LMFDB - Embedded newform 1380.2.i.a.1241.13

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			1241.13
Root			$-1.32140 - 1.11978i$ of defining polynomial
Character	$\chi$	$=$	1380.1241
Dual form			1380.2.i.a.1241.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.32140 - 1.11978i) q^{3} -1.00000 q^{5} +1.85076i q^{7} +(0.492177 - 2.95935i) q^{9} +O(q^{10})$	\(q+(1.32140 - 1.11978i) q^{3} -1.00000 q^{5} +1.85076i q^{7} +(0.492177 - 2.95935i) q^{9} -5.93302 q^{11} -1.57259 q^{13} +(-1.32140 + 1.11978i) q^{15} -4.36283 q^{17} +0.298508i q^{19} +(2.07244 + 2.44558i) q^{21} +(1.25887 + 4.62766i) q^{23} +1.00000 q^{25} +(-2.66347 - 4.46161i) q^{27} +6.89025i q^{29} -9.74432 q^{31} +(-7.83987 + 6.64369i) q^{33} -1.85076i q^{35} +7.74614i q^{37} +(-2.07801 + 1.76095i) q^{39} -7.49393i q^{41} -4.96783i q^{43} +(-0.492177 + 2.95935i) q^{45} -4.73134i q^{47} +3.57470 q^{49} +(-5.76503 + 4.88542i) q^{51} -2.15842 q^{53} +5.93302 q^{55} +(0.334264 + 0.394447i) q^{57} -3.27536i q^{59} -12.6813i q^{61} +(5.47704 + 0.910901i) q^{63} +1.57259 q^{65} +13.5300i q^{67} +(6.84544 + 4.70532i) q^{69} +1.09936i q^{71} -9.96226 q^{73} +(1.32140 - 1.11978i) q^{75} -10.9806i q^{77} +5.00684i q^{79} +(-8.51552 - 2.91305i) q^{81} -5.01676 q^{83} +4.36283 q^{85} +(7.71558 + 9.10476i) q^{87} -8.95408 q^{89} -2.91048i q^{91} +(-12.8761 + 10.9115i) q^{93} -0.298508i q^{95} -11.2114i q^{97} +(-2.92010 + 17.5579i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.32140	−	1.11978i	0.762909	−	0.646506i
$3$	1.32140	−	1.11978i	0.762909	−	0.646506i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.85076i			0.699520i	0.936839	+	0.349760i	$0.113737\pi$
$7$			1.85076i			0.699520i	−0.936839	+	0.349760i	$0.886263\pi$
$8$		0			0
$9$	0.492177	−	2.95935i	0.164059	−	0.986451i
$10$		0			0
$11$	−5.93302			−1.78887			−0.894436	−	0.447196i	$-0.852423\pi$
$11$	−5.93302			−1.78887			−0.894436	+	0.447196i	$0.852423\pi$
$12$		0			0
$13$	−1.57259			−0.436157			−0.218079	−	0.975931i	$-0.569979\pi$
$13$	−1.57259			−0.436157			−0.218079	+	0.975931i	$0.569979\pi$
$14$		0			0
$15$	−1.32140	+	1.11978i	−0.341183	+	0.289126i
$16$		0			0
$17$	−4.36283			−1.05814			−0.529071	−	0.848578i	$-0.677459\pi$
$17$	−4.36283			−1.05814			−0.529071	+	0.848578i	$0.677459\pi$
$18$		0			0
$19$			0.298508i			0.0684824i	0.999414	+	0.0342412i	$0.0109014\pi$
$19$			0.298508i			0.0684824i	−0.999414	+	0.0342412i	$0.989099\pi$
$20$		0			0
$21$	2.07244	+	2.44558i	0.452244	+	0.533670i
$22$		0			0
$23$	1.25887	+	4.62766i	0.262493	+	0.964934i
$23$	1.25887	+	4.62766i	0.262493	+	0.964934i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−2.66347	−	4.46161i	−0.512584	−	0.858637i
$28$		0			0
$29$			6.89025i			1.27949i	0.768588	+	0.639744i	$0.220960\pi$
$29$			6.89025i			1.27949i	−0.768588	+	0.639744i	$0.779040\pi$
$30$		0			0
$31$	−9.74432			−1.75013			−0.875066	−	0.484004i	$-0.839182\pi$
$31$	−9.74432			−1.75013			−0.875066	+	0.484004i	$0.839182\pi$
$32$		0			0
$33$	−7.83987	+	6.64369i	−1.36475	+	1.15652i
$34$		0			0
$35$		−	1.85076i		−	0.312835i
$36$		0			0
$37$			7.74614i			1.27346i	0.771088	+	0.636729i	$0.219713\pi$
$37$			7.74614i			1.27346i	−0.771088	+	0.636729i	$0.780287\pi$
$38$		0			0
$39$	−2.07801	+	1.76095i	−0.332748	+	0.281978i
$40$		0			0
$41$		−	7.49393i		−	1.17036i	−0.810905	−	0.585178i	$-0.801025\pi$
$41$		−	7.49393i		−	1.17036i	0.810905	−	0.585178i	$-0.198975\pi$
$42$		0			0
$43$		−	4.96783i		−	0.757588i	−0.925481	−	0.378794i	$-0.876339\pi$
$43$		−	4.96783i		−	0.757588i	0.925481	−	0.378794i	$-0.123661\pi$
$44$		0			0
$45$	−0.492177	+	2.95935i	−0.0733695	+	0.441154i
$46$		0			0
$47$		−	4.73134i		−	0.690136i	−0.938578	−	0.345068i	$-0.887856\pi$
$47$		−	4.73134i		−	0.690136i	0.938578	−	0.345068i	$-0.112144\pi$
$48$		0			0
$49$	3.57470			0.510671
$50$		0			0
$51$	−5.76503	+	4.88542i	−0.807265	+	0.684095i
$52$		0			0
$53$	−2.15842			−0.296482			−0.148241	−	0.988951i	$-0.547361\pi$
$53$	−2.15842			−0.296482			−0.148241	+	0.988951i	$0.547361\pi$
$54$		0			0
$55$	5.93302			0.800008
$56$		0			0
$57$	0.334264	+	0.394447i	0.0442743	+	0.0522458i
$58$		0			0
$59$		−	3.27536i		−	0.426415i	−0.977007	−	0.213208i	$-0.931609\pi$
$59$		−	3.27536i		−	0.426415i	0.977007	−	0.213208i	$-0.0683911\pi$
$60$		0			0
$61$		−	12.6813i		−	1.62368i	−0.583883	−	0.811838i	$-0.698467\pi$
$61$		−	12.6813i		−	1.62368i	0.583883	−	0.811838i	$-0.301533\pi$
$62$		0			0
$63$	5.47704	+	0.910901i	0.690042	+	0.114763i
$64$		0			0
$65$	1.57259			0.195055
$66$		0			0
$67$			13.5300i			1.65295i	0.562972	+	0.826476i	$0.309658\pi$
$67$			13.5300i			1.65295i	−0.562972	+	0.826476i	$0.690342\pi$
$68$		0			0
$69$	6.84544	+	4.70532i	0.824094	+	0.566453i
$70$		0			0
$71$			1.09936i			0.130469i	0.997870	+	0.0652347i	$0.0207796\pi$
$71$			1.09936i			0.130469i	−0.997870	+	0.0652347i	$0.979220\pi$
$72$		0			0
$73$	−9.96226			−1.16599			−0.582997	−	0.812474i	$-0.698120\pi$
$73$	−9.96226			−1.16599			−0.582997	+	0.812474i	$0.698120\pi$
$74$		0			0
$75$	1.32140	−	1.11978i	0.152582	−	0.129301i
$76$		0			0
$77$		−	10.9806i		−	1.25135i
$78$		0			0
$79$			5.00684i			0.563314i	0.959515	+	0.281657i	$0.0908840\pi$
$79$			5.00684i			0.563314i	−0.959515	+	0.281657i	$0.909116\pi$
$80$		0			0
$81$	−8.51552	−	2.91305i	−0.946169	−	0.323672i
$82$		0			0
$83$	−5.01676			−0.550661			−0.275330	−	0.961350i	$-0.588787\pi$
$83$	−5.01676			−0.550661			−0.275330	+	0.961350i	$0.588787\pi$
$84$		0			0
$85$	4.36283			0.473215
$86$		0			0
$87$	7.71558	+	9.10476i	0.827197	+	0.976132i
$88$		0			0
