LMFDB - Embedded newform 5472.2.k.d.2431.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(2431,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5472.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.6
Character	$\chi$	$=$	5472.2431
Dual form			5472.2.k.d.2431.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62229 q^{5} -3.45599i q^{7} +O(q^{10})$	$q-2.62229 q^{5} -3.45599i q^{7} -3.79091i q^{11} -0.312566i q^{13} -4.13983 q^{17} +(3.41509 - 2.70871i) q^{19} -1.93245i q^{23} +1.87642 q^{25} -5.99971i q^{29} +9.38220 q^{31} +9.06261i q^{35} +6.64930i q^{37} -12.1267i q^{41} -10.2763i q^{43} -2.71541i q^{47} -4.94386 q^{49} +7.29368i q^{53} +9.94086i q^{55} +10.4650 q^{59} -2.54101 q^{61} +0.819640i q^{65} -11.2157 q^{67} -8.48104 q^{71} +16.2733 q^{73} -13.1013 q^{77} -2.45868 q^{79} +4.38764i q^{83} +10.8558 q^{85} -5.30976i q^{89} -1.08023 q^{91} +(-8.95537 + 7.10304i) q^{95} -10.6290i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 40 q^{25} - 24 q^{49} + 48 q^{61} + 16 q^{73} - 16 q^{85}+O(q^{100}) $ 40 * q + 40 * q^25 - 24 * q^49 + 48 * q^61 + 16 * q^73 - 16 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.62229			−1.17272			−0.586362	−	0.810049i	$-0.699441\pi$
$5$	−2.62229			−1.17272			−0.586362	+	0.810049i	$0.699441\pi$
$6$		0			0
$7$		−	3.45599i		−	1.30624i	−0.757254	−	0.653120i	$-0.773460\pi$
$7$		−	3.45599i		−	1.30624i	0.757254	−	0.653120i	$-0.226540\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.79091i		−	1.14300i	−0.820602	−	0.571500i	$-0.806362\pi$
$11$		−	3.79091i		−	1.14300i	0.820602	−	0.571500i	$-0.193638\pi$
$12$		0			0
$13$		−	0.312566i		−	0.0866903i	−0.999060	−	0.0433451i	$-0.986198\pi$
$13$		−	0.312566i		−	0.0866903i	0.999060	−	0.0433451i	$-0.0138015\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.13983			−1.00406			−0.502028	−	0.864851i	$-0.667412\pi$
$17$	−4.13983			−1.00406			−0.502028	+	0.864851i	$0.667412\pi$
$18$		0			0
$19$	3.41509	−	2.70871i	0.783476	−	0.621422i
$19$	3.41509	−	2.70871i	0.783476	−	0.621422i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.93245i		−	0.402943i	−0.979494	−	0.201472i	$-0.935428\pi$
$23$		−	1.93245i		−	0.402943i	0.979494	−	0.201472i	$-0.0645724\pi$
$24$		0			0
$25$	1.87642			0.375283
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.99971i		−	1.11412i	−0.830473	−	0.557059i	$-0.811930\pi$
$29$		−	5.99971i		−	1.11412i	0.830473	−	0.557059i	$-0.188070\pi$
$30$		0			0
$31$	9.38220			1.68509			0.842546	−	0.538624i	$-0.181056\pi$
$31$	9.38220			1.68509			0.842546	+	0.538624i	$0.181056\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			9.06261i			1.53186i
$36$		0			0
$37$			6.64930i			1.09314i	0.837414	+	0.546569i	$0.184067\pi$
$37$			6.64930i			1.09314i	−0.837414	+	0.546569i	$0.815933\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	12.1267i		−	1.89387i	−0.321432	−	0.946933i	$-0.604164\pi$
$41$		−	12.1267i		−	1.89387i	0.321432	−	0.946933i	$-0.395836\pi$
$42$		0			0
$43$		−	10.2763i		−	1.56712i	−0.621319	−	0.783558i	$-0.713403\pi$
$43$		−	10.2763i		−	1.56712i	0.621319	−	0.783558i	$-0.286597\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.71541i		−	0.396083i	−0.980194	−	0.198041i	$-0.936542\pi$
$47$		−	2.71541i		−	0.396083i	0.980194	−	0.198041i	$-0.0634580\pi$
$48$		0			0
$49$	−4.94386			−0.706266
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.29368i			1.00186i	0.865487	+	0.500932i	$0.167009\pi$
$53$			7.29368i			1.00186i	−0.865487	+	0.500932i	$0.832991\pi$
$54$		0			0
$55$			9.94086i			1.34043i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.4650			1.36242			0.681211	−	0.732087i	$-0.261454\pi$
$59$	10.4650			1.36242			0.681211	+	0.732087i	$0.261454\pi$
$60$		0			0
$61$	−2.54101			−0.325343			−0.162672	−	0.986680i	$-0.552011\pi$
$61$	−2.54101			−0.325343			−0.162672	+	0.986680i	$0.552011\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.819640i			0.101664i
$66$		0			0
$67$	−11.2157			−1.37022			−0.685111	−	0.728439i	$-0.740246\pi$
$67$	−11.2157			−1.37022			−0.685111	+	0.728439i	$0.740246\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.48104			−1.00651			−0.503257	−	0.864137i	$-0.667865\pi$
$71$	−8.48104			−1.00651			−0.503257	+	0.864137i	$0.667865\pi$
$72$		0			0
$73$	16.2733			1.90464			0.952321	−	0.305098i	$-0.0986893\pi$
$73$	16.2733			1.90464			0.952321	+	0.305098i	$0.0986893\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−13.1013			−1.49303
$78$		0			0
$79$	−2.45868			−0.276624			−0.138312	−	0.990389i	$-0.544168\pi$
$79$	−2.45868			−0.276624			−0.138312	+	0.990389i	$0.544168\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			4.38764i			0.481606i	0.970574	+	0.240803i	$0.0774107\pi$
$83$			4.38764i			0.481606i	−0.970574	+	0.240803i	$0.922589\pi$
$84$		0			0
$85$	10.8558			1.17748
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.30976i		−	0.562833i	−0.959586	−	0.281417i	$-0.909196\pi$
