LMFDB - Embedded newform 531.5.b.a.296.39

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			296.39
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0693431i q^{2} +15.9952 q^{4} +17.8774i q^{5} +33.2548 q^{7} +2.21865i q^{8} +O(q^{10})$	\(q+0.0693431i q^{2} +15.9952 q^{4} +17.8774i q^{5} +33.2548 q^{7} +2.21865i q^{8} -1.23968 q^{10} +97.9283i q^{11} +231.659 q^{13} +2.30599i q^{14} +255.769 q^{16} -451.486i q^{17} -70.8782 q^{19} +285.953i q^{20} -6.79066 q^{22} -161.063i q^{23} +305.398 q^{25} +16.0639i q^{26} +531.917 q^{28} +775.727i q^{29} -502.632 q^{31} +53.2342i q^{32} +31.3074 q^{34} +594.510i q^{35} +1357.98 q^{37} -4.91492i q^{38} -39.6637 q^{40} +827.853i q^{41} +1753.74 q^{43} +1566.38i q^{44} +11.1686 q^{46} -1739.53i q^{47} -1295.12 q^{49} +21.1773i q^{50} +3705.43 q^{52} -516.094i q^{53} -1750.70 q^{55} +73.7806i q^{56} -53.7914 q^{58} -453.188i q^{59} -453.778 q^{61} -34.8541i q^{62} +4088.62 q^{64} +4141.46i q^{65} -8498.61 q^{67} -7221.60i q^{68} -41.2252 q^{70} +5832.47i q^{71} -3346.36 q^{73} +94.1663i q^{74} -1133.71 q^{76} +3256.58i q^{77} +6626.76 q^{79} +4572.49i q^{80} -57.4059 q^{82} +2434.22i q^{83} +8071.40 q^{85} +121.610i q^{86} -217.268 q^{88} +10343.2i q^{89} +7703.76 q^{91} -2576.24i q^{92} +120.625 q^{94} -1267.12i q^{95} +5154.24 q^{97} -89.8077i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0693431i	0.0173358i	0.999962	+	0.00866789i	$0.00275911\pi$
$2$	0.0693431i	0.0173358i	−0.999962	+	0.00866789i	$0.997241\pi$
$3$	0	0
$3$	0	0
$4$	15.9952	0.999699
$5$	17.8774i	0.715097i	0.933895	+	0.357548i	$0.116387\pi$
$5$	17.8774i	0.715097i	−0.933895	+	0.357548i	$0.883613\pi$
$6$	0	0
$7$	33.2548	0.678669	0.339335	−	0.940666i	$-0.389798\pi$
$7$	33.2548	0.678669	0.339335	+	0.940666i	$0.389798\pi$
$8$	2.21865i	0.0346664i
$9$	0	0
$10$	−1.23968	−0.0123968
$11$	97.9283i	0.809325i	0.914466	+	0.404662i	$0.132611\pi$
$11$	97.9283i	0.809325i	−0.914466	+	0.404662i	$0.867389\pi$
$12$	0	0
$13$	231.659	1.37076	0.685381	−	0.728185i	$-0.259636\pi$
$13$	231.659	1.37076	0.685381	+	0.728185i	$0.259636\pi$
$14$	2.30599i	0.0117653i
$15$	0	0
$16$	255.769	0.999099
$17$	− 451.486i	− 1.56223i	−0.624385	−	0.781117i	$-0.714650\pi$
$17$	− 451.486i	− 1.56223i	0.624385	−	0.781117i	$-0.285350\pi$
$18$	0	0
$19$	−70.8782	−0.196338	−0.0981692	−	0.995170i	$-0.531299\pi$
$19$	−70.8782	−0.196338	−0.0981692	+	0.995170i	$0.531299\pi$
$20$	285.953i	0.714882i
$21$	0	0
$22$	−6.79066	−0.0140303
$23$	− 161.063i	− 0.304467i	−0.988345	−	0.152234i	$-0.951353\pi$
$23$	− 161.063i	− 0.304467i	0.988345	−	0.152234i	$-0.0486466\pi$
$24$	0	0
$25$	305.398	0.488637
$26$	16.0639i	0.0237632i
$27$	0	0
$28$	531.917	0.678465
$29$	775.727i	0.922387i	0.887300	+	0.461193i	$0.152579\pi$
$29$	775.727i	0.922387i	−0.887300	+	0.461193i	$0.847421\pi$
$30$	0	0
$31$	−502.632	−0.523031	−0.261515	−	0.965199i	$-0.584222\pi$
$31$	−502.632	−0.523031	−0.261515	+	0.965199i	$0.584222\pi$
$32$	53.2342i	0.0519865i
$33$	0	0
$34$	31.3074	0.0270826
$35$	594.510i	0.485314i
$36$	0	0
$37$	1357.98	0.991947	0.495973	−	0.868338i	$-0.334811\pi$
$37$	1357.98	0.991947	0.495973	+	0.868338i	$0.334811\pi$
$38$	− 4.91492i	− 0.00340368i
$39$	0	0
$40$	−39.6637	−0.0247898
$41$	827.853i	0.492476i	0.969209	+	0.246238i	$0.0791946\pi$
$41$	827.853i	0.492476i	−0.969209	+	0.246238i	$0.920805\pi$
$42$	0	0
$43$	1753.74	0.948482	0.474241	−	0.880395i	$-0.342723\pi$
$43$	1753.74	0.948482	0.474241	+	0.880395i	$0.342723\pi$
$44$	1566.38i	0.809082i
$45$	0	0
$46$	11.1686	0.00527818
$47$	− 1739.53i	− 0.787476i	−0.919223	−	0.393738i	$-0.871182\pi$
$47$	− 1739.53i	− 0.787476i	0.919223	−	0.393738i	$-0.128818\pi$
$48$	0	0
$49$	−1295.12	−0.539408
$50$	21.1773i	0.00847091i
$51$	0	0
$52$	3705.43	1.37035
$53$	− 516.094i	− 0.183729i	−0.995772	−	0.0918643i	$-0.970717\pi$
$53$	− 516.094i	− 0.183729i	0.995772	−	0.0918643i	$-0.0292826\pi$
$54$	0	0
$55$	−1750.70	−0.578745
$56$	73.7806i	0.0235270i
$57$	0	0
$58$	−53.7914	−0.0159903
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	−453.778	−0.121950	−0.0609752	−	0.998139i	$-0.519421\pi$
$61$	−453.778	−0.121950	−0.0609752	+	0.998139i	$0.519421\pi$
$62$	− 34.8541i	− 0.00906715i
$63$	0	0
$64$	4088.62	0.998197
$65$	4141.46i	0.980227i
$66$	0	0
$67$	−8498.61	−1.89321	−0.946604	−	0.322397i	$-0.895511\pi$
$67$	−8498.61	−1.89321	−0.946604	+	0.322397i	$0.895511\pi$
$68$	− 7221.60i	− 1.56176i
$69$	0	0
$70$	−41.2252	−0.00841330
$71$	5832.47i	1.15701i	0.815680	+	0.578504i	$0.196363\pi$
$71$	5832.47i	1.15701i	−0.815680	+	0.578504i	$0.803637\pi$
$72$	0	0
$73$	−3346.36	−0.627953	−0.313977	−	0.949431i	$-0.601661\pi$
$73$	−3346.36	−0.627953	−0.313977	+	0.949431i	$0.601661\pi$
$74$	94.1663i	0.0171962i
$75$	0	0
$76$	−1133.71	−0.196279
$77$	3256.58i	0.549264i
$78$	0	0
$79$	6626.76	1.06181	0.530905	−	0.847431i	$-0.321852\pi$
$79$	6626.76	1.06181	0.530905	+	0.847431i	$0.321852\pi$
$80$	4572.49i	0.714452i
$81$	0	0
$82$	−57.4059	−0.00853747
$83$	2434.22i	0.353349i	0.984269	+	0.176675i	$0.0565340\pi$
