LMFDB - Embedded newform 531.5.b.a.296.1

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			296.1
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.76

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.89147i q^{2} -46.2753 q^{4} -30.7341i q^{5} -38.0387 q^{7} +238.917i q^{8} +O(q^{10})$	\(q-7.89147i q^{2} -46.2753 q^{4} -30.7341i q^{5} -38.0387 q^{7} +238.917i q^{8} -242.537 q^{10} +97.9176i q^{11} +266.782 q^{13} +300.181i q^{14} +1145.00 q^{16} -405.373i q^{17} +569.314 q^{19} +1422.23i q^{20} +772.714 q^{22} +551.399i q^{23} -319.583 q^{25} -2105.30i q^{26} +1760.25 q^{28} +1575.91i q^{29} -590.227 q^{31} -5213.07i q^{32} -3198.99 q^{34} +1169.08i q^{35} +1146.09 q^{37} -4492.73i q^{38} +7342.89 q^{40} -627.996i q^{41} +545.964 q^{43} -4531.17i q^{44} +4351.35 q^{46} +831.478i q^{47} -954.056 q^{49} +2521.98i q^{50} -12345.4 q^{52} +4600.66i q^{53} +3009.41 q^{55} -9088.09i q^{56} +12436.3 q^{58} +453.188i q^{59} +4840.16 q^{61} +4657.76i q^{62} -22818.8 q^{64} -8199.30i q^{65} -2866.26 q^{67} +18758.8i q^{68} +9225.80 q^{70} +5595.36i q^{71} +7462.48 q^{73} -9044.36i q^{74} -26345.2 q^{76} -3724.66i q^{77} -57.5610 q^{79} -35190.5i q^{80} -4955.81 q^{82} -7578.24i q^{83} -12458.8 q^{85} -4308.46i q^{86} -23394.2 q^{88} -4901.93i q^{89} -10148.0 q^{91} -25516.2i q^{92} +6561.59 q^{94} -17497.3i q^{95} +4492.34 q^{97} +7528.90i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 7.89147i	− 1.97287i	−0.164159	−	0.986434i	$-0.552491\pi$
$2$	− 7.89147i	− 1.97287i	0.164159	−	0.986434i	$-0.447509\pi$
$3$	0	0
$3$	0	0
$4$	−46.2753	−2.89221
$5$	− 30.7341i	− 1.22936i	−0.788775	−	0.614682i	$-0.789285\pi$
$5$	− 30.7341i	− 1.22936i	0.788775	−	0.614682i	$-0.210715\pi$
$6$	0	0
$7$	−38.0387	−0.776300	−0.388150	−	0.921596i	$-0.626886\pi$
$7$	−38.0387	−0.776300	−0.388150	+	0.921596i	$0.626886\pi$
$8$	238.917i	3.73308i
$9$	0	0
$10$	−242.537	−2.42537
$11$	97.9176i	0.809236i	0.914486	+	0.404618i	$0.132596\pi$
$11$	97.9176i	0.809236i	−0.914486	+	0.404618i	$0.867404\pi$
$12$	0	0
$13$	266.782	1.57859	0.789296	−	0.614013i	$-0.210446\pi$
$13$	266.782	1.57859	0.789296	+	0.614013i	$0.210446\pi$
$14$	300.181i	1.53154i
$15$	0	0
$16$	1145.00	4.47266
$17$	− 405.373i	− 1.40268i	−0.712829	−	0.701338i	$-0.752586\pi$
$17$	− 405.373i	− 1.40268i	0.712829	−	0.701338i	$-0.247414\pi$
$18$	0	0
$19$	569.314	1.57705	0.788524	−	0.615005i	$-0.210846\pi$
$19$	569.314	1.57705	0.788524	+	0.615005i	$0.210846\pi$
$20$	1422.23i	3.55557i
$21$	0	0
$22$	772.714	1.59652
$23$	551.399i	1.04234i	0.853452	+	0.521171i	$0.174505\pi$
$23$	551.399i	1.04234i	−0.853452	+	0.521171i	$0.825495\pi$
$24$	0	0
$25$	−319.583	−0.511333
$26$	− 2105.30i	− 3.11435i
$27$	0	0
$28$	1760.25	2.24522
$29$	1575.91i	1.87386i	0.349522	+	0.936928i	$0.386344\pi$
$29$	1575.91i	1.87386i	−0.349522	+	0.936928i	$0.613656\pi$
$30$	0	0
$31$	−590.227	−0.614180	−0.307090	−	0.951681i	$-0.599355\pi$
$31$	−590.227	−0.614180	−0.307090	+	0.951681i	$0.599355\pi$
$32$	− 5213.07i	− 5.09089i
$33$	0	0
$34$	−3198.99	−2.76729
$35$	1169.08i	0.954355i
$36$	0	0
$37$	1146.09	0.837175	0.418588	−	0.908176i	$-0.362525\pi$
$37$	1146.09	0.837175	0.418588	+	0.908176i	$0.362525\pi$
$38$	− 4492.73i	− 3.11131i
$39$	0	0
$40$	7342.89	4.58931
$41$	− 627.996i	− 0.373585i	−0.982399	−	0.186792i	$-0.940191\pi$
$41$	− 627.996i	− 0.373585i	0.982399	−	0.186792i	$-0.0598092\pi$
$42$	0	0
$43$	545.964	0.295275	0.147638	−	0.989042i	$-0.452833\pi$
$43$	545.964	0.295275	0.147638	+	0.989042i	$0.452833\pi$
$44$	− 4531.17i	− 2.34048i
$45$	0	0
$46$	4351.35	2.05640
$47$	831.478i	0.376405i	0.982130	+	0.188202i	$0.0602661\pi$
$47$	831.478i	0.376405i	−0.982130	+	0.188202i	$0.939734\pi$
$48$	0	0
$49$	−954.056	−0.397358
$50$	2521.98i	1.00879i
$51$	0	0
$52$	−12345.4	−4.56562
$53$	4600.66i	1.63783i	0.573915	+	0.818915i	$0.305424\pi$
$53$	4600.66i	1.63783i	−0.573915	+	0.818915i	$0.694576\pi$
$54$	0	0
$55$	3009.41	0.994845
$56$	− 9088.09i	− 2.89799i
$57$	0	0
$58$	12436.3	3.69687
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	4840.16	1.30077	0.650384	−	0.759606i	$-0.274608\pi$
$61$	4840.16	1.30077	0.650384	+	0.759606i	$0.274608\pi$
$62$	4657.76i	1.21170i
$63$	0	0
$64$	−22818.8	−5.57099
$65$	− 8199.30i	− 1.94066i
$66$	0	0
$67$	−2866.26	−0.638508	−0.319254	−	0.947669i	$-0.603432\pi$
$67$	−2866.26	−0.638508	−0.319254	+	0.947669i	$0.603432\pi$
$68$	18758.8i	4.05683i
$69$	0	0
$70$	9225.80	1.88282
$71$	5595.36i	1.10997i	0.831860	+	0.554986i	$0.187276\pi$
$71$	5595.36i	1.10997i	−0.831860	+	0.554986i	$0.812724\pi$
$72$	0	0
$73$	7462.48	1.40035	0.700176	−	0.713970i	$-0.253105\pi$
$73$	7462.48	1.40035	0.700176	+	0.713970i	$0.253105\pi$
$74$	− 9044.36i	− 1.65164i
$75$	0	0
$76$	−26345.2	−4.56115
$77$	− 3724.66i	− 0.628211i
$78$	0	0
$79$	−57.5610	−0.00922303	−0.00461152	−	0.999989i	$-0.501468\pi$
$79$	−57.5610	−0.00922303	−0.00461152	+	0.999989i	$0.501468\pi$
$80$	− 35190.5i	− 5.49852i
$81$	0	0
$82$	−4955.81	−0.737033
$83$	− 7578.24i	− 1.10005i	−0.835148	−	0.550025i	$-0.814618\pi$
