LMFDB - Embedded newform 531.5.b.a.296.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(296,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("531.296");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.19
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.58

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.51417i q^{2} -4.37776 q^{4} -15.3988i q^{5} +19.6389 q^{7} -52.4648i q^{8} +O(q^{10})$	\(q-4.51417i q^{2} -4.37776 q^{4} -15.3988i q^{5} +19.6389 q^{7} -52.4648i q^{8} -69.5129 q^{10} +179.769i q^{11} -125.037 q^{13} -88.6533i q^{14} -306.879 q^{16} +43.9122i q^{17} -32.3433 q^{19} +67.4123i q^{20} +811.507 q^{22} +791.044i q^{23} +387.877 q^{25} +564.438i q^{26} -85.9743 q^{28} +430.311i q^{29} -1590.79 q^{31} +545.870i q^{32} +198.227 q^{34} -302.415i q^{35} -1366.37 q^{37} +146.003i q^{38} -807.895 q^{40} +578.465i q^{41} -1702.47 q^{43} -786.984i q^{44} +3570.91 q^{46} +207.099i q^{47} -2015.31 q^{49} -1750.94i q^{50} +547.382 q^{52} +3568.34i q^{53} +2768.22 q^{55} -1030.35i q^{56} +1942.50 q^{58} -453.188i q^{59} +6522.69 q^{61} +7181.12i q^{62} -2445.92 q^{64} +1925.42i q^{65} -1152.48 q^{67} -192.237i q^{68} -1365.16 q^{70} +54.5133i q^{71} -8114.08 q^{73} +6168.01i q^{74} +141.591 q^{76} +3530.45i q^{77} +2730.39 q^{79} +4725.58i q^{80} +2611.29 q^{82} +3278.11i q^{83} +676.195 q^{85} +7685.25i q^{86} +9431.53 q^{88} +173.670i q^{89} -2455.58 q^{91} -3463.00i q^{92} +934.881 q^{94} +498.048i q^{95} +773.040 q^{97} +9097.48i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) $ 76 * q - 632 * q^4 - 48 * q^7	$ 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) $ 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 4.51417i	− 1.12854i	−0.825589	−	0.564272i	$-0.809157\pi$
$2$	− 4.51417i	− 1.12854i	0.825589	−	0.564272i	$-0.190843\pi$
$3$	0	0
$3$	0	0
$4$	−4.37776	−0.273610
$5$	− 15.3988i	− 0.615952i	−0.951394	−	0.307976i	$-0.900348\pi$
$5$	− 15.3988i	− 0.615952i	0.951394	−	0.307976i	$-0.0996517\pi$
$6$	0	0
$7$	19.6389	0.400793	0.200397	−	0.979715i	$-0.435777\pi$
$7$	19.6389	0.400793	0.200397	+	0.979715i	$0.435777\pi$
$8$	− 52.4648i	− 0.819762i
$9$	0	0
$10$	−69.5129	−0.695129
$11$	179.769i	1.48569i	0.669463	+	0.742846i	$0.266524\pi$
$11$	179.769i	1.48569i	−0.669463	+	0.742846i	$0.733476\pi$
$12$	0	0
$13$	−125.037	−0.739863	−0.369932	−	0.929059i	$-0.620619\pi$
$13$	−125.037	−0.739863	−0.369932	+	0.929059i	$0.620619\pi$
$14$	− 88.6533i	− 0.452313i
$15$	0	0
$16$	−306.879	−1.19875
$17$	43.9122i	0.151945i	0.997110	+	0.0759726i	$0.0242062\pi$
$17$	43.9122i	0.151945i	−0.997110	+	0.0759726i	$0.975794\pi$
$18$	0	0
$19$	−32.3433	−0.0895935	−0.0447968	−	0.998996i	$-0.514264\pi$
$19$	−32.3433	−0.0895935	−0.0447968	+	0.998996i	$0.514264\pi$
$20$	67.4123i	0.168531i
$21$	0	0
$22$	811.507	1.67667
$23$	791.044i	1.49536i	0.664061	+	0.747678i	$0.268831\pi$
$23$	791.044i	1.49536i	−0.664061	+	0.747678i	$0.731169\pi$
$24$	0	0
$25$	387.877	0.620603
$26$	564.438i	0.834968i
$27$	0	0
$28$	−85.9743	−0.109661
$29$	430.311i	0.511666i	0.966721	+	0.255833i	$0.0823497\pi$
$29$	430.311i	0.511666i	−0.966721	+	0.255833i	$0.917650\pi$
$30$	0	0
$31$	−1590.79	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$31$	−1590.79	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$32$	545.870i	0.533076i
$33$	0	0
$34$	198.227	0.171477
$35$	− 302.415i	− 0.246870i
$36$	0	0
$37$	−1366.37	−0.998076	−0.499038	−	0.866580i	$-0.666313\pi$
$37$	−1366.37	−0.998076	−0.499038	+	0.866580i	$0.666313\pi$
$38$	146.003i	0.101110i
$39$	0	0
$40$	−807.895	−0.504935
$41$	578.465i	0.344120i	0.985086	+	0.172060i	$0.0550423\pi$
$41$	578.465i	0.344120i	−0.985086	+	0.172060i	$0.944958\pi$
$42$	0	0
$43$	−1702.47	−0.920753	−0.460376	−	0.887724i	$-0.652286\pi$
$43$	−1702.47	−0.920753	−0.460376	+	0.887724i	$0.652286\pi$
$44$	− 786.984i	− 0.406500i
$45$	0	0
$46$	3570.91	1.68757
$47$	207.099i	0.0937524i	0.998901	+	0.0468762i	$0.0149266\pi$
$47$	207.099i	0.0937524i	−0.998901	+	0.0468762i	$0.985073\pi$
$48$	0	0
$49$	−2015.31	−0.839365
$50$	− 1750.94i	− 0.700377i
$51$	0	0
$52$	547.382	0.202434
$53$	3568.34i	1.27033i	0.772378	+	0.635163i	$0.219067\pi$
$53$	3568.34i	1.27033i	−0.772378	+	0.635163i	$0.780933\pi$
$54$	0	0
$55$	2768.22	0.915115
$56$	− 1030.35i	− 0.328555i
$57$	0	0
$58$	1942.50	0.577437
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	6522.69	1.75294	0.876469	−	0.481458i	$-0.159892\pi$
$61$	6522.69	1.75294	0.876469	+	0.481458i	$0.159892\pi$
$62$	7181.12i	1.86814i
$63$	0	0
$64$	−2445.92	−0.597148
$65$	1925.42i	0.455720i
$66$	0	0
$67$	−1152.48	−0.256734	−0.128367	−	0.991727i	$-0.540974\pi$
$67$	−1152.48	−0.256734	−0.128367	+	0.991727i	$0.540974\pi$
$68$	− 192.237i	− 0.0415738i
$69$	0	0
$70$	−1365.16	−0.278603
$71$	54.5133i	0.0108140i	0.999985	+	0.00540700i	$0.00172111\pi$
$71$	54.5133i	0.0108140i	−0.999985	+	0.00540700i	$0.998279\pi$
$72$	0	0
$73$	−8114.08	−1.52263	−0.761313	−	0.648384i	$-0.775445\pi$
$73$	−8114.08	−1.52263	−0.761313	+	0.648384i	$0.775445\pi$
$74$	6168.01i	1.12637i
