LMFDB - Embedded newform 1380.2.i.a.1241.9

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			1241.9
Root			$-0.208374 - 1.71947i$ of defining polynomial
Character	$\chi$	$=$	1380.1241
Dual form			1380.2.i.a.1241.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.208374 - 1.71947i) q^{3} -1.00000 q^{5} +0.187545i q^{7} +(-2.91316 - 0.716587i) q^{9} +O(q^{10})$	\(q+(0.208374 - 1.71947i) q^{3} -1.00000 q^{5} +0.187545i q^{7} +(-2.91316 - 0.716587i) q^{9} +4.36350 q^{11} +6.75087 q^{13} +(-0.208374 + 1.71947i) q^{15} -0.956365 q^{17} +5.93504i q^{19} +(0.322479 + 0.0390797i) q^{21} +(4.02726 - 2.60406i) q^{23} +1.00000 q^{25} +(-1.83918 + 4.85978i) q^{27} -4.44484i q^{29} -3.97602 q^{31} +(0.909242 - 7.50292i) q^{33} -0.187545i q^{35} +9.51993i q^{37} +(1.40671 - 11.6079i) q^{39} -8.47654i q^{41} -7.63510i q^{43} +(2.91316 + 0.716587i) q^{45} -7.92127i q^{47} +6.96483 q^{49} +(-0.199282 + 1.64444i) q^{51} +6.24442 q^{53} -4.36350 q^{55} +(10.2051 + 1.23671i) q^{57} +0.525495i q^{59} -2.02279i q^{61} +(0.134393 - 0.546350i) q^{63} -6.75087 q^{65} -7.87613i q^{67} +(-3.63843 - 7.46738i) q^{69} +2.38671i q^{71} -4.28743 q^{73} +(0.208374 - 1.71947i) q^{75} +0.818355i q^{77} -13.4888i q^{79} +(7.97301 + 4.17507i) q^{81} +4.08526 q^{83} +0.956365 q^{85} +(-7.64278 - 0.926191i) q^{87} +1.73121 q^{89} +1.26610i q^{91} +(-0.828501 + 6.83665i) q^{93} -5.93504i q^{95} +12.4359i q^{97} +(-12.7116 - 3.12683i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.208374	−	1.71947i	0.120305	−	0.992737i
$3$	0.208374	−	1.71947i	0.120305	−	0.992737i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			0.187545i			0.0708855i	0.999372	+	0.0354428i	$0.0112841\pi$
$7$			0.187545i			0.0708855i	−0.999372	+	0.0354428i	$0.988716\pi$
$8$		0			0
$9$	−2.91316	−	0.716587i	−0.971053	−	0.238862i
$10$		0			0
$11$	4.36350			1.31565			0.657823	−	0.753173i	$-0.271478\pi$
$11$	4.36350			1.31565			0.657823	+	0.753173i	$0.271478\pi$
$12$		0			0
$13$	6.75087			1.87235			0.936177	−	0.351528i	$-0.114338\pi$
$13$	6.75087			1.87235			0.936177	+	0.351528i	$0.114338\pi$
$14$		0			0
$15$	−0.208374	+	1.71947i	−0.0538020	+	0.443965i
$16$		0			0
$17$	−0.956365			−0.231953			−0.115976	−	0.993252i	$-0.537000\pi$
$17$	−0.956365			−0.231953			−0.115976	+	0.993252i	$0.537000\pi$
$18$		0			0
$19$			5.93504i			1.36159i	0.732473	+	0.680796i	$0.238366\pi$
$19$			5.93504i			1.36159i	−0.732473	+	0.680796i	$0.761634\pi$
$20$		0			0
$21$	0.322479	+	0.0390797i	0.0703707	+	0.00852788i
$22$		0			0
$23$	4.02726	−	2.60406i	0.839743	−	0.542984i
$23$	4.02726	−	2.60406i	0.839743	−	0.542984i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.83918	+	4.85978i	−0.353950	+	0.935264i
$28$		0			0
$29$		−	4.44484i		−	0.825387i	−0.910870	−	0.412693i	$-0.864588\pi$
$29$		−	4.44484i		−	0.825387i	0.910870	−	0.412693i	$-0.135412\pi$
$30$		0			0
$31$	−3.97602			−0.714115			−0.357057	−	0.934082i	$-0.616220\pi$
$31$	−3.97602			−0.714115			−0.357057	+	0.934082i	$0.616220\pi$
$32$		0			0
$33$	0.909242	−	7.50292i	0.158279	−	1.30609i
$34$		0			0
$35$		−	0.187545i		−	0.0317010i
$36$		0			0
$37$			9.51993i			1.56507i	0.622609	+	0.782533i	$0.286073\pi$
$37$			9.51993i			1.56507i	−0.622609	+	0.782533i	$0.713927\pi$
$38$		0			0
$39$	1.40671	−	11.6079i	0.225254	−	1.85876i
$40$		0			0
$41$		−	8.47654i		−	1.32381i	−0.749586	−	0.661907i	$-0.769748\pi$
$41$		−	8.47654i		−	1.32381i	0.749586	−	0.661907i	$-0.230252\pi$
$42$		0			0
$43$		−	7.63510i		−	1.16434i	−0.813066	−	0.582171i	$-0.802203\pi$
$43$		−	7.63510i		−	1.16434i	0.813066	−	0.582171i	$-0.197797\pi$
$44$		0			0
$45$	2.91316	+	0.716587i	0.434268	+	0.106823i
$46$		0			0
$47$		−	7.92127i		−	1.15544i	−0.816236	−	0.577718i	$-0.803943\pi$
$47$		−	7.92127i		−	1.15544i	0.816236	−	0.577718i	$-0.196057\pi$
$48$		0			0
$49$	6.96483			0.994975
$50$		0			0
$51$	−0.199282	+	1.64444i	−0.0279051	+	0.230268i
$52$		0			0
$53$	6.24442			0.857737			0.428869	−	0.903367i	$-0.358912\pi$
$53$	6.24442			0.857737			0.428869	+	0.903367i	$0.358912\pi$
$54$		0			0
$55$	−4.36350			−0.588375
$56$		0			0
$57$	10.2051	+	1.23671i	1.35170	+	0.163806i
$58$		0			0
$59$			0.525495i			0.0684136i	0.999415	+	0.0342068i	$0.0108905\pi$
$59$			0.525495i			0.0684136i	−0.999415	+	0.0342068i	$0.989110\pi$
$60$		0			0
$61$		−	2.02279i		−	0.258992i	−0.991580	−	0.129496i	$-0.958664\pi$
$61$		−	2.02279i		−	0.258992i	0.991580	−	0.129496i	$-0.0413359\pi$
$62$		0			0
$63$	0.134393	−	0.546350i	0.0169319	−	0.0688336i
$64$		0			0
$65$	−6.75087			−0.837343
$66$		0			0
$67$		−	7.87613i		−	0.962222i	−0.876660	−	0.481111i	$-0.840233\pi$
$67$		−	7.87613i		−	0.962222i	0.876660	−	0.481111i	$-0.159767\pi$
$68$		0			0
$69$	−3.63843	−	7.46738i	−0.438015	−	0.898967i
$70$		0			0
$71$			2.38671i			0.283251i	0.989920	+	0.141625i	$0.0452329\pi$
$71$			2.38671i			0.283251i	−0.989920	+	0.141625i	$0.954767\pi$
$72$		0			0
$73$	−4.28743			−0.501806			−0.250903	−	0.968012i	$-0.580727\pi$
$73$	−4.28743			−0.501806			−0.250903	+	0.968012i	$0.580727\pi$
$74$		0			0
$75$	0.208374	−	1.71947i	0.0240610	−	0.198547i
$76$		0			0
$77$			0.818355i			0.0932603i
$78$		0			0
$79$		−	13.4888i		−	1.51761i	−0.651316	−	0.758807i	$-0.725783\pi$
$79$		−	13.4888i		−	1.51761i	0.651316	−	0.758807i	$-0.274217\pi$
$80$		0			0
$81$	7.97301	+	4.17507i	0.885890	+	0.463896i
$82$		0			0
$83$	4.08526			0.448416			0.224208	−	0.974541i	$-0.428021\pi$
$83$	4.08526			0.448416			0.224208	+	0.974541i	$0.428021\pi$
$84$		0			0
$85$	0.956365			0.103732
$86$		0			0
$87$	−7.64278	−	0.926191i	−0.819392	−	0.0992981i
$88$		0			0
