LMFDB - Embedded newform 3344.2.o.b.1519.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1519,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.o (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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$64$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.12
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.b.1519.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16100 q^{3} -1.12654 q^{5} +1.31746i q^{7} +1.66990 q^{9} +O(q^{10})$	$q-2.16100 q^{3} -1.12654 q^{5} +1.31746i q^{7} +1.66990 q^{9} +1.00000i q^{11} +7.16895i q^{13} +2.43446 q^{15} -5.19429 q^{17} +(0.322044 - 4.34699i) q^{19} -2.84703i q^{21} +4.67510i q^{23} -3.73090 q^{25} +2.87433 q^{27} +1.14874i q^{29} -3.75302 q^{31} -2.16100i q^{33} -1.48418i q^{35} +9.89841i q^{37} -15.4921i q^{39} +2.84818i q^{41} -7.64380i q^{43} -1.88122 q^{45} +7.31677i q^{47} +5.26429 q^{49} +11.2248 q^{51} +6.58207i q^{53} -1.12654i q^{55} +(-0.695936 + 9.39382i) q^{57} +12.8125 q^{59} +2.11178 q^{61} +2.20004i q^{63} -8.07613i q^{65} +0.581157 q^{67} -10.1029i q^{69} +3.44165 q^{71} -7.69997 q^{73} +8.06246 q^{75} -1.31746 q^{77} -4.18208 q^{79} -11.2211 q^{81} -7.86021i q^{83} +5.85160 q^{85} -2.48242i q^{87} +4.17089i q^{89} -9.44482 q^{91} +8.11027 q^{93} +(-0.362797 + 4.89707i) q^{95} -4.63855i q^{97} +1.66990i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 56 q^{9}+O(q^{10}) $ 64 * q + 56 * q^9	$ 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) $ 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times$.

$n$	$705$	$837$	$2433$	$2927$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−2.16100			−1.24765			−0.623826	−	0.781563i	$-0.714423\pi$
$3$	−2.16100			−1.24765			−0.623826	+	0.781563i	$0.714423\pi$
$4$		0			0
$5$	−1.12654			−0.503806			−0.251903	−	0.967753i	$-0.581056\pi$
$5$	−1.12654			−0.503806			−0.251903	+	0.967753i	$0.581056\pi$
$6$		0			0
$7$			1.31746i			0.497954i	0.968509	+	0.248977i	$0.0800944\pi$
$7$			1.31746i			0.497954i	−0.968509	+	0.248977i	$0.919906\pi$
$8$		0			0
$9$	1.66990			0.556635
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			7.16895i			1.98831i	0.107972	+	0.994154i	$0.465564\pi$
$13$			7.16895i			1.98831i	−0.107972	+	0.994154i	$0.534436\pi$
$14$		0			0
$15$	2.43446			0.628574
$16$		0			0
$17$	−5.19429			−1.25980			−0.629901	−	0.776676i	$-0.716904\pi$
$17$	−5.19429			−1.25980			−0.629901	+	0.776676i	$0.716904\pi$
$18$		0			0
$19$	0.322044	−	4.34699i	0.0738819	−	0.997267i
$19$	0.322044	−	4.34699i	0.0738819	−	0.997267i
$20$		0			0
$21$		−	2.84703i		−	0.621273i
$22$		0			0
$23$			4.67510i			0.974826i	0.873172	+	0.487413i	$0.162059\pi$
$23$			4.67510i			0.974826i	−0.873172	+	0.487413i	$0.837941\pi$
$24$		0			0
$25$	−3.73090			−0.746180
$26$		0			0
$27$	2.87433			0.553166
$28$		0			0
$29$			1.14874i			0.213316i	0.994296	+	0.106658i	$0.0340149\pi$
$29$			1.14874i			0.213316i	−0.994296	+	0.106658i	$0.965985\pi$
$30$		0			0
$31$	−3.75302			−0.674063			−0.337031	−	0.941493i	$-0.609423\pi$
$31$	−3.75302			−0.674063			−0.337031	+	0.941493i	$0.609423\pi$
$32$		0			0
$33$		−	2.16100i		−	0.376181i
$34$		0			0
$35$		−	1.48418i		−	0.250872i
$36$		0			0
$37$			9.89841i			1.62729i	0.581363	+	0.813644i	$0.302520\pi$
$37$			9.89841i			1.62729i	−0.581363	+	0.813644i	$0.697480\pi$
$38$		0			0
$39$		−	15.4921i		−	2.48072i
$40$		0			0
$41$			2.84818i			0.444811i	0.974954	+	0.222406i	$0.0713909\pi$
$41$			2.84818i			0.444811i	−0.974954	+	0.222406i	$0.928609\pi$
$42$		0			0
$43$		−	7.64380i		−	1.16567i	−0.812591	−	0.582834i	$-0.801944\pi$
$43$		−	7.64380i		−	1.16567i	0.812591	−	0.582834i	$-0.198056\pi$
$44$		0			0
$45$	−1.88122			−0.280436
$46$		0			0
$47$			7.31677i			1.06726i	0.845718	+	0.533630i	$0.179173\pi$
$47$			7.31677i			1.06726i	−0.845718	+	0.533630i	$0.820827\pi$
$48$		0			0
$49$	5.26429			0.752042
$50$		0			0
$51$	11.2248			1.57179
$52$		0			0
$53$			6.58207i			0.904117i	0.891988	+	0.452059i	$0.149310\pi$
$53$			6.58207i			0.904117i	−0.891988	+	0.452059i	$0.850690\pi$
$54$		0			0
$55$		−	1.12654i		−	0.151903i
$56$		0			0
$57$	−0.695936	+	9.39382i	−0.0921789	+	1.24424i
$58$		0			0
$59$	12.8125			1.66804			0.834020	−	0.551734i	$-0.186034\pi$
$59$	12.8125			1.66804			0.834020	+	0.551734i	$0.186034\pi$
$60$		0			0
$61$	2.11178			0.270386			0.135193	−	0.990819i	$-0.456835\pi$
$61$	2.11178			0.270386			0.135193	+	0.990819i	$0.456835\pi$
$62$		0			0
$63$			2.20004i			0.277178i
$64$		0			0
$65$		−	8.07613i		−	1.00172i
$66$		0			0
$67$	0.581157			0.0709996			0.0354998	−	0.999370i	$-0.488698\pi$
$67$	0.581157			0.0709996			0.0354998	+	0.999370i	$0.488698\pi$
$68$		0			0
$69$		−	10.1029i		−	1.21624i
$70$		0			0
$71$	3.44165			0.408449			0.204224	−	0.978924i	$-0.434533\pi$
$71$	3.44165			0.408449			0.204224	+	0.978924i	$0.434533\pi$
$72$		0			0
$73$	−7.69997			−0.901213			−0.450607	−	0.892723i	$-0.648792\pi$
$73$	−7.69997			−0.901213			−0.450607	+	0.892723i	$0.648792\pi$
$74$		0			0
$75$	8.06246			0.930972
$76$		0			0
$77$	−1.31746			−0.150139
$78$		0			0
$79$	−4.18208			−0.470521			−0.235260	−	0.971932i	$-0.575594\pi$
$79$	−4.18208			−0.470521			−0.235260	+	0.971932i	$0.575594\pi$
