LMFDB - Embedded newform 1464.2.l.b.121.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1464,2,Mod(121,1464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1464.121"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1464 = 2^{3} \cdot 3 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1464.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] 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:	$11.6900988559$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 35x^{14} + 443x^{12} + 2546x^{10} + 6883x^{8} + 8967x^{6} + 5233x^{4} + 984x^{2} + 4 $ x^16 + 35x^14 + 443x^12 + 2546x^10 + 6883x^8 + 8967x^6 + 5233x^4 + 984*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{61}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			121.3
Root			$2.73537i$ of defining polynomial
Character	$\chi$	$=$	1464.121
Dual form			1464.2.l.b.121.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.81251 q^{5} -4.23671i q^{7} +1.00000 q^{9} +4.52763i q^{11} +2.04174 q^{13} +1.81251 q^{15} +4.13734i q^{17} +6.74245 q^{19} +4.23671i q^{21} -3.52566i q^{23} -1.71479 q^{25} -1.00000 q^{27} -4.81823i q^{29} -7.91580i q^{31} -4.52763i q^{33} +7.67909i q^{35} -2.88820i q^{37} -2.04174 q^{39} -7.10131 q^{41} +1.77862i q^{43} -1.81251 q^{45} -3.94611 q^{47} -10.9497 q^{49} -4.13734i q^{51} -5.09757i q^{53} -8.20640i q^{55} -6.74245 q^{57} -10.3600i q^{59} +(-7.58886 - 1.84641i) q^{61} -4.23671i q^{63} -3.70067 q^{65} -1.13899i q^{67} +3.52566i q^{69} +5.23506i q^{71} +10.0194 q^{73} +1.71479 q^{75} +19.1823 q^{77} -12.4137i q^{79} +1.00000 q^{81} +5.80862 q^{83} -7.49898i q^{85} +4.81823i q^{87} -6.80415i q^{89} -8.65024i q^{91} +7.91580i q^{93} -12.2208 q^{95} -17.4036 q^{97} +4.52763i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 4 q^{5} + 16 q^{9} - 6 q^{13} - 4 q^{15} - 4 q^{19} + 8 q^{25} - 16 q^{27} + 6 q^{39} - 18 q^{41} + 4 q^{45} - 8 q^{47} - 22 q^{49} + 4 q^{57} + 26 q^{61} + 10 q^{65} + 36 q^{73} - 8 q^{75}+ \cdots - 10 q^{97}+O(q^{100}) $ 16 * q - 16 * q^3 + 4 * q^5 + 16 * q^9 - 6 * q^13 - 4 * q^15 - 4 * q^19 + 8 * q^25 - 16 * q^27 + 6 * q^39 - 18 * q^41 + 4 * q^45 - 8 * q^47 - 22 * q^49 + 4 * q^57 + 26 * q^61 + 10 * q^65 + 36 * q^73 - 8 * q^75 + 46 * q^77 + 16 * q^81 + 14 * q^83 - 44 * q^95 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1464\mathbb{Z}\right)^\times$.

$n$	$367$	$673$	$733$	$977$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−1.81251			−0.810581			−0.405290	−	0.914188i	$-0.632830\pi$
$5$	−1.81251			−0.810581			−0.405290	+	0.914188i	$0.632830\pi$
$6$		0			0
$7$		−	4.23671i		−	1.60133i	−0.599116	−	0.800663i	$-0.704481\pi$
$7$		−	4.23671i		−	1.60133i	0.599116	−	0.800663i	$-0.295519\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			4.52763i			1.36513i	0.730823	+	0.682567i	$0.239136\pi$
$11$			4.52763i			1.36513i	−0.730823	+	0.682567i	$0.760864\pi$
$12$		0			0
$13$	2.04174			0.566275			0.283138	−	0.959079i	$-0.408625\pi$
$13$	2.04174			0.566275			0.283138	+	0.959079i	$0.408625\pi$
$14$		0			0
$15$	1.81251			0.467989
$16$		0			0
$17$			4.13734i			1.00345i	0.865027	+	0.501726i	$0.167301\pi$
$17$			4.13734i			1.00345i	−0.865027	+	0.501726i	$0.832699\pi$
$18$		0			0
$19$	6.74245			1.54682			0.773412	−	0.633904i	$-0.218548\pi$
$19$	6.74245			1.54682			0.773412	+	0.633904i	$0.218548\pi$
$20$		0			0
$21$			4.23671i			0.924526i
$22$		0			0
$23$		−	3.52566i		−	0.735150i	−0.929994	−	0.367575i	$-0.880188\pi$
$23$		−	3.52566i		−	0.735150i	0.929994	−	0.367575i	$-0.119812\pi$
$24$		0			0
$25$	−1.71479			−0.342959
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	4.81823i		−	0.894724i	−0.894353	−	0.447362i	$-0.852364\pi$
$29$		−	4.81823i		−	0.894724i	0.894353	−	0.447362i	$-0.147636\pi$
$30$		0			0
$31$		−	7.91580i		−	1.42172i	−0.703334	−	0.710860i	$-0.748306\pi$
$31$		−	7.91580i		−	1.42172i	0.703334	−	0.710860i	$-0.251694\pi$
$32$		0			0
$33$		−	4.52763i		−	0.788160i
$34$		0			0
$35$			7.67909i			1.29800i
$36$		0			0
$37$		−	2.88820i		−	0.474817i	−0.971410	−	0.237409i	$-0.923702\pi$
$37$		−	2.88820i		−	0.474817i	0.971410	−	0.237409i	$-0.0762980\pi$
$38$		0			0
$39$	−2.04174			−0.326939
$40$		0			0
$41$	−7.10131			−1.10904			−0.554519	−	0.832171i	$-0.687098\pi$
$41$	−7.10131			−1.10904			−0.554519	+	0.832171i	$0.687098\pi$
$42$		0			0
$43$			1.77862i			0.271237i	0.990761	+	0.135618i	$0.0433021\pi$
$43$			1.77862i			0.271237i	−0.990761	+	0.135618i	$0.956698\pi$
$44$		0			0
$45$	−1.81251			−0.270194
$46$		0			0
$47$	−3.94611			−0.575599			−0.287800	−	0.957691i	$-0.592924\pi$
$47$	−3.94611			−0.575599			−0.287800	+	0.957691i	$0.592924\pi$
$48$		0			0
$49$	−10.9497			−1.56424
$50$		0			0
$51$		−	4.13734i		−	0.579343i
$52$		0			0
$53$		−	5.09757i		−	0.700204i	−0.936711	−	0.350102i	$-0.886147\pi$
$53$		−	5.09757i		−	0.700204i	0.936711	−	0.350102i	$-0.113853\pi$
$54$		0			0
$55$		−	8.20640i		−	1.10655i
$56$		0			0
$57$	−6.74245			−0.893059
$58$		0			0
$59$		−	10.3600i		−	1.34875i	−0.738388	−	0.674376i	$-0.764413\pi$
$59$		−	10.3600i		−	1.34875i	0.738388	−	0.674376i	$-0.235587\pi$
$60$		0			0
$61$	−7.58886	−	1.84641i	−0.971654	−	0.236409i
$61$	−7.58886	−	1.84641i	−0.971654	−	0.236409i
$62$		0			0
$63$		−	4.23671i		−	0.533775i
$64$		0			0
$65$	−3.70067			−0.459012
$66$		0			0