$83$	− 7578.24i	− 1.10005i	0.835148	−	0.550025i	$-0.185382\pi$
$84$	0	0
$85$	−12458.8	−1.72440
$86$	− 4308.46i	− 0.582539i
$87$	0	0
$88$	−23394.2	−3.02094
$89$	− 4901.93i	− 0.618852i	−0.950923	−	0.309426i	$-0.899863\pi$
$89$	− 4901.93i	− 0.618852i	0.950923	−	0.309426i	$-0.100137\pi$
$90$	0	0
$91$	−10148.0	−1.22546
$92$	− 25516.2i	− 3.01467i
$93$	0	0
$94$	6561.59	0.742597
$95$	− 17497.3i	− 1.93876i
$96$	0	0
$97$	4492.34	0.477452	0.238726	−	0.971087i	$-0.423270\pi$
$97$	4492.34	0.477452	0.238726	+	0.971087i	$0.423270\pi$
$98$	7528.90i	0.783934i
$99$	0	0
$100$	14788.8	1.47888
$101$	4511.14i	0.442225i	0.975248	+	0.221112i	$0.0709688\pi$
$101$	4511.14i	0.442225i	−0.975248	+	0.221112i	$0.929031\pi$
$102$	0	0
$103$	−5735.81	−0.540655	−0.270328	−	0.962768i	$-0.587132\pi$
$103$	−5735.81	−0.540655	−0.270328	+	0.962768i	$0.587132\pi$
$104$	63738.7i	5.89300i
$105$	0	0
$106$	36306.0	3.23122
$107$	19027.5i	1.66193i	0.556322	+	0.830967i	$0.312212\pi$
$107$	19027.5i	1.66193i	−0.556322	+	0.830967i	$0.687788\pi$
$108$	0	0
$109$	−872.407	−0.0734287	−0.0367144	−	0.999326i	$-0.511689\pi$
$109$	−872.407	−0.0734287	−0.0367144	+	0.999326i	$0.511689\pi$
$110$	− 23748.6i	− 1.96270i
$111$	0	0
$112$	−43554.4	−3.47213
$113$	− 17682.3i	− 1.38478i	−0.721522	−	0.692391i	$-0.756557\pi$
$113$	− 17682.3i	− 1.38478i	0.721522	−	0.692391i	$-0.243443\pi$
$114$	0	0
$115$	16946.7	1.28142
$116$	− 72925.9i	− 5.41958i
$117$	0	0
$118$	3576.32	0.256846
$119$	15419.9i	1.08890i
$120$	0	0
$121$	5053.14	0.345137
$122$	− 38196.0i	− 2.56624i
$123$	0	0
$124$	27312.9	1.77634
$125$	− 9386.70i	− 0.600749i
$126$	0	0
$127$	−3736.99	−0.231694	−0.115847	−	0.993267i	$-0.536958\pi$
$127$	−3736.99	−0.231694	−0.115847	+	0.993267i	$0.536958\pi$
$128$	96664.6i	5.89994i
$129$	0	0
$130$	−64704.5	−3.82867
$131$	15398.3i	0.897284i	0.893712	+	0.448642i	$0.148092\pi$
$131$	15398.3i	0.897284i	−0.893712	+	0.448642i	$0.851908\pi$
$132$	0	0
$133$	−21656.0	−1.22426
$134$	22619.0i	1.25969i
$135$	0	0
$136$	96850.5	5.23630
$137$	− 5681.19i	− 0.302690i	−0.988481	−	0.151345i	$-0.951640\pi$
$137$	− 5681.19i	− 0.302690i	0.988481	−	0.151345i	$-0.0483605\pi$
$138$	0	0
$139$	−5777.15	−0.299009	−0.149504	−	0.988761i	$-0.547768\pi$
$139$	−5777.15	−0.299009	−0.149504	+	0.988761i	$0.547768\pi$
$140$	− 54099.8i	− 2.76019i
$141$	0	0
$142$	44155.7	2.18983
$143$	26122.7i	1.27745i
$144$	0	0
$145$	48434.2	2.30365
$146$	− 58889.9i	− 2.76271i
$147$	0	0
$148$	−53035.8	−2.42128
$149$	23565.8i	1.06147i	0.847537	+	0.530737i	$0.178085\pi$
$149$	23565.8i	1.06147i	−0.847537	+	0.530737i	$0.821915\pi$
$150$	0	0
$151$	−11254.5	−0.493598	−0.246799	−	0.969067i	$-0.579379\pi$
$151$	−11254.5	−0.493598	−0.246799	+	0.969067i	$0.579379\pi$
$152$	136019.i	5.88724i
$153$	0	0
$154$	−29393.1	−1.23938
$155$	18140.1i	0.755050i
$156$	0	0
$157$	−13876.3	−0.562955	−0.281478	−	0.959568i	$-0.590825\pi$
$157$	−13876.3	−0.562955	−0.281478	+	0.959568i	$0.590825\pi$
$158$	454.241i	0.0181958i
$159$	0	0
$160$	−160219.	−6.25855
$161$	− 20974.5i	− 0.809170i
$162$	0	0
$163$	26838.6	1.01015	0.505073	−	0.863077i	$-0.331466\pi$
$163$	26838.6	1.01015	0.505073	+	0.863077i	$0.331466\pi$
$164$	29060.7i	1.08048i
$165$	0	0
$166$	−59803.5	−2.17025
$167$	35538.6i	1.27429i	0.770745	+	0.637143i	$0.219884\pi$
$167$	35538.6i	1.27429i	−0.770745	+	0.637143i	$0.780116\pi$
$168$	0	0
$169$	42611.7	1.49195
$170$	98318.1i	3.40201i
$171$	0	0
$172$	−25264.7	−0.853998
$173$	− 24487.9i	− 0.818200i	−0.912489	−	0.409100i	$-0.865843\pi$
$173$	− 24487.9i	− 0.818200i	0.912489	−	0.409100i	$-0.134157\pi$
$174$	0	0
$175$	12156.5	0.396948
$176$	112116.i	3.61944i
$177$	0	0
$178$	−38683.4	−1.22091
$179$	− 13416.1i	− 0.418717i	−0.977839	−	0.209358i	$-0.932862\pi$
$179$	− 13416.1i	− 0.418717i	0.977839	−	0.209358i	$-0.0671375\pi$
$180$	0	0
$181$	21580.7	0.658732	0.329366	−	0.944202i	$-0.393165\pi$
$181$	21580.7	0.658732	0.329366	+	0.944202i	$0.393165\pi$
$182$	80083.0i	2.41767i
$183$	0	0
$184$	−131738.	−3.89114
$185$	− 35224.1i	− 1.02919i
$186$	0	0
$187$	39693.2	1.13510
$188$	− 38476.9i	− 1.08864i
$189$	0	0
$190$	−138080.	−3.82492
$191$	− 20485.4i	− 0.561536i	−0.959776	−	0.280768i	$-0.909411\pi$
$191$	− 20485.4i	− 0.561536i	0.959776	−	0.280768i	$-0.0905892\pi$
$192$	0	0
$193$	13149.3	0.353011	0.176506	−	0.984300i	$-0.443521\pi$
$193$	13149.3	0.353011	0.176506	+	0.984300i	$0.443521\pi$
$194$	− 35451.2i	− 0.941949i
$195$	0	0
$196$	44149.2	1.14924
$197$	− 51059.8i	− 1.31567i	−0.753163	−	0.657834i	$-0.771473\pi$
$197$	− 51059.8i	− 1.31567i	0.753163	−	0.657834i	$-0.228527\pi$
$198$	0	0
$199$	26985.3	0.681429	0.340715	−	0.940167i	$-0.389331\pi$
$199$	26985.3	0.681429	0.340715	+	0.940167i	$0.389331\pi$
$200$	− 76353.9i	− 1.90885i
$201$	0	0
$202$	35599.5	0.872451
$203$	− 59945.7i	− 1.45468i
$204$	0	0
$205$	−19300.9	−0.459271
$206$	45264.0i	1.06664i
$207$	0	0
$208$	305466.	7.06050
$209$	55745.9i	1.27620i
$210$	0	0
$211$	−24609.2	−0.552756	−0.276378	−	0.961049i	$-0.589134\pi$
