LMFDB - Embedded newform 531.5.b.a.296.57

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			296.57
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.32230i q^{2} -2.68230 q^{4} +35.4048i q^{5} +80.8997 q^{7} +57.5631i q^{8} +O(q^{10})$	\(q+4.32230i q^{2} -2.68230 q^{4} +35.4048i q^{5} +80.8997 q^{7} +57.5631i q^{8} -153.030 q^{10} +74.8463i q^{11} -179.122 q^{13} +349.673i q^{14} -291.722 q^{16} -35.1211i q^{17} -52.2004 q^{19} -94.9664i q^{20} -323.509 q^{22} -217.263i q^{23} -628.499 q^{25} -774.219i q^{26} -216.997 q^{28} +557.425i q^{29} +342.028 q^{31} -339.901i q^{32} +151.804 q^{34} +2864.24i q^{35} -940.422 q^{37} -225.626i q^{38} -2038.01 q^{40} +2609.49i q^{41} +200.649 q^{43} -200.761i q^{44} +939.075 q^{46} -250.942i q^{47} +4143.76 q^{49} -2716.56i q^{50} +480.459 q^{52} +763.735i q^{53} -2649.92 q^{55} +4656.84i q^{56} -2409.36 q^{58} +453.188i q^{59} +5637.34 q^{61} +1478.35i q^{62} -3198.40 q^{64} -6341.77i q^{65} -937.190 q^{67} +94.2054i q^{68} -12380.1 q^{70} -6906.38i q^{71} -4216.58 q^{73} -4064.79i q^{74} +140.017 q^{76} +6055.04i q^{77} -672.685 q^{79} -10328.4i q^{80} -11279.0 q^{82} -3210.06i q^{83} +1243.45 q^{85} +867.264i q^{86} -4308.39 q^{88} +10513.8i q^{89} -14490.9 q^{91} +582.764i q^{92} +1084.65 q^{94} -1848.14i q^{95} +4428.89 q^{97} +17910.6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.32230i	1.08058i	0.841480	+	0.540288i	$0.181685\pi$
$2$	4.32230i	1.08058i	−0.841480	+	0.540288i	$0.818315\pi$
$3$	0	0
$3$	0	0
$4$	−2.68230	−0.167644
$5$	35.4048i	1.41619i	0.706116	+	0.708096i	$0.250446\pi$
$5$	35.4048i	1.41619i	−0.706116	+	0.708096i	$0.749554\pi$
$6$	0	0
$7$	80.8997	1.65101	0.825507	−	0.564392i	$-0.190889\pi$
$7$	80.8997	1.65101	0.825507	+	0.564392i	$0.190889\pi$
$8$	57.5631i	0.899424i
$9$	0	0
$10$	−153.030	−1.53030
$11$	74.8463i	0.618565i	0.950970	+	0.309282i	$0.100089\pi$
$11$	74.8463i	0.618565i	−0.950970	+	0.309282i	$0.899911\pi$
$12$	0	0
$13$	−179.122	−1.05989	−0.529947	−	0.848031i	$-0.677788\pi$
$13$	−179.122	−1.05989	−0.529947	+	0.848031i	$0.677788\pi$
$14$	349.673i	1.78405i
$15$	0	0
$16$	−291.722	−1.13954
$17$	− 35.1211i	− 0.121526i	−0.998152	−	0.0607631i	$-0.980647\pi$
$17$	− 35.1211i	− 0.121526i	0.998152	−	0.0607631i	$-0.0193534\pi$
$18$	0	0
$19$	−52.2004	−0.144599	−0.0722997	−	0.997383i	$-0.523034\pi$
$19$	−52.2004	−0.144599	−0.0722997	+	0.997383i	$0.523034\pi$
$20$	− 94.9664i	− 0.237416i
$21$	0	0
$22$	−323.509	−0.668406
$23$	− 217.263i	− 0.410704i	−0.978688	−	0.205352i	$-0.934166\pi$
$23$	− 217.263i	− 0.410704i	0.978688	−	0.205352i	$-0.0658340\pi$
$24$	0	0
$25$	−628.499	−1.00560
$26$	− 774.219i	− 1.14529i
$27$	0	0
$28$	−216.997	−0.276782
$29$	557.425i	0.662812i	0.943488	+	0.331406i	$0.107523\pi$
$29$	557.425i	0.662812i	−0.943488	+	0.331406i	$0.892477\pi$
$30$	0	0
$31$	342.028	0.355909	0.177954	−	0.984039i	$-0.443052\pi$
$31$	342.028	0.355909	0.177954	+	0.984039i	$0.443052\pi$
$32$	− 339.901i	− 0.331935i
$33$	0	0
$34$	151.804	0.131318
$35$	2864.24i	2.33815i
$36$	0	0
$37$	−940.422	−0.686941	−0.343470	−	0.939164i	$-0.611602\pi$
$37$	−940.422	−0.686941	−0.343470	+	0.939164i	$0.611602\pi$
$38$	− 225.626i	− 0.156251i
$39$	0	0
$40$	−2038.01	−1.27376
$41$	2609.49i	1.55234i	0.630521	+	0.776172i	$0.282841\pi$
$41$	2609.49i	1.55234i	−0.630521	+	0.776172i	$0.717159\pi$
$42$	0	0
$43$	200.649	0.108517	0.0542587	−	0.998527i	$-0.482720\pi$
$43$	200.649	0.108517	0.0542587	+	0.998527i	$0.482720\pi$
$44$	− 200.761i	− 0.103699i
$45$	0	0
$46$	939.075	0.443797
$47$	− 250.942i	− 0.113600i	−0.998386	−	0.0567999i	$-0.981910\pi$
$47$	− 250.942i	− 0.113600i	0.998386	−	0.0567999i	$-0.0180897\pi$
$48$	0	0
$49$	4143.76	1.72585
$50$	− 2716.56i	− 1.08663i
$51$	0	0
$52$	480.459	0.177685
$53$	763.735i	0.271889i	0.990716	+	0.135944i	$0.0434068\pi$
$53$	763.735i	0.271889i	−0.990716	+	0.135944i	$0.956593\pi$
$54$	0	0
$55$	−2649.92	−0.876006
$56$	4656.84i	1.48496i
$57$	0	0
$58$	−2409.36	−0.716218
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	5637.34	1.51501	0.757504	−	0.652831i	$-0.226419\pi$
$61$	5637.34	1.51501	0.757504	+	0.652831i	$0.226419\pi$
$62$	1478.35i	0.384586i
$63$	0	0
$64$	−3198.40	−0.780859
$65$	− 6341.77i	− 1.50101i
$66$	0	0
$67$	−937.190	−0.208775	−0.104387	−	0.994537i	$-0.533288\pi$
$67$	−937.190	−0.208775	−0.104387	+	0.994537i	$0.533288\pi$
$68$	94.2054i	0.0203731i
$69$	0	0
$70$	−12380.1	−2.52655
$71$	− 6906.38i	− 1.37004i	−0.728523	−	0.685021i	$-0.759793\pi$
$71$	− 6906.38i	− 1.37004i	0.728523	−	0.685021i	$-0.240207\pi$
$72$	0	0
$73$	−4216.58	−0.791251	−0.395626	−	0.918412i	$-0.629472\pi$
$73$	−4216.58	−0.791251	−0.395626	+	0.918412i	$0.629472\pi$
$74$	− 4064.79i	− 0.742291i
$75$	0	0
$76$	140.017	0.0242412
$77$	6055.04i	1.02126i
$78$	0	0
$79$	−672.685	−0.107785	−0.0538924	−	0.998547i	$-0.517163\pi$
$79$	−672.685	−0.107785	−0.0538924	+	0.998547i	$0.517163\pi$
$80$	− 10328.4i	− 1.61381i
$81$	0	0
$82$	−11279.0	−1.67743
$83$	− 3210.06i	− 0.465970i	−0.972480	−	0.232985i	$-0.925151\pi$
