LMFDB - Embedded newform 1380.2.i.a.1241.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1241,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1380.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.i (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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + \cdots + 6561 $ x^16 - x^14 - 4x^13 - 2x^12 + 12x^11 + 17x^10 + 16x^9 - 110x^8 + 48x^7 + 153x^6 + 324x^5 - 162x^4 - 972x^3 - 729x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.12
Root			$-1.04974 + 1.37769i$ of defining polynomial
Character	$\chi$	$=$	1380.1241
Dual form			1380.2.i.a.1241.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.04974 + 1.37769i) q^{3} -1.00000 q^{5} -2.10004i q^{7} +(-0.796081 + 2.89245i) q^{9} +O(q^{10})$	\(q+(1.04974 + 1.37769i) q^{3} -1.00000 q^{5} -2.10004i q^{7} +(-0.796081 + 2.89245i) q^{9} +2.45920 q^{11} -5.70869 q^{13} +(-1.04974 - 1.37769i) q^{15} +6.65577 q^{17} +4.23245i q^{19} +(2.89321 - 2.20450i) q^{21} +(3.63955 + 3.12309i) q^{23} +1.00000 q^{25} +(-4.82059 + 1.93957i) q^{27} +10.0198i q^{29} +4.21597 q^{31} +(2.58152 + 3.38802i) q^{33} +2.10004i q^{35} +8.14987i q^{37} +(-5.99266 - 7.86483i) q^{39} +3.58027i q^{41} +2.94492i q^{43} +(0.796081 - 2.89245i) q^{45} -6.21937i q^{47} +2.58984 q^{49} +(6.98685 + 9.16962i) q^{51} -11.4974 q^{53} -2.45920 q^{55} +(-5.83102 + 4.44298i) q^{57} +6.31669i q^{59} -6.35077i q^{61} +(6.07425 + 1.67180i) q^{63} +5.70869 q^{65} -11.5057i q^{67} +(-0.482075 + 8.29262i) q^{69} +4.67888i q^{71} +16.6582 q^{73} +(1.04974 + 1.37769i) q^{75} -5.16440i q^{77} -8.63724i q^{79} +(-7.73251 - 4.60525i) q^{81} +0.357391 q^{83} -6.65577 q^{85} +(-13.8043 + 10.5182i) q^{87} -10.1485 q^{89} +11.9885i q^{91} +(4.42568 + 5.80831i) q^{93} -4.23245i q^{95} -3.20088i q^{97} +(-1.95772 + 7.11310i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 + 2 * q^9	$ 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) $ 16 * q - 16 * q^5 + 2 * q^9 - 4 * q^21 + 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 + 2 * q^33 - 14 * q^39 - 2 * q^45 + 8 * q^49 - 2 * q^51 + 4 * q^53 - 2 * q^57 + 30 * q^63 - 4 * q^73 + 10 * q^81 - 4 * q^83 - 10 * q^87 - 28 * q^89 - 10 * q^93 - 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	1.04974	+	1.37769i	0.606069	+	0.795412i
$3$	1.04974	+	1.37769i	0.606069	+	0.795412i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	2.10004i		−	0.793740i	−0.917875	−	0.396870i	$-0.870096\pi$
$7$		−	2.10004i		−	0.793740i	0.917875	−	0.396870i	$-0.129904\pi$
$8$		0			0
$9$	−0.796081	+	2.89245i	−0.265360	+	0.964149i
$10$		0			0
$11$	2.45920			0.741476			0.370738	−	0.928738i	$-0.379105\pi$
$11$	2.45920			0.741476			0.370738	+	0.928738i	$0.379105\pi$
$12$		0			0
$13$	−5.70869			−1.58331			−0.791653	−	0.610971i	$-0.790779\pi$
$13$	−5.70869			−1.58331			−0.791653	+	0.610971i	$0.790779\pi$
$14$		0			0
$15$	−1.04974	−	1.37769i	−0.271042	−	0.355719i
$16$		0			0
$17$	6.65577			1.61426			0.807131	−	0.590372i	$-0.201019\pi$
$17$	6.65577			1.61426			0.807131	+	0.590372i	$0.201019\pi$
$18$		0			0
$19$			4.23245i			0.970990i	0.874239	+	0.485495i	$0.161361\pi$
$19$			4.23245i			0.970990i	−0.874239	+	0.485495i	$0.838639\pi$
$20$		0			0
$21$	2.89321	−	2.20450i	0.631350	−	0.481061i
$22$		0			0
$23$	3.63955	+	3.12309i	0.758898	+	0.651209i
$23$	3.63955	+	3.12309i	0.758898	+	0.651209i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−4.82059	+	1.93957i	−0.927723	+	0.373270i
$28$		0			0
$29$			10.0198i			1.86064i	0.366754	+	0.930318i	$0.380469\pi$
$29$			10.0198i			1.86064i	−0.366754	+	0.930318i	$0.619531\pi$
$30$		0			0
$31$	4.21597			0.757210			0.378605	−	0.925558i	$-0.376404\pi$
$31$	4.21597			0.757210			0.378605	+	0.925558i	$0.376404\pi$
$32$		0			0
$33$	2.58152	+	3.38802i	0.449385	+	0.589779i
$34$		0			0
$35$			2.10004i			0.354971i
$36$		0			0
$37$			8.14987i			1.33983i	0.742438	+	0.669915i	$0.233670\pi$
$37$			8.14987i			1.33983i	−0.742438	+	0.669915i	$0.766330\pi$
$38$		0			0
$39$	−5.99266	−	7.86483i	−0.959593	−	1.25938i
$40$		0			0
$41$			3.58027i			0.559145i	0.960125	+	0.279572i	$0.0901927\pi$
$41$			3.58027i			0.559145i	−0.960125	+	0.279572i	$0.909807\pi$
$42$		0			0
$43$			2.94492i			0.449097i	0.974463	+	0.224548i	$0.0720907\pi$
$43$			2.94492i			0.449097i	−0.974463	+	0.224548i	$0.927909\pi$
$44$		0			0
$45$	0.796081	−	2.89245i	0.118673	−	0.431181i
$46$		0			0
$47$		−	6.21937i		−	0.907188i	−0.891208	−	0.453594i	$-0.850142\pi$
$47$		−	6.21937i		−	0.907188i	0.891208	−	0.453594i	$-0.149858\pi$
$48$		0			0
$49$	2.58984			0.369977
$50$		0			0
$51$	6.98685	+	9.16962i	0.978354	+	1.28400i
$52$		0			0
$53$	−11.4974			−1.57929			−0.789647	−	0.613562i	$-0.789736\pi$
$53$	−11.4974			−1.57929			−0.789647	+	0.613562i	$0.789736\pi$
$54$		0			0
$55$	−2.45920			−0.331598
$56$		0			0
$57$	−5.83102	+	4.44298i	−0.772337	+	0.588487i
$58$		0			0
$59$			6.31669i			0.822363i	0.911553	+	0.411182i	$0.134884\pi$
$59$			6.31669i			0.822363i	−0.911553	+	0.411182i	$0.865116\pi$
$60$		0			0
$61$		−	6.35077i		−	0.813132i	−0.913621	−	0.406566i	$-0.866726\pi$
$61$		−	6.35077i		−	0.813132i	0.913621	−	0.406566i	$-0.133274\pi$
$62$		0			0
$63$	6.07425	+	1.67180i	0.765284	+	0.210627i
$64$		0			0
$65$	5.70869			0.708076
$66$		0			0
$67$		−	11.5057i		−	1.40565i	−0.711362	−	0.702825i	$-0.751922\pi$
$67$		−	11.5057i		−	1.40565i	0.711362	−	0.702825i	$-0.248078\pi$
$68$		0			0
$69$	−0.482075	+	8.29262i	−0.0580350	+	0.998315i
$70$		0			0
$71$			4.67888i			0.555280i	0.960685	+	0.277640i	$0.0895523\pi$
$71$			4.67888i			0.555280i	−0.960685	+	0.277640i	$0.910448\pi$
$72$		0			0
$73$	16.6582			1.94969			0.974845	−	0.222884i	$-0.0715473\pi$
