LMFDB - Embedded newform 5472.2.k.d.2431.14

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			2431.14
Character	$\chi$	$=$	5472.2431
Dual form			5472.2.k.d.2431.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32365 q^{5} -0.339676i q^{7} +O(q^{10})$	$q-1.32365 q^{5} -0.339676i q^{7} +0.576869i q^{11} +2.03883i q^{13} +6.46996 q^{17} +(-2.76466 + 3.36996i) q^{19} -1.23648i q^{23} -3.24795 q^{25} +2.19433i q^{29} +4.72657 q^{31} +0.449613i q^{35} -6.47522i q^{37} -4.29355i q^{41} -10.4722i q^{43} +11.2538i q^{47} +6.88462 q^{49} +10.9968i q^{53} -0.763574i q^{55} -11.6309 q^{59} -8.98787 q^{61} -2.69870i q^{65} +8.33409 q^{67} -12.2304 q^{71} -1.82405 q^{73} +0.195949 q^{77} +8.98300 q^{79} +6.79316i q^{83} -8.56397 q^{85} +12.8645i q^{89} +0.692542 q^{91} +(3.65945 - 4.46066i) q^{95} -17.6073i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.32365			−0.591955			−0.295978	−	0.955195i	$-0.595645\pi$
$5$	−1.32365			−0.591955			−0.295978	+	0.955195i	$0.595645\pi$
$6$		0			0
$7$		−	0.339676i		−	0.128386i	−0.997938	−	0.0641928i	$-0.979553\pi$
$7$		−	0.339676i		−	0.128386i	0.997938	−	0.0641928i	$-0.0204473\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.576869i			0.173933i	0.996211	+	0.0869663i	$0.0277172\pi$
$11$			0.576869i			0.173933i	−0.996211	+	0.0869663i	$0.972283\pi$
$12$		0			0
$13$			2.03883i			0.565470i	0.959198	+	0.282735i	$0.0912416\pi$
$13$			2.03883i			0.565470i	−0.959198	+	0.282735i	$0.908758\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.46996			1.56920			0.784598	−	0.620005i	$-0.212870\pi$
$17$	6.46996			1.56920			0.784598	+	0.620005i	$0.212870\pi$
$18$		0			0
$19$	−2.76466	+	3.36996i	−0.634257	+	0.773122i
$19$	−2.76466	+	3.36996i	−0.634257	+	0.773122i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.23648i		−	0.257825i	−0.991656	−	0.128912i	$-0.958851\pi$
$23$		−	1.23648i		−	0.257825i	0.991656	−	0.128912i	$-0.0411486\pi$
$24$		0			0
$25$	−3.24795			−0.649589
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.19433i			0.407477i	0.979025	+	0.203738i	$0.0653092\pi$
$29$			2.19433i			0.407477i	−0.979025	+	0.203738i	$0.934691\pi$
$30$		0			0
$31$	4.72657			0.848917			0.424459	−	0.905447i	$-0.360464\pi$
$31$	4.72657			0.848917			0.424459	+	0.905447i	$0.360464\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			0.449613i			0.0759985i
$36$		0			0
$37$		−	6.47522i		−	1.06452i	−0.846581	−	0.532260i	$-0.821343\pi$
$37$		−	6.47522i		−	1.06452i	0.846581	−	0.532260i	$-0.178657\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.29355i		−	0.670540i	−0.942122	−	0.335270i	$-0.891173\pi$
$41$		−	4.29355i		−	0.670540i	0.942122	−	0.335270i	$-0.108827\pi$
$42$		0			0
$43$		−	10.4722i		−	1.59700i	−0.601994	−	0.798501i	$-0.705627\pi$
$43$		−	10.4722i		−	1.59700i	0.601994	−	0.798501i	$-0.294373\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			11.2538i			1.64154i	0.571260	+	0.820769i	$0.306455\pi$
$47$			11.2538i			1.64154i	−0.571260	+	0.820769i	$0.693545\pi$
$48$		0			0
$49$	6.88462			0.983517
$50$		0			0
$51$		0			0
$52$		0			0
$53$			10.9968i			1.51053i	0.655420	+	0.755264i	$0.272491\pi$
$53$			10.9968i			1.51053i	−0.655420	+	0.755264i	$0.727509\pi$
$54$		0			0
$55$		−	0.763574i		−	0.102960i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.6309			−1.51422			−0.757108	−	0.653290i	$-0.773388\pi$
$59$	−11.6309			−1.51422			−0.757108	+	0.653290i	$0.773388\pi$
$60$		0			0
$61$	−8.98787			−1.15078			−0.575389	−	0.817880i	$-0.695149\pi$
$61$	−8.98787			−1.15078			−0.575389	+	0.817880i	$0.695149\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.69870i		−	0.334733i
$66$		0			0
$67$	8.33409			1.01817			0.509085	−	0.860716i	$-0.329984\pi$
$67$	8.33409			1.01817			0.509085	+	0.860716i	$0.329984\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.2304			−1.45148			−0.725740	−	0.687969i	$-0.758503\pi$
$71$	−12.2304			−1.45148			−0.725740	+	0.687969i	$0.758503\pi$
$72$		0			0
$73$	−1.82405			−0.213489			−0.106744	−	0.994286i	$-0.534043\pi$
$73$	−1.82405			−0.213489			−0.106744	+	0.994286i	$0.534043\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.195949			0.0223304
$78$		0			0
$79$	8.98300			1.01067			0.505333	−	0.862924i	$-0.331370\pi$
$79$	8.98300			1.01067			0.505333	+	0.862924i	$0.331370\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			6.79316i			0.745646i	0.927902	+	0.372823i	$0.121610\pi$
$83$			6.79316i			0.745646i	−0.927902	+	0.372823i	$0.878390\pi$
$84$		0			0
$85$	−8.56397			−0.928893
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.8645i			1.36363i	0.731523	+	0.681817i	$0.238810\pi$
$89$			12.8645i			1.36363i	−0.731523	+	0.681817i	$0.761190\pi$
$90$		0			0
$91$	0.692542			0.0725981
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.65945	−	4.46066i	0.375452	−	0.457654i
$96$		0			0
$97$		−	17.6073i		−	1.78775i	−0.448320	−	0.893873i	$-0.647977\pi$
