LMFDB - Embedded newform 531.5.b.a.296.62

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			296.62
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65016i q^{2} -15.9243 q^{4} -29.6313i q^{5} -73.1972 q^{7} +0.427826i q^{8} +O(q^{10})$	\(q+5.65016i q^{2} -15.9243 q^{4} -29.6313i q^{5} -73.1972 q^{7} +0.427826i q^{8} +167.422 q^{10} +200.465i q^{11} +240.983 q^{13} -413.576i q^{14} -257.206 q^{16} +296.739i q^{17} -180.279 q^{19} +471.857i q^{20} -1132.66 q^{22} -767.217i q^{23} -253.014 q^{25} +1361.59i q^{26} +1165.61 q^{28} +1238.56i q^{29} -1520.40 q^{31} -1446.41i q^{32} -1676.62 q^{34} +2168.93i q^{35} +1520.34 q^{37} -1018.61i q^{38} +12.6771 q^{40} -1554.94i q^{41} -793.101 q^{43} -3192.27i q^{44} +4334.90 q^{46} -3964.42i q^{47} +2956.83 q^{49} -1429.57i q^{50} -3837.48 q^{52} -4917.99i q^{53} +5940.05 q^{55} -31.3157i q^{56} -6998.05 q^{58} +453.188i q^{59} -2641.40 q^{61} -8590.51i q^{62} +4057.14 q^{64} -7140.65i q^{65} -3648.98 q^{67} -4725.36i q^{68} -12254.8 q^{70} +1839.07i q^{71} +473.457 q^{73} +8590.15i q^{74} +2870.82 q^{76} -14673.5i q^{77} +3999.13 q^{79} +7621.34i q^{80} +8785.68 q^{82} -619.441i q^{83} +8792.77 q^{85} -4481.15i q^{86} -85.7643 q^{88} -12045.9i q^{89} -17639.3 q^{91} +12217.4i q^{92} +22399.6 q^{94} +5341.91i q^{95} -9026.76 q^{97} +16706.5i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.65016i	1.41254i	0.707943	+	0.706270i	$0.249623\pi$
$2$	5.65016i	1.41254i	−0.707943	+	0.706270i	$0.750377\pi$
$3$	0	0
$3$	0	0
$4$	−15.9243	−0.995268
$5$	− 29.6313i	− 1.18525i	−0.805478	−	0.592626i	$-0.798091\pi$
$5$	− 29.6313i	− 1.18525i	0.805478	−	0.592626i	$-0.201909\pi$
$6$	0	0
$7$	−73.1972	−1.49382	−0.746910	−	0.664925i	$-0.768463\pi$
$7$	−73.1972	−1.49382	−0.746910	+	0.664925i	$0.768463\pi$
$8$	0.427826i	0.00668479i
$9$	0	0
$10$	167.422	1.67422
$11$	200.465i	1.65674i	0.560183	+	0.828369i	$0.310731\pi$
$11$	200.465i	1.65674i	−0.560183	+	0.828369i	$0.689269\pi$
$12$	0	0
$13$	240.983	1.42594	0.712968	−	0.701196i	$-0.247350\pi$
$13$	240.983	1.42594	0.712968	+	0.701196i	$0.247350\pi$
$14$	− 413.576i	− 2.11008i
$15$	0	0
$16$	−257.206	−1.00471
$17$	296.739i	1.02678i	0.858156	+	0.513390i	$0.171610\pi$
$17$	296.739i	1.02678i	−0.858156	+	0.513390i	$0.828390\pi$
$18$	0	0
$19$	−180.279	−0.499388	−0.249694	−	0.968325i	$-0.580330\pi$
$19$	−180.279	−0.499388	−0.249694	+	0.968325i	$0.580330\pi$
$20$	471.857i	1.17964i
$21$	0	0
$22$	−1132.66	−2.34021
$23$	− 767.217i	− 1.45032i	−0.688583	−	0.725158i	$-0.741767\pi$
$23$	− 767.217i	− 1.45032i	0.688583	−	0.725158i	$-0.258233\pi$
$24$	0	0
$25$	−253.014	−0.404823
$26$	1361.59i	2.01419i
$27$	0	0
$28$	1165.61	1.48675
$29$	1238.56i	1.47272i	0.676589	+	0.736361i	$0.263457\pi$
$29$	1238.56i	1.47272i	−0.676589	+	0.736361i	$0.736543\pi$
$30$	0	0
$31$	−1520.40	−1.58210	−0.791052	−	0.611749i	$-0.790466\pi$
$31$	−1520.40	−1.58210	−0.791052	+	0.611749i	$0.790466\pi$
$32$	− 1446.41i	− 1.41251i
$33$	0	0
$34$	−1676.62	−1.45037
$35$	2168.93i	1.77055i
$36$	0	0
$37$	1520.34	1.11055	0.555273	−	0.831668i	$-0.312614\pi$
$37$	1520.34	1.11055	0.555273	+	0.831668i	$0.312614\pi$
$38$	− 1018.61i	− 0.705406i
$39$	0	0
$40$	12.6771	0.00792316
$41$	− 1554.94i	− 0.925012i	−0.886616	−	0.462506i	$-0.846950\pi$
$41$	− 1554.94i	− 0.925012i	0.886616	−	0.462506i	$-0.153050\pi$
$42$	0	0
$43$	−793.101	−0.428935	−0.214468	−	0.976731i	$-0.568802\pi$
$43$	−793.101	−0.428935	−0.214468	+	0.976731i	$0.568802\pi$
$44$	− 3192.27i	− 1.64890i
$45$	0	0
$46$	4334.90	2.04863
$47$	− 3964.42i	− 1.79467i	−0.441355	−	0.897333i	$-0.645502\pi$
$47$	− 3964.42i	− 1.79467i	0.441355	−	0.897333i	$-0.354498\pi$
$48$	0	0
$49$	2956.83	1.23150
$50$	− 1429.57i	− 0.571828i
$51$	0	0
$52$	−3837.48	−1.41919
$53$	− 4917.99i	− 1.75080i	−0.483403	−	0.875398i	$-0.660599\pi$
$53$	− 4917.99i	− 1.75080i	0.483403	−	0.875398i	$-0.339401\pi$
$54$	0	0
$55$	5940.05	1.96365
$56$	− 31.3157i	− 0.00998587i
$57$	0	0
$58$	−6998.05	−2.08028
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	−2641.40	−0.709862	−0.354931	−	0.934892i	$-0.615496\pi$
$61$	−2641.40	−0.709862	−0.354931	+	0.934892i	$0.615496\pi$
$62$	− 8590.51i	− 2.23478i
$63$	0	0
$64$	4057.14	0.990513
$65$	− 7140.65i	− 1.69009i
$66$	0	0
$67$	−3648.98	−0.812871	−0.406435	−	0.913679i	$-0.633228\pi$
$67$	−3648.98	−0.812871	−0.406435	+	0.913679i	$0.633228\pi$
$68$	− 4725.36i	− 1.02192i
$69$	0	0
$70$	−12254.8	−2.50098
$71$	1839.07i	0.364822i	0.983222	+	0.182411i	$0.0583902\pi$
$71$	1839.07i	0.364822i	−0.983222	+	0.182411i	$0.941610\pi$
$72$	0	0
$73$	473.457	0.0888453	0.0444226	−	0.999013i	$-0.485855\pi$
$73$	473.457	0.0888453	0.0444226	+	0.999013i	$0.485855\pi$
$74$	8590.15i	1.56869i
$75$	0	0
$76$	2870.82	0.497025
$77$	− 14673.5i	− 2.47487i
$78$	0	0
$79$	3999.13	0.640784	0.320392	−	0.947285i	$-0.396185\pi$
$79$	3999.13	0.640784	0.320392	+	0.947285i	$0.396185\pi$
$80$	7621.34i	1.19083i
$81$	0	0
$82$	8785.68	1.30662
$83$	− 619.441i	− 0.0899175i	−0.998989	−	0.0449587i	$-0.985684\pi$
