LMFDB - Embedded newform 1815.2.c.d.364.6

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			364.6
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1815.364
Dual form			1815.2.c.d.364.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +(-1.48119 + 1.67513i) q^{5} +2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +(-1.48119 + 1.67513i) q^{5} +2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +(-4.48119 - 3.96239i) q^{10} +5.15633i q^{12} +5.11871i q^{13} -7.50659 q^{14} +(1.67513 + 1.48119i) q^{15} +12.2750 q^{16} +4.54420i q^{17} -2.67513i q^{18} -4.57452 q^{19} +(7.63752 - 8.63752i) q^{20} +2.80606 q^{21} +4.00000i q^{23} -8.44358 q^{24} +(-0.612127 - 4.96239i) q^{25} -13.6932 q^{26} +1.00000i q^{27} -14.4690i q^{28} -2.38787 q^{29} +(-3.96239 + 4.48119i) q^{30} -0.962389 q^{31} +15.9502i q^{32} -12.1563 q^{34} +(-4.70052 - 4.15633i) q^{35} +5.15633 q^{36} -1.61213i q^{37} -12.2374i q^{38} +5.11871 q^{39} +(14.1441 + 12.5066i) q^{40} +2.38787 q^{41} +7.50659i q^{42} -2.80606i q^{43} +(1.48119 - 1.67513i) q^{45} -10.7005 q^{46} +4.31265i q^{47} -12.2750i q^{48} -0.873992 q^{49} +(13.2750 - 1.63752i) q^{50} +4.54420 q^{51} -26.3938i q^{52} -6.57452i q^{53} -2.67513 q^{54} +23.6932 q^{56} +4.57452i q^{57} -6.38787i q^{58} +13.2750 q^{59} +(-8.63752 - 7.63752i) q^{60} -7.92478 q^{61} -2.57452i q^{62} -2.80606i q^{63} -18.1187 q^{64} +(-8.57452 - 7.58181i) q^{65} -10.7005i q^{67} -23.4314i q^{68} +4.00000 q^{69} +(11.1187 - 12.5745i) q^{70} -7.35026 q^{71} +8.44358i q^{72} -6.41819i q^{73} +4.31265 q^{74} +(-4.96239 + 0.612127i) q^{75} +23.5877 q^{76} +13.6932i q^{78} +1.35026 q^{79} +(-18.1817 + 20.5623i) q^{80} +1.00000 q^{81} +6.38787i q^{82} +0.806063i q^{83} -14.4690 q^{84} +(-7.61213 - 6.73084i) q^{85} +7.50659 q^{86} +2.38787i q^{87} +2.96239 q^{89} +(4.48119 + 3.96239i) q^{90} -14.3634 q^{91} -20.6253i q^{92} +0.962389i q^{93} -11.5369 q^{94} +(6.77575 - 7.66291i) q^{95} +15.9502 q^{96} +9.92478i q^{97} -2.33804i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			2.67513i			1.89160i	0.324745	+	0.945802i	$0.394721\pi$
$2$			2.67513i			1.89160i	−0.324745	+	0.945802i	$0.605279\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−5.15633			−2.57816
$5$	−1.48119	+	1.67513i	−0.662410	+	0.749141i
$5$	−1.48119	+	1.67513i	−0.662410	+	0.749141i
$6$	2.67513			1.09212
$7$			2.80606i			1.06059i	0.847812	+	0.530296i	$0.177919\pi$
$7$			2.80606i			1.06059i	−0.847812	+	0.530296i	$0.822081\pi$
$8$		−	8.44358i		−	2.98526i
$9$	−1.00000			−0.333333
$10$	−4.48119	−	3.96239i	−1.41708	−	1.25302i
$11$		0			0
$11$		0			0
$12$			5.15633i			1.48850i
$13$			5.11871i			1.41968i	0.704365	+	0.709838i	$0.251232\pi$
$13$			5.11871i			1.41968i	−0.704365	+	0.709838i	$0.748768\pi$
$14$	−7.50659			−2.00622
$15$	1.67513	+	1.48119i	0.432517	+	0.382443i
$16$	12.2750			3.06876
$17$			4.54420i			1.10213i	0.834462	+	0.551065i	$0.185778\pi$
$17$			4.54420i			1.10213i	−0.834462	+	0.551065i	$0.814222\pi$
$18$		−	2.67513i		−	0.630534i
$19$	−4.57452			−1.04947			−0.524733	−	0.851267i	$-0.675835\pi$
$19$	−4.57452			−1.04947			−0.524733	+	0.851267i	$0.675835\pi$
$20$	7.63752	−	8.63752i	1.70780	−	1.93141i
$21$	2.80606			0.612333
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−8.44358			−1.72354
$25$	−0.612127	−	4.96239i	−0.122425	−	0.992478i
$26$	−13.6932			−2.68546
$27$			1.00000i			0.192450i
$28$		−	14.4690i		−	2.73438i
$29$	−2.38787			−0.443417			−0.221708	−	0.975113i	$-0.571163\pi$
$29$	−2.38787			−0.443417			−0.221708	+	0.975113i	$0.571163\pi$
$30$	−3.96239	+	4.48119i	−0.723430	+	0.818150i
$31$	−0.962389			−0.172850			−0.0864250	−	0.996258i	$-0.527544\pi$
$31$	−0.962389			−0.172850			−0.0864250	+	0.996258i	$0.527544\pi$
$32$			15.9502i			2.81962i
$33$		0			0
$34$	−12.1563			−2.08479
$35$	−4.70052	−	4.15633i	−0.794533	−	0.702547i
$36$	5.15633			0.859388
$37$		−	1.61213i		−	0.265032i	−0.991181	−	0.132516i	$-0.957694\pi$
$37$		−	1.61213i		−	0.265032i	0.991181	−	0.132516i	$-0.0423056\pi$
$38$		−	12.2374i		−	1.98517i
$39$	5.11871			0.819650
$40$	14.1441	+	12.5066i	2.23638	+	1.97747i
$41$	2.38787			0.372923			0.186462	−	0.982462i	$-0.440298\pi$
$41$	2.38787			0.372923			0.186462	+	0.982462i	$0.440298\pi$
$42$			7.50659i			1.15829i
$43$		−	2.80606i		−	0.427921i	−0.976842	−	0.213960i	$-0.931364\pi$
$43$		−	2.80606i		−	0.427921i	0.976842	−	0.213960i	$-0.0686363\pi$
$44$		0			0
$45$	1.48119	−	1.67513i	0.220803	−	0.249714i
$46$	−10.7005			−1.57771
$47$			4.31265i			0.629065i	0.949247	+	0.314532i	$0.101848\pi$
$47$			4.31265i			0.629065i	−0.949247	+	0.314532i	$0.898152\pi$
$48$		−	12.2750i		−	1.77175i
$49$	−0.873992			−0.124856
$50$	13.2750	−	1.63752i	1.87737	−	0.231580i
$51$	4.54420			0.636315
$52$		−	26.3938i		−	3.66015i
$53$		−	6.57452i		−	0.903079i	−0.892251	−	0.451540i	$-0.850875\pi$
$53$		−	6.57452i		−	0.903079i	0.892251	−	0.451540i	$-0.149125\pi$
$54$	−2.67513			−0.364039
$55$		0			0
$56$	23.6932			3.16614
$57$			4.57452i			0.605909i
$58$		−	6.38787i		−	0.838769i
$59$	13.2750			1.72826			0.864131	−	0.503266i	$-0.167868\pi$
$59$	13.2750			1.72826			0.864131	+	0.503266i	$0.167868\pi$
$60$	−8.63752	−	7.63752i	−1.11510	−	0.986000i
