LMFDB - Embedded newform 1519.4.a.h.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1519,4,Mod(1,1519)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1519, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1519.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1519.a (trivial)

Newform invariants

comment:select newform

sage:traces = [23,5,-6,91,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.6239012987$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	1519.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.43344 q^{2} +7.14275 q^{3} +3.78854 q^{4} +20.1541 q^{5} -24.5242 q^{6} +14.4598 q^{8} +24.0189 q^{9} -69.1980 q^{10} -45.2815 q^{11} +27.0606 q^{12} -48.8977 q^{13} +143.956 q^{15} -79.9553 q^{16} -129.447 q^{17} -82.4675 q^{18} +7.66371 q^{19} +76.3546 q^{20} +155.472 q^{22} +75.5895 q^{23} +103.283 q^{24} +281.188 q^{25} +167.888 q^{26} -21.2934 q^{27} +122.016 q^{29} -494.264 q^{30} -31.0000 q^{31} +158.844 q^{32} -323.435 q^{33} +444.451 q^{34} +90.9965 q^{36} +38.0857 q^{37} -26.3129 q^{38} -349.264 q^{39} +291.424 q^{40} -459.195 q^{41} -162.854 q^{43} -171.551 q^{44} +484.079 q^{45} -259.532 q^{46} -463.507 q^{47} -571.101 q^{48} -965.442 q^{50} -924.611 q^{51} -185.251 q^{52} +411.623 q^{53} +73.1096 q^{54} -912.608 q^{55} +54.7400 q^{57} -418.937 q^{58} +658.985 q^{59} +545.382 q^{60} +46.8510 q^{61} +106.437 q^{62} +94.2618 q^{64} -985.490 q^{65} +1110.49 q^{66} +206.607 q^{67} -490.417 q^{68} +539.917 q^{69} -860.793 q^{71} +347.309 q^{72} -1151.60 q^{73} -130.765 q^{74} +2008.45 q^{75} +29.0343 q^{76} +1199.18 q^{78} -316.963 q^{79} -1611.43 q^{80} -800.603 q^{81} +1576.62 q^{82} -27.1396 q^{83} -2608.90 q^{85} +559.149 q^{86} +871.533 q^{87} -654.762 q^{88} -616.740 q^{89} -1662.06 q^{90} +286.374 q^{92} -221.425 q^{93} +1591.43 q^{94} +154.455 q^{95} +1134.58 q^{96} -898.006 q^{97} -1087.61 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) $ 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * 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 $

$2$	−3.43344	−1.21391	−0.606953	−	0.794738i	$-0.707608\pi$
$2$	−3.43344	−1.21391	−0.606953	+	0.794738i	$0.707608\pi$
$3$	7.14275	1.37462	0.687311	−	0.726363i	$-0.258791\pi$
$3$	7.14275	1.37462	0.687311	+	0.726363i	$0.258791\pi$
$4$	3.78854	0.473568
$5$	20.1541	1.80264	0.901319	−	0.433157i	$-0.142600\pi$
$5$	20.1541	1.80264	0.901319	+	0.433157i	$0.142600\pi$
$6$	−24.5242	−1.66866
$7$	0	0
$7$	0	0
$8$	14.4598	0.639039
$9$	24.0189	0.889588
$10$	−69.1980	−2.18823
$11$	−45.2815	−1.24117	−0.620586	−	0.784139i	$-0.713105\pi$
$11$	−45.2815	−1.24117	−0.620586	+	0.784139i	$0.713105\pi$
$12$	27.0606	0.650977
$13$	−48.8977	−1.04322	−0.521608	−	0.853186i	$-0.674667\pi$
$13$	−48.8977	−1.04322	−0.521608	+	0.853186i	$0.674667\pi$
$14$	0	0
$15$	143.956	2.47795
$16$	−79.9553	−1.24930
$17$	−129.447	−1.84680	−0.923400	−	0.383838i	$-0.874602\pi$
$17$	−129.447	−1.84680	−0.923400	+	0.383838i	$0.874602\pi$
$18$	−82.4675	−1.07988
$19$	7.66371	0.0925356	0.0462678	−	0.998929i	$-0.485267\pi$
$19$	7.66371	0.0925356	0.0462678	+	0.998929i	$0.485267\pi$
$20$	76.3546	0.853671
$21$	0	0
$22$	155.472	1.50667
$23$	75.5895	0.685282	0.342641	−	0.939466i	$-0.388678\pi$
$23$	75.5895	0.685282	0.342641	+	0.939466i	$0.388678\pi$
$24$	103.283	0.878438
$25$	281.188	2.24950
$26$	167.888	1.26636
$27$	−21.2934	−0.151774
$28$	0	0
$29$	122.016	0.781306	0.390653	−	0.920538i	$-0.372249\pi$
$29$	122.016	0.781306	0.390653	+	0.920538i	$0.372249\pi$
$30$	−494.264	−3.00799
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	158.844	0.877495
$33$	−323.435	−1.70614
$34$	444.451	2.24184
$35$	0	0
$36$	90.9965	0.421280
$37$	38.0857	0.169223	0.0846115	−	0.996414i	$-0.473035\pi$
$37$	38.0857	0.169223	0.0846115	+	0.996414i	$0.473035\pi$
$38$	−26.3129	−0.112329
$39$	−349.264	−1.43403
$40$	291.424	1.15196
$41$	−459.195	−1.74913	−0.874564	−	0.484909i	$-0.838853\pi$
$41$	−459.195	−1.74913	−0.874564	+	0.484909i	$0.838853\pi$
$42$	0	0
$43$	−162.854	−0.577557	−0.288779	−	0.957396i	$-0.593249\pi$
$43$	−162.854	−0.577557	−0.288779	+	0.957396i	$0.593249\pi$
$44$	−171.551	−0.587779
$45$	484.079	1.60360
$46$	−259.532	−0.831868
$47$	−463.507	−1.43850	−0.719250	−	0.694752i	$-0.755514\pi$
$47$	−463.507	−1.43850	−0.719250	+	0.694752i	$0.755514\pi$
$48$	−571.101	−1.71732
$49$	0	0
$50$	−965.442	−2.73068
$51$	−924.611	−2.53865
$52$	−185.251	−0.494033
$53$	411.623	1.06681	0.533404	−	0.845861i	$-0.320913\pi$
$53$	411.623	1.06681	0.533404	+	0.845861i	$0.320913\pi$
$54$	73.1096	0.184240
$55$	−912.608	−2.23738
$56$	0	0
$57$	54.7400	0.127202
$58$	−418.937	−0.948432
$59$	658.985	1.45411	0.727055	−	0.686579i	$-0.240889\pi$
$59$	658.985	1.45411	0.727055	+	0.686579i	$0.240889\pi$
$60$	545.382	1.17348
$61$	46.8510	0.0983386	0.0491693	−	0.998790i	$-0.484343\pi$
$61$	46.8510	0.0983386	0.0491693	+	0.998790i	$0.484343\pi$
$62$	106.437	0.218024
$63$	0	0
$64$	94.2618	0.184105
$65$	−985.490	−1.88054
$66$	1110.49	2.07110
$67$	206.607	0.376733	0.188366	−	0.982099i	$-0.439681\pi$
$67$	206.607	0.376733	0.188366	+	0.982099i	$0.439681\pi$
$68$	−490.417	−0.874585
$69$	539.917	0.942005
$70$	0	0
$71$	−860.793	−1.43883	−0.719417	−	0.694578i	$-0.755591\pi$
