LMFDB - Embedded newform 531.5.c.d.235.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,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, 5, names="a")

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.15
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33963i q^{2} +10.5261 q^{4} -30.0439 q^{5} -41.7287 q^{7} -62.0613i q^{8} +O(q^{10})$	\(q-2.33963i q^{2} +10.5261 q^{4} -30.0439 q^{5} -41.7287 q^{7} -62.0613i q^{8} +70.2916i q^{10} -117.370i q^{11} -118.348i q^{13} +97.6298i q^{14} +23.2175 q^{16} -263.868 q^{17} +373.890 q^{19} -316.246 q^{20} -274.603 q^{22} -503.177i q^{23} +277.635 q^{25} -276.891 q^{26} -439.242 q^{28} +549.815 q^{29} +450.839i q^{31} -1047.30i q^{32} +617.355i q^{34} +1253.69 q^{35} +1430.88i q^{37} -874.765i q^{38} +1864.56i q^{40} -1441.61 q^{41} +3202.06i q^{43} -1235.45i q^{44} -1177.25 q^{46} +967.133i q^{47} -659.713 q^{49} -649.564i q^{50} -1245.75i q^{52} -1662.12 q^{53} +3526.26i q^{55} +2589.74i q^{56} -1286.36i q^{58} +(34.3306 + 3480.83i) q^{59} -1712.94i q^{61} +1054.80 q^{62} -2078.82 q^{64} +3555.63i q^{65} +5062.54i q^{67} -2777.51 q^{68} -2933.18i q^{70} -6150.31 q^{71} -4170.58i q^{73} +3347.74 q^{74} +3935.62 q^{76} +4897.71i q^{77} -5990.22 q^{79} -697.544 q^{80} +3372.84i q^{82} -2937.78i q^{83} +7927.64 q^{85} +7491.63 q^{86} -7284.15 q^{88} -3877.47i q^{89} +4938.51i q^{91} -5296.50i q^{92} +2262.73 q^{94} -11233.1 q^{95} +2340.59i q^{97} +1543.48i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7	$ 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * 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 $

$2$		−	2.33963i		−	0.584908i	−0.956280	−	0.292454i	$-0.905528\pi$
$2$		−	2.33963i		−	0.584908i	0.956280	−	0.292454i	$-0.0944718\pi$
$3$		0			0
$3$		0			0
$4$	10.5261			0.657883
$5$	−30.0439			−1.20176			−0.600878	−	0.799341i	$-0.705182\pi$
$5$	−30.0439			−1.20176			−0.600878	+	0.799341i	$0.705182\pi$
$6$		0			0
$7$	−41.7287			−0.851607			−0.425803	−	0.904816i	$-0.640008\pi$
$7$	−41.7287			−0.851607			−0.425803	+	0.904816i	$0.640008\pi$
$8$		−	62.0613i		−	0.969708i
$9$		0			0
$10$			70.2916i			0.702916i
$11$		−	117.370i		−	0.970001i	−0.874514	−	0.485000i	$-0.838819\pi$
$11$		−	117.370i		−	0.970001i	0.874514	−	0.485000i	$-0.161181\pi$
$12$		0			0
$13$		−	118.348i		−	0.700284i	−0.936697	−	0.350142i	$-0.886133\pi$
$13$		−	118.348i		−	0.700284i	0.936697	−	0.350142i	$-0.113867\pi$
$14$			97.6298i			0.498111i
$15$		0			0
$16$	23.2175			0.0906934
$17$	−263.868			−0.913040			−0.456520	−	0.889713i	$-0.650904\pi$
$17$	−263.868			−0.913040			−0.456520	+	0.889713i	$0.650904\pi$
$18$		0			0
$19$	373.890			1.03571			0.517854	−	0.855469i	$-0.326731\pi$
$19$	373.890			1.03571			0.517854	+	0.855469i	$0.326731\pi$
$20$	−316.246			−0.790615
$21$		0			0
$22$	−274.603			−0.567361
$23$		−	503.177i		−	0.951185i	−0.879666	−	0.475592i	$-0.842234\pi$
$23$		−	503.177i		−	0.951185i	0.879666	−	0.475592i	$-0.157766\pi$
$24$		0			0
$25$	277.635			0.444217
$26$	−276.891			−0.409601
$27$		0			0
$28$	−439.242			−0.560258
$29$	549.815			0.653763			0.326882	−	0.945065i	$-0.394002\pi$
$29$	549.815			0.653763			0.326882	+	0.945065i	$0.394002\pi$
$30$		0			0
$31$			450.839i			0.469135i	0.972100	+	0.234568i	$0.0753674\pi$
$31$			450.839i			0.469135i	−0.972100	+	0.234568i	$0.924633\pi$
$32$		−	1047.30i		−	1.02276i
$33$		0			0
$34$			617.355i			0.534044i
$35$	1253.69			1.02342
$36$		0			0
$37$			1430.88i			1.04520i	0.852577	+	0.522601i	$0.175038\pi$
$37$			1430.88i			1.04520i	−0.852577	+	0.522601i	$0.824962\pi$
$38$		−	874.765i		−	0.605793i
$39$		0			0
$40$			1864.56i			1.16535i
$41$	−1441.61			−0.857591			−0.428796	−	0.903401i	$-0.641062\pi$
$41$	−1441.61			−0.857591			−0.428796	+	0.903401i	$0.641062\pi$
$42$		0			0
$43$			3202.06i			1.73178i	0.500236	+	0.865889i	$0.333247\pi$
$43$			3202.06i			1.73178i	−0.500236	+	0.865889i	$0.666753\pi$
$44$		−	1235.45i		−	0.638147i
$45$		0			0
$46$	−1177.25			−0.556355
$47$			967.133i			0.437815i	0.975746	+	0.218907i	$0.0702493\pi$
$47$			967.133i			0.437815i	−0.975746	+	0.218907i	$0.929751\pi$
$48$		0			0
$49$	−659.713			−0.274766
$50$		−	649.564i		−	0.259826i
$51$		0			0
$52$		−	1245.75i		−	0.460705i
$53$	−1662.12			−0.591713			−0.295856	−	0.955232i	$-0.595605\pi$
$53$	−1662.12			−0.591713			−0.295856	+	0.955232i	$0.595605\pi$
$54$		0			0
$55$			3526.26i			1.16570i
$56$			2589.74i			0.825810i
$57$		0			0
$58$		−	1286.36i		−	0.382391i
$59$	34.3306	+	3480.83i	0.00986228	+	0.999951i
$59$	34.3306	+	3480.83i	0.00986228	+	0.999951i
$60$		0			0
$61$		−	1712.94i		−	0.460343i	−0.973150	−	0.230172i	$-0.926071\pi$
$61$		−	1712.94i		−	0.460343i	0.973150	−	0.230172i	$-0.0739288\pi$
$62$	1054.80			0.274401
$63$		0			0
$64$	−2078.82			−0.507524
$65$			3555.63i			0.841570i
$66$		0			0
$67$			5062.54i			1.12777i	0.825854	+	0.563883i	$0.190693\pi$
$67$			5062.54i			1.12777i	−0.825854	+	0.563883i	$0.809307\pi$
$68$	−2777.51			−0.600673
$69$		0			0
$70$		−	2933.18i		−	0.598608i
$71$	−6150.31			−1.22006			−0.610028	−	0.792380i	$-0.708842\pi$
$71$	−6150.31			−1.22006			−0.610028	+	0.792380i	$0.708842\pi$
$72$		0			0
$73$		−	4170.58i		−	0.782619i	−0.920259	−	0.391309i	$-0.872022\pi$
$73$		−	4170.58i		−	0.782619i	0.920259	−	0.391309i	$-0.127978\pi$
$74$	3347.74			0.611347
$75$		0			0
$76$	3935.62			0.681374
$77$			4897.71i			0.826059i
$78$		0			0
$79$	−5990.22			−0.959818			−0.479909	−	0.877318i	$-0.659330\pi$
$79$	−5990.22			−0.959818			−0.479909	+	0.877318i	$0.659330\pi$
$80$	−697.544			−0.108991
