LMFDB - Embedded newform 740.2.d.a.149.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [740,2,Mod(149,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("740.149"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 740 = 2^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	740.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$5.90892974957$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 26 x^{16} + 283 x^{14} + 1674 x^{12} + 5841 x^{10} + 12196 x^{8} + 14736 x^{6} + 9408 x^{4} + \cdots + 144 $ x^18 + 26x^16 + 283x^14 + 1674x^12 + 5841x^10 + 12196x^8 + 14736x^6 + 9408x^4 + 2592x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.12
Root			$1.78123i$ of defining polynomial
Character	$\chi$	$=$	740.149
Dual form			740.2.d.a.149.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.06683i q^{3} +(1.49239 + 1.66516i) q^{5} -0.712098i q^{7} +1.86188 q^{9} +2.90228 q^{11} -6.88922i q^{13} +(-1.77644 + 1.59213i) q^{15} -0.433830i q^{17} +7.42477 q^{19} +0.759686 q^{21} +5.73141i q^{23} +(-0.545534 + 4.97015i) q^{25} +5.18679i q^{27} -10.5952 q^{29} +0.00620693 q^{31} +3.09624i q^{33} +(1.18576 - 1.06273i) q^{35} +1.00000i q^{37} +7.34962 q^{39} -3.79738 q^{41} -4.77044i q^{43} +(2.77865 + 3.10033i) q^{45} +3.37418i q^{47} +6.49292 q^{49} +0.462823 q^{51} +0.266177i q^{53} +(4.33134 + 4.83277i) q^{55} +7.92096i q^{57} +4.17923 q^{59} -6.75494 q^{61} -1.32584i q^{63} +(11.4717 - 10.2814i) q^{65} +11.4728i q^{67} -6.11443 q^{69} -12.9624 q^{71} -11.1454i q^{73} +(-5.30230 - 0.581991i) q^{75} -2.06671i q^{77} +4.76967 q^{79} +0.0522097 q^{81} +1.61080i q^{83} +(0.722398 - 0.647445i) q^{85} -11.3032i q^{87} -7.65696 q^{89} -4.90579 q^{91} +0.00662174i q^{93} +(11.0807 + 12.3635i) q^{95} -1.08422i q^{97} +5.40368 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{5} - 18 q^{9} + 6 q^{15} + 4 q^{19} - 16 q^{21} - 2 q^{25} - 4 q^{29} + 8 q^{31} - 2 q^{35} + 8 q^{39} - 4 q^{41} + 8 q^{45} + 6 q^{49} - 40 q^{51} - 6 q^{55} + 8 q^{59} - 12 q^{65} + 28 q^{69}+ \cdots + 16 q^{99}+O(q^{100}) $ 18 * q + 2 * q^5 - 18 * q^9 + 6 * q^15 + 4 * q^19 - 16 * q^21 - 2 * q^25 - 4 * q^29 + 8 * q^31 - 2 * q^35 + 8 * q^39 - 4 * q^41 + 8 * q^45 + 6 * q^49 - 40 * q^51 - 6 * q^55 + 8 * q^59 - 12 * q^65 + 28 * q^69 - 24 * q^71 - 16 * q^75 - 24 * q^79 + 34 * q^81 + 36 * q^91 + 24 * q^95 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$.

$n$	$261$	$297$	$371$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			1.06683i			0.615934i	0.951397	+	0.307967i	$0.0996486\pi$
$3$			1.06683i			0.615934i	−0.951397	+	0.307967i	$0.900351\pi$
$4$		0			0
$5$	1.49239	+	1.66516i	0.667418	+	0.744683i
$5$	1.49239	+	1.66516i	0.667418	+	0.744683i
$6$		0			0
$7$		−	0.712098i		−	0.269148i	−0.990904	−	0.134574i	$-0.957033\pi$
$7$		−	0.712098i		−	0.269148i	0.990904	−	0.134574i	$-0.0429665\pi$
$8$		0			0
$9$	1.86188			0.620625
$10$		0			0
$11$	2.90228			0.875070			0.437535	−	0.899201i	$-0.355852\pi$
$11$	2.90228			0.875070			0.437535	+	0.899201i	$0.355852\pi$
$12$		0			0
$13$		−	6.88922i		−	1.91072i	−0.295437	−	0.955362i	$-0.595465\pi$
$13$		−	6.88922i		−	1.91072i	0.295437	−	0.955362i	$-0.404535\pi$
$14$		0			0
$15$	−1.77644	+	1.59213i	−0.458676	+	0.411085i
$16$		0			0
$17$		−	0.433830i		−	0.105219i	−0.998615	−	0.0526097i	$-0.983246\pi$
$17$		−	0.433830i		−	0.105219i	0.998615	−	0.0526097i	$-0.0167539\pi$
$18$		0			0
$19$	7.42477			1.70336			0.851680	−	0.524062i	$-0.175584\pi$
$19$	7.42477			1.70336			0.851680	+	0.524062i	$0.175584\pi$
$20$		0			0
$21$	0.759686			0.165777
$22$		0			0
$23$			5.73141i			1.19508i	0.801839	+	0.597540i	$0.203855\pi$
$23$			5.73141i			1.19508i	−0.801839	+	0.597540i	$0.796145\pi$
$24$		0			0
$25$	−0.545534	+	4.97015i	−0.109107	+	0.994030i
$26$		0			0
$27$			5.18679i			0.998198i
$28$		0			0
$29$	−10.5952			−1.96747			−0.983736	−	0.179623i	$-0.942512\pi$
$29$	−10.5952			−1.96747			−0.983736	+	0.179623i	$0.942512\pi$
$30$		0			0
$31$	0.00620693			0.00111480			0.000557399	−	1.00000i	$-0.499823\pi$
$31$	0.00620693			0.00111480			0.000557399		1.00000i	$0.499823\pi$
$32$		0			0
$33$			3.09624i			0.538986i
$34$		0			0
$35$	1.18576	−	1.06273i	0.200430	−	0.179634i
$36$		0			0
$37$			1.00000i			0.164399i
$37$			1.00000i			0.164399i
$38$		0			0
$39$	7.34962			1.17688
$40$		0			0
$41$	−3.79738			−0.593051			−0.296526	−	0.955025i	$-0.595828\pi$
$41$	−3.79738			−0.593051			−0.296526	+	0.955025i	$0.595828\pi$
$42$		0			0
$43$		−	4.77044i		−	0.727485i	−0.931500	−	0.363742i	$-0.881499\pi$
$43$		−	4.77044i		−	0.727485i	0.931500	−	0.363742i	$-0.118501\pi$
$44$		0			0
$45$	2.77865	+	3.10033i	0.414216	+	0.462169i
$46$		0			0
$47$			3.37418i			0.492174i	0.969248	+	0.246087i	$0.0791449\pi$
$47$			3.37418i			0.492174i	−0.969248	+	0.246087i	$0.920855\pi$
$48$		0			0
$49$	6.49292			0.927560
$50$		0			0
$51$	0.462823			0.0648082
$52$		0			0
$53$			0.266177i			0.0365622i	0.999833	+	0.0182811i	$0.00581938\pi$
$53$			0.266177i			0.0365622i	−0.999833	+	0.0182811i	$0.994181\pi$
$54$		0			0
$55$	4.33134	+	4.83277i	0.584038	+	0.651650i
$56$		0			0
$57$			7.92096i			1.04916i
$58$		0			0
$59$	4.17923			0.544090			0.272045	−	0.962285i	$-0.412300\pi$
$59$	4.17923			0.544090			0.272045	+	0.962285i	$0.412300\pi$
$60$		0			0
$61$	−6.75494			−0.864882			−0.432441	−	0.901662i	$-0.642348\pi$
$61$	−6.75494			−0.864882			−0.432441	+	0.901662i	$0.642348\pi$
$62$		0			0
$63$		−	1.32584i		−	0.167040i
$64$		0			0
$65$	11.4717	−	10.2814i	1.42288	−	1.27525i
$66$		0			0
$67$			11.4728i			1.40162i	0.713347	+	0.700811i	$0.247179\pi$
