LMFDB - Embedded newform 531.5.c.d.235.9

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			235.9
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.85825i q^{2} -7.60257 q^{4} +12.5087 q^{5} +50.4755 q^{7} -40.7968i q^{8} +O(q^{10})$	\(q-4.85825i q^{2} -7.60257 q^{4} +12.5087 q^{5} +50.4755 q^{7} -40.7968i q^{8} -60.7705i q^{10} -191.226i q^{11} -0.640162i q^{13} -245.222i q^{14} -319.842 q^{16} +549.428 q^{17} +524.497 q^{19} -95.0984 q^{20} -929.025 q^{22} +543.113i q^{23} -468.532 q^{25} -3.11006 q^{26} -383.743 q^{28} +296.169 q^{29} +1078.43i q^{31} +901.123i q^{32} -2669.26i q^{34} +631.384 q^{35} -1168.43i q^{37} -2548.14i q^{38} -510.316i q^{40} +1128.78 q^{41} +1522.74i q^{43} +1453.81i q^{44} +2638.58 q^{46} -103.190i q^{47} +146.777 q^{49} +2276.24i q^{50} +4.86687i q^{52} +2811.88 q^{53} -2392.00i q^{55} -2059.24i q^{56} -1438.86i q^{58} +(2562.55 - 2355.99i) q^{59} -4909.65i q^{61} +5239.30 q^{62} -739.596 q^{64} -8.00760i q^{65} -5854.39i q^{67} -4177.06 q^{68} -3067.42i q^{70} -5828.87 q^{71} +6751.02i q^{73} -5676.53 q^{74} -3987.52 q^{76} -9652.25i q^{77} -9333.98 q^{79} -4000.81 q^{80} -5483.90i q^{82} -1839.42i q^{83} +6872.64 q^{85} +7397.84 q^{86} -7801.43 q^{88} -5734.95i q^{89} -32.3125i q^{91} -4129.05i q^{92} -501.323 q^{94} +6560.79 q^{95} +15272.6i q^{97} -713.079i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	4.85825i		−	1.21456i	−0.794487	−	0.607281i	$-0.792260\pi$
$2$		−	4.85825i		−	1.21456i	0.794487	−	0.607281i	$-0.207740\pi$
$3$		0			0
$3$		0			0
$4$	−7.60257			−0.475160
$5$	12.5087			0.500349			0.250174	−	0.968201i	$-0.419512\pi$
$5$	12.5087			0.500349			0.250174	+	0.968201i	$0.419512\pi$
$6$		0			0
$7$	50.4755			1.03011			0.515056	−	0.857156i	$-0.327771\pi$
$7$	50.4755			1.03011			0.515056	+	0.857156i	$0.327771\pi$
$8$		−	40.7968i		−	0.637450i
$9$		0			0
$10$		−	60.7705i		−	0.607705i
$11$		−	191.226i		−	1.58038i	−0.612860	−	0.790192i	$-0.709981\pi$
$11$		−	191.226i		−	1.58038i	0.612860	−	0.790192i	$-0.290019\pi$
$12$		0			0
$13$		−	0.640162i		−	0.00378794i	−0.999998	−	0.00189397i	$-0.999397\pi$
$13$		−	0.640162i		−	0.00378794i	0.999998	−	0.00189397i	$-0.000602869\pi$
$14$		−	245.222i		−	1.25114i
$15$		0			0
$16$	−319.842			−1.24938
$17$	549.428			1.90113			0.950567	−	0.310520i	$-0.100503\pi$
$17$	549.428			1.90113			0.950567	+	0.310520i	$0.100503\pi$
$18$		0			0
$19$	524.497			1.45290			0.726450	−	0.687219i	$-0.241169\pi$
$19$	524.497			1.45290			0.726450	+	0.687219i	$0.241169\pi$
$20$	−95.0984			−0.237746
$21$		0			0
$22$	−929.025			−1.91947
$23$			543.113i			1.02668i	0.858186	+	0.513339i	$0.171592\pi$
$23$			543.113i			1.02668i	−0.858186	+	0.513339i	$0.828408\pi$
$24$		0			0
$25$	−468.532			−0.749651
$26$	−3.11006			−0.00460068
$27$		0			0
$28$	−383.743			−0.489469
$29$	296.169			0.352163			0.176081	−	0.984376i	$-0.443658\pi$
$29$	296.169			0.352163			0.176081	+	0.984376i	$0.443658\pi$
$30$		0			0
$31$			1078.43i			1.12220i	0.827748	+	0.561100i	$0.189622\pi$
$31$			1078.43i			1.12220i	−0.827748	+	0.561100i	$0.810378\pi$
$32$			901.123i			0.880003i
$33$		0			0
$34$		−	2669.26i		−	2.30904i
$35$	631.384			0.515416
$36$		0			0
$37$		−	1168.43i		−	0.853493i	−0.904371	−	0.426747i	$-0.859660\pi$
$37$		−	1168.43i		−	0.853493i	0.904371	−	0.426747i	$-0.140340\pi$
$38$		−	2548.14i		−	1.76464i
$39$		0			0
$40$		−	510.316i		−	0.318947i
$41$	1128.78			0.671494			0.335747	−	0.941952i	$-0.391011\pi$
$41$	1128.78			0.671494			0.335747	+	0.941952i	$0.391011\pi$
$42$		0			0
$43$			1522.74i			0.823547i	0.911286	+	0.411773i	$0.135090\pi$
$43$			1522.74i			0.823547i	−0.911286	+	0.411773i	$0.864910\pi$
$44$			1453.81i			0.750936i
$45$		0			0
$46$	2638.58			1.24696
$47$		−	103.190i		−	0.0467135i	−0.999727	−	0.0233567i	$-0.992565\pi$
$47$		−	103.190i		−	0.0467135i	0.999727	−	0.0233567i	$-0.00743536\pi$
$48$		0			0
$49$	146.777			0.0611316
$50$			2276.24i			0.910498i
$51$		0			0
$52$			4.86687i			0.00179988i
$53$	2811.88			1.00103			0.500513	−	0.865729i	$-0.333145\pi$
$53$	2811.88			1.00103			0.500513	+	0.865729i	$0.333145\pi$
$54$		0			0
$55$		−	2392.00i		−	0.790743i
$56$		−	2059.24i		−	0.656645i
$57$		0			0
$58$		−	1438.86i		−	0.427723i
$59$	2562.55	−	2355.99i	0.736155	−	0.676813i
$59$	2562.55	−	2355.99i	0.736155	−	0.676813i
$60$		0			0
$61$		−	4909.65i		−	1.31944i	−0.751510	−	0.659722i	$-0.770674\pi$
$61$		−	4909.65i		−	1.31944i	0.751510	−	0.659722i	$-0.229326\pi$
$62$	5239.30			1.36298
$63$		0			0
$64$	−739.596			−0.180565
$65$		−	8.00760i		−	0.00189529i
$66$		0			0
$67$		−	5854.39i		−	1.30416i	−0.758149	−	0.652082i	$-0.773896\pi$
$67$		−	5854.39i		−	1.30416i	0.758149	−	0.652082i	$-0.226104\pi$
$68$	−4177.06			−0.903343
$69$		0			0
$70$		−	3067.42i		−	0.626004i
$71$	−5828.87			−1.15629			−0.578146	−	0.815933i	$-0.696224\pi$
$71$	−5828.87			−1.15629			−0.578146	+	0.815933i	$0.696224\pi$
$72$		0			0
$73$			6751.02i			1.26684i	0.773806	+	0.633422i	$0.218350\pi$
$73$			6751.02i			1.26684i	−0.773806	+	0.633422i	$0.781650\pi$
$74$	−5676.53			−1.03662
$75$		0			0
$76$	−3987.52			−0.690361
$77$		−	9652.25i		−	1.62797i
$78$		0			0
$79$	−9333.98			−1.49559			−0.747795	−	0.663929i	$-0.768888\pi$
$79$	−9333.98			−1.49559			−0.747795	+	0.663929i	$0.768888\pi$
$80$	−4000.81			−0.625127
$81$		0			0
$82$		−	5483.90i		−	0.815571i
$83$		−	1839.42i		−	0.267008i	−0.991048	−	0.133504i	$-0.957377\pi$
$83$		−	1839.42i		−	0.267008i	0.991048	−	0.133504i	$-0.0426229\pi$
$84$		0			0
$85$	6872.64			0.951230