$75$	0	0
$76$	141.591	0.0245137
$77$	3530.45i	0.595455i
$78$	0	0
$79$	2730.39	0.437492	0.218746	−	0.975782i	$-0.429803\pi$
$79$	2730.39	0.437492	0.218746	+	0.975782i	$0.429803\pi$
$80$	4725.58i	0.738371i
$81$	0	0
$82$	2611.29	0.388354
$83$	3278.11i	0.475847i	0.971284	+	0.237924i	$0.0764668\pi$
$83$	3278.11i	0.475847i	−0.971284	+	0.237924i	$0.923533\pi$
$84$	0	0
$85$	676.195	0.0935910
$86$	7685.25i	1.03911i
$87$	0	0
$88$	9431.53	1.21791
$89$	173.670i	0.0219252i	0.999940	+	0.0109626i	$0.00348958\pi$
$89$	173.670i	0.0219252i	−0.999940	+	0.0109626i	$0.996510\pi$
$90$	0	0
$91$	−2455.58	−0.296532
$92$	− 3463.00i	− 0.409145i
$93$	0	0
$94$	934.881	0.105804
$95$	498.048i	0.0551853i
$96$	0	0
$97$	773.040	0.0821597	0.0410798	−	0.999156i	$-0.486920\pi$
$97$	773.040	0.0821597	0.0410798	+	0.999156i	$0.486920\pi$
$98$	9097.48i	0.947259i
$99$	0	0
$100$	−1698.03	−0.169803
$101$	11463.2i	1.12373i	0.827229	+	0.561866i	$0.189916\pi$
$101$	11463.2i	1.12373i	−0.827229	+	0.561866i	$0.810084\pi$
$102$	0	0
$103$	14241.8	1.34243	0.671215	−	0.741263i	$-0.265773\pi$
$103$	14241.8	1.34243	0.671215	+	0.741263i	$0.265773\pi$
$104$	6560.03i	0.606512i
$105$	0	0
$106$	16108.1	1.43362
$107$	− 16837.5i	− 1.47065i	−0.677714	−	0.735325i	$-0.737029\pi$
$107$	− 16837.5i	− 1.47065i	0.677714	−	0.735325i	$-0.262971\pi$
$108$	0	0
$109$	13596.4	1.14438	0.572191	−	0.820120i	$-0.306094\pi$
$109$	13596.4	1.14438	0.572191	+	0.820120i	$0.306094\pi$
$110$	− 12496.2i	− 1.03275i
$111$	0	0
$112$	−6026.77	−0.480450
$113$	19889.3i	1.55762i	0.627259	+	0.778811i	$0.284177\pi$
$113$	19889.3i	1.55762i	−0.627259	+	0.778811i	$0.715823\pi$
$114$	0	0
$115$	12181.1	0.921068
$116$	− 1883.80i	− 0.139997i
$117$	0	0
$118$	−2045.77	−0.146924
$119$	862.386i	0.0608987i
$120$	0	0
$121$	−17675.8	−1.20728
$122$	− 29444.5i	− 1.97827i
$123$	0	0
$124$	6964.11	0.452921
$125$	− 15597.1i	− 0.998214i
$126$	0	0
$127$	2314.16	0.143478	0.0717392	−	0.997423i	$-0.477145\pi$
$127$	2314.16	0.143478	0.0717392	+	0.997423i	$0.477145\pi$
$128$	19775.2i	1.20698i
$129$	0	0
$130$	8691.67	0.514300
$131$	2362.45i	0.137664i	0.997628	+	0.0688320i	$0.0219273\pi$
$131$	2362.45i	0.137664i	−0.997628	+	0.0688320i	$0.978073\pi$
$132$	0	0
$133$	−635.185	−0.0359085
$134$	5202.50i	0.289736i
$135$	0	0
$136$	2303.84	0.124559
$137$	− 27948.6i	− 1.48908i	−0.667576	−	0.744541i	$-0.732668\pi$
$137$	− 27948.6i	− 1.48908i	0.667576	−	0.744541i	$-0.267332\pi$
$138$	0	0
$139$	5161.08	0.267123	0.133561	−	0.991041i	$-0.457359\pi$
$139$	5161.08	0.267123	0.133561	+	0.991041i	$0.457359\pi$
$140$	1323.90i	0.0675460i
$141$	0	0
$142$	246.083	0.0122041
$143$	− 22477.7i	− 1.09921i
$144$	0	0
$145$	6626.27	0.315162
$146$	36628.3i	1.71835i
$147$	0	0
$148$	5981.62	0.273084
$149$	9670.82i	0.435603i	0.975993	+	0.217802i	$0.0698885\pi$
$149$	9670.82i	0.435603i	−0.975993	+	0.217802i	$0.930111\pi$
$150$	0	0
$151$	36444.5	1.59837	0.799185	−	0.601085i	$-0.205265\pi$
$151$	36444.5	1.59837	0.799185	+	0.601085i	$0.205265\pi$
$152$	1696.88i	0.0734454i
$153$	0	0
$154$	15937.1	0.671997
$155$	24496.3i	1.01962i
$156$	0	0
$157$	−42927.1	−1.74153	−0.870767	−	0.491696i	$-0.836377\pi$
$157$	−42927.1	−1.74153	−0.870767	+	0.491696i	$0.836377\pi$
$158$	− 12325.4i	− 0.493729i
$159$	0	0
$160$	8405.75	0.328350
$161$	15535.2i	0.599329i
$162$	0	0
$163$	10002.2	0.376461	0.188231	−	0.982125i	$-0.439725\pi$
$163$	10002.2	0.376461	0.188231	+	0.982125i	$0.439725\pi$
$164$	− 2532.38i	− 0.0941547i
$165$	0	0
$166$	14798.0	0.537014
$167$	− 28397.2i	− 1.01822i	−0.860701	−	0.509111i	$-0.829974\pi$
$167$	− 28397.2i	− 1.01822i	0.860701	−	0.509111i	$-0.170026\pi$
$168$	0	0
$169$	−12926.8	−0.452602
$170$	− 3052.46i	− 0.105622i
$171$	0	0
$172$	7453.02	0.251927
$173$	19756.3i	0.660106i	0.943962	+	0.330053i	$0.107067\pi$
$173$	19756.3i	0.660106i	−0.943962	+	0.330053i	$0.892933\pi$
$174$	0	0
$175$	7617.46	0.248733
$176$	− 55167.3i	− 1.78097i
$177$	0	0
$178$	783.975	0.0247436
$179$	17996.3i	0.561664i	0.959757	+	0.280832i	$0.0906104\pi$
$179$	17996.3i	0.561664i	−0.959757	+	0.280832i	$0.909390\pi$
$180$	0	0
$181$	8632.39	0.263496	0.131748	−	0.991283i	$-0.457941\pi$
$181$	8632.39	0.263496	0.131748	+	0.991283i	$0.457941\pi$
$182$	11084.9i	0.334650i
$183$	0	0
$184$	41501.9	1.22584
$185$	21040.4i	0.614767i
$186$	0	0
$187$	−7894.03	−0.225744
$188$	− 906.630i	− 0.0256516i
$189$	0	0
$190$	2248.27	0.0622791
$191$	31341.7i	0.859125i	0.903037	+	0.429562i	$0.141332\pi$
$191$	31341.7i	0.859125i	−0.903037	+	0.429562i	$0.858668\pi$
$192$	0	0
$193$	30504.1	0.818925	0.409462	−	0.912327i	$-0.365716\pi$
$193$	30504.1	0.818925	0.409462	+	0.912327i	$0.365716\pi$
$194$	− 3489.64i	− 0.0927208i
$195$	0	0
$196$	8822.57	0.229659
$197$	− 56881.1i	− 1.46567i	−0.680407	−	0.732834i	$-0.738197\pi$
$197$	− 56881.1i	− 1.46567i	0.680407	−	0.732834i	$-0.261803\pi$
$198$	0	0
$199$	−32464.4	−0.819788	−0.409894	−	0.912133i	$-0.634434\pi$
$199$	−32464.4	−0.819788	−0.409894	+	0.912133i	$0.634434\pi$
$200$	− 20349.9i	− 0.508747i
$201$	0	0
$202$	51746.8	1.26818
$203$	8450.82i	0.205072i