$83$	− 619.441i	− 0.0899175i	0.998989	−	0.0449587i	$-0.0143156\pi$
$84$	0	0
$85$	8792.77	1.21699
$86$	− 4481.15i	− 0.605888i
$87$	0	0
$88$	−85.7643	−0.0110749
$89$	− 12045.9i	− 1.52075i	−0.649482	−	0.760377i	$-0.725014\pi$
$89$	− 12045.9i	− 1.52075i	0.649482	−	0.760377i	$-0.274986\pi$
$90$	0	0
$91$	−17639.3	−2.13009
$92$	12217.4i	1.44345i
$93$	0	0
$94$	22399.6	2.53504
$95$	5341.91i	0.591901i
$96$	0	0
$97$	−9026.76	−0.959375	−0.479688	−	0.877439i	$-0.659250\pi$
$97$	−9026.76	−0.959375	−0.479688	+	0.877439i	$0.659250\pi$
$98$	16706.5i	1.73954i
$99$	0	0
$100$	4029.07	0.402907
$101$	− 3813.41i	− 0.373827i	−0.982376	−	0.186913i	$-0.940152\pi$
$101$	− 3813.41i	− 0.373827i	0.982376	−	0.186913i	$-0.0598484\pi$
$102$	0	0
$103$	1640.57	0.154639	0.0773196	−	0.997006i	$-0.475364\pi$
$103$	1640.57	0.154639	0.0773196	+	0.997006i	$0.475364\pi$
$104$	103.099i	0.00953208i
$105$	0	0
$106$	27787.4	2.47307
$107$	− 17517.7i	− 1.53006i	−0.643992	−	0.765032i	$-0.722723\pi$
$107$	− 17517.7i	− 1.53006i	0.643992	−	0.765032i	$-0.277277\pi$
$108$	0	0
$109$	14381.5	1.21046	0.605230	−	0.796050i	$-0.293081\pi$
$109$	14381.5	1.21046	0.605230	+	0.796050i	$0.293081\pi$
$110$	33562.2i	2.77374i
$111$	0	0
$112$	18826.7	1.50086
$113$	− 1751.62i	− 0.137177i	−0.997645	−	0.0685887i	$-0.978150\pi$
$113$	− 1751.62i	− 0.137177i	0.997645	−	0.0685887i	$-0.0218496\pi$
$114$	0	0
$115$	−22733.6	−1.71899
$116$	− 19723.2i	− 1.46575i
$117$	0	0
$118$	−2560.58	−0.183897
$119$	− 21720.5i	− 1.53382i
$120$	0	0
$121$	−25545.4	−1.74478
$122$	− 14924.3i	− 1.00271i
$123$	0	0
$124$	24211.3	1.57462
$125$	− 11022.4i	− 0.705435i
$126$	0	0
$127$	2340.12	0.145088	0.0725440	−	0.997365i	$-0.476888\pi$
$127$	2340.12	0.145088	0.0725440	+	0.997365i	$0.476888\pi$
$128$	− 219.045i	− 0.0133694i
$129$	0	0
$130$	40345.8	2.38732
$131$	− 19211.1i	− 1.11946i	−0.828675	−	0.559730i	$-0.810905\pi$
$131$	− 19211.1i	− 1.11946i	0.828675	−	0.559730i	$-0.189095\pi$
$132$	0	0
$133$	13195.9	0.745996
$134$	− 20617.3i	− 1.14821i
$135$	0	0
$136$	−126.953	−0.00686380
$137$	30365.8i	1.61787i	0.587900	+	0.808934i	$0.299955\pi$
$137$	30365.8i	1.61787i	−0.587900	+	0.808934i	$0.700045\pi$
$138$	0	0
$139$	4258.62	0.220414	0.110207	−	0.993909i	$-0.464849\pi$
$139$	4258.62	0.220414	0.110207	+	0.993909i	$0.464849\pi$
$140$	− 34538.6i	− 1.76217i
$141$	0	0
$142$	−10391.0	−0.515326
$143$	48308.8i	2.36240i
$144$	0	0
$145$	36700.1	1.74555
$146$	2675.10i	0.125497i
$147$	0	0
$148$	−24210.3	−1.10529
$149$	41968.2i	1.89037i	0.326528	+	0.945187i	$0.394121\pi$
$149$	41968.2i	1.89037i	−0.326528	+	0.945187i	$0.605879\pi$
$150$	0	0
$151$	−19041.9	−0.835135	−0.417568	−	0.908646i	$-0.637117\pi$
$151$	−19041.9	−0.835135	−0.417568	+	0.908646i	$0.637117\pi$
$152$	− 77.1282i	− 0.00333830i
$153$	0	0
$154$	82907.6	3.49585
$155$	45051.5i	1.87519i
$156$	0	0
$157$	−20506.8	−0.831952	−0.415976	−	0.909376i	$-0.636560\pi$
$157$	−20506.8	−0.831952	−0.415976	+	0.909376i	$0.636560\pi$
$158$	22595.7i	0.905133i
$159$	0	0
$160$	−42859.0	−1.67418
$161$	56158.1i	2.16651i
$162$	0	0
$163$	−10616.4	−0.399578	−0.199789	−	0.979839i	$-0.564026\pi$
$163$	−10616.4	−0.399578	−0.199789	+	0.979839i	$0.564026\pi$
$164$	24761.4i	0.920634i
$165$	0	0
$166$	3499.94	0.127012
$167$	15119.6i	0.542136i	0.962560	+	0.271068i	$0.0873769\pi$
$167$	15119.6i	0.542136i	−0.962560	+	0.271068i	$0.912623\pi$
$168$	0	0
$169$	29511.9	1.03329
$170$	49680.5i	1.71905i
$171$	0	0
$172$	12629.6	0.426905
$173$	− 17021.4i	− 0.568727i	−0.958717	−	0.284364i	$-0.908218\pi$
$173$	− 17021.4i	− 0.568727i	0.958717	−	0.284364i	$-0.0917823\pi$
$174$	0	0
$175$	18519.9	0.604733
$176$	− 51560.8i	− 1.66454i
$177$	0	0
$178$	68061.2	2.14812
$179$	− 32065.6i	− 1.00077i	−0.865804	−	0.500383i	$-0.833192\pi$
$179$	− 32065.6i	− 1.00077i	0.865804	−	0.500383i	$-0.166808\pi$
$180$	0	0
$181$	−41357.7	−1.26241	−0.631203	−	0.775618i	$-0.717438\pi$
$181$	−41357.7	−1.26241	−0.631203	+	0.775618i	$0.717438\pi$
$182$	− 99664.8i	− 3.00884i
$183$	0	0
$184$	328.236	0.00969505
$185$	− 45049.6i	− 1.31628i
$186$	0	0
$187$	−59485.9	−1.70110
$188$	63130.5i	1.78617i
$189$	0	0
$190$	−30182.6	−0.836084
$191$	− 6563.44i	− 0.179914i	−0.995946	−	0.0899569i	$-0.971327\pi$
$191$	− 6563.44i	− 0.179914i	0.995946	−	0.0899569i	$-0.0286729\pi$
$192$	0	0
$193$	32559.7	0.874110	0.437055	−	0.899435i	$-0.356021\pi$
$193$	32559.7	0.874110	0.437055	+	0.899435i	$0.356021\pi$
$194$	− 51002.6i	− 1.35516i
$195$	0	0
$196$	−47085.4	−1.22567
$197$	4434.46i	0.114264i	0.998367	+	0.0571319i	$0.0181956\pi$
$197$	4434.46i	0.114264i	−0.998367	+	0.0571319i	$0.981804\pi$
$198$	0	0
$199$	−11765.9	−0.297112	−0.148556	−	0.988904i	$-0.547462\pi$
$199$	−11765.9	−0.297112	−0.148556	+	0.988904i	$0.547462\pi$
$200$	− 108.246i	− 0.00270615i
$201$	0	0
$202$	21546.3	0.528045
$203$	− 90659.0i	− 2.19998i
$204$	0	0
$205$	−46075.0	−1.09637
$206$	9269.46i	0.218434i
$207$	0	0
$208$	−61982.3	−1.43265
$209$	− 36139.7i	− 0.827356i
$210$	0	0
