LMFDB - Embedded newform 531.5.c.d.235.14

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			235.14
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43119i q^{2} +10.0893 q^{4} -25.9319 q^{5} -17.4595 q^{7} -63.4281i q^{8} +O(q^{10})$	\(q-2.43119i q^{2} +10.0893 q^{4} -25.9319 q^{5} -17.4595 q^{7} -63.4281i q^{8} +63.0454i q^{10} -44.7152i q^{11} +157.364i q^{13} +42.4475i q^{14} +7.22282 q^{16} +328.532 q^{17} +678.379 q^{19} -261.635 q^{20} -108.711 q^{22} +424.331i q^{23} +47.4624 q^{25} +382.582 q^{26} -176.154 q^{28} -870.549 q^{29} -1380.41i q^{31} -1032.41i q^{32} -798.725i q^{34} +452.758 q^{35} +2568.09i q^{37} -1649.27i q^{38} +1644.81i q^{40} -503.324 q^{41} -2003.40i q^{43} -451.145i q^{44} +1031.63 q^{46} -2422.01i q^{47} -2096.17 q^{49} -115.390i q^{50} +1587.69i q^{52} +1526.26 q^{53} +1159.55i q^{55} +1107.42i q^{56} +2116.47i q^{58} +(1105.23 - 3300.88i) q^{59} -5708.35i q^{61} -3356.05 q^{62} -2394.42 q^{64} -4080.74i q^{65} -4773.92i q^{67} +3314.66 q^{68} -1100.74i q^{70} +2795.15 q^{71} -4719.76i q^{73} +6243.53 q^{74} +6844.37 q^{76} +780.705i q^{77} +8669.75 q^{79} -187.301 q^{80} +1223.68i q^{82} -7458.93i q^{83} -8519.45 q^{85} -4870.65 q^{86} -2836.20 q^{88} +4001.99i q^{89} -2747.50i q^{91} +4281.21i q^{92} -5888.37 q^{94} -17591.6 q^{95} -5912.30i q^{97} +5096.18i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.43119i		−	0.607798i	−0.952704	−	0.303899i	$-0.901711\pi$
$2$		−	2.43119i		−	0.607798i	0.952704	−	0.303899i	$-0.0982886\pi$
$3$		0			0
$3$		0			0
$4$	10.0893			0.630581
$5$	−25.9319			−1.03728			−0.518638	−	0.854994i	$-0.673561\pi$
$5$	−25.9319			−1.03728			−0.518638	+	0.854994i	$0.673561\pi$
$6$		0			0
$7$	−17.4595			−0.356317			−0.178158	−	0.984002i	$-0.557014\pi$
$7$	−17.4595			−0.356317			−0.178158	+	0.984002i	$0.557014\pi$
$8$		−	63.4281i		−	0.991064i
$9$		0			0
$10$			63.0454i			0.630454i
$11$		−	44.7152i		−	0.369547i	−0.982781	−	0.184773i	$-0.940845\pi$
$11$		−	44.7152i		−	0.369547i	0.982781	−	0.184773i	$-0.0591551\pi$
$12$		0			0
$13$			157.364i			0.931148i	0.885009	+	0.465574i	$0.154152\pi$
$13$			157.364i			0.931148i	−0.885009	+	0.465574i	$0.845848\pi$
$14$			42.4475i			0.216569i
$15$		0			0
$16$	7.22282			0.0282141
$17$	328.532			1.13679			0.568395	−	0.822756i	$-0.307565\pi$
$17$	328.532			1.13679			0.568395	+	0.822756i	$0.307565\pi$
$18$		0			0
$19$	678.379			1.87917			0.939583	−	0.342322i	$-0.111213\pi$
$19$	678.379			1.87917			0.939583	+	0.342322i	$0.111213\pi$
$20$	−261.635			−0.654086
$21$		0			0
$22$	−108.711			−0.224610
$23$			424.331i			0.802139i	0.916048	+	0.401069i	$0.131361\pi$
$23$			424.331i			0.802139i	−0.916048	+	0.401069i	$0.868639\pi$
$24$		0			0
$25$	47.4624			0.0759398
$26$	382.582			0.565950
$27$		0			0
$28$	−176.154			−0.224687
$29$	−870.549			−1.03514			−0.517568	−	0.855642i	$-0.673163\pi$
$29$	−870.549			−1.03514			−0.517568	+	0.855642i	$0.673163\pi$
$30$		0			0
$31$		−	1380.41i		−	1.43643i	−0.695820	−	0.718216i	$-0.744959\pi$
$31$		−	1380.41i		−	1.43643i	0.695820	−	0.718216i	$-0.255041\pi$
$32$		−	1032.41i		−	1.00821i
$33$		0			0
$34$		−	798.725i		−	0.690938i
$35$	452.758			0.369599
$36$		0			0
$37$			2568.09i			1.87589i	0.346784	+	0.937945i	$0.387274\pi$
$37$			2568.09i			1.87589i	−0.346784	+	0.937945i	$0.612726\pi$
$38$		−	1649.27i		−	1.14215i
$39$		0			0
$40$			1644.81i			1.02801i
$41$	−503.324			−0.299419			−0.149710	−	0.988730i	$-0.547834\pi$
$41$	−503.324			−0.299419			−0.149710	+	0.988730i	$0.547834\pi$
$42$		0			0
$43$		−	2003.40i		−	1.08350i	−0.840538	−	0.541752i	$-0.817761\pi$
$43$		−	2003.40i		−	1.08350i	0.840538	−	0.541752i	$-0.182239\pi$
$44$		−	451.145i		−	0.233029i
$45$		0			0
$46$	1031.63			0.487538
$47$		−	2422.01i		−	1.09643i	−0.836338	−	0.548214i	$-0.815308\pi$
$47$		−	2422.01i		−	1.09643i	0.836338	−	0.548214i	$-0.184692\pi$
$48$		0			0
$49$	−2096.17			−0.873038
$50$		−	115.390i		−	0.0461561i
$51$		0			0
$52$			1587.69i			0.587164i
$53$	1526.26			0.543347			0.271674	−	0.962389i	$-0.412423\pi$
$53$	1526.26			0.543347			0.271674	+	0.962389i	$0.412423\pi$
$54$		0			0
$55$			1159.55i			0.383322i
$56$			1107.42i			0.353133i
$57$		0			0
$58$			2116.47i			0.629154i
$59$	1105.23	−	3300.88i	0.317504	−	0.948257i
$59$	1105.23	−	3300.88i	0.317504	−	0.948257i
$60$		0			0
$61$		−	5708.35i		−	1.53409i	−0.641594	−	0.767045i	$-0.721726\pi$
$61$		−	5708.35i		−	1.53409i	0.641594	−	0.767045i	$-0.278274\pi$
$62$	−3356.05			−0.873061
$63$		0			0
$64$	−2394.42			−0.584576
$65$		−	4080.74i		−	0.965857i
$66$		0			0
$67$		−	4773.92i		−	1.06347i	−0.846911	−	0.531735i	$-0.821540\pi$
$67$		−	4773.92i		−	1.06347i	0.846911	−	0.531735i	$-0.178460\pi$
$68$	3314.66			0.716838
$69$		0			0
$70$		−	1100.74i		−	0.224641i
$71$	2795.15			0.554484			0.277242	−	0.960800i	$-0.410580\pi$
$71$	2795.15			0.554484			0.277242	+	0.960800i	$0.410580\pi$
$72$		0			0
$73$		−	4719.76i		−	0.885674i	−0.896602	−	0.442837i	$-0.853972\pi$
$73$		−	4719.76i		−	0.885674i	0.896602	−	0.442837i	$-0.146028\pi$
$74$	6243.53			1.14016
$75$		0			0
$76$	6844.37			1.18497
$77$			780.705i			0.131676i
$78$		0			0
$79$	8669.75			1.38916			0.694580	−	0.719415i	$-0.255590\pi$
$79$	8669.75			1.38916			0.694580	+	0.719415i	$0.255590\pi$
$80$	−187.301			−0.0292658
$81$		0			0
$82$			1223.68i			0.181987i
$83$		−	7458.93i		−	1.08273i	−0.840787	−	0.541365i	$-0.817908\pi$
$83$		−	7458.93i		−	1.08273i	0.840787	−	0.541365i	$-0.182092\pi$
$84$		0			0
$85$	−8519.45			−1.17916