$83$	− 3210.06i	− 0.465970i	0.972480	−	0.232985i	$-0.0748492\pi$
$84$	0	0
$85$	1243.45	0.172104
$86$	867.264i	0.117261i
$87$	0	0
$88$	−4308.39	−0.556352
$89$	10513.8i	1.32733i	0.748029	+	0.663666i	$0.231000\pi$
$89$	10513.8i	1.32733i	−0.748029	+	0.663666i	$0.769000\pi$
$90$	0	0
$91$	−14490.9	−1.74990
$92$	582.764i	0.0688521i
$93$	0	0
$94$	1084.65	0.122753
$95$	− 1848.14i	− 0.204781i
$96$	0	0
$97$	4428.89	0.470708	0.235354	−	0.971910i	$-0.424375\pi$
$97$	4428.89	0.470708	0.235354	+	0.971910i	$0.424375\pi$
$98$	17910.6i	1.86491i
$99$	0	0
$100$	1685.83	0.168583
$101$	− 15117.0i	− 1.48191i	−0.671553	−	0.740956i	$-0.734373\pi$
$101$	− 15117.0i	− 1.48191i	0.671553	−	0.740956i	$-0.265627\pi$
$102$	0	0
$103$	−15057.9	−1.41935	−0.709674	−	0.704530i	$-0.751158\pi$
$103$	−15057.9	−1.41935	−0.709674	+	0.704530i	$0.751158\pi$
$104$	− 10310.8i	− 0.953293i
$105$	0	0
$106$	−3301.09	−0.293796
$107$	− 12041.3i	− 1.05173i	−0.850568	−	0.525865i	$-0.823742\pi$
$107$	− 12041.3i	− 1.05173i	0.850568	−	0.525865i	$-0.176258\pi$
$108$	0	0
$109$	−1786.57	−0.150372	−0.0751859	−	0.997170i	$-0.523955\pi$
$109$	−1786.57	−0.150372	−0.0751859	+	0.997170i	$0.523955\pi$
$110$	− 11453.8i	− 0.946591i
$111$	0	0
$112$	−23600.2	−1.88140
$113$	17352.1i	1.35893i	0.733709	+	0.679464i	$0.237787\pi$
$113$	17352.1i	1.35893i	−0.733709	+	0.679464i	$0.762213\pi$
$114$	0	0
$115$	7692.14	0.581636
$116$	− 1495.18i	− 0.111116i
$117$	0	0
$118$	−1958.81	−0.140679
$119$	− 2841.28i	− 0.200641i
$120$	0	0
$121$	9039.03	0.617378
$122$	24366.3i	1.63708i
$123$	0	0
$124$	−917.423	−0.0596659
$125$	− 123.890i	− 0.00792893i
$126$	0	0
$127$	30453.1	1.88810	0.944048	−	0.329808i	$-0.106984\pi$
$127$	30453.1	1.88810	0.944048	+	0.329808i	$0.106984\pi$
$128$	− 19262.9i	− 1.17571i
$129$	0	0
$130$	27411.1	1.62196
$131$	6070.47i	0.353736i	0.984235	+	0.176868i	$0.0565966\pi$
$131$	6070.47i	0.353736i	−0.984235	+	0.176868i	$0.943403\pi$
$132$	0	0
$133$	−4223.00	−0.238736
$134$	− 4050.82i	− 0.225597i
$135$	0	0
$136$	2021.68	0.109304
$137$	− 18575.1i	− 0.989669i	−0.868987	−	0.494834i	$-0.835229\pi$
$137$	− 18575.1i	− 0.989669i	0.868987	−	0.494834i	$-0.164771\pi$
$138$	0	0
$139$	−32151.8	−1.66408	−0.832042	−	0.554713i	$-0.812828\pi$
$139$	−32151.8	−1.66408	−0.832042	+	0.554713i	$0.812828\pi$
$140$	− 7682.75i	− 0.391977i
$141$	0	0
$142$	29851.5	1.48043
$143$	− 13406.6i	− 0.655612i
$144$	0	0
$145$	−19735.5	−0.938668
$146$	− 18225.3i	− 0.855007i
$147$	0	0
$148$	2522.50	0.115161
$149$	26594.9i	1.19792i	0.800780	+	0.598958i	$0.204418\pi$
$149$	26594.9i	1.19792i	−0.800780	+	0.598958i	$0.795582\pi$
$150$	0	0
$151$	−1326.20	−0.0581643	−0.0290822	−	0.999577i	$-0.509258\pi$
$151$	−1326.20	−0.0581643	−0.0290822	+	0.999577i	$0.509258\pi$
$152$	− 3004.82i	− 0.130056i
$153$	0	0
$154$	−26171.7	−1.10355
$155$	12109.4i	0.504035i
$156$	0	0
$157$	16095.7	0.652997	0.326498	−	0.945198i	$-0.394131\pi$
$157$	16095.7	0.652997	0.326498	+	0.945198i	$0.394131\pi$
$158$	− 2907.55i	− 0.116470i
$159$	0	0
$160$	12034.1	0.470084
$161$	− 17576.5i	− 0.678078i
$162$	0	0
$163$	−21946.2	−0.826008	−0.413004	−	0.910729i	$-0.635520\pi$
$163$	−21946.2	−0.826008	−0.413004	+	0.910729i	$0.635520\pi$
$164$	− 6999.45i	− 0.260241i
$165$	0	0
$166$	13874.9	0.503515
$167$	− 35703.7i	− 1.28021i	−0.768288	−	0.640104i	$-0.778891\pi$
$167$	− 35703.7i	− 1.28021i	0.768288	−	0.640104i	$-0.221109\pi$
$168$	0	0
$169$	3523.66	0.123373
$170$	5374.58i	0.185972i
$171$	0	0
$172$	−538.200	−0.0181923
$173$	− 12071.2i	− 0.403327i	−0.979455	−	0.201663i	$-0.935365\pi$
$173$	− 12071.2i	− 0.403327i	0.979455	−	0.201663i	$-0.0646347\pi$
$174$	0	0
$175$	−50845.4	−1.66026
$176$	− 21834.3i	− 0.704879i
$177$	0	0
$178$	−45443.8	−1.43428
$179$	− 17945.5i	− 0.560080i	−0.959988	−	0.280040i	$-0.909652\pi$
$179$	− 17945.5i	− 0.560080i	0.959988	−	0.280040i	$-0.0903477\pi$
$180$	0	0
$181$	7643.60	0.233314	0.116657	−	0.993172i	$-0.462782\pi$
$181$	7643.60	0.233314	0.116657	+	0.993172i	$0.462782\pi$
$182$	− 62634.1i	− 1.89090i
$183$	0	0
$184$	12506.3	0.369397
$185$	− 33295.4i	− 0.972840i
$186$	0	0
$187$	2628.68	0.0751718
$188$	673.102i	0.0190443i
$189$	0	0
$190$	7988.24	0.221281
$191$	− 38231.7i	− 1.04799i	−0.851722	−	0.523995i	$-0.824441\pi$
$191$	− 38231.7i	− 1.04799i	0.851722	−	0.523995i	$-0.175559\pi$
$192$	0	0
$193$	−13657.7	−0.366660	−0.183330	−	0.983051i	$-0.558688\pi$
$193$	−13657.7	−0.366660	−0.183330	+	0.983051i	$0.558688\pi$
$194$	19143.0i	0.508635i
$195$	0	0
$196$	−11114.8	−0.289328
$197$	31208.5i	0.804156i	0.915605	+	0.402078i	$0.131712\pi$
$197$	31208.5i	0.804156i	−0.915605	+	0.402078i	$0.868288\pi$
$198$	0	0
$199$	54577.6	1.37819	0.689093	−	0.724673i	$-0.258009\pi$
$199$	54577.6	1.37819	0.689093	+	0.724673i	$0.258009\pi$
$200$	− 36178.4i	− 0.904459i
$201$	0	0
$202$	65340.2	1.60132
$203$	45095.5i	1.09431i
$204$	0	0
$205$	−92388.5	−2.19842
$206$	− 65084.7i	− 1.53371i
$207$	0	0
$208$	52253.8	1.20779
$209$	− 3907.01i	− 0.0894441i
$210$	0	0
