LMFDB - Embedded newform 2380.2.m.d.169.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2380,2,Mod(169,2380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2380.169"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2380.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [26,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$19.0043956811$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.16
Character	$\chi$	$=$	2380.169
Dual form			2380.2.m.d.169.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.807362 q^{3} +(-2.19195 - 0.442004i) q^{5} +1.00000 q^{7} -2.34817 q^{9} +0.748338i q^{11} -2.28620i q^{13} +(-1.76969 - 0.356857i) q^{15} +(0.793630 + 4.04600i) q^{17} +0.712341 q^{19} +0.807362 q^{21} +2.80575 q^{23} +(4.60927 + 1.93770i) q^{25} -4.31791 q^{27} -5.30790i q^{29} +4.52957i q^{31} +0.604179i q^{33} +(-2.19195 - 0.442004i) q^{35} -9.33442 q^{37} -1.84579i q^{39} -1.92803i q^{41} +6.53531i q^{43} +(5.14706 + 1.03790i) q^{45} +8.50706i q^{47} +1.00000 q^{49} +(0.640747 + 3.26659i) q^{51} +13.3808i q^{53} +(0.330768 - 1.64032i) q^{55} +0.575117 q^{57} +6.74114 q^{59} -1.53774i q^{61} -2.34817 q^{63} +(-1.01051 + 5.01124i) q^{65} +4.60728i q^{67} +2.26525 q^{69} +12.8158i q^{71} -3.39600 q^{73} +(3.72134 + 1.56442i) q^{75} +0.748338i q^{77} +5.22533i q^{79} +3.55839 q^{81} +6.69151i q^{83} +(0.0487536 - 9.21942i) q^{85} -4.28539i q^{87} +0.864356 q^{89} -2.28620i q^{91} +3.65700i q^{93} +(-1.56141 - 0.314858i) q^{95} +7.00477 q^{97} -1.75722i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q + 6 q^{3} - 4 q^{5} + 26 q^{7} + 32 q^{9} + 10 q^{15} - 11 q^{17} + 6 q^{21} - 16 q^{23} + 4 q^{25} + 6 q^{27} - 4 q^{35} - 20 q^{37} - 18 q^{45} + 26 q^{49} + 9 q^{51} - 14 q^{55} + 32 q^{59} + 32 q^{63}+ \cdots - 46 q^{97}+O(q^{100}) $ 26 * q + 6 * q^3 - 4 * q^5 + 26 * q^7 + 32 * q^9 + 10 * q^15 - 11 * q^17 + 6 * q^21 - 16 * q^23 + 4 * q^25 + 6 * q^27 - 4 * q^35 - 20 * q^37 - 18 * q^45 + 26 * q^49 + 9 * q^51 - 14 * q^55 + 32 * q^59 + 32 * q^63 - 42 * q^65 - 8 * q^69 - 12 * q^73 + 8 * q^75 + 66 * q^81 + 11 * q^85 + 4 * q^89 + 20 * q^95 - 46 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2380\mathbb{Z}\right)^\times$.

$n$	$477$	$1191$	$1261$	$1361$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0.807362			0.466130			0.233065	−	0.972461i	$-0.425124\pi$
$3$	0.807362			0.466130			0.233065	+	0.972461i	$0.425124\pi$
$4$		0			0
$5$	−2.19195	−	0.442004i	−0.980269	−	0.197670i
$5$	−2.19195	−	0.442004i	−0.980269	−	0.197670i
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$	−2.34817			−0.782722
$10$		0			0
$11$			0.748338i			0.225632i	0.993616	+	0.112816i	$0.0359871\pi$
$11$			0.748338i			0.225632i	−0.993616	+	0.112816i	$0.964013\pi$
$12$		0			0
$13$		−	2.28620i		−	0.634079i	−0.948412	−	0.317039i	$-0.897311\pi$
$13$		−	2.28620i		−	0.634079i	0.948412	−	0.317039i	$-0.102689\pi$
$14$		0			0
$15$	−1.76969	−	0.356857i	−0.456933	−	0.0921400i
$16$		0			0
$17$	0.793630	+	4.04600i	0.192484	+	0.981300i
$17$	0.793630	+	4.04600i	0.192484	+	0.981300i
$18$		0			0
$19$	0.712341			0.163422			0.0817112	−	0.996656i	$-0.473962\pi$
$19$	0.712341			0.163422			0.0817112	+	0.996656i	$0.473962\pi$
$20$		0			0
$21$	0.807362			0.176181
$22$		0			0
$23$	2.80575			0.585039			0.292519	−	0.956260i	$-0.405506\pi$
$23$	2.80575			0.585039			0.292519	+	0.956260i	$0.405506\pi$
$24$		0			0
$25$	4.60927	+	1.93770i	0.921853	+	0.387540i
$26$		0			0
$27$	−4.31791			−0.830981
$28$		0			0
$29$		−	5.30790i		−	0.985652i	−0.870128	−	0.492826i	$-0.835964\pi$
$29$		−	5.30790i		−	0.985652i	0.870128	−	0.492826i	$-0.164036\pi$
$30$		0			0
$31$			4.52957i			0.813535i	0.913532	+	0.406768i	$0.133344\pi$
$31$			4.52957i			0.813535i	−0.913532	+	0.406768i	$0.866656\pi$
$32$		0			0
$33$			0.604179i			0.105174i
$34$		0			0
$35$	−2.19195	−	0.442004i	−0.370507	−	0.0747123i
$36$		0			0
$37$	−9.33442			−1.53457			−0.767284	−	0.641307i	$-0.778393\pi$
$37$	−9.33442			−1.53457			−0.767284	+	0.641307i	$0.778393\pi$
$38$		0			0
$39$		−	1.84579i		−	0.295563i
$40$		0			0
$41$		−	1.92803i		−	0.301108i	−0.988602	−	0.150554i	$-0.951894\pi$
$41$		−	1.92803i		−	0.301108i	0.988602	−	0.150554i	$-0.0481058\pi$
$42$		0			0
$43$			6.53531i			0.996626i	0.866997	+	0.498313i	$0.166047\pi$
$43$			6.53531i			0.996626i	−0.866997	+	0.498313i	$0.833953\pi$
$44$		0			0
$45$	5.14706	+	1.03790i	0.767278	+	0.154721i
$46$		0			0
$47$			8.50706i			1.24088i	0.784253	+	0.620441i	$0.213046\pi$
$47$			8.50706i			1.24088i	−0.784253	+	0.620441i	$0.786954\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	0.640747	+	3.26659i	0.0897225	+	0.457414i
$52$		0			0
$53$			13.3808i			1.83799i	0.394268	+	0.918996i	$0.370998\pi$
$53$			13.3808i			1.83799i	−0.394268	+	0.918996i	$0.629002\pi$
$54$		0			0
$55$	0.330768	−	1.64032i	0.0446008	−	0.221180i
$56$		0			0
$57$	0.575117			0.0761761
$58$		0			0
$59$	6.74114			0.877622			0.438811	−	0.898579i	$-0.355400\pi$
$59$	6.74114			0.877622			0.438811	+	0.898579i	$0.355400\pi$
$60$		0			0
$61$		−	1.53774i		−	0.196888i	−0.995143	−	0.0984440i	$-0.968613\pi$
$61$		−	1.53774i		−	0.196888i	0.995143	−	0.0984440i	$-0.0313865\pi$
$62$		0			0
$63$	−2.34817			−0.295841
$64$		0			0
$65$	−1.01051	+	5.01124i	−0.125338	+	0.621567i
$66$		0			0
$67$			4.60728i			0.562868i	0.959581	+	0.281434i	$0.0908101\pi$
$67$			4.60728i			0.562868i	−0.959581	+	0.281434i	$0.909190\pi$
$68$		0			0
$69$	2.26525			0.272704
$70$		0			0
$71$			12.8158i			1.52096i	0.649362	+	0.760479i	$0.275036\pi$
$71$			12.8158i			1.52096i	−0.649362	+	0.760479i	$0.724964\pi$
$72$		0			0
