LMFDB - Embedded newform 637.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,4,Mod(1,637)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("637.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	637.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$37.5842166737$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -24.1771 q^{8} -13.4233 q^{9} +O(q^{10})$	\(q+2.56155 q^{2} +3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -24.1771 q^{8} -13.4233 q^{9} -1.43845 q^{10} +64.7386 q^{11} -5.30019 q^{12} +13.0000 q^{13} -2.06913 q^{15} -50.4233 q^{16} +25.5464 q^{17} -34.3845 q^{18} +107.970 q^{19} +0.807764 q^{20} +165.831 q^{22} +73.2614 q^{23} -89.0843 q^{24} -124.685 q^{25} +33.3002 q^{26} -148.946 q^{27} +175.909 q^{29} -5.30019 q^{30} +113.093 q^{31} +64.2547 q^{32} +238.540 q^{33} +65.4384 q^{34} +19.3087 q^{36} +114.808 q^{37} +276.570 q^{38} +47.9006 q^{39} +13.5767 q^{40} +69.6458 q^{41} +438.302 q^{43} -93.1231 q^{44} +7.53789 q^{45} +187.663 q^{46} +31.9479 q^{47} -185.793 q^{48} -319.386 q^{50} +94.1298 q^{51} -18.6998 q^{52} +2.84658 q^{53} -381.533 q^{54} -36.3542 q^{55} +397.831 q^{57} +450.600 q^{58} -71.6325 q^{59} +2.97633 q^{60} +920.695 q^{61} +289.693 q^{62} +567.978 q^{64} -7.30019 q^{65} +611.032 q^{66} -444.280 q^{67} -36.7471 q^{68} +269.943 q^{69} -541.719 q^{71} +324.536 q^{72} -764.004 q^{73} +294.086 q^{74} -459.420 q^{75} -155.309 q^{76} +122.700 q^{78} -421.538 q^{79} +28.3153 q^{80} -186.386 q^{81} +178.401 q^{82} -603.797 q^{83} -14.3457 q^{85} +1122.73 q^{86} +648.165 q^{87} -1565.19 q^{88} +1159.88 q^{89} +19.3087 q^{90} -105.383 q^{92} +416.708 q^{93} +81.8362 q^{94} -60.6307 q^{95} +236.757 q^{96} -583.269 q^{97} -869.006 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9}+O(q^{10}) $ 2 * q + q^2 - 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 - 3 * q^8 + 35 * q^9	\( 2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9} - 7 q^{10} + 80 q^{11} + 43 q^{12} + 26 q^{13} - 33 q^{15} - 39 q^{16} - 19 q^{17} - 110 q^{18} + 84 q^{19} - 19 q^{20} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 13 q^{26} - 335 q^{27} - 44 q^{29} + 43 q^{30} + 86 q^{31} - 123 q^{32} + 106 q^{33} + 135 q^{34} - 250 q^{36} + 209 q^{37} + 314 q^{38} - 65 q^{39} + 89 q^{40} + 230 q^{41} + 287 q^{43} - 178 q^{44} + 180 q^{45} - 4 q^{46} - 435 q^{47} - 285 q^{48} - 144 q^{50} + 481 q^{51} - 91 q^{52} - 118 q^{53} - 91 q^{54} + 18 q^{55} + 606 q^{57} + 794 q^{58} + 368 q^{59} + 175 q^{60} + 1058 q^{61} + 332 q^{62} + 769 q^{64} + 39 q^{65} + 818 q^{66} + 68 q^{67} + 211 q^{68} - 796 q^{69} - 131 q^{71} + 1350 q^{72} - 456 q^{73} + 147 q^{74} + 516 q^{75} - 22 q^{76} + 299 q^{78} - 1008 q^{79} + 69 q^{80} + 122 q^{81} - 72 q^{82} - 1958 q^{83} - 173 q^{85} + 1359 q^{86} + 2558 q^{87} - 1242 q^{88} + 720 q^{89} - 250 q^{90} - 788 q^{92} + 652 q^{93} + 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} - 130 q^{99}+O(q^{100}) \) 2 * q + q^2 - 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 - 3 * q^8 + 35 * q^9 - 7 * q^10 + 80 * q^11 + 43 * q^12 + 26 * q^13 - 33 * q^15 - 39 * q^16 - 19 * q^17 - 110 * q^18 + 84 * q^19 - 19 * q^20 + 142 * q^22 + 196 * q^23 - 273 * q^24 - 237 * q^25 + 13 * q^26 - 335 * q^27 - 44 * q^29 + 43 * q^30 + 86 * q^31 - 123 * q^32 + 106 * q^33 + 135 * q^34 - 250 * q^36 + 209 * q^37 + 314 * q^38 - 65 * q^39 + 89 * q^40 + 230 * q^41 + 287 * q^43 - 178 * q^44 + 180 * q^45 - 4 * q^46 - 435 * q^47 - 285 * q^48 - 144 * q^50 + 481 * q^51 - 91 * q^52 - 118 * q^53 - 91 * q^54 + 18 * q^55 + 606 * q^57 + 794 * q^58 + 368 * q^59 + 175 * q^60 + 1058 * q^61 + 332 * q^62 + 769 * q^64 + 39 * q^65 + 818 * q^66 + 68 * q^67 + 211 * q^68 - 796 * q^69 - 131 * q^71 + 1350 * q^72 - 456 * q^73 + 147 * q^74 + 516 * q^75 - 22 * q^76 + 299 * q^78 - 1008 * q^79 + 69 * q^80 + 122 * q^81 - 72 * q^82 - 1958 * q^83 - 173 * q^85 + 1359 * q^86 + 2558 * q^87 - 1242 * q^88 + 720 * q^89 - 250 * q^90 - 788 * q^92 + 652 * q^93 + 811 * q^94 - 146 * q^95 + 1863 * q^96 + 928 * q^97 - 130 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	2.56155	0.905646	0.452823	−	0.891601i	$-0.350417\pi$
$2$	2.56155	0.905646	0.452823	+	0.891601i	$0.350417\pi$
$3$	3.68466	0.709113	0.354556	−	0.935035i	$-0.384632\pi$
$3$	3.68466	0.709113	0.354556	+	0.935035i	$0.384632\pi$
$4$	−1.43845	−0.179806
$5$	−0.561553	−0.0502268	−0.0251134	−	0.999685i	$-0.507995\pi$
$5$	−0.561553	−0.0502268	−0.0251134	+	0.999685i	$0.507995\pi$
$6$	9.43845	0.642205
$7$	0	0
$7$	0	0
$8$	−24.1771	−1.06849
$9$	−13.4233	−0.497159
$10$	−1.43845	−0.0454877
$11$	64.7386	1.77449	0.887247	−	0.461295i	$-0.152615\pi$
$11$	64.7386	1.77449	0.887247	+	0.461295i	$0.152615\pi$
$12$	−5.30019	−0.127503
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−2.06913	−0.0356165
$16$	−50.4233	−0.787864
$17$	25.5464	0.364465	0.182233	−	0.983255i	$-0.441668\pi$
$17$	25.5464	0.364465	0.182233	+	0.983255i	$0.441668\pi$
$18$	−34.3845	−0.450250
$19$	107.970	1.30368	0.651841	−	0.758356i	$-0.273997\pi$
$19$	107.970	1.30368	0.651841	+	0.758356i	$0.273997\pi$
$20$	0.807764	0.00903108
$21$	0	0
$22$	165.831	1.60706
$23$	73.2614	0.664176	0.332088	−	0.943248i	$-0.392247\pi$
$23$	73.2614	0.664176	0.332088	+	0.943248i	$0.392247\pi$
$24$	−89.0843	−0.757677
$25$	−124.685	−0.997477
$26$	33.3002	0.251181
$27$	−148.946	−1.06165
$28$	0	0
$29$	175.909	1.12640	0.563198	−	0.826322i	$-0.309571\pi$
$29$	175.909	1.12640	0.563198	+	0.826322i	$0.309571\pi$
$30$	−5.30019	−0.0322559
$31$	113.093	0.655228	0.327614	−	0.944812i	$-0.393755\pi$
$31$	113.093	0.655228	0.327614	+	0.944812i	$0.393755\pi$
$32$	64.2547	0.354961
$33$	238.540	1.25832
$34$	65.4384	0.330077
$35$	0	0
$36$	19.3087	0.0893921
$37$	114.808	0.510116	0.255058	−	0.966926i	$-0.417905\pi$
$37$	114.808	0.510116	0.255058	+	0.966926i	$0.417905\pi$
$38$	276.570	1.18067
$39$	47.9006	0.196673
$40$	13.5767	0.0536666
$41$	69.6458	0.265289	0.132645	−	0.991164i	$-0.457653\pi$
$41$	69.6458	0.265289	0.132645	+	0.991164i	$0.457653\pi$