$61$	−7.92478			−1.01466			−0.507332	−	0.861751i	$-0.669368\pi$
$61$	−7.92478			−1.01466			−0.507332	+	0.861751i	$0.669368\pi$
$62$		−	2.57452i		−	0.326964i
$63$		−	2.80606i		−	0.353531i
$64$	−18.1187			−2.26484
$65$	−8.57452	−	7.58181i	−1.06354	−	0.940408i
$66$		0			0
$67$		−	10.7005i		−	1.30728i	−0.756807	−	0.653639i	$-0.773242\pi$
$67$		−	10.7005i		−	1.30728i	0.756807	−	0.653639i	$-0.226758\pi$
$68$		−	23.4314i		−	2.84147i
$69$	4.00000			0.481543
$70$	11.1187	−	12.5745i	1.32894	−	1.50294i
$71$	−7.35026			−0.872316			−0.436158	−	0.899870i	$-0.643661\pi$
$71$	−7.35026			−0.872316			−0.436158	+	0.899870i	$0.643661\pi$
$72$			8.44358i			0.995086i
$73$		−	6.41819i		−	0.751192i	−0.926783	−	0.375596i	$-0.877438\pi$
$73$		−	6.41819i		−	0.751192i	0.926783	−	0.375596i	$-0.122562\pi$
$74$	4.31265			0.501335
$75$	−4.96239	+	0.612127i	−0.573007	+	0.0706823i
$76$	23.5877			2.70569
$77$		0			0
$78$			13.6932i			1.55045i
$79$	1.35026			0.151916			0.0759582	−	0.997111i	$-0.475798\pi$
$79$	1.35026			0.151916			0.0759582	+	0.997111i	$0.475798\pi$
$80$	−18.1817	+	20.5623i	−2.03278	+	2.29893i
$81$	1.00000			0.111111
$82$			6.38787i			0.705423i
$83$			0.806063i			0.0884770i	0.999021	+	0.0442385i	$0.0140861\pi$
$83$			0.806063i			0.0884770i	−0.999021	+	0.0442385i	$0.985914\pi$
$84$	−14.4690			−1.57869
$85$	−7.61213	−	6.73084i	−0.825651	−	0.730062i
$86$	7.50659			0.809456
$87$			2.38787i			0.256007i
$88$		0			0
$89$	2.96239			0.314013			0.157006	−	0.987598i	$-0.449816\pi$
$89$	2.96239			0.314013			0.157006	+	0.987598i	$0.449816\pi$
$90$	4.48119	+	3.96239i	0.472359	+	0.417672i
$91$	−14.3634			−1.50570
$92$		−	20.6253i		−	2.15034i
$93$			0.962389i			0.0997950i
$94$	−11.5369			−1.18994
$95$	6.77575	−	7.66291i	0.695177	−	0.786198i
$96$	15.9502			1.62791
$97$			9.92478i			1.00771i	0.863789	+	0.503854i	$0.168085\pi$
$97$			9.92478i			1.00771i	−0.863789	+	0.503854i	$0.831915\pi$
$98$		−	2.33804i		−	0.236178i
$99$		0			0
$100$	3.15633	+	25.5877i	0.315633	+	2.55877i
$101$	13.6121			1.35446			0.677229	−	0.735773i	$-0.263181\pi$
$101$	13.6121			1.35446			0.677229	+	0.735773i	$0.263181\pi$
$102$			12.1563i			1.20366i
$103$			16.3127i			1.60733i	0.595080	+	0.803667i	$0.297120\pi$
$103$			16.3127i			1.60733i	−0.595080	+	0.803667i	$0.702880\pi$
$104$	43.2203			4.23810
$105$	−4.15633	+	4.70052i	−0.405616	+	0.458724i
$106$	17.5877			1.70827
$107$		−	9.43136i		−	0.911764i	−0.890040	−	0.455882i	$-0.849324\pi$
$107$		−	9.43136i		−	0.911764i	0.890040	−	0.455882i	$-0.150676\pi$
$108$		−	5.15633i		−	0.496168i
$109$	−15.4010			−1.47515			−0.737576	−	0.675264i	$-0.764030\pi$
$109$	−15.4010			−1.47515			−0.737576	+	0.675264i	$0.764030\pi$
$110$		0			0
$111$	−1.61213			−0.153016
$112$			34.4445i			3.25470i
$113$		−	13.7381i		−	1.29238i	−0.763179	−	0.646188i	$-0.776362\pi$
$113$		−	13.7381i		−	1.29238i	0.763179	−	0.646188i	$-0.223638\pi$
$114$	−12.2374			−1.14614
$115$	−6.70052	−	5.92478i	−0.624827	−	0.552488i
$116$	12.3127			1.14320
$117$		−	5.11871i		−	0.473225i
$118$			35.5125i			3.26919i
$119$	−12.7513			−1.16891
$120$	12.5066	−	14.1441i	1.14169	−	1.29117i
$121$		0			0
$122$		−	21.1998i		−	1.91934i
$123$		−	2.38787i		−	0.215307i
$124$	4.96239			0.445636
$125$	9.21933	+	6.32487i	0.824602	+	0.565713i
$126$	7.50659			0.668740
$127$			12.4182i			1.10194i	0.834526	+	0.550968i	$0.185741\pi$
$127$			12.4182i			1.10194i	−0.834526	+	0.550968i	$0.814259\pi$
$128$		−	16.5696i		−	1.46456i
$129$	−2.80606			−0.247060
$130$	20.2823	−	22.9380i	1.77888	−	2.01179i
$131$	−5.92478			−0.517650			−0.258825	−	0.965924i	$-0.583335\pi$
$131$	−5.92478			−0.517650			−0.258825	+	0.965924i	$0.583335\pi$
$132$		0			0
$133$		−	12.8364i		−	1.11306i
$134$	28.6253			2.47285
$135$	−1.67513	−	1.48119i	−0.144172	−	0.127481i
$136$	38.3693			3.29014
$137$		−	9.79877i		−	0.837165i	−0.908179	−	0.418583i	$-0.862527\pi$
$137$		−	9.79877i		−	0.837165i	0.908179	−	0.418583i	$-0.137473\pi$
$138$			10.7005i			0.910889i
$139$	−2.12601			−0.180326			−0.0901628	−	0.995927i	$-0.528739\pi$
$139$	−2.12601			−0.180326			−0.0901628	+	0.995927i	$0.528739\pi$
$140$	24.2374	+	21.4314i	2.04844	+	1.81128i
$141$	4.31265			0.363191
$142$		−	19.6629i		−	1.65007i
$143$		0			0
$144$	−12.2750			−1.02292
$145$	3.53690	−	4.00000i	0.293724	−	0.332182i
$146$	17.1695			1.42096
$147$			0.873992i			0.0720856i
$148$			8.31265i			0.683296i
$149$	8.31265			0.680999			0.340499	−	0.940245i	$-0.389404\pi$
$149$	8.31265			0.680999			0.340499	+	0.940245i	$0.389404\pi$
$150$	−1.63752	−	13.2750i	−0.133703	−	1.08390i
$151$	15.2750			1.24307			0.621533	−	0.783388i	$-0.286510\pi$
$151$	15.2750			1.24307			0.621533	+	0.783388i	$0.286510\pi$
$152$			38.6253i			3.13293i
$153$		−	4.54420i		−	0.367377i
$154$		0			0
$155$	1.42548	−	1.61213i	0.114498	−	0.129489i
$156$	−26.3938			−2.11319
$157$		−	7.01317i		−	0.559712i	−0.960042	−	0.279856i	$-0.909713\pi$
$157$		−	7.01317i		−	0.559712i	0.960042	−	0.279856i	$-0.0902868\pi$
$158$			3.61213i			0.287365i
$159$	−6.57452			−0.521393
$160$	−26.7186	−	23.6253i	−2.11229	−	1.86774i
$161$	−11.2243			−0.884595
$162$			2.67513i			0.210178i
$163$			6.76116i			0.529575i	0.964307	+	0.264787i	$0.0853018\pi$
