LMFDB - Embedded newform 5472.2.f.c.1025.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5472,2,Mod(1025,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5472.1025"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,28,0,0,0,-16,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(59)] == traces)

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

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 38 x^{18} + 528 x^{16} + 3442 x^{14} + 11480 x^{12} + 20550 x^{10} + 20369 x^{8} + 11136 x^{6} + \cdots + 16 $ x^20 + 38x^18 + 528x^16 + 3442x^14 + 11480x^12 + 20550x^10 + 20369x^8 + 11136x^6 + 3192x^4 + 416*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.7
Root			$-3.15931i$ of defining polynomial
Character	$\chi$	$=$	5472.1025
Dual form			5472.2.f.c.1025.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.06784i q^{5} -2.36143 q^{7} +1.12815i q^{11} +4.80373i q^{13} +4.69501i q^{17} +(-3.63064 - 2.41215i) q^{19} -7.88327i q^{23} +3.85973 q^{25} -3.53792 q^{29} +2.74877i q^{31} +2.52162i q^{35} -3.24552i q^{37} -4.61197 q^{41} +3.12136 q^{43} -5.36165i q^{47} -1.42363 q^{49} -1.59167 q^{53} +1.20468 q^{55} +9.17796 q^{59} +0.269865 q^{61} +5.12959 q^{65} -3.18840i q^{67} +16.3102 q^{71} -13.1634 q^{73} -2.66405i q^{77} -7.11877i q^{79} +0.310200i q^{83} +5.01349 q^{85} -2.18154 q^{89} -11.3437i q^{91} +(-2.57578 + 3.87692i) q^{95} +9.96747i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{25} - 16 q^{41} + 28 q^{49} - 16 q^{53} + 16 q^{65} + 16 q^{73} - 32 q^{85} + 48 q^{89}+O(q^{100}) $ 20 * q - 4 * q^25 - 16 * q^41 + 28 * q^49 - 16 * q^53 + 16 * q^65 + 16 * q^73 - 32 * q^85 + 48 * q^89

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
$5$		−	1.06784i		−	0.477550i	−0.971075	−	0.238775i	$-0.923254\pi$
$5$		−	1.06784i		−	0.477550i	0.971075	−	0.238775i	$-0.0767459\pi$
$6$		0			0
$7$	−2.36143			−0.892538			−0.446269	−	0.894899i	$-0.647248\pi$
$7$	−2.36143			−0.892538			−0.446269	+	0.894899i	$0.647248\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.12815i			0.340150i	0.985431	+	0.170075i	$0.0544010\pi$
$11$			1.12815i			0.340150i	−0.985431	+	0.170075i	$0.945599\pi$
$12$		0			0
$13$			4.80373i			1.33232i	0.745811	+	0.666158i	$0.232062\pi$
$13$			4.80373i			1.33232i	−0.745811	+	0.666158i	$0.767938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.69501i			1.13871i	0.822093	+	0.569353i	$0.192806\pi$
$17$			4.69501i			1.13871i	−0.822093	+	0.569353i	$0.807194\pi$
$18$		0			0
$19$	−3.63064	−	2.41215i	−0.832926	−	0.553385i
$19$	−3.63064	−	2.41215i	−0.832926	−	0.553385i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.88327i		−	1.64378i	−0.569649	−	0.821888i	$-0.692921\pi$
$23$		−	7.88327i		−	1.64378i	0.569649	−	0.821888i	$-0.307079\pi$
$24$		0			0
$25$	3.85973			0.771946
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.53792			−0.656975			−0.328488	−	0.944508i	$-0.606539\pi$
$29$	−3.53792			−0.656975			−0.328488	+	0.944508i	$0.606539\pi$
$30$		0			0
$31$			2.74877i			0.493694i	0.969054	+	0.246847i	$0.0793945\pi$
$31$			2.74877i			0.493694i	−0.969054	+	0.246847i	$0.920605\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.52162i			0.426232i
$36$		0			0
$37$		−	3.24552i		−	0.533561i	−0.963757	−	0.266780i	$-0.914040\pi$
$37$		−	3.24552i		−	0.533561i	0.963757	−	0.266780i	$-0.0859598\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.61197			−0.720268			−0.360134	−	0.932901i	$-0.617269\pi$
$41$	−4.61197			−0.720268			−0.360134	+	0.932901i	$0.617269\pi$
$42$		0			0
$43$	3.12136			0.476003			0.238002	−	0.971265i	$-0.423508\pi$
$43$	3.12136			0.476003			0.238002	+	0.971265i	$0.423508\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	5.36165i		−	0.782077i	−0.920374	−	0.391038i	$-0.872116\pi$
$47$		−	5.36165i		−	0.782077i	0.920374	−	0.391038i	$-0.127884\pi$
$48$		0			0
$49$	−1.42363			−0.203375
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.59167			−0.218633			−0.109317	−	0.994007i	$-0.534866\pi$
$53$	−1.59167			−0.218633			−0.109317	+	0.994007i	$0.534866\pi$
$54$		0			0
$55$	1.20468			0.162439
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.17796			1.19487			0.597434	−	0.801918i	$-0.296187\pi$
$59$	9.17796			1.19487			0.597434	+	0.801918i	$0.296187\pi$
$60$		0			0
$61$	0.269865			0.0345527			0.0172763	−	0.999851i	$-0.494500\pi$
$61$	0.269865			0.0345527			0.0172763	+	0.999851i	$0.494500\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.12959			0.636248
$66$		0			0
$67$		−	3.18840i		−	0.389525i	−0.980850	−	0.194762i	$-0.937606\pi$
$67$		−	3.18840i		−	0.389525i	0.980850	−	0.194762i	$-0.0623936\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	16.3102			1.93566			0.967832	−	0.251599i	$-0.0809565\pi$
$71$	16.3102			1.93566			0.967832	+	0.251599i	$0.0809565\pi$
$72$		0			0
$73$	−13.1634			−1.54066			−0.770328	−	0.637647i	$-0.779908\pi$
$73$	−13.1634			−1.54066			−0.770328	+	0.637647i	$0.779908\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.66405i		−	0.303597i
