LMFDB - Embedded newform 531.5.b.a.296.5

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			296.5
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.72

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.18445i q^{2} -35.6164 q^{4} +15.5237i q^{5} -43.6466 q^{7} +140.933i q^{8} +O(q^{10})$	\(q-7.18445i q^{2} -35.6164 q^{4} +15.5237i q^{5} -43.6466 q^{7} +140.933i q^{8} +111.529 q^{10} +1.21721i q^{11} -174.225 q^{13} +313.577i q^{14} +442.665 q^{16} +372.430i q^{17} +70.8089 q^{19} -552.898i q^{20} +8.74502 q^{22} +65.1472i q^{23} +384.015 q^{25} +1251.71i q^{26} +1554.53 q^{28} -412.612i q^{29} +474.497 q^{31} -925.376i q^{32} +2675.71 q^{34} -677.557i q^{35} +403.733 q^{37} -508.723i q^{38} -2187.80 q^{40} -1860.58i q^{41} -261.076 q^{43} -43.3528i q^{44} +468.047 q^{46} -2249.20i q^{47} -495.975 q^{49} -2758.94i q^{50} +6205.28 q^{52} +879.546i q^{53} -18.8957 q^{55} -6151.25i q^{56} -2964.39 q^{58} +453.188i q^{59} +718.974 q^{61} -3409.00i q^{62} +434.314 q^{64} -2704.62i q^{65} -2833.84 q^{67} -13264.6i q^{68} -4867.87 q^{70} -7631.49i q^{71} -6752.18 q^{73} -2900.60i q^{74} -2521.96 q^{76} -53.1273i q^{77} +1263.31 q^{79} +6871.79i q^{80} -13367.2 q^{82} +2471.89i q^{83} -5781.49 q^{85} +1875.69i q^{86} -171.546 q^{88} +1245.27i q^{89} +7604.35 q^{91} -2320.31i q^{92} -16159.2 q^{94} +1099.22i q^{95} +13115.9 q^{97} +3563.31i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 7.18445i	− 1.79611i	−0.439880	−	0.898057i	$-0.644979\pi$
$2$	− 7.18445i	− 1.79611i	0.439880	−	0.898057i	$-0.355021\pi$
$3$	0	0
$3$	0	0
$4$	−35.6164	−2.22602
$5$	15.5237i	0.620948i	0.950582	+	0.310474i	$0.100488\pi$
$5$	15.5237i	0.620948i	−0.950582	+	0.310474i	$0.899512\pi$
$6$	0	0
$7$	−43.6466	−0.890747	−0.445373	−	0.895345i	$-0.646929\pi$
$7$	−43.6466	−0.890747	−0.445373	+	0.895345i	$0.646929\pi$
$8$	140.933i	2.20208i
$9$	0	0
$10$	111.529	1.11529
$11$	1.21721i	0.0100596i	0.999987	+	0.00502981i	$0.00160105\pi$
$11$	1.21721i	0.0100596i	−0.999987	+	0.00502981i	$0.998399\pi$
$12$	0	0
$13$	−174.225	−1.03092	−0.515460	−	0.856914i	$-0.672379\pi$
$13$	−174.225	−1.03092	−0.515460	+	0.856914i	$0.672379\pi$
$14$	313.577i	1.59988i
$15$	0	0
$16$	442.665	1.72916
$17$	372.430i	1.28869i	0.764737	+	0.644343i	$0.222869\pi$
$17$	372.430i	1.28869i	−0.764737	+	0.644343i	$0.777131\pi$
$18$	0	0
$19$	70.8089	0.196147	0.0980733	−	0.995179i	$-0.468732\pi$
$19$	70.8089	0.196147	0.0980733	+	0.995179i	$0.468732\pi$
$20$	− 552.898i	− 1.38225i
$21$	0	0
$22$	8.74502	0.0180682
$23$	65.1472i	0.123152i	0.998102	+	0.0615758i	$0.0196126\pi$
$23$	65.1472i	0.123152i	−0.998102	+	0.0615758i	$0.980387\pi$
$24$	0	0
$25$	384.015	0.614424
$26$	1251.71i	1.85165i
$27$	0	0
$28$	1554.53	1.98282
$29$	− 412.612i	− 0.490621i	−0.969445	−	0.245310i	$-0.921110\pi$
$29$	− 412.612i	− 0.490621i	0.969445	−	0.245310i	$-0.0788899\pi$
$30$	0	0
$31$	474.497	0.493753	0.246877	−	0.969047i	$-0.420596\pi$
$31$	474.497	0.493753	0.246877	+	0.969047i	$0.420596\pi$
$32$	− 925.376i	− 0.903687i
$33$	0	0
$34$	2675.71	2.31463
$35$	− 677.557i	− 0.553107i
$36$	0	0
$37$	403.733	0.294911	0.147455	−	0.989069i	$-0.452892\pi$
$37$	403.733	0.294911	0.147455	+	0.989069i	$0.452892\pi$
$38$	− 508.723i	− 0.352301i
$39$	0	0
$40$	−2187.80	−1.36738
$41$	− 1860.58i	− 1.10683i	−0.832906	−	0.553414i	$-0.813325\pi$
$41$	− 1860.58i	− 1.10683i	0.832906	−	0.553414i	$-0.186675\pi$
$42$	0	0
$43$	−261.076	−0.141199	−0.0705994	−	0.997505i	$-0.522491\pi$
$43$	−261.076	−0.141199	−0.0705994	+	0.997505i	$0.522491\pi$
$44$	− 43.3528i	− 0.0223930i
$45$	0	0
$46$	468.047	0.221194
$47$	− 2249.20i	− 1.01820i	−0.860708	−	0.509098i	$-0.829979\pi$
$47$	− 2249.20i	− 1.01820i	0.860708	−	0.509098i	$-0.170021\pi$
$48$	0	0
$49$	−495.975	−0.206570
$50$	− 2758.94i	− 1.10357i
$51$	0	0
$52$	6205.28	2.29485
$53$	879.546i	0.313117i	0.987669	+	0.156559i	$0.0500400\pi$
$53$	879.546i	0.313117i	−0.987669	+	0.156559i	$0.949960\pi$
$54$	0	0
$55$	−18.8957	−0.00624650
$56$	− 6151.25i	− 1.96149i
$57$	0	0
$58$	−2964.39	−0.881211
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	718.974	0.193221	0.0966103	−	0.995322i	$-0.469200\pi$
$61$	718.974	0.193221	0.0966103	+	0.995322i	$0.469200\pi$
$62$	− 3409.00i	− 0.886837i
$63$	0	0
$64$	434.314	0.106034
$65$	− 2704.62i	− 0.640148i
$66$	0	0
$67$	−2833.84	−0.631286	−0.315643	−	0.948878i	$-0.602220\pi$
$67$	−2833.84	−0.631286	−0.315643	+	0.948878i	$0.602220\pi$
$68$	− 13264.6i	− 2.86864i
$69$	0	0
$70$	−4867.87	−0.993444
$71$	− 7631.49i	− 1.51388i	−0.653483	−	0.756942i	$-0.726693\pi$
$71$	− 7631.49i	− 1.51388i	0.653483	−	0.756942i	$-0.273307\pi$
$72$	0	0
$73$	−6752.18	−1.26706	−0.633532	−	0.773717i	$-0.718395\pi$
$73$	−6752.18	−1.26706	−0.633532	+	0.773717i	$0.718395\pi$
$74$	− 2900.60i	− 0.529693i
$75$	0	0
$76$	−2521.96	−0.436627
$77$	− 53.1273i	− 0.00896058i
$78$	0	0
$79$	1263.31	0.202421	0.101211	−	0.994865i	$-0.467728\pi$
$79$	1263.31	0.202421	0.101211	+	0.994865i	$0.467728\pi$
$80$	6871.79i	1.07372i
$81$	0	0