$89$	1.73121			0.183508			0.0917542	−	0.995782i	$-0.470753\pi$
$89$	1.73121			0.183508			0.0917542	+	0.995782i	$0.470753\pi$
$90$		0			0
$91$			1.26610i			0.132723i
$92$		0			0
$93$	−0.828501	+	6.83665i	−0.0859116	+	0.708928i
$94$		0			0
$95$		−	5.93504i		−	0.608923i
$96$		0			0
$97$			12.4359i			1.26267i	0.775509	+	0.631337i	$0.217493\pi$
$97$			12.4359i			1.26267i	−0.775509	+	0.631337i	$0.782507\pi$
$98$		0			0
$99$	−12.7116	−	3.12683i	−1.27756	−	0.314258i
$100$		0			0
$101$			0.126407i			0.0125780i	0.999980	+	0.00628900i	$0.00200186\pi$
$101$			0.126407i			0.0125780i	−0.999980	+	0.00628900i	$0.997998\pi$
$102$		0			0
$103$			11.2600i			1.10948i	0.832024	+	0.554740i	$0.187182\pi$
$103$			11.2600i			1.10948i	−0.832024	+	0.554740i	$0.812818\pi$
$104$		0			0
$105$	−0.322479	−	0.0390797i	−0.0314707	−	0.00381378i
$106$		0			0
$107$	−5.76751			−0.557566			−0.278783	−	0.960354i	$-0.589931\pi$
$107$	−5.76751			−0.557566			−0.278783	+	0.960354i	$0.589931\pi$
$108$		0			0
$109$		−	4.83222i		−	0.462842i	−0.972854	−	0.231421i	$-0.925662\pi$
$109$		−	4.83222i		−	0.462842i	0.972854	−	0.231421i	$-0.0743375\pi$
$110$		0			0
$111$	16.3692	+	1.98371i	1.55370	+	0.188285i
$112$		0			0
$113$	−13.3323			−1.25420			−0.627101	−	0.778938i	$-0.715759\pi$
$113$	−13.3323			−1.25420			−0.627101	+	0.778938i	$0.715759\pi$
$114$		0			0
$115$	−4.02726	+	2.60406i	−0.375544	+	0.242830i
$116$		0			0
$117$	−19.6664	−	4.83759i	−1.81816	−	0.447235i
$118$		0			0
$119$		−	0.179362i		−	0.0164421i
$120$		0			0
$121$	8.04016			0.730924
$122$		0			0
$123$	−14.5752	−	1.76629i	−1.31420	−	0.159261i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	3.62198			0.321398			0.160699	−	0.987003i	$-0.448625\pi$
$127$	3.62198			0.321398			0.160699	+	0.987003i	$0.448625\pi$
$128$		0			0
$129$	−13.1283	−	1.59096i	−1.15589	−	0.140076i
$130$		0			0
$131$			2.51312i			0.219573i	0.993955	+	0.109786i	$0.0350166\pi$
$131$			2.51312i			0.219573i	−0.993955	+	0.109786i	$0.964983\pi$
$132$		0			0
$133$	−1.11309			−0.0965172
$134$		0			0
$135$	1.83918	−	4.85978i	0.158291	−	0.418263i
$136$		0			0
$137$	18.4039			1.57235			0.786176	−	0.618003i	$-0.212058\pi$
$137$	18.4039			1.57235			0.786176	+	0.618003i	$0.212058\pi$
$138$		0			0
$139$	1.21774			0.103287			0.0516436	−	0.998666i	$-0.483554\pi$
$139$	1.21774			0.103287			0.0516436	+	0.998666i	$0.483554\pi$
$140$		0			0
$141$	−13.6204	−	1.65059i	−1.14704	−	0.139005i
$142$		0			0
$143$	29.4575			2.46336
$144$		0			0
$145$			4.44484i			0.369124i
$146$		0			0
$147$	1.45129	−	11.9758i	0.119700	−	0.987749i
$148$		0			0
$149$	−16.1340			−1.32175			−0.660876	−	0.750495i	$-0.729815\pi$
$149$	−16.1340			−1.32175			−0.660876	+	0.750495i	$0.729815\pi$
$150$		0			0
$151$	−11.4844			−0.934590			−0.467295	−	0.884102i	$-0.654771\pi$
$151$	−11.4844			−0.934590			−0.467295	+	0.884102i	$0.654771\pi$
$152$		0			0
$153$	2.78605	+	0.685319i	0.225238	+	0.0554048i
$154$		0			0
$155$	3.97602			0.319362
$156$		0			0
$157$			22.3016i			1.77986i	0.456097	+	0.889930i	$0.349247\pi$
$157$			22.3016i			1.77986i	−0.456097	+	0.889930i	$0.650753\pi$
$158$		0			0
$159$	1.30118	−	10.7371i	0.103190	−	0.851507i
$160$		0			0
$161$	0.488380	+	0.755295i	0.0384897	+	0.0595256i
$162$		0			0
$163$	12.2300			0.957928			0.478964	−	0.877834i	$-0.341012\pi$
$163$	12.2300			0.957928			0.478964	+	0.877834i	$0.341012\pi$
$164$		0			0
$165$	−0.909242	+	7.50292i	−0.0707844	+	0.584101i
$166$		0			0
$167$		−	9.12412i		−	0.706046i	−0.935615	−	0.353023i	$-0.885154\pi$
$167$		−	9.12412i		−	0.706046i	0.935615	−	0.353023i	$-0.114846\pi$
$168$		0			0
$169$	32.5743			2.50571
$170$		0			0
$171$	4.25298	−	17.2897i	0.325233	−	1.32218i
$172$		0			0
$173$			12.6174i			0.959284i	0.877464	+	0.479642i	$0.159234\pi$
$173$			12.6174i			0.959284i	−0.877464	+	0.479642i	$0.840766\pi$
$174$		0			0
$175$			0.187545i			0.0141771i
$176$		0			0
$177$	0.903573	+	0.109500i	0.0679167	+	0.00823050i
$178$		0			0
$179$		−	20.2204i		−	1.51134i	−0.654951	−	0.755671i	$-0.727311\pi$
$179$		−	20.2204i		−	1.51134i	0.654951	−	0.755671i	$-0.272689\pi$
$180$		0			0
$181$		−	13.9913i		−	1.03996i	−0.854177	−	0.519982i	$-0.825939\pi$
$181$		−	13.9913i		−	1.03996i	0.854177	−	0.519982i	$-0.174061\pi$
$182$		0			0
$183$	−3.47813	−	0.421497i	−0.257110	−	0.0311580i
$184$		0			0
$185$		−	9.51993i		−	0.699919i
$186$		0			0
$187$	−4.17310			−0.305168
$188$		0			0
$189$	−0.911429	−	0.344930i	−0.0662967	−	0.0250899i
$190$		0			0
$191$	−14.1977			−1.02731			−0.513653	−	0.857998i	$-0.671708\pi$
$191$	−14.1977			−1.02731			−0.513653	+	0.857998i	$0.671708\pi$
$192$		0			0
$193$	13.1913			0.949528			0.474764	−	0.880113i	$-0.342533\pi$
$193$	13.1913			0.949528			0.474764	+	0.880113i	$0.342533\pi$
$194$		0			0
$195$	−1.40671	+	11.6079i	−0.100736	+	0.831261i
$196$		0			0
$197$			21.5576i			1.53592i	0.640499	+	0.767959i	$0.278728\pi$
$197$			21.5576i			1.53592i	−0.640499	+	0.767959i	$0.721272\pi$
$198$		0			0
$199$			1.91585i			0.135811i	0.997692	+	0.0679055i	$0.0216316\pi$
$199$			1.91585i			0.135811i	−0.997692	+	0.0679055i	$0.978368\pi$
$200$		0			0
$201$	−13.5428	−	1.64118i	−0.955233	−	0.115760i
$202$		0			0
$203$	0.833610			0.0585080
$204$		0			0
$205$			8.47654i			0.592027i
$206$		0			0
$207$	−13.5981	+	4.70016i	−0.945134	+	0.326684i
$208$		0			0
$209$			25.8976i			1.79137i
$210$		0			0
$211$	20.3873			1.40352			0.701761	−	0.712412i	$-0.252397\pi$
$211$	20.3873			1.40352			0.701761	+	0.712412i	$0.252397\pi$
$212$		0			0
$213$	4.10389	+	0.497330i	0.281194	+	0.0340765i
$214$		0			0
$215$			7.63510i			0.520710i
$216$		0			0
$217$		−	0.745685i		−	0.0506204i
