LMFDB - Embedded newform 1824.2.j.d.1103.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1824,2,Mod(1103,1824)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1824, 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("1824.1103");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1824 = 2^{5} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1824.j (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:	$14.5647133287$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1103.21
Character	$\chi$	$=$	1824.1103
Dual form			1824.2.j.d.1103.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.59320 - 0.679496i) q^{3} -2.84644 q^{5} -0.466225i q^{7} +(2.07657 - 2.16515i) q^{9} +O(q^{10})$	\(q+(1.59320 - 0.679496i) q^{3} -2.84644 q^{5} -0.466225i q^{7} +(2.07657 - 2.16515i) q^{9} +4.27688i q^{11} -4.00344i q^{13} +(-4.53494 + 1.93414i) q^{15} -3.48404i q^{17} -1.00000 q^{19} +(-0.316798 - 0.742789i) q^{21} -1.49782 q^{23} +3.10219 q^{25} +(1.83719 - 4.86053i) q^{27} -8.54193 q^{29} -7.64656i q^{31} +(2.90612 + 6.81392i) q^{33} +1.32708i q^{35} +0.360333i q^{37} +(-2.72032 - 6.37829i) q^{39} -3.88529i q^{41} -6.31001 q^{43} +(-5.91082 + 6.16295i) q^{45} +3.96323 q^{47} +6.78263 q^{49} +(-2.36739 - 5.55077i) q^{51} -1.30160 q^{53} -12.1739i q^{55} +(-1.59320 + 0.679496i) q^{57} -3.00712i q^{59} -2.20911i q^{61} +(-1.00944 - 0.968149i) q^{63} +11.3955i q^{65} -11.7881 q^{67} +(-2.38632 + 1.01776i) q^{69} -6.80967 q^{71} -1.40984 q^{73} +(4.94241 - 2.10793i) q^{75} +1.99399 q^{77} +14.6441i q^{79} +(-0.375707 - 8.99215i) q^{81} -14.6883i q^{83} +9.91710i q^{85} +(-13.6090 + 5.80420i) q^{87} -15.8556i q^{89} -1.86651 q^{91} +(-5.19580 - 12.1825i) q^{93} +2.84644 q^{95} -14.3334 q^{97} +(9.26006 + 8.88124i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{3} - 6 q^{9}+O(q^{10}) $ 24 * q - 6 * q^3 - 6 * q^9	$ 24 q - 6 q^{3} - 6 q^{9} - 24 q^{19} + 96 q^{25} + 24 q^{33} - 16 q^{43} - 36 q^{49} - 38 q^{51} + 6 q^{57} - 20 q^{67} - 12 q^{73} + 30 q^{75} + 34 q^{81} + 36 q^{91} - 104 q^{97} - 56 q^{99}+O(q^{100}) $ 24 * q - 6 * q^3 - 6 * q^9 - 24 * q^19 + 96 * q^25 + 24 * q^33 - 16 * q^43 - 36 * q^49 - 38 * q^51 + 6 * q^57 - 20 * q^67 - 12 * q^73 + 30 * q^75 + 34 * q^81 + 36 * q^91 - 104 * q^97 - 56 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1824\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$799$	$1217$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	1.59320	−	0.679496i	0.919834	−	0.392307i
$3$	1.59320	−	0.679496i	0.919834	−	0.392307i
$4$		0			0
$5$	−2.84644			−1.27296			−0.636482	−	0.771291i	$-0.719611\pi$
$5$	−2.84644			−1.27296			−0.636482	+	0.771291i	$0.719611\pi$
$6$		0			0
$7$		−	0.466225i		−	0.176216i	−0.996111	−	0.0881082i	$-0.971918\pi$
$7$		−	0.466225i		−	0.176216i	0.996111	−	0.0881082i	$-0.0280821\pi$
$8$		0			0
$9$	2.07657	−	2.16515i	0.692190	−	0.721715i
$10$		0			0
$11$			4.27688i			1.28953i	0.764382	+	0.644764i	$0.223044\pi$
$11$			4.27688i			1.28953i	−0.764382	+	0.644764i	$0.776956\pi$
$12$		0			0
$13$		−	4.00344i		−	1.11036i	−0.831732	−	0.555178i	$-0.812650\pi$
$13$		−	4.00344i		−	1.11036i	0.831732	−	0.555178i	$-0.187350\pi$
$14$		0			0
$15$	−4.53494	+	1.93414i	−1.17092	+	0.499393i
$16$		0			0
$17$		−	3.48404i		−	0.845004i	−0.906362	−	0.422502i	$-0.861152\pi$
$17$		−	3.48404i		−	0.845004i	0.906362	−	0.422502i	$-0.138848\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	−0.316798	−	0.742789i	−0.0691309	−	0.162090i
$22$		0			0
$23$	−1.49782			−0.312317			−0.156158	−	0.987732i	$-0.549911\pi$
$23$	−1.49782			−0.312317			−0.156158	+	0.987732i	$0.549911\pi$
$24$		0			0
$25$	3.10219			0.620439
$26$		0			0
$27$	1.83719	−	4.86053i	0.353566	−	0.935409i
$28$		0			0
$29$	−8.54193			−1.58620			−0.793098	−	0.609094i	$-0.791533\pi$
$29$	−8.54193			−1.58620			−0.793098	+	0.609094i	$0.791533\pi$
$30$		0			0
$31$		−	7.64656i		−	1.37336i	−0.726959	−	0.686681i	$-0.759067\pi$
$31$		−	7.64656i		−	1.37336i	0.726959	−	0.686681i	$-0.240933\pi$
$32$		0			0
$33$	2.90612	+	6.81392i	0.505891	+	1.18615i
$34$		0			0
$35$			1.32708i			0.224317i
$36$		0			0
$37$			0.360333i			0.0592385i	0.999561	+	0.0296192i	$0.00942947\pi$
$37$			0.360333i			0.0592385i	−0.999561	+	0.0296192i	$0.990571\pi$
$38$		0			0
$39$	−2.72032	−	6.37829i	−0.435600	−	1.02134i
$40$		0			0
$41$		−	3.88529i		−	0.606780i	−0.952867	−	0.303390i	$-0.901882\pi$
$41$		−	3.88529i		−	0.606780i	0.952867	−	0.303390i	$-0.0981184\pi$
$42$		0			0
$43$	−6.31001			−0.962268			−0.481134	−	0.876647i	$-0.659775\pi$
$43$	−6.31001			−0.962268			−0.481134	+	0.876647i	$0.659775\pi$
$44$		0			0
$45$	−5.91082	+	6.16295i	−0.881134	+	0.918718i
$46$		0			0
$47$	3.96323			0.578097			0.289048	−	0.957314i	$-0.406661\pi$
$47$	3.96323			0.578097			0.289048	+	0.957314i	$0.406661\pi$
$48$		0			0
$49$	6.78263			0.968948
$50$		0			0
$51$	−2.36739	−	5.55077i	−0.331501	−	0.777264i
$52$		0			0
$53$	−1.30160			−0.178788			−0.0893940	−	0.995996i	$-0.528493\pi$
$53$	−1.30160			−0.178788			−0.0893940	+	0.995996i	$0.528493\pi$
$54$		0			0
$55$		−	12.1739i		−	1.64152i
$56$		0			0
$57$	−1.59320	+	0.679496i	−0.211024	+	0.0900014i
$58$		0			0
$59$		−	3.00712i		−	0.391494i	−0.980654	−	0.195747i	$-0.937287\pi$
$59$		−	3.00712i		−	0.391494i	0.980654	−	0.195747i	$-0.0627132\pi$
$60$		0			0
$61$		−	2.20911i		−	0.282848i	−0.989949	−	0.141424i	$-0.954832\pi$
$61$		−	2.20911i		−	0.282848i	0.989949	−	0.141424i	$-0.0451681\pi$
$62$		0			0
$63$	−1.00944	−	0.968149i	−0.127178	−	0.121975i
$64$		0			0
$65$			11.3955i			1.41344i
$66$		0			0
$67$	−11.7881			−1.44015			−0.720073	−	0.693898i	$-0.755892\pi$
$67$	−11.7881			−1.44015			−0.720073	+	0.693898i	$0.755892\pi$
$68$		0			0
$69$	−2.38632	+	1.01776i	−0.287280	+	0.122524i
$70$		0			0
