LMFDB - Embedded newform 2070.4.a.bi.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2070,4,Mod(1,2070)] 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("2070.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2070.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-8,0,16,20,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$122.133953712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.92791$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5524 q^{7} -8.00000 q^{8} -10.0000 q^{10} -1.63537 q^{11} -88.6599 q^{13} -47.1048 q^{14} +16.0000 q^{16} +7.31435 q^{17} -6.53738 q^{19} +20.0000 q^{20} +3.27074 q^{22} -23.0000 q^{23} +25.0000 q^{25} +177.320 q^{26} +94.2096 q^{28} -119.240 q^{29} -156.456 q^{31} -32.0000 q^{32} -14.6287 q^{34} +117.762 q^{35} -293.173 q^{37} +13.0748 q^{38} -40.0000 q^{40} -74.3404 q^{41} +468.081 q^{43} -6.54148 q^{44} +46.0000 q^{46} +393.971 q^{47} +211.715 q^{49} -50.0000 q^{50} -354.639 q^{52} -233.171 q^{53} -8.17685 q^{55} -188.419 q^{56} +238.481 q^{58} +766.648 q^{59} +178.365 q^{61} +312.912 q^{62} +64.0000 q^{64} -443.299 q^{65} +246.904 q^{67} +29.2574 q^{68} -235.524 q^{70} +650.678 q^{71} +695.444 q^{73} +586.346 q^{74} -26.1495 q^{76} -38.5169 q^{77} +660.717 q^{79} +80.0000 q^{80} +148.681 q^{82} +1328.39 q^{83} +36.5718 q^{85} -936.163 q^{86} +13.0830 q^{88} +824.702 q^{89} -2088.15 q^{91} -92.0000 q^{92} -787.942 q^{94} -32.6869 q^{95} +383.833 q^{97} -423.430 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} + 20 q^{5} + 26 q^{7} - 32 q^{8} - 40 q^{10} - 93 q^{11} + 32 q^{13} - 52 q^{14} + 64 q^{16} - 108 q^{17} + 185 q^{19} + 80 q^{20} + 186 q^{22} - 92 q^{23} + 100 q^{25} - 64 q^{26}+ \cdots - 1200 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 + 20 * q^5 + 26 * q^7 - 32 * q^8 - 40 * q^10 - 93 * q^11 + 32 * q^13 - 52 * q^14 + 64 * q^16 - 108 * q^17 + 185 * q^19 + 80 * q^20 + 186 * q^22 - 92 * q^23 + 100 * q^25 - 64 * q^26 + 104 * q^28 - 294 * q^29 - 211 * q^31 - 128 * q^32 + 216 * q^34 + 130 * q^35 + 5 * q^37 - 370 * q^38 - 160 * q^40 + 369 * q^41 - 100 * q^43 - 372 * q^44 + 184 * q^46 + 363 * q^47 + 600 * q^49 - 200 * q^50 + 128 * q^52 - 21 * q^53 - 465 * q^55 - 208 * q^56 + 588 * q^58 + 33 * q^59 - 307 * q^61 + 422 * q^62 + 256 * q^64 + 160 * q^65 + 725 * q^67 - 432 * q^68 - 260 * q^70 + 1257 * q^71 + 509 * q^73 - 10 * q^74 + 740 * q^76 + 1962 * q^77 + 1202 * q^79 + 320 * q^80 - 738 * q^82 + 1377 * q^83 - 540 * q^85 + 200 * q^86 + 744 * q^88 + 984 * q^89 - 995 * q^91 - 368 * q^92 - 726 * q^94 + 925 * q^95 + 137 * q^97 - 1200 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	23.5524	1.27171	0.635855	−	0.771809i	$-0.280648\pi$
$7$	23.5524	1.27171	0.635855	+	0.771809i	$0.280648\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	−1.63537	−0.0448257	−0.0224129	−	0.999749i	$-0.507135\pi$
$11$	−1.63537	−0.0448257	−0.0224129	+	0.999749i	$0.507135\pi$
$12$	0	0
$13$	−88.6599	−1.89152	−0.945762	−	0.324860i	$-0.894683\pi$
$13$	−88.6599	−1.89152	−0.945762	+	0.324860i	$0.894683\pi$
$14$	−47.1048	−0.899234
$15$	0	0
$16$	16.0000	0.250000
$17$	7.31435	0.104352	0.0521762	−	0.998638i	$-0.483384\pi$
$17$	7.31435	0.104352	0.0521762	+	0.998638i	$0.483384\pi$
$18$	0	0
$19$	−6.53738	−0.0789357	−0.0394678	−	0.999221i	$-0.512566\pi$
$19$	−6.53738	−0.0789357	−0.0394678	+	0.999221i	$0.512566\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	3.27074	0.0316966
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	177.320	1.33751
$27$	0	0
$28$	94.2096	0.635855
$29$	−119.240	−0.763530	−0.381765	−	0.924259i	$-0.624684\pi$
$29$	−119.240	−0.763530	−0.381765	+	0.924259i	$0.624684\pi$
$30$	0	0
$31$	−156.456	−0.906462	−0.453231	−	0.891393i	$-0.649729\pi$
$31$	−156.456	−0.906462	−0.453231	+	0.891393i	$0.649729\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−14.6287	−0.0737883
$35$	117.762	0.568726
$36$	0	0
$37$	−293.173	−1.30263	−0.651315	−	0.758807i	$-0.725782\pi$
$37$	−293.173	−1.30263	−0.651315	+	0.758807i	$0.725782\pi$
$38$	13.0748	0.0558159
$39$	0	0
$40$	−40.0000	−0.158114
$41$	−74.3404	−0.283171	−0.141586	−	0.989926i	$-0.545220\pi$
$41$	−74.3404	−0.283171	−0.141586	+	0.989926i	$0.545220\pi$
$42$	0	0
$43$	468.081	1.66004	0.830020	−	0.557733i	$-0.188329\pi$
$43$	468.081	1.66004	0.830020	+	0.557733i	$0.188329\pi$
$44$	−6.54148	−0.0224129
$45$	0	0
$46$	46.0000	0.147442
$47$	393.971	1.22269	0.611346	−	0.791363i	$-0.290628\pi$
$47$	393.971	1.22269	0.611346	+	0.791363i	$0.290628\pi$
$48$	0	0
$49$	211.715	0.617245
$50$	−50.0000	−0.141421
$51$	0	0
$52$	−354.639	−0.945762
$53$	−233.171	−0.604311	−0.302155	−	0.953259i	$-0.597706\pi$
$53$	−233.171	−0.604311	−0.302155	+	0.953259i	$0.597706\pi$
$54$	0	0
$55$	−8.17685	−0.0200467
$56$	−188.419	−0.449617
$57$	0	0
$58$	238.481	0.539897
$59$	766.648	1.69168	0.845840	−	0.533437i	$-0.179100\pi$
$59$	766.648	1.69168	0.845840	+	0.533437i	$0.179100\pi$
$60$	0	0
$61$	178.365	0.374383	0.187191	−	0.982323i	$-0.440062\pi$
$61$	178.365	0.374383	0.187191	+	0.982323i	$0.440062\pi$
$62$	312.912	0.640965
$63$	0	0
$64$	64.0000	0.125000
$65$	−443.299	−0.845915