$82$	−13367.2	−1.98799
$83$	2471.89i	0.358817i	0.983775	+	0.179409i	$0.0574184\pi$
$83$	2471.89i	0.358817i	−0.983775	+	0.179409i	$0.942582\pi$
$84$	0	0
$85$	−5781.49	−0.800207
$86$	1875.69i	0.253609i
$87$	0	0
$88$	−171.546	−0.0221521
$89$	1245.27i	0.157211i	0.996906	+	0.0786053i	$0.0250467\pi$
$89$	1245.27i	0.157211i	−0.996906	+	0.0786053i	$0.974953\pi$
$90$	0	0
$91$	7604.35	0.918288
$92$	− 2320.31i	− 0.274139i
$93$	0	0
$94$	−16159.2	−1.82880
$95$	1099.22i	0.121797i
$96$	0	0
$97$	13115.9	1.39397	0.696986	−	0.717084i	$-0.254524\pi$
$97$	13115.9	1.39397	0.696986	+	0.717084i	$0.254524\pi$
$98$	3563.31i	0.371024i
$99$	0	0
$100$	−13677.2	−1.36772
$101$	− 7695.62i	− 0.754399i	−0.926132	−	0.377199i	$-0.876887\pi$
$101$	− 7695.62i	− 0.754399i	0.926132	−	0.377199i	$-0.123113\pi$
$102$	0	0
$103$	8710.01	0.821002	0.410501	−	0.911860i	$-0.365354\pi$
$103$	8710.01	0.821002	0.410501	+	0.911860i	$0.365354\pi$
$104$	− 24554.1i	− 2.27017i
$105$	0	0
$106$	6319.06	0.562394
$107$	− 3999.35i	− 0.349319i	−0.984629	−	0.174659i	$-0.944118\pi$
$107$	− 3999.35i	− 0.349319i	0.984629	−	0.174659i	$-0.0558824\pi$
$108$	0	0
$109$	17980.5	1.51338	0.756692	−	0.653772i	$-0.226814\pi$
$109$	17980.5	1.51338	0.756692	+	0.653772i	$0.226814\pi$
$110$	135.755i	0.0112194i
$111$	0	0
$112$	−19320.8	−1.54024
$113$	− 4534.59i	− 0.355125i	−0.984110	−	0.177562i	$-0.943179\pi$
$113$	− 4534.59i	− 0.355125i	0.984110	−	0.177562i	$-0.0568212\pi$
$114$	0	0
$115$	−1011.33	−0.0764708
$116$	14695.8i	1.09213i
$117$	0	0
$118$	3255.91	0.233834
$119$	− 16255.3i	− 1.14789i
$120$	0	0
$121$	14639.5	0.999899
$122$	− 5165.43i	− 0.347046i
$123$	0	0
$124$	−16899.9	−1.09911
$125$	15663.6i	1.00247i
$126$	0	0
$127$	−567.971	−0.0352143	−0.0176071	−	0.999845i	$-0.505605\pi$
$127$	−567.971	−0.0352143	−0.0176071	+	0.999845i	$0.505605\pi$
$128$	− 17926.3i	− 1.09414i
$129$	0	0
$130$	−19431.2	−1.14978
$131$	− 12776.2i	− 0.744488i	−0.928135	−	0.372244i	$-0.878588\pi$
$131$	− 12776.2i	− 0.744488i	0.928135	−	0.372244i	$-0.121412\pi$
$132$	0	0
$133$	−3090.57	−0.174717
$134$	20359.6i	1.13386i
$135$	0	0
$136$	−52487.7	−2.83779
$137$	15857.6i	0.844881i	0.906391	+	0.422441i	$0.138827\pi$
$137$	15857.6i	0.844881i	−0.906391	+	0.422441i	$0.861173\pi$
$138$	0	0
$139$	1922.39	0.0994976	0.0497488	−	0.998762i	$-0.484158\pi$
$139$	1922.39	0.0994976	0.0497488	+	0.998762i	$0.484158\pi$
$140$	24132.1i	1.23123i
$141$	0	0
$142$	−54828.1	−2.71911
$143$	− 212.070i	− 0.0103707i
$144$	0	0
$145$	6405.27	0.304650
$146$	48510.7i	2.27579i
$147$	0	0
$148$	−14379.5	−0.656479
$149$	8962.59i	0.403702i	0.979416	+	0.201851i	$0.0646957\pi$
$149$	8962.59i	0.403702i	−0.979416	+	0.201851i	$0.935304\pi$
$150$	0	0
$151$	20077.3	0.880544	0.440272	−	0.897864i	$-0.354882\pi$
$151$	20077.3	0.880544	0.440272	+	0.897864i	$0.354882\pi$
$152$	9979.31i	0.431930i
$153$	0	0
$154$	−381.690	−0.0160942
$155$	7365.95i	0.306595i
$156$	0	0
$157$	−27050.6	−1.09743	−0.548716	−	0.836009i	$-0.684883\pi$
$157$	−27050.6	−1.09743	−0.548716	+	0.836009i	$0.684883\pi$
$158$	− 9076.21i	− 0.363572i
$159$	0	0
$160$	14365.3	0.561143
$161$	− 2843.45i	− 0.109697i
$162$	0	0
$163$	49117.8	1.84869	0.924344	−	0.381560i	$-0.124613\pi$
$163$	49117.8	1.84869	0.924344	+	0.381560i	$0.124613\pi$
$164$	66267.1i	2.46383i
$165$	0	0
$166$	17759.2	0.644477
$167$	− 10973.9i	− 0.393485i	−0.980455	−	0.196742i	$-0.936964\pi$
$167$	− 10973.9i	− 0.393485i	0.980455	−	0.196742i	$-0.0630363\pi$
$168$	0	0
$169$	1793.50	0.0627956
$170$	41536.9i	1.43726i
$171$	0	0
$172$	9298.60	0.314312
$173$	18112.6i	0.605186i	0.953120	+	0.302593i	$0.0978523\pi$
$173$	18112.6i	0.605186i	−0.953120	+	0.302593i	$0.902148\pi$
$174$	0	0
$175$	−16760.9	−0.547296
$176$	538.818i	0.0173947i
$177$	0	0
$178$	8946.55	0.282368
$179$	25646.4i	0.800423i	0.916423	+	0.400212i	$0.131063\pi$
$179$	25646.4i	0.800423i	−0.916423	+	0.400212i	$0.868937\pi$
$180$	0	0
$181$	−7034.92	−0.214735	−0.107367	−	0.994219i	$-0.534242\pi$
$181$	−7034.92	−0.214735	−0.107367	+	0.994219i	$0.534242\pi$
$182$	− 54633.1i	− 1.64935i
$183$	0	0
$184$	−9181.40	−0.271190
$185$	6267.43i	0.183124i
$186$	0	0
$187$	−453.327	−0.0129637
$188$	80108.2i	2.26653i
$189$	0	0
$190$	7897.27	0.218761
$191$	4050.45i	0.111029i	0.998458	+	0.0555146i	$0.0176799\pi$
$191$	4050.45i	0.111029i	−0.998458	+	0.0555146i	$0.982320\pi$
$192$	0	0
$193$	−20075.0	−0.538940	−0.269470	−	0.963009i	$-0.586848\pi$
$193$	−20075.0	−0.538940	−0.269470	+	0.963009i	$0.586848\pi$
$194$	− 94230.5i	− 2.50373i
$195$	0	0
$196$	17664.8	0.459830
$197$	− 38082.6i	− 0.981283i	−0.871361	−	0.490642i	$-0.836762\pi$
$197$	− 38082.6i	− 0.981283i	0.871361	−	0.490642i	$-0.163238\pi$
$198$	0	0
$199$	25403.5	0.641487	0.320743	−	0.947166i	$-0.396067\pi$
$199$	25403.5	0.641487	0.320743	+	0.947166i	$0.396067\pi$
$200$	54120.4i	1.35301i
$201$	0	0
$202$	−55288.8	−1.35499
$203$	18009.1i	0.437019i
$204$	0	0
$205$	28883.1	0.687283
$206$	− 62576.7i	− 1.47461i
$207$	0	0
$208$	−77123.5	−1.78262
$209$	86.1896i	0.00197316i
$210$	0	0