$86$	7397.84			1.00025
$87$		0			0
$88$	−7801.43			−1.00742
$89$		−	5734.95i		−	0.724018i	−0.932175	−	0.362009i	$-0.882091\pi$
$89$		−	5734.95i		−	0.724018i	0.932175	−	0.362009i	$-0.117909\pi$
$90$		0			0
$91$		−	32.3125i		−	0.00390200i
$92$		−	4129.05i		−	0.487837i
$93$		0			0
$94$	−501.323			−0.0567364
$95$	6560.79			0.726957
$96$		0			0
$97$			15272.6i			1.62319i	0.584223	+	0.811593i	$0.301399\pi$
$97$			15272.6i			1.62319i	−0.584223	+	0.811593i	$0.698601\pi$
$98$		−	713.079i		−	0.0742481i
$99$		0			0
$100$	3562.04			0.356204
$101$		−	13635.2i		−	1.33665i	−0.743868	−	0.668327i	$-0.767011\pi$
$101$		−	13635.2i		−	1.33665i	0.743868	−	0.668327i	$-0.232989\pi$
$102$		0			0
$103$			4773.40i			0.449939i	0.974366	+	0.224969i	$0.0722282\pi$
$103$			4773.40i			0.449939i	−0.974366	+	0.224969i	$0.927772\pi$
$104$	−26.1166			−0.00241462
$105$		0			0
$106$		−	13660.8i		−	1.21581i
$107$	−10911.5			−0.953051			−0.476525	−	0.879161i	$-0.658104\pi$
$107$	−10911.5			−0.953051			−0.476525	+	0.879161i	$0.658104\pi$
$108$		0			0
$109$			1327.72i			0.111752i	0.998438	+	0.0558758i	$0.0177951\pi$
$109$			1327.72i			0.111752i	−0.998438	+	0.0558758i	$0.982205\pi$
$110$	−11620.9			−0.960406
$111$		0			0
$112$	−16144.2			−1.28700
$113$			7206.57i			0.564380i	0.959359	+	0.282190i	$0.0910608\pi$
$113$			7206.57i			0.564380i	−0.959359	+	0.282190i	$0.908939\pi$
$114$		0			0
$115$			6793.65i			0.513697i
$116$	−2251.64			−0.167334
$117$		0			0
$118$	−11446.0	−	12449.5i	−0.822031	−	0.894105i
$119$	27732.6			1.95838
$120$		0			0
$121$	−21926.5			−1.49761
$122$	−23852.3			−1.60255
$123$		0			0
$124$		−	8198.86i		−	0.533225i
$125$	−13678.7			−0.875436
$126$		0			0
$127$	30152.5			1.86946			0.934729	−	0.355361i	$-0.115642\pi$
$127$	30152.5			1.86946			0.934729	+	0.355361i	$0.115642\pi$
$128$			18011.1i			1.09931i
$129$		0			0
$130$	−38.9029			−0.00230195
$131$		−	30591.4i		−	1.78261i	−0.453404	−	0.891305i	$-0.649790\pi$
$131$		−	30591.4i		−	1.78261i	0.453404	−	0.891305i	$-0.350210\pi$
$132$		0			0
$133$	26474.3			1.49665
$134$	−28442.1			−1.58399
$135$		0			0
$136$		−	22414.9i		−	1.21188i
$137$	−28251.4			−1.50521			−0.752607	−	0.658470i	$-0.771204\pi$
$137$	−28251.4			−1.50521			−0.752607	+	0.658470i	$0.771204\pi$
$138$		0			0
$139$	6125.81			0.317054			0.158527	−	0.987355i	$-0.449325\pi$
$139$	6125.81			0.317054			0.158527	+	0.987355i	$0.449325\pi$
$140$	−4800.14			−0.244905
$141$		0			0
$142$			28318.1i			1.40439i
$143$	−122.416			−0.00598640
$144$		0			0
$145$	3704.69			0.176204
$146$	32798.1			1.53866
$147$		0			0
$148$			8883.08i			0.405546i
$149$			35941.9i			1.61893i	0.587166	+	0.809467i	$0.300244\pi$
$149$			35941.9i			1.61893i	−0.587166	+	0.809467i	$0.699756\pi$
$150$		0			0
$151$			3815.72i			0.167349i	0.996493	+	0.0836745i	$0.0266656\pi$
$151$			3815.72i			0.167349i	−0.996493	+	0.0836745i	$0.973334\pi$
$152$		−	21397.8i		−	0.926152i
$153$		0			0
$154$	−46893.0			−1.97727
$155$			13489.8i			0.561491i
$156$		0			0
$157$		−	14168.1i		−	0.574795i	−0.957811	−	0.287398i	$-0.907210\pi$
$157$		−	14168.1i		−	0.574795i	0.957811	−	0.287398i	$-0.0927901\pi$
$158$			45346.8i			1.81649i
$159$		0			0
$160$			11271.9i			0.440308i
$161$			27413.9i			1.05759i
$162$		0			0
$163$	−5436.37			−0.204613			−0.102307	−	0.994753i	$-0.532622\pi$
$163$	−5436.37			−0.204613			−0.102307	+	0.994753i	$0.532622\pi$
$164$	−8581.64			−0.319068
$165$		0			0
$166$	−8936.35			−0.324298
$167$	−10745.5			−0.385296			−0.192648	−	0.981268i	$-0.561707\pi$
$167$	−10745.5			−0.385296			−0.192648	+	0.981268i	$0.561707\pi$
$168$		0			0
$169$	28560.6			0.999986
$170$		−	33389.0i		−	1.15533i
$171$		0			0
$172$		−	11576.7i		−	0.391317i
$173$			27249.2i			0.910463i	0.890373	+	0.455231i	$0.150443\pi$
$173$			27249.2i			0.910463i	−0.890373	+	0.455231i	$0.849557\pi$
$174$		0			0
$175$	−23649.4			−0.772225
$176$			61162.3i			1.97450i
$177$		0			0
$178$	−27861.8			−0.879365
$179$		−	60313.1i		−	1.88237i	−0.337890	−	0.941186i	$-0.609713\pi$
$179$		−	60313.1i		−	1.88237i	0.337890	−	0.941186i	$-0.390287\pi$
$180$		0			0
$181$	−27872.8			−0.850793			−0.425396	−	0.905007i	$-0.639865\pi$
$181$	−27872.8			−0.850793			−0.425396	+	0.905007i	$0.639865\pi$
$182$	−156.982			−0.00473922
$183$		0			0
$184$	22157.3			0.654456
$185$		−	14615.6i		−	0.427044i
$186$		0			0
$187$		−	105065.i		−	3.00452i
$188$			784.509i			0.0221964i
$189$		0			0
$190$		−	31873.9i		−	0.882934i
$191$			32832.5i			0.899989i	0.893031	+	0.449995i	$0.148574\pi$
$191$			32832.5i			0.899989i	−0.893031	+	0.449995i	$0.851426\pi$
$192$		0			0
$193$	−51056.4			−1.37068			−0.685339	−	0.728224i	$-0.740346\pi$
$193$	−51056.4			−1.37068			−0.685339	+	0.728224i	$0.740346\pi$
$194$	74197.9			1.97146
$195$		0			0
$196$	−1115.88			−0.0290473
$197$	−52427.0			−1.35090			−0.675449	−	0.737407i	$-0.736050\pi$
$197$	−52427.0			−1.35090			−0.675449	+	0.737407i	$0.736050\pi$
$198$		0			0
$199$	−21283.2			−0.537440			−0.268720	−	0.963218i	$-0.586601\pi$
$199$	−21283.2			−0.537440			−0.268720	+	0.963218i	$0.586601\pi$
$200$			19114.6i			0.477865i
$201$		0			0
$202$	−66243.2			−1.62345
$203$	14949.3			0.362767
$204$		0			0
$205$	14119.6			0.335981
$206$	23190.3			0.546478
$207$		0			0
$208$			204.751i			0.00473259i
$209$		−	100298.i		−	2.29614i
$210$		0			0
$211$			67797.8i			1.52283i	0.648267	+	0.761413i	$0.275494\pi$
$211$			67797.8i			1.52283i	−0.648267	+	0.761413i	$0.724506\pi$
$212$	−21377.5			−0.475648
$213$		0			0
$214$			53010.7i			1.15754i