$80$		0			0
$81$	−11.2211			−1.24679
$82$		0			0
$83$		−	7.86021i		−	0.862770i	−0.902168	−	0.431385i	$-0.858025\pi$
$83$		−	7.86021i		−	0.862770i	0.902168	−	0.431385i	$-0.141975\pi$
$84$		0			0
$85$	5.85160			0.634695
$86$		0			0
$87$		−	2.48242i		−	0.266144i
$88$		0			0
$89$			4.17089i			0.442114i	0.975261	+	0.221057i	$0.0709506\pi$
$89$			4.17089i			0.442114i	−0.975261	+	0.221057i	$0.929049\pi$
$90$		0			0
$91$	−9.44482			−0.990086
$92$		0			0
$93$	8.11027			0.840995
$94$		0			0
$95$	−0.362797	+	4.89707i	−0.0372221	+	0.502429i
$96$		0			0
$97$		−	4.63855i		−	0.470974i	−0.971878	−	0.235487i	$-0.924332\pi$
$97$		−	4.63855i		−	0.470974i	0.971878	−	0.235487i	$-0.0756684\pi$
$98$		0			0
$99$			1.66990i			0.167832i
$100$		0			0
$101$	−4.87020			−0.484603			−0.242302	−	0.970201i	$-0.577902\pi$
$101$	−4.87020			−0.484603			−0.242302	+	0.970201i	$0.577902\pi$
$102$		0			0
$103$	−8.64944			−0.852255			−0.426127	−	0.904663i	$-0.640123\pi$
$103$	−8.64944			−0.852255			−0.426127	+	0.904663i	$0.640123\pi$
$104$		0			0
$105$			3.20731i			0.313001i
$106$		0			0
$107$	−3.88077			−0.375168			−0.187584	−	0.982249i	$-0.560066\pi$
$107$	−3.88077			−0.375168			−0.187584	+	0.982249i	$0.560066\pi$
$108$		0			0
$109$		−	1.62410i		−	0.155561i	−0.996971	−	0.0777803i	$-0.975217\pi$
$109$		−	1.62410i		−	0.155561i	0.996971	−	0.0777803i	$-0.0247833\pi$
$110$		0			0
$111$		−	21.3904i		−	2.03029i
$112$		0			0
$113$		−	3.34564i		−	0.314732i	−0.987540	−	0.157366i	$-0.949700\pi$
$113$		−	3.34564i		−	0.314732i	0.987540	−	0.157366i	$-0.0503002\pi$
$114$		0			0
$115$		−	5.26671i		−	0.491123i
$116$		0			0
$117$			11.9714i			1.10676i
$118$		0			0
$119$		−	6.84329i		−	0.627323i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	6.15491i		−	0.554970i
$124$		0			0
$125$	9.83574			0.879735
$126$		0			0
$127$	−14.4289			−1.28035			−0.640177	−	0.768227i	$-0.721139\pi$
$127$	−14.4289			−1.28035			−0.640177	+	0.768227i	$0.721139\pi$
$128$		0			0
$129$			16.5182i			1.45435i
$130$		0			0
$131$			2.74733i			0.240035i	0.992772	+	0.120018i	$0.0382951\pi$
$131$			2.74733i			0.240035i	−0.992772	+	0.120018i	$0.961705\pi$
$132$		0			0
$133$	5.72699	+	0.424281i	0.496593	+	0.0367898i
$134$		0			0
$135$	−3.23806			−0.278688
$136$		0			0
$137$	−5.88488			−0.502779			−0.251390	−	0.967886i	$-0.580888\pi$
$137$	−5.88488			−0.502779			−0.251390	+	0.967886i	$0.580888\pi$
$138$		0			0
$139$		−	15.5275i		−	1.31703i	−0.752569	−	0.658514i	$-0.771185\pi$
$139$		−	15.5275i		−	1.31703i	0.752569	−	0.658514i	$-0.228815\pi$
$140$		0			0
$141$		−	15.8115i		−	1.33157i
$142$		0			0
$143$	−7.16895			−0.599497
$144$		0			0
$145$		−	1.29411i		−	0.107470i
$146$		0			0
$147$	−11.3761			−0.938286
$148$		0			0
$149$	21.0292			1.72278			0.861388	−	0.507947i	$-0.169595\pi$
$149$	21.0292			1.72278			0.861388	+	0.507947i	$0.169595\pi$
$150$		0			0
$151$	9.75405			0.793774			0.396887	−	0.917868i	$-0.370091\pi$
$151$	9.75405			0.793774			0.396887	+	0.917868i	$0.370091\pi$
$152$		0			0
$153$	−8.67397			−0.701249
$154$		0			0
$155$	4.22794			0.339597
$156$		0			0
$157$	−18.0284			−1.43882			−0.719410	−	0.694585i	$-0.755588\pi$
$157$	−18.0284			−1.43882			−0.719410	+	0.694585i	$0.755588\pi$
$158$		0			0
$159$		−	14.2238i		−	1.12802i
$160$		0			0
$161$	−6.15927			−0.485418
$162$		0			0
$163$			3.05574i			0.239344i	0.992813	+	0.119672i	$0.0381843\pi$
$163$			3.05574i			0.239344i	−0.992813	+	0.119672i	$0.961816\pi$
$164$		0			0
$165$			2.43446i			0.189522i
$166$		0			0
$167$	−8.29763			−0.642090			−0.321045	−	0.947064i	$-0.604034\pi$
$167$	−8.29763			−0.642090			−0.321045	+	0.947064i	$0.604034\pi$
$168$		0			0
$169$	−38.3938			−2.95337
$170$		0			0
$171$	0.537782	−	7.25905i	0.0411252	−	0.555113i
$172$		0			0
$173$		−	6.92196i		−	0.526267i	−0.964759	−	0.263133i	$-0.915244\pi$
$173$		−	6.92196i		−	0.526267i	0.964759	−	0.263133i	$-0.0847559\pi$
$174$		0			0
$175$		−	4.91532i		−	0.371563i
$176$		0			0
$177$	−27.6877			−2.08113
$178$		0			0
$179$	−7.49351			−0.560092			−0.280046	−	0.959987i	$-0.590350\pi$
$179$	−7.49351			−0.560092			−0.280046	+	0.959987i	$0.590350\pi$
$180$		0			0
$181$			5.94813i			0.442121i	0.975260	+	0.221061i	$0.0709519\pi$
$181$			5.94813i			0.442121i	−0.975260	+	0.221061i	$0.929048\pi$
$182$		0			0
$183$	−4.56354			−0.337347
$184$		0			0
$185$		−	11.1510i		−	0.819837i
$186$		0			0
$187$		−	5.19429i		−	0.379844i
$188$		0			0
$189$			3.78683i			0.275451i
$190$		0			0
$191$		−	9.71747i		−	0.703132i	−0.936163	−	0.351566i	$-0.885649\pi$
$191$		−	9.71747i		−	0.703132i	0.936163	−	0.351566i	$-0.114351\pi$
$192$		0			0
$193$		−	6.84665i		−	0.492833i	−0.969164	−	0.246416i	$-0.920747\pi$
$193$		−	6.84665i		−	0.492833i	0.969164	−	0.246416i	$-0.0792531\pi$
$194$		0			0
$195$			17.4525i			1.24980i
$196$		0			0
$197$	−7.81046			−0.556473			−0.278236	−	0.960513i	$-0.589750\pi$
$197$	−7.81046			−0.556473			−0.278236	+	0.960513i	$0.589750\pi$
$198$		0			0
$199$		−	19.3707i		−	1.37315i	−0.727059	−	0.686575i	$-0.759114\pi$
$199$		−	19.3707i		−	1.37315i	0.727059	−	0.686575i	$-0.240886\pi$
$200$		0			0
$201$	−1.25588			−0.0885828
$202$		0			0
$203$	−1.51342			−0.106221
$204$		0			0