$204$	0	0
$205$	8907.68	0.211961
$206$	− 64290.1i	− 1.51499i
$207$	0	0
$208$	38371.2	0.886909
$209$	− 5814.31i	− 0.133108i
$210$	0	0
$211$	43473.9	0.976480	0.488240	−	0.872709i	$-0.337639\pi$
$211$	43473.9	0.976480	0.488240	+	0.872709i	$0.337639\pi$
$212$	− 15621.4i	− 0.347574i
$213$	0	0
$214$	−76007.3	−1.65969
$215$	26216.0i	0.567140i
$216$	0	0
$217$	−31241.4	−0.663454
$218$	− 61376.5i	− 1.29149i
$219$	0	0
$220$	−12118.6	−0.250385
$221$	− 5490.64i	− 0.112419i
$222$	0	0
$223$	−36276.6	−0.729485	−0.364742	−	0.931108i	$-0.618843\pi$
$223$	−36276.6	−0.729485	−0.364742	+	0.931108i	$0.618843\pi$
$224$	10720.3i	0.213653i
$225$	0	0
$226$	89783.6	1.75784
$227$	− 36940.1i	− 0.716880i	−0.933553	−	0.358440i	$-0.883309\pi$
$227$	− 36940.1i	− 0.716880i	0.933553	−	0.358440i	$-0.116691\pi$
$228$	0	0
$229$	−72109.5	−1.37506	−0.687529	−	0.726156i	$-0.741305\pi$
$229$	−72109.5	−1.37506	−0.687529	+	0.726156i	$0.741305\pi$
$230$	− 54987.7i	− 1.03947i
$231$	0	0
$232$	22576.2	0.419444
$233$	26807.4i	0.493791i	0.969042	+	0.246896i	$0.0794105\pi$
$233$	26807.4i	0.493791i	−0.969042	+	0.246896i	$0.920590\pi$
$234$	0	0
$235$	3189.08	0.0577470
$236$	1983.95i	0.0356210i
$237$	0	0
$238$	3892.96	0.0687268
$239$	57224.4i	1.00181i	0.865502	+	0.500905i	$0.166999\pi$
$239$	57224.4i	1.00181i	−0.865502	+	0.500905i	$0.833001\pi$
$240$	0	0
$241$	−89841.9	−1.54684	−0.773419	−	0.633895i	$-0.781455\pi$
$241$	−89841.9	−1.54684	−0.773419	+	0.633895i	$0.781455\pi$
$242$	79791.5i	1.36247i
$243$	0	0
$244$	−28554.8	−0.479622
$245$	31033.4i	0.517009i
$246$	0	0
$247$	4044.10	0.0662870
$248$	83460.6i	1.35700i
$249$	0	0
$250$	−70408.0	−1.12653
$251$	46772.5i	0.742409i	0.928551	+	0.371204i	$0.121055\pi$
$251$	46772.5i	0.742409i	−0.928551	+	0.371204i	$0.878945\pi$
$252$	0	0
$253$	−142205.	−2.22164
$254$	− 10446.5i	− 0.161922i
$255$	0	0
$256$	50134.1	0.764985
$257$	97883.3i	1.48198i	0.671516	+	0.740990i	$0.265644\pi$
$257$	97883.3i	1.48198i	−0.671516	+	0.740990i	$0.734356\pi$
$258$	0	0
$259$	−26833.9	−0.400022
$260$	− 8429.03i	− 0.124690i
$261$	0	0
$262$	10664.5	0.155360
$263$	− 98074.0i	− 1.41789i	−0.705264	−	0.708945i	$-0.749172\pi$
$263$	− 98074.0i	− 1.41789i	0.705264	−	0.708945i	$-0.250828\pi$
$264$	0	0
$265$	54948.3	0.782460
$266$	2867.34i	0.0405243i
$267$	0	0
$268$	5045.29	0.0702451
$269$	− 61941.2i	− 0.856003i	−0.903778	−	0.428001i	$-0.859218\pi$
$269$	− 61941.2i	− 0.856003i	0.903778	−	0.428001i	$-0.140782\pi$
$270$	0	0
$271$	−79161.0	−1.07789	−0.538943	−	0.842342i	$-0.681176\pi$
$271$	−79161.0	−1.07789	−0.538943	+	0.842342i	$0.681176\pi$
$272$	− 13475.7i	− 0.182144i
$273$	0	0
$274$	−126165.	−1.68049
$275$	69728.1i	0.922024i
$276$	0	0
$277$	−69964.5	−0.911839	−0.455919	−	0.890021i	$-0.650689\pi$
$277$	−69964.5	−0.911839	−0.455919	+	0.890021i	$0.650689\pi$
$278$	− 23298.0i	− 0.301460i
$279$	0	0
$280$	−15866.2	−0.202374
$281$	129903.i	1.64516i	0.568650	+	0.822579i	$0.307466\pi$
$281$	129903.i	1.64516i	−0.568650	+	0.822579i	$0.692534\pi$
$282$	0	0
$283$	−115050.	−1.43652	−0.718261	−	0.695774i	$-0.755061\pi$
$283$	−115050.	−1.43652	−0.718261	+	0.695774i	$0.755061\pi$
$284$	− 238.646i	− 0.00295882i
$285$	0	0
$286$	−101468.	−1.24050
$287$	11360.4i	0.137921i
$288$	0	0
$289$	81592.7	0.976913
$290$	− 29912.1i	− 0.355674i
$291$	0	0
$292$	35521.5	0.416606
$293$	135254.i	1.57549i	0.616002	+	0.787744i	$0.288751\pi$
$293$	135254.i	1.57549i	−0.616002	+	0.787744i	$0.711249\pi$
$294$	0	0
$295$	−6978.55	−0.0801902
$296$	71686.1i	0.818185i
$297$	0	0
$298$	43655.8	0.491597
$299$	− 98909.6i	− 1.10636i
$300$	0	0
$301$	−33434.6	−0.369032
$302$	− 164517.i	− 1.80383i
$303$	0	0
$304$	9925.48	0.107400
$305$	− 100442.i	− 1.07973i
$306$	0	0
$307$	15959.4	0.169332	0.0846659	−	0.996409i	$-0.473018\pi$
$307$	15959.4	0.169332	0.0846659	+	0.996409i	$0.473018\pi$
$308$	− 15455.5i	− 0.162923i
$309$	0	0
$310$	110581.	1.15068
$311$	− 138076.i	− 1.42757i	−0.700365	−	0.713785i	$-0.746980\pi$
$311$	− 138076.i	− 1.42757i	0.700365	−	0.713785i	$-0.253020\pi$
$312$	0	0
$313$	−110443.	−1.12733	−0.563665	−	0.826003i	$-0.690609\pi$
$313$	−110443.	−1.12733	−0.563665	+	0.826003i	$0.690609\pi$
$314$	193780.i	1.96540i
$315$	0	0
$316$	−11953.0	−0.119702
$317$	− 35537.1i	− 0.353641i	−0.984243	−	0.176821i	$-0.943419\pi$
$317$	− 35537.1i	− 0.353641i	0.984243	−	0.176821i	$-0.0565813\pi$
$318$	0	0
$319$	−77356.4	−0.760177
$320$	37664.2i	0.367815i
$321$	0	0
$322$	70128.6	0.676369
$323$	− 1420.26i	− 0.0136133i
$324$	0	0
$325$	−48498.9	−0.459161
$326$	− 45151.7i	− 0.424853i
$327$	0	0
$328$	30349.1	0.282096
$329$	4067.19i	0.0375753i
$330$	0	0
$331$	−122330.	−1.11655	−0.558274	−	0.829657i	$-0.688536\pi$
$331$	−122330.	−1.11655	−0.558274	+	0.829657i	$0.688536\pi$
$332$	− 14350.8i	− 0.130197i
$333$	0	0
$334$	−128190.	−1.14911
$335$	17746.8i	0.158136i
$336$	0	0
$337$	201962.	1.77832	0.889160	−	0.457596i	$-0.151289\pi$
$337$	201962.	1.77832	0.889160	+	0.457596i	$0.151289\pi$
$338$	58353.7i	0.510781i
$339$	0	0
$340$	−2960.22	−0.0256075
$341$	− 285975.i	− 2.45934i