$211$	−24609.2	−0.552756	−0.276378	+	0.961049i	$0.589134\pi$
$212$	− 212897.i	− 4.73694i
$213$	0	0
$214$	150155.	3.27878
$215$	− 16779.7i	− 0.363001i
$216$	0	0
$217$	22451.5	0.476788
$218$	6884.57i	0.144865i
$219$	0	0
$220$	−139261.	−2.87730
$221$	− 108146.i	− 2.21425i
$222$	0	0
$223$	82301.1	1.65499	0.827496	−	0.561471i	$-0.189764\pi$
$223$	82301.1	1.65499	0.827496	+	0.561471i	$0.189764\pi$
$224$	198298.i	3.95206i
$225$	0	0
$226$	−139539.	−2.73199
$227$	− 33434.3i	− 0.648844i	−0.945913	−	0.324422i	$-0.894830\pi$
$227$	− 33434.3i	− 0.648844i	0.945913	−	0.324422i	$-0.105170\pi$
$228$	0	0
$229$	−28896.0	−0.551020	−0.275510	−	0.961298i	$-0.588847\pi$
$229$	−28896.0	−0.551020	−0.275510	+	0.961298i	$0.588847\pi$
$230$	− 133735.i	− 2.52806i
$231$	0	0
$232$	−376512.	−6.99525
$233$	− 81940.3i	− 1.50934i	−0.656107	−	0.754668i	$-0.727798\pi$
$233$	− 81940.3i	− 1.50934i	0.656107	−	0.754668i	$-0.272202\pi$
$234$	0	0
$235$	25554.7	0.462738
$236$	− 20971.4i	− 0.376533i
$237$	0	0
$238$	121686.	2.14825
$239$	23743.6i	0.415673i	0.978164	+	0.207836i	$0.0666422\pi$
$239$	23743.6i	0.415673i	−0.978164	+	0.207836i	$0.933358\pi$
$240$	0	0
$241$	36976.2	0.636631	0.318315	−	0.947985i	$-0.396883\pi$
$241$	36976.2	0.636631	0.318315	+	0.947985i	$0.396883\pi$
$242$	− 39876.7i	− 0.680909i
$243$	0	0
$244$	−223980.	−3.76209
$245$	29322.0i	0.488497i
$246$	0	0
$247$	151883.	2.48951
$248$	− 141015.i	− 2.29278i
$249$	0	0
$250$	−74074.8	−1.18520
$251$	11789.2i	0.187127i	0.995613	+	0.0935637i	$0.0298259\pi$
$251$	11789.2i	0.187127i	−0.995613	+	0.0935637i	$0.970174\pi$
$252$	0	0
$253$	−53991.6	−0.843501
$254$	29490.3i	0.457101i
$255$	0	0
$256$	397726.	6.06881
$257$	− 34248.5i	− 0.518532i	−0.965806	−	0.259266i	$-0.916519\pi$
$257$	− 34248.5i	− 0.518532i	0.965806	−	0.259266i	$-0.0834806\pi$
$258$	0	0
$259$	−43595.9	−0.649899
$260$	379425.i	5.61280i
$261$	0	0
$262$	121515.	1.77022
$263$	83903.7i	1.21302i	0.795074	+	0.606512i	$0.207432\pi$
$263$	83903.7i	1.21302i	−0.795074	+	0.606512i	$0.792568\pi$
$264$	0	0
$265$	141397.	2.01349
$266$	170898.i	2.41531i
$267$	0	0
$268$	132637.	1.84670
$269$	50613.0i	0.699451i	0.936852	+	0.349726i	$0.113725\pi$
$269$	50613.0i	0.699451i	−0.936852	+	0.349726i	$0.886275\pi$
$270$	0	0
$271$	108503.	1.47742	0.738709	−	0.674025i	$-0.235436\pi$
$271$	108503.	1.47742	0.738709	+	0.674025i	$0.235436\pi$
$272$	− 464153.i	− 6.27369i
$273$	0	0
$274$	−44832.9	−0.597168
$275$	− 31292.8i	− 0.413790i
$276$	0	0
$277$	−48391.4	−0.630679	−0.315339	−	0.948979i	$-0.602118\pi$
$277$	−48391.4	−0.630679	−0.315339	+	0.948979i	$0.602118\pi$
$278$	45590.2i	0.589905i
$279$	0	0
$280$	−279314.	−3.56268
$281$	− 104837.i	− 1.32771i	−0.747860	−	0.663856i	$-0.768919\pi$
$281$	− 104837.i	− 1.32771i	0.747860	−	0.663856i	$-0.231081\pi$
$282$	0	0
$283$	14243.6	0.177847	0.0889236	−	0.996038i	$-0.471657\pi$
$283$	14243.6	0.177847	0.0889236	+	0.996038i	$0.471657\pi$
$284$	− 258927.i	− 3.21027i
$285$	0	0
$286$	206146.	2.52025
$287$	23888.2i	0.290014i
$288$	0	0
$289$	−80806.6	−0.967500
$290$	− 382217.i	− 4.54480i
$291$	0	0
$292$	−345329.	−4.05011
$293$	68975.2i	0.803448i	0.915761	+	0.401724i	$0.131589\pi$
$293$	68975.2i	0.803448i	−0.915761	+	0.401724i	$0.868411\pi$
$294$	0	0
$295$	13928.3	0.160049
$296$	273821.i	3.12524i
$297$	0	0
$298$	185969.	2.09415
$299$	147103.i	1.64543i
$300$	0	0
$301$	−20767.8	−0.229222
$302$	88814.8i	0.973803i
$303$	0	0
$304$	651865.	7.05359
$305$	− 148758.i	− 1.59912i
$306$	0	0
$307$	−93690.0	−0.994069	−0.497034	−	0.867731i	$-0.665578\pi$
$307$	−93690.0	−0.994069	−0.497034	+	0.867731i	$0.665578\pi$
$308$	172360.i	1.81692i
$309$	0	0
$310$	143152.	1.48961
$311$	22556.7i	0.233214i	0.993178	+	0.116607i	$0.0372019\pi$
$311$	22556.7i	0.233214i	−0.993178	+	0.116607i	$0.962798\pi$
$312$	0	0
$313$	58390.8	0.596013	0.298006	−	0.954564i	$-0.403678\pi$
$313$	58390.8	0.596013	0.298006	+	0.954564i	$0.403678\pi$
$314$	109504.i	1.11064i
$315$	0	0
$316$	2663.65	0.0266749
$317$	− 55440.6i	− 0.551708i	−0.961199	−	0.275854i	$-0.911039\pi$
$317$	− 55440.6i	− 0.551708i	0.961199	−	0.275854i	$-0.0889607\pi$
$318$	0	0
$319$	−154310.	−1.51639
$320$	701314.i	6.84877i
$321$	0	0
$322$	−165520.	−1.59639
$323$	− 230785.i	− 2.21209i
$324$	0	0
$325$	−85259.1	−0.807187
$326$	− 211796.i	− 1.99288i
$327$	0	0
$328$	150039.	1.39462
$329$	− 31628.4i	− 0.292203i
$330$	0	0
$331$	−110835.	−1.01163	−0.505814	−	0.862643i	$-0.668808\pi$
$331$	−110835.	−1.01163	−0.505814	+	0.862643i	$0.668808\pi$
$332$	350686.i	3.18157i
$333$	0	0
$334$	280452.	2.51400
$335$	88092.0i	0.784958i
$336$	0	0
$337$	−53449.4	−0.470634	−0.235317	−	0.971919i	$-0.575613\pi$
$337$	−53449.4	−0.470634	−0.235317	+	0.971919i	$0.575613\pi$
$338$	− 336269.i	− 2.94343i
$339$	0	0
$340$	576534.	4.98732
$341$	− 57793.6i	− 0.497016i
$342$	0	0
$343$	127622.	1.08477
$344$	130440.i	1.10229i
$345$	0	0
$346$	−193246.	−1.61420
$347$	187145.i	1.55424i	0.629350	+	0.777122i	$0.283321\pi$
$347$	187145.i	1.55424i	−0.629350	+	0.777122i	$0.716679\pi$
$348$	0	0