$205$		−	3.20860i		−	0.224099i
$206$		0			0
$207$			7.80697i			0.542622i
$208$		0			0
$209$	4.34699	+	0.322044i	0.300687	+	0.0222762i
$210$		0			0
$211$	6.02593			0.414842			0.207421	−	0.978252i	$-0.433493\pi$
$211$	6.02593			0.414842			0.207421	+	0.978252i	$0.433493\pi$
$212$		0			0
$213$	−7.43739			−0.509602
$214$		0			0
$215$			8.61107i			0.587270i
$216$		0			0
$217$		−	4.94447i		−	0.335652i
$218$		0			0
$219$	16.6396			1.12440
$220$		0			0
$221$		−	37.2376i		−	2.50487i
$222$		0			0
$223$	−22.9751			−1.53853			−0.769263	−	0.638933i	$-0.779376\pi$
$223$	−22.9751			−1.53853			−0.769263	+	0.638933i	$0.779376\pi$
$224$		0			0
$225$	−6.23024			−0.415349
$226$		0			0
$227$	20.9065			1.38762			0.693808	−	0.720160i	$-0.255932\pi$
$227$	20.9065			1.38762			0.693808	+	0.720160i	$0.255932\pi$
$228$		0			0
$229$	−1.42226			−0.0939857			−0.0469929	−	0.998895i	$-0.514964\pi$
$229$	−1.42226			−0.0939857			−0.0469929	+	0.998895i	$0.514964\pi$
$230$		0			0
$231$	2.84703			0.187321
$232$		0			0
$233$	21.8726			1.43292			0.716461	−	0.697627i	$-0.245761\pi$
$233$	21.8726			1.43292			0.716461	+	0.697627i	$0.245761\pi$
$234$		0			0
$235$		−	8.24266i		−	0.537692i
$236$		0			0
$237$	9.03746			0.587046
$238$		0			0
$239$		−	16.4595i		−	1.06467i	−0.846533	−	0.532337i	$-0.821314\pi$
$239$		−	16.4595i		−	1.06467i	0.846533	−	0.532337i	$-0.178686\pi$
$240$		0			0
$241$		−	25.8311i		−	1.66393i	−0.554830	−	0.831964i	$-0.687217\pi$
$241$		−	25.8311i		−	1.66393i	0.554830	−	0.831964i	$-0.312783\pi$
$242$		0			0
$243$	15.6258			1.00240
$244$		0			0
$245$	−5.93046			−0.378883
$246$		0			0
$247$	31.1633	+	2.30872i	1.98287	+	0.146900i
$248$		0			0
$249$			16.9859i			1.07644i
$250$		0			0
$251$		−	19.7119i		−	1.24420i	−0.782936	−	0.622102i	$-0.786279\pi$
$251$		−	19.7119i		−	1.24420i	0.782936	−	0.622102i	$-0.213721\pi$
$252$		0			0
$253$	−4.67510			−0.293921
$254$		0			0
$255$	−12.6453			−0.791879
$256$		0			0
$257$			5.59676i			0.349116i	0.984647	+	0.174558i	$0.0558497\pi$
$257$			5.59676i			0.349116i	−0.984647	+	0.174558i	$0.944150\pi$
$258$		0			0
$259$	−13.0408			−0.810315
$260$		0			0
$261$			1.91828i			0.118739i
$262$		0			0
$263$			20.8283i			1.28433i	0.766567	+	0.642164i	$0.221963\pi$
$263$			20.8283i			1.28433i	−0.766567	+	0.642164i	$0.778037\pi$
$264$		0			0
$265$		−	7.41499i		−	0.455499i
$266$		0			0
$267$		−	9.01328i		−	0.551604i
$268$		0			0
$269$		−	20.3709i		−	1.24204i	−0.783796	−	0.621018i	$-0.786719\pi$
$269$		−	20.3709i		−	1.24204i	0.783796	−	0.621018i	$-0.213281\pi$
$270$		0			0
$271$			10.8633i			0.659896i	0.943999	+	0.329948i	$0.107031\pi$
$271$			10.8633i			0.659896i	−0.943999	+	0.329948i	$0.892969\pi$
$272$		0			0
$273$	20.4102			1.23528
$274$		0			0
$275$		−	3.73090i		−	0.224982i
$276$		0			0
$277$	0.634182			0.0381043			0.0190521	−	0.999818i	$-0.493935\pi$
$277$	0.634182			0.0381043			0.0190521	+	0.999818i	$0.493935\pi$
$278$		0			0
$279$	−6.26719			−0.375207
$280$		0			0
$281$			22.8541i			1.36336i	0.731651	+	0.681679i	$0.238750\pi$
$281$			22.8541i			1.36336i	−0.731651	+	0.681679i	$0.761250\pi$
$282$		0			0
$283$		−	9.55066i		−	0.567728i	−0.958865	−	0.283864i	$-0.908384\pi$
$283$		−	9.55066i		−	0.567728i	0.958865	−	0.283864i	$-0.0916164\pi$
$284$		0			0
$285$	0.784002	−	10.5826i	0.0464403	−	0.626856i
$286$		0			0
$287$	−3.75237			−0.221496
$288$		0			0
$289$	9.98069			0.587099
$290$		0			0
$291$			10.0239i			0.587611i
$292$		0			0
$293$		−	5.56946i		−	0.325371i	−0.986678	−	0.162686i	$-0.947984\pi$
$293$		−	5.56946i		−	0.325371i	0.986678	−	0.162686i	$-0.0520156\pi$
$294$		0			0
$295$	−14.4338			−0.840369
$296$		0			0
$297$			2.87433i			0.166786i
$298$		0			0
$299$	−33.5155			−1.93825
$300$		0			0
$301$	10.0704			0.580449
$302$		0			0
$303$	10.5245			0.604616
$304$		0			0
$305$	−2.37901			−0.136222
$306$		0			0
$307$	−19.9760			−1.14009			−0.570045	−	0.821614i	$-0.693074\pi$
$307$	−19.9760			−1.14009			−0.570045	+	0.821614i	$0.693074\pi$
$308$		0			0
$309$	18.6914			1.06332
$310$		0			0
$311$		−	1.48464i		−	0.0841861i	−0.999114	−	0.0420930i	$-0.986597\pi$
$311$		−	1.48464i		−	0.0841861i	0.999114	−	0.0420930i	$-0.0134026\pi$
$312$		0			0
$313$	−20.0345			−1.13242			−0.566210	−	0.824261i	$-0.691591\pi$
$313$	−20.0345			−1.13242			−0.566210	+	0.824261i	$0.691591\pi$
$314$		0			0
$315$		−	2.47844i		−	0.139644i
$316$		0			0
$317$		−	7.54218i		−	0.423611i	−0.977312	−	0.211806i	$-0.932066\pi$
$317$		−	7.54218i		−	0.423611i	0.977312	−	0.211806i	$-0.0679344\pi$
$318$		0			0
$319$	−1.14874			−0.0643171
$320$		0			0
$321$	8.38632			0.468079
$322$		0			0
$323$	−1.67279	+	22.5795i	−0.0930766	+	1.25636i
$324$		0			0
$325$		−	26.7466i		−	1.48364i
$326$		0			0
$327$			3.50967i			0.194085i
$328$		0			0
$329$	−9.63957			−0.531447
$330$		0			0
$331$	33.3566			1.83344			0.916721	−	0.399527i	$-0.130826\pi$
$331$	33.3566			1.83344			0.916721	+	0.399527i	$0.130826\pi$
$332$		0			0
$333$			16.5294i			0.905805i
$334$		0			0
$335$	−0.654699			−0.0357700
$336$		0			0
$337$		−	12.9681i		−	0.706416i	−0.935545	−	0.353208i	$-0.885091\pi$
$337$		−	12.9681i		−	0.706416i	0.935545	−	0.353208i	$-0.114909\pi$
$338$		0			0
$339$			7.22992i			0.392675i
$340$		0			0
