LMFDB - Embedded newform 531.5.c.d.235.20

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			235.20
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.389117i q^{2} +15.8486 q^{4} +17.9988 q^{5} -47.2538 q^{7} -12.3928i q^{8} +O(q^{10})$	\(q-0.389117i q^{2} +15.8486 q^{4} +17.9988 q^{5} -47.2538 q^{7} -12.3928i q^{8} -7.00365i q^{10} +197.341i q^{11} -176.297i q^{13} +18.3873i q^{14} +248.755 q^{16} -486.600 q^{17} +56.2166 q^{19} +285.256 q^{20} +76.7889 q^{22} +848.854i q^{23} -301.043 q^{25} -68.6003 q^{26} -748.906 q^{28} -275.393 q^{29} +843.954i q^{31} -295.080i q^{32} +189.344i q^{34} -850.512 q^{35} +1855.12i q^{37} -21.8749i q^{38} -223.056i q^{40} -1479.48 q^{41} -2758.38i q^{43} +3127.58i q^{44} +330.304 q^{46} +1437.11i q^{47} -168.080 q^{49} +117.141i q^{50} -2794.06i q^{52} +1119.21 q^{53} +3551.91i q^{55} +585.608i q^{56} +107.160i q^{58} +(1640.41 + 3070.25i) q^{59} -115.162i q^{61} +328.397 q^{62} +3865.26 q^{64} -3173.14i q^{65} -49.5250i q^{67} -7711.92 q^{68} +330.949i q^{70} +2085.72 q^{71} -209.953i q^{73} +721.859 q^{74} +890.954 q^{76} -9325.12i q^{77} -6542.76 q^{79} +4477.30 q^{80} +575.691i q^{82} -6802.27i q^{83} -8758.22 q^{85} -1073.33 q^{86} +2445.62 q^{88} -9119.14i q^{89} +8330.70i q^{91} +13453.1i q^{92} +559.206 q^{94} +1011.83 q^{95} -2946.12i q^{97} +65.4030i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	0.389117i		−	0.0972793i	−0.998816	−	0.0486397i	$-0.984511\pi$
$2$		−	0.389117i		−	0.0972793i	0.998816	−	0.0486397i	$-0.0154886\pi$
$3$		0			0
$3$		0			0
$4$	15.8486			0.990537
$5$	17.9988			0.719952			0.359976	−	0.932962i	$-0.382785\pi$
$5$	17.9988			0.719952			0.359976	+	0.932962i	$0.382785\pi$
$6$		0			0
$7$	−47.2538			−0.964363			−0.482181	−	0.876071i	$-0.660155\pi$
$7$	−47.2538			−0.964363			−0.482181	+	0.876071i	$0.660155\pi$
$8$		−	12.3928i		−	0.193638i
$9$		0			0
$10$		−	7.00365i		−	0.0700365i
$11$			197.341i			1.63092i	0.578814	+	0.815460i	$0.303516\pi$
$11$			197.341i			1.63092i	−0.578814	+	0.815460i	$0.696484\pi$
$12$		0			0
$13$		−	176.297i		−	1.04318i	−0.853197	−	0.521589i	$-0.825339\pi$
$13$		−	176.297i		−	1.04318i	0.853197	−	0.521589i	$-0.174661\pi$
$14$			18.3873i			0.0938126i
$15$		0			0
$16$	248.755			0.971700
$17$	−486.600			−1.68374			−0.841868	−	0.539683i	$-0.818544\pi$
$17$	−486.600			−1.68374			−0.841868	+	0.539683i	$0.818544\pi$
$18$		0			0
$19$	56.2166			0.155725			0.0778624	−	0.996964i	$-0.475191\pi$
$19$	56.2166			0.155725			0.0778624	+	0.996964i	$0.475191\pi$
$20$	285.256			0.713139
$21$		0			0
$22$	76.7889			0.158655
$23$			848.854i			1.60464i	0.596895	+	0.802319i	$0.296401\pi$
$23$			848.854i			1.60464i	−0.596895	+	0.802319i	$0.703599\pi$
$24$		0			0
$25$	−301.043			−0.481669
$26$	−68.6003			−0.101480
$27$		0			0
$28$	−748.906			−0.955237
$29$	−275.393			−0.327459			−0.163729	−	0.986505i	$-0.552352\pi$
$29$	−275.393			−0.327459			−0.163729	+	0.986505i	$0.552352\pi$
$30$		0			0
$31$			843.954i			0.878204i	0.898437	+	0.439102i	$0.144703\pi$
$31$			843.954i			0.878204i	−0.898437	+	0.439102i	$0.855297\pi$
$32$		−	295.080i		−	0.288164i
$33$		0			0
$34$			189.344i			0.163793i
$35$	−850.512			−0.694295
$36$		0			0
$37$			1855.12i			1.35509i	0.735481	+	0.677546i	$0.236956\pi$
$37$			1855.12i			1.35509i	−0.735481	+	0.677546i	$0.763044\pi$
$38$		−	21.8749i		−	0.0151488i
$39$		0			0
$40$		−	223.056i		−	0.139410i
$41$	−1479.48			−0.880118			−0.440059	−	0.897969i	$-0.645042\pi$
$41$	−1479.48			−0.880118			−0.440059	+	0.897969i	$0.645042\pi$
$42$		0			0
$43$		−	2758.38i		−	1.49182i	−0.666045	−	0.745912i	$-0.732014\pi$
$43$		−	2758.38i		−	1.49182i	0.666045	−	0.745912i	$-0.267986\pi$
$44$			3127.58i			1.61549i
$45$		0			0
$46$	330.304			0.156098
$47$			1437.11i			0.650572i	0.945616	+	0.325286i	$0.105461\pi$
$47$			1437.11i			0.650572i	−0.945616	+	0.325286i	$0.894539\pi$
$48$		0			0
$49$	−168.080			−0.0700043
$50$			117.141i			0.0468564i
$51$		0			0
$52$		−	2794.06i		−	1.03331i
$53$	1119.21			0.398439			0.199219	−	0.979955i	$-0.436159\pi$
$53$	1119.21			0.398439			0.199219	+	0.979955i	$0.436159\pi$
$54$		0			0
$55$			3551.91i			1.17418i
$56$			585.608i			0.186737i
$57$		0			0
$58$			107.160i			0.0318550i
$59$	1640.41	+	3070.25i	0.471247	+	0.882001i
$59$	1640.41	+	3070.25i	0.471247	+	0.882001i
$60$		0			0
$61$		−	115.162i		−	0.0309492i	−0.999880	−	0.0154746i	$-0.995074\pi$
$61$		−	115.162i		−	0.0309492i	0.999880	−	0.0154746i	$-0.00492592\pi$
$62$	328.397			0.0854311
$63$		0			0
$64$	3865.26			0.943667
$65$		−	3173.14i		−	0.751038i
$66$		0			0
$67$		−	49.5250i		−	0.0110325i	−0.999985	−	0.00551626i	$-0.998244\pi$
$67$		−	49.5250i		−	0.0110325i	0.999985	−	0.00551626i	$-0.00175589\pi$
$68$	−7711.92			−1.66780
$69$		0			0
$70$			330.949i			0.0675406i
$71$	2085.72			0.413751			0.206876	−	0.978367i	$-0.433670\pi$
$71$	2085.72			0.413751			0.206876	+	0.978367i	$0.433670\pi$
$72$		0			0
$73$		−	209.953i		−	0.0393982i	−0.999806	−	0.0196991i	$-0.993729\pi$
$73$		−	209.953i		−	0.0393982i	0.999806	−	0.0196991i	$-0.00627083\pi$
$74$	721.859			0.131822
$75$		0			0
$76$	890.954			0.154251
$77$		−	9325.12i		−	1.57280i
$78$		0			0
$79$	−6542.76			−1.04835			−0.524176	−	0.851610i	$-0.675626\pi$
$79$	−6542.76			−1.04835			−0.524176	+	0.851610i	$0.675626\pi$
$80$	4477.30			0.699577
$81$		0			0
$82$			575.691i			0.0856173i
$83$		−	6802.27i		−	0.987410i	−0.869629	−	0.493705i	$-0.835642\pi$
$83$		−	6802.27i		−	0.987410i	0.869629	−	0.493705i	$-0.164358\pi$