$67$			11.4728i			1.40162i	−0.713347	+	0.700811i	$0.752821\pi$
$68$		0			0
$69$	−6.11443			−0.736091
$70$		0			0
$71$	−12.9624			−1.53836			−0.769178	−	0.639034i	$-0.779334\pi$
$71$	−12.9624			−1.53836			−0.769178	+	0.639034i	$0.779334\pi$
$72$		0			0
$73$		−	11.1454i		−	1.30447i	−0.758019	−	0.652233i	$-0.773832\pi$
$73$		−	11.1454i		−	1.30447i	0.758019	−	0.652233i	$-0.226168\pi$
$74$		0			0
$75$	−5.30230	−	0.581991i	−0.612257	−	0.0672025i
$76$		0			0
$77$		−	2.06671i		−	0.235523i
$78$		0			0
$79$	4.76967			0.536629			0.268315	−	0.963331i	$-0.413533\pi$
$79$	4.76967			0.536629			0.268315	+	0.963331i	$0.413533\pi$
$80$		0			0
$81$	0.0522097			0.00580108
$82$		0			0
$83$			1.61080i			0.176808i	0.996085	+	0.0884042i	$0.0281767\pi$
$83$			1.61080i			0.176808i	−0.996085	+	0.0884042i	$0.971823\pi$
$84$		0			0
$85$	0.722398	−	0.647445i	0.0783551	−	0.0702253i
$86$		0			0
$87$		−	11.3032i		−	1.21183i
$88$		0			0
$89$	−7.65696			−0.811636			−0.405818	−	0.913954i	$-0.633013\pi$
$89$	−7.65696			−0.811636			−0.405818	+	0.913954i	$0.633013\pi$
$90$		0			0
$91$	−4.90579			−0.514267
$92$		0			0
$93$			0.00662174i			0.000686642i
$94$		0			0
$95$	11.0807	+	12.3635i	1.13685	+	1.26846i
$96$		0			0
$97$		−	1.08422i		−	0.110086i	−0.998484	−	0.0550432i	$-0.982470\pi$
$97$		−	1.08422i		−	0.110086i	0.998484	−	0.0550432i	$-0.0175296\pi$
$98$		0			0
$99$	5.40368			0.543091
$100$		0			0
$101$	15.2062			1.51307			0.756536	−	0.653952i	$-0.226890\pi$
$101$	15.2062			1.51307			0.756536	+	0.653952i	$0.226890\pi$
$102$		0			0
$103$		−	8.78922i		−	0.866028i	−0.901387	−	0.433014i	$-0.857450\pi$
$103$		−	8.78922i		−	0.866028i	0.901387	−	0.433014i	$-0.142550\pi$
$104$		0			0
$105$	1.13375	+	1.26500i	0.110643	+	0.123452i
$106$		0			0
$107$			11.0469i			1.06794i	0.845503	+	0.533971i	$0.179301\pi$
$107$			11.0469i			1.06794i	−0.845503	+	0.533971i	$0.820699\pi$
$108$		0			0
$109$	−3.74658			−0.358858			−0.179429	−	0.983771i	$-0.557425\pi$
$109$	−3.74658			−0.358858			−0.179429	+	0.983771i	$0.557425\pi$
$110$		0			0
$111$	−1.06683			−0.101259
$112$		0			0
$113$		−	9.22516i		−	0.867830i	−0.900954	−	0.433915i	$-0.857132\pi$
$113$		−	9.22516i		−	0.867830i	0.900954	−	0.433915i	$-0.142868\pi$
$114$		0			0
$115$	−9.54372	+	8.55350i	−0.889957	+	0.797618i
$116$		0			0
$117$		−	12.8269i		−	1.18584i
$118$		0			0
$119$	−0.308930			−0.0283195
$120$		0			0
$121$	−2.57677			−0.234252
$122$		0			0
$123$		−	4.05115i		−	0.365280i
$124$		0			0
$125$	−9.09026	+	6.50901i	−0.813057	+	0.582183i
$126$		0			0
$127$		−	13.1782i		−	1.16938i	−0.811257	−	0.584690i	$-0.801216\pi$
$127$		−	13.1782i		−	1.16938i	0.811257	−	0.584690i	$-0.198784\pi$
$128$		0			0
$129$	5.08924			0.448083
$130$		0			0
$131$	9.86252			0.861693			0.430846	−	0.902425i	$-0.358215\pi$
$131$	9.86252			0.861693			0.430846	+	0.902425i	$0.358215\pi$
$132$		0			0
$133$		−	5.28716i		−	0.458455i
$134$		0			0
$135$	−8.63685	+	7.74072i	−0.743342	+	0.666215i
$136$		0			0
$137$		−	17.5546i		−	1.49979i	−0.661554	−	0.749897i	$-0.730103\pi$
$137$		−	17.5546i		−	1.49979i	0.661554	−	0.749897i	$-0.269897\pi$
$138$		0			0
$139$	−20.5440			−1.74252			−0.871261	−	0.490820i	$-0.836697\pi$
$139$	−20.5440			−1.74252			−0.871261	+	0.490820i	$0.836697\pi$
$140$		0			0
$141$	−3.59967			−0.303147
$142$		0			0
$143$		−	19.9944i		−	1.67202i
$144$		0			0
$145$	−15.8121	−	17.6427i	−1.31313	−	1.46514i
$146$		0			0
$147$			6.92683i			0.571315i
$148$		0			0
$149$	5.09279			0.417218			0.208609	−	0.977999i	$-0.433106\pi$
$149$	5.09279			0.417218			0.208609	+	0.977999i	$0.433106\pi$
$150$		0			0
$151$	−4.79117			−0.389900			−0.194950	−	0.980813i	$-0.562454\pi$
$151$	−4.79117			−0.389900			−0.194950	+	0.980813i	$0.562454\pi$
$152$		0			0
$153$		−	0.807738i		−	0.0653018i
$154$		0			0
$155$	0.00926318	+	0.0103356i	0.000744036	+	0.000830172i
$156$		0			0
$157$			16.2565i			1.29741i	0.761040	+	0.648705i	$0.224689\pi$
$157$			16.2565i			1.29741i	−0.761040	+	0.648705i	$0.775311\pi$
$158$		0			0
$159$	−0.283965			−0.0225199
$160$		0			0
$161$	4.08132			0.321653
$162$		0			0
$163$		−	16.5283i		−	1.29460i	−0.762237	−	0.647298i	$-0.775899\pi$
$163$		−	16.5283i		−	1.29460i	0.762237	−	0.647298i	$-0.224101\pi$
$164$		0			0
$165$	−5.15574	+	4.62080i	−0.401374	+	0.359729i
$166$		0			0
$167$			12.1706i			0.941787i	0.882190	+	0.470894i	$0.156068\pi$
$167$			12.1706i			0.941787i	−0.882190	+	0.470894i	$0.843932\pi$
$168$		0			0
$169$	−34.4613			−2.65087
$170$		0			0
$171$	13.8240			1.05715
$172$		0			0
$173$		−	2.52257i		−	0.191787i	−0.995392	−	0.0958936i	$-0.969429\pi$
$173$		−	2.52257i		−	0.191787i	0.995392	−	0.0958936i	$-0.0305709\pi$
$174$		0			0
$175$	3.53923	+	0.388473i	0.267541	+	0.0293658i
$176$		0			0
$177$			4.45853i			0.335123i
$178$		0			0
$179$	−23.0731			−1.72456			−0.862282	−	0.506428i	$-0.830966\pi$
$179$	−23.0731			−1.72456			−0.862282	+	0.506428i	$0.830966\pi$
$180$		0			0
$181$	−6.88080			−0.511446			−0.255723	−	0.966750i	$-0.582314\pi$
$181$	−6.88080			−0.511446			−0.255723	+	0.966750i	$0.582314\pi$
$182$		0			0
$183$		−	7.20637i		−	0.532710i
$184$		0			0
$185$	−1.66516	+	1.49239i	−0.122425	+	0.109723i
$186$		0			0
$187$		−	1.25910i		−	0.0920743i
$188$		0			0
$189$	3.69350			0.268663
$190$		0			0
$191$	−3.23320			−0.233946			−0.116973	−	0.993135i	$-0.537319\pi$
$191$	−3.23320			−0.233946			−0.116973	+	0.993135i	$0.537319\pi$
$192$		0			0
$193$		−	14.3709i		−	1.03444i	−0.855853	−	0.517219i	$-0.826967\pi$
$193$		−	14.3709i		−	1.03444i	0.855853	−	0.517219i	$-0.173033\pi$
$194$		0			0
$195$	10.9685	+	12.2383i	0.785471	+	0.876403i
