LMFDB - Embedded newform 1456.2.r.j.417.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(417,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))

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

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

chi := DirichletCharacter("1456.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.r (of order $3$, degree $2$, 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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			417.2
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	1456.417
Dual form			1456.2.r.j.625.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.11803 - 1.93649i) q^{3} +(1.11803 + 1.93649i) q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{9} +O(q^{10})$	\(q+(1.11803 - 1.93649i) q^{3} +(1.11803 + 1.93649i) q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{9} +(-1.50000 + 2.59808i) q^{11} -1.00000 q^{13} +5.00000 q^{15} +(3.73607 - 6.47106i) q^{17} +(1.50000 + 2.59808i) q^{19} +(-1.11803 - 5.80948i) q^{21} +(-1.88197 - 3.25966i) q^{23} +2.23607 q^{27} -4.47214 q^{29} +(2.50000 - 4.33013i) q^{31} +(3.35410 + 5.80948i) q^{33} +(5.59017 + 1.93649i) q^{35} +(4.35410 + 7.54153i) q^{37} +(-1.11803 + 1.93649i) q^{39} +4.47214 q^{41} +8.00000 q^{43} +(2.23607 - 3.87298i) q^{45} +(0.736068 + 1.27491i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-8.35410 - 14.4697i) q^{51} +(-0.736068 + 1.27491i) q^{53} -6.70820 q^{55} +6.70820 q^{57} +(3.73607 - 6.47106i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(-5.00000 - 1.73205i) q^{63} +(-1.11803 - 1.93649i) q^{65} +(-1.50000 + 2.59808i) q^{67} -8.41641 q^{69} -8.94427 q^{71} +(-5.35410 + 9.27358i) q^{73} +(1.50000 + 7.79423i) q^{77} +(5.35410 + 9.27358i) q^{79} +(5.50000 - 9.52628i) q^{81} +16.7082 q^{85} +(-5.00000 + 8.66025i) q^{87} +(1.11803 + 1.93649i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(-5.59017 - 9.68246i) q^{93} +(-3.35410 + 5.80948i) q^{95} -17.4164 q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 8 * q^7 - 4 * q^9	$ 4 q + 8 q^{7} - 4 q^{9} - 6 q^{11} - 4 q^{13} + 20 q^{15} + 6 q^{17} + 6 q^{19} - 12 q^{23} + 10 q^{31} + 4 q^{37} + 32 q^{43} - 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} + 6 q^{59} - 6 q^{61} - 20 q^{63} - 6 q^{67} + 20 q^{69} - 8 q^{73} + 6 q^{77} + 8 q^{79} + 22 q^{81} + 40 q^{85} - 20 q^{87} - 8 q^{91} - 16 q^{97} + 24 q^{99}+O(q^{100}) $ 4 * q + 8 * q^7 - 4 * q^9 - 6 * q^11 - 4 * q^13 + 20 * q^15 + 6 * q^17 + 6 * q^19 - 12 * q^23 + 10 * q^31 + 4 * q^37 + 32 * q^43 - 6 * q^47 + 4 * q^49 - 20 * q^51 + 6 * q^53 + 6 * q^59 - 6 * q^61 - 20 * q^63 - 6 * q^67 + 20 * q^69 - 8 * q^73 + 6 * q^77 + 8 * q^79 + 22 * q^81 + 40 * q^85 - 20 * q^87 - 8 * q^91 - 16 * q^97 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	1.11803	−	1.93649i	0.645497	−	1.11803i	−0.338689	−	0.940898i	$-0.609984\pi$
$3$	1.11803	−	1.93649i	0.645497	−	1.11803i	0.984186	−	0.177136i	$-0.0566831\pi$
$4$		0			0
$5$	1.11803	+	1.93649i	0.500000	+	0.866025i	1.00000			$0$
$5$	1.11803	+	1.93649i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	−1.00000	−	1.73205i	−0.333333	−	0.577350i
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	5.00000			1.29099
$16$		0			0
$17$	3.73607	−	6.47106i	0.906130	−	1.56946i	0.0867359	−	0.996231i	$-0.472356\pi$
$17$	3.73607	−	6.47106i	0.906130	−	1.56946i	0.819394	−	0.573231i	$-0.194310\pi$
$18$		0			0
$19$	1.50000	+	2.59808i	0.344124	+	0.596040i	0.985194	−	0.171442i	$-0.0548427\pi$
$19$	1.50000	+	2.59808i	0.344124	+	0.596040i	−0.641071	+	0.767482i	$0.721509\pi$
$20$		0			0
$21$	−1.11803	−	5.80948i	−0.243975	−	1.26773i
$22$		0			0
$23$	−1.88197	−	3.25966i	−0.392417	−	0.679686i	0.600351	−	0.799737i	$-0.295028\pi$
$23$	−1.88197	−	3.25966i	−0.392417	−	0.679686i	−0.992768	+	0.120051i	$0.961694\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.23607			0.430331
$28$		0			0
$29$	−4.47214			−0.830455			−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214			−0.830455			−0.415227	+	0.909718i	$0.636298\pi$
$30$		0			0
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	0.998322	+	0.0579057i	$0.0184423\pi$
$32$		0			0
$33$	3.35410	+	5.80948i	0.583874	+	1.01130i
$34$		0			0
$35$	5.59017	+	1.93649i	0.944911	+	0.327327i
$36$		0			0
$37$	4.35410	+	7.54153i	0.715810	+	1.23982i	0.962646	+	0.270762i	$0.0872757\pi$
$37$	4.35410	+	7.54153i	0.715810	+	1.23982i	−0.246836	+	0.969057i	$0.579391\pi$
$38$		0			0
$39$	−1.11803	+	1.93649i	−0.179029	+	0.310087i
$40$		0			0
$41$	4.47214			0.698430			0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214			0.698430			0.349215	+	0.937043i	$0.386448\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$	2.23607	−	3.87298i	0.333333	−	0.577350i
$46$		0			0
$47$	0.736068	+	1.27491i	0.107367	+	0.185964i	0.914703	−	0.404128i	$-0.132425\pi$
$47$	0.736068	+	1.27491i	0.107367	+	0.185964i	−0.807336	+	0.590092i	$0.799091\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	−8.35410	−	14.4697i	−1.16981	−	2.02617i
$52$		0			0
$53$	−0.736068	+	1.27491i	−0.101107	+	0.175122i	−0.912141	−	0.409877i	$-0.865572\pi$
