LMFDB - Embedded newform 2940.2.f.a.1469.32

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(1469,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.1469"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1469.32
Character	$\chi$	$=$	2940.1469
Dual form			2940.2.f.a.1469.30

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.69264 + 0.367385i) q^{3} +(2.03939 + 0.916997i) q^{5} +(2.73006 + 1.24370i) q^{9} -5.87781i q^{11} -3.54004 q^{13} +(3.11506 + 2.30139i) q^{15} +3.10812i q^{17} -4.13866i q^{19} +7.15173 q^{23} +(3.31823 + 3.74023i) q^{25} +(4.16408 + 3.10812i) q^{27} +6.84761i q^{29} -4.50776i q^{31} +(2.15942 - 9.94902i) q^{33} +1.97369i q^{37} +(-5.99202 - 1.30056i) q^{39} +6.33142 q^{41} -3.88979i q^{43} +(4.42718 + 5.03985i) q^{45} +1.83399i q^{47} +(-1.14188 + 5.26093i) q^{51} +13.8078 q^{53} +(5.38994 - 11.9872i) q^{55} +(1.52048 - 7.00526i) q^{57} +4.65168 q^{59} -0.810766i q^{61} +(-7.21953 - 3.24621i) q^{65} -10.2094i q^{67} +(12.1053 + 2.62744i) q^{69} +1.18684i q^{71} -2.17045 q^{73} +(4.24247 + 7.54993i) q^{75} +10.1046 q^{79} +(5.90641 + 6.79075i) q^{81} +5.31243i q^{83} +(-2.85014 + 6.33868i) q^{85} +(-2.51571 + 11.5905i) q^{87} -12.5797 q^{89} +(1.65609 - 7.63002i) q^{93} +(3.79514 - 8.44035i) q^{95} -4.50686 q^{97} +(7.31025 - 16.0468i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2 q^{15} - 12 q^{25} - 48 q^{39} + 20 q^{51} + 56 q^{79} + 40 q^{81} - 4 q^{85} - 28 q^{99}+O(q^{100}) $ 32 * q + 2 * q^15 - 12 * q^25 - 48 * q^39 + 20 * q^51 + 56 * q^79 + 40 * q^81 - 4 * q^85 - 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	1.69264	+	0.367385i	0.977246	+	0.212110i
$3$	1.69264	+	0.367385i	0.977246	+	0.212110i
$4$		0			0
$5$	2.03939	+	0.916997i	0.912044	+	0.410093i
$5$	2.03939	+	0.916997i	0.912044	+	0.410093i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.73006	+	1.24370i	0.910019	+	0.414567i
$10$		0			0
$11$		−	5.87781i		−	1.77223i	−0.463467	−	0.886114i	$-0.653395\pi$
$11$		−	5.87781i		−	1.77223i	0.463467	−	0.886114i	$-0.346605\pi$
$12$		0			0
$13$	−3.54004			−0.981831			−0.490916	−	0.871207i	$-0.663338\pi$
$13$	−3.54004			−0.981831			−0.490916	+	0.871207i	$0.663338\pi$
$14$		0			0
$15$	3.11506	+	2.30139i	0.804306	+	0.594216i
$16$		0			0
$17$			3.10812i			0.753830i	0.926248	+	0.376915i	$0.123015\pi$
$17$			3.10812i			0.753830i	−0.926248	+	0.376915i	$0.876985\pi$
$18$		0			0
$19$		−	4.13866i		−	0.949474i	−0.880128	−	0.474737i	$-0.842543\pi$
$19$		−	4.13866i		−	0.949474i	0.880128	−	0.474737i	$-0.157457\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.15173			1.49124			0.745620	−	0.666372i	$-0.232154\pi$
$23$	7.15173			1.49124			0.745620	+	0.666372i	$0.232154\pi$
$24$		0			0
$25$	3.31823	+	3.74023i	0.663647	+	0.748046i
$26$		0			0
$27$	4.16408	+	3.10812i	0.801378	+	0.598158i
$28$		0			0
$29$			6.84761i			1.27157i	0.771867	+	0.635785i	$0.219323\pi$
$29$			6.84761i			1.27157i	−0.771867	+	0.635785i	$0.780677\pi$
$30$		0			0
$31$		−	4.50776i		−	0.809618i	−0.914401	−	0.404809i	$-0.867338\pi$
$31$		−	4.50776i		−	0.809618i	0.914401	−	0.404809i	$-0.132662\pi$
$32$		0			0
$33$	2.15942	−	9.94902i	0.375907	−	1.73190i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			1.97369i			0.324472i	0.986752	+	0.162236i	$0.0518706\pi$
$37$			1.97369i			0.324472i	−0.986752	+	0.162236i	$0.948129\pi$
$38$		0			0
$39$	−5.99202	−	1.30056i	−0.959490	−	0.208256i
$40$		0			0
$41$	6.33142			0.988801			0.494401	−	0.869234i	$-0.335388\pi$
$41$	6.33142			0.988801			0.494401	+	0.869234i	$0.335388\pi$
$42$		0			0
$43$		−	3.88979i		−	0.593187i	−0.955004	−	0.296594i	$-0.904149\pi$
$43$		−	3.88979i		−	0.593187i	0.955004	−	0.296594i	$-0.0958507\pi$
$44$		0			0
$45$	4.42718	+	5.03985i	0.659965	+	0.751296i
$46$		0			0
$47$			1.83399i			0.267515i	0.991014	+	0.133758i	$0.0427044\pi$
$47$			1.83399i			0.267515i	−0.991014	+	0.133758i	$0.957296\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−1.14188	+	5.26093i	−0.159895	+	0.736677i
$52$		0			0
$53$	13.8078			1.89665			0.948325	−	0.317302i	$-0.102777\pi$
$53$	13.8078			1.89665			0.948325	+	0.317302i	$0.102777\pi$
$54$		0			0
$55$	5.38994	−	11.9872i	0.726779	−	1.61635i
$56$		0			0
$57$	1.52048	−	7.00526i	0.201393	−	0.927869i
$58$		0			0
$59$	4.65168			0.605598			0.302799	−	0.953054i	$-0.402079\pi$
$59$	4.65168			0.605598			0.302799	+	0.953054i	$0.402079\pi$
$60$		0			0
$61$		−	0.810766i		−	0.103808i	−0.998652	−	0.0519040i	$-0.983471\pi$
$61$		−	0.810766i		−	0.103808i	0.998652	−	0.0519040i	$-0.0165290\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.21953	−	3.24621i	−0.895473	−	0.402643i
$66$		0			0
$67$		−	10.2094i		−	1.24727i	−0.781714	−	0.623637i	$-0.785654\pi$
$67$		−	10.2094i		−	1.24727i	0.781714	−	0.623637i	$-0.214346\pi$
$68$		0			0
$69$	12.1053	+	2.62744i	1.45731	+	0.316307i
$70$		0			0
$71$			1.18684i			0.140852i	0.997517	+	0.0704262i	$0.0224359\pi$
$71$			1.18684i			0.140852i	−0.997517	+	0.0704262i	$0.977564\pi$
$72$		0			0
$73$	−2.17045			−0.254032			−0.127016	−	0.991901i	$-0.540540\pi$
$73$	−2.17045			−0.254032			−0.127016	+	0.991901i	$0.540540\pi$
$74$		0			0
$75$	4.24247	+	7.54993i	0.489878	+	0.871791i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	10.1046			1.13685			0.568426	−	0.822734i	$-0.307553\pi$
$79$	10.1046			1.13685			0.568426	+	0.822734i	$0.307553\pi$
$80$		0			0
$81$	5.90641	+	6.79075i	0.656268	+	0.754528i
$82$		0			0
$83$			5.31243i			0.583115i	0.956553	+	0.291558i	$0.0941735\pi$
$83$			5.31243i			0.583115i	−0.956553	+	0.291558i	$0.905826\pi$
$84$		0			0
$85$	−2.85014	+	6.33868i	−0.309141	+	0.687526i
$86$		0			0
$87$	−2.51571	+	11.5905i	−0.269713	+	1.24264i
$88$		0			0
$89$	−12.5797			−1.33345			−0.666724	−	0.745305i	$-0.732304\pi$
