LMFDB - Embedded newform 2401.4.a.f.1.55

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2401,4,Mod(1,2401)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2401, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2401.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2401.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:	$141.663585924$
Analytic rank:	$1$
Dimension:	$78$
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.55
Character	$\chi$	$=$	2401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.50263 q^{2} -1.91363 q^{3} -1.73685 q^{4} +0.617821 q^{5} -4.78909 q^{6} -24.3677 q^{8} -23.3380 q^{9} +O(q^{10})$	\(q+2.50263 q^{2} -1.91363 q^{3} -1.73685 q^{4} +0.617821 q^{5} -4.78909 q^{6} -24.3677 q^{8} -23.3380 q^{9} +1.54618 q^{10} -0.980419 q^{11} +3.32369 q^{12} -17.2201 q^{13} -1.18228 q^{15} -47.0885 q^{16} +91.7828 q^{17} -58.4064 q^{18} +129.179 q^{19} -1.07306 q^{20} -2.45362 q^{22} +8.42602 q^{23} +46.6307 q^{24} -124.618 q^{25} -43.0955 q^{26} +96.3282 q^{27} +271.857 q^{29} -2.95880 q^{30} -172.856 q^{31} +77.0967 q^{32} +1.87615 q^{33} +229.698 q^{34} +40.5347 q^{36} +31.6376 q^{37} +323.287 q^{38} +32.9528 q^{39} -15.0549 q^{40} +186.358 q^{41} +89.7114 q^{43} +1.70284 q^{44} -14.4187 q^{45} +21.0872 q^{46} -325.239 q^{47} +90.1098 q^{48} -311.873 q^{50} -175.638 q^{51} +29.9088 q^{52} +465.832 q^{53} +241.074 q^{54} -0.605724 q^{55} -247.200 q^{57} +680.357 q^{58} -380.173 q^{59} +2.05344 q^{60} -216.463 q^{61} -432.595 q^{62} +569.653 q^{64} -10.6389 q^{65} +4.69532 q^{66} -395.364 q^{67} -159.413 q^{68} -16.1242 q^{69} -467.435 q^{71} +568.695 q^{72} -599.902 q^{73} +79.1772 q^{74} +238.473 q^{75} -224.365 q^{76} +82.4686 q^{78} +3.23666 q^{79} -29.0923 q^{80} +445.791 q^{81} +466.384 q^{82} -532.464 q^{83} +56.7054 q^{85} +224.514 q^{86} -520.232 q^{87} +23.8906 q^{88} +1141.31 q^{89} -36.0847 q^{90} -14.6348 q^{92} +330.783 q^{93} -813.951 q^{94} +79.8095 q^{95} -147.534 q^{96} -224.251 q^{97} +22.8810 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q - q^{2} - 35 q^{3} + 287 q^{4} - 63 q^{5} - 70 q^{6} + 15 q^{8} + 579 q^{9}+O(q^{10}) $ 78 * q - q^2 - 35 * q^3 + 287 * q^4 - 63 * q^5 - 70 * q^6 + 15 * q^8 + 579 * q^9	\( 78 q - q^{2} - 35 q^{3} + 287 q^{4} - 63 q^{5} - 70 q^{6} + 15 q^{8} + 579 q^{9} - 126 q^{10} - q^{11} - 532 q^{12} - 196 q^{13} + 164 q^{15} + 979 q^{16} - 497 q^{17} + 16 q^{18} - 490 q^{19} - 868 q^{20} + 64 q^{22} + 69 q^{23} - 1365 q^{24} + 1419 q^{25} - 1176 q^{26} - 1547 q^{27} - 54 q^{29} + 39 q^{30} - 1162 q^{31} + 1624 q^{32} + 161 q^{33} - 154 q^{34} + 1813 q^{36} - 323 q^{37} - 882 q^{38} - 126 q^{39} - 2723 q^{40} - 1610 q^{41} + 2 q^{43} + 140 q^{44} - 1736 q^{45} + 1478 q^{46} - 2765 q^{47} - 4802 q^{48} - 3092 q^{50} + 2125 q^{51} + 581 q^{52} - 2213 q^{53} - 3514 q^{54} - 3115 q^{55} + 73 q^{57} - 2403 q^{58} - 5061 q^{59} + 1918 q^{60} - 1925 q^{61} - 3178 q^{62} + 2101 q^{64} - 1876 q^{65} - 5670 q^{66} + 202 q^{67} - 5635 q^{68} - 7595 q^{69} + 1262 q^{71} + 4072 q^{72} - 4837 q^{73} - 3821 q^{74} - 7882 q^{75} - 5817 q^{76} + 6692 q^{78} + 608 q^{79} - 12348 q^{80} + 2850 q^{81} + 5208 q^{82} - 3668 q^{83} - 3091 q^{85} + 3610 q^{86} - 938 q^{87} + 3702 q^{88} - 8925 q^{89} + 2198 q^{90} + 2940 q^{92} + 4435 q^{93} - 4102 q^{94} + 3665 q^{95} - 9800 q^{96} - 5558 q^{97} - 5402 q^{99}+O(q^{100}) \) 78 * q - q^2 - 35 * q^3 + 287 * q^4 - 63 * q^5 - 70 * q^6 + 15 * q^8 + 579 * q^9 - 126 * q^10 - q^11 - 532 * q^12 - 196 * q^13 + 164 * q^15 + 979 * q^16 - 497 * q^17 + 16 * q^18 - 490 * q^19 - 868 * q^20 + 64 * q^22 + 69 * q^23 - 1365 * q^24 + 1419 * q^25 - 1176 * q^26 - 1547 * q^27 - 54 * q^29 + 39 * q^30 - 1162 * q^31 + 1624 * q^32 + 161 * q^33 - 154 * q^34 + 1813 * q^36 - 323 * q^37 - 882 * q^38 - 126 * q^39 - 2723 * q^40 - 1610 * q^41 + 2 * q^43 + 140 * q^44 - 1736 * q^45 + 1478 * q^46 - 2765 * q^47 - 4802 * q^48 - 3092 * q^50 + 2125 * q^51 + 581 * q^52 - 2213 * q^53 - 3514 * q^54 - 3115 * q^55 + 73 * q^57 - 2403 * q^58 - 5061 * q^59 + 1918 * q^60 - 1925 * q^61 - 3178 * q^62 + 2101 * q^64 - 1876 * q^65 - 5670 * q^66 + 202 * q^67 - 5635 * q^68 - 7595 * q^69 + 1262 * q^71 + 4072 * q^72 - 4837 * q^73 - 3821 * q^74 - 7882 * q^75 - 5817 * q^76 + 6692 * q^78 + 608 * q^79 - 12348 * q^80 + 2850 * q^81 + 5208 * q^82 - 3668 * q^83 - 3091 * q^85 + 3610 * q^86 - 938 * q^87 + 3702 * q^88 - 8925 * q^89 + 2198 * q^90 + 2940 * q^92 + 4435 * q^93 - 4102 * q^94 + 3665 * q^95 - 9800 * q^96 - 5558 * q^97 - 5402 * 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 $

$2$	2.50263	0.884813	0.442406	−	0.896815i	$-0.354125\pi$
$2$	2.50263	0.884813	0.442406	+	0.896815i	$0.354125\pi$
$3$	−1.91363	−0.368277	−0.184139	−	0.982900i	$-0.558950\pi$
$3$	−1.91363	−0.368277	−0.184139	+	0.982900i	$0.558950\pi$
$4$	−1.73685	−0.217107
$5$	0.617821	0.0552596	0.0276298	−	0.999618i	$-0.491204\pi$
$5$	0.617821	0.0552596	0.0276298	+	0.999618i	$0.491204\pi$
$6$	−4.78909	−0.325856
$7$	0	0
$7$	0	0
$8$	−24.3677	−1.07691
$9$	−23.3380	−0.864372
$10$	1.54618	0.0488944
$11$	−0.980419	−0.0268734	−0.0134367	−	0.999910i	$-0.504277\pi$
$11$	−0.980419	−0.0268734	−0.0134367	+	0.999910i	$0.504277\pi$
$12$	3.32369	0.0799555
$13$	−17.2201	−0.367384	−0.183692	−	0.982984i	$-0.558805\pi$
$13$	−17.2201	−0.367384	−0.183692	+	0.982984i	$0.558805\pi$
$14$	0	0
$15$	−1.18228	−0.0203509
$16$	−47.0885	−0.735758
$17$	91.7828	1.30945	0.654723	−	0.755868i	$-0.272785\pi$
$17$	91.7828	1.30945	0.654723	+	0.755868i	$0.272785\pi$
$18$	−58.4064	−0.764807
$19$	129.179	1.55977	0.779886	−	0.625921i	$-0.215277\pi$
$19$	129.179	1.55977	0.779886	+	0.625921i	$0.215277\pi$
$20$	−1.07306	−0.0119972
$21$	0	0
$22$	−2.45362	−0.0237779
$23$	8.42602	0.0763890	0.0381945	−	0.999270i	$-0.487839\pi$
$23$	8.42602	0.0763890	0.0381945	+	0.999270i	$0.487839\pi$
$24$	46.6307	0.396602
$25$	−124.618	−0.996946
$26$	−43.0955	−0.325066
$27$	96.3282	0.686606
$28$	0	0
$29$	271.857	1.74078	0.870389	−	0.492365i	$-0.163867\pi$