$211$	−34422.9	−0.773184	−0.386592	−	0.922251i	$-0.626348\pi$
$211$	−34422.9	−0.773184	−0.386592	+	0.922251i	$0.626348\pi$
$212$	78315.4i	1.74251i
$213$	0	0
$214$	98977.8	2.16128
$215$	23500.6i	0.508396i
$216$	0	0
$217$	111289.	2.36338
$218$	81257.6i	1.70982i
$219$	0	0
$220$	−94591.0	−1.95436
$221$	71509.2i	1.46412i
$222$	0	0
$223$	−51274.7	−1.03108	−0.515541	−	0.856865i	$-0.672409\pi$
$223$	−51274.7	−1.03108	−0.515541	+	0.856865i	$0.672409\pi$
$224$	105873.i	2.11003i
$225$	0	0
$226$	9896.92	0.193769
$227$	− 85013.7i	− 1.64982i	−0.565263	−	0.824911i	$-0.691225\pi$
$227$	− 85013.7i	− 1.64982i	0.565263	−	0.824911i	$-0.308775\pi$
$228$	0	0
$229$	98044.0	1.86961	0.934803	−	0.355166i	$-0.115576\pi$
$229$	98044.0	1.86961	0.934803	+	0.355166i	$0.115576\pi$
$230$	− 128449.i	− 2.42814i
$231$	0	0
$232$	−529.888	−0.00984483
$233$	− 7375.60i	− 0.135858i	−0.997690	−	0.0679291i	$-0.978361\pi$
$233$	− 7375.60i	− 0.135858i	0.997690	−	0.0679291i	$-0.0216392\pi$
$234$	0	0
$235$	−117471.	−2.12713
$236$	− 7216.69i	− 0.129573i
$237$	0	0
$238$	122724.	2.16659
$239$	93774.0i	1.64167i	0.571163	+	0.820837i	$0.306492\pi$
$239$	93774.0i	1.64167i	−0.571163	+	0.820837i	$0.693508\pi$
$240$	0	0
$241$	22969.1	0.395467	0.197733	−	0.980256i	$-0.436642\pi$
$241$	22969.1	0.395467	0.197733	+	0.980256i	$0.436642\pi$
$242$	− 144335.i	− 2.46457i
$243$	0	0
$244$	42062.3	0.706503
$245$	− 87614.7i	− 1.45964i
$246$	0	0
$247$	−43444.3	−0.712096
$248$	− 650.468i	− 0.0105760i
$249$	0	0
$250$	62278.4	0.996455
$251$	− 103132.i	− 1.63699i	−0.574517	−	0.818493i	$-0.694810\pi$
$251$	− 103132.i	− 1.63699i	0.574517	−	0.818493i	$-0.305190\pi$
$252$	0	0
$253$	153800.	2.40279
$254$	13222.1i	0.204942i
$255$	0	0
$256$	66151.9	1.00940
$257$	103550.i	1.56778i	0.620898	+	0.783891i	$0.286768\pi$
$257$	103550.i	1.56778i	−0.620898	+	0.783891i	$0.713232\pi$
$258$	0	0
$259$	−111284.	−1.65896
$260$	113710.i	1.68210i
$261$	0	0
$262$	108546.	1.58128
$263$	30192.4i	0.436502i	0.975893	+	0.218251i	$0.0700352\pi$
$263$	30192.4i	0.436502i	−0.975893	+	0.218251i	$0.929965\pi$
$264$	0	0
$265$	−145726.	−2.07513
$266$	74559.1i	1.05375i
$267$	0	0
$268$	58107.3	0.809024
$269$	17120.7i	0.236602i	0.992978	+	0.118301i	$0.0377447\pi$
$269$	17120.7i	0.236602i	−0.992978	+	0.118301i	$0.962255\pi$
$270$	0	0
$271$	−114809.	−1.56328	−0.781641	−	0.623729i	$-0.785617\pi$
$271$	−114809.	−1.56328	−0.781641	+	0.623729i	$0.785617\pi$
$272$	− 76323.1i	− 1.03162i
$273$	0	0
$274$	−171571.	−2.28530
$275$	− 50720.6i	− 0.670686i
$276$	0	0
$277$	35284.0	0.459852	0.229926	−	0.973208i	$-0.426152\pi$
$277$	35284.0	0.459852	0.229926	+	0.973208i	$0.426152\pi$
$278$	24061.9i	0.311343i
$279$	0	0
$280$	−927.925	−0.0118358
$281$	− 93441.5i	− 1.18339i	−0.806162	−	0.591694i	$-0.798459\pi$
$281$	− 93441.5i	− 1.18339i	0.806162	−	0.591694i	$-0.201541\pi$
$282$	0	0
$283$	113570.	1.41805	0.709025	−	0.705183i	$-0.249135\pi$
$283$	113570.	1.41805	0.709025	+	0.705183i	$0.249135\pi$
$284$	− 29285.8i	− 0.363096i
$285$	0	0
$286$	−272952.	−3.33699
$287$	113818.i	1.38180i
$288$	0	0
$289$	−4533.19	−0.0542761
$290$	207361.i	2.46565i
$291$	0	0
$292$	−7539.46	−0.0884248
$293$	81971.1i	0.954829i	0.878678	+	0.477415i	$0.158426\pi$
$293$	81971.1i	0.954829i	−0.878678	+	0.477415i	$0.841574\pi$
$294$	0	0
$295$	13428.5	0.154307
$296$	650.440i	0.00742376i
$297$	0	0
$298$	−237127.	−2.67023
$299$	− 184886.i	− 2.06806i
$300$	0	0
$301$	58052.8	0.640752
$302$	− 107590.i	− 1.17966i
$303$	0	0
$304$	46368.8	0.501740
$305$	78268.0i	0.841366i
$306$	0	0
$307$	70774.1	0.750927	0.375463	−	0.926837i	$-0.377484\pi$
$307$	70774.1	0.750927	0.375463	+	0.926837i	$0.377484\pi$
$308$	233665.i	2.46316i
$309$	0	0
$310$	−254548.	−2.64878
$311$	− 134887.i	− 1.39460i	−0.716780	−	0.697299i	$-0.754385\pi$
$311$	− 134887.i	− 1.39460i	0.716780	−	0.697299i	$-0.245615\pi$
$312$	0	0
$313$	−109303.	−1.11569	−0.557845	−	0.829945i	$-0.688372\pi$
$313$	−109303.	−1.11569	−0.557845	+	0.829945i	$0.688372\pi$
$314$	− 115867.i	− 1.17516i
$315$	0	0
$316$	−63683.3	−0.637752
$317$	− 167573.i	− 1.66757i	−0.552087	−	0.833787i	$-0.686168\pi$
$317$	− 167573.i	− 1.66757i	0.552087	−	0.833787i	$-0.313832\pi$
$318$	0	0
$319$	−248288.	−2.43991
$320$	− 120218.i	− 1.17401i
$321$	0	0
$322$	−317302.	−3.06028
$323$	− 53495.9i	− 0.512762i
$324$	0	0
$325$	−60972.2	−0.577252
$326$	− 59984.2i	− 0.564419i
$327$	0	0
$328$	665.246	0.00618350
$329$	290184.i	2.68091i
$330$	0	0
$331$	−94707.3	−0.864425	−0.432212	−	0.901772i	$-0.642267\pi$
$331$	−94707.3	−0.864425	−0.432212	+	0.901772i	$0.642267\pi$
$332$	9864.16i	0.0894919i
$333$	0	0
$334$	−85428.3	−0.765789
$335$	108124.i	0.963457i
$336$	0	0
$337$	−132591.	−1.16749	−0.583747	−	0.811935i	$-0.698414\pi$
$337$	−132591.	−1.16749	−0.583747	+	0.811935i	$0.698414\pi$
$338$	166747.i	1.45957i
$339$	0	0
$340$	−140019.	−1.21123
$341$	− 304788.i	− 2.62113i
$342$	0	0
$343$	−40685.1	−0.345817
$344$	− 339.309i	− 0.00286734i
$345$	0	0
$346$	96173.8	0.803349
$347$	114355.i	0.949721i	0.880061	+	0.474861i	$0.157502\pi$