$81$		0			0
$82$			3372.84i			0.501612i
$83$		−	2937.78i		−	0.426445i	−0.977004	−	0.213222i	$-0.931604\pi$
$83$		−	2937.78i		−	0.426445i	0.977004	−	0.213222i	$-0.0683959\pi$
$84$		0			0
$85$	7927.64			1.09725
$86$	7491.63			1.01293
$87$		0			0
$88$	−7284.15			−0.940618
$89$		−	3877.47i		−	0.489518i	−0.969584	−	0.244759i	$-0.921291\pi$
$89$		−	3877.47i		−	0.489518i	0.969584	−	0.244759i	$-0.0787088\pi$
$90$		0			0
$91$			4938.51i			0.596367i
$92$		−	5296.50i		−	0.625769i
$93$		0			0
$94$	2262.73			0.256081
$95$	−11233.1			−1.24467
$96$		0			0
$97$			2340.59i			0.248760i	0.992235	+	0.124380i	$0.0396942\pi$
$97$			2340.59i			0.248760i	−0.992235	+	0.124380i	$0.960306\pi$
$98$			1543.48i			0.160713i
$99$		0			0
$100$	2922.43			0.292243
$101$			18031.3i			1.76761i	0.467860	+	0.883803i	$0.345025\pi$
$101$			18031.3i			1.76761i	−0.467860	+	0.883803i	$0.654975\pi$
$102$		0			0
$103$		−	12277.9i		−	1.15731i	−0.815572	−	0.578656i	$-0.803577\pi$
$103$		−	12277.9i		−	1.15731i	0.815572	−	0.578656i	$-0.196423\pi$
$104$	−7344.83			−0.679071
$105$		0			0
$106$			3888.75i			0.346097i
$107$	−7815.43			−0.682630			−0.341315	−	0.939949i	$-0.610872\pi$
$107$	−7815.43			−0.682630			−0.341315	+	0.939949i	$0.610872\pi$
$108$		0			0
$109$		−	4707.99i		−	0.396262i	−0.980176	−	0.198131i	$-0.936513\pi$
$109$		−	4707.99i		−	0.396262i	0.980176	−	0.198131i	$-0.0634872\pi$
$110$	8250.13			0.681829
$111$		0			0
$112$	−968.837			−0.0772351
$113$			8859.89i			0.693860i	0.937891	+	0.346930i	$0.112776\pi$
$113$			8859.89i			0.693860i	−0.937891	+	0.346930i	$0.887224\pi$
$114$		0			0
$115$			15117.4i			1.14309i
$116$	5787.42			0.430100
$117$		0			0
$118$	8143.86	−	80.3209i	0.584879	−	0.00576852i
$119$	11010.9			0.777551
$120$		0			0
$121$	865.256			0.0590981
$122$	−4007.64			−0.269258
$123$		0			0
$124$			4745.59i			0.308636i
$125$	10436.2			0.667916
$126$		0			0
$127$	−7759.81			−0.481109			−0.240554	−	0.970636i	$-0.577329\pi$
$127$	−7759.81			−0.481109			−0.240554	+	0.970636i	$0.577329\pi$
$128$		−	11893.2i		−	0.725901i
$129$		0			0
$130$	8318.87			0.492241
$131$			26570.9i			1.54833i	0.632984	+	0.774165i	$0.281830\pi$
$131$			26570.9i			1.54833i	−0.632984	+	0.774165i	$0.718170\pi$
$132$		0			0
$133$	−15602.0			−0.882015
$134$	11844.5			0.659639
$135$		0			0
$136$			16376.0i			0.885382i
$137$	6370.46			0.339414			0.169707	−	0.985495i	$-0.445718\pi$
$137$	6370.46			0.339414			0.169707	+	0.985495i	$0.445718\pi$
$138$		0			0
$139$	26487.6			1.37092			0.685461	−	0.728109i	$-0.259601\pi$
$139$	26487.6			1.37092			0.685461	+	0.728109i	$0.259601\pi$
$140$	13196.5			0.673293
$141$		0			0
$142$			14389.4i			0.713620i
$143$	−13890.5			−0.679276
$144$		0			0
$145$	−16518.6			−0.785664
$146$	−9757.60			−0.457760
$147$		0			0
$148$			15061.7i			0.687621i
$149$			37732.0i			1.69956i	0.527136	+	0.849781i	$0.323266\pi$
$149$			37732.0i			1.69956i	−0.527136	+	0.849781i	$0.676734\pi$
$150$		0			0
$151$			19425.6i			0.851961i	0.904732	+	0.425981i	$0.140071\pi$
$151$			19425.6i			0.851961i	−0.904732	+	0.425981i	$0.859929\pi$
$152$		−	23204.1i		−	1.00433i
$153$		0			0
$154$	11458.8			0.483168
$155$		−	13545.0i		−	0.563786i
$156$		0			0
$157$		−	40542.8i		−	1.64480i	−0.568907	−	0.822402i	$-0.692634\pi$
$157$		−	40542.8i		−	1.64480i	0.568907	−	0.822402i	$-0.307366\pi$
$158$			14014.9i			0.561405i
$159$		0			0
$160$			31465.0i			1.22910i
$161$			20996.9i			0.810036i
$162$		0			0
$163$	−6024.65			−0.226755			−0.113377	−	0.993552i	$-0.536167\pi$
$163$	−6024.65			−0.226755			−0.113377	+	0.993552i	$0.536167\pi$
$164$	−15174.6			−0.564195
$165$		0			0
$166$	−6873.32			−0.249431
$167$	38348.4			1.37504			0.687519	−	0.726167i	$-0.258700\pi$
$167$	38348.4			1.37504			0.687519	+	0.726167i	$0.258700\pi$
$168$		0			0
$169$	14554.8			0.509602
$170$		−	18547.7i		−	0.641790i
$171$		0			0
$172$			33705.3i			1.13931i
$173$		−	6642.40i		−	0.221938i	−0.993824	−	0.110969i	$-0.964604\pi$
$173$		−	6642.40i		−	0.221938i	0.993824	−	0.110969i	$-0.0353955\pi$
$174$		0			0
$175$	−11585.4			−0.378298
$176$		−	2725.04i		−	0.0879727i
$177$		0			0
$178$	−9071.85			−0.286323
$179$			15505.0i			0.483910i	0.970287	+	0.241955i	$0.0777887\pi$
$179$			15505.0i			0.483910i	−0.970287	+	0.241955i	$0.922211\pi$
$180$		0			0
$181$	−7207.84			−0.220013			−0.110006	−	0.993931i	$-0.535087\pi$
$181$	−7207.84			−0.220013			−0.110006	+	0.993931i	$0.535087\pi$
$182$	11554.3			0.348819
$183$		0			0
$184$	−31227.8			−0.922372
$185$		−	42989.3i		−	1.25608i
$186$		0			0
$187$			30970.3i			0.885649i
$188$			10180.2i			0.288031i
$189$		0			0
$190$			26281.3i			0.728015i
$191$			24422.5i			0.669459i	0.942314	+	0.334729i	$0.108645\pi$
$191$			24422.5i			0.669459i	−0.942314	+	0.334729i	$0.891355\pi$
$192$		0			0
$193$	−42315.7			−1.13602			−0.568011	−	0.823021i	$-0.692287\pi$
$193$	−42315.7			−1.13602			−0.568011	+	0.823021i	$0.692287\pi$
$194$	5476.11			0.145502
$195$		0			0
$196$	−6944.23			−0.180764
$197$	3081.53			0.0794024			0.0397012	−	0.999212i	$-0.487359\pi$
$197$	3081.53			0.0794024			0.0397012	+	0.999212i	$0.487359\pi$
$198$		0			0
$199$	71081.9			1.79495			0.897476	−	0.441064i	$-0.145399\pi$
$199$	71081.9			1.79495			0.897476	+	0.441064i	$0.145399\pi$
$200$		−	17230.4i		−	0.430761i
$201$		0			0
$202$	42186.7			1.03389
$203$	−22943.1			−0.556749
$204$		0			0
$205$	43311.6			1.03062
$206$	−28725.8			−0.676920
$207$		0			0
$208$		−	2747.74i		−	0.0635111i
$209$		−	43883.5i		−	1.00464i
$210$		0			0
$211$		−	31417.1i		−	0.705670i	−0.935686	−	0.352835i	$-0.885218\pi$