$86$	−4870.65			−0.658552
$87$		0			0
$88$	−2836.20			−0.366245
$89$			4001.99i			0.505238i	0.967566	+	0.252619i	$0.0812920\pi$
$89$			4001.99i			0.505238i	−0.967566	+	0.252619i	$0.918708\pi$
$90$		0			0
$91$		−	2747.50i		−	0.331784i
$92$			4281.21i			0.505814i
$93$		0			0
$94$	−5888.37			−0.666407
$95$	−17591.6			−1.94921
$96$		0			0
$97$		−	5912.30i		−	0.628367i	−0.949362	−	0.314183i	$-0.898269\pi$
$97$		−	5912.30i		−	0.628367i	0.949362	−	0.314183i	$-0.101731\pi$
$98$			5096.18i			0.530631i
$99$		0			0
$100$	478.862			0.0478862
$101$			14483.7i			1.41983i	0.704286	+	0.709916i	$0.251267\pi$
$101$			14483.7i			1.41983i	−0.704286	+	0.709916i	$0.748733\pi$
$102$		0			0
$103$			211.537i			0.0199394i	0.999950	+	0.00996968i	$0.00317350\pi$
$103$			211.537i			0.0199394i	−0.999950	+	0.00996968i	$0.996826\pi$
$104$	9981.30			0.922828
$105$		0			0
$106$		−	3710.64i		−	0.330246i
$107$	12751.0			1.11373			0.556863	−	0.830605i	$-0.312005\pi$
$107$	12751.0			1.11373			0.556863	+	0.830605i	$0.312005\pi$
$108$		0			0
$109$		−	18050.8i		−	1.51930i	−0.650334	−	0.759648i	$-0.725371\pi$
$109$		−	18050.8i		−	1.51930i	0.650334	−	0.759648i	$-0.274629\pi$
$110$	2819.09			0.232982
$111$		0			0
$112$	−126.107			−0.0100532
$113$			9631.35i			0.754276i	0.926157	+	0.377138i	$0.123092\pi$
$113$			9631.35i			0.754276i	−0.926157	+	0.377138i	$0.876908\pi$
$114$		0			0
$115$		−	11003.7i		−	0.832039i
$116$	−8783.24			−0.652737
$117$		0			0
$118$	−8025.08	−	2687.03i	−0.576349	−	0.192979i
$119$	−5736.01			−0.405057
$120$		0			0
$121$	12641.6			0.863435
$122$	−13878.1			−0.932417
$123$		0			0
$124$		−	13927.4i		−	0.905788i
$125$	14976.6			0.958505
$126$		0			0
$127$	−25023.8			−1.55148			−0.775741	−	0.631052i	$-0.782624\pi$
$127$	−25023.8			−1.55148			−0.775741	+	0.631052i	$0.782624\pi$
$128$		−	10697.3i		−	0.652909i
$129$		0			0
$130$	−9921.08			−0.587046
$131$		−	2066.15i		−	0.120398i	−0.998186	−	0.0601990i	$-0.980826\pi$
$131$		−	2066.15i		−	0.120398i	0.998186	−	0.0601990i	$-0.0191735\pi$
$132$		0			0
$133$	−11844.2			−0.669578
$134$	−11606.3			−0.646375
$135$		0			0
$136$		−	20838.2i		−	1.12663i
$137$	−8060.06			−0.429434			−0.214717	−	0.976676i	$-0.568883\pi$
$137$	−8060.06			−0.429434			−0.214717	+	0.976676i	$0.568883\pi$
$138$		0			0
$139$	−8208.19			−0.424833			−0.212416	−	0.977179i	$-0.568133\pi$
$139$	−8208.19			−0.424833			−0.212416	+	0.977179i	$0.568133\pi$
$140$	4568.01			0.233062
$141$		0			0
$142$		−	6795.56i		−	0.337014i
$143$	7036.56			0.344103
$144$		0			0
$145$	22575.0			1.07372
$146$	−11474.6			−0.538311
$147$		0			0
$148$			25910.3i			1.18290i
$149$		−	37339.6i		−	1.68189i	−0.541122	−	0.840944i	$-0.682000\pi$
$149$		−	37339.6i		−	1.68189i	0.541122	−	0.840944i	$-0.318000\pi$
$150$		0			0
$151$		−	10313.8i		−	0.452340i	−0.974088	−	0.226170i	$-0.927379\pi$
$151$		−	10313.8i		−	0.452340i	0.974088	−	0.226170i	$-0.0726205\pi$
$152$		−	43028.3i		−	1.86237i
$153$		0			0
$154$	1898.04			0.0800323
$155$			35796.7i			1.48998i
$156$		0			0
$157$		−	10989.2i		−	0.445826i	−0.974838	−	0.222913i	$-0.928443\pi$
$157$		−	10989.2i		−	0.445826i	0.974838	−	0.222913i	$-0.0715566\pi$
$158$		−	21077.8i		−	0.844329i
$159$		0			0
$160$			26772.3i			1.04579i
$161$		−	7408.62i		−	0.285815i
$162$		0			0
$163$	1659.70			0.0624676			0.0312338	−	0.999512i	$-0.490056\pi$
$163$	1659.70			0.0624676			0.0312338	+	0.999512i	$0.490056\pi$
$164$	−5078.19			−0.188808
$165$		0			0
$166$	−18134.1			−0.658082
$167$	5744.24			0.205968			0.102984	−	0.994683i	$-0.467161\pi$
$167$	5744.24			0.205968			0.102984	+	0.994683i	$0.467161\pi$
$168$		0			0
$169$	3797.58			0.132964
$170$			20712.4i			0.716693i
$171$		0			0
$172$		−	20212.9i		−	0.683238i
$173$			48337.4i			1.61507i	0.589821	+	0.807534i	$0.299198\pi$
$173$			48337.4i			1.61507i	−0.589821	+	0.807534i	$0.700802\pi$
$174$		0			0
$175$	−828.671			−0.0270586
$176$		−	322.969i		−	0.0104264i
$177$		0			0
$178$	9729.62			0.307083
$179$		−	8489.09i		−	0.264945i	−0.991187	−	0.132472i	$-0.957708\pi$
$179$		−	8489.09i		−	0.264945i	0.991187	−	0.132472i	$-0.0422916\pi$
$180$		0			0
$181$	−63579.4			−1.94070			−0.970352	−	0.241696i	$-0.922296\pi$
$181$	−63579.4			−1.94070			−0.970352	+	0.241696i	$0.922296\pi$
$182$	−6679.70			−0.201657
$183$		0			0
$184$	26914.5			0.794971
$185$		−	66595.5i		−	1.94581i
$186$		0			0
$187$		−	14690.4i		−	0.420097i
$188$		−	24436.4i		−	0.691387i
$189$		0			0
$190$			42768.7i			1.18473i
$191$		−	36006.4i		−	0.986991i	−0.869748	−	0.493495i	$-0.835719\pi$
$191$		−	36006.4i		−	0.986991i	0.869748	−	0.493495i	$-0.164281\pi$
$192$		0			0
$193$	7371.06			0.197886			0.0989431	−	0.995093i	$-0.468454\pi$
$193$	7371.06			0.197886			0.0989431	+	0.995093i	$0.468454\pi$
$194$	−14373.9			−0.381920
$195$		0			0
$196$	−21148.8			−0.550522
$197$	−62886.8			−1.62042			−0.810209	−	0.586141i	$-0.800647\pi$
$197$	−62886.8			−1.62042			−0.810209	+	0.586141i	$0.800647\pi$
$198$		0			0
$199$	24801.5			0.626285			0.313143	−	0.949706i	$-0.398618\pi$
$199$	24801.5			0.626285			0.313143	+	0.949706i	$0.398618\pi$
$200$		−	3010.45i		−	0.0752613i
$201$		0			0
$202$	35212.7			0.862972
$203$	15199.4			0.368836
$204$		0			0
$205$	13052.1			0.310580
$206$	514.286			0.0121191
$207$		0			0
$208$			1136.61i			0.0262715i
$209$		−	30333.8i		−	0.694440i
$210$		0			0
$211$		−	17502.7i		−	0.393134i	−0.980490	−	0.196567i	$-0.937021\pi$
$211$		−	17502.7i		−	0.393134i	0.980490	−	0.196567i	$-0.0629794\pi$
$212$	15398.9			0.342625
$213$		0			0