$211$	−29794.0	−0.669211	−0.334606	−	0.942358i	$-0.608603\pi$
$211$	−29794.0	−0.669211	−0.334606	+	0.942358i	$0.608603\pi$
$212$	− 2048.57i	− 0.0455805i
$213$	0	0
$214$	52046.0	1.13647
$215$	7103.92i	0.153681i
$216$	0	0
$217$	27670.0	0.587610
$218$	− 7722.09i	− 0.162488i
$219$	0	0
$220$	7107.89	0.146857
$221$	6290.95i	0.128805i
$222$	0	0
$223$	42536.2	0.855361	0.427680	−	0.903930i	$-0.359331\pi$
$223$	42536.2	0.855361	0.427680	+	0.903930i	$0.359331\pi$
$224$	− 27497.9i	− 0.548029i
$225$	0	0
$226$	−75001.2	−1.46842
$227$	71712.1i	1.39168i	0.718195	+	0.695842i	$0.244969\pi$
$227$	71712.1i	1.39168i	−0.718195	+	0.695842i	$0.755031\pi$
$228$	0	0
$229$	69410.5	1.32359	0.661796	−	0.749684i	$-0.269795\pi$
$229$	69410.5	1.32359	0.661796	+	0.749684i	$0.269795\pi$
$230$	33247.7i	0.628502i
$231$	0	0
$232$	−32087.1	−0.596149
$233$	93296.6i	1.71852i	0.511541	+	0.859259i	$0.329075\pi$
$233$	93296.6i	1.71852i	−0.511541	+	0.859259i	$0.670925\pi$
$234$	0	0
$235$	8884.54	0.160879
$236$	− 1215.59i	− 0.0218254i
$237$	0	0
$238$	12280.9	0.216808
$239$	− 49917.2i	− 0.873885i	−0.899489	−	0.436943i	$-0.856061\pi$
$239$	− 49917.2i	− 0.873885i	0.899489	−	0.436943i	$-0.143939\pi$
$240$	0	0
$241$	33916.5	0.583951	0.291976	−	0.956426i	$-0.405687\pi$
$241$	33916.5	0.583951	0.291976	+	0.956426i	$0.405687\pi$
$242$	39069.4i	0.667123i
$243$	0	0
$244$	−15121.1	−0.253982
$245$	146709.i	2.44413i
$246$	0	0
$247$	9350.24	0.153260
$248$	19688.2i	0.320113i
$249$	0	0
$250$	535.488	0.00856781
$251$	52128.7i	0.827427i	0.910407	+	0.413714i	$0.135768\pi$
$251$	52128.7i	0.827427i	−0.910407	+	0.413714i	$0.864232\pi$
$252$	0	0
$253$	16261.3	0.254047
$254$	131628.i	2.04023i
$255$	0	0
$256$	32085.6	0.489587
$257$	39878.2i	0.603767i	0.953345	+	0.301883i	$0.0976153\pi$
$257$	39878.2i	0.603767i	−0.953345	+	0.301883i	$0.902385\pi$
$258$	0	0
$259$	−76079.8	−1.13415
$260$	17010.6i	0.251636i
$261$	0	0
$262$	−26238.4	−0.382239
$263$	− 20967.7i	− 0.303137i	−0.988447	−	0.151568i	$-0.951568\pi$
$263$	− 20967.7i	− 0.303137i	0.988447	−	0.151568i	$-0.0484324\pi$
$264$	0	0
$265$	−27039.9	−0.385046
$266$	− 18253.1i	− 0.257972i
$267$	0	0
$268$	2513.83	0.0349998
$269$	97957.2i	1.35373i	0.736107	+	0.676865i	$0.236662\pi$
$269$	97957.2i	1.35373i	−0.736107	+	0.676865i	$0.763338\pi$
$270$	0	0
$271$	−127544.	−1.73669	−0.868343	−	0.495964i	$-0.834815\pi$
$271$	−127544.	−1.73669	−0.868343	+	0.495964i	$0.834815\pi$
$272$	10245.6i	0.138484i
$273$	0	0
$274$	80287.2	1.06941
$275$	− 47040.9i	− 0.622028i
$276$	0	0
$277$	125419.	1.63457	0.817284	−	0.576235i	$-0.195479\pi$
$277$	125419.	1.63457	0.817284	+	0.576235i	$0.195479\pi$
$278$	− 138970.i	− 1.79817i
$279$	0	0
$280$	−164874.	−2.10299
$281$	123857.i	1.56858i	0.620394	+	0.784291i	$0.286973\pi$
$281$	123857.i	1.56858i	−0.620394	+	0.784291i	$0.713027\pi$
$282$	0	0
$283$	65236.8	0.814554	0.407277	−	0.913305i	$-0.366478\pi$
$283$	65236.8	0.814554	0.407277	+	0.913305i	$0.366478\pi$
$284$	18525.0i	0.229679i
$285$	0	0
$286$	57947.5	0.708439
$287$	211107.i	2.56294i
$288$	0	0
$289$	82287.5	0.985231
$290$	− 85302.8i	− 1.01430i
$291$	0	0
$292$	11310.1	0.132648
$293$	− 35520.8i	− 0.413759i	−0.978366	−	0.206880i	$-0.933669\pi$
$293$	− 35520.8i	− 0.413759i	0.978366	−	0.206880i	$-0.0663309\pi$
$294$	0	0
$295$	−16045.0	−0.184372
$296$	− 54133.6i	− 0.617851i
$297$	0	0
$298$	−114951.	−1.29444
$299$	38916.5i	0.435303i
$300$	0	0
$301$	16232.4	0.179164
$302$	− 5732.26i	− 0.0628510i
$303$	0	0
$304$	15228.0	0.164777
$305$	199589.i	2.14554i
$306$	0	0
$307$	−61923.0	−0.657015	−0.328507	−	0.944501i	$-0.606546\pi$
$307$	−61923.0	−0.657015	−0.328507	+	0.944501i	$0.606546\pi$
$308$	− 16241.5i	− 0.171208i
$309$	0	0
$310$	−52340.6	−0.544648
$311$	− 123642.i	− 1.27833i	−0.769068	−	0.639167i	$-0.779279\pi$
$311$	− 123642.i	− 1.27833i	0.769068	−	0.639167i	$-0.220721\pi$
$312$	0	0
$313$	−6179.96	−0.0630808	−0.0315404	−	0.999502i	$-0.510041\pi$
$313$	−6179.96	−0.0630808	−0.0315404	+	0.999502i	$0.510041\pi$
$314$	69570.6i	0.705612i
$315$	0	0
$316$	1804.35	0.0180695
$317$	141752.i	1.41062i	0.708898	+	0.705311i	$0.249193\pi$
$317$	141752.i	1.41062i	−0.708898	+	0.705311i	$0.750807\pi$
$318$	0	0
$319$	−41721.2	−0.409992
$320$	− 113239.i	− 1.10585i
$321$	0	0
$322$	75970.8	0.732715
$323$	1833.33i	0.0175726i
$324$	0	0
$325$	112578.	1.06583
$326$	− 94858.1i	− 0.892564i
$327$	0	0
$328$	−150210.	−1.39622
$329$	− 20301.1i	− 0.187555i
$330$	0	0
$331$	106533.	0.972365	0.486183	−	0.873857i	$-0.338389\pi$
$331$	106533.	0.972365	0.486183	+	0.873857i	$0.338389\pi$
$332$	8610.37i	0.0781170i
$333$	0	0
$334$	154322.	1.38336
$335$	− 33181.0i	− 0.295665i
$336$	0	0
$337$	−71154.3	−0.626529	−0.313265	−	0.949666i	$-0.601423\pi$
$337$	−71154.3	−0.626529	−0.313265	+	0.949666i	$0.601423\pi$
$338$	15230.3i	0.133314i
$339$	0	0
$340$	−3335.32	−0.0288523
$341$	25599.6i	0.220152i
$342$	0	0
$343$	140988.	1.19838
$344$	11550.0i	0.0976031i
$345$	0	0
$346$	52175.2	0.435825
$347$	159624.i	1.32568i	0.748760	+	0.662841i	$0.230650\pi$