$215$			19047.5i			0.412061i
$216$		0			0
$217$			54434.5i			1.15599i
$218$	6450.40			0.135729
$219$		0			0
$220$			18185.3i			0.375730i
$221$		−	351.722i		−	0.00720138i
$222$		0			0
$223$	45072.8			0.906368			0.453184	−	0.891417i	$-0.350288\pi$
$223$	45072.8			0.906368			0.453184	+	0.891417i	$0.350288\pi$
$224$			45484.6i			0.906502i
$225$		0			0
$226$	35011.3			0.685474
$227$		−	13186.6i		−	0.255906i	−0.991780	−	0.127953i	$-0.959159\pi$
$227$		−	13186.6i		−	0.255906i	0.991780	−	0.127953i	$-0.0408407\pi$
$228$		0			0
$229$			5098.55i			0.0972244i	0.998818	+	0.0486122i	$0.0154799\pi$
$229$			5098.55i			0.0972244i	−0.998818	+	0.0486122i	$0.984520\pi$
$230$	33005.2			0.623917
$231$		0			0
$232$		−	12082.7i		−	0.224486i
$233$		−	45461.0i		−	0.837390i	−0.908127	−	0.418695i	$-0.862488\pi$
$233$		−	45461.0i		−	0.837390i	0.908127	−	0.418695i	$-0.137512\pi$
$234$		0			0
$235$		−	1290.78i		−	0.0233730i
$236$	−19482.0	+	17911.5i	−0.349792	+	0.321595i
$237$		0			0
$238$		−	134732.i		−	2.37858i
$239$	80213.6			1.40427			0.702137	−	0.712041i	$-0.252229\pi$
$239$	80213.6			1.40427			0.702137	+	0.712041i	$0.252229\pi$
$240$		0			0
$241$	27279.2			0.469676			0.234838	−	0.972035i	$-0.424544\pi$
$241$	27279.2			0.469676			0.234838	+	0.972035i	$0.424544\pi$
$242$			106525.i			1.81894i
$243$		0			0
$244$			37325.9i			0.626947i
$245$	1835.99			0.0305871
$246$		0			0
$247$		−	335.763i		−	0.00550350i
$248$	43996.7			0.715346
$249$		0			0
$250$			66454.4i			1.06327i
$251$	87290.9			1.38555			0.692774	−	0.721155i	$-0.256388\pi$
$251$	87290.9			1.38555			0.692774	+	0.721155i	$0.256388\pi$
$252$		0			0
$253$	103858.			1.62255
$254$		−	146488.i		−	2.27057i
$255$		0			0
$256$	75668.9			1.15462
$257$	1669.20			0.0252721			0.0126361	−	0.999920i	$-0.495978\pi$
$257$	1669.20			0.0252721			0.0126361	+	0.999920i	$0.495978\pi$
$258$		0			0
$259$		−	58977.2i		−	0.879194i
$260$			60.8783i			0.000900567i
$261$		0			0
$262$	−148620.			−2.16509
$263$	−99916.5			−1.44453			−0.722264	−	0.691618i	$-0.756898\pi$
$263$	−99916.5			−1.44453			−0.722264	+	0.691618i	$0.756898\pi$
$264$		0			0
$265$	35173.1			0.500862
$266$		−	128618.i		−	1.81777i
$267$		0			0
$268$			44508.4i			0.619687i
$269$			24594.1i			0.339881i	0.985454	+	0.169940i	$0.0543575\pi$
$269$			24594.1i			0.339881i	−0.985454	+	0.169940i	$0.945642\pi$
$270$		0			0
$271$	12306.3			0.167567			0.0837836	−	0.996484i	$-0.473300\pi$
$271$	12306.3			0.167567			0.0837836	+	0.996484i	$0.473300\pi$
$272$	−175730.			−2.37524
$273$		0			0
$274$			137252.i			1.82818i
$275$			89595.7i			1.18474i
$276$		0			0
$277$	30853.9			0.402115			0.201058	−	0.979579i	$-0.435562\pi$
$277$	30853.9			0.402115			0.201058	+	0.979579i	$0.435562\pi$
$278$		−	29760.7i		−	0.385082i
$279$		0			0
$280$		−	25758.5i		−	0.328552i
$281$	−6311.40			−0.0799306			−0.0399653	−	0.999201i	$-0.512725\pi$
$281$	−6311.40			−0.0799306			−0.0399653	+	0.999201i	$0.512725\pi$
$282$		0			0
$283$		−	37474.9i		−	0.467916i	−0.972247	−	0.233958i	$-0.924832\pi$
$283$		−	37474.9i		−	0.467916i	0.972247	−	0.233958i	$-0.0751678\pi$
$284$	44314.3			0.549424
$285$		0			0
$286$			594.726i			0.00727085i
$287$	56975.9			0.691715
$288$		0			0
$289$	218350.			2.61431
$290$		−	17998.3i		−	0.214011i
$291$		0			0
$292$		−	51325.0i		−	0.601954i
$293$	−13284.3			−0.154740			−0.0773700	−	0.997002i	$-0.524652\pi$
$293$	−13284.3			−0.154740			−0.0773700	+	0.997002i	$0.524652\pi$
$294$		0			0
$295$	32054.3	−	29470.4i	0.368334	−	0.338643i
$296$	−47668.3			−0.544059
$297$		0			0
$298$	174615.			1.96629
$299$	347.680			0.00388899
$300$		0			0
$301$			76861.0i			0.848346i
$302$	18537.7			0.203256
$303$		0			0
$304$	−167756.			−1.81523
$305$		−	61413.4i		−	0.660182i
$306$		0			0
$307$	27275.0			0.289393			0.144697	−	0.989476i	$-0.453779\pi$
$307$	27275.0			0.289393			0.144697	+	0.989476i	$0.453779\pi$
$308$			73381.9i			0.773548i
$309$		0			0
$310$	65536.9			0.681966
$311$	−74737.1			−0.772708			−0.386354	−	0.922351i	$-0.626266\pi$
$311$	−74737.1			−0.772708			−0.386354	+	0.922351i	$0.626266\pi$
$312$		0			0
$313$			147610.i			1.50670i	0.657618	+	0.753352i	$0.271564\pi$
$313$			147610.i			1.50670i	−0.657618	+	0.753352i	$0.728436\pi$
$314$	−68832.3			−0.698124
$315$		0			0
$316$	70962.2			0.710645
$317$	75632.0			0.752640			0.376320	−	0.926490i	$-0.377189\pi$
$317$	75632.0			0.752640			0.376320	+	0.926490i	$0.377189\pi$
$318$		0			0
$319$		−	56635.3i		−	0.556552i
$320$	−9251.40			−0.0903457
$321$		0			0
$322$	133183.			1.28451
$323$	288173.			2.76216
$324$		0			0
$325$			299.936i			0.00283963i
$326$			26411.2i			0.248515i
$327$		0			0
$328$		−	46050.7i		−	0.428044i
$329$		−	5208.57i		−	0.0481201i
$330$		0			0
$331$	43603.6			0.397985			0.198992	−	0.980001i	$-0.436233\pi$
$331$	43603.6			0.397985			0.198992	+	0.980001i	$0.436233\pi$
$332$			13984.3i			0.126872i
$333$		0			0
$334$			52204.3i			0.467965i
$335$		−	73230.9i		−	0.652537i
$336$		0			0
$337$			111943.i			0.985682i	0.870119	+	0.492841i	$0.164042\pi$
$337$			111943.i			0.985682i	−0.870119	+	0.492841i	$0.835958\pi$
$338$		−	138754.i		−	1.21454i
$339$		0			0
$340$	−52249.7			−0.451987
$341$	206225.			1.77351
$342$		0			0
$343$	−113783.			−0.967140
$344$	62122.9			0.524970
$345$		0			0
$346$	132384.			1.10581
$347$			43440.9i			0.360778i	0.983595	+	0.180389i	$0.0577357\pi$
$347$			43440.9i			0.360778i	−0.983595	+	0.180389i	$0.942264\pi$
$348$		0			0
$349$		−	63411.1i		−	0.520612i	−0.965526	−	0.260306i	$-0.916177\pi$