$196$		0			0
$197$			19.1861i			1.36696i	0.729971	+	0.683478i	$0.239533\pi$
$197$			19.1861i			1.36696i	−0.729971	+	0.683478i	$0.760467\pi$
$198$		0			0
$199$	21.5450			1.52728			0.763641	−	0.645641i	$-0.223410\pi$
$199$	21.5450			1.52728			0.763641	+	0.645641i	$0.223410\pi$
$200$		0			0
$201$	−12.2395			−0.863307
$202$		0			0
$203$			7.54479i			0.529540i
$204$		0			0
$205$	−5.66718	−	6.32325i	−0.395813	−	0.441635i
$206$		0			0
$207$			10.6712i			0.741697i
$208$		0			0
$209$	21.5488			1.49056
$210$		0			0
$211$	12.6715			0.872344			0.436172	−	0.899863i	$-0.356334\pi$
$211$	12.6715			0.872344			0.436172	+	0.899863i	$0.356334\pi$
$212$		0			0
$213$		−	13.8287i		−	0.947526i
$214$		0			0
$215$	7.94355	−	7.11936i	0.541746	−	0.485536i
$216$		0			0
$217$		−	0.00441994i		−	0.000300045i
$218$		0			0
$219$	11.8902			0.803465
$220$		0			0
$221$	−2.98875			−0.201045
$222$		0			0
$223$		−	17.8258i		−	1.19370i	−0.802352	−	0.596852i	$-0.796418\pi$
$223$		−	17.8258i		−	1.19370i	0.802352	−	0.596852i	$-0.203582\pi$
$224$		0			0
$225$	−1.01572	+	9.25380i	−0.0677144	+	0.616920i
$226$		0			0
$227$		−	16.7949i		−	1.11472i	−0.830272	−	0.557359i	$-0.811815\pi$
$227$		−	16.7949i		−	1.11472i	0.830272	−	0.557359i	$-0.188185\pi$
$228$		0			0
$229$	−6.27187			−0.414457			−0.207228	−	0.978293i	$-0.566444\pi$
$229$	−6.27187			−0.414457			−0.207228	+	0.978293i	$0.566444\pi$
$230$		0			0
$231$	2.20482			0.145067
$232$		0			0
$233$		−	16.1095i		−	1.05537i	−0.849441	−	0.527683i	$-0.823061\pi$
$233$		−	16.1095i		−	1.05537i	0.849441	−	0.527683i	$-0.176939\pi$
$234$		0			0
$235$	−5.61855	+	5.03559i	−0.366514	+	0.328486i
$236$		0			0
$237$			5.08842i			0.330528i
$238$		0			0
$239$	−7.74007			−0.500663			−0.250332	−	0.968160i	$-0.580540\pi$
$239$	−7.74007			−0.500663			−0.250332	+	0.968160i	$0.580540\pi$
$240$		0			0
$241$	−4.58711			−0.295482			−0.147741	−	0.989026i	$-0.547200\pi$
$241$	−4.58711			−0.295482			−0.147741	+	0.989026i	$0.547200\pi$
$242$		0			0
$243$			15.6161i			1.00177i
$244$		0			0
$245$	9.68998	+	10.8118i	0.619070	+	0.690738i
$246$		0			0
$247$		−	51.1509i		−	3.25465i
$248$		0			0
$249$	−1.71845			−0.108902
$250$		0			0
$251$	15.8129			0.998103			0.499051	−	0.866572i	$-0.333682\pi$
$251$	15.8129			0.998103			0.499051	+	0.866572i	$0.333682\pi$
$252$		0			0
$253$			16.6341i			1.04578i
$254$		0			0
$255$	0.690713	+	0.770675i	0.0432541	+	0.0482616i
$256$		0			0
$257$			27.7461i			1.73075i	0.501122	+	0.865377i	$0.332921\pi$
$257$			27.7461i			1.73075i	−0.501122	+	0.865377i	$0.667079\pi$
$258$		0			0
$259$	0.712098			0.0442476
$260$		0			0
$261$	−19.7269			−1.22106
$262$		0			0
$263$			16.8648i			1.03993i	0.854187	+	0.519965i	$0.174055\pi$
$263$			16.8648i			1.03993i	−0.854187	+	0.519965i	$0.825945\pi$
$264$		0			0
$265$	−0.443227	+	0.397240i	−0.0272272	+	0.0244023i
$266$		0			0
$267$		−	8.16867i		−	0.499914i
$268$		0			0
$269$	−23.3683			−1.42479			−0.712396	−	0.701778i	$-0.752390\pi$
$269$	−23.3683			−1.42479			−0.712396	+	0.701778i	$0.752390\pi$
$270$		0			0
$271$	−20.6086			−1.25188			−0.625941	−	0.779871i	$-0.715285\pi$
$271$	−20.6086			−1.25188			−0.625941	+	0.779871i	$0.715285\pi$
$272$		0			0
$273$		−	5.23364i		−	0.316755i
$274$		0			0
$275$	−1.58329	+	14.4248i	−0.0954761	+	0.869846i
$276$		0			0
$277$		−	8.14619i		−	0.489457i	−0.969592	−	0.244729i	$-0.921301\pi$
$277$		−	8.14619i		−	0.489457i	0.969592	−	0.244729i	$-0.0786988\pi$
$278$		0			0
$279$	0.0115565			0.000691872
$280$		0			0
$281$	14.5230			0.866367			0.433183	−	0.901306i	$-0.357390\pi$
$281$	14.5230			0.866367			0.433183	+	0.901306i	$0.357390\pi$
$282$		0			0
$283$			31.8286i			1.89202i	0.324142	+	0.946008i	$0.394924\pi$
$283$			31.8286i			1.89202i	−0.324142	+	0.946008i	$0.605076\pi$
$284$		0			0
$285$	−13.1897	+	11.8212i	−0.781290	+	0.700226i
$286$		0			0
$287$			2.70410i			0.159618i
$288$		0			0
$289$	16.8118			0.988929
$290$		0			0
$291$	1.15668			0.0678059
$292$		0			0
$293$		−	30.8815i		−	1.80411i	−0.431617	−	0.902057i	$-0.642057\pi$
$293$		−	30.8815i		−	1.80411i	0.431617	−	0.902057i	$-0.357943\pi$
$294$		0			0
$295$	6.23705	+	6.95910i	0.363135	+	0.405175i
$296$		0			0
$297$			15.0535i			0.873494i
$298$		0			0
$299$	39.4849			2.28347
$300$		0			0
$301$	−3.39702			−0.195801
$302$		0			0
$303$			16.2224i			0.931953i
$304$		0			0
$305$	−10.0810	−	11.2481i	−0.577238	−	0.644063i
$306$		0			0
$307$			5.84499i			0.333591i	0.985991	+	0.166796i	$0.0533420\pi$
$307$			5.84499i			0.333591i	−0.985991	+	0.166796i	$0.946658\pi$
$308$		0			0
$309$	9.37660			0.533416
$310$		0			0
$311$	−5.67788			−0.321963			−0.160981	−	0.986957i	$-0.551466\pi$
$311$	−5.67788			−0.321963			−0.160981	+	0.986957i	$0.551466\pi$
$312$		0			0
$313$			23.2016i			1.31143i	0.755007	+	0.655717i	$0.227633\pi$
$313$			23.2016i			1.31143i	−0.755007	+	0.655717i	$0.772367\pi$
$314$		0			0
$315$	2.20774	−	1.97867i	0.124392	−	0.111485i
$316$		0			0
$317$		−	10.6935i		−	0.600605i	−0.953844	−	0.300303i	$-0.902912\pi$
$317$		−	10.6935i		−	0.600605i	0.953844	−	0.300303i	$-0.0970877\pi$
$318$		0			0
$319$	−30.7501			−1.72168
$320$		0			0
$321$	−11.7851			−0.657781
$322$		0			0
$323$		−	3.22109i		−	0.179226i
$324$		0			0
$325$	34.2404	+	3.75830i	1.89932	+	0.208473i
$326$		0			0
$327$		−	3.99696i		−	0.221033i
$328$		0			0
$329$	2.40274			0.132467
$330$		0			0
$331$	4.02026			0.220973			0.110487	−	0.993878i	$-0.464759\pi$
$331$	4.02026			0.220973			0.110487	+	0.993878i	$0.464759\pi$
$332$		0			0
$333$			1.86188i			0.102030i
$334$		0			0
$335$	−19.1040	+	17.1219i	−1.04377	+	0.935468i
