LMFDB - Embedded newform 1932.2.k.a.1609.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1609,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1932.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.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:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.3
Character	$\chi$	$=$	1932.1609
Dual form			1932.2.k.a.1609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.11684 q^{5} +(2.00032 + 1.73168i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.11684 q^{5} +(2.00032 + 1.73168i) q^{7} -1.00000 q^{9} -4.26413i q^{11} +2.13093i q^{13} -1.11684i q^{15} -7.30657 q^{17} +1.25349 q^{19} +(-1.73168 + 2.00032i) q^{21} +(2.60180 + 4.02873i) q^{23} -3.75266 q^{25} -1.00000i q^{27} -4.69379 q^{29} +10.0969i q^{31} +4.26413 q^{33} +(-2.23404 - 1.93401i) q^{35} +3.75864i q^{37} -2.13093 q^{39} -4.12042i q^{41} -4.35420i q^{43} +1.11684 q^{45} +8.59820i q^{47} +(1.00258 + 6.92783i) q^{49} -7.30657i q^{51} -6.83736i q^{53} +4.76236i q^{55} +1.25349i q^{57} +3.01977i q^{59} -5.54919 q^{61} +(-2.00032 - 1.73168i) q^{63} -2.37991i q^{65} -7.79556i q^{67} +(-4.02873 + 2.60180i) q^{69} -16.3741 q^{71} +1.71890i q^{73} -3.75266i q^{75} +(7.38409 - 8.52963i) q^{77} -6.31656i q^{79} +1.00000 q^{81} -10.2991 q^{83} +8.16028 q^{85} -4.69379i q^{87} +10.5105 q^{89} +(-3.69008 + 4.26254i) q^{91} -10.0969 q^{93} -1.39995 q^{95} -17.4486 q^{97} +4.26413i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{9}+O(q^{10}) $ 32 * q - 32 * q^9	$ 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) $ 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times$.

$n$	$829$	$925$	$967$	$1289$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.11684			−0.499467			−0.249733	−	0.968315i	$-0.580343\pi$
$5$	−1.11684			−0.499467			−0.249733	+	0.968315i	$0.580343\pi$
$6$		0			0
$7$	2.00032	+	1.73168i	0.756051	+	0.654513i
$7$	2.00032	+	1.73168i	0.756051	+	0.654513i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	4.26413i		−	1.28568i	−0.765999	−	0.642841i	$-0.777756\pi$
$11$		−	4.26413i		−	1.28568i	0.765999	−	0.642841i	$-0.222244\pi$
$12$		0			0
$13$			2.13093i			0.591012i	0.955341	+	0.295506i	$0.0954883\pi$
$13$			2.13093i			0.591012i	−0.955341	+	0.295506i	$0.904512\pi$
$14$		0			0
$15$		−	1.11684i		−	0.288367i
$16$		0			0
$17$	−7.30657			−1.77210			−0.886052	−	0.463586i	$-0.846562\pi$
$17$	−7.30657			−1.77210			−0.886052	+	0.463586i	$0.846562\pi$
$18$		0			0
$19$	1.25349			0.287570			0.143785	−	0.989609i	$-0.454073\pi$
$19$	1.25349			0.287570			0.143785	+	0.989609i	$0.454073\pi$
$20$		0			0
$21$	−1.73168	+	2.00032i	−0.377883	+	0.436506i
$22$		0			0
$23$	2.60180	+	4.02873i	0.542512	+	0.840048i
$23$	2.60180	+	4.02873i	0.542512	+	0.840048i
$24$		0			0
$25$	−3.75266			−0.750533
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.69379			−0.871614			−0.435807	−	0.900040i	$-0.643537\pi$
$29$	−4.69379			−0.871614			−0.435807	+	0.900040i	$0.643537\pi$
$30$		0			0
$31$			10.0969i			1.81345i	0.421720	+	0.906726i	$0.361427\pi$
$31$			10.0969i			1.81345i	−0.421720	+	0.906726i	$0.638573\pi$
$32$		0			0
$33$	4.26413			0.742289
$34$		0			0
$35$	−2.23404	−	1.93401i	−0.377622	−	0.326908i
$36$		0			0
$37$			3.75864i			0.617917i	0.951076	+	0.308958i	$0.0999804\pi$
$37$			3.75864i			0.617917i	−0.951076	+	0.308958i	$0.900020\pi$
$38$		0			0
$39$	−2.13093			−0.341221
$40$		0			0
$41$		−	4.12042i		−	0.643501i	−0.946824	−	0.321751i	$-0.895729\pi$
$41$		−	4.12042i		−	0.643501i	0.946824	−	0.321751i	$-0.104271\pi$
$42$		0			0
$43$		−	4.35420i		−	0.664009i	−0.943278	−	0.332005i	$-0.892275\pi$
$43$		−	4.35420i		−	0.664009i	0.943278	−	0.332005i	$-0.107725\pi$
$44$		0			0
$45$	1.11684			0.166489
$46$		0			0
$47$			8.59820i			1.25418i	0.778948	+	0.627088i	$0.215753\pi$
$47$			8.59820i			1.25418i	−0.778948	+	0.627088i	$0.784247\pi$
$48$		0			0
$49$	1.00258	+	6.92783i	0.143226	+	0.989690i
$50$		0			0
$51$		−	7.30657i		−	1.02312i
$52$		0			0
$53$		−	6.83736i		−	0.939184i	−0.882884	−	0.469592i	$-0.844401\pi$
$53$		−	6.83736i		−	0.939184i	0.882884	−	0.469592i	$-0.155599\pi$
$54$		0			0
$55$			4.76236i			0.642156i
$56$		0			0
$57$			1.25349i			0.166028i
$58$		0			0
$59$			3.01977i			0.393141i	0.980490	+	0.196570i	$0.0629804\pi$
$59$			3.01977i			0.393141i	−0.980490	+	0.196570i	$0.937020\pi$
$60$		0			0
$61$	−5.54919			−0.710500			−0.355250	−	0.934771i	$-0.615604\pi$
$61$	−5.54919			−0.710500			−0.355250	+	0.934771i	$0.615604\pi$
$62$		0			0
$63$	−2.00032	−	1.73168i	−0.252017	−	0.218171i
$64$		0			0
$65$		−	2.37991i		−	0.295191i
$66$		0			0
$67$		−	7.79556i		−	0.952380i	−0.879343	−	0.476190i	$-0.842018\pi$
$67$		−	7.79556i		−	0.952380i	0.879343	−	0.476190i	$-0.157982\pi$
$68$		0			0
$69$	−4.02873	+	2.60180i	−0.485002	+	0.313220i
$70$		0			0
$71$	−16.3741			−1.94325			−0.971625	−	0.236528i	$-0.923991\pi$
$71$	−16.3741			−1.94325			−0.971625	+	0.236528i	$0.923991\pi$
$72$		0			0
$73$			1.71890i			0.201182i	0.994928	+	0.100591i	$0.0320734\pi$
$73$			1.71890i			0.201182i	−0.994928	+	0.100591i	$0.967927\pi$
$74$		0			0
$75$		−	3.75266i		−	0.433320i
$76$		0			0
$77$	7.38409	−	8.52963i	0.841496	−	0.972041i
$78$		0			0
$79$		−	6.31656i		−	0.710669i	−0.934739	−	0.355334i	$-0.884367\pi$
$79$		−	6.31656i		−	0.710669i	0.934739	−	0.355334i	$-0.115633\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−10.2991			−1.13048			−0.565238	−	0.824928i	$-0.691216\pi$