$349$	55841.1	0.458462	0.229231	−	0.973372i	$-0.426379\pi$
$349$	55841.1	0.458462	0.229231	+	0.973372i	$0.426379\pi$
$350$	− 95933.0i	− 0.783127i
$351$	0	0
$352$	510451.	4.11973
$353$	− 22199.8i	− 0.178155i	−0.996025	−	0.0890777i	$-0.971608\pi$
$353$	− 22199.8i	− 0.178155i	0.996025	−	0.0890777i	$-0.0283919\pi$
$354$	0	0
$355$	171968.	1.36456
$356$	226838.i	1.78985i
$357$	0	0
$358$	−105873.	−0.826073
$359$	− 245560.i	− 1.90532i	−0.304034	−	0.952661i	$-0.598334\pi$
$359$	− 245560.i	− 1.90532i	0.304034	−	0.952661i	$-0.401666\pi$
$360$	0	0
$361$	193797.	1.48708
$362$	− 170304.i	− 1.29959i
$363$	0	0
$364$	469604.	3.54429
$365$	− 229352.i	− 1.72154i
$366$	0	0
$367$	144668.	1.07409	0.537045	−	0.843554i	$-0.319541\pi$
$367$	144668.	1.07409	0.537045	+	0.843554i	$0.319541\pi$
$368$	631352.i	4.66204i
$369$	0	0
$370$	−277970.	−2.03046
$371$	− 175003.i	− 1.27145i
$372$	0	0
$373$	14333.8	0.103025	0.0515127	−	0.998672i	$-0.483596\pi$
$373$	14333.8	0.103025	0.0515127	+	0.998672i	$0.483596\pi$
$374$	− 313238.i	− 2.23940i
$375$	0	0
$376$	−198654.	−1.40515
$377$	420425.i	2.95805i
$378$	0	0
$379$	−143982.	−1.00238	−0.501188	−	0.865338i	$-0.667104\pi$
$379$	−143982.	−1.00238	−0.501188	+	0.865338i	$0.667104\pi$
$380$	809695.i	5.60731i
$381$	0	0
$382$	−161660.	−1.10784
$383$	− 232495.i	− 1.58495i	−0.609903	−	0.792476i	$-0.708791\pi$
$383$	− 232495.i	− 1.58495i	0.609903	−	0.792476i	$-0.291209\pi$
$384$	0	0
$385$	−114474.	−0.772299
$386$	− 103768.i	− 0.696445i
$387$	0	0
$388$	−207885.	−1.38089
$389$	138944.i	0.918206i	0.888383	+	0.459103i	$0.151829\pi$
$389$	138944.i	0.918206i	−0.888383	+	0.459103i	$0.848171\pi$
$390$	0	0
$391$	223522.	1.46207
$392$	− 227940.i	− 1.48337i
$393$	0	0
$394$	−402937.	−2.59564
$395$	1769.08i	0.0113385i
$396$	0	0
$397$	145759.	0.924815	0.462407	−	0.886668i	$-0.346986\pi$
$397$	145759.	0.924815	0.462407	+	0.886668i	$0.346986\pi$
$398$	− 212954.i	− 1.34437i
$399$	0	0
$400$	−365923.	−2.28702
$401$	147498.i	0.917273i	0.888624	+	0.458637i	$0.151662\pi$
$401$	147498.i	0.917273i	−0.888624	+	0.458637i	$0.848338\pi$
$402$	0	0
$403$	−157462.	−0.969539
$404$	− 208754.i	− 1.27901i
$405$	0	0
$406$	−473060.	−2.86988
$407$	112223.i	0.677473i
$408$	0	0
$409$	−199915.	−1.19509	−0.597543	−	0.801837i	$-0.703856\pi$
$409$	−199915.	−1.19509	−0.597543	+	0.801837i	$0.703856\pi$
$410$	152312.i	0.906082i
$411$	0	0
$412$	265427.	1.56369
$413$	− 17238.7i	− 0.101066i
$414$	0	0
$415$	−232910.	−1.35236
$416$	− 1.39075e6i	− 8.03643i
$417$	0	0
$418$	439917.	2.51778
$419$	− 58490.7i	− 0.333165i	−0.986028	−	0.166582i	$-0.946727\pi$
$419$	− 58490.7i	− 0.333165i	0.986028	−	0.166582i	$-0.0532732\pi$
$420$	0	0
$421$	37928.6	0.213994	0.106997	−	0.994259i	$-0.465876\pi$
$421$	37928.6	0.213994	0.106997	+	0.994259i	$0.465876\pi$
$422$	194203.i	1.09051i
$423$	0	0
$424$	−1.09918e6	−6.11414
$425$	129551.i	0.717235i
$426$	0	0
$427$	−184113.	−1.00979
$428$	− 880503.i	− 4.80666i
$429$	0	0
$430$	−132417.	−0.716152
$431$	175808.i	0.946419i	0.880950	+	0.473209i	$0.156905\pi$
$431$	175808.i	0.946419i	−0.880950	+	0.473209i	$0.843095\pi$
$432$	0	0
$433$	266918.	1.42365	0.711823	−	0.702359i	$-0.247870\pi$
$433$	266918.	1.42365	0.711823	+	0.702359i	$0.247870\pi$
$434$	− 177175.i	− 0.940640i
$435$	0	0
$436$	40370.9	0.212371
$437$	313919.i	1.64382i
$438$	0	0
$439$	−195586.	−1.01487	−0.507434	−	0.861690i	$-0.669406\pi$
$439$	−195586.	−1.01487	−0.507434	+	0.861690i	$0.669406\pi$
$440$	718998.i	3.71383i
$441$	0	0
$442$	−853434.	−4.36843
$443$	− 43264.2i	− 0.220455i	−0.993906	−	0.110228i	$-0.964842\pi$
$443$	− 43264.2i	− 0.220455i	0.993906	−	0.110228i	$-0.0351580\pi$
$444$	0	0
$445$	−150656.	−0.760794
$446$	− 649477.i	− 3.26508i
$447$	0	0
$448$	867997.	4.32476
$449$	− 12614.8i	− 0.0625731i	−0.999510	−	0.0312866i	$-0.990040\pi$
$449$	− 12614.8i	− 0.0625731i	0.999510	−	0.0312866i	$-0.00996045\pi$
$450$	0	0
$451$	61491.9	0.302318
$452$	818253.i	4.00508i
$453$	0	0
$454$	−263846.	−1.28008
$455$	311891.i	1.50654i
$456$	0	0
$457$	309312.	1.48103	0.740516	−	0.672039i	$-0.234581\pi$
$457$	309312.	1.48103	0.740516	+	0.672039i	$0.234581\pi$
$458$	228032.i	1.08709i
$459$	0	0
$460$	−784215.	−3.70612
$461$	54977.6i	0.258692i	0.991599	+	0.129346i	$0.0412879\pi$
$461$	54977.6i	0.258692i	−0.991599	+	0.129346i	$0.958712\pi$
$462$	0	0
$463$	−252184.	−1.17640	−0.588202	−	0.808714i	$-0.700164\pi$
$463$	−252184.	−1.17640	−0.588202	+	0.808714i	$0.700164\pi$
$464$	1.80442e6i	8.38112i
$465$	0	0
$466$	−646630.	−2.97772
$467$	214523.i	0.983649i	0.870694	+	0.491825i	$0.163670\pi$
$467$	214523.i	0.983649i	−0.870694	+	0.491825i	$0.836330\pi$
$468$	0	0
$469$	109029.	0.495674
$470$	− 201664.i	− 0.912921i
$471$	0	0
$472$	−108274.	−0.486005
$473$	53459.5i	0.238948i
$474$	0	0
$475$	−181943.	−0.806397
$476$	− 713560.i	− 3.14932i
$477$	0	0
$478$	187372.	0.820068
$479$	236791.i	1.03203i	0.856578	+	0.516017i	$0.172586\pi$
$479$	236791.i	1.03203i	−0.856578	+	0.516017i	$0.827414\pi$
$480$	0	0
$481$	305757.	1.32156