$214$		−	31000.2i		−	0.676920i
$215$			51951.9i			1.12389i
$216$		0			0
$217$			24101.3i			0.511825i
$218$	−43884.9			−0.923426
$219$		0			0
$220$			11699.0i			0.241716i
$221$			51699.1i			1.05852i
$222$		0			0
$223$	54042.6			1.08674			0.543371	−	0.839493i	$-0.317148\pi$
$223$	54042.6			1.08674			0.543371	+	0.839493i	$0.317148\pi$
$224$			18025.4i			0.359243i
$225$		0			0
$226$	23415.7			0.458448
$227$		−	24892.4i		−	0.483076i	−0.970391	−	0.241538i	$-0.922348\pi$
$227$		−	24892.4i		−	0.483076i	0.970391	−	0.241538i	$-0.0776518\pi$
$228$		0			0
$229$			81045.6i			1.54546i	0.634733	+	0.772731i	$0.281110\pi$
$229$			81045.6i			1.54546i	−0.634733	+	0.772731i	$0.718890\pi$
$230$	−26752.1			−0.505712
$231$		0			0
$232$			55217.3i			1.02589i
$233$			92463.9i			1.70318i	0.524209	+	0.851589i	$0.324361\pi$
$233$			92463.9i			1.70318i	−0.524209	+	0.851589i	$0.675639\pi$
$234$		0			0
$235$			62807.2i			1.13730i
$236$	11151.0	−	33303.6i	0.200212	−	0.597953i
$237$		0			0
$238$			13945.4i			0.246193i
$239$	66752.2			1.16861			0.584305	−	0.811534i	$-0.301367\pi$
$239$	66752.2			1.16861			0.584305	+	0.811534i	$0.301367\pi$
$240$		0			0
$241$	−57886.0			−0.996643			−0.498322	−	0.866992i	$-0.666050\pi$
$241$	−57886.0			−0.996643			−0.498322	+	0.866992i	$0.666050\pi$
$242$		−	30734.1i		−	0.524794i
$243$		0			0
$244$		−	57593.2i		−	0.967368i
$245$	54357.5			0.905581
$246$		0			0
$247$			106752.i			1.74978i
$248$	−87556.9			−1.42360
$249$		0			0
$250$		−	36411.1i		−	0.582577i
$251$	1482.64			0.0235336			0.0117668	−	0.999931i	$-0.496254\pi$
$251$	1482.64			0.0235336			0.0117668	+	0.999931i	$0.496254\pi$
$252$		0			0
$253$	18974.0			0.296428
$254$			60837.8i			0.942988i
$255$		0			0
$256$	−64317.9			−0.981413
$257$	63595.3			0.962850			0.481425	−	0.876487i	$-0.340119\pi$
$257$	63595.3			0.962850			0.481425	+	0.876487i	$0.340119\pi$
$258$		0			0
$259$		−	44837.7i		−	0.668411i
$260$		−	41171.9i		−	0.609051i
$261$		0			0
$262$	−5023.21			−0.0731777
$263$	79407.3			1.14802			0.574010	−	0.818849i	$-0.305387\pi$
$263$	79407.3			1.14802			0.574010	+	0.818849i	$0.305387\pi$
$264$		0			0
$265$	−39578.9			−0.563601
$266$			28795.5i			0.406968i
$267$		0			0
$268$		−	48165.5i		−	0.670605i
$269$			103489.i			1.43018i	0.699031	+	0.715091i	$0.253615\pi$
$269$			103489.i			1.43018i	−0.699031	+	0.715091i	$0.746385\pi$
$270$		0			0
$271$	144360.			1.96566			0.982830	−	0.184513i	$-0.0590709\pi$
$271$	144360.			1.96566			0.982830	+	0.184513i	$0.0590709\pi$
$272$	2372.93			0.0320735
$273$		0			0
$274$			19595.5i			0.261010i
$275$		−	2122.29i		−	0.0280633i
$276$		0			0
$277$	47254.9			0.615867			0.307934	−	0.951408i	$-0.400363\pi$
$277$	47254.9			0.615867			0.307934	+	0.951408i	$0.400363\pi$
$278$			19955.7i			0.258213i
$279$		0			0
$280$		−	28717.6i		−	0.366296i
$281$	31731.5			0.401863			0.200931	−	0.979605i	$-0.435603\pi$
$281$	31731.5			0.401863			0.200931	+	0.979605i	$0.435603\pi$
$282$		0			0
$283$		−	12999.5i		−	0.162313i	−0.996701	−	0.0811563i	$-0.974139\pi$
$283$		−	12999.5i		−	0.162313i	0.996701	−	0.0811563i	$-0.0258613\pi$
$284$	28201.1			0.349647
$285$		0			0
$286$		−	17107.2i		−	0.209145i
$287$	8787.80			0.106688
$288$		0			0
$289$	24412.3			0.292289
$290$		−	54884.1i		−	0.652606i
$291$		0			0
$292$		−	47619.0i		−	0.558489i
$293$	27627.7			0.321817			0.160909	−	0.986969i	$-0.448558\pi$
$293$	27627.7			0.321817			0.160909	+	0.986969i	$0.448558\pi$
$294$		0			0
$295$	−28660.8	+	85598.1i	−0.329339	+	0.983603i
$296$	162889.			1.85913
$297$		0			0
$298$	−90779.8			−1.02225
$299$	−66774.5			−0.746910
$300$		0			0
$301$			34978.4i			0.386071i
$302$	−25074.9			−0.274932
$303$		0			0
$304$	4899.81			0.0530190
$305$			148028.i			1.59127i
$306$		0			0
$307$	−25754.0			−0.273255			−0.136627	−	0.990623i	$-0.543626\pi$
$307$	−25754.0			−0.273255			−0.136627	+	0.990623i	$0.543626\pi$
$308$			7876.77i			0.0830322i
$309$		0			0
$310$	87028.6			0.905605
$311$	−123728.			−1.27923			−0.639613	−	0.768697i	$-0.720905\pi$
$311$	−123728.			−1.27923			−0.639613	+	0.768697i	$0.720905\pi$
$312$		0			0
$313$		−	12824.2i		−	0.130900i	−0.997856	−	0.0654502i	$-0.979152\pi$
$313$		−	12824.2i		−	0.130900i	0.997856	−	0.0654502i	$-0.0208484\pi$
$314$	−26716.8			−0.270972
$315$		0			0
$316$	87471.7			0.875978
$317$	136329.			1.35666			0.678329	−	0.734759i	$-0.262705\pi$
$317$	136329.			1.35666			0.678329	+	0.734759i	$0.262705\pi$
$318$		0			0
$319$			38926.8i			0.382531i
$320$	62091.9			0.606366
$321$		0			0
$322$	−18011.8			−0.173718
$323$	222869.			2.13622
$324$		0			0
$325$			7468.87i			0.0707112i
$326$		−	4035.05i		−	0.0379677i
$327$		0			0
$328$			31924.9i			0.296744i
$329$			42287.1i			0.390676i
$330$		0			0
$331$	−153145.			−1.39781			−0.698904	−	0.715216i	$-0.746328\pi$
$331$	−153145.			−1.39781			−0.698904	+	0.715216i	$0.746328\pi$
$332$		−	75255.4i		−	0.682750i
$333$		0			0
$334$		−	13965.4i		−	0.125187i
$335$			123797.i			1.10311i
$336$		0			0
$337$		−	60949.0i		−	0.536669i	−0.963326	−	0.268335i	$-0.913527\pi$
$337$		−	60949.0i		−	0.536669i	0.963326	−	0.268335i	$-0.0864733\pi$
$338$		−	9232.64i		−	0.0808151i
$339$		0			0
$340$	−85955.3			−0.743558
$341$	−61725.3			−0.530829
$342$		0			0
$343$	78518.3			0.667395
$344$	−127072.			−1.07382
$345$		0			0
$346$	117517.			0.981635
$347$		−	150274.i		−	1.24803i	−0.781412	−	0.624015i	$-0.785500\pi$
$347$		−	150274.i		−	1.24803i	0.781412	−	0.624015i	$-0.214500\pi$
$348$		0			0
$349$			4128.34i			0.0338942i	0.999856	+	0.0169471i	$0.00539468\pi$