$349$		−	63411.1i		−	0.520612i	0.965526	−	0.260306i	$-0.0838235\pi$
$350$			114895.i			0.937915i
$351$		0			0
$352$	172318.			1.39074
$353$		−	152287.i		−	1.22212i	−0.791585	−	0.611059i	$-0.790744\pi$
$353$		−	152287.i		−	1.22212i	0.791585	−	0.611059i	$-0.209256\pi$
$354$		0			0
$355$	−72911.7			−0.578549
$356$			43600.3i			0.344025i
$357$		0			0
$358$	−293016.			−2.28626
$359$	−38536.4			−0.299008			−0.149504	−	0.988761i	$-0.547768\pi$
$359$	−38536.4			−0.299008			−0.149504	+	0.988761i	$0.547768\pi$
$360$		0			0
$361$	144776.			1.11092
$362$			135413.i			1.03334i
$363$		0			0
$364$			245.658i			0.00185408i
$365$			84446.6i			0.633864i
$366$		0			0
$367$			127968.i			0.950102i	0.879958	+	0.475051i	$0.157570\pi$
$367$			127968.i			0.950102i	−0.879958	+	0.475051i	$0.842430\pi$
$368$		−	173710.i		−	1.28271i
$369$		0			0
$370$	−71006.1			−0.518672
$371$	141931.			1.03117
$372$		0			0
$373$	−181481.			−1.30441			−0.652203	−	0.758044i	$-0.726155\pi$
$373$	−181481.			−1.30441			−0.652203	+	0.758044i	$0.726155\pi$
$374$	−510432.			−3.64918
$375$		0			0
$376$	−4209.83			−0.0297775
$377$		−	189.596i		−	0.00133397i
$378$		0			0
$379$	−52087.4			−0.362622			−0.181311	−	0.983426i	$-0.558034\pi$
$379$	−52087.4			−0.362622			−0.181311	+	0.983426i	$0.558034\pi$
$380$	−49878.8			−0.345421
$381$		0			0
$382$	159508.			1.09309
$383$	129457.			0.882527			0.441263	−	0.897378i	$-0.354530\pi$
$383$	129457.			0.882527			0.441263	+	0.897378i	$0.354530\pi$
$384$		0			0
$385$		−	120737.i		−	0.814554i
$386$			248045.i			1.66477i
$387$		0			0
$388$		−	116111.i		−	0.771274i
$389$	239714.			1.58414			0.792070	−	0.610430i	$-0.209003\pi$
$389$	239714.			1.58414			0.792070	+	0.610430i	$0.209003\pi$
$390$		0			0
$391$			298401.i			1.95185i
$392$		−	5988.03i		−	0.0389684i
$393$		0			0
$394$			254703.i			1.64075i
$395$	−116756.			−0.748317
$396$		0			0
$397$		−	63125.7i		−	0.400521i	−0.979743	−	0.200260i	$-0.935821\pi$
$397$		−	63125.7i		−	0.400521i	0.979743	−	0.200260i	$-0.0641788\pi$
$398$			103399.i			0.652755i
$399$		0			0
$400$	149856.			0.936601
$401$		−	5633.87i		−	0.0350363i	−0.999847	−	0.0175181i	$-0.994424\pi$
$401$		−	5633.87i		−	0.0350363i	0.999847	−	0.0175181i	$-0.00557648\pi$
$402$		0			0
$403$	690.372			0.00425082
$404$			103663.i			0.635125i
$405$		0			0
$406$		−	72627.2i		−	0.440603i
$407$	−223435.			−1.34885
$408$		0			0
$409$			208211.i			1.24468i	0.782747	+	0.622340i	$0.213818\pi$
$409$			208211.i			1.24468i	−0.782747	+	0.622340i	$0.786182\pi$
$410$		−	68596.6i		−	0.408070i
$411$		0			0
$412$		−	36290.1i		−	0.213793i
$413$	129346.	−	118920.i	0.758322	−	0.697194i
$414$		0			0
$415$		−	23008.8i		−	0.133597i
$416$	576.864			0.00333340
$417$		0			0
$418$	−487271.			−2.78880
$419$		−	29129.4i		−	0.165922i	−0.996553	−	0.0829610i	$-0.973562\pi$
$419$		−	29129.4i		−	0.165922i	0.996553	−	0.0829610i	$-0.0264377\pi$
$420$		0			0
$421$			297451.i			1.67823i	0.543956	+	0.839113i	$0.316926\pi$
$421$			297451.i			1.67823i	−0.543956	+	0.839113i	$0.683074\pi$
$422$	329378.			1.84957
$423$		0			0
$424$		−	114716.i		−	0.638105i
$425$	−257424.			−1.42519
$426$		0			0
$427$		−	247817.i		−	1.35917i
$428$	82955.2			0.452852
$429$		0			0
$430$	92537.5			0.500473
$431$			340194.i			1.83135i	0.401917	+	0.915676i	$0.368344\pi$
$431$			340194.i			1.83135i	−0.401917	+	0.915676i	$0.631656\pi$
$432$		0			0
$433$	−19705.6			−0.105103			−0.0525514	−	0.998618i	$-0.516735\pi$
$433$	−19705.6			−0.105103			−0.0525514	+	0.998618i	$0.516735\pi$
$434$	264456.			1.40402
$435$		0			0
$436$		−	10094.1i		−	0.0530999i
$437$			284861.i			1.49166i
$438$		0			0
$439$	29208.1			0.151557			0.0757783	−	0.997125i	$-0.475856\pi$
$439$	29208.1			0.151557			0.0757783	+	0.997125i	$0.475856\pi$
$440$	−97585.9			−0.504059
$441$		0			0
$442$	−1708.75			−0.00874652
$443$			136923.i			0.697698i	0.937179	+	0.348849i	$0.113427\pi$
$443$			136923.i			0.697698i	−0.937179	+	0.348849i	$0.886573\pi$
$444$		0			0
$445$		−	71736.9i		−	0.362262i
$446$		−	218975.i		−	1.10084i
$447$		0			0
$448$	−37331.5			−0.186003
$449$	−324943.			−1.61181			−0.805906	−	0.592043i	$-0.798322\pi$
$449$	−324943.			−1.61181			−0.805906	+	0.592043i	$0.798322\pi$
$450$		0			0
$451$		−	215853.i		−	1.06122i
$452$		−	54788.4i		−	0.268171i
$453$		0			0
$454$	−64063.8			−0.310814
$455$		−	404.188i		−	0.00195236i
$456$		0			0
$457$		−	142683.i		−	0.683189i	−0.939847	−	0.341595i	$-0.889033\pi$
$457$		−	142683.i		−	0.683189i	0.939847	−	0.341595i	$-0.110967\pi$
$458$	24770.0			0.118085
$459$		0			0
$460$		−	51649.1i		−	0.244089i
$461$	−107011.			−0.503533			−0.251767	−	0.967788i	$-0.581012\pi$
$461$	−107011.			−0.503533			−0.251767	+	0.967788i	$0.581012\pi$
$462$		0			0
$463$			344520.i			1.60714i	0.595213	+	0.803568i	$0.297068\pi$
$463$			344520.i			1.60714i	−0.595213	+	0.803568i	$0.702932\pi$
$464$	−94727.2			−0.439986
$465$		0			0
$466$	−220861.			−1.01706
$467$		−	12985.4i		−	0.0595419i	−0.999557	−	0.0297709i	$-0.990522\pi$
$467$		−	12985.4i		−	0.0595419i	0.999557	−	0.0297709i	$-0.00947778\pi$
$468$		0			0
$469$		−	295503.i		−	1.34343i
$470$	−6270.91			−0.0283880
$471$		0			0
$472$	−96116.8	−	104544.i	−0.431435	−	0.469262i
$473$	291188.			1.30152
$474$		0			0
$475$	−245744.			−1.08917
$476$	−210839.			−0.930545
$477$		0			0
$478$		−	389697.i		−	1.70558i
$479$	−148631.			−0.647798			−0.323899	−	0.946092i	$-0.604994\pi$
$479$	−148631.			−0.647798			−0.323899	+	0.946092i	$0.604994\pi$
$480$		0			0
$481$	−747.985			−0.00323298