$211$		−	31417.1i		−	0.705670i	0.935686	−	0.352835i	$-0.114782\pi$
$212$	−17495.7			−0.389278
$213$		0			0
$214$			18285.2i			0.399275i
$215$		−	96202.3i		−	2.08117i
$216$		0			0
$217$		−	18812.9i		−	0.399519i
$218$	−11015.0			−0.231777
$219$		0			0
$220$			37117.8i			0.766897i
$221$			31228.3i			0.639387i
$222$		0			0
$223$	−55061.3			−1.10723			−0.553613	−	0.832774i	$-0.686751\pi$
$223$	−55061.3			−1.10723			−0.553613	+	0.832774i	$0.686751\pi$
$224$			43702.6i			0.870986i
$225$		0			0
$226$	20728.9			0.405844
$227$		−	23259.5i		−	0.451386i	−0.974198	−	0.225693i	$-0.927535\pi$
$227$		−	23259.5i		−	0.451386i	0.974198	−	0.225693i	$-0.0724646\pi$
$228$		0			0
$229$		−	16859.6i		−	0.321496i	−0.986995	−	0.160748i	$-0.948609\pi$
$229$		−	16859.6i		−	0.321496i	0.986995	−	0.160748i	$-0.0513907\pi$
$230$	35369.1			0.668603
$231$		0			0
$232$		−	34122.3i		−	0.633960i
$233$		−	56400.3i		−	1.03889i	−0.854504	−	0.519445i	$-0.826139\pi$
$233$		−	56400.3i		−	1.03889i	0.854504	−	0.519445i	$-0.173861\pi$
$234$		0			0
$235$		−	29056.4i		−	0.526147i
$236$	361.368	+	36639.7i	0.00648823	+	0.657851i
$237$		0			0
$238$		−	25761.4i		−	0.454795i
$239$	−17569.0			−0.307575			−0.153788	−	0.988104i	$-0.549147\pi$
$239$	−17569.0			−0.307575			−0.153788	+	0.988104i	$0.549147\pi$
$240$		0			0
$241$	97271.4			1.67475			0.837377	−	0.546625i	$-0.184088\pi$
$241$	97271.4			1.67475			0.837377	+	0.546625i	$0.184088\pi$
$242$		−	2024.38i		−	0.0345670i
$243$		0			0
$244$		−	18030.6i		−	0.302852i
$245$	19820.3			0.330202
$246$		0			0
$247$		−	44249.2i		−	0.725289i
$248$	27979.7			0.454924
$249$		0			0
$250$		−	24416.8i		−	0.390669i
$251$	41339.8			0.656177			0.328089	−	0.944647i	$-0.393596\pi$
$251$	41339.8			0.656177			0.328089	+	0.944647i	$0.393596\pi$
$252$		0			0
$253$	−59057.9			−0.922650
$254$			18155.1i			0.281404i
$255$		0			0
$256$	−61086.7			−0.932109
$257$	14776.8			0.223725			0.111863	−	0.993724i	$-0.464318\pi$
$257$	14776.8			0.223725			0.111863	+	0.993724i	$0.464318\pi$
$258$		0			0
$259$		−	59708.9i		−	0.890102i
$260$			37427.1i			0.553655i
$261$		0			0
$262$	62166.1			0.905630
$263$	−46511.4			−0.672431			−0.336216	−	0.941785i	$-0.609147\pi$
$263$	−46511.4			−0.672431			−0.336216	+	0.941785i	$0.609147\pi$
$264$		0			0
$265$	49936.6			0.711094
$266$			36502.8i			0.515897i
$267$		0			0
$268$			53289.0i			0.741939i
$269$		−	52167.3i		−	0.720931i	−0.932773	−	0.360465i	$-0.882618\pi$
$269$		−	52167.3i		−	0.720931i	0.932773	−	0.360465i	$-0.117382\pi$
$270$		0			0
$271$	−118272.			−1.61044			−0.805220	−	0.592977i	$-0.797953\pi$
$271$	−118272.			−1.61044			−0.805220	+	0.592977i	$0.797953\pi$
$272$	−6126.37			−0.0828066
$273$		0			0
$274$		−	14904.5i		−	0.198526i
$275$		−	32586.1i		−	0.430891i
$276$		0			0
$277$	−116769.			−1.52184			−0.760919	−	0.648847i	$-0.775252\pi$
$277$	−116769.			−1.52184			−0.760919	+	0.648847i	$0.775252\pi$
$278$		−	61971.2i		−	0.801863i
$279$		0			0
$280$		−	77805.9i		−	0.992422i
$281$	−148473.			−1.88034			−0.940169	−	0.340710i	$-0.889333\pi$
$281$	−148473.			−1.88034			−0.940169	+	0.340710i	$0.889333\pi$
$282$		0			0
$283$		−	30655.6i		−	0.382769i	−0.981515	−	0.191385i	$-0.938702\pi$
$283$		−	30655.6i		−	0.382769i	0.981515	−	0.191385i	$-0.0612978\pi$
$284$	−64738.9			−0.802655
$285$		0			0
$286$			32498.7i			0.397314i
$287$	60156.6			0.730331
$288$		0			0
$289$	−13894.4			−0.166359
$290$			38647.4i			0.459541i
$291$		0			0
$292$		−	43900.0i		−	0.514872i
$293$	−77404.6			−0.901636			−0.450818	−	0.892616i	$-0.648868\pi$
$293$	−77404.6			−0.901636			−0.450818	+	0.892616i	$0.648868\pi$
$294$		0			0
$295$	−1031.42	−	104578.i	−0.0118520	−	1.20170i
$296$	88802.5			1.01354
$297$		0			0
$298$	88278.8			0.994086
$299$	−59550.0			−0.666100
$300$		0			0
$301$		−	133618.i		−	1.47479i
$302$	45448.7			0.498319
$303$		0			0
$304$	8680.80			0.0939318
$305$			51463.3i			0.553220i
$306$		0			0
$307$	−124048.			−1.31617			−0.658085	−	0.752943i	$-0.728633\pi$
$307$	−124048.			−1.31617			−0.658085	+	0.752943i	$0.728633\pi$
$308$			51553.9i			0.543451i
$309$		0			0
$310$	−31690.2			−0.329762
$311$	3890.57			0.0402247			0.0201123	−	0.999798i	$-0.493598\pi$
$311$	3890.57			0.0402247			0.0201123	+	0.999798i	$0.493598\pi$
$312$		0			0
$313$			18870.8i			0.192620i	0.995351	+	0.0963099i	$0.0307040\pi$
$313$			18870.8i			0.192620i	−0.995351	+	0.0963099i	$0.969296\pi$
$314$	−94855.1			−0.962058
$315$		0			0
$316$	−63053.9			−0.631448
$317$	31502.4			0.313491			0.156745	−	0.987639i	$-0.449900\pi$
$317$	31502.4			0.313491			0.156745	+	0.987639i	$0.449900\pi$
$318$		0			0
$319$		−	64531.9i		−	0.634151i
$320$	62455.8			0.609920
$321$		0			0
$322$	49125.1			0.473796
$323$	−98657.8			−0.945642
$324$		0			0
$325$		−	32857.6i		−	0.311078i
$326$			14095.4i			0.132631i
$327$		0			0
$328$			89468.3i			0.831614i
$329$		−	40357.2i		−	0.372846i
$330$		0			0
$331$	4457.32			0.0406834			0.0203417	−	0.999793i	$-0.493525\pi$
$331$	4457.32			0.0406834			0.0203417	+	0.999793i	$0.493525\pi$
$332$		−	30923.4i		−	0.280551i
$333$		0			0
$334$		−	89721.1i		−	0.804270i
$335$		−	152099.i		−	1.35530i
$336$		0			0
$337$		−	46555.0i		−	0.409927i	−0.978770	−	0.204964i	$-0.934292\pi$
$337$		−	46555.0i		−	0.409927i	0.978770	−	0.204964i	$-0.0657076\pi$
$338$		−	34052.7i		−	0.298070i
$339$		0			0
$340$	83447.3			0.721863
$341$	52915.0			0.455061
$342$		0			0
$343$	127720.			1.08560
$344$	198724.			1.67932
$345$		0			0
$346$	−15540.8			−0.129814
