LMFDB - Embedded newform 1503.2.c.a.1502.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1502,1503)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1503.1502");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1502.3
Character	$\chi$	$=$	1503.1502
Dual form			1503.2.c.a.1502.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66906i q^{2} -5.12388 q^{4} -3.16492 q^{5} -0.263490 q^{7} +8.33782i q^{8} +O(q^{10})$	\(q-2.66906i q^{2} -5.12388 q^{4} -3.16492 q^{5} -0.263490 q^{7} +8.33782i q^{8} +8.44736i q^{10} +0.255224i q^{11} +3.30041i q^{13} +0.703271i q^{14} +12.0064 q^{16} -0.0330469 q^{17} +3.25126 q^{19} +16.2167 q^{20} +0.681208 q^{22} -6.64578 q^{23} +5.01673 q^{25} +8.80899 q^{26} +1.35009 q^{28} -6.88330i q^{29} +4.02950 q^{31} -15.3701i q^{32} +0.0882042i q^{34} +0.833926 q^{35} +1.76737i q^{37} -8.67780i q^{38} -26.3886i q^{40} +7.12477 q^{41} +1.29567i q^{43} -1.30774i q^{44} +17.7380i q^{46} -5.98565i q^{47} -6.93057 q^{49} -13.3899i q^{50} -16.9109i q^{52} -3.33373 q^{53} -0.807764i q^{55} -2.19694i q^{56} -18.3719 q^{58} +13.5617 q^{59} +11.8500 q^{61} -10.7550i q^{62} -17.0110 q^{64} -10.4455i q^{65} +8.31866i q^{67} +0.169328 q^{68} -2.22580i q^{70} -0.716835 q^{71} +11.0907i q^{73} +4.71723 q^{74} -16.6591 q^{76} -0.0672491i q^{77} +2.66308i q^{79} -37.9993 q^{80} -19.0164i q^{82} -0.600591 q^{83} +0.104591 q^{85} +3.45821 q^{86} -2.12801 q^{88} +5.08137i q^{89} -0.869626i q^{91} +34.0522 q^{92} -15.9761 q^{94} -10.2900 q^{95} -9.53461 q^{97} +18.4981i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 48 q^{4}+O(q^{10}) $ 56 * q - 48 * q^4	$ 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) $ 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.66906i		−	1.88731i	−0.330930	−	0.943655i	$-0.607363\pi$
$2$		−	2.66906i		−	1.88731i	0.330930	−	0.943655i	$-0.392637\pi$
$3$		0			0
$3$		0			0
$4$	−5.12388			−2.56194
$5$	−3.16492			−1.41540			−0.707698	−	0.706515i	$-0.750266\pi$
$5$	−3.16492			−1.41540			−0.707698	+	0.706515i	$0.750266\pi$
$6$		0			0
$7$	−0.263490			−0.0995900			−0.0497950	−	0.998759i	$-0.515857\pi$
$7$	−0.263490			−0.0995900			−0.0497950	+	0.998759i	$0.515857\pi$
$8$			8.33782i			2.94787i
$9$		0			0
$10$			8.44736i			2.67129i
$11$			0.255224i			0.0769530i	0.999260	+	0.0384765i	$0.0122505\pi$
$11$			0.255224i			0.0769530i	−0.999260	+	0.0384765i	$0.987750\pi$
$12$		0			0
$13$			3.30041i			0.915369i	0.889115	+	0.457685i	$0.151321\pi$
$13$			3.30041i			0.915369i	−0.889115	+	0.457685i	$0.848679\pi$
$14$			0.703271i			0.187957i
$15$		0			0
$16$	12.0064			3.00160
$17$	−0.0330469			−0.00801505			−0.00400753	−	0.999992i	$-0.501276\pi$
$17$	−0.0330469			−0.00801505			−0.00400753	+	0.999992i	$0.501276\pi$
$18$		0			0
$19$	3.25126			0.745890			0.372945	−	0.927854i	$-0.378348\pi$
$19$	3.25126			0.745890			0.372945	+	0.927854i	$0.378348\pi$
$20$	16.2167			3.62616
$21$		0			0
$22$	0.681208			0.145234
$23$	−6.64578			−1.38574			−0.692871	−	0.721062i	$-0.743654\pi$
$23$	−6.64578			−1.38574			−0.692871	+	0.721062i	$0.743654\pi$
$24$		0			0
$25$	5.01673			1.00335
$26$	8.80899			1.72759
$27$		0			0
$28$	1.35009			0.255144
$29$		−	6.88330i		−	1.27820i	−0.769125	−	0.639098i	$-0.779308\pi$
$29$		−	6.88330i		−	1.27820i	0.769125	−	0.639098i	$-0.220692\pi$
$30$		0			0
$31$	4.02950			0.723719			0.361859	−	0.932233i	$-0.382142\pi$
$31$	4.02950			0.723719			0.361859	+	0.932233i	$0.382142\pi$
$32$		−	15.3701i		−	2.71708i
$33$		0			0
$34$			0.0882042i			0.0151269i
$35$	0.833926			0.140959
$36$		0			0
$37$			1.76737i			0.290554i	0.989391	+	0.145277i	$0.0464074\pi$
$37$			1.76737i			0.290554i	−0.989391	+	0.145277i	$0.953593\pi$
$38$		−	8.67780i		−	1.40773i
$39$		0			0
$40$		−	26.3886i		−	4.17240i
$41$	7.12477			1.11270			0.556351	−	0.830948i	$-0.312201\pi$
$41$	7.12477			1.11270			0.556351	+	0.830948i	$0.312201\pi$
$42$		0			0
$43$			1.29567i			0.197587i	0.995108	+	0.0987937i	$0.0314984\pi$
$43$			1.29567i			0.197587i	−0.995108	+	0.0987937i	$0.968502\pi$
$44$		−	1.30774i		−	0.197149i
$45$		0			0
$46$			17.7380i			2.61532i
$47$		−	5.98565i		−	0.873097i	−0.899681	−	0.436549i	$-0.856201\pi$
$47$		−	5.98565i		−	0.873097i	0.899681	−	0.436549i	$-0.143799\pi$
$48$		0			0
$49$	−6.93057			−0.990082
$50$		−	13.3899i		−	1.89362i
$51$		0			0
$52$		−	16.9109i		−	2.34512i
$53$	−3.33373			−0.457924			−0.228962	−	0.973435i	$-0.573533\pi$
$53$	−3.33373			−0.457924			−0.228962	+	0.973435i	$0.573533\pi$
$54$		0			0
$55$		−	0.807764i		−	0.108919i
$56$		−	2.19694i		−	0.293578i
$57$		0			0
$58$	−18.3719			−2.41235
$59$	13.5617			1.76558			0.882792	−	0.469763i	$-0.155661\pi$
$59$	13.5617			1.76558			0.882792	+	0.469763i	$0.155661\pi$
$60$		0			0
$61$	11.8500			1.51724			0.758620	−	0.651534i	$-0.225874\pi$
$61$	11.8500			1.51724			0.758620	+	0.651534i	$0.225874\pi$
$62$		−	10.7550i		−	1.36588i
$63$		0			0
$64$	−17.0110			−2.12638
$65$		−	10.4455i		−	1.29561i
$66$		0			0
$67$			8.31866i			1.01629i	0.861273	+	0.508143i	$0.169668\pi$
$67$			8.31866i			1.01629i	−0.861273	+	0.508143i	$0.830332\pi$
$68$	0.169328			0.0205341
$69$		0			0
$70$		−	2.22580i		−	0.266034i
$71$	−0.716835			−0.0850727			−0.0425363	−	0.999095i	$-0.513544\pi$
$71$	−0.716835			−0.0850727			−0.0425363	+	0.999095i	$0.513544\pi$
$72$		0			0
$73$			11.0907i			1.29807i	0.760757	+	0.649037i	$0.224828\pi$
$73$			11.0907i			1.29807i	−0.760757	+	0.649037i	$0.775172\pi$
$74$	4.71723			0.548366
$75$		0			0
$76$	−16.6591			−1.91092
$77$		−	0.0672491i		−	0.00766374i
$78$		0			0