$78$		0			0
$79$		−	7.11877i		−	0.800924i	−0.916313	−	0.400462i	$-0.868850\pi$
$79$		−	7.11877i		−	0.800924i	0.916313	−	0.400462i	$-0.131150\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			0.310200i			0.0340489i	0.999855	+	0.0170244i	$0.00541931\pi$
$83$			0.310200i			0.0340489i	−0.999855	+	0.0170244i	$0.994581\pi$
$84$		0			0
$85$	5.01349			0.543789
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.18154			−0.231242			−0.115621	−	0.993293i	$-0.536886\pi$
$89$	−2.18154			−0.231242			−0.115621	+	0.993293i	$0.536886\pi$
$90$		0			0
$91$		−	11.3437i		−	1.18914i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.57578	+	3.87692i	−0.264269	+	0.397764i
$96$		0			0
$97$			9.96747i			1.01204i	0.862521	+	0.506022i	$0.168884\pi$
$97$			9.96747i			1.01204i	−0.862521	+	0.506022i	$0.831116\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			3.35024i			0.333361i	0.986011	+	0.166681i	$0.0533049\pi$
$101$			3.35024i			0.333361i	−0.986011	+	0.166681i	$0.946695\pi$
$102$		0			0
$103$			4.60362i			0.453608i	0.973940	+	0.226804i	$0.0728277\pi$
$103$			4.60362i			0.453608i	−0.973940	+	0.226804i	$0.927172\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	15.2190			1.47128			0.735639	−	0.677374i	$-0.236882\pi$
$107$	15.2190			1.47128			0.735639	+	0.677374i	$0.236882\pi$
$108$		0			0
$109$		−	12.0581i		−	1.15495i	−0.816407	−	0.577477i	$-0.804037\pi$
$109$		−	12.0581i		−	1.15495i	0.816407	−	0.577477i	$-0.195963\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.3314			1.34819			0.674093	−	0.738646i	$-0.264534\pi$
$113$	14.3314			1.34819			0.674093	+	0.738646i	$0.264534\pi$
$114$		0			0
$115$	−8.41803			−0.784986
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	11.0869i		−	1.01634i
$120$		0			0
$121$	9.72728			0.884298
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	9.46073i		−	0.846193i
$126$		0			0
$127$		−	21.9532i		−	1.94803i	−0.226477	−	0.974016i	$-0.572721\pi$
$127$		−	21.9532i		−	1.94803i	0.226477	−	0.974016i	$-0.427279\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.21142i		−	0.193213i	−0.995323	−	0.0966064i	$-0.969201\pi$
$131$		−	2.21142i		−	0.193213i	0.995323	−	0.0966064i	$-0.0307988\pi$
$132$		0			0
$133$	8.57351	+	5.69613i	0.743418	+	0.493917i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	11.5677i		−	0.988295i	−0.869378	−	0.494148i	$-0.835480\pi$
$137$		−	11.5677i		−	0.988295i	0.869378	−	0.494148i	$-0.164520\pi$
$138$		0			0
$139$	−0.573802			−0.0486692			−0.0243346	−	0.999704i	$-0.507747\pi$
$139$	−0.573802			−0.0486692			−0.0243346	+	0.999704i	$0.507747\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.41933			−0.453187
$144$		0			0
$145$			3.77792i			0.313739i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	6.02231i		−	0.493367i	−0.969096	−	0.246684i	$-0.920659\pi$
$149$		−	6.02231i		−	0.493367i	0.969096	−	0.246684i	$-0.0793408\pi$
$150$		0			0
$151$		−	20.0984i		−	1.63558i	−0.575515	−	0.817791i	$-0.695198\pi$
$151$		−	20.0984i		−	1.63558i	0.575515	−	0.817791i	$-0.304802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.93524			0.235764
$156$		0			0
$157$	−3.77321			−0.301135			−0.150567	−	0.988600i	$-0.548110\pi$
$157$	−3.77321			−0.301135			−0.150567	+	0.988600i	$0.548110\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			18.6158i			1.46713i
$162$		0			0
$163$	−21.8586			−1.71210			−0.856048	−	0.516897i	$-0.827087\pi$
$163$	−21.8586			−1.71210			−0.856048	+	0.516897i	$0.827087\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.6237			−1.44115			−0.720573	−	0.693380i	$-0.756121\pi$
$167$	−18.6237			−1.44115			−0.720573	+	0.693380i	$0.756121\pi$
$168$		0			0
$169$	−10.0758			−0.775065
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.9117			1.28578			0.642888	−	0.765960i	$-0.277736\pi$
$173$	16.9117			1.28578			0.642888	+	0.765960i	$0.277736\pi$
$174$		0			0
$175$	−9.11450			−0.688991
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.79969			0.732463			0.366231	−	0.930524i	$-0.380648\pi$
$179$	9.79969			0.732463			0.366231	+	0.930524i	$0.380648\pi$
$180$		0			0
$181$		−	8.93364i		−	0.664032i	−0.943274	−	0.332016i	$-0.892271\pi$
$181$		−	8.93364i		−	0.664032i	0.943274	−	0.332016i	$-0.107729\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.46568			−0.254802
$186$		0			0
$187$	−5.29667			−0.387331
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4.28695i		−	0.310193i	−0.987899	−	0.155096i	$-0.950431\pi$
$191$		−	4.28695i		−	0.310193i	0.987899	−	0.155096i	$-0.0495688\pi$
$192$		0			0
$193$		−	18.9215i		−	1.36200i	−0.732285	−	0.680998i	$-0.761546\pi$
$193$		−	18.9215i		−	1.36200i	0.732285	−	0.680998i	$-0.238454\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			15.8647i			1.13031i	0.824984	+	0.565156i	$0.191184\pi$
$197$			15.8647i			1.13031i	−0.824984	+	0.565156i	$0.808816\pi$
$198$		0			0
$199$	−11.4759			−0.813507			−0.406754	−	0.913538i	$-0.633339\pi$
