LMFDB - Embedded newform 531.3.c.c.235.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,3,Mod(235,531)]

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([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	531.c (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:	$14.4687020375$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 $ x^20 + 60x^18 + 1522x^16 + 21286x^14 + 179593x^12 + 941588x^10 + 3057025x^8 + 5962230x^6 + 6517782x^4 + 3428244*x^2 + 570861
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.7
Root			$-1.61360i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.c.235.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61360i q^{2} +1.39629 q^{4} +6.36659 q^{5} +6.66564 q^{7} -8.70746i q^{8} +O(q^{10})$	\(q-1.61360i q^{2} +1.39629 q^{4} +6.36659 q^{5} +6.66564 q^{7} -8.70746i q^{8} -10.2731i q^{10} -16.0509i q^{11} +7.84026i q^{13} -10.7557i q^{14} -8.46523 q^{16} -18.9063 q^{17} +7.10898 q^{19} +8.88958 q^{20} -25.8997 q^{22} +33.6730i q^{23} +15.5334 q^{25} +12.6511 q^{26} +9.30714 q^{28} +46.2052 q^{29} -29.5207i q^{31} -21.1703i q^{32} +30.5072i q^{34} +42.4373 q^{35} -1.91983i q^{37} -11.4711i q^{38} -55.4368i q^{40} -46.4596 q^{41} +21.6675i q^{43} -22.4116i q^{44} +54.3348 q^{46} +75.3742i q^{47} -4.56931 q^{49} -25.0647i q^{50} +10.9473i q^{52} -19.5526 q^{53} -102.189i q^{55} -58.0408i q^{56} -74.5569i q^{58} +(4.68014 + 58.8141i) q^{59} -41.0613i q^{61} -47.6346 q^{62} -68.0214 q^{64} +49.9157i q^{65} -90.1839i q^{67} -26.3986 q^{68} -68.4770i q^{70} -57.1922 q^{71} +69.6739i q^{73} -3.09784 q^{74} +9.92617 q^{76} -106.989i q^{77} +118.749 q^{79} -53.8946 q^{80} +74.9673i q^{82} -86.3181i q^{83} -120.368 q^{85} +34.9627 q^{86} -139.762 q^{88} -38.4935i q^{89} +52.2603i q^{91} +47.0172i q^{92} +121.624 q^{94} +45.2599 q^{95} -24.0445i q^{97} +7.37305i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7	$ 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

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 $

$2$		−	1.61360i		−	0.806801i	−0.915023	−	0.403401i	$-0.867828\pi$
$2$		−	1.61360i		−	0.806801i	0.915023	−	0.403401i	$-0.132172\pi$
$3$		0			0
$3$		0			0
$4$	1.39629			0.349072
$5$	6.36659			1.27332			0.636659	−	0.771146i	$-0.280316\pi$
$5$	6.36659			1.27332			0.636659	+	0.771146i	$0.280316\pi$
$6$		0			0
$7$	6.66564			0.952234			0.476117	−	0.879382i	$-0.342044\pi$
$7$	6.66564			0.952234			0.476117	+	0.879382i	$0.342044\pi$
$8$		−	8.70746i		−	1.08843i
$9$		0			0
$10$		−	10.2731i		−	1.02731i
$11$		−	16.0509i		−	1.45917i	−0.683890	−	0.729586i	$-0.739713\pi$
$11$		−	16.0509i		−	1.45917i	0.683890	−	0.729586i	$-0.260287\pi$
$12$		0			0
$13$			7.84026i			0.603097i	0.953451	+	0.301548i	$0.0975035\pi$
$13$			7.84026i			0.603097i	−0.953451	+	0.301548i	$0.902497\pi$
$14$		−	10.7557i		−	0.768263i
$15$		0			0
$16$	−8.46523			−0.529077
$17$	−18.9063			−1.11213			−0.556067	−	0.831137i	$-0.687690\pi$
$17$	−18.9063			−1.11213			−0.556067	+	0.831137i	$0.687690\pi$
$18$		0			0
$19$	7.10898			0.374157			0.187078	−	0.982345i	$-0.440098\pi$
$19$	7.10898			0.374157			0.187078	+	0.982345i	$0.440098\pi$
$20$	8.88958			0.444479
$21$		0			0
$22$	−25.8997			−1.17726
$23$			33.6730i			1.46404i	0.681282	+	0.732021i	$0.261423\pi$
$23$			33.6730i			1.46404i	−0.681282	+	0.732021i	$0.738577\pi$
$24$		0			0
$25$	15.5334			0.621336
$26$	12.6511			0.486579
$27$		0			0
$28$	9.30714			0.332398
$29$	46.2052			1.59328			0.796642	−	0.604451i	$-0.206608\pi$
$29$	46.2052			1.59328			0.796642	+	0.604451i	$0.206608\pi$
$30$		0			0
$31$		−	29.5207i		−	0.952280i	−0.879369	−	0.476140i	$-0.842036\pi$
$31$		−	29.5207i		−	0.952280i	0.879369	−	0.476140i	$-0.157964\pi$
$32$		−	21.1703i		−	0.661573i
$33$		0			0
$34$			30.5072i			0.897271i
$35$	42.4373			1.21250
$36$		0			0
$37$		−	1.91983i		−	0.0518873i	−0.999663	−	0.0259437i	$-0.991741\pi$
$37$		−	1.91983i		−	0.0518873i	0.999663	−	0.0259437i	$-0.00825905\pi$
$38$		−	11.4711i		−	0.301870i
$39$		0			0
$40$		−	55.4368i		−	1.38592i
$41$	−46.4596			−1.13316			−0.566580	−	0.824007i	$-0.691734\pi$
$41$	−46.4596			−1.13316			−0.566580	+	0.824007i	$0.691734\pi$
$42$		0			0
$43$			21.6675i			0.503895i	0.967741	+	0.251947i	$0.0810710\pi$
$43$			21.6675i			0.503895i	−0.967741	+	0.251947i	$0.918929\pi$
$44$		−	22.4116i		−	0.509355i
$45$		0			0
$46$	54.3348			1.18119
$47$			75.3742i			1.60371i	0.597521	+	0.801853i	$0.296152\pi$
$47$			75.3742i			1.60371i	−0.597521	+	0.801853i	$0.703848\pi$
$48$		0			0
$49$	−4.56931			−0.0932512
$50$		−	25.0647i		−	0.501295i
$51$		0			0
$52$			10.9473i			0.210524i
$53$	−19.5526			−0.368916			−0.184458	−	0.982840i	$-0.559053\pi$
$53$	−19.5526			−0.368916			−0.184458	+	0.982840i	$0.559053\pi$
$54$		0			0
$55$		−	102.189i		−	1.85799i
$56$		−	58.0408i		−	1.03644i
$57$		0			0
$58$		−	74.5569i		−	1.28546i
$59$	4.68014	+	58.8141i	0.0793243	+	0.996849i
$59$	4.68014	+	58.8141i	0.0793243	+	0.996849i
$60$		0			0
$61$		−	41.0613i		−	0.673136i	−0.941659	−	0.336568i	$-0.890734\pi$
$61$		−	41.0613i		−	0.673136i	0.941659	−	0.336568i	$-0.109266\pi$
$62$	−47.6346			−0.768301
$63$		0			0
$64$	−68.0214			−1.06283
$65$			49.9157i			0.767934i
$66$		0			0
$67$		−	90.1839i		−	1.34603i	−0.739630	−	0.673014i	$-0.764999\pi$
$67$		−	90.1839i		−	1.34603i	0.739630	−	0.673014i	$-0.235001\pi$
$68$	−26.3986			−0.388215
$69$		0			0
$70$		−	68.4770i		−	0.978243i
$71$	−57.1922			−0.805523			−0.402762	−	0.915305i	$-0.631950\pi$
$71$	−57.1922			−0.805523			−0.402762	+	0.915305i	$0.631950\pi$
$72$		0			0