$342$	0	0
$343$	−86731.5	−0.737205
$344$	89319.8i	0.754799i
$345$	0	0
$346$	89183.4	0.744958
$347$	29020.7i	0.241018i	0.992712	+	0.120509i	$0.0384526\pi$
$347$	29020.7i	0.241018i	−0.992712	+	0.120509i	$0.961547\pi$
$348$	0	0
$349$	−123313.	−1.01241	−0.506205	−	0.862413i	$-0.668952\pi$
$349$	−123313.	−1.01241	−0.506205	+	0.862413i	$0.668952\pi$
$350$	− 34386.5i	− 0.280707i
$351$	0	0
$352$	−98130.3	−0.791987
$353$	− 144505.i	− 1.15967i	−0.814734	−	0.579835i	$-0.803117\pi$
$353$	− 144505.i	− 1.15967i	0.814734	−	0.579835i	$-0.196883\pi$
$354$	0	0
$355$	839.440	0.00666090
$356$	− 760.285i	− 0.00599897i
$357$	0	0
$358$	81238.3	0.633862
$359$	− 44975.7i	− 0.348971i	−0.984660	−	0.174485i	$-0.944174\pi$
$359$	− 44975.7i	− 0.348971i	0.984660	−	0.174485i	$-0.0558261\pi$
$360$	0	0
$361$	−129275.	−0.991973
$362$	− 38968.1i	− 0.297367i
$363$	0	0
$364$	10750.0	0.0811342
$365$	124947.i	0.937865i
$366$	0	0
$367$	−108087.	−0.802494	−0.401247	−	0.915970i	$-0.631423\pi$
$367$	−108087.	−0.802494	−0.401247	+	0.915970i	$0.631423\pi$
$368$	− 242755.i	− 1.79256i
$369$	0	0
$370$	94980.0	0.693791
$371$	70078.3i	0.509138i
$372$	0	0
$373$	39038.4	0.280591	0.140296	−	0.990110i	$-0.455195\pi$
$373$	39038.4	0.280591	0.140296	+	0.990110i	$0.455195\pi$
$374$	35635.0i	0.254762i
$375$	0	0
$376$	10865.4	0.0768547
$377$	− 53804.7i	− 0.378563i
$378$	0	0
$379$	−112968.	−0.786460	−0.393230	−	0.919440i	$-0.628642\pi$
$379$	−112968.	−0.786460	−0.393230	+	0.919440i	$0.628642\pi$
$380$	− 2180.33i	− 0.0150993i
$381$	0	0
$382$	141482.	0.969560
$383$	− 181513.i	− 1.23740i	−0.785628	−	0.618699i	$-0.787660\pi$
$383$	− 181513.i	− 1.23740i	0.785628	−	0.618699i	$-0.212340\pi$
$384$	0	0
$385$	54364.8	0.366772
$386$	− 137701.i	− 0.924192i
$387$	0	0
$388$	−3384.19	−0.0224797
$389$	199879.i	1.32089i	0.750873	+	0.660447i	$0.229633\pi$
$389$	199879.i	1.32089i	−0.750873	+	0.660447i	$0.770367\pi$
$390$	0	0
$391$	−34736.4	−0.227212
$392$	105733.i	0.688080i
$393$	0	0
$394$	−256771.	−1.65407
$395$	− 42044.7i	− 0.269474i
$396$	0	0
$397$	58756.8	0.372801	0.186401	−	0.982474i	$-0.440318\pi$
$397$	58756.8	0.372801	0.186401	+	0.982474i	$0.440318\pi$
$398$	146550.i	0.925166i
$399$	0	0
$400$	−119031.	−0.743946
$401$	233357.i	1.45121i	0.688110	+	0.725606i	$0.258441\pi$
$401$	233357.i	1.45121i	−0.688110	+	0.725606i	$0.741559\pi$
$402$	0	0
$403$	198908.	1.22473
$404$	− 50183.1i	− 0.307464i
$405$	0	0
$406$	38148.5	0.231433
$407$	− 245630.i	− 1.48283i
$408$	0	0
$409$	36418.4	0.217708	0.108854	−	0.994058i	$-0.465282\pi$
$409$	36418.4	0.217708	0.108854	+	0.994058i	$0.465282\pi$
$410$	− 40210.8i	− 0.239208i
$411$	0	0
$412$	−62347.4	−0.367302
$413$	− 8900.10i	− 0.0521789i
$414$	0	0
$415$	50479.0	0.293099
$416$	− 68253.9i	− 0.394403i
$417$	0	0
$418$	−26246.8	−0.150219
$419$	49083.7i	0.279582i	0.990181	+	0.139791i	$0.0446430\pi$
$419$	49083.7i	0.279582i	−0.990181	+	0.139791i	$0.955357\pi$
$420$	0	0
$421$	−41803.6	−0.235857	−0.117929	−	0.993022i	$-0.537625\pi$
$421$	−41803.6	−0.235857	−0.117929	+	0.993022i	$0.537625\pi$
$422$	− 196249.i	− 1.10200i
$423$	0	0
$424$	187212.	1.04137
$425$	17032.5i	0.0942976i
$426$	0	0
$427$	128098.	0.702566
$428$	73710.5i	0.402385i
$429$	0	0
$430$	118344.	0.640042
$431$	24480.2i	0.131783i	0.997827	+	0.0658916i	$0.0209892\pi$
$431$	24480.2i	0.131783i	−0.997827	+	0.0658916i	$0.979011\pi$
$432$	0	0
$433$	107487.	0.573299	0.286649	−	0.958036i	$-0.407459\pi$
$433$	107487.	0.573299	0.286649	+	0.958036i	$0.407459\pi$
$434$	141029.i	0.748737i
$435$	0	0
$436$	−59521.8	−0.313115
$437$	− 25584.9i	− 0.133974i
$438$	0	0
$439$	106710.	0.553701	0.276850	−	0.960913i	$-0.410709\pi$
$439$	106710.	0.553701	0.276850	+	0.960913i	$0.410709\pi$
$440$	− 145234.i	− 0.750177i
$441$	0	0
$442$	−24785.7	−0.126869
$443$	− 248355.i	− 1.26551i	−0.774353	−	0.632754i	$-0.781924\pi$
$443$	− 248355.i	− 1.26551i	0.774353	−	0.632754i	$-0.218076\pi$
$444$	0	0
$445$	2674.31	0.0135049
$446$	163759.i	0.823255i
$447$	0	0
$448$	−48035.1	−0.239333
$449$	7717.95i	0.0382833i	0.999817	+	0.0191416i	$0.00609335\pi$
$449$	7717.95i	0.0382833i	−0.999817	+	0.0191416i	$0.993907\pi$
$450$	0	0
$451$	−103990.	−0.511256
$452$	− 87070.5i	− 0.426181i
$453$	0	0
$454$	−166754.	−0.809031
$455$	37813.1i	0.182650i
$456$	0	0
$457$	−172040.	−0.823755	−0.411877	−	0.911239i	$-0.635127\pi$
$457$	−172040.	−0.823755	−0.411877	+	0.911239i	$0.635127\pi$
$458$	325515.i	1.55181i
$459$	0	0
$460$	−53326.1	−0.252014
$461$	− 222593.i	− 1.04739i	−0.851904	−	0.523697i	$-0.824552\pi$
$461$	− 222593.i	− 1.04739i	0.851904	−	0.523697i	$-0.175448\pi$
$462$	0	0
$463$	328016.	1.53015	0.765074	−	0.643943i	$-0.222702\pi$
$463$	328016.	1.53015	0.765074	+	0.643943i	$0.222702\pi$
$464$	− 132054.i	− 0.613358i
$465$	0	0
$466$	121013.	0.557265
$467$	89422.4i	0.410027i	0.978759	+	0.205014i	$0.0657238\pi$
$467$	89422.4i	0.410027i	−0.978759	+	0.205014i	$0.934276\pi$
$468$	0	0
$469$	−22633.4	−0.102897
$470$	− 14396.1i	− 0.0651700i
$471$	0	0
$472$	−23776.4	−0.106724
$473$	− 306051.i	− 1.36795i
$474$	0	0