$73$	−3.39600			−0.397472			−0.198736	−	0.980053i	$-0.563684\pi$
$73$	−3.39600			−0.397472			−0.198736	+	0.980053i	$0.563684\pi$
$74$		0			0
$75$	3.72134	+	1.56442i	0.429704	+	0.180644i
$76$		0			0
$77$			0.748338i			0.0852810i
$78$		0			0
$79$			5.22533i			0.587895i	0.955822	+	0.293948i	$0.0949691\pi$
$79$			5.22533i			0.587895i	−0.955822	+	0.293948i	$0.905031\pi$
$80$		0			0
$81$	3.55839			0.395377
$82$		0			0
$83$			6.69151i			0.734488i	0.930125	+	0.367244i	$0.119699\pi$
$83$			6.69151i			0.734488i	−0.930125	+	0.367244i	$0.880301\pi$
$84$		0			0
$85$	0.0487536	−	9.21942i	0.00528807	−	0.999986i
$86$		0			0
$87$		−	4.28539i		−	0.459443i
$88$		0			0
$89$	0.864356			0.0916216			0.0458108	−	0.998950i	$-0.485413\pi$
$89$	0.864356			0.0916216			0.0458108	+	0.998950i	$0.485413\pi$
$90$		0			0
$91$		−	2.28620i		−	0.239659i
$92$		0			0
$93$			3.65700i			0.379214i
$94$		0			0
$95$	−1.56141	−	0.314858i	−0.160198	−	0.0323037i
$96$		0			0
$97$	7.00477			0.711226			0.355613	−	0.934633i	$-0.384272\pi$
$97$	7.00477			0.711226			0.355613	+	0.934633i	$0.384272\pi$
$98$		0			0
$99$		−	1.75722i		−	0.176608i
$100$		0			0
$101$	−13.9278			−1.38587			−0.692935	−	0.721000i	$-0.743683\pi$
$101$	−13.9278			−1.38587			−0.692935	+	0.721000i	$0.743683\pi$
$102$		0			0
$103$			6.50691i			0.641145i	0.947224	+	0.320573i	$0.103875\pi$
$103$			6.50691i			0.641145i	−0.947224	+	0.320573i	$0.896125\pi$
$104$		0			0
$105$	−1.76969	−	0.356857i	−0.172704	−	0.0348257i
$106$		0			0
$107$	2.71392			0.262364			0.131182	−	0.991358i	$-0.458123\pi$
$107$	2.71392			0.262364			0.131182	+	0.991358i	$0.458123\pi$
$108$		0			0
$109$		−	9.90433i		−	0.948663i	−0.880346	−	0.474332i	$-0.842690\pi$
$109$		−	9.90433i		−	0.948663i	0.880346	−	0.474332i	$-0.157310\pi$
$110$		0			0
$111$	−7.53625			−0.715309
$112$		0			0
$113$	−0.715790			−0.0673359			−0.0336679	−	0.999433i	$-0.510719\pi$
$113$	−0.715790			−0.0673359			−0.0336679	+	0.999433i	$0.510719\pi$
$114$		0			0
$115$	−6.15005	−	1.24015i	−0.573495	−	0.115645i
$116$		0			0
$117$			5.36839i			0.496307i
$118$		0			0
$119$	0.793630	+	4.04600i	0.0727520	+	0.370897i
$120$		0			0
$121$	10.4400			0.949090
$122$		0			0
$123$		−	1.55662i		−	0.140356i
$124$		0			0
$125$	−9.24680	−	6.28464i	−0.827059	−	0.562116i
$126$		0			0
$127$		−	1.67651i		−	0.148766i	−0.997230	−	0.0743829i	$-0.976301\pi$
$127$		−	1.67651i		−	0.148766i	0.997230	−	0.0743829i	$-0.0236987\pi$
$128$		0			0
$129$			5.27636i			0.464558i
$130$		0			0
$131$			16.9660i			1.48233i	0.671326	+	0.741163i	$0.265725\pi$
$131$			16.9660i			1.48233i	−0.671326	+	0.741163i	$0.734275\pi$
$132$		0			0
$133$	0.712341			0.0617678
$134$		0			0
$135$	9.46462	+	1.90853i	0.814585	+	0.164260i
$136$		0			0
$137$			21.2590i			1.81628i	0.418672	+	0.908138i	$0.362496\pi$
$137$			21.2590i			1.81628i	−0.418672	+	0.908138i	$0.637504\pi$
$138$		0			0
$139$		−	12.8748i		−	1.09203i	−0.837776	−	0.546015i	$-0.816144\pi$
$139$		−	12.8748i		−	1.09203i	0.837776	−	0.546015i	$-0.183856\pi$
$140$		0			0
$141$			6.86828i			0.578413i
$142$		0			0
$143$	1.71085			0.143069
$144$		0			0
$145$	−2.34611	+	11.6346i	−0.194834	+	0.966204i
$146$		0			0
$147$	0.807362			0.0665901
$148$		0			0
$149$	−4.71300			−0.386104			−0.193052	−	0.981189i	$-0.561839\pi$
$149$	−4.71300			−0.386104			−0.193052	+	0.981189i	$0.561839\pi$
$150$		0			0
$151$	8.59803			0.699698			0.349849	−	0.936806i	$-0.386233\pi$
$151$	8.59803			0.699698			0.349849	+	0.936806i	$0.386233\pi$
$152$		0			0
$153$	−1.86358	−	9.50069i	−0.150661	−	0.768086i
$154$		0			0
$155$	2.00209	−	9.92859i	0.160812	−	0.797483i
$156$		0			0
$157$			9.86515i			0.787324i	0.919255	+	0.393662i	$0.128792\pi$
$157$			9.86515i			0.787324i	−0.919255	+	0.393662i	$0.871208\pi$
$158$		0			0
$159$			10.8031i			0.856744i
$160$		0			0
$161$	2.80575			0.221124
$162$		0			0
$163$	8.17360			0.640206			0.320103	−	0.947383i	$-0.396283\pi$
$163$	8.17360			0.640206			0.320103	+	0.947383i	$0.396283\pi$
$164$		0			0
$165$	0.267050	−	1.32433i	0.0207898	−	0.103099i
$166$		0			0
$167$	3.21694			0.248935			0.124467	−	0.992224i	$-0.460278\pi$
$167$	3.21694			0.248935			0.124467	+	0.992224i	$0.460278\pi$
$168$		0			0
$169$	7.77328			0.597944
$170$		0			0
$171$	−1.67270			−0.127914
$172$		0			0
$173$	−19.0720			−1.45002			−0.725009	−	0.688740i	$-0.758164\pi$
$173$	−19.0720			−1.45002			−0.725009	+	0.688740i	$0.758164\pi$
$174$		0			0
$175$	4.60927	+	1.93770i	0.348428	+	0.146476i
$176$		0			0
$177$	5.44254			0.409086
$178$		0			0
$179$	−3.91871			−0.292898			−0.146449	−	0.989218i	$-0.546784\pi$
$179$	−3.91871			−0.292898			−0.146449	+	0.989218i	$0.546784\pi$
$180$		0			0
$181$		−	15.6263i		−	1.16149i	−0.814085	−	0.580746i	$-0.802761\pi$
$181$		−	15.6263i		−	1.16149i	0.814085	−	0.580746i	$-0.197239\pi$
$182$		0			0
$183$		−	1.24152i		−	0.0917755i
$184$		0			0
$185$	20.4606	+	4.12585i	1.50429	+	0.303338i
$186$		0			0
$187$	−3.02778	+	0.593904i	−0.221413	+	0.0434305i
$188$		0			0
$189$	−4.31791			−0.314081
$190$		0			0
$191$	−12.5499			−0.908077			−0.454038	−	0.890982i	$-0.650017\pi$
$191$	−12.5499			−0.908077			−0.454038	+	0.890982i	$0.650017\pi$
$192$		0			0
$193$	−20.0850			−1.44575			−0.722874	−	0.690980i	$-0.757179\pi$
$193$	−20.0850			−1.44575			−0.722874	+	0.690980i	$0.757179\pi$
$194$		0			0
$195$	−0.815847	+	4.04588i	−0.0584240	+	0.289731i
$196$		0			0
$197$	11.3948			0.811848			0.405924	−	0.913907i	$-0.366950\pi$
$197$	11.3948			0.811848			0.405924	+	0.913907i	$0.366950\pi$
$198$		0			0
$199$		−	26.0963i		−	1.84992i	−0.380068	−	0.924959i	$-0.624099\pi$