$29$	271.857	1.74078	0.870389	+	0.492365i	$0.163867\pi$
$30$	−2.95880	−0.0180067
$31$	−172.856	−1.00148	−0.500741	−	0.865597i	$-0.666939\pi$
$31$	−172.856	−1.00148	−0.500741	+	0.865597i	$0.666939\pi$
$32$	77.0967	0.425903
$33$	1.87615	0.00989686
$34$	229.698	1.15862
$35$	0	0
$36$	40.5347	0.187661
$37$	31.6376	0.140573	0.0702864	−	0.997527i	$-0.477609\pi$
$37$	31.6376	0.140573	0.0702864	+	0.997527i	$0.477609\pi$
$38$	323.287	1.38011
$39$	32.9528	0.135299
$40$	−15.0549	−0.0595097
$41$	186.358	0.709859	0.354929	−	0.934893i	$-0.384505\pi$
$41$	186.358	0.709859	0.354929	+	0.934893i	$0.384505\pi$
$42$	0	0
$43$	89.7114	0.318160	0.159080	−	0.987266i	$-0.449147\pi$
$43$	89.7114	0.318160	0.159080	+	0.987266i	$0.449147\pi$
$44$	1.70284	0.00583439
$45$	−14.4187	−0.0477649
$46$	21.0872	0.0675899
$47$	−325.239	−1.00938	−0.504691	−	0.863300i	$-0.668393\pi$
$47$	−325.239	−1.00938	−0.504691	+	0.863300i	$0.668393\pi$
$48$	90.1098	0.270963
$49$	0	0
$50$	−311.873	−0.882111
$51$	−175.638	−0.482240
$52$	29.9088	0.0797616
$53$	465.832	1.20730	0.603651	−	0.797249i	$-0.293712\pi$
$53$	465.832	1.20730	0.603651	+	0.797249i	$0.293712\pi$
$54$	241.074	0.607518
$55$	−0.605724	−0.00148501
$56$	0	0
$57$	−247.200	−0.574429
$58$	680.357	1.54026
$59$	−380.173	−0.838886	−0.419443	−	0.907782i	$-0.637775\pi$
$59$	−380.173	−0.838886	−0.419443	+	0.907782i	$0.637775\pi$
$60$	2.05344	0.00441831
$61$	−216.463	−0.454349	−0.227175	−	0.973854i	$-0.572949\pi$
$61$	−216.463	−0.454349	−0.227175	+	0.973854i	$0.572949\pi$
$62$	−432.595	−0.886124
$63$	0	0
$64$	569.653	1.11260
$65$	−10.6389	−0.0203015
$66$	4.69532	0.00875687
$67$	−395.364	−0.720916	−0.360458	−	0.932775i	$-0.617380\pi$
$67$	−395.364	−0.720916	−0.360458	+	0.932775i	$0.617380\pi$
$68$	−159.413	−0.284290
$69$	−16.1242	−0.0281323
$70$	0	0
$71$	−467.435	−0.781329	−0.390664	−	0.920533i	$-0.627755\pi$
$71$	−467.435	−0.781329	−0.390664	+	0.920533i	$0.627755\pi$
$72$	568.695	0.930852
$73$	−599.902	−0.961824	−0.480912	−	0.876769i	$-0.659694\pi$
$73$	−599.902	−0.961824	−0.480912	+	0.876769i	$0.659694\pi$
$74$	79.1772	0.124381
$75$	238.473	0.367153
$76$	−224.365	−0.338637
$77$	0	0
$78$	82.4686	0.119715
$79$	3.23666	0.00460953	0.00230476	−	0.999997i	$-0.499266\pi$
$79$	3.23666	0.00460953	0.00230476	+	0.999997i	$0.499266\pi$
$80$	−29.0923	−0.0406577
$81$	445.791	0.611510
$82$	466.384	0.628092
$83$	−532.464	−0.704163	−0.352082	−	0.935969i	$-0.614526\pi$
$83$	−532.464	−0.704163	−0.352082	+	0.935969i	$0.614526\pi$
$84$	0	0
$85$	56.7054	0.0723595
$86$	224.514	0.281512
$87$	−520.232	−0.641089
$88$	23.8906	0.0289403
$89$	1141.31	1.35931	0.679655	−	0.733532i	$-0.262130\pi$
$89$	1141.31	1.35931	0.679655	+	0.733532i	$0.262130\pi$
$90$	−36.0847	−0.0422630
$91$	0	0
$92$	−14.6348	−0.0165846
$93$	330.783	0.368823
$94$	−813.951	−0.893113
$95$	79.8095	0.0861924
$96$	−147.534	−0.156851
$97$	−224.251	−0.234734	−0.117367	−	0.993089i	$-0.537445\pi$
$97$	−224.251	−0.234734	−0.117367	+	0.993089i	$0.537445\pi$
$98$	0	0
$99$	22.8810	0.0232286
$100$	216.444	0.216444
$101$	−380.000	−0.374371	−0.187185	−	0.982325i	$-0.559936\pi$
$101$	−380.000	−0.374371	−0.187185	+	0.982325i	$0.559936\pi$
$102$	−439.556	−0.426692
$103$	−1295.60	−1.23941	−0.619707	−	0.784833i	$-0.712748\pi$
$103$	−1295.60	−1.23941	−0.619707	+	0.784833i	$0.712748\pi$
$104$	419.615	0.395640
$105$	0	0
$106$	1165.81	1.06824
$107$	−868.797	−0.784951	−0.392476	−	0.919762i	$-0.628381\pi$
$107$	−868.797	−0.784951	−0.392476	+	0.919762i	$0.628381\pi$
$108$	−167.308	−0.149067
$109$	−1589.12	−1.39643	−0.698213	−	0.715890i	$-0.746021\pi$
$109$	−1589.12	−1.39643	−0.698213	+	0.715890i	$0.746021\pi$
$110$	−1.51590	−0.00131396
$111$	−60.5426	−0.0517698
$112$	0	0
$113$	−960.090	−0.799271	−0.399636	−	0.916674i	$-0.630863\pi$
$113$	−960.090	−0.799271	−0.399636	+	0.916674i	$0.630863\pi$
$114$	−618.650	−0.508262
$115$	5.20577	0.00422123
$116$	−472.176	−0.377934
$117$	401.883	0.317557
$118$	−951.431	−0.742257
$119$	0	0
$120$	28.8094	0.0219161
$121$	−1330.04	−0.999278
$122$	−541.727	−0.402014
$123$	−356.619	−0.261425
$124$	300.226	0.217428
$125$	−154.220	−0.110351
$126$	0	0
$127$	148.945	0.104069	0.0520343	−	0.998645i	$-0.483429\pi$
$127$	148.945	0.104069	0.0520343	+	0.998645i	$0.483429\pi$
$128$	808.855	0.558542
$129$	−171.674	−0.117171
$130$	−26.6253	−0.0179630
$131$	2176.32	1.45149	0.725747	−	0.687961i	$-0.241494\pi$
$131$	2176.32	1.45149	0.725747	+	0.687961i	$0.241494\pi$
$132$	−3.25860	−0.00214867
$133$	0	0
$134$	−989.448	−0.637875
$135$	59.5136	0.0379416
$136$	−2236.54	−1.41016
$137$	2719.06	1.69566	0.847828	−	0.530272i	$-0.177910\pi$
$137$	2719.06	1.69566	0.847828	+	0.530272i	$0.177910\pi$
$138$	−40.3530	−0.0248918
$139$	−2943.20	−1.79596	−0.897981	−	0.440034i	$-0.854966\pi$
$139$	−2943.20	−1.79596	−0.897981	+	0.440034i	$0.854966\pi$
$140$	0	0
$141$	622.385	0.371732
$142$	−1169.82	−0.691330
$143$	16.8829	0.00987286
$144$	1098.95	0.635969
$145$	167.959	0.0961947
$146$	−1501.33	−0.851034
$147$	0	0
$148$	−54.9499	−0.0305193
$149$	1271.97	0.699352	0.349676	−	0.936871i	$-0.386292\pi$
$149$	1271.97	0.699352	0.349676	+	0.936871i	$0.386292\pi$
$150$	596.809	0.324861
$151$	−2881.68	−1.55303	−0.776516	−	0.630097i	$-0.783015\pi$
$151$	−2881.68	−1.55303	−0.776516	+	0.630097i	$0.783015\pi$
$152$	−3147.80	−1.67974
$153$	−2142.03	−1.13185
$154$	0	0
$155$	−106.794	−0.0553415
$156$	−57.2342	−0.0293744
$157$	−2792.45	−1.41950	−0.709752	−	0.704452i	$-0.751193\pi$
$157$	−2792.45	−1.41950	−0.709752	+	0.704452i	$0.751193\pi$
$158$	8.10016	0.00407857
$159$	−891.429	−0.444622
$160$	47.6320	0.0235353
$161$	0	0
$162$	1115.65	0.541072
$163$	3670.44	1.76375	0.881873	−	0.471486i	$-0.156282\pi$
$163$	3670.44	1.76375	0.881873	+	0.471486i	$0.156282\pi$
$164$	−323.676	−0.154115
$165$	1.15913	0.000546897 0
$166$	−1332.56	−0.623053
$167$	604.411	0.280064	0.140032	−	0.990147i	$-0.455279\pi$
$167$	604.411	0.280064	0.140032	+	0.990147i	$0.455279\pi$
$168$	0	0