$71$	−6.80967			−0.808159			−0.404079	−	0.914724i	$-0.632408\pi$
$71$	−6.80967			−0.808159			−0.404079	+	0.914724i	$0.632408\pi$
$72$		0			0
$73$	−1.40984			−0.165009			−0.0825044	−	0.996591i	$-0.526292\pi$
$73$	−1.40984			−0.165009			−0.0825044	+	0.996591i	$0.526292\pi$
$74$		0			0
$75$	4.94241	−	2.10793i	0.570701	−	0.243403i
$76$		0			0
$77$	1.99399			0.227236
$78$		0			0
$79$			14.6441i			1.64759i	0.566890	+	0.823794i	$0.308147\pi$
$79$			14.6441i			1.64759i	−0.566890	+	0.823794i	$0.691853\pi$
$80$		0			0
$81$	−0.375707	−	8.99215i	−0.0417452	−	0.999128i
$82$		0			0
$83$		−	14.6883i		−	1.61225i	−0.591743	−	0.806126i	$-0.701560\pi$
$83$		−	14.6883i		−	1.61225i	0.591743	−	0.806126i	$-0.298440\pi$
$84$		0			0
$85$			9.91710i			1.07566i
$86$		0			0
$87$	−13.6090	+	5.80420i	−1.45904	+	0.622276i
$88$		0			0
$89$		−	15.8556i		−	1.68069i	−0.542050	−	0.840346i	$-0.682352\pi$
$89$		−	15.8556i		−	1.68069i	0.542050	−	0.840346i	$-0.317648\pi$
$90$		0			0
$91$	−1.86651			−0.195663
$92$		0			0
$93$	−5.19580	−	12.1825i	−0.538780	−	1.26327i
$94$		0			0
$95$	2.84644			0.292038
$96$		0			0
$97$	−14.3334			−1.45534			−0.727668	−	0.685930i	$-0.759396\pi$
$97$	−14.3334			−1.45534			−0.727668	+	0.685930i	$0.759396\pi$
$98$		0			0
$99$	9.26006	+	8.88124i	0.930671	+	0.892598i
$100$		0			0
$101$	0.908973			0.0904462			0.0452231	−	0.998977i	$-0.485600\pi$
$101$	0.908973			0.0904462			0.0452231	+	0.998977i	$0.485600\pi$
$102$		0			0
$103$		−	19.6382i		−	1.93501i	−0.252849	−	0.967506i	$-0.581368\pi$
$103$		−	19.6382i		−	1.93501i	0.252849	−	0.967506i	$-0.418632\pi$
$104$		0			0
$105$	0.901744	+	2.11430i	0.0880012	+	0.206335i
$106$		0			0
$107$		−	5.24428i		−	0.506983i	−0.967338	−	0.253492i	$-0.918421\pi$
$107$		−	5.24428i		−	0.506983i	0.967338	−	0.253492i	$-0.0815791\pi$
$108$		0			0
$109$		−	0.135163i		−	0.0129463i	−0.999979	−	0.00647315i	$-0.997940\pi$
$109$		−	0.135163i		−	0.0129463i	0.999979	−	0.00647315i	$-0.00206048\pi$
$110$		0			0
$111$	0.244845	+	0.574083i	0.0232397	+	0.0544896i
$112$		0			0
$113$			16.7462i			1.57535i	0.616092	+	0.787674i	$0.288715\pi$
$113$			16.7462i			1.57535i	−0.616092	+	0.787674i	$0.711285\pi$
$114$		0			0
$115$	4.26344			0.397568
$116$		0			0
$117$	−8.66804	−	8.31344i	−0.801361	−	0.768578i
$118$		0			0
$119$	−1.62435			−0.148904
$120$		0			0
$121$	−7.29169			−0.662881
$122$		0			0
$123$	−2.64004	−	6.19004i	−0.238044	−	0.558137i
$124$		0			0
$125$	5.40198			0.483168
$126$		0			0
$127$			2.61709i			0.232229i	0.993236	+	0.116115i	$0.0370440\pi$
$127$			2.61709i			0.232229i	−0.993236	+	0.116115i	$0.962956\pi$
$128$		0			0
$129$	−10.0531	+	4.28763i	−0.885127	+	0.377505i
$130$		0			0
$131$			8.90568i			0.778093i	0.921218	+	0.389046i	$0.127195\pi$
$131$			8.90568i			0.778093i	−0.921218	+	0.389046i	$0.872805\pi$
$132$		0			0
$133$			0.466225i			0.0404268i
$134$		0			0
$135$	−5.22943	+	13.8352i	−0.450078	+	1.19074i
$136$		0			0
$137$		−	6.78619i		−	0.579783i	−0.957059	−	0.289892i	$-0.906381\pi$
$137$		−	6.78619i		−	0.579783i	0.957059	−	0.289892i	$-0.0936193\pi$
$138$		0			0
$139$	−14.1008			−1.19601			−0.598005	−	0.801492i	$-0.704040\pi$
$139$	−14.1008			−1.19601			−0.598005	+	0.801492i	$0.704040\pi$
$140$		0			0
$141$	6.31422	−	2.69300i	0.531753	−	0.226791i
$142$		0			0
$143$	17.1222			1.43183
$144$		0			0
$145$	24.3140			2.01917
$146$		0			0
$147$	10.8061	−	4.60877i	0.891271	−	0.380125i
$148$		0			0
$149$	11.5732			0.948110			0.474055	−	0.880495i	$-0.342790\pi$
$149$	11.5732			0.948110			0.474055	+	0.880495i	$0.342790\pi$
$150$		0			0
$151$			12.6407i			1.02869i	0.857585	+	0.514343i	$0.171964\pi$
$151$			12.6407i			1.02869i	−0.857585	+	0.514343i	$0.828036\pi$
$152$		0			0
$153$	−7.54346	−	7.23486i	−0.609852	−	0.584904i
$154$		0			0
$155$			21.7654i			1.74824i
$156$		0			0
$157$			0.932449i			0.0744176i	0.999308	+	0.0372088i	$0.0118467\pi$
$157$			0.932449i			0.0744176i	−0.999308	+	0.0372088i	$0.988153\pi$
$158$		0			0
$159$	−2.07370	+	0.884429i	−0.164455	+	0.0701398i
$160$		0			0
$161$			0.698320i			0.0550353i
$162$		0			0
$163$	23.1233			1.81115			0.905577	−	0.424182i	$-0.139438\pi$
$163$	23.1233			1.81115			0.905577	+	0.424182i	$0.139438\pi$
$164$		0			0
$165$	−8.27209	−	19.3954i	−0.643981	−	1.50993i
$166$		0			0
$167$	−18.6761			−1.44520			−0.722599	−	0.691268i	$-0.757053\pi$
$167$	−18.6761			−1.44520			−0.722599	+	0.691268i	$0.757053\pi$
$168$		0			0
$169$	−3.02757			−0.232890
$170$		0			0
$171$	−2.07657	+	2.16515i	−0.158799	+	0.165573i
$172$		0			0
$173$	15.6538			1.19014			0.595069	−	0.803675i	$-0.297125\pi$
$173$	15.6538			1.19014			0.595069	+	0.803675i	$0.297125\pi$
$174$		0			0
$175$		−	1.44632i		−	0.109331i
$176$		0			0
$177$	−2.04333	−	4.79095i	−0.153586	−	0.360110i
$178$		0			0
$179$		−	1.31831i		−	0.0985351i	−0.998786	−	0.0492676i	$-0.984311\pi$
$179$		−	1.31831i		−	0.0985351i	0.998786	−	0.0492676i	$-0.0156887\pi$
$180$		0			0
$181$			21.7722i			1.61832i	0.587591	+	0.809158i	$0.300076\pi$
$181$			21.7722i			1.61832i	−0.587591	+	0.809158i	$0.699924\pi$
$182$		0			0
$183$	−1.50108	−	3.51956i	−0.110963	−	0.260173i
$184$		0			0
$185$		−	1.02567i		−	0.0754085i
$186$		0			0
$187$	14.9008			1.08966
$188$		0			0
$189$	−2.26610	−	0.856541i	−0.164834	−	0.0623042i
$190$		0			0
$191$	−3.41715			−0.247256			−0.123628	−	0.992329i	$-0.539453\pi$
$191$	−3.41715			−0.247256			−0.123628	+	0.992329i	$0.539453\pi$
$192$		0			0
$193$	22.3929			1.61187			0.805937	−	0.592002i	$-0.201662\pi$
$193$	22.3929			1.61187			0.805937	+	0.592002i	$0.201662\pi$
$194$		0			0
$195$	7.74323	+	18.1554i	0.554504	+	1.30013i
$196$		0			0
$197$	18.8339			1.34186			0.670930	−	0.741521i	$-0.265895\pi$
$197$	18.8339			1.34186			0.670930	+	0.741521i	$0.265895\pi$
$198$		0			0
$199$			19.2624i			1.36547i	0.730664	+	0.682737i	$0.239211\pi$