$89$	−12.5797			−1.33345			−0.666724	+	0.745305i	$0.732304\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.65609	−	7.63002i	0.171728	−	0.791196i
$94$		0			0
$95$	3.79514	−	8.44035i	0.389373	−	0.865962i
$96$		0			0
$97$	−4.50686			−0.457602			−0.228801	−	0.973473i	$-0.573481\pi$
$97$	−4.50686			−0.457602			−0.228801	+	0.973473i	$0.573481\pi$
$98$		0			0
$99$	7.31025	−	16.0468i	0.734708	−	1.61276i
$100$		0			0
$101$	1.79358			0.178468			0.0892340	−	0.996011i	$-0.471558\pi$
$101$	1.79358			0.178468			0.0892340	+	0.996011i	$0.471558\pi$
$102$		0			0
$103$	−12.6138			−1.24287			−0.621436	−	0.783465i	$-0.713450\pi$
$103$	−12.6138			−1.24287			−0.621436	+	0.783465i	$0.713450\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.47951			0.336377			0.168189	−	0.985755i	$-0.446208\pi$
$107$	3.47951			0.336377			0.168189	+	0.985755i	$0.446208\pi$
$108$		0			0
$109$	−9.97605			−0.955532			−0.477766	−	0.878487i	$-0.658553\pi$
$109$	−9.97605			−0.955532			−0.477766	+	0.878487i	$0.658553\pi$
$110$		0			0
$111$	−0.725103	+	3.34074i	−0.0688238	+	0.317089i
$112$		0			0
$113$	−2.54532			−0.239444			−0.119722	−	0.992807i	$-0.538200\pi$
$113$	−2.54532			−0.239444			−0.119722	+	0.992807i	$0.538200\pi$
$114$		0			0
$115$	14.5852	+	6.55811i	1.36007	+	0.611547i
$116$		0			0
$117$	−9.66452	−	4.40276i	−0.893485	−	0.407035i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−23.5487			−2.14079
$122$		0			0
$123$	10.7168	+	2.32607i	0.966302	+	0.209735i
$124$		0			0
$125$	3.33740	+	10.6706i	0.298506	+	0.954408i
$126$		0			0
$127$			12.6616i			1.12353i	0.827295	+	0.561767i	$0.189878\pi$
$127$			12.6616i			1.12353i	−0.827295	+	0.561767i	$0.810122\pi$
$128$		0			0
$129$	1.42905	−	6.58401i	0.125821	−	0.579690i
$130$		0			0
$131$	−12.6954			−1.10920			−0.554601	−	0.832116i	$-0.687129\pi$
$131$	−12.6954			−1.10920			−0.554601	+	0.832116i	$0.687129\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	5.64206	+	10.1571i	0.485591	+	0.874186i
$136$		0			0
$137$	5.24782			0.448352			0.224176	−	0.974549i	$-0.428031\pi$
$137$	5.24782			0.448352			0.224176	+	0.974549i	$0.428031\pi$
$138$		0			0
$139$			14.2842i			1.21157i	0.795627	+	0.605787i	$0.207142\pi$
$139$			14.2842i			1.21157i	−0.795627	+	0.605787i	$0.792858\pi$
$140$		0			0
$141$	−0.673782	+	3.10429i	−0.0567427	+	0.261428i
$142$		0			0
$143$			20.8077i			1.74003i
$144$		0			0
$145$	−6.27924	+	13.9650i	−0.521462	+	1.15973i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	15.0991i		−	1.23697i	−0.785798	−	0.618484i	$-0.787747\pi$
$149$		−	15.0991i		−	1.23697i	0.785798	−	0.618484i	$-0.212253\pi$
$150$		0			0
$151$	−8.33958			−0.678665			−0.339333	−	0.940666i	$-0.610201\pi$
$151$	−8.33958			−0.678665			−0.339333	+	0.940666i	$0.610201\pi$
$152$		0			0
$153$	−3.86558	+	8.48535i	−0.312513	+	0.686000i
$154$		0			0
$155$	4.13361	−	9.19310i	0.332019	−	0.738407i
$156$		0			0
$157$	0.960596			0.0766639			0.0383319	−	0.999265i	$-0.487796\pi$
$157$	0.960596			0.0766639			0.0383319	+	0.999265i	$0.487796\pi$
$158$		0			0
$159$	23.3717	+	5.07279i	1.85349	+	0.402298i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	18.6713i		−	1.46245i	−0.682137	−	0.731224i	$-0.738949\pi$
$163$		−	18.6713i		−	1.46245i	0.682137	−	0.731224i	$-0.261051\pi$
$164$		0			0
$165$	13.5271	−	18.3098i	1.05309	−	1.42541i
$166$		0			0
$167$			14.2212i			1.10047i	0.835011	+	0.550233i	$0.185461\pi$
$167$			14.2212i			1.10047i	−0.835011	+	0.550233i	$0.814539\pi$
$168$		0			0
$169$	−0.468096			−0.0360074
$170$		0			0
$171$	5.14726	−	11.2988i	0.393621	−	0.864039i
$172$		0			0
$173$		−	0.714262i		−	0.0543043i	−0.999631	−	0.0271522i	$-0.991356\pi$
$173$		−	0.714262i		−	0.0543043i	0.999631	−	0.0271522i	$-0.00864386\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.87362	+	1.70896i	0.591818	+	0.128453i
$178$		0			0
$179$		−	11.1834i		−	0.835886i	−0.908473	−	0.417943i	$-0.862751\pi$
$179$		−	11.1834i		−	0.835886i	0.908473	−	0.417943i	$-0.137249\pi$
$180$		0			0
$181$		−	3.69700i		−	0.274796i	−0.990516	−	0.137398i	$-0.956126\pi$
$181$		−	3.69700i		−	0.274796i	0.990516	−	0.137398i	$-0.0438739\pi$
$182$		0			0
$183$	0.297864	−	1.37233i	0.0220187	−	0.101446i
$184$		0			0
$185$	−1.80986	+	4.02512i	−0.133064	+	0.295933i
$186$		0			0
$187$	18.2690			1.33596
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.3883i		−	0.751670i	−0.926687	−	0.375835i	$-0.877356\pi$
$191$		−	10.3883i		−	0.751670i	0.926687	−	0.375835i	$-0.122644\pi$
$192$		0			0
$193$			5.92106i			0.426207i	0.977030	+	0.213104i	$0.0683572\pi$
$193$			5.92106i			0.426207i	−0.977030	+	0.213104i	$0.931643\pi$
$194$		0			0
$195$	−11.0275	−	8.14701i	−0.789693	−	0.583420i
$196$		0			0
$197$	11.1053			0.791221			0.395610	−	0.918418i	$-0.370533\pi$
$197$	11.1053			0.791221			0.395610	+	0.918418i	$0.370533\pi$
$198$		0			0
$199$			13.2406i			0.938600i	0.883039	+	0.469300i	$0.155494\pi$
$199$			13.2406i			0.938600i	−0.883039	+	0.469300i	$0.844506\pi$
$200$		0			0
$201$	3.75077	−	17.2808i	0.264559	−	1.21889i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	12.9122	+	5.80589i	0.901830	+	0.405501i
$206$		0			0
$207$	19.5246	+	8.89462i	1.35706	+	0.618219i
$208$		0			0
$209$	−24.3263			−1.68268
$210$		0			0
$211$	−19.9122			−1.37082			−0.685408	−	0.728160i	$-0.740376\pi$
$211$	−19.9122			−1.37082			−0.685408	+	0.728160i	$0.740376\pi$
$212$		0			0
$213$	−0.436029	+	2.00890i	−0.0298762	+	0.137647i
$214$		0			0
$215$	3.56692	−	7.93280i	0.243262	−	0.541013i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−3.67378	−	0.797390i	−0.248251	−	0.0538826i
$220$		0			0
$221$		−	11.0029i		−	0.740134i
$222$		0			0
$223$	−10.4924			−0.702623			−0.351311	−	0.936259i	$-0.614264\pi$
$223$	−10.4924			−0.702623			−0.351311	+	0.936259i	$0.614264\pi$