$73$			69.6739i			0.954436i	0.878785	+	0.477218i	$0.158355\pi$
$73$			69.6739i			0.954436i	−0.878785	+	0.477218i	$0.841645\pi$
$74$	−3.09784			−0.0418628
$75$		0			0
$76$	9.92617			0.130608
$77$		−	106.989i		−	1.38947i
$78$		0			0
$79$	118.749			1.50315			0.751575	−	0.659647i	$-0.229294\pi$
$79$	118.749			1.50315			0.751575	+	0.659647i	$0.229294\pi$
$80$	−53.8946			−0.673683
$81$		0			0
$82$			74.9673i			0.914235i
$83$		−	86.3181i		−	1.03998i	−0.854173	−	0.519988i	$-0.825936\pi$
$83$		−	86.3181i		−	1.03998i	0.854173	−	0.519988i	$-0.174064\pi$
$84$		0			0
$85$	−120.368			−1.41610
$86$	34.9627			0.406543
$87$		0			0
$88$	−139.762			−1.58821
$89$		−	38.4935i		−	0.432512i	−0.976337	−	0.216256i	$-0.930615\pi$
$89$		−	38.4935i		−	0.432512i	0.976337	−	0.216256i	$-0.0693845\pi$
$90$		0			0
$91$			52.2603i			0.574289i
$92$			47.0172i			0.511056i
$93$		0			0
$94$	121.624			1.29387
$95$	45.2599			0.476420
$96$		0			0
$97$		−	24.0445i		−	0.247881i	−0.992290	−	0.123941i	$-0.960447\pi$
$97$		−	24.0445i		−	0.247881i	0.992290	−	0.123941i	$-0.0395532\pi$
$98$			7.37305i			0.0752352i
$99$		0			0
$100$	21.6891			0.216891
$101$		−	56.4784i		−	0.559192i	−0.960118	−	0.279596i	$-0.909799\pi$
$101$		−	56.4784i		−	0.559192i	0.960118	−	0.279596i	$-0.0902006\pi$
$102$		0			0
$103$			99.6967i			0.967929i	0.875088	+	0.483964i	$0.160804\pi$
$103$			99.6967i			0.967929i	−0.875088	+	0.483964i	$0.839196\pi$
$104$	68.2688			0.656430
$105$		0			0
$106$			31.5500i			0.297642i
$107$	193.034			1.80406			0.902030	−	0.431673i	$-0.142076\pi$
$107$	193.034			1.80406			0.902030	+	0.431673i	$0.142076\pi$
$108$		0			0
$109$		−	65.4956i		−	0.600877i	−0.953801	−	0.300439i	$-0.902867\pi$
$109$		−	65.4956i		−	0.600877i	0.953801	−	0.300439i	$-0.0971330\pi$
$110$	−164.893			−1.49903
$111$		0			0
$112$	−56.4262			−0.503805
$113$			169.975i			1.50420i	0.659048	+	0.752101i	$0.270959\pi$
$113$			169.975i			1.50420i	−0.659048	+	0.752101i	$0.729041\pi$
$114$		0			0
$115$			214.382i			1.86419i
$116$	64.5158			0.556171
$117$		0			0
$118$	94.9025	−	7.55188i	0.804259	−	0.0639990i
$119$	−126.022			−1.05901
$120$		0			0
$121$	−136.631			−1.12918
$122$	−66.2566			−0.543087
$123$		0			0
$124$		−	41.2194i		−	0.332414i
$125$	−60.2699			−0.482159
$126$		0			0
$127$	−166.750			−1.31299			−0.656495	−	0.754331i	$-0.727962\pi$
$127$	−166.750			−1.31299			−0.656495	+	0.754331i	$0.727962\pi$
$128$			25.0782i			0.195924i
$129$		0			0
$130$	80.5441			0.619570
$131$		−	45.5267i		−	0.347532i	−0.984787	−	0.173766i	$-0.944406\pi$
$131$		−	45.5267i		−	0.347532i	0.984787	−	0.173766i	$-0.0555936\pi$
$132$		0			0
$133$	47.3859			0.356285
$134$	−145.521			−1.08598
$135$		0			0
$136$			164.626i			1.21048i
$137$	106.697			0.778807			0.389404	−	0.921067i	$-0.372681\pi$
$137$	106.697			0.778807			0.389404	+	0.921067i	$0.372681\pi$
$138$		0			0
$139$	−133.637			−0.961414			−0.480707	−	0.876881i	$-0.659620\pi$
$139$	−133.637			−0.961414			−0.480707	+	0.876881i	$0.659620\pi$
$140$	59.2547			0.423248
$141$		0			0
$142$			92.2854i			0.649897i
$143$	125.843			0.880022
$144$		0			0
$145$	294.170			2.02876
$146$	112.426			0.770040
$147$		0			0
$148$		−	2.68064i		−	0.0181124i
$149$			118.228i			0.793478i	0.917931	+	0.396739i	$0.129858\pi$
$149$			118.228i			0.793478i	−0.917931	+	0.396739i	$0.870142\pi$
$150$		0			0
$151$			221.371i			1.46603i	0.680212	+	0.733015i	$0.261888\pi$
$151$			221.371i			1.46603i	−0.680212	+	0.733015i	$0.738112\pi$
$152$		−	61.9012i		−	0.407244i
$153$		0			0
$154$	−172.638			−1.12103
$155$		−	187.946i		−	1.21255i
$156$		0			0
$157$			13.3063i			0.0847535i	0.999102	+	0.0423768i	$0.0134930\pi$
$157$			13.3063i			0.0847535i	−0.999102	+	0.0423768i	$0.986507\pi$
$158$		−	191.614i		−	1.21274i
$159$		0			0
$160$		−	134.783i		−	0.842392i
$161$			224.452i			1.39411i
$162$		0			0
$163$	227.187			1.39378			0.696891	−	0.717177i	$-0.254566\pi$
$163$	227.187			1.39378			0.696891	+	0.717177i	$0.254566\pi$
$164$	−64.8709			−0.395554
$165$		0			0
$166$	−139.283			−0.839055
$167$	−176.740			−1.05832			−0.529162	−	0.848521i	$-0.677494\pi$
$167$	−176.740			−1.05832			−0.529162	+	0.848521i	$0.677494\pi$
$168$		0			0
$169$	107.530			0.636274
$170$			194.227i			1.14251i
$171$		0			0
$172$			30.2540i			0.175895i
$173$		−	184.896i		−	1.06876i	−0.845244	−	0.534381i	$-0.820545\pi$
$173$		−	184.896i		−	1.06876i	0.845244	−	0.534381i	$-0.179455\pi$
$174$		0			0
$175$	103.540			0.591657
$176$			135.874i			0.772014i
$177$		0			0
$178$	−62.1133			−0.348951
$179$			290.925i			1.62528i	0.582765	+	0.812641i	$0.301971\pi$
$179$			290.925i			1.62528i	−0.582765	+	0.812641i	$0.698029\pi$
$180$		0			0
$181$	71.9948			0.397761			0.198881	−	0.980024i	$-0.436269\pi$
$181$	71.9948			0.397761			0.198881	+	0.980024i	$0.436269\pi$
$182$	84.3274			0.463337
$183$		0			0
$184$	293.206			1.59351
$185$		−	12.2228i		−	0.0660690i
$186$		0			0
$187$			303.463i			1.62279i
$188$			105.244i			0.559809i
$189$		0			0
$190$		−	73.0315i		−	0.384376i
$191$			169.923i			0.889651i	0.895617	+	0.444825i	$0.146734\pi$
$191$			169.923i			0.889651i	−0.895617	+	0.444825i	$0.853266\pi$
$192$		0			0
$193$	−15.6413			−0.0810432			−0.0405216	−	0.999179i	$-0.512902\pi$
$193$	−15.6413			−0.0810432			−0.0405216	+	0.999179i	$0.512902\pi$
$194$	−38.7982			−0.199991
$195$		0			0
$196$	−6.38007			−0.0325514
$197$	100.597			0.510645			0.255323	−	0.966856i	$-0.417818\pi$
$197$	100.597			0.510645			0.255323	+	0.966856i	$0.417818\pi$
$198$		0			0
$199$	−84.4909			−0.424577			−0.212289	−	0.977207i	$-0.568092\pi$