$53$	−0.736068	+	1.27491i	−0.101107	+	0.175122i	0.811034	+	0.584999i	$0.198905\pi$
$54$		0			0
$55$	−6.70820			−0.904534
$56$		0			0
$57$	6.70820			0.888523
$58$		0			0
$59$	3.73607	−	6.47106i	0.486395	−	0.842460i	−0.513483	−	0.858100i	$-0.671645\pi$
$59$	3.73607	−	6.47106i	0.486395	−	0.842460i	0.999878	+	0.0156395i	$0.00497842\pi$
$60$		0			0
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	0.753876	−	0.657017i	$-0.228182\pi$
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	−0.945931	+	0.324367i	$0.894849\pi$
$62$		0			0
$63$	−5.00000	−	1.73205i	−0.629941	−	0.218218i
$64$		0			0
$65$	−1.11803	−	1.93649i	−0.138675	−	0.240192i
$66$		0			0
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	−0.942987	−	0.332830i	$-0.891996\pi$
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	0.759733	+	0.650236i	$0.225330\pi$
$68$		0			0
$69$	−8.41641			−1.01322
$70$		0			0
$71$	−8.94427			−1.06149			−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427			−1.06149			−0.530745	+	0.847532i	$0.678088\pi$
$72$		0			0
$73$	−5.35410	+	9.27358i	−0.626650	+	1.08539i	0.361569	+	0.932345i	$0.382241\pi$
$73$	−5.35410	+	9.27358i	−0.626650	+	1.08539i	−0.988219	+	0.153045i	$0.951092\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.50000	+	7.79423i	0.170941	+	0.888235i
$78$		0			0
$79$	5.35410	+	9.27358i	0.602384	+	1.04336i	0.992459	+	0.122576i	$0.0391155\pi$
$79$	5.35410	+	9.27358i	0.602384	+	1.04336i	−0.390076	+	0.920783i	$0.627551\pi$
$80$		0			0
$81$	5.50000	−	9.52628i	0.611111	−	1.05848i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	16.7082			1.81226
$86$		0			0
$87$	−5.00000	+	8.66025i	−0.536056	+	0.928477i
$88$		0			0
$89$	1.11803	+	1.93649i	0.118511	+	0.205268i	0.919178	−	0.393842i	$-0.128854\pi$
$89$	1.11803	+	1.93649i	0.118511	+	0.205268i	−0.800667	+	0.599110i	$0.795521\pi$
$90$		0			0
$91$	−2.00000	+	1.73205i	−0.209657	+	0.181568i
$92$		0			0
$93$	−5.59017	−	9.68246i	−0.579674	−	1.00402i
$94$		0			0
$95$	−3.35410	+	5.80948i	−0.344124	+	0.596040i
$96$		0			0
$97$	−17.4164			−1.76837			−0.884184	−	0.467139i	$-0.845285\pi$
$97$	−17.4164			−1.76837			−0.884184	+	0.467139i	$0.845285\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	−0.550474	−	0.834853i	$-0.685553\pi$
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	0.998240	+	0.0592978i	$0.0188862\pi$
$102$		0			0
$103$	5.35410	+	9.27358i	0.527555	+	0.913753i	0.999484	+	0.0321160i	$0.0102246\pi$
$103$	5.35410	+	9.27358i	0.527555	+	0.913753i	−0.471929	+	0.881637i	$0.656442\pi$
$104$		0			0
$105$	10.0000	−	8.66025i	0.975900	−	0.845154i
$106$		0			0
$107$	−7.11803	−	12.3288i	−0.688126	−	1.19187i	−0.972443	−	0.233139i	$-0.925100\pi$
$107$	−7.11803	−	12.3288i	−0.688126	−	1.19187i	0.284317	−	0.958730i	$-0.408233\pi$
$108$		0			0
$109$	−5.35410	+	9.27358i	−0.512830	+	0.888248i	0.487059	+	0.873369i	$0.338070\pi$
$109$	−5.35410	+	9.27358i	−0.512830	+	0.888248i	−0.999889	+	0.0148787i	$0.995264\pi$
$110$		0			0
$111$	19.4721			1.84821
$112$		0			0
$113$	−14.9443			−1.40584			−0.702919	−	0.711269i	$-0.748121\pi$
$113$	−14.9443			−1.40584			−0.702919	+	0.711269i	$0.748121\pi$
$114$		0			0
$115$	4.20820	−	7.28882i	0.392417	−	0.679686i
$116$		0			0
$117$	1.00000	+	1.73205i	0.0924500	+	0.160128i
$118$		0			0
$119$	−3.73607	−	19.4132i	−0.342485	−	1.77960i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$	5.00000	−	8.66025i	0.450835	−	0.780869i
$124$		0			0
$125$	11.1803			1.00000
$126$		0			0
$127$	−15.4164			−1.36798			−0.683992	−	0.729489i	$-0.739758\pi$
$127$	−15.4164			−1.36798			−0.683992	+	0.729489i	$0.739758\pi$
$128$		0			0
$129$	8.94427	−	15.4919i	0.787499	−	1.36399i
$130$		0			0
$131$	−1.88197	−	3.25966i	−0.164428	−	0.284798i	0.772024	−	0.635593i	$-0.219245\pi$
$131$	−1.88197	−	3.25966i	−0.164428	−	0.284798i	−0.936452	+	0.350796i	$0.885911\pi$
$132$		0			0
$133$	7.50000	+	2.59808i	0.650332	+	0.225282i
$134$		0			0
$135$	2.50000	+	4.33013i	0.215166	+	0.372678i
$136$		0			0
$137$	1.88197	−	3.25966i	0.160787	−	0.278492i	−0.774364	−	0.632740i	$-0.781930\pi$
$137$	1.88197	−	3.25966i	0.160787	−	0.278492i	0.935151	+	0.354249i	$0.115263\pi$
$138$		0			0
$139$	−3.41641			−0.289776			−0.144888	−	0.989448i	$-0.546282\pi$
$139$	−3.41641			−0.289776			−0.144888	+	0.989448i	$0.546282\pi$
$140$		0			0
$141$	3.29180			0.277219
$142$		0			0
$143$	1.50000	−	2.59808i	0.125436	−	0.217262i
$144$		0			0
$145$	−5.00000	−	8.66025i	−0.415227	−	0.719195i
$146$		0			0
$147$	−12.2984	−	9.68246i	−1.01435	−	0.798596i
$148$		0			0
$149$	6.35410	+	11.0056i	0.520548	+	0.901616i	0.999715	+	0.0238920i	$0.00760577\pi$
$149$	6.35410	+	11.0056i	0.520548	+	0.901616i	−0.479166	+	0.877724i	$0.659061\pi$
$150$		0			0
$151$	−3.20820	+	5.55677i	−0.261080	+	0.452204i	−0.966529	−	0.256557i	$-0.917412\pi$
$151$	−3.20820	+	5.55677i	−0.261080	+	0.452204i	0.705449	+	0.708760i	$0.250745\pi$
$152$		0			0
$153$	−14.9443			−1.20817
$154$		0			0
$155$	11.1803			0.898027
$156$		0			0
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	−0.971219	−	0.238190i	$-0.923446\pi$
