LMFDB - Embedded newform 552.2.m.b.137.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(137,552)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("552.137");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.m (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.40774219157$
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} - 2x^{14} - 2x^{12} + 8x^{10} - 8x^{8} + 32x^{6} - 32x^{4} - 128x^{2} + 256 $ x^16 - 2x^14 - 2x^12 + 8x^10 - 8x^8 + 32x^6 - 32x^4 - 128*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			137.7
Root			$0.883519 - 1.10426i$ of defining polynomial
Character	$\chi$	$=$	552.137
Dual form			552.2.m.b.137.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.356193 + 1.69503i) q^{3} -0.441484 q^{5} -3.97556i q^{7} +(-2.74625 - 1.20752i) q^{9} +O(q^{10})$	\(q+(-0.356193 + 1.69503i) q^{3} -0.441484 q^{5} -3.97556i q^{7} +(-2.74625 - 1.20752i) q^{9} +4.64330 q^{11} +3.73736 q^{13} +(0.157254 - 0.748329i) q^{15} +0.441484 q^{17} -2.47890i q^{19} +(6.73870 + 1.41607i) q^{21} +(4.64330 - 1.19990i) q^{23} -4.80509 q^{25} +(3.02497 - 4.22487i) q^{27} +6.41503i q^{29} +4.71239 q^{31} +(-1.65391 + 7.87053i) q^{33} +1.75515i q^{35} +7.33743i q^{37} +(-1.33122 + 6.33493i) q^{39} -0.365088i q^{41} -7.69067i q^{43} +(1.21243 + 0.533099i) q^{45} +2.36509i q^{47} -8.80509 q^{49} +(-0.157254 + 0.748329i) q^{51} +8.70712 q^{53} -2.04994 q^{55} +(4.20182 + 0.882968i) q^{57} +2.20489i q^{59} -13.1629i q^{61} +(-4.80055 + 10.9179i) q^{63} -1.64998 q^{65} -8.93337i q^{67} +(0.379955 + 8.29793i) q^{69} -8.22012i q^{71} +10.9173 q^{73} +(1.71154 - 8.14477i) q^{75} -18.4597i q^{77} +2.02639i q^{79} +(6.08381 + 6.63229i) q^{81} -11.2589 q^{83} -0.194908 q^{85} +(-10.8737 - 2.28499i) q^{87} -17.3647 q^{89} -14.8581i q^{91} +(-1.67852 + 7.98764i) q^{93} +1.09440i q^{95} +5.82545i q^{97} +(-12.7517 - 5.60686i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 16 * q + 4 * q^3 + 4 * q^9	$ 16 q + 4 q^{3} + 4 q^{9} + 8 q^{13} + 32 q^{25} + 16 q^{27} + 56 q^{31} - 44 q^{39} - 32 q^{49} + 32 q^{55} + 4 q^{69} + 40 q^{73} - 20 q^{75} + 4 q^{81} - 112 q^{85} - 36 q^{87} - 36 q^{93}+O(q^{100}) $ 16 * q + 4 * q^3 + 4 * q^9 + 8 * q^13 + 32 * q^25 + 16 * q^27 + 56 * q^31 - 44 * q^39 - 32 * q^49 + 32 * q^55 + 4 * q^69 + 40 * q^73 - 20 * q^75 + 4 * q^81 - 112 * q^85 - 36 * q^87 - 36 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.356193	+	1.69503i	−0.205648	+	0.978626i
$3$	−0.356193	+	1.69503i	−0.205648	+	0.978626i
$4$		0			0
$5$	−0.441484			−0.197438			−0.0987189	−	0.995115i	$-0.531474\pi$
$5$	−0.441484			−0.197438			−0.0987189	+	0.995115i	$0.531474\pi$
$6$		0			0
$7$		−	3.97556i		−	1.50262i	−0.659949	−	0.751311i	$-0.729422\pi$
$7$		−	3.97556i		−	1.50262i	0.659949	−	0.751311i	$-0.270578\pi$
$8$		0			0
$9$	−2.74625	−	1.20752i	−0.915418	−	0.402505i
$10$		0			0
$11$	4.64330			1.40001			0.700004	−	0.714139i	$-0.253182\pi$
$11$	4.64330			1.40001			0.700004	+	0.714139i	$0.253182\pi$
$12$		0			0
$13$	3.73736			1.03656			0.518278	−	0.855212i	$-0.326573\pi$
$13$	3.73736			1.03656			0.518278	+	0.855212i	$0.326573\pi$
$14$		0			0
$15$	0.157254	−	0.748329i	0.0406027	−	0.193218i
$16$		0			0
$17$	0.441484			0.107076			0.0535378	−	0.998566i	$-0.482950\pi$
$17$	0.441484			0.107076			0.0535378	+	0.998566i	$0.482950\pi$
$18$		0			0
$19$		−	2.47890i		−	0.568700i	−0.958721	−	0.284350i	$-0.908222\pi$
$19$		−	2.47890i		−	0.568700i	0.958721	−	0.284350i	$-0.0917777\pi$
$20$		0			0
$21$	6.73870	+	1.41607i	1.47050	+	0.309011i
$22$		0			0
$23$	4.64330	−	1.19990i	0.968195	−	0.250196i
$23$	4.64330	−	1.19990i	0.968195	−	0.250196i
$24$		0			0
$25$	−4.80509			−0.961018
$26$		0			0
$27$	3.02497	−	4.22487i	0.582156	−	0.813077i
$28$		0			0
$29$			6.41503i			1.19124i	0.803266	+	0.595621i	$0.203094\pi$
$29$			6.41503i			1.19124i	−0.803266	+	0.595621i	$0.796906\pi$
$30$		0			0
$31$	4.71239			0.846370			0.423185	−	0.906043i	$-0.360912\pi$
$31$	4.71239			0.846370			0.423185	+	0.906043i	$0.360912\pi$
$32$		0			0
$33$	−1.65391	+	7.87053i	−0.287909	+	1.37008i
$34$		0			0
$35$			1.75515i			0.296674i
$36$		0			0
$37$			7.33743i			1.20627i	0.797640	+	0.603133i	$0.206081\pi$
$37$			7.33743i			1.20627i	−0.797640	+	0.603133i	$0.793919\pi$
$38$		0			0
$39$	−1.33122	+	6.33493i	−0.213166	+	1.01440i
$40$		0			0
$41$		−	0.365088i		−	0.0570172i	−0.999594	−	0.0285086i	$-0.990924\pi$
$41$		−	0.365088i		−	0.0570172i	0.999594	−	0.0285086i	$-0.00907579\pi$
$42$		0			0
$43$		−	7.69067i		−	1.17282i	−0.810016	−	0.586408i	$-0.800542\pi$
$43$		−	7.69067i		−	1.17282i	0.810016	−	0.586408i	$-0.199458\pi$
$44$		0			0
$45$	1.21243	+	0.533099i	0.180738	+	0.0794697i
$46$		0			0
$47$			2.36509i			0.344984i	0.985011	+	0.172492i	$0.0551818\pi$
$47$			2.36509i			0.344984i	−0.985011	+	0.172492i	$0.944818\pi$
$48$		0			0
$49$	−8.80509			−1.25787
$50$		0			0
$51$	−0.157254	+	0.748329i	−0.0220199	+	0.104787i
$52$		0			0
$53$	8.70712			1.19601			0.598007	−	0.801491i	$-0.295959\pi$
$53$	8.70712			1.19601			0.598007	+	0.801491i	$0.295959\pi$
$54$		0			0
$55$	−2.04994			−0.276414
$56$		0			0
$57$	4.20182	+	0.882968i	0.556544	+	0.116952i
$58$		0			0
$59$			2.20489i			0.287053i	0.989647	+	0.143526i	$0.0458441\pi$
$59$			2.20489i			0.287053i	−0.989647	+	0.143526i	$0.954156\pi$
$60$		0			0
$61$		−	13.1629i		−	1.68533i	−0.538434	−	0.842667i	$-0.680984\pi$
$61$		−	13.1629i		−	1.68533i	0.538434	−	0.842667i	$-0.319016\pi$
$62$		0			0
$63$	−4.80055	+	10.9179i	−0.604813	+	1.37553i
$64$		0			0
$65$	−1.64998			−0.204655
$66$		0			0
$67$		−	8.93337i		−	1.09138i	−0.837986	−	0.545692i	$-0.816267\pi$
$67$		−	8.93337i		−	1.09138i	0.837986	−	0.545692i	$-0.183733\pi$
$68$		0			0
$69$	0.379955	+	8.29793i	0.0457412	+	0.998953i
$70$		0			0