$89$		−	5.30976i		−	0.562833i	0.959586	−	0.281417i	$-0.0908043\pi$
$90$		0			0
$91$	−1.08023			−0.113238
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.95537	+	7.10304i	−0.918802	+	0.728757i
$96$		0			0
$97$		−	10.6290i		−	1.07921i	−0.841919	−	0.539604i	$-0.818574\pi$
$97$		−	10.6290i		−	1.07921i	0.841919	−	0.539604i	$-0.181426\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−16.6077			−1.65253			−0.826263	−	0.563284i	$-0.809538\pi$
$101$	−16.6077			−1.65253			−0.826263	+	0.563284i	$0.809538\pi$
$102$		0			0
$103$	2.91378			0.287103			0.143552	−	0.989643i	$-0.454148\pi$
$103$	2.91378			0.287103			0.143552	+	0.989643i	$0.454148\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−17.3067			−1.67310			−0.836552	−	0.547887i	$-0.815432\pi$
$107$	−17.3067			−1.67310			−0.836552	+	0.547887i	$0.815432\pi$
$108$		0			0
$109$			19.4387i			1.86189i	0.365156	+	0.930946i	$0.381016\pi$
$109$			19.4387i			1.86189i	−0.365156	+	0.930946i	$0.618984\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.49012i		−	0.422395i	−0.977443	−	0.211197i	$-0.932264\pi$
$113$		−	4.49012i		−	0.422395i	0.977443	−	0.211197i	$-0.0677363\pi$
$114$		0			0
$115$			5.06744i			0.472541i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			14.3072i			1.31154i
$120$		0			0
$121$	−3.37096			−0.306451
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.19095			0.732621
$126$		0			0
$127$	−5.62177			−0.498851			−0.249426	−	0.968394i	$-0.580242\pi$
$127$	−5.62177			−0.498851			−0.249426	+	0.968394i	$0.580242\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			13.4502i			1.17515i	0.809168	+	0.587577i	$0.199918\pi$
$131$			13.4502i			1.17515i	−0.809168	+	0.587577i	$0.800082\pi$
$132$		0			0
$133$	−9.36129	−	11.8025i	−0.811727	−	1.02341i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	17.1524			1.46543			0.732716	−	0.680534i	$-0.238252\pi$
$137$	17.1524			1.46543			0.732716	+	0.680534i	$0.238252\pi$
$138$		0			0
$139$			5.20882i			0.441806i	0.975296	+	0.220903i	$0.0709005\pi$
$139$			5.20882i			0.441806i	−0.975296	+	0.220903i	$0.929100\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.18491			−0.0990870
$144$		0			0
$145$			15.7330i			1.30655i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0179			−0.820696			−0.410348	−	0.911929i	$-0.634593\pi$
$149$	−10.0179			−0.820696			−0.410348	+	0.911929i	$0.634593\pi$
$150$		0			0
$151$	15.7573			1.28231			0.641155	−	0.767411i	$-0.278456\pi$
$151$	15.7573			1.28231			0.641155	+	0.767411i	$0.278456\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−24.6029			−1.97615
$156$		0			0
$157$	−19.1497			−1.52831			−0.764156	−	0.645032i	$-0.776844\pi$
$157$	−19.1497			−1.52831			−0.764156	+	0.645032i	$0.776844\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.67852			−0.526341
$162$		0			0
$163$		−	10.4849i		−	0.821239i	−0.911807	−	0.410619i	$-0.865313\pi$
$163$		−	10.4849i		−	0.821239i	0.911807	−	0.410619i	$-0.134687\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.41213			0.418803			0.209402	−	0.977830i	$-0.432848\pi$
$167$	5.41213			0.418803			0.209402	+	0.977830i	$0.432848\pi$
$168$		0			0
$169$	12.9023			0.992485
$170$		0			0
$171$		0			0
$172$		0			0
$173$			7.29368i			0.554528i	0.960794	+	0.277264i	$0.0894277\pi$
$173$			7.29368i			0.554528i	−0.960794	+	0.277264i	$0.910572\pi$
$174$		0			0
$175$		−	6.48487i		−	0.490210i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.0690			0.827335			0.413667	−	0.910428i	$-0.364248\pi$
$179$	11.0690			0.827335			0.413667	+	0.910428i	$0.364248\pi$
$180$		0			0
$181$		−	11.1282i		−	0.827152i	−0.910470	−	0.413576i	$-0.864280\pi$
$181$		−	11.1282i		−	0.827152i	0.910470	−	0.413576i	$-0.135720\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	17.4364i		−	1.28195i
$186$		0			0
$187$			15.6937i			1.14764i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			11.6929i			0.846069i	0.906114	+	0.423034i	$0.139035\pi$
$191$			11.6929i			0.846069i	−0.906114	+	0.423034i	$0.860965\pi$
$192$		0			0
$193$		−	21.7958i		−	1.56890i	−0.620195	−	0.784448i	$-0.712946\pi$
$193$		−	21.7958i		−	1.56890i	0.620195	−	0.784448i	$-0.287054\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.47914			−0.319125			−0.159563	−	0.987188i	$-0.551008\pi$
$197$	−4.47914			−0.319125			−0.159563	+	0.987188i	$0.551008\pi$
$198$		0			0
$199$			3.45599i			0.244989i	0.992469	+	0.122494i	$0.0390893\pi$
$199$			3.45599i			0.244989i	−0.992469	+	0.122494i	$0.960911\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.7349			−1.45531
$204$		0			0
$205$			31.7996i			2.22098i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.2685	−	12.9463i	−0.710286	−	0.895514i
$210$		0			0
$211$	−19.5970			−1.34911			−0.674556	−	0.738224i	$-0.735665\pi$
$211$	−19.5970			−1.34911			−0.674556	+	0.738224i	$0.735665\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			26.9474i			1.83780i