$67$		−	1.13899i		−	0.139150i	−0.997577	−	0.0695748i	$-0.977836\pi$
$67$		−	1.13899i		−	0.139150i	0.997577	−	0.0695748i	$-0.0221643\pi$
$68$		0			0
$69$			3.52566i			0.424439i
$70$		0			0
$71$			5.23506i			0.621287i	0.950527	+	0.310643i	$0.100544\pi$
$71$			5.23506i			0.621287i	−0.950527	+	0.310643i	$0.899456\pi$
$72$		0			0
$73$	10.0194			1.17268			0.586341	−	0.810065i	$-0.300568\pi$
$73$	10.0194			1.17268			0.586341	+	0.810065i	$0.300568\pi$
$74$		0			0
$75$	1.71479			0.198007
$76$		0			0
$77$	19.1823			2.18602
$78$		0			0
$79$		−	12.4137i		−	1.39665i	−0.715781	−	0.698325i	$-0.753929\pi$
$79$		−	12.4137i		−	1.39665i	0.715781	−	0.698325i	$-0.246071\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	5.80862			0.637579			0.318789	−	0.947826i	$-0.396724\pi$
$83$	5.80862			0.637579			0.318789	+	0.947826i	$0.396724\pi$
$84$		0			0
$85$		−	7.49898i		−	0.813378i
$86$		0			0
$87$			4.81823i			0.516569i
$88$		0			0
$89$		−	6.80415i		−	0.721239i	−0.932713	−	0.360619i	$-0.882565\pi$
$89$		−	6.80415i		−	0.721239i	0.932713	−	0.360619i	$-0.117435\pi$
$90$		0			0
$91$		−	8.65024i		−	0.906791i
$92$		0			0
$93$			7.91580i			0.820830i
$94$		0			0
$95$	−12.2208			−1.25383
$96$		0			0
$97$	−17.4036			−1.76707			−0.883536	−	0.468364i	$-0.844844\pi$
$97$	−17.4036			−1.76707			−0.883536	+	0.468364i	$0.844844\pi$
$98$		0			0
$99$			4.52763i			0.455044i
$100$		0			0
$101$		−	12.0672i		−	1.20073i	−0.799725	−	0.600366i	$-0.795021\pi$
$101$		−	12.0672i		−	1.20073i	0.799725	−	0.600366i	$-0.204979\pi$
$102$		0			0
$103$	13.1329			1.29402			0.647012	−	0.762479i	$-0.276018\pi$
$103$	13.1329			1.29402			0.647012	+	0.762479i	$0.276018\pi$
$104$		0			0
$105$		−	7.67909i		−	0.749403i
$106$		0			0
$107$	17.9691			1.73714			0.868569	−	0.495569i	$-0.165040\pi$
$107$	17.9691			1.73714			0.868569	+	0.495569i	$0.165040\pi$
$108$		0			0
$109$	−16.7816			−1.60738			−0.803691	−	0.595047i	$-0.797134\pi$
$109$	−16.7816			−1.60738			−0.803691	+	0.595047i	$0.797134\pi$
$110$		0			0
$111$			2.88820i			0.274136i
$112$		0			0
$113$	−8.64682			−0.813425			−0.406712	−	0.913556i	$-0.633325\pi$
$113$	−8.64682			−0.813425			−0.406712	+	0.913556i	$0.633325\pi$
$114$		0			0
$115$			6.39030i			0.595898i
$116$		0			0
$117$	2.04174			0.188758
$118$		0			0
$119$	17.5287			1.60685
$120$		0			0
$121$	−9.49947			−0.863588
$122$		0			0
$123$	7.10131			0.640303
$124$		0			0
$125$	12.1707			1.08858
$126$		0			0
$127$	−2.37012			−0.210314			−0.105157	−	0.994456i	$-0.533534\pi$
$127$	−2.37012			−0.210314			−0.105157	+	0.994456i	$0.533534\pi$
$128$		0			0
$129$		−	1.77862i		−	0.156599i
$130$		0			0
$131$	−0.440404			−0.0384783			−0.0192391	−	0.999815i	$-0.506124\pi$
$131$	−0.440404			−0.0384783			−0.0192391	+	0.999815i	$0.506124\pi$
$132$		0			0
$133$		−	28.5658i		−	2.47697i
$134$		0			0
$135$	1.81251			0.155996
$136$		0			0
$137$	7.39773			0.632031			0.316015	−	0.948754i	$-0.397655\pi$
$137$	7.39773			0.632031			0.316015	+	0.948754i	$0.397655\pi$
$138$		0			0
$139$		−	9.16474i		−	0.777343i	−0.921376	−	0.388671i	$-0.872934\pi$
$139$		−	9.16474i		−	0.777343i	0.921376	−	0.388671i	$-0.127066\pi$
$140$		0			0
$141$	3.94611			0.332322
$142$		0			0
$143$			9.24423i			0.773041i
$144$		0			0
$145$			8.73312i			0.725246i
$146$		0			0
$147$	10.9497			0.903116
$148$		0			0
$149$	−11.1990			−0.917457			−0.458728	−	0.888577i	$-0.651695\pi$
$149$	−11.1990			−0.917457			−0.458728	+	0.888577i	$0.651695\pi$
$150$		0			0
$151$		−	9.88970i		−	0.804812i	−0.915461	−	0.402406i	$-0.868174\pi$
$151$		−	9.88970i		−	0.804812i	0.915461	−	0.402406i	$-0.131826\pi$
$152$		0			0
$153$			4.13734i			0.334484i
$154$		0			0
$155$			14.3475i			1.15242i
$156$		0			0
$157$			13.5131i			1.07846i	0.842159	+	0.539230i	$0.181285\pi$
$157$			13.5131i			1.07846i	−0.842159	+	0.539230i	$0.818715\pi$
$158$		0			0
$159$			5.09757i			0.404263i
$160$		0			0
$161$	−14.9372			−1.17721
$162$		0			0
$163$	20.3250			1.59198			0.795989	−	0.605311i	$-0.206951\pi$
$163$	20.3250			1.59198			0.795989	+	0.605311i	$0.206951\pi$
$164$		0			0
$165$			8.20640i			0.638867i
$166$		0			0
$167$	−18.8905			−1.46179			−0.730895	−	0.682490i	$-0.760897\pi$
$167$	−18.8905			−1.46179			−0.730895	+	0.682490i	$0.760897\pi$
$168$		0			0
$169$	−8.83132			−0.679332
$170$		0			0
$171$	6.74245			0.515608
$172$		0			0
$173$			15.7597i			1.19819i	0.800678	+	0.599095i	$0.204473\pi$
$173$			15.7597i			1.19819i	−0.800678	+	0.599095i	$0.795527\pi$
$174$		0			0
$175$			7.26508i			0.549189i
$176$		0			0
$177$			10.3600i			0.778703i
$178$		0			0
$179$	−7.17041			−0.535942			−0.267971	−	0.963427i	$-0.586353\pi$
$179$	−7.17041			−0.535942			−0.267971	+	0.963427i	$0.586353\pi$
$180$		0			0
$181$			9.81675i			0.729674i	0.931071	+	0.364837i	$0.118875\pi$
$181$			9.81675i			0.729674i	−0.931071	+	0.364837i	$0.881125\pi$
$182$		0			0
$183$	7.58886	+	1.84641i	0.560985	+	0.136491i
$184$		0			0
$185$			5.23490i			0.384878i
$186$		0			0
$187$	−18.7323			−1.36984
$188$		0			0
$189$			4.23671i			0.308175i
$190$		0			0
$191$		−	16.8420i		−	1.21864i	−0.792923	−	0.609322i	$-0.791442\pi$
$191$		−	16.8420i		−	1.21864i	0.792923	−	0.609322i	$-0.208558\pi$
$192$		0			0
$193$		−	22.2191i		−	1.59936i	−0.600423	−	0.799682i	$-0.705001\pi$
$193$		−	22.2191i		−	1.59936i	0.600423	−	0.799682i	$-0.294999\pi$
$194$		0			0
