LMFDB - Embedded newform 1380.2.f.a.829.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(829,1380)]

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, 0, 1, 0]))

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

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

chi := DirichletCharacter("1380.829");

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.f (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:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.5
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	1380.829
Dual form			1380.2.f.a.829.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.539189 - 2.17009i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-0.539189 - 2.17009i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -0.539189 q^{11} +2.24846i q^{13} +(2.17009 - 0.539189i) q^{15} +3.24846i q^{17} -2.24846 q^{19} -1.00000 q^{21} +1.00000i q^{23} +(-4.41855 + 2.34017i) q^{25} -1.00000i q^{27} +8.51026 q^{29} -6.70928 q^{31} -0.539189i q^{33} +(2.17009 - 0.539189i) q^{35} +7.04945i q^{37} -2.24846 q^{39} +5.14116 q^{41} +4.97107i q^{43} +(0.539189 + 2.17009i) q^{45} +8.87936i q^{47} +6.00000 q^{49} -3.24846 q^{51} +4.80098i q^{53} +(0.290725 + 1.17009i) q^{55} -2.24846i q^{57} +0.170086 q^{59} -2.67316 q^{61} -1.00000i q^{63} +(4.87936 - 1.21235i) q^{65} +8.31124i q^{67} -1.00000 q^{69} -11.2979 q^{71} +8.09171i q^{73} +(-2.34017 - 4.41855i) q^{75} -0.539189i q^{77} -5.57531 q^{79} +1.00000 q^{81} -0.411363i q^{83} +(7.04945 - 1.75154i) q^{85} +8.51026i q^{87} +6.68035 q^{89} -2.24846 q^{91} -6.70928i q^{93} +(1.21235 + 4.87936i) q^{95} -14.8865i q^{97} +0.539189 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 2 q^{15} + 4 q^{19} - 6 q^{21} + 2 q^{25} + 18 q^{29} - 26 q^{31} + 2 q^{35} + 4 q^{39} - 10 q^{41} + 36 q^{49} - 2 q^{51} + 16 q^{55} - 10 q^{59} - 40 q^{61} + 4 q^{65} - 6 q^{69} - 14 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 6 q^{85} - 4 q^{89} + 4 q^{91} + 28 q^{95}+O(q^{100}) $ 6 * q - 6 * q^9 + 2 * q^15 + 4 * q^19 - 6 * q^21 + 2 * q^25 + 18 * q^29 - 26 * q^31 + 2 * q^35 + 4 * q^39 - 10 * q^41 + 36 * q^49 - 2 * q^51 + 16 * q^55 - 10 * q^59 - 40 * q^61 + 4 * q^65 - 6 * q^69 - 14 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 6 * q^85 - 4 * q^89 + 4 * q^91 + 28 * q^95

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−0.539189	−	2.17009i	−0.241133	−	0.970492i
$5$	−0.539189	−	2.17009i	−0.241133	−	0.970492i
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.539189			−0.162572			−0.0812858	−	0.996691i	$-0.525903\pi$
$11$	−0.539189			−0.162572			−0.0812858	+	0.996691i	$0.525903\pi$
$12$		0			0
$13$			2.24846i			0.623612i	0.950146	+	0.311806i	$0.100934\pi$
$13$			2.24846i			0.623612i	−0.950146	+	0.311806i	$0.899066\pi$
$14$		0			0
$15$	2.17009	−	0.539189i	0.560314	−	0.139218i
$16$		0			0
$17$			3.24846i			0.787868i	0.919139	+	0.393934i	$0.128886\pi$
$17$			3.24846i			0.787868i	−0.919139	+	0.393934i	$0.871114\pi$
$18$		0			0
$19$	−2.24846			−0.515833			−0.257917	−	0.966167i	$-0.583036\pi$
$19$	−2.24846			−0.515833			−0.257917	+	0.966167i	$0.583036\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−4.41855	+	2.34017i	−0.883710	+	0.468035i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	8.51026			1.58032			0.790158	−	0.612904i	$-0.209999\pi$
$29$	8.51026			1.58032			0.790158	+	0.612904i	$0.209999\pi$
$30$		0			0
$31$	−6.70928			−1.20502			−0.602511	−	0.798111i	$-0.705833\pi$
$31$	−6.70928			−1.20502			−0.602511	+	0.798111i	$0.705833\pi$
$32$		0			0
$33$		−	0.539189i		−	0.0938607i
$34$		0			0
$35$	2.17009	−	0.539189i	0.366812	−	0.0911396i
$36$		0			0
$37$			7.04945i			1.15892i	0.815000	+	0.579461i	$0.196737\pi$
$37$			7.04945i			1.15892i	−0.815000	+	0.579461i	$0.803263\pi$
$38$		0			0
$39$	−2.24846			−0.360042
$40$		0			0
$41$	5.14116			0.802914			0.401457	−	0.915878i	$-0.368504\pi$
$41$	5.14116			0.802914			0.401457	+	0.915878i	$0.368504\pi$
$42$		0			0
$43$			4.97107i			0.758081i	0.925380	+	0.379041i	$0.123746\pi$
$43$			4.97107i			0.758081i	−0.925380	+	0.379041i	$0.876254\pi$
$44$		0			0
$45$	0.539189	+	2.17009i	0.0803775	+	0.323497i
$46$		0			0
$47$			8.87936i			1.29519i	0.761986	+	0.647594i	$0.224225\pi$
$47$			8.87936i			1.29519i	−0.761986	+	0.647594i	$0.775775\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$	−3.24846			−0.454876
$52$		0			0
$53$			4.80098i			0.659466i	0.944074	+	0.329733i	$0.106959\pi$
$53$			4.80098i			0.659466i	−0.944074	+	0.329733i	$0.893041\pi$
$54$		0			0
$55$	0.290725	+	1.17009i	0.0392013	+	0.157774i
$56$		0			0
$57$		−	2.24846i		−	0.297816i
$58$		0			0
$59$	0.170086			0.0221434			0.0110717	−	0.999939i	$-0.496476\pi$
$59$	0.170086			0.0221434			0.0110717	+	0.999939i	$0.496476\pi$
$60$		0			0
$61$	−2.67316			−0.342263			−0.171131	−	0.985248i	$-0.554742\pi$
$61$	−2.67316			−0.342263			−0.171131	+	0.985248i	$0.554742\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	4.87936	−	1.21235i	0.605210	−	0.150373i
$66$		0			0
$67$			8.31124i			1.01538i	0.861540	+	0.507690i	$0.169500\pi$
$67$			8.31124i			1.01538i	−0.861540	+	0.507690i	$0.830500\pi$
$68$		0			0
$69$	−1.00000			−0.120386
$70$		0			0
$71$	−11.2979			−1.34082			−0.670408	−	0.741993i	$-0.733881\pi$
$71$	−11.2979			−1.34082			−0.670408	+	0.741993i	$0.733881\pi$
$72$		0			0
$73$			8.09171i			0.947063i	0.880777	+	0.473531i	$0.157021\pi$
$73$			8.09171i			0.947063i	−0.880777	+	0.473531i	$0.842979\pi$
$74$		0			0
$75$	−2.34017	−	4.41855i	−0.270220	−	0.510210i
$76$		0			0