$199$			19.2624i			1.36547i	−0.730664	+	0.682737i	$0.760789\pi$
$200$		0			0
$201$	−18.7808	+	8.00997i	−1.32470	+	0.564980i
$202$		0			0
$203$			3.98246i			0.279514i
$204$		0			0
$205$			11.0592i			0.772409i
$206$		0			0
$207$	−3.11033	+	3.24299i	−0.216183	+	0.225404i
$208$		0			0
$209$		−	4.27688i		−	0.295838i
$210$		0			0
$211$	14.6747			1.01025			0.505124	−	0.863047i	$-0.331447\pi$
$211$	14.6747			1.01025			0.505124	+	0.863047i	$0.331447\pi$
$212$		0			0
$213$	−10.8492	+	4.62714i	−0.743372	+	0.317046i
$214$		0			0
$215$	17.9610			1.22493
$216$		0			0
$217$	−3.56501			−0.242009
$218$		0			0
$219$	−2.24615	+	0.957977i	−0.151781	+	0.0647341i
$220$		0			0
$221$	−13.9482			−0.938255
$222$		0			0
$223$		−	11.7498i		−	0.786827i	−0.919362	−	0.393414i	$-0.871294\pi$
$223$		−	11.7498i		−	0.786827i	0.919362	−	0.393414i	$-0.128706\pi$
$224$		0			0
$225$	6.44192	−	6.71670i	0.429462	−	0.447780i
$226$		0			0
$227$		−	1.12266i		−	0.0745135i	−0.999306	−	0.0372568i	$-0.988138\pi$
$227$		−	1.12266i		−	0.0745135i	0.999306	−	0.0372568i	$-0.0118619\pi$
$228$		0			0
$229$		−	25.6416i		−	1.69445i	−0.531236	−	0.847224i	$-0.678272\pi$
$229$		−	25.6416i		−	1.69445i	0.531236	−	0.847224i	$-0.321728\pi$
$230$		0			0
$231$	3.17682	−	1.35491i	0.209019	−	0.0891462i
$232$		0			0
$233$			9.09416i			0.595778i	0.954600	+	0.297889i	$0.0962826\pi$
$233$			9.09416i			0.595778i	−0.954600	+	0.297889i	$0.903717\pi$
$234$		0			0
$235$	−11.2811			−0.735897
$236$		0			0
$237$	9.95059	+	23.3309i	0.646360	+	1.51551i
$238$		0			0
$239$	4.14167			0.267903			0.133951	−	0.990988i	$-0.457233\pi$
$239$	4.14167			0.267903			0.133951	+	0.990988i	$0.457233\pi$
$240$		0			0
$241$	−8.61961			−0.555238			−0.277619	−	0.960691i	$-0.589545\pi$
$241$	−8.61961			−0.555238			−0.277619	+	0.960691i	$0.589545\pi$
$242$		0			0
$243$	−6.70871	−	14.0710i	−0.430364	−	0.902656i
$244$		0			0
$245$	−19.3063			−1.23344
$246$		0			0
$247$			4.00344i			0.254733i
$248$		0			0
$249$	−9.98065	−	23.4014i	−0.632498	−	1.48301i
$250$		0			0
$251$			3.05061i			0.192553i	0.995355	+	0.0962763i	$0.0306932\pi$
$251$			3.05061i			0.192553i	−0.995355	+	0.0962763i	$0.969307\pi$
$252$		0			0
$253$		−	6.40599i		−	0.402741i
$254$		0			0
$255$	6.73863	+	15.7999i	0.421989	+	0.989429i
$256$		0			0
$257$		−	4.18627i		−	0.261133i	−0.991440	−	0.130566i	$-0.958320\pi$
$257$		−	4.18627i		−	0.261133i	0.991440	−	0.130566i	$-0.0416795\pi$
$258$		0			0
$259$	0.167996			0.0104388
$260$		0			0
$261$	−17.7379	+	18.4945i	−1.09795	+	1.14478i
$262$		0			0
$263$	21.1464			1.30394			0.651971	−	0.758244i	$-0.273943\pi$
$263$	21.1464			1.30394			0.651971	+	0.758244i	$0.273943\pi$
$264$		0			0
$265$	3.70491			0.227591
$266$		0			0
$267$	−10.7738	−	25.2612i	−0.659348	−	1.54596i
$268$		0			0
$269$	−7.73388			−0.471543			−0.235771	−	0.971809i	$-0.575762\pi$
$269$	−7.73388			−0.471543			−0.235771	+	0.971809i	$0.575762\pi$
$270$		0			0
$271$			3.05116i			0.185345i	0.995697	+	0.0926723i	$0.0295409\pi$
$271$			3.05116i			0.185345i	−0.995697	+	0.0926723i	$0.970459\pi$
$272$		0			0
$273$	−2.97372	+	1.26828i	−0.179977	+	0.0767599i
$274$		0			0
$275$			13.2677i			0.800073i
$276$		0			0
$277$		−	18.8564i		−	1.13297i	−0.824071	−	0.566487i	$-0.808302\pi$
$277$		−	18.8564i		−	1.13297i	0.824071	−	0.566487i	$-0.191698\pi$
$278$		0			0
$279$	−16.5559	−	15.8786i	−0.991176	−	0.950628i
$280$		0			0
$281$			15.4367i			0.920876i	0.887692	+	0.460438i	$0.152308\pi$
$281$			15.4367i			0.920876i	−0.887692	+	0.460438i	$0.847692\pi$
$282$		0			0
$283$	−16.4832			−0.979823			−0.489911	−	0.871772i	$-0.662971\pi$
$283$	−16.4832			−0.979823			−0.489911	+	0.871772i	$0.662971\pi$
$284$		0			0
$285$	4.53494	−	1.93414i	0.268627	−	0.114569i
$286$		0			0
$287$	−1.81142			−0.106925
$288$		0			0
$289$	4.86145			0.285968
$290$		0			0
$291$	−22.8360	+	9.73948i	−1.33867	+	0.570938i
$292$		0			0
$293$	29.2733			1.71016			0.855082	−	0.518493i	$-0.173507\pi$
$293$	29.2733			1.71016			0.855082	+	0.518493i	$0.173507\pi$
$294$		0			0
$295$			8.55958i			0.498358i
$296$		0			0
$297$	20.7879	+	7.85742i	1.20624	+	0.455934i
$298$		0			0
$299$			5.99643i			0.346783i
$300$		0			0
$301$			2.94188i			0.169567i
$302$		0			0
$303$	1.44818	−	0.617643i	0.0831955	−	0.0354827i
$304$		0			0
$305$			6.28810i			0.360055i
$306$		0			0
$307$	21.4068			1.22175			0.610875	−	0.791727i	$-0.290818\pi$
$307$	21.4068			1.22175			0.610875	+	0.791727i	$0.290818\pi$
$308$		0			0
$309$	−13.3441	−	31.2876i	−0.759119	−	1.77989i
$310$		0			0
$311$	8.16716			0.463117			0.231559	−	0.972821i	$-0.425617\pi$
$311$	8.16716			0.463117			0.231559	+	0.972821i	$0.425617\pi$
$312$		0			0
$313$	−17.6187			−0.995866			−0.497933	−	0.867215i	$-0.665907\pi$
$313$	−17.6187			−0.995866			−0.497933	+	0.867215i	$0.665907\pi$
$314$		0			0
$315$	2.87332	+	2.75577i	0.161893	+	0.155270i
$316$		0			0
$317$	−2.52671			−0.141914			−0.0709571	−	0.997479i	$-0.522605\pi$
$317$	−2.52671			−0.141914			−0.0709571	+	0.997479i	$0.522605\pi$
$318$		0			0
$319$		−	36.5328i		−	2.04544i
$320$		0			0
$321$	−3.56346	−	8.35518i	−0.198893	−	0.466341i
$322$		0			0
$323$			3.48404i			0.193857i
$324$		0			0
$325$		−	12.4195i		−	0.688908i
$326$		0			0
$327$	−0.0918429	−	0.215342i	−0.00507892	−	0.0119084i
$328$		0			0
$329$		−	1.84776i		−	0.101870i
$330$		0			0
$331$	−19.9850			−1.09848			−0.549238	−	0.835666i	$-0.685082\pi$
$331$	−19.9850			−1.09848			−0.549238	+	0.835666i	$0.685082\pi$
$332$		0			0
$333$	0.780174	+	0.748258i	0.0427533	+	0.0410043i
$334$		0			0
$335$	33.5541			1.83325
$336$		0			0
$337$	11.8512			0.645578			0.322789	−	0.946471i	$-0.395380\pi$
$337$	11.8512			0.645578			0.322789	+	0.946471i	$0.395380\pi$
$338$		0			0
$339$	11.3790	+	26.6800i	0.618021	+	1.44906i
$340$		0			0