$347$	114355.i	0.949721i	−0.880061	+	0.474861i	$0.842498\pi$
$348$	0	0
$349$	36161.0	0.296886	0.148443	−	0.988921i	$-0.452574\pi$
$349$	36161.0	0.296886	0.148443	+	0.988921i	$0.452574\pi$
$350$	104641.i	0.854209i
$351$	0	0
$352$	289955.	2.34016
$353$	− 146266.i	− 1.17380i	−0.809658	−	0.586902i	$-0.800347\pi$
$353$	− 146266.i	− 1.17380i	0.809658	−	0.586902i	$-0.199653\pi$
$354$	0	0
$355$	54494.0	0.432406
$356$	191822.i	1.51356i
$357$	0	0
$358$	181176.	1.41362
$359$	− 119880.i	− 0.930157i	−0.885269	−	0.465079i	$-0.846026\pi$
$359$	− 119880.i	− 0.930157i	0.885269	−	0.465079i	$-0.153974\pi$
$360$	0	0
$361$	−97820.4	−0.750611
$362$	− 233677.i	− 1.78320i
$363$	0	0
$364$	280893.	2.12001
$365$	− 14029.1i	− 0.105304i
$366$	0	0
$367$	168395.	1.25025	0.625127	−	0.780523i	$-0.285047\pi$
$367$	168395.	1.25025	0.625127	+	0.780523i	$0.285047\pi$
$368$	197333.i	1.45715i
$369$	0	0
$370$	254537.	1.85929
$371$	359983.i	2.61537i
$372$	0	0
$373$	−38763.0	−0.278612	−0.139306	−	0.990249i	$-0.544487\pi$
$373$	−38763.0	−0.278612	−0.139306	+	0.990249i	$0.544487\pi$
$374$	− 336105.i	− 2.40288i
$375$	0	0
$376$	1696.08	0.0119970
$377$	298472.i	2.10001i
$378$	0	0
$379$	−155522.	−1.08271	−0.541357	−	0.840793i	$-0.682089\pi$
$379$	−155522.	−1.08271	−0.541357	+	0.840793i	$0.682089\pi$
$380$	− 85066.0i	− 0.589100i
$381$	0	0
$382$	37084.5	0.254135
$383$	− 112983.i	− 0.770223i	−0.922870	−	0.385111i	$-0.874163\pi$
$383$	− 112983.i	− 0.770223i	0.922870	−	0.385111i	$-0.125837\pi$
$384$	0	0
$385$	−434795.	−2.93334
$386$	183968.i	1.23472i
$387$	0	0
$388$	143745.	0.954835
$389$	183484.i	1.21255i	0.795257	+	0.606273i	$0.207336\pi$
$389$	183484.i	1.21255i	−0.795257	+	0.606273i	$0.792664\pi$
$390$	0	0
$391$	227663.	1.48915
$392$	1265.01i	0.00823230i
$393$	0	0
$394$	−25055.4	−0.161402
$395$	− 118500.i	− 0.759491i
$396$	0	0
$397$	−19110.1	−0.121250	−0.0606251	−	0.998161i	$-0.519309\pi$
$397$	−19110.1	−0.121250	−0.0606251	+	0.998161i	$0.519309\pi$
$398$	− 66479.3i	− 0.419682i
$399$	0	0
$400$	65076.7	0.406730
$401$	− 25817.0i	− 0.160553i	−0.996773	−	0.0802764i	$-0.974420\pi$
$401$	− 25817.0i	− 0.160553i	0.996773	−	0.0802764i	$-0.0255803\pi$
$402$	0	0
$403$	−366391.	−2.25598
$404$	60725.7i	0.372058i
$405$	0	0
$406$	512238.	3.10756
$407$	304775.i	1.83988i
$408$	0	0
$409$	−15849.8	−0.0947496	−0.0473748	−	0.998877i	$-0.515086\pi$
$409$	−15849.8	−0.0947496	−0.0473748	+	0.998877i	$0.515086\pi$
$410$	− 260331.i	− 1.54867i
$411$	0	0
$412$	−26124.8	−0.153907
$413$	− 33172.1i	− 0.194479i
$414$	0	0
$415$	−18354.9	−0.106575
$416$	− 348560.i	− 2.01415i
$417$	0	0
$418$	204195.	1.16867
$419$	− 24434.2i	− 0.139178i	−0.997576	−	0.0695890i	$-0.977831\pi$
$419$	− 24434.2i	− 0.139178i	0.997576	−	0.0695890i	$-0.0221688\pi$
$420$	0	0
$421$	206276.	1.16382	0.581908	−	0.813255i	$-0.302307\pi$
$421$	206276.	1.16382	0.581908	+	0.813255i	$0.302307\pi$
$422$	− 194495.i	− 1.09215i
$423$	0	0
$424$	2104.04	0.0117037
$425$	− 75079.3i	− 0.415664i
$426$	0	0
$427$	193343.	1.06041
$428$	278957.i	1.52282i
$429$	0	0
$430$	−132782.	−0.718130
$431$	119649.i	0.644100i	0.946723	+	0.322050i	$0.104372\pi$
$431$	119649.i	0.644100i	−0.946723	+	0.322050i	$0.895628\pi$
$432$	0	0
$433$	118357.	0.631277	0.315638	−	0.948880i	$-0.397781\pi$
$433$	118357.	0.631277	0.315638	+	0.948880i	$0.397781\pi$
$434$	628801.i	3.33837i
$435$	0	0
$436$	−229015.	−1.20473
$437$	138313.i	0.724271i
$438$	0	0
$439$	36146.8	0.187560	0.0937801	−	0.995593i	$-0.470105\pi$
$439$	36146.8	0.187560	0.0937801	+	0.995593i	$0.470105\pi$
$440$	2541.31i	0.0131266i
$441$	0	0
$442$	−404038.	−2.06813
$443$	− 115036.i	− 0.586175i	−0.956086	−	0.293087i	$-0.905317\pi$
$443$	− 115036.i	− 0.586175i	0.956086	−	0.293087i	$-0.0946827\pi$
$444$	0	0
$445$	−356935.	−1.80248
$446$	− 289710.i	− 1.45644i
$447$	0	0
$448$	−296971.	−1.47965
$449$	− 179682.i	− 0.891275i	−0.895214	−	0.445637i	$-0.852977\pi$
$449$	− 179682.i	− 0.891275i	0.895214	−	0.445637i	$-0.147023\pi$
$450$	0	0
$451$	311712.	1.53250
$452$	27893.3i	0.136528i
$453$	0	0
$454$	480341.	2.33044
$455$	522675.i	2.52470i
$456$	0	0
$457$	34984.2	0.167509	0.0837547	−	0.996486i	$-0.473309\pi$
$457$	34984.2	0.167509	0.0837547	+	0.996486i	$0.473309\pi$
$458$	553964.i	2.64089i
$459$	0	0
$460$	362017.	1.71085
$461$	109450.i	0.515010i	0.966277	+	0.257505i	$0.0829003\pi$
$461$	109450.i	0.515010i	−0.966277	+	0.257505i	$0.917100\pi$
$462$	0	0
$463$	49530.8	0.231054	0.115527	−	0.993304i	$-0.463144\pi$
$463$	49530.8	0.231054	0.115527	+	0.993304i	$0.463144\pi$
$464$	− 318565.i	− 1.47966i
$465$	0	0
$466$	41673.3	0.191905
$467$	− 197321.i	− 0.904772i	−0.891822	−	0.452386i	$-0.850573\pi$
$467$	− 197321.i	− 0.904772i	0.891822	−	0.452386i	$-0.149427\pi$
$468$	0	0
$469$	267095.	1.21428
$470$	− 663729.i	− 3.00466i
$471$	0	0
$472$	−193.886	−0.000870285 0
$473$	− 158989.i	− 0.710633i
$474$	0	0
$475$	45613.2	0.202164
$476$	345883.i	1.52657i
$477$	0	0
$478$	−529838.	−2.31893
$479$	− 35787.9i	− 0.155979i	−0.996954	−	0.0779894i	$-0.975150\pi$