$199$	−11.4759			−0.813507			−0.406754	+	0.913538i	$0.633339\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.35457			0.586376
$204$		0			0
$205$			4.92482i			0.343964i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.72127	−	4.09591i	0.188234	−	0.283320i
$210$		0			0
$211$		−	3.18840i		−	0.219498i	−0.993959	−	0.109749i	$-0.964995\pi$
$211$		−	3.18840i		−	0.219498i	0.993959	−	0.109749i	$-0.0350048\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.33310i		−	0.227316i
$216$		0			0
$217$		−	6.49105i		−	0.440641i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.5535			−1.51712
$222$		0			0
$223$		−	8.75467i		−	0.586256i	−0.956073	−	0.293128i	$-0.905304\pi$
$223$		−	8.75467i		−	0.586256i	0.956073	−	0.293128i	$-0.0946962\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.24273			−0.414344			−0.207172	−	0.978305i	$-0.566426\pi$
$227$	−6.24273			−0.414344			−0.207172	+	0.978305i	$0.566426\pi$
$228$		0			0
$229$	−20.9327			−1.38327			−0.691636	−	0.722246i	$-0.743110\pi$
$229$	−20.9327			−1.38327			−0.691636	+	0.722246i	$0.743110\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	14.3025i		−	0.936986i	−0.883467	−	0.468493i	$-0.844797\pi$
$233$		−	14.3025i		−	0.936986i	0.883467	−	0.468493i	$-0.155203\pi$
$234$		0			0
$235$	−5.72536			−0.373481
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	14.2971i		−	0.924802i	−0.886671	−	0.462401i	$-0.846988\pi$
$239$		−	14.2971i		−	0.924802i	0.886671	−	0.462401i	$-0.153012\pi$
$240$		0			0
$241$		−	10.2609i		−	0.660965i	−0.943812	−	0.330483i	$-0.892788\pi$
$241$		−	10.2609i		−	0.660965i	0.943812	−	0.330483i	$-0.107212\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			1.52020i			0.0971220i
$246$		0			0
$247$	11.5873	−	17.4406i	0.737283	−	1.10972i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			15.2588i			0.963126i	0.876411	+	0.481563i	$0.159931\pi$
$251$			15.2588i			0.963126i	−0.876411	+	0.481563i	$0.840069\pi$
$252$		0			0
$253$	8.89351			0.559130
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.6849			−1.29029			−0.645146	−	0.764059i	$-0.723203\pi$
$257$	−20.6849			−1.29029			−0.645146	+	0.764059i	$0.723203\pi$
$258$		0			0
$259$			7.66409i			0.476223i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.9388i		−	1.41446i	−0.706981	−	0.707232i	$-0.749943\pi$
$263$		−	22.9388i		−	1.41446i	0.706981	−	0.707232i	$-0.250057\pi$
$264$		0			0
$265$			1.69964i			0.104408i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.14139			−0.496389			−0.248195	−	0.968710i	$-0.579837\pi$
$269$	−8.14139			−0.496389			−0.248195	+	0.968710i	$0.579837\pi$
$270$		0			0
$271$	−18.3559			−1.11504			−0.557521	−	0.830163i	$-0.688248\pi$
$271$	−18.3559			−1.11504			−0.557521	+	0.830163i	$0.688248\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			4.35435i			0.262577i
$276$		0			0
$277$	−22.5411			−1.35436			−0.677181	−	0.735817i	$-0.736799\pi$
$277$	−22.5411			−1.35436			−0.677181	+	0.735817i	$0.736799\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.42329			−0.263871			−0.131936	−	0.991258i	$-0.542119\pi$
$281$	−4.42329			−0.263871			−0.131936	+	0.991258i	$0.542119\pi$
$282$		0			0
$283$	−28.4078			−1.68867			−0.844335	−	0.535815i	$-0.820004\pi$
$283$	−28.4078			−1.68867			−0.844335	+	0.535815i	$0.820004\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.8909			0.642867
$288$		0			0
$289$	−5.04307			−0.296651
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.0306			1.22862			0.614310	−	0.789065i	$-0.289434\pi$
$293$	21.0306			1.22862			0.614310	+	0.789065i	$0.289434\pi$
$294$		0			0
$295$		−	9.80055i		−	0.570610i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	37.8691			2.19003
$300$		0			0
$301$	−7.37089			−0.424851
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	0.288171i		−	0.0165006i
$306$		0			0
$307$		−	24.7020i		−	1.40982i	−0.709298	−	0.704908i	$-0.750988\pi$
$307$		−	24.7020i		−	1.40982i	0.709298	−	0.704908i	$-0.249012\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	16.2510i		−	0.921512i	−0.887527	−	0.460756i	$-0.847578\pi$
$311$		−	16.2510i		−	0.921512i	0.887527	−	0.460756i	$-0.152422\pi$
$312$		0			0
$313$	29.8868			1.68930			0.844650	−	0.535318i	$-0.179808\pi$
$313$	29.8868			1.68930			0.844650	+	0.535318i	$0.179808\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	28.7351			1.61392			0.806962	−	0.590604i	$-0.201110\pi$
$317$	28.7351			1.61392			0.806962	+	0.590604i	$0.201110\pi$
$318$		0			0
$319$		−	3.99131i		−	0.223470i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.3251	−	17.0459i	0.630143	−	0.948457i
$324$		0			0
$325$			18.5411i			1.02848i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			12.6612i			0.698034i
$330$		0			0
$331$			14.9942i			0.824155i	0.911149	+	0.412078i	$0.135197\pi$
$331$			14.9942i			0.824155i	−0.911149	+	0.412078i	$0.864803\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.40468			−0.186018
$336$		0			0
$337$			15.4370i			0.840908i	0.907314	+	0.420454i	$0.138129\pi$
