LMFDB - Embedded newform 1815.2.c.d.364.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,2,Mod(364,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1815.364");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1815.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:	$14.4928479669$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			364.2
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	1815.364
Dual form			1815.2.c.d.364.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53919i q^{2} +1.00000i q^{3} -0.369102 q^{4} +(2.17009 - 0.539189i) q^{5} +1.53919 q^{6} -0.290725i q^{7} -2.51026i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.53919i q^{2} +1.00000i q^{3} -0.369102 q^{4} +(2.17009 - 0.539189i) q^{5} +1.53919 q^{6} -0.290725i q^{7} -2.51026i q^{8} -1.00000 q^{9} +(-0.829914 - 3.34017i) q^{10} -0.369102i q^{12} +6.97107i q^{13} -0.447480 q^{14} +(0.539189 + 2.17009i) q^{15} -4.60197 q^{16} -4.78765i q^{17} +1.53919i q^{18} +7.75872 q^{19} +(-0.800984 + 0.199016i) q^{20} +0.290725 q^{21} -4.00000i q^{23} +2.51026 q^{24} +(4.41855 - 2.34017i) q^{25} +10.7298 q^{26} -1.00000i q^{27} +0.107307i q^{28} -7.41855 q^{29} +(3.34017 - 0.829914i) q^{30} +6.34017 q^{31} +2.06278i q^{32} -7.36910 q^{34} +(-0.156755 - 0.630898i) q^{35} +0.369102 q^{36} -3.41855i q^{37} -11.9421i q^{38} -6.97107 q^{39} +(-1.35350 - 5.44748i) q^{40} +7.41855 q^{41} -0.447480i q^{42} +0.290725i q^{43} +(-2.17009 + 0.539189i) q^{45} -6.15676 q^{46} +5.26180i q^{47} -4.60197i q^{48} +6.91548 q^{49} +(-3.60197 - 6.80098i) q^{50} +4.78765 q^{51} -2.57304i q^{52} -5.75872i q^{53} -1.53919 q^{54} -0.729794 q^{56} +7.75872i q^{57} +11.4186i q^{58} -3.60197 q^{59} +(-0.199016 - 0.800984i) q^{60} +6.68035 q^{61} -9.75872i q^{62} +0.290725i q^{63} -6.02893 q^{64} +(3.75872 + 15.1278i) q^{65} +6.15676i q^{67} +1.76713i q^{68} +4.00000 q^{69} +(-0.971071 + 0.241276i) q^{70} -5.07838 q^{71} +2.51026i q^{72} -1.12783i q^{73} -5.26180 q^{74} +(2.34017 + 4.41855i) q^{75} -2.86376 q^{76} +10.7298i q^{78} -0.921622 q^{79} +(-9.98667 + 2.48133i) q^{80} +1.00000 q^{81} -11.4186i q^{82} +1.70928i q^{83} -0.107307 q^{84} +(-2.58145 - 10.3896i) q^{85} +0.447480 q^{86} -7.41855i q^{87} -4.34017 q^{89} +(0.829914 + 3.34017i) q^{90} +2.02666 q^{91} +1.47641i q^{92} +6.34017i q^{93} +8.09890 q^{94} +(16.8371 - 4.18342i) q^{95} -2.06278 q^{96} +4.68035i q^{97} -10.6442i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 10 * q^4 + 2 * q^5 + 6 * q^6 - 6 * q^9	$ 6 q - 10 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{9} - 16 q^{10} - 4 q^{14} + 10 q^{16} - 4 q^{19} + 14 q^{20} + 16 q^{21} - 18 q^{24} - 2 q^{25} - 16 q^{26} - 16 q^{29} - 2 q^{30} + 16 q^{31} - 52 q^{34} + 12 q^{35} + 10 q^{36} - 12 q^{39} + 12 q^{40} + 16 q^{41} - 2 q^{45} - 24 q^{46} - 22 q^{49} + 16 q^{50} + 8 q^{51} - 6 q^{54} + 76 q^{56} + 16 q^{59} - 20 q^{60} - 4 q^{61} - 66 q^{64} - 28 q^{65} + 24 q^{69} + 24 q^{70} - 24 q^{71} - 16 q^{74} - 8 q^{75} + 36 q^{76} - 12 q^{79} - 58 q^{80} + 6 q^{81} - 24 q^{84} - 44 q^{85} + 4 q^{86} - 4 q^{89} + 16 q^{90} + 16 q^{91} - 24 q^{94} + 44 q^{95} + 22 q^{96}+O(q^{100}) $ 6 * q - 10 * q^4 + 2 * q^5 + 6 * q^6 - 6 * q^9 - 16 * q^10 - 4 * q^14 + 10 * q^16 - 4 * q^19 + 14 * q^20 + 16 * q^21 - 18 * q^24 - 2 * q^25 - 16 * q^26 - 16 * q^29 - 2 * q^30 + 16 * q^31 - 52 * q^34 + 12 * q^35 + 10 * q^36 - 12 * q^39 + 12 * q^40 + 16 * q^41 - 2 * q^45 - 24 * q^46 - 22 * q^49 + 16 * q^50 + 8 * q^51 - 6 * q^54 + 76 * q^56 + 16 * q^59 - 20 * q^60 - 4 * q^61 - 66 * q^64 - 28 * q^65 + 24 * q^69 + 24 * q^70 - 24 * q^71 - 16 * q^74 - 8 * q^75 + 36 * q^76 - 12 * q^79 - 58 * q^80 + 6 * q^81 - 24 * q^84 - 44 * q^85 + 4 * q^86 - 4 * q^89 + 16 * q^90 + 16 * q^91 - 24 * q^94 + 44 * q^95 + 22 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$.

$n$	$727$	$1211$	$1696$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	1.53919i		−	1.08837i	−0.838965	−	0.544185i	$-0.816839\pi$
$2$		−	1.53919i		−	1.08837i	0.838965	−	0.544185i	$-0.183161\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−0.369102			−0.184551
$5$	2.17009	−	0.539189i	0.970492	−	0.241133i
$5$	2.17009	−	0.539189i	0.970492	−	0.241133i
$6$	1.53919			0.628371
$7$		−	0.290725i		−	0.109884i	−0.998490	−	0.0549418i	$-0.982503\pi$
$7$		−	0.290725i		−	0.109884i	0.998490	−	0.0549418i	$-0.0174973\pi$
$8$		−	2.51026i		−	0.887511i
$9$	−1.00000			−0.333333
$10$	−0.829914	−	3.34017i	−0.262442	−	1.05626i
$11$		0			0
$11$		0			0
$12$		−	0.369102i		−	0.106551i
$13$			6.97107i			1.93343i	0.255861	+	0.966714i	$0.417641\pi$
$13$			6.97107i			1.93343i	−0.255861	+	0.966714i	$0.582359\pi$
$14$	−0.447480			−0.119594
$15$	0.539189	+	2.17009i	0.139218	+	0.560314i
$16$	−4.60197			−1.15049
$17$		−	4.78765i		−	1.16118i	−0.814197	−	0.580588i	$-0.802823\pi$
$17$		−	4.78765i		−	1.16118i	0.814197	−	0.580588i	$-0.197177\pi$
$18$			1.53919i			0.362790i
$19$	7.75872			1.77997			0.889987	−	0.455987i	$-0.150714\pi$
$19$	7.75872			1.77997			0.889987	+	0.455987i	$0.150714\pi$
$20$	−0.800984	+	0.199016i	−0.179105	+	0.0445013i
$21$	0.290725			0.0634413
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	2.51026			0.512405
$25$	4.41855	−	2.34017i	0.883710	−	0.468035i
$26$	10.7298			2.10429
$27$		−	1.00000i		−	0.192450i
$28$			0.107307i			0.0202791i
$29$	−7.41855			−1.37759			−0.688795	−	0.724956i	$-0.741860\pi$
$29$	−7.41855			−1.37759			−0.688795	+	0.724956i	$0.741860\pi$
$30$	3.34017	−	0.829914i	0.609829	−	0.151521i
$31$	6.34017			1.13873			0.569364	−	0.822085i	$-0.307189\pi$
$31$	6.34017			1.13873			0.569364	+	0.822085i	$0.307189\pi$
$32$			2.06278i			0.364651i
$33$		0			0