$199$	−84.4909			−0.424577			−0.212289	+	0.977207i	$0.568092\pi$
$200$		−	135.257i		−	0.676283i
$201$		0			0
$202$	−91.1337			−0.451157
$203$	307.987			1.51718
$204$		0			0
$205$	−295.789			−1.44287
$206$	160.871			0.780926
$207$		0			0
$208$		−	66.3696i		−	0.319085i
$209$		−	114.105i		−	0.545959i
$210$		0			0
$211$			307.104i			1.45547i	0.685859	+	0.727735i	$0.259427\pi$
$211$			307.104i			1.45547i	−0.685859	+	0.727735i	$0.740573\pi$
$212$	−27.3010			−0.128778
$213$		0			0
$214$		−	311.481i		−	1.45552i
$215$			137.948i			0.641618i
$216$		0			0
$217$		−	196.774i		−	0.906793i
$218$	−105.684			−0.484788
$219$		0			0
$220$		−	142.686i		−	0.648571i
$221$		−	148.230i		−	0.670725i
$222$		0			0
$223$	−334.093			−1.49818			−0.749088	−	0.662471i	$-0.769508\pi$
$223$	−334.093			−1.49818			−0.749088	+	0.662471i	$0.769508\pi$
$224$		−	141.114i		−	0.629972i
$225$		0			0
$226$	274.272			1.21359
$227$		−	194.419i		−	0.856471i	−0.903667	−	0.428236i	$-0.859135\pi$
$227$		−	194.419i		−	0.856471i	0.903667	−	0.428236i	$-0.140865\pi$
$228$		0			0
$229$		−	198.825i		−	0.868230i	−0.900857	−	0.434115i	$-0.857061\pi$
$229$		−	198.825i		−	0.868230i	0.900857	−	0.434115i	$-0.142939\pi$
$230$	345.927			1.50403
$231$		0			0
$232$		−	402.330i		−	1.73418i
$233$			268.492i			1.15233i	0.817335	+	0.576163i	$0.195450\pi$
$233$			268.492i			1.15233i	−0.817335	+	0.576163i	$0.804550\pi$
$234$		0			0
$235$			479.876i			2.04203i
$236$	6.53481	+	82.1213i	0.0276899	+	0.347972i
$237$		0			0
$238$			203.350i			0.854412i
$239$	−321.465			−1.34504			−0.672522	−	0.740077i	$-0.734789\pi$
$239$	−321.465			−1.34504			−0.672522	+	0.740077i	$0.734789\pi$
$240$		0			0
$241$	−44.9975			−0.186712			−0.0933559	−	0.995633i	$-0.529759\pi$
$241$	−44.9975			−0.186712			−0.0933559	+	0.995633i	$0.529759\pi$
$242$			220.468i			0.911024i
$243$		0			0
$244$		−	57.3334i		−	0.234973i
$245$	−29.0909			−0.118738
$246$		0			0
$247$			55.7362i			0.225653i
$248$	−257.050			−1.03649
$249$		0			0
$250$			97.2516i			0.389006i
$251$	−307.277			−1.22421			−0.612105	−	0.790776i	$-0.709677\pi$
$251$	−307.277			−1.22421			−0.612105	+	0.790776i	$0.709677\pi$
$252$		0			0
$253$	540.481			2.13629
$254$			269.068i			1.05932i
$255$		0			0
$256$	−231.619			−0.904763
$257$	269.259			1.04770			0.523850	−	0.851811i	$-0.324495\pi$
$257$	269.259			1.04770			0.523850	+	0.851811i	$0.324495\pi$
$258$		0			0
$259$		−	12.7969i		−	0.0494089i
$260$			69.6966i			0.268064i
$261$		0			0
$262$	−73.4619			−0.280389
$263$	294.101			1.11825			0.559127	−	0.829082i	$-0.311136\pi$
$263$	294.101			1.11825			0.559127	+	0.829082i	$0.311136\pi$
$264$		0			0
$265$	−124.483			−0.469747
$266$		−	76.4619i		−	0.287451i
$267$		0			0
$268$		−	125.923i		−	0.469860i
$269$			52.5633i			0.195403i	0.995216	+	0.0977013i	$0.0311490\pi$
$269$			52.5633i			0.195403i	−0.995216	+	0.0977013i	$0.968851\pi$
$270$		0			0
$271$	−63.5650			−0.234557			−0.117279	−	0.993099i	$-0.537417\pi$
$271$	−63.5650			−0.234557			−0.117279	+	0.993099i	$0.537417\pi$
$272$	160.046			0.588405
$273$		0			0
$274$		−	172.166i		−	0.628343i
$275$		−	249.325i		−	0.906636i
$276$		0			0
$277$	−76.0155			−0.274424			−0.137212	−	0.990542i	$-0.543814\pi$
$277$	−76.0155			−0.274424			−0.137212	+	0.990542i	$0.543814\pi$
$278$			215.636i			0.775670i
$279$		0			0
$280$		−	369.521i		−	1.31972i
$281$	−146.718			−0.522127			−0.261063	−	0.965322i	$-0.584073\pi$
$281$	−146.718			−0.522127			−0.261063	+	0.965322i	$0.584073\pi$
$282$		0			0
$283$			497.505i			1.75797i	0.476850	+	0.878985i	$0.341779\pi$
$283$			497.505i			1.75797i	−0.476850	+	0.878985i	$0.658221\pi$
$284$	−79.8567			−0.281185
$285$		0			0
$286$		−	203.061i		−	0.710002i
$287$	−309.683			−1.07903
$288$		0			0
$289$	68.4476			0.236843
$290$		−	474.673i		−	1.63680i
$291$		0			0
$292$			97.2847i			0.333167i
$293$	−158.624			−0.541378			−0.270689	−	0.962667i	$-0.587251\pi$
$293$	−158.624			−0.541378			−0.270689	+	0.962667i	$0.587251\pi$
$294$		0			0
$295$	29.7965	+	374.445i	0.101005	+	1.26930i
$296$	−16.7169			−0.0564759
$297$		0			0
$298$	190.773			0.640179
$299$	−264.005			−0.882960
$300$		0			0
$301$			144.428i			0.479826i
$302$	357.204			1.18280
$303$		0			0
$304$	−60.1792			−0.197958
$305$		−	261.420i		−	0.857116i
$306$		0			0
$307$	302.624			0.985747			0.492873	−	0.870101i	$-0.335947\pi$
$307$	302.624			0.985747			0.492873	+	0.870101i	$0.335947\pi$
$308$		−	149.388i		−	0.485025i
$309$		0			0
$310$	−303.270			−0.978290
$311$	−545.048			−1.75257			−0.876283	−	0.481798i	$-0.839984\pi$
$311$	−545.048			−1.75257			−0.876283	+	0.481798i	$0.839984\pi$
$312$		0			0
$313$		−	378.973i		−	1.21078i	−0.795931	−	0.605388i	$-0.793018\pi$
$313$		−	378.973i		−	1.21078i	0.795931	−	0.605388i	$-0.206982\pi$
$314$	21.4711			0.0683792
$315$		0			0
$316$	165.808			0.524708
$317$	400.912			1.26471			0.632354	−	0.774680i	$-0.282089\pi$
$317$	400.912			1.26471			0.632354	+	0.774680i	$0.282089\pi$
$318$		0			0
$319$		−	741.635i		−	2.32487i
$320$	−433.064			−1.35333
$321$		0			0
$322$	362.176			1.12477
$323$	−134.404			−0.416113
$324$		0			0
$325$			121.786i			0.374726i
$326$		−	366.589i		−	1.12451i
$327$		0			0
$328$			404.545i			1.23337i
$329$			502.417i			1.52710i
$330$		0			0
$331$	−256.922			−0.776199			−0.388099	−	0.921618i	$-0.626868\pi$
$331$	−256.922			−0.776199			−0.388099	+	0.921618i	$0.626868\pi$
$332$		−	120.525i		−	0.363027i
$333$		0			0
$334$			285.188i			0.853857i
$335$		−	574.163i		−	1.71392i
$336$		0			0
$337$			519.832i			1.54253i	0.636516	+	0.771264i	$0.280375\pi$