$224$		0			0
$225$	4.40723	+	14.3379i	0.293815	+	0.955862i
$226$		0			0
$227$			16.4255i			1.09020i	0.838372	+	0.545098i	$0.183508\pi$
$227$			16.4255i			1.09020i	−0.838372	+	0.545098i	$0.816492\pi$
$228$		0			0
$229$			1.04366i			0.0689672i	0.999405	+	0.0344836i	$0.0109786\pi$
$229$			1.04366i			0.0689672i	−0.999405	+	0.0344836i	$0.989021\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−28.9238			−1.89486			−0.947431	−	0.319960i	$-0.896331\pi$
$233$	−28.9238			−1.89486			−0.947431	+	0.319960i	$0.896331\pi$
$234$		0			0
$235$	−1.68177	+	3.74023i	−0.109706	+	0.243986i
$236$		0			0
$237$	17.1034	+	3.71227i	1.11098	+	0.241138i
$238$		0			0
$239$			5.69731i			0.368528i	0.982877	+	0.184264i	$0.0589902\pi$
$239$			5.69731i			0.368528i	−0.982877	+	0.184264i	$0.941010\pi$
$240$		0			0
$241$		−	7.05058i		−	0.454168i	−0.973875	−	0.227084i	$-0.927081\pi$
$241$		−	7.05058i		−	0.454168i	0.973875	−	0.227084i	$-0.0729192\pi$
$242$		0			0
$243$	7.50260	+	13.6642i	0.481292	+	0.876560i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			14.6510i			0.932223i
$248$		0			0
$249$	−1.95171	+	8.99203i	−0.123685	+	0.569847i
$250$		0			0
$251$	−3.93237			−0.248209			−0.124105	−	0.992269i	$-0.539606\pi$
$251$	−3.93237			−0.248209			−0.124105	+	0.992269i	$0.539606\pi$
$252$		0			0
$253$		−	42.0366i		−	2.64282i
$254$		0			0
$255$	−7.15299	+	9.68199i	−0.447938	+	0.606310i
$256$		0			0
$257$		−	15.3057i		−	0.954746i	−0.878701	−	0.477373i	$-0.841589\pi$
$257$		−	15.3057i		−	0.954746i	0.878701	−	0.477373i	$-0.158411\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−8.51639	+	18.6944i	−0.527151	+	1.15715i
$262$		0			0
$263$	−19.5195			−1.20363			−0.601813	−	0.798637i	$-0.705555\pi$
$263$	−19.5195			−1.20363			−0.601813	+	0.798637i	$0.705555\pi$
$264$		0			0
$265$	28.1595	+	12.6617i	1.72983	+	0.777803i
$266$		0			0
$267$	−21.2929	−	4.62161i	−1.30311	−	0.282838i
$268$		0			0
$269$	9.83730			0.599791			0.299895	−	0.953972i	$-0.403048\pi$
$269$	9.83730			0.599791			0.299895	+	0.953972i	$0.403048\pi$
$270$		0			0
$271$		−	20.8793i		−	1.26833i	−0.773198	−	0.634164i	$-0.781344\pi$
$271$		−	20.8793i		−	1.26833i	0.773198	−	0.634164i	$-0.218656\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21.9844	−	19.5040i	1.32571	−	1.17613i
$276$		0			0
$277$		−	8.50375i		−	0.510941i	−0.966817	−	0.255470i	$-0.917770\pi$
$277$		−	8.50375i		−	0.510941i	0.966817	−	0.255470i	$-0.0822303\pi$
$278$		0			0
$279$	5.60632	−	12.3064i	0.335641	−	0.736768i
$280$		0			0
$281$		−	7.63690i		−	0.455579i	−0.973710	−	0.227790i	$-0.926850\pi$
$281$		−	7.63690i		−	0.455579i	0.973710	−	0.227790i	$-0.0731499\pi$
$282$		0			0
$283$	−9.00750			−0.535440			−0.267720	−	0.963497i	$-0.586270\pi$
$283$	−9.00750			−0.535440			−0.267720	+	0.963497i	$0.586270\pi$
$284$		0			0
$285$	9.52466	−	12.8922i	0.564192	−	0.763667i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.33958			0.431740
$290$		0			0
$291$	−7.62848	−	1.65575i	−0.447190	−	0.0970620i
$292$		0			0
$293$			27.6102i			1.61301i	0.591231	+	0.806503i	$0.298642\pi$
$293$			27.6102i			1.61301i	−0.591231	+	0.806503i	$0.701358\pi$
$294$		0			0
$295$	9.48660	+	4.26558i	0.552331	+	0.248352i
$296$		0			0
$297$	18.2690	−	24.4757i	1.06007	−	1.42022i
$298$		0			0
$299$	−25.3174			−1.46415
$300$		0			0
$301$		0			0
$302$		0			0
$303$	3.03589	+	0.658936i	0.174407	+	0.0378549i
$304$		0			0
$305$	0.743470	−	1.65347i	0.0425710	−	0.0946774i
$306$		0			0
$307$	9.99609			0.570507			0.285253	−	0.958452i	$-0.407922\pi$
$307$	9.99609			0.570507			0.285253	+	0.958452i	$0.407922\pi$
$308$		0			0
$309$	−21.3506	−	4.63411i	−1.21459	−	0.263625i
$310$		0			0
$311$	−23.8672			−1.35339			−0.676693	−	0.736265i	$-0.736588\pi$
$311$	−23.8672			−1.35339			−0.676693	+	0.736265i	$0.736588\pi$
$312$		0			0
$313$	5.59519			0.316259			0.158129	−	0.987418i	$-0.449454\pi$
$313$	5.59519			0.316259			0.158129	+	0.987418i	$0.449454\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.7366			−0.659191			−0.329596	−	0.944122i	$-0.606912\pi$
$317$	−11.7366			−0.659191			−0.329596	+	0.944122i	$0.606912\pi$
$318$		0			0
$319$	40.2490			2.25351
$320$		0			0
$321$	5.88956	+	1.27832i	0.328723	+	0.0713490i
$322$		0			0
$323$	12.8635			0.715742
$324$		0			0
$325$	−11.7467	−	13.2406i	−0.651589	−	0.734455i
$326$		0			0
$327$	−16.8859	−	3.66505i	−0.933790	−	0.202678i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.52907			0.139010			0.0695050	−	0.997582i	$-0.477858\pi$
$331$	2.52907			0.139010			0.0695050	+	0.997582i	$0.477858\pi$
$332$		0			0
$333$	−2.45468	+	5.38827i	−0.134515	+	0.295276i
$334$		0			0
$335$	9.36196	−	20.8209i	0.511499	−	1.13757i
$336$		0			0
$337$			34.3299i			1.87007i	0.354560	+	0.935033i	$0.384631\pi$
$337$			34.3299i			1.87007i	−0.354560	+	0.935033i	$0.615369\pi$
$338$		0			0
$339$	−4.30831	−	0.935114i	−0.233995	−	0.0507884i
$340$		0			0
$341$	−26.4958			−1.43483
$342$		0			0
$343$		0			0
$344$		0			0
$345$	22.2781	+	16.4589i	1.19941	+	0.886118i
$346$		0			0
$347$	0.595727			0.0319803			0.0159902	−	0.999872i	$-0.494910\pi$
$347$	0.595727			0.0319803			0.0159902	+	0.999872i	$0.494910\pi$
$348$		0			0
$349$		−	23.0392i		−	1.23326i	−0.787253	−	0.616630i	$-0.788497\pi$
$349$		−	23.0392i		−	1.23326i	0.787253	−	0.616630i	$-0.211503\pi$
$350$		0			0
$351$	−14.7410	−	11.0029i	−0.786818	−	0.587290i
$352$		0			0
$353$		−	10.8971i		−	0.579995i	−0.957027	−	0.289997i	$-0.906346\pi$
$353$		−	10.8971i		−	0.579995i	0.957027	−	0.289997i	$-0.0936544\pi$
$354$		0			0
$355$	−1.08833	+	2.42044i	−0.0577627	+	0.128464i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	10.4306i		−	0.550508i	−0.961372	−	0.275254i	$-0.911238\pi$
