LMFDB - Embedded newform 810.5.d.c.161.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,5,Mod(161,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, 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("810.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	810.d (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:	$83.7296700979$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.12
Character	$\chi$	$=$	810.161
Dual form			810.5.d.c.161.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +21.7826 q^{7} -22.6274i q^{8} +O(q^{10})$	\(q+2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +21.7826 q^{7} -22.6274i q^{8} +31.6228 q^{10} -9.40670i q^{11} -58.6807 q^{13} +61.6104i q^{14} +64.0000 q^{16} -410.280i q^{17} +410.417 q^{19} +89.4427i q^{20} +26.6062 q^{22} +652.270i q^{23} -125.000 q^{25} -165.974i q^{26} -174.261 q^{28} -227.096i q^{29} -1369.93 q^{31} +181.019i q^{32} +1160.45 q^{34} -243.537i q^{35} +558.643 q^{37} +1160.83i q^{38} -252.982 q^{40} -636.000i q^{41} +1071.38 q^{43} +75.2536i q^{44} -1844.90 q^{46} -909.165i q^{47} -1926.52 q^{49} -353.553i q^{50} +469.446 q^{52} -1794.11i q^{53} -105.170 q^{55} -492.883i q^{56} +642.325 q^{58} +4244.43i q^{59} +2972.01 q^{61} -3874.73i q^{62} -512.000 q^{64} +656.070i q^{65} -5471.41 q^{67} +3282.24i q^{68} +688.826 q^{70} +6487.87i q^{71} -247.606 q^{73} +1580.08i q^{74} -3283.34 q^{76} -204.902i q^{77} -11667.5 q^{79} -715.542i q^{80} +1798.88 q^{82} -8349.97i q^{83} -4587.07 q^{85} +3030.33i q^{86} -212.849 q^{88} -14454.4i q^{89} -1278.22 q^{91} -5218.16i q^{92} +2571.51 q^{94} -4588.60i q^{95} -15852.1 q^{97} -5449.02i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) $ 32 * q - 256 * q^4 - 104 * q^7	$ 32 q - 256 q^{4} - 104 q^{7} + 40 q^{13} + 2048 q^{16} - 200 q^{19} + 1344 q^{22} - 4000 q^{25} + 832 q^{28} - 1472 q^{31} - 384 q^{34} - 4136 q^{37} - 272 q^{43} - 2112 q^{46} + 22296 q^{49} - 320 q^{52} - 12344 q^{61} - 16384 q^{64} + 40936 q^{67} + 4800 q^{70} - 41432 q^{73} + 1600 q^{76} - 14048 q^{79} - 17664 q^{82} - 17400 q^{85} - 10752 q^{88} + 69392 q^{91} + 1344 q^{94} + 36664 q^{97}+O(q^{100}) $ 32 * q - 256 * q^4 - 104 * q^7 + 40 * q^13 + 2048 * q^16 - 200 * q^19 + 1344 * q^22 - 4000 * q^25 + 832 * q^28 - 1472 * q^31 - 384 * q^34 - 4136 * q^37 - 272 * q^43 - 2112 * q^46 + 22296 * q^49 - 320 * q^52 - 12344 * q^61 - 16384 * q^64 + 40936 * q^67 + 4800 * q^70 - 41432 * q^73 + 1600 * q^76 - 14048 * q^79 - 17664 * q^82 - 17400 * q^85 - 10752 * q^88 + 69392 * q^91 + 1344 * q^94 + 36664 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	21.7826	0.444542	0.222271	−	0.974985i	$-0.428653\pi$
$7$	21.7826	0.444542	0.222271	+	0.974985i	$0.428653\pi$
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	31.6228	0.316228
$11$	− 9.40670i	− 0.0777413i	−0.999244	−	0.0388707i	$-0.987624\pi$
$11$	− 9.40670i	− 0.0777413i	0.999244	−	0.0388707i	$-0.0123760\pi$
$12$	0	0
$13$	−58.6807	−0.347223	−0.173612	−	0.984814i	$-0.555544\pi$
$13$	−58.6807	−0.347223	−0.173612	+	0.984814i	$0.555544\pi$
$14$	61.6104i	0.314339i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 410.280i	− 1.41965i	−0.704377	−	0.709827i	$-0.748773\pi$
$17$	− 410.280i	− 1.41965i	0.704377	−	0.709827i	$-0.251227\pi$
$18$	0	0
$19$	410.417	1.13689	0.568444	−	0.822722i	$-0.307545\pi$
$19$	410.417	1.13689	0.568444	+	0.822722i	$0.307545\pi$
$20$	89.4427i	0.223607i
$21$	0	0
$22$	26.6062	0.0549714
$23$	652.270i	1.23302i	0.787346	+	0.616512i	$0.211455\pi$
$23$	652.270i	1.23302i	−0.787346	+	0.616512i	$0.788545\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	− 165.974i	− 0.245524i
$27$	0	0
$28$	−174.261	−0.222271
$29$	− 227.096i	− 0.270031i	−0.990843	−	0.135016i	$-0.956892\pi$
$29$	− 227.096i	− 0.270031i	0.990843	−	0.135016i	$-0.0431085\pi$
$30$	0	0
$31$	−1369.93	−1.42552	−0.712760	−	0.701408i	$-0.752555\pi$
$31$	−1369.93	−1.42552	−0.712760	+	0.701408i	$0.752555\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	1160.45	1.00385
$35$	− 243.537i	− 0.198805i
$36$	0	0
$37$	558.643	0.408066	0.204033	−	0.978964i	$-0.434595\pi$
$37$	558.643	0.408066	0.204033	+	0.978964i	$0.434595\pi$
$38$	1160.83i	0.803902i
$39$	0	0
$40$	−252.982	−0.158114
$41$	− 636.000i	− 0.378346i	−0.981944	−	0.189173i	$-0.939419\pi$
$41$	− 636.000i	− 0.378346i	0.981944	−	0.189173i	$-0.0605807\pi$
$42$	0	0
$43$	1071.38	0.579439	0.289719	−	0.957112i	$-0.406438\pi$
$43$	1071.38	0.579439	0.289719	+	0.957112i	$0.406438\pi$
$44$	75.2536i	0.0388707i
$45$	0	0
$46$	−1844.90	−0.871880
$47$	− 909.165i	− 0.411573i	−0.978597	−	0.205787i	$-0.934025\pi$
$47$	− 909.165i	− 0.411573i	0.978597	−	0.205787i	$-0.0659753\pi$
$48$	0	0
$49$	−1926.52	−0.802382
$50$	− 353.553i	− 0.141421i
$51$	0	0
$52$	469.446	0.173612
$53$	− 1794.11i	− 0.638699i	−0.947637	−	0.319350i	$-0.896536\pi$
$53$	− 1794.11i	− 0.638699i	0.947637	−	0.319350i	$-0.103464\pi$
$54$	0	0
$55$	−105.170	−0.0347670
$56$	− 492.883i	− 0.157169i
$57$	0	0
$58$	642.325	0.190941
$59$	4244.43i	1.21931i	0.792666	+	0.609656i	$0.208693\pi$
$59$	4244.43i	1.21931i	−0.792666	+	0.609656i	$0.791307\pi$
$60$	0	0
$61$	2972.01	0.798712	0.399356	−	0.916796i	$-0.369234\pi$
$61$	2972.01	0.798712	0.399356	+	0.916796i	$0.369234\pi$
$62$	− 3874.73i	− 1.00800i
$63$	0	0
$64$	−512.000	−0.125000
