LMFDB - Embedded newform 2401.4.a.c.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2401,4,Mod(1,2401)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2401.a (trivial)

Newform invariants

comment:select newform

sage:traces = [39,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.663585924$
Analytic rank:	$1$
Dimension:	$39$
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	\(q-4.44388 q^{2} -6.83693 q^{3} +11.7481 q^{4} +2.20264 q^{5} +30.3825 q^{6} -16.6559 q^{8} +19.7437 q^{9} -9.78829 q^{10} -3.73267 q^{11} -80.3207 q^{12} -89.7963 q^{13} -15.0593 q^{15} -19.9676 q^{16} +55.9124 q^{17} -87.7385 q^{18} +27.6480 q^{19} +25.8768 q^{20} +16.5875 q^{22} +55.7657 q^{23} +113.875 q^{24} -120.148 q^{25} +399.044 q^{26} +49.6110 q^{27} +99.5539 q^{29} +66.9219 q^{30} -99.2310 q^{31} +221.981 q^{32} +25.5200 q^{33} -248.468 q^{34} +231.950 q^{36} -358.528 q^{37} -122.865 q^{38} +613.931 q^{39} -36.6870 q^{40} +333.568 q^{41} +425.796 q^{43} -43.8516 q^{44} +43.4883 q^{45} -247.816 q^{46} -483.393 q^{47} +136.517 q^{48} +533.925 q^{50} -382.269 q^{51} -1054.93 q^{52} +407.896 q^{53} -220.465 q^{54} -8.22175 q^{55} -189.028 q^{57} -442.405 q^{58} -572.404 q^{59} -176.918 q^{60} +153.944 q^{61} +440.970 q^{62} -826.715 q^{64} -197.789 q^{65} -113.408 q^{66} +370.306 q^{67} +656.862 q^{68} -381.266 q^{69} -386.755 q^{71} -328.849 q^{72} -678.209 q^{73} +1593.25 q^{74} +821.446 q^{75} +324.811 q^{76} -2728.24 q^{78} -677.093 q^{79} -43.9816 q^{80} -872.267 q^{81} -1482.33 q^{82} +792.337 q^{83} +123.155 q^{85} -1892.19 q^{86} -680.643 q^{87} +62.1710 q^{88} +128.629 q^{89} -193.257 q^{90} +655.138 q^{92} +678.436 q^{93} +2148.14 q^{94} +60.8988 q^{95} -1517.67 q^{96} -430.011 q^{97} -73.6966 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) $ 39 * q + q^2 - q^3 + 145 * q^4 - 27 * q^5 - 41 * q^6 - 12 * q^8 + 312 * q^9 - 78 * q^10 + q^11 + 91 * q^12 - 77 * q^13 - 161 * q^15 + 461 * q^16 - 211 * q^17 + 8 * q^18 - 314 * q^19 - 476 * q^20 - 61 * q^22 - 69 * q^23 - 330 * q^24 + 606 * q^25 - 504 * q^26 + 50 * q^27 + 57 * q^29 + 42 * q^30 - 638 * q^31 - 1600 * q^32 - 1574 * q^33 - 1343 * q^34 + 782 * q^36 + 71 * q^37 - 1359 * q^38 - 84 * q^39 + 155 * q^40 - 1393 * q^41 - 125 * q^43 + 52 * q^44 - 1129 * q^45 - 1454 * q^46 - 1483 * q^47 + 974 * q^48 + 3074 * q^50 - 2044 * q^51 - 3899 * q^52 + 2213 * q^53 - 1142 * q^54 - 1604 * q^55 + 98 * q^57 + 2403 * q^58 - 2073 * q^59 - 1519 * q^60 - 2575 * q^61 - 1742 * q^62 + 1358 * q^64 + 1876 * q^65 + 48 * q^66 + 176 * q^67 - 3038 * q^68 + 638 * q^69 - 1259 * q^71 - 3799 * q^72 + 307 * q^73 + 3845 * q^74 - 131 * q^75 - 1974 * q^76 - 6041 * q^78 + 22 * q^79 + 804 * q^80 + 795 * q^81 - 8043 * q^82 - 6349 * q^83 + 3094 * q^85 + 1745 * q^86 - 9508 * q^87 + 1299 * q^88 - 2253 * q^89 - 11156 * q^90 + 1284 * q^92 - 3430 * q^93 - 2738 * q^94 - 3290 * q^95 - 3031 * q^96 + 770 * q^97 + 5384 * q^99

Coefficient data

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

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

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

$2$	−4.44388	−1.57115	−0.785574	−	0.618768i	$-0.787632\pi$
$2$	−4.44388	−1.57115	−0.785574	+	0.618768i	$0.787632\pi$
$3$	−6.83693	−1.31577	−0.657884	−	0.753119i	$-0.728548\pi$
$3$	−6.83693	−1.31577	−0.657884	+	0.753119i	$0.728548\pi$
$4$	11.7481	1.46851
$5$	2.20264	0.197011	0.0985053	−	0.995137i	$-0.468594\pi$
$5$	2.20264	0.197011	0.0985053	+	0.995137i	$0.468594\pi$
$6$	30.3825	2.06727
$7$	0	0
$7$	0	0
$8$	−16.6559	−0.736094
$9$	19.7437	0.731247
$10$	−9.78829	−0.309533
$11$	−3.73267	−0.102313	−0.0511565	−	0.998691i	$-0.516291\pi$
$11$	−3.73267	−0.102313	−0.0511565	+	0.998691i	$0.516291\pi$
$12$	−80.3207	−1.93222
$13$	−89.7963	−1.91577	−0.957885	−	0.287153i	$-0.907291\pi$
$13$	−89.7963	−1.91577	−0.957885	+	0.287153i	$0.907291\pi$
$14$	0	0
$15$	−15.0593	−0.259220
$16$	−19.9676	−0.311994
$17$	55.9124	0.797691	0.398846	−	0.917018i	$-0.369411\pi$
$17$	55.9124	0.797691	0.398846	+	0.917018i	$0.369411\pi$
$18$	−87.7385	−1.14890
$19$	27.6480	0.333837	0.166918	−	0.985971i	$-0.446618\pi$
$19$	27.6480	0.333837	0.166918	+	0.985971i	$0.446618\pi$
$20$	25.8768	0.289311
$21$	0	0
$22$	16.5875	0.160749
$23$	55.7657	0.505563	0.252781	−	0.967523i	$-0.418655\pi$
$23$	55.7657	0.505563	0.252781	+	0.967523i	$0.418655\pi$
$24$	113.875	0.968529
$25$	−120.148	−0.961187
$26$	399.044	3.00996
$27$	49.6110	0.353616
$28$	0	0
$29$	99.5539	0.637472	0.318736	−	0.947844i	$-0.396742\pi$
$29$	99.5539	0.637472	0.318736	+	0.947844i	$0.396742\pi$
$30$	66.9219	0.407274
$31$	−99.2310	−0.574916	−0.287458	−	0.957793i	$-0.592810\pi$
$31$	−99.2310	−0.574916	−0.287458	+	0.957793i	$0.592810\pi$
$32$	221.981	1.22628
$33$	25.5200	0.134620
$34$	−248.468	−1.25329
$35$	0	0
$36$	231.950	1.07384
$37$	−358.528	−1.59302	−0.796508	−	0.604627i	$-0.793322\pi$
$37$	−358.528	−1.59302	−0.796508	+	0.604627i	$0.793322\pi$
$38$	−122.865	−0.524507
$39$	613.931	2.52071
$40$	−36.6870	−0.145018
$41$	333.568	1.27060	0.635299	−	0.772266i	$-0.280877\pi$
$41$	333.568	1.27060	0.635299	+	0.772266i	$0.280877\pi$
$42$	0	0
$43$	425.796	1.51008	0.755039	−	0.655680i	$-0.227618\pi$
$43$	425.796	1.51008	0.755039	+	0.655680i	$0.227618\pi$
$44$	−43.8516	−0.150247
$45$	43.4883	0.144063
$46$	−247.816	−0.794314
$47$	−483.393	−1.50022	−0.750108	−	0.661315i	$-0.769998\pi$
$47$	−483.393	−1.50022	−0.750108	+	0.661315i	$0.769998\pi$
$48$	136.517	0.410512
$49$	0	0
$50$	533.925	1.51017
$51$	−382.269	−1.04958
$52$	−1054.93	−2.81332