$71$	−860.793	−1.43883	−0.719417	+	0.694578i	$0.755591\pi$
$72$	347.309	0.568482
$73$	−1151.60	−1.84637	−0.923185	−	0.384356i	$-0.874424\pi$
$73$	−1151.60	−1.84637	−0.923185	+	0.384356i	$0.874424\pi$
$74$	−130.765	−0.205421
$75$	2008.45	3.09221
$76$	29.0343	0.0438219
$77$	0	0
$78$	1199.18	1.74077
$79$	−316.963	−0.451407	−0.225704	−	0.974196i	$-0.572468\pi$
$79$	−316.963	−0.451407	−0.225704	+	0.974196i	$0.572468\pi$
$80$	−1611.43	−2.25204
$81$	−800.603	−1.09822
$82$	1576.62	2.12328
$83$	−27.1396	−0.0358910	−0.0179455	−	0.999839i	$-0.505713\pi$
$83$	−27.1396	−0.0358910	−0.0179455	+	0.999839i	$0.505713\pi$
$84$	0	0
$85$	−2608.90	−3.32911
$86$	559.149	0.701100
$87$	871.533	1.07400
$88$	−654.762	−0.793158
$89$	−616.740	−0.734543	−0.367272	−	0.930114i	$-0.619708\pi$
$89$	−616.740	−0.734543	−0.367272	+	0.930114i	$0.619708\pi$
$90$	−1662.06	−1.94663
$91$	0	0
$92$	286.374	0.324528
$93$	−221.425	−0.246890
$94$	1591.43	1.74620
$95$	154.455	0.166808
$96$	1134.58	1.20622
$97$	−898.006	−0.939986	−0.469993	−	0.882670i	$-0.655744\pi$
$97$	−898.006	−0.939986	−0.469993	+	0.882670i	$0.655744\pi$
$98$	0	0
$99$	−1087.61	−1.10413
$100$	1065.29	1.06529
$101$	−102.450	−0.100932	−0.0504661	−	0.998726i	$-0.516071\pi$
$101$	−102.450	−0.100932	−0.0504661	+	0.998726i	$0.516071\pi$
$102$	3174.60	3.08169
$103$	781.745	0.747841	0.373921	−	0.927461i	$-0.378013\pi$
$103$	781.745	0.747841	0.373921	+	0.927461i	$0.378013\pi$
$104$	−707.052	−0.666655
$105$	0	0
$106$	−1413.29	−1.29500
$107$	636.318	0.574909	0.287454	−	0.957794i	$-0.407191\pi$
$107$	636.318	0.574909	0.287454	+	0.957794i	$0.407191\pi$
$108$	−80.6708	−0.0718755
$109$	−1841.67	−1.61835	−0.809174	−	0.587569i	$-0.800085\pi$
$109$	−1841.67	−1.61835	−0.809174	+	0.587569i	$0.800085\pi$
$110$	3133.39	2.71597
$111$	272.037	0.232618
$112$	0	0
$113$	2017.48	1.67955	0.839774	−	0.542936i	$-0.182688\pi$
$113$	2017.48	1.67955	0.839774	+	0.542936i	$0.182688\pi$
$114$	−187.947	−0.154411
$115$	1523.44	1.23532
$116$	462.264	0.370001
$117$	−1174.47	−0.928032
$118$	−2262.59	−1.76515
$119$	0	0
$120$	2081.57	1.58351
$121$	719.415	0.540507
$122$	−160.860	−0.119374
$123$	−3279.92	−2.40439
$124$	−117.445	−0.0850553
$125$	3147.82	2.25240
$126$	0	0
$127$	−1811.68	−1.26583	−0.632916	−	0.774220i	$-0.718142\pi$
$127$	−1811.68	−1.26583	−0.632916	+	0.774220i	$0.718142\pi$
$128$	−1594.39	−1.10098
$129$	−1163.22	−0.793923
$130$	3383.62	2.28280
$131$	−1871.87	−1.24845	−0.624223	−	0.781246i	$-0.714584\pi$
$131$	−1871.87	−1.24845	−0.624223	+	0.781246i	$0.714584\pi$
$132$	−1225.34	−0.807974
$133$	0	0
$134$	−709.375	−0.457318
$135$	−429.148	−0.273594
$136$	−1871.79	−1.18018
$137$	1147.41	0.715549	0.357775	−	0.933808i	$-0.383536\pi$
$137$	1147.41	0.715549	0.357775	+	0.933808i	$0.383536\pi$
$138$	−1853.77	−1.14351
$139$	−1158.56	−0.706960	−0.353480	−	0.935442i	$-0.615002\pi$
$139$	−1158.56	−0.706960	−0.353480	+	0.935442i	$0.615002\pi$
$140$	0	0
$141$	−3310.72	−1.97739
$142$	2955.48	1.74661
$143$	2214.16	1.29481
$144$	−1920.44	−1.11136
$145$	2459.13	1.40841
$146$	3953.97	2.24132
$147$	0	0
$148$	144.289	0.0801386
$149$	2994.34	1.64635	0.823173	−	0.567790i	$-0.192201\pi$
$149$	2994.34	1.64635	0.823173	+	0.567790i	$0.192201\pi$
$150$	−6895.91	−3.75366
$151$	1318.63	0.710653	0.355327	−	0.934742i	$-0.384370\pi$
$151$	1318.63	0.710653	0.355327	+	0.934742i	$0.384370\pi$
$152$	110.816	0.0591339
$153$	−3109.18	−1.64289
$154$	0	0
$155$	−624.777	−0.323763
$156$	−1323.20	−0.679109
$157$	−266.197	−0.135317	−0.0676586	−	0.997709i	$-0.521553\pi$
$157$	−266.197	−0.135317	−0.0676586	+	0.997709i	$0.521553\pi$
$158$	1088.28	0.547966
$159$	2940.12	1.46646
$160$	3201.35	1.58180
$161$	0	0
$162$	2748.83	1.33314
$163$	−1658.19	−0.796807	−0.398403	−	0.917210i	$-0.630436\pi$
$163$	−1658.19	−0.796807	−0.398403	+	0.917210i	$0.630436\pi$
$164$	−1739.68	−0.828331
$165$	−6518.53	−3.07556
$166$	93.1822	0.0435683
$167$	−1905.59	−0.882989	−0.441494	−	0.897264i	$-0.645551\pi$
$167$	−1905.59	−0.882989	−0.441494	+	0.897264i	$0.645551\pi$
$168$	0	0
$169$	193.990	0.0882976
$170$	8957.50	4.04123
$171$	184.074	0.0823186
$172$	−616.978	−0.273512
$173$	−1720.26	−0.756006	−0.378003	−	0.925804i	$-0.623389\pi$
$173$	−1720.26	−0.756006	−0.378003	+	0.925804i	$0.623389\pi$
$174$	−2992.36	−1.30374
$175$	0	0
$176$	3620.50	1.55060
$177$	4706.96	1.99885
$178$	2117.54	0.891667
$179$	1163.37	0.485780	0.242890	−	0.970054i	$-0.421905\pi$
$179$	1163.37	0.485780	0.242890	+	0.970054i	$0.421905\pi$
$180$	1833.95	0.759415
$181$	1340.88	0.550647	0.275323	−	0.961352i	$-0.411215\pi$
$181$	1340.88	0.550647	0.275323	+	0.961352i	$0.411215\pi$
$182$	0	0
$183$	334.645	0.135178
$184$	1093.01	0.437923
$185$	767.583	0.305048
$186$	760.251	0.299701
$187$	5861.58	2.29220
$188$	−1756.02	−0.681227
$189$	0	0
$190$	−530.313	−0.202489
$191$	−1727.63	−0.654487	−0.327243	−	0.944940i	$-0.606120\pi$
$191$	−1727.63	−0.654487	−0.327243	+	0.944940i	$0.606120\pi$
$192$	673.288	0.253075
$193$	3142.56	1.17205	0.586027	−	0.810291i	$-0.300691\pi$
$193$	3142.56	1.17205	0.586027	+	0.810291i	$0.300691\pi$
$194$	3083.25	1.14105
$195$	−7039.11	−2.58503
$196$	0	0
$197$	592.207	0.214178	0.107089	−	0.994249i	$-0.465847\pi$
$197$	592.207	0.214178	0.107089	+	0.994249i	$0.465847\pi$
$198$	3734.25	1.34031