$336$		0			0
$337$		−	8.37957i		−	0.456464i	−0.973607	−	0.228232i	$-0.926705\pi$
$337$		−	8.37957i		−	0.456464i	0.973607	−	0.228232i	$-0.0732945\pi$
$338$		0			0
$339$	9.84167			0.534526
$340$		0			0
$341$	0.0180143			0.000975527
$342$		0			0
$343$		−	9.60827i		−	0.518798i
$344$		0			0
$345$	−9.12512	−	10.1815i	−0.491280	−	0.548155i
$346$		0			0
$347$		−	6.89482i		−	0.370134i	−0.982726	−	0.185067i	$-0.940750\pi$
$347$		−	6.89482i		−	0.370134i	0.982726	−	0.185067i	$-0.0592501\pi$
$348$		0			0
$349$	−0.448004			−0.0239811			−0.0119906	−	0.999928i	$-0.503817\pi$
$349$	−0.448004			−0.0239811			−0.0119906	+	0.999928i	$0.503817\pi$
$350$		0			0
$351$	35.7329			1.90728
$352$		0			0
$353$		−	19.2281i		−	1.02341i	−0.859162	−	0.511704i	$-0.829014\pi$
$353$		−	19.2281i		−	1.02341i	0.859162	−	0.511704i	$-0.170986\pi$
$354$		0			0
$355$	−19.3450	−	21.5845i	−1.02673	−	1.14559i
$356$		0			0
$357$		−	0.329575i		−	0.0174430i
$358$		0			0
$359$	22.3320			1.17864			0.589319	−	0.807900i	$-0.299396\pi$
$359$	22.3320			1.17864			0.589319	+	0.807900i	$0.299396\pi$
$360$		0			0
$361$	36.1272			1.90143
$362$		0			0
$363$		−	2.74897i		−	0.144284i
$364$		0			0
$365$	18.5588	−	16.6332i	0.971414	−	0.870624i
$366$		0			0
$367$			22.1230i			1.15481i	0.816458	+	0.577405i	$0.195935\pi$
$367$			22.1230i			1.15481i	−0.816458	+	0.577405i	$0.804065\pi$
$368$		0			0
$369$	−7.07025			−0.368062
$370$		0			0
$371$	0.189544			0.00984062
$372$		0			0
$373$		−	15.1576i		−	0.784830i	−0.919788	−	0.392415i	$-0.871640\pi$
$373$		−	15.1576i		−	0.784830i	0.919788	−	0.392415i	$-0.128360\pi$
$374$		0			0
$375$	−6.94400	−	9.69775i	−0.358587	−	0.500790i
$376$		0			0
$377$			72.9923i			3.75930i
$378$		0			0
$379$	−27.2193			−1.39816			−0.699081	−	0.715043i	$-0.746407\pi$
$379$	−27.2193			−1.39816			−0.699081	+	0.715043i	$0.746407\pi$
$380$		0			0
$381$	14.0589			0.720261
$382$		0			0
$383$			8.95280i			0.457467i	0.973489	+	0.228733i	$0.0734584\pi$
$383$			8.95280i			0.457467i	−0.973489	+	0.228733i	$0.926542\pi$
$384$		0			0
$385$	3.44140	−	3.08434i	0.175390	−	0.157192i
$386$		0			0
$387$		−	8.88196i		−	0.451495i
$388$		0			0
$389$	4.53175			0.229769			0.114885	−	0.993379i	$-0.463350\pi$
$389$	4.53175			0.229769			0.114885	+	0.993379i	$0.463350\pi$
$390$		0			0
$391$	2.48646			0.125746
$392$		0			0
$393$			10.5216i			0.530746i
$394$		0			0
$395$	7.11821	+	7.94227i	0.358156	+	0.399619i
$396$		0			0
$397$			16.6964i			0.837967i	0.907994	+	0.418984i	$0.137614\pi$
$397$			16.6964i			0.837967i	−0.907994	+	0.418984i	$0.862386\pi$
$398$		0			0
$399$	5.64050			0.282378
$400$		0			0
$401$	22.0622			1.10174			0.550868	−	0.834592i	$-0.314297\pi$
$401$	22.0622			1.10174			0.550868	+	0.834592i	$0.314297\pi$
$402$		0			0
$403$		−	0.0427609i		−	0.00213007i
$404$		0			0
$405$	0.0779174	+	0.0869377i	0.00387175	+	0.00431997i
$406$		0			0
$407$			2.90228i			0.143861i
$408$		0			0
$409$	−9.66990			−0.478146			−0.239073	−	0.971002i	$-0.576843\pi$
$409$	−9.66990			−0.478146			−0.239073	+	0.971002i	$0.576843\pi$
$410$		0			0
$411$	18.7278			0.923774
$412$		0			0
$413$		−	2.97602i		−	0.146440i
$414$		0			0
$415$	−2.68225	+	2.40395i	−0.131666	+	0.118005i
$416$		0			0
$417$		−	21.9170i		−	1.07328i
$418$		0			0
$419$	29.4554			1.43899			0.719494	−	0.694498i	$-0.244374\pi$
$419$	29.4554			1.43899			0.719494	+	0.694498i	$0.244374\pi$
$420$		0			0
$421$	19.6889			0.959578			0.479789	−	0.877384i	$-0.340713\pi$
$421$	19.6889			0.959578			0.479789	+	0.877384i	$0.340713\pi$
$422$		0			0
$423$			6.28230i			0.305456i
$424$		0			0
$425$	2.15620	+	0.236669i	0.104591	+	0.0114801i
$426$		0			0
$427$			4.81018i			0.232781i
$428$		0			0
$429$	21.3306			1.02985
$430$		0			0
$431$	−6.22182			−0.299694			−0.149847	−	0.988709i	$-0.547878\pi$
$431$	−6.22182			−0.299694			−0.149847	+	0.988709i	$0.547878\pi$
$432$		0			0
$433$			25.2855i			1.21515i	0.794264	+	0.607573i	$0.207857\pi$
$433$			25.2855i			1.21515i	−0.794264	+	0.607573i	$0.792143\pi$
$434$		0			0
$435$	18.8217	−	16.8688i	0.902431	−	0.808799i
$436$		0			0
$437$			42.5544i			2.03565i
$438$		0			0
$439$	−30.8836			−1.47399			−0.736996	−	0.675897i	$-0.763756\pi$
$439$	−30.8836			−1.47399			−0.736996	+	0.675897i	$0.763756\pi$
$440$		0			0
$441$	12.0890			0.575667
$442$		0			0
$443$			32.2620i			1.53281i	0.642355	+	0.766407i	$0.277958\pi$
$443$			32.2620i			1.53281i	−0.642355	+	0.766407i	$0.722042\pi$
$444$		0			0
$445$	−11.4272	−	12.7501i	−0.541701	−	0.604412i
$446$		0			0
$447$			5.43314i			0.256979i
$448$		0			0
$449$	−10.5328			−0.497074			−0.248537	−	0.968622i	$-0.579950\pi$
$449$	−10.5328			−0.497074			−0.248537	+	0.968622i	$0.579950\pi$
$450$		0			0
$451$	−11.0211			−0.518961
$452$		0			0
$453$		−	5.11136i		−	0.240153i
$454$		0			0
$455$	−7.32137	−	8.16895i	−0.343231	−	0.382966i
$456$		0			0
$457$		−	22.4029i		−	1.04796i	−0.851730	−	0.523982i	$-0.824446\pi$
$457$		−	22.4029i		−	1.04796i	0.851730	−	0.523982i	$-0.175554\pi$
$458$		0			0
$459$	2.25019			0.105030
$460$		0			0
$461$	−7.47407			−0.348102			−0.174051	−	0.984737i	$-0.555686\pi$
$461$	−7.47407			−0.348102			−0.174051	+	0.984737i	$0.555686\pi$
$462$		0			0
$463$		−	24.7307i		−	1.14933i	−0.818387	−	0.574667i	$-0.805131\pi$
$463$		−	24.7307i		−	1.14933i	0.818387	−	0.574667i	$-0.194869\pi$
$464$		0			0
$465$	−0.0110263	+	0.00988223i	−0.000511331	+	0.000458277i
$466$		0			0
$467$		−	18.6718i		−	0.864028i	−0.901867	−	0.432014i	$-0.857803\pi$
$467$		−	18.6718i		−	0.864028i	0.901867	−	0.432014i	$-0.142197\pi$
$468$		0			0
$469$	8.16974			0.377243
$470$		0			0