$347$	159624.i	1.32568i	−0.748760	+	0.662841i	$0.769350\pi$
$348$	0	0
$349$	−185862.	−1.52595	−0.762973	−	0.646431i	$-0.776261\pi$
$349$	−185862.	−1.52595	−0.762973	+	0.646431i	$0.776261\pi$
$350$	− 219769.i	− 1.79403i
$351$	0	0
$352$	25440.4	0.205323
$353$	16071.9i	0.128978i	0.997918	+	0.0644892i	$0.0205418\pi$
$353$	16071.9i	0.128978i	−0.997918	+	0.0644892i	$0.979458\pi$
$354$	0	0
$355$	244519.	1.94024
$356$	− 28201.2i	− 0.222519i
$357$	0	0
$358$	77565.9	0.605208
$359$	31888.6i	0.247426i	0.992318	+	0.123713i	$0.0394803\pi$
$359$	31888.6i	0.247426i	−0.992318	+	0.123713i	$0.960520\pi$
$360$	0	0
$361$	−127596.	−0.979091
$362$	33038.0i	0.252114i
$363$	0	0
$364$	38869.0	0.293360
$365$	− 149287.i	− 1.12056i
$366$	0	0
$367$	22943.5	0.170344	0.0851720	−	0.996366i	$-0.472856\pi$
$367$	22943.5	0.170344	0.0851720	+	0.996366i	$0.472856\pi$
$368$	63380.3i	0.468014i
$369$	0	0
$370$	143913.	1.05123
$371$	61785.9i	0.448892i
$372$	0	0
$373$	261236.	1.87765	0.938826	−	0.344391i	$-0.111915\pi$
$373$	261236.	1.87765	0.938826	+	0.344391i	$0.111915\pi$
$374$	11362.0i	0.0812288i
$375$	0	0
$376$	14445.0	0.102174
$377$	− 99847.0i	− 0.702510i
$378$	0	0
$379$	66351.7	0.461928	0.230964	−	0.972962i	$-0.425812\pi$
$379$	66351.7	0.461928	0.230964	+	0.972962i	$0.425812\pi$
$380$	4957.28i	0.0343302i
$381$	0	0
$382$	165249.	1.13243
$383$	− 134431.i	− 0.916437i	−0.888840	−	0.458218i	$-0.848488\pi$
$383$	− 134431.i	− 0.916437i	0.888840	−	0.458218i	$-0.151512\pi$
$384$	0	0
$385$	−214378.	−1.44630
$386$	− 59032.7i	− 0.396204i
$387$	0	0
$388$	−11879.6	−0.0789113
$389$	121476.i	0.802770i	0.915910	+	0.401385i	$0.131471\pi$
$389$	121476.i	0.802770i	−0.915910	+	0.401385i	$0.868529\pi$
$390$	0	0
$391$	−7630.49	−0.0499113
$392$	238528.i	1.55227i
$393$	0	0
$394$	−134893.	−0.868952
$395$	− 23816.3i	− 0.152644i
$396$	0	0
$397$	4596.75	0.0291655	0.0145828	−	0.999894i	$-0.495358\pi$
$397$	4596.75	0.0291655	0.0145828	+	0.999894i	$0.495358\pi$
$398$	235901.i	1.48924i
$399$	0	0
$400$	183347.	1.14592
$401$	89891.5i	0.559023i	0.960142	+	0.279512i	$0.0901726\pi$
$401$	89891.5i	0.559023i	−0.960142	+	0.279512i	$0.909827\pi$
$402$	0	0
$403$	−61264.7	−0.377225
$404$	40548.3i	0.248434i
$405$	0	0
$406$	−194916.	−1.18249
$407$	− 70387.1i	− 0.424917i
$408$	0	0
$409$	298599.	1.78501	0.892507	−	0.451033i	$-0.148944\pi$
$409$	298599.	1.78501	0.892507	+	0.451033i	$0.148944\pi$
$410$	− 399331.i	− 2.37556i
$411$	0	0
$412$	40389.8	0.237945
$413$	36662.7i	0.214944i
$414$	0	0
$415$	113652.	0.659902
$416$	60883.8i	0.351816i
$417$	0	0
$418$	16887.3	0.0966512
$419$	− 223820.i	− 1.27489i	−0.770497	−	0.637443i	$-0.779992\pi$
$419$	− 223820.i	− 1.27489i	0.770497	−	0.637443i	$-0.220008\pi$
$420$	0	0
$421$	−159956.	−0.902478	−0.451239	−	0.892403i	$-0.649018\pi$
$421$	−159956.	−0.902478	−0.451239	+	0.892403i	$0.649018\pi$
$422$	− 128779.i	− 0.723134i
$423$	0	0
$424$	−43963.0	−0.244543
$425$	22073.6i	0.122207i
$426$	0	0
$427$	456059.	2.50130
$428$	32298.3i	0.176316i
$429$	0	0
$430$	−30705.3	−0.166064
$431$	− 49406.5i	− 0.265968i	−0.991118	−	0.132984i	$-0.957544\pi$
$431$	− 49406.5i	− 0.265968i	0.991118	−	0.132984i	$-0.0424559\pi$
$432$	0	0
$433$	185818.	0.991089	0.495545	−	0.868583i	$-0.334969\pi$
$433$	185818.	0.991089	0.495545	+	0.868583i	$0.334969\pi$
$434$	119598.i	0.634957i
$435$	0	0
$436$	4792.12	0.0252089
$437$	11341.2i	0.0593876i
$438$	0	0
$439$	267337.	1.38717	0.693585	−	0.720374i	$-0.256030\pi$
$439$	267337.	1.38717	0.693585	+	0.720374i	$0.256030\pi$
$440$	− 152538.i	− 0.787901i
$441$	0	0
$442$	−27191.4	−0.139183
$443$	− 132301.i	− 0.674146i	−0.941478	−	0.337073i	$-0.890563\pi$
$443$	− 132301.i	− 0.674146i	0.941478	−	0.337073i	$-0.109437\pi$
$444$	0	0
$445$	−372239.	−1.87976
$446$	183854.i	0.924282i
$447$	0	0
$448$	−258749.	−1.28921
$449$	137875.i	0.683899i	0.939718	+	0.341949i	$0.111087\pi$
$449$	137875.i	0.683899i	−0.939718	+	0.341949i	$0.888913\pi$
$450$	0	0
$451$	−195311.	−0.960225
$452$	− 46543.7i	− 0.227816i
$453$	0	0
$454$	−309961.	−1.50382
$455$	− 513047.i	− 2.47819i
$456$	0	0
$457$	−412401.	−1.97464	−0.987318	−	0.158758i	$-0.949251\pi$
$457$	−412401.	−1.97464	−0.987318	+	0.158758i	$0.949251\pi$
$458$	300013.i	1.43024i
$459$	0	0
$460$	−20632.6	−0.0975078
$461$	75628.8i	0.355865i	0.984043	+	0.177932i	$0.0569409\pi$
$461$	75628.8i	0.355865i	−0.984043	+	0.177932i	$0.943059\pi$
$462$	0	0
$463$	270017.	1.25959	0.629796	−	0.776761i	$-0.283139\pi$
$463$	270017.	1.25959	0.629796	+	0.776761i	$0.283139\pi$
$464$	− 162613.i	− 0.755300i
$465$	0	0
$466$	−403256.	−1.85699
$467$	− 22216.8i	− 0.101870i	−0.998702	−	0.0509352i	$-0.983780\pi$
$467$	− 22216.8i	− 0.101870i	0.998702	−	0.0509352i	$-0.0162202\pi$
$468$	0	0
$469$	−75818.4	−0.344690
$470$	38401.7i	0.173842i
$471$	0	0
$472$	−26086.9	−0.117095
$473$	15017.8i	0.0671250i
$474$	0	0
$475$	32807.9	0.145409
$476$	7621.18i	0.0336363i
$477$	0	0
$478$	215757.	0.944299
$479$	402052.i	1.75231i	0.482029	+	0.876155i	$0.339900\pi$