$66$	0	0
$67$	246.904	0.450211	0.225105	−	0.974334i	$-0.427727\pi$
$67$	246.904	0.450211	0.225105	+	0.974334i	$0.427727\pi$
$68$	29.2574	0.0521762
$69$	0	0
$70$	−235.524	−0.402150
$71$	650.678	1.08762	0.543812	−	0.839207i	$-0.316980\pi$
$71$	650.678	1.08762	0.543812	+	0.839207i	$0.316980\pi$
$72$	0	0
$73$	695.444	1.11501	0.557504	−	0.830174i	$-0.311759\pi$
$73$	695.444	1.11501	0.557504	+	0.830174i	$0.311759\pi$
$74$	586.346	0.921099
$75$	0	0
$76$	−26.1495	−0.0394678
$77$	−38.5169	−0.0570053
$78$	0	0
$79$	660.717	0.940967	0.470484	−	0.882409i	$-0.344079\pi$
$79$	660.717	0.940967	0.470484	+	0.882409i	$0.344079\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	148.681	0.200232
$83$	1328.39	1.75674	0.878372	−	0.477977i	$-0.158630\pi$
$83$	1328.39	1.75674	0.878372	+	0.477977i	$0.158630\pi$
$84$	0	0
$85$	36.5718	0.0466678
$86$	−936.163	−1.17383
$87$	0	0
$88$	13.0830	0.0158483
$89$	824.702	0.982227	0.491113	−	0.871096i	$-0.336590\pi$
$89$	824.702	0.982227	0.491113	+	0.871096i	$0.336590\pi$
$90$	0	0
$91$	−2088.15	−2.40547
$92$	−92.0000	−0.104257
$93$	0	0
$94$	−787.942	−0.864574
$95$	−32.6869	−0.0353011
$96$	0	0
$97$	383.833	0.401776	0.200888	−	0.979614i	$-0.435617\pi$
$97$	383.833	0.401776	0.200888	+	0.979614i	$0.435617\pi$
$98$	−423.430	−0.436458
$99$	0	0
$100$	100.000	0.100000
$101$	674.579	0.664586	0.332293	−	0.943176i	$-0.392178\pi$
$101$	674.579	0.664586	0.332293	+	0.943176i	$0.392178\pi$
$102$	0	0
$103$	169.297	0.161955	0.0809773	−	0.996716i	$-0.474196\pi$
$103$	169.297	0.161955	0.0809773	+	0.996716i	$0.474196\pi$
$104$	709.279	0.668755
$105$	0	0
$106$	466.342	0.427312
$107$	1533.31	1.38533	0.692665	−	0.721259i	$-0.256436\pi$
$107$	1533.31	1.38533	0.692665	+	0.721259i	$0.256436\pi$
$108$	0	0
$109$	1239.75	1.08941	0.544707	−	0.838626i	$-0.316641\pi$
$109$	1239.75	1.08941	0.544707	+	0.838626i	$0.316641\pi$
$110$	16.3537	0.0141751
$111$	0	0
$112$	376.838	0.317927
$113$	−2299.49	−1.91432	−0.957159	−	0.289561i	$-0.906491\pi$
$113$	−2299.49	−1.91432	−0.957159	+	0.289561i	$0.906491\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	−476.961	−0.381765
$117$	0	0
$118$	−1533.30	−1.19620
$119$	172.270	0.132706
$120$	0	0
$121$	−1328.33	−0.997991
$122$	−356.731	−0.264729
$123$	0	0
$124$	−625.824	−0.453231
$125$	125.000	0.0894427
$126$	0	0
$127$	−1177.82	−0.822948	−0.411474	−	0.911422i	$-0.634986\pi$
$127$	−1177.82	−0.822948	−0.411474	+	0.911422i	$0.634986\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	886.599	0.598153
$131$	−2385.56	−1.59105	−0.795525	−	0.605921i	$-0.792805\pi$
$131$	−2385.56	−1.59105	−0.795525	+	0.605921i	$0.792805\pi$
$132$	0	0
$133$	−153.971	−0.100383
$134$	−493.808	−0.318347
$135$	0	0
$136$	−58.5148	−0.0368941
$137$	−1123.22	−0.700460	−0.350230	−	0.936664i	$-0.613897\pi$
$137$	−1123.22	−0.700460	−0.350230	+	0.936664i	$0.613897\pi$
$138$	0	0
$139$	−1801.75	−1.09944	−0.549721	−	0.835348i	$-0.685266\pi$
$139$	−1801.75	−1.09944	−0.549721	+	0.835348i	$0.685266\pi$
$140$	471.048	0.284363
$141$	0	0
$142$	−1301.36	−0.769067
$143$	144.992	0.0847889
$144$	0	0
$145$	−596.202	−0.341461
$146$	−1390.89	−0.788429
$147$	0	0
$148$	−1172.69	−0.651315
$149$	−663.138	−0.364607	−0.182303	−	0.983242i	$-0.558355\pi$
$149$	−663.138	−0.364607	−0.182303	+	0.983242i	$0.558355\pi$
$150$	0	0
$151$	1617.66	0.871808	0.435904	−	0.899993i	$-0.356429\pi$
$151$	1617.66	0.871808	0.435904	+	0.899993i	$0.356429\pi$
$152$	52.2990	0.0279080
$153$	0	0
$154$	77.0338	0.0403088
$155$	−782.280	−0.405382
$156$	0	0
$157$	875.080	0.444834	0.222417	−	0.974952i	$-0.428605\pi$
$157$	875.080	0.444834	0.222417	+	0.974952i	$0.428605\pi$
$158$	−1321.43	−0.665364
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−541.705	−0.265170
$162$	0	0
$163$	2532.39	1.21689	0.608443	−	0.793597i	$-0.291794\pi$
$163$	2532.39	1.21689	0.608443	+	0.793597i	$0.291794\pi$
$164$	−297.361	−0.141586
$165$	0	0
$166$	−2656.78	−1.24221
$167$	1131.65	0.524371	0.262186	−	0.965017i	$-0.415557\pi$
$167$	1131.65	0.524371	0.262186	+	0.965017i	$0.415557\pi$
$168$	0	0
$169$	5663.57	2.57786
$170$	−73.1435	−0.0329991
$171$	0	0
$172$	1872.33	0.830020
$173$	3186.23	1.40026	0.700128	−	0.714017i	$-0.253126\pi$
$173$	3186.23	1.40026	0.700128	+	0.714017i	$0.253126\pi$
$174$	0	0
$175$	588.810	0.254342
$176$	−26.1659	−0.0112064
$177$	0	0
$178$	−1649.40	−0.694539
$179$	−1663.48	−0.694606	−0.347303	−	0.937753i	$-0.612902\pi$
$179$	−1663.48	−0.694606	−0.347303	+	0.937753i	$0.612902\pi$
$180$	0	0
$181$	−3747.35	−1.53889	−0.769443	−	0.638716i	$-0.779466\pi$
$181$	−3747.35	−1.53889	−0.769443	+	0.638716i	$0.779466\pi$
$182$	4176.30	1.70092
$183$	0	0
$184$	184.000	0.0737210
$185$	−1465.86	−0.582554
$186$	0	0
$187$	−11.9617	−0.00467767
$188$	1575.88	0.611346
$189$	0	0
$190$	65.3738	0.0249616
$191$	−2262.97	−0.857291	−0.428645	−	0.903473i	$-0.641009\pi$
$191$	−2262.97	−0.857291	−0.428645	+	0.903473i	$0.641009\pi$
$192$	0	0
$193$	−2001.25	−0.746390	−0.373195	−	0.927753i	$-0.621738\pi$
$193$	−2001.25	−0.746390	−0.373195	+	0.927753i	$0.621738\pi$
$194$	−767.665	−0.284099
$195$	0	0
$196$	846.860	0.308622