$341$		−	3.75302i		−	0.203238i
$342$		0			0
$343$			16.1577i			0.872436i
$344$		0			0
$345$			11.3813i			0.612750i
$346$		0			0
$347$			11.7076i			0.628497i	0.949341	+	0.314248i	$0.101752\pi$
$347$			11.7076i			0.628497i	−0.949341	+	0.314248i	$0.898248\pi$
$348$		0			0
$349$	−34.0343			−1.82181			−0.910907	−	0.412611i	$-0.864617\pi$
$349$	−34.0343			−1.82181			−0.910907	+	0.412611i	$0.864617\pi$
$350$		0			0
$351$			20.6059i			1.09986i
$352$		0			0
$353$	12.1583			0.647118			0.323559	−	0.946208i	$-0.395121\pi$
$353$	12.1583			0.647118			0.323559	+	0.946208i	$0.395121\pi$
$354$		0			0
$355$	−3.87717			−0.205779
$356$		0			0
$357$			14.7883i			0.782681i
$358$		0			0
$359$			8.14616i			0.429938i	0.976621	+	0.214969i	$0.0689650\pi$
$359$			8.14616i			0.429938i	−0.976621	+	0.214969i	$0.931035\pi$
$360$		0			0
$361$	−18.7926	−	2.79984i	−0.989083	−	0.147360i
$362$		0			0
$363$	2.16100			0.113423
$364$		0			0
$365$	8.67436			0.454037
$366$		0			0
$367$			6.31739i			0.329765i	0.986313	+	0.164883i	$0.0527245\pi$
$367$			6.31739i			0.329765i	−0.986313	+	0.164883i	$0.947275\pi$
$368$		0			0
$369$			4.75619i			0.247597i
$370$		0			0
$371$	−8.67163			−0.450209
$372$		0			0
$373$			9.48528i			0.491129i	0.969380	+	0.245565i	$0.0789733\pi$
$373$			9.48528i			0.491129i	−0.969380	+	0.245565i	$0.921027\pi$
$374$		0			0
$375$	−21.2550			−1.09760
$376$		0			0
$377$	−8.23525			−0.424137
$378$		0			0
$379$	−7.11688			−0.365569			−0.182785	−	0.983153i	$-0.558511\pi$
$379$	−7.11688			−0.365569			−0.182785	+	0.983153i	$0.558511\pi$
$380$		0			0
$381$	31.1807			1.59744
$382$		0			0
$383$	10.6413			0.543746			0.271873	−	0.962333i	$-0.412357\pi$
$383$	10.6413			0.543746			0.271873	+	0.962333i	$0.412357\pi$
$384$		0			0
$385$	1.48418			0.0756408
$386$		0			0
$387$		−	12.7644i		−	0.648851i
$388$		0			0
$389$	−25.2714			−1.28131			−0.640655	−	0.767829i	$-0.721337\pi$
$389$	−25.2714			−1.28131			−0.640655	+	0.767829i	$0.721337\pi$
$390$		0			0
$391$		−	24.2838i		−	1.22809i
$392$		0			0
$393$		−	5.93697i		−	0.299481i
$394$		0			0
$395$	4.71130			0.237051
$396$		0			0
$397$	20.2110			1.01436			0.507180	−	0.861840i	$-0.330688\pi$
$397$	20.2110			1.01436			0.507180	+	0.861840i	$0.330688\pi$
$398$		0			0
$399$	−12.3760	−	0.916869i	−0.619575	−	0.0459009i
$400$		0			0
$401$		−	32.9604i		−	1.64596i	−0.568067	−	0.822982i	$-0.692309\pi$
$401$		−	32.9604i		−	1.64596i	0.568067	−	0.822982i	$-0.307691\pi$
$402$		0			0
$403$		−	26.9052i		−	1.34024i
$404$		0			0
$405$	12.6411			0.628141
$406$		0			0
$407$	−9.89841			−0.490646
$408$		0			0
$409$			24.3474i			1.20390i	0.798533	+	0.601951i	$0.205610\pi$
$409$			24.3474i			1.20390i	−0.798533	+	0.601951i	$0.794390\pi$
$410$		0			0
$411$	12.7172			0.627293
$412$		0			0
$413$			16.8799i			0.830608i
$414$		0			0
$415$			8.85487i			0.434669i
$416$		0			0
$417$			33.5549i			1.64319i
$418$		0			0
$419$			26.0578i			1.27300i	0.771275	+	0.636502i	$0.219619\pi$
$419$			26.0578i			1.27300i	−0.771275	+	0.636502i	$0.780381\pi$
$420$		0			0
$421$			19.6389i			0.957142i	0.878049	+	0.478571i	$0.158845\pi$
$421$			19.6389i			0.957142i	−0.878049	+	0.478571i	$0.841155\pi$
$422$		0			0
$423$			12.2183i			0.594074i
$424$		0			0
$425$	19.3794			0.940038
$426$		0			0
$427$			2.78219i			0.134640i
$428$		0			0
$429$	15.4921			0.747964
$430$		0			0
$431$	−31.2941			−1.50738			−0.753692	−	0.657228i	$-0.771729\pi$
$431$	−31.2941			−1.50738			−0.753692	+	0.657228i	$0.771729\pi$
$432$		0			0
$433$		−	24.4822i		−	1.17654i	−0.808665	−	0.588269i	$-0.799810\pi$
$433$		−	24.4822i		−	1.17654i	0.808665	−	0.588269i	$-0.200190\pi$
$434$		0			0
$435$			2.79656i			0.134085i
$436$		0			0
$437$	20.3226	+	1.50559i	0.972162	+	0.0720220i
$438$		0			0
$439$	30.5221			1.45674			0.728371	−	0.685183i	$-0.240278\pi$
$439$	30.5221			1.45674			0.728371	+	0.685183i	$0.240278\pi$
$440$		0			0
$441$	8.79086			0.418612
$442$		0			0
$443$			4.20719i			0.199890i	0.994993	+	0.0999448i	$0.0318666\pi$
$443$			4.20719i			0.199890i	−0.994993	+	0.0999448i	$0.968133\pi$
$444$		0			0
$445$		−	4.69869i		−	0.222739i
$446$		0			0
$447$	−45.4439			−2.14943
$448$		0			0
$449$			30.6581i			1.44685i	0.690405	+	0.723423i	$0.257432\pi$
$449$			30.6581i			1.44685i	−0.690405	+	0.723423i	$0.742568\pi$
$450$		0			0
$451$	−2.84818			−0.134116
$452$		0			0
$453$	−21.0785			−0.990353
$454$		0			0
$455$	10.6400			0.498811
$456$		0			0
$457$	26.5508			1.24199			0.620997	−	0.783813i	$-0.286728\pi$
$457$	26.5508			1.24199			0.620997	+	0.783813i	$0.286728\pi$
$458$		0			0
$459$	−14.9301			−0.696879
$460$		0			0
$461$	−8.50808			−0.396261			−0.198130	−	0.980176i	$-0.563487\pi$
$461$	−8.50808			−0.396261			−0.198130	+	0.980176i	$0.563487\pi$
$462$		0			0
$463$			8.69188i			0.403946i	0.979391	+	0.201973i	$0.0647353\pi$
$463$			8.69188i			0.403946i	−0.979391	+	0.201973i	$0.935265\pi$
$464$		0			0
$465$	−9.13657			−0.423698
$466$		0			0
$467$		−	18.6134i		−	0.861327i	−0.902513	−	0.430664i	$-0.858280\pi$
$467$		−	18.6134i		−	0.861327i	0.902513	−	0.430664i	$-0.141720\pi$
$468$		0			0
$469$			0.765652i			0.0353545i
$470$		0			0
$471$	38.9592			1.79515
$472$		0			0
$473$	7.64380			0.351462
$474$		0			0