$53$	407.896	1.05715	0.528573	−	0.848888i	$-0.322727\pi$
$53$	407.896	1.05715	0.528573	+	0.848888i	$0.322727\pi$
$54$	−220.465	−0.555584
$55$	−8.22175	−0.0201567
$56$	0	0
$57$	−189.028	−0.439252
$58$	−442.405	−1.00156
$59$	−572.404	−1.26306	−0.631531	−	0.775350i	$-0.717573\pi$
$59$	−572.404	−1.26306	−0.631531	+	0.775350i	$0.717573\pi$
$60$	−176.918	−0.380667
$61$	153.944	0.323124	0.161562	−	0.986863i	$-0.448347\pi$
$61$	153.944	0.323124	0.161562	+	0.986863i	$0.448347\pi$
$62$	440.970	0.903279
$63$	0	0
$64$	−826.715	−1.61468
$65$	−197.789	−0.377427
$66$	−113.408	−0.211508
$67$	370.306	0.675225	0.337612	−	0.941285i	$-0.390381\pi$
$67$	370.306	0.675225	0.337612	+	0.941285i	$0.390381\pi$
$68$	656.862	1.17142
$69$	−381.266	−0.665204
$70$	0	0
$71$	−386.755	−0.646470	−0.323235	−	0.946319i	$-0.604770\pi$
$71$	−386.755	−0.646470	−0.323235	+	0.946319i	$0.604770\pi$
$72$	−328.849	−0.538267
$73$	−678.209	−1.08737	−0.543687	−	0.839288i	$-0.682972\pi$
$73$	−678.209	−1.08737	−0.543687	+	0.839288i	$0.682972\pi$
$74$	1593.25	2.50287
$75$	821.446	1.26470
$76$	324.811	0.490241
$77$	0	0
$78$	−2728.24	−3.96041
$79$	−677.093	−0.964290	−0.482145	−	0.876091i	$-0.660142\pi$
$79$	−677.093	−0.964290	−0.482145	+	0.876091i	$0.660142\pi$
$80$	−43.9816	−0.0614661
$81$	−872.267	−1.19652
$82$	−1482.33	−1.99630
$83$	792.337	1.04784	0.523918	−	0.851769i	$-0.324470\pi$
$83$	792.337	1.04784	0.523918	+	0.851769i	$0.324470\pi$
$84$	0	0
$85$	123.155	0.157154
$86$	−1892.19	−2.37256
$87$	−680.643	−0.838766
$88$	62.1710	0.0753119
$89$	128.629	0.153198	0.0765991	−	0.997062i	$-0.475594\pi$
$89$	128.629	0.153198	0.0765991	+	0.997062i	$0.475594\pi$
$90$	−193.257	−0.226345
$91$	0	0
$92$	655.138	0.742423
$93$	678.436	0.756457
$94$	2148.14	2.35706
$95$	60.8988	0.0657693
$96$	−1517.67	−1.61350
$97$	−430.011	−0.450114	−0.225057	−	0.974346i	$-0.572257\pi$
$97$	−430.011	−0.450114	−0.225057	+	0.974346i	$0.572257\pi$
$98$	0	0
$99$	−73.6966	−0.0748161
$100$	−1411.51	−1.41151
$101$	−1247.01	−1.22854	−0.614270	−	0.789096i	$-0.710549\pi$
$101$	−1247.01	−1.22854	−0.614270	+	0.789096i	$0.710549\pi$
$102$	1698.76	1.64904
$103$	649.847	0.621663	0.310832	−	0.950465i	$-0.399392\pi$
$103$	649.847	0.621663	0.310832	+	0.950465i	$0.399392\pi$
$104$	1495.64	1.41019
$105$	0	0
$106$	−1812.64	−1.66093
$107$	386.102	0.348841	0.174420	−	0.984671i	$-0.444195\pi$
$107$	386.102	0.348841	0.174420	+	0.984671i	$0.444195\pi$
$108$	582.833	0.519288
$109$	−542.672	−0.476867	−0.238434	−	0.971159i	$-0.576634\pi$
$109$	−542.672	−0.476867	−0.238434	+	0.971159i	$0.576634\pi$
$110$	36.5364	0.0316692
$111$	2451.23	2.09604
$112$	0	0
$113$	2356.81	1.96203	0.981017	−	0.193923i	$-0.0621211\pi$
$113$	2356.81	1.96203	0.981017	+	0.193923i	$0.0621211\pi$
$114$	840.017	0.690130
$115$	122.832	0.0996012
$116$	1169.56	0.936132
$117$	−1772.91	−1.40090
$118$	2543.70	1.98446
$119$	0	0
$120$	250.827	0.190811
$121$	−1317.07	−0.989532
$122$	−684.110	−0.507675
$123$	−2280.58	−1.67181
$124$	−1165.77	−0.844269
$125$	−539.975	−0.386375
$126$	0	0
$127$	1608.58	1.12393	0.561963	−	0.827162i	$-0.310046\pi$
$127$	1608.58	1.12393	0.561963	+	0.827162i	$0.310046\pi$
$128$	1897.98	1.31062
$129$	−2911.14	−1.98691
$130$	878.952	0.592993
$131$	−354.977	−0.236752	−0.118376	−	0.992969i	$-0.537769\pi$
$131$	−354.977	−0.236752	−0.118376	+	0.992969i	$0.537769\pi$
$132$	299.811	0.197691
$133$	0	0
$134$	−1645.59	−1.06088
$135$	109.275	0.0696662
$136$	−931.272	−0.587176
$137$	2722.48	1.69779	0.848894	−	0.528563i	$-0.177269\pi$
$137$	2722.48	1.69779	0.848894	+	0.528563i	$0.177269\pi$
$138$	1694.30	1.04513
$139$	1236.80	0.754707	0.377353	−	0.926069i	$-0.376834\pi$
$139$	1236.80	0.754707	0.377353	+	0.926069i	$0.376834\pi$
$140$	0	0
$141$	3304.93	1.97394
$142$	1718.69	1.01570
$143$	335.180	0.196008
$144$	−394.234	−0.228145
$145$	219.282	0.125589
$146$	3013.88	1.70843
$147$	0	0
$148$	−4212.01	−2.33936
$149$	2425.58	1.33363	0.666817	−	0.745222i	$-0.267656\pi$
$149$	2425.58	1.33363	0.666817	+	0.745222i	$0.267656\pi$
$150$	−3650.41	−1.98703
$151$	−554.107	−0.298627	−0.149313	−	0.988790i	$-0.547706\pi$
$151$	−554.107	−0.298627	−0.149313	+	0.988790i	$0.547706\pi$
$152$	−460.503	−0.245735
$153$	1103.92	0.583310
$154$	0	0
$155$	−218.571	−0.113265
$156$	7212.50	3.70168
$157$	893.163	0.454027	0.227013	−	0.973892i	$-0.427104\pi$
$157$	893.163	0.454027	0.227013	+	0.973892i	$0.427104\pi$
$158$	3008.92	1.51504
$159$	−2788.76	−1.39096
$160$	488.945	0.241591
$161$	0	0
$162$	3876.25	1.87992
$163$	1156.98	0.555963	0.277982	−	0.960586i	$-0.410335\pi$
$163$	1156.98	0.555963	0.277982	+	0.960586i	$0.410335\pi$
$164$	3918.77	1.86588
$165$	56.2115	0.0265216
$166$	−3521.05	−1.64630
$167$	3603.38	1.66969	0.834844	−	0.550487i	$-0.185558\pi$
$167$	3603.38	1.66969	0.834844	+	0.550487i	$0.185558\pi$
$168$	0	0
$169$	5866.37	2.67017
$170$	−547.287	−0.246912
$171$	545.874	0.244117
$172$	5002.28	2.21756
$173$	−1175.65	−0.516664	−0.258332	−	0.966056i	$-0.583173\pi$
$173$	−1175.65	−0.516664	−0.258332	+	0.966056i	$0.583173\pi$
$174$	3024.70	1.31783
$175$	0	0
$176$	74.5326	0.0319210
$177$	3913.49	1.66190
$178$	−571.612	−0.240697
$179$	1255.13	0.524095	0.262047	−	0.965055i	$-0.415602\pi$
$179$	1255.13	0.524095	0.262047	+	0.965055i	$0.415602\pi$
$180$	510.903	0.211558
$181$	−1550.86	−0.636876	−0.318438	−	0.947944i	$-0.603158\pi$
$181$	−1550.86	−0.636876	−0.318438	+	0.947944i	$0.603158\pi$
$182$	0	0
$183$	−1052.51	−0.425156
$184$	−928.828	−0.372142
$185$	−789.710	−0.313841
$186$	−3014.89	−1.18851
$187$	−208.703	−0.0816141
$188$	−5678.93	−2.20308
$189$	0	0