$89$	−8.95408			−0.949131			−0.474565	−	0.880220i	$-0.657395\pi$
$89$	−8.95408			−0.949131			−0.474565	+	0.880220i	$0.657395\pi$
$90$		0			0
$91$		−	2.91048i		−	0.305101i
$92$		0			0
$93$	−12.8761	+	10.9115i	−1.33519	+	1.13147i
$94$		0			0
$95$		−	0.298508i		−	0.0306263i
$96$		0			0
$97$		−	11.2114i		−	1.13834i	−0.822220	−	0.569170i	$-0.807265\pi$
$97$		−	11.2114i		−	1.13834i	0.822220	−	0.569170i	$-0.192735\pi$
$98$		0			0
$99$	−2.92010	+	17.5579i	−0.293481	+	1.76463i
$100$		0			0
$101$			13.3200i			1.32539i	0.748888	+	0.662697i	$0.230588\pi$
$101$			13.3200i			1.32539i	−0.748888	+	0.662697i	$0.769412\pi$
$102$		0			0
$103$		−	2.67492i		−	0.263568i	−0.991278	−	0.131784i	$-0.957930\pi$
$103$		−	2.67492i		−	0.263568i	0.991278	−	0.131784i	$-0.0420705\pi$
$104$		0			0
$105$	−2.07244	−	2.44558i	−0.202250	−	0.238665i
$106$		0			0
$107$	3.17299			0.306744			0.153372	−	0.988168i	$-0.450987\pi$
$107$	3.17299			0.306744			0.153372	+	0.988168i	$0.450987\pi$
$108$		0			0
$109$			3.94338i			0.377708i	0.982005	+	0.188854i	$0.0604772\pi$
$109$			3.94338i			0.377708i	−0.982005	+	0.188854i	$0.939523\pi$
$110$		0			0
$111$	8.67399	+	10.2357i	0.823299	+	0.971532i
$112$		0			0
$113$	2.45864			0.231289			0.115645	−	0.993291i	$-0.463107\pi$
$113$	2.45864			0.231289			0.115645	+	0.993291i	$0.463107\pi$
$114$		0			0
$115$	−1.25887	−	4.62766i	−0.117390	−	0.431532i
$116$		0			0
$117$	−0.773992	+	4.65384i	−0.0715556	+	0.430248i
$118$		0			0
$119$		−	8.07454i		−	0.740191i
$120$		0			0
$121$	24.2007			2.20006
$122$		0			0
$123$	−8.39157	−	9.90246i	−0.756642	−	0.892875i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	7.80700			0.692759			0.346380	−	0.938094i	$-0.387411\pi$
$127$	7.80700			0.692759			0.346380	+	0.938094i	$0.387411\pi$
$128$		0			0
$129$	−5.56289	−	6.56448i	−0.489785	−	0.577970i
$130$		0			0
$131$			14.4194i			1.25983i	0.776665	+	0.629914i	$0.216910\pi$
$131$			14.4194i			1.25983i	−0.776665	+	0.629914i	$0.783090\pi$
$132$		0			0
$133$	−0.552466			−0.0479049
$134$		0			0
$135$	2.66347	+	4.46161i	0.229235	+	0.383994i
$136$		0			0
$137$	21.3155			1.82111			0.910554	−	0.413390i	$-0.135655\pi$
$137$	21.3155			1.82111			0.910554	+	0.413390i	$0.135655\pi$
$138$		0			0
$139$	6.99913			0.593658			0.296829	−	0.954931i	$-0.404071\pi$
$139$	6.99913			0.593658			0.296829	+	0.954931i	$0.404071\pi$
$140$		0			0
$141$	−5.29806	−	6.25197i	−0.446177	−	0.526511i
$142$		0			0
$143$	9.33019			0.780230
$144$		0			0
$145$		−	6.89025i		−	0.572204i
$146$		0			0
$147$	4.72359	−	4.00288i	0.389596	−	0.330152i
$148$		0			0
$149$	0.124112			0.0101677			0.00508383	−	0.999987i	$-0.498382\pi$
$149$	0.124112			0.0101677			0.00508383	+	0.999987i	$0.498382\pi$
$150$		0			0
$151$	−11.3148			−0.920788			−0.460394	−	0.887715i	$-0.652292\pi$
$151$	−11.3148			−0.920788			−0.460394	+	0.887715i	$0.652292\pi$
$152$		0			0
$153$	−2.14729	+	12.9111i	−0.173598	+	1.04380i
$154$		0			0
$155$	9.74432			0.782683
$156$		0			0
$157$		−	16.4553i		−	1.31328i	−0.754204	−	0.656640i	$-0.771977\pi$
$157$		−	16.4553i		−	1.31328i	0.754204	−	0.656640i	$-0.228023\pi$
$158$		0			0
$159$	−2.85213	+	2.41696i	−0.226189	+	0.191677i
$160$		0			0
$161$	−8.56468	+	2.32986i	−0.674991	+	0.183619i
$162$		0			0
$163$	−20.3222			−1.59176			−0.795879	−	0.605455i	$-0.792991\pi$
$163$	−20.3222			−1.59176			−0.795879	+	0.605455i	$0.792991\pi$
$164$		0			0
$165$	7.83987	−	6.64369i	0.610333	−	0.517210i
$166$		0			0
$167$		−	20.1479i		−	1.55909i	−0.626346	−	0.779546i	$-0.715450\pi$
$167$		−	20.1479i		−	1.55909i	0.626346	−	0.779546i	$-0.284550\pi$
$168$		0			0
$169$	−10.5270			−0.809767
$170$		0			0
$171$	0.883390	+	0.146919i	0.0675545	+	0.0112352i
$172$		0			0
$173$			8.70765i			0.662031i	0.943625	+	0.331015i	$0.107391\pi$
$173$			8.70765i			0.662031i	−0.943625	+	0.331015i	$0.892609\pi$
$174$		0			0
$175$			1.85076i			0.139904i
$176$		0			0
$177$	−3.66769	−	4.32805i	−0.275680	−	0.325316i
$178$		0			0
$179$			21.9724i			1.64229i	0.570716	+	0.821147i	$0.306666\pi$
$179$			21.9724i			1.64229i	−0.570716	+	0.821147i	$0.693334\pi$
$180$		0			0
$181$			6.57440i			0.488671i	0.969691	+	0.244336i	$0.0785699\pi$
$181$			6.57440i			0.488671i	−0.969691	+	0.244336i	$0.921430\pi$
$182$		0			0
$183$	−14.2003	−	16.7570i	−1.04972	−	1.23872i
$184$		0			0
$185$		−	7.74614i		−	0.569508i
$186$		0			0
$187$	25.8847			1.89288
$188$		0			0
$189$	8.25735	−	4.92943i	0.600634	−	0.358563i
$190$		0			0
$191$	−7.77801			−0.562797			−0.281399	−	0.959591i	$-0.590798\pi$
$191$	−7.77801			−0.562797			−0.281399	+	0.959591i	$0.590798\pi$
$192$		0			0
$193$	5.83354			0.419907			0.209954	−	0.977711i	$-0.432669\pi$
$193$	5.83354			0.419907			0.209954	+	0.977711i	$0.432669\pi$
$194$		0			0
$195$	2.07801	−	1.76095i	0.148809	−	0.126105i
$196$		0			0
$197$		−	20.6734i		−	1.47292i	−0.676482	−	0.736459i	$-0.736496\pi$
$197$		−	20.6734i		−	1.47292i	0.676482	−	0.736459i	$-0.263504\pi$
$198$		0			0
$199$		−	1.22251i		−	0.0866617i	−0.999061	−	0.0433308i	$-0.986203\pi$
$199$		−	1.22251i		−	0.0866617i	0.999061	−	0.0433308i	$-0.0137970\pi$
$200$		0			0
$201$	15.1506	+	17.8785i	1.06864	+	1.26105i
$202$		0			0
$203$	−12.7522			−0.895028
$204$		0			0
$205$			7.49393i			0.523399i
$206$		0			0
$207$	14.3145	−	1.44781i	0.994924	−	0.100630i
$208$		0			0
$209$		−	1.77105i		−	0.122506i
$210$		0			0
$211$	−2.01055			−0.138412			−0.0692059	−	0.997602i	$-0.522047\pi$
$211$	−2.01055			−0.138412			−0.0692059	+	0.997602i	$0.522047\pi$
$212$		0			0
$213$	1.23104	+	1.45268i	0.0843493	+	0.0995363i
$214$		0			0
$215$			4.96783i			0.338804i
$216$		0			0