$169$	−1900.47	−0.865029
$170$	141.912	0.0640246
$171$	−3014.78	−1.34822
$172$	−155.816	−0.0690746
$173$	1657.39	0.728375	0.364188	−	0.931326i	$-0.381347\pi$
$173$	1657.39	0.728375	0.364188	+	0.931326i	$0.381347\pi$
$174$	−1301.95	−0.567244
$175$	0	0
$176$	46.1665	0.0197723
$177$	727.508	0.308943
$178$	2856.27	1.20273
$179$	105.379	0.0440024	0.0220012	−	0.999758i	$-0.492996\pi$
$179$	105.379	0.0440024	0.0220012	+	0.999758i	$0.492996\pi$
$180$	25.0432	0.0103701
$181$	−3436.29	−1.41114	−0.705572	−	0.708638i	$-0.749310\pi$
$181$	−3436.29	−1.41114	−0.705572	+	0.708638i	$0.749310\pi$
$182$	0	0
$183$	414.230	0.167326
$184$	−205.323	−0.0822642
$185$	19.5464	0.00776800
$186$	827.826	0.326339
$187$	−89.9856	−0.0351893
$188$	564.892	0.219143
$189$	0	0
$190$	199.733	0.0762642
$191$	1157.70	0.438576	0.219288	−	0.975660i	$-0.429627\pi$
$191$	1157.70	0.438576	0.219288	+	0.975660i	$0.429627\pi$
$192$	−1090.10	−0.409746
$193$	4537.19	1.69220	0.846098	−	0.533027i	$-0.178946\pi$
$193$	4537.19	1.69220	0.846098	+	0.533027i	$0.178946\pi$
$194$	−561.217	−0.207696
$195$	20.3590	0.00747659
$196$	0	0
$197$	−1841.84	−0.666119	−0.333059	−	0.942906i	$-0.608081\pi$
$197$	−1841.84	−0.666119	−0.333059	+	0.942906i	$0.608081\pi$
$198$	57.2627	0.0205530
$199$	1008.78	0.359348	0.179674	−	0.983726i	$-0.442496\pi$
$199$	1008.78	0.359348	0.179674	+	0.983726i	$0.442496\pi$
$200$	3036.66	1.07362
$201$	756.578	0.265497
$202$	−950.999	−0.331248
$203$	0	0
$204$	305.057	0.104697
$205$	115.136	0.0392265
$206$	−3242.41	−1.09665
$207$	−196.647	−0.0660285
$208$	810.869	0.270306
$209$	−126.649	−0.0419164
$210$	0	0
$211$	3064.23	0.999763	0.499882	−	0.866094i	$-0.333377\pi$
$211$	3064.23	0.999763	0.499882	+	0.866094i	$0.333377\pi$
$212$	−809.082	−0.262113
$213$	894.496	0.287746
$214$	−2174.28	−0.694535
$215$	55.4256	0.0175814
$216$	−2347.30	−0.739414
$217$	0	0
$218$	−3976.99	−1.23558
$219$	1147.99	0.354218
$220$	1.05205	0.000322406 0
$221$	−1580.51	−0.481070
$222$	−151.516	−0.0458066
$223$	−337.755	−0.101425	−0.0507125	−	0.998713i	$-0.516149\pi$
$223$	−337.755	−0.101425	−0.0507125	+	0.998713i	$0.516149\pi$
$224$	0	0
$225$	2908.35	0.861732
$226$	−2402.75	−0.707205
$227$	−3290.71	−0.962168	−0.481084	−	0.876674i	$-0.659757\pi$
$227$	−3290.71	−0.962168	−0.481084	+	0.876674i	$0.659757\pi$
$228$	429.350	0.124712
$229$	3607.98	1.04115	0.520573	−	0.853817i	$-0.325719\pi$
$229$	3607.98	1.04115	0.520573	+	0.853817i	$0.325719\pi$
$230$	13.0281	0.00373499
$231$	0	0
$232$	−6624.53	−1.87466
$233$	−4.04598	−0.00113760	−0.000568800	−	1.00000i	$-0.500181\pi$
$233$	−4.04598	−0.00113760	−0.000568800		1.00000i	$0.500181\pi$
$234$	1005.76	0.280978
$235$	−200.939	−0.0557780
$236$	660.304	0.182128
$237$	−6.19376	−0.00169759
$238$	0	0
$239$	6351.34	1.71897	0.859485	−	0.511161i	$-0.170785\pi$
$239$	6351.34	1.71897	0.859485	+	0.511161i	$0.170785\pi$
$240$	55.6718	0.0149733
$241$	−6827.49	−1.82488	−0.912442	−	0.409205i	$-0.865806\pi$
$241$	−6827.49	−1.82488	−0.912442	+	0.409205i	$0.865806\pi$
$242$	−3328.59	−0.884174
$243$	−3453.94	−0.911811
$244$	375.965	0.0986422
$245$	0	0
$246$	−892.485	−0.231312
$247$	−2224.47	−0.573036
$248$	4212.12	1.07851
$249$	1018.94	0.259327
$250$	−385.954	−0.0976395
$251$	784.315	0.197233	0.0986166	−	0.995126i	$-0.468558\pi$
$251$	784.315	0.197233	0.0986166	+	0.995126i	$0.468558\pi$
$252$	0	0
$253$	−8.26103	−0.00205283
$254$	372.753	0.0920811
$255$	−108.513	−0.0266484
$256$	−2532.96	−0.618398
$257$	−2360.35	−0.572897	−0.286449	−	0.958096i	$-0.592475\pi$
$257$	−2360.35	−0.572897	−0.286449	+	0.958096i	$0.592475\pi$
$258$	−429.636	−0.103674
$259$	0	0
$260$	18.4783	0.00440759
$261$	−6344.61	−1.50468
$262$	5446.51	1.28430
$263$	−5348.84	−1.25408	−0.627041	−	0.778986i	$-0.715734\pi$
$263$	−5348.84	−1.25408	−0.627041	+	0.778986i	$0.715734\pi$
$264$	−45.7176	−0.0106580
$265$	287.801	0.0667150
$266$	0	0
$267$	−2184.04	−0.500603
$268$	686.689	0.156516
$269$	−4209.98	−0.954227	−0.477114	−	0.878842i	$-0.658317\pi$
$269$	−4209.98	−0.954227	−0.477114	+	0.878842i	$0.658317\pi$
$270$	148.940	0.0335712
$271$	2557.10	0.573183	0.286591	−	0.958053i	$-0.407478\pi$
$271$	2557.10	0.573183	0.286591	+	0.958053i	$0.407478\pi$
$272$	−4321.92	−0.963436
$273$	0	0
$274$	6804.79	1.50034
$275$	122.178	0.0267913
$276$	28.0054	0.00610772
$277$	−1499.60	−0.325279	−0.162639	−	0.986686i	$-0.552001\pi$
$277$	−1499.60	−0.325279	−0.162639	+	0.986686i	$0.552001\pi$
$278$	−7365.73	−1.58909
$279$	4034.13	0.865653
$280$	0	0
$281$	−4956.74	−1.05229	−0.526146	−	0.850394i	$-0.676364\pi$
$281$	−4956.74	−1.05229	−0.526146	+	0.850394i	$0.676364\pi$
$282$	1557.60	0.328913
$283$	1980.65	0.416032	0.208016	−	0.978125i	$-0.433299\pi$
$283$	1980.65	0.416032	0.208016	+	0.978125i	$0.433299\pi$
$284$	811.866	0.169632
$285$	−152.725	−0.0317427
$286$	42.2516	0.00873563
$287$	0	0
$288$	−1799.29	−0.368139
$289$	3511.08	0.714651
$290$	420.339	0.0851143
$291$	429.132	0.0864474
$292$	1041.94	0.208818
$293$	−2385.58	−0.475656	−0.237828	−	0.971307i	$-0.576436\pi$
$293$	−2385.58	−0.475656	−0.237828	+	0.971307i	$0.576436\pi$
$294$	0	0
$295$	−234.879	−0.0463565
$296$	−770.937	−0.151384
$297$	−94.4419	−0.0184514
$298$	3183.26	0.618796
$299$	−145.097	−0.0280641
$300$	−414.192	−0.0797113
$301$	0	0
$302$	−7211.78	−1.37414
$303$	727.178	0.137872
$304$	−6082.84	−1.14762
$305$	−133.736	−0.0251072
$306$	−5360.71	−1.00147
$307$	−8210.96	−1.52646	−0.763232	−	0.646125i	$-0.776389\pi$
$307$	−8210.96	−1.52646	−0.763232	+	0.646125i	$0.776389\pi$
$308$	0	0
$309$	2479.30	0.456448
$310$	−267.267	−0.0489669
$311$	9189.53	1.67553	0.837766	−	0.546029i	$-0.183861\pi$
$311$	9189.53	1.67553	0.837766	+	0.546029i	$0.183861\pi$
$312$	−802.985	−0.145705
$313$	9235.77	1.66785	0.833925	−	0.551879i	$-0.186089\pi$
$313$	9235.77	1.66785	0.833925	+	0.551879i	$0.186089\pi$
$314$	−6988.47	−1.25599
$315$	0	0
$316$	−5.62160	−0.00100076
$317$	17.2422	0.00305495	0.00152747	−	0.999999i	$-0.499514\pi$
$317$	17.2422	0.00305495	0.00152747	+	0.999999i	$0.499514\pi$