$163$			6.76116i			0.529575i	−0.964307	+	0.264787i	$0.914698\pi$
$164$	−12.3127			−0.961456
$165$		0			0
$166$	−2.15633			−0.167363
$167$		−	11.8192i		−	0.914600i	−0.889312	−	0.457300i	$-0.848817\pi$
$167$		−	11.8192i		−	0.914600i	0.889312	−	0.457300i	$-0.151183\pi$
$168$		−	23.6932i		−	1.82797i
$169$	−13.2012			−1.01548
$170$	18.0059	−	20.3634i	1.38099	−	1.56180i
$171$	4.57452			0.349822
$172$			14.4690i			1.10325i
$173$		−	6.99271i		−	0.531646i	−0.964022	−	0.265823i	$-0.914356\pi$
$173$		−	6.99271i		−	0.531646i	0.964022	−	0.265823i	$-0.0856436\pi$
$174$	−6.38787			−0.484263
$175$	13.9248	−	1.71767i	1.05261	−	0.129843i
$176$		0			0
$177$		−	13.2750i		−	0.997813i
$178$			7.92478i			0.593987i
$179$	2.70052			0.201847			0.100923	−	0.994894i	$-0.467820\pi$
$179$	2.70052			0.201847			0.100923	+	0.994894i	$0.467820\pi$
$180$	−7.63752	+	8.63752i	−0.569267	+	0.643803i
$181$	−21.1998			−1.57577			−0.787885	−	0.615822i	$-0.788824\pi$
$181$	−21.1998			−1.57577			−0.787885	+	0.615822i	$0.788824\pi$
$182$		−	38.4241i		−	2.84818i
$183$			7.92478i			0.585816i
$184$	33.7743			2.48988
$185$	2.70052	+	2.38787i	0.198546	+	0.175560i
$186$	−2.57452			−0.188773
$187$		0			0
$188$		−	22.2374i		−	1.62183i
$189$	−2.80606			−0.204111
$190$	20.4993	+	18.1260i	1.48717	+	1.31500i
$191$	13.1490			0.951430			0.475715	−	0.879599i	$-0.342189\pi$
$191$	13.1490			0.951430			0.475715	+	0.879599i	$0.342189\pi$
$192$			18.1187i			1.30761i
$193$		−	5.89446i		−	0.424293i	−0.977238	−	0.212146i	$-0.931955\pi$
$193$		−	5.89446i		−	0.424293i	0.977238	−	0.212146i	$-0.0680453\pi$
$194$	−26.5501			−1.90618
$195$	−7.58181	+	8.57452i	−0.542945	+	0.614034i
$196$	4.50659			0.321899
$197$			20.3938i			1.45299i	0.687169	+	0.726497i	$0.258853\pi$
$197$			20.3938i			1.45299i	−0.687169	+	0.726497i	$0.741147\pi$
$198$		0			0
$199$	−17.4010			−1.23353			−0.616764	−	0.787148i	$-0.711557\pi$
$199$	−17.4010			−1.23353			−0.616764	+	0.787148i	$0.711557\pi$
$200$	−41.9003	+	5.16854i	−2.96280	+	0.365471i
$201$	−10.7005			−0.754757
$202$			36.4142i			2.56210i
$203$		−	6.70052i		−	0.470285i
$204$	−23.4314			−1.64052
$205$	−3.53690	+	4.00000i	−0.247028	+	0.279372i
$206$	−43.6385			−3.04044
$207$		−	4.00000i		−	0.278019i
$208$			62.8324i			4.35664i
$209$		0			0
$210$	−12.5745	−	11.1187i	−0.867724	−	0.767264i
$211$	−18.1260			−1.24785			−0.623923	−	0.781486i	$-0.714462\pi$
$211$	−18.1260			−1.24785			−0.623923	+	0.781486i	$0.714462\pi$
$212$			33.9003i			2.32828i
$213$			7.35026i			0.503632i
$214$	25.2301			1.72470
$215$	4.70052	+	4.15633i	0.320573	+	0.283459i
$216$	8.44358			0.574513
$217$		−	2.70052i		−	0.183323i
$218$		−	41.1998i		−	2.79040i
$219$	−6.41819			−0.433701
$220$		0			0
$221$	−23.2605			−1.56467
$222$		−	4.31265i		−	0.289446i
$223$			23.6385i			1.58295i	0.611202	+	0.791475i	$0.290686\pi$
$223$			23.6385i			1.58295i	−0.611202	+	0.791475i	$0.709314\pi$
$224$	−44.7572			−2.99047
$225$	0.612127	+	4.96239i	0.0408085	+	0.330826i
$226$	36.7513			2.44466
$227$		−	1.26916i		−	0.0842371i	−0.999113	−	0.0421185i	$-0.986589\pi$
$227$		−	1.26916i		−	0.0842371i	0.999113	−	0.0421185i	$-0.0134107\pi$
$228$		−	23.5877i		−	1.56213i
$229$	−16.1768			−1.06899			−0.534496	−	0.845171i	$-0.679499\pi$
$229$	−16.1768			−1.06899			−0.534496	+	0.845171i	$0.679499\pi$
$230$	15.8496	−	17.9248i	1.04509	−	1.18192i
$231$		0			0
$232$			20.1622i			1.32371i
$233$			18.4690i			1.20994i	0.796247	+	0.604971i	$0.206815\pi$
$233$			18.4690i			1.20994i	−0.796247	+	0.604971i	$0.793185\pi$
$234$	13.6932			0.895154
$235$	−7.22425	−	6.38787i	−0.471258	−	0.416699i
$236$	−68.4504			−4.45574
$237$		−	1.35026i		−	0.0877089i
$238$		−	34.1114i		−	2.21111i
$239$	19.3258			1.25008			0.625042	−	0.780591i	$-0.285082\pi$
$239$	19.3258			1.25008			0.625042	+	0.780591i	$0.285082\pi$
$240$	20.5623	+	18.1817i	1.32729	+	1.17362i
$241$	−28.5501			−1.83907			−0.919536	−	0.393006i	$-0.871435\pi$
$241$	−28.5501			−1.83907			−0.919536	+	0.393006i	$0.871435\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$	40.8627			2.61597
$245$	1.29455	−	1.46405i	0.0827059	−	0.0935348i
$246$	6.38787			0.407276
$247$		−	23.4156i		−	1.48990i
$248$			8.12601i			0.516002i
$249$	0.806063			0.0510822
$250$	−16.9199	+	24.6629i	−1.07011	+	1.55982i
$251$	23.1998			1.46436			0.732180	−	0.681112i	$-0.238503\pi$
$251$	23.1998			1.46436			0.732180	+	0.681112i	$0.238503\pi$
$252$			14.4690i			0.911460i
$253$		0			0
$254$	−33.2203			−2.08443
$255$	−6.73084	+	7.61213i	−0.421502	+	0.476690i
$256$	8.08840			0.505525
$257$			10.8872i			0.679123i	0.940584	+	0.339561i	$0.110279\pi$
$257$			10.8872i			0.679123i	−0.940584	+	0.339561i	$0.889721\pi$
$258$		−	7.50659i		−	0.467340i
$259$	4.52373			0.281091
$260$	44.2130	+	39.0943i	2.74197	+	2.42452i
$261$	2.38787			0.147806
$262$		−	15.8496i		−	0.979189i
$263$		−	9.11871i		−	0.562284i	−0.959666	−	0.281142i	$-0.909287\pi$
$263$		−	9.11871i		−	0.562284i	0.959666	−	0.281142i	$-0.0907132\pi$
$264$		0			0
$265$	11.0132	+	9.73813i	0.676534	+	0.598209i
$266$	34.3390			2.10546
$267$		−	2.96239i		−	0.181295i
$268$			55.1754i			3.37037i
$269$	11.2995			0.688941			0.344471	−	0.938797i	$-0.388058\pi$