$199$		−	26.0963i		−	1.84992i	0.380068	−	0.924959i	$-0.375901\pi$
$200$		0			0
$201$			3.71974i			0.262370i
$202$		0			0
$203$		−	5.30790i		−	0.372542i
$204$		0			0
$205$	−0.852197	+	4.22615i	−0.0595201	+	0.295167i
$206$		0			0
$207$	−6.58837			−0.457923
$208$		0			0
$209$			0.533072i			0.0368734i
$210$		0			0
$211$			15.4488i			1.06354i	0.846889	+	0.531769i	$0.178473\pi$
$211$			15.4488i			1.06354i	−0.846889	+	0.531769i	$0.821527\pi$
$212$		0			0
$213$			10.3470i			0.708965i
$214$		0			0
$215$	2.88863	−	14.3251i	0.197003	−	0.976961i
$216$		0			0
$217$			4.52957i			0.307487i
$218$		0			0
$219$	−2.74180			−0.185274
$220$		0			0
$221$	9.24999	−	1.81440i	0.622221	−	0.122050i
$222$		0			0
$223$		−	24.6699i		−	1.65202i	−0.563655	−	0.826010i	$-0.690605\pi$
$223$		−	24.6699i		−	1.65202i	0.563655	−	0.826010i	$-0.309395\pi$
$224$		0			0
$225$	−10.8233	−	4.55004i	−0.721555	−	0.303336i
$226$		0			0
$227$	−21.4843			−1.42596			−0.712981	−	0.701183i	$-0.752656\pi$
$227$	−21.4843			−1.42596			−0.712981	+	0.701183i	$0.752656\pi$
$228$		0			0
$229$	25.2840			1.67081			0.835407	−	0.549632i	$-0.185232\pi$
$229$	25.2840			1.67081			0.835407	+	0.549632i	$0.185232\pi$
$230$		0			0
$231$			0.604179i			0.0397521i
$232$		0			0
$233$	10.3072			0.675246			0.337623	−	0.941282i	$-0.390377\pi$
$233$	10.3072			0.675246			0.337623	+	0.941282i	$0.390377\pi$
$234$		0			0
$235$	3.76015	−	18.6470i	0.245285	−	1.21640i
$236$		0			0
$237$			4.21873i			0.274036i
$238$		0			0
$239$	−10.1266			−0.655032			−0.327516	−	0.944846i	$-0.606212\pi$
$239$	−10.1266			−0.655032			−0.327516	+	0.944846i	$0.606212\pi$
$240$		0			0
$241$			7.72491i			0.497605i	0.968554	+	0.248803i	$0.0800371\pi$
$241$			7.72491i			0.497605i	−0.968554	+	0.248803i	$0.919963\pi$
$242$		0			0
$243$	15.8266			1.01528
$244$		0			0
$245$	−2.19195	−	0.442004i	−0.140038	−	0.0282386i
$246$		0			0
$247$		−	1.62856i		−	0.103623i
$248$		0			0
$249$			5.40247i			0.342367i
$250$		0			0
$251$	19.3215			1.21956			0.609780	−	0.792571i	$-0.291258\pi$
$251$	19.3215			1.21956			0.609780	+	0.792571i	$0.291258\pi$
$252$		0			0
$253$			2.09965i			0.132004i
$254$		0			0
$255$	0.0393618	−	7.44340i	0.00246493	−	0.466124i
$256$		0			0
$257$			19.3313i			1.20585i	0.797796	+	0.602927i	$0.205999\pi$
$257$			19.3313i			1.20585i	−0.797796	+	0.602927i	$0.794001\pi$
$258$		0			0
$259$	−9.33442			−0.580013
$260$		0			0
$261$			12.4638i			0.771492i
$262$		0			0
$263$			1.77501i			0.109452i	0.998501	+	0.0547259i	$0.0174285\pi$
$263$			1.77501i			0.109452i	−0.998501	+	0.0547259i	$0.982571\pi$
$264$		0			0
$265$	5.91435	−	29.3300i	0.363316	−	1.80173i
$266$		0			0
$267$	0.697848			0.0427076
$268$		0			0
$269$		−	3.34737i		−	0.204093i	−0.994780	−	0.102046i	$-0.967461\pi$
$269$		−	3.34737i		−	0.204093i	0.994780	−	0.102046i	$-0.0325390\pi$
$270$		0			0
$271$	−16.1661			−0.982018			−0.491009	−	0.871154i	$-0.663372\pi$
$271$	−16.1661			−0.982018			−0.491009	+	0.871154i	$0.663372\pi$
$272$		0			0
$273$		−	1.84579i		−	0.111712i
$274$		0			0
$275$	−1.45005	+	3.44929i	−0.0874415	+	0.208000i
$276$		0			0
$277$	−19.8690			−1.19381			−0.596906	−	0.802311i	$-0.703603\pi$
$277$	−19.8690			−1.19381			−0.596906	+	0.802311i	$0.703603\pi$
$278$		0			0
$279$		−	10.6362i		−	0.636772i
$280$		0			0
$281$	−5.83444			−0.348053			−0.174027	−	0.984741i	$-0.555678\pi$
$281$	−5.83444			−0.348053			−0.174027	+	0.984741i	$0.555678\pi$
$282$		0			0
$283$	−14.4435			−0.858578			−0.429289	−	0.903167i	$-0.641236\pi$
$283$	−14.4435			−0.858578			−0.429289	+	0.903167i	$0.641236\pi$
$284$		0			0
$285$	−1.26063	−	0.254204i	−0.0746731	−	0.0150577i
$286$		0			0
$287$		−	1.92803i		−	0.113808i
$288$		0			0
$289$	−15.7403	+	6.42206i	−0.925900	+	0.377768i
$290$		0			0
$291$	5.65538			0.331524
$292$		0			0
$293$			11.1472i			0.651228i	0.945503	+	0.325614i	$0.105571\pi$
$293$			11.1472i			0.651228i	−0.945503	+	0.325614i	$0.894429\pi$
$294$		0			0
$295$	−14.7762	−	2.97961i	−0.860305	−	0.173480i
$296$		0			0
$297$		−	3.23125i		−	0.187496i
$298$		0			0
$299$		−	6.41451i		−	0.370961i
$300$		0			0
$301$			6.53531i			0.376689i
$302$		0			0
$303$	−11.2448			−0.645997
$304$		0			0
$305$	−0.679689	+	3.37066i	−0.0389189	+	0.193003i
$306$		0			0
$307$		−	16.4732i		−	0.940175i	−0.882620	−	0.470088i	$-0.844222\pi$
$307$		−	16.4732i		−	0.940175i	0.882620	−	0.470088i	$-0.155778\pi$
$308$		0			0
$309$			5.25343i			0.298857i
$310$		0			0
$311$		−	14.5026i		−	0.822368i	−0.911552	−	0.411184i	$-0.865115\pi$
$311$		−	14.5026i		−	0.822368i	0.911552	−	0.411184i	$-0.134885\pi$
$312$		0			0
$313$	12.8807			0.728060			0.364030	−	0.931387i	$-0.381401\pi$
$313$	12.8807			0.728060			0.364030	+	0.931387i	$0.381401\pi$
$314$		0			0
$315$	5.14706	+	1.03790i	0.290004	+	0.0584790i
$316$		0			0
$317$	22.9542			1.28924			0.644618	−	0.764505i	$-0.277017\pi$
$317$	22.9542			1.28924			0.644618	+	0.764505i	$0.277017\pi$
$318$		0			0
$319$	3.97210			0.222395
$320$		0			0
$321$	2.19111			0.122296
$322$		0			0
$323$	0.565336	+	2.88214i	0.0314561	+	0.160366i
$324$		0			0
$325$	4.42997	−	10.5377i	0.245730	−	0.584527i
$326$		0			0
$327$		−	7.99638i		−	0.442201i
$328$		0			0
$329$			8.50706i			0.469010i
$330$		0			0
$331$	−31.5703			−1.73526			−0.867629	−	0.497212i	$-0.834357\pi$
$331$	−31.5703			−1.73526			−0.867629	+	0.497212i	$0.834357\pi$
$332$		0			0
$333$	21.9188			1.20114
$334$		0			0
$335$	2.03643	−	10.0989i	0.111262	−	0.551762i
$336$		0			0
$337$	−7.01847			−0.382321			−0.191160	−	0.981559i	$-0.561225\pi$
$337$	−7.01847			−0.382321			−0.191160	+	0.981559i	$0.561225\pi$
$338$		0			0
$339$	−0.577901			−0.0313873