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	0.691888	+	0.722005i	$0.256779\pi$
$158$		0			0
$159$	1.64590	+	2.85078i	0.130528	+	0.226081i
$160$		0			0
$161$	−9.40983	−	3.25966i	−0.741598	−	0.256897i
$162$		0			0
$163$	5.20820	+	9.02087i	0.407938	+	0.706569i	0.994659	−	0.103220i	$-0.0329146\pi$
$163$	5.20820	+	9.02087i	0.407938	+	0.706569i	−0.586721	+	0.809789i	$0.699581\pi$
$164$		0			0
$165$	−7.50000	+	12.9904i	−0.583874	+	1.01130i
$166$		0			0
$167$	−13.5279			−1.04682			−0.523409	−	0.852082i	$-0.675340\pi$
$167$	−13.5279			−1.04682			−0.523409	+	0.852082i	$0.675340\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	3.00000	−	5.19615i	0.229416	−	0.397360i
$172$		0			0
$173$	−5.20820	−	9.02087i	−0.395972	−	0.685844i	0.597252	−	0.802053i	$-0.296259\pi$
$173$	−5.20820	−	9.02087i	−0.395972	−	0.685844i	−0.993225	+	0.116209i	$0.962926\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−8.35410	−	14.4697i	−0.627933	−	1.08761i
$178$		0			0
$179$	10.0623	−	17.4284i	0.752092	−	1.30266i	−0.194715	−	0.980860i	$-0.562378\pi$
$179$	10.0623	−	17.4284i	0.752092	−	1.30266i	0.946807	−	0.321802i	$-0.104288\pi$
$180$		0			0
$181$	1.41641			0.105281			0.0526404	−	0.998614i	$-0.483236\pi$
$181$	1.41641			0.105281			0.0526404	+	0.998614i	$0.483236\pi$
$182$		0			0
$183$	−6.70820			−0.495885
$184$		0			0
$185$	−9.73607	+	16.8634i	−0.715810	+	1.23982i
$186$		0			0
$187$	11.2082	+	19.4132i	0.819625	+	1.41963i
$188$		0			0
$189$	4.47214	−	3.87298i	0.325300	−	0.281718i
$190$		0			0
$191$	−5.59017	−	9.68246i	−0.404491	−	0.700598i	0.589772	−	0.807570i	$-0.299218\pi$
$191$	−5.59017	−	9.68246i	−0.404491	−	0.700598i	−0.994262	+	0.106972i	$0.965884\pi$
$192$		0			0
$193$	6.35410	−	11.0056i	0.457378	−	0.792202i	−0.541443	−	0.840737i	$-0.682122\pi$
$193$	6.35410	−	11.0056i	0.457378	−	0.792202i	0.998821	+	0.0485349i	$0.0154552\pi$
$194$		0			0
$195$	−5.00000			−0.358057
$196$		0			0
$197$	−26.9443			−1.91970			−0.959850	−	0.280514i	$-0.909495\pi$
$197$	−26.9443			−1.91970			−0.959850	+	0.280514i	$0.909495\pi$
$198$		0			0
$199$	−3.64590	+	6.31488i	−0.258451	+	0.447650i	−0.965827	−	0.259187i	$-0.916545\pi$
$199$	−3.64590	+	6.31488i	−0.258451	+	0.447650i	0.707376	+	0.706837i	$0.249879\pi$
$200$		0			0
$201$	3.35410	+	5.80948i	0.236580	+	0.409769i
$202$		0			0
$203$	−8.94427	+	7.74597i	−0.627765	+	0.543660i
$204$		0			0
$205$	5.00000	+	8.66025i	0.349215	+	0.604858i
$206$		0			0
$207$	−3.76393	+	6.51932i	−0.261611	+	0.453124i
$208$		0			0
$209$	−9.00000			−0.622543
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	−10.0000	+	17.3205i	−0.685189	+	1.18678i
$214$		0			0
$215$	8.94427	+	15.4919i	0.609994	+	1.05654i
$216$		0			0
$217$	−2.50000	−	12.9904i	−0.169711	−	0.881845i
$218$		0			0
$219$	11.9721	+	20.7363i	0.809002	+	1.40123i
$220$		0			0
$221$	−3.73607	+	6.47106i	−0.251315	+	0.435291i
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.97214	+	10.3440i	−0.396385	+	0.686558i	−0.993277	−	0.115763i	$-0.963069\pi$
$227$	−5.97214	+	10.3440i	−0.396385	+	0.686558i	0.596892	+	0.802321i	$0.296402\pi$
$228$		0			0
$229$	8.06231	+	13.9643i	0.532772	+	0.922788i	0.999268	+	0.0382649i	$0.0121831\pi$
$229$	8.06231	+	13.9643i	0.532772	+	0.922788i	−0.466495	+	0.884524i	$0.654484\pi$
$230$		0			0
$231$	16.7705	+	5.80948i	1.10342	+	0.382235i
$232$		0			0
$233$	2.97214	+	5.14789i	0.194711	+	0.337250i	0.946806	−	0.321806i	$-0.104290\pi$
$233$	2.97214	+	5.14789i	0.194711	+	0.337250i	−0.752095	+	0.659055i	$0.770956\pi$
$234$		0			0
$235$	−1.64590	+	2.85078i	−0.107367	+	0.185964i
$236$		0			0
$237$	23.9443			1.55535
$238$		0			0
$239$	7.41641			0.479728			0.239864	−	0.970807i	$-0.422897\pi$
$239$	7.41641			0.479728			0.239864	+	0.970807i	$0.422897\pi$
$240$		0			0
$241$	4.35410	−	7.54153i	0.280472	−	0.485792i	−0.691029	−	0.722827i	$-0.742842\pi$
$241$	4.35410	−	7.54153i	0.280472	−	0.485792i	0.971501	+	0.237035i	$0.0761756\pi$
$242$		0			0
$243$	−8.94427	−	15.4919i	−0.573775	−	0.993808i
$244$		0			0
$245$	14.5344	−	5.80948i	0.928571	−	0.371154i
$246$		0			0
$247$	−1.50000	−	2.59808i	−0.0954427	−	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.4721			−0.660995			−0.330498	−	0.943807i	$-0.607217\pi$
$251$	−10.4721			−0.660995			−0.330498	+	0.943807i	$0.607217\pi$
$252$		0			0
$253$	11.2918			0.709909
$254$		0			0
$255$	18.6803	−	32.3553i	1.16981	−	2.02617i
$256$		0			0
$257$	8.97214	+	15.5402i	0.559666	+	0.969371i	0.997524	+	0.0703264i	$0.0224041\pi$
$257$	8.97214	+	15.5402i	0.559666	+	0.969371i	−0.437858	+	0.899044i	$0.644263\pi$
$258$		0			0
$259$	21.7705	+	7.54153i	1.35275	+	0.468608i
$260$		0			0
$261$	4.47214	+	7.74597i	0.276818	+	0.479463i
$262$		0			0
$263$	−7.06231	+	12.2323i	−0.435480	+	0.754274i	−0.997335	−	0.0729620i	$-0.976755\pi$
$263$	−7.06231	+	12.2323i	−0.435480	+	0.754274i	0.561854	+	0.827236i	$0.310088\pi$
$264$		0			0
$265$	−3.29180			−0.202213
$266$		0			0
$267$	5.00000			0.305995
$268$		0			0
$269$	−2.26393	+	3.92125i	−0.138034	+	0.239083i	−0.926753	−	0.375672i	$-0.877412\pi$