$359$		−	10.4306i		−	0.550508i	0.961372	−	0.275254i	$-0.0887620\pi$
$360$		0			0
$361$	1.87149			0.0984992
$362$		0			0
$363$	−39.8595	−	8.65145i	−2.09208	−	0.454083i
$364$		0			0
$365$	−4.42639	−	1.99029i	−0.231688	−	0.104177i
$366$		0			0
$367$	8.54942			0.446276			0.223138	−	0.974787i	$-0.428370\pi$
$367$	8.54942			0.446276			0.223138	+	0.974787i	$0.428370\pi$
$368$		0			0
$369$	17.2851	+	7.87440i	0.899828	+	0.409925i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			6.62947i			0.343261i	0.985161	+	0.171630i	$0.0549035\pi$
$373$			6.62947i			0.343261i	−0.985161	+	0.171630i	$0.945096\pi$
$374$		0			0
$375$	1.72879	+	19.2876i	0.0892741	+	0.996007i
$376$		0			0
$377$		−	24.2408i		−	1.24847i
$378$		0			0
$379$	22.3165			1.14632			0.573162	−	0.819442i	$-0.305717\pi$
$379$	22.3165			1.14632			0.573162	+	0.819442i	$0.305717\pi$
$380$		0			0
$381$	−4.65168	+	21.4315i	−0.238313	+	1.09797i
$382$		0			0
$383$			28.1890i			1.44039i	0.693771	+	0.720196i	$0.255948\pi$
$383$			28.1890i			1.44039i	−0.693771	+	0.720196i	$0.744052\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.83774	−	10.6193i	0.245916	−	0.539811i
$388$		0			0
$389$		−	4.13277i		−	0.209540i	−0.994496	−	0.104770i	$-0.966589\pi$
$389$		−	4.13277i		−	0.209540i	0.994496	−	0.104770i	$-0.0334106\pi$
$390$		0			0
$391$			22.2284i			1.12414i
$392$		0			0
$393$	−21.4887	−	4.66411i	−1.08396	−	0.235273i
$394$		0			0
$395$	20.6072	+	9.26585i	1.03686	+	0.466216i
$396$		0			0
$397$	13.2378			0.664386			0.332193	−	0.943211i	$-0.392211\pi$
$397$	13.2378			0.664386			0.332193	+	0.943211i	$0.392211\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			0.775246i			0.0387140i	0.999813	+	0.0193570i	$0.00616190\pi$
$401$			0.775246i			0.0387140i	−0.999813	+	0.0193570i	$0.993838\pi$
$402$		0			0
$403$			15.9577i			0.794909i
$404$		0			0
$405$	5.81839	+	19.2652i	0.289118	+	0.957294i
$406$		0			0
$407$	11.6010			0.575038
$408$		0			0
$409$			4.58524i			0.226726i	0.993554	+	0.113363i	$0.0361622\pi$
$409$			4.58524i			0.226726i	−0.993554	+	0.113363i	$0.963838\pi$
$410$		0			0
$411$	8.88267	+	1.92797i	0.438150	+	0.0950999i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.87149	+	10.8341i	−0.239132	+	0.531827i
$416$		0			0
$417$	−5.24782	+	24.1781i	−0.256987	+	1.18401i
$418$		0			0
$419$	11.9761			0.585070			0.292535	−	0.956255i	$-0.405501\pi$
$419$	11.9761			0.585070			0.292535	+	0.956255i	$0.405501\pi$
$420$		0			0
$421$	25.5487			1.24517			0.622584	−	0.782553i	$-0.286083\pi$
$421$	25.5487			1.24517			0.622584	+	0.782553i	$0.286083\pi$
$422$		0			0
$423$	−2.28094	+	5.00691i	−0.110903	+	0.243444i
$424$		0			0
$425$	−11.6251	+	10.3135i	−0.563900	+	0.500277i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−7.64445	+	35.2200i	−0.369078	+	1.70044i
$430$		0			0
$431$			14.5070i			0.698778i	0.936978	+	0.349389i	$0.113611\pi$
$431$			14.5070i			0.698778i	−0.936978	+	0.349389i	$0.886389\pi$
$432$		0			0
$433$	−40.0655			−1.92543			−0.962713	−	0.270524i	$-0.912803\pi$
$433$	−40.0655			−1.92543			−0.962713	+	0.270524i	$0.912803\pi$
$434$		0			0
$435$	−15.7590	+	21.3307i	−0.755586	+	1.02273i
$436$		0			0
$437$		−	29.5986i		−	1.41589i
$438$		0			0
$439$			31.3891i			1.49812i	0.662502	+	0.749060i	$0.269494\pi$
$439$			31.3891i			1.49812i	−0.662502	+	0.749060i	$0.730506\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.0355			−0.476802			−0.238401	−	0.971167i	$-0.576623\pi$
$443$	−10.0355			−0.476802			−0.238401	+	0.971167i	$0.576623\pi$
$444$		0			0
$445$	−25.6550	−	11.5356i	−1.21616	−	0.546838i
$446$		0			0
$447$	5.54719	−	25.5574i	0.262373	−	1.20882i
$448$		0			0
$449$			16.7060i			0.788405i	0.919024	+	0.394203i	$0.128979\pi$
$449$			16.7060i			0.788405i	−0.919024	+	0.394203i	$0.871021\pi$
$450$		0			0
$451$		−	37.2149i		−	1.75238i
$452$		0			0
$453$	−14.1159	−	3.06384i	−0.663223	−	0.143952i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	14.1481i		−	0.661821i	−0.943662	−	0.330910i	$-0.892644\pi$
$457$		−	14.1481i		−	0.661821i	0.943662	−	0.330910i	$-0.107356\pi$
$458$		0			0
$459$	−9.66042	+	12.9425i	−0.450910	+	0.604103i
$460$		0			0
$461$	−4.22519			−0.196787			−0.0983935	−	0.995148i	$-0.531370\pi$
$461$	−4.22519			−0.196787			−0.0983935	+	0.995148i	$0.531370\pi$
$462$		0			0
$463$		−	33.2490i		−	1.54521i	−0.634886	−	0.772606i	$-0.718953\pi$
$463$		−	33.2490i		−	1.54521i	0.634886	−	0.772606i	$-0.281047\pi$
$464$		0			0
$465$	10.3741	−	14.0420i	0.481088	−	0.651181i
$466$		0			0
$467$			36.8629i			1.70581i	0.522066	+	0.852905i	$0.325161\pi$
$467$			36.8629i			1.70581i	−0.522066	+	0.852905i	$0.674839\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.62594	+	0.352909i	0.0749195	+	0.0162612i
$472$		0			0
$473$	−22.8635			−1.05126
$474$		0			0
$475$	15.4795	−	13.7330i	0.710250	−	0.630115i
$476$		0			0
$477$	37.6961	+	17.1728i	1.72599	+	0.786289i
$478$		0			0
$479$	−9.33593			−0.426570			−0.213285	−	0.976990i	$-0.568416\pi$
$479$	−9.33593			−0.426570			−0.213285	+	0.976990i	$0.568416\pi$
$480$		0			0
$481$		−	6.98693i		−	0.318577i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.19124	−	4.13277i	−0.417353	−	0.187660i
$486$		0			0
$487$		−	25.3389i		−	1.14822i	−0.818779	−	0.574109i	$-0.805349\pi$
$487$		−	25.3389i		−	1.14822i	0.818779	−	0.574109i	$-0.194651\pi$
$488$		0			0
$489$	6.85956	−	31.6038i	0.310200	−	1.42917i
$490$		0			0
$491$			35.4614i			1.60035i	0.599765	+	0.800176i	$0.295261\pi$
$491$			35.4614i			1.60035i	−0.599765	+	0.800176i	$0.704739\pi$
$492$		0			0
$493$	−21.2832			−0.958547
$494$		0			0
$495$	29.6233	−	26.0222i	1.33147	−	1.16961i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−29.4174			−1.31690			−0.658451	−	0.752624i	$-0.728788\pi$
$499$	−29.4174			−1.31690			−0.658451	+	0.752624i	$0.728788\pi$