$475$	−1.20151	+	16.2182i	−0.0551292	+	0.744140i
$476$		0			0
$477$			10.9914i			0.503263i
$478$		0			0
$479$		−	9.16394i		−	0.418711i	−0.977840	−	0.209356i	$-0.932863\pi$
$479$		−	9.16394i		−	0.418711i	0.977840	−	0.209356i	$-0.0671366\pi$
$480$		0			0
$481$	−70.9612			−3.23555
$482$		0			0
$483$	13.3102			0.605633
$484$		0			0
$485$			5.22553i			0.237279i
$486$		0			0
$487$	6.61902			0.299936			0.149968	−	0.988691i	$-0.452083\pi$
$487$	6.61902			0.299936			0.149968	+	0.988691i	$0.452083\pi$
$488$		0			0
$489$		−	6.60344i		−	0.298618i
$490$		0			0
$491$			20.2413i			0.913476i	0.889601	+	0.456738i	$0.150982\pi$
$491$			20.2413i			0.913476i	−0.889601	+	0.456738i	$0.849018\pi$
$492$		0			0
$493$		−	5.96689i		−	0.268735i
$494$		0			0
$495$		−	1.88122i		−	0.0845545i
$496$		0			0
$497$			4.53424i			0.203389i
$498$		0			0
$499$		−	13.2577i		−	0.593498i	−0.954956	−	0.296749i	$-0.904098\pi$
$499$		−	13.2577i		−	0.593498i	0.954956	−	0.296749i	$-0.0959024\pi$
$500$		0			0
$501$	17.9311			0.801104
$502$		0			0
$503$		−	19.7203i		−	0.879283i	−0.898173	−	0.439642i	$-0.855105\pi$
$503$		−	19.7203i		−	0.879283i	0.898173	−	0.439642i	$-0.144895\pi$
$504$		0			0
$505$	5.48650			0.244146
$506$		0			0
$507$	82.9688			3.68477
$508$		0			0
$509$			37.7278i			1.67225i	0.548536	+	0.836127i	$0.315185\pi$
$509$			37.7278i			1.67225i	−0.548536	+	0.836127i	$0.684815\pi$
$510$		0			0
$511$		−	10.1444i		−	0.448763i
$512$		0			0
$513$	0.925661	−	12.4947i	0.0408689	−	0.551654i
$514$		0			0
$515$	9.74398			0.429371
$516$		0			0
$517$	−7.31677			−0.321791
$518$		0			0
$519$			14.9583i			0.656597i
$520$		0			0
$521$		−	33.4085i		−	1.46365i	−0.681491	−	0.731827i	$-0.738668\pi$
$521$		−	33.4085i		−	1.46365i	0.681491	−	0.731827i	$-0.261332\pi$
$522$		0			0
$523$	11.3806			0.497641			0.248820	−	0.968550i	$-0.419957\pi$
$523$	11.3806			0.497641			0.248820	+	0.968550i	$0.419957\pi$
$524$		0			0
$525$			10.6220i			0.463581i
$526$		0			0
$527$	19.4943			0.849185
$528$		0			0
$529$	1.14343			0.0497145
$530$		0			0
$531$	21.3956			0.928489
$532$		0			0
$533$	−20.4185			−0.884422
$534$		0			0
$535$	4.37186			0.189012
$536$		0			0
$537$	16.1935			0.698799
$538$		0			0
$539$			5.26429i			0.226749i
$540$		0			0
$541$	33.9807			1.46095			0.730473	−	0.682942i	$-0.239300\pi$
$541$	33.9807			1.46095			0.730473	+	0.682942i	$0.239300\pi$
$542$		0			0
$543$		−	12.8539i		−	0.551613i
$544$		0			0
$545$			1.82962i			0.0783723i
$546$		0			0
$547$	−25.9126			−1.10794			−0.553972	−	0.832536i	$-0.686888\pi$
$547$	−25.9126			−1.10794			−0.553972	+	0.832536i	$0.686888\pi$
$548$		0			0
$549$	3.52647			0.150506
$550$		0			0
$551$	4.99356	+	0.369945i	0.212733	+	0.0157602i
$552$		0			0
$553$		−	5.50973i		−	0.234298i
$554$		0			0
$555$			24.0973i			1.02287i
$556$		0			0
$557$	31.8569			1.34982			0.674910	−	0.737900i	$-0.264183\pi$
$557$	31.8569			1.34982			0.674910	+	0.737900i	$0.264183\pi$
$558$		0			0
$559$	54.7980			2.31771
$560$		0			0
$561$			11.2248i			0.473913i
$562$		0			0
$563$	7.76178			0.327120			0.163560	−	0.986533i	$-0.447702\pi$
$563$	7.76178			0.327120			0.163560	+	0.986533i	$0.447702\pi$
$564$		0			0
$565$			3.76901i			0.158564i
$566$		0			0
$567$		−	14.7834i		−	0.620845i
$568$		0			0
$569$			5.24153i			0.219736i	0.993946	+	0.109868i	$0.0350429\pi$
$569$			5.24153i			0.219736i	−0.993946	+	0.109868i	$0.964957\pi$
$570$		0			0
$571$			8.03573i			0.336285i	0.985763	+	0.168142i	$0.0537769\pi$
$571$			8.03573i			0.336285i	−0.985763	+	0.168142i	$0.946223\pi$
$572$		0			0
$573$			20.9994i			0.877263i
$574$		0			0
$575$		−	17.4423i		−	0.727395i
$576$		0			0
$577$	1.74173			0.0725090			0.0362545	−	0.999343i	$-0.488457\pi$
$577$	1.74173			0.0725090			0.0362545	+	0.999343i	$0.488457\pi$
$578$		0			0
$579$			14.7956i			0.614884i
$580$		0			0
$581$	10.3555			0.429620
$582$		0			0
$583$	−6.58207			−0.272602
$584$		0			0
$585$		−	13.4864i		−	0.557593i
$586$		0			0
$587$			41.4088i			1.70912i	0.519350	+	0.854562i	$0.326174\pi$
$587$			41.4088i			1.70912i	−0.519350	+	0.854562i	$0.673826\pi$
$588$		0			0
$589$	−1.20864	+	16.3143i	−0.0498011	+	0.672220i
$590$		0			0
$591$	16.8784			0.694284
$592$		0			0
$593$	−3.71806			−0.152682			−0.0763411	−	0.997082i	$-0.524324\pi$
$593$	−3.71806			−0.152682			−0.0763411	+	0.997082i	$0.524324\pi$
$594$		0			0
$595$			7.70926i			0.316049i
$596$		0			0
$597$			41.8599i			1.71321i
$598$		0			0
$599$	−15.2624			−0.623606			−0.311803	−	0.950147i	$-0.600933\pi$
$599$	−15.2624			−0.623606			−0.311803	+	0.950147i	$0.600933\pi$
$600$		0			0
$601$		−	48.3234i		−	1.97115i	−0.169232	−	0.985576i	$-0.554129\pi$
$601$		−	48.3234i		−	1.97115i	0.169232	−	0.985576i	$-0.445871\pi$
$602$		0			0
$603$	0.970476			0.0395208
$604$		0			0
$605$	1.12654			0.0458005
$606$		0			0
$607$	−22.7236			−0.922322			−0.461161	−	0.887316i	$-0.652567\pi$
$607$	−22.7236			−0.922322			−0.461161	+	0.887316i	$0.652567\pi$
$608$		0			0
$609$	3.27050			0.132527
$610$		0			0
$611$	−52.4535			−2.12204
$612$		0			0
$613$	24.2165			0.978094			0.489047	−	0.872257i	$-0.337345\pi$
$613$	24.2165			0.978094			0.489047	+	0.872257i	$0.337345\pi$
$614$		0			0
$615$			6.93378i			0.279597i
$616$		0			0