$77$		−	0.539189i		−	0.0614463i
$78$		0			0
$79$	−5.57531			−0.627271			−0.313635	−	0.949543i	$-0.601547\pi$
$79$	−5.57531			−0.627271			−0.313635	+	0.949543i	$0.601547\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	0.411363i		−	0.0451529i	−0.999745	−	0.0225765i	$-0.992813\pi$
$83$		−	0.411363i		−	0.0451529i	0.999745	−	0.0225765i	$-0.00718692\pi$
$84$		0			0
$85$	7.04945	−	1.75154i	0.764620	−	0.189981i
$86$		0			0
$87$			8.51026i			0.912396i
$88$		0			0
$89$	6.68035			0.708115			0.354058	−	0.935224i	$-0.384802\pi$
$89$	6.68035			0.708115			0.354058	+	0.935224i	$0.384802\pi$
$90$		0			0
$91$	−2.24846			−0.235703
$92$		0			0
$93$		−	6.70928i		−	0.695719i
$94$		0			0
$95$	1.21235	+	4.87936i	0.124384	+	0.500612i
$96$		0			0
$97$		−	14.8865i		−	1.51150i	−0.654860	−	0.755750i	$-0.727272\pi$
$97$		−	14.8865i		−	1.51150i	0.654860	−	0.755750i	$-0.272728\pi$
$98$		0			0
$99$	0.539189			0.0541905
$100$		0			0
$101$	−2.12064			−0.211011			−0.105506	−	0.994419i	$-0.533646\pi$
$101$	−2.12064			−0.211011			−0.105506	+	0.994419i	$0.533646\pi$
$102$		0			0
$103$			0.290725i			0.0286460i	0.999897	+	0.0143230i	$0.00455930\pi$
$103$			0.290725i			0.0286460i	−0.999897	+	0.0143230i	$0.995441\pi$
$104$		0			0
$105$	0.539189	+	2.17009i	0.0526194	+	0.211779i
$106$		0			0
$107$			12.3763i			1.19646i	0.801324	+	0.598231i	$0.204129\pi$
$107$			12.3763i			1.19646i	−0.801324	+	0.598231i	$0.795871\pi$
$108$		0			0
$109$	−12.1906			−1.16765			−0.583824	−	0.811880i	$-0.698444\pi$
$109$	−12.1906			−1.16765			−0.583824	+	0.811880i	$0.698444\pi$
$110$		0			0
$111$	−7.04945			−0.669104
$112$		0			0
$113$		−	1.72261i		−	0.162049i	−0.996712	−	0.0810246i	$-0.974181\pi$
$113$		−	1.72261i		−	0.162049i	0.996712	−	0.0810246i	$-0.0258192\pi$
$114$		0			0
$115$	2.17009	−	0.539189i	0.202362	−	0.0502796i
$116$		0			0
$117$		−	2.24846i		−	0.207871i
$118$		0			0
$119$	−3.24846			−0.297786
$120$		0			0
$121$	−10.7093			−0.973570
$122$		0			0
$123$			5.14116i			0.463563i
$124$		0			0
$125$	7.46081	+	8.32684i	0.667315	+	0.744775i
$126$		0			0
$127$		−	12.1639i		−	1.07938i	−0.841865	−	0.539688i	$-0.818542\pi$
$127$		−	12.1639i		−	1.07938i	0.841865	−	0.539688i	$-0.181458\pi$
$128$		0			0
$129$	−4.97107			−0.437678
$130$		0			0
$131$	7.78539			0.680212			0.340106	−	0.940387i	$-0.389537\pi$
$131$	7.78539			0.680212			0.340106	+	0.940387i	$0.389537\pi$
$132$		0			0
$133$		−	2.24846i		−	0.194967i
$134$		0			0
$135$	−2.17009	+	0.539189i	−0.186771	+	0.0464060i
$136$		0			0
$137$			9.86603i			0.842912i	0.906849	+	0.421456i	$0.138481\pi$
$137$			9.86603i			0.842912i	−0.906849	+	0.421456i	$0.861519\pi$
$138$		0			0
$139$	−4.07838			−0.345923			−0.172962	−	0.984929i	$-0.555334\pi$
$139$	−4.07838			−0.345923			−0.172962	+	0.984929i	$0.555334\pi$
$140$		0			0
$141$	−8.87936			−0.747777
$142$		0			0
$143$		−	1.21235i		−	0.101382i
$144$		0			0
$145$	−4.58864	−	18.4680i	−0.381066	−	1.53368i
$146$		0			0
$147$			6.00000i			0.494872i
$148$		0			0
$149$	8.53919			0.699558			0.349779	−	0.936832i	$-0.386257\pi$
$149$	8.53919			0.699558			0.349779	+	0.936832i	$0.386257\pi$
$150$		0			0
$151$	−2.63090			−0.214099			−0.107050	−	0.994254i	$-0.534140\pi$
$151$	−2.63090			−0.214099			−0.107050	+	0.994254i	$0.534140\pi$
$152$		0			0
$153$		−	3.24846i		−	0.262623i
$154$		0			0
$155$	3.61757	+	14.5597i	0.290570	+	1.16946i
$156$		0			0
$157$			16.7587i			1.33749i	0.743491	+	0.668746i	$0.233169\pi$
$157$			16.7587i			1.33749i	−0.743491	+	0.668746i	$0.766831\pi$
$158$		0			0
$159$	−4.80098			−0.380743
$160$		0			0
$161$	−1.00000			−0.0788110
$162$		0			0
$163$			5.26180i			0.412136i	0.978538	+	0.206068i	$0.0660667\pi$
$163$			5.26180i			0.412136i	−0.978538	+	0.206068i	$0.933933\pi$
$164$		0			0
$165$	−1.17009	+	0.290725i	−0.0910911	+	0.0226329i
$166$		0			0
$167$		−	19.4752i		−	1.50704i	−0.657428	−	0.753518i	$-0.728356\pi$
$167$		−	19.4752i		−	1.50704i	0.657428	−	0.753518i	$-0.271644\pi$
$168$		0			0
$169$	7.94441			0.611108
$170$		0			0
$171$	2.24846			0.171944
$172$		0			0
$173$		−	20.4391i		−	1.55395i	−0.629529	−	0.776977i	$-0.716752\pi$
$173$		−	20.4391i		−	1.55395i	0.629529	−	0.776977i	$-0.283248\pi$
$174$		0			0
$175$	−2.34017	−	4.41855i	−0.176900	−	0.334011i
$176$		0			0
$177$			0.170086i			0.0127845i
$178$		0			0
$179$	−2.78765			−0.208359			−0.104179	−	0.994559i	$-0.533222\pi$
$179$	−2.78765			−0.208359			−0.104179	+	0.994559i	$0.533222\pi$
$180$		0			0
$181$	−15.9155			−1.18299			−0.591494	−	0.806309i	$-0.701462\pi$
$181$	−15.9155			−1.18299			−0.591494	+	0.806309i	$0.701462\pi$
$182$		0			0
$183$		−	2.67316i		−	0.197606i
$184$		0			0
$185$	15.2979	−	3.80098i	1.12472	−	0.279454i
$186$		0			0
$187$		−	1.75154i		−	0.128085i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	−1.12064			−0.0810865			−0.0405433	−	0.999178i	$-0.512909\pi$
$191$	−1.12064			−0.0810865			−0.0405433	+	0.999178i	$0.512909\pi$
$192$		0			0
$193$		−	12.8638i		−	0.925954i	−0.886370	−	0.462977i	$-0.846781\pi$
$193$		−	12.8638i		−	0.925954i	0.886370	−	0.462977i	$-0.153219\pi$
$194$		0			0
$195$	1.21235	+	4.87936i	0.0868180	+	0.349418i
$196$		0			0
$197$		−	11.7854i		−	0.839674i	−0.907600	−	0.419837i	$-0.862087\pi$
$197$		−	11.7854i		−	0.839674i	0.907600	−	0.419837i	$-0.137913\pi$
$198$		0			0
$199$	25.1194			1.78067			0.890334	−	0.455308i	$-0.150471\pi$
$199$	25.1194			1.78067			0.890334	+	0.455308i	$0.150471\pi$
$200$		0			0
$201$	−8.31124			−0.586230
$202$		0			0
$203$			8.51026i			0.597303i
$204$		0			0
$205$	−2.77205	−	11.1568i	−0.193609	−	0.779222i