$83$	−10.2991			−1.13048			−0.565238	+	0.824928i	$0.691216\pi$
$84$		0			0
$85$	8.16028			0.885107
$86$		0			0
$87$		−	4.69379i		−	0.503227i
$88$		0			0
$89$	10.5105			1.11411			0.557057	−	0.830474i	$-0.311930\pi$
$89$	10.5105			1.11411			0.557057	+	0.830474i	$0.311930\pi$
$90$		0			0
$91$	−3.69008	+	4.26254i	−0.386825	+	0.446835i
$92$		0			0
$93$	−10.0969			−1.04700
$94$		0			0
$95$	−1.39995			−0.143632
$96$		0			0
$97$	−17.4486			−1.77164			−0.885821	−	0.464027i	$-0.846404\pi$
$97$	−17.4486			−1.77164			−0.885821	+	0.464027i	$0.846404\pi$
$98$		0			0
$99$			4.26413i			0.428561i
$100$		0			0
$101$			15.5766i			1.54993i	0.632007	+	0.774963i	$0.282231\pi$
$101$			15.5766i			1.54993i	−0.632007	+	0.774963i	$0.717769\pi$
$102$		0			0
$103$	−3.11101			−0.306537			−0.153268	−	0.988185i	$-0.548980\pi$
$103$	−3.11101			−0.306537			−0.153268	+	0.988185i	$0.548980\pi$
$104$		0			0
$105$	1.93401	−	2.23404i	0.188740	−	0.218020i
$106$		0			0
$107$		−	6.30059i		−	0.609101i	−0.952496	−	0.304550i	$-0.901494\pi$
$107$		−	6.30059i		−	0.609101i	0.952496	−	0.304550i	$-0.0985062\pi$
$108$		0			0
$109$			17.3438i			1.66124i	0.556843	+	0.830618i	$0.312012\pi$
$109$			17.3438i			1.66124i	−0.556843	+	0.830618i	$0.687988\pi$
$110$		0			0
$111$	−3.75864			−0.356754
$112$		0			0
$113$		−	11.4420i		−	1.07638i	−0.842825	−	0.538188i	$-0.819109\pi$
$113$		−	11.4420i		−	1.07638i	0.842825	−	0.538188i	$-0.180891\pi$
$114$		0			0
$115$	−2.90580	−	4.49945i	−0.270967	−	0.419576i
$116$		0			0
$117$		−	2.13093i		−	0.197004i
$118$		0			0
$119$	−14.6155	−	12.6526i	−1.33980	−	1.15986i
$120$		0			0
$121$	−7.18277			−0.652979
$122$		0			0
$123$	4.12042			0.371525
$124$		0			0
$125$	9.77534			0.874333
$126$		0			0
$127$	−20.7384			−1.84024			−0.920119	−	0.391638i	$-0.871908\pi$
$127$	−20.7384			−1.84024			−0.920119	+	0.391638i	$0.871908\pi$
$128$		0			0
$129$	4.35420			0.383366
$130$		0			0
$131$			4.08054i			0.356518i	0.983984	+	0.178259i	$0.0570465\pi$
$131$			4.08054i			0.356518i	−0.983984	+	0.178259i	$0.942953\pi$
$132$		0			0
$133$	2.50738	+	2.17064i	0.217417	+	0.188218i
$134$		0			0
$135$			1.11684i			0.0961225i
$136$		0			0
$137$			0.207316i			0.0177122i	0.999961	+	0.00885610i	$0.00281902\pi$
$137$			0.207316i			0.0177122i	−0.999961	+	0.00885610i	$0.997181\pi$
$138$		0			0
$139$			6.02685i			0.511191i	0.966784	+	0.255595i	$0.0822715\pi$
$139$			6.02685i			0.511191i	−0.966784	+	0.255595i	$0.917729\pi$
$140$		0			0
$141$	−8.59820			−0.724099
$142$		0			0
$143$	9.08653			0.759854
$144$		0			0
$145$	5.24222			0.435342
$146$		0			0
$147$	−6.92783	+	1.00258i	−0.571398	+	0.0826915i
$148$		0			0
$149$			9.26845i			0.759301i	0.925130	+	0.379650i	$0.123956\pi$
$149$			9.26845i			0.759301i	−0.925130	+	0.379650i	$0.876044\pi$
$150$		0			0
$151$	8.20437			0.667662			0.333831	−	0.942633i	$-0.391658\pi$
$151$	8.20437			0.667662			0.333831	+	0.942633i	$0.391658\pi$
$152$		0			0
$153$	7.30657			0.590701
$154$		0			0
$155$		−	11.2766i		−	0.905760i
$156$		0			0
$157$	4.87797			0.389304			0.194652	−	0.980872i	$-0.437642\pi$
$157$	4.87797			0.389304			0.194652	+	0.980872i	$0.437642\pi$
$158$		0			0
$159$	6.83736			0.542238
$160$		0			0
$161$	−1.77202	+	12.5642i	−0.139655	+	0.990200i
$162$		0			0
$163$	−17.4972			−1.37048			−0.685242	−	0.728315i	$-0.740304\pi$
$163$	−17.4972			−1.37048			−0.685242	+	0.728315i	$0.740304\pi$
$164$		0			0
$165$	−4.76236			−0.370749
$166$		0			0
$167$		−	10.5885i		−	0.819363i	−0.912229	−	0.409682i	$-0.865640\pi$
$167$		−	10.5885i		−	0.819363i	0.912229	−	0.409682i	$-0.134360\pi$
$168$		0			0
$169$	8.45916			0.650704
$170$		0			0
$171$	−1.25349			−0.0958566
$172$		0			0
$173$			3.32588i			0.252862i	0.991975	+	0.126431i	$0.0403522\pi$
$173$			3.32588i			0.252862i	−0.991975	+	0.126431i	$0.959648\pi$
$174$		0			0
$175$	−7.50654	−	6.49841i	−0.567441	−	0.491233i
$176$		0			0
$177$	−3.01977			−0.226980
$178$		0			0
$179$	10.0777			0.753246			0.376623	−	0.926367i	$-0.377085\pi$
$179$	10.0777			0.753246			0.376623	+	0.926367i	$0.377085\pi$
$180$		0			0
$181$	−18.7769			−1.39568			−0.697838	−	0.716256i	$-0.745854\pi$
$181$	−18.7769			−1.39568			−0.697838	+	0.716256i	$0.745854\pi$
$182$		0			0
$183$		−	5.54919i		−	0.410208i
$184$		0			0
$185$		−	4.19781i		−	0.308629i
$186$		0			0
$187$			31.1561i			2.27836i
$188$		0			0
$189$	1.73168	−	2.00032i	0.125961	−	0.145502i
$190$		0			0
$191$			6.32358i			0.457558i	0.973478	+	0.228779i	$0.0734733\pi$
$191$			6.32358i			0.457558i	−0.973478	+	0.228779i	$0.926527\pi$
$192$		0			0
$193$	14.9762			1.07801			0.539004	−	0.842303i	$-0.318801\pi$
$193$	14.9762			1.07801			0.539004	+	0.842303i	$0.318801\pi$
$194$		0			0
$195$	2.37991			0.170429
$196$		0			0
$197$	13.5382			0.964554			0.482277	−	0.876019i	$-0.339810\pi$
$197$	13.5382			0.964554			0.482277	+	0.876019i	$0.339810\pi$
$198$		0			0
$199$	6.78370			0.480884			0.240442	−	0.970664i	$-0.422708\pi$
$199$	6.78370			0.480884			0.240442	+	0.970664i	$0.422708\pi$
$200$		0			0
$201$	7.79556			0.549857
$202$		0			0
$203$	−9.38909	−	8.12813i	−0.658985	−	0.570483i
$204$		0			0
$205$			4.60186i			0.321408i
$206$		0			0
$207$	−2.60180	−	4.02873i	−0.180837	−	0.280016i
$208$		0			0
$209$		−	5.34503i		−	0.369723i
$210$		0			0
$211$	26.2518			1.80725			0.903625	−	0.428325i	$-0.140896\pi$