$83$	2434.22i	0.353349i	−0.984269	+	0.176675i	$0.943466\pi$
$84$	0	0
$85$	8071.40	1.11715
$86$	121.610i	0.0164427i
$87$	0	0
$88$	−217.268	−0.0280564
$89$	10343.2i	1.30579i	0.757449	+	0.652894i	$0.226445\pi$
$89$	10343.2i	1.30579i	−0.757449	+	0.652894i	$0.773555\pi$
$90$	0	0
$91$	7703.76	0.930294
$92$	− 2576.24i	− 0.304376i
$93$	0	0
$94$	120.625	0.0136515
$95$	− 1267.12i	− 0.140401i
$96$	0	0
$97$	5154.24	0.547799	0.273899	−	0.961758i	$-0.411686\pi$
$97$	5154.24	0.547799	0.273899	+	0.961758i	$0.411686\pi$
$98$	− 89.8077i	− 0.00935107i
$99$	0	0
$100$	4884.90	0.488490
$101$	10768.8i	1.05566i	0.849350	+	0.527830i	$0.176994\pi$
$101$	10768.8i	1.05566i	−0.849350	+	0.527830i	$0.823006\pi$
$102$	0	0
$103$	−2728.82	−0.257218	−0.128609	−	0.991695i	$-0.541051\pi$
$103$	−2728.82	−0.257218	−0.128609	+	0.991695i	$0.541051\pi$
$104$	513.969i	0.0475193i
$105$	0	0
$106$	35.7876	0.00318508
$107$	− 15714.4i	− 1.37256i	−0.727338	−	0.686279i	$-0.759243\pi$
$107$	− 15714.4i	− 1.37256i	0.727338	−	0.686279i	$-0.240757\pi$
$108$	0	0
$109$	13882.4	1.16845	0.584227	−	0.811590i	$-0.301398\pi$
$109$	13882.4	1.16845	0.584227	+	0.811590i	$0.301398\pi$
$110$	− 121.399i	− 0.0100330i
$111$	0	0
$112$	8505.55	0.678057
$113$	− 17770.3i	− 1.39167i	−0.718201	−	0.695836i	$-0.755034\pi$
$113$	− 17770.3i	− 1.39167i	0.718201	−	0.695836i	$-0.244966\pi$
$114$	0	0
$115$	2879.39	0.217723
$116$	12407.9i	0.922110i
$117$	0	0
$118$	31.4255	0.00225693
$119$	− 15014.1i	− 1.06024i
$120$	0	0
$121$	5051.05	0.344993
$122$	− 31.4664i	− 0.00211411i
$123$	0	0
$124$	−8039.70	−0.522873
$125$	16633.1i	1.06452i
$126$	0	0
$127$	21946.7	1.36070	0.680350	−	0.732887i	$-0.261828\pi$
$127$	21946.7	1.36070	0.680350	+	0.732887i	$0.261828\pi$
$128$	1135.26i	0.0692911i
$129$	0	0
$130$	−287.182	−0.0169930
$131$	6710.47i	0.391030i	0.980701	+	0.195515i	$0.0626379\pi$
$131$	6710.47i	0.391030i	−0.980701	+	0.195515i	$0.937362\pi$
$132$	0	0
$133$	−2357.04	−0.133249
$134$	− 589.321i	− 0.0328203i
$135$	0	0
$136$	1001.69	0.0541570
$137$	15294.2i	0.814867i	0.913235	+	0.407433i	$0.133576\pi$
$137$	15294.2i	0.814867i	−0.913235	+	0.407433i	$0.866424\pi$
$138$	0	0
$139$	19274.5	0.997594	0.498797	−	0.866719i	$-0.333775\pi$
$139$	19274.5	0.997594	0.498797	+	0.866719i	$0.333775\pi$
$140$	9509.29i	0.485168i
$141$	0	0
$142$	−404.442	−0.0200576
$143$	22686.0i	1.10939i
$144$	0	0
$145$	−13868.0	−0.659596
$146$	− 232.047i	− 0.0108861i
$147$	0	0
$148$	21721.1	0.991649
$149$	28141.0i	1.26756i	0.773515	+	0.633778i	$0.218497\pi$
$149$	28141.0i	1.26756i	−0.773515	+	0.633778i	$0.781503\pi$
$150$	0	0
$151$	−6333.54	−0.277774	−0.138887	−	0.990308i	$-0.544353\pi$
$151$	−6333.54	−0.277774	−0.138887	+	0.990308i	$0.544353\pi$
$152$	− 157.254i	− 0.00680634i
$153$	0	0
$154$	−225.822	−0.00952192
$155$	− 8985.77i	− 0.374017i
$156$	0	0
$157$	13288.3	0.539102	0.269551	−	0.962986i	$-0.413125\pi$
$157$	13288.3	0.539102	0.269551	+	0.962986i	$0.413125\pi$
$158$	459.520i	0.0184073i
$159$	0	0
$160$	−951.690	−0.0371754
$161$	− 5356.12i	− 0.206632i
$162$	0	0
$163$	3434.63	0.129272	0.0646360	−	0.997909i	$-0.479411\pi$
$163$	3434.63	0.129272	0.0646360	+	0.997909i	$0.479411\pi$
$164$	13241.7i	0.492328i
$165$	0	0
$166$	−168.797	−0.00612559
$167$	− 3311.72i	− 0.118746i	−0.998236	−	0.0593732i	$-0.981090\pi$
$167$	− 3311.72i	− 0.118746i	0.998236	−	0.0593732i	$-0.0189102\pi$
$168$	0	0
$169$	25104.8	0.878988
$170$	559.696i	0.0193666i
$171$	0	0
$172$	28051.4	0.948197
$173$	436.587i	0.0145874i	0.999973	+	0.00729371i	$0.00232168\pi$
$173$	436.587i	0.0145874i	−0.999973	+	0.00729371i	$0.997678\pi$
$174$	0	0
$175$	10155.9	0.331623
$176$	25047.0i	0.808595i
$177$	0	0
$178$	−717.227	−0.0226369
$179$	910.807i	0.0284263i	0.999899	+	0.0142131i	$0.00452434\pi$
$179$	910.807i	0.0284263i	−0.999899	+	0.0142131i	$0.995476\pi$
$180$	0	0
$181$	−15307.2	−0.467238	−0.233619	−	0.972328i	$-0.575057\pi$
$181$	−15307.2	−0.467238	−0.233619	+	0.972328i	$0.575057\pi$
$182$	534.203i	0.0161274i
$183$	0	0
$184$	357.342	0.0105548
$185$	24277.1i	0.709338i
$186$	0	0
$187$	44213.2	1.26436
$188$	− 27824.2i	− 0.787239i
$189$	0	0
$190$	87.8660	0.00243396
$191$	56.6355i	0.00155247i	1.00000		0.000776233i	$0.000247083\pi$
$191$	56.6355i	0.00155247i	−1.00000		0.000776233i	$0.999753\pi$
$192$	0	0
$193$	−25737.6	−0.690961	−0.345481	−	0.938426i	$-0.612284\pi$
$193$	−25737.6	−0.690961	−0.345481	+	0.938426i	$0.612284\pi$
$194$	357.411i	0.00949652i
$195$	0	0
$196$	−20715.7	−0.539246
$197$	2547.14i	0.0656326i	0.999461	+	0.0328163i	$0.0104476\pi$
$197$	2547.14i	0.0656326i	−0.999461	+	0.0328163i	$0.989552\pi$
$198$	0	0
$199$	−55707.0	−1.40671	−0.703353	−	0.710840i	$-0.748315\pi$
$199$	−55707.0	−1.40671	−0.703353	+	0.710840i	$0.748315\pi$
$200$	677.571i	0.0169393i
$201$	0	0
$202$	−746.742	−0.0183007
$203$	25796.6i	0.625995i
$204$	0	0
$205$	−14799.9	−0.352168
$206$	− 189.225i	− 0.00445907i
$207$	0	0
$208$	59251.2	1.36953
$209$	− 6940.98i	− 0.158902i
$210$	0	0
$211$	−37691.6	−0.846603	−0.423301	−	0.905989i	$-0.639129\pi$
$211$	−37691.6	−0.846603	−0.423301	+	0.905989i	$0.639129\pi$