$65$	656.070i	0.155283i
$66$	0	0
$67$	−5471.41	−1.21885	−0.609424	−	0.792845i	$-0.708599\pi$
$67$	−5471.41	−1.21885	−0.609424	+	0.792845i	$0.708599\pi$
$68$	3282.24i	0.709827i
$69$	0	0
$70$	688.826	0.140577
$71$	6487.87i	1.28702i	0.765438	+	0.643510i	$0.222522\pi$
$71$	6487.87i	1.28702i	−0.765438	+	0.643510i	$0.777478\pi$
$72$	0	0
$73$	−247.606	−0.0464638	−0.0232319	−	0.999730i	$-0.507396\pi$
$73$	−247.606	−0.0464638	−0.0232319	+	0.999730i	$0.507396\pi$
$74$	1580.08i	0.288546i
$75$	0	0
$76$	−3283.34	−0.568444
$77$	− 204.902i	− 0.0345593i
$78$	0	0
$79$	−11667.5	−1.86949	−0.934746	−	0.355316i	$-0.884373\pi$
$79$	−11667.5	−1.86949	−0.934746	+	0.355316i	$0.884373\pi$
$80$	− 715.542i	− 0.111803i
$81$	0	0
$82$	1798.88	0.267531
$83$	− 8349.97i	− 1.21207i	−0.795437	−	0.606036i	$-0.792759\pi$
$83$	− 8349.97i	− 1.21207i	0.795437	−	0.606036i	$-0.207241\pi$
$84$	0	0
$85$	−4587.07	−0.634888
$86$	3030.33i	0.409725i
$87$	0	0
$88$	−212.849	−0.0274857
$89$	− 14454.4i	− 1.82482i	−0.409273	−	0.912412i	$-0.634218\pi$
$89$	− 14454.4i	− 1.82482i	0.409273	−	0.912412i	$-0.365782\pi$
$90$	0	0
$91$	−1278.22	−0.154355
$92$	− 5218.16i	− 0.616512i
$93$	0	0
$94$	2571.51	0.291026
$95$	− 4588.60i	− 0.508432i
$96$	0	0
$97$	−15852.1	−1.68478	−0.842391	−	0.538867i	$-0.818852\pi$
$97$	−15852.1	−1.68478	−0.842391	+	0.538867i	$0.818852\pi$
$98$	− 5449.02i	− 0.567370i
$99$	0	0
$100$	1000.00	0.100000
$101$	− 9941.19i	− 0.974531i	−0.873254	−	0.487266i	$-0.837994\pi$
$101$	− 9941.19i	− 0.974531i	0.873254	−	0.487266i	$-0.162006\pi$
$102$	0	0
$103$	−1922.26	−0.181191	−0.0905955	−	0.995888i	$-0.528877\pi$
$103$	−1922.26	−0.181191	−0.0905955	+	0.995888i	$0.528877\pi$
$104$	1327.79i	0.122762i
$105$	0	0
$106$	5074.50	0.451629
$107$	3591.35i	0.313683i	0.987624	+	0.156841i	$0.0501311\pi$
$107$	3591.35i	0.313683i	−0.987624	+	0.156841i	$0.949869\pi$
$108$	0	0
$109$	7956.77	0.669705	0.334853	−	0.942271i	$-0.391314\pi$
$109$	7956.77	0.669705	0.334853	+	0.942271i	$0.391314\pi$
$110$	− 297.466i	− 0.0245840i
$111$	0	0
$112$	1394.08	0.111136
$113$	− 20206.6i	− 1.58247i	−0.611512	−	0.791235i	$-0.709438\pi$
$113$	− 20206.6i	− 1.58247i	0.611512	−	0.791235i	$-0.290562\pi$
$114$	0	0
$115$	7292.60	0.551425
$116$	1816.77i	0.135016i
$117$	0	0
$118$	−12005.1	−0.862184
$119$	− 8936.95i	− 0.631096i
$120$	0	0
$121$	14552.5	0.993956
$122$	8406.10i	0.564775i
$123$	0	0
$124$	10959.4	0.712760
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	−9963.27	−0.617724	−0.308862	−	0.951107i	$-0.599948\pi$
$127$	−9963.27	−0.617724	−0.308862	+	0.951107i	$0.599948\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	−1855.65	−0.109802
$131$	− 30361.5i	− 1.76922i	−0.466336	−	0.884608i	$-0.654426\pi$
$131$	− 30361.5i	− 1.76922i	0.466336	−	0.884608i	$-0.345574\pi$
$132$	0	0
$133$	8939.94	0.505395
$134$	− 15475.5i	− 0.861856i
$135$	0	0
$136$	−9283.57	−0.501923
$137$	8569.90i	0.456599i	0.973591	+	0.228299i	$0.0733165\pi$
$137$	8569.90i	0.456599i	−0.973591	+	0.228299i	$0.926684\pi$
$138$	0	0
$139$	−11787.6	−0.610094	−0.305047	−	0.952337i	$-0.598672\pi$
$139$	−11787.6	−0.610094	−0.305047	+	0.952337i	$0.598672\pi$
$140$	1948.29i	0.0994027i
$141$	0	0
$142$	−18350.5	−0.910060
$143$	551.992i	0.0269936i
$144$	0	0
$145$	−2539.01	−0.120762
$146$	− 700.335i	− 0.0328549i
$147$	0	0
$148$	−4469.14	−0.204033
$149$	− 26217.4i	− 1.18091i	−0.807070	−	0.590456i	$-0.798948\pi$
$149$	− 26217.4i	− 1.18091i	0.807070	−	0.590456i	$-0.201052\pi$
$150$	0	0
$151$	−16770.6	−0.735520	−0.367760	−	0.929921i	$-0.619875\pi$
$151$	−16770.6	−0.735520	−0.367760	+	0.929921i	$0.619875\pi$
$152$	− 9286.67i	− 0.401951i
$153$	0	0
$154$	579.551	0.0244371
$155$	15316.2i	0.637512i
$156$	0	0
$157$	15422.5	0.625683	0.312842	−	0.949805i	$-0.398719\pi$
$157$	15422.5	0.625683	0.312842	+	0.949805i	$0.398719\pi$
$158$	− 33000.7i	− 1.32193i
$159$	0	0
$160$	2023.86	0.0790569
$161$	14208.1i	0.548131i
$162$	0	0
$163$	−12736.4	−0.479369	−0.239685	−	0.970851i	$-0.577044\pi$
$163$	−12736.4	−0.479369	−0.239685	+	0.970851i	$0.577044\pi$
$164$	5088.00i	0.189173i
$165$	0	0
$166$	23617.3	0.857065
$167$	− 7333.57i	− 0.262955i	−0.991319	−	0.131478i	$-0.958028\pi$
$167$	− 7333.57i	− 0.262955i	0.991319	−	0.131478i	$-0.0419722\pi$
$168$	0	0
$169$	−25117.6	−0.879436
$170$	− 12974.2i	− 0.448934i
$171$	0	0
$172$	−8571.06	−0.289719
$173$	801.220i	0.0267707i	0.999910	+	0.0133853i	$0.00426081\pi$
$173$	801.220i	0.0267707i	−0.999910	+	0.0133853i	$0.995739\pi$
$174$	0	0
$175$	−2722.82	−0.0889085
$176$	− 602.029i	− 0.0194353i
$177$	0	0
$178$	40883.3	1.29035
$179$	− 4488.67i	− 0.140091i	−0.997544	−	0.0700456i	$-0.977686\pi$
$179$	− 4488.67i	− 0.140091i	0.997544	−	0.0700456i	$-0.0223145\pi$
$180$	0	0
$181$	17747.9	0.541737	0.270869	−	0.962616i	$-0.412689\pi$
$181$	17747.9	0.541737	0.270869	+	0.962616i	$0.412689\pi$
$182$	− 3615.34i	− 0.109146i
$183$	0	0
$184$	14759.2	0.435940
$185$	− 6245.82i	− 0.182493i
$186$	0	0
$187$	−3859.38	−0.110366
$188$	7273.32i	0.205787i
$189$	0	0
$190$	12978.5	0.359516
$191$	− 66365.6i	− 1.81918i	−0.415506	−	0.909591i	$-0.636395\pi$
$191$	− 66365.6i	− 1.81918i	0.415506	−	0.909591i	$-0.363605\pi$
$192$	0	0
$193$	−10122.7	−0.271757	−0.135878	−	0.990726i	$-0.543386\pi$
$193$	−10122.7	−0.271757	−0.135878	+	0.990726i	$0.543386\pi$
$194$	− 44836.5i	− 1.19132i