$217$		−	18.0344i		−	1.22425i
$218$		0			0
$219$	−13.1641	+	11.1556i	−0.889547	+	0.753823i
$220$		0			0
$221$	6.86093			0.461516
$222$		0			0
$223$	12.4673			0.834873			0.417436	−	0.908706i	$-0.362929\pi$
$223$	12.4673			0.834873			0.417436	+	0.908706i	$0.362929\pi$
$224$		0			0
$225$	0.492177	−	2.95935i	0.0328118	−	0.197290i
$226$		0			0
$227$	−24.4163			−1.62057			−0.810283	−	0.586039i	$-0.800687\pi$
$227$	−24.4163			−1.62057			−0.810283	+	0.586039i	$0.800687\pi$
$228$		0			0
$229$		−	29.4233i		−	1.94435i	−0.234262	−	0.972173i	$-0.575267\pi$
$229$		−	29.4233i		−	1.94435i	0.234262	−	0.972173i	$-0.424733\pi$
$230$		0			0
$231$	−12.2958	−	14.5097i	−0.809007	−	0.954668i
$232$		0			0
$233$		−	2.46663i		−	0.161594i	−0.996731	−	0.0807971i	$-0.974253\pi$
$233$		−	2.46663i		−	0.161594i	0.996731	−	0.0807971i	$-0.0257466\pi$
$234$		0			0
$235$			4.73134i			0.308638i
$236$		0			0
$237$	5.60657	+	6.61602i	0.364186	+	0.429757i
$238$		0			0
$239$		−	8.04122i		−	0.520143i	−0.965589	−	0.260071i	$-0.916254\pi$
$239$		−	8.04122i		−	0.520143i	0.965589	−	0.260071i	$-0.0837461\pi$
$240$		0			0
$241$		−	22.8993i		−	1.47507i	−0.675306	−	0.737537i	$-0.735989\pi$
$241$		−	22.8993i		−	1.47507i	0.675306	−	0.737537i	$-0.264011\pi$
$242$		0			0
$243$	−14.5144	+	5.68623i	−0.931097	+	0.364772i
$244$		0			0
$245$	−3.57470			−0.228379
$246$		0			0
$247$		−	0.469430i		−	0.0298691i
$248$		0			0
$249$	−6.62913	+	5.61767i	−0.420104	+	0.356006i
$250$		0			0
$251$	−7.28474			−0.459809			−0.229904	−	0.973213i	$-0.573841\pi$
$251$	−7.28474			−0.459809			−0.229904	+	0.973213i	$0.573841\pi$
$252$		0			0
$253$	−7.46890	−	27.4560i	−0.469566	−	1.72614i
$254$		0			0
$255$	5.76503	−	4.88542i	0.361020	−	0.305937i
$256$		0			0
$257$		−	0.181856i		−	0.0113439i	−0.999984	−	0.00567193i	$-0.998195\pi$
$257$		−	0.181856i		−	0.0113439i	0.999984	−	0.00567193i	$-0.00180544\pi$
$258$		0			0
$259$	−14.3362			−0.890810
$260$		0			0
$261$	20.3907	+	3.39123i	1.26215	+	0.209912i
$262$		0			0
$263$	12.7020			0.783239			0.391619	−	0.920127i	$-0.371915\pi$
$263$	12.7020			0.783239			0.391619	+	0.920127i	$0.371915\pi$
$264$		0			0
$265$	2.15842			0.132591
$266$		0			0
$267$	−11.8319	+	10.0266i	−0.724100	+	0.613619i
$268$		0			0
$269$			29.8595i			1.82056i	0.413989	+	0.910282i	$0.364135\pi$
$269$			29.8595i			1.82056i	−0.413989	+	0.910282i	$0.635865\pi$
$270$		0			0
$271$	−10.1037			−0.613757			−0.306878	−	0.951749i	$-0.599284\pi$
$271$	−10.1037			−0.613757			−0.306878	+	0.951749i	$0.599284\pi$
$272$		0			0
$273$	−3.25910	−	3.84589i	−0.197250	−	0.232764i
$274$		0			0
$275$	−5.93302			−0.357774
$276$		0			0
$277$	18.4889			1.11089			0.555445	−	0.831553i	$-0.312548\pi$
$277$	18.4889			1.11089			0.555445	+	0.831553i	$0.312548\pi$
$278$		0			0
$279$	−4.79594	+	28.8369i	−0.287125	+	1.72642i
$280$		0			0
$281$	−2.56349			−0.152925			−0.0764624	−	0.997072i	$-0.524363\pi$
$281$	−2.56349			−0.152925			−0.0764624	+	0.997072i	$0.524363\pi$
$282$		0			0
$283$			18.2025i			1.08202i	0.841015	+	0.541012i	$0.181959\pi$
$283$			18.2025i			1.08202i	−0.841015	+	0.541012i	$0.818041\pi$
$284$		0			0
$285$	−0.334264	−	0.394447i	−0.0198001	−	0.0233651i
$286$		0			0
$287$	13.8694			0.818688
$288$		0			0
$289$	2.03428			0.119663
$290$		0			0
$291$	−12.5543	−	14.8146i	−0.735944	−	0.868450i
$292$		0			0
$293$	19.4784			1.13794			0.568969	−	0.822359i	$-0.307342\pi$
$293$	19.4784			1.13794			0.568969	+	0.822359i	$0.307342\pi$
$294$		0			0
$295$			3.27536i			0.190699i
$296$		0			0
$297$	15.8024	+	26.4708i	0.916948	+	1.53599i
$298$		0			0
$299$	−1.97968	−	7.27740i	−0.114488	−	0.420863i
$300$		0			0
$301$	9.19425			0.529948
$302$		0			0
$303$	14.9155	+	17.6011i	0.856875	+	1.01115i
$304$		0			0
$305$			12.6813i			0.726130i
$306$		0			0
$307$	−8.53645			−0.487201			−0.243600	−	0.969876i	$-0.578329\pi$
$307$	−8.53645			−0.487201			−0.243600	+	0.969876i	$0.578329\pi$
$308$		0			0
$309$	−2.99533	−	3.53463i	−0.170398	−	0.201078i
$310$		0			0
$311$			15.1958i			0.861674i	0.902430	+	0.430837i	$0.141782\pi$
$311$			15.1958i			0.861674i	−0.902430	+	0.430837i	$0.858218\pi$
$312$		0			0
$313$		−	33.6979i		−	1.90472i	−0.304983	−	0.952358i	$-0.598651\pi$
$313$		−	33.6979i		−	1.90472i	0.304983	−	0.952358i	$-0.401349\pi$
$314$		0			0
$315$	−5.47704	−	0.910901i	−0.308596	−	0.0513235i
$316$		0			0
$317$		−	11.3860i		−	0.639501i	−0.947502	−	0.319751i	$-0.896401\pi$
$317$		−	11.3860i		−	0.639501i	0.947502	−	0.319751i	$-0.103599\pi$
$318$		0			0
$319$		−	40.8800i		−	2.28884i
$320$		0			0
$321$	4.19277	−	3.55305i	0.234018	−	0.198312i
$322$		0			0
$323$		−	1.30234i		−	0.0724641i
$324$		0			0
$325$	−1.57259			−0.0872315
$326$		0			0
$327$	4.41573	+	5.21077i	0.244190	+	0.288156i
$328$		0			0
$329$	8.75655			0.482764
$330$		0			0
$331$	−20.0299			−1.10094			−0.550471	−	0.834854i	$-0.685552\pi$
$331$	−20.0299			−1.10094			−0.550471	+	0.834854i	$0.685552\pi$
$332$		0			0
$333$	22.9236	+	3.81248i	1.25620	+	0.208922i
$334$		0			0
$335$		−	13.5300i		−	0.739222i
$336$		0			0
$337$			0.0706679i			0.00384952i	0.999998	+	0.00192476i	$0.000612671\pi$
$337$			0.0706679i			0.00384952i	−0.999998	+	0.00192476i	$0.999387\pi$
$338$		0			0
$339$	3.24883	−	2.75314i	0.176452	−	0.149530i
$340$		0			0
$341$	57.8132			3.13076
$342$		0			0
$343$			19.5712i			1.05675i
$344$		0			0
$345$	−6.84544	−	4.70532i	−0.368546	−	0.253326i
$346$		0			0
$347$			5.39728i			0.289741i	0.989451	+	0.144871i	$0.0462766\pi$
$347$			5.39728i			0.289741i	−0.989451	+	0.144871i	$0.953723\pi$
$348$		0			0
$349$	17.3846			0.930576			0.465288	−	0.885159i	$-0.345951\pi$