$337$			519.832i			1.54253i	−0.636516	+	0.771264i	$0.719625\pi$
$338$		−	173.511i		−	0.513347i
$339$		0			0
$340$	−168.069			−0.494320
$341$	−473.833			−1.38954
$342$		0			0
$343$	−357.073			−1.04103
$344$	188.669			0.548456
$345$		0			0
$346$	−298.349			−0.862279
$347$			144.710i			0.417032i	0.978019	+	0.208516i	$0.0668633\pi$
$347$			144.710i			0.417032i	−0.978019	+	0.208516i	$0.933137\pi$
$348$		0			0
$349$		−	444.542i		−	1.27376i	−0.770963	−	0.636880i	$-0.780225\pi$
$349$		−	444.542i		−	1.27376i	0.770963	−	0.636880i	$-0.219775\pi$
$350$		−	167.072i		−	0.477350i
$351$		0			0
$352$	−339.802			−0.965348
$353$		−	625.820i		−	1.77286i	−0.462863	−	0.886430i	$-0.653178\pi$
$353$		−	625.820i		−	1.77286i	0.462863	−	0.886430i	$-0.346822\pi$
$354$		0			0
$355$	−364.119			−1.02569
$356$		−	53.7480i		−	0.150978i
$357$		0			0
$358$	469.438			1.31128
$359$	351.287			0.978517			0.489258	−	0.872139i	$-0.337268\pi$
$359$	351.287			0.978517			0.489258	+	0.872139i	$0.337268\pi$
$360$		0			0
$361$	−310.462			−0.860007
$362$		−	116.171i		−	0.320914i
$363$		0			0
$364$			72.9704i			0.200468i
$365$			443.585i			1.21530i
$366$		0			0
$367$			221.302i			0.603002i	0.953466	+	0.301501i	$0.0974877\pi$
$367$			221.302i			0.603002i	−0.953466	+	0.301501i	$0.902512\pi$
$368$		−	285.050i		−	0.774592i
$369$		0			0
$370$	−19.7227			−0.0533046
$371$	−130.330			−0.351294
$372$		0			0
$373$	−159.549			−0.427745			−0.213873	−	0.976862i	$-0.568608\pi$
$373$	−159.549			−0.427745			−0.213873	+	0.976862i	$0.568608\pi$
$374$	489.668			1.30927
$375$		0			0
$376$	656.318			1.74553
$377$			362.261i			0.960905i
$378$		0			0
$379$	264.069			0.696752			0.348376	−	0.937355i	$-0.386733\pi$
$379$	264.069			0.696752			0.348376	+	0.937355i	$0.386733\pi$
$380$	63.1958			0.166305
$381$		0			0
$382$	274.189			0.717771
$383$	89.2279			0.232971			0.116486	−	0.993192i	$-0.462837\pi$
$383$	89.2279			0.232971			0.116486	+	0.993192i	$0.462837\pi$
$384$		0			0
$385$		−	681.157i		−	1.76924i
$386$			25.2389i			0.0653857i
$387$		0			0
$388$		−	33.5730i		−	0.0865283i
$389$	161.318			0.414699			0.207350	−	0.978267i	$-0.433516\pi$
$389$	161.318			0.414699			0.207350	+	0.978267i	$0.433516\pi$
$390$		0			0
$391$		−	636.631i		−	1.62821i
$392$			39.7871i			0.101498i
$393$		0			0
$394$		−	162.324i		−	0.411989i
$395$	756.025			1.91399
$396$		0			0
$397$		−	519.713i		−	1.30910i	−0.756018	−	0.654551i	$-0.772858\pi$
$397$		−	519.713i		−	1.30910i	0.756018	−	0.654551i	$-0.227142\pi$
$398$			136.335i			0.342549i
$399$		0			0
$400$	−131.494			−0.328735
$401$		−	550.540i		−	1.37292i	−0.727169	−	0.686459i	$-0.759164\pi$
$401$		−	550.540i		−	1.37292i	0.727169	−	0.686459i	$-0.240836\pi$
$402$		0			0
$403$	231.450			0.574317
$404$		−	78.8601i		−	0.195198i
$405$		0			0
$406$		−	496.969i		−	1.22406i
$407$	−30.8150			−0.0757125
$408$		0			0
$409$		−	17.8723i		−	0.0436976i	−0.999761	−	0.0218488i	$-0.993045\pi$
$409$		−	17.8723i		−	0.0436976i	0.999761	−	0.0218488i	$-0.00695524\pi$
$410$			477.286i			1.16411i
$411$		0			0
$412$			139.205i			0.337877i
$413$	31.1961	+	392.033i	0.0755353	+	0.949233i
$414$		0			0
$415$		−	549.551i		−	1.32422i
$416$	165.981			0.398992
$417$		0			0
$418$	−184.121			−0.440480
$419$			18.7816i			0.0448248i	0.999749	+	0.0224124i	$0.00713469\pi$
$419$			18.7816i			0.0448248i	−0.999749	+	0.0224124i	$0.992865\pi$
$420$		0			0
$421$		−	561.743i		−	1.33431i	−0.744920	−	0.667153i	$-0.767513\pi$
$421$		−	561.743i		−	1.33431i	0.744920	−	0.667153i	$-0.232487\pi$
$422$	495.544			1.17427
$423$		0			0
$424$			170.253i			0.401540i
$425$	−293.679			−0.691010
$426$		0			0
$427$		−	273.700i		−	0.640983i
$428$	269.532			0.629747
$429$		0			0
$430$	222.593			0.517658
$431$		−	43.8318i		−	0.101698i	−0.998706	−	0.0508490i	$-0.983807\pi$
$431$		−	43.8318i		−	0.101698i	0.998706	−	0.0508490i	$-0.0161927\pi$
$432$		0			0
$433$	100.552			0.232221			0.116111	−	0.993236i	$-0.462957\pi$
$433$	100.552			0.232221			0.116111	+	0.993236i	$0.462957\pi$
$434$	−317.515			−0.731602
$435$		0			0
$436$		−	91.4507i		−	0.209749i
$437$			239.381i			0.547781i
$438$		0			0
$439$	580.171			1.32157			0.660787	−	0.750573i	$-0.270223\pi$
$439$	580.171			1.32157			0.660787	+	0.750573i	$0.270223\pi$
$440$	−889.810			−2.02229
$441$		0			0
$442$	−239.185			−0.541142
$443$		−	763.409i		−	1.72327i	−0.507528	−	0.861635i	$-0.669441\pi$
$443$		−	763.409i		−	1.72327i	0.507528	−	0.861635i	$-0.330559\pi$
$444$		0			0
$445$		−	245.072i		−	0.550724i
$446$			539.093i			1.20873i
$447$		0			0
$448$	−453.406			−1.01207
$449$	−350.584			−0.780810			−0.390405	−	0.920643i	$-0.627665\pi$
$449$	−350.584			−0.780810			−0.390405	+	0.920643i	$0.627665\pi$
$450$		0			0
$451$			745.717i			1.65348i
$452$			237.334i			0.525074i
$453$		0			0
$454$	−313.715			−0.691002
$455$			332.720i			0.731252i
$456$		0			0
$457$			627.986i			1.37415i	0.726587	+	0.687075i	$0.241106\pi$
$457$			627.986i			1.37415i	−0.726587	+	0.687075i	$0.758894\pi$
$458$	−320.824			−0.700489
$459$		0			0
$460$			299.339i			0.650736i
$461$	−602.743			−1.30747			−0.653735	−	0.756724i	$-0.726799\pi$
$461$	−602.743			−1.30747			−0.653735	+	0.756724i	$0.726799\pi$
$462$		0			0
$463$			340.754i			0.735969i	0.929832	+	0.367985i	$0.119952\pi$
$463$			340.754i			0.735969i	−0.929832	+	0.367985i	$0.880048\pi$
$464$	−391.138			−0.842970
$465$		0			0
$466$	433.239			0.929697
$467$			682.672i			1.46182i	0.682471	+	0.730912i	$0.260905\pi$
$467$			682.672i			1.46182i	−0.682471	+	0.730912i	$0.739095\pi$
$468$		0			0
$469$		−	601.133i		−	1.28173i