$349$			4128.34i			0.0338942i	−0.999856	+	0.0169471i	$0.994605\pi$
$350$			2014.66i			0.0164462i
$351$		0			0
$352$	−46164.4			−0.372582
$353$		−	126346.i		−	1.01394i	−0.861964	−	0.506969i	$-0.830766\pi$
$353$		−	126346.i		−	1.01394i	0.861964	−	0.506969i	$-0.169234\pi$
$354$		0			0
$355$	−72483.6			−0.575152
$356$			40377.3i			0.318594i
$357$		0			0
$358$	−20638.6			−0.161033
$359$	−182628.			−1.41703			−0.708514	−	0.705697i	$-0.750634\pi$
$359$	−182628.			−1.41703			−0.708514	+	0.705697i	$0.750634\pi$
$360$		0			0
$361$	329877.			2.53126
$362$			154574.i			1.17956i
$363$		0			0
$364$		−	27720.4i		−	0.209217i
$365$			122392.i			0.918688i
$366$		0			0
$367$			8500.43i			0.0631116i	0.999502	+	0.0315558i	$0.0100462\pi$
$367$			8500.43i			0.0631116i	−0.999502	+	0.0315558i	$0.989954\pi$
$368$			3064.87i			0.0226316i
$369$		0			0
$370$	−161907.			−1.18266
$371$	−26647.8			−0.193604
$372$		0			0
$373$	82283.7			0.591420			0.295710	−	0.955278i	$-0.404444\pi$
$373$	82283.7			0.591420			0.295710	+	0.955278i	$0.404444\pi$
$374$	−35715.1			−0.255334
$375$		0			0
$376$	−153623.			−1.08663
$377$		−	136993.i		−	0.963865i
$378$		0			0
$379$	2637.31			0.0183604			0.00918022	−	0.999958i	$-0.497078\pi$
$379$	2637.31			0.0183604			0.00918022	+	0.999958i	$0.497078\pi$
$380$	−177487.			−1.22914
$381$		0			0
$382$	−87538.5			−0.599891
$383$	−45438.5			−0.309761			−0.154880	−	0.987933i	$-0.549499\pi$
$383$	−45438.5			−0.309761			−0.154880	+	0.987933i	$0.549499\pi$
$384$		0			0
$385$		−	20245.2i		−	0.136584i
$386$		−	17920.5i		−	0.120275i
$387$		0			0
$388$		−	59651.0i		−	0.396236i
$389$	36809.8			0.243257			0.121628	−	0.992576i	$-0.461188\pi$
$389$	36809.8			0.243257			0.121628	+	0.992576i	$0.461188\pi$
$390$		0			0
$391$			139406.i			0.911862i
$392$			132956.i			0.865237i
$393$		0			0
$394$			152890.i			0.984888i
$395$	−224823.			−1.44094
$396$		0			0
$397$			184237.i			1.16895i	0.811412	+	0.584475i	$0.198700\pi$
$397$			184237.i			1.16895i	−0.811412	+	0.584475i	$0.801300\pi$
$398$		−	60297.3i		−	0.380655i
$399$		0			0
$400$	342.812			0.00214258
$401$		−	243995.i		−	1.51738i	−0.651455	−	0.758688i	$-0.725841\pi$
$401$		−	243995.i		−	1.51738i	0.651455	−	0.758688i	$-0.274159\pi$
$402$		0			0
$403$	217227.			1.33753
$404$			146131.i			0.895320i
$405$		0			0
$406$		−	36952.6i		−	0.224178i
$407$	114833.			0.693229
$408$		0			0
$409$			39142.7i			0.233994i	0.993132	+	0.116997i	$0.0373268\pi$
$409$			39142.7i			0.233994i	−0.993132	+	0.116997i	$0.962673\pi$
$410$		−	31732.3i		−	0.188770i
$411$		0			0
$412$			2134.26i			0.0125734i
$413$	−19296.8	+	57631.8i	−0.113132	+	0.337880i
$414$		0			0
$415$			193424.i			1.12309i
$416$	162464.			0.938795
$417$		0			0
$418$	−73747.4			−0.422079
$419$		−	148939.i		−	0.848362i	−0.905577	−	0.424181i	$-0.860562\pi$
$419$		−	148939.i		−	0.848362i	0.905577	−	0.424181i	$-0.139438\pi$
$420$		0			0
$421$			258237.i			1.45698i	0.685055	+	0.728492i	$0.259778\pi$
$421$			258237.i			1.45698i	−0.685055	+	0.728492i	$0.740222\pi$
$422$	−42552.5			−0.238946
$423$		0			0
$424$		−	96808.0i		−	0.538492i
$425$	15592.9			0.0863276
$426$		0			0
$427$			99665.0i			0.546622i
$428$	128649.			0.702295
$429$		0			0
$430$	126305.			0.683100
$431$		−	260135.i		−	1.40037i	−0.713960	−	0.700187i	$-0.753100\pi$
$431$		−	260135.i		−	1.40037i	0.713960	−	0.700187i	$-0.246900\pi$
$432$		0			0
$433$	281301.			1.50036			0.750181	−	0.661232i	$-0.229966\pi$
$433$	281301.			1.50036			0.750181	+	0.661232i	$0.229966\pi$
$434$	58595.0			0.311086
$435$		0			0
$436$		−	182120.i		−	0.958040i
$437$			287857.i			1.50735i
$438$		0			0
$439$	−226376.			−1.17463			−0.587316	−	0.809357i	$-0.699816\pi$
$439$	−226376.			−1.17463			−0.587316	+	0.809357i	$0.699816\pi$
$440$	73548.0			0.379897
$441$		0			0
$442$	125691.			0.643366
$443$		−	191214.i		−	0.974346i	−0.873305	−	0.487173i	$-0.838028\pi$
$443$		−	191214.i		−	0.974346i	0.873305	−	0.487173i	$-0.161972\pi$
$444$		0			0
$445$		−	103779.i		−	0.524071i
$446$		−	131388.i		−	0.660519i
$447$		0			0
$448$	41805.5			0.208294
$449$	−205380.			−1.01874			−0.509372	−	0.860546i	$-0.670122\pi$
$449$	−205380.			−1.01874			−0.509372	+	0.860546i	$0.670122\pi$
$450$		0			0
$451$			22506.2i			0.110649i
$452$			97173.6i			0.475633i
$453$		0			0
$454$	−60518.3			−0.293613
$455$			71247.8i			0.344151i
$456$		0			0
$457$			145715.i			0.697703i	0.937178	+	0.348852i	$0.113428\pi$
$457$			145715.i			0.697703i	−0.937178	+	0.348852i	$0.886572\pi$
$458$	197038.			0.939329
$459$		0			0
$460$		−	111020.i		−	0.524668i
$461$	91731.3			0.431634			0.215817	−	0.976434i	$-0.430759\pi$
$461$	91731.3			0.431634			0.215817	+	0.976434i	$0.430759\pi$
$462$		0			0
$463$			363576.i			1.69603i	0.529974	+	0.848014i	$0.322202\pi$
$463$			363576.i			1.69603i	−0.529974	+	0.848014i	$0.677798\pi$
$464$	−6287.82			−0.0292055
$465$		0			0
$466$	224797.			1.03519
$467$			107740.i			0.494019i	0.969013	+	0.247009i	$0.0794478\pi$
$467$			107740.i			0.494019i	−0.969013	+	0.247009i	$0.920552\pi$
$468$		0			0
$469$			83350.3i			0.378932i
$470$	152697.			0.691247
$471$		0			0
$472$	−209369.	−	70102.9i	−0.939784	−	0.314667i
$473$	−89582.3			−0.400406
$474$		0			0
$475$	32197.5			0.142704
$476$	−57872.4			−0.255421
$477$		0			0
$478$		−	162288.i		−	0.710280i
$479$	−148486.			−0.647165			−0.323582	−	0.946200i	$-0.604887\pi$
$479$	−148486.			−0.647165			−0.323582	+	0.946200i	$0.604887\pi$
$480$		0			0
$481$	−404125.			−1.74673