$347$			173485.i			1.44080i	0.693560	+	0.720399i	$0.256041\pi$
$347$			173485.i			1.44080i	−0.693560	+	0.720399i	$0.743959\pi$
$348$		0			0
$349$			160924.i			1.32120i	0.750737	+	0.660601i	$0.229699\pi$
$349$			160924.i			1.32120i	−0.750737	+	0.660601i	$0.770301\pi$
$350$			27105.5i			0.221269i
$351$		0			0
$352$	−122922.			−0.992074
$353$			109289.i			0.877055i	0.898718	+	0.438527i	$0.144500\pi$
$353$			109289.i			0.877055i	−0.898718	+	0.438527i	$0.855500\pi$
$354$		0			0
$355$	184779.			1.46621
$356$		−	40814.8i		−	0.322045i
$357$		0			0
$358$	36275.9			0.283043
$359$	−127764.			−0.991336			−0.495668	−	0.868512i	$-0.665077\pi$
$359$	−127764.			−0.991336			−0.495668	+	0.868512i	$0.665077\pi$
$360$		0			0
$361$	9472.92			0.0726891
$362$			16863.7i			0.128687i
$363$		0			0
$364$			51983.4i			0.392339i
$365$			125300.i			0.940516i
$366$		0			0
$367$		−	131014.i		−	0.972718i	−0.873759	−	0.486359i	$-0.838325\pi$
$367$		−	131014.i		−	0.972718i	0.873759	−	0.486359i	$-0.161675\pi$
$368$		−	11682.5i		−	0.0862662i
$369$		0			0
$370$	−100579.			−0.734690
$371$	69358.2			0.503907
$372$		0			0
$373$	162049.			1.16474			0.582369	−	0.812924i	$-0.302126\pi$
$373$	162049.			1.16474			0.582369	+	0.812924i	$0.302126\pi$
$374$	72459.0			0.518023
$375$		0			0
$376$	60021.6			0.424553
$377$		−	65069.5i		−	0.457820i
$378$		0			0
$379$	−170314.			−1.18569			−0.592847	−	0.805315i	$-0.701996\pi$
$379$	−170314.			−1.18569			−0.592847	+	0.805315i	$0.701996\pi$
$380$	−118241.			−0.818845
$381$		0			0
$382$	57139.7			0.391572
$383$	−197908.			−1.34916			−0.674582	−	0.738200i	$-0.735676\pi$
$383$	−197908.			−1.34916			−0.674582	+	0.738200i	$0.735676\pi$
$384$		0			0
$385$		−	147146.i		−	0.992722i
$386$			99003.0i			0.664468i
$387$		0			0
$388$			24637.3i			0.163655i
$389$	130696.			0.863699			0.431849	−	0.901946i	$-0.357861\pi$
$389$	130696.			0.863699			0.431849	+	0.901946i	$0.357861\pi$
$390$		0			0
$391$			132772.i			0.868470i
$392$			40942.7i			0.266443i
$393$		0			0
$394$		−	7209.64i		−	0.0464431i
$395$	179970.			1.15347
$396$		0			0
$397$			35127.3i			0.222876i	0.993771	+	0.111438i	$0.0355457\pi$
$397$			35127.3i			0.222876i	−0.993771	+	0.111438i	$0.964454\pi$
$398$		−	166305.i		−	1.04988i
$399$		0			0
$400$	6446.00			0.0402875
$401$		−	177132.i		−	1.10156i	−0.834650	−	0.550780i	$-0.814330\pi$
$401$		−	177132.i		−	1.10156i	0.834650	−	0.550780i	$-0.185670\pi$
$402$		0			0
$403$	53355.9			0.328528
$404$			189800.i			1.16288i
$405$		0			0
$406$			53678.3i			0.325647i
$407$	167943.			1.01385
$408$		0			0
$409$		−	194928.i		−	1.16527i	−0.812733	−	0.582636i	$-0.802021\pi$
$409$		−	194928.i		−	1.16527i	0.812733	−	0.582636i	$-0.197979\pi$
$410$		−	101333.i		−	0.602815i
$411$		0			0
$412$		−	129239.i		−	0.761376i
$413$	−1432.57	−	145251.i	−0.00839878	−	0.851565i
$414$		0			0
$415$			88262.3i			0.512483i
$416$	−123946.			−0.716219
$417$		0			0
$418$	−102671.			−0.587620
$419$		−	86241.5i		−	0.491234i	−0.969367	−	0.245617i	$-0.921009\pi$
$419$		−	86241.5i		−	0.491234i	0.969367	−	0.245617i	$-0.0789906\pi$
$420$		0			0
$421$		−	51752.7i		−	0.291990i	−0.989285	−	0.145995i	$-0.953362\pi$
$421$		−	51752.7i		−	0.291990i	0.989285	−	0.145995i	$-0.0466384\pi$
$422$	−73504.5			−0.412752
$423$		0			0
$424$			103153.i			0.573789i
$425$	−73259.2			−0.405587
$426$		0			0
$427$			71478.7i			0.392031i
$428$	−82266.2			−0.449091
$429$		0			0
$430$	−225078.			−1.21729
$431$		−	184933.i		−	0.995545i	−0.867308	−	0.497773i	$-0.834151\pi$
$431$		−	184933.i		−	0.995545i	0.867308	−	0.497773i	$-0.165849\pi$
$432$		0			0
$433$	−201660.			−1.07558			−0.537790	−	0.843079i	$-0.680741\pi$
$433$	−201660.			−1.07558			−0.537790	+	0.843079i	$0.680741\pi$
$434$	−44015.3			−0.233681
$435$		0			0
$436$		−	49556.9i		−	0.260694i
$437$		−	188133.i		−	0.985149i
$438$		0			0
$439$	264954.			1.37480			0.687402	−	0.726277i	$-0.258751\pi$
$439$	264954.			1.37480			0.687402	+	0.726277i	$0.258751\pi$
$440$	218844.			1.13039
$441$		0			0
$442$	73062.7			0.373982
$443$		−	377029.i		−	1.92117i	−0.277978	−	0.960587i	$-0.589664\pi$
$443$		−	377029.i		−	1.92117i	0.277978	−	0.960587i	$-0.410336\pi$
$444$		0			0
$445$			116494.i			0.588281i
$446$			128823.i			0.647625i
$447$		0			0
$448$	86746.5			0.432211
$449$	−108089.			−0.536155			−0.268078	−	0.963397i	$-0.586388\pi$
$449$	−108089.			−0.536155			−0.268078	+	0.963397i	$0.586388\pi$
$450$		0			0
$451$			169202.i			0.831864i
$452$			93260.4i			0.456479i
$453$		0			0
$454$	−54418.5			−0.264019
$455$		−	148372.i		−	0.716687i
$456$		0			0
$457$			265515.i			1.27132i	0.771968	+	0.635662i	$0.219273\pi$
$457$			265515.i			1.27132i	−0.771968	+	0.635662i	$0.780727\pi$
$458$	−39445.2			−0.188046
$459$		0			0
$460$			159128.i			0.752021i
$461$	−216820.			−1.02023			−0.510114	−	0.860107i	$-0.670397\pi$
$461$	−216820.			−1.02023			−0.510114	+	0.860107i	$0.670397\pi$
$462$		0			0
$463$			245917.i			1.14716i	0.819148	+	0.573582i	$0.194447\pi$
$463$			245917.i			1.14716i	−0.819148	+	0.573582i	$0.805553\pi$
$464$	12765.3			0.0592920
$465$		0			0
$466$	−131956.			−0.607654
$467$			198167.i			0.908651i	0.890836	+	0.454326i	$0.150120\pi$
$467$			198167.i			0.908651i	−0.890836	+	0.454326i	$0.849880\pi$
$468$		0			0
$469$		−	211254.i		−	0.960414i
$470$	−67981.3			−0.307747
$471$		0			0
$472$	216025.	−	2130.60i	0.969661	−	0.00956353i
$473$	375826.			1.67983
$474$		0			0
$475$	103805.			0.460078
$476$	115902.			0.511538
$477$		0			0
$478$			41105.0i			0.179903i
$479$	419986.			1.83048			0.915238	−	0.402915i	$-0.132003\pi$