$340$		0			0
$341$	−3.38965			−0.183560
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$	−4.96532	−	1.00125i	−0.267324	−	0.0539055i
$346$		0			0
$347$	17.2589			0.926504			0.463252	−	0.886227i	$-0.346683\pi$
$347$	17.2589			0.926504			0.463252	+	0.886227i	$0.346683\pi$
$348$		0			0
$349$	−9.22975			−0.494057			−0.247028	−	0.969008i	$-0.579454\pi$
$349$	−9.22975			−0.494057			−0.247028	+	0.969008i	$0.579454\pi$
$350$		0			0
$351$			9.87161i			0.526907i
$352$		0			0
$353$			5.66145i			0.301328i	0.988585	+	0.150664i	$0.0481412\pi$
$353$			5.66145i			0.301328i	−0.988585	+	0.150664i	$0.951859\pi$
$354$		0			0
$355$	5.66464	−	28.0916i	0.300648	−	1.49095i
$356$		0			0
$357$	0.640747	+	3.26659i	0.0339119	+	0.172886i
$358$		0			0
$359$	36.1283			1.90678			0.953388	−	0.301746i	$-0.0975695\pi$
$359$	36.1283			1.90678			0.953388	+	0.301746i	$0.0975695\pi$
$360$		0			0
$361$	−18.4926			−0.973293
$362$		0			0
$363$	8.42885			0.442400
$364$		0			0
$365$	7.44385	+	1.50104i	0.389629	+	0.0785683i
$366$		0			0
$367$	22.2824			1.16313			0.581566	−	0.813499i	$-0.302440\pi$
$367$	22.2824			1.16313			0.581566	+	0.813499i	$0.302440\pi$
$368$		0			0
$369$			4.52734i			0.235684i
$370$		0			0
$371$			13.3808i			0.694695i
$372$		0			0
$373$		−	22.2799i		−	1.15361i	−0.816882	−	0.576805i	$-0.804299\pi$
$373$		−	22.2799i		−	1.15361i	0.816882	−	0.576805i	$-0.195701\pi$
$374$		0			0
$375$	−7.46551	−	5.07398i	−0.385517	−	0.262019i
$376$		0			0
$377$	−12.1349			−0.624981
$378$		0			0
$379$			24.4108i			1.25390i	0.779060	+	0.626950i	$0.215697\pi$
$379$			24.4108i			1.25390i	−0.779060	+	0.626950i	$0.784303\pi$
$380$		0			0
$381$		−	1.35355i		−	0.0693443i
$382$		0			0
$383$		−	24.1204i		−	1.23250i	−0.787552	−	0.616248i	$-0.788652\pi$
$383$		−	24.1204i		−	1.23250i	0.787552	−	0.616248i	$-0.211348\pi$
$384$		0			0
$385$	0.330768	−	1.64032i	0.0168575	−	0.0835983i
$386$		0			0
$387$		−	15.3460i		−	0.780081i
$388$		0			0
$389$	13.6717			0.693181			0.346590	−	0.938017i	$-0.387339\pi$
$389$	13.6717			0.693181			0.346590	+	0.938017i	$0.387339\pi$
$390$		0			0
$391$	2.22673	+	11.3521i	0.112610	+	0.574099i
$392$		0			0
$393$			13.6977i			0.690957i
$394$		0			0
$395$	2.30961	−	11.4536i	0.116209	−	0.576295i
$396$		0			0
$397$	−14.5962			−0.732562			−0.366281	−	0.930504i	$-0.619369\pi$
$397$	−14.5962			−0.732562			−0.366281	+	0.930504i	$0.619369\pi$
$398$		0			0
$399$	0.575117			0.0287919
$400$		0			0
$401$			33.9031i			1.69304i	0.532358	+	0.846519i	$0.321306\pi$
$401$			33.9031i			1.69304i	−0.532358	+	0.846519i	$0.678694\pi$
$402$		0			0
$403$	10.3555			0.515845
$404$		0			0
$405$	−7.79980	−	1.57282i	−0.387575	−	0.0781541i
$406$		0			0
$407$		−	6.98530i		−	0.346248i
$408$		0			0
$409$	−18.4514			−0.912365			−0.456182	−	0.889886i	$-0.650784\pi$
$409$	−18.4514			−0.912365			−0.456182	+	0.889886i	$0.650784\pi$
$410$		0			0
$411$			17.1637i			0.846621i
$412$		0			0
$413$	6.74114			0.331710
$414$		0			0
$415$	2.95767	−	14.6674i	0.145186	−	0.719996i
$416$		0			0
$417$		−	10.3947i		−	0.509028i
$418$		0			0
$419$		−	29.1254i		−	1.42287i	−0.702752	−	0.711435i	$-0.748046\pi$
$419$		−	29.1254i		−	1.42287i	0.702752	−	0.711435i	$-0.251954\pi$
$420$		0			0
$421$	7.57872			0.369364			0.184682	−	0.982798i	$-0.440874\pi$
$421$	7.57872			0.369364			0.184682	+	0.982798i	$0.440874\pi$
$422$		0			0
$423$		−	19.9760i		−	0.971267i
$424$		0			0
$425$	−4.18188	+	20.1869i	−0.202851	+	0.979210i
$426$		0			0
$427$		−	1.53774i		−	0.0744167i
$428$		0			0
$429$	1.38128			0.0666887
$430$		0			0
$431$			19.1063i			0.920320i	0.887836	+	0.460160i	$0.152208\pi$
$431$			19.1063i			0.920320i	−0.887836	+	0.460160i	$0.847792\pi$
$432$		0			0
$433$		−	16.9823i		−	0.816117i	−0.912956	−	0.408058i	$-0.866206\pi$
$433$		−	16.9823i		−	0.816117i	0.912956	−	0.408058i	$-0.133794\pi$
$434$		0			0
$435$	−1.89416	+	9.39336i	−0.0908180	+	0.450377i
$436$		0			0
$437$	1.99865			0.0956084
$438$		0			0
$439$		−	0.404036i		−	0.0192836i	−0.999954	−	0.00964180i	$-0.996931\pi$
$439$		−	0.404036i		−	0.0192836i	0.999954	−	0.00964180i	$-0.00306913\pi$
$440$		0			0
$441$	−2.34817			−0.111817
$442$		0			0
$443$		−	4.75167i		−	0.225759i	−0.993609	−	0.112879i	$-0.963993\pi$
$443$		−	4.75167i		−	0.225759i	0.993609	−	0.112879i	$-0.0360074\pi$
$444$		0			0
$445$	−1.89462	−	0.382049i	−0.0898138	−	0.0181108i
$446$		0			0
$447$	−3.80509			−0.179975
$448$		0			0
$449$			27.9118i			1.31724i	0.752476	+	0.658620i	$0.228860\pi$
$449$			27.9118i			1.31724i	−0.752476	+	0.658620i	$0.771140\pi$
$450$		0			0
$451$	1.44282			0.0679397
$452$		0			0
$453$	6.94172			0.326150
$454$		0			0
$455$	−1.01051	+	5.01124i	−0.0473734	+	0.234930i
$456$		0			0
$457$		−	30.5973i		−	1.43128i	−0.698469	−	0.715640i	$-0.746135\pi$
$457$		−	30.5973i		−	1.43128i	0.698469	−	0.715640i	$-0.253865\pi$
$458$		0			0
$459$	−3.42682	−	17.4703i	−0.159950	−	0.815442i
$460$		0			0
$461$	6.89878			0.321308			0.160654	−	0.987011i	$-0.448640\pi$
$461$	6.89878			0.321308			0.160654	+	0.987011i	$0.448640\pi$
$462$		0			0
$463$			4.55142i			0.211523i	0.994392	+	0.105761i	$0.0337279\pi$
$463$			4.55142i			0.211523i	−0.994392	+	0.105761i	$0.966272\pi$
$464$		0			0
$465$	1.61641	−	8.01596i	0.0749592	−	0.371731i
$466$		0			0
$467$			0.723439i			0.0334768i	0.999860	+	0.0167384i	$0.00532824\pi$
$467$			0.723439i			0.0334768i	−0.999860	+	0.0167384i	$0.994672\pi$
$468$		0			0
$469$			4.60728i			0.212744i
$470$		0			0
$471$			7.96474i			0.366996i
$472$		0			0
$473$	−4.89062			−0.224871
$474$		0			0
$475$	3.28337	+	1.38030i	0.150651	+	0.0633326i
$476$		0			0
$477$		−	31.4203i		−	1.43864i
