LMFDB - Embedded newform 1815.2.c.d.364.5

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

$q$-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	$0.183161\pi$
$2$			1.53919i			1.08837i	−0.838965	+	0.544185i	$0.816839\pi$
$3$		−	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	$0.0174973\pi$
$7$			0.290725i			0.109884i	−0.998490	+	0.0549418i	$0.982503\pi$
$8$			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	$-0.582359\pi$
$13$		−	6.97107i		−	1.93343i	0.255861	−	0.966714i	$-0.417641\pi$
$14$	−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	$0.197177\pi$
$17$			4.78765i			1.16118i	−0.814197	+	0.580588i	$0.802823\pi$
$18$		−	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	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	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	$0.0906671\pi$
$37$			3.41855i			0.562006i	−0.959707	+	0.281003i	$0.909333\pi$
$38$			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	$-0.992943\pi$
$43$		−	0.290725i		−	0.0443351i	0.999754	−	0.0221675i	$-0.00705673\pi$
$44$		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	$-0.874630\pi$
$47$		−	5.26180i		−	0.767512i	0.923435	−	0.383756i	$-0.125370\pi$
$48$			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	$0.129432\pi$
$53$			5.75872i			0.791022i	−0.918461	+	0.395511i	$0.870568\pi$
$54$	−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	$-0.877271\pi$
$67$		−	6.15676i		−	0.752167i	0.926586	−	0.376084i	$-0.122729\pi$
$68$		−	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	$0.0210241\pi$
$73$			1.12783i			0.132002i	−0.997820	+	0.0660010i	$0.978976\pi$
$74$	−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	$-0.970096\pi$
$83$		−	1.70928i		−	0.187617i	0.995590	−	0.0938087i	$-0.0299042\pi$
$84$	−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	$-0.923636\pi$
$97$		−	4.68035i		−	0.475217i	0.971361	−	0.237609i	$-0.0763635\pi$
$98$			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	$0.107712\pi$
$103$			6.73820i			0.663935i	−0.943291	+	0.331968i	$0.892288\pi$
$104$	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	$0.201384\pi$
$107$			12.2329i			1.18260i	−0.806453	+	0.591298i	$0.798616\pi$
$108$		−	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	$-0.717271\pi$
$113$		−	16.4969i		−	1.55190i	0.630794	−	0.775950i	$-0.282729\pi$
$114$	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	$0.0693558\pi$
$127$			4.87217i			0.432336i	−0.976356	+	0.216168i	$0.930644\pi$
$128$		−	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	$0.180845\pi$
$137$			12.5958i			1.07613i	−0.842902	+	0.538067i	$0.819155\pi$
$138$			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	$0.0915029\pi$
$157$			7.10504i			0.567044i	−0.958966	+	0.283522i	$0.908497\pi$
$158$		−	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	$-0.644844\pi$
$163$		−	22.9360i		−	1.79649i	0.439499	−	0.898243i	$-0.355156\pi$
$164$	−2.73820			−0.213818
$165$		0			0
$166$	2.63090			0.204197
$167$			4.81432i			0.372543i	0.982498	+	0.186271i	$0.0596404\pi$
$167$			4.81432i			0.372543i	−0.982498	+	0.186271i	$0.940360\pi$
$168$			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	$0.162957\pi$
$173$			12.8865i			0.979746i	−0.871794	+	0.489873i	$0.837043\pi$
$174$	−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	$-0.955566\pi$
$193$		−	3.86603i		−	0.278283i	0.990273	−	0.139141i	$-0.0444343\pi$
$194$	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	$-0.901209\pi$
$197$		−	8.57304i		−	0.610804i	0.952224	−	0.305402i	$-0.0987908\pi$
$198$		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	$-0.895514\pi$
$223$		−	9.62863i		−	0.644781i	0.946607	−	0.322390i	$-0.104486\pi$
$224$	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	$-0.791057\pi$
$227$		−	18.3896i		−	1.22056i	0.792185	−	0.610281i	$-0.208943\pi$
$228$			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	$0.0429555\pi$
$233$			4.10731i			0.269079i	−0.990908	+	0.134539i	$0.957045\pi$
$234$	−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	$-0.888313\pi$
$257$		−	11.0205i		−	0.687441i	0.939072	−	0.343721i	$-0.111687\pi$
$258$		−	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	$0.0291988\pi$
$263$			2.97107i			0.183204i	−0.995796	+	0.0916020i	$0.970801\pi$
$264$		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	$0.0492319\pi$
$277$			5.12783i			0.308101i	−0.988063	+	0.154051i	$0.950768\pi$
$278$		−	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	$-0.719393\pi$
$283$		−	25.9649i		−	1.54345i	0.635954	−	0.771727i	$-0.280607\pi$
$284$	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	$-0.901590\pi$
$293$		−	10.4163i		−	0.608526i	0.952588	−	0.304263i	$-0.0984101\pi$
$294$	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	$-0.688594\pi$
$307$		−	29.0700i		−	1.65911i	0.558425	−	0.829555i	$-0.311406\pi$
$308$		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	$0.250927\pi$
$313$			25.0928i			1.41833i	−0.705044	+	0.709163i	$0.749073\pi$
$314$	−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	$-0.797264\pi$
$317$		−	21.1773i		−	1.18943i	0.803935	−	0.594717i	$-0.202736\pi$
$318$			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	$0.223927\pi$