$482$			140732.i			0.605758i
$483$		0			0
$484$	127544.			0.544466
$485$			153317.i			0.651789i
$486$		0			0
$487$	−3404.16			−0.0143533			−0.00717666	−	0.999974i	$-0.502284\pi$
$487$	−3404.16			−0.0143533			−0.00717666	+	0.999974i	$0.502284\pi$
$488$	−362070.			−1.52038
$489$		0			0
$490$		−	132154.i		−	0.550411i
$491$	−326718.			−1.35522			−0.677611	−	0.735420i	$-0.736985\pi$
$491$	−326718.			−1.35522			−0.677611	+	0.735420i	$0.736985\pi$
$492$		0			0
$493$	−286003.			−1.17673
$494$	259536.			1.06351
$495$		0			0
$496$		−	9970.46i		−	0.0405277i
$497$	−48802.0			−0.197572
$498$		0			0
$499$	170255.			0.683754			0.341877	−	0.939745i	$-0.388937\pi$
$499$	170255.			0.683754			0.341877	+	0.939745i	$0.388937\pi$
$500$	151104.			0.604415
$501$		0			0
$502$		−	3604.58i		−	0.0143037i
$503$		−	198224.i		−	0.783466i	−0.920079	−	0.391733i	$-0.871876\pi$
$503$		−	198224.i		−	0.783466i	0.920079	−	0.391733i	$-0.128124\pi$
$504$		0			0
$505$		−	375590.i		−	1.47276i
$506$		−	46129.6i		−	0.180168i
$507$		0			0
$508$	−252473.			−0.978335
$509$			252370.i			0.974099i	0.873374	+	0.487049i	$0.161927\pi$
$509$			252370.i			0.974099i	−0.873374	+	0.487049i	$0.838073\pi$
$510$		0			0
$511$			82404.7i			0.315580i
$512$		−	14787.0i		−	0.0564079i
$513$		0			0
$514$		−	154612.i		−	0.585219i
$515$		−	5485.54i		−	0.0206826i
$516$		0			0
$517$	−108301.			−0.405181
$518$	−109009.			−0.406259
$519$		0			0
$520$	−258834.			−0.957226
$521$	372104.			1.37085			0.685424	−	0.728144i	$-0.259617\pi$
$521$	372104.			1.37085			0.685424	+	0.728144i	$0.259617\pi$
$522$		0			0
$523$	−68686.7			−0.251113			−0.125557	−	0.992086i	$-0.540072\pi$
$523$	−68686.7			−0.251113			−0.125557	+	0.992086i	$0.540072\pi$
$524$		−	20846.0i		−	0.0759207i
$525$		0			0
$526$		−	193055.i		−	0.697764i
$527$		−	453510.i		−	1.63292i
$528$		0			0
$529$	99783.9			0.356574
$530$			96223.8i			0.342556i
$531$		0			0
$532$	−119499.			−0.422223
$533$		−	79205.1i		−	0.278804i
$534$		0			0
$535$	−330659.			−1.15524
$536$	−302801.			−1.05397
$537$		0			0
$538$	251603.			0.869262
$539$			93730.4i			0.322629i
$540$		0			0
$541$		−	51022.3i		−	0.174327i	−0.996194	−	0.0871637i	$-0.972220\pi$
$541$		−	51022.3i		−	0.174327i	0.996194	−	0.0871637i	$-0.0277803\pi$
$542$		−	350967.i		−	1.19472i
$543$		0			0
$544$		−	339180.i		−	1.14613i
$545$			468090.i			1.57593i
$546$		0			0
$547$	116201.			0.388362			0.194181	−	0.980966i	$-0.437795\pi$
$547$	116201.			0.388362			0.194181	+	0.980966i	$0.437795\pi$
$548$	−81320.3			−0.270793
$549$		0			0
$550$	−5159.69			−0.0170568
$551$	−590562.			−1.94519
$552$		0			0
$553$	−151370.			−0.494981
$554$		−	114886.i		−	0.374323i
$555$		0			0
$556$	−82814.9			−0.267892
$557$	428570.			1.38137			0.690687	−	0.723154i	$-0.257308\pi$
$557$	428570.			1.38137			0.690687	+	0.723154i	$0.257308\pi$
$558$		0			0
$559$	315263.			1.00890
$560$	3270.19			0.0104279
$561$		0			0
$562$		−	77145.4i		−	0.244251i
$563$			131577.i			0.415110i	0.978223	+	0.207555i	$0.0665506\pi$
$563$			131577.i			0.415110i	−0.978223	+	0.207555i	$0.933449\pi$
$564$		0			0
$565$		−	249759.i		−	0.782392i
$566$	−31604.2			−0.0986533
$567$		0			0
$568$		−	177291.i		−	0.549529i
$569$		−	399439.i		−	1.23375i	−0.787063	−	0.616873i	$-0.788399\pi$
$569$		−	399439.i		−	1.23375i	0.787063	−	0.616873i	$-0.211601\pi$
$570$		0			0
$571$		−	131744.i		−	0.404072i	−0.979378	−	0.202036i	$-0.935244\pi$
$571$		−	131744.i		−	0.404072i	0.979378	−	0.202036i	$-0.0647558\pi$
$572$	70993.9			0.216985
$573$		0			0
$574$		−	21364.8i		−	0.0648449i
$575$			20139.8i			0.0609143i
$576$		0			0
$577$	−160918.			−0.483339			−0.241670	−	0.970359i	$-0.577695\pi$
$577$	−160918.			−0.483339			−0.241670	+	0.970359i	$0.577695\pi$
$578$		−	59351.0i		−	0.177653i
$579$		0			0
$580$	227766.			0.677068
$581$			130229.i			0.385795i
$582$		0			0
$583$		−	68247.1i		−	0.200792i
$584$	−299365.			−0.877760
$585$		0			0
$586$		−	67168.2i		−	0.195600i
$587$		−	171759.i		−	0.498474i	−0.968443	−	0.249237i	$-0.919820\pi$
$587$		−	171759.i		−	0.498474i	0.968443	−	0.249237i	$-0.0801798\pi$
$588$		0			0
$589$		−	936442.i		−	2.69929i
$590$	208105.	+	69679.9i	0.597832	+	0.200172i
$591$		0			0
$592$			18548.9i			0.0529266i
$593$	−457675.			−1.30151			−0.650755	−	0.759288i	$-0.725547\pi$
$593$	−457675.			−1.30151			−0.650755	+	0.759288i	$0.725547\pi$
$594$		0			0
$595$	148746.			0.420156
$596$		−	376731.i		−	1.06057i
$597$		0			0
$598$			162342.i			0.453970i
$599$	450231.			1.25482			0.627410	−	0.778689i	$-0.284115\pi$
$599$	450231.			1.25482			0.627410	+	0.778689i	$0.284115\pi$
$600$		0			0
$601$		−	188572.i		−	0.522071i	−0.965329	−	0.261035i	$-0.915936\pi$
$601$		−	188572.i		−	0.522071i	0.965329	−	0.261035i	$-0.0840639\pi$
$602$	85039.2			0.234653
$603$		0			0
$604$		−	104059.i		−	0.285237i
$605$	−327819.			−0.895620
$606$		0			0
$607$	266272.			0.722682			0.361341	−	0.932434i	$-0.382319\pi$
$607$	266272.			0.722682			0.361341	+	0.932434i	$0.382319\pi$
$608$		−	700365.i		−	1.89460i
$609$		0			0
$610$	359885.			0.967173
$611$	381137.			1.02094
$612$		0			0
$613$		−	331432.i		−	0.882010i	−0.897505	−	0.441005i	$-0.854622\pi$
$613$		−	331432.i		−	0.882010i	0.897505	−	0.441005i	$-0.145378\pi$
$614$			62612.9i			0.166084i
$615$		0			0
$616$	49518.7			0.130499
$617$	341099.			0.896005			0.448002	−	0.894032i	$-0.352136\pi$
$617$	341099.			0.896005			0.448002	+	0.894032i	$0.352136\pi$
$618$		0			0
$619$	−101548.			−0.265027			−0.132513	−	0.991181i	$-0.542305\pi$