$71$		−	8.22012i		−	0.975549i	−0.872970	−	0.487775i	$-0.837809\pi$
$71$		−	8.22012i		−	0.975549i	0.872970	−	0.487775i	$-0.162191\pi$
$72$		0			0
$73$	10.9173			1.27777			0.638885	−	0.769302i	$-0.279396\pi$
$73$	10.9173			1.27777			0.638885	+	0.769302i	$0.279396\pi$
$74$		0			0
$75$	1.71154	−	8.14477i	0.197632	−	0.940478i
$76$		0			0
$77$		−	18.4597i		−	2.10368i
$78$		0			0
$79$			2.02639i			0.227987i	0.993481	+	0.113994i	$0.0363643\pi$
$79$			2.02639i			0.227987i	−0.993481	+	0.113994i	$0.963636\pi$
$80$		0			0
$81$	6.08381	+	6.63229i	0.675979	+	0.736921i
$82$		0			0
$83$	−11.2589			−1.23583			−0.617915	−	0.786245i	$-0.712022\pi$
$83$	−11.2589			−1.23583			−0.617915	+	0.786245i	$0.712022\pi$
$84$		0			0
$85$	−0.194908			−0.0211408
$86$		0			0
$87$	−10.8737	−	2.28499i	−1.16578	−	0.244977i
$88$		0			0
$89$	−17.3647			−1.84065			−0.920327	−	0.391149i	$-0.872078\pi$
$89$	−17.3647			−1.84065			−0.920327	+	0.391149i	$0.872078\pi$
$90$		0			0
$91$		−	14.8581i		−	1.55755i
$92$		0			0
$93$	−1.67852	+	7.98764i	−0.174054	+	0.828279i
$94$		0			0
$95$			1.09440i			0.112283i
$96$		0			0
$97$			5.82545i			0.591485i	0.955268	+	0.295742i	$0.0955670\pi$
$97$			5.82545i			0.591485i	−0.955268	+	0.295742i	$0.904433\pi$
$98$		0			0
$99$	−12.7517	−	5.60686i	−1.28159	−	0.563511i
$100$		0			0
$101$			6.04994i			0.601992i	0.953625	+	0.300996i	$0.0973191\pi$
$101$			6.04994i			0.601992i	−0.953625	+	0.300996i	$0.902681\pi$
$102$		0			0
$103$			12.3792i			1.21976i	0.792494	+	0.609879i	$0.208782\pi$
$103$			12.3792i			1.21976i	−0.792494	+	0.609879i	$0.791218\pi$
$104$		0			0
$105$	−2.97503	−	0.625172i	−0.290333	−	0.0610105i
$106$		0			0
$107$	−4.32879			−0.418480			−0.209240	−	0.977864i	$-0.567099\pi$
$107$	−4.32879			−0.418480			−0.209240	+	0.977864i	$0.567099\pi$
$108$		0			0
$109$		−	7.17625i		−	0.687360i	−0.939087	−	0.343680i	$-0.888326\pi$
$109$		−	7.17625i		−	0.687360i	0.939087	−	0.343680i	$-0.111674\pi$
$110$		0			0
$111$	−12.4372	−	2.61354i	−1.18048	−	0.248067i
$112$		0			0
$113$	−5.39929			−0.507923			−0.253961	−	0.967214i	$-0.581734\pi$
$113$	−5.39929			−0.507923			−0.253961	+	0.967214i	$0.581734\pi$
$114$		0			0
$115$	−2.04994	+	0.529737i	−0.191158	+	0.0493982i
$116$		0			0
$117$	−10.2637	−	4.51292i	−0.948882	−	0.417220i
$118$		0			0
$119$		−	1.75515i		−	0.160894i
$120$		0			0
$121$	10.5602			0.960022
$122$		0			0
$123$	0.618835	+	0.130042i	0.0557985	+	0.0117255i
$124$		0			0
$125$	4.32879			0.387179
$126$		0			0
$127$	13.7473			1.21988			0.609940	−	0.792448i	$-0.291194\pi$
$127$	13.7473			1.21988			0.609940	+	0.792448i	$0.291194\pi$
$128$		0			0
$129$	13.0359	+	2.73936i	1.14775	+	0.241187i
$130$		0			0
$131$			19.7998i			1.72992i	0.501840	+	0.864960i	$0.332656\pi$
$131$			19.7998i			1.72992i	−0.501840	+	0.864960i	$0.667344\pi$
$132$		0			0
$133$	−9.85504			−0.854540
$134$		0			0
$135$	−1.33548	+	1.86521i	−0.114940	+	0.160532i
$136$		0			0
$137$	−12.1529			−1.03830			−0.519148	−	0.854684i	$-0.673751\pi$
$137$	−12.1529			−1.03830			−0.519148	+	0.854684i	$0.673751\pi$
$138$		0			0
$139$	21.8828			1.85608			0.928038	−	0.372486i	$-0.121495\pi$
$139$	21.8828			1.85608			0.928038	+	0.372486i	$0.121495\pi$
$140$		0			0
$141$	−4.00889	−	0.842428i	−0.337610	−	0.0709452i
$142$		0			0
$143$	17.3537			1.45119
$144$		0			0
$145$		−	2.83214i		−	0.235196i
$146$		0			0
$147$	3.13631	−	14.9249i	0.258679	−	1.23098i
$148$		0			0
$149$	−17.1107			−1.40177			−0.700884	−	0.713276i	$-0.747211\pi$
$149$	−17.1107			−1.40177			−0.700884	+	0.713276i	$0.747211\pi$
$150$		0			0
$151$	−18.2726			−1.48701			−0.743503	−	0.668733i	$-0.766837\pi$
$151$	−18.2726			−1.48701			−0.743503	+	0.668733i	$0.766837\pi$
$152$		0			0
$153$	−1.21243	−	0.533099i	−0.0980189	−	0.0430985i
$154$		0			0
$155$	−2.08044			−0.167105
$156$		0			0
$157$			10.1696i			0.811620i	0.913957	+	0.405810i	$0.133011\pi$
$157$			10.1696i			0.811620i	−0.913957	+	0.405810i	$0.866989\pi$
$158$		0			0
$159$	−3.10141	+	14.7588i	−0.245958	+	1.17045i
$160$		0			0
$161$	−4.77028	−	18.4597i	−0.375951	−	1.45483i
$162$		0			0
$163$	−13.4730			−1.05529			−0.527644	−	0.849465i	$-0.676925\pi$
$163$	−13.4730			−1.05529			−0.527644	+	0.849465i	$0.676925\pi$
$164$		0			0
$165$	0.730176	−	3.47472i	0.0568441	−	0.270506i
$166$		0			0
$167$			5.60020i			0.433356i	0.976243	+	0.216678i	$0.0695222\pi$
$167$			5.60020i			0.433356i	−0.976243	+	0.216678i	$0.930478\pi$
$168$		0			0
$169$	0.967846			0.0744497
$170$		0			0
$171$	−2.99332	+	6.80770i	−0.228905	+	0.520598i
$172$		0			0
$173$			1.56024i			0.118623i	0.998240	+	0.0593114i	$0.0188905\pi$
$173$			1.56024i			0.118623i	−0.998240	+	0.0593114i	$0.981110\pi$
$174$		0			0
$175$			19.1029i			1.44405i
$176$		0			0
$177$	−3.73736	−	0.785367i	−0.280917	−	0.0590318i
$178$		0			0
$179$		−	15.1952i		−	1.13574i	−0.823119	−	0.567869i	$-0.807768\pi$
$179$		−	15.1952i		−	1.13574i	0.823119	−	0.567869i	$-0.192232\pi$
$180$		0			0
$181$		−	10.3307i		−	0.767879i	−0.923358	−	0.383939i	$-0.874567\pi$
$181$		−	10.3307i		−	0.767879i	0.923358	−	0.383939i	$-0.125433\pi$
$182$		0			0
$183$	22.3115	+	4.68853i	1.64931	+	0.346586i
$184$		0			0
$185$		−	3.23936i		−	0.238163i
$186$		0			0
$187$	2.04994			0.149907
$188$		0			0
$189$	−16.7962	−	12.0260i	−1.22175	−	0.874760i
$190$		0			0
$191$	−2.08044			−0.150536			−0.0752678	−	0.997163i	$-0.523981\pi$
$191$	−2.08044			−0.150536			−0.0752678	+	0.997163i	$0.523981\pi$
$192$		0			0
$193$	−3.77294			−0.271582			−0.135791	−	0.990738i	$-0.543358\pi$
$193$	−3.77294			−0.271582			−0.135791	+	0.990738i	$0.543358\pi$
$194$		0			0
$195$	0.587713	−	2.79677i	0.0420870	−	0.200281i
$196$		0			0
$197$			22.8942i			1.63115i	0.578653	+	0.815574i	$0.303578\pi$
$197$			22.8942i			1.63115i	−0.578653	+	0.815574i	$0.696422\pi$
$198$		0			0