$34$	−7.36910			−1.26379
$35$	−0.156755	−	0.630898i	−0.0264965	−	0.106641i
$36$	0.369102			0.0615171
$37$		−	3.41855i		−	0.562006i	−0.959707	−	0.281003i	$-0.909333\pi$
$37$		−	3.41855i		−	0.562006i	0.959707	−	0.281003i	$-0.0906671\pi$
$38$		−	11.9421i		−	1.93727i
$39$	−6.97107			−1.11626
$40$	−1.35350	−	5.44748i	−0.214008	−	0.861322i
$41$	7.41855			1.15858			0.579291	−	0.815120i	$-0.303329\pi$
$41$	7.41855			1.15858			0.579291	+	0.815120i	$0.303329\pi$
$42$		−	0.447480i		−	0.0690477i
$43$			0.290725i			0.0443351i	0.999754	+	0.0221675i	$0.00705673\pi$
$43$			0.290725i			0.0443351i	−0.999754	+	0.0221675i	$0.992943\pi$
$44$		0			0
$45$	−2.17009	+	0.539189i	−0.323497	+	0.0803775i
$46$	−6.15676			−0.907764
$47$			5.26180i			0.767512i	0.923435	+	0.383756i	$0.125370\pi$
$47$			5.26180i			0.767512i	−0.923435	+	0.383756i	$0.874630\pi$
$48$		−	4.60197i		−	0.664237i
$49$	6.91548			0.987926
$50$	−3.60197	−	6.80098i	−0.509395	−	0.961804i
$51$	4.78765			0.670406
$52$		−	2.57304i		−	0.356816i
$53$		−	5.75872i		−	0.791022i	−0.918461	−	0.395511i	$-0.870568\pi$
$53$		−	5.75872i		−	0.791022i	0.918461	−	0.395511i	$-0.129432\pi$
$54$	−1.53919			−0.209457
$55$		0			0
$56$	−0.729794			−0.0975229
$57$			7.75872i			1.02767i
$58$			11.4186i			1.49933i
$59$	−3.60197			−0.468936			−0.234468	−	0.972124i	$-0.575335\pi$
$59$	−3.60197			−0.468936			−0.234468	+	0.972124i	$0.575335\pi$
$60$	−0.199016	−	0.800984i	−0.0256928	−	0.103407i
$61$	6.68035			0.855331			0.427665	−	0.903937i	$-0.359336\pi$
$61$	6.68035			0.855331			0.427665	+	0.903937i	$0.359336\pi$
$62$		−	9.75872i		−	1.23936i
$63$			0.290725i			0.0366279i
$64$	−6.02893			−0.753616
$65$	3.75872	+	15.1278i	0.466212	+	1.87638i
$66$		0			0
$67$			6.15676i			0.752167i	0.926586	+	0.376084i	$0.122729\pi$
$67$			6.15676i			0.752167i	−0.926586	+	0.376084i	$0.877271\pi$
$68$			1.76713i			0.214296i
$69$	4.00000			0.481543
$70$	−0.971071	+	0.241276i	−0.116065	+	0.0288380i
$71$	−5.07838			−0.602693			−0.301346	−	0.953515i	$-0.597436\pi$
$71$	−5.07838			−0.602693			−0.301346	+	0.953515i	$0.597436\pi$
$72$			2.51026i			0.295837i
$73$		−	1.12783i		−	0.132002i	−0.997820	−	0.0660010i	$-0.978976\pi$
$73$		−	1.12783i		−	0.132002i	0.997820	−	0.0660010i	$-0.0210241\pi$
$74$	−5.26180			−0.611671
$75$	2.34017	+	4.41855i	0.270220	+	0.510210i
$76$	−2.86376			−0.328496
$77$		0			0
$78$			10.7298i			1.21491i
$79$	−0.921622			−0.103691			−0.0518453	−	0.998655i	$-0.516510\pi$
$79$	−0.921622			−0.103691			−0.0518453	+	0.998655i	$0.516510\pi$
$80$	−9.98667	+	2.48133i	−1.11654	+	0.277421i
$81$	1.00000			0.111111
$82$		−	11.4186i		−	1.26097i
$83$			1.70928i			0.187617i	0.995590	+	0.0938087i	$0.0299042\pi$
$83$			1.70928i			0.187617i	−0.995590	+	0.0938087i	$0.970096\pi$
$84$	−0.107307			−0.0117082
$85$	−2.58145	−	10.3896i	−0.279997	−	1.12691i
$86$	0.447480			0.0482530
$87$		−	7.41855i		−	0.795352i
$88$		0			0
$89$	−4.34017			−0.460057			−0.230029	−	0.973184i	$-0.573882\pi$
$89$	−4.34017			−0.460057			−0.230029	+	0.973184i	$0.573882\pi$
$90$	0.829914	+	3.34017i	0.0874806	+	0.352085i
$91$	2.02666			0.212452
$92$			1.47641i			0.153926i
$93$			6.34017i			0.657445i
$94$	8.09890			0.835337
$95$	16.8371	−	4.18342i	1.72745	−	0.429210i
$96$	−2.06278			−0.210532
$97$			4.68035i			0.475217i	0.971361	+	0.237609i	$0.0763635\pi$
$97$			4.68035i			0.475217i	−0.971361	+	0.237609i	$0.923636\pi$
$98$		−	10.6442i		−	1.07523i
$99$		0			0
$100$	−1.63090	+	0.863763i	−0.163090	+	0.0863763i
$101$	8.58145			0.853886			0.426943	−	0.904279i	$-0.359590\pi$
$101$	8.58145			0.853886			0.426943	+	0.904279i	$0.359590\pi$
$102$		−	7.36910i		−	0.729650i
$103$		−	6.73820i		−	0.663935i	−0.943291	−	0.331968i	$-0.892288\pi$
$103$		−	6.73820i		−	0.663935i	0.943291	−	0.331968i	$-0.107712\pi$
$104$	17.4992			1.71594
$105$	0.630898	−	0.156755i	0.0615693	−	0.0152978i
$106$	−8.86376			−0.860925
$107$		−	12.2329i		−	1.18260i	−0.806453	−	0.591298i	$-0.798616\pi$
$107$		−	12.2329i		−	1.18260i	0.806453	−	0.591298i	$-0.201384\pi$
$108$			0.369102i			0.0355169i
$109$	−6.31351			−0.604725			−0.302362	−	0.953193i	$-0.597775\pi$
$109$	−6.31351			−0.604725			−0.302362	+	0.953193i	$0.597775\pi$
$110$		0			0
$111$	3.41855			0.324474
$112$			1.33791i			0.126420i
$113$			16.4969i			1.55190i	0.630794	+	0.775950i	$0.282729\pi$
$113$			16.4969i			1.55190i	−0.630794	+	0.775950i	$0.717271\pi$
$114$	11.9421			1.11848
$115$	−2.15676	−	8.68035i	−0.201118	−	0.809446i
$116$	2.73820			0.254236
$117$		−	6.97107i		−	0.644476i
$118$			5.54411i			0.510377i
$119$	−1.39189			−0.127594
$120$	5.44748	−	1.35350i	0.497285	−	0.123557i
$121$		0			0
$122$		−	10.2823i		−	0.930917i
$123$			7.41855i			0.668908i
$124$	−2.34017			−0.210154
$125$	8.32684	−	7.46081i	0.744775	−	0.667315i
$126$	0.447480			0.0398647
$127$		−	4.87217i		−	0.432336i	−0.976356	−	0.216168i	$-0.930644\pi$
$127$		−	4.87217i		−	0.432336i	0.976356	−	0.216168i	$-0.0693558\pi$
$128$			13.4052i			1.18487i
$129$	−0.290725			−0.0255969
$130$	23.2846	−	5.78539i	2.04219	−	0.507412i
$131$	8.68035			0.758405			0.379203	−	0.925314i	$-0.376198\pi$
$131$	8.68035			0.758405			0.379203	+	0.925314i	$0.376198\pi$
$132$		0			0
$133$		−	2.25565i		−	0.195590i
$134$	9.47641			0.818637
$135$	−0.539189	−	2.17009i	−0.0464060	−	0.186771i
$136$	−12.0183			−1.03056
$137$		−	12.5958i		−	1.07613i	−0.842902	−	0.538067i	$-0.819155\pi$
$137$		−	12.5958i		−	1.07613i	0.842902	−	0.538067i	$-0.180845\pi$
$138$		−	6.15676i		−	0.524098i
$139$	−9.91548			−0.841020			−0.420510	−	0.907288i	$-0.638149\pi$
$139$	−9.91548			−0.841020			−0.420510	+	0.907288i	$0.638149\pi$
$140$	0.0578588	+	0.232866i	0.00488996	+	0.0196808i
