LMFDB - Embedded newform 2400.2.b.i.2351.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(2351,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2400.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.b (of order $2$, degree $1$, not 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:	$19.1640964851$
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} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 $ x^16 + 24x^14 + 192x^12 + 672x^10 + 1092x^8 + 880x^6 + 352x^4 + 64*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.16
Root			$2.13875i$ of defining polynomial
Character	$\chi$	$=$	2400.2351
Dual form			2400.2.b.i.2351.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.64533 + 0.541196i) q^{3} +3.29066i q^{7} +(2.41421 + 1.78089i) q^{9} +O(q^{10})$	$q+(1.64533 + 0.541196i) q^{3} +3.29066i q^{7} +(2.41421 + 1.78089i) q^{9} +2.51856i q^{11} -4.65369i q^{13} +3.69552i q^{17} +0.828427 q^{19} +(-1.78089 + 5.41421i) q^{21} +2.61313 q^{23} +(3.00836 + 4.23671i) q^{27} -6.08034 q^{29} +1.17157i q^{31} +(-1.36303 + 4.14386i) q^{33} +1.92762i q^{37} +(2.51856 - 7.65685i) q^{39} +8.59890i q^{41} +6.01673 q^{43} +2.61313 q^{47} -3.82843 q^{49} +(-2.00000 + 6.08034i) q^{51} -4.59220 q^{53} +(1.36303 + 0.448342i) q^{57} +2.51856i q^{59} -8.48528i q^{61} +(-5.86030 + 7.94435i) q^{63} +3.29066 q^{67} +(4.29945 + 1.41421i) q^{69} -7.12356 q^{71} -6.58132 q^{73} -8.28772 q^{77} +16.4853i q^{79} +(2.65685 + 8.59890i) q^{81} -9.37011i q^{83} +(-10.0042 - 3.29066i) q^{87} +5.03712i q^{89} +15.3137 q^{91} +(-0.634051 + 1.92762i) q^{93} -2.72607 q^{97} +(-4.48528 + 6.08034i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} - 32 q^{19} - 16 q^{49} - 32 q^{51} - 48 q^{81} + 64 q^{91} + 64 q^{99}+O(q^{100}) $ 16 * q + 16 * q^9 - 32 * q^19 - 16 * q^49 - 32 * q^51 - 48 * q^81 + 64 * q^91 + 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	1.64533	+	0.541196i	0.949931	+	0.312460i
$3$	1.64533	+	0.541196i	0.949931	+	0.312460i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			3.29066i			1.24375i	0.783116	+	0.621876i	$0.213629\pi$
$7$			3.29066i			1.24375i	−0.783116	+	0.621876i	$0.786371\pi$
$8$		0			0
$9$	2.41421	+	1.78089i	0.804738	+	0.593630i
$10$		0			0
$11$			2.51856i			0.759374i	0.925115	+	0.379687i	$0.123968\pi$
$11$			2.51856i			0.759374i	−0.925115	+	0.379687i	$0.876032\pi$
$12$		0			0
$13$		−	4.65369i		−	1.29070i	−0.763886	−	0.645351i	$-0.776711\pi$
$13$		−	4.65369i		−	1.29070i	0.763886	−	0.645351i	$-0.223289\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.69552i			0.896295i	0.893960	+	0.448147i	$0.147916\pi$
$17$			3.69552i			0.896295i	−0.893960	+	0.448147i	$0.852084\pi$
$18$		0			0
$19$	0.828427			0.190054			0.0950271	−	0.995475i	$-0.469706\pi$
$19$	0.828427			0.190054			0.0950271	+	0.995475i	$0.469706\pi$
$20$		0			0
$21$	−1.78089	+	5.41421i	−0.388622	+	1.18148i
$22$		0			0
$23$	2.61313			0.544874			0.272437	−	0.962174i	$-0.412170\pi$
$23$	2.61313			0.544874			0.272437	+	0.962174i	$0.412170\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.00836	+	4.23671i	0.578960	+	0.815356i
$28$		0			0
$29$	−6.08034			−1.12909			−0.564546	−	0.825402i	$-0.690949\pi$
$29$	−6.08034			−1.12909			−0.564546	+	0.825402i	$0.690949\pi$
$30$		0			0
$31$			1.17157i			0.210421i	0.994450	+	0.105210i	$0.0335516\pi$
$31$			1.17157i			0.210421i	−0.994450	+	0.105210i	$0.966448\pi$
$32$		0			0
$33$	−1.36303	+	4.14386i	−0.237274	+	0.721353i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			1.92762i			0.316899i	0.987367	+	0.158450i	$0.0506495\pi$
$37$			1.92762i			0.316899i	−0.987367	+	0.158450i	$0.949350\pi$
$38$		0			0
$39$	2.51856	−	7.65685i	0.403292	−	1.22608i
$40$		0			0
$41$			8.59890i			1.34292i	0.741039	+	0.671461i	$0.234333\pi$
$41$			8.59890i			1.34292i	−0.741039	+	0.671461i	$0.765667\pi$
$42$		0			0
$43$	6.01673			0.917542			0.458771	−	0.888554i	$-0.348290\pi$
$43$	6.01673			0.917542			0.458771	+	0.888554i	$0.348290\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.61313			0.381164			0.190582	−	0.981671i	$-0.438963\pi$
$47$	2.61313			0.381164			0.190582	+	0.981671i	$0.438963\pi$
$48$		0			0
$49$	−3.82843			−0.546918
$50$		0			0
$51$	−2.00000	+	6.08034i	−0.280056	+	0.851418i
$52$		0			0
$53$	−4.59220			−0.630787			−0.315394	−	0.948961i	$-0.602137\pi$
$53$	−4.59220			−0.630787			−0.315394	+	0.948961i	$0.602137\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.36303	+	0.448342i	0.180538	+	0.0593843i
$58$		0			0
$59$			2.51856i			0.327889i	0.986470	+	0.163944i	$0.0524217\pi$
$59$			2.51856i			0.327889i	−0.986470	+	0.163944i	$0.947578\pi$
$60$		0			0
$61$		−	8.48528i		−	1.08643i	−0.839594	−	0.543214i	$-0.817207\pi$
$61$		−	8.48528i		−	1.08643i	0.839594	−	0.543214i	$-0.182793\pi$
$62$		0			0
$63$	−5.86030	+	7.94435i	−0.738329	+	1.00089i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.29066			0.402018			0.201009	−	0.979589i	$-0.435578\pi$
$67$	3.29066			0.402018			0.201009	+	0.979589i	$0.435578\pi$
$68$		0			0
$69$	4.29945	+	1.41421i	0.517593	+	0.170251i
$70$		0			0
$71$	−7.12356			−0.845412			−0.422706	−	0.906267i	$-0.638920\pi$
$71$	−7.12356			−0.845412			−0.422706	+	0.906267i	$0.638920\pi$
$72$		0			0
$73$	−6.58132			−0.770285			−0.385142	−	0.922857i	$-0.625848\pi$
$73$	−6.58132			−0.770285			−0.385142	+	0.922857i	$0.625848\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.28772			−0.944473
$78$		0			0
$79$			16.4853i			1.85474i	0.374147	+	0.927370i	$0.377936\pi$
$79$			16.4853i			1.85474i	−0.374147	+	0.927370i	$0.622064\pi$
$80$		0			0
$81$	2.65685	+	8.59890i	0.295206	+	0.955434i
$82$		0			0