$206$		0			0
$207$		−	1.00000i		−	0.0695048i
$208$		0			0
$209$	1.21235			0.0838598
$210$		0			0
$211$	4.51745			0.310994			0.155497	−	0.987836i	$-0.450302\pi$
$211$	4.51745			0.310994			0.155497	+	0.987836i	$0.450302\pi$
$212$		0			0
$213$		−	11.2979i		−	0.774120i
$214$		0			0
$215$	10.7877	−	2.68035i	0.735712	−	0.182798i
$216$		0			0
$217$		−	6.70928i		−	0.455455i
$218$		0			0
$219$	−8.09171			−0.546787
$220$		0			0
$221$	−7.30406			−0.491324
$222$		0			0
$223$			3.31965i			0.222300i	0.993804	+	0.111150i	$0.0354535\pi$
$223$			3.31965i			0.222300i	−0.993804	+	0.111150i	$0.964547\pi$
$224$		0			0
$225$	4.41855	−	2.34017i	0.294570	−	0.156012i
$226$		0			0
$227$			19.5174i			1.29542i	0.761888	+	0.647709i	$0.224273\pi$
$227$			19.5174i			1.29542i	−0.761888	+	0.647709i	$0.775727\pi$
$228$		0			0
$229$	9.39189			0.620633			0.310317	−	0.950633i	$-0.399565\pi$
$229$	9.39189			0.620633			0.310317	+	0.950633i	$0.399565\pi$
$230$		0			0
$231$	0.539189			0.0354760
$232$		0			0
$233$			15.1506i			0.992550i	0.868165	+	0.496275i	$0.165299\pi$
$233$			15.1506i			0.992550i	−0.868165	+	0.496275i	$0.834701\pi$
$234$		0			0
$235$	19.2690	−	4.78765i	1.25697	−	0.312312i
$236$		0			0
$237$		−	5.57531i		−	0.362155i
$238$		0			0
$239$	27.5536			1.78229			0.891146	−	0.453717i	$-0.149902\pi$
$239$	27.5536			1.78229			0.891146	+	0.453717i	$0.149902\pi$
$240$		0			0
$241$	−18.6153			−1.19912			−0.599558	−	0.800331i	$-0.704657\pi$
$241$	−18.6153			−1.19912			−0.599558	+	0.800331i	$0.704657\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−3.23513	−	13.0205i	−0.206685	−	0.831850i
$246$		0			0
$247$		−	5.05559i		−	0.321680i
$248$		0			0
$249$	0.411363			0.0260691
$250$		0			0
$251$	−4.18342			−0.264055			−0.132027	−	0.991246i	$-0.542149\pi$
$251$	−4.18342			−0.264055			−0.132027	+	0.991246i	$0.542149\pi$
$252$		0			0
$253$		−	0.539189i		−	0.0338985i
$254$		0			0
$255$	1.75154	+	7.04945i	0.109685	+	0.441454i
$256$		0			0
$257$		−	12.9060i		−	0.805056i	−0.915408	−	0.402528i	$-0.868132\pi$
$257$		−	12.9060i		−	0.805056i	0.915408	−	0.402528i	$-0.131868\pi$
$258$		0			0
$259$	−7.04945			−0.438031
$260$		0			0
$261$	−8.51026			−0.526772
$262$		0			0
$263$		−	4.70209i		−	0.289943i	−0.989436	−	0.144972i	$-0.953691\pi$
$263$		−	4.70209i		−	0.289943i	0.989436	−	0.144972i	$-0.0463091\pi$
$264$		0			0
$265$	10.4186	−	2.58864i	0.640006	−	0.159019i
$266$		0			0
$267$			6.68035i			0.408831i
$268$		0			0
$269$	30.5246			1.86112			0.930560	−	0.366140i	$-0.119321\pi$
$269$	30.5246			1.86112			0.930560	+	0.366140i	$0.119321\pi$
$270$		0			0
$271$	−2.65983			−0.161573			−0.0807865	−	0.996731i	$-0.525743\pi$
$271$	−2.65983			−0.161573			−0.0807865	+	0.996731i	$0.525743\pi$
$272$		0			0
$273$		−	2.24846i		−	0.136083i
$274$		0			0
$275$	2.38243	−	1.26180i	0.143666	−	0.0760891i
$276$		0			0
$277$			0.581449i			0.0349359i	0.999847	+	0.0174680i	$0.00556050\pi$
$277$			0.581449i			0.0349359i	−0.999847	+	0.0174680i	$0.994439\pi$
$278$		0			0
$279$	6.70928			0.401674
$280$		0			0
$281$	29.3907			1.75330			0.876650	−	0.481129i	$-0.159773\pi$
$281$	29.3907			1.75330			0.876650	+	0.481129i	$0.159773\pi$
$282$		0			0
$283$		−	2.23513i		−	0.132865i	−0.997791	−	0.0664324i	$-0.978838\pi$
$283$		−	2.23513i		−	0.132865i	0.997791	−	0.0664324i	$-0.0211617\pi$
$284$		0			0
$285$	−4.87936	+	1.21235i	−0.289028	+	0.0718132i
$286$		0			0
$287$			5.14116i			0.303473i
$288$		0			0
$289$	6.44748			0.379264
$290$		0			0
$291$	14.8865			0.872665
$292$		0			0
$293$		−	6.01333i		−	0.351303i	−0.984452	−	0.175651i	$-0.943797\pi$
$293$		−	6.01333i		−	0.351303i	0.984452	−	0.175651i	$-0.0562031\pi$
$294$		0			0
$295$	−0.0917087	−	0.369102i	−0.00533949	−	0.0214900i
$296$		0			0
$297$			0.539189i			0.0312869i
$298$		0			0
$299$	−2.24846			−0.130032
$300$		0			0
$301$	−4.97107			−0.286528
$302$		0			0
$303$		−	2.12064i		−	0.121827i
$304$		0			0
$305$	1.44134	+	5.80098i	0.0825307	+	0.332163i
$306$		0			0
$307$		−	0.516403i		−	0.0294726i	−0.999891	−	0.0147363i	$-0.995309\pi$
$307$		−	0.516403i		−	0.0294726i	0.999891	−	0.0147363i	$-0.00469089\pi$
$308$		0			0
$309$	−0.290725			−0.0165387
$310$		0			0
$311$	−11.5031			−0.652279			−0.326140	−	0.945322i	$-0.605748\pi$
$311$	−11.5031			−0.652279			−0.326140	+	0.945322i	$0.605748\pi$
$312$		0			0
$313$		−	3.70313i		−	0.209313i	−0.994508	−	0.104657i	$-0.966626\pi$
$313$		−	3.70313i		−	0.209313i	0.994508	−	0.104657i	$-0.0333744\pi$
$314$		0			0
$315$	−2.17009	+	0.539189i	−0.122271	+	0.0303799i
$316$		0			0
$317$		−	18.9204i		−	1.06268i	−0.847160	−	0.531338i	$-0.821690\pi$
$317$		−	18.9204i		−	1.06268i	0.847160	−	0.531338i	$-0.178310\pi$
$318$		0			0
$319$	−4.58864			−0.256914
$320$		0			0
$321$	−12.3763			−0.690777
$322$		0			0
$323$		−	7.30406i		−	0.406409i
$324$		0			0
$325$	−5.26180	−	9.93495i	−0.291872	−	0.551092i
$326$		0			0
$327$		−	12.1906i		−	0.674142i
$328$		0			0
$329$	−8.87936			−0.489535
$330$		0			0
$331$	1.29072			0.0709446			0.0354723	−	0.999371i	$-0.488706\pi$
$331$	1.29072			0.0709446			0.0354723	+	0.999371i	$0.488706\pi$
$332$		0			0
$333$		−	7.04945i		−	0.386307i
$334$		0			0
$335$	18.0361	−	4.48133i	0.985418	−	0.244841i
$336$		0			0
$337$		−	12.5236i		−	0.682203i	−0.940026	−	0.341102i	$-0.889200\pi$
$337$		−	12.5236i		−	0.682203i	0.940026	−	0.341102i	$-0.110800\pi$
$338$		0			0
$339$	1.72261			0.0935591
$340$		0			0
$341$	3.61757			0.195902
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$	0.539189	+	2.17009i	0.0290290	+	0.116834i