$190$	−270.627	−0.103333
$191$	−2294.47	−0.869224	−0.434612	−	0.900618i	$-0.643115\pi$
$191$	−2294.47	−0.869224	−0.434612	+	0.900618i	$0.643115\pi$
$192$	5652.20	2.12454
$193$	−646.341	−0.241060	−0.120530	−	0.992710i	$-0.538459\pi$
$193$	−646.341	−0.241060	−0.120530	+	0.992710i	$0.538459\pi$
$194$	1910.92	0.707195
$195$	1352.27	0.496606
$196$	0	0
$197$	−3165.25	−1.14475	−0.572373	−	0.819993i	$-0.693977\pi$
$197$	−3165.25	−1.14475	−0.572373	+	0.819993i	$0.693977\pi$
$198$	327.499	0.117547
$199$	−201.450	−0.0717608	−0.0358804	−	0.999356i	$-0.511424\pi$
$199$	−201.450	−0.0717608	−0.0358804	+	0.999356i	$0.511424\pi$
$200$	2001.18	0.707524
$201$	−2531.76	−0.888440
$202$	5541.58	1.93022
$203$	0	0
$204$	−4490.92	−1.54131
$205$	734.731	0.250321
$206$	−2887.84	−0.976725
$207$	1101.02	0.369691
$208$	1793.02	0.597709
$209$	−103.201	−0.0341558
$210$	0	0
$211$	2723.81	0.888696	0.444348	−	0.895854i	$-0.353435\pi$
$211$	2723.81	0.888696	0.444348	+	0.895854i	$0.353435\pi$
$212$	4791.98	1.55243
$213$	2644.22	0.850605
$214$	−1715.79	−0.548080
$215$	937.878	0.297501
$216$	−826.316	−0.260295
$217$	0	0
$218$	2411.57	0.749229
$219$	4636.87	1.43073
$220$	−96.5895	−0.0296003
$221$	−5020.73	−1.52819
$222$	−10893.0	−3.29319
$223$	−1587.31	−0.476654	−0.238327	−	0.971185i	$-0.576599\pi$
$223$	−1587.31	−0.476654	−0.238327	+	0.971185i	$0.576599\pi$
$224$	0	0
$225$	−2372.17	−0.702865
$226$	−10473.4	−3.08265
$227$	−2876.55	−0.841072	−0.420536	−	0.907276i	$-0.638158\pi$
$227$	−2876.55	−0.841072	−0.420536	+	0.907276i	$0.638158\pi$
$228$	−2220.71	−0.645044
$229$	1187.01	0.342532	0.171266	−	0.985225i	$-0.445214\pi$
$229$	1187.01	0.342532	0.171266	+	0.985225i	$0.445214\pi$
$230$	−545.850	−0.156488
$231$	0	0
$232$	−1658.16	−0.469239
$233$	736.827	0.207172	0.103586	−	0.994620i	$-0.466968\pi$
$233$	736.827	0.207172	0.103586	+	0.994620i	$0.466968\pi$
$234$	7878.59	2.20102
$235$	−1064.74	−0.295558
$236$	−6724.64	−1.85482
$237$	4629.24	1.26878
$238$	0	0
$239$	−1048.13	−0.283673	−0.141837	−	0.989890i	$-0.545301\pi$
$239$	−1048.13	−0.283673	−0.141837	+	0.989890i	$0.545301\pi$
$240$	300.699	0.0808752
$241$	4186.88	1.11909	0.559546	−	0.828800i	$-0.310976\pi$
$241$	4186.88	1.11909	0.559546	+	0.828800i	$0.310976\pi$
$242$	5852.89	1.55470
$243$	4624.13	1.22073
$244$	1808.55	0.474510
$245$	0	0
$246$	10134.6	2.62667
$247$	−2482.69	−0.639554
$248$	1652.78	0.423193
$249$	−5417.16	−1.37871
$250$	2399.58	0.607052
$251$	−3902.91	−0.981473	−0.490736	−	0.871308i	$-0.663272\pi$
$251$	−3902.91	−0.981473	−0.490736	+	0.871308i	$0.663272\pi$
$252$	0	0
$253$	−208.155	−0.0517256
$254$	−7148.34	−1.76585
$255$	−842.004	−0.206778
$256$	−1820.65	−0.444494
$257$	−505.770	−0.122759	−0.0613795	−	0.998115i	$-0.519550\pi$
$257$	−505.770	−0.122759	−0.0613795	+	0.998115i	$0.519550\pi$
$258$	12936.8	3.12174
$259$	0	0
$260$	−2323.64	−0.554254
$261$	1965.56	0.466150
$262$	1577.47	0.371972
$263$	5882.69	1.37925	0.689623	−	0.724168i	$-0.257776\pi$
$263$	5882.69	1.37925	0.689623	+	0.724168i	$0.257776\pi$
$264$	−425.059	−0.0990931
$265$	898.449	0.208269
$266$	0	0
$267$	−879.428	−0.201573
$268$	4350.38	0.991573
$269$	808.874	0.183338	0.0916690	−	0.995790i	$-0.470780\pi$
$269$	808.874	0.183338	0.0916690	+	0.995790i	$0.470780\pi$
$270$	−485.607	−0.109456
$271$	6567.64	1.47216	0.736080	−	0.676894i	$-0.236675\pi$
$271$	6567.64	1.47216	0.736080	+	0.676894i	$0.236675\pi$
$272$	−1116.44	−0.248875
$273$	0	0
$274$	−12098.4	−2.66748
$275$	448.474	0.0983419
$276$	−4479.14	−0.976856
$277$	5972.29	1.29545	0.647726	−	0.761873i	$-0.275720\pi$
$277$	5972.29	1.29545	0.647726	+	0.761873i	$0.275720\pi$
$278$	−5496.20	−1.18576
$279$	−1959.18	−0.420406
$280$	0	0
$281$	5025.55	1.06690	0.533450	−	0.845831i	$-0.320895\pi$
$281$	5025.55	1.06690	0.533450	+	0.845831i	$0.320895\pi$
$282$	−14686.7	−3.10135
$283$	2009.99	0.422195	0.211098	−	0.977465i	$-0.432296\pi$
$283$	2009.99	0.422195	0.211098	+	0.977465i	$0.432296\pi$
$284$	−4543.62	−0.949346
$285$	−416.361	−0.0865372
$286$	−1489.50	−0.307958
$287$	0	0
$288$	4382.72	0.896716
$289$	−1786.80	−0.363689
$290$	−974.462	−0.197318
$291$	2939.96	0.592246
$292$	−7967.64	−1.59682
$293$	−5065.33	−1.00997	−0.504983	−	0.863129i	$-0.668501\pi$
$293$	−5065.33	−1.00997	−0.504983	+	0.863129i	$0.668501\pi$
$294$	0	0
$295$	−1260.80	−0.248837
$296$	5971.61	1.17261
$297$	−185.182	−0.0361795
$298$	−10779.0	−2.09534
$299$	−5007.55	−0.968542
$300$	9650.40	1.85722
$301$	0	0
$302$	2462.39	0.469187
$303$	8525.75	1.61647
$304$	−552.066	−0.104155
$305$	339.085	0.0636588
$306$	−4905.67	−0.916466
$307$	345.396	0.0642111	0.0321055	−	0.999484i	$-0.489779\pi$
$307$	345.396	0.0642111	0.0321055	+	0.999484i	$0.489779\pi$
$308$	0	0
$309$	−4442.96	−0.817965
$310$	971.301	0.177955
$311$	6330.00	1.15415	0.577076	−	0.816690i	$-0.304193\pi$
$311$	6330.00	1.15415	0.577076	+	0.816690i	$0.304193\pi$
$312$	−10225.6	−1.85548
$313$	4465.91	0.806480	0.403240	−	0.915094i	$-0.367884\pi$
$313$	4465.91	0.806480	0.403240	+	0.915094i	$0.367884\pi$
$314$	−3969.11	−0.713343
$315$	0	0
$316$	−7954.53	−1.41607
$317$	−6559.23	−1.16215	−0.581077	−	0.813848i	$-0.697369\pi$
$317$	−6559.23	−1.16215	−0.581077	+	0.813848i	$0.697369\pi$
$318$	12392.9	2.18541
$319$	−371.602	−0.0652216
$320$	−1820.96	−0.318109
$321$	−2639.76	−0.458993
$322$	0	0
$323$	1545.87	0.266299
$324$	−10247.4	−1.75710
$325$	10788.9	1.84141
$326$	−5141.50	−0.873501
$327$	3710.21	0.627447
$328$	−5555.87	−0.935279
$329$	0	0
$330$	−249.797	−0.0416694
$331$	4262.89	0.707885	0.353942	−	0.935267i	$-0.384841\pi$