$478$		0			0
$479$		−	26.9152i		−	1.22979i	−0.788609	−	0.614895i	$-0.789199\pi$
$479$		−	26.9152i		−	1.22979i	0.788609	−	0.614895i	$-0.210801\pi$
$480$		0			0
$481$			21.3404i			0.973037i
$482$		0			0
$483$	2.26525			0.103073
$484$		0			0
$485$	−15.3541	−	3.09613i	−0.697193	−	0.140588i
$486$		0			0
$487$	−5.93638			−0.269003			−0.134502	−	0.990913i	$-0.542943\pi$
$487$	−5.93638			−0.269003			−0.134502	+	0.990913i	$0.542943\pi$
$488$		0			0
$489$	6.59905			0.298419
$490$		0			0
$491$	−16.4842			−0.743924			−0.371962	−	0.928248i	$-0.621315\pi$
$491$	−16.4842			−0.743924			−0.371962	+	0.928248i	$0.621315\pi$
$492$		0			0
$493$	21.4758	−	4.21251i	0.967221	−	0.189722i
$494$		0			0
$495$	−0.776699	+	3.85174i	−0.0349100	+	0.173123i
$496$		0			0
$497$			12.8158i			0.574868i
$498$		0			0
$499$			11.9978i			0.537098i	0.963266	+	0.268549i	$0.0865440\pi$
$499$			11.9978i			0.537098i	−0.963266	+	0.268549i	$0.913456\pi$
$500$		0			0
$501$	2.59724			0.116036
$502$		0			0
$503$	1.68306			0.0750441			0.0375221	−	0.999296i	$-0.488054\pi$
$503$	1.68306			0.0750441			0.0375221	+	0.999296i	$0.488054\pi$
$504$		0			0
$505$	30.5291	+	6.15615i	1.35853	+	0.273945i
$506$		0			0
$507$	6.27585			0.278720
$508$		0			0
$509$	−3.98212			−0.176504			−0.0882522	−	0.996098i	$-0.528128\pi$
$509$	−3.98212			−0.176504			−0.0882522	+	0.996098i	$0.528128\pi$
$510$		0			0
$511$	−3.39600			−0.150230
$512$		0			0
$513$	−3.07582			−0.135801
$514$		0			0
$515$	2.87608	−	14.2628i	0.126735	−	0.628495i
$516$		0			0
$517$	−6.36616			−0.279983
$518$		0			0
$519$	−15.3980			−0.675897
$520$		0			0
$521$			18.7826i			0.822879i	0.911437	+	0.411439i	$0.134974\pi$
$521$			18.7826i			0.822879i	−0.911437	+	0.411439i	$0.865026\pi$
$522$		0			0
$523$		−	31.2301i		−	1.36559i	−0.730608	−	0.682797i	$-0.760763\pi$
$523$		−	31.2301i		−	1.36559i	0.730608	−	0.682797i	$-0.239237\pi$
$524$		0			0
$525$	3.72134	+	1.56442i	0.162413	+	0.0682770i
$526$		0			0
$527$	−18.3267	+	3.59481i	−0.798322	+	0.156592i
$528$		0			0
$529$	−15.1278			−0.657730
$530$		0			0
$531$	−15.8293			−0.686934
$532$		0			0
$533$	−4.40787			−0.190926
$534$		0			0
$535$	−5.94877	−	1.19956i	−0.257188	−	0.0518616i
$536$		0			0
$537$	−3.16382			−0.136529
$538$		0			0
$539$			0.748338i			0.0322332i
$540$		0			0
$541$			28.0864i			1.20753i	0.797163	+	0.603764i	$0.206333\pi$
$541$			28.0864i			1.20753i	−0.797163	+	0.603764i	$0.793667\pi$
$542$		0			0
$543$		−	12.6161i		−	0.541407i
$544$		0			0
$545$	−4.37775	+	21.7098i	−0.187522	+	0.929945i
$546$		0			0
$547$	19.4131			0.830045			0.415023	−	0.909811i	$-0.363774\pi$
$547$	19.4131			0.830045			0.415023	+	0.909811i	$0.363774\pi$
$548$		0			0
$549$			3.61088i			0.154109i
$550$		0			0
$551$		−	3.78104i		−	0.161078i
$552$		0			0
$553$			5.22533i			0.222204i
$554$		0			0
$555$	16.5191	+	3.33105i	0.701195	+	0.141395i
$556$		0			0
$557$		−	25.0570i		−	1.06170i	−0.847466	−	0.530850i	$-0.821873\pi$
$557$		−	25.0570i		−	1.06170i	0.847466	−	0.530850i	$-0.178127\pi$
$558$		0			0
$559$	14.9410			0.631939
$560$		0			0
$561$	−2.44451	+	0.479495i	−0.103207	+	0.0202443i
$562$		0			0
$563$			8.18528i			0.344969i	0.985012	+	0.172484i	$0.0551794\pi$
$563$			8.18528i			0.344969i	−0.985012	+	0.172484i	$0.944821\pi$
$564$		0			0
$565$	1.56897	+	0.316382i	0.0660072	+	0.0133103i
$566$		0			0
$567$	3.55839			0.149438
$568$		0			0
$569$	−14.3787			−0.602788			−0.301394	−	0.953500i	$-0.597452\pi$
$569$	−14.3787			−0.602788			−0.301394	+	0.953500i	$0.597452\pi$
$570$		0			0
$571$		−	2.69245i		−	0.112676i	−0.998412	−	0.0563378i	$-0.982058\pi$
$571$		−	2.69245i		−	0.112676i	0.998412	−	0.0563378i	$-0.0179424\pi$
$572$		0			0
$573$	−10.1323			−0.423282
$574$		0			0
$575$	12.9324	+	5.43669i	0.539320	+	0.226726i
$576$		0			0
$577$			28.3361i			1.17965i	0.807532	+	0.589824i	$0.200803\pi$
$577$			28.3361i			1.17965i	−0.807532	+	0.589824i	$0.799197\pi$
$578$		0			0
$579$	−16.2158			−0.673907
$580$		0			0
$581$			6.69151i			0.277610i
$582$		0			0
$583$	−10.0133			−0.414710
$584$		0			0
$585$	2.37285	−	11.7672i	0.0981051	−	0.486515i
$586$		0			0
$587$		−	17.8657i		−	0.737397i	−0.929549	−	0.368699i	$-0.879803\pi$
$587$		−	17.8657i		−	0.737397i	0.929549	−	0.368699i	$-0.120197\pi$
$588$		0			0
$589$			3.22660i			0.132950i
$590$		0			0
$591$	9.19975			0.378427
$592$		0			0
$593$			29.7093i			1.22001i	0.792396	+	0.610007i	$0.208833\pi$
$593$			29.7093i			1.22001i	−0.792396	+	0.610007i	$0.791167\pi$
$594$		0			0
$595$	0.0487536	−	9.21942i	0.00199870	−	0.377959i
$596$		0			0
$597$		−	21.0691i		−	0.862303i
$598$		0			0
$599$	−23.8491			−0.974448			−0.487224	−	0.873277i	$-0.661990\pi$
$599$	−23.8491			−0.974448			−0.487224	+	0.873277i	$0.661990\pi$
$600$		0			0
$601$		−	14.6210i		−	0.596404i	−0.954503	−	0.298202i	$-0.903613\pi$
$601$		−	14.6210i		−	0.596404i	0.954503	−	0.298202i	$-0.0963869\pi$
$602$		0			0
$603$		−	10.8187i		−	0.440570i
$604$		0			0
$605$	−22.8839	−	4.61451i	−0.930363	−	0.187607i
$606$		0			0
$607$	11.6231			0.471766			0.235883	−	0.971781i	$-0.424202\pi$
$607$	11.6231			0.471766			0.235883	+	0.971781i	$0.424202\pi$
$608$		0			0
$609$		−	4.28539i		−	0.173653i
$610$		0			0
$611$	19.4489			0.786817
$612$		0			0
$613$		−	31.9544i		−	1.29063i	−0.763918	−	0.645314i	$-0.776727\pi$
$613$		−	31.9544i		−	1.29063i	0.763918	−	0.645314i	$-0.223273\pi$
$614$		0			0
$615$	−0.688032	+	3.41203i	−0.0277441	+	0.137586i
$616$		0			0
$617$	−23.1594			−0.932361			−0.466181	−	0.884690i	$-0.654370\pi$
$617$	−23.1594			−0.932361			−0.466181	+	0.884690i	$0.654370\pi$
$618$		0			0