$318$	−2230.91	−0.393407
$319$	−266.534	−0.0467806
$320$	351.944	0.0614820
$321$	1662.55	0.289080
$322$	0	0
$323$	11856.4	2.04244
$324$	−774.273	−0.132763
$325$	2145.94	0.366262
$326$	9185.74	1.56059
$327$	3040.99	0.514272
$328$	−4541.12	−0.764455
$329$	0	0
$330$	2.90087	0.000483901 0
$331$	−4920.25	−0.817044	−0.408522	−	0.912749i	$-0.633956\pi$
$331$	−4920.25	−0.817044	−0.408522	+	0.912749i	$0.633956\pi$
$332$	924.812	0.152879
$333$	−738.360	−0.121507
$334$	1512.62	0.247804
$335$	−244.264	−0.0398375
$336$	0	0
$337$	3429.43	0.554341	0.277171	−	0.960821i	$-0.410603\pi$
$337$	3429.43	0.554341	0.277171	+	0.960821i	$0.410603\pi$
$338$	−4756.17	−0.765388
$339$	1837.25	0.294354
$340$	−98.4889	−0.0157097
$341$	169.472	0.0269132
$342$	−7544.88	−1.19292
$343$	0	0
$344$	−2186.06	−0.342630
$345$	−9.96190	−0.00155458
$346$	4147.83	0.644475
$347$	−5465.44	−0.845533	−0.422767	−	0.906239i	$-0.638941\pi$
$347$	−5465.44	−0.845533	−0.422767	+	0.906239i	$0.638941\pi$
$348$	903.567	0.139185
$349$	−12680.7	−1.94493	−0.972465	−	0.233049i	$-0.925130\pi$
$349$	−12680.7	−1.94493	−0.972465	+	0.233049i	$0.925130\pi$
$350$	0	0
$351$	−1658.78	−0.252248
$352$	−75.5871	−0.0114455
$353$	−10666.4	−1.60826	−0.804130	−	0.594454i	$-0.797368\pi$
$353$	−10666.4	−1.60826	−0.804130	+	0.594454i	$0.797368\pi$
$354$	1820.68	0.273356
$355$	−288.791	−0.0431759
$356$	−1982.29	−0.295115
$357$	0	0
$358$	263.726	0.0389339
$359$	4862.00	0.714782	0.357391	−	0.933955i	$-0.383666\pi$
$359$	4862.00	0.714782	0.357391	+	0.933955i	$0.383666\pi$
$360$	351.352	0.0514385
$361$	9828.19	1.43289
$362$	−8599.75	−1.24860
$363$	2545.20	0.368011
$364$	0	0
$365$	−370.632	−0.0531500
$366$	1036.66	0.148053
$367$	−7806.58	−1.11036	−0.555178	−	0.831732i	$-0.687350\pi$
$367$	−7806.58	−1.11036	−0.555178	+	0.831732i	$0.687350\pi$
$368$	−396.769	−0.0562038
$369$	−4349.23	−0.613582
$370$	48.9174	0.00687323
$371$	0	0
$372$	−574.521	−0.0800740
$373$	−6220.29	−0.863470	−0.431735	−	0.902001i	$-0.642098\pi$
$373$	−6220.29	−0.863470	−0.431735	+	0.902001i	$0.642098\pi$
$374$	−225.200	−0.0311359
$375$	295.118	0.0406396
$376$	7925.32	1.08701
$377$	−4681.40	−0.639535
$378$	0	0
$379$	−6155.24	−0.834230	−0.417115	−	0.908854i	$-0.636959\pi$
$379$	−6155.24	−0.834230	−0.417115	+	0.908854i	$0.636959\pi$
$380$	−138.617	−0.0187129
$381$	−285.024	−0.0383261
$382$	2897.29	0.388058
$383$	−2968.21	−0.396001	−0.198000	−	0.980202i	$-0.563445\pi$
$383$	−2968.21	−0.396001	−0.198000	+	0.980202i	$0.563445\pi$
$384$	−1547.84	−0.205698
$385$	0	0
$386$	11354.9	1.49728
$387$	−2093.69	−0.275008
$388$	389.491	0.0509624
$389$	−6871.39	−0.895613	−0.447806	−	0.894131i	$-0.647795\pi$
$389$	−6871.39	−0.895613	−0.447806	+	0.894131i	$0.647795\pi$
$390$	50.9509	0.00661538
$391$	773.364	0.100027
$392$	0	0
$393$	−4164.66	−0.534553
$394$	−4609.43	−0.589390
$395$	1.99968	0.000254721 0
$396$	−39.7410	−0.00504308
$397$	7158.99	0.905036	0.452518	−	0.891755i	$-0.350526\pi$
$397$	7158.99	0.905036	0.452518	+	0.891755i	$0.350526\pi$
$398$	2524.59	0.317956
$399$	0	0
$400$	5868.09	0.733511
$401$	−5130.78	−0.638950	−0.319475	−	0.947595i	$-0.603507\pi$
$401$	−5130.78	−0.638950	−0.319475	+	0.947595i	$0.603507\pi$
$402$	1893.43	0.234915
$403$	2976.61	0.367929
$404$	660.004	0.0812783
$405$	275.419	0.0337918
$406$	0	0
$407$	−31.0181	−0.00377767
$408$	4279.90	0.519329
$409$	965.388	0.116712	0.0583562	−	0.998296i	$-0.481414\pi$
$409$	965.388	0.116712	0.0583562	+	0.998296i	$0.481414\pi$
$410$	288.142	0.0347081
$411$	−5203.26	−0.624472
$412$	2250.27	0.269085
$413$	0	0
$414$	−492.134	−0.0584228
$415$	−328.968	−0.0389118
$416$	−1327.61	−0.156470
$417$	5632.18	0.661412
$418$	−316.956	−0.0370881
$419$	−13111.9	−1.52878	−0.764392	−	0.644752i	$-0.776961\pi$
$419$	−13111.9	−1.52878	−0.764392	+	0.644752i	$0.776961\pi$
$420$	0	0
$421$	−3191.23	−0.369432	−0.184716	−	0.982792i	$-0.559137\pi$
$421$	−3191.23	−0.369432	−0.184716	+	0.982792i	$0.559137\pi$
$422$	7668.62	0.884603
$423$	7590.43	0.872481
$424$	−11351.3	−1.30016
$425$	−11437.8	−1.30545
$426$	2238.59	0.254601
$427$	0	0
$428$	1508.97	0.170418
$429$	−32.3076	−0.00363595
$430$	138.710	0.0155562
$431$	−12101.7	−1.35248	−0.676240	−	0.736681i	$-0.736392\pi$
$431$	−12101.7	−1.35248	−0.676240	+	0.736681i	$0.736392\pi$
$432$	−4535.95	−0.505176
$433$	−8574.80	−0.951683	−0.475841	−	0.879531i	$-0.657856\pi$
$433$	−8574.80	−0.951683	−0.475841	+	0.879531i	$0.657856\pi$
$434$	0	0
$435$	−321.411	−0.0354264
$436$	2760.07	0.303173
$437$	1088.46	0.119149
$438$	2872.98	0.313417
$439$	10956.3	1.19116	0.595578	−	0.803297i	$-0.296923\pi$
$439$	10956.3	1.19116	0.595578	+	0.803297i	$0.296923\pi$
$440$	14.7601	0.00159923
$441$	0	0
$442$	−3955.43	−0.425657
$443$	9259.98	0.993126	0.496563	−	0.868001i	$-0.334595\pi$
$443$	9259.98	0.993126	0.496563	+	0.868001i	$0.334595\pi$
$444$	105.154	0.0112396
$445$	705.126	0.0751150
$446$	−845.276	−0.0897421
$447$	−2434.07	−0.257556
$448$	0	0
$449$	3737.33	0.392818	0.196409	−	0.980522i	$-0.437072\pi$
$449$	3737.33	0.392818	0.196409	+	0.980522i	$0.437072\pi$
$450$	7278.51	0.762472
$451$	−182.709	−0.0190763
$452$	1667.53	0.173527
$453$	5514.46	0.571947
$454$	−8235.43	−0.851339
$455$	0	0
$456$	6023.70	0.618609
$457$	−7668.63	−0.784952	−0.392476	−	0.919762i	$-0.628381\pi$
$457$	−7668.63	−0.784952	−0.392476	+	0.919762i	$0.628381\pi$
$458$	9029.44	0.921218
$459$	8841.27	0.899074
$460$	−9.04167	−0.000916456 0
$461$	−13700.1	−1.38411	−0.692055	−	0.721844i	$-0.743295\pi$
$461$	−13700.1	−1.38411	−0.692055	+	0.721844i	$0.743295\pi$
$462$	0	0
$463$	−5090.74	−0.510986	−0.255493	−	0.966811i	$-0.582238\pi$
$463$	−5090.74	−0.510986	−0.255493	+	0.966811i	$0.582238\pi$
$464$	−12801.3	−1.28079
$465$	204.365	0.0203810
$466$	−10.1256	−0.00100656
$467$	689.574	0.0683291	0.0341645	−	0.999416i	$-0.489123\pi$
$467$	689.574	0.0683291	0.0341645	+	0.999416i	$0.489123\pi$
$468$	−698.012	−0.0689436
$469$	0	0
$470$	−502.876	−0.0493531
$471$	5343.71	0.522771
$472$	9263.94	0.903405