$479$	− 35787.9i	− 0.155979i	0.996954	−	0.0779894i	$-0.0248500\pi$
$480$	0	0
$481$	366376.	1.58357
$482$	129779.i	0.558612i
$483$	0	0
$484$	406791.	1.73652
$485$	267475.i	1.13710i
$486$	0	0
$487$	−107035.	−0.451303	−0.225652	−	0.974208i	$-0.572451\pi$
$487$	−107035.	−0.451303	−0.225652	+	0.974208i	$0.572451\pi$
$488$	− 1130.06i	− 0.00474528i
$489$	0	0
$490$	495037.	2.06179
$491$	323373.i	1.34134i	0.741754	+	0.670672i	$0.233994\pi$
$491$	323373.i	1.34134i	−0.741754	+	0.670672i	$0.766006\pi$
$492$	0	0
$493$	−367529.	−1.51216
$494$	− 245467.i	− 1.00586i
$495$	0	0
$496$	391056.	1.58956
$497$	− 134615.i	− 0.544979i
$498$	0	0
$499$	401801.	1.61365	0.806827	−	0.590788i	$-0.201183\pi$
$499$	401801.	1.61365	0.806827	+	0.590788i	$0.201183\pi$
$500$	175524.i	0.702097i
$501$	0	0
$502$	582710.	2.31231
$503$	12278.9i	0.0485315i	0.999706	+	0.0242657i	$0.00772479\pi$
$503$	12278.9i	0.0485315i	−0.999706	+	0.0242657i	$0.992275\pi$
$504$	0	0
$505$	−112996.	−0.443079
$506$	868997.i	3.39404i
$507$	0	0
$508$	−37264.8	−0.144401
$509$	107505.i	0.414949i	0.978240	+	0.207475i	$0.0665244\pi$
$509$	107505.i	0.414949i	−0.978240	+	0.207475i	$0.933476\pi$
$510$	0	0
$511$	−34655.7	−0.132719
$512$	370264.i	1.41244i
$513$	0	0
$514$	−585076.	−2.21455
$515$	− 48612.1i	− 0.183286i
$516$	0	0
$517$	794728.	2.97329
$518$	− 628775.i	− 2.34334i
$519$	0	0
$520$	3054.96	0.0112979
$521$	456127.i	1.68039i	0.542282	+	0.840196i	$0.317560\pi$
$521$	456127.i	1.68039i	−0.542282	+	0.840196i	$0.682440\pi$
$522$	0	0
$523$	−37681.7	−0.137761	−0.0688807	−	0.997625i	$-0.521943\pi$
$523$	−37681.7	−0.137761	−0.0688807	+	0.997625i	$0.521943\pi$
$524$	305922.i	1.11416i
$525$	0	0
$526$	−170592.	−0.616576
$527$	− 451163.i	− 1.62447i
$528$	0	0
$529$	−308781.	−1.10342
$530$	− 823377.i	− 2.93121i
$531$	0	0
$532$	−210136.	−0.742466
$533$	− 374716.i	− 1.31901i
$534$	0	0
$535$	−519073.	−1.81351
$536$	− 1561.13i	− 0.00543387i
$537$	0	0
$538$	−96734.9	−0.334209
$539$	592742.i	2.04027i
$540$	0	0
$541$	−115879.	−0.395922	−0.197961	−	0.980210i	$-0.563432\pi$
$541$	−115879.	−0.395922	−0.197961	+	0.980210i	$0.563432\pi$
$542$	− 648689.i	− 2.20820i
$543$	0	0
$544$	429206.	1.45033
$545$	− 426142.i	− 1.43470i
$546$	0	0
$547$	−379915.	−1.26973	−0.634865	−	0.772623i	$-0.718944\pi$
$547$	−379915.	−1.26973	−0.634865	+	0.772623i	$0.718944\pi$
$548$	− 483553.i	− 1.61021i
$549$	0	0
$550$	286579.	0.947370
$551$	− 223286.i	− 0.735460i
$552$	0	0
$553$	−292725.	−0.957216
$554$	199360.i	0.649559i
$555$	0	0
$556$	−67815.4	−0.219371
$557$	411151.i	1.32523i	0.748961	+	0.662614i	$0.230553\pi$
$557$	411151.i	1.32523i	−0.748961	+	0.662614i	$0.769447\pi$
$558$	0	0
$559$	−191124.	−0.611634
$560$	− 557861.i	− 1.77889i
$561$	0	0
$562$	527959.	1.67158
$563$	− 318311.i	− 1.00423i	−0.864800	−	0.502117i	$-0.832555\pi$
$563$	− 318311.i	− 1.00423i	0.864800	−	0.502117i	$-0.167445\pi$
$564$	0	0
$565$	−51902.8	−0.162590
$566$	641690.i	2.00305i
$567$	0	0
$568$	−786.802	−0.00243876
$569$	− 115576.i	− 0.356979i	−0.983942	−	0.178490i	$-0.942879\pi$
$569$	− 115576.i	− 0.356979i	0.983942	−	0.178490i	$-0.0571211\pi$
$570$	0	0
$571$	−307964.	−0.944555	−0.472278	−	0.881450i	$-0.656568\pi$
$571$	−307964.	−0.944555	−0.472278	+	0.881450i	$0.656568\pi$
$572$	− 769283.i	− 2.35122i
$573$	0	0
$574$	−643087.	−1.95185
$575$	194117.i	0.587121i
$576$	0	0
$577$	536105.	1.61027	0.805134	−	0.593093i	$-0.202093\pi$
$577$	536105.	1.61027	0.805134	+	0.593093i	$0.202093\pi$
$578$	− 25613.2i	− 0.0766671i
$579$	0	0
$580$	−584423.	−1.73729
$581$	45341.4i	0.134321i
$582$	0	0
$583$	985886.	2.90061
$584$	202.557i	0 0.000593912i
$585$	0	0
$586$	−463150.	−1.34873
$587$	26181.0i	0.0759818i	0.999278	+	0.0379909i	$0.0120958\pi$
$587$	26181.0i	0.0759818i	−0.999278	+	0.0379909i	$0.987904\pi$
$588$	0	0
$589$	274097.	0.790084
$590$	75873.4i	0.217964i
$591$	0	0
$592$	−391040.	−1.11578
$593$	− 251440.i	− 0.715030i	−0.933907	−	0.357515i	$-0.883624\pi$
$593$	− 251440.i	− 0.715030i	0.933907	−	0.357515i	$-0.116376\pi$
$594$	0	0
$595$	−643606.	−1.81797
$596$	− 668314.i	− 1.88143i
$597$	0	0
$598$	1.04464e6	2.92121
$599$	− 55718.5i	− 0.155291i	−0.996981	−	0.0776455i	$-0.975260\pi$
$599$	− 55718.5i	− 0.155291i	0.996981	−	0.0776455i	$-0.0247402\pi$
$600$	0	0
$601$	−544332.	−1.50700	−0.753502	−	0.657445i	$-0.771637\pi$
$601$	−544332.	−1.50700	−0.753502	+	0.657445i	$0.771637\pi$
$602$	328007.i	0.905087i
$603$	0	0
$604$	303229.	0.831183
$605$	756942.i	2.06801i
$606$	0	0
$607$	−218253.	−0.592355	−0.296177	−	0.955133i	$-0.595712\pi$
$607$	−218253.	−0.592355	−0.296177	+	0.955133i	$0.595712\pi$
$608$	260757.i	0.705390i
$609$	0	0
$610$	−442227.	−1.18846
$611$	− 955358.i	− 2.55908i
$612$	0	0
$613$	90885.5	0.241865	0.120933	−	0.992661i	$-0.461411\pi$
$613$	90885.5	0.241865	0.120933	+	0.992661i	$0.461411\pi$
$614$	399885.i	1.06071i
$615$	0	0
$616$	6277.71	0.0165440
$617$	260137.i	0.683333i	0.939821	+	0.341667i	$0.110991\pi$
$617$	260137.i	0.683333i	−0.939821	+	0.341667i	$0.889009\pi$