$216$		0			0
$217$		−	32.4248i		−	2.20114i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.29397i			0.0870418i
$222$		0			0
$223$	−14.9621			−1.00194			−0.500969	−	0.865465i	$-0.667023\pi$
$223$	−14.9621			−1.00194			−0.500969	+	0.865465i	$0.667023\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.82419			−0.187448			−0.0937240	−	0.995598i	$-0.529877\pi$
$227$	−2.82419			−0.187448			−0.0937240	+	0.995598i	$0.529877\pi$
$228$		0			0
$229$	−6.72095			−0.444133			−0.222066	−	0.975032i	$-0.571280\pi$
$229$	−6.72095			−0.444133			−0.222066	+	0.975032i	$0.571280\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−25.0979			−1.64422			−0.822109	−	0.569330i	$-0.807203\pi$
$233$	−25.0979			−1.64422			−0.822109	+	0.569330i	$0.807203\pi$
$234$		0			0
$235$			7.12059i			0.464496i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.66116i		−	0.172136i	−0.996289	−	0.0860681i	$-0.972570\pi$
$239$		−	2.66116i		−	0.172136i	0.996289	−	0.0860681i	$-0.0274303\pi$
$240$		0			0
$241$			4.47889i			0.288511i	0.989541	+	0.144255i	$0.0460786\pi$
$241$			4.47889i			0.288511i	−0.989541	+	0.144255i	$0.953921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.9642			0.828255
$246$		0			0
$247$	−0.846653	−	1.06744i	−0.0538712	−	0.0679198i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			19.3927i			1.22405i	0.790837	+	0.612027i	$0.209646\pi$
$251$			19.3927i			1.22405i	−0.790837	+	0.612027i	$0.790354\pi$
$252$		0			0
$253$	−7.32573			−0.460564
$254$		0			0
$255$		0			0
$256$		0			0
$257$			8.58341i			0.535419i	0.963500	+	0.267709i	$0.0862667\pi$
$257$			8.58341i			0.535419i	−0.963500	+	0.267709i	$0.913733\pi$
$258$		0			0
$259$	22.9799			1.42790
$260$		0			0
$261$		0			0
$262$		0			0
$263$			10.2237i			0.630423i	0.949021	+	0.315212i	$0.102076\pi$
$263$			10.2237i			0.630423i	−0.949021	+	0.315212i	$0.897924\pi$
$264$		0			0
$265$		−	19.1262i		−	1.17491i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	4.96021i		−	0.302429i	−0.988501	−	0.151215i	$-0.951682\pi$
$269$		−	4.96021i		−	0.302429i	0.988501	−	0.151215i	$-0.0483185\pi$
$270$		0			0
$271$			7.15568i			0.434676i	0.976096	+	0.217338i	$0.0697374\pi$
$271$			7.15568i			0.434676i	−0.976096	+	0.217338i	$0.930263\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	7.11332i		−	0.428949i
$276$		0			0
$277$	−10.0501			−0.603855			−0.301927	−	0.953331i	$-0.597630\pi$
$277$	−10.0501			−0.603855			−0.301927	+	0.953331i	$0.597630\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			13.0802i			0.780301i	0.920751	+	0.390150i	$0.127577\pi$
$281$			13.0802i			0.780301i	−0.920751	+	0.390150i	$0.872423\pi$
$282$		0			0
$283$			6.29684i			0.374308i	0.982331	+	0.187154i	$0.0599264\pi$
$283$			6.29684i			0.374308i	−0.982331	+	0.187154i	$0.940074\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−41.9096			−2.47384
$288$		0			0
$289$	0.138169			0.00812760
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	3.06646i		−	0.179145i	−0.995980	−	0.0895723i	$-0.971450\pi$
$293$		−	3.06646i		−	0.179145i	0.995980	−	0.0895723i	$-0.0285500\pi$
$294$		0			0
$295$	−27.4422			−1.59775
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.604018			−0.0349313
$300$		0			0
$301$	−35.5147			−2.04703
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.66328			0.381538
$306$		0			0
$307$	−2.58688			−0.147641			−0.0738205	−	0.997272i	$-0.523519\pi$
$307$	−2.58688			−0.147641			−0.0738205	+	0.997272i	$0.523519\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7.46437i		−	0.423266i	−0.977349	−	0.211633i	$-0.932122\pi$
$311$		−	7.46437i		−	0.423266i	0.977349	−	0.211633i	$-0.0678781\pi$
$312$		0			0
$313$	−23.2062			−1.31169			−0.655847	−	0.754894i	$-0.727688\pi$
$313$	−23.2062			−1.31169			−0.655847	+	0.754894i	$0.727688\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	31.2066i		−	1.75274i	−0.481641	−	0.876368i	$-0.659959\pi$
$317$		−	31.2066i		−	1.75274i	0.481641	−	0.876368i	$-0.340041\pi$
$318$		0			0
$319$	−22.7443			−1.27344
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−14.1379	+	11.2136i	−0.786654	+	0.623942i
$324$		0			0
$325$		−	0.586504i		−	0.0325334i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.38441			−0.517379
$330$		0			0
$331$	33.5050			1.84160			0.920800	−	0.390034i	$-0.127537\pi$
$331$	33.5050			1.84160			0.920800	+	0.390034i	$0.127537\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	29.4110			1.60689
$336$		0			0
$337$		−	4.17629i		−	0.227497i	−0.993510	−	0.113748i	$-0.963714\pi$
$337$		−	4.17629i		−	0.227497i	0.993510	−	0.113748i	$-0.0362858\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	35.5670i		−	1.92606i
$342$		0			0
$343$		−	7.10600i		−	0.383688i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			3.84515i			0.206418i	0.994660	+	0.103209i	$0.0329111\pi$
$347$			3.84515i			0.206418i	−0.994660	+	0.103209i	$0.967089\pi$
$348$		0			0
