LMFDB - Embedded newform 2040.2.m.i.409.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,2,Mod(409,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.409"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2040.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-12,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.2894820123$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 2 x^{10} + 6 x^{9} - 14 x^{8} - 34 x^{7} + 114 x^{6} - 290 x^{5} + 529 x^{4} + \cdots + 64 $ x^12 - 2x^11 + 2x^10 + 6x^9 - 14x^8 - 34x^7 + 114x^6 - 290x^5 + 529x^4 - 456x^3 + 288x^2 - 192*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			409.5
Root			$-0.0924738 - 1.64386i$ of defining polynomial
Character	$\chi$	$=$	2040.409
Dual form			2040.2.m.i.409.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.42993 - 1.71910i) q^{5} -2.41349i q^{7} -1.00000 q^{9} -4.65192 q^{11} +3.62721i q^{13} +(-1.71910 - 1.42993i) q^{15} +1.00000i q^{17} -6.65192 q^{19} -2.41349 q^{21} +2.60832i q^{23} +(-0.910600 - 4.91638i) q^{25} +1.00000i q^{27} +1.21372 q^{29} -7.43531 q^{31} +4.65192i q^{33} +(-4.14903 - 3.45112i) q^{35} +3.46526i q^{37} +3.62721 q^{39} -10.0276 q^{41} -2.94823i q^{43} +(-1.42993 + 1.71910i) q^{45} +7.64325i q^{47} +1.17506 q^{49} +1.00000 q^{51} -6.35097i q^{53} +(-6.65192 + 7.99711i) q^{55} +6.65192i q^{57} +4.15614 q^{59} +1.31605 q^{61} +2.41349i q^{63} +(6.23554 + 5.18666i) q^{65} -6.78389i q^{67} +2.60832 q^{69} +16.5594 q^{71} -1.73243i q^{73} +(-4.91638 + 0.910600i) q^{75} +11.2274i q^{77} -11.1579 q^{79} +1.00000 q^{81} -4.11304i q^{83} +(1.71910 + 1.42993i) q^{85} -1.21372i q^{87} +5.54381 q^{89} +8.75425 q^{91} +7.43531i q^{93} +(-9.51178 + 11.4353i) q^{95} +8.66795i q^{97} +4.65192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9} + 4 q^{11} - 20 q^{19} + 4 q^{21} - 4 q^{29} + 16 q^{31} - 12 q^{35} - 8 q^{39} - 4 q^{41} + 8 q^{49} + 12 q^{51} - 20 q^{55} + 8 q^{59} + 8 q^{61} - 16 q^{65} - 8 q^{69} + 32 q^{71} - 4 q^{75}+ \cdots - 4 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 + 4 * q^11 - 20 * q^19 + 4 * q^21 - 4 * q^29 + 16 * q^31 - 12 * q^35 - 8 * q^39 - 4 * q^41 + 8 * q^49 + 12 * q^51 - 20 * q^55 + 8 * q^59 + 8 * q^61 - 16 * q^65 - 8 * q^69 + 32 * q^71 - 4 * q^75 - 24 * q^79 + 12 * q^81 + 40 * q^89 + 56 * q^91 - 20 * q^95 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$5$	1.42993	−	1.71910i	0.639484	−	0.768804i
$5$	1.42993	−	1.71910i	0.639484	−	0.768804i
$6$		0			0
$7$		−	2.41349i		−	0.912214i	−0.889925	−	0.456107i	$-0.849243\pi$
$7$		−	2.41349i		−	0.912214i	0.889925	−	0.456107i	$-0.150757\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−4.65192			−1.40261			−0.701303	−	0.712863i	$-0.747398\pi$
$11$	−4.65192			−1.40261			−0.701303	+	0.712863i	$0.747398\pi$
$12$		0			0
$13$			3.62721i			1.00601i	0.864284	+	0.503004i	$0.167772\pi$
$13$			3.62721i			1.00601i	−0.864284	+	0.503004i	$0.832228\pi$
$14$		0			0
$15$	−1.71910	−	1.42993i	−0.443869	−	0.369206i
$16$		0			0
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$	−6.65192			−1.52605			−0.763027	−	0.646366i	$-0.776288\pi$
$19$	−6.65192			−1.52605			−0.763027	+	0.646366i	$0.776288\pi$
$20$		0			0
$21$	−2.41349			−0.526667
$22$		0			0
$23$			2.60832i			0.543873i	0.962315	+	0.271937i	$0.0876641\pi$
$23$			2.60832i			0.543873i	−0.962315	+	0.271937i	$0.912336\pi$
$24$		0			0
$25$	−0.910600	−	4.91638i	−0.182120	−	0.983276i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	1.21372			0.225383			0.112691	−	0.993630i	$-0.464053\pi$
$29$	1.21372			0.225383			0.112691	+	0.993630i	$0.464053\pi$
$30$		0			0
$31$	−7.43531			−1.33542			−0.667710	−	0.744421i	$-0.732725\pi$
$31$	−7.43531			−1.33542			−0.667710	+	0.744421i	$0.732725\pi$
$32$		0			0
$33$			4.65192i			0.809795i
$34$		0			0
$35$	−4.14903	−	3.45112i	−0.701314	−	0.583346i
$36$		0			0
$37$			3.46526i			0.569685i	0.958574	+	0.284842i	$0.0919413\pi$
$37$			3.46526i			0.569685i	−0.958574	+	0.284842i	$0.908059\pi$
$38$		0			0
$39$	3.62721			0.580819
$40$		0			0
$41$	−10.0276			−1.56605			−0.783024	−	0.621992i	$-0.786324\pi$
$41$	−10.0276			−1.56605			−0.783024	+	0.621992i	$0.786324\pi$
$42$		0			0
$43$		−	2.94823i		−	0.449601i	−0.974405	−	0.224801i	$-0.927827\pi$
$43$		−	2.94823i		−	0.449601i	0.974405	−	0.224801i	$-0.0721731\pi$
$44$		0			0
$45$	−1.42993	+	1.71910i	−0.213161	+	0.256268i
$46$		0			0
$47$			7.64325i			1.11488i	0.830217	+	0.557441i	$0.188217\pi$
$47$			7.64325i			1.11488i	−0.830217	+	0.557441i	$0.811783\pi$
$48$		0			0
$49$	1.17506			0.167866
$50$		0			0
$51$	1.00000			0.140028
$52$		0			0
$53$		−	6.35097i		−	0.872373i	−0.899856	−	0.436186i	$-0.856329\pi$
$53$		−	6.35097i		−	0.872373i	0.899856	−	0.436186i	$-0.143671\pi$
$54$		0			0
$55$	−6.65192	+	7.99711i	−0.896945	+	1.07833i
$56$		0			0
$57$			6.65192i			0.881068i
$58$		0			0
$59$	4.15614			0.541083			0.270541	−	0.962708i	$-0.412797\pi$
$59$	4.15614			0.541083			0.270541	+	0.962708i	$0.412797\pi$
$60$		0			0
$61$	1.31605			0.168503			0.0842513	−	0.996445i	$-0.473150\pi$
$61$	1.31605			0.168503			0.0842513	+	0.996445i	$0.473150\pi$
$62$		0			0
$63$			2.41349i			0.304071i
$64$		0			0
$65$	6.23554	+	5.18666i	0.773423	+	0.643326i
$66$		0			0
$67$		−	6.78389i		−	0.828784i	−0.910099	−	0.414392i	$-0.863994\pi$
$67$		−	6.78389i		−	0.828784i	0.910099	−	0.414392i	$-0.136006\pi$
$68$		0			0
$69$	2.60832			0.314005
$70$		0			0
$71$	16.5594			1.96524			0.982622	−	0.185616i	$-0.0594281\pi$
$71$	16.5594			1.96524			0.982622	+	0.185616i	$0.0594281\pi$