$42$	0	0
$43$	438.302	1.55443	0.777214	−	0.629236i	$-0.216632\pi$
$43$	438.302	1.55443	0.777214	+	0.629236i	$0.216632\pi$
$44$	−93.1231	−0.319064
$45$	7.53789	0.0249707
$46$	187.663	0.601508
$47$	31.9479	0.0991506	0.0495753	−	0.998770i	$-0.484213\pi$
$47$	31.9479	0.0991506	0.0495753	+	0.998770i	$0.484213\pi$
$48$	−185.793	−0.558684
$49$	0	0
$50$	−319.386	−0.903361
$51$	94.1298	0.258447
$52$	−18.6998	−0.0498692
$53$	2.84658	0.00737752	0.00368876	−	0.999993i	$-0.498826\pi$
$53$	2.84658	0.00737752	0.00368876	+	0.999993i	$0.498826\pi$
$54$	−381.533	−0.961483
$55$	−36.3542	−0.0891272
$56$	0	0
$57$	397.831	0.924457
$58$	450.600	1.02012
$59$	−71.6325	−0.158064	−0.0790319	−	0.996872i	$-0.525183\pi$
$59$	−71.6325	−0.158064	−0.0790319	+	0.996872i	$0.525183\pi$
$60$	2.97633	0.00640405
$61$	920.695	1.93251	0.966253	−	0.257593i	$-0.0829295\pi$
$61$	920.695	1.93251	0.966253	+	0.257593i	$0.0829295\pi$
$62$	289.693	0.593404
$63$	0	0
$64$	567.978	1.10933
$65$	−7.30019	−0.0139304
$66$	611.032	1.13959
$67$	−444.280	−0.810112	−0.405056	−	0.914292i	$-0.632748\pi$
$67$	−444.280	−0.810112	−0.405056	+	0.914292i	$0.632748\pi$
$68$	−36.7471	−0.0655330
$69$	269.943	0.470976
$70$	0	0
$71$	−541.719	−0.905496	−0.452748	−	0.891639i	$-0.649556\pi$
$71$	−541.719	−0.905496	−0.452748	+	0.891639i	$0.649556\pi$
$72$	324.536	0.531207
$73$	−764.004	−1.22493	−0.612465	−	0.790498i	$-0.709822\pi$
$73$	−764.004	−1.22493	−0.612465	+	0.790498i	$0.709822\pi$
$74$	294.086	0.461984
$75$	−459.420	−0.707324
$76$	−155.309	−0.234410
$77$	0	0
$78$	122.700	0.178116
$79$	−421.538	−0.600338	−0.300169	−	0.953886i	$-0.597043\pi$
$79$	−421.538	−0.600338	−0.300169	+	0.953886i	$0.597043\pi$
$80$	28.3153	0.0395719
$81$	−186.386	−0.255674
$82$	178.401	0.240258
$83$	−603.797	−0.798498	−0.399249	−	0.916842i	$-0.630729\pi$
$83$	−603.797	−0.798498	−0.399249	+	0.916842i	$0.630729\pi$
$84$	0	0
$85$	−14.3457	−0.0183059
$86$	1122.73	1.40776
$87$	648.165	0.798742
$88$	−1565.19	−1.89602
$89$	1159.88	1.38143	0.690715	−	0.723127i	$-0.257296\pi$
$89$	1159.88	1.38143	0.690715	+	0.723127i	$0.257296\pi$
$90$	19.3087	0.0226146
$91$	0	0
$92$	−105.383	−0.119423
$93$	416.708	0.464631
$94$	81.8362	0.0897953
$95$	−60.6307	−0.0654798
$96$	236.757	0.251707
$97$	−583.269	−0.610536	−0.305268	−	0.952267i	$-0.598746\pi$
$97$	−583.269	−0.610536	−0.305268	+	0.952267i	$0.598746\pi$
$98$	0	0
$99$	−869.006	−0.882206
$100$	179.352	0.179352
$101$	−921.740	−0.908085	−0.454043	−	0.890980i	$-0.650019\pi$
$101$	−921.740	−0.908085	−0.454043	+	0.890980i	$0.650019\pi$
$102$	241.118	0.234061
$103$	930.712	0.890347	0.445174	−	0.895444i	$-0.353142\pi$
$103$	930.712	0.890347	0.445174	+	0.895444i	$0.353142\pi$
$104$	−314.302	−0.296345
$105$	0	0
$106$	7.29168	0.00668142
$107$	857.383	0.774638	0.387319	−	0.921946i	$-0.373401\pi$
$107$	857.383	0.774638	0.387319	+	0.921946i	$0.373401\pi$
$108$	214.251	0.190892
$109$	671.853	0.590384	0.295192	−	0.955438i	$-0.404616\pi$
$109$	671.853	0.590384	0.295192	+	0.955438i	$0.404616\pi$
$110$	−93.1231	−0.0807176
$111$	423.027	0.361730
$112$	0	0
$113$	641.474	0.534024	0.267012	−	0.963693i	$-0.413964\pi$
$113$	641.474	0.534024	0.267012	+	0.963693i	$0.413964\pi$
$114$	1019.07	0.837231
$115$	−41.1401	−0.0333594
$116$	−253.036	−0.202533
$117$	−174.503	−0.137887
$118$	−183.491	−0.143150
$119$	0	0
$120$	50.0255	0.0380557
$121$	2860.09	2.14883
$122$	2358.41	1.75017
$123$	256.621	0.188120
$124$	−162.678	−0.117814
$125$	140.211	0.100327
$126$	0	0
$127$	−553.174	−0.386506	−0.193253	−	0.981149i	$-0.561904\pi$
$127$	−553.174	−0.386506	−0.193253	+	0.981149i	$0.561904\pi$
$128$	940.868	0.649702
$129$	1614.99	1.10227
$130$	−18.6998	−0.0126160
$131$	−2056.40	−1.37152	−0.685758	−	0.727830i	$-0.740529\pi$
$131$	−2056.40	−1.37152	−0.685758	+	0.727830i	$0.740529\pi$
$132$	−343.127	−0.226253
$133$	0	0
$134$	−1138.05	−0.733674
$135$	83.6411	0.0533235
$136$	−617.637	−0.389426
$137$	−1808.57	−1.12786	−0.563928	−	0.825824i	$-0.690710\pi$
$137$	−1808.57	−1.12786	−0.563928	+	0.825824i	$0.690710\pi$
$138$	691.474	0.426537
$139$	−1493.64	−0.911428	−0.455714	−	0.890126i	$-0.650616\pi$
$139$	−1493.64	−0.911428	−0.455714	+	0.890126i	$0.650616\pi$
$140$	0	0
$141$	117.717	0.0703090
$142$	−1387.64	−0.820058
$143$	841.602	0.492156
$144$	676.847	0.391694
$145$	−98.7822	−0.0565753
$146$	−1957.04	−1.10935
$147$	0	0
$148$	−165.145	−0.0917218
$149$	−2759.02	−1.51696	−0.758482	−	0.651694i	$-0.774059\pi$
$149$	−2759.02	−1.51696	−0.758482	+	0.651694i	$0.774059\pi$
$150$	−1176.83	−0.640585
$151$	−976.355	−0.526190	−0.263095	−	0.964770i	$-0.584743\pi$
$151$	−976.355	−0.526190	−0.263095	+	0.964770i	$0.584743\pi$
$152$	−2610.39	−1.39297
$153$	−342.917	−0.181197
$154$	0	0
$155$	−63.5076	−0.0329100
$156$	−68.9024	−0.0353629
$157$	564.875	0.287146	0.143573	−	0.989640i	$-0.454141\pi$
$157$	564.875	0.287146	0.143573	+	0.989640i	$0.454141\pi$
$158$	−1079.79	−0.543694
$159$	10.4887	0.00523149
$160$	−36.0824	−0.0178285
$161$	0	0
$162$	−477.438	−0.231550
$163$	1508.53	0.724892	0.362446	−	0.932005i	$-0.381942\pi$
$163$	1508.53	0.724892	0.362446	+	0.932005i	$0.381942\pi$
$164$	−100.182	−0.0477005
$165$	−133.953	−0.0632012
$166$	−1546.66	−0.723157
$167$	−592.521	−0.274555	−0.137277	−	0.990533i	$-0.543835\pi$
$167$	−592.521	−0.274555	−0.137277	+	0.990533i	$0.543835\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−36.7471	−0.0165787
$171$	−1449.31	−0.648137
$172$	−630.474	−0.279495
$173$	4495.57	1.97568	0.987838	−	0.155488i	$-0.0496952\pi$
$173$	4495.57	1.97568	0.987838	+	0.155488i	$0.0496952\pi$
$174$	1660.31	0.723377
$175$	0	0
$176$	−3264.34	−1.39806
$177$	−263.941	−0.112085
$178$	2971.10	1.25109
$179$	−154.285	−0.0644235	−0.0322117	−	0.999481i	$-0.510255\pi$
$179$	−154.285	−0.0644235	−0.0322117	+	0.999481i	$0.510255\pi$
$180$	−10.8429	−0.00448988
$181$	−1071.35	−0.439959	−0.219979	−	0.975505i	$-0.570599\pi$
$181$	−1071.35	−0.439959	−0.219979	+	0.975505i	$0.570599\pi$
$182$	0	0
$183$	3392.45	1.37037
$184$	−1771.25	−0.709663
$185$	−64.4706	−0.0256215