$617$	27.7350			1.11657			0.558285	−	0.829649i	$-0.311459\pi$
$617$	27.7350			1.11657			0.558285	+	0.829649i	$0.311459\pi$
$618$		0			0
$619$			26.2236i			1.05402i	0.849860	+	0.527009i	$0.176686\pi$
$619$			26.2236i			1.05402i	−0.849860	+	0.527009i	$0.823314\pi$
$620$		0			0
$621$			13.4378i			0.539240i
$622$		0			0
$623$	−5.49499			−0.220152
$624$		0			0
$625$	7.57410			0.302964
$626$		0			0
$627$	−9.39382	−	0.695936i	−0.375153	−	0.0277930i
$628$		0			0
$629$		−	51.4152i		−	2.05006i
$630$		0			0
$631$		−	11.5532i		−	0.459925i	−0.973199	−	0.229963i	$-0.926140\pi$
$631$		−	11.5532i		−	0.459925i	0.973199	−	0.229963i	$-0.0738604\pi$
$632$		0			0
$633$	−13.0220			−0.517579
$634$		0			0
$635$	16.2547			0.645050
$636$		0			0
$637$			37.7394i			1.49529i
$638$		0			0
$639$	5.74722			0.227357
$640$		0			0
$641$			26.6987i			1.05453i	0.849700	+	0.527267i	$0.176783\pi$
$641$			26.6987i			1.05453i	−0.849700	+	0.527267i	$0.823217\pi$
$642$		0			0
$643$			5.16292i			0.203606i	0.994805	+	0.101803i	$0.0324611\pi$
$643$			5.16292i			0.203606i	−0.994805	+	0.101803i	$0.967539\pi$
$644$		0			0
$645$		−	18.6085i		−	0.732709i
$646$		0			0
$647$		−	46.6948i		−	1.83576i	−0.396857	−	0.917880i	$-0.629899\pi$
$647$		−	46.6948i		−	1.83576i	0.396857	−	0.917880i	$-0.370101\pi$
$648$		0			0
$649$			12.8125i			0.502933i
$650$		0			0
$651$			10.6850i			0.418777i
$652$		0			0
$653$	31.1128			1.21754			0.608769	−	0.793347i	$-0.291664\pi$
$653$	31.1128			1.21754			0.608769	+	0.793347i	$0.291664\pi$
$654$		0			0
$655$		−	3.09499i		−	0.120931i
$656$		0			0
$657$	−12.8582			−0.501647
$658$		0			0
$659$	−1.27840			−0.0497994			−0.0248997	−	0.999690i	$-0.507927\pi$
$659$	−1.27840			−0.0497994			−0.0248997	+	0.999690i	$0.507927\pi$
$660$		0			0
$661$		−	40.7377i		−	1.58451i	−0.610189	−	0.792256i	$-0.708907\pi$
$661$		−	40.7377i		−	1.58451i	0.610189	−	0.792256i	$-0.291093\pi$
$662$		0			0
$663$			80.4703i			3.12521i
$664$		0			0
$665$	−6.45171	−	0.477971i	−0.250186	−	0.0185349i
$666$		0			0
$667$	−5.37047			−0.207946
$668$		0			0
$669$	49.6491			1.91954
$670$		0			0
$671$			2.11178i			0.0815243i
$672$		0			0
$673$			34.0935i			1.31421i	0.753799	+	0.657105i	$0.228219\pi$
$673$			34.0935i			1.31421i	−0.753799	+	0.657105i	$0.771781\pi$
$674$		0			0
$675$	−10.7238			−0.412761
$676$		0			0
$677$			19.2280i			0.738992i	0.929232	+	0.369496i	$0.120470\pi$
$677$			19.2280i			0.738992i	−0.929232	+	0.369496i	$0.879530\pi$
$678$		0			0
$679$	6.11112			0.234523
$680$		0			0
$681$	−45.1790			−1.73126
$682$		0			0
$683$	−11.0580			−0.423123			−0.211562	−	0.977365i	$-0.567855\pi$
$683$	−11.0580			−0.423123			−0.211562	+	0.977365i	$0.567855\pi$
$684$		0			0
$685$	6.62958			0.253303
$686$		0			0
$687$	3.07350			0.117261
$688$		0			0
$689$	−47.1865			−1.79766
$690$		0			0
$691$		−	27.7366i		−	1.05515i	−0.849508	−	0.527575i	$-0.823101\pi$
$691$		−	27.7366i		−	1.05515i	0.849508	−	0.527575i	$-0.176899\pi$
$692$		0			0
$693$	−2.20004			−0.0835724
$694$		0			0
$695$			17.4924i			0.663526i
$696$		0			0
$697$		−	14.7943i		−	0.560374i
$698$		0			0
$699$	−47.2666			−1.78779
$700$		0			0
$701$	4.53110			0.171137			0.0855686	−	0.996332i	$-0.472729\pi$
$701$	4.53110			0.171137			0.0855686	+	0.996332i	$0.472729\pi$
$702$		0			0
$703$	43.0282	+	3.18772i	1.62284	+	0.120227i
$704$		0			0
$705$			17.8124i			0.670852i
$706$		0			0
$707$		−	6.41631i		−	0.241310i
$708$		0			0
$709$	−22.4048			−0.841432			−0.420716	−	0.907192i	$-0.638221\pi$
$709$	−22.4048			−0.841432			−0.420716	+	0.907192i	$0.638221\pi$
$710$		0			0
$711$	−6.98367			−0.261908
$712$		0			0
$713$		−	17.5458i		−	0.657094i
$714$		0			0
$715$	8.07613			0.302030
$716$		0			0
$717$			35.5688i			1.32834i
$718$		0			0
$719$			37.3452i			1.39274i	0.717682	+	0.696371i	$0.245203\pi$
$719$			37.3452i			1.39274i	−0.717682	+	0.696371i	$0.754797\pi$
$720$		0			0
$721$		−	11.3953i		−	0.424384i
$722$		0			0
$723$			55.8209i			2.07600i
$724$		0			0
$725$		−	4.28583i		−	0.159172i
$726$		0			0
$727$		−	27.2034i		−	1.00892i	−0.863435	−	0.504460i	$-0.831692\pi$
$727$		−	27.2034i		−	1.00892i	0.863435	−	0.504460i	$-0.168308\pi$
$728$		0			0
$729$	−0.103944			−0.00384979
$730$		0			0
$731$			39.7041i			1.46851i
$732$		0			0
$733$	13.0356			0.481481			0.240741	−	0.970589i	$-0.422610\pi$
$733$	13.0356			0.481481			0.240741	+	0.970589i	$0.422610\pi$
$734$		0			0
$735$	12.8157			0.472714
$736$		0			0
$737$			0.581157i			0.0214072i
$738$		0			0
$739$		−	40.8547i		−	1.50286i	−0.659810	−	0.751432i	$-0.729363\pi$
$739$		−	40.8547i		−	1.50286i	0.659810	−	0.751432i	$-0.270637\pi$
$740$		0			0
$741$	−67.3438	−	4.98912i	−2.47394	−	0.183280i
$742$		0			0
$743$	41.2171			1.51211			0.756054	−	0.654509i	$-0.227124\pi$
$743$	41.2171			1.51211			0.756054	+	0.654509i	$0.227124\pi$
$744$		0			0
$745$	−23.6903			−0.867945
$746$		0			0
$747$		−	13.1258i		−	0.480248i
$748$		0			0
$749$		−	5.11277i		−	0.186816i
$750$		0			0
$751$	−37.1533			−1.35574			−0.677872	−	0.735180i	$-0.737098\pi$
$751$	−37.1533			−1.35574			−0.677872	+	0.735180i	$0.737098\pi$
$752$		0			0
$753$			42.5973i			1.55233i
$754$		0			0
$755$	−10.9884			−0.399908
$756$		0			0
$757$	−5.17379			−0.188045			−0.0940223	−	0.995570i	$-0.529973\pi$
