LMFDB - Embedded newform 129.4.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [129,4,Mod(1,129)] 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("129.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(129, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 129 = 3 \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	129.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-3,-15,19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.61124639074$
Analytic rank:	$1$
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} - 2x^{4} - 27x^{3} + 20x^{2} + 162x + 104 $ x^5 - 2x^4 - 27x^3 + 20x^2 + 162x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.798544$ of defining polynomial
Character	$\chi$	$=$	129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.79854 q^{2} -3.00000 q^{3} -4.76524 q^{4} +15.8745 q^{5} +5.39563 q^{6} -13.8629 q^{7} +22.9588 q^{8} +9.00000 q^{9} -28.5510 q^{10} -19.4212 q^{11} +14.2957 q^{12} +29.8244 q^{13} +24.9330 q^{14} -47.6235 q^{15} -3.17059 q^{16} -86.7593 q^{17} -16.1869 q^{18} -139.656 q^{19} -75.6458 q^{20} +41.5887 q^{21} +34.9299 q^{22} -182.120 q^{23} -68.8765 q^{24} +127.000 q^{25} -53.6405 q^{26} -27.0000 q^{27} +66.0600 q^{28} -21.4997 q^{29} +85.6530 q^{30} +52.7481 q^{31} -177.968 q^{32} +58.2636 q^{33} +156.041 q^{34} -220.067 q^{35} -42.8871 q^{36} -52.1158 q^{37} +251.177 q^{38} -89.4731 q^{39} +364.460 q^{40} -114.113 q^{41} -74.7991 q^{42} +43.0000 q^{43} +92.5467 q^{44} +142.871 q^{45} +327.551 q^{46} +395.042 q^{47} +9.51177 q^{48} -150.820 q^{49} -228.415 q^{50} +260.278 q^{51} -142.120 q^{52} -176.074 q^{53} +48.5607 q^{54} -308.302 q^{55} -318.276 q^{56} +418.967 q^{57} +38.6681 q^{58} -636.427 q^{59} +226.937 q^{60} +635.503 q^{61} -94.8698 q^{62} -124.766 q^{63} +345.449 q^{64} +473.447 q^{65} -104.790 q^{66} -709.583 q^{67} +413.429 q^{68} +546.360 q^{69} +395.799 q^{70} +377.804 q^{71} +206.630 q^{72} +539.211 q^{73} +93.7326 q^{74} -381.000 q^{75} +665.493 q^{76} +269.234 q^{77} +160.921 q^{78} -398.534 q^{79} -50.3316 q^{80} +81.0000 q^{81} +205.237 q^{82} +280.011 q^{83} -198.180 q^{84} -1377.26 q^{85} -77.3374 q^{86} +64.4990 q^{87} -445.889 q^{88} -926.460 q^{89} -256.959 q^{90} -413.452 q^{91} +867.845 q^{92} -158.244 q^{93} -710.500 q^{94} -2216.97 q^{95} +533.905 q^{96} -805.757 q^{97} +271.257 q^{98} -174.791 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} - 15 q^{3} + 19 q^{4} - 12 q^{5} + 9 q^{6} - 18 q^{7} - 15 q^{8} + 45 q^{9} - 32 q^{10} - 60 q^{11} - 57 q^{12} + 42 q^{13} - 246 q^{14} + 36 q^{15} - 93 q^{16} - 72 q^{17} - 27 q^{18} - 102 q^{19}+ \cdots - 540 q^{99}+O(q^{100}) $ 5 * q - 3 * q^2 - 15 * q^3 + 19 * q^4 - 12 * q^5 + 9 * q^6 - 18 * q^7 - 15 * q^8 + 45 * q^9 - 32 * q^10 - 60 * q^11 - 57 * q^12 + 42 * q^13 - 246 * q^14 + 36 * q^15 - 93 * q^16 - 72 * q^17 - 27 * q^18 - 102 * q^19 - 420 * q^20 + 54 * q^21 - 86 * q^22 - 414 * q^23 + 45 * q^24 + 89 * q^25 - 144 * q^26 - 135 * q^27 - 34 * q^28 - 588 * q^29 + 96 * q^30 - 370 * q^31 - 363 * q^32 + 180 * q^33 + 562 * q^34 + 42 * q^35 + 171 * q^36 + 242 * q^37 + 486 * q^38 - 126 * q^39 + 508 * q^40 + 60 * q^41 + 738 * q^42 + 215 * q^43 + 318 * q^44 - 108 * q^45 + 388 * q^46 - 354 * q^47 + 279 * q^48 + 359 * q^49 + 753 * q^50 + 216 * q^51 + 1748 * q^52 - 264 * q^53 + 81 * q^54 - 294 * q^55 + 198 * q^56 + 306 * q^57 + 924 * q^58 - 1500 * q^59 + 1260 * q^60 + 498 * q^61 - 312 * q^62 - 162 * q^63 + 303 * q^64 - 1566 * q^65 + 258 * q^66 + 186 * q^67 + 570 * q^68 + 1242 * q^69 + 3062 * q^70 - 1680 * q^71 - 135 * q^72 - 250 * q^73 - 498 * q^74 - 267 * q^75 - 646 * q^76 + 318 * q^77 + 432 * q^78 - 1568 * q^79 + 216 * q^80 + 405 * q^81 + 3206 * q^82 - 840 * q^83 + 102 * q^84 - 1332 * q^85 - 129 * q^86 + 1764 * q^87 - 1202 * q^88 - 1782 * q^89 - 288 * q^90 - 2322 * q^91 + 240 * q^92 + 1110 * q^93 - 1206 * q^94 - 2634 * q^95 + 1089 * q^96 + 1726 * q^97 + 1599 * q^98 - 540 * 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 $

$2$	−1.79854	−0.635881	−0.317941	−	0.948111i	$-0.602991\pi$
$2$	−1.79854	−0.635881	−0.317941	+	0.948111i	$0.602991\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.76524	−0.595655
$5$	15.8745	1.41986	0.709929	−	0.704273i	$-0.248727\pi$
$5$	15.8745	1.41986	0.709929	+	0.704273i	$0.248727\pi$
$6$	5.39563	0.367126
$7$	−13.8629	−0.748526	−0.374263	−	0.927323i	$-0.622104\pi$
$7$	−13.8629	−0.748526	−0.374263	+	0.927323i	$0.622104\pi$
$8$	22.9588	1.01465
$9$	9.00000	0.333333
$10$	−28.5510	−0.902862
$11$	−19.4212	−0.532338	−0.266169	−	0.963926i	$-0.585758\pi$
$11$	−19.4212	−0.532338	−0.266169	+	0.963926i	$0.585758\pi$
$12$	14.2957	0.343901
$13$	29.8244	0.636292	0.318146	−	0.948042i	$-0.396940\pi$
$13$	29.8244	0.636292	0.318146	+	0.948042i	$0.396940\pi$
$14$	24.9330	0.475974
$15$	−47.6235	−0.819756
$16$	−3.17059	−0.0495405
$17$	−86.7593	−1.23778	−0.618889	−	0.785478i	$-0.712417\pi$
$17$	−86.7593	−1.23778	−0.618889	+	0.785478i	$0.712417\pi$
$18$	−16.1869	−0.211960
$19$	−139.656	−1.68627	−0.843137	−	0.537698i	$-0.819294\pi$
$19$	−139.656	−1.68627	−0.843137	+	0.537698i	$0.819294\pi$
$20$	−75.6458	−0.845746
$21$	41.5887	0.432161
$22$	34.9299	0.338504
$23$	−182.120	−1.65107	−0.825536	−	0.564350i	$-0.809127\pi$
$23$	−182.120	−1.65107	−0.825536	+	0.564350i	$0.809127\pi$
$24$	−68.8765	−0.585807
$25$	127.000	1.01600
$26$	−53.6405	−0.404606
$27$	−27.0000	−0.192450
$28$	66.0600	0.445863
$29$	−21.4997	−0.137669	−0.0688343	−	0.997628i	$-0.521928\pi$
$29$	−21.4997	−0.137669	−0.0688343	+	0.997628i	$0.521928\pi$
$30$	85.6530	0.521268
$31$	52.7481	0.305608	0.152804	−	0.988257i	$-0.451170\pi$
$31$	52.7481	0.305608	0.152804	+	0.988257i	$0.451170\pi$
$32$	−177.968	−0.983145
$33$	58.2636	0.307345
$34$	156.041	0.787080
$35$	−220.067	−1.06280
$36$	−42.8871	−0.198552
$37$	−52.1158	−0.231562	−0.115781	−	0.993275i	$-0.536937\pi$
$37$	−52.1158	−0.231562	−0.115781	+	0.993275i	$0.536937\pi$
$38$	251.177	1.07227
$39$	−89.4731	−0.367363
$40$	364.460	1.44066