$218$		0			0
$219$	−0.893390	+	7.37211i	−0.0603697	+	0.498161i
$220$		0			0
$221$	−6.45630			−0.434298
$222$		0			0
$223$	−2.96138			−0.198309			−0.0991543	−	0.995072i	$-0.531614\pi$
$223$	−2.96138			−0.198309			−0.0991543	+	0.995072i	$0.531614\pi$
$224$		0			0
$225$	−2.91316	−	0.716587i	−0.194211	−	0.0477725i
$226$		0			0
$227$	16.6922			1.10790			0.553949	−	0.832551i	$-0.313120\pi$
$227$	16.6922			1.10790			0.553949	+	0.832551i	$0.313120\pi$
$228$		0			0
$229$		−	0.105691i		−	0.00698424i	−0.999994	−	0.00349212i	$-0.998888\pi$
$229$		−	0.105691i		−	0.00698424i	0.999994	−	0.00349212i	$-0.00111158\pi$
$230$		0			0
$231$	1.40714	+	0.170524i	0.0925829	+	0.0112197i
$232$		0			0
$233$			22.0306i			1.44327i	0.692273	+	0.721636i	$0.256610\pi$
$233$			22.0306i			1.44327i	−0.692273	+	0.721636i	$0.743390\pi$
$234$		0			0
$235$			7.92127i			0.516727i
$236$		0			0
$237$	−23.1937	−	2.81073i	−1.50659	−	0.182576i
$238$		0			0
$239$			18.8622i			1.22009i	0.792365	+	0.610047i	$0.208849\pi$
$239$			18.8622i			1.22009i	−0.792365	+	0.610047i	$0.791151\pi$
$240$		0			0
$241$			14.1539i			0.911733i	0.890048	+	0.455867i	$0.150671\pi$
$241$			14.1539i			0.911733i	−0.890048	+	0.455867i	$0.849329\pi$
$242$		0			0
$243$	8.84028	−	12.8394i	0.567104	−	0.823646i
$244$		0			0
$245$	−6.96483			−0.444966
$246$		0			0
$247$			40.0667i			2.54938i
$248$		0			0
$249$	0.851264	−	7.02449i	0.0539466	−	0.445159i
$250$		0			0
$251$	−8.81212			−0.556216			−0.278108	−	0.960550i	$-0.589707\pi$
$251$	−8.81212			−0.556216			−0.278108	+	0.960550i	$0.589707\pi$
$252$		0			0
$253$	17.5730	−	11.3628i	1.10480	−	0.714375i
$254$		0			0
$255$	0.199282	−	1.64444i	0.0124795	−	0.102979i
$256$		0			0
$257$			21.7148i			1.35453i	0.735739	+	0.677265i	$0.236835\pi$
$257$			21.7148i			1.35453i	−0.735739	+	0.677265i	$0.763165\pi$
$258$		0			0
$259$	−1.78542			−0.110941
$260$		0			0
$261$	−3.18512	+	12.9485i	−0.197154	+	0.801494i
$262$		0			0
$263$	−30.3401			−1.87085			−0.935424	−	0.353527i	$-0.884982\pi$
$263$	−30.3401			−1.87085			−0.935424	+	0.353527i	$0.884982\pi$
$264$		0			0
$265$	−6.24442			−0.383592
$266$		0			0
$267$	0.360741	−	2.97677i	0.0220770	−	0.182176i
$268$		0			0
$269$		−	20.9344i		−	1.27639i	−0.769874	−	0.638196i	$-0.779681\pi$
$269$		−	20.9344i		−	1.27639i	0.769874	−	0.638196i	$-0.220319\pi$
$270$		0			0
$271$	20.6077			1.25183			0.625915	−	0.779891i	$-0.284726\pi$
$271$	20.6077			1.25183			0.625915	+	0.779891i	$0.284726\pi$
$272$		0			0
$273$	2.17701	+	0.263822i	0.131759	+	0.0159672i
$274$		0			0
$275$	4.36350			0.263129
$276$		0			0
$277$	−8.09324			−0.486276			−0.243138	−	0.969992i	$-0.578177\pi$
$277$	−8.09324			−0.486276			−0.243138	+	0.969992i	$0.578177\pi$
$278$		0			0
$279$	11.5828	+	2.84917i	0.693444	+	0.170575i
$280$		0			0
$281$	6.04239			0.360459			0.180229	−	0.983625i	$-0.442316\pi$
$281$	6.04239			0.360459			0.180229	+	0.983625i	$0.442316\pi$
$282$		0			0
$283$		−	6.14109i		−	0.365050i	−0.983201	−	0.182525i	$-0.941573\pi$
$283$		−	6.14109i		−	0.365050i	0.983201	−	0.182525i	$-0.0584270\pi$
$284$		0			0
$285$	−10.2051	−	1.23671i	−0.604500	−	0.0732564i
$286$		0			0
$287$	1.58974			0.0938392
$288$		0			0
$289$	−16.0854			−0.946198
$290$		0			0
$291$	21.3832	+	2.59132i	1.25350	+	0.151906i
$292$		0			0
$293$	−19.7278			−1.15251			−0.576255	−	0.817270i	$-0.695486\pi$
$293$	−19.7278			−1.15251			−0.576255	+	0.817270i	$0.695486\pi$
$294$		0			0
$295$		−	0.525495i		−	0.0305955i
$296$		0			0
$297$	−8.02526	+	21.2056i	−0.465673	+	1.23048i
$298$		0			0
$299$	27.1875	−	17.5797i	1.57230	−	1.01666i
$300$		0			0
$301$	1.43193			0.0825350
$302$		0			0
$303$	0.217354	+	0.0263401i	0.0124867	+	0.00151320i
$304$		0			0
$305$			2.02279i			0.115825i
$306$		0			0
$307$	−4.60843			−0.263017			−0.131509	−	0.991315i	$-0.541982\pi$
$307$	−4.60843			−0.263017			−0.131509	+	0.991315i	$0.541982\pi$
$308$		0			0
$309$	19.3612	+	2.34629i	1.10142	+	0.133476i
$310$		0			0
$311$		−	11.8041i		−	0.669349i	−0.942334	−	0.334674i	$-0.891374\pi$
$311$		−	11.8041i		−	0.669349i	0.942334	−	0.334674i	$-0.108626\pi$
$312$		0			0
$313$		−	21.0273i		−	1.18853i	−0.804268	−	0.594266i	$-0.797443\pi$
$313$		−	21.0273i		−	1.18853i	0.804268	−	0.594266i	$-0.202557\pi$
$314$		0			0
$315$	−0.134393	+	0.546350i	−0.00757217	+	0.0307833i
$316$		0			0
$317$			27.3475i			1.53599i	0.640455	+	0.767996i	$0.278746\pi$
$317$			27.3475i			1.53599i	−0.640455	+	0.767996i	$0.721254\pi$
$318$		0			0
$319$		−	19.3951i		−	1.08592i
$320$		0			0
$321$	−1.20180	+	9.91706i	−0.0670779	+	0.553516i
$322$		0			0
$323$		−	5.67607i		−	0.315825i
$324$		0			0
$325$	6.75087			0.374471
$326$		0			0
$327$	−8.30886	−	1.00691i	−0.459481	−	0.0556822i
$328$		0			0
$329$	1.48560			0.0819037
$330$		0			0
$331$	16.1655			0.888534			0.444267	−	0.895894i	$-0.353464\pi$
$331$	16.1655			0.888534			0.444267	+	0.895894i	$0.353464\pi$
$332$		0			0
$333$	6.82186	−	27.7331i	0.373836	−	1.51976i
$334$		0			0
$335$			7.87613i			0.430319i
$336$		0			0
$337$		−	12.4862i		−	0.680168i	−0.940395	−	0.340084i	$-0.889545\pi$
$337$		−	12.4862i		−	0.680168i	0.940395	−	0.340084i	$-0.110455\pi$
$338$		0			0
$339$	−2.77812	+	22.9246i	−0.150887	+	1.24509i
$340$		0			0
$341$	−17.3494			−0.939522
$342$		0			0
$343$			2.61904i			0.141415i
$344$		0			0
$345$	3.63843	+	7.46738i	0.195886	+	0.402030i
$346$		0			0
$347$		−	7.95560i		−	0.427079i	−0.976934	−	0.213539i	$-0.931501\pi$
$347$		−	7.95560i		−	0.427079i	0.976934	−	0.213539i	$-0.0684992\pi$
$348$		0			0
$349$	10.5081			0.562487			0.281244	−	0.959636i	$-0.409253\pi$