$211$	−45286.4	−1.01719	−0.508596	−	0.861005i	$-0.669835\pi$
$211$	−45286.4	−1.01719	−0.508596	+	0.861005i	$0.669835\pi$
$212$	− 31326.2i	− 0.697006i
$213$	0	0
$214$	−28733.2	−0.627416
$215$	− 4052.87i	− 0.0876771i
$216$	0	0
$217$	−20710.2	−0.439809
$218$	− 129180.i	− 2.71821i
$219$	0	0
$220$	672.996	0.0139049
$221$	− 64886.8i	− 1.32853i
$222$	0	0
$223$	8518.20	0.171293	0.0856463	−	0.996326i	$-0.472705\pi$
$223$	8518.20	0.171293	0.0856463	+	0.996326i	$0.472705\pi$
$224$	40389.5i	0.804957i
$225$	0	0
$226$	−32578.6	−0.637845
$227$	− 84296.7i	− 1.63591i	−0.575284	−	0.817954i	$-0.695108\pi$
$227$	− 84296.7i	− 1.63591i	0.575284	−	0.817954i	$-0.304892\pi$
$228$	0	0
$229$	61059.9	1.16435	0.582177	−	0.813062i	$-0.302201\pi$
$229$	61059.9	1.16435	0.582177	+	0.813062i	$0.302201\pi$
$230$	7265.83i	0.137350i
$231$	0	0
$232$	58150.7	1.08039
$233$	25669.4i	0.472828i	0.971652	+	0.236414i	$0.0759722\pi$
$233$	25669.4i	0.472828i	−0.971652	+	0.236414i	$0.924028\pi$
$234$	0	0
$235$	34915.8	0.632247
$236$	− 16140.9i	− 0.289804i
$237$	0	0
$238$	−116785.	−2.06174
$239$	− 12241.5i	− 0.214309i	−0.994242	−	0.107154i	$-0.965826\pi$
$239$	− 12241.5i	− 0.214309i	0.994242	−	0.107154i	$-0.0341739\pi$
$240$	0	0
$241$	19062.8	0.328211	0.164106	−	0.986443i	$-0.447526\pi$
$241$	19062.8	0.328211	0.164106	+	0.986443i	$0.447526\pi$
$242$	− 105177.i	− 1.79593i
$243$	0	0
$244$	−25607.2	−0.430114
$245$	− 7699.37i	− 0.128269i
$246$	0	0
$247$	−12336.7	−0.202211
$248$	66872.3i	1.08728i
$249$	0	0
$250$	112535.	1.80056
$251$	− 52574.4i	− 0.834501i	−0.908792	−	0.417250i	$-0.862994\pi$
$251$	− 52574.4i	− 0.834501i	0.908792	−	0.417250i	$-0.137006\pi$
$252$	0	0
$253$	−79.2982	−0.00123886
$254$	4080.56i	0.0632488i
$255$	0	0
$256$	−121842.	−1.85916
$257$	− 108470.i	− 1.64226i	−0.570740	−	0.821131i	$-0.693344\pi$
$257$	− 108470.i	− 1.64226i	0.570740	−	0.821131i	$-0.306656\pi$
$258$	0	0
$259$	−17621.6	−0.262691
$260$	96328.9i	1.42498i
$261$	0	0
$262$	−91789.8	−1.33719
$263$	− 16502.0i	− 0.238575i	−0.992860	−	0.119287i	$-0.961939\pi$
$263$	− 16502.0i	− 0.238575i	0.992860	−	0.119287i	$-0.0380610\pi$
$264$	0	0
$265$	−13653.8	−0.194429
$266$	22204.0i	0.313811i
$267$	0	0
$268$	100931.	1.40526
$269$	117780.i	1.62767i	0.581094	+	0.813836i	$0.302625\pi$
$269$	117780.i	1.62767i	−0.581094	+	0.813836i	$0.697375\pi$
$270$	0	0
$271$	102972.	1.40210	0.701049	−	0.713113i	$-0.252715\pi$
$271$	102972.	1.40210	0.701049	+	0.713113i	$0.252715\pi$
$272$	164862.i	2.22834i
$273$	0	0
$274$	113928.	1.51750
$275$	467.428i	0.00618087i
$276$	0	0
$277$	82086.3	1.06982	0.534911	−	0.844909i	$-0.320345\pi$
$277$	82086.3	1.06982	0.534911	+	0.844909i	$0.320345\pi$
$278$	− 13811.3i	− 0.178709i
$279$	0	0
$280$	95490.1	1.21799
$281$	− 130095.i	− 1.64758i	−0.566892	−	0.823792i	$-0.691855\pi$
$281$	− 130095.i	− 1.64758i	0.566892	−	0.823792i	$-0.308145\pi$
$282$	0	0
$283$	124067.	1.54912	0.774558	−	0.632503i	$-0.217973\pi$
$283$	124067.	1.54912	0.774558	+	0.632503i	$0.217973\pi$
$284$	271806.i	3.36994i
$285$	0	0
$286$	−1523.61	−0.0186269
$287$	81207.9i	0.985904i
$288$	0	0
$289$	−55183.2	−0.660710
$290$	− 46018.4i	− 0.547186i
$291$	0	0
$292$	240488.	2.82051
$293$	− 49630.4i	− 0.578113i	−0.957312	−	0.289057i	$-0.906658\pi$
$293$	− 49630.4i	− 0.578113i	0.957312	−	0.289057i	$-0.0933416\pi$
$294$	0	0
$295$	−7035.15	−0.0808405
$296$	56899.3i	0.649417i
$297$	0	0
$298$	64391.3	0.725095
$299$	− 11350.3i	− 0.126959i
$300$	0	0
$301$	11395.1	0.125772
$302$	− 144244.i	− 1.58156i
$303$	0	0
$304$	31344.6	0.339169
$305$	11161.1i	0.119980i
$306$	0	0
$307$	4876.14	0.0517368	0.0258684	−	0.999665i	$-0.491765\pi$
$307$	4876.14	0.0517368	0.0258684	+	0.999665i	$0.491765\pi$
$308$	1892.20i	0.0199465i
$309$	0	0
$310$	52920.3	0.550680
$311$	28435.8i	0.293999i	0.989137	+	0.146999i	$0.0469615\pi$
$311$	28435.8i	0.293999i	−0.989137	+	0.146999i	$0.953038\pi$
$312$	0	0
$313$	33035.5	0.337204	0.168602	−	0.985684i	$-0.446075\pi$
$313$	33035.5	0.337204	0.168602	+	0.985684i	$0.446075\pi$
$314$	194344.i	1.97111i
$315$	0	0
$316$	−44994.6	−0.450595
$317$	− 60035.9i	− 0.597438i	−0.954341	−	0.298719i	$-0.903441\pi$
$317$	− 60035.9i	− 0.597438i	0.954341	−	0.298719i	$-0.0965593\pi$
$318$	0	0
$319$	502.238	0.00493546
$320$	6742.16i	0.0658414i
$321$	0	0
$322$	−20428.7	−0.197028
$323$	26371.4i	0.252771i
$324$	0	0
$325$	−66905.1	−0.633422
$326$	− 352885.i	− 3.32045i
$327$	0	0
$328$	262217.	2.43732
$329$	98169.7i	0.906955i
$330$	0	0
$331$	185100.	1.68947	0.844736	−	0.535184i	$-0.179758\pi$
$331$	185100.	1.68947	0.844736	+	0.535184i	$0.179758\pi$
$332$	− 88039.9i	− 0.798736i
$333$	0	0
$334$	−78841.5	−0.706743
$335$	− 43991.7i	− 0.391996i
$336$	0	0
$337$	−56283.4	−0.495587	−0.247794	−	0.968813i	$-0.579706\pi$
$337$	−56283.4	−0.495587	−0.247794	+	0.968813i	$0.579706\pi$
$338$	− 12885.4i	− 0.112788i
$339$	0	0
$340$	205916.	1.78128
$341$	577.565i	0.00496698i
$342$	0	0
$343$	126443.	1.07475
$344$	− 36794.3i	− 0.310931i
$345$	0	0
$346$	130129.	1.08698