$619$	−101548.			−0.265027			−0.132513	+	0.991181i	$0.542305\pi$
$620$			361163.i			0.939551i
$621$		0			0
$622$			300807.i			0.777511i
$623$		−	69872.9i		−	0.180025i
$624$		0			0
$625$	−418036.			−1.07017
$626$	−31178.1			−0.0795611
$627$		0			0
$628$		−	110873.i		−	0.281130i
$629$			843701.i			2.13249i
$630$		0			0
$631$	−278409.			−0.699238			−0.349619	−	0.936892i	$-0.613689\pi$
$631$	−278409.			−0.699238			−0.349619	+	0.936892i	$0.613689\pi$
$632$		−	549906.i		−	1.37675i
$633$		0			0
$634$		−	331442.i		−	0.824574i
$635$	648915.			1.60931
$636$		0			0
$637$		−	329861.i		−	0.812928i
$638$	94638.5			0.232502
$639$		0			0
$640$			277400.i			0.677246i
$641$	−267843.			−0.651874			−0.325937	−	0.945391i	$-0.605680\pi$
$641$	−267843.			−0.651874			−0.325937	+	0.945391i	$0.605680\pi$
$642$		0			0
$643$	247371.			0.598311			0.299155	−	0.954204i	$-0.403295\pi$
$643$	247371.			0.598311			0.299155	+	0.954204i	$0.403295\pi$
$644$		−	74747.8i		−	0.180230i
$645$		0			0
$646$		−	541838.i		−	1.29839i
$647$	66763.5			0.159489			0.0797445	−	0.996815i	$-0.474590\pi$
$647$	66763.5			0.159489			0.0797445	+	0.996815i	$0.474590\pi$
$648$		0			0
$649$	−147599.	−	49420.7i	−0.350425	−	0.117333i
$650$	18158.3			0.0429781
$651$		0			0
$652$	16745.2			0.0393909
$653$	585647.			1.37344			0.686719	−	0.726923i	$-0.259050\pi$
$653$	585647.			1.37344			0.686719	+	0.726923i	$0.259050\pi$
$654$		0			0
$655$			53579.2i			0.124886i
$656$	−3635.42			−0.00844786
$657$		0			0
$658$	102808.			0.237452
$659$		−	34391.7i		−	0.0791922i	−0.999216	−	0.0395961i	$-0.987393\pi$
$659$		−	34391.7i		−	0.0791922i	0.999216	−	0.0395961i	$-0.0126071\pi$
$660$		0			0
$661$	−151480.			−0.346698			−0.173349	−	0.984860i	$-0.555459\pi$
$661$	−151480.			−0.346698			−0.173349	+	0.984860i	$0.555459\pi$
$662$			372325.i			0.849585i
$663$		0			0
$664$	−473106.			−1.07306
$665$	307142.			0.694537
$666$		0			0
$667$		−	369401.i		−	0.830323i
$668$	57955.4			0.129880
$669$		0			0
$670$	300974.			0.670469
$671$	−255250.			−0.566918
$672$		0			0
$673$			355378.i			0.784622i	0.919833	+	0.392311i	$0.128324\pi$
$673$			355378.i			0.784622i	−0.919833	+	0.392311i	$0.871676\pi$
$674$	−148179.			−0.326187
$675$		0			0
$676$	38314.9			0.0838444
$677$	−670210.			−1.46229			−0.731145	−	0.682222i	$-0.761014\pi$
$677$	−670210.			−1.46229			−0.731145	+	0.682222i	$0.761014\pi$
$678$		0			0
$679$			103226.i			0.223898i
$680$			540373.i			1.16863i
$681$		0			0
$682$			150066.i			0.322637i
$683$			237295.i			0.508683i	0.967114	+	0.254341i	$0.0818587\pi$
$683$			237295.i			0.508683i	−0.967114	+	0.254341i	$0.918141\pi$
$684$		0			0
$685$	209012.			0.445442
$686$		−	190893.i		−	0.405641i
$687$		0			0
$688$		−	14470.2i		−	0.0305701i
$689$			240179.i			0.505937i
$690$		0			0
$691$			40265.6i			0.0843293i	0.999111	+	0.0421646i	$0.0134254\pi$
$691$			40265.6i			0.0843293i	−0.999111	+	0.0421646i	$0.986575\pi$
$692$			487690.i			1.01843i
$693$		0			0
$694$	−365345.			−0.758551
$695$	212854.			0.440668
$696$		0			0
$697$	−165358.			−0.340377
$698$	10036.8			0.0206008
$699$		0			0
$700$	−8360.71			−0.0170627
$701$		−	55287.6i		−	0.112510i	−0.998416	−	0.0562551i	$-0.982084\pi$
$701$		−	55287.6i		−	0.112510i	0.998416	−	0.0562551i	$-0.0179160\pi$
$702$		0			0
$703$			1.74214e6i			3.52511i
$704$			107067.i			0.216028i
$705$		0			0
$706$	−307171.			−0.616270
$707$		−	252879.i		−	0.505910i
$708$		0			0
$709$	91346.8			0.181719			0.0908596	−	0.995864i	$-0.471039\pi$
$709$	91346.8			0.181719			0.0908596	+	0.995864i	$0.471039\pi$
$710$			176222.i			0.349577i
$711$		0			0
$712$	253839.			0.500724
$713$	585752.			1.15222
$714$		0			0
$715$	−182471.			−0.356929
$716$		−	85649.0i		−	0.167069i
$717$		0			0
$718$			444004.i			0.861267i
$719$			25854.5i			0.0500124i	0.999687	+	0.0250062i	$0.00796055\pi$
$719$			25854.5i			0.0500124i	−0.999687	+	0.0250062i	$0.992039\pi$
$720$		0			0
$721$		−	3693.33i		−	0.00710473i
$722$		−	801994.i		−	1.53850i
$723$		0			0
$724$	−641472.			−1.22377
$725$	−41318.4			−0.0786081
$726$		0			0
$727$	441000.			0.834391			0.417196	−	0.908817i	$-0.363013\pi$
$727$	441000.			0.834391			0.417196	+	0.908817i	$0.363013\pi$
$728$	−174269.			−0.328819
$729$		0			0
$730$	297559.			0.558377
$731$		−	658181.i		−	1.23172i
$732$		0			0
$733$	426201.			0.793244			0.396622	−	0.917982i	$-0.370182\pi$
$733$	426201.			0.793244			0.396622	+	0.917982i	$0.370182\pi$
$734$	20666.2			0.0383591
$735$		0			0
$736$	438084.			0.808727
$737$	−213467.			−0.393002
$738$		0			0
$739$		−	91383.8i		−	0.167333i	−0.996494	−	0.0836663i	$-0.973337\pi$
$739$		−	91383.8i		−	0.167333i	0.996494	−	0.0836663i	$-0.0266630\pi$
$740$		−	671902.i		−	1.22699i
$741$		0			0
$742$			64786.0i			0.117672i
$743$	−309686.			−0.560975			−0.280488	−	0.959858i	$-0.590496\pi$
$743$	−309686.			−0.560975			−0.280488	+	0.959858i	$0.590496\pi$
$744$		0			0
$745$			968286.i			1.74458i
$746$		−	200048.i		−	0.359464i
$747$		0			0
$748$		−	148215.i		−	0.264905i
$749$	−222627.			−0.396839
$750$		0			0
$751$			218166.i			0.386819i	0.981118	+	0.193409i	$0.0619545\pi$
$751$			218166.i			0.386819i	−0.981118	+	0.193409i	$0.938045\pi$
$752$		−	17493.7i		−	0.0309348i
$753$		0			0
$754$	−333057.			−0.585835
$755$			267457.i			0.469201i
$756$		0			0
$757$	48000.1			0.0837627			0.0418813	−	0.999123i	$-0.486665\pi$
$757$	48000.1			0.0837627			0.0418813	+	0.999123i	$0.486665\pi$
$758$		−	6411.81i		−	0.0111594i
$759$		0			0
$760$			1.11580e6i			1.93179i