$349$	17.3846			0.930576			0.465288	+	0.885159i	$0.345951\pi$
$350$		0			0
$351$	4.18853	+	7.01627i	0.223567	+	0.374501i
$352$		0			0
$353$			16.8977i			0.899374i	0.893186	+	0.449687i	$0.148464\pi$
$353$			16.8977i			0.899374i	−0.893186	+	0.449687i	$0.851536\pi$
$354$		0			0
$355$		−	1.09936i		−	0.0583477i
$356$		0			0
$357$	−9.04172	−	10.6697i	−0.478538	−	0.564698i
$358$		0			0
$359$	−9.73212			−0.513642			−0.256821	−	0.966459i	$-0.582675\pi$
$359$	−9.73212			−0.513642			−0.256821	+	0.966459i	$0.582675\pi$
$360$		0			0
$361$	18.9109			0.995310
$362$		0			0
$363$	31.9787	−	27.0995i	1.67845	−	1.42236i
$364$		0			0
$365$	9.96226			0.521449
$366$		0			0
$367$			14.7900i			0.772032i	0.922492	+	0.386016i	$0.126149\pi$
$367$			14.7900i			0.772032i	−0.922492	+	0.386016i	$0.873851\pi$
$368$		0			0
$369$	−22.1772	−	3.68835i	−1.15450	−	0.192008i
$370$		0			0
$371$		−	3.99471i		−	0.207395i
$372$		0			0
$373$		−	11.6524i		−	0.603337i	−0.953413	−	0.301669i	$-0.902456\pi$
$373$		−	11.6524i		−	0.603337i	0.953413	−	0.301669i	$-0.0975436\pi$
$374$		0			0
$375$	−1.32140	+	1.11978i	−0.0682366	+	0.0578253i
$376$		0			0
$377$		−	10.8355i		−	0.558058i
$378$		0			0
$379$		−	7.72950i		−	0.397038i	−0.980097	−	0.198519i	$-0.936387\pi$
$379$		−	7.72950i		−	0.397038i	0.980097	−	0.198519i	$-0.0636131\pi$
$380$		0			0
$381$	10.3161	−	8.74213i	0.528512	−	0.447873i
$382$		0			0
$383$	−5.48133			−0.280083			−0.140041	−	0.990146i	$-0.544724\pi$
$383$	−5.48133			−0.280083			−0.140041	+	0.990146i	$0.544724\pi$
$384$		0			0
$385$			10.9806i			0.559622i
$386$		0			0
$387$	−14.7016	−	2.44506i	−0.747323	−	0.124289i
$388$		0			0
$389$	31.2241			1.58313			0.791563	−	0.611087i	$-0.209268\pi$
$389$	31.2241			1.58313			0.791563	+	0.611087i	$0.209268\pi$
$390$		0			0
$391$	−5.49223	−	20.1897i	−0.277754	−	1.02104i
$392$		0			0
$393$	16.1466	+	19.0537i	0.814487	+	0.961134i
$394$		0			0
$395$		−	5.00684i		−	0.251922i
$396$		0			0
$397$	3.73676			0.187543			0.0937713	−	0.995594i	$-0.470108\pi$
$397$	3.73676			0.187543			0.0937713	+	0.995594i	$0.470108\pi$
$398$		0			0
$399$	−0.730026	+	0.618641i	−0.0365470	+	0.0309708i
$400$		0			0
$401$	−19.5706			−0.977307			−0.488654	−	0.872478i	$-0.662512\pi$
$401$	−19.5706			−0.977307			−0.488654	+	0.872478i	$0.662512\pi$
$402$		0			0
$403$	15.3238			0.763333
$404$		0			0
$405$	8.51552	+	2.91305i	0.423140	+	0.144751i
$406$		0			0
$407$		−	45.9580i		−	2.27805i
$408$		0			0
$409$	−31.7691			−1.57088			−0.785441	−	0.618936i	$-0.787564\pi$
$409$	−31.7691			−1.57088			−0.785441	+	0.618936i	$0.787564\pi$
$410$		0			0
$411$	28.1663	−	23.8687i	1.38934	−	1.17736i
$412$		0			0
$413$	6.06189			0.298286
$414$		0			0
$415$	5.01676			0.246263
$416$		0			0
$417$	9.24862	−	7.83749i	0.452907	−	0.383804i
$418$		0			0
$419$	−5.72503			−0.279686			−0.139843	−	0.990174i	$-0.544660\pi$
$419$	−5.72503			−0.279686			−0.139843	+	0.990174i	$0.544660\pi$
$420$		0			0
$421$			13.7796i			0.671574i	0.941938	+	0.335787i	$0.109002\pi$
$421$			13.7796i			0.671574i	−0.941938	+	0.335787i	$0.890998\pi$
$422$		0			0
$423$	−14.0017	−	2.32866i	−0.680785	−	0.113223i
$424$		0			0
$425$	−4.36283			−0.211628
$426$		0			0
$427$	23.4700			1.13579
$428$		0			0
$429$	12.3289	−	10.4478i	0.595244	−	0.504423i
$430$		0			0
$431$	−9.49753			−0.457480			−0.228740	−	0.973488i	$-0.573461\pi$
$431$	−9.49753			−0.457480			−0.228740	+	0.973488i	$0.573461\pi$
$432$		0			0
$433$			19.7577i			0.949495i	0.880122	+	0.474747i	$0.157461\pi$
$433$			19.7577i			0.949495i	−0.880122	+	0.474747i	$0.842539\pi$
$434$		0			0
$435$	−7.71558	−	9.10476i	−0.369934	−	0.436540i
$436$		0			0
$437$	−1.38139	+	0.375783i	−0.0660810	+	0.0179761i
$438$		0			0
$439$	−23.6715			−1.12978			−0.564890	−	0.825166i	$-0.691082\pi$
$439$	−23.6715			−1.12978			−0.564890	+	0.825166i	$0.691082\pi$
$440$		0			0
$441$	1.75939	−	10.5788i	0.0837803	−	0.503752i
$442$		0			0
$443$		−	18.4755i		−	0.877797i	−0.898537	−	0.438899i	$-0.855369\pi$
$443$		−	18.4755i		−	0.877797i	0.898537	−	0.438899i	$-0.144631\pi$
$444$		0			0
$445$	8.95408			0.424464
$446$		0			0
$447$	0.164001	−	0.138979i	0.00775700	−	0.00657346i
$448$		0			0
$449$			16.9642i			0.800588i	0.916387	+	0.400294i	$0.131092\pi$
$449$			16.9642i			0.800588i	−0.916387	+	0.400294i	$0.868908\pi$
$450$		0			0
$451$			44.4616i			2.09362i
$452$		0			0
$453$	−14.9514	+	12.6701i	−0.702477	+	0.595295i
$454$		0			0
$455$			2.91048i			0.136445i
$456$		0			0
$457$		−	37.7577i		−	1.76623i	−0.469156	−	0.883116i	$-0.655442\pi$
$457$		−	37.7577i		−	1.76623i	0.469156	−	0.883116i	$-0.344558\pi$
$458$		0			0
$459$	11.6202	+	19.4652i	0.542387	+	0.908559i
$460$		0			0
$461$			28.3574i			1.32074i	0.750942	+	0.660368i	$0.229600\pi$
$461$			28.3574i			1.32074i	−0.750942	+	0.660368i	$0.770400\pi$
$462$		0			0
$463$	25.7040			1.19457			0.597284	−	0.802030i	$-0.296247\pi$
$463$	25.7040			1.19457			0.597284	+	0.802030i	$0.296247\pi$
$464$		0			0
$465$	12.8761	−	10.9115i	0.597115	−	0.506009i
$466$		0			0
$467$	30.3971			1.40661			0.703306	−	0.710888i	$-0.251707\pi$
$467$	30.3971			1.40661			0.703306	+	0.710888i	$0.251707\pi$
$468$		0			0
$469$	−25.0407			−1.15627
$470$		0			0
$471$	−18.4264	−	21.7440i	−0.849043	−	1.00191i
$472$		0			0
$473$			29.4743i			1.35523i
$474$		0			0
$475$			0.298508i			0.0136965i
$476$		0			0
$477$	−1.06233	+	6.38753i	−0.0486406	+	0.292465i
$478$		0			0
$479$	17.8861			0.817236			0.408618	−	0.912705i	$-0.366011\pi$
$479$	17.8861			0.817236			0.408618	+	0.912705i	$0.366011\pi$
$480$		0			0
$481$		−	12.1815i		−	0.555428i