$199$	3234.34	1.15214	0.576072	−	0.817399i	$-0.304585\pi$
$199$	3234.34	1.15214	0.576072	+	0.817399i	$0.304585\pi$
$200$	4065.92	1.43752
$201$	1475.74	0.517866
$202$	351.756	0.122522
$203$	0	0
$204$	−3502.93	−1.20222
$205$	−9254.67	−3.15304
$206$	−2684.08	−0.907809
$207$	1815.58	0.609619
$208$	3909.63	1.30329
$209$	−347.024	−0.114853
$210$	0	0
$211$	685.823	0.223763	0.111881	−	0.993722i	$-0.464312\pi$
$211$	685.823	0.223763	0.111881	+	0.993722i	$0.464312\pi$
$212$	1559.45	0.505206
$213$	−6148.43	−1.97786
$214$	−2184.76	−0.697885
$215$	−3282.17	−1.04113
$216$	−307.898	−0.0969898
$217$	0	0
$218$	6323.27	1.96452
$219$	−8225.62	−2.53806
$220$	−3457.45	−1.05955
$221$	6329.69	1.92661
$222$	−934.023	−0.282376
$223$	−3511.63	−1.05451	−0.527257	−	0.849706i	$-0.676779\pi$
$223$	−3511.63	−1.05451	−0.527257	+	0.849706i	$0.676779\pi$
$224$	0	0
$225$	6753.81	2.00113
$226$	−6926.92	−2.03881
$227$	−1339.76	−0.391730	−0.195865	−	0.980631i	$-0.562752\pi$
$227$	−1339.76	−0.391730	−0.195865	+	0.980631i	$0.562752\pi$
$228$	207.385	0.0602385
$229$	−1381.53	−0.398664	−0.199332	−	0.979932i	$-0.563877\pi$
$229$	−1381.53	−0.398664	−0.199332	+	0.979932i	$0.563877\pi$
$230$	−5230.64	−1.49956
$231$	0	0
$232$	1764.33	0.499285
$233$	1264.99	0.355676	0.177838	−	0.984060i	$-0.443090\pi$
$233$	1264.99	0.355676	0.177838	+	0.984060i	$0.443090\pi$
$234$	4032.48	1.12654
$235$	−9341.57	−2.59309
$236$	2496.59	0.688619
$237$	−2263.99	−0.620514
$238$	0	0
$239$	1778.75	0.481412	0.240706	−	0.970598i	$-0.422621\pi$
$239$	1778.75	0.481412	0.240706	+	0.970598i	$0.422621\pi$
$240$	−11510.0	−3.09570
$241$	−4920.45	−1.31516	−0.657582	−	0.753383i	$-0.728421\pi$
$241$	−4920.45	−1.31516	−0.657582	+	0.753383i	$0.728421\pi$
$242$	−2470.07	−0.656125
$243$	−5143.59	−1.35787
$244$	177.497	0.0465700
$245$	0	0
$246$	11261.4	2.91871
$247$	−374.738	−0.0965345
$248$	−448.254	−0.114775
$249$	−193.851	−0.0493366
$250$	−10807.9	−2.73420
$251$	−5897.79	−1.48313	−0.741565	−	0.670881i	$-0.765916\pi$
$251$	−5897.79	−1.48313	−0.741565	+	0.670881i	$0.765916\pi$
$252$	0	0
$253$	−3422.81	−0.850553
$254$	6220.31	1.53660
$255$	−18634.7	−4.57627
$256$	4720.16	1.15238
$257$	6593.02	1.60024	0.800120	−	0.599841i	$-0.204769\pi$
$257$	6593.02	1.60024	0.800120	+	0.599841i	$0.204769\pi$
$258$	3993.86	0.963748
$259$	0	0
$260$	−3733.57	−0.890562
$261$	2930.70	0.695041
$262$	6426.98	1.51550
$263$	111.057	0.0260383	0.0130191	−	0.999915i	$-0.495856\pi$
$263$	111.057	0.0260383	0.0130191	+	0.999915i	$0.495856\pi$
$264$	−4676.80	−1.09029
$265$	8295.90	1.92307
$266$	0	0
$267$	−4405.22	−1.00972
$268$	782.741	0.178409
$269$	4754.87	1.07773	0.538865	−	0.842392i	$-0.318853\pi$
$269$	4754.87	1.07773	0.538865	+	0.842392i	$0.318853\pi$
$270$	1473.46	0.332118
$271$	−1084.76	−0.243154	−0.121577	−	0.992582i	$-0.538795\pi$
$271$	−1084.76	−0.243154	−0.121577	+	0.992582i	$0.538795\pi$
$272$	10350.0	2.30721
$273$	0	0
$274$	−3939.58	−0.868610
$275$	−12732.6	−2.79202
$276$	2045.50	0.446103
$277$	−111.129	−0.0241050	−0.0120525	−	0.999927i	$-0.503837\pi$
$277$	−111.129	−0.0241050	−0.0120525	+	0.999927i	$0.503837\pi$
$278$	3977.84	0.858183
$279$	−744.585	−0.159775
$280$	0	0
$281$	−6678.95	−1.41791	−0.708955	−	0.705254i	$-0.750833\pi$
$281$	−6678.95	−1.41791	−0.708955	+	0.705254i	$0.750833\pi$
$282$	11367.2	2.40037
$283$	7132.95	1.49827	0.749135	−	0.662418i	$-0.230470\pi$
$283$	7132.95	1.49827	0.749135	+	0.662418i	$0.230470\pi$
$284$	−3261.15	−0.681386
$285$	1103.23	0.229298
$286$	−7602.21	−1.57178
$287$	0	0
$288$	3815.24	0.780609
$289$	11843.6	2.41067
$290$	−8443.29	−1.70968
$291$	−6414.23	−1.29213
$292$	−4362.90	−0.874381
$293$	−1089.85	−0.217303	−0.108652	−	0.994080i	$-0.534653\pi$
$293$	−1089.85	−0.217303	−0.108652	+	0.994080i	$0.534653\pi$
$294$	0	0
$295$	13281.2	2.62123
$296$	550.712	0.108140
$297$	964.196	0.188378
$298$	−10280.9	−1.99851
$299$	−3696.16	−0.714897
$300$	7609.10	1.46437
$301$	0	0
$302$	−4527.44	−0.862666
$303$	−731.774	−0.138744
$304$	−612.754	−0.115605
$305$	944.239	0.177269
$306$	10675.2	1.99432
$307$	1800.98	0.334812	0.167406	−	0.985888i	$-0.446461\pi$
$307$	1800.98	0.334812	0.167406	+	0.985888i	$0.446461\pi$
$308$	0	0
$309$	5583.81	1.02800
$310$	2145.14	0.393018
$311$	−2437.50	−0.444431	−0.222215	−	0.974998i	$-0.571329\pi$
$311$	−2437.50	−0.444431	−0.222215	+	0.974998i	$0.571329\pi$
$312$	−5050.30	−0.916400
$313$	−393.707	−0.0710978	−0.0355489	−	0.999368i	$-0.511318\pi$
$313$	−393.707	−0.0710978	−0.0355489	+	0.999368i	$0.511318\pi$
$314$	913.971	0.164262
$315$	0	0
$316$	−1200.83	−0.213772
$317$	−10858.2	−1.92383	−0.961916	−	0.273344i	$-0.911870\pi$
$317$	−10858.2	−1.92383	−0.961916	+	0.273344i	$0.911870\pi$
$318$	−10094.8	−1.78014
$319$	−5525.09	−0.969735
$320$	1899.76	0.331875
$321$	4545.06	0.790283
$322$	0	0
$323$	−992.048	−0.170895
$324$	−3033.12	−0.520082
$325$	−13749.4	−2.34671
$326$	5693.31	0.967249
$327$	−13154.6	−2.22462
$328$	−6639.88	−1.11776
$329$	0	0
$330$	22381.0	3.73344
$331$	6793.88	1.12817	0.564086	−	0.825716i	$-0.309228\pi$
$331$	6793.88	1.12817	0.564086	+	0.825716i	$0.309228\pi$
$332$	−102.819	−0.0169968
$333$	914.776	0.150539
$334$	6542.74	1.07187
$335$	4163.98	0.679113
$336$	0	0
$337$	1194.77	0.193126	0.0965630	−	0.995327i	$-0.469215\pi$