$211$	26.2518			1.80725			0.903625	+	0.428325i	$0.140896\pi$
$212$		0			0
$213$		−	16.3741i		−	1.12194i
$214$		0			0
$215$			4.86296i			0.331651i
$216$		0			0
$217$	−17.4845	+	20.1970i	−1.18693	+	1.37106i
$218$		0			0
$219$	−1.71890			−0.116153
$220$		0			0
$221$		−	15.5698i		−	1.04733i
$222$		0			0
$223$		−	17.6775i		−	1.18377i	−0.806021	−	0.591887i	$-0.798383\pi$
$223$		−	17.6775i		−	1.18377i	0.806021	−	0.591887i	$-0.201617\pi$
$224$		0			0
$225$	3.75266			0.250178
$226$		0			0
$227$	7.76448			0.515346			0.257673	−	0.966232i	$-0.417044\pi$
$227$	7.76448			0.515346			0.257673	+	0.966232i	$0.417044\pi$
$228$		0			0
$229$	15.5815			1.02966			0.514828	−	0.857294i	$-0.327856\pi$
$229$	15.5815			1.02966			0.514828	+	0.857294i	$0.327856\pi$
$230$		0			0
$231$	8.52963	+	7.38409i	0.561208	+	0.485838i
$232$		0			0
$233$	−23.7789			−1.55781			−0.778905	−	0.627142i	$-0.784224\pi$
$233$	−23.7789			−1.55781			−0.778905	+	0.627142i	$0.784224\pi$
$234$		0			0
$235$		−	9.60283i		−	0.626420i
$236$		0			0
$237$	6.31656			0.410305
$238$		0			0
$239$	−22.0583			−1.42683			−0.713416	−	0.700740i	$-0.752853\pi$
$239$	−22.0583			−1.42683			−0.713416	+	0.700740i	$0.752853\pi$
$240$		0			0
$241$	−3.72756			−0.240113			−0.120057	−	0.992767i	$-0.538308\pi$
$241$	−3.72756			−0.240113			−0.120057	+	0.992767i	$0.538308\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−1.11973	−	7.73729i	−0.0715366	−	0.494317i
$246$		0			0
$247$			2.67109i			0.169957i
$248$		0			0
$249$		−	10.2991i		−	0.652681i
$250$		0			0
$251$	−4.07189			−0.257015			−0.128508	−	0.991709i	$-0.541019\pi$
$251$	−4.07189			−0.257015			−0.128508	+	0.991709i	$0.541019\pi$
$252$		0			0
$253$	17.1790	−	11.0944i	1.08003	−	0.697499i
$254$		0			0
$255$			8.16028i			0.511017i
$256$		0			0
$257$		−	16.7873i		−	1.04717i	−0.851975	−	0.523583i	$-0.824595\pi$
$257$		−	16.7873i		−	1.04717i	0.851975	−	0.523583i	$-0.175405\pi$
$258$		0			0
$259$	−6.50876	+	7.51850i	−0.404435	+	0.467177i
$260$		0			0
$261$	4.69379			0.290538
$262$		0			0
$263$			11.4236i			0.704409i	0.935923	+	0.352204i	$0.114568\pi$
$263$			11.4236i			0.704409i	−0.935923	+	0.352204i	$0.885432\pi$
$264$		0			0
$265$			7.63625i			0.469091i
$266$		0			0
$267$			10.5105i			0.643234i
$268$		0			0
$269$		−	20.3524i		−	1.24091i	−0.784242	−	0.620455i	$-0.786948\pi$
$269$		−	20.3524i		−	1.24091i	0.784242	−	0.620455i	$-0.213052\pi$
$270$		0			0
$271$			7.87620i			0.478445i	0.970965	+	0.239223i	$0.0768926\pi$
$271$			7.87620i			0.478445i	−0.970965	+	0.239223i	$0.923107\pi$
$272$		0			0
$273$	−4.26254	−	3.69008i	−0.257981	−	0.223334i
$274$		0			0
$275$			16.0018i			0.964947i
$276$		0			0
$277$	25.7568			1.54758			0.773788	−	0.633444i	$-0.218359\pi$
$277$	25.7568			1.54758			0.773788	+	0.633444i	$0.218359\pi$
$278$		0			0
$279$		−	10.0969i		−	0.604484i
$280$		0			0
$281$			26.6580i			1.59028i	0.606423	+	0.795142i	$0.292604\pi$
$281$			26.6580i			1.59028i	−0.606423	+	0.795142i	$0.707396\pi$
$282$		0			0
$283$	−18.6392			−1.10798			−0.553992	−	0.832522i	$-0.686896\pi$
$283$	−18.6392			−1.10798			−0.553992	+	0.832522i	$0.686896\pi$
$284$		0			0
$285$		−	1.39995i		−	0.0829257i
$286$		0			0
$287$	7.13524	−	8.24216i	0.421180	−	0.486520i
$288$		0			0
$289$	36.3860			2.14035
$290$		0			0
$291$		−	17.4486i		−	1.02286i
$292$		0			0
$293$	9.47142			0.553326			0.276663	−	0.960967i	$-0.410771\pi$
$293$	9.47142			0.553326			0.276663	+	0.960967i	$0.410771\pi$
$294$		0			0
$295$		−	3.37261i		−	0.196361i
$296$		0			0
$297$	−4.26413			−0.247430
$298$		0			0
$299$	−8.58492	+	5.54424i	−0.496479	+	0.320631i
$300$		0			0
$301$	7.54007	−	8.70981i	0.434603	−	0.502025i
$302$		0			0
$303$	−15.5766			−0.894850
$304$		0			0
$305$	6.19756			0.354871
$306$		0			0
$307$			0.933162i			0.0532584i	0.999645	+	0.0266292i	$0.00847733\pi$
$307$			0.933162i			0.0532584i	−0.999645	+	0.0266292i	$0.991523\pi$
$308$		0			0
$309$		−	3.11101i		−	0.176979i
$310$		0			0
$311$			11.2774i			0.639481i	0.947505	+	0.319741i	$0.103596\pi$
$311$			11.2774i			0.639481i	−0.947505	+	0.319741i	$0.896404\pi$
$312$		0			0
$313$	8.29856			0.469062			0.234531	−	0.972109i	$-0.424644\pi$
$313$	8.29856			0.469062			0.234531	+	0.972109i	$0.424644\pi$
$314$		0			0
$315$	2.23404	+	1.93401i	0.125874	+	0.108969i
$316$		0			0
$317$	25.3696			1.42490			0.712450	−	0.701723i	$-0.247585\pi$
$317$	25.3696			1.42490			0.712450	+	0.701723i	$0.247585\pi$
$318$		0			0
$319$			20.0149i			1.12062i
$320$		0			0
$321$	6.30059			0.351665
$322$		0			0
$323$	−9.15869			−0.509603
$324$		0			0
$325$		−	7.99665i		−	0.443574i
$326$		0			0
$327$	−17.3438			−0.959115
$328$		0			0
$329$	−14.8893	+	17.1992i	−0.820874	+	0.948221i
$330$		0			0
$331$	9.99467			0.549357			0.274678	−	0.961536i	$-0.411429\pi$
$331$	9.99467			0.549357			0.274678	+	0.961536i	$0.411429\pi$
$332$		0			0
$333$		−	3.75864i		−	0.205972i
$334$		0			0
$335$			8.70641i			0.475682i
$336$		0			0
$337$		−	23.2921i		−	1.26880i	−0.773003	−	0.634402i	$-0.781246\pi$
$337$		−	23.2921i		−	1.26880i	0.773003	−	0.634402i	$-0.218754\pi$
$338$		0			0
$339$	11.4420			0.621446
$340$		0			0
$341$	43.0544			2.33152
$342$		0			0
$343$	−9.99129	+	15.5940i	−0.539479	+	0.841999i
$344$		0			0
$345$	4.49945	−	2.90580i	0.242242	−	0.156443i
$346$		0			0
$347$	−1.26457			−0.0678855			−0.0339427	−	0.999424i	$-0.510806\pi$
$347$	−1.26457			−0.0678855			−0.0339427	+	0.999424i	$0.510806\pi$
$348$		0			0