$346$		0			0
$347$			5.44521i			0.292314i	0.989261	+	0.146157i	$0.0466905\pi$
$347$			5.44521i			0.292314i	−0.989261	+	0.146157i	$0.953309\pi$
$348$		0			0
$349$	5.39803			0.288950			0.144475	−	0.989508i	$-0.453851\pi$
$349$	5.39803			0.288950			0.144475	+	0.989508i	$0.453851\pi$
$350$		0			0
$351$	2.24846			0.120014
$352$		0			0
$353$		−	14.0566i		−	0.748159i	−0.927397	−	0.374080i	$-0.877959\pi$
$353$		−	14.0566i		−	0.748159i	0.927397	−	0.374080i	$-0.122041\pi$
$354$		0			0
$355$	6.09171	+	24.5174i	0.323314	+	1.30125i
$356$		0			0
$357$		−	3.24846i		−	0.171927i
$358$		0			0
$359$	−27.9143			−1.47326			−0.736629	−	0.676297i	$-0.763584\pi$
$359$	−27.9143			−1.47326			−0.736629	+	0.676297i	$0.763584\pi$
$360$		0			0
$361$	−13.9444			−0.733916
$362$		0			0
$363$		−	10.7093i		−	0.562091i
$364$		0			0
$365$	17.5597	−	4.36296i	0.919117	−	0.228368i
$366$		0			0
$367$		−	20.6248i		−	1.07660i	−0.842752	−	0.538302i	$-0.819066\pi$
$367$		−	20.6248i		−	1.07660i	0.842752	−	0.538302i	$-0.180934\pi$
$368$		0			0
$369$	−5.14116			−0.267638
$370$		0			0
$371$	−4.80098			−0.249255
$372$		0			0
$373$		−	1.22672i		−	0.0635173i	−0.999496	−	0.0317586i	$-0.989889\pi$
$373$		−	1.22672i		−	0.0635173i	0.999496	−	0.0317586i	$-0.0101108\pi$
$374$		0			0
$375$	−8.32684	+	7.46081i	−0.429996	+	0.385275i
$376$		0			0
$377$			19.1350i			0.985503i
$378$		0			0
$379$	−21.3607			−1.09723			−0.548613	−	0.836077i	$-0.684844\pi$
$379$	−21.3607			−1.09723			−0.548613	+	0.836077i	$0.684844\pi$
$380$		0			0
$381$	12.1639			0.623178
$382$		0			0
$383$		−	31.9698i		−	1.63358i	−0.576933	−	0.816791i	$-0.695751\pi$
$383$		−	31.9698i		−	1.63358i	0.576933	−	0.816791i	$-0.304249\pi$
$384$		0			0
$385$	−1.17009	+	0.290725i	−0.0596331	+	0.0148167i
$386$		0			0
$387$		−	4.97107i		−	0.252694i
$388$		0			0
$389$	−18.3545			−0.930613			−0.465306	−	0.885150i	$-0.654056\pi$
$389$	−18.3545			−0.930613			−0.465306	+	0.885150i	$0.654056\pi$
$390$		0			0
$391$	−3.24846			−0.164282
$392$		0			0
$393$			7.78539i			0.392721i
$394$		0			0
$395$	3.00614	+	12.0989i	0.151255	+	0.608762i
$396$		0			0
$397$		−	19.5753i		−	0.982456i	−0.871031	−	0.491228i	$-0.836548\pi$
$397$		−	19.5753i		−	0.982456i	0.871031	−	0.491228i	$-0.163452\pi$
$398$		0			0
$399$	2.24846			0.112564
$400$		0			0
$401$	−9.50307			−0.474561			−0.237280	−	0.971441i	$-0.576256\pi$
$401$	−9.50307			−0.474561			−0.237280	+	0.971441i	$0.576256\pi$
$402$		0			0
$403$		−	15.0856i		−	0.751466i
$404$		0			0
$405$	−0.539189	−	2.17009i	−0.0267925	−	0.107832i
$406$		0			0
$407$		−	3.80098i		−	0.188408i
$408$		0			0
$409$	29.8059			1.47381			0.736904	−	0.675998i	$-0.236287\pi$
$409$	29.8059			1.47381			0.736904	+	0.675998i	$0.236287\pi$
$410$		0			0
$411$	−9.86603			−0.486655
$412$		0			0
$413$			0.170086i			0.00836941i
$414$		0			0
$415$	−0.892693	+	0.221802i	−0.0438206	+	0.0108878i
$416$		0			0
$417$		−	4.07838i		−	0.199719i
$418$		0			0
$419$	2.39681			0.117092			0.0585459	−	0.998285i	$-0.481354\pi$
$419$	2.39681			0.117092			0.0585459	+	0.998285i	$0.481354\pi$
$420$		0			0
$421$	−29.2111			−1.42366			−0.711832	−	0.702350i	$-0.752134\pi$
$421$	−29.2111			−1.42366			−0.711832	+	0.702350i	$0.752134\pi$
$422$		0			0
$423$		−	8.87936i		−	0.431729i
$424$		0			0
$425$	−7.60197	−	14.3535i	−0.368750	−	0.696247i
$426$		0			0
$427$		−	2.67316i		−	0.129363i
$428$		0			0
$429$	1.21235			0.0585327
$430$		0			0
$431$	−12.5236			−0.603240			−0.301620	−	0.953428i	$-0.597527\pi$
$431$	−12.5236			−0.603240			−0.301620	+	0.953428i	$0.597527\pi$
$432$		0			0
$433$			28.0928i			1.35005i	0.737794	+	0.675026i	$0.235868\pi$
$433$			28.0928i			1.35005i	−0.737794	+	0.675026i	$0.764132\pi$
$434$		0			0
$435$	18.4680	−	4.58864i	0.885473	−	0.220008i
$436$		0			0
$437$		−	2.24846i		−	0.107559i
$438$		0			0
$439$	−18.5997			−0.887715			−0.443858	−	0.896097i	$-0.646390\pi$
$439$	−18.5997			−0.887715			−0.443858	+	0.896097i	$0.646390\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$		−	17.1084i		−	0.812842i	−0.913686	−	0.406421i	$-0.866777\pi$
$443$		−	17.1084i		−	0.812842i	0.913686	−	0.406421i	$-0.133223\pi$
$444$		0			0
$445$	−3.60197	−	14.4969i	−0.170750	−	0.687220i
$446$		0			0
$447$			8.53919i			0.403890i
$448$		0			0
$449$	20.9083			0.986723			0.493362	−	0.869824i	$-0.335768\pi$
$449$	20.9083			0.986723			0.493362	+	0.869824i	$0.335768\pi$
$450$		0			0
$451$	−2.77205			−0.130531
$452$		0			0
$453$		−	2.63090i		−	0.123610i
$454$		0			0
$455$	1.21235	+	4.87936i	0.0568357	+	0.228748i
$456$		0			0
$457$		−	8.52586i		−	0.398823i	−0.979916	−	0.199411i	$-0.936097\pi$
$457$		−	8.52586i		−	0.398823i	0.979916	−	0.199411i	$-0.0639030\pi$
$458$		0			0
$459$	3.24846			0.151625
$460$		0			0
$461$	−8.73820			−0.406979			−0.203489	−	0.979077i	$-0.565228\pi$
$461$	−8.73820			−0.406979			−0.203489	+	0.979077i	$0.565228\pi$
$462$		0			0
$463$			33.8960i			1.57528i	0.616135	+	0.787640i	$0.288698\pi$
$463$			33.8960i			1.57528i	−0.616135	+	0.787640i	$0.711302\pi$
$464$		0			0
$465$	−14.5597	+	3.61757i	−0.675190	+	0.167761i
$466$		0			0
$467$		−	2.00946i		−	0.0929865i	−0.998919	−	0.0464933i	$-0.985195\pi$
$467$		−	2.00946i		−	0.0929865i	0.998919	−	0.0464933i	$-0.0148046\pi$
$468$		0			0
$469$	−8.31124			−0.383778
$470$		0			0
$471$	−16.7587			−0.772201
$472$		0			0
$473$		−	2.68035i		−	0.123242i
$474$		0			0
$475$	9.93495	−	5.26180i	0.455847	−	0.241428i
$476$		0			0
$477$		−	4.80098i		−	0.219822i
$478$		0			0
$479$	−10.5392			−0.481548			−0.240774	−	0.970581i	$-0.577401\pi$