$269$	−2.26393	+	3.92125i	−0.138034	+	0.239083i	0.788718	+	0.614755i	$0.210745\pi$
$270$		0			0
$271$	3.20820	+	5.55677i	0.194885	+	0.337550i	0.946863	−	0.321638i	$-0.104233\pi$
$271$	3.20820	+	5.55677i	0.194885	+	0.337550i	−0.751978	+	0.659188i	$0.770900\pi$
$272$		0			0
$273$	1.11803	+	5.80948i	0.0676665	+	0.351605i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.2082	+	22.8773i	−0.793604	+	1.37456i	0.130118	+	0.991499i	$0.458464\pi$
$277$	−13.2082	+	22.8773i	−0.793604	+	1.37456i	−0.923722	+	0.383064i	$0.874869\pi$
$278$		0			0
$279$	−10.0000			−0.598684
$280$		0			0
$281$	9.05573			0.540219			0.270110	−	0.962830i	$-0.412940\pi$
$281$	9.05573			0.540219			0.270110	+	0.962830i	$0.412940\pi$
$282$		0			0
$283$	−7.06231	+	12.2323i	−0.419811	+	0.727133i	−0.995920	−	0.0902393i	$-0.971237\pi$
$283$	−7.06231	+	12.2323i	−0.419811	+	0.727133i	0.576110	+	0.817372i	$0.304570\pi$
$284$		0			0
$285$	7.50000	+	12.9904i	0.444262	+	0.769484i
$286$		0			0
$287$	8.94427	−	7.74597i	0.527964	−	0.457230i
$288$		0			0
$289$	−19.4164	−	33.6302i	−1.14214	−	1.97825i
$290$		0			0
$291$	−19.4721	+	33.7267i	−1.14148	+	1.97710i
$292$		0			0
$293$	2.94427			0.172006			0.0860031	−	0.996295i	$-0.472591\pi$
$293$	2.94427			0.172006			0.0860031	+	0.996295i	$0.472591\pi$
$294$		0			0
$295$	16.7082			0.972789
$296$		0			0
$297$	−3.35410	+	5.80948i	−0.194625	+	0.337100i
$298$		0			0
$299$	1.88197	+	3.25966i	0.108837	+	0.188511i
$300$		0			0
$301$	16.0000	−	13.8564i	0.922225	−	0.798670i
$302$		0			0
$303$	−10.0623	−	17.4284i	−0.578064	−	1.00124i
$304$		0			0
$305$	3.35410	−	5.80948i	0.192055	−	0.332650i
$306$		0			0
$307$	7.41641			0.423277			0.211638	−	0.977348i	$-0.432120\pi$
$307$	7.41641			0.423277			0.211638	+	0.977348i	$0.432120\pi$
$308$		0			0
$309$	23.9443			1.36214
$310$		0			0
$311$	−16.1180	+	27.9173i	−0.913970	+	1.58304i	−0.105567	+	0.994412i	$0.533666\pi$
$311$	−16.1180	+	27.9173i	−0.913970	+	1.58304i	−0.808403	+	0.588630i	$0.799667\pi$
$312$		0			0
$313$	16.2082	+	28.0734i	0.916142	+	1.58680i	0.805221	+	0.592975i	$0.202047\pi$
$313$	16.2082	+	28.0734i	0.916142	+	1.58680i	0.110921	+	0.993829i	$0.464620\pi$
$314$		0			0
$315$	−2.23607	−	11.6190i	−0.125988	−	0.654654i
$316$		0			0
$317$	1.88197	+	3.25966i	0.105702	+	0.183081i	0.914025	−	0.405659i	$-0.132958\pi$
$317$	1.88197	+	3.25966i	0.105702	+	0.183081i	−0.808323	+	0.588739i	$0.799624\pi$
$318$		0			0
$319$	6.70820	−	11.6190i	0.375587	−	0.650536i
$320$		0			0
$321$	−31.8328			−1.77673
$322$		0			0
$323$	22.4164			1.24728
$324$		0			0
$325$		0			0
$326$		0			0
$327$	11.9721	+	20.7363i	0.662061	+	1.14672i
$328$		0			0
$329$	3.68034	+	1.27491i	0.202904	+	0.0702879i
$330$		0			0
$331$	−14.2082	−	24.6093i	−0.780954	−	1.35265i	−0.931387	−	0.364031i	$-0.881400\pi$
$331$	−14.2082	−	24.6093i	−0.780954	−	1.35265i	0.150433	−	0.988620i	$-0.451933\pi$
$332$		0			0
$333$	8.70820	−	15.0831i	0.477207	−	0.826546i
$334$		0			0
$335$	−6.70820			−0.366508
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$		0			0
$339$	−16.7082	+	28.9395i	−0.907465	+	1.57178i
$340$		0			0
$341$	7.50000	+	12.9904i	0.406148	+	0.703469i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	−9.40983	−	16.2983i	−0.506608	−	0.877471i
$346$		0			0
$347$	−17.5344	+	30.3705i	−0.941298	+	1.63038i	−0.178299	+	0.983976i	$0.557060\pi$
$347$	−17.5344	+	30.3705i	−0.941298	+	1.63038i	−0.762999	+	0.646400i	$0.776274\pi$
$348$		0			0
$349$	−2.58359			−0.138297			−0.0691483	−	0.997606i	$-0.522028\pi$
$349$	−2.58359			−0.138297			−0.0691483	+	0.997606i	$0.522028\pi$
$350$		0			0
$351$	−2.23607			−0.119352
$352$		0			0
$353$	−15.3541	+	26.5941i	−0.817216	+	1.41546i	0.0905091	+	0.995896i	$0.471151\pi$
$353$	−15.3541	+	26.5941i	−0.817216	+	1.41546i	−0.907725	+	0.419565i	$0.862183\pi$
$354$		0			0
$355$	−10.0000	−	17.3205i	−0.530745	−	0.919277i
$356$		0			0
$357$	−41.7705	−	14.4697i	−2.21073	−	0.765819i
$358$		0			0
$359$	−2.97214	−	5.14789i	−0.156863	−	0.271695i	0.776873	−	0.629658i	$-0.216805\pi$
$359$	−2.97214	−	5.14789i	−0.156863	−	0.271695i	−0.933736	+	0.357963i	$0.883472\pi$
$360$		0			0
$361$	5.00000	−	8.66025i	0.263158	−	0.455803i
$362$		0			0
$363$	4.47214			0.234726
$364$		0			0
$365$	−23.9443			−1.25330
$366$		0			0
$367$	0.354102	−	0.613323i	0.0184840	−	0.0320152i	−0.856635	−	0.515922i	$-0.827449\pi$
$367$	0.354102	−	0.613323i	0.0184840	−	0.0320152i	0.875119	+	0.483907i	$0.160783\pi$
$368$		0			0
$369$	−4.47214	−	7.74597i	−0.232810	−	0.403239i
$370$		0			0
$371$	0.736068	+	3.82472i	0.0382147	+	0.198570i
$372$		0			0
$373$	14.2082	+	24.6093i	0.735673	+	1.27422i	0.954427	+	0.298443i	$0.0964673\pi$
$373$	14.2082	+	24.6093i	0.735673	+	1.27422i	−0.218755	+	0.975780i	$0.570199\pi$
$374$		0			0
$375$	12.5000	−	21.6506i	0.645497	−	1.11803i
$376$		0			0
$377$	4.47214			0.230327
$378$		0			0
$379$	−11.4164			−0.586421			−0.293211	−	0.956048i	$-0.594724\pi$
$379$	−11.4164			−0.586421			−0.293211	+	0.956048i	$0.594724\pi$
$380$		0			0
$381$	−17.2361	+	29.8537i	−0.883031	+	1.52945i
$382$		0			0