$337$	1194.77	0.193126	0.0965630	+	0.995327i	$0.469215\pi$
$338$	−666.053	−0.107185
$339$	14410.4	2.30874
$340$	−9883.91	−1.57656
$341$	1403.73	0.222921
$342$	−632.007	−0.0999270
$343$	0	0
$344$	−2354.83	−0.369082
$345$	10881.5	1.69809
$346$	5906.42	0.917721
$347$	8195.06	1.26782	0.633910	−	0.773407i	$-0.281449\pi$
$347$	8195.06	1.26782	0.633910	+	0.773407i	$0.281449\pi$
$348$	3301.84	0.508612
$349$	−176.910	−0.0271341	−0.0135670	−	0.999908i	$-0.504319\pi$
$349$	−176.910	−0.0271341	−0.0135670	+	0.999908i	$0.504319\pi$
$350$	0	0
$351$	1041.20	0.158333
$352$	−7192.68	−1.08912
$353$	−7535.84	−1.13624	−0.568120	−	0.822946i	$-0.692329\pi$
$353$	−7535.84	−1.13624	−0.568120	+	0.822946i	$0.692329\pi$
$354$	−16161.1	−2.42642
$355$	−17348.5	−2.59370
$356$	−2336.55	−0.347856
$357$	0	0
$358$	−3994.38	−0.589691
$359$	11893.9	1.74857	0.874287	−	0.485409i	$-0.161329\pi$
$359$	11893.9	1.74857	0.874287	+	0.485409i	$0.161329\pi$
$360$	6999.69	1.02477
$361$	−6800.27	−0.991437
$362$	−4603.85	−0.668433
$363$	5138.60	0.742994
$364$	0	0
$365$	−23209.5	−3.32833
$366$	−1148.98	−0.164094
$367$	−32.1253	−0.00456928	−0.00228464	−	0.999997i	$-0.500727\pi$
$367$	−32.1253	−0.00456928	−0.00228464	+	0.999997i	$0.500727\pi$
$368$	−6043.78	−0.856124
$369$	−11029.4	−1.55600
$370$	−2635.45	−0.370299
$371$	0	0
$372$	−838.879	−0.116919
$373$	9694.12	1.34569	0.672845	−	0.739783i	$-0.265072\pi$
$373$	9694.12	1.34569	0.672845	+	0.739783i	$0.265072\pi$
$374$	−20125.4	−2.78251
$375$	22484.1	3.09619
$376$	−6702.23	−0.919258
$377$	−5966.33	−0.815070
$378$	0	0
$379$	1157.26	0.156846	0.0784229	−	0.996920i	$-0.475012\pi$
$379$	1157.26	0.156846	0.0784229	+	0.996920i	$0.475012\pi$
$380$	585.160	0.0789949
$381$	−12940.4	−1.74004
$382$	5931.73	0.794485
$383$	−9682.74	−1.29181	−0.645907	−	0.763416i	$-0.723521\pi$
$383$	−9682.74	−1.29181	−0.645907	+	0.763416i	$0.723521\pi$
$384$	−11388.3	−1.51343
$385$	0	0
$386$	−10789.8	−1.42276
$387$	−3911.56	−0.513788
$388$	−3402.13	−0.445147
$389$	−3160.32	−0.411914	−0.205957	−	0.978561i	$-0.566031\pi$
$389$	−3160.32	−0.411914	−0.205957	+	0.978561i	$0.566031\pi$
$390$	24168.4	3.13798
$391$	−9784.86	−1.26558
$392$	0	0
$393$	−13370.3	−1.71614
$394$	−2033.31	−0.259992
$395$	−6388.11	−0.813723
$396$	−4120.46	−0.522881
$397$	−12660.0	−1.60047	−0.800236	−	0.599685i	$-0.795292\pi$
$397$	−12660.0	−1.60047	−0.800236	+	0.599685i	$0.795292\pi$
$398$	−11104.9	−1.39859
$399$	0	0
$400$	−22482.4	−2.81030
$401$	7200.69	0.896721	0.448360	−	0.893853i	$-0.352008\pi$
$401$	7200.69	0.896721	0.448360	+	0.893853i	$0.352008\pi$
$402$	−5066.89	−0.628640
$403$	1515.83	0.187367
$404$	−388.136	−0.0477982
$405$	−16135.4	−1.97969
$406$	0	0
$407$	−1724.58	−0.210035
$408$	−13369.7	−1.62230
$409$	5732.94	0.693094	0.346547	−	0.938033i	$-0.387354\pi$
$409$	5732.94	0.693094	0.346547	+	0.938033i	$0.387354\pi$
$410$	31775.4	3.82750
$411$	8195.70	0.983610
$412$	2961.67	0.354154
$413$	0	0
$414$	−6233.68	−0.740021
$415$	−546.973	−0.0646985
$416$	−7767.09	−0.915416
$417$	−8275.28	−0.971804
$418$	1191.49	0.139420
$419$	−6793.44	−0.792080	−0.396040	−	0.918233i	$-0.629616\pi$
$419$	−6793.44	−0.792080	−0.396040	+	0.918233i	$0.629616\pi$
$420$	0	0
$421$	−16394.5	−1.89791	−0.948956	−	0.315409i	$-0.897858\pi$
$421$	−16394.5	−1.89791	−0.948956	+	0.315409i	$0.897858\pi$
$422$	−2354.73	−0.271627
$423$	−11132.9	−1.27967
$424$	5952.00	0.681732
$425$	−36399.0	−4.15438
$426$	21110.3	2.40093
$427$	0	0
$428$	2410.72	0.272258
$429$	15815.2	1.77987
$430$	11269.1	1.26383
$431$	7563.78	0.845324	0.422662	−	0.906287i	$-0.361096\pi$
$431$	7563.78	0.845324	0.422662	+	0.906287i	$0.361096\pi$
$432$	1702.52	0.189612
$433$	5214.40	0.578725	0.289362	−	0.957220i	$-0.406557\pi$
$433$	5214.40	0.578725	0.289362	+	0.957220i	$0.406557\pi$
$434$	0	0
$435$	17565.0	1.93603
$436$	−6977.24	−0.766397
$437$	579.296	0.0634130
$438$	28242.2	3.08097
$439$	10204.0	1.10937	0.554683	−	0.832062i	$-0.312839\pi$
$439$	10204.0	1.10937	0.554683	+	0.832062i	$0.312839\pi$
$440$	−13196.1	−1.42978
$441$	0	0
$442$	−21732.6	−2.33872
$443$	−10816.9	−1.16011	−0.580053	−	0.814578i	$-0.696968\pi$
$443$	−10816.9	−1.16011	−0.580053	+	0.814578i	$0.696968\pi$
$444$	1030.62	0.110160
$445$	−12429.8	−1.32412
$446$	12057.0	1.28008
$447$	21387.8	2.26311
$448$	0	0
$449$	−10060.4	−1.05742	−0.528708	−	0.848804i	$-0.677323\pi$
$449$	−10060.4	−1.05742	−0.528708	+	0.848804i	$0.677323\pi$
$450$	−23188.8	−2.42918
$451$	20793.1	2.17097
$452$	7643.32	0.795379
$453$	9418.65	0.976880
$454$	4599.98	0.475524
$455$	0	0
$456$	791.530	0.0812868
$457$	13568.3	1.38883	0.694417	−	0.719573i	$-0.255662\pi$
$457$	13568.3	1.38883	0.694417	+	0.719573i	$0.255662\pi$
$458$	4743.40	0.483940
$459$	2756.37	0.280297
$460$	5771.61	0.585005
$461$	2265.96	0.228929	0.114465	−	0.993427i	$-0.463485\pi$
$461$	2265.96	0.228929	0.114465	+	0.993427i	$0.463485\pi$
$462$	0	0
$463$	−15276.2	−1.53336	−0.766678	−	0.642031i	$-0.778092\pi$
$463$	−15276.2	−1.53336	−0.766678	+	0.642031i	$0.778092\pi$
$464$	−9755.86	−0.976087
$465$	−4462.63	−0.445052
$466$	−4343.28	−0.431757
$467$	9287.49	0.920287	0.460143	−	0.887845i	$-0.347798\pi$
$467$	9287.49	0.920287	0.460143	+	0.887845i	$0.347798\pi$
$468$	−4449.53	−0.439486
$469$	0	0
$470$	32073.8	3.14777