$471$	−17.3429			−0.799119
$472$		0			0
$473$		−	13.8451i		−	0.636600i
$474$		0			0
$475$	−4.05046	+	36.9022i	−0.185848	+	1.69319i
$476$		0			0
$477$			0.495588i			0.0226914i
$478$		0			0
$479$	−18.6251			−0.851002			−0.425501	−	0.904958i	$-0.639902\pi$
$479$	−18.6251			−0.851002			−0.425501	+	0.904958i	$0.639902\pi$
$480$		0			0
$481$	6.88922			0.314121
$482$		0			0
$483$			4.35407i			0.198117i
$484$		0			0
$485$	1.80541	−	1.61809i	0.0819795	−	0.0734736i
$486$		0			0
$487$		−	20.0450i		−	0.908328i	−0.890918	−	0.454164i	$-0.849938\pi$
$487$		−	20.0450i		−	0.908328i	0.890918	−	0.454164i	$-0.150062\pi$
$488$		0			0
$489$	17.6329			0.797385
$490$		0			0
$491$	1.75002			0.0789771			0.0394885	−	0.999220i	$-0.487427\pi$
$491$	1.75002			0.0789771			0.0394885	+	0.999220i	$0.487427\pi$
$492$		0			0
$493$			4.59650i			0.207016i
$494$		0			0
$495$	8.06441	+	8.99801i	0.362468	+	0.404431i
$496$		0			0
$497$			9.23051i			0.414045i
$498$		0			0
$499$	9.99742			0.447546			0.223773	−	0.974641i	$-0.428163\pi$
$499$	9.99742			0.447546			0.223773	+	0.974641i	$0.428163\pi$
$500$		0			0
$501$	−12.9839			−0.580079
$502$		0			0
$503$		−	31.0214i		−	1.38318i	−0.722292	−	0.691588i	$-0.756911\pi$
$503$		−	31.0214i		−	1.38318i	0.722292	−	0.691588i	$-0.243089\pi$
$504$		0			0
$505$	22.6936	+	25.3208i	1.00985	+	1.12676i
$506$		0			0
$507$		−	36.7643i		−	1.63276i
$508$		0			0
$509$	13.6334			0.604291			0.302145	−	0.953262i	$-0.402297\pi$
$509$	13.6334			0.604291			0.302145	+	0.953262i	$0.402297\pi$
$510$		0			0
$511$	−7.93658			−0.351094
$512$		0			0
$513$			38.5107i			1.70029i
$514$		0			0
$515$	14.6355	−	13.1170i	0.644917	−	0.578003i
$516$		0			0
$517$			9.79280i			0.430687i
$518$		0			0
$519$	2.69115			0.118128
$520$		0			0
$521$	11.4516			0.501702			0.250851	−	0.968026i	$-0.419290\pi$
$521$	11.4516			0.501702			0.250851	+	0.968026i	$0.419290\pi$
$522$		0			0
$523$			7.38651i			0.322990i	0.986874	+	0.161495i	$0.0516315\pi$
$523$			7.38651i			0.322990i	−0.986874	+	0.161495i	$0.948369\pi$
$524$		0			0
$525$	−0.414434	+	3.77576i	−0.0180874	+	0.164787i
$526$		0			0
$527$		−	0.00269276i		−	0.000117298i
$528$		0			0
$529$	−9.84901			−0.428218
$530$		0			0
$531$	7.78121			0.337676
$532$		0			0
$533$			26.1610i			1.13316i
$534$		0			0
$535$	−18.3948	+	16.4863i	−0.795278	+	0.712763i
$536$		0			0
$537$		−	24.6151i		−	1.06222i
$538$		0			0
$539$	18.8443			0.811680
$540$		0			0
$541$	−20.0037			−0.860028			−0.430014	−	0.902822i	$-0.641491\pi$
$541$	−20.0037			−0.860028			−0.430014	+	0.902822i	$0.641491\pi$
$542$		0			0
$543$		−	7.34064i		−	0.315017i
$544$		0			0
$545$	−5.59137	−	6.23867i	−0.239508	−	0.267235i
$546$		0			0
$547$		−	6.90350i		−	0.295172i	−0.989049	−	0.147586i	$-0.952850\pi$
$547$		−	6.90350i		−	0.295172i	0.989049	−	0.147586i	$-0.0471503\pi$
$548$		0			0
$549$	−12.5769			−0.536767
$550$		0			0
$551$	−78.6666			−3.35131
$552$		0			0
$553$		−	3.39647i		−	0.144433i
$554$		0			0
$555$	−1.59213	−	1.77644i	−0.0675820	−	0.0754058i
$556$		0			0
$557$		−	10.6028i		−	0.449256i	−0.974445	−	0.224628i	$-0.927883\pi$
$557$		−	10.6028i		−	0.449256i	0.974445	−	0.224628i	$-0.0721166\pi$
$558$		0			0
$559$	−32.8646			−1.39002
$560$		0			0
$561$	1.34324			0.0567117
$562$		0			0
$563$			35.8274i			1.50995i	0.655756	+	0.754973i	$0.272350\pi$
$563$			35.8274i			1.50995i	−0.655756	+	0.754973i	$0.727650\pi$
$564$		0			0
$565$	15.3614	−	13.7675i	0.646258	−	0.579205i
$566$		0			0
$567$		−	0.0371784i		−	0.00156135i
$568$		0			0
$569$	−14.1461			−0.593034			−0.296517	−	0.955028i	$-0.595825\pi$
$569$	−14.1461			−0.593034			−0.296517	+	0.955028i	$0.595825\pi$
$570$		0			0
$571$	−28.4581			−1.19094			−0.595468	−	0.803379i	$-0.703033\pi$
$571$	−28.4581			−1.19094			−0.595468	+	0.803379i	$0.703033\pi$
$572$		0			0
$573$		−	3.44927i		−	0.144095i
$574$		0			0
$575$	−28.4859	−	3.12667i	−1.18795	−	0.130391i
$576$		0			0
$577$			3.25563i			0.135534i	0.997701	+	0.0677669i	$0.0215874\pi$
$577$			3.25563i			0.135534i	−0.997701	+	0.0677669i	$0.978413\pi$
$578$		0			0
$579$	15.3313			0.637146
$580$		0			0
$581$	1.14705			0.0475876
$582$		0			0
$583$			0.772519i			0.0319945i
$584$		0			0
$585$	21.3588	−	19.1427i	0.883078	−	0.791454i
$586$		0			0
$587$			24.7049i			1.01968i	0.860269	+	0.509840i	$0.170295\pi$
$587$			24.7049i			1.01968i	−0.860269	+	0.509840i	$0.829705\pi$
$588$		0			0
$589$	0.0460851			0.00189890
$590$		0			0
$591$	−20.4683			−0.841954
$592$		0			0
$593$		−	8.74194i		−	0.358989i	−0.983759	−	0.179494i	$-0.942554\pi$
$593$		−	8.74194i		−	0.358989i	0.983759	−	0.179494i	$-0.0574461\pi$
$594$		0			0
$595$	−0.461044	−	0.514418i	−0.0189010	−	0.0210891i
$596$		0			0
$597$			22.9848i			0.940705i
$598$		0			0
$599$	−11.3662			−0.464411			−0.232205	−	0.972667i	$-0.574594\pi$
$599$	−11.3662			−0.464411			−0.232205	+	0.972667i	$0.574594\pi$
$600$		0			0
$601$	−38.8180			−1.58342			−0.791710	−	0.610897i	$-0.790809\pi$
$601$	−38.8180			−1.58342			−0.791710	+	0.610897i	$0.790809\pi$
$602$		0			0
$603$			21.3609i			0.869883i
$604$		0			0
$605$	−3.84555	−	4.29074i	−0.156344	−	0.174444i
$606$		0			0
$607$		−	14.6825i		−	0.595945i	−0.954574	−	0.297972i	$-0.903690\pi$
$607$		−	14.6825i		−	0.595945i	0.954574	−	0.297972i	$-0.0963103\pi$
$608$		0			0
$609$	−8.04900			−0.326162
$610$		0			0
$611$	23.2454			0.940409
$612$		0			0
$613$		−	21.7660i		−	0.879120i	−0.898213	−	0.439560i	$-0.855134\pi$
$613$		−	21.7660i		−	0.879120i	0.898213	−	0.439560i	$-0.144866\pi$
$614$		0			0
$615$	6.74583	−	6.04591i	0.272018	−	0.243795i
$616$		0			0