$383$	7.50000	+	12.9904i	0.383232	+	0.663777i	0.991522	−	0.129937i	$-0.0414776\pi$
$383$	7.50000	+	12.9904i	0.383232	+	0.663777i	−0.608290	+	0.793715i	$0.708144\pi$
$384$		0			0
$385$	−13.4164	+	11.6190i	−0.683763	+	0.592157i
$386$		0			0
$387$	−8.00000	−	13.8564i	−0.406663	−	0.704361i
$388$		0			0
$389$	3.73607	−	6.47106i	0.189426	−	0.328096i	−0.755633	−	0.654995i	$-0.772671\pi$
$389$	3.73607	−	6.47106i	0.189426	−	0.328096i	0.945059	+	0.326900i	$0.106004\pi$
$390$		0			0
$391$	−28.1246			−1.42232
$392$		0			0
$393$	−8.41641			−0.424552
$394$		0			0
$395$	−11.9721	+	20.7363i	−0.602384	+	1.04336i
$396$		0			0
$397$	−7.06231	−	12.2323i	−0.354447	−	0.613920i	0.632576	−	0.774498i	$-0.281998\pi$
$397$	−7.06231	−	12.2323i	−0.354447	−	0.613920i	−0.987023	+	0.160578i	$0.948664\pi$
$398$		0			0
$399$	13.4164	−	11.6190i	0.671660	−	0.581675i
$400$		0			0
$401$	−4.88197	−	8.45581i	−0.243794	−	0.422263i	0.717998	−	0.696045i	$-0.245059\pi$
$401$	−4.88197	−	8.45581i	−0.243794	−	0.422263i	−0.961792	+	0.273782i	$0.911725\pi$
$402$		0			0
$403$	−2.50000	+	4.33013i	−0.124534	+	0.215699i
$404$		0			0
$405$	24.5967			1.22222
$406$		0			0
$407$	−26.1246			−1.29495
$408$		0			0
$409$	2.35410	−	4.07742i	0.116403	−	0.201616i	−0.801937	−	0.597409i	$-0.796197\pi$
$409$	2.35410	−	4.07742i	0.116403	−	0.201616i	0.918340	+	0.395793i	$0.129530\pi$
$410$		0			0
$411$	−4.20820	−	7.28882i	−0.207575	−	0.359531i
$412$		0			0
$413$	−3.73607	−	19.4132i	−0.183840	−	0.955260i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.81966	+	6.61585i	−0.187050	+	0.323979i
$418$		0			0
$419$	−15.0557			−0.735520			−0.367760	−	0.929921i	$-0.619875\pi$
$419$	−15.0557			−0.735520			−0.367760	+	0.929921i	$0.619875\pi$
$420$		0			0
$421$	−13.4164			−0.653876			−0.326938	−	0.945046i	$-0.606017\pi$
$421$	−13.4164			−0.653876			−0.326938	+	0.945046i	$0.606017\pi$
$422$		0			0
$423$	1.47214	−	2.54981i	0.0715777	−	0.123976i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.50000	−	2.59808i	−0.362950	−	0.125730i
$428$		0			0
$429$	−3.35410	−	5.80948i	−0.161938	−	0.280484i
$430$		0			0
$431$	6.68034	−	11.5707i	0.321781	−	0.557340i	−0.659075	−	0.752077i	$-0.729052\pi$
$431$	6.68034	−	11.5707i	0.321781	−	0.557340i	0.980856	+	0.194737i	$0.0623853\pi$
$432$		0			0
$433$	2.58359			0.124160			0.0620798	−	0.998071i	$-0.480227\pi$
$433$	2.58359			0.124160			0.0620798	+	0.998071i	$0.480227\pi$
$434$		0			0
$435$	−22.3607			−1.07211
$436$		0			0
$437$	5.64590	−	9.77898i	0.270080	−	0.467792i
$438$		0			0
$439$	−8.06231	−	13.9643i	−0.384793	−	0.666481i	0.606948	−	0.794742i	$-0.292394\pi$
$439$	−8.06231	−	13.9643i	−0.384793	−	0.666481i	−0.991740	+	0.128261i	$0.959060\pi$
$440$		0			0
$441$	−13.0000	+	5.19615i	−0.619048	+	0.247436i
$442$		0			0
$443$	−1.11803	−	1.93649i	−0.0531194	−	0.0920055i	0.838243	−	0.545297i	$-0.183583\pi$
$443$	−1.11803	−	1.93649i	−0.0531194	−	0.0920055i	−0.891362	+	0.453291i	$0.850250\pi$
$444$		0			0
$445$	−2.50000	+	4.33013i	−0.118511	+	0.205268i
$446$		0			0
$447$	28.4164			1.34405
$448$		0			0
$449$	10.3607			0.488951			0.244475	−	0.969656i	$-0.421384\pi$
$449$	10.3607			0.488951			0.244475	+	0.969656i	$0.421384\pi$
$450$		0			0
$451$	−6.70820	+	11.6190i	−0.315877	+	0.547115i
$452$		0			0
$453$	7.17376	+	12.4253i	0.337053	+	0.583792i
$454$		0			0
$455$	−5.59017	−	1.93649i	−0.262071	−	0.0907841i
$456$		0			0
$457$	−17.0623	−	29.5528i	−0.798141	−	1.38242i	−0.920825	−	0.389975i	$-0.872484\pi$
$457$	−17.0623	−	29.5528i	−0.798141	−	1.38242i	0.122684	−	0.992446i	$-0.460850\pi$
$458$		0			0
$459$	8.35410	−	14.4697i	0.389936	−	0.675389i
$460$		0			0
$461$	−10.3607			−0.482545			−0.241272	−	0.970457i	$-0.577565\pi$
$461$	−10.3607			−0.482545			−0.241272	+	0.970457i	$0.577565\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$		0			0
$465$	12.5000	−	21.6506i	0.579674	−	1.00402i
$466$		0			0
$467$	−10.8262	−	18.7516i	−0.500979	−	0.867720i	−0.999999	−	0.00113029i	$-0.999640\pi$
$467$	−10.8262	−	18.7516i	−0.500979	−	0.867720i	0.499021	−	0.866590i	$-0.333693\pi$
$468$		0			0
$469$	1.50000	+	7.79423i	0.0692636	+	0.359904i
$470$		0			0
$471$	7.82624	+	13.5554i	0.360614	+	0.624602i
$472$		0			0
$473$	−12.0000	+	20.7846i	−0.551761	+	0.955677i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	2.94427			0.134809
$478$		0			0
$479$	−14.9164	+	25.8360i	−0.681548	+	1.18048i	0.292960	+	0.956125i	$0.405360\pi$
$479$	−14.9164	+	25.8360i	−0.681548	+	1.18048i	−0.974508	+	0.224351i	$0.927974\pi$
$480$		0			0
$481$	−4.35410	−	7.54153i	−0.198530	−	0.343864i
$482$		0			0
$483$	−16.8328	+	14.5776i	−0.765920	+	0.663306i
$484$		0			0
$485$	−19.4721	−	33.7267i	−0.884184	−	1.53145i
$486$		0			0
$487$	15.9164	−	27.5680i	0.721241	−	1.24923i	−0.239261	−	0.970955i	$-0.576905\pi$
$487$	15.9164	−	27.5680i	0.721241	−	1.24923i	0.960503	−	0.278271i	$-0.0897614\pi$
$488$		0			0
$489$	23.2918			1.05329
$490$		0			0
$491$	34.4721			1.55571			0.777853	−	0.628446i	$-0.216309\pi$
$491$	34.4721			1.55571			0.777853	+	0.628446i	$0.216309\pi$
$492$		0			0
$493$	−16.7082	+	28.9395i	−0.752500	+	1.30337i