$475$	−12545.2	−0.0556020
$476$	− 3775.32i	− 0.0166625i
$477$	0	0
$478$	258321.	1.13059
$479$	299550.i	1.30556i	0.757547	+	0.652781i	$0.226398\pi$
$479$	299550.i	1.30556i	−0.757547	+	0.652781i	$0.773602\pi$
$480$	0	0
$481$	170846.	0.738440
$482$	405562.i	1.74567i
$483$	0	0
$484$	77380.3	0.330324
$485$	− 11903.9i	− 0.0506064i
$486$	0	0
$487$	−154661.	−0.652113	−0.326057	−	0.945350i	$-0.605720\pi$
$487$	−154661.	−0.652113	−0.326057	+	0.945350i	$0.605720\pi$
$488$	− 342211.i	− 1.43699i
$489$	0	0
$490$	140090.	0.583467
$491$	− 293886.i	− 1.21903i	−0.792773	−	0.609517i	$-0.791363\pi$
$491$	− 293886.i	− 1.21903i	0.792773	−	0.609517i	$-0.208637\pi$
$492$	0	0
$493$	−18895.9	−0.0777452
$494$	− 18255.8i	− 0.0748077i
$495$	0	0
$496$	488182.	1.98435
$497$	1070.58i	0.00433418i
$498$	0	0
$499$	313019.	1.25710	0.628549	−	0.777770i	$-0.283649\pi$
$499$	313019.	1.25710	0.628549	+	0.777770i	$0.283649\pi$
$500$	68280.4i	0.273121i
$501$	0	0
$502$	211139.	0.837840
$503$	388086.i	1.53388i	0.641718	+	0.766941i	$0.278222\pi$
$503$	388086.i	1.53388i	−0.641718	+	0.766941i	$0.721778\pi$
$504$	0	0
$505$	176519.	0.692165
$506$	641937.i	2.50722i
$507$	0	0
$508$	−10130.9	−0.0392571
$509$	− 352135.i	− 1.35917i	−0.733597	−	0.679585i	$-0.762160\pi$
$509$	− 352135.i	− 1.35917i	0.733597	−	0.679585i	$-0.237840\pi$
$510$	0	0
$511$	−159351.	−0.610259
$512$	90089.6i	0.343664i
$513$	0	0
$514$	441862.	1.67248
$515$	− 219307.i	− 0.826873i
$516$	0	0
$517$	−37229.9	−0.139287
$518$	121133.i	0.451442i
$519$	0	0
$520$	101017.	0.373583
$521$	33158.3i	0.122156i	0.998133	+	0.0610782i	$0.0194539\pi$
$521$	33158.3i	0.122156i	−0.998133	+	0.0610782i	$0.980546\pi$
$522$	0	0
$523$	−127108.	−0.464695	−0.232348	−	0.972633i	$-0.574641\pi$
$523$	−127108.	−0.464695	−0.232348	+	0.972633i	$0.574641\pi$
$524$	− 10342.3i	− 0.0376663i
$525$	0	0
$526$	−442723.	−1.60015
$527$	− 69855.2i	− 0.251523i
$528$	0	0
$529$	−345909.	−1.23609
$530$	− 248046.i	− 0.883040i
$531$	0	0
$532$	2780.69	0.00982493
$533$	− 72329.5i	− 0.254602i
$534$	0	0
$535$	−259277.	−0.905851
$536$	60464.6i	0.210461i
$537$	0	0
$538$	−279613.	−0.966036
$539$	− 362290.i	− 1.24704i
$540$	0	0
$541$	−110260.	−0.376724	−0.188362	−	0.982100i	$-0.560318\pi$
$541$	−110260.	−0.376724	−0.188362	+	0.982100i	$0.560318\pi$
$542$	357347.i	1.21644i
$543$	0	0
$544$	−23970.3	−0.0809984
$545$	− 209368.i	− 0.704885i
$546$	0	0
$547$	−465135.	−1.55455	−0.777275	−	0.629162i	$-0.783398\pi$
$547$	−465135.	−1.55455	−0.777275	+	0.629162i	$0.783398\pi$
$548$	122352.i	0.407428i
$549$	0	0
$550$	314765.	1.04054
$551$	− 13917.7i	− 0.0458419i
$552$	0	0
$553$	53621.7	0.175344
$554$	315832.i	1.02905i
$555$	0	0
$556$	−22594.0	−0.0730876
$557$	19475.6i	0.0627740i	0.999507	+	0.0313870i	$0.00999243\pi$
$557$	19475.6i	0.0627740i	−0.999507	+	0.0313870i	$0.990008\pi$
$558$	0	0
$559$	212872.	0.681231
$560$	92805.0i	0.295934i
$561$	0	0
$562$	586406.	1.85663
$563$	− 380491.i	− 1.20040i	−0.799849	−	0.600202i	$-0.795087\pi$
$563$	− 380491.i	− 1.20040i	0.799849	−	0.600202i	$-0.204913\pi$
$564$	0	0
$565$	306271.	0.959420
$566$	519354.i	1.62118i
$567$	0	0
$568$	2860.03	0.00886491
$569$	− 222474.i	− 0.687155i	−0.939124	−	0.343578i	$-0.888361\pi$
$569$	− 222474.i	− 0.687155i	0.939124	−	0.343578i	$-0.111639\pi$
$570$	0	0
$571$	176925.	0.542648	0.271324	−	0.962488i	$-0.412539\pi$
$571$	176925.	0.542648	0.271324	+	0.962488i	$0.412539\pi$
$572$	98402.1i	0.300755i
$573$	0	0
$574$	51282.9	0.155650
$575$	306827.i	0.928022i
$576$	0	0
$577$	428908.	1.28829	0.644143	−	0.764905i	$-0.277214\pi$
$577$	428908.	1.28829	0.644143	+	0.764905i	$0.277214\pi$
$578$	− 368324.i	− 1.10249i
$579$	0	0
$580$	−29008.2	−0.0862314
$581$	64378.4i	0.190716i
$582$	0	0
$583$	−641476.	−1.88731
$584$	425703.i	1.24819i
$585$	0	0
$586$	610561.	1.77801
$587$	− 143526.i	− 0.416538i	−0.978072	−	0.208269i	$-0.933217\pi$
$587$	− 143526.i	− 0.416538i	0.978072	−	0.208269i	$-0.0667829\pi$
$588$	0	0
$589$	51451.4	0.148309
$590$	31502.4i	0.0904981i
$591$	0	0
$592$	419310.	1.19644
$593$	9519.94i	0.0270723i	0.999908	+	0.0135361i	$0.00430882\pi$
$593$	9519.94i	0.0270723i	−0.999908	+	0.0135361i	$0.995691\pi$
$594$	0	0
$595$	13279.7	0.0375107
$596$	− 42336.6i	− 0.119185i
$597$	0	0
$598$	−446495.	−1.24857
$599$	356820.i	0.994480i	0.867613	+	0.497240i	$0.165653\pi$
$599$	356820.i	0.994480i	−0.867613	+	0.497240i	$0.834347\pi$
$600$	0	0
$601$	−100436.	−0.278062	−0.139031	−	0.990288i	$-0.544399\pi$
$601$	−100436.	−0.278062	−0.139031	+	0.990288i	$0.544399\pi$
$602$	150930.i	0.416468i
$603$	0	0
$604$	−159545.	−0.437330
$605$	272186.i	0.743626i
$606$	0	0
$607$	343874.	0.933303	0.466651	−	0.884441i	$-0.345460\pi$
$607$	343874.	0.933303	0.466651	+	0.884441i	$0.345460\pi$
$608$	− 17655.2i	− 0.0477602i
$609$	0	0
$610$	−453411.	−1.21852
$611$	− 25895.0i	− 0.0693639i
$612$	0	0
$613$	289580.	0.770632	0.385316	−	0.922785i	$-0.374092\pi$
$613$	289580.	0.770632	0.385316	+	0.922785i	$0.374092\pi$
$614$	− 72043.3i	− 0.191098i
$615$	0	0
$616$	185225.	0.488132