$83$		−	9.37011i		−	1.02850i	−0.857639	−	0.514252i	$-0.828070\pi$
$83$		−	9.37011i		−	1.02850i	0.857639	−	0.514252i	$-0.171930\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−10.0042	−	3.29066i	−1.07256	−	0.352796i
$88$		0			0
$89$			5.03712i			0.533934i	0.963706	+	0.266967i	$0.0860214\pi$
$89$			5.03712i			0.533934i	−0.963706	+	0.266967i	$0.913979\pi$
$90$		0			0
$91$	15.3137			1.60531
$92$		0			0
$93$	−0.634051	+	1.92762i	−0.0657480	+	0.199885i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.72607			−0.276790			−0.138395	−	0.990377i	$-0.544194\pi$
$97$	−2.72607			−0.276790			−0.138395	+	0.990377i	$0.544194\pi$
$98$		0			0
$99$	−4.48528	+	6.08034i	−0.450788	+	0.611097i
$100$		0			0
$101$	−13.2039			−1.31384			−0.656919	−	0.753961i	$-0.728141\pi$
$101$	−13.2039			−1.31384			−0.656919	+	0.753961i	$0.728141\pi$
$102$		0			0
$103$			6.01673i			0.592846i	0.955057	+	0.296423i	$0.0957938\pi$
$103$			6.01673i			0.592846i	−0.955057	+	0.296423i	$0.904206\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.47343i		−	0.819157i	−0.912275	−	0.409579i	$-0.865676\pi$
$107$		−	8.47343i		−	0.819157i	0.912275	−	0.409579i	$-0.134324\pi$
$108$		0			0
$109$		−	0.485281i		−	0.0464815i	−0.999730	−	0.0232408i	$-0.992602\pi$
$109$		−	0.485281i		−	0.0464815i	0.999730	−	0.0232408i	$-0.00739843\pi$
$110$		0			0
$111$	−1.04322	+	3.17157i	−0.0990182	+	0.301032i
$112$		0			0
$113$		−	8.92177i		−	0.839290i	−0.907688	−	0.419645i	$-0.862155\pi$
$113$		−	8.92177i		−	0.839290i	0.907688	−	0.419645i	$-0.137845\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	8.28772	−	11.2350i	0.766200	−	1.03868i
$118$		0			0
$119$	−12.1607			−1.11477
$120$		0			0
$121$	4.65685			0.423350
$122$		0			0
$123$	−4.65369	+	14.1480i	−0.419609	+	1.27568i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	3.29066i		−	0.291999i	−0.989285	−	0.145999i	$-0.953360\pi$
$127$		−	3.29066i		−	0.291999i	0.989285	−	0.145999i	$-0.0466397\pi$
$128$		0			0
$129$	9.89949	+	3.25623i	0.871602	+	0.286695i
$130$		0			0
$131$			21.8028i			1.90492i	0.304663	+	0.952460i	$0.401456\pi$
$131$			21.8028i			1.90492i	−0.304663	+	0.952460i	$0.598544\pi$
$132$		0			0
$133$			2.72607i			0.236380i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	11.9832i		−	1.02380i	−0.859046	−	0.511899i	$-0.828942\pi$
$137$		−	11.9832i		−	1.02380i	0.859046	−	0.511899i	$-0.171058\pi$
$138$		0			0
$139$	14.4853			1.22863			0.614313	−	0.789063i	$-0.289433\pi$
$139$	14.4853			1.22863			0.614313	+	0.789063i	$0.289433\pi$
$140$		0			0
$141$	4.29945	+	1.41421i	0.362079	+	0.119098i
$142$		0			0
$143$	11.7206			0.980126
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.29902	−	2.07193i	−0.519535	−	0.170890i
$148$		0			0
$149$	14.6792			1.20257			0.601285	−	0.799034i	$-0.294656\pi$
$149$	14.6792			1.20257			0.601285	+	0.799034i	$0.294656\pi$
$150$		0			0
$151$		−	2.82843i		−	0.230174i	−0.993355	−	0.115087i	$-0.963285\pi$
$151$		−	2.82843i		−	0.230174i	0.993355	−	0.115087i	$-0.0367147\pi$
$152$		0			0
$153$	−6.58132	+	8.92177i	−0.532068	+	0.721282i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	5.78287i		−	0.461523i	−0.973010	−	0.230762i	$-0.925878\pi$
$157$		−	5.78287i		−	0.461523i	0.973010	−	0.230762i	$-0.0741217\pi$
$158$		0			0
$159$	−7.55568	−	2.48528i	−0.599204	−	0.197096i
$160$		0			0
$161$			8.59890i			0.677688i
$162$		0			0
$163$	−6.01673			−0.471266			−0.235633	−	0.971842i	$-0.575716\pi$
$163$	−6.01673			−0.471266			−0.235633	+	0.971842i	$0.575716\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	17.3952			1.34608			0.673040	−	0.739606i	$-0.264988\pi$
$167$	17.3952			1.34608			0.673040	+	0.739606i	$0.264988\pi$
$168$		0			0
$169$	−8.65685			−0.665912
$170$		0			0
$171$	2.00000	+	1.47534i	0.152944	+	0.112822i
$172$		0			0
$173$	21.5391			1.63758			0.818792	−	0.574090i	$-0.194644\pi$
$173$	21.5391			1.63758			0.818792	+	0.574090i	$0.194644\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.36303	+	4.14386i	−0.102452	+	0.311472i
$178$		0			0
$179$			9.64212i			0.720686i	0.932820	+	0.360343i	$0.117340\pi$
$179$			9.64212i			0.720686i	−0.932820	+	0.360343i	$0.882660\pi$
$180$		0			0
$181$			10.3431i			0.768800i	0.923167	+	0.384400i	$0.125592\pi$
$181$			10.3431i			0.768800i	−0.923167	+	0.384400i	$0.874408\pi$
$182$		0			0
$183$	4.59220	−	13.9611i	0.339465	−	1.03203i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−9.30739			−0.680623
$188$		0			0
$189$	−13.9416	+	9.89949i	−1.01410	+	0.720082i
$190$		0			0
$191$	−5.03712			−0.364473			−0.182237	−	0.983255i	$-0.558334\pi$
$191$	−5.03712			−0.364473			−0.182237	+	0.983255i	$0.558334\pi$
$192$		0			0
$193$	19.7439			1.42120			0.710600	−	0.703596i	$-0.248424\pi$
$193$	19.7439			1.42120			0.710600	+	0.703596i	$0.248424\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.9832			0.853770			0.426885	−	0.904306i	$-0.359611\pi$
$197$	11.9832			0.853770			0.426885	+	0.904306i	$0.359611\pi$
$198$		0			0
$199$		−	9.17157i		−	0.650156i	−0.945687	−	0.325078i	$-0.894610\pi$
$199$		−	9.17157i		−	0.650156i	0.945687	−	0.325078i	$-0.105390\pi$
$200$		0			0
$201$	5.41421	+	1.78089i	0.381889	+	0.125614i
$202$		0			0
$203$		−	20.0083i		−	1.40431i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	6.30864	+	4.65369i	0.438481	+	0.323454i
$208$		0			0
$209$			2.08644i			0.144322i
$210$		0			0
$211$	−18.4853			−1.27258			−0.636290	−	0.771450i	$-0.719532\pi$
$211$	−18.4853			−1.27258			−0.636290	+	0.771450i	$0.719532\pi$
$212$		0			0
$213$	−11.7206	−	3.85525i	−0.803083	−	0.264157i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.85525			−0.261711
$218$		0			0