$212$	− 8255.02i	− 0.183673i
$213$	0	0
$214$	1089.69	0.0237944
$215$	31352.4i	0.678256i
$216$	0	0
$217$	−16714.9	−0.354965
$218$	962.649i	0.0202561i
$219$	0	0
$220$	−28002.9	−0.578571
$221$	− 104591.i	− 2.14145i
$222$	0	0
$223$	−73270.2	−1.47339	−0.736695	−	0.676225i	$-0.763615\pi$
$223$	−73270.2	−1.47339	−0.736695	+	0.676225i	$0.763615\pi$
$224$	1770.29i	0.0352816i
$225$	0	0
$226$	1232.25	0.0241257
$227$	− 16511.5i	− 0.320431i	−0.987082	−	0.160216i	$-0.948781\pi$
$227$	− 16511.5i	− 0.320431i	0.987082	−	0.160216i	$-0.0512189\pi$
$228$	0	0
$229$	−2087.46	−0.0398059	−0.0199030	−	0.999802i	$-0.506336\pi$
$229$	−2087.46	−0.0398059	−0.0199030	+	0.999802i	$0.506336\pi$
$230$	199.666i	0.00377441i
$231$	0	0
$232$	−1721.07	−0.0319758
$233$	− 18658.1i	− 0.343681i	−0.985125	−	0.171840i	$-0.945029\pi$
$233$	− 18658.1i	− 0.343681i	0.985125	−	0.171840i	$-0.0549713\pi$
$234$	0	0
$235$	31098.4	0.563121
$236$	− 7248.82i	− 0.130150i
$237$	0	0
$238$	1041.12	0.0183801
$239$	− 54862.6i	− 0.960462i	−0.877142	−	0.480231i	$-0.840553\pi$
$239$	− 54862.6i	− 0.960462i	0.877142	−	0.480231i	$-0.159447\pi$
$240$	0	0
$241$	−98742.7	−1.70009	−0.850043	−	0.526713i	$-0.823424\pi$
$241$	−98742.7	−1.70009	−0.850043	+	0.526713i	$0.823424\pi$
$242$	350.255i	0.00598073i
$243$	0	0
$244$	−7258.26	−0.121914
$245$	− 23153.4i	− 0.385729i
$246$	0	0
$247$	−16419.6	−0.269133
$248$	− 1115.16i	− 0.0181316i
$249$	0	0
$250$	−1153.39	−0.0184543
$251$	− 66987.6i	− 1.06328i	−0.846971	−	0.531639i	$-0.821576\pi$
$251$	− 66987.6i	− 1.06328i	0.846971	−	0.531639i	$-0.178424\pi$
$252$	0	0
$253$	15772.6	0.246413
$254$	1521.86i	0.0235888i
$255$	0	0
$256$	65339.1	0.996996
$257$	− 92088.8i	− 1.39425i	−0.716949	−	0.697125i	$-0.754462\pi$
$257$	− 92088.8i	− 1.39425i	0.716949	−	0.697125i	$-0.245538\pi$
$258$	0	0
$259$	45159.2	0.673204
$260$	66243.4i	0.979933i
$261$	0	0
$262$	−465.325	−0.00677882
$263$	− 1564.33i	− 0.0226160i	−0.999936	−	0.0113080i	$-0.996400\pi$
$263$	− 1564.33i	− 0.0226160i	0.999936	−	0.0113080i	$-0.00359952\pi$
$264$	0	0
$265$	9226.42	0.131384
$266$	− 163.444i	− 0.00230997i
$267$	0	0
$268$	−135937.	−1.89264
$269$	− 36825.8i	− 0.508917i	−0.967084	−	0.254459i	$-0.918103\pi$
$269$	− 36825.8i	− 0.508917i	0.967084	−	0.254459i	$-0.0818972\pi$
$270$	0	0
$271$	23521.2	0.320273	0.160137	−	0.987095i	$-0.448807\pi$
$271$	23521.2	0.320273	0.160137	+	0.987095i	$0.448807\pi$
$272$	− 115476.i	− 1.56083i
$273$	0	0
$274$	−1060.55	−0.0141264
$275$	29907.1i	0.395466i
$276$	0	0
$277$	24795.2	0.323153	0.161577	−	0.986860i	$-0.448342\pi$
$277$	24795.2	0.323153	0.161577	+	0.986860i	$0.448342\pi$
$278$	1336.56i	0.0172941i
$279$	0	0
$280$	−1319.01	−0.0168241
$281$	116182.i	1.47138i	0.677318	+	0.735690i	$0.263142\pi$
$281$	116182.i	1.47138i	−0.677318	+	0.735690i	$0.736858\pi$
$282$	0	0
$283$	−117218.	−1.46360	−0.731798	−	0.681521i	$-0.761319\pi$
$283$	−117218.	−1.46360	−0.731798	+	0.681521i	$0.761319\pi$
$284$	93291.5i	1.15666i
$285$	0	0
$286$	−1573.12	−0.0192322
$287$	27530.1i	0.334229i
$288$	0	0
$289$	−120318.	−1.44058
$290$	− 961.651i	− 0.0114346i
$291$	0	0
$292$	−53525.7	−0.627764
$293$	− 109666.i	− 1.27743i	−0.769445	−	0.638714i	$-0.779467\pi$
$293$	− 109666.i	− 1.27743i	0.769445	−	0.638714i	$-0.220533\pi$
$294$	0	0
$295$	8101.82	0.0930976
$296$	3012.87i	0.0343872i
$297$	0	0
$298$	−1951.39	−0.0219741
$299$	− 37311.7i	− 0.417352i
$300$	0	0
$301$	58320.3	0.643705
$302$	− 439.187i	− 0.00481544i
$303$	0	0
$304$	−18128.5	−0.196161
$305$	− 8112.37i	− 0.0872064i
$306$	0	0
$307$	88981.9	0.944115	0.472058	−	0.881568i	$-0.343511\pi$
$307$	88981.9	0.944115	0.472058	+	0.881568i	$0.343511\pi$
$308$	52089.7i	0.549099i
$309$	0	0
$310$	623.101	0.00648389
$311$	114635.i	1.18522i	0.805491	+	0.592608i	$0.201902\pi$
$311$	114635.i	1.18522i	−0.805491	+	0.592608i	$0.798098\pi$
$312$	0	0
$313$	119096.	1.21565	0.607827	−	0.794070i	$-0.292041\pi$
$313$	119096.	1.21565	0.607827	+	0.794070i	$0.292041\pi$
$314$	921.454i	0.00934576i
$315$	0	0
$316$	105996.	1.06149
$317$	− 160058.i	− 1.59279i	−0.604775	−	0.796397i	$-0.706737\pi$
$317$	− 160058.i	− 1.59279i	0.604775	−	0.796397i	$-0.293263\pi$
$318$	0	0
$319$	−75965.7	−0.746511
$320$	73093.9i	0.713807i
$321$	0	0
$322$	371.410	0.00358214
$323$	32000.5i	0.306727i
$324$	0	0
$325$	70748.1	0.669805
$326$	238.168i	0.00224103i
$327$	0	0
$328$	−1836.71	−0.0170724
$329$	− 57847.8i	− 0.534435i
$330$	0	0
$331$	−134843.	−1.23076	−0.615379	−	0.788231i	$-0.710997\pi$
$331$	−134843.	−1.23076	−0.615379	+	0.788231i	$0.710997\pi$
$332$	38935.9i	0.353243i
$333$	0	0
$334$	229.645	0.00205856
$335$	− 151933.i	− 1.35383i
$336$	0	0
$337$	−158850.	−1.39871	−0.699354	−	0.714775i	$-0.746529\pi$
$337$	−158850.	−1.39871	−0.699354	+	0.714775i	$0.746529\pi$
$338$	1740.84i	0.0152380i
$339$	0	0
$340$	129104.	1.11681
$341$	− 49221.9i	− 0.423302i
$342$	0	0
$343$	−122914.	−1.04475
$344$	3890.94i	0.0328804i
$345$	0	0
$346$	−30.2743	−0.000252884 0
$347$	− 97525.4i	− 0.809951i	−0.914327	−	0.404976i	$-0.867280\pi$