$500$		0			0
$501$	−5.22464	+	24.0713i	−0.233420	+	1.07543i
$502$		0			0
$503$		−	20.9709i		−	0.935046i	−0.883981	−	0.467523i	$-0.845147\pi$
$503$		−	20.9709i		−	0.935046i	0.883981	−	0.467523i	$-0.154853\pi$
$504$		0			0
$505$	3.65781	+	1.64471i	0.162771	+	0.0731886i
$506$		0			0
$507$	−0.792318	−	0.171972i	−0.0351881	−	0.00763753i
$508$		0			0
$509$	7.43642			0.329613			0.164807	−	0.986326i	$-0.447300\pi$
$509$	7.43642			0.329613			0.164807	+	0.986326i	$0.447300\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	12.8635	−	17.2337i	0.567936	−	0.760888i
$514$		0			0
$515$	−25.7244	−	11.5668i	−1.13355	−	0.509693i
$516$		0			0
$517$	10.7799			0.474098
$518$		0			0
$519$	0.262409	−	1.20899i	0.0115185	−	0.0530687i
$520$		0			0
$521$	5.10250			0.223545			0.111772	−	0.993734i	$-0.464347\pi$
$521$	5.10250			0.223545			0.111772	+	0.993734i	$0.464347\pi$
$522$		0			0
$523$	−16.4966			−0.721345			−0.360673	−	0.932692i	$-0.617453\pi$
$523$	−16.4966			−0.721345			−0.360673	+	0.932692i	$0.617453\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.0107			0.610315
$528$		0			0
$529$	28.1473			1.22379
$530$		0			0
$531$	12.6994	+	5.78531i	0.551105	+	0.251061i
$532$		0			0
$533$	−22.4135			−0.970836
$534$		0			0
$535$	7.09609	+	3.19070i	0.306791	+	0.137946i
$536$		0			0
$537$	4.10861	−	18.9294i	0.177300	−	0.816866i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	20.6154			0.886323			0.443162	−	0.896442i	$-0.353857\pi$
$541$	20.6154			0.886323			0.443162	+	0.896442i	$0.353857\pi$
$542$		0			0
$543$	1.35822	−	6.25769i	0.0582869	−	0.268543i
$544$		0			0
$545$	−20.3451	−	9.14800i	−0.871487	−	0.391857i
$546$		0			0
$547$		−	25.6070i		−	1.09488i	−0.836846	−	0.547438i	$-0.815603\pi$
$547$		−	25.6070i		−	1.09488i	0.836846	−	0.547438i	$-0.184397\pi$
$548$		0			0
$549$	1.00835	−	2.21344i	0.0430354	−	0.0944672i
$550$		0			0
$551$	28.3399			1.20732
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.54222	+	6.14816i	−0.192806	+	0.260975i
$556$		0			0
$557$	−36.1972			−1.53372			−0.766862	−	0.641812i	$-0.778183\pi$
$557$	−36.1972			−1.53372			−0.766862	+	0.641812i	$0.778183\pi$
$558$		0			0
$559$			13.7700i			0.582410i
$560$		0			0
$561$	30.9228	+	6.71175i	1.30556	+	0.283370i
$562$		0			0
$563$		−	30.6466i		−	1.29160i	−0.763506	−	0.645801i	$-0.776524\pi$
$563$		−	30.6466i		−	1.29160i	0.763506	−	0.645801i	$-0.223476\pi$
$564$		0			0
$565$	−5.19090	−	2.33405i	−0.218383	−	0.0981943i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			15.2598i			0.639722i	0.947464	+	0.319861i	$0.103636\pi$
$569$			15.2598i			0.639722i	−0.947464	+	0.319861i	$0.896364\pi$
$570$		0			0
$571$	−18.1046			−0.757652			−0.378826	−	0.925468i	$-0.623672\pi$
$571$	−18.1046			−0.757652			−0.378826	+	0.925468i	$0.623672\pi$
$572$		0			0
$573$	3.81650	−	17.5836i	0.159437	−	0.734566i
$574$		0			0
$575$	23.7311	+	26.7491i	0.989656	+	1.11552i
$576$		0			0
$577$	−6.56672			−0.273376			−0.136688	−	0.990614i	$-0.543646\pi$
$577$	−6.56672			−0.273376			−0.136688	+	0.990614i	$0.543646\pi$
$578$		0			0
$579$	−2.17531	+	10.0222i	−0.0904028	+	0.416509i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	81.1598i		−	3.36129i
$584$		0			0
$585$	−15.6724	−	17.8413i	−0.647975	−	0.737646i
$586$		0			0
$587$		−	9.35074i		−	0.385946i	−0.981204	−	0.192973i	$-0.938187\pi$
$587$		−	9.35074i		−	0.385946i	0.981204	−	0.192973i	$-0.0618130\pi$
$588$		0			0
$589$	−18.6561			−0.768712
$590$		0			0
$591$	18.7973	+	4.07993i	0.773217	+	0.167826i
$592$		0			0
$593$		−	26.9976i		−	1.10866i	−0.832298	−	0.554329i	$-0.812975\pi$
$593$		−	26.9976i		−	1.10866i	0.832298	−	0.554329i	$-0.187025\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.86439	+	22.4115i	−0.199086	+	0.917243i
$598$		0			0
$599$		−	22.7726i		−	0.930461i	−0.885189	−	0.465231i	$-0.845971\pi$
$599$		−	22.7726i		−	0.930461i	0.885189	−	0.465231i	$-0.154029\pi$
$600$		0			0
$601$		−	5.26871i		−	0.214915i	−0.994210	−	0.107458i	$-0.965729\pi$
$601$		−	5.26871i		−	0.214915i	0.994210	−	0.107458i	$-0.0342710\pi$
$602$		0			0
$603$	12.6974	−	27.8722i	0.517079	−	1.13504i
$604$		0			0
$605$	−48.0250	−	21.5941i	−1.95250	−	0.877925i
$606$		0			0
$607$	9.35499			0.379707			0.189854	−	0.981812i	$-0.439199\pi$
$607$	9.35499			0.379707			0.189854	+	0.981812i	$0.439199\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	6.49242i		−	0.262655i
$612$		0			0
$613$			31.5591i			1.27466i	0.770591	+	0.637330i	$0.219961\pi$
$613$			31.5591i			1.27466i	−0.770591	+	0.637330i	$0.780039\pi$
$614$		0			0
$615$	19.7228	+	14.5710i	0.795299	+	0.587561i
$616$		0			0
$617$	−39.6450			−1.59605			−0.798023	−	0.602626i	$-0.794121\pi$
$617$	−39.6450			−1.59605			−0.798023	+	0.602626i	$0.794121\pi$
$618$		0			0
$619$			38.3829i			1.54274i	0.636387	+	0.771370i	$0.280428\pi$
$619$			38.3829i			1.54274i	−0.636387	+	0.771370i	$0.719572\pi$
$620$		0			0
$621$	29.7804	+	22.2284i	1.19505	+	0.891997i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.97865	+	24.8219i	−0.119146	+	0.992877i
$626$		0			0
$627$	−41.1756	−	8.93712i	−1.64440	−	0.356914i
$628$		0			0
$629$	−6.13446			−0.244597
$630$		0			0
$631$	1.06381			0.0423495			0.0211748	−	0.999776i	$-0.493259\pi$
$631$	1.06381			0.0423495			0.0211748	+	0.999776i	$0.493259\pi$
$632$		0			0
$633$	−33.7042	−	7.31547i	−1.33962	−	0.290764i
$634$		0			0
$635$	−11.6106	+	25.8219i	−0.460754	+	1.02471i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−1.47608	+	3.24015i	−0.0583928	+	0.128178i
$640$		0			0
$641$		−	29.9812i		−	1.18419i	−0.805870	−	0.592093i	$-0.798302\pi$
$641$		−	29.9812i		−	1.18419i	0.805870	−	0.592093i	$-0.201698\pi$
$642$		0			0
$643$	−16.0861			−0.634373			−0.317187	−	0.948363i	$-0.602738\pi$