$84$		0			0
$85$	−8758.22			−1.21221
$86$	−1073.33			−0.145124
$87$		0			0
$88$	2445.62			0.315808
$89$		−	9119.14i		−	1.15126i	−0.817710	−	0.575630i	$-0.804757\pi$
$89$		−	9119.14i		−	1.15126i	0.817710	−	0.575630i	$-0.195243\pi$
$90$		0			0
$91$			8330.70i			1.00600i
$92$			13453.1i			1.58945i
$93$		0			0
$94$	559.206			0.0632872
$95$	1011.83			0.112114
$96$		0			0
$97$		−	2946.12i		−	0.313117i	−0.987669	−	0.156559i	$-0.949960\pi$
$97$		−	2946.12i		−	0.313117i	0.987669	−	0.156559i	$-0.0500400\pi$
$98$			65.4030i			0.00680997i
$99$		0			0
$100$	−4771.11			−0.477111
$101$			4577.99i			0.448779i	0.974500	+	0.224389i	$0.0720388\pi$
$101$			4577.99i			0.448779i	−0.974500	+	0.224389i	$0.927961\pi$
$102$		0			0
$103$			14158.2i			1.33455i	0.744812	+	0.667274i	$0.232539\pi$
$103$			14158.2i			1.33455i	−0.744812	+	0.667274i	$0.767461\pi$
$104$	−2184.82			−0.201999
$105$		0			0
$106$		−	435.506i		−	0.0387598i
$107$	−2298.00			−0.200717			−0.100358	−	0.994951i	$-0.531999\pi$
$107$	−2298.00			−0.200717			−0.100358	+	0.994951i	$0.531999\pi$
$108$		0			0
$109$			18428.3i			1.55107i	0.631302	+	0.775537i	$0.282521\pi$
$109$			18428.3i			1.55107i	−0.631302	+	0.775537i	$0.717479\pi$
$110$	1382.11			0.114224
$111$		0			0
$112$	−11754.6			−0.937071
$113$			15385.4i			1.20490i	0.798156	+	0.602451i	$0.205809\pi$
$113$			15385.4i			1.20490i	−0.798156	+	0.602451i	$0.794191\pi$
$114$		0			0
$115$			15278.4i			1.15526i
$116$	−4364.59			−0.324360
$117$		0			0
$118$	1194.69	−	638.312i	0.0858005	−	0.0458426i
$119$	22993.7			1.62373
$120$		0			0
$121$	−24302.6			−1.65990
$122$	−44.8115			−0.00301072
$123$		0			0
$124$			13375.5i			0.869893i
$125$	−16667.7			−1.06673
$126$		0			0
$127$	28666.4			1.77732			0.888661	−	0.458564i	$-0.151636\pi$
$127$	28666.4			1.77732			0.888661	+	0.458564i	$0.151636\pi$
$128$		−	6225.33i		−	0.379964i
$129$		0			0
$130$	−1234.72			−0.0730605
$131$			19845.6i			1.15643i	0.815883	+	0.578217i	$0.196251\pi$
$131$			19845.6i			1.15643i	−0.815883	+	0.578217i	$0.803749\pi$
$132$		0			0
$133$	−2656.45			−0.150175
$134$	−19.2710			−0.00107324
$135$		0			0
$136$			6030.35i			0.326035i
$137$	−19165.2			−1.02111			−0.510555	−	0.859845i	$-0.670560\pi$
$137$	−19165.2			−1.02111			−0.510555	+	0.859845i	$0.670560\pi$
$138$		0			0
$139$	−20252.2			−1.04820			−0.524099	−	0.851658i	$-0.675598\pi$
$139$	−20252.2			−1.04820			−0.524099	+	0.851658i	$0.675598\pi$
$140$	−13479.4			−0.687725
$141$		0			0
$142$		−	811.590i		−	0.0402495i
$143$	34790.7			1.70134
$144$		0			0
$145$	−4956.74			−0.235755
$146$	−81.6964			−0.00383263
$147$		0			0
$148$			29401.0i			1.34227i
$149$			8521.84i			0.383849i	0.981410	+	0.191925i	$0.0614729\pi$
$149$			8521.84i			0.383849i	−0.981410	+	0.191925i	$0.938527\pi$
$150$		0			0
$151$			43196.8i			1.89452i	0.320473	+	0.947258i	$0.396158\pi$
$151$			43196.8i			1.89452i	−0.320473	+	0.947258i	$0.603842\pi$
$152$		−	696.684i		−	0.0301542i
$153$		0			0
$154$	−3628.57			−0.153001
$155$			15190.2i			0.632265i
$156$		0			0
$157$		−	7274.02i		−	0.295104i	−0.989054	−	0.147552i	$-0.952861\pi$
$157$		−	7274.02i		−	0.295104i	0.989054	−	0.147552i	$-0.0471394\pi$
$158$			2545.90i			0.101983i
$159$		0			0
$160$		−	5311.09i		−	0.207465i
$161$		−	40111.6i		−	1.54745i
$162$		0			0
$163$	−15415.8			−0.580219			−0.290109	−	0.956994i	$-0.593692\pi$
$163$	−15415.8			−0.580219			−0.290109	+	0.956994i	$0.593692\pi$
$164$	−23447.6			−0.871789
$165$		0			0
$166$	−2646.88			−0.0960546
$167$	−37730.3			−1.35288			−0.676438	−	0.736500i	$-0.736477\pi$
$167$	−37730.3			−1.35288			−0.676438	+	0.736500i	$0.736477\pi$
$168$		0			0
$169$	−2519.67			−0.0882205
$170$			3407.97i			0.117923i
$171$		0			0
$172$		−	43716.4i		−	1.47771i
$173$		−	12104.2i		−	0.404430i	−0.979341	−	0.202215i	$-0.935186\pi$
$173$		−	12104.2i		−	0.404430i	0.979341	−	0.202215i	$-0.0648139\pi$
$174$		0			0
$175$	14225.4			0.464503
$176$			49089.7i			1.58476i
$177$		0			0
$178$	−3548.41			−0.111994
$179$			288.509i			0.00900436i	0.999990	+	0.00450218i	$0.00143309\pi$
$179$			288.509i			0.00900436i	−0.999990	+	0.00450218i	$0.998567\pi$
$180$		0			0
$181$	−12771.6			−0.389841			−0.194921	−	0.980819i	$-0.562445\pi$
$181$	−12771.6			−0.389841			−0.194921	+	0.980819i	$0.562445\pi$
$182$	3241.62			0.0978632
$183$		0			0
$184$	10519.7			0.310719
$185$			33389.9i			0.975601i
$186$		0			0
$187$		−	96026.2i		−	2.74604i
$188$			22776.2i			0.644415i
$189$		0			0
$190$		−	393.722i		−	0.0109064i
$191$		−	49064.3i		−	1.34493i	−0.740130	−	0.672464i	$-0.765236\pi$
$191$		−	49064.3i		−	1.34493i	0.740130	−	0.672464i	$-0.234764\pi$
$192$		0			0
$193$	51351.0			1.37859			0.689294	−	0.724481i	$-0.257921\pi$
$193$	51351.0			1.37859			0.689294	+	0.724481i	$0.257921\pi$
$194$	−1146.39			−0.0304598
$195$		0			0
$196$	−2663.84			−0.0693418
$197$	30154.4			0.776996			0.388498	−	0.921450i	$-0.372994\pi$
$197$	30154.4			0.776996			0.388498	+	0.921450i	$0.372994\pi$
$198$		0			0
$199$	67941.6			1.71565			0.857827	−	0.513939i	$-0.171814\pi$
$199$	67941.6			1.71565			0.857827	+	0.513939i	$0.171814\pi$
$200$			3730.78i			0.0932694i
$201$		0			0
$202$	1781.38			0.0436569
$203$	13013.4			0.315789
$204$		0			0
$205$	−26628.8			−0.633643
$206$	5509.21			0.129824
$207$		0			0
$208$		−	43854.8i		−	1.01366i
$209$			11093.9i			0.253975i
$210$		0			0
$211$		−	2303.99i		−	0.0517507i	−0.999665	−	0.0258754i	$-0.991763\pi$
$211$		−	2303.99i		−	0.0517507i	0.999665	−	0.0258754i	$-0.00823730\pi$
$212$	17738.0			0.394668