$97$		−	17.6073i		−	1.78775i	0.448320	−	0.893873i	$-0.352023\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.89438			−0.686017			−0.343008	−	0.939332i	$-0.611446\pi$
$101$	−6.89438			−0.686017			−0.343008	+	0.939332i	$0.611446\pi$
$102$		0			0
$103$	−13.7532			−1.35514			−0.677572	−	0.735457i	$-0.736968\pi$
$103$	−13.7532			−1.35514			−0.677572	+	0.735457i	$0.736968\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.79792			0.463832			0.231916	−	0.972736i	$-0.425500\pi$
$107$	4.79792			0.463832			0.231916	+	0.972736i	$0.425500\pi$
$108$		0			0
$109$			12.5171i			1.19892i	0.800404	+	0.599461i	$0.204619\pi$
$109$			12.5171i			1.19892i	−0.800404	+	0.599461i	$0.795381\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.1658i			0.956317i	0.878274	+	0.478159i	$0.158696\pi$
$113$			10.1658i			0.956317i	−0.878274	+	0.478159i	$0.841304\pi$
$114$		0			0
$115$			1.63667i			0.152621i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.19769i		−	0.201462i
$120$		0			0
$121$	10.6672			0.969747
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.9174			0.976483
$126$		0			0
$127$	−11.6090			−1.03013			−0.515066	−	0.857151i	$-0.672233\pi$
$127$	−11.6090			−1.03013			−0.515066	+	0.857151i	$0.672233\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.24278i			0.632804i	0.948625	+	0.316402i	$0.102475\pi$
$131$			7.24278i			0.632804i	−0.948625	+	0.316402i	$0.897525\pi$
$132$		0			0
$133$	1.14470	+	0.939090i	0.0992577	+	0.0814294i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	17.1914			1.46876			0.734381	−	0.678737i	$-0.237473\pi$
$137$	17.1914			1.46876			0.734381	+	0.678737i	$0.237473\pi$
$138$		0			0
$139$			8.83557i			0.749423i	0.927141	+	0.374712i	$0.122258\pi$
$139$			8.83557i			0.749423i	−0.927141	+	0.374712i	$0.877742\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.17614			−0.0983535
$144$		0			0
$145$		−	2.90453i		−	0.241208i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.3829			1.42407			0.712033	−	0.702146i	$-0.247775\pi$
$149$	17.3829			1.42407			0.712033	+	0.702146i	$0.247775\pi$
$150$		0			0
$151$	3.96745			0.322866			0.161433	−	0.986884i	$-0.448388\pi$
$151$	3.96745			0.322866			0.161433	+	0.986884i	$0.448388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.25634			−0.502521
$156$		0			0
$157$	4.07199			0.324980			0.162490	−	0.986710i	$-0.448047\pi$
$157$	4.07199			0.324980			0.162490	+	0.986710i	$0.448047\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.420004			−0.0331010
$162$		0			0
$163$			5.10325i			0.399717i	0.979825	+	0.199859i	$0.0640483\pi$
$163$			5.10325i			0.399717i	−0.979825	+	0.199859i	$0.935952\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−19.8087			−1.53285			−0.766423	−	0.642336i	$-0.777965\pi$
$167$	−19.8087			−1.53285			−0.766423	+	0.642336i	$0.777965\pi$
$168$		0			0
$169$	8.84317			0.680244
$170$		0			0
$171$		0			0
$172$		0			0
$173$			10.9968i			0.836072i	0.908430	+	0.418036i	$0.137281\pi$
$173$			10.9968i			0.836072i	−0.908430	+	0.418036i	$0.862719\pi$
$174$		0			0
$175$			1.10325i			0.0833978i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.1519			−1.05776			−0.528881	−	0.848696i	$-0.677388\pi$
$179$	−14.1519			−1.05776			−0.528881	+	0.848696i	$0.677388\pi$
$180$		0			0
$181$			22.9094i			1.70284i	0.524485	+	0.851420i	$0.324258\pi$
$181$			22.9094i			1.70284i	−0.524485	+	0.851420i	$0.675742\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			8.57094i			0.630148i
$186$		0			0
$187$			3.73232i			0.272934i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			15.7798i			1.14178i	0.821026	+	0.570891i	$0.193402\pi$
$191$			15.7798i			1.14178i	−0.821026	+	0.570891i	$0.806598\pi$
$192$		0			0
$193$			16.0018i			1.15183i	0.817509	+	0.575916i	$0.195354\pi$
$193$			16.0018i			1.15183i	−0.817509	+	0.575916i	$0.804646\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.0003			0.783740			0.391870	−	0.920021i	$-0.371828\pi$
$197$	11.0003			0.783740			0.391870	+	0.920021i	$0.371828\pi$
$198$		0			0
$199$			0.339676i			0.0240790i	0.999928	+	0.0120395i	$0.00383239\pi$
$199$			0.339676i			0.0240790i	−0.999928	+	0.0120395i	$0.996168\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.745361			0.0523141
$204$		0			0
$205$			5.68316i			0.396929i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.94403	−	1.59485i	−0.134471	−	0.110318i
$210$		0			0
$211$	−22.5824			−1.55463			−0.777317	−	0.629109i	$-0.783420\pi$
$211$	−22.5824			−1.55463			−0.777317	+	0.629109i	$0.783420\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			13.8616i			0.945353i
$216$		0			0
$217$		−	1.60550i		−	0.108989i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			13.1911i			0.887332i
$222$		0			0
$223$	8.95807			0.599877			0.299938	−	0.953959i	$-0.403034\pi$
