.610097
$982$	1443.91	0.0469216
$983$	−821.880	−0.0266672	−0.0133336	−	0.999911i	$-0.504244\pi$
$983$	−821.880	−0.0266672	−0.0133336	+	0.999911i	$0.504244\pi$
$984$	−15415.6	−0.499422
$985$	994.656	0.0321750
$986$	−47096.4	−1.52115
$987$	−6058.28	−0.195377
$988$	952.641	0.0306756
$989$	−12818.4	−0.412134
$990$	−899.028	−0.0288616
$991$	1358.94	0.0435603	0.0217801	−	0.999763i	$-0.493067\pi$
$991$	1358.94	0.0435603	0.0217801	+	0.999763i	$0.493067\pi$
$992$	−5362.17	−0.171622
$993$	−9816.62	−0.313717
$994$	21219.2	0.677095
$995$	−6793.42	−0.216448
$996$	2116.12	0.0673212
$997$	−20623.2	−0.655108	−0.327554	−	0.944833i	$-0.606224\pi$
$997$	−20623.2	−0.655108	−0.327554	+	0.944833i	$0.606224\pi$
$998$	33218.0	1.05360
$999$	10820.5	0.342687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.4.a.b.1.3	✓	8
3.2	odd	2		1449.4.a.i.1.6		8
7.6	odd	2		1127.4.a.e.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.a.b.1.3	✓	8	1.1	even	1	trivial
1127.4.a.e.1.3		8	7.6	odd	2
1449.4.a.i.1.6		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	161.a (trivial)

\(f(q)\)	\(=\)	\(q-2.91250 q^{2} +3.10924 q^{3} +0.482647 q^{4} +2.69280 q^{5} -9.05565 q^{6} -7.00000 q^{7} +21.8943 q^{8} -17.3326 q^{9} -7.84277 q^{10} -6.61362 q^{11} +1.50066 q^{12} +21.3608 q^{13} +20.3875 q^{14} +8.37256 q^{15} -67.6282 q^{16} -117.295 q^{17} +50.4813 q^{18} +92.4022 q^{19} +1.29967 q^{20} -21.7647 q^{21} +19.2621 q^{22} +23.0000 q^{23} +68.0746 q^{24} -117.749 q^{25} -62.2133 q^{26} -137.841 q^{27} -3.37853 q^{28} -137.861 q^{29} -24.3851 q^{30} -245.826 q^{31} +21.8129 q^{32} -20.5633 q^{33} +341.621 q^{34} -18.8496 q^{35} -8.36554 q^{36} -78.4997 q^{37} -269.121 q^{38} +66.4158 q^{39} +58.9569 q^{40} -226.452 q^{41} +63.3896 q^{42} -557.321 q^{43} -3.19204 q^{44} -46.6733 q^{45} -66.9875 q^{46} +278.354 q^{47} -210.272 q^{48} +49.0000 q^{49} +342.943 q^{50} -364.698 q^{51} +10.3097 q^{52} +114.979 q^{53} +401.461 q^{54} -17.8091 q^{55} -153.260 q^{56} +287.301 q^{57} +401.521 q^{58} +311.388 q^{59} +4.04099 q^{60} +449.956 q^{61} +715.967 q^{62} +121.328 q^{63} +477.496 q^{64} +57.5203 q^{65} +59.8906 q^{66} -881.349 q^{67} -56.6120 q^{68} +71.5125 q^{69} +54.8994 q^{70} +1040.80 q^{71} -379.485 q^{72} +83.8276 q^{73} +228.630 q^{74} -366.109 q^{75} +44.5976 q^{76} +46.2953 q^{77} -193.436 q^{78} -622.942 q^{79} -182.109 q^{80} +39.4012 q^{81} +659.540 q^{82} +1410.12 q^{83} -10.5046 q^{84} -315.852 q^{85} +1623.20 q^{86} -428.644 q^{87} -144.800 q^{88} +1240.47 q^{89} +135.936 q^{90} -149.526 q^{91} +11.1009 q^{92} -764.331 q^{93} -810.705 q^{94} +248.821 q^{95} +67.8214 q^{96} +727.667 q^{97} -142.712 q^{98} +114.631 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{3} + 24 q^{4} - 24 q^{5} - 41 q^{6} - 56 q^{7} - 69 q^{8} + 95 q^{9} - 30 q^{10} - 98 q^{11} - 131 q^{12} - 145 q^{13} - 232 q^{15} - 76 q^{16} - 96 q^{17} - 69 q^{18} - 226 q^{19} - 22 q^{20}+ \cdots - 1676 q^{99}+O(q^{100}) \) 8 * q - 3 * q^3 + 24 * q^4 - 24 * q^5 - 41 * q^6 - 56 * q^7 - 69 * q^8 + 95 * q^9 - 30 * q^10 - 98 * q^11 - 131 * q^12 - 145 * q^13 - 232 * q^15 - 76 * q^16 - 96 * q^17 - 69 * q^18 - 226 * q^19 - 22 * q^20 + 21 * q^21 - 98 * q^22 + 184 * q^23 - 216 * q^24 + 60 * q^25 - 185 * q^26 - 531 * q^27 - 168 * q^28 + 73 * q^29 - 416 * q^30 - 987 * q^31 + 118 * q^32 + 16 * q^33 - 846 * q^34 + 168 * q^35 - 69 * q^36 - 2 * q^37 - 454 * q^38 - 675 * q^39 - 640 * q^40 - 907 * q^41 + 287 * q^42 + 636 * q^43 + 126 * q^44 - 126 * q^45 + 421 * q^47 + 2165 * q^48 + 392 * q^49 + 2308 * q^50 - 408 * q^51 - 931 * q^52 + 138 * q^53 + 1717 * q^54 - 90 * q^55 + 483 * q^56 + 984 * q^57 + 1329 * q^58 - 924 * q^59 + 2642 * q^60 - 890 * q^61 + 2175 * q^62 - 665 * q^63 + 303 * q^64 - 392 * q^65 + 408 * q^66 - 1392 * q^67 + 2592 * q^68 - 69 * q^69 + 210 * q^70 + 259 * q^71 + 141 * q^72 - 297 * q^73 + 2314 * q^74 - 851 * q^75 - 428 * q^76 + 686 * q^77 + 6789 * q^78 - 1788 * q^79 - 2104 * q^80 + 2300 * q^81 - 219 * q^82 + 2114 * q^83 + 917 * q^84 + 528 * q^85 - 2360 * q^86 - 2441 * q^87 + 2144 * q^88 - 1040 * q^89 + 4872 * q^90 + 1015 * q^91 + 552 * q^92 - 621 * q^93 - 1009 * q^94 + 1268 * q^95 - 1163 * q^96 - 1452 * q^97 - 1676 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more