$341$	32.7034			1.77099
$342$		0			0
$343$		−	6.42581i		−	0.346961i
$344$		0			0
$345$	6.79252	−	2.89699i	0.365697	−	0.155969i
$346$		0			0
$347$		−	22.9770i		−	1.23347i	−0.787170	−	0.616736i	$-0.788455\pi$
$347$		−	22.9770i		−	1.23347i	0.787170	−	0.616736i	$-0.211545\pi$
$348$		0			0
$349$		−	21.1528i		−	1.13228i	−0.824308	−	0.566141i	$-0.808436\pi$
$349$		−	21.1528i		−	1.13228i	0.824308	−	0.566141i	$-0.191564\pi$
$350$		0			0
$351$	−19.4589	−	7.35507i	−1.03864	−	0.392585i
$352$		0			0
$353$		−	18.5723i		−	0.988504i	−0.869319	−	0.494252i	$-0.835442\pi$
$353$		−	18.5723i		−	0.988504i	0.869319	−	0.494252i	$-0.164558\pi$
$354$		0			0
$355$	19.3833			1.02876
$356$		0			0
$357$	−2.58791	+	1.10374i	−0.136967	+	0.0584159i
$358$		0			0
$359$	−7.11376			−0.375450			−0.187725	−	0.982222i	$-0.560111\pi$
$359$	−7.11376			−0.375450			−0.187725	+	0.982222i	$0.560111\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$	−11.6171	+	4.95467i	−0.609741	+	0.260053i
$364$		0			0
$365$	4.01301			0.210050
$366$		0			0
$367$			0.517055i			0.0269900i	0.999909	+	0.0134950i	$0.00429573\pi$
$367$			0.517055i			0.0269900i	−0.999909	+	0.0134950i	$0.995704\pi$
$368$		0			0
$369$	−8.41221	−	8.06807i	−0.437922	−	0.420007i
$370$		0			0
$371$			0.606837i			0.0315054i
$372$		0			0
$373$			27.1084i			1.40362i	0.712364	+	0.701810i	$0.247624\pi$
$373$			27.1084i			1.40362i	−0.712364	+	0.701810i	$0.752376\pi$
$374$		0			0
$375$	8.60644	−	3.67062i	0.444435	−	0.189550i
$376$		0			0
$377$			34.1971i			1.76124i
$378$		0			0
$379$	4.55933			0.234197			0.117098	−	0.993120i	$-0.462641\pi$
$379$	4.55933			0.234197			0.117098	+	0.993120i	$0.462641\pi$
$380$		0			0
$381$	1.77830	+	4.16955i	0.0911052	+	0.213612i
$382$		0			0
$383$	−8.31238			−0.424743			−0.212371	−	0.977189i	$-0.568119\pi$
$383$	−8.31238			−0.424743			−0.212371	+	0.977189i	$0.568119\pi$
$384$		0			0
$385$	−5.67575			−0.289263
$386$		0			0
$387$	−13.1032	+	13.6621i	−0.666073	+	0.694483i
$388$		0			0
$389$	18.9309			0.959835			0.479918	−	0.877314i	$-0.340667\pi$
$389$	18.9309			0.959835			0.479918	+	0.877314i	$0.340667\pi$
$390$		0			0
$391$			5.21846i			0.263909i
$392$		0			0
$393$	6.05137	+	14.1885i	0.305251	+	0.715717i
$394$		0			0
$395$		−	41.6834i		−	2.09732i
$396$		0			0
$397$		−	19.7700i		−	0.992227i	−0.868258	−	0.496113i	$-0.834760\pi$
$397$		−	19.7700i		−	0.992227i	0.868258	−	0.496113i	$-0.165240\pi$
$398$		0			0
$399$	0.316798	+	0.742789i	0.0158597	+	0.0371860i
$400$		0			0
$401$			8.03205i			0.401102i	0.979683	+	0.200551i	$0.0642732\pi$
$401$			8.03205i			0.401102i	−0.979683	+	0.200551i	$0.935727\pi$
$402$		0			0
$403$	−30.6126			−1.52492
$404$		0			0
$405$	1.06942	+	25.5956i	0.0531401	+	1.27185i
$406$		0			0
$407$	−1.54110			−0.0763896
$408$		0			0
$409$	−29.4278			−1.45511			−0.727556	−	0.686048i	$-0.759344\pi$
$409$	−29.4278			−1.45511			−0.727556	+	0.686048i	$0.759344\pi$
$410$		0			0
$411$	−4.61119	−	10.8118i	−0.227453	−	0.533305i
$412$		0			0
$413$	−1.40200			−0.0689877
$414$		0			0
$415$			41.8094i			2.05234i
$416$		0			0
$417$	−22.4653	+	9.58140i	−1.10013	+	0.469203i
$418$		0			0
$419$		−	12.7128i		−	0.621062i	−0.950563	−	0.310531i	$-0.899493\pi$
$419$		−	12.7128i		−	0.621062i	0.950563	−	0.310531i	$-0.100507\pi$
$420$		0			0
$421$			26.4745i			1.29029i	0.764061	+	0.645143i	$0.223202\pi$
$421$			26.4745i			1.29029i	−0.764061	+	0.645143i	$0.776798\pi$
$422$		0			0
$423$	8.22993	−	8.58097i	0.400153	−	0.417221i
$424$		0			0
$425$		−	10.8082i		−	0.524273i
$426$		0			0
$427$	−1.02994			−0.0498424
$428$		0			0
$429$	27.2792	−	11.6345i	1.31705	−	0.561719i
$430$		0			0
$431$	−4.48203			−0.215892			−0.107946	−	0.994157i	$-0.534427\pi$
$431$	−4.48203			−0.215892			−0.107946	+	0.994157i	$0.534427\pi$
$432$		0			0
$433$	31.5853			1.51789			0.758947	−	0.651152i	$-0.225714\pi$
$433$	31.5853			1.51789			0.758947	+	0.651152i	$0.225714\pi$
$434$		0			0
$435$	38.7371	−	16.5213i	1.85730	−	0.792135i
$436$		0			0
$437$	1.49782			0.0716504
$438$		0			0
$439$			30.8696i			1.47333i	0.676259	+	0.736664i	$0.263600\pi$
$439$			30.8696i			1.47333i	−0.676259	+	0.736664i	$0.736400\pi$
$440$		0			0
$441$	14.0846	−	14.6854i	0.670696	−	0.699304i
$442$		0			0
$443$		−	13.5157i		−	0.642152i	−0.947053	−	0.321076i	$-0.895955\pi$
$443$		−	13.5157i		−	0.642152i	0.947053	−	0.321076i	$-0.104045\pi$
$444$		0			0
$445$			45.1320i			2.13946i
$446$		0			0
$447$	18.4384	−	7.86392i	0.872105	−	0.371950i
$448$		0			0
$449$			39.6229i			1.86992i	0.354753	+	0.934960i	$0.384565\pi$
$449$			39.6229i			1.86992i	−0.354753	+	0.934960i	$0.615435\pi$
$450$		0			0
$451$	16.6169			0.782459
$452$		0			0
$453$	8.58930	+	20.1392i	0.403561	+	0.946220i
$454$		0			0
$455$	5.31289			0.249072
$456$		0			0
$457$	−35.9360			−1.68102			−0.840508	−	0.541799i	$-0.817743\pi$
$457$	−35.9360			−1.68102			−0.840508	+	0.541799i	$0.817743\pi$
$458$		0			0
$459$	−16.9343	−	6.40083i	−0.790425	−	0.298765i
$460$		0			0
$461$	40.7343			1.89719			0.948593	−	0.316499i	$-0.102507\pi$
$461$	40.7343			1.89719			0.948593	+	0.316499i	$0.102507\pi$
$462$		0			0
$463$		−	11.2414i		−	0.522433i	−0.965280	−	0.261216i	$-0.915876\pi$
$463$		−	11.2414i		−	0.522433i	0.965280	−	0.261216i	$-0.0841236\pi$
$464$		0			0
$465$	14.7895	+	34.6767i	0.685847	+	1.60809i
$466$		0			0
$467$			22.1470i			1.02484i	0.858734	+	0.512421i	$0.171251\pi$
$467$			22.1470i			1.02484i	−0.858734	+	0.512421i	$0.828749\pi$
$468$		0			0
$469$			5.49591i			0.253777i
$470$		0			0
$471$	0.633596	+	1.48558i	0.0291945	+	0.0684518i
$472$		0			0
$473$		−	26.9872i		−	1.24087i
$474$		0			0
$475$	−3.10219			−0.142338
$476$		0			0
$477$	−2.70286	+	2.81815i	−0.123755	+	0.129034i
$478$		0			0
$479$	14.4920			0.662158			0.331079	−	0.943603i	$-0.392587\pi$