$482$		−	132529.i		−	0.570450i
$483$		0			0
$484$	166698.			0.711606
$485$			191040.i			0.812159i
$486$		0			0
$487$	174834.			0.737169			0.368584	−	0.929594i	$-0.379843\pi$
$487$	174834.			0.737169			0.368584	+	0.929594i	$0.379843\pi$
$488$	−200298.			−0.841079
$489$		0			0
$490$		−	8919.71i		−	0.0371500i
$491$	289993.			1.20288			0.601442	−	0.798916i	$-0.294593\pi$
$491$	289993.			1.20288			0.601442	+	0.798916i	$0.294593\pi$
$492$		0			0
$493$	162723.			0.669508
$494$	−1631.22			−0.00668434
$495$		0			0
$496$		−	344928.i		−	1.40206i
$497$	−294215.			−1.19111
$498$		0			0
$499$	−337039.			−1.35356			−0.676782	−	0.736183i	$-0.736626\pi$
$499$	−337039.			−1.35356			−0.676782	+	0.736183i	$0.736626\pi$
$500$	103993.			0.415972
$501$		0			0
$502$		−	424081.i		−	1.68283i
$503$			424514.i			1.67786i	0.544238	+	0.838931i	$0.316819\pi$
$503$			424514.i			1.67786i	−0.544238	+	0.838931i	$0.683181\pi$
$504$		0			0
$505$		−	170559.i		−	0.668793i
$506$		−	504566.i		−	1.97068i
$507$		0			0
$508$	−229236.			−0.888292
$509$		−	166873.i		−	0.644095i	−0.946724	−	0.322047i	$-0.895629\pi$
$509$		−	166873.i		−	0.644095i	0.946724	−	0.322047i	$-0.104371\pi$
$510$		0			0
$511$			340761.i			1.30499i
$512$		−	79440.3i		−	0.303041i
$513$		0			0
$514$		−	8109.37i		−	0.0306945i
$515$			59709.1i			0.225126i
$516$		0			0
$517$	−19732.7			−0.0738252
$518$	−286526.			−1.06784
$519$		0			0
$520$	−326.685			−0.00120815
$521$	−234510.			−0.863943			−0.431972	−	0.901887i	$-0.642182\pi$
$521$	−234510.			−0.863943			−0.431972	+	0.901887i	$0.642182\pi$
$522$		0			0
$523$	133011.			0.486277			0.243138	−	0.969992i	$-0.421823\pi$
$523$	133011.			0.486277			0.243138	+	0.969992i	$0.421823\pi$
$524$			232573.i			0.847026i
$525$		0			0
$526$			485419.i			1.75447i
$527$			592521.i			2.13345i
$528$		0			0
$529$	−15130.5			−0.0540683
$530$		−	170879.i		−	0.608328i
$531$		0			0
$532$	−201272.			−0.711149
$533$		−	722.603i		−	0.00254358i
$534$		0			0
$535$	−136489.			−0.476858
$536$	−238840.			−0.831339
$537$		0			0
$538$	119484.			0.412806
$539$		−	28067.6i		−	0.0966114i
$540$		0			0
$541$			37399.8i			0.127784i	0.997957	+	0.0638918i	$0.0203512\pi$
$541$			37399.8i			0.127784i	−0.997957	+	0.0638918i	$0.979649\pi$
$542$		−	59787.0i		−	0.203521i
$543$		0			0
$544$			495102.i			1.67300i
$545$			16608.1i			0.0559148i
$546$		0			0
$547$	−3623.89			−0.0121116			−0.00605579	−	0.999982i	$-0.501928\pi$
$547$	−3623.89			−0.0121116			−0.00605579	+	0.999982i	$0.501928\pi$
$548$	214783.			0.715218
$549$		0			0
$550$	435278.			1.43894
$551$	155340.			0.511657
$552$		0			0
$553$	−471138.			−1.54063
$554$		−	149896.i		−	0.488394i
$555$		0			0
$556$	−46571.8			−0.150652
$557$	−223715.			−0.721082			−0.360541	−	0.932743i	$-0.617408\pi$
$557$	−223715.			−0.721082			−0.360541	+	0.932743i	$0.617408\pi$
$558$		0			0
$559$	974.798			0.00311954
$560$	−201943.			−0.643951
$561$		0			0
$562$			30662.3i			0.0970807i
$563$			74625.8i			0.235436i	0.993047	+	0.117718i	$0.0375578\pi$
$563$			74625.8i			0.235436i	−0.993047	+	0.117718i	$0.962442\pi$
$564$		0			0
$565$			90144.9i			0.282387i
$566$	−182062.			−0.568313
$567$		0			0
$568$			237799.i			0.737079i
$569$		−	319733.i		−	0.987558i	−0.869588	−	0.493779i	$-0.835615\pi$
$569$		−	319733.i		−	0.987558i	0.869588	−	0.493779i	$-0.164385\pi$
$570$		0			0
$571$		−	25255.1i		−	0.0774598i	−0.999250	−	0.0387299i	$-0.987669\pi$
$571$		−	25255.1i		−	0.0774598i	0.999250	−	0.0387299i	$-0.0123312\pi$
$572$	930.674			0.00284450
$573$		0			0
$574$		−	276803.i		−	0.840130i
$575$		−	254466.i		−	0.769650i
$576$		0			0
$577$	442259.			1.32839			0.664194	−	0.747560i	$-0.268775\pi$
$577$	442259.			1.32839			0.664194	+	0.747560i	$0.268775\pi$
$578$		−	1.06080e6i		−	3.17524i
$579$		0			0
$580$	−28165.2			−0.0837252
$581$		−	92845.6i		−	0.275048i
$582$		0			0
$583$		−	537706.i		−	1.58201i
$584$	275420.			0.807551
$585$		0			0
$586$			64538.3i			0.187941i
$587$			122543.i			0.355641i	0.984063	+	0.177821i	$0.0569047\pi$
$587$			122543.i			0.355641i	−0.984063	+	0.177821i	$0.943095\pi$
$588$		0			0
$589$			565635.i			1.63044i
$590$	−143174.	−	155728.i	−0.411302	−	0.447365i
$591$		0			0
$592$			373714.i			1.06634i
$593$	531802.			1.51231			0.756154	−	0.654393i	$-0.227076\pi$
$593$	531802.			1.51231			0.756154	+	0.654393i	$0.227076\pi$
$594$		0			0
$595$	346900.			0.979874
$596$		−	273251.i		−	0.769253i
$597$		0			0
$598$		−	1689.11i		−	0.00472342i
$599$	−55605.2			−0.154975			−0.0774875	−	0.996993i	$-0.524690\pi$
$599$	−55605.2			−0.154975			−0.0774875	+	0.996993i	$0.524690\pi$
$600$		0			0
$601$			351895.i			0.974236i	0.873336	+	0.487118i	$0.161952\pi$
$601$			351895.i			0.974236i	−0.873336	+	0.487118i	$0.838048\pi$
$602$	373410.			1.03037
$603$		0			0
$604$		−	29009.3i		−	0.0795176i
$605$	−274273.			−0.749329
$606$		0			0
$607$	130700.			0.354731			0.177365	−	0.984145i	$-0.443243\pi$
$607$	130700.			0.354731			0.177365	+	0.984145i	$0.443243\pi$
$608$			472636.i			1.27856i
$609$		0			0
$610$	−298362.			−0.801832
$611$	−66.0583			−0.000176948
$612$		0			0
$613$		−	509749.i		−	1.35655i	−0.734808	−	0.678275i	$-0.762728\pi$
$613$		−	509749.i		−	1.35655i	0.734808	−	0.678275i	$-0.237272\pi$
$614$		−	132509.i		−	0.351486i
$615$		0			0
$616$	−393781.			−1.03775
$617$	77698.9			0.204101			0.102050	−	0.994779i	$-0.467460\pi$
$617$	77698.9			0.204101			0.102050	+	0.994779i	$0.467460\pi$
$618$		0			0
$619$	221586.			0.578309			0.289155	−	0.957282i	$-0.406626\pi$