$269$	11.2995			0.688941			0.344471	+	0.938797i	$0.388058\pi$
$270$	3.96239	−	4.48119i	0.241143	−	0.272717i
$271$	−25.1998			−1.53078			−0.765390	−	0.643567i	$-0.777454\pi$
$271$	−25.1998			−1.53078			−0.765390	+	0.643567i	$0.777454\pi$
$272$			55.7802i			3.38217i
$273$			14.3634i			0.869315i
$274$	26.2130			1.58358
$275$		0			0
$276$	−20.6253			−1.24150
$277$		−	2.41819i		−	0.145295i	−0.997358	−	0.0726475i	$-0.976855\pi$
$277$		−	2.41819i		−	0.145295i	0.997358	−	0.0726475i	$-0.0231448\pi$
$278$		−	5.68735i		−	0.341105i
$279$	0.962389			0.0576167
$280$	−35.0943	+	39.6893i	−2.09728	+	2.37189i
$281$	−30.4894			−1.81885			−0.909424	−	0.415870i	$-0.863477\pi$
$281$	−30.4894			−1.81885			−0.909424	+	0.415870i	$0.863477\pi$
$282$			11.5369i			0.687013i
$283$		−	8.35756i		−	0.496805i	−0.968657	−	0.248403i	$-0.920094\pi$
$283$		−	8.35756i		−	0.496805i	0.968657	−	0.248403i	$-0.0799056\pi$
$284$	37.9003			2.24897
$285$	−7.66291	−	6.77575i	−0.453912	−	0.401361i
$286$		0			0
$287$			6.70052i			0.395519i
$288$		−	15.9502i		−	0.939873i
$289$	−3.64974			−0.214690
$290$	10.7005	+	9.46168i	0.628356	+	0.555609i
$291$	9.92478			0.581801
$292$			33.0943i			1.93670i
$293$			23.0943i			1.34918i	0.738192	+	0.674591i	$0.235680\pi$
$293$			23.0943i			1.34918i	−0.738192	+	0.674591i	$0.764320\pi$
$294$	−2.33804			−0.136357
$295$	−19.6629	+	22.2374i	−1.14482	+	1.29471i
$296$	−13.6121			−0.791189
$297$		0			0
$298$			22.2374i			1.28818i
$299$	−20.4749			−1.18409
$300$	25.5877	−	3.15633i	1.47731	−	0.182231i
$301$	7.87399			0.453849
$302$			40.8627i			2.35139i
$303$		−	13.6121i		−	0.781996i
$304$	−56.1524			−3.22056
$305$	11.7381	−	13.2750i	0.672124	−	0.760127i
$306$	12.1563			0.694931
$307$			2.65562i			0.151564i	0.997124	+	0.0757821i	$0.0241453\pi$
$307$			2.65562i			0.151564i	−0.997124	+	0.0757821i	$0.975855\pi$
$308$		0			0
$309$	16.3127			0.927994
$310$	4.31265	+	3.81336i	0.244942	+	0.216584i
$311$	−15.9756			−0.905891			−0.452946	−	0.891538i	$-0.649627\pi$
$311$	−15.9756			−0.905891			−0.452946	+	0.891538i	$0.649627\pi$
$312$		−	43.2203i		−	2.44687i
$313$		−	0.0606343i		−	0.00342726i	−0.999999	−	0.00171363i	$-0.999455\pi$
$313$		−	0.0606343i		−	0.00342726i	0.999999	−	0.00171363i	$-0.000545465\pi$
$314$	18.7612			1.05875
$315$	4.70052	+	4.15633i	0.264844	+	0.234182i
$316$	−6.96239			−0.391665
$317$		−	3.81336i		−	0.214180i	−0.994249	−	0.107090i	$-0.965847\pi$
$317$		−	3.81336i		−	0.214180i	0.994249	−	0.107090i	$-0.0341532\pi$
$318$		−	17.5877i		−	0.986269i
$319$		0			0
$320$	26.8373	−	30.3512i	1.50025	−	1.69668i
$321$	−9.43136			−0.526407
$322$		−	30.0263i		−	1.67330i
$323$		−	20.7875i		−	1.15665i
$324$	−5.15633			−0.286463
$325$	25.4010	−	3.13330i	1.40900	−	0.173804i
$326$	−18.0870			−1.00175
$327$			15.4010i			0.851680i
$328$		−	20.1622i		−	1.11327i
$329$	−12.1016			−0.667181
$330$		0			0
$331$	24.2882			1.33500			0.667500	−	0.744609i	$-0.267364\pi$
$331$	24.2882			1.33500			0.667500	+	0.744609i	$0.267364\pi$
$332$		−	4.15633i		−	0.228108i
$333$			1.61213i			0.0883440i
$334$	31.6180			1.73006
$335$	17.9248	+	15.8496i	0.979335	+	0.865954i
$336$	34.4445			1.87910
$337$		−	22.5804i		−	1.23003i	−0.788514	−	0.615016i	$-0.789149\pi$
$337$		−	22.5804i		−	1.23003i	0.788514	−	0.615016i	$-0.210851\pi$
$338$		−	35.3150i		−	1.92088i
$339$	−13.7381			−0.746153
$340$	39.2506	+	34.7064i	2.12866	+	1.88222i
$341$		0			0
$342$			12.2374i			0.661724i
$343$			17.1900i			0.928171i
$344$	−23.6932			−1.27745
$345$	−5.92478	+	6.70052i	−0.318979	+	0.360744i
$346$	18.7064			1.00566
$347$			20.1925i			1.08399i	0.840381	+	0.541996i	$0.182331\pi$
$347$			20.1925i			1.08399i	−0.840381	+	0.541996i	$0.817669\pi$
$348$		−	12.3127i		−	0.660027i
$349$	−27.2506			−1.45869			−0.729346	−	0.684145i	$-0.760175\pi$
$349$	−27.2506			−1.45869			−0.729346	+	0.684145i	$0.760175\pi$
$350$	4.59498	+	37.2506i	0.245612	+	1.99113i
$351$	−5.11871			−0.273217
$352$		0			0
$353$			5.02302i			0.267349i	0.991025	+	0.133674i	$0.0426776\pi$
$353$			5.02302i			0.267349i	−0.991025	+	0.133674i	$0.957322\pi$
$354$	35.5125			1.88747
$355$	10.8872	−	12.3127i	0.577831	−	0.653488i
$356$	−15.2750			−0.809575
$357$			12.7513i			0.674871i
$358$			7.22425i			0.381814i
$359$	−18.1768			−0.959334			−0.479667	−	0.877450i	$-0.659243\pi$
$359$	−18.1768			−0.959334			−0.479667	+	0.877450i	$0.659243\pi$
$360$	−14.1441	−	12.5066i	−0.745460	−	0.659155i
$361$	1.92619			0.101379
$362$		−	56.7123i		−	2.98073i
$363$		0			0
$364$	74.0625			3.88193
$365$	10.7513	+	9.50659i	0.562749	+	0.497598i
$366$	−21.1998			−1.10813
$367$			8.56467i			0.447072i	0.974696	+	0.223536i	$0.0717600\pi$
$367$			8.56467i			0.447072i	−0.974696	+	0.223536i	$0.928240\pi$
$368$			49.1002i			2.55952i
$369$	−2.38787			−0.124308
$370$	−6.38787	+	7.22425i	−0.332090	+	0.375571i
$371$	18.4485			0.957799
$372$		−	4.96239i		−	0.257288i
$373$		−	7.81924i		−	0.404865i	−0.979296	−	0.202432i	$-0.935115\pi$
$373$		−	7.81924i		−	0.404865i	0.979296	−	0.202432i	$-0.0648846\pi$
$374$		0			0
$375$	6.32487	−	9.21933i	0.326615	−	0.476084i
$376$	36.4142			1.87792
$377$		−	12.2228i		−	0.629508i
$378$		−	7.50659i		−	0.386097i