$199$			10.1607i			0.720276i	0.932899	+	0.360138i	$0.117270\pi$
$199$			10.1607i			0.720276i	−0.932899	+	0.360138i	$0.882730\pi$
$200$		0			0
$201$	15.1423	+	3.18200i	1.06806	+	0.224441i
$202$		0			0
$203$	25.5034			1.78998
$204$		0			0
$205$			0.161181i			0.0112573i
$206$		0			0
$207$	−14.2006	−	2.31163i	−0.987008	−	0.160669i
$208$		0			0
$209$		−	11.5103i		−	0.796184i
$210$		0			0
$211$	−17.3093			−1.19162			−0.595810	−	0.803126i	$-0.703169\pi$
$211$	−17.3093			−1.19162			−0.595810	+	0.803126i	$0.703169\pi$
$212$		0			0
$213$	13.9334	+	2.92795i	0.954698	+	0.200620i
$214$		0			0
$215$			3.39531i			0.231558i
$216$		0			0
$217$		−	18.7344i		−	1.27177i
$218$		0			0
$219$	−3.88866	+	18.5051i	−0.262771	+	1.25046i
$220$		0			0
$221$	1.64998			0.110990
$222$		0			0
$223$	−17.0705			−1.14313			−0.571564	−	0.820558i	$-0.693663\pi$
$223$	−17.0705			−1.14313			−0.571564	+	0.820558i	$0.693663\pi$
$224$		0			0
$225$	13.1960	+	5.80222i	0.879733	+	0.386815i
$226$		0			0
$227$	24.9824			1.65814			0.829071	−	0.559143i	$-0.188870\pi$
$227$	24.9824			1.65814			0.829071	+	0.559143i	$0.188870\pi$
$228$		0			0
$229$			11.9355i			0.788720i	0.918956	+	0.394360i	$0.129034\pi$
$229$			11.9355i			0.788720i	−0.918956	+	0.394360i	$0.870966\pi$
$230$		0			0
$231$	31.2898	+	6.57523i	2.05872	+	0.432618i
$232$		0			0
$233$			15.6549i			1.02559i	0.858513	+	0.512793i	$0.171389\pi$
$233$			15.6549i			1.02559i	−0.858513	+	0.512793i	$0.828611\pi$
$234$		0			0
$235$		−	1.04415i		−	0.0681128i
$236$		0			0
$237$	−3.43480	−	0.721788i	−0.223114	−	0.0468852i
$238$		0			0
$239$		−	13.7804i		−	0.891378i	−0.895188	−	0.445689i	$-0.852959\pi$
$239$		−	13.7804i		−	0.891378i	0.895188	−	0.445689i	$-0.147041\pi$
$240$		0			0
$241$			1.06620i			0.0686799i	0.999410	+	0.0343399i	$0.0109329\pi$
$241$			1.06620i			0.0686799i	−0.999410	+	0.0343399i	$0.989067\pi$
$242$		0			0
$243$	−13.4089	+	7.94987i	−0.860184	+	0.509984i
$244$		0			0
$245$	3.88731			0.248351
$246$		0			0
$247$		−	9.26455i		−	0.589489i
$248$		0			0
$249$	4.01036	−	19.0843i	0.254146	−	1.20942i
$250$		0			0
$251$	−9.49301			−0.599193			−0.299597	−	0.954066i	$-0.596852\pi$
$251$	−9.49301			−0.599193			−0.299597	+	0.954066i	$0.596852\pi$
$252$		0			0
$253$	21.5602	−	5.57150i	1.35548	−	0.350277i
$254$		0			0
$255$	0.0694250	−	0.330375i	0.00434756	−	0.0206889i
$256$		0			0
$257$			5.51554i			0.344050i	0.985093	+	0.172025i	$0.0550310\pi$
$257$			5.51554i			0.344050i	−0.985093	+	0.172025i	$0.944969\pi$
$258$		0			0
$259$	29.1704			1.81256
$260$		0			0
$261$	7.74625	−	17.6173i	0.479481	−	1.09048i
$262$		0			0
$263$	16.5313			1.01936			0.509681	−	0.860364i	$-0.329764\pi$
$263$	16.5313			1.01936			0.509681	+	0.860364i	$0.329764\pi$
$264$		0			0
$265$	−3.84405			−0.236138
$266$		0			0
$267$	6.18519	−	29.4337i	0.378527	−	1.80131i
$268$		0			0
$269$			3.11474i			0.189909i	0.995482	+	0.0949547i	$0.0302706\pi$
$269$			3.11474i			0.189909i	−0.995482	+	0.0949547i	$0.969729\pi$
$270$		0			0
$271$	−11.1205			−0.675521			−0.337760	−	0.941232i	$-0.609669\pi$
$271$	−11.1205			−0.675521			−0.337760	+	0.941232i	$0.609669\pi$
$272$		0			0
$273$	25.1849	+	5.29235i	1.52426	+	0.320308i
$274$		0			0
$275$	−22.3115			−1.34543
$276$		0			0
$277$	16.1676			0.971418			0.485709	−	0.874121i	$-0.338561\pi$
$277$	16.1676			0.971418			0.485709	+	0.874121i	$0.338561\pi$
$278$		0			0
$279$	−12.9414	−	5.69028i	−0.774782	−	0.340668i
$280$		0			0
$281$	−0.695438			−0.0414864			−0.0207432	−	0.999785i	$-0.506603\pi$
$281$	−0.695438			−0.0414864			−0.0207432	+	0.999785i	$0.506603\pi$
$282$		0			0
$283$			24.3147i			1.44536i	0.691183	+	0.722679i	$0.257090\pi$
$283$			24.3147i			1.44536i	−0.691183	+	0.722679i	$0.742910\pi$
$284$		0			0
$285$	−1.85504	−	0.389817i	−0.109883	−	0.0230907i
$286$		0			0
$287$	−1.45143			−0.0856752
$288$		0			0
$289$	−16.8051			−0.988535
$290$		0			0
$291$	−9.87431	−	2.07499i	−0.578843	−	0.121638i
$292$		0			0
$293$	14.5479			0.849897			0.424948	−	0.905218i	$-0.360292\pi$
$293$	14.5479			0.849897			0.424948	+	0.905218i	$0.360292\pi$
$294$		0			0
$295$		−	0.973425i		−	0.0566750i
$296$		0			0
$297$	14.0459	−	19.6174i	0.815023	−	1.13831i
$298$		0			0
$299$	17.3537	−	4.48446i	1.00359	−	0.259343i
$300$		0			0
$301$	−30.5747			−1.76230
$302$		0			0
$303$	−10.2548	−	2.15495i	−0.589125	−	0.123799i
$304$		0			0
$305$			5.81121i			0.332749i
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$	−20.9831	−	4.40938i	−1.19369	−	0.250841i
$310$		0			0
$311$			15.9253i			0.903042i	0.892260	+	0.451521i	$0.149118\pi$
$311$			15.9253i			0.903042i	−0.892260	+	0.451521i	$0.850882\pi$
$312$		0			0
$313$		−	28.8112i		−	1.62850i	−0.580511	−	0.814252i	$-0.697147\pi$
$313$		−	28.8112i		−	1.62850i	0.580511	−	0.814252i	$-0.302853\pi$
$314$		0			0
$315$	2.11937	−	4.82008i	0.119413	−	0.271581i
$316$		0			0
$317$		−	29.6601i		−	1.66588i	−0.553364	−	0.832939i	$-0.686656\pi$
$317$		−	29.6601i		−	1.66588i	0.553364	−	0.832939i	$-0.313344\pi$
$318$		0			0
$319$			29.7869i			1.66775i
$320$		0			0
$321$	1.54189	−	7.33743i	0.0860597	−	0.409536i
$322$		0			0
$323$		−	1.09440i		−	0.0608939i
$324$		0			0
$325$	−17.9583			−0.996150
$326$		0			0
$327$	12.1640	+	2.55613i	0.672669	+	0.141354i
$328$		0			0
$329$	9.40255			0.518380
$330$		0			0
$331$	12.8479			0.706182			0.353091	−	0.935589i	$-0.385131\pi$
$331$	12.8479			0.706182			0.353091	+	0.935589i	$0.385131\pi$
$332$		0			0
$333$	8.86007	−	20.1505i	0.485529	−	1.10424i
$334$		0			0
$335$			3.94394i			0.215481i
$336$		0			0
$337$			18.5732i			1.01175i	0.862608	+	0.505873i	$0.168830\pi$
$337$			18.5732i			1.01175i	−0.862608	+	0.505873i	$0.831170\pi$
$338$		0			0
$339$	1.92319	−	9.15196i	0.104453	−	0.497066i
$340$		0			0
$341$	21.8810			1.18492
$342$		0			0