$72$		0			0
$73$		−	1.73243i		−	0.202766i	−0.994847	−	0.101383i	$-0.967673\pi$
$73$		−	1.73243i		−	0.202766i	0.994847	−	0.101383i	$-0.0323267\pi$
$74$		0			0
$75$	−4.91638	+	0.910600i	−0.567695	+	0.105147i
$76$		0			0
$77$			11.2274i			1.27948i
$78$		0			0
$79$	−11.1579			−1.25536			−0.627682	−	0.778470i	$-0.715996\pi$
$79$	−11.1579			−1.25536			−0.627682	+	0.778470i	$0.715996\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.11304i		−	0.451465i	−0.974189	−	0.225733i	$-0.927522\pi$
$83$		−	4.11304i		−	0.451465i	0.974189	−	0.225733i	$-0.0724775\pi$
$84$		0			0
$85$	1.71910	+	1.42993i	0.186462	+	0.155098i
$86$		0			0
$87$		−	1.21372i		−	0.130125i
$88$		0			0
$89$	5.54381			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$89$	5.54381			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$90$		0			0
$91$	8.75425			0.917694
$92$		0			0
$93$			7.43531i			0.771005i
$94$		0			0
$95$	−9.51178	+	11.4353i	−0.975888	+	1.17324i
$96$		0			0
$97$			8.66795i			0.880097i	0.897974	+	0.440049i	$0.145039\pi$
$97$			8.66795i			0.880097i	−0.897974	+	0.440049i	$0.854961\pi$
$98$		0			0
$99$	4.65192			0.467535
$100$		0			0
$101$	−6.53235			−0.649993			−0.324997	−	0.945715i	$-0.605363\pi$
$101$	−6.53235			−0.649993			−0.324997	+	0.945715i	$0.605363\pi$
$102$		0			0
$103$		−	12.1945i		−	1.20156i	−0.799414	−	0.600781i	$-0.794856\pi$
$103$		−	12.1945i		−	1.20156i	0.799414	−	0.600781i	$-0.205144\pi$
$104$		0			0
$105$	−3.45112	+	4.14903i	−0.336795	+	0.404904i
$106$		0			0
$107$		−	5.03163i		−	0.486426i	−0.969973	−	0.243213i	$-0.921799\pi$
$107$		−	5.03163i		−	0.486426i	0.969973	−	0.243213i	$-0.0782015\pi$
$108$		0			0
$109$	−1.90349			−0.182321			−0.0911607	−	0.995836i	$-0.529058\pi$
$109$	−1.90349			−0.182321			−0.0911607	+	0.995836i	$0.529058\pi$
$110$		0			0
$111$	3.46526			0.328908
$112$		0			0
$113$			3.20320i			0.301331i	0.988585	+	0.150666i	$0.0481417\pi$
$113$			3.20320i			0.301331i	−0.988585	+	0.150666i	$0.951858\pi$
$114$		0			0
$115$	4.48397	+	3.72972i	0.418132	+	0.347798i
$116$		0			0
$117$		−	3.62721i		−	0.335336i
$118$		0			0
$119$	2.41349			0.221244
$120$		0			0
$121$	10.6404			0.967305
$122$		0			0
$123$			10.0276i			0.904158i
$124$		0			0
$125$	−9.75384	−	5.46467i	−0.872410	−	0.488775i
$126$		0			0
$127$		−	3.59322i		−	0.318847i	−0.987210	−	0.159424i	$-0.949036\pi$
$127$		−	3.59322i		−	0.318847i	0.987210	−	0.159424i	$-0.0509636\pi$
$128$		0			0
$129$	−2.94823			−0.259577
$130$		0			0
$131$	−17.4984			−1.52884			−0.764419	−	0.644719i	$-0.776974\pi$
$131$	−17.4984			−1.52884			−0.764419	+	0.644719i	$0.776974\pi$
$132$		0			0
$133$			16.0543i			1.39209i
$134$		0			0
$135$	1.71910	+	1.42993i	0.147956	+	0.123069i
$136$		0			0
$137$		−	10.1460i		−	0.866827i	−0.901195	−	0.433414i	$-0.857309\pi$
$137$		−	10.1460i		−	0.866827i	0.901195	−	0.433414i	$-0.142691\pi$
$138$		0			0
$139$	−18.1308			−1.53784			−0.768918	−	0.639347i	$-0.779205\pi$
$139$	−18.1308			−1.53784			−0.768918	+	0.639347i	$0.779205\pi$
$140$		0			0
$141$	7.64325			0.643677
$142$		0			0
$143$		−	16.8735i		−	1.41103i
$144$		0			0
$145$	1.73554	−	2.08651i	0.144129	−	0.173275i
$146$		0			0
$147$		−	1.17506i		−	0.0969175i
$148$		0			0
$149$	0.271307			0.0222263			0.0111132	−	0.999938i	$-0.496462\pi$
$149$	0.271307			0.0222263			0.0111132	+	0.999938i	$0.496462\pi$
$150$		0			0
$151$	−4.39749			−0.357863			−0.178931	−	0.983862i	$-0.557264\pi$
$151$	−4.39749			−0.357863			−0.178931	+	0.983862i	$0.557264\pi$
$152$		0			0
$153$		−	1.00000i		−	0.0808452i
$154$		0			0
$155$	−10.6320	+	12.7820i	−0.853980	+	1.02668i
$156$		0			0
$157$		−	4.88453i		−	0.389828i	−0.980820	−	0.194914i	$-0.937557\pi$
$157$		−	4.88453i		−	0.389828i	0.980820	−	0.194914i	$-0.0624428\pi$
$158$		0			0
$159$	−6.35097			−0.503665
$160$		0			0
$161$	6.29517			0.496129
$162$		0			0
$163$		−	3.73622i		−	0.292643i	−0.989237	−	0.146322i	$-0.953257\pi$
$163$		−	3.73622i		−	0.292643i	0.989237	−	0.146322i	$-0.0467435\pi$
$164$		0			0
$165$	7.99711	+	6.65192i	0.622574	+	0.517851i
$166$		0			0
$167$			18.6763i			1.44521i	0.691259	+	0.722607i	$0.257056\pi$
$167$			18.6763i			1.44521i	−0.691259	+	0.722607i	$0.742944\pi$
$168$		0			0
$169$	−0.156674			−0.0120519
$170$		0			0
$171$	6.65192			0.508685
$172$		0			0
$173$		−	3.39168i		−	0.257864i	−0.991653	−	0.128932i	$-0.958845\pi$
$173$		−	3.39168i		−	0.257864i	0.991653	−	0.128932i	$-0.0411550\pi$
$174$		0			0
$175$	−11.8656	+	2.19772i	−0.896958	+	0.166132i
$176$		0			0
$177$		−	4.15614i		−	0.312394i
$178$		0			0
$179$	−10.0412			−0.750515			−0.375257	−	0.926921i	$-0.622446\pi$
$179$	−10.0412			−0.750515			−0.375257	+	0.926921i	$0.622446\pi$
$180$		0			0
$181$	−5.34811			−0.397522			−0.198761	−	0.980048i	$-0.563692\pi$
$181$	−5.34811			−0.397522			−0.198761	+	0.980048i	$0.563692\pi$
$182$		0			0
$183$		−	1.31605i		−	0.0972851i
$184$		0			0
$185$	5.95712	+	4.95508i	0.437976	+	0.364305i
$186$		0			0
$187$		−	4.65192i		−	0.340182i
$188$		0			0
$189$	2.41349			0.175556
$190$		0			0
$191$	4.01445			0.290476			0.145238	−	0.989397i	$-0.453605\pi$
$191$	4.01445			0.290476			0.145238	+	0.989397i	$0.453605\pi$
$192$		0			0
$193$		−	22.6795i		−	1.63251i	−0.577694	−	0.816254i	$-0.696047\pi$
$193$		−	22.6795i		−	1.63251i	0.577694	−	0.816254i	$-0.303953\pi$
$194$		0			0
$195$	5.18666	−	6.23554i	0.371425	−	0.446536i
$196$		0			0
$197$			25.4440i			1.81281i	0.422412	+	0.906404i	$0.361184\pi$
$197$			25.4440i			1.81281i	−0.422412	+	0.906404i	$0.638816\pi$
$198$		0			0