$195$	3.70067			0.265011
$196$		0			0
$197$	3.28437			0.234002			0.117001	−	0.993132i	$-0.462672\pi$
$197$	3.28437			0.234002			0.117001	+	0.993132i	$0.462672\pi$
$198$		0			0
$199$	−5.40407			−0.383084			−0.191542	−	0.981484i	$-0.561349\pi$
$199$	−5.40407			−0.383084			−0.191542	+	0.981484i	$0.561349\pi$
$200$		0			0
$201$			1.13899i			0.0803381i
$202$		0			0
$203$	−20.4135			−1.43274
$204$		0			0
$205$	12.8712			0.898965
$206$		0			0
$207$		−	3.52566i		−	0.245050i
$208$		0			0
$209$			30.5273i			2.11162i
$210$		0			0
$211$			5.72067i			0.393827i	0.980421	+	0.196914i	$0.0630918\pi$
$211$			5.72067i			0.393827i	−0.980421	+	0.196914i	$0.936908\pi$
$212$		0			0
$213$		−	5.23506i		−	0.358700i
$214$		0			0
$215$		−	3.22377i		−	0.219859i
$216$		0			0
$217$	−33.5369			−2.27664
$218$		0			0
$219$	−10.0194			−0.677048
$220$		0			0
$221$			8.44734i			0.568230i
$222$		0			0
$223$			21.0105i			1.40697i	0.710711	+	0.703484i	$0.248374\pi$
$223$			21.0105i			1.40697i	−0.710711	+	0.703484i	$0.751626\pi$
$224$		0			0
$225$	−1.71479			−0.114320
$226$		0			0
$227$		−	16.8763i		−	1.12012i	−0.828453	−	0.560058i	$-0.810779\pi$
$227$		−	16.8763i		−	1.12012i	0.828453	−	0.560058i	$-0.189221\pi$
$228$		0			0
$229$	11.2187			0.741352			0.370676	−	0.928762i	$-0.379126\pi$
$229$	11.2187			0.741352			0.370676	+	0.928762i	$0.379126\pi$
$230$		0			0
$231$	−19.1823			−1.26210
$232$		0			0
$233$			6.01278i			0.393910i	0.980412	+	0.196955i	$0.0631053\pi$
$233$			6.01278i			0.393910i	−0.980412	+	0.196955i	$0.936895\pi$
$234$		0			0
$235$	7.15238			0.466570
$236$		0			0
$237$			12.4137i			0.806357i
$238$		0			0
$239$	−6.11143			−0.395316			−0.197658	−	0.980271i	$-0.563334\pi$
$239$	−6.11143			−0.395316			−0.197658	+	0.980271i	$0.563334\pi$
$240$		0			0
$241$	23.0547			1.48508			0.742540	−	0.669801i	$-0.233621\pi$
$241$	23.0547			1.48508			0.742540	+	0.669801i	$0.233621\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	19.8465			1.26794
$246$		0			0
$247$	13.7663			0.875928
$248$		0			0
$249$	−5.80862			−0.368106
$250$		0			0
$251$		−	17.6245i		−	1.11245i	−0.831033	−	0.556223i	$-0.812250\pi$
$251$		−	17.6245i		−	1.11245i	0.831033	−	0.556223i	$-0.187750\pi$
$252$		0			0
$253$	15.9629			1.00358
$254$		0			0
$255$			7.49898i			0.469604i
$256$		0			0
$257$	13.5412			0.844680			0.422340	−	0.906438i	$-0.361209\pi$
$257$	13.5412			0.844680			0.422340	+	0.906438i	$0.361209\pi$
$258$		0			0
$259$	−12.2365			−0.760337
$260$		0			0
$261$		−	4.81823i		−	0.298241i
$262$		0			0
$263$	19.7299			1.21660			0.608298	−	0.793709i	$-0.291852\pi$
$263$	19.7299			1.21660			0.608298	+	0.793709i	$0.291852\pi$
$264$		0			0
$265$			9.23941i			0.567572i
$266$		0			0
$267$			6.80415i			0.416407i
$268$		0			0
$269$	−14.4303			−0.879830			−0.439915	−	0.898039i	$-0.644991\pi$
$269$	−14.4303			−0.879830			−0.439915	+	0.898039i	$0.644991\pi$
$270$		0			0
$271$	−17.3192			−1.05206			−0.526032	−	0.850465i	$-0.676321\pi$
$271$	−17.3192			−1.05206			−0.526032	+	0.850465i	$0.676321\pi$
$272$		0			0
$273$			8.65024i			0.523536i
$274$		0			0
$275$		−	7.76396i		−	0.468184i
$276$		0			0
$277$		−	5.25294i		−	0.315619i	−0.987470	−	0.157809i	$-0.949557\pi$
$277$		−	5.25294i		−	0.315619i	0.987470	−	0.157809i	$-0.0504431\pi$
$278$		0			0
$279$		−	7.91580i		−	0.473907i
$280$		0			0
$281$			13.2334i			0.789437i	0.918802	+	0.394719i	$0.129158\pi$
$281$			13.2334i			0.789437i	−0.918802	+	0.394719i	$0.870842\pi$
$282$		0			0
$283$	14.3228			0.851403			0.425701	−	0.904864i	$-0.360027\pi$
$283$	14.3228			0.851403			0.425701	+	0.904864i	$0.360027\pi$
$284$		0			0
$285$	12.2208			0.723897
$286$		0			0
$287$			30.0862i			1.77593i
$288$		0			0
$289$	−0.117549			−0.00691466
$290$		0			0
$291$	17.4036			1.02022
$292$		0			0
$293$	10.1488			0.592902			0.296451	−	0.955048i	$-0.404197\pi$
$293$	10.1488			0.592902			0.296451	+	0.955048i	$0.404197\pi$
$294$		0			0
$295$			18.7776i			1.09327i
$296$		0			0
$297$		−	4.52763i		−	0.262720i
$298$		0			0
$299$		−	7.19845i		−	0.416297i
$300$		0			0
$301$	7.53548			0.434338
$302$		0			0
$303$			12.0672i			0.693243i
$304$		0			0
$305$	13.7549	+	3.34664i	0.787604	+	0.191628i
$306$		0			0
$307$		−	18.9992i		−	1.08434i	−0.840269	−	0.542170i	$-0.817603\pi$
$307$		−	18.9992i		−	1.08434i	0.840269	−	0.542170i	$-0.182397\pi$
$308$		0			0
$309$	−13.1329			−0.747106
$310$		0			0
$311$		−	3.51553i		−	0.199348i	−0.995020	−	0.0996738i	$-0.968220\pi$
$311$		−	3.51553i		−	0.199348i	0.995020	−	0.0996738i	$-0.0317799\pi$
$312$		0			0
$313$		−	11.4881i		−	0.649345i	−0.945826	−	0.324673i	$-0.894746\pi$
$313$		−	11.4881i		−	0.649345i	0.945826	−	0.324673i	$-0.105254\pi$
$314$		0			0
$315$			7.67909i			0.432668i
$316$		0			0
$317$	10.5307			0.591464			0.295732	−	0.955271i	$-0.404437\pi$
$317$	10.5307			0.591464			0.295732	+	0.955271i	$0.404437\pi$
$318$		0			0
$319$	21.8152			1.22142
$320$		0			0
$321$	−17.9691			−1.00294
$322$		0			0
$323$			27.8958i			1.55216i
$324$		0			0
$325$	−3.50116			−0.194209
$326$		0			0
$327$	16.7816			0.928023
$328$		0			0
$329$			16.7185i			0.921721i
$330$		0			0
$331$		−	14.8177i		−	0.814456i	−0.913327	−	0.407228i	$-0.866495\pi$
$331$		−	14.8177i		−	0.814456i	0.913327	−	0.407228i	$-0.133505\pi$
$332$		0			0
$333$		−	2.88820i		−	0.158272i
$334$		0			0