$482$		0			0
$483$	−8.70840	+	12.6692i	−0.396246	+	0.576470i
$484$		0			0
$485$			11.2114i			0.509081i
$486$		0			0
$487$	7.41803			0.336143			0.168072	−	0.985775i	$-0.446246\pi$
$487$	7.41803			0.336143			0.168072	+	0.985775i	$0.446246\pi$
$488$		0			0
$489$	−26.8537	+	22.7564i	−1.21437	+	1.02908i
$490$		0			0
$491$			17.2580i			0.778841i	0.921060	+	0.389420i	$0.127325\pi$
$491$			17.2580i			0.778841i	−0.921060	+	0.389420i	$0.872675\pi$
$492$		0			0
$493$		−	30.0610i		−	1.35388i
$494$		0			0
$495$	2.92010	−	17.5579i	0.131249	−	0.789168i
$496$		0			0
$497$	−2.03464			−0.0912660
$498$		0			0
$499$	−11.2939			−0.505584			−0.252792	−	0.967521i	$-0.581349\pi$
$499$	−11.2939			−0.505584			−0.252792	+	0.967521i	$0.581349\pi$
$500$		0			0
$501$	−22.5612	−	26.6234i	−1.00796	−	1.18944i
$502$		0			0
$503$	−36.9693			−1.64838			−0.824190	−	0.566313i	$-0.808369\pi$
$503$	−36.9693			−1.64838			−0.824190	+	0.566313i	$0.808369\pi$
$504$		0			0
$505$		−	13.3200i		−	0.592734i
$506$		0			0
$507$	−13.9103	+	11.7879i	−0.617778	+	0.523519i
$508$		0			0
$509$		−	5.40302i		−	0.239485i	−0.992805	−	0.119742i	$-0.961793\pi$
$509$		−	5.40302i		−	0.239485i	0.992805	−	0.119742i	$-0.0382069\pi$
$510$		0			0
$511$		−	18.4377i		−	0.815637i
$512$		0			0
$513$	1.33183	−	0.795066i	0.0588016	−	0.0351030i
$514$		0			0
$515$			2.67492i			0.117871i
$516$		0			0
$517$			28.0711i			1.23457i
$518$		0			0
$519$	9.75067	+	11.5063i	0.428007	+	0.505069i
$520$		0			0
$521$	−43.5433			−1.90766			−0.953832	−	0.300340i	$-0.902900\pi$
$521$	−43.5433			−1.90766			−0.953832	+	0.300340i	$0.902900\pi$
$522$		0			0
$523$			37.3274i			1.63222i	0.577900	+	0.816108i	$0.303872\pi$
$523$			37.3274i			1.63222i	−0.577900	+	0.816108i	$0.696128\pi$
$524$		0			0
$525$	2.07244	+	2.44558i	0.0904489	+	0.106734i
$526$		0			0
$527$	42.5128			1.85189
$528$		0			0
$529$	−19.8305	+	11.6512i	−0.862195	+	0.506576i
$530$		0			0
$531$	−9.69294	−	1.61206i	−0.420638	−	0.0699574i
$532$		0			0
$533$			11.7849i			0.510459i
$534$		0			0
$535$	−3.17299			−0.137180
$536$		0			0
$537$	24.6043	+	29.0343i	1.06175	+	1.25292i
$538$		0			0
$539$	−21.2088			−0.913526
$540$		0			0
$541$	8.09169			0.347889			0.173945	−	0.984755i	$-0.444349\pi$
$541$	8.09169			0.347889			0.173945	+	0.984755i	$0.444349\pi$
$542$		0			0
$543$	7.36190	+	8.68739i	0.315929	+	0.372812i
$544$		0			0
$545$		−	3.94338i		−	0.168916i
$546$		0			0
$547$	3.43983			0.147077			0.0735384	−	0.997292i	$-0.476571\pi$
$547$	3.43983			0.147077			0.0735384	+	0.997292i	$0.476571\pi$
$548$		0			0
$549$	−37.5285	−	6.24146i	−1.60168	−	0.266379i
$550$		0			0
$551$	−2.05680			−0.0876225
$552$		0			0
$553$	−9.26645			−0.394049
$554$		0			0
$555$	−8.67399	−	10.2357i	−0.368190	−	0.434482i
$556$		0			0
$557$	17.7505			0.752112			0.376056	−	0.926597i	$-0.377280\pi$
$557$	17.7505			0.752112			0.376056	+	0.926597i	$0.377280\pi$
$558$		0			0
$559$			7.81235i			0.330427i
$560$		0			0
$561$	34.2040	−	28.9853i	1.44409	−	1.22376i
$562$		0			0
$563$	−16.1525			−0.680747			−0.340374	−	0.940290i	$-0.610554\pi$
$563$	−16.1525			−0.680747			−0.340374	+	0.940290i	$0.610554\pi$
$564$		0			0
$565$	−2.45864			−0.103436
$566$		0			0
$567$	5.39135	−	15.7602i	0.226415	−	0.661865i
$568$		0			0
$569$	−19.5974			−0.821564			−0.410782	−	0.911734i	$-0.634744\pi$
$569$	−19.5974			−0.821564			−0.410782	+	0.911734i	$0.634744\pi$
$570$		0			0
$571$			34.2808i			1.43460i	0.696762	+	0.717302i	$0.254623\pi$
$571$			34.2808i			1.43460i	−0.696762	+	0.717302i	$0.745377\pi$
$572$		0			0
$573$	−10.2778	+	8.70967i	−0.429363	+	0.363852i
$574$		0			0
$575$	1.25887	+	4.62766i	0.0524985	+	0.192987i
$576$		0			0
$577$	35.3396			1.47121			0.735603	−	0.677413i	$-0.236899\pi$
$577$	35.3396			1.47121			0.735603	+	0.677413i	$0.236899\pi$
$578$		0			0
$579$	7.70842	−	6.53229i	0.320351	−	0.271473i
$580$		0			0
$581$		−	9.28480i		−	0.385198i
$582$		0			0
$583$	12.8060			0.530368
$584$		0			0
$585$	0.773992	−	4.65384i	0.0320006	−	0.192413i
$586$		0			0
$587$		−	21.2965i		−	0.879001i	−0.898242	−	0.439500i	$-0.855156\pi$
$587$		−	21.2965i		−	0.879001i	0.898242	−	0.439500i	$-0.144844\pi$
$588$		0			0
$589$		−	2.90876i		−	0.119853i
$590$		0			0
$591$	−23.1497	−	27.3178i	−0.952251	−	1.12370i
$592$		0			0
$593$		−	2.73683i		−	0.112388i	−0.998420	−	0.0561941i	$-0.982103\pi$
$593$		−	2.73683i		−	0.112388i	0.998420	−	0.0561941i	$-0.0178966\pi$
$594$		0			0
$595$			8.07454i			0.331024i
$596$		0			0
$597$	−1.36895	−	1.61542i	−0.0560273	−	0.0661149i
$598$		0			0
$599$			13.5256i			0.552642i	0.961065	+	0.276321i	$0.0891153\pi$
$599$			13.5256i			0.552642i	−0.961065	+	0.276321i	$0.910885\pi$
$600$		0			0
$601$	−16.6058			−0.677366			−0.338683	−	0.940900i	$-0.609982\pi$
$601$	−16.6058			−0.677366			−0.338683	+	0.940900i	$0.609982\pi$
$602$		0			0
$603$	40.0400	+	6.65916i	1.63055	+	0.271182i
$604$		0			0
$605$	−24.2007			−0.983898
$606$		0			0
$607$	−8.18331			−0.332151			−0.166075	−	0.986113i	$-0.553109\pi$
$607$	−8.18331			−0.332151			−0.166075	+	0.986113i	$0.553109\pi$
$608$		0			0
$609$	−16.8507	+	14.2797i	−0.682825	+	0.578641i
$610$		0			0
$611$			7.44044i			0.301008i
$612$		0			0
$613$			6.47985i			0.261719i	0.991401	+	0.130859i	$0.0417736\pi$
$613$			6.47985i			0.261719i	−0.991401	+	0.130859i	$0.958226\pi$
$614$		0			0
$615$	8.39157	+	9.90246i	0.338381	+	0.399306i
$616$		0			0
$617$	6.81838			0.274498			0.137249	−	0.990537i	$-0.456174\pi$
$617$	6.81838			0.274498			0.137249	+	0.990537i	$0.456174\pi$
$618$		0			0
$619$			35.5394i			1.42845i	0.699917	+	0.714224i	$0.253220\pi$