$471$	−1901.38	−0.186010
$472$	9528.79	0.929233
$473$	7374.26	0.716848
$474$	7773.28	0.753246
$475$	2154.94	0.208159
$476$	0	0
$477$	9886.74	0.949020
$478$	−6107.22	−0.584389
$479$	1181.38	0.112690	0.0563451	−	0.998411i	$-0.482055\pi$
$479$	1181.38	0.112690	0.0563451	+	0.998411i	$0.482055\pi$
$480$	22866.4	2.17439
$481$	−1862.31	−0.176536
$482$	16894.1	1.59648
$483$	0	0
$484$	2725.53	0.255967
$485$	−18098.5	−1.69445
$486$	17660.2	1.64832
$487$	−12878.9	−1.19835	−0.599176	−	0.800618i	$-0.704505\pi$
$487$	−12878.9	−1.19835	−0.599176	+	0.800618i	$0.704505\pi$
$488$	677.456	0.0628422
$489$	−11844.0	−1.09531
$490$	0	0
$491$	−4684.26	−0.430545	−0.215273	−	0.976554i	$-0.569064\pi$
$491$	−4684.26	−0.430545	−0.215273	+	0.976554i	$0.569064\pi$
$492$	−12426.1	−1.13864
$493$	−15794.7	−1.44292
$494$	1286.64	0.117184
$495$	−21919.8	−1.99035
$496$	2478.61	0.224381
$497$	0	0
$498$	665.577	0.0598900
$499$	20385.2	1.82879	0.914394	−	0.404826i	$-0.132668\pi$
$499$	20385.2	1.82879	0.914394	+	0.404826i	$0.132668\pi$
$500$	11925.6	1.06666
$501$	−13611.2	−1.21378
$502$	20249.7	1.80038
$503$	6735.74	0.597081	0.298540	−	0.954397i	$-0.403500\pi$
$503$	6735.74	0.597081	0.298540	+	0.954397i	$0.403500\pi$
$504$	0	0
$505$	−2064.79	−0.181944
$506$	11752.0	1.03249
$507$	1385.62	0.121376
$508$	−6863.63	−0.599457
$509$	9255.00	0.805934	0.402967	−	0.915214i	$-0.367979\pi$
$509$	9255.00	0.805934	0.402967	+	0.915214i	$0.367979\pi$
$510$	63981.2	5.55517
$511$	0	0
$512$	−3451.28	−0.297903
$513$	−163.186	−0.0140445
$514$	−22636.8	−1.94254
$515$	15755.4	1.34809
$516$	−4406.92	−0.375976
$517$	20988.3	1.78542
$518$	0	0
$519$	−12287.4	−1.03922
$520$	−14250.0	−1.20174
$521$	22973.1	1.93180	0.965900	−	0.258916i	$-0.0833652\pi$
$521$	22973.1	1.93180	0.965900	+	0.258916i	$0.0833652\pi$
$522$	−10062.4	−0.843714
$523$	−18656.5	−1.55983	−0.779915	−	0.625885i	$-0.784738\pi$
$523$	−18656.5	−1.55983	−0.779915	+	0.625885i	$0.784738\pi$
$524$	−7091.67	−0.591224
$525$	0	0
$526$	−381.308	−0.0316080
$527$	4012.87	0.331695
$528$	25860.3	2.13149
$529$	−6453.23	−0.530388
$530$	−28483.5	−2.33442
$531$	15828.1	1.29356
$532$	0	0
$533$	22453.6	1.82472
$534$	15125.1	1.22571
$535$	12824.4	1.03635
$536$	2987.50	0.240747
$537$	8309.68	0.667764
$538$	−16325.6	−1.30826
$539$	0	0
$540$	−1625.85	−0.129565
$541$	−10162.5	−0.807614	−0.403807	−	0.914844i	$-0.632313\pi$
$541$	−10162.5	−0.807614	−0.403807	+	0.914844i	$0.632313\pi$
$542$	3724.48	0.295166
$543$	9577.59	0.756931
$544$	−20561.9	−1.62056
$545$	−37117.2	−2.91729
$546$	0	0
$547$	−10318.7	−0.806570	−0.403285	−	0.915074i	$-0.632132\pi$
$547$	−10318.7	−0.806570	−0.403285	+	0.915074i	$0.632132\pi$
$548$	4347.03	0.338861
$549$	1125.31	0.0874808
$550$	43716.7	3.38924
$551$	935.099	0.0722986
$552$	7807.10	0.601978
$553$	0	0
$554$	381.555	0.0292613
$555$	5482.65	0.419326
$556$	−4389.24	−0.334793
$557$	−16895.5	−1.28525	−0.642624	−	0.766182i	$-0.722154\pi$
$557$	−16895.5	−1.28525	−0.642624	+	0.766182i	$0.722154\pi$
$558$	2556.49	0.193952
$559$	7963.18	0.602516
$560$	0	0
$561$	41867.8	3.15091
$562$	22931.8	1.72121
$563$	11013.2	0.824423	0.412211	−	0.911088i	$-0.364757\pi$
$563$	11013.2	0.824423	0.412211	+	0.911088i	$0.364757\pi$
$564$	−12542.8	−0.936430
$565$	40660.6	3.02761
$566$	−24490.6	−1.81876
$567$	0	0
$568$	−12446.9	−0.919472
$569$	5014.70	0.369468	0.184734	−	0.982789i	$-0.440858\pi$
$569$	5014.70	0.369468	0.184734	+	0.982789i	$0.440858\pi$
$570$	−3787.90	−0.278346
$571$	9669.39	0.708671	0.354336	−	0.935118i	$-0.384707\pi$
$571$	9669.39	0.708671	0.354336	+	0.935118i	$0.384707\pi$
$572$	8388.45	0.613180
$573$	−12340.0	−0.899673
$574$	0	0
$575$	21254.8	1.54154
$576$	2264.06	0.163778
$577$	18084.1	1.30477	0.652383	−	0.757889i	$-0.273769\pi$
$577$	18084.1	1.30477	0.652383	+	0.757889i	$0.273769\pi$
$578$	−40664.5	−2.92633
$579$	22446.5	1.61113
$580$	9316.52	0.666978
$581$	0	0
$582$	22022.9	1.56852
$583$	−18638.9	−1.32409
$584$	−16652.0	−1.17990
$585$	−23670.4	−1.67290
$586$	3741.95	0.263786
$587$	17325.8	1.21825	0.609125	−	0.793074i	$-0.291521\pi$
$587$	17325.8	1.21825	0.609125	+	0.793074i	$0.291521\pi$
$588$	0	0
$589$	−237.575	−0.0166199
$590$	−45600.4	−3.18193
$591$	4229.99	0.294414
$592$	−3045.15	−0.211411
$593$	3261.52	0.225860	0.112930	−	0.993603i	$-0.463976\pi$
$593$	3261.52	0.225860	0.112930	+	0.993603i	$0.463976\pi$
$594$	−3310.51	−0.228673
$595$	0	0
$596$	11344.2	0.779656
$597$	23102.1	1.58376
$598$	12690.5	0.867818
$599$	1032.57	0.0704332	0.0352166	−	0.999380i	$-0.488788\pi$
$599$	1032.57	0.0704332	0.0352166	+	0.999380i	$0.488788\pi$
$600$	29041.8	1.97605
$601$	17047.4	1.15703	0.578516	−	0.815671i	$-0.303632\pi$
$601$	17047.4	1.15703	0.578516	+	0.815671i	$0.303632\pi$
$602$	0	0
$603$	4962.48	0.335137
$604$	4995.68	0.336542
$605$	14499.2	0.974338
$606$	2512.51	0.168422
$607$	16096.1	1.07631	0.538154	−	0.842847i	$-0.319122\pi$
$607$	16096.1	1.07631	0.538154	+	0.842847i	$0.319122\pi$
$608$	1217.33	0.0811995
$609$	0	0
$610$	−3241.99	−0.215188
$611$	22664.5	1.50066
$612$	−11779.3	−0.778021
$613$	16465.1	1.08486	0.542431	−	0.840100i	$-0.317504\pi$
$613$	16465.1	1.08486	0.542431	+	0.840100i	$0.317504\pi$
$614$	−6183.57	−0.406431
$615$	−66103.8	−4.33425
$616$	0	0
