LMFDB - Embedded newform 531.5.b.a.296.18

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			296.18
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.59

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.71716i q^{2} -6.25160 q^{4} +39.7285i q^{5} -15.3622 q^{7} -45.9848i q^{8} +O(q^{10})$	\(q-4.71716i q^{2} -6.25160 q^{4} +39.7285i q^{5} -15.3622 q^{7} -45.9848i q^{8} +187.405 q^{10} +168.118i q^{11} -143.677 q^{13} +72.4659i q^{14} -316.943 q^{16} -26.0234i q^{17} +401.301 q^{19} -248.366i q^{20} +793.042 q^{22} +224.799i q^{23} -953.350 q^{25} +677.748i q^{26} +96.0382 q^{28} -1435.43i q^{29} -552.559 q^{31} +759.315i q^{32} -122.756 q^{34} -610.316i q^{35} +927.545 q^{37} -1893.00i q^{38} +1826.90 q^{40} +856.977i q^{41} -1232.75 q^{43} -1051.01i q^{44} +1060.41 q^{46} -2243.89i q^{47} -2165.00 q^{49} +4497.11i q^{50} +898.212 q^{52} -1637.68i q^{53} -6679.09 q^{55} +706.427i q^{56} -6771.13 q^{58} -453.188i q^{59} -5431.99 q^{61} +2606.51i q^{62} -1489.28 q^{64} -5708.07i q^{65} -6119.94 q^{67} +162.688i q^{68} -2878.96 q^{70} -1426.30i q^{71} -7015.38 q^{73} -4375.38i q^{74} -2508.77 q^{76} -2582.67i q^{77} -433.212 q^{79} -12591.7i q^{80} +4042.50 q^{82} +2164.70i q^{83} +1033.87 q^{85} +5815.09i q^{86} +7730.89 q^{88} -13938.4i q^{89} +2207.19 q^{91} -1405.35i q^{92} -10584.8 q^{94} +15943.1i q^{95} +12915.8 q^{97} +10212.7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 4.71716i	− 1.17929i	−0.807663	−	0.589645i	$-0.799268\pi$
$2$	− 4.71716i	− 1.17929i	0.807663	−	0.589645i	$-0.200732\pi$
$3$	0	0
$3$	0	0
$4$	−6.25160	−0.390725
$5$	39.7285i	1.58914i	0.607174	+	0.794569i	$0.292303\pi$
$5$	39.7285i	1.58914i	−0.607174	+	0.794569i	$0.707697\pi$
$6$	0	0
$7$	−15.3622	−0.313514	−0.156757	−	0.987637i	$-0.550104\pi$
$7$	−15.3622	−0.313514	−0.156757	+	0.987637i	$0.550104\pi$
$8$	− 45.9848i	− 0.718512i
$9$	0	0
$10$	187.405	1.87405
$11$	168.118i	1.38941i	0.719295	+	0.694704i	$0.244465\pi$
$11$	168.118i	1.38941i	−0.719295	+	0.694704i	$0.755535\pi$
$12$	0	0
$13$	−143.677	−0.850161	−0.425080	−	0.905156i	$-0.639754\pi$
$13$	−143.677	−0.850161	−0.425080	+	0.905156i	$0.639754\pi$
$14$	72.4659i	0.369724i
$15$	0	0
$16$	−316.943	−1.23806
$17$	− 26.0234i	− 0.0900463i	−0.998986	−	0.0450231i	$-0.985664\pi$
$17$	− 26.0234i	− 0.0900463i	0.998986	−	0.0450231i	$-0.0143362\pi$
$18$	0	0
$19$	401.301	1.11164	0.555819	−	0.831303i	$-0.312405\pi$
$19$	401.301	1.11164	0.555819	+	0.831303i	$0.312405\pi$
$20$	− 248.366i	− 0.620916i
$21$	0	0
$22$	793.042	1.63852
$23$	224.799i	0.424950i	0.977167	+	0.212475i	$0.0681524\pi$
$23$	224.799i	0.424950i	−0.977167	+	0.212475i	$0.931848\pi$
$24$	0	0
$25$	−953.350	−1.52536
$26$	677.748i	1.00259i
$27$	0	0
$28$	96.0382	0.122498
$29$	− 1435.43i	− 1.70681i	−0.521250	−	0.853404i	$-0.674534\pi$
$29$	− 1435.43i	− 1.70681i	0.521250	−	0.853404i	$-0.325466\pi$
$30$	0	0
$31$	−552.559	−0.574983	−0.287492	−	0.957783i	$-0.592821\pi$
$31$	−552.559	−0.574983	−0.287492	+	0.957783i	$0.592821\pi$
$32$	759.315i	0.741519i
$33$	0	0
$34$	−122.756	−0.106191
$35$	− 610.316i	− 0.498217i
$36$	0	0
$37$	927.545	0.677535	0.338767	−	0.940870i	$-0.389990\pi$
$37$	927.545	0.677535	0.338767	+	0.940870i	$0.389990\pi$
$38$	− 1893.00i	− 1.31094i
$39$	0	0
$40$	1826.90	1.14181
$41$	856.977i	0.509802i	0.966967	+	0.254901i	$0.0820428\pi$
$41$	856.977i	0.509802i	−0.966967	+	0.254901i	$0.917957\pi$
$42$	0	0
$43$	−1232.75	−0.666713	−0.333357	−	0.942801i	$-0.608181\pi$
$43$	−1232.75	−0.666713	−0.333357	+	0.942801i	$0.608181\pi$
$44$	− 1051.01i	− 0.542877i
$45$	0	0
$46$	1060.41	0.501139
$47$	− 2243.89i	− 1.01580i	−0.861417	−	0.507898i	$-0.830423\pi$
$47$	− 2243.89i	− 1.01580i	0.861417	−	0.507898i	$-0.169577\pi$
$48$	0	0
$49$	−2165.00	−0.901709
$50$	4497.11i	1.79884i
$51$	0	0
$52$	898.212	0.332179
$53$	− 1637.68i	− 0.583012i	−0.956569	−	0.291506i	$-0.905844\pi$
$53$	− 1637.68i	− 0.583012i	0.956569	−	0.291506i	$-0.0941564\pi$
$54$	0	0
$55$	−6679.09	−2.20796
$56$	706.427i	0.225264i
$57$	0	0
$58$	−6771.13	−2.01282
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	−5431.99	−1.45982	−0.729909	−	0.683544i	$-0.760438\pi$
$61$	−5431.99	−1.45982	−0.729909	+	0.683544i	$0.760438\pi$
$62$	2606.51i	0.678072i
$63$	0	0
$64$	−1489.28	−0.363593
$65$	− 5708.07i	− 1.35102i
$66$	0	0
$67$	−6119.94	−1.36332	−0.681660	−	0.731669i	$-0.738742\pi$
$67$	−6119.94	−1.36332	−0.681660	+	0.731669i	$0.738742\pi$
$68$	162.688i	0.0351833i
$69$	0	0
$70$	−2878.96	−0.587542
$71$	− 1426.30i	− 0.282941i	−0.989942	−	0.141470i	$-0.954817\pi$
$71$	− 1426.30i	− 0.282941i	0.989942	−	0.141470i	$-0.0451830\pi$
$72$	0	0
$73$	−7015.38	−1.31645	−0.658226	−	0.752820i	$-0.728693\pi$
$73$	−7015.38	−1.31645	−0.658226	+	0.752820i	$0.728693\pi$
$74$	− 4375.38i	− 0.799010i
$75$	0	0
$76$	−2508.77	−0.434345
$77$	− 2582.67i	− 0.435599i
$78$	0	0
$79$	−433.212	−0.0694138	−0.0347069	−	0.999398i	$-0.511050\pi$
$79$	−433.212	−0.0694138	−0.0347069	+	0.999398i	$0.511050\pi$