$197$	4804.08	1.73744	0.868722	−	0.495300i	$-0.164942\pi$
$197$	4804.08	1.73744	0.868722	+	0.495300i	$0.164942\pi$
$198$	0	0
$199$	3885.03	1.38393	0.691967	−	0.721929i	$-0.256744\pi$
$199$	3885.03	1.38393	0.691967	+	0.721929i	$0.256744\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−1349.16	−0.469933
$203$	−2808.39	−0.970989
$204$	0	0
$205$	−371.702	−0.126638
$206$	−338.594	−0.114519
$207$	0	0
$208$	−1418.56	−0.472881
$209$	10.6910	0.00353835
$210$	0	0
$211$	4568.39	1.49052	0.745262	−	0.666772i	$-0.232324\pi$
$211$	4568.39	1.49052	0.745262	+	0.666772i	$0.232324\pi$
$212$	−932.683	−0.302155
$213$	0	0
$214$	−3066.61	−0.979576
$215$	2340.41	0.742393
$216$	0	0
$217$	−3684.91	−1.15276
$218$	−2479.49	−0.770332
$219$	0	0
$220$	−32.7074	−0.0100233
$221$	−648.489	−0.197385
$222$	0	0
$223$	3081.07	0.925218	0.462609	−	0.886563i	$-0.346913\pi$
$223$	3081.07	0.925218	0.462609	+	0.886563i	$0.346913\pi$
$224$	−753.676	−0.224809
$225$	0	0
$226$	4598.98	1.35363
$227$	−678.971	−0.198524	−0.0992618	−	0.995061i	$-0.531648\pi$
$227$	−678.971	−0.198524	−0.0992618	+	0.995061i	$0.531648\pi$
$228$	0	0
$229$	6399.27	1.84662	0.923310	−	0.384056i	$-0.125473\pi$
$229$	6399.27	1.84662	0.923310	+	0.384056i	$0.125473\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	953.923	0.269949
$233$	−6083.30	−1.71043	−0.855215	−	0.518274i	$-0.826575\pi$
$233$	−6083.30	−1.71043	−0.855215	+	0.518274i	$0.826575\pi$
$234$	0	0
$235$	1969.85	0.546805
$236$	3066.59	0.845840
$237$	0	0
$238$	−344.541	−0.0938372
$239$	−438.313	−0.118628	−0.0593140	−	0.998239i	$-0.518891\pi$
$239$	−438.313	−0.118628	−0.0593140	+	0.998239i	$0.518891\pi$
$240$	0	0
$241$	2854.41	0.762941	0.381471	−	0.924381i	$-0.375418\pi$
$241$	2854.41	0.762941	0.381471	+	0.924381i	$0.375418\pi$
$242$	2656.65	0.705686
$243$	0	0
$244$	713.462	0.187191
$245$	1058.57	0.276040
$246$	0	0
$247$	579.603	0.149309
$248$	1251.65	0.320483
$249$	0	0
$250$	−250.000	−0.0632456
$251$	2292.93	0.576607	0.288303	−	0.957539i	$-0.406909\pi$
$251$	2292.93	0.576607	0.288303	+	0.957539i	$0.406909\pi$
$252$	0	0
$253$	37.6135	0.00934681
$254$	2355.63	0.581912
$255$	0	0
$256$	256.000	0.0625000
$257$	−3547.59	−0.861060	−0.430530	−	0.902576i	$-0.641673\pi$
$257$	−3547.59	−0.861060	−0.430530	+	0.902576i	$0.641673\pi$
$258$	0	0
$259$	−6904.92	−1.65657
$260$	−1773.20	−0.422958
$261$	0	0
$262$	4771.12	1.12504
$263$	2278.35	0.534178	0.267089	−	0.963672i	$-0.413938\pi$
$263$	2278.35	0.534178	0.267089	+	0.963672i	$0.413938\pi$
$264$	0	0
$265$	−1165.85	−0.270256
$266$	307.942	0.0709817
$267$	0	0
$268$	987.616	0.225105
$269$	−3915.36	−0.887450	−0.443725	−	0.896163i	$-0.646343\pi$
$269$	−3915.36	−0.887450	−0.443725	+	0.896163i	$0.646343\pi$
$270$	0	0
$271$	6252.71	1.40157	0.700784	−	0.713374i	$-0.252834\pi$
$271$	6252.71	1.40157	0.700784	+	0.713374i	$0.252834\pi$
$272$	117.030	0.0260881
$273$	0	0
$274$	2246.44	0.495300
$275$	−40.8843	−0.00896514
$276$	0	0
$277$	−4700.53	−1.01959	−0.509797	−	0.860295i	$-0.670279\pi$
$277$	−4700.53	−1.01959	−0.509797	+	0.860295i	$0.670279\pi$
$278$	3603.50	0.777423
$279$	0	0
$280$	−942.096	−0.201075
$281$	−5508.07	−1.16934	−0.584669	−	0.811272i	$-0.698776\pi$
$281$	−5508.07	−1.16934	−0.584669	+	0.811272i	$0.698776\pi$
$282$	0	0
$283$	6639.20	1.39456	0.697278	−	0.716801i	$-0.254394\pi$
$283$	6639.20	1.39456	0.697278	+	0.716801i	$0.254394\pi$
$284$	2602.71	0.543812
$285$	0	0
$286$	−289.983	−0.0599548
$287$	−1750.89	−0.360111
$288$	0	0
$289$	−4859.50	−0.989111
$290$	1192.40	0.241449
$291$	0	0
$292$	2781.78	0.557504
$293$	2120.85	0.422872	0.211436	−	0.977392i	$-0.432186\pi$
$293$	2120.85	0.422872	0.211436	+	0.977392i	$0.432186\pi$
$294$	0	0
$295$	3833.24	0.756542
$296$	2345.38	0.460550
$297$	0	0
$298$	1326.28	0.257816
$299$	2039.18	0.394410
$300$	0	0
$301$	11024.4	2.11109
$302$	−3235.31	−0.616461
$303$	0	0
$304$	−104.598	−0.0197339
$305$	891.827	0.167429
$306$	0	0
$307$	7422.87	1.37995	0.689977	−	0.723832i	$-0.257621\pi$
$307$	7422.87	1.37995	0.689977	+	0.723832i	$0.257621\pi$
$308$	−154.068	−0.0285026
$309$	0	0
$310$	1564.56	0.286648
$311$	2563.97	0.467490	0.233745	−	0.972298i	$-0.424902\pi$
$311$	2563.97	0.467490	0.233745	+	0.972298i	$0.424902\pi$
$312$	0	0
$313$	9403.53	1.69814	0.849072	−	0.528277i	$-0.177162\pi$
$313$	9403.53	1.69814	0.849072	+	0.528277i	$0.177162\pi$
$314$	−1750.16	−0.314545
$315$	0	0
$316$	2642.87	0.470484
$317$	−407.110	−0.0721312	−0.0360656	−	0.999349i	$-0.511483\pi$
$317$	−407.110	−0.0721312	−0.0360656	+	0.999349i	$0.511483\pi$
$318$	0	0
$319$	195.002	0.0342258
$320$	320.000	0.0559017
$321$	0	0
$322$	1083.41	0.187503
$323$	−47.8167	−0.00823713
$324$	0	0
$325$	−2216.50	−0.378305
$326$	−5064.79	−0.860469
$327$	0	0
$328$	594.723	0.100116
$329$	9278.95	1.55491
$330$	0	0
$331$	−7122.33	−1.18272	−0.591358	−	0.806409i	$-0.701408\pi$
$331$	−7122.33	−1.18272	−0.591358	+	0.806409i	$0.701408\pi$
$332$	5313.56	0.878372
$333$	0	0
$334$	−2263.31	−0.370786
$335$	1234.52	0.201340
$336$	0	0
$337$	−8267.78	−1.33642	−0.668212	−	0.743971i	$-0.732940\pi$
$337$	−8267.78	−1.33642	−0.668212	+	0.743971i	$0.732940\pi$
$338$	−11327.1	−1.82283