$219$	−10.8284	−	3.56178i	−0.731717	−	0.240683i
$220$		0			0
$221$	17.1978			1.15685
$222$		0			0
$223$		−	21.9054i		−	1.46690i	−0.679746	−	0.733448i	$-0.737910\pi$
$223$		−	21.9054i		−	1.46690i	0.679746	−	0.733448i	$-0.262090\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	4.14386i		−	0.275038i	−0.990499	−	0.137519i	$-0.956087\pi$
$227$		−	4.14386i		−	0.275038i	0.990499	−	0.137519i	$-0.0439128\pi$
$228$		0			0
$229$		−	19.3137i		−	1.27629i	−0.769918	−	0.638143i	$-0.779703\pi$
$229$		−	19.3137i		−	1.27629i	0.769918	−	0.638143i	$-0.220297\pi$
$230$		0			0
$231$	−13.6360	−	4.48528i	−0.897184	−	0.295110i
$232$		0			0
$233$		−	15.9414i		−	1.04436i	−0.852837	−	0.522178i	$-0.825120\pi$
$233$		−	15.9414i		−	1.04436i	0.852837	−	0.522178i	$-0.174880\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−8.92177	+	27.1237i	−0.579531	+	1.76187i
$238$		0			0
$239$	22.2349			1.43826			0.719129	−	0.694877i	$-0.244541\pi$
$239$	22.2349			1.43826			0.719129	+	0.694877i	$0.244541\pi$
$240$		0			0
$241$	−6.48528			−0.417754			−0.208877	−	0.977942i	$-0.566981\pi$
$241$	−6.48528			−0.417754			−0.208877	+	0.977942i	$0.566981\pi$
$242$		0			0
$243$	−0.282294	+	15.5859i	−0.0181092	+	0.999836i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	3.85525i		−	0.245303i
$248$		0			0
$249$	5.07107	−	15.4169i	0.321366	−	0.977007i
$250$		0			0
$251$		−	11.7286i		−	0.740301i	−0.928972	−	0.370150i	$-0.879306\pi$
$251$		−	11.7286i		−	0.740301i	0.928972	−	0.370150i	$-0.120694\pi$
$252$		0			0
$253$			6.58132i			0.413764i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	16.3128i		−	1.01756i	−0.860895	−	0.508782i	$-0.830096\pi$
$257$		−	16.3128i		−	1.01756i	0.860895	−	0.508782i	$-0.169904\pi$
$258$		0			0
$259$	−6.34315			−0.394144
$260$		0			0
$261$	−14.6792	−	10.8284i	−0.908622	−	0.670263i
$262$		0			0
$263$	−22.2500			−1.37200			−0.685998	−	0.727604i	$-0.740634\pi$
$263$	−22.2500			−1.37200			−0.685998	+	0.727604i	$0.740634\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.72607	+	8.28772i	−0.166833	+	0.507200i
$268$		0			0
$269$	−9.64212			−0.587891			−0.293945	−	0.955822i	$-0.594968\pi$
$269$	−9.64212			−0.587891			−0.293945	+	0.955822i	$0.594968\pi$
$270$		0			0
$271$			14.1421i			0.859074i	0.903049	+	0.429537i	$0.141323\pi$
$271$			14.1421i			0.859074i	−0.903049	+	0.429537i	$0.858677\pi$
$272$		0			0
$273$	25.1961	+	8.28772i	1.52494	+	0.501596i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			15.0903i			0.906685i	0.891336	+	0.453343i	$0.149769\pi$
$277$			15.0903i			0.906685i	−0.891336	+	0.453343i	$0.850231\pi$
$278$		0			0
$279$	−2.08644	+	2.82843i	−0.124912	+	0.169334i
$280$		0			0
$281$		−	3.56178i		−	0.212478i	−0.994341	−	0.106239i	$-0.966119\pi$
$281$		−	3.56178i		−	0.212478i	0.994341	−	0.106239i	$-0.0338809\pi$
$282$		0			0
$283$	−18.0502			−1.07297			−0.536486	−	0.843909i	$-0.680249\pi$
$283$	−18.0502			−1.07297			−0.536486	+	0.843909i	$0.680249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−28.2960			−1.67026
$288$		0			0
$289$	3.34315			0.196656
$290$		0			0
$291$	−4.48528	−	1.47534i	−0.262932	−	0.0864859i
$292$		0			0
$293$	−2.42742			−0.141811			−0.0709056	−	0.997483i	$-0.522589\pi$
$293$	−2.42742			−0.141811			−0.0709056	+	0.997483i	$0.522589\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−10.6704	+	7.57675i	−0.619161	+	0.439647i
$298$		0			0
$299$		−	12.1607i		−	0.703271i
$300$		0			0
$301$			19.7990i			1.14119i
$302$		0			0
$303$	−21.7248	−	7.14590i	−1.24806	−	0.410521i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−7.14590			−0.407838			−0.203919	−	0.978988i	$-0.565368\pi$
$307$	−7.14590			−0.407838			−0.203919	+	0.978988i	$0.565368\pi$
$308$		0			0
$309$	−3.25623	+	9.89949i	−0.185240	+	0.563163i
$310$		0			0
$311$	34.3956			1.95040			0.975198	−	0.221334i	$-0.0710411\pi$
$311$	34.3956			1.95040			0.975198	+	0.221334i	$0.0710411\pi$
$312$		0			0
$313$	−17.0179			−0.961907			−0.480954	−	0.876746i	$-0.659709\pi$
$313$	−17.0179			−0.961907			−0.480954	+	0.876746i	$0.659709\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.2513			0.744269			0.372135	−	0.928179i	$-0.378626\pi$
$317$	13.2513			0.744269			0.372135	+	0.928179i	$0.378626\pi$
$318$		0			0
$319$		−	15.3137i		−	0.857403i
$320$		0			0
$321$	4.58579	−	13.9416i	0.255954	−	0.778143i
$322$		0			0
$323$			3.06147i			0.170345i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.262632	−	0.798447i	0.0145236	−	0.0441542i
$328$		0			0
$329$			8.59890i			0.474073i
$330$		0			0
$331$	18.4853			1.01604			0.508021	−	0.861344i	$-0.330377\pi$
$331$	18.4853			1.01604			0.508021	+	0.861344i	$0.330377\pi$
$332$		0			0
$333$	−3.43289	+	4.65369i	−0.188121	+	0.255021i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	27.9222			1.52102			0.760508	−	0.649328i	$-0.224950\pi$
$337$	27.9222			1.52102			0.760508	+	0.649328i	$0.224950\pi$
$338$		0			0
$339$	4.82843	−	14.6792i	0.262244	−	0.797267i
$340$		0			0
$341$	−2.95068			−0.159788
$342$		0			0
$343$			10.4366i			0.563521i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			0.185709i			0.00996939i	0.999988	+	0.00498469i	$0.00158668\pi$
$347$			0.185709i			0.00996939i	−0.999988	+	0.00498469i	$0.998413\pi$
$348$		0			0
$349$		−	2.34315i		−	0.125426i	−0.998032	−	0.0627129i	$-0.980025\pi$
$349$		−	2.34315i		−	0.125426i	0.998032	−	0.0627129i	$-0.0199752\pi$
$350$		0			0
$351$	19.7164	−	14.0000i	1.05238	−	0.747265i
$352$		0			0
$353$		−	32.3630i		−	1.72251i	−0.508175	−	0.861254i	$-0.669680\pi$
$353$		−	32.3630i		−	1.72251i	0.508175	−	0.861254i	$-0.330320\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−20.0083	−	6.58132i	−1.05895	−	0.348320i