$617$	175732.i	0.461616i	0.972999	+	0.230808i	$0.0741370\pi$
$617$	175732.i	0.461616i	−0.972999	+	0.230808i	$0.925863\pi$
$618$	0	0
$619$	546063.	1.42515	0.712577	−	0.701594i	$-0.247528\pi$
$619$	546063.	1.42515	0.712577	+	0.701594i	$0.247528\pi$
$620$	− 107239.i	− 0.278978i
$621$	0	0
$622$	−623299.	−1.61107
$623$	3410.68i	0.00878749i
$624$	0	0
$625$	2246.30	0.00575053
$626$	498561.i	1.27224i
$627$	0	0
$628$	187924.	0.476501
$629$	− 60000.1i	− 0.151653i
$630$	0	0
$631$	−124186.	−0.311900	−0.155950	−	0.987765i	$-0.549844\pi$
$631$	−124186.	−0.311900	−0.155950	+	0.987765i	$0.549844\pi$
$632$	− 143249.i	− 0.358639i
$633$	0	0
$634$	−160421.	−0.399100
$635$	− 35635.3i	− 0.0883758i
$636$	0	0
$637$	251989.	0.621015
$638$	349200.i	0.857893i
$639$	0	0
$640$	304515.	0.743444
$641$	457732.i	1.11403i	0.830504	+	0.557013i	$0.188053\pi$
$641$	457732.i	1.11403i	−0.830504	+	0.557013i	$0.811947\pi$
$642$	0	0
$643$	−584712.	−1.41423	−0.707115	−	0.707098i	$-0.750004\pi$
$643$	−584712.	−1.41423	−0.707115	+	0.707098i	$0.750004\pi$
$644$	− 68009.5i	− 0.163983i
$645$	0	0
$646$	−6411.31	−0.0153632
$647$	− 573738.i	− 1.37058i	−0.728270	−	0.685290i	$-0.759675\pi$
$647$	− 573738.i	− 1.37058i	0.728270	−	0.685290i	$-0.240325\pi$
$648$	0	0
$649$	81468.9	0.193421
$650$	218932.i	0.518183i
$651$	0	0
$652$	−43787.2	−0.103004
$653$	447594.i	1.04968i	0.851200	+	0.524842i	$0.175876\pi$
$653$	447594.i	1.04968i	−0.851200	+	0.524842i	$0.824124\pi$
$654$	0	0
$655$	36379.0	0.0847945
$656$	− 177519.i	− 0.412513i
$657$	0	0
$658$	18360.0	0.0424054
$659$	288724.i	0.664833i	0.943133	+	0.332416i	$0.107864\pi$
$659$	288724.i	0.664833i	−0.943133	+	0.332416i	$0.892136\pi$
$660$	0	0
$661$	404588.	0.925999	0.462999	−	0.886359i	$-0.346773\pi$
$661$	404588.	0.925999	0.462999	+	0.886359i	$0.346773\pi$
$662$	552219.i	1.26007i
$663$	0	0
$664$	171985.	0.390082
$665$	9781.10i	0.0221179i
$666$	0	0
$667$	−340395.	−0.765123
$668$	124316.i	0.278596i
$669$	0	0
$670$	80112.2	0.178463
$671$	1.17257e6i	2.60433i
$672$	0	0
$673$	−597937.	−1.32016	−0.660078	−	0.751197i	$-0.729477\pi$
$673$	−597937.	−1.32016	−0.660078	+	0.751197i	$0.729477\pi$
$674$	− 911692.i	− 2.00691i
$675$	0	0
$676$	56590.4	0.123837
$677$	55247.2i	0.120540i	0.998182	+	0.0602702i	$0.0191962\pi$
$677$	55247.2i	0.120540i	−0.998182	+	0.0602702i	$0.980804\pi$
$678$	0	0
$679$	15181.6	0.0329291
$680$	− 35476.4i	− 0.0767224i
$681$	0	0
$682$	−1.29094e6	−2.77547
$683$	184882.i	0.396326i	0.980169	+	0.198163i	$0.0634976\pi$
$683$	184882.i	0.396326i	−0.980169	+	0.198163i	$0.936502\pi$
$684$	0	0
$685$	−430375.	−0.917204
$686$	391521.i	0.831968i
$687$	0	0
$688$	522454.	1.10375
$689$	− 446175.i	− 0.939867i
$690$	0	0
$691$	89761.4	0.187989	0.0939947	−	0.995573i	$-0.470036\pi$
$691$	89761.4	0.187989	0.0939947	+	0.995573i	$0.470036\pi$
$692$	− 86488.4i	− 0.180612i
$693$	0	0
$694$	131004.	0.271999
$695$	− 79474.5i	− 0.164535i
$696$	0	0
$697$	−25401.7	−0.0522874
$698$	556654.i	1.14255i
$699$	0	0
$700$	−33347.4	−0.0680560
$701$	637414.i	1.29714i	0.761156	+	0.648569i	$0.224632\pi$
$701$	637414.i	1.29714i	−0.761156	+	0.648569i	$0.775368\pi$
$702$	0	0
$703$	44192.7	0.0894211
$704$	− 439699.i	− 0.887178i
$705$	0	0
$706$	−652323.	−1.30874
$707$	225124.i	0.450384i
$708$	0	0
$709$	−509846.	−1.01425	−0.507127	−	0.861871i	$-0.669292\pi$
$709$	−509846.	−1.01425	−0.507127	+	0.861871i	$0.669292\pi$
$710$	− 3789.38i	− 0.00751712i
$711$	0	0
$712$	9111.55	0.0179735
$713$	− 1.25839e6i	− 2.47534i
$714$	0	0
$715$	−346130.	−0.677060
$716$	− 78783.4i	− 0.153677i
$717$	0	0
$718$	−203028.	−0.393828
$719$	− 831097.i	− 1.60766i	−0.594859	−	0.803830i	$-0.702792\pi$
$719$	− 831097.i	− 1.60766i	0.594859	−	0.803830i	$-0.297208\pi$
$720$	0	0
$721$	279694.	0.538037
$722$	583569.i	1.11948i
$723$	0	0
$724$	−37790.6	−0.0720952
$725$	166908.i	0.317541i
$726$	0	0
$727$	238001.	0.450309	0.225154	−	0.974323i	$-0.427711\pi$
$727$	238001.	0.450309	0.225154	+	0.974323i	$0.427711\pi$
$728$	128832.i	0.243086i
$729$	0	0
$730$	564033.	1.05842
$731$	− 74759.2i	− 0.139904i
$732$	0	0
$733$	194098.	0.361255	0.180628	−	0.983552i	$-0.442187\pi$
$733$	194098.	0.361255	0.180628	+	0.983552i	$0.442187\pi$
$734$	487924.i	0.905650i
$735$	0	0
$736$	−431807.	−0.797139
$737$	− 207180.i	− 0.381428i
$738$	0	0
$739$	276770.	0.506792	0.253396	−	0.967363i	$-0.418452\pi$
$739$	276770.	0.506792	0.253396	+	0.967363i	$0.418452\pi$
$740$	− 92109.9i	− 0.168207i
$741$	0	0
$742$	316346.	0.574584
$743$	456121.i	0.826233i	0.910678	+	0.413116i	$0.135560\pi$
$743$	456121.i	0.826233i	−0.910678	+	0.413116i	$0.864440\pi$
$744$	0	0
$745$	148919.	0.268311
$746$	− 176226.i	− 0.316659i
$747$	0	0
$748$	34558.2	0.0617658
$749$	− 330669.i	− 0.589427i
$750$	0	0
$751$	655460.	1.16216	0.581080	−	0.813846i	$-0.302630\pi$
$751$	655460.	1.16216	0.581080	+	0.813846i	$0.302630\pi$
$752$	− 63554.4i	− 0.112385i
$753$	0	0
$754$	−242884.	−0.427224
$755$	− 561201.i	− 0.984520i
$756$	0	0
$757$	751680.	1.31172	0.655860	−	0.754883i	$-0.272306\pi$
$757$	751680.	1.31172	0.655860	+	0.754883i	$0.272306\pi$