$494$		0			0
$495$	6.70820	+	11.6190i	0.301511	+	0.522233i
$496$		0			0
$497$	−17.8885	+	15.4919i	−0.802411	+	0.694908i
$498$		0			0
$499$	−0.208204	−	0.360620i	−0.00932049	−	0.0161436i	0.861328	−	0.508050i	$-0.169634\pi$
$499$	−0.208204	−	0.360620i	−0.00932049	−	0.0161436i	−0.870648	+	0.491907i	$0.836300\pi$
$500$		0			0
$501$	−15.1246	+	26.1966i	−0.675718	+	1.17038i
$502$		0			0
$503$	−3.05573			−0.136248			−0.0681241	−	0.997677i	$-0.521701\pi$
$503$	−3.05573			−0.136248			−0.0681241	+	0.997677i	$0.521701\pi$
$504$		0			0
$505$	20.1246			0.895533
$506$		0			0
$507$	1.11803	−	1.93649i	0.0496536	−	0.0860026i
$508$		0			0
$509$	7.88197	+	13.6520i	0.349362	+	0.605113i	0.986136	−	0.165938i	$-0.0530650\pi$
$509$	7.88197	+	13.6520i	0.349362	+	0.605113i	−0.636774	+	0.771050i	$0.719732\pi$
$510$		0			0
$511$	5.35410	+	27.8207i	0.236852	+	1.23072i
$512$		0			0
$513$	3.35410	+	5.80948i	0.148087	+	0.256495i
$514$		0			0
$515$	−11.9721	+	20.7363i	−0.527555	+	0.913753i
$516$		0			0
$517$	−4.41641			−0.194233
$518$		0			0
$519$	−23.2918			−1.02240
$520$		0			0
$521$	0.0278640	−	0.0482619i	0.00122075	−	0.00211439i	−0.865414	−	0.501057i	$-0.832945\pi$
$521$	0.0278640	−	0.0482619i	0.00122075	−	0.00211439i	0.866635	+	0.498942i	$0.166278\pi$
$522$		0			0
$523$	9.64590	+	16.7072i	0.421786	+	0.730554i	0.996114	−	0.0880707i	$-0.0280701\pi$
$523$	9.64590	+	16.7072i	0.421786	+	0.730554i	−0.574329	+	0.818625i	$0.694737\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.6803	−	32.3553i	−0.813728	−	1.40942i
$528$		0			0
$529$	4.41641	−	7.64944i	0.192018	−	0.332584i
$530$		0			0
$531$	−14.9443			−0.648526
$532$		0			0
$533$	−4.47214			−0.193710
$534$		0			0
$535$	15.9164	−	27.5680i	0.688126	−	1.19187i
$536$		0			0
$537$	−22.5000	−	38.9711i	−0.970947	−	1.68173i
$538$		0			0
$539$	16.5000	+	12.9904i	0.710705	+	0.559535i
$540$		0			0
$541$	−7.35410	−	12.7377i	−0.316178	−	0.547636i	0.663510	−	0.748168i	$-0.269066\pi$
$541$	−7.35410	−	12.7377i	−0.316178	−	0.547636i	−0.979687	+	0.200532i	$0.935733\pi$
$542$		0			0
$543$	1.58359	−	2.74286i	0.0679584	−	0.117707i
$544$		0			0
$545$	−23.9443			−1.02566
$546$		0			0
$547$	31.4164			1.34327			0.671634	−	0.740883i	$-0.265593\pi$
$547$	31.4164			1.34327			0.671634	+	0.740883i	$0.265593\pi$
$548$		0			0
$549$	−3.00000	+	5.19615i	−0.128037	+	0.221766i
$550$		0			0
$551$	−6.70820	−	11.6190i	−0.285779	−	0.494984i
$552$		0			0
$553$	26.7705	+	9.27358i	1.13840	+	0.394353i
$554$		0			0
$555$	21.7705	+	37.7076i	0.924107	+	1.60060i
$556$		0			0
$557$	2.64590	−	4.58283i	0.112110	−	0.194181i	−0.804511	−	0.593938i	$-0.797572\pi$
$557$	2.64590	−	4.58283i	0.112110	−	0.194181i	0.916621	+	0.399758i	$0.130906\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$	50.1246			2.11626
$562$		0			0
$563$	−18.2984	+	31.6937i	−0.771185	+	1.33573i	0.165730	+	0.986171i	$0.447002\pi$
$563$	−18.2984	+	31.6937i	−0.771185	+	1.33573i	−0.936914	+	0.349560i	$0.886331\pi$
$564$		0			0
$565$	−16.7082	−	28.9395i	−0.702919	−	1.21749i
$566$		0			0
$567$	−5.50000	−	28.5788i	−0.230978	−	1.20020i
$568$		0			0
$569$	−8.26393	−	14.3136i	−0.346442	−	0.600055i	0.639173	−	0.769063i	$-0.279277\pi$
$569$	−8.26393	−	14.3136i	−0.346442	−	0.600055i	−0.985615	+	0.169008i	$0.945944\pi$
$570$		0			0
$571$	2.06231	−	3.57202i	0.0863048	−	0.149484i	−0.819642	−	0.572877i	$-0.805827\pi$
$571$	2.06231	−	3.57202i	0.0863048	−	0.149484i	0.905946	+	0.423392i	$0.139161\pi$
$572$		0			0
$573$	−25.0000			−1.04439
$574$		0			0
$575$		0			0
$576$		0			0
$577$	16.3541	−	28.3261i	0.680830	−	1.17923i	−0.293898	−	0.955837i	$-0.594953\pi$
$577$	16.3541	−	28.3261i	0.680830	−	1.17923i	0.974728	−	0.223396i	$-0.0717142\pi$
$578$		0			0
$579$	−14.2082	−	24.6093i	−0.590473	−	1.02273i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.20820	−	3.82472i	−0.0914545	−	0.158404i
$584$		0			0
$585$	−2.23607	+	3.87298i	−0.0924500	+	0.160128i
$586$		0			0
$587$	41.8885			1.72893			0.864463	−	0.502697i	$-0.167659\pi$
$587$	41.8885			1.72893			0.864463	+	0.502697i	$0.167659\pi$
$588$		0			0
$589$	15.0000			0.618064
$590$		0			0
$591$	−30.1246	+	52.1774i	−1.23916	+	2.14629i
$592$		0			0
$593$	−16.1180	−	27.9173i	−0.661888	−	1.14642i	−0.980119	−	0.198411i	$-0.936422\pi$
$593$	−16.1180	−	27.9173i	−0.661888	−	1.14642i	0.318231	−	0.948013i	$-0.396911\pi$
$594$		0			0
$595$	33.4164	−	28.9395i	1.36994	−	1.18640i
$596$		0			0
$597$	8.15248	+	14.1205i	0.333659	+	0.577914i
$598$		0			0
$599$	20.5344	−	35.5667i	0.839015	−	1.45322i	−0.0517049	−	0.998662i	$-0.516466\pi$
$599$	20.5344	−	35.5667i	0.839015	−	1.45322i	0.890719	−	0.454553i	$-0.150201\pi$
$600$		0			0
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$		0			0
$603$	6.00000			0.244339
$604$		0			0
$605$	−2.23607	+	3.87298i	−0.0909091	+	0.157459i
$606$		0			0
$607$	−8.06231	−	13.9643i	−0.327239	−	0.566794i	0.654724	−	0.755868i	$-0.272785\pi$
$607$	−8.06231	−	13.9643i	−0.327239	−	0.566794i	−0.981963	+	0.189074i	$0.939452\pi$
$608$		0			0
$609$	5.00000	+	25.9808i	0.202610	+	1.05279i
$610$		0			0