$617$		−	24.6742i		−	0.993348i	−0.867937	−	0.496674i	$-0.834555\pi$
$617$		−	24.6742i		−	0.993348i	0.867937	−	0.496674i	$-0.165445\pi$
$618$		0			0
$619$	−6.02265			−0.242071			−0.121035	−	0.992648i	$-0.538621\pi$
$619$	−6.02265			−0.242071			−0.121035	+	0.992648i	$0.538621\pi$
$620$		0			0
$621$	−29.7276			−1.19293
$622$		0			0
$623$			5.45250i			0.218450i
$624$		0			0
$625$	−24.4048	−	5.42277i	−0.976191	−	0.216911i
$626$		0			0
$627$			22.9888i			0.918086i
$628$		0			0
$629$	0.433830			0.0172979
$630$		0			0
$631$	8.11252			0.322954			0.161477	−	0.986876i	$-0.448374\pi$
$631$	8.11252			0.322954			0.161477	+	0.986876i	$0.448374\pi$
$632$		0			0
$633$			13.5184i			0.537306i
$634$		0			0
$635$	21.9439	−	19.6671i	0.870818	−	0.780465i
$636$		0			0
$637$		−	44.7311i		−	1.77231i
$638$		0			0
$639$	−24.1344			−0.954743
$640$		0			0
$641$	13.8587			0.547384			0.273692	−	0.961817i	$-0.411755\pi$
$641$	13.8587			0.547384			0.273692	+	0.961817i	$0.411755\pi$
$642$		0			0
$643$			11.6697i			0.460207i	0.973166	+	0.230103i	$0.0739065\pi$
$643$			11.6697i			0.460207i	−0.973166	+	0.230103i	$0.926094\pi$
$644$		0			0
$645$	7.59514	+	8.47441i	0.299058	+	0.333680i
$646$		0			0
$647$		−	17.5481i		−	0.689889i	−0.938623	−	0.344944i	$-0.887898\pi$
$647$		−	17.5481i		−	0.689889i	0.938623	−	0.344944i	$-0.112102\pi$
$648$		0			0
$649$	12.1293			0.476117
$650$		0			0
$651$	0.00471532			0.000184808
$652$		0			0
$653$			9.84974i			0.385450i	0.981253	+	0.192725i	$0.0617325\pi$
$653$			9.84974i			0.385450i	−0.981253	+	0.192725i	$0.938267\pi$
$654$		0			0
$655$	14.7187	+	16.4227i	0.575109	+	0.641688i
$656$		0			0
$657$		−	20.7513i		−	0.809584i
$658$		0			0
$659$	−0.135284			−0.00526992			−0.00263496	−	0.999997i	$-0.500839\pi$
$659$	−0.135284			−0.00526992			−0.00263496	+	0.999997i	$0.500839\pi$
$660$		0			0
$661$	8.33911			0.324354			0.162177	−	0.986762i	$-0.448148\pi$
$661$	8.33911			0.324354			0.162177	+	0.986762i	$0.448148\pi$
$662$		0			0
$663$		−	3.18849i		−	0.123831i
$664$		0			0
$665$	8.80399	−	7.89052i	0.341404	−	0.305981i
$666$		0			0
$667$		−	60.7251i		−	2.35129i
$668$		0			0
$669$	19.0171			0.735243
$670$		0			0
$671$	−19.6047			−0.756832
$672$		0			0
$673$			0.594681i			0.0229233i	0.999934	+	0.0114616i	$0.00364843\pi$
$673$			0.594681i			0.0229233i	−0.999934	+	0.0114616i	$0.996352\pi$
$674$		0			0
$675$	−25.7791	−	2.82957i	−0.992239	−	0.108910i
$676$		0			0
$677$			39.0573i			1.50109i	0.660817	+	0.750547i	$0.270210\pi$
$677$			39.0573i			1.50109i	−0.660817	+	0.750547i	$0.729790\pi$
$678$		0			0
$679$	−0.772074			−0.0296295
$680$		0			0
$681$	17.9173			0.686593
$682$		0			0
$683$			7.10636i			0.271917i	0.990715	+	0.135959i	$0.0434115\pi$
$683$			7.10636i			0.271917i	−0.990715	+	0.135959i	$0.956589\pi$
$684$		0			0
$685$	29.2313	−	26.1984i	1.11687	−	1.00099i
$686$		0			0
$687$		−	6.69101i		−	0.255278i
$688$		0			0
$689$	1.83375			0.0698603
$690$		0			0
$691$	−0.828863			−0.0315314			−0.0157657	−	0.999876i	$-0.505019\pi$
$691$	−0.828863			−0.0315314			−0.0157657	+	0.999876i	$0.505019\pi$
$692$		0			0
$693$		−	3.84795i		−	0.146172i
$694$		0			0
$695$	−30.6597	−	34.2092i	−1.16299	−	1.29763i
$696$		0			0
$697$			1.64742i			0.0624004i
$698$		0			0
$699$	17.1860			0.650036
$700$		0			0
$701$	−14.1819			−0.535641			−0.267821	−	0.963469i	$-0.586303\pi$
$701$	−14.1819			−0.535641			−0.267821	+	0.963469i	$0.586303\pi$
$702$		0			0
$703$			7.42477i			0.280031i
$704$		0			0
$705$	−5.37212	−	5.99403i	−0.202326	−	0.225748i
$706$		0			0
$707$		−	10.8283i		−	0.407240i
$708$		0			0
$709$	30.3888			1.14128			0.570638	−	0.821202i	$-0.306696\pi$
$709$	30.3888			1.14128			0.570638	+	0.821202i	$0.306696\pi$
$710$		0			0
$711$	8.88053			0.333046
$712$		0			0
$713$			0.0355745i			0.00133227i
$714$		0			0
$715$	33.2940	−	29.8395i	1.24512	−	1.11593i
$716$		0			0
$717$		−	8.25733i		−	0.308376i
$718$		0			0
$719$	−18.0469			−0.673037			−0.336518	−	0.941677i	$-0.609249\pi$
$719$	−18.0469			−0.673037			−0.336518	+	0.941677i	$0.609249\pi$
$720$		0			0
$721$	−6.25879			−0.233089
$722$		0			0
$723$		−	4.89366i		−	0.181997i
$724$		0			0
$725$	5.78001	−	52.6595i	0.214664	−	1.95573i
$726$		0			0
$727$		−	50.1159i		−	1.85870i	−0.369205	−	0.929348i	$-0.620370\pi$
$727$		−	50.1159i		−	1.85870i	0.369205	−	0.929348i	$-0.379630\pi$
$728$		0			0
$729$	−16.5030			−0.611224
$730$		0			0
$731$	−2.06956			−0.0765454
$732$		0			0
$733$			17.9931i			0.664592i	0.943175	+	0.332296i	$0.107823\pi$
$733$			17.9931i			0.664592i	−0.943175	+	0.332296i	$0.892177\pi$
$734$		0			0
$735$	−11.5343	+	10.3375i	−0.425449	+	0.381306i
$736$		0			0
$737$			33.2972i			1.22652i
$738$		0			0
$739$	−5.07934			−0.186847			−0.0934233	−	0.995626i	$-0.529781\pi$
$739$	−5.07934			−0.186847			−0.0934233	+	0.995626i	$0.529781\pi$
$740$		0			0
$741$	54.5692			2.00465
$742$		0			0
$743$		−	1.25458i		−	0.0460262i	−0.999735	−	0.0230131i	$-0.992674\pi$
$743$		−	1.25458i		−	0.0460262i	0.999735	−	0.0230131i	$-0.00732595\pi$
$744$		0			0
$745$	7.60044	+	8.48033i	0.278459	+	0.310695i
$746$		0			0
$747$			2.99911i			0.109732i
$748$		0			0
$749$	7.86645			0.287434
$750$		0			0
$751$	5.54949			0.202504			0.101252	−	0.994861i	$-0.467715\pi$
$751$	5.54949			0.202504			0.101252	+	0.994861i	$0.467715\pi$
$752$		0			0
$753$			16.8697i			0.614765i
$754$		0			0
$755$	−7.15030	−	7.97808i	−0.260226	−	0.290352i
$756$		0			0
$757$			10.3643i			0.376697i	0.982102	+	0.188349i	$0.0603135\pi$
$757$			10.3643i			0.376697i	−0.982102	+	0.188349i	$0.939687\pi$
$758$		0			0
$759$	−17.7458			−0.644131
$760$		0			0