$343$			7.17625i			0.387481i
$344$		0			0
$345$	−0.167744	−	3.66340i	−0.00903105	−	0.197231i
$346$		0			0
$347$			21.0544i			1.13026i	0.825001	+	0.565131i	$0.191174\pi$
$347$			21.0544i			1.13026i	−0.825001	+	0.565131i	$0.808826\pi$
$348$		0			0
$349$	17.7074			0.947854			0.473927	−	0.880564i	$-0.342836\pi$
$349$	17.7074			0.947854			0.473927	+	0.880564i	$0.342836\pi$
$350$		0			0
$351$	11.3054	−	15.7899i	0.603438	−	0.842800i
$352$		0			0
$353$			11.6654i			0.620885i	0.950592	+	0.310443i	$0.100477\pi$
$353$			11.6654i			0.620885i	−0.950592	+	0.310443i	$0.899523\pi$
$354$		0			0
$355$			3.62905i			0.192610i
$356$		0			0
$357$	2.97503	+	0.625172i	0.157455	+	0.0330876i
$358$		0			0
$359$	−2.67095			−0.140968			−0.0704838	−	0.997513i	$-0.522454\pi$
$359$	−2.67095			−0.140968			−0.0704838	+	0.997513i	$0.522454\pi$
$360$		0			0
$361$	12.8550			0.676581
$362$		0			0
$363$	−3.76148	+	17.8999i	−0.197427	+	0.939502i
$364$		0			0
$365$	−4.81981			−0.252280
$366$		0			0
$367$			27.0541i			1.41221i	0.708107	+	0.706105i	$0.249549\pi$
$367$			27.0541i			1.41221i	−0.708107	+	0.706105i	$0.750451\pi$
$368$		0			0
$369$	−0.440849	+	1.00262i	−0.0229497	+	0.0521945i
$370$		0			0
$371$		−	34.6157i		−	1.79716i
$372$		0			0
$373$			16.7145i			0.865444i	0.901528	+	0.432722i	$0.142447\pi$
$373$			16.7145i			0.865444i	−0.901528	+	0.432722i	$0.857553\pi$
$374$		0			0
$375$	−1.54189	+	7.33743i	−0.0796227	+	0.378903i
$376$		0			0
$377$			23.9753i			1.23479i
$378$		0			0
$379$		−	26.2639i		−	1.34908i	−0.738236	−	0.674542i	$-0.764341\pi$
$379$		−	26.2639i		−	1.34908i	0.738236	−	0.674542i	$-0.235659\pi$
$380$		0			0
$381$	−4.89671	+	23.3022i	−0.250866	+	1.19381i
$382$		0			0
$383$	11.0525			0.564758			0.282379	−	0.959303i	$-0.408876\pi$
$383$	11.0525			0.564758			0.282379	+	0.959303i	$0.408876\pi$
$384$		0			0
$385$			8.14968i			0.415346i
$386$		0			0
$387$	−9.28660	+	21.1205i	−0.472064	+	1.07362i
$388$		0			0
$389$	−14.1728			−0.718591			−0.359296	−	0.933224i	$-0.616983\pi$
$389$	−14.1728			−0.718591			−0.359296	+	0.933224i	$0.616983\pi$
$390$		0			0
$391$	2.04994	−	0.529737i	0.103670	−	0.0267900i
$392$		0			0
$393$	−33.5613	−	7.05257i	−1.69295	−	0.355755i
$394$		0			0
$395$		−	0.894621i		−	0.0450133i
$396$		0			0
$397$	28.4775			1.42925			0.714623	−	0.699510i	$-0.246598\pi$
$397$	28.4775			1.42925			0.714623	+	0.699510i	$0.246598\pi$
$398$		0			0
$399$	3.51030	−	16.7046i	0.175735	−	0.836275i
$400$		0			0
$401$	7.05713			0.352416			0.176208	−	0.984353i	$-0.443617\pi$
$401$	7.05713			0.352416			0.176208	+	0.984353i	$0.443617\pi$
$402$		0			0
$403$	17.6119			0.877310
$404$		0			0
$405$	−2.68591	−	2.92805i	−0.133464	−	0.145496i
$406$		0			0
$407$			34.0699i			1.68878i
$408$		0			0
$409$	−6.21270			−0.307198			−0.153599	−	0.988133i	$-0.549086\pi$
$409$	−6.21270			−0.307198			−0.153599	+	0.988133i	$0.549086\pi$
$410$		0			0
$411$	4.32879	−	20.5996i	0.213524	−	1.01610i
$412$		0			0
$413$	8.76568			0.431331
$414$		0			0
$415$	4.97065			0.243999
$416$		0			0
$417$	−7.79451	+	37.0920i	−0.381699	+	1.81640i
$418$		0			0
$419$	−31.9126			−1.55903			−0.779516	−	0.626382i	$-0.784535\pi$
$419$	−31.9126			−1.55903			−0.779516	+	0.626382i	$0.784535\pi$
$420$		0			0
$421$			17.7610i			0.865616i	0.901486	+	0.432808i	$0.142477\pi$
$421$			17.7610i			0.865616i	−0.901486	+	0.432808i	$0.857523\pi$
$422$		0			0
$423$	2.85588	−	6.49513i	0.138858	−	0.315804i
$424$		0			0
$425$	−2.12137			−0.102902
$426$		0			0
$427$	−52.3299			−2.53242
$428$		0			0
$429$	−6.18126	+	29.4150i	−0.298434	+	1.42017i
$430$		0			0
$431$	−14.1363			−0.680922			−0.340461	−	0.940259i	$-0.610583\pi$
$431$	−14.1363			−0.680922			−0.340461	+	0.940259i	$0.610583\pi$
$432$		0			0
$433$			11.8427i			0.569125i	0.958657	+	0.284563i	$0.0918484\pi$
$433$			11.8427i			0.569125i	−0.958657	+	0.284563i	$0.908152\pi$
$434$		0			0
$435$	4.80055	+	1.00879i	0.230169	+	0.0483676i
$436$		0			0
$437$	−2.97444	−	11.5103i	−0.142287	−	0.550612i
$438$		0			0
$439$	−10.4880			−0.500566			−0.250283	−	0.968173i	$-0.580524\pi$
$439$	−10.4880			−0.500566			−0.250283	+	0.968173i	$0.580524\pi$
$440$		0			0
$441$	24.1810	+	10.6323i	1.15148	+	0.506299i
$442$		0			0
$443$		−	25.5294i		−	1.21294i	−0.795107	−	0.606469i	$-0.792585\pi$
$443$		−	25.5294i		−	1.21294i	0.795107	−	0.606469i	$-0.207415\pi$
$444$		0			0
$445$	7.66624			0.363415
$446$		0			0
$447$	6.09473	−	29.0032i	0.288271	−	1.37181i
$448$		0			0
$449$		−	30.6308i		−	1.44556i	−0.691080	−	0.722778i	$-0.742865\pi$
$449$		−	30.6308i		−	1.44556i	0.691080	−	0.722778i	$-0.257135\pi$
$450$		0			0
$451$		−	1.69521i		−	0.0798245i
$452$		0			0
$453$	6.50858	−	30.9726i	0.305800	−	1.45522i
$454$		0			0
$455$			6.55962i			0.307520i
$456$		0			0
$457$			21.7277i			1.01638i	0.861245	+	0.508189i	$0.169685\pi$
$457$			21.7277i			1.01638i	−0.861245	+	0.508189i	$0.830315\pi$
$458$		0			0
$459$	1.33548	−	1.86521i	0.0623347	−	0.0870608i
$460$		0			0
$461$			6.34561i			0.295544i	0.989021	+	0.147772i	$0.0472102\pi$
$461$			6.34561i			0.295544i	−0.989021	+	0.147772i	$0.952790\pi$
$462$		0			0
$463$	−27.2593			−1.26685			−0.633424	−	0.773805i	$-0.718351\pi$
$463$	−27.2593			−1.26685			−0.633424	+	0.773805i	$0.718351\pi$
$464$		0			0
$465$	0.741040	−	3.52642i	0.0343649	−	0.163534i
$466$		0			0
$467$	10.9444			0.506448			0.253224	−	0.967408i	$-0.418509\pi$
$467$	10.9444			0.506448			0.253224	+	0.967408i	$0.418509\pi$
$468$		0			0
$469$	−35.5152			−1.63994
$470$		0			0
$471$	−17.2377	−	3.62233i	−0.794272	−	0.166908i
$472$		0			0
$473$		−	35.7101i		−	1.64195i
$474$		0			0
$475$			11.9114i			0.546531i
$476$		0			0
$477$	−23.9119	−	10.5140i	−1.09485	−	0.481402i
$478$		0			0
$479$	−15.5877			−0.712222			−0.356111	−	0.934444i	$-0.615898\pi$