$79$			2.66308i			0.299619i	0.988715	+	0.149810i	$0.0478661\pi$
$79$			2.66308i			0.299619i	−0.988715	+	0.149810i	$0.952134\pi$
$80$	−37.9993			−4.24845
$81$		0			0
$82$		−	19.0164i		−	2.10001i
$83$	−0.600591			−0.0659234			−0.0329617	−	0.999457i	$-0.510494\pi$
$83$	−0.600591			−0.0659234			−0.0329617	+	0.999457i	$0.510494\pi$
$84$		0			0
$85$	0.104591			0.0113445
$86$	3.45821			0.372909
$87$		0			0
$88$	−2.12801			−0.226847
$89$			5.08137i			0.538625i	0.963053	+	0.269312i	$0.0867964\pi$
$89$			5.08137i			0.538625i	−0.963053	+	0.269312i	$0.913204\pi$
$90$		0			0
$91$		−	0.869626i		−	0.0911616i
$92$	34.0522			3.55019
$93$		0			0
$94$	−15.9761			−1.64781
$95$	−10.2900			−1.05573
$96$		0			0
$97$	−9.53461			−0.968093			−0.484046	−	0.875042i	$-0.660833\pi$
$97$	−9.53461			−0.968093			−0.484046	+	0.875042i	$0.660833\pi$
$98$			18.4981i			1.86859i
$99$		0			0
$100$	−25.7051			−2.57051
$101$	−14.2610			−1.41902			−0.709512	−	0.704693i	$-0.751085\pi$
$101$	−14.2610			−1.41902			−0.709512	+	0.704693i	$0.751085\pi$
$102$		0			0
$103$		−	19.6710i		−	1.93824i	−0.246592	−	0.969119i	$-0.579311\pi$
$103$		−	19.6710i		−	1.93824i	0.246592	−	0.969119i	$-0.420689\pi$
$104$	−27.5182			−2.69839
$105$		0			0
$106$			8.89794i			0.864244i
$107$			1.13521i			0.109745i	0.998493	+	0.0548724i	$0.0174752\pi$
$107$			1.13521i			0.109745i	−0.998493	+	0.0548724i	$0.982525\pi$
$108$		0			0
$109$		−	14.8611i		−	1.42343i	−0.702466	−	0.711717i	$-0.747918\pi$
$109$		−	14.8611i		−	1.42343i	0.702466	−	0.711717i	$-0.252082\pi$
$110$	−2.15597			−0.205564
$111$		0			0
$112$	−3.16357			−0.298929
$113$	13.4346			1.26382			0.631911	−	0.775041i	$-0.282271\pi$
$113$	13.4346			1.26382			0.631911	+	0.775041i	$0.282271\pi$
$114$		0			0
$115$	21.0334			1.96137
$116$			35.2692i			3.27466i
$117$		0			0
$118$		−	36.1970i		−	3.33221i
$119$	0.00870754			0.000798219
$120$		0			0
$121$	10.9349			0.994078
$122$		−	31.6284i		−	2.86350i
$123$		0			0
$124$	−20.6467			−1.85412
$125$	−0.0529361			−0.00473475
$126$		0			0
$127$	18.4150			1.63407			0.817033	−	0.576591i	$-0.195617\pi$
$127$	18.4150			1.63407			0.817033	+	0.576591i	$0.195617\pi$
$128$			14.6631i			1.29605i
$129$		0			0
$130$	−27.8798			−2.44522
$131$	2.68250			0.234371			0.117186	−	0.993110i	$-0.462613\pi$
$131$	2.68250			0.234371			0.117186	+	0.993110i	$0.462613\pi$
$132$		0			0
$133$	−0.856675			−0.0742831
$134$	22.2030			1.91805
$135$		0			0
$136$		−	0.275539i		−	0.0236273i
$137$			5.90038i			0.504103i	0.967714	+	0.252052i	$0.0811053\pi$
$137$			5.90038i			0.504103i	−0.967714	+	0.252052i	$0.918895\pi$
$138$		0			0
$139$		−	18.3993i		−	1.56060i	−0.625402	−	0.780302i	$-0.715065\pi$
$139$		−	18.3993i		−	1.56060i	0.625402	−	0.780302i	$-0.284935\pi$
$140$	−4.27294			−0.361129
$141$		0			0
$142$			1.91328i			0.160559i
$143$	−0.842344			−0.0704404
$144$		0			0
$145$			21.7851i			1.80915i
$146$	29.6019			2.44987
$147$		0			0
$148$		−	9.05581i		−	0.744383i
$149$	20.0577			1.64319			0.821597	−	0.570069i	$-0.193083\pi$
$149$	20.0577			1.64319			0.821597	+	0.570069i	$0.193083\pi$
$150$		0			0
$151$			12.7120i			1.03449i	0.855838	+	0.517244i	$0.173042\pi$
$151$			12.7120i			1.03449i	−0.855838	+	0.517244i	$0.826958\pi$
$152$			27.1084i			2.19878i
$153$		0			0
$154$	−0.179492			−0.0144639
$155$	−12.7530			−1.02435
$156$		0			0
$157$	9.46619			0.755485			0.377742	−	0.925911i	$-0.376701\pi$
$157$	9.46619			0.755485			0.377742	+	0.925911i	$0.376701\pi$
$158$	7.10791			0.565475
$159$		0			0
$160$			48.6452i			3.84574i
$161$	1.75110			0.138006
$162$		0			0
$163$			13.5831i			1.06391i	0.846774	+	0.531953i	$0.178542\pi$
$163$			13.5831i			1.06391i	−0.846774	+	0.531953i	$0.821458\pi$
$164$	−36.5064			−2.85067
$165$		0			0
$166$			1.60301i			0.124418i
$167$	−6.60284	−	11.1087i	−0.510943	−	0.859614i
$167$	−6.60284	−	11.1087i	−0.510943	−	0.859614i
$168$		0			0
$169$	2.10729			0.162099
$170$		−	0.279159i		−	0.0214105i
$171$		0			0
$172$		−	6.63885i		−	0.506207i
$173$			10.1735i			0.773473i	0.922190	+	0.386737i	$0.126398\pi$
$173$			10.1735i			0.773473i	−0.922190	+	0.386737i	$0.873602\pi$
$174$		0			0
$175$	−1.32186			−0.0999231
$176$			3.06432i			0.230982i
$177$		0			0
$178$	13.5625			1.01655
$179$			10.7841i			0.806040i	0.915191	+	0.403020i	$0.132040\pi$
$179$			10.7841i			0.806040i	−0.915191	+	0.403020i	$0.867960\pi$
$180$		0			0
$181$	7.86216			0.584390			0.292195	−	0.956359i	$-0.405614\pi$
$181$	7.86216			0.584390			0.292195	+	0.956359i	$0.405614\pi$
$182$	−2.32108			−0.172050
$183$		0			0
$184$		−	55.4113i		−	4.08498i
$185$		−	5.59360i		−	0.411250i
$186$		0			0
$187$		−	0.00843437i		−	0.000616782i
$188$			30.6698i			2.23682i
$189$		0			0
$190$			27.4646i			1.99249i
$191$		−	13.5386i		−	0.979619i	−0.871829	−	0.489810i	$-0.837066\pi$
$191$		−	13.5386i		−	0.979619i	0.871829	−	0.489810i	$-0.162934\pi$
$192$		0			0
$193$			10.1347i			0.729512i	0.931103	+	0.364756i	$0.118848\pi$
$193$			10.1347i			0.729512i	−0.931103	+	0.364756i	$0.881152\pi$
$194$			25.4484i			1.82709i
$195$		0			0
$196$	35.5114			2.53653
$197$	21.6938			1.54562			0.772809	−	0.634639i	$-0.218851\pi$
$197$	21.6938			1.54562			0.772809	+	0.634639i	$0.218851\pi$
$198$		0			0
$199$	21.9727			1.55761			0.778803	−	0.627269i	$-0.215827\pi$
$199$	21.9727			1.55761			0.778803	+	0.627269i	$0.215827\pi$
$200$			41.8286i			2.95773i
$201$		0			0
$202$			38.0635i			2.67814i
$203$			1.81368i			0.127296i
$204$		0			0
$205$	−22.5493			−1.57491
$206$	−52.5030			−3.65806
$207$		0			0
$208$			39.6260i			2.74757i