$141$	−5.26180			−0.443123
$142$			7.81658i			0.655953i
$143$		0			0
$144$	4.60197			0.383497
$145$	−16.0989	+	4.00000i	−1.33694	+	0.332182i
$146$	−1.73594			−0.143667
$147$			6.91548i			0.570379i
$148$			1.26180i			0.103719i
$149$	−1.26180			−0.103370			−0.0516851	−	0.998663i	$-0.516459\pi$
$149$	−1.26180			−0.103370			−0.0516851	+	0.998663i	$0.516459\pi$
$150$	6.80098	−	3.60197i	0.555298	−	0.294099i
$151$	−1.60197			−0.130366			−0.0651832	−	0.997873i	$-0.520763\pi$
$151$	−1.60197			−0.130366			−0.0651832	+	0.997873i	$0.520763\pi$
$152$		−	19.4764i		−	1.57975i
$153$			4.78765i			0.387059i
$154$		0			0
$155$	13.7587	−	3.41855i	1.10513	−	0.274585i
$156$	2.57304			0.206008
$157$		−	7.10504i		−	0.567044i	−0.958966	−	0.283522i	$-0.908497\pi$
$157$		−	7.10504i		−	0.567044i	0.958966	−	0.283522i	$-0.0915029\pi$
$158$			1.41855i			0.112854i
$159$	5.75872			0.456696
$160$	1.11223	+	4.47641i	0.0879293	+	0.353891i
$161$	−1.16290			−0.0916492
$162$		−	1.53919i		−	0.120930i
$163$			22.9360i			1.79649i	0.439499	+	0.898243i	$0.355156\pi$
$163$			22.9360i			1.79649i	−0.439499	+	0.898243i	$0.644844\pi$
$164$	−2.73820			−0.213818
$165$		0			0
$166$	2.63090			0.204197
$167$		−	4.81432i		−	0.372543i	−0.982498	−	0.186271i	$-0.940360\pi$
$167$		−	4.81432i		−	0.372543i	0.982498	−	0.186271i	$-0.0596404\pi$
$168$		−	0.729794i		−	0.0563049i
$169$	−35.5958			−2.73814
$170$	−15.9916	+	3.97334i	−1.22650	+	0.304741i
$171$	−7.75872			−0.593324
$172$		−	0.107307i		−	0.00818209i
$173$		−	12.8865i		−	0.979746i	−0.871794	−	0.489873i	$-0.837043\pi$
$173$		−	12.8865i		−	0.979746i	0.871794	−	0.489873i	$-0.162957\pi$
$174$	−11.4186			−0.865638
$175$	−0.680346	−	1.28458i	−0.0514293	−	0.0971052i
$176$		0			0
$177$		−	3.60197i		−	0.270741i
$178$			6.68035i			0.500713i
$179$	−1.84324			−0.137771			−0.0688853	−	0.997625i	$-0.521944\pi$
$179$	−1.84324			−0.137771			−0.0688853	+	0.997625i	$0.521944\pi$
$180$	0.800984	−	0.199016i	0.0597018	−	0.0148338i
$181$	10.2823			0.764278			0.382139	−	0.924105i	$-0.375187\pi$
$181$	10.2823			0.764278			0.382139	+	0.924105i	$0.375187\pi$
$182$		−	3.11942i		−	0.231226i
$183$			6.68035i			0.493825i
$184$	−10.0410			−0.740235
$185$	−1.84324	−	7.41855i	−0.135518	−	0.545423i
$186$	9.75872			0.715544
$187$		0			0
$188$		−	1.94214i		−	0.141645i
$189$	−0.290725			−0.0211471
$190$	−6.43907	−	25.9155i	−0.467139	−	1.88011i
$191$	−11.5174			−0.833373			−0.416687	−	0.909050i	$-0.636809\pi$
$191$	−11.5174			−0.833373			−0.416687	+	0.909050i	$0.636809\pi$
$192$		−	6.02893i		−	0.435101i
$193$			3.86603i			0.278283i	0.990273	+	0.139141i	$0.0444343\pi$
$193$			3.86603i			0.278283i	−0.990273	+	0.139141i	$0.955566\pi$
$194$	7.20394			0.517212
$195$	−15.1278	+	3.75872i	−1.08333	+	0.269168i
$196$	−2.55252			−0.182323
$197$			8.57304i			0.610804i	0.952224	+	0.305402i	$0.0987908\pi$
$197$			8.57304i			0.610804i	−0.952224	+	0.305402i	$0.901209\pi$
$198$		0			0
$199$	−8.31351			−0.589329			−0.294665	−	0.955601i	$-0.595208\pi$
$199$	−8.31351			−0.589329			−0.294665	+	0.955601i	$0.595208\pi$
$200$	−5.87444	−	11.0917i	−0.415386	−	0.784302i
$201$	−6.15676			−0.434264
$202$		−	13.2085i		−	0.929345i
$203$			2.15676i			0.151375i
$204$	−1.76713			−0.123724
$205$	16.0989	−	4.00000i	1.12440	−	0.279372i
$206$	−10.3714			−0.722608
$207$			4.00000i			0.278019i
$208$		−	32.0806i		−	2.22439i
$209$		0			0
$210$	−0.241276	−	0.971071i	−0.0166496	−	0.0670102i
$211$	−25.9155			−1.78410			−0.892048	−	0.451941i	$-0.850732\pi$
$211$	−25.9155			−1.78410			−0.892048	+	0.451941i	$0.850732\pi$
$212$			2.12556i			0.145984i
$213$		−	5.07838i		−	0.347965i
$214$	−18.8287			−1.28710
$215$	0.156755	+	0.630898i	0.0106906	+	0.0430269i
$216$	−2.51026			−0.170802
$217$		−	1.84324i		−	0.125128i
$218$			9.71769i			0.658165i
$219$	1.12783			0.0762114
$220$		0			0
$221$	33.3751			2.24505
$222$		−	5.26180i		−	0.353149i
$223$			9.62863i			0.644781i	0.946607	+	0.322390i	$0.104486\pi$
$223$			9.62863i			0.644781i	−0.946607	+	0.322390i	$0.895514\pi$
$224$	0.599701			0.0400692
$225$	−4.41855	+	2.34017i	−0.294570	+	0.156012i
$226$	25.3919			1.68904
$227$			18.3896i			1.22056i	0.792185	+	0.610281i	$0.208943\pi$
$227$			18.3896i			1.22056i	−0.792185	+	0.610281i	$0.791057\pi$
$228$		−	2.86376i		−	0.189657i
$229$	−17.1506			−1.13334			−0.566672	−	0.823943i	$-0.691769\pi$
$229$	−17.1506			−1.13334			−0.566672	+	0.823943i	$0.691769\pi$
$230$	−13.3607	+	3.31965i	−0.880978	+	0.218892i
$231$		0			0
$232$			18.6225i			1.22263i
$233$		−	4.10731i		−	0.269079i	−0.990908	−	0.134539i	$-0.957045\pi$
$233$		−	4.10731i		−	0.269079i	0.990908	−	0.134539i	$-0.0429555\pi$
$234$	−10.7298			−0.701429
$235$	2.83710	+	11.4186i	0.185072	+	0.744864i
$236$	1.32950			0.0865428
$237$		−	0.921622i		−	0.0598658i
$238$			2.14238i			0.138870i
$239$	−4.36683			−0.282467			−0.141234	−	0.989976i	$-0.545107\pi$
$239$	−4.36683			−0.282467			−0.141234	+	0.989976i	$0.545107\pi$
$240$	−2.48133	−	9.98667i	−0.160169	−	0.644637i
$241$	5.20394			0.335215			0.167608	−	0.985854i	$-0.446396\pi$
$241$	5.20394			0.335215			0.167608	+	0.985854i	$0.446396\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$	−2.46573			−0.157852
$245$	15.0072	−	3.72875i	0.958774	−	0.238221i
$246$	11.4186			0.728020
$247$			54.0866i			3.44145i
$248$		−	15.9155i		−	1.01063i
$249$	−1.70928			−0.108321
$250$	−11.4836	−	12.8166i	−0.726286	−	0.810592i
$251$	−8.28231			−0.522775			−0.261388	−	0.965234i	$-0.584180\pi$
$251$	−8.28231			−0.522775			−0.261388	+	0.965234i	$0.584180\pi$
$252$		−	0.107307i		−	0.00675972i
$253$		0			0
$254$	−7.49920			−0.470541
$255$	10.3896	−	2.58145i	0.650623	−	0.161657i
$256$	8.57531			0.535957