$758$	509957.i	0.887554i
$759$	0	0
$760$	26130.0	0.0452389
$761$	597522.i	1.03177i	0.856657	+	0.515887i	$0.172538\pi$
$761$	597522.i	1.03177i	−0.856657	+	0.515887i	$0.827462\pi$
$762$	0	0
$763$	267018.	0.458661
$764$	− 137207.i	− 0.235065i
$765$	0	0
$766$	−819379.	−1.39646
$767$	56665.2i	0.0963220i
$768$	0	0
$769$	727106.	1.22955	0.614774	−	0.788704i	$-0.289247\pi$
$769$	727106.	1.22955	0.614774	+	0.788704i	$0.289247\pi$
$770$	− 245412.i	− 0.413918i
$771$	0	0
$772$	−133540.	−0.224066
$773$	89174.4i	0.149239i	0.997212	+	0.0746193i	$0.0237742\pi$
$773$	89174.4i	0.149239i	−0.997212	+	0.0746193i	$0.976226\pi$
$774$	0	0
$775$	−617032.	−1.02732
$776$	− 40557.4i	− 0.0673514i
$777$	0	0
$778$	902288.	1.49069
$779$	− 18709.5i	− 0.0308309i
$780$	0	0
$781$	−9799.79	−0.0160663
$782$	156806.i	0.256419i
$783$	0	0
$784$	618458.	1.00619
$785$	661025.i	1.07270i
$786$	0	0
$787$	−108299.	−0.174854	−0.0874270	−	0.996171i	$-0.527864\pi$
$787$	−108299.	−0.174854	−0.0874270	+	0.996171i	$0.527864\pi$
$788$	249012.i	0.401022i
$789$	0	0
$790$	−189797.	−0.304113
$791$	390603.i	0.624284i
$792$	0	0
$793$	−815576.	−1.29693
$794$	− 265238.i	− 0.420722i
$795$	0	0
$796$	142121.	0.224302
$797$	346101.i	0.544861i	0.962175	+	0.272431i	$0.0878276\pi$
$797$	346101.i	0.544861i	−0.962175	+	0.272431i	$0.912172\pi$
$798$	0	0
$799$	−9094.17	−0.0142452
$800$	211730.i	0.330829i
$801$	0	0
$802$	1.05341e6	1.63776
$803$	− 1.45866e6i	− 2.26215i
$804$	0	0
$805$	239224.	0.369158
$806$	− 897904.i	− 1.38217i
$807$	0	0
$808$	601413.	0.921193
$809$	− 426856.i	− 0.652206i	−0.945334	−	0.326103i	$-0.894264\pi$
$809$	− 426856.i	− 0.652206i	0.945334	−	0.326103i	$-0.105736\pi$
$810$	0	0
$811$	−529804.	−0.805514	−0.402757	−	0.915307i	$-0.631948\pi$
$811$	−529804.	−0.805514	−0.402757	+	0.915307i	$0.631948\pi$
$812$	− 36995.7i	− 0.0561098i
$813$	0	0
$814$	−1.10882e6	−1.67344
$815$	− 154022.i	− 0.231882i
$816$	0	0
$817$	55063.5	0.0824935
$818$	− 164399.i	− 0.245693i
$819$	0	0
$820$	−38995.7	−0.0579948
$821$	− 1.01564e6i	− 1.50680i	−0.657563	−	0.753400i	$-0.728413\pi$
$821$	− 1.01564e6i	− 1.50680i	0.657563	−	0.753400i	$-0.271587\pi$
$822$	0	0
$823$	−665731.	−0.982877	−0.491438	−	0.870912i	$-0.663529\pi$
$823$	−665731.	−0.982877	−0.491438	+	0.870912i	$0.663529\pi$
$824$	− 747195.i	− 1.10047i
$825$	0	0
$826$	−40176.6	−0.0588861
$827$	− 1.13151e6i	− 1.65443i	−0.561889	−	0.827213i	$-0.689925\pi$
$827$	− 1.13151e6i	− 1.65443i	0.561889	−	0.827213i	$-0.310075\pi$
$828$	0	0
$829$	−1.18587e6	−1.72555	−0.862776	−	0.505586i	$-0.831276\pi$
$829$	−1.18587e6	−1.72555	−0.862776	+	0.505586i	$0.831276\pi$
$830$	− 227871.i	− 0.330775i
$831$	0	0
$832$	305830.	0.441808
$833$	− 88496.8i	− 0.127537i
$834$	0	0
$835$	−437283.	−0.627176
$836$	25453.6i	0.0364198i
$837$	0	0
$838$	221572.	0.315520
$839$	58164.1i	0.0826287i	0.999146	+	0.0413143i	$0.0131545\pi$
$839$	58164.1i	0.0826287i	−0.999146	+	0.0413143i	$0.986845\pi$
$840$	0	0
$841$	522114.	0.738198
$842$	188709.i	0.266175i
$843$	0	0
$844$	−190318.	−0.267175
$845$	199057.i	0.278782i
$846$	0	0
$847$	−347132.	−0.483869
$848$	− 1.09505e6i	− 1.52280i
$849$	0	0
$850$	76887.7	0.106419
$851$	− 1.08086e6i	− 1.49248i
$852$	0	0
$853$	−1.00054e6	−1.37510	−0.687550	−	0.726137i	$-0.741314\pi$
$853$	−1.00054e6	−1.37510	−0.687550	+	0.726137i	$0.741314\pi$
$854$	− 578258.i	− 0.792877i
$855$	0	0
$856$	−883375.	−1.20558
$857$	574607.i	0.782365i	0.920313	+	0.391182i	$0.127934\pi$
$857$	574607.i	0.782365i	−0.920313	+	0.391182i	$0.872066\pi$
$858$	0	0
$859$	180215.	0.244234	0.122117	−	0.992516i	$-0.461032\pi$
$859$	180215.	0.244234	0.122117	+	0.992516i	$0.461032\pi$
$860$	− 114768.i	− 0.155175i
$861$	0	0
$862$	110508.	0.148723
$863$	− 318719.i	− 0.427943i	−0.976840	−	0.213972i	$-0.931360\pi$
$863$	− 318719.i	− 0.427943i	0.976840	−	0.213972i	$-0.0686400\pi$
$864$	0	0
$865$	304224.	0.406594
$866$	− 485216.i	− 0.646993i
$867$	0	0
$868$	136767.	0.181528
$869$	490838.i	0.649978i
$870$	0	0
$871$	144103.	0.189948
$872$	− 713333.i	− 0.938122i
$873$	0	0
$874$	−115495.	−0.151196
$875$	− 306309.i	− 0.400078i
$876$	0	0
$877$	−495975.	−0.644853	−0.322426	−	0.946595i	$-0.604498\pi$
$877$	−495975.	−0.644853	−0.322426	+	0.946595i	$0.604498\pi$
$878$	− 481707.i	− 0.624876i
$879$	0	0
$880$	−849511.	−1.09699
$881$	195953.i	0.252464i	0.992001	+	0.126232i	$0.0402884\pi$
$881$	195953.i	0.252464i	−0.992001	+	0.126232i	$0.959712\pi$
$882$	0	0
$883$	620866.	0.796300	0.398150	−	0.917320i	$-0.369652\pi$
$883$	620866.	0.796300	0.398150	+	0.917320i	$0.369652\pi$
$884$	24036.7i	0.0307589i
$885$	0	0
$886$	−1.12112e6	−1.42818
$887$	− 65081.0i	− 0.0827193i	−0.999144	−	0.0413597i	$-0.986831\pi$
$887$	− 65081.0i	− 0.0827193i	0.999144	−	0.0413597i	$-0.0131689\pi$
$888$	0	0
$889$	45447.5	0.0575052
$890$	− 12072.3i	− 0.0152409i
$891$	0	0
$892$	158810.	0.199594
$893$	− 6698.26i	− 0.00839961i
$894$	0	0
$895$	277121.	0.345958
$896$	388363.i	0.483751i
$897$	0	0
$898$	34840.2	0.0432043
$899$	− 684535.i	− 0.846987i
$900$	0	0
$901$	−156694.	−0.193020