$209$			0.829799i			0.0573984i
$210$		0			0
$211$	15.4107			1.06091			0.530457	−	0.847712i	$-0.322020\pi$
$211$	15.4107			1.06091			0.530457	+	0.847712i	$0.322020\pi$
$212$	17.0817			1.17317
$213$		0			0
$214$	3.02994			0.207123
$215$		−	4.10069i		−	0.279664i
$216$		0			0
$217$	−1.06173			−0.0720751
$218$	−39.6651			−2.68646
$219$		0			0
$220$			4.13889i			0.279044i
$221$		−	0.109068i		−	0.00733673i
$222$		0			0
$223$	16.1848			1.08382			0.541908	−	0.840438i	$-0.317702\pi$
$223$	16.1848			1.08382			0.541908	+	0.840438i	$0.317702\pi$
$224$			4.04988i			0.270594i
$225$		0			0
$226$		−	35.8578i		−	2.38522i
$227$	−9.31736			−0.618415			−0.309208	−	0.950995i	$-0.600064\pi$
$227$	−9.31736			−0.618415			−0.309208	+	0.950995i	$0.600064\pi$
$228$		0			0
$229$	−26.1929			−1.73088			−0.865438	−	0.501017i	$-0.832959\pi$
$229$	−26.1929			−1.73088			−0.865438	+	0.501017i	$0.832959\pi$
$230$		−	56.1393i		−	3.70172i
$231$		0			0
$232$	57.3918			3.76795
$233$			3.77826i			0.247522i	0.992312	+	0.123761i	$0.0394956\pi$
$233$			3.77826i			0.247522i	−0.992312	+	0.123761i	$0.960504\pi$
$234$		0			0
$235$			18.9441i			1.23578i
$236$	−69.4886			−4.52332
$237$		0			0
$238$		−	0.0232409i		−	0.00150649i
$239$			18.0275i			1.16610i	0.812435	+	0.583051i	$0.198141\pi$
$239$			18.0275i			1.16610i	−0.812435	+	0.583051i	$0.801859\pi$
$240$		0			0
$241$			14.9449i			0.962685i	0.876533	+	0.481342i	$0.159851\pi$
$241$			14.9449i			0.962685i	−0.876533	+	0.481342i	$0.840149\pi$
$242$		−	29.1858i		−	1.87613i
$243$		0			0
$244$	−60.7181			−3.88708
$245$	21.9347			1.40136
$246$		0			0
$247$			10.7305i			0.682764i
$248$			33.5972i			2.13343i
$249$		0			0
$250$			0.141290i			0.00893594i
$251$		−	13.4301i		−	0.847703i	−0.905732	−	0.423851i	$-0.860678\pi$
$251$		−	13.4301i		−	0.847703i	0.905732	−	0.423851i	$-0.139322\pi$
$252$		0			0
$253$		−	1.69616i		−	0.106637i
$254$		−	49.1507i		−	3.08399i
$255$		0			0
$256$	5.11478			0.319674
$257$	21.4002			1.33491			0.667453	−	0.744652i	$-0.267384\pi$
$257$	21.4002			1.33491			0.667453	+	0.744652i	$0.267384\pi$
$258$		0			0
$259$		−	0.465686i		−	0.0289363i
$260$			53.5217i			3.31927i
$261$		0			0
$262$		−	7.15976i		−	0.442331i
$263$			20.8030i			1.28277i	0.767221	+	0.641383i	$0.221639\pi$
$263$			20.8030i			1.28277i	−0.767221	+	0.641383i	$0.778361\pi$
$264$		0			0
$265$	10.5510			0.648143
$266$			2.28652i			0.140195i
$267$		0			0
$268$		−	42.6238i		−	2.60367i
$269$	−1.79118			−0.109210			−0.0546052	−	0.998508i	$-0.517390\pi$
$269$	−1.79118			−0.109210			−0.0546052	+	0.998508i	$0.517390\pi$
$270$		0			0
$271$		−	12.1748i		−	0.739565i	−0.929118	−	0.369783i	$-0.879432\pi$
$271$		−	12.1748i		−	0.739565i	0.929118	−	0.369783i	$-0.120568\pi$
$272$	−0.396774			−0.0240580
$273$		0			0
$274$	15.7485			0.951400
$275$			1.28039i			0.0772104i
$276$		0			0
$277$		−	16.3538i		−	0.982606i	−0.870989	−	0.491303i	$-0.836521\pi$
$277$		−	16.3538i		−	0.982606i	0.870989	−	0.491303i	$-0.163479\pi$
$278$	−49.1087			−2.94535
$279$		0			0
$280$			6.95313i			0.415529i
$281$			22.9080i			1.36658i	0.730149	+	0.683288i	$0.239451\pi$
$281$			22.9080i			1.36658i	−0.730149	+	0.683288i	$0.760549\pi$
$282$		0			0
$283$	−2.86854			−0.170517			−0.0852585	−	0.996359i	$-0.527172\pi$
$283$	−2.86854			−0.170517			−0.0852585	+	0.996359i	$0.527172\pi$
$284$	3.67298			0.217951
$285$		0			0
$286$			2.24827i			0.132943i
$287$	−1.87731			−0.110814
$288$		0			0
$289$	−16.9989			−0.999936
$290$	58.1458			3.41444
$291$		0			0
$292$		−	56.8277i		−	3.32559i
$293$		−	1.08052i		−	0.0631246i	−0.999502	−	0.0315623i	$-0.989952\pi$
$293$		−	1.08052i		−	0.0631246i	0.999502	−	0.0315623i	$-0.0100483\pi$
$294$		0			0
$295$	−42.9218			−2.49900
$296$	−14.7361			−0.856516
$297$		0			0
$298$		−	53.5353i		−	3.10122i
$299$		−	21.9338i		−	1.26846i
$300$		0			0
$301$		−	0.341396i		−	0.0196777i
$302$	33.9291			1.95240
$303$		0			0
$304$	39.0359			2.23886
$305$	−37.5044			−2.14749
$306$		0			0
$307$			13.8657i			0.791359i	0.918389	+	0.395680i	$0.129491\pi$
$307$			13.8657i			0.791359i	−0.918389	+	0.395680i	$0.870509\pi$
$308$			0.344576i			0.0196340i
$309$		0			0
$310$			34.0386i			1.93326i
$311$			16.3123i			0.924987i	0.886623	+	0.462493i	$0.153045\pi$
$311$			16.3123i			0.924987i	−0.886623	+	0.462493i	$0.846955\pi$
$312$		0			0
$313$			25.0621i			1.41659i	0.705915	+	0.708296i	$0.250536\pi$
$313$			25.0621i			1.41659i	−0.705915	+	0.708296i	$0.749464\pi$
$314$		−	25.2658i		−	1.42583i
$315$		0			0
$316$		−	13.6453i		−	0.767607i
$317$			12.2233i			0.686530i	0.939239	+	0.343265i	$0.111533\pi$
$317$			12.2233i			0.686530i	−0.939239	+	0.343265i	$0.888467\pi$
$318$		0			0
$319$	1.75678			0.0983610
$320$	53.8385			3.00966
$321$		0			0
$322$		−	4.67379i		−	0.260460i
$323$	−0.107444			−0.00597834
$324$		0			0
$325$			16.5573i			0.918431i
$326$	36.2540			2.00792
$327$		0			0
$328$			59.4050i			3.28009i
$329$			1.57716i			0.0869517i
$330$		0			0
$331$		−	9.57839i		−	0.526476i	−0.964731	−	0.263238i	$-0.915210\pi$
$331$		−	9.57839i		−	0.526476i	0.964731	−	0.263238i	$-0.0847904\pi$
$332$	3.07736			0.168892
$333$		0			0
$334$	−29.6497	+	17.6234i	−1.62236	+	0.964308i
$335$		−	26.3279i		−	1.43845i
$336$		0			0
$337$	33.4441			1.82182			0.910908	−	0.412609i	$-0.135382\pi$
$337$	33.4441			1.82182			0.910908	+	0.412609i	$0.135382\pi$
$338$		−	5.62448i		−	0.305932i
$339$		0			0
$340$	−0.535911			−0.0290639
$341$			1.02842i			0.0556923i
$342$		0			0
$343$	3.67057			0.198192
$344$	−10.8030			−0.582461
$345$		0			0