$347$	− 97525.4i	− 0.809951i	0.914327	−	0.404976i	$-0.132720\pi$
$348$	0	0
$349$	24598.5	0.201956	0.100978	−	0.994889i	$-0.467803\pi$
$349$	24598.5	0.201956	0.100978	+	0.994889i	$0.467803\pi$
$350$	704.245i	0.00574894i
$351$	0	0
$352$	−5213.14	−0.0420740
$353$	− 43593.6i	− 0.349843i	−0.984582	−	0.174922i	$-0.944033\pi$
$353$	− 43593.6i	− 0.349843i	0.984582	−	0.174922i	$-0.0559672\pi$
$354$	0	0
$355$	−104270.	−0.827372
$356$	165441.i	1.30540i
$357$	0	0
$358$	−63.1582	−0.000492792 0
$359$	27533.0i	0.213631i	0.994279	+	0.106816i	$0.0340655\pi$
$359$	27533.0i	0.213631i	−0.994279	+	0.106816i	$0.965935\pi$
$360$	0	0
$361$	−125297.	−0.961451
$362$	− 1061.45i	− 0.00809993i
$363$	0	0
$364$	123223.	0.930014
$365$	− 59824.3i	− 0.449047i
$366$	0	0
$367$	−190657.	−1.41554	−0.707769	−	0.706444i	$-0.750298\pi$
$367$	−190657.	−1.41554	−0.707769	+	0.706444i	$0.750298\pi$
$368$	− 41195.0i	− 0.304193i
$369$	0	0
$370$	−1683.45	−0.0122969
$371$	− 17162.6i	− 0.124691i
$372$	0	0
$373$	82522.4	0.593136	0.296568	−	0.955012i	$-0.404158\pi$
$373$	82522.4	0.593136	0.296568	+	0.955012i	$0.404158\pi$
$374$	3065.88i	0.0219186i
$375$	0	0
$376$	3859.41	0.0272989
$377$	179704.i	1.26437i
$378$	0	0
$379$	86143.5	0.599714	0.299857	−	0.953984i	$-0.403061\pi$
$379$	86143.5	0.599714	0.299857	+	0.953984i	$0.403061\pi$
$380$	− 20267.8i	− 0.140359i
$381$	0	0
$382$	−3.92728	−2.69132e−5 0
$383$	193979.i	1.32238i	0.750218	+	0.661191i	$0.229949\pi$
$383$	193979.i	1.32238i	−0.750218	+	0.661191i	$0.770051\pi$
$384$	0	0
$385$	−58219.3	−0.392777
$386$	− 1784.73i	− 0.0119784i
$387$	0	0
$388$	82443.0	0.547634
$389$	− 190614.i	− 1.25967i	−0.776731	−	0.629833i	$-0.783123\pi$
$389$	− 190614.i	− 1.25967i	0.776731	−	0.629833i	$-0.216877\pi$
$390$	0	0
$391$	−72717.7	−0.475649
$392$	− 2873.41i	− 0.0186993i
$393$	0	0
$394$	−176.626	−0.00113779
$395$	118469.i	0.759297i
$396$	0	0
$397$	−198399.	−1.25880	−0.629401	−	0.777081i	$-0.716700\pi$
$397$	−198399.	−1.25880	−0.629401	+	0.777081i	$0.716700\pi$
$398$	− 3862.90i	− 0.0243864i
$399$	0	0
$400$	78111.4	0.488196
$401$	− 272593.i	− 1.69522i	−0.530622	−	0.847608i	$-0.678042\pi$
$401$	− 272593.i	− 1.69522i	0.530622	−	0.847608i	$-0.321958\pi$
$402$	0	0
$403$	−116439.	−0.716950
$404$	172249.i	1.05534i
$405$	0	0
$406$	−1788.82	−0.0108521
$407$	132984.i	0.802807i
$408$	0	0
$409$	−278058.	−1.66222	−0.831109	−	0.556109i	$-0.812294\pi$
$409$	−278058.	−1.66222	−0.831109	+	0.556109i	$0.812294\pi$
$410$	− 1026.27i	− 0.00610511i
$411$	0	0
$412$	−43648.0	−0.257140
$413$	− 15070.7i	− 0.0883552i
$414$	0	0
$415$	−43517.6	−0.252679
$416$	12332.2i	0.0712611i
$417$	0	0
$418$	481.309	0.00275468
$419$	54413.5i	0.309941i	0.987919	+	0.154970i	$0.0495282\pi$
$419$	54413.5i	0.309941i	−0.987919	+	0.154970i	$0.950472\pi$
$420$	0	0
$421$	98005.5	0.552951	0.276475	−	0.961021i	$-0.410834\pi$
$421$	98005.5	0.552951	0.276475	+	0.961021i	$0.410834\pi$
$422$	− 2613.65i	− 0.0146765i
$423$	0	0
$424$	1145.03	0.00636920
$425$	− 137883.i	− 0.763365i
$426$	0	0
$427$	−15090.3	−0.0827640
$428$	− 251355.i	− 1.37215i
$429$	0	0
$430$	−2174.07	−0.0117581
$431$	− 205949.i	− 1.10868i	−0.832292	−	0.554338i	$-0.812972\pi$
$431$	− 205949.i	− 1.10868i	0.832292	−	0.554338i	$-0.187028\pi$
$432$	0	0
$433$	−49020.0	−0.261455	−0.130728	−	0.991418i	$-0.541731\pi$
$433$	−49020.0	−0.261455	−0.130728	+	0.991418i	$0.541731\pi$
$434$	− 1159.07i	− 0.00615359i
$435$	0	0
$436$	222052.	1.16810
$437$	11415.9i	0.0597786i
$438$	0	0
$439$	195356.	1.01367	0.506836	−	0.862043i	$-0.330815\pi$
$439$	195356.	1.01367	0.506836	+	0.862043i	$0.330815\pi$
$440$	− 3884.20i	− 0.0200630i
$441$	0	0
$442$	7252.64	0.0371237
$443$	201289.i	1.02568i	0.858483	+	0.512841i	$0.171407\pi$
$443$	201289.i	1.02568i	−0.858483	+	0.512841i	$0.828593\pi$
$444$	0	0
$445$	−184909.	−0.933765
$446$	− 5080.79i	− 0.0255424i
$447$	0	0
$448$	135966.	0.677446
$449$	− 175637.i	− 0.871211i	−0.900138	−	0.435606i	$-0.856534\pi$
$449$	− 175637.i	− 0.871211i	0.900138	−	0.435606i	$-0.143466\pi$
$450$	0	0
$451$	−81070.2	−0.398573
$452$	− 284239.i	− 1.39125i
$453$	0	0
$454$	1144.96	0.00555492
$455$	137723.i	0.665250i
$456$	0	0
$457$	−109722.	−0.525368	−0.262684	−	0.964882i	$-0.584608\pi$
$457$	−109722.	−0.525368	−0.262684	+	0.964882i	$0.584608\pi$
$458$	− 144.751i	0 0.000690067i
$459$	0	0
$460$	46056.4	0.217658
$461$	− 366048.i	− 1.72241i	−0.508257	−	0.861205i	$-0.669710\pi$
$461$	− 366048.i	− 1.72241i	0.508257	−	0.861205i	$-0.330290\pi$
$462$	0	0
$463$	344234.	1.60580	0.802901	−	0.596112i	$-0.203289\pi$
$463$	344234.	1.60580	0.802901	+	0.596112i	$0.203289\pi$
$464$	198407.i	0.921555i
$465$	0	0
$466$	1293.81	0.00595797
$467$	− 362925.i	− 1.66411i	−0.554690	−	0.832057i	$-0.687163\pi$
$467$	− 362925.i	− 1.66411i	0.554690	−	0.832057i	$-0.312837\pi$
$468$	0	0
$469$	−282620.	−1.28486
$470$	2156.46i	0.00976215i
$471$	0	0
$472$	1005.46	0.00451318
$473$	171741.i	0.767630i
$474$	0	0
$475$	−21646.1	−0.0959382
$476$	− 240153.i	− 1.05992i
$477$	0	0
$478$	3804.34	0.0166504
$479$	− 93414.0i	− 0.407138i	−0.979061	−	0.203569i	$-0.934746\pi$