$199$	−17.9355			−1.27141			−0.635706	−	0.771931i	$-0.719291\pi$
$199$	−17.9355			−1.27141			−0.635706	+	0.771931i	$0.719291\pi$
$200$		0			0
$201$	−6.78389			−0.478499
$202$		0			0
$203$		−	2.92931i		−	0.205597i
$204$		0			0
$205$	−14.3388	+	17.2384i	−1.00146	+	1.20398i
$206$		0			0
$207$		−	2.60832i		−	0.181291i
$208$		0			0
$209$	30.9442			2.14045
$210$		0			0
$211$	−12.6240			−0.869074			−0.434537	−	0.900654i	$-0.643088\pi$
$211$	−12.6240			−0.869074			−0.434537	+	0.900654i	$0.643088\pi$
$212$		0			0
$213$		−	16.5594i		−	1.13463i
$214$		0			0
$215$	−5.06830	−	4.21577i	−0.345655	−	0.287513i
$216$		0			0
$217$			17.9450i			1.21819i
$218$		0			0
$219$	−1.73243			−0.117067
$220$		0			0
$221$	−3.62721			−0.243993
$222$		0			0
$223$		−	16.0440i		−	1.07439i	−0.843460	−	0.537193i	$-0.819485\pi$
$223$		−	16.0440i		−	1.07439i	0.843460	−	0.537193i	$-0.180515\pi$
$224$		0			0
$225$	0.910600	+	4.91638i	0.0607067	+	0.327759i
$226$		0			0
$227$			14.9392i			0.991549i	0.868451	+	0.495774i	$0.165116\pi$
$227$			14.9392i			0.991549i	−0.868451	+	0.495774i	$0.834884\pi$
$228$		0			0
$229$	−19.7555			−1.30548			−0.652739	−	0.757583i	$-0.726380\pi$
$229$	−19.7555			−1.30548			−0.652739	+	0.757583i	$0.726380\pi$
$230$		0			0
$231$	11.2274			0.738706
$232$		0			0
$233$			19.1079i			1.25180i	0.779904	+	0.625899i	$0.215268\pi$
$233$			19.1079i			1.25180i	−0.779904	+	0.625899i	$0.784732\pi$
$234$		0			0
$235$	13.1395	+	10.9293i	0.857126	+	0.712949i
$236$		0			0
$237$			11.1579i			0.724785i
$238$		0			0
$239$	−7.23997			−0.468315			−0.234157	−	0.972199i	$-0.575233\pi$
$239$	−7.23997			−0.468315			−0.234157	+	0.972199i	$0.575233\pi$
$240$		0			0
$241$	−25.2468			−1.62629			−0.813145	−	0.582061i	$-0.802247\pi$
$241$	−25.2468			−1.62629			−0.813145	+	0.582061i	$0.802247\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	1.68026	−	2.02005i	0.107348	−	0.129056i
$246$		0			0
$247$		−	24.1279i		−	1.53522i
$248$		0			0
$249$	−4.11304			−0.260654
$250$		0			0
$251$	14.9474			0.943471			0.471736	−	0.881740i	$-0.343628\pi$
$251$	14.9474			0.943471			0.471736	+	0.881740i	$0.343628\pi$
$252$		0			0
$253$		−	12.1337i		−	0.762840i
$254$		0			0
$255$	1.42993	−	1.71910i	0.0895457	−	0.107654i
$256$		0			0
$257$			11.8165i			0.737092i	0.929609	+	0.368546i	$0.120144\pi$
$257$			11.8165i			0.737092i	−0.929609	+	0.368546i	$0.879856\pi$
$258$		0			0
$259$	8.36337			0.519674
$260$		0			0
$261$	−1.21372			−0.0751275
$262$		0			0
$263$		−	29.9008i		−	1.84376i	−0.387473	−	0.921881i	$-0.626652\pi$
$263$		−	29.9008i		−	1.84376i	0.387473	−	0.921881i	$-0.373348\pi$
$264$		0			0
$265$	−10.9179	−	9.08144i	−0.670684	−	0.557869i
$266$		0			0
$267$		−	5.54381i		−	0.339276i
$268$		0			0
$269$	15.7400			0.959684			0.479842	−	0.877355i	$-0.340694\pi$
$269$	15.7400			0.959684			0.479842	+	0.877355i	$0.340694\pi$
$270$		0			0
$271$	27.0474			1.64301			0.821505	−	0.570201i	$-0.193135\pi$
$271$	27.0474			1.64301			0.821505	+	0.570201i	$0.193135\pi$
$272$		0			0
$273$		−	8.75425i		−	0.529831i
$274$		0			0
$275$	4.23604	+	22.8706i	0.255443	+	1.37915i
$276$		0			0
$277$		−	11.4045i		−	0.685230i	−0.939476	−	0.342615i	$-0.888687\pi$
$277$		−	11.4045i		−	0.685230i	0.939476	−	0.342615i	$-0.111313\pi$
$278$		0			0
$279$	7.43531			0.445140
$280$		0			0
$281$	31.2791			1.86595			0.932976	−	0.359938i	$-0.117202\pi$
$281$	31.2791			1.86595			0.932976	+	0.359938i	$0.117202\pi$
$282$		0			0
$283$			2.75051i			0.163501i	0.996653	+	0.0817503i	$0.0260510\pi$
$283$			2.75051i			0.163501i	−0.996653	+	0.0817503i	$0.973949\pi$
$284$		0			0
$285$	11.4353	+	9.51178i	0.677369	+	0.563429i
$286$		0			0
$287$			24.2015i			1.42857i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	8.66795			0.508124
$292$		0			0
$293$		−	29.9097i		−	1.74734i	−0.486518	−	0.873671i	$-0.661733\pi$
$293$		−	29.9097i		−	1.74734i	0.486518	−	0.873671i	$-0.338267\pi$
$294$		0			0
$295$	5.94299	−	7.14481i	0.346014	−	0.415987i
$296$		0			0
$297$		−	4.65192i		−	0.269932i
$298$		0			0
$299$	−9.46095			−0.547141
$300$		0			0
$301$	−7.11553			−0.410132
$302$		0			0
$303$			6.53235i			0.375274i
$304$		0			0
$305$	1.88186	−	2.26242i	0.107755	−	0.129546i
$306$		0			0
$307$		−	13.5938i		−	0.775837i	−0.921694	−	0.387918i	$-0.873194\pi$
$307$		−	13.5938i		−	0.775837i	0.921694	−	0.387918i	$-0.126806\pi$
$308$		0			0
$309$	−12.1945			−0.693722
$310$		0			0
$311$	8.30743			0.471071			0.235535	−	0.971866i	$-0.424316\pi$
$311$	8.30743			0.471071			0.235535	+	0.971866i	$0.424316\pi$
$312$		0			0
$313$		−	1.35440i		−	0.0765552i	−0.999267	−	0.0382776i	$-0.987813\pi$
$313$		−	1.35440i		−	0.0765552i	0.999267	−	0.0382776i	$-0.0121871\pi$
$314$		0			0
$315$	4.14903	+	3.45112i	0.233771	+	0.194449i
$316$		0			0
$317$			2.02208i			0.113571i	0.998386	+	0.0567857i	$0.0180852\pi$
$317$			2.02208i			0.113571i	−0.998386	+	0.0567857i	$0.981915\pi$
$318$		0			0
$319$	−5.64614			−0.316123
$320$		0			0
$321$	−5.03163			−0.280838
$322$		0			0
$323$		−	6.65192i		−	0.370123i
$324$		0			0
$325$	17.8328	−	3.30294i	0.989184	−	0.183214i
$326$		0			0
$327$			1.90349i			0.105263i
$328$		0			0
$329$	18.4469			1.01701
$330$		0			0
$331$	−6.58722			−0.362066			−0.181033	−	0.983477i	$-0.557944\pi$
$331$	−6.58722			−0.362066			−0.181033	+	0.983477i	$0.557944\pi$
$332$		0			0
$333$		−	3.46526i		−	0.189895i
$334$		0			0
$335$	−11.6622	−	9.70048i	−0.637172	−	0.529994i
$336$		0			0
$337$		−	1.11090i		−	0.0605143i	−0.999542	−	0.0302572i	$-0.990367\pi$