$346$	27.1536			1.45978
$347$	−19.3489			−1.03870			−0.519351	−	0.854561i	$-0.673826\pi$
$347$	−19.3489			−1.03870			−0.519351	+	0.854561i	$0.673826\pi$
$348$		0			0
$349$		−	5.49409i		−	0.294092i	−0.989130	−	0.147046i	$-0.953023\pi$
$349$		−	5.49409i		−	0.294092i	0.989130	−	0.147046i	$-0.0469765\pi$
$350$			3.52812i			0.188586i
$351$		0			0
$352$	3.92283			0.209087
$353$		−	10.2441i		−	0.545237i	−0.962122	−	0.272618i	$-0.912110\pi$
$353$		−	10.2441i		−	0.545237i	0.962122	−	0.272618i	$-0.0878896\pi$
$354$		0			0
$355$	2.26873			0.120412
$356$		−	26.0364i		−	1.37992i
$357$		0			0
$358$	28.7834			1.52125
$359$		−	7.78801i		−	0.411036i	−0.978653	−	0.205518i	$-0.934112\pi$
$359$		−	7.78801i		−	0.411036i	0.978653	−	0.205518i	$-0.0658879\pi$
$360$		0			0
$361$	−8.42932			−0.443649
$362$		−	20.9846i		−	1.10292i
$363$		0			0
$364$			4.45586i			0.233551i
$365$		−	35.1013i		−	1.83729i
$366$		0			0
$367$	17.7132			0.924620			0.462310	−	0.886718i	$-0.347021\pi$
$367$	17.7132			0.924620			0.462310	+	0.886718i	$0.347021\pi$
$368$	−79.7918			−4.15944
$369$		0			0
$370$	−14.9297			−0.776156
$371$	0.878407			0.0456046
$372$		0			0
$373$		−	30.6132i		−	1.58509i	−0.609813	−	0.792546i	$-0.708755\pi$
$373$		−	30.6132i		−	1.58509i	0.609813	−	0.792546i	$-0.291245\pi$
$374$	−0.0225118			−0.00116406
$375$		0			0
$376$	49.9073			2.57377
$377$	22.7177			1.17002
$378$		0			0
$379$			6.62770i			0.340442i	0.985406	+	0.170221i	$0.0544482\pi$
$379$			6.62770i			0.340442i	−0.985406	+	0.170221i	$0.945552\pi$
$380$	52.7246			2.70471
$381$		0			0
$382$	−36.1354			−1.84885
$383$			25.8991i			1.32338i	0.749777	+	0.661690i	$0.230161\pi$
$383$			25.8991i			1.32338i	−0.749777	+	0.661690i	$0.769839\pi$
$384$		0			0
$385$			0.212838i			0.0108472i
$386$	27.0501			1.37682
$387$		0			0
$388$	48.8542			2.48020
$389$	−25.2244			−1.27893			−0.639465	−	0.768821i	$-0.720844\pi$
$389$	−25.2244			−1.27893			−0.639465	+	0.768821i	$0.720844\pi$
$390$		0			0
$391$	0.219622			0.0111068
$392$		−	57.7859i		−	2.91863i
$393$		0			0
$394$		−	57.9020i		−	2.91706i
$395$		−	8.42842i		−	0.424080i
$396$		0			0
$397$	−11.2378			−0.564010			−0.282005	−	0.959413i	$-0.590999\pi$
$397$	−11.2378			−0.564010			−0.282005	+	0.959413i	$0.590999\pi$
$398$		−	58.6465i		−	2.93969i
$399$		0			0
$400$	60.2328			3.01164
$401$	21.1057			1.05397			0.526984	−	0.849875i	$-0.323323\pi$
$401$	21.1057			1.05397			0.526984	+	0.849875i	$0.323323\pi$
$402$		0			0
$403$			13.2990i			0.662470i
$404$	73.0717			3.63545
$405$		0			0
$406$	4.84083			0.240246
$407$	−0.451076			−0.0223590
$408$		0			0
$409$	−23.9652			−1.18501			−0.592503	−	0.805569i	$-0.701860\pi$
$409$	−23.9652			−1.18501			−0.592503	+	0.805569i	$0.701860\pi$
$410$			60.1855i			2.97235i
$411$		0			0
$412$			100.792i			4.96565i
$413$	−3.57338			−0.175835
$414$		0			0
$415$	1.90082			0.0933077
$416$	50.7277			2.48713
$417$		0			0
$418$	2.21478			0.108329
$419$			18.9705i			0.926771i	0.886157	+	0.463385i	$0.153365\pi$
$419$			18.9705i			0.926771i	−0.886157	+	0.463385i	$0.846635\pi$
$420$		0			0
$421$	0.836551			0.0407710			0.0203855	−	0.999792i	$-0.493511\pi$
$421$	0.836551			0.0407710			0.0203855	+	0.999792i	$0.493511\pi$
$422$		−	41.1320i		−	2.00227i
$423$		0			0
$424$		−	27.7961i		−	1.34990i
$425$	−0.165787			−0.00804186
$426$		0			0
$427$	−3.12236			−0.151102
$428$		−	5.81668i		−	0.281160i
$429$		0			0
$430$	−10.9450			−0.527814
$431$		−	26.9491i		−	1.29809i	−0.760749	−	0.649047i	$-0.775168\pi$
$431$		−	26.9491i		−	1.29809i	0.760749	−	0.649047i	$-0.224832\pi$
$432$		0			0
$433$	−16.2067			−0.778846			−0.389423	−	0.921059i	$-0.627325\pi$
$433$	−16.2067			−0.778846			−0.389423	+	0.921059i	$0.627325\pi$
$434$			2.83383i			0.136028i
$435$		0			0
$436$			76.1465i			3.64675i
$437$	−21.6071			−1.03361
$438$		0			0
$439$			21.2271i			1.01312i	0.862206	+	0.506558i	$0.169082\pi$
$439$			21.2271i			1.01312i	−0.862206	+	0.506558i	$0.830918\pi$
$440$	6.73499			0.321078
$441$		0			0
$442$	−0.291110			−0.0138467
$443$	−30.2353			−1.43652			−0.718262	−	0.695772i	$-0.755062\pi$
$443$	−30.2353			−1.43652			−0.718262	+	0.695772i	$0.755062\pi$
$444$		0			0
$445$		−	16.0821i		−	0.762367i
$446$		−	43.1983i		−	2.04550i
$447$		0			0
$448$	4.48223			0.211766
$449$		−	11.2259i		−	0.529782i	−0.964278	−	0.264891i	$-0.914664\pi$
$449$		−	11.2259i		−	0.529782i	0.964278	−	0.264891i	$-0.0853359\pi$
$450$		0			0
$451$			1.81841i			0.0856257i
$452$	−68.8373			−3.23783
$453$		0			0
$454$			24.8686i			1.16714i
$455$			2.75230i			0.129030i
$456$		0			0
$457$		−	15.3329i		−	0.717241i	−0.933484	−	0.358620i	$-0.883247\pi$
$457$		−	15.3329i		−	0.717241i	0.933484	−	0.358620i	$-0.116753\pi$
$458$			69.9104i			3.26670i
$459$		0			0
$460$	−107.772			−5.02492
$461$		−	33.5860i		−	1.56426i	−0.623118	−	0.782128i	$-0.714134\pi$
$461$		−	33.5860i		−	1.56426i	0.623118	−	0.782128i	$-0.285866\pi$
$462$		0			0
$463$			19.0309i			0.884440i	0.896907	+	0.442220i	$0.145809\pi$
$463$			19.0309i			0.884440i	−0.896907	+	0.442220i	$0.854191\pi$
$464$		−	82.6436i		−	3.83663i
$465$		0			0
$466$	10.0844			0.467151
$467$			38.7583i			1.79352i	0.442515	+	0.896761i	$0.354086\pi$
$467$			38.7583i			1.79352i	−0.442515	+	0.896761i	$0.645914\pi$
$468$		0			0
$469$		−	2.19189i		−	0.101212i
$470$	50.5630			2.33230
$471$		0			0
$472$			113.075i			5.20471i
$473$	−0.330686			−0.0152049
$474$		0			0
$475$	16.3107			0.748385
$476$	−0.0446164			−0.00204499
$477$		0			0
$478$	48.1165			2.20080
$479$	13.6750			0.624826			0.312413	−	0.949946i	$-0.398863\pi$