$331$	4262.89	0.707885	0.353942	+	0.935267i	$0.384841\pi$
$332$	9308.42	1.53875
$333$	−7078.66	−1.16489
$334$	−16013.0	−2.62333
$335$	815.653	0.133026
$336$	0	0
$337$	2557.21	0.413353	0.206677	−	0.978409i	$-0.433735\pi$
$337$	2557.21	0.413353	0.206677	+	0.978409i	$0.433735\pi$
$338$	−26069.4	−4.19524
$339$	−16113.3	−2.58158
$340$	1446.83	0.230781
$341$	370.396	0.0588214
$342$	−2425.80	−0.383544
$343$	0	0
$344$	−7092.03	−1.11156
$345$	−839.794	−0.131052
$346$	5224.43	0.811755
$347$	5603.22	0.866848	0.433424	−	0.901190i	$-0.357305\pi$
$347$	5603.22	0.866848	0.433424	+	0.901190i	$0.357305\pi$
$348$	−7996.23	−1.23173
$349$	8313.90	1.27517	0.637583	−	0.770382i	$-0.279934\pi$
$349$	8313.90	1.27517	0.637583	+	0.770382i	$0.279934\pi$
$350$	0	0
$351$	−4454.88	−0.677448
$352$	−828.582	−0.125465
$353$	−7272.00	−1.09646	−0.548229	−	0.836328i	$-0.684698\pi$
$353$	−7272.00	−1.09646	−0.548229	+	0.836328i	$0.684698\pi$
$354$	−17391.1	−2.61109
$355$	−851.884	−0.127361
$356$	1511.14	0.224973
$357$	0	0
$358$	−5577.65	−0.823431
$359$	−2327.90	−0.342233	−0.171116	−	0.985251i	$-0.554737\pi$
$359$	−2327.90	−0.342233	−0.171116	+	0.985251i	$0.554737\pi$
$360$	−724.337	−0.106044
$361$	−6094.59	−0.888553
$362$	6891.83	1.00063
$363$	9004.70	1.30200
$364$	0	0
$365$	−1493.85	−0.214224
$366$	4677.21	0.667983
$367$	−203.670	−0.0289686	−0.0144843	−	0.999895i	$-0.504611\pi$
$367$	−203.670	−0.0289686	−0.0144843	+	0.999895i	$0.504611\pi$
$368$	−1113.51	−0.157733
$369$	6585.85	0.929121
$370$	3509.37	0.493091
$371$	0	0
$372$	7970.30	1.11086
$373$	−2267.07	−0.314703	−0.157352	−	0.987543i	$-0.550296\pi$
$373$	−2267.07	−0.314703	−0.157352	+	0.987543i	$0.550296\pi$
$374$	927.449	0.128228
$375$	3691.77	0.508379
$376$	8051.35	1.10430
$377$	−8939.56	−1.22125
$378$	0	0
$379$	2332.08	0.316071	0.158036	−	0.987433i	$-0.449484\pi$
$379$	2332.08	0.316071	0.158036	+	0.987433i	$0.449484\pi$
$380$	715.443	0.0965827
$381$	−10997.8	−1.47883
$382$	10196.3	1.36568
$383$	−2701.43	−0.360408	−0.180204	−	0.983629i	$-0.557676\pi$
$383$	−2701.43	−0.360408	−0.180204	+	0.983629i	$0.557676\pi$
$384$	−12976.3	−1.72447
$385$	0	0
$386$	2872.26	0.378741
$387$	8406.79	1.10424
$388$	−5051.80	−0.660995
$389$	−5766.95	−0.751660	−0.375830	−	0.926689i	$-0.622642\pi$
$389$	−5766.95	−0.751660	−0.375830	+	0.926689i	$0.622642\pi$
$390$	−6009.33	−0.780242
$391$	3117.99	0.403283
$392$	0	0
$393$	2426.95	0.311510
$394$	14066.0	1.79857
$395$	−1491.40	−0.189975
$396$	−865.792	−0.109868
$397$	−11836.2	−1.49633	−0.748163	−	0.663515i	$-0.769064\pi$
$397$	−11836.2	−1.49633	−0.748163	+	0.663515i	$0.769064\pi$
$398$	895.218	0.112747
$399$	0	0
$400$	2399.08	0.299885
$401$	1018.84	0.126879	0.0634393	−	0.997986i	$-0.479793\pi$
$401$	1018.84	0.126879	0.0634393	+	0.997986i	$0.479793\pi$
$402$	11250.8	1.39587
$403$	8910.57	1.10141
$404$	−14650.0	−1.80412
$405$	−1921.29	−0.235728
$406$	0	0
$407$	1338.27	0.162986
$408$	6367.04	0.772587
$409$	−8330.82	−1.00717	−0.503585	−	0.863946i	$-0.667986\pi$
$409$	−8330.82	−1.00717	−0.503585	+	0.863946i	$0.667986\pi$
$410$	−3265.05	−0.393292
$411$	−18613.4	−2.23390
$412$	7634.44	0.912917
$413$	0	0
$414$	−4892.80	−0.580840
$415$	1745.24	0.206435
$416$	−19933.1	−2.34928
$417$	−8455.94	−0.993020
$418$	458.613	0.0536638
$419$	−3781.96	−0.440957	−0.220479	−	0.975392i	$-0.570762\pi$
$419$	−3781.96	−0.440957	−0.220479	+	0.975392i	$0.570762\pi$
$420$	0	0
$421$	10382.3	1.20191	0.600954	−	0.799283i	$-0.294787\pi$
$421$	10382.3	1.20191	0.600954	+	0.799283i	$0.294787\pi$
$422$	−12104.3	−1.39627
$423$	−9543.96	−1.09703
$424$	−6793.87	−0.778160
$425$	−6717.78	−0.766730
$426$	−11750.6	−1.33643
$427$	0	0
$428$	4535.95	0.512275
$429$	−2291.60	−0.257901
$430$	−4167.82	−0.467419
$431$	3284.34	0.367056	0.183528	−	0.983014i	$-0.441248\pi$
$431$	3284.34	0.367056	0.183528	+	0.983014i	$0.441248\pi$
$432$	−990.614	−0.110326
$433$	−4684.24	−0.519885	−0.259943	−	0.965624i	$-0.583704\pi$
$433$	−4684.24	−0.519885	−0.259943	+	0.965624i	$0.583704\pi$
$434$	0	0
$435$	−1499.22	−0.165246
$436$	−6375.34	−0.700283
$437$	1541.81	0.168775
$438$	−20605.7	−2.24790
$439$	−15685.9	−1.70535	−0.852674	−	0.522444i	$-0.825020\pi$
$439$	−15685.9	−1.70535	−0.852674	+	0.522444i	$0.825020\pi$
$440$	136.941	0.0148372
$441$	0	0
$442$	22311.5	2.40102
$443$	8431.92	0.904318	0.452159	−	0.891937i	$-0.350654\pi$
$443$	8431.92	0.904318	0.452159	+	0.891937i	$0.350654\pi$
$444$	28797.2	3.07805
$445$	283.324	0.0301817
$446$	7053.79	0.748894
$447$	−16583.5	−1.75475
$448$	0	0
$449$	7280.78	0.765259	0.382629	−	0.923902i	$-0.375019\pi$
$449$	7280.78	0.765259	0.382629	+	0.923902i	$0.375019\pi$
$450$	10541.6	1.10431
$451$	−1245.10	−0.129999
$452$	27687.9	2.88126
$453$	3788.40	0.392924
$454$	12783.0	1.32145
$455$	0	0
$456$	3148.43	0.323331
$457$	10271.1	1.05134	0.525669	−	0.850689i	$-0.323815\pi$
$457$	10271.1	1.05134	0.525669	+	0.850689i	$0.323815\pi$
$458$	−5274.93	−0.538169
$459$	2773.87	0.282077
$460$	1443.04	0.146265
$461$	−1401.39	−0.141582	−0.0707908	−	0.997491i	$-0.522552\pi$
$461$	−1401.39	−0.141582	−0.0707908	+	0.997491i	$0.522552\pi$
$462$	0	0
$463$	1931.78	0.193903	0.0969517	−	0.995289i	$-0.469091\pi$
$463$	1931.78	0.193903	0.0969517	+	0.995289i	$0.469091\pi$
$464$	−1987.85	−0.198888
$465$	1494.35	0.149030
$466$	−3274.37	−0.325498
$467$	−527.449	−0.0522643	−0.0261321	−	0.999658i	$-0.508319\pi$
$467$	−527.449	−0.0522643	−0.0261321	+	0.999658i	$0.508319\pi$
$468$	−20828.2	−2.05723
$469$	0	0
$470$	4731.59	0.464366
$471$	−6106.50	−0.597394
$472$	9533.91	0.929733
$473$	−1589.36	−0.154501