$611$	−0.736068	−	1.27491i	−0.0297781	−	0.0515772i
$612$		0			0
$613$	−11.0623	+	19.1605i	−0.446802	+	0.773884i	−0.998176	−	0.0603742i	$-0.980771\pi$
$613$	−11.0623	+	19.1605i	−0.446802	+	0.773884i	0.551373	+	0.834259i	$0.314104\pi$
$614$		0			0
$615$	22.3607			0.901670
$616$		0			0
$617$	4.47214			0.180041			0.0900207	−	0.995940i	$-0.471307\pi$
$617$	4.47214			0.180041			0.0900207	+	0.995940i	$0.471307\pi$
$618$		0			0
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	0.984738	+	0.174042i	$0.0556830\pi$
$620$		0			0
$621$	−4.20820	−	7.28882i	−0.168869	−	0.292490i
$622$		0			0
$623$	5.59017	+	1.93649i	0.223965	+	0.0775839i
$624$		0			0
$625$	12.5000	+	21.6506i	0.500000	+	0.866025i
$626$		0			0
$627$	−10.0623	+	17.4284i	−0.401850	+	0.696024i
$628$		0			0
$629$	65.0689			2.59447
$630$		0			0
$631$	30.8328			1.22744			0.613718	−	0.789526i	$-0.289673\pi$
$631$	30.8328			1.22744			0.613718	+	0.789526i	$0.289673\pi$
$632$		0			0
$633$	−4.47214	+	7.74597i	−0.177751	+	0.307875i
$634$		0			0
$635$	−17.2361	−	29.8537i	−0.683992	−	1.18471i
$636$		0			0
$637$	−1.00000	+	6.92820i	−0.0396214	+	0.274505i
$638$		0			0
$639$	8.94427	+	15.4919i	0.353830	+	0.612851i
$640$		0			0
$641$	2.97214	−	5.14789i	0.117392	−	0.203329i	−0.801341	−	0.598208i	$-0.795880\pi$
$641$	2.97214	−	5.14789i	0.117392	−	0.203329i	0.918734	+	0.394878i	$0.129213\pi$
$642$		0			0
$643$	18.8328			0.742694			0.371347	−	0.928494i	$-0.378896\pi$
$643$	18.8328			0.742694			0.371347	+	0.928494i	$0.378896\pi$
$644$		0			0
$645$	40.0000			1.57500
$646$		0			0
$647$	7.88197	−	13.6520i	0.309872	−	0.536714i	−0.668462	−	0.743746i	$-0.733047\pi$
$647$	7.88197	−	13.6520i	0.309872	−	0.536714i	0.978334	+	0.207032i	$0.0663804\pi$
$648$		0			0
$649$	11.2082	+	19.4132i	0.439960	+	0.762034i
$650$		0			0
$651$	−27.9508	−	9.68246i	−1.09548	−	0.379485i
$652$		0			0
$653$	−6.73607	−	11.6672i	−0.263603	−	0.456573i	0.703594	−	0.710602i	$-0.251577\pi$
$653$	−6.73607	−	11.6672i	−0.263603	−	0.456573i	−0.967197	+	0.254029i	$0.918244\pi$
$654$		0			0
$655$	4.20820	−	7.28882i	0.164428	−	0.284798i
$656$		0			0
$657$	21.4164			0.835534
$658$		0			0
$659$	−8.94427			−0.348419			−0.174210	−	0.984709i	$-0.555737\pi$
$659$	−8.94427			−0.348419			−0.174210	+	0.984709i	$0.555737\pi$
$660$		0			0
$661$	−3.35410	+	5.80948i	−0.130459	+	0.225962i	−0.923854	−	0.382746i	$-0.874979\pi$
$661$	−3.35410	+	5.80948i	−0.130459	+	0.225962i	0.793394	+	0.608708i	$0.208312\pi$
$662$		0			0
$663$	8.35410	+	14.4697i	0.324446	+	0.561958i
$664$		0			0
$665$	3.35410	+	17.4284i	0.130066	+	0.675845i
$666$		0			0
$667$	8.41641	+	14.5776i	0.325885	+	0.564449i
$668$		0			0
$669$	4.47214	−	7.74597i	0.172903	−	0.299476i
$670$		0			0
$671$	9.00000			0.347441
$672$		0			0
$673$	17.4164			0.671353			0.335677	−	0.941977i	$-0.391035\pi$
$673$	17.4164			0.671353			0.335677	+	0.941977i	$0.391035\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.4443	−	28.4823i	−0.632005	−	1.09466i	−0.987141	−	0.159850i	$-0.948899\pi$
$677$	−16.4443	−	28.4823i	−0.632005	−	1.09466i	0.355137	−	0.934814i	$-0.384434\pi$
$678$		0			0
$679$	−34.8328	+	30.1661i	−1.33676	+	1.15767i
$680$		0			0
$681$	13.3541	+	23.1300i	0.511730	+	0.886343i
$682$		0			0
$683$	−2.26393	+	3.92125i	−0.0866270	+	0.150042i	−0.906083	−	0.423099i	$-0.860942\pi$
$683$	−2.26393	+	3.92125i	−0.0866270	+	0.150042i	0.819456	+	0.573142i	$0.194275\pi$
$684$		0			0
$685$	8.41641			0.321574
$686$		0			0
$687$	36.0557			1.37561
$688$		0			0
$689$	0.736068	−	1.27491i	0.0280420	−	0.0485701i
$690$		0			0
$691$	0.916408	+	1.58726i	0.0348618	+	0.0603824i	0.882930	−	0.469505i	$-0.155568\pi$
$691$	0.916408	+	1.58726i	0.0348618	+	0.0603824i	−0.848068	+	0.529887i	$0.822234\pi$
$692$		0			0
$693$	12.0000	−	10.3923i	0.455842	−	0.394771i
$694$		0			0
$695$	−3.81966	−	6.61585i	−0.144888	−	0.250953i
$696$		0			0
$697$	16.7082	−	28.9395i	0.632868	−	1.09616i
$698$		0			0
$699$	13.2918			0.502742
$700$		0			0
$701$	22.3607			0.844551			0.422276	−	0.906467i	$-0.361231\pi$
$701$	22.3607			0.844551			0.422276	+	0.906467i	$0.361231\pi$
$702$		0			0
$703$	−13.0623	+	22.6246i	−0.492654	+	0.853302i
$704$		0			0
$705$	3.68034	+	6.37454i	0.138610	+	0.240079i
$706$		0			0
$707$	−4.50000	−	23.3827i	−0.169240	−	0.879396i
$708$		0			0
$709$	4.93769	+	8.55234i	0.185439	+	0.321190i	0.943724	−	0.330733i	$-0.107296\pi$
$709$	4.93769	+	8.55234i	0.185439	+	0.321190i	−0.758285	+	0.651923i	$0.773963\pi$
$710$		0			0
$711$	10.7082	−	18.5472i	0.401589	−	0.695573i
$712$		0			0
$713$	−18.8197			−0.704802
$714$		0			0
$715$	6.70820			0.250873
$716$		0			0
$717$	8.29180	−	14.3618i	0.309663	−	0.536352i
$718$		0			0
$719$	5.64590	+	9.77898i	0.210556	+	0.364694i	0.951889	−	0.306444i	$-0.0991391\pi$
$719$	5.64590	+	9.77898i	0.210556	+	0.364694i	−0.741332	+	0.671138i	$0.765806\pi$
$720$		0			0
$721$	26.7705	+	9.27358i	0.996986	+	0.345366i
$722$		0			0
$723$	−9.73607	−	16.8634i	−0.362088	−	0.627155i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−14.8328			−0.550119			−0.275059	−	0.961427i	$-0.588698\pi$