$80$	− 12591.7i	− 1.96745i
$81$	0	0
$82$	4042.50	0.601204
$83$	2164.70i	0.314226i	0.987581	+	0.157113i	$0.0502187\pi$
$83$	2164.70i	0.314226i	−0.987581	+	0.157113i	$0.949781\pi$
$84$	0	0
$85$	1033.87	0.143096
$86$	5815.09i	0.786248i
$87$	0	0
$88$	7730.89	0.998307
$89$	− 13938.4i	− 1.75967i	−0.475277	−	0.879836i	$-0.657652\pi$
$89$	− 13938.4i	− 1.75967i	0.475277	−	0.879836i	$-0.342348\pi$
$90$	0	0
$91$	2207.19	0.266537
$92$	− 1405.35i	− 0.166039i
$93$	0	0
$94$	−10584.8	−1.19792
$95$	15943.1i	1.76655i
$96$	0	0
$97$	12915.8	1.37271	0.686355	−	0.727267i	$-0.259210\pi$
$97$	12915.8	1.37271	0.686355	+	0.727267i	$0.259210\pi$
$98$	10212.7i	1.06338i
$99$	0	0
$100$	5959.96	0.595996
$101$	3387.81i	0.332106i	0.986117	+	0.166053i	$0.0531022\pi$
$101$	3387.81i	0.332106i	−0.986117	+	0.166053i	$0.946898\pi$
$102$	0	0
$103$	−11000.4	−1.03689	−0.518447	−	0.855110i	$-0.673490\pi$
$103$	−11000.4	−1.03689	−0.518447	+	0.855110i	$0.673490\pi$
$104$	6606.96i	0.610851i
$105$	0	0
$106$	−7725.21	−0.687541
$107$	7290.49i	0.636780i	0.947960	+	0.318390i	$0.103142\pi$
$107$	7290.49i	0.636780i	−0.947960	+	0.318390i	$0.896858\pi$
$108$	0	0
$109$	−3765.53	−0.316937	−0.158469	−	0.987364i	$-0.550656\pi$
$109$	−3765.53	−0.316937	−0.158469	+	0.987364i	$0.550656\pi$
$110$	31506.3i	2.60383i
$111$	0	0
$112$	4868.94	0.388149
$113$	− 2544.40i	− 0.199264i	−0.995024	−	0.0996319i	$-0.968233\pi$
$113$	− 2544.40i	− 0.199264i	0.995024	−	0.0996319i	$-0.0317665\pi$
$114$	0	0
$115$	−8930.90	−0.675304
$116$	8973.70i	0.666892i
$117$	0	0
$118$	−2137.76	−0.153530
$119$	399.776i	0.0282308i
$120$	0	0
$121$	−13622.8	−0.930456
$122$	25623.5i	1.72155i
$123$	0	0
$124$	3454.38	0.224660
$125$	− 13044.8i	− 0.834870i
$126$	0	0
$127$	25071.5	1.55444	0.777218	−	0.629232i	$-0.216630\pi$
$127$	25071.5	1.55444	0.777218	+	0.629232i	$0.216630\pi$
$128$	19174.2i	1.17030i
$129$	0	0
$130$	−26925.9	−1.59325
$131$	20082.8i	1.17026i	0.810941	+	0.585128i	$0.198956\pi$
$131$	20082.8i	1.17026i	−0.810941	+	0.585128i	$0.801044\pi$
$132$	0	0
$133$	−6164.86	−0.348514
$134$	28868.7i	1.60775i
$135$	0	0
$136$	−1196.68	−0.0646993
$137$	− 9740.02i	− 0.518942i	−0.965751	−	0.259471i	$-0.916452\pi$
$137$	− 9740.02i	− 0.518942i	0.965751	−	0.259471i	$-0.0835482\pi$
$138$	0	0
$139$	−16122.4	−0.834448	−0.417224	−	0.908804i	$-0.636997\pi$
$139$	−16122.4	−0.834448	−0.417224	+	0.908804i	$0.636997\pi$
$140$	3815.45i	0.194666i
$141$	0	0
$142$	−6728.11	−0.333669
$143$	− 24154.8i	− 1.18122i
$144$	0	0
$145$	57027.2	2.71235
$146$	33092.7i	1.55248i
$147$	0	0
$148$	−5798.64	−0.264730
$149$	3356.87i	0.151204i	0.997138	+	0.0756019i	$0.0240878\pi$
$149$	3356.87i	0.151204i	−0.997138	+	0.0756019i	$0.975912\pi$
$150$	0	0
$151$	19133.2	0.839138	0.419569	−	0.907723i	$-0.362181\pi$
$151$	19133.2	0.839138	0.419569	+	0.907723i	$0.362181\pi$
$152$	− 18453.7i	− 0.798725i
$153$	0	0
$154$	−12182.9	−0.513698
$155$	− 21952.3i	− 0.913728i
$156$	0	0
$157$	18452.5	0.748609	0.374304	−	0.927306i	$-0.377882\pi$
$157$	18452.5	0.748609	0.374304	+	0.927306i	$0.377882\pi$
$158$	2043.53i	0.0818590i
$159$	0	0
$160$	−30166.4	−1.17838
$161$	− 3453.40i	− 0.133228i
$162$	0	0
$163$	−29399.7	−1.10654	−0.553271	−	0.833001i	$-0.686621\pi$
$163$	−29399.7	−1.10654	−0.553271	+	0.833001i	$0.686621\pi$
$164$	− 5357.47i	− 0.199192i
$165$	0	0
$166$	10211.2	0.370563
$167$	13819.7i	0.495524i	0.968821	+	0.247762i	$0.0796952\pi$
$167$	13819.7i	0.495524i	−0.968821	+	0.247762i	$0.920305\pi$
$168$	0	0
$169$	−7917.88	−0.277227
$170$	− 4876.92i	− 0.168752i
$171$	0	0
$172$	7706.67	0.260501
$173$	527.364i	0.0176205i	0.999961	+	0.00881025i	$0.00280443\pi$
$173$	527.364i	0.0176205i	−0.999961	+	0.00881025i	$0.997196\pi$
$174$	0	0
$175$	14645.5	0.478222
$176$	− 53284.0i	− 1.72017i
$177$	0	0
$178$	−65749.5	−2.07516
$179$	17702.7i	0.552500i	0.961086	+	0.276250i	$0.0890918\pi$
$179$	17702.7i	0.552500i	−0.961086	+	0.276250i	$0.910908\pi$
$180$	0	0
$181$	−48312.1	−1.47468	−0.737341	−	0.675520i	$-0.763919\pi$
$181$	−48312.1	−1.47468	−0.737341	+	0.675520i	$0.763919\pi$
$182$	− 10411.7i	− 0.314325i
$183$	0	0
$184$	10337.3	0.305332
$185$	36849.9i	1.07670i
$186$	0	0
$187$	4375.01	0.125111
$188$	14027.9i	0.396897i
$189$	0	0
$190$	75206.0	2.08327
$191$	− 55991.9i	− 1.53482i	−0.641154	−	0.767412i	$-0.721544\pi$
$191$	− 55991.9i	− 1.53482i	0.641154	−	0.767412i	$-0.278456\pi$
$192$	0	0
$193$	27532.7	0.739152	0.369576	−	0.929201i	$-0.379503\pi$
$193$	27532.7	0.739152	0.369576	+	0.929201i	$0.379503\pi$
$194$	− 60926.0i	− 1.61882i
$195$	0	0
$196$	13534.7	0.352320
$197$	1162.14i	0.0299452i	0.999888	+	0.0149726i	$0.00476611\pi$
$197$	1162.14i	0.0299452i	−0.999888	+	0.0149726i	$0.995234\pi$
$198$	0	0
$199$	297.778	0.00751944	0.00375972	−	0.999993i	$-0.498803\pi$
$199$	297.778	0.00751944	0.00375972	+	0.999993i	$0.498803\pi$
$200$	43839.6i	1.09599i
$201$	0	0
$202$	15980.8	0.391649
$203$	22051.3i	0.535108i
$204$	0	0
$205$	−34046.4	−0.810145
$206$	51890.7i	1.22280i
$207$	0	0
$208$	45537.5	1.05255