$349$	10.5081			0.562487			0.281244	+	0.959636i	$0.409253\pi$
$350$		0			0
$351$	−12.4161	+	32.8077i	−0.662720	+	1.75115i
$352$		0			0
$353$		−	14.0694i		−	0.748839i	−0.927259	−	0.374420i	$-0.877842\pi$
$353$		−	14.0694i		−	0.748839i	0.927259	−	0.374420i	$-0.122158\pi$
$354$		0			0
$355$		−	2.38671i		−	0.126674i
$356$		0			0
$357$	−0.308408	−	0.0373744i	−0.0163227	−	0.00197806i
$358$		0			0
$359$	−31.4414			−1.65941			−0.829707	−	0.558198i	$-0.811493\pi$
$359$	−31.4414			−1.65941			−0.829707	+	0.558198i	$0.811493\pi$
$360$		0			0
$361$	−16.2247			−0.853933
$362$		0			0
$363$	1.67536	−	13.8248i	0.0879338	−	0.725615i
$364$		0			0
$365$	4.28743			0.224414
$366$		0			0
$367$			14.5851i			0.761335i	0.924712	+	0.380668i	$0.124306\pi$
$367$			14.5851i			0.761335i	−0.924712	+	0.380668i	$0.875694\pi$
$368$		0			0
$369$	−6.07418	+	24.6935i	−0.316209	+	1.28549i
$370$		0			0
$371$			1.17111i			0.0608011i
$372$		0			0
$373$		−	2.87471i		−	0.148847i	−0.997227	−	0.0744234i	$-0.976288\pi$
$373$		−	2.87471i		−	0.148847i	0.997227	−	0.0744234i	$-0.0237116\pi$
$374$		0			0
$375$	−0.208374	+	1.71947i	−0.0107604	+	0.0887931i
$376$		0			0
$377$		−	30.0066i		−	1.54542i
$378$		0			0
$379$		−	24.7462i		−	1.27113i	−0.772048	−	0.635564i	$-0.780768\pi$
$379$		−	24.7462i		−	1.27113i	0.772048	−	0.635564i	$-0.219232\pi$
$380$		0			0
$381$	0.754727	−	6.22788i	0.0386658	−	0.319064i
$382$		0			0
$383$	−25.3468			−1.29516			−0.647580	−	0.761997i	$-0.724219\pi$
$383$	−25.3468			−1.29516			−0.647580	+	0.761997i	$0.724219\pi$
$384$		0			0
$385$		−	0.818355i		−	0.0417073i
$386$		0			0
$387$	−5.47122	+	22.2423i	−0.278118	+	1.13064i
$388$		0			0
$389$	1.48056			0.0750676			0.0375338	−	0.999295i	$-0.488050\pi$
$389$	1.48056			0.0750676			0.0375338	+	0.999295i	$0.488050\pi$
$390$		0			0
$391$	−3.85154	+	2.49043i	−0.194781	+	0.125947i
$392$		0			0
$393$	4.32124	+	0.523670i	0.217978	+	0.0264157i
$394$		0			0
$395$			13.4888i			0.678697i
$396$		0			0
$397$	−32.5206			−1.63216			−0.816082	−	0.577937i	$-0.803858\pi$
$397$	−32.5206			−1.63216			−0.816082	+	0.577937i	$0.803858\pi$
$398$		0			0
$399$	−0.231939	+	1.91393i	−0.0116115	+	0.0958162i
$400$		0			0
$401$	−10.9854			−0.548586			−0.274293	−	0.961646i	$-0.588444\pi$
$401$	−10.9854			−0.548586			−0.274293	+	0.961646i	$0.588444\pi$
$402$		0			0
$403$	−26.8416			−1.33708
$404$		0			0
$405$	−7.97301	−	4.17507i	−0.396182	−	0.207461i
$406$		0			0
$407$			41.5402i			2.05907i
$408$		0			0
$409$	13.4281			0.663978			0.331989	−	0.943283i	$-0.392280\pi$
$409$	13.4281			0.663978			0.331989	+	0.943283i	$0.392280\pi$
$410$		0			0
$411$	3.83490	−	31.6450i	0.189162	−	1.56093i
$412$		0			0
$413$	−0.0985542			−0.00484953
$414$		0			0
$415$	−4.08526			−0.200538
$416$		0			0
$417$	0.253746	−	2.09387i	0.0124260	−	0.102537i
$418$		0			0
$419$	9.50684			0.464440			0.232220	−	0.972663i	$-0.425401\pi$
$419$	9.50684			0.464440			0.232220	+	0.972663i	$0.425401\pi$
$420$		0			0
$421$			28.7212i			1.39978i	0.714249	+	0.699892i	$0.246768\pi$
$421$			28.7212i			1.39978i	−0.714249	+	0.699892i	$0.753232\pi$
$422$		0			0
$423$	−5.67628	+	23.0759i	−0.275990	+	1.12199i
$424$		0			0
$425$	−0.956365			−0.0463905
$426$		0			0
$427$	0.379365			0.0183588
$428$		0			0
$429$	6.13818	−	50.6512i	0.296354	−	2.44546i
$430$		0			0
$431$	−24.4182			−1.17619			−0.588093	−	0.808794i	$-0.700121\pi$
$431$	−24.4182			−1.17619			−0.588093	+	0.808794i	$0.700121\pi$
$432$		0			0
$433$			6.86449i			0.329887i	0.986303	+	0.164943i	$0.0527441\pi$
$433$			6.86449i			0.329887i	−0.986303	+	0.164943i	$0.947256\pi$
$434$		0			0
$435$	7.64278	+	0.926191i	0.366443	+	0.0444075i
$436$		0			0
$437$	15.4552	+	23.9020i	0.739323	+	1.14339i
$438$		0			0
$439$	35.7576			1.70662			0.853308	−	0.521407i	$-0.174593\pi$
$439$	35.7576			1.70662			0.853308	+	0.521407i	$0.174593\pi$
$440$		0			0
$441$	−20.2897	−	4.99091i	−0.966174	−	0.237662i
$442$		0			0
$443$			9.12257i			0.433426i	0.976235	+	0.216713i	$0.0695336\pi$
$443$			9.12257i			0.433426i	−0.976235	+	0.216713i	$0.930466\pi$
$444$		0			0
$445$	−1.73121			−0.0820674
$446$		0			0
$447$	−3.36192	+	27.7420i	−0.159013	+	1.31215i
$448$		0			0
$449$			33.7685i			1.59363i	0.604221	+	0.796817i	$0.293484\pi$
$449$			33.7685i			1.59363i	−0.604221	+	0.796817i	$0.706516\pi$
$450$		0			0
$451$		−	36.9874i		−	1.74167i
$452$		0			0
$453$	−2.39306	+	19.7471i	−0.112436	+	0.927802i
$454$		0			0
$455$		−	1.26610i		−	0.0593555i
$456$		0			0
$457$			26.5785i			1.24329i	0.783299	+	0.621645i	$0.213535\pi$
$457$			26.5785i			1.24329i	−0.783299	+	0.621645i	$0.786465\pi$
$458$		0			0
$459$	1.75893	−	4.64772i	0.0820997	−	0.216937i
$460$		0			0
$461$			11.6065i			0.540566i	0.962781	+	0.270283i	$0.0871173\pi$
$461$			11.6065i			0.540566i	−0.962781	+	0.270283i	$0.912883\pi$
$462$		0			0
$463$	3.61792			0.168139			0.0840695	−	0.996460i	$-0.473208\pi$
$463$	3.61792			0.168139			0.0840695	+	0.996460i	$0.473208\pi$
$464$		0			0
$465$	0.828501	−	6.83665i	0.0384208	−	0.317042i
$466$		0			0
$467$	3.03785			0.140575			0.0702874	−	0.997527i	$-0.477608\pi$
$467$	3.03785			0.140575			0.0702874	+	0.997527i	$0.477608\pi$
$468$		0			0
$469$	1.47713			0.0682076
$470$		0			0
$471$	38.3469	+	4.64708i	1.76693	+	0.214126i
$472$		0			0
$473$		−	33.3158i		−	1.53186i
$474$		0			0
$475$			5.93504i			0.272318i
$476$		0			0
$477$	−18.1910	−	4.47467i	−0.832908	−	0.204881i
$478$		0			0
$479$	−0.0908651			−0.00415173			−0.00207587	−	0.999998i	$-0.500661\pi$
$479$	−0.0908651			−0.00415173			−0.00207587	+	0.999998i	$0.500661\pi$
$480$		0			0
$481$			64.2678i			2.93036i
$482$		0			0