$479$	− 93414.0i	− 0.407138i	0.979061	−	0.203569i	$-0.0652540\pi$
$480$	0	0
$481$	314587.	1.35972
$482$	− 6847.13i	− 0.0294723i
$483$	0	0
$484$	80792.4	0.344890
$485$	92144.4i	0.391729i
$486$	0	0
$487$	28198.6	0.118897	0.0594483	−	0.998231i	$-0.481066\pi$
$487$	28198.6	0.118897	0.0594483	+	0.998231i	$0.481066\pi$
$488$	− 1006.77i	− 0.00422758i
$489$	0	0
$490$	1605.53	0.00668692
$491$	763.533i	0.00316712i	0.999999	+	0.00158356i	$0.000504064\pi$
$491$	763.533i	0.00316712i	−0.999999	+	0.00158356i	$0.999496\pi$
$492$	0	0
$493$	350230.	1.44098
$494$	− 1138.58i	− 0.00466564i
$495$	0	0
$496$	−128558.	−0.522559
$497$	193958.i	0.785225i
$498$	0	0
$499$	−302263.	−1.21390	−0.606952	−	0.794739i	$-0.707608\pi$
$499$	−302263.	−1.21390	−0.606952	+	0.794739i	$0.707608\pi$
$500$	266050.i	1.06420i
$501$	0	0
$502$	4645.13	0.0184328
$503$	387875.i	1.53305i	0.642216	+	0.766523i	$0.278015\pi$
$503$	387875.i	1.53305i	−0.642216	+	0.766523i	$0.721985\pi$
$504$	0	0
$505$	−192518.	−0.754899
$506$	1093.72i	0.00427176i
$507$	0	0
$508$	351042.	1.36029
$509$	499623.i	1.92844i	0.265101	+	0.964221i	$0.414595\pi$
$509$	499623.i	1.92844i	−0.265101	+	0.964221i	$0.585405\pi$
$510$	0	0
$511$	−111283.	−0.426172
$512$	22695.1i	0.0865748i
$513$	0	0
$514$	6385.73	0.0241704
$515$	− 48784.3i	− 0.183936i
$516$	0	0
$517$	170350.	0.637324
$518$	3131.48i	0.0116705i
$519$	0	0
$520$	−9188.44	−0.0339809
$521$	357536.i	1.31718i	0.752503	+	0.658588i	$0.228846\pi$
$521$	357536.i	1.31718i	−0.752503	+	0.658588i	$0.771154\pi$
$522$	0	0
$523$	178973.	0.654310	0.327155	−	0.944971i	$-0.393910\pi$
$523$	178973.	0.654310	0.327155	+	0.944971i	$0.393910\pi$
$524$	107335.i	0.390913i
$525$	0	0
$526$	108.475	0.000392066 0
$527$	226931.i	0.817096i
$528$	0	0
$529$	253900.	0.907300
$530$	639.789i	0.00227764i
$531$	0	0
$532$	−37701.3	−0.133209
$533$	191779.i	0.675068i
$534$	0	0
$535$	280933.	0.981512
$536$	− 18855.4i	− 0.0656307i
$537$	0	0
$538$	2553.61	0.00882248
$539$	− 126829.i	− 0.436557i
$540$	0	0
$541$	334475.	1.14280	0.571399	−	0.820672i	$-0.306401\pi$
$541$	334475.	1.14280	0.571399	+	0.820672i	$0.306401\pi$
$542$	1631.03i	0.00555219i
$543$	0	0
$544$	24034.5	0.0812151
$545$	248181.i	0.835557i
$546$	0	0
$547$	−159416.	−0.532791	−0.266396	−	0.963864i	$-0.585833\pi$
$547$	−159416.	−0.532791	−0.266396	+	0.963864i	$0.585833\pi$
$548$	244634.i	0.814622i
$549$	0	0
$550$	−2073.85	−0.00685571
$551$	− 54982.1i	− 0.181100i
$552$	0	0
$553$	220371.	0.720618
$554$	1719.38i	0.00560212i
$555$	0	0
$556$	308300.	0.997294
$557$	1422.26i	0.00458424i	0.999997	+	0.00229212i	$0.000729605\pi$
$557$	1422.26i	0.00458424i	−0.999997	+	0.00229212i	$0.999270\pi$
$558$	0	0
$559$	406270.	1.30014
$560$	152057.i	0.484876i
$561$	0	0
$562$	−8056.40	−0.0255075
$563$	− 140333.i	− 0.442733i	−0.975191	−	0.221366i	$-0.928948\pi$
$563$	− 140333.i	− 0.442733i	0.975191	−	0.221366i	$-0.0710517\pi$
$564$	0	0
$565$	317686.	0.995180
$566$	− 8128.27i	− 0.0253726i
$567$	0	0
$568$	−12940.2	−0.0401092
$569$	− 81694.5i	− 0.252330i	−0.992009	−	0.126165i	$-0.959733\pi$
$569$	− 81694.5i	− 0.252330i	0.992009	−	0.126165i	$-0.0402668\pi$
$570$	0	0
$571$	−336620.	−1.03245	−0.516223	−	0.856454i	$-0.672662\pi$
$571$	−336620.	−1.03245	−0.516223	+	0.856454i	$0.672662\pi$
$572$	362866.i	1.10906i
$573$	0	0
$574$	−1909.02	−0.00579411
$575$	− 49188.4i	− 0.148774i
$576$	0	0
$577$	−423305.	−1.27146	−0.635729	−	0.771913i	$-0.719300\pi$
$577$	−423305.	−1.27146	−0.635729	+	0.771913i	$0.719300\pi$
$578$	− 8343.25i	− 0.0249735i
$579$	0	0
$580$	−221821.	−0.659397
$581$	80949.6i	0.239807i
$582$	0	0
$583$	50540.2	0.148696
$584$	− 7424.40i	− 0.0217688i
$585$	0	0
$586$	7604.58	0.0221452
$587$	72339.4i	0.209942i	0.994475	+	0.104971i	$0.0334749\pi$
$587$	72339.4i	0.209942i	−0.994475	+	0.104971i	$0.966525\pi$
$588$	0	0
$589$	35625.7	0.102691
$590$	561.806i	0.00161392i
$591$	0	0
$592$	347328.	0.991053
$593$	− 505864.i	− 1.43855i	−0.694726	−	0.719275i	$-0.744474\pi$
$593$	− 505864.i	− 1.43855i	0.694726	−	0.719275i	$-0.255526\pi$
$594$	0	0
$595$	268413.	0.758174
$596$	450121.i	1.26718i
$597$	0	0
$598$	2587.31	0.00723512
$599$	− 326376.i	− 0.909630i	−0.890586	−	0.454815i	$-0.849705\pi$
$599$	− 326376.i	− 0.909630i	0.890586	−	0.454815i	$-0.150295\pi$
$600$	0	0
$601$	130149.	0.360323	0.180162	−	0.983637i	$-0.442338\pi$
$601$	130149.	0.360323	0.180162	+	0.983637i	$0.442338\pi$
$602$	4044.12i	0.0111591i
$603$	0	0
$604$	−101306.	−0.277691
$605$	90299.6i	0.246703i
$606$	0	0
$607$	53264.3	0.144563	0.0722817	−	0.997384i	$-0.476972\pi$
$607$	53264.3	0.144563	0.0722817	+	0.997384i	$0.476972\pi$
$608$	− 3773.14i	− 0.0102070i
$609$	0	0
$610$	562.537	0.00151179
$611$	− 402978.i	− 1.07944i
$612$	0	0
$613$	500886.	1.33296	0.666481	−	0.745522i	$-0.267800\pi$
$613$	500886.	1.33296	0.666481	+	0.745522i	$0.267800\pi$
$614$	6170.29i	0.0163670i
$615$	0	0
$616$	−7225.21	−0.0190410
$617$	363017.i	0.953578i	0.879018	+	0.476789i	$0.158199\pi$
$617$	363017.i	0.953578i	−0.879018	+	0.476789i	$0.841801\pi$
$618$	0	0