$617$	−7750.51	−0.505711	−0.252856	−	0.967504i	$-0.581370\pi$
$617$	−7750.51	−0.505711	−0.252856	+	0.967504i	$0.581370\pi$
$618$	−19171.7	−1.24790
$619$	20355.8	1.32176	0.660878	−	0.750493i	$-0.270184\pi$
$619$	20355.8	1.32176	0.660878	+	0.750493i	$0.270184\pi$
$620$	−2366.99	−0.153324
$621$	−1609.55	−0.104008
$622$	8369.02	0.539497
$623$	0	0
$624$	27925.5	1.79153
$625$	28293.0	1.81075
$626$	1351.77	0.0863060
$627$	−2478.71	−0.157879
$628$	−1008.50	−0.0640818
$629$	−4930.10	−0.312521
$630$	0	0
$631$	23215.4	1.46465	0.732323	−	0.680958i	$-0.238436\pi$
$631$	23215.4	1.46465	0.732323	+	0.680958i	$0.238436\pi$
$632$	−4583.23	−0.288467
$633$	4898.66	0.307590
$634$	37280.9	2.33535
$635$	−36512.8	−2.28184
$636$	11138.8	0.694467
$637$	0	0
$638$	18970.1	1.17717
$639$	−20675.3	−1.27997
$640$	−32133.5	−1.98467
$641$	−14526.8	−0.895124	−0.447562	−	0.894253i	$-0.647708\pi$
$641$	−14526.8	−0.895124	−0.447562	+	0.894253i	$0.647708\pi$
$642$	−15605.2	−0.959329
$643$	28240.2	1.73201	0.866007	−	0.500031i	$-0.166678\pi$
$643$	28240.2	1.73201	0.866007	+	0.500031i	$0.166678\pi$
$644$	0	0
$645$	−23443.7	−1.43116
$646$	3406.14	0.207450
$647$	−19897.9	−1.20907	−0.604534	−	0.796579i	$-0.706641\pi$
$647$	−19897.9	−1.20907	−0.604534	+	0.796579i	$0.706641\pi$
$648$	−11576.6	−0.701806
$649$	−29839.8	−1.80480
$650$	47207.9	2.84869
$651$	0	0
$652$	−6282.12	−0.377342
$653$	6834.90	0.409602	0.204801	−	0.978804i	$-0.434345\pi$
$653$	6834.90	0.409602	0.204801	+	0.978804i	$0.434345\pi$
$654$	45165.5	2.70048
$655$	−37725.9	−2.25049
$656$	36715.1	2.18519
$657$	−27660.2	−1.64251
$658$	0	0
$659$	−25450.7	−1.50443	−0.752215	−	0.658918i	$-0.771015\pi$
$659$	−25450.7	−1.50443	−0.752215	+	0.658918i	$0.771015\pi$
$660$	−24695.7	−1.45648
$661$	5696.97	0.335229	0.167615	−	0.985853i	$-0.446394\pi$
$661$	5696.97	0.335229	0.167615	+	0.985853i	$0.446394\pi$
$662$	−23326.4	−1.36950
$663$	45211.4	2.64836
$664$	−392.433	−0.0229358
$665$	0	0
$666$	−3140.83	−0.182740
$667$	9223.16	0.535416
$668$	−7219.41	−0.418155
$669$	−25082.7	−1.44956
$670$	−14296.8	−0.824379
$671$	−2121.48	−0.122055
$672$	0	0
$673$	−1003.65	−0.0574855	−0.0287427	−	0.999587i	$-0.509150\pi$
$673$	−1003.65	−0.0574855	−0.0287427	+	0.999587i	$0.509150\pi$
$674$	−4102.19	−0.234437
$675$	−5987.43	−0.341417
$676$	734.938	0.0418149
$677$	−7108.38	−0.403541	−0.201770	−	0.979433i	$-0.564670\pi$
$677$	−7108.38	−0.403541	−0.201770	+	0.979433i	$0.564670\pi$
$678$	−49477.2	−2.80260
$679$	0	0
$680$	−37724.1	−2.12743
$681$	−9569.55	−0.538482
$682$	−4819.62	−0.270605
$683$	5410.24	0.303100	0.151550	−	0.988450i	$-0.451574\pi$
$683$	5410.24	0.303100	0.151550	+	0.988450i	$0.451574\pi$
$684$	697.371	0.0389834
$685$	23125.1	1.28988
$686$	0	0
$687$	−9867.91	−0.548012
$688$	13021.0	0.721543
$689$	−20127.5	−1.11291
$690$	−37361.1	−2.06133
$691$	12652.6	0.696565	0.348282	−	0.937390i	$-0.386765\pi$
$691$	12652.6	0.696565	0.348282	+	0.937390i	$0.386765\pi$
$692$	−6517.28	−0.358020
$693$	0	0
$694$	−28137.3	−1.53902
$695$	−23349.7	−1.27439
$696$	12602.2	0.686329
$697$	59441.7	3.23029
$698$	607.411	0.0329382
$699$	9035.52	0.488920
$700$	0	0
$701$	13078.1	0.704640	0.352320	−	0.935880i	$-0.385393\pi$
$701$	13078.1	0.704640	0.352320	+	0.935880i	$0.385393\pi$
$702$	−3574.89	−0.192202
$703$	291.878	0.0156592
$704$	−4268.32	−0.228506
$705$	−66724.5	−3.56452
$706$	25873.9	1.37929
$707$	0	0
$708$	17832.5	0.946592
$709$	−1757.47	−0.0930935	−0.0465467	−	0.998916i	$-0.514822\pi$
$709$	−1757.47	−0.0930935	−0.0465467	+	0.998916i	$0.514822\pi$
$710$	59565.1	3.14850
$711$	−7613.10	−0.401566
$712$	−8917.95	−0.469402
$713$	−2343.27	−0.123080
$714$	0	0
$715$	44624.5	2.33407
$716$	4407.49	0.230050
$717$	12705.1	0.661760
$718$	−40837.2	−2.12260
$719$	−20874.1	−1.08271	−0.541357	−	0.840793i	$-0.682089\pi$
$719$	−20874.1	−1.08271	−0.541357	+	0.840793i	$0.682089\pi$
$720$	−38704.7	−2.00339
$721$	0	0
$722$	23348.3	1.20351
$723$	−35145.6	−1.80785
$724$	5079.99	0.260768
$725$	34309.5	1.75755
$726$	−17643.1	−0.901924
$727$	424.484	0.0216551	0.0108275	−	0.999941i	$-0.496553\pi$
$727$	424.484	0.0216551	0.0108275	+	0.999941i	$0.496553\pi$
$728$	0	0
$729$	−15123.1	−0.768332
$730$	79688.6	4.04029
$731$	21081.0	1.06663
$732$	1267.82	0.0640161
$733$	−20852.7	−1.05077	−0.525384	−	0.850865i	$-0.676078\pi$
$733$	−20852.7	−1.05077	−0.525384	+	0.850865i	$0.676078\pi$
$734$	110.300	0.00554667
$735$	0	0
$736$	12006.9	0.601332
$737$	−9355.49	−0.467590
$738$	37868.7	1.88884
$739$	−3608.77	−0.179636	−0.0898178	−	0.995958i	$-0.528628\pi$
$739$	−3608.77	−0.179636	−0.0898178	+	0.995958i	$0.528628\pi$
$740$	2908.02	0.144461
$741$	−2676.66	−0.132699
$742$	0	0
$743$	23195.7	1.14531	0.572656	−	0.819796i	$-0.305913\pi$
$743$	23195.7	1.14531	0.572656	+	0.819796i	$0.305913\pi$
$744$	−3201.77	−0.157772
$745$	60348.2	2.96777
$746$	−33284.2	−1.63354
$747$	−651.862	−0.0319282
$748$	22206.8	1.08551
$749$	0	0
$750$	−77197.8	−3.75849
$751$	14566.2	0.707761	0.353881	−	0.935291i	$-0.384862\pi$
$751$	14566.2	0.707761	0.353881	+	0.935291i	$0.384862\pi$
$752$	37059.8	1.79712
$753$	−42126.5	−2.03874
$754$	20485.1	0.989419
$755$	26575.8	1.28105
$756$	0	0
$757$	−15259.8	−0.732664	−0.366332	−	0.930484i	$-0.619387\pi$
$757$	−15259.8	−0.732664	−0.366332	+	0.930484i	$0.619387\pi$