$479$	402052.i	1.75231i	−0.482029	+	0.876155i	$0.660100\pi$
$480$	0	0
$481$	168450.	0.728084
$482$	146597.i	0.631003i
$483$	0	0
$484$	−24245.4	−0.103500
$485$	156804.i	0.666612i
$486$	0	0
$487$	−262091.	−1.10508	−0.552541	−	0.833486i	$-0.686342\pi$
$487$	−262091.	−1.10508	−0.552541	+	0.833486i	$0.686342\pi$
$488$	324503.i	1.36263i
$489$	0	0
$490$	−634120.	−2.64107
$491$	− 109085.i	− 0.452481i	−0.974071	−	0.226240i	$-0.927357\pi$
$491$	− 109085.i	− 0.452481i	0.974071	−	0.226240i	$-0.0726435\pi$
$492$	0	0
$493$	19577.3	0.0805490
$494$	40414.6i	0.165609i
$495$	0	0
$496$	−99777.2	−0.405572
$497$	− 558724.i	− 2.26196i
$498$	0	0
$499$	191287.	0.768218	0.384109	−	0.923288i	$-0.374509\pi$
$499$	191287.	0.768218	0.384109	+	0.923288i	$0.374509\pi$
$500$	332.309i	0.00132924i
$501$	0	0
$502$	−225316.	−0.894098
$503$	− 435166.i	− 1.71996i	−0.510326	−	0.859981i	$-0.670475\pi$
$503$	− 435166.i	− 1.71996i	0.510326	−	0.859981i	$-0.329525\pi$
$504$	0	0
$505$	535214.	2.09867
$506$	70286.3i	0.274517i
$507$	0	0
$508$	−81684.5	−0.316528
$509$	272661.i	1.05242i	0.850356	+	0.526208i	$0.176387\pi$
$509$	272661.i	1.05242i	−0.850356	+	0.526208i	$0.823613\pi$
$510$	0	0
$511$	−341120.	−1.30637
$512$	− 169522.i	− 0.646676i
$513$	0	0
$514$	−172366.	−0.652416
$515$	− 533121.i	− 2.01007i
$516$	0	0
$517$	18782.1	0.0702688
$518$	− 328840.i	− 1.22553i
$519$	0	0
$520$	365052.	1.35005
$521$	− 188991.i	− 0.696252i	−0.937448	−	0.348126i	$-0.886818\pi$
$521$	− 188991.i	− 0.696252i	0.937448	−	0.348126i	$-0.113182\pi$
$522$	0	0
$523$	539131.	1.97102	0.985510	−	0.169615i	$-0.0542523\pi$
$523$	539131.	1.97102	0.985510	+	0.169615i	$0.0542523\pi$
$524$	− 16282.8i	− 0.0593018i
$525$	0	0
$526$	90628.6	0.327562
$527$	− 12012.4i	− 0.0432522i
$528$	0	0
$529$	232638.	0.831322
$530$	− 116875.i	− 0.416072i
$531$	0	0
$532$	11327.4	0.0400226
$533$	− 467417.i	− 1.64532i
$534$	0	0
$535$	426318.	1.48945
$536$	− 53947.6i	− 0.187777i
$537$	0	0
$538$	−423401.	−1.46281
$539$	310145.i	1.06755i
$540$	0	0
$541$	−23563.5	−0.0805090	−0.0402545	−	0.999189i	$-0.512817\pi$
$541$	−23563.5	−0.0805090	−0.0402545	+	0.999189i	$0.512817\pi$
$542$	− 551284.i	− 1.87662i
$543$	0	0
$544$	−11937.7	−0.0403388
$545$	− 63253.0i	− 0.212955i
$546$	0	0
$547$	216153.	0.722415	0.361208	−	0.932485i	$-0.382365\pi$
$547$	216153.	0.722415	0.361208	+	0.932485i	$0.382365\pi$
$548$	49824.0i	0.165912i
$549$	0	0
$550$	203325.	0.672148
$551$	− 29097.8i	− 0.0958422i
$552$	0	0
$553$	−54420.0	−0.177954
$554$	542098.i	1.76627i
$555$	0	0
$556$	86240.8	0.278974
$557$	168431.i	0.542890i	0.962454	+	0.271445i	$0.0875015\pi$
$557$	168431.i	0.542890i	−0.962454	+	0.271445i	$0.912498\pi$
$558$	0	0
$559$	−35940.6	−0.115017
$560$	− 835561.i	− 2.66442i
$561$	0	0
$562$	−535346.	−1.69497
$563$	143313.i	0.452135i	0.974112	+	0.226068i	$0.0725870\pi$
$563$	143313.i	0.452135i	−0.974112	+	0.226068i	$0.927413\pi$
$564$	0	0
$565$	−614349.	−1.92450
$566$	281973.i	0.880187i
$567$	0	0
$568$	397553.	1.23225
$569$	− 65667.9i	− 0.202828i	−0.994844	−	0.101414i	$-0.967663\pi$
$569$	− 65667.9i	− 0.202828i	0.994844	−	0.101414i	$-0.0323367\pi$
$570$	0	0
$571$	−470888.	−1.44426	−0.722129	−	0.691758i	$-0.756837\pi$
$571$	−470888.	−1.44426	−0.722129	+	0.691758i	$0.756837\pi$
$572$	35960.6i	0.109909i
$573$	0	0
$574$	−912468.	−2.76945
$575$	136549.i	0.413004i
$576$	0	0
$577$	609690.	1.83129	0.915645	−	0.401987i	$-0.131680\pi$
$577$	609690.	1.83129	0.915645	+	0.401987i	$0.131680\pi$
$578$	355672.i	1.06462i
$579$	0	0
$580$	52936.6	0.157362
$581$	− 259693.i	− 0.769322i
$582$	0	0
$583$	−57162.8	−0.168181
$584$	− 242719.i	− 0.711670i
$585$	0	0
$586$	153532.	0.447098
$587$	467266.i	1.35609i	0.735021	+	0.678044i	$0.237172\pi$
$587$	467266.i	1.35609i	−0.735021	+	0.678044i	$0.762828\pi$
$588$	0	0
$589$	−17854.0	−0.0514642
$590$	− 69351.4i	− 0.199228i
$591$	0	0
$592$	274342.	0.782796
$593$	− 578586.i	− 1.64535i	−0.568512	−	0.822675i	$-0.692481\pi$
$593$	− 578586.i	− 1.64535i	0.568512	−	0.822675i	$-0.307519\pi$
$594$	0	0
$595$	100595.	0.284147
$596$	− 71335.7i	− 0.200823i
$597$	0	0
$598$	−168209.	−0.470377
$599$	− 432215.i	− 1.20461i	−0.798266	−	0.602304i	$-0.794249\pi$
$599$	− 432215.i	− 1.20461i	0.798266	−	0.602304i	$-0.205751\pi$
$600$	0	0
$601$	−210477.	−0.582715	−0.291358	−	0.956614i	$-0.594107\pi$
$601$	−210477.	−0.582715	−0.291358	+	0.956614i	$0.594107\pi$
$602$	70161.4i	0.193600i
$603$	0	0
$604$	3557.28	0.00975090
$605$	320025.i	0.874325i
$606$	0	0
$607$	83046.2	0.225394	0.112697	−	0.993629i	$-0.464051\pi$
$607$	83046.2	0.225394	0.112697	+	0.993629i	$0.464051\pi$
$608$	17743.0i	0.0479976i
$609$	0	0
$610$	−862684.	−2.31842
$611$	44949.2i	0.120404i
$612$	0	0
$613$	127458.	0.339192	0.169596	−	0.985514i	$-0.445754\pi$
$613$	127458.	0.339192	0.169596	+	0.985514i	$0.445754\pi$
$614$	− 267650.i	− 0.709954i
$615$	0	0
$616$	−348547.	−0.918544
$617$	− 643606.i	− 1.69063i	−0.534265	−	0.845317i	$-0.679412\pi$
$617$	− 643606.i	− 1.69063i	0.534265	−	0.845317i	$-0.320588\pi$
$618$	0	0