$358$		0			0
$359$	9.21001			0.486086			0.243043	−	0.970016i	$-0.421854\pi$
$359$	9.21001			0.486086			0.243043	+	0.970016i	$0.421854\pi$
$360$		0			0
$361$	−18.3137			−0.963879
$362$		0			0
$363$	7.66206	+	2.52027i	0.402154	+	0.132280i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	2.16148i		−	0.112828i	−0.998407	−	0.0564142i	$-0.982033\pi$
$367$		−	2.16148i		−	0.112828i	0.998407	−	0.0564142i	$-0.0179667\pi$
$368$		0			0
$369$	−15.3137	+	20.7596i	−0.797200	+	1.08070i
$370$		0			0
$371$		−	15.1114i		−	0.784543i
$372$		0			0
$373$			24.3976i			1.26326i	0.775269	+	0.631631i	$0.217614\pi$
$373$			24.3976i			1.26326i	−0.775269	+	0.631631i	$0.782386\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			28.2960i			1.45732i
$378$		0			0
$379$	20.8284			1.06988			0.534942	−	0.844889i	$-0.320333\pi$
$379$	20.8284			1.06988			0.534942	+	0.844889i	$0.320333\pi$
$380$		0			0
$381$	1.78089	−	5.41421i	0.0912378	−	0.277379i
$382$		0			0
$383$	29.1158			1.48775			0.743874	−	0.668320i	$-0.232986\pi$
$383$	29.1158			1.48775			0.743874	+	0.668320i	$0.232986\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	14.5257	+	10.7151i	0.738381	+	0.544681i
$388$		0			0
$389$	−7.55568			−0.383088			−0.191544	−	0.981484i	$-0.561350\pi$
$389$	−7.55568			−0.383088			−0.191544	+	0.981484i	$0.561350\pi$
$390$		0			0
$391$			9.65685i			0.488368i
$392$		0			0
$393$	−11.7996	+	35.8728i	−0.595211	+	1.80954i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	27.1237i		−	1.36130i	−0.732609	−	0.680650i	$-0.761697\pi$
$397$		−	27.1237i		−	1.36130i	0.732609	−	0.680650i	$-0.238303\pi$
$398$		0			0
$399$	−1.47534	+	4.48528i	−0.0738593	+	0.224545i
$400$		0			0
$401$		−	15.1114i		−	0.754625i	−0.926086	−	0.377313i	$-0.876848\pi$
$401$		−	15.1114i		−	0.754625i	0.926086	−	0.377313i	$-0.123152\pi$
$402$		0			0
$403$	5.45214			0.271590
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.85483			−0.240645
$408$		0			0
$409$	12.8284			0.634325			0.317162	−	0.948371i	$-0.397270\pi$
$409$	12.8284			0.634325			0.317162	+	0.948371i	$0.397270\pi$
$410$		0			0
$411$	6.48528	−	19.7164i	0.319895	−	0.972537i
$412$		0			0
$413$	−8.28772			−0.407812
$414$		0			0
$415$		0			0
$416$		0			0
$417$	23.8331	+	7.83938i	1.16711	+	0.383896i
$418$		0			0
$419$		−	2.51856i		−	0.123040i	−0.998106	−	0.0615199i	$-0.980405\pi$
$419$		−	2.51856i		−	0.123040i	0.998106	−	0.0615199i	$-0.0195948\pi$
$420$		0			0
$421$		−	34.8284i		−	1.69743i	−0.528848	−	0.848717i	$-0.677376\pi$
$421$		−	34.8284i		−	1.69743i	0.528848	−	0.848717i	$-0.322624\pi$
$422$		0			0
$423$	6.30864	+	4.65369i	0.306737	+	0.226270i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	27.9222			1.35125
$428$		0			0
$429$	19.2842	+	6.34315i	0.931052	+	0.306250i
$430$		0			0
$431$	−27.2720			−1.31365			−0.656824	−	0.754044i	$-0.728101\pi$
$431$	−27.2720			−1.31365			−0.656824	+	0.754044i	$0.728101\pi$
$432$		0			0
$433$	23.5992			1.13410			0.567052	−	0.823682i	$-0.308084\pi$
$433$	23.5992			1.13410			0.567052	+	0.823682i	$0.308084\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.16478			0.103556
$438$		0			0
$439$			28.4853i			1.35953i	0.733431	+	0.679764i	$0.237918\pi$
$439$			28.4853i			1.35953i	−0.733431	+	0.679764i	$0.762082\pi$
$440$		0			0
$441$	−9.24264	−	6.81801i	−0.440126	−	0.324667i
$442$		0			0
$443$		−	1.60766i		−	0.0763821i	−0.999270	−	0.0381910i	$-0.987840\pi$
$443$		−	1.60766i		−	0.0763821i	0.999270	−	0.0381910i	$-0.0121595\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	24.1522	+	7.94435i	1.14236	+	0.375755i
$448$		0			0
$449$		−	35.0067i		−	1.65207i	−0.563620	−	0.826035i	$-0.690592\pi$
$449$		−	35.0067i		−	1.65207i	0.563620	−	0.826035i	$-0.309408\pi$
$450$		0			0
$451$	−21.6569			−1.01978
$452$		0			0
$453$	1.53073	−	4.65369i	0.0719201	−	0.218650i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18.6148			−0.870762			−0.435381	−	0.900246i	$-0.643386\pi$
$457$	−18.6148			−0.870762			−0.435381	+	0.900246i	$0.643386\pi$
$458$		0			0
$459$	−15.6569	+	11.1175i	−0.730799	+	0.518919i
$460$		0			0
$461$	−3.99390			−0.186014			−0.0930072	−	0.995665i	$-0.529648\pi$
$461$	−3.99390			−0.186014			−0.0930072	+	0.995665i	$0.529648\pi$
$462$		0			0
$463$			25.7607i			1.19720i	0.801048	+	0.598600i	$0.204276\pi$
$463$			25.7607i			1.19720i	−0.801048	+	0.598600i	$0.795724\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.8645i			0.734120i	0.930197	+	0.367060i	$0.119636\pi$
$467$			15.8645i			0.734120i	−0.930197	+	0.367060i	$0.880364\pi$
$468$		0			0
$469$			10.8284i			0.500010i
$470$		0			0
$471$	3.12967	−	9.51472i	0.144207	−	0.438415i
$472$		0			0
$473$			15.1535i			0.696758i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−11.0866	−	8.17821i	−0.507618	−	0.374455i
$478$		0			0
$479$	−19.2842			−0.881120			−0.440560	−	0.897723i	$-0.645220\pi$
$479$	−19.2842			−0.881120			−0.440560	+	0.897723i	$0.645220\pi$
$480$		0			0
$481$	8.97056			0.409022
$482$		0			0
$483$	−4.65369	+	14.1480i	−0.211750	+	0.643757i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	8.74280i		−	0.396174i	−0.980184	−	0.198087i	$-0.936527\pi$
$487$		−	8.74280i		−	0.396174i	0.980184	−	0.198087i	$-0.0634729\pi$
$488$		0			0
$489$	−9.89949	−	3.25623i	−0.447671	−	0.147252i
$490$		0			0
$491$		−	36.9142i		−	1.66591i	−0.553338	−	0.832957i	$-0.686646\pi$
$491$		−	36.9142i		−	1.66591i	0.553338	−	0.832957i	$-0.313354\pi$
$492$		0			0
$493$		−	22.4700i		−	1.01200i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	23.4412i		−	1.05148i
$498$		0			0
$499$	−37.1127			−1.66139			−0.830696	−	0.556726i	$-0.812057\pi$