$758$	−3973.40	−0.190396
$759$	−24448.2	−1.16919
$760$	2233.39	0.106597
$761$	−22098.8	−1.05267	−0.526334	−	0.850278i	$-0.676434\pi$
$761$	−22098.8	−1.05267	−0.526334	+	0.850278i	$0.676434\pi$
$762$	44430.1	2.11225
$763$	0	0
$764$	−6545.20	−0.309944
$765$	−62662.8	−2.96154
$766$	33245.1	1.56814
$767$	−32222.9	−1.51695
$768$	33714.9	1.58409
$769$	22359.3	1.04850	0.524251	−	0.851564i	$-0.324345\pi$
$769$	22359.3	1.04850	0.524251	+	0.851564i	$0.324345\pi$
$770$	0	0
$771$	47092.3	2.19973
$772$	11905.7	0.555047
$773$	−4262.45	−0.198331	−0.0991654	−	0.995071i	$-0.531617\pi$
$773$	−4262.45	−0.198331	−0.0991654	+	0.995071i	$0.531617\pi$
$774$	13430.1	0.623690
$775$	−8716.81	−0.404022
$776$	−12985.0	−0.600688
$777$	0	0
$778$	10850.8	0.500025
$779$	−3519.14	−0.161857
$780$	−26668.0	−1.22419
$781$	38978.0	1.78584
$782$	33595.8	1.53630
$783$	−2598.14	−0.118582
$784$	0	0
$785$	−5364.95	−0.243928
$786$	45906.3	2.08324
$787$	−30196.6	−1.36772	−0.683858	−	0.729615i	$-0.739699\pi$
$787$	−30196.6	−1.36772	−0.683858	+	0.729615i	$0.739699\pi$
$788$	2243.60	0.101428
$789$	793.253	0.0357928
$790$	21933.2	0.987783
$791$	0	0
$792$	−15726.7	−0.705584
$793$	−2290.91	−0.102588
$794$	43467.4	1.94282
$795$	59255.5	2.64349
$796$	12253.4	0.545618
$797$	−15172.7	−0.674335	−0.337167	−	0.941445i	$-0.609469\pi$
$797$	−15172.7	−0.674335	−0.337167	+	0.941445i	$0.609469\pi$
$798$	0	0
$799$	59999.8	2.65662
$800$	44664.8	1.97393
$801$	−14813.4	−0.653441
$802$	−24723.2	−1.08853
$803$	52146.3	2.29166
$804$	5590.92	0.245244
$805$	0	0
$806$	−5204.52	−0.227446
$807$	33962.9	1.48147
$808$	−1481.41	−0.0644996
$809$	22827.9	0.992071	0.496036	−	0.868302i	$-0.334789\pi$
$809$	22827.9	0.992071	0.496036	+	0.868302i	$0.334789\pi$
$810$	55400.1	2.40316
$811$	−2583.24	−0.111849	−0.0559247	−	0.998435i	$-0.517811\pi$
$811$	−2583.24	−0.111849	−0.0559247	+	0.998435i	$0.517811\pi$
$812$	0	0
$813$	−7748.21	−0.334245
$814$	5921.25	0.254963
$815$	−33419.3	−1.43635
$816$	73927.5	3.17154
$817$	−1248.06	−0.0534446
$818$	−19683.7	−0.841351
$819$	0	0
$820$	−35061.7	−1.49318
$821$	−24183.3	−1.02802	−0.514009	−	0.857785i	$-0.671840\pi$
$821$	−24183.3	−1.02802	−0.514009	+	0.857785i	$0.671840\pi$
$822$	−28139.5	−1.19401
$823$	−44461.0	−1.88313	−0.941563	−	0.336838i	$-0.890643\pi$
$823$	−44461.0	−1.88313	−0.941563	+	0.336838i	$0.890643\pi$
$824$	11303.9	0.477900
$825$	−90945.8	−3.83797
$826$	0	0
$827$	−5900.10	−0.248085	−0.124043	−	0.992277i	$-0.539586\pi$
$827$	−5900.10	−0.248085	−0.124043	+	0.992277i	$0.539586\pi$
$828$	6878.38	0.288696
$829$	−16755.6	−0.701984	−0.350992	−	0.936378i	$-0.614156\pi$
$829$	−16755.6	−0.701984	−0.350992	+	0.936378i	$0.614156\pi$
$830$	1878.00	0.0785378
$831$	−793.767	−0.0331354
$832$	−4609.19	−0.192061
$833$	0	0
$834$	28412.7	1.17968
$835$	−38405.5	−1.59171
$836$	−1314.72	−0.0543904
$837$	660.094	0.0272595
$838$	23324.9	0.961510
$839$	−24739.5	−1.01800	−0.509000	−	0.860766i	$-0.669985\pi$
$839$	−24739.5	−1.01800	−0.509000	+	0.860766i	$0.669985\pi$
$840$	0	0
$841$	−9500.99	−0.389561
$842$	56289.7	2.30389
$843$	−47706.0	−1.94909
$844$	2598.27	0.105967
$845$	3909.69	0.159169
$846$	38224.3	1.55340
$847$	0	0
$848$	−32911.5	−1.33276
$849$	50948.9	2.05956
$850$	124974.	5.04303
$851$	2878.88	0.115966
$852$	−23293.6	−0.936648
$853$	−14922.0	−0.598969	−0.299485	−	0.954101i	$-0.596815\pi$
$853$	−14922.0	−0.598969	−0.299485	+	0.954101i	$0.596815\pi$
$854$	0	0
$855$	3709.84	0.148391
$856$	9201.04	0.367389
$857$	−42423.2	−1.69096	−0.845478	−	0.534010i	$-0.820685\pi$
$857$	−42423.2	−1.69096	−0.845478	+	0.534010i	$0.820685\pi$
$858$	−54300.7	−2.16060
$859$	−21763.6	−0.864450	−0.432225	−	0.901766i	$-0.642271\pi$
$859$	−21763.6	−0.864450	−0.432225	+	0.901766i	$0.642271\pi$
$860$	−12434.6	−0.493044
$861$	0	0
$862$	−25969.8	−1.02614
$863$	−16069.6	−0.633853	−0.316926	−	0.948450i	$-0.602651\pi$
$863$	−16069.6	−0.633853	−0.316926	+	0.948450i	$0.602651\pi$
$864$	−3382.31	−0.133181
$865$	−34670.3	−1.36281
$866$	−17903.3	−0.702518
$867$	84596.1	3.31377
$868$	0	0
$869$	14352.6	0.560274
$870$	−60308.3	−2.35016
$871$	−10102.6	−0.393013
$872$	−26630.2	−1.03419
$873$	−21569.1	−0.836201
$874$	−1988.98	−0.0769774
$875$	0	0
$876$	−31163.1	−1.20194
$877$	18778.4	0.723034	0.361517	−	0.932365i	$-0.382259\pi$
$877$	18778.4	0.723034	0.361517	+	0.932365i	$0.382259\pi$
$878$	−35035.0	−1.34667
$879$	−7784.55	−0.298710
$880$	72967.8	2.79516
$881$	28981.8	1.10831	0.554155	−	0.832414i	$-0.313042\pi$
$881$	28981.8	1.10831	0.554155	+	0.832414i	$0.313042\pi$
$882$	0	0
$883$	30172.4	1.14992	0.574962	−	0.818180i	$-0.305017\pi$
$883$	30172.4	1.14992	0.574962	+	0.818180i	$0.305017\pi$
$884$	23980.3	0.912380
$885$	94864.6	3.60321
$886$	37139.3	1.40826
$887$	8572.71	0.324514	0.162257	−	0.986749i	$-0.448123\pi$
$887$	8572.71	0.324514	0.162257	+	0.986749i	$0.448123\pi$
$888$	3933.60	0.148652
$889$	0	0
$890$	42677.2	1.60735
$891$	36252.5	1.36308
$892$	−13304.0	−0.499383
$893$	−3552.19	−0.133112
$894$	−73433.8	−2.74720
$895$	23446.7	0.875685
$896$	0	0
$897$	−26400.7	−0.982714
$898$	34541.8	1.28360
$899$	−3782.51	−0.140327
$900$	25587.1	0.947670
$901$	−53283.6	−1.97018
$902$	−71391.8	−2.63535
$903$	0	0
$904$	29172.4	1.07330
$905$	27024.3	0.992616
$906$	−32338.4	−1.18584