$619$			35.5394i			1.42845i	−0.699917	+	0.714224i	$0.746780\pi$
$620$		0			0
$621$	17.2939	−	17.9422i	0.693978	−	0.719996i
$622$		0			0
$623$		−	16.5718i		−	0.663936i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−1.98319	−	2.34026i	−0.0792011	−	0.0934611i
$628$		0			0
$629$		−	33.7951i		−	1.34750i
$630$		0			0
$631$			30.8509i			1.22816i	0.789245	+	0.614078i	$0.210472\pi$
$631$			30.8509i			1.22816i	−0.789245	+	0.614078i	$0.789528\pi$
$632$		0			0
$633$	−2.65673	+	2.25137i	−0.105596	+	0.0894841i
$634$		0			0
$635$	−7.80700			−0.309811
$636$		0			0
$637$	−5.62153			−0.222733
$638$		0			0
$639$	3.25338	+	0.541078i	0.128702	+	0.0214047i
$640$		0			0
$641$	31.9128			1.26048			0.630241	−	0.776400i	$-0.282956\pi$
$641$	31.9128			1.26048			0.630241	+	0.776400i	$0.282956\pi$
$642$		0			0
$643$			16.3951i			0.646559i	0.946304	+	0.323279i	$0.104785\pi$
$643$			16.3951i			0.646559i	−0.946304	+	0.323279i	$0.895215\pi$
$644$		0			0
$645$	5.56289	+	6.56448i	0.219039	+	0.258476i
$646$		0			0
$647$			17.8559i			0.701989i	0.936378	+	0.350994i	$0.114156\pi$
$647$			17.8559i			0.701989i	−0.936378	+	0.350994i	$0.885844\pi$
$648$		0			0
$649$			19.4328i			0.762803i
$650$		0			0
$651$	−20.1946	−	23.8306i	−0.791487	−	0.933993i
$652$		0			0
$653$		−	11.7658i		−	0.460431i	−0.973140	−	0.230216i	$-0.926057\pi$
$653$		−	11.7658i		−	0.460431i	0.973140	−	0.230216i	$-0.0739431\pi$
$654$		0			0
$655$		−	14.4194i		−	0.563412i
$656$		0			0
$657$	−4.90320	+	29.4818i	−0.191292	+	1.15020i
$658$		0			0
$659$	−42.6370			−1.66090			−0.830451	−	0.557091i	$-0.811917\pi$
$659$	−42.6370			−1.66090			−0.830451	+	0.557091i	$0.811917\pi$
$660$		0			0
$661$			44.8493i			1.74443i	0.489119	+	0.872217i	$0.337318\pi$
$661$			44.8493i			1.74443i	−0.489119	+	0.872217i	$0.662682\pi$
$662$		0			0
$663$	9.06601	−	7.68274i	0.352095	−	0.298373i
$664$		0			0
$665$	0.552466			0.0214237
$666$		0			0
$667$	−31.8858	+	8.67393i	−1.23462	+	0.335856i
$668$		0			0
$669$	16.4743	−	13.9607i	0.636932	−	0.539751i
$670$		0			0
$671$			75.2384i			2.90455i
$672$		0			0
$673$	−7.57903			−0.292150			−0.146075	−	0.989273i	$-0.546664\pi$
$673$	−7.57903			−0.292150			−0.146075	+	0.989273i	$0.546664\pi$
$674$		0			0
$675$	−2.66347	−	4.46161i	−0.102517	−	0.171727i
$676$		0			0
$677$	−37.5395			−1.44276			−0.721381	−	0.692539i	$-0.756492\pi$
$677$	−37.5395			−1.44276			−0.721381	+	0.692539i	$0.756492\pi$
$678$		0			0
$679$	20.7495			0.796292
$680$		0			0
$681$	−32.2636	+	27.3409i	−1.23634	+	1.04771i
$682$		0			0
$683$		−	25.9774i		−	0.993996i	−0.867752	−	0.496998i	$-0.834436\pi$
$683$		−	25.9774i		−	0.993996i	0.867752	−	0.496998i	$-0.165564\pi$
$684$		0			0
$685$	−21.3155			−0.814424
$686$		0			0
$687$	−32.9477	−	38.8799i	−1.25703	−	1.48336i
$688$		0			0
$689$	3.39431			0.129313
$690$		0			0
$691$	42.8858			1.63145			0.815726	−	0.578439i	$-0.196338\pi$
$691$	42.8858			1.63145			0.815726	+	0.578439i	$0.196338\pi$
$692$		0			0
$693$	−32.4954	−	5.40439i	−1.23440	−	0.205296i
$694$		0			0
$695$	−6.99913			−0.265492
$696$		0			0
$697$			32.6948i			1.23840i
$698$		0			0
$699$	−2.76209	−	3.25939i	−0.104472	−	0.123282i
$700$		0			0
$701$	−41.2754			−1.55895			−0.779474	−	0.626434i	$-0.784514\pi$
$701$	−41.2754			−1.55895			−0.779474	+	0.626434i	$0.784514\pi$
$702$		0			0
$703$	−2.31229			−0.0872095
$704$		0			0
$705$	5.29806	+	6.25197i	0.199537	+	0.235463i
$706$		0			0
$707$	−24.6522			−0.927140
$708$		0			0
$709$			26.3547i			0.989771i	0.868958	+	0.494885i	$0.164790\pi$
$709$			26.3547i			0.989771i	−0.868958	+	0.494885i	$0.835210\pi$
$710$		0			0
$711$	14.8170	+	2.46425i	0.555681	+	0.0924168i
$712$		0			0
$713$	−12.2668	−	45.0934i	−0.459397	−	1.68876i
$714$		0			0
$715$	−9.33019			−0.348929
$716$		0			0
$717$	−9.00441	−	10.6256i	−0.336276	−	0.396822i
$718$		0			0
$719$			5.86671i			0.218791i	0.993998	+	0.109396i	$0.0348916\pi$
$719$			5.86671i			0.218791i	−0.993998	+	0.109396i	$0.965108\pi$
$720$		0			0
$721$	4.95063			0.184371
$722$		0			0
$723$	−25.6422	−	30.2591i	−0.953645	−	1.12535i
$724$		0			0
$725$			6.89025i			0.255898i
$726$		0			0
$727$		−	32.5270i		−	1.20636i	−0.797605	−	0.603181i	$-0.793900\pi$
$727$		−	32.5270i		−	1.20636i	0.797605	−	0.603181i	$-0.206100\pi$
$728$		0			0
$729$	−12.8119	+	23.7667i	−0.474515	+	0.880248i
$730$		0			0
$731$			21.6738i			0.801635i
$732$		0			0
$733$			4.37557i			0.161615i	0.996730	+	0.0808077i	$0.0257500\pi$
$733$			4.37557i			0.161615i	−0.996730	+	0.0808077i	$0.974250\pi$
$734$		0			0
$735$	−4.72359	+	4.00288i	−0.174232	+	0.147649i
$736$		0			0
$737$		−	80.2737i		−	2.95692i
$738$		0			0
$739$	−30.0183			−1.10424			−0.552121	−	0.833764i	$-0.686181\pi$
$739$	−30.0183			−1.10424			−0.552121	+	0.833764i	$0.686181\pi$
$740$		0			0
$741$	−0.525659	−	0.620303i	−0.0193106	−	0.0227874i
$742$		0			0
$743$	−26.8252			−0.984121			−0.492061	−	0.870561i	$-0.663756\pi$
$743$	−26.8252			−0.984121			−0.492061	+	0.870561i	$0.663756\pi$
$744$		0			0
$745$	−0.124112			−0.00454712
$746$		0			0
$747$	−2.46913	+	14.8463i	−0.0903409	+	0.543199i
$748$		0			0
$749$			5.87243i			0.214574i
$750$		0			0
$751$		−	19.7339i		−	0.720100i	−0.932933	−	0.360050i	$-0.882760\pi$
$751$		−	19.7339i		−	0.720100i	0.932933	−	0.360050i	$-0.117240\pi$
$752$		0			0
$753$	−9.62604	+	8.15732i	−0.350792	+	0.297269i
$754$		0			0
$755$	11.3148			0.411789
$756$		0			0
$757$			25.0898i			0.911905i	0.890004	+	0.455953i	$0.150701\pi$
$757$			25.0898i			0.911905i	−0.890004	+	0.455953i	$0.849299\pi$
$758$		0			0
$759$	−40.6141	−	27.9167i	−1.47420	−	1.01331i