$223$	8.95807			0.599877			0.299938	+	0.953959i	$0.403034\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.57354			−0.436301			−0.218151	−	0.975915i	$-0.570002\pi$
$227$	−6.57354			−0.436301			−0.218151	+	0.975915i	$0.570002\pi$
$228$		0			0
$229$	−0.709376			−0.0468769			−0.0234385	−	0.999725i	$-0.507461\pi$
$229$	−0.709376			−0.0468769			−0.0234385	+	0.999725i	$0.507461\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.50916			0.622966			0.311483	−	0.950252i	$-0.399174\pi$
$233$	9.50916			0.622966			0.311483	+	0.950252i	$0.399174\pi$
$234$		0			0
$235$		−	14.8961i		−	0.971717i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			18.6014i			1.20322i	0.798788	+	0.601612i	$0.205475\pi$
$239$			18.6014i			1.20322i	−0.798788	+	0.601612i	$0.794525\pi$
$240$		0			0
$241$		−	16.4341i		−	1.05862i	−0.848430	−	0.529308i	$-0.822452\pi$
$241$		−	16.4341i		−	1.05862i	0.848430	−	0.529308i	$-0.177548\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.11284			−0.582198
$246$		0			0
$247$	−6.87078	−	5.63667i	−0.437177	−	0.358653i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			18.6891i			1.17965i	0.807532	+	0.589824i	$0.200803\pi$
$251$			18.6891i			1.17965i	−0.807532	+	0.589824i	$0.799197\pi$
$252$		0			0
$253$	0.713289			0.0448441
$254$		0			0
$255$		0			0
$256$		0			0
$257$			20.4428i			1.27519i	0.770372	+	0.637595i	$0.220071\pi$
$257$			20.4428i			1.27519i	−0.770372	+	0.637595i	$0.779929\pi$
$258$		0			0
$259$	−2.19948			−0.136669
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.7881i		−	1.52850i	−0.644921	−	0.764249i	$-0.723110\pi$
$263$		−	24.7881i		−	1.52850i	0.644921	−	0.764249i	$-0.276890\pi$
$264$		0			0
$265$		−	14.5560i		−	0.894165i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	1.97894i		−	0.120658i	−0.998179	−	0.0603291i	$-0.980785\pi$
$269$		−	1.97894i		−	0.120658i	0.998179	−	0.0603291i	$-0.0192150\pi$
$270$		0			0
$271$			21.3684i			1.29804i	0.760773	+	0.649018i	$0.224820\pi$
$271$			21.3684i			1.29804i	−0.760773	+	0.649018i	$0.775180\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1.87364i		−	0.112985i
$276$		0			0
$277$	−27.1810			−1.63315			−0.816574	−	0.577240i	$-0.804130\pi$
$277$	−27.1810			−1.63315			−0.816574	+	0.577240i	$0.804130\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8.02890i		−	0.478964i	−0.970901	−	0.239482i	$-0.923022\pi$
$281$		−	8.02890i		−	0.478964i	0.970901	−	0.239482i	$-0.0769776\pi$
$282$		0			0
$283$			0.156215i			0.00928603i	0.999989	+	0.00464301i	$0.00147792\pi$
$283$			0.156215i			0.00928603i	−0.999989	+	0.00464301i	$0.998522\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.45842			−0.0860876
$288$		0			0
$289$	24.8604			1.46237
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.98807i			0.583509i	0.956493	+	0.291755i	$0.0942391\pi$
$293$			9.98807i			0.583509i	−0.956493	+	0.291755i	$0.905761\pi$
$294$		0			0
$295$	15.3953			0.896348
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.52098			0.145792
$300$		0			0
$301$	−3.55717			−0.205032
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.8968			0.681209
$306$		0			0
$307$	25.4288			1.45130			0.725648	−	0.688066i	$-0.241540\pi$
$307$	25.4288			1.45130			0.725648	+	0.688066i	$0.241540\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			14.5475i			0.824915i	0.910977	+	0.412457i	$0.135329\pi$
$311$			14.5475i			0.824915i	−0.910977	+	0.412457i	$0.864671\pi$
$312$		0			0
$313$	24.5472			1.38749			0.693745	−	0.720220i	$-0.255959\pi$
$313$	24.5472			1.38749			0.693745	+	0.720220i	$0.255959\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5.92968i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$317$			5.92968i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$318$		0			0
$319$	−1.26584			−0.0708734
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−17.8872	+	21.8035i	−0.995273	+	1.21318i
$324$		0			0
$325$		−	6.62201i		−	0.367323i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.82265			0.210750
$330$		0			0
$331$	−2.29805			−0.126312			−0.0631560	−	0.998004i	$-0.520117\pi$
$331$	−2.29805			−0.126312			−0.0631560	+	0.998004i	$0.520117\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.0314			−0.602711
$336$		0			0
$337$			5.82148i			0.317116i	0.987350	+	0.158558i	$0.0506845\pi$
$337$			5.82148i			0.317116i	−0.987350	+	0.158558i	$0.949315\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.72661i			0.147654i
$342$		0			0
$343$		−	4.71627i		−	0.254655i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6.77070i			0.363470i	0.983348	+	0.181735i	$0.0581714\pi$
$347$			6.77070i			0.363470i	−0.983348	+	0.181735i	$0.941829\pi$
$348$		0			0
$349$	−21.4025			−1.14565			−0.572825	−	0.819677i	$-0.694153\pi$
$349$	−21.4025			−1.14565			−0.572825	+	0.819677i	$0.694153\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.7518			0.731935			0.365968	−	0.930628i	$-0.380738\pi$