$73$	16.6582			1.94969			0.974845	+	0.222884i	$0.0715473\pi$
$74$		0			0
$75$	1.04974	+	1.37769i	0.121214	+	0.159082i
$76$		0			0
$77$		−	5.16440i		−	0.588539i
$78$		0			0
$79$		−	8.63724i		−	0.971765i	−0.874024	−	0.485883i	$-0.838498\pi$
$79$		−	8.63724i		−	0.971765i	0.874024	−	0.485883i	$-0.161502\pi$
$80$		0			0
$81$	−7.73251	−	4.60525i	−0.859168	−	0.511694i
$82$		0			0
$83$	0.357391			0.0392287			0.0196144	−	0.999808i	$-0.493756\pi$
$83$	0.357391			0.0392287			0.0196144	+	0.999808i	$0.493756\pi$
$84$		0			0
$85$	−6.65577			−0.721920
$86$		0			0
$87$	−13.8043	+	10.5182i	−1.47997	+	1.12767i
$88$		0			0
$89$	−10.1485			−1.07574			−0.537869	−	0.843028i	$-0.680771\pi$
$89$	−10.1485			−1.07574			−0.537869	+	0.843028i	$0.680771\pi$
$90$		0			0
$91$			11.9885i			1.25673i
$92$		0			0
$93$	4.42568	+	5.80831i	0.458921	+	0.602294i
$94$		0			0
$95$		−	4.23245i		−	0.434240i
$96$		0			0
$97$		−	3.20088i		−	0.325000i	−0.986709	−	0.162500i	$-0.948044\pi$
$97$		−	3.20088i		−	0.325000i	0.986709	−	0.162500i	$-0.0519558\pi$
$98$		0			0
$99$	−1.95772	+	7.11310i	−0.196758	+	0.714893i
$100$		0			0
$101$			5.63022i			0.560228i	0.959967	+	0.280114i	$0.0903722\pi$
$101$			5.63022i			0.560228i	−0.959967	+	0.280114i	$0.909628\pi$
$102$		0			0
$103$			17.6884i			1.74289i	0.490492	+	0.871446i	$0.336817\pi$
$103$			17.6884i			1.74289i	−0.490492	+	0.871446i	$0.663183\pi$
$104$		0			0
$105$	−2.89321	+	2.20450i	−0.282348	+	0.215137i
$106$		0			0
$107$	9.14858			0.884426			0.442213	−	0.896910i	$-0.354193\pi$
$107$	9.14858			0.884426			0.442213	+	0.896910i	$0.354193\pi$
$108$		0			0
$109$		−	17.5308i		−	1.67915i	−0.543244	−	0.839575i	$-0.682804\pi$
$109$		−	17.5308i		−	1.67915i	0.543244	−	0.839575i	$-0.317196\pi$
$110$		0			0
$111$	−11.2280	+	8.55526i	−1.06572	+	0.812030i
$112$		0			0
$113$	4.36358			0.410491			0.205245	−	0.978711i	$-0.434201\pi$
$113$	4.36358			0.410491			0.205245	+	0.978711i	$0.434201\pi$
$114$		0			0
$115$	−3.63955	−	3.12309i	−0.339390	−	0.291230i
$116$		0			0
$117$	4.54458	−	16.5121i	0.420147	−	1.52654i
$118$		0			0
$119$		−	13.9774i		−	1.28130i
$120$		0			0
$121$	−4.95235			−0.450214
$122$		0			0
$123$	−4.93252	+	3.75837i	−0.444750	+	0.338880i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−12.7219			−1.12888			−0.564441	−	0.825473i	$-0.690908\pi$
$127$	−12.7219			−1.12888			−0.564441	+	0.825473i	$0.690908\pi$
$128$		0			0
$129$	−4.05720	+	3.09141i	−0.357217	+	0.272184i
$130$		0			0
$131$			10.3091i			0.900710i	0.892850	+	0.450355i	$0.148703\pi$
$131$			10.3091i			0.900710i	−0.892850	+	0.450355i	$0.851297\pi$
$132$		0			0
$133$	8.88830			0.770714
$134$		0			0
$135$	4.82059	−	1.93957i	0.414890	−	0.166932i
$136$		0			0
$137$	18.9056			1.61521			0.807607	−	0.589721i	$-0.200762\pi$
$137$	18.9056			1.61521			0.807607	+	0.589721i	$0.200762\pi$
$138$		0			0
$139$	−17.5128			−1.48542			−0.742710	−	0.669614i	$-0.766460\pi$
$139$	−17.5128			−1.48542			−0.742710	+	0.669614i	$0.766460\pi$
$140$		0			0
$141$	8.56838	−	6.52873i	0.721588	−	0.549819i
$142$		0			0
$143$	−14.0388			−1.17398
$144$		0			0
$145$		−	10.0198i		−	0.832102i
$146$		0			0
$147$	2.71867	+	3.56801i	0.224232	+	0.294284i
$148$		0			0
$149$	−1.19031			−0.0975141			−0.0487570	−	0.998811i	$-0.515526\pi$
$149$	−1.19031			−0.0975141			−0.0487570	+	0.998811i	$0.515526\pi$
$150$		0			0
$151$	7.99999			0.651030			0.325515	−	0.945537i	$-0.394462\pi$
$151$	7.99999			0.651030			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	−5.29853	+	19.2515i	−0.428361	+	1.55639i
$154$		0			0
$155$	−4.21597			−0.338635
$156$		0			0
$157$		−	20.1432i		−	1.60760i	−0.594899	−	0.803800i	$-0.702808\pi$
$157$		−	20.1432i		−	1.60760i	0.594899	−	0.803800i	$-0.297192\pi$
$158$		0			0
$159$	−12.0693	−	15.8399i	−0.957161	−	1.25619i
$160$		0			0
$161$	6.55861	−	7.64319i	0.516891	−	0.602368i
$162$		0			0
$163$	−6.91704			−0.541784			−0.270892	−	0.962610i	$-0.587319\pi$
$163$	−6.91704			−0.541784			−0.270892	+	0.962610i	$0.587319\pi$
$164$		0			0
$165$	−2.58152	−	3.38802i	−0.200971	−	0.263757i
$166$		0			0
$167$		−	6.19080i		−	0.479058i	−0.970889	−	0.239529i	$-0.923007\pi$
$167$		−	6.19080i		−	0.479058i	0.970889	−	0.239529i	$-0.0769931\pi$
$168$		0			0
$169$	19.5892			1.50686
$170$		0			0
$171$	−12.2421	−	3.36937i	−0.936180	−	0.257662i
$172$		0			0
$173$		−	24.7750i		−	1.88361i	−0.336159	−	0.941805i	$-0.609128\pi$
$173$		−	24.7750i		−	1.88361i	0.336159	−	0.941805i	$-0.390872\pi$
$174$		0			0
$175$		−	2.10004i		−	0.158748i
$176$		0			0
$177$	−8.70247	+	6.63090i	−0.654118	+	0.498409i
$178$		0			0
$179$			9.97353i			0.745456i	0.927941	+	0.372728i	$0.121578\pi$
$179$			9.97353i			0.745456i	−0.927941	+	0.372728i	$0.878422\pi$
$180$		0			0
$181$			7.84144i			0.582849i	0.956594	+	0.291425i	$0.0941293\pi$
$181$			7.84144i			0.582849i	−0.956594	+	0.291425i	$0.905871\pi$
$182$		0			0
$183$	8.74941	−	6.66667i	0.646775	−	0.492814i
$184$		0			0
$185$		−	8.14987i		−	0.599190i
$186$		0			0
$187$	16.3679			1.19694
$188$		0			0
$189$	4.07317	+	10.1234i	0.296280	+	0.736370i
$190$		0			0
$191$	−0.480474			−0.0347659			−0.0173829	−	0.999849i	$-0.505533\pi$
$191$	−0.480474			−0.0347659			−0.0173829	+	0.999849i	$0.505533\pi$
$192$		0			0
$193$	8.72137			0.627778			0.313889	−	0.949460i	$-0.398368\pi$
$193$	8.72137			0.627778			0.313889	+	0.949460i	$0.398368\pi$
$194$		0			0
$195$	5.99266	+	7.86483i	0.429143	+	0.563212i
$196$		0			0
$197$			4.62001i			0.329162i	0.986364	+	0.164581i	$0.0526272\pi$
$197$			4.62001i			0.329162i	−0.986364	+	0.164581i	$0.947373\pi$
$198$		0			0
$199$			22.4544i			1.59175i	0.605461	+	0.795875i	$0.292989\pi$
$199$			22.4544i			1.59175i	−0.605461	+	0.795875i	$0.707011\pi$
$200$		0			0