$41$	−114.113	−0.434669	−0.217334	−	0.976097i	$-0.569736\pi$
$41$	−114.113	−0.434669	−0.217334	+	0.976097i	$0.569736\pi$
$42$	−74.7991	−0.274803
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	92.5467	0.317090
$45$	142.871	0.473286
$46$	327.551	1.04989
$47$	395.042	1.22602	0.613008	−	0.790077i	$-0.289959\pi$
$47$	395.042	1.22602	0.613008	+	0.790077i	$0.289959\pi$
$48$	9.51177	0.0286022
$49$	−150.820	−0.439709
$50$	−228.415	−0.646055
$51$	260.278	0.714632
$52$	−142.120	−0.379010
$53$	−176.074	−0.456333	−0.228167	−	0.973622i	$-0.573273\pi$
$53$	−176.074	−0.456333	−0.228167	+	0.973622i	$0.573273\pi$
$54$	48.5607	0.122375
$55$	−308.302	−0.755845
$56$	−318.276	−0.759489
$57$	418.967	0.973571
$58$	38.6681	0.0875409
$59$	−636.427	−1.40433	−0.702167	−	0.712012i	$-0.747784\pi$
$59$	−636.427	−1.40433	−0.702167	+	0.712012i	$0.747784\pi$
$60$	226.937	0.488292
$61$	635.503	1.33390	0.666950	−	0.745103i	$-0.267600\pi$
$61$	635.503	1.33390	0.666950	+	0.745103i	$0.267600\pi$
$62$	−94.8698	−0.194330
$63$	−124.766	−0.249509
$64$	345.449	0.674704
$65$	473.447	0.903445
$66$	−104.790	−0.195435
$67$	−709.583	−1.29387	−0.646935	−	0.762545i	$-0.723950\pi$
$67$	−709.583	−1.29387	−0.646935	+	0.762545i	$0.723950\pi$
$68$	413.429	0.737289
$69$	546.360	0.953247
$70$	395.799	0.675815
$71$	377.804	0.631508	0.315754	−	0.948841i	$-0.397743\pi$
$71$	377.804	0.631508	0.315754	+	0.948841i	$0.397743\pi$
$72$	206.630	0.338216
$73$	539.211	0.864519	0.432259	−	0.901749i	$-0.357717\pi$
$73$	539.211	0.864519	0.432259	+	0.901749i	$0.357717\pi$
$74$	93.7326	0.147246
$75$	−381.000	−0.586588
$76$	665.493	1.00444
$77$	269.234	0.398468
$78$	160.921	0.233600
$79$	−398.534	−0.567577	−0.283788	−	0.958887i	$-0.591591\pi$
$79$	−398.534	−0.567577	−0.283788	+	0.958887i	$0.591591\pi$
$80$	−50.3316	−0.0703405
$81$	81.0000	0.111111
$82$	205.237	0.276398
$83$	280.011	0.370303	0.185152	−	0.982710i	$-0.440722\pi$
$83$	280.011	0.370303	0.185152	+	0.982710i	$0.440722\pi$
$84$	−198.180	−0.257419
$85$	−1377.26	−1.75747
$86$	−77.3374	−0.0969710
$87$	64.4990	0.0794830
$88$	−445.889	−0.540135
$89$	−926.460	−1.10342	−0.551711	−	0.834035i	$-0.686025\pi$
$89$	−926.460	−1.10342	−0.551711	+	0.834035i	$0.686025\pi$
$90$	−256.959	−0.300954
$91$	−413.452	−0.476281
$92$	867.845	0.983469
$93$	−158.244	−0.176443
$94$	−710.500	−0.779601
$95$	−2216.97	−2.39427
$96$	533.905	0.567619
$97$	−805.757	−0.843425	−0.421713	−	0.906730i	$-0.638571\pi$
$97$	−805.757	−0.843425	−0.421713	+	0.906730i	$0.638571\pi$
$98$	271.257	0.279603
$99$	−174.791	−0.177446
$100$	−605.185	−0.605185
$101$	−1916.83	−1.88843	−0.944215	−	0.329330i	$-0.893177\pi$
$101$	−1916.83	−1.88843	−0.944215	+	0.329330i	$0.893177\pi$
$102$	−468.122	−0.454421
$103$	862.225	0.824830	0.412415	−	0.910996i	$-0.364685\pi$
$103$	862.225	0.824830	0.412415	+	0.910996i	$0.364685\pi$
$104$	684.733	0.645612
$105$	660.200	0.613608
$106$	316.677	0.290174
$107$	−281.764	−0.254572	−0.127286	−	0.991866i	$-0.540627\pi$
$107$	−281.764	−0.254572	−0.127286	+	0.991866i	$0.540627\pi$
$108$	128.661	0.114634
$109$	1500.26	1.31834	0.659169	−	0.751995i	$-0.270908\pi$
$109$	1500.26	1.31834	0.659169	+	0.751995i	$0.270908\pi$
$110$	554.495	0.480627
$111$	156.347	0.133692
$112$	43.9536	0.0370823
$113$	864.470	0.719668	0.359834	−	0.933016i	$-0.382833\pi$
$113$	864.470	0.719668	0.359834	+	0.933016i	$0.382833\pi$
$114$	−753.531	−0.619076
$115$	−2891.06	−2.34429
$116$	102.451	0.0820029
$117$	268.419	0.212097
$118$	1144.64	0.892990
$119$	1202.74	0.926509
$120$	−1093.38	−0.831763
$121$	−953.817	−0.716617
$122$	−1142.98	−0.848202
$123$	342.338	0.250956
$124$	−251.357	−0.182037
$125$	31.7479	0.0227170
$126$	224.397	0.158658
$127$	2115.68	1.47823	0.739117	−	0.673577i	$-0.235243\pi$
$127$	2115.68	1.47823	0.739117	+	0.673577i	$0.235243\pi$
$128$	802.442	0.554113
$129$	−129.000	−0.0880451
$130$	−851.516	−0.574484
$131$	−2746.65	−1.83188	−0.915938	−	0.401319i	$-0.868552\pi$
$131$	−2746.65	−1.83188	−0.915938	+	0.401319i	$0.868552\pi$
$132$	−277.640	−0.183072
$133$	1936.03	1.26222
$134$	1276.22	0.822748
$135$	−428.612	−0.273252
$136$	−1991.89	−1.25591
$137$	549.998	0.342989	0.171494	−	0.985185i	$-0.445140\pi$
$137$	549.998	0.342989	0.171494	+	0.985185i	$0.445140\pi$
$138$	−982.653	−0.606152
$139$	2882.76	1.75908	0.879542	−	0.475821i	$-0.157849\pi$
$139$	2882.76	1.75908	0.879542	+	0.475821i	$0.157849\pi$
$140$	1048.67	0.633062
$141$	−1185.12	−0.707840
$142$	−679.497	−0.401564
$143$	−579.226	−0.338722
$144$	−28.5353	−0.0165135
$145$	−341.297	−0.195470
$146$	−969.795	−0.549731
$147$	452.461	0.253866
$148$	248.344	0.137931
$149$	1607.75	0.883972	0.441986	−	0.897022i	$-0.354274\pi$
$149$	1607.75	0.883972	0.441986	+	0.897022i	$0.354274\pi$
$150$	685.245	0.373000
$151$	1000.01	0.538938	0.269469	−	0.963009i	$-0.413152\pi$
$151$	1000.01	0.538938	0.269469	+	0.963009i	$0.413152\pi$
$152$	−3206.33	−1.71097
$153$	−780.834	−0.412593
$154$	−484.229	−0.253379
$155$	837.350	0.433920
$156$	426.361	0.218822
$157$	−1452.58	−0.738396	−0.369198	−	0.929351i	$-0.620368\pi$
$157$	−1452.58	−0.738396	−0.369198	+	0.929351i	$0.620368\pi$
$158$	716.781	0.360911
$159$	528.223	0.263464
$160$	−2825.16	−1.39593
$161$	2524.71	1.23587
$162$	−145.682	−0.0706535
$163$	2940.80	1.41314	0.706568	−	0.707645i	$-0.250243\pi$
$163$	2940.80	1.41314	0.706568	+	0.707645i	$0.250243\pi$
$164$	543.775	0.258913
$165$	924.906	0.436387
$166$	−503.612	−0.235469
$167$	2979.84	1.38076	0.690379	−	0.723448i	$-0.257444\pi$
$167$	2979.84	1.38076	0.690379	+	0.723448i	$0.257444\pi$
$168$	954.828	0.438491
$169$	−1307.51	−0.595133
$170$	2477.07	1.11754
$171$	−1256.90	−0.562092
$172$	−204.905	−0.0908365
$173$	3026.47	1.33005	0.665023	−	0.746823i	$-0.268422\pi$
$173$	3026.47	1.33005	0.665023	+	0.746823i	$0.268422\pi$
$174$	−116.004	−0.0505417
$175$	−1760.59	−0.760502
$176$	61.5767	0.0263723
$177$	1909.28	0.810793
$178$	1666.28	0.701646
$179$	−3156.81	−1.31816	−0.659081	−	0.752072i	$-0.729055\pi$
$179$	−3156.81	−1.31816	−0.659081	+	0.752072i	$0.729055\pi$