$349$	15.8449			0.848160			0.424080	−	0.905625i	$-0.360598\pi$
$349$	15.8449			0.848160			0.424080	+	0.905625i	$0.360598\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−27.8038			−1.47985			−0.739923	−	0.672691i	$-0.765138\pi$
$353$	−27.8038			−1.47985			−0.739923	+	0.672691i	$0.765138\pi$
$354$		0			0
$355$	22.2398			1.18036
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	9.61537i		−	0.507480i	−0.967272	−	0.253740i	$-0.918339\pi$
$359$		−	9.61537i		−	0.507480i	0.967272	−	0.253740i	$-0.0816607\pi$
$360$		0			0
$361$	4.32573	−	18.5010i	0.227670	−	0.973738i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−42.6733			−2.23362
$366$		0			0
$367$		−	9.99054i		−	0.521502i	−0.965406	−	0.260751i	$-0.916030\pi$
$367$		−	9.99054i		−	0.521502i	0.965406	−	0.260751i	$-0.0839701\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	25.2069			1.30868
$372$		0			0
$373$		−	5.14265i		−	0.266276i	−0.991097	−	0.133138i	$-0.957495\pi$
$373$		−	5.14265i		−	0.266276i	0.991097	−	0.133138i	$-0.0425054\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.87531			−0.0965832
$378$		0			0
$379$	2.41684			0.124145			0.0620724	−	0.998072i	$-0.480229\pi$
$379$	2.41684			0.124145			0.0620724	+	0.998072i	$0.480229\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.27695			−0.218542			−0.109271	−	0.994012i	$-0.534852\pi$
$383$	−4.27695			−0.218542			−0.109271	+	0.994012i	$0.534852\pi$
$384$		0			0
$385$	34.3555			1.75092
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.10750			−0.258960			−0.129480	−	0.991582i	$-0.541331\pi$
$389$	−5.10750			−0.258960			−0.129480	+	0.991582i	$0.541331\pi$
$390$		0			0
$391$			8.00000i			0.404577i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.44739			0.324403
$396$		0			0
$397$	2.02098			0.101430			0.0507150	−	0.998713i	$-0.483850\pi$
$397$	2.02098			0.101430			0.0507150	+	0.998713i	$0.483850\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	5.17583i		−	0.258469i	−0.991614	−	0.129234i	$-0.958748\pi$
$401$		−	5.17583i		−	0.258469i	0.991614	−	0.129234i	$-0.0412520\pi$
$402$		0			0
$403$		−	2.93256i		−	0.146081i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25.2069			1.24946
$408$		0			0
$409$		−	12.0483i		−	0.595752i	−0.954605	−	0.297876i	$-0.903722\pi$
$409$		−	12.0483i		−	0.595752i	0.954605	−	0.297876i	$-0.0962782\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	36.1668i		−	1.77965i
$414$		0			0
$415$		−	11.5057i		−	0.564791i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	14.2684i		−	0.697058i	−0.937298	−	0.348529i	$-0.886681\pi$
$419$		−	14.2684i		−	0.697058i	0.937298	−	0.348529i	$-0.113319\pi$
$420$		0			0
$421$			28.5324i			1.39058i	0.718728	+	0.695292i	$0.244725\pi$
$421$			28.5324i			1.39058i	−0.718728	+	0.695292i	$0.755275\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.76804			−0.376805
$426$		0			0
$427$			8.78172i			0.424977i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−22.3742			−1.07773			−0.538864	−	0.842393i	$-0.681146\pi$
$431$	−22.3742			−1.07773			−0.538864	+	0.842393i	$0.681146\pi$
$432$		0			0
$433$			25.6495i			1.23264i	0.787496	+	0.616319i	$0.211377\pi$
$433$			25.6495i			1.23264i	−0.787496	+	0.616319i	$0.788623\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.23445	−	6.59949i	−0.250398	−	0.315696i
$438$		0			0
$439$	−18.1044			−0.864075			−0.432038	−	0.901856i	$-0.642205\pi$
$439$	−18.1044			−0.864075			−0.432038	+	0.901856i	$0.642205\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	24.9715i		−	1.18643i	−0.805044	−	0.593215i	$-0.797858\pi$
$443$		−	24.9715i		−	1.18643i	0.805044	−	0.593215i	$-0.202142\pi$
$444$		0			0
$445$			13.9237i			0.660049i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7.89770i			0.372715i	0.982482	+	0.186358i	$0.0596683\pi$
$449$			7.89770i			0.372715i	−0.982482	+	0.186358i	$0.940332\pi$
$450$		0			0
$451$	−45.9710			−2.16469
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.83267			0.132797
$456$		0			0
$457$	35.8646			1.67768			0.838839	−	0.544380i	$-0.183235\pi$
$457$	35.8646			1.67768			0.838839	+	0.544380i	$0.183235\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	26.1038			1.21578			0.607888	−	0.794023i	$-0.292017\pi$
$461$	26.1038			1.21578			0.607888	+	0.794023i	$0.292017\pi$
$462$		0			0
$463$			22.9205i			1.06521i	0.846365	+	0.532603i	$0.178786\pi$
$463$			22.9205i			1.06521i	−0.846365	+	0.532603i	$0.821214\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			40.7435i			1.88538i	0.333666	+	0.942691i	$0.391714\pi$
$467$			40.7435i			1.88538i	−0.333666	+	0.942691i	$0.608286\pi$
$468$		0			0
$469$			38.7615i			1.78984i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−38.9563			−1.79121
$474$		0			0
$475$	6.40814	−	5.08268i	0.294025	−	0.233209i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	35.6052i		−	1.62684i	−0.581675	−	0.813422i	$-0.697602\pi$