$201$	15.8514	−	12.0781i	1.11807	−	0.851922i
$202$		0			0
$203$	21.0420			1.47686
$204$		0			0
$205$		−	3.58027i		−	0.250057i
$206$		0			0
$207$	−11.9308	+	8.04097i	−0.829245	+	0.558886i
$208$		0			0
$209$			10.4084i			0.719966i
$210$		0			0
$211$	9.68297			0.666603			0.333302	−	0.942820i	$-0.391837\pi$
$211$	9.68297			0.666603			0.333302	+	0.942820i	$0.391837\pi$
$212$		0			0
$213$	−6.44606	+	4.91162i	−0.441677	+	0.336538i
$214$		0			0
$215$		−	2.94492i		−	0.200842i
$216$		0			0
$217$		−	8.85369i		−	0.601027i
$218$		0			0
$219$	17.4868	+	22.9498i	1.18165	+	1.55081i
$220$		0			0
$221$	−37.9958			−2.55587
$222$		0			0
$223$	22.0343			1.47553			0.737764	−	0.675059i	$-0.235882\pi$
$223$	22.0343			1.47553			0.737764	+	0.675059i	$0.235882\pi$
$224$		0			0
$225$	−0.796081	+	2.89245i	−0.0530721	+	0.192830i
$226$		0			0
$227$	7.42749			0.492980			0.246490	−	0.969145i	$-0.420723\pi$
$227$	7.42749			0.492980			0.246490	+	0.969145i	$0.420723\pi$
$228$		0			0
$229$			0.528930i			0.0349527i	0.999847	+	0.0174763i	$0.00556318\pi$
$229$			0.528930i			0.0349527i	−0.999847	+	0.0174763i	$0.994437\pi$
$230$		0			0
$231$	7.11497	−	5.42130i	0.468131	−	0.356695i
$232$		0			0
$233$			9.01673i			0.590706i	0.955388	+	0.295353i	$0.0954372\pi$
$233$			9.01673i			0.590706i	−0.955388	+	0.295353i	$0.904563\pi$
$234$		0			0
$235$			6.21937i			0.405707i
$236$		0			0
$237$	11.8995	−	9.06688i	0.772954	−	0.588957i
$238$		0			0
$239$			19.6206i			1.26915i	0.772860	+	0.634577i	$0.218826\pi$
$239$			19.6206i			1.26915i	−0.772860	+	0.634577i	$0.781174\pi$
$240$		0			0
$241$		−	20.8794i		−	1.34496i	−0.740115	−	0.672480i	$-0.765229\pi$
$241$		−	20.8794i		−	1.34496i	0.740115	−	0.672480i	$-0.234771\pi$
$242$		0			0
$243$	−1.77253	−	15.4874i	−0.113708	−	0.993514i
$244$		0			0
$245$	−2.58984			−0.165459
$246$		0			0
$247$		−	24.1617i		−	1.53738i
$248$		0			0
$249$	0.375168	+	0.492375i	0.0237753	+	0.0312030i
$250$		0			0
$251$	4.59024			0.289734			0.144867	−	0.989451i	$-0.453725\pi$
$251$	4.59024			0.289734			0.144867	+	0.989451i	$0.453725\pi$
$252$		0			0
$253$	8.95036	+	7.68029i	0.562704	+	0.482856i
$254$		0			0
$255$	−6.98685	−	9.16962i	−0.437533	−	0.574224i
$256$		0			0
$257$			8.76463i			0.546723i	0.961911	+	0.273361i	$0.0881355\pi$
$257$			8.76463i			0.546723i	−0.961911	+	0.273361i	$0.911865\pi$
$258$		0			0
$259$	17.1150			1.06348
$260$		0			0
$261$	−28.9818	−	7.97660i	−1.79393	−	0.493739i
$262$		0			0
$263$	11.9427			0.736418			0.368209	−	0.929743i	$-0.379971\pi$
$263$	11.9427			0.736418			0.368209	+	0.929743i	$0.379971\pi$
$264$		0			0
$265$	11.4974			0.706282
$266$		0			0
$267$	−10.6533	−	13.9815i	−0.651972	−	0.855655i
$268$		0			0
$269$		−	26.8539i		−	1.63731i	−0.574283	−	0.818657i	$-0.694719\pi$
$269$		−	26.8539i		−	1.63731i	0.574283	−	0.818657i	$-0.305281\pi$
$270$		0			0
$271$	−2.38396			−0.144815			−0.0724076	−	0.997375i	$-0.523068\pi$
$271$	−2.38396			−0.144815			−0.0724076	+	0.997375i	$0.523068\pi$
$272$		0			0
$273$	−16.5164	+	12.5848i	−0.999620	+	0.761667i
$274$		0			0
$275$	2.45920			0.148295
$276$		0			0
$277$	−2.65253			−0.159375			−0.0796877	−	0.996820i	$-0.525392\pi$
$277$	−2.65253			−0.159375			−0.0796877	+	0.996820i	$0.525392\pi$
$278$		0			0
$279$	−3.35625	+	12.1945i	−0.200933	+	0.730063i
$280$		0			0
$281$	−10.8118			−0.644981			−0.322490	−	0.946573i	$-0.604520\pi$
$281$	−10.8118			−0.644981			−0.322490	+	0.946573i	$0.604520\pi$
$282$		0			0
$283$		−	25.8630i		−	1.53740i	−0.639612	−	0.768698i	$-0.720905\pi$
$283$		−	25.8630i		−	1.53740i	0.639612	−	0.768698i	$-0.279095\pi$
$284$		0			0
$285$	5.83102	−	4.44298i	0.345400	−	0.263180i
$286$		0			0
$287$	7.51871			0.443815
$288$		0			0
$289$	27.2993			1.60584
$290$		0			0
$291$	4.40983	−	3.36010i	0.258509	−	0.196972i
$292$		0			0
$293$	−17.5838			−1.02725			−0.513627	−	0.858014i	$-0.671698\pi$
$293$	−17.5838			−1.02725			−0.513627	+	0.858014i	$0.671698\pi$
$294$		0			0
$295$		−	6.31669i		−	0.367772i
$296$		0			0
$297$	−11.8548	+	4.76978i	−0.687884	+	0.276771i
$298$		0			0
$299$	−20.7771	−	17.8288i	−1.20157	−	1.03106i
$300$		0			0
$301$	6.18445			0.356466
$302$		0			0
$303$	−7.75672	+	5.91028i	−0.445612	+	0.339537i
$304$		0			0
$305$			6.35077i			0.363644i
$306$		0			0
$307$	−20.5653			−1.17373			−0.586863	−	0.809686i	$-0.699637\pi$
$307$	−20.5653			−1.17373			−0.586863	+	0.809686i	$0.699637\pi$
$308$		0			0
$309$	−24.3692	+	18.5683i	−1.38632	+	1.05631i
$310$		0			0
$311$		−	10.4170i		−	0.590692i	−0.955390	−	0.295346i	$-0.904565\pi$
$311$		−	10.4170i		−	0.590692i	0.955390	−	0.295346i	$-0.0954348\pi$
$312$		0			0
$313$		−	23.2950i		−	1.31671i	−0.752708	−	0.658355i	$-0.771253\pi$
$313$		−	23.2950i		−	1.31671i	0.752708	−	0.658355i	$-0.228747\pi$
$314$		0			0
$315$	−6.07425	−	1.67180i	−0.342245	−	0.0941953i
$316$		0			0
$317$		−	21.8902i		−	1.22948i	−0.788731	−	0.614739i	$-0.789261\pi$
$317$		−	21.8902i		−	1.22948i	0.788731	−	0.614739i	$-0.210739\pi$
$318$		0			0
$319$			24.6407i			1.37962i
$320$		0			0
$321$	9.60365	+	12.6039i	0.536024	+	0.703483i
$322$		0			0
$323$			28.1702i			1.56743i
$324$		0			0
$325$	−5.70869			−0.316661
$326$		0			0
$327$	24.1521	−	18.4029i	1.33562	−	1.01768i
$328$		0			0
$329$	−13.0609			−0.720071
$330$		0			0
$331$	−19.5946			−1.07702			−0.538508	−	0.842621i	$-0.681012\pi$
$331$	−19.5946			−1.07702			−0.538508	+	0.842621i	$0.681012\pi$
$332$		0			0
$333$	−23.5731	−	6.48796i	−1.29180	−	0.355538i
$334$		0			0
$335$			11.5057i			0.628626i
$336$		0			0
$337$		−	14.6291i		−	0.796897i	−0.917191	−	0.398448i	$-0.869549\pi$
$337$		−	14.6291i		−	0.796897i	0.917191	−	0.398448i	$-0.130451\pi$
$338$		0			0
$339$	4.58063	+	6.01167i	0.248786	+	0.326509i