$180$	−680.812	−0.281915
$181$	−449.095	−0.184425	−0.0922126	−	0.995739i	$-0.529394\pi$
$181$	−449.095	−0.184425	−0.0922126	+	0.995739i	$0.529394\pi$
$182$	743.612	0.302858
$183$	−1906.51	−0.770127
$184$	−4181.27	−1.67526
$185$	−827.313	−0.328785
$186$	284.609	0.112197
$187$	1684.97	0.658916
$188$	−1882.47	−0.730282
$189$	374.298	0.144054
$190$	3987.31	1.52247
$191$	3042.71	1.15268	0.576342	−	0.817209i	$-0.304480\pi$
$191$	3042.71	1.15268	0.576342	+	0.817209i	$0.304480\pi$
$192$	−1036.35	−0.389541
$193$	2992.68	1.11615	0.558077	−	0.829789i	$-0.311539\pi$
$193$	2992.68	1.11615	0.558077	+	0.829789i	$0.311539\pi$
$194$	1449.19	0.536318
$195$	−1420.34	−0.521604
$196$	718.695	0.261915
$197$	−742.880	−0.268670	−0.134335	−	0.990936i	$-0.542890\pi$
$197$	−742.880	−0.268670	−0.134335	+	0.990936i	$0.542890\pi$
$198$	314.369	0.112835
$199$	−2837.30	−1.01071	−0.505353	−	0.862912i	$-0.668638\pi$
$199$	−2837.30	−1.01071	−0.505353	+	0.862912i	$0.668638\pi$
$200$	2915.77	1.03088
$201$	2128.75	0.747016
$202$	3447.50	1.20082
$203$	298.047	0.103048
$204$	−1240.29	−0.425674
$205$	−1811.48	−0.617168
$206$	−1550.75	−0.524494
$207$	−1639.08	−0.550357
$208$	−94.5609	−0.0315222
$209$	2712.28	0.897668
$210$	−1187.40	−0.390182
$211$	−1142.10	−0.372631	−0.186315	−	0.982490i	$-0.559655\pi$
$211$	−1142.10	−0.372631	−0.186315	+	0.982490i	$0.559655\pi$
$212$	839.036	0.271817
$213$	−1133.41	−0.364601
$214$	506.765	0.161877
$215$	682.604	0.216526
$216$	−619.889	−0.195269
$217$	−731.241	−0.228755
$218$	−2698.28	−0.838306
$219$	−1617.63	−0.499130
$220$	1469.13	0.450222
$221$	−2587.54	−0.787588
$222$	−281.198	−0.0850124
$223$	−560.071	−0.168185	−0.0840923	−	0.996458i	$-0.526799\pi$
$223$	−560.071	−0.168185	−0.0840923	+	0.996458i	$0.526799\pi$
$224$	2467.16	0.735910
$225$	1143.00	0.338666
$226$	−1554.79	−0.457623
$227$	−3639.04	−1.06402	−0.532008	−	0.846739i	$-0.678562\pi$
$227$	−3639.04	−1.06402	−0.532008	+	0.846739i	$0.678562\pi$
$228$	−1996.48	−0.579912
$229$	3825.22	1.10383	0.551917	−	0.833899i	$-0.313897\pi$
$229$	3825.22	1.10383	0.551917	+	0.833899i	$0.313897\pi$
$230$	5199.71	1.49069
$231$	−807.702	−0.230056
$232$	−493.608	−0.139685
$233$	−1493.65	−0.419966	−0.209983	−	0.977705i	$-0.567341\pi$
$233$	−1493.65	−0.419966	−0.209983	+	0.977705i	$0.567341\pi$
$234$	−482.764	−0.134869
$235$	6271.09	1.74077
$236$	3032.73	0.836499
$237$	1195.60	0.327690
$238$	−2163.17	−0.589150
$239$	−3975.45	−1.07594	−0.537972	−	0.842963i	$-0.680809\pi$
$239$	−3975.45	−1.07594	−0.537972	+	0.842963i	$0.680809\pi$
$240$	150.995	0.0406111
$241$	−6797.14	−1.81677	−0.908386	−	0.418133i	$-0.862685\pi$
$241$	−6797.14	−1.81677	−0.908386	+	0.418133i	$0.862685\pi$
$242$	1715.48	0.455683
$243$	−243.000	−0.0641500
$244$	−3028.32	−0.794543
$245$	−2394.20	−0.624325
$246$	−615.711	−0.159578
$247$	−4165.15	−1.07296
$248$	1211.04	0.310084
$249$	−840.032	−0.213795
$250$	−57.1000	−0.0144453
$251$	2598.82	0.653531	0.326765	−	0.945105i	$-0.394041\pi$
$251$	2598.82	0.653531	0.326765	+	0.945105i	$0.394041\pi$
$252$	594.540	0.148621
$253$	3536.99	0.878928
$254$	−3805.14	−0.939982
$255$	4131.78	1.01468
$256$	−4206.82	−1.02705
$257$	−2589.69	−0.628563	−0.314282	−	0.949330i	$-0.601764\pi$
$257$	−2589.69	−0.628563	−0.314282	+	0.949330i	$0.601764\pi$
$258$	232.012	0.0559862
$259$	722.476	0.173330
$260$	−2256.09	−0.538141
$261$	−193.497	−0.0458895
$262$	4939.97	1.16486
$263$	2868.27	0.672490	0.336245	−	0.941775i	$-0.390843\pi$
$263$	2868.27	0.672490	0.336245	+	0.941775i	$0.390843\pi$
$264$	1337.67	0.311847
$265$	−2795.09	−0.647929
$266$	−3482.04	−0.802622
$267$	2779.38	0.637061
$268$	3381.33	0.770700
$269$	3284.38	0.744432	0.372216	−	0.928146i	$-0.378598\pi$
$269$	3284.38	0.744432	0.372216	+	0.928146i	$0.378598\pi$
$270$	770.877	0.173756
$271$	4402.44	0.986823	0.493411	−	0.869796i	$-0.335750\pi$
$271$	4402.44	0.986823	0.493411	+	0.869796i	$0.335750\pi$
$272$	275.078	0.0613201
$273$	1240.36	0.274981
$274$	−989.195	−0.218100
$275$	−2466.49	−0.540855
$276$	−2603.54	−0.567806
$277$	4129.43	0.895717	0.447858	−	0.894104i	$-0.352187\pi$
$277$	4129.43	0.895717	0.447858	+	0.894104i	$0.352187\pi$
$278$	−5184.78	−1.11857
$279$	474.733	0.101869
$280$	−5052.47	−1.07837
$281$	−1035.87	−0.219909	−0.109955	−	0.993937i	$-0.535071\pi$
$281$	−1035.87	−0.219909	−0.109955	+	0.993937i	$0.535071\pi$
$282$	2131.50	0.450103
$283$	−727.753	−0.152864	−0.0764319	−	0.997075i	$-0.524353\pi$
$283$	−727.753	−0.152864	−0.0764319	+	0.997075i	$0.524353\pi$
$284$	−1800.33	−0.376161
$285$	6650.90	1.38233
$286$	1041.76	0.215387
$287$	1581.93	0.325361
$288$	−1601.71	−0.327715
$289$	2614.18	0.532095
$290$	613.837	0.124296
$291$	2417.27	0.486952
$292$	−2569.47	−0.514955
$293$	−6566.10	−1.30920	−0.654600	−	0.755976i	$-0.727163\pi$
$293$	−6566.10	−1.30920	−0.654600	+	0.755976i	$0.727163\pi$
$294$	−813.771	−0.161429
$295$	−10103.0	−1.99396
$296$	−1196.52	−0.234954
$297$	524.373	0.102448
$298$	−2891.61	−0.562102
$299$	−5431.62	−1.05056
$300$	1815.55	0.349404
$301$	−596.104	−0.114149
$302$	−1798.56	−0.342701
$303$	5750.48	1.09029
$304$	442.791	0.0835389
$305$	10088.3	1.89395
$306$	1404.36	0.262360
$307$	−9731.07	−1.80906	−0.904531	−	0.426409i	$-0.859778\pi$
$307$	−9731.07	−1.80906	−0.904531	+	0.426409i	$0.859778\pi$
$308$	−1282.96	−0.237350
$309$	−2586.67	−0.476216
$310$	−1506.01	−0.275922
$311$	−6608.94	−1.20501	−0.602506	−	0.798115i	$-0.705831\pi$
$311$	−6608.94	−1.20501	−0.602506	+	0.798115i	$0.705831\pi$
$312$	−2054.20	−0.372744
$313$	602.437	0.108791	0.0543957	−	0.998519i	$-0.482677\pi$
$313$	602.437	0.108791	0.0543957	+	0.998519i	$0.482677\pi$
$314$	2612.52	0.469532
$315$	−1980.60	−0.354267
$316$	1899.11	0.338080
$317$	−4530.62	−0.802728	−0.401364	−	0.915919i	$-0.631464\pi$
$317$	−4530.62	−0.802728	−0.401364	+	0.915919i	$0.631464\pi$
$318$	−950.032	−0.167532
$319$	417.550	0.0732862
$320$	5483.83	0.957985
$321$	845.293	0.146977
$322$	−4540.80	−0.785866
$323$	12116.4	2.08723
$324$	−385.984	−0.0661839