$335$			2.06443i			0.112792i
$336$		0			0
$337$		−	27.2208i		−	1.48281i	−0.671056	−	0.741406i	$-0.734159\pi$
$337$		−	27.2208i		−	1.48281i	0.671056	−	0.741406i	$-0.265841\pi$
$338$		0			0
$339$	8.64682			0.469631
$340$		0			0
$341$	35.8398			1.94084
$342$		0			0
$343$			16.7337i			0.903535i
$344$		0			0
$345$		−	6.39030i		−	0.344042i
$346$		0			0
$347$	−33.4564			−1.79604			−0.898018	−	0.439959i	$-0.854993\pi$
$347$	−33.4564			−1.79604			−0.898018	+	0.439959i	$0.854993\pi$
$348$		0			0
$349$			14.7928i			0.791842i	0.918285	+	0.395921i	$0.129575\pi$
$349$			14.7928i			0.791842i	−0.918285	+	0.395921i	$0.870425\pi$
$350$		0			0
$351$	−2.04174			−0.108980
$352$		0			0
$353$	17.6004			0.936773			0.468387	−	0.883524i	$-0.344835\pi$
$353$	17.6004			0.936773			0.468387	+	0.883524i	$0.344835\pi$
$354$		0			0
$355$		−	9.48861i		−	0.503603i
$356$		0			0
$357$	−17.5287			−0.927716
$358$		0			0
$359$			25.9202i			1.36801i	0.729476	+	0.684007i	$0.239764\pi$
$359$			25.9202i			1.36801i	−0.729476	+	0.684007i	$0.760236\pi$
$360$		0			0
$361$	26.4606			1.39266
$362$		0			0
$363$	9.49947			0.498593
$364$		0			0
$365$	−18.1603			−0.950553
$366$		0			0
$367$	21.4544			1.11991			0.559956	−	0.828522i	$-0.310818\pi$
$367$	21.4544			1.11991			0.559956	+	0.828522i	$0.310818\pi$
$368$		0			0
$369$	−7.10131			−0.369679
$370$		0			0
$371$	−21.5969			−1.12125
$372$		0			0
$373$		−	9.33221i		−	0.483203i	−0.970376	−	0.241602i	$-0.922327\pi$
$373$		−	9.33221i		−	0.483203i	0.970376	−	0.241602i	$-0.0776727\pi$
$374$		0			0
$375$	−12.1707			−0.628490
$376$		0			0
$377$		−	9.83756i		−	0.506660i
$378$		0			0
$379$	−8.24656			−0.423597			−0.211799	−	0.977313i	$-0.567932\pi$
$379$	−8.24656			−0.423597			−0.211799	+	0.977313i	$0.567932\pi$
$380$		0			0
$381$	2.37012			0.121425
$382$		0			0
$383$			4.49840i			0.229858i	0.993374	+	0.114929i	$0.0366640\pi$
$383$			4.49840i			0.229858i	−0.993374	+	0.114929i	$0.963336\pi$
$384$		0			0
$385$	−34.7681			−1.77195
$386$		0			0
$387$			1.77862i			0.0904122i
$388$		0			0
$389$		−	23.9619i		−	1.21492i	−0.794352	−	0.607458i	$-0.792189\pi$
$389$		−	23.9619i		−	1.21492i	0.794352	−	0.607458i	$-0.207811\pi$
$390$		0			0
$391$	14.5868			0.737687
$392$		0			0
$393$	0.440404			0.0222155
$394$		0			0
$395$			22.5000i			1.13210i
$396$		0			0
$397$			6.58429i			0.330456i	0.986255	+	0.165228i	$0.0528360\pi$
$397$			6.58429i			0.330456i	−0.986255	+	0.165228i	$0.947164\pi$
$398$		0			0
$399$			28.5658i			1.43008i
$400$		0			0
$401$			9.83908i			0.491340i	0.969353	+	0.245670i	$0.0790080\pi$
$401$			9.83908i			0.491340i	−0.969353	+	0.245670i	$0.920992\pi$
$402$		0			0
$403$		−	16.1620i		−	0.805085i
$404$		0			0
$405$	−1.81251			−0.0900645
$406$		0			0
$407$	13.0767			0.648189
$408$		0			0
$409$			8.75842i			0.433076i	0.976274	+	0.216538i	$0.0694765\pi$
$409$			8.75842i			0.433076i	−0.976274	+	0.216538i	$0.930523\pi$
$410$		0			0
$411$	−7.39773			−0.364903
$412$		0			0
$413$	−43.8921			−2.15979
$414$		0			0
$415$	−10.5282			−0.516809
$416$		0			0
$417$			9.16474i			0.448799i
$418$		0			0
$419$			27.4460i			1.34083i	0.741988	+	0.670413i	$0.233883\pi$
$419$			27.4460i			1.34083i	−0.741988	+	0.670413i	$0.766117\pi$
$420$		0			0
$421$			5.57132i			0.271529i	0.990741	+	0.135765i	$0.0433491\pi$
$421$			5.57132i			0.271529i	−0.990741	+	0.135765i	$0.956651\pi$
$422$		0			0
$423$	−3.94611			−0.191866
$424$		0			0
$425$		−	7.09468i		−	0.344143i
$426$		0			0
$427$	−7.82270	+	32.1518i	−0.378567	+	1.55593i
$428$		0			0
$429$		−	9.24423i		−	0.446316i
$430$		0			0
$431$	−37.2843			−1.79592			−0.897961	−	0.440076i	$-0.854952\pi$
$431$	−37.2843			−1.79592			−0.897961	+	0.440076i	$0.854952\pi$
$432$		0			0
$433$			33.5732i			1.61343i	0.590943	+	0.806713i	$0.298756\pi$
$433$			33.5732i			1.61343i	−0.590943	+	0.806713i	$0.701244\pi$
$434$		0			0
$435$		−	8.73312i		−	0.418721i
$436$		0			0
$437$		−	23.7715i		−	1.13715i
$438$		0			0
$439$	−3.80497			−0.181601			−0.0908006	−	0.995869i	$-0.528943\pi$
$439$	−3.80497			−0.181601			−0.0908006	+	0.995869i	$0.528943\pi$
$440$		0			0
$441$	−10.9497			−0.521414
$442$		0			0
$443$	13.7992			0.655618			0.327809	−	0.944744i	$-0.393690\pi$
$443$	13.7992			0.655618			0.327809	+	0.944744i	$0.393690\pi$
$444$		0			0
$445$			12.3326i			0.584622i
$446$		0			0
$447$	11.1990			0.529694
$448$		0			0
$449$	11.1227			0.524915			0.262458	−	0.964944i	$-0.415467\pi$
$449$	11.1227			0.524915			0.262458	+	0.964944i	$0.415467\pi$
$450$		0			0
$451$		−	32.1521i		−	1.51398i
$452$		0			0
$453$			9.88970i			0.464659i
$454$		0			0
$455$			15.6787i			0.735027i
$456$		0			0
$457$		−	15.4899i		−	0.724588i	−0.932064	−	0.362294i	$-0.881994\pi$
$457$		−	15.4899i		−	0.724588i	0.932064	−	0.362294i	$-0.118006\pi$
$458$		0			0
$459$		−	4.13734i		−	0.193114i
$460$		0			0
$461$	15.9253			0.741715			0.370858	−	0.928690i	$-0.379064\pi$
$461$	15.9253			0.741715			0.370858	+	0.928690i	$0.379064\pi$
$462$		0			0
$463$	−18.9210			−0.879332			−0.439666	−	0.898161i	$-0.644903\pi$
$463$	−18.9210			−0.879332			−0.439666	+	0.898161i	$0.644903\pi$
$464$		0			0
$465$		−	14.3475i		−	0.665349i
$466$		0			0
$467$			16.1497i			0.747320i	0.927566	+	0.373660i	$0.121897\pi$
$467$			16.1497i			0.747320i	−0.927566	+	0.373660i	$0.878103\pi$
$468$		0			0
$469$	−4.82556			−0.222824
$470$		0			0