$470$	774.329			1.64751
$471$		0			0
$472$	512.121	−	40.7521i	1.08500	−	0.0863392i
$473$	347.782			0.735269
$474$		0			0
$475$	110.427			0.232477
$476$	−175.963			−0.369671
$477$		0			0
$478$			518.717i			1.08518i
$479$	757.928			1.58231			0.791156	−	0.611614i	$-0.209480\pi$
$479$	757.928			1.58231			0.791156	+	0.611614i	$0.209480\pi$
$480$		0			0
$481$	15.0520			0.0312931
$482$			72.6081i			0.150639i
$483$		0			0
$484$	−190.776			−0.394165
$485$		−	153.081i		−	0.315631i
$486$		0			0
$487$	−105.431			−0.216491			−0.108246	−	0.994124i	$-0.534523\pi$
$487$	−105.431			−0.216491			−0.108246	+	0.994124i	$0.534523\pi$
$488$	−357.540			−0.732663
$489$		0			0
$490$			46.9411i			0.0957983i
$491$	603.562			1.22925			0.614625	−	0.788820i	$-0.289307\pi$
$491$	603.562			1.22925			0.614625	+	0.788820i	$0.289307\pi$
$492$		0			0
$493$	−873.570			−1.77195
$494$	89.9361			0.182057
$495$		0			0
$496$			249.899i			0.503830i
$497$	−381.222			−0.767046
$498$		0			0
$499$	308.346			0.617928			0.308964	−	0.951074i	$-0.400018\pi$
$499$	308.346			0.617928			0.308964	+	0.951074i	$0.400018\pi$
$500$	−84.1540			−0.168308
$501$		0			0
$502$			495.823i			0.987695i
$503$			6.39623i			0.0127162i	0.999980	+	0.00635809i	$0.00202386\pi$
$503$			6.39623i			0.0127162i	−0.999980	+	0.00635809i	$0.997976\pi$
$504$		0			0
$505$		−	359.575i		−	0.712029i
$506$		−	872.122i		−	1.72356i
$507$		0			0
$508$	−232.830			−0.458328
$509$		−	243.213i		−	0.477826i	−0.971041	−	0.238913i	$-0.923209\pi$
$509$		−	243.213i		−	0.477826i	0.971041	−	0.238913i	$-0.0767911\pi$
$510$		0			0
$511$			464.420i			0.908846i
$512$			474.055i			0.925888i
$513$		0			0
$514$		−	434.477i		−	0.845285i
$515$			634.727i			1.23248i
$516$		0			0
$517$	1209.82			2.34008
$518$	−20.6491			−0.0398631
$519$		0			0
$520$	434.639			0.835844
$521$	903.968			1.73506			0.867532	−	0.497382i	$-0.165705\pi$
$521$	903.968			1.73506			0.867532	+	0.497382i	$0.165705\pi$
$522$		0			0
$523$	−860.311			−1.64495			−0.822477	−	0.568798i	$-0.807408\pi$
$523$	−860.311			−1.64495			−0.822477	+	0.568798i	$0.807408\pi$
$524$		−	63.5683i		−	0.121314i
$525$		0			0
$526$		−	474.562i		−	0.902209i
$527$			558.126i			1.05906i
$528$		0			0
$529$	−604.870			−1.14342
$530$			200.866i			0.378993i
$531$		0			0
$532$	66.1643			0.124369
$533$		−	364.255i		−	0.683406i
$534$		0			0
$535$	1228.97			2.29714
$536$	−785.272			−1.46506
$537$		0			0
$538$	84.8163			0.157651
$539$			73.3414i			0.136069i
$540$		0			0
$541$		−	562.764i		−	1.04023i	−0.854097	−	0.520115i	$-0.825889\pi$
$541$		−	562.764i		−	1.04023i	0.854097	−	0.520115i	$-0.174111\pi$
$542$			102.569i			0.189241i
$543$		0			0
$544$			400.252i			0.735758i
$545$		−	416.983i		−	0.765107i
$546$		0			0
$547$	−186.037			−0.340104			−0.170052	−	0.985435i	$-0.554394\pi$
$547$	−186.037			−0.340104			−0.170052	+	0.985435i	$0.554394\pi$
$548$	148.979			0.271860
$549$		0			0
$550$	−402.311			−0.731475
$551$	328.472			0.596138
$552$		0			0
$553$	791.537			1.43135
$554$			122.659i			0.221406i
$555$		0			0
$556$	−186.595			−0.335602
$557$	206.473			0.370687			0.185343	−	0.982674i	$-0.440660\pi$
$557$	206.473			0.370687			0.185343	+	0.982674i	$0.440660\pi$
$558$		0			0
$559$	−169.879			−0.303897
$560$	−359.242			−0.641503
$561$		0			0
$562$			236.744i			0.421252i
$563$		−	560.271i		−	0.995152i	−0.867420	−	0.497576i	$-0.834224\pi$
$563$		−	560.271i		−	0.995152i	0.867420	−	0.497576i	$-0.165776\pi$
$564$		0			0
$565$			1082.16i			1.91533i
$566$	802.776			1.41833
$567$		0			0
$568$			497.999i			0.876758i
$569$			694.519i			1.22060i	0.792172	+	0.610298i	$0.208950\pi$
$569$			694.519i			1.22060i	−0.792172	+	0.610298i	$0.791050\pi$
$570$		0			0
$571$			344.549i			0.603414i	0.953401	+	0.301707i	$0.0975564\pi$
$571$			344.549i			0.603414i	−0.953401	+	0.301707i	$0.902444\pi$
$572$	175.713			0.307191
$573$		0			0
$574$			499.705i			0.870566i
$575$			523.056i			0.909663i
$576$		0			0
$577$	−543.385			−0.941742			−0.470871	−	0.882202i	$-0.656060\pi$
$577$	−543.385			−0.941742			−0.470871	+	0.882202i	$0.656060\pi$
$578$		−	110.447i		−	0.191085i
$579$		0			0
$580$	410.745			0.708182
$581$		−	575.365i		−	0.990301i
$582$		0			0
$583$			313.836i			0.538312i
$584$	606.682			1.03884
$585$		0			0
$586$			255.956i			0.436784i
$587$		−	995.683i		−	1.69622i	−0.529818	−	0.848111i	$-0.677740\pi$
$587$		−	995.683i		−	1.69622i	0.529818	−	0.848111i	$-0.322260\pi$
$588$		0			0
$589$		−	209.862i		−	0.356302i
$590$	604.205	−	48.0797i	1.02408	−	0.0814910i
$591$		0			0
$592$			16.2518i			0.0274524i
$593$	−503.068			−0.848345			−0.424172	−	0.905581i	$-0.639435\pi$
$593$	−503.068			−0.848345			−0.424172	+	0.905581i	$0.639435\pi$
$594$		0			0
$595$	−802.332			−1.34846
$596$			165.081i			0.276981i
$597$		0			0
$598$			425.999i			0.712373i
$599$	475.339			0.793554			0.396777	−	0.917915i	$-0.370129\pi$
$599$	475.339			0.793554			0.396777	+	0.917915i	$0.370129\pi$
$600$		0			0
$601$		−	288.933i		−	0.480754i	−0.970680	−	0.240377i	$-0.922729\pi$
$601$		−	288.933i		−	0.480754i	0.970680	−	0.240377i	$-0.0772710\pi$
$602$	233.049			0.387124
$603$		0			0
$604$			309.097i			0.511750i
$605$	−869.872			−1.43780
$606$		0			0
$607$	76.8532			0.126612			0.0633058	−	0.997994i	$-0.479836\pi$
$607$	76.8532			0.126612			0.0633058	+	0.997994i	$0.479836\pi$
$608$		−	150.499i		−	0.247532i
$609$		0			0
$610$	−421.828			−0.691522
$611$	−590.953			−0.967190
$612$		0			0
$613$		−	518.309i		−	0.845529i	−0.906240	−	0.422765i	$-0.861060\pi$
$613$		−	518.309i		−	0.845529i	0.906240	−	0.422765i	$-0.138940\pi$