$195$	0	0
$196$	15412.2	0.401191
$197$	26543.2i	0.683945i	0.939710	+	0.341972i	$0.111095\pi$
$197$	26543.2i	0.683945i	−0.939710	+	0.341972i	$0.888905\pi$
$198$	0	0
$199$	44723.6	1.12936	0.564678	−	0.825311i	$-0.309000\pi$
$199$	44723.6	1.12936	0.564678	+	0.825311i	$0.309000\pi$
$200$	2828.43i	0.0707107i
$201$	0	0
$202$	28117.9	0.689098
$203$	− 4946.74i	− 0.120040i
$204$	0	0
$205$	−7110.69	−0.169202
$206$	− 5436.96i	− 0.128121i
$207$	0	0
$208$	−3755.57	−0.0868058
$209$	− 3860.67i	− 0.0883833i
$210$	0	0
$211$	68331.1	1.53481	0.767403	−	0.641165i	$-0.221548\pi$
$211$	68331.1	1.53481	0.767403	+	0.641165i	$0.221548\pi$
$212$	14352.9i	0.319350i
$213$	0	0
$214$	−10157.9	−0.221807
$215$	− 11978.4i	− 0.259133i
$216$	0	0
$217$	−29840.5	−0.633704
$218$	22505.1i	0.473553i
$219$	0	0
$220$	841.361	0.0173835
$221$	24075.5i	0.492936i
$222$	0	0
$223$	22439.6	0.451238	0.225619	−	0.974216i	$-0.427559\pi$
$223$	22439.6	0.451238	0.225619	+	0.974216i	$0.427559\pi$
$224$	3943.07i	0.0785847i
$225$	0	0
$226$	57152.8	1.11898
$227$	− 88009.4i	− 1.70796i	−0.520307	−	0.853979i	$-0.674182\pi$
$227$	− 88009.4i	− 1.70796i	0.520307	−	0.853979i	$-0.325818\pi$
$228$	0	0
$229$	−41547.6	−0.792274	−0.396137	−	0.918192i	$-0.629649\pi$
$229$	−41547.6	−0.792274	−0.396137	+	0.918192i	$0.629649\pi$
$230$	20626.6i	0.389916i
$231$	0	0
$232$	−5138.60	−0.0954704
$233$	− 86321.1i	− 1.59003i	−0.606591	−	0.795014i	$-0.707463\pi$
$233$	− 86321.1i	− 1.59003i	0.606591	−	0.795014i	$-0.292537\pi$
$234$	0	0
$235$	−10164.8	−0.184061
$236$	− 33955.4i	− 0.609656i
$237$	0	0
$238$	25277.5	0.446252
$239$	62834.2i	1.10002i	0.835158	+	0.550009i	$0.185376\pi$
$239$	62834.2i	1.10002i	−0.835158	+	0.550009i	$0.814624\pi$
$240$	0	0
$241$	−93603.2	−1.61160	−0.805799	−	0.592189i	$-0.798264\pi$
$241$	−93603.2	−1.61160	−0.805799	+	0.592189i	$0.798264\pi$
$242$	41160.7i	0.702833i
$243$	0	0
$244$	−23776.1	−0.399356
$245$	21539.1i	0.358836i
$246$	0	0
$247$	−24083.6	−0.394754
$248$	30997.9i	0.503998i
$249$	0	0
$250$	−3952.85	−0.0632456
$251$	− 9680.26i	− 0.153652i	−0.997044	−	0.0768262i	$-0.975521\pi$
$251$	− 9680.26i	− 0.153652i	0.997044	−	0.0768262i	$-0.0244787\pi$
$252$	0	0
$253$	6135.71	0.0958569
$254$	− 28180.4i	− 0.436797i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 36145.9i	− 0.547259i	−0.961835	−	0.273630i	$-0.911776\pi$
$257$	− 36145.9i	− 0.547259i	0.961835	−	0.273630i	$-0.0882242\pi$
$258$	0	0
$259$	12168.7	0.181403
$260$	− 5248.56i	− 0.0776414i
$261$	0	0
$262$	85875.3	1.25102
$263$	107432.i	1.55318i	0.630009	+	0.776588i	$0.283051\pi$
$263$	107432.i	1.55318i	−0.630009	+	0.776588i	$0.716949\pi$
$264$	0	0
$265$	−20058.7	−0.285635
$266$	25286.0i	0.357368i
$267$	0	0
$268$	43771.3	0.609424
$269$	− 36125.0i	− 0.499233i	−0.968345	−	0.249616i	$-0.919695\pi$
$269$	− 36125.0i	− 0.499233i	0.968345	−	0.249616i	$-0.0803045\pi$
$270$	0	0
$271$	−12964.6	−0.176530	−0.0882651	−	0.996097i	$-0.528132\pi$
$271$	−12964.6	−0.176530	−0.0882651	+	0.996097i	$0.528132\pi$
$272$	− 26257.9i	− 0.354913i
$273$	0	0
$274$	−24239.3	−0.322864
$275$	1175.84i	0.0155483i
$276$	0	0
$277$	122051.	1.59067	0.795336	−	0.606168i	$-0.207294\pi$
$277$	122051.	1.59067	0.795336	+	0.606168i	$0.207294\pi$
$278$	− 33340.5i	− 0.431402i
$279$	0	0
$280$	−5510.60	−0.0702883
$281$	− 51587.3i	− 0.653326i	−0.945141	−	0.326663i	$-0.894076\pi$
$281$	− 51587.3i	− 0.653326i	0.945141	−	0.326663i	$-0.105924\pi$
$282$	0	0
$283$	−109975.	−1.37316	−0.686582	−	0.727052i	$-0.740890\pi$
$283$	−109975.	−1.37316	−0.686582	+	0.727052i	$0.740890\pi$
$284$	− 51902.9i	− 0.643510i
$285$	0	0
$286$	−1561.27	−0.0190873
$287$	− 13853.7i	− 0.168191i
$288$	0	0
$289$	−84808.5	−1.01541
$290$	− 7181.41i	− 0.0853914i
$291$	0	0
$292$	1980.85	0.0232319
$293$	− 65788.3i	− 0.766326i	−0.923681	−	0.383163i	$-0.874835\pi$
$293$	− 65788.3i	− 0.766326i	0.923681	−	0.383163i	$-0.125165\pi$
$294$	0	0
$295$	47454.1	0.545293
$296$	− 12640.6i	− 0.144273i
$297$	0	0
$298$	74154.1	0.835031
$299$	− 38275.6i	− 0.428134i
$300$	0	0
$301$	23337.5	0.257585
$302$	− 47434.4i	− 0.520091i
$303$	0	0
$304$	26266.7	0.284222
$305$	− 33228.0i	− 0.357195i
$306$	0	0
$307$	−72401.7	−0.768196	−0.384098	−	0.923292i	$-0.625487\pi$
$307$	−72401.7	−0.768196	−0.384098	+	0.923292i	$0.625487\pi$
$308$	1639.22i	0.0172797i
$309$	0	0
$310$	−43320.9	−0.450789
$311$	101726.i	1.05175i	0.850563	+	0.525874i	$0.176262\pi$
$311$	101726.i	1.05175i	−0.850563	+	0.525874i	$0.823738\pi$
$312$	0	0
$313$	183253.	1.87052	0.935259	−	0.353963i	$-0.115166\pi$
$313$	183253.	1.87052	0.935259	+	0.353963i	$0.115166\pi$
$314$	43621.3i	0.442425i
$315$	0	0
$316$	93340.0	0.934746
$317$	120863.i	1.20275i	0.798967	+	0.601374i	$0.205380\pi$
$317$	120863.i	1.20275i	−0.798967	+	0.601374i	$0.794620\pi$
$318$	0	0
$319$	−2136.23	−0.0209926
$320$	5724.33i	0.0559017i
$321$	0	0
$322$	−40186.6	−0.387587
$323$	− 168386.i	− 1.61399i
$324$	0	0
$325$	7335.09	0.0694446
$326$	− 36023.9i	− 0.338965i
$327$	0	0
$328$	−14391.0	−0.133766
$329$	− 19804.0i	− 0.182962i
$330$	0	0
$331$	−42990.8	−0.392392	−0.196196	−	0.980565i	$-0.562859\pi$
$331$	−42990.8	−0.392392	−0.196196	+	0.980565i	$0.562859\pi$
$332$	66799.8i	0.606036i
$333$	0	0
$334$	20742.5	0.185938
$335$	61172.2i	0.545085i
$336$	0	0
$337$	−136787.	−1.20444	−0.602219	−	0.798331i	$-0.705717\pi$