$471$		−	13.5131i		−	0.622649i
$472$		0			0
$473$	−8.05293			−0.370274
$474$		0			0
$475$	−11.5619			−0.530497
$476$		0			0
$477$		−	5.09757i		−	0.233401i
$478$		0			0
$479$	−28.8605			−1.31867			−0.659335	−	0.751850i	$-0.729162\pi$
$479$	−28.8605			−1.31867			−0.659335	+	0.751850i	$0.729162\pi$
$480$		0			0
$481$		−	5.89694i		−	0.268877i
$482$		0			0
$483$	14.9372			0.679665
$484$		0			0
$485$	31.5443			1.43235
$486$		0			0
$487$	−12.7695			−0.578643			−0.289321	−	0.957232i	$-0.593430\pi$
$487$	−12.7695			−0.578643			−0.289321	+	0.957232i	$0.593430\pi$
$488$		0			0
$489$	−20.3250			−0.919129
$490$		0			0
$491$	29.4941			1.33105			0.665525	−	0.746375i	$-0.268208\pi$
$491$	29.4941			1.33105			0.665525	+	0.746375i	$0.268208\pi$
$492$		0			0
$493$	19.9347			0.897812
$494$		0			0
$495$		−	8.20640i		−	0.368850i
$496$		0			0
$497$	22.1794			0.994882
$498$		0			0
$499$			19.8664i			0.889342i	0.895694	+	0.444671i	$0.146679\pi$
$499$			19.8664i			0.889342i	−0.895694	+	0.444671i	$0.853321\pi$
$500$		0			0
$501$	18.8905			0.843964
$502$		0			0
$503$	28.9167			1.28933			0.644665	−	0.764465i	$-0.276997\pi$
$503$	28.9167			1.28933			0.644665	+	0.764465i	$0.276997\pi$
$504$		0			0
$505$			21.8720i			0.973291i
$506$		0			0
$507$	8.83132			0.392213
$508$		0			0
$509$			27.2976i			1.20995i	0.796246	+	0.604973i	$0.206816\pi$
$509$			27.2976i			1.20995i	−0.796246	+	0.604973i	$0.793184\pi$
$510$		0			0
$511$		−	42.4492i		−	1.87784i
$512$		0			0
$513$	−6.74245			−0.297686
$514$		0			0
$515$	−23.8036			−1.04891
$516$		0			0
$517$		−	17.8665i		−	0.785769i
$518$		0			0
$519$		−	15.7597i		−	0.691775i
$520$		0			0
$521$		−	29.7312i		−	1.30255i	−0.758843	−	0.651273i	$-0.774235\pi$
$521$		−	29.7312i		−	1.30255i	0.758843	−	0.651273i	$-0.225765\pi$
$522$		0			0
$523$			40.0813i			1.75263i	0.481734	+	0.876317i	$0.340007\pi$
$523$			40.0813i			1.75263i	−0.481734	+	0.876317i	$0.659993\pi$
$524$		0			0
$525$		−	7.26508i		−	0.317074i
$526$		0			0
$527$	32.7503			1.42663
$528$		0			0
$529$	10.5698			0.459555
$530$		0			0
$531$		−	10.3600i		−	0.449584i
$532$		0			0
$533$	−14.4990			−0.628021
$534$		0			0
$535$	−32.5692			−1.40809
$536$		0			0
$537$	7.17041			0.309426
$538$		0			0
$539$		−	49.5762i		−	2.13540i
$540$		0			0
$541$			25.0211i			1.07574i	0.843027	+	0.537871i	$0.180771\pi$
$541$			25.0211i			1.07574i	−0.843027	+	0.537871i	$0.819229\pi$
$542$		0			0
$543$		−	9.81675i		−	0.421277i
$544$		0			0
$545$	30.4168			1.30291
$546$		0			0
$547$		−	44.2112i		−	1.89033i	−0.326588	−	0.945167i	$-0.605899\pi$
$547$		−	44.2112i		−	1.89033i	0.326588	−	0.945167i	$-0.394101\pi$
$548$		0			0
$549$	−7.58886	−	1.84641i	−0.323885	−	0.0788028i
$550$		0			0
$551$		−	32.4867i		−	1.38398i
$552$		0			0
$553$	−52.5932			−2.23649
$554$		0			0
$555$		−	5.23490i		−	0.222209i
$556$		0			0
$557$		−	2.76685i		−	0.117235i	−0.998281	−	0.0586177i	$-0.981331\pi$
$557$		−	2.76685i		−	0.117235i	0.998281	−	0.0586177i	$-0.0186693\pi$
$558$		0			0
$559$			3.63147i			0.153595i
$560$		0			0
$561$	18.7323			0.790880
$562$		0			0
$563$	−20.8439			−0.878465			−0.439232	−	0.898374i	$-0.644749\pi$
$563$	−20.8439			−0.878465			−0.439232	+	0.898374i	$0.644749\pi$
$564$		0			0
$565$	15.6725			0.659346
$566$		0			0
$567$		−	4.23671i		−	0.177925i
$568$		0			0
$569$	−42.8616			−1.79685			−0.898425	−	0.439127i	$-0.855288\pi$
$569$	−42.8616			−1.79685			−0.898425	+	0.439127i	$0.855288\pi$
$570$		0			0
$571$	34.0053			1.42308			0.711538	−	0.702647i	$-0.247999\pi$
$571$	34.0053			1.42308			0.711538	+	0.702647i	$0.247999\pi$
$572$		0			0
$573$			16.8420i			0.703584i
$574$		0			0
$575$			6.04577i			0.252126i
$576$		0			0
$577$			24.0173i			0.999853i	0.866068	+	0.499926i	$0.166640\pi$
$577$			24.0173i			0.999853i	−0.866068	+	0.499926i	$0.833360\pi$
$578$		0			0
$579$			22.2191i			0.923394i
$580$		0			0
$581$		−	24.6094i		−	1.02097i
$582$		0			0
$583$	23.0799			0.955872
$584$		0			0
$585$	−3.70067			−0.153004
$586$		0			0
$587$			24.8423i			1.02535i	0.858582	+	0.512676i	$0.171346\pi$
$587$			24.8423i			1.02535i	−0.858582	+	0.512676i	$0.828654\pi$
$588$		0			0
$589$		−	53.3719i		−	2.19915i
$590$		0			0
$591$	−3.28437			−0.135101
$592$		0			0
$593$			9.27503i			0.380880i	0.981699	+	0.190440i	$0.0609914\pi$
$593$			9.27503i			0.380880i	−0.981699	+	0.190440i	$0.939009\pi$
$594$		0			0
$595$	−31.7710			−1.30248
$596$		0			0
$597$	5.40407			0.221174
$598$		0			0
$599$		−	19.2210i		−	0.785351i	−0.919677	−	0.392675i	$-0.871550\pi$
$599$		−	19.2210i		−	0.785351i	0.919677	−	0.392675i	$-0.128450\pi$
$600$		0			0
$601$	−19.7975			−0.807556			−0.403778	−	0.914857i	$-0.632303\pi$
$601$	−19.7975			−0.807556			−0.403778	+	0.914857i	$0.632303\pi$
$602$		0			0
$603$		−	1.13899i		−	0.0463832i
$604$		0			0
$605$	17.2179			0.700008
$606$		0			0
$607$	−22.6670			−0.920024			−0.460012	−	0.887913i	$-0.652155\pi$
$607$	−22.6670			−0.920024			−0.460012	+	0.887913i	$0.652155\pi$
$608$		0			0
$609$	20.4135			0.827195
$610$		0			0
$611$	−8.05691			−0.325948
$612$		0			0
$613$	31.1484			1.25807			0.629037	−	0.777375i	$-0.283449\pi$
$613$	31.1484			1.25807			0.629037	+	0.777375i	$0.283449\pi$
$614$		0			0
$615$	−12.8712			−0.519018
$616$		0			0
$617$		−	20.1656i		−	0.811838i	−0.913909	−	0.405919i	$-0.866951\pi$