$902$	469429.i	0.576974i
$903$	0	0
$904$	1.04349e6	1.27688
$905$	− 132929.i	− 0.162301i
$906$	0	0
$907$	−551565.	−0.670474	−0.335237	−	0.942134i	$-0.608816\pi$
$907$	−551565.	−0.670474	−0.335237	+	0.942134i	$0.608816\pi$
$908$	161715.i	0.196146i
$909$	0	0
$910$	170695.	0.206128
$911$	302394.i	0.364365i	0.983265	+	0.182182i	$0.0583161\pi$
$911$	302394.i	0.364365i	−0.983265	+	0.182182i	$0.941684\pi$
$912$	0	0
$913$	−589302.	−0.706962
$914$	776620.i	0.929643i
$915$	0	0
$916$	315678.	0.376230
$917$	46395.9i	0.0551749i
$918$	0	0
$919$	239555.	0.283644	0.141822	−	0.989892i	$-0.454704\pi$
$919$	239555.	0.283644	0.141822	+	0.989892i	$0.454704\pi$
$920$	− 639080.i	− 0.755057i
$921$	0	0
$922$	−1.00483e6	−1.18203
$923$	− 6816.18i	− 0.00800088i
$924$	0	0
$925$	−529982.	−0.619409
$926$	− 1.48072e6i	− 1.72684i
$927$	0	0
$928$	−234894.	−0.272757
$929$	996631.i	1.15479i	0.816465	+	0.577395i	$0.195931\pi$
$929$	996631.i	1.15479i	−0.816465	+	0.577395i	$0.804069\pi$
$930$	0	0
$931$	65181.9	0.0752017
$932$	− 117357.i	− 0.135106i
$933$	0	0
$934$	403668.	0.462733
$935$	121559.i	0.139047i
$936$	0	0
$937$	1.17812e6	1.34187	0.670937	−	0.741514i	$-0.265892\pi$
$937$	1.17812e6	1.34187	0.670937	+	0.741514i	$0.265892\pi$
$938$	102171.i	0.116124i
$939$	0	0
$940$	−13961.0	−0.0158002
$941$	808862.i	0.913472i	0.889602	+	0.456736i	$0.150982\pi$
$941$	808862.i	0.913472i	−0.889602	+	0.456736i	$0.849018\pi$
$942$	0	0
$943$	−457591.	−0.514582
$944$	139074.i	0.156064i
$945$	0	0
$946$	−1.38157e6	−1.54380
$947$	− 1.23676e6i	− 1.37906i	−0.724255	−	0.689532i	$-0.757816\pi$
$947$	− 1.23676e6i	− 1.37906i	0.724255	−	0.689532i	$-0.242184\pi$
$948$	0	0
$949$	1.01456e6	1.12654
$950$	56631.2i	0.0627493i
$951$	0	0
$952$	45244.9	0.0499224
$953$	693976.i	0.764115i	0.924139	+	0.382057i	$0.124784\pi$
$953$	693976.i	0.764115i	−0.924139	+	0.382057i	$0.875216\pi$
$954$	0	0
$955$	482625.	0.529180
$956$	− 250515.i	− 0.274105i
$957$	0	0
$958$	1.35222e6	1.47338
$959$	− 548879.i	− 0.596814i
$960$	0	0
$961$	1.60710e6	1.74019
$962$	− 771229.i	− 0.833361i
$963$	0	0
$964$	393307.	0.423231
$965$	− 469727.i	− 0.504419i
$966$	0	0
$967$	−334109.	−0.357302	−0.178651	−	0.983913i	$-0.557173\pi$
$967$	−334109.	−0.357302	−0.178651	+	0.983913i	$0.557173\pi$
$968$	927356.i	0.989682i
$969$	0	0
$970$	−53736.3	−0.0571116
$971$	− 168226.i	− 0.178425i	−0.996013	−	0.0892125i	$-0.971565\pi$
$971$	− 168226.i	− 0.178425i	0.996013	−	0.0892125i	$-0.0284350\pi$
$972$	0	0
$973$	101358.	0.107061
$974$	698167.i	0.735938i
$975$	0	0
$976$	−2.00168e6	−2.10133
$977$	650514.i	0.681503i	0.940153	+	0.340751i	$0.110681\pi$
$977$	650514.i	0.681503i	−0.940153	+	0.340751i	$0.889319\pi$
$978$	0	0
$979$	−31220.4	−0.0325741
$980$	− 135857.i	− 0.141459i
$981$	0	0
$982$	−1.32665e6	−1.37573
$983$	1.29506e6i	1.34024i	0.742254	+	0.670119i	$0.233757\pi$
$983$	1.29506e6i	1.34024i	−0.742254	+	0.670119i	$0.766243\pi$
$984$	0	0
$985$	−875902.	−0.902782
$986$	85299.3i	0.0877388i
$987$	0	0
$988$	−17704.1	−0.0181368
$989$	− 1.34673e6i	− 1.37685i
$990$	0	0
$991$	−1.12305e6	−1.14354	−0.571772	−	0.820413i	$-0.693744\pi$
$991$	−1.12305e6	−1.14354	−0.571772	+	0.820413i	$0.693744\pi$
$992$	− 868366.i	− 0.882429i
$993$	0	0
$994$	4832.79	0.00489131
$995$	499913.i	0.504950i
$996$	0	0
$997$	1.11322e6	1.11993	0.559966	−	0.828516i	$-0.310814\pi$
$997$	1.11322e6	1.11993	0.559966	+	0.828516i	$0.310814\pi$
$998$	− 1.41302e6i	− 1.41869i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.19	✓	76
3.2	odd	2	inner	531.5.b.a.296.58	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.19	✓	76	1.1	even	1	trivial
531.5.b.a.296.58	yes	76	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.51417i q^{2} -4.37776 q^{4} -15.3988i q^{5} +19.6389 q^{7} -52.4648i q^{8} +O(q^{10})\)	\(q-4.51417i q^{2} -4.37776 q^{4} -15.3988i q^{5} +19.6389 q^{7} -52.4648i q^{8} -69.5129 q^{10} +179.769i q^{11} -125.037 q^{13} -88.6533i q^{14} -306.879 q^{16} +43.9122i q^{17} -32.3433 q^{19} +67.4123i q^{20} +811.507 q^{22} +791.044i q^{23} +387.877 q^{25} +564.438i q^{26} -85.9743 q^{28} +430.311i q^{29} -1590.79 q^{31} +545.870i q^{32} +198.227 q^{34} -302.415i q^{35} -1366.37 q^{37} +146.003i q^{38} -807.895 q^{40} +578.465i q^{41} -1702.47 q^{43} -786.984i q^{44} +3570.91 q^{46} +207.099i q^{47} -2015.31 q^{49} -1750.94i q^{50} +547.382 q^{52} +3568.34i q^{53} +2768.22 q^{55} -1030.35i q^{56} +1942.50 q^{58} -453.188i q^{59} +6522.69 q^{61} +7181.12i q^{62} -2445.92 q^{64} +1925.42i q^{65} -1152.48 q^{67} -192.237i q^{68} -1365.16 q^{70} +54.5133i q^{71} -8114.08 q^{73} +6168.01i q^{74} +141.591 q^{76} +3530.45i q^{77} +2730.39 q^{79} +4725.58i q^{80} +2611.29 q^{82} +3278.11i q^{83} +676.195 q^{85} +7685.25i q^{86} +9431.53 q^{88} +173.670i q^{89} -2455.58 q^{91} -3463.00i q^{92} +934.881 q^{94} +498.048i q^{95} +773.040 q^{97} +9097.48i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 76 * q - 632 * q^4 - 48 * q^7	\( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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