$761$	−17.9530			−0.650794			−0.325397	−	0.945577i	$-0.605498\pi$
$761$	−17.9530			−0.650794			−0.325397	+	0.945577i	$0.605498\pi$
$762$		0			0
$763$			2.66793i			0.0965857i
$764$		0			0
$765$	1.34502	−	1.20546i	0.0486291	−	0.0435836i
$766$		0			0
$767$		−	28.7916i		−	1.03961i
$768$		0			0
$769$	53.6887			1.93606			0.968032	−	0.250826i	$-0.0807021\pi$
$769$	53.6887			1.93606			0.968032	+	0.250826i	$0.0807021\pi$
$770$		0			0
$771$	−29.6003			−1.06603
$772$		0			0
$773$			49.5152i			1.78094i	0.455044	+	0.890469i	$0.349624\pi$
$773$			49.5152i			1.78094i	−0.455044	+	0.890469i	$0.650376\pi$
$774$		0			0
$775$	−0.00338609	+	0.0308494i	−0.000121632	+	0.00110814i
$776$		0			0
$777$			0.759686i			0.0272536i
$778$		0			0
$779$	−28.1947			−1.01018
$780$		0			0
$781$	−37.6206			−1.34617
$782$		0			0
$783$		−	54.9548i		−	1.96393i
$784$		0			0
$785$	−27.0697	+	24.2611i	−0.966159	+	0.865914i
$786$		0			0
$787$			39.3331i			1.40207i	0.713125	+	0.701036i	$0.247279\pi$
$787$			39.3331i			1.40207i	−0.713125	+	0.701036i	$0.752721\pi$
$788$		0			0
$789$	−17.9919			−0.640529
$790$		0			0
$791$	−6.56921			−0.233574
$792$		0			0
$793$			46.5363i			1.65255i
$794$		0			0
$795$	−0.423787	−	0.472848i	−0.0150302	−	0.0167702i
$796$		0			0
$797$			3.54138i			0.125442i	0.998031	+	0.0627211i	$0.0199779\pi$
$797$			3.54138i			0.125442i	−0.998031	+	0.0627211i	$0.980022\pi$
$798$		0			0
$799$	1.46382			0.0517862
$800$		0			0
$801$	−14.2563			−0.503722
$802$		0			0
$803$		−	32.3470i		−	1.14150i
$804$		0			0
$805$	6.09093	+	6.79606i	0.214677	+	0.239530i
$806$		0			0
$807$		−	24.9300i		−	0.877577i
$808$		0			0
$809$	−16.8600			−0.592765			−0.296382	−	0.955069i	$-0.595780\pi$
$809$	−16.8600			−0.592765			−0.296382	+	0.955069i	$0.595780\pi$
$810$		0			0
$811$	−52.9317			−1.85868			−0.929342	−	0.369220i	$-0.879625\pi$
$811$	−52.9317			−1.85868			−0.929342	+	0.369220i	$0.879625\pi$
$812$		0			0
$813$		−	21.9858i		−	0.771076i
$814$		0			0
$815$	27.5223	−	24.6667i	0.964064	−	0.864036i
$816$		0			0
$817$		−	35.4194i		−	1.23917i
$818$		0			0
$819$	−9.13398			−0.319167
$820$		0			0
$821$	−5.39834			−0.188403			−0.0942017	−	0.995553i	$-0.530030\pi$
$821$	−5.39834			−0.188403			−0.0942017	+	0.995553i	$0.530030\pi$
$822$		0			0
$823$			31.4399i			1.09593i	0.836503	+	0.547963i	$0.184597\pi$
$823$			31.4399i			1.09593i	−0.836503	+	0.547963i	$0.815403\pi$
$824$		0			0
$825$	−15.3888	−	1.68910i	−0.535768	−	0.0588069i
$826$		0			0
$827$			7.70172i			0.267815i	0.990994	+	0.133908i	$0.0427525\pi$
$827$			7.70172i			0.267815i	−0.990994	+	0.133908i	$0.957247\pi$
$828$		0			0
$829$	−0.202796			−0.00704338			−0.00352169	−	0.999994i	$-0.501121\pi$
$829$	−0.202796			−0.00704338			−0.00352169	+	0.999994i	$0.501121\pi$
$830$		0			0
$831$	8.69059			0.301473
$832$		0			0
$833$		−	2.81682i		−	0.0975972i
$834$		0			0
$835$	−20.2660	+	18.1633i	−0.701333	+	0.628566i
$836$		0			0
$837$			0.0321941i			0.00111279i
$838$		0			0
$839$	7.29847			0.251971			0.125986	−	0.992032i	$-0.459791\pi$
$839$	7.29847			0.251971			0.125986	+	0.992032i	$0.459791\pi$
$840$		0			0
$841$	83.2573			2.87094
$842$		0			0
$843$			15.4935i			0.533625i
$844$		0			0
$845$	−51.4298	−	57.3837i	−1.76924	−	1.97406i
$846$		0			0
$847$			1.83491i			0.0630484i
$848$		0			0
$849$	−33.9557			−1.16536
$850$		0			0
$851$	−5.73141			−0.196470
$852$		0			0
$853$			43.4552i			1.48788i	0.668248	+	0.743938i	$0.267044\pi$
$853$			43.4552i			1.48788i	−0.668248	+	0.743938i	$0.732956\pi$
$854$		0			0
$855$	20.6308	+	23.0192i	0.705560	+	0.787241i
$856$		0			0
$857$		−	9.44057i		−	0.322484i	−0.986915	−	0.161242i	$-0.948450\pi$
$857$		−	9.44057i		−	0.322484i	0.986915	−	0.161242i	$-0.0515499\pi$
$858$		0			0
$859$	26.6063			0.907797			0.453898	−	0.891054i	$-0.350033\pi$
$859$	26.6063			0.907797			0.453898	+	0.891054i	$0.350033\pi$
$860$		0			0
$861$	−2.88482			−0.0983143
$862$		0			0
$863$		−	10.7086i		−	0.364526i	−0.983250	−	0.182263i	$-0.941658\pi$
$863$		−	10.7086i		−	0.364526i	0.983250	−	0.182263i	$-0.0583422\pi$
$864$		0			0
$865$	4.20049	−	3.76466i	0.142821	−	0.128002i
$866$		0			0
$867$			17.9353i			0.609115i
$868$		0			0
$869$	13.8429			0.469588
$870$		0			0
$871$	79.0384			2.67812
$872$		0			0
$873$		−	2.01869i		−	0.0683224i
$874$		0			0
$875$	4.63505	+	6.47315i	0.156693	+	0.218832i
$876$		0			0
$877$			25.7701i			0.870195i	0.900383	+	0.435097i	$0.143286\pi$
$877$			25.7701i			0.870195i	−0.900383	+	0.435097i	$0.856714\pi$
$878$		0			0
$879$	32.9452			1.11122
$880$		0			0
$881$	−31.3316			−1.05559			−0.527794	−	0.849372i	$-0.676981\pi$
$881$	−31.3316			−1.05559			−0.527794	+	0.849372i	$0.676981\pi$
$882$		0			0
$883$		−	41.0469i		−	1.38134i	−0.723171	−	0.690669i	$-0.757316\pi$
$883$		−	41.0469i		−	1.38134i	0.723171	−	0.690669i	$-0.242684\pi$
$884$		0			0
$885$	−7.42417	+	6.65387i	−0.249561	+	0.223667i
$886$		0			0
$887$		−	0.130253i		−	0.00437345i	−0.999998	−	0.00218673i	$-0.999304\pi$
$887$		−	0.130253i		−	0.00437345i	0.999998	−	0.00218673i	$-0.000696057\pi$
$888$		0			0
$889$	−9.38420			−0.314736
$890$		0			0
$891$	0.151527			0.00507635
$892$		0			0
$893$			25.0525i			0.838350i
$894$		0			0
$895$	−34.4341	−	38.4205i	−1.15101	−	1.28425i
$896$		0			0
$897$			42.1236i			1.40647i
$898$		0			0
$899$	−0.0657634			−0.00219333
$900$		0			0
$901$	0.115476			0.00384705
$902$		0			0
$903$		−	3.62404i		−	0.120600i
$904$		0			0
$905$	−10.2689	−	11.4577i	−0.341348	−	0.380865i
$906$		0			0
$907$		−	29.8074i		−	0.989738i	−0.868968	−	0.494869i	$-0.835216\pi$
$907$		−	29.8074i		−	0.989738i	0.868968	−	0.494869i	$-0.164784\pi$