$353$	13.7518			0.731935			0.365968	+	0.930628i	$0.380738\pi$
$354$		0			0
$355$	16.1888			0.859211
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.80637i		−	0.359226i	−0.983737	−	0.179613i	$-0.942515\pi$
$359$		−	6.80637i		−	0.359226i	0.983737	−	0.179613i	$-0.0574846\pi$
$360$		0			0
$361$	−3.71329	−	18.6336i	−0.195436	−	0.980716i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.41440			0.126376
$366$		0			0
$367$		−	15.8885i		−	0.829375i	−0.909964	−	0.414687i	$-0.863891\pi$
$367$		−	15.8885i		−	0.829375i	0.909964	−	0.414687i	$-0.136109\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.73536			0.193930
$372$		0			0
$373$			31.2115i			1.61607i	0.589136	+	0.808034i	$0.299468\pi$
$373$			31.2115i			1.61607i	−0.589136	+	0.808034i	$0.700532\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.47386			−0.230416
$378$		0			0
$379$	−34.2766			−1.76067			−0.880336	−	0.474350i	$-0.842683\pi$
$379$	−34.2766			−1.76067			−0.880336	+	0.474350i	$0.842683\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−26.9972			−1.37949			−0.689746	−	0.724052i	$-0.742278\pi$
$383$	−26.9972			−1.37949			−0.689746	+	0.724052i	$0.742278\pi$
$384$		0			0
$385$	−0.259368			−0.0132186
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.26959			0.317881			0.158940	−	0.987288i	$-0.449192\pi$
$389$	6.26959			0.317881			0.158940	+	0.987288i	$0.449192\pi$
$390$		0			0
$391$		−	8.00000i		−	0.404577i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.8904			−0.598269
$396$		0			0
$397$	−20.0438			−1.00597			−0.502985	−	0.864295i	$-0.667765\pi$
$397$	−20.0438			−1.00597			−0.502985	+	0.864295i	$0.667765\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.24074i			0.161835i	0.996721	+	0.0809174i	$0.0257850\pi$
$401$			3.24074i			0.161835i	−0.996721	+	0.0809174i	$0.974215\pi$
$402$		0			0
$403$			9.63667i			0.480037i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.73536			0.185155
$408$		0			0
$409$			4.79513i			0.237104i	0.992948	+	0.118552i	$0.0378252\pi$
$409$			4.79513i			0.237104i	−0.992948	+	0.118552i	$0.962175\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			3.95074i			0.194403i
$414$		0			0
$415$		−	8.99178i		−	0.441389i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	33.0930i		−	1.61670i	−0.588703	−	0.808349i	$-0.700361\pi$
$419$		−	33.0930i		−	1.61670i	0.588703	−	0.808349i	$-0.299639\pi$
$420$		0			0
$421$			24.6616i			1.20193i	0.799273	+	0.600967i	$0.205218\pi$
$421$			24.6616i			1.20193i	−0.799273	+	0.600967i	$0.794782\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−21.0141			−1.01933
$426$		0			0
$427$			3.05297i			0.147743i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.65204			−0.224081			−0.112040	−	0.993704i	$-0.535739\pi$
$431$	−4.65204			−0.224081			−0.112040	+	0.993704i	$0.535739\pi$
$432$		0			0
$433$		−	28.3582i		−	1.36281i	−0.731907	−	0.681405i	$-0.761369\pi$
$433$		−	28.3582i		−	1.36281i	0.731907	−	0.681405i	$-0.238631\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.16690	+	3.41846i	0.199330	+	0.163527i
$438$		0			0
$439$	−41.0602			−1.95969			−0.979847	−	0.199749i	$-0.935987\pi$
$439$	−41.0602			−1.95969			−0.979847	+	0.199749i	$0.935987\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	2.28116i		−	0.108381i	−0.998531	−	0.0541906i	$-0.982742\pi$
$443$		−	2.28116i		−	0.108381i	0.998531	−	0.0541906i	$-0.0172579\pi$
$444$		0			0
$445$		−	17.0281i		−	0.807210i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			13.5178i			0.637944i	0.947764	+	0.318972i	$0.103338\pi$
$449$			13.5178i			0.637944i	−0.947764	+	0.318972i	$0.896662\pi$
$450$		0			0
$451$	2.47681			0.116629
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.916684			−0.0429748
$456$		0			0
$457$	11.9338			0.558238			0.279119	−	0.960256i	$-0.409958\pi$
$457$	11.9338			0.558238			0.279119	+	0.960256i	$0.409958\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.6512			0.775525			0.387762	−	0.921759i	$-0.373248\pi$
$461$	16.6512			0.775525			0.387762	+	0.921759i	$0.373248\pi$
$462$		0			0
$463$			29.9635i			1.39252i	0.717788	+	0.696261i	$0.245154\pi$
$463$			29.9635i			1.39252i	−0.717788	+	0.696261i	$0.754846\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	31.6208i		−	1.46323i	−0.681716	−	0.731617i	$-0.738766\pi$
$467$		−	31.6208i		−	1.46323i	0.681716	−	0.731617i	$-0.261234\pi$
$468$		0			0
$469$		−	2.83089i		−	0.130718i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.04111			0.277771
$474$		0			0
$475$	8.97947	−	10.9455i	0.412006	−	0.502212i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	14.0541i		−	0.642150i	−0.947054	−	0.321075i	$-0.895956\pi$
$479$		−	14.0541i		−	0.642150i	0.947054	−	0.321075i	$-0.104044\pi$
$480$		0			0
$481$	13.2019			0.601954
$482$		0			0
$483$		0			0
$484$		0			0
$485$			23.3059i			1.05827i
$486$		0			0
$487$	22.3600			1.01323			0.506613	−	0.862173i	$-0.330897\pi$