$479$	−10.5392			−0.481548			−0.240774	+	0.970581i	$0.577401\pi$
$480$		0			0
$481$	−15.8504			−0.722718
$482$		0			0
$483$		−	1.00000i		−	0.0455016i
$484$		0			0
$485$	−32.3051	+	8.02666i	−1.46690	+	0.364472i
$486$		0			0
$487$			8.09171i			0.366670i	0.983050	+	0.183335i	$0.0586894\pi$
$487$			8.09171i			0.366670i	−0.983050	+	0.183335i	$0.941311\pi$
$488$		0			0
$489$	−5.26180			−0.237947
$490$		0			0
$491$	−26.9455			−1.21603			−0.608016	−	0.793925i	$-0.708034\pi$
$491$	−26.9455			−1.21603			−0.608016	+	0.793925i	$0.708034\pi$
$492$		0			0
$493$			27.6453i			1.24508i
$494$		0			0
$495$	−0.290725	−	1.17009i	−0.0130671	−	0.0525915i
$496$		0			0
$497$		−	11.2979i		−	0.506781i
$498$		0			0
$499$	25.8554			1.15744			0.578722	−	0.815525i	$-0.303552\pi$
$499$	25.8554			1.15744			0.578722	+	0.815525i	$0.303552\pi$
$500$		0			0
$501$	19.4752			0.870087
$502$		0			0
$503$			6.08557i			0.271342i	0.990754	+	0.135671i	$0.0433190\pi$
$503$			6.08557i			0.271342i	−0.990754	+	0.135671i	$0.956681\pi$
$504$		0			0
$505$	1.14342	+	4.60197i	0.0508817	+	0.204785i
$506$		0			0
$507$			7.94441i			0.352824i
$508$		0			0
$509$	29.3835			1.30240			0.651200	−	0.758906i	$-0.274266\pi$
$509$	29.3835			1.30240			0.651200	+	0.758906i	$0.274266\pi$
$510$		0			0
$511$	−8.09171			−0.357956
$512$		0			0
$513$			2.24846i			0.0992721i
$514$		0			0
$515$	0.630898	−	0.156755i	0.0278007	−	0.00690747i
$516$		0			0
$517$		−	4.78765i		−	0.210561i
$518$		0			0
$519$	20.4391			0.897176
$520$		0			0
$521$	22.6647			0.992961			0.496480	−	0.868048i	$-0.334625\pi$
$521$	22.6647			0.992961			0.496480	+	0.868048i	$0.334625\pi$
$522$		0			0
$523$			37.5441i			1.64169i	0.571152	+	0.820845i	$0.306497\pi$
$523$			37.5441i			1.64169i	−0.571152	+	0.820845i	$0.693503\pi$
$524$		0			0
$525$	4.41855	−	2.34017i	0.192841	−	0.102134i
$526$		0			0
$527$		−	21.7948i		−	0.949398i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−0.170086			−0.00738112
$532$		0			0
$533$			11.5597i			0.500707i
$534$		0			0
$535$	26.8576	−	6.67316i	1.16116	−	0.288506i
$536$		0			0
$537$		−	2.78765i		−	0.120296i
$538$		0			0
$539$	−3.23513			−0.139347
$540$		0			0
$541$	−28.2329			−1.21383			−0.606913	−	0.794768i	$-0.707592\pi$
$541$	−28.2329			−1.21383			−0.606913	+	0.794768i	$0.707592\pi$
$542$		0			0
$543$		−	15.9155i		−	0.682999i
$544$		0			0
$545$	6.57304	+	26.4547i	0.281558	+	1.13319i
$546$		0			0
$547$			32.4534i			1.38761i	0.720163	+	0.693805i	$0.244067\pi$
$547$			32.4534i			1.38761i	−0.720163	+	0.693805i	$0.755933\pi$
$548$		0			0
$549$	2.67316			0.114088
$550$		0			0
$551$	−19.1350			−0.815179
$552$		0			0
$553$		−	5.57531i		−	0.237086i
$554$		0			0
$555$	3.80098	+	15.2979i	0.161343	+	0.649360i
$556$		0			0
$557$		−	20.4341i		−	0.865823i	−0.901437	−	0.432911i	$-0.857486\pi$
$557$		−	20.4341i		−	0.865823i	0.901437	−	0.432911i	$-0.142514\pi$
$558$		0			0
$559$	−11.1773			−0.472748
$560$		0			0
$561$	1.75154			0.0739499
$562$		0			0
$563$			20.0316i			0.844231i	0.906542	+	0.422115i	$0.138712\pi$
$563$			20.0316i			0.844231i	−0.906542	+	0.422115i	$0.861288\pi$
$564$		0			0
$565$	−3.73820	+	0.928810i	−0.157267	+	0.0390753i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	43.2762			1.81423			0.907116	−	0.420881i	$-0.138279\pi$
$569$	43.2762			1.81423			0.907116	+	0.420881i	$0.138279\pi$
$570$		0			0
$571$	24.0605			1.00690			0.503451	−	0.864024i	$-0.332064\pi$
$571$	24.0605			1.00690			0.503451	+	0.864024i	$0.332064\pi$
$572$		0			0
$573$		−	1.12064i		−	0.0468153i
$574$		0			0
$575$	−2.34017	−	4.41855i	−0.0975920	−	0.184266i
$576$		0			0
$577$		−	35.9877i		−	1.49819i	−0.662464	−	0.749094i	$-0.730489\pi$
$577$		−	35.9877i		−	1.49819i	0.662464	−	0.749094i	$-0.269511\pi$
$578$		0			0
$579$	12.8638			0.534600
$580$		0			0
$581$	0.411363			0.0170662
$582$		0			0
$583$		−	2.58864i		−	0.107210i
$584$		0			0
$585$	−4.87936	+	1.21235i	−0.201737	+	0.0501244i
$586$		0			0
$587$			11.2618i			0.464824i	0.972617	+	0.232412i	$0.0746618\pi$
$587$			11.2618i			0.464824i	−0.972617	+	0.232412i	$0.925338\pi$
$588$		0			0
$589$	15.0856			0.621590
$590$		0			0
$591$	11.7854			0.484786
$592$		0			0
$593$			1.56424i			0.0642357i	0.999484	+	0.0321179i	$0.0102252\pi$
$593$			1.56424i			0.0642357i	−0.999484	+	0.0321179i	$0.989775\pi$
$594$		0			0
$595$	1.75154	+	7.04945i	0.0718060	+	0.288999i
$596$		0			0
$597$			25.1194i			1.02807i
$598$		0			0
$599$	12.2290			0.499663			0.249831	−	0.968289i	$-0.419625\pi$
$599$	12.2290			0.499663			0.249831	+	0.968289i	$0.419625\pi$
$600$		0			0
$601$	24.6658			1.00614			0.503069	−	0.864246i	$-0.332204\pi$
$601$	24.6658			1.00614			0.503069	+	0.864246i	$0.332204\pi$
$602$		0			0
$603$		−	8.31124i		−	0.338460i
$604$		0			0
$605$	5.77432	+	23.2401i	0.234760	+	0.944843i
$606$		0			0
$607$		−	20.5574i		−	0.834401i	−0.908815	−	0.417200i	$-0.863011\pi$
$607$		−	20.5574i		−	0.834401i	0.908815	−	0.417200i	$-0.136989\pi$
$608$		0			0
$609$	−8.51026			−0.344853
$610$		0			0
$611$	−19.9649			−0.807695
$612$		0			0
$613$			40.7526i			1.64598i	0.568055	+	0.822991i	$0.307696\pi$
$613$			40.7526i			1.64598i	−0.568055	+	0.822991i	$0.692304\pi$
$614$		0			0
$615$	11.1568	−	2.77205i	0.449884	−	0.111780i
$616$		0			0
$617$			31.7019i			1.27627i	0.769924	+	0.638135i	$0.220294\pi$
$617$			31.7019i			1.27627i	−0.769924	+	0.638135i	$0.779706\pi$
$618$		0			0
$619$	−40.5692			−1.63061			−0.815306	−	0.579030i	$-0.803431\pi$
$619$	−40.5692			−1.63061			−0.815306	+	0.579030i	$0.803431\pi$
$620$		0			0
$621$	1.00000			0.0401286
$622$		0			0