$257$			11.0205i			0.687441i	0.939072	+	0.343721i	$0.111687\pi$
$257$			11.0205i			0.687441i	−0.939072	+	0.343721i	$0.888313\pi$
$258$			0.447480i			0.0278589i
$259$	−0.993857			−0.0617553
$260$	−1.38735	−	5.58372i	−0.0860400	−	0.346287i
$261$	7.41855			0.459197
$262$		−	13.3607i		−	0.825426i
$263$		−	2.97107i		−	0.183204i	−0.995796	−	0.0916020i	$-0.970801\pi$
$263$		−	2.97107i		−	0.183204i	0.995796	−	0.0916020i	$-0.0291988\pi$
$264$		0			0
$265$	−3.10504	−	12.4969i	−0.190741	−	0.767680i
$266$	−3.47187			−0.212874
$267$		−	4.34017i		−	0.265614i
$268$		−	2.27247i		−	0.138813i
$269$	15.8432			0.965980			0.482990	−	0.875626i	$-0.339551\pi$
$269$	15.8432			0.965980			0.482990	+	0.875626i	$0.339551\pi$
$270$	−3.34017	+	0.829914i	−0.203276	+	0.0505069i
$271$	6.28231			0.381623			0.190812	−	0.981627i	$-0.438888\pi$
$271$	6.28231			0.381623			0.190812	+	0.981627i	$0.438888\pi$
$272$			22.0326i			1.33592i
$273$			2.02666i			0.122659i
$274$	−19.3874			−1.17123
$275$		0			0
$276$	−1.47641			−0.0888694
$277$		−	5.12783i		−	0.308101i	−0.988063	−	0.154051i	$-0.950768\pi$
$277$		−	5.12783i		−	0.308101i	0.988063	−	0.154051i	$-0.0492319\pi$
$278$			15.2618i			0.915342i
$279$	−6.34017			−0.379576
$280$	−1.58372	+	0.393497i	−0.0946452	+	0.0235159i
$281$	−21.8888			−1.30578			−0.652889	−	0.757454i	$-0.726443\pi$
$281$	−21.8888			−1.30578			−0.652889	+	0.757454i	$0.726443\pi$
$282$			8.09890i			0.482282i
$283$			25.9649i			1.54345i	0.635954	+	0.771727i	$0.280607\pi$
$283$			25.9649i			1.54345i	−0.635954	+	0.771727i	$0.719393\pi$
$284$	1.87444			0.111228
$285$	4.18342	+	16.8371i	0.247804	+	0.997344i
$286$		0			0
$287$		−	2.15676i		−	0.127309i
$288$		−	2.06278i		−	0.121550i
$289$	−5.92162			−0.348331
$290$	6.15676	+	24.7792i	0.361537	+	1.45509i
$291$	−4.68035			−0.274367
$292$			0.416283i			0.0243611i
$293$			10.4163i			0.608526i	0.952588	+	0.304263i	$0.0984101\pi$
$293$			10.4163i			0.608526i	−0.952588	+	0.304263i	$0.901590\pi$
$294$	10.6442			0.620784
$295$	−7.81658	+	1.94214i	−0.455099	+	0.113076i
$296$	−8.58145			−0.498787
$297$		0			0
$298$			1.94214i			0.112505i
$299$	27.8843			1.61259
$300$	−0.863763	−	1.63090i	−0.0498694	−	0.0941599i
$301$	0.0845208			0.00487170
$302$			2.46573i			0.141887i
$303$			8.58145i			0.492991i
$304$	−35.7054			−2.04785
$305$	14.4969	−	3.60197i	0.830092	−	0.206248i
$306$	7.36910			0.421264
$307$			29.0700i			1.65911i	0.558425	+	0.829555i	$0.311406\pi$
$307$			29.0700i			1.65911i	−0.558425	+	0.829555i	$0.688594\pi$
$308$		0			0
$309$	6.73820			0.383323
$310$	−5.26180	−	21.1773i	−0.298850	−	1.20279i
$311$	5.44521			0.308770			0.154385	−	0.988011i	$-0.450660\pi$
$311$	5.44521			0.308770			0.154385	+	0.988011i	$0.450660\pi$
$312$			17.4992i			0.990697i
$313$		−	25.0928i		−	1.41833i	−0.705044	−	0.709163i	$-0.749073\pi$
$313$		−	25.0928i		−	1.41833i	0.705044	−	0.709163i	$-0.250927\pi$
$314$	−10.9360			−0.617154
$315$	0.156755	+	0.630898i	0.00883217	+	0.0355471i
$316$	0.340173			0.0191362
$317$			21.1773i			1.18943i	0.803935	+	0.594717i	$0.202736\pi$
$317$			21.1773i			1.18943i	−0.803935	+	0.594717i	$0.797264\pi$
$318$		−	8.86376i		−	0.497055i
$319$		0			0
$320$	−13.0833	+	3.25073i	−0.731379	+	0.181721i
$321$	12.2329			0.682772
$322$			1.78992i			0.0997484i
$323$		−	37.1461i		−	2.06686i
$324$	−0.369102			−0.0205057
$325$	16.3135	+	30.8020i	0.904911	+	1.70859i
$326$	35.3028			1.95524
$327$		−	6.31351i		−	0.349138i
$328$		−	18.6225i		−	1.02825i
$329$	1.52973			0.0843369
$330$		0			0
$331$	−6.70701			−0.368650			−0.184325	−	0.982865i	$-0.559010\pi$
$331$	−6.70701			−0.368650			−0.184325	+	0.982865i	$0.559010\pi$
$332$		−	0.630898i		−	0.0346250i
$333$			3.41855i			0.187335i
$334$	−7.41014			−0.405465
$335$	3.31965	+	13.3607i	0.181372	+	0.729973i
$336$	−1.33791			−0.0729887
$337$		−	23.7503i		−	1.29376i	−0.762591	−	0.646881i	$-0.776073\pi$
$337$		−	23.7503i		−	1.29376i	0.762591	−	0.646881i	$-0.223927\pi$
$338$			54.7887i			2.98011i
$339$	−16.4969			−0.895990
$340$	0.952819	+	3.83483i	0.0516739	+	0.207973i
$341$		0			0
$342$			11.9421i			0.645757i
$343$		−	4.04557i		−	0.218440i
$344$	0.729794			0.0393479
$345$	8.68035	−	2.15676i	0.467334	−	0.116116i
$346$	−19.8348			−1.06633
$347$			31.1689i			1.67323i	0.547790	+	0.836616i	$0.315469\pi$
$347$			31.1689i			1.67323i	−0.547790	+	0.836616i	$0.684531\pi$
$348$			2.73820i			0.146783i
$349$	11.0472			0.591342			0.295671	−	0.955290i	$-0.404457\pi$
$349$	11.0472			0.591342			0.295671	+	0.955290i	$0.404457\pi$
$350$	−1.97721	+	1.04718i	−0.105687	+	0.0559742i
$351$	6.97107			0.372088
$352$		0			0
$353$			27.4329i			1.46011i	0.683390	+	0.730054i	$0.260505\pi$
$353$			27.4329i			1.46011i	−0.683390	+	0.730054i	$0.739495\pi$
$354$	−5.54411			−0.294666
$355$	−11.0205	+	2.73820i	−0.584908	+	0.145329i
$356$	1.60197			0.0849041
$357$		−	1.39189i		−	0.0736666i
$358$			2.83710i			0.149945i
$359$	−19.1506			−1.01073			−0.505365	−	0.862905i	$-0.668642\pi$
$359$	−19.1506			−1.01073			−0.505365	+	0.862905i	$0.668642\pi$
$360$	1.35350	+	5.44748i	0.0713359	+	0.287107i
$361$	41.1978			2.16830
$362$		−	15.8264i		−	0.831818i
$363$		0			0
$364$	−0.748046			−0.0392083
$365$	−0.608111	−	2.44748i	−0.0318300	−	0.128107i
$366$	10.2823			0.537465
$367$		−	14.5692i		−	0.760504i	−0.924883	−	0.380252i	$-0.875837\pi$
$367$		−	14.5692i		−	0.760504i	0.924883	−	0.380252i	$-0.124163\pi$
$368$			18.4079i			0.959577i
$369$	−7.41855			−0.386194
$370$	−11.4186	+	2.83710i	−0.593622	+	0.147494i
$371$	−1.67420			−0.0869203
$372$		−	2.34017i		−	0.121332i
$373$		−	8.81432i		−	0.456388i	−0.973616	−	0.228194i	$-0.926718\pi$
$373$		−	8.81432i		−	0.456388i	0.973616	−	0.228194i	$-0.0732820\pi$