$337$			15.4370i			0.840908i	−0.907314	+	0.420454i	$0.861871\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.10103			−0.167930
$342$		0			0
$343$	19.8918			1.07406
$344$		0			0
$345$		0			0
$346$		0			0
$347$			32.7596i			1.75863i	0.476245	+	0.879313i	$0.341998\pi$
$347$			32.7596i			1.75863i	−0.476245	+	0.879313i	$0.658002\pi$
$348$		0			0
$349$	18.0106			0.964087			0.482044	−	0.876147i	$-0.339895\pi$
$349$	18.0106			0.964087			0.482044	+	0.876147i	$0.339895\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	19.1862i		−	1.02118i	−0.859824	−	0.510590i	$-0.829427\pi$
$353$		−	19.1862i		−	1.02118i	0.859824	−	0.510590i	$-0.170573\pi$
$354$		0			0
$355$		−	17.4166i		−	0.924377i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	5.08985i		−	0.268632i	−0.990939	−	0.134316i	$-0.957116\pi$
$359$		−	5.08985i		−	0.268632i	0.990939	−	0.134316i	$-0.0428838\pi$
$360$		0			0
$361$	7.36307	+	17.5153i	0.387530	+	0.921857i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			14.0563i			0.735741i
$366$		0			0
$367$	−9.44574			−0.493063			−0.246532	−	0.969135i	$-0.579291\pi$
$367$	−9.44574			−0.493063			−0.246532	+	0.969135i	$0.579291\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.75863			0.195138
$372$		0			0
$373$			8.14674i			0.421822i	0.977505	+	0.210911i	$0.0676431\pi$
$373$			8.14674i			0.421822i	−0.977505	+	0.210911i	$0.932357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	16.9952i		−	0.875298i
$378$		0			0
$379$		−	23.0661i		−	1.18483i	−0.805635	−	0.592413i	$-0.798176\pi$
$379$		−	23.0661i		−	1.18483i	0.805635	−	0.592413i	$-0.201824\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.51050			0.332671			0.166336	−	0.986069i	$-0.446807\pi$
$383$	6.51050			0.332671			0.166336	+	0.986069i	$0.446807\pi$
$384$		0			0
$385$	−2.84477			−0.144983
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	9.85875i		−	0.499858i	−0.968264	−	0.249929i	$-0.919593\pi$
$389$		−	9.85875i		−	0.499858i	0.968264	−	0.249929i	$-0.0804073\pi$
$390$		0			0
$391$	37.0120			1.87178
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.60167			−0.382482
$396$		0			0
$397$	27.4226			1.37630			0.688150	−	0.725569i	$-0.258423\pi$
$397$	27.4226			1.37630			0.688150	+	0.725569i	$0.258423\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.80530			0.289903			0.144951	−	0.989439i	$-0.453697\pi$
$401$	5.80530			0.289903			0.144951	+	0.989439i	$0.453697\pi$
$402$		0			0
$403$	−13.2044			−0.657756
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.66144			0.181491
$408$		0			0
$409$		−	27.8428i		−	1.37674i	−0.725362	−	0.688368i	$-0.758327\pi$
$409$		−	27.8428i		−	1.37674i	0.725362	−	0.688368i	$-0.241673\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−21.6732			−1.06647
$414$		0			0
$415$	0.331242			0.0162600
$416$		0			0
$417$		0			0
$418$		0			0
$419$			3.65834i			0.178722i	0.995999	+	0.0893609i	$0.0284824\pi$
$419$			3.65834i			0.178722i	−0.995999	+	0.0893609i	$0.971518\pi$
$420$		0			0
$421$			6.88200i			0.335408i	0.985837	+	0.167704i	$0.0536353\pi$
$421$			6.88200i			0.335408i	−0.985837	+	0.167704i	$0.946365\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			18.1214i			0.879019i
$426$		0			0
$427$	−0.637268			−0.0308396
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.4868			0.745973			0.372986	−	0.927837i	$-0.378334\pi$
$431$	15.4868			0.745973			0.372986	+	0.927837i	$0.378334\pi$
$432$		0			0
$433$			23.3016i			1.11980i	0.828560	+	0.559901i	$0.189161\pi$
$433$			23.3016i			1.11980i	−0.828560	+	0.559901i	$0.810839\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−19.0156	+	28.6213i	−0.909641	+	1.36914i
$438$		0			0
$439$		−	5.57562i		−	0.266110i	−0.991109	−	0.133055i	$-0.957521\pi$
$439$		−	5.57562i		−	0.266110i	0.991109	−	0.133055i	$-0.0424787\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	1.33217i		−	0.0632934i	−0.999499	−	0.0316467i	$-0.989925\pi$
$443$		−	1.33217i		−	0.0632934i	0.999499	−	0.0316467i	$-0.0100751\pi$
$444$		0			0
$445$			2.32952i			0.110430i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.72795			0.459091			0.229545	−	0.973298i	$-0.426276\pi$
$449$	9.72795			0.459091			0.229545	+	0.973298i	$0.426276\pi$
$450$		0			0
$451$		−	5.20299i		−	0.244999i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.1132			−0.567875
$456$		0			0
$457$	−4.67561			−0.218716			−0.109358	−	0.994002i	$-0.534879\pi$
$457$	−4.67561			−0.218716			−0.109358	+	0.994002i	$0.534879\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			14.3475i			0.668232i	0.942532	+	0.334116i	$0.108438\pi$
$461$			14.3475i			0.668232i	−0.942532	+	0.334116i	$0.891562\pi$
$462$		0			0
$463$	−7.41555			−0.344630			−0.172315	−	0.985042i	$-0.555125\pi$
$463$	−7.41555			−0.344630			−0.172315	+	0.985042i	$0.555125\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	0.659703i		−	0.0305274i	−0.999884	−	0.0152637i	$-0.995141\pi$
$467$		−	0.659703i		−	0.0305274i	0.999884	−	0.0152637i	$-0.00485877\pi$
$468$		0			0
$469$			7.52919i			0.347666i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			3.52137i			0.161913i