$186$	1067.42	0.420791
$187$	1653.84	0.646742
$188$	−45.9554	−0.0178279
$189$	0	0
$190$	−155.309	−0.0593015
$191$	677.203	0.256548	0.128274	−	0.991739i	$-0.459056\pi$
$191$	677.203	0.256548	0.128274	+	0.991739i	$0.459056\pi$
$192$	2092.81	0.786642
$193$	1321.68	0.492936	0.246468	−	0.969151i	$-0.420730\pi$
$193$	1321.68	0.492936	0.246468	+	0.969151i	$0.420730\pi$
$194$	−1494.07	−0.552929
$195$	−26.8987	−0.00987823
$196$	0	0
$197$	1267.37	0.458356	0.229178	−	0.973385i	$-0.426396\pi$
$197$	1267.37	0.458356	0.229178	+	0.973385i	$0.426396\pi$
$198$	−2226.00	−0.798966
$199$	−2396.24	−0.853593	−0.426796	−	0.904348i	$-0.640358\pi$
$199$	−2396.24	−0.853593	−0.426796	+	0.904348i	$0.640358\pi$
$200$	3014.51	1.06579
$201$	−1637.02	−0.574460
$202$	−2361.09	−0.822403
$203$	0	0
$204$	−135.401	−0.0464703
$205$	−39.1098	−0.0133246
$206$	2384.07	0.806339
$207$	−983.409	−0.330201
$208$	−655.503	−0.218514
$209$	6989.81	2.31337
$210$	0	0
$211$	−91.5539	−0.0298712	−0.0149356	−	0.999888i	$-0.504754\pi$
$211$	−91.5539	−0.0298712	−0.0149356	+	0.999888i	$0.504754\pi$
$212$	−4.09466	−0.00132652
$213$	−1996.05	−0.642099
$214$	2196.23	0.701548
$215$	−246.130	−0.0780740
$216$	3601.08	1.13436
$217$	0	0
$218$	1720.99	0.534679
$219$	−2815.09	−0.868613
$220$	52.2935	0.0160256
$221$	332.103	0.101085
$222$	1083.61	0.327599
$223$	−1235.42	−0.370985	−0.185493	−	0.982646i	$-0.559388\pi$
$223$	−1235.42	−0.370985	−0.185493	+	0.982646i	$0.559388\pi$
$224$	0	0
$225$	1673.68	0.495905
$226$	1643.17	0.483637
$227$	−3301.66	−0.965370	−0.482685	−	0.875794i	$-0.660338\pi$
$227$	−3301.66	−0.965370	−0.482685	+	0.875794i	$0.660338\pi$
$228$	−572.260	−0.166223
$229$	−211.283	−0.0609694	−0.0304847	−	0.999535i	$-0.509705\pi$
$229$	−211.283	−0.0609694	−0.0304847	+	0.999535i	$0.509705\pi$
$230$	−105.383	−0.0302118
$231$	0	0
$232$	−4252.97	−1.20354
$233$	−256.724	−0.0721827	−0.0360913	−	0.999348i	$-0.511491\pi$
$233$	−256.724	−0.0721827	−0.0360913	+	0.999348i	$0.511491\pi$
$234$	−446.998	−0.124877
$235$	−17.9404	−0.00498002
$236$	103.040	0.0284208
$237$	−1553.22	−0.425708
$238$	0	0
$239$	−3549.62	−0.960694	−0.480347	−	0.877078i	$-0.659489\pi$
$239$	−3549.62	−0.960694	−0.480347	+	0.877078i	$0.659489\pi$
$240$	104.332	0.0280609
$241$	5030.10	1.34447	0.672235	−	0.740338i	$-0.265335\pi$
$241$	5030.10	1.34447	0.672235	+	0.740338i	$0.265335\pi$
$242$	7326.27	1.94608
$243$	3334.77	0.880353
$244$	−1324.37	−0.347476
$245$	0	0
$246$	657.349	0.170370
$247$	1403.61	0.361576
$248$	−2734.25	−0.700102
$249$	−2224.79	−0.566226
$250$	359.158	0.0908606
$251$	718.784	0.180754	0.0903770	−	0.995908i	$-0.471193\pi$
$251$	718.784	0.180754	0.0903770	+	0.995908i	$0.471193\pi$
$252$	0	0
$253$	4742.84	1.17858
$254$	−1416.98	−0.350038
$255$	−52.8588	−0.0129810
$256$	−2133.74	−0.520933
$257$	−1280.79	−0.310871	−0.155435	−	0.987846i	$-0.549678\pi$
$257$	−1280.79	−0.310871	−0.155435	+	0.987846i	$0.549678\pi$
$258$	4136.89	0.998262
$259$	0	0
$260$	10.5009	0.00250477
$261$	−2361.28	−0.559998
$262$	−5267.58	−1.24211
$263$	5225.55	1.22517	0.612587	−	0.790403i	$-0.290129\pi$
$263$	5225.55	1.22517	0.612587	+	0.790403i	$0.290129\pi$
$264$	−5767.19	−1.34449
$265$	−1.59851	−0.000370549 0
$266$	0	0
$267$	4273.77	0.979590
$268$	639.074	0.145663
$269$	−6443.80	−1.46054	−0.730270	−	0.683158i	$-0.760606\pi$
$269$	−6443.80	−1.46054	−0.730270	+	0.683158i	$0.760606\pi$
$270$	214.251	0.0482922
$271$	3929.93	0.880909	0.440455	−	0.897775i	$-0.354817\pi$
$271$	3929.93	0.880909	0.440455	+	0.897775i	$0.354817\pi$
$272$	−1288.13	−0.287149
$273$	0	0
$274$	−4632.74	−1.02144
$275$	−8071.91	−1.77002
$276$	−388.299	−0.0846842
$277$	−5884.40	−1.27639	−0.638194	−	0.769876i	$-0.720318\pi$
$277$	−5884.40	−1.27639	−0.638194	+	0.769876i	$0.720318\pi$
$278$	−3826.03	−0.825431
$279$	−1518.08	−0.325752
$280$	0	0
$281$	3529.79	0.749358	0.374679	−	0.927155i	$-0.377753\pi$
$281$	3529.79	0.749358	0.374679	+	0.927155i	$0.377753\pi$
$282$	301.538	0.0636750
$283$	2611.00	0.548438	0.274219	−	0.961667i	$-0.411581\pi$
$283$	2611.00	0.548438	0.274219	+	0.961667i	$0.411581\pi$
$284$	779.234	0.162813
$285$	−223.403	−0.0464325
$286$	2155.81	0.445719
$287$	0	0
$288$	−862.510	−0.176472
$289$	−4260.38	−0.867165
$290$	−253.036	−0.0512372
$291$	−2149.15	−0.432939
$292$	1098.98	0.220250
$293$	5491.03	1.09484	0.547422	−	0.836857i	$-0.315609\pi$
$293$	5491.03	1.09484	0.547422	+	0.836857i	$0.315609\pi$
$294$	0	0
$295$	40.2255	0.00793904
$296$	−2775.72	−0.545052
$297$	−9642.56	−1.88390
$298$	−7067.37	−1.37383
$299$	952.398	0.184209
$300$	660.852	0.127181
$301$	0	0
$302$	−2500.99	−0.476542
$303$	−3396.30	−0.643935
$304$	−5444.19	−1.02712
$305$	−517.019	−0.0970637
$306$	−878.399	−0.164100
$307$	7307.59	1.35852	0.679261	−	0.733897i	$-0.262300\pi$
$307$	7307.59	1.35852	0.679261	+	0.733897i	$0.262300\pi$
$308$	0	0
$309$	3429.36	0.631357
$310$	−162.678	−0.0298048
$311$	−7904.92	−1.44131	−0.720654	−	0.693295i	$-0.756158\pi$
$311$	−7904.92	−1.44131	−0.720654	+	0.693295i	$0.756158\pi$
$312$	−1158.10	−0.210142
$313$	−10002.4	−1.80629	−0.903145	−	0.429336i	$-0.858748\pi$
$313$	−10002.4	−1.80629	−0.903145	+	0.429336i	$0.858748\pi$
$314$	1446.96	0.260053
$315$	0	0
$316$	606.360	0.107944
$317$	−6230.81	−1.10397	−0.551983	−	0.833856i	$-0.686129\pi$
$317$	−6230.81	−1.10397	−0.551983	+	0.833856i	$0.686129\pi$
$318$	26.8673	0.00473788
$319$	11388.1	1.99878
$320$	−318.950	−0.0557182
$321$	3159.16	0.549306
$322$	0	0
$323$	2758.24	0.475147
$324$	268.107	0.0459717
$325$	−1620.90	−0.276650
$326$	3864.19	0.656495
$327$	2475.55	0.418649
$328$	−1683.83	−0.283458
$329$	0	0
$330$	−343.127	−0.0572379
$331$	−4634.51	−0.769594	−0.384797	−	0.923001i	$-0.625729\pi$
$331$	−4634.51	−0.769594	−0.384797	+	0.923001i	$0.625729\pi$
$332$	868.531	0.143575
$333$	−1541.10	−0.253609
$334$	−1517.77	−0.248649
$335$	249.487	0.0406893
$336$	0	0
$337$	3029.82	0.489747	0.244874	−	0.969555i	$-0.421254\pi$
$337$	3029.82	0.489747	0.244874	+	0.969555i	$0.421254\pi$
$338$	432.902	0.0696651
$339$	2363.61	0.378684