$479$	419986.			1.83048			0.915238	+	0.402915i	$0.132003\pi$
$480$		0			0
$481$	169342.			0.731939
$482$		−	227579.i		−	0.979577i
$483$		0			0
$484$	9107.80			0.0388797
$485$		−	70320.3i		−	0.298949i
$486$		0			0
$487$	−336768.			−1.41995			−0.709975	−	0.704227i	$-0.751294\pi$
$487$	−336768.			−1.41995			−0.709975	+	0.704227i	$0.751294\pi$
$488$	−106307.			−0.446399
$489$		0			0
$490$		−	46372.3i		−	0.193137i
$491$	−279399.			−1.15894			−0.579471	−	0.814993i	$-0.696741\pi$
$491$	−279399.			−1.15894			−0.579471	+	0.814993i	$0.696741\pi$
$492$		0			0
$493$	−145079.			−0.596912
$494$	−103527.			−0.424227
$495$		0			0
$496$			10467.4i			0.0425474i
$497$	256644.			1.03901
$498$		0			0
$499$	23800.2			0.0955826			0.0477913	−	0.998857i	$-0.484782\pi$
$499$	23800.2			0.0955826			0.0477913	+	0.998857i	$0.484782\pi$
$500$	109853.			0.439410
$501$		0			0
$502$		−	96719.9i		−	0.383803i
$503$			25529.1i			0.100902i	0.998727	+	0.0504509i	$0.0160659\pi$
$503$			25529.1i			0.100902i	−0.998727	+	0.0504509i	$0.983934\pi$
$504$		0			0
$505$		−	541732.i		−	2.12423i
$506$			138174.i			0.539665i
$507$		0			0
$508$	−81680.7			−0.316513
$509$			6537.03i			0.0252316i	0.999920	+	0.0126158i	$0.00401584\pi$
$509$			6537.03i			0.0252316i	−0.999920	+	0.0126158i	$0.995984\pi$
$510$		0			0
$511$			174033.i			0.666483i
$512$		−	47370.3i		−	0.180703i
$513$		0			0
$514$		−	34572.3i		−	0.130859i
$515$			368877.i			1.39081i
$516$		0			0
$517$	113513.			0.424681
$518$	−139697.			−0.520627
$519$		0			0
$520$	220667.			0.816078
$521$	307294.			1.13208			0.566042	−	0.824376i	$-0.308474\pi$
$521$	307294.			1.13208			0.566042	+	0.824376i	$0.308474\pi$
$522$		0			0
$523$	−92199.8			−0.337075			−0.168537	−	0.985695i	$-0.553904\pi$
$523$	−92199.8			−0.337075			−0.168537	+	0.985695i	$0.553904\pi$
$524$			279689.i			1.01862i
$525$		0			0
$526$			108819.i			0.393310i
$527$		−	118962.i		−	0.428339i
$528$		0			0
$529$	26654.1			0.0952472
$530$		−	116833.i		−	0.415924i
$531$		0			0
$532$	−164228.			−0.580263
$533$			170612.i			0.600557i
$534$		0			0
$535$	234806.			0.820354
$536$	314188.			1.09360
$537$		0			0
$538$	−122052.			−0.421678
$539$			77430.6i			0.266523i
$540$		0			0
$541$		−	234603.i		−	0.801564i	−0.916173	−	0.400782i	$-0.868738\pi$
$541$		−	234603.i		−	0.801564i	0.916173	−	0.400782i	$-0.131262\pi$
$542$			276713.i			0.941958i
$543$		0			0
$544$			276350.i			0.933816i
$545$			141446.i			0.476210i
$546$		0			0
$547$	−246372.			−0.823411			−0.411706	−	0.911317i	$-0.635067\pi$
$547$	−246372.			−0.823411			−0.411706	+	0.911317i	$0.635067\pi$
$548$	67056.3			0.223295
$549$		0			0
$550$	−76239.4			−0.252031
$551$	205570.			0.677107
$552$		0			0
$553$	249964.			0.817387
$554$			273197.i			0.890134i
$555$		0			0
$556$	278812.			0.901907
$557$	−567300.			−1.82853			−0.914266	−	0.405115i	$-0.867231\pi$
$557$	−567300.			−1.82853			−0.914266	+	0.405115i	$0.867231\pi$
$558$		0			0
$559$	378957.			1.21274
$560$	29107.6			0.0928177
$561$		0			0
$562$			347373.i			1.09982i
$563$			327570.i			1.03344i	0.856153	+	0.516722i	$0.172848\pi$
$563$			327570.i			1.03344i	−0.856153	+	0.516722i	$0.827152\pi$
$564$		0			0
$565$		−	266186.i		−	0.833850i
$566$	−71722.8			−0.223885
$567$		0			0
$568$			381696.i			1.18310i
$569$			379301.i			1.17154i	0.810476	+	0.585772i	$0.199209\pi$
$569$			379301.i			1.17154i	−0.810476	+	0.585772i	$0.800791\pi$
$570$		0			0
$571$			580504.i			1.78046i	0.455509	+	0.890231i	$0.349457\pi$
$571$			580504.i			1.78046i	−0.455509	+	0.890231i	$0.650543\pi$
$572$	−146213.			−0.446884
$573$		0			0
$574$		−	140744.i		−	0.427176i
$575$		−	139700.i		−	0.422532i
$576$		0			0
$577$	−34030.7			−0.102216			−0.0511080	−	0.998693i	$-0.516275\pi$
$577$	−34030.7			−0.102216			−0.0511080	+	0.998693i	$0.516275\pi$
$578$			32507.8i			0.0973044i
$579$		0			0
$580$	−173877.			−0.516875
$581$			122590.i			0.363163i
$582$		0			0
$583$			195083.i			0.573962i
$584$	−258831.			−0.758912
$585$		0			0
$586$			181098.i			0.527374i
$587$			360445.i			1.04607i	0.852310	+	0.523037i	$0.175201\pi$
$587$			360445.i			1.04607i	−0.852310	+	0.523037i	$0.824799\pi$
$588$		0			0
$589$			168564.i			0.485886i
$590$	−244673.	+	2413.15i	−0.702882	+	0.00693235i
$591$		0			0
$592$			33221.5i			0.0947930i
$593$	−203460.			−0.578589			−0.289294	−	0.957240i	$-0.593421\pi$
$593$	−203460.			−0.578589			−0.289294	+	0.957240i	$0.593421\pi$
$594$		0			0
$595$	−330810.			−0.934426
$596$			397172.i			1.11811i
$597$		0			0
$598$			139325.i			0.389607i
$599$	−68798.6			−0.191746			−0.0958729	−	0.995394i	$-0.530564\pi$
$599$	−68798.6			−0.191746			−0.0958729	+	0.995394i	$0.530564\pi$
$600$		0			0
$601$			549707.i			1.52189i	0.648818	+	0.760943i	$0.275263\pi$
$601$			549707.i			1.52189i	−0.648818	+	0.760943i	$0.724737\pi$
$602$	−312616.			−0.862618
$603$		0			0
$604$			204476.i			0.560491i
$605$	−25995.7			−0.0710215
$606$		0			0
$607$	−350392.			−0.950991			−0.475495	−	0.879718i	$-0.657731\pi$
$607$	−350392.			−0.950991			−0.475495	+	0.879718i	$0.657731\pi$
$608$		−	391576.i		−	1.05928i
$609$		0			0
$610$	120405.			0.323583
$611$	114458.			0.306595
$612$		0			0
$613$		−	450003.i		−	1.19755i	−0.800917	−	0.598776i	$-0.795654\pi$
$613$		−	450003.i		−	1.19755i	0.800917	−	0.598776i	$-0.204346\pi$
$614$			290226.i			0.769838i
$615$		0			0
$616$	303958.			0.801037
$617$	−165810.			−0.435552			−0.217776	−	0.975999i	$-0.569880\pi$
$617$	−165810.			−0.435552			−0.217776	+	0.975999i	$0.569880\pi$
$618$		0			0