$337$	−136787.	−1.20444	−0.602219	+	0.798331i	$0.705717\pi$
$338$	− 71043.2i	− 0.621855i
$339$	0	0
$340$	36696.5	0.317444
$341$	12886.5i	0.110822i
$342$	0	0
$343$	−94264.5	−0.801235
$344$	− 24242.6i	− 0.204863i
$345$	0	0
$346$	−2266.19	−0.0189297
$347$	172814.i	1.43523i	0.696442	+	0.717613i	$0.254765\pi$
$347$	172814.i	1.43523i	−0.696442	+	0.717613i	$0.745235\pi$
$348$	0	0
$349$	−34809.2	−0.285787	−0.142894	−	0.989738i	$-0.545641\pi$
$349$	−34809.2	−0.285787	−0.142894	+	0.989738i	$0.545641\pi$
$350$	− 7701.30i	− 0.0628678i
$351$	0	0
$352$	1702.79	0.0137429
$353$	25433.9i	0.204110i	0.994779	+	0.102055i	$0.0325418\pi$
$353$	25433.9i	0.204110i	−0.994779	+	0.102055i	$0.967458\pi$
$354$	0	0
$355$	72536.6	0.575573
$356$	115635.i	0.912412i
$357$	0	0
$358$	12695.9	0.0990595
$359$	− 96097.7i	− 0.745632i	−0.927905	−	0.372816i	$-0.878392\pi$
$359$	− 96097.7i	− 0.745632i	0.927905	−	0.372816i	$-0.121608\pi$
$360$	0	0
$361$	38121.1	0.292517
$362$	50198.5i	0.383066i
$363$	0	0
$364$	10225.7	0.0771777
$365$	2768.32i	0.0207793i
$366$	0	0
$367$	−196065.	−1.45569	−0.727844	−	0.685742i	$-0.759478\pi$
$367$	−196065.	−1.45569	−0.727844	+	0.685742i	$0.759478\pi$
$368$	41745.3i	0.308256i
$369$	0	0
$370$	17665.8	0.129042
$371$	− 39080.3i	− 0.283929i
$372$	0	0
$373$	−237124.	−1.70434	−0.852172	−	0.523263i	$-0.824715\pi$
$373$	−237124.	−1.70434	−0.852172	+	0.523263i	$0.824715\pi$
$374$	− 10916.0i	− 0.0780403i
$375$	0	0
$376$	−20572.1	−0.145513
$377$	13326.2i	0.0937611i
$378$	0	0
$379$	−284664.	−1.98177	−0.990887	−	0.134694i	$-0.956995\pi$
$379$	−284664.	−1.98177	−0.990887	+	0.134694i	$0.956995\pi$
$380$	36708.8i	0.254216i
$381$	0	0
$382$	187710.	1.28636
$383$	95518.7i	0.651165i	0.945514	+	0.325582i	$0.105560\pi$
$383$	95518.7i	0.651165i	−0.945514	+	0.325582i	$0.894440\pi$
$384$	0	0
$385$	−2290.88	−0.0154554
$386$	− 28631.2i	− 0.192161i
$387$	0	0
$388$	126817.	0.842391
$389$	− 183273.i	− 1.21115i	−0.795788	−	0.605576i	$-0.792943\pi$
$389$	− 183273.i	− 1.21115i	0.795788	−	0.605576i	$-0.207057\pi$
$390$	0	0
$391$	267613.	1.75047
$392$	43592.2i	0.283685i
$393$	0	0
$394$	−75075.6	−0.483622
$395$	130447.i	0.836062i
$396$	0	0
$397$	149785.	0.950357	0.475178	−	0.879889i	$-0.342384\pi$
$397$	149785.	0.950357	0.475178	+	0.879889i	$0.342384\pi$
$398$	126498.i	0.798575i
$399$	0	0
$400$	−8000.00	−0.0500000
$401$	− 246440.i	− 1.53258i	−0.642495	−	0.766290i	$-0.722101\pi$
$401$	− 246440.i	− 1.53258i	0.642495	−	0.766290i	$-0.277899\pi$
$402$	0	0
$403$	80388.2	0.494974
$404$	79529.6i	0.487266i
$405$	0	0
$406$	13991.5	0.0848813
$407$	− 5254.99i	− 0.0317236i
$408$	0	0
$409$	−112555.	−0.672852	−0.336426	−	0.941710i	$-0.609218\pi$
$409$	−112555.	−0.672852	−0.336426	+	0.941710i	$0.609218\pi$
$410$	− 20112.1i	− 0.119644i
$411$	0	0
$412$	15378.0	0.0905955
$413$	92454.6i	0.542036i
$414$	0	0
$415$	−93355.5	−0.542055
$416$	− 10622.3i	− 0.0613810i
$417$	0	0
$418$	10919.6	0.0624964
$419$	9103.40i	0.0518532i	0.999664	+	0.0259266i	$0.00825362\pi$
$419$	9103.40i	0.0518532i	−0.999664	+	0.0259266i	$0.991746\pi$
$420$	0	0
$421$	−233667.	−1.31836	−0.659179	−	0.751986i	$-0.729096\pi$
$421$	−233667.	−1.31836	−0.659179	+	0.751986i	$0.729096\pi$
$422$	193270.i	1.08527i
$423$	0	0
$424$	−40596.0	−0.225814
$425$	51285.0i	0.283931i
$426$	0	0
$427$	64738.0	0.355061
$428$	− 28730.8i	− 0.156841i
$429$	0	0
$430$	33880.1	0.183235
$431$	201805.i	1.08637i	0.839613	+	0.543185i	$0.182782\pi$
$431$	201805.i	1.08637i	−0.839613	+	0.543185i	$0.817218\pi$
$432$	0	0
$433$	179801.	0.958994	0.479497	−	0.877543i	$-0.340819\pi$
$433$	179801.	0.958994	0.479497	+	0.877543i	$0.340819\pi$
$434$	− 84401.7i	− 0.448097i
$435$	0	0
$436$	−63654.1	−0.334853
$437$	267702.i	1.40181i
$438$	0	0
$439$	−227320.	−1.17953	−0.589764	−	0.807576i	$-0.700779\pi$
$439$	−227320.	−1.17953	−0.589764	+	0.807576i	$0.700779\pi$
$440$	2379.73i	0.0122920i
$441$	0	0
$442$	−68095.8	−0.348559
$443$	277613.i	1.41460i	0.706916	+	0.707298i	$0.250086\pi$
$443$	277613.i	1.41460i	−0.706916	+	0.707298i	$0.749914\pi$
$444$	0	0
$445$	−161605.	−0.816086
$446$	63468.9i	0.319074i
$447$	0	0
$448$	−11152.7	−0.0555678
$449$	210853.i	1.04589i	0.852365	+	0.522947i	$0.175167\pi$
$449$	210853.i	1.04589i	−0.852365	+	0.522947i	$0.824833\pi$
$450$	0	0
$451$	−5982.66	−0.0294131
$452$	161653.i	0.791235i
$453$	0	0
$454$	248928.	1.20771
$455$	14290.9i	0.0690298i
$456$	0	0
$457$	−134886.	−0.645856	−0.322928	−	0.946424i	$-0.604667\pi$
$457$	−134886.	−0.645856	−0.322928	+	0.946424i	$0.604667\pi$
$458$	− 117514.i	− 0.560222i
$459$	0	0
$460$	−58340.8	−0.275713
$461$	140246.i	0.659915i	0.943996	+	0.329957i	$0.107034\pi$
$461$	140246.i	0.659915i	−0.943996	+	0.329957i	$0.892966\pi$
$462$	0	0
$463$	393363.	1.83498	0.917489	−	0.397761i	$-0.130213\pi$
$463$	393363.	1.83498	0.917489	+	0.397761i	$0.130213\pi$
$464$	− 14534.2i	− 0.0675078i
$465$	0	0
$466$	244153.	1.12432
$467$	167700.i	0.768952i	0.923135	+	0.384476i	$0.125618\pi$
$467$	167700.i	0.768952i	−0.923135	+	0.384476i	$0.874382\pi$
$468$	0	0
$469$	−119181.	−0.541830
$470$	− 28750.3i	− 0.130151i
$471$	0	0
$472$	96040.4	0.431092
$473$	− 10078.2i	− 0.0450463i
$474$	0	0
$475$	−51302.1	−0.227378
$476$	71495.6i	0.315548i
$477$	0	0
$478$	−177722.	−0.777831
$479$	− 77365.4i	− 0.337191i	−0.985685	−	0.168595i	$-0.946077\pi$