$339$	0	0
$340$	146.287	0.0233339
$341$	255.864	0.0406328
$342$	0	0
$343$	−3092.08	−0.486753
$344$	−3744.65	−0.586913
$345$	0	0
$346$	−6372.45	−0.990130
$347$	−1048.83	−0.162260	−0.0811302	−	0.996704i	$-0.525853\pi$
$347$	−1048.83	−0.162260	−0.0811302	+	0.996704i	$0.525853\pi$
$348$	0	0
$349$	−9135.87	−1.40124	−0.700619	−	0.713535i	$-0.747093\pi$
$349$	−9135.87	−1.40124	−0.700619	+	0.713535i	$0.747093\pi$
$350$	−1177.62	−0.179847
$351$	0	0
$352$	52.3319	0.00792414
$353$	−5253.06	−0.792046	−0.396023	−	0.918240i	$-0.629610\pi$
$353$	−5253.06	−0.792046	−0.396023	+	0.918240i	$0.629610\pi$
$354$	0	0
$355$	3253.39	0.486400
$356$	3298.81	0.491113
$357$	0	0
$358$	3326.96	0.491160
$359$	11269.7	1.65680	0.828400	−	0.560137i	$-0.189252\pi$
$359$	11269.7	1.65680	0.828400	+	0.560137i	$0.189252\pi$
$360$	0	0
$361$	−6816.26	−0.993769
$362$	7494.70	1.08816
$363$	0	0
$364$	−8352.60	−1.20273
$365$	3477.22	0.498646
$366$	0	0
$367$	−5675.46	−0.807238	−0.403619	−	0.914927i	$-0.632248\pi$
$367$	−5675.46	−0.807238	−0.403619	+	0.914927i	$0.632248\pi$
$368$	−368.000	−0.0521286
$369$	0	0
$370$	2931.73	0.411928
$371$	−5491.73	−0.768508
$372$	0	0
$373$	3662.20	0.508369	0.254185	−	0.967156i	$-0.418193\pi$
$373$	3662.20	0.508369	0.254185	+	0.967156i	$0.418193\pi$
$374$	23.9233	0.00330761
$375$	0	0
$376$	−3151.77	−0.432287
$377$	10571.8	1.44424
$378$	0	0
$379$	5520.89	0.748256	0.374128	−	0.927377i	$-0.377942\pi$
$379$	5520.89	0.748256	0.374128	+	0.927377i	$0.377942\pi$
$380$	−130.748	−0.0176506
$381$	0	0
$382$	4525.93	0.606196
$383$	2309.13	0.308071	0.154036	−	0.988065i	$-0.450773\pi$
$383$	2309.13	0.308071	0.154036	+	0.988065i	$0.450773\pi$
$384$	0	0
$385$	−192.584	−0.0254935
$386$	4002.50	0.527777
$387$	0	0
$388$	1535.33	0.200888
$389$	8194.51	1.06807	0.534034	−	0.845463i	$-0.320676\pi$
$389$	8194.51	1.06807	0.534034	+	0.845463i	$0.320676\pi$
$390$	0	0
$391$	−168.230	−0.0217590
$392$	−1693.72	−0.218229
$393$	0	0
$394$	−9608.16	−1.22856
$395$	3303.58	0.420813
$396$	0	0
$397$	7273.90	0.919564	0.459782	−	0.888032i	$-0.347928\pi$
$397$	7273.90	0.919564	0.459782	+	0.888032i	$0.347928\pi$
$398$	−7770.07	−0.978589
$399$	0	0
$400$	400.000	0.0500000
$401$	1835.81	0.228618	0.114309	−	0.993445i	$-0.463535\pi$
$401$	1835.81	0.228618	0.114309	+	0.993445i	$0.463535\pi$
$402$	0	0
$403$	13871.4	1.71460
$404$	2698.32	0.332293
$405$	0	0
$406$	5616.79	0.686593
$407$	479.446	0.0583914
$408$	0	0
$409$	−2865.80	−0.346466	−0.173233	−	0.984881i	$-0.555421\pi$
$409$	−2865.80	−0.346466	−0.173233	+	0.984881i	$0.555421\pi$
$410$	743.404	0.0895466
$411$	0	0
$412$	677.187	0.0809773
$413$	18056.4	2.15132
$414$	0	0
$415$	6641.95	0.785640
$416$	2837.12	0.334377
$417$	0	0
$418$	−21.3821	−0.00250199
$419$	−11044.0	−1.28767	−0.643836	−	0.765164i	$-0.722658\pi$
$419$	−11044.0	−1.28767	−0.643836	+	0.765164i	$0.722658\pi$
$420$	0	0
$421$	6343.24	0.734324	0.367162	−	0.930157i	$-0.380329\pi$
$421$	6343.24	0.734324	0.367162	+	0.930157i	$0.380329\pi$
$422$	−9136.77	−1.05396
$423$	0	0
$424$	1865.37	0.213656
$425$	182.859	0.0208705
$426$	0	0
$427$	4200.93	0.476106
$428$	6133.23	0.692665
$429$	0	0
$430$	−4680.81	−0.524951
$431$	4117.01	0.460115	0.230058	−	0.973177i	$-0.426109\pi$
$431$	4117.01	0.460115	0.230058	+	0.973177i	$0.426109\pi$
$432$	0	0
$433$	−758.791	−0.0842151	−0.0421076	−	0.999113i	$-0.513407\pi$
$433$	−758.791	−0.0842151	−0.0421076	+	0.999113i	$0.513407\pi$
$434$	7369.82	0.815122
$435$	0	0
$436$	4958.98	0.544707
$437$	150.360	0.0164592
$438$	0	0
$439$	4150.10	0.451193	0.225596	−	0.974221i	$-0.427567\pi$
$439$	4150.10	0.451193	0.225596	+	0.974221i	$0.427567\pi$
$440$	65.4148	0.00708757
$441$	0	0
$442$	1296.98	0.139572
$443$	4597.24	0.493051	0.246525	−	0.969136i	$-0.420711\pi$
$443$	4597.24	0.493051	0.246525	+	0.969136i	$0.420711\pi$
$444$	0	0
$445$	4123.51	0.439265
$446$	−6162.13	−0.654228
$447$	0	0
$448$	1507.35	0.158964
$449$	−6402.58	−0.672954	−0.336477	−	0.941692i	$-0.609235\pi$
$449$	−6402.58	−0.672954	−0.336477	+	0.941692i	$0.609235\pi$
$450$	0	0
$451$	121.574	0.0126933
$452$	−9197.97	−0.957159
$453$	0	0
$454$	1357.94	0.140377
$455$	−10440.8	−1.07576
$456$	0	0
$457$	551.499	0.0564508	0.0282254	−	0.999602i	$-0.491014\pi$
$457$	551.499	0.0564508	0.0282254	+	0.999602i	$0.491014\pi$
$458$	−12798.5	−1.30576
$459$	0	0
$460$	−460.000	−0.0466252
$461$	3901.96	0.394213	0.197107	−	0.980382i	$-0.436846\pi$
$461$	3901.96	0.394213	0.197107	+	0.980382i	$0.436846\pi$
$462$	0	0
$463$	4828.15	0.484628	0.242314	−	0.970198i	$-0.422094\pi$
$463$	4828.15	0.484628	0.242314	+	0.970198i	$0.422094\pi$
$464$	−1907.85	−0.190883
$465$	0	0
$466$	12166.6	1.20946
$467$	14136.3	1.40075	0.700375	−	0.713775i	$-0.253016\pi$
$467$	14136.3	1.40075	0.700375	+	0.713775i	$0.253016\pi$
$468$	0	0
$469$	5815.18	0.572538
$470$	−3939.71	−0.386649
$471$	0	0
$472$	−6133.19	−0.598099
$473$	−765.487	−0.0744125
$474$	0	0
$475$	−163.434	−0.0157871
$476$	689.082	0.0663530
$477$	0	0
$478$	876.626	0.0838827
$479$	−15246.1	−1.45430	−0.727152	−	0.686476i	$-0.759157\pi$
$479$	−15246.1	−1.45430	−0.727152	+	0.686476i	$0.759157\pi$
$480$	0	0