$374$		0			0
$375$	7.46081	+	8.32684i	0.385275	+	0.429996i
$376$	13.2085			0.681175
$377$		−	51.7152i		−	2.66347i
$378$			0.447480i			0.0230159i
$379$	23.5174			1.20801			0.604005	−	0.796980i	$-0.293571\pi$
$379$	23.5174			1.20801			0.604005	+	0.796980i	$0.293571\pi$
$380$	−6.21461	+	1.54411i	−0.318803	+	0.0792111i
$381$	4.87217			0.249609
$382$			17.7275i			0.907019i
$383$			19.3028i			0.986329i	0.869936	+	0.493164i	$0.164160\pi$
$383$			19.3028i			0.986329i	−0.869936	+	0.493164i	$0.835840\pi$
$384$	−13.4052			−0.684082
$385$		0			0
$386$	5.95055			0.302875
$387$		−	0.290725i		−	0.0147784i
$388$		−	1.72753i		−	0.0877019i
$389$	20.0410			1.01612			0.508060	−	0.861321i	$-0.330363\pi$
$389$	20.0410			1.01612			0.508060	+	0.861321i	$0.330363\pi$
$390$	5.78539	+	23.2846i	0.292954	+	1.17906i
$391$	−19.1506			−0.968488
$392$		−	17.3596i		−	0.876795i
$393$			8.68035i			0.437866i
$394$	13.1955			0.664781
$395$	−2.00000	+	0.496928i	−0.100631	+	0.0250032i
$396$		0			0
$397$		−	33.3607i		−	1.67433i	−0.546954	−	0.837163i	$-0.684213\pi$
$397$		−	33.3607i		−	1.67433i	0.546954	−	0.837163i	$-0.315787\pi$
$398$			12.7961i			0.641409i
$399$	2.25565			0.112924
$400$	−20.3340	+	10.7694i	−1.01670	+	0.538470i
$401$	−37.6475			−1.88003			−0.940014	−	0.341135i	$-0.889189\pi$
$401$	−37.6475			−1.88003			−0.940014	+	0.341135i	$0.889189\pi$
$402$			9.47641i			0.472640i
$403$			44.1978i			2.20165i
$404$	−3.16743			−0.157586
$405$	2.17009	−	0.539189i	0.107832	−	0.0267925i
$406$	3.31965			0.164752
$407$		0			0
$408$		−	12.0183i		−	0.594992i
$409$	−25.2039			−1.24625			−0.623127	−	0.782120i	$-0.714138\pi$
$409$	−25.2039			−1.24625			−0.623127	+	0.782120i	$0.714138\pi$
$410$	−6.15676	−	24.7792i	−0.304060	−	1.22376i
$411$	12.5958			0.621306
$412$			2.48709i			0.122530i
$413$			1.04718i			0.0515284i
$414$	6.15676			0.302588
$415$	0.921622	+	3.70928i	0.0452407	+	0.182081i
$416$	−14.3798			−0.705027
$417$		−	9.91548i		−	0.485563i
$418$		0			0
$419$	−17.8432			−0.871700			−0.435850	−	0.900019i	$-0.643552\pi$
$419$	−17.8432			−0.871700			−0.435850	+	0.900019i	$0.643552\pi$
$420$	−0.232866	+	0.0578588i	−0.0113627	+	0.00282322i
$421$	−11.8120			−0.575684			−0.287842	−	0.957678i	$-0.592938\pi$
$421$	−11.8120			−0.575684			−0.287842	+	0.957678i	$0.592938\pi$
$422$			39.8888i			1.94176i
$423$		−	5.26180i		−	0.255837i
$424$	−14.4559			−0.702040
$425$	−11.2039	−	21.1545i	−0.543471	−	1.02614i
$426$	−7.81658			−0.378715
$427$		−	1.94214i		−	0.0939868i
$428$			4.51518i			0.218249i
$429$		0			0
$430$	0.971071	−	0.241276i	0.0468292	−	0.0116354i
$431$	−16.6803			−0.803464			−0.401732	−	0.915757i	$-0.631592\pi$
$431$	−16.6803			−0.803464			−0.401732	+	0.915757i	$0.631592\pi$
$432$			4.60197i			0.221412i
$433$			28.3135i			1.36066i	0.732906	+	0.680330i	$0.238164\pi$
$433$			28.3135i			1.36066i	−0.732906	+	0.680330i	$0.761836\pi$
$434$	−2.83710			−0.136185
$435$	−4.00000	−	16.0989i	−0.191785	−	0.771883i
$436$	2.33033			0.111603
$437$		−	31.0349i		−	1.48460i
$438$		−	1.73594i		−	0.0829463i
$439$	−27.4452			−1.30989			−0.654944	−	0.755677i	$-0.727308\pi$
$439$	−27.4452			−1.30989			−0.654944	+	0.755677i	$0.727308\pi$
$440$		0			0
$441$	−6.91548			−0.329309
$442$		−	51.3705i		−	2.44345i
$443$		−	0.412408i		−	0.0195941i	−0.999952	−	0.00979704i	$-0.996881\pi$
$443$		−	0.412408i		−	0.0195941i	0.999952	−	0.00979704i	$-0.00311854\pi$
$444$	−1.26180			−0.0598822
$445$	−9.41855	+	2.34017i	−0.446482	+	0.110935i
$446$	14.8203			0.701761
$447$		−	1.26180i		−	0.0596809i
$448$			1.75276i			0.0828100i
$449$	−1.33403			−0.0629568			−0.0314784	−	0.999504i	$-0.510022\pi$
$449$	−1.33403			−0.0629568			−0.0314784	+	0.999504i	$0.510022\pi$
$450$	3.60197	+	6.80098i	0.169798	+	0.320601i
$451$		0			0
$452$		−	6.08906i		−	0.286405i
$453$		−	1.60197i		−	0.0752670i
$454$	28.3051			1.32842
$455$	4.39803	−	1.09275i	0.206183	−	0.0512291i
$456$	19.4764			0.912066
$457$		−	11.6514i		−	0.545030i	−0.962151	−	0.272515i	$-0.912145\pi$
$457$		−	11.6514i		−	0.545030i	0.962151	−	0.272515i	$-0.0878555\pi$
$458$			26.3980i			1.23350i
$459$	−4.78765			−0.223469
$460$	0.796064	+	3.20394i	0.0371167	+	0.149384i
$461$	2.05786			0.0958440			0.0479220	−	0.998851i	$-0.484740\pi$
$461$	2.05786			0.0958440			0.0479220	+	0.998851i	$0.484740\pi$
$462$		0			0
$463$		−	28.7792i		−	1.33748i	−0.743494	−	0.668742i	$-0.766833\pi$
$463$		−	28.7792i		−	1.33748i	0.743494	−	0.668742i	$-0.233167\pi$
$464$	34.1399			1.58491
$465$	3.41855	+	13.7587i	0.158531	+	0.638046i
$466$	−6.32192			−0.292857
$467$			1.84324i			0.0852952i	0.999090	+	0.0426476i	$0.0135793\pi$
$467$			1.84324i			0.0852952i	−0.999090	+	0.0426476i	$0.986421\pi$
$468$			2.57304i			0.118939i
$469$	1.78992			0.0826509
$470$	17.5753	−	4.36683i	0.810688	−	0.201427i
$471$	7.10504			0.327383
$472$			9.04187i			0.416186i
$473$		0			0
$474$	−1.41855			−0.0651562
$475$	34.2823	−	18.1568i	1.57298	−	0.833089i
$476$	0.513749			0.0235477
$477$			5.75872i			0.263674i
$478$			6.72138i			0.307429i
$479$	26.8371			1.22622			0.613109	−	0.789998i	$-0.289919\pi$
$479$	26.8371			1.22622			0.613109	+	0.789998i	$0.289919\pi$
$480$	−4.47641	+	1.11223i	−0.204319	+	0.0507660i
$481$	23.8310			1.08660
$482$		−	8.00984i		−	0.364838i
$483$		−	1.16290i		−	0.0529137i
$484$		0			0
$485$	2.52359	+	10.1568i	0.114590	+	0.461195i
$486$	1.53919			0.0698190
$487$			28.5646i			1.29439i	0.762326	+	0.647193i	$0.224057\pi$
$487$			28.5646i			1.29439i	−0.762326	+	0.647193i	$0.775943\pi$
$488$		−	16.7694i		−	0.759115i
$489$	−22.9360			−1.03720
$490$	−5.73925	−	23.0989i	−0.259273	−	1.04350i
$491$	−25.9877			−1.17281			−0.586405	−	0.810018i	$-0.699457\pi$