$379$	−1.14903			−0.0590218			−0.0295109	−	0.999564i	$-0.509395\pi$
$379$	−1.14903			−0.0590218			−0.0295109	+	0.999564i	$0.509395\pi$
$380$	−34.9380	+	39.5125i	−1.79228	+	2.02695i
$381$	12.4182			0.636203
$382$			35.1754i			1.79973i
$383$			34.0870i			1.74176i	0.491493	+	0.870882i	$0.336451\pi$
$383$			34.0870i			1.74176i	−0.491493	+	0.870882i	$0.663549\pi$
$384$	−16.5696			−0.845563
$385$		0			0
$386$	15.7685			0.802593
$387$			2.80606i			0.142640i
$388$		−	51.1754i		−	2.59804i
$389$	−23.7743			−1.20541			−0.602703	−	0.797965i	$-0.705910\pi$
$389$	−23.7743			−1.20541			−0.602703	+	0.797965i	$0.705910\pi$
$390$	−22.9380	−	20.2823i	−1.16151	−	1.02704i
$391$	−18.1768			−0.919240
$392$			7.37962i			0.372727i
$393$			5.92478i			0.298865i
$394$	−54.5560			−2.74849
$395$	−2.00000	+	2.26187i	−0.100631	+	0.113807i
$396$		0			0
$397$			4.15045i			0.208305i	0.994561	+	0.104152i	$0.0332130\pi$
$397$			4.15045i			0.208305i	−0.994561	+	0.104152i	$0.966787\pi$
$398$		−	46.5501i		−	2.33334i
$399$	−12.8364			−0.642623
$400$	−7.51388	−	60.9135i	−0.375694	−	3.04568i
$401$	−33.9149			−1.69363			−0.846815	−	0.531887i	$-0.821483\pi$
$401$	−33.9149			−1.69363			−0.846815	+	0.531887i	$0.821483\pi$
$402$		−	28.6253i		−	1.42770i
$403$		−	4.92619i		−	0.245391i
$404$	−70.1886			−3.49201
$405$	−1.48119	+	1.67513i	−0.0736011	+	0.0832379i
$406$	17.9248			0.889592
$407$		0			0
$408$		−	38.3693i		−	1.89956i
$409$	8.55008			0.422774			0.211387	−	0.977402i	$-0.432202\pi$
$409$	8.55008			0.422774			0.211387	+	0.977402i	$0.432202\pi$
$410$	−10.7005	−	9.46168i	−0.528461	−	0.467279i
$411$	−9.79877			−0.483338
$412$		−	84.1133i		−	4.14397i
$413$			37.2506i			1.83298i
$414$	10.7005			0.525902
$415$	−1.35026	−	1.19394i	−0.0662817	−	0.0586080i
$416$	−81.6444			−4.00294
$417$			2.12601i			0.104111i
$418$		0			0
$419$	−13.2995			−0.649722			−0.324861	−	0.945762i	$-0.605318\pi$
$419$	−13.2995			−0.649722			−0.324861	+	0.945762i	$0.605318\pi$
$420$	21.4314	−	24.2374i	1.04574	−	1.18267i
$421$	33.3014			1.62301			0.811505	−	0.584345i	$-0.198649\pi$
$421$	33.3014			1.62301			0.811505	+	0.584345i	$0.198649\pi$
$422$		−	48.4894i		−	2.36043i
$423$		−	4.31265i		−	0.209688i
$424$	−55.5125			−2.69592
$425$	22.5501	−	2.78163i	1.09384	−	0.134929i
$426$	−19.6629			−0.952671
$427$		−	22.2374i		−	1.07614i
$428$			48.6312i			2.35068i
$429$		0			0
$430$	−11.1187	+	12.5745i	−0.536192	+	0.606397i
$431$	−2.07522			−0.0999600			−0.0499800	−	0.998750i	$-0.515916\pi$
$431$	−2.07522			−0.0999600			−0.0499800	+	0.998750i	$0.515916\pi$
$432$			12.2750i			0.590583i
$433$		−	37.4010i		−	1.79738i	−0.438585	−	0.898690i	$-0.644520\pi$
$433$		−	37.4010i		−	1.79738i	0.438585	−	0.898690i	$-0.355480\pi$
$434$	7.22425			0.346775
$435$	−4.00000	−	3.53690i	−0.191785	−	0.169582i
$436$	79.4128			3.80318
$437$		−	18.2981i		−	0.875315i
$438$		−	17.1695i		−	0.820390i
$439$	−6.02444			−0.287531			−0.143765	−	0.989612i	$-0.545921\pi$
$439$	−6.02444			−0.287531			−0.143765	+	0.989612i	$0.545921\pi$
$440$		0			0
$441$	0.873992			0.0416187
$442$		−	62.2247i		−	2.95973i
$443$		−	10.1359i		−	0.481569i	−0.970579	−	0.240785i	$-0.922595\pi$
$443$		−	10.1359i		−	0.481569i	0.970579	−	0.240785i	$-0.0774047\pi$
$444$	8.31265			0.394501
$445$	−4.38787	+	4.96239i	−0.208005	+	0.235240i
$446$	−63.2360			−2.99431
$447$		−	8.31265i		−	0.393175i
$448$		−	50.8423i		−	2.40207i
$449$	11.4861			0.542063			0.271032	−	0.962570i	$-0.412635\pi$
$449$	11.4861			0.542063			0.271032	+	0.962570i	$0.412635\pi$
$450$	−13.2750	+	1.63752i	−0.625791	+	0.0771934i
$451$		0			0
$452$			70.8383i			3.33195i
$453$		−	15.2750i		−	0.717684i
$454$	3.39517			0.159343
$455$	21.2750	−	24.0606i	0.997389	−	1.12798i
$456$	38.6253			1.80880
$457$		−	15.0435i		−	0.703705i	−0.936055	−	0.351852i	$-0.885552\pi$
$457$		−	15.0435i		−	0.703705i	0.936055	−	0.351852i	$-0.114448\pi$
$458$		−	43.2750i		−	2.02211i
$459$	−4.54420			−0.212105
$460$	34.5501	+	30.5501i	1.61091	+	1.42440i
$461$	26.2374			1.22200			0.610999	−	0.791631i	$-0.290768\pi$
$461$	26.2374			1.22200			0.610999	+	0.791631i	$0.290768\pi$
$462$		0			0
$463$		−	5.46168i		−	0.253826i	−0.991914	−	0.126913i	$-0.959493\pi$
$463$		−	5.46168i		−	0.253826i	0.991914	−	0.126913i	$-0.0405069\pi$
$464$	−29.3112			−1.36074
$465$	−1.61213	−	1.42548i	−0.0747606	−	0.0661053i
$466$	−49.4069			−2.28873
$467$			2.70052i			0.124965i	0.998046	+	0.0624827i	$0.0199018\pi$
$467$			2.70052i			0.124965i	−0.998046	+	0.0624827i	$0.980098\pi$
$468$			26.3938i			1.22005i
$469$	30.0263			1.38649
$470$	17.0884	−	19.3258i	0.788229	−	0.891434i
$471$	−7.01317			−0.323150
$472$		−	112.089i		−	5.15931i
$473$		0			0
$474$	3.61213			0.165910
$475$	2.80018	+	22.7005i	0.128481	+	1.04157i
$476$	65.7499			3.01364
$477$			6.57452i			0.301026i
$478$			51.6991i			2.36466i
$479$	16.7757			0.766503			0.383252	−	0.923644i	$-0.374804\pi$
$479$	16.7757			0.766503			0.383252	+	0.923644i	$0.374804\pi$
$480$	−23.6253	+	26.7186i	−1.07834	+	1.21953i
$481$	8.25202			0.376260
$482$		−	76.3752i		−	3.47879i
$483$			11.2243i			0.510721i
$484$		0			0
$485$	−16.6253	−	14.7005i	−0.754916	−	0.667516i
$486$	2.67513			0.121346
$487$			34.3996i			1.55880i	0.626529	+	0.779398i	$0.284475\pi$