$473$	−87.9547	−0.00855003
$474$	−15.5007	−0.00150204
$475$	−16098.1	−1.55501
$476$	0	0
$477$	−10871.6	−1.04356
$478$	15895.0	1.52097
$479$	−13509.8	−1.28868	−0.644342	−	0.764738i	$-0.722869\pi$
$479$	−13509.8	−1.28868	−0.644342	+	0.764738i	$0.722869\pi$
$480$	−91.1498	−0.00866750
$481$	−544.803	−0.0516442
$482$	−17086.7	−1.61468
$483$	0	0
$484$	2310.08	0.216950
$485$	−138.547	−0.0129713
$486$	−8643.92	−0.806782
$487$	−8925.67	−0.830514	−0.415257	−	0.909704i	$-0.636308\pi$
$487$	−8925.67	−0.830514	−0.415257	+	0.909704i	$0.636308\pi$
$488$	5274.72	0.489294
$489$	−7023.84	−0.649548
$490$	0	0
$491$	19193.8	1.76416	0.882082	−	0.471095i	$-0.156141\pi$
$491$	19193.8	1.76416	0.882082	+	0.471095i	$0.156141\pi$
$492$	619.395	0.0567571
$493$	24951.8	2.27946
$494$	−5567.03	−0.507029
$495$	14.1364	0.00128360
$496$	8139.56	0.736848
$497$	0	0
$498$	2550.02	0.229456
$499$	−8308.78	−0.745395	−0.372697	−	0.927953i	$-0.621567\pi$
$499$	−8308.78	−0.745395	−0.372697	+	0.927953i	$0.621567\pi$
$500$	267.857	0.0239578
$501$	−1156.62	−0.103141
$502$	1962.85	0.174514
$503$	11354.3	1.00649	0.503243	−	0.864145i	$-0.332140\pi$
$503$	11354.3	1.00649	0.503243	+	0.864145i	$0.332140\pi$
$504$	0	0
$505$	−234.772	−0.0206876
$506$	−20.6743	−0.00181637
$507$	3636.78	0.318571
$508$	−258.695	−0.0225940
$509$	−1474.26	−0.128380	−0.0641902	−	0.997938i	$-0.520446\pi$
$509$	−1474.26	−0.128380	−0.0641902	+	0.997938i	$0.520446\pi$
$510$	−271.567	−0.0235788
$511$	0	0
$512$	−12809.9	−1.10571
$513$	12443.6	1.07095
$514$	−5907.08	−0.506907
$515$	−800.452	−0.0684895
$516$	298.173	0.0254386
$517$	318.870	0.0271255
$518$	0	0
$519$	−3171.62	−0.268244
$520$	259.247	0.0218629
$521$	17425.6	1.46532	0.732658	−	0.680596i	$-0.238279\pi$
$521$	17425.6	1.46532	0.732658	+	0.680596i	$0.238279\pi$
$522$	−15878.2	−1.33136
$523$	−19324.7	−1.61570	−0.807851	−	0.589387i	$-0.799369\pi$
$523$	−19324.7	−1.61570	−0.807851	+	0.589387i	$0.799369\pi$
$524$	−3779.94	−0.315129
$525$	0	0
$526$	−13386.2	−1.10963
$527$	−15865.3	−1.31139
$528$	−88.3453	−0.00728170
$529$	−12096.0	−0.994165
$530$	720.259	0.0590303
$531$	8872.48	0.725109
$532$	0	0
$533$	−3209.10	−0.260791
$534$	−5465.84	−0.442940
$535$	−536.761	−0.0433761
$536$	9634.11	0.776362
$537$	−201.657	−0.0162051
$538$	−10536.0	−0.844312
$539$	0	0
$540$	−103.366	−0.00823737
$541$	−5700.64	−0.453030	−0.226515	−	0.974008i	$-0.572733\pi$
$541$	−5700.64	−0.453030	−0.226515	+	0.974008i	$0.572733\pi$
$542$	6399.46	0.507159
$543$	6575.77	0.519693
$544$	7076.15	0.557698
$545$	−981.795	−0.0771660
$546$	0	0
$547$	−1893.79	−0.148031	−0.0740153	−	0.997257i	$-0.523581\pi$
$547$	−1893.79	−0.148031	−0.0740153	+	0.997257i	$0.523581\pi$
$548$	−4722.60	−0.368138
$549$	5051.83	0.392726
$550$	305.766	0.0237053
$551$	35118.2	2.71522
$552$	392.911	0.0302960
$553$	0	0
$554$	−3752.94	−0.287811
$555$	−37.4045	−0.00286078
$556$	5111.90	0.389915
$557$	6220.29	0.473181	0.236591	−	0.971609i	$-0.423970\pi$
$557$	6220.29	0.473181	0.236591	+	0.971609i	$0.423970\pi$
$558$	10095.9	0.765940
$559$	−1544.84	−0.116887
$560$	0	0
$561$	172.199	0.0129594
$562$	−12404.9	−0.931082
$563$	18935.0	1.41743	0.708717	−	0.705493i	$-0.249274\pi$
$563$	18935.0	1.41743	0.708717	+	0.705493i	$0.249274\pi$
$564$	−1080.99	−0.0807056
$565$	−593.164	−0.0441674
$566$	4956.82	0.368111
$567$	0	0
$568$	11390.3	0.841422
$569$	−9590.04	−0.706565	−0.353283	−	0.935517i	$-0.614935\pi$
$569$	−9590.04	−0.706565	−0.353283	+	0.935517i	$0.614935\pi$
$570$	−382.215	−0.0280864
$571$	3021.11	0.221417	0.110709	−	0.993853i	$-0.464688\pi$
$571$	3021.11	0.221417	0.110709	+	0.993853i	$0.464688\pi$
$572$	−29.3231	−0.00214346
$573$	−2215.40	−0.161518
$574$	0	0
$575$	−1050.04	−0.0761557
$576$	−13294.6	−0.961702
$577$	24222.8	1.74768	0.873838	−	0.486217i	$-0.161624\pi$
$577$	24222.8	1.74768	0.873838	+	0.486217i	$0.161624\pi$
$578$	8786.93	0.632333
$579$	−8682.48	−0.623197
$580$	−291.720	−0.0208845
$581$	0	0
$582$	1073.96	0.0764897
$583$	−456.711	−0.0324443
$584$	14618.2	1.03580
$585$	248.292	0.0175481
$586$	−5970.23	−0.420867
$587$	21773.6	1.53099	0.765496	−	0.643441i	$-0.222494\pi$
$587$	21773.6	1.53099	0.765496	+	0.643441i	$0.222494\pi$
$588$	0	0
$589$	−22329.4	−1.56208
$590$	−587.814	−0.0410168
$591$	3524.58	0.245316
$592$	−1489.77	−0.103428
$593$	2087.65	0.144569	0.0722845	−	0.997384i	$-0.476971\pi$
$593$	2087.65	0.144569	0.0722845	+	0.997384i	$0.476971\pi$
$594$	−236.353	−0.0163261
$595$	0	0
$596$	−2209.22	−0.151834
$597$	−1930.42	−0.132340
$598$	−363.123	−0.0248315
$599$	27542.8	1.87875	0.939373	−	0.342898i	$-0.111408\pi$
$599$	27542.8	1.87875	0.939373	+	0.342898i	$0.111408\pi$
$600$	−5811.04	−0.395391
$601$	−9008.25	−0.611404	−0.305702	−	0.952127i	$-0.598891\pi$
$601$	−9008.25	−0.611404	−0.305702	+	0.952127i	$0.598891\pi$
$602$	0	0
$603$	9227.01	0.623139
$604$	5005.06	0.337174
$605$	−821.726	−0.0552197
$606$	1819.86	0.121991
$607$	9839.03	0.657914	0.328957	−	0.944345i	$-0.393303\pi$
$607$	9839.03	0.657914	0.328957	+	0.944345i	$0.393303\pi$
$608$	9959.27	0.664312
$609$	0	0
$610$	−334.691	−0.0222151
$611$	5600.64	0.370831
$612$	3720.39	0.245732
$613$	14774.9	0.973493	0.486747	−	0.873543i	$-0.338183\pi$
$613$	14774.9	0.973493	0.486747	+	0.873543i	$0.338183\pi$
$614$	−20549.0	−1.35063
$615$	−220.327	−0.0144462
$616$	0	0
$617$	13249.6	0.864521	0.432260	−	0.901749i	$-0.357716\pi$
$617$	13249.6	0.864521	0.432260	+	0.901749i	$0.357716\pi$
$618$	6204.77	0.403871
$619$	−17781.8	−1.15462	−0.577312	−	0.816524i	$-0.695898\pi$
$619$	−17781.8	−1.15462	−0.577312	+	0.816524i	$0.695898\pi$
$620$	185.486	0.0120150
$621$	811.663	0.0524491
$622$	22998.0	1.48253
$623$	0	0
$624$	−1551.70	−0.0995476
$625$	15482.0	0.990848
$626$	23113.7	1.47573
$627$	242.360	0.0154369
$628$	4850.08	0.308184
$629$	2903.79	0.184073
$630$	0	0
$631$	8439.17	0.532421	0.266211	−	0.963915i	$-0.414228\pi$
$631$	8439.17	0.532421	0.266211	+	0.963915i	$0.414228\pi$
$632$	−78.8700	−0.00496405
$633$	−5863.78	−0.368190