$474$		0			0
$475$	−14.0133	−	9.31024i	−0.642973	−	0.427183i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	13.6397i		−	0.623212i	−0.950211	−	0.311606i	$-0.899133\pi$
$479$		−	13.6397i		−	0.623212i	0.950211	−	0.311606i	$-0.100867\pi$
$480$		0			0
$481$	15.5906			0.710871
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.6436			0.483302
$486$		0			0
$487$		−	27.0721i		−	1.22675i	−0.789791	−	0.613376i	$-0.789811\pi$
$487$		−	27.0721i		−	1.22675i	0.789791	−	0.613376i	$-0.210189\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			33.5861i			1.51572i	0.652418	+	0.757859i	$0.273755\pi$
$491$			33.5861i			1.51572i	−0.652418	+	0.757859i	$0.726245\pi$
$492$		0			0
$493$		−	16.6106i		−	0.748102i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−38.5154			−1.72765
$498$		0			0
$499$	3.18483			0.142573			0.0712863	−	0.997456i	$-0.477290\pi$
$499$	3.18483			0.142573			0.0712863	+	0.997456i	$0.477290\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	10.5504i		−	0.470418i	−0.971945	−	0.235209i	$-0.924423\pi$
$503$		−	10.5504i		−	0.470418i	0.971945	−	0.235209i	$-0.0755774\pi$
$504$		0			0
$505$	3.57750			0.159197
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.598801			0.0265414			0.0132707	−	0.999912i	$-0.495776\pi$
$509$	0.598801			0.0265414			0.0132707	+	0.999912i	$0.495776\pi$
$510$		0			0
$511$	31.0845			1.37510
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.91591			0.216621
$516$		0			0
$517$	6.04874			0.266023
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.88603			−0.433115			−0.216557	−	0.976270i	$-0.569483\pi$
$521$	−9.88603			−0.433115			−0.216557	+	0.976270i	$0.569483\pi$
$522$		0			0
$523$			41.2487i			1.80368i	0.432072	+	0.901839i	$0.357783\pi$
$523$			41.2487i			1.80368i	−0.432072	+	0.901839i	$0.642217\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.9055			−0.562173
$528$		0			0
$529$	−39.1460			−1.70200
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	22.1547i		−	0.959624i
$534$		0			0
$535$		−	16.2514i		−	0.702609i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.60607i		−	0.0691781i
$540$		0			0
$541$	−16.6928			−0.717680			−0.358840	−	0.933399i	$-0.616828\pi$
$541$	−16.6928			−0.717680			−0.358840	+	0.933399i	$0.616828\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.8760			−0.551549
$546$		0			0
$547$			45.3457i			1.93884i	0.245405	+	0.969421i	$0.421079\pi$
$547$			45.3457i			1.93884i	−0.245405	+	0.969421i	$0.578921\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.8449	+	8.53399i	0.547212	+	0.363560i
$552$		0			0
$553$			16.8105i			0.714856i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	16.0114i		−	0.678426i	−0.940710	−	0.339213i	$-0.889839\pi$
$557$		−	16.0114i		−	0.678426i	0.940710	−	0.339213i	$-0.110161\pi$
$558$		0			0
$559$			14.9942i			0.634187i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.9583			−0.883288			−0.441644	−	0.897190i	$-0.645605\pi$
$563$	−20.9583			−0.883288			−0.441644	+	0.897190i	$0.645605\pi$
$564$		0			0
$565$		−	15.3036i		−	0.643827i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	35.6361			1.49394			0.746971	−	0.664856i	$-0.231507\pi$
$569$	35.6361			1.49394			0.746971	+	0.664856i	$0.231507\pi$
$570$		0			0
$571$	−45.8247			−1.91771			−0.958853	−	0.283903i	$-0.908370\pi$
$571$	−45.8247			−1.91771			−0.958853	+	0.283903i	$0.908370\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	30.4273i		−	1.26891i
$576$		0			0
$577$	2.03946			0.0849036			0.0424518	−	0.999099i	$-0.486483\pi$
$577$	2.03946			0.0849036			0.0424518	+	0.999099i	$0.486483\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	0.732517i		−	0.0303899i
$582$		0			0
$583$		−	1.79565i		−	0.0743681i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	9.86694i		−	0.407252i	−0.979049	−	0.203626i	$-0.934727\pi$
$587$		−	9.86694i		−	0.407252i	0.979049	−	0.203626i	$-0.0652727\pi$
$588$		0			0
$589$	6.63045	−	9.97980i	0.273203	−	0.411211i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			16.1272i			0.662265i	0.943584	+	0.331133i	$0.107431\pi$
$593$			16.1272i			0.662265i	−0.943584	+	0.331133i	$0.892569\pi$
$594$		0			0
$595$	−11.8390			−0.485353
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.2898			−0.461289			−0.230644	−	0.973038i	$-0.574083\pi$
$599$	−11.2898			−0.461289			−0.230644	+	0.973038i	$0.574083\pi$
$600$		0			0
$601$		−	18.9215i		−	0.771822i	−0.922536	−	0.385911i	$-0.873887\pi$
$601$		−	18.9215i		−	0.771822i	0.922536	−	0.385911i	$-0.126113\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	10.3871i		−	0.422297i
$606$		0			0
$607$			14.0925i			0.571996i	0.958230	+	0.285998i	$0.0923250\pi$
$607$			14.0925i			0.571996i	−0.958230	+	0.285998i	$0.907675\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.7559			1.04197
$612$		0			0
$613$	−19.7088			−0.796030			−0.398015	−	0.917379i	$-0.630301\pi$