$487$			34.3996i			1.55880i	−0.626529	+	0.779398i	$0.715525\pi$
$488$			66.9135i			3.02903i
$489$	6.76116			0.305750
$490$	3.91653	+	3.46310i	0.176931	+	0.156447i
$491$	−14.9525			−0.674799			−0.337399	−	0.941362i	$-0.609547\pi$
$491$	−14.9525			−0.674799			−0.337399	+	0.941362i	$0.609547\pi$
$492$			12.3127i			0.555097i
$493$		−	10.8510i		−	0.488703i
$494$	62.6399			2.81830
$495$		0			0
$496$	−11.8134			−0.530435
$497$		−	20.6253i		−	0.925171i
$498$			2.15633i			0.0966272i
$499$	2.85097			0.127627			0.0638135	−	0.997962i	$-0.479674\pi$
$499$	2.85097			0.127627			0.0638135	+	0.997962i	$0.479674\pi$
$500$	−47.5379	−	32.6131i	−2.12596	−	1.45850i
$501$	−11.8192			−0.528045
$502$			62.0625i			2.76999i
$503$		−	1.48024i		−	0.0660006i	−0.999455	−	0.0330003i	$-0.989494\pi$
$503$		−	1.48024i		−	0.0660006i	0.999455	−	0.0330003i	$-0.0105062\pi$
$504$	−23.6932			−1.05538
$505$	−20.1622	+	22.8021i	−0.897206	+	1.01468i
$506$		0			0
$507$			13.2012i			0.586287i
$508$		−	64.0322i		−	2.84097i
$509$	23.2995			1.03273			0.516366	−	0.856368i	$-0.327285\pi$
$509$	23.2995			1.03273			0.516366	+	0.856368i	$0.327285\pi$
$510$	−20.3634	−	18.0059i	−0.901708	−	0.797314i
$511$	18.0098			0.796709
$512$		−	11.5017i		−	0.508306i
$513$		−	4.57452i		−	0.201970i
$514$	−29.1246			−1.28463
$515$	−27.3258	−	24.1622i	−1.20412	−	1.06471i
$516$	14.4690			0.636961
$517$		0			0
$518$			12.1016i			0.531712i
$519$	−6.99271			−0.306946
$520$	−64.0176	+	72.3996i	−2.80736	+	3.17493i
$521$	−6.81194			−0.298437			−0.149218	−	0.988804i	$-0.547676\pi$
$521$	−6.81194			−0.298437			−0.149218	+	0.988804i	$0.547676\pi$
$522$			6.38787i			0.279590i
$523$			16.0567i			0.702109i	0.936355	+	0.351054i	$0.114177\pi$
$523$			16.0567i			0.702109i	−0.936355	+	0.351054i	$0.885823\pi$
$524$	30.5501			1.33459
$525$	−1.71767	−	13.9248i	−0.0749651	−	0.607727i
$526$	24.3938			1.06362
$527$		−	4.37328i		−	0.190503i
$528$		0			0
$529$	7.00000			0.304348
$530$	−26.0508	+	29.4617i	−1.13157	+	1.27973i
$531$	−13.2750			−0.576088
$532$			66.1886i			2.86964i
$533$			12.2228i			0.529430i
$534$	7.92478			0.342939
$535$	15.7988	+	13.9697i	0.683040	+	0.603962i
$536$	−90.3508			−3.90256
$537$		−	2.70052i		−	0.116536i
$538$			30.2276i			1.30320i
$539$		0			0
$540$	8.63752	+	7.63752i	0.371700	+	0.328667i
$541$	−6.62530			−0.284844			−0.142422	−	0.989806i	$-0.545489\pi$
$541$	−6.62530			−0.284844			−0.142422	+	0.989806i	$0.545489\pi$
$542$		−	67.4128i		−	2.89563i
$543$			21.1998i			0.909771i
$544$	−72.4807			−3.10759
$545$	22.8119	−	25.7988i	0.977156	−	1.10510i
$546$	−38.4241			−1.64440
$547$		−	5.75860i		−	0.246220i	−0.992393	−	0.123110i	$-0.960713\pi$
$547$		−	5.75860i		−	0.246220i	0.992393	−	0.123110i	$-0.0392868\pi$
$548$			50.5256i			2.15835i
$549$	7.92478			0.338221
$550$		0			0
$551$	10.9234			0.465351
$552$		−	33.7743i		−	1.43753i
$553$			3.78892i			0.161121i
$554$	6.46898			0.274840
$555$	2.38787	−	2.70052i	0.101360	−	0.114631i
$556$	10.9624			0.464909
$557$			19.8700i			0.841920i	0.907079	+	0.420960i	$0.138307\pi$
$557$			19.8700i			0.841920i	−0.907079	+	0.420960i	$0.861693\pi$
$558$			2.57452i			0.108988i
$559$	14.3634			0.607509
$560$	−57.6991	−	51.0191i	−2.43823	−	2.15595i
$561$		0			0
$562$		−	81.5633i		−	3.44054i
$563$		−	5.83383i		−	0.245866i	−0.992415	−	0.122933i	$-0.960770\pi$
$563$		−	5.83383i		−	0.245866i	0.992415	−	0.122933i	$-0.0392301\pi$
$564$	−22.2374			−0.936365
$565$	23.0132	+	20.3488i	0.968171	+	0.856082i
$566$	22.3576			0.939758
$567$			2.80606i			0.117844i
$568$			62.0625i			2.60409i
$569$	46.7123			1.95828			0.979140	−	0.203185i	$-0.0651293\pi$
$569$	46.7123			1.95828			0.979140	+	0.203185i	$0.0651293\pi$
$570$	18.1260	−	20.4993i	0.759215	−	0.858621i
$571$	24.4241			1.02212			0.511058	−	0.859546i	$-0.329254\pi$
$571$	24.4241			1.02212			0.511058	+	0.859546i	$0.329254\pi$
$572$		0			0
$573$		−	13.1490i		−	0.549309i
$574$	−17.9248			−0.748166
$575$	19.8496	−	2.44851i	0.827784	−	0.102110i
$576$	18.1187			0.754946
$577$		−	16.5647i		−	0.689596i	−0.938677	−	0.344798i	$-0.887947\pi$
$577$		−	16.5647i		−	0.689596i	0.938677	−	0.344798i	$-0.112053\pi$
$578$		−	9.76353i		−	0.406109i
$579$	−5.89446			−0.244965
$580$	−18.2374	+	20.6253i	−0.757268	+	0.856419i
$581$	−2.26187			−0.0938380
$582$			26.5501i			1.10054i
$583$		0			0
$584$	−54.1925			−2.24250
$585$	8.57452	+	7.58181i	0.354513	+	0.313469i
$586$	−61.7802			−2.55212
$587$		−	27.3258i		−	1.12786i	−0.825823	−	0.563929i	$-0.809289\pi$
$587$		−	27.3258i		−	1.12786i	0.825823	−	0.563929i	$-0.190711\pi$
$588$		−	4.50659i		−	0.185849i
$589$	4.40246			0.181400
$590$	−59.4880	−	52.6009i	−2.44908	−	2.16554i
$591$	20.3938			0.838887
$592$		−	19.7889i		−	0.813320i
$593$			12.6048i			0.517618i	0.965928	+	0.258809i	$0.0833301\pi$
$593$			12.6048i			0.517618i	−0.965928	+	0.258809i	$0.916670\pi$
$594$		0			0
$595$	18.8872	−	21.3601i	0.774298	−	0.875679i
$596$	−42.8627			−1.75573
$597$			17.4010i			0.712177i
$598$		−	54.7729i		−	2.23983i
$599$	−10.5990			−0.433061			−0.216531	−	0.976276i	$-0.569474\pi$
$599$	−10.5990			−0.433061			−0.216531	+	0.976276i	$0.569474\pi$
$600$	5.16854	+	41.9003i	0.211005	+	1.71057i