$499$	−37.1127			−1.66139			−0.830696	+	0.556726i	$0.812057\pi$
$500$		0			0
$501$	28.6208	+	9.41421i	1.27868	+	0.420596i
$502$		0			0
$503$	9.63274			0.429503			0.214751	−	0.976669i	$-0.431106\pi$
$503$	9.63274			0.429503			0.214751	+	0.976669i	$0.431106\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−14.2434	−	4.68506i	−0.632570	−	0.208071i
$508$		0			0
$509$	13.2039			0.585253			0.292626	−	0.956227i	$-0.405471\pi$
$509$	13.2039			0.585253			0.292626	+	0.956227i	$0.405471\pi$
$510$		0			0
$511$		−	21.6569i		−	0.958043i
$512$		0			0
$513$	2.49221	+	3.50981i	0.110034	+	0.154962i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			6.58132i			0.289446i
$518$		0			0
$519$	35.4388	+	11.6569i	1.55559	+	0.511679i
$520$		0			0
$521$		−	34.3956i		−	1.50690i	−0.657506	−	0.753450i	$-0.728388\pi$
$521$		−	34.3956i		−	1.50690i	0.657506	−	0.753450i	$-0.271612\pi$
$522$		0			0
$523$	12.5980			0.550874			0.275437	−	0.961319i	$-0.411177\pi$
$523$	12.5980			0.550874			0.275437	+	0.961319i	$0.411177\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.32957			−0.188599
$528$		0			0
$529$	−16.1716			−0.703112
$530$		0			0
$531$	−4.48528	+	6.08034i	−0.194645	+	0.263864i
$532$		0			0
$533$	40.0166			1.73331
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−5.21828	+	15.8645i	−0.225185	+	0.684602i
$538$		0			0
$539$		−	9.64212i		−	0.415316i
$540$		0			0
$541$		−	16.0000i		−	0.687894i	−0.938989	−	0.343947i	$-0.888236\pi$
$541$		−	16.0000i		−	0.687894i	0.938989	−	0.343947i	$-0.111764\pi$
$542$		0			0
$543$	−5.59767	+	17.0179i	−0.240219	+	0.730307i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	11.0011			0.470375			0.235188	−	0.971950i	$-0.424430\pi$
$547$	11.0011			0.470375			0.235188	+	0.971950i	$0.424430\pi$
$548$		0			0
$549$	15.1114	−	20.4853i	0.644937	−	0.874291i
$550$		0			0
$551$	−5.03712			−0.214589
$552$		0			0
$553$	−54.2474			−2.30683
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.96362			−0.210315			−0.105158	−	0.994456i	$-0.533535\pi$
$557$	−4.96362			−0.210315			−0.105158	+	0.994456i	$0.533535\pi$
$558$		0			0
$559$		−	28.0000i		−	1.18427i
$560$		0			0
$561$	−15.3137	−	5.03712i	−0.646545	−	0.212667i
$562$		0			0
$563$		−	21.9874i		−	0.926658i	−0.886186	−	0.463329i	$-0.846655\pi$
$563$		−	21.9874i		−	0.926658i	0.886186	−	0.463329i	$-0.153345\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−28.2960	+	8.74280i	−1.18832	+	0.367163i
$568$		0			0
$569$		−	15.7225i		−	0.659120i	−0.944135	−	0.329560i	$-0.893100\pi$
$569$		−	15.7225i		−	0.659120i	0.944135	−	0.329560i	$-0.106900\pi$
$570$		0			0
$571$	−21.1127			−0.883539			−0.441769	−	0.897129i	$-0.645649\pi$
$571$	−21.1127			−0.883539			−0.441769	+	0.897129i	$0.645649\pi$
$572$		0			0
$573$	−8.28772	−	2.72607i	−0.346224	−	0.113883i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.85525			0.160496			0.0802480	−	0.996775i	$-0.474429\pi$
$577$	3.85525			0.160496			0.0802480	+	0.996775i	$0.474429\pi$
$578$		0			0
$579$	32.4853	+	10.6853i	1.35004	+	0.444068i
$580$		0			0
$581$	30.8338			1.27920
$582$		0			0
$583$		−	11.5657i		−	0.479004i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	7.20533i		−	0.297396i	−0.988883	−	0.148698i	$-0.952492\pi$
$587$		−	7.20533i		−	0.297396i	0.988883	−	0.148698i	$-0.0475082\pi$
$588$		0			0
$589$			0.970563i			0.0399913i
$590$		0			0
$591$	19.7164	+	6.48528i	0.811023	+	0.266769i
$592$		0			0
$593$		−	1.00547i		−	0.0412897i	−0.999787	−	0.0206448i	$-0.993428\pi$
$593$		−	1.00547i		−	0.0412897i	0.999787	−	0.0206448i	$-0.00657192\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.96362	−	15.0903i	0.203147	−	0.617603i
$598$		0			0
$599$	−32.3092			−1.32012			−0.660058	−	0.751214i	$-0.729468\pi$
$599$	−32.3092			−1.32012			−0.660058	+	0.751214i	$0.729468\pi$
$600$		0			0
$601$	13.5147			0.551277			0.275638	−	0.961261i	$-0.411111\pi$
$601$	13.5147			0.551277			0.275638	+	0.961261i	$0.411111\pi$
$602$		0			0
$603$	7.94435	+	5.86030i	0.323519	+	0.238650i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			4.88755i			0.198380i	0.995069	+	0.0991898i	$0.0316251\pi$
$607$			4.88755i			0.198380i	−0.995069	+	0.0991898i	$0.968375\pi$
$608$		0			0
$609$	10.8284	−	32.9203i	0.438790	−	1.33400i
$610$		0			0
$611$		−	12.1607i		−	0.491969i
$612$		0			0
$613$		−	23.2685i		−	0.939804i	−0.882718	−	0.469902i	$-0.844289\pi$
$613$		−	23.2685i		−	0.939804i	0.882718	−	0.469902i	$-0.155711\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.55035i		−	0.344224i	−0.985077	−	0.172112i	$-0.944941\pi$
$617$		−	8.55035i		−	0.344224i	0.985077	−	0.172112i	$-0.0550591\pi$
$618$		0			0
$619$	−2.48528			−0.0998919			−0.0499459	−	0.998752i	$-0.515905\pi$
$619$	−2.48528			−0.0998919			−0.0499459	+	0.998752i	$0.515905\pi$
$620$		0			0
$621$	7.86123	+	11.0711i	0.315460	+	0.444267i
$622$		0			0
$623$	−16.5754			−0.664081
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1.12918	+	3.43289i	−0.0450949	+	0.137096i
$628$		0			0
$629$	−7.12356			−0.284035
$630$		0			0
$631$			2.14214i			0.0852771i	0.999091	+	0.0426385i	$0.0135764\pi$
$631$			2.14214i			0.0852771i	−0.999091	+	0.0426385i	$0.986424\pi$
$632$		0			0
$633$	−30.4144	−	10.0042i	−1.20886	−	0.397630i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			17.8163i			0.705908i
$638$		0			0
$639$	−17.1978	−	12.6863i	−0.680335	−	0.501862i
$640$		0			0
$641$		−	35.0067i		−	1.38268i	−0.722529	−	0.691341i	$-0.757020\pi$
$641$		−	35.0067i		−	1.38268i	0.722529	−	0.691341i	$-0.242980\pi$
$642$		0			0
$643$	35.0681			1.38295			0.691475	−	0.722401i	$-0.256961\pi$