$613$	−19.7088			−0.796030			−0.398015	+	0.917379i	$0.630301\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			0.721275i			0.0290374i	0.999895	+	0.0145187i	$0.00462161\pi$
$617$			0.721275i			0.0290374i	−0.999895	+	0.0145187i	$0.995378\pi$
$618$		0			0
$619$	−12.8736			−0.517435			−0.258718	−	0.965953i	$-0.583300\pi$
$619$	−12.8736			−0.517435			−0.258718	+	0.965953i	$0.583300\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.15155			0.206393
$624$		0			0
$625$	9.19614			0.367846
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15.2377			0.607569
$630$		0			0
$631$	25.4402			1.01276			0.506380	−	0.862311i	$-0.330983\pi$
$631$	25.4402			1.01276			0.506380	+	0.862311i	$0.330983\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−23.4424			−0.930284
$636$		0			0
$637$		−	6.83873i		−	0.270960i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	42.9562			1.69667			0.848333	−	0.529463i	$-0.177606\pi$
$641$	42.9562			1.69667			0.848333	+	0.529463i	$0.177606\pi$
$642$		0			0
$643$	−13.6583			−0.538630			−0.269315	−	0.963052i	$-0.586797\pi$
$643$	−13.6583			−0.538630			−0.269315	+	0.963052i	$0.586797\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.1800i		−	1.02924i	−0.857418	−	0.514621i	$-0.827933\pi$
$647$		−	26.1800i		−	1.02924i	0.857418	−	0.514621i	$-0.172067\pi$
$648$		0			0
$649$			10.3541i			0.406435i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	14.5958i		−	0.571177i	−0.958352	−	0.285588i	$-0.907811\pi$
$653$		−	14.5958i		−	0.571177i	0.958352	−	0.285588i	$-0.0921890\pi$
$654$		0			0
$655$	−2.36143			−0.0922689
$656$		0			0
$657$		0			0
$658$		0			0
$659$	31.1753			1.21442			0.607208	−	0.794543i	$-0.292290\pi$
$659$	31.1753			1.21442			0.607208	+	0.794543i	$0.292290\pi$
$660$		0			0
$661$			7.62562i			0.296602i	0.988942	+	0.148301i	$0.0473805\pi$
$661$			7.62562i			0.296602i	−0.988942	+	0.148301i	$0.952620\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.08253	−	9.15510i	0.235870	−	0.355020i
$666$		0			0
$667$			27.8904i			1.07992i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			0.304448i			0.0117531i
$672$		0			0
$673$		−	10.3195i		−	0.397786i	−0.980021	−	0.198893i	$-0.936265\pi$
$673$		−	10.3195i		−	0.397786i	0.980021	−	0.198893i	$-0.0637346\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.0089			−0.461541			−0.230770	−	0.973008i	$-0.574125\pi$
$677$	−12.0089			−0.461541			−0.230770	+	0.973008i	$0.574125\pi$
$678$		0			0
$679$		−	23.5375i		−	0.903288i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.7105			1.02205			0.511024	−	0.859566i	$-0.329266\pi$
$683$	26.7105			1.02205			0.511024	+	0.859566i	$0.329266\pi$
$684$		0			0
$685$	−12.3524			−0.471961
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	7.64597i		−	0.291288i
$690$		0			0
$691$	30.9230			1.17637			0.588184	−	0.808727i	$-0.299843\pi$
$691$	30.9230			1.17637			0.588184	+	0.808727i	$0.299843\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			0.612726i			0.0232420i
$696$		0			0
$697$		−	21.6532i		−	0.820174i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	15.8974i		−	0.600438i	−0.953870	−	0.300219i	$-0.902940\pi$
$701$		−	15.8974i		−	0.600438i	0.953870	−	0.300219i	$-0.0970597\pi$
$702$		0			0
$703$	−7.82869	+	11.7833i	−0.295265	+	0.444416i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.91137i		−	0.297538i
$708$		0			0
$709$	−4.99467			−0.187579			−0.0937893	−	0.995592i	$-0.529898\pi$
$709$	−4.99467			−0.187579			−0.0937893	+	0.995592i	$0.529898\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.6693			0.811523
$714$		0			0
$715$			5.78695i			0.216420i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	22.9681i		−	0.856566i	−0.903645	−	0.428283i	$-0.859119\pi$
$719$		−	22.9681i		−	0.856566i	0.903645	−	0.428283i	$-0.140881\pi$
$720$		0			0
$721$		−	10.8711i		−	0.404863i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13.6554			−0.507149
$726$		0			0
$727$	14.1434			0.524549			0.262275	−	0.964993i	$-0.415527\pi$
$727$	14.1434			0.524549			0.262275	+	0.964993i	$0.415527\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			14.6548i			0.542028i
$732$		0			0
$733$	−39.4162			−1.45587			−0.727935	−	0.685646i	$-0.759520\pi$
$733$	−39.4162			−1.45587			−0.727935	+	0.685646i	$0.759520\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.59699			0.132497
$738$		0			0
$739$	19.5611			0.719567			0.359784	−	0.933036i	$-0.382850\pi$
$739$	19.5611			0.719567			0.359784	+	0.933036i	$0.382850\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−37.3997			−1.37206			−0.686031	−	0.727573i	$-0.740648\pi$
$743$	−37.3997			−1.37206			−0.686031	+	0.727573i	$0.740648\pi$
$744$		0			0
$745$	−6.43084			−0.235608
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−35.9387			−1.31317
$750$		0			0
$751$			21.7343i			0.793095i	0.918014	+	0.396547i	$0.129792\pi$
$751$			21.7343i			0.793095i	−0.918014	+	0.396547i	$0.870208\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.4617			−0.781073
$756$		0			0
$757$	51.2731			1.86355			0.931776	−	0.363034i	$-0.118259\pi$