$213$		0			0
$214$			894.193i			0.0195256i
$215$		−	49647.6i		−	1.07404i
$216$		0			0
$217$		−	39880.0i		−	0.846907i
$218$	7170.78			0.150888
$219$		0			0
$220$			56292.7i			1.16307i
$221$			85786.1i			1.75644i
$222$		0			0
$223$	28222.6			0.567527			0.283764	−	0.958894i	$-0.408417\pi$
$223$	28222.6			0.567527			0.283764	+	0.958894i	$0.408417\pi$
$224$			13943.7i			0.277895i
$225$		0			0
$226$	5986.73			0.117212
$227$		−	63511.2i		−	1.23253i	−0.787538	−	0.616267i	$-0.788644\pi$
$227$		−	63511.2i		−	1.23253i	0.787538	−	0.616267i	$-0.211356\pi$
$228$		0			0
$229$			36930.9i			0.704236i	0.935956	+	0.352118i	$0.114538\pi$
$229$			36930.9i			0.704236i	−0.935956	+	0.352118i	$0.885462\pi$
$230$	5945.07			0.112383
$231$		0			0
$232$			3412.90i			0.0634085i
$233$		−	53127.8i		−	0.978610i	−0.872113	−	0.489305i	$-0.837250\pi$
$233$		−	53127.8i		−	0.978610i	0.872113	−	0.489305i	$-0.162750\pi$
$234$		0			0
$235$			25866.3i			0.468381i
$236$	25998.2	+	48659.1i	0.466787	+	0.873655i
$237$		0			0
$238$		−	8947.24i		−	0.157956i
$239$	24487.8			0.428700			0.214350	−	0.976757i	$-0.431237\pi$
$239$	24487.8			0.428700			0.214350	+	0.976757i	$0.431237\pi$
$240$		0			0
$241$	12746.1			0.219454			0.109727	−	0.993962i	$-0.465002\pi$
$241$	12746.1			0.219454			0.109727	+	0.993962i	$0.465002\pi$
$242$			9456.55i			0.161474i
$243$		0			0
$244$		−	1825.16i		−	0.0306563i
$245$	−3025.25			−0.0503998
$246$		0			0
$247$		−	9910.83i		−	0.162449i
$248$	10459.0			0.170054
$249$		0			0
$250$			6485.68i			0.103771i
$251$	−44798.5			−0.711076			−0.355538	−	0.934662i	$-0.615702\pi$
$251$	−44798.5			−0.711076			−0.355538	+	0.934662i	$0.615702\pi$
$252$		0			0
$253$	−167514.			−2.61704
$254$		−	11154.6i		−	0.172897i
$255$		0			0
$256$	59421.8			0.906705
$257$	−105733.			−1.60083			−0.800413	−	0.599449i	$-0.795386\pi$
$257$	−105733.			−1.60083			−0.800413	+	0.599449i	$0.795386\pi$
$258$		0			0
$259$		−	87661.4i		−	1.30680i
$260$		−	50289.7i		−	0.743931i
$261$		0			0
$262$	7722.25			0.112497
$263$	−9304.75			−0.134522			−0.0672610	−	0.997735i	$-0.521426\pi$
$263$	−9304.75			−0.134522			−0.0672610	+	0.997735i	$0.521426\pi$
$264$		0			0
$265$	20144.5			0.286857
$266$			1033.67i			0.0146089i
$267$		0			0
$268$		−	784.901i		−	0.0109281i
$269$			6683.88i			0.0923685i	0.998933	+	0.0461842i	$0.0147061\pi$
$269$			6683.88i			0.0923685i	−0.998933	+	0.0461842i	$0.985294\pi$
$270$		0			0
$271$	−137964.			−1.87856			−0.939282	−	0.343145i	$-0.888508\pi$
$271$	−137964.			−1.87856			−0.939282	+	0.343145i	$0.888508\pi$
$272$	−121044.			−1.63609
$273$		0			0
$274$			7457.51i			0.0993329i
$275$		−	59408.2i		−	0.785563i
$276$		0			0
$277$	96139.5			1.25298			0.626488	−	0.779431i	$-0.284492\pi$
$277$	96139.5			1.25298			0.626488	+	0.779431i	$0.284492\pi$
$278$			7880.49i			0.101968i
$279$		0			0
$280$			10540.3i			0.134442i
$281$	13544.9			0.171539			0.0857696	−	0.996315i	$-0.472665\pi$
$281$	13544.9			0.171539			0.0857696	+	0.996315i	$0.472665\pi$
$282$		0			0
$283$		−	147321.i		−	1.83946i	−0.392547	−	0.919732i	$-0.628406\pi$
$283$		−	147321.i		−	1.83946i	0.392547	−	0.919732i	$-0.371594\pi$
$284$	33055.7			0.409836
$285$		0			0
$286$		−	13537.7i		−	0.165505i
$287$	69910.9			0.848753
$288$		0			0
$289$	153258.			1.83497
$290$			1928.76i			0.0229341i
$291$		0			0
$292$		−	3327.46i		−	0.0390254i
$293$	−21384.4			−0.249094			−0.124547	−	0.992214i	$-0.539748\pi$
$293$	−21384.4			−0.249094			−0.124547	+	0.992214i	$0.539748\pi$
$294$		0			0
$295$	29525.4	+	55260.8i	0.339275	+	0.634999i
$296$	22990.2			0.262397
$297$		0			0
$298$	3316.00			0.0373406
$299$	149650.			1.67392
$300$		0			0
$301$			130344.i			1.43866i
$302$	16808.6			0.184297
$303$		0			0
$304$	13984.2			0.151318
$305$		−	2072.78i		−	0.0222820i
$306$		0			0
$307$	−89703.8			−0.951775			−0.475887	−	0.879506i	$-0.657873\pi$
$307$	−89703.8			−0.951775			−0.475887	+	0.879506i	$0.657873\pi$
$308$		−	147790.i		−	1.55791i
$309$		0			0
$310$	5910.76			0.0615063
$311$	23438.7			0.242333			0.121166	−	0.992632i	$-0.461337\pi$
$311$	23438.7			0.242333			0.121166	+	0.992632i	$0.461337\pi$
$312$		0			0
$313$		−	48687.8i		−	0.496971i	−0.968636	−	0.248486i	$-0.920067\pi$
$313$		−	48687.8i		−	0.496971i	0.968636	−	0.248486i	$-0.0799329\pi$
$314$	−2830.45			−0.0287075
$315$		0			0
$316$	−103694.			−1.03843
$317$	126653.			1.26036			0.630182	−	0.776448i	$-0.282980\pi$
$317$	126653.			1.26036			0.630182	+	0.776448i	$0.282980\pi$
$318$		0			0
$319$		−	54346.4i		−	0.534059i
$320$	69570.1			0.679395
$321$		0			0
$322$	−15608.1			−0.150535
$323$	−27355.0			−0.262199
$324$		0			0
$325$			53073.0i			0.502466i
$326$			5998.57i			0.0564433i
$327$		0			0
$328$			18334.9i			0.170424i
$329$		−	67909.0i		−	0.627387i
$330$		0			0
$331$	177452.			1.61966			0.809831	−	0.586663i	$-0.199559\pi$
$331$	177452.			1.61966			0.809831	+	0.586663i	$0.199559\pi$
$332$		−	107806.i		−	0.978066i
$333$		0			0
$334$			14681.5i			0.131607i
$335$		−	891.390i		−	0.00794289i
$336$		0			0
$337$			88144.0i			0.776127i	0.921633	+	0.388064i	$0.126856\pi$
$337$			88144.0i			0.776127i	−0.921633	+	0.388064i	$0.873144\pi$
$338$			980.446i			0.00858203i
$339$		0			0
$340$	−138805.			−1.20074
$341$	−166547.			−1.43228
$342$		0			0
$343$	121399.			1.03187
$344$	−34184.2			−0.288874
$345$		0			0
$346$	−4709.95			−0.0393427
$347$		−	79473.9i		−	0.660033i	−0.943975	−	0.330016i	$-0.892946\pi$
$347$		−	79473.9i		−	0.660033i	0.943975	−	0.330016i	$-0.107054\pi$