$614$		−	488.315i		−	0.795302i
$615$		0			0
$616$	−931.605			−1.51235
$617$	440.167			0.713399			0.356699	−	0.934219i	$-0.383902\pi$
$617$	440.167			0.713399			0.356699	+	0.934219i	$0.383902\pi$
$618$		0			0
$619$	624.254			1.00849			0.504244	−	0.863561i	$-0.331771\pi$
$619$	624.254			1.00849			0.504244	+	0.863561i	$0.331771\pi$
$620$		−	262.427i		−	0.423269i
$621$		0			0
$622$			879.490i			1.41397i
$623$		−	256.584i		−	0.411852i
$624$		0			0
$625$	−772.048			−1.23528
$626$	−611.511			−0.976855
$627$		0			0
$628$			18.5794i			0.0295851i
$629$			36.2969i			0.0577057i
$630$		0			0
$631$	−647.088			−1.02550			−0.512748	−	0.858539i	$-0.671372\pi$
$631$	−647.088			−1.02550			−0.512748	+	0.858539i	$0.671372\pi$
$632$		−	1034.00i		−	1.63608i
$633$		0			0
$634$		−	646.913i		−	1.02037i
$635$	−1061.63			−1.67185
$636$		0			0
$637$		−	35.8246i		−	0.0562395i
$638$	−1196.70			−1.87571
$639$		0			0
$640$			159.663i			0.249473i
$641$	−463.987			−0.723848			−0.361924	−	0.932208i	$-0.617880\pi$
$641$	−463.987			−0.723848			−0.361924	+	0.932208i	$0.617880\pi$
$642$		0			0
$643$	−12.4294			−0.0193304			−0.00966518	−	0.999953i	$-0.503077\pi$
$643$	−12.4294			−0.0193304			−0.00966518	+	0.999953i	$0.503077\pi$
$644$			313.399i			0.486645i
$645$		0			0
$646$			216.875i			0.335720i
$647$	−879.244			−1.35895			−0.679477	−	0.733696i	$-0.737793\pi$
$647$	−879.244			−1.35895			−0.679477	+	0.733696i	$0.737793\pi$
$648$		0			0
$649$	944.018	−	75.1203i	1.45457	−	0.115748i
$650$	196.514			0.302329
$651$		0			0
$652$	317.218			0.486530
$653$	−200.318			−0.306766			−0.153383	−	0.988167i	$-0.549017\pi$
$653$	−200.318			−0.306766			−0.153383	+	0.988167i	$0.549017\pi$
$654$		0			0
$655$		−	289.849i		−	0.442518i
$656$	393.291			0.599529
$657$		0			0
$658$	810.701			1.23207
$659$			652.030i			0.989423i	0.869057	+	0.494712i	$0.164726\pi$
$659$			652.030i			0.989423i	−0.869057	+	0.494712i	$0.835274\pi$
$660$		0			0
$661$	−440.130			−0.665855			−0.332928	−	0.942952i	$-0.608036\pi$
$661$	−440.130			−0.665855			−0.332928	+	0.942952i	$0.608036\pi$
$662$			414.570i			0.626238i
$663$		0			0
$664$	−751.611			−1.13194
$665$	301.686			0.453663
$666$		0			0
$667$			1555.87i			2.33264i
$668$	−246.780			−0.369431
$669$		0			0
$670$	−926.471			−1.38279
$671$	−659.070			−0.982221
$672$		0			0
$673$			178.659i			0.265466i	0.991152	+	0.132733i	$0.0423754\pi$
$673$			178.659i			0.265466i	−0.991152	+	0.132733i	$0.957625\pi$
$674$	838.802			1.24451
$675$		0			0
$676$	150.143			0.222105
$677$	890.544			1.31543			0.657713	−	0.753268i	$-0.271524\pi$
$677$	890.544			1.31543			0.657713	+	0.753268i	$0.271524\pi$
$678$		0			0
$679$		−	160.272i		−	0.236041i
$680$			1048.10i			1.54133i
$681$		0			0
$682$			764.578i			1.12108i
$683$			230.871i			0.338025i	0.985614	+	0.169013i	$0.0540578\pi$
$683$			230.871i			0.338025i	−0.985614	+	0.169013i	$0.945942\pi$
$684$		0			0
$685$	679.293			0.991669
$686$			576.175i			0.839905i
$687$		0			0
$688$		−	183.420i		−	0.266599i
$689$		−	153.297i		−	0.222492i
$690$		0			0
$691$		−	305.505i		−	0.442120i	−0.975260	−	0.221060i	$-0.929048\pi$
$691$		−	305.505i		−	0.442120i	0.975260	−	0.221060i	$-0.0709516\pi$
$692$		−	258.168i		−	0.373075i
$693$		0			0
$694$	233.504			0.336462
$695$	−850.808			−1.22418
$696$		0			0
$697$	878.378			1.26023
$698$	−717.315			−1.02767
$699$		0			0
$700$	144.572			0.206531
$701$		−	309.337i		−	0.441280i	−0.975355	−	0.220640i	$-0.929185\pi$
$701$		−	309.337i		−	0.441280i	0.975355	−	0.220640i	$-0.0708146\pi$
$702$		0			0
$703$		−	13.6480i		−	0.0194140i
$704$			1091.80i			1.55086i
$705$		0			0
$706$	−1009.82			−1.43035
$707$		−	376.465i		−	0.532482i
$708$		0			0
$709$	−1363.47			−1.92309			−0.961543	−	0.274655i	$-0.911436\pi$
$709$	−1363.47			−1.92309			−0.961543	+	0.274655i	$0.911436\pi$
$710$			587.543i			0.827525i
$711$		0			0
$712$	−335.181			−0.470760
$713$	994.049			1.39418
$714$		0			0
$715$	801.191			1.12055
$716$			406.215i			0.567340i
$717$		0			0
$718$		−	566.838i		−	0.789468i
$719$		−	358.898i		−	0.499162i	−0.968354	−	0.249581i	$-0.919707\pi$
$719$		−	358.898i		−	0.499162i	0.968354	−	0.249581i	$-0.0802929\pi$
$720$		0			0
$721$			664.542i			0.921694i
$722$			500.963i			0.693854i
$723$		0			0
$724$	100.525			0.138847
$725$	717.725			0.989966
$726$		0			0
$727$	123.204			0.169469			0.0847345	−	0.996404i	$-0.472996\pi$
$727$	123.204			0.169469			0.0847345	+	0.996404i	$0.472996\pi$
$728$	455.055			0.625075
$729$		0			0
$730$	715.769			0.980506
$731$		−	409.652i		−	0.560399i
$732$		0			0
$733$	505.875			0.690143			0.345071	−	0.938576i	$-0.387855\pi$
$733$	505.875			0.690143			0.345071	+	0.938576i	$0.387855\pi$
$734$	357.093			0.486503
$735$		0			0
$736$	712.868			0.968571
$737$	−1447.53			−1.96408
$738$		0			0
$739$		−	214.823i		−	0.290694i	−0.989381	−	0.145347i	$-0.953570\pi$
$739$		−	214.823i		−	0.290694i	0.989381	−	0.145347i	$-0.0464298\pi$
$740$		−	17.0665i		−	0.0230628i
$741$		0			0
$742$			210.301i			0.283425i
$743$	919.142			1.23707			0.618534	−	0.785758i	$-0.287727\pi$
$743$	919.142			1.23707			0.618534	+	0.785758i	$0.287727\pi$
$744$		0			0
$745$			752.710i			1.01035i
$746$			257.448i			0.345105i
$747$		0			0
$748$			423.721i			0.566472i
$749$	1286.70			1.71789
$750$		0			0
$751$		−	448.537i		−	0.597254i	−0.954370	−	0.298627i	$-0.903471\pi$
$751$		−	448.537i		−	0.597254i	0.954370	−	0.298627i	$-0.0965286\pi$
$752$		−	638.060i		−	0.848484i
$753$		0			0
$754$	584.545			0.775259
$755$			1409.37i			1.86672i
$756$		0			0