$757$	51.2731			1.86355			0.931776	+	0.363034i	$0.118259\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			10.2733i			0.372407i	0.982511	+	0.186203i	$0.0596184\pi$
$761$			10.2733i			0.372407i	−0.982511	+	0.186203i	$0.940382\pi$
$762$		0			0
$763$			28.4743i			1.03084i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			44.0885i			1.59194i
$768$		0			0
$769$	6.40479			0.230963			0.115481	−	0.993310i	$-0.463159\pi$
$769$	6.40479			0.230963			0.115481	+	0.993310i	$0.463159\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	29.4767			1.06020			0.530101	−	0.847935i	$-0.322154\pi$
$773$	29.4767			1.06020			0.530101	+	0.847935i	$0.322154\pi$
$774$		0			0
$775$			10.6095i			0.381105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.7444	+	11.1248i	0.599930	+	0.398586i
$780$		0			0
$781$			18.4003i			0.658416i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			4.02917i			0.143807i
$786$		0			0
$787$			22.7715i			0.811717i	0.913936	+	0.405859i	$0.133028\pi$
$787$			22.7715i			0.811717i	−0.913936	+	0.405859i	$0.866972\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−33.8427			−1.20331
$792$		0			0
$793$			1.29636i			0.0460351i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.4065			0.439462			0.219731	−	0.975561i	$-0.429482\pi$
$797$	12.4065			0.439462			0.219731	+	0.975561i	$0.429482\pi$
$798$		0			0
$799$	25.1730			0.890556
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	14.8503i		−	0.524054i
$804$		0			0
$805$	19.8786			0.700630
$806$		0			0
$807$		0			0
$808$		0			0
$809$			29.9907i			1.05442i	0.849736	+	0.527209i	$0.176762\pi$
$809$			29.9907i			1.05442i	−0.849736	+	0.527209i	$0.823238\pi$
$810$		0			0
$811$		−	23.8227i		−	0.836529i	−0.908325	−	0.418265i	$-0.862638\pi$
$811$		−	23.8227i		−	0.836529i	0.908325	−	0.418265i	$-0.137362\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			23.3413i			0.817612i
$816$		0			0
$817$	−11.3325	−	7.52919i	−0.396475	−	0.263413i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	15.9274i		−	0.555870i	−0.960600	−	0.277935i	$-0.910350\pi$
$821$		−	15.9274i		−	0.555870i	0.960600	−	0.277935i	$-0.0896499\pi$
$822$		0			0
$823$	36.1294			1.25939			0.629696	−	0.776842i	$-0.283179\pi$
$823$	36.1294			1.25939			0.629696	+	0.776842i	$0.283179\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.3970			0.848366			0.424183	−	0.905576i	$-0.360561\pi$
$827$	24.3970			0.848366			0.424183	+	0.905576i	$0.360561\pi$
$828$		0			0
$829$		−	49.1297i		−	1.70634i	−0.521630	−	0.853172i	$-0.674676\pi$
$829$		−	49.1297i		−	1.70634i	0.521630	−	0.853172i	$-0.325324\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	6.68394i		−	0.231585i
$834$		0			0
$835$			19.8870i			0.688219i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.31351			−0.0798713			−0.0399357	−	0.999202i	$-0.512715\pi$
$839$	−2.31351			−0.0798713			−0.0399357	+	0.999202i	$0.512715\pi$
$840$		0			0
$841$	−16.4831			−0.568383
$842$		0			0
$843$		0			0
$844$		0			0
$845$			10.7593i			0.370132i
$846$		0			0
$847$	−22.9703			−0.789270
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25.5853			−0.877054
$852$		0			0
$853$	−0.259187			−0.00887438			−0.00443719	−	0.999990i	$-0.501412\pi$
$853$	−0.259187			−0.00887438			−0.00443719	+	0.999990i	$0.501412\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.6511			−1.28614			−0.643068	−	0.765809i	$-0.722339\pi$
$857$	−37.6511			−1.28614			−0.643068	+	0.765809i	$0.722339\pi$
$858$		0			0
$859$	18.7464			0.639618			0.319809	−	0.947482i	$-0.396381\pi$
$859$	18.7464			0.639618			0.319809	+	0.947482i	$0.396381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−40.0854			−1.36452			−0.682262	−	0.731108i	$-0.739004\pi$
$863$	−40.0854			−1.36452			−0.682262	+	0.731108i	$0.739004\pi$
$864$		0			0
$865$		−	18.0589i		−	0.614023i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.03104			0.272434
$870$		0			0
$871$	15.3162			0.518970
$872$		0			0
$873$		0			0
$874$		0			0
$875$			22.3409i			0.755260i
$876$		0			0
$877$			56.1594i			1.89637i	0.317721	+	0.948184i	$0.397082\pi$
$877$			56.1594i			1.89637i	−0.317721	+	0.948184i	$0.602918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	0.653276i		−	0.0220094i	−0.999939	−	0.0110047i	$-0.996497\pi$
$881$		−	0.653276i		−	0.0220094i	0.999939	−	0.0110047i	$-0.00350298\pi$
$882$		0			0
$883$	47.6956			1.60509			0.802543	−	0.596595i	$-0.203480\pi$
$883$	47.6956			1.60509			0.802543	+	0.596595i	$0.203480\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−43.3746			−1.45638			−0.728189	−	0.685377i	$-0.759638\pi$
$887$	−43.3746			−1.45638			−0.728189	+	0.685377i	$0.759638\pi$
$888$		0			0
$889$			51.8411i			1.73869i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.9331	+	19.4662i	−0.432790	+	0.651412i
$894$		0			0
$895$		−	10.4644i		−	0.349788i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	9.72494i		−	0.324345i
$900$		0			0
$901$		−	7.47291i		−	0.248959i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.53965			−0.317109