$348$		0			0
$349$		−	24382.5i		−	0.200183i	−0.994978	−	0.100091i	$-0.968086\pi$
$349$		−	24382.5i		−	0.200183i	0.994978	−	0.100091i	$-0.0319135\pi$
$350$		−	5535.36i		−	0.0451866i
$351$		0			0
$352$	58231.5			0.469973
$353$			61148.2i			0.490720i	0.969432	+	0.245360i	$0.0789062\pi$
$353$			61148.2i			0.490720i	−0.969432	+	0.245360i	$0.921094\pi$
$354$		0			0
$355$	37540.5			0.297881
$356$		−	144525.i		−	1.14037i
$357$		0			0
$358$	112.264			0.000875939
$359$	241084.			1.87059			0.935297	−	0.353864i	$-0.115132\pi$
$359$	241084.			1.87059			0.935297	+	0.353864i	$0.115132\pi$
$360$		0			0
$361$	−127161.			−0.975750
$362$			4969.64i			0.0379235i
$363$		0			0
$364$			132030.i			0.996482i
$365$		−	3778.91i		−	0.0283648i
$366$		0			0
$367$		−	237472.i		−	1.76311i	−0.472078	−	0.881557i	$-0.656496\pi$
$367$		−	237472.i		−	1.76311i	0.472078	−	0.881557i	$-0.343504\pi$
$368$			211157.i			1.55923i
$369$		0			0
$370$	12992.6			0.0949058
$371$	−52887.1			−0.384239
$372$		0			0
$373$	−168379.			−1.21023			−0.605117	−	0.796137i	$-0.706874\pi$
$373$	−168379.			−1.21023			−0.605117	+	0.796137i	$0.706874\pi$
$374$	−37365.5			−0.267133
$375$		0			0
$376$	17809.9			0.125976
$377$			48551.0i			0.341598i
$378$		0			0
$379$	234352.			1.63152			0.815758	−	0.578394i	$-0.196320\pi$
$379$	234352.			1.63152			0.815758	+	0.578394i	$0.196320\pi$
$380$	16036.1			0.111053
$381$		0			0
$382$	−19091.8			−0.130834
$383$	−114382.			−0.779762			−0.389881	−	0.920865i	$-0.627484\pi$
$383$	−114382.			−0.779762			−0.389881	+	0.920865i	$0.627484\pi$
$384$		0			0
$385$		−	167841.i		−	1.13234i
$386$		−	19981.6i		−	0.134108i
$387$		0			0
$388$		−	46691.9i		−	0.310154i
$389$	248829.			1.64438			0.822189	−	0.569215i	$-0.192753\pi$
$389$	248829.			1.64438			0.822189	+	0.569215i	$0.192753\pi$
$390$		0			0
$391$		−	413052.i		−	2.70179i
$392$			2082.99i			0.0135555i
$393$		0			0
$394$		−	11733.6i		−	0.0755857i
$395$	−117762.			−0.754763
$396$		0			0
$397$		−	178881.i		−	1.13497i	−0.823385	−	0.567483i	$-0.807917\pi$
$397$		−	178881.i		−	1.13497i	0.823385	−	0.567483i	$-0.192083\pi$
$398$		−	26437.2i		−	0.166898i
$399$		0			0
$400$	−74886.0			−0.468037
$401$		−	79311.1i		−	0.493225i	−0.969114	−	0.246613i	$-0.920682\pi$
$401$		−	79311.1i		−	0.493225i	0.969114	−	0.246613i	$-0.0793175\pi$
$402$		0			0
$403$	148787.			0.916123
$404$			72554.7i			0.444532i
$405$		0			0
$406$		−	5063.72i		−	0.0307198i
$407$	−366092.			−2.21004
$408$		0			0
$409$		−	85968.6i		−	0.513917i	−0.966422	−	0.256959i	$-0.917280\pi$
$409$		−	85968.6i		−	0.513917i	0.966422	−	0.256959i	$-0.0827204\pi$
$410$			10361.7i			0.0616404i
$411$		0			0
$412$			224388.i			1.32192i
$413$	−77515.6	−	145081.i	−0.454453	−	0.850569i
$414$		0			0
$415$		−	122433.i		−	0.710888i
$416$	−52021.8			−0.300607
$417$		0			0
$418$	4316.81			0.0247065
$419$			282200.i			1.60742i	0.595023	+	0.803709i	$0.297143\pi$
$419$			282200.i			1.60742i	−0.595023	+	0.803709i	$0.702857\pi$
$420$		0			0
$421$			254214.i			1.43428i	0.696928	+	0.717141i	$0.254550\pi$
$421$			254214.i			1.43428i	−0.696928	+	0.717141i	$0.745450\pi$
$422$	−896.524			−0.00503427
$423$		0			0
$424$		−	13870.2i		−	0.0771529i
$425$	146487.			0.811003
$426$		0			0
$427$			5441.84i			0.0298463i
$428$	−36420.1			−0.198817
$429$		0			0
$430$	−19318.7			−0.104482
$431$			196721.i			1.05900i	0.848310	+	0.529500i	$0.177620\pi$
$431$			196721.i			1.05900i	−0.848310	+	0.529500i	$0.822380\pi$
$432$		0			0
$433$	246611.			1.31533			0.657667	−	0.753309i	$-0.271543\pi$
$433$	246611.			1.31533			0.657667	+	0.753309i	$0.271543\pi$
$434$	−15518.0			−0.0823866
$435$		0			0
$436$			292063.i			1.53640i
$437$			47719.7i			0.249882i
$438$		0			0
$439$	73628.2			0.382046			0.191023	−	0.981586i	$-0.438820\pi$
$439$	73628.2			0.382046			0.191023	+	0.981586i	$0.438820\pi$
$440$	44018.2			0.227367
$441$		0			0
$442$	33380.9			0.170865
$443$		−	21033.6i		−	0.107178i	−0.998563	−	0.0535890i	$-0.982934\pi$
$443$		−	21033.6i		−	0.107178i	0.998563	−	0.0535890i	$-0.0170661\pi$
$444$		0			0
$445$		−	164134.i		−	0.828853i
$446$		−	10981.9i		−	0.0552087i
$447$		0			0
$448$	−182648.			−0.910038
$449$	−99536.9			−0.493732			−0.246866	−	0.969050i	$-0.579401\pi$
$449$	−99536.9			−0.493732			−0.246866	+	0.969050i	$0.579401\pi$
$450$		0			0
$451$		−	291962.i		−	1.43540i
$452$			243837.i			1.19350i
$453$		0			0
$454$	−24713.3			−0.119900
$455$			149943.i			0.724274i
$456$		0			0
$457$			282977.i			1.35494i	0.735551	+	0.677469i	$0.236923\pi$
$457$			282977.i			1.35494i	−0.735551	+	0.677469i	$0.763077\pi$
$458$	14370.4			0.0685077
$459$		0			0
$460$			242140.i			1.14433i
$461$	−3519.59			−0.0165611			−0.00828057	−	0.999966i	$-0.502636\pi$
$461$	−3519.59			−0.0165611			−0.00828057	+	0.999966i	$0.502636\pi$
$462$		0			0
$463$		−	295702.i		−	1.37940i	−0.724093	−	0.689702i	$-0.757741\pi$
$463$		−	295702.i		−	1.37940i	0.724093	−	0.689702i	$-0.242259\pi$
$464$	−68505.4			−0.318192
$465$		0			0
$466$	−20672.9			−0.0951986
$467$			77026.5i			0.353189i	0.984284	+	0.176594i	$0.0565081\pi$
$467$			77026.5i			0.353189i	−0.984284	+	0.176594i	$0.943492\pi$
$468$		0			0
$469$			2340.24i			0.0106394i
$470$	10065.0			0.0455638
$471$		0			0
$472$	38049.1	−	20329.3i	0.170789	−	0.0912513i
$473$	544342.			2.43304
$474$		0			0
$475$	−16923.6			−0.0750077
$476$	364417.			1.60837
$477$		0			0
$478$		−	9528.62i		−	0.0417037i
$479$	−76085.5			−0.331612			−0.165806	−	0.986158i	$-0.553023\pi$
$479$	−76085.5			−0.331612			−0.165806	+	0.986158i	$0.553023\pi$