$474$	−20571.8	−1.99345
$475$	−3321.87	−0.320879
$476$	0	0
$477$	8053.36	0.773036
$478$	4657.76	0.445693
$479$	8007.38	0.763813	0.381907	−	0.924201i	$-0.375268\pi$
$479$	8007.38	0.763813	0.381907	+	0.924201i	$0.375268\pi$
$480$	−3342.89	−0.317877
$481$	32194.5	3.05185
$482$	−18606.0	−1.75826
$483$	0	0
$484$	−15473.0	−1.45313
$485$	−947.162	−0.0886772
$486$	−20549.1	−1.91795
$487$	−399.582	−0.0371802	−0.0185901	−	0.999827i	$-0.505918\pi$
$487$	−399.582	−0.0371802	−0.0185901	+	0.999827i	$0.505918\pi$
$488$	−2564.08	−0.237850
$489$	−7910.23	−0.731519
$490$	0	0
$491$	−20175.1	−1.85436	−0.927179	−	0.374619i	$-0.877773\pi$
$491$	−20175.1	−1.85436	−0.927179	+	0.374619i	$0.877773\pi$
$492$	−26792.4	−2.45507
$493$	5566.30	0.508506
$494$	11032.8	1.00483
$495$	−162.327	−0.0147396
$496$	1981.41	0.179371
$497$	0	0
$498$	24073.2	2.16616
$499$	−7128.18	−0.639481	−0.319741	−	0.947505i	$-0.603596\pi$
$499$	−7128.18	−0.639481	−0.319741	+	0.947505i	$0.603596\pi$
$500$	−6343.65	−0.567394
$501$	−24636.1	−2.19692
$502$	17344.1	1.54204
$503$	−12392.7	−1.09853	−0.549266	−	0.835648i	$-0.685093\pi$
$503$	−12392.7	−1.09853	−0.549266	+	0.835648i	$0.685093\pi$
$504$	0	0
$505$	−2746.73	−0.242035
$506$	925.015	0.0812686
$507$	−40108.0	−3.51333
$508$	18897.7	1.65049
$509$	−15904.7	−1.38500	−0.692500	−	0.721417i	$-0.743491\pi$
$509$	−15904.7	−1.38500	−0.692500	+	0.721417i	$0.743491\pi$
$510$	3741.76	0.324879
$511$	0	0
$512$	−7093.06	−0.612250
$513$	1371.65	0.118050
$514$	2247.58	0.192872
$515$	1431.38	0.122474
$516$	−34200.3	−2.91780
$517$	1804.35	0.153491
$518$	0	0
$519$	8037.82	0.679810
$520$	3294.36	0.277822
$521$	3469.67	0.291764	0.145882	−	0.989302i	$-0.453398\pi$
$521$	3469.67	0.291764	0.145882	+	0.989302i	$0.453398\pi$
$522$	−8734.71	−0.732390
$523$	−17900.9	−1.49666	−0.748330	−	0.663327i	$-0.769144\pi$
$523$	−17900.9	−1.49666	−0.748330	+	0.663327i	$0.769144\pi$
$524$	−4170.29	−0.347672
$525$	0	0
$526$	−26141.9	−2.16700
$527$	−5548.24	−0.458606
$528$	−509.574	−0.0420007
$529$	−9057.19	−0.744406
$530$	−3992.60	−0.327222
$531$	−11301.4	−0.923611
$532$	0	0
$533$	−29953.1	−2.43417
$534$	3908.07	0.316702
$535$	850.447	0.0687253
$536$	−6167.78	−0.497029
$537$	−8581.25	−0.689587
$538$	−3594.54	−0.288051
$539$	0	0
$540$	1283.77	0.102305
$541$	12364.0	0.982565	0.491283	−	0.871000i	$-0.336528\pi$
$541$	12364.0	0.982565	0.491283	+	0.871000i	$0.336528\pi$
$542$	−29185.8	−2.31298
$543$	10603.1	0.837981
$544$	12411.5	0.978195
$545$	−1195.31	−0.0939479
$546$	0	0
$547$	−10662.8	−0.833468	−0.416734	−	0.909028i	$-0.636825\pi$
$547$	−10662.8	−0.833468	−0.416734	+	0.909028i	$0.636825\pi$
$548$	31983.8	2.49321
$549$	3039.43	0.236283
$550$	−1992.96	−0.154510
$551$	2752.47	0.212811
$552$	6350.33	0.489652
$553$	0	0
$554$	−26540.1	−2.03535
$555$	5399.19	0.412942
$556$	14530.0	1.10829
$557$	24249.3	1.84466	0.922331	−	0.386401i	$-0.126282\pi$
$557$	24249.3	1.84466	0.922331	+	0.386401i	$0.126282\pi$
$558$	8706.38	0.660520
$559$	−38234.9	−2.89296
$560$	0	0
$561$	1426.89	0.107385
$562$	−22332.9	−1.67626
$563$	−8700.36	−0.651290	−0.325645	−	0.945492i	$-0.605581\pi$
$563$	−8700.36	−0.651290	−0.325645	+	0.945492i	$0.605581\pi$
$564$	38826.5	2.89874
$565$	5191.21	0.386541
$566$	−8932.13	−0.663331
$567$	0	0
$568$	6441.75	0.475863
$569$	10763.3	0.793007	0.396504	−	0.918033i	$-0.370223\pi$
$569$	10763.3	0.793007	0.396504	+	0.918033i	$0.370223\pi$
$570$	1850.26	0.135963
$571$	−7634.99	−0.559570	−0.279785	−	0.960063i	$-0.590263\pi$
$571$	−7634.99	−0.559570	−0.279785	+	0.960063i	$0.590263\pi$
$572$	3937.71	0.287839
$573$	15687.1	1.14370
$574$	0	0
$575$	−6700.15	−0.485940
$576$	−16322.4	−1.18073
$577$	−27425.2	−1.97873	−0.989365	−	0.145452i	$-0.953536\pi$
$577$	−27425.2	−1.97873	−0.989365	+	0.145452i	$0.953536\pi$
$578$	7940.33	0.571409
$579$	4418.99	0.317180
$580$	2576.13	0.184428
$581$	0	0
$582$	−13064.8	−0.930506
$583$	−1522.54	−0.108160
$584$	11296.2	0.800410
$585$	−3905.09	−0.275992
$586$	22509.7	1.58681
$587$	5616.82	0.394942	0.197471	−	0.980309i	$-0.436727\pi$
$587$	5616.82	0.394942	0.197471	+	0.980309i	$0.436727\pi$
$588$	0	0
$589$	−2743.54	−0.191928
$590$	5602.86	0.390959
$591$	21640.6	1.50622
$592$	7158.95	0.497012
$593$	13893.2	0.962099	0.481049	−	0.876693i	$-0.340256\pi$
$593$	13893.2	0.962099	0.481049	+	0.876693i	$0.340256\pi$
$594$	822.924	0.0568434
$595$	0	0
$596$	28495.9	1.95845
$597$	1377.30	0.0944206
$598$	22252.9	1.52172
$599$	−24927.7	−1.70036	−0.850182	−	0.526488i	$-0.823508\pi$
$599$	−24927.7	−1.70036	−0.850182	+	0.526488i	$0.823508\pi$
$600$	−13681.9	−0.930938
$601$	19529.7	1.32551	0.662756	−	0.748835i	$-0.269386\pi$
$601$	19529.7	1.32551	0.662756	+	0.748835i	$0.269386\pi$
$602$	0	0
$603$	7311.20	0.493756
$604$	−6509.69	−0.438535
$605$	−2901.03	−0.194948
$606$	−37887.4	−2.53972
$607$	−22682.1	−1.51670	−0.758350	−	0.651848i	$-0.773994\pi$
$607$	−22682.1	−1.51670	−0.758350	+	0.651848i	$0.773994\pi$
$608$	6137.34	0.409378
$609$	0	0
$610$	−1506.85	−0.100017
$611$	43406.9	2.87407
$612$	12968.9	0.856594
$613$	748.937	0.0493463	0.0246732	−	0.999696i	$-0.492145\pi$
$613$	748.937	0.0493463	0.0246732	+	0.999696i	$0.492145\pi$
$614$	−1534.90	−0.100885
$615$	−5023.31	−0.329365
$616$	0	0
$617$	9220.40	0.601620	0.300810	−	0.953684i	$-0.402743\pi$
$617$	9220.40	0.601620	0.300810	+	0.953684i	$0.402743\pi$
$618$	19744.0	1.28514
$619$	1085.86	0.0705082	0.0352541	−	0.999378i	$-0.488776\pi$
$619$	1085.86	0.0705082	0.0352541	+	0.999378i	$0.488776\pi$
$620$	−2567.78	−0.166330
$621$	2766.59	0.178775
$622$	−28129.8	−1.81335