$482$	− 291796.i	− 1.25599i
$483$	0	0
$484$	−233836.	−0.998207
$485$	− 138068.i	− 0.586961i
$486$	0	0
$487$	287933.	1.21404	0.607022	−	0.794685i	$-0.292364\pi$
$487$	287933.	1.21404	0.607022	+	0.794685i	$0.292364\pi$
$488$	1.15639e6i	4.85586i
$489$	0	0
$490$	231394.	0.963740
$491$	65201.6i	0.270455i	0.990815	+	0.135228i	$0.0431766\pi$
$491$	65201.6i	0.270455i	−0.990815	+	0.135228i	$0.956823\pi$
$492$	0	0
$493$	638833.	2.62841
$494$	− 1.19858e6i	− 4.91148i
$495$	0	0
$496$	−675810.	−2.74702
$497$	− 212841.i	− 0.861671i
$498$	0	0
$499$	−255807.	−1.02733	−0.513666	−	0.857990i	$-0.671713\pi$
$499$	−255807.	−1.02733	−0.513666	+	0.857990i	$0.671713\pi$
$500$	434372.i	1.73749i
$501$	0	0
$502$	93034.3	0.369178
$503$	175779.i	0.694753i	0.937726	+	0.347376i	$0.112927\pi$
$503$	175779.i	0.694753i	−0.937726	+	0.347376i	$0.887073\pi$
$504$	0	0
$505$	138646.	0.543655
$506$	426073.i	1.66412i
$507$	0	0
$508$	172930.	0.670106
$509$	32249.5i	0.124476i	0.998061	+	0.0622382i	$0.0198239\pi$
$509$	32249.5i	0.124476i	−0.998061	+	0.0622382i	$0.980176\pi$
$510$	0	0
$511$	−283863.	−1.08709
$512$	− 1.59201e6i	− 6.07303i
$513$	0	0
$514$	−270271.	−1.02300
$515$	176285.i	0.664662i
$516$	0	0
$517$	−81416.3	−0.304600
$518$	344036.i	1.28217i
$519$	0	0
$520$	1.95895e6	7.24464
$521$	− 454411.i	− 1.67407i	−0.547149	−	0.837035i	$-0.684287\pi$
$521$	− 454411.i	− 1.67407i	0.547149	−	0.837035i	$-0.315713\pi$
$522$	0	0
$523$	353992.	1.29417	0.647083	−	0.762420i	$-0.275989\pi$
$523$	353992.	1.29417	0.647083	+	0.762420i	$0.275989\pi$
$524$	− 712561.i	− 2.59513i
$525$	0	0
$526$	662123.	2.39314
$527$	239262.i	0.861495i
$528$	0	0
$529$	−24199.5	−0.0864758
$530$	− 1.11583e6i	− 3.97234i
$531$	0	0
$532$	1.00214e6	3.54082
$533$	− 167538.i	− 0.589738i
$534$	0	0
$535$	584792.	2.04312
$536$	− 684799.i	− 2.38360i
$537$	0	0
$538$	399411.	1.37992
$539$	− 93418.8i	− 0.321556i
$540$	0	0
$541$	135948.	0.464494	0.232247	−	0.972657i	$-0.425392\pi$
$541$	135948.	0.464494	0.232247	+	0.972657i	$0.425392\pi$
$542$	− 856248.i	− 2.91475i
$543$	0	0
$544$	−2.11324e6	−7.14087
$545$	26812.6i	0.0902706i
$546$	0	0
$547$	270604.	0.904398	0.452199	−	0.891917i	$-0.350640\pi$
$547$	270604.	0.904398	0.452199	+	0.891917i	$0.350640\pi$
$548$	262899.i	0.875443i
$549$	0	0
$550$	−246947.	−0.816352
$551$	897189.i	2.95516i
$552$	0	0
$553$	2189.54	0.00715984
$554$	381879.i	1.24425i
$555$	0	0
$556$	267339.	0.864796
$557$	349893.i	1.12778i	0.825849	+	0.563891i	$0.190696\pi$
$557$	349893.i	1.12778i	−0.825849	+	0.563891i	$0.809304\pi$
$558$	0	0
$559$	145653.	0.466119
$560$	1.33860e6i	4.26850i
$561$	0	0
$562$	−827322.	−2.61940
$563$	326045.i	1.02863i	0.857601	+	0.514316i	$0.171954\pi$
$563$	326045.i	1.02863i	−0.857601	+	0.514316i	$0.828046\pi$
$564$	0	0
$565$	−543449.	−1.70240
$566$	− 112403.i	− 0.350869i
$567$	0	0
$568$	−1.33683e6	−4.14361
$569$	11943.9i	0.0368912i	0.999830	+	0.0184456i	$0.00587176\pi$
$569$	11943.9i	0.0368912i	−0.999830	+	0.0184456i	$0.994128\pi$
$570$	0	0
$571$	−19985.7	−0.0612980	−0.0306490	−	0.999530i	$-0.509757\pi$
$571$	−19985.7	−0.0612980	−0.0306490	+	0.999530i	$0.509757\pi$
$572$	− 1.20883e6i	− 3.69466i
$573$	0	0
$574$	188513.	0.572159
$575$	− 176218.i	− 0.532984i
$576$	0	0
$577$	117550.	0.353078	0.176539	−	0.984294i	$-0.443510\pi$
$577$	117550.	0.353078	0.176539	+	0.984294i	$0.443510\pi$
$578$	637683.i	1.90875i
$579$	0	0
$580$	−2.24131e6	−6.66263
$581$	288267.i	0.853969i
$582$	0	0
$583$	−450486.	−1.32539
$584$	1.78291e6i	5.22762i
$585$	0	0
$586$	544316.	1.58510
$587$	285221.i	0.827761i	0.910331	+	0.413881i	$0.135827\pi$
$587$	285221.i	0.827761i	−0.910331	+	0.413881i	$0.864173\pi$
$588$	0	0
$589$	−336024.	−0.968590
$590$	− 109915.i	− 0.315756i
$591$	0	0
$592$	1.31228e6	3.74440
$593$	86944.4i	0.247248i	0.992329	+	0.123624i	$0.0394516\pi$
$593$	86944.4i	0.247248i	−0.992329	+	0.123624i	$0.960548\pi$
$594$	0	0
$595$	473916.	1.33865
$596$	− 1.09051e6i	− 3.07000i
$597$	0	0
$598$	1.16086e6	3.24622
$599$	− 216807.i	− 0.604253i	−0.953268	−	0.302127i	$-0.902303\pi$
$599$	− 216807.i	− 0.604253i	0.953268	−	0.302127i	$-0.0976966\pi$
$600$	0	0
$601$	−69954.0	−0.193671	−0.0968353	−	0.995300i	$-0.530872\pi$
$601$	−69954.0	−0.193671	−0.0968353	+	0.995300i	$0.530872\pi$
$602$	163888.i	0.452226i
$603$	0	0
$604$	520807.	1.42759
$605$	− 155304.i	− 0.424298i
$606$	0	0
$607$	−301673.	−0.818766	−0.409383	−	0.912363i	$-0.634256\pi$
$607$	−301673.	−0.818766	−0.409383	+	0.912363i	$0.634256\pi$
$608$	− 2.96787e6i	− 8.02857i
$609$	0	0
$610$	−1.17392e6	−3.15484
$611$	221823.i	0.594190i
$612$	0	0
$613$	447345.	1.19048	0.595240	−	0.803548i	$-0.297057\pi$
$613$	447345.	1.19048	0.595240	+	0.803548i	$0.297057\pi$
$614$	739352.i	1.96117i
$615$	0	0
$616$	889884.	2.34516
$617$	130266.i	0.342185i	0.985255	+	0.171093i	$0.0547297\pi$
$617$	130266.i	0.342185i	−0.985255	+	0.171093i	$0.945270\pi$
$618$	0	0
$619$	367325.	0.958670	0.479335	−	0.877632i	$-0.340878\pi$
$619$	367325.	0.958670	0.479335	+	0.877632i	$0.340878\pi$
$620$	− 839438.i	− 2.18376i