$479$	− 77365.4i	− 0.337191i	0.985685	−	0.168595i	$-0.0539232\pi$
$480$	0	0
$481$	−32781.6	−0.141690
$482$	− 264750.i	− 1.13957i
$483$	0	0
$484$	−116420.	−0.496978
$485$	177232.i	0.753457i
$486$	0	0
$487$	55724.3	0.234956	0.117478	−	0.993075i	$-0.462519\pi$
$487$	55724.3	0.234956	0.117478	+	0.993075i	$0.462519\pi$
$488$	− 67248.8i	− 0.282387i
$489$	0	0
$490$	−60921.9	−0.253735
$491$	− 175351.i	− 0.727354i	−0.931525	−	0.363677i	$-0.881521\pi$
$491$	− 175351.i	− 0.727354i	0.931525	−	0.363677i	$-0.118479\pi$
$492$	0	0
$493$	−93173.0	−0.383351
$494$	− 68118.6i	− 0.279133i
$495$	0	0
$496$	−87675.2	−0.356380
$497$	141322.i	0.572135i
$498$	0	0
$499$	−73481.6	−0.295106	−0.147553	−	0.989054i	$-0.547140\pi$
$499$	−73481.6	−0.295106	−0.147553	+	0.989054i	$0.547140\pi$
$500$	− 11180.3i	− 0.0447214i
$501$	0	0
$502$	27379.9	0.108649
$503$	− 88732.0i	− 0.350707i	−0.984506	−	0.175353i	$-0.943893\pi$
$503$	− 88732.0i	− 0.350707i	0.984506	−	0.175353i	$-0.0561068\pi$
$504$	0	0
$505$	−111146.	−0.435824
$506$	17354.4i	0.0677811i
$507$	0	0
$508$	79706.2	0.308862
$509$	− 300835.i	− 1.16116i	−0.814202	−	0.580582i	$-0.802825\pi$
$509$	− 300835.i	− 1.16116i	0.814202	−	0.580582i	$-0.197175\pi$
$510$	0	0
$511$	−5393.49	−0.0206551
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	102236.	0.386971
$515$	21491.5i	0.0810311i
$516$	0	0
$517$	−8552.25	−0.0319963
$518$	34418.2i	0.128271i
$519$	0	0
$520$	14845.2	0.0549008
$521$	466381.i	1.71817i	0.511837	+	0.859083i	$0.328965\pi$
$521$	466381.i	1.71817i	−0.511837	+	0.859083i	$0.671035\pi$
$522$	0	0
$523$	−19673.6	−0.0719250	−0.0359625	−	0.999353i	$-0.511450\pi$
$523$	−19673.6	−0.0719250	−0.0359625	+	0.999353i	$0.511450\pi$
$524$	242892.i	0.884608i
$525$	0	0
$526$	−303863.	−1.09826
$527$	562053.i	2.02375i
$528$	0	0
$529$	−145615.	−0.520348
$530$	− 56734.6i	− 0.201975i
$531$	0	0
$532$	−71519.5	−0.252698
$533$	37320.9i	0.131371i
$534$	0	0
$535$	40152.5	0.140283
$536$	123804.i	0.430928i
$537$	0	0
$538$	102177.	0.353011
$539$	18122.2i	0.0623782i
$540$	0	0
$541$	−529746.	−1.80998	−0.904989	−	0.425435i	$-0.860121\pi$
$541$	−529746.	−1.80998	−0.904989	+	0.425435i	$0.860121\pi$
$542$	− 36669.3i	− 0.124826i
$543$	0	0
$544$	74268.6	0.250962
$545$	− 88959.3i	− 0.299501i
$546$	0	0
$547$	118865.	0.397264	0.198632	−	0.980074i	$-0.436350\pi$
$547$	118865.	0.397264	0.198632	+	0.980074i	$0.436350\pi$
$548$	− 68559.2i	− 0.228299i
$549$	0	0
$550$	−3325.77	−0.0109943
$551$	− 93204.1i	− 0.306995i
$552$	0	0
$553$	−254148.	−0.831069
$554$	345212.i	1.12478i
$555$	0	0
$556$	94301.1	0.305047
$557$	− 37863.2i	− 0.122041i	−0.998136	−	0.0610207i	$-0.980564\pi$
$557$	− 37863.2i	− 0.122041i	0.998136	−	0.0610207i	$-0.0194356\pi$
$558$	0	0
$559$	−62869.5	−0.201195
$560$	− 15586.3i	− 0.0497013i
$561$	0	0
$562$	145911.	0.461971
$563$	− 76068.2i	− 0.239986i	−0.992775	−	0.119993i	$-0.961713\pi$
$563$	− 76068.2i	− 0.239986i	0.992775	−	0.119993i	$-0.0382873\pi$
$564$	0	0
$565$	−225916.	−0.707702
$566$	− 311057.i	− 0.970974i
$567$	0	0
$568$	146804.	0.455030
$569$	520887.i	1.60886i	0.594046	+	0.804431i	$0.297530\pi$
$569$	520887.i	1.60886i	−0.594046	+	0.804431i	$0.702470\pi$
$570$	0	0
$571$	−126163.	−0.386954	−0.193477	−	0.981105i	$-0.561976\pi$
$571$	−126163.	−0.386954	−0.193477	+	0.981105i	$0.561976\pi$
$572$	− 4415.93i	− 0.0134968i
$573$	0	0
$574$	39184.2	0.118929
$575$	− 81533.7i	− 0.246605i
$576$	0	0
$577$	385377.	1.15753	0.578767	−	0.815493i	$-0.303534\pi$
$577$	385377.	1.15753	0.578767	+	0.815493i	$0.303534\pi$
$578$	− 239875.i	− 0.718007i
$579$	0	0
$580$	20312.1	0.0603808
$581$	− 181884.i	− 0.538818i
$582$	0	0
$583$	−16876.6	−0.0496533
$584$	5602.68i	0.0164274i
$585$	0	0
$586$	186078.	0.541875
$587$	− 20288.3i	− 0.0588801i	−0.999567	−	0.0294401i	$-0.990628\pi$
$587$	− 20288.3i	− 0.0588801i	0.999567	−	0.0294401i	$-0.00937241\pi$
$588$	0	0
$589$	−562241.	−1.62066
$590$	134221.i	0.385581i
$591$	0	0
$592$	35753.1	0.102017
$593$	147727.i	0.420097i	0.977691	+	0.210048i	$0.0673621\pi$
$593$	147727.i	0.420097i	−0.977691	+	0.210048i	$0.932638\pi$
$594$	0	0
$595$	−99918.1	−0.282235
$596$	209739.i	0.590456i
$597$	0	0
$598$	108260.	0.302737
$599$	197930.i	0.551642i	0.961209	+	0.275821i	$0.0889497\pi$
$599$	197930.i	0.551642i	−0.961209	+	0.275821i	$0.911050\pi$
$600$	0	0
$601$	263353.	0.729104	0.364552	−	0.931183i	$-0.381222\pi$
$601$	263353.	0.729104	0.364552	+	0.931183i	$0.381222\pi$
$602$	66008.3i	0.182140i
$603$	0	0
$604$	134165.	0.367760
$605$	− 162702.i	− 0.444511i
$606$	0	0
$607$	−164622.	−0.446798	−0.223399	−	0.974727i	$-0.571715\pi$
$607$	−164622.	−0.446798	−0.223399	+	0.974727i	$0.571715\pi$
$608$	74293.4i	0.200975i
$609$	0	0
$610$	93983.1	0.252575
$611$	53350.5i	0.142908i
$612$	0	0
$613$	570191.	1.51740	0.758698	−	0.651442i	$-0.225836\pi$
$613$	570191.	1.51740	0.758698	+	0.651442i	$0.225836\pi$
$614$	− 204783.i	− 0.543196i
$615$	0	0
$616$	−4636.41	−0.0122186
$617$	153122.i	0.402224i	0.979568	+	0.201112i	$0.0644555\pi$
$617$	153122.i	0.402224i	−0.979568	+	0.201112i	$0.935544\pi$
$618$	0	0
$619$	418936.	1.09337	0.546683	−	0.837339i	$-0.315890\pi$
$619$	418936.	1.09337	0.546683	+	0.837339i	$0.315890\pi$
$620$	− 122530.i	− 0.318756i
$621$	0	0
$622$	−287725.	−0.743698
$623$	− 314855.i	− 0.811212i
$624$	0	0