$487$	22.3600			1.01323			0.506613	+	0.862173i	$0.330897\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			42.2760i			1.90789i	0.299985	+	0.953944i	$0.403018\pi$
$491$			42.2760i			1.90789i	−0.299985	+	0.953944i	$0.596982\pi$
$492$		0			0
$493$			14.1972i			0.639410i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			4.15437i			0.186349i
$498$		0			0
$499$			23.1043i			1.03429i	0.855898	+	0.517145i	$0.173005\pi$
$499$			23.1043i			1.03429i	−0.855898	+	0.517145i	$0.826995\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			35.4501i			1.58064i	0.612692	+	0.790321i	$0.290086\pi$
$503$			35.4501i			1.58064i	−0.612692	+	0.790321i	$0.709914\pi$
$504$		0			0
$505$	9.12576			0.406091
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	40.2458i		−	1.78386i	−0.452171	−	0.891931i	$-0.649350\pi$
$509$		−	40.2458i		−	1.78386i	0.452171	−	0.891931i	$-0.350650\pi$
$510$		0			0
$511$			0.619586i			0.0274089i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.2044			0.802184
$516$		0			0
$517$	−6.49198			−0.285517
$518$		0			0
$519$		0			0
$520$		0			0
$521$			40.9529i			1.79418i	0.441849	+	0.897089i	$0.354323\pi$
$521$			40.9529i			1.79418i	−0.441849	+	0.897089i	$0.645677\pi$
$522$		0			0
$523$	4.73822			0.207188			0.103594	−	0.994620i	$-0.466966\pi$
$523$	4.73822			0.207188			0.103594	+	0.994620i	$0.466966\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	30.5807			1.33212
$528$		0			0
$529$	21.4711			0.933526
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8.75381			0.379170
$534$		0			0
$535$	−6.35078			−0.274568
$536$		0			0
$537$		0			0
$538$		0			0
$539$			3.97152i			0.171066i
$540$		0			0
$541$	−8.05718			−0.346405			−0.173203	−	0.984886i	$-0.555412\pi$
$541$	−8.05718			−0.346405			−0.173203	+	0.984886i	$0.555412\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	16.5683i		−	0.709709i
$546$		0			0
$547$	9.63897			0.412133			0.206066	−	0.978538i	$-0.433934\pi$
$547$	9.63897			0.412133			0.206066	+	0.978538i	$0.433934\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.39481	−	6.06658i	−0.315029	−	0.258445i
$552$		0			0
$553$		−	3.05131i		−	0.129755i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.8657			0.926477			0.463239	−	0.886234i	$-0.346687\pi$
$557$	21.8657			0.926477			0.463239	+	0.886234i	$0.346687\pi$
$558$		0			0
$559$	21.3511			0.903056
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.8371			−1.55250			−0.776248	−	0.630427i	$-0.782880\pi$
$563$	−36.8371			−1.55250			−0.776248	+	0.630427i	$0.782880\pi$
$564$		0			0
$565$		−	13.4560i		−	0.566097i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			0.700989i			0.0293870i	0.999892	+	0.0146935i	$0.00467725\pi$
$569$			0.700989i			0.0293870i	−0.999892	+	0.0146935i	$0.995323\pi$
$570$		0			0
$571$		−	35.0051i		−	1.46492i	−0.680812	−	0.732458i	$-0.738373\pi$
$571$		−	35.0051i		−	1.46492i	0.680812	−	0.732458i	$-0.261627\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			4.01603i			0.167480i
$576$		0			0
$577$	−1.96586			−0.0818396			−0.0409198	−	0.999162i	$-0.513029\pi$
$577$	−1.96586			−0.0818396			−0.0409198	+	0.999162i	$0.513029\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.30748			0.0957302
$582$		0			0
$583$	−6.34372			−0.262730
$584$		0			0
$585$		0			0
$586$		0			0
$587$			21.3645i			0.881809i	0.897554	+	0.440904i	$0.145342\pi$
$587$			21.3645i			0.881809i	−0.897554	+	0.440904i	$0.854658\pi$
$588$		0			0
$589$	−13.0674	+	15.9284i	−0.538432	+	0.656317i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−31.5944			−1.29743			−0.648714	−	0.761032i	$-0.724693\pi$
$593$	−31.5944			−1.29743			−0.648714	+	0.761032i	$0.724693\pi$
$594$		0			0
$595$			2.90898i			0.119256i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.2252			0.662944			0.331472	−	0.943465i	$-0.392455\pi$
$599$	16.2252			0.662944			0.331472	+	0.943465i	$0.392455\pi$
$600$		0			0
$601$		−	10.9107i		−	0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$601$		−	10.9107i		−	0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.1197			−0.574047
$606$		0			0
$607$	9.71719			0.394409			0.197204	−	0.980362i	$-0.436814\pi$
$607$	9.71719			0.394409			0.197204	+	0.980362i	$0.436814\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−22.9446			−0.928240
$612$		0			0
$613$	−29.2530			−1.18152			−0.590759	−	0.806848i	$-0.701172\pi$
$613$	−29.2530			−1.18152			−0.590759	+	0.806848i	$0.701172\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.92916			−0.399733			−0.199867	−	0.979823i	$-0.564051\pi$
$617$	−9.92916			−0.399733			−0.199867	+	0.979823i	$0.564051\pi$
$618$		0			0
$619$		−	0.946424i		−	0.0380400i	−0.999819	−	0.0190200i	$-0.993945\pi$
$619$		−	0.946424i		−	0.0380400i	0.999819	−	0.0190200i	$-0.00605462\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.36976			0.175071
$624$		0			0