$623$			6.68035i			0.267642i
$624$		0			0
$625$	14.0472	−	20.6803i	0.561887	−	0.827214i
$626$		0			0
$627$			1.21235i			0.0484165i
$628$		0			0
$629$	−22.8999			−0.913078
$630$		0			0
$631$	22.6297			0.900873			0.450437	−	0.892808i	$-0.351268\pi$
$631$	22.6297			0.900873			0.450437	+	0.892808i	$0.351268\pi$
$632$		0			0
$633$			4.51745i			0.179552i
$634$		0			0
$635$	−26.3968	+	6.55866i	−1.04753	+	0.260273i
$636$		0			0
$637$			13.4908i			0.534524i
$638$		0			0
$639$	11.2979			0.446939
$640$		0			0
$641$	8.50799			0.336045			0.168023	−	0.985783i	$-0.446262\pi$
$641$	8.50799			0.336045			0.168023	+	0.985783i	$0.446262\pi$
$642$		0			0
$643$			25.1795i			0.992984i	0.868041	+	0.496492i	$0.165379\pi$
$643$			25.1795i			0.992984i	−0.868041	+	0.496492i	$0.834621\pi$
$644$		0			0
$645$	2.68035	+	10.7877i	0.105539	+	0.424763i
$646$		0			0
$647$			4.53919i			0.178454i	0.996011	+	0.0892270i	$0.0284396\pi$
$647$			4.53919i			0.178454i	−0.996011	+	0.0892270i	$0.971560\pi$
$648$		0			0
$649$	−0.0917087			−0.00359988
$650$		0			0
$651$	6.70928			0.262957
$652$		0			0
$653$			11.1038i			0.434526i	0.976113	+	0.217263i	$0.0697129\pi$
$653$			11.1038i			0.434526i	−0.976113	+	0.217263i	$0.930287\pi$
$654$		0			0
$655$	−4.19779	−	16.8950i	−0.164021	−	0.660141i
$656$		0			0
$657$		−	8.09171i		−	0.315688i
$658$		0			0
$659$	−18.2713			−0.711747			−0.355873	−	0.934534i	$-0.615817\pi$
$659$	−18.2713			−0.711747			−0.355873	+	0.934534i	$0.615817\pi$
$660$		0			0
$661$	31.2306			1.21473			0.607365	−	0.794423i	$-0.292227\pi$
$661$	31.2306			1.21473			0.607365	+	0.794423i	$0.292227\pi$
$662$		0			0
$663$		−	7.30406i		−	0.283666i
$664$		0			0
$665$	−4.87936	+	1.21235i	−0.189214	+	0.0470128i
$666$		0			0
$667$			8.51026i			0.329519i
$668$		0			0
$669$	−3.31965			−0.128345
$670$		0			0
$671$	1.44134			0.0556422
$672$		0			0
$673$			31.3535i			1.20859i	0.796761	+	0.604294i	$0.206545\pi$
$673$			31.3535i			1.20859i	−0.796761	+	0.604294i	$0.793455\pi$
$674$		0			0
$675$	2.34017	+	4.41855i	0.0900733	+	0.170070i
$676$		0			0
$677$			27.8215i			1.06927i	0.845084	+	0.534634i	$0.179550\pi$
$677$			27.8215i			1.06927i	−0.845084	+	0.534634i	$0.820450\pi$
$678$		0			0
$679$	14.8865			0.571293
$680$		0			0
$681$	−19.5174			−0.747910
$682$		0			0
$683$			38.2544i			1.46377i	0.681431	+	0.731883i	$0.261358\pi$
$683$			38.2544i			1.46377i	−0.681431	+	0.731883i	$0.738642\pi$
$684$		0			0
$685$	21.4101	−	5.31965i	0.818039	−	0.203254i
$686$		0			0
$687$			9.39189i			0.358323i
$688$		0			0
$689$	−10.7948			−0.411251
$690$		0			0
$691$	13.9506			0.530704			0.265352	−	0.964152i	$-0.414512\pi$
$691$	13.9506			0.530704			0.265352	+	0.964152i	$0.414512\pi$
$692$		0			0
$693$			0.539189i			0.0204821i
$694$		0			0
$695$	2.19902	+	8.85043i	0.0834134	+	0.335716i
$696$		0			0
$697$			16.7009i			0.632590i
$698$		0			0
$699$	−15.1506			−0.573049
$700$		0			0
$701$	−20.1678			−0.761728			−0.380864	−	0.924631i	$-0.624373\pi$
$701$	−20.1678			−0.761728			−0.380864	+	0.924631i	$0.624373\pi$
$702$		0			0
$703$		−	15.8504i		−	0.597810i
$704$		0			0
$705$	4.78765	+	19.2690i	0.180313	+	0.725712i
$706$		0			0
$707$		−	2.12064i		−	0.0797548i
$708$		0			0
$709$	−15.3724			−0.577323			−0.288662	−	0.957431i	$-0.593210\pi$
$709$	−15.3724			−0.577323			−0.288662	+	0.957431i	$0.593210\pi$
$710$		0			0
$711$	5.57531			0.209090
$712$		0			0
$713$		−	6.70928i		−	0.251264i
$714$		0			0
$715$	−2.63090	+	0.653684i	−0.0983900	+	0.0244464i
$716$		0			0
$717$			27.5536i			1.02901i
$718$		0			0
$719$	−38.2378			−1.42603			−0.713014	−	0.701149i	$-0.752671\pi$
$719$	−38.2378			−1.42603			−0.713014	+	0.701149i	$0.752671\pi$
$720$		0			0
$721$	−0.290725			−0.0108272
$722$		0			0
$723$		−	18.6153i		−	0.692310i
$724$		0			0
$725$	−37.6030	+	19.9155i	−1.39654	+	0.739642i
$726$		0			0
$727$		−	22.8888i		−	0.848899i	−0.905452	−	0.424450i	$-0.860468\pi$
$727$		−	22.8888i		−	0.848899i	0.905452	−	0.424450i	$-0.139532\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−16.1483			−0.597268
$732$		0			0
$733$			3.00841i			0.111118i	0.998455	+	0.0555591i	$0.0176941\pi$
$733$			3.00841i			0.111118i	−0.998455	+	0.0555591i	$0.982306\pi$
$734$		0			0
$735$	13.0205	−	3.23513i	0.480269	−	0.119330i
$736$		0			0
$737$		−	4.48133i		−	0.165072i
$738$		0			0
$739$	8.15449			0.299968			0.149984	−	0.988688i	$-0.452078\pi$
$739$	8.15449			0.299968			0.149984	+	0.988688i	$0.452078\pi$
$740$		0			0
$741$	5.05559			0.185722
$742$		0			0
$743$		−	42.8827i		−	1.57321i	−0.617455	−	0.786606i	$-0.711836\pi$
$743$		−	42.8827i		−	1.57321i	0.617455	−	0.786606i	$-0.288164\pi$
$744$		0			0
$745$	−4.60424	−	18.5308i	−0.168686	−	0.678915i
$746$		0			0
$747$			0.411363i			0.0150510i
$748$		0			0
$749$	−12.3763			−0.452220
$750$		0			0
$751$	49.9637			1.82320			0.911601	−	0.411077i	$-0.134847\pi$
$751$	49.9637			1.82320			0.911601	+	0.411077i	$0.134847\pi$
$752$		0			0
$753$		−	4.18342i		−	0.152452i
$754$		0			0
$755$	1.41855	+	5.70928i	0.0516263	+	0.207782i
$756$		0			0
$757$			8.62863i			0.313613i	0.987629	+	0.156806i	$0.0501199\pi$
$757$			8.62863i			0.313613i	−0.987629	+	0.156806i	$0.949880\pi$
$758$		0			0
$759$	0.539189			0.0195713
$760$		0			0
$761$	−37.3617			−1.35436			−0.677181	−	0.735817i	$-0.736799\pi$
$761$	−37.3617			−1.35436			−0.677181	+	0.735817i	$0.736799\pi$
$762$		0			0
$763$		−	12.1906i		−	0.441330i
$764$		0			0
$765$	−7.04945	+	1.75154i	−0.254873	+	0.0633269i
$766$		0			0
$767$			0.382433i			0.0138089i
$768$		0			0
$769$	−22.4775			−0.810558			−0.405279	−	0.914193i	$-0.632826\pi$