$479$		−	35.6052i		−	1.62684i	0.581675	−	0.813422i	$-0.302398\pi$
$480$		0			0
$481$	2.07835			0.0947645
$482$		0			0
$483$		0			0
$484$		0			0
$485$			27.8723i			1.26561i
$486$		0			0
$487$	41.9003			1.89868			0.949342	−	0.314246i	$-0.101752\pi$
$487$	41.9003			1.89868			0.949342	+	0.314246i	$0.101752\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	10.4460i		−	0.471424i	−0.971823	−	0.235712i	$-0.924258\pi$
$491$		−	10.4460i		−	0.471424i	0.971823	−	0.235712i	$-0.0757421\pi$
$492$		0			0
$493$			24.8378i			1.11864i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			29.3104i			1.31475i
$498$		0			0
$499$			26.2351i			1.17444i	0.809426	+	0.587222i	$0.199779\pi$
$499$			26.2351i			1.17444i	−0.809426	+	0.587222i	$0.800221\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			28.9607i			1.29129i	0.763637	+	0.645646i	$0.223412\pi$
$503$			28.9607i			1.29129i	−0.763637	+	0.645646i	$0.776588\pi$
$504$		0			0
$505$	43.5502			1.93796
$506$		0			0
$507$		0			0
$508$		0			0
$509$			17.5728i			0.778900i	0.921048	+	0.389450i	$0.127335\pi$
$509$			17.5728i			0.778900i	−0.921048	+	0.389450i	$0.872665\pi$
$510$		0			0
$511$		−	56.2402i		−	2.48792i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.64078			−0.336693
$516$		0			0
$517$	−10.2938			−0.452723
$518$		0			0
$519$		0			0
$520$		0			0
$521$			27.7881i			1.21742i	0.793393	+	0.608710i	$0.208313\pi$
$521$			27.7881i			1.21742i	−0.793393	+	0.608710i	$0.791687\pi$
$522$		0			0
$523$	4.77511			0.208801			0.104401	−	0.994535i	$-0.466708\pi$
$523$	4.77511			0.208801			0.104401	+	0.994535i	$0.466708\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−38.8407			−1.69193
$528$		0			0
$529$	19.2656			0.837637
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.79038			−0.164180
$534$		0			0
$535$	45.3833			1.96209
$536$		0			0
$537$		0			0
$538$		0			0
$539$			18.7417i			0.807262i
$540$		0			0
$541$	−7.43964			−0.319855			−0.159927	−	0.987129i	$-0.551126\pi$
$541$	−7.43964			−0.319855			−0.159927	+	0.987129i	$0.551126\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	50.9740i		−	2.18349i
$546$		0			0
$547$	−11.4353			−0.488936			−0.244468	−	0.969657i	$-0.578613\pi$
$547$	−11.4353			−0.488936			−0.244468	+	0.969657i	$0.578613\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−16.2515	−	20.4896i	−0.692337	−	0.872885i
$552$		0			0
$553$			8.49719i			0.361337i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.9954			0.847233			0.423617	−	0.905842i	$-0.360760\pi$
$557$	19.9954			0.847233			0.423617	+	0.905842i	$0.360760\pi$
$558$		0			0
$559$	−3.21201			−0.135854
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.2378			0.600052			0.300026	−	0.953931i	$-0.403005\pi$
$563$	14.2378			0.600052			0.300026	+	0.953931i	$0.403005\pi$
$564$		0			0
$565$			11.7744i			0.495353i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.5935i		−	1.11486i	−0.830225	−	0.557428i	$-0.811788\pi$
$569$		−	26.5935i		−	1.11486i	0.830225	−	0.557428i	$-0.188212\pi$
$570$		0			0
$571$		−	0.223119i		−	0.00933725i	−0.999989	−	0.00466863i	$-0.998514\pi$
$571$		−	0.223119i		−	0.00933725i	0.999989	−	0.00466863i	$-0.00148608\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.62608i		−	0.151218i
$576$		0			0
$577$	−10.6893			−0.445000			−0.222500	−	0.974933i	$-0.571422\pi$
$577$	−10.6893			−0.445000			−0.222500	+	0.974933i	$0.571422\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.1636			0.629093
$582$		0			0
$583$	27.6497			1.14513
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.7331i		−	0.773197i	−0.922248	−	0.386598i	$-0.873650\pi$
$587$		−	18.7331i		−	0.773197i	0.922248	−	0.386598i	$-0.126350\pi$
$588$		0			0
$589$	32.0411	−	25.4137i	1.32023	−	1.04715i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.608378			−0.0249831			−0.0124915	−	0.999922i	$-0.503976\pi$
$593$	−0.608378			−0.0249831			−0.0124915	+	0.999922i	$0.503976\pi$
$594$		0			0
$595$		−	37.5176i		−	1.53807i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	37.7055			1.54060			0.770302	−	0.637679i	$-0.220105\pi$
$599$	37.7055			1.54060			0.770302	+	0.637679i	$0.220105\pi$
$600$		0			0
$601$		−	37.0682i		−	1.51204i	−0.654547	−	0.756022i	$-0.727140\pi$
$601$		−	37.0682i		−	1.51204i	0.654547	−	0.756022i	$-0.272860\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.83965			0.359383
$606$		0			0
$607$	−21.3372			−0.866051			−0.433025	−	0.901382i	$-0.642554\pi$
$607$	−21.3372			−0.866051			−0.433025	+	0.901382i	$0.642554\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.848744			−0.0343365
$612$		0			0
$613$	11.0995			0.448306			0.224153	−	0.974554i	$-0.428038\pi$
$613$	11.0995			0.448306			0.224153	+	0.974554i	$0.428038\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.4194			0.741536			0.370768	−	0.928725i	$-0.379094\pi$
$617$	18.4194			0.741536			0.370768	+	0.928725i	$0.379094\pi$
$618$		0			0
$619$			10.3872i			0.417496i	0.977969	+	0.208748i	$0.0669388\pi$