$619$		−	35.0960i		−	1.41063i	−0.708896	−	0.705313i	$-0.750806\pi$
$619$		−	35.0960i		−	1.41063i	0.708896	−	0.705313i	$-0.249194\pi$
$620$		0			0
$621$	−12.1150			−0.486156
$622$		0			0
$623$	0.864356			0.0346297
$624$		0			0
$625$	17.4907	+	17.8627i	0.699626	+	0.714509i
$626$		0			0
$627$			0.430382i			0.0171878i
$628$		0			0
$629$	−7.40808	−	37.7671i	−0.295379	−	1.50587i
$630$		0			0
$631$	21.4433			0.853643			0.426821	−	0.904336i	$-0.359633\pi$
$631$	21.4433			0.853643			0.426821	+	0.904336i	$0.359633\pi$
$632$		0			0
$633$			12.4728i			0.495748i
$634$		0			0
$635$	−0.741021	+	3.67481i	−0.0294065	+	0.145830i
$636$		0			0
$637$		−	2.28620i		−	0.0905826i
$638$		0			0
$639$		−	30.0937i		−	1.19049i
$640$		0			0
$641$		−	12.1278i		−	0.479020i	−0.970894	−	0.239510i	$-0.923013\pi$
$641$		−	12.1278i		−	0.479020i	0.970894	−	0.239510i	$-0.0769868\pi$
$642$		0			0
$643$	−31.7218			−1.25099			−0.625493	−	0.780230i	$-0.715102\pi$
$643$	−31.7218			−1.25099			−0.625493	+	0.780230i	$0.715102\pi$
$644$		0			0
$645$	2.33217	−	11.5655i	0.0918291	−	0.455391i
$646$		0			0
$647$			8.99066i			0.353459i	0.984259	+	0.176730i	$0.0565518\pi$
$647$			8.99066i			0.353459i	−0.984259	+	0.176730i	$0.943448\pi$
$648$		0			0
$649$			5.04465i			0.198020i
$650$		0			0
$651$			3.65700i			0.143329i
$652$		0			0
$653$	−7.02683			−0.274981			−0.137491	−	0.990503i	$-0.543904\pi$
$653$	−7.02683			−0.274981			−0.137491	+	0.990503i	$0.543904\pi$
$654$		0			0
$655$	7.49903	−	37.1886i	0.293011	−	1.45308i
$656$		0			0
$657$	7.97438			0.311110
$658$		0			0
$659$	32.2230			1.25523			0.627615	−	0.778524i	$-0.284031\pi$
$659$	32.2230			1.25523			0.627615	+	0.778524i	$0.284031\pi$
$660$		0			0
$661$	26.9697			1.04900			0.524499	−	0.851411i	$-0.324253\pi$
$661$	26.9697			1.04900			0.524499	+	0.851411i	$0.324253\pi$
$662$		0			0
$663$	7.46808	−	1.46488i	0.290036	−	0.0568911i
$664$		0			0
$665$	−1.56141	−	0.314858i	−0.0605491	−	0.0122097i
$666$		0			0
$667$		−	14.8926i		−	0.576645i
$668$		0			0
$669$		−	19.9175i		−	0.770057i
$670$		0			0
$671$	1.15075			0.0444243
$672$		0			0
$673$	2.49807			0.0962933			0.0481467	−	0.998840i	$-0.484669\pi$
$673$	2.49807			0.0962933			0.0481467	+	0.998840i	$0.484669\pi$
$674$		0			0
$675$	−19.9024	−	8.36679i	−0.766043	−	0.322038i
$676$		0			0
$677$	32.7874			1.26012			0.630061	−	0.776545i	$-0.283030\pi$
$677$	32.7874			1.26012			0.630061	+	0.776545i	$0.283030\pi$
$678$		0			0
$679$	7.00477			0.268818
$680$		0			0
$681$	−17.3456			−0.664684
$682$		0			0
$683$	−35.0076			−1.33953			−0.669764	−	0.742574i	$-0.733605\pi$
$683$	−35.0076			−1.33953			−0.669764	+	0.742574i	$0.733605\pi$
$684$		0			0
$685$	9.39654	−	46.5985i	0.359023	−	1.78044i
$686$		0			0
$687$	20.4133			0.778817
$688$		0			0
$689$	30.5912			1.16543
$690$		0			0
$691$			27.3482i			1.04037i	0.854052	+	0.520187i	$0.174138\pi$
$691$			27.3482i			1.04037i	−0.854052	+	0.520187i	$0.825862\pi$
$692$		0			0
$693$		−	1.75722i		−	0.0667514i
$694$		0			0
$695$	−5.69073	+	28.2210i	−0.215862	+	1.07048i
$696$		0			0
$697$	7.80083	−	1.53014i	0.295477	−	0.0579584i
$698$		0			0
$699$	8.32162			0.314753
$700$		0			0
$701$	−23.3065			−0.880273			−0.440137	−	0.897931i	$-0.645070\pi$
$701$	−23.3065			−0.880273			−0.440137	+	0.897931i	$0.645070\pi$
$702$		0			0
$703$	−6.64929			−0.250783
$704$		0			0
$705$	3.03580	−	15.0549i	0.114335	−	0.567000i
$706$		0			0
$707$	−13.9278			−0.523810
$708$		0			0
$709$			1.55904i			0.0585509i	0.999571	+	0.0292755i	$0.00932000\pi$
$709$			1.55904i			0.0585509i	−0.999571	+	0.0292755i	$0.990680\pi$
$710$		0			0
$711$		−	12.2699i		−	0.460159i
$712$		0			0
$713$			12.7088i			0.475950i
$714$		0			0
$715$	−3.75010	−	0.756203i	−0.140246	−	0.0282804i
$716$		0			0
$717$	−8.17579			−0.305330
$718$		0			0
$719$		−	30.0897i		−	1.12216i	−0.827763	−	0.561078i	$-0.810387\pi$
$719$		−	30.0897i		−	1.12216i	0.827763	−	0.561078i	$-0.189613\pi$
$720$		0			0
$721$			6.50691i			0.242330i
$722$		0			0
$723$			6.23680i			0.231949i
$724$		0			0
$725$	10.2851	−	24.4655i	0.381979	−	0.908627i
$726$		0			0
$727$		−	7.43734i		−	0.275836i	−0.990444	−	0.137918i	$-0.955959\pi$
$727$		−	7.43734i		−	0.275836i	0.990444	−	0.137918i	$-0.0440410\pi$
$728$		0			0
$729$	2.10264			0.0778754
$730$		0			0
$731$	−26.4419	+	5.18662i	−0.977989	+	0.191834i
$732$		0			0
$733$		−	38.5114i		−	1.42245i	−0.702965	−	0.711225i	$-0.748141\pi$
$733$		−	38.5114i		−	1.42245i	0.702965	−	0.711225i	$-0.251859\pi$
$734$		0			0
$735$	−1.76969	−	0.356857i	−0.0652762	−	0.0131629i
$736$		0			0
$737$	−3.44780			−0.127001
$738$		0			0
$739$	11.0777			0.407501			0.203751	−	0.979023i	$-0.434687\pi$
$739$	11.0777			0.407501			0.203751	+	0.979023i	$0.434687\pi$
$740$		0			0
$741$		−	1.31483i		−	0.0483016i
$742$		0			0
$743$	20.1707			0.739989			0.369995	−	0.929034i	$-0.379359\pi$
$743$	20.1707			0.739989			0.369995	+	0.929034i	$0.379359\pi$
$744$		0			0
$745$	10.3306	+	2.08316i	0.378485	+	0.0763212i
$746$		0			0
$747$		−	15.7128i		−	0.574900i
$748$		0			0
$749$	2.71392			0.0991645
$750$		0			0
$751$			28.1753i			1.02813i	0.857751	+	0.514065i	$0.171861\pi$
$751$			28.1753i			1.02813i	−0.857751	+	0.514065i	$0.828139\pi$
$752$		0			0
$753$	15.5994			0.568474
$754$		0			0
$755$	−18.8464	−	3.80036i	−0.685892	−	0.138309i
$756$		0			0
$757$			11.4237i			0.415200i	0.978214	+	0.207600i	$0.0665653\pi$
$757$			11.4237i			0.415200i	−0.978214	+	0.207600i	$0.933435\pi$
$758$		0			0
$759$			1.69518i			0.0615310i
$760$		0			0
$761$	39.6513			1.43736			0.718679	−	0.695342i	$-0.244747\pi$
$761$	39.6513			1.43736			0.718679	+	0.695342i	$0.244747\pi$