$349$			1.40224i			0.0750599i	0.999296	+	0.0375300i	$0.0119490\pi$
$349$			1.40224i			0.0750599i	−0.999296	+	0.0375300i	$0.988051\pi$
$350$		0			0
$351$	2.13093			0.113740
$352$		0			0
$353$		−	22.3559i		−	1.18989i	−0.803768	−	0.594943i	$-0.797175\pi$
$353$		−	22.3559i		−	1.18989i	0.803768	−	0.594943i	$-0.202825\pi$
$354$		0			0
$355$	18.2873			0.970589
$356$		0			0
$357$	12.6526	−	14.6155i	0.669648	−	0.773534i
$358$		0			0
$359$			9.25509i			0.488465i	0.969717	+	0.244232i	$0.0785360\pi$
$359$			9.25509i			0.488465i	−0.969717	+	0.244232i	$0.921464\pi$
$360$		0			0
$361$	−17.4288			−0.917304
$362$		0			0
$363$		−	7.18277i		−	0.376998i
$364$		0			0
$365$		−	1.91974i		−	0.100484i
$366$		0			0
$367$	32.3658			1.68948			0.844740	−	0.535176i	$-0.179755\pi$
$367$	32.3658			1.68948			0.844740	+	0.535176i	$0.179755\pi$
$368$		0			0
$369$			4.12042i			0.214500i
$370$		0			0
$371$	11.8401	−	13.6769i	0.614708	−	0.710071i
$372$		0			0
$373$			7.27294i			0.376579i	0.982114	+	0.188289i	$0.0602943\pi$
$373$			7.27294i			0.376579i	−0.982114	+	0.188289i	$0.939706\pi$
$374$		0			0
$375$			9.77534i			0.504797i
$376$		0			0
$377$		−	10.0021i		−	0.515135i
$378$		0			0
$379$			29.4300i			1.51172i	0.654734	+	0.755860i	$0.272781\pi$
$379$			29.4300i			1.51172i	−0.654734	+	0.755860i	$0.727219\pi$
$380$		0			0
$381$		−	20.7384i		−	1.06246i
$382$		0			0
$383$	4.38338			0.223981			0.111990	−	0.993709i	$-0.464277\pi$
$383$	4.38338			0.223981			0.111990	+	0.993709i	$0.464277\pi$
$384$		0			0
$385$	−8.24687	+	9.52625i	−0.420299	+	0.485503i
$386$		0			0
$387$			4.35420i			0.221336i
$388$		0			0
$389$			11.1485i			0.565254i	0.959230	+	0.282627i	$0.0912058\pi$
$389$			11.1485i			0.565254i	−0.959230	+	0.282627i	$0.908794\pi$
$390$		0			0
$391$	−19.0102	−	29.4362i	−0.961388	−	1.48865i
$392$		0			0
$393$	−4.08054			−0.205836
$394$		0			0
$395$			7.05460i			0.354956i
$396$		0			0
$397$			37.4232i			1.87821i	0.343624	+	0.939107i	$0.388346\pi$
$397$			37.4232i			1.87821i	−0.343624	+	0.939107i	$0.611654\pi$
$398$		0			0
$399$	−2.17064	+	2.50738i	−0.108668	+	0.125526i
$400$		0			0
$401$		−	1.59208i		−	0.0795048i	−0.999210	−	0.0397524i	$-0.987343\pi$
$401$		−	1.59208i		−	0.0795048i	0.999210	−	0.0397524i	$-0.0126569\pi$
$402$		0			0
$403$	−21.5157			−1.07177
$404$		0			0
$405$	−1.11684			−0.0554963
$406$		0			0
$407$	16.0273			0.794445
$408$		0			0
$409$			35.2915i			1.74505i	0.488567	+	0.872526i	$0.337520\pi$
$409$			35.2915i			1.74505i	−0.488567	+	0.872526i	$0.662480\pi$
$410$		0			0
$411$	−0.207316			−0.0102261
$412$		0			0
$413$	−5.22927	+	6.04052i	−0.257316	+	0.297234i
$414$		0			0
$415$	11.5025			0.564636
$416$		0			0
$417$	−6.02685			−0.295136
$418$		0			0
$419$	−31.6106			−1.54428			−0.772139	−	0.635454i	$-0.780813\pi$
$419$	−31.6106			−1.54428			−0.772139	+	0.635454i	$0.780813\pi$
$420$		0			0
$421$			25.9368i			1.26408i	0.774935	+	0.632042i	$0.217783\pi$
$421$			25.9368i			1.26408i	−0.774935	+	0.632042i	$0.782217\pi$
$422$		0			0
$423$		−	8.59820i		−	0.418059i
$424$		0			0
$425$	27.4191			1.33002
$426$		0			0
$427$	−11.1002	−	9.60940i	−0.537174	−	0.465032i
$428$		0			0
$429$			9.08653i			0.438702i
$430$		0			0
$431$			8.79274i			0.423531i	0.977320	+	0.211766i	$0.0679213\pi$
$431$			8.79274i			0.423531i	−0.977320	+	0.211766i	$0.932079\pi$
$432$		0			0
$433$	−7.01748			−0.337238			−0.168619	−	0.985681i	$-0.553931\pi$
$433$	−7.01748			−0.337238			−0.168619	+	0.985681i	$0.553931\pi$
$434$		0			0
$435$			5.24222i			0.251345i
$436$		0			0
$437$	3.26132	+	5.04996i	0.156010	+	0.241572i
$438$		0			0
$439$			10.2670i			0.490019i	0.969521	+	0.245009i	$0.0787911\pi$
$439$			10.2670i			0.490019i	−0.969521	+	0.245009i	$0.921209\pi$
$440$		0			0
$441$	−1.00258	−	6.92783i	−0.0477420	−	0.329897i
$442$		0			0
$443$	2.59327			0.123210			0.0616050	−	0.998101i	$-0.480378\pi$
$443$	2.59327			0.123210			0.0616050	+	0.998101i	$0.480378\pi$
$444$		0			0
$445$	−11.7386			−0.556463
$446$		0			0
$447$	−9.26845			−0.438382
$448$		0			0
$449$	11.9114			0.562133			0.281066	−	0.959688i	$-0.409312\pi$
$449$	11.9114			0.562133			0.281066	+	0.959688i	$0.409312\pi$
$450$		0			0
$451$	−17.5700			−0.827338
$452$		0			0
$453$			8.20437i			0.385475i
$454$		0			0
$455$	4.12123	−	4.76058i	0.193206	−	0.223180i
$456$		0			0
$457$			10.0223i			0.468823i	0.972137	+	0.234411i	$0.0753163\pi$
$457$			10.0223i			0.468823i	−0.972137	+	0.234411i	$0.924684\pi$
$458$		0			0
$459$			7.30657i			0.341041i
$460$		0			0
$461$		−	7.66691i		−	0.357084i	−0.983932	−	0.178542i	$-0.942862\pi$
$461$		−	7.66691i		−	0.357084i	0.983932	−	0.178542i	$-0.0571380\pi$
$462$		0			0
$463$	6.92356			0.321765			0.160883	−	0.986974i	$-0.448566\pi$
$463$	6.92356			0.321765			0.160883	+	0.986974i	$0.448566\pi$
$464$		0			0
$465$	11.2766			0.522941
$466$		0			0
$467$	29.2830			1.35506			0.677529	−	0.735496i	$-0.263051\pi$
$467$	29.2830			1.35506			0.677529	+	0.735496i	$0.263051\pi$
$468$		0			0
$469$	13.4994	−	15.5936i	0.623345	−	0.720047i
$470$		0			0
$471$			4.87797i			0.224765i
$472$		0			0
$473$	−18.5669			−0.853705
$474$		0			0
$475$	−4.70392			−0.215831
$476$		0			0
$477$			6.83736i			0.313061i
$478$		0			0
$479$	−13.7917			−0.630158			−0.315079	−	0.949065i	$-0.602031\pi$
$479$	−13.7917			−0.630158			−0.315079	+	0.949065i	$0.602031\pi$
$480$		0			0
$481$	−8.00938			−0.365196
$482$		0			0