$479$	−15.5877			−0.712222			−0.356111	+	0.934444i	$0.615898\pi$
$480$		0			0
$481$			27.4226i			1.25036i
$482$		0			0
$483$	32.9889	−	1.51054i	1.50105	−	0.0687317i
$484$		0			0
$485$		−	2.57184i		−	0.116781i
$486$		0			0
$487$	31.0333			1.40625			0.703126	−	0.711066i	$-0.251787\pi$
$487$	31.0333			1.40625			0.703126	+	0.711066i	$0.251787\pi$
$488$		0			0
$489$	4.79900	−	22.8372i	0.217018	−	1.03273i
$490$		0			0
$491$		−	1.77376i		−	0.0800488i	−0.999199	−	0.0400244i	$-0.987256\pi$
$491$		−	1.77376i		−	0.0800488i	0.999199	−	0.0400244i	$-0.0127436\pi$
$492$		0			0
$493$			2.83214i			0.127553i
$494$		0			0
$495$	5.62966	+	2.47534i	0.253035	+	0.111258i
$496$		0			0
$497$	−32.6796			−1.46588
$498$		0			0
$499$	−17.3376			−0.776136			−0.388068	−	0.921631i	$-0.626857\pi$
$499$	−17.3376			−0.776136			−0.388068	+	0.921631i	$0.626857\pi$
$500$		0			0
$501$	−9.49251	−	1.99475i	−0.424094	−	0.0891190i
$502$		0			0
$503$	−38.6363			−1.72271			−0.861355	−	0.508004i	$-0.830383\pi$
$503$	−38.6363			−1.72271			−0.861355	+	0.508004i	$0.830383\pi$
$504$		0			0
$505$		−	2.67095i		−	0.118856i
$506$		0			0
$507$	−0.344740	+	1.64053i	−0.0153104	+	0.0728584i
$508$		0			0
$509$			12.8443i			0.569313i	0.958630	+	0.284657i	$0.0918796\pi$
$509$			12.8443i			0.569313i	−0.958630	+	0.284657i	$0.908120\pi$
$510$		0			0
$511$		−	43.4023i		−	1.92001i
$512$		0			0
$513$	−10.4731	−	7.49861i	−0.462397	−	0.331072i
$514$		0			0
$515$		−	5.46522i		−	0.240826i
$516$		0			0
$517$			10.9818i			0.482980i
$518$		0			0
$519$	−2.64465	−	0.555747i	−0.116087	−	0.0243946i
$520$		0			0
$521$	14.7158			0.644711			0.322355	−	0.946619i	$-0.395525\pi$
$521$	14.7158			0.644711			0.322355	+	0.946619i	$0.395525\pi$
$522$		0			0
$523$		−	17.5910i		−	0.769199i	−0.923084	−	0.384599i	$-0.874340\pi$
$523$		−	17.5910i		−	0.769199i	0.923084	−	0.384599i	$-0.125660\pi$
$524$		0			0
$525$	−32.3801	−	6.80433i	−1.41318	−	0.296966i
$526$		0			0
$527$	2.08044			0.0906256
$528$		0			0
$529$	20.1205	−	11.1430i	0.874803	−	0.484478i
$530$		0			0
$531$	2.66244	−	6.05519i	0.115540	−	0.262773i
$532$		0			0
$533$		−	1.36446i		−	0.0591015i
$534$		0			0
$535$	1.91109			0.0826238
$536$		0			0
$537$	25.7562	+	5.41241i	1.11146	+	0.233563i
$538$		0			0
$539$	−40.8847			−1.76103
$540$		0			0
$541$	−23.3120			−1.00226			−0.501130	−	0.865372i	$-0.667082\pi$
$541$	−23.3120			−1.00226			−0.501130	+	0.865372i	$0.667082\pi$
$542$		0			0
$543$	17.5109	+	3.67974i	0.751466	+	0.157913i
$544$		0			0
$545$			3.16820i			0.135711i
$546$		0			0
$547$	4.84785			0.207279			0.103640	−	0.994615i	$-0.466951\pi$
$547$	4.84785			0.207279			0.103640	+	0.994615i	$0.466951\pi$
$548$		0			0
$549$	−15.8944	+	36.1486i	−0.678356	+	1.54279i
$550$		0			0
$551$	15.9022			0.677459
$552$		0			0
$553$	8.05606			0.342578
$554$		0			0
$555$	5.49081	+	1.15384i	0.233072	+	0.0489777i
$556$		0			0
$557$	30.0971			1.27526			0.637628	−	0.770344i	$-0.279916\pi$
$557$	30.0971			1.27526			0.637628	+	0.770344i	$0.279916\pi$
$558$		0			0
$559$		−	28.7428i		−	1.21569i
$560$		0			0
$561$	−0.730176	+	3.47472i	−0.0308280	+	0.146703i
$562$		0			0
$563$	12.4785			0.525905			0.262952	−	0.964809i	$-0.415304\pi$
$563$	12.4785			0.525905			0.262952	+	0.964809i	$0.415304\pi$
$564$		0			0
$565$	2.38370			0.100283
$566$		0			0
$567$	26.3671	−	24.1866i	1.10731	−	1.01574i
$568$		0			0
$569$	−33.8881			−1.42066			−0.710332	−	0.703867i	$-0.751455\pi$
$569$	−33.8881			−1.42066			−0.710332	+	0.703867i	$0.751455\pi$
$570$		0			0
$571$			10.4454i			0.437124i	0.975823	+	0.218562i	$0.0701367\pi$
$571$			10.4454i			0.437124i	−0.975823	+	0.218562i	$0.929863\pi$
$572$		0			0
$573$	0.741040	−	3.52642i	0.0309574	−	0.147318i
$574$		0			0
$575$	−22.3115	+	5.76563i	−0.930453	+	0.240443i
$576$		0			0
$577$	−37.8428			−1.57542			−0.787709	−	0.616047i	$-0.788733\pi$
$577$	−37.8428			−1.57542			−0.787709	+	0.616047i	$0.788733\pi$
$578$		0			0
$579$	1.34389	−	6.39524i	0.0558503	−	0.265777i
$580$		0			0
$581$			44.7606i			1.85698i
$582$		0			0
$583$	40.4298			1.67443
$584$		0			0
$585$	4.53128	+	1.99238i	0.187345	+	0.0823749i
$586$		0			0
$587$			13.0502i			0.538639i	0.963051	+	0.269319i	$0.0867987\pi$
$587$			13.0502i			0.538639i	−0.963051	+	0.269319i	$0.913201\pi$
$588$		0			0
$589$		−	11.6816i		−	0.481330i
$590$		0			0
$591$	−38.8064	−	8.15477i	−1.59628	−	0.335442i
$592$		0			0
$593$			6.35984i			0.261167i	0.991437	+	0.130584i	$0.0416851\pi$
$593$			6.35984i			0.261167i	−0.991437	+	0.130584i	$0.958315\pi$
$594$		0			0
$595$			0.774870i			0.0317666i
$596$		0			0
$597$	−17.2228	−	3.61919i	−0.704881	−	0.148124i
$598$		0			0
$599$		−	47.3193i		−	1.93341i	−0.255889	−	0.966706i	$-0.582368\pi$
$599$		−	47.3193i		−	1.93341i	0.255889	−	0.966706i	$-0.417632\pi$
$600$		0			0
$601$	−18.5835			−0.758038			−0.379019	−	0.925389i	$-0.623738\pi$
$601$	−18.5835			−0.758038			−0.379019	+	0.925389i	$0.623738\pi$
$602$		0			0
$603$	−10.7872	+	24.5333i	−0.439288	+	0.999073i
$604$		0			0
$605$	−4.66218			−0.189545
$606$		0			0
$607$	9.91109			0.402279			0.201139	−	0.979563i	$-0.435536\pi$
$607$	9.91109			0.402279			0.201139	+	0.979563i	$0.435536\pi$
$608$		0			0
$609$	−9.08412	+	43.2290i	−0.368107	+	1.75173i
$610$		0			0
$611$			8.83918i			0.357595i
$612$		0			0
$613$		−	16.3480i		−	0.660291i	−0.943930	−	0.330145i	$-0.892902\pi$
$613$		−	16.3480i		−	0.660291i	0.943930	−	0.330145i	$-0.107098\pi$
$614$		0			0
$615$	−0.273206	−	0.0574114i	−0.0110167	−	0.00231505i
$616$		0			0
$617$	−42.5535			−1.71314			−0.856571	−	0.516030i	$-0.827409\pi$
$617$	−42.5535			−1.71314			−0.856571	+	0.516030i	$0.827409\pi$
$618$		0			0
$619$		−	32.2505i		−	1.29626i	−0.761531	−	0.648129i	$-0.775552\pi$
$619$		−	32.2505i		−	1.29626i	0.761531	−	0.648129i	$-0.224448\pi$