$727$	−14.8328			−0.550119			−0.275059	+	0.961427i	$0.588698\pi$
$728$		0			0
$729$	−7.00000			−0.259259
$730$		0			0
$731$	29.8885	−	51.7685i	1.10547	−	1.91473i
$732$		0			0
$733$	−7.64590	−	13.2431i	−0.282408	−	0.489144i	0.689570	−	0.724219i	$-0.257800\pi$
$733$	−7.64590	−	13.2431i	−0.282408	−	0.489144i	−0.971977	+	0.235075i	$0.924466\pi$
$734$		0			0
$735$	5.00000	−	34.6410i	0.184428	−	1.27775i
$736$		0			0
$737$	−4.50000	−	7.79423i	−0.165760	−	0.287104i
$738$		0			0
$739$	17.9164	−	31.0321i	0.659066	−	1.14154i	−0.321792	−	0.946810i	$-0.604285\pi$
$739$	17.9164	−	31.0321i	0.659066	−	1.14154i	0.980858	−	0.194725i	$-0.0623814\pi$
$740$		0			0
$741$	−6.70820			−0.246432
$742$		0			0
$743$	−15.0557			−0.552341			−0.276171	−	0.961109i	$-0.589065\pi$
$743$	−15.0557			−0.552341			−0.276171	+	0.961109i	$0.589065\pi$
$744$		0			0
$745$	−14.2082	+	24.6093i	−0.520548	+	0.901616i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−35.5902	−	12.3288i	−1.30044	−	0.450484i
$750$		0			0
$751$	15.0623	+	26.0887i	0.549631	+	0.951989i	0.998300	+	0.0582911i	$0.0185651\pi$
$751$	15.0623	+	26.0887i	0.549631	+	0.951989i	−0.448668	+	0.893698i	$0.648102\pi$
$752$		0			0
$753$	−11.7082	+	20.2792i	−0.426671	+	0.739015i
$754$		0			0
$755$	−14.3475			−0.522160
$756$		0			0
$757$	0.832816			0.0302692			0.0151346	−	0.999885i	$-0.495182\pi$
$757$	0.832816			0.0302692			0.0151346	+	0.999885i	$0.495182\pi$
$758$		0			0
$759$	12.6246	−	21.8665i	0.458244	−	0.793703i
$760$		0			0
$761$	−16.7705	−	29.0474i	−0.607931	−	1.05297i	−0.991581	−	0.129488i	$-0.958667\pi$
$761$	−16.7705	−	29.0474i	−0.607931	−	1.05297i	0.383650	−	0.923478i	$-0.374667\pi$
$762$		0			0
$763$	5.35410	+	27.8207i	0.193832	+	1.00718i
$764$		0			0
$765$	−16.7082	−	28.9395i	−0.604086	−	1.04631i
$766$		0			0
$767$	−3.73607	+	6.47106i	−0.134902	+	0.233656i
$768$		0			0
$769$	46.0000			1.65880			0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000			1.65880			0.829401	+	0.558653i	$0.188682\pi$
$770$		0			0
$771$	40.1246			1.44505
$772$		0			0
$773$	−5.53444	+	9.58593i	−0.199060	+	0.344782i	−0.948224	−	0.317603i	$-0.897122\pi$
$773$	−5.53444	+	9.58593i	−0.199060	+	0.344782i	0.749164	+	0.662385i	$0.230456\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	38.9443	−	33.7267i	1.39712	−	1.20994i
$778$		0			0
$779$	6.70820	+	11.6190i	0.240346	+	0.416292i
$780$		0			0
$781$	13.4164	−	23.2379i	0.480077	−	0.831517i
$782$		0			0
$783$	−10.0000			−0.357371
$784$		0			0
$785$	−15.6525			−0.558661
$786$		0			0
$787$	−3.20820	+	5.55677i	−0.114360	+	0.198078i	−0.917524	−	0.397681i	$-0.869815\pi$
$787$	−3.20820	+	5.55677i	−0.114360	+	0.198078i	0.803164	+	0.595758i	$0.203148\pi$
$788$		0			0
$789$	15.7918	+	27.3522i	0.562203	+	0.973764i
$790$		0			0

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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.11803 - 1.93649i) q^{3} +(1.11803 + 1.93649i) q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{9} +O(q^{10})\)	\(q+(1.11803 - 1.93649i) q^{3} +(1.11803 + 1.93649i) q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{9} +(-1.50000 + 2.59808i) q^{11} -1.00000 q^{13} +5.00000 q^{15} +(3.73607 - 6.47106i) q^{17} +(1.50000 + 2.59808i) q^{19} +(-1.11803 - 5.80948i) q^{21} +(-1.88197 - 3.25966i) q^{23} +2.23607 q^{27} -4.47214 q^{29} +(2.50000 - 4.33013i) q^{31} +(3.35410 + 5.80948i) q^{33} +(5.59017 + 1.93649i) q^{35} +(4.35410 + 7.54153i) q^{37} +(-1.11803 + 1.93649i) q^{39} +4.47214 q^{41} +8.00000 q^{43} +(2.23607 - 3.87298i) q^{45} +(0.736068 + 1.27491i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-8.35410 - 14.4697i) q^{51} +(-0.736068 + 1.27491i) q^{53} -6.70820 q^{55} +6.70820 q^{57} +(3.73607 - 6.47106i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(-5.00000 - 1.73205i) q^{63} +(-1.11803 - 1.93649i) q^{65} +(-1.50000 + 2.59808i) q^{67} -8.41641 q^{69} -8.94427 q^{71} +(-5.35410 + 9.27358i) q^{73} +(1.50000 + 7.79423i) q^{77} +(5.35410 + 9.27358i) q^{79} +(5.50000 - 9.52628i) q^{81} +16.7082 q^{85} +(-5.00000 + 8.66025i) q^{87} +(1.11803 + 1.93649i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(-5.59017 - 9.68246i) q^{93} +(-3.35410 + 5.80948i) q^{95} -17.4164 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 8 * q^7 - 4 * q^9	\( 4 q + 8 q^{7} - 4 q^{9} - 6 q^{11} - 4 q^{13} + 20 q^{15} + 6 q^{17} + 6 q^{19} - 12 q^{23} + 10 q^{31} + 4 q^{37} + 32 q^{43} - 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} + 6 q^{59} - 6 q^{61} - 20 q^{63} - 6 q^{67} + 20 q^{69} - 8 q^{73} + 6 q^{77} + 8 q^{79} + 22 q^{81} + 40 q^{85} - 20 q^{87} - 8 q^{91} - 16 q^{97} + 24 q^{99}+O(q^{100}) \) 4 * q + 8 * q^7 - 4 * q^9 - 6 * q^11 - 4 * q^13 + 20 * q^15 + 6 * q^17 + 6 * q^19 - 12 * q^23 + 10 * q^31 + 4 * q^37 + 32 * q^43 - 6 * q^47 + 4 * q^49 - 20 * q^51 + 6 * q^53 + 6 * q^59 - 6 * q^61 - 20 * q^63 - 6 * q^67 + 20 * q^69 - 8 * q^73 + 6 * q^77 + 8 * q^79 + 22 * q^81 + 40 * q^85 - 20 * q^87 - 8 * q^91 - 16 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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