$643$	−16.0861			−0.634373			−0.317187	+	0.948363i	$0.602738\pi$
$644$		0			0
$645$	8.95191	−	12.1169i	0.352481	−	0.477104i
$646$		0			0
$647$		−	24.9639i		−	0.981431i	−0.871320	−	0.490716i	$-0.836735\pi$
$647$		−	24.9639i		−	0.981431i	0.871320	−	0.490716i	$-0.163265\pi$
$648$		0			0
$649$		−	27.3417i		−	1.07326i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.39213			−0.171877			−0.0859387	−	0.996300i	$-0.527389\pi$
$653$	−4.39213			−0.171877			−0.0859387	+	0.996300i	$0.527389\pi$
$654$		0			0
$655$	−25.8909	−	11.6416i	−1.01164	−	0.454877i
$656$		0			0
$657$	−5.92544	−	2.69939i	−0.231173	−	0.105313i
$658$		0			0
$659$			1.93959i			0.0755557i	0.999286	+	0.0377778i	$0.0120279\pi$
$659$			1.93959i			0.0755557i	−0.999286	+	0.0377778i	$0.987972\pi$
$660$		0			0
$661$			24.9454i			0.970265i	0.874441	+	0.485132i	$0.161229\pi$
$661$			24.9454i			0.970265i	−0.874441	+	0.485132i	$0.838771\pi$
$662$		0			0
$663$	4.04230	−	18.6239i	0.156990	−	0.723293i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			48.9723i			1.89621i
$668$		0			0
$669$	−17.7598	−	3.85475i	−0.686635	−	0.149033i
$670$		0			0
$671$	−4.76553			−0.183971
$672$		0			0
$673$			6.21519i			0.239578i	0.992799	+	0.119789i	$0.0382218\pi$
$673$			6.21519i			0.239578i	−0.992799	+	0.119789i	$0.961778\pi$
$674$		0			0
$675$	2.19231	+	25.8881i	0.0843819	+	0.996433i
$676$		0			0
$677$			16.6603i			0.640309i	0.947365	+	0.320154i	$0.103735\pi$
$677$			16.6603i			0.640309i	−0.947365	+	0.320154i	$0.896265\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.03448	+	27.8024i	−0.231242	+	1.06539i
$682$		0			0
$683$	22.2677			0.852051			0.426025	−	0.904711i	$-0.359913\pi$
$683$	22.2677			0.852051			0.426025	+	0.904711i	$0.359913\pi$
$684$		0			0
$685$	10.7024	+	4.81224i	0.408916	+	0.183866i
$686$		0			0
$687$	−0.383426	+	1.76654i	−0.0146286	+	0.0673979i
$688$		0			0
$689$	−48.8803			−1.86219
$690$		0			0
$691$			27.7696i			1.05640i	0.849119	+	0.528202i	$0.177134\pi$
$691$			27.7696i			1.05640i	−0.849119	+	0.528202i	$0.822866\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.0986	+	29.1312i	−0.496858	+	1.10501i
$696$		0			0
$697$			19.6788i			0.745388i
$698$		0			0
$699$	−48.9576	−	10.6262i	−1.85175	−	0.401919i
$700$		0			0
$701$			0.789291i			0.0298111i	0.999889	+	0.0149056i	$0.00474476\pi$
$701$			0.789291i			0.0298111i	−0.999889	+	0.0149056i	$0.995255\pi$
$702$		0			0
$703$	8.16842			0.308078
$704$		0			0
$705$	−4.22073	+	5.71301i	−0.158962	+	0.215164i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	15.8686			0.595960			0.297980	−	0.954572i	$-0.403687\pi$
$709$	15.8686			0.595960			0.297980	+	0.954572i	$0.403687\pi$
$710$		0			0
$711$	27.5860	+	12.5671i	1.03456	+	0.471302i
$712$		0			0
$713$		−	32.2383i		−	1.20733i
$714$		0			0
$715$	−19.0806	+	42.4351i	−0.713574	+	1.58698i
$716$		0			0
$717$	−2.09311	+	9.64349i	−0.0781686	+	0.360143i
$718$		0			0
$719$	45.2605			1.68793			0.843967	−	0.536396i	$-0.180215\pi$
$719$	45.2605			1.68793			0.843967	+	0.536396i	$0.180215\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	2.59028	−	11.9341i	0.0963336	−	0.443834i
$724$		0			0
$725$	−25.6116	+	22.7220i	−0.951192	+	0.843873i
$726$		0			0
$727$	−21.4724			−0.796369			−0.398185	−	0.917305i	$-0.630360\pi$
$727$	−21.4724			−0.796369			−0.398185	+	0.917305i	$0.630360\pi$
$728$		0			0
$729$	7.67916	+	25.8849i	0.284413	+	0.958702i
$730$		0			0
$731$	12.0899			0.447162
$732$		0			0
$733$	−14.7624			−0.545263			−0.272631	−	0.962119i	$-0.587894\pi$
$733$	−14.7624			−0.545263			−0.272631	+	0.962119i	$0.587894\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−60.0088			−2.21045
$738$		0			0
$739$	−2.44131			−0.0898049			−0.0449024	−	0.998991i	$-0.514298\pi$
$739$	−2.44131			−0.0898049			−0.0449024	+	0.998991i	$0.514298\pi$
$740$		0			0
$741$	−5.38258	+	24.7989i	−0.197734	+	0.911011i
$742$		0			0
$743$	−32.2677			−1.18379			−0.591894	−	0.806016i	$-0.701620\pi$
$743$	−32.2677			−1.18379			−0.591894	+	0.806016i	$0.701620\pi$
$744$		0			0
$745$	13.8458	−	30.7930i	0.507272	−	1.12817i
$746$		0			0
$747$	−6.60708	+	14.5032i	−0.241741	+	0.530646i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	14.0168			0.511480			0.255740	−	0.966746i	$-0.417681\pi$
$751$	14.0168			0.511480			0.255740	+	0.966746i	$0.417681\pi$
$752$		0			0
$753$	−6.65608	−	1.44470i	−0.242561	−	0.0526476i
$754$		0			0
$755$	−17.0077	−	7.64737i	−0.618972	−	0.278316i
$756$		0			0
$757$			41.4500i			1.50653i	0.657720	+	0.753263i	$0.271521\pi$
$757$			41.4500i			1.50653i	−0.657720	+	0.753263i	$0.728479\pi$
$758$		0			0
$759$	15.4436	−	71.1527i	0.560568	−	2.58268i
$760$		0			0
$761$	−40.1172			−1.45425			−0.727124	−	0.686506i	$-0.759144\pi$
$761$	−40.1172			−1.45425			−0.727124	+	0.686506i	$0.759144\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−15.6645	+	13.7602i	−0.566350	+	0.497502i
$766$		0			0
$767$	−16.4672			−0.594595
$768$		0			0
$769$			22.4441i			0.809355i	0.914459	+	0.404677i	$0.132616\pi$
$769$			22.4441i			0.809355i	−0.914459	+	0.404677i	$0.867384\pi$
$770$		0			0
$771$	5.62310	−	25.9071i	0.202511	−	0.933021i
$772$		0			0
$773$		−	29.0928i		−	1.04640i	−0.852211	−	0.523198i	$-0.824739\pi$
$773$		−	29.0928i		−	1.04640i	0.852211	−	0.523198i	$-0.175261\pi$
$774$		0			0
$775$	16.8601	−	14.9578i	0.605632	−	0.537301i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	26.2036i		−	0.938841i
$780$		0			0
$781$	6.97605			0.249623
$782$		0			0
$783$	−21.2832	+	28.5140i	−0.760600	+	1.01901i
$784$		0			0
$785$	1.95903	+	0.880863i	0.0699208	+	0.0314394i
$786$		0			0
$787$	−33.8923			−1.20813			−0.604064	−	0.796936i	$-0.706453\pi$
$787$	−33.8923			−1.20813			−0.604064	+	0.796936i	$0.706453\pi$
$788$		0			0