$347$	73372.2i	0.609359i	0.952455	+	0.304679i	$0.0985493\pi$
$347$	73372.2i	0.609359i	−0.952455	+	0.304679i	$0.901451\pi$
$348$	0	0
$349$	208998.	1.71590	0.857948	−	0.513737i	$-0.171739\pi$
$349$	208998.	1.71590	0.857948	+	0.513737i	$0.171739\pi$
$350$	120418.i	0.983006i
$351$	0	0
$352$	1126.38	0.00909076
$353$	− 40934.8i	− 0.328506i	−0.986418	−	0.164253i	$-0.947479\pi$
$353$	− 40934.8i	− 0.328506i	0.986418	−	0.164253i	$-0.0525213\pi$
$354$	0	0
$355$	118469.	0.940043
$356$	− 44351.9i	− 0.349955i
$357$	0	0
$358$	184255.	1.43765
$359$	166730.i	1.29367i	0.762629	+	0.646836i	$0.223908\pi$
$359$	166730.i	1.29367i	−0.762629	+	0.646836i	$0.776092\pi$
$360$	0	0
$361$	−125307.	−0.961527
$362$	50542.1i	0.385688i
$363$	0	0
$364$	−270839.	−2.04413
$365$	− 104819.i	− 0.786781i
$366$	0	0
$367$	−56232.0	−0.417495	−0.208747	−	0.977970i	$-0.566939\pi$
$367$	−56232.0	−0.417495	−0.208747	+	0.977970i	$0.566939\pi$
$368$	28838.4i	0.212949i
$369$	0	0
$370$	45028.1	0.328912
$371$	− 38389.2i	− 0.278908i
$372$	0	0
$373$	48971.9	0.351989	0.175995	−	0.984391i	$-0.443686\pi$
$373$	48971.9	0.351989	0.175995	+	0.984391i	$0.443686\pi$
$374$	3256.91i	0.0232843i
$375$	0	0
$376$	316986.	2.24215
$377$	71887.6i	0.505791i
$378$	0	0
$379$	−38517.2	−0.268149	−0.134074	−	0.990971i	$-0.542806\pi$
$379$	−38517.2	−0.268149	−0.134074	+	0.990971i	$0.542806\pi$
$380$	− 39150.1i	− 0.271123i
$381$	0	0
$382$	29100.3	0.199421
$383$	185606.i	1.26531i	0.774436	+	0.632653i	$0.218034\pi$
$383$	185606.i	1.26531i	−0.774436	+	0.632653i	$0.781966\pi$
$384$	0	0
$385$	824.732	0.00556405
$386$	144228.i	0.967997i
$387$	0	0
$388$	−467140.	−3.10302
$389$	− 151369.i	− 1.00032i	−0.865934	−	0.500159i	$-0.833275\pi$
$389$	− 151369.i	− 1.00032i	0.865934	−	0.500159i	$-0.166725\pi$
$390$	0	0
$391$	−24262.8	−0.158704
$392$	− 69899.3i	− 0.454884i
$393$	0	0
$394$	−273603.	−1.76250
$395$	19611.3i	0.125693i
$396$	0	0
$397$	6978.50	0.0442773	0.0221387	−	0.999755i	$-0.492952\pi$
$397$	6978.50	0.0442773	0.0221387	+	0.999755i	$0.492952\pi$
$398$	− 182510.i	− 1.15218i
$399$	0	0
$400$	169990.	1.06244
$401$	150562.i	0.936327i	0.883642	+	0.468163i	$0.155084\pi$
$401$	150562.i	0.936327i	−0.883642	+	0.468163i	$0.844916\pi$
$402$	0	0
$403$	−82669.5	−0.509020
$404$	274090.i	1.67931i
$405$	0	0
$406$	129386.	0.784936
$407$	491.430i	0.00296669i
$408$	0	0
$409$	−236736.	−1.41520	−0.707601	−	0.706612i	$-0.750223\pi$
$409$	−236736.	−1.41520	−0.707601	+	0.706612i	$0.750223\pi$
$410$	− 207509.i	− 1.23444i
$411$	0	0
$412$	−310219.	−1.82757
$413$	− 19780.1i	− 0.115965i
$414$	0	0
$415$	−38372.9	−0.222807
$416$	161224.i	0.931629i
$417$	0	0
$418$	619.225	0.00354402
$419$	− 242122.i	− 1.37913i	−0.724222	−	0.689566i	$-0.757801\pi$
$419$	− 242122.i	− 1.37913i	0.724222	−	0.689566i	$-0.242199\pi$
$420$	0	0
$421$	8180.60	0.0461552	0.0230776	−	0.999734i	$-0.492654\pi$
$421$	8180.60	0.0461552	0.0230776	+	0.999734i	$0.492654\pi$
$422$	325358.i	1.82699i
$423$	0	0
$424$	−123957.	−0.689508
$425$	143019.i	0.791799i
$426$	0	0
$427$	−31380.7	−0.172111
$428$	142442.i	0.777592i
$429$	0	0
$430$	−29117.7	−0.157478
$431$	− 282305.i	− 1.51972i	−0.650085	−	0.759861i	$-0.725267\pi$
$431$	− 282305.i	− 1.51972i	0.650085	−	0.759861i	$-0.274733\pi$
$432$	0	0
$433$	−154639.	−0.824790	−0.412395	−	0.911005i	$-0.635308\pi$
$433$	−154639.	−0.824790	−0.412395	+	0.911005i	$0.635308\pi$
$434$	148791.i	0.789947i
$435$	0	0
$436$	−640401.	−3.36883
$437$	4613.00i	0.0241558i
$438$	0	0
$439$	210628.	1.09292	0.546458	−	0.837487i	$-0.315976\pi$
$439$	210628.	1.09292	0.546458	+	0.837487i	$0.315976\pi$
$440$	− 2663.02i	− 0.0137553i
$441$	0	0
$442$	−466176.	−2.38619
$443$	225809.i	1.15062i	0.817934	+	0.575312i	$0.195119\pi$
$443$	225809.i	1.15062i	−0.817934	+	0.575312i	$0.804881\pi$
$444$	0	0
$445$	−19331.1	−0.0976196
$446$	− 61198.7i	− 0.307661i
$447$	0	0
$448$	−18956.3	−0.0944491
$449$	3377.28i	0.0167523i	0.999965	+	0.00837615i	$0.00266624\pi$
$449$	3377.28i	0.0167523i	−0.999965	+	0.00837615i	$0.997334\pi$
$450$	0	0
$451$	2264.72	0.0111343
$452$	161506.i	0.790517i
$453$	0	0
$454$	−605626.	−2.93828
$455$	118048.i	0.570209i
$456$	0	0
$457$	−223785.	−1.07151	−0.535757	−	0.844372i	$-0.679974\pi$
$457$	−223785.	−1.07151	−0.535757	+	0.844372i	$0.679974\pi$
$458$	− 438682.i	− 2.09131i
$459$	0	0
$460$	36019.8	0.170226
$461$	238279.i	1.12120i	0.828086	+	0.560601i	$0.189430\pi$
$461$	238279.i	1.12120i	−0.828086	+	0.560601i	$0.810570\pi$
$462$	0	0
$463$	338112.	1.57724	0.788621	−	0.614880i	$-0.210796\pi$
$463$	338112.	1.57724	0.788621	+	0.614880i	$0.210796\pi$
$464$	− 182649.i	− 0.848362i
$465$	0	0
$466$	184420.	0.849253
$467$	122007.i	0.559435i	0.960082	+	0.279718i	$0.0902408\pi$
$467$	122007.i	0.559435i	−0.960082	+	0.279718i	$0.909759\pi$
$468$	0	0
$469$	123688.	0.562316
$470$	− 250851.i	− 1.13559i
$471$	0	0
$472$	−63869.1	−0.286686
$473$	− 317.786i	− 0.00142041i
$474$	0	0
$475$	27191.7	0.120517
$476$	578955.i	2.55524i
$477$	0	0
$478$	−87948.8	−0.384923
$479$	29224.9i	0.127374i	0.997970	+	0.0636872i	$0.0202860\pi$