$619$			10.3872i			0.417496i	−0.977969	+	0.208748i	$0.933061\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.3505			−0.735196
$624$		0			0
$625$	−30.8611			−1.23445
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	27.5270i		−	1.09757i
$630$		0			0
$631$			28.3486i			1.12854i	0.825590	+	0.564270i	$0.190842\pi$
$631$			28.3486i			1.12854i	−0.825590	+	0.564270i	$0.809158\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.7419			0.585015
$636$		0			0
$637$			1.54528i			0.0612263i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			8.71908i			0.344383i	0.985064	+	0.172191i	$0.0550848\pi$
$641$			8.71908i			0.344383i	−0.985064	+	0.172191i	$0.944915\pi$
$642$		0			0
$643$		−	44.9723i		−	1.77353i	−0.462218	−	0.886767i	$-0.652946\pi$
$643$		−	44.9723i		−	1.77353i	0.462218	−	0.886767i	$-0.347054\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			37.1977i			1.46239i	0.682167	+	0.731197i	$0.261038\pi$
$647$			37.1977i			1.46239i	−0.682167	+	0.731197i	$0.738962\pi$
$648$		0			0
$649$		−	39.6717i		−	1.55725i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.6923			0.574955			0.287478	−	0.957787i	$-0.407183\pi$
$653$	14.6923			0.574955			0.287478	+	0.957787i	$0.407183\pi$
$654$		0			0
$655$		−	35.2705i		−	1.37813i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.08649			0.276050			0.138025	−	0.990429i	$-0.455925\pi$
$659$	7.08649			0.276050			0.138025	+	0.990429i	$0.455925\pi$
$660$		0			0
$661$			23.6437i			0.919632i	0.888014	+	0.459816i	$0.152085\pi$
$661$			23.6437i			0.919632i	−0.888014	+	0.459816i	$0.847915\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	24.5480	+	30.9497i	0.951932	+	1.20018i
$666$		0			0
$667$	−11.5941			−0.448926
$668$		0			0
$669$		0			0
$670$		0			0
$671$			9.63274i			0.371868i
$672$		0			0
$673$		−	26.3731i		−	1.01661i	−0.861178	−	0.508304i	$-0.830273\pi$
$673$		−	26.3731i		−	1.01661i	0.861178	−	0.508304i	$-0.169727\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			28.6187i			1.09990i	0.835196	+	0.549952i	$0.185354\pi$
$677$			28.6187i			1.09990i	−0.835196	+	0.549952i	$0.814646\pi$
$678$		0			0
$679$	−36.7336			−1.40971
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.54086			0.288543			0.144271	−	0.989538i	$-0.453916\pi$
$683$	7.54086			0.288543			0.144271	+	0.989538i	$0.453916\pi$
$684$		0			0
$685$	−44.9787			−1.71855
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.27976			0.0868519
$690$		0			0
$691$		−	26.2643i		−	0.999140i	−0.866273	−	0.499570i	$-0.833491\pi$
$691$		−	26.2643i		−	0.999140i	0.866273	−	0.499570i	$-0.166509\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	13.6590i		−	0.518117i
$696$		0			0
$697$			50.2023i			1.90155i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.89410			0.260387			0.130193	−	0.991489i	$-0.458440\pi$
$701$	6.89410			0.260387			0.130193	+	0.991489i	$0.458440\pi$
$702$		0			0
$703$	18.0111	+	22.7080i	0.679300	+	0.856448i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			57.3960i			2.15860i
$708$		0			0
$709$	5.47996			0.205804			0.102902	−	0.994691i	$-0.467187\pi$
$709$	5.47996			0.205804			0.102902	+	0.994691i	$0.467187\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	18.1306i		−	0.678997i
$714$		0			0
$715$	3.10718			0.116202
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	20.0108i		−	0.746277i	−0.927776	−	0.373139i	$-0.878282\pi$
$719$		−	20.0108i		−	0.746277i	0.927776	−	0.373139i	$-0.121718\pi$
$720$		0			0
$721$		−	10.0700i		−	0.375026i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	11.2580i		−	0.418110i
$726$		0			0
$727$			11.4852i			0.425961i	0.977056	+	0.212980i	$0.0683171\pi$
$727$			11.4852i			0.425961i	−0.977056	+	0.212980i	$0.931683\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			42.5420i			1.57347i
$732$		0			0
$733$	−39.4684			−1.45780			−0.728899	−	0.684621i	$-0.759968\pi$
$733$	−39.4684			−1.45780			−0.728899	+	0.684621i	$0.759968\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			42.5178i			1.56616i
$738$		0			0
$739$			23.5271i			0.865459i	0.901524	+	0.432729i	$0.142449\pi$
$739$			23.5271i			0.865459i	−0.901524	+	0.432729i	$0.857551\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	36.0080			1.32101			0.660503	−	0.750824i	$-0.270343\pi$
$743$	36.0080			1.32101			0.660503	+	0.750824i	$0.270343\pi$
$744$		0			0
$745$	26.2698			0.962451
$746$		0			0
$747$		0			0
$748$		0			0
$749$			59.8119i			2.18548i
$750$		0			0
$751$	27.1440			0.990497			0.495249	−	0.868751i	$-0.335077\pi$
$751$	27.1440			0.990497			0.495249	+	0.868751i	$0.335077\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−41.3202			−1.50380
$756$		0			0
$757$	−20.5449			−0.746717			−0.373359	−	0.927687i	$-0.621794\pi$
$757$	−20.5449			−0.746717			−0.373359	+	0.927687i	$0.621794\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	43.3551			1.57162			0.785811	−	0.618467i	$-0.212246\pi$