$340$		0			0
$341$	10.3679			0.561453
$342$		0			0
$343$		−	20.1390i		−	1.08741i
$344$		0			0
$345$	0.482075	−	8.29262i	0.0259540	−	0.446460i
$346$		0			0
$347$		−	32.2411i		−	1.73079i	−0.501089	−	0.865396i	$-0.667067\pi$
$347$		−	32.2411i		−	1.73079i	0.501089	−	0.865396i	$-0.332933\pi$
$348$		0			0
$349$	32.8575			1.75882			0.879411	−	0.476063i	$-0.157937\pi$
$349$	32.8575			1.75882			0.879411	+	0.476063i	$0.157937\pi$
$350$		0			0
$351$	27.5193	−	11.0724i	1.46887	−	0.591001i
$352$		0			0
$353$		−	6.01588i		−	0.320193i	−0.987101	−	0.160096i	$-0.948819\pi$
$353$		−	6.01588i		−	0.320193i	0.987101	−	0.160096i	$-0.0511805\pi$
$354$		0			0
$355$		−	4.67888i		−	0.248329i
$356$		0			0
$357$	19.2565	−	14.6726i	1.01916	−	0.776559i
$358$		0			0
$359$	10.9826			0.579639			0.289820	−	0.957081i	$-0.406405\pi$
$359$	10.9826			0.579639			0.289820	+	0.957081i	$0.406405\pi$
$360$		0			0
$361$	1.08638			0.0571777
$362$		0			0
$363$	−5.19870	−	6.82283i	−0.272861	−	0.358106i
$364$		0			0
$365$	−16.6582			−0.871928
$366$		0			0
$367$			8.78011i			0.458318i	0.973389	+	0.229159i	$0.0735975\pi$
$367$			8.78011i			0.458318i	−0.973389	+	0.229159i	$0.926402\pi$
$368$		0			0
$369$	−10.3558	−	2.85019i	−0.539099	−	0.148375i
$370$		0			0
$371$			24.1450i			1.25355i
$372$		0			0
$373$			12.4920i			0.646809i	0.946261	+	0.323405i	$0.104828\pi$
$373$			12.4920i			0.646809i	−0.946261	+	0.323405i	$0.895172\pi$
$374$		0			0
$375$	−1.04974	−	1.37769i	−0.0542085	−	0.0711438i
$376$		0			0
$377$		−	57.2001i		−	2.94596i
$378$		0			0
$379$			19.7833i			1.01620i	0.861299	+	0.508099i	$0.169652\pi$
$379$			19.7833i			1.01620i	−0.861299	+	0.508099i	$0.830348\pi$
$380$		0			0
$381$	−13.3547	−	17.5268i	−0.684181	−	0.897926i
$382$		0			0
$383$	−36.8745			−1.88420			−0.942100	−	0.335333i	$-0.891151\pi$
$383$	−36.8745			−1.88420			−0.942100	+	0.335333i	$0.891151\pi$
$384$		0			0
$385$			5.16440i			0.263202i
$386$		0			0
$387$	−8.51804	−	2.34440i	−0.432996	−	0.119172i
$388$		0			0
$389$	−0.556164			−0.0281986			−0.0140993	−	0.999901i	$-0.504488\pi$
$389$	−0.556164			−0.0281986			−0.0140993	+	0.999901i	$0.504488\pi$
$390$		0			0
$391$	24.2240	+	20.7866i	1.22506	+	1.05122i
$392$		0			0
$393$	−14.2028	+	10.8219i	−0.716435	+	0.545892i
$394$		0			0
$395$			8.63724i			0.434587i
$396$		0			0
$397$	2.49892			0.125417			0.0627086	−	0.998032i	$-0.480026\pi$
$397$	2.49892			0.125417			0.0627086	+	0.998032i	$0.480026\pi$
$398$		0			0
$399$	9.33043	+	12.2454i	0.467106	+	0.613035i
$400$		0			0
$401$	−18.7476			−0.936209			−0.468105	−	0.883673i	$-0.655063\pi$
$401$	−18.7476			−0.936209			−0.468105	+	0.883673i	$0.655063\pi$
$402$		0			0
$403$	−24.0677			−1.19890
$404$		0			0
$405$	7.73251	+	4.60525i	0.384232	+	0.228836i
$406$		0			0
$407$			20.0421i			0.993451i
$408$		0			0
$409$	11.5081			0.569039			0.284519	−	0.958670i	$-0.408166\pi$
$409$	11.5081			0.569039			0.284519	+	0.958670i	$0.408166\pi$
$410$		0			0
$411$	19.8460	+	26.0461i	0.978931	+	1.28476i
$412$		0			0
$413$	13.2653			0.652742
$414$		0			0
$415$	−0.357391			−0.0175436
$416$		0			0
$417$	−18.3840	−	24.1273i	−0.900267	−	1.18152i
$418$		0			0
$419$	33.3841			1.63092			0.815459	−	0.578814i	$-0.196484\pi$
$419$	33.3841			1.63092			0.815459	+	0.578814i	$0.196484\pi$
$420$		0			0
$421$		−	1.48324i		−	0.0722887i	−0.999347	−	0.0361443i	$-0.988492\pi$
$421$		−	1.48324i		−	0.0722887i	0.999347	−	0.0361443i	$-0.0115076\pi$
$422$		0			0
$423$	17.9892	+	4.95112i	0.874665	+	0.240732i
$424$		0			0
$425$	6.65577			0.322852
$426$		0			0
$427$	−13.3368			−0.645415
$428$		0			0
$429$	−14.7371	−	19.3412i	−0.711515	−	0.933800i
$430$		0			0
$431$	−3.81450			−0.183738			−0.0918690	−	0.995771i	$-0.529284\pi$
$431$	−3.81450			−0.183738			−0.0918690	+	0.995771i	$0.529284\pi$
$432$		0			0
$433$		−	28.4844i		−	1.36887i	−0.729072	−	0.684437i	$-0.760048\pi$
$433$		−	28.4844i		−	1.36887i	0.729072	−	0.684437i	$-0.239952\pi$
$434$		0			0
$435$	13.8043	−	10.5182i	0.661864	−	0.504311i
$436$		0			0
$437$	−13.2183	+	15.4042i	−0.632318	+	0.736883i
$438$		0			0
$439$	−3.17785			−0.151671			−0.0758353	−	0.997120i	$-0.524162\pi$
$439$	−3.17785			−0.151671			−0.0758353	+	0.997120i	$0.524162\pi$
$440$		0			0
$441$	−2.06172	+	7.49098i	−0.0981773	+	0.356713i
$442$		0			0
$443$		−	11.7560i		−	0.558547i	−0.960212	−	0.279273i	$-0.909906\pi$
$443$		−	11.7560i		−	0.558547i	0.960212	−	0.279273i	$-0.0900936\pi$
$444$		0			0
$445$	10.1485			0.481085
$446$		0			0
$447$	−1.24952	−	1.63988i	−0.0591003	−	0.0775639i
$448$		0			0
$449$		−	21.6298i		−	1.02077i	−0.859945	−	0.510387i	$-0.829502\pi$
$449$		−	21.6298i		−	1.02077i	0.859945	−	0.510387i	$-0.170498\pi$
$450$		0			0
$451$			8.80459i			0.414592i
$452$		0			0
$453$	8.39793	+	11.0215i	0.394569	+	0.517837i
$454$		0			0
$455$		−	11.9885i		−	0.562028i
$456$		0			0
$457$			18.1916i			0.850967i	0.904966	+	0.425484i	$0.139896\pi$
$457$			18.1916i			0.850967i	−0.904966	+	0.425484i	$0.860104\pi$
$458$		0			0
$459$	−32.0847	+	12.9093i	−1.49759	+	0.602556i
$460$		0			0
$461$			9.80601i			0.456711i	0.973578	+	0.228356i	$0.0733349\pi$
$461$			9.80601i			0.456711i	−0.973578	+	0.228356i	$0.926665\pi$
$462$		0			0
$463$	1.37625			0.0639596			0.0319798	−	0.999489i	$-0.489819\pi$
$463$	1.37625			0.0639596			0.0319798	+	0.999489i	$0.489819\pi$
$464$		0			0
$465$	−4.42568	−	5.80831i	−0.205236	−	0.269354i
$466$		0			0
$467$	−22.9520			−1.06209			−0.531045	−	0.847344i	$-0.678200\pi$
$467$	−22.9520			−1.06209			−0.531045	+	0.847344i	$0.678200\pi$
$468$		0			0
$469$	−24.1625			−1.11572
$470$		0			0
$471$	27.7511	−	21.1452i	1.27870	−	0.974317i
$472$		0			0
$473$			7.24215i			0.332994i
$474$		0			0
$475$			4.23245i			0.194198i