$769$	−22.4775			−0.810558			−0.405279	+	0.914193i	$0.632826\pi$
$770$		0			0
$771$	12.9060			0.464799
$772$		0			0
$773$		−	8.28685i		−	0.298057i	−0.988833	−	0.149029i	$-0.952385\pi$
$773$		−	8.28685i		−	0.298057i	0.988833	−	0.149029i	$-0.0476147\pi$
$774$		0			0
$775$	29.6453	−	15.7009i	1.06489	−	0.563992i
$776$		0			0
$777$		−	7.04945i		−	0.252898i
$778$		0			0
$779$	−11.5597			−0.414170
$780$		0			0
$781$	6.09171			0.217978
$782$		0			0
$783$		−	8.51026i		−	0.304132i
$784$		0			0
$785$	36.3679	−	9.03612i	1.29803	−	0.322513i
$786$		0			0
$787$			20.1734i			0.719104i	0.933125	+	0.359552i	$0.117070\pi$
$787$			20.1734i			0.719104i	−0.933125	+	0.359552i	$0.882930\pi$
$788$		0			0
$789$	4.70209			0.167399
$790$		0			0
$791$	1.72261			0.0612488
$792$		0			0
$793$		−	6.01050i		−	0.213439i
$794$		0			0
$795$	2.58864	+	10.4186i	0.0918095	+	0.369508i
$796$		0			0
$797$			0.832181i			0.0294774i	0.999891	+	0.0147387i	$0.00469164\pi$
$797$			0.832181i			0.0294774i	−0.999891	+	0.0147387i	$0.995308\pi$
$798$		0			0
$799$	−28.8443			−1.02044
$800$		0			0
$801$	−6.68035			−0.236038
$802$		0			0
$803$		−	4.36296i		−	0.153965i
$804$		0			0
$805$	0.539189	+	2.17009i	0.0190039	+	0.0764855i
$806$		0			0
$807$			30.5246i			1.07452i
$808$		0			0
$809$	9.01106			0.316812			0.158406	−	0.987374i	$-0.449364\pi$
$809$	9.01106			0.316812			0.158406	+	0.987374i	$0.449364\pi$
$810$		0			0
$811$	−50.7152			−1.78085			−0.890426	−	0.455127i	$-0.849594\pi$
$811$	−50.7152			−1.78085			−0.890426	+	0.455127i	$0.849594\pi$
$812$		0			0
$813$		−	2.65983i		−	0.0932842i
$814$		0			0
$815$	11.4186	−	2.83710i	0.399974	−	0.0993793i
$816$		0			0
$817$		−	11.1773i		−	0.391043i
$818$		0			0
$819$	2.24846			0.0785677
$820$		0			0
$821$	16.0761			0.561060			0.280530	−	0.959845i	$-0.409490\pi$
$821$	16.0761			0.561060			0.280530	+	0.959845i	$0.409490\pi$
$822$		0			0
$823$			39.0928i			1.36269i	0.731963	+	0.681344i	$0.238604\pi$
$823$			39.0928i			1.36269i	−0.731963	+	0.681344i	$0.761396\pi$
$824$		0			0
$825$	1.26180	+	2.38243i	0.0439301	+	0.0829457i
$826$		0			0
$827$		−	16.8732i		−	0.586739i	−0.955999	−	0.293370i	$-0.905223\pi$
$827$		−	16.8732i		−	0.586739i	0.955999	−	0.293370i	$-0.0947767\pi$
$828$		0			0
$829$	36.8492			1.27983			0.639913	−	0.768447i	$-0.278970\pi$
$829$	36.8492			1.27983			0.639913	+	0.768447i	$0.278970\pi$
$830$		0			0
$831$	−0.581449			−0.0201703
$832$		0			0
$833$			19.4908i			0.675316i
$834$		0			0
$835$	−42.2628	+	10.5008i	−1.46257	+	0.363395i
$836$		0			0
$837$			6.70928i			0.231906i
$838$		0			0
$839$	−6.70701			−0.231552			−0.115776	−	0.993275i	$-0.536935\pi$
$839$	−6.70701			−0.231552			−0.115776	+	0.993275i	$0.536935\pi$
$840$		0			0
$841$	43.4245			1.49740
$842$		0			0
$843$			29.3907i			1.01227i
$844$		0			0
$845$	−4.28354	−	17.2401i	−0.147358	−	0.593076i
$846$		0			0
$847$		−	10.7093i		−	0.367975i
$848$		0			0
$849$	2.23513			0.0767096
$850$		0			0
$851$	−7.04945			−0.241652
$852$		0			0
$853$		−	4.96719i		−	0.170074i	−0.996378	−	0.0850368i	$-0.972899\pi$
$853$		−	4.96719i		−	0.170074i	0.996378	−	0.0850368i	$-0.0271008\pi$
$854$		0			0
$855$	−1.21235	−	4.87936i	−0.0414614	−	0.166871i
$856$		0			0
$857$			29.3751i			1.00343i	0.865032	+	0.501717i	$0.167298\pi$
$857$			29.3751i			1.00343i	−0.865032	+	0.501717i	$0.832702\pi$
$858$		0			0
$859$	36.8720			1.25806			0.629028	−	0.777383i	$-0.283453\pi$
$859$	36.8720			1.25806			0.629028	+	0.777383i	$0.283453\pi$
$860$		0			0
$861$	−5.14116			−0.175210
$862$		0			0
$863$		−	22.7838i		−	0.775569i	−0.921750	−	0.387784i	$-0.873241\pi$
$863$		−	22.7838i		−	0.775569i	0.921750	−	0.387784i	$-0.126759\pi$
$864$		0			0
$865$	−44.3545	+	11.0205i	−1.50810	+	0.374709i
$866$		0			0
$867$			6.44748i			0.218968i
$868$		0			0
$869$	3.00614			0.101976
$870$		0			0
$871$	−18.6875			−0.633203
$872$		0			0
$873$			14.8865i			0.503833i
$874$		0			0
$875$	−8.32684	+	7.46081i	−0.281499	+	0.252221i
$876$		0			0
$877$			8.34471i			0.281781i	0.990025	+	0.140890i	$0.0449965\pi$
$877$			8.34471i			0.281781i	−0.990025	+	0.140890i	$0.955003\pi$
$878$		0			0
$879$	6.01333			0.202825
$880$		0			0
$881$	−26.9926			−0.909405			−0.454702	−	0.890643i	$-0.650254\pi$
$881$	−26.9926			−0.909405			−0.454702	+	0.890643i	$0.650254\pi$
$882$		0			0
$883$		−	6.30179i		−	0.212072i	−0.994362	−	0.106036i	$-0.966184\pi$
$883$		−	6.30179i		−	0.212072i	0.994362	−	0.106036i	$-0.0338159\pi$
$884$		0			0
$885$	0.369102	−	0.0917087i	0.0124072	−	0.00308276i
$886$		0			0
$887$			4.82273i			0.161931i	0.996717	+	0.0809656i	$0.0258004\pi$
$887$			4.82273i			0.161931i	−0.996717	+	0.0809656i	$0.974200\pi$
$888$		0			0
$889$	12.1639			0.407966
$890$		0			0
$891$	−0.539189			−0.0180635
$892$		0			0
$893$		−	19.9649i		−	0.668101i
$894$		0			0
$895$	1.50307	+	6.04945i	0.0502421	+	0.202211i
$896$		0			0
$897$		−	2.24846i		−	0.0750740i
$898$		0			0
$899$	−57.0977			−1.90431
$900$		0			0
$901$	−15.5958			−0.519572
$902$		0			0
$903$		−	4.97107i		−	0.165427i
$904$		0			0
$905$	8.58145	+	34.5380i	0.285257	+	1.14808i
$906$		0			0
$907$			11.4163i			0.379071i	0.981874	+	0.189536i	$0.0606983\pi$
$907$			11.4163i			0.379071i	−0.981874	+	0.189536i	$0.939302\pi$
$908$		0			0
$909$	2.12064			0.0703371
$910$		0			0
$911$	−42.1933			−1.39793			−0.698963	−	0.715158i	$-0.746355\pi$
$911$	−42.1933			−1.39793			−0.698963	+	0.715158i	$0.746355\pi$
$912$		0			0
$913$			0.221802i			0.00734058i
$914$		0			0
$915$	−5.80098	+	1.44134i	−0.191775	+	0.0476491i