$479$	14.4920			0.662158			0.331079	+	0.943603i	$0.392587\pi$
$480$		0			0
$481$	1.44258			0.0657758
$482$		0			0
$483$	0.474505	+	1.11256i	0.0215907	+	0.0506234i
$484$		0			0
$485$	40.7991			1.85259
$486$		0			0
$487$		−	21.3493i		−	0.967431i	−0.875225	−	0.483716i	$-0.839287\pi$
$487$		−	21.3493i		−	0.967431i	0.875225	−	0.483716i	$-0.160713\pi$
$488$		0			0
$489$	36.8400	−	15.7122i	1.66596	−	0.710529i
$490$		0			0
$491$		−	18.0569i		−	0.814896i	−0.913228	−	0.407448i	$-0.866419\pi$
$491$		−	18.0569i		−	0.814896i	0.913228	−	0.407448i	$-0.133581\pi$
$492$		0			0
$493$			29.7604i			1.34034i
$494$		0			0
$495$	−26.3582	−	25.2799i	−1.18471	−	1.13625i
$496$		0			0
$497$			3.17484i			0.142411i
$498$		0			0
$499$	−2.30123			−0.103017			−0.0515086	−	0.998673i	$-0.516403\pi$
$499$	−2.30123			−0.103017			−0.0515086	+	0.998673i	$0.516403\pi$
$500$		0			0
$501$	−29.7547	+	12.6903i	−1.32934	+	0.566961i
$502$		0			0
$503$	42.1587			1.87976			0.939881	−	0.341502i	$-0.110936\pi$
$503$	42.1587			1.87976			0.939881	+	0.341502i	$0.110936\pi$
$504$		0			0
$505$	−2.58733			−0.115135
$506$		0			0
$507$	−4.82353	+	2.05722i	−0.214220	+	0.0913644i
$508$		0			0
$509$	−39.0804			−1.73221			−0.866104	−	0.499863i	$-0.833384\pi$
$509$	−39.0804			−1.73221			−0.866104	+	0.499863i	$0.833384\pi$
$510$		0			0
$511$			0.657300i			0.0290772i
$512$		0			0
$513$	−1.83719	+	4.86053i	−0.0811137	+	0.214598i
$514$		0			0
$515$			55.8989i			2.46320i
$516$		0			0
$517$			16.9503i			0.745472i
$518$		0			0
$519$	24.9397	−	10.6367i	1.09473	−	0.466899i
$520$		0			0
$521$			26.5526i			1.16329i	0.813443	+	0.581645i	$0.197591\pi$
$521$			26.5526i			1.16329i	−0.813443	+	0.581645i	$0.802409\pi$
$522$		0			0
$523$	−30.8111			−1.34728			−0.673639	−	0.739061i	$-0.735270\pi$
$523$	−30.8111			−1.34728			−0.673639	+	0.739061i	$0.735270\pi$
$524$		0			0
$525$	−0.982768	−	2.30428i	−0.0428915	−	0.100567i
$526$		0			0
$527$	−26.6409			−1.16050
$528$		0			0
$529$	−20.7565			−0.902458
$530$		0			0
$531$	−6.51086	−	6.24451i	−0.282547	−	0.270989i
$532$		0			0
$533$	−15.5545			−0.673741
$534$		0			0
$535$			14.9275i			0.645372i
$536$		0			0
$537$	−0.895786	−	2.10033i	−0.0386560	−	0.0906360i
$538$		0			0
$539$			29.0085i			1.24948i
$540$		0			0
$541$		−	23.5736i		−	1.01351i	−0.862091	−	0.506753i	$-0.830845\pi$
$541$		−	23.5736i		−	1.01351i	0.862091	−	0.506753i	$-0.169155\pi$
$542$		0			0
$543$	14.7941	+	34.6875i	0.634877	+	1.48858i
$544$		0			0
$545$			0.384734i			0.0164802i
$546$		0			0
$547$	6.72108			0.287373			0.143686	−	0.989623i	$-0.454104\pi$
$547$	6.72108			0.287373			0.143686	+	0.989623i	$0.454104\pi$
$548$		0			0
$549$	−4.78305	−	4.58738i	−0.204136	−	0.195785i
$550$		0			0
$551$	8.54193			0.363898
$552$		0			0
$553$	6.82743			0.290332
$554$		0			0
$555$	−0.696936	−	1.63409i	−0.0295833	−	0.0693633i
$556$		0			0
$557$	−16.6408			−0.705093			−0.352547	−	0.935794i	$-0.614684\pi$
$557$	−16.6408			−0.705093			−0.352547	+	0.935794i	$0.614684\pi$
$558$		0			0
$559$			25.2618i			1.06846i
$560$		0			0
$561$	23.7400	−	10.1250i	1.00230	−	0.427480i
$562$		0			0
$563$			36.2254i			1.52672i	0.645973	+	0.763360i	$0.276452\pi$
$563$			36.2254i			1.52672i	−0.645973	+	0.763360i	$0.723548\pi$
$564$		0			0
$565$		−	47.6669i		−	2.00536i
$566$		0			0
$567$	−4.19236	+	0.175164i	−0.176063	+	0.00735618i
$568$		0			0
$569$		−	17.6565i		−	0.740200i	−0.928992	−	0.370100i	$-0.879323\pi$
$569$		−	17.6565i		−	0.740200i	0.928992	−	0.370100i	$-0.120677\pi$
$570$		0			0
$571$	−24.7456			−1.03557			−0.517786	−	0.855510i	$-0.673243\pi$
$571$	−24.7456			−1.03557			−0.517786	+	0.855510i	$0.673243\pi$
$572$		0			0
$573$	−5.44420	+	2.32194i	−0.227435	+	0.0970003i
$574$		0			0
$575$	−4.64652			−0.193773
$576$		0			0
$577$	17.4071			0.724669			0.362335	−	0.932048i	$-0.381980\pi$
$577$	17.4071			0.724669			0.362335	+	0.932048i	$0.381980\pi$
$578$		0			0
$579$	35.6763	−	15.2159i	1.48266	−	0.632349i
$580$		0			0
$581$	−6.84806			−0.284105
$582$		0			0
$583$		−	5.56677i		−	0.230552i
$584$		0			0
$585$	24.6730	+	23.6637i	1.02010	+	0.978372i
$586$		0			0
$587$			15.1652i			0.625936i	0.949764	+	0.312968i	$0.101323\pi$
$587$			15.1652i			0.625936i	−0.949764	+	0.312968i	$0.898677\pi$
$588$		0			0
$589$			7.64656i			0.315071i
$590$		0			0
$591$	30.0062	−	12.7976i	1.23429	−	0.526421i
$592$		0			0
$593$		−	2.75839i		−	0.113274i	−0.998395	−	0.0566368i	$-0.981962\pi$
$593$		−	2.75839i		−	0.113274i	0.998395	−	0.0566368i	$-0.0180377\pi$
$594$		0			0
$595$	4.62360			0.189549
$596$		0			0
$597$	13.0887	+	30.6888i	0.535685	+	1.25601i
$598$		0			0
$599$	10.8449			0.443110			0.221555	−	0.975148i	$-0.428887\pi$
$599$	10.8449			0.443110			0.221555	+	0.975148i	$0.428887\pi$
$600$		0			0
$601$	18.9772			0.774096			0.387048	−	0.922060i	$-0.373495\pi$
$601$	18.9772			0.774096			0.387048	+	0.922060i	$0.373495\pi$
$602$		0			0
$603$	−24.4788	+	25.5230i	−0.996855	+	1.03938i
$604$		0			0
$605$	20.7553			0.843824
$606$		0			0
$607$			11.5022i			0.466860i	0.972374	+	0.233430i	$0.0749950\pi$
$607$			11.5022i			0.466860i	−0.972374	+	0.233430i	$0.925005\pi$
$608$		0			0
$609$	2.70606	+	6.34485i	0.109655	+	0.257106i
$610$		0			0
$611$		−	15.8666i		−	0.641893i
$612$		0			0
$613$			27.9553i			1.12911i	0.825397	+	0.564553i	$0.190951\pi$
$613$			27.9553i			1.12911i	−0.825397	+	0.564553i	$0.809049\pi$
$614$		0			0
$615$	7.51469	+	17.6195i	0.303022	+	0.710488i
$616$		0			0
$617$			8.63436i			0.347606i	0.984780	+	0.173803i	$0.0556056\pi$
$617$			8.63436i			0.347606i	−0.984780	+	0.173803i	$0.944394\pi$
$618$		0			0
$619$	23.5198			0.945342			0.472671	−	0.881239i	$-0.343290\pi$
$619$	23.5198			0.945342			0.472671	+	0.881239i	$0.343290\pi$
$620$		0			0
$621$	−2.75177	+	7.28019i	−0.110425	+	0.292144i
$622$		0			0
$623$	−7.39228			−0.296166
$624$		0			0