$491$	−25.9877			−1.17281			−0.586405	+	0.810018i	$0.699457\pi$
$492$		−	2.73820i		−	0.123448i
$493$			35.5174i			1.59963i
$494$	83.2495			3.74557
$495$		0			0
$496$	−29.1773			−1.31010
$497$			1.47641i			0.0662260i
$498$			2.63090i			0.117893i
$499$	27.5174			1.23185			0.615925	−	0.787805i	$-0.288782\pi$
$499$	27.5174			1.23185			0.615925	+	0.787805i	$0.288782\pi$
$500$	−3.07346	+	2.75380i	−0.137449	+	0.123154i
$501$	4.81432			0.215088
$502$			12.7480i			0.568973i
$503$			22.6576i			1.01025i	0.863046	+	0.505125i	$0.168554\pi$
$503$			22.6576i			1.01025i	−0.863046	+	0.505125i	$0.831446\pi$
$504$	0.729794			0.0325076
$505$	18.6225	−	4.62702i	0.828690	−	0.205900i
$506$		0			0
$507$		−	35.5958i		−	1.58087i
$508$			1.79833i			0.0797880i
$509$	27.8432			1.23413			0.617065	−	0.786912i	$-0.288322\pi$
$509$	27.8432			1.23413			0.617065	+	0.786912i	$0.288322\pi$
$510$	−3.97334	−	15.9916i	−0.175942	−	0.708119i
$511$	−0.327887			−0.0145049
$512$			13.6114i			0.601546i
$513$		−	7.75872i		−	0.342556i
$514$	16.9627			0.748191
$515$	−3.63317	−	14.6225i	−0.160096	−	0.644344i
$516$	0.107307			0.00472393
$517$		0			0
$518$			1.52973i			0.0672126i
$519$	12.8865			0.565657
$520$	37.9748	−	9.43537i	1.66530	−	0.413768i
$521$	29.7009			1.30122			0.650609	−	0.759413i	$-0.274514\pi$
$521$	29.7009			1.30122			0.650609	+	0.759413i	$0.274514\pi$
$522$		−	11.4186i		−	0.499776i
$523$			24.7565i			1.08252i	0.840854	+	0.541262i	$0.182053\pi$
$523$			24.7565i			1.08252i	−0.840854	+	0.541262i	$0.817947\pi$
$524$	−3.20394			−0.139965
$525$	1.28458	−	0.680346i	0.0560637	−	0.0296927i
$526$	−4.57304			−0.199394
$527$		−	30.3545i		−	1.32226i
$528$		0			0
$529$	7.00000			0.304348
$530$	−19.2351	+	4.77924i	−0.835521	+	0.207597i
$531$	3.60197			0.156312
$532$			0.832567i			0.0360963i
$533$			51.7152i			2.24004i
$534$	−6.68035			−0.289087
$535$	−6.59583	−	26.5464i	−0.285162	−	1.14770i
$536$	15.4551			0.667557
$537$		−	1.84324i		−	0.0795419i
$538$		−	24.3857i		−	1.05134i
$539$		0			0
$540$	0.199016	+	0.800984i	0.00856428	+	0.0344689i
$541$	12.5236			0.538431			0.269216	−	0.963080i	$-0.413236\pi$
$541$	12.5236			0.538431			0.269216	+	0.963080i	$0.413236\pi$
$542$		−	9.66967i		−	0.415348i
$543$			10.2823i			0.441256i
$544$	9.87587			0.423425
$545$	−13.7009	+	3.40417i	−0.586881	+	0.145819i
$546$	3.11942			0.133499
$547$			14.2784i			0.610502i	0.952272	+	0.305251i	$0.0987404\pi$
$547$			14.2784i			0.610502i	−0.952272	+	0.305251i	$0.901260\pi$
$548$			4.64915i			0.198602i
$549$	−6.68035			−0.285110
$550$		0			0
$551$	−57.5585			−2.45207
$552$		−	10.0410i		−	0.427375i
$553$			0.267938i			0.0113939i
$554$	−7.89269			−0.335328
$555$	7.41855	−	1.84324i	0.314900	−	0.0782414i
$556$	3.65983			0.155211
$557$			3.57918i			0.151655i	0.997121	+	0.0758274i	$0.0241598\pi$
$557$			3.57918i			0.151655i	−0.997121	+	0.0758274i	$0.975840\pi$
$558$			9.75872i			0.413120i
$559$	−2.02666			−0.0857187
$560$	0.721384	+	2.90337i	0.0304840	+	0.122690i
$561$		0			0
$562$			33.6910i			1.42117i
$563$			28.9588i			1.22047i	0.792222	+	0.610234i	$0.208924\pi$
$563$			28.9588i			1.22047i	−0.792222	+	0.610234i	$0.791076\pi$
$564$	1.94214			0.0817789
$565$	8.89496	+	35.7998i	0.374214	+	1.50611i
$566$	39.9649			1.67985
$567$		−	0.290725i		−	0.0122093i
$568$			12.7480i			0.534896i
$569$	−25.8264			−1.08270			−0.541350	−	0.840797i	$-0.682087\pi$
$569$	−25.8264			−1.08270			−0.541350	+	0.840797i	$0.682087\pi$
$570$	25.9155	−	6.43907i	1.08548	−	0.269703i
$571$	−17.1194			−0.716425			−0.358213	−	0.933640i	$-0.616614\pi$
$571$	−17.1194			−0.716425			−0.358213	+	0.933640i	$0.616614\pi$
$572$		0			0
$573$		−	11.5174i		−	0.481148i
$574$	−3.31965			−0.138560
$575$	−9.36069	−	17.6742i	−0.390368	−	0.737065i
$576$	6.02893			0.251205
$577$			22.5692i			0.939567i	0.882782	+	0.469783i	$0.155668\pi$
$577$			22.5692i			0.939567i	−0.882782	+	0.469783i	$0.844332\pi$
$578$			9.11450i			0.379113i
$579$	−3.86603			−0.160667
$580$	5.94214	−	1.47641i	0.246734	−	0.0613046i
$581$	0.496928			0.0206161
$582$			7.20394i			0.298613i
$583$		0			0
$584$	−2.83114			−0.117153
$585$	−3.75872	−	15.1278i	−0.155404	−	0.625459i
$586$	16.0326			0.662302
$587$			3.63317i			0.149957i	0.997185	+	0.0749784i	$0.0238888\pi$
$587$			3.63317i			0.149957i	−0.997185	+	0.0749784i	$0.976111\pi$
$588$		−	2.55252i		−	0.105264i
$589$	49.1917			2.02691
$590$	2.98932	+	12.0312i	0.123068	+	0.495317i
$591$	−8.57304			−0.352648
$592$			15.7321i			0.646584i
$593$			12.3051i			0.505310i	0.967556	+	0.252655i	$0.0813037\pi$
$593$			12.3051i			0.505310i	−0.967556	+	0.252655i	$0.918696\pi$
$594$		0			0
$595$	−3.02052	+	0.750491i	−0.123829	+	0.0307671i
$596$	0.465732			0.0190771
$597$		−	8.31351i		−	0.340249i
$598$		−	42.9192i		−	1.75510i
$599$	−19.6865			−0.804368			−0.402184	−	0.915559i	$-0.631749\pi$
$599$	−19.6865			−0.804368			−0.402184	+	0.915559i	$0.631749\pi$
$600$	11.0917	−	5.87444i	0.452817	−	0.239823i
$601$	−25.8843			−1.05584			−0.527921	−	0.849294i	$-0.677028\pi$
$601$	−25.8843			−1.05584			−0.527921	+	0.849294i	$0.677028\pi$
$602$		−	0.130094i		−	0.00530222i
$603$		−	6.15676i		−	0.250722i
$604$	0.591290			0.0240593
$605$		0			0
$606$	13.2085			0.536557
$607$			41.5357i			1.68588i	0.538006	+	0.842941i	$0.319178\pi$
$607$			41.5357i			1.68588i	−0.538006	+	0.842941i	$0.680822\pi$
$608$			16.0045i			0.649070i
$609$	−2.15676			−0.0873961
$610$	−5.54411	−	22.3135i	−0.224474	−	0.903448i
$611$	−36.6803			−1.48393
$612$		−	1.76713i		−	0.0714322i
$613$		−	20.6453i		−	0.833855i	−0.908940	−	0.416927i	$-0.863107\pi$
$613$		−	20.6453i		−	0.833855i	0.908940	−	0.416927i	$-0.136893\pi$
$614$	44.7442			1.80573