$762$		0			0
$763$		−	9.90433i		−	0.358561i
$764$		0			0
$765$	−0.114482	+	21.6487i	−0.00413909	+	0.782711i
$766$		0			0
$767$		−	15.4116i		−	0.556481i
$768$		0			0
$769$	23.5728			0.850057			0.425028	−	0.905180i	$-0.360264\pi$
$769$	23.5728			0.850057			0.425028	+	0.905180i	$0.360264\pi$
$770$		0			0
$771$			15.6074i			0.562086i
$772$		0			0
$773$		−	4.41262i		−	0.158711i	−0.996846	−	0.0793555i	$-0.974714\pi$
$773$		−	4.41262i		−	0.158711i	0.996846	−	0.0793555i	$-0.0252862\pi$
$774$		0			0
$775$	−8.77694	+	20.8780i	−0.315277	+	0.749960i
$776$		0			0
$777$	−7.53625			−0.270362
$778$		0			0
$779$		−	1.37342i		−	0.0492078i
$780$		0			0
$781$	−9.59057			−0.343178
$782$		0			0
$783$			22.9190i			0.819059i
$784$		0			0
$785$	4.36043	−	21.6239i	0.155630	−	0.771789i
$786$		0			0
$787$	35.2683			1.25718			0.628589	−	0.777737i	$-0.283633\pi$
$787$	35.2683			1.25718			0.628589	+	0.777737i	$0.283633\pi$
$788$		0			0
$789$			1.43308i			0.0510189i
$790$		0			0
$791$	−0.715790			−0.0254506
$792$		0			0
$793$	−3.51560			−0.124842
$794$		0			0
$795$	4.77502	−	23.6799i	0.169353	−	0.839839i
$796$		0			0
$797$		−	4.97423i		−	0.176196i	−0.996112	−	0.0880982i	$-0.971921\pi$
$797$		−	4.97423i		−	0.176196i	0.996112	−	0.0880982i	$-0.0280789\pi$
$798$		0			0
$799$	−34.4196	+	6.75146i	−1.21768	+	0.238850i
$800$		0			0
$801$	−2.02965			−0.0717143
$802$		0			0
$803$		−	2.54136i		−	0.0896825i
$804$		0			0
$805$	−6.15005	−	1.24015i	−0.216761	−	0.0437096i
$806$		0			0
$807$		−	2.70254i		−	0.0951337i
$808$		0			0
$809$		−	30.3565i		−	1.06728i	−0.845713	−	0.533638i	$-0.820824\pi$
$809$		−	30.3565i		−	1.06728i	0.845713	−	0.533638i	$-0.179176\pi$
$810$		0			0
$811$		−	26.9435i		−	0.946115i	−0.881032	−	0.473058i	$-0.843150\pi$
$811$		−	26.9435i		−	0.946115i	0.881032	−	0.473058i	$-0.156850\pi$
$812$		0			0
$813$	−13.0519			−0.457749
$814$		0			0
$815$	−17.9161	−	3.61276i	−0.627574	−	0.126550i
$816$		0			0
$817$			4.65537i			0.162871i
$818$		0			0
$819$			5.36839i			0.187587i
$820$		0			0
$821$			17.6829i			0.617137i	0.951202	+	0.308568i	$0.0998499\pi$
$821$			17.6829i			0.617137i	−0.951202	+	0.308568i	$0.900150\pi$
$822$		0			0
$823$	23.0591			0.803791			0.401895	−	0.915686i	$-0.368352\pi$
$823$	23.0591			0.803791			0.401895	+	0.915686i	$0.368352\pi$
$824$		0			0
$825$	−1.17072	+	2.78482i	−0.0407591	+	0.0969551i
$826$		0			0
$827$	−14.8622			−0.516809			−0.258405	−	0.966037i	$-0.583197\pi$
$827$	−14.8622			−0.516809			−0.258405	+	0.966037i	$0.583197\pi$
$828$		0			0
$829$	8.88012			0.308419			0.154210	−	0.988038i	$-0.450717\pi$
$829$	8.88012			0.308419			0.154210	+	0.988038i	$0.450717\pi$
$830$		0			0
$831$	−16.0415			−0.556472
$832$		0			0
$833$	0.793630	+	4.04600i	0.0274977	+	0.140186i
$834$		0			0
$835$	−7.05137	−	1.42190i	−0.244023	−	0.0492069i
$836$		0			0
$837$		−	19.5583i		−	0.676033i
$838$		0			0
$839$		−	41.6530i		−	1.43802i	−0.695000	−	0.719010i	$-0.744596\pi$
$839$		−	41.6530i		−	1.43802i	0.695000	−	0.719010i	$-0.255404\pi$
$840$		0			0
$841$	0.826198			0.0284896
$842$		0			0
$843$	−4.71050			−0.162238
$844$		0			0
$845$	−17.0386	−	3.43582i	−0.586146	−	0.118196i
$846$		0			0
$847$	10.4400			0.358722
$848$		0			0
$849$	−11.6611			−0.400209
$850$		0			0
$851$	−26.1900			−0.897782
$852$		0			0
$853$	−9.19446			−0.314812			−0.157406	−	0.987534i	$-0.550313\pi$
$853$	−9.19446			−0.314812			−0.157406	+	0.987534i	$0.550313\pi$
$854$		0			0
$855$	3.66646	+	0.739338i	0.125390	+	0.0252848i
$856$		0			0
$857$	−31.7838			−1.08571			−0.542857	−	0.839825i	$-0.682657\pi$
$857$	−31.7838			−1.08571			−0.542857	+	0.839825i	$0.682657\pi$
$858$		0			0
$859$	−36.7960			−1.25546			−0.627731	−	0.778430i	$-0.716016\pi$
$859$	−36.7960			−1.25546			−0.627731	+	0.778430i	$0.716016\pi$
$860$		0			0
$861$		−	1.55662i		−	0.0530494i
$862$		0			0
$863$			11.6255i			0.395736i	0.980229	+	0.197868i	$0.0634017\pi$
$863$			11.6255i			0.395736i	−0.980229	+	0.197868i	$0.936598\pi$
$864$		0			0
$865$	41.8048	+	8.42990i	1.42141	+	0.286625i
$866$		0			0
$867$	−12.7081	+	5.18493i	−0.431590	+	0.176089i
$868$		0			0
$869$	−3.91031			−0.132648
$870$		0			0
$871$	10.5332			0.356903
$872$		0			0
$873$	−16.4484			−0.556693
$874$		0			0
$875$	−9.24680	−	6.28464i	−0.312599	−	0.212460i
$876$		0			0
$877$	35.6436			1.20360			0.601799	−	0.798648i	$-0.294451\pi$
$877$	35.6436			1.20360			0.601799	+	0.798648i	$0.294451\pi$
$878$		0			0
$879$			8.99984i			0.303557i
$880$		0			0
$881$		−	40.5282i		−	1.36543i	−0.730685	−	0.682715i	$-0.760799\pi$
$881$		−	40.5282i		−	1.36543i	0.730685	−	0.682715i	$-0.239201\pi$
$882$		0			0
$883$			48.7052i			1.63906i	0.573035	+	0.819531i	$0.305766\pi$
$883$			48.7052i			1.63906i	−0.573035	+	0.819531i	$0.694234\pi$
$884$		0			0
$885$	−11.9298	−	2.40562i	−0.401014	−	0.0808641i
$886$		0			0
$887$	−46.5884			−1.56429			−0.782143	−	0.623098i	$-0.785874\pi$
$887$	−46.5884			−1.56429			−0.782143	+	0.623098i	$0.785874\pi$
$888$		0			0
$889$		−	1.67651i		−	0.0562282i
$890$		0			0
$891$			2.66288i			0.0892098i
$892$		0			0
$893$			6.05993i			0.202788i
$894$		0			0
$895$	8.58960	+	1.73208i	0.287119	+	0.0578972i
$896$		0			0
$897$		−	5.17883i		−	0.172916i
$898$		0			0
$899$	24.0425			0.801863
$900$		0			0
$901$	−54.1387	+	10.6194i	−1.80362	+	0.353783i
$902$		0			0
$903$			5.27636i			0.175586i
$904$		0			0
$905$	−6.90688	+	34.2520i	−0.229592	+	1.13857i
$906$		0			0
$907$	−11.9355			−0.396313			−0.198157	−	0.980170i	$-0.563495\pi$
$907$	−11.9355			−0.396313			−0.198157	+	0.980170i	$0.563495\pi$
$908$		0			0
$909$	32.7049			1.08475
$910$		0			0