$476$		0			0
$477$	9.15289	−	33.2557i	0.419082	−	1.52267i
$478$		0			0
$479$	14.8455			0.678310			0.339155	−	0.940730i	$-0.389859\pi$
$479$	14.8455			0.678310			0.339155	+	0.940730i	$0.389859\pi$
$480$		0			0
$481$		−	46.5251i		−	2.12136i
$482$		0			0
$483$	17.4148	+	1.01237i	0.792402	+	0.0460647i
$484$		0			0
$485$			3.20088i			0.145344i
$486$		0			0
$487$	−26.0973			−1.18258			−0.591292	−	0.806458i	$-0.701382\pi$
$487$	−26.0973			−1.18258			−0.591292	+	0.806458i	$0.701382\pi$
$488$		0			0
$489$	−7.26111	−	9.52956i	−0.328359	−	0.430942i
$490$		0			0
$491$			4.88534i			0.220472i	0.993905	+	0.110236i	$0.0351607\pi$
$491$			4.88534i			0.220472i	−0.993905	+	0.110236i	$0.964839\pi$
$492$		0			0
$493$			66.6897i			3.00355i
$494$		0			0
$495$	1.95772	−	7.11310i	0.0879929	−	0.319710i
$496$		0			0
$497$	9.82582			0.440748
$498$		0			0
$499$	−8.88179			−0.397603			−0.198802	−	0.980040i	$-0.563705\pi$
$499$	−8.88179			−0.397603			−0.198802	+	0.980040i	$0.563705\pi$
$500$		0			0
$501$	8.52902	−	6.49874i	0.381049	−	0.290342i
$502$		0			0
$503$	−23.4487			−1.04552			−0.522762	−	0.852479i	$-0.675098\pi$
$503$	−23.4487			−1.04552			−0.522762	+	0.852479i	$0.675098\pi$
$504$		0			0
$505$		−	5.63022i		−	0.250541i
$506$		0			0
$507$	20.5636	+	26.9879i	0.913261	+	1.19857i
$508$		0			0
$509$		−	13.0350i		−	0.577766i	−0.957364	−	0.288883i	$-0.906716\pi$
$509$		−	13.0350i		−	0.577766i	0.957364	−	0.288883i	$-0.0932839\pi$
$510$		0			0
$511$		−	34.9828i		−	1.54755i
$512$		0			0
$513$	−8.20913	−	20.4029i	−0.362442	−	0.900810i
$514$		0			0
$515$		−	17.6884i		−	0.779445i
$516$		0			0
$517$		−	15.2946i		−	0.672658i
$518$		0			0
$519$	34.1324	−	26.0074i	1.49825	−	1.14160i
$520$		0			0
$521$	0.150584			0.00659722			0.00329861	−	0.999995i	$-0.498950\pi$
$521$	0.150584			0.00659722			0.00329861	+	0.999995i	$0.498950\pi$
$522$		0			0
$523$			16.3478i			0.714841i	0.933944	+	0.357420i	$0.116344\pi$
$523$			16.3478i			0.714841i	−0.933944	+	0.357420i	$0.883656\pi$
$524$		0			0
$525$	2.89321	−	2.20450i	0.126270	−	0.0962122i
$526$		0			0
$527$	28.0605			1.22233
$528$		0			0
$529$	3.49261	+	22.7333i	0.151853	+	0.988403i
$530$		0			0
$531$	−18.2707	−	5.02860i	−0.792881	−	0.218223i
$532$		0			0
$533$		−	20.4387i		−	0.885297i
$534$		0			0
$535$	−9.14858			−0.395528
$536$		0			0
$537$	−13.7405	+	10.4696i	−0.592945	+	0.451798i
$538$		0			0
$539$	6.36893			0.274329
$540$		0			0
$541$	1.94558			0.0836470			0.0418235	−	0.999125i	$-0.486683\pi$
$541$	1.94558			0.0836470			0.0418235	+	0.999125i	$0.486683\pi$
$542$		0			0
$543$	−10.8031	+	8.23149i	−0.463605	+	0.353247i
$544$		0			0
$545$			17.5308i			0.750938i
$546$		0			0
$547$	−12.6623			−0.541402			−0.270701	−	0.962664i	$-0.587255\pi$
$547$	−12.6623			−0.541402			−0.270701	+	0.962664i	$0.587255\pi$
$548$		0			0
$549$	18.3693	+	5.05572i	0.783981	+	0.215773i
$550$		0			0
$551$	−42.4084			−1.80666
$552$		0			0
$553$	−18.1385			−0.771329
$554$		0			0
$555$	11.2280	−	8.55526i	0.476603	−	0.363151i
$556$		0			0
$557$	26.1748			1.10906			0.554531	−	0.832163i	$-0.312898\pi$
$557$	26.1748			1.10906			0.554531	+	0.832163i	$0.312898\pi$
$558$		0			0
$559$		−	16.8117i		−	0.711058i
$560$		0			0
$561$	17.1820	+	22.5499i	0.725426	+	0.952057i
$562$		0			0
$563$	16.2745			0.685889			0.342944	−	0.939356i	$-0.388576\pi$
$563$	16.2745			0.685889			0.342944	+	0.939356i	$0.388576\pi$
$564$		0			0
$565$	−4.36358			−0.183577
$566$		0			0
$567$	−9.67119	+	16.2386i	−0.406152	+	0.681956i
$568$		0			0
$569$	−36.6423			−1.53613			−0.768063	−	0.640374i	$-0.778779\pi$
$569$	−36.6423			−1.53613			−0.768063	+	0.640374i	$0.778779\pi$
$570$		0			0
$571$			13.6412i			0.570866i	0.958399	+	0.285433i	$0.0921375\pi$
$571$			13.6412i			0.570866i	−0.958399	+	0.285433i	$0.907863\pi$
$572$		0			0
$573$	−0.504374	−	0.661946i	−0.0210705	−	0.0276532i
$574$		0			0
$575$	3.63955	+	3.12309i	0.151780	+	0.130242i
$576$		0			0
$577$	−7.78424			−0.324062			−0.162031	−	0.986786i	$-0.551804\pi$
$577$	−7.78424			−0.324062			−0.162031	+	0.986786i	$0.551804\pi$
$578$		0			0
$579$	9.15520	+	12.0154i	0.380477	+	0.499342i
$580$		0			0
$581$		−	0.750534i		−	0.0311374i
$582$		0			0
$583$	−28.2744			−1.17101
$584$		0			0
$585$	−4.54458	+	16.5121i	−0.187895	+	0.682691i
$586$		0			0
$587$			26.9231i			1.11124i	0.831438	+	0.555618i	$0.187518\pi$
$587$			26.9231i			1.11124i	−0.831438	+	0.555618i	$0.812482\pi$
$588$		0			0
$589$			17.8439i			0.735243i
$590$		0			0
$591$	−6.36496	+	4.84982i	−0.261819	+	0.199495i
$592$		0			0
$593$			15.5959i			0.640446i	0.947342	+	0.320223i	$0.103758\pi$
$593$			15.5959i			0.640446i	−0.947342	+	0.320223i	$0.896242\pi$
$594$		0			0
$595$			13.9774i			0.573016i
$596$		0			0
$597$	−30.9353	+	23.5713i	−1.26610	+	0.964711i
$598$		0			0
$599$			27.5270i			1.12472i	0.826891	+	0.562362i	$0.190107\pi$
$599$			27.5270i			1.12472i	−0.826891	+	0.562362i	$0.809893\pi$
$600$		0			0
$601$	−17.0167			−0.694127			−0.347064	−	0.937842i	$-0.612821\pi$
$601$	−17.0167			−0.694127			−0.347064	+	0.937842i	$0.612821\pi$
$602$		0			0
$603$	33.2798	+	9.15951i	1.35526	+	0.373004i
$604$		0			0
$605$	4.95235			0.201342
$606$		0			0
$607$	33.3885			1.35520			0.677599	−	0.735432i	$-0.263021\pi$
$607$	33.3885			1.35520			0.677599	+	0.735432i	$0.263021\pi$
$608$		0			0
$609$	22.0887	+	28.9895i	0.895080	+	1.17471i
$610$		0			0
$611$			35.5045i			1.43636i
$612$		0			0
$613$			5.49000i			0.221739i	0.993835	+	0.110870i	$0.0353636\pi$
$613$			5.49000i			0.221739i	−0.993835	+	0.110870i	$0.964636\pi$
$614$		0			0
$615$	4.93252	−	3.75837i	0.198898	−	0.151552i
$616$		0			0
$617$	39.7896			1.60187			0.800934	−	0.598752i	$-0.204337\pi$
$617$	39.7896			1.60187			0.800934	+	0.598752i	$0.204337\pi$