$916$		0			0
$917$			7.78539i			0.257096i
$918$		0			0
$919$	15.7731			0.520307			0.260153	−	0.965567i	$-0.416227\pi$
$919$	15.7731			0.520307			0.260153	+	0.965567i	$0.416227\pi$
$920$		0			0
$921$	0.516403			0.0170160
$922$		0			0
$923$		−	25.4030i		−	0.836148i
$924$		0			0
$925$	−16.4969	−	31.1483i	−0.542416	−	1.02415i
$926$		0			0
$927$		−	0.290725i		−	0.00954865i
$928$		0			0
$929$	20.0133			0.656616			0.328308	−	0.944571i	$-0.393522\pi$
$929$	20.0133			0.656616			0.328308	+	0.944571i	$0.393522\pi$
$930$		0			0
$931$	−13.4908			−0.442143
$932$		0			0
$933$		−	11.5031i		−	0.376594i
$934$		0			0
$935$	−3.80098	+	0.944409i	−0.124305	+	0.0308855i
$936$		0			0
$937$			21.0433i			0.687455i	0.939069	+	0.343727i	$0.111690\pi$
$937$			21.0433i			0.687455i	−0.939069	+	0.343727i	$0.888310\pi$
$938$		0			0
$939$	3.70313			0.120847
$940$		0			0
$941$	−24.9204			−0.812382			−0.406191	−	0.913788i	$-0.633143\pi$
$941$	−24.9204			−0.812382			−0.406191	+	0.913788i	$0.633143\pi$
$942$		0			0
$943$			5.14116i			0.167419i
$944$		0			0
$945$	−0.539189	−	2.17009i	−0.0175398	−	0.0705929i
$946$		0			0
$947$			39.9421i			1.29795i	0.760812	+	0.648973i	$0.224801\pi$
$947$			39.9421i			1.29795i	−0.760812	+	0.648973i	$0.775199\pi$
$948$		0			0
$949$	−18.1939			−0.590600
$950$		0			0
$951$	18.9204			0.613536
$952$		0			0
$953$		−	25.8310i		−	0.836747i	−0.908275	−	0.418373i	$-0.862600\pi$
$953$		−	25.8310i		−	0.836747i	0.908275	−	0.418373i	$-0.137400\pi$
$954$		0			0
$955$	0.604236	+	2.43188i	0.0195526	+	0.0786938i
$956$		0			0
$957$		−	4.58864i		−	0.148330i
$958$		0			0
$959$	−9.86603			−0.318591
$960$		0			0
$961$	14.0144			0.452077
$962$		0			0
$963$		−	12.3763i		−	0.398820i
$964$		0			0
$965$	−27.9155	+	6.93600i	−0.898631	+	0.223278i
$966$		0			0
$967$		−	17.3802i		−	0.558908i	−0.960159	−	0.279454i	$-0.909846\pi$
$967$		−	17.3802i		−	0.558908i	0.960159	−	0.279454i	$-0.0901535\pi$
$968$		0			0
$969$	7.30406			0.234640
$970$		0			0
$971$	61.4908			1.97333			0.986667	−	0.162754i	$-0.0520378\pi$
$971$	61.4908			1.97333			0.986667	+	0.162754i	$0.0520378\pi$
$972$		0			0
$973$		−	4.07838i		−	0.130747i
$974$		0			0
$975$	9.93495	−	5.26180i	0.318173	−	0.168512i
$976$		0			0
$977$		−	11.2940i		−	0.361328i	−0.983545	−	0.180664i	$-0.942175\pi$
$977$		−	11.2940i		−	0.361328i	0.983545	−	0.180664i	$-0.0578247\pi$
$978$		0			0
$979$	−3.60197			−0.115119
$980$		0			0
$981$	12.1906			0.389216
$982$		0			0
$983$		−	34.0010i		−	1.08446i	−0.840229	−	0.542232i	$-0.817579\pi$
$983$		−	34.0010i		−	1.08446i	0.840229	−	0.542232i	$-0.182421\pi$
$984$		0			0
$985$	−25.5753	+	6.35455i	−0.814897	+	0.202473i
$986$		0			0
$987$		−	8.87936i		−	0.282633i
$988$		0			0
$989$	−4.97107			−0.158071
$990$		0			0
$991$	35.2023			1.11824			0.559119	−	0.829087i	$-0.311139\pi$
$991$	35.2023			1.11824			0.559119	+	0.829087i	$0.311139\pi$
$992$		0			0
$993$			1.29072i			0.0409599i
$994$		0			0
$995$	−13.5441	−	54.5113i	−0.429377	−	1.72812i
$996$		0			0
$997$		−	1.91548i		−	0.0606638i	−0.999540	−	0.0303319i	$-0.990344\pi$
$997$		−	1.91548i		−	0.0606638i	0.999540	−	0.0303319i	$-0.00965643\pi$
$998$		0			0
$999$	7.04945			0.223035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.f.a.829.5	yes	6
3.2	odd	2		4140.2.f.a.829.4		6
5.2	odd	4		6900.2.a.z.1.2		3
5.3	odd	4		6900.2.a.y.1.2		3
5.4	even	2	inner	1380.2.f.a.829.2	✓	6
15.14	odd	2		4140.2.f.a.829.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.a.829.2	✓	6	5.4	even	2	inner
1380.2.f.a.829.5	yes	6	1.1	even	1	trivial
4140.2.f.a.829.3		6	15.14	odd	2
4140.2.f.a.829.4		6	3.2	odd	2
6900.2.a.y.1.2		3	5.3	odd	4
6900.2.a.z.1.2		3	5.2	odd	4

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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.539189 - 2.17009i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-0.539189 - 2.17009i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -0.539189 q^{11} +2.24846i q^{13} +(2.17009 - 0.539189i) q^{15} +3.24846i q^{17} -2.24846 q^{19} -1.00000 q^{21} +1.00000i q^{23} +(-4.41855 + 2.34017i) q^{25} -1.00000i q^{27} +8.51026 q^{29} -6.70928 q^{31} -0.539189i q^{33} +(2.17009 - 0.539189i) q^{35} +7.04945i q^{37} -2.24846 q^{39} +5.14116 q^{41} +4.97107i q^{43} +(0.539189 + 2.17009i) q^{45} +8.87936i q^{47} +6.00000 q^{49} -3.24846 q^{51} +4.80098i q^{53} +(0.290725 + 1.17009i) q^{55} -2.24846i q^{57} +0.170086 q^{59} -2.67316 q^{61} -1.00000i q^{63} +(4.87936 - 1.21235i) q^{65} +8.31124i q^{67} -1.00000 q^{69} -11.2979 q^{71} +8.09171i q^{73} +(-2.34017 - 4.41855i) q^{75} -0.539189i q^{77} -5.57531 q^{79} +1.00000 q^{81} -0.411363i q^{83} +(7.04945 - 1.75154i) q^{85} +8.51026i q^{87} +6.68035 q^{89} -2.24846 q^{91} -6.70928i q^{93} +(1.21235 + 4.87936i) q^{95} -14.8865i q^{97} +0.539189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 2 q^{15} + 4 q^{19} - 6 q^{21} + 2 q^{25} + 18 q^{29} - 26 q^{31} + 2 q^{35} + 4 q^{39} - 10 q^{41} + 36 q^{49} - 2 q^{51} + 16 q^{55} - 10 q^{59} - 40 q^{61} + 4 q^{65} - 6 q^{69} - 14 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 6 q^{85} - 4 q^{89} + 4 q^{91} + 28 q^{95}+O(q^{100}) \) 6 * q - 6 * q^9 + 2 * q^15 + 4 * q^19 - 6 * q^21 + 2 * q^25 + 18 * q^29 - 26 * q^31 + 2 * q^35 + 4 * q^39 - 10 * q^41 + 36 * q^49 - 2 * q^51 + 16 * q^55 - 10 * q^59 - 40 * q^61 + 4 * q^65 - 6 * q^69 - 14 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 6 * q^85 - 4 * q^89 + 4 * q^91 + 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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