$325$	3787.69	0.646472
$326$	−5289.16	−0.898587
$327$	−4500.78	−0.761142
$328$	−2619.90	−0.441035
$329$	−5476.42	−0.917704
$330$	−1663.48	−0.277490
$331$	−8363.35	−1.38879	−0.694397	−	0.719592i	$-0.744329\pi$
$331$	−8363.35	−1.38879	−0.694397	+	0.719592i	$0.744329\pi$
$332$	−1334.32	−0.220573
$333$	−469.042	−0.0771873
$334$	−5359.37	−0.877999
$335$	−11264.3	−1.83711
$336$	−131.861	−0.0214095
$337$	6673.33	1.07869	0.539347	−	0.842084i	$-0.318671\pi$
$337$	6673.33	1.07869	0.539347	+	0.842084i	$0.318671\pi$
$338$	2351.61	0.378434
$339$	−2593.41	−0.415500
$340$	6562.98	1.04685
$341$	−1024.43	−0.162687
$342$	2260.59	0.357424
$343$	6845.78	1.07766
$344$	987.230	0.154732
$345$	8673.19	1.35348
$346$	−5443.23	−0.845751
$347$	−4057.98	−0.627792	−0.313896	−	0.949457i	$-0.601634\pi$
$347$	−4057.98	−0.627792	−0.313896	+	0.949457i	$0.601634\pi$
$348$	−307.353	−0.0473444
$349$	8984.32	1.37799	0.688997	−	0.724764i	$-0.258051\pi$
$349$	8984.32	1.37799	0.688997	+	0.724764i	$0.258051\pi$
$350$	3166.49	0.483589
$351$	−805.258	−0.122454
$352$	3456.36	0.523365
$353$	961.720	0.145006	0.0725031	−	0.997368i	$-0.476901\pi$
$353$	961.720	0.145006	0.0725031	+	0.997368i	$0.476901\pi$
$354$	−3433.93	−0.515568
$355$	5997.45	0.896652
$356$	4414.80	0.657259
$357$	−3608.21	−0.534920
$358$	5677.66	0.838195
$359$	−652.646	−0.0959480	−0.0479740	−	0.998849i	$-0.515276\pi$
$359$	−652.646	−0.0959480	−0.0479740	+	0.998849i	$0.515276\pi$
$360$	3280.14	0.480219
$361$	12644.7	1.84352
$362$	807.717	0.117273
$363$	2861.45	0.413739
$364$	1970.20	0.283699
$365$	8559.71	1.22749
$366$	3428.94	0.489709
$367$	9202.96	1.30897	0.654483	−	0.756077i	$-0.272886\pi$
$367$	9202.96	1.30897	0.654483	+	0.756077i	$0.272886\pi$
$368$	577.428	0.0817949
$369$	−1027.01	−0.144890
$370$	1487.96	0.209068
$371$	2440.90	0.341577
$372$	754.072	0.105099
$373$	7807.24	1.08376	0.541881	−	0.840455i	$-0.317712\pi$
$373$	7807.24	1.08376	0.541881	+	0.840455i	$0.317712\pi$
$374$	−3030.50	−0.418993
$375$	−95.2437	−0.0131156
$376$	9069.70	1.24397
$377$	−641.214	−0.0875974
$378$	−673.192	−0.0916012
$379$	−10505.3	−1.42380	−0.711900	−	0.702281i	$-0.752165\pi$
$379$	−10505.3	−1.42380	−0.711900	+	0.702281i	$0.752165\pi$
$380$	10564.4	1.42616
$381$	−6347.03	−0.853459
$382$	−5472.44	−0.732970
$383$	−2477.18	−0.330491	−0.165246	−	0.986252i	$-0.552842\pi$
$383$	−2477.18	−0.330491	−0.165246	+	0.986252i	$0.552842\pi$
$384$	−2407.33	−0.319918
$385$	4273.96	0.565769
$386$	−5382.47	−0.709742
$387$	387.000	0.0508329
$388$	3839.62	0.502390
$389$	−3558.54	−0.463817	−0.231909	−	0.972738i	$-0.574497\pi$
$389$	−3558.54	−0.463817	−0.231909	+	0.972738i	$0.574497\pi$
$390$	2554.55	0.331678
$391$	15800.6	2.04366
$392$	−3462.66	−0.446150
$393$	8239.95	1.05763
$394$	1336.10	0.170842
$395$	−6326.53	−0.805879
$396$	832.920	0.105697
$397$	1649.11	0.208479	0.104240	−	0.994552i	$-0.466759\pi$
$397$	1649.11	0.208479	0.104240	+	0.994552i	$0.466759\pi$
$398$	5103.00	0.642690
$399$	−5808.10	−0.728743
$400$	−402.665	−0.0503331
$401$	−12735.1	−1.58593	−0.792967	−	0.609264i	$-0.791465\pi$
$401$	−12735.1	−1.58593	−0.792967	+	0.609264i	$0.791465\pi$
$402$	−3828.65	−0.475014
$403$	1573.18	0.194456
$404$	9134.14	1.12485
$405$	1285.83	0.157762
$406$	−536.052	−0.0655266
$407$	1012.15	0.123269
$408$	5975.68	0.725099
$409$	−13949.9	−1.68650	−0.843252	−	0.537519i	$-0.819362\pi$
$409$	−13949.9	−1.68650	−0.843252	+	0.537519i	$0.819362\pi$
$410$	3258.03	0.392446
$411$	−1649.99	−0.198025
$412$	−4108.71	−0.491314
$413$	8822.72	1.05118
$414$	2947.96	0.349962
$415$	4445.03	0.525778
$416$	−5307.80	−0.625567
$417$	−8648.29	−1.01561
$418$	−4878.16	−0.570810
$419$	−15462.2	−1.80281	−0.901404	−	0.432979i	$-0.857462\pi$
$419$	−15462.2	−1.80281	−0.901404	+	0.432979i	$0.857462\pi$
$420$	−3146.01	−0.365499
$421$	3424.97	0.396492	0.198246	−	0.980152i	$-0.436476\pi$
$421$	3424.97	0.396492	0.198246	+	0.980152i	$0.436476\pi$
$422$	2054.11	0.236949
$423$	3555.37	0.408672
$424$	−4042.46	−0.463017
$425$	−11018.4	−1.25758
$426$	2038.49	0.231843
$427$	−8809.91	−0.998458
$428$	1342.67	0.151637
$429$	1737.68	0.195561
$430$	−1227.69	−0.137685
$431$	−2700.98	−0.301860	−0.150930	−	0.988544i	$-0.548227\pi$
$431$	−2700.98	−0.301860	−0.150930	+	0.988544i	$0.548227\pi$
$432$	85.6060	0.00953407
$433$	13315.1	1.47778	0.738892	−	0.673823i	$-0.235349\pi$
$433$	13315.1	1.47778	0.738892	+	0.673823i	$0.235349\pi$
$434$	1315.17	0.145461
$435$	1023.89	0.112855
$436$	−7149.09	−0.785274
$437$	25434.1	2.78416
$438$	2909.38	0.317388
$439$	−2137.64	−0.232401	−0.116201	−	0.993226i	$-0.537072\pi$
$439$	−2137.64	−0.232401	−0.116201	+	0.993226i	$0.537072\pi$
$440$	−7078.26	−0.766916
$441$	−1357.38	−0.146570
$442$	4653.81	0.500813
$443$	12413.5	1.33134	0.665669	−	0.746247i	$-0.268146\pi$
$443$	12413.5	1.33134	0.665669	+	0.746247i	$0.268146\pi$
$444$	−745.033	−0.0796345
$445$	−14707.1	−1.56670
$446$	1007.31	0.106945
$447$	−4823.25	−0.510362
$448$	−4788.92	−0.505034
$449$	−9692.45	−1.01874	−0.509371	−	0.860547i	$-0.670122\pi$
$449$	−9692.45	−1.01874	−0.509371	+	0.860547i	$0.670122\pi$
$450$	−2055.73	−0.215352
$451$	2216.21	0.231391
$452$	−4119.40	−0.428674
$453$	−3000.03	−0.311156
$454$	6544.97	0.676588
$455$	−6563.35	−0.676252
$456$	9619.00	0.987831
$457$	−12760.5	−1.30615	−0.653077	−	0.757292i	$-0.726522\pi$
$457$	−12760.5	−1.30615	−0.653077	+	0.757292i	$0.726522\pi$
$458$	−6879.83	−0.701907
$459$	2342.50	0.238211
$460$	13776.6	1.39639
$461$	−5418.09	−0.547387	−0.273694	−	0.961817i	$-0.588245\pi$
$461$	−5418.09	−0.547387	−0.273694	+	0.961817i	$0.588245\pi$
$462$	1452.69	0.146288
$463$	−12664.5	−1.27121	−0.635606	−	0.772013i	$-0.719250\pi$
$463$	−12664.5	−1.27121	−0.635606	+	0.772013i	$0.719250\pi$
$464$	68.1666	0.00682017
$465$	−2512.05	−0.250524
$466$	2686.39	0.267049
$467$	8797.04	0.871688	0.435844	−	0.900022i	$-0.356450\pi$
$467$	8797.04	0.871688	0.435844	+	0.900022i	$0.356450\pi$
$468$	−1279.08	−0.126337
$469$	9836.86	0.968495
$470$	−11278.8	−1.10692
$471$	4357.73	0.426313