$619$	−197298.	−0.514922	−0.257461	−	0.966289i	$-0.582886\pi$
$619$	−197298.	−0.514922	−0.257461	+	0.966289i	$0.582886\pi$
$620$	− 143729.i	− 0.373905i
$621$	0	0
$622$	−7949.17	−0.0205466
$623$	343959.i	0.886199i
$624$	0	0
$625$	−106483.	−0.272597
$626$	8258.52i	0.0210743i
$627$	0	0
$628$	212549.	0.538940
$629$	− 613106.i	− 1.54965i
$630$	0	0
$631$	−579068.	−1.45436	−0.727178	−	0.686449i	$-0.759168\pi$
$631$	−579068.	−1.45436	−0.727178	+	0.686449i	$0.759168\pi$
$632$	14702.4i	0.0368091i
$633$	0	0
$634$	11098.9	0.0276123
$635$	392351.i	0.973032i
$636$	0	0
$637$	−300026.	−0.739400
$638$	− 5267.70i	− 0.0129414i
$639$	0	0
$640$	−20295.6	−0.0495498
$641$	595413.i	1.44911i	0.689215	+	0.724557i	$0.257956\pi$
$641$	595413.i	1.44911i	−0.689215	+	0.724557i	$0.742044\pi$
$642$	0	0
$643$	−381497.	−0.922717	−0.461359	−	0.887214i	$-0.652638\pi$
$643$	−381497.	−0.922717	−0.461359	+	0.887214i	$0.652638\pi$
$644$	− 85672.1i	− 0.206570i
$645$	0	0
$646$	−2219.01	−0.00531735
$647$	− 522270.i	− 1.24763i	−0.781571	−	0.623816i	$-0.785581\pi$
$647$	− 522270.i	− 1.24763i	0.781571	−	0.623816i	$-0.214419\pi$
$648$	0	0
$649$	44379.9	0.105365
$650$	4905.90i	0.0116116i
$651$	0	0
$652$	54937.5	0.129233
$653$	686769.i	1.61059i	0.592877	+	0.805293i	$0.297992\pi$
$653$	686769.i	1.61059i	−0.592877	+	0.805293i	$0.702008\pi$
$654$	0	0
$655$	−119966.	−0.279624
$656$	211739.i	0.492032i
$657$	0	0
$658$	4011.35	0.00926486
$659$	− 416511.i	− 0.959082i	−0.877519	−	0.479541i	$-0.840803\pi$
$659$	− 416511.i	− 0.959082i	0.877519	−	0.479541i	$-0.159197\pi$
$660$	0	0
$661$	691316.	1.58224	0.791122	−	0.611658i	$-0.209497\pi$
$661$	691316.	1.58224	0.791122	+	0.611658i	$0.209497\pi$
$662$	− 9350.45i	− 0.0213362i
$663$	0	0
$664$	−5400.68	−0.0122493
$665$	− 42137.8i	− 0.0952858i
$666$	0	0
$667$	124941.	0.280836
$668$	− 52971.6i	− 0.118711i
$669$	0	0
$670$	10535.5	0.0234697
$671$	− 44437.7i	− 0.0986976i
$672$	0	0
$673$	126365.	0.278995	0.139497	−	0.990222i	$-0.455451\pi$
$673$	126365.	0.278995	0.139497	+	0.990222i	$0.455451\pi$
$674$	− 11015.2i	− 0.0242477i
$675$	0	0
$676$	401556.	0.878724
$677$	357573.i	0.780167i	0.920780	+	0.390083i	$0.127554\pi$
$677$	357573.i	0.780167i	−0.920780	+	0.390083i	$0.872446\pi$
$678$	0	0
$679$	171403.	0.371774
$680$	17907.6i	0.0387275i
$681$	0	0
$682$	3413.20	0.00733827
$683$	− 814280.i	− 1.74555i	−0.488123	−	0.872775i	$-0.662318\pi$
$683$	− 814280.i	− 1.74555i	0.488123	−	0.872775i	$-0.337682\pi$
$684$	0	0
$685$	−273421.	−0.582709
$686$	− 8523.22i	− 0.0181115i
$687$	0	0
$688$	448553.	0.947627
$689$	− 119558.i	− 0.251848i
$690$	0	0
$691$	391286.	0.819480	0.409740	−	0.912202i	$-0.365619\pi$
$691$	391286.	0.819480	0.409740	+	0.912202i	$0.365619\pi$
$692$	6983.29i	0.0145830i
$693$	0	0
$694$	6762.72	0.0140411
$695$	344579.i	0.713376i
$696$	0	0
$697$	373764.	0.769364
$698$	1705.73i	0.00350107i
$699$	0	0
$700$	162446.	0.331523
$701$	− 70213.5i	− 0.142884i	−0.997445	−	0.0714422i	$-0.977240\pi$
$701$	− 70213.5i	− 0.142884i	0.997445	−	0.0714422i	$-0.0227601\pi$
$702$	0	0
$703$	−96250.8	−0.194757
$704$	400391.i	0.807866i
$705$	0	0
$706$	3022.92	0.00606481
$707$	358114.i	0.716444i
$708$	0	0
$709$	151383.	0.301152	0.150576	−	0.988598i	$-0.451887\pi$
$709$	151383.	0.301152	0.150576	+	0.988598i	$0.451887\pi$
$710$	− 7230.38i	− 0.0143431i
$711$	0	0
$712$	−22947.8	−0.0452670
$713$	80955.5i	0.159246i
$714$	0	0
$715$	−405566.	−0.793322
$716$	14568.5i	0.0284178i
$717$	0	0
$718$	−1909.23	−0.00370347
$719$	− 536883.i	− 1.03854i	−0.854611	−	0.519268i	$-0.826205\pi$
$719$	− 536883.i	− 1.03854i	0.854611	−	0.519268i	$-0.173795\pi$
$720$	0	0
$721$	−90746.4	−0.174566
$722$	− 8688.51i	− 0.0166675i
$723$	0	0
$724$	−244841.	−0.467097
$725$	236906.i	0.450712i
$726$	0	0
$727$	−341812.	−0.646723	−0.323361	−	0.946276i	$-0.604813\pi$
$727$	−341812.	−0.646723	−0.323361	+	0.946276i	$0.604813\pi$
$728$	17091.9i	0.0322499i
$729$	0	0
$730$	4148.40	0.00778458
$731$	− 791790.i	− 1.48175i
$732$	0	0
$733$	−716427.	−1.33341	−0.666705	−	0.745321i	$-0.732296\pi$
$733$	−716427.	−1.33341	−0.666705	+	0.745321i	$0.732296\pi$
$734$	− 13220.8i	− 0.0245395i
$735$	0	0
$736$	8574.07	0.0158282
$737$	− 832255.i	− 1.53222i
$738$	0	0
$739$	−544732.	−0.997456	−0.498728	−	0.866759i	$-0.666199\pi$
$739$	−544732.	−0.997456	−0.498728	+	0.866759i	$0.666199\pi$
$740$	388317.i	0.709125i
$741$	0	0
$742$	1190.11	0.00216162
$743$	657748.i	1.19147i	0.803182	+	0.595733i	$0.203138\pi$
$743$	657748.i	1.19147i	−0.803182	+	0.595733i	$0.796862\pi$
$744$	0	0
$745$	−503089.	−0.906426
$746$	5722.36i	0.0102825i
$747$	0	0
$748$	707199.	1.26398
$749$	− 522580.i	− 0.931513i
$750$	0	0
$751$	−910314.	−1.61403	−0.807014	−	0.590532i	$-0.798918\pi$
$751$	−910314.	−1.61403	−0.807014	+	0.590532i	$0.798918\pi$
$752$	− 444919.i	− 0.786766i
$753$	0	0
$754$	−12461.2	−0.0219189
$755$	− 113227.i	− 0.198636i
$756$	0	0
$757$	909040.	1.58632	0.793161	−	0.609012i	$-0.208434\pi$
$757$	909040.	1.58632	0.793161	+	0.609012i	$0.208434\pi$
$758$	5973.46i	0.0103965i
$759$	0	0
$760$	2811.29	0.00486719