$625$	−30.8874			−1.23549
$626$		0			0
$627$	−2.90612	−	6.81392i	−0.116059	−	0.272122i
$628$		0			0
$629$	1.25542			0.0500568
$630$		0			0
$631$		−	17.9945i		−	0.716351i	−0.933654	−	0.358176i	$-0.883399\pi$
$631$		−	17.9945i		−	0.716351i	0.933654	−	0.358176i	$-0.116601\pi$
$632$		0			0
$633$	23.3797	−	9.97140i	0.929261	−	0.396328i
$634$		0			0
$635$		−	7.44938i		−	0.295620i
$636$		0			0
$637$		−	27.1539i		−	1.07588i
$638$		0			0
$639$	−14.1408	+	14.7439i	−0.559400	+	0.583260i
$640$		0			0
$641$		−	19.8114i		−	0.782505i	−0.920283	−	0.391253i	$-0.872042\pi$
$641$		−	19.8114i		−	0.782505i	0.920283	−	0.391253i	$-0.127958\pi$
$642$		0			0
$643$	14.6875			0.579219			0.289610	−	0.957145i	$-0.406474\pi$
$643$	14.6875			0.579219			0.289610	+	0.957145i	$0.406474\pi$
$644$		0			0
$645$	28.6155	−	12.2045i	1.12674	−	0.480550i
$646$		0			0
$647$	−12.3332			−0.484868			−0.242434	−	0.970168i	$-0.577946\pi$
$647$	−12.3332			−0.484868			−0.242434	+	0.970168i	$0.577946\pi$
$648$		0			0
$649$	12.8611			0.504843
$650$		0			0
$651$	−5.67978	+	2.42241i	−0.222608	+	0.0949418i
$652$		0			0
$653$	−9.77352			−0.382467			−0.191234	−	0.981545i	$-0.561249\pi$
$653$	−9.77352			−0.382467			−0.191234	+	0.981545i	$0.561249\pi$
$654$		0			0
$655$		−	25.3494i		−	0.990485i
$656$		0			0
$657$	−2.92762	+	3.05250i	−0.114217	+	0.119089i
$658$		0			0
$659$		−	26.5173i		−	1.03297i	−0.856297	−	0.516484i	$-0.827241\pi$
$659$		−	26.5173i		−	1.03297i	0.856297	−	0.516484i	$-0.172759\pi$
$660$		0			0
$661$		−	43.9961i		−	1.71125i	−0.517597	−	0.855625i	$-0.673173\pi$
$661$		−	43.9961i		−	1.71125i	0.517597	−	0.855625i	$-0.326827\pi$
$662$		0			0
$663$	−22.2222	+	9.47772i	−0.863040	+	0.368084i
$664$		0			0
$665$		−	1.32708i		−	0.0514619i
$666$		0			0
$667$	12.7943			0.495396
$668$		0			0
$669$	−7.98397	−	18.7198i	−0.308678	−	0.723751i
$670$		0			0
$671$	9.44811			0.364740
$672$		0			0
$673$	7.50506			0.289299			0.144649	−	0.989483i	$-0.453795\pi$
$673$	7.50506			0.289299			0.144649	+	0.989483i	$0.453795\pi$
$674$		0			0
$675$	5.69930	−	15.0783i	0.219366	−	0.580364i
$676$		0			0
$677$	3.17793			0.122138			0.0610690	−	0.998134i	$-0.480549\pi$
$677$	3.17793			0.122138			0.0610690	+	0.998134i	$0.480549\pi$
$678$		0			0
$679$			6.68258i			0.256454i
$680$		0			0
$681$	−0.762842	−	1.78862i	−0.0292322	−	0.0685401i
$682$		0			0
$683$		−	16.3890i		−	0.627107i	−0.949571	−	0.313554i	$-0.898480\pi$
$683$		−	16.3890i		−	0.627107i	0.949571	−	0.313554i	$-0.101520\pi$
$684$		0			0
$685$			19.3164i			0.738044i
$686$		0			0
$687$	−17.4234	−	40.8523i	−0.664744	−	1.55861i
$688$		0			0
$689$			5.21087i			0.198518i
$690$		0			0
$691$	13.2798			0.505186			0.252593	−	0.967573i	$-0.418717\pi$
$691$	13.2798			0.505186			0.252593	+	0.967573i	$0.418717\pi$
$692$		0			0
$693$	4.14065	−	4.31727i	0.157290	−	0.164000i
$694$		0			0
$695$	40.1369			1.52248
$696$		0			0
$697$	−13.5365			−0.512731
$698$		0			0
$699$	6.17944	+	14.4888i	0.233728	+	0.548017i
$700$		0			0
$701$	−26.1554			−0.987875			−0.493938	−	0.869497i	$-0.664443\pi$
$701$	−26.1554			−0.987875			−0.493938	+	0.869497i	$0.664443\pi$
$702$		0			0
$703$		−	0.360333i		−	0.0135902i
$704$		0			0
$705$	−17.9730	+	7.66545i	−0.676903	+	0.288698i
$706$		0			0
$707$		−	0.423786i		−	0.0159381i
$708$		0			0
$709$			3.81405i			0.143240i	0.997432	+	0.0716198i	$0.0228168\pi$
$709$			3.81405i			0.143240i	−0.997432	+	0.0716198i	$0.977183\pi$
$710$		0			0
$711$	31.7066	+	30.4095i	1.18909	+	1.14044i
$712$		0			0
$713$			11.4532i			0.428924i
$714$		0			0
$715$	−48.7374			−1.82267
$716$		0			0
$717$	6.59851	−	2.81425i	0.246426	−	0.105100i
$718$		0			0
$719$	−35.8564			−1.33722			−0.668609	−	0.743614i	$-0.733110\pi$
$719$	−35.8564			−1.33722			−0.668609	+	0.743614i	$0.733110\pi$
$720$		0			0
$721$	−9.15582			−0.340981
$722$		0			0
$723$	−13.7328	+	5.85699i	−0.510727	+	0.217824i
$724$		0			0
$725$	−26.4987			−0.984138
$726$		0			0
$727$		−	31.0603i		−	1.15196i	−0.817463	−	0.575982i	$-0.804620\pi$
$727$		−	31.0603i		−	1.15196i	0.817463	−	0.575982i	$-0.195380\pi$
$728$		0			0
$729$	−20.2495	−	17.8594i	−0.749982	−	0.661459i
$730$		0			0
$731$			21.9844i			0.813121i
$732$		0			0
$733$			11.3722i			0.420043i	0.977697	+	0.210022i	$0.0673534\pi$
$733$			11.3722i			0.420043i	−0.977697	+	0.210022i	$0.932647\pi$
$734$		0			0
$735$	−30.7588	+	13.1186i	−1.13456	+	0.483886i
$736$		0			0
$737$		−	50.4163i		−	1.85711i
$738$		0			0
$739$	14.8112			0.544840			0.272420	−	0.962178i	$-0.412176\pi$
$739$	14.8112			0.544840			0.272420	+	0.962178i	$0.412176\pi$
$740$		0			0
$741$	2.72032	+	6.37829i	0.0999336	+	0.234312i
$742$		0			0
$743$	14.6124			0.536075			0.268038	−	0.963408i	$-0.413625\pi$
$743$	14.6124			0.536075			0.268038	+	0.963408i	$0.413625\pi$
$744$		0			0
$745$	−32.9423			−1.20691
$746$		0			0
$747$	−31.8024	−	30.5013i	−1.16359	−	1.11599i
$748$		0			0
$749$	−2.44501			−0.0893388
$750$		0			0
$751$			16.2098i			0.591503i	0.955265	+	0.295751i	$0.0955700\pi$
$751$			16.2098i			0.591503i	−0.955265	+	0.295751i	$0.904430\pi$
$752$		0			0
$753$	2.07287	+	4.86022i	0.0755397	+	0.177116i
$754$		0			0
$755$		−	35.9809i		−	1.30948i
$756$		0			0
$757$		−	25.0524i		−	0.910547i	−0.890352	−	0.455273i	$-0.849542\pi$
$757$		−	25.0524i		−	0.910547i	0.890352	−	0.455273i	$-0.150458\pi$
$758$		0			0
$759$	−4.35284	−	10.2060i	−0.157998	−	0.370455i
$760$		0			0
$761$		−	5.42352i		−	0.196602i	−0.995157	−	0.0983012i	$-0.968659\pi$
$761$		−	5.42352i		−	0.196602i	0.995157	−	0.0983012i	$-0.0313408\pi$
$762$		0			0
$763$	−0.0630165			−0.00228135
$764$		0			0
$765$	21.4720	+	20.5936i	0.776320	+	0.744562i
$766$		0			0
$767$	−12.0389			−0.434698
$768$		0			0
$769$	−29.2196			−1.05369			−0.526843	−	0.849963i	$-0.676624\pi$
$769$	−29.2196			−1.05369			−0.526843	+	0.849963i	$0.676624\pi$