$618$		0			0
$619$			13.2084i			0.530891i	0.964126	+	0.265445i	$0.0855190\pi$
$619$			13.2084i			0.530891i	−0.964126	+	0.265445i	$0.914481\pi$
$620$		0			0
$621$	−23.6022	−	7.99598i	−0.947124	−	0.320867i
$622$		0			0
$623$			21.3122i			0.853856i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−14.3396	+	10.9262i	−0.572669	+	0.436349i
$628$		0			0
$629$			54.2437i			2.16284i
$630$		0			0
$631$			35.1881i			1.40082i	0.713743	+	0.700408i	$0.246999\pi$
$631$			35.1881i			1.40082i	−0.713743	+	0.700408i	$0.753001\pi$
$632$		0			0
$633$	10.1646	+	13.3402i	0.404008	+	0.530224i
$634$		0			0
$635$	12.7219			0.504851
$636$		0			0
$637$	−14.7846			−0.585788
$638$		0			0
$639$	−13.5334	−	3.72476i	−0.535373	−	0.147349i
$640$		0			0
$641$	1.21185			0.0478650			0.0239325	−	0.999714i	$-0.492381\pi$
$641$	1.21185			0.0478650			0.0239325	+	0.999714i	$0.492381\pi$
$642$		0			0
$643$		−	29.9576i		−	1.18141i	−0.806887	−	0.590706i	$-0.798849\pi$
$643$		−	29.9576i		−	1.18141i	0.806887	−	0.590706i	$-0.201151\pi$
$644$		0			0
$645$	4.05720	−	3.09141i	0.159752	−	0.121724i
$646$		0			0
$647$		−	17.1022i		−	0.672356i	−0.941798	−	0.336178i	$-0.890866\pi$
$647$		−	17.1022i		−	0.672356i	0.941798	−	0.336178i	$-0.109134\pi$
$648$		0			0
$649$			15.5340i			0.609762i
$650$		0			0
$651$	12.1977	−	9.29409i	0.478064	−	0.364264i
$652$		0			0
$653$			33.0755i			1.29435i	0.762343	+	0.647173i	$0.224049\pi$
$653$			33.0755i			1.29435i	−0.762343	+	0.647173i	$0.775951\pi$
$654$		0			0
$655$		−	10.3091i		−	0.402810i
$656$		0			0
$657$	−13.2612	+	48.1828i	−0.517370	+	1.87979i
$658$		0			0
$659$	−1.56671			−0.0610305			−0.0305152	−	0.999534i	$-0.509715\pi$
$659$	−1.56671			−0.0610305			−0.0305152	+	0.999534i	$0.509715\pi$
$660$		0			0
$661$		−	38.7068i		−	1.50552i	−0.658295	−	0.752760i	$-0.728722\pi$
$661$		−	38.7068i		−	1.50552i	0.658295	−	0.752760i	$-0.271278\pi$
$662$		0			0
$663$	−39.8858	−	52.3465i	−1.54903	−	2.03297i
$664$		0			0
$665$	−8.88830			−0.344674
$666$		0			0
$667$	−31.2928	+	36.4676i	−1.21166	+	1.41203i
$668$		0			0
$669$	23.1304	+	30.3566i	0.894272	+	1.17365i
$670$		0			0
$671$		−	15.6178i		−	0.602918i
$672$		0			0
$673$	14.4281			0.556163			0.278081	−	0.960557i	$-0.410301\pi$
$673$	14.4281			0.556163			0.278081	+	0.960557i	$0.410301\pi$
$674$		0			0
$675$	−4.82059	+	1.93957i	−0.185545	+	0.0746541i
$676$		0			0
$677$	23.1400			0.889344			0.444672	−	0.895693i	$-0.353320\pi$
$677$	23.1400			0.889344			0.444672	+	0.895693i	$0.353320\pi$
$678$		0			0
$679$	−6.72197			−0.257965
$680$		0			0
$681$	7.79696	+	10.2328i	0.298780	+	0.392122i
$682$		0			0
$683$		−	30.5623i		−	1.16943i	−0.811238	−	0.584717i	$-0.801206\pi$
$683$		−	30.5623i		−	1.16943i	0.811238	−	0.584717i	$-0.198794\pi$
$684$		0			0
$685$	−18.9056			−0.722346
$686$		0			0
$687$	−0.728704	+	0.555240i	−0.0278018	+	0.0211837i
$688$		0			0
$689$	65.6353			2.50051
$690$		0			0
$691$	14.6786			0.558402			0.279201	−	0.960233i	$-0.409930\pi$
$691$	14.6786			0.558402			0.279201	+	0.960233i	$0.409930\pi$
$692$		0			0
$693$	14.9378	+	4.11128i	0.567439	+	0.156175i
$694$		0			0
$695$	17.5128			0.664300
$696$		0			0
$697$			23.8295i			0.902606i
$698$		0			0
$699$	−12.4223	+	9.46525i	−0.469854	+	0.358008i
$700$		0			0
$701$	−6.81567			−0.257424			−0.128712	−	0.991682i	$-0.541084\pi$
$701$	−6.81567			−0.257424			−0.128712	+	0.991682i	$0.541084\pi$
$702$		0			0
$703$	−34.4939			−1.30096
$704$		0			0
$705$	−8.56838	+	6.52873i	−0.322704	+	0.245886i
$706$		0			0
$707$	11.8237			0.444675
$708$		0			0
$709$		−	17.7493i		−	0.666590i	−0.942823	−	0.333295i	$-0.891839\pi$
$709$		−	17.7493i		−	0.666590i	0.942823	−	0.333295i	$-0.108161\pi$
$710$		0			0
$711$	24.9828	+	6.87594i	0.936927	+	0.257868i
$712$		0			0
$713$	15.3442	+	13.1668i	0.574645	+	0.493102i
$714$		0			0
$715$	14.0388			0.525021
$716$		0			0
$717$	−27.0312	+	20.5966i	−1.00950	+	0.769195i
$718$		0			0
$719$		−	18.5271i		−	0.690945i	−0.938429	−	0.345473i	$-0.887719\pi$
$719$		−	18.5271i		−	0.690945i	0.938429	−	0.345473i	$-0.112281\pi$
$720$		0			0
$721$	37.1463			1.38340
$722$		0			0
$723$	28.7654	−	21.9180i	1.06980	−	0.815139i
$724$		0			0
$725$			10.0198i			0.372127i
$726$		0			0
$727$		−	49.0709i		−	1.81994i	−0.414674	−	0.909970i	$-0.636104\pi$
$727$		−	49.0709i		−	1.81994i	0.414674	−	0.909970i	$-0.363896\pi$
$728$		0			0
$729$	19.4761	−	18.6997i	0.721338	−	0.692583i
$730$		0			0
$731$			19.6007i			0.724960i
$732$		0			0
$733$			1.28988i			0.0476428i	0.999716	+	0.0238214i	$0.00758330\pi$
$733$			1.28988i			0.0476428i	−0.999716	+	0.0238214i	$0.992417\pi$
$734$		0			0
$735$	−2.71867	−	3.56801i	−0.100280	−	0.131608i
$736$		0			0
$737$		−	28.2949i		−	1.04226i
$738$		0			0
$739$	−8.26951			−0.304199			−0.152099	−	0.988365i	$-0.548603\pi$
$739$	−8.26951			−0.304199			−0.152099	+	0.988365i	$0.548603\pi$
$740$		0			0
$741$	33.2875	−	25.3636i	1.22285	−	0.931756i
$742$		0			0
$743$	14.3974			0.528191			0.264096	−	0.964497i	$-0.414927\pi$
$743$	14.3974			0.528191			0.264096	+	0.964497i	$0.414927\pi$
$744$		0			0
$745$	1.19031			0.0436096
$746$		0			0
$747$	−0.284512	+	1.03373i	−0.0104098	+	0.0378224i
$748$		0			0
$749$		−	19.2124i		−	0.702004i
$750$		0			0
$751$			34.9350i			1.27480i	0.770534	+	0.637399i	$0.219990\pi$
$751$			34.9350i			1.27480i	−0.770534	+	0.637399i	$0.780010\pi$
$752$		0			0
$753$	4.81858	+	6.32395i	0.175599	+	0.230458i
$754$		0			0
$755$	−7.99999			−0.291149
$756$		0			0
$757$		−	3.01036i		−	0.109414i	−0.998502	−	0.0547068i	$-0.982578\pi$
$757$		−	3.01036i		−	0.109414i	0.998502	−	0.0547068i	$-0.0174224\pi$
$758$		0			0
$759$	−1.18552	+	20.3932i	−0.0430315	+	0.740226i
$760$		0			0