$789$	−33.0396	−	7.17120i	−1.17624	−	0.255301i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			2.87015i			0.101922i
$794$		0			0
$795$	43.0122	+	31.7771i	1.52549	+	1.12702i
$796$		0			0
$797$			30.8362i			1.09227i	0.837696	+	0.546137i	$0.183902\pi$
$797$			30.8362i			1.09227i	−0.837696	+	0.546137i	$0.816098\pi$
$798$		0			0
$799$	−5.70028			−0.201661
$800$		0			0
$801$	−34.3433	−	15.6454i	−1.21346	−	0.552804i
$802$		0			0
$803$			12.7575i			0.450202i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	16.6510	+	3.61408i	0.586143	+	0.127222i
$808$		0			0
$809$			37.1049i			1.30454i	0.757988	+	0.652269i	$0.226183\pi$
$809$			37.1049i			1.30454i	−0.757988	+	0.652269i	$0.773817\pi$
$810$		0			0
$811$		−	33.9956i		−	1.19375i	−0.802336	−	0.596873i	$-0.796410\pi$
$811$		−	33.9956i		−	1.19375i	0.802336	−	0.596873i	$-0.203590\pi$
$812$		0			0
$813$	7.67076	−	35.3412i	0.269025	−	1.23947i
$814$		0			0
$815$	17.1215	−	38.0781i	0.599741	−	1.33382i
$816$		0			0
$817$	−16.0985			−0.563216
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	46.2531i		−	1.61424i	−0.590385	−	0.807122i	$-0.701024\pi$
$821$		−	46.2531i		−	1.61424i	0.590385	−	0.807122i	$-0.298976\pi$
$822$		0			0
$823$			47.0648i			1.64058i	0.571951	+	0.820288i	$0.306187\pi$
$823$			47.0648i			1.64058i	−0.571951	+	0.820288i	$0.693813\pi$
$824$		0			0
$825$	44.3771	−	24.9364i	1.54501	−	0.868175i
$826$		0			0
$827$	14.6102			0.508046			0.254023	−	0.967198i	$-0.418246\pi$
$827$	14.6102			0.508046			0.254023	+	0.967198i	$0.418246\pi$
$828$		0			0
$829$			32.3173i			1.12243i	0.827672	+	0.561213i	$0.189665\pi$
$829$			32.3173i			1.12243i	−0.827672	+	0.561213i	$0.810335\pi$
$830$		0			0
$831$	3.12415	−	14.3938i	0.108376	−	0.499315i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−13.0408	+	29.0025i	−0.451294	+	1.00367i
$836$		0			0
$837$	14.0107	−	18.7707i	0.484280	−	0.648810i
$838$		0			0
$839$	15.0113			0.518249			0.259124	−	0.965844i	$-0.416566\pi$
$839$	15.0113			0.518249			0.259124	+	0.965844i	$0.416566\pi$
$840$		0			0
$841$	−17.8898			−0.616888
$842$		0			0
$843$	2.80569	−	12.9265i	0.0966329	−	0.445213i
$844$		0			0
$845$	−0.954631	−	0.429242i	−0.0328403	−	0.0147664i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−15.2464	−	3.30922i	−0.523257	−	0.113572i
$850$		0			0
$851$			14.1153i			0.483865i
$852$		0			0
$853$	−43.2807			−1.48190			−0.740952	−	0.671558i	$-0.765626\pi$
$853$	−43.2807			−1.48190			−0.740952	+	0.671558i	$0.765626\pi$
$854$		0			0
$855$	20.8582	−	18.3226i	0.713336	−	0.626620i
$856$		0			0
$857$			21.1604i			0.722827i	0.932406	+	0.361413i	$0.117706\pi$
$857$			21.1604i			0.722827i	−0.932406	+	0.361413i	$0.882294\pi$
$858$		0			0
$859$		−	42.2472i		−	1.44146i	−0.693218	−	0.720728i	$-0.743808\pi$
$859$		−	42.2472i		−	1.44146i	0.693218	−	0.720728i	$-0.256192\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	35.6914			1.21495			0.607475	−	0.794339i	$-0.292183\pi$
$863$	35.6914			1.21495			0.607475	+	0.794339i	$0.292183\pi$
$864$		0			0
$865$	0.654976	−	1.45666i	0.0222698	−	0.0495279i
$866$		0			0
$867$	12.4233	+	2.69645i	0.421916	+	0.0915764i
$868$		0			0
$869$		−	59.3927i		−	2.01476i
$870$		0			0
$871$			36.1416i			1.22461i
$872$		0			0
$873$	−12.3040	−	5.60519i	−0.416426	−	0.189707i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	2.19992i		−	0.0742860i	−0.999310	−	0.0371430i	$-0.988174\pi$
$877$		−	2.19992i		−	0.0742860i	0.999310	−	0.0371430i	$-0.0118257\pi$
$878$		0			0
$879$	−10.1436	+	46.7341i	−0.342135	+	1.57630i
$880$		0			0
$881$	−29.2075			−0.984025			−0.492013	−	0.870588i	$-0.663739\pi$
$881$	−29.2075			−0.984025			−0.492013	+	0.870588i	$0.663739\pi$
$882$		0			0
$883$			2.21069i			0.0743956i	0.999308	+	0.0371978i	$0.0118432\pi$
$883$			2.21069i			0.0743956i	−0.999308	+	0.0371978i	$0.988157\pi$
$884$		0			0
$885$	14.4903	+	10.7053i	0.487086	+	0.359856i
$886$		0			0
$887$			13.3627i			0.448675i	0.974512	+	0.224337i	$0.0720217\pi$
$887$			13.3627i			0.448675i	−0.974512	+	0.224337i	$0.927978\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	39.9148	−	34.7168i	1.33720	−	1.16306i
$892$		0			0
$893$	7.59028			0.253999
$894$		0			0
$895$	10.2551	−	22.8073i	0.342791	−	0.762364i
$896$		0			0
$897$	−42.8533	−	9.30126i	−1.43083	−	0.310560i
$898$		0			0
$899$	30.8674			1.02949
$900$		0			0
$901$			42.9164i			1.42975i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.39014	−	7.53963i	0.112692	−	0.250626i
$906$		0			0
$907$		−	4.18890i		−	0.139090i	−0.997579	−	0.0695451i	$-0.977845\pi$
$907$		−	4.18890i		−	0.139090i	0.997579	−	0.0695451i	$-0.0221548\pi$
$908$		0			0
$909$	4.89658	+	2.23068i	0.162409	+	0.0739870i
$910$		0			0
$911$		−	22.5190i		−	0.746087i	−0.927814	−	0.373043i	$-0.878314\pi$
$911$		−	22.5190i		−	0.746087i	0.927814	−	0.373043i	$-0.121686\pi$
$912$		0			0
$913$	31.2255			1.03341
$914$		0			0
$915$	1.86589	−	2.52559i	0.0616843	−	0.0834933i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	14.5291			0.479270			0.239635	−	0.970863i	$-0.422972\pi$
$919$	14.5291			0.479270			0.239635	+	0.970863i	$0.422972\pi$
$920$		0			0
$921$	16.9198	+	3.67242i	0.557526	+	0.121010i
$922$		0			0
$923$		−	4.20148i		−	0.138293i
$924$		0			0
$925$	−7.38204	+	6.54915i	−0.242720	+	0.215335i
$926$		0			0
$927$	−34.4363	−	15.6878i	−1.13104	−	0.515254i
$928$		0			0
$929$	−41.5577			−1.36346			−0.681732	−	0.731602i	$-0.738773\pi$
$929$	−41.5577			−1.36346			−0.681732	+	0.731602i	$0.738773\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−40.3986	−	8.76847i	−1.32259	−	0.287067i
$934$		0			0
$935$	37.2576	+	16.7526i	1.21845	+	0.547868i
$936$		0			0
$937$	45.5010			1.48645			0.743226	−	0.669040i	$-0.233295\pi$
$937$	45.5010			1.48645			0.743226	+	0.669040i	$0.233295\pi$
$938$		0			0