$643$	35.0681			1.38295			0.691475	+	0.722401i	$0.256961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	28.3730			1.11546			0.557728	−	0.830024i	$-0.311673\pi$
$647$	28.3730			1.11546			0.557728	+	0.830024i	$0.311673\pi$
$648$		0			0
$649$	−6.34315			−0.248990
$650$		0			0
$651$	−6.34315	−	2.08644i	−0.248607	−	0.0817742i
$652$		0			0
$653$	−24.2291			−0.948158			−0.474079	−	0.880482i	$-0.657219\pi$
$653$	−24.2291			−0.948158			−0.474079	+	0.880482i	$0.657219\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−15.8887	−	11.7206i	−0.619877	−	0.457264i
$658$		0			0
$659$		−	14.6792i		−	0.571822i	−0.958256	−	0.285911i	$-0.907704\pi$
$659$		−	14.6792i		−	0.571822i	0.958256	−	0.285911i	$-0.0922962\pi$
$660$		0			0
$661$			44.7696i			1.74133i	0.491873	+	0.870667i	$0.336312\pi$
$661$			44.7696i			1.74133i	−0.491873	+	0.870667i	$0.663688\pi$
$662$		0			0
$663$	28.2960	+	9.30739i	1.09893	+	0.361469i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−15.8887			−0.615213
$668$		0			0
$669$	11.8551	−	36.0416i	0.458346	−	1.39345i
$670$		0			0
$671$	21.3707			0.825006
$672$		0			0
$673$	47.6661			1.83739			0.918697	−	0.394964i	$-0.129243\pi$
$673$	47.6661			1.83739			0.918697	+	0.394964i	$0.129243\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.00547			0.0386433			0.0193217	−	0.999813i	$-0.493849\pi$
$677$	1.00547			0.0386433			0.0193217	+	0.999813i	$0.493849\pi$
$678$		0			0
$679$		−	8.97056i		−	0.344259i
$680$		0			0
$681$	2.24264	−	6.81801i	0.0859382	−	0.261267i
$682$		0			0
$683$		−	14.9678i		−	0.572726i	−0.958121	−	0.286363i	$-0.907554\pi$
$683$		−	14.9678i		−	0.572726i	0.958121	−	0.286363i	$-0.0924464\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	10.4525	−	31.7774i	0.398788	−	1.21238i
$688$		0			0
$689$			21.3707i			0.814159i
$690$		0			0
$691$	7.17157			0.272819			0.136410	−	0.990653i	$-0.456444\pi$
$691$	7.17157			0.272819			0.136410	+	0.990653i	$0.456444\pi$
$692$		0			0
$693$	−20.0083	−	14.7595i	−0.760053	−	0.560668i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−31.7774			−1.20365
$698$		0			0
$699$	8.62742	−	26.2288i	0.326319	−	0.992065i
$700$		0			0
$701$	−26.8399			−1.01373			−0.506865	−	0.862025i	$-0.669196\pi$
$701$	−26.8399			−1.01373			−0.506865	+	0.862025i	$0.669196\pi$
$702$		0			0
$703$			1.59689i			0.0602280i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	43.4495i		−	1.63409i
$708$		0			0
$709$			36.2843i			1.36268i	0.731965	+	0.681342i	$0.238603\pi$
$709$			36.2843i			1.36268i	−0.731965	+	0.681342i	$0.761397\pi$
$710$		0			0
$711$	−29.3585	+	39.7990i	−1.10103	+	1.49258i
$712$		0			0
$713$			3.06147i			0.114653i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	36.5838	+	12.0335i	1.36625	+	0.449398i
$718$		0			0
$719$	2.95068			0.110042			0.0550208	−	0.998485i	$-0.482477\pi$
$719$	2.95068			0.110042			0.0550208	+	0.998485i	$0.482477\pi$
$720$		0			0
$721$	−19.7990			−0.737353
$722$		0			0
$723$	−10.6704	−	3.50981i	−0.396837	−	0.130531i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			47.1015i			1.74690i	0.486915	+	0.873449i	$0.338122\pi$
$727$			47.1015i			1.74690i	−0.486915	+	0.873449i	$0.661878\pi$
$728$		0			0
$729$	−8.89949	+	25.4912i	−0.329611	+	0.944117i
$730$		0			0
$731$			22.2349i			0.822388i
$732$		0			0
$733$		−	9.63811i		−	0.355992i	−0.984031	−	0.177996i	$-0.943039\pi$
$733$		−	9.63811i		−	0.355992i	0.984031	−	0.177996i	$-0.0569614\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			8.28772i			0.305282i
$738$		0			0
$739$	−27.1716			−0.999522			−0.499761	−	0.866163i	$-0.666579\pi$
$739$	−27.1716			−0.999522			−0.499761	+	0.866163i	$0.666579\pi$
$740$		0			0
$741$	2.08644	−	6.34315i	0.0766474	−	0.233021i
$742$		0			0
$743$	−15.2304			−0.558750			−0.279375	−	0.960182i	$-0.590127\pi$
$743$	−15.2304			−0.558750			−0.279375	+	0.960182i	$0.590127\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	16.6871	−	22.6215i	0.610551	−	0.827676i
$748$		0			0
$749$	27.8832			1.01883
$750$		0			0
$751$		−	35.1127i		−	1.28128i	−0.767841	−	0.640640i	$-0.778669\pi$
$751$		−	35.1127i		−	1.28128i	0.767841	−	0.640640i	$-0.221331\pi$
$752$		0			0
$753$	6.34746	−	19.2974i	0.231314	−	0.703235i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			12.8319i			0.466383i	0.972431	+	0.233192i	$0.0749170\pi$
$757$			12.8319i			0.466383i	−0.972431	+	0.233192i	$0.925083\pi$
$758$		0			0
$759$	−3.56178	+	10.8284i	−0.129284	+	0.393047i
$760$		0			0
$761$			17.1978i			0.623420i	0.950177	+	0.311710i	$0.100902\pi$
$761$			17.1978i			0.623420i	−0.950177	+	0.311710i	$0.899098\pi$
$762$		0			0
$763$	1.59689			0.0578115
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.7206			0.423207
$768$		0			0
$769$	49.5980			1.78855			0.894274	−	0.447519i	$-0.147692\pi$
$769$	49.5980			1.78855			0.894274	+	0.447519i	$0.147692\pi$
$770$		0			0
$771$	8.82843	−	26.8399i	0.317948	−	0.966616i
$772$		0			0
$773$	−18.8490			−0.677952			−0.338976	−	0.940795i	$-0.610081\pi$
$773$	−18.8490			−0.677952			−0.338976	+	0.940795i	$0.610081\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−10.4366	−	3.43289i	−0.374410	−	0.123154i
$778$		0			0
$779$			7.12356i			0.255228i
$780$		0			0
$781$		−	17.9411i		−	0.641984i
$782$		0			0
$783$	−18.2919	−	25.7607i	−0.653699	−	0.920611i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−48.2307			−1.71924			−0.859619	−	0.510935i	$-0.829299\pi$
$787$	−48.2307			−1.71924			−0.859619	+	0.510935i	$0.829299\pi$
$788$		0			0
$789$	−36.6086	−	12.0416i	−1.30330	−	0.428693i
$790$		0			0
$791$	29.3585			1.04387
$792$		0			0
$793$	−39.4879			−1.40226
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.92177			0.316025			0.158013	−	0.987437i	$-0.449491\pi$