$907$	2936.66	0.107508	0.0537542	−	0.998554i	$-0.482881\pi$
$907$	2936.66	0.107508	0.0537542	+	0.998554i	$0.482881\pi$
$908$	−5075.72	−0.185511
$909$	−2460.73	−0.0897881
$910$	0	0
$911$	−7759.84	−0.282212	−0.141106	−	0.989995i	$-0.545066\pi$
$911$	−7759.84	−0.282212	−0.141106	+	0.989995i	$0.545066\pi$
$912$	−4376.75	−0.158913
$913$	1228.92	0.0445469
$914$	−46585.9	−1.68591
$915$	6744.46	0.243678
$916$	−5233.98	−0.188794
$917$	0	0
$918$	−9463.85	−0.340254
$919$	−6009.88	−0.215721	−0.107861	−	0.994166i	$-0.534400\pi$
$919$	−6009.88	−0.215721	−0.107861	+	0.994166i	$0.534400\pi$
$920$	22028.6	0.789415
$921$	12864.0	0.460241
$922$	−7780.05	−0.277898
$923$	42090.8	1.50101
$924$	0	0
$925$	10709.2	0.380667
$926$	52449.9	1.86135
$927$	18776.7	0.665271
$928$	19381.5	0.685592
$929$	3369.07	0.118983	0.0594917	−	0.998229i	$-0.481052\pi$
$929$	3369.07	0.118983	0.0594917	+	0.998229i	$0.481052\pi$
$930$	15322.2	0.540252
$931$	0	0
$932$	4792.47	0.168436
$933$	−17410.5	−0.610925
$934$	−31888.1	−1.11714
$935$	118135.	4.13200
$936$	−16982.6	−0.593049
$937$	−393.560	−0.0137215	−0.00686076	−	0.999976i	$-0.502184\pi$
$937$	−393.560	−0.0137215	−0.00686076	+	0.999976i	$0.502184\pi$
$938$	0	0
$939$	−2812.15	−0.0977327
$940$	−35390.9	−1.22800
$941$	−13063.5	−0.452560	−0.226280	−	0.974062i	$-0.572656\pi$
$941$	−13063.5	−0.452560	−0.226280	+	0.974062i	$0.572656\pi$
$942$	6528.27	0.225799
$943$	−34710.3	−1.19865
$944$	−52689.3	−1.81662
$945$	0	0
$946$	−25319.1	−0.870186
$947$	8946.95	0.307008	0.153504	−	0.988148i	$-0.450944\pi$
$947$	8946.95	0.307008	0.153504	+	0.988148i	$0.450944\pi$
$948$	−8577.22	−0.293856
$949$	56310.8	1.92616
$950$	−7398.87	−0.252685
$951$	−77557.1	−2.64454
$952$	0	0
$953$	−53087.3	−1.80448	−0.902238	−	0.431239i	$-0.858076\pi$
$953$	−53087.3	−1.80448	−0.902238	+	0.431239i	$0.858076\pi$
$954$	−33945.6	−1.15202
$955$	−34818.8	−1.17980
$956$	6738.85	0.227981
$957$	−39464.3	−1.33302
$958$	−4056.20	−0.136795
$959$	0	0
$960$	13569.5	0.456202
$961$	961.000	0.0322581
$962$	6394.12	0.214298
$963$	15283.7	0.511432
$964$	−18641.3	−0.622819
$965$	63335.5	2.11279
$966$	0	0
$967$	500.465	0.0166431	0.00832154	−	0.999965i	$-0.497351\pi$
$967$	500.465	0.0166431	0.00832154	+	0.999965i	$0.497351\pi$
$968$	10402.6	0.345405
$969$	−7085.95	−0.234916
$970$	62140.2	2.05691
$971$	−52229.7	−1.72619	−0.863095	−	0.505041i	$-0.831477\pi$
$971$	−52229.7	−1.72619	−0.863095	+	0.505041i	$0.831477\pi$
$972$	−19486.7	−0.643041
$973$	0	0
$974$	44218.9	1.45469
$975$	−98208.8	−3.22584
$976$	−3745.98	−0.122854
$977$	12340.5	0.404103	0.202051	−	0.979375i	$-0.435239\pi$
$977$	12340.5	0.404103	0.202051	+	0.979375i	$0.435239\pi$
$978$	40665.9	1.32960
$979$	27926.9	0.911694
$980$	0	0
$981$	−44234.9	−1.43966
$982$	16083.2	0.522641
$983$	−5979.47	−0.194014	−0.0970068	−	0.995284i	$-0.530927\pi$
$983$	−5979.47	−0.194014	−0.0970068	+	0.995284i	$0.530927\pi$
$984$	−47427.0	−1.53650
$985$	11935.4	0.386085
$986$	54230.3	1.75157
$987$	0	0
$988$	−1419.71	−0.0457156
$989$	−12310.0	−0.395790
$990$	75260.5	2.41610
$991$	46918.4	1.50395	0.751974	−	0.659193i	$-0.229102\pi$
$991$	46918.4	1.50395	0.751974	+	0.659193i	$0.229102\pi$
$992$	−4924.15	−0.157603
$993$	48527.0	1.55081
$994$	0	0
$995$	65185.3	2.07690
$996$	−734.413	−0.0233642
$997$	−28526.2	−0.906153	−0.453076	−	0.891472i	$-0.649674\pi$
$997$	−28526.2	−0.906153	−0.453076	+	0.891472i	$0.649674\pi$
$998$	−69991.3	−2.21998
$999$	−810.973	−0.0256837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.5	✓	23
7.6	odd	2		1519.4.a.i.1.5	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.5	✓	23	1.1	even	1	trivial
1519.4.a.i.1.5	yes	23	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:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1519.a (trivial)

\(f(q)\)	\(=\)	\(q-3.43344 q^{2} +7.14275 q^{3} +3.78854 q^{4} +20.1541 q^{5} -24.5242 q^{6} +14.4598 q^{8} +24.0189 q^{9} -69.1980 q^{10} -45.2815 q^{11} +27.0606 q^{12} -48.8977 q^{13} +143.956 q^{15} -79.9553 q^{16} -129.447 q^{17} -82.4675 q^{18} +7.66371 q^{19} +76.3546 q^{20} +155.472 q^{22} +75.5895 q^{23} +103.283 q^{24} +281.188 q^{25} +167.888 q^{26} -21.2934 q^{27} +122.016 q^{29} -494.264 q^{30} -31.0000 q^{31} +158.844 q^{32} -323.435 q^{33} +444.451 q^{34} +90.9965 q^{36} +38.0857 q^{37} -26.3129 q^{38} -349.264 q^{39} +291.424 q^{40} -459.195 q^{41} -162.854 q^{43} -171.551 q^{44} +484.079 q^{45} -259.532 q^{46} -463.507 q^{47} -571.101 q^{48} -965.442 q^{50} -924.611 q^{51} -185.251 q^{52} +411.623 q^{53} +73.1096 q^{54} -912.608 q^{55} +54.7400 q^{57} -418.937 q^{58} +658.985 q^{59} +545.382 q^{60} +46.8510 q^{61} +106.437 q^{62} +94.2618 q^{64} -985.490 q^{65} +1110.49 q^{66} +206.607 q^{67} -490.417 q^{68} +539.917 q^{69} -860.793 q^{71} +347.309 q^{72} -1151.60 q^{73} -130.765 q^{74} +2008.45 q^{75} +29.0343 q^{76} +1199.18 q^{78} -316.963 q^{79} -1611.43 q^{80} -800.603 q^{81} +1576.62 q^{82} -27.1396 q^{83} -2608.90 q^{85} +559.149 q^{86} +871.533 q^{87} -654.762 q^{88} -616.740 q^{89} -1662.06 q^{90} +286.374 q^{92} -221.425 q^{93} +1591.43 q^{94} +154.455 q^{95} +1134.58 q^{96} -898.006 q^{97} -1087.61 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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