$209$	67466.1i	1.54452i
$210$	0	0
$211$	−7597.21	−0.170643	−0.0853216	−	0.996353i	$-0.527192\pi$
$211$	−7597.21	−0.170643	−0.0853216	+	0.996353i	$0.527192\pi$
$212$	10238.1i	0.227797i
$213$	0	0
$214$	34390.4	0.750948
$215$	− 48975.3i	− 1.05950i
$216$	0	0
$217$	8488.51	0.180265
$218$	17762.6i	0.373761i
$219$	0	0
$220$	41755.0	0.862706
$221$	3738.96i	0.0765538i
$222$	0	0
$223$	−78202.6	−1.57258	−0.786288	−	0.617861i	$-0.788000\pi$
$223$	−78202.6	−1.57258	−0.786288	+	0.617861i	$0.788000\pi$
$224$	− 11664.7i	− 0.232476i
$225$	0	0
$226$	−12002.3	−0.234990
$227$	27144.6i	0.526783i	0.964689	+	0.263391i	$0.0848411\pi$
$227$	27144.6i	0.526783i	−0.964689	+	0.263391i	$0.915159\pi$
$228$	0	0
$229$	−85956.0	−1.63910	−0.819549	−	0.573009i	$-0.805776\pi$
$229$	−85956.0	−1.63910	−0.819549	+	0.573009i	$0.805776\pi$
$230$	42128.5i	0.796380i
$231$	0	0
$232$	−66007.7	−1.22636
$233$	63161.0i	1.16342i	0.813396	+	0.581711i	$0.197616\pi$
$233$	63161.0i	1.16342i	−0.813396	+	0.581711i	$0.802384\pi$
$234$	0	0
$235$	89146.5	1.61424
$236$	2833.15i	0.0508681i
$237$	0	0
$238$	1885.81	0.0332923
$239$	− 77441.4i	− 1.35574i	−0.735181	−	0.677871i	$-0.762903\pi$
$239$	− 77441.4i	− 1.35574i	0.735181	−	0.677871i	$-0.237097\pi$
$240$	0	0
$241$	−55966.7	−0.963597	−0.481799	−	0.876282i	$-0.660016\pi$
$241$	−55966.7	−0.963597	−0.481799	+	0.876282i	$0.660016\pi$
$242$	64261.0i	1.09728i
$243$	0	0
$244$	33958.6	0.570388
$245$	− 86012.2i	− 1.43294i
$246$	0	0
$247$	−57657.8	−0.945070
$248$	25409.3i	0.413132i
$249$	0	0
$250$	−61534.6	−0.984554
$251$	− 115267.i	− 1.82961i	−0.403899	−	0.914804i	$-0.632345\pi$
$251$	− 115267.i	− 1.82961i	0.403899	−	0.914804i	$-0.367655\pi$
$252$	0	0
$253$	−37792.8	−0.590429
$254$	− 118266.i	− 1.83313i
$255$	0	0
$256$	66619.3	1.01653
$257$	− 18252.5i	− 0.276347i	−0.990408	−	0.138174i	$-0.955877\pi$
$257$	− 18252.5i	− 0.276347i	0.990408	−	0.138174i	$-0.0441232\pi$
$258$	0	0
$259$	−14249.1	−0.212417
$260$	35684.6i	0.527878i
$261$	0	0
$262$	94733.7	1.38007
$263$	57935.7i	0.837597i	0.908079	+	0.418798i	$0.137549\pi$
$263$	57935.7i	0.837597i	−0.908079	+	0.418798i	$0.862451\pi$
$264$	0	0
$265$	65062.6	0.926487
$266$	29080.6i	0.410999i
$267$	0	0
$268$	38259.4	0.532683
$269$	71703.8i	0.990918i	0.868631	+	0.495459i	$0.165000\pi$
$269$	71703.8i	0.990918i	−0.868631	+	0.495459i	$0.835000\pi$
$270$	0	0
$271$	83867.4	1.14197	0.570985	−	0.820961i	$-0.306562\pi$
$271$	83867.4	1.14197	0.570985	+	0.820961i	$0.306562\pi$
$272$	8247.93i	0.111483i
$273$	0	0
$274$	−45945.2	−0.611983
$275$	− 160276.i	− 2.11935i
$276$	0	0
$277$	125501.	1.63564	0.817818	−	0.575477i	$-0.195183\pi$
$277$	125501.	1.63564	0.817818	+	0.575477i	$0.195183\pi$
$278$	76051.8i	0.984057i
$279$	0	0
$280$	−28065.2	−0.357975
$281$	− 68672.1i	− 0.869697i	−0.900504	−	0.434848i	$-0.856802\pi$
$281$	− 68672.1i	− 0.869697i	0.900504	−	0.434848i	$-0.143198\pi$
$282$	0	0
$283$	137883.	1.72162	0.860811	−	0.508924i	$-0.169957\pi$
$283$	137883.	1.72162	0.860811	+	0.508924i	$0.169957\pi$
$284$	8916.68i	0.110552i
$285$	0	0
$286$	−113942.	−1.39300
$287$	− 13165.0i	− 0.159830i
$288$	0	0
$289$	82843.8	0.991892
$290$	− 269007.i	− 3.19865i
$291$	0	0
$292$	43857.3	0.514371
$293$	− 43367.4i	− 0.505159i	−0.967576	−	0.252579i	$-0.918721\pi$
$293$	− 43367.4i	− 0.505159i	0.967576	−	0.252579i	$-0.0812789\pi$
$294$	0	0
$295$	18004.4	0.206888
$296$	− 42652.9i	− 0.486817i
$297$	0	0
$298$	15834.9	0.178313
$299$	− 32298.4i	− 0.361276i
$300$	0	0
$301$	18937.8	0.209024
$302$	− 90254.3i	− 0.989587i
$303$	0	0
$304$	−127190.	−1.37627
$305$	− 215804.i	− 2.31985i
$306$	0	0
$307$	−162153.	−1.72047	−0.860236	−	0.509896i	$-0.829684\pi$
$307$	−162153.	−1.72047	−0.860236	+	0.509896i	$0.829684\pi$
$308$	16145.8i	0.170199i
$309$	0	0
$310$	−103553.	−1.07755
$311$	76969.0i	0.795784i	0.917432	+	0.397892i	$0.130258\pi$
$311$	76969.0i	0.795784i	−0.917432	+	0.397892i	$0.869742\pi$
$312$	0	0
$313$	36866.3	0.376306	0.188153	−	0.982140i	$-0.439750\pi$
$313$	36866.3	0.376306	0.188153	+	0.982140i	$0.439750\pi$
$314$	− 87043.2i	− 0.882827i
$315$	0	0
$316$	2708.27	0.0271217
$317$	172200.i	1.71362i	0.515635	+	0.856808i	$0.327556\pi$
$317$	172200.i	1.71362i	−0.515635	+	0.856808i	$0.672444\pi$
$318$	0	0
$319$	241321.	2.37145
$320$	− 59166.8i	− 0.577800i
$321$	0	0
$322$	−16290.2	−0.157114
$323$	− 10443.2i	− 0.100099i
$324$	0	0
$325$	136975.	1.29680
$326$	138683.i	1.30493i
$327$	0	0
$328$	39407.9	0.366299
$329$	34471.1i	0.318466i
$330$	0	0
$331$	−97270.4	−0.887819	−0.443910	−	0.896072i	$-0.646409\pi$
$331$	−97270.4	−0.887819	−0.443910	+	0.896072i	$0.646409\pi$
$332$	− 13532.8i	− 0.122776i
$333$	0	0
$334$	65189.6	0.584367
$335$	− 243136.i	− 2.16650i
$336$	0	0
$337$	47739.4	0.420356	0.210178	−	0.977663i	$-0.432596\pi$
$337$	47739.4	0.420356	0.210178	+	0.977663i	$0.432596\pi$
$338$	37349.9i	0.326931i
$339$	0	0
$340$	−6463.33	−0.0559112
$341$	− 92895.3i	− 0.798887i
$342$	0	0
$343$	70143.8	0.596212
$344$	56687.8i	0.479041i
$345$	0	0