$337$		−	1.11090i		−	0.0605143i	0.999542	−	0.0302572i	$-0.00963262\pi$
$338$		0			0
$339$	3.20320			0.173974
$340$		0			0
$341$	34.5884			1.87307
$342$		0			0
$343$		−	19.7304i		−	1.06534i
$344$		0			0
$345$	3.72972	−	4.48397i	0.200801	−	0.241409i
$346$		0			0
$347$		−	10.5874i		−	0.568360i	−0.958771	−	0.284180i	$-0.908279\pi$
$347$		−	10.5874i		−	0.568360i	0.958771	−	0.284180i	$-0.0917213\pi$
$348$		0			0
$349$	−24.1172			−1.29096			−0.645482	−	0.763775i	$-0.723344\pi$
$349$	−24.1172			−1.29096			−0.645482	+	0.763775i	$0.723344\pi$
$350$		0			0
$351$	−3.62721			−0.193606
$352$		0			0
$353$			20.8929i			1.11202i	0.831177	+	0.556008i	$0.187668\pi$
$353$			20.8929i			1.11202i	−0.831177	+	0.556008i	$0.812332\pi$
$354$		0			0
$355$	23.6788	−	28.4673i	1.25674	−	1.51089i
$356$		0			0
$357$		−	2.41349i		−	0.127735i
$358$		0			0
$359$	12.8218			0.676709			0.338354	−	0.941019i	$-0.390130\pi$
$359$	12.8218			0.676709			0.338354	+	0.941019i	$0.390130\pi$
$360$		0			0
$361$	25.2480			1.32884
$362$		0			0
$363$		−	10.6404i		−	0.558474i
$364$		0			0
$365$	−2.97822	−	2.47725i	−0.155887	−	0.129665i
$366$		0			0
$367$		−	12.2329i		−	0.638551i	−0.947662	−	0.319276i	$-0.896560\pi$
$367$		−	12.2329i		−	0.638551i	0.947662	−	0.319276i	$-0.103440\pi$
$368$		0			0
$369$	10.0276			0.522016
$370$		0			0
$371$	−15.3280			−0.795791
$372$		0			0
$373$		−	32.7589i		−	1.69619i	−0.529844	−	0.848095i	$-0.677750\pi$
$373$		−	32.7589i		−	1.69619i	0.529844	−	0.848095i	$-0.322250\pi$
$374$		0			0
$375$	−5.46467	+	9.75384i	−0.282194	+	0.503686i
$376$		0			0
$377$			4.40243i			0.226737i
$378$		0			0
$379$	−19.8034			−1.01723			−0.508617	−	0.860993i	$-0.669843\pi$
$379$	−19.8034			−1.01723			−0.508617	+	0.860993i	$0.669843\pi$
$380$		0			0
$381$	−3.59322			−0.184086
$382$		0			0
$383$			10.8115i			0.552441i	0.961094	+	0.276220i	$0.0890820\pi$
$383$			10.8115i			0.552441i	−0.961094	+	0.276220i	$0.910918\pi$
$384$		0			0
$385$	19.3009	+	16.0543i	0.983667	+	0.818205i
$386$		0			0
$387$			2.94823i			0.149867i
$388$		0			0
$389$	−15.2105			−0.771203			−0.385601	−	0.922665i	$-0.626006\pi$
$389$	−15.2105			−0.771203			−0.385601	+	0.922665i	$0.626006\pi$
$390$		0			0
$391$	−2.60832			−0.131909
$392$		0			0
$393$			17.4984i			0.882676i
$394$		0			0
$395$	−15.9550	+	19.1816i	−0.802785	+	0.965129i
$396$		0			0
$397$			7.36643i			0.369711i	0.982766	+	0.184855i	$0.0591817\pi$
$397$			7.36643i			0.369711i	−0.982766	+	0.184855i	$0.940818\pi$
$398$		0			0
$399$	16.0543			0.803723
$400$		0			0
$401$	−32.4755			−1.62175			−0.810874	−	0.585221i	$-0.801008\pi$
$401$	−32.4755			−1.62175			−0.810874	+	0.585221i	$0.801008\pi$
$402$		0			0
$403$		−	26.9694i		−	1.34344i
$404$		0			0
$405$	1.42993	−	1.71910i	0.0710538	−	0.0854227i
$406$		0			0
$407$		−	16.1201i		−	0.799044i
$408$		0			0
$409$	23.9935			1.18640			0.593202	−	0.805054i	$-0.297864\pi$
$409$	23.9935			1.18640			0.593202	+	0.805054i	$0.297864\pi$
$410$		0			0
$411$	−10.1460			−0.500463
$412$		0			0
$413$		−	10.0308i		−	0.493583i
$414$		0			0
$415$	−7.07073	−	5.88136i	−0.347088	−	0.288705i
$416$		0			0
$417$			18.1308i			0.887870i
$418$		0			0
$419$	−15.9394			−0.778691			−0.389346	−	0.921092i	$-0.627299\pi$
$419$	−15.9394			−0.778691			−0.389346	+	0.921092i	$0.627299\pi$
$420$		0			0
$421$	−26.8290			−1.30757			−0.653783	−	0.756682i	$-0.726819\pi$
$421$	−26.8290			−1.30757			−0.653783	+	0.756682i	$0.726819\pi$
$422$		0			0
$423$		−	7.64325i		−	0.371627i
$424$		0			0
$425$	4.91638	−	0.910600i	0.238480	−	0.0441706i
$426$		0			0
$427$		−	3.17627i		−	0.153710i
$428$		0			0
$429$	−16.8735			−0.814660
$430$		0			0
$431$	−16.5673			−0.798017			−0.399009	−	0.916947i	$-0.630646\pi$
$431$	−16.5673			−0.798017			−0.399009	+	0.916947i	$0.630646\pi$
$432$		0			0
$433$		−	8.35634i		−	0.401580i	−0.979634	−	0.200790i	$-0.935649\pi$
$433$		−	8.35634i		−	0.401580i	0.979634	−	0.200790i	$-0.0643509\pi$
$434$		0			0
$435$	−2.08651	−	1.73554i	−0.100040	−	0.0832127i
$436$		0			0
$437$		−	17.3504i		−	0.829980i
$438$		0			0
$439$	37.0665			1.76909			0.884545	−	0.466455i	$-0.154469\pi$
$439$	37.0665			1.76909			0.884545	+	0.466455i	$0.154469\pi$
$440$		0			0
$441$	−1.17506			−0.0559553
$442$		0			0
$443$			41.3532i			1.96475i	0.186915	+	0.982376i	$0.440151\pi$
$443$			41.3532i			1.96475i	−0.186915	+	0.982376i	$0.559849\pi$
$444$		0			0
$445$	7.92726	−	9.53036i	0.375788	−	0.451782i
$446$		0			0
$447$		−	0.271307i		−	0.0128324i
$448$		0			0
$449$	24.0906			1.13691			0.568453	−	0.822716i	$-0.307542\pi$
$449$	24.0906			1.13691			0.568453	+	0.822716i	$0.307542\pi$
$450$		0			0
$451$	46.6476			2.19655
$452$		0			0
$453$			4.39749i			0.206612i
$454$		0			0
$455$	12.5180	−	15.0494i	0.586851	−	0.705527i
$456$		0			0
$457$		−	12.5492i		−	0.587028i	−0.955955	−	0.293514i	$-0.905175\pi$
$457$		−	12.5492i		−	0.587028i	0.955955	−	0.293514i	$-0.0948248\pi$
$458$		0			0
$459$	−1.00000			−0.0466760
$460$		0			0
$461$	−6.94125			−0.323286			−0.161643	−	0.986849i	$-0.551679\pi$
$461$	−6.94125			−0.323286			−0.161643	+	0.986849i	$0.551679\pi$
$462$		0			0
$463$		−	30.9138i		−	1.43669i	−0.695689	−	0.718343i	$-0.744901\pi$
$463$		−	30.9138i		−	1.43669i	0.695689	−	0.718343i	$-0.255099\pi$
$464$		0			0
$465$	12.7820	+	10.6320i	0.592752	+	0.493046i
$466$		0			0
$467$		−	10.0979i		−	0.467275i	−0.972324	−	0.233638i	$-0.924937\pi$
$467$		−	10.0979i		−	0.467275i	0.972324	−	0.233638i	$-0.0750630\pi$
$468$		0			0
$469$	−16.3728			−0.756028
$470$		0			0
$471$	−4.88453			−0.225067
$472$		0			0