$634$	43.1508	0.00270306
$635$	92.0212	0.00575079
$636$	1548.28	0.0965304
$637$	0	0
$638$	−667.034	−0.0413921
$639$	10909.0	0.675359
$640$	499.728	0.0308648
$641$	4344.30	0.267690	0.133845	−	0.991002i	$-0.457268\pi$
$641$	4344.30	0.267690	0.133845	+	0.991002i	$0.457268\pi$
$642$	4160.75	0.255781
$643$	−11712.4	−0.718338	−0.359169	−	0.933273i	$-0.616940\pi$
$643$	−11712.4	−0.718338	−0.359169	+	0.933273i	$0.616940\pi$
$644$	0	0
$645$	−106.064	−0.00647482
$646$	29672.2	1.80718
$647$	−16059.4	−0.975828	−0.487914	−	0.872892i	$-0.662242\pi$
$647$	−16059.4	−0.975828	−0.487914	+	0.872892i	$0.662242\pi$
$648$	−10862.9	−0.658542
$649$	372.728	0.0225437
$650$	5370.49	0.324074
$651$	0	0
$652$	−6375.01	−0.382921
$653$	14273.9	0.855408	0.427704	−	0.903919i	$-0.359322\pi$
$653$	14273.9	0.855408	0.427704	+	0.903919i	$0.359322\pi$
$654$	7610.46	0.455034
$655$	1344.58	0.0802090
$656$	−8775.31	−0.522284
$657$	14000.5	0.831374
$658$	0	0
$659$	−15861.1	−0.937575	−0.468787	−	0.883311i	$-0.655309\pi$
$659$	−15861.1	−0.937575	−0.468787	+	0.883311i	$0.655309\pi$
$660$	−2.01323	−0.000118735 0
$661$	−8525.87	−0.501692	−0.250846	−	0.968027i	$-0.580709\pi$
$661$	−8525.87	−0.501692	−0.250846	+	0.968027i	$0.580709\pi$
$662$	−12313.6	−0.722931
$663$	3024.50	0.177167
$664$	12974.9	0.758321
$665$	0	0
$666$	−1847.84	−0.107511
$667$	2290.67	0.132976
$668$	−1049.77	−0.0608038
$669$	646.337	0.0373525
$670$	−611.302	−0.0352488
$671$	212.225	0.0122099
$672$	0	0
$673$	18410.4	1.05448	0.527242	−	0.849715i	$-0.323226\pi$
$673$	18410.4	1.05448	0.527242	+	0.849715i	$0.323226\pi$
$674$	8582.59	0.490488
$675$	−12004.2	−0.684509
$676$	3300.83	0.187803
$677$	−11951.5	−0.678486	−0.339243	−	0.940699i	$-0.610171\pi$
$677$	−11951.5	−0.678486	−0.339243	+	0.940699i	$0.610171\pi$
$678$	4597.96	0.260448
$679$	0	0
$680$	−1381.78	−0.0779248
$681$	6297.19	0.354345
$682$	424.125	0.0238132
$683$	−26227.5	−1.46935	−0.734677	−	0.678417i	$-0.762666\pi$
$683$	−26227.5	−1.46935	−0.734677	+	0.678417i	$0.762666\pi$
$684$	5236.23	0.292708
$685$	1679.89	0.0937013
$686$	0	0
$687$	−6904.33	−0.383430
$688$	−4224.38	−0.234088
$689$	−8021.68	−0.443544
$690$	−24.9309	−0.00137551
$691$	15022.5	0.827039	0.413520	−	0.910495i	$-0.364299\pi$
$691$	15022.5	0.827039	0.413520	+	0.910495i	$0.364299\pi$
$692$	−2878.64	−0.158135
$693$	0	0
$694$	−13678.0	−0.748138
$695$	−1818.37	−0.0992442
$696$	12676.9	0.690396
$697$	17104.4	0.929522
$698$	−31735.0	−1.72090
$699$	7.74249	0.000418953 0
$700$	0	0
$701$	21035.4	1.13338	0.566688	−	0.823932i	$-0.308224\pi$
$701$	21035.4	1.13338	0.566688	+	0.823932i	$0.308224\pi$
$702$	−4151.31	−0.223192
$703$	4086.91	0.219262
$704$	−558.498	−0.0298994
$705$	384.523	0.0205418
$706$	−26694.1	−1.42301
$707$	0	0
$708$	−1263.57	−0.0670735
$709$	21847.1	1.15724	0.578621	−	0.815596i	$-0.303591\pi$
$709$	21847.1	1.15724	0.578621	+	0.815596i	$0.303591\pi$
$710$	−722.738	−0.0382026
$711$	−75.5373	−0.00398435
$712$	−27811.1	−1.46386
$713$	−1456.49	−0.0765022
$714$	0	0
$715$	10.4306	0.000545571 0
$716$	−183.029	−0.00955322
$717$	−12154.1	−0.633058
$718$	12167.8	0.632448
$719$	8554.60	0.443717	0.221859	−	0.975079i	$-0.428788\pi$
$719$	8554.60	0.443717	0.221859	+	0.975079i	$0.428788\pi$
$720$	678.957	0.0351434
$721$	0	0
$722$	24596.3	1.26784
$723$	13065.3	0.672064
$724$	5968.33	0.306369
$725$	−33878.3	−1.73546
$726$	6369.68	0.325621
$727$	13069.1	0.666722	0.333361	−	0.942799i	$-0.391817\pi$
$727$	13069.1	0.666722	0.333361	+	0.942799i	$0.391817\pi$
$728$	0	0
$729$	−5426.82	−0.275711
$730$	−927.554	−0.0470278
$731$	8233.96	0.416613
$732$	−719.456	−0.0363277
$733$	−21302.1	−1.07341	−0.536706	−	0.843769i	$-0.680332\pi$
$733$	−21302.1	−1.07341	−0.536706	+	0.843769i	$0.680332\pi$
$734$	−19537.0	−0.982456
$735$	0	0
$736$	649.618	0.0325343
$737$	387.622	0.0193735
$738$	−10884.5	−0.542905
$739$	−497.719	−0.0247752	−0.0123876	−	0.999923i	$-0.503943\pi$
$739$	−497.719	−0.0247752	−0.0123876	+	0.999923i	$0.503943\pi$
$740$	−33.9492	−0.00168648
$741$	4256.81	0.211036
$742$	0	0
$743$	−5284.63	−0.260934	−0.130467	−	0.991453i	$-0.541648\pi$
$743$	−5284.63	−0.260934	−0.130467	+	0.991453i	$0.541648\pi$
$744$	−8060.42	−0.397190
$745$	785.848	0.0386459
$746$	−15567.1	−0.764009
$747$	12426.7	0.608659
$748$	156.292	0.00763983
$749$	0	0
$750$	738.572	0.0359584
$751$	−23507.0	−1.14219	−0.571094	−	0.820884i	$-0.693481\pi$
$751$	−23507.0	−1.14219	−0.571094	+	0.820884i	$0.693481\pi$
$752$	15315.0	0.742661
$753$	−1500.89	−0.0726365
$754$	−11715.8	−0.565868
$755$	−1780.36	−0.0858200
$756$	0	0
$757$	34242.2	1.64406	0.822030	−	0.569444i	$-0.192841\pi$
$757$	34242.2	1.64406	0.822030	+	0.569444i	$0.192841\pi$
$758$	−15404.3	−0.738137
$759$	15.8085	0.000756011 0
$760$	−1944.78	−0.0928216
$761$	17434.3	0.830478	0.415239	−	0.909712i	$-0.363698\pi$
$761$	17434.3	0.830478	0.415239	+	0.909712i	$0.363698\pi$
$762$	−713.310	−0.0339114
$763$	0	0
$764$	−2010.75	−0.0952178
$765$	−1323.39	−0.0625456
$766$	−7428.32	−0.350387
$767$	6546.61	0.308193
$768$	4847.13	0.227742
$769$	11600.9	0.544003	0.272002	−	0.962297i	$-0.412314\pi$
$769$	11600.9	0.544003	0.272002	+	0.962297i	$0.412314\pi$
$770$	0	0
$771$	4516.83	0.210985
$772$	−7880.43	−0.367387
$773$	−24014.9	−1.11741	−0.558703	−	0.829368i	$-0.688701\pi$
$773$	−24014.9	−1.11741	−0.558703	+	0.829368i	$0.688701\pi$
$774$	−5239.72	−0.243331
$775$	21541.1	0.998424
$776$	5464.49	0.252788
$777$	0	0
$778$	−17196.5	−0.792449
$779$	24073.5	1.10722
$780$	−35.3605	−0.00162322
$781$	458.282	0.0209970
$782$	1935.44	0.0885054
$783$	26187.5	1.19523
$784$	0	0
$785$	−1725.24	−0.0784412
$786$	−10422.6	−0.472979
$787$	−37375.0	−1.69285	−0.846425	−	0.532508i	$-0.821250\pi$
$787$	−37375.0	−1.69285	−0.846425	+	0.532508i	$0.821250\pi$
$788$	3199.00	0.144619
$789$	10235.7	0.461850
$790$	5.00445	0.000225380 0
$791$	0	0
$792$	−557.559	−0.0250151
$793$	3727.52	0.166921
$794$	17916.3	0.800788
$795$	−550.744	−0.0245696
$796$	−1752.10	−0.0780169
$797$	−17581.5	−0.781393	−0.390696	−	0.920520i	$-0.627766\pi$