$625$	15625.0	0.0400000
$626$	518317.i	1.32266i
$627$	0	0
$628$	−123380.	−0.312842
$629$	− 229200.i	− 0.579313i
$630$	0	0
$631$	596394.	1.49787	0.748936	−	0.662642i	$-0.230565\pi$
$631$	596394.	1.49787	0.748936	+	0.662642i	$0.230565\pi$
$632$	264005.i	0.660965i
$633$	0	0
$634$	−341852.	−0.850472
$635$	111393.i	0.276255i
$636$	0	0
$637$	113050.	0.278606
$638$	− 6042.16i	− 0.0148440i
$639$	0	0
$640$	−16190.9	−0.0395285
$641$	579352.i	1.41002i	0.709196	+	0.705011i	$0.249058\pi$
$641$	579352.i	1.41002i	−0.709196	+	0.705011i	$0.750942\pi$
$642$	0	0
$643$	761394.	1.84157	0.920784	−	0.390073i	$-0.127550\pi$
$643$	761394.	1.84157	0.920784	+	0.390073i	$0.127550\pi$
$644$	− 113665.i	− 0.274066i
$645$	0	0
$646$	476267.	1.14126
$647$	− 508688.i	− 1.21519i	−0.794249	−	0.607593i	$-0.792135\pi$
$647$	− 508688.i	− 1.21519i	0.794249	−	0.607593i	$-0.207865\pi$
$648$	0	0
$649$	39926.1	0.0947910
$650$	20746.8i	0.0491048i
$651$	0	0
$652$	101891.	0.239685
$653$	232211.i	0.544573i	0.962216	+	0.272286i	$0.0877798\pi$
$653$	232211.i	0.544573i	−0.962216	+	0.272286i	$0.912220\pi$
$654$	0	0
$655$	−339452.	−0.791217
$656$	− 40704.0i	− 0.0945865i
$657$	0	0
$658$	56014.1	0.129374
$659$	− 373686.i	− 0.860470i	−0.902717	−	0.430235i	$-0.858431\pi$
$659$	− 373686.i	− 0.860470i	0.902717	−	0.430235i	$-0.141569\pi$
$660$	0	0
$661$	152104.	0.348127	0.174064	−	0.984734i	$-0.444310\pi$
$661$	152104.	0.348127	0.174064	+	0.984734i	$0.444310\pi$
$662$	− 121596.i	− 0.277463i
$663$	0	0
$664$	−188938.	−0.428532
$665$	− 99951.5i	− 0.226020i
$666$	0	0
$667$	148128.	0.332955
$668$	58668.5i	0.131478i
$669$	0	0
$670$	−173021.	−0.385434
$671$	− 27956.8i	− 0.0620929i
$672$	0	0
$673$	560347.	1.23716	0.618582	−	0.785721i	$-0.287708\pi$
$673$	560347.	1.23716	0.618582	+	0.785721i	$0.287708\pi$
$674$	− 386892.i	− 0.851666i
$675$	0	0
$676$	200941.	0.439718
$677$	− 38475.9i	− 0.0839483i	−0.999119	−	0.0419741i	$-0.986635\pi$
$677$	− 38475.9i	− 0.0839483i	0.999119	−	0.0419741i	$-0.0133647\pi$
$678$	0	0
$679$	−345300.	−0.748957
$680$	103793.i	0.224467i
$681$	0	0
$682$	−36448.5	−0.0783629
$683$	583852.i	1.25159i	0.779989	+	0.625794i	$0.215225\pi$
$683$	583852.i	1.25159i	−0.779989	+	0.625794i	$0.784775\pi$
$684$	0	0
$685$	95814.4	0.204197
$686$	− 266620.i	− 0.566559i
$687$	0	0
$688$	68568.5	0.144860
$689$	105279.i	0.221771i
$690$	0	0
$691$	81953.8	0.171638	0.0858189	−	0.996311i	$-0.472649\pi$
$691$	81953.8	0.171638	0.0858189	+	0.996311i	$0.472649\pi$
$692$	− 6409.76i	− 0.0133853i
$693$	0	0
$694$	−488792.	−1.01486
$695$	131790.i	0.272842i
$696$	0	0
$697$	−260938.	−0.537120
$698$	− 98455.2i	− 0.202082i
$699$	0	0
$700$	21782.6	0.0444542
$701$	− 599196.i	− 1.21936i	−0.792647	−	0.609681i	$-0.791298\pi$
$701$	− 599196.i	− 1.21936i	0.792647	−	0.609681i	$-0.208702\pi$
$702$	0	0
$703$	229276.	0.463926
$704$	4816.23i	0.00971767i
$705$	0	0
$706$	−71938.0	−0.144327
$707$	− 216545.i	− 0.433220i
$708$	0	0
$709$	−372084.	−0.740198	−0.370099	−	0.928992i	$-0.620676\pi$
$709$	−372084.	−0.740198	−0.370099	+	0.928992i	$0.620676\pi$
$710$	205164.i	0.406991i
$711$	0	0
$712$	−327066.	−0.645173
$713$	− 893561.i	− 1.75770i
$714$	0	0
$715$	6171.46	0.0120719
$716$	35909.3i	0.0700456i
$717$	0	0
$718$	271805.	0.527241
$719$	− 405471.i	− 0.784335i	−0.919894	−	0.392168i	$-0.871725\pi$
$719$	− 405471.i	− 0.784335i	0.919894	−	0.392168i	$-0.128275\pi$
$720$	0	0
$721$	−41871.7	−0.0805471
$722$	107823.i	0.206840i
$723$	0	0
$724$	−141983.	−0.270869
$725$	28387.0i	0.0540062i
$726$	0	0
$727$	960112.	1.81657	0.908287	−	0.418347i	$-0.137390\pi$
$727$	960112.	1.81657	0.908287	+	0.418347i	$0.137390\pi$
$728$	28922.7i	0.0545729i
$729$	0	0
$730$	−7829.98	−0.0146932
$731$	− 439566.i	− 0.822602i
$732$	0	0
$733$	687333.	1.27926	0.639631	−	0.768682i	$-0.279087\pi$
$733$	687333.	1.27926	0.639631	+	0.768682i	$0.279087\pi$
$734$	− 554556.i	− 1.02933i
$735$	0	0
$736$	−118073.	−0.217970
$737$	51467.9i	0.0947548i
$738$	0	0
$739$	114763.	0.210143	0.105071	−	0.994465i	$-0.466493\pi$
$739$	114763.	0.210143	0.105071	+	0.994465i	$0.466493\pi$
$740$	49966.5i	0.0912464i
$741$	0	0
$742$	110536.	0.200768
$743$	− 51183.8i	− 0.0927161i	−0.998925	−	0.0463580i	$-0.985238\pi$
$743$	− 51183.8i	− 0.0927161i	0.998925	−	0.0463580i	$-0.0147615\pi$
$744$	0	0
$745$	−293120.	−0.528120
$746$	− 670687.i	− 1.20515i
$747$	0	0
$748$	30875.0	0.0551829
$749$	78228.9i	0.139445i
$750$	0	0
$751$	622876.	1.10439	0.552194	−	0.833716i	$-0.313791\pi$
$751$	622876.	1.10439	0.552194	+	0.833716i	$0.313791\pi$
$752$	− 58186.6i	− 0.102893i
$753$	0	0
$754$	−37692.1	−0.0662991
$755$	187501.i	0.328934i
$756$	0	0
$757$	−419466.	−0.731990	−0.365995	−	0.930617i	$-0.619271\pi$
$757$	−419466.	−0.731990	−0.365995	+	0.930617i	$0.619271\pi$
$758$	− 805152.i	− 1.40133i
$759$	0	0
$760$	−103828.	−0.179758
$761$	− 728967.i	− 1.25875i	−0.777103	−	0.629374i	$-0.783311\pi$
$761$	− 728967.i	− 1.25875i	0.777103	−	0.629374i	$-0.216689\pi$
$762$	0	0
$763$	173319.	0.297712
$764$	530924.i	0.909591i
$765$	0	0
$766$	−270168.	−0.460443
$767$	− 249066.i	− 0.423374i
$768$	0	0
$769$	−276442.	−0.467467	−0.233734	−	0.972301i	$-0.575094\pi$
$769$	−276442.	−0.467467	−0.233734	+	0.972301i	$0.575094\pi$
$770$	− 6479.58i	− 0.0109286i
$771$	0	0
$772$	80981.3	0.135878
$773$	1.06081e6i	1.77533i	0.460487	+	0.887666i	$0.347675\pi$