$472$	−14611.6	−1.42490
$473$	−835.112	−0.0811807
$474$	−2150.34	−0.208372
$475$	−17736.3	−1.71325
$476$	−5731.32	−0.551879
$477$	−1584.67	−0.152111
$478$	7150.03	0.684173
$479$	13666.2	1.30360	0.651800	−	0.758391i	$-0.274014\pi$
$479$	13666.2	1.30360	0.651800	+	0.758391i	$0.274014\pi$
$480$	8475.48	0.805939
$481$	−1554.32	−0.147341
$482$	12225.0	1.15525
$483$	−7574.13	−0.713530
$484$	4545.16	0.426856
$485$	−12791.0	−1.19754
$486$	437.046	0.0407918
$487$	2132.79	0.198451	0.0992257	−	0.995065i	$-0.468363\pi$
$487$	2132.79	0.198451	0.0992257	+	0.995065i	$0.468363\pi$
$488$	14590.4	1.35344
$489$	−8822.39	−0.815874
$490$	4306.07	0.396997
$491$	8153.74	0.749436	0.374718	−	0.927139i	$-0.377739\pi$
$491$	8153.74	0.749436	0.374718	+	0.927139i	$0.377739\pi$
$492$	−1631.32	−0.149483
$493$	1865.30	0.170403
$494$	7491.20	0.682277
$495$	−2774.72	−0.251948
$496$	−167.243	−0.0151400
$497$	−5237.45	−0.472700
$498$	1510.83	0.135948
$499$	−20090.5	−1.80235	−0.901176	−	0.433452i	$-0.857295\pi$
$499$	−20090.5	−1.80235	−0.901176	+	0.433452i	$0.857295\pi$
$500$	−151.286	−0.0135315
$501$	−8939.51	−0.797181
$502$	−4674.10	−0.415568
$503$	−11273.2	−0.999295	−0.499647	−	0.866229i	$-0.666537\pi$
$503$	−11273.2	−0.999295	−0.499647	+	0.866229i	$0.666537\pi$
$504$	−2864.48	−0.253163
$505$	−30428.7	−2.68130
$506$	−6361.43	−0.558894
$507$	3922.52	0.343600
$508$	−10081.7	−0.880518
$509$	−4063.75	−0.353876	−0.176938	−	0.984222i	$-0.556619\pi$
$509$	−4063.75	−0.353876	−0.176938	+	0.984222i	$0.556619\pi$
$510$	−7431.20	−0.645214
$511$	−7475.02	−0.647114
$512$	1146.61	0.0989716
$513$	3770.70	0.324524
$514$	4657.68	0.399692
$515$	13687.4	1.17114
$516$	614.716	0.0524445
$517$	−7672.19	−0.652654
$518$	−1299.40	−0.110217
$519$	−9079.40	−0.767902
$520$	10869.8	0.916678
$521$	3516.11	0.295669	0.147834	−	0.989012i	$-0.452770\pi$
$521$	3516.11	0.295669	0.147834	+	0.989012i	$0.452770\pi$
$522$	348.013	0.0291803
$523$	12440.1	1.04009	0.520046	−	0.854138i	$-0.325915\pi$
$523$	12440.1	1.04009	0.520046	+	0.854138i	$0.325915\pi$
$524$	13088.4	1.09117
$525$	5281.76	0.439076
$526$	−5158.70	−0.427624
$527$	−4576.39	−0.378275
$528$	−184.730	−0.0152260
$529$	21000.7	1.72604
$530$	5027.09	0.412006
$531$	−5727.84	−0.468112
$532$	−9225.65	−0.751847
$533$	−3403.34	−0.276576
$534$	−4998.84	−0.405095
$535$	−4472.87	−0.361456
$536$	−16291.2	−1.31282
$537$	9470.43	0.761042
$538$	−5907.10	−0.473370
$539$	2929.11	0.234074
$540$	2042.44	0.162764
$541$	15406.7	1.22438	0.612188	−	0.790712i	$-0.290290\pi$
$541$	15406.7	1.22438	0.612188	+	0.790712i	$0.290290\pi$
$542$	−7917.98	−0.627502
$543$	1347.28	0.106478
$544$	15440.4	1.21692
$545$	23815.9	1.87185
$546$	−2230.84	−0.174855
$547$	168.517	0.0131723	0.00658617	−	0.999978i	$-0.497904\pi$
$547$	168.517	0.0131723	0.00658617	+	0.999978i	$0.497904\pi$
$548$	−2620.87	−0.204303
$549$	5719.53	0.444633
$550$	4436.10	0.343920
$551$	3002.55	0.232147
$552$	12543.8	0.967209
$553$	5524.83	0.424846
$554$	−7426.97	−0.569570
$555$	2481.94	0.189824
$556$	−13737.1	−1.04781
$557$	13682.0	1.04080	0.520401	−	0.853922i	$-0.325783\pi$
$557$	13682.0	1.04080	0.520401	+	0.853922i	$0.325783\pi$
$558$	−853.828	−0.0647768
$559$	1282.45	0.0970336
$560$	697.741	0.0526517
$561$	−5054.91	−0.380425
$562$	1863.05	0.139836
$563$	10203.3	0.763794	0.381897	−	0.924205i	$-0.375271\pi$
$563$	10203.3	0.763794	0.381897	+	0.924205i	$0.375271\pi$
$564$	5647.40	0.421629
$565$	13723.0	1.02183
$566$	1308.90	0.0972032
$567$	−1122.89	−0.0831695
$568$	8673.94	0.640758
$569$	11342.0	0.835642	0.417821	−	0.908529i	$-0.362794\pi$
$569$	11342.0	0.835642	0.417821	+	0.908529i	$0.362794\pi$
$570$	−11961.9	−0.879000
$571$	−18747.7	−1.37402	−0.687012	−	0.726646i	$-0.741078\pi$
$571$	−18747.7	−1.37402	−0.687012	+	0.726646i	$0.741078\pi$
$572$	2760.15	0.201762
$573$	−9128.12	−0.665502
$574$	−2845.18	−0.206891
$575$	−23129.2	−1.67749
$576$	3109.04	0.224901
$577$	−25011.9	−1.80461	−0.902304	−	0.431101i	$-0.858125\pi$
$577$	−25011.9	−1.80461	−0.902304	+	0.431101i	$0.858125\pi$
$578$	−4701.72	−0.338349
$579$	−8978.04	−0.644412
$580$	1626.36	0.116433
$581$	−3881.76	−0.277181
$582$	−4347.57	−0.309644
$583$	3419.57	0.242923
$584$	12379.7	0.877181
$585$	4261.03	0.301148
$586$	11809.4	0.832496
$587$	24860.2	1.74802	0.874011	−	0.485906i	$-0.161510\pi$
$587$	24860.2	1.74802	0.874011	+	0.485906i	$0.161510\pi$
$588$	−2156.08	−0.151217
$589$	−7366.58	−0.515339
$590$	18170.6	1.26792
$591$	2228.64	0.155117
$592$	165.238	0.0114717
$593$	−12475.8	−0.863946	−0.431973	−	0.901886i	$-0.642182\pi$
$593$	−12475.8	−0.863946	−0.431973	+	0.901886i	$0.642182\pi$
$594$	−943.107	−0.0651451
$595$	19092.8	1.31551
$596$	−7661.30	−0.526542
$597$	8511.89	0.583532
$598$	9769.00	0.668034
$599$	7451.41	0.508275	0.254137	−	0.967168i	$-0.418208\pi$
$599$	7451.41	0.508275	0.254137	+	0.967168i	$0.418208\pi$
$600$	−8747.32	−0.595179
$601$	−15678.6	−1.06413	−0.532067	−	0.846702i	$-0.678585\pi$
$601$	−15678.6	−1.06413	−0.532067	+	0.846702i	$0.678585\pi$
$602$	1072.12	0.0725853
$603$	−6386.24	−0.431290
$604$	−4765.29	−0.321021
$605$	−15141.4	−1.01749
$606$	−10342.5	−0.693292
$607$	−4861.24	−0.325060	−0.162530	−	0.986704i	$-0.551965\pi$
$607$	−4861.24	−0.325060	−0.162530	+	0.986704i	$0.551965\pi$
$608$	24854.3	1.65785
$609$	−894.142	−0.0594950
$610$	−18144.3	−1.20433
$611$	11781.9	0.780104
$612$	3720.86	0.245763
$613$	6442.10	0.424460	0.212230	−	0.977220i	$-0.431927\pi$
$613$	6442.10	0.424460	0.212230	+	0.977220i	$0.431927\pi$
$614$	17501.8	1.15035
$615$	5434.45	0.356322
$616$	6181.30	0.404305
$617$	−15406.4	−1.00525	−0.502625	−	0.864504i	$-0.667632\pi$
$617$	−15406.4	−1.00525	−0.502625	+	0.864504i	$0.667632\pi$
$618$	4652.25	0.302817
$619$	20117.9	1.30631	0.653157	−	0.757223i	$-0.273444\pi$
$619$	20117.9	1.30631	0.653157	+	0.757223i	$0.273444\pi$
$620$	−3990.17	−0.258466
$621$	4917.24	0.317749
$622$	11886.5	0.766244
$623$	12843.4	0.825940
$624$	283.683	0.0181994
$625$	−15371.0	−0.983745
$626$	−1083.51	−0.0691785
$627$	−8136.85	−0.518269