$483$	−12.5642	−	1.77202i	−0.571692	−	0.0806299i
$484$		0			0
$485$	19.4874			0.884876
$486$		0			0
$487$	5.07490			0.229966			0.114983	−	0.993367i	$-0.463319\pi$
$487$	5.07490			0.229966			0.114983	+	0.993367i	$0.463319\pi$
$488$		0			0
$489$		−	17.4972i		−	0.791249i
$490$		0			0
$491$	9.87680			0.445734			0.222867	−	0.974849i	$-0.428458\pi$
$491$	9.87680			0.445734			0.222867	+	0.974849i	$0.428458\pi$
$492$		0			0
$493$	34.2955			1.54459
$494$		0			0
$495$		−	4.76236i		−	0.214052i
$496$		0			0
$497$	−32.7535	−	28.3547i	−1.46920	−	1.27188i
$498$		0			0
$499$	13.5716			0.607550			0.303775	−	0.952744i	$-0.401753\pi$
$499$	13.5716			0.607550			0.303775	+	0.952744i	$0.401753\pi$
$500$		0			0
$501$	10.5885			0.473060
$502$		0			0
$503$	−16.9527			−0.755882			−0.377941	−	0.925830i	$-0.623368\pi$
$503$	−16.9527			−0.755882			−0.377941	+	0.925830i	$0.623368\pi$
$504$		0			0
$505$		−	17.3966i		−	0.774137i
$506$		0			0
$507$			8.45916i			0.375684i
$508$		0			0
$509$		−	42.6305i		−	1.88956i	−0.327699	−	0.944782i	$-0.606273\pi$
$509$		−	42.6305i		−	1.88956i	0.327699	−	0.944782i	$-0.393727\pi$
$510$		0			0
$511$	−2.97659	+	3.43836i	−0.131676	+	0.152104i
$512$		0			0
$513$		−	1.25349i		−	0.0553428i
$514$		0			0
$515$	3.47451			0.153105
$516$		0			0
$517$	36.6638			1.61247
$518$		0			0
$519$	−3.32588			−0.145990
$520$		0			0
$521$	−16.8790			−0.739481			−0.369740	−	0.929135i	$-0.620553\pi$
$521$	−16.8790			−0.739481			−0.369740	+	0.929135i	$0.620553\pi$
$522$		0			0
$523$	20.2369			0.884897			0.442449	−	0.896794i	$-0.354110\pi$
$523$	20.2369			0.884897			0.442449	+	0.896794i	$0.354110\pi$
$524$		0			0
$525$	6.49841	−	7.50654i	0.283614	−	0.327612i
$526$		0			0
$527$		−	73.7735i		−	3.21363i
$528$		0			0
$529$	−9.46129	+	20.9639i	−0.411361	+	0.911473i
$530$		0			0
$531$		−	3.01977i		−	0.131047i
$532$		0			0
$533$	8.78030			0.380317
$534$		0			0
$535$			7.03676i			0.304226i
$536$		0			0
$537$			10.0777i			0.434886i
$538$		0			0
$539$	29.5411	−	4.27513i	1.27243	−	0.184143i
$540$		0			0
$541$	26.1500			1.12428			0.562139	−	0.827043i	$-0.309979\pi$
$541$	26.1500			1.12428			0.562139	+	0.827043i	$0.309979\pi$
$542$		0			0
$543$		−	18.7769i		−	0.805794i
$544$		0			0
$545$		−	19.3703i		−	0.829732i
$546$		0			0
$547$	37.0838			1.58559			0.792795	−	0.609488i	$-0.208625\pi$
$547$	37.0838			1.58559			0.792795	+	0.609488i	$0.208625\pi$
$548$		0			0
$549$	5.54919			0.236833
$550$		0			0
$551$	−5.88360			−0.250650
$552$		0			0
$553$	10.9383	−	12.6352i	0.465142	−	0.537302i
$554$		0			0
$555$	4.19781			0.178187
$556$		0			0
$557$		−	32.2160i		−	1.36503i	−0.730870	−	0.682517i	$-0.760885\pi$
$557$		−	32.2160i		−	1.36503i	0.730870	−	0.682517i	$-0.239115\pi$
$558$		0			0
$559$	9.27848			0.392438
$560$		0			0
$561$	−31.1561			−1.31541
$562$		0			0
$563$	27.2272			1.14749			0.573744	−	0.819034i	$-0.305490\pi$
$563$	27.2272			1.14749			0.573744	+	0.819034i	$0.305490\pi$
$564$		0			0
$565$			12.7790i			0.537615i
$566$		0			0
$567$	2.00032	+	1.73168i	0.0840057	+	0.0727236i
$568$		0			0
$569$			36.6125i			1.53488i	0.641123	+	0.767438i	$0.278469\pi$
$569$			36.6125i			1.53488i	−0.641123	+	0.767438i	$0.721531\pi$
$570$		0			0
$571$		−	35.7443i		−	1.49585i	−0.663781	−	0.747927i	$-0.731049\pi$
$571$		−	35.7443i		−	1.49585i	0.663781	−	0.747927i	$-0.268951\pi$
$572$		0			0
$573$	−6.32358			−0.264171
$574$		0			0
$575$	−9.76367	−	15.1185i	−0.407173	−	0.630483i
$576$		0			0
$577$		−	20.7507i		−	0.863861i	−0.901907	−	0.431931i	$-0.857832\pi$
$577$		−	20.7507i		−	0.863861i	0.901907	−	0.431931i	$-0.142168\pi$
$578$		0			0
$579$			14.9762i			0.622389i
$580$		0			0
$581$	−20.6016	−	17.8348i	−0.854698	−	0.739911i
$582$		0			0
$583$	−29.1554			−1.20749
$584$		0			0
$585$			2.37991i			0.0983970i
$586$		0			0
$587$			11.0312i			0.455306i	0.973742	+	0.227653i	$0.0731052\pi$
$587$			11.0312i			0.455306i	−0.973742	+	0.227653i	$0.926895\pi$
$588$		0			0
$589$			12.6563i			0.521494i
$590$		0			0
$591$			13.5382i			0.556886i
$592$		0			0
$593$			5.32183i			0.218541i	0.994012	+	0.109271i	$0.0348515\pi$
$593$			5.32183i			0.218541i	−0.994012	+	0.109271i	$0.965149\pi$
$594$		0			0
$595$	16.3232	+	14.1310i	0.669186	+	0.579314i
$596$		0			0
$597$			6.78370i			0.277638i
$598$		0			0
$599$	46.7811			1.91142			0.955712	−	0.294303i	$-0.0950874\pi$
$599$	46.7811			1.91142			0.955712	+	0.294303i	$0.0950874\pi$
$600$		0			0
$601$			6.17568i			0.251911i	0.992036	+	0.125956i	$0.0401997\pi$
$601$			6.17568i			0.251911i	−0.992036	+	0.125956i	$0.959800\pi$
$602$		0			0
$603$			7.79556i			0.317460i
$604$		0			0
$605$	8.02202			0.326142
$606$		0			0
$607$		−	33.5563i		−	1.36201i	−0.732280	−	0.681004i	$-0.761544\pi$
$607$		−	33.5563i		−	1.36201i	0.732280	−	0.681004i	$-0.238456\pi$
$608$		0			0
$609$	8.12813	−	9.38909i	0.329368	−	0.380465i
$610$		0			0
$611$	−18.3221			−0.741234
$612$		0			0
$613$			11.8339i			0.477965i	0.971024	+	0.238983i	$0.0768139\pi$
$613$			11.8339i			0.477965i	−0.971024	+	0.238983i	$0.923186\pi$
$614$		0			0
$615$	−4.60186			−0.185565
$616$		0			0
$617$			17.0912i			0.688067i	0.938957	+	0.344033i	$0.111793\pi$
$617$			17.0912i			0.688067i	−0.938957	+	0.344033i	$0.888207\pi$
$618$		0			0
$619$	−4.44408			−0.178623			−0.0893113	−	0.996004i	$-0.528467\pi$
$619$	−4.44408			−0.178623			−0.0893113	+	0.996004i	$0.528467\pi$
$620$		0			0
$621$	4.02873	−	2.60180i	0.161667	−	0.104407i