$480$		0			0
$481$	327052.			1.41360
$482$		−	4959.73i		−	0.0213483i
$483$		0			0
$484$	−385161.			−1.64419
$485$		−	53026.7i		−	0.225430i
$486$		0			0
$487$	174246.			0.734690			0.367345	−	0.930085i	$-0.380267\pi$
$487$	174246.			0.734690			0.367345	+	0.930085i	$0.380267\pi$
$488$	−1427.18			−0.00599295
$489$		0			0
$490$			1177.18i			0.00490285i
$491$	−328811.			−1.36390			−0.681951	−	0.731398i	$-0.738868\pi$
$491$	−328811.			−1.36390			−0.681951	+	0.731398i	$0.738868\pi$
$492$		0			0
$493$	134006.			0.551354
$494$	−3856.48			−0.0158029
$495$		0			0
$496$			209938.i			0.853351i
$497$	−98558.2			−0.399006
$498$		0			0
$499$	−289611.			−1.16309			−0.581546	−	0.813513i	$-0.697552\pi$
$499$	−289611.			−1.16309			−0.581546	+	0.813513i	$0.697552\pi$
$500$	−264159.			−1.05664
$501$		0			0
$502$			17431.9i			0.0691730i
$503$		−	453397.i		−	1.79202i	−0.444036	−	0.896009i	$-0.646454\pi$
$503$		−	453397.i		−	1.79202i	0.444036	−	0.896009i	$-0.353546\pi$
$504$		0			0
$505$			82398.4i			0.323099i
$506$			65182.6i			0.254584i
$507$		0			0
$508$	454323.			1.76050
$509$		−	427793.i		−	1.65119i	−0.564260	−	0.825597i	$-0.690839\pi$
$509$		−	427793.i		−	1.65119i	0.564260	−	0.825597i	$-0.309161\pi$
$510$		0			0
$511$			9921.08i			0.0379942i
$512$		−	122727.i		−	0.468167i
$513$		0			0
$514$			41142.5i			0.155727i
$515$			254831.i			0.960811i
$516$		0			0
$517$	−283602.			−1.06103
$518$	−34110.6			−0.127125
$519$		0			0
$520$	−39324.2			−0.145430
$521$	142162.			0.523730			0.261865	−	0.965105i	$-0.415663\pi$
$521$	142162.			0.523730			0.261865	+	0.965105i	$0.415663\pi$
$522$		0			0
$523$	−197294.			−0.721289			−0.360645	−	0.932703i	$-0.617443\pi$
$523$	−197294.			−0.721289			−0.360645	+	0.932703i	$0.617443\pi$
$524$			314524.i			1.14549i
$525$		0			0
$526$			3620.64i			0.0130862i
$527$		−	410668.i		−	1.47866i
$528$		0			0
$529$	−440712.			−1.57487
$530$		−	7838.58i		−	0.0279052i
$531$		0			0
$532$	−42101.0			−0.148754
$533$			260828.i			0.918120i
$534$		0			0
$535$	−41361.3			−0.144506
$536$	−613.755			−0.00213632
$537$		0			0
$538$	2600.81			0.00898554
$539$		−	33169.2i		−	0.114171i
$540$		0			0
$541$			225666.i			0.771029i	0.922702	+	0.385515i	$0.125976\pi$
$541$			225666.i			0.771029i	−0.922702	+	0.385515i	$0.874024\pi$
$542$			53684.1i			0.182746i
$543$		0			0
$544$			143586.i			0.485193i
$545$			331688.i			1.11670i
$546$		0			0
$547$	648.964			0.00216893			0.00108447	−	0.999999i	$-0.499655\pi$
$547$	648.964			0.00216893			0.00108447	+	0.999999i	$0.499655\pi$
$548$	−303741.			−1.01145
$549$		0			0
$550$	−23116.8			−0.0764190
$551$	−15481.7			−0.0509935
$552$		0			0
$553$	309170.			1.01099
$554$		−	37409.6i		−	0.121889i
$555$		0			0
$556$	−320969.			−1.03828
$557$	−179490.			−0.578536			−0.289268	−	0.957248i	$-0.593412\pi$
$557$	−179490.			−0.578536			−0.289268	+	0.957248i	$0.593412\pi$
$558$		0			0
$559$	−486295.			−1.55624
$560$	−211569.			−0.674647
$561$		0			0
$562$		−	5270.56i		−	0.0166872i
$563$			56816.3i			0.179249i	0.995976	+	0.0896243i	$0.0285666\pi$
$563$			56816.3i			0.179249i	−0.995976	+	0.0896243i	$0.971433\pi$
$564$		0			0
$565$			276919.i			0.867472i
$566$	−57325.1			−0.178942
$567$		0			0
$568$		−	25848.0i		−	0.0801180i
$569$		−	361962.i		−	1.11799i	−0.829170	−	0.558996i	$-0.811187\pi$
$569$		−	361962.i		−	1.11799i	0.829170	−	0.558996i	$-0.188813\pi$
$570$		0			0
$571$			496112.i			1.52163i	0.648972	+	0.760813i	$0.275199\pi$
$571$			496112.i			1.52163i	−0.648972	+	0.760813i	$0.724801\pi$
$572$	551383.			1.68524
$573$		0			0
$574$		−	27203.6i		−	0.0825661i
$575$		−	255541.i		−	0.772904i
$576$		0			0
$577$	−9847.40			−0.0295781			−0.0147890	−	0.999891i	$-0.504708\pi$
$577$	−9847.40			−0.0295781			−0.0147890	+	0.999891i	$0.504708\pi$
$578$		−	59635.5i		−	0.178504i
$579$		0			0
$580$	−78557.4			−0.233524
$581$			321433.i			0.952222i
$582$		0			0
$583$			220867.i			0.649821i
$584$	−2601.92			−0.00762900
$585$		0			0
$586$			8321.06i			0.0242317i
$587$			550647.i			1.59807i	0.601282	+	0.799037i	$0.294657\pi$
$587$			550647.i			1.59807i	−0.601282	+	0.799037i	$0.705343\pi$
$588$		0			0
$589$			47444.3i			0.136758i
$590$	21502.9	−	11488.9i	0.0617723	−	0.0330045i
$591$		0			0
$592$			461471.i			1.31674i
$593$	455801.			1.29618			0.648090	−	0.761563i	$-0.275568\pi$
$593$	455801.			1.29618			0.648090	+	0.761563i	$0.275568\pi$
$594$		0			0
$595$	413859.			1.16901
$596$			135059.i			0.380217i
$597$		0			0
$598$		−	58231.6i		−	0.162838i
$599$	−556678.			−1.55149			−0.775747	−	0.631044i	$-0.782627\pi$
$599$	−556678.			−1.55149			−0.775747	+	0.631044i	$0.782627\pi$
$600$		0			0
$601$		−	157038.i		−	0.434766i	−0.976086	−	0.217383i	$-0.930248\pi$
$601$		−	157038.i		−	0.434766i	0.976086	−	0.217383i	$-0.0697520\pi$
$602$	50719.1			0.139952
$603$		0			0
$604$			684609.i			1.87659i
$605$	−437417.			−1.19505
$606$		0			0
$607$	−550937.			−1.49529			−0.747644	−	0.664100i	$-0.768815\pi$
$607$	−550937.			−1.49529			−0.747644	+	0.664100i	$0.768815\pi$
$608$		−	16588.4i		−	0.0448743i
$609$		0			0
$610$	−806.554			−0.00216757
$611$	253359.			0.678662
$612$		0			0
$613$			582881.i			1.55117i	0.631244	+	0.775585i	$0.282545\pi$
$613$			582881.i			1.55117i	−0.631244	+	0.775585i	$0.717455\pi$
$614$			34905.3i			0.0925880i
$615$		0			0
$616$	−115565.			−0.304554
$617$	498475.			1.30940			0.654701	−	0.755888i	$-0.272795\pi$
$617$	498475.			1.30940			0.654701	+	0.755888i	$0.272795\pi$
$618$		0			0