$757$	−250.783			−0.331286			−0.165643	−	0.986186i	$-0.552970\pi$
$757$	−250.783			−0.331286			−0.165643	+	0.986186i	$0.552970\pi$
$758$		−	426.102i		−	0.562140i
$759$		0			0
$760$		−	394.099i		−	0.518551i
$761$	−746.517			−0.980968			−0.490484	−	0.871450i	$-0.663180\pi$
$761$	−746.517			−0.980968			−0.490484	+	0.871450i	$0.663180\pi$
$762$		0			0
$763$		−	436.570i		−	0.572175i
$764$			237.262i			0.310552i
$765$		0			0
$766$		−	143.978i		−	0.187961i
$767$	−461.118	+	36.6935i	−0.601196	+	0.0478403i
$768$		0			0
$769$		−	1189.01i		−	1.54618i	−0.634296	−	0.773091i	$-0.718710\pi$
$769$		−	1189.01i		−	1.54618i	0.634296	−	0.773091i	$-0.281290\pi$
$770$	−1099.12			−1.42742
$771$		0			0
$772$	−21.8398			−0.0282899
$773$			900.789i			1.16532i	0.812717	+	0.582658i	$0.197987\pi$
$773$			900.789i			1.16532i	−0.812717	+	0.582658i	$0.802013\pi$
$774$		0			0
$775$		−	458.557i		−	0.591686i
$776$	−209.366			−0.269802
$777$		0			0
$778$		−	260.303i		−	0.334580i
$779$	−330.280			−0.423980
$780$		0			0
$781$			917.985i			1.17540i
$782$	−1027.27			−1.31364
$783$		0			0
$784$	38.6803			0.0493371
$785$			84.7157i			0.107918i
$786$		0			0
$787$	−212.704			−0.270272			−0.135136	−	0.990827i	$-0.543147\pi$
$787$	−212.704			−0.270272			−0.135136	+	0.990827i	$0.543147\pi$
$788$	140.463			0.178252
$789$		0			0
$790$		−	1219.92i		−	1.54421i
$791$			1132.99i			1.43235i
$792$		0			0
$793$	321.931			0.405966
$794$	−838.610			−1.05618
$795$		0			0
$796$	−117.974			−0.148208
$797$			487.589i			0.611780i	0.952067	+	0.305890i	$0.0989540\pi$
$797$			487.589i			0.611780i	−0.952067	+	0.305890i	$0.901046\pi$
$798$		0			0
$799$		−	1425.05i		−	1.78354i
$800$		−	328.847i		−	0.411059i
$801$		0			0
$802$	−888.353			−1.10767
$803$	1118.33			1.39269
$804$		0			0
$805$			1428.99i			1.77515i
$806$		−	373.468i		−	0.463360i
$807$		0			0
$808$	−491.784			−0.608643
$809$		−	913.419i		−	1.12907i	−0.825408	−	0.564536i	$-0.809055\pi$
$809$		−	913.419i		−	1.12907i	0.825408	−	0.564536i	$-0.190945\pi$
$810$		0			0
$811$		−	569.765i		−	0.702547i	−0.936273	−	0.351273i	$-0.885749\pi$
$811$		−	569.765i		−	0.702547i	0.936273	−	0.351273i	$-0.114251\pi$
$812$	430.039			0.529604
$813$		0			0
$814$			49.7231i			0.0610849i
$815$	1446.40			1.77473
$816$		0			0
$817$			154.034i			0.188536i
$818$	−28.8388			−0.0352553
$819$		0			0
$820$	−413.006			−0.503666
$821$		−	1353.27i		−	1.64832i	−0.566359	−	0.824158i	$-0.691649\pi$
$821$		−	1353.27i		−	1.64832i	0.566359	−	0.824158i	$-0.308351\pi$
$822$		0			0
$823$		−	1255.98i		−	1.52610i	−0.646340	−	0.763049i	$-0.723701\pi$
$823$		−	1255.98i		−	1.52610i	0.646340	−	0.763049i	$-0.276299\pi$
$824$	868.105			1.05353
$825$		0			0
$826$	632.586	−	50.3381i	0.765842	−	0.0609420i
$827$	1318.17			1.59392			0.796961	−	0.604031i	$-0.206440\pi$
$827$	1318.17			1.59392			0.796961	+	0.604031i	$0.206440\pi$
$828$		0			0
$829$	−329.831			−0.397866			−0.198933	−	0.980013i	$-0.563748\pi$
$829$	−329.831			−0.397866			−0.198933	+	0.980013i	$0.563748\pi$
$830$	−886.758			−1.06838
$831$		0			0
$832$		−	533.306i		−	0.640992i
$833$	86.3887			0.103708
$834$		0			0
$835$	−1125.23			−1.34758
$836$		−	159.324i		−	0.190579i
$837$		0			0
$838$	30.3060			0.0361647
$839$		−	949.502i		−	1.13171i	−0.824506	−	0.565853i	$-0.808547\pi$
$839$		−	949.502i		−	1.13171i	0.824506	−	0.565853i	$-0.191453\pi$
$840$		0			0
$841$	1293.92			1.53856
$842$	−906.430			−1.07652
$843$		0			0
$844$			428.805i			0.508063i
$845$	684.601			0.810179
$846$		0			0
$847$	−910.731			−1.07524
$848$	165.517			0.195185
$849$		0			0
$850$			473.881i			0.557507i
$851$	64.6464			0.0759653
$852$		0			0
$853$	442.888			0.519213			0.259606	−	0.965715i	$-0.416407\pi$
$853$	442.888			0.519213			0.259606	+	0.965715i	$0.416407\pi$
$854$	−441.642			−0.517146
$855$		0			0
$856$		−	1680.84i		−	1.96360i
$857$			1181.30i			1.37841i	0.724567	+	0.689204i	$0.242040\pi$
$857$			1181.30i			1.37841i	−0.724567	+	0.689204i	$0.757960\pi$
$858$		0			0
$859$		−	822.613i		−	0.957641i	−0.877913	−	0.478820i	$-0.841065\pi$
$859$		−	822.613i		−	0.957641i	0.877913	−	0.478820i	$-0.158935\pi$
$860$			192.615i			0.223971i
$861$		0			0
$862$	−70.7271			−0.0820500
$863$			220.298i			0.255270i	0.991821	+	0.127635i	$0.0407385\pi$
$863$			220.298i			0.255270i	−0.991821	+	0.127635i	$0.959261\pi$
$864$		0			0
$865$		−	1177.16i		−	1.36087i
$866$		−	162.251i		−	0.187356i
$867$		0			0
$868$		−	274.753i		−	0.316536i
$869$		−	1906.02i		−	2.19335i
$870$		0			0
$871$	707.065			0.811785
$872$	−570.300			−0.654014
$873$		0			0
$874$	386.265			0.441951
$875$	−401.737			−0.459128
$876$		0			0
$877$	985.610			1.12384			0.561921	−	0.827191i	$-0.310062\pi$
$877$	985.610			1.12384			0.561921	+	0.827191i	$0.310062\pi$
$878$		−	936.165i		−	1.06625i
$879$		0			0
$880$			865.056i			0.983019i
$881$		−	444.402i		−	0.504429i	−0.967671	−	0.252215i	$-0.918841\pi$
$881$		−	444.402i		−	0.504429i	0.967671	−	0.252215i	$-0.0811589\pi$
$882$		0			0
$883$	268.071			0.303592			0.151796	−	0.988412i	$-0.451494\pi$
$883$	268.071			0.303592			0.151796	+	0.988412i	$0.451494\pi$
$884$		−	206.972i		−	0.234131i
$885$		0			0
$886$	−1231.84			−1.39034
$887$		−	231.757i		−	0.261282i	−0.991430	−	0.130641i	$-0.958297\pi$
$887$		−	231.757i		−	0.261282i	0.991430	−	0.130641i	$-0.0417035\pi$
$888$		0			0
$889$	−1111.49			−1.25027
$890$	−395.449			−0.444325
$891$		0			0
$892$	−466.490			−0.522971
$893$			535.833i			0.600037i
$894$		0			0
$895$			1852.20i			2.06950i
$896$			167.162i			0.186565i
$897$		0			0
$898$			565.703i			0.629958i