$473$			13.7149i			0.630614i
$474$		0			0
$475$	6.05724	+	32.7034i	0.277925	+	1.50053i
$476$		0			0
$477$			6.35097i			0.290791i
$478$		0			0
$479$	33.2723			1.52025			0.760126	−	0.649776i	$-0.225137\pi$
$479$	33.2723			1.52025			0.760126	+	0.649776i	$0.225137\pi$
$480$		0			0
$481$	−12.5692			−0.573108
$482$		0			0
$483$		−	6.29517i		−	0.286440i
$484$		0			0
$485$	14.9011	+	12.3946i	0.676623	+	0.562808i
$486$		0			0
$487$			25.7880i			1.16857i	0.811550	+	0.584283i	$0.198624\pi$
$487$			25.7880i			1.16857i	−0.811550	+	0.584283i	$0.801376\pi$
$488$		0			0
$489$	−3.73622			−0.168958
$490$		0			0
$491$	26.0082			1.17373			0.586867	−	0.809683i	$-0.300361\pi$
$491$	26.0082			1.17373			0.586867	+	0.809683i	$0.300361\pi$
$492$		0			0
$493$			1.21372i			0.0546633i
$494$		0			0
$495$	6.65192	−	7.99711i	0.298982	−	0.359443i
$496$		0			0
$497$		−	39.9661i		−	1.79272i
$498$		0			0
$499$	23.3224			1.04405			0.522027	−	0.852929i	$-0.325176\pi$
$499$	23.3224			1.04405			0.522027	+	0.852929i	$0.325176\pi$
$500$		0			0
$501$	18.6763			0.834395
$502$		0			0
$503$		−	11.9436i		−	0.532537i	−0.963899	−	0.266269i	$-0.914209\pi$
$503$		−	11.9436i		−	0.532537i	0.963899	−	0.266269i	$-0.0857908\pi$
$504$		0			0
$505$	−9.34080	+	11.2298i	−0.415660	+	0.499718i
$506$		0			0
$507$			0.156674i			0.00695814i
$508$		0			0
$509$	1.17863			0.0522417			0.0261209	−	0.999659i	$-0.491685\pi$
$509$	1.17863			0.0522417			0.0261209	+	0.999659i	$0.491685\pi$
$510$		0			0
$511$	−4.18121			−0.184966
$512$		0			0
$513$		−	6.65192i		−	0.293689i
$514$		0			0
$515$	−20.9636	−	17.4373i	−0.923766	−	0.768380i
$516$		0			0
$517$		−	35.5558i		−	1.56374i
$518$		0			0
$519$	−3.39168			−0.148878
$520$		0			0
$521$	24.0747			1.05473			0.527366	−	0.849638i	$-0.323180\pi$
$521$	24.0747			1.05473			0.527366	+	0.849638i	$0.323180\pi$
$522$		0			0
$523$			13.8144i			0.604063i	0.953298	+	0.302031i	$0.0976647\pi$
$523$			13.8144i			0.604063i	−0.953298	+	0.302031i	$0.902335\pi$
$524$		0			0
$525$	2.19772	+	11.8656i	0.0959166	+	0.517859i
$526$		0			0
$527$		−	7.43531i		−	0.323887i
$528$		0			0
$529$	16.1966			0.704202
$530$		0			0
$531$	−4.15614			−0.180361
$532$		0			0
$533$		−	36.3722i		−	1.57546i
$534$		0			0
$535$	−8.64987	−	7.19488i	−0.373967	−	0.311062i
$536$		0			0
$537$			10.0412i			0.433310i
$538$		0			0
$539$	−5.46629			−0.235450
$540$		0			0
$541$	−31.6533			−1.36088			−0.680441	−	0.732803i	$-0.738212\pi$
$541$	−31.6533			−1.36088			−0.680441	+	0.732803i	$0.738212\pi$
$542$		0			0
$543$			5.34811i			0.229510i
$544$		0			0
$545$	−2.72186	+	3.27229i	−0.116592	+	0.140169i
$546$		0			0
$547$		−	10.1899i		−	0.435691i	−0.975983	−	0.217845i	$-0.930097\pi$
$547$		−	10.1899i		−	0.435691i	0.975983	−	0.217845i	$-0.0699028\pi$
$548$		0			0
$549$	−1.31605			−0.0561676
$550$		0			0
$551$	−8.07358			−0.343946
$552$		0			0
$553$			26.9295i			1.14516i
$554$		0			0
$555$	4.95508	−	5.95712i	0.210331	−	0.252866i
$556$		0			0
$557$		−	26.6108i		−	1.12753i	−0.825934	−	0.563767i	$-0.809352\pi$
$557$		−	26.6108i		−	1.12753i	0.825934	−	0.563767i	$-0.190648\pi$
$558$		0			0
$559$	10.6939			0.452302
$560$		0			0
$561$	−4.65192			−0.196404
$562$		0			0
$563$		−	20.6769i		−	0.871429i	−0.900085	−	0.435714i	$-0.856496\pi$
$563$		−	20.6769i		−	0.871429i	0.900085	−	0.435714i	$-0.143504\pi$
$564$		0			0
$565$	5.50661	+	4.58035i	0.231665	+	0.192697i
$566$		0			0
$567$		−	2.41349i		−	0.101357i
$568$		0			0
$569$	−24.5961			−1.03112			−0.515561	−	0.856853i	$-0.672417\pi$
$569$	−24.5961			−1.03112			−0.515561	+	0.856853i	$0.672417\pi$
$570$		0			0
$571$	2.13842			0.0894902			0.0447451	−	0.998998i	$-0.485752\pi$
$571$	2.13842			0.0894902			0.0447451	+	0.998998i	$0.485752\pi$
$572$		0			0
$573$		−	4.01445i		−	0.167706i
$574$		0			0
$575$	12.8235	−	2.37514i	0.534778	−	0.0990502i
$576$		0			0
$577$			38.1523i			1.58830i	0.607721	+	0.794151i	$0.292084\pi$
$577$			38.1523i			1.58830i	−0.607721	+	0.794151i	$0.707916\pi$
$578$		0			0
$579$	−22.6795			−0.942529
$580$		0			0
$581$	−9.92679			−0.411833
$582$		0			0
$583$			29.5442i			1.22360i
$584$		0			0
$585$	−6.23554	−	5.18666i	−0.257808	−	0.214442i
$586$		0			0
$587$			0.574837i			0.0237261i	0.999930	+	0.0118630i	$0.00377621\pi$
$587$			0.574837i			0.0237261i	−0.999930	+	0.0118630i	$0.996224\pi$
$588$		0			0
$589$	49.4591			2.03792
$590$		0			0
$591$	25.4440			1.04663
$592$		0			0
$593$		−	11.9977i		−	0.492687i	−0.969183	−	0.246344i	$-0.920771\pi$
$593$		−	11.9977i		−	0.492687i	0.969183	−	0.246344i	$-0.0792292\pi$
$594$		0			0
$595$	3.45112	−	4.14903i	0.141482	−	0.170094i
$596$		0			0
$597$			17.9355i			0.734051i
$598$		0			0
$599$	33.4831			1.36808			0.684041	−	0.729444i	$-0.260221\pi$
$599$	33.4831			1.36808			0.684041	+	0.729444i	$0.260221\pi$
$600$		0			0
$601$	6.45177			0.263173			0.131587	−	0.991305i	$-0.457993\pi$
$601$	6.45177			0.263173			0.131587	+	0.991305i	$0.457993\pi$
$602$		0			0
$603$			6.78389i			0.276261i
$604$		0			0
$605$	15.2150	−	18.2918i	0.618576	−	0.743668i
$606$		0			0
$607$			37.5464i			1.52396i	0.647601	+	0.761980i	$0.275773\pi$
$607$			37.5464i			1.52396i	−0.647601	+	0.761980i	$0.724227\pi$
$608$		0			0
$609$	−2.92931			−0.118702
$610$		0			0
$611$	−27.7237			−1.12158
$612$		0			0
$613$		−	34.4269i		−	1.39049i	−0.718773	−	0.695244i	$-0.755296\pi$
$613$		−	34.4269i		−	1.39049i	0.718773	−	0.695244i	$-0.244704\pi$
$614$		0			0
$615$	17.2384	+	14.3388i	0.695120	+	0.578195i
$616$		0			0
$617$			46.5507i			1.87406i	0.349250	+	0.937030i	$0.386436\pi$