$340$	20.6355	0.00329151
$341$	7321.47	1.16270
$342$	−3712.48	−0.586982
$343$	0	0
$344$	−10596.9	−1.66089
$345$	−151.587	−0.0236556
$346$	11515.6	1.78926
$347$	2841.60	0.439611	0.219805	−	0.975544i	$-0.429458\pi$
$347$	2841.60	0.439611	0.219805	+	0.975544i	$0.429458\pi$
$348$	−932.351	−0.143619
$349$	−7565.68	−1.16040	−0.580202	−	0.814472i	$-0.697027\pi$
$349$	−7565.68	−1.16040	−0.580202	+	0.814472i	$0.697027\pi$
$350$	0	0
$351$	−1936.30	−0.294450
$352$	4159.76	0.629875
$353$	2339.44	0.352736	0.176368	−	0.984324i	$-0.443565\pi$
$353$	2339.44	0.352736	0.176368	+	0.984324i	$0.443565\pi$
$354$	−676.100	−0.101509
$355$	304.204	0.0454802
$356$	−1668.43	−0.248389
$357$	0	0
$358$	−395.209	−0.0583449
$359$	−2531.68	−0.372192	−0.186096	−	0.982532i	$-0.559583\pi$
$359$	−2531.68	−0.372192	−0.186096	+	0.982532i	$0.559583\pi$
$360$	−182.244	−0.0266809
$361$	4798.45	0.699585
$362$	−2744.31	−0.398447
$363$	10538.5	1.52376
$364$	0	0
$365$	429.028	0.0615243
$366$	8689.93	1.24107
$367$	−6577.81	−0.935583	−0.467792	−	0.883839i	$-0.654950\pi$
$367$	−6577.81	−0.935583	−0.467792	+	0.883839i	$0.654950\pi$
$368$	−3694.08	−0.523280
$369$	−934.876	−0.131891
$370$	−165.145	−0.0232040
$371$	0	0
$372$	−599.413	−0.0835433
$373$	2902.72	0.402942	0.201471	−	0.979495i	$-0.435428\pi$
$373$	2902.72	0.402942	0.201471	+	0.979495i	$0.435428\pi$
$374$	4236.40	0.585719
$375$	516.630	0.0711431
$376$	−772.407	−0.105941
$377$	2286.82	0.312406
$378$	0	0
$379$	−1865.73	−0.252866	−0.126433	−	0.991975i	$-0.540353\pi$
$379$	−1865.73	−0.252866	−0.126433	+	0.991975i	$0.540353\pi$
$380$	87.2140	0.0117736
$381$	−2038.26	−0.274076
$382$	1734.69	0.232342
$383$	−10836.0	−1.44567	−0.722837	−	0.691019i	$-0.757162\pi$
$383$	−10836.0	−1.44567	−0.722837	+	0.691019i	$0.757162\pi$
$384$	3466.78	0.460712
$385$	0	0
$386$	3385.55	0.446425
$387$	−5883.46	−0.772798
$388$	839.001	0.109778
$389$	−9520.34	−1.24088	−0.620438	−	0.784256i	$-0.713045\pi$
$389$	−9520.34	−1.24088	−0.620438	+	0.784256i	$0.713045\pi$
$390$	−68.9024	−0.00894618
$391$	1871.56	0.242069
$392$	0	0
$393$	−7577.13	−0.972559
$394$	3246.43	0.415108
$395$	236.716	0.0301531
$396$	1250.02	0.158626
$397$	10108.8	1.27796	0.638978	−	0.769225i	$-0.279358\pi$
$397$	10108.8	1.27796	0.638978	+	0.769225i	$0.279358\pi$
$398$	−6138.10	−0.773053
$399$	0	0
$400$	6287.01	0.785876
$401$	2084.38	0.259573	0.129787	−	0.991542i	$-0.458571\pi$
$401$	2084.38	0.259573	0.129787	+	0.991542i	$0.458571\pi$
$402$	−4193.32	−0.520258
$403$	1470.21	0.181728
$404$	1325.88	0.163279
$405$	104.666	0.0128417
$406$	0	0
$407$	7432.50	0.905197
$408$	−2275.78	−0.276147
$409$	9716.53	1.17470	0.587349	−	0.809334i	$-0.300172\pi$
$409$	9716.53	1.17470	0.587349	+	0.809334i	$0.300172\pi$
$410$	−100.182	−0.0120674
$411$	−6663.95	−0.799777
$412$	−1338.78	−0.160090
$413$	0	0
$414$	−2519.05	−0.299045
$415$	339.064	0.0401060
$416$	835.311	0.0984483
$417$	−5503.54	−0.646305
$418$	17904.8	2.09510
$419$	−13381.9	−1.56026	−0.780129	−	0.625619i	$-0.784847\pi$
$419$	−13381.9	−1.56026	−0.780129	+	0.625619i	$0.784847\pi$
$420$	0	0
$421$	−9463.37	−1.09553	−0.547763	−	0.836633i	$-0.684521\pi$
$421$	−9463.37	−1.09553	−0.547763	+	0.836633i	$0.684521\pi$
$422$	−234.520	−0.0270527
$423$	−428.846	−0.0492936
$424$	−68.8221	−0.00788278
$425$	−3185.24	−0.363546
$426$	−5112.98	−0.581514
$427$	0	0
$428$	−1233.30	−0.139285
$429$	3101.02	0.348994
$430$	−630.474	−0.0707074
$431$	4852.28	0.542288	0.271144	−	0.962539i	$-0.412598\pi$
$431$	4852.28	0.542288	0.271144	+	0.962539i	$0.412598\pi$
$432$	7510.35	0.836439
$433$	8208.00	0.910973	0.455486	−	0.890243i	$-0.349465\pi$
$433$	8208.00	0.910973	0.455486	+	0.890243i	$0.349465\pi$
$434$	0	0
$435$	−363.979	−0.0401183
$436$	−966.425	−0.106155
$437$	7910.01	0.865874
$438$	−7211.01	−0.786656
$439$	2993.80	0.325481	0.162741	−	0.986669i	$-0.447967\pi$
$439$	2993.80	0.325481	0.162741	+	0.986669i	$0.447967\pi$
$440$	878.938	0.0952311
$441$	0	0
$442$	850.700	0.0915468
$443$	9743.67	1.04500	0.522501	−	0.852639i	$-0.324999\pi$
$443$	9743.67	1.04500	0.522501	+	0.852639i	$0.324999\pi$
$444$	−608.503	−0.0650411
$445$	−651.335	−0.0693848
$446$	−3164.59	−0.335981
$447$	−10166.0	−1.07570
$448$	0	0
$449$	−561.459	−0.0590131	−0.0295065	−	0.999565i	$-0.509394\pi$
$449$	−561.459	−0.0590131	−0.0295065	+	0.999565i	$0.509394\pi$
$450$	4287.22	0.449114
$451$	4508.78	0.470754
$452$	−922.726	−0.0960207
$453$	−3597.54	−0.373128
$454$	−8457.38	−0.874283
$455$	0	0
$456$	−9618.40	−0.987770
$457$	13758.4	1.40830	0.704148	−	0.710054i	$-0.251329\pi$
$457$	13758.4	1.40830	0.704148	+	0.710054i	$0.251329\pi$
$458$	−541.213	−0.0552166
$459$	−3805.03	−0.386936
$460$	59.1779	0.00599823
$461$	−12009.2	−1.21329	−0.606644	−	0.794974i	$-0.707485\pi$
$461$	−12009.2	−1.21329	−0.606644	+	0.794974i	$0.707485\pi$
$462$	0	0
$463$	13635.7	1.36870	0.684348	−	0.729156i	$-0.260087\pi$
$463$	13635.7	1.36870	0.684348	+	0.729156i	$0.260087\pi$
$464$	−8869.91	−0.887447
$465$	−234.004	−0.0233369
$466$	−657.613	−0.0653719
$467$	−8821.95	−0.874157	−0.437079	−	0.899423i	$-0.643987\pi$
$467$	−8821.95	−0.874157	−0.437079	+	0.899423i	$0.643987\pi$
$468$	251.013	0.0247929
$469$	0	0
$470$	−45.9554	−0.00451013
$471$	2081.37	0.203619
$472$	1731.87	0.168889
$473$	28375.1	2.75832
$474$	−3978.66	−0.385540
$475$	−13462.2	−1.30039
$476$	0	0
$477$	−38.2105	−0.00366780
$478$	−9092.54	−0.870049
$479$	14620.0	1.39459	0.697293	−	0.716786i	$-0.254388\pi$
$479$	14620.0	1.39459	0.697293	+	0.716786i	$0.254388\pi$
$480$	−132.951	−0.0126424
$481$	1492.50	0.141481
$482$	12884.9	1.21761
$483$	0	0
$484$	−4114.09	−0.386372
$485$	327.536	0.0306653
$486$	8542.20	0.797288
$487$	−9798.86	−0.911763	−0.455882	−	0.890040i	$-0.650676\pi$
$487$	−9798.86	−0.911763	−0.455882	+	0.890040i	$0.650676\pi$
$488$	−22259.7	−2.06486
$489$	5558.43	0.514030
$490$	0	0
$491$	−10836.1	−0.995977	−0.497989	−	0.867184i	$-0.665928\pi$
$491$	−10836.1	−0.995977	−0.497989	+	0.867184i	$0.665928\pi$
$492$	−369.136	−0.0338251
$493$	4493.84	0.410532
$494$	3595.41	0.327460