$617$		−	20.1656i		−	0.811838i	0.913909	−	0.405919i	$-0.133049\pi$
$618$		0			0
$619$	38.2387			1.53694			0.768472	−	0.639884i	$-0.221018\pi$
$619$	38.2387			1.53694			0.768472	+	0.639884i	$0.221018\pi$
$620$		0			0
$621$			3.52566i			0.141480i
$622$		0			0
$623$	−28.8272			−1.15494
$624$		0			0
$625$	−13.4855			−0.539420
$626$		0			0
$627$		−	30.5273i		−	1.21914i
$628$		0			0
$629$	11.9495			0.476456
$630$		0			0
$631$		−	32.8632i		−	1.30826i	−0.756380	−	0.654132i	$-0.773034\pi$
$631$		−	32.8632i		−	1.30826i	0.756380	−	0.654132i	$-0.226966\pi$
$632$		0			0
$633$		−	5.72067i		−	0.227376i
$634$		0			0
$635$	4.29587			0.170476
$636$		0			0
$637$	−22.3564			−0.885792
$638$		0			0
$639$			5.23506i			0.207096i
$640$		0			0
$641$		−	10.0840i		−	0.398293i	−0.979970	−	0.199147i	$-0.936183\pi$
$641$		−	10.0840i		−	0.398293i	0.979970	−	0.199147i	$-0.0638170\pi$
$642$		0			0
$643$			13.8970i			0.548046i	0.961723	+	0.274023i	$0.0883544\pi$
$643$			13.8970i			0.548046i	−0.961723	+	0.274023i	$0.911646\pi$
$644$		0			0
$645$			3.22377i			0.126936i
$646$		0			0
$647$		−	25.4097i		−	0.998958i	−0.866326	−	0.499479i	$-0.833525\pi$
$647$		−	25.4097i		−	0.998958i	0.866326	−	0.499479i	$-0.166475\pi$
$648$		0			0
$649$	46.9061			1.84123
$650$		0			0
$651$	33.5369			1.31442
$652$		0			0
$653$		−	20.3517i		−	0.796424i	−0.917293	−	0.398212i	$-0.869631\pi$
$653$		−	20.3517i		−	0.796424i	0.917293	−	0.398212i	$-0.130369\pi$
$654$		0			0
$655$	0.798239			0.0311898
$656$		0			0
$657$	10.0194			0.390894
$658$		0			0
$659$	20.6370			0.803902			0.401951	−	0.915661i	$-0.368332\pi$
$659$	20.6370			0.803902			0.401951	+	0.915661i	$0.368332\pi$
$660$		0			0
$661$			39.3646i			1.53110i	0.643374	+	0.765552i	$0.277534\pi$
$661$			39.3646i			1.53110i	−0.643374	+	0.765552i	$0.722466\pi$
$662$		0			0
$663$		−	8.44734i		−	0.328068i
$664$		0			0
$665$			51.7759i			2.00778i
$666$		0			0
$667$	−16.9874			−0.657756
$668$		0			0
$669$		−	21.0105i		−	0.812314i
$670$		0			0
$671$	8.35987	−	34.3596i	0.322729	−	1.32644i
$672$		0			0
$673$			6.52383i			0.251475i	0.992064	+	0.125738i	$0.0401298\pi$
$673$			6.52383i			0.251475i	−0.992064	+	0.125738i	$0.959870\pi$
$674$		0			0
$675$	1.71479			0.0660025
$676$		0			0
$677$			32.9581i			1.26668i	0.773873	+	0.633341i	$0.218317\pi$
$677$			32.9581i			1.26668i	−0.773873	+	0.633341i	$0.781683\pi$
$678$		0			0
$679$			73.7341i			2.82966i
$680$		0			0
$681$			16.8763i			0.646700i
$682$		0			0
$683$	−45.2420			−1.73114			−0.865569	−	0.500790i	$-0.833043\pi$
$683$	−45.2420			−1.73114			−0.865569	+	0.500790i	$0.833043\pi$
$684$		0			0
$685$	−13.4085			−0.512312
$686$		0			0
$687$	−11.2187			−0.428020
$688$		0			0
$689$		−	10.4079i		−	0.396509i
$690$		0			0
$691$	−3.03046			−0.115284			−0.0576420	−	0.998337i	$-0.518358\pi$
$691$	−3.03046			−0.115284			−0.0576420	+	0.998337i	$0.518358\pi$
$692$		0			0
$693$	19.1823			0.728674
$694$		0			0
$695$			16.6112i			0.630099i
$696$		0			0
$697$		−	29.3805i		−	1.11287i
$698$		0			0
$699$		−	6.01278i		−	0.227424i
$700$		0			0
$701$			5.38212i			0.203280i	0.994821	+	0.101640i	$0.0324090\pi$
$701$			5.38212i			0.203280i	−0.994821	+	0.101640i	$0.967591\pi$
$702$		0			0
$703$		−	19.4735i		−	0.734458i
$704$		0			0
$705$	−7.15238			−0.269374
$706$		0			0
$707$	−51.1253			−1.92276
$708$		0			0
$709$			17.6399i			0.662480i	0.943546	+	0.331240i	$0.107467\pi$
$709$			17.6399i			0.662480i	−0.943546	+	0.331240i	$0.892533\pi$
$710$		0			0
$711$		−	12.4137i		−	0.465550i
$712$		0			0
$713$	−27.9084			−1.04518
$714$		0			0
$715$		−	16.7553i		−	0.626613i
$716$		0			0
$717$	6.11143			0.228236
$718$		0			0
$719$	48.8685			1.82249			0.911244	−	0.411868i	$-0.135123\pi$
$719$	48.8685			1.82249			0.911244	+	0.411868i	$0.135123\pi$
$720$		0			0
$721$		−	55.6403i		−	2.07215i
$722$		0			0
$723$	−23.0547			−0.857412
$724$		0			0
$725$			8.26228i			0.306853i
$726$		0			0
$727$	27.5815			1.02294			0.511471	−	0.859301i	$-0.329101\pi$
$727$	27.5815			1.02294			0.511471	+	0.859301i	$0.329101\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−7.35874			−0.272173
$732$		0			0
$733$	9.09642			0.335984			0.167992	−	0.985788i	$-0.446272\pi$
$733$	9.09642			0.335984			0.167992	+	0.985788i	$0.446272\pi$
$734$		0			0
$735$	−19.8465			−0.732048
$736$		0			0
$737$	5.15692			0.189958
$738$		0			0
$739$		−	18.4730i		−	0.679538i	−0.940509	−	0.339769i	$-0.889651\pi$
$739$		−	18.4730i		−	0.679538i	0.940509	−	0.339769i	$-0.110349\pi$
$740$		0			0
$741$	−13.7663			−0.505717
$742$		0			0
$743$		−	44.0004i		−	1.61422i	−0.590402	−	0.807109i	$-0.701031\pi$
$743$		−	44.0004i		−	1.61422i	0.590402	−	0.807109i	$-0.298969\pi$
$744$		0			0
$745$	20.2983			0.743673
$746$		0			0
$747$	5.80862			0.212526
$748$		0			0
$749$		−	76.1298i		−	2.78172i
$750$		0			0
$751$	−54.6020			−1.99245			−0.996227	−	0.0867863i	$-0.972340\pi$
$751$	−54.6020			−1.99245			−0.996227	+	0.0867863i	$0.972340\pi$
$752$		0			0
$753$			17.6245i			0.642271i
$754$		0			0
$755$			17.9252i			0.652365i
$756$		0			0
$757$	−8.66174			−0.314816			−0.157408	−	0.987534i	$-0.550314\pi$
$757$	−8.66174			−0.314816			−0.157408	+	0.987534i	$0.550314\pi$
$758$		0			0
$759$	−15.9629			−0.579416
$760$		0			0
$761$		−	54.3501i		−	1.97019i	−0.172016	−	0.985094i	$-0.555028\pi$
$761$		−	54.3501i		−	1.97019i	0.172016	−	0.985094i	$-0.444972\pi$