$617$			46.5507i			1.87406i	−0.349250	+	0.937030i	$0.613564\pi$
$618$		0			0
$619$	−10.4479			−0.419936			−0.209968	−	0.977708i	$-0.567336\pi$
$619$	−10.4479			−0.419936			−0.209968	+	0.977708i	$0.567336\pi$
$620$		0			0
$621$	−2.60832			−0.104668
$622$		0			0
$623$		−	13.3799i		−	0.536056i
$624$		0			0
$625$	−23.3416	+	8.95371i	−0.933665	+	0.358149i
$626$		0			0
$627$		−	30.9442i		−	1.23579i
$628$		0			0
$629$	−3.46526			−0.138169
$630$		0			0
$631$	−32.5916			−1.29745			−0.648725	−	0.761023i	$-0.724698\pi$
$631$	−32.5916			−1.29745			−0.648725	+	0.761023i	$0.724698\pi$
$632$		0			0
$633$			12.6240i			0.501760i
$634$		0			0
$635$	−6.17711	−	5.13806i	−0.245131	−	0.203898i
$636$		0			0
$637$			4.26220i			0.168875i
$638$		0			0
$639$	−16.5594			−0.655082
$640$		0			0
$641$	−20.5801			−0.812864			−0.406432	−	0.913681i	$-0.633227\pi$
$641$	−20.5801			−0.812864			−0.406432	+	0.913681i	$0.633227\pi$
$642$		0			0
$643$		−	16.0608i		−	0.633377i	−0.948530	−	0.316688i	$-0.897429\pi$
$643$		−	16.0608i		−	0.633377i	0.948530	−	0.316688i	$-0.102571\pi$
$644$		0			0
$645$	−4.21577	+	5.06830i	−0.165996	+	0.199564i
$646$		0			0
$647$		−	14.9857i		−	0.589149i	−0.955629	−	0.294575i	$-0.904822\pi$
$647$		−	14.9857i		−	0.589149i	0.955629	−	0.294575i	$-0.0951779\pi$
$648$		0			0
$649$	−19.3340			−0.758926
$650$		0			0
$651$	17.9450			0.703322
$652$		0			0
$653$		−	27.0095i		−	1.05696i	−0.848944	−	0.528482i	$-0.822761\pi$
$653$		−	27.0095i		−	1.05696i	0.848944	−	0.528482i	$-0.177239\pi$
$654$		0			0
$655$	−25.0214	+	30.0814i	−0.977668	+	1.17538i
$656$		0			0
$657$			1.73243i			0.0675886i
$658$		0			0
$659$	−27.0266			−1.05281			−0.526404	−	0.850235i	$-0.676460\pi$
$659$	−27.0266			−1.05281			−0.526404	+	0.850235i	$0.676460\pi$
$660$		0			0
$661$	42.0903			1.63712			0.818562	−	0.574419i	$-0.194772\pi$
$661$	42.0903			1.63712			0.818562	+	0.574419i	$0.194772\pi$
$662$		0			0
$663$			3.62721i			0.140869i
$664$		0			0
$665$	27.5990	+	22.9566i	1.07024	+	0.890218i
$666$		0			0
$667$			3.16578i			0.122580i
$668$		0			0
$669$	−16.0440			−0.620297
$670$		0			0
$671$	−6.12215			−0.236343
$672$		0			0
$673$		−	42.5546i		−	1.64036i	−0.572105	−	0.820181i	$-0.693873\pi$
$673$		−	42.5546i		−	1.64036i	0.572105	−	0.820181i	$-0.306127\pi$
$674$		0			0
$675$	4.91638	−	0.910600i	0.189232	−	0.0350490i
$676$		0			0
$677$		−	4.44192i		−	0.170717i	−0.996350	−	0.0853585i	$-0.972796\pi$
$677$		−	4.44192i		−	0.170717i	0.996350	−	0.0853585i	$-0.0272036\pi$
$678$		0			0
$679$	20.9200			0.802837
$680$		0			0
$681$	14.9392			0.572471
$682$		0			0
$683$			50.4204i			1.92928i	0.263568	+	0.964641i	$0.415101\pi$
$683$			50.4204i			1.92928i	−0.263568	+	0.964641i	$0.584899\pi$
$684$		0			0
$685$	−17.4419	−	14.5080i	−0.666421	−	0.554322i
$686$		0			0
$687$			19.7555i			0.753718i
$688$		0			0
$689$	23.0363			0.877614
$690$		0			0
$691$	41.0903			1.56315			0.781575	−	0.623812i	$-0.214417\pi$
$691$	41.0903			1.56315			0.781575	+	0.623812i	$0.214417\pi$
$692$		0			0
$693$		−	11.2274i		−	0.426492i
$694$		0			0
$695$	−25.9258	+	31.1687i	−0.983422	+	1.18229i
$696$		0			0
$697$		−	10.0276i		−	0.379822i
$698$		0			0
$699$	19.1079			0.722726
$700$		0			0
$701$	−3.65598			−0.138084			−0.0690422	−	0.997614i	$-0.521994\pi$
$701$	−3.65598			−0.138084			−0.0690422	+	0.997614i	$0.521994\pi$
$702$		0			0
$703$		−	23.0506i		−	0.869371i
$704$		0			0
$705$	10.9293	−	13.1395i	0.411622	−	0.494862i
$706$		0			0
$707$			15.7658i			0.592933i
$708$		0			0
$709$	−0.347352			−0.0130451			−0.00652253	−	0.999979i	$-0.502076\pi$
$709$	−0.347352			−0.0130451			−0.00652253	+	0.999979i	$0.502076\pi$
$710$		0			0
$711$	11.1579			0.418455
$712$		0			0
$713$		−	19.3937i		−	0.726299i
$714$		0			0
$715$	−29.0072	−	24.1279i	−1.08481	−	0.902333i
$716$		0			0
$717$			7.23997i			0.270382i
$718$		0			0
$719$	41.8701			1.56149			0.780746	−	0.624849i	$-0.214839\pi$
$719$	41.8701			1.56149			0.780746	+	0.624849i	$0.214839\pi$
$720$		0			0
$721$	−29.4314			−1.09608
$722$		0			0
$723$			25.2468i			0.938940i
$724$		0			0
$725$	−1.10522	−	5.96712i	−0.0410467	−	0.221613i
$726$		0			0
$727$			30.6150i			1.13545i	0.823220	+	0.567723i	$0.192176\pi$
$727$			30.6150i			1.13545i	−0.823220	+	0.567723i	$0.807824\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	2.94823			0.109044
$732$		0			0
$733$			37.7122i			1.39293i	0.717590	+	0.696466i	$0.245245\pi$
$733$			37.7122i			1.39293i	−0.717590	+	0.696466i	$0.754755\pi$
$734$		0			0
$735$	−2.02005	−	1.68026i	−0.0745106	−	0.0619772i
$736$		0			0
$737$			31.5581i			1.16246i
$738$		0			0
$739$	−28.1417			−1.03521			−0.517605	−	0.855619i	$-0.673176\pi$
$739$	−28.1417			−1.03521			−0.517605	+	0.855619i	$0.673176\pi$
$740$		0			0
$741$	−24.1279			−0.886362
$742$		0			0
$743$			20.1838i			0.740471i	0.928938	+	0.370235i	$0.120723\pi$
$743$			20.1838i			0.740471i	−0.928938	+	0.370235i	$0.879277\pi$
$744$		0			0
$745$	0.387950	−	0.466404i	0.0142134	−	0.0170877i
$746$		0			0
$747$			4.11304i			0.150488i
$748$		0			0
$749$	−12.1438			−0.443725
$750$		0			0
$751$	−6.27963			−0.229147			−0.114574	−	0.993415i	$-0.536550\pi$
$751$	−6.27963			−0.229147			−0.114574	+	0.993415i	$0.536550\pi$
$752$		0			0
$753$		−	14.9474i		−	0.544713i
$754$		0			0
$755$	−6.28811	+	7.55972i	−0.228848	+	0.275126i
$756$		0			0
$757$		−	0.0623304i		−	0.00226544i	−0.999999	−	0.00113272i	$-0.999639\pi$
$757$		−	0.0623304i		−	0.00226544i	0.999999	−	0.00113272i	$-0.000360556\pi$
$758$		0			0
$759$	−12.1337			−0.440426