$495$	487.993	0.0443104
$496$	−5702.51	−0.516230
$497$	0	0
$498$	−5698.91	−0.512800
$499$	2589.96	0.232349	0.116175	−	0.993229i	$-0.462937\pi$
$499$	2589.96	0.232349	0.116175	+	0.993229i	$0.462937\pi$
$500$	−201.686	−0.0180394
$501$	−2183.24	−0.194690
$502$	1841.20	0.163699
$503$	17067.5	1.51292	0.756462	−	0.654038i	$-0.226926\pi$
$503$	17067.5	1.51292	0.756462	+	0.654038i	$0.226926\pi$
$504$	0	0
$505$	517.606	0.0456102
$506$	12149.0	1.06737
$507$	622.707	0.0545471
$508$	795.712	0.0694961
$509$	1012.89	0.0882038	0.0441019	−	0.999027i	$-0.485957\pi$
$509$	1012.89	0.0882038	0.0441019	+	0.999027i	$0.485957\pi$
$510$	−135.401	−0.0117562
$511$	0	0
$512$	−12992.6	−1.12148
$513$	−16081.7	−1.38406
$514$	−3280.82	−0.281539
$515$	−522.644	−0.0447193
$516$	−2323.08	−0.198194
$517$	2068.26	0.175942
$518$	0	0
$519$	16564.6	1.40098
$520$	176.497	0.0148845
$521$	14367.7	1.20818	0.604089	−	0.796917i	$-0.293537\pi$
$521$	14367.7	1.20818	0.604089	+	0.796917i	$0.293537\pi$
$522$	−6048.54	−0.507160
$523$	16219.9	1.35611	0.678057	−	0.735010i	$-0.262822\pi$
$523$	16219.9	1.35611	0.678057	+	0.735010i	$0.262822\pi$
$524$	2958.02	0.246607
$525$	0	0
$526$	13385.5	1.10957
$527$	2889.11	0.238808
$528$	−12028.0	−0.991382
$529$	−6799.77	−0.558870
$530$	−4.09466	−0.000335586 0
$531$	961.545	0.0785828
$532$	0	0
$533$	905.396	0.0735780
$534$	10947.5	0.887161
$535$	−481.466	−0.0389076
$536$	10741.4	0.865593
$537$	−568.488	−0.0456835
$538$	−16506.1	−1.32273
$539$	0	0
$540$	−120.313	−0.00958788
$541$	17592.2	1.39806	0.699029	−	0.715094i	$-0.253616\pi$
$541$	17592.2	1.39806	0.699029	+	0.715094i	$0.253616\pi$
$542$	10066.7	0.797792
$543$	−3947.55	−0.311980
$544$	1641.48	0.129371
$545$	−377.281	−0.0296531
$546$	0	0
$547$	10504.6	0.821103	0.410552	−	0.911837i	$-0.365336\pi$
$547$	10504.6	0.821103	0.410552	+	0.911837i	$0.365336\pi$
$548$	2601.53	0.202795
$549$	−12358.8	−0.960763
$550$	−20676.6	−1.60301
$551$	18992.8	1.46846
$552$	−6526.44	−0.503231
$553$	0	0
$554$	−15073.2	−1.15596
$555$	−237.552	−0.0181685
$556$	2148.52	0.163880
$557$	−507.558	−0.0386102	−0.0193051	−	0.999814i	$-0.506145\pi$
$557$	−507.558	−0.0386102	−0.0193051	+	0.999814i	$0.506145\pi$
$558$	−3888.64	−0.295016
$559$	5697.93	0.431121
$560$	0	0
$561$	6093.83	0.458613
$562$	9041.75	0.678653
$563$	3443.14	0.257746	0.128873	−	0.991661i	$-0.458864\pi$
$563$	3443.14	0.257746	0.128873	+	0.991661i	$0.458864\pi$
$564$	−169.330	−0.0126420
$565$	−360.221	−0.0268223
$566$	6688.21	0.496690
$567$	0	0
$568$	13097.2	0.967509
$569$	23972.2	1.76620	0.883098	−	0.469189i	$-0.155454\pi$
$569$	23972.2	1.76620	0.883098	+	0.469189i	$0.155454\pi$
$570$	−572.260	−0.0420514
$571$	−7458.32	−0.546622	−0.273311	−	0.961926i	$-0.588119\pi$
$571$	−7458.32	−0.546622	−0.273311	+	0.961926i	$0.588119\pi$
$572$	−1210.60	−0.0884926
$573$	2495.26	0.181922
$574$	0	0
$575$	−9134.57	−0.662501
$576$	−7624.14	−0.551515
$577$	−5669.57	−0.409059	−0.204530	−	0.978860i	$-0.565566\pi$
$577$	−5669.57	−0.409059	−0.204530	+	0.978860i	$0.565566\pi$
$578$	−10913.2	−0.785344
$579$	4869.94	0.349547
$580$	142.093	0.0101726
$581$	0	0
$582$	−5505.15	−0.392089
$583$	184.284	0.0130914
$584$	18471.4	1.30882
$585$	97.9925	0.00692563
$586$	14065.6	0.991541
$587$	−1017.39	−0.0715371	−0.0357685	−	0.999360i	$-0.511388\pi$
$587$	−1017.39	−0.0715371	−0.0357685	+	0.999360i	$0.511388\pi$
$588$	0	0
$589$	12210.6	0.854208
$590$	103.040	0.00718996
$591$	4669.81	0.325026
$592$	−5788.99	−0.401902
$593$	10198.2	0.706221	0.353111	−	0.935582i	$-0.385124\pi$
$593$	10198.2	0.706221	0.353111	+	0.935582i	$0.385124\pi$
$594$	−24699.9	−1.70615
$595$	0	0
$596$	3968.70	0.272759
$597$	−8829.33	−0.605294
$598$	2439.62	0.166828
$599$	12516.3	0.853763	0.426881	−	0.904308i	$-0.359612\pi$
$599$	12516.3	0.853763	0.426881	+	0.904308i	$0.359612\pi$
$600$	11107.4	0.755766
$601$	−9627.46	−0.653431	−0.326716	−	0.945123i	$-0.605942\pi$
$601$	−9627.46	−0.653431	−0.326716	+	0.945123i	$0.605942\pi$
$602$	0	0
$603$	5963.70	0.402754
$604$	1404.44	0.0946120
$605$	−1606.09	−0.107929
$606$	−8699.80	−0.583177
$607$	−6667.20	−0.445821	−0.222910	−	0.974839i	$-0.571556\pi$
$607$	−6667.20	−0.445821	−0.222910	+	0.974839i	$0.571556\pi$
$608$	6937.56	0.462755
$609$	0	0
$610$	−1324.37	−0.0879053
$611$	415.323	0.0274994
$612$	493.268	0.0325803
$613$	−23085.4	−1.52106	−0.760530	−	0.649302i	$-0.775061\pi$
$613$	−23085.4	−1.52106	−0.760530	+	0.649302i	$0.775061\pi$
$614$	18718.8	1.23034
$615$	−144.106	−0.00944866
$616$	0	0
$617$	3049.24	0.198959	0.0994796	−	0.995040i	$-0.468282\pi$
$617$	3049.24	0.198959	0.0994796	+	0.995040i	$0.468282\pi$
$618$	8784.48	0.571786
$619$	−7296.58	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$619$	−7296.58	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$620$	91.3523	0.00591741
$621$	−10912.0	−0.705126
$622$	−20248.9	−1.30531
$623$	0	0
$624$	−2415.30	−0.154951
$625$	15506.8	0.992438
$626$	−25621.7	−1.63586
$627$	25755.1	1.64044
$628$	−812.543	−0.0516305
$629$	2932.92	0.185920
$630$	0	0
$631$	−23829.5	−1.50339	−0.751694	−	0.659512i	$-0.770763\pi$
$631$	−23829.5	−1.50339	−0.751694	+	0.659512i	$0.770763\pi$
$632$	10191.6	0.641453
$633$	−337.345	−0.0211821
$634$	−15960.5	−0.999801
$635$	310.637	0.0194130
$636$	−15.0874	−0.000940653 0
$637$	0	0
$638$	29171.3	1.81019
$639$	7271.65	0.450175
$640$	−528.347	−0.0326324
$641$	13405.3	0.826016	0.413008	−	0.910727i	$-0.364478\pi$
$641$	13405.3	0.826016	0.413008	+	0.910727i	$0.364478\pi$
$642$	8092.36	0.497477
$643$	−5251.51	−0.322083	−0.161042	−	0.986948i	$-0.551485\pi$
$643$	−5251.51	−0.322083	−0.161042	+	0.986948i	$0.551485\pi$
$644$	0	0
$645$	−906.904	−0.0553633
$646$	7065.37	0.430315
$647$	−21611.4	−1.31319	−0.656595	−	0.754244i	$-0.728004\pi$
$647$	−21611.4	−1.31319	−0.656595	+	0.754244i	$0.728004\pi$
$648$	4506.28	0.273184
$649$	−4637.39	−0.280483
$650$	−4152.02	−0.250547
$651$	0	0
$652$	−2169.94	−0.130340
$653$	−21595.8	−1.29420	−0.647099	−	0.762406i	$-0.724018\pi$
$653$	−21595.8	−1.29420	−0.647099	+	0.762406i	$0.724018\pi$