$481$	25992.7	2.46396
$482$	−5708.83	−0.539481
$483$	0	0
$484$	−5313.30	−0.498995
$485$	1919.16	0.179680
$486$	0	0
$487$	3876.39	0.360690	0.180345	−	0.983603i	$-0.442279\pi$
$487$	3876.39	0.360690	0.180345	+	0.983603i	$0.442279\pi$
$488$	−1426.92	−0.132364
$489$	0	0
$490$	−2117.15	−0.195190
$491$	5027.14	0.462061	0.231030	−	0.972947i	$-0.425790\pi$
$491$	5027.14	0.462061	0.231030	+	0.972947i	$0.425790\pi$
$492$	0	0
$493$	−872.166	−0.0796762
$494$	−1159.21	−0.105577
$495$	0	0
$496$	−2503.30	−0.226616
$497$	15325.0	1.38314
$498$	0	0
$499$	7178.10	0.643960	0.321980	−	0.946747i	$-0.395652\pi$
$499$	7178.10	0.643960	0.321980	+	0.946747i	$0.395652\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−4585.86	−0.407723
$503$	18169.6	1.61063	0.805313	−	0.592850i	$-0.201997\pi$
$503$	18169.6	1.61063	0.805313	+	0.592850i	$0.201997\pi$
$504$	0	0
$505$	3372.90	0.297212
$506$	−75.2271	−0.00660919
$507$	0	0
$508$	−4711.27	−0.411474
$509$	−2296.14	−0.199950	−0.0999750	−	0.994990i	$-0.531876\pi$
$509$	−2296.14	−0.199950	−0.0999750	+	0.994990i	$0.531876\pi$
$510$	0	0
$511$	16379.4	1.41797
$512$	−512.000	−0.0441942
$513$	0	0
$514$	7095.18	0.608862
$515$	846.484	0.0724283
$516$	0	0
$517$	−644.288	−0.0548081
$518$	13809.8	1.17137
$519$	0	0
$520$	3546.39	0.299076
$521$	−12945.5	−1.08859	−0.544293	−	0.838895i	$-0.683202\pi$
$521$	−12945.5	−1.08859	−0.544293	+	0.838895i	$0.683202\pi$
$522$	0	0
$523$	−16093.6	−1.34555	−0.672775	−	0.739847i	$-0.734898\pi$
$523$	−16093.6	−1.34555	−0.672775	+	0.739847i	$0.734898\pi$
$524$	−9542.25	−0.795525
$525$	0	0
$526$	−4556.69	−0.377721
$527$	−1144.37	−0.0945915
$528$	0	0
$529$	529.000	0.0434783
$530$	2331.71	0.191100
$531$	0	0
$532$	−615.884	−0.0501916
$533$	6591.01	0.535625
$534$	0	0
$535$	7666.53	0.619538
$536$	−1975.23	−0.159174
$537$	0	0
$538$	7830.73	0.627522
$539$	−346.232	−0.0276684
$540$	0	0
$541$	−15424.7	−1.22580	−0.612900	−	0.790161i	$-0.709997\pi$
$541$	−15424.7	−1.22580	−0.612900	+	0.790161i	$0.709997\pi$
$542$	−12505.4	−0.991058
$543$	0	0
$544$	−234.059	−0.0184471
$545$	6198.73	0.487201
$546$	0	0
$547$	5307.37	0.414857	0.207428	−	0.978250i	$-0.433491\pi$
$547$	5307.37	0.414857	0.207428	+	0.978250i	$0.433491\pi$
$548$	−4492.87	−0.350230
$549$	0	0
$550$	81.7685	0.00633931
$551$	779.519	0.0602698
$552$	0	0
$553$	15561.5	1.19664
$554$	9401.06	0.720961
$555$	0	0
$556$	−7207.00	−0.549721
$557$	13530.9	1.02930	0.514651	−	0.857400i	$-0.327921\pi$
$557$	13530.9	1.02930	0.514651	+	0.857400i	$0.327921\pi$
$558$	0	0
$559$	−41500.0	−3.14001
$560$	1884.19	0.142181
$561$	0	0
$562$	11016.1	0.826847
$563$	18109.0	1.35560	0.677799	−	0.735248i	$-0.262934\pi$
$563$	18109.0	1.35560	0.677799	+	0.735248i	$0.262934\pi$
$564$	0	0
$565$	−11497.5	−0.856109
$566$	−13278.4	−0.986100
$567$	0	0
$568$	−5205.43	−0.384533
$569$	908.991	0.0669717	0.0334858	−	0.999439i	$-0.489339\pi$
$569$	908.991	0.0669717	0.0334858	+	0.999439i	$0.489339\pi$
$570$	0	0
$571$	−8894.71	−0.651895	−0.325948	−	0.945388i	$-0.605683\pi$
$571$	−8894.71	−0.651895	−0.325948	+	0.945388i	$0.605683\pi$
$572$	579.967	0.0423945
$573$	0	0
$574$	3501.79	0.254637
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−12947.3	−0.934148	−0.467074	−	0.884218i	$-0.654692\pi$
$577$	−12947.3	−0.934148	−0.467074	+	0.884218i	$0.654692\pi$
$578$	9719.00	0.699407
$579$	0	0
$580$	−2384.81	−0.170731
$581$	31286.8	2.23407
$582$	0	0
$583$	381.321	0.0270887
$584$	−5563.55	−0.394215
$585$	0	0
$586$	−4241.71	−0.299016
$587$	19404.1	1.36438	0.682190	−	0.731175i	$-0.261028\pi$
$587$	19404.1	1.36438	0.682190	+	0.731175i	$0.261028\pi$
$588$	0	0
$589$	1022.81	0.0715522
$590$	−7666.48	−0.534956
$591$	0	0
$592$	−4690.77	−0.325658
$593$	13633.9	0.944141	0.472070	−	0.881561i	$-0.343507\pi$
$593$	13633.9	0.944141	0.472070	+	0.881561i	$0.343507\pi$
$594$	0	0
$595$	861.352	0.0593479
$596$	−2652.55	−0.182303
$597$	0	0
$598$	−4078.35	−0.278890
$599$	−11276.5	−0.769191	−0.384595	−	0.923085i	$-0.625659\pi$
$599$	−11276.5	−0.769191	−0.384595	+	0.923085i	$0.625659\pi$
$600$	0	0
$601$	−380.766	−0.0258432	−0.0129216	−	0.999917i	$-0.504113\pi$
$601$	−380.766	−0.0258432	−0.0129216	+	0.999917i	$0.504113\pi$
$602$	−22048.9	−1.49277
$603$	0	0
$604$	6470.63	0.435904
$605$	−6641.63	−0.446315
$606$	0	0
$607$	−21669.2	−1.44897	−0.724485	−	0.689290i	$-0.757923\pi$
$607$	−21669.2	−1.44897	−0.724485	+	0.689290i	$0.757923\pi$
$608$	209.196	0.0139540
$609$	0	0
$610$	−1783.65	−0.118390
$611$	−34929.4	−2.31275
$612$	0	0
$613$	−936.980	−0.0617362	−0.0308681	−	0.999523i	$-0.509827\pi$
$613$	−936.980	−0.0617362	−0.0308681	+	0.999523i	$0.509827\pi$
$614$	−14845.7	−0.975774
$615$	0	0
$616$	308.135	0.0201544
$617$	17539.2	1.14441	0.572205	−	0.820111i	$-0.306088\pi$
$617$	17539.2	1.14441	0.572205	+	0.820111i	$0.306088\pi$
$618$	0	0
$619$	−4740.20	−0.307794	−0.153897	−	0.988087i	$-0.549182\pi$
$619$	−4740.20	−0.307794	−0.153897	+	0.988087i	$0.549182\pi$
$620$	−3129.12	−0.202691
$621$	0	0
$622$	−5127.94	−0.330565
$623$	19423.7	1.24911
$624$	0	0
$625$	625.000	0.0400000
$626$	−18807.1	−1.20077
$627$	0	0
$628$	3500.32	0.222417
$629$	−2144.37	−0.135933