$906$		0			0
$907$		−	40.4286i		−	1.34241i	−0.741272	−	0.671205i	$-0.765777\pi$
$907$		−	40.4286i		−	1.34241i	0.741272	−	0.671205i	$-0.234223\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.5325			0.580879			0.290439	−	0.956893i	$-0.406199\pi$
$911$	17.5325			0.580879			0.290439	+	0.956893i	$0.406199\pi$
$912$		0			0
$913$	−0.349952			−0.0115817
$914$		0			0
$915$		0			0
$916$		0			0
$917$			5.22213i			0.172450i
$918$		0			0
$919$	−21.3638			−0.704726			−0.352363	−	0.935863i	$-0.614622\pi$
$919$	−21.3638			−0.704726			−0.352363	+	0.935863i	$0.614622\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			78.3498i			2.57891i
$924$		0			0
$925$		−	12.5268i		−	0.411880i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	0.941448i		−	0.0308879i	−0.999881	−	0.0154439i	$-0.995084\pi$
$929$		−	0.941448i		−	0.0308879i	0.999881	−	0.0154439i	$-0.00491616\pi$
$930$		0			0
$931$	5.16868	+	3.43400i	0.169397	+	0.112545i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5.65597i			0.184970i
$936$		0			0
$937$	−26.4685			−0.864688			−0.432344	−	0.901709i	$-0.642313\pi$
$937$	−26.4685			−0.864688			−0.432344	+	0.901709i	$0.642313\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	35.8796			1.16964			0.584821	−	0.811162i	$-0.301165\pi$
$941$	35.8796			1.16964			0.584821	+	0.811162i	$0.301165\pi$
$942$		0			0
$943$			36.3574i			1.18396i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			24.0432i			0.781298i	0.920540	+	0.390649i	$0.127749\pi$
$947$			24.0432i			0.781298i	−0.920540	+	0.390649i	$0.872251\pi$
$948$		0			0
$949$		−	63.2333i		−	2.05264i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−43.6479			−1.41389			−0.706947	−	0.707267i	$-0.749928\pi$
$953$	−43.6479			−1.41389			−0.706947	+	0.707267i	$0.749928\pi$
$954$		0			0
$955$	−4.57775			−0.148133
$956$		0			0
$957$		0			0
$958$		0			0
$959$			27.3164i			0.882091i
$960$		0			0
$961$	23.4442			0.756266
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−20.2050			−0.650422
$966$		0			0
$967$	−21.1057			−0.678713			−0.339356	−	0.940658i	$-0.610209\pi$
$967$	−21.1057			−0.678713			−0.339356	+	0.940658i	$0.610209\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.98067			−0.0635627			−0.0317814	−	0.999495i	$-0.510118\pi$
$971$	−1.98067			−0.0635627			−0.0317814	+	0.999495i	$0.510118\pi$
$972$		0			0
$973$	1.35500			0.0434392
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.0586			0.641732			0.320866	−	0.947125i	$-0.396026\pi$
$977$	20.0586			0.641732			0.320866	+	0.947125i	$0.396026\pi$
$978$		0			0
$979$		−	2.46110i		−	0.0786571i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−27.0644			−0.863222			−0.431611	−	0.902060i	$-0.642055\pi$
$983$	−27.0644			−0.863222			−0.431611	+	0.902060i	$0.642055\pi$
$984$		0			0
$985$	16.9409			0.539781
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	24.6065i		−	0.782443i
$990$		0			0
$991$			6.52113i			0.207151i	0.994622	+	0.103575i	$0.0330283\pi$
$991$			6.52113i			0.207151i	−0.994622	+	0.103575i	$0.966972\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			12.2544i			0.388491i
$996$		0			0
$997$	−7.46523			−0.236426			−0.118213	−	0.992988i	$-0.537717\pi$
$997$	−7.46523			−0.236426			−0.118213	+	0.992988i	$0.537717\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.f.c.1025.7	✓	20
3.2	odd	2		5472.2.f.d.1025.13	yes	20
4.3	odd	2	inner	5472.2.f.c.1025.8	yes	20
12.11	even	2		5472.2.f.d.1025.14	yes	20
19.18	odd	2		5472.2.f.d.1025.7	yes	20
57.56	even	2	inner	5472.2.f.c.1025.13	yes	20
76.75	even	2		5472.2.f.d.1025.8	yes	20
228.227	odd	2	inner	5472.2.f.c.1025.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.f.c.1025.7	✓	20	1.1	even	1	trivial
5472.2.f.c.1025.8	yes	20	4.3	odd	2	inner
5472.2.f.c.1025.13	yes	20	57.56	even	2	inner
5472.2.f.c.1025.14	yes	20	228.227	odd	2	inner
5472.2.f.d.1025.7	yes	20	19.18	odd	2
5472.2.f.d.1025.8	yes	20	76.75	even	2
5472.2.f.d.1025.13	yes	20	3.2	odd	2
5472.2.f.d.1025.14	yes	20	12.11	even	2

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.06784i q^{5} -2.36143 q^{7} +1.12815i q^{11} +4.80373i q^{13} +4.69501i q^{17} +(-3.63064 - 2.41215i) q^{19} -7.88327i q^{23} +3.85973 q^{25} -3.53792 q^{29} +2.74877i q^{31} +2.52162i q^{35} -3.24552i q^{37} -4.61197 q^{41} +3.12136 q^{43} -5.36165i q^{47} -1.42363 q^{49} -1.59167 q^{53} +1.20468 q^{55} +9.17796 q^{59} +0.269865 q^{61} +5.12959 q^{65} -3.18840i q^{67} +16.3102 q^{71} -13.1634 q^{73} -2.66405i q^{77} -7.11877i q^{79} +0.310200i q^{83} +5.01349 q^{85} -2.18154 q^{89} -11.3437i q^{91} +(-2.57578 + 3.87692i) q^{95} +9.96747i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{25} - 16 q^{41} + 28 q^{49} - 16 q^{53} + 16 q^{65} + 16 q^{73} - 32 q^{85} + 48 q^{89}+O(q^{100}) \) 20 * q - 4 * q^25 - 16 * q^41 + 28 * q^49 - 16 * q^53 + 16 * q^65 + 16 * q^73 - 32 * q^85 + 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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