$621$	0	0
$622$	178006.	0.460101
$623$	186463.i	0.480415i
$624$	0	0
$625$	−488231.	−1.24987
$626$	− 460789.i	− 1.17585i
$627$	0	0
$628$	642130.	1.62818
$629$	− 464595.i	− 1.17429i
$630$	0	0
$631$	−1458.86	−0.00366399	−0.00183199	−	0.999998i	$-0.500583\pi$
$631$	−1458.86	−0.00366399	−0.00183199	+	0.999998i	$0.500583\pi$
$632$	− 13752.3i	− 0.0344303i
$633$	0	0
$634$	−437508.	−1.08845
$635$	114853.i	0.284835i
$636$	0	0
$637$	−254525.	−0.627266
$638$	1.21773e6i	2.99164i
$639$	0	0
$640$	2.97090e6	7.25317
$641$	633002.i	1.54060i	0.637683	+	0.770299i	$0.279893\pi$
$641$	633002.i	1.54060i	−0.637683	+	0.770299i	$0.720107\pi$
$642$	0	0
$643$	−466214.	−1.12762	−0.563811	−	0.825904i	$-0.690665\pi$
$643$	−466214.	−1.12762	−0.563811	+	0.825904i	$0.690665\pi$
$644$	970602.i	2.34029i
$645$	0	0
$646$	−1.82123e6	−4.36415
$647$	− 657916.i	− 1.57167i	−0.618435	−	0.785836i	$-0.712233\pi$
$647$	− 657916.i	− 1.57167i	0.618435	−	0.785836i	$-0.287767\pi$
$648$	0	0
$649$	−44375.0	−0.105354
$650$	672820.i	1.59247i
$651$	0	0
$652$	−1.24196e6	−2.92155
$653$	185217.i	0.434366i	0.976131	+	0.217183i	$0.0696868\pi$
$653$	185217.i	0.434366i	−0.976131	+	0.217183i	$0.930313\pi$
$654$	0	0
$655$	473252.	1.10309
$656$	− 719056.i	− 1.67092i
$657$	0	0
$658$	−249594.	−0.576478
$659$	68433.8i	0.157580i	0.996891	+	0.0787898i	$0.0251056\pi$
$659$	68433.8i	0.157580i	−0.996891	+	0.0787898i	$0.974894\pi$
$660$	0	0
$661$	−650096.	−1.48790	−0.743951	−	0.668234i	$-0.767051\pi$
$661$	−650096.	−1.48790	−0.743951	+	0.668234i	$0.767051\pi$
$662$	874651.i	1.99581i
$663$	0	0
$664$	1.81057e6	4.10657
$665$	665576.i	1.50506i
$666$	0	0
$667$	−868956.	−1.95320
$668$	− 1.64456e6i	− 3.68550i
$669$	0	0
$670$	695175.	1.54862
$671$	473936.i	1.05263i
$672$	0	0
$673$	−179348.	−0.395973	−0.197987	−	0.980205i	$-0.563440\pi$
$673$	−179348.	−0.395973	−0.197987	+	0.980205i	$0.563440\pi$
$674$	421794.i	0.928498i
$675$	0	0
$676$	−1.97187e6	−4.31504
$677$	− 676369.i	− 1.47573i	−0.674949	−	0.737864i	$-0.735834\pi$
$677$	− 676369.i	− 1.47573i	0.674949	−	0.737864i	$-0.264166\pi$
$678$	0	0
$679$	−170883.	−0.370646
$680$	− 2.97661e6i	− 6.43731i
$681$	0	0
$682$	−456076.	−0.980548
$683$	− 369112.i	− 0.791255i	−0.918411	−	0.395628i	$-0.870527\pi$
$683$	− 369112.i	− 0.791255i	0.918411	−	0.395628i	$-0.129473\pi$
$684$	0	0
$685$	−174606.	−0.372116
$686$	− 1.00713e6i	− 2.14011i
$687$	0	0
$688$	625129.	1.32067
$689$	1.22737e6i	2.58546i
$690$	0	0
$691$	274828.	0.575579	0.287789	−	0.957694i	$-0.407080\pi$
$691$	274828.	0.575579	0.287789	+	0.957694i	$0.407080\pi$
$692$	1.13319e6i	2.36641i
$693$	0	0
$694$	1.47685e6	3.06632
$695$	177555.i	0.367590i
$696$	0	0
$697$	−254573.	−0.524018
$698$	− 440668.i	− 0.904484i
$699$	0	0
$700$	−562548.	−1.14806
$701$	869321.i	1.76907i	0.466478	+	0.884533i	$0.345523\pi$
$701$	869321.i	1.76907i	−0.466478	+	0.884533i	$0.654477\pi$
$702$	0	0
$703$	652487.	1.32026
$704$	− 2.23436e6i	− 4.50825i
$705$	0	0
$706$	−175189.	−0.351477
$707$	− 171598.i	− 0.343299i
$708$	0	0
$709$	−476697.	−0.948310	−0.474155	−	0.880441i	$-0.657246\pi$
$709$	−476697.	−0.948310	−0.474155	+	0.880441i	$0.657246\pi$
$710$	− 1.35708e6i	− 2.69209i
$711$	0	0
$712$	1.17115e6	2.31022
$713$	− 325450.i	− 0.640185i
$714$	0	0
$715$	802856.	1.57045
$716$	620834.i	1.21102i
$717$	0	0
$718$	−1.93783e6	−3.75895
$719$	− 497774.i	− 0.962885i	−0.876478	−	0.481443i	$-0.840113\pi$
$719$	− 497774.i	− 0.962885i	0.876478	−	0.481443i	$-0.159887\pi$
$720$	0	0
$721$	218183.	0.419711
$722$	− 1.52935e6i	− 2.93381i
$723$	0	0
$724$	−998655.	−1.90519
$725$	− 503636.i	− 0.958165i
$726$	0	0
$727$	389597.	0.737134	0.368567	−	0.929601i	$-0.379848\pi$
$727$	389597.	0.737134	0.368567	+	0.929601i	$0.379848\pi$
$728$	− 2.42454e6i	− 4.57474i
$729$	0	0
$730$	−1.80993e6	−3.39637
$731$	− 221319.i	− 0.414176i
$732$	0	0
$733$	224707.	0.418224	0.209112	−	0.977892i	$-0.432943\pi$
$733$	224707.	0.418224	0.209112	+	0.977892i	$0.432943\pi$
$734$	− 1.14164e6i	− 2.11904i
$735$	0	0
$736$	2.87448e6	5.30644
$737$	− 280658.i	− 0.516704i
$738$	0	0
$739$	−908286.	−1.66316	−0.831579	−	0.555406i	$-0.812563\pi$
$739$	−908286.	−1.66316	−0.831579	+	0.555406i	$0.812563\pi$
$740$	1.63001e6i	2.97664i
$741$	0	0
$742$	−1.38103e6	−2.50840
$743$	− 957884.i	− 1.73514i	−0.497312	−	0.867572i	$-0.665679\pi$
$743$	− 957884.i	− 1.73514i	0.497312	−	0.867572i	$-0.334321\pi$
$744$	0	0
$745$	724272.	1.30494
$746$	− 113115.i	− 0.203256i
$747$	0	0
$748$	−1.83682e6	−3.28293
$749$	− 723781.i	− 1.29016i
$750$	0	0
$751$	603841.	1.07064	0.535319	−	0.844650i	$-0.320191\pi$
$751$	603841.	1.07064	0.535319	+	0.844650i	$0.320191\pi$
$752$	952043.i	1.68353i
$753$	0	0
$754$	3.31777e6	5.83585
$755$	345897.i	0.606811i
$756$	0	0
$757$	−322660.	−0.563058	−0.281529	−	0.959553i	$-0.590842\pi$
$757$	−322660.	−0.563058	−0.281529	+	0.959553i	$0.590842\pi$
$758$	1.13623e6i	1.97756i
$759$	0	0
$760$	4.18041e6	7.23755
$761$	− 884497.i	− 1.52731i	−0.645624	−	0.763655i	$-0.723403\pi$