$773$	1.06081e6i	1.77533i	−0.460487	+	0.887666i	$0.652325\pi$
$774$	0	0
$775$	171241.	0.285104
$776$	358692.i	0.595660i
$777$	0	0
$778$	518373.	0.856413
$779$	− 261025.i	− 0.430138i
$780$	0	0
$781$	61029.4	0.100055
$782$	756924.i	1.23777i
$783$	0	0
$784$	−123297.	−0.200596
$785$	− 172428.i	− 0.279814i
$786$	0	0
$787$	−678210.	−1.09500	−0.547501	−	0.836805i	$-0.684421\pi$
$787$	−678210.	−1.09500	−0.547501	+	0.836805i	$0.684421\pi$
$788$	− 212346.i	− 0.341972i
$789$	0	0
$790$	−368959.	−0.591185
$791$	− 440151.i	− 0.703475i
$792$	0	0
$793$	−174399.	−0.277331
$794$	423655.i	0.672004i
$795$	0	0
$796$	−357789.	−0.564678
$797$	− 383250.i	− 0.603345i	−0.953412	−	0.301672i	$-0.902455\pi$
$797$	− 383250.i	− 0.603345i	0.953412	−	0.301672i	$-0.0975449\pi$
$798$	0	0
$799$	−373012.	−0.584291
$800$	− 22627.4i	− 0.0353553i
$801$	0	0
$802$	697039.	1.08370
$803$	2329.15i	0.00361216i
$804$	0	0
$805$	158852.	0.245132
$806$	227372.i	0.349999i
$807$	0	0
$808$	−224944.	−0.344549
$809$	− 1.15507e6i	− 1.76486i	−0.470441	−	0.882431i	$-0.655905\pi$
$809$	− 1.15507e6i	− 1.76486i	0.470441	−	0.882431i	$-0.344095\pi$
$810$	0	0
$811$	173673.	0.264053	0.132026	−	0.991246i	$-0.457852\pi$
$811$	173673.	0.264053	0.132026	+	0.991246i	$0.457852\pi$
$812$	39573.9i	0.0600202i
$813$	0	0
$814$	14863.3	0.0224320
$815$	142397.i	0.214380i
$816$	0	0
$817$	439713.	0.658758
$818$	− 318355.i	− 0.475778i
$819$	0	0
$820$	56885.6	0.0846008
$821$	247035.i	0.366498i	0.983067	+	0.183249i	$0.0586615\pi$
$821$	247035.i	0.366498i	−0.983067	+	0.183249i	$0.941338\pi$
$822$	0	0
$823$	864386.	1.27617	0.638084	−	0.769967i	$-0.279727\pi$
$823$	864386.	1.27617	0.638084	+	0.769967i	$0.279727\pi$
$824$	43495.7i	0.0640607i
$825$	0	0
$826$	−261501.	−0.383277
$827$	− 28103.8i	− 0.0410917i	−0.999789	−	0.0205459i	$-0.993460\pi$
$827$	− 28103.8i	− 0.0410917i	0.999789	−	0.0205459i	$-0.00654041\pi$
$828$	0	0
$829$	324316.	0.471910	0.235955	−	0.971764i	$-0.424178\pi$
$829$	324316.	0.471910	0.235955	+	0.971764i	$0.424178\pi$
$830$	− 264049.i	− 0.383291i
$831$	0	0
$832$	30044.5	0.0434029
$833$	790412.i	1.13910i
$834$	0	0
$835$	−81991.8	−0.117597
$836$	30885.4i	0.0441916i
$837$	0	0
$838$	−25748.3	−0.0366657
$839$	− 489743.i	− 0.695736i	−0.937543	−	0.347868i	$-0.886906\pi$
$839$	− 489743.i	− 0.695736i	0.937543	−	0.347868i	$-0.113094\pi$
$840$	0	0
$841$	655708.	0.927083
$842$	− 660910.i	− 0.932219i
$843$	0	0
$844$	−546649.	−0.767403
$845$	280823.i	0.393296i
$846$	0	0
$847$	316991.	0.441856
$848$	− 114823.i	− 0.159675i
$849$	0	0
$850$	−145056.	−0.200769
$851$	364386.i	0.503156i
$852$	0	0
$853$	−1.09030e6	−1.49848	−0.749238	−	0.662301i	$-0.769580\pi$
$853$	−1.09030e6	−1.49848	−0.749238	+	0.662301i	$0.769580\pi$
$854$	183107.i	0.251066i
$855$	0	0
$856$	81263.0	0.110904
$857$	400823.i	0.545747i	0.962050	+	0.272874i	$0.0879741\pi$
$857$	400823.i	0.545747i	−0.962050	+	0.272874i	$0.912026\pi$
$858$	0	0
$859$	−36202.8	−0.0490632	−0.0245316	−	0.999699i	$-0.507809\pi$
$859$	−36202.8	−0.0490632	−0.0245316	+	0.999699i	$0.507809\pi$
$860$	95827.4i	0.129566i
$861$	0	0
$862$	−570791.	−0.768179
$863$	− 516556.i	− 0.693579i	−0.937943	−	0.346790i	$-0.887272\pi$
$863$	− 516556.i	− 0.693579i	0.937943	−	0.346790i	$-0.112728\pi$
$864$	0	0
$865$	8957.91	0.0119722
$866$	508554.i	0.678111i
$867$	0	0
$868$	238724.	0.316852
$869$	109753.i	0.145337i
$870$	0	0
$871$	321066.	0.423212
$872$	− 180041.i	− 0.236776i
$873$	0	0
$874$	−757177.	−0.991230
$875$	30442.1i	0.0397611i
$876$	0	0
$877$	−4876.09	−0.00633976	−0.00316988	−	0.999995i	$-0.501009\pi$
$877$	−4876.09	−0.00633976	−0.00316988	+	0.999995i	$0.501009\pi$
$878$	− 642958.i	− 0.834052i
$879$	0	0
$880$	−6730.89	−0.00869174
$881$	− 1.15317e6i	− 1.48573i	−0.669440	−	0.742866i	$-0.733466\pi$
$881$	− 1.15317e6i	− 1.48573i	0.669440	−	0.742866i	$-0.266534\pi$
$882$	0	0
$883$	441172.	0.565831	0.282915	−	0.959145i	$-0.408698\pi$
$883$	441172.	0.565831	0.282915	+	0.959145i	$0.408698\pi$
$884$	− 192604.i	− 0.246468i
$885$	0	0
$886$	−785208.	−1.00027
$887$	− 1.26908e6i	− 1.61303i	−0.591213	−	0.806516i	$-0.701351\pi$
$887$	− 1.26908e6i	− 1.61303i	0.591213	−	0.806516i	$-0.298649\pi$
$888$	0	0
$889$	−217026.	−0.274604
$890$	− 457089.i	− 0.577060i
$891$	0	0
$892$	−179517.	−0.225619
$893$	− 373137.i	− 0.467913i
$894$	0	0
$895$	−50184.8	−0.0626507
$896$	− 31544.5i	− 0.0392924i
$897$	0	0
$898$	−596383.	−0.739558
$899$	311105.i	0.384935i
$900$	0	0
$901$	−736086.	−0.906732
$902$	− 16921.5i	− 0.0207982i
$903$	0	0
$904$	−457222.	−0.559488
$905$	− 198427.i	− 0.242272i
$906$	0	0
$907$	774689.	0.941701	0.470851	−	0.882213i	$-0.343947\pi$
$907$	774689.	0.941701	0.470851	+	0.882213i	$0.343947\pi$
$908$	704075.i	0.853979i
$909$	0	0
$910$	−40420.8	−0.0488115
$911$	74061.3i	0.0892390i	0.999004	+	0.0446195i	$0.0142075\pi$
$911$	74061.3i	0.0892390i	−0.999004	+	0.0446195i	$0.985792\pi$
$912$	0	0
$913$	−78545.7	−0.0942282
$914$	− 381516.i	− 0.456689i
$915$	0	0
$916$	332381.	0.396137
$917$	− 661352.i	− 0.786491i
$918$	0	0
$919$	−1.38059e6	−1.63468	−0.817341	−	0.576155i	$-0.804553\pi$
$919$	−1.38059e6	−1.63468	−0.817341	+	0.576155i	$0.804553\pi$
$920$	− 165013.i	− 0.194958i
$921$	0	0
$922$	−396675.	−0.466630
$923$	− 380713.i	− 0.446883i
$924$	0	0
$925$	−69830.4	−0.0816133
$926$	1.11260e6i	1.29753i