$479$	13.6750			0.624826			0.312413	+	0.949946i	$0.398863\pi$
$480$		0			0
$481$	−5.83306			−0.265965
$482$	39.8888			1.81689
$483$		0			0
$484$	−56.0289			−2.54677
$485$	30.1763			1.37023
$486$		0			0
$487$		−	18.6315i		−	0.844274i	−0.906532	−	0.422137i	$-0.861280\pi$
$487$		−	18.6315i		−	0.844274i	0.906532	−	0.422137i	$-0.138720\pi$
$488$			98.8034i			4.47262i
$489$		0			0
$490$		−	58.5451i		−	2.64480i
$491$			9.40322i			0.424361i	0.977230	+	0.212181i	$0.0680565\pi$
$491$			9.40322i			0.424361i	−0.977230	+	0.212181i	$0.931943\pi$
$492$		0			0
$493$			0.227472i			0.0102448i
$494$	28.6403			1.28859
$495$		0			0
$496$	48.3797			2.17231
$497$	0.188879			0.00847238
$498$		0			0
$499$			19.5453i			0.874969i	0.899226	+	0.437485i	$0.144131\pi$
$499$			19.5453i			0.874969i	−0.899226	+	0.437485i	$0.855869\pi$
$500$	0.271238			0.0121301
$501$		0			0
$502$	−35.8458			−1.59988
$503$			31.2910i			1.39520i	0.716489	+	0.697598i	$0.245748\pi$
$503$			31.2910i			1.39520i	−0.716489	+	0.697598i	$0.754252\pi$
$504$		0			0
$505$	45.1350			2.00848
$506$	−4.52716			−0.201257
$507$		0			0
$508$	−94.3562			−4.18638
$509$		−	12.3803i		−	0.548749i	−0.961623	−	0.274375i	$-0.911529\pi$
$509$		−	12.3803i		−	0.548749i	0.961623	−	0.274375i	$-0.0884708\pi$
$510$		0			0
$511$		−	2.92230i		−	0.129275i
$512$			15.6746i			0.692727i
$513$		0			0
$514$		−	57.1184i		−	2.51938i
$515$			62.2571i			2.74337i
$516$		0			0
$517$	1.52768			0.0671874
$518$	−1.24294			−0.0546118
$519$		0			0
$520$	87.0931			3.81928
$521$	8.90185			0.389997			0.194999	−	0.980804i	$-0.437530\pi$
$521$	8.90185			0.389997			0.194999	+	0.980804i	$0.437530\pi$
$522$		0			0
$523$	22.1740			0.969599			0.484800	−	0.874625i	$-0.338892\pi$
$523$	22.1740			0.969599			0.484800	+	0.874625i	$0.338892\pi$
$524$	−13.7448			−0.600445
$525$		0			0
$526$	55.5244			2.42098
$527$	−0.133162			−0.00580064
$528$		0			0
$529$	21.1664			0.920278
$530$		−	28.1613i		−	1.22325i
$531$		0			0
$532$	4.38950			0.190309
$533$			23.5147i			1.01853i
$534$		0			0
$535$		−	3.59285i		−	0.155332i
$536$	−69.3595			−2.99588
$537$		0			0
$538$			4.78077i			0.206114i
$539$		−	1.76885i		−	0.0761897i
$540$		0			0
$541$		−	26.7991i		−	1.15218i	−0.817385	−	0.576091i	$-0.804577\pi$
$541$		−	26.7991i		−	1.15218i	0.817385	−	0.576091i	$-0.195423\pi$
$542$	−32.4952			−1.39579
$543$		0			0
$544$			0.507935i			0.0217775i
$545$			47.0342i			2.01472i
$546$		0			0
$547$		−	31.0103i		−	1.32590i	−0.748662	−	0.662952i	$-0.769303\pi$
$547$		−	31.0103i		−	1.32590i	0.748662	−	0.662952i	$-0.230697\pi$
$548$		−	30.2328i		−	1.29148i
$549$		0			0
$550$	3.41744			0.145720
$551$		−	22.3794i		−	0.953394i
$552$		0			0
$553$		−	0.701694i		−	0.0298391i
$554$	−43.6493			−1.85448
$555$		0			0
$556$			94.2756i			3.99818i
$557$			6.92206i			0.293297i	0.989189	+	0.146649i	$0.0468486\pi$
$557$			6.92206i			0.293297i	−0.989189	+	0.146649i	$0.953151\pi$
$558$		0			0
$559$	−4.27623			−0.180865
$560$	10.0124			0.423103
$561$		0			0
$562$	61.1428			2.57915
$563$			2.42694i			0.102283i	0.998691	+	0.0511416i	$0.0162860\pi$
$563$			2.42694i			0.102283i	−0.998691	+	0.0511416i	$0.983714\pi$
$564$		0			0
$565$	−42.5195			−1.78881
$566$			7.65631i			0.321819i
$567$		0			0
$568$		−	5.97684i		−	0.250783i
$569$	−26.6574			−1.11754			−0.558769	−	0.829323i	$-0.688726\pi$
$569$	−26.6574			−1.11754			−0.558769	+	0.829323i	$0.688726\pi$
$570$		0			0
$571$		−	32.4787i		−	1.35919i	−0.733587	−	0.679595i	$-0.762155\pi$
$571$		−	32.4787i		−	1.35919i	0.733587	−	0.679595i	$-0.237845\pi$
$572$	4.31607			0.180464
$573$		0			0
$574$			5.01064i			0.209140i
$575$	−33.3401			−1.39038
$576$		0			0
$577$	−14.3641			−0.597985			−0.298992	−	0.954255i	$-0.596651\pi$
$577$	−14.3641			−0.597985			−0.298992	+	0.954255i	$0.596651\pi$
$578$			45.3711i			1.88719i
$579$		0			0
$580$		−	111.624i		−	4.63495i
$581$	0.158250			0.00656531
$582$		0			0
$583$		−	0.850849i		−	0.0352386i
$584$	−92.4727			−3.82655
$585$		0			0
$586$	−2.88397			−0.119136
$587$	−21.0683			−0.869580			−0.434790	−	0.900532i	$-0.643177\pi$
$587$	−21.0683			−0.869580			−0.434790	+	0.900532i	$0.643177\pi$
$588$		0			0
$589$	13.1009			0.539814
$590$			114.561i			4.71639i
$591$		0			0
$592$			21.2198i			0.872128i
$593$	8.22178			0.337628			0.168814	−	0.985648i	$-0.446006\pi$
$593$	8.22178			0.337628			0.168814	+	0.985648i	$0.446006\pi$
$594$		0			0
$595$	−0.0275587			−0.00112980
$596$	−102.773			−4.20976
$597$		0			0
$598$	−58.5426			−2.39399
$599$			20.7650i			0.848435i	0.905560	+	0.424217i	$0.139451\pi$
$599$			20.7650i			0.848435i	−0.905560	+	0.424217i	$0.860549\pi$
$600$		0			0
$601$	−35.5012			−1.44813			−0.724063	−	0.689734i	$-0.757728\pi$
$601$	−35.5012			−1.44813			−0.724063	+	0.689734i	$0.757728\pi$
$602$	−0.911206			−0.0371380
$603$		0			0
$604$		−	65.1348i		−	2.65030i
$605$	−34.6080			−1.40701
$606$		0			0
$607$			0.997724i			0.0404964i	0.999795	+	0.0202482i	$0.00644564\pi$
$607$			0.997724i			0.0404964i	−0.999795	+	0.0202482i	$0.993554\pi$
$608$		−	49.9722i		−	2.02664i
$609$		0			0
$610$			100.101i			4.05299i
$611$	19.7551			0.799206
$612$		0			0
$613$	29.3460			1.18527			0.592636	−	0.805470i	$-0.298087\pi$
$613$	29.3460			1.18527			0.592636	+	0.805470i	$0.298087\pi$
$614$	37.0085			1.49354
$615$		0			0
$616$	0.560711			0.0225917
$617$			28.2248i			1.13629i	0.822929	+	0.568144i	$0.192338\pi$
$617$			28.2248i			1.13629i	−0.822929	+	0.568144i	$0.807662\pi$
$618$		0			0
$619$			0.928948i			0.0373376i	0.999826	+	0.0186688i	$0.00594281\pi$