$761$	− 884497.i	− 1.52731i	0.645624	−	0.763655i	$-0.276597\pi$
$762$	0	0
$763$	33185.2	0.0570028
$764$	947968.i	1.62408i
$765$	0	0
$766$	−1.83473e6	−3.12690
$767$	120902.i	0.205515i
$768$	0	0
$769$	−998022.	−1.68767	−0.843835	−	0.536603i	$-0.819707\pi$
$769$	−998022.	−1.68767	−0.843835	+	0.536603i	$0.819707\pi$
$770$	903368.i	1.52364i
$771$	0	0
$772$	−608489.	−1.02098
$773$	− 401722.i	− 0.672305i	−0.941808	−	0.336153i	$-0.890874\pi$
$773$	− 401722.i	− 0.672305i	0.941808	−	0.336153i	$-0.109126\pi$
$774$	0	0
$775$	188627.	0.314051
$776$	1.07330e6i	1.78236i
$777$	0	0
$778$	1.09647e6	1.81150
$779$	− 357527.i	− 0.589161i
$780$	0	0
$781$	−547885.	−0.898229
$782$	− 1.76392e6i	− 2.88447i
$783$	0	0
$784$	−1.09239e6	−1.77724
$785$	426475.i	0.692077i
$786$	0	0
$787$	−119699.	−0.193259	−0.0966296	−	0.995320i	$-0.530806\pi$
$787$	−119699.	−0.193259	−0.0966296	+	0.995320i	$0.530806\pi$
$788$	2.36281e6i	3.80519i
$789$	0	0
$790$	13960.7	0.0223693
$791$	672611.i	1.07501i
$792$	0	0
$793$	1.29127e6	2.05338
$794$	− 1.15025e6i	− 1.82454i
$795$	0	0
$796$	−1.24875e6	−1.97084
$797$	572276.i	0.900925i	0.892795	+	0.450462i	$0.148741\pi$
$797$	572276.i	0.900925i	−0.892795	+	0.450462i	$0.851259\pi$
$798$	0	0
$799$	337059.	0.527974
$800$	1.66601e6i	2.60314i
$801$	0	0
$802$	1.16398e6	1.80966
$803$	730708.i	1.13322i
$804$	0	0
$805$	−644632.	−0.994764
$806$	1.24261e6i	1.91277i
$807$	0	0
$808$	−1.07779e6	−1.65086
$809$	641287.i	0.979841i	0.871767	+	0.489920i	$0.162974\pi$
$809$	641287.i	0.979841i	−0.871767	+	0.489920i	$0.837026\pi$
$810$	0	0
$811$	−768329.	−1.16817	−0.584084	−	0.811693i	$-0.698546\pi$
$811$	−768329.	−1.16817	−0.584084	+	0.811693i	$0.698546\pi$
$812$	2.77401e6i	4.20722i
$813$	0	0
$814$	885602.	1.33656
$815$	− 824859.i	− 1.24184i
$816$	0	0
$817$	310825.	0.465663
$818$	1.57763e6i	2.35775i
$819$	0	0
$820$	893154.	1.32831
$821$	− 520639.i	− 0.772414i	−0.922412	−	0.386207i	$-0.873785\pi$
$821$	− 520639.i	− 0.772414i	0.922412	−	0.386207i	$-0.126215\pi$
$822$	0	0
$823$	1.20606e6	1.78061	0.890304	−	0.455367i	$-0.150492\pi$
$823$	1.20606e6	1.78061	0.890304	+	0.455367i	$0.150492\pi$
$824$	− 1.37038e6i	− 2.01831i
$825$	0	0
$826$	−136039.	−0.199389
$827$	840904.i	1.22952i	0.788715	+	0.614759i	$0.210747\pi$
$827$	840904.i	1.22952i	−0.788715	+	0.614759i	$0.789253\pi$
$828$	0	0
$829$	−440575.	−0.641079	−0.320539	−	0.947235i	$-0.603864\pi$
$829$	−440575.	−0.641079	−0.320539	+	0.947235i	$0.603864\pi$
$830$	1.83801e6i	2.66803i
$831$	0	0
$832$	−6.08764e6	−8.79432
$833$	386749.i	0.557364i
$834$	0	0
$835$	1.09225e6	1.56656
$836$	− 2.57966e6i	− 3.69105i
$837$	0	0
$838$	−461578.	−0.657290
$839$	− 201331.i	− 0.286014i	−0.989722	−	0.143007i	$-0.954323\pi$
$839$	− 201331.i	− 0.286014i	0.989722	−	0.143007i	$-0.0456771\pi$
$840$	0	0
$841$	−1.77622e6	−2.51134
$842$	− 299312.i	− 0.422183i
$843$	0	0
$844$	1.13880e6	1.59869
$845$	− 1.30963e6i	− 1.83415i
$846$	0	0
$847$	−192215.	−0.267930
$848$	5.26776e6i	7.32545i
$849$	0	0
$850$	1.02234e6	1.41501
$851$	631954.i	0.872622i
$852$	0	0
$853$	736483.	1.01220	0.506098	−	0.862476i	$-0.331087\pi$
$853$	736483.	1.01220	0.506098	+	0.862476i	$0.331087\pi$
$854$	1.45293e6i	1.99218i
$855$	0	0
$856$	−4.54599e6	−6.20413
$857$	946536.i	1.28877i	0.764701	+	0.644385i	$0.222887\pi$
$857$	946536.i	1.28877i	−0.764701	+	0.644385i	$0.777113\pi$
$858$	0	0
$859$	851046.	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$859$	851046.	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$860$	776486.i	1.04987i
$861$	0	0
$862$	1.38738e6	1.86716
$863$	650921.i	0.873990i	0.899464	+	0.436995i	$0.143957\pi$
$863$	650921.i	0.873990i	−0.899464	+	0.436995i	$0.856043\pi$
$864$	0	0
$865$	−752613.	−1.00587
$866$	− 2.10637e6i	− 2.80866i
$867$	0	0
$868$	−1.03895e6	−1.37897
$869$	− 5636.23i	− 0.00746361i
$870$	0	0
$871$	−764668.	−1.00794
$872$	− 208433.i	− 0.274115i
$873$	0	0
$874$	2.47728e6	3.24304
$875$	357058.i	0.466361i
$876$	0	0
$877$	−635629.	−0.826427	−0.413214	−	0.910634i	$-0.635594\pi$
$877$	−635629.	−0.826427	−0.413214	+	0.910634i	$0.635594\pi$
$878$	1.54347e6i	2.00220i
$879$	0	0
$880$	3.44577e6	4.44960
$881$	− 639569.i	− 0.824016i	−0.911180	−	0.412008i	$-0.864828\pi$
$881$	− 639569.i	− 0.824016i	0.911180	−	0.412008i	$-0.135172\pi$
$882$	0	0
$883$	−778181.	−0.998066	−0.499033	−	0.866583i	$-0.666311\pi$
$883$	−778181.	−0.998066	−0.499033	+	0.866583i	$0.666311\pi$
$884$	5.00451e6i	6.40408i
$885$	0	0
$886$	−341418.	−0.434930
$887$	− 1.21372e6i	− 1.54267i	−0.636432	−	0.771333i	$-0.719590\pi$
$887$	− 1.21372e6i	− 1.54267i	0.636432	−	0.771333i	$-0.280410\pi$
$888$	0	0
$889$	142150.	0.179864
$890$	1.18890e6i	1.50095i
$891$	0	0
$892$	−3.80851e6	−4.78658
$893$	473372.i	0.593608i
$894$	0	0
$895$	−412331.	−0.514755
$896$	− 3.67700e6i	− 4.58013i
$897$	0	0
$898$	−99549.4	−0.123449
$899$	− 930146.i	− 1.15088i
$900$	0	0
$901$	1.86499e6	2.29734
$902$	− 485261.i	− 0.596434i
$903$	0	0
$904$	4.22460e6	5.16950
$905$	− 663263.i	− 0.809821i
$906$	0	0
$907$	−1.06908e6	−1.29955	−0.649776	−	0.760126i	$-0.725137\pi$