$346$	2487.66	0.0207797
$347$	72783.1i	0.604466i	0.953234	+	0.302233i	$0.0977320\pi$
$347$	72783.1i	0.604466i	−0.953234	+	0.302233i	$0.902268\pi$
$348$	0	0
$349$	−227998.	−1.87189	−0.935945	−	0.352148i	$-0.885452\pi$
$349$	−227998.	−1.87189	−0.935945	+	0.352148i	$0.885452\pi$
$350$	− 69085.4i	− 0.563962i
$351$	0	0
$352$	−127655.	−1.03027
$353$	69850.2i	0.560555i	0.959919	+	0.280278i	$0.0904265\pi$
$353$	69850.2i	0.560555i	−0.959919	+	0.280278i	$0.909574\pi$
$354$	0	0
$355$	56664.9	0.449632
$356$	87137.0i	0.687548i
$357$	0	0
$358$	83506.3	0.651558
$359$	− 211958.i	− 1.64460i	−0.569051	−	0.822302i	$-0.692689\pi$
$359$	− 211958.i	− 1.64460i	0.569051	−	0.822302i	$-0.307311\pi$
$360$	0	0
$361$	30721.6	0.235738
$362$	227896.i	1.73908i
$363$	0	0
$364$	−13798.5	−0.104143
$365$	− 278710.i	− 2.09203i
$366$	0	0
$367$	26200.6	0.194526	0.0972632	−	0.995259i	$-0.468991\pi$
$367$	26200.6	0.194526	0.0972632	+	0.995259i	$0.468991\pi$
$368$	− 71248.3i	− 0.526113i
$369$	0	0
$370$	173827.	1.26974
$371$	25158.4i	0.182783i
$372$	0	0
$373$	−118816.	−0.853995	−0.426998	−	0.904253i	$-0.640429\pi$
$373$	−118816.	−0.853995	−0.426998	+	0.904253i	$0.640429\pi$
$374$	− 20637.6i	− 0.147542i
$375$	0	0
$376$	−103185.	−0.729862
$377$	206238.i	1.45106i
$378$	0	0
$379$	−86883.5	−0.604866	−0.302433	−	0.953171i	$-0.597799\pi$
$379$	−86883.5	−0.604866	−0.302433	+	0.953171i	$0.597799\pi$
$380$	− 99669.7i	− 0.690234i
$381$	0	0
$382$	−264123.	−1.81000
$383$	− 242509.i	− 1.65322i	−0.562775	−	0.826610i	$-0.690266\pi$
$383$	− 242509.i	− 1.65322i	0.562775	−	0.826610i	$-0.309734\pi$
$384$	0	0
$385$	102605.	0.692227
$386$	− 129876.i	− 0.871674i
$387$	0	0
$388$	−80744.5	−0.536352
$389$	92933.0i	0.614144i	0.951686	+	0.307072i	$0.0993493\pi$
$389$	92933.0i	0.614144i	−0.951686	+	0.307072i	$0.900651\pi$
$390$	0	0
$391$	5850.02	0.0382652
$392$	99557.2i	0.647889i
$393$	0	0
$394$	5482.02	0.0353141
$395$	− 17210.8i	− 0.110308i
$396$	0	0
$397$	78212.9	0.496246	0.248123	−	0.968729i	$-0.420186\pi$
$397$	78212.9	0.496246	0.248123	+	0.968729i	$0.420186\pi$
$398$	− 1404.66i	− 0.00886761i
$399$	0	0
$400$	302158.	1.88849
$401$	176734.i	1.09908i	0.835467	+	0.549541i	$0.185197\pi$
$401$	176734.i	1.09908i	−0.835467	+	0.549541i	$0.814803\pi$
$402$	0	0
$403$	79390.1	0.488828
$404$	− 21179.2i	− 0.129762i
$405$	0	0
$406$	104019.	0.631048
$407$	155937.i	0.941372i
$408$	0	0
$409$	−151830.	−0.907636	−0.453818	−	0.891094i	$-0.649938\pi$
$409$	−151830.	−0.907636	−0.453818	+	0.891094i	$0.649938\pi$
$410$	160602.i	0.955396i
$411$	0	0
$412$	68770.1	0.405140
$413$	6961.95i	0.0408160i
$414$	0	0
$415$	−86000.3	−0.499348
$416$	− 109096.i	− 0.630410i
$417$	0	0
$418$	318249.	1.82144
$419$	3665.57i	0.0208792i	0.999946	+	0.0104396i	$0.00332309\pi$
$419$	3665.57i	0.0208792i	−0.999946	+	0.0104396i	$0.996677\pi$
$420$	0	0
$421$	290205.	1.63735	0.818675	−	0.574257i	$-0.194709\pi$
$421$	290205.	1.63735	0.818675	+	0.574257i	$0.194709\pi$
$422$	35837.2i	0.201238i
$423$	0	0
$424$	−75308.4	−0.418901
$425$	24809.4i	0.137353i
$426$	0	0
$427$	83447.2	0.457674
$428$	− 45577.2i	− 0.248806i
$429$	0	0
$430$	−231025.	−1.24946
$431$	− 234541.i	− 1.26259i	−0.775541	−	0.631297i	$-0.782523\pi$
$431$	− 234541.i	− 1.26259i	0.775541	−	0.631297i	$-0.217477\pi$
$432$	0	0
$433$	89811.0	0.479020	0.239510	−	0.970894i	$-0.423013\pi$
$433$	89811.0	0.479020	0.239510	+	0.970894i	$0.423013\pi$
$434$	− 40041.7i	− 0.212585i
$435$	0	0
$436$	23540.6	0.123835
$437$	90211.9i	0.472390i
$438$	0	0
$439$	−219329.	−1.13807	−0.569033	−	0.822315i	$-0.692682\pi$
$439$	−219329.	−1.13807	−0.569033	+	0.822315i	$0.692682\pi$
$440$	307136.i	1.58645i
$441$	0	0
$442$	17637.3	0.0902791
$443$	144315.i	0.735367i	0.929951	+	0.367684i	$0.119849\pi$
$443$	144315.i	0.735367i	−0.929951	+	0.367684i	$0.880151\pi$
$444$	0	0
$445$	553750.	2.79636
$446$	368894.i	1.85452i
$447$	0	0
$448$	22878.6	0.113992
$449$	− 290762.i	− 1.44227i	−0.692797	−	0.721133i	$-0.743622\pi$
$449$	− 290762.i	− 1.44227i	0.692797	−	0.721133i	$-0.256378\pi$
$450$	0	0
$451$	−144074.	−0.708323
$452$	15906.6i	0.0778574i
$453$	0	0
$454$	128045.	0.621230
$455$	87688.4i	0.423565i
$456$	0	0
$457$	69959.4	0.334976	0.167488	−	0.985874i	$-0.446434\pi$
$457$	69959.4	0.334976	0.167488	+	0.985874i	$0.446434\pi$
$458$	405468.i	1.93297i
$459$	0	0
$460$	55832.4	0.263858
$461$	243557.i	1.14604i	0.819542	+	0.573018i	$0.194228\pi$
$461$	243557.i	1.14604i	−0.819542	+	0.573018i	$0.805772\pi$
$462$	0	0
$463$	95691.4	0.446387	0.223193	−	0.974774i	$-0.428352\pi$
$463$	95691.4	0.446387	0.223193	+	0.974774i	$0.428352\pi$
$464$	454948.i	2.11313i
$465$	0	0
$466$	297940.	1.37201
$467$	− 321286.i	− 1.47319i	−0.676336	−	0.736593i	$-0.736433\pi$
$467$	− 321286.i	− 1.47319i	0.676336	−	0.736593i	$-0.263567\pi$
$468$	0	0
$469$	94015.7	0.427420
$470$	− 420518.i	− 1.90366i
$471$	0	0
$472$	−20839.7	−0.0935423
$473$	− 207248.i	− 0.926337i
$474$	0	0
$475$	−382581.	−1.69565
$476$	− 2499.24i	− 0.0110305i
$477$	0	0
$478$	−365303.	−1.59881