$619$	−286451.			−0.747599			−0.373799	−	0.927510i	$-0.621945\pi$
$619$	−286451.			−0.747599			−0.373799	+	0.927510i	$0.621945\pi$
$620$		−	142576.i		−	0.370905i
$621$		0			0
$622$		−	9102.49i		−	0.0235277i
$623$			161802.i			0.416877i
$624$		0			0
$625$	−487066.			−1.24689
$626$	44150.6			0.112665
$627$		0			0
$628$		−	426758.i		−	1.08209i
$629$		−	377565.i		−	0.954312i
$630$		0			0
$631$	94171.7			0.236517			0.118258	−	0.992983i	$-0.462269\pi$
$631$	94171.7			0.236517			0.118258	+	0.992983i	$0.462269\pi$
$632$			371761.i			0.930743i
$633$		0			0
$634$		−	73703.9i		−	0.183363i
$635$	233135.			0.578175
$636$		0			0
$637$			78075.7i			0.192414i
$638$	−150981.			−0.370920
$639$		0			0
$640$			357317.i			0.872356i
$641$	−739487.			−1.79976			−0.899880	−	0.436137i	$-0.856346\pi$
$641$	−739487.			−1.79976			−0.899880	+	0.436137i	$0.856346\pi$
$642$		0			0
$643$	446727.			1.08049			0.540244	−	0.841508i	$-0.318332\pi$
$643$	446727.			1.08049			0.540244	+	0.841508i	$0.318332\pi$
$644$			221016.i			0.532909i
$645$		0			0
$646$			230823.i			0.553113i
$647$	370357.			0.884732			0.442366	−	0.896835i	$-0.354139\pi$
$647$	370357.			0.884732			0.442366	+	0.896835i	$0.354139\pi$
$648$		0			0
$649$	408546.	−	4029.38i	0.969954	−	0.00956642i
$650$	−76874.6			−0.181952
$651$		0			0
$652$	−63416.2			−0.149178
$653$	339613.			0.796449			0.398224	−	0.917288i	$-0.369627\pi$
$653$	339613.			0.796449			0.398224	+	0.917288i	$0.369627\pi$
$654$		0			0
$655$		−	798293.i		−	1.86071i
$656$	−33470.6			−0.0777779
$657$		0			0
$658$	−94421.0			−0.218081
$659$			387109.i			0.891378i	0.895188	+	0.445689i	$0.147041\pi$
$659$			387109.i			0.891378i	−0.895188	+	0.445689i	$0.852959\pi$
$660$		0			0
$661$	−367793.			−0.841784			−0.420892	−	0.907111i	$-0.638283\pi$
$661$	−367793.			−0.841784			−0.420892	+	0.907111i	$0.638283\pi$
$662$		−	10428.5i		−	0.0237961i
$663$		0			0
$664$	−182322.			−0.413527
$665$	468744.			1.05997
$666$		0			0
$667$		−	276654.i		−	0.621850i
$668$	403660.			0.904614
$669$		0			0
$670$	−355854.			−0.792725
$671$	−201048.			−0.446533
$672$		0			0
$673$		−	356274.i		−	0.786600i	−0.919410	−	0.393300i	$-0.871333\pi$
$673$		−	356274.i		−	0.786600i	0.919410	−	0.393300i	$-0.128667\pi$
$674$	−108922.			−0.239770
$675$		0			0
$676$	153205.			0.335259
$677$	−427232.			−0.932152			−0.466076	−	0.884745i	$-0.654333\pi$
$677$	−427232.			−0.932152			−0.466076	+	0.884745i	$0.654333\pi$
$678$		0			0
$679$		−	97669.7i		−	0.211846i
$680$		−	492000.i		−	1.06401i
$681$		0			0
$682$		−	123802.i		−	0.266169i
$683$		−	816072.i		−	1.74939i	−0.484673	−	0.874695i	$-0.661061\pi$
$683$		−	816072.i		−	1.74939i	0.484673	−	0.874695i	$-0.338939\pi$
$684$		0			0
$685$	−191393.			−0.407893
$686$		−	298817.i		−	0.634975i
$687$		0			0
$688$			74343.8i			0.157061i
$689$			196709.i			0.414367i
$690$		0			0
$691$			101566.i			0.212712i	0.994328	+	0.106356i	$0.0339183\pi$
$691$			101566.i			0.212712i	−0.994328	+	0.106356i	$0.966082\pi$
$692$		−	69918.7i		−	0.146010i
$693$		0			0
$694$	405891.			0.842733
$695$	−795790.			−1.64751
$696$		0			0
$697$	380396.			0.783015
$698$	376502.			0.772782
$699$		0			0
$700$	−121949.			−0.248876
$701$			265473.i			0.540236i	0.962827	+	0.270118i	$0.0870628\pi$
$701$			265473.i			0.540236i	−0.962827	+	0.270118i	$0.912937\pi$
$702$		0			0
$703$			534993.i			1.08252i
$704$			243991.i			0.492299i
$705$		0			0
$706$	255696.			0.512996
$707$		−	752425.i		−	1.50530i
$708$		0			0
$709$	−380605.			−0.757150			−0.378575	−	0.925571i	$-0.623586\pi$
$709$	−380605.			−0.757150			−0.378575	+	0.925571i	$0.623586\pi$
$710$		−	432315.i		−	0.857597i
$711$		0			0
$712$	−240641.			−0.474689
$713$	226852.			0.446234
$714$		0			0
$715$	417325.			0.816324
$716$			163207.i			0.318356i
$717$		0			0
$718$			298921.i			0.579840i
$719$		−	356832.i		−	0.690249i	−0.938557	−	0.345124i	$-0.887837\pi$
$719$		−	356832.i		−	0.690249i	0.938557	−	0.345124i	$-0.112163\pi$
$720$		0			0
$721$			512342.i			0.985575i
$722$		−	22163.1i		−	0.0425164i
$723$		0			0
$724$	−75870.6			−0.144743
$725$	152648.			0.290413
$726$		0			0
$727$	84925.0			0.160682			0.0803409	−	0.996767i	$-0.474399\pi$
$727$	84925.0			0.160682			0.0803409	+	0.996767i	$0.474399\pi$
$728$	306491.			0.578302
$729$		0			0
$730$	293156.			0.550115
$731$		−	844922.i		−	1.58118i
$732$		0			0
$733$	−359849.			−0.669750			−0.334875	−	0.942263i	$-0.608694\pi$
$733$	−359849.			−0.669750			−0.334875	+	0.942263i	$0.608694\pi$
$734$	−306525.			−0.568950
$735$		0			0
$736$	−526978.			−0.972830
$737$	594191.			1.09393
$738$		0			0
$739$		−	813633.i		−	1.48984i	−0.667154	−	0.744920i	$-0.732488\pi$
$739$		−	813633.i		−	1.48984i	0.667154	−	0.744920i	$-0.267512\pi$
$740$		−	452511.i		−	0.826353i
$741$		0			0
$742$		−	162273.i		−	0.294739i
$743$	407788.			0.738680			0.369340	−	0.929294i	$-0.379584\pi$
$743$	407788.			0.738680			0.369340	+	0.929294i	$0.379584\pi$
$744$		0			0
$745$		−	1.13361e6i		−	2.04246i
$746$		−	379135.i		−	0.681265i
$747$		0			0
$748$			325997.i			0.582654i
$749$	326128.			0.581332
$750$		0			0
$751$			726139.i			1.28748i	0.765245	+	0.643739i	$0.222618\pi$
$751$			726139.i			1.28748i	−0.765245	+	0.643739i	$0.777382\pi$
$752$			22454.4i			0.0397069i
$753$		0			0
$754$	−152239.			−0.267782
$755$		−	583620.i		−	1.02385i
$756$		0			0
$757$	226073.			0.394508			0.197254	−	0.980352i	$-0.436798\pi$
$757$	226073.			0.394508			0.197254	+	0.980352i	$0.436798\pi$
$758$			398472.i			0.693521i
$759$		0			0
$760$			697142.i			1.20696i
$761$	32895.1			0.0568018			0.0284009	−	0.999597i	$-0.490958\pi$