$615$	4.00000	+	16.0989i	0.161296	+	0.649170i
$616$		0			0
$617$			8.69472i			0.350036i	0.984565	+	0.175018i	$0.0559984\pi$
$617$			8.69472i			0.350036i	−0.984565	+	0.175018i	$0.944002\pi$
$618$		−	10.3714i		−	0.417198i
$619$	−2.65368			−0.106661			−0.0533303	−	0.998577i	$-0.516984\pi$
$619$	−2.65368			−0.106661			−0.0533303	+	0.998577i	$0.516984\pi$
$620$	−5.07838	+	1.26180i	−0.203953	+	0.0506749i
$621$	−4.00000			−0.160514
$622$		−	8.38121i		−	0.336056i
$623$			1.26180i			0.0505528i
$624$	32.0806			1.28425
$625$	14.0472	−	20.6803i	0.561887	−	0.827214i
$626$	−38.6225			−1.54367
$627$		0			0
$628$			2.62249i			0.104649i
$629$	−16.3668			−0.652588
$630$	0.971071	−	0.241276i	0.0386884	−	0.00961268i
$631$	−10.6393			−0.423544			−0.211772	−	0.977319i	$-0.567923\pi$
$631$	−10.6393			−0.423544			−0.211772	+	0.977319i	$0.567923\pi$
$632$			2.31351i			0.0920265i
$633$		−	25.9155i		−	1.03005i
$634$	32.5958			1.29455
$635$	−2.62702	−	10.5730i	−0.104250	−	0.419578i
$636$	−2.12556			−0.0842839
$637$			48.2083i			1.91008i
$638$		0			0
$639$	5.07838			0.200898
$640$	7.22795	+	29.0905i	0.285710	+	1.14990i
$641$	−35.8576			−1.41629			−0.708145	−	0.706067i	$-0.750468\pi$
$641$	−35.8576			−1.41629			−0.708145	+	0.706067i	$0.750468\pi$
$642$		−	18.8287i		−	0.743109i
$643$		−	12.2146i		−	0.481697i	−0.970563	−	0.240849i	$-0.922574\pi$
$643$		−	12.2146i		−	0.481697i	0.970563	−	0.240849i	$-0.0774258\pi$
$644$	0.429229			0.0169140
$645$	−0.630898	+	0.156755i	−0.0248416	+	0.00617224i
$646$	−57.1748			−2.24951
$647$			25.9421i			1.01989i	0.860207	+	0.509945i	$0.170334\pi$
$647$			25.9421i			1.01989i	−0.860207	+	0.509945i	$0.829666\pi$
$648$		−	2.51026i		−	0.0986123i
$649$		0			0
$650$	47.4101	−	25.1096i	1.85958	−	0.984879i
$651$	1.84324			0.0722424
$652$		−	8.46573i		−	0.331544i
$653$			14.3402i			0.561174i	0.959829	+	0.280587i	$0.0905292\pi$
$653$			14.3402i			0.561174i	−0.959829	+	0.280587i	$0.909471\pi$
$654$	−9.71769			−0.379992
$655$	18.8371	−	4.68035i	0.736026	−	0.182876i
$656$	−34.1399			−1.33294
$657$			1.12783i			0.0440007i
$658$		−	2.35455i		−	0.0917899i
$659$	6.52359			0.254123			0.127062	−	0.991895i	$-0.459445\pi$
$659$	6.52359			0.254123			0.127062	+	0.991895i	$0.459445\pi$
$660$		0			0
$661$	3.16290			0.123022			0.0615112	−	0.998106i	$-0.480408\pi$
$661$	3.16290			0.123022			0.0615112	+	0.998106i	$0.480408\pi$
$662$			10.3234i			0.401228i
$663$			33.3751i			1.29618i
$664$	4.29072			0.166512
$665$	−1.21622	−	4.89496i	−0.0471631	−	0.189818i
$666$	5.26180			0.203890
$667$			29.6742i			1.14899i
$668$			1.77698i			0.0687532i
$669$	−9.62863			−0.372264
$670$	20.5646	−	5.10957i	0.794481	−	0.197400i
$671$		0			0
$672$			0.599701i			0.0231340i
$673$		−	18.4885i		−	0.712680i	−0.934356	−	0.356340i	$-0.884024\pi$
$673$		−	18.4885i		−	0.712680i	0.934356	−	0.356340i	$-0.115976\pi$
$674$	−36.5562			−1.40809
$675$	−2.34017	−	4.41855i	−0.0900733	−	0.170070i
$676$	13.1385			0.505327
$677$		−	15.0966i		−	0.580211i	−0.956995	−	0.290105i	$-0.906310\pi$
$677$		−	15.0966i		−	0.580211i	0.956995	−	0.290105i	$-0.0936903\pi$
$678$			25.3919i			0.975170i
$679$	1.36069			0.0522186
$680$	−26.0806	+	6.48011i	−1.00015	+	0.248501i
$681$	−18.3896			−0.704692
$682$		0			0
$683$			12.4657i			0.476988i	0.971144	+	0.238494i	$0.0766537\pi$
$683$			12.4657i			0.476988i	−0.971144	+	0.238494i	$0.923346\pi$
$684$	2.86376			0.109499
$685$	−6.79153	−	27.3340i	−0.259491	−	1.04438i
$686$	−6.22690			−0.237744
$687$		−	17.1506i		−	0.654337i
$688$		−	1.33791i		−	0.0510072i
$689$	40.1445			1.52938
$690$	−3.31965	−	13.3607i	−0.126377	−	0.508633i
$691$	−30.7214			−1.16870			−0.584348	−	0.811503i	$-0.698650\pi$
$691$	−30.7214			−1.16870			−0.584348	+	0.811503i	$0.698650\pi$
$692$			4.75646i			0.180813i
$693$		0			0
$694$	47.9748			1.82110
$695$	−21.5174	+	5.34632i	−0.816203	+	0.202797i
$696$	−18.6225			−0.705884
$697$		−	35.5174i		−	1.34532i
$698$		−	17.0037i		−	0.643599i
$699$	4.10731			0.155353
$700$	0.251117	+	0.474142i	0.00949134	+	0.0179209i
$701$	−17.9955			−0.679679			−0.339840	−	0.940483i	$-0.610373\pi$
$701$	−17.9955			−0.679679			−0.339840	+	0.940483i	$0.610373\pi$
$702$		−	10.7298i		−	0.404970i
$703$		−	26.5236i		−	1.00036i
$704$		0			0
$705$	−11.4186	+	2.83710i	−0.430047	+	0.106851i
$706$	42.2245			1.58914
$707$		−	2.49484i		−	0.0938281i
$708$			1.32950i			0.0499655i
$709$	−23.5897			−0.885929			−0.442965	−	0.896539i	$-0.646073\pi$
$709$	−23.5897			−0.885929			−0.442965	+	0.896539i	$0.646073\pi$
$710$	4.21461	+	16.9627i	0.158172	+	0.636597i
$711$	0.921622			0.0345635
$712$			10.8950i			0.408306i
$713$		−	25.3607i		−	0.949765i
$714$	−2.14238			−0.0801765
$715$		0			0
$716$	0.680346			0.0254257
$717$		−	4.36683i		−	0.163082i
$718$			29.4764i			1.10005i
$719$	−24.5646			−0.916106			−0.458053	−	0.888925i	$-0.651453\pi$
$719$	−24.5646			−0.916106			−0.458053	+	0.888925i	$0.651453\pi$
$720$	9.98667	−	2.48133i	0.372181	−	0.0924737i
$721$	−1.95896			−0.0729556
$722$		−	63.4112i		−	2.35992i
$723$			5.20394i			0.193536i
$724$	−3.79523			−0.141048
$725$	−32.7792	+	17.3607i	−1.21739	+	0.644760i
$726$		0			0
$727$			8.51130i			0.315667i	0.987466	+	0.157833i	$0.0504509\pi$
$727$			8.51130i			0.315667i	−0.987466	+	0.157833i	$0.949549\pi$
$728$		−	5.08745i		−	0.188553i
$729$	−1.00000			−0.0370370
$730$	−3.76713	+	0.935998i	−0.139428	+	0.0346428i
$731$	1.39189			0.0514809
$732$		−	2.46573i		−	0.0911361i
$733$		−	29.8615i		−	1.10296i	−0.834188	−	0.551480i	$-0.814063\pi$
$733$		−	29.8615i		−	1.10296i	0.834188	−	0.551480i	$-0.185937\pi$
$734$	−22.4247			−0.827711
$735$	3.72875	+	15.0072i	0.137537	+	0.553548i
$736$	8.25112			0.304140
$737$		0			0