$760$		0			0
$761$		−	8.95788i		−	0.324723i	−0.986731	−	0.162361i	$-0.948089\pi$
$761$		−	8.95788i		−	0.324723i	0.986731	−	0.162361i	$-0.0519111\pi$
$762$		0			0
$763$	−7.29825			−0.264214
$764$		0			0
$765$	2.14729	−	12.9111i	0.0776353	−	0.466803i
$766$		0			0
$767$			5.15079i			0.185984i
$768$		0			0
$769$		−	46.5681i		−	1.67929i	−0.543137	−	0.839644i	$-0.682764\pi$
$769$		−	46.5681i		−	1.67929i	0.543137	−	0.839644i	$-0.317236\pi$
$770$		0			0
$771$	−0.203639	−	0.240304i	−0.00733387	−	0.00865432i
$772$		0			0
$773$	−33.1150			−1.19106			−0.595532	−	0.803332i	$-0.703059\pi$
$773$	−33.1150			−1.19106			−0.595532	+	0.803332i	$0.703059\pi$
$774$		0			0
$775$	−9.74432			−0.350026
$776$		0			0
$777$	−18.9438	+	16.0534i	−0.679607	+	0.575914i
$778$		0			0
$779$	2.23700			0.0801488
$780$		0			0
$781$		−	6.52249i		−	0.233393i
$782$		0			0
$783$	30.7416	−	18.3520i	1.09862	−	0.655845i
$784$		0			0
$785$			16.4553i			0.587316i
$786$		0			0
$787$			25.2040i			0.898427i	0.893424	+	0.449213i	$0.148296\pi$
$787$			25.2040i			0.898427i	−0.893424	+	0.449213i	$0.851704\pi$
$788$		0			0
$789$	16.7844	−	14.2235i	0.597540	−	0.506369i
$790$		0			0
$791$			4.55034i			0.161791i
$792$		0			0
$793$			19.9425i			0.708178i
$794$		0			0
$795$	2.85213	−	2.41696i	0.101155	−	0.0857208i
$796$		0			0
$797$	−9.49657			−0.336386			−0.168193	−	0.985754i	$-0.553793\pi$
$797$	−9.49657			−0.336386			−0.168193	+	0.985754i	$0.553793\pi$
$798$		0			0
$799$			20.6420i			0.730262i
$800$		0			0
$801$	−4.40700	+	26.4983i	−0.155714	+	0.936270i
$802$		0			0
$803$	59.1063			2.08582
$804$		0			0
$805$	8.56468	−	2.32986i	0.301865	−	0.0821169i
$806$		0			0
$807$	33.4361	+	39.4562i	1.17701	+	1.38892i
$808$		0			0
$809$		−	22.2991i		−	0.783994i	−0.919966	−	0.391997i	$-0.871784\pi$
$809$		−	22.2991i		−	0.783994i	0.919966	−	0.391997i	$-0.128216\pi$
$810$		0			0
$811$	−29.6511			−1.04119			−0.520596	−	0.853803i	$-0.674290\pi$
$811$	−29.6511			−1.04119			−0.520596	+	0.853803i	$0.674290\pi$
$812$		0			0
$813$	−13.3510	+	11.3140i	−0.468240	+	0.396798i
$814$		0			0
$815$	20.3222			0.711856
$816$		0			0
$817$	1.48294			0.0518815
$818$		0			0
$819$	−8.61312	−	1.43247i	−0.300967	−	0.0500546i
$820$		0			0
$821$		−	31.5677i		−	1.10172i	−0.834598	−	0.550860i	$-0.814300\pi$
$821$		−	31.5677i		−	1.10172i	0.834598	−	0.550860i	$-0.185700\pi$
$822$		0			0
$823$	−56.4408			−1.96740			−0.983702	−	0.179808i	$-0.942452\pi$
$823$	−56.4408			−1.96740			−0.983702	+	0.179808i	$0.942452\pi$
$824$		0			0
$825$	−7.83987	+	6.64369i	−0.272949	+	0.231303i
$826$		0			0
$827$	−9.20307			−0.320022			−0.160011	−	0.987115i	$-0.551153\pi$
$827$	−9.20307			−0.320022			−0.160011	+	0.987115i	$0.551153\pi$
$828$		0			0
$829$	−4.92979			−0.171219			−0.0856093	−	0.996329i	$-0.527284\pi$
$829$	−4.92979			−0.171219			−0.0856093	+	0.996329i	$0.527284\pi$
$830$		0			0
$831$	24.4312	−	20.7035i	0.847508	−	0.718198i
$832$		0			0
$833$	−15.5958			−0.540362
$834$		0			0
$835$			20.1479i			0.697247i
$836$		0			0
$837$	25.9537	+	43.4753i	0.897090	+	1.50273i
$838$		0			0
$839$	46.2990			1.59842			0.799210	−	0.601051i	$-0.205251\pi$
$839$	46.2990			1.59842			0.799210	+	0.601051i	$0.205251\pi$
$840$		0			0
$841$	−18.4756			−0.637089
$842$		0			0
$843$	−3.38738	+	2.87055i	−0.116668	+	0.0988669i
$844$		0			0
$845$	10.5270			0.362139
$846$		0			0
$847$			44.7896i			1.53899i
$848$		0			0
$849$	20.3828	+	24.0527i	0.699536	+	0.825486i
$850$		0			0
$851$	−35.8465	+	9.75139i	−1.22880	+	0.334273i
$852$		0			0
$853$	−17.5885			−0.602220			−0.301110	−	0.953589i	$-0.597357\pi$
$853$	−17.5885			−0.602220			−0.301110	+	0.953589i	$0.597357\pi$
$854$		0			0
$855$	−0.883390	−	0.146919i	−0.0302113	−	0.00502452i
$856$		0			0
$857$		−	32.4538i		−	1.10860i	−0.832317	−	0.554300i	$-0.812986\pi$
$857$		−	32.4538i		−	1.10860i	0.832317	−	0.554300i	$-0.187014\pi$
$858$		0			0
$859$	21.2404			0.724712			0.362356	−	0.932040i	$-0.381972\pi$
$859$	21.2404			0.724712			0.362356	+	0.932040i	$0.381972\pi$
$860$		0			0
$861$	18.3270	−	15.5308i	0.624584	−	0.529287i
$862$		0			0
$863$			38.5182i			1.31117i	0.755120	+	0.655587i	$0.227579\pi$
$863$			38.5182i			1.31117i	−0.755120	+	0.655587i	$0.772421\pi$
$864$		0			0
$865$		−	8.70765i		−	0.296069i
$866$		0			0
$867$	2.68809	−	2.27795i	0.0912922	−	0.0773631i
$868$		0			0
$869$		−	29.7057i		−	1.00770i
$870$		0			0
$871$		−	21.2771i		−	0.720947i
$872$		0			0
$873$	−33.1783	−	5.51797i	−1.12292	−	0.186755i
$874$		0			0
$875$		−	1.85076i		−	0.0625670i
$876$		0			0
$877$	−26.7942			−0.904775			−0.452388	−	0.891821i	$-0.649428\pi$
$877$	−26.7942			−0.904775			−0.452388	+	0.891821i	$0.649428\pi$
$878$		0			0
$879$	25.7386	−	21.8115i	0.868143	−	0.735684i
$880$		0			0
$881$	19.6957			0.663566			0.331783	−	0.943356i	$-0.392350\pi$
$881$	19.6957			0.663566			0.331783	+	0.943356i	$0.392350\pi$
$882$		0			0
$883$	21.2358			0.714642			0.357321	−	0.933982i	$-0.383690\pi$
$883$	21.2358			0.714642			0.357321	+	0.933982i	$0.383690\pi$
$884$		0			0
$885$	3.66769	+	4.32805i	0.123288	+	0.145486i
$886$		0			0
$887$		−	26.6361i		−	0.894352i	−0.894446	−	0.447176i	$-0.852430\pi$
$887$		−	26.6361i		−	0.894352i	0.894446	−	0.447176i	$-0.147570\pi$
$888$		0			0
$889$			14.4489i			0.484599i
$890$		0			0
$891$	50.5227	+	17.2832i	1.69258	+	0.579009i
$892$		0			0
$893$	1.41234			0.0472622
$894$		0			0
$895$		−	21.9724i		−	0.734457i
$896$		0			0
$897$	−10.7650	−	7.39952i	−0.359434	−	0.247063i
$898$		0			0
$899$		−	67.1408i		−	2.23927i
$900$		0			0
$901$	9.41682			0.313720
$902$		0			0
$903$	12.1493	−	10.2956i	0.404302	−	0.342615i