$623$	0	0
$624$	−12258.7	−0.786447
$625$	13829.2	0.885067
$626$	−19846.0	−1.26710
$627$	705.579	0.0449411
$628$	10492.9	0.666741
$629$	−20046.2	−1.27074
$630$	0	0
$631$	−4920.34	−0.310421	−0.155211	−	0.987881i	$-0.549606\pi$
$631$	−4920.34	−0.310421	−0.155211	+	0.987881i	$0.549606\pi$
$632$	11277.6	0.709808
$633$	−18622.5	−1.16932
$634$	29148.4	1.82592
$635$	3543.14	0.221425
$636$	−32762.5	−2.04264
$637$	0	0
$638$	1651.35	0.102473
$639$	−7635.96	−0.472729
$640$	4180.57	0.258205
$641$	−13320.2	−0.820772	−0.410386	−	0.911912i	$-0.634606\pi$
$641$	−13320.2	−0.820772	−0.410386	+	0.911912i	$0.634606\pi$
$642$	11730.8	0.721147
$643$	1511.07	0.0926761	0.0463381	−	0.998926i	$-0.485245\pi$
$643$	1511.07	0.0926761	0.0463381	+	0.998926i	$0.485245\pi$
$644$	0	0
$645$	−6412.21	−0.391443
$646$	−6869.65	−0.418395
$647$	−12741.8	−0.774238	−0.387119	−	0.922030i	$-0.626530\pi$
$647$	−12741.8	−0.774238	−0.387119	+	0.922030i	$0.626530\pi$
$648$	14528.4	0.880755
$649$	2136.60	0.129228
$650$	−47944.4	−2.89313
$651$	0	0
$652$	13592.3	0.816436
$653$	16209.8	0.971420	0.485710	−	0.874120i	$-0.338561\pi$
$653$	16209.8	0.971420	0.485710	+	0.874120i	$0.338561\pi$
$654$	−16487.7	−0.985813
$655$	−781.888	−0.0466426
$656$	−6660.55	−0.396419
$657$	−13390.3	−0.795140
$658$	0	0
$659$	−18808.8	−1.11182	−0.555909	−	0.831243i	$-0.687630\pi$
$659$	−18808.8	−1.11182	−0.555909	+	0.831243i	$0.687630\pi$
$660$	660.376	0.0389471
$661$	−15084.1	−0.887598	−0.443799	−	0.896126i	$-0.646370\pi$
$661$	−15084.1	−0.887598	−0.443799	+	0.896126i	$0.646370\pi$
$662$	−18943.8	−1.11219
$663$	34326.4	2.01075
$664$	−13197.1	−0.771305
$665$	0	0
$666$	31456.7	1.83021
$667$	5551.69	0.322282
$668$	42332.7	2.45195
$669$	10852.3	0.627166
$670$	−3624.66	−0.209004
$671$	−574.623	−0.0330597
$672$	0	0
$673$	27763.4	1.59019	0.795095	−	0.606484i	$-0.207421\pi$
$673$	27763.4	1.59019	0.795095	+	0.606484i	$0.207421\pi$
$674$	−11363.9	−0.649440
$675$	−5960.68	−0.339891
$676$	68918.4	3.92117
$677$	−2746.12	−0.155896	−0.0779482	−	0.996957i	$-0.524837\pi$
$677$	−2746.12	−0.155896	−0.0779482	+	0.996957i	$0.524837\pi$
$678$	71605.7	4.05605
$679$	0	0
$680$	−2051.26	−0.115680
$681$	19666.8	1.10666
$682$	−1646.00	−0.0924171
$683$	22096.8	1.23794	0.618968	−	0.785416i	$-0.287551\pi$
$683$	22096.8	1.23794	0.618968	+	0.785416i	$0.287551\pi$
$684$	6412.96	0.358488
$685$	5996.65	0.334482
$686$	0	0
$687$	−8115.52	−0.450693
$688$	−8502.14	−0.471135
$689$	−36627.5	−2.02525
$690$	3731.94	0.205902
$691$	−11422.7	−0.628855	−0.314427	−	0.949281i	$-0.601813\pi$
$691$	−11422.7	−0.628855	−0.314427	+	0.949281i	$0.601813\pi$
$692$	−13811.6	−0.758724
$693$	0	0
$694$	−24900.0	−1.36195
$695$	2724.24	0.148685
$696$	11336.7	0.617410
$697$	18650.6	1.01354
$698$	−36946.0	−2.00347
$699$	−5037.64	−0.272591
$700$	0	0
$701$	11747.6	0.632953	0.316476	−	0.948600i	$-0.397500\pi$
$701$	11747.6	0.632953	0.316476	+	0.948600i	$0.397500\pi$
$702$	19797.0	1.06437
$703$	−9912.59	−0.531807
$704$	3085.86	0.165203
$705$	7279.58	0.388886
$706$	32315.9	1.72270
$707$	0	0
$708$	45975.9	2.44051
$709$	−22217.1	−1.17684	−0.588420	−	0.808556i	$-0.700250\pi$
$709$	−22217.1	−1.17684	−0.588420	+	0.808556i	$0.700250\pi$
$710$	3785.67	0.200104
$711$	−13368.3	−0.705134
$712$	−2142.43	−0.112768
$713$	−5533.68	−0.290656
$714$	0	0
$715$	738.282	0.0386156
$716$	14745.4	0.769637
$717$	7166.00	0.373248
$718$	10344.9	0.537699
$719$	20507.3	1.06369	0.531846	−	0.846841i	$-0.321498\pi$
$719$	20507.3	1.06369	0.531846	+	0.846841i	$0.321498\pi$
$720$	−868.358	−0.0449469
$721$	0	0
$722$	27083.6	1.39605
$723$	−28625.5	−1.47246
$724$	−18219.6	−0.935257
$725$	−11961.2	−0.612730
$726$	−40015.8	−2.04563
$727$	−35457.6	−1.80887	−0.904437	−	0.426608i	$-0.859708\pi$
$727$	−35457.6	−1.80887	−0.904437	+	0.426608i	$0.859708\pi$
$728$	0	0
$729$	−8063.69	−0.409678
$730$	6638.50	0.336578
$731$	23807.3	1.20458
$732$	−12364.9	−0.624345
$733$	28862.7	1.45439	0.727196	−	0.686430i	$-0.240823\pi$
$733$	28862.7	1.45439	0.727196	+	0.686430i	$0.240823\pi$
$734$	905.083	0.0455139
$735$	0	0
$736$	12378.9	0.619963
$737$	−1382.23	−0.0690843
$738$	−29266.7	−1.45979
$739$	28103.1	1.39890	0.699452	−	0.714679i	$-0.253427\pi$
$739$	28103.1	1.39890	0.699452	+	0.714679i	$0.253427\pi$
$740$	−9277.55	−0.460878
$741$	16974.0	0.841505
$742$	0	0
$743$	23389.8	1.15490	0.577448	−	0.816427i	$-0.304048\pi$
$743$	23389.8	1.15490	0.577448	+	0.816427i	$0.304048\pi$
$744$	−11300.0	−0.556824
$745$	5342.69	0.262740
$746$	10074.6	0.494445
$747$	15643.7	0.766227
$748$	−2451.85	−0.119851
$749$	0	0
$750$	−16405.8	−0.798740
$751$	−33032.6	−1.60503	−0.802515	−	0.596632i	$-0.796505\pi$
$751$	−33032.6	−1.60503	−0.802515	+	0.596632i	$0.796505\pi$
$752$	9652.21	0.468059
$753$	26684.0	1.29139
$754$	39726.3	1.91876
$755$	−1220.50	−0.0588326
$756$	0	0
$757$	7058.64	0.338904	0.169452	−	0.985538i	$-0.445800\pi$
$757$	7058.64	0.338904	0.169452	+	0.985538i	$0.445800\pi$
$758$	−10363.5	−0.496595
$759$	1423.14	0.0680589
$760$	−1014.33	−0.0484124
$761$	15115.1	0.720003	0.360001	−	0.932952i	$-0.382776\pi$
$761$	15115.1	0.720003	0.360001	+	0.932952i	$0.382776\pi$
$762$	48872.8	2.32346
$763$	0	0
$764$	−26955.5	−1.27646
$765$	2431.54	0.114918
$766$	12004.8	0.566255
$767$	51399.8	2.41974
$768$	12447.6	0.584851
$769$	−11309.9	−0.530357	−0.265178	−	0.964199i	$-0.585431\pi$
$769$	−11309.9	−0.530357	−0.265178	+	0.964199i	$0.585431\pi$
$770$	0	0
$771$	3457.91	0.161522
$772$	−7593.25	−0.353999
$773$	7891.72	0.367200	0.183600	−	0.983001i	$-0.441225\pi$
$773$	7891.72	0.367200	0.183600	+	0.983001i	$0.441225\pi$