$761$	43.3551			1.57162			0.785811	+	0.618467i	$0.212246\pi$
$762$		0			0
$763$	67.1800			2.43208
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.27099i		−	0.118109i
$768$		0			0
$769$	23.3068			0.840464			0.420232	−	0.907417i	$-0.361949\pi$
$769$	23.3068			0.840464			0.420232	+	0.907417i	$0.361949\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	28.1874i		−	1.01383i	−0.861996	−	0.506915i	$-0.830786\pi$
$773$		−	28.1874i		−	1.01383i	0.861996	−	0.506915i	$-0.169214\pi$
$774$		0			0
$775$	17.6049			0.632387
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−32.8477	−	41.4137i	−1.17689	−	1.48380i
$780$		0			0
$781$			32.1508i			1.15045i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	50.2161			1.79229
$786$		0			0
$787$	−29.2617			−1.04307			−0.521533	−	0.853231i	$-0.674640\pi$
$787$	−29.2617			−1.04307			−0.521533	+	0.853231i	$0.674640\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.5178			−0.551749
$792$		0			0
$793$			0.794235i			0.0282041i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			32.5731i			1.15380i	0.816815	+	0.576899i	$0.195738\pi$
$797$			32.5731i			1.15380i	−0.816815	+	0.576899i	$0.804262\pi$
$798$		0			0
$799$			11.2413i			0.397689i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	61.6904i		−	2.17701i
$804$		0			0
$805$	17.5130			0.617253
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.4739			−0.368244			−0.184122	−	0.982903i	$-0.558944\pi$
$809$	−10.4739			−0.368244			−0.184122	+	0.982903i	$0.558944\pi$
$810$		0			0
$811$	−16.9823			−0.596330			−0.298165	−	0.954514i	$-0.596375\pi$
$811$	−16.9823			−0.596330			−0.298165	+	0.954514i	$0.596375\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			27.4944i			0.963087i
$816$		0			0
$817$	−27.8355	−	35.0944i	−0.973840	−	1.22780i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.9411			0.556348			0.278174	−	0.960531i	$-0.410271\pi$
$821$	15.9411			0.556348			0.278174	+	0.960531i	$0.410271\pi$
$822$		0			0
$823$			49.9892i			1.74251i	0.490828	+	0.871256i	$0.336694\pi$
$823$			49.9892i			1.74251i	−0.490828	+	0.871256i	$0.663306\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.6964			1.17174			0.585869	−	0.810406i	$-0.300753\pi$
$827$	33.6964			1.17174			0.585869	+	0.810406i	$0.300753\pi$
$828$		0			0
$829$			10.0314i			0.348403i	0.984710	+	0.174202i	$0.0557345\pi$
$829$			10.0314i			0.348403i	−0.984710	+	0.174202i	$0.944266\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	20.4667			0.709130
$834$		0			0
$835$	−14.1922			−0.491141
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−40.4884			−1.39782			−0.698908	−	0.715212i	$-0.746330\pi$
$839$	−40.4884			−1.39782			−0.698908	+	0.715212i	$0.746330\pi$
$840$		0			0
$841$	−6.99653			−0.241260
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−33.8336			−1.16391
$846$		0			0
$847$			11.6500i			0.400299i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.8494			0.440473
$852$		0			0
$853$	−32.6103			−1.11656			−0.558278	−	0.829654i	$-0.688538\pi$
$853$	−32.6103			−1.11656			−0.558278	+	0.829654i	$0.688538\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			45.9254i			1.56878i	0.620267	+	0.784391i	$0.287024\pi$
$857$			45.9254i			1.56878i	−0.620267	+	0.784391i	$0.712976\pi$
$858$		0			0
$859$		−	13.4992i		−	0.460586i	−0.973121	−	0.230293i	$-0.926032\pi$
$859$		−	13.4992i		−	0.460586i	0.973121	−	0.230293i	$-0.0739684\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	53.4329			1.81888			0.909439	−	0.415837i	$-0.136511\pi$
$863$	53.4329			1.81888			0.909439	+	0.415837i	$0.136511\pi$
$864$		0			0
$865$		−	19.1262i		−	0.650309i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9.32064i			0.316181i
$870$		0			0
$871$			3.50566i			0.118785i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	28.3078i		−	0.956979i
$876$		0			0
$877$		−	36.9423i		−	1.24745i	−0.781643	−	0.623726i	$-0.785618\pi$
$877$		−	36.9423i		−	1.24745i	0.781643	−	0.623726i	$-0.214382\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−56.2710			−1.89582			−0.947908	−	0.318543i	$-0.896807\pi$
$881$	−56.2710			−1.89582			−0.947908	+	0.318543i	$0.896807\pi$
$882$		0			0
$883$		−	13.3449i		−	0.449093i	−0.974463	−	0.224546i	$-0.927910\pi$
$883$		−	13.3449i		−	0.449093i	0.974463	−	0.224546i	$-0.0720900\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.90158			0.198156			0.0990778	−	0.995080i	$-0.468411\pi$
$887$	5.90158			0.198156			0.0990778	+	0.995080i	$0.468411\pi$
$888$		0			0
$889$			19.4288i			0.651620i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.35526	−	9.27336i	−0.246134	−	0.310321i
$894$		0			0
$895$	−29.0261			−0.970236
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	56.2905i		−	1.87739i
$900$		0			0
$901$		−	30.1946i		−	1.00593i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			29.1814i			0.970021i
$906$		0			0