$757$	−5.17379			−0.188045			−0.0940223	+	0.995570i	$0.529973\pi$
$758$		0			0
$759$	10.1029			0.366711
$760$		0			0
$761$	−45.1457			−1.63653			−0.818266	−	0.574840i	$-0.805064\pi$
$761$	−45.1457			−1.63653			−0.818266	+	0.574840i	$0.805064\pi$
$762$		0			0
$763$	2.13969			0.0774620
$764$		0			0
$765$	9.77161			0.353293
$766$		0			0
$767$			91.8519i			3.31658i
$768$		0			0
$769$	35.8023			1.29106			0.645532	−	0.763733i	$-0.276636\pi$
$769$	35.8023			1.29106			0.645532	+	0.763733i	$0.276636\pi$
$770$		0			0
$771$		−	12.0946i		−	0.435575i
$772$		0			0
$773$			30.2891i			1.08942i	0.838624	+	0.544711i	$0.183361\pi$
$773$			30.2891i			1.08942i	−0.838624	+	0.544711i	$0.816639\pi$
$774$		0			0
$775$	14.0021			0.502972
$776$		0			0
$777$	28.1811			1.01099
$778$		0			0
$779$	12.3810	+	0.917240i	0.443596	+	0.0328635i
$780$		0			0
$781$			3.44165i			0.123152i
$782$		0			0
$783$			3.30186i			0.117999i
$784$		0			0
$785$	20.3098			0.724886
$786$		0			0
$787$	−6.56422			−0.233989			−0.116995	−	0.993133i	$-0.537326\pi$
$787$	−6.56422			−0.233989			−0.116995	+	0.993133i	$0.537326\pi$
$788$		0			0
$789$		−	45.0099i		−	1.60239i
$790$		0			0
$791$	4.40776			0.156722
$792$		0			0
$793$			15.1392i			0.537610i
$794$		0			0
$795$			16.0238i			0.568305i
$796$		0			0
$797$		−	31.5063i		−	1.11601i	−0.829838	−	0.558005i	$-0.811567\pi$
$797$		−	31.5063i		−	1.11601i	0.829838	−	0.558005i	$-0.188433\pi$
$798$		0			0
$799$		−	38.0054i		−	1.34454i
$800$		0			0
$801$			6.96499i			0.246096i
$802$		0			0
$803$		−	7.69997i		−	0.271726i
$804$		0			0
$805$	6.93869			0.244557
$806$		0			0
$807$			44.0215i			1.54963i
$808$		0			0
$809$	48.1906			1.69429			0.847146	−	0.531360i	$-0.178319\pi$
$809$	48.1906			1.69429			0.847146	+	0.531360i	$0.178319\pi$
$810$		0			0
$811$	−29.7246			−1.04377			−0.521885	−	0.853016i	$-0.674771\pi$
$811$	−29.7246			−1.04377			−0.521885	+	0.853016i	$0.674771\pi$
$812$		0			0
$813$		−	23.4754i		−	0.823320i
$814$		0			0
$815$		−	3.44243i		−	0.120583i
$816$		0			0
$817$	−33.2275	−	2.46164i	−1.16248	−	0.0861218i
$818$		0			0
$819$	−15.7719			−0.551116
$820$		0			0
$821$	28.5258			0.995556			0.497778	−	0.867304i	$-0.334149\pi$
$821$	28.5258			0.995556			0.497778	+	0.867304i	$0.334149\pi$
$822$		0			0
$823$		−	7.75616i		−	0.270363i	−0.990821	−	0.135181i	$-0.956838\pi$
$823$		−	7.75616i		−	0.270363i	0.990821	−	0.135181i	$-0.0431617\pi$
$824$		0			0
$825$			8.06246i			0.280699i
$826$		0			0
$827$	−34.4596			−1.19828			−0.599139	−	0.800645i	$-0.704491\pi$
$827$	−34.4596			−1.19828			−0.599139	+	0.800645i	$0.704491\pi$
$828$		0			0
$829$			16.2894i			0.565755i	0.959156	+	0.282878i	$0.0912891\pi$
$829$			16.2894i			0.565755i	−0.959156	+	0.282878i	$0.908711\pi$
$830$		0			0
$831$	−1.37046			−0.0475409
$832$		0			0
$833$	−27.3443			−0.947423
$834$		0			0
$835$	9.34764			0.323488
$836$		0			0
$837$	−10.7874			−0.372868
$838$		0			0
$839$	3.36741			0.116256			0.0581279	−	0.998309i	$-0.481487\pi$
$839$	3.36741			0.116256			0.0581279	+	0.998309i	$0.481487\pi$
$840$		0			0
$841$	27.6804			0.954496
$842$		0			0
$843$		−	49.3875i		−	1.70100i
$844$		0			0
$845$	43.2523			1.48792
$846$		0			0
$847$		−	1.31746i		−	0.0452685i
$848$		0			0
$849$			20.6389i			0.708327i
$850$		0			0
$851$	−46.2761			−1.58632
$852$		0			0
$853$	49.7655			1.70394			0.851969	−	0.523592i	$-0.175408\pi$
$853$	49.7655			1.70394			0.851969	+	0.523592i	$0.175408\pi$
$854$		0			0
$855$	−0.605835	+	8.17764i	−0.0207191	+	0.279669i
$856$		0			0
$857$			23.0898i			0.788732i	0.918953	+	0.394366i	$0.129036\pi$
$857$			23.0898i			0.788732i	−0.918953	+	0.394366i	$0.870964\pi$
$858$		0			0
$859$		−	4.34393i		−	0.148213i	−0.997250	−	0.0741066i	$-0.976390\pi$
$859$		−	4.34393i		−	0.148213i	0.997250	−	0.0741066i	$-0.0236105\pi$
$860$		0			0
$861$	8.10886			0.276349
$862$		0			0
$863$	47.7761			1.62632			0.813159	−	0.582042i	$-0.197746\pi$
$863$	47.7761			1.62632			0.813159	+	0.582042i	$0.197746\pi$
$864$		0			0
$865$			7.79789i			0.265136i
$866$		0			0
$867$	−21.5682			−0.732496
$868$		0			0
$869$		−	4.18208i		−	0.141867i
$870$		0			0
$871$			4.16628i			0.141169i
$872$		0			0
$873$		−	7.74593i		−	0.262160i
$874$		0			0
$875$			12.9582i			0.438068i
$876$		0			0
$877$		−	16.3168i		−	0.550979i	−0.961304	−	0.275490i	$-0.911160\pi$
$877$		−	16.3168i		−	0.550979i	0.961304	−	0.275490i	$-0.0888400\pi$
$878$		0			0
$879$			12.0356i			0.405950i
$880$		0			0
$881$	−54.4910			−1.83585			−0.917925	−	0.396755i	$-0.870136\pi$
$881$	−54.4910			−1.83585			−0.917925	+	0.396755i	$0.870136\pi$
$882$		0			0
$883$			20.9224i			0.704096i	0.935982	+	0.352048i	$0.114515\pi$
$883$			20.9224i			0.704096i	−0.935982	+	0.352048i	$0.885485\pi$
$884$		0			0
$885$	31.1914			1.04849
$886$		0			0
$887$	44.3979			1.49074			0.745368	−	0.666653i	$-0.232274\pi$
$887$	44.3979			1.49074			0.745368	+	0.666653i	$0.232274\pi$
$888$		0			0
$889$		−	19.0095i		−	0.637557i
$890$		0			0
$891$		−	11.2211i		−	0.375922i
$892$		0			0
$893$	31.8059	+	2.35632i	1.06434	+	0.0788513i
$894$		0			0
$895$	8.44177			0.282177
$896$		0			0
$897$	72.4270			2.41827
$898$		0			0
$899$		−	4.31125i		−	0.143788i
$900$		0			0