$619$	221586.			0.578309			0.289155	+	0.957282i	$0.406626\pi$
$620$		−	102557.i		−	0.266798i
$621$		0			0
$622$			363091.i			0.938502i
$623$		−	289475.i		−	0.745820i
$624$		0			0
$625$	121730.			0.311628
$626$	717127.			1.82998
$627$		0			0
$628$			107714.i			0.273120i
$629$		−	641969.i		−	1.62260i
$630$		0			0
$631$	2526.90			0.00634642			0.00317321	−	0.999995i	$-0.498990\pi$
$631$	2526.90			0.00634642			0.00317321	+	0.999995i	$0.498990\pi$
$632$			380797.i			0.953365i
$633$		0			0
$634$		−	367439.i		−	0.914127i
$635$	377169.			0.935381
$636$		0			0
$637$		−	93.9610i		−	0.000231563i
$638$	−275148.			−0.675967
$639$		0			0
$640$			225296.i			0.550039i
$641$	−359210.			−0.874244			−0.437122	−	0.899402i	$-0.644002\pi$
$641$	−359210.			−0.874244			−0.437122	+	0.899402i	$0.644002\pi$
$642$		0			0
$643$	512961.			1.24069			0.620344	−	0.784330i	$-0.286993\pi$
$643$	512961.			1.24069			0.620344	+	0.784330i	$0.286993\pi$
$644$		−	208416.i		−	0.502527i
$645$		0			0
$646$		−	1.40002e6i		−	3.35481i
$647$	621636.			1.48500			0.742502	−	0.669844i	$-0.233639\pi$
$647$	621636.			1.48500			0.742502	+	0.669844i	$0.233639\pi$
$648$		0			0
$649$	−450527.	−	490028.i	−1.06962	−	1.16341i
$650$	1457.16			0.00344891
$651$		0			0
$652$	41330.3			0.0972241
$653$	−194566.			−0.456291			−0.228145	−	0.973627i	$-0.573266\pi$
$653$	−194566.			−0.456291			−0.228145	+	0.973627i	$0.573266\pi$
$654$		0			0
$655$		−	382659.i		−	0.891927i
$656$	−361032.			−0.838954
$657$		0			0
$658$	−25304.5			−0.0584449
$659$		−	481512.i		−	1.10876i	−0.832265	−	0.554378i	$-0.812956\pi$
$659$		−	481512.i		−	1.10876i	0.832265	−	0.554378i	$-0.187044\pi$
$660$		0			0
$661$	340031.			0.778244			0.389122	−	0.921186i	$-0.372778\pi$
$661$	340031.			0.778244			0.389122	+	0.921186i	$0.372778\pi$
$662$		−	211837.i		−	0.483377i
$663$		0			0
$664$	−75042.4			−0.170204
$665$	331159.			0.748847
$666$		0			0
$667$			160853.i			0.361558i
$668$	81693.4			0.183077
$669$		0			0
$670$	−355774.			−0.792546
$671$	−938855.			−2.08523
$672$		0			0
$673$		−	413955.i		−	0.913951i	−0.889479	−	0.456976i	$-0.848933\pi$
$673$		−	413955.i		−	0.913951i	0.889479	−	0.456976i	$-0.151067\pi$
$674$	543846.			1.19717
$675$		0			0
$676$	−217134.			−0.475153
$677$	374555.			0.817219			0.408609	−	0.912709i	$-0.366014\pi$
$677$	374555.			0.817219			0.408609	+	0.912709i	$0.366014\pi$
$678$		0			0
$679$			770890.i			1.67206i
$680$		−	280382.i		−	0.606362i
$681$		0			0
$682$		−	1.00189e6i		−	2.15403i
$683$			648839.i			1.39090i	0.718576	+	0.695449i	$0.244794\pi$
$683$			648839.i			1.39090i	−0.718576	+	0.695449i	$0.755206\pi$
$684$		0			0
$685$	−353388.			−0.753132
$686$			552786.i			1.17465i
$687$		0			0
$688$		−	487036.i		−	1.02893i
$689$		−	1800.06i		−	0.00379183i
$690$		0			0
$691$			622050.i			1.30277i	0.758746	+	0.651387i	$0.225813\pi$
$691$			622050.i			1.30277i	−0.758746	+	0.651387i	$0.774187\pi$
$692$		−	207164.i		−	0.432616i
$693$		0			0
$694$	211047.			0.438187
$695$	76626.0			0.158638
$696$		0			0
$697$	620184.			1.27660
$698$	−308067.			−0.632316
$699$		0			0
$700$	179796.			0.366931
$701$			98359.5i			0.200161i	0.994979	+	0.100081i	$0.0319101\pi$
$701$			98359.5i			0.200161i	−0.994979	+	0.100081i	$0.968090\pi$
$702$		0			0
$703$		−	612839.i		−	1.24004i
$704$			141430.i			0.285363i
$705$		0			0
$706$	−739847.			−1.48434
$707$		−	688244.i		−	1.37690i
$708$		0			0
$709$	−772872.			−1.53750			−0.768750	−	0.639549i	$-0.779121\pi$
$709$	−772872.			−1.53750			−0.768750	+	0.639549i	$0.779121\pi$
$710$			354223.i			0.702684i
$711$		0			0
$712$	−233968.			−0.461526
$713$	−585711.			−1.15214
$714$		0			0
$715$	−1531.27			−0.00299529
$716$			458534.i			0.894428i
$717$		0			0
$718$			187220.i			0.363164i
$719$			273060.i			0.528202i	0.964495	+	0.264101i	$0.0850752\pi$
$719$			273060.i			0.528202i	−0.964495	+	0.264101i	$0.914925\pi$
$720$		0			0
$721$			240940.i			0.463487i
$722$		−	703358.i		−	1.34928i
$723$		0			0
$724$	211905.			0.404263
$725$	−138764.			−0.263999
$726$		0			0
$727$	31372.5			0.0593581			0.0296791	−	0.999559i	$-0.490551\pi$
$727$	31372.5			0.0593581			0.0296791	+	0.999559i	$0.490551\pi$
$728$	−1318.25			−0.00248733
$729$		0			0
$730$	410262.			0.769867
$731$			836634.i			1.56567i
$732$		0			0
$733$	67831.8			0.126248			0.0631241	−	0.998006i	$-0.479894\pi$
$733$	67831.8			0.126248			0.0631241	+	0.998006i	$0.479894\pi$
$734$	621701.			1.15396
$735$		0			0
$736$	−489411.			−0.903480
$737$	−1.11951e6			−2.06108
$738$		0			0
$739$			788958.i			1.44466i	0.691549	+	0.722329i	$0.256929\pi$
$739$			788958.i			1.44466i	−0.691549	+	0.722329i	$0.743071\pi$
$740$			111116.i			0.202914i
$741$		0			0
$742$		−	689537.i		−	1.25242i
$743$	−50905.9			−0.0922127			−0.0461064	−	0.998937i	$-0.514681\pi$
$743$	−50905.9			−0.0922127			−0.0461064	+	0.998937i	$0.514681\pi$
$744$		0			0
$745$			449588.i			0.810031i
$746$			881678.i			1.58428i
$747$		0			0
$748$			798764.i			1.42763i
$749$	−550762.			−0.981749
$750$		0			0
$751$		−	473502.i		−	0.839541i	−0.907630	−	0.419770i	$-0.862111\pi$
$751$		−	473502.i		−	0.839541i	0.907630	−	0.419770i	$-0.137889\pi$
$752$			33004.5i			0.0583630i
$753$		0			0
$754$	−921.103			−0.00162019
$755$			47729.8i			0.0837329i
$756$		0			0
$757$	−5901.48			−0.0102984			−0.00514919	−	0.999987i	$-0.501639\pi$
$757$	−5901.48			−0.0102984			−0.00514919	+	0.999987i	$0.501639\pi$
$758$			253053.i			0.440427i
$759$		0			0
$760$		−	267659.i		−	0.463399i
$761$	−559827.			−0.966684			−0.483342	−	0.875431i	$-0.660577\pi$