$619$	469002.			1.22403			0.612017	−	0.790844i	$-0.290358\pi$
$619$	469002.			1.22403			0.612017	+	0.790844i	$0.290358\pi$
$620$			240743.i			0.626282i
$621$		0			0
$622$		−	9120.40i		−	0.0235740i
$623$			430914.i			1.11023i
$624$		0			0
$625$	−111846.			−0.286327
$626$	−18945.3			−0.0483450
$627$		0			0
$628$		−	115283.i		−	0.292311i
$629$		−	902701.i		−	2.28162i
$630$		0			0
$631$	−55781.7			−0.140098			−0.0700492	−	0.997544i	$-0.522316\pi$
$631$	−55781.7			−0.140098			−0.0700492	+	0.997544i	$0.522316\pi$
$632$			81083.4i			0.203001i
$633$		0			0
$634$		−	49282.7i		−	0.122607i
$635$	515962.			1.27959
$636$		0			0
$637$			29632.1i			0.0730269i
$638$	−21147.1			−0.0519529
$639$		0			0
$640$		−	112048.i		−	0.273556i
$641$	77645.7			0.188974			0.0944869	−	0.995526i	$-0.469879\pi$
$641$	77645.7			0.188974			0.0944869	+	0.995526i	$0.469879\pi$
$642$		0			0
$643$	295599.			0.714960			0.357480	−	0.933921i	$-0.383636\pi$
$643$	295599.			0.714960			0.357480	+	0.933921i	$0.383636\pi$
$644$		−	635711.i		−	1.53281i
$645$		0			0
$646$			10644.3i			0.0255066i
$647$	392444.			0.937496			0.468748	−	0.883332i	$-0.344705\pi$
$647$	392444.			0.937496			0.468748	+	0.883332i	$0.344705\pi$
$648$		0			0
$649$	−605886.	+	323721.i	−1.43847	+	0.768565i
$650$	20651.6			0.0488796
$651$		0			0
$652$	−244319.			−0.574728
$653$	225024.			0.527720			0.263860	−	0.964561i	$-0.415004\pi$
$653$	225024.			0.527720			0.263860	+	0.964561i	$0.415004\pi$
$654$		0			0
$655$			357197.i			0.832577i
$656$	−368028.			−0.855210
$657$		0			0
$658$	−26424.6			−0.0610318
$659$		−	490926.i		−	1.13044i	−0.824942	−	0.565218i	$-0.808792\pi$
$659$		−	490926.i		−	1.13044i	0.824942	−	0.565218i	$-0.191208\pi$
$660$		0			0
$661$	−209902.			−0.480412			−0.240206	−	0.970722i	$-0.577215\pi$
$661$	−209902.			−0.480412			−0.240206	+	0.970722i	$0.577215\pi$
$662$		−	69049.6i		−	0.157560i
$663$		0			0
$664$	−84299.4			−0.191200
$665$	−47812.9			−0.108119
$666$		0			0
$667$		−	233768.i		−	0.525453i
$668$	−597973.			−1.34007
$669$		0			0
$670$	−346.855			−0.000772679
$671$	22726.2			0.0504757
$672$		0			0
$673$		−	535935.i		−	1.18327i	−0.806208	−	0.591633i	$-0.798484\pi$
$673$		−	535935.i		−	1.18327i	0.806208	−	0.591633i	$-0.201516\pi$
$674$	34298.4			0.0755012
$675$		0			0
$676$	−39933.1			−0.0873857
$677$	−589822.			−1.28690			−0.643448	−	0.765489i	$-0.722497\pi$
$677$	−589822.			−1.28690			−0.643448	+	0.765489i	$0.722497\pi$
$678$		0			0
$679$			139215.i			0.301959i
$680$			108539.i			0.234730i
$681$		0			0
$682$			64806.3i			0.139331i
$683$			569992.i			1.22188i	0.791678	+	0.610939i	$0.209208\pi$
$683$			569992.i			1.22188i	−0.791678	+	0.610939i	$0.790792\pi$
$684$		0			0
$685$	−344951.			−0.735150
$686$		−	47238.4i		−	0.100380i
$687$		0			0
$688$		−	686161.i		−	1.44960i
$689$		−	197314.i		−	0.415642i
$690$		0			0
$691$		−	151727.i		−	0.317765i	−0.987297	−	0.158883i	$-0.949211\pi$
$691$		−	151727.i		−	0.317765i	0.987297	−	0.158883i	$-0.0507891\pi$
$692$		−	191834.i		−	0.400602i
$693$		0			0
$694$	−30924.7			−0.0642076
$695$	−364516.			−0.754652
$696$		0			0
$697$	719914.			1.48189
$698$	−9487.65			−0.0194737
$699$		0			0
$700$	225453.			0.460108
$701$			288172.i			0.586429i	0.956047	+	0.293214i	$0.0947250\pi$
$701$			288172.i			0.586429i	−0.956047	+	0.293214i	$0.905275\pi$
$702$		0			0
$703$			104289.i			0.211021i
$704$			762776.i			1.53905i
$705$		0			0
$706$	23793.8			0.0477369
$707$		−	216328.i		−	0.432786i
$708$		0			0
$709$	−17367.9			−0.0345506			−0.0172753	−	0.999851i	$-0.505499\pi$
$709$	−17367.9			−0.0345506			−0.0172753	+	0.999851i	$0.505499\pi$
$710$		−	14607.7i		−	0.0289777i
$711$		0			0
$712$	−113012.			−0.222928
$713$	−716394.			−1.40920
$714$		0			0
$715$	626191.			1.22488
$716$			4572.46i			0.00891915i
$717$		0			0
$718$		−	93810.0i		−	0.181970i
$719$		−	524931.i		−	1.01542i	−0.861529	−	0.507708i	$-0.830493\pi$
$719$		−	524931.i		−	1.01542i	0.861529	−	0.507708i	$-0.169507\pi$
$720$		0			0
$721$		−	669029.i		−	1.28699i
$722$			49480.4i			0.0949203i
$723$		0			0
$724$	−202412.			−0.386152
$725$	82905.1			0.157727
$726$		0			0
$727$	416443.			0.787928			0.393964	−	0.919126i	$-0.371103\pi$
$727$	416443.			0.787928			0.393964	+	0.919126i	$0.371103\pi$
$728$	103241.			0.194800
$729$		0			0
$730$	−1470.44			−0.00275931
$731$			1.34223e6i			2.51184i
$732$		0			0
$733$	−22984.6			−0.0427788			−0.0213894	−	0.999771i	$-0.506809\pi$
$733$	−22984.6			−0.0427788			−0.0213894	+	0.999771i	$0.506809\pi$
$734$	−92404.5			−0.171515
$735$		0			0
$736$	250480.			0.462400
$737$	9773.32			0.0179931
$738$		0			0
$739$		−	339929.i		−	0.622442i	−0.950337	−	0.311221i	$-0.899262\pi$
$739$		−	339929.i		−	0.622442i	0.950337	−	0.311221i	$-0.100738\pi$
$740$			529184.i			0.966369i
$741$		0			0
$742$			20579.3i			0.0373786i
$743$	28658.0			0.0519121			0.0259560	−	0.999663i	$-0.491737\pi$
$743$	28658.0			0.0519121			0.0259560	+	0.999663i	$0.491737\pi$
$744$		0			0
$745$			153383.i			0.276353i
$746$			65519.0i			0.117731i
$747$		0			0
$748$		−	1.52188e6i		−	2.72005i
$749$	108589.			0.193564
$750$		0			0
$751$		−	398692.i		−	0.706899i	−0.935454	−	0.353449i	$-0.885009\pi$
$751$		−	398692.i		−	0.706899i	0.935454	−	0.353449i	$-0.114991\pi$
$752$			357489.i			0.632161i
$753$		0			0
$754$	18892.0			0.0332304
$755$			777492.i			1.36396i
$756$		0			0
$757$	55848.3			0.0974581			0.0487290	−	0.998812i	$-0.484483\pi$
$757$	55848.3			0.0974581			0.0487290	+	0.998812i	$0.484483\pi$
$758$		−	91190.6i		−	0.158713i