$654$	6341.25	0.379148
$655$	1154.78	0.0688869
$656$	−3511.77	−0.209012
$657$	10255.4	0.608985
$658$	0	0
$659$	−16642.6	−0.983768	−0.491884	−	0.870661i	$-0.663692\pi$
$659$	−16642.6	−0.983768	−0.491884	+	0.870661i	$0.663692\pi$
$660$	192.684	0.0113640
$661$	−26981.1	−1.58766	−0.793831	−	0.608139i	$-0.791916\pi$
$661$	−26981.1	−1.58766	−0.793831	+	0.608139i	$0.791916\pi$
$662$	−11871.5	−0.696980
$663$	1223.69	0.0716803
$664$	14598.1	0.853185
$665$	0	0
$666$	−3947.60	−0.229680
$667$	12887.3	0.748126
$668$	852.310	0.0493665
$669$	−4552.09	−0.263070
$670$	639.074	0.0368501
$671$	59604.5	3.42922
$672$	0	0
$673$	11149.2	0.638591	0.319296	−	0.947655i	$-0.396554\pi$
$673$	11149.2	0.638591	0.319296	+	0.947655i	$0.396554\pi$
$674$	7761.04	0.443537
$675$	18571.3	1.05898
$676$	−243.098	−0.0138312
$677$	−3314.33	−0.188154	−0.0940769	−	0.995565i	$-0.529990\pi$
$677$	−3314.33	−0.188154	−0.0940769	+	0.995565i	$0.529990\pi$
$678$	6054.51	0.342953
$679$	0	0
$680$	346.836	0.0195596
$681$	−12165.5	−0.684556
$682$	18754.3	1.05299
$683$	24505.2	1.37287	0.686433	−	0.727193i	$-0.259176\pi$
$683$	24505.2	1.37287	0.686433	+	0.727193i	$0.259176\pi$
$684$	2084.75	0.116539
$685$	1015.61	0.0566486
$686$	0	0
$687$	−778.506	−0.0432342
$688$	−22100.6	−1.22468
$689$	37.0056	0.00204616
$690$	−388.299	−0.0214236
$691$	21752.8	1.19756	0.598782	−	0.800912i	$-0.295652\pi$
$691$	21752.8	1.19756	0.598782	+	0.800912i	$0.295652\pi$
$692$	−6466.64	−0.355238
$693$	0	0
$694$	7278.90	0.398132
$695$	838.755	0.0457781
$696$	−15670.7	−0.853445
$697$	1779.20	0.0966887
$698$	−19379.9	−1.05092
$699$	−945.941	−0.0511857
$700$	0	0
$701$	34250.9	1.84542	0.922709	−	0.385496i	$-0.125970\pi$
$701$	34250.9	1.84542	0.922709	+	0.385496i	$0.125970\pi$
$702$	−4959.93	−0.266667
$703$	12395.8	0.665028
$704$	36770.1	1.96850
$705$	−66.1043	−0.00353140
$706$	5992.59	0.319454
$707$	0	0
$708$	379.666	0.0201536
$709$	−5527.11	−0.292771	−0.146386	−	0.989228i	$-0.546764\pi$
$709$	−5527.11	−0.292771	−0.146386	+	0.989228i	$0.546764\pi$
$710$	779.234	0.0411889
$711$	5658.43	0.298464
$712$	−28042.6	−1.47604
$713$	8285.33	0.435187
$714$	0	0
$715$	−472.604	−0.0247194
$716$	221.931	0.0115837
$717$	−13079.1	−0.681241
$718$	−6485.02	−0.337074
$719$	3777.78	0.195949	0.0979745	−	0.995189i	$-0.468764\pi$
$719$	3777.78	0.195949	0.0979745	+	0.995189i	$0.468764\pi$
$720$	−380.085	−0.0196735
$721$	0	0
$722$	12291.5	0.633576
$723$	18534.2	0.953380
$724$	1541.08	0.0791072
$725$	−21933.2	−1.12355
$726$	26994.8	1.37999
$727$	−19076.8	−0.973204	−0.486602	−	0.873624i	$-0.661764\pi$
$727$	−19076.8	−0.973204	−0.486602	+	0.873624i	$0.661764\pi$
$728$	0	0
$729$	17319.9	0.879944
$730$	1098.98	0.0557192
$731$	11197.0	0.566535
$732$	−4879.86	−0.246400
$733$	−7997.30	−0.402984	−0.201492	−	0.979490i	$-0.564579\pi$
$733$	−7997.30	−0.402984	−0.201492	+	0.979490i	$0.564579\pi$
$734$	−16849.4	−0.847307
$735$	0	0
$736$	4707.39	0.235756
$737$	−28762.1	−1.43754
$738$	−2394.74	−0.119446
$739$	28983.6	1.44273	0.721367	−	0.692553i	$-0.243514\pi$
$739$	28983.6	1.44273	0.721367	+	0.692553i	$0.243514\pi$
$740$	92.7376	0.00460689
$741$	5171.81	0.256398
$742$	0	0
$743$	−19145.4	−0.945324	−0.472662	−	0.881244i	$-0.656707\pi$
$743$	−19145.4	−0.945324	−0.472662	+	0.881244i	$0.656707\pi$
$744$	−10074.8	−0.496451
$745$	1549.33	0.0761923
$746$	7435.47	0.364922
$747$	8104.95	0.396981
$748$	−2378.96	−0.116288
$749$	0	0
$750$	1323.38	0.0644304
$751$	−25516.9	−1.23985	−0.619923	−	0.784663i	$-0.712836\pi$
$751$	−25516.9	−1.23985	−0.619923	+	0.784663i	$0.712836\pi$
$752$	−1610.92	−0.0781172
$753$	2648.47	0.128175
$754$	5857.80	0.282929
$755$	548.275	0.0264288
$756$	0	0
$757$	−17230.6	−0.827289	−0.413645	−	0.910438i	$-0.635744\pi$
$757$	−17230.6	−0.827289	−0.413645	+	0.910438i	$0.635744\pi$
$758$	−4779.17	−0.229007
$759$	17475.7	0.835744
$760$	1465.87	0.0699642
$761$	2343.06	0.111611	0.0558053	−	0.998442i	$-0.482227\pi$
$761$	2343.06	0.111611	0.0558053	+	0.998442i	$0.482227\pi$
$762$	−5221.11	−0.248216
$763$	0	0
$764$	−974.121	−0.0461289
$765$	192.566	0.00910096
$766$	−27756.9	−1.30927
$767$	−931.223	−0.0438390
$768$	−7862.11	−0.369400
$769$	7100.18	0.332950	0.166475	−	0.986046i	$-0.446761\pi$
$769$	7100.18	0.332950	0.166475	+	0.986046i	$0.446761\pi$
$770$	0	0
$771$	−4719.29	−0.220442
$772$	−1901.17	−0.0886328
$773$	−12270.4	−0.570940	−0.285470	−	0.958388i	$-0.592150\pi$
$773$	−12270.4	−0.570940	−0.285470	+	0.958388i	$0.592150\pi$
$774$	−15070.8	−0.699881
$775$	−14100.9	−0.653575
$776$	14101.7	0.652349
$777$	0	0
$778$	−24386.9	−1.12379
$779$	7519.64	0.345852
$780$	38.6924	0.00177616
$781$	−35070.1	−1.60680
$782$	4794.11	0.219229
$783$	−26201.0	−1.19584
$784$	0	0
$785$	−317.207	−0.0144224
$786$	−19409.2	−0.880794
$787$	−3425.04	−0.155133	−0.0775663	−	0.996987i	$-0.524715\pi$
$787$	−3425.04	−0.155133	−0.0775663	+	0.996987i	$0.524715\pi$
$788$	−1823.04	−0.0824151
$789$	19254.3	0.868787
$790$	606.360	0.0273080
$791$	0	0
$792$	21010.0	0.942624
$793$	11969.0	0.535981
$794$	25894.3	1.15737
$795$	−5.88995	−0.000262761 0
$796$	3446.87	0.153481
$797$	11781.1	0.523600	0.261800	−	0.965122i	$-0.415684\pi$
$797$	11781.1	0.523600	0.261800	+	0.965122i	$0.415684\pi$
$798$	0	0
$799$	816.154	0.0361370
$800$	−8011.58	−0.354065
$801$	−15569.4	−0.686790
$802$	5339.24	0.235081
$803$	−49460.6	−2.17363
$804$	2354.77	0.103291
$805$	0	0
$806$	3766.01	0.164581
$807$	−23743.2	−1.03569
$808$	22285.0	0.970276
$809$	18910.1	0.821810	0.410905	−	0.911678i	$-0.365213\pi$
$809$	18910.1	0.821810	0.410905	+	0.911678i	$0.365213\pi$
$810$	268.107	0.0116300
$811$	−12803.3	−0.554359	−0.277180	−	0.960818i	$-0.589400\pi$
$811$	−12803.3	−0.554359	−0.277180	+	0.960818i	$0.589400\pi$
$812$	0	0
$813$	14480.5	0.624664
$814$	19038.7	0.819788
$815$	−847.121	−0.0364090
$816$	−4746.33	−0.203621
$817$	47323.3	2.02648
$818$	24889.4	1.06386
$819$	0	0
$820$	56.2574	0.00239585
$821$	19335.1	0.821923	0.410962	−	0.911653i	$-0.365193\pi$
$821$	19335.1	0.821923	0.410962	+	0.911653i	$0.365193\pi$