$939$	9.47064	+	2.05559i	0.309063	+	0.0670817i
$940$		0			0
$941$	−20.0487			−0.653569			−0.326785	−	0.945099i	$-0.605965\pi$
$941$	−20.0487			−0.653569			−0.326785	+	0.945099i	$0.605965\pi$
$942$		0			0
$943$	45.2806			1.47454
$944$		0			0
$945$		0			0
$946$		0			0
$947$	29.4980			0.958555			0.479278	−	0.877663i	$-0.340899\pi$
$947$	29.4980			0.958555			0.479278	+	0.877663i	$0.340899\pi$
$948$		0			0
$949$	7.68347			0.249416
$950$		0			0
$951$	−19.8658	−	4.31184i	−0.644192	−	0.139821i
$952$		0			0
$953$	1.26250			0.0408964			0.0204482	−	0.999791i	$-0.493491\pi$
$953$	1.26250			0.0408964			0.0204482	+	0.999791i	$0.493491\pi$
$954$		0			0
$955$	9.52602	−	21.1858i	0.308255	−	0.685555i
$956$		0			0
$957$	68.1270	+	14.7869i	2.20223	+	0.477992i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	10.6801			0.344518
$962$		0			0
$963$	9.49927	+	4.32748i	0.306110	+	0.139451i
$964$		0			0
$965$	−5.42959	+	12.0754i	−0.174785	+	0.388719i
$966$		0			0
$967$		−	26.0942i		−	0.839132i	−0.907725	−	0.419566i	$-0.862182\pi$
$967$		−	26.0942i		−	0.839132i	0.907725	−	0.419566i	$-0.137818\pi$
$968$		0			0
$969$	21.7732	+	4.72585i	0.699456	+	0.151816i
$970$		0			0
$971$	13.9063			0.446276			0.223138	−	0.974787i	$-0.428370\pi$
$971$	13.9063			0.446276			0.223138	+	0.974787i	$0.428370\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−15.0185	−	26.7271i	−0.480977	−	0.855952i
$976$		0			0
$977$	−2.83810			−0.0907990			−0.0453995	−	0.998969i	$-0.514456\pi$
$977$	−2.83810			−0.0907990			−0.0453995	+	0.998969i	$0.514456\pi$
$978$		0			0
$979$			73.9413i			2.36317i
$980$		0			0
$981$	−27.2352	−	12.4072i	−0.869552	−	0.396132i
$982$		0			0
$983$			49.1410i			1.56735i	0.621169	+	0.783677i	$0.286658\pi$
$983$			49.1410i			1.56735i	−0.621169	+	0.783677i	$0.713342\pi$
$984$		0			0
$985$	22.6481	+	10.1835i	0.721628	+	0.324475i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	27.8187i		−	0.884584i
$990$		0			0
$991$	20.6561			0.656163			0.328081	−	0.944649i	$-0.393598\pi$
$991$	20.6561			0.656163			0.328081	+	0.944649i	$0.393598\pi$
$992$		0			0
$993$	4.28080	+	0.929142i	0.135847	+	0.0294854i
$994$		0			0
$995$	−12.1416	+	27.0027i	−0.384914	+	0.856044i
$996$		0			0
$997$	41.6906			1.32035			0.660177	−	0.751110i	$-0.270481\pi$
$997$	41.6906			1.32035			0.660177	+	0.751110i	$0.270481\pi$
$998$		0			0
$999$	−6.13446	+	8.21859i	−0.194086	+	0.260025i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.f.a.1469.32		32
3.2	odd	2	inner	2940.2.f.a.1469.29		32
5.4	even	2	inner	2940.2.f.a.1469.2		32
7.4	even	3		420.2.bn.a.89.7	yes	32
7.5	odd	6		420.2.bn.a.269.12	yes	32
7.6	odd	2	inner	2940.2.f.a.1469.1		32
15.14	odd	2	inner	2940.2.f.a.1469.3		32
21.5	even	6		420.2.bn.a.269.10	yes	32
21.11	odd	6		420.2.bn.a.89.5	✓	32
21.20	even	2	inner	2940.2.f.a.1469.4		32
35.4	even	6		420.2.bn.a.89.10	yes	32
35.12	even	12		2100.2.bi.n.101.4		32
35.18	odd	12		2100.2.bi.n.1601.15		32
35.19	odd	6		420.2.bn.a.269.5	yes	32
35.32	odd	12		2100.2.bi.n.1601.2		32
35.33	even	12		2100.2.bi.n.101.13		32
35.34	odd	2	inner	2940.2.f.a.1469.31		32
105.32	even	12		2100.2.bi.n.1601.4		32
105.47	odd	12		2100.2.bi.n.101.2		32
105.53	even	12		2100.2.bi.n.1601.13		32
105.68	odd	12		2100.2.bi.n.101.15		32
105.74	odd	6		420.2.bn.a.89.12	yes	32
105.89	even	6		420.2.bn.a.269.7	yes	32
105.104	even	2	inner	2940.2.f.a.1469.30		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bn.a.89.5	✓	32	21.11	odd	6
420.2.bn.a.89.7	yes	32	7.4	even	3
420.2.bn.a.89.10	yes	32	35.4	even	6
420.2.bn.a.89.12	yes	32	105.74	odd	6
420.2.bn.a.269.5	yes	32	35.19	odd	6
420.2.bn.a.269.7	yes	32	105.89	even	6
420.2.bn.a.269.10	yes	32	21.5	even	6
420.2.bn.a.269.12	yes	32	7.5	odd	6
2100.2.bi.n.101.2		32	105.47	odd	12
2100.2.bi.n.101.4		32	35.12	even	12
2100.2.bi.n.101.13		32	35.33	even	12
2100.2.bi.n.101.15		32	105.68	odd	12
2100.2.bi.n.1601.2		32	35.32	odd	12
2100.2.bi.n.1601.4		32	105.32	even	12
2100.2.bi.n.1601.13		32	105.53	even	12
2100.2.bi.n.1601.15		32	35.18	odd	12
2940.2.f.a.1469.1		32	7.6	odd	2	inner
2940.2.f.a.1469.2		32	5.4	even	2	inner
2940.2.f.a.1469.3		32	15.14	odd	2	inner
2940.2.f.a.1469.4		32	21.20	even	2	inner
2940.2.f.a.1469.29		32	3.2	odd	2	inner
2940.2.f.a.1469.30		32	105.104	even	2	inner
2940.2.f.a.1469.31		32	35.34	odd	2	inner
2940.2.f.a.1469.32		32	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:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.69264 + 0.367385i) q^{3} +(2.03939 + 0.916997i) q^{5} +(2.73006 + 1.24370i) q^{9} -5.87781i q^{11} -3.54004 q^{13} +(3.11506 + 2.30139i) q^{15} +3.10812i q^{17} -4.13866i q^{19} +7.15173 q^{23} +(3.31823 + 3.74023i) q^{25} +(4.16408 + 3.10812i) q^{27} +6.84761i q^{29} -4.50776i q^{31} +(2.15942 - 9.94902i) q^{33} +1.97369i q^{37} +(-5.99202 - 1.30056i) q^{39} +6.33142 q^{41} -3.88979i q^{43} +(4.42718 + 5.03985i) q^{45} +1.83399i q^{47} +(-1.14188 + 5.26093i) q^{51} +13.8078 q^{53} +(5.38994 - 11.9872i) q^{55} +(1.52048 - 7.00526i) q^{57} +4.65168 q^{59} -0.810766i q^{61} +(-7.21953 - 3.24621i) q^{65} -10.2094i q^{67} +(12.1053 + 2.62744i) q^{69} +1.18684i q^{71} -2.17045 q^{73} +(4.24247 + 7.54993i) q^{75} +10.1046 q^{79} +(5.90641 + 6.79075i) q^{81} +5.31243i q^{83} +(-2.85014 + 6.33868i) q^{85} +(-2.51571 + 11.5905i) q^{87} -12.5797 q^{89} +(1.65609 - 7.63002i) q^{93} +(3.79514 - 8.44035i) q^{95} -4.50686 q^{97} +(7.31025 - 16.0468i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{15} - 12 q^{25} - 48 q^{39} + 20 q^{51} + 56 q^{79} + 40 q^{81} - 4 q^{85} - 28 q^{99}+O(q^{100}) \) 32 * q + 2 * q^15 - 12 * q^25 - 48 * q^39 + 20 * q^51 + 56 * q^79 + 40 * q^81 - 4 * q^85 - 28 * q^99

Introduction

L-functions

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