$625$	1.78888			0.0715552
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	41.8944i		−	1.67044i
$630$		0			0
$631$		−	24.0260i		−	0.956461i	−0.878234	−	0.478230i	$-0.841278\pi$
$631$		−	24.0260i		−	0.956461i	0.878234	−	0.478230i	$-0.158722\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.3663			0.609792
$636$		0			0
$637$			14.0366i			0.556149i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	19.3900i		−	0.765860i	−0.923777	−	0.382930i	$-0.874915\pi$
$641$		−	19.3900i		−	0.765860i	0.923777	−	0.382930i	$-0.125085\pi$
$642$		0			0
$643$		−	5.78127i		−	0.227991i	−0.993481	−	0.113996i	$-0.963635\pi$
$643$		−	5.78127i		−	0.227991i	0.993481	−	0.113996i	$-0.0363650\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			2.07799i			0.0816944i	0.999165	+	0.0408472i	$0.0130057\pi$
$647$			2.07799i			0.0816944i	−0.999165	+	0.0408472i	$0.986994\pi$
$648$		0			0
$649$		−	6.70951i		−	0.263371i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20.3701			−0.797143			−0.398571	−	0.917137i	$-0.630494\pi$
$653$	−20.3701			−0.797143			−0.398571	+	0.917137i	$0.630494\pi$
$654$		0			0
$655$		−	9.58691i		−	0.374592i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	32.2986			1.25817			0.629087	−	0.777335i	$-0.283429\pi$
$659$	32.2986			1.25817			0.629087	+	0.777335i	$0.283429\pi$
$660$		0			0
$661$		−	12.5778i		−	0.489222i	−0.969621	−	0.244611i	$-0.921340\pi$
$661$		−	12.5778i		−	0.489222i	0.969621	−	0.244611i	$-0.0786602\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.51518	−	1.24303i	−0.0587561	−	0.0482026i
$666$		0			0
$667$	2.71325			0.105058
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	5.18482i		−	0.200158i
$672$		0			0
$673$		−	19.6600i		−	0.757836i	−0.925430	−	0.378918i	$-0.876296\pi$
$673$		−	19.6600i		−	0.757836i	0.925430	−	0.378918i	$-0.123704\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	32.3120i		−	1.24185i	−0.783870	−	0.620925i	$-0.786757\pi$
$677$		−	32.3120i		−	1.24185i	0.783870	−	0.620925i	$-0.213243\pi$
$678$		0			0
$679$	−5.98077			−0.229521
$680$		0			0
$681$		0			0
$682$		0			0
$683$	36.5199			1.39739			0.698697	−	0.715418i	$-0.253764\pi$
$683$	36.5199			1.39739			0.698697	+	0.715418i	$0.253764\pi$
$684$		0			0
$685$	−22.7555			−0.869441
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22.4206			−0.854158
$690$		0			0
$691$			16.1205i			0.613254i	0.951830	+	0.306627i	$0.0992004\pi$
$691$			16.1205i			0.613254i	−0.951830	+	0.306627i	$0.900800\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	11.6952i		−	0.443625i
$696$		0			0
$697$		−	27.7791i		−	1.05221i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.0616			0.833255			0.416628	−	0.909077i	$-0.363212\pi$
$701$	22.0616			0.833255			0.416628	+	0.909077i	$0.363212\pi$
$702$		0			0
$703$	21.8213	+	17.9018i	0.823004	+	0.675179i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			2.34186i			0.0880746i
$708$		0			0
$709$	−23.0317			−0.864973			−0.432487	−	0.901640i	$-0.642364\pi$
$709$	−23.0317			−0.864973			−0.432487	+	0.901640i	$0.642364\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	5.84433i		−	0.218872i
$714$		0			0
$715$	1.55680			0.0582209
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	2.75966i		−	0.102918i	−0.998675	−	0.0514590i	$-0.983613\pi$
$719$		−	2.75966i		−	0.102918i	0.998675	−	0.0514590i	$-0.0163871\pi$
$720$		0			0
$721$			4.67163i			0.173981i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	7.12706i		−	0.264692i
$726$		0			0
$727$		−	46.8852i		−	1.73887i	−0.494044	−	0.869437i	$-0.664482\pi$
$727$		−	46.8852i		−	1.73887i	0.494044	−	0.869437i	$-0.335518\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	67.7550i		−	2.50601i
$732$		0			0
$733$	37.2610			1.37627			0.688134	−	0.725584i	$-0.258430\pi$
$733$	37.2610			1.37627			0.688134	+	0.725584i	$0.258430\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			4.80768i			0.177093i
$738$		0			0
$739$		−	14.7975i		−	0.544335i	−0.962250	−	0.272168i	$-0.912259\pi$
$739$		−	14.7975i		−	0.544335i	0.962250	−	0.272168i	$-0.0877405\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−29.6640			−1.08827			−0.544134	−	0.838998i	$-0.683142\pi$
$743$	−29.6640			−1.08827			−0.544134	+	0.838998i	$0.683142\pi$
$744$		0			0
$745$	−23.0090			−0.842983
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1.62974i		−	0.0595494i
$750$		0			0
$751$	−7.79736			−0.284530			−0.142265	−	0.989829i	$-0.545438\pi$
$751$	−7.79736			−0.284530			−0.142265	+	0.989829i	$0.545438\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.25152			−0.191122
$756$		0			0
$757$	0.649328			0.0236002			0.0118001	−	0.999930i	$-0.496244\pi$
$757$	0.649328			0.0236002			0.0118001	+	0.999930i	$0.496244\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.7995			0.608982			0.304491	−	0.952515i	$-0.401514\pi$
$761$	16.7995			0.608982			0.304491	+	0.952515i	$0.401514\pi$