$759$		0			0
$760$		−	12539.5i		−	0.0217096i
$761$	31452.4			0.0543106			0.0271553	−	0.999631i	$-0.491355\pi$
$761$	31452.4			0.0543106			0.0271553	+	0.999631i	$0.491355\pi$
$762$		0			0
$763$		−	870808.i		−	1.49580i
$764$		−	777600.i		−	1.33220i
$765$		0			0
$766$			44508.2i			0.0758547i
$767$	541276.	−	289199.i	0.920085	−	0.491594i
$768$		0			0
$769$			1.10780e6i			1.87330i	0.350263	+	0.936651i	$0.386092\pi$
$769$			1.10780e6i			1.87330i	−0.350263	+	0.936651i	$0.613908\pi$
$770$	−65309.9			−0.110153
$771$		0			0
$772$	813842.			1.36554
$773$		−	392220.i		−	0.656403i	−0.944608	−	0.328201i	$-0.893558\pi$
$773$		−	392220.i		−	0.656403i	0.944608	−	0.328201i	$-0.106442\pi$
$774$		0			0
$775$		−	254066.i		−	0.423003i
$776$	−36510.8			−0.0606314
$777$		0			0
$778$		−	96823.6i		−	0.159964i
$779$	−83171.3			−0.137056
$780$		0			0
$781$			411599.i			0.674795i
$782$	−160726.			−0.262828
$783$		0			0
$784$	−41810.8			−0.0680232
$785$		−	130924.i		−	0.212461i
$786$		0			0
$787$	208960.			0.337376			0.168688	−	0.985670i	$-0.446047\pi$
$787$	208960.			0.337376			0.168688	+	0.985670i	$0.446047\pi$
$788$	477905.			0.769643
$789$		0			0
$790$			45823.2i			0.0734228i
$791$		−	727018.i		−	1.16196i
$792$		0			0
$793$	−20302.7			−0.0322855
$794$	−69605.7			−0.110409
$795$		0			0
$796$	1.07678e6			1.69942
$797$		−	12012.2i		−	0.0189107i	−0.999955	−	0.00945534i	$-0.996990\pi$
$797$		−	12012.2i		−	0.0189107i	0.999955	−	0.00945534i	$-0.00300977\pi$
$798$		0			0
$799$		−	699299.i		−	1.09539i
$800$			88831.9i			0.138800i
$801$		0			0
$802$	−30861.3			−0.0479806
$803$	41432.4			0.0642553
$804$		0			0
$805$		−	721960.i		−	1.11409i
$806$		−	57895.5i		−	0.0891199i
$807$		0			0
$808$	56734.3			0.0869007
$809$		−	864652.i		−	1.32113i	−0.750770	−	0.660563i	$-0.770317\pi$
$809$		−	864652.i		−	1.32113i	0.750770	−	0.660563i	$-0.229683\pi$
$810$		0			0
$811$			547154.i			0.831893i	0.909389	+	0.415947i	$0.136550\pi$
$811$			547154.i			0.831893i	−0.909389	+	0.415947i	$0.863450\pi$
$812$	206243.			0.312801
$813$		0			0
$814$			142453.i			0.214992i
$815$	−277467.			−0.417730
$816$		0			0
$817$		−	155067.i		−	0.232314i
$818$	−33451.9			−0.0499935
$819$		0			0
$820$	−422030.			−0.627647
$821$			1.15886e6i			1.71928i	0.510901	+	0.859639i	$0.329312\pi$
$821$			1.15886e6i			1.71928i	−0.510901	+	0.859639i	$0.670688\pi$
$822$		0			0
$823$			563170.i			0.831457i	0.909489	+	0.415728i	$0.136473\pi$
$823$			563170.i			0.831457i	−0.909489	+	0.415728i	$0.863527\pi$
$824$	175461.			0.258419
$825$		0			0
$826$	−56453.4	+	30162.7i	−0.0827428	+	0.0442089i
$827$	1.33840e6			1.95692			0.978461	−	0.206431i	$-0.0661850\pi$
$827$	1.33840e6			1.95692			0.978461	+	0.206431i	$0.0661850\pi$
$828$		0			0
$829$	−153786.			−0.223773			−0.111887	−	0.993721i	$-0.535689\pi$
$829$	−153786.			−0.223773			−0.111887	+	0.993721i	$0.535689\pi$
$830$	−47640.7			−0.0691547
$831$		0			0
$832$		−	681434.i		−	0.984413i
$833$	81787.8			0.117869
$834$		0			0
$835$	−679101.			−0.974006
$836$			175822.i			0.251571i
$837$		0			0
$838$	109809.			0.156369
$839$			234998.i			0.333841i	0.985970	+	0.166920i	$0.0533823\pi$
$839$			234998.i			0.333841i	−0.985970	+	0.166920i	$0.946618\pi$
$840$		0			0
$841$	−631440.			−0.892771
$842$	98919.0			0.139526
$843$		0			0
$844$		−	36515.0i		−	0.0512610i
$845$	−45351.0			−0.0635146
$846$		0			0
$847$	1.14839e6			1.60074
$848$	278410.			0.387163
$849$		0			0
$850$		−	57000.8i		−	0.0788938i
$851$	−1.57473e6			−2.17443
$852$		0			0
$853$	−544134.			−0.747839			−0.373919	−	0.927461i	$-0.621986\pi$
$853$	−544134.			−0.747839			−0.373919	+	0.927461i	$0.621986\pi$
$854$	2117.52			0.00290343
$855$		0			0
$856$			28478.8i			0.0388664i
$857$			301859.i			0.411000i	0.978657	+	0.205500i	$0.0658821\pi$
$857$			301859.i			0.411000i	−0.978657	+	0.205500i	$0.934118\pi$
$858$		0			0
$859$			1.32940e6i			1.80164i	0.434191	+	0.900821i	$0.357034\pi$
$859$			1.32940e6i			1.80164i	−0.434191	+	0.900821i	$0.642966\pi$
$860$		−	786844.i		−	1.06388i
$861$		0			0
$862$	76547.5			0.103019
$863$			707688.i			0.950212i	0.879929	+	0.475106i	$0.157590\pi$
$863$			707688.i			0.950212i	−0.879929	+	0.475106i	$0.842410\pi$
$864$		0			0
$865$		−	217861.i		−	0.291170i
$866$		−	95960.5i		−	0.127955i
$867$		0			0
$868$		−	632042.i		−	0.838893i
$869$		−	1.29116e6i		−	1.70978i
$870$		0			0
$871$	−8731.11			−0.0115089
$872$	228379.			0.300347
$873$		0			0
$874$	18568.6			0.0243084
$875$	787610.			1.02872
$876$		0			0
$877$	−423150.			−0.550168			−0.275084	−	0.961420i	$-0.588706\pi$
$877$	−423150.			−0.550168			−0.275084	+	0.961420i	$0.588706\pi$
$878$		−	28650.0i		−	0.0371652i
$879$		0			0
$880$			883555.i			1.14095i
$881$		−	1.11419e6i		−	1.43551i	−0.696296	−	0.717755i	$-0.745170\pi$
$881$		−	1.11419e6i		−	1.43551i	0.696296	−	0.717755i	$-0.254830\pi$
$882$		0			0
$883$	−25385.3			−0.0325583			−0.0162792	−	0.999867i	$-0.505182\pi$
$883$	−25385.3			−0.0325583			−0.0162792	+	0.999867i	$0.505182\pi$
$884$			1.35959e6i			1.73982i
$885$		0			0
$886$	−8184.52			−0.0104262
$887$			70599.4i			0.0897334i	0.998993	+	0.0448667i	$0.0142863\pi$
$887$			70599.4i			0.0897334i	−0.998993	+	0.0448667i	$0.985714\pi$
$888$		0			0
$889$	−1.35460e6			−1.71398
$890$	−63867.2			−0.0806303
$891$		0			0
$892$	447288.			0.562156
$893$			80789.7i			0.101310i
$894$		0			0
$895$			5192.82i			0.00648271i
$896$			294170.i			0.366423i
$897$		0			0
$898$			38731.5i			0.0480299i
$899$		−	232419.i		−	0.287576i