$622$		0			0
$623$	21.0245	+	18.2009i	0.842327	+	0.729202i
$624$		0			0
$625$	7.84580			0.313832
$626$		0			0
$627$	5.34503			0.213460
$628$		0			0
$629$		−	27.4628i		−	1.09501i
$630$		0			0
$631$		−	38.5406i		−	1.53428i	−0.641481	−	0.767139i	$-0.721680\pi$
$631$		−	38.5406i		−	1.53428i	0.641481	−	0.767139i	$-0.278320\pi$
$632$		0			0
$633$			26.2518i			1.04342i
$634$		0			0
$635$	23.1616			0.919138
$636$		0			0
$637$	−14.7627	+	2.13643i	−0.584919	+	0.0846483i
$638$		0			0
$639$	16.3741			0.647750
$640$		0			0
$641$		−	22.2676i		−	0.879518i	−0.898116	−	0.439759i	$-0.855064\pi$
$641$		−	22.2676i		−	0.879518i	0.898116	−	0.439759i	$-0.144936\pi$
$642$		0			0
$643$	−38.8253			−1.53112			−0.765560	−	0.643364i	$-0.777538\pi$
$643$	−38.8253			−1.53112			−0.765560	+	0.643364i	$0.777538\pi$
$644$		0			0
$645$	−4.86296			−0.191479
$646$		0			0
$647$			17.0277i			0.669427i	0.942320	+	0.334714i	$0.108640\pi$
$647$			17.0277i			0.669427i	−0.942320	+	0.334714i	$0.891360\pi$
$648$		0			0
$649$	12.8767			0.505454
$650$		0			0
$651$	−20.1970	−	17.4845i	−0.791583	−	0.685273i
$652$		0			0
$653$	−22.8547			−0.894374			−0.447187	−	0.894441i	$-0.647574\pi$
$653$	−22.8547			−0.894374			−0.447187	+	0.894441i	$0.647574\pi$
$654$		0			0
$655$		−	4.55731i		−	0.178069i
$656$		0			0
$657$		−	1.71890i		−	0.0670608i
$658$		0			0
$659$			7.62653i			0.297087i	0.988906	+	0.148544i	$0.0474586\pi$
$659$			7.62653i			0.297087i	−0.988906	+	0.148544i	$0.952541\pi$
$660$		0			0
$661$	−27.7634			−1.07987			−0.539935	−	0.841707i	$-0.681551\pi$
$661$	−27.7634			−1.07987			−0.539935	+	0.841707i	$0.681551\pi$
$662$		0			0
$663$	15.5698			0.604679
$664$		0			0
$665$	−2.80035	−	2.42426i	−0.108593	−	0.0940087i
$666$		0			0
$667$	−12.2123	−	18.9100i	−0.472861	−	0.732198i
$668$		0			0
$669$	17.6775			0.683452
$670$		0			0
$671$			23.6624i			0.913478i
$672$		0			0
$673$	12.3131			0.474636			0.237318	−	0.971432i	$-0.423732\pi$
$673$	12.3131			0.474636			0.237318	+	0.971432i	$0.423732\pi$
$674$		0			0
$675$			3.75266i			0.144440i
$676$		0			0
$677$	31.8859			1.22548			0.612738	−	0.790286i	$-0.290068\pi$
$677$	31.8859			1.22548			0.612738	+	0.790286i	$0.290068\pi$
$678$		0			0
$679$	−34.9029	−	30.2154i	−1.33945	−	1.15956i
$680$		0			0
$681$			7.76448i			0.297535i
$682$		0			0
$683$	−22.7315			−0.869798			−0.434899	−	0.900479i	$-0.643216\pi$
$683$	−22.7315			−0.869798			−0.434899	+	0.900479i	$0.643216\pi$
$684$		0			0
$685$		−	0.231539i		−	0.00884666i
$686$		0			0
$687$			15.5815i			0.594472i
$688$		0			0
$689$	14.5699			0.555069
$690$		0			0
$691$			11.3547i			0.431952i	0.976399	+	0.215976i	$0.0692934\pi$
$691$			11.3547i			0.431952i	−0.976399	+	0.215976i	$0.930707\pi$
$692$		0			0
$693$	−7.38409	+	8.52963i	−0.280499	+	0.324014i
$694$		0			0
$695$		−	6.73104i		−	0.255323i
$696$		0			0
$697$			30.1061i			1.14035i
$698$		0			0
$699$		−	23.7789i		−	0.899402i
$700$		0			0
$701$			44.1431i			1.66726i	0.552321	+	0.833631i	$0.313742\pi$
$701$			44.1431i			1.66726i	−0.552321	+	0.833631i	$0.686258\pi$
$702$		0			0
$703$			4.71141i			0.177694i
$704$		0			0
$705$	9.60283			0.361664
$706$		0			0
$707$	−26.9736	+	31.1581i	−1.01445	+	1.17182i
$708$		0			0
$709$			6.02652i			0.226331i	0.993576	+	0.113165i	$0.0360990\pi$
$709$			6.02652i			0.226331i	−0.993576	+	0.113165i	$0.963901\pi$
$710$		0			0
$711$			6.31656i			0.236890i
$712$		0			0
$713$	−40.6776	+	26.2700i	−1.52339	+	0.983821i
$714$		0			0
$715$	−10.1482			−0.379522
$716$		0			0
$717$		−	22.0583i		−	0.823782i
$718$		0			0
$719$		−	12.3124i		−	0.459176i	−0.973288	−	0.229588i	$-0.926262\pi$
$719$		−	12.3124i		−	0.459176i	0.973288	−	0.229588i	$-0.0737379\pi$
$720$		0			0
$721$	−6.22302	−	5.38727i	−0.231757	−	0.200632i
$722$		0			0
$723$		−	3.72756i		−	0.138629i
$724$		0			0
$725$	17.6142			0.654175
$726$		0			0
$727$	22.3999			0.830767			0.415384	−	0.909646i	$-0.363647\pi$
$727$	22.3999			0.830767			0.415384	+	0.909646i	$0.363647\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			31.8143i			1.17669i
$732$		0			0
$733$	−1.57056			−0.0580100			−0.0290050	−	0.999579i	$-0.509234\pi$
$733$	−1.57056			−0.0580100			−0.0290050	+	0.999579i	$0.509234\pi$
$734$		0			0
$735$	7.73729	−	1.11973i	0.285394	−	0.0413017i
$736$		0			0
$737$	−33.2413			−1.22446
$738$		0			0
$739$	2.04395			0.0751881			0.0375940	−	0.999293i	$-0.488031\pi$
$739$	2.04395			0.0751881			0.0375940	+	0.999293i	$0.488031\pi$
$740$		0			0
$741$	−2.67109			−0.0981249
$742$		0			0
$743$			45.4311i			1.66671i	0.552740	+	0.833353i	$0.313582\pi$
$743$			45.4311i			1.66671i	−0.552740	+	0.833353i	$0.686418\pi$
$744$		0			0
$745$		−	10.3514i		−	0.379246i
$746$		0			0
$747$	10.2991			0.376825
$748$		0			0
$749$	10.9106	−	12.6032i	0.398664	−	0.460511i
$750$		0			0
$751$		−	24.2305i		−	0.884184i	−0.896970	−	0.442092i	$-0.854236\pi$
$751$		−	24.2305i		−	0.884184i	0.896970	−	0.442092i	$-0.145764\pi$
$752$		0			0
$753$		−	4.07189i		−	0.148388i
$754$		0			0
$755$	−9.16299			−0.333475
$756$		0			0
$757$		−	46.1434i		−	1.67711i	−0.544816	−	0.838555i	$-0.683401\pi$
$757$		−	46.1434i		−	1.67711i	0.544816	−	0.838555i	$-0.316599\pi$
$758$		0			0
$759$	11.0944	+	17.1790i	0.402701	+	0.623558i
$760$		0			0
$761$			36.6804i			1.32967i	0.746992	+	0.664833i	$0.231497\pi$
$761$			36.6804i			1.32967i	−0.746992	+	0.664833i	$0.768503\pi$
$762$		0			0
$763$	−30.0339	+	34.6932i	−1.08730	+	1.25598i