$770$		0			0
$771$	−2.84456	−	6.66957i	−0.102444	−	0.240199i
$772$		0			0
$773$	−14.1117			−0.507564			−0.253782	−	0.967262i	$-0.581675\pi$
$773$	−14.1117			−0.507564			−0.253782	+	0.967262i	$0.581675\pi$
$774$		0			0
$775$		−	23.7211i		−	0.852087i
$776$		0			0
$777$	0.267652	−	0.114153i	0.00960195	−	0.00409521i
$778$		0			0
$779$			3.88529i			0.139205i
$780$		0			0
$781$		−	29.1241i		−	1.04214i
$782$		0			0
$783$	−15.6931	+	41.5183i	−0.560826	+	1.48374i
$784$		0			0
$785$		−	2.65416i		−	0.0947309i
$786$		0			0
$787$	−1.16977			−0.0416978			−0.0208489	−	0.999783i	$-0.506637\pi$
$787$	−1.16977			−0.0416978			−0.0208489	+	0.999783i	$0.506637\pi$
$788$		0			0
$789$	33.6904	−	14.3689i	1.19941	−	0.511545i
$790$		0			0
$791$	7.80749			0.277602
$792$		0			0
$793$	−8.84406			−0.314062
$794$		0			0
$795$	5.90266	−	2.51747i	0.209346	−	0.0892855i
$796$		0			0
$797$	−27.3497			−0.968776			−0.484388	−	0.874853i	$-0.660958\pi$
$797$	−27.3497			−0.968776			−0.484388	+	0.874853i	$0.660958\pi$
$798$		0			0
$799$		−	13.8081i		−	0.488494i
$800$		0			0
$801$	−34.3297	−	32.9253i	−1.21298	−	1.16336i
$802$		0			0
$803$		−	6.02970i		−	0.212783i
$804$		0			0
$805$		−	1.98772i		−	0.0700580i
$806$		0			0
$807$	−12.3216	+	5.25514i	−0.433741	+	0.184990i
$808$		0			0
$809$		−	20.4291i		−	0.718249i	−0.933290	−	0.359124i	$-0.883075\pi$
$809$		−	20.4291i		−	0.718249i	0.933290	−	0.359124i	$-0.116925\pi$
$810$		0			0
$811$	15.2642			0.535998			0.267999	−	0.963419i	$-0.413638\pi$
$811$	15.2642			0.535998			0.267999	+	0.963419i	$0.413638\pi$
$812$		0			0
$813$	2.07325	+	4.86110i	0.0727120	+	0.170486i
$814$		0			0
$815$	−65.8189			−2.30554
$816$		0			0
$817$	6.31001			0.220759
$818$		0			0
$819$	−3.87593	+	4.04125i	−0.135436	+	0.141213i
$820$		0			0
$821$	−22.7493			−0.793956			−0.396978	−	0.917828i	$-0.629941\pi$
$821$	−22.7493			−0.793956			−0.396978	+	0.917828i	$0.629941\pi$
$822$		0			0
$823$			10.0940i			0.351854i	0.984403	+	0.175927i	$0.0562923\pi$
$823$			10.0940i			0.351854i	−0.984403	+	0.175927i	$0.943708\pi$
$824$		0			0
$825$	9.01535	+	21.1381i	0.313874	+	0.735934i
$826$		0			0
$827$		−	8.22962i		−	0.286172i	−0.989710	−	0.143086i	$-0.954297\pi$
$827$		−	8.22962i		−	0.286172i	0.989710	−	0.143086i	$-0.0457026\pi$
$828$		0			0
$829$			26.0198i			0.903706i	0.892092	+	0.451853i	$0.149237\pi$
$829$			26.0198i			0.903706i	−0.892092	+	0.451853i	$0.850763\pi$
$830$		0			0
$831$	−12.8129	−	30.0421i	−0.444474	−	1.04215i
$832$		0			0
$833$		−	23.6310i		−	0.818765i
$834$		0			0
$835$	53.1602			1.83969
$836$		0			0
$837$	−37.1663	−	14.0481i	−1.28466	−	0.485575i
$838$		0			0
$839$	−37.2701			−1.28671			−0.643353	−	0.765569i	$-0.722457\pi$
$839$	−37.2701			−1.28671			−0.643353	+	0.765569i	$0.722457\pi$
$840$		0			0
$841$	43.9645			1.51602
$842$		0			0
$843$	10.4892	+	24.5937i	0.361266	+	0.847053i
$844$		0			0
$845$	8.61779			0.296461
$846$		0			0
$847$			3.39957i			0.116810i
$848$		0			0
$849$	−26.2610	+	11.2002i	−0.901275	+	0.384391i
$850$		0			0
$851$		−	0.539714i		−	0.0185012i
$852$		0			0
$853$		−	49.0827i		−	1.68056i	−0.542153	−	0.840280i	$-0.682391\pi$
$853$		−	49.0827i		−	1.68056i	0.542153	−	0.840280i	$-0.317609\pi$
$854$		0			0
$855$	5.91082	−	6.16295i	0.202146	−	0.210768i
$856$		0			0
$857$		−	23.7686i		−	0.811919i	−0.913891	−	0.405959i	$-0.866937\pi$
$857$		−	23.7686i		−	0.811919i	0.913891	−	0.405959i	$-0.133063\pi$
$858$		0			0
$859$	3.42519			0.116866			0.0584330	−	0.998291i	$-0.481390\pi$
$859$	3.42519			0.116866			0.0584330	+	0.998291i	$0.481390\pi$
$860$		0			0
$861$	−2.88595	+	1.23085i	−0.0983528	+	0.0419472i
$862$		0			0
$863$	46.0387			1.56717			0.783587	−	0.621282i	$-0.213388\pi$
$863$	46.0387			1.56717			0.783587	+	0.621282i	$0.213388\pi$
$864$		0			0
$865$	−44.5576			−1.51500
$866$		0			0
$867$	7.74526	−	3.30334i	0.263043	−	0.112187i
$868$		0			0
$869$	−62.6310			−2.12461
$870$		0			0
$871$			47.1930i			1.59907i
$872$		0			0
$873$	−29.7643	+	31.0339i	−1.00737	+	1.05034i
$874$		0			0
$875$		−	2.51854i		−	0.0851421i
$876$		0			0
$877$		−	41.7008i		−	1.40814i	−0.710132	−	0.704068i	$-0.751365\pi$
$877$		−	41.7008i		−	1.40814i	0.710132	−	0.704068i	$-0.248635\pi$
$878$		0			0
$879$	46.6382	−	19.8911i	1.57307	−	0.670909i
$880$		0			0
$881$			30.3385i			1.02213i	0.859542	+	0.511065i	$0.170749\pi$
$881$			30.3385i			1.02213i	−0.859542	+	0.511065i	$0.829251\pi$
$882$		0			0
$883$	−30.8512			−1.03823			−0.519113	−	0.854706i	$-0.673738\pi$
$883$	−30.8512			−1.03823			−0.519113	+	0.854706i	$0.673738\pi$
$884$		0			0
$885$	5.81620	+	13.6371i	0.195509	+	0.458407i
$886$		0			0
$887$	−49.1511			−1.65033			−0.825166	−	0.564890i	$-0.808919\pi$
$887$	−49.1511			−1.65033			−0.825166	+	0.564890i	$0.808919\pi$
$888$		0			0
$889$	1.22015			0.0409226
$890$		0			0
$891$	38.4584	−	1.60685i	1.28840	−	0.0538316i
$892$		0			0
$893$	−3.96323			−0.132625
$894$		0			0
$895$			3.75248i			0.125432i
$896$		0			0
$897$	4.07455	+	9.55351i	0.136045	+	0.318983i
$898$		0			0
$899$			65.3163i			2.17842i
$900$		0			0
$901$			4.53482i			0.151077i
$902$		0			0
$903$	1.99900	+	4.68701i	0.0665225	+	0.155974i
$904$		0			0
$905$		−	61.9732i		−	2.06006i
$906$		0			0
$907$	0.778059			0.0258350			0.0129175	−	0.999917i	$-0.495888\pi$
$907$	0.778059			0.0258350			0.0129175	+	0.999917i	$0.495888\pi$
$908$		0			0
$909$	1.88755	−	1.96806i	0.0626060	−	0.0652764i
$910$		0			0
$911$	−30.2563			−1.00243			−0.501217	−	0.865321i	$-0.667114\pi$
$911$	−30.2563			−1.00243			−0.501217	+	0.865321i	$0.667114\pi$
$912$		0			0
$913$	62.8202			2.07904
$914$		0			0
$915$	4.27274	+	10.0182i	0.141252	+	0.331191i
$916$		0			0
$917$	4.15205			0.137113
$918$		0			0
$919$		−	3.71572i		−	0.122570i	−0.998120	−	0.0612851i	$-0.980480\pi$