$479$	29224.9i	0.127374i	−0.997970	+	0.0636872i	$0.979714\pi$
$480$	0	0
$481$	−70340.6	−0.304029
$482$	− 136956.i	− 0.589505i
$483$	0	0
$484$	−521407.	−2.22580
$485$	203607.i	0.865584i
$486$	0	0
$487$	−38979.0	−0.164351	−0.0821756	−	0.996618i	$-0.526187\pi$
$487$	−38979.0	−0.164351	−0.0821756	+	0.996618i	$0.526187\pi$
$488$	101327.i	0.425487i
$489$	0	0
$490$	−55315.8	−0.230386
$491$	− 166060.i	− 0.688813i	−0.938821	−	0.344407i	$-0.888080\pi$
$491$	− 166060.i	− 0.688813i	0.938821	−	0.344407i	$-0.111920\pi$
$492$	0	0
$493$	153669.	0.632256
$494$	88632.5i	0.363195i
$495$	0	0
$496$	210043.	0.853778
$497$	333088.i	1.34849i
$498$	0	0
$499$	−279840.	−1.12385	−0.561925	−	0.827189i	$-0.689939\pi$
$499$	−279840.	−1.12385	−0.561925	+	0.827189i	$0.689939\pi$
$500$	− 557882.i	− 2.23153i
$501$	0	0
$502$	−377718.	−1.49886
$503$	− 130740.i	− 0.516740i	−0.966046	−	0.258370i	$-0.916815\pi$
$503$	− 130740.i	− 0.516740i	0.966046	−	0.258370i	$-0.0831854\pi$
$504$	0	0
$505$	119465.	0.468442
$506$	569.714i	0.00222513i
$507$	0	0
$508$	20229.1	0.0783878
$509$	− 133364.i	− 0.514757i	−0.966311	−	0.257378i	$-0.917141\pi$
$509$	− 133364.i	− 0.514757i	0.966311	−	0.257378i	$-0.0828587\pi$
$510$	0	0
$511$	294710.	1.12863
$512$	588546.i	2.24512i
$513$	0	0
$514$	−779296.	−2.94969
$515$	135212.i	0.509800i
$516$	0	0
$517$	2737.75	0.0102427
$518$	126601.i	0.471823i
$519$	0	0
$520$	381171.	1.40966
$521$	− 514542.i	− 1.89560i	−0.318873	−	0.947798i	$-0.603304\pi$
$521$	− 514542.i	− 1.89560i	0.318873	−	0.947798i	$-0.396696\pi$
$522$	0	0
$523$	223325.	0.816460	0.408230	−	0.912879i	$-0.366146\pi$
$523$	223325.	0.816460	0.408230	+	0.912879i	$0.366146\pi$
$524$	455041.i	1.65725i
$525$	0	0
$526$	−118558.	−0.428507
$527$	176717.i	0.636293i
$528$	0	0
$529$	275597.	0.984834
$530$	98095.1i	0.349217i
$531$	0	0
$532$	110075.	0.388924
$533$	324160.i	1.14105i
$534$	0	0
$535$	62084.7	0.216909
$536$	− 399382.i	− 1.39014i
$537$	0	0
$538$	846185.	2.92349
$539$	− 603.708i	− 0.00207802i
$540$	0	0
$541$	−79784.0	−0.272597	−0.136299	−	0.990668i	$-0.543521\pi$
$541$	−79784.0	−0.272597	−0.136299	+	0.990668i	$0.543521\pi$
$542$	− 739794.i	− 2.51833i
$543$	0	0
$544$	344638.	1.16457
$545$	279124.i	0.939732i
$546$	0	0
$547$	153107.	0.511707	0.255853	−	0.966716i	$-0.417644\pi$
$547$	153107.	0.511707	0.255853	+	0.966716i	$0.417644\pi$
$548$	− 564790.i	− 1.88073i
$549$	0	0
$550$	3358.22	0.0111015
$551$	− 29216.6i	− 0.0962336i
$552$	0	0
$553$	−55139.3	−0.180306
$554$	− 589745.i	− 1.92152i
$555$	0	0
$556$	−68468.7	−0.221484
$557$	255519.i	0.823593i	0.911276	+	0.411797i	$0.135099\pi$
$557$	255519.i	0.823593i	−0.911276	+	0.411797i	$0.864901\pi$
$558$	0	0
$559$	45486.2	0.145565
$560$	− 299930.i	− 0.956411i
$561$	0	0
$562$	−934660.	−2.95925
$563$	47289.3i	0.149192i	0.997214	+	0.0745961i	$0.0237667\pi$
$563$	47289.3i	0.149192i	−0.997214	+	0.0745961i	$0.976233\pi$
$564$	0	0
$565$	70393.6	0.220514
$566$	− 891354.i	− 2.78239i
$567$	0	0
$568$	1.07553e6	3.33369
$569$	− 116940.i	− 0.361192i	−0.983557	−	0.180596i	$-0.942197\pi$
$569$	− 116940.i	− 0.361192i	0.983557	−	0.180596i	$-0.0578027\pi$
$570$	0	0
$571$	−79161.2	−0.242795	−0.121398	−	0.992604i	$-0.538738\pi$
$571$	−79161.2	−0.242795	−0.121398	+	0.992604i	$0.538738\pi$
$572$	7553.16i	0.0230854i
$573$	0	0
$574$	583435.	1.77080
$575$	25017.5i	0.0756673i
$576$	0	0
$577$	326369.	0.980295	0.490147	−	0.871640i	$-0.336943\pi$
$577$	326369.	0.980295	0.490147	+	0.871640i	$0.336943\pi$
$578$	396461.i	1.18671i
$579$	0	0
$580$	−228132.	−0.678158
$581$	− 107890.i	− 0.319615i
$582$	0	0
$583$	−1070.60	−0.00314984
$584$	− 951605.i	− 2.79017i
$585$	0	0
$586$	−356568.	−1.03836
$587$	− 139294.i	− 0.404255i	−0.979359	−	0.202128i	$-0.935214\pi$
$587$	− 139294.i	− 0.404255i	0.979359	−	0.202128i	$-0.0647856\pi$
$588$	0	0
$589$	33598.6	0.0968480
$590$	50543.7i	0.145199i
$591$	0	0
$592$	178718.	0.509948
$593$	− 124472.i	− 0.353967i	−0.984214	−	0.176983i	$-0.943366\pi$
$593$	− 124472.i	− 0.353967i	0.984214	−	0.176983i	$-0.0566339\pi$
$594$	0	0
$595$	252342.	0.712781
$596$	− 319215.i	− 0.898650i
$597$	0	0
$598$	−81545.8	−0.228034
$599$	589020.i	1.64163i	0.571191	+	0.820817i	$0.306482\pi$
$599$	589020.i	1.64163i	−0.571191	+	0.820817i	$0.693518\pi$
$600$	0	0
$601$	−556200.	−1.53986	−0.769931	−	0.638127i	$-0.779710\pi$
$601$	−556200.	−1.53986	−0.769931	+	0.638127i	$0.779710\pi$
$602$	− 81867.5i	− 0.225901i
$603$	0	0
$604$	−715080.	−1.96011
$605$	227259.i	0.620885i
$606$	0	0
$607$	105145.	0.285372	0.142686	−	0.989768i	$-0.454426\pi$
$607$	105145.	0.285372	0.142686	+	0.989768i	$0.454426\pi$
$608$	− 65524.8i	− 0.177255i
$609$	0	0
$610$	80186.6	0.215498
$611$	391867.i	1.04968i
$612$	0	0
$613$	221000.	0.588128	0.294064	−	0.955786i	$-0.404992\pi$
$613$	221000.	0.588128	0.294064	+	0.955786i	$0.404992\pi$
$614$	− 35032.4i	− 0.0929252i
$615$	0	0
$616$	7487.39	0.0197319
$617$	625197.i	1.64228i	0.570728	+	0.821139i	$0.306661\pi$
$617$	625197.i	1.64228i	−0.570728	+	0.821139i	$0.693339\pi$