$907$	8.69861			0.288833			0.144416	−	0.989517i	$-0.453870\pi$
$907$	8.69861			0.288833			0.144416	+	0.989517i	$0.453870\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.6963			0.851357			0.425679	−	0.904874i	$-0.360035\pi$
$911$	25.6963			0.851357			0.425679	+	0.904874i	$0.360035\pi$
$912$		0			0
$913$	16.6331			0.550476
$914$		0			0
$915$		0			0
$916$		0			0
$917$	46.4839			1.53503
$918$		0			0
$919$		−	55.0734i		−	1.81670i	−0.418208	−	0.908351i	$-0.637342\pi$
$919$		−	55.0734i		−	1.81670i	0.418208	−	0.908351i	$-0.362658\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.65089i			0.0872550i
$924$		0			0
$925$			12.4769i			0.410237i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	34.2386			1.12333			0.561666	−	0.827364i	$-0.310161\pi$
$929$	34.2386			1.12333			0.561666	+	0.827364i	$0.310161\pi$
$930$		0			0
$931$	−16.8837	+	13.3915i	−0.553342	+	0.438889i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	41.1534i		−	1.34586i
$936$		0			0
$937$	−39.3300			−1.28486			−0.642428	−	0.766346i	$-0.722073\pi$
$937$	−39.3300			−1.28486			−0.642428	+	0.766346i	$0.722073\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	40.9670i		−	1.33548i	−0.744393	−	0.667742i	$-0.767261\pi$
$941$		−	40.9670i		−	1.33548i	0.744393	−	0.667742i	$-0.232739\pi$
$942$		0			0
$943$	−23.4341			−0.763120
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	23.7808i		−	0.772773i	−0.922337	−	0.386387i	$-0.873723\pi$
$947$		−	23.7808i		−	0.772773i	0.922337	−	0.386387i	$-0.126277\pi$
$948$		0			0
$949$		−	5.08647i		−	0.165114i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	20.4489i		−	0.662405i	−0.943560	−	0.331203i	$-0.892546\pi$
$953$		−	20.4489i		−	0.662405i	0.943560	−	0.331203i	$-0.107454\pi$
$954$		0			0
$955$		−	30.6622i		−	0.992206i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	59.2787i		−	1.91421i
$960$		0			0
$961$	57.0256			1.83954
$962$		0			0
$963$		0			0
$964$		0			0
$965$			57.1549i			1.83988i
$966$		0			0
$967$			21.6384i			0.695846i	0.937523	+	0.347923i	$0.113113\pi$
$967$			21.6384i			0.695846i	−0.937523	+	0.347923i	$0.886887\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36.5206			−1.17200			−0.586001	−	0.810311i	$-0.699298\pi$
$971$	−36.5206			−1.17200			−0.586001	+	0.810311i	$0.699298\pi$
$972$		0			0
$973$	18.0016			0.577106
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	15.4088i		−	0.492971i	−0.969147	−	0.246485i	$-0.920724\pi$
$977$		−	15.4088i		−	0.492971i	0.969147	−	0.246485i	$-0.0792757\pi$
$978$		0			0
$979$	−20.1288			−0.643319
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.7027			0.532734			0.266367	−	0.963872i	$-0.414177\pi$
$983$	16.7027			0.532734			0.266367	+	0.963872i	$0.414177\pi$
$984$		0			0
$985$	11.7456			0.374246
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.8583			−0.631459
$990$		0			0
$991$	−6.67333			−0.211985			−0.105993	−	0.994367i	$-0.533802\pi$
$991$	−6.67333			−0.211985			−0.105993	+	0.994367i	$0.533802\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	9.06261i		−	0.287304i
$996$		0			0
$997$	13.6012			0.430755			0.215378	−	0.976531i	$-0.430902\pi$
$997$	13.6012			0.430755			0.215378	+	0.976531i	$0.430902\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.k.d.2431.6	yes	40
3.2	odd	2	inner	5472.2.k.d.2431.33	yes	40
4.3	odd	2	inner	5472.2.k.d.2431.8	yes	40
12.11	even	2	inner	5472.2.k.d.2431.34	yes	40
19.18	odd	2	inner	5472.2.k.d.2431.5	✓	40
57.56	even	2	inner	5472.2.k.d.2431.35	yes	40
76.75	even	2	inner	5472.2.k.d.2431.7	yes	40
228.227	odd	2	inner	5472.2.k.d.2431.36	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.k.d.2431.5	✓	40	19.18	odd	2	inner
5472.2.k.d.2431.6	yes	40	1.1	even	1	trivial
5472.2.k.d.2431.7	yes	40	76.75	even	2	inner
5472.2.k.d.2431.8	yes	40	4.3	odd	2	inner
5472.2.k.d.2431.33	yes	40	3.2	odd	2	inner
5472.2.k.d.2431.34	yes	40	12.11	even	2	inner
5472.2.k.d.2431.35	yes	40	57.56	even	2	inner
5472.2.k.d.2431.36	yes	40	228.227	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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.62229 q^{5} -3.45599i q^{7} +O(q^{10})\)	\(q-2.62229 q^{5} -3.45599i q^{7} -3.79091i q^{11} -0.312566i q^{13} -4.13983 q^{17} +(3.41509 - 2.70871i) q^{19} -1.93245i q^{23} +1.87642 q^{25} -5.99971i q^{29} +9.38220 q^{31} +9.06261i q^{35} +6.64930i q^{37} -12.1267i q^{41} -10.2763i q^{43} -2.71541i q^{47} -4.94386 q^{49} +7.29368i q^{53} +9.94086i q^{55} +10.4650 q^{59} -2.54101 q^{61} +0.819640i q^{65} -11.2157 q^{67} -8.48104 q^{71} +16.2733 q^{73} -13.1013 q^{77} -2.45868 q^{79} +4.38764i q^{83} +10.8558 q^{85} -5.30976i q^{89} -1.08023 q^{91} +(-8.95537 + 7.10304i) q^{95} -10.6290i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 40 q^{25} - 24 q^{49} + 48 q^{61} + 16 q^{73} - 16 q^{85}+O(q^{100}) \) 40 * q + 40 * q^25 - 24 * q^49 + 48 * q^61 + 16 * q^73 - 16 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more