$797$	−17581.5	−0.781393	−0.390696	+	0.920520i	$0.627766\pi$
$798$	0	0
$799$	−29851.3	−1.32173
$800$	−9607.66	−0.424603
$801$	−26635.9	−1.17495
$802$	−12840.4	−0.565351
$803$	588.155	0.0258475
$804$	−1314.06	−0.0576412
$805$	0	0
$806$	7449.34	0.325548
$807$	8056.33	0.351420
$808$	9259.74	0.403164
$809$	21260.5	0.923953	0.461977	−	0.886892i	$-0.347140\pi$
$809$	21260.5	0.923953	0.461977	+	0.886892i	$0.347140\pi$
$810$	689.272	0.0298994
$811$	−6985.63	−0.302464	−0.151232	−	0.988498i	$-0.548324\pi$
$811$	−6985.63	−0.302464	−0.151232	+	0.988498i	$0.548324\pi$
$812$	0	0
$813$	−4893.32	−0.211090
$814$	−77.6268	−0.00334253
$815$	2267.67	0.0974640
$816$	8270.53	0.354812
$817$	11588.8	0.496256
$818$	2416.01	0.103269
$819$	0	0
$820$	−199.974	−0.00851634
$821$	−10375.6	−0.441059	−0.220529	−	0.975380i	$-0.570779\pi$
$821$	−10375.6	−0.441059	−0.220529	+	0.975380i	$0.570779\pi$
$822$	−13021.8	−0.552540
$823$	7702.99	0.326257	0.163128	−	0.986605i	$-0.447842\pi$
$823$	7702.99	0.326257	0.163128	+	0.986605i	$0.447842\pi$
$824$	31570.9	1.33474
$825$	−233.803	−0.00986664
$826$	0	0
$827$	−15757.1	−0.662547	−0.331274	−	0.943535i	$-0.607478\pi$
$827$	−15757.1	−0.662547	−0.331274	+	0.943535i	$0.607478\pi$
$828$	341.547	0.0143352
$829$	−25243.7	−1.05760	−0.528799	−	0.848747i	$-0.677358\pi$
$829$	−25243.7	−1.05760	−0.528799	+	0.848747i	$0.677358\pi$
$830$	−823.284	−0.0344297
$831$	2869.67	0.119793
$832$	−9809.47	−0.408753
$833$	0	0
$834$	14095.2	0.585226
$835$	373.418	0.0154763
$836$	219.971	0.00910032
$837$	−16650.9	−0.687623
$838$	−32814.3	−1.35269
$839$	29603.8	1.21816	0.609081	−	0.793108i	$-0.291538\pi$
$839$	29603.8	1.21816	0.609081	+	0.793108i	$0.291538\pi$
$840$	0	0
$841$	49517.2	2.03031
$842$	−7986.46	−0.326878
$843$	9485.34	0.387536
$844$	−5322.11	−0.217055
$845$	−1174.15	−0.0478012
$846$	18996.0	0.771982
$847$	0	0
$848$	−21935.4	−0.888282
$849$	−3790.21	−0.153215
$850$	−28624.6	−1.15508
$851$	266.579	0.0107382
$852$	−1553.61	−0.0624715
$853$	38078.8	1.52848	0.764239	−	0.644933i	$-0.223115\pi$
$853$	38078.8	1.52848	0.764239	+	0.644933i	$0.223115\pi$
$854$	0	0
$855$	−1862.60	−0.0745023
$856$	21170.6	0.845323
$857$	683.831	0.0272570	0.0136285	−	0.999907i	$-0.495662\pi$
$857$	683.831	0.0272570	0.0136285	+	0.999907i	$0.495662\pi$
$858$	−80.8538	−0.00321714
$859$	18033.8	0.716303	0.358151	−	0.933664i	$-0.383407\pi$
$859$	18033.8	0.716303	0.358151	+	0.933664i	$0.383407\pi$
$860$	−96.2662	−0.00381703
$861$	0	0
$862$	−30286.1	−1.19669
$863$	−18408.7	−0.726116	−0.363058	−	0.931766i	$-0.618267\pi$
$863$	−18408.7	−0.726116	−0.363058	+	0.931766i	$0.618267\pi$
$864$	7426.59	0.292428
$865$	1023.97	0.0402497
$866$	−21459.5	−0.842061
$867$	−6718.90	−0.263190
$868$	0	0
$869$	−3.17328	−0.000123874 0
$870$	−804.371	−0.0313457
$871$	6808.20	0.264853
$872$	38723.3	1.50383
$873$	5233.58	0.202898
$874$	2724.02	0.105425
$875$	0	0
$876$	−1993.88	−0.0769031
$877$	−10955.9	−0.421841	−0.210921	−	0.977503i	$-0.567646\pi$
$877$	−10955.9	−0.421841	−0.210921	+	0.977503i	$0.567646\pi$
$878$	27419.7	1.05395
$879$	4565.11	0.175173
$880$	28.5226	0.00109261
$881$	8478.85	0.324245	0.162122	−	0.986771i	$-0.448166\pi$
$881$	8478.85	0.324245	0.162122	+	0.986771i	$0.448166\pi$
$882$	0	0
$883$	13031.0	0.496633	0.248316	−	0.968679i	$-0.420123\pi$
$883$	13031.0	0.496633	0.248316	+	0.968679i	$0.420123\pi$
$884$	2745.11	0.104444
$885$	449.470	0.0170721
$886$	23174.3	0.878730
$887$	−29764.6	−1.12671	−0.563357	−	0.826213i	$-0.690491\pi$
$887$	−29764.6	−1.12671	−0.563357	+	0.826213i	$0.690491\pi$
$888$	1475.28	0.0557515
$889$	0	0
$890$	1764.67	0.0664627
$891$	−437.062	−0.0164334
$892$	586.631	0.0220200
$893$	−42014.0	−1.57441
$894$	−6091.56	−0.227888
$895$	65.1057	0.00243156
$896$	0	0
$897$	277.661	0.0103354
$898$	9353.14	0.347571
$899$	−46992.2	−1.74336
$900$	−5051.37	−0.187088
$901$	42755.4	1.58090
$902$	−457.252	−0.0168790
$903$	0	0
$904$	23395.2	0.860744
$905$	−2123.01	−0.0779793
$906$	13800.6	0.506066
$907$	−19639.6	−0.718989	−0.359494	−	0.933147i	$-0.617051\pi$
$907$	−19639.6	−0.718989	−0.359494	+	0.933147i	$0.617051\pi$
$908$	5715.48	0.208893
$909$	8868.46	0.323595
$910$	0	0
$911$	−46796.5	−1.70191	−0.850953	−	0.525242i	$-0.823975\pi$
$911$	−46796.5	−1.70191	−0.850953	+	0.525242i	$0.823975\pi$
$912$	11640.3	0.422641
$913$	522.038	0.0189233
$914$	−19191.7	−0.694536
$915$	255.920	0.00924640
$916$	−6266.54	−0.226040
$917$	0	0
$918$	22126.4	0.795512
$919$	−37534.7	−1.34729	−0.673644	−	0.739056i	$-0.735272\pi$
$919$	−37534.7	−1.34729	−0.673644	+	0.739056i	$0.735272\pi$
$920$	−126.853	−0.00454589
$921$	15712.7	0.562162
$922$	−34286.1	−1.22468
$923$	8049.28	0.287048
$924$	0	0
$925$	−3942.63	−0.140144
$926$	−12740.2	−0.452127
$927$	30236.8	1.07131
$928$	20959.3	0.741403
$929$	3490.91	0.123287	0.0616433	−	0.998098i	$-0.480366\pi$
$929$	3490.91	0.123287	0.0616433	+	0.998098i	$0.480366\pi$
$930$	511.448	0.0180334
$931$	0	0
$932$	7.02727	0.000246981 0
$933$	−17585.3	−0.617061
$934$	1725.75	0.0604584
$935$	−55.5950	−0.00194455
$936$	−9792.98	−0.341980
$937$	−34852.1	−1.21512	−0.607561	−	0.794273i	$-0.707852\pi$
$937$	−34852.1	−1.21512	−0.607561	+	0.794273i	$0.707852\pi$
$938$	0	0
$939$	−17673.8	−0.614231
$940$	349.002	0.0121098
$941$	34551.1	1.19695	0.598476	−	0.801140i	$-0.295773\pi$
$941$	34551.1	1.19695	0.598476	+	0.801140i	$0.295773\pi$
$942$	13373.3	0.462554
$943$	1570.25	0.0542254
$944$	17901.8	0.617217
$945$	0	0
$946$	−220.118	−0.00756517
$947$	50504.1	1.73301	0.866506	−	0.499166i	$-0.166360\pi$
$947$	50504.1	1.73301	0.866506	+	0.499166i	$0.166360\pi$
$948$	10.7576	0.000368557 0
$949$	10330.4	0.353359
$950$	−40287.4	−1.37589
$951$	−32.9951	−0.00112507
$952$	0	0
$953$	32154.1	1.09294	0.546471	−	0.837478i	$-0.315971\pi$
$953$	32154.1	1.09294	0.546471	+	0.837478i	$0.315971\pi$
$954$	−27207.6	−0.923353
$955$	715.250	0.0242356
$956$	−11031.3	−0.373200
$957$	510.045	0.0172282
$958$	−33810.0	−1.14024
$959$	0	0
$960$	−673.488	−0.0226424