$927$	0	0
$928$	41108.8	0.0477352
$929$	508818.i	0.589564i	0.955565	+	0.294782i	$0.0952471\pi$
$929$	508818.i	0.589564i	−0.955565	+	0.294782i	$0.904753\pi$
$930$	0	0
$931$	−790676.	−0.912219
$932$	690569.i	0.795014i
$933$	0	0
$934$	−474327.	−0.543731
$935$	43149.2i	0.0493570i
$936$	0	0
$937$	−903136.	−1.02867	−0.514333	−	0.857591i	$-0.671960\pi$
$937$	−903136.	−1.02867	−0.514333	+	0.857591i	$0.671960\pi$
$938$	− 337096.i	− 0.383131i
$939$	0	0
$940$	81318.2	0.0920306
$941$	− 1.48547e6i	− 1.67759i	−0.544449	−	0.838794i	$-0.683261\pi$
$941$	− 1.48547e6i	− 1.67759i	0.544449	−	0.838794i	$-0.316739\pi$
$942$	0	0
$943$	414843.	0.466510
$944$	271643.i	0.304828i
$945$	0	0
$946$	28505.4	0.0318526
$947$	555552.i	0.619477i	0.950822	+	0.309738i	$0.100241\pi$
$947$	555552.i	0.619477i	−0.950822	+	0.309738i	$0.899759\pi$
$948$	0	0
$949$	14529.7	0.0161333
$950$	− 145104.i	− 0.160780i
$951$	0	0
$952$	−202220.	−0.223126
$953$	227966.i	0.251007i	0.992093	+	0.125503i	$0.0400546\pi$
$953$	227966.i	0.251007i	−0.992093	+	0.125503i	$0.959945\pi$
$954$	0	0
$955$	−741989.	−0.813563
$956$	− 502673.i	− 0.550009i
$957$	0	0
$958$	218823.	0.238430
$959$	186675.i	0.202977i
$960$	0	0
$961$	953175.	1.03211
$962$	− 92720.2i	− 0.100190i
$963$	0	0
$964$	748826.	0.805799
$965$	113175.i	0.121533i
$966$	0	0
$967$	1.52355e6	1.62932	0.814658	−	0.579942i	$-0.196925\pi$
$967$	1.52355e6	1.62932	0.814658	+	0.579942i	$0.196925\pi$
$968$	− 329286.i	− 0.351417i
$969$	0	0
$970$	−501288.	−0.532775
$971$	759197.i	0.805223i	0.915371	+	0.402611i	$0.131897\pi$
$971$	759197.i	0.805223i	−0.915371	+	0.402611i	$0.868103\pi$
$972$	0	0
$973$	−256765.	−0.271213
$974$	157612.i	0.166139i
$975$	0	0
$976$	190208.	0.199678
$977$	− 243840.i	− 0.255456i	−0.991809	−	0.127728i	$-0.959232\pi$
$977$	− 243840.i	− 0.255456i	0.991809	−	0.127728i	$-0.0407684\pi$
$978$	0	0
$979$	−135969.	−0.141864
$980$	− 172313.i	− 0.179418i
$981$	0	0
$982$	495968.	0.514317
$983$	203673.i	0.210778i	0.994431	+	0.105389i	$0.0336088\pi$
$983$	203673.i	0.210778i	−0.994431	+	0.105389i	$0.966391\pi$
$984$	0	0
$985$	296762.	0.305869
$986$	− 263533.i	− 0.271070i
$987$	0	0
$988$	192668.	0.197377
$989$	698830.i	0.714462i
$990$	0	0
$991$	713627.	0.726648	0.363324	−	0.931663i	$-0.381642\pi$
$991$	713627.	0.726648	0.363324	+	0.931663i	$0.381642\pi$
$992$	− 247983.i	− 0.251999i
$993$	0	0
$994$	−399720.	−0.404560
$995$	− 500025.i	− 0.505063i
$996$	0	0
$997$	−60853.0	−0.0612198	−0.0306099	−	0.999531i	$-0.509745\pi$
$997$	−60853.0	−0.0612198	−0.0306099	+	0.999531i	$0.509745\pi$
$998$	− 207837.i	− 0.208671i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.5.d.c.161.12		32
3.2	odd	2	inner	810.5.d.c.161.11		32
9.2	odd	6		90.5.h.a.41.2	yes	32
9.4	even	3		90.5.h.a.11.2	✓	32
9.5	odd	6		270.5.h.a.251.12		32
9.7	even	3		270.5.h.a.71.12		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.5.h.a.11.2	✓	32	9.4	even	3
90.5.h.a.41.2	yes	32	9.2	odd	6
270.5.h.a.71.12		32	9.7	even	3
270.5.h.a.251.12		32	9.5	odd	6
810.5.d.c.161.11		32	3.2	odd	2	inner
810.5.d.c.161.12		32	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	810.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +21.7826 q^{7} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +21.7826 q^{7} -22.6274i q^{8} +31.6228 q^{10} -9.40670i q^{11} -58.6807 q^{13} +61.6104i q^{14} +64.0000 q^{16} -410.280i q^{17} +410.417 q^{19} +89.4427i q^{20} +26.6062 q^{22} +652.270i q^{23} -125.000 q^{25} -165.974i q^{26} -174.261 q^{28} -227.096i q^{29} -1369.93 q^{31} +181.019i q^{32} +1160.45 q^{34} -243.537i q^{35} +558.643 q^{37} +1160.83i q^{38} -252.982 q^{40} -636.000i q^{41} +1071.38 q^{43} +75.2536i q^{44} -1844.90 q^{46} -909.165i q^{47} -1926.52 q^{49} -353.553i q^{50} +469.446 q^{52} -1794.11i q^{53} -105.170 q^{55} -492.883i q^{56} +642.325 q^{58} +4244.43i q^{59} +2972.01 q^{61} -3874.73i q^{62} -512.000 q^{64} +656.070i q^{65} -5471.41 q^{67} +3282.24i q^{68} +688.826 q^{70} +6487.87i q^{71} -247.606 q^{73} +1580.08i q^{74} -3283.34 q^{76} -204.902i q^{77} -11667.5 q^{79} -715.542i q^{80} +1798.88 q^{82} -8349.97i q^{83} -4587.07 q^{85} +3030.33i q^{86} -212.849 q^{88} -14454.4i q^{89} -1278.22 q^{91} -5218.16i q^{92} +2571.51 q^{94} -4588.60i q^{95} -15852.1 q^{97} -5449.02i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) \) 32 * q - 256 * q^4 - 104 * q^7	\( 32 q - 256 q^{4} - 104 q^{7} + 40 q^{13} + 2048 q^{16} - 200 q^{19} + 1344 q^{22} - 4000 q^{25} + 832 q^{28} - 1472 q^{31} - 384 q^{34} - 4136 q^{37} - 272 q^{43} - 2112 q^{46} + 22296 q^{49} - 320 q^{52} - 12344 q^{61} - 16384 q^{64} + 40936 q^{67} + 4800 q^{70} - 41432 q^{73} + 1600 q^{76} - 14048 q^{79} - 17664 q^{82} - 17400 q^{85} - 10752 q^{88} + 69392 q^{91} + 1344 q^{94} + 36664 q^{97}+O(q^{100}) \) 32 * q - 256 * q^4 - 104 * q^7 + 40 * q^13 + 2048 * q^16 - 200 * q^19 + 1344 * q^22 - 4000 * q^25 + 832 * q^28 - 1472 * q^31 - 384 * q^34 - 4136 * q^37 - 272 * q^43 - 2112 * q^46 + 22296 * q^49 - 320 * q^52 - 12344 * q^61 - 16384 * q^64 + 40936 * q^67 + 4800 * q^70 - 41432 * q^73 + 1600 * q^76 - 14048 * q^79 - 17664 * q^82 - 17400 * q^85 - 10752 * q^88 + 69392 * q^91 + 1344 * q^94 + 36664 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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