$479$	247123.i	1.07706i	0.842605	+	0.538532i	$0.181021\pi$
$479$	247123.i	1.07706i	−0.842605	+	0.538532i	$0.818979\pi$
$480$	0	0
$481$	−133267.	−0.576013
$482$	264004.i	1.13636i
$483$	0	0
$484$	85164.3	0.363552
$485$	513126.i	2.18142i
$486$	0	0
$487$	−210963.	−0.889506	−0.444753	−	0.895653i	$-0.646709\pi$
$487$	−210963.	−0.889506	−0.444753	+	0.895653i	$0.646709\pi$
$488$	249789.i	1.04890i
$489$	0	0
$490$	−405733.	−1.68985
$491$	157816.i	0.654617i	0.944918	+	0.327308i	$0.106142\pi$
$491$	157816.i	0.654617i	−0.944918	+	0.327308i	$0.893858\pi$
$492$	0	0
$493$	−37354.6	−0.153692
$494$	271981.i	1.11451i
$495$	0	0
$496$	175130.	0.711863
$497$	21911.2i	0.0887059i
$498$	0	0
$499$	108519.	0.435818	0.217909	−	0.975969i	$-0.430076\pi$
$499$	108519.	0.435818	0.217909	+	0.975969i	$0.430076\pi$
$500$	81551.2i	0.326205i
$501$	0	0
$502$	−543733.	−2.15764
$503$	− 248477.i	− 0.982087i	−0.871135	−	0.491044i	$-0.836616\pi$
$503$	− 248477.i	− 0.982087i	0.871135	−	0.491044i	$-0.163384\pi$
$504$	0	0
$505$	−134592.	−0.527762
$506$	178275.i	0.696287i
$507$	0	0
$508$	−156737.	−0.607357
$509$	384431.i	1.48382i	0.670497	+	0.741912i	$0.266081\pi$
$509$	384431.i	1.48382i	−0.670497	+	0.741912i	$0.733919\pi$
$510$	0	0
$511$	107772.	0.412726
$512$	− 7466.79i	− 0.0284835i
$513$	0	0
$514$	−86099.7	−0.325893
$515$	− 437029.i	− 1.64777i
$516$	0	0
$517$	377240.	1.41136
$518$	67215.4i	0.250501i
$519$	0	0
$520$	−262484.	−0.970726
$521$	481611.i	1.77427i	0.461506	+	0.887137i	$0.347309\pi$
$521$	481611.i	1.77427i	−0.461506	+	0.887137i	$0.652691\pi$
$522$	0	0
$523$	9616.62	0.0351576	0.0175788	−	0.999845i	$-0.494404\pi$
$523$	9616.62	0.0351576	0.0175788	+	0.999845i	$0.494404\pi$
$524$	− 125549.i	− 0.457249i
$525$	0	0
$526$	273292.	0.987769
$527$	14379.4i	0.0517751i
$528$	0	0
$529$	229307.	0.819418
$530$	− 306910.i	− 1.09260i
$531$	0	0
$532$	38540.3	0.136173
$533$	− 123128.i	− 0.433413i
$534$	0	0
$535$	−289640.	−1.01193
$536$	281424.i	0.979561i
$537$	0	0
$538$	338238.	1.16858
$539$	− 363977.i	− 1.25284i
$540$	0	0
$541$	−137373.	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$541$	−137373.	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$542$	− 395616.i	− 1.34671i
$543$	0	0
$544$	19759.9	0.0667710
$545$	− 149599.i	− 0.503657i
$546$	0	0
$547$	498380.	1.66566	0.832829	−	0.553530i	$-0.186720\pi$
$547$	498380.	1.66566	0.832829	+	0.553530i	$0.186720\pi$
$548$	60890.7i	0.202764i
$549$	0	0
$550$	−756046.	−2.49933
$551$	− 576038.i	− 1.89735i
$552$	0	0
$553$	6655.08	0.0217622
$554$	− 592007.i	− 1.92889i
$555$	0	0
$556$	100791.	0.326040
$557$	291991.i	0.941151i	0.882360	+	0.470576i	$0.155954\pi$
$557$	291991.i	0.941151i	−0.882360	+	0.470576i	$0.844046\pi$
$558$	0	0
$559$	177118.	0.566813
$560$	193435.i	0.616822i
$561$	0	0
$562$	−323937.	−1.02562
$563$	− 70114.0i	− 0.221202i	−0.993865	−	0.110601i	$-0.964723\pi$
$563$	− 70114.0i	− 0.221202i	0.993865	−	0.110601i	$-0.0352775\pi$
$564$	0	0
$565$	101085.	0.316658
$566$	− 650416.i	− 2.03029i
$567$	0	0
$568$	−65588.3	−0.203296
$569$	825.132i	0.00254858i	0.999999	+	0.00127429i	$0.000405620\pi$
$569$	825.132i	0.00254858i	−0.999999	+	0.00127429i	$0.999594\pi$
$570$	0	0
$571$	−224711.	−0.689212	−0.344606	−	0.938747i	$-0.611988\pi$
$571$	−224711.	−0.689212	−0.344606	+	0.938747i	$0.611988\pi$
$572$	151006.i	0.461532i
$573$	0	0
$574$	−62101.6	−0.188486
$575$	− 214312.i	− 0.648202i
$576$	0	0
$577$	−370062.	−1.11153	−0.555767	−	0.831338i	$-0.687575\pi$
$577$	−370062.	−1.11153	−0.555767	+	0.831338i	$0.687575\pi$
$578$	− 390787.i	− 1.16973i
$579$	0	0
$580$	−356511.	−1.05978
$581$	− 33254.5i	− 0.0985142i
$582$	0	0
$583$	275324.	0.810042
$584$	322600.i	0.945887i
$585$	0	0
$586$	−204571.	−0.595729
$587$	392041.i	1.13777i	0.822417	+	0.568886i	$0.192625\pi$
$587$	392041.i	1.13777i	−0.822417	+	0.568886i	$0.807375\pi$
$588$	0	0
$589$	−221743.	−0.639173
$590$	− 84929.8i	− 0.243981i
$591$	0	0
$592$	−293979.	−0.838828
$593$	− 207400.i	− 0.589791i	−0.955529	−	0.294896i	$-0.904715\pi$
$593$	− 207400.i	− 0.589791i	0.955529	−	0.294896i	$-0.0952849\pi$
$594$	0	0
$595$	−15882.5	−0.0448626
$596$	− 20985.8i	− 0.0590791i
$597$	0	0
$598$	−152357.	−0.426049
$599$	− 120553.i	− 0.335988i	−0.985788	−	0.167994i	$-0.946271\pi$
$599$	− 120553.i	− 0.335988i	0.985788	−	0.167994i	$-0.0537289\pi$
$600$	0	0
$601$	−612682.	−1.69624	−0.848118	−	0.529807i	$-0.822264\pi$
$601$	−612682.	−1.69624	−0.848118	+	0.529807i	$0.822264\pi$
$602$	− 89332.5i	− 0.246500i
$603$	0	0
$604$	−119613.	−0.327872
$605$	− 541213.i	− 1.47862i
$606$	0	0
$607$	283900.	0.770528	0.385264	−	0.922806i	$-0.374110\pi$
$607$	283900.	0.770528	0.385264	+	0.922806i	$0.374110\pi$
$608$	304714.i	0.824300i
$609$	0	0
$610$	−1.01798e6	−2.73578
$611$	322396.i	0.863590i
$612$	0	0
$613$	566889.	1.50861	0.754305	−	0.656525i	$-0.227974\pi$
$613$	566889.	1.50861	0.754305	+	0.656525i	$0.227974\pi$
$614$	764901.i	2.02894i
$615$	0	0
$616$	−118763.	−0.312983