$630$	0	0
$631$	−18505.6	−1.16751	−0.583754	−	0.811930i	$-0.698417\pi$
$631$	−18505.6	−1.16751	−0.583754	+	0.811930i	$0.698417\pi$
$632$	−5285.73	−0.332682
$633$	0	0
$634$	814.221	0.0510045
$635$	−5889.09	−0.368033
$636$	0	0
$637$	−18770.6	−1.16753
$638$	−390.004	−0.0242013
$639$	0	0
$640$	−640.000	−0.0395285
$641$	−10163.6	−0.626266	−0.313133	−	0.949709i	$-0.601379\pi$
$641$	−10163.6	−0.626266	−0.313133	+	0.949709i	$0.601379\pi$
$642$	0	0
$643$	7132.33	0.437436	0.218718	−	0.975788i	$-0.429812\pi$
$643$	7132.33	0.437436	0.218718	+	0.975788i	$0.429812\pi$
$644$	−2166.82	−0.132585
$645$	0	0
$646$	95.6334	0.00582453
$647$	−14536.9	−0.883315	−0.441657	−	0.897184i	$-0.645609\pi$
$647$	−14536.9	−0.883315	−0.441657	+	0.897184i	$0.645609\pi$
$648$	0	0
$649$	−1253.75	−0.0758307
$650$	4432.99	0.267502
$651$	0	0
$652$	10129.6	0.608443
$653$	−3318.33	−0.198861	−0.0994305	−	0.995045i	$-0.531702\pi$
$653$	−3318.33	−0.198861	−0.0994305	+	0.995045i	$0.531702\pi$
$654$	0	0
$655$	−11927.8	−0.711539
$656$	−1189.45	−0.0707928
$657$	0	0
$658$	−18557.9	−1.09949
$659$	−3088.17	−0.182546	−0.0912730	−	0.995826i	$-0.529094\pi$
$659$	−3088.17	−0.182546	−0.0912730	+	0.995826i	$0.529094\pi$
$660$	0	0
$661$	27602.0	1.62420	0.812098	−	0.583521i	$-0.198325\pi$
$661$	27602.0	1.62420	0.812098	+	0.583521i	$0.198325\pi$
$662$	14244.7	0.836306
$663$	0	0
$664$	−10627.1	−0.621103
$665$	−769.854	−0.0448927
$666$	0	0
$667$	2742.53	0.159207
$668$	4526.61	0.262186
$669$	0	0
$670$	−2469.04	−0.142369
$671$	−291.694	−0.0167820
$672$	0	0
$673$	8091.33	0.463444	0.231722	−	0.972782i	$-0.425564\pi$
$673$	8091.33	0.463444	0.231722	+	0.972782i	$0.425564\pi$
$674$	16535.6	0.944994
$675$	0	0
$676$	22654.3	1.28893
$677$	9515.37	0.540185	0.270093	−	0.962834i	$-0.412946\pi$
$677$	9515.37	0.540185	0.270093	+	0.962834i	$0.412946\pi$
$678$	0	0
$679$	9040.18	0.510943
$680$	−292.574	−0.0164996
$681$	0	0
$682$	−511.727	−0.0287317
$683$	630.346	0.0353141	0.0176570	−	0.999844i	$-0.494379\pi$
$683$	630.346	0.0353141	0.0176570	+	0.999844i	$0.494379\pi$
$684$	0	0
$685$	−5616.09	−0.313255
$686$	6184.15	0.344187
$687$	0	0
$688$	7489.30	0.415010
$689$	20672.9	1.14307
$690$	0	0
$691$	−6515.92	−0.358723	−0.179361	−	0.983783i	$-0.557403\pi$
$691$	−6515.92	−0.358723	−0.179361	+	0.983783i	$0.557403\pi$
$692$	12744.9	0.700128
$693$	0	0
$694$	2097.67	0.114735
$695$	−9008.76	−0.491686
$696$	0	0
$697$	−543.751	−0.0295496
$698$	18271.7	0.990825
$699$	0	0
$700$	2355.24	0.127171
$701$	32850.2	1.76995	0.884975	−	0.465638i	$-0.154175\pi$
$701$	32850.2	1.76995	0.884975	+	0.465638i	$0.154175\pi$
$702$	0	0
$703$	1916.58	0.102824
$704$	−104.664	−0.00560321
$705$	0	0
$706$	10506.1	0.560061
$707$	15888.0	0.845160
$708$	0	0
$709$	−21841.2	−1.15693	−0.578466	−	0.815707i	$-0.696348\pi$
$709$	−21841.2	−1.15693	−0.578466	+	0.815707i	$0.696348\pi$
$710$	−6506.78	−0.343937
$711$	0	0
$712$	−6597.61	−0.347270
$713$	3598.49	0.189010
$714$	0	0
$715$	724.959	0.0379188
$716$	−6653.93	−0.347303
$717$	0	0
$718$	−22539.4	−1.17153
$719$	−25807.3	−1.33859	−0.669296	−	0.742996i	$-0.733404\pi$
$719$	−25807.3	−1.33859	−0.669296	+	0.742996i	$0.733404\pi$
$720$	0	0
$721$	3987.34	0.205959
$722$	13632.5	0.702701
$723$	0	0
$724$	−14989.4	−0.769443
$725$	−2981.01	−0.152706
$726$	0	0
$727$	14264.0	0.727679	0.363839	−	0.931462i	$-0.381466\pi$
$727$	14264.0	0.727679	0.363839	+	0.931462i	$0.381466\pi$
$728$	16705.2	0.850462
$729$	0	0
$730$	−6954.44	−0.352596
$731$	3423.71	0.173229
$732$	0	0
$733$	811.706	0.0409018	0.0204509	−	0.999791i	$-0.493490\pi$
$733$	811.706	0.0409018	0.0204509	+	0.999791i	$0.493490\pi$
$734$	11350.9	0.570804
$735$	0	0
$736$	736.000	0.0368605
$737$	−403.780	−0.0201810
$738$	0	0
$739$	−30696.9	−1.52802	−0.764008	−	0.645207i	$-0.776771\pi$
$739$	−30696.9	−1.52802	−0.764008	+	0.645207i	$0.776771\pi$
$740$	−5863.46	−0.291277
$741$	0	0
$742$	10983.5	0.543417
$743$	26935.6	1.32998	0.664988	−	0.746854i	$-0.268437\pi$
$743$	26935.6	1.32998	0.664988	+	0.746854i	$0.268437\pi$
$744$	0	0
$745$	−3315.69	−0.163057
$746$	−7324.40	−0.359471
$747$	0	0
$748$	−47.8467	−0.00233883
$749$	36113.0	1.76174
$750$	0	0
$751$	22371.2	1.08700	0.543501	−	0.839409i	$-0.317098\pi$
$751$	22371.2	1.08700	0.543501	+	0.839409i	$0.317098\pi$
$752$	6303.53	0.305673
$753$	0	0
$754$	−21143.7	−1.02123
$755$	8088.28	0.389884
$756$	0	0
$757$	697.182	0.0334736	0.0167368	−	0.999860i	$-0.494672\pi$
$757$	697.182	0.0334736	0.0167368	+	0.999860i	$0.494672\pi$
$758$	−11041.8	−0.529097
$759$	0	0
$760$	261.495	0.0124808
$761$	−39756.9	−1.89380	−0.946902	−	0.321522i	$-0.895806\pi$
$761$	−39756.9	−1.89380	−0.946902	+	0.321522i	$0.895806\pi$
$762$	0	0
$763$	29199.0	1.38542
$764$	−9051.87	−0.428645
$765$	0	0
$766$	−4618.27	−0.217839
$767$	−67970.9	−3.19985
$768$	0	0
$769$	−12574.8	−0.589673	−0.294836	−	0.955548i	$-0.595265\pi$
$769$	−12574.8	−0.589673	−0.294836	+	0.955548i	$0.595265\pi$
$770$	385.169	0.0180267
$771$	0	0
$772$	−8005.00	−0.373195
$773$	−2478.65	−0.115331	−0.0576654	−	0.998336i	$-0.518366\pi$
$773$	−2478.65	−0.115331	−0.0576654	+	0.998336i	$0.518366\pi$
$774$	0	0