$483$	1.40047	−	0.682371i	0.0637238	−	0.0310490i
$484$		0			0
$485$		−	12.4359i		−	0.564685i
$486$		0			0
$487$	−20.2817			−0.919054			−0.459527	−	0.888164i	$-0.651981\pi$
$487$	−20.2817			−0.919054			−0.459527	+	0.888164i	$0.651981\pi$
$488$		0			0
$489$	2.54842	−	21.0291i	0.115244	−	0.950971i
$490$		0			0
$491$		−	27.9168i		−	1.25987i	−0.776648	−	0.629935i	$-0.783082\pi$
$491$		−	27.9168i		−	1.25987i	0.776648	−	0.629935i	$-0.216918\pi$
$492$		0			0
$493$			4.25089i			0.191451i
$494$		0			0
$495$	12.7116	+	3.12683i	0.571343	+	0.140541i
$496$		0			0
$497$	−0.447618			−0.0200784
$498$		0			0
$499$	−26.7631			−1.19808			−0.599041	−	0.800719i	$-0.704451\pi$
$499$	−26.7631			−1.19808			−0.599041	+	0.800719i	$0.704451\pi$
$500$		0			0
$501$	−15.6887	−	1.90123i	−0.700918	−	0.0849408i
$502$		0			0
$503$	11.3660			0.506784			0.253392	−	0.967364i	$-0.418454\pi$
$503$	11.3660			0.506784			0.253392	+	0.967364i	$0.418454\pi$
$504$		0			0
$505$		−	0.126407i		−	0.00562506i
$506$		0			0
$507$	6.78764	−	56.0105i	0.301450	−	2.48751i
$508$		0			0
$509$			26.6388i			1.18074i	0.807131	+	0.590372i	$0.201019\pi$
$509$			26.6388i			1.18074i	−0.807131	+	0.590372i	$0.798981\pi$
$510$		0			0
$511$		−	0.804088i		−	0.0355708i
$512$		0			0
$513$	−28.8430	−	10.9156i	−1.27345	−	0.481936i
$514$		0			0
$515$		−	11.2600i		−	0.496175i
$516$		0			0
$517$		−	34.5645i		−	1.52014i
$518$		0			0
$519$	21.6953	+	2.62915i	0.952317	+	0.115407i
$520$		0			0
$521$	−39.3888			−1.72565			−0.862827	−	0.505499i	$-0.831308\pi$
$521$	−39.3888			−1.72565			−0.862827	+	0.505499i	$0.831308\pi$
$522$		0			0
$523$			39.2737i			1.71732i	0.512545	+	0.858661i	$0.328703\pi$
$523$			39.2737i			1.71732i	−0.512545	+	0.858661i	$0.671297\pi$
$524$		0			0
$525$	0.322479	+	0.0390797i	0.0140741	+	0.00170558i
$526$		0			0
$527$	3.80253			0.165641
$528$		0			0
$529$	9.43773	−	20.9745i	0.410336	−	0.911934i
$530$		0			0
$531$	0.376563	−	1.53085i	0.0163414	−	0.0664333i
$532$		0			0
$533$		−	57.2240i		−	2.47865i
$534$		0			0
$535$	5.76751			0.249351
$536$		0			0
$537$	−34.7683	−	4.21341i	−1.50036	−	0.181822i
$538$		0			0
$539$	30.3910			1.30903
$540$		0			0
$541$	−35.5155			−1.52693			−0.763465	−	0.645849i	$-0.776504\pi$
$541$	−35.5155			−1.52693			−0.763465	+	0.645849i	$0.776504\pi$
$542$		0			0
$543$	−24.0576	−	2.91543i	−1.03241	−	0.125113i
$544$		0			0
$545$			4.83222i			0.206989i
$546$		0			0
$547$	−12.2488			−0.523721			−0.261860	−	0.965106i	$-0.584336\pi$
$547$	−12.2488			−0.523721			−0.261860	+	0.965106i	$0.584336\pi$
$548$		0			0
$549$	−1.44950	+	5.89271i	−0.0618633	+	0.251495i
$550$		0			0
$551$	26.3803			1.12384
$552$		0			0
$553$	2.52977			0.107577
$554$		0			0
$555$	−16.3692	−	1.98371i	−0.694836	−	0.0842038i
$556$		0			0
$557$	40.4280			1.71299			0.856496	−	0.516154i	$-0.172637\pi$
$557$	40.4280			1.71299			0.856496	+	0.516154i	$0.172637\pi$
$558$		0			0
$559$		−	51.5436i		−	2.18006i
$560$		0			0
$561$	−0.869568	+	7.17553i	−0.0367132	+	0.302951i
$562$		0			0
$563$	29.6387			1.24912			0.624561	−	0.780976i	$-0.285278\pi$
$563$	29.6387			1.24912			0.624561	+	0.780976i	$0.285278\pi$
$564$		0			0
$565$	13.3323			0.560896
$566$		0			0
$567$	−0.783015	+	1.49530i	−0.0328835	+	0.0627967i
$568$		0			0
$569$	19.1081			0.801054			0.400527	−	0.916285i	$-0.368827\pi$
$569$	19.1081			0.801054			0.400527	+	0.916285i	$0.368827\pi$
$570$		0			0
$571$		−	36.6457i		−	1.53357i	−0.641902	−	0.766787i	$-0.721854\pi$
$571$		−	36.6457i		−	1.53357i	0.641902	−	0.766787i	$-0.278146\pi$
$572$		0			0
$573$	−2.95843	+	24.4125i	−0.123590	+	1.01985i
$574$		0			0
$575$	4.02726	−	2.60406i	0.167949	−	0.108597i
$576$		0			0
$577$	3.56680			0.148488			0.0742438	−	0.997240i	$-0.476346\pi$
$577$	3.56680			0.148488			0.0742438	+	0.997240i	$0.476346\pi$
$578$		0			0
$579$	2.74872	−	22.6820i	0.114233	−	0.942631i
$580$		0			0
$581$			0.766172i			0.0317862i
$582$		0			0
$583$	27.2475			1.12848
$584$		0			0
$585$	19.6664	+	4.83759i	0.813104	+	0.200010i
$586$		0			0
$587$		−	39.2111i		−	1.61841i	−0.587524	−	0.809207i	$-0.699897\pi$
$587$		−	39.2111i		−	1.61841i	0.587524	−	0.809207i	$-0.300103\pi$
$588$		0			0
$589$		−	23.5979i		−	0.972333i
$590$		0			0
$591$	37.0677	+	4.49206i	1.52476	+	0.184779i
$592$		0			0
$593$			42.6434i			1.75116i	0.483076	+	0.875578i	$0.339519\pi$
$593$			42.6434i			1.75116i	−0.483076	+	0.875578i	$0.660481\pi$
$594$		0			0
$595$			0.179362i			0.00735312i
$596$		0			0
$597$	3.29425	+	0.399214i	0.134825	+	0.0163387i
$598$		0			0
$599$			23.2440i			0.949724i	0.880060	+	0.474862i	$0.157502\pi$
$599$			23.2440i			0.949724i	−0.880060	+	0.474862i	$0.842498\pi$
$600$		0			0
$601$	31.3605			1.27922			0.639610	−	0.768699i	$-0.279096\pi$
$601$	31.3605			1.27922			0.639610	+	0.768699i	$0.279096\pi$
$602$		0			0
$603$	−5.64393	+	22.9444i	−0.229839	+	0.934369i
$604$		0			0
$605$	−8.04016			−0.326879
$606$		0			0
$607$	−41.9418			−1.70237			−0.851183	−	0.524869i	$-0.824114\pi$
$607$	−41.9418			−1.70237			−0.851183	+	0.524869i	$0.824114\pi$
$608$		0			0
$609$	0.173703	−	1.43337i	0.00703880	−	0.0580830i
$610$		0			0
$611$		−	53.4755i		−	2.16339i
$612$		0			0
$613$		−	44.3655i		−	1.79190i	−0.444151	−	0.895952i	$-0.646495\pi$
$613$		−	44.3655i		−	1.79190i	0.444151	−	0.895952i	$-0.353505\pi$
$614$		0			0
$615$	14.5752	+	1.76629i	0.587727	+	0.0712238i
$616$		0			0
$617$	11.8190			0.475816			0.237908	−	0.971288i	$-0.423538\pi$
$617$	11.8190			0.475816			0.237908	+	0.971288i	$0.423538\pi$
$618$		0			0
$619$			43.3150i			1.74098i	0.492188	+	0.870489i	$0.336197\pi$
$619$			43.3150i			1.74098i	−0.492188	+	0.870489i	$0.663803\pi$