$901$		−	34.1892i		−	1.13901i
$902$		0			0
$903$	−21.7621			−0.724198
$904$		0			0
$905$		−	6.70083i		−	0.222743i
$906$		0			0
$907$	23.4091			0.777287			0.388643	−	0.921388i	$-0.372944\pi$
$907$	23.4091			0.777287			0.388643	+	0.921388i	$0.372944\pi$
$908$		0			0
$909$	−8.13277			−0.269747
$910$		0			0
$911$	−55.9336			−1.85316			−0.926582	−	0.376094i	$-0.877267\pi$
$911$	−55.9336			−1.85316			−0.926582	+	0.376094i	$0.877267\pi$
$912$		0			0
$913$	7.86021			0.260135
$914$		0			0
$915$	5.14103			0.169957
$916$		0			0
$917$	−3.61951			−0.119527
$918$		0			0
$919$			24.7952i			0.817917i	0.912553	+	0.408959i	$0.134108\pi$
$919$			24.7952i			0.817917i	−0.912553	+	0.408959i	$0.865892\pi$
$920$		0			0
$921$	43.1680			1.42243
$922$		0			0
$923$			24.6730i			0.812122i
$924$		0			0
$925$		−	36.9300i		−	1.21425i
$926$		0			0
$927$	−14.4437			−0.474395
$928$		0			0
$929$	−17.4418			−0.572245			−0.286123	−	0.958193i	$-0.592366\pi$
$929$	−17.4418			−0.572245			−0.286123	+	0.958193i	$0.592366\pi$
$930$		0			0
$931$	1.69533	−	22.8838i	0.0555623	−	0.749986i
$932$		0			0
$933$			3.20830i			0.105035i
$934$		0			0
$935$			5.85160i			0.191368i
$936$		0			0
$937$	−31.7219			−1.03631			−0.518154	−	0.855287i	$-0.673381\pi$
$937$	−31.7219			−1.03631			−0.518154	+	0.855287i	$0.673381\pi$
$938$		0			0
$939$	43.2946			1.41286
$940$		0			0
$941$		−	25.9353i		−	0.845468i	−0.906254	−	0.422734i	$-0.861071\pi$
$941$		−	25.9353i		−	0.845468i	0.906254	−	0.422734i	$-0.138929\pi$
$942$		0			0
$943$	−13.3155			−0.433614
$944$		0			0
$945$		−	4.26603i		−	0.138774i
$946$		0			0
$947$			39.3326i			1.27814i	0.769149	+	0.639069i	$0.220680\pi$
$947$			39.3326i			1.27814i	−0.769149	+	0.639069i	$0.779320\pi$
$948$		0			0
$949$		−	55.2007i		−	1.79189i
$950$		0			0
$951$			16.2986i			0.528519i
$952$		0			0
$953$			32.8172i			1.06305i	0.847041	+	0.531527i	$0.178382\pi$
$953$			32.8172i			1.06305i	−0.847041	+	0.531527i	$0.821618\pi$
$954$		0			0
$955$			10.9472i			0.354242i
$956$		0			0
$957$	2.48242			0.0802453
$958$		0			0
$959$		−	7.75311i		−	0.250361i
$960$		0			0
$961$	−16.9148			−0.545639
$962$		0			0
$963$	−6.48051			−0.208831
$964$		0			0
$965$			7.71305i			0.248292i
$966$		0			0
$967$		−	46.0017i		−	1.47932i	−0.672983	−	0.739658i	$-0.734987\pi$
$967$		−	46.0017i		−	1.47932i	0.672983	−	0.739658i	$-0.265013\pi$
$968$		0			0
$969$	3.61489	−	48.7943i	0.116127	−	1.56750i
$970$		0			0
$971$	19.7085			0.632474			0.316237	−	0.948680i	$-0.397580\pi$
$971$	19.7085			0.632474			0.316237	+	0.948680i	$0.397580\pi$
$972$		0			0
$973$	20.4569			0.655819
$974$		0			0
$975$			57.7993i			1.85106i
$976$		0			0
$977$		−	37.8818i		−	1.21195i	−0.795485	−	0.605973i	$-0.792784\pi$
$977$		−	37.8818i		−	1.21195i	0.795485	−	0.605973i	$-0.207216\pi$
$978$		0			0
$979$	−4.17089			−0.133302
$980$		0			0
$981$		−	2.71209i		−	0.0865904i
$982$		0			0
$983$	−26.3524			−0.840511			−0.420256	−	0.907406i	$-0.638060\pi$
$983$	−26.3524			−0.840511			−0.420256	+	0.907406i	$0.638060\pi$
$984$		0			0
$985$	8.79883			0.280354
$986$		0			0
$987$	20.8311			0.663060
$988$		0			0
$989$	35.7355			1.13632
$990$		0			0
$991$	−21.6340			−0.687227			−0.343614	−	0.939111i	$-0.611651\pi$
$991$	−21.6340			−0.687227			−0.343614	+	0.939111i	$0.611651\pi$
$992$		0			0
$993$	−72.0834			−2.28750
$994$		0			0
$995$			21.8219i			0.691800i
$996$		0			0
$997$	−10.0114			−0.317064			−0.158532	−	0.987354i	$-0.550676\pi$
$997$	−10.0114			−0.317064			−0.158532	+	0.987354i	$0.550676\pi$
$998$		0			0
$999$			28.4513i			0.900160i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.b.1519.12	yes	64
4.3	odd	2	inner	3344.2.o.b.1519.54	yes	64
19.18	odd	2	inner	3344.2.o.b.1519.53	yes	64
76.75	even	2	inner	3344.2.o.b.1519.11	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.b.1519.11	✓	64	76.75	even	2	inner
3344.2.o.b.1519.12	yes	64	1.1	even	1	trivial
3344.2.o.b.1519.53	yes	64	19.18	odd	2	inner
3344.2.o.b.1519.54	yes	64	4.3	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.16100 q^{3} -1.12654 q^{5} +1.31746i q^{7} +1.66990 q^{9} +O(q^{10})\)	\(q-2.16100 q^{3} -1.12654 q^{5} +1.31746i q^{7} +1.66990 q^{9} +1.00000i q^{11} +7.16895i q^{13} +2.43446 q^{15} -5.19429 q^{17} +(0.322044 - 4.34699i) q^{19} -2.84703i q^{21} +4.67510i q^{23} -3.73090 q^{25} +2.87433 q^{27} +1.14874i q^{29} -3.75302 q^{31} -2.16100i q^{33} -1.48418i q^{35} +9.89841i q^{37} -15.4921i q^{39} +2.84818i q^{41} -7.64380i q^{43} -1.88122 q^{45} +7.31677i q^{47} +5.26429 q^{49} +11.2248 q^{51} +6.58207i q^{53} -1.12654i q^{55} +(-0.695936 + 9.39382i) q^{57} +12.8125 q^{59} +2.11178 q^{61} +2.20004i q^{63} -8.07613i q^{65} +0.581157 q^{67} -10.1029i q^{69} +3.44165 q^{71} -7.69997 q^{73} +8.06246 q^{75} -1.31746 q^{77} -4.18208 q^{79} -11.2211 q^{81} -7.86021i q^{83} +5.85160 q^{85} -2.48242i q^{87} +4.17089i q^{89} -9.44482 q^{91} +8.11027 q^{93} +(-0.362797 + 4.89707i) q^{95} -4.63855i q^{97} +1.66990i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 56 q^{9}+O(q^{10}) \) 64 * q + 56 * q^9	\( 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) \) 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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