$775$	−3911.40	−0.181292
$776$	−3070.66	−0.142049
$777$	0	0
$778$	−16389.0	−0.755238
$779$	485.991	0.0223523
$780$	0	0
$781$	−1064.10	−0.0487535
$782$	336.460	0.0153859
$783$	0	0
$784$	3387.44	0.154311
$785$	4375.40	0.198936
$786$	0	0
$787$	−7716.40	−0.349504	−0.174752	−	0.984612i	$-0.555912\pi$
$787$	−7716.40	−0.349504	−0.174752	+	0.984612i	$0.555912\pi$
$788$	19216.3	0.868722
$789$	0	0
$790$	−6607.17	−0.297560
$791$	−54158.5	−2.43446
$792$	0	0
$793$	−15813.9	−0.708154
$794$	−14547.8	−0.650230
$795$	0	0
$796$	15540.1	0.691967
$797$	26798.1	1.19101	0.595507	−	0.803350i	$-0.296951\pi$
$797$	26798.1	1.19101	0.595507	+	0.803350i	$0.296951\pi$
$798$	0	0
$799$	2881.64	0.127591
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−3671.61	−0.161657
$803$	−1137.31	−0.0499810
$804$	0	0
$805$	−2708.52	−0.118588
$806$	−27742.7	−1.21240
$807$	0	0
$808$	−5396.63	−0.234967
$809$	−45278.0	−1.96772	−0.983862	−	0.178927i	$-0.942738\pi$
$809$	−45278.0	−1.96772	−0.983862	+	0.178927i	$0.942738\pi$
$810$	0	0
$811$	−24889.6	−1.07767	−0.538836	−	0.842411i	$-0.681136\pi$
$811$	−24889.6	−1.07767	−0.538836	+	0.842411i	$0.681136\pi$
$812$	−11233.6	−0.485494
$813$	0	0
$814$	−958.893	−0.0412889
$815$	12662.0	0.544208
$816$	0	0
$817$	−3060.03	−0.131036
$818$	5731.60	0.244989
$819$	0	0
$820$	−1486.81	−0.0633190
$821$	−19274.0	−0.819326	−0.409663	−	0.912237i	$-0.634354\pi$
$821$	−19274.0	−0.819326	−0.409663	+	0.912237i	$0.634354\pi$
$822$	0	0
$823$	−21620.8	−0.915740	−0.457870	−	0.889019i	$-0.651387\pi$
$823$	−21620.8	−0.915740	−0.457870	+	0.889019i	$0.651387\pi$
$824$	−1354.37	−0.0572596
$825$	0	0
$826$	−36112.8	−1.52122
$827$	9012.81	0.378967	0.189484	−	0.981884i	$-0.439319\pi$
$827$	9012.81	0.378967	0.189484	+	0.981884i	$0.439319\pi$
$828$	0	0
$829$	26574.7	1.11336	0.556682	−	0.830726i	$-0.312074\pi$
$829$	26574.7	1.11336	0.556682	+	0.830726i	$0.312074\pi$
$830$	−13283.9	−0.555531
$831$	0	0
$832$	−5674.23	−0.236441
$833$	1548.56	0.0644110
$834$	0	0
$835$	5658.27	0.234506
$836$	42.7642	0.00176917
$837$	0	0
$838$	22088.0	0.910521
$839$	1261.30	0.0519009	0.0259504	−	0.999663i	$-0.491739\pi$
$839$	1261.30	0.0519009	0.0259504	+	0.999663i	$0.491739\pi$
$840$	0	0
$841$	−10170.7	−0.417022
$842$	−12686.5	−0.519245
$843$	0	0
$844$	18273.5	0.745262
$845$	28317.8	1.15286
$846$	0	0
$847$	−31285.2	−1.26915
$848$	−3730.73	−0.151078
$849$	0	0
$850$	−365.718	−0.0147577
$851$	6742.98	0.271617
$852$	0	0
$853$	32765.1	1.31519	0.657594	−	0.753373i	$-0.271574\pi$
$853$	32765.1	1.31519	0.657594	+	0.753373i	$0.271574\pi$
$854$	−8401.86	−0.336658
$855$	0	0
$856$	−12266.5	−0.489788
$857$	−722.029	−0.0287795	−0.0143898	−	0.999896i	$-0.504581\pi$
$857$	−722.029	−0.0287795	−0.0143898	+	0.999896i	$0.504581\pi$
$858$	0	0
$859$	−36626.9	−1.45482	−0.727411	−	0.686202i	$-0.759277\pi$
$859$	−36626.9	−1.45482	−0.727411	+	0.686202i	$0.759277\pi$
$860$	9361.63	0.371196
$861$	0	0
$862$	−8234.03	−0.325351
$863$	33207.9	1.30986	0.654931	−	0.755688i	$-0.272698\pi$
$863$	33207.9	1.30986	0.654931	+	0.755688i	$0.272698\pi$
$864$	0	0
$865$	15931.1	0.626214
$866$	1517.58	0.0595491
$867$	0	0
$868$	−14739.6	−0.576378
$869$	−1080.52	−0.0421795
$870$	0	0
$871$	−21890.5	−0.851585
$872$	−9917.97	−0.385166
$873$	0	0
$874$	−300.719	−0.0116384
$875$	2944.05	0.113745
$876$	0	0
$877$	−39507.6	−1.52118	−0.760592	−	0.649230i	$-0.775091\pi$
$877$	−39507.6	−1.52118	−0.760592	+	0.649230i	$0.775091\pi$
$878$	−8300.21	−0.319041
$879$	0	0
$880$	−130.830	−0.00501167
$881$	−47928.6	−1.83287	−0.916434	−	0.400187i	$-0.868945\pi$
$881$	−47928.6	−1.83287	−0.916434	+	0.400187i	$0.868945\pi$
$882$	0	0
$883$	48176.2	1.83608	0.918040	−	0.396488i	$-0.129771\pi$
$883$	48176.2	1.83608	0.918040	+	0.396488i	$0.129771\pi$
$884$	−2593.96	−0.0986925
$885$	0	0
$886$	−9194.48	−0.348640
$887$	−31421.7	−1.18944	−0.594722	−	0.803932i	$-0.702738\pi$
$887$	−31421.7	−1.18944	−0.594722	+	0.803932i	$0.702738\pi$
$888$	0	0
$889$	−27740.4	−1.04655
$890$	−8247.02	−0.310607
$891$	0	0
$892$	12324.3	0.462609
$893$	−2575.54	−0.0965141
$894$	0	0
$895$	−8317.41	−0.310637
$896$	−3014.71	−0.112404
$897$	0	0
$898$	12805.2	0.475851
$899$	18655.9	0.692111
$900$	0	0
$901$	−1705.49	−0.0630613
$902$	−243.148	−0.00897555
$903$	0	0
$904$	18395.9	0.676814
$905$	−18736.7	−0.688211
$906$	0	0
$907$	27694.1	1.01385	0.506927	−	0.861989i	$-0.330781\pi$
$907$	27694.1	1.01385	0.506927	+	0.861989i	$0.330781\pi$
$908$	−2715.88	−0.0992618
$909$	0	0
$910$	20881.5	0.760676
$911$	27325.1	0.993764	0.496882	−	0.867818i	$-0.334478\pi$
$911$	27325.1	0.993764	0.496882	+	0.867818i	$0.334478\pi$
$912$	0	0
$913$	−2172.41	−0.0787473
$914$	−1103.00	−0.0399168
$915$	0	0
$916$	25597.1	0.923310
$917$	−56185.7	−2.02335
$918$	0	0
$919$	28135.6	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$919$	28135.6	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	−7803.91	−0.278751
$923$	−57689.1	−2.05727
$924$	0	0
$925$	−7329.32	−0.260526
$926$	−9656.29	−0.342684
$927$	0	0
$928$	3815.69	0.134974
$929$	−32497.6	−1.14770	−0.573850	−	0.818961i	$-0.694551\pi$
$929$	−32497.6	−1.14770	−0.573850	+	0.818961i	$0.694551\pi$