$619$			0.928948i			0.0373376i	−0.999826	+	0.0186688i	$0.994057\pi$
$620$	65.3450			2.62432
$621$		0			0
$622$	43.5385			1.74574
$623$		−	1.33889i		−	0.0536416i
$624$		0			0
$625$	−24.9161			−0.996644
$626$	66.8922			2.67355
$627$		0			0
$628$	−48.5036			−1.93551
$629$		−	0.0584062i		−	0.00232881i
$630$		0			0
$631$	4.01365			0.159781			0.0798904	−	0.996804i	$-0.474543\pi$
$631$	4.01365			0.159781			0.0798904	+	0.996804i	$0.474543\pi$
$632$	−22.2043			−0.883238
$633$		0			0
$634$	32.6248			1.29569
$635$	−58.2820			−2.31285
$636$		0			0
$637$		−	22.8737i		−	0.906290i
$638$		−	4.68896i		−	0.185638i
$639$		0			0
$640$		−	46.4077i		−	1.83442i
$641$	23.4922			0.927884			0.463942	−	0.885865i	$-0.346435\pi$
$641$	23.4922			0.927884			0.463942	+	0.885865i	$0.346435\pi$
$642$		0			0
$643$		−	8.71323i		−	0.343616i	−0.985130	−	0.171808i	$-0.945039\pi$
$643$		−	8.71323i		−	0.343616i	0.985130	−	0.171808i	$-0.0549609\pi$
$644$	−8.97242			−0.353563
$645$		0			0
$646$			0.286774i			0.0112830i
$647$	−26.2460			−1.03184			−0.515918	−	0.856638i	$-0.672549\pi$
$647$	−26.2460			−1.03184			−0.515918	+	0.856638i	$0.672549\pi$
$648$		0			0
$649$			3.46128i			0.135867i
$650$	44.1923			1.73336
$651$		0			0
$652$		−	69.5979i		−	2.72567i
$653$		−	18.3945i		−	0.719832i	−0.932985	−	0.359916i	$-0.882805\pi$
$653$		−	18.3945i		−	0.719832i	0.932985	−	0.359916i	$-0.117195\pi$
$654$		0			0
$655$	−8.48990			−0.331728
$656$	85.5427			3.33988
$657$		0			0
$658$	4.20954			0.164105
$659$	39.4887			1.53826			0.769131	−	0.639091i	$-0.220689\pi$
$659$	39.4887			1.53826			0.769131	+	0.639091i	$0.220689\pi$
$660$		0			0
$661$		−	18.3170i		−	0.712449i	−0.934400	−	0.356224i	$-0.884064\pi$
$661$		−	18.3170i		−	0.712449i	0.934400	−	0.356224i	$-0.115936\pi$
$662$	−25.5653			−0.993623
$663$		0			0
$664$		−	5.00762i		−	0.194333i
$665$	2.71131			0.105140
$666$		0			0
$667$			45.7449i			1.77125i
$668$	33.8322	+	56.9195i	1.30901	+	2.20228i
$669$		0			0
$670$	−70.2708			−2.71480
$671$			3.02441i			0.116756i
$672$		0			0
$673$		−	44.1504i		−	1.70187i	−0.525268	−	0.850937i	$-0.676035\pi$
$673$		−	44.1504i		−	1.70187i	0.525268	−	0.850937i	$-0.323965\pi$
$674$		−	89.2643i		−	3.43833i
$675$		0			0
$676$	−10.7975			−0.415289
$677$			5.66967i			0.217903i	0.994047	+	0.108952i	$0.0347494\pi$
$677$			5.66967i			0.217903i	−0.994047	+	0.108952i	$0.965251\pi$
$678$		0			0
$679$	2.51228			0.0964123
$680$			0.872060i			0.0334420i
$681$		0			0
$682$	2.74493			0.105109
$683$	5.50528			0.210654			0.105327	−	0.994438i	$-0.466411\pi$
$683$	5.50528			0.210654			0.105327	+	0.994438i	$0.466411\pi$
$684$		0			0
$685$		−	18.6742i		−	0.713506i
$686$		−	9.79697i		−	0.374050i
$687$		0			0
$688$			15.5563i			0.593078i
$689$		−	11.0027i		−	0.419169i
$690$		0			0
$691$		−	31.9679i		−	1.21612i	−0.793893	−	0.608058i	$-0.791949\pi$
$691$		−	31.9679i		−	1.21612i	0.793893	−	0.608058i	$-0.208051\pi$
$692$		−	52.1276i		−	1.98159i
$693$		0			0
$694$			51.6433i			1.96035i
$695$			58.2322i			2.20887i
$696$		0			0
$697$	−0.235451			−0.00891836
$698$	−14.6641			−0.555043
$699$		0			0
$700$	6.77304			0.255997
$701$			1.91434i			0.0723035i	0.999346	+	0.0361518i	$0.0115100\pi$
$701$			1.91434i			0.0723035i	−0.999346	+	0.0361518i	$0.988490\pi$
$702$		0			0
$703$			5.74619i			0.216722i
$704$		−	4.34162i		−	0.163631i
$705$		0			0
$706$	−27.3420			−1.02903
$707$	3.75764			0.141321
$708$		0			0
$709$			17.1538i			0.644225i	0.946701	+	0.322112i	$0.104393\pi$
$709$			17.1538i			0.644225i	−0.946701	+	0.322112i	$0.895607\pi$
$710$		−	6.05537i		−	0.227254i
$711$		0			0
$712$	−42.3676			−1.58779
$713$	−26.7791			−1.00289
$714$		0			0
$715$	2.66595			0.0997010
$716$		−	55.2564i		−	2.06503i
$717$		0			0
$718$	−20.7867			−0.775752
$719$	28.0696			1.04682			0.523409	−	0.852082i	$-0.324660\pi$
$719$	28.0696			1.04682			0.523409	+	0.852082i	$0.324660\pi$
$720$		0			0
$721$			5.18311i			0.193029i
$722$			22.4984i			0.837302i
$723$		0			0
$724$	−40.2848			−1.49717
$725$		−	34.5316i		−	1.28247i
$726$		0			0
$727$			5.82205i			0.215928i	0.994155	+	0.107964i	$0.0344331\pi$
$727$			5.82205i			0.215928i	−0.994155	+	0.107964i	$0.965567\pi$
$728$	7.25079			0.268732
$729$		0			0
$730$	−93.6876			−3.46753
$731$		−	0.0428178i		−	0.00158367i
$732$		0			0
$733$	13.2069			0.487808			0.243904	−	0.969799i	$-0.421572\pi$
$733$	13.2069			0.487808			0.243904	+	0.969799i	$0.421572\pi$
$734$		−	47.2775i		−	1.74505i
$735$		0			0
$736$			102.146i			3.76517i
$737$	−2.12312			−0.0782062
$738$		0			0
$739$			4.79210i			0.176280i	0.996108	+	0.0881402i	$0.0280923\pi$
$739$			4.79210i			0.176280i	−0.996108	+	0.0881402i	$0.971908\pi$
$740$			28.6609i			1.05360i
$741$		0			0
$742$		−	2.34452i		−	0.0860700i
$743$		−	53.5192i		−	1.96343i	−0.190361	−	0.981714i	$-0.560966\pi$
$743$		−	53.5192i		−	1.96343i	0.190361	−	0.981714i	$-0.439034\pi$
$744$		0			0
$745$	−63.4812			−2.32577
$746$	−81.7085			−2.99156
$747$		0			0
$748$			0.0432167i			0.00158016i
$749$		−	0.299117i		−	0.0109295i
$750$		0			0
$751$			36.3279i			1.32562i	0.748786	+	0.662812i	$0.230637\pi$
$751$			36.3279i			1.32562i	−0.748786	+	0.662812i	$0.769363\pi$
$752$		−	71.8661i		−	2.62069i
$753$		0			0
$754$		−	60.6350i		−	2.20819i
$755$		−	40.2325i		−	1.46421i
$756$		0			0
$757$	2.19043			0.0796124			0.0398062	−	0.999207i	$-0.487326\pi$
$757$	2.19043			0.0796124			0.0398062	+	0.999207i	$0.487326\pi$
$758$	17.6897			0.642520
$759$		0			0
$760$		−	85.7960i		−	3.11215i
$761$			51.6855i			1.87360i	0.349870	+	0.936798i	$0.386226\pi$