$797$	8.92177			0.316025			0.158013	+	0.987437i	$0.449491\pi$
$798$		0			0
$799$			9.65685i			0.341635i
$800$		0			0
$801$	−8.97056	+	12.1607i	−0.316959	+	0.429677i
$802$		0			0
$803$		−	16.5754i		−	0.584935i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−15.8645	−	5.21828i	−0.558456	−	0.183692i
$808$		0			0
$809$			4.17289i			0.146711i	0.997306	+	0.0733555i	$0.0233708\pi$
$809$			4.17289i			0.146711i	−0.997306	+	0.0733555i	$0.976629\pi$
$810$		0			0
$811$	14.4853			0.508647			0.254324	−	0.967119i	$-0.418147\pi$
$811$	14.4853			0.508647			0.254324	+	0.967119i	$0.418147\pi$
$812$		0			0
$813$	−7.65367	+	23.2685i	−0.268426	+	0.816061i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	4.98442			0.174383
$818$		0			0
$819$	36.9706	+	27.2720i	1.29186	+	0.952962i
$820$		0			0
$821$	23.8893			0.833741			0.416870	−	0.908966i	$-0.363127\pi$
$821$	23.8893			0.833741			0.416870	+	0.908966i	$0.363127\pi$
$822$		0			0
$823$		−	23.5023i		−	0.819239i	−0.912256	−	0.409620i	$-0.865661\pi$
$823$		−	23.5023i		−	0.819239i	0.912256	−	0.409620i	$-0.134339\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			35.5014i			1.23450i	0.786766	+	0.617252i	$0.211754\pi$
$827$			35.5014i			1.23450i	−0.786766	+	0.617252i	$0.788246\pi$
$828$		0			0
$829$			5.17157i			0.179616i	0.995959	+	0.0898081i	$0.0286254\pi$
$829$			5.17157i			0.179616i	−0.995959	+	0.0898081i	$0.971375\pi$
$830$		0			0
$831$	−8.16679	+	24.8284i	−0.283303	+	0.861289i
$832$		0			0
$833$		−	14.1480i		−	0.490200i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.96362	+	3.52452i	−0.171568	+	0.121825i
$838$		0			0
$839$	53.6799			1.85323			0.926617	−	0.376006i	$-0.122703\pi$
$839$	53.6799			1.85323			0.926617	+	0.376006i	$0.122703\pi$
$840$		0			0
$841$	7.97056			0.274847
$842$		0			0
$843$	1.92762	−	5.86030i	0.0663908	−	0.201840i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			15.3241i			0.526543i
$848$		0			0
$849$	−29.6985	−	9.76869i	−1.01925	−	0.335261i
$850$		0			0
$851$			5.03712i			0.172670i
$852$		0			0
$853$		−	36.4311i		−	1.24738i	−0.781673	−	0.623688i	$-0.785633\pi$
$853$		−	36.4311i		−	1.24738i	0.781673	−	0.623688i	$-0.214367\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	28.5587i		−	0.975546i	−0.872971	−	0.487773i	$-0.837809\pi$
$857$		−	28.5587i		−	0.975546i	0.872971	−	0.487773i	$-0.162191\pi$
$858$		0			0
$859$	3.85786			0.131629			0.0658143	−	0.997832i	$-0.479035\pi$
$859$	3.85786			0.131629			0.0658143	+	0.997832i	$0.479035\pi$
$860$		0			0
$861$	−46.5563	−	15.3137i	−1.58663	−	0.521890i
$862$		0			0
$863$	20.0852			0.683710			0.341855	−	0.939753i	$-0.388945\pi$
$863$	20.0852			0.683710			0.341855	+	0.939753i	$0.388945\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	5.50057	+	1.80930i	0.186809	+	0.0614470i
$868$		0			0
$869$	−41.5192			−1.40844
$870$		0			0
$871$		−	15.3137i		−	0.518885i
$872$		0			0
$873$	−6.58132	−	4.85483i	−0.222744	−	0.164311i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	35.3019i		−	1.19206i	−0.802962	−	0.596031i	$-0.796744\pi$
$877$		−	35.3019i		−	1.19206i	0.802962	−	0.596031i	$-0.203256\pi$
$878$		0			0
$879$	−3.99390	−	1.31371i	−0.134711	−	0.0443103i
$880$		0			0
$881$			42.9945i			1.44852i	0.689526	+	0.724261i	$0.257819\pi$
$881$			42.9945i			1.44852i	−0.689526	+	0.724261i	$0.742181\pi$
$882$		0			0
$883$	−37.7941			−1.27187			−0.635937	−	0.771741i	$-0.719386\pi$
$883$	−37.7941			−1.27187			−0.635937	+	0.771741i	$0.719386\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.5404			0.421064			0.210532	−	0.977587i	$-0.432480\pi$
$887$	12.5404			0.421064			0.210532	+	0.977587i	$0.432480\pi$
$888$		0			0
$889$	10.8284			0.363174
$890$		0			0
$891$	−21.6569	+	6.69145i	−0.725532	+	0.224172i
$892$		0			0
$893$	2.16478			0.0724417
$894$		0			0
$895$		0			0
$896$		0			0
$897$	6.58132	−	20.0083i	0.219744	−	0.668059i
$898$		0			0
$899$		−	7.12356i		−	0.237584i
$900$		0			0
$901$		−	16.9706i		−	0.565371i
$902$		0			0
$903$	−10.7151	+	32.5758i	−0.356577	+	1.08406i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−7.61362			−0.252806			−0.126403	−	0.991979i	$-0.540343\pi$
$907$	−7.61362			−0.252806			−0.126403	+	0.991979i	$0.540343\pi$
$908$		0			0
$909$	−31.8771	−	23.5147i	−1.05730	−	0.779934i
$910$		0			0
$911$	2.95068			0.0977603			0.0488801	−	0.998805i	$-0.484435\pi$
$911$	2.95068			0.0977603			0.0488801	+	0.998805i	$0.484435\pi$
$912$		0			0
$913$	23.5992			0.781019
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−71.7456			−2.36925
$918$		0			0
$919$		−	6.14214i		−	0.202610i	−0.994855	−	0.101305i	$-0.967698\pi$
$919$		−	6.14214i		−	0.202610i	0.994855	−	0.101305i	$-0.0323019\pi$
$920$		0			0
$921$	−11.7574	−	3.86733i	−0.387418	−	0.127433i
$922$		0			0
$923$			33.1509i			1.09117i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−10.7151	+	14.5257i	−0.351931	+	0.477085i
$928$		0			0
$929$			26.6609i			0.874717i	0.899287	+	0.437359i	$0.144086\pi$
$929$			26.6609i			0.874717i	−0.899287	+	0.437359i	$0.855914\pi$
$930$		0			0
$931$	−3.17157			−0.103944
$932$		0			0
$933$	56.5921	+	18.6148i	1.85274	+	0.609420i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	8.17821			0.267170			0.133585	−	0.991037i	$-0.457351\pi$
$937$	8.17821			0.267170			0.133585	+	0.991037i	$0.457351\pi$
$938$		0			0
$939$	−28.0000	−	9.21001i	−0.913745	−	0.300557i
$940$		0			0
$941$	−23.2781			−0.758846			−0.379423	−	0.925223i	$-0.623877\pi$
$941$	−23.2781			−0.758846			−0.379423	+	0.925223i	$0.623877\pi$
$942$		0			0
$943$			22.4700i			0.731724i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	21.6160i		−	0.702425i	−0.936296	−	0.351213i	$-0.885769\pi$