$930$	0	0
$931$	−1384.06	−0.0487226
$932$	−24333.2	−0.855215
$933$	0	0
$934$	−28272.6	−0.990479
$935$	−59.8084	−0.00209192
$936$	0	0
$937$	−12209.3	−0.425678	−0.212839	−	0.977087i	$-0.568271\pi$
$937$	−12209.3	−0.425678	−0.212839	+	0.977087i	$0.568271\pi$
$938$	−11630.4	−0.404845
$939$	0	0
$940$	7879.42	0.273402
$941$	1317.78	0.0456518	0.0228259	−	0.999739i	$-0.492734\pi$
$941$	1317.78	0.0456518	0.0228259	+	0.999739i	$0.492734\pi$
$942$	0	0
$943$	1709.83	0.0590453
$944$	12266.4	0.422920
$945$	0	0
$946$	1530.97	0.0526176
$947$	11954.8	0.410219	0.205110	−	0.978739i	$-0.434245\pi$
$947$	11954.8	0.410219	0.205110	+	0.978739i	$0.434245\pi$
$948$	0	0
$949$	−61657.9	−2.10906
$950$	326.869	0.0111632
$951$	0	0
$952$	−1378.16	−0.0469186
$953$	19658.8	0.668216	0.334108	−	0.942535i	$-0.391565\pi$
$953$	19658.8	0.668216	0.334108	+	0.942535i	$0.391565\pi$
$954$	0	0
$955$	−11314.8	−0.383392
$956$	−1753.25	−0.0593140
$957$	0	0
$958$	30492.2	1.02835
$959$	−26454.5	−0.890782
$960$	0	0
$961$	−5312.52	−0.178326
$962$	−51985.3	−1.74228
$963$	0	0
$964$	11417.7	0.381471
$965$	−10006.3	−0.333796
$966$	0	0
$967$	−7604.58	−0.252892	−0.126446	−	0.991973i	$-0.540357\pi$
$967$	−7604.58	−0.252892	−0.126446	+	0.991973i	$0.540357\pi$
$968$	10626.6	0.352843
$969$	0	0
$970$	−3838.33	−0.127053
$971$	14731.2	0.486866	0.243433	−	0.969918i	$-0.421726\pi$
$971$	14731.2	0.486866	0.243433	+	0.969918i	$0.421726\pi$
$972$	0	0
$973$	−42435.5	−1.39817
$974$	−7752.77	−0.255046
$975$	0	0
$976$	2853.85	0.0935957
$977$	50584.6	1.65644	0.828222	−	0.560400i	$-0.189353\pi$
$977$	50584.6	1.65644	0.828222	+	0.560400i	$0.189353\pi$
$978$	0	0
$979$	−1348.69	−0.0440290
$980$	4234.30	0.138020
$981$	0	0
$982$	−10054.3	−0.326726
$983$	−15625.9	−0.507007	−0.253503	−	0.967334i	$-0.581583\pi$
$983$	−15625.9	−0.507007	−0.253503	+	0.967334i	$0.581583\pi$
$984$	0	0
$985$	24020.4	0.777009
$986$	1744.33	0.0563396
$987$	0	0
$988$	2318.41	0.0746544
$989$	−10765.9	−0.346142
$990$	0	0
$991$	18267.8	0.585567	0.292783	−	0.956179i	$-0.405419\pi$
$991$	18267.8	0.585567	0.292783	+	0.956179i	$0.405419\pi$
$992$	5006.59	0.160241
$993$	0	0
$994$	−30650.1	−0.978029
$995$	19425.2	0.618914
$996$	0	0
$997$	21580.5	0.685519	0.342760	−	0.939423i	$-0.388638\pi$
$997$	21580.5	0.685519	0.342760	+	0.939423i	$0.388638\pi$
$998$	−14356.2	−0.455348
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bi.1.3		4
3.2	odd	2		230.4.a.i.1.4	✓	4
12.11	even	2		1840.4.a.l.1.1		4
15.2	even	4		1150.4.b.m.599.5		8
15.8	even	4		1150.4.b.m.599.4		8
15.14	odd	2		1150.4.a.o.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.4	✓	4	3.2	odd	2
1150.4.a.o.1.1		4	15.14	odd	2
1150.4.b.m.599.4		8	15.8	even	4
1150.4.b.m.599.5		8	15.2	even	4
1840.4.a.l.1.1		4	12.11	even	2
2070.4.a.bi.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5524 q^{7} -8.00000 q^{8} -10.0000 q^{10} -1.63537 q^{11} -88.6599 q^{13} -47.1048 q^{14} +16.0000 q^{16} +7.31435 q^{17} -6.53738 q^{19} +20.0000 q^{20} +3.27074 q^{22} -23.0000 q^{23} +25.0000 q^{25} +177.320 q^{26} +94.2096 q^{28} -119.240 q^{29} -156.456 q^{31} -32.0000 q^{32} -14.6287 q^{34} +117.762 q^{35} -293.173 q^{37} +13.0748 q^{38} -40.0000 q^{40} -74.3404 q^{41} +468.081 q^{43} -6.54148 q^{44} +46.0000 q^{46} +393.971 q^{47} +211.715 q^{49} -50.0000 q^{50} -354.639 q^{52} -233.171 q^{53} -8.17685 q^{55} -188.419 q^{56} +238.481 q^{58} +766.648 q^{59} +178.365 q^{61} +312.912 q^{62} +64.0000 q^{64} -443.299 q^{65} +246.904 q^{67} +29.2574 q^{68} -235.524 q^{70} +650.678 q^{71} +695.444 q^{73} +586.346 q^{74} -26.1495 q^{76} -38.5169 q^{77} +660.717 q^{79} +80.0000 q^{80} +148.681 q^{82} +1328.39 q^{83} +36.5718 q^{85} -936.163 q^{86} +13.0830 q^{88} +824.702 q^{89} -2088.15 q^{91} -92.0000 q^{92} -787.942 q^{94} -32.6869 q^{95} +383.833 q^{97} -423.430 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} + 20 q^{5} + 26 q^{7} - 32 q^{8} - 40 q^{10} - 93 q^{11} + 32 q^{13} - 52 q^{14} + 64 q^{16} - 108 q^{17} + 185 q^{19} + 80 q^{20} + 186 q^{22} - 92 q^{23} + 100 q^{25} - 64 q^{26}+ \cdots - 1200 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 + 20 * q^5 + 26 * q^7 - 32 * q^8 - 40 * q^10 - 93 * q^11 + 32 * q^13 - 52 * q^14 + 64 * q^16 - 108 * q^17 + 185 * q^19 + 80 * q^20 + 186 * q^22 - 92 * q^23 + 100 * q^25 - 64 * q^26 + 104 * q^28 - 294 * q^29 - 211 * q^31 - 128 * q^32 + 216 * q^34 + 130 * q^35 + 5 * q^37 - 370 * q^38 - 160 * q^40 + 369 * q^41 - 100 * q^43 - 372 * q^44 + 184 * q^46 + 363 * q^47 + 600 * q^49 - 200 * q^50 + 128 * q^52 - 21 * q^53 - 465 * q^55 - 208 * q^56 + 588 * q^58 + 33 * q^59 - 307 * q^61 + 422 * q^62 + 256 * q^64 + 160 * q^65 + 725 * q^67 - 432 * q^68 - 260 * q^70 + 1257 * q^71 + 509 * q^73 - 10 * q^74 + 740 * q^76 + 1962 * q^77 + 1202 * q^79 + 320 * q^80 - 738 * q^82 + 1377 * q^83 - 540 * q^85 + 200 * q^86 + 744 * q^88 + 984 * q^89 - 995 * q^91 - 368 * q^92 - 726 * q^94 + 925 * q^95 + 137 * q^97 - 1200 * q^98

Introduction

L-functions

Modular forms

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Groups

Database

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