$628$	6921.87	0.439829
$629$	4521.53	0.286622
$630$	3562.19	0.225272
$631$	−27335.4	−1.72457	−0.862286	−	0.506422i	$-0.830968\pi$
$631$	−27335.4	−1.72457	−0.862286	+	0.506422i	$0.830968\pi$
$632$	−9149.87	−0.575890
$633$	3426.29	0.215139
$634$	8148.51	0.510440
$635$	33585.3	2.09888
$636$	−2517.11	−0.156934
$637$	−4498.12	−0.279784
$638$	−750.981	−0.0466013
$639$	3400.23	0.210503
$640$	12738.4	0.786763
$641$	23746.8	1.46325	0.731624	−	0.681708i	$-0.238763\pi$
$641$	23746.8	1.46325	0.731624	+	0.681708i	$0.238763\pi$
$642$	−1520.30	−0.0934600
$643$	−10330.7	−0.633596	−0.316798	−	0.948493i	$-0.602608\pi$
$643$	−10330.7	−0.633596	−0.316798	+	0.948493i	$0.602608\pi$
$644$	−12030.8	−0.736152
$645$	−2047.81	−0.125012
$646$	−21792.0	−1.32723
$647$	−23240.7	−1.41219	−0.706094	−	0.708118i	$-0.749545\pi$
$647$	−23240.7	−1.41219	−0.706094	+	0.708118i	$0.749545\pi$
$648$	1859.67	0.112739
$649$	12360.2	0.747580
$650$	−6812.34	−0.411080
$651$	2193.72	0.132072
$652$	−14013.6	−0.841741
$653$	−8881.27	−0.532237	−0.266119	−	0.963940i	$-0.585741\pi$
$653$	−8881.27	−0.532237	−0.266119	+	0.963940i	$0.585741\pi$
$654$	8094.85	0.483996
$655$	−43601.7	−2.60101
$656$	361.805	0.0215337
$657$	4852.90	0.288173
$658$	9849.58	0.583551
$659$	20220.2	1.19525	0.597623	−	0.801777i	$-0.296112\pi$
$659$	20220.2	1.19525	0.597623	+	0.801777i	$0.296112\pi$
$660$	−4407.40	−0.259936
$661$	21173.9	1.24595	0.622973	−	0.782244i	$-0.285925\pi$
$661$	21173.9	1.24595	0.622973	+	0.782244i	$0.285925\pi$
$662$	15041.9	0.883109
$663$	7762.63	0.454714
$664$	6428.72	0.375727
$665$	30733.6	1.79217
$666$	843.594	0.0490820
$667$	3915.52	0.227301
$668$	−14199.6	−0.822456
$669$	1680.21	0.0971014
$670$	20259.3	1.16819
$671$	−12342.2	−0.710085
$672$	−7401.47	−0.424878
$673$	−5831.33	−0.333999	−0.167000	−	0.985957i	$-0.553408\pi$
$673$	−5831.33	−0.333999	−0.167000	+	0.985957i	$0.553408\pi$
$674$	−12002.3	−0.685921
$675$	−3429.00	−0.195529
$676$	6230.58	0.354494
$677$	−17236.1	−0.978486	−0.489243	−	0.872147i	$-0.662727\pi$
$677$	−17236.1	−0.978486	−0.489243	+	0.872147i	$0.662727\pi$
$678$	4664.36	0.264209
$679$	11170.1	0.631325
$680$	−31620.3	−1.78321
$681$	10917.1	0.614310
$682$	1842.49	0.103449
$683$	7667.39	0.429553	0.214776	−	0.976663i	$-0.431098\pi$
$683$	7667.39	0.429553	0.214776	+	0.976663i	$0.431098\pi$
$684$	5989.44	0.334813
$685$	8730.94	0.486996
$686$	−12312.4	−0.685264
$687$	−11475.7	−0.637298
$688$	−136.335	−0.00755485
$689$	−5251.30	−0.290361
$690$	−15599.1	−0.860650
$691$	30169.5	1.66093	0.830465	−	0.557071i	$-0.188075\pi$
$691$	30169.5	1.66093	0.830465	+	0.557071i	$0.188075\pi$
$692$	−14421.8	−0.792248
$693$	2423.11	0.132823
$694$	7298.45	0.399201
$695$	45762.4	2.49765
$696$	1480.82	0.0806472
$697$	9900.35	0.538024
$698$	−16158.7	−0.876240
$699$	4480.94	0.242467
$700$	8389.61	0.452996
$701$	−20355.0	−1.09671	−0.548357	−	0.836244i	$-0.684746\pi$
$701$	−20355.0	−1.09671	−0.548357	+	0.836244i	$0.684746\pi$
$702$	1448.29	0.0778665
$703$	7278.27	0.390477
$704$	−6709.03	−0.359171
$705$	−18813.3	−1.00503
$706$	−1729.70	−0.0922068
$707$	26572.8	1.41354
$708$	−9098.18	−0.482953
$709$	−9215.45	−0.488143	−0.244072	−	0.969757i	$-0.578483\pi$
$709$	−9215.45	−0.488143	−0.244072	+	0.969757i	$0.578483\pi$
$710$	−10786.7	−0.570164
$711$	−3586.80	−0.189192
$712$	−21270.5	−1.11958
$713$	−9606.48	−0.504580
$714$	6489.52	0.340146
$715$	−9194.92	−0.480938
$716$	15043.0	0.785170
$717$	11926.4	0.621197
$718$	1173.81	0.0610115
$719$	8522.47	0.442051	0.221025	−	0.975268i	$-0.429060\pi$
$719$	8522.47	0.442051	0.221025	+	0.975268i	$0.429060\pi$
$720$	−452.984	−0.0234468
$721$	−11952.9	−0.617407
$722$	−22742.1	−1.17226
$723$	20391.4	1.04891
$724$	2140.04	0.109854
$725$	−2730.46	−0.139871
$726$	−5146.44	−0.263089
$727$	−8716.00	−0.444647	−0.222324	−	0.974973i	$-0.571364\pi$
$727$	−8716.00	−0.444647	−0.222324	+	0.974973i	$0.571364\pi$
$728$	−9492.38	−0.483257
$729$	729.000	0.0370370
$730$	−15395.0	−0.780541
$731$	−3730.65	−0.188759
$732$	9084.97	0.458730
$733$	11410.4	0.574968	0.287484	−	0.957785i	$-0.407181\pi$
$733$	11410.4	0.574968	0.287484	+	0.957785i	$0.407181\pi$
$734$	−16551.9	−0.832347
$735$	7182.59	0.360454
$736$	32411.6	1.62324
$737$	13781.0	0.688776
$738$	1847.13	0.0921326
$739$	−19087.3	−0.950121	−0.475061	−	0.879953i	$-0.657574\pi$
$739$	−19087.3	−0.950121	−0.475061	+	0.879953i	$0.657574\pi$
$740$	3942.34	0.195842
$741$	12495.4	0.619476
$742$	−4390.06	−0.217202
$743$	−28348.4	−1.39974	−0.699868	−	0.714272i	$-0.746758\pi$
$743$	−28348.4	−1.39974	−0.699868	+	0.714272i	$0.746758\pi$
$744$	−3633.11	−0.179027
$745$	25522.2	1.25512
$746$	−14041.7	−0.689145
$747$	2520.10	0.123434
$748$	−8029.29	−0.392487
$749$	3906.07	0.190553
$750$	171.300	0.00833999
$751$	5307.71	0.257898	0.128949	−	0.991651i	$-0.458840\pi$
$751$	5307.71	0.257898	0.128949	+	0.991651i	$0.458840\pi$
$752$	−1252.52	−0.0607374
$753$	−7796.47	−0.377316
$754$	1153.25	0.0557015
$755$	15874.7	0.765216
$756$	−1783.62	−0.0858064
$757$	17117.3	0.821846	0.410923	−	0.911670i	$-0.365206\pi$
$757$	17117.3	0.821846	0.410923	+	0.911670i	$0.365206\pi$
$758$	18894.2	0.905368
$759$	−10611.0	−0.507449
$760$	−50899.0	−2.42934
$761$	−11428.0	−0.544370	−0.272185	−	0.962245i	$-0.587746\pi$
$761$	−11428.0	−0.544370	−0.272185	+	0.962245i	$0.587746\pi$
$762$	11415.4	0.542699
$763$	−20797.9	−0.986809
$764$	−14499.2	−0.686602
$765$	−12395.4	−0.585824
$766$	4455.33	0.210153
$767$	−18981.0	−0.893567
$768$	12620.4	0.592970
$769$	4584.57	0.214986	0.107493	−	0.994206i	$-0.465718\pi$
$769$	4584.57	0.214986	0.107493	+	0.994206i	$0.465718\pi$
$770$	−7686.90	−0.359762
$771$	7769.08	0.362901
$772$	−14260.8	−0.664843
$773$	−29603.2	−1.37743	−0.688716	−	0.725031i	$-0.741825\pi$
$773$	−29603.2	−1.37743	−0.688716	+	0.725031i	$0.741825\pi$
$774$	−696.037	−0.0323237
$775$	6699.01	0.310497
$776$	−18499.3	−0.855779
$777$	−2167.43	−0.100072
$778$	6400.18	0.294933
$779$	15936.5	0.732971
$780$	6768.27	0.310696
$781$	−7337.41	−0.336176