$601$	22.4749			0.916768			0.458384	−	0.888754i	$-0.348428\pi$
$601$	22.4749			0.916768			0.458384	+	0.888754i	$0.348428\pi$
$602$			21.0640i			0.858503i
$603$			10.7005i			0.435759i
$604$	−78.7631			−3.20482
$605$		0			0
$606$	36.4142			1.47923
$607$			33.5183i			1.36047i	0.732995	+	0.680234i	$0.238122\pi$
$607$			33.5183i			1.36047i	−0.732995	+	0.680234i	$0.761878\pi$
$608$		−	72.9643i		−	2.95909i
$609$	−6.70052			−0.271519
$610$	35.5125	+	31.4010i	1.43786	+	1.27139i
$611$	−22.0752			−0.893068
$612$			23.4314i			0.947157i
$613$		−	11.5672i		−	0.467196i	−0.972333	−	0.233598i	$-0.924950\pi$
$613$		−	11.5672i		−	0.467196i	0.972333	−	0.233598i	$-0.0750499\pi$
$614$	−7.10413			−0.286699
$615$	4.00000	+	3.53690i	0.161296	+	0.142622i
$616$		0			0
$617$			33.3357i			1.34204i	0.741438	+	0.671022i	$0.234144\pi$
$617$			33.3357i			1.34204i	−0.741438	+	0.671022i	$0.765856\pi$
$618$			43.6385i			1.75540i
$619$	−4.43866			−0.178405			−0.0892024	−	0.996014i	$-0.528432\pi$
$619$	−4.43866			−0.178405			−0.0892024	+	0.996014i	$0.528432\pi$
$620$	−7.35026	+	8.31265i	−0.295194	+	0.333844i
$621$	−4.00000			−0.160514
$622$		−	42.7367i		−	1.71359i
$623$			8.31265i			0.333039i
$624$	62.8324			2.51531
$625$	−24.2506	+	6.07522i	−0.970024	+	0.243009i
$626$	0.162205			0.00648301
$627$		0			0
$628$			36.1622i			1.44303i
$629$	7.32582			0.292100
$630$	−11.1187	+	12.5745i	−0.442980	+	0.500981i
$631$	−39.8496			−1.58639			−0.793193	−	0.608971i	$-0.791583\pi$
$631$	−39.8496			−1.58639			−0.793193	+	0.608971i	$0.791583\pi$
$632$		−	11.4010i		−	0.453509i
$633$			18.1260i			0.720444i
$634$	10.2012			0.405143
$635$	−20.8021	−	18.3938i	−0.825506	−	0.729934i
$636$	33.9003			1.34424
$637$		−	4.47371i		−	0.177255i
$638$		0			0
$639$	7.35026			0.290772
$640$	27.7562	+	24.5428i	1.09716	+	0.970139i
$641$	−3.88858			−0.153590			−0.0767948	−	0.997047i	$-0.524469\pi$
$641$	−3.88858			−0.153590			−0.0767948	+	0.997047i	$0.524469\pi$
$642$		−	25.2301i		−	0.995754i
$643$			40.9380i			1.61444i	0.590254	+	0.807218i	$0.299028\pi$
$643$			40.9380i			1.61444i	−0.590254	+	0.807218i	$0.700972\pi$
$644$	57.8759			2.28063
$645$	4.15633	−	4.70052i	0.163655	−	0.185083i
$646$	55.6093			2.18792
$647$		−	1.76257i		−	0.0692939i	−0.999400	−	0.0346469i	$-0.988969\pi$
$647$		−	1.76257i		−	0.0692939i	0.999400	−	0.0346469i	$-0.0110307\pi$
$648$		−	8.44358i		−	0.331695i
$649$		0			0
$650$	8.38199	+	67.9511i	0.328769	+	2.66526i
$651$	−2.70052			−0.105842
$652$		−	34.8627i		−	1.36533i
$653$		−	7.03761i		−	0.275403i	−0.990474	−	0.137702i	$-0.956029\pi$
$653$		−	7.03761i		−	0.275403i	0.990474	−	0.137702i	$-0.0439715\pi$
$654$	−41.1998			−1.61104
$655$	8.77575	−	9.92478i	0.342897	−	0.387793i
$656$	29.3112			1.14441
$657$			6.41819i			0.250397i
$658$		−	32.3733i		−	1.26204i
$659$	−12.6253			−0.491812			−0.245906	−	0.969294i	$-0.579085\pi$
$659$	−12.6253			−0.491812			−0.245906	+	0.969294i	$0.579085\pi$
$660$		0			0
$661$	13.2243			0.514364			0.257182	−	0.966363i	$-0.417206\pi$
$661$	13.2243			0.514364			0.257182	+	0.966363i	$0.417206\pi$
$662$			64.9741i			2.52529i
$663$			23.2605i			0.903361i
$664$	6.80606			0.264126
$665$	21.5026	+	19.0132i	0.833836	+	0.737299i
$666$	−4.31265			−0.167112
$667$		−	9.55149i		−	0.369835i
$668$			60.9438i			2.35799i
$669$	23.6385			0.913916
$670$	−42.3996	+	47.9511i	−1.63804	+	1.85251i
$671$		0			0
$672$			44.7572i			1.72655i
$673$		−	18.2677i		−	0.704170i	−0.935968	−	0.352085i	$-0.885473\pi$
$673$		−	18.2677i		−	0.704170i	0.935968	−	0.352085i	$-0.114527\pi$
$674$	60.4055			2.32673
$675$	4.96239	−	0.612127i	0.191002	−	0.0235608i
$676$	68.0698			2.61807
$677$		−	33.0191i		−	1.26903i	−0.772912	−	0.634513i	$-0.781201\pi$
$677$		−	33.0191i		−	1.26903i	0.772912	−	0.634513i	$-0.218799\pi$
$678$		−	36.7513i		−	1.41143i
$679$	−27.8496			−1.06877
$680$	−56.8324	+	64.2736i	−2.17942	+	2.46478i
$681$	−1.26916			−0.0486343
$682$		0			0
$683$			30.8627i			1.18093i	0.807063	+	0.590465i	$0.201056\pi$
$683$			30.8627i			1.18093i	−0.807063	+	0.590465i	$0.798944\pi$
$684$	−23.5877			−0.901898
$685$	16.4142	+	14.5139i	0.627155	+	0.554547i
$686$	−45.9854			−1.75573
$687$			16.1768i			0.617183i
$688$		−	34.4445i		−	1.31319i
$689$	33.6531			1.28208
$690$	−17.9248	−	15.8496i	−0.682385	−	0.603382i
$691$	27.6991			1.05372			0.526862	−	0.849951i	$-0.323369\pi$
$691$	27.6991			1.05372			0.526862	+	0.849951i	$0.323369\pi$
$692$			36.0567i			1.37067i
$693$		0			0
$694$	−54.0176			−2.05048
$695$	3.14903	−	3.56134i	0.119450	−	0.135089i
$696$	20.1622			0.764246
$697$			10.8510i			0.411010i
$698$		−	72.8989i		−	2.75927i
$699$	18.4690			0.698561
$700$	−71.8007	+	8.85685i	−2.71381	+	0.334757i
$701$	38.9643			1.47166			0.735831	−	0.677166i	$-0.236792\pi$
$701$	38.9643			1.47166			0.735831	+	0.677166i	$0.236792\pi$
$702$		−	13.6932i		−	0.516818i
$703$			7.37470i			0.278142i
$704$		0			0
$705$	−6.38787	+	7.22425i	−0.240581	+	0.272081i
$706$	−13.4372			−0.505717
$707$			38.1965i			1.43653i
$708$			68.4504i			2.57252i
$709$	4.32250			0.162335			0.0811674	−	0.996700i	$-0.474135\pi$
$709$	4.32250			0.162335			0.0811674	+	0.996700i	$0.474135\pi$
$710$	32.9380	+	29.1246i	1.23614	+	1.09303i