$760$		0			0
$761$	−5.39043			−0.195403			−0.0977015	−	0.995216i	$-0.531149\pi$
$761$	−5.39043			−0.195403			−0.0977015	+	0.995216i	$0.531149\pi$
$762$		0			0
$763$			4.59406i			0.166316i
$764$		0			0
$765$	−1.71910	−	1.42993i	−0.0621541	−	0.0516992i
$766$		0			0
$767$			15.0752i			0.544334i
$768$		0			0
$769$	10.7412			0.387339			0.193669	−	0.981067i	$-0.437961\pi$
$769$	10.7412			0.387339			0.193669	+	0.981067i	$0.437961\pi$
$770$		0			0
$771$	11.8165			0.425561
$772$		0			0
$773$		−	50.8382i		−	1.82852i	−0.405126	−	0.914261i	$-0.632772\pi$
$773$		−	50.8382i		−	1.82852i	0.405126	−	0.914261i	$-0.367228\pi$
$774$		0			0
$775$	6.77059	+	36.5548i	0.243207	+	1.31309i
$776$		0			0
$777$		−	8.36337i		−	0.300034i
$778$		0			0
$779$	66.7028			2.38987
$780$		0			0
$781$	−77.0332			−2.75646
$782$		0			0
$783$			1.21372i			0.0433749i
$784$		0			0
$785$	−8.39699	−	6.98454i	−0.299701	−	0.249289i
$786$		0			0
$787$			47.5523i			1.69506i	0.530749	+	0.847529i	$0.321911\pi$
$787$			47.5523i			1.69506i	−0.530749	+	0.847529i	$0.678089\pi$
$788$		0			0
$789$	−29.9008			−1.06450
$790$		0			0
$791$	7.73088			0.274879
$792$		0			0
$793$			4.77359i			0.169515i
$794$		0			0
$795$	−9.08144	+	10.9179i	−0.322086	+	0.387220i
$796$		0			0
$797$		−	23.2489i		−	0.823517i	−0.911293	−	0.411759i	$-0.864915\pi$
$797$		−	23.2489i		−	0.823517i	0.911293	−	0.411759i	$-0.135085\pi$
$798$		0			0
$799$	−7.64325			−0.270399
$800$		0			0
$801$	−5.54381			−0.195881
$802$		0			0
$803$			8.05913i			0.284400i
$804$		0			0
$805$	9.00165	−	10.8220i	0.317266	−	0.381426i
$806$		0			0
$807$		−	15.7400i		−	0.554074i
$808$		0			0
$809$	48.4670			1.70401			0.852005	−	0.523534i	$-0.175387\pi$
$809$	48.4670			1.70401			0.852005	+	0.523534i	$0.175387\pi$
$810$		0			0
$811$	27.5894			0.968796			0.484398	−	0.874848i	$-0.339039\pi$
$811$	27.5894			0.968796			0.484398	+	0.874848i	$0.339039\pi$
$812$		0			0
$813$		−	27.0474i		−	0.948592i
$814$		0			0
$815$	−6.42293	−	5.34253i	−0.224986	−	0.187141i
$816$		0			0
$817$			19.6114i			0.686116i
$818$		0			0
$819$	−8.75425			−0.305898
$820$		0			0
$821$	2.76951			0.0966566			0.0483283	−	0.998832i	$-0.484611\pi$
$821$	2.76951			0.0966566			0.0483283	+	0.998832i	$0.484611\pi$
$822$		0			0
$823$			20.4456i			0.712687i	0.934355	+	0.356344i	$0.115977\pi$
$823$			20.4456i			0.712687i	−0.934355	+	0.356344i	$0.884023\pi$
$824$		0			0
$825$	22.8706	−	4.23604i	0.796252	−	0.147480i
$826$		0			0
$827$			29.7493i			1.03448i	0.855839	+	0.517242i	$0.173041\pi$
$827$			29.7493i			1.03448i	−0.855839	+	0.517242i	$0.826959\pi$
$828$		0			0
$829$	−8.81029			−0.305994			−0.152997	−	0.988227i	$-0.548892\pi$
$829$	−8.81029			−0.305994			−0.152997	+	0.988227i	$0.548892\pi$
$830$		0			0
$831$	−11.4045			−0.395618
$832$		0			0
$833$			1.17506i			0.0407135i
$834$		0			0
$835$	32.1064	+	26.7058i	1.11109	+	0.924191i
$836$		0			0
$837$		−	7.43531i		−	0.257002i
$838$		0			0
$839$	−30.3579			−1.04807			−0.524036	−	0.851696i	$-0.675574\pi$
$839$	−30.3579			−1.04807			−0.524036	+	0.851696i	$0.675574\pi$
$840$		0			0
$841$	−27.5269			−0.949203
$842$		0			0
$843$		−	31.2791i		−	1.07731i
$844$		0			0
$845$	−0.224033	+	0.269338i	−0.00770697	+	0.00926552i
$846$		0			0
$847$		−	25.6804i		−	0.882389i
$848$		0			0
$849$	2.75051			0.0943972
$850$		0			0
$851$	−9.03852			−0.309836
$852$		0			0
$853$			9.91876i			0.339612i	0.985478	+	0.169806i	$0.0543141\pi$
$853$			9.91876i			0.339612i	−0.985478	+	0.169806i	$0.945686\pi$
$854$		0			0
$855$	9.51178	−	11.4353i	0.325296	−	0.391079i
$856$		0			0
$857$		−	40.5012i		−	1.38350i	−0.722139	−	0.691748i	$-0.756841\pi$
$857$		−	40.5012i		−	1.38350i	0.722139	−	0.691748i	$-0.243159\pi$
$858$		0			0
$859$	10.4163			0.355400			0.177700	−	0.984085i	$-0.443134\pi$
$859$	10.4163			0.355400			0.177700	+	0.984085i	$0.443134\pi$
$860$		0			0
$861$	24.2015			0.824785
$862$		0			0
$863$			29.9264i			1.01871i	0.860558	+	0.509353i	$0.170115\pi$
$863$			29.9264i			1.01871i	−0.860558	+	0.509353i	$0.829885\pi$
$864$		0			0
$865$	−5.83063	−	4.84986i	−0.198247	−	0.164900i
$866$		0			0
$867$			1.00000i			0.0339618i
$868$		0			0
$869$	51.9057			1.76078
$870$		0			0
$871$	24.6066			0.833763
$872$		0			0
$873$		−	8.66795i		−	0.293366i
$874$		0			0
$875$	−13.1889	+	23.5408i	−0.445867	+	0.795824i
$876$		0			0
$877$		−	49.5750i		−	1.67403i	−0.547180	−	0.837015i	$-0.684299\pi$
$877$		−	49.5750i		−	1.67403i	0.547180	−	0.837015i	$-0.315701\pi$
$878$		0			0
$879$	−29.9097			−1.00883
$880$		0			0
$881$	−41.4197			−1.39547			−0.697733	−	0.716358i	$-0.745808\pi$
$881$	−41.4197			−1.39547			−0.697733	+	0.716358i	$0.745808\pi$
$882$		0			0
$883$		−	57.6726i		−	1.94084i	−0.241426	−	0.970419i	$-0.577615\pi$
$883$		−	57.6726i		−	1.94084i	0.241426	−	0.970419i	$-0.422385\pi$
$884$		0			0
$885$	−7.14481	−	5.94299i	−0.240170	−	0.199771i
$886$		0			0
$887$		−	27.9702i		−	0.939147i	−0.882893	−	0.469573i	$-0.844408\pi$
$887$		−	27.9702i		−	0.939147i	0.882893	−	0.469573i	$-0.155592\pi$
$888$		0			0
$889$	−8.67222			−0.290857
$890$		0			0
$891$	−4.65192			−0.155845
$892$		0			0
$893$		−	50.8423i		−	1.70137i
$894$		0			0
$895$	−14.3582	+	17.2618i	−0.479942	+	0.576999i
$896$		0			0
$897$			9.46095i			0.315892i
$898$		0			0
$899$	−9.02440			−0.300980
$900$		0			0
$901$	6.35097			0.211581
$902$		0			0
$903$			7.11553i			0.236790i
$904$		0			0
$905$	−7.64743	+	9.19394i	−0.254209	+	0.305617i
$906$		0			0
$907$			53.1133i			1.76360i	0.471626	+	0.881799i	$0.343667\pi$