$617$	247342.i	0.649722i	0.945762	+	0.324861i	$0.105318\pi$
$617$	247342.i	0.649722i	−0.945762	+	0.324861i	$0.894682\pi$
$618$	0	0
$619$	−155617.	−0.406140	−0.203070	−	0.979164i	$-0.565092\pi$
$619$	−155617.	−0.406140	−0.203070	+	0.979164i	$0.565092\pi$
$620$	137237.i	0.357016i
$621$	0	0
$622$	363075.	0.938460
$623$	214124.i	0.551682i
$624$	0	0
$625$	−77592.2	−0.198636
$626$	− 173904.i	− 0.443774i
$627$	0	0
$628$	−115357.	−0.292500
$629$	− 24137.8i	− 0.0610095i
$630$	0	0
$631$	−194435.	−0.488332	−0.244166	−	0.969733i	$-0.578514\pi$
$631$	−194435.	−0.488332	−0.244166	+	0.969733i	$0.578514\pi$
$632$	19921.1i	0.0498747i
$633$	0	0
$634$	812293.	2.02085
$635$	996052.i	2.47021i
$636$	0	0
$637$	311061.	0.766597
$638$	− 1.13835e6i	− 2.79663i
$639$	0	0
$640$	−761762.	−1.85977
$641$	646139.i	1.57257i	0.617864	+	0.786285i	$0.287998\pi$
$641$	646139.i	1.57257i	−0.617864	+	0.786285i	$0.712002\pi$
$642$	0	0
$643$	625417.	1.51268	0.756341	−	0.654177i	$-0.226985\pi$
$643$	625417.	1.51268	0.756341	+	0.654177i	$0.226985\pi$
$644$	21589.3i	0.0520554i
$645$	0	0
$646$	−49262.3	−0.118046
$647$	89142.3i	0.212949i	0.994315	+	0.106474i	$0.0339562\pi$
$647$	89142.3i	0.212949i	−0.994315	+	0.106474i	$0.966044\pi$
$648$	0	0
$649$	76189.2	0.180886
$650$	− 646131.i	− 1.52930i
$651$	0	0
$652$	183795.	0.432353
$653$	− 391081.i	− 0.917150i	−0.888656	−	0.458575i	$-0.848360\pi$
$653$	− 391081.i	− 0.917150i	0.888656	−	0.458575i	$-0.151640\pi$
$654$	0	0
$655$	−797858.	−1.85970
$656$	− 271613.i	− 0.631165i
$657$	0	0
$658$	162606.	0.375564
$659$	530577.i	1.22174i	0.791732	+	0.610868i	$0.209179\pi$
$659$	530577.i	1.22174i	−0.791732	+	0.610868i	$0.790821\pi$
$660$	0	0
$661$	−732931.	−1.67749	−0.838745	−	0.544524i	$-0.816710\pi$
$661$	−732931.	−1.67749	−0.838745	+	0.544524i	$0.816710\pi$
$662$	458840.i	1.04700i
$663$	0	0
$664$	99543.3	0.225775
$665$	− 244921.i	− 0.553837i
$666$	0	0
$667$	322681.	0.725308
$668$	− 86395.1i	− 0.193614i
$669$	0	0
$670$	−1.14691e6	−2.55494
$671$	− 913217.i	− 2.02828i
$672$	0	0
$673$	−613715.	−1.35499	−0.677496	−	0.735527i	$-0.736935\pi$
$673$	−613715.	−1.35499	−0.677496	+	0.735527i	$0.736935\pi$
$674$	− 225194.i	− 0.495721i
$675$	0	0
$676$	49499.4	0.108320
$677$	− 312265.i	− 0.681312i	−0.940188	−	0.340656i	$-0.889351\pi$
$677$	− 312265.i	− 0.681312i	0.940188	−	0.340656i	$-0.110649\pi$
$678$	0	0
$679$	−198415.	−0.430364
$680$	− 47542.2i	− 0.102816i
$681$	0	0
$682$	−438202.	−0.942119
$683$	256229.i	0.549272i	0.961548	+	0.274636i	$0.0885574\pi$
$683$	256229.i	0.549272i	−0.961548	+	0.274636i	$0.911443\pi$
$684$	0	0
$685$	386956.	0.824670
$686$	− 330879.i	− 0.703107i
$687$	0	0
$688$	390712.	0.825430
$689$	235297.i	0.495654i
$690$	0	0
$691$	460834.	0.965136	0.482568	−	0.875859i	$-0.339704\pi$
$691$	460834.	0.965136	0.482568	+	0.875859i	$0.339704\pi$
$692$	− 3296.87i	− 0.00688477i
$693$	0	0
$694$	343329.	0.712840
$695$	− 640517.i	− 1.32605i
$696$	0	0
$697$	22301.4	0.0459057
$698$	1.07550e6i	2.20750i
$699$	0	0
$700$	−91558.1	−0.186853
$701$	248601.i	0.505902i	0.967479	+	0.252951i	$0.0814011\pi$
$701$	248601.i	0.505902i	−0.967479	+	0.252951i	$0.918599\pi$
$702$	0	0
$703$	372225.	0.753173
$704$	− 250375.i	− 0.505180i
$705$	0	0
$706$	329495.	0.661057
$707$	− 52044.1i	− 0.104120i
$708$	0	0
$709$	521993.	1.03842	0.519209	−	0.854648i	$-0.326227\pi$
$709$	521993.	1.03842	0.519209	+	0.854648i	$0.326227\pi$
$710$	− 267297.i	− 0.530247i
$711$	0	0
$712$	−640952.	−1.26435
$713$	− 124214.i	− 0.244339i
$714$	0	0
$715$	959632.	1.87712
$716$	− 110670.i	− 0.215876i
$717$	0	0
$718$	−999841.	−1.93947
$719$	945475.i	1.82891i	0.404689	+	0.914454i	$0.367380\pi$
$719$	945475.i	1.82891i	−0.404689	+	0.914454i	$0.632620\pi$
$720$	0	0
$721$	168990.	0.325081
$722$	− 144919.i	− 0.278004i
$723$	0	0
$724$	302028.	0.576195
$725$	1.36846e6i	2.60350i
$726$	0	0
$727$	103354.	0.195551	0.0977756	−	0.995208i	$-0.468827\pi$
$727$	103354.	0.195551	0.0977756	+	0.995208i	$0.468827\pi$
$728$	− 101497.i	− 0.191510i
$729$	0	0
$730$	−1.31472e6	−2.46710
$731$	32080.4i	0.0600350i
$732$	0	0
$733$	97835.4	0.182091	0.0910454	−	0.995847i	$-0.470979\pi$
$733$	97835.4	0.182091	0.0910454	+	0.995847i	$0.470979\pi$
$734$	− 123592.i	− 0.229403i
$735$	0	0
$736$	−170693.	−0.315108
$737$	− 1.02887e6i	− 1.89421i
$738$	0	0
$739$	−439363.	−0.804516	−0.402258	−	0.915526i	$-0.631774\pi$
$739$	−439363.	−0.804516	−0.402258	+	0.915526i	$0.631774\pi$
$740$	− 230371.i	− 0.420692i
$741$	0	0
$742$	118676.	0.215554
$743$	210703.i	0.381675i	0.981622	+	0.190837i	$0.0611203\pi$
$743$	210703.i	0.381675i	−0.981622	+	0.190837i	$0.938880\pi$
$744$	0	0
$745$	−133363.	−0.240284
$746$	560472.i	1.00711i
$747$	0	0
$748$	−27350.8	−0.0488840
$749$	− 111998.i	− 0.199639i
$750$	0	0
$751$	238624.	0.423091	0.211546	−	0.977368i	$-0.432150\pi$
$751$	238624.	0.423091	0.211546	+	0.977368i	$0.432150\pi$
$752$	711187.i	1.25762i
$753$	0	0
$754$	972857.	1.71122
$755$	760132.i	1.33351i
$756$	0	0
$757$	790008.	1.37860	0.689302	−	0.724474i	$-0.257917\pi$