$762$		0			0
$763$			71.0986i			2.57394i
$764$		0			0
$765$		−	7.49898i		−	0.271126i
$766$		0			0
$767$		−	21.1523i		−	0.763765i
$768$		0			0
$769$			26.6158i			0.959790i	0.877326	+	0.479895i	$0.159325\pi$
$769$			26.6158i			0.959790i	−0.877326	+	0.479895i	$0.840675\pi$
$770$		0			0
$771$	−13.5412			−0.487676
$772$		0			0
$773$	35.5326			1.27802			0.639009	−	0.769199i	$-0.279345\pi$
$773$	35.5326			1.27802			0.639009	+	0.769199i	$0.279345\pi$
$774$		0			0
$775$			13.5740i			0.487591i
$776$		0			0
$777$	12.2365			0.438981
$778$		0			0
$779$	−47.8802			−1.71549
$780$		0			0
$781$	−23.7024			−0.848139
$782$		0			0
$783$			4.81823i			0.172190i
$784$		0			0
$785$		−	24.4926i		−	0.874179i
$786$		0			0
$787$			24.2867i			0.865728i	0.901459	+	0.432864i	$0.142497\pi$
$787$			24.2867i			0.865728i	−0.901459	+	0.432864i	$0.857503\pi$
$788$		0			0
$789$	−19.7299			−0.702402
$790$		0			0
$791$			36.6341i			1.30256i
$792$		0			0
$793$	−15.4944	−	3.76988i	−0.550224	−	0.133872i
$794$		0			0
$795$		−	9.23941i		−	0.327688i
$796$		0			0
$797$	24.8841			0.881441			0.440720	−	0.897644i	$-0.354723\pi$
$797$	24.8841			0.881441			0.440720	+	0.897644i	$0.354723\pi$
$798$		0			0
$799$		−	16.3264i		−	0.577586i
$800$		0			0
$801$		−	6.80415i		−	0.240413i
$802$		0			0
$803$			45.3642i			1.60087i
$804$		0			0
$805$	27.0738			0.954227
$806$		0			0
$807$	14.4303			0.507970
$808$		0			0
$809$	−6.35029			−0.223264			−0.111632	−	0.993750i	$-0.535608\pi$
$809$	−6.35029			−0.223264			−0.111632	+	0.993750i	$0.535608\pi$
$810$		0			0
$811$		−	4.84985i		−	0.170301i	−0.996368	−	0.0851506i	$-0.972863\pi$
$811$		−	4.84985i		−	0.170301i	0.996368	−	0.0851506i	$-0.0271371\pi$
$812$		0			0
$813$	17.3192			0.607410
$814$		0			0
$815$	−36.8394			−1.29043
$816$		0			0
$817$			11.9922i			0.419555i
$818$		0			0
$819$		−	8.65024i		−	0.302264i
$820$		0			0
$821$		−	20.8192i		−	0.726594i	−0.931673	−	0.363297i	$-0.881651\pi$
$821$		−	20.8192i		−	0.726594i	0.931673	−	0.363297i	$-0.118349\pi$
$822$		0			0
$823$		−	37.5878i		−	1.31023i	−0.755529	−	0.655115i	$-0.772620\pi$
$823$		−	37.5878i		−	1.31023i	0.755529	−	0.655115i	$-0.227380\pi$
$824$		0			0
$825$			7.76396i			0.270306i
$826$		0			0
$827$	−0.873028			−0.0303581			−0.0151791	−	0.999885i	$-0.504832\pi$
$827$	−0.873028			−0.0303581			−0.0151791	+	0.999885i	$0.504832\pi$
$828$		0			0
$829$	42.5423			1.47755			0.738777	−	0.673950i	$-0.235404\pi$
$829$	42.5423			1.47755			0.738777	+	0.673950i	$0.235404\pi$
$830$		0			0
$831$			5.25294i			0.182222i
$832$		0			0
$833$		−	45.3026i		−	1.56964i
$834$		0			0
$835$	34.2393			1.18490
$836$		0			0
$837$			7.91580i			0.273610i
$838$		0			0
$839$	41.9142			1.44704			0.723519	−	0.690305i	$-0.242523\pi$
$839$	41.9142			1.44704			0.723519	+	0.690305i	$0.242523\pi$
$840$		0			0
$841$	5.78462			0.199470
$842$		0			0
$843$		−	13.2334i		−	0.455782i
$844$		0			0
$845$	16.0069			0.550654
$846$		0			0
$847$			40.2465i			1.38289i
$848$		0			0
$849$	−14.3228			−0.491558
$850$		0			0
$851$	−10.1828			−0.349062
$852$		0			0
$853$	−15.9105			−0.544765			−0.272382	−	0.962189i	$-0.587812\pi$
$853$	−15.9105			−0.544765			−0.272382	+	0.962189i	$0.587812\pi$
$854$		0			0
$855$	−12.2208			−0.417942
$856$		0			0
$857$	−29.6408			−1.01251			−0.506256	−	0.862384i	$-0.668971\pi$
$857$	−29.6408			−1.01251			−0.506256	+	0.862384i	$0.668971\pi$
$858$		0			0
$859$	−12.1491			−0.414523			−0.207262	−	0.978286i	$-0.566455\pi$
$859$	−12.1491			−0.414523			−0.207262	+	0.978286i	$0.566455\pi$
$860$		0			0
$861$		−	30.0862i		−	1.02533i
$862$		0			0
$863$	33.3890			1.13658			0.568288	−	0.822830i	$-0.307606\pi$
$863$	33.3890			1.13658			0.568288	+	0.822830i	$0.307606\pi$
$864$		0			0
$865$		−	28.5647i		−	0.971229i
$866$		0			0
$867$	0.117549			0.00399218
$868$		0			0
$869$	56.2047			1.90661
$870$		0			0
$871$		−	2.32551i		−	0.0787970i
$872$		0			0
$873$	−17.4036			−0.589024
$874$		0			0
$875$		−	51.5635i		−	1.74317i
$876$		0			0
$877$			25.5630i			0.863201i	0.902065	+	0.431601i	$0.142051\pi$
$877$			25.5630i			0.863201i	−0.902065	+	0.431601i	$0.857949\pi$
$878$		0			0
$879$	−10.1488			−0.342312
$880$		0			0
$881$	−26.7517			−0.901286			−0.450643	−	0.892704i	$-0.648805\pi$
$881$	−26.7517			−0.901286			−0.450643	+	0.892704i	$0.648805\pi$
$882$		0			0
$883$			22.9828i			0.773433i	0.922199	+	0.386717i	$0.126391\pi$
$883$			22.9828i			0.773433i	−0.922199	+	0.386717i	$0.873609\pi$
$884$		0			0
$885$		−	18.7776i		−	0.631201i
$886$		0			0
$887$			35.4293i			1.18960i	0.803873	+	0.594800i	$0.202769\pi$
$887$			35.4293i			1.18960i	−0.803873	+	0.594800i	$0.797231\pi$
$888$		0			0
$889$			10.0415i			0.336781i
$890$		0			0
$891$			4.52763i			0.151681i
$892$		0			0
$893$	−26.6064			−0.890350
$894$		0			0
$895$	12.9965			0.434424
$896$		0			0
$897$			7.19845i			0.240349i
$898$		0			0
$899$	−38.1402			−1.27205
$900$		0			0
$901$	21.0903			0.702621
$902$		0			0
$903$	−7.53548			−0.250765
$904$		0			0
$905$		−	17.7930i		−	0.591459i
$906$		0			0
$907$			22.5061i			0.747303i	0.927569	+	0.373652i	$0.121894\pi$
$907$			22.5061i			0.747303i	−0.927569	+	0.373652i	$0.878106\pi$
$908$		0			0
$909$		−	12.0672i		−	0.400244i
$910$		0			0
$911$	−48.6779			−1.61277			−0.806385	−	0.591390i	$-0.798579\pi$