$618$	0	0
$619$	−24931.6	−0.0650681	−0.0325341	−	0.999471i	$-0.510358\pi$
$619$	−24931.6	−0.0650681	−0.0325341	+	0.999471i	$0.510358\pi$
$620$	− 717413.i	− 1.86632i
$621$	0	0
$622$	762132.	1.96992
$623$	881725.i	2.27173i
$624$	0	0
$625$	−484743.	−1.24094
$626$	− 617580.i	− 1.57596i
$627$	0	0
$628$	326556.	0.828015
$629$	451144.i	1.14029i
$630$	0	0
$631$	−160104.	−0.402110	−0.201055	−	0.979580i	$-0.564437\pi$
$631$	−160104.	−0.402110	−0.201055	+	0.979580i	$0.564437\pi$
$632$	1710.93i	0.00428351i
$633$	0	0
$634$	946812.	2.35551
$635$	− 69340.9i	− 0.171966i
$636$	0	0
$637$	712546.	1.75604
$638$	− 1.40287e6i	− 3.44648i
$639$	0	0
$640$	−6490.58	−0.0158461
$641$	208302.i	0.506965i	0.967340	+	0.253482i	$0.0815759\pi$
$641$	208302.i	0.506965i	−0.967340	+	0.253482i	$0.918424\pi$
$642$	0	0
$643$	−548208.	−1.32594	−0.662969	−	0.748647i	$-0.730704\pi$
$643$	−548208.	−1.32594	−0.662969	+	0.748647i	$0.730704\pi$
$644$	− 894278.i	− 2.15626i
$645$	0	0
$646$	302260.	0.724296
$647$	94064.3i	0.224707i	0.993668	+	0.112353i	$0.0358389\pi$
$647$	94064.3i	0.224707i	−0.993668	+	0.112353i	$0.964161\pi$
$648$	0	0
$649$	−90848.4	−0.215689
$650$	− 344503.i	− 0.815391i
$651$	0	0
$652$	169058.	0.397687
$653$	23584.6i	0.0553098i	0.999618	+	0.0276549i	$0.00880394\pi$
$653$	23584.6i	0.0553098i	−0.999618	+	0.0276549i	$0.991196\pi$
$654$	0	0
$655$	−569249.	−1.32684
$656$	399941.i	0.929368i
$657$	0	0
$658$	−1.63959e6	−3.78689
$659$	87963.4i	0.202549i	0.994858	+	0.101275i	$0.0322921\pi$
$659$	87963.4i	0.202549i	−0.994858	+	0.101275i	$0.967708\pi$
$660$	0	0
$661$	−508795.	−1.16450	−0.582250	−	0.813010i	$-0.697828\pi$
$661$	−508795.	−1.16450	−0.582250	+	0.813010i	$0.697828\pi$
$662$	− 535111.i	− 1.22103i
$663$	0	0
$664$	265.013	0.000601079 0
$665$	− 391013.i	− 0.884194i
$666$	0	0
$667$	950244.	2.13591
$668$	− 240769.i	− 0.539571i
$669$	0	0
$670$	−610918.	−1.36092
$671$	− 529508.i	− 1.17606i
$672$	0	0
$673$	−38331.0	−0.0846292	−0.0423146	−	0.999104i	$-0.513473\pi$
$673$	−38331.0	−0.0846292	−0.0423146	+	0.999104i	$0.513473\pi$
$674$	− 749161.i	− 1.64913i
$675$	0	0
$676$	−469956.	−1.02840
$677$	307437.i	0.670779i	0.942080	+	0.335389i	$0.108868\pi$
$677$	307437.i	0.670779i	−0.942080	+	0.335389i	$0.891132\pi$
$678$	0	0
$679$	660734.	1.43313
$680$	3761.78i	0.00813534i
$681$	0	0
$682$	1.72210e6	3.70245
$683$	475290.i	1.01887i	0.860510	+	0.509434i	$0.170145\pi$
$683$	475290.i	1.01887i	−0.860510	+	0.509434i	$0.829855\pi$
$684$	0	0
$685$	899777.	1.91758
$686$	− 229877.i	− 0.488481i
$687$	0	0
$688$	203990.	0.430955
$689$	− 1.18515e6i	− 2.49652i
$690$	0	0
$691$	75405.9	0.157924	0.0789622	−	0.996878i	$-0.474839\pi$
$691$	75405.9	0.157924	0.0789622	+	0.996878i	$0.474839\pi$
$692$	271054.i	0.566036i
$693$	0	0
$694$	−646124.	−1.34152
$695$	− 126188.i	− 0.261246i
$696$	0	0
$697$	461413.	0.949783
$698$	204315.i	0.419363i
$699$	0	0
$700$	−294917.	−0.601871
$701$	− 463036.i	− 0.942278i	−0.882059	−	0.471139i	$-0.843843\pi$
$701$	− 463036.i	− 0.942278i	0.882059	−	0.471139i	$-0.156157\pi$
$702$	0	0
$703$	−274085.	−0.554594
$704$	813316.i	1.64102i
$705$	0	0
$706$	826429.	1.65804
$707$	279131.i	0.558430i
$708$	0	0
$709$	−703784.	−1.40006	−0.700030	−	0.714113i	$-0.746830\pi$
$709$	−703784.	−1.40006	−0.700030	+	0.714113i	$0.746830\pi$
$710$	307900.i	0.610791i
$711$	0	0
$712$	5153.55	0.0101659
$713$	1.16648e6i	2.29455i
$714$	0	0
$715$	1.43145e6	2.80004
$716$	510621.i	0.996031i
$717$	0	0
$718$	677339.	1.31388
$719$	252336.i	0.488115i	0.969761	+	0.244057i	$0.0784785\pi$
$719$	252336.i	0.488115i	−0.969761	+	0.244057i	$0.921521\pi$
$720$	0	0
$721$	−120085.	−0.231003
$722$	− 552701.i	− 1.06027i
$723$	0	0
$724$	658591.	1.25643
$725$	− 313373.i	− 0.596192i
$726$	0	0
$727$	449742.	0.850931	0.425466	−	0.904975i	$-0.360110\pi$
$727$	449742.	0.850931	0.425466	+	0.904975i	$0.360110\pi$
$728$	− 7546.55i	− 0.0142392i
$729$	0	0
$730$	79266.8	0.148746
$731$	− 235344.i	− 0.440422i
$732$	0	0
$733$	−365740.	−0.680713	−0.340357	−	0.940296i	$-0.610548\pi$
$733$	−365740.	−0.680713	−0.340357	+	0.940296i	$0.610548\pi$
$734$	951461.i	1.76603i
$735$	0	0
$736$	−1.10971e6	−2.04858
$737$	− 731494.i	− 1.34671i
$738$	0	0
$739$	−953965.	−1.74680	−0.873401	−	0.487002i	$-0.838090\pi$
$739$	−953965.	−1.74680	−0.873401	+	0.487002i	$0.838090\pi$
$740$	717382.i	1.31005i
$741$	0	0
$742$	−2.03396e6	−3.69432
$743$	543606.i	0.984706i	0.870396	+	0.492353i	$0.163863\pi$
$743$	543606.i	0.984706i	−0.870396	+	0.492353i	$0.836137\pi$
$744$	0	0
$745$	1.24357e6	2.24057
$746$	− 219017.i	− 0.393551i
$747$	0	0
$748$	947271.	1.69305
$749$	1.28225e6i	2.28564i
$750$	0	0
$751$	−635254.	−1.12633	−0.563167	−	0.826343i	$-0.690417\pi$
$751$	−635254.	−1.12633	−0.563167	+	0.826343i	$0.690417\pi$
$752$	1.01967e6i	1.80312i
$753$	0	0
$754$	−1.68641e6	−2.96634
$755$	564237.i	0.989846i
$756$	0	0
$757$	241849.	0.422039	0.211019	−	0.977482i	$-0.432322\pi$
$757$	241849.	0.422039	0.211019	+	0.977482i	$0.432322\pi$