$620$		0			0
$621$	8.97643	−	23.2470i	0.360212	−	0.932871i
$622$		0			0
$623$			69.0344i			2.76581i
$624$		0			0
$625$	22.1144			0.884575
$626$		0			0
$627$	19.5103	+	4.09989i	0.779166	+	0.163734i
$628$		0			0
$629$			3.23936i			0.129162i
$630$		0			0
$631$		−	11.3350i		−	0.451241i	−0.974215	−	0.225621i	$-0.927559\pi$
$631$		−	11.3350i		−	0.451241i	0.974215	−	0.225621i	$-0.0724409\pi$
$632$		0			0
$633$	6.16544	−	29.3397i	0.245054	−	1.16615i
$634$		0			0
$635$	−6.06923			−0.240850
$636$		0			0
$637$	−32.9078			−1.30385
$638$		0			0
$639$	−9.92593	+	22.5745i	−0.392664	+	0.893035i
$640$		0			0
$641$	3.47330			0.137187			0.0685936	−	0.997645i	$-0.478149\pi$
$641$	3.47330			0.137187			0.0685936	+	0.997645i	$0.478149\pi$
$642$		0			0
$643$			10.6309i			0.419242i	0.977783	+	0.209621i	$0.0672230\pi$
$643$			10.6309i			0.419242i	−0.977783	+	0.209621i	$0.932777\pi$
$644$		0			0
$645$	−5.75515	−	1.20939i	−0.226609	−	0.0476195i
$646$		0			0
$647$		−	46.0447i		−	1.81020i	−0.425195	−	0.905102i	$-0.639794\pi$
$647$		−	46.0447i		−	1.81020i	0.425195	−	0.905102i	$-0.360206\pi$
$648$		0			0
$649$			10.2380i			0.401876i
$650$		0			0
$651$	31.7553	+	6.67306i	1.24459	+	0.261538i
$652$		0			0
$653$			15.4856i			0.605997i	0.952991	+	0.302999i	$0.0979877\pi$
$653$			15.4856i			0.605997i	−0.952991	+	0.302999i	$0.902012\pi$
$654$		0			0
$655$		−	8.74132i		−	0.341552i
$656$		0			0
$657$	−29.9816	−	13.1828i	−1.16969	−	0.514310i
$658$		0			0
$659$	−12.7930			−0.498344			−0.249172	−	0.968459i	$-0.580158\pi$
$659$	−12.7930			−0.498344			−0.249172	+	0.968459i	$0.580158\pi$
$660$		0			0
$661$			45.8724i			1.78423i	0.451809	+	0.892115i	$0.350779\pi$
$661$			45.8724i			1.78423i	−0.451809	+	0.892115i	$0.649221\pi$
$662$		0			0
$663$	−0.587713	+	2.79677i	−0.0228249	+	0.108618i
$664$		0			0
$665$	4.35084			0.168718
$666$		0			0
$667$	7.69740	+	29.7869i	0.298044	+	1.15335i
$668$		0			0
$669$	6.08041	−	28.9351i	0.235082	−	1.11869i
$670$		0			0
$671$		−	61.1192i		−	2.35948i
$672$		0			0
$673$	12.4371			0.479414			0.239707	−	0.970845i	$-0.422949\pi$
$673$	12.4371			0.479414			0.239707	+	0.970845i	$0.422949\pi$
$674$		0			0
$675$	−14.5353	+	20.3009i	−0.559463	+	0.781382i
$676$		0			0
$677$	38.3329			1.47325			0.736626	−	0.676301i	$-0.236418\pi$
$677$	38.3329			1.47325			0.736626	+	0.676301i	$0.236418\pi$
$678$		0			0
$679$	23.1594			0.888778
$680$		0			0
$681$	−8.89857	+	42.3460i	−0.340994	+	1.62270i
$682$		0			0
$683$			5.42004i			0.207392i	0.994609	+	0.103696i	$0.0330669\pi$
$683$			5.42004i			0.207392i	−0.994609	+	0.103696i	$0.966933\pi$
$684$		0			0
$685$	5.36533			0.204999
$686$		0			0
$687$	−20.2310	−	4.25134i	−0.771862	−	0.162199i
$688$		0			0
$689$	32.5416			1.23974
$690$		0			0
$691$	41.3093			1.57148			0.785739	−	0.618558i	$-0.212283\pi$
$691$	41.3093			1.57148			0.785739	+	0.618558i	$0.212283\pi$
$692$		0			0
$693$	−22.2904	+	50.6951i	−0.846743	+	1.92575i
$694$		0			0
$695$	−9.66091			−0.366459
$696$		0			0
$697$		−	0.161181i		−	0.00610515i
$698$		0			0
$699$	−26.5355	−	5.57616i	−1.00366	−	0.210910i
$700$		0			0
$701$	44.1815			1.66871			0.834356	−	0.551226i	$-0.185840\pi$
$701$	44.1815			1.66871			0.834356	+	0.551226i	$0.185840\pi$
$702$		0			0
$703$	18.1888			0.686003
$704$		0			0
$705$	1.76986	+	0.371919i	0.0666569	+	0.0140073i
$706$		0			0
$707$	24.0519			0.904566
$708$		0			0
$709$		−	16.9000i		−	0.634694i	−0.948309	−	0.317347i	$-0.897208\pi$
$709$		−	16.9000i		−	0.634694i	0.948309	−	0.317347i	$-0.102792\pi$
$710$		0			0
$711$	2.44690	−	5.56499i	0.0917661	−	0.208704i
$712$		0			0
$713$	21.8810	−	5.65439i	0.819451	−	0.211759i
$714$		0			0
$715$	−7.66137			−0.286519
$716$		0			0
$717$	23.3581	+	4.90847i	0.872325	+	0.183310i
$718$		0			0
$719$		−	3.80409i		−	0.141869i	−0.997481	−	0.0709344i	$-0.977402\pi$
$719$		−	3.80409i		−	0.141869i	0.997481	−	0.0709344i	$-0.0225981\pi$
$720$		0			0
$721$	49.2143			1.83283
$722$		0			0
$723$	−1.80724	−	0.379773i	−0.0672119	−	0.0141239i
$724$		0			0
$725$		−	30.8248i		−	1.14480i
$726$		0			0
$727$		−	5.87165i		−	0.217767i	−0.994055	−	0.108884i	$-0.965272\pi$
$727$		−	5.87165i		−	0.217767i	0.994055	−	0.108884i	$-0.0347276\pi$
$728$		0			0
$729$	−8.69909	−	25.5602i	−0.322188	−	0.946676i
$730$		0			0
$731$		−	3.39531i		−	0.125580i
$732$		0			0
$733$		−	35.6101i		−	1.31529i	−0.753329	−	0.657644i	$-0.771553\pi$
$733$		−	35.6101i		−	1.31529i	0.753329	−	0.657644i	$-0.228447\pi$
$734$		0			0
$735$	−1.38463	+	6.58911i	−0.0510729	+	0.243043i
$736$		0			0
$737$		−	41.4803i		−	1.52795i
$738$		0			0
$739$	−45.4574			−1.67218			−0.836089	−	0.548594i	$-0.815163\pi$
$739$	−45.4574			−1.67218			−0.836089	+	0.548594i	$0.815163\pi$
$740$		0			0
$741$	15.7037	+	3.29997i	0.576890	+	0.121227i
$742$		0			0
$743$	7.83517			0.287445			0.143722	−	0.989618i	$-0.454093\pi$
$743$	7.83517			0.287445			0.143722	+	0.989618i	$0.454093\pi$
$744$		0			0
$745$	7.55413			0.276762
$746$		0			0
$747$	30.9199	+	13.5954i	1.13130	+	0.497428i
$748$		0			0
$749$			17.2094i			0.628817i
$750$		0			0
$751$		−	14.3284i		−	0.522849i	−0.965224	−	0.261425i	$-0.915808\pi$
$751$		−	14.3284i		−	0.522849i	0.965224	−	0.261425i	$-0.0841923\pi$
$752$		0			0
$753$	3.38134	−	16.0909i	0.123223	−	0.586386i
$754$		0			0
$755$	8.06708			0.293591
$756$		0			0
$757$			14.2291i			0.517165i	0.965989	+	0.258582i	$0.0832554\pi$
$757$			14.2291i			0.517165i	−0.965989	+	0.258582i	$0.916745\pi$
$758$		0			0
$759$	1.76425	+	38.5298i	0.0640381	+	1.39854i
$760$		0			0
$761$			35.2346i			1.27725i	0.769517	+	0.638627i	$0.220497\pi$
$761$			35.2346i			1.27725i	−0.769517	+	0.638627i	$0.779503\pi$
$762$		0			0
$763$	−28.5296			−1.03284
$764$		0			0
$765$	0.535268	+	0.235355i	0.0193526	+	0.00850927i
$766$		0			0
$767$			8.24047i			0.297546i