$900$		0			0
$901$	−544609.			−0.670866
$902$	−113608.			−0.139635
$903$		0			0
$904$	190669.			0.233315
$905$	−229873.			−0.280667
$906$		0			0
$907$	−575833.			−0.699975			−0.349987	−	0.936754i	$-0.613814\pi$
$907$	−575833.			−0.699975			−0.349987	+	0.936754i	$0.613814\pi$
$908$		−	1.00656e6i		−	1.22087i
$909$		0			0
$910$	58345.3			0.0704569
$911$	−805946.			−0.971112			−0.485556	−	0.874206i	$-0.661383\pi$
$911$	−805946.			−0.971112			−0.485556	+	0.874206i	$0.661383\pi$
$912$		0			0
$913$	1.34237e6			1.61039
$914$	110111.			0.131807
$915$		0			0
$916$			585302.i			0.697572i
$917$		−	937778.i		−	1.11522i
$918$		0			0
$919$		−	176120.i		−	0.208534i	−0.994549	−	0.104267i	$-0.966750\pi$
$919$		−	176120.i		−	0.208534i	0.994549	−	0.104267i	$-0.0332496\pi$
$920$	189342.			0.223703
$921$		0			0
$922$			1369.53i			0.00161106i
$923$		−	367706.i		−	0.431616i
$924$		0			0
$925$		−	558471.i		−	0.652705i
$926$	−115063.			−0.134188
$927$		0			0
$928$			81263.0i			0.0943620i
$929$			398648.i			0.461911i	0.972964	+	0.230956i	$0.0741852\pi$
$929$			398648.i			0.461911i	−0.972964	+	0.230956i	$0.925815\pi$
$930$		0			0
$931$	−9448.91			−0.0109014
$932$		−	842000.i		−	0.969350i
$933$		0			0
$934$	29972.4			0.0343580
$935$		−	1.72836e6i		−	1.97702i
$936$		0			0
$937$			115590.i			0.131656i	0.997831	+	0.0658282i	$0.0209689\pi$
$937$			115590.i			0.131656i	−0.997831	+	0.0658282i	$0.979031\pi$
$938$	910.629			0.00103499
$939$		0			0
$940$			409945.i			0.463948i
$941$		−	31726.3i		−	0.0358294i	−0.999840	−	0.0179147i	$-0.994297\pi$
$941$		−	31726.3i		−	0.0358294i	0.999840	−	0.0179147i	$-0.00570273\pi$
$942$		0			0
$943$		−	1.25586e6i		−	1.41227i
$944$	408060.	+	763740.i	0.457910	+	0.857041i
$945$		0			0
$946$		−	211813.i		−	0.236685i
$947$	−1.57588e6			−1.75721			−0.878605	−	0.477549i	$-0.841525\pi$
$947$	−1.57588e6			−1.75721			−0.878605	+	0.477549i	$0.841525\pi$
$948$		0			0
$949$	−37014.1			−0.0410994
$950$			6585.27i			0.00729670i
$951$		0			0
$952$		−	284957.i		−	0.314417i
$953$	−490152.			−0.539691			−0.269845	−	0.962904i	$-0.586973\pi$
$953$	−490152.			−0.539691			−0.269845	+	0.962904i	$0.586973\pi$
$954$		0			0
$955$		−	883099.i		−	0.968284i
$956$	388097.			0.424643
$957$		0			0
$958$			29606.2i			0.0322590i
$959$	905628.			0.984720
$960$		0			0
$961$	211263.			0.228758
$962$		−	127262.i		−	0.137514i
$963$		0			0
$964$	202008.			0.217377
$965$	924258.			0.992518
$966$		0			0
$967$			112759.i			0.120586i	0.998181	+	0.0602931i	$0.0192035\pi$
$967$			112759.i			0.120586i	−0.998181	+	0.0602931i	$0.980796\pi$
$968$			301178.i			0.321420i
$969$		0			0
$970$	−20633.6			−0.0219296
$971$	1.66125e6			1.76196			0.880978	−	0.473156i	$-0.156885\pi$
$971$	1.66125e6			1.76196			0.880978	+	0.473156i	$0.156885\pi$
$972$		0			0
$973$	956994.			1.01084
$974$		−	67802.0i		−	0.0714701i
$975$		0			0
$976$		−	28647.1i		−	0.0300733i
$977$		−	356838.i		−	0.373837i	−0.982375	−	0.186918i	$-0.940150\pi$
$977$		−	356838.i		−	0.373837i	0.982375	−	0.186918i	$-0.0598500\pi$
$978$		0			0
$979$	1.79958e6			1.87761
$980$	−47945.9			−0.0499228
$981$		0			0
$982$			127946.i			0.132679i
$983$			577236.i			0.597374i	0.954351	+	0.298687i	$0.0965486\pi$
$983$			577236.i			0.597374i	−0.954351	+	0.298687i	$0.903451\pi$
$984$		0			0
$985$	542744.			0.559400
$986$		−	52144.1i		−	0.0536354i
$987$		0			0
$988$		−	157073.i		−	0.160911i
$989$	2.34146e6			2.39384
$990$		0			0
$991$		−	929225.i		−	0.946180i	−0.881014	−	0.473090i	$-0.843139\pi$
$991$		−	929225.i		−	0.946180i	0.881014	−	0.473090i	$-0.156861\pi$
$992$	249034.			0.253067
$993$		0			0
$994$			38350.7i			0.0388151i
$995$	1.22287e6			1.23519
$996$		0			0
$997$	977120.			0.983009			0.491504	−	0.870875i	$-0.336447\pi$
$997$	977120.			0.983009			0.491504	+	0.870875i	$0.336447\pi$
$998$			112693.i			0.113145i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.20		40
3.2	odd	2		177.5.c.a.58.21	yes	40
59.58	odd	2	inner	531.5.c.d.235.21		40
177.176	even	2		177.5.c.a.58.20	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.20	✓	40	177.176	even	2
177.5.c.a.58.21	yes	40	3.2	odd	2
531.5.c.d.235.20		40	1.1	even	1	trivial
531.5.c.d.235.21		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)

\(f(q)\)	\(=\)	\(q-0.389117i q^{2} +15.8486 q^{4} +17.9988 q^{5} -47.2538 q^{7} -12.3928i q^{8} +O(q^{10})\)	\(q-0.389117i q^{2} +15.8486 q^{4} +17.9988 q^{5} -47.2538 q^{7} -12.3928i q^{8} -7.00365i q^{10} +197.341i q^{11} -176.297i q^{13} +18.3873i q^{14} +248.755 q^{16} -486.600 q^{17} +56.2166 q^{19} +285.256 q^{20} +76.7889 q^{22} +848.854i q^{23} -301.043 q^{25} -68.6003 q^{26} -748.906 q^{28} -275.393 q^{29} +843.954i q^{31} -295.080i q^{32} +189.344i q^{34} -850.512 q^{35} +1855.12i q^{37} -21.8749i q^{38} -223.056i q^{40} -1479.48 q^{41} -2758.38i q^{43} +3127.58i q^{44} +330.304 q^{46} +1437.11i q^{47} -168.080 q^{49} +117.141i q^{50} -2794.06i q^{52} +1119.21 q^{53} +3551.91i q^{55} +585.608i q^{56} +107.160i q^{58} +(1640.41 + 3070.25i) q^{59} -115.162i q^{61} +328.397 q^{62} +3865.26 q^{64} -3173.14i q^{65} -49.5250i q^{67} -7711.92 q^{68} +330.949i q^{70} +2085.72 q^{71} -209.953i q^{73} +721.859 q^{74} +890.954 q^{76} -9325.12i q^{77} -6542.76 q^{79} +4477.30 q^{80} +575.691i q^{82} -6802.27i q^{83} -8758.22 q^{85} -1073.33 q^{86} +2445.62 q^{88} -9119.14i q^{89} +8330.70i q^{91} +13453.1i q^{92} +559.206 q^{94} +1011.83 q^{95} -2946.12i q^{97} +65.4030i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more