$762$		0			0
$763$	4.25177			0.153924
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	23.7134i		−	0.856243i
$768$		0			0
$769$	−14.0281			−0.505866			−0.252933	−	0.967484i	$-0.581395\pi$
$769$	−14.0281			−0.505866			−0.252933	+	0.967484i	$0.581395\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			21.8726i			0.786703i	0.919388	+	0.393352i	$0.128684\pi$
$773$			21.8726i			0.786703i	−0.919388	+	0.393352i	$0.871316\pi$
$774$		0			0
$775$	−15.3516			−0.551447
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.4691	+	11.8702i	0.518409	+	0.425294i
$780$		0			0
$781$		−	7.05533i		−	0.252460i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.38990			−0.192374
$786$		0			0
$787$	22.1975			0.791255			0.395628	−	0.918411i	$-0.370527\pi$
$787$	22.1975			0.791255			0.395628	+	0.918411i	$0.370527\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.45308			0.122777
$792$		0			0
$793$		−	18.3247i		−	0.650730i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			52.0226i			1.84273i	0.388694	+	0.921367i	$0.372926\pi$
$797$			52.0226i			1.84273i	−0.388694	+	0.921367i	$0.627074\pi$
$798$		0			0
$799$			72.8117i			2.57589i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.05224i		−	0.0371326i
$804$		0			0
$805$	0.555939			0.0195943
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.7714			−0.589652			−0.294826	−	0.955551i	$-0.595262\pi$
$809$	−16.7714			−0.589652			−0.294826	+	0.955551i	$0.595262\pi$
$810$		0			0
$811$	−16.4591			−0.577956			−0.288978	−	0.957336i	$-0.593315\pi$
$811$	−16.4591			−0.577956			−0.288978	+	0.957336i	$0.593315\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	6.75493i		−	0.236615i
$816$		0			0
$817$	35.2911	+	28.9522i	1.23468	+	1.01291i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−23.7963			−0.830498			−0.415249	−	0.909708i	$-0.636306\pi$
$821$	−23.7963			−0.830498			−0.415249	+	0.909708i	$0.636306\pi$
$822$		0			0
$823$		−	11.7609i		−	0.409959i	−0.978766	−	0.204979i	$-0.934287\pi$
$823$		−	11.7609i		−	0.409959i	0.978766	−	0.204979i	$-0.0657128\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.47553			0.294723			0.147361	−	0.989083i	$-0.452922\pi$
$827$	8.47553			0.294723			0.147361	+	0.989083i	$0.452922\pi$
$828$		0			0
$829$			6.02803i			0.209362i	0.994506	+	0.104681i	$0.0333822\pi$
$829$			6.02803i			0.209362i	−0.994506	+	0.104681i	$0.966618\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	44.5432			1.54333
$834$		0			0
$835$	26.2199			0.907376
$836$		0			0
$837$		0			0
$838$		0			0
$839$	32.6735			1.12802			0.564008	−	0.825769i	$-0.309259\pi$
$839$	32.6735			1.12802			0.564008	+	0.825769i	$0.309259\pi$
$840$		0			0
$841$	24.1849			0.833963
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.7053			−0.402674
$846$		0			0
$847$		−	3.62340i		−	0.124502i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00651			−0.274460
$852$		0			0
$853$	12.0586			0.412880			0.206440	−	0.978459i	$-0.433812\pi$
$853$	12.0586			0.412880			0.206440	+	0.978459i	$0.433812\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	27.5140i		−	0.939862i	−0.882703	−	0.469931i	$-0.844279\pi$
$857$		−	27.5140i		−	0.939862i	0.882703	−	0.469931i	$-0.155721\pi$
$858$		0			0
$859$		−	13.0662i		−	0.445814i	−0.974840	−	0.222907i	$-0.928445\pi$
$859$		−	13.0662i		−	0.445814i	0.974840	−	0.222907i	$-0.0715547\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.47221			−0.288397			−0.144199	−	0.989549i	$-0.546060\pi$
$863$	−8.47221			−0.288397			−0.144199	+	0.989549i	$0.546060\pi$
$864$		0			0
$865$		−	14.5560i		−	0.494917i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			5.18201i			0.175788i
$870$		0			0
$871$			16.9918i			0.575745i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	3.70838i		−	0.125366i
$876$		0			0
$877$			25.5283i			0.862029i	0.902345	+	0.431015i	$0.141844\pi$
$877$			25.5283i			0.862029i	−0.902345	+	0.431015i	$0.858156\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25.0875			0.845221			0.422610	−	0.906312i	$-0.361114\pi$
$881$	25.0875			0.845221			0.422610	+	0.906312i	$0.361114\pi$
$882$		0			0
$883$			7.25216i			0.244055i	0.992527	+	0.122027i	$0.0389396\pi$
$883$			7.25216i			0.244055i	−0.992527	+	0.122027i	$0.961060\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	31.1225			1.04499			0.522495	−	0.852642i	$-0.325001\pi$
$887$	31.1225			1.04499			0.522495	+	0.852642i	$0.325001\pi$
$888$		0			0
$889$			3.94330i			0.132254i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−37.9249	−	31.1130i	−1.26911	−	1.04116i
$894$		0			0
$895$	18.7322			0.626148
$896$		0			0
$897$		0			0
$898$		0			0
$899$			10.3717i			0.345914i
$900$		0			0
$901$			71.1489i			2.37031i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	30.3240i		−	1.00800i
$906$		0			0
$907$	−54.2747			−1.80216			−0.901080	−	0.433653i	$-0.857225\pi$