$761$	− 186991.i	− 0.322888i	−0.986882	−	0.161444i	$-0.948385\pi$
$761$	− 186991.i	− 0.322888i	0.986882	−	0.161444i	$-0.0516150\pi$
$762$	0	0
$763$	461656.	0.792993
$764$	905.896i	0.00155200i
$765$	0	0
$766$	−13451.1	−0.0229245
$767$	− 104985.i	− 0.178458i
$768$	0	0
$769$	−247137.	−0.417912	−0.208956	−	0.977925i	$-0.567007\pi$
$769$	−247137.	−0.417912	−0.208956	+	0.977925i	$0.567007\pi$
$770$	− 4037.11i	− 0.00680909i
$771$	0	0
$772$	−411678.	−0.690754
$773$	− 448251.i	− 0.750175i	−0.926989	−	0.375087i	$-0.877613\pi$
$773$	− 448251.i	− 0.750175i	0.926989	−	0.375087i	$-0.122387\pi$
$774$	0	0
$775$	−153503.	−0.255572
$776$	11435.4i	0.0189902i
$777$	0	0
$778$	13217.8	0.0218373
$779$	− 58676.7i	− 0.0966921i
$780$	0	0
$781$	−571164.	−0.936395
$782$	− 5042.47i	− 0.00824575i
$783$	0	0
$784$	−331252.	−0.538922
$785$	237561.i	0.385510i
$786$	0	0
$787$	227267.	0.366933	0.183467	−	0.983026i	$-0.441268\pi$
$787$	227267.	0.366933	0.183467	+	0.983026i	$0.441268\pi$
$788$	40741.9i	0.0656129i
$789$	0	0
$790$	−8215.04	−0.0131630
$791$	− 590946.i	− 0.944485i
$792$	0	0
$793$	−105122.	−0.167165
$794$	− 13757.6i	− 0.0218223i
$795$	0	0
$796$	−891044.	−1.40628
$797$	− 499378.i	− 0.786163i	−0.919504	−	0.393082i	$-0.871409\pi$
$797$	− 499378.i	− 0.786163i	0.919504	−	0.393082i	$-0.128591\pi$
$798$	0	0
$799$	−785375.	−1.23022
$800$	16257.6i	0.0254025i
$801$	0	0
$802$	18902.4	0.0293879
$803$	− 327704.i	− 0.508218i
$804$	0	0
$805$	95753.5	0.147762
$806$	− 8074.26i	− 0.0124289i
$807$	0	0
$808$	−23892.2	−0.0365959
$809$	157681.i	0.240926i	0.992718	+	0.120463i	$0.0384379\pi$
$809$	157681.i	0.240926i	−0.992718	+	0.120463i	$0.961562\pi$
$810$	0	0
$811$	−455209.	−0.692101	−0.346051	−	0.938216i	$-0.612477\pi$
$811$	−455209.	−0.692101	−0.346051	+	0.938216i	$0.612477\pi$
$812$	412622.i	0.625807i
$813$	0	0
$814$	−9221.55	−0.0139173
$815$	61402.3i	0.0924420i
$816$	0	0
$817$	−124302.	−0.186223
$818$	− 19281.4i	− 0.0288159i
$819$	0	0
$820$	−236727.	−0.352062
$821$	504392.i	0.748310i	0.927366	+	0.374155i	$0.122067\pi$
$821$	504392.i	0.748310i	−0.927366	+	0.374155i	$0.877933\pi$
$822$	0	0
$823$	710647.	1.04919	0.524595	−	0.851352i	$-0.324217\pi$
$823$	710647.	1.04919	0.524595	+	0.851352i	$0.324217\pi$
$824$	− 6054.30i	− 0.00891680i
$825$	0	0
$826$	1045.05	0.00153171
$827$	332545.i	0.486228i	0.969998	+	0.243114i	$0.0781688\pi$
$827$	332545.i	0.486228i	−0.969998	+	0.243114i	$0.921831\pi$
$828$	0	0
$829$	673348.	0.979784	0.489892	−	0.871783i	$-0.337036\pi$
$829$	673348.	0.979784	0.489892	+	0.871783i	$0.337036\pi$
$830$	− 3017.65i	− 0.00438039i
$831$	0	0
$832$	947164.	1.36829
$833$	584728.i	0.842682i
$834$	0	0
$835$	59205.0	0.0849152
$836$	− 111022.i	− 0.158854i
$837$	0	0
$838$	−3773.20	−0.00537307
$839$	− 10538.6i	− 0.0149712i	−0.999972	−	0.00748562i	$-0.997617\pi$
$839$	− 10538.6i	− 0.0149712i	0.999972	−	0.00748562i	$-0.00238277\pi$
$840$	0	0
$841$	105528.	0.149202
$842$	6796.01i	0.00958583i
$843$	0	0
$844$	−602884.	−0.846348
$845$	448809.i	0.628562i
$846$	0	0
$847$	167971.	0.234136
$848$	− 132001.i	− 0.183563i
$849$	0	0
$850$	9561.23	0.0132335
$851$	− 218720.i	− 0.302015i
$852$	0	0
$853$	179481.	0.246672	0.123336	−	0.992365i	$-0.460641\pi$
$853$	179481.	0.246672	0.123336	+	0.992365i	$0.460641\pi$
$854$	− 1046.41i	− 0.00143478i
$855$	0	0
$856$	34864.8	0.0475816
$857$	− 328799.i	− 0.447682i	−0.974626	−	0.223841i	$-0.928140\pi$
$857$	− 328799.i	− 0.447682i	0.974626	−	0.223841i	$-0.0718596\pi$
$858$	0	0
$859$	237394.	0.321724	0.160862	−	0.986977i	$-0.448573\pi$
$859$	237394.	0.321724	0.160862	+	0.986977i	$0.448573\pi$
$860$	501487.i	0.678052i
$861$	0	0
$862$	14281.1	0.0192198
$863$	− 263334.i	− 0.353578i	−0.984249	−	0.176789i	$-0.943429\pi$
$863$	− 263334.i	− 0.353578i	0.984249	−	0.176789i	$-0.0565711\pi$
$864$	0	0
$865$	−7805.04	−0.0104314
$866$	− 3399.20i	− 0.00453253i
$867$	0	0
$868$	−267359.	−0.354858
$869$	648947.i	0.859350i
$870$	0	0
$871$	−1.96878e6	−2.59514
$872$	30800.2i	0.0405060i
$873$	0	0
$874$	−791.612	−0.00103631
$875$	553131.i	0.722456i
$876$	0	0
$877$	659388.	0.857317	0.428659	−	0.903467i	$-0.358986\pi$
$877$	659388.	0.857317	0.428659	+	0.903467i	$0.358986\pi$
$878$	13546.6i	0.0175728i
$879$	0	0
$880$	−447776.	−0.578224
$881$	− 615974.i	− 0.793616i	−0.917902	−	0.396808i	$-0.870118\pi$
$881$	− 615974.i	− 0.793616i	0.917902	−	0.396808i	$-0.129882\pi$
$882$	0	0
$883$	−933378.	−1.19712	−0.598558	−	0.801080i	$-0.704259\pi$
$883$	−933378.	−1.19712	−0.598558	+	0.801080i	$0.704259\pi$
$884$	− 1.67295e6i	− 2.14081i
$885$	0	0
$886$	−13958.0	−0.0177810
$887$	149731.i	0.190311i	0.995462	+	0.0951556i	$0.0303349\pi$
$887$	149731.i	0.190311i	−0.995462	+	0.0951556i	$0.969665\pi$
$888$	0	0
$889$	729834.	0.923465
$890$	− 12822.2i	− 0.0161876i
$891$	0	0
$892$	−1.17197e6	−1.47295
$893$	123295.i	0.154612i
$894$	0	0
$895$	−16282.9	−0.0203275
$896$	37753.0i	0.0470257i
$897$	0	0
$898$	12179.2	0.0151031
$899$	− 389906.i	− 0.482437i
$900$	0	0
$901$	−233009.	−0.287027
$902$	− 5621.67i	− 0.00690958i
$903$	0	0