$907$			53.1133i			1.76360i	−0.471626	+	0.881799i	$0.656333\pi$
$908$		0			0
$909$	6.53235			0.216664
$910$		0			0
$911$	−3.23393			−0.107145			−0.0535724	−	0.998564i	$-0.517061\pi$
$911$	−3.23393			−0.107145			−0.0535724	+	0.998564i	$0.517061\pi$
$912$		0			0
$913$			19.1335i			0.633228i
$914$		0			0
$915$	−2.26242	−	1.88186i	−0.0747932	−	0.0622123i
$916$		0			0
$917$			42.2321i			1.39463i
$918$		0			0
$919$	−49.8990			−1.64602			−0.823008	−	0.568030i	$-0.807706\pi$
$919$	−49.8990			−1.64602			−0.823008	+	0.568030i	$0.807706\pi$
$920$		0			0
$921$	−13.5938			−0.447930
$922$		0			0
$923$			60.0646i			1.97705i
$924$		0			0
$925$	17.0365	−	3.15546i	0.560158	−	0.103751i
$926$		0			0
$927$			12.1945i			0.400521i
$928$		0			0
$929$	12.7837			0.419418			0.209709	−	0.977764i	$-0.432748\pi$
$929$	12.7837			0.419418			0.209709	+	0.977764i	$0.432748\pi$
$930$		0			0
$931$	−7.81642			−0.256173
$932$		0			0
$933$		−	8.30743i		−	0.271973i
$934$		0			0
$935$	−7.99711	−	6.65192i	−0.261533	−	0.217541i
$936$		0			0
$937$			48.3516i			1.57958i	0.613380	+	0.789788i	$0.289810\pi$
$937$			48.3516i			1.57958i	−0.613380	+	0.789788i	$0.710190\pi$
$938$		0			0
$939$	−1.35440			−0.0441991
$940$		0			0
$941$	−47.1424			−1.53680			−0.768399	−	0.639971i	$-0.778946\pi$
$941$	−47.1424			−1.53680			−0.768399	+	0.639971i	$0.778946\pi$
$942$		0			0
$943$		−	26.1552i		−	0.851731i
$944$		0			0
$945$	3.45112	−	4.14903i	0.112265	−	0.134968i
$946$		0			0
$947$		−	35.4078i		−	1.15060i	−0.817943	−	0.575299i	$-0.804886\pi$
$947$		−	35.4078i		−	1.15060i	0.817943	−	0.575299i	$-0.195114\pi$
$948$		0			0
$949$	6.28389			0.203984
$950$		0			0
$951$	2.02208			0.0655704
$952$		0			0
$953$		−	46.1846i		−	1.49606i	−0.663662	−	0.748032i	$-0.730999\pi$
$953$		−	46.1846i		−	1.49606i	0.663662	−	0.748032i	$-0.269001\pi$
$954$		0			0
$955$	5.74039	−	6.90124i	0.185755	−	0.223319i
$956$		0			0
$957$			5.64614i			0.182514i
$958$		0			0
$959$	−24.4872			−0.790732
$960$		0			0
$961$	24.2838			0.783347
$962$		0			0
$963$			5.03163i			0.162142i
$964$		0			0
$965$	−38.9883	−	32.4301i	−1.25508	−	1.04396i
$966$		0			0
$967$			54.0845i			1.73924i	0.493721	+	0.869620i	$0.335636\pi$
$967$			54.0845i			1.73924i	−0.493721	+	0.869620i	$0.664364\pi$
$968$		0			0
$969$	−6.65192			−0.213690
$970$		0			0
$971$	30.9595			0.993536			0.496768	−	0.867883i	$-0.334520\pi$
$971$	30.9595			0.993536			0.496768	+	0.867883i	$0.334520\pi$
$972$		0			0
$973$			43.7586i			1.40284i
$974$		0			0
$975$	−3.30294	−	17.8328i	−0.105779	−	0.571105i
$976$		0			0
$977$		−	6.78722i		−	0.217142i	−0.994089	−	0.108571i	$-0.965372\pi$
$977$		−	6.78722i		−	0.217142i	0.994089	−	0.108571i	$-0.0346275\pi$
$978$		0			0
$979$	−25.7894			−0.824232
$980$		0			0
$981$	1.90349			0.0607738
$982$		0			0
$983$		−	18.1512i		−	0.578934i	−0.957188	−	0.289467i	$-0.906522\pi$
$983$		−	18.1512i		−	0.578934i	0.957188	−	0.289467i	$-0.0934781\pi$
$984$		0			0
$985$	43.7407	+	36.3831i	1.39369	+	1.15926i
$986$		0			0
$987$		−	18.4469i		−	0.587171i
$988$		0			0
$989$	7.68994			0.244526
$990$		0			0
$991$	60.4926			1.92161			0.960805	−	0.277226i	$-0.0894152\pi$
$991$	60.4926			1.92161			0.960805	+	0.277226i	$0.0894152\pi$
$992$		0			0
$993$			6.58722i			0.209039i
$994$		0			0
$995$	−25.6465	+	30.8329i	−0.813048	+	0.977468i
$996$		0			0
$997$		−	23.5773i		−	0.746700i	−0.927691	−	0.373350i	$-0.878209\pi$
$997$		−	23.5773i		−	0.746700i	0.927691	−	0.373350i	$-0.121791\pi$
$998$		0			0
$999$	−3.46526			−0.109636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.2.m.i.409.5	✓	12
4.3	odd	2		4080.2.m.t.2449.11		12
5.4	even	2	inner	2040.2.m.i.409.11	yes	12
20.19	odd	2		4080.2.m.t.2449.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.m.i.409.5	✓	12	1.1	even	1	trivial
2040.2.m.i.409.11	yes	12	5.4	even	2	inner
4080.2.m.t.2449.5		12	20.19	odd	2
4080.2.m.t.2449.11		12	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2040.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.42993 - 1.71910i) q^{5} -2.41349i q^{7} -1.00000 q^{9} -4.65192 q^{11} +3.62721i q^{13} +(-1.71910 - 1.42993i) q^{15} +1.00000i q^{17} -6.65192 q^{19} -2.41349 q^{21} +2.60832i q^{23} +(-0.910600 - 4.91638i) q^{25} +1.00000i q^{27} +1.21372 q^{29} -7.43531 q^{31} +4.65192i q^{33} +(-4.14903 - 3.45112i) q^{35} +3.46526i q^{37} +3.62721 q^{39} -10.0276 q^{41} -2.94823i q^{43} +(-1.42993 + 1.71910i) q^{45} +7.64325i q^{47} +1.17506 q^{49} +1.00000 q^{51} -6.35097i q^{53} +(-6.65192 + 7.99711i) q^{55} +6.65192i q^{57} +4.15614 q^{59} +1.31605 q^{61} +2.41349i q^{63} +(6.23554 + 5.18666i) q^{65} -6.78389i q^{67} +2.60832 q^{69} +16.5594 q^{71} -1.73243i q^{73} +(-4.91638 + 0.910600i) q^{75} +11.2274i q^{77} -11.1579 q^{79} +1.00000 q^{81} -4.11304i q^{83} +(1.71910 + 1.42993i) q^{85} -1.21372i q^{87} +5.54381 q^{89} +8.75425 q^{91} +7.43531i q^{93} +(-9.51178 + 11.4353i) q^{95} +8.66795i q^{97} +4.65192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9} + 4 q^{11} - 20 q^{19} + 4 q^{21} - 4 q^{29} + 16 q^{31} - 12 q^{35} - 8 q^{39} - 4 q^{41} + 8 q^{49} + 12 q^{51} - 20 q^{55} + 8 q^{59} + 8 q^{61} - 16 q^{65} - 8 q^{69} + 32 q^{71} - 4 q^{75}+ \cdots - 4 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 + 4 * q^11 - 20 * q^19 + 4 * q^21 - 4 * q^29 + 16 * q^31 - 12 * q^35 - 8 * q^39 - 4 * q^41 + 8 * q^49 + 12 * q^51 - 20 * q^55 + 8 * q^59 + 8 * q^61 - 16 * q^65 - 8 * q^69 + 32 * q^71 - 4 * q^75 - 24 * q^79 + 12 * q^81 + 40 * q^89 + 56 * q^91 - 20 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more