$907$	−1.06908e6	−1.29955	−0.649776	+	0.760126i	$0.725137\pi$
$908$	1.54718e6i	1.87659i
$909$	0	0
$910$	2.46128e6	2.97220
$911$	− 132856.i	− 0.160082i	−0.996792	−	0.0800411i	$-0.974495\pi$
$911$	− 132856.i	− 0.160082i	0.996792	−	0.0800411i	$-0.0255051\pi$
$912$	0	0
$913$	742043.	0.890200
$914$	− 2.44093e6i	− 2.92188i
$915$	0	0
$916$	1.33717e6	1.59366
$917$	− 585731.i	− 0.696562i
$918$	0	0
$919$	26758.0	0.0316828	0.0158414	−	0.999875i	$-0.494957\pi$
$919$	26758.0	0.0316828	0.0158414	+	0.999875i	$0.494957\pi$
$920$	4.04886e6i	4.78362i
$921$	0	0
$922$	433854.	0.510366
$923$	1.49274e6i	1.75219i
$924$	0	0
$925$	−366272.	−0.428076
$926$	1.99011e6i	2.32089i
$927$	0	0
$928$	8.21534e6	9.53959
$929$	− 546397.i	− 0.633107i	−0.948575	−	0.316553i	$-0.897474\pi$
$929$	− 546397.i	− 0.633107i	0.948575	−	0.316553i	$-0.102526\pi$
$930$	0	0
$931$	−543157.	−0.626652
$932$	3.79182e6i	4.36531i
$933$	0	0
$934$	1.69290e6	1.94061
$935$	− 1.21993e6i	− 1.39545i
$936$	0	0
$937$	−1.07946e6	−1.22949	−0.614747	−	0.788725i	$-0.710742\pi$
$937$	−1.07946e6	−1.22949	−0.614747	+	0.788725i	$0.710742\pi$
$938$	− 860399.i	− 0.977900i
$939$	0	0
$940$	−1.18255e6	−1.33833
$941$	7295.07i	0.00823854i	0.999992	+	0.00411927i	$0.00131121\pi$
$941$	7295.07i	0.00823854i	−0.999992	+	0.00411927i	$0.998689\pi$
$942$	0	0
$943$	346276.	0.389403
$944$	518900.i	0.582290i
$945$	0	0
$946$	421874.	0.471412
$947$	708201.i	0.789690i	0.918748	+	0.394845i	$0.129202\pi$
$947$	708201.i	0.789690i	−0.918748	+	0.394845i	$0.870798\pi$
$948$	0	0
$949$	1.99086e6	2.21059
$950$	1.43580e6i	1.59091i
$951$	0	0
$952$	−3.68407e6	−4.06494
$953$	1.78259e6i	1.96275i	0.192107	+	0.981374i	$0.438468\pi$
$953$	1.78259e6i	1.96275i	−0.192107	+	0.981374i	$0.561532\pi$
$954$	0	0
$955$	−629600.	−0.690332
$956$	− 1.09874e6i	− 1.20221i
$957$	0	0
$958$	1.86863e6	2.03607
$959$	216105.i	0.234978i
$960$	0	0
$961$	−575154.	−0.622783
$962$	− 2.41287e6i	− 2.60726i
$963$	0	0
$964$	−1.71108e6	−1.84127
$965$	− 404132.i	− 0.433979i
$966$	0	0
$967$	794491.	0.849642	0.424821	−	0.905277i	$-0.360337\pi$
$967$	794491.	0.849642	0.424821	+	0.905277i	$0.360337\pi$
$968$	1.20728e6i	1.28842i
$969$	0	0
$970$	−1.08956e6	−1.15800
$971$	− 1.80995e6i	− 1.91967i	−0.280559	−	0.959837i	$-0.590520\pi$
$971$	− 1.80995e6i	− 1.91967i	0.280559	−	0.959837i	$-0.409480\pi$
$972$	0	0
$973$	219755.	0.232121
$974$	− 2.27222e6i	− 2.39515i
$975$	0	0
$976$	5.54198e6	5.81789
$977$	− 1.21769e6i	− 1.27570i	−0.770161	−	0.637849i	$-0.779824\pi$
$977$	− 1.21769e6i	− 1.27570i	0.770161	−	0.637849i	$-0.220176\pi$
$978$	0	0
$979$	479985.	0.500798
$980$	− 1.35689e6i	− 1.41283i
$981$	0	0
$982$	514537.	0.533572
$983$	− 1.69305e6i	− 1.75211i	−0.482209	−	0.876056i	$-0.660165\pi$
$983$	− 1.69305e6i	− 1.75211i	0.482209	−	0.876056i	$-0.339835\pi$
$984$	0	0
$985$	−1.56928e6	−1.61743
$986$	− 5.04133e6i	− 5.18551i
$987$	0	0
$988$	−7.02842e6	−7.20019
$989$	301044.i	0.307778i
$990$	0	0
$991$	392325.	0.399483	0.199742	−	0.979849i	$-0.435990\pi$
$991$	392325.	0.399483	0.199742	+	0.979849i	$0.435990\pi$
$992$	3.07689e6i	3.12672i
$993$	0	0
$994$	−1.67962e6	−1.69996
$995$	− 829368.i	− 0.837724i
$996$	0	0
$997$	−1.14366e6	−1.15055	−0.575277	−	0.817958i	$-0.695106\pi$
$997$	−1.14366e6	−1.15055	−0.575277	+	0.817958i	$0.695106\pi$
$998$	2.01869e6i	2.02679i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.1	✓	76
3.2	odd	2	inner	531.5.b.a.296.76	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.1	✓	76	1.1	even	1	trivial
531.5.b.a.296.76	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)

\(f(q)\)	\(=\)	\(q-7.89147i q^{2} -46.2753 q^{4} -30.7341i q^{5} -38.0387 q^{7} +238.917i q^{8} +O(q^{10})\)	\(q-7.89147i q^{2} -46.2753 q^{4} -30.7341i q^{5} -38.0387 q^{7} +238.917i q^{8} -242.537 q^{10} +97.9176i q^{11} +266.782 q^{13} +300.181i q^{14} +1145.00 q^{16} -405.373i q^{17} +569.314 q^{19} +1422.23i q^{20} +772.714 q^{22} +551.399i q^{23} -319.583 q^{25} -2105.30i q^{26} +1760.25 q^{28} +1575.91i q^{29} -590.227 q^{31} -5213.07i q^{32} -3198.99 q^{34} +1169.08i q^{35} +1146.09 q^{37} -4492.73i q^{38} +7342.89 q^{40} -627.996i q^{41} +545.964 q^{43} -4531.17i q^{44} +4351.35 q^{46} +831.478i q^{47} -954.056 q^{49} +2521.98i q^{50} -12345.4 q^{52} +4600.66i q^{53} +3009.41 q^{55} -9088.09i q^{56} +12436.3 q^{58} +453.188i q^{59} +4840.16 q^{61} +4657.76i q^{62} -22818.8 q^{64} -8199.30i q^{65} -2866.26 q^{67} +18758.8i q^{68} +9225.80 q^{70} +5595.36i q^{71} +7462.48 q^{73} -9044.36i q^{74} -26345.2 q^{76} -3724.66i q^{77} -57.5610 q^{79} -35190.5i q^{80} -4955.81 q^{82} -7578.24i q^{83} -12458.8 q^{85} -4308.46i q^{86} -23394.2 q^{88} -4901.93i q^{89} -10148.0 q^{91} -25516.2i q^{92} +6561.59 q^{94} -17497.3i q^{95} +4492.34 q^{97} +7528.90i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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

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Groups

Database

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