$761$	−236294.			−0.408021			−0.204011	−	0.978969i	$-0.565398\pi$
$761$	−236294.			−0.408021			−0.204011	+	0.978969i	$0.565398\pi$
$762$		0			0
$763$			315158.i			0.541351i
$764$		−	363279.i		−	0.622378i
$765$		0			0
$766$			110470.i			0.188272i
$767$	519440.	+	173924.i	0.882967	+	0.295644i
$768$		0			0
$769$		−	110994.i		−	0.187692i	−0.995587	−	0.0938459i	$-0.970084\pi$
$769$		−	110994.i		−	0.187692i	0.995587	−	0.0938459i	$-0.0299161\pi$
$770$	−49219.9			−0.0830155
$771$		0			0
$772$	74368.9			0.124783
$773$		−	164246.i		−	0.274875i	−0.990510	−	0.137437i	$-0.956113\pi$
$773$		−	164246.i		−	0.274875i	0.990510	−	0.137437i	$-0.0438866\pi$
$774$		0			0
$775$		−	65517.7i		−	0.109082i
$776$	−375006.			−0.622752
$777$		0			0
$778$		−	89491.8i		−	0.147851i
$779$	−341444.			−0.562659
$780$		0			0
$781$		−	124986.i		−	0.204908i
$782$	338924.			0.554228
$783$		0			0
$784$	−15140.2			−0.0246320
$785$			284970.i			0.462445i
$786$		0			0
$787$	909477.			1.46839			0.734196	−	0.678937i	$-0.237559\pi$
$787$	909477.			1.46839			0.734196	+	0.678937i	$0.237559\pi$
$788$	−634484.			−1.02181
$789$		0			0
$790$			546588.i			0.875802i
$791$		−	168159.i		−	0.268761i
$792$		0			0
$793$	898288.			1.42846
$794$	447916.			0.710486
$795$		0			0
$796$	250230.			0.394924
$797$			686063.i			1.08006i	0.841646	+	0.540029i	$0.181587\pi$
$797$			686063.i			1.08006i	−0.841646	+	0.540029i	$0.818413\pi$
$798$		0			0
$799$		−	795708.i		−	1.24641i
$800$		−	49000.7i		−	0.0765635i
$801$		0			0
$802$	−593200.			−0.922258
$803$	−211045.			−0.327298
$804$		0			0
$805$			192119.i			0.296469i
$806$		−	528121.i		−	0.812949i
$807$		0			0
$808$	918675.			1.40715
$809$			187550.i			0.286563i	0.989682	+	0.143282i	$0.0457654\pi$
$809$			187550.i			0.286563i	−0.989682	+	0.143282i	$0.954235\pi$
$810$		0			0
$811$			752282.i			1.14377i	0.820333	+	0.571886i	$0.193788\pi$
$811$			752282.i			1.14377i	−0.820333	+	0.571886i	$0.806212\pi$
$812$	153351.			0.232581
$813$		0			0
$814$		−	279181.i		−	0.421343i
$815$	−43039.2			−0.0647961
$816$		0			0
$817$		−	1.35906e6i		−	2.03608i
$818$	95163.6			0.142221
$819$		0			0
$820$	131687.			0.195846
$821$			763503.i			1.13272i	0.824156	+	0.566362i	$0.191650\pi$
$821$			763503.i			1.13272i	−0.824156	+	0.566362i	$0.808350\pi$
$822$		0			0
$823$			1.04357e6i			1.54071i	0.637616	+	0.770355i	$0.279921\pi$
$823$			1.04357e6i			1.54071i	−0.637616	+	0.770355i	$0.720079\pi$
$824$	13417.4			0.0197612
$825$		0			0
$826$	140114.	+	46914.3i	0.205363	+	0.0687615i
$827$	291033.			0.425531			0.212766	−	0.977103i	$-0.431753\pi$
$827$	291033.			0.425531			0.212766	+	0.977103i	$0.431753\pi$
$828$		0			0
$829$	−1.12966e6			−1.64376			−0.821882	−	0.569658i	$-0.807076\pi$
$829$	−1.12966e6			−1.64376			−0.821882	+	0.569658i	$0.807076\pi$
$830$	470251.			0.682612
$831$		0			0
$832$		−	376796.i		−	0.544327i
$833$	−688657.			−0.992461
$834$		0			0
$835$	−148959.			−0.213645
$836$		−	306047.i		−	0.437901i
$837$		0			0
$838$	−362100.			−0.515633
$839$			415299.i			0.589979i	0.955500	+	0.294990i	$0.0953162\pi$
$839$			415299.i			0.589979i	−0.955500	+	0.294990i	$0.904684\pi$
$840$		0			0
$841$	50575.2			0.0715066
$842$	627824.			0.885552
$843$		0			0
$844$		−	176590.i		−	0.247903i
$845$	−98478.3			−0.137920
$846$		0			0
$847$	−220715.			−0.307656
$848$	11023.9			0.0153301
$849$		0			0
$850$		−	37909.4i		−	0.0524697i
$851$	−1.08972e6			−1.50472
$852$		0			0
$853$	−1.12470e6			−1.54575			−0.772876	−	0.634558i	$-0.781182\pi$
$853$	−1.12470e6			−1.54575			−0.772876	+	0.634558i	$0.781182\pi$
$854$	242305.			0.332236
$855$		0			0
$856$		−	808775.i		−	1.10377i
$857$		−	1.37093e6i		−	1.86661i	−0.359085	−	0.933305i	$-0.616911\pi$
$857$		−	1.37093e6i		−	1.86661i	0.359085	−	0.933305i	$-0.383089\pi$
$858$		0			0
$859$			763286.i			1.03443i	0.855855	+	0.517215i	$0.173031\pi$
$859$			763286.i			1.03443i	−0.855855	+	0.517215i	$0.826969\pi$
$860$			524159.i			0.708706i
$861$		0			0
$862$	−632438.			−0.851145
$863$			850435.i			1.14188i	0.820992	+	0.570939i	$0.193421\pi$
$863$			850435.i			1.14188i	−0.820992	+	0.570939i	$0.806579\pi$
$864$		0			0
$865$		−	1.25348e6i		−	1.67527i
$866$		−	683898.i		−	0.911918i
$867$		0			0
$868$			243166.i			0.322747i
$869$		−	387669.i		−	0.513360i
$870$		0			0
$871$	751243.			0.990248
$872$	−1.14493e6			−1.50572
$873$		0			0
$874$	699837.			0.916166
$875$	−261485.			−0.341531
$876$		0			0
$877$	−1.17892e6			−1.53280			−0.766400	−	0.642364i	$-0.777954\pi$
$877$	−1.17892e6			−1.53280			−0.766400	+	0.642364i	$0.777954\pi$
$878$			550365.i			0.713940i
$879$		0			0
$880$			8375.21i			0.0108151i
$881$			932263.i			1.20112i	0.799579	+	0.600560i	$0.205056\pi$
$881$			932263.i			1.20112i	−0.799579	+	0.600560i	$0.794944\pi$
$882$		0			0
$883$	569874.			0.730900			0.365450	−	0.930831i	$-0.380915\pi$
$883$	569874.			0.730900			0.365450	+	0.930831i	$0.380915\pi$
$884$			521608.i			0.667482i
$885$		0			0
$886$	−464879.			−0.592206
$887$		−	147269.i		−	0.187183i	−0.995611	−	0.0935913i	$-0.970165\pi$
$887$		−	147269.i		−	0.187183i	0.995611	−	0.0935913i	$-0.0298347\pi$
$888$		0			0
$889$	436904.			0.552819
$890$	−252307.			−0.318530
$891$		0			0
$892$	545252.			0.685279
$893$		−	1.64304e6i		−	2.06037i
$894$		0			0
$895$			220138.i			0.274820i
$896$			186769.i			0.232642i
$897$		0			0
$898$			499318.i			0.619191i
$899$			1.20172e6i			1.48690i
$900$		0			0
$901$	501426.			0.617671
$902$	54716.9			0.0672526
$903$		0			0
$904$	610899.			0.747536
$905$	1.64873e6			2.01304
$906$		0			0