$758$	− 878725.i	− 1.52938i
$759$	0	0
$760$	−2285.41	−0.00395673
$761$	478315.i	0.825934i	0.910746	+	0.412967i	$0.135507\pi$
$761$	478315.i	0.825934i	−0.910746	+	0.412967i	$0.864493\pi$
$762$	0	0
$763$	−1.05268e6	−1.80821
$764$	104518.i	0.179062i
$765$	0	0
$766$	638373.	1.08797
$767$	109211.i	0.185641i
$768$	0	0
$769$	−487066.	−0.823636	−0.411818	−	0.911266i	$-0.635106\pi$
$769$	−487066.	−0.823636	−0.411818	+	0.911266i	$0.635106\pi$
$770$	− 2.45666e6i	− 4.14346i
$771$	0	0
$772$	−518490.	−0.869974
$773$	− 356896.i	− 0.597287i	−0.954365	−	0.298643i	$-0.903466\pi$
$773$	− 356896.i	− 0.597287i	0.954365	−	0.298643i	$-0.0965341\pi$
$774$	0	0
$775$	384683.	0.640472
$776$	− 3861.89i	− 0.00641322i
$777$	0	0
$778$	−1.03671e6	−1.71277
$779$	280324.i	0.461940i
$780$	0	0
$781$	−368670.	−0.604415
$782$	1.28633e6i	2.10349i
$783$	0	0
$784$	−760513.	−1.23730
$785$	607643.i	0.986073i
$786$	0	0
$787$	−400982.	−0.647404	−0.323702	−	0.946159i	$-0.604928\pi$
$787$	−400982.	−0.647404	−0.323702	+	0.946159i	$0.604928\pi$
$788$	− 70615.6i	− 0.113723i
$789$	0	0
$790$	669541.	1.07281
$791$	128214.i	0.204918i
$792$	0	0
$793$	−636532.	−1.01222
$794$	− 107975.i	− 0.171271i
$795$	0	0
$796$	187364.	0.295706
$797$	265746.i	0.418360i	0.977877	+	0.209180i	$0.0670794\pi$
$797$	265746.i	0.418360i	−0.977877	+	0.209180i	$0.932921\pi$
$798$	0	0
$799$	1.17640e6	1.84273
$800$	365962.i	0.571815i
$801$	0	0
$802$	145870.	0.226787
$803$	94911.6i	0.147193i
$804$	0	0
$805$	1.66404e6	2.56786
$806$	− 2.07017e6i	− 3.18666i
$807$	0	0
$808$	1631.48	0.00249895
$809$	− 941347.i	− 1.43831i	−0.694849	−	0.719155i	$-0.744529\pi$
$809$	− 941347.i	− 1.43831i	0.694849	−	0.719155i	$-0.255471\pi$
$810$	0	0
$811$	1.03268e6	1.57009	0.785045	−	0.619438i	$-0.212640\pi$
$811$	1.03268e6	1.57009	0.785045	+	0.619438i	$0.212640\pi$
$812$	1.44368e6i	2.18957i
$813$	0	0
$814$	−1.72203e6	−2.59891
$815$	314577.i	0.473600i
$816$	0	0
$817$	142980.	0.214205
$818$	− 89553.9i	− 0.133838i
$819$	0	0
$820$	733712.	1.09118
$821$	− 1.24333e6i	− 1.84458i	−0.386494	−	0.922292i	$-0.626314\pi$
$821$	− 1.24333e6i	− 1.84458i	0.386494	−	0.922292i	$-0.373686\pi$
$822$	0	0
$823$	−565950.	−0.835562	−0.417781	−	0.908548i	$-0.637192\pi$
$823$	−565950.	−0.835562	−0.417781	+	0.908548i	$0.637192\pi$
$824$	701.878i	0.00103373i
$825$	0	0
$826$	187427.	0.274709
$827$	− 953131.i	− 1.39361i	−0.717260	−	0.696806i	$-0.754604\pi$
$827$	− 953131.i	− 1.39361i	0.717260	−	0.696806i	$-0.245396\pi$
$828$	0	0
$829$	−415415.	−0.604468	−0.302234	−	0.953234i	$-0.597732\pi$
$829$	−415415.	−0.604468	−0.302234	+	0.953234i	$0.597732\pi$
$830$	− 103708.i	− 0.150541i
$831$	0	0
$832$	977703.	1.41241
$833$	877407.i	1.26448i
$834$	0	0
$835$	448015.	0.642568
$836$	575499.i	0.823440i
$837$	0	0
$838$	138057.	0.196595
$839$	774323.i	1.10001i	0.835160	+	0.550007i	$0.185375\pi$
$839$	774323.i	1.10001i	−0.835160	+	0.550007i	$0.814625\pi$
$840$	0	0
$841$	−826748.	−1.16891
$842$	1.16549e6i	1.64393i
$843$	0	0
$844$	548160.	0.769525
$845$	− 874476.i	− 1.22471i
$846$	0	0
$847$	1.86985e6	2.60639
$848$	1.26493e6i	1.75904i
$849$	0	0
$850$	424210.	0.587141
$851$	− 1.16643e6i	− 1.61064i
$852$	0	0
$853$	−163949.	−0.225326	−0.112663	−	0.993633i	$-0.535938\pi$
$853$	−163949.	−0.225326	−0.112663	+	0.993633i	$0.535938\pi$
$854$	1.09242e6i	1.49787i
$855$	0	0
$856$	7494.54	0.0102282
$857$	50524.8i	0.0687928i	0.999408	+	0.0343964i	$0.0109509\pi$
$857$	50524.8i	0.0687928i	−0.999408	+	0.0343964i	$0.989049\pi$
$858$	0	0
$859$	−437616.	−0.593071	−0.296536	−	0.955022i	$-0.595831\pi$
$859$	−437616.	−0.593071	−0.296536	+	0.955022i	$0.595831\pi$
$860$	− 374230.i	− 0.505990i
$861$	0	0
$862$	−676034.	−0.909816
$863$	419251.i	0.562928i	0.959572	+	0.281464i	$0.0908199\pi$
$863$	419251.i	0.562928i	−0.959572	+	0.281464i	$0.909180\pi$
$864$	0	0
$865$	−504367.	−0.674085
$866$	668738.i	0.891703i
$867$	0	0
$868$	−1.77220e6	−2.35219
$869$	801688.i	1.06161i
$870$	0	0
$871$	−879342.	−1.15910
$872$	6152.78i	0.00809167i
$873$	0	0
$874$	−781492.	−1.02306
$875$	806810.i	1.05379i
$876$	0	0
$877$	648530.	0.843201	0.421600	−	0.906782i	$-0.361469\pi$
$877$	648530.	0.843201	0.421600	+	0.906782i	$0.361469\pi$
$878$	204235.i	0.264936i
$879$	0	0
$880$	−1.52782e6	−1.97290
$881$	− 1.28477e6i	− 1.65529i	−0.561249	−	0.827647i	$-0.689679\pi$
$881$	− 1.28477e6i	− 1.65529i	0.561249	−	0.827647i	$-0.310321\pi$
$882$	0	0
$883$	719446.	0.922735	0.461367	−	0.887209i	$-0.347359\pi$
$883$	719446.	0.922735	0.461367	+	0.887209i	$0.347359\pi$
$884$	− 1.13873e6i	− 1.45719i
$885$	0	0
$886$	649973.	0.827995
$887$	93097.0i	0.118328i	0.998248	+	0.0591641i	$0.0188435\pi$
$887$	93097.0i	0.118328i	−0.998248	+	0.0591641i	$0.981156\pi$
$888$	0	0
$889$	−171290.	−0.216735
$890$	− 2.01674e6i	− 2.54607i
$891$	0	0
$892$	816513.	1.02620
$893$	714702.i	0.896235i
$894$	0	0
$895$	−950145.	−1.18616
$896$	16033.4i	0.0199715i
$897$	0	0
$898$	1.01523e6	1.25896
$899$	− 1.88311e6i	− 2.33000i
$900$	0	0
$901$	1.45936e6	1.79768
$902$	1.76122e6i	2.16472i