$822$	−17070.1	−0.724315
$823$	−2125.90	−0.0900417	−0.0450209	−	0.998986i	$-0.514335\pi$
$823$	−2125.90	−0.0900417	−0.0450209	+	0.998986i	$0.514335\pi$
$824$	−22501.9	−0.951324
$825$	−29742.2	−1.25514
$826$	0	0
$827$	−6989.24	−0.293881	−0.146941	−	0.989145i	$-0.546943\pi$
$827$	−6989.24	−0.293881	−0.146941	+	0.989145i	$0.546943\pi$
$828$	1414.58	0.0593721
$829$	32649.7	1.36788	0.683938	−	0.729540i	$-0.260266\pi$
$829$	32649.7	1.36788	0.683938	+	0.729540i	$0.260266\pi$
$830$	868.531	0.0363219
$831$	−21682.0	−0.905103
$832$	7383.72	0.307673
$833$	0	0
$834$	−14097.6	−0.585324
$835$	332.732	0.0137900
$836$	−10054.5	−0.415958
$837$	−16844.7	−0.695626
$838$	−34278.4	−1.41304
$839$	4038.23	0.166168	0.0830841	−	0.996543i	$-0.473523\pi$
$839$	4038.23	0.166168	0.0830841	+	0.996543i	$0.473523\pi$
$840$	0	0
$841$	6555.00	0.268769
$842$	−24240.9	−0.992159
$843$	13006.1	0.531380
$844$	131.695	0.00537102
$845$	−94.9024	−0.00386360
$846$	−1098.51	−0.0446426
$847$	0	0
$848$	−143.534	−0.00581248
$849$	9620.64	0.388904
$850$	−8159.17	−0.329244
$851$	8410.97	0.338807
$852$	2871.21	0.115453
$853$	−8114.12	−0.325700	−0.162850	−	0.986651i	$-0.552069\pi$
$853$	−8114.12	−0.325700	−0.162850	+	0.986651i	$0.552069\pi$
$854$	0	0
$855$	813.863	0.0325538
$856$	−20729.0	−0.827690
$857$	22298.1	0.888786	0.444393	−	0.895832i	$-0.353419\pi$
$857$	22298.1	0.888786	0.444393	+	0.895832i	$0.353419\pi$
$858$	7943.42	0.316065
$859$	−33550.5	−1.33263	−0.666315	−	0.745670i	$-0.732130\pi$
$859$	−33550.5	−1.33263	−0.666315	+	0.745670i	$0.732130\pi$
$860$	354.045	0.0140382
$861$	0	0
$862$	12429.4	0.491121
$863$	−14120.5	−0.556972	−0.278486	−	0.960440i	$-0.589833\pi$
$863$	−14120.5	−0.556972	−0.278486	+	0.960440i	$0.589833\pi$
$864$	−9570.49	−0.376846
$865$	−2524.50	−0.0992319
$866$	21025.2	0.825018
$867$	−15698.1	−0.614918
$868$	0	0
$869$	−27289.8	−1.06530
$870$	−932.351	−0.0363329
$871$	−5775.64	−0.224685
$872$	−16243.4	−0.630817
$873$	7829.39	0.303533
$874$	20261.9	0.784175
$875$	0	0
$876$	4049.36	0.156182
$877$	−1941.69	−0.0747619	−0.0373809	−	0.999301i	$-0.511901\pi$
$877$	−1941.69	−0.0747619	−0.0373809	+	0.999301i	$0.511901\pi$
$878$	7668.78	0.294771
$879$	20232.6	0.776368
$880$	1833.10	0.0702201
$881$	790.231	0.0302197	0.0151099	−	0.999886i	$-0.495190\pi$
$881$	790.231	0.0302197	0.0151099	+	0.999886i	$0.495190\pi$
$882$	0	0
$883$	−36638.6	−1.39636	−0.698180	−	0.715922i	$-0.746007\pi$
$883$	−36638.6	−1.39636	−0.698180	+	0.715922i	$0.746007\pi$
$884$	−477.713	−0.0181756
$885$	148.217	0.00562968
$886$	24958.9	0.946401
$887$	40686.3	1.54015	0.770075	−	0.637954i	$-0.220219\pi$
$887$	40686.3	1.54015	0.770075	+	0.637954i	$0.220219\pi$
$888$	−10227.6	−0.386503
$889$	0	0
$890$	−1668.43	−0.0628381
$891$	−12066.4	−0.453692
$892$	1777.08	0.0667053
$893$	3449.40	0.129261
$894$	−26040.9	−0.974202
$895$	86.6392	0.00323579
$896$	0	0
$897$	3509.26	0.130625
$898$	−1438.21	−0.0534449
$899$	19894.0	0.738046
$900$	−2407.50	−0.0891666
$901$	72.7200	0.00268885
$902$	11549.5	0.426336
$903$	0	0
$904$	−15509.0	−0.570598
$905$	601.618	0.0220977
$906$	−9215.28	−0.337922
$907$	−10464.4	−0.383093	−0.191547	−	0.981484i	$-0.561350\pi$
$907$	−10464.4	−0.383093	−0.191547	+	0.981484i	$0.561350\pi$
$908$	4749.26	0.173579
$909$	12372.8	0.451463
$910$	0	0
$911$	−35611.5	−1.29513	−0.647563	−	0.762011i	$-0.724212\pi$
$911$	−35611.5	−1.29513	−0.647563	+	0.762011i	$0.724212\pi$
$912$	−20060.0	−0.728346
$913$	−39089.0	−1.41693
$914$	35242.9	1.27542
$915$	−1905.04	−0.0688291
$916$	303.920	0.0109627
$917$	0	0
$918$	−9746.80	−0.350427
$919$	1077.25	0.0386674	0.0193337	−	0.999813i	$-0.493846\pi$
$919$	1077.25	0.0386674	0.0193337	+	0.999813i	$0.493846\pi$
$920$	994.648	0.0356441
$921$	26926.0	0.963346
$922$	−30762.3	−1.09881
$923$	−7042.34	−0.251139
$924$	0	0
$925$	−14314.8	−0.508829
$926$	34928.6	1.23955
$927$	−12493.2	−0.442644
$928$	11303.0	0.399826
$929$	−55733.8	−1.96832	−0.984159	−	0.177290i	$-0.943267\pi$
$929$	−55733.8	−1.96832	−0.984159	+	0.177290i	$0.943267\pi$
$930$	−599.413	−0.0211350
$931$	0	0
$932$	369.284	0.0129789
$933$	−29126.9	−1.02205
$934$	−22597.9	−0.791677
$935$	−928.718	−0.0324838
$936$	4218.97	0.147330
$937$	3198.60	0.111519	0.0557596	−	0.998444i	$-0.482242\pi$
$937$	3198.60	0.111519	0.0557596	+	0.998444i	$0.482242\pi$
$938$	0	0
$939$	−36855.4	−1.28086
$940$	25.8064	0.000895437 0
$941$	−8823.35	−0.305667	−0.152834	−	0.988252i	$-0.548840\pi$
$941$	−8823.35	−0.305667	−0.152834	+	0.988252i	$0.548840\pi$
$942$	5331.54	0.184407
$943$	5102.35	0.176199
$944$	3611.95	0.124533
$945$	0	0
$946$	72684.3	2.49806
$947$	28290.4	0.970766	0.485383	−	0.874301i	$-0.338680\pi$
$947$	28290.4	0.970766	0.485383	+	0.874301i	$0.338680\pi$
$948$	2234.23	0.0765447
$949$	−9932.05	−0.339734
$950$	−34484.0	−1.17769
$951$	−22958.4	−0.782836
$952$	0	0
$953$	−12399.0	−0.421452	−0.210726	−	0.977545i	$-0.567583\pi$
$953$	−12399.0	−0.421452	−0.210726	+	0.977545i	$0.567583\pi$
$954$	−97.8783	−0.00332173
$955$	−380.285	−0.0128856
$956$	5105.94	0.172739
$957$	41961.3	1.41736
$958$	37450.0	1.26300
$959$	0	0
$960$	−1175.22	−0.0395105
$961$	−17001.0	−0.570676
$962$	3823.12	0.128131
$963$	−11508.9	−0.385118
$964$	−7235.53	−0.241744
$965$	−742.193	−0.0247586
$966$	0	0
$967$	−26667.1	−0.886820	−0.443410	−	0.896319i	$-0.646231\pi$
$967$	−26667.1	−0.886820	−0.443410	+	0.896319i	$0.646231\pi$
$968$	−69148.6	−2.29599
$969$	10163.2	0.336933
$970$	839.001	0.0277719
$971$	−49420.7	−1.63335	−0.816676	−	0.577096i	$-0.804186\pi$
$971$	−49420.7	−1.63335	−0.816676	+	0.577096i	$0.804186\pi$
$972$	−4796.89	−0.158293
$973$	0	0
$974$	−25100.3	−0.825735
$975$	−5972.46	−0.196176
$976$	−46424.5	−1.52255
$977$	778.759	0.0255012	0.0127506	−	0.999919i	$-0.495941\pi$
$977$	778.759	0.0255012	0.0127506	+	0.999919i	$0.495941\pi$
$978$	14238.2	0.465529
$979$	75089.2	2.45134
$980$	0	0
$981$	−9018.48	−0.293515
$982$	−27757.2	−0.902003
$983$	−5997.90	−0.194612	−0.0973059	−	0.995255i	$-0.531023\pi$
$983$	−5997.90	−0.194612	−0.0973059	+	0.995255i	$0.531023\pi$