$782$	−28418.1	−1.29953
$783$	580.491	0.0264943
$784$	478.190	0.0217834
$785$	−23058.9	−1.04842
$786$	−14819.9	−0.672530
$787$	29561.6	1.33895	0.669477	−	0.742833i	$-0.266518\pi$
$787$	29561.6	1.33895	0.669477	+	0.742833i	$0.266518\pi$
$788$	3540.00	0.160035
$789$	−8604.80	−0.388262
$790$	11378.5	0.512443
$791$	−11984.0	−0.538690
$792$	−4013.00	−0.180045
$793$	18953.5	0.848749
$794$	−2965.99	−0.132568
$795$	8385.27	0.374082
$796$	13520.4	0.602032
$797$	−3060.54	−0.136022	−0.0680112	−	0.997685i	$-0.521665\pi$
$797$	−3060.54	−0.136022	−0.0680112	+	0.997685i	$0.521665\pi$
$798$	10446.1	0.463394
$799$	−34273.5	−1.51754
$800$	−22602.0	−0.998875
$801$	−8338.14	−0.367807
$802$	22904.6	1.00847
$803$	−10472.1	−0.460216
$804$	−10144.0	−0.444964
$805$	40078.5	1.75476
$806$	−2829.43	−0.123651
$807$	−9853.14	−0.429798
$808$	−44008.1	−1.91609
$809$	−25313.6	−1.10010	−0.550049	−	0.835132i	$-0.685391\pi$
$809$	−25313.6	−1.10010	−0.550049	+	0.835132i	$0.685391\pi$
$810$	−2312.63	−0.100318
$811$	11217.5	0.485696	0.242848	−	0.970064i	$-0.421918\pi$
$811$	11217.5	0.485696	0.242848	+	0.970064i	$0.421918\pi$
$812$	−1420.27	−0.0613813
$813$	−13207.3	−0.569742
$814$	−1820.40	−0.0783845
$815$	46683.7	2.00645
$816$	−825.235	−0.0354032
$817$	−6005.20	−0.257155
$818$	25089.6	1.07242
$819$	−3721.07	−0.158760
$820$	8632.15	0.367619
$821$	−4267.25	−0.181398	−0.0906992	−	0.995878i	$-0.528910\pi$
$821$	−4267.25	−0.181398	−0.0906992	+	0.995878i	$0.528910\pi$
$822$	2967.59	0.125920
$823$	−3901.96	−0.165266	−0.0826330	−	0.996580i	$-0.526333\pi$
$823$	−3901.96	−0.165266	−0.0826330	+	0.996580i	$0.526333\pi$
$824$	19795.7	0.836912
$825$	7399.48	0.312263
$826$	−15868.0	−0.668426
$827$	22918.9	0.963687	0.481844	−	0.876257i	$-0.339967\pi$
$827$	22918.9	0.963687	0.481844	+	0.876257i	$0.339967\pi$
$828$	7810.61	0.327823
$829$	6746.93	0.282666	0.141333	−	0.989962i	$-0.454861\pi$
$829$	6746.93	0.282666	0.141333	+	0.989962i	$0.454861\pi$
$830$	−7994.59	−0.334333
$831$	−12388.3	−0.517142
$832$	10302.8	0.429309
$833$	13085.1	0.544263
$834$	15554.3	0.645806
$835$	47303.4	1.96048
$836$	−12924.7	−0.534700
$837$	−1424.20	−0.0588142
$838$	27809.4	1.14637
$839$	32066.9	1.31952	0.659758	−	0.751478i	$-0.270659\pi$
$839$	32066.9	1.31952	0.659758	+	0.751478i	$0.270659\pi$
$840$	15157.4	0.622596
$841$	−23926.8	−0.981047
$842$	−6159.96	−0.252122
$843$	3107.60	0.126965
$844$	5442.36	0.221959
$845$	−20756.0	−0.845004
$846$	−6394.50	−0.259867
$847$	13222.7	0.536406
$848$	558.259	0.0226070
$849$	2183.26	0.0882559
$850$	19817.1	0.799673
$851$	9491.33	0.382325
$852$	5400.98	0.217177
$853$	−3613.74	−0.145055	−0.0725276	−	0.997366i	$-0.523107\pi$
$853$	−3613.74	−0.145055	−0.0725276	+	0.997366i	$0.523107\pi$
$854$	15845.0	0.634901
$855$	−19952.7	−0.798091
$856$	−6468.98	−0.258301
$857$	−44182.4	−1.76108	−0.880538	−	0.473975i	$-0.842819\pi$
$857$	−44182.4	−1.76108	−0.880538	+	0.473975i	$0.842819\pi$
$858$	−3125.29	−0.124354
$859$	−150.821	−0.00599061	−0.00299530	−	0.999996i	$-0.500953\pi$
$859$	−150.821	−0.00599061	−0.00299530	+	0.999996i	$0.500953\pi$
$860$	−3252.77	−0.128975
$861$	−4745.80	−0.187847
$862$	4857.84	0.191947
$863$	−32760.7	−1.29222	−0.646111	−	0.763243i	$-0.723606\pi$
$863$	−32760.7	−1.29222	−0.646111	+	0.763243i	$0.723606\pi$
$864$	4805.14	0.189206
$865$	48043.6	1.88848
$866$	−23947.7	−0.939696
$867$	−7842.55	−0.307205
$868$	3484.54	0.136259
$869$	7740.01	0.302142
$870$	−1841.51	−0.0717621
$871$	−21162.9	−0.823279
$872$	34444.2	1.33765
$873$	−7251.81	−0.281142
$874$	−45744.4	−1.77040
$875$	−440.118	−0.0170042
$876$	7708.41	0.297309
$877$	−39537.4	−1.52233	−0.761165	−	0.648558i	$-0.775372\pi$
$877$	−39537.4	−1.52233	−0.761165	+	0.648558i	$0.775372\pi$
$878$	3844.64	0.147780
$879$	19698.3	0.755867
$880$	977.500	0.0374449
$881$	25803.5	0.986769	0.493384	−	0.869811i	$-0.335760\pi$
$881$	25803.5	0.986769	0.493384	+	0.869811i	$0.335760\pi$
$882$	2441.31	0.0932010
$883$	−8861.95	−0.337744	−0.168872	−	0.985638i	$-0.554013\pi$
$883$	−8861.95	−0.337744	−0.168872	+	0.985638i	$0.554013\pi$
$884$	12330.3	0.469131
$885$	30308.9	1.15121
$886$	−22326.2	−0.846573
$887$	−17420.1	−0.659423	−0.329712	−	0.944082i	$-0.606951\pi$
$887$	−17420.1	−0.659423	−0.329712	+	0.944082i	$0.606951\pi$
$888$	3589.56	0.135651
$889$	−29329.4	−1.10650
$890$	26451.4	0.996238
$891$	−1573.12	−0.0591486
$892$	2668.87	0.100180
$893$	−55169.8	−2.06740
$894$	8674.82	0.324529
$895$	−50112.8	−1.87161
$896$	−11124.2	−0.414768
$897$	16294.8	0.606543
$898$	17432.3	0.647799
$899$	−1134.07	−0.0420726
$900$	−5446.66	−0.201728
$901$	15276.1	0.564839
$902$	−3985.95	−0.147137
$903$	1788.31	0.0659040
$904$	19847.2	0.730209
$905$	−7129.16	−0.261858
$906$	5395.69	0.197858
$907$	6120.82	0.224078	0.112039	−	0.993704i	$-0.464262\pi$
$907$	6120.82	0.224078	0.112039	+	0.993704i	$0.464262\pi$
$908$	17340.9	0.633786
$909$	−17251.4	−0.629477
$910$	11804.5	0.430016
$911$	−11586.9	−0.421394	−0.210697	−	0.977551i	$-0.567573\pi$
$911$	−11586.9	−0.421394	−0.210697	+	0.977551i	$0.567573\pi$
$912$	−1328.37	−0.0482312
$913$	−5438.15	−0.197126
$914$	22950.4	0.830559
$915$	−30264.9	−1.09347
$916$	−18228.1	−0.657504
$917$	38076.5	1.37121
$918$	−4213.09	−0.151474
$919$	−35347.8	−1.26879	−0.634394	−	0.773010i	$-0.718750\pi$
$919$	−35347.8	−1.26879	−0.634394	+	0.773010i	$0.718750\pi$
$920$	−66375.5	−2.37863
$921$	29193.2	1.04446
$922$	9744.68	0.348074
$923$	11267.8	0.401823
$924$	3848.89	0.137034
$925$	−6618.71	−0.235267
$926$	22777.7	0.808340
$927$	7760.02	0.274943
$928$	3826.26	0.135348
$929$	−11502.8	−0.406236	−0.203118	−	0.979154i	$-0.565108\pi$
$929$	−11502.8	−0.406236	−0.203118	+	0.979154i	$0.565108\pi$
$930$	4518.03	0.159303
$931$	21062.9	0.741471
$932$	7117.58	0.250155
$933$	19826.8	0.695713
$934$	−15821.9	−0.554290
$935$	26748.1	0.935568
$936$	6162.60	0.215204
$937$	−6562.11	−0.228788	−0.114394	−	0.993435i	$-0.536493\pi$
$937$	−6562.11	−0.228788	−0.114394	+	0.993435i	$0.536493\pi$
$938$	−17692.0	−0.615848
$939$	−1807.31	−0.0628108