$337$			23.7503i			1.29376i	−0.762591	+	0.646881i	$0.776073\pi$
$338$		−	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	$-0.684531\pi$
$347$		−	31.1689i		−	1.67323i	0.547790	−	0.836616i	$-0.315469\pi$
$348$		−	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	$-0.739495\pi$
$353$		−	27.4329i		−	1.46011i	0.683390	−	0.730054i	$-0.260505\pi$
$354$	−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	$0.124163\pi$
$367$			14.5692i			0.760504i	−0.924883	+	0.380252i	$0.875837\pi$
$368$		−	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	$0.0732820\pi$
$373$			8.81432i			0.456388i	−0.973616	+	0.228194i	$0.926718\pi$
$374$		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	$-0.835840\pi$
$383$		−	19.3028i		−	0.986329i	0.869936	−	0.493164i	$-0.164160\pi$
$384$	−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	$0.315787\pi$
$397$			33.3607i			1.67433i	−0.546954	+	0.837163i	$0.684213\pi$
$398$		−	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	$-0.761836\pi$
$433$		−	28.3135i		−	1.36066i	0.732906	−	0.680330i	$-0.238164\pi$
$434$	−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	$0.00311854\pi$
$443$			0.412408i			0.0195941i	−0.999952	+	0.00979704i	$0.996881\pi$
$444$	−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	$0.0878555\pi$
$457$			11.6514i			0.545030i	−0.962151	+	0.272515i	$0.912145\pi$
$458$		−	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	$0.233167\pi$
$463$			28.7792i			1.33748i	−0.743494	+	0.668742i	$0.766833\pi$
$464$	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	$-0.986421\pi$
$467$		−	1.84324i		−	0.0852952i	0.999090	−	0.0426476i	$-0.0135793\pi$
$468$		−	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	$-0.775943\pi$
$487$		−	28.5646i		−	1.29439i	0.762326	−	0.647193i	$-0.224057\pi$
$488$			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	$-0.831446\pi$
$503$		−	22.6576i		−	1.01025i	0.863046	−	0.505125i	$-0.168554\pi$
$504$	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	$-0.817947\pi$
$523$		−	24.7565i		−	1.08252i	0.840854	−	0.541262i	$-0.182053\pi$
$524$	−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	$-0.901260\pi$
$547$		−	14.2784i		−	0.610502i	0.952272	−	0.305251i	$-0.0987404\pi$
$548$		−	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	$-0.975840\pi$
$557$		−	3.57918i		−	0.151655i	0.997121	−	0.0758274i	$-0.0241598\pi$
$558$		−	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	$-0.791076\pi$
$563$		−	28.9588i		−	1.22047i	0.792222	−	0.610234i	$-0.208924\pi$
$564$	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	$-0.844332\pi$
$577$		−	22.5692i		−	0.939567i	0.882782	−	0.469783i	$-0.155668\pi$
$578$		−	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	$-0.976111\pi$
$587$		−	3.63317i		−	0.149957i	0.997185	−	0.0749784i	$-0.0238888\pi$
$588$			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	$-0.918696\pi$
$593$		−	12.3051i		−	0.505310i	0.967556	−	0.252655i	$-0.0813037\pi$
$594$		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	$-0.680822\pi$
$607$		−	41.5357i		−	1.68588i	0.538006	−	0.842941i	$-0.319178\pi$
$608$		−	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	$0.136893\pi$
$613$			20.6453i			0.833855i	−0.908940	+	0.416927i	$0.863107\pi$
$614$	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	$-0.944002\pi$
$617$		−	8.69472i		−	0.350036i	0.984565	−	0.175018i	$-0.0559984\pi$
$618$			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	$0.0774258\pi$
$643$			12.2146i			0.481697i	−0.970563	+	0.240849i	$0.922574\pi$
$644$	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	$-0.829666\pi$
$647$		−	25.9421i		−	1.01989i	0.860207	−	0.509945i	$-0.170334\pi$
$648$			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	$-0.909471\pi$
$653$		−	14.3402i		−	0.561174i	0.959829	−	0.280587i	$-0.0905292\pi$
$654$	−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	$0.115976\pi$
$673$			18.4885i			0.712680i	−0.934356	+	0.356340i	$0.884024\pi$
$674$	−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	$0.0936903\pi$
$677$			15.0966i			0.580211i	−0.956995	+	0.290105i	$0.906310\pi$
$678$		−	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	$-0.923346\pi$
$683$		−	12.4657i		−	0.476988i	0.971144	−	0.238494i	$-0.0766537\pi$
$684$	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	$-0.949549\pi$
$727$		−	8.51130i		−	0.315667i	0.987466	−	0.157833i	$-0.0504509\pi$
$728$			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	$0.185937\pi$
$733$			29.8615i			1.10296i	−0.834188	+	0.551480i	$0.814063\pi$
$734$	−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	$0.0127029\pi$
$743$			2.17501i			0.0797933i	−0.999204	+	0.0398966i	$0.987297\pi$
$744$	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	$0.159440\pi$
$757$			26.4247i			0.960422i	−0.877153	+	0.480211i	$0.840560\pi$
$758$			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	$-0.894909\pi$
$773$		−	18.0267i		−	0.648374i	0.945993	−	0.324187i	$-0.105091\pi$
$774$	−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

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

Modular forms

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Database

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