$899$		−	1364.01i		−	1.51725i
$900$		0			0
$901$	369.666			0.410284
$902$	1203.29			1.33403
$903$		0			0
$904$	1480.05			1.63722
$905$	458.361			0.506476
$906$		0			0
$907$	569.019			0.627364			0.313682	−	0.949528i	$-0.398437\pi$
$907$	569.019			0.627364			0.313682	+	0.949528i	$0.398437\pi$
$908$		−	271.465i		−	0.298970i
$909$		0			0
$910$	536.877			0.589975
$911$	557.041			0.611461			0.305730	−	0.952118i	$-0.401099\pi$
$911$	557.041			0.611461			0.305730	+	0.952118i	$0.401099\pi$
$912$		0			0
$913$	−1385.48			−1.51750
$914$	1013.32			1.10867
$915$		0			0
$916$		−	277.616i		−	0.303075i
$917$		−	303.464i		−	0.330931i
$918$		0			0
$919$			389.979i			0.424351i	0.977232	+	0.212176i	$0.0680549\pi$
$919$			389.979i			0.424351i	−0.977232	+	0.212176i	$0.931945\pi$
$920$	1866.72			2.02905
$921$		0			0
$922$			972.588i			1.05487i
$923$		−	448.401i		−	0.485809i
$924$		0			0
$925$		−	29.8215i		−	0.0322395i
$926$	549.841			0.593781
$927$		0			0
$928$		−	978.180i		−	1.05407i
$929$			235.870i			0.253897i	0.991909	+	0.126948i	$0.0405183\pi$
$929$			235.870i			0.253897i	−0.991909	+	0.126948i	$0.959482\pi$
$930$		0			0
$931$	−32.4831			−0.0348906
$932$			374.892i			0.402244i
$933$		0			0
$934$	1101.56			1.17940
$935$			1932.02i			2.06633i
$936$		0			0
$937$		−	772.809i		−	0.824770i	−0.911010	−	0.412385i	$-0.864696\pi$
$937$		−	772.809i		−	0.824770i	0.911010	−	0.412385i	$-0.135304\pi$
$938$	−969.989			−1.03410
$939$		0			0
$940$			670.045i			0.712814i
$941$		−	1275.95i		−	1.35595i	−0.735086	−	0.677974i	$-0.762858\pi$
$941$		−	1275.95i		−	1.35595i	0.735086	−	0.677974i	$-0.237142\pi$
$942$		0			0
$943$		−	1564.43i		−	1.65900i
$944$	−39.6184	−	497.875i	−0.0419687	−	0.527410i
$945$		0			0
$946$		−	561.182i		−	0.593216i
$947$	−1385.44			−1.46297			−0.731487	−	0.681855i	$-0.761173\pi$
$947$	−1385.44			−1.46297			−0.731487	+	0.681855i	$0.761173\pi$
$948$		0			0
$949$	−546.261			−0.575618
$950$		−	178.185i		−	0.187563i
$951$		0			0
$952$			1097.34i			1.15266i
$953$	1150.91			1.20767			0.603835	−	0.797109i	$-0.293638\pi$
$953$	1150.91			1.20767			0.603835	+	0.797109i	$0.293638\pi$
$954$		0			0
$955$			1081.83i			1.13281i
$956$	−448.858			−0.469517
$957$		0			0
$958$		−	1222.99i		−	1.27661i
$959$	711.200			0.741606
$960$		0			0
$961$	89.5293			0.0931626
$962$		−	24.2879i		−	0.0252473i
$963$		0			0
$964$	−62.8295			−0.0651758
$965$	−99.5819			−0.103194
$966$		0			0
$967$			876.840i			0.906763i	0.891317	+	0.453382i	$0.149782\pi$
$967$			876.840i			0.906763i	−0.891317	+	0.453382i	$0.850218\pi$
$968$			1189.71i			1.22904i
$969$		0			0
$970$	−247.012			−0.254652
$971$	−794.804			−0.818542			−0.409271	−	0.912413i	$-0.634217\pi$
$971$	−794.804			−0.818542			−0.409271	+	0.912413i	$0.634217\pi$
$972$		0			0
$973$	−890.772			−0.915490
$974$			170.124i			0.174665i
$975$		0			0
$976$			347.594i			0.356141i
$977$		−	1736.90i		−	1.77779i	−0.458114	−	0.888893i	$-0.651475\pi$
$977$		−	1736.90i		−	1.77779i	0.458114	−	0.888893i	$-0.348525\pi$
$978$		0			0
$979$	−617.855			−0.631108
$980$	−40.6192			−0.0414482
$981$		0			0
$982$		−	973.908i		−	0.991760i
$983$		−	427.042i		−	0.434427i	−0.976124	−	0.217214i	$-0.930303\pi$
$983$		−	427.042i		−	0.434427i	0.976124	−	0.217214i	$-0.0696968\pi$
$984$		0			0
$985$	640.460			0.650214
$986$			1409.59i			1.42961i
$987$		0			0
$988$			77.8238i			0.0787690i
$989$	−729.609			−0.737724
$990$		0			0
$991$			1352.88i			1.36517i	0.730806	+	0.682585i	$0.239144\pi$
$991$			1352.88i			1.36517i	−0.730806	+	0.682585i	$0.760856\pi$
$992$	−624.963			−0.630003
$993$		0			0
$994$			615.141i			0.618854i
$995$	−537.918			−0.540621
$996$		0			0
$997$	1441.02			1.44535			0.722676	−	0.691187i	$-0.242912\pi$
$997$	1441.02			1.44535			0.722676	+	0.691187i	$0.242912\pi$
$998$		−	497.548i		−	0.498545i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.c.235.7		20
3.2	odd	2		177.3.c.a.58.14	yes	20
59.58	odd	2	inner	531.3.c.c.235.14		20
177.176	even	2		177.3.c.a.58.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.7	✓	20	177.176	even	2
177.3.c.a.58.14	yes	20	3.2	odd	2
531.3.c.c.235.7		20	1.1	even	1	trivial
531.3.c.c.235.14		20	59.58	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.61360i q^{2} +1.39629 q^{4} +6.36659 q^{5} +6.66564 q^{7} -8.70746i q^{8} +O(q^{10})\)	\(q-1.61360i q^{2} +1.39629 q^{4} +6.36659 q^{5} +6.66564 q^{7} -8.70746i q^{8} -10.2731i q^{10} -16.0509i q^{11} +7.84026i q^{13} -10.7557i q^{14} -8.46523 q^{16} -18.9063 q^{17} +7.10898 q^{19} +8.88958 q^{20} -25.8997 q^{22} +33.6730i q^{23} +15.5334 q^{25} +12.6511 q^{26} +9.30714 q^{28} +46.2052 q^{29} -29.5207i q^{31} -21.1703i q^{32} +30.5072i q^{34} +42.4373 q^{35} -1.91983i q^{37} -11.4711i q^{38} -55.4368i q^{40} -46.4596 q^{41} +21.6675i q^{43} -22.4116i q^{44} +54.3348 q^{46} +75.3742i q^{47} -4.56931 q^{49} -25.0647i q^{50} +10.9473i q^{52} -19.5526 q^{53} -102.189i q^{55} -58.0408i q^{56} -74.5569i q^{58} +(4.68014 + 58.8141i) q^{59} -41.0613i q^{61} -47.6346 q^{62} -68.0214 q^{64} +49.9157i q^{65} -90.1839i q^{67} -26.3986 q^{68} -68.4770i q^{70} -57.1922 q^{71} +69.6739i q^{73} -3.09784 q^{74} +9.92617 q^{76} -106.989i q^{77} +118.749 q^{79} -53.8946 q^{80} +74.9673i q^{82} -86.3181i q^{83} -120.368 q^{85} +34.9627 q^{86} -139.762 q^{88} -38.4935i q^{89} +52.2603i q^{91} +47.0172i q^{92} +121.624 q^{94} +45.2599 q^{95} -24.0445i q^{97} +7.37305i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7	\( 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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