$947$		−	21.6160i		−	0.702425i	0.936296	−	0.351213i	$-0.114231\pi$
$948$		0			0
$949$			30.6274i			0.994208i
$950$		0			0
$951$	21.8028	+	7.17157i	0.707005	+	0.232554i
$952$		0			0
$953$			44.0836i			1.42801i	0.700142	+	0.714004i	$0.253120\pi$
$953$			44.0836i			1.42801i	−0.700142	+	0.714004i	$0.746880\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	8.28772	−	25.1961i	0.267904	−	0.814474i
$958$		0			0
$959$	39.4327			1.27335
$960$		0			0
$961$	29.6274			0.955723
$962$		0			0
$963$	15.0903	−	20.4567i	0.486277	−	0.659207i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	55.2797i		−	1.77768i	−0.458222	−	0.888838i	$-0.651513\pi$
$967$		−	55.2797i		−	1.77768i	0.458222	−	0.888838i	$-0.348487\pi$
$968$		0			0
$969$	−1.65685	+	5.03712i	−0.0532258	+	0.161816i
$970$		0			0
$971$			13.8150i			0.443345i	0.975121	+	0.221672i	$0.0711516\pi$
$971$			13.8150i			0.443345i	−0.975121	+	0.221672i	$0.928848\pi$
$972$		0			0
$973$			47.6661i			1.52811i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			33.2597i			1.06407i	0.846722	+	0.532035i	$0.178573\pi$
$977$			33.2597i			1.06407i	−0.846722	+	0.532035i	$0.821427\pi$
$978$		0			0
$979$	−12.6863			−0.405456
$980$		0			0
$981$	0.864233	−	1.17157i	0.0275928	−	0.0374054i
$982$		0			0
$983$	10.0042			0.319083			0.159542	−	0.987191i	$-0.448998\pi$
$983$	10.0042			0.319083			0.159542	+	0.987191i	$0.448998\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−4.65369	+	14.1480i	−0.148129	+	0.450336i
$988$		0			0
$989$	15.7225			0.499945
$990$		0			0
$991$			19.7990i			0.628936i	0.949268	+	0.314468i	$0.101826\pi$
$991$			19.7990i			0.628936i	−0.949268	+	0.314468i	$0.898174\pi$
$992$		0			0
$993$	30.4144	+	10.0042i	0.965171	+	0.317472i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	8.50894i		−	0.269481i	−0.990881	−	0.134740i	$-0.956980\pi$
$997$		−	8.50894i		−	0.269481i	0.990881	−	0.134740i	$-0.0430200\pi$
$998$		0			0
$999$	−8.16679	+	5.79899i	−0.258386	+	0.183472i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.b.i.2351.16		16
3.2	odd	2	inner	2400.2.b.i.2351.14		16
4.3	odd	2		600.2.b.i.251.5		16
5.2	odd	4		480.2.m.b.239.9		16
5.3	odd	4		480.2.m.b.239.7		16
5.4	even	2	inner	2400.2.b.i.2351.1		16
8.3	odd	2	inner	2400.2.b.i.2351.15		16
8.5	even	2		600.2.b.i.251.9		16
12.11	even	2		600.2.b.i.251.11		16
15.2	even	4		480.2.m.b.239.6		16
15.8	even	4		480.2.m.b.239.12		16
15.14	odd	2	inner	2400.2.b.i.2351.3		16
20.3	even	4		120.2.m.b.59.3	yes	16
20.7	even	4		120.2.m.b.59.14	yes	16
20.19	odd	2		600.2.b.i.251.12		16
24.5	odd	2		600.2.b.i.251.7		16
24.11	even	2	inner	2400.2.b.i.2351.13		16
40.3	even	4		480.2.m.b.239.8		16
40.13	odd	4		120.2.m.b.59.1	✓	16
40.19	odd	2	inner	2400.2.b.i.2351.2		16
40.27	even	4		480.2.m.b.239.10		16
40.29	even	2		600.2.b.i.251.8		16
40.37	odd	4		120.2.m.b.59.16	yes	16
60.23	odd	4		120.2.m.b.59.13	yes	16
60.47	odd	4		120.2.m.b.59.4	yes	16
60.59	even	2		600.2.b.i.251.6		16
120.29	odd	2		600.2.b.i.251.10		16
120.53	even	4		120.2.m.b.59.15	yes	16
120.59	even	2	inner	2400.2.b.i.2351.4		16
120.77	even	4		120.2.m.b.59.2	yes	16
120.83	odd	4		480.2.m.b.239.11		16
120.107	odd	4		480.2.m.b.239.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.m.b.59.1	✓	16	40.13	odd	4
120.2.m.b.59.2	yes	16	120.77	even	4
120.2.m.b.59.3	yes	16	20.3	even	4
120.2.m.b.59.4	yes	16	60.47	odd	4
120.2.m.b.59.13	yes	16	60.23	odd	4
120.2.m.b.59.14	yes	16	20.7	even	4
120.2.m.b.59.15	yes	16	120.53	even	4
120.2.m.b.59.16	yes	16	40.37	odd	4
480.2.m.b.239.5		16	120.107	odd	4
480.2.m.b.239.6		16	15.2	even	4
480.2.m.b.239.7		16	5.3	odd	4
480.2.m.b.239.8		16	40.3	even	4
480.2.m.b.239.9		16	5.2	odd	4
480.2.m.b.239.10		16	40.27	even	4
480.2.m.b.239.11		16	120.83	odd	4
480.2.m.b.239.12		16	15.8	even	4
600.2.b.i.251.5		16	4.3	odd	2
600.2.b.i.251.6		16	60.59	even	2
600.2.b.i.251.7		16	24.5	odd	2
600.2.b.i.251.8		16	40.29	even	2
600.2.b.i.251.9		16	8.5	even	2
600.2.b.i.251.10		16	120.29	odd	2
600.2.b.i.251.11		16	12.11	even	2
600.2.b.i.251.12		16	20.19	odd	2
2400.2.b.i.2351.1		16	5.4	even	2	inner
2400.2.b.i.2351.2		16	40.19	odd	2	inner
2400.2.b.i.2351.3		16	15.14	odd	2	inner
2400.2.b.i.2351.4		16	120.59	even	2	inner
2400.2.b.i.2351.13		16	24.11	even	2	inner
2400.2.b.i.2351.14		16	3.2	odd	2	inner
2400.2.b.i.2351.15		16	8.3	odd	2	inner
2400.2.b.i.2351.16		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.64533 + 0.541196i) q^{3} +3.29066i q^{7} +(2.41421 + 1.78089i) q^{9} +O(q^{10})\)	\(q+(1.64533 + 0.541196i) q^{3} +3.29066i q^{7} +(2.41421 + 1.78089i) q^{9} +2.51856i q^{11} -4.65369i q^{13} +3.69552i q^{17} +0.828427 q^{19} +(-1.78089 + 5.41421i) q^{21} +2.61313 q^{23} +(3.00836 + 4.23671i) q^{27} -6.08034 q^{29} +1.17157i q^{31} +(-1.36303 + 4.14386i) q^{33} +1.92762i q^{37} +(2.51856 - 7.65685i) q^{39} +8.59890i q^{41} +6.01673 q^{43} +2.61313 q^{47} -3.82843 q^{49} +(-2.00000 + 6.08034i) q^{51} -4.59220 q^{53} +(1.36303 + 0.448342i) q^{57} +2.51856i q^{59} -8.48528i q^{61} +(-5.86030 + 7.94435i) q^{63} +3.29066 q^{67} +(4.29945 + 1.41421i) q^{69} -7.12356 q^{71} -6.58132 q^{73} -8.28772 q^{77} +16.4853i q^{79} +(2.65685 + 8.59890i) q^{81} -9.37011i q^{83} +(-10.0042 - 3.29066i) q^{87} +5.03712i q^{89} +15.3137 q^{91} +(-0.634051 + 1.92762i) q^{93} -2.72607 q^{97} +(-4.48528 + 6.08034i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} - 32 q^{19} - 16 q^{49} - 32 q^{51} - 48 q^{81} + 64 q^{91} + 64 q^{99}+O(q^{100}) \) 16 * q + 16 * q^9 - 32 * q^19 - 16 * q^49 - 32 * q^51 - 48 * q^81 + 64 * q^91 + 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more