$911$		−	12.6691i		−	0.419746i	−0.977729	−	0.209873i	$-0.932695\pi$
$911$		−	12.6691i		−	0.419746i	0.977729	−	0.209873i	$-0.0673050\pi$
$912$		0			0
$913$	−5.00751			−0.165724
$914$		0			0
$915$	−0.548755	+	2.72134i	−0.0181413	+	0.0899647i
$916$		0			0
$917$			16.9660i			0.560266i
$918$		0			0
$919$	−34.8061			−1.14815			−0.574074	−	0.818804i	$-0.694638\pi$
$919$	−34.8061			−1.14815			−0.574074	+	0.818804i	$0.694638\pi$
$920$		0			0
$921$		−	13.2998i		−	0.438244i
$922$		0			0
$923$	29.2996			0.964407
$924$		0			0
$925$	−43.0248	−	18.0873i	−1.41465	−	0.594706i
$926$		0			0
$927$		−	15.2793i		−	0.501839i
$928$		0			0
$929$			27.3173i			0.896250i	0.893971	+	0.448125i	$0.147908\pi$
$929$			27.3173i			0.896250i	−0.893971	+	0.448125i	$0.852092\pi$
$930$		0			0
$931$	0.712341			0.0233460
$932$		0			0
$933$		−	11.7089i		−	0.383331i
$934$		0			0
$935$	6.89924	+	0.0364842i	0.225629	+	0.00119316i
$936$		0			0
$937$			16.7006i			0.545585i	0.962073	+	0.272792i	$0.0879472\pi$
$937$			16.7006i			0.545585i	−0.962073	+	0.272792i	$0.912053\pi$
$938$		0			0
$939$	10.3994			0.339371
$940$		0			0
$941$			47.8707i			1.56054i	0.625444	+	0.780269i	$0.284918\pi$
$941$			47.8707i			1.56054i	−0.625444	+	0.780269i	$0.715082\pi$
$942$		0			0
$943$		−	5.40957i		−	0.176160i
$944$		0			0
$945$	9.46462	+	1.90853i	0.307884	+	0.0620845i
$946$		0			0
$947$	57.2021			1.85882			0.929409	−	0.369052i	$-0.120318\pi$
$947$	57.2021			1.85882			0.929409	+	0.369052i	$0.120318\pi$
$948$		0			0
$949$			7.76395i			0.252028i
$950$		0			0
$951$	18.5323			0.600952
$952$		0			0
$953$			42.7545i			1.38495i	0.721440	+	0.692477i	$0.243481\pi$
$953$			42.7545i			1.38495i	−0.721440	+	0.692477i	$0.756519\pi$
$954$		0			0
$955$	27.5087	+	5.54709i	0.890159	+	0.179500i
$956$		0			0
$957$	3.20692			0.103665
$958$		0			0
$959$			21.2590i			0.686488i
$960$		0			0
$961$	10.4830			0.338160
$962$		0			0
$963$	−6.37274			−0.205359
$964$		0			0
$965$	44.0252	+	8.87764i	1.41722	+	0.285781i
$966$		0			0
$967$		−	5.83813i		−	0.187742i	−0.995584	−	0.0938708i	$-0.970076\pi$
$967$		−	5.83813i		−	0.187742i	0.995584	−	0.0938708i	$-0.0299241\pi$
$968$		0			0
$969$	0.456430	+	2.32693i	0.0146627	+	0.0747516i
$970$		0			0
$971$	16.9513			0.543993			0.271996	−	0.962298i	$-0.412316\pi$
$971$	16.9513			0.543993			0.271996	+	0.962298i	$0.412316\pi$
$972$		0			0
$973$		−	12.8748i		−	0.412748i
$974$		0			0
$975$	3.57659	−	8.50775i	0.114542	−	0.272466i
$976$		0			0
$977$			49.4664i			1.58257i	0.611448	+	0.791284i	$0.290587\pi$
$977$			49.4664i			1.58257i	−0.611448	+	0.791284i	$0.709413\pi$
$978$		0			0
$979$			0.646831i			0.0206728i
$980$		0			0
$981$			23.2570i			0.742540i
$982$		0			0
$983$	0.404420			0.0128990			0.00644950	−	0.999979i	$-0.497947\pi$
$983$	0.404420			0.0128990			0.00644950	+	0.999979i	$0.497947\pi$
$984$		0			0
$985$	−24.9769	−	5.03656i	−0.795829	−	0.160478i
$986$		0			0
$987$			6.86828i			0.218620i
$988$		0			0
$989$			18.3364i			0.583065i
$990$		0			0
$991$			7.48474i			0.237761i	0.992909	+	0.118880i	$0.0379305\pi$
$991$			7.48474i			0.237761i	−0.992909	+	0.118880i	$0.962070\pi$
$992$		0			0
$993$	−25.4886			−0.808857
$994$		0			0
$995$	−11.5347	+	57.2017i	−0.365673	+	1.81342i
$996$		0			0
$997$	−14.8925			−0.471649			−0.235825	−	0.971796i	$-0.575779\pi$
$997$	−14.8925			−0.471649			−0.235825	+	0.971796i	$0.575779\pi$
$998$		0			0
$999$	40.3051			1.27520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2380.2.m.d.169.16	yes	26
5.4	even	2		2380.2.m.c.169.12	yes	26
17.16	even	2		2380.2.m.c.169.11	✓	26
85.84	even	2	inner	2380.2.m.d.169.15	yes	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2380.2.m.c.169.11	✓	26	17.16	even	2
2380.2.m.c.169.12	yes	26	5.4	even	2
2380.2.m.d.169.15	yes	26	85.84	even	2	inner
2380.2.m.d.169.16	yes	26	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2380.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.807362 q^{3} +(-2.19195 - 0.442004i) q^{5} +1.00000 q^{7} -2.34817 q^{9} +0.748338i q^{11} -2.28620i q^{13} +(-1.76969 - 0.356857i) q^{15} +(0.793630 + 4.04600i) q^{17} +0.712341 q^{19} +0.807362 q^{21} +2.80575 q^{23} +(4.60927 + 1.93770i) q^{25} -4.31791 q^{27} -5.30790i q^{29} +4.52957i q^{31} +0.604179i q^{33} +(-2.19195 - 0.442004i) q^{35} -9.33442 q^{37} -1.84579i q^{39} -1.92803i q^{41} +6.53531i q^{43} +(5.14706 + 1.03790i) q^{45} +8.50706i q^{47} +1.00000 q^{49} +(0.640747 + 3.26659i) q^{51} +13.3808i q^{53} +(0.330768 - 1.64032i) q^{55} +0.575117 q^{57} +6.74114 q^{59} -1.53774i q^{61} -2.34817 q^{63} +(-1.01051 + 5.01124i) q^{65} +4.60728i q^{67} +2.26525 q^{69} +12.8158i q^{71} -3.39600 q^{73} +(3.72134 + 1.56442i) q^{75} +0.748338i q^{77} +5.22533i q^{79} +3.55839 q^{81} +6.69151i q^{83} +(0.0487536 - 9.21942i) q^{85} -4.28539i q^{87} +0.864356 q^{89} -2.28620i q^{91} +3.65700i q^{93} +(-1.56141 - 0.314858i) q^{95} +7.00477 q^{97} -1.75722i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q + 6 q^{3} - 4 q^{5} + 26 q^{7} + 32 q^{9} + 10 q^{15} - 11 q^{17} + 6 q^{21} - 16 q^{23} + 4 q^{25} + 6 q^{27} - 4 q^{35} - 20 q^{37} - 18 q^{45} + 26 q^{49} + 9 q^{51} - 14 q^{55} + 32 q^{59} + 32 q^{63}+ \cdots - 46 q^{97}+O(q^{100}) \) 26 * q + 6 * q^3 - 4 * q^5 + 26 * q^7 + 32 * q^9 + 10 * q^15 - 11 * q^17 + 6 * q^21 - 16 * q^23 + 4 * q^25 + 6 * q^27 - 4 * q^35 - 20 * q^37 - 18 * q^45 + 26 * q^49 + 9 * q^51 - 14 * q^55 + 32 * q^59 + 32 * q^63 - 42 * q^65 - 8 * q^69 - 12 * q^73 + 8 * q^75 + 66 * q^81 + 11 * q^85 + 4 * q^89 + 20 * q^95 - 46 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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