$961$	88.3619	0.00296606
$962$	−1363.44	−0.0456955
$963$	20276.0	0.678490
$964$	11858.3	0.396195
$965$	2803.17	0.0935101
$966$	0	0
$967$	36096.8	1.20041	0.600204	−	0.799847i	$-0.295086\pi$
$967$	36096.8	1.20041	0.600204	+	0.799847i	$0.295086\pi$
$968$	32410.0	1.07613
$969$	−22688.7	−0.752184
$970$	−346.732	−0.0114772
$971$	−34492.8	−1.13999	−0.569993	−	0.821649i	$-0.693054\pi$
$971$	−34492.8	−1.13999	−0.569993	+	0.821649i	$0.693054\pi$
$972$	5998.98	0.197960
$973$	0	0
$974$	−22337.6	−0.734850
$975$	−4106.52	−0.134886
$976$	10192.9	0.334291
$977$	28450.2	0.931630	0.465815	−	0.884882i	$-0.345761\pi$
$977$	28450.2	0.931630	0.465815	+	0.884882i	$0.345761\pi$
$978$	−17578.1	−0.574728
$979$	−1118.96	−0.0365293
$980$	0	0
$981$	37087.0	1.20703
$982$	48035.0	1.56096
$983$	−45303.0	−1.46993	−0.734966	−	0.678104i	$-0.762802\pi$
$983$	−45303.0	−1.46993	−0.734966	+	0.678104i	$0.762802\pi$
$984$	8689.99	0.281531
$985$	−1137.93	−0.0368095
$986$	62445.1	2.01689
$987$	0	0
$988$	3863.58	0.124410
$989$	755.910	0.0243039
$990$	35.3781	0.00113575
$991$	−18124.2	−0.580964	−0.290482	−	0.956880i	$-0.593816\pi$
$991$	−18124.2	−0.580964	−0.290482	+	0.956880i	$0.593816\pi$
$992$	−13326.7	−0.426534
$993$	9415.52	0.300899
$994$	0	0
$995$	623.244	0.0198575
$996$	−1769.74	−0.0563017
$997$	−7619.05	−0.242024	−0.121012	−	0.992651i	$-0.538614\pi$
$997$	−7619.05	−0.242024	−0.121012	+	0.992651i	$0.538614\pi$
$998$	−20793.8	−0.659535
$999$	3047.59	0.0965181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.4.a.f.1.55		78
7.6	odd	2		2401.4.a.g.1.55		78
49.3	odd	42		49.4.g.a.9.4	✓	156
49.33	odd	42		49.4.g.a.11.4	yes	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.4	✓	156	49.3	odd	42
49.4.g.a.11.4	yes	156	49.33	odd	42
2401.4.a.f.1.55		78	1.1	even	1	trivial
2401.4.a.g.1.55		78	7.6	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:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q+2.50263 q^{2} -1.91363 q^{3} -1.73685 q^{4} +0.617821 q^{5} -4.78909 q^{6} -24.3677 q^{8} -23.3380 q^{9} +O(q^{10})\)	\(q+2.50263 q^{2} -1.91363 q^{3} -1.73685 q^{4} +0.617821 q^{5} -4.78909 q^{6} -24.3677 q^{8} -23.3380 q^{9} +1.54618 q^{10} -0.980419 q^{11} +3.32369 q^{12} -17.2201 q^{13} -1.18228 q^{15} -47.0885 q^{16} +91.7828 q^{17} -58.4064 q^{18} +129.179 q^{19} -1.07306 q^{20} -2.45362 q^{22} +8.42602 q^{23} +46.6307 q^{24} -124.618 q^{25} -43.0955 q^{26} +96.3282 q^{27} +271.857 q^{29} -2.95880 q^{30} -172.856 q^{31} +77.0967 q^{32} +1.87615 q^{33} +229.698 q^{34} +40.5347 q^{36} +31.6376 q^{37} +323.287 q^{38} +32.9528 q^{39} -15.0549 q^{40} +186.358 q^{41} +89.7114 q^{43} +1.70284 q^{44} -14.4187 q^{45} +21.0872 q^{46} -325.239 q^{47} +90.1098 q^{48} -311.873 q^{50} -175.638 q^{51} +29.9088 q^{52} +465.832 q^{53} +241.074 q^{54} -0.605724 q^{55} -247.200 q^{57} +680.357 q^{58} -380.173 q^{59} +2.05344 q^{60} -216.463 q^{61} -432.595 q^{62} +569.653 q^{64} -10.6389 q^{65} +4.69532 q^{66} -395.364 q^{67} -159.413 q^{68} -16.1242 q^{69} -467.435 q^{71} +568.695 q^{72} -599.902 q^{73} +79.1772 q^{74} +238.473 q^{75} -224.365 q^{76} +82.4686 q^{78} +3.23666 q^{79} -29.0923 q^{80} +445.791 q^{81} +466.384 q^{82} -532.464 q^{83} +56.7054 q^{85} +224.514 q^{86} -520.232 q^{87} +23.8906 q^{88} +1141.31 q^{89} -36.0847 q^{90} -14.6348 q^{92} +330.783 q^{93} -813.951 q^{94} +79.8095 q^{95} -147.534 q^{96} -224.251 q^{97} +22.8810 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - q^{2} - 35 q^{3} + 287 q^{4} - 63 q^{5} - 70 q^{6} + 15 q^{8} + 579 q^{9}+O(q^{10}) \) 78 * q - q^2 - 35 * q^3 + 287 * q^4 - 63 * q^5 - 70 * q^6 + 15 * q^8 + 579 * q^9	\( 78 q - q^{2} - 35 q^{3} + 287 q^{4} - 63 q^{5} - 70 q^{6} + 15 q^{8} + 579 q^{9} - 126 q^{10} - q^{11} - 532 q^{12} - 196 q^{13} + 164 q^{15} + 979 q^{16} - 497 q^{17} + 16 q^{18} - 490 q^{19} - 868 q^{20} + 64 q^{22} + 69 q^{23} - 1365 q^{24} + 1419 q^{25} - 1176 q^{26} - 1547 q^{27} - 54 q^{29} + 39 q^{30} - 1162 q^{31} + 1624 q^{32} + 161 q^{33} - 154 q^{34} + 1813 q^{36} - 323 q^{37} - 882 q^{38} - 126 q^{39} - 2723 q^{40} - 1610 q^{41} + 2 q^{43} + 140 q^{44} - 1736 q^{45} + 1478 q^{46} - 2765 q^{47} - 4802 q^{48} - 3092 q^{50} + 2125 q^{51} + 581 q^{52} - 2213 q^{53} - 3514 q^{54} - 3115 q^{55} + 73 q^{57} - 2403 q^{58} - 5061 q^{59} + 1918 q^{60} - 1925 q^{61} - 3178 q^{62} + 2101 q^{64} - 1876 q^{65} - 5670 q^{66} + 202 q^{67} - 5635 q^{68} - 7595 q^{69} + 1262 q^{71} + 4072 q^{72} - 4837 q^{73} - 3821 q^{74} - 7882 q^{75} - 5817 q^{76} + 6692 q^{78} + 608 q^{79} - 12348 q^{80} + 2850 q^{81} + 5208 q^{82} - 3668 q^{83} - 3091 q^{85} + 3610 q^{86} - 938 q^{87} + 3702 q^{88} - 8925 q^{89} + 2198 q^{90} + 2940 q^{92} + 4435 q^{93} - 4102 q^{94} + 3665 q^{95} - 9800 q^{96} - 5558 q^{97} - 5402 q^{99}+O(q^{100}) \) 78 * q - q^2 - 35 * q^3 + 287 * q^4 - 63 * q^5 - 70 * q^6 + 15 * q^8 + 579 * q^9 - 126 * q^10 - q^11 - 532 * q^12 - 196 * q^13 + 164 * q^15 + 979 * q^16 - 497 * q^17 + 16 * q^18 - 490 * q^19 - 868 * q^20 + 64 * q^22 + 69 * q^23 - 1365 * q^24 + 1419 * q^25 - 1176 * q^26 - 1547 * q^27 - 54 * q^29 + 39 * q^30 - 1162 * q^31 + 1624 * q^32 + 161 * q^33 - 154 * q^34 + 1813 * q^36 - 323 * q^37 - 882 * q^38 - 126 * q^39 - 2723 * q^40 - 1610 * q^41 + 2 * q^43 + 140 * q^44 - 1736 * q^45 + 1478 * q^46 - 2765 * q^47 - 4802 * q^48 - 3092 * q^50 + 2125 * q^51 + 581 * q^52 - 2213 * q^53 - 3514 * q^54 - 3115 * q^55 + 73 * q^57 - 2403 * q^58 - 5061 * q^59 + 1918 * q^60 - 1925 * q^61 - 3178 * q^62 + 2101 * q^64 - 1876 * q^65 - 5670 * q^66 + 202 * q^67 - 5635 * q^68 - 7595 * q^69 + 1262 * q^71 + 4072 * q^72 - 4837 * q^73 - 3821 * q^74 - 7882 * q^75 - 5817 * q^76 + 6692 * q^78 + 608 * q^79 - 12348 * q^80 + 2850 * q^81 + 5208 * q^82 - 3668 * q^83 - 3091 * q^85 + 3610 * q^86 - 938 * q^87 + 3702 * q^88 - 8925 * q^89 + 2198 * q^90 + 2940 * q^92 + 4435 * q^93 - 4102 * q^94 + 3665 * q^95 - 9800 * q^96 - 5558 * q^97 - 5402 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more