$764$		0			0
$765$	−8.16028			−0.295036
$766$		0			0
$767$	−6.43491			−0.232351
$768$		0			0
$769$	−8.26961			−0.298210			−0.149105	−	0.988821i	$-0.547639\pi$
$769$	−8.26961			−0.298210			−0.149105	+	0.988821i	$0.547639\pi$
$770$		0			0
$771$	16.7873			0.604581
$772$		0			0
$773$	−8.44084			−0.303596			−0.151798	−	0.988412i	$-0.548506\pi$
$773$	−8.44084			−0.303596			−0.151798	+	0.988412i	$0.548506\pi$
$774$		0			0
$775$		−	37.8902i		−	1.36106i
$776$		0			0
$777$	−7.51850	−	6.50876i	−0.269725	−	0.233500i
$778$		0			0
$779$		−	5.16489i		−	0.185051i
$780$		0			0
$781$			69.8213i			2.49840i
$782$		0			0
$783$			4.69379i			0.167742i
$784$		0			0
$785$	−5.44792			−0.194445
$786$		0			0
$787$	−35.0164			−1.24820			−0.624100	−	0.781344i	$-0.714534\pi$
$787$	−35.0164			−1.24820			−0.624100	+	0.781344i	$0.714534\pi$
$788$		0			0
$789$	−11.4236			−0.406691
$790$		0			0
$791$	19.8139	−	22.8878i	0.704502	−	0.813795i
$792$		0			0
$793$		−	11.8249i		−	0.419914i
$794$		0			0
$795$	−7.63625			−0.270830
$796$		0			0
$797$	−27.5582			−0.976160			−0.488080	−	0.872799i	$-0.662303\pi$
$797$	−27.5582			−0.976160			−0.488080	+	0.872799i	$0.662303\pi$
$798$		0			0
$799$		−	62.8233i		−	2.22253i
$800$		0			0
$801$	−10.5105			−0.371372
$802$		0			0
$803$	7.32962			0.258657
$804$		0			0
$805$	1.97907	−	14.0323i	0.0697531	−	0.494572i
$806$		0			0
$807$	20.3524			0.716440
$808$		0			0
$809$	−22.8010			−0.801641			−0.400821	−	0.916157i	$-0.631275\pi$
$809$	−22.8010			−0.801641			−0.400821	+	0.916157i	$0.631275\pi$
$810$		0			0
$811$		−	14.3204i		−	0.502858i	−0.967876	−	0.251429i	$-0.919099\pi$
$811$		−	14.3204i		−	0.502858i	0.967876	−	0.251429i	$-0.0809006\pi$
$812$		0			0
$813$	−7.87620			−0.276230
$814$		0			0
$815$	19.5416			0.684512
$816$		0			0
$817$		−	5.45794i		−	0.190949i
$818$		0			0
$819$	3.69008	−	4.26254i	0.128942	−	0.148945i
$820$		0			0
$821$	−17.4471			−0.608907			−0.304453	−	0.952527i	$-0.598474\pi$
$821$	−17.4471			−0.608907			−0.304453	+	0.952527i	$0.598474\pi$
$822$		0			0
$823$	−32.5078			−1.13315			−0.566575	−	0.824010i	$-0.691732\pi$
$823$	−32.5078			−1.13315			−0.566575	+	0.824010i	$0.691732\pi$
$824$		0			0
$825$	−16.0018			−0.557112
$826$		0			0
$827$			25.4274i			0.884196i	0.896967	+	0.442098i	$0.145766\pi$
$827$			25.4274i			0.884196i	−0.896967	+	0.442098i	$0.854234\pi$
$828$		0			0
$829$		−	18.0516i		−	0.626958i	−0.949595	−	0.313479i	$-0.898506\pi$
$829$		−	18.0516i		−	0.626958i	0.949595	−	0.313479i	$-0.101494\pi$
$830$		0			0
$831$			25.7568i			0.893494i
$832$		0			0
$833$	−7.32543	−	50.6187i	−0.253811	−	1.75383i
$834$		0			0
$835$			11.8257i			0.409245i
$836$		0			0
$837$	10.0969			0.348999
$838$		0			0
$839$	−50.3284			−1.73753			−0.868765	−	0.495224i	$-0.835086\pi$
$839$	−50.3284			−1.73753			−0.868765	+	0.495224i	$0.835086\pi$
$840$		0			0
$841$	−6.96838			−0.240289
$842$		0			0
$843$	−26.6580			−0.918151
$844$		0			0
$845$	−9.44754			−0.325005
$846$		0			0
$847$	−14.3679	−	12.4382i	−0.493686	−	0.427383i
$848$		0			0
$849$		−	18.6392i		−	0.639695i
$850$		0			0
$851$	−15.1425	+	9.77923i	−0.519080	+	0.335228i
$852$		0			0
$853$			3.89420i			0.133335i	0.997775	+	0.0666675i	$0.0212367\pi$
$853$			3.89420i			0.133335i	−0.997775	+	0.0666675i	$0.978763\pi$
$854$		0			0
$855$	1.39995			0.0478772
$856$		0			0
$857$			43.6608i			1.49143i	0.666268	+	0.745713i	$0.267891\pi$
$857$			43.6608i			1.49143i	−0.666268	+	0.745713i	$0.732109\pi$
$858$		0			0
$859$			15.5366i			0.530104i	0.964234	+	0.265052i	$0.0853891\pi$
$859$			15.5366i			0.530104i	−0.964234	+	0.265052i	$0.914611\pi$
$860$		0			0
$861$	8.24216	+	7.13524i	0.280892	+	0.243168i
$862$		0			0
$863$	−24.7206			−0.841500			−0.420750	−	0.907177i	$-0.638233\pi$
$863$	−24.7206			−0.841500			−0.420750	+	0.907177i	$0.638233\pi$
$864$		0			0
$865$		−	3.71448i		−	0.126296i
$866$		0			0
$867$			36.3860i			1.23573i
$868$		0			0
$869$	−26.9346			−0.913694
$870$		0			0
$871$	16.6118			0.562868
$872$		0			0
$873$	17.4486			0.590547
$874$		0			0
$875$	19.5538	+	16.9277i	0.661040	+	0.572262i
$876$		0			0
$877$	34.5175			1.16557			0.582787	−	0.812625i	$-0.301962\pi$
$877$	34.5175			1.16557			0.582787	+	0.812625i	$0.301962\pi$
$878$		0			0
$879$			9.47142i			0.319463i
$880$		0			0
$881$	47.5502			1.60201			0.801004	−	0.598659i	$-0.204300\pi$
$881$	47.5502			1.60201			0.801004	+	0.598659i	$0.204300\pi$
$882$		0			0
$883$	19.0828			0.642188			0.321094	−	0.947047i	$-0.395949\pi$
$883$	19.0828			0.642188			0.321094	+	0.947047i	$0.395949\pi$
$884$		0			0
$885$	3.37261			0.113369
$886$		0			0
$887$			47.9700i			1.61068i	0.592816	+	0.805338i	$0.298016\pi$
$887$			47.9700i			1.61068i	−0.592816	+	0.805338i	$0.701984\pi$
$888$		0			0
$889$	−41.4836	−	35.9123i	−1.39131	−	1.20446i
$890$		0			0
$891$		−	4.26413i		−	0.142854i
$892$		0			0
$893$			10.7777i			0.360663i
$894$		0			0
$895$	−11.2552			−0.376221
$896$		0			0
$897$	−5.54424	−	8.58492i	−0.185117	−	0.286642i
$898$		0			0
$899$		−	47.3926i		−	1.58063i
$900$		0			0
$901$			49.9576i			1.66433i
$902$		0			0
$903$	8.70981	+	7.54007i	0.289844	+	0.250918i
$904$		0			0
$905$	20.9708			0.697094
$906$		0			0
$907$			59.4370i			1.97357i	0.162029	+	0.986786i	$0.448196\pi$
$907$			59.4370i			1.97357i	−0.162029	+	0.986786i	$0.551804\pi$
$908$		0			0
$909$		−	15.5766i		−	0.516642i
$910$		0			0