$911$	−48.6779			−1.61277			−0.806385	+	0.591390i	$0.798579\pi$
$912$		0			0
$913$			26.2993i			0.870380i
$914$		0			0
$915$	−13.7549	−	3.34664i	−0.454723	−	0.110637i
$916$		0			0
$917$			1.86586i			0.0616163i
$918$		0			0
$919$	35.1684			1.16010			0.580049	−	0.814582i	$-0.303034\pi$
$919$	35.1684			1.16010			0.580049	+	0.814582i	$0.303034\pi$
$920$		0			0
$921$			18.9992i			0.626043i
$922$		0			0
$923$			10.6886i			0.351819i
$924$		0			0
$925$			4.95267i			0.162843i
$926$		0			0
$927$	13.1329			0.431342
$928$		0			0
$929$	−24.4672			−0.802742			−0.401371	−	0.915916i	$-0.631466\pi$
$929$	−24.4672			−0.802742			−0.401371	+	0.915916i	$0.631466\pi$
$930$		0			0
$931$	−73.8278			−2.41961
$932$		0			0
$933$			3.51553i			0.115093i
$934$		0			0
$935$	33.9526			1.11037
$936$		0			0
$937$	57.6008			1.88174			0.940868	−	0.338773i	$-0.110012\pi$
$937$	57.6008			1.88174			0.940868	+	0.338773i	$0.110012\pi$
$938$		0			0
$939$			11.4881i			0.374900i
$940$		0			0
$941$		−	44.2803i		−	1.44350i	−0.692155	−	0.721749i	$-0.743339\pi$
$941$		−	44.2803i		−	1.44350i	0.692155	−	0.721749i	$-0.256661\pi$
$942$		0			0
$943$			25.0368i			0.815309i
$944$		0			0
$945$		−	7.67909i		−	0.249801i
$946$		0			0
$947$		−	39.3019i		−	1.27714i	−0.769564	−	0.638570i	$-0.779526\pi$
$947$		−	39.3019i		−	1.27714i	0.769564	−	0.638570i	$-0.220474\pi$
$948$		0			0
$949$	20.4569			0.664061
$950$		0			0
$951$	−10.5307			−0.341482
$952$		0			0
$953$			26.1922i			0.848450i	0.905557	+	0.424225i	$0.139453\pi$
$953$			26.1922i			0.848450i	−0.905557	+	0.424225i	$0.860547\pi$
$954$		0			0
$955$			30.5263i			0.987809i
$956$		0			0
$957$	−21.8152			−0.705185
$958$		0			0
$959$		−	31.3420i		−	1.01209i
$960$		0			0
$961$	−31.6599			−1.02129
$962$		0			0
$963$	17.9691			0.579046
$964$		0			0
$965$			40.2724i			1.29641i
$966$		0			0
$967$	−8.03129			−0.258269			−0.129134	−	0.991627i	$-0.541220\pi$
$967$	−8.03129			−0.258269			−0.129134	+	0.991627i	$0.541220\pi$
$968$		0			0
$969$		−	27.8958i		−	0.896141i
$970$		0			0
$971$	−50.1464			−1.60927			−0.804637	−	0.593767i	$-0.797640\pi$
$971$	−50.1464			−1.60927			−0.804637	+	0.593767i	$0.797640\pi$
$972$		0			0
$973$	−38.8283			−1.24478
$974$		0			0
$975$	3.50116			0.112127
$976$		0			0
$977$	7.12753			0.228030			0.114015	−	0.993479i	$-0.463629\pi$
$977$	7.12753			0.228030			0.114015	+	0.993479i	$0.463629\pi$
$978$		0			0
$979$	30.8067			0.984587
$980$		0			0
$981$	−16.7816			−0.535794
$982$		0			0
$983$			17.6969i			0.564442i	0.959349	+	0.282221i	$0.0910711\pi$
$983$			17.6969i			0.564442i	−0.959349	+	0.282221i	$0.908929\pi$
$984$		0			0
$985$	−5.95297			−0.189677
$986$		0			0
$987$		−	16.7185i		−	0.532156i
$988$		0			0
$989$	6.27079			0.199400
$990$		0			0
$991$	−19.5719			−0.621721			−0.310861	−	0.950455i	$-0.600617\pi$
$991$	−19.5719			−0.621721			−0.310861	+	0.950455i	$0.600617\pi$
$992$		0			0
$993$			14.8177i			0.470226i
$994$		0			0
$995$	9.79494			0.310521
$996$		0			0
$997$			46.9225i			1.48605i	0.669264	+	0.743025i	$0.266610\pi$
$997$			46.9225i			1.48605i	−0.669264	+	0.743025i	$0.733390\pi$
$998$		0			0
$999$			2.88820i			0.0913786i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1464.2.l.b.121.3	✓	16
3.2	odd	2		4392.2.l.b.1585.13		16
4.3	odd	2		2928.2.l.j.1585.4		16
61.60	even	2	inner	1464.2.l.b.121.4	yes	16
183.182	odd	2		4392.2.l.b.1585.14		16
244.243	odd	2		2928.2.l.j.1585.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1464.2.l.b.121.3	✓	16	1.1	even	1	trivial
1464.2.l.b.121.4	yes	16	61.60	even	2	inner
2928.2.l.j.1585.3		16	244.243	odd	2
2928.2.l.j.1585.4		16	4.3	odd	2
4392.2.l.b.1585.13		16	3.2	odd	2
4392.2.l.b.1585.14		16	183.182	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:	\( N \)	\(=\)	\( 1464 = 2^{3} \cdot 3 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1464.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.81251 q^{5} -4.23671i q^{7} +1.00000 q^{9} +4.52763i q^{11} +2.04174 q^{13} +1.81251 q^{15} +4.13734i q^{17} +6.74245 q^{19} +4.23671i q^{21} -3.52566i q^{23} -1.71479 q^{25} -1.00000 q^{27} -4.81823i q^{29} -7.91580i q^{31} -4.52763i q^{33} +7.67909i q^{35} -2.88820i q^{37} -2.04174 q^{39} -7.10131 q^{41} +1.77862i q^{43} -1.81251 q^{45} -3.94611 q^{47} -10.9497 q^{49} -4.13734i q^{51} -5.09757i q^{53} -8.20640i q^{55} -6.74245 q^{57} -10.3600i q^{59} +(-7.58886 - 1.84641i) q^{61} -4.23671i q^{63} -3.70067 q^{65} -1.13899i q^{67} +3.52566i q^{69} +5.23506i q^{71} +10.0194 q^{73} +1.71479 q^{75} +19.1823 q^{77} -12.4137i q^{79} +1.00000 q^{81} +5.80862 q^{83} -7.49898i q^{85} +4.81823i q^{87} -6.80415i q^{89} -8.65024i q^{91} +7.91580i q^{93} -12.2208 q^{95} -17.4036 q^{97} +4.52763i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 4 q^{5} + 16 q^{9} - 6 q^{13} - 4 q^{15} - 4 q^{19} + 8 q^{25} - 16 q^{27} + 6 q^{39} - 18 q^{41} + 4 q^{45} - 8 q^{47} - 22 q^{49} + 4 q^{57} + 26 q^{61} + 10 q^{65} + 36 q^{73} - 8 q^{75}+ \cdots - 10 q^{97}+O(q^{100}) \) 16 * q - 16 * q^3 + 4 * q^5 + 16 * q^9 - 6 * q^13 - 4 * q^15 - 4 * q^19 + 8 * q^25 - 16 * q^27 + 6 * q^39 - 18 * q^41 + 4 * q^45 - 8 * q^47 - 22 * q^49 + 4 * q^57 + 26 * q^61 + 10 * q^65 + 36 * q^73 - 8 * q^75 + 46 * q^77 + 16 * q^81 + 14 * q^83 - 44 * q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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