$907$	−54.2747			−1.80216			−0.901080	+	0.433653i	$0.857225\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	54.6666			1.81118			0.905592	−	0.424150i	$-0.139427\pi$
$911$	54.6666			1.81118			0.905592	+	0.424150i	$0.139427\pi$
$912$		0			0
$913$	−3.91876			−0.129692
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.46020			0.0812429
$918$		0			0
$919$		−	30.6325i		−	1.01047i	−0.862981	−	0.505236i	$-0.831405\pi$
$919$		−	30.6325i		−	1.01047i	0.862981	−	0.505236i	$-0.168595\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	24.9357i		−	0.820768i
$924$		0			0
$925$			21.0312i			0.691501i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−54.1192			−1.77559			−0.887797	−	0.460236i	$-0.847765\pi$
$929$	−54.1192			−1.77559			−0.887797	+	0.460236i	$0.847765\pi$
$930$		0			0
$931$	−19.0336	+	23.2009i	−0.623803	+	0.760379i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	4.94029i		−	0.161565i
$936$		0			0
$937$	32.7103			1.06860			0.534300	−	0.845295i	$-0.320575\pi$
$937$	32.7103			1.06860			0.534300	+	0.845295i	$0.320575\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			3.69681i			0.120513i	0.998183	+	0.0602563i	$0.0191918\pi$
$941$			3.69681i			0.120513i	−0.998183	+	0.0602563i	$0.980808\pi$
$942$		0			0
$943$	−5.30890			−0.172882
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	16.1516i		−	0.524857i	−0.964951	−	0.262428i	$-0.915477\pi$
$947$		−	16.1516i		−	0.524857i	0.964951	−	0.262428i	$-0.0845233\pi$
$948$		0			0
$949$		−	3.71892i		−	0.120721i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	25.6779i		−	0.831790i	−0.909413	−	0.415895i	$-0.863468\pi$
$953$		−	25.6779i		−	0.831790i	0.909413	−	0.415895i	$-0.136532\pi$
$954$		0			0
$955$		−	20.8869i		−	0.675884i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	5.83952i		−	0.188568i
$960$		0			0
$961$	−8.65952			−0.279339
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	21.1808i		−	0.681833i
$966$		0			0
$967$			34.6059i			1.11285i	0.830897	+	0.556426i	$0.187828\pi$
$967$			34.6059i			1.11285i	−0.830897	+	0.556426i	$0.812172\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.0491			−0.482947			−0.241474	−	0.970407i	$-0.577631\pi$
$971$	−15.0491			−0.482947			−0.241474	+	0.970407i	$0.577631\pi$
$972$		0			0
$973$	3.00123			0.0962151
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	45.0911i		−	1.44259i	−0.692627	−	0.721296i	$-0.743547\pi$
$977$		−	45.0911i		−	1.44259i	0.692627	−	0.721296i	$-0.256453\pi$
$978$		0			0
$979$	−7.42112			−0.237180
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.27694			−0.0726231			−0.0363115	−	0.999341i	$-0.511561\pi$
$983$	−2.27694			−0.0726231			−0.0363115	+	0.999341i	$0.511561\pi$
$984$		0			0
$985$	−14.5606			−0.463939
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.9488			−0.411747
$990$		0			0
$991$	8.35704			0.265470			0.132735	−	0.991152i	$-0.457624\pi$
$991$	8.35704			0.265470			0.132735	+	0.991152i	$0.457624\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	0.449613i		−	0.0142537i
$996$		0			0
$997$	−2.71350			−0.0859373			−0.0429687	−	0.999076i	$-0.513682\pi$
$997$	−2.71350			−0.0859373			−0.0429687	+	0.999076i	$0.513682\pi$
$998$		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.2431.14	yes	40
3.2	odd	2	inner	5472.2.k.d.2431.28	yes	40
4.3	odd	2	inner	5472.2.k.d.2431.16	yes	40
12.11	even	2	inner	5472.2.k.d.2431.26	yes	40
19.18	odd	2	inner	5472.2.k.d.2431.13	✓	40
57.56	even	2	inner	5472.2.k.d.2431.27	yes	40
76.75	even	2	inner	5472.2.k.d.2431.15	yes	40
228.227	odd	2	inner	5472.2.k.d.2431.25	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.k.d.2431.13	✓	40	19.18	odd	2	inner
5472.2.k.d.2431.14	yes	40	1.1	even	1	trivial
5472.2.k.d.2431.15	yes	40	76.75	even	2	inner
5472.2.k.d.2431.16	yes	40	4.3	odd	2	inner
5472.2.k.d.2431.25	yes	40	228.227	odd	2	inner
5472.2.k.d.2431.26	yes	40	12.11	even	2	inner
5472.2.k.d.2431.27	yes	40	57.56	even	2	inner
5472.2.k.d.2431.28	yes	40	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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.32365 q^{5} -0.339676i q^{7} +O(q^{10})\)	\(q-1.32365 q^{5} -0.339676i q^{7} +0.576869i q^{11} +2.03883i q^{13} +6.46996 q^{17} +(-2.76466 + 3.36996i) q^{19} -1.23648i q^{23} -3.24795 q^{25} +2.19433i q^{29} +4.72657 q^{31} +0.449613i q^{35} -6.47522i q^{37} -4.29355i q^{41} -10.4722i q^{43} +11.2538i q^{47} +6.88462 q^{49} +10.9968i q^{53} -0.763574i q^{55} -11.6309 q^{59} -8.98787 q^{61} -2.69870i q^{65} +8.33409 q^{67} -12.2304 q^{71} -1.82405 q^{73} +0.195949 q^{77} +8.98300 q^{79} +6.79316i q^{83} -8.56397 q^{85} +12.8645i q^{89} +0.692542 q^{91} +(3.65945 - 4.46066i) q^{95} -17.6073i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 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

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