$903$	0	0
$904$	749.389	0.000917002 0
$905$	1.22548e6i	1.49627i
$906$	0	0
$907$	−912557.	−1.10929	−0.554645	−	0.832087i	$-0.687146\pi$
$907$	−912557.	−1.10929	−0.554645	+	0.832087i	$0.687146\pi$
$908$	1.35378e6i	1.64201i
$909$	0	0
$910$	−2.95320e6	−3.56623
$911$	670058.i	0.807376i	0.914897	+	0.403688i	$0.132272\pi$
$911$	670058.i	0.807376i	−0.914897	+	0.403688i	$0.867728\pi$
$912$	0	0
$913$	124177.	0.148970
$914$	197666.i	0.236614i
$915$	0	0
$916$	−1.56128e6	−1.86076
$917$	1.40620e6i	1.67227i
$918$	0	0
$919$	775354.	0.918055	0.459028	−	0.888422i	$-0.348198\pi$
$919$	775354.	0.918055	0.459028	+	0.888422i	$0.348198\pi$
$920$	− 9726.05i	− 0.0114911i
$921$	0	0
$922$	−618412.	−0.727471
$923$	443185.i	0.520213i
$924$	0	0
$925$	−384667.	−0.449574
$926$	279857.i	0.326373i
$927$	0	0
$928$	1.79146e6	2.08023
$929$	66335.0i	0.0768619i	0.999261	+	0.0384310i	$0.0122360\pi$
$929$	66335.0i	0.0768619i	−0.999261	+	0.0384310i	$0.987764\pi$
$930$	0	0
$931$	−533055.	−0.614996
$932$	117451.i	0.135215i
$933$	0	0
$934$	1.11489e6	1.27803
$935$	1.76265e6i	2.01624i
$936$	0	0
$937$	−676749.	−0.770812	−0.385406	−	0.922747i	$-0.625939\pi$
$937$	−676749.	−0.770812	−0.385406	+	0.922747i	$0.625939\pi$
$938$	1.50913e6i	1.71522i
$939$	0	0
$940$	1.87064e6	2.11706
$941$	76394.8i	0.0862749i	0.999069	+	0.0431375i	$0.0137353\pi$
$941$	76394.8i	0.0862749i	−0.999069	+	0.0431375i	$0.986265\pi$
$942$	0	0
$943$	−1.19298e6	−1.34156
$944$	− 116562.i	− 0.130802i
$945$	0	0
$946$	898314.	1.00380
$947$	534361.i	0.595847i	0.954590	+	0.297924i	$0.0962941\pi$
$947$	534361.i	0.595847i	−0.954590	+	0.297924i	$0.903706\pi$
$948$	0	0
$949$	114095.	0.126688
$950$	257722.i	0.285564i
$951$	0	0
$952$	9292.59	0.0102533
$953$	643029.i	0.708019i	0.935242	+	0.354009i	$0.115182\pi$
$953$	643029.i	0.708019i	−0.935242	+	0.354009i	$0.884818\pi$
$954$	0	0
$955$	−194483.	−0.213243
$956$	− 1.49328e6i	− 1.63390i
$957$	0	0
$958$	202207.	0.220326
$959$	− 2.22269e6i	− 2.41680i
$960$	0	0
$961$	1.38810e6	1.50305
$962$	2.07008e6i	2.23685i
$963$	0	0
$964$	−365766.	−0.393595
$965$	− 964787.i	− 1.03604i
$966$	0	0
$967$	−235274.	−0.251606	−0.125803	−	0.992055i	$-0.540151\pi$
$967$	−235274.	−0.251606	−0.125803	+	0.992055i	$0.540151\pi$
$968$	− 10929.0i	− 0.0116635i
$969$	0	0
$970$	−1.51127e6	−1.60620
$971$	17230.1i	0.0182746i	0.999958	+	0.00913731i	$0.00290854\pi$
$971$	17230.1i	0.0182746i	−0.999958	+	0.00913731i	$0.997091\pi$
$972$	0	0
$973$	−311719.	−0.329259
$974$	− 604765.i	− 0.637484i
$975$	0	0
$976$	679383.	0.713206
$977$	1.15856e6i	1.21375i	0.794796	+	0.606877i	$0.207578\pi$
$977$	1.15856e6i	1.21375i	−0.794796	+	0.606877i	$0.792422\pi$
$978$	0	0
$979$	2.41478e6	2.51949
$980$	1.39520e6i	1.45273i
$981$	0	0
$982$	−1.82711e6	−1.89470
$983$	− 593353.i	− 0.614053i	−0.951701	−	0.307027i	$-0.900666\pi$
$983$	− 593353.i	− 0.614053i	0.951701	−	0.307027i	$-0.0993340\pi$
$984$	0	0
$985$	131399.	0.135431
$986$	− 2.07660e6i	− 2.13599i
$987$	0	0
$988$	691819.	0.708726
$989$	608481.i	0.622091i
$990$	0	0
$991$	−1.32432e6	−1.34849	−0.674244	−	0.738509i	$-0.735530\pi$
$991$	−1.32432e6	−1.34849	−0.674244	+	0.738509i	$0.735530\pi$
$992$	2.19912e6i	2.23473i
$993$	0	0
$994$	760594.	0.769804
$995$	348640.i	0.352152i
$996$	0	0
$997$	−1.09398e6	−1.10057	−0.550287	−	0.834976i	$-0.685482\pi$
$997$	−1.09398e6	−1.10057	−0.550287	+	0.834976i	$0.685482\pi$
$998$	2.27024e6i	2.27935i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.62	yes	76
3.2	odd	2	inner	531.5.b.a.296.15	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.15	✓	76	3.2	odd	2	inner
531.5.b.a.296.62	yes	76	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65016i q^{2} -15.9243 q^{4} -29.6313i q^{5} -73.1972 q^{7} +0.427826i q^{8} +O(q^{10})\)	\(q+5.65016i q^{2} -15.9243 q^{4} -29.6313i q^{5} -73.1972 q^{7} +0.427826i q^{8} +167.422 q^{10} +200.465i q^{11} +240.983 q^{13} -413.576i q^{14} -257.206 q^{16} +296.739i q^{17} -180.279 q^{19} +471.857i q^{20} -1132.66 q^{22} -767.217i q^{23} -253.014 q^{25} +1361.59i q^{26} +1165.61 q^{28} +1238.56i q^{29} -1520.40 q^{31} -1446.41i q^{32} -1676.62 q^{34} +2168.93i q^{35} +1520.34 q^{37} -1018.61i q^{38} +12.6771 q^{40} -1554.94i q^{41} -793.101 q^{43} -3192.27i q^{44} +4334.90 q^{46} -3964.42i q^{47} +2956.83 q^{49} -1429.57i q^{50} -3837.48 q^{52} -4917.99i q^{53} +5940.05 q^{55} -31.3157i q^{56} -6998.05 q^{58} +453.188i q^{59} -2641.40 q^{61} -8590.51i q^{62} +4057.14 q^{64} -7140.65i q^{65} -3648.98 q^{67} -4725.36i q^{68} -12254.8 q^{70} +1839.07i q^{71} +473.457 q^{73} +8590.15i q^{74} +2870.82 q^{76} -14673.5i q^{77} +3999.13 q^{79} +7621.34i q^{80} +8785.68 q^{82} -619.441i q^{83} +8792.77 q^{85} -4481.15i q^{86} -85.7643 q^{88} -12045.9i q^{89} -17639.3 q^{91} +12217.4i q^{92} +22399.6 q^{94} +5341.91i q^{95} -9026.76 q^{97} +16706.5i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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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