$911$			24.3776i			0.807666i	0.914833	+	0.403833i	$0.132322\pi$
$911$			24.3776i			0.807666i	−0.914833	+	0.403833i	$0.867678\pi$
$912$		0			0
$913$			43.9168i			1.45343i
$914$		0			0
$915$			6.19756i			0.204885i
$916$		0			0
$917$	−7.06618	+	8.16239i	−0.233346	+	0.269546i
$918$		0			0
$919$		−	5.79957i		−	0.191310i	−0.995415	−	0.0956551i	$-0.969505\pi$
$919$		−	5.79957i		−	0.191310i	0.995415	−	0.0956551i	$-0.0304946\pi$
$920$		0			0
$921$	−0.933162			−0.0307487
$922$		0			0
$923$		−	34.8920i		−	1.14848i
$924$		0			0
$925$		−	14.1049i		−	0.463767i
$926$		0			0
$927$	3.11101			0.102179
$928$		0			0
$929$			17.3328i			0.568672i	0.958725	+	0.284336i	$0.0917731\pi$
$929$			17.3328i			0.568672i	−0.958725	+	0.284336i	$0.908227\pi$
$930$		0			0
$931$	1.25672	+	8.68395i	0.0411874	+	0.284605i
$932$		0			0
$933$	−11.2774			−0.369205
$934$		0			0
$935$		−	34.7965i		−	1.13797i
$936$		0			0
$937$	27.4866			0.897949			0.448975	−	0.893544i	$-0.351789\pi$
$937$	27.4866			0.897949			0.448975	+	0.893544i	$0.351789\pi$
$938$		0			0
$939$			8.29856i			0.270813i
$940$		0			0
$941$	−30.2794			−0.987079			−0.493540	−	0.869723i	$-0.664297\pi$
$941$	−30.2794			−0.987079			−0.493540	+	0.869723i	$0.664297\pi$
$942$		0			0
$943$	16.6000	−	10.7205i	0.540572	−	0.349107i
$944$		0			0
$945$	−1.93401	+	2.23404i	−0.0629134	+	0.0726735i
$946$		0			0
$947$	−3.95672			−0.128576			−0.0642881	−	0.997931i	$-0.520478\pi$
$947$	−3.95672			−0.128576			−0.0642881	+	0.997931i	$0.520478\pi$
$948$		0			0
$949$	−3.66285			−0.118901
$950$		0			0
$951$			25.3696i			0.822667i
$952$		0			0
$953$		−	42.0942i		−	1.36357i	−0.731555	−	0.681783i	$-0.761205\pi$
$953$		−	42.0942i		−	1.36357i	0.731555	−	0.681783i	$-0.238795\pi$
$954$		0			0
$955$		−	7.06244i		−	0.228535i
$956$		0			0
$957$	−20.0149			−0.646990
$958$		0			0
$959$	−0.359005	+	0.414699i	−0.0115929	+	0.0133913i
$960$		0			0
$961$	−70.9469			−2.28861
$962$		0			0
$963$			6.30059i			0.203034i
$964$		0			0
$965$	−16.7260			−0.538430
$966$		0			0
$967$	−43.8588			−1.41040			−0.705202	−	0.709007i	$-0.749144\pi$
$967$	−43.8588			−1.41040			−0.705202	+	0.709007i	$0.749144\pi$
$968$		0			0
$969$		−	9.15869i		−	0.294220i
$970$		0			0
$971$	−23.5077			−0.754398			−0.377199	−	0.926132i	$-0.623113\pi$
$971$	−23.5077			−0.754398			−0.377199	+	0.926132i	$0.623113\pi$
$972$		0			0
$973$	−10.4366	+	12.0556i	−0.334581	+	0.386486i
$974$		0			0
$975$	7.99665			0.256098
$976$		0			0
$977$		−	10.6578i		−	0.340975i	−0.985360	−	0.170487i	$-0.945466\pi$
$977$		−	10.6578i		−	0.340975i	0.985360	−	0.170487i	$-0.0545342\pi$
$978$		0			0
$979$		−	44.8182i		−	1.43240i
$980$		0			0
$981$		−	17.3438i		−	0.553745i
$982$		0			0
$983$	−9.85308			−0.314264			−0.157132	−	0.987578i	$-0.550225\pi$
$983$	−9.85308			−0.314264			−0.157132	+	0.987578i	$0.550225\pi$
$984$		0			0
$985$	−15.1200			−0.481763
$986$		0			0
$987$	−17.1992	−	14.8893i	−0.547456	−	0.473932i
$988$		0			0
$989$	17.5419	−	11.3288i	0.557800	−	0.360233i
$990$		0			0
$991$	−31.9643			−1.01538			−0.507690	−	0.861540i	$-0.669501\pi$
$991$	−31.9643			−1.01538			−0.507690	+	0.861540i	$0.669501\pi$
$992$		0			0
$993$			9.99467i			0.317171i
$994$		0			0
$995$	−7.57632			−0.240185
$996$		0			0
$997$		−	38.4257i		−	1.21695i	−0.793571	−	0.608477i	$-0.791781\pi$
$997$		−	38.4257i		−	1.21695i	0.793571	−	0.608477i	$-0.208219\pi$
$998$		0			0
$999$	3.75864			0.118918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.k.a.1609.3	yes	32
3.2	odd	2		5796.2.k.d.5473.19		32
7.6	odd	2	inner	1932.2.k.a.1609.2	yes	32
21.20	even	2		5796.2.k.d.5473.13		32
23.22	odd	2	inner	1932.2.k.a.1609.4	yes	32
69.68	even	2		5796.2.k.d.5473.14		32
161.160	even	2	inner	1932.2.k.a.1609.1	✓	32
483.482	odd	2		5796.2.k.d.5473.20		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.1	✓	32	161.160	even	2	inner
1932.2.k.a.1609.2	yes	32	7.6	odd	2	inner
1932.2.k.a.1609.3	yes	32	1.1	even	1	trivial
1932.2.k.a.1609.4	yes	32	23.22	odd	2	inner
5796.2.k.d.5473.13		32	21.20	even	2
5796.2.k.d.5473.14		32	69.68	even	2
5796.2.k.d.5473.19		32	3.2	odd	2
5796.2.k.d.5473.20		32	483.482	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.11684 q^{5} +(2.00032 + 1.73168i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.11684 q^{5} +(2.00032 + 1.73168i) q^{7} -1.00000 q^{9} -4.26413i q^{11} +2.13093i q^{13} -1.11684i q^{15} -7.30657 q^{17} +1.25349 q^{19} +(-1.73168 + 2.00032i) q^{21} +(2.60180 + 4.02873i) q^{23} -3.75266 q^{25} -1.00000i q^{27} -4.69379 q^{29} +10.0969i q^{31} +4.26413 q^{33} +(-2.23404 - 1.93401i) q^{35} +3.75864i q^{37} -2.13093 q^{39} -4.12042i q^{41} -4.35420i q^{43} +1.11684 q^{45} +8.59820i q^{47} +(1.00258 + 6.92783i) q^{49} -7.30657i q^{51} -6.83736i q^{53} +4.76236i q^{55} +1.25349i q^{57} +3.01977i q^{59} -5.54919 q^{61} +(-2.00032 - 1.73168i) q^{63} -2.37991i q^{65} -7.79556i q^{67} +(-4.02873 + 2.60180i) q^{69} -16.3741 q^{71} +1.71890i q^{73} -3.75266i q^{75} +(7.38409 - 8.52963i) q^{77} -6.31656i q^{79} +1.00000 q^{81} -10.2991 q^{83} +8.16028 q^{85} -4.69379i q^{87} +10.5105 q^{89} +(-3.69008 + 4.26254i) q^{91} -10.0969 q^{93} -1.39995 q^{95} -17.4486 q^{97} +4.26413i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9}+O(q^{10}) \) 32 * q - 32 * q^9	\( 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) \) 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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