$774$	−37358.7	−1.73493
$775$	11922.4	0.552602
$776$	7162.23	0.331326
$777$	0	0
$778$	25627.6	1.18097
$779$	9222.49	0.424172
$780$	15886.6	0.729270
$781$	1443.63	0.0661422
$782$	−13856.0	−0.633617
$783$	4938.97	0.225421
$784$	0	0
$785$	1967.32	0.0894480
$786$	−10785.1	−0.489429
$787$	33014.4	1.49534	0.747672	−	0.664068i	$-0.231171\pi$
$787$	33014.4	1.49534	0.747672	+	0.664068i	$0.231171\pi$
$788$	−37185.6	−1.68107
$789$	−40219.5	−1.81477
$790$	6627.58	0.298479
$791$	0	0
$792$	1227.48	0.0550717
$793$	−13823.6	−0.619031
$794$	52598.6	2.35095
$795$	−6142.64	−0.274034
$796$	−2366.64	−0.105381
$797$	−20725.0	−0.921099	−0.460549	−	0.887634i	$-0.652348\pi$
$797$	−20725.0	−0.921099	−0.460549	+	0.887634i	$0.652348\pi$
$798$	0	0
$799$	−27027.7	−1.19671
$800$	−26670.6	−1.17869
$801$	2539.61	0.112026
$802$	−4527.59	−0.199345
$803$	2531.53	0.111253
$804$	−29743.2	−1.30468
$805$	0	0
$806$	−39597.5	−1.73047
$807$	−5530.22	−0.241230
$808$	20770.1	0.904321
$809$	23575.3	1.02455	0.512276	−	0.858821i	$-0.328803\pi$
$809$	23575.3	1.02455	0.512276	+	0.858821i	$0.328803\pi$
$810$	8537.99	0.370364
$811$	40751.9	1.76448	0.882239	−	0.470802i	$-0.156035\pi$
$811$	40751.9	1.76448	0.882239	+	0.470802i	$0.156035\pi$
$812$	0	0
$813$	−44902.5	−1.93702
$814$	−5947.09	−0.256076
$815$	2548.43	0.109531
$816$	7633.01	0.327462
$817$	11772.4	0.504119
$818$	37021.2	1.58241
$819$	0	0
$820$	8631.66	0.367598
$821$	20443.5	0.869041	0.434520	−	0.900662i	$-0.356918\pi$
$821$	20443.5	0.869041	0.434520	+	0.900662i	$0.356918\pi$
$822$	82715.7	3.50978
$823$	−40351.9	−1.70909	−0.854543	−	0.519381i	$-0.826163\pi$
$823$	−40351.9	−1.70909	−0.854543	+	0.519381i	$0.826163\pi$
$824$	−10823.8	−0.457603
$825$	−3066.19	−0.129395
$826$	0	0
$827$	10793.9	0.453860	0.226930	−	0.973911i	$-0.427131\pi$
$827$	10793.9	0.453860	0.226930	+	0.973911i	$0.427131\pi$
$828$	12934.8	0.542894
$829$	−14556.3	−0.609846	−0.304923	−	0.952377i	$-0.598631\pi$
$829$	−14556.3	−0.609846	−0.304923	+	0.952377i	$0.598631\pi$
$830$	−7755.62	−0.324339
$831$	−40832.2	−1.70452
$832$	74236.0	3.09335
$833$	0	0
$834$	37577.2	1.56018
$835$	7936.97	0.328946
$836$	−1212.41	−0.0501580
$837$	−4922.95	−0.203300
$838$	16806.6	0.692809
$839$	−22647.3	−0.931908	−0.465954	−	0.884809i	$-0.654289\pi$
$839$	−22647.3	−0.931908	−0.465954	+	0.884809i	$0.654289\pi$
$840$	0	0
$841$	−14478.0	−0.593629
$842$	−46137.8	−1.88838
$843$	−34359.3	−1.40379
$844$	31999.5	1.30506
$845$	12921.5	0.526052
$846$	42412.2	1.72359
$847$	0	0
$848$	−8144.71	−0.329824
$849$	−13742.1	−0.555511
$850$	29853.0	1.20465
$851$	−19993.5	−0.805370
$852$	31064.4	1.24912
$853$	−2750.34	−0.110399	−0.0551993	−	0.998475i	$-0.517579\pi$
$853$	−2750.34	−0.110399	−0.0551993	+	0.998475i	$0.517579\pi$
$854$	0	0
$855$	1202.37	0.0480936
$856$	−6430.89	−0.256779
$857$	19173.5	0.764240	0.382120	−	0.924113i	$-0.375194\pi$
$857$	19173.5	0.764240	0.382120	+	0.924113i	$0.375194\pi$
$858$	10183.6	0.405201
$859$	−23561.1	−0.935850	−0.467925	−	0.883768i	$-0.654998\pi$
$859$	−23561.1	−0.935850	−0.467925	+	0.883768i	$0.654998\pi$
$860$	11018.2	0.436883
$861$	0	0
$862$	−14595.2	−0.576699
$863$	−17600.4	−0.694236	−0.347118	−	0.937821i	$-0.612840\pi$
$863$	−17600.4	−0.694236	−0.347118	+	0.937821i	$0.612840\pi$
$864$	11012.7	0.433634
$865$	−2589.53	−0.101788
$866$	20816.2	0.816817
$867$	12216.3	0.478530
$868$	0	0
$869$	2527.36	0.0986593
$870$	6662.33	0.259625
$871$	−33252.1	−1.29358
$872$	9038.69	0.351019
$873$	−8490.00	−0.329144
$874$	−6851.62	−0.265171
$875$	0	0
$876$	54474.2	2.10104
$877$	21056.6	0.810754	0.405377	−	0.914150i	$-0.367140\pi$
$877$	21056.6	0.810754	0.405377	+	0.914150i	$0.367140\pi$
$878$	69706.3	2.67935
$879$	34631.3	1.32888
$880$	164.169	0.00628878
$881$	−48164.8	−1.84190	−0.920949	−	0.389682i	$-0.872585\pi$
$881$	−48164.8	−1.84190	−0.920949	+	0.389682i	$0.872585\pi$
$882$	0	0
$883$	−13429.6	−0.511825	−0.255912	−	0.966700i	$-0.582376\pi$
$883$	−13429.6	−0.511825	−0.255912	+	0.966700i	$0.582376\pi$
$884$	−58983.8	−2.24416
$885$	8620.03	0.327412
$886$	−37470.4	−1.42082
$887$	−13331.1	−0.504639	−0.252320	−	0.967644i	$-0.581193\pi$
$887$	−13331.1	−0.504639	−0.252320	+	0.967644i	$0.581193\pi$
$888$	−40827.5	−1.54288
$889$	0	0
$890$	−1259.06	−0.0474199
$891$	3255.88	0.122420
$892$	−18647.8	−0.699970
$893$	−13364.9	−0.500827
$894$	73695.2	2.75698
$895$	2764.61	0.103252
$896$	0	0
$897$	34236.3	1.27438
$898$	−32354.9	−1.20234
$899$	−9878.83	−0.366493
$900$	−27868.4	−1.03216
$901$	22806.4	0.843277
$902$	5533.06	0.204247
$903$	0	0
$904$	−39254.8	−1.44424
$905$	−3415.99	−0.125471
$906$	−16835.2	−0.617341
$907$	−19946.9	−0.730237	−0.365118	−	0.930961i	$-0.618971\pi$
$907$	−19946.9	−0.730237	−0.365118	+	0.930961i	$0.618971\pi$
$908$	−33793.9	−1.23512
$909$	−24620.6	−0.898366
$910$	0	0
$911$	−22860.7	−0.831402	−0.415701	−	0.909501i	$-0.636464\pi$
$911$	−22860.7	−0.831402	−0.415701	+	0.909501i	$0.636464\pi$
$912$	3774.44	0.137044
$913$	−2957.53	−0.107207
$914$	−45643.5	−1.65181
$915$	−2318.30	−0.0837603
$916$	13945.1	0.503011
$917$	0	0
$918$	−12326.7	−0.443184
$919$	−4413.38	−0.158416	−0.0792078	−	0.996858i	$-0.525239\pi$
$919$	−4413.38	−0.158416	−0.0792078	+	0.996858i	$0.525239\pi$
$920$	−2045.88	−0.0733159
$921$	−2361.45	−0.0844869
$922$	6227.60	0.222446
$923$	34729.1	1.23849
$924$	0	0
$925$	43076.5	1.53119
$926$	−8584.58	−0.304651
$927$	12830.4	0.454590
$928$	22099.1	0.781721
$929$	4486.97	0.158464	0.0792319	−	0.996856i	$-0.474753\pi$
$929$	4486.97	0.158464	0.0792319	+	0.996856i	$0.474753\pi$