$761$	−559827.			−0.966684			−0.483342	+	0.875431i	$0.660577\pi$
$762$		0			0
$763$			67017.4i			0.115117i
$764$		−	249611.i		−	0.427639i
$765$		0			0
$766$		−	628934.i		−	1.07188i
$767$	−1508.21	−	1640.45i	−0.00256373	−	0.00278851i
$768$		0			0
$769$		−	752520.i		−	1.27252i	−0.771473	−	0.636262i	$-0.780480\pi$
$769$		−	752520.i		−	1.27252i	0.771473	−	0.636262i	$-0.219520\pi$
$770$	−586572.			−0.989327
$771$		0			0
$772$	388160.			0.651292
$773$			293795.i			0.491684i	0.969310	+	0.245842i	$0.0790644\pi$
$773$			293795.i			0.491684i	−0.969310	+	0.245842i	$0.920936\pi$
$774$		0			0
$775$		−	505281.i		−	0.841258i
$776$	623072.			1.03470
$777$		0			0
$778$		−	1.16459e6i		−	1.92404i
$779$	592043.			0.975614
$780$		0			0
$781$			1.11463e6i			1.82738i
$782$	1.44971e6			2.37065
$783$		0			0
$784$	−46945.5			−0.0763768
$785$		−	177225.i		−	0.287598i
$786$		0			0
$787$	642350.			1.03710			0.518552	−	0.855046i	$-0.326471\pi$
$787$	642350.			1.03710			0.518552	+	0.855046i	$0.326471\pi$
$788$	398579.			0.641893
$789$		0			0
$790$			567230.i			0.908877i
$791$			363755.i			0.581375i
$792$		0			0
$793$	−3142.97			−0.00499797
$794$	−306680.			−0.486457
$795$		0			0
$796$	161807.			0.255370
$797$		−	766042.i		−	1.20597i	−0.797753	−	0.602984i	$-0.793978\pi$
$797$		−	766042.i		−	1.20597i	0.797753	−	0.602984i	$-0.206022\pi$
$798$		0			0
$799$		−	56695.5i		−	0.0888086i
$800$		−	422205.i		−	0.659695i
$801$		0			0
$802$	−27370.7			−0.0425537
$803$	1.29097e6			2.00210
$804$		0			0
$805$			342913.i			0.529166i
$806$		−	3354.00i		−	0.00516289i
$807$		0			0
$808$	−556273.			−0.852050
$809$		−	605837.i		−	0.925675i	−0.886443	−	0.462837i	$-0.846831\pi$
$809$		−	605837.i		−	0.925675i	0.886443	−	0.462837i	$-0.153169\pi$
$810$		0			0
$811$		−	98920.8i		−	0.150399i	−0.997168	−	0.0751996i	$-0.976041\pi$
$811$		−	98920.8i		−	0.150399i	0.997168	−	0.0751996i	$-0.0239594\pi$
$812$	−113653.			−0.172372
$813$		0			0
$814$			1.08550e6i			1.63826i
$815$	−68002.0			−0.102378
$816$		0			0
$817$			798671.i			1.19653i
$818$	1.01154e6			1.51174
$819$		0			0
$820$	−107345.			−0.159645
$821$			412837.i			0.612480i	0.951954	+	0.306240i	$0.0990710\pi$
$821$			412837.i			0.612480i	−0.951954	+	0.306240i	$0.900929\pi$
$822$		0			0
$823$			1.01994e6i			1.50583i	0.658118	+	0.752915i	$0.271353\pi$
$823$			1.01994e6i			1.50583i	−0.658118	+	0.752915i	$0.728647\pi$
$824$	194739.			0.286813
$825$		0			0
$826$	−577741.	−	628396.i	−0.846785	−	0.921029i
$827$	−396788.			−0.580159			−0.290080	−	0.957003i	$-0.593682\pi$
$827$	−396788.			−0.580159			−0.290080	+	0.957003i	$0.593682\pi$
$828$		0			0
$829$	573455.			0.834431			0.417216	−	0.908808i	$-0.363006\pi$
$829$	573455.			0.834431			0.417216	+	0.908808i	$0.363006\pi$
$830$	−111782.			−0.162262
$831$		0			0
$832$			473.461i			0.000683971i
$833$	80643.4			0.116219
$834$		0			0
$835$	−134413.			−0.192782
$836$			762520.i			1.09103i
$837$		0			0
$838$	−141518.			−0.201522
$839$		−	1.02305e6i		−	1.45337i	−0.686973	−	0.726683i	$-0.741061\pi$
$839$		−	1.02305e6i		−	1.45337i	0.686973	−	0.726683i	$-0.258939\pi$
$840$		0			0
$841$	−619565.			−0.875982
$842$	1.44509e6			2.03831
$843$		0			0
$844$		−	515437.i		−	0.723587i
$845$	357256.			0.500342
$846$		0			0
$847$	−1.10675e6			−1.54271
$848$	−899359.			−1.25067
$849$		0			0
$850$			1.25063e6i			1.73098i
$851$	634590.			0.876263
$852$		0			0
$853$	−101559.			−0.139580			−0.0697898	−	0.997562i	$-0.522233\pi$
$853$	−101559.			−0.139580			−0.0697898	+	0.997562i	$0.522233\pi$
$854$	−1.20396e6			−1.65080
$855$		0			0
$856$			445154.i			0.607522i
$857$			1.31133e6i			1.78545i	0.450598	+	0.892727i	$0.351211\pi$
$857$			1.31133e6i			1.78545i	−0.450598	+	0.892727i	$0.648789\pi$
$858$		0			0
$859$			815255.i			1.10486i	0.833559	+	0.552430i	$0.186299\pi$
$859$			815255.i			1.10486i	−0.833559	+	0.552430i	$0.813701\pi$
$860$		−	144810.i		−	0.195795i
$861$		0			0
$862$	1.65275e6			2.22429
$863$			797750.i			1.07114i	0.844492	+	0.535569i	$0.179903\pi$
$863$			797750.i			1.07114i	−0.844492	+	0.535569i	$0.820097\pi$
$864$		0			0
$865$			340853.i			0.455549i
$866$			95734.8i			0.127654i
$867$		0			0
$868$		−	413842.i		−	0.549281i
$869$			1.78490e6i			2.36361i
$870$		0			0
$871$	−3747.75			−0.00494009
$872$	54166.8			0.0712361
$873$		0			0
$874$	1.38393e6			1.81171
$875$	−690439.			−0.901797
$876$		0			0
$877$	−542419.			−0.705238			−0.352619	−	0.935767i	$-0.614709\pi$
$877$	−542419.			−0.705238			−0.352619	+	0.935767i	$0.614709\pi$
$878$		−	141900.i		−	0.184075i
$879$		0			0
$880$			765062.i			0.987941i
$881$			449854.i			0.579588i	0.957089	+	0.289794i	$0.0935868\pi$
$881$			449854.i			0.579588i	−0.957089	+	0.289794i	$0.906413\pi$
$882$		0			0
$883$	−469650.			−0.602355			−0.301178	−	0.953568i	$-0.597380\pi$
$883$	−469650.			−0.602355			−0.301178	+	0.953568i	$0.597380\pi$
$884$			2673.99i			0.00342181i
$885$		0			0
$886$	665203.			0.847397
$887$			126942.i			0.161346i	0.996741	+	0.0806729i	$0.0257069\pi$
$887$			126942.i			0.161346i	−0.996741	+	0.0806729i	$0.974293\pi$
$888$		0			0
$889$	1.52196e6			1.92575
$890$	−348516.			−0.439989
$891$		0			0
$892$	−342669.			−0.430670
$893$		−	54122.9i		−	0.0678700i
$894$		0			0
$895$		−	754439.i		−	0.941842i
$896$			909120.i			1.13241i
$897$		0			0
$898$			1.57865e6i			1.95765i
$899$			319398.i			0.395197i
$900$		0			0
$901$	1.54493e6			1.90309
$902$	−1.04867e6			−1.28892
$903$		0			0
$904$	294005.			0.359764
$905$	−348653.			−0.425693
$906$		0			0