$907$	−546243.			−0.664005			−0.332003	−	0.943278i	$-0.607724\pi$
$907$	−546243.			−0.664005			−0.332003	+	0.943278i	$0.607724\pi$
$908$		−	251147.i		−	0.304619i
$909$		0			0
$910$	173217.			0.209174
$911$	−819476.			−0.987414			−0.493707	−	0.869628i	$-0.664359\pi$
$911$	−819476.			−0.987414			−0.493707	+	0.869628i	$0.664359\pi$
$912$		0			0
$913$	−333527.			−0.400120
$914$	354260.			0.424063
$915$		0			0
$916$			817694.i			0.974540i
$917$			36074.0i			0.0428998i
$918$		0			0
$919$		−	1.44513e6i		−	1.71111i	−0.517715	−	0.855553i	$-0.673217\pi$
$919$		−	1.44513e6i		−	1.71111i	0.517715	−	0.855553i	$-0.326783\pi$
$920$	−697945.			−0.824604
$921$		0			0
$922$		−	223016.i		−	0.262346i
$923$			439857.i			0.516307i
$924$		0			0
$925$			121888.i			0.142455i
$926$	883923.			1.03084
$927$		0			0
$928$			898764.i			1.04364i
$929$		−	69295.6i		−	0.0802923i	−0.999194	−	0.0401462i	$-0.987218\pi$
$929$		−	69295.6i		−	0.0802923i	0.999194	−	0.0401462i	$-0.0127824\pi$
$930$		0			0
$931$	−1.42199e6			−1.64058
$932$			932896.i			1.07399i
$933$		0			0
$934$	261937.			0.300264
$935$			380949.i			0.435756i
$936$		0			0
$937$			403625.i			0.459726i	0.973223	+	0.229863i	$0.0738278\pi$
$937$			403625.i			0.459726i	−0.973223	+	0.229863i	$0.926172\pi$
$938$	202641.			0.230314
$939$		0			0
$940$			633681.i			0.717159i
$941$			496072.i			0.560229i	0.959967	+	0.280114i	$0.0903724\pi$
$941$			496072.i			0.560229i	−0.959967	+	0.280114i	$0.909628\pi$
$942$		0			0
$943$		−	213576.i		−	0.240176i
$944$	7982.90	−	23841.7i	0.00895811	−	0.0267542i
$945$		0			0
$946$			217792.i			0.243366i
$947$	−145308.			−0.162028			−0.0810139	−	0.996713i	$-0.525816\pi$
$947$	−145308.			−0.162028			−0.0810139	+	0.996713i	$0.525816\pi$
$948$		0			0
$949$	742720.			0.824693
$950$		−	78278.3i		−	0.0867350i
$951$		0			0
$952$			363824.i			0.401438i
$953$	−1.62459e6			−1.78879			−0.894394	−	0.447280i	$-0.852393\pi$
$953$	−1.62459e6			−1.78879			−0.894394	+	0.447280i	$0.852393\pi$
$954$		0			0
$955$			933714.i			1.02378i
$956$	673483.			0.736904
$957$		0			0
$958$			360998.i			0.393346i
$959$	140725.			0.153015
$960$		0			0
$961$	−982016.			−1.06334
$962$			982507.i			1.06166i
$963$		0			0
$964$	−584030.			−0.628465
$965$	−191146.			−0.205262
$966$		0			0
$967$		−	54787.5i		−	0.0585907i	−0.999571	−	0.0292953i	$-0.990674\pi$
$967$		−	54787.5i		−	0.0585907i	0.999571	−	0.0292953i	$-0.00932633\pi$
$968$		−	801830.i		−	0.855720i
$969$		0			0
$970$	372743.			0.396156
$971$	1.29608e6			1.37466			0.687328	−	0.726347i	$-0.258784\pi$
$971$	1.29608e6			1.37466			0.687328	+	0.726347i	$0.258784\pi$
$972$		0			0
$973$	143311.			0.151375
$974$			8276.18i			0.00872393i
$975$		0			0
$976$		−	41230.3i		−	0.0432830i
$977$			944291.i			0.989275i	0.869099	+	0.494637i	$0.164699\pi$
$977$			944291.i			0.989275i	−0.869099	+	0.494637i	$0.835301\pi$
$978$		0			0
$979$	178950.			0.186709
$980$	548429.			0.571043
$981$		0			0
$982$			794315.i			0.823702i
$983$			274967.i			0.284560i	0.989826	+	0.142280i	$0.0454433\pi$
$983$			274967.i			0.284560i	−0.989826	+	0.142280i	$0.954557\pi$
$984$		0			0
$985$	1.63077e6			1.68082
$986$			695329.i			0.715215i
$987$		0			0
$988$			1.07706e6i			1.10338i
$989$	850105.			0.869121
$990$		0			0
$991$		−	1.10811e6i		−	1.12833i	−0.825661	−	0.564166i	$-0.809198\pi$
$991$		−	1.10811e6i		−	1.12833i	0.825661	−	0.564166i	$-0.190802\pi$
$992$	−1.42515e6			−1.44823
$993$		0			0
$994$			118647.i			0.120084i
$995$	−643150.			−0.649630
$996$		0			0
$997$	34272.3			0.0344788			0.0172394	−	0.999851i	$-0.494512\pi$
$997$	34272.3			0.0344788			0.0172394	+	0.999851i	$0.494512\pi$
$998$		−	413924.i		−	0.415584i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.14		40
3.2	odd	2		177.5.c.a.58.27	yes	40
59.58	odd	2	inner	531.5.c.d.235.27		40
177.176	even	2		177.5.c.a.58.14	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.14	✓	40	177.176	even	2
177.5.c.a.58.27	yes	40	3.2	odd	2
531.5.c.d.235.14		40	1.1	even	1	trivial
531.5.c.d.235.27		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)

\(f(q)\)	\(=\)	\(q-2.43119i q^{2} +10.0893 q^{4} -25.9319 q^{5} -17.4595 q^{7} -63.4281i q^{8} +O(q^{10})\)	\(q-2.43119i q^{2} +10.0893 q^{4} -25.9319 q^{5} -17.4595 q^{7} -63.4281i q^{8} +63.0454i q^{10} -44.7152i q^{11} +157.364i q^{13} +42.4475i q^{14} +7.22282 q^{16} +328.532 q^{17} +678.379 q^{19} -261.635 q^{20} -108.711 q^{22} +424.331i q^{23} +47.4624 q^{25} +382.582 q^{26} -176.154 q^{28} -870.549 q^{29} -1380.41i q^{31} -1032.41i q^{32} -798.725i q^{34} +452.758 q^{35} +2568.09i q^{37} -1649.27i q^{38} +1644.81i q^{40} -503.324 q^{41} -2003.40i q^{43} -451.145i q^{44} +1031.63 q^{46} -2422.01i q^{47} -2096.17 q^{49} -115.390i q^{50} +1587.69i q^{52} +1526.26 q^{53} +1159.55i q^{55} +1107.42i q^{56} +2116.47i q^{58} +(1105.23 - 3300.88i) q^{59} -5708.35i q^{61} -3356.05 q^{62} -2394.42 q^{64} -4080.74i q^{65} -4773.92i q^{67} +3314.66 q^{68} -1100.74i q^{70} +2795.15 q^{71} -4719.76i q^{73} +6243.53 q^{74} +6844.37 q^{76} +780.705i q^{77} +8669.75 q^{79} -187.301 q^{80} +1223.68i q^{82} -7458.93i q^{83} -8519.45 q^{85} -4870.65 q^{86} -2836.20 q^{88} +4001.99i q^{89} -2747.50i q^{91} +4281.21i q^{92} -5888.37 q^{94} -17591.6 q^{95} -5912.30i q^{97} +5096.18i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 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

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Database

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