$940$	−29883.2	−1.03690
$941$	10971.1	0.380072	0.190036	−	0.981777i	$-0.439140\pi$
$941$	10971.1	0.380072	0.190036	+	0.981777i	$0.439140\pi$
$942$	−7837.57	−0.271085
$943$	20782.2	0.717669
$944$	2017.85	0.0695714
$945$	5941.80	0.204536
$946$	1501.99	0.0516213
$947$	−5475.36	−0.187883	−0.0939416	−	0.995578i	$-0.529947\pi$
$947$	−5475.36	−0.187883	−0.0939416	+	0.995578i	$0.529947\pi$
$948$	−5697.33	−0.195190
$949$	16081.6	0.550086
$950$	31899.5	1.08943
$951$	13591.8	0.463455
$952$	27613.4	0.940080
$953$	5188.24	0.176352	0.0881761	−	0.996105i	$-0.471896\pi$
$953$	5188.24	0.176352	0.0881761	+	0.996105i	$0.471896\pi$
$954$	2850.10	0.0967246
$955$	48301.5	1.63665
$956$	18944.0	0.640892
$957$	−1252.65	−0.0423118
$958$	−24579.2	−0.828935
$959$	−7624.56	−0.256736
$960$	−16451.5	−0.553093
$961$	−27008.6	−0.906604
$962$	2795.52	0.0936914
$963$	−2535.88	−0.0848573
$964$	32390.0	1.08217
$965$	47507.3	1.58478
$966$	13622.4	0.453720
$967$	−20369.6	−0.677398	−0.338699	−	0.940895i	$-0.609987\pi$
$967$	−20369.6	−0.677398	−0.338699	+	0.940895i	$0.609987\pi$
$968$	−21898.5	−0.727113
$969$	−36349.3	−1.20507
$970$	23005.2	0.761496
$971$	36713.8	1.21339	0.606695	−	0.794935i	$-0.292495\pi$
$971$	36713.8	1.21339	0.606695	+	0.794935i	$0.292495\pi$
$972$	1157.95	0.0382113
$973$	−39963.4	−1.31672
$974$	−3835.91	−0.126192
$975$	−11363.1	−0.373241
$976$	−2014.92	−0.0660820
$977$	21533.7	0.705142	0.352571	−	0.935785i	$-0.385308\pi$
$977$	21533.7	0.705142	0.352571	+	0.935785i	$0.385308\pi$
$978$	15867.5	0.518799
$979$	17993.0	0.587393
$980$	11408.9	0.371882
$981$	13502.3	0.439446
$982$	−14664.9	−0.476553
$983$	6781.79	0.220046	0.110023	−	0.993929i	$-0.464907\pi$
$983$	6781.79	0.220046	0.110023	+	0.993929i	$0.464907\pi$
$984$	7859.69	0.254632
$985$	−11792.9	−0.381474
$986$	−3354.82	−0.108356
$987$	16429.3	0.529837
$988$	19847.9	0.639116
$989$	−7831.16	−0.251786
$990$	4990.45	0.160209
$991$	−28393.7	−0.910147	−0.455074	−	0.890454i	$-0.650387\pi$
$991$	−28393.7	−0.910147	−0.455074	+	0.890454i	$0.650387\pi$
$992$	−9387.49	−0.300457
$993$	25090.0	0.801821
$994$	9419.79	0.300581
$995$	−45040.7	−1.43506
$996$	4002.95	0.127348
$997$	−50484.8	−1.60368	−0.801841	−	0.597538i	$-0.796146\pi$
$997$	−50484.8	−1.60368	−0.801841	+	0.597538i	$0.796146\pi$
$998$	36133.6	1.14608
$999$	1407.13	0.0445641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.4.a.d.1.3	✓	5
3.2	odd	2		387.4.a.f.1.3		5
4.3	odd	2		2064.4.a.r.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.4.a.d.1.3	✓	5	1.1	even	1	trivial
387.4.a.f.1.3		5	3.2	odd	2
2064.4.a.r.1.5		5	4.3	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:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	129.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79854 q^{2} -3.00000 q^{3} -4.76524 q^{4} +15.8745 q^{5} +5.39563 q^{6} -13.8629 q^{7} +22.9588 q^{8} +9.00000 q^{9} -28.5510 q^{10} -19.4212 q^{11} +14.2957 q^{12} +29.8244 q^{13} +24.9330 q^{14} -47.6235 q^{15} -3.17059 q^{16} -86.7593 q^{17} -16.1869 q^{18} -139.656 q^{19} -75.6458 q^{20} +41.5887 q^{21} +34.9299 q^{22} -182.120 q^{23} -68.8765 q^{24} +127.000 q^{25} -53.6405 q^{26} -27.0000 q^{27} +66.0600 q^{28} -21.4997 q^{29} +85.6530 q^{30} +52.7481 q^{31} -177.968 q^{32} +58.2636 q^{33} +156.041 q^{34} -220.067 q^{35} -42.8871 q^{36} -52.1158 q^{37} +251.177 q^{38} -89.4731 q^{39} +364.460 q^{40} -114.113 q^{41} -74.7991 q^{42} +43.0000 q^{43} +92.5467 q^{44} +142.871 q^{45} +327.551 q^{46} +395.042 q^{47} +9.51177 q^{48} -150.820 q^{49} -228.415 q^{50} +260.278 q^{51} -142.120 q^{52} -176.074 q^{53} +48.5607 q^{54} -308.302 q^{55} -318.276 q^{56} +418.967 q^{57} +38.6681 q^{58} -636.427 q^{59} +226.937 q^{60} +635.503 q^{61} -94.8698 q^{62} -124.766 q^{63} +345.449 q^{64} +473.447 q^{65} -104.790 q^{66} -709.583 q^{67} +413.429 q^{68} +546.360 q^{69} +395.799 q^{70} +377.804 q^{71} +206.630 q^{72} +539.211 q^{73} +93.7326 q^{74} -381.000 q^{75} +665.493 q^{76} +269.234 q^{77} +160.921 q^{78} -398.534 q^{79} -50.3316 q^{80} +81.0000 q^{81} +205.237 q^{82} +280.011 q^{83} -198.180 q^{84} -1377.26 q^{85} -77.3374 q^{86} +64.4990 q^{87} -445.889 q^{88} -926.460 q^{89} -256.959 q^{90} -413.452 q^{91} +867.845 q^{92} -158.244 q^{93} -710.500 q^{94} -2216.97 q^{95} +533.905 q^{96} -805.757 q^{97} +271.257 q^{98} -174.791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} - 15 q^{3} + 19 q^{4} - 12 q^{5} + 9 q^{6} - 18 q^{7} - 15 q^{8} + 45 q^{9} - 32 q^{10} - 60 q^{11} - 57 q^{12} + 42 q^{13} - 246 q^{14} + 36 q^{15} - 93 q^{16} - 72 q^{17} - 27 q^{18} - 102 q^{19}+ \cdots - 540 q^{99}+O(q^{100}) \) 5 * q - 3 * q^2 - 15 * q^3 + 19 * q^4 - 12 * q^5 + 9 * q^6 - 18 * q^7 - 15 * q^8 + 45 * q^9 - 32 * q^10 - 60 * q^11 - 57 * q^12 + 42 * q^13 - 246 * q^14 + 36 * q^15 - 93 * q^16 - 72 * q^17 - 27 * q^18 - 102 * q^19 - 420 * q^20 + 54 * q^21 - 86 * q^22 - 414 * q^23 + 45 * q^24 + 89 * q^25 - 144 * q^26 - 135 * q^27 - 34 * q^28 - 588 * q^29 + 96 * q^30 - 370 * q^31 - 363 * q^32 + 180 * q^33 + 562 * q^34 + 42 * q^35 + 171 * q^36 + 242 * q^37 + 486 * q^38 - 126 * q^39 + 508 * q^40 + 60 * q^41 + 738 * q^42 + 215 * q^43 + 318 * q^44 - 108 * q^45 + 388 * q^46 - 354 * q^47 + 279 * q^48 + 359 * q^49 + 753 * q^50 + 216 * q^51 + 1748 * q^52 - 264 * q^53 + 81 * q^54 - 294 * q^55 + 198 * q^56 + 306 * q^57 + 924 * q^58 - 1500 * q^59 + 1260 * q^60 + 498 * q^61 - 312 * q^62 - 162 * q^63 + 303 * q^64 - 1566 * q^65 + 258 * q^66 + 186 * q^67 + 570 * q^68 + 1242 * q^69 + 3062 * q^70 - 1680 * q^71 - 135 * q^72 - 250 * q^73 - 498 * q^74 - 267 * q^75 - 646 * q^76 + 318 * q^77 + 432 * q^78 - 1568 * q^79 + 216 * q^80 + 405 * q^81 + 3206 * q^82 - 840 * q^83 + 102 * q^84 - 1332 * q^85 - 129 * q^86 + 1764 * q^87 - 1202 * q^88 - 1782 * q^89 - 288 * q^90 - 2322 * q^91 + 240 * q^92 + 1110 * q^93 - 1206 * q^94 - 2634 * q^95 + 1089 * q^96 + 1726 * q^97 + 1599 * q^98 - 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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