$904$	39425.9	0.0482442
$905$	− 273653.i	− 0.334120i
$906$	0	0
$907$	1.24275e6	1.51067	0.755334	−	0.655340i	$-0.227474\pi$
$907$	1.24275e6	1.51067	0.755334	+	0.655340i	$0.227474\pi$
$908$	− 264104.i	− 0.320335i
$909$	0	0
$910$	−9550.17	−0.0115326
$911$	− 547445.i	− 0.659636i	−0.944045	−	0.329818i	$-0.893013\pi$
$911$	− 547445.i	− 0.659636i	0.944045	−	0.329818i	$-0.106987\pi$
$912$	0	0
$913$	−238379.	−0.285974
$914$	− 7608.50i	− 0.00910766i
$915$	0	0
$916$	−33389.4	−0.0397940
$917$	223155.i	0.265380i
$918$	0	0
$919$	346600.	0.410390	0.205195	−	0.978721i	$-0.434217\pi$
$919$	346600.	0.410390	0.205195	+	0.978721i	$0.434217\pi$
$920$	6388.35i	0.00754768i
$921$	0	0
$922$	25382.9	0.0298593
$923$	1.35114e6i	1.58598i
$924$	0	0
$925$	414723.	0.484702
$926$	23870.3i	0.0278379i
$927$	0	0
$928$	−41295.2	−0.0479517
$929$	332911.i	0.385742i	0.981224	+	0.192871i	$0.0617799\pi$
$929$	332911.i	0.385742i	−0.981224	+	0.192871i	$0.938220\pi$
$930$	0	0
$931$	91795.7	0.105907
$932$	− 298439.i	− 0.343577i
$933$	0	0
$934$	25166.4	0.0288487
$935$	790418.i	0.904136i
$936$	0	0
$937$	−401391.	−0.457181	−0.228591	−	0.973523i	$-0.573412\pi$
$937$	−401391.	−0.457181	−0.228591	+	0.973523i	$0.573412\pi$
$938$	− 19597.7i	− 0.0222741i
$939$	0	0
$940$	497424.	0.562952
$941$	892267.i	1.00766i	0.863802	+	0.503832i	$0.168077\pi$
$941$	892267.i	1.00766i	−0.863802	+	0.503832i	$0.831923\pi$
$942$	0	0
$943$	133337.	0.149943
$944$	− 115911.i	− 0.130072i
$945$	0	0
$946$	−11909.1	−0.0133075
$947$	− 578173.i	− 0.644701i	−0.946620	−	0.322350i	$-0.895527\pi$
$947$	− 578173.i	− 0.644701i	0.946620	−	0.322350i	$-0.104473\pi$
$948$	0	0
$949$	−775214.	−0.860774
$950$	− 1501.01i	− 0.00166316i
$951$	0	0
$952$	33310.9	0.0367547
$953$	− 734097.i	− 0.808291i	−0.914695	−	0.404146i	$-0.867569\pi$
$953$	− 734097.i	− 0.808291i	0.914695	−	0.404146i	$-0.132431\pi$
$954$	0	0
$955$	−1012.50	−0.00111016
$956$	− 877537.i	− 0.960173i
$957$	0	0
$958$	6477.62	0.00705805
$959$	508607.i	0.553025i
$960$	0	0
$961$	−670882.	−0.726439
$962$	21814.4i	0.0235719i
$963$	0	0
$964$	−1.57941e6	−1.69958
$965$	− 460122.i	− 0.494104i
$966$	0	0
$967$	−1.64678e6	−1.76109	−0.880547	−	0.473958i	$-0.842825\pi$
$967$	−1.64678e6	−1.76109	−0.880547	+	0.473958i	$0.842825\pi$
$968$	11206.5i	0.0119597i
$969$	0	0
$970$	−6389.59	−0.00679093
$971$	− 1.30962e6i	− 1.38901i	−0.719487	−	0.694506i	$-0.755623\pi$
$971$	− 1.30962e6i	− 1.38901i	0.719487	−	0.694506i	$-0.244377\pi$
$972$	0	0
$973$	640970.	0.677036
$974$	1955.38i	0.00206117i
$975$	0	0
$976$	−116062.	−0.121841
$977$	1.17419e6i	1.23013i	0.788478	+	0.615063i	$0.210870\pi$
$977$	1.17419e6i	1.23013i	−0.788478	+	0.615063i	$0.789130\pi$
$978$	0	0
$979$	−1.01289e6	−1.05681
$980$	− 370343.i	− 0.385613i
$981$	0	0
$982$	−52.9458	−5.49046e−5 0
$983$	− 1.66138e6i	− 1.71934i	−0.510850	−	0.859670i	$-0.670669\pi$
$983$	− 1.66138e6i	− 1.71934i	0.510850	−	0.859670i	$-0.329331\pi$
$984$	0	0
$985$	−45536.2	−0.0469337
$986$	24286.0i	0.0249806i
$987$	0	0
$988$	−262634.	−0.269052
$989$	− 282463.i	− 0.288781i
$990$	0	0
$991$	1.40233e6	1.42792	0.713960	−	0.700187i	$-0.246900\pi$
$991$	1.40233e6	1.42792	0.713960	+	0.700187i	$0.246900\pi$
$992$	− 26757.2i	− 0.0271905i
$993$	0	0
$994$	−13449.6	−0.0136125
$995$	− 995897.i	− 1.00593i
$996$	0	0
$997$	215760.	0.217060	0.108530	−	0.994093i	$-0.465386\pi$
$997$	215760.	0.217060	0.108530	+	0.994093i	$0.465386\pi$
$998$	− 20959.9i	− 0.0210440i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.39	yes	76
3.2	odd	2	inner	531.5.b.a.296.38	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.38	✓	76	3.2	odd	2	inner
531.5.b.a.296.39	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)

\(f(q)\)	\(=\)	\(q+0.0693431i q^{2} +15.9952 q^{4} +17.8774i q^{5} +33.2548 q^{7} +2.21865i q^{8} +O(q^{10})\)	\(q+0.0693431i q^{2} +15.9952 q^{4} +17.8774i q^{5} +33.2548 q^{7} +2.21865i q^{8} -1.23968 q^{10} +97.9283i q^{11} +231.659 q^{13} +2.30599i q^{14} +255.769 q^{16} -451.486i q^{17} -70.8782 q^{19} +285.953i q^{20} -6.79066 q^{22} -161.063i q^{23} +305.398 q^{25} +16.0639i q^{26} +531.917 q^{28} +775.727i q^{29} -502.632 q^{31} +53.2342i q^{32} +31.3074 q^{34} +594.510i q^{35} +1357.98 q^{37} -4.91492i q^{38} -39.6637 q^{40} +827.853i q^{41} +1753.74 q^{43} +1566.38i q^{44} +11.1686 q^{46} -1739.53i q^{47} -1295.12 q^{49} +21.1773i q^{50} +3705.43 q^{52} -516.094i q^{53} -1750.70 q^{55} +73.7806i q^{56} -53.7914 q^{58} -453.188i q^{59} -453.778 q^{61} -34.8541i q^{62} +4088.62 q^{64} +4141.46i q^{65} -8498.61 q^{67} -7221.60i q^{68} -41.2252 q^{70} +5832.47i q^{71} -3346.36 q^{73} +94.1663i q^{74} -1133.71 q^{76} +3256.58i q^{77} +6626.76 q^{79} +4572.49i q^{80} -57.4059 q^{82} +2434.22i q^{83} +8071.40 q^{85} +121.610i q^{86} -217.268 q^{88} +10343.2i q^{89} +7703.76 q^{91} -2576.24i q^{92} +120.625 q^{94} -1267.12i q^{95} +5154.24 q^{97} -89.8077i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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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