$761$			28.9870i			1.05078i	0.850862	+	0.525389i	$0.176080\pi$
$761$			28.9870i			1.05078i	−0.850862	+	0.525389i	$0.823920\pi$
$762$		0			0
$763$	−36.8154			−1.33281
$764$		0			0
$765$	5.29853	−	19.2515i	0.191569	−	0.696039i
$766$		0			0
$767$		−	36.0601i		−	1.30205i
$768$		0			0
$769$			27.0243i			0.974522i	0.873256	+	0.487261i	$0.162004\pi$
$769$			27.0243i			0.974522i	−0.873256	+	0.487261i	$0.837996\pi$
$770$		0			0
$771$	−12.0750	+	9.20061i	−0.434870	+	0.331352i
$772$		0			0
$773$	26.9565			0.969557			0.484778	−	0.874637i	$-0.338900\pi$
$773$	26.9565			0.969557			0.484778	+	0.874637i	$0.338900\pi$
$774$		0			0
$775$	4.21597			0.151442
$776$		0			0
$777$	17.9664	+	23.5793i	0.644540	+	0.845902i
$778$		0			0
$779$	−15.1533			−0.542924
$780$		0			0
$781$			11.5063i			0.411727i
$782$		0			0
$783$	−19.4342	−	48.3015i	−0.694520	−	1.72615i
$784$		0			0
$785$			20.1432i			0.718941i
$786$		0			0
$787$			4.31890i			0.153952i	0.997033	+	0.0769760i	$0.0245265\pi$
$787$			4.31890i			0.153952i	−0.997033	+	0.0769760i	$0.975474\pi$
$788$		0			0
$789$	12.5368	+	16.4534i	0.446320	+	0.585756i
$790$		0			0
$791$		−	9.16368i		−	0.325823i
$792$		0			0
$793$			36.2546i			1.28744i
$794$		0			0
$795$	12.0693	+	15.8399i	0.428055	+	0.561785i
$796$		0			0
$797$	−26.2051			−0.928232			−0.464116	−	0.885774i	$-0.653628\pi$
$797$	−26.2051			−0.928232			−0.464116	+	0.885774i	$0.653628\pi$
$798$		0			0
$799$		−	41.3947i		−	1.46444i
$800$		0			0
$801$	8.07903	−	29.3540i	0.285458	−	1.03717i
$802$		0			0
$803$	40.9657			1.44565
$804$		0			0
$805$	−6.55861	+	7.64319i	−0.231161	+	0.269387i
$806$		0			0
$807$	36.9965	−	28.1897i	1.30234	−	0.992325i
$808$		0			0
$809$			11.9573i			0.420396i	0.977659	+	0.210198i	$0.0674108\pi$
$809$			11.9573i			0.420396i	−0.977659	+	0.210198i	$0.932589\pi$
$810$		0			0
$811$	−21.7481			−0.763680			−0.381840	−	0.924229i	$-0.624709\pi$
$811$	−21.7481			−0.763680			−0.381840	+	0.924229i	$0.624709\pi$
$812$		0			0
$813$	−2.50254	−	3.28437i	−0.0877680	−	0.115188i
$814$		0			0
$815$	6.91704			0.242293
$816$		0			0
$817$	−12.4642			−0.436069
$818$		0			0
$819$	−34.6760	−	9.54379i	−1.21168	−	0.333487i
$820$		0			0
$821$		−	18.9499i		−	0.661355i	−0.943744	−	0.330678i	$-0.892723\pi$
$821$		−	18.9499i		−	0.661355i	0.943744	−	0.330678i	$-0.107277\pi$
$822$		0			0
$823$	43.6239			1.52064			0.760318	−	0.649552i	$-0.225043\pi$
$823$	43.6239			1.52064			0.760318	+	0.649552i	$0.225043\pi$
$824$		0			0
$825$	2.58152	+	3.38802i	0.0898771	+	0.117956i
$826$		0			0
$827$	25.3388			0.881118			0.440559	−	0.897724i	$-0.354780\pi$
$827$	25.3388			0.881118			0.440559	+	0.897724i	$0.354780\pi$
$828$		0			0
$829$	2.67930			0.0930558			0.0465279	−	0.998917i	$-0.485184\pi$
$829$	2.67930			0.0930558			0.0465279	+	0.998917i	$0.485184\pi$
$830$		0			0
$831$	−2.78448	−	3.65438i	−0.0965925	−	0.126769i
$832$		0			0
$833$	17.2374			0.597240
$834$		0			0
$835$			6.19080i			0.214241i
$836$		0			0
$837$	−20.3234	+	8.17716i	−0.702481	+	0.282644i
$838$		0			0
$839$	−31.6194			−1.09162			−0.545811	−	0.837908i	$-0.683778\pi$
$839$	−31.6194			−1.09162			−0.545811	+	0.837908i	$0.683778\pi$
$840$		0			0
$841$	−71.3970			−2.46197
$842$		0			0
$843$	−11.3497	−	14.8954i	−0.390903	−	0.513025i
$844$		0			0
$845$	−19.5892			−0.673888
$846$		0			0
$847$			10.4001i			0.357353i
$848$		0			0
$849$	35.6313	−	27.1495i	1.22286	−	0.931769i
$850$		0			0
$851$	−25.4528	+	29.6618i	−0.872510	+	1.01679i
$852$		0			0
$853$	44.3716			1.51926			0.759628	−	0.650358i	$-0.225381\pi$
$853$	44.3716			1.51926			0.759628	+	0.650358i	$0.225381\pi$
$854$		0			0
$855$	12.2421	+	3.36937i	0.418672	+	0.115230i
$856$		0			0
$857$			46.9550i			1.60395i	0.597356	+	0.801976i	$0.296218\pi$
$857$			46.9550i			1.60395i	−0.597356	+	0.801976i	$0.703782\pi$
$858$		0			0
$859$	19.1700			0.654071			0.327036	−	0.945012i	$-0.393950\pi$
$859$	19.1700			0.654071			0.327036	+	0.945012i	$0.393950\pi$
$860$		0			0
$861$	7.89271	+	10.3585i	0.268983	+	0.353016i
$862$		0			0
$863$			23.1614i			0.788422i	0.919020	+	0.394211i	$0.128982\pi$
$863$			23.1614i			0.788422i	−0.919020	+	0.394211i	$0.871018\pi$
$864$		0			0
$865$			24.7750i			0.842376i
$866$		0			0
$867$	28.6572	+	37.6101i	0.973251	+	1.27731i
$868$		0			0
$869$		−	21.2407i		−	0.720540i
$870$		0			0
$871$			65.6828i			2.22558i
$872$		0			0
$873$	9.25838	+	2.54816i	0.313349	+	0.0862421i
$874$		0			0
$875$			2.10004i			0.0709942i
$876$		0			0
$877$	52.1228			1.76006			0.880030	−	0.474917i	$-0.157522\pi$
$877$	52.1228			1.76006			0.880030	+	0.474917i	$0.157522\pi$
$878$		0			0
$879$	−18.4584	−	24.2250i	−0.622587	−	0.817090i
$880$		0			0
$881$	−1.21055			−0.0407844			−0.0203922	−	0.999792i	$-0.506491\pi$
$881$	−1.21055			−0.0407844			−0.0203922	+	0.999792i	$0.506491\pi$
$882$		0			0
$883$	−40.0713			−1.34851			−0.674253	−	0.738501i	$-0.735534\pi$
$883$	−40.0713			−1.34851			−0.674253	+	0.738501i	$0.735534\pi$
$884$		0			0
$885$	8.70247	−	6.63090i	0.292530	−	0.222895i
$886$		0			0
$887$		−	24.3166i		−	0.816471i	−0.912877	−	0.408236i	$-0.866144\pi$
$887$		−	24.3166i		−	0.816471i	0.912877	−	0.408236i	$-0.133856\pi$
$888$		0			0
$889$			26.7164i			0.896038i
$890$		0			0
$891$	−19.0158	−	11.3252i	−0.637052	−	0.379409i
$892$		0			0
$893$	26.3232			0.880871
$894$		0			0
$895$		−	9.97353i		−	0.333378i
$896$		0			0
$897$	2.75202	−	47.3400i	0.0918871	−	1.58064i
$898$		0			0
$899$			42.2433i			1.40889i
$900$		0			0
$901$	−76.5243			−2.54939
$902$		0			0
$903$	6.49208	+	8.52028i	0.216043	+	0.283537i
$904$		0			0
$905$		−	7.84144i		−	0.260658i
$906$		0			0
$907$			6.57864i			0.218440i	0.994018	+	0.109220i	$0.0348353\pi$
$907$			6.57864i			0.218440i	−0.994018	+	0.109220i	$0.965165\pi$