$761$			51.6855i			1.87360i	−0.349870	+	0.936798i	$0.613774\pi$
$762$		0			0
$763$			3.91575i			0.141760i
$764$			69.3702i			2.50973i
$765$		0			0
$766$	69.1261			2.49763
$767$			44.7592i			1.61616i
$768$		0			0
$769$		−	50.8989i		−	1.83546i	−0.397203	−	0.917731i	$-0.630019\pi$
$769$		−	50.8989i		−	1.83546i	0.397203	−	0.917731i	$-0.369981\pi$
$770$	0.568077			0.0204721
$771$		0			0
$772$		−	51.9290i		−	1.86897i
$773$	−22.2062			−0.798702			−0.399351	−	0.916798i	$-0.630764\pi$
$773$	−22.2062			−0.798702			−0.399351	+	0.916798i	$0.630764\pi$
$774$		0			0
$775$	20.2149			0.726140
$776$		−	79.4979i		−	2.85381i
$777$		0			0
$778$			67.3255i			2.41374i
$779$	23.1645			0.829953
$780$		0			0
$781$		−	0.182954i		−	0.00654659i
$782$		−	0.586185i		−	0.0209619i
$783$		0			0
$784$	−83.2112			−2.97183
$785$	−29.9598			−1.06931
$786$		0			0
$787$		−	16.3293i		−	0.582077i	−0.956711	−	0.291039i	$-0.905999\pi$
$787$		−	16.3293i		−	0.582077i	0.956711	−	0.291039i	$-0.0940008\pi$
$788$	−111.156			−3.95978
$789$		0			0
$790$	−22.4960			−0.800371
$791$	−3.53989			−0.125864
$792$		0			0
$793$			39.1099i			1.38883i
$794$			29.9944i			1.06446i
$795$		0			0
$796$	−112.586			−3.99049
$797$	−51.0117			−1.80693			−0.903463	−	0.428666i	$-0.858984\pi$
$797$	−51.0117			−1.80693			−0.903463	+	0.428666i	$0.858984\pi$
$798$		0			0
$799$			0.197807i			0.00699792i
$800$		−	77.1077i		−	2.72617i
$801$		0			0
$802$		−	56.3324i		−	1.98917i
$803$	−2.83063			−0.0998906
$804$		0			0
$805$	−5.54209			−0.195333
$806$	35.4958			1.25029
$807$		0			0
$808$		−	118.906i		−	4.18309i
$809$		−	48.3819i		−	1.70102i	−0.525961	−	0.850509i	$-0.676294\pi$
$809$		−	48.3819i		−	1.70102i	0.525961	−	0.850509i	$-0.323706\pi$
$810$		0			0
$811$		−	7.24062i		−	0.254253i	−0.991887	−	0.127126i	$-0.959425\pi$
$811$		−	7.24062i		−	0.254253i	0.991887	−	0.127126i	$-0.0405754\pi$
$812$		−	9.29309i		−	0.326124i
$813$		0			0
$814$			1.20395i			0.0421984i
$815$		−	42.9893i		−	1.50585i
$816$		0			0
$817$			4.21255i			0.147378i
$818$			63.9647i			2.23647i
$819$		0			0
$820$	115.540			4.03483
$821$	13.4855			0.470646			0.235323	−	0.971917i	$-0.424385\pi$
$821$	13.4855			0.470646			0.235323	+	0.971917i	$0.424385\pi$
$822$		0			0
$823$		−	26.0344i		−	0.907504i	−0.891128	−	0.453752i	$-0.850085\pi$
$823$		−	26.0344i		−	0.907504i	0.891128	−	0.453752i	$-0.149915\pi$
$824$	164.013			5.71367
$825$		0			0
$826$			9.53756i			0.331854i
$827$	30.6842			1.06699			0.533496	−	0.845802i	$-0.320878\pi$
$827$	30.6842			1.06699			0.533496	+	0.845802i	$0.320878\pi$
$828$		0			0
$829$			21.4967i			0.746611i	0.927709	+	0.373305i	$0.121776\pi$
$829$			21.4967i			0.746611i	−0.927709	+	0.373305i	$0.878224\pi$
$830$		−	5.07341i		−	0.176101i
$831$		0			0
$832$		−	56.1433i		−	1.94642i
$833$	0.229034			0.00793556
$834$		0			0
$835$	20.8975	+	35.1581i	0.723187	+	1.21669i
$836$		−	4.25179i		−	0.147051i
$837$		0			0
$838$	50.6335			1.74910
$839$			19.0492i			0.657652i	0.944391	+	0.328826i	$0.106653\pi$
$839$			19.0492i			0.657652i	−0.944391	+	0.328826i	$0.893347\pi$
$840$		0			0
$841$	−18.3798			−0.633788
$842$		−	2.23281i		−	0.0769475i
$843$		0			0
$844$	−78.9624			−2.71800
$845$	−6.66941			−0.229435
$846$		0			0
$847$	−2.88123			−0.0990002
$848$	−40.0261			−1.37450
$849$		0			0
$850$			0.442496i			0.0151775i
$851$		−	11.7456i		−	0.402633i
$852$		0			0
$853$	26.9492			0.922722			0.461361	−	0.887212i	$-0.347361\pi$
$853$	26.9492			0.922722			0.461361	+	0.887212i	$0.347361\pi$
$854$			8.33378i			0.285176i
$855$		0			0
$856$	−9.46517			−0.323513
$857$			24.5599i			0.838951i	0.907767	+	0.419475i	$0.137786\pi$
$857$			24.5599i			0.838951i	−0.907767	+	0.419475i	$0.862214\pi$
$858$		0			0
$859$	−2.25741			−0.0770219			−0.0385109	−	0.999258i	$-0.512261\pi$
$859$	−2.25741			−0.0770219			−0.0385109	+	0.999258i	$0.512261\pi$
$860$			21.0114i			0.716484i
$861$		0			0
$862$	−71.9288			−2.44990
$863$			26.6938i			0.908668i	0.890832	+	0.454334i	$0.150123\pi$
$863$			26.6938i			0.908668i	−0.890832	+	0.454334i	$0.849877\pi$
$864$		0			0
$865$		−	32.1982i		−	1.09477i
$866$			43.2567i			1.46992i
$867$		0			0
$868$	5.44019			0.184652
$869$	−0.679681			−0.0230566
$870$		0			0
$871$	−27.4550			−0.930277
$872$	123.909			4.19609
$873$		0			0
$874$			57.6708i			1.95074i
$875$	0.0139481			0.000471533
$876$		0			0
$877$	−30.2985			−1.02311			−0.511554	−	0.859251i	$-0.670930\pi$
$877$	−30.2985			−1.02311			−0.511554	+	0.859251i	$0.670930\pi$
$878$	56.6565			1.91206
$879$		0			0
$880$		−	9.69833i		−	0.326931i
$881$	−19.6792			−0.663010			−0.331505	−	0.943453i	$-0.607556\pi$
$881$	−19.6792			−0.663010			−0.331505	+	0.943453i	$0.607556\pi$
$882$		0			0
$883$	−3.86894			−0.130200			−0.0651001	−	0.997879i	$-0.520737\pi$
$883$	−3.86894			−0.130200			−0.0651001	+	0.997879i	$0.520737\pi$
$884$			0.558853i			0.0187963i
$885$		0			0
$886$			80.7000i			2.71117i
$887$	20.8727			0.700837			0.350418	−	0.936593i	$-0.386039\pi$
$887$	20.8727			0.700837			0.350418	+	0.936593i	$0.386039\pi$
$888$		0			0
$889$	−4.85217			−0.162737
$890$	−42.9242			−1.43882
$891$		0			0
$892$	−82.9291			−2.77667
$893$		−	19.4609i		−	0.651234i
$894$		0			0
$895$		−	34.1308i		−	1.14087i
$896$		−	3.86359i		−	0.129074i
$897$		0			0
$898$	−29.9625			−0.999862
$899$		−	27.7362i		−	0.925055i
$900$		0			0
$901$	0.110170			0.00367028
$902$	4.85345			0.161602
$903$		0			0
$904$			112.015i			3.72558i
$905$	−24.8831			−0.827143
$906$		0			0
$907$	−22.0032			−0.730605			−0.365303	−	0.930889i	$-0.619034\pi$