$619$	379883.	0.991445	0.495722	−	0.868481i	$-0.334903\pi$
$619$	379883.	0.991445	0.495722	+	0.868481i	$0.334903\pi$
$620$	− 32481.2i	− 0.0844984i
$621$	0	0
$622$	534417.	1.38134
$623$	850563.i	2.19144i
$624$	0	0
$625$	−388426.	−0.994370
$626$	− 26711.7i	− 0.0681636i
$627$	0	0
$628$	−43173.6	−0.109471
$629$	33028.6i	0.0834813i
$630$	0	0
$631$	−577341.	−1.45002	−0.725010	−	0.688738i	$-0.758165\pi$
$631$	−577341.	−1.45002	−0.725010	+	0.688738i	$0.758165\pi$
$632$	− 38721.8i	− 0.0969442i
$633$	0	0
$634$	−612695.	−1.52428
$635$	1.07819e6i	2.67391i
$636$	0	0
$637$	−742238.	−1.82921
$638$	− 180332.i	− 0.443027i
$639$	0	0
$640$	681998.	1.66503
$641$	569828.i	1.38685i	0.720531	+	0.693423i	$0.243898\pi$
$641$	569828.i	1.38685i	−0.720531	+	0.693423i	$0.756102\pi$
$642$	0	0
$643$	−601978.	−1.45599	−0.727995	−	0.685582i	$-0.759548\pi$
$643$	−601978.	−1.45599	−0.727995	+	0.685582i	$0.759548\pi$
$644$	47145.4i	0.113676i
$645$	0	0
$646$	−7924.22	−0.0189885
$647$	545071.i	1.30210i	0.759034	+	0.651051i	$0.225672\pi$
$647$	545071.i	1.30210i	−0.759034	+	0.651051i	$0.774328\pi$
$648$	0	0
$649$	−33919.4	−0.0805303
$650$	486596.i	1.15171i
$651$	0	0
$652$	58866.4	0.138475
$653$	282580.i	0.662696i	0.943509	+	0.331348i	$0.107503\pi$
$653$	282580.i	0.662696i	−0.943509	+	0.331348i	$0.892497\pi$
$654$	0	0
$655$	−214924.	−0.500959
$656$	− 761246.i	− 1.76896i
$657$	0	0
$658$	87747.5	0.202667
$659$	244946.i	0.564026i	0.959411	+	0.282013i	$0.0910021\pi$
$659$	244946.i	0.564026i	−0.959411	+	0.282013i	$0.908998\pi$
$660$	0	0
$661$	−191463.	−0.438210	−0.219105	−	0.975701i	$-0.570314\pi$
$661$	−191463.	−0.438210	−0.219105	+	0.975701i	$0.570314\pi$
$662$	460469.i	1.05071i
$663$	0	0
$664$	184781.	0.419104
$665$	− 149514.i	− 0.338095i
$666$	0	0
$667$	121108.	0.272220
$668$	95768.2i	0.214619i
$669$	0	0
$670$	143418.	0.319489
$671$	421935.i	0.937131i
$672$	0	0
$673$	−493313.	−1.08916	−0.544581	−	0.838708i	$-0.683311\pi$
$673$	−493313.	−1.08916	−0.544581	+	0.838708i	$0.683311\pi$
$674$	− 307550.i	− 0.677012i
$675$	0	0
$676$	−9451.53	−0.0206828
$677$	514657.i	1.12290i	0.827511	+	0.561449i	$0.189756\pi$
$677$	514657.i	1.12290i	−0.827511	+	0.561449i	$0.810244\pi$
$678$	0	0
$679$	358296.	0.777145
$680$	71577.1i	0.154795i
$681$	0	0
$682$	−110649.	−0.237891
$683$	503702.i	1.07977i	0.841738	+	0.539887i	$0.181533\pi$
$683$	503702.i	1.07977i	−0.841738	+	0.539887i	$0.818467\pi$
$684$	0	0
$685$	657647.	1.40156
$686$	609394.i	1.29494i
$687$	0	0
$688$	−58533.6	−0.123660
$689$	− 136802.i	− 0.288173i
$690$	0	0
$691$	650009.	1.36133	0.680665	−	0.732595i	$-0.261691\pi$
$691$	650009.	1.36133	0.680665	+	0.732595i	$0.261691\pi$
$692$	32378.5i	0.0676153i
$693$	0	0
$694$	−689944.	−1.43250
$695$	− 1.13833e6i	− 2.35666i
$696$	0	0
$697$	91648.1	0.188650
$698$	− 803350.i	− 1.64890i
$699$	0	0
$700$	136383.	0.278332
$701$	847827.i	1.72533i	0.505779	+	0.862663i	$0.331205\pi$
$701$	847827.i	1.72533i	−0.505779	+	0.862663i	$0.668795\pi$
$702$	0	0
$703$	49090.4	0.0993312
$704$	− 239388.i	− 0.483012i
$705$	0	0
$706$	−69467.5	−0.139371
$707$	− 1.22296e6i	− 2.44666i
$708$	0	0
$709$	289988.	0.576884	0.288442	−	0.957497i	$-0.406863\pi$
$709$	289988.	0.576884	0.288442	+	0.957497i	$0.406863\pi$
$710$	1.05689e6i	2.09658i
$711$	0	0
$712$	−605207.	−1.19383
$713$	− 74309.9i	− 0.146173i
$714$	0	0
$715$	474659.	0.928473
$716$	48135.3i	0.0938940i
$717$	0	0
$718$	−137832.	−0.267363
$719$	− 89042.8i	− 0.172243i	−0.996285	−	0.0861214i	$-0.972553\pi$
$719$	− 89042.8i	− 0.172243i	0.996285	−	0.0861214i	$-0.0274473\pi$
$720$	0	0
$721$	−1.21818e6	−2.34336
$722$	− 551509.i	− 1.05798i
$723$	0	0
$724$	−20502.5	−0.0391137
$725$	− 350341.i	− 0.666523i
$726$	0	0
$727$	−513696.	−0.971935	−0.485967	−	0.873977i	$-0.661533\pi$
$727$	−513696.	−0.971935	−0.485967	+	0.873977i	$0.661533\pi$
$728$	− 834142.i	− 1.57390i
$729$	0	0
$730$	645264.	1.21085
$731$	− 7046.99i	− 0.0131877i
$732$	0	0
$733$	964932.	1.79593	0.897963	−	0.440070i	$-0.145046\pi$
$733$	964932.	1.79593	0.897963	+	0.440070i	$0.145046\pi$
$734$	99168.6i	0.184070i
$735$	0	0
$736$	−73847.9	−0.136327
$737$	− 70145.2i	− 0.129141i
$738$	0	0
$739$	−430132.	−0.787613	−0.393806	−	0.919193i	$-0.628842\pi$
$739$	−430132.	−0.787613	−0.393806	+	0.919193i	$0.628842\pi$
$740$	89308.5i	0.163091i
$741$	0	0
$742$	−267057.	−0.485062
$743$	115592.i	0.209386i	0.994505	+	0.104693i	$0.0333861\pi$
$743$	115592.i	0.209386i	−0.994505	+	0.104693i	$0.966614\pi$
$744$	0	0
$745$	−941588.	−1.69648
$746$	1.12914e6i	2.02895i
$747$	0	0
$748$	−7050.93	−0.0126021
$749$	− 974134.i	− 1.73642i
$750$	0	0
$751$	105239.	0.186594	0.0932971	−	0.995638i	$-0.470259\pi$
$751$	105239.	0.186594	0.0932971	+	0.995638i	$0.470259\pi$
$752$	73205.3i	0.129451i
$753$	0	0
$754$	431569.	0.759115
$755$	− 46954.0i	− 0.0823718i
$756$	0	0
$757$	−142474.	−0.248625	−0.124312	−	0.992243i	$-0.539672\pi$
$757$	−142474.	−0.248625	−0.124312	+	0.992243i	$0.539672\pi$
$758$	286792.i	0.499148i
$759$	0	0
$760$	106385.	0.184184