$711$	−1.35026			−0.0506388
$712$		−	25.0132i		−	0.937408i
$713$		−	3.84955i		−	0.144167i
$714$	−34.1114			−1.27659
$715$		0			0
$716$	−13.9248			−0.520393
$717$		−	19.3258i		−	0.721736i
$718$		−	48.6253i		−	1.81468i
$719$	38.3996			1.43206			0.716032	−	0.698067i	$-0.245956\pi$
$719$	38.3996			1.43206			0.716032	+	0.698067i	$0.245956\pi$
$720$	18.1817	−	20.5623i	0.677593	−	0.766312i
$721$	−45.7743			−1.70473
$722$			5.15282i			0.191768i
$723$			28.5501i			1.06179i
$724$	109.313			4.06259
$725$	1.46168	+	11.8496i	0.0542855	+	0.440081i
$726$		0			0
$727$			21.6728i			0.803798i	0.915684	+	0.401899i	$0.131650\pi$
$727$			21.6728i			0.803798i	−0.915684	+	0.401899i	$0.868350\pi$
$728$			121.279i			4.49489i
$729$	−1.00000			−0.0370370
$730$	−25.4314	+	28.7612i	−0.941257	+	1.06450i
$731$	12.7513			0.471624
$732$		−	40.8627i		−	1.51033i
$733$		−	25.0698i		−	0.925976i	−0.886365	−	0.462988i	$-0.846777\pi$
$733$		−	25.0698i		−	0.925976i	0.886365	−	0.462988i	$-0.153223\pi$
$734$	−22.9116			−0.845683
$735$	−1.46405	−	1.29455i	−0.0540023	−	0.0477503i
$736$	−63.8007			−2.35172
$737$		0			0
$738$		−	6.38787i		−	0.235141i
$739$	−18.9018			−0.695312			−0.347656	−	0.937622i	$-0.613022\pi$
$739$	−18.9018			−0.695312			−0.347656	+	0.937622i	$0.613022\pi$
$740$	−13.9248	−	12.3127i	−0.511885	−	0.452622i
$741$	−23.4156			−0.860195
$742$			49.3522i			1.81178i
$743$		−	43.6688i		−	1.60205i	−0.598629	−	0.801026i	$-0.704288\pi$
$743$		−	43.6688i		−	1.60205i	0.598629	−	0.801026i	$-0.295712\pi$
$744$	8.12601			0.297914
$745$	−12.3127	+	13.9248i	−0.451101	+	0.510164i
$746$	20.9175			0.765843
$747$		−	0.806063i		−	0.0294923i
$748$		0			0
$749$	26.4650			0.967010
$750$	24.6629	+	16.9199i	0.900562	+	0.617826i
$751$	−47.0541			−1.71703			−0.858514	−	0.512789i	$-0.828612\pi$
$751$	−47.0541			−1.71703			−0.858514	+	0.512789i	$0.828612\pi$
$752$			52.9380i			1.93045i
$753$		−	23.1998i		−	0.845448i
$754$	32.6977			1.19078
$755$	−22.6253	+	25.5877i	−0.823419	+	0.931231i
$756$	14.4690			0.526232
$757$			26.9116i			0.978119i	0.872250	+	0.489059i	$0.162660\pi$
$757$			26.9116i			0.978119i	−0.872250	+	0.489059i	$0.837340\pi$
$758$		−	3.07381i		−	0.111646i
$759$		0			0
$760$	−64.7024	−	57.2116i	−2.34700	−	2.07528i
$761$	−16.6859			−0.604865			−0.302432	−	0.953171i	$-0.597799\pi$
$761$	−16.6859			−0.604865			−0.302432	+	0.953171i	$0.597799\pi$
$762$			33.2203i			1.20344i
$763$		−	43.2163i		−	1.56454i
$764$	−67.8007			−2.45294
$765$	7.61213	+	6.73084i	0.275217	+	0.243354i
$766$	−91.1871			−3.29473
$767$			67.9511i			2.45357i
$768$		−	8.08840i		−	0.291865i
$769$	23.2995			0.840201			0.420100	−	0.907478i	$-0.361995\pi$
$769$	23.2995			0.840201			0.420100	+	0.907478i	$0.361995\pi$
$770$		0			0
$771$	10.8872			0.392092
$772$			30.3938i			1.09390i
$773$		−	1.63656i		−	0.0588631i	−0.999567	−	0.0294316i	$-0.990630\pi$
$773$		−	1.63656i		−	0.0588631i	0.999567	−	0.0294316i	$-0.00936971\pi$
$774$	−7.50659			−0.269819
$775$	0.589104	+	4.77575i	0.0211612	+	0.171550i
$776$	83.8007			3.00827
$777$		−	4.52373i		−	0.162288i
$778$		−	63.5994i		−	2.28015i
$779$	−10.9234			−0.391370
$780$	39.0943	−	44.2130i	1.39980	−	1.58308i
$781$		0			0
$782$		−	48.6253i		−	1.73884i
$783$		−	2.38787i		−

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)

\(f(q)\)	\(=\)	\(q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +(-1.48119 + 1.67513i) q^{5} +2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +(-1.48119 + 1.67513i) q^{5} +2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +(-4.48119 - 3.96239i) q^{10} +5.15633i q^{12} +5.11871i q^{13} -7.50659 q^{14} +(1.67513 + 1.48119i) q^{15} +12.2750 q^{16} +4.54420i q^{17} -2.67513i q^{18} -4.57452 q^{19} +(7.63752 - 8.63752i) q^{20} +2.80606 q^{21} +4.00000i q^{23} -8.44358 q^{24} +(-0.612127 - 4.96239i) q^{25} -13.6932 q^{26} +1.00000i q^{27} -14.4690i q^{28} -2.38787 q^{29} +(-3.96239 + 4.48119i) q^{30} -0.962389 q^{31} +15.9502i q^{32} -12.1563 q^{34} +(-4.70052 - 4.15633i) q^{35} +5.15633 q^{36} -1.61213i q^{37} -12.2374i q^{38} +5.11871 q^{39} +(14.1441 + 12.5066i) q^{40} +2.38787 q^{41} +7.50659i q^{42} -2.80606i q^{43} +(1.48119 - 1.67513i) q^{45} -10.7005 q^{46} +4.31265i q^{47} -12.2750i q^{48} -0.873992 q^{49} +(13.2750 - 1.63752i) q^{50} +4.54420 q^{51} -26.3938i q^{52} -6.57452i q^{53} -2.67513 q^{54} +23.6932 q^{56} +4.57452i q^{57} -6.38787i q^{58} +13.2750 q^{59} +(-8.63752 - 7.63752i) q^{60} -7.92478 q^{61} -2.57452i q^{62} -2.80606i q^{63} -18.1187 q^{64} +(-8.57452 - 7.58181i) q^{65} -10.7005i q^{67} -23.4314i q^{68} +4.00000 q^{69} +(11.1187 - 12.5745i) q^{70} -7.35026 q^{71} +8.44358i q^{72} -6.41819i q^{73} +4.31265 q^{74} +(-4.96239 + 0.612127i) q^{75} +23.5877 q^{76} +13.6932i q^{78} +1.35026 q^{79} +(-18.1817 + 20.5623i) q^{80} +1.00000 q^{81} +6.38787i q^{82} +0.806063i q^{83} -14.4690 q^{84} +(-7.61213 - 6.73084i) q^{85} +7.50659 q^{86} +2.38787i q^{87} +2.96239 q^{89} +(4.48119 + 3.96239i) q^{90} -14.3634 q^{91} -20.6253i q^{92} +0.962389i q^{93} -11.5369 q^{94} +(6.77575 - 7.66291i) q^{95} +15.9502 q^{96} +9.92478i q^{97} -2.33804i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more