$907$	−22.0032			−0.730605			−0.365303	+	0.930889i	$0.619034\pi$
$908$	47.7411			1.58434
$909$		0			0
$910$	7.34605			0.243519
$911$			38.7865i			1.28506i	0.766262	+	0.642528i	$0.222114\pi$
$911$			38.7865i			1.28506i	−0.766262	+	0.642528i	$0.777886\pi$
$912$		0			0
$913$		−	0.153285i		−	0.00507300i
$914$	−40.9243			−1.35366
$915$		0			0
$916$	134.209			4.43440
$917$	−0.706813			−0.0233410
$918$		0			0
$919$	12.3085			0.406019			0.203009	−	0.979177i	$-0.434928\pi$
$919$	12.3085			0.406019			0.203009	+	0.979177i	$0.434928\pi$
$920$			175.373i			5.78186i
$921$		0			0
$922$	−89.6430			−2.95224
$923$		−	2.36585i		−	0.0778729i
$924$		0			0
$925$			8.86643i			0.291526i
$926$	50.7945			1.66921
$927$		0			0
$928$	−105.797			−3.47296
$929$		−	14.3617i		−	0.471193i	−0.971851	−	0.235597i	$-0.924296\pi$
$929$		−	14.3617i		−	0.471193i	0.971851	−	0.235597i	$-0.0757044\pi$
$930$		0			0
$931$	−22.5331			−0.738492
$932$		−	19.3594i		−	0.634137i
$933$		0			0
$934$	103.448			3.38493
$935$			0.0266941i			0.000872990i
$936$		0			0
$937$			8.37181i			0.273495i	0.990606	+	0.136748i	$0.0436649\pi$
$937$			8.37181i			0.273495i	−0.990606	+	0.136748i	$0.956335\pi$
$938$	−5.85028			−0.191018
$939$		0			0
$940$		−	97.0674i		−	3.16599i
$941$	16.4631			0.536682			0.268341	−	0.963324i	$-0.413525\pi$
$941$	16.4631			0.536682			0.268341	+	0.963324i	$0.413525\pi$
$942$		0			0
$943$	−47.3496			−1.54192
$944$	162.827			5.29957
$945$		0			0
$946$			0.882619i			0.0286964i
$947$			47.2501i			1.53542i	0.640797	+	0.767710i	$0.278604\pi$
$947$			47.2501i			1.53542i	−0.640797	+	0.767710i	$0.721396\pi$
$948$		0			0
$949$	−36.6040			−1.18822
$950$		−	43.5342i		−	1.41243i
$951$		0			0
$952$			0.0726019i			0.00235304i
$953$	27.6233			0.894806			0.447403	−	0.894333i	$-0.352349\pi$
$953$	27.6233			0.894806			0.447403	+	0.894333i	$0.352349\pi$
$954$		0			0
$955$			42.8486i			1.38655i
$956$		−	92.3708i		−	2.98749i
$957$		0			0
$958$		−	36.4993i		−	1.17924i
$959$		−	1.55469i		−	0.0502036i
$960$		0			0
$961$	−14.7632			−0.476231
$962$			15.5688i			0.501958i
$963$		0			0
$964$		−	76.5758i		−	2.46634i
$965$		−	32.0755i		−	1.03255i
$966$		0			0
$967$	−2.74580			−0.0882990			−0.0441495	−	0.999025i	$-0.514058\pi$
$967$	−2.74580			−0.0882990			−0.0441495	+	0.999025i	$0.514058\pi$
$968$			91.1729i			2.93041i
$969$		0			0
$970$		−	80.5423i		−	2.58606i
$971$	16.9504			0.543964			0.271982	−	0.962302i	$-0.412321\pi$
$971$	16.9504			0.543964			0.271982	+	0.962302i	$0.412321\pi$
$972$		0			0
$973$			4.84803i			0.155421i
$974$	−49.7286			−1.59341
$975$		0			0
$976$	142.276			4.55414
$977$	−36.3709			−1.16361			−0.581805	−	0.813328i	$-0.697653\pi$
$977$	−36.3709			−1.16361			−0.581805	+	0.813328i	$0.697653\pi$
$978$		0			0
$979$	−1.29689			−0.0414488
$980$	−112.391			−3.59019
$981$		0			0
$982$	25.0978			0.800902
$983$	53.8846			1.71865			0.859326	−	0.511429i	$-0.170884\pi$
$983$	53.8846			1.71865			0.859326	+	0.511429i	$0.170884\pi$
$984$		0			0
$985$	−68.6591			−2.18766
$986$	0.607136			0.0193351
$987$		0			0
$988$		−	54.9817i		−	1.74920i
$989$		−	8.61072i		−	0.273805i
$990$		0			0
$991$			55.2505i			1.75509i	0.479494	+	0.877545i	$0.340820\pi$
$991$			55.2505i			1.75509i	−0.479494	+	0.877545i	$0.659180\pi$
$992$		−	61.9338i		−	1.96640i
$993$		0			0
$994$		−	0.504130i		−	0.0159900i
$995$	−69.5420			−2.20463
$996$		0			0
$997$	−21.2566			−0.673202			−0.336601	−	0.941647i	$-0.609277\pi$
$997$	−21.2566			−0.673202			−0.336601	+	0.941647i	$0.609277\pi$
$998$	52.1677			1.65134
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.c.a.1502.3	✓	56
3.2	odd	2	inner	1503.2.c.a.1502.54	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.53	yes	56
501.500	even	2	inner	1503.2.c.a.1502.4	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.2.c.a.1502.3	✓	56	1.1	even	1	trivial
1503.2.c.a.1502.4	yes	56	501.500	even	2	inner
1503.2.c.a.1502.53	yes	56	167.166	odd	2	inner
1503.2.c.a.1502.54	yes	56	3.2	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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.66906i q^{2} -5.12388 q^{4} -3.16492 q^{5} -0.263490 q^{7} +8.33782i q^{8} +O(q^{10})\)	\(q-2.66906i q^{2} -5.12388 q^{4} -3.16492 q^{5} -0.263490 q^{7} +8.33782i q^{8} +8.44736i q^{10} +0.255224i q^{11} +3.30041i q^{13} +0.703271i q^{14} +12.0064 q^{16} -0.0330469 q^{17} +3.25126 q^{19} +16.2167 q^{20} +0.681208 q^{22} -6.64578 q^{23} +5.01673 q^{25} +8.80899 q^{26} +1.35009 q^{28} -6.88330i q^{29} +4.02950 q^{31} -15.3701i q^{32} +0.0882042i q^{34} +0.833926 q^{35} +1.76737i q^{37} -8.67780i q^{38} -26.3886i q^{40} +7.12477 q^{41} +1.29567i q^{43} -1.30774i q^{44} +17.7380i q^{46} -5.98565i q^{47} -6.93057 q^{49} -13.3899i q^{50} -16.9109i q^{52} -3.33373 q^{53} -0.807764i q^{55} -2.19694i q^{56} -18.3719 q^{58} +13.5617 q^{59} +11.8500 q^{61} -10.7550i q^{62} -17.0110 q^{64} -10.4455i q^{65} +8.31866i q^{67} +0.169328 q^{68} -2.22580i q^{70} -0.716835 q^{71} +11.0907i q^{73} +4.71723 q^{74} -16.6591 q^{76} -0.0672491i q^{77} +2.66308i q^{79} -37.9993 q^{80} -19.0164i q^{82} -0.600591 q^{83} +0.104591 q^{85} +3.45821 q^{86} -2.12801 q^{88} +5.08137i q^{89} -0.869626i q^{91} +34.0522 q^{92} -15.9761 q^{94} -10.2900 q^{95} -9.53461 q^{97} +18.4981i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 48 q^{4}+O(q^{10}) \) 56 * q - 48 * q^4	\( 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) \) 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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