$904$		0			0
$905$		−	6.57440i		−	0.218541i
$906$		0			0
$907$			19.5210i			0.648184i	0.946026	+	0.324092i	$0.105059\pi$
$907$			19.5210i			0.648184i	−0.946026	+	0.324092i	$0.894941\pi$
$908$		0			0
$909$	39.4187	+	6.55582i	1.30743	+	0.217443i
$910$		0			0
$911$	34.2098			1.13342			0.566710	−	0.823917i	$-0.308216\pi$
$911$	34.2098			1.13342			0.566710	+	0.823917i	$0.308216\pi$
$912$		0			0
$913$	29.7645			0.985062
$914$		0			0
$915$	14.2003	+	16.7570i	0.469447	+	0.553971i
$916$		0			0
$917$	−26.6868			−0.881276
$918$		0			0
$919$			42.8057i			1.41203i	0.708196	+	0.706016i	$0.249509\pi$
$919$			42.8057i			1.41203i	−0.708196	+	0.706016i	$0.750491\pi$
$920$		0			0
$921$	−11.2800	+	9.55896i	−0.371690	+	0.314978i
$922$		0			0
$923$		−	1.72883i		−	0.0569052i
$924$		0			0
$925$			7.74614i			0.254692i
$926$		0			0
$927$	−7.91603	−	1.31654i	−0.259997	−	0.0432407i
$928$		0			0
$929$			32.8154i			1.07664i	0.842741	+	0.538319i	$0.180940\pi$
$929$			32.8154i			1.07664i	−0.842741	+	0.538319i	$0.819060\pi$
$930$		0			0
$931$			1.06708i			0.0349720i
$932$		0			0
$933$	17.0160	+	20.0797i	0.557078	+	0.657379i
$934$		0			0
$935$	−25.8847			−0.846522
$936$		0			0
$937$			33.0144i			1.07853i	0.842135	+	0.539267i	$0.181299\pi$
$937$			33.0144i			1.07853i	−0.842135	+	0.539267i	$0.818701\pi$
$938$		0			0
$939$	−37.7342	−	44.5282i	−1.23141	−	1.45312i
$940$		0			0
$941$	−29.2954			−0.955003			−0.477502	−	0.878631i	$-0.658458\pi$
$941$	−29.2954			−0.955003			−0.477502	+	0.878631i	$0.658458\pi$
$942$		0			0
$943$	34.6794	−	9.43389i	1.12932	−	0.307210i
$944$		0			0
$945$	−8.25735	+	4.92943i	−0.268612	+	0.160354i
$946$		0			0
$947$			9.16441i			0.297803i	0.988852	+	0.148902i	$0.0475738\pi$
$947$			9.16441i			0.297803i	−0.988852	+	0.148902i	$0.952426\pi$
$948$		0			0
$949$	15.6665			0.508557
$950$		0			0
$951$	−12.7498	−	15.0454i	−0.413442	−	0.487881i
$952$		0			0
$953$	55.1358			1.78602			0.893011	−	0.450034i	$-0.148588\pi$
$953$	55.1358			1.78602			0.893011	+	0.450034i	$0.148588\pi$
$954$		0			0
$955$	7.77801			0.251691
$956$		0			0
$957$	−45.7767	−	54.0187i	−1.47975	−	1.74618i
$958$		0			0
$959$			39.4499i			1.27390i
$960$		0			0
$961$	63.9518			2.06296
$962$		0			0
$963$	1.56167	−	9.38999i	0.0503242	−	0.302588i
$964$		0			0
$965$	−5.83354			−0.187788
$966$		0			0
$967$	−37.1029			−1.19315			−0.596574	−	0.802558i	$-0.703472\pi$
$967$	−37.1029			−1.19315			−0.596574	+	0.802558i	$0.703472\pi$
$968$		0			0
$969$	−1.45834	−	1.72091i	−0.0468485	−	0.0552835i
$970$		0			0
$971$	49.2063			1.57911			0.789553	−	0.613683i	$-0.210313\pi$
$971$	49.2063			1.57911			0.789553	+	0.613683i	$0.210313\pi$
$972$		0			0
$973$			12.9537i			0.415276i
$974$		0			0
$975$	−2.07801	+	1.76095i	−0.0665496	+	0.0563957i
$976$		0			0
$977$	16.4555			0.526459			0.263230	−	0.964733i	$-0.415212\pi$
$977$	16.4555			0.526459			0.263230	+	0.964733i	$0.415212\pi$
$978$		0			0
$979$	53.1247			1.69787
$980$		0			0
$981$	11.6699	+	1.94084i	0.372590	+	0.0619664i
$982$		0			0
$983$	30.0293			0.957786			0.478893	−	0.877873i	$-0.341038\pi$
$983$	30.0293			0.957786			0.478893	+	0.877873i	$0.341038\pi$
$984$		0			0
$985$			20.6734i			0.658709i
$986$		0			0
$987$	11.5709	−	9.80543i	0.368305	−	0.312110i
$988$		0			0
$989$	22.9895	−	6.25386i	0.731022	−	0.198861i
$990$		0			0
$991$	−41.5047			−1.31844			−0.659220	−	0.751950i	$-0.729113\pi$
$991$	−41.5047			−1.31844			−0.659220	+	0.751950i	$0.729113\pi$
$992$		0			0
$993$	−26.4674	+	22.4291i	−0.839918	+	0.711766i
$994$		0			0
$995$			1.22251i			0.0387563i
$996$		0			0
$997$	−32.5391			−1.03052			−0.515262	−	0.857032i	$-0.672306\pi$
$997$	−32.5391			−1.03052			−0.515262	+	0.857032i	$0.672306\pi$
$998$		0			0
$999$	34.5603	−	20.6316i	1.09344	−	0.652755i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.i.a.1241.13	✓	16
3.2	odd	2		1380.2.i.b.1241.14	yes	16
23.22	odd	2		1380.2.i.b.1241.13	yes	16
69.68	even	2	inner	1380.2.i.a.1241.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.i.a.1241.13	✓	16	1.1	even	1	trivial
1380.2.i.a.1241.14	yes	16	69.68	even	2	inner
1380.2.i.b.1241.13	yes	16	23.22	odd	2
1380.2.i.b.1241.14	yes	16	3.2	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)

\(f(q)\)	\(=\)	\(q+(1.32140 - 1.11978i) q^{3} -1.00000 q^{5} +1.85076i q^{7} +(0.492177 - 2.95935i) q^{9} +O(q^{10})\)	\(q+(1.32140 - 1.11978i) q^{3} -1.00000 q^{5} +1.85076i q^{7} +(0.492177 - 2.95935i) q^{9} -5.93302 q^{11} -1.57259 q^{13} +(-1.32140 + 1.11978i) q^{15} -4.36283 q^{17} +0.298508i q^{19} +(2.07244 + 2.44558i) q^{21} +(1.25887 + 4.62766i) q^{23} +1.00000 q^{25} +(-2.66347 - 4.46161i) q^{27} +6.89025i q^{29} -9.74432 q^{31} +(-7.83987 + 6.64369i) q^{33} -1.85076i q^{35} +7.74614i q^{37} +(-2.07801 + 1.76095i) q^{39} -7.49393i q^{41} -4.96783i q^{43} +(-0.492177 + 2.95935i) q^{45} -4.73134i q^{47} +3.57470 q^{49} +(-5.76503 + 4.88542i) q^{51} -2.15842 q^{53} +5.93302 q^{55} +(0.334264 + 0.394447i) q^{57} -3.27536i q^{59} -12.6813i q^{61} +(5.47704 + 0.910901i) q^{63} +1.57259 q^{65} +13.5300i q^{67} +(6.84544 + 4.70532i) q^{69} +1.09936i q^{71} -9.96226 q^{73} +(1.32140 - 1.11978i) q^{75} -10.9806i q^{77} +5.00684i q^{79} +(-8.51552 - 2.91305i) q^{81} -5.01676 q^{83} +4.36283 q^{85} +(7.71558 + 9.10476i) q^{87} -8.95408 q^{89} -2.91048i q^{91} +(-12.8761 + 10.9115i) q^{93} -0.298508i q^{95} -11.2114i q^{97} +(-2.92010 + 17.5579i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) \) 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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