$907$	396178.			0.481588			0.240794	−	0.970576i	$-0.422592\pi$
$907$	396178.			0.481588			0.240794	+	0.970576i	$0.422592\pi$
$908$			100252.i			0.121597i
$909$		0			0
$910$	−1963.64			−0.00237126
$911$	1.15177e6			1.38780			0.693902	−	0.720070i	$-0.255890\pi$
$911$	1.15177e6			1.38780			0.693902	+	0.720070i	$0.255890\pi$
$912$		0			0
$913$	−351745.			−0.421975
$914$	−693191.			−0.829776
$915$		0			0
$916$		−	38762.0i		−	0.0461972i
$917$		−	1.54412e6i		−	1.83629i
$918$		0			0
$919$			675878.i			0.800271i	0.916456	+	0.400136i	$0.131037\pi$
$919$			675878.i			0.800271i	−0.916456	+	0.400136i	$0.868963\pi$
$920$	277159.			0.327456
$921$		0			0
$922$			519888.i			0.611572i
$923$			3731.42i			0.00437996i
$924$		0			0
$925$			547448.i			0.639822i
$926$	1.67376e6			1.95197
$927$		0			0
$928$			266884.i			0.309904i
$929$		−	729975.i		−	0.845818i	−0.906172	−	0.422909i	$-0.861009\pi$
$929$		−	729975.i		−	0.845818i	0.906172	−	0.422909i	$-0.138991\pi$
$930$		0			0
$931$	76984.1			0.0888181
$932$			345621.i			0.397894i
$933$		0			0
$934$	−63086.4			−0.0723173
$935$		−	1.31423e6i		−	1.50331i
$936$		0			0
$937$			487935.i			0.555754i	0.960617	+	0.277877i	$0.0896308\pi$
$937$			487935.i			0.555754i	−0.960617	+	0.277877i	$0.910369\pi$
$938$	−1.43563e6			−1.63168
$939$		0			0
$940$			9813.21i			0.0111059i
$941$			51925.4i			0.0586409i	0.999570	+	0.0293204i	$0.00933433\pi$
$941$			51925.4i			0.0586409i	−0.999570	+	0.0293204i	$0.990666\pi$
$942$		0			0
$943$			613056.i			0.689409i
$944$	−819613.	+	753544.i	−0.919739	+	0.845599i
$945$		0			0
$946$		−	1.41466e6i		−	1.58078i
$947$	−1.51811e6			−1.69279			−0.846393	−	0.532559i	$-0.821231\pi$
$947$	−1.51811e6			−1.69279			−0.846393	+	0.532559i	$0.821231\pi$
$948$		0			0
$949$	4321.74			0.00479873
$950$			1.19388e6i			1.32286i
$951$		0			0
$952$		−	1.13140e6i		−	1.24837i
$953$	1.18731e6			1.30731			0.653653	−	0.756794i	$-0.273236\pi$
$953$	1.18731e6			1.30731			0.653653	+	0.756794i	$0.273236\pi$
$954$		0			0
$955$			410693.i			0.450309i
$956$	−609829.			−0.667256
$957$		0			0
$958$			722088.i			0.786790i
$959$	−1.42600e6			−1.55054
$960$		0			0
$961$	−239498.			−0.259332
$962$			3633.90i			0.00392665i
$963$		0			0
$964$	−207392.			−0.223171
$965$	−638650.			−0.685817
$966$		0			0
$967$		−	1.72012e6i		−	1.83953i	−0.392475	−	0.919763i	$-0.628381\pi$
$967$		−	1.72012e6i		−	1.83953i	0.392475	−	0.919763i	$-0.371619\pi$
$968$			894533.i			0.954654i
$969$		0			0
$970$	928120.			0.986418
$971$	88363.1			0.0937200			0.0468600	−	0.998901i	$-0.485079\pi$
$971$	88363.1			0.0937200			0.0468600	+	0.998901i	$0.485079\pi$
$972$		0			0
$973$	309203.			0.326602
$974$		−	849385.i		−	0.895337i
$975$		0			0
$976$			1.57031e6i			1.64849i
$977$			205443.i			0.215229i	0.994193	+	0.107615i	$0.0343213\pi$
$977$			205443.i			0.215229i	−0.994193	+	0.107615i	$0.965679\pi$
$978$		0			0
$979$	−1.09667e6			−1.14423
$980$	−13958.3			−0.0145338
$981$		0			0
$982$		−	1.40886e6i		−	1.46098i
$983$		−	1.79432e6i		−	1.85692i	−0.371435	−	0.928459i	$-0.621134\pi$
$983$		−	1.79432e6i		−	1.85692i	0.371435	−	0.928459i	$-0.378866\pi$
$984$		0			0
$985$	−655794.			−0.675920
$986$		−	790550.i		−	0.813159i
$987$		0			0
$988$			2552.66i			0.00261504i
$989$	−827018.			−0.845517
$990$		0			0
$991$			660202.i			0.672248i	0.941818	+	0.336124i	$0.109116\pi$
$991$			660202.i			0.672248i	−0.941818	+	0.336124i	$0.890884\pi$
$992$	−971801.			−0.987538
$993$		0			0
$994$			1.42937e6i			1.44668i
$995$	−266225.			−0.268908
$996$		0			0
$997$	−684976.			−0.689105			−0.344552	−	0.938767i	$-0.611969\pi$
$997$	−684976.			−0.689105			−0.344552	+	0.938767i	$0.611969\pi$
$998$			1.63742e6i			1.64399i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.9		40
3.2	odd	2		177.5.c.a.58.32	yes	40
59.58	odd	2	inner	531.5.c.d.235.32		40
177.176	even	2		177.5.c.a.58.9	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.9	✓	40	177.176	even	2
177.5.c.a.58.32	yes	40	3.2	odd	2
531.5.c.d.235.9		40	1.1	even	1	trivial
531.5.c.d.235.32		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)

\(f(q)\)	\(=\)	\(q-4.85825i q^{2} -7.60257 q^{4} +12.5087 q^{5} +50.4755 q^{7} -40.7968i q^{8} +O(q^{10})\)	\(q-4.85825i q^{2} -7.60257 q^{4} +12.5087 q^{5} +50.4755 q^{7} -40.7968i q^{8} -60.7705i q^{10} -191.226i q^{11} -0.640162i q^{13} -245.222i q^{14} -319.842 q^{16} +549.428 q^{17} +524.497 q^{19} -95.0984 q^{20} -929.025 q^{22} +543.113i q^{23} -468.532 q^{25} -3.11006 q^{26} -383.743 q^{28} +296.169 q^{29} +1078.43i q^{31} +901.123i q^{32} -2669.26i q^{34} +631.384 q^{35} -1168.43i q^{37} -2548.14i q^{38} -510.316i q^{40} +1128.78 q^{41} +1522.74i q^{43} +1453.81i q^{44} +2638.58 q^{46} -103.190i q^{47} +146.777 q^{49} +2276.24i q^{50} +4.86687i q^{52} +2811.88 q^{53} -2392.00i q^{55} -2059.24i q^{56} -1438.86i q^{58} +(2562.55 - 2355.99i) q^{59} -4909.65i q^{61} +5239.30 q^{62} -739.596 q^{64} -8.00760i q^{65} -5854.39i q^{67} -4177.06 q^{68} -3067.42i q^{70} -5828.87 q^{71} +6751.02i q^{73} -5676.53 q^{74} -3987.52 q^{76} -9652.25i q^{77} -9333.98 q^{79} -4000.81 q^{80} -5483.90i q^{82} -1839.42i q^{83} +6872.64 q^{85} +7397.84 q^{86} -7801.43 q^{88} -5734.95i q^{89} -32.3125i q^{91} -4129.05i q^{92} -501.323 q^{94} +6560.79 q^{95} +15272.6i q^{97} -713.079i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 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

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Groups

Database

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