$761$	− 108327.i	− 0.187055i	−0.995617	−	0.0935273i	$-0.970186\pi$
$761$	− 108327.i	− 0.187055i	0.995617	−	0.0935273i	$-0.0298142\pi$
$762$	0	0
$763$	−144533.	−0.248266
$764$	102549.i	0.175689i
$765$	0	0
$766$	581052.	0.990279
$767$	− 81175.8i	− 0.137986i
$768$	0	0
$769$	−61612.6	−0.104188	−0.0520939	−	0.998642i	$-0.516590\pi$
$769$	−61612.6	−0.104188	−0.0520939	+	0.998642i	$0.516590\pi$
$770$	− 926605.i	− 1.56283i
$771$	0	0
$772$	36634.1	0.0614683
$773$	− 162930.i	− 0.272673i	−0.990663	−	0.136337i	$-0.956467\pi$
$773$	− 162930.i	− 0.272673i	0.990663	−	0.136337i	$-0.0435329\pi$
$774$	0	0
$775$	−214964.	−0.357901
$776$	254941.i	0.423366i
$777$	0	0
$778$	−525056.	−0.867453
$779$	− 136216.i	− 0.224468i
$780$	0	0
$781$	516917.	0.847460
$782$	− 32981.3i	− 0.0539330i
$783$	0	0
$784$	−1.20883e6	−1.96667
$785$	569865.i	0.924769i
$786$	0	0
$787$	958257.	1.54715	0.773575	−	0.633705i	$-0.218466\pi$
$787$	958257.	1.54715	0.773575	+	0.633705i	$0.218466\pi$
$788$	− 83710.7i	− 0.134812i
$789$	0	0
$790$	102941.	0.164943
$791$	1.40378e6i	2.24361i
$792$	0	0
$793$	−1.00977e6	−1.60575
$794$	19868.5i	0.0315156i
$795$	0	0
$796$	−146394.	−0.231045
$797$	499593.i	0.786502i	0.919431	+	0.393251i	$0.128650\pi$
$797$	499593.i	0.786502i	−0.919431	+	0.393251i	$0.871350\pi$
$798$	0	0
$799$	−8813.34	−0.0138053
$800$	213628.i	0.333793i
$801$	0	0
$802$	−388538.	−0.604067
$803$	− 315595.i	− 0.489440i
$804$	0	0
$805$	622291.	0.960289
$806$	− 264805.i	− 0.407620i
$807$	0	0
$808$	870181.	1.33287
$809$	− 881779.i	− 1.34730i	−0.739053	−	0.673648i	$-0.764727\pi$
$809$	− 881779.i	− 1.34730i	0.739053	−	0.673648i	$-0.235273\pi$
$810$	0	0
$811$	−601631.	−0.914721	−0.457361	−	0.889281i	$-0.651205\pi$
$811$	−601631.	−0.914721	−0.457361	+	0.889281i	$0.651205\pi$
$812$	− 120960.i	− 0.183455i
$813$	0	0
$814$	304234.	0.459155
$815$	− 777001.i	− 1.16979i
$816$	0	0
$817$	−10473.9	−0.0156915
$818$	1.29064e6i	1.92884i
$819$	0	0
$820$	247814.	0.368551
$821$	− 949142.i	− 1.40814i	−0.710132	−	0.704068i	$-0.751365\pi$
$821$	− 949142.i	− 1.40814i	0.710132	−	0.704068i	$-0.248635\pi$
$822$	0	0
$823$	1.28137e6	1.89180	0.945900	−	0.324459i	$-0.105182\pi$
$823$	1.28137e6	1.89180	0.945900	+	0.324459i	$0.105182\pi$
$824$	− 866778.i	− 1.27660i
$825$	0	0
$826$	−158467.	−0.232263
$827$	− 1712.52i	− 0.00250395i	−0.999999	−	0.00125197i	$-0.999601\pi$
$827$	− 1712.52i	− 0.00250395i	0.999999	−	0.00125197i	$-0.000398515\pi$
$828$	0	0
$829$	773201.	1.12508	0.562540	−	0.826770i	$-0.309824\pi$
$829$	773201.	1.12508	0.562540	+	0.826770i	$0.309824\pi$
$830$	491237.i	0.713074i
$831$	0	0
$832$	572903.	0.827627
$833$	− 145533.i	− 0.209735i
$834$	0	0
$835$	1.26408e6	1.81302
$836$	10479.8i	0.0149948i
$837$	0	0
$838$	967420.	1.37761
$839$	1.14963e6i	1.63317i	0.577223	+	0.816587i	$0.304137\pi$
$839$	1.14963e6i	1.63317i	−0.577223	+	0.816587i	$0.695863\pi$
$840$	0	0
$841$	396559.	0.560681
$842$	− 691379.i	− 0.975196i
$843$	0	0
$844$	79916.5	0.112189
$845$	124755.i	0.174720i
$846$	0	0
$847$	731254.	1.01930
$848$	− 222798.i	− 0.309828i
$849$	0	0
$850$	−95408.6	−0.132053
$851$	204318.i	0.282129i
$852$	0	0
$853$	316153.	0.434510	0.217255	−	0.976115i	$-0.430290\pi$
$853$	316153.	0.434510	0.217255	+	0.976115i	$0.430290\pi$
$854$	1.97123e6i	2.70284i
$855$	0	0
$856$	693132.	0.945951
$857$	− 218053.i	− 0.296894i	−0.988920	−	0.148447i	$-0.952573\pi$
$857$	− 218053.i	− 0.296894i	0.988920	−	0.148447i	$-0.0474274\pi$
$858$	0	0
$859$	217908.	0.295316	0.147658	−	0.989039i	$-0.452827\pi$
$859$	217908.	0.295316	0.147658	+	0.989039i	$0.452827\pi$
$860$	− 19054.9i	− 0.0257638i
$861$	0	0
$862$	213550.	0.287399
$863$	609960.i	0.818992i	0.912312	+	0.409496i	$0.134295\pi$
$863$	609960.i	0.818992i	−0.912312	+	0.409496i	$0.865705\pi$
$864$	0	0
$865$	427377.	0.571188
$866$	803163.i	1.07095i
$867$	0	0
$868$	−74219.2	−0.0985092
$869$	− 50348.0i	− 0.0666719i
$870$	0	0
$871$	167871.	0.221279
$872$	− 102840.i	− 0.135248i
$873$	0	0
$874$	−49020.1	−0.0641728
$875$	− 10022.6i	− 0.0130908i
$876$	0	0
$877$	531190.	0.690638	0.345319	−	0.938485i	$-0.387771\pi$
$877$	531190.	0.690638	0.345319	+	0.938485i	$0.387771\pi$
$878$	1.15551e6i	1.49894i
$879$	0	0
$880$	773040.	0.998244
$881$	− 788972.i	− 1.01651i	−0.861208	−	0.508253i	$-0.830292\pi$
$881$	− 788972.i	− 1.01651i	0.861208	−	0.508253i	$-0.169708\pi$
$882$	0	0
$883$	316017.	0.405311	0.202656	−	0.979250i	$-0.435043\pi$
$883$	316017.	0.405311	0.202656	+	0.979250i	$0.435043\pi$
$884$	− 16874.2i	− 0.0215933i
$885$	0	0
$886$	571843.	0.728466
$887$	− 554734.i	− 0.705078i	−0.935797	−	0.352539i	$-0.885318\pi$
$887$	− 554734.i	− 0.705078i	0.935797	−	0.352539i	$-0.114682\pi$
$888$	0	0
$889$	2.46365e6	3.11727
$890$	− 1.60893e6i	− 2.03122i
$891$	0	0
$892$	−114095.	−0.143396
$893$	13099.3i	0.0164265i
$894$	0	0
$895$	635357.	0.793180
$896$	− 1.55836e6i	− 1.94112i
$897$	0	0
$898$	−595936.	−0.739004
$899$	190655.i	0.235900i
$900$	0	0
$901$	26823.2	0.0330416
$902$	− 844193.i	− 1.03760i
$903$	0	0
$904$	−998843.	−1.22225