$618$	0	0
$619$	−573732.	−1.49736	−0.748682	−	0.662929i	$-0.769313\pi$
$619$	−573732.	−1.49736	−0.748682	+	0.662929i	$0.769313\pi$
$620$	− 262348.i	− 0.682488i
$621$	0	0
$622$	204296.	0.528055
$623$	− 54351.6i	− 0.140035i
$624$	0	0
$625$	−3148.40	−0.00805990
$626$	− 237342.i	− 0.605656i
$627$	0	0
$628$	963445.	2.44291
$629$	150362.i	0.380047i
$630$	0	0
$631$	−119692.	−0.300611	−0.150306	−	0.988640i	$-0.548026\pi$
$631$	−119692.	−0.300611	−0.150306	+	0.988640i	$0.548026\pi$
$632$	178042.i	0.445748i
$633$	0	0
$634$	−431325.	−1.07307
$635$	− 8817.01i	− 0.0218662i
$636$	0	0
$637$	86411.5	0.212957
$638$	− 3608.30i	− 0.00886465i
$639$	0	0
$640$	278283.	0.679401
$641$	520671.i	1.26721i	0.773658	+	0.633603i	$0.218425\pi$
$641$	520671.i	1.26721i	−0.773658	+	0.633603i	$0.781575\pi$
$642$	0	0
$643$	−94998.2	−0.229770	−0.114885	−	0.993379i	$-0.536650\pi$
$643$	−94998.2	−0.229770	−0.114885	+	0.993379i	$0.536650\pi$
$644$	101274.i	0.244188i
$645$	0	0
$646$	189464.	0.454006
$647$	− 161214.i	− 0.385119i	−0.981285	−	0.192559i	$-0.938321\pi$
$647$	− 161214.i	− 0.385119i	0.981285	−	0.192559i	$-0.0616788\pi$
$648$	0	0
$649$	−551.627	−0.00130965
$650$	480677.i	1.13770i
$651$	0	0
$652$	−1.74940e6	−4.11522
$653$	446530.i	1.04719i	0.851968	+	0.523594i	$0.175409\pi$
$653$	446530.i	1.04719i	−0.851968	+	0.523594i	$0.824591\pi$
$654$	0	0
$655$	198333.	0.462288
$656$	− 823613.i	− 1.91388i
$657$	0	0
$658$	705296.	1.62899
$659$	− 177777.i	− 0.409360i	−0.978829	−	0.204680i	$-0.934385\pi$
$659$	− 177777.i	− 0.409360i	0.978829	−	0.204680i	$-0.0656154\pi$
$660$	0	0
$661$	−417499.	−0.955548	−0.477774	−	0.878483i	$-0.658556\pi$
$661$	−417499.	−0.955548	−0.477774	+	0.878483i	$0.658556\pi$
$662$	− 1.32984e6i	− 3.03448i
$663$	0	0
$664$	−348371.	−0.790144
$665$	− 47977.0i	− 0.108490i
$666$	0	0
$667$	26880.5	0.0604208
$668$	390851.i	0.875907i
$669$	0	0
$670$	−316057.	−0.704069
$671$	875.145i	0.00194373i
$672$	0	0
$673$	91589.8	0.202217	0.101108	−	0.994875i	$-0.467761\pi$
$673$	91589.8	0.202217	0.101108	+	0.994875i	$0.467761\pi$
$674$	404365.i	0.890131i
$675$	0	0
$676$	−63878.2	−0.139785
$677$	424628.i	0.926469i	0.886236	+	0.463235i	$0.153311\pi$
$677$	424628.i	0.926469i	−0.886236	+	0.463235i	$0.846689\pi$
$678$	0	0
$679$	−572464.	−1.24168
$680$	− 814803.i	− 1.76212i
$681$	0	0
$682$	4149.49	0.00892125
$683$	− 91836.1i	− 0.196867i	−0.995144	−	0.0984333i	$-0.968617\pi$
$683$	− 91836.1i	− 0.196867i	0.995144	−	0.0984333i	$-0.0313831\pi$
$684$	0	0
$685$	−246168.	−0.524627
$686$	− 908425.i	− 1.93037i
$687$	0	0
$688$	−115569.	−0.244155
$689$	− 153239.i	− 0.322799i
$690$	0	0
$691$	−584417.	−1.22396	−0.611979	−	0.790874i	$-0.709626\pi$
$691$	−584417.	−1.22396	−0.611979	+	0.790874i	$0.709626\pi$
$692$	− 645106.i	− 1.34716i
$693$	0	0
$694$	527140.	1.09448
$695$	29842.6i	0.0617828i
$696$	0	0
$697$	692936.	1.42635
$698$	− 1.50153e6i	− 3.08194i
$699$	0	0
$700$	596964.	1.21829
$701$	− 962035.i	− 1.95774i	−0.204486	−	0.978869i	$-0.565552\pi$
$701$	− 962035.i	− 1.95774i	0.204486	−	0.978869i	$-0.434448\pi$
$702$	0	0
$703$	28587.9	0.0578457
$704$	528.653i	0.00106666i
$705$	0	0
$706$	−294094.	−0.590034
$707$	335888.i	0.671978i
$708$	0	0
$709$	331501.	0.659466	0.329733	−	0.944074i	$-0.393041\pi$
$709$	331501.	0.659466	0.329733	+	0.944074i	$0.393041\pi$
$710$	− 851134.i	− 1.68842i
$711$	0	0
$712$	−175499.	−0.346190
$713$	30912.2i	0.0608066i
$714$	0	0
$715$	3292.11	0.00643964
$716$	− 913431.i	− 1.78176i
$717$	0	0
$718$	1.19786e6	2.32358
$719$	− 900767.i	− 1.74243i	−0.490903	−	0.871214i	$-0.663333\pi$
$719$	− 900767.i	− 1.74243i	0.490903	−	0.871214i	$-0.336667\pi$
$720$	0	0
$721$	−380162.	−0.731305
$722$	900263.i	1.72701i
$723$	0	0
$724$	250558.	0.478005
$725$	− 158449.i	− 0.301449i
$726$	0	0
$727$	−847695.	−1.60388	−0.801938	−	0.597407i	$-0.796198\pi$
$727$	−847695.	−1.60388	−0.801938	+	0.597407i	$0.796198\pi$
$728$	1.07170e6i	2.02214i
$729$	0	0
$730$	−753066.	−1.41315
$731$	− 97232.7i	− 0.181961i
$732$	0	0
$733$	166042.	0.309037	0.154518	−	0.987990i	$-0.450617\pi$
$733$	166042.	0.309037	0.154518	+	0.987990i	$0.450617\pi$
$734$	403996.i	0.749868i
$735$	0	0
$736$	60285.7	0.111291
$737$	− 3449.40i	− 0.00635050i
$738$	0	0
$739$	724070.	1.32584	0.662921	−	0.748690i	$-0.269317\pi$
$739$	724070.	1.32584	0.662921	+	0.748690i	$0.269317\pi$
$740$	− 223223.i	− 0.407639i
$741$	0	0
$742$	−275805.	−0.500951
$743$	665006.i	1.20461i	0.798265	+	0.602307i	$0.205752\pi$
$743$	665006.i	1.20461i	−0.798265	+	0.602307i	$0.794248\pi$
$744$	0	0
$745$	−139133.	−0.250678
$746$	− 351836.i	− 0.632212i
$747$	0	0
$748$	16145.9	0.0288575
$749$	174558.i	0.311155i
$750$	0	0
$751$	380983.	0.675501	0.337750	−	0.941236i	$-0.390334\pi$
$751$	380983.	0.675501	0.337750	+	0.941236i	$0.390334\pi$
$752$	− 995640.i	− 1.76062i
$753$	0	0
$754$	516473.	0.908458
$755$	311674.i	0.546772i
$756$	0	0
$757$	−459466.	−0.801791	−0.400896	−	0.916124i	$-0.631301\pi$
$757$	−459466.	−0.801791	−0.400896	+	0.916124i	$0.631301\pi$
$758$	276725.i	0.481626i