$620$		0			0
$621$	5.24830	+	24.3609i	0.210607	+	0.977571i
$622$		0			0
$623$			0.324681i			0.0130081i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	44.5301	+	5.39639i	1.77836	+	0.215511i
$628$		0			0
$629$		−	9.10453i		−	0.363021i
$630$		0			0
$631$			5.77912i			0.230063i	0.993362	+	0.115032i	$0.0366969\pi$
$631$			5.77912i			0.230063i	−0.993362	+	0.115032i	$0.963303\pi$
$632$		0			0
$633$	4.24820	−	35.0554i	0.168851	−	1.39333i
$634$		0			0
$635$	−3.62198			−0.143734
$636$		0			0
$637$	47.0187			1.86295
$638$		0			0
$639$	1.71029	−	6.95288i	0.0676580	−	0.275052i
$640$		0			0
$641$	1.52676			0.0603036			0.0301518	−	0.999545i	$-0.490401\pi$
$641$	1.52676			0.0603036			0.0301518	+	0.999545i	$0.490401\pi$
$642$		0			0
$643$			11.6023i			0.457551i	0.973479	+	0.228776i	$0.0734722\pi$
$643$			11.6023i			0.457551i	−0.973479	+	0.228776i	$0.926528\pi$
$644$		0			0
$645$	13.1283	+	1.59096i	0.516928	+	0.0626440i
$646$		0			0
$647$		−	12.9339i		−	0.508485i	−0.967141	−	0.254242i	$-0.918174\pi$
$647$		−	12.9339i		−	0.508485i	0.967141	−	0.254242i	$-0.0818261\pi$
$648$		0			0
$649$			2.29300i			0.0900081i
$650$		0			0
$651$	−1.28218	−	0.155382i	−0.0502527	−	0.00608989i
$652$		0			0
$653$			3.54433i			0.138700i	0.997592	+	0.0693502i	$0.0220926\pi$
$653$			3.54433i			0.138700i	−0.997592	+	0.0693502i	$0.977907\pi$
$654$		0			0
$655$		−	2.51312i		−	0.0981958i
$656$		0			0
$657$	12.4900	+	3.07232i	0.487280	+	0.119863i
$658$		0			0
$659$	−29.8500			−1.16279			−0.581395	−	0.813622i	$-0.697493\pi$
$659$	−29.8500			−1.16279			−0.581395	+	0.813622i	$0.697493\pi$
$660$		0			0
$661$		−	31.0421i		−	1.20740i	−0.797212	−	0.603700i	$-0.793693\pi$
$661$		−	31.0421i		−	1.20740i	0.797212	−	0.603700i	$-0.206307\pi$
$662$		0			0
$663$	−1.34533	+	11.1014i	−0.0522482	+	0.431143i
$664$		0			0
$665$	1.11309			0.0431638
$666$		0			0
$667$	−11.5746	−	17.9006i	−0.448172	−	0.693112i
$668$		0			0
$669$	−0.617076	+	5.09201i	−0.0238575	+	0.196868i
$670$		0			0
$671$		−	8.82645i		−	0.340741i
$672$		0			0
$673$	9.14264			0.352423			0.176211	−	0.984352i	$-0.443616\pi$
$673$	9.14264			0.352423			0.176211	+	0.984352i	$0.443616\pi$
$674$		0			0
$675$	−1.83918	+	4.85978i	−0.0707900	+	0.187053i
$676$		0			0
$677$	10.0265			0.385350			0.192675	−	0.981263i	$-0.438284\pi$
$677$	10.0265			0.385350			0.192675	+	0.981263i	$0.438284\pi$
$678$		0			0
$679$	−2.33230			−0.0895053
$680$		0			0
$681$	3.47822	−	28.7017i	0.133286	−	1.09985i
$682$		0			0
$683$		−	14.7246i		−	0.563421i	−0.959499	−	0.281710i	$-0.909098\pi$
$683$		−	14.7246i		−	0.563421i	0.959499	−	0.281710i	$-0.0909018\pi$
$684$		0			0
$685$	−18.4039			−0.703177
$686$		0			0
$687$	−0.181732	−	0.0220232i	−0.00693351	−	0.000840239i
$688$		0			0
$689$	42.1553			1.60599
$690$		0			0
$691$	−12.4472			−0.473515			−0.236757	−	0.971569i	$-0.576085\pi$
$691$	−12.4472			−0.473515			−0.236757	+	0.971569i	$0.576085\pi$
$692$		0			0
$693$	0.586423	−	2.38400i	0.0222764	−	0.0905607i
$694$		0			0
$695$	−1.21774			−0.0461915
$696$		0			0
$697$			8.10667i			0.307062i
$698$		0			0
$699$	37.8810	+	4.59061i	1.43279	+	0.173633i
$700$		0			0
$701$	−43.8908			−1.65773			−0.828867	−	0.559446i	$-0.811014\pi$
$701$	−43.8908			−1.65773			−0.828867	+	0.559446i	$0.811014\pi$
$702$		0			0
$703$	−56.5012			−2.13098
$704$		0			0
$705$	13.6204	+	1.65059i	0.512974	+	0.0621648i
$706$		0			0
$707$	−0.0237071			−0.000891599
$708$		0			0
$709$			49.5649i			1.86145i	0.365720	+	0.930725i	$0.380823\pi$
$709$			49.5649i			1.86145i	−0.365720	+	0.930725i	$0.619177\pi$
$710$		0			0
$711$	−9.66593	+	39.2952i	−0.362501	+	1.47368i
$712$		0			0
$713$	−16.0125	+	10.3538i	−0.599673	+	0.387753i
$714$		0			0
$715$	−29.4575			−1.10165
$716$		0			0
$717$	32.4330	+	3.93040i	1.21123	+	0.146783i
$718$		0			0
$719$			8.72405i			0.325352i	0.986680	+	0.162676i	$0.0520125\pi$
$719$			8.72405i			0.325352i	−0.986680	+	0.162676i	$0.947987\pi$
$720$		0			0
$721$	−2.11176			−0.0786461
$722$		0			0
$723$	24.3372	+	2.94931i	0.905111	+	0.109686i
$724$		0			0
$725$		−	4.44484i		−	0.165077i
$726$		0			0
$727$		−	12.5678i		−	0.466113i	−0.972463	−	0.233057i	$-0.925127\pi$
$727$		−	12.5678i		−	0.466113i	0.972463	−	0.233057i	$-0.0748727\pi$
$728$		0			0
$729$	−20.2348	−	17.8760i	−0.749439	−	0.662074i
$730$		0			0
$731$			7.30194i			0.270072i
$732$		0			0
$733$		−	10.5708i		−	0.390440i	−0.980759	−	0.195220i	$-0.937458\pi$
$733$		−	10.5708i		−	0.390440i	0.980759	−	0.195220i	$-0.0625421\pi$
$734$		0			0
$735$	−1.45129	+	11.9758i	−0.0535317	+	0.441735i
$736$		0			0
$737$		−	34.3675i		−	1.26594i
$738$		0			0
$739$	−39.3127			−1.44614			−0.723070	−	0.690774i	$-0.757270\pi$
$739$	−39.3127			−1.44614			−0.723070	+	0.690774i	$0.757270\pi$
$740$		0			0
$741$	68.8935	+	8.34887i	2.53087	+	0.306704i
$742$		0			0
$743$	−5.69891			−0.209073			−0.104536	−	0.994521i	$-0.533336\pi$
$743$	−5.69891			−0.209073			−0.104536	+	0.994521i	$0.533336\pi$
$744$		0			0
$745$	16.1340			0.591105
$746$		0			0
$747$	−11.9010	−	2.92745i	−0.435436	−	0.107110i
$748$		0			0
$749$		−	1.08167i		−	0.0395234i
$750$		0			0
$751$		−	38.2980i		−	1.39751i	−0.715359	−	0.698757i	$-0.753737\pi$
$751$		−	38.2980i		−	1.39751i	0.715359	−	0.698757i	$-0.246263\pi$
$752$		0			0
$753$	−1.83622	+	15.1522i	−0.0669156	+	0.552176i
$754$		0			0
$755$	11.4844			0.417961
$756$		0			0
$757$			13.1087i			0.476444i	0.971211	+	0.238222i	$0.0765645\pi$
$757$			13.1087i			0.476444i	−0.971211	+	0.238222i	$0.923435\pi$
$758$		0			0
$759$	−15.8763	−	32.5840i	−0.576273	−	1.18272i
$760$		0			0
$761$		−	39.6953i		−	1.43895i	−0.694516	−	0.719477i	$-0.744381\pi$