$768$		0			0
$769$			32.5483i			1.17372i	0.809687	+	0.586861i	$0.199637\pi$
$769$			32.5483i			1.17372i	−0.809687	+	0.586861i	$0.800363\pi$
$770$		0			0
$771$	−9.34901	−	1.96460i	−0.336696	−	0.0707533i
$772$		0			0
$773$	48.9373			1.76015			0.880076	−	0.474833i	$-0.157491\pi$
$773$	48.9373			1.76015			0.880076	+	0.474833i	$0.157491\pi$
$774$		0			0
$775$	−22.6434			−0.813377
$776$		0			0
$777$	−10.3903	+	49.4447i	−0.372750	+	1.77382i
$778$		0			0
$779$	−0.905018			−0.0324256
$780$		0			0
$781$		−	38.1685i		−	1.36578i
$782$		0			0
$783$	27.1027	+	19.4053i	0.968571	+	0.693489i
$784$		0			0
$785$		−	4.48970i		−	0.160244i
$786$		0			0
$787$			5.40381i			0.192625i	0.995351	+	0.0963126i	$0.0307048\pi$
$787$			5.40381i			0.192625i	−0.995351	+	0.0963126i	$0.969295\pi$
$788$		0			0
$789$	−5.88832	+	28.0210i	−0.209630	+	0.997573i
$790$		0			0
$791$			21.4652i			0.763215i
$792$		0			0
$793$		−	49.1944i		−	1.74694i
$794$		0			0
$795$	1.36923	−	6.51579i	0.0485614	−	0.231091i
$796$		0			0
$797$	−3.48235			−0.123351			−0.0616755	−	0.998096i	$-0.519644\pi$
$797$	−3.48235			−0.123351			−0.0616755	+	0.998096i	$0.519644\pi$
$798$		0			0
$799$			1.04415i			0.0369393i
$800$		0			0
$801$	47.6879	+	20.9682i	1.68497	+	0.740873i
$802$		0			0
$803$	50.6922			1.78889
$804$		0			0
$805$	2.10600	+	8.14968i	0.0742268	+	0.287238i
$806$		0			0
$807$	−5.27959	−	1.10945i	−0.185850	−	0.0390545i
$808$		0			0
$809$			49.0905i			1.72593i	0.505264	+	0.862965i	$0.331395\pi$
$809$			49.0905i			1.72593i	−0.505264	+	0.862965i	$0.668605\pi$
$810$		0			0
$811$	4.88343			0.171481			0.0857403	−	0.996318i	$-0.472674\pi$
$811$	4.88343			0.171481			0.0857403	+	0.996318i	$0.472674\pi$
$812$		0			0
$813$	3.96104	−	18.8495i	0.138920	−	0.661082i
$814$		0			0
$815$	5.94813			0.208354
$816$		0			0
$817$	−19.0644			−0.666980
$818$		0			0
$819$	−17.9414	+	40.8041i	−0.626923	+	1.42581i
$820$		0			0
$821$			6.82958i			0.238354i	0.992873	+	0.119177i	$0.0380256\pi$
$821$			6.82958i			0.238354i	−0.992873	+	0.119177i	$0.961974\pi$
$822$		0			0
$823$	−25.6674			−0.894710			−0.447355	−	0.894356i	$-0.647634\pi$
$823$	−25.6674			−0.894710			−0.447355	+	0.894356i	$0.647634\pi$
$824$		0			0
$825$	7.94720	−	37.8186i	0.276686	−	1.31668i
$826$		0			0
$827$	−51.6227			−1.79510			−0.897549	−	0.440915i	$-0.854654\pi$
$827$	−51.6227			−1.79510			−0.897549	+	0.440915i	$0.854654\pi$
$828$		0			0
$829$	27.4043			0.951790			0.475895	−	0.879502i	$-0.342124\pi$
$829$	27.4043			0.951790			0.475895	+	0.879502i	$0.342124\pi$
$830$		0			0
$831$	−5.75880	+	27.4046i	−0.199770	+	0.950655i
$832$		0			0
$833$	−3.88731			−0.134687
$834$		0			0
$835$		−	2.47240i		−	0.0855609i
$836$		0			0
$837$	14.2548	−	19.9092i	0.492719	−	0.688164i
$838$		0			0
$839$	−43.8006			−1.51216			−0.756082	−	0.654477i	$-0.772889\pi$
$839$	−43.8006			−1.51216			−0.756082	+	0.654477i	$0.772889\pi$
$840$		0			0
$841$	−12.1526			−0.419056
$842$		0			0
$843$	0.247710	−	1.17879i	0.00853159	−	0.0405996i
$844$		0			0
$845$	−0.427289			−0.0146992
$846$		0			0
$847$		−	41.9829i		−	1.44255i
$848$		0			0
$849$	−41.2141	−	8.66073i	−1.41447	−	0.297235i
$850$		0			0
$851$	8.80419	+	34.0699i	0.301804	+	1.16790i
$852$		0			0
$853$	17.9610			0.614974			0.307487	−	0.951552i	$-0.400512\pi$
$853$	17.9610			0.614974			0.307487	+	0.951552i	$0.400512\pi$
$854$		0			0
$855$	1.32150	−	3.00549i	0.0451944	−	0.102786i
$856$		0			0
$857$		−	10.2335i		−	0.349569i	−0.984607	−	0.174785i	$-0.944077\pi$
$857$		−	10.2335i		−	0.349569i	0.984607	−	0.174785i	$-0.0559229\pi$
$858$		0			0
$859$	7.28761			0.248650			0.124325	−	0.992242i	$-0.460323\pi$
$859$	7.28761			0.248650			0.124325	+	0.992242i	$0.460323\pi$
$860$		0			0
$861$	0.516989	−	2.46022i	0.0176189	−	0.0838440i
$862$		0			0
$863$		−	6.04470i		−	0.205764i	−0.994694	−	0.102882i	$-0.967194\pi$
$863$		−	6.04470i		−	0.205764i	0.994694	−	0.102882i	$-0.0328064\pi$
$864$		0			0
$865$		−	0.688821i		−	0.0234206i
$866$		0			0
$867$	5.98586	−	28.4851i	0.203290	−	0.967406i
$868$		0			0
$869$			9.40916i			0.319184i
$870$		0			0
$871$		−	33.3872i		−	1.13128i
$872$		0			0
$873$	7.03432	−	15.9982i	0.238076	−	0.541456i
$874$		0			0
$875$		−	17.2094i		−	0.581783i
$876$		0			0
$877$	21.3592			0.721250			0.360625	−	0.932711i	$-0.382564\pi$
$877$	21.3592			0.721250			0.360625	+	0.932711i	$0.382564\pi$
$878$		0			0
$879$	−5.18186	+	24.6591i	−0.174780	+	0.831731i
$880$		0			0
$881$	−18.4847			−0.622766			−0.311383	−	0.950284i	$-0.600792\pi$
$881$	−18.4847			−0.622766			−0.311383	+	0.950284i	$0.600792\pi$
$882$		0			0
$883$	49.5090			1.66611			0.833056	−	0.553189i	$-0.186589\pi$
$883$	49.5090			1.66611			0.833056	+	0.553189i	$0.186589\pi$
$884$		0			0
$885$	1.64998	+	0.346727i	0.0554636	+	0.0116551i
$886$		0			0
$887$		−	43.8047i		−	1.47082i	−0.677623	−	0.735409i	$-0.736990\pi$
$887$		−	43.8047i		−	1.47082i	0.677623	−	0.735409i	$-0.263010\pi$
$888$		0			0
$889$		−	54.6534i		−	1.83302i
$890$		0			0
$891$	28.2490	+	30.7957i	0.946376	+	1.03170i
$892$		0			0
$893$	5.86283			0.196192
$894$		0			0
$895$			6.70842i			0.224238i
$896$		0			0
$897$	1.42003	+	31.0123i	0.0474134	+	1.03547i
$898$		0			0
$899$			30.2301i			1.00823i
$900$		0			0
$901$	3.84405			0.128064
$902$		0			0
$903$	10.8905	−	51.8251i	0.362413	−	1.72463i
$904$		0			0
$905$			4.56086i			0.151608i
$906$		0			0
$907$			52.8742i			1.75566i	0.478972	+	0.877830i	$0.341010\pi$
$907$			52.8742i			1.75566i	−0.478972	+	0.877830i	$0.658990\pi$
$908$		0			0
$909$	7.30540	−	16.6147i	0.242305	−	0.551074i
$910$		0			0
$911$	44.6671			1.47989			0.739943	−	0.672670i	$-0.234852\pi$
$911$	44.6671			1.47989			0.739943	+	0.672670i	$0.234852\pi$
$912$		0			0
$913$	−52.2787			−1.73017
$914$		0			0
$915$	−9.85017	−	2.06991i	−0.325637	−	0.0684292i