$984$	−6204.35	−0.201004
$985$	−711.693	−0.0230218
$986$	11511.2	0.371797
$987$	0	0
$988$	−2019.01	−0.0650135
$989$	32110.6	1.03241
$990$	1250.02	0.0401295
$991$	8974.94	0.287688	0.143844	−	0.989600i	$-0.454054\pi$
$991$	8974.94	0.287688	0.143844	+	0.989600i	$0.454054\pi$
$992$	7266.75	0.232580
$993$	−17076.6	−0.545729
$994$	0	0
$995$	1345.62	0.0428732
$996$	3200.24	0.101811
$997$	−28530.2	−0.906280	−0.453140	−	0.891439i	$-0.649696\pi$
$997$	−28530.2	−0.906280	−0.453140	+	0.891439i	$0.649696\pi$
$998$	6634.31	0.210426
$999$	−17100.2	−0.541567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.4.a.b.1.2		2
7.6	odd	2		13.4.a.b.1.2	✓	2
21.20	even	2		117.4.a.d.1.1		2
28.27	even	2		208.4.a.h.1.2		2
35.13	even	4		325.4.b.e.274.1		4
35.27	even	4		325.4.b.e.274.4		4
35.34	odd	2		325.4.a.f.1.1		2
56.13	odd	2		832.4.a.s.1.2		2
56.27	even	2		832.4.a.z.1.1		2
77.76	even	2		1573.4.a.b.1.1		2
84.83	odd	2		1872.4.a.bb.1.1		2
91.6	even	12		169.4.e.f.23.1		8
91.20	even	12		169.4.e.f.23.4		8
91.34	even	4		169.4.b.f.168.4		4
91.41	even	12		169.4.e.f.147.4		8
91.48	odd	6		169.4.c.g.146.1		4
91.55	odd	6		169.4.c.g.22.1		4
91.62	odd	6		169.4.c.j.22.2		4
91.69	odd	6		169.4.c.j.146.2		4
91.76	even	12		169.4.e.f.147.1		8
91.83	even	4		169.4.b.f.168.1		4
91.90	odd	2		169.4.a.g.1.1		2
273.272	even	2		1521.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.2	✓	2	7.6	odd	2
117.4.a.d.1.1		2	21.20	even	2
169.4.a.g.1.1		2	91.90	odd	2
169.4.b.f.168.1		4	91.83	even	4
169.4.b.f.168.4		4	91.34	even	4
169.4.c.g.22.1		4	91.55	odd	6
169.4.c.g.146.1		4	91.48	odd	6
169.4.c.j.22.2		4	91.62	odd	6
169.4.c.j.146.2		4	91.69	odd	6
169.4.e.f.23.1		8	91.6	even	12
169.4.e.f.23.4		8	91.20	even	12
169.4.e.f.147.1		8	91.76	even	12
169.4.e.f.147.4		8	91.41	even	12
208.4.a.h.1.2		2	28.27	even	2
325.4.a.f.1.1		2	35.34	odd	2
325.4.b.e.274.1		4	35.13	even	4
325.4.b.e.274.4		4	35.27	even	4
637.4.a.b.1.2		2	1.1	even	1	trivial
832.4.a.s.1.2		2	56.13	odd	2
832.4.a.z.1.1		2	56.27	even	2
1521.4.a.r.1.2		2	273.272	even	2
1573.4.a.b.1.1		2	77.76	even	2
1872.4.a.bb.1.1		2	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} +3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -24.1771 q^{8} -13.4233 q^{9} +O(q^{10})\)	\(q+2.56155 q^{2} +3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -24.1771 q^{8} -13.4233 q^{9} -1.43845 q^{10} +64.7386 q^{11} -5.30019 q^{12} +13.0000 q^{13} -2.06913 q^{15} -50.4233 q^{16} +25.5464 q^{17} -34.3845 q^{18} +107.970 q^{19} +0.807764 q^{20} +165.831 q^{22} +73.2614 q^{23} -89.0843 q^{24} -124.685 q^{25} +33.3002 q^{26} -148.946 q^{27} +175.909 q^{29} -5.30019 q^{30} +113.093 q^{31} +64.2547 q^{32} +238.540 q^{33} +65.4384 q^{34} +19.3087 q^{36} +114.808 q^{37} +276.570 q^{38} +47.9006 q^{39} +13.5767 q^{40} +69.6458 q^{41} +438.302 q^{43} -93.1231 q^{44} +7.53789 q^{45} +187.663 q^{46} +31.9479 q^{47} -185.793 q^{48} -319.386 q^{50} +94.1298 q^{51} -18.6998 q^{52} +2.84658 q^{53} -381.533 q^{54} -36.3542 q^{55} +397.831 q^{57} +450.600 q^{58} -71.6325 q^{59} +2.97633 q^{60} +920.695 q^{61} +289.693 q^{62} +567.978 q^{64} -7.30019 q^{65} +611.032 q^{66} -444.280 q^{67} -36.7471 q^{68} +269.943 q^{69} -541.719 q^{71} +324.536 q^{72} -764.004 q^{73} +294.086 q^{74} -459.420 q^{75} -155.309 q^{76} +122.700 q^{78} -421.538 q^{79} +28.3153 q^{80} -186.386 q^{81} +178.401 q^{82} -603.797 q^{83} -14.3457 q^{85} +1122.73 q^{86} +648.165 q^{87} -1565.19 q^{88} +1159.88 q^{89} +19.3087 q^{90} -105.383 q^{92} +416.708 q^{93} +81.8362 q^{94} -60.6307 q^{95} +236.757 q^{96} -583.269 q^{97} -869.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9}+O(q^{10}) \) 2 * q + q^2 - 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 - 3 * q^8 + 35 * q^9	\( 2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9} - 7 q^{10} + 80 q^{11} + 43 q^{12} + 26 q^{13} - 33 q^{15} - 39 q^{16} - 19 q^{17} - 110 q^{18} + 84 q^{19} - 19 q^{20} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 13 q^{26} - 335 q^{27} - 44 q^{29} + 43 q^{30} + 86 q^{31} - 123 q^{32} + 106 q^{33} + 135 q^{34} - 250 q^{36} + 209 q^{37} + 314 q^{38} - 65 q^{39} + 89 q^{40} + 230 q^{41} + 287 q^{43} - 178 q^{44} + 180 q^{45} - 4 q^{46} - 435 q^{47} - 285 q^{48} - 144 q^{50} + 481 q^{51} - 91 q^{52} - 118 q^{53} - 91 q^{54} + 18 q^{55} + 606 q^{57} + 794 q^{58} + 368 q^{59} + 175 q^{60} + 1058 q^{61} + 332 q^{62} + 769 q^{64} + 39 q^{65} + 818 q^{66} + 68 q^{67} + 211 q^{68} - 796 q^{69} - 131 q^{71} + 1350 q^{72} - 456 q^{73} + 147 q^{74} + 516 q^{75} - 22 q^{76} + 299 q^{78} - 1008 q^{79} + 69 q^{80} + 122 q^{81} - 72 q^{82} - 1958 q^{83} - 173 q^{85} + 1359 q^{86} + 2558 q^{87} - 1242 q^{88} + 720 q^{89} - 250 q^{90} - 788 q^{92} + 652 q^{93} + 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} - 130 q^{99}+O(q^{100}) \) 2 * q + q^2 - 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 - 3 * q^8 + 35 * q^9 - 7 * q^10 + 80 * q^11 + 43 * q^12 + 26 * q^13 - 33 * q^15 - 39 * q^16 - 19 * q^17 - 110 * q^18 + 84 * q^19 - 19 * q^20 + 142 * q^22 + 196 * q^23 - 273 * q^24 - 237 * q^25 + 13 * q^26 - 335 * q^27 - 44 * q^29 + 43 * q^30 + 86 * q^31 - 123 * q^32 + 106 * q^33 + 135 * q^34 - 250 * q^36 + 209 * q^37 + 314 * q^38 - 65 * q^39 + 89 * q^40 + 230 * q^41 + 287 * q^43 - 178 * q^44 + 180 * q^45 - 4 * q^46 - 435 * q^47 - 285 * q^48 - 144 * q^50 + 481 * q^51 - 91 * q^52 - 118 * q^53 - 91 * q^54 + 18 * q^55 + 606 * q^57 + 794 * q^58 + 368 * q^59 + 175 * q^60 + 1058 * q^61 + 332 * q^62 + 769 * q^64 + 39 * q^65 + 818 * q^66 + 68 * q^67 + 211 * q^68 - 796 * q^69 - 131 * q^71 + 1350 * q^72 - 456 * q^73 + 147 * q^74 + 516 * q^75 - 22 * q^76 + 299 * q^78 - 1008 * q^79 + 69 * q^80 + 122 * q^81 - 72 * q^82 - 1958 * q^83 - 173 * q^85 + 1359 * q^86 + 2558 * q^87 - 1242 * q^88 + 720 * q^89 - 250 * q^90 - 788 * q^92 + 652 * q^93 + 811 * q^94 - 146 * q^95 + 1863 * q^96 + 928 * q^97 - 130 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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