$908$		0			0
$909$	−16.2851	−	4.48211i	−0.540143	−	0.148662i
$910$		0			0
$911$	−58.0247			−1.92244			−0.961221	−	0.275778i	$-0.911065\pi$
$911$	−58.0247			−1.92244			−0.961221	+	0.275778i	$0.911065\pi$
$912$		0			0
$913$	0.878894			0.0290871
$914$		0			0
$915$	−8.74941	+	6.66667i	−0.289247	+	0.220393i
$916$		0			0
$917$	21.6495			0.714929
$918$		0			0
$919$			15.0368i			0.496018i	0.968758	+	0.248009i	$0.0797762\pi$
$919$			15.0368i			0.496018i	−0.968758	+	0.248009i	$0.920224\pi$
$920$		0			0
$921$	−21.5883	−	28.3327i	−0.711359	−	0.933596i
$922$		0			0
$923$		−	26.7103i		−	0.879179i
$924$		0			0
$925$			8.14987i			0.267966i
$926$		0			0
$927$	−51.1628	−	14.0814i	−1.68041	−	0.462494i
$928$		0			0
$929$		−	0.289490i		−	0.00949786i	−0.999989	−	0.00474893i	$-0.998488\pi$
$929$		−	0.289490i		−	0.00949786i	0.999989	−	0.00474893i	$-0.00151164\pi$
$930$		0			0
$931$			10.9614i			0.359244i
$932$		0			0
$933$	14.3514	−	10.9351i	0.469843	−	0.358000i
$934$		0			0
$935$	−16.3679			−0.535286
$936$		0			0
$937$		−	17.4492i		−	0.570041i	−0.958521	−	0.285020i	$-0.908000\pi$
$937$		−	17.4492i		−	0.570041i	0.958521	−	0.285020i	$-0.0920004\pi$
$938$		0			0
$939$	32.0933	−	24.4537i	1.04733	−	0.798017i
$940$		0			0
$941$	37.3268			1.21682			0.608410	−	0.793623i	$-0.291808\pi$
$941$	37.3268			1.21682			0.608410	+	0.793623i	$0.291808\pi$
$942$		0			0
$943$	−11.1815	+	13.0306i	−0.364120	+	0.424334i
$944$		0			0
$945$	−4.07317	−	10.1234i	−0.132500	−	0.329315i
$946$		0			0
$947$			19.9705i			0.648954i	0.945894	+	0.324477i	$0.105188\pi$
$947$			19.9705i			0.648954i	−0.945894	+	0.324477i	$0.894812\pi$
$948$		0			0
$949$	−95.0963			−3.08696
$950$		0			0
$951$	30.1580	−	22.9791i	0.977942	−	0.745149i
$952$		0			0
$953$	−51.9935			−1.68423			−0.842117	−	0.539295i	$-0.818691\pi$
$953$	−51.9935			−1.68423			−0.842117	+	0.539295i	$0.818691\pi$
$954$		0			0
$955$	0.480474			0.0155478
$956$		0			0
$957$	−33.9474	+	25.8664i	−1.09736	+	0.836143i
$958$		0			0
$959$		−	39.7025i		−	1.28206i
$960$		0			0
$961$	−13.2256			−0.426633
$962$		0			0
$963$	−7.28301	+	26.4618i	−0.234692	+	0.852719i
$964$		0			0
$965$	−8.72137			−0.280751
$966$		0			0
$967$	16.6488			0.535389			0.267694	−	0.963504i	$-0.413738\pi$
$967$	16.6488			0.535389			0.267694	+	0.963504i	$0.413738\pi$
$968$		0			0
$969$	−38.8099	+	29.5715i	−1.24675	+	0.949973i
$970$		0			0
$971$	−26.7522			−0.858518			−0.429259	−	0.903181i	$-0.641225\pi$
$971$	−26.7522			−0.858518			−0.429259	+	0.903181i	$0.641225\pi$
$972$		0			0
$973$			36.7776i			1.17904i
$974$		0			0
$975$	−5.99266	−	7.86483i	−0.191919	−	0.251876i
$976$		0			0
$977$	−37.1398			−1.18821			−0.594104	−	0.804389i	$-0.702493\pi$
$977$	−37.1398			−1.18821			−0.594104	+	0.804389i	$0.702493\pi$
$978$		0			0
$979$	−24.9572			−0.797634
$980$		0			0
$981$	50.7070	+	13.9560i	1.61895	+	0.445580i
$982$		0			0
$983$	22.5055			0.717815			0.358907	−	0.933373i	$-0.383149\pi$
$983$	22.5055			0.717815			0.358907	+	0.933373i	$0.383149\pi$
$984$		0			0
$985$		−	4.62001i		−	0.147206i
$986$		0			0
$987$	−13.7106	−	17.9939i	−0.436413	−	0.572753i
$988$		0			0
$989$	−9.19727	+	10.7182i	−0.292456	+	0.340819i
$990$		0			0
$991$	−25.7925			−0.819325			−0.409662	−	0.912237i	$-0.634353\pi$
$991$	−25.7925			−0.819325			−0.409662	+	0.912237i	$0.634353\pi$
$992$		0			0
$993$	−20.5693	−	26.9953i	−0.652746	−	0.856671i
$994$		0			0
$995$		−	22.4544i		−	0.711852i
$996$		0			0
$997$	2.61017			0.0826648			0.0413324	−	0.999145i	$-0.486840\pi$
$997$	2.61017			0.0826648			0.0413324	+	0.999145i	$0.486840\pi$
$998$		0			0
$999$	−15.8072	−	39.2872i	−0.500119	−	1.24299i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.i.a.1241.12	yes	16
3.2	odd	2		1380.2.i.b.1241.11	yes	16
23.22	odd	2		1380.2.i.b.1241.12	yes	16
69.68	even	2	inner	1380.2.i.a.1241.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.i.a.1241.11	✓	16	69.68	even	2	inner
1380.2.i.a.1241.12	yes	16	1.1	even	1	trivial
1380.2.i.b.1241.11	yes	16	3.2	odd	2
1380.2.i.b.1241.12	yes	16	23.22	odd	2

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.04974 + 1.37769i) q^{3} -1.00000 q^{5} -2.10004i q^{7} +(-0.796081 + 2.89245i) q^{9} +O(q^{10})\)	\(q+(1.04974 + 1.37769i) q^{3} -1.00000 q^{5} -2.10004i q^{7} +(-0.796081 + 2.89245i) q^{9} +2.45920 q^{11} -5.70869 q^{13} +(-1.04974 - 1.37769i) q^{15} +6.65577 q^{17} +4.23245i q^{19} +(2.89321 - 2.20450i) q^{21} +(3.63955 + 3.12309i) q^{23} +1.00000 q^{25} +(-4.82059 + 1.93957i) q^{27} +10.0198i q^{29} +4.21597 q^{31} +(2.58152 + 3.38802i) q^{33} +2.10004i q^{35} +8.14987i q^{37} +(-5.99266 - 7.86483i) q^{39} +3.58027i q^{41} +2.94492i q^{43} +(0.796081 - 2.89245i) q^{45} -6.21937i q^{47} +2.58984 q^{49} +(6.98685 + 9.16962i) q^{51} -11.4974 q^{53} -2.45920 q^{55} +(-5.83102 + 4.44298i) q^{57} +6.31669i q^{59} -6.35077i q^{61} +(6.07425 + 1.67180i) q^{63} +5.70869 q^{65} -11.5057i q^{67} +(-0.482075 + 8.29262i) q^{69} +4.67888i q^{71} +16.6582 q^{73} +(1.04974 + 1.37769i) q^{75} -5.16440i q^{77} -8.63724i q^{79} +(-7.73251 - 4.60525i) q^{81} +0.357391 q^{83} -6.65577 q^{85} +(-13.8043 + 10.5182i) q^{87} -10.1485 q^{89} +11.9885i q^{91} +(4.42568 + 5.80831i) q^{93} -4.23245i q^{95} -3.20088i q^{97} +(-1.95772 + 7.11310i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 + 2 * q^9	\( 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) \) 16 * q - 16 * q^5 + 2 * q^9 - 4 * q^21 + 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 + 2 * q^33 - 14 * q^39 - 2 * q^45 + 8 * q^49 - 2 * q^51 + 4 * q^53 - 2 * q^57 + 30 * q^63 - 4 * q^73 + 10 * q^81 - 4 * q^83 - 10 * q^87 - 28 * q^89 - 10 * q^93 - 26 * q^99

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