$761$	32895.1			0.0568018			0.0284009	+	0.999597i	$0.490958\pi$
$762$		0			0
$763$			196458.i			0.337459i
$764$			257075.i			0.440426i
$765$		0			0
$766$			463031.i			0.789137i
$767$	411949.	−	4062.96i	0.700250	−	0.00690639i
$768$		0			0
$769$		−	954320.i		−	1.61377i	−0.590710	−	0.806884i	$-0.701152\pi$
$769$		−	954320.i		−	1.61377i	0.590710	−	0.806884i	$-0.298848\pi$
$770$	−344268.			−0.580650
$771$		0			0
$772$	−445420.			−0.747370
$773$		−	430986.i		−	0.721280i	−0.932705	−	0.360640i	$-0.882558\pi$
$773$		−	430986.i		−	0.721280i	0.932705	−	0.360640i	$-0.117442\pi$
$774$		0			0
$775$			125169.i			0.208398i
$776$	145260.			0.241225
$777$		0			0
$778$		−	305780.i		−	0.505184i
$779$	−539004.			−0.888213
$780$		0			0
$781$			721862.i			1.18346i
$782$	310639.			0.507974
$783$		0			0
$784$	−15316.9			−0.0249195
$785$			1.21806e6i			1.97665i
$786$		0			0
$787$	449468.			0.725687			0.362844	−	0.931850i	$-0.381806\pi$
$787$	449468.			0.725687			0.362844	+	0.931850i	$0.381806\pi$
$788$	32436.6			0.0522375
$789$		0			0
$790$		−	421062.i		−	0.674671i
$791$		−	369712.i		−	0.590895i
$792$		0			0
$793$	−202723.			−0.322371
$794$	82184.9			0.130362
$795$		0			0
$796$	748217.			1.18087
$797$		−	300325.i		−	0.472797i	−0.971656	−	0.236398i	$-0.924033\pi$
$797$		−	300325.i		−	0.472797i	0.971656	−	0.236398i	$-0.0759671\pi$
$798$		0			0
$799$		−	255196.i		−	0.399742i
$800$		−	290768.i		−	0.454325i
$801$		0			0
$802$	−414424.			−0.644311
$803$	−489501.			−0.759141
$804$		0			0
$805$		−	630830.i		−	0.973465i
$806$		−	124833.i		−	0.192158i
$807$		0			0
$808$	1.11905e6			1.71406
$809$			676692.i			1.03394i	0.856005	+	0.516968i	$0.172940\pi$
$809$			676692.i			1.03394i	−0.856005	+	0.516968i	$0.827060\pi$
$810$		0			0
$811$		−	435345.i		−	0.661899i	−0.943648	−	0.330950i	$-0.892631\pi$
$811$		−	435345.i		−	0.661899i	0.943648	−	0.330950i	$-0.107369\pi$
$812$	−241502.			−0.366276
$813$		0			0
$814$		−	392924.i		−	0.593007i
$815$	181004.			0.272504
$816$		0			0
$817$			1.19722e6i			1.79362i
$818$	−456059.			−0.681576
$819$		0			0
$820$	455904.			0.678024
$821$			1.03403e6i			1.53408i	0.641599	+	0.767040i	$0.278271\pi$
$821$			1.03403e6i			1.53408i	−0.641599	+	0.767040i	$0.721729\pi$
$822$		0			0
$823$		−	1.06922e6i		−	1.57858i	−0.614020	−	0.789291i	$-0.710449\pi$
$823$		−	1.06922e6i		−	1.57858i	0.614020	−	0.789291i	$-0.289551\pi$
$824$	−761984.			−1.12226
$825$		0			0
$826$	−339833.	+	3351.69i	−0.498087	+	0.00491251i
$827$	−344792.			−0.504135			−0.252067	−	0.967710i	$-0.581110\pi$
$827$	−344792.			−0.504135			−0.252067	+	0.967710i	$0.581110\pi$
$828$		0			0
$829$	71136.1			0.103510			0.0517549	−	0.998660i	$-0.483519\pi$
$829$	71136.1			0.103510			0.0517549	+	0.998660i	$0.483519\pi$
$830$	206501.			0.299755
$831$		0			0
$832$			246024.i			0.355411i
$833$	174077.			0.250872
$834$		0			0
$835$	−1.15214e6			−1.65246
$836$		−	461924.i		−	0.660934i
$837$		0			0
$838$	−201773.			−0.287326
$839$		−	708706.i		−	1.00680i	−0.864054	−	0.503399i	$-0.832083\pi$
$839$		−	708706.i		−	1.00680i	0.864054	−	0.503399i	$-0.167917\pi$
$840$		0			0
$841$	−404984.			−0.572593
$842$	−121082.			−0.170787
$843$		0			0
$844$		−	330701.i		−	0.464248i
$845$	−437281.			−0.612418
$846$		0			0
$847$	−36106.0			−0.0503284
$848$	−38590.3			−0.0536644
$849$		0			0
$850$			171400.i			0.237231i
$851$	719987.			0.994181
$852$		0			0
$853$	1.25574e6			1.72584			0.862922	−	0.505337i	$-0.168632\pi$
$853$	1.25574e6			1.72584			0.862922	+	0.505337i	$0.168632\pi$
$854$	167234.			0.229302
$855$		0			0
$856$			485036.i			0.661952i
$857$			799600.i			1.08871i	0.838856	+	0.544353i	$0.183225\pi$
$857$			799600.i			1.08871i	−0.838856	+	0.544353i	$0.816775\pi$
$858$		0			0
$859$			853825.i			1.15713i	0.815636	+	0.578565i	$0.196387\pi$
$859$			853825.i			1.15713i	−0.815636	+	0.578565i	$0.803613\pi$
$860$		−	1.01264e6i		−	1.36917i
$861$		0			0
$862$	−432676.			−0.582302
$863$			365706.i			0.491033i	0.969392	+	0.245516i	$0.0789575\pi$
$863$			365706.i			0.491033i	−0.969392	+	0.245516i	$0.921043\pi$
$864$		0			0
$865$			199563.i			0.266716i
$866$			471809.i			0.629115i
$867$		0			0
$868$		−	198027.i		−	0.262837i
$869$			703073.i			0.931024i
$870$		0			0
$871$	599142.			0.789757
$872$	−292184.			−0.384259
$873$		0			0
$874$	−440161.			−0.576221
$875$	−435489.			−0.568802
$876$		0			0
$877$	538704.			0.700408			0.350204	−	0.936674i	$-0.386112\pi$
$877$	538704.			0.700408			0.350204	+	0.936674i	$0.386112\pi$
$878$		−	619894.i		−	0.804134i
$879$		0			0
$880$			81870.8i			0.105722i
$881$		−	131412.i		−	0.169311i	−0.996410	−	0.0846554i	$-0.973021\pi$
$881$		−	131412.i		−	0.169311i	0.996410	−	0.0846554i	$-0.0269789\pi$
$882$		0			0
$883$	152864.			0.196058			0.0980289	−	0.995184i	$-0.468746\pi$
$883$	152864.			0.196058			0.0980289	+	0.995184i	$0.468746\pi$
$884$			328713.i			0.420642i
$885$		0			0
$886$	−882108.			−1.12371
$887$			437411.i			0.555959i	0.960587	+	0.277980i	$0.0896648\pi$
$887$			437411.i			0.555959i	−0.960587	+	0.277980i	$0.910335\pi$
$888$		0			0
$889$	323807.			0.409716
$890$	272554.			0.344090
$891$		0			0
$892$	−579582.			−0.728426
$893$			361602.i			0.453448i
$894$		0			0
$895$		−	465830.i		−	0.581542i
$896$			496286.i			0.618182i
$897$		0			0
$898$			252889.i			0.313601i
$899$			247878.i			0.306703i
$900$		0			0
$901$	438581.			0.540257
$902$	395870.			0.486564
$903$		0			0
$904$	549857.			0.672841
$905$	216551.			0.264402
$906$		0			0
$907$	1.20250e6			1.46174			0.730870	−	0.682517i	$-0.239115\pi$