$930$	−6640.72	−0.234148
$931$	0	0
$932$	8656.29	0.304234
$933$	−43277.8	−1.51860
$934$	2343.92	0.0821149
$935$	−459.698	−0.0160788
$936$	29529.4	1.03119
$937$	27737.5	0.967070	0.483535	−	0.875325i	$-0.339353\pi$
$937$	27737.5	0.967070	0.483535	+	0.875325i	$0.339353\pi$
$938$	0	0
$939$	−30533.2	−1.06114
$940$	−12508.7	−0.434030
$941$	−13038.1	−0.451679	−0.225840	−	0.974164i	$-0.572513\pi$
$941$	−13038.1	−0.451679	−0.225840	+	0.974164i	$0.572513\pi$
$942$	27136.5	0.938595
$943$	18601.6	0.642367
$944$	11429.6	0.394068
$945$	0	0
$946$	7062.91	0.242743
$947$	17168.9	0.589139	0.294570	−	0.955630i	$-0.404824\pi$
$947$	17168.9	0.589139	0.294570	+	0.955630i	$0.404824\pi$
$948$	54384.6	1.86322
$949$	60900.6	2.08316
$950$	14762.0	0.504149
$951$	44845.0	1.52913
$952$	0	0
$953$	−31785.3	−1.08040	−0.540202	−	0.841535i	$-0.681652\pi$
$953$	−31785.3	−1.08040	−0.540202	+	0.841535i	$0.681652\pi$
$954$	−35788.2	−1.21455
$955$	−5053.90	−0.171246
$956$	−12313.5	−0.416576
$957$	2540.62	0.0858166
$958$	−35583.8	−1.20006
$959$	0	0
$960$	12449.8	0.418557
$961$	−19944.2	−0.669471
$962$	−143068.	−4.79491
$963$	7623.08	0.255089
$964$	49187.8	1.64339
$965$	−1423.66	−0.0474914
$966$	0	0
$967$	−32468.2	−1.07974	−0.539869	−	0.841749i	$-0.681526\pi$
$967$	−32468.2	−1.07974	−0.539869	+	0.841749i	$0.681526\pi$
$968$	21936.9	0.728389
$969$	−10569.0	−0.350387
$970$	4209.07	0.139325
$971$	−44926.1	−1.48481	−0.742404	−	0.669953i	$-0.766314\pi$
$971$	−44926.1	−1.48481	−0.742404	+	0.669953i	$0.766314\pi$
$972$	54324.6	1.79266
$973$	0	0
$974$	1775.69	0.0584157
$975$	−73762.8	−2.42287
$976$	−3073.90	−0.100813
$977$	−7929.70	−0.259666	−0.129833	−	0.991536i	$-0.541444\pi$
$977$	−7929.70	−0.259666	−0.129833	+	0.991536i	$0.541444\pi$
$978$	35152.1	1.14932
$979$	−480.130	−0.0156742
$980$	0	0
$981$	−10714.3	−0.348708
$982$	89655.7	2.91347
$983$	−41903.4	−1.35963	−0.679813	−	0.733386i	$-0.737939\pi$
$983$	−41903.4	−1.35963	−0.679813	+	0.733386i	$0.737939\pi$
$984$	37985.1	1.23061
$985$	−6971.93	−0.225527
$986$	−24735.9	−0.798938
$987$	0	0
$988$	−29166.8	−0.939190
$989$	23744.8	0.763439
$990$	721.364	0.0231580
$991$	−27271.8	−0.874186	−0.437093	−	0.899416i	$-0.643992\pi$
$991$	−27271.8	−0.874186	−0.437093	+	0.899416i	$0.643992\pi$
$992$	−22027.4	−0.705010
$993$	−29145.1	−0.931412
$994$	0	0
$995$	−443.722	−0.0141376
$996$	−63641.1	−2.02464
$997$	296.153	0.00940749	0.00470374	−	0.999989i	$-0.498503\pi$
$997$	296.153	0.00940749	0.00470374	+	0.999989i	$0.498503\pi$
$998$	31676.8	1.00472
$999$	−17786.9	−0.563317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.4.a.c.1.5		39
7.6	odd	2		2401.4.a.d.1.5		39
49.20	odd	14		49.4.e.a.8.2	✓	78
49.27	odd	14		49.4.e.a.43.2	yes	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.2	✓	78	49.20	odd	14
49.4.e.a.43.2	yes	78	49.27	odd	14
2401.4.a.c.1.5		39	1.1	even	1	trivial
2401.4.a.d.1.5		39	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q-4.44388 q^{2} -6.83693 q^{3} +11.7481 q^{4} +2.20264 q^{5} +30.3825 q^{6} -16.6559 q^{8} +19.7437 q^{9} -9.78829 q^{10} -3.73267 q^{11} -80.3207 q^{12} -89.7963 q^{13} -15.0593 q^{15} -19.9676 q^{16} +55.9124 q^{17} -87.7385 q^{18} +27.6480 q^{19} +25.8768 q^{20} +16.5875 q^{22} +55.7657 q^{23} +113.875 q^{24} -120.148 q^{25} +399.044 q^{26} +49.6110 q^{27} +99.5539 q^{29} +66.9219 q^{30} -99.2310 q^{31} +221.981 q^{32} +25.5200 q^{33} -248.468 q^{34} +231.950 q^{36} -358.528 q^{37} -122.865 q^{38} +613.931 q^{39} -36.6870 q^{40} +333.568 q^{41} +425.796 q^{43} -43.8516 q^{44} +43.4883 q^{45} -247.816 q^{46} -483.393 q^{47} +136.517 q^{48} +533.925 q^{50} -382.269 q^{51} -1054.93 q^{52} +407.896 q^{53} -220.465 q^{54} -8.22175 q^{55} -189.028 q^{57} -442.405 q^{58} -572.404 q^{59} -176.918 q^{60} +153.944 q^{61} +440.970 q^{62} -826.715 q^{64} -197.789 q^{65} -113.408 q^{66} +370.306 q^{67} +656.862 q^{68} -381.266 q^{69} -386.755 q^{71} -328.849 q^{72} -678.209 q^{73} +1593.25 q^{74} +821.446 q^{75} +324.811 q^{76} -2728.24 q^{78} -677.093 q^{79} -43.9816 q^{80} -872.267 q^{81} -1482.33 q^{82} +792.337 q^{83} +123.155 q^{85} -1892.19 q^{86} -680.643 q^{87} +62.1710 q^{88} +128.629 q^{89} -193.257 q^{90} +655.138 q^{92} +678.436 q^{93} +2148.14 q^{94} +60.8988 q^{95} -1517.67 q^{96} -430.011 q^{97} -73.6966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 39 * q + q^2 - q^3 + 145 * q^4 - 27 * q^5 - 41 * q^6 - 12 * q^8 + 312 * q^9 - 78 * q^10 + q^11 + 91 * q^12 - 77 * q^13 - 161 * q^15 + 461 * q^16 - 211 * q^17 + 8 * q^18 - 314 * q^19 - 476 * q^20 - 61 * q^22 - 69 * q^23 - 330 * q^24 + 606 * q^25 - 504 * q^26 + 50 * q^27 + 57 * q^29 + 42 * q^30 - 638 * q^31 - 1600 * q^32 - 1574 * q^33 - 1343 * q^34 + 782 * q^36 + 71 * q^37 - 1359 * q^38 - 84 * q^39 + 155 * q^40 - 1393 * q^41 - 125 * q^43 + 52 * q^44 - 1129 * q^45 - 1454 * q^46 - 1483 * q^47 + 974 * q^48 + 3074 * q^50 - 2044 * q^51 - 3899 * q^52 + 2213 * q^53 - 1142 * q^54 - 1604 * q^55 + 98 * q^57 + 2403 * q^58 - 2073 * q^59 - 1519 * q^60 - 2575 * q^61 - 1742 * q^62 + 1358 * q^64 + 1876 * q^65 + 48 * q^66 + 176 * q^67 - 3038 * q^68 + 638 * q^69 - 1259 * q^71 - 3799 * q^72 + 307 * q^73 + 3845 * q^74 - 131 * q^75 - 1974 * q^76 - 6041 * q^78 + 22 * q^79 + 804 * q^80 + 795 * q^81 - 8043 * q^82 - 6349 * q^83 + 3094 * q^85 + 1745 * q^86 - 9508 * q^87 + 1299 * q^88 - 2253 * q^89 - 11156 * q^90 + 1284 * q^92 - 3430 * q^93 - 2738 * q^94 - 3290 * q^95 - 3031 * q^96 + 770 * q^97 + 5384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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