$757$	790008.	1.37860	0.689302	+	0.724474i	$0.257917\pi$
$758$	409843.i	0.713312i
$759$	0	0
$760$	733139.	1.26928
$761$	− 406758.i	− 0.702372i	−0.936306	−	0.351186i	$-0.885779\pi$
$761$	− 406758.i	− 0.702372i	0.936306	−	0.351186i	$-0.114221\pi$
$762$	0	0
$763$	57846.8	0.0993643
$764$	350039.i	0.599694i
$765$	0	0
$766$	−1.14396e6	−1.94963
$767$	65112.7i	0.110681i
$768$	0	0
$769$	−518218.	−0.876314	−0.438157	−	0.898899i	$-0.644369\pi$
$769$	−518218.	−0.876314	−0.438157	+	0.898899i	$0.644369\pi$
$770$	− 484006.i	− 0.816337i
$771$	0	0
$772$	−172123.	−0.288805
$773$	522217.i	0.873961i	0.899471	+	0.436980i	$0.143952\pi$
$773$	522217.i	0.873961i	−0.899471	+	0.436980i	$0.856048\pi$
$774$	0	0
$775$	526782.	0.877057
$776$	− 593931.i	− 0.986308i
$777$	0	0
$778$	438380.	0.724254
$779$	343906.i	0.566715i
$780$	0	0
$781$	239788.	0.393120
$782$	− 27595.5i	− 0.0451257i
$783$	0	0
$784$	686183.	1.11637
$785$	733088.i	1.18964i
$786$	0	0
$787$	−34131.4	−0.0551067	−0.0275533	−	0.999620i	$-0.508772\pi$
$787$	−34131.4	−0.0551067	−0.0275533	+	0.999620i	$0.508772\pi$
$788$	− 7265.26i	− 0.0117004i
$789$	0	0
$790$	−81186.2	−0.130085
$791$	39087.6i	0.0624720i
$792$	0	0
$793$	780452.	1.24108
$794$	− 368943.i	− 0.585218i
$795$	0	0
$796$	−1861.59	−0.00293803
$797$	− 472219.i	− 0.743407i	−0.928351	−	0.371704i	$-0.878774\pi$
$797$	− 472219.i	− 0.743407i	0.928351	−	0.371704i	$-0.121226\pi$
$798$	0	0
$799$	−58393.7	−0.0914687
$800$	− 723893.i	− 1.13108i
$801$	0	0
$802$	833680.	1.29614
$803$	− 1.17941e6i	− 1.82909i
$804$	0	0
$805$	137198.	0.211717
$806$	− 374496.i	− 0.576470i
$807$	0	0
$808$	155788.	0.238622
$809$	− 1.21258e6i	− 1.85274i	−0.376612	−	0.926371i	$-0.622911\pi$
$809$	− 1.21258e6i	− 1.85274i	0.376612	−	0.926371i	$-0.377089\pi$
$810$	0	0
$811$	428967.	0.652202	0.326101	−	0.945335i	$-0.394265\pi$
$811$	428967.	0.652202	0.326101	+	0.945335i	$0.394265\pi$
$812$	− 137856.i	− 0.209080i
$813$	0	0
$814$	735582.	1.11015
$815$	− 1.16801e6i	− 1.75845i
$816$	0	0
$817$	−494705.	−0.741143
$818$	716208.i	1.07037i
$819$	0	0
$820$	212844.	0.316544
$821$	− 1.10246e6i	− 1.63560i	−0.575500	−	0.817802i	$-0.695192\pi$
$821$	− 1.10246e6i	− 1.63560i	0.575500	−	0.817802i	$-0.304808\pi$
$822$	0	0
$823$	763238.	1.12684	0.563418	−	0.826172i	$-0.309486\pi$
$823$	763238.	1.12684	0.563418	+	0.826172i	$0.309486\pi$
$824$	505851.i	0.745020i
$825$	0	0
$826$	32840.6	0.0481340
$827$	1.03868e6i	1.51869i	0.650688	+	0.759345i	$0.274481\pi$
$827$	1.03868e6i	1.51869i	−0.650688	+	0.759345i	$0.725519\pi$
$828$	0	0
$829$	−373826.	−0.543952	−0.271976	−	0.962304i	$-0.587677\pi$
$829$	−373826.	−0.543952	−0.271976	+	0.962304i	$0.587677\pi$
$830$	405677.i	0.588876i
$831$	0	0
$832$	213975.	0.309113
$833$	56340.7i	0.0811955i
$834$	0	0
$835$	−549035.	−0.787457
$836$	− 421771.i	− 0.603482i
$837$	0	0
$838$	17291.1	0.0246226
$839$	1.14414e6i	1.62538i	0.582698	+	0.812688i	$0.301997\pi$
$839$	1.14414e6i	1.62538i	−0.582698	+	0.812688i	$0.698003\pi$
$840$	0	0
$841$	−1.35316e6	−1.91319
$842$	− 1.36895e6i	− 1.93091i
$843$	0	0
$844$	47494.7	0.0666745
$845$	− 314565.i	− 0.440552i
$846$	0	0
$847$	209276.	0.291711
$848$	519052.i	0.721804i
$849$	0	0
$850$	117030.	0.161979
$851$	208511.i	0.287918i
$852$	0	0
$853$	634403.	0.871901	0.435951	−	0.899971i	$-0.356412\pi$
$853$	634403.	0.871901	0.435951	+	0.899971i	$0.356412\pi$
$854$	− 393634.i	− 0.539730i
$855$	0	0
$856$	335252.	0.457534
$857$	105371.i	0.143470i	0.997424	+	0.0717348i	$0.0228535\pi$
$857$	105371.i	0.143470i	−0.997424	+	0.0717348i	$0.977146\pi$
$858$	0	0
$859$	271071.	0.367365	0.183682	−	0.982986i	$-0.441198\pi$
$859$	271071.	0.367365	0.183682	+	0.982986i	$0.441198\pi$
$860$	306174.i	0.413973i
$861$	0	0
$862$	−1.10637e6	−1.48897
$863$	1.28995e6i	1.73201i	0.500033	+	0.866006i	$0.333321\pi$
$863$	1.28995e6i	1.73201i	−0.500033	+	0.866006i	$0.666679\pi$
$864$	0	0
$865$	−20951.4	−0.0280014
$866$	− 423653.i	− 0.564903i
$867$	0	0
$868$	−53066.8	−0.0704341
$869$	− 72830.9i	− 0.0964441i
$870$	0	0
$871$	879296.	1.15904
$872$	173157.i	0.227723i
$873$	0	0
$874$	425544.	0.557085
$875$	200397.i	0.261744i
$876$	0	0
$877$	−756910.	−0.984113	−0.492057	−	0.870563i	$-0.663755\pi$
$877$	−756910.	−0.984113	−0.492057	+	0.870563i	$0.663755\pi$
$878$	1.03461e6i	1.34211i
$879$	0	0
$880$	2.11689e6	2.73359
$881$	927554.i	1.19505i	0.801849	+	0.597527i	$0.203850\pi$
$881$	927554.i	1.19505i	−0.801849	+	0.597527i	$0.796150\pi$
$882$	0	0
$883$	−884320.	−1.13420	−0.567098	−	0.823650i	$-0.691934\pi$
$883$	−884320.	−1.13420	−0.567098	+	0.823650i	$0.691934\pi$
$884$	− 23374.5i	− 0.0299115i
$885$	0	0
$886$	680757.	0.867211
$887$	309689.i	0.393621i	0.980442	+	0.196810i	$0.0630584\pi$
$887$	309689.i	0.393621i	−0.980442	+	0.196810i	$0.936942\pi$
$888$	0	0
$889$	−385153.	−0.487337
$890$	− 2.61213e6i	− 3.29772i
$891$	0	0
$892$	488891.	0.614444
$893$	− 900478.i	− 1.12920i
$894$	0	0
$895$	−703299.	−0.877999
$896$	− 294558.i	− 0.366906i
$897$	0	0
$898$	−1.37157e6	−1.70085