$738$			11.4186i			0.420323i
$739$	−36.7526			−1.35197			−0.675983	−	0.736917i	$-0.736281\pi$
$739$	−36.7526			−1.35197			−0.675983	+	0.736917i	$0.736281\pi$
$740$	0.680346	+	2.73820i	0.0250100	+	0.100658i
$741$	−54.0866			−1.98692
$742$			2.57691i			0.0946015i
$743$		−	2.17501i		−	0.0797933i	−0.999204	−	0.0398966i	$-0.987297\pi$
$743$		−	2.17501i		−	0.0797933i	0.999204	−	0.0398966i	$-0.0127029\pi$
$744$	15.9155			0.583490
$745$	−2.73820	+	0.680346i	−0.100320	+	0.0249259i
$746$	−13.5669			−0.496719
$747$		−	1.70928i		−	0.0625391i
$748$		0			0
$749$	−3.55640			−0.129948
$750$	12.8166	−	11.4836i	0.467995	−	0.419322i
$751$	−44.4580			−1.62229			−0.811147	−	0.584842i	$-0.801157\pi$
$751$	−44.4580			−1.62229			−0.811147	+	0.584842i	$0.801157\pi$
$752$		−	24.2146i		−	0.883016i
$753$		−	8.28231i		−	0.301824i
$754$	−79.5995			−2.89884
$755$	−3.47641	+	0.863763i	−0.126519	+	0.0314356i
$756$	0.107307			0.00390272
$757$		−	26.4247i		−	0.960422i	−0.877153	−	0.480211i	$-0.840560\pi$
$757$		−	26.4247i		−	0.960422i	0.877153	−	0.480211i	$-0.159440\pi$
$758$		−	36.1978i		−	1.31476i
$759$		0			0
$760$	−10.5015	−	42.2655i	−0.380928	−	1.53313i
$761$	27.6163			1.00109			0.500546	−	0.865710i	$-0.333133\pi$
$761$	27.6163			1.00109			0.500546	+	0.865710i	$0.333133\pi$
$762$		−	7.49920i		−	0.271667i
$763$			1.83549i			0.0664493i
$764$	4.25112			0.153800
$765$	2.58145	+	10.3896i	0.0933325	+	0.375638i
$766$	29.7107			1.07349
$767$		−	25.1096i		−	0.906654i
$768$			8.57531i			0.309435i
$769$	27.8432			1.00405			0.502027	−	0.864852i	$-0.332588\pi$
$769$	27.8432			1.00405			0.502027	+	0.864852i	$0.332588\pi$
$770$		0			0
$771$	−11.0205			−0.396894
$772$		−	1.42696i		−	0.0513575i
$773$			18.0267i			0.648374i	0.945993	+	0.324187i	$0.105091\pi$
$773$			18.0267i			0.648374i	−0.945993	+	0.324187i	$0.894909\pi$
$774$	−0.447480			−0.0160843
$775$	28.0144	−	14.8371i	1.00631	−	0.532964i
$776$	11.7489			0.421760
$777$		−	0.993857i		−	0.0356544i
$778$		−	30.8469i		−	1.10592i
$779$	57.5585			2.06225
$780$	5.58372	−	1.38735i	0.199929	−	0.0496752i
$781$		0			0
$782$			29.4764i			1.05407i
$783$

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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1815.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.53919i q^{2} +1.00000i q^{3} -0.369102 q^{4} +(2.17009 - 0.539189i) q^{5} +1.53919 q^{6} -0.290725i q^{7} -2.51026i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.53919i q^{2} +1.00000i q^{3} -0.369102 q^{4} +(2.17009 - 0.539189i) q^{5} +1.53919 q^{6} -0.290725i q^{7} -2.51026i q^{8} -1.00000 q^{9} +(-0.829914 - 3.34017i) q^{10} -0.369102i q^{12} +6.97107i q^{13} -0.447480 q^{14} +(0.539189 + 2.17009i) q^{15} -4.60197 q^{16} -4.78765i q^{17} +1.53919i q^{18} +7.75872 q^{19} +(-0.800984 + 0.199016i) q^{20} +0.290725 q^{21} -4.00000i q^{23} +2.51026 q^{24} +(4.41855 - 2.34017i) q^{25} +10.7298 q^{26} -1.00000i q^{27} +0.107307i q^{28} -7.41855 q^{29} +(3.34017 - 0.829914i) q^{30} +6.34017 q^{31} +2.06278i q^{32} -7.36910 q^{34} +(-0.156755 - 0.630898i) q^{35} +0.369102 q^{36} -3.41855i q^{37} -11.9421i q^{38} -6.97107 q^{39} +(-1.35350 - 5.44748i) q^{40} +7.41855 q^{41} -0.447480i q^{42} +0.290725i q^{43} +(-2.17009 + 0.539189i) q^{45} -6.15676 q^{46} +5.26180i q^{47} -4.60197i q^{48} +6.91548 q^{49} +(-3.60197 - 6.80098i) q^{50} +4.78765 q^{51} -2.57304i q^{52} -5.75872i q^{53} -1.53919 q^{54} -0.729794 q^{56} +7.75872i q^{57} +11.4186i q^{58} -3.60197 q^{59} +(-0.199016 - 0.800984i) q^{60} +6.68035 q^{61} -9.75872i q^{62} +0.290725i q^{63} -6.02893 q^{64} +(3.75872 + 15.1278i) q^{65} +6.15676i q^{67} +1.76713i q^{68} +4.00000 q^{69} +(-0.971071 + 0.241276i) q^{70} -5.07838 q^{71} +2.51026i q^{72} -1.12783i q^{73} -5.26180 q^{74} +(2.34017 + 4.41855i) q^{75} -2.86376 q^{76} +10.7298i q^{78} -0.921622 q^{79} +(-9.98667 + 2.48133i) q^{80} +1.00000 q^{81} -11.4186i q^{82} +1.70928i q^{83} -0.107307 q^{84} +(-2.58145 - 10.3896i) q^{85} +0.447480 q^{86} -7.41855i q^{87} -4.34017 q^{89} +(0.829914 + 3.34017i) q^{90} +2.02666 q^{91} +1.47641i q^{92} +6.34017i q^{93} +8.09890 q^{94} +(16.8371 - 4.18342i) q^{95} -2.06278 q^{96} +4.68035i q^{97} -10.6442i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 10 * q^4 + 2 * q^5 + 6 * q^6 - 6 * q^9	\( 6 q - 10 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{9} - 16 q^{10} - 4 q^{14} + 10 q^{16} - 4 q^{19} + 14 q^{20} + 16 q^{21} - 18 q^{24} - 2 q^{25} - 16 q^{26} - 16 q^{29} - 2 q^{30} + 16 q^{31} - 52 q^{34} + 12 q^{35} + 10 q^{36} - 12 q^{39} + 12 q^{40} + 16 q^{41} - 2 q^{45} - 24 q^{46} - 22 q^{49} + 16 q^{50} + 8 q^{51} - 6 q^{54} + 76 q^{56} + 16 q^{59} - 20 q^{60} - 4 q^{61} - 66 q^{64} - 28 q^{65} + 24 q^{69} + 24 q^{70} - 24 q^{71} - 16 q^{74} - 8 q^{75} + 36 q^{76} - 12 q^{79} - 58 q^{80} + 6 q^{81} - 24 q^{84} - 44 q^{85} + 4 q^{86} - 4 q^{89} + 16 q^{90} + 16 q^{91} - 24 q^{94} + 44 q^{95} + 22 q^{96}+O(q^{100}) \) 6 * q - 10 * q^4 + 2 * q^5 + 6 * q^6 - 6 * q^9 - 16 * q^10 - 4 * q^14 + 10 * q^16 - 4 * q^19 + 14 * q^20 + 16 * q^21 - 18 * q^24 - 2 * q^25 - 16 * q^26 - 16 * q^29 - 2 * q^30 + 16 * q^31 - 52 * q^34 + 12 * q^35 + 10 * q^36 - 12 * q^39 + 12 * q^40 + 16 * q^41 - 2 * q^45 - 24 * q^46 - 22 * q^49 + 16 * q^50 + 8 * q^51 - 6 * q^54 + 76 * q^56 + 16 * q^59 - 20 * q^60 - 4 * q^61 - 66 * q^64 - 28 * q^65 + 24 * q^69 + 24 * q^70 - 24 * q^71 - 16 * q^74 - 8 * q^75 + 36 * q^76 - 12 * q^79 - 58 * q^80 + 6 * q^81 - 24 * q^84 - 44 * q^85 + 4 * q^86 - 4 * q^89 + 16 * q^90 + 16 * q^91 - 24 * q^94 + 44 * q^95 + 22 * q^96

Introduction

L-functions

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