$907$	1.20250e6			1.46174			0.730870	+	0.682517i	$0.239115\pi$
$908$		−	244832.i		−	0.296959i
$909$		0			0
$910$	−347136.			−0.419196
$911$	1.26490e6			1.52412			0.762059	−	0.647508i	$-0.224189\pi$
$911$	1.26490e6			1.52412			0.762059	+	0.647508i	$0.224189\pi$
$912$		0			0
$913$	−344807.			−0.413652
$914$	621206.			0.743607
$915$		0			0
$916$		−	177466.i		−	0.211507i
$917$		−	1.10877e6i		−	1.31857i
$918$		0			0
$919$		−	831045.i		−	0.983996i	−0.870596	−	0.491998i	$-0.836267\pi$
$919$		−	831045.i		−	0.983996i	0.870596	−	0.491998i	$-0.163733\pi$
$920$	938205.			1.10847
$921$		0			0
$922$			507279.i			0.596740i
$923$			727876.i			0.854386i
$924$		0			0
$925$			397264.i			0.464297i
$926$	575354.			0.670985
$927$		0			0
$928$		−	575822.i		−	0.668640i
$929$			1.33453e6i			1.54631i	0.634217	+	0.773155i	$0.281323\pi$
$929$			1.33453e6i			1.54631i	−0.634217	+	0.773155i	$0.718677\pi$
$930$		0			0
$931$	−246660.			−0.284577
$932$		−	593677.i		−	0.683468i
$933$		0			0
$934$	463637.			0.531477
$935$		−	930468.i		−	1.06433i
$936$		0			0
$937$		−	1.31851e6i		−	1.50177i	−0.660431	−	0.750887i	$-0.729626\pi$
$937$		−	1.31851e6i		−	1.50177i	0.660431	−	0.750887i	$-0.270374\pi$
$938$	−494255.			−0.561753
$939$		0			0
$940$		−	305852.i		−	0.346143i
$941$		−	1.26151e6i		−	1.42466i	−0.701842	−	0.712332i	$-0.747639\pi$
$941$		−	1.26151e6i		−	1.42466i	0.701842	−	0.712332i	$-0.252361\pi$
$942$		0			0
$943$			725385.i			0.815728i
$944$	797.070	+	80816.2i	0.000894443	+	0.0906890i
$945$		0			0
$946$		−	879294.i		−	0.982543i
$947$	−1.34445e6			−1.49914			−0.749572	−	0.661923i	$-0.769741\pi$
$947$	−1.34445e6			−1.49914			−0.749572	+	0.661923i	$0.769741\pi$
$948$		0			0
$949$	−493579.			−0.548055
$950$		−	242866.i		−	0.269103i
$951$		0			0
$952$		−	683351.i		−	0.753997i
$953$	−654376.			−0.720513			−0.360256	−	0.932853i	$-0.617311\pi$
$953$	−654376.			−0.720513			−0.360256	+	0.932853i	$0.617311\pi$
$954$		0			0
$955$		−	733748.i		−	0.804526i
$956$	−184934.			−0.202349
$957$		0			0
$958$		−	982612.i		−	1.07066i
$959$	−265831.			−0.289047
$960$		0			0
$961$	720265.			0.779912
$962$		−	396198.i		−	0.428117i
$963$		0			0
$964$	1.02389e6			1.10179
$965$	1.27133e6			1.36522
$966$		0			0
$967$		−	1.67352e6i		−	1.78969i	−0.446373	−	0.894847i	$-0.647284\pi$
$967$		−	1.67352e6i		−	1.78969i	0.446373	−	0.894847i	$-0.352716\pi$
$968$		−	53698.9i		−	0.0573080i
$969$		0			0
$970$	−164524.			−0.174858
$971$	1.34581e6			1.42740			0.713700	−	0.700451i	$-0.247018\pi$
$971$	1.34581e6			1.42740			0.713700	+	0.700451i	$0.247018\pi$
$972$		0			0
$973$	−1.10529e6			−1.16749
$974$			787913.i			0.830539i
$975$		0			0
$976$		−	39770.1i		−	0.0417501i
$977$			108100.i			0.113250i	0.998396	+	0.0566248i	$0.0180339\pi$
$977$			108100.i			0.113250i	−0.998396	+	0.0566248i	$0.981966\pi$
$978$		0			0
$979$	−455099.			−0.474833
$980$	208632.			0.217234
$981$		0			0
$982$			653690.i			0.677874i
$983$		−	412747.i		−	0.427147i	−0.976927	−	0.213573i	$-0.931490\pi$
$983$		−	412747.i		−	0.427147i	0.976927	−	0.213573i	$-0.0685103\pi$
$984$		0			0
$985$	−92581.1			−0.0954223
$986$			339431.i			0.349138i
$987$		0			0
$988$		−	465772.i		−	0.477155i
$989$	1.61120e6			1.64724
$990$		0			0
$991$		−	1.11405e6i		−	1.13438i	−0.823588	−	0.567188i	$-0.808031\pi$
$991$		−	1.11405e6i		−	1.13438i	0.823588	−	0.567188i	$-0.191969\pi$
$992$	472164.			0.479810
$993$		0			0
$994$		−	600453.i		−	0.607724i
$995$	−2.13558e6			−2.15709
$996$		0			0
$997$	−289466.			−0.291210			−0.145605	−	0.989343i	$-0.546513\pi$
$997$	−289466.			−0.291210			−0.145605	+	0.989343i	$0.546513\pi$
$998$		−	55683.6i		−	0.0559070i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.15		40
3.2	odd	2		177.5.c.a.58.26	yes	40
59.58	odd	2	inner	531.5.c.d.235.26		40
177.176	even	2		177.5.c.a.58.15	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.15	✓	40	177.176	even	2
177.5.c.a.58.26	yes	40	3.2	odd	2
531.5.c.d.235.15		40	1.1	even	1	trivial
531.5.c.d.235.26		40	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.33963i q^{2} +10.5261 q^{4} -30.0439 q^{5} -41.7287 q^{7} -62.0613i q^{8} +O(q^{10})\)	\(q-2.33963i q^{2} +10.5261 q^{4} -30.0439 q^{5} -41.7287 q^{7} -62.0613i q^{8} +70.2916i q^{10} -117.370i q^{11} -118.348i q^{13} +97.6298i q^{14} +23.2175 q^{16} -263.868 q^{17} +373.890 q^{19} -316.246 q^{20} -274.603 q^{22} -503.177i q^{23} +277.635 q^{25} -276.891 q^{26} -439.242 q^{28} +549.815 q^{29} +450.839i q^{31} -1047.30i q^{32} +617.355i q^{34} +1253.69 q^{35} +1430.88i q^{37} -874.765i q^{38} +1864.56i q^{40} -1441.61 q^{41} +3202.06i q^{43} -1235.45i q^{44} -1177.25 q^{46} +967.133i q^{47} -659.713 q^{49} -649.564i q^{50} -1245.75i q^{52} -1662.12 q^{53} +3526.26i q^{55} +2589.74i q^{56} -1286.36i q^{58} +(34.3306 + 3480.83i) q^{59} -1712.94i q^{61} +1054.80 q^{62} -2078.82 q^{64} +3555.63i q^{65} +5062.54i q^{67} -2777.51 q^{68} -2933.18i q^{70} -6150.31 q^{71} -4170.58i q^{73} +3347.74 q^{74} +3935.62 q^{76} +4897.71i q^{77} -5990.22 q^{79} -697.544 q^{80} +3372.84i q^{82} -2937.78i q^{83} +7927.64 q^{85} +7491.63 q^{86} -7284.15 q^{88} -3877.47i q^{89} +4938.51i q^{91} -5296.50i q^{92} +2262.73 q^{94} -11233.1 q^{95} +2340.59i q^{97} +1543.48i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7	\( 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Introduction

L-functions

Modular forms

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Groups

Database

Properties

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