$899$	793157.i	0.981386i
$900$	0	0
$901$	−42618.0	−0.0524981
$902$	679618.i	0.835318i
$903$	0	0
$904$	−117004.	−0.143173
$905$	− 1.91936e6i	− 2.34348i
$906$	0	0
$907$	540225.	0.656690	0.328345	−	0.944558i	$-0.393509\pi$
$907$	540225.	0.656690	0.328345	+	0.944558i	$0.393509\pi$
$908$	− 169697.i	− 0.205827i
$909$	0	0
$910$	413640.	0.499505
$911$	− 1.01438e6i	− 1.22226i	−0.791530	−	0.611130i	$-0.790715\pi$
$911$	− 1.01438e6i	− 1.22226i	0.791530	−	0.611130i	$-0.209285\pi$
$912$	0	0
$913$	−363926.	−0.436588
$914$	− 330009.i	− 0.395034i
$915$	0	0
$916$	537362.	0.640437
$917$	− 308515.i	− 0.366892i
$918$	0	0
$919$	295984.	0.350459	0.175229	−	0.984528i	$-0.443933\pi$
$919$	295984.	0.350459	0.175229	+	0.984528i	$0.443933\pi$
$920$	410685.i	0.485214i
$921$	0	0
$922$	1.14890e6	1.35151
$923$	204927.i	0.240545i
$924$	0	0
$925$	−884275.	−1.03348
$926$	− 451392.i	− 0.526419i
$927$	0	0
$928$	1.08994e6	1.26563
$929$	539338.i	0.624927i	0.949930	+	0.312464i	$0.101154\pi$
$929$	539338.i	0.624927i	−0.949930	+	0.312464i	$0.898846\pi$
$930$	0	0
$931$	−868818.	−1.00237
$932$	− 394857.i	− 0.454578i
$933$	0	0
$934$	−1.51556e6	−1.73731
$935$	173812.i	0.198819i
$936$	0	0
$937$	651647.	0.742221	0.371110	−	0.928589i	$-0.378977\pi$
$937$	651647.	0.742221	0.371110	+	0.928589i	$0.378977\pi$
$938$	− 443487.i	− 0.504052i
$939$	0	0
$940$	−557308.	−0.630724
$941$	1.75031e6i	1.97668i	0.152267	+	0.988339i	$0.451343\pi$
$941$	1.75031e6i	1.97668i	−0.152267	+	0.988339i	$0.548657\pi$
$942$	0	0
$943$	−192647.	−0.216640
$944$	143635.i	0.161182i
$945$	0	0
$946$	−977624.	−1.09242
$947$	− 1.09623e6i	− 1.22237i	−0.791487	−	0.611186i	$-0.790693\pi$
$947$	− 1.09623e6i	− 1.22237i	0.791487	−	0.611186i	$-0.209307\pi$
$948$	0	0
$949$	1.00795e6	1.11920
$950$	1.80469e6i	1.99966i
$951$	0	0
$952$	18383.6	0.0202841
$953$	− 991768.i	− 1.09200i	−0.837784	−	0.546002i	$-0.816149\pi$
$953$	− 991768.i	− 1.09200i	0.837784	−	0.546002i	$-0.183851\pi$
$954$	0	0
$955$	2.22447e6	2.43905
$956$	484132.i	0.529722i
$957$	0	0
$958$	1.16572e6	1.27017
$959$	149628.i	0.162696i
$960$	0	0
$961$	−618200.	−0.669394
$962$	628642.i	0.679287i
$963$	0	0
$964$	349881.	0.376501
$965$	1.09383e6i	1.17461i
$966$	0	0
$967$	−1.77859e6	−1.90205	−0.951025	−	0.309113i	$-0.899968\pi$
$967$	−1.77859e6	−1.90205	−0.951025	+	0.309113i	$0.899968\pi$
$968$	626442.i	0.668544i
$969$	0	0
$970$	2.42050e6	2.57253
$971$	− 1.04249e6i	− 1.10569i	−0.833284	−	0.552845i	$-0.813542\pi$
$971$	− 1.04249e6i	− 1.10569i	0.833284	−	0.552845i	$-0.186458\pi$
$972$	0	0
$973$	247675.	0.261611
$974$	995147.i	1.04899i
$975$	0	0
$976$	1.72163e6	1.80734
$977$	224683.i	0.235386i	0.993050	+	0.117693i	$0.0375498\pi$
$977$	224683.i	0.235386i	−0.993050	+	0.117693i	$0.962450\pi$
$978$	0	0
$979$	2.34330e6	2.44490
$980$	537714.i	0.559886i
$981$	0	0
$982$	744442.	0.771983
$983$	− 429387.i	− 0.444367i	−0.975005	−	0.222183i	$-0.928682\pi$
$983$	− 429387.i	− 0.444367i	0.975005	−	0.222183i	$-0.0713184\pi$
$984$	0	0
$985$	−46170.2	−0.0475871
$986$	176208.i	0.181247i
$987$	0	0
$988$	360453.	0.369263
$989$	− 277121.i	− 0.283320i
$990$	0	0
$991$	1.54173e6	1.56986	0.784932	−	0.619582i	$-0.212698\pi$
$991$	1.54173e6	1.56986	0.784932	+	0.619582i	$0.212698\pi$
$992$	− 419566.i	− 0.426361i
$993$	0	0
$994$	103358.	0.104610
$995$	11830.2i	0.0119494i
$996$	0	0
$997$	1.23140e6	1.23883	0.619413	−	0.785065i	$-0.287371\pi$
$997$	1.23140e6	1.23883	0.619413	+	0.785065i	$0.287371\pi$
$998$	− 511902.i	− 0.513955i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.18	✓	76
3.2	odd	2	inner	531.5.b.a.296.59	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.18	✓	76	1.1	even	1	trivial
531.5.b.a.296.59	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)

\(f(q)\)	\(=\)	\(q-4.71716i q^{2} -6.25160 q^{4} +39.7285i q^{5} -15.3622 q^{7} -45.9848i q^{8} +O(q^{10})\)	\(q-4.71716i q^{2} -6.25160 q^{4} +39.7285i q^{5} -15.3622 q^{7} -45.9848i q^{8} +187.405 q^{10} +168.118i q^{11} -143.677 q^{13} +72.4659i q^{14} -316.943 q^{16} -26.0234i q^{17} +401.301 q^{19} -248.366i q^{20} +793.042 q^{22} +224.799i q^{23} -953.350 q^{25} +677.748i q^{26} +96.0382 q^{28} -1435.43i q^{29} -552.559 q^{31} +759.315i q^{32} -122.756 q^{34} -610.316i q^{35} +927.545 q^{37} -1893.00i q^{38} +1826.90 q^{40} +856.977i q^{41} -1232.75 q^{43} -1051.01i q^{44} +1060.41 q^{46} -2243.89i q^{47} -2165.00 q^{49} +4497.11i q^{50} +898.212 q^{52} -1637.68i q^{53} -6679.09 q^{55} +706.427i q^{56} -6771.13 q^{58} -453.188i q^{59} -5431.99 q^{61} +2606.51i q^{62} -1489.28 q^{64} -5708.07i q^{65} -6119.94 q^{67} +162.688i q^{68} -2878.96 q^{70} -1426.30i q^{71} -7015.38 q^{73} -4375.38i q^{74} -2508.77 q^{76} -2582.67i q^{77} -433.212 q^{79} -12591.7i q^{80} +4042.50 q^{82} +2164.70i q^{83} +1033.87 q^{85} +5815.09i q^{86} +7730.89 q^{88} -13938.4i q^{89} +2207.19 q^{91} -1405.35i q^{92} -10584.8 q^{94} +15943.1i q^{95} +12915.8 q^{97} +10212.7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more