$759$	0	0
$760$	−154916.	−0.268206
$761$	331477.i	0.572380i	0.958173	+	0.286190i	$0.0923889\pi$
$761$	331477.i	0.572380i	−0.958173	+	0.286190i	$0.907611\pi$
$762$	0	0
$763$	−784788.	−1.34804
$764$	− 144263.i	− 0.247154i
$765$	0	0
$766$	1.33348e6	2.27263
$767$	− 78956.8i	− 0.134214i
$768$	0	0
$769$	97668.8	0.165159	0.0825797	−	0.996584i	$-0.473684\pi$
$769$	97668.8	0.165159	0.0825797	+	0.996584i	$0.473684\pi$
$770$	− 5925.25i	− 0.00999367i
$771$	0	0
$772$	714998.	1.19969
$773$	915611.i	1.53233i	0.642645	+	0.766164i	$0.277837\pi$
$773$	915611.i	1.53233i	−0.642645	+	0.766164i	$0.722163\pi$
$774$	0	0
$775$	182214.	0.303374
$776$	1.84846e6i	3.06964i
$777$	0	0
$778$	−1.08750e6	−1.79668
$779$	− 131746.i	− 0.217101i
$780$	0	0
$781$	9289.16	0.0152291
$782$	174315.i	0.285050i
$783$	0	0
$784$	−219551.	−0.357193
$785$	− 419925.i	− 0.681448i
$786$	0	0
$787$	−599735.	−0.968300	−0.484150	−	0.874985i	$-0.660871\pi$
$787$	−599735.	−0.968300	−0.484150	+	0.874985i	$0.660871\pi$
$788$	1.35637e6i	2.18436i
$789$	0	0
$790$	140896.	0.225759
$791$	197919.i	0.316326i
$792$	0	0
$793$	−125264.	−0.199195
$794$	− 50136.7i	− 0.0795271i
$795$	0	0
$796$	−904781.	−1.42796
$797$	− 602942.i	− 0.949203i	−0.880201	−	0.474601i	$-0.842592\pi$
$797$	− 602942.i	− 0.949203i	0.880201	−	0.474601i	$-0.157408\pi$
$798$	0	0
$799$	837668.	1.31213
$800$	− 355358.i	− 0.555247i
$801$	0	0
$802$	1.08171e6	1.68175
$803$	− 8218.86i	− 0.0127462i
$804$	0	0
$805$	44140.9	0.0681161
$806$	593935.i	0.914258i
$807$	0	0
$808$	1.08457e6	1.66125
$809$	− 862359.i	− 1.31762i	−0.752308	−	0.658811i	$-0.771060\pi$
$809$	− 862359.i	− 1.31762i	0.752308	−	0.658811i	$-0.228940\pi$
$810$	0	0
$811$	−1.04152e6	−1.58352	−0.791762	−	0.610829i	$-0.790836\pi$
$811$	−1.04152e6	−1.58352	−0.791762	+	0.610829i	$0.790836\pi$
$812$	− 641420.i	− 0.972815i
$813$	0	0
$814$	3530.65	0.00532852
$815$	762490.i	1.14794i
$816$	0	0
$817$	−18486.5	−0.0276956
$818$	1.70082e6i	2.54186i
$819$	0	0
$820$	−1.02871e6	−1.52991
$821$	939216.i	1.39341i	0.717358	+	0.696705i	$0.245351\pi$
$821$	939216.i	1.39341i	−0.717358	+	0.696705i	$0.754649\pi$
$822$	0	0
$823$	415137.	0.612903	0.306451	−	0.951886i	$-0.400858\pi$
$823$	415137.	0.612903	0.306451	+	0.951886i	$0.400858\pi$
$824$	1.22753e6i	1.80791i
$825$	0	0
$826$	−142109.	−0.208287
$827$	108681.i	0.158907i	0.996839	+	0.0794537i	$0.0253176\pi$
$827$	108681.i	0.158907i	−0.996839	+	0.0794537i	$0.974682\pi$
$828$	0	0
$829$	−289557.	−0.421332	−0.210666	−	0.977558i	$-0.567563\pi$
$829$	−289557.	−0.421332	−0.210666	+	0.977558i	$0.567563\pi$
$830$	275688.i	0.400187i
$831$	0	0
$832$	−75668.5	−0.109312
$833$	− 184716.i	− 0.266204i
$834$	0	0
$835$	170355.	0.244334
$836$	− 3069.76i	− 0.00439230i
$837$	0	0
$838$	−1.73951e6	−2.47708
$839$	314352.i	0.446573i	0.974753	+	0.223287i	$0.0716786\pi$
$839$	314352.i	0.446573i	−0.974753	+	0.223287i	$0.928321\pi$
$840$	0	0
$841$	537032.	0.759291
$842$	− 58773.2i	− 0.0829001i
$843$	0	0
$844$	1.61294e6	2.26430
$845$	27841.8i	0.0389928i
$846$	0	0
$847$	−638965.	−0.890657
$848$	389344.i	0.541429i
$849$	0	0
$850$	1.02751e6	1.42216
$851$	26302.1i	0.0363188i
$852$	0	0
$853$	−127995.	−0.175912	−0.0879562	−	0.996124i	$-0.528034\pi$
$853$	−127995.	−0.175912	−0.0879562	+	0.996124i	$0.528034\pi$
$854$	225454.i	0.309130i
$855$	0	0
$856$	563641.	0.769228
$857$	− 756967.i	− 1.03066i	−0.856992	−	0.515330i	$-0.827669\pi$
$857$	− 756967.i	− 1.03066i	0.856992	−	0.515330i	$-0.172331\pi$
$858$	0	0
$859$	974149.	1.32020	0.660099	−	0.751178i	$-0.270514\pi$
$859$	974149.	1.32020	0.660099	+	0.751178i	$0.270514\pi$
$860$	144349.i	0.195171i
$861$	0	0
$862$	−2.02821e6	−2.72959
$863$	641927.i	0.861914i	0.902372	+	0.430957i	$0.141824\pi$
$863$	641927.i	0.861914i	−0.902372	+	0.430957i	$0.858176\pi$
$864$	0	0
$865$	−281175.	−0.375789
$866$	1.11100e6i	1.48142i
$867$	0	0
$868$	737622.	0.979026
$869$	1537.72i	0.00203628i
$870$	0	0
$871$	493728.	0.650805
$872$	2.53405e6i	3.33259i
$873$	0	0
$874$	33141.9	0.0433865
$875$	− 683665.i	− 0.892950i
$876$	0	0
$877$	239271.	0.311093	0.155547	−	0.987829i	$-0.450286\pi$
$877$	239271.	0.311093	0.155547	+	0.987829i	$0.450286\pi$
$878$	− 1.51324e6i	− 1.96300i
$879$	0	0
$880$	−8364.45	−0.0108012
$881$	− 1.31329e6i	− 1.69203i	−0.533159	−	0.846015i	$-0.678995\pi$
$881$	− 1.31329e6i	− 1.69203i	0.533159	−	0.846015i	$-0.321005\pi$
$882$	0	0
$883$	775089.	0.994100	0.497050	−	0.867722i	$-0.334417\pi$
$883$	775089.	0.994100	0.497050	+	0.867722i	$0.334417\pi$
$884$	2.31103e6i	2.95734i
$885$	0	0
$886$	1.62231e6	2.06665
$887$	542678.i	0.689755i	0.938648	+	0.344878i	$0.112080\pi$
$887$	542678.i	0.689755i	−0.938648	+	0.344878i	$0.887920\pi$
$888$	0	0
$889$	24790.0	0.0313670
$890$	138884.i	0.175336i
$891$	0	0
$892$	−303388.	−0.381301
$893$	− 159263.i	− 0.199716i
$894$	0	0
$895$	−398126.	−0.497021
$896$	782423.i	0.974598i
$897$	0	0
$898$	24263.9	0.0300890
$899$	− 195783.i	− 0.242246i
$900$	0	0
$901$	−327569.	−0.403509
$902$	− 16270.8i	− 0.0199984i
$903$	0	0