$905$	270620.i	0.330417i
$906$	0	0
$907$	29557.3	0.0359294	0.0179647	−	0.999839i	$-0.494281\pi$
$907$	29557.3	0.0359294	0.0179647	+	0.999839i	$0.494281\pi$
$908$	− 192354.i	− 0.233307i
$909$	0	0
$910$	2.21755e6	2.67787
$911$	− 619357.i	− 0.746284i	−0.927774	−	0.373142i	$-0.878280\pi$
$911$	− 619357.i	− 0.746284i	0.927774	−	0.373142i	$-0.121720\pi$
$912$	0	0
$913$	240262.	0.288232
$914$	− 1.78252e6i	− 2.13374i
$915$	0	0
$916$	−186180.	−0.221892
$917$	491099.i	0.584024i
$918$	0	0
$919$	−749127.	−0.887001	−0.443501	−	0.896274i	$-0.646264\pi$
$919$	−749127.	−0.887001	−0.443501	+	0.896274i	$0.646264\pi$
$920$	442783.i	0.523137i
$921$	0	0
$922$	−326890.	−0.384539
$923$	1.23708e6i	1.45210i
$924$	0	0
$925$	591054.	0.690787
$926$	1.16710e6i	1.36108i
$927$	0	0
$928$	189469.	0.220010
$929$	1.00435e6i	1.16373i	0.813285	+	0.581865i	$0.197677\pi$
$929$	1.00435e6i	1.16373i	−0.813285	+	0.581865i	$0.802323\pi$
$930$	0	0
$931$	−216306.	−0.249556
$932$	− 250250.i	− 0.288099i
$933$	0	0
$934$	96027.8	0.110079
$935$	93068.0i	0.106458i
$936$	0	0
$937$	−1.37731e6	−1.56875	−0.784373	−	0.620289i	$-0.787015\pi$
$937$	−1.37731e6	−1.56875	−0.784373	+	0.620289i	$0.787015\pi$
$938$	− 327710.i	− 0.372464i
$939$	0	0
$940$	−23831.0	−0.0269704
$941$	− 442815.i	− 0.500084i	−0.968235	−	0.250042i	$-0.919556\pi$
$941$	− 442815.i	− 0.500084i	0.968235	−	0.250042i	$-0.0804444\pi$
$942$	0	0
$943$	566945.	0.637554
$944$	− 132205.i	− 0.148355i
$945$	0	0
$946$	−64911.5	−0.0725336
$947$	1.42993e6i	1.59446i	0.603675	+	0.797230i	$0.293702\pi$
$947$	1.42993e6i	1.59446i	−0.603675	+	0.797230i	$0.706298\pi$
$948$	0	0
$949$	755281.	0.838642
$950$	141806.i	0.157125i
$951$	0	0
$952$	163553.	0.180462
$953$	− 661295.i	− 0.728130i	−0.931373	−	0.364065i	$-0.881389\pi$
$953$	− 661295.i	− 0.728130i	0.931373	−	0.364065i	$-0.118611\pi$
$954$	0	0
$955$	1.35359e6	1.48415
$956$	133893.i	0.146502i
$957$	0	0
$958$	−1.73779e6	−1.89350
$959$	− 1.50272e6i	− 1.63396i
$960$	0	0
$961$	−806538.	−0.873329
$962$	728093.i	0.786749i
$963$	0	0
$964$	−90974.3	−0.0978959
$965$	− 483548.i	− 0.519260i
$966$	0	0
$967$	−767667.	−0.820956	−0.410478	−	0.911871i	$-0.634638\pi$
$967$	−767667.	−0.820956	−0.410478	+	0.911871i	$0.634638\pi$
$968$	520315.i	0.555284i
$969$	0	0
$970$	−677754.	−0.720325
$971$	815289.i	0.864715i	0.901702	+	0.432358i	$0.142318\pi$
$971$	815289.i	0.864715i	−0.901702	+	0.432358i	$0.857682\pi$
$972$	0	0
$973$	−2.60107e6	−2.74742
$974$	− 1.13284e6i	− 1.19412i
$975$	0	0
$976$	−1.64454e6	−1.72641
$977$	− 693287.i	− 0.726313i	−0.931728	−	0.363156i	$-0.881699\pi$
$977$	− 693287.i	− 0.726313i	0.931728	−	0.363156i	$-0.118301\pi$
$978$	0	0
$979$	−786919.	−0.821041
$980$	− 393518.i	− 0.409743i
$981$	0	0
$982$	471496.	0.488940
$983$	− 1.02128e6i	− 1.05691i	−0.848960	−	0.528457i	$-0.822771\pi$
$983$	− 1.02128e6i	− 1.05691i	0.848960	−	0.528457i	$-0.177229\pi$
$984$	0	0
$985$	−1.10493e6	−1.13884
$986$	84619.2i	0.0870393i
$987$	0	0
$988$	−25080.2	−0.0256931
$989$	− 43593.4i	− 0.0445685i
$990$	0	0
$991$	−1.13517e6	−1.15589	−0.577943	−	0.816077i	$-0.696145\pi$
$991$	−1.13517e6	−1.15589	−0.577943	+	0.816077i	$0.696145\pi$
$992$	− 116256.i	− 0.118138i
$993$	0	0
$994$	2.41497e6	2.44422
$995$	1.93231e6i	1.95178i
$996$	0	0
$997$	−1.33077e6	−1.33879	−0.669394	−	0.742907i	$-0.733446\pi$
$997$	−1.33077e6	−1.33879	−0.669394	+	0.742907i	$0.733446\pi$
$998$	826801.i	0.830118i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.57	yes	76
3.2	odd	2	inner	531.5.b.a.296.20	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.20	✓	76	3.2	odd	2	inner
531.5.b.a.296.57	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)

\(f(q)\)	\(=\)	\(q+4.32230i q^{2} -2.68230 q^{4} +35.4048i q^{5} +80.8997 q^{7} +57.5631i q^{8} +O(q^{10})\)	\(q+4.32230i q^{2} -2.68230 q^{4} +35.4048i q^{5} +80.8997 q^{7} +57.5631i q^{8} -153.030 q^{10} +74.8463i q^{11} -179.122 q^{13} +349.673i q^{14} -291.722 q^{16} -35.1211i q^{17} -52.2004 q^{19} -94.9664i q^{20} -323.509 q^{22} -217.263i q^{23} -628.499 q^{25} -774.219i q^{26} -216.997 q^{28} +557.425i q^{29} +342.028 q^{31} -339.901i q^{32} +151.804 q^{34} +2864.24i q^{35} -940.422 q^{37} -225.626i q^{38} -2038.01 q^{40} +2609.49i q^{41} +200.649 q^{43} -200.761i q^{44} +939.075 q^{46} -250.942i q^{47} +4143.76 q^{49} -2716.56i q^{50} +480.459 q^{52} +763.735i q^{53} -2649.92 q^{55} +4656.84i q^{56} -2409.36 q^{58} +453.188i q^{59} +5637.34 q^{61} +1478.35i q^{62} -3198.40 q^{64} -6341.77i q^{65} -937.190 q^{67} +94.2054i q^{68} -12380.1 q^{70} -6906.38i q^{71} -4216.58 q^{73} -4064.79i q^{74} +140.017 q^{76} +6055.04i q^{77} -672.685 q^{79} -10328.4i q^{80} -11279.0 q^{82} -3210.06i q^{83} +1243.45 q^{85} +867.264i q^{86} -4308.39 q^{88} +10513.8i q^{89} -14490.9 q^{91} +582.764i q^{92} +1084.65 q^{94} -1848.14i q^{95} +4428.89 q^{97} +17910.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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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