$916$		0			0
$917$	78.7155			2.59941
$918$		0			0
$919$		−	45.8812i		−	1.51348i	−0.653715	−	0.756741i	$-0.726790\pi$
$919$		−	45.8812i		−	1.51348i	0.653715	−	0.756741i	$-0.273210\pi$
$920$		0			0
$921$	−1.42477	+	6.78012i	−0.0469478	+	0.223413i
$922$		0			0
$923$		−	30.7215i		−	1.01121i
$924$		0			0
$925$		−	35.2570i		−	1.15924i
$926$		0			0
$927$	14.9481	−	33.9964i	0.490959	−	1.11659i
$928$		0			0
$929$			50.7554i			1.66523i	0.553852	+	0.832615i	$0.313157\pi$
$929$			50.7554i			1.66523i	−0.553852	+	0.832615i	$0.686843\pi$
$930$		0			0
$931$			21.8270i			0.715350i
$932$		0			0
$933$	−26.9939	−	5.67249i	−0.883741	−	0.185709i
$934$		0			0
$935$	−0.905018			−0.0295972
$936$		0			0
$937$		−	17.2908i		−	0.564866i	−0.959287	−	0.282433i	$-0.908859\pi$
$937$		−	17.2908i		−	0.564866i	0.959287	−	0.282433i	$-0.0911415\pi$
$938$		0			0
$939$	48.8358	+	10.2623i	1.59370	+	0.334899i
$940$		0			0
$941$	7.94010			0.258840			0.129420	−	0.991590i	$-0.458688\pi$
$941$	7.94010			0.258840			0.129420	+	0.991590i	$0.458688\pi$
$942$		0			0
$943$	−0.438069	−	1.69521i	−0.0142655	−	0.0552037i
$944$		0			0
$945$	7.41528	+	5.30927i	0.241219	+	0.172711i
$946$		0			0
$947$		−	0.655503i		−	0.0213010i	−0.999943	−	0.0106505i	$-0.996610\pi$
$947$		−	0.655503i		−	0.0213010i	0.999943	−	0.0106505i	$-0.00339022\pi$
$948$		0			0
$949$	40.8018			1.32448
$950$		0			0
$951$	50.2748	+	10.5647i	1.63027	+	0.342585i
$952$		0			0
$953$	−52.2530			−1.69264			−0.846320	−	0.532675i	$-0.821187\pi$
$953$	−52.2530			−1.69264			−0.846320	+	0.532675i	$0.821187\pi$
$954$		0			0
$955$	0.918483			0.0297214
$956$		0			0
$957$	−50.4897	−	10.6099i	−1.63210	−	0.342969i
$958$		0			0
$959$			48.3148i			1.56017i
$960$		0			0
$961$	−8.79342			−0.283659
$962$		0			0
$963$	11.8880	+	5.22709i	0.383084	+	0.168441i
$964$		0			0
$965$	1.66569			0.0536205
$966$		0			0
$967$	−12.7124			−0.408803			−0.204401	−	0.978887i	$-0.565525\pi$
$967$	−12.7124			−0.408803			−0.204401	+	0.978887i	$0.565525\pi$
$968$		0			0
$969$	1.85504	+	0.389817i	0.0595923	+	0.0125227i
$970$		0			0
$971$	−18.2842			−0.586768			−0.293384	−	0.955995i	$-0.594781\pi$
$971$	−18.2842			−0.586768			−0.293384	+	0.955995i	$0.594781\pi$
$972$		0			0
$973$		−	86.9965i		−	2.78898i
$974$		0			0
$975$	6.39664	−	30.4399i	0.204856	−	0.974858i
$976$		0			0
$977$	7.18728			0.229941			0.114971	−	0.993369i	$-0.463323\pi$
$977$	7.18728			0.229941			0.114971	+	0.993369i	$0.463323\pi$
$978$		0			0
$979$	−80.6295			−2.57693
$980$		0			0
$981$	−8.66544	+	19.7078i	−0.276666	+	0.629222i
$982$		0			0
$983$	33.5704			1.07073			0.535365	−	0.844621i	$-0.320174\pi$
$983$	33.5704			1.07073			0.535365	+	0.844621i	$0.320174\pi$
$984$		0			0
$985$		−	10.1074i		−	0.322050i
$986$		0			0
$987$	−3.34912	+	15.9376i	−0.106604	+	0.507300i
$988$		0			0
$989$	−9.22803	−	35.7101i	−0.293434	−	1.13551i
$990$		0			0
$991$	5.30927			0.168655			0.0843273	−	0.996438i	$-0.473126\pi$
$991$	5.30927			0.168655			0.0843273	+	0.996438i	$0.473126\pi$
$992$		0			0
$993$	−4.57632	+	21.7775i	−0.145225	+	0.691088i
$994$		0			0
$995$		−	4.48581i		−	0.142210i
$996$		0			0
$997$	−3.89400			−0.123324			−0.0616621	−	0.998097i	$-0.519640\pi$
$997$	−3.89400			−0.123324			−0.0616621	+	0.998097i	$0.519640\pi$
$998$		0			0
$999$	30.9997	+	22.1955i	0.980788	+	0.702236i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.m.b.137.7	yes	16
3.2	odd	2	inner	552.2.m.b.137.6	yes	16
4.3	odd	2		1104.2.m.e.689.9		16
12.11	even	2		1104.2.m.e.689.12		16
23.22	odd	2	inner	552.2.m.b.137.8	yes	16
69.68	even	2	inner	552.2.m.b.137.5	✓	16
92.91	even	2		1104.2.m.e.689.10		16
276.275	odd	2		1104.2.m.e.689.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.m.b.137.5	✓	16	69.68	even	2	inner
552.2.m.b.137.6	yes	16	3.2	odd	2	inner
552.2.m.b.137.7	yes	16	1.1	even	1	trivial
552.2.m.b.137.8	yes	16	23.22	odd	2	inner
1104.2.m.e.689.9		16	4.3	odd	2
1104.2.m.e.689.10		16	92.91	even	2
1104.2.m.e.689.11		16	276.275	odd	2
1104.2.m.e.689.12		16	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.356193 + 1.69503i) q^{3} -0.441484 q^{5} -3.97556i q^{7} +(-2.74625 - 1.20752i) q^{9} +O(q^{10})\)	\(q+(-0.356193 + 1.69503i) q^{3} -0.441484 q^{5} -3.97556i q^{7} +(-2.74625 - 1.20752i) q^{9} +4.64330 q^{11} +3.73736 q^{13} +(0.157254 - 0.748329i) q^{15} +0.441484 q^{17} -2.47890i q^{19} +(6.73870 + 1.41607i) q^{21} +(4.64330 - 1.19990i) q^{23} -4.80509 q^{25} +(3.02497 - 4.22487i) q^{27} +6.41503i q^{29} +4.71239 q^{31} +(-1.65391 + 7.87053i) q^{33} +1.75515i q^{35} +7.33743i q^{37} +(-1.33122 + 6.33493i) q^{39} -0.365088i q^{41} -7.69067i q^{43} +(1.21243 + 0.533099i) q^{45} +2.36509i q^{47} -8.80509 q^{49} +(-0.157254 + 0.748329i) q^{51} +8.70712 q^{53} -2.04994 q^{55} +(4.20182 + 0.882968i) q^{57} +2.20489i q^{59} -13.1629i q^{61} +(-4.80055 + 10.9179i) q^{63} -1.64998 q^{65} -8.93337i q^{67} +(0.379955 + 8.29793i) q^{69} -8.22012i q^{71} +10.9173 q^{73} +(1.71154 - 8.14477i) q^{75} -18.4597i q^{77} +2.02639i q^{79} +(6.08381 + 6.63229i) q^{81} -11.2589 q^{83} -0.194908 q^{85} +(-10.8737 - 2.28499i) q^{87} -17.3647 q^{89} -14.8581i q^{91} +(-1.67852 + 7.98764i) q^{93} +1.09440i q^{95} +5.82545i q^{97} +(-12.7517 - 5.60686i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 16 * q + 4 * q^3 + 4 * q^9	\( 16 q + 4 q^{3} + 4 q^{9} + 8 q^{13} + 32 q^{25} + 16 q^{27} + 56 q^{31} - 44 q^{39} - 32 q^{49} + 32 q^{55} + 4 q^{69} + 40 q^{73} - 20 q^{75} + 4 q^{81} - 112 q^{85} - 36 q^{87} - 36 q^{93}+O(q^{100}) \) 16 * q + 4 * q^3 + 4 * q^9 + 8 * q^13 + 32 * q^25 + 16 * q^27 + 56 * q^31 - 44 * q^39 - 32 * q^49 + 32 * q^55 + 4 * q^69 + 40 * q^73 - 20 * q^75 + 4 * q^81 - 112 * q^85 - 36 * q^87 - 36 * q^93

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