$761$		−	39.6953i		−	1.43895i	0.694516	−	0.719477i	$-0.255619\pi$
$762$		0			0
$763$	0.906260			0.0328088
$764$		0			0
$765$	−2.78605	−	0.685319i	−0.100730	−	0.0247778i
$766$		0			0
$767$			3.54755i			0.128095i
$768$		0			0
$769$			11.0014i			0.396719i	0.980129	+	0.198360i	$0.0635614\pi$
$769$			11.0014i			0.396719i	−0.980129	+	0.198360i	$0.936439\pi$
$770$		0			0
$771$	37.3379	+	4.52480i	1.34469	+	0.162957i
$772$		0			0
$773$	43.1581			1.55229			0.776144	−	0.630556i	$-0.217173\pi$
$773$	43.1581			1.55229			0.776144	+	0.630556i	$0.217173\pi$
$774$		0			0
$775$	−3.97602			−0.142823
$776$		0			0
$777$	−0.372036	+	3.06998i	−0.0133467	+	0.110135i
$778$		0			0
$779$	50.3086			1.80249
$780$		0			0
$781$			10.4144i			0.372658i
$782$		0			0
$783$	21.6009	+	8.17486i	0.771955	+	0.292146i
$784$		0			0
$785$		−	22.3016i		−	0.795978i
$786$		0			0
$787$			33.8343i			1.20606i	0.797717	+	0.603031i	$0.206041\pi$
$787$			33.8343i			1.20606i	−0.797717	+	0.603031i	$0.793959\pi$
$788$		0			0
$789$	−6.32209	+	52.1689i	−0.225072	+	1.85726i
$790$		0			0
$791$		−	2.50042i		−	0.0889048i
$792$		0			0
$793$		−	13.6556i		−	0.484924i
$794$		0			0
$795$	−1.30118	+	10.7371i	−0.0461480	+	0.380806i
$796$		0			0
$797$	4.52419			0.160255			0.0801275	−	0.996785i	$-0.474467\pi$
$797$	4.52419			0.160255			0.0801275	+	0.996785i	$0.474467\pi$
$798$		0			0
$799$			7.57563i			0.268006i
$800$		0			0
$801$	−5.04331	−	1.24057i	−0.178196	−	0.0438333i
$802$		0			0
$803$	−18.7082			−0.660199
$804$		0			0
$805$	−0.488380	−	0.755295i	−0.0172131	−	0.0266207i
$806$		0			0
$807$	−35.9961	−	4.36219i	−1.26712	−	0.153556i
$808$		0			0
$809$		−	1.57880i		−	0.0555077i	−0.999615	−	0.0277539i	$-0.991165\pi$
$809$		−	1.57880i		−	0.0555077i	0.999615	−	0.0277539i	$-0.00883546\pi$
$810$		0			0
$811$	−3.58195			−0.125779			−0.0628897	−	0.998020i	$-0.520032\pi$
$811$	−3.58195			−0.125779			−0.0628897	+	0.998020i	$0.520032\pi$
$812$		0			0
$813$	4.29412	−	35.4344i	0.150601	−	1.24274i
$814$		0			0
$815$	−12.2300			−0.428399
$816$		0			0
$817$	45.3146			1.58536
$818$		0			0
$819$	0.907268	−	3.68834i	0.0317025	−	0.128881i
$820$		0			0
$821$		−	25.4983i		−	0.889898i	−0.895556	−	0.444949i	$-0.853222\pi$
$821$		−	25.4983i		−	0.889898i	0.895556	−	0.444949i	$-0.146778\pi$
$822$		0			0
$823$	0.365706			0.0127477			0.00637386	−	0.999980i	$-0.497971\pi$
$823$	0.365706			0.0127477			0.00637386	+	0.999980i	$0.497971\pi$
$824$		0			0
$825$	0.909242	−	7.50292i	0.0316557	−	0.261218i
$826$		0			0
$827$	−20.7234			−0.720623			−0.360311	−	0.932832i	$-0.617330\pi$
$827$	−20.7234			−0.720623			−0.360311	+	0.932832i	$0.617330\pi$
$828$		0			0
$829$	−14.7977			−0.513946			−0.256973	−	0.966419i	$-0.582725\pi$
$829$	−14.7977			−0.513946			−0.256973	+	0.966419i	$0.582725\pi$
$830$		0			0
$831$	−1.68642	+	13.9161i	−0.0585014	+	0.482744i
$832$		0			0
$833$	−6.66092			−0.230787
$834$		0			0
$835$			9.12412i			0.315753i
$836$		0			0
$837$	7.31262	−	19.3226i	0.252761	−	0.667886i
$838$		0			0
$839$	19.8342			0.684754			0.342377	−	0.939563i	$-0.388768\pi$
$839$	19.8342			0.684754			0.342377	+	0.939563i	$0.388768\pi$
$840$		0			0
$841$	9.24337			0.318737
$842$		0			0
$843$	1.25908	−	10.3897i	0.0433650	−	0.357841i
$844$		0			0
$845$	−32.5743			−1.12059
$846$		0			0
$847$			1.50790i			0.0518119i
$848$		0			0
$849$	−10.5594	−	1.27965i	−0.362399	−	0.0439173i
$850$		0			0
$851$	24.7905	+	38.3393i	0.849807	+	1.31425i
$852$		0			0
$853$	−15.7004			−0.537572			−0.268786	−	0.963200i	$-0.586623\pi$
$853$	−15.7004			−0.537572			−0.268786	+	0.963200i	$0.586623\pi$
$854$		0			0
$855$	−4.25298	+	17.2897i	−0.145449	+	0.591296i
$856$		0			0
$857$		−	15.5570i		−	0.531418i	−0.964053	−	0.265709i	$-0.914394\pi$
$857$		−	15.5570i		−	0.531418i	0.964053	−	0.265709i	$-0.0856061\pi$
$858$		0			0
$859$	−18.1242			−0.618390			−0.309195	−	0.950999i	$-0.600060\pi$
$859$	−18.1242			−0.618390			−0.309195	+	0.950999i	$0.600060\pi$
$860$		0			0
$861$	0.331260	−	2.73351i	0.0112893	−	0.0931577i
$862$		0			0
$863$		−	2.51915i		−	0.0857527i	−0.999080	−	0.0428764i	$-0.986348\pi$
$863$		−	2.51915i		−	0.0857527i	0.999080	−	0.0428764i	$-0.0136522\pi$
$864$		0			0
$865$		−	12.6174i		−	0.429005i
$866$		0			0
$867$	−3.35178	+	27.6583i	−0.113832	+	0.939326i
$868$		0			0
$869$		−	58.8586i		−	1.99664i
$870$		0			0
$871$		−	53.1707i		−	1.80162i
$872$		0			0
$873$	8.91140	−	36.2277i	0.301605	−	1.22612i
$874$		0			0
$875$		−	0.187545i		−	0.00634019i
$876$		0			0
$877$	−18.5977			−0.627999			−0.313999	−	0.949423i	$-0.601669\pi$
$877$	−18.5977			−0.627999			−0.313999	+	0.949423i	$0.601669\pi$
$878$		0			0
$879$	−4.11076	+	33.9213i	−0.138653	+	1.14414i
$880$		0			0
$881$	17.8225			0.600456			0.300228	−	0.953868i	$-0.402937\pi$
$881$	17.8225			0.600456			0.300228	+	0.953868i	$0.402937\pi$
$882$		0			0
$883$	−20.1025			−0.676502			−0.338251	−	0.941056i	$-0.609835\pi$
$883$	−20.1025			−0.676502			−0.338251	+	0.941056i	$0.609835\pi$
$884$		0			0
$885$	−0.903573	−	0.109500i	−0.0303733	−	0.00368079i
$886$		0			0
$887$			35.9425i			1.20683i	0.797427	+	0.603416i	$0.206194\pi$
$887$			35.9425i			1.20683i	−0.797427	+	0.603416i	$0.793806\pi$
$888$		0			0
$889$			0.679285i			0.0227825i
$890$		0			0
$891$	34.7902	+	18.2179i	1.16552	+	0.610323i
$892$		0			0
$893$	47.0131			1.57323
$894$		0			0
$895$			20.2204i			0.675893i
$896$		0			0
$897$	−24.5626	−	50.4114i	−0.820120	−	1.68319i
$898$		0			0
$899$			17.6728i			0.589421i
$900$		0			0
$901$	−5.97195			−0.198954
$902$		0			0
$903$	0.298377	−	2.46216i	0.00992937	−	0.0819355i
$904$		0			0