$919$		−	3.71572i		−	0.122570i	0.998120	−	0.0612851i	$-0.0195199\pi$
$920$		0			0
$921$	34.1053	−	14.5458i	1.12381	−	0.479301i
$922$		0			0
$923$			27.2621i			0.897344i
$924$		0			0
$925$			1.11782i			0.0367538i
$926$		0			0
$927$	−42.5196	−	40.7802i	−1.39653	−	1.33940i
$928$		0			0
$929$		−	47.2890i		−	1.55150i	−0.631039	−	0.775751i	$-0.717371\pi$
$929$		−	47.2890i		−	1.55150i	0.631039	−	0.775751i	$-0.282629\pi$
$930$		0			0
$931$	−6.78263			−0.222292
$932$		0			0
$933$	13.0119	−	5.54955i	0.425991	−	0.181684i
$934$		0			0
$935$	−42.4142			−1.38709
$936$		0			0
$937$	14.3287			0.468099			0.234049	−	0.972225i	$-0.424802\pi$
$937$	14.3287			0.468099			0.234049	+	0.972225i	$0.424802\pi$
$938$		0			0
$939$	−28.0701	+	11.9718i	−0.916032	+	0.390685i
$940$		0			0
$941$	−12.3295			−0.401931			−0.200965	−	0.979598i	$-0.564408\pi$
$941$	−12.3295			−0.401931			−0.200965	+	0.979598i	$0.564408\pi$
$942$		0			0
$943$			5.81945i			0.189507i
$944$		0			0
$945$	6.45030	+	2.43809i	0.209828	+	0.0793110i
$946$		0			0
$947$			22.0884i			0.717776i	0.933381	+	0.358888i	$0.116844\pi$
$947$			22.0884i			0.717776i	−0.933381	+	0.358888i	$0.883156\pi$
$948$		0			0
$949$			5.64420i			0.183218i
$950$		0			0
$951$	−4.02556	+	1.71689i	−0.130538	+	0.0556740i
$952$		0			0
$953$			51.7883i			1.67759i	0.544449	+	0.838794i	$0.316739\pi$
$953$			51.7883i			1.67759i	−0.544449	+	0.838794i	$0.683261\pi$
$954$		0			0
$955$	9.72669			0.314748
$956$		0			0
$957$	−24.8239	−	58.2040i	−0.802442	−	1.88147i
$958$		0			0
$959$	−3.16389			−0.102167
$960$		0			0
$961$	−27.4698			−0.886123
$962$		0			0
$963$	−11.3546	−	10.8901i	−0.365898	−	0.350929i
$964$		0			0
$965$	−63.7398			−2.05186
$966$		0			0
$967$			7.70555i			0.247794i	0.992295	+	0.123897i	$0.0395392\pi$
$967$			7.70555i			0.247794i	−0.992295	+	0.123897i	$0.960461\pi$
$968$		0			0
$969$	2.36739	+	5.55077i	0.0760516	+	0.178317i
$970$		0			0
$971$		−	56.4878i		−	1.81278i	−0.422443	−	0.906389i	$-0.638827\pi$
$971$		−	56.4878i		−	1.81278i	0.422443	−	0.906389i	$-0.361173\pi$
$972$		0			0
$973$			6.57412i			0.210757i
$974$		0			0
$975$	−8.43897	−	19.7867i	−0.270263	−	0.633681i
$976$		0			0
$977$		−	15.5457i		−	0.497352i	−0.968587	−	0.248676i	$-0.920005\pi$
$977$		−	15.5457i		−	0.497352i	0.968587	−	0.248676i	$-0.0799954\pi$
$978$		0			0
$979$	67.8126			2.16730
$980$		0			0
$981$	−0.292648	−	0.280676i	−0.00934354	−	0.00896130i
$982$		0			0
$983$	−18.2981			−0.583618			−0.291809	−	0.956477i	$-0.594257\pi$
$983$	−18.2981			−0.583618			−0.291809	+	0.956477i	$0.594257\pi$
$984$		0			0
$985$	−53.6095			−1.70814
$986$		0			0
$987$	−1.25554	−	2.94385i	−0.0399644	−	0.0937036i
$988$		0			0
$989$	9.45125			0.300532
$990$		0			0
$991$		−	57.6946i		−	1.83273i	−0.400346	−	0.916364i	$-0.631110\pi$
$991$		−	57.6946i		−	1.83273i	0.400346	−	0.916364i	$-0.368890\pi$
$992$		0			0
$993$	−31.8401	+	13.5797i	−1.01042	+	0.430940i
$994$		0			0
$995$		−	54.8291i		−	1.73820i
$996$		0			0
$997$		−	36.9642i		−	1.17067i	−0.810793	−	0.585333i	$-0.800964\pi$
$997$		−	36.9642i		−	1.17067i	0.810793	−	0.585333i	$-0.199036\pi$
$998$		0			0
$999$	1.75141	+	0.661999i	0.0554122	+	0.0209447i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1824.2.j.d.1103.21		24
3.2	odd	2	inner	1824.2.j.d.1103.24		24
4.3	odd	2		456.2.j.e.419.23	yes	24
8.3	odd	2	inner	1824.2.j.d.1103.22		24
8.5	even	2		456.2.j.e.419.1	✓	24
12.11	even	2		456.2.j.e.419.2	yes	24
24.5	odd	2		456.2.j.e.419.24	yes	24
24.11	even	2	inner	1824.2.j.d.1103.23		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.j.e.419.1	✓	24	8.5	even	2
456.2.j.e.419.2	yes	24	12.11	even	2
456.2.j.e.419.23	yes	24	4.3	odd	2
456.2.j.e.419.24	yes	24	24.5	odd	2
1824.2.j.d.1103.21		24	1.1	even	1	trivial
1824.2.j.d.1103.22		24	8.3	odd	2	inner
1824.2.j.d.1103.23		24	24.11	even	2	inner
1824.2.j.d.1103.24		24	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1824 = 2^{5} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1824.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.59320 - 0.679496i) q^{3} -2.84644 q^{5} -0.466225i q^{7} +(2.07657 - 2.16515i) q^{9} +O(q^{10})\)	\(q+(1.59320 - 0.679496i) q^{3} -2.84644 q^{5} -0.466225i q^{7} +(2.07657 - 2.16515i) q^{9} +4.27688i q^{11} -4.00344i q^{13} +(-4.53494 + 1.93414i) q^{15} -3.48404i q^{17} -1.00000 q^{19} +(-0.316798 - 0.742789i) q^{21} -1.49782 q^{23} +3.10219 q^{25} +(1.83719 - 4.86053i) q^{27} -8.54193 q^{29} -7.64656i q^{31} +(2.90612 + 6.81392i) q^{33} +1.32708i q^{35} +0.360333i q^{37} +(-2.72032 - 6.37829i) q^{39} -3.88529i q^{41} -6.31001 q^{43} +(-5.91082 + 6.16295i) q^{45} +3.96323 q^{47} +6.78263 q^{49} +(-2.36739 - 5.55077i) q^{51} -1.30160 q^{53} -12.1739i q^{55} +(-1.59320 + 0.679496i) q^{57} -3.00712i q^{59} -2.20911i q^{61} +(-1.00944 - 0.968149i) q^{63} +11.3955i q^{65} -11.7881 q^{67} +(-2.38632 + 1.01776i) q^{69} -6.80967 q^{71} -1.40984 q^{73} +(4.94241 - 2.10793i) q^{75} +1.99399 q^{77} +14.6441i q^{79} +(-0.375707 - 8.99215i) q^{81} -14.6883i q^{83} +9.91710i q^{85} +(-13.6090 + 5.80420i) q^{87} -15.8556i q^{89} -1.86651 q^{91} +(-5.19580 - 12.1825i) q^{93} +2.84644 q^{95} -14.3334 q^{97} +(9.26006 + 8.88124i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} - 6 q^{9}+O(q^{10}) \) 24 * q - 6 * q^3 - 6 * q^9	\( 24 q - 6 q^{3} - 6 q^{9} - 24 q^{19} + 96 q^{25} + 24 q^{33} - 16 q^{43} - 36 q^{49} - 38 q^{51} + 6 q^{57} - 20 q^{67} - 12 q^{73} + 30 q^{75} + 34 q^{81} + 36 q^{91} - 104 q^{97} - 56 q^{99}+O(q^{100}) \) 24 * q - 6 * q^3 - 6 * q^9 - 24 * q^19 + 96 * q^25 + 24 * q^33 - 16 * q^43 - 36 * q^49 - 38 * q^51 + 6 * q^57 - 20 * q^67 - 12 * q^73 + 30 * q^75 + 34 * q^81 + 36 * q^91 - 104 * q^97 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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