$908$		0			0
$909$	28.3120			0.939051
$910$		0			0
$911$	30.0475			0.995517			0.497759	−	0.867316i	$-0.334157\pi$
$911$	30.0475			0.995517			0.497759	+	0.867316i	$0.334157\pi$
$912$		0			0
$913$			4.67500i			0.154720i
$914$		0			0
$915$	11.9998	−	10.7547i	0.396700	−	0.355540i
$916$		0			0
$917$		−	7.02308i		−	0.231923i
$918$		0			0
$919$	40.6134			1.33971			0.669857	−	0.742490i	$-0.266355\pi$
$919$	40.6134			1.33971			0.669857	+	0.742490i	$0.266355\pi$
$920$		0			0
$921$	−6.23561			−0.205470
$922$		0			0
$923$			89.3009i			2.93938i
$924$		0			0
$925$	−4.97015	−	0.545534i	−0.163418	−	0.0179370i
$926$		0			0
$927$		−	16.3644i		−	0.537479i
$928$		0			0
$929$	−51.8039			−1.69963			−0.849816	−	0.527079i	$-0.823287\pi$
$929$	−51.8039			−1.69963			−0.849816	+	0.527079i	$0.823287\pi$
$930$		0			0
$931$	48.2084			1.57997
$932$		0			0
$933$		−	6.05733i		−	0.198308i
$934$		0			0
$935$	2.09660	−	1.87907i	0.0685662	−	0.0614520i
$936$		0			0
$937$			33.2111i			1.08496i	0.840069	+	0.542480i	$0.182515\pi$
$937$			33.2111i			1.08496i	−0.840069	+	0.542480i	$0.817485\pi$
$938$		0			0
$939$	−24.7522			−0.807756
$940$		0			0
$941$	−17.5055			−0.570664			−0.285332	−	0.958429i	$-0.592104\pi$
$941$	−17.5055			−0.570664			−0.285332	+	0.958429i	$0.592104\pi$
$942$		0			0
$943$		−	21.7643i		−	0.708744i
$944$		0			0
$945$	5.51215	+	6.15028i	0.179310	+	0.200069i
$946$		0			0
$947$		−	1.04714i		−	0.0340275i	−0.999855	−	0.0170138i	$-0.994584\pi$
$947$		−	1.04714i		−	0.0340275i	0.999855	−	0.0170138i	$-0.00541591\pi$
$948$		0			0
$949$	−76.7828			−2.49247
$950$		0			0
$951$	11.4081			0.369933
$952$		0			0
$953$		−	34.0009i		−	1.10140i	−0.834705	−	0.550698i	$-0.814362\pi$
$953$		−	34.0009i		−	1.10140i	0.834705	−	0.550698i	$-0.185638\pi$
$954$		0			0
$955$	−4.82520	−	5.38380i	−0.156140	−	0.174216i
$956$		0			0
$957$		−	32.8051i		−	1.06044i
$958$		0			0
$959$	−12.5006			−0.403666
$960$		0			0
$961$	−31.0000			−0.999999
$962$		0			0
$963$			20.5679i			0.662791i
$964$		0			0
$965$	23.9298	−	21.4470i	0.770329	−	0.690402i
$966$		0			0
$967$			47.7649i			1.53602i	0.640440	+	0.768009i	$0.278752\pi$
$967$			47.7649i			1.53602i	−0.640440	+	0.768009i	$0.721248\pi$
$968$		0			0
$969$	3.43635			0.110392
$970$		0			0
$971$	28.5287			0.915530			0.457765	−	0.889073i	$-0.348650\pi$
$971$	28.5287			0.915530			0.457765	+	0.889073i	$0.348650\pi$
$972$		0			0
$973$			14.6294i			0.468996i
$974$		0			0
$975$	−4.00946	+	36.5287i	−0.128406	+	1.16985i
$976$		0			0
$977$			20.1807i			0.645638i	0.946461	+	0.322819i	$0.104631\pi$
$977$			20.1807i			0.645638i	−0.946461	+	0.322819i	$0.895369\pi$
$978$		0			0
$979$	−22.2226			−0.710239
$980$		0			0
$981$	−6.97567			−0.222716
$982$		0			0
$983$			37.2253i			1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$983$			37.2253i			1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$984$		0			0
$985$	−31.9480	+	28.6332i	−1.01795	+	0.912330i
$986$		0			0
$987$			2.56332i			0.0815912i
$988$		0			0
$989$	27.3413			0.869403
$990$		0			0
$991$	−14.4987			−0.460565			−0.230283	−	0.973124i	$-0.573965\pi$
$991$	−14.4987			−0.460565			−0.230283	+	0.973124i	$0.573965\pi$
$992$		0			0
$993$			4.28893i			0.136105i
$994$		0			0
$995$	32.1535	+	35.8759i	1.01934	+	1.13734i
$996$		0			0
$997$			17.0704i			0.540624i	0.962773	+	0.270312i	$0.0871269\pi$
$997$			17.0704i			0.540624i	−0.962773	+	0.270312i	$0.912873\pi$
$998$		0			0
$999$	−5.18679			−0.164103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	740.2.d.a.149.12	yes	18
3.2	odd	2		6660.2.f.c.5329.5		18
5.2	odd	4		3700.2.a.o.1.6		9
5.3	odd	4		3700.2.a.p.1.4		9
5.4	even	2	inner	740.2.d.a.149.7	✓	18
15.14	odd	2		6660.2.f.c.5329.6		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.d.a.149.7	✓	18	5.4	even	2	inner
740.2.d.a.149.12	yes	18	1.1	even	1	trivial
3700.2.a.o.1.6		9	5.2	odd	4
3700.2.a.p.1.4		9	5.3	odd	4
6660.2.f.c.5329.5		18	3.2	odd	2
6660.2.f.c.5329.6		18	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.06683i q^{3} +(1.49239 + 1.66516i) q^{5} -0.712098i q^{7} +1.86188 q^{9} +2.90228 q^{11} -6.88922i q^{13} +(-1.77644 + 1.59213i) q^{15} -0.433830i q^{17} +7.42477 q^{19} +0.759686 q^{21} +5.73141i q^{23} +(-0.545534 + 4.97015i) q^{25} +5.18679i q^{27} -10.5952 q^{29} +0.00620693 q^{31} +3.09624i q^{33} +(1.18576 - 1.06273i) q^{35} +1.00000i q^{37} +7.34962 q^{39} -3.79738 q^{41} -4.77044i q^{43} +(2.77865 + 3.10033i) q^{45} +3.37418i q^{47} +6.49292 q^{49} +0.462823 q^{51} +0.266177i q^{53} +(4.33134 + 4.83277i) q^{55} +7.92096i q^{57} +4.17923 q^{59} -6.75494 q^{61} -1.32584i q^{63} +(11.4717 - 10.2814i) q^{65} +11.4728i q^{67} -6.11443 q^{69} -12.9624 q^{71} -11.1454i q^{73} +(-5.30230 - 0.581991i) q^{75} -2.06671i q^{77} +4.76967 q^{79} +0.0522097 q^{81} +1.61080i q^{83} +(0.722398 - 0.647445i) q^{85} -11.3032i q^{87} -7.65696 q^{89} -4.90579 q^{91} +0.00662174i q^{93} +(11.0807 + 12.3635i) q^{95} -1.08422i q^{97} +5.40368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{5} - 18 q^{9} + 6 q^{15} + 4 q^{19} - 16 q^{21} - 2 q^{25} - 4 q^{29} + 8 q^{31} - 2 q^{35} + 8 q^{39} - 4 q^{41} + 8 q^{45} + 6 q^{49} - 40 q^{51} - 6 q^{55} + 8 q^{59} - 12 q^{65} + 28 q^{69}+ \cdots + 16 q^{99}+O(q^{100}) \) 18 * q + 2 * q^5 - 18 * q^9 + 6 * q^15 + 4 * q^19 - 16 * q^21 - 2 * q^25 - 4 * q^29 + 8 * q^31 - 2 * q^35 + 8 * q^39 - 4 * q^41 + 8 * q^45 + 6 * q^49 - 40 * q^51 - 6 * q^55 + 8 * q^59 - 12 * q^65 + 28 * q^69 - 24 * q^71 - 16 * q^75 - 24 * q^79 + 34 * q^81 + 36 * q^91 + 24 * q^95 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more