$905$			13.9913i			0.465086i
$906$		0			0
$907$		−	44.2167i		−	1.46819i	−0.679046	−	0.734096i	$-0.737606\pi$
$907$		−	44.2167i		−	1.46819i	0.679046	−	0.734096i	$-0.262394\pi$
$908$		0			0
$909$	0.0905819	−	0.368245i	0.00300441	−	0.0122139i
$910$		0			0
$911$	−40.0784			−1.32786			−0.663928	−	0.747797i	$-0.731112\pi$
$911$	−40.0784			−1.32786			−0.663928	+	0.747797i	$0.731112\pi$
$912$		0			0
$913$	17.8261			0.589956
$914$		0			0
$915$	3.47813	+	0.421497i	0.114983	+	0.0139343i
$916$		0			0
$917$	−0.471325			−0.0155645
$918$		0			0
$919$		−	27.8185i		−	0.917649i	−0.888527	−	0.458825i	$-0.848271\pi$
$919$		−	27.8185i		−	0.917649i	0.888527	−	0.458825i	$-0.151729\pi$
$920$		0			0
$921$	−0.960279	+	7.92407i	−0.0316423	+	0.261107i
$922$		0			0
$923$			16.1124i			0.530346i
$924$		0			0
$925$			9.51993i			0.313013i
$926$		0			0
$927$	8.06877	−	32.8022i	0.265013	−	1.07736i
$928$		0			0
$929$			8.55183i			0.280576i	0.990111	+	0.140288i	$0.0448029\pi$
$929$			8.55183i			0.280576i	−0.990111	+	0.140288i	$0.955197\pi$
$930$		0			0
$931$			41.3365i			1.35475i
$932$		0			0
$933$	−20.2968	−	2.45967i	−0.664487	−	0.0805260i
$934$		0			0
$935$	4.17310			0.136475
$936$		0			0
$937$			30.7135i			1.00337i	0.865051	+	0.501684i	$0.167286\pi$
$937$			30.7135i			1.00337i	−0.865051	+	0.501684i	$0.832714\pi$
$938$		0			0
$939$	−36.1558	−	4.38155i	−1.17990	−	0.142986i
$940$		0			0
$941$	24.5302			0.799661			0.399831	−	0.916589i	$-0.369069\pi$
$941$	24.5302			0.799661			0.399831	+	0.916589i	$0.369069\pi$
$942$		0			0
$943$	−22.0734	−	34.1373i	−0.718810	−	1.11166i
$944$		0			0
$945$	0.911429	+	0.344930i	0.0296488	+	0.0112206i
$946$		0			0
$947$			7.96398i			0.258795i	0.991593	+	0.129397i	$0.0413043\pi$
$947$			7.96398i			0.258795i	−0.991593	+	0.129397i	$0.958696\pi$
$948$		0			0
$949$	−28.9439			−0.939558
$950$		0			0
$951$	47.0233	+	5.69853i	1.52484	+	0.184787i
$952$		0			0
$953$	35.0590			1.13567			0.567836	−	0.823142i	$-0.307781\pi$
$953$	35.0590			1.13567			0.567836	+	0.823142i	$0.307781\pi$
$954$		0			0
$955$	14.1977			0.459425
$956$		0			0
$957$	−33.3493	−	4.04144i	−1.07803	−	0.130641i
$958$		0			0
$959$			3.45157i			0.111457i
$960$		0			0
$961$	−15.1912			−0.490040
$962$		0			0
$963$	16.8017	+	4.13292i	0.541426	+	0.133182i
$964$		0			0
$965$	−13.1913			−0.424642
$966$		0			0
$967$	52.2914			1.68158			0.840789	−	0.541363i	$-0.182091\pi$
$967$	52.2914			1.68158			0.840789	+	0.541363i	$0.182091\pi$
$968$		0			0
$969$	−9.75983	−	1.18275i	−0.313531	−	0.0379953i
$970$		0			0
$971$	−37.2490			−1.19538			−0.597689	−	0.801728i	$-0.703914\pi$
$971$	−37.2490			−1.19538			−0.597689	+	0.801728i	$0.703914\pi$
$972$		0			0
$973$			0.228381i			0.00732157i
$974$		0			0
$975$	1.40671	−	11.6079i	0.0450507	−	0.371751i
$976$		0			0
$977$	−22.9791			−0.735167			−0.367583	−	0.929991i	$-0.619815\pi$
$977$	−22.9791			−0.735167			−0.367583	+	0.929991i	$0.619815\pi$
$978$		0			0
$979$	7.55416			0.241432
$980$		0			0
$981$	−3.46270	+	14.0770i	−0.110556	+	0.449445i
$982$		0			0
$983$	−45.9968			−1.46707			−0.733536	−	0.679651i	$-0.762131\pi$
$983$	−45.9968			−1.46707			−0.733536	+	0.679651i	$0.762131\pi$
$984$		0			0
$985$		−	21.5576i		−	0.686884i
$986$		0			0
$987$	0.309561	−	2.55444i	0.00985343	−	0.0813089i
$988$		0			0
$989$	−19.8823	−	30.7486i	−0.632220	−	0.977748i
$990$		0			0
$991$	−47.8353			−1.51954			−0.759768	−	0.650194i	$-0.774688\pi$
$991$	−47.8353			−1.51954			−0.759768	+	0.650194i	$0.774688\pi$
$992$		0			0
$993$	3.36847	−	27.7960i	0.106895	−	0.882081i
$994$		0			0
$995$		−	1.91585i		−	0.0607365i
$996$		0			0
$997$	−32.1839			−1.01927			−0.509637	−	0.860390i	$-0.670220\pi$
$997$	−32.1839			−1.01927			−0.509637	+	0.860390i	$0.670220\pi$
$998$		0			0
$999$	−46.2647	−	17.5089i	−1.46375	−	0.553956i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.i.a.1241.9	✓	16
3.2	odd	2		1380.2.i.b.1241.10	yes	16
23.22	odd	2		1380.2.i.b.1241.9	yes	16
69.68	even	2	inner	1380.2.i.a.1241.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.i.a.1241.9	✓	16	1.1	even	1	trivial
1380.2.i.a.1241.10	yes	16	69.68	even	2	inner
1380.2.i.b.1241.9	yes	16	23.22	odd	2
1380.2.i.b.1241.10	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)

\(f(q)\)	\(=\)	\(q+(0.208374 - 1.71947i) q^{3} -1.00000 q^{5} +0.187545i q^{7} +(-2.91316 - 0.716587i) q^{9} +O(q^{10})\)	\(q+(0.208374 - 1.71947i) q^{3} -1.00000 q^{5} +0.187545i q^{7} +(-2.91316 - 0.716587i) q^{9} +4.36350 q^{11} +6.75087 q^{13} +(-0.208374 + 1.71947i) q^{15} -0.956365 q^{17} +5.93504i q^{19} +(0.322479 + 0.0390797i) q^{21} +(4.02726 - 2.60406i) q^{23} +1.00000 q^{25} +(-1.83918 + 4.85978i) q^{27} -4.44484i q^{29} -3.97602 q^{31} +(0.909242 - 7.50292i) q^{33} -0.187545i q^{35} +9.51993i q^{37} +(1.40671 - 11.6079i) q^{39} -8.47654i q^{41} -7.63510i q^{43} +(2.91316 + 0.716587i) q^{45} -7.92127i q^{47} +6.96483 q^{49} +(-0.199282 + 1.64444i) q^{51} +6.24442 q^{53} -4.36350 q^{55} +(10.2051 + 1.23671i) q^{57} +0.525495i q^{59} -2.02279i q^{61} +(0.134393 - 0.546350i) q^{63} -6.75087 q^{65} -7.87613i q^{67} +(-3.63843 - 7.46738i) q^{69} +2.38671i q^{71} -4.28743 q^{73} +(0.208374 - 1.71947i) q^{75} +0.818355i q^{77} -13.4888i q^{79} +(7.97301 + 4.17507i) q^{81} +4.08526 q^{83} +0.956365 q^{85} +(-7.64278 - 0.926191i) q^{87} +1.73121 q^{89} +1.26610i q^{91} +(-0.828501 + 6.83665i) q^{93} -5.93504i q^{95} +12.4359i q^{97} +(-12.7116 - 3.12683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) \) 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

Introduction

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