$904$	639074.	0.782013
$905$	− 109208.i	− 0.133339i
$906$	0	0
$907$	−1.26966e6	−1.54338	−0.771691	−	0.635998i	$-0.780589\pi$
$907$	−1.26966e6	−1.54338	−0.771691	+	0.635998i	$0.780589\pi$
$908$	3.00234e6i	3.64157i
$909$	0	0
$910$	848107.	1.02416
$911$	223102.i	0.268823i	0.990926	+	0.134412i	$0.0429145\pi$
$911$	223102.i	0.268823i	−0.990926	+	0.134412i	$0.957086\pi$
$912$	0	0
$913$	−3008.82	−0.00360957
$914$	1.60777e6i	1.92456i
$915$	0	0
$916$	−2.17473e6	−2.59188
$917$	557636.i	0.663150i
$918$	0	0
$919$	−207173.	−0.245303	−0.122652	−	0.992450i	$-0.539140\pi$
$919$	−207173.	−0.245303	−0.122652	+	0.992450i	$0.539140\pi$
$920$	− 142529.i	− 0.168395i
$921$	0	0
$922$	1.71191e6	2.01381
$923$	1.32960e6i	1.56069i
$924$	0	0
$925$	155039.	0.181200
$926$	− 2.42915e6i	− 2.83290i
$927$	0	0
$928$	−381821.	−0.443368
$929$	− 307542.i	− 0.356347i	−0.983999	−	0.178174i	$-0.942981\pi$
$929$	− 307542.i	− 0.356347i	0.983999	−	0.178174i	$-0.0570188\pi$
$930$	0	0
$931$	−35119.5	−0.0405180
$932$	− 914250.i	− 1.05253i
$933$	0	0
$934$	876551.	1.00481
$935$	− 7037.32i	− 0.00804978i
$936$	0	0
$937$	1.16285e6	1.32448	0.662238	−	0.749294i	$-0.269607\pi$
$937$	1.16285e6	1.32448	0.662238	+	0.749294i	$0.269607\pi$
$938$	− 888628.i	− 1.00998i
$939$	0	0
$940$	−1.24358e6	−1.40740
$941$	− 1.28513e6i	− 1.45133i	−0.688047	−	0.725666i	$-0.741532\pi$
$941$	− 1.28513e6i	− 1.45133i	0.688047	−	0.725666i	$-0.258468\pi$
$942$	0	0
$943$	121212.	0.136308
$944$	200610.i	0.225117i
$945$	0	0
$946$	−2283.12	−0.00255121
$947$	1.30996e6i	1.46069i	0.683076	+	0.730347i	$0.260642\pi$
$947$	1.30996e6i	1.46069i	−0.683076	+	0.730347i	$0.739358\pi$
$948$	0	0
$949$	1.17640e6	1.30624
$950$	− 195357.i	− 0.216462i
$951$	0	0
$952$	2.29091e6	2.52775
$953$	365773.i	0.402741i	0.979515	+	0.201370i	$0.0645395\pi$
$953$	365773.i	0.402741i	−0.979515	+	0.201370i	$0.935461\pi$
$954$	0	0
$955$	−62878.0	−0.0689433
$956$	435999.i	0.477057i
$957$	0	0
$958$	209965.	0.228779
$959$	− 692129.i	− 0.752575i
$960$	0	0
$961$	−698374.	−0.756208
$962$	505359.i	0.546071i
$963$	0	0
$964$	−678950.	−0.730606
$965$	− 311638.i	− 0.334653i
$966$	0	0
$967$	1.16976e6	1.25096	0.625478	−	0.780241i	$-0.284904\pi$
$967$	1.16976e6	1.25096	0.625478	+	0.780241i	$0.284904\pi$
$968$	2.06319e6i	2.20186i
$969$	0	0
$970$	1.46281e6	1.55469
$971$	− 1.72227e6i	− 1.82668i	−0.407195	−	0.913341i	$-0.633493\pi$
$971$	− 1.72227e6i	− 1.82668i	0.407195	−	0.913341i	$-0.366507\pi$
$972$	0	0
$973$	−83905.9	−0.0886271
$974$	280043.i	0.295193i
$975$	0	0
$976$	318264.	0.334109
$977$	− 630328.i	− 0.660355i	−0.943919	−	0.330177i	$-0.892891\pi$
$977$	− 630328.i	− 0.660355i	0.943919	−	0.330177i	$-0.107109\pi$
$978$	0	0
$979$	−1515.76	−0.00158148
$980$	274224.i	0.285531i
$981$	0	0
$982$	−1.19305e6	−1.23719
$983$	1.45145e6i	1.50209i	0.660253	+	0.751043i	$0.270449\pi$
$983$	1.45145e6i	1.50209i	−0.660253	+	0.751043i	$0.729551\pi$
$984$	0	0
$985$	591183.	0.609326
$986$	− 1.10403e6i	− 1.13560i
$987$	0	0
$988$	439389.	0.450127
$989$	− 17008.4i	− 0.0173889i
$990$	0	0
$991$	1.85789e6	1.89179	0.945897	−	0.324467i	$-0.105185\pi$
$991$	1.85789e6	1.89179	0.945897	+	0.324467i	$0.105185\pi$
$992$	− 439088.i	− 0.446199i
$993$	0	0
$994$	2.39306e6	2.42204
$995$	394357.i	0.398330i
$996$	0	0
$997$	−1.42325e6	−1.43182	−0.715912	−	0.698191i	$-0.753989\pi$
$997$	−1.42325e6	−1.43182	−0.715912	+	0.698191i	$0.753989\pi$
$998$	2.01049e6i	2.01856i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.5	✓	76
3.2	odd	2	inner	531.5.b.a.296.72	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.5	✓	76	1.1	even	1	trivial
531.5.b.a.296.72	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)

\(f(q)\)	\(=\)	\(q-7.18445i q^{2} -35.6164 q^{4} +15.5237i q^{5} -43.6466 q^{7} +140.933i q^{8} +O(q^{10})\)	\(q-7.18445i q^{2} -35.6164 q^{4} +15.5237i q^{5} -43.6466 q^{7} +140.933i q^{8} +111.529 q^{10} +1.21721i q^{11} -174.225 q^{13} +313.577i q^{14} +442.665 q^{16} +372.430i q^{17} +70.8089 q^{19} -552.898i q^{20} +8.74502 q^{22} +65.1472i q^{23} +384.015 q^{25} +1251.71i q^{26} +1554.53 q^{28} -412.612i q^{29} +474.497 q^{31} -925.376i q^{32} +2675.71 q^{34} -677.557i q^{35} +403.733 q^{37} -508.723i q^{38} -2187.80 q^{40} -1860.58i q^{41} -261.076 q^{43} -43.3528i q^{44} +468.047 q^{46} -2249.20i q^{47} -495.975 q^{49} -2758.94i q^{50} +6205.28 q^{52} +879.546i q^{53} -18.8957 q^{55} -6151.25i q^{56} -2964.39 q^{58} +453.188i q^{59} +718.974 q^{61} -3409.00i q^{62} +434.314 q^{64} -2704.62i q^{65} -2833.84 q^{67} -13264.6i q^{68} -4867.87 q^{70} -7631.49i q^{71} -6752.18 q^{73} -2900.60i q^{74} -2521.96 q^{76} -53.1273i q^{77} +1263.31 q^{79} +6871.79i q^{80} -13367.2 q^{82} +2471.89i q^{83} -5781.49 q^{85} +1875.69i q^{86} -171.546 q^{88} +1245.27i q^{89} +7604.35 q^{91} -2320.31i q^{92} -16159.2 q^{94} +1099.22i q^{95} +13115.9 q^{97} +3563.31i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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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