LMFDB - Embedded newform 3344.2.o.a.1519.14

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:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.14
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.a.1519.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.963498 q^{3} +3.73336 q^{5} -0.145369i q^{7} -2.07167 q^{9} +O(q^{10})$	$q-0.963498 q^{3} +3.73336 q^{5} -0.145369i q^{7} -2.07167 q^{9} +1.00000i q^{11} +5.37224i q^{13} -3.59709 q^{15} +1.20440 q^{17} +(-3.41798 - 2.70507i) q^{19} +0.140063i q^{21} -6.87893i q^{23} +8.93797 q^{25} +4.88655 q^{27} -1.91582i q^{29} +6.32456 q^{31} -0.963498i q^{33} -0.542716i q^{35} +8.26609i q^{37} -5.17614i q^{39} +7.55484i q^{41} +3.30043i q^{43} -7.73429 q^{45} +9.21781i q^{47} +6.97887 q^{49} -1.16044 q^{51} +0.682633i q^{53} +3.73336i q^{55} +(3.29322 + 2.60633i) q^{57} -10.4693 q^{59} +2.20573 q^{61} +0.301157i q^{63} +20.0565i q^{65} +8.98497 q^{67} +6.62784i q^{69} +10.6511 q^{71} +9.94951 q^{73} -8.61172 q^{75} +0.145369 q^{77} -16.0101 q^{79} +1.50683 q^{81} +1.70000i q^{83} +4.49647 q^{85} +1.84589i q^{87} +15.4603i q^{89} +0.780958 q^{91} -6.09370 q^{93} +(-12.7606 - 10.0990i) q^{95} -8.95894i q^{97} -2.07167i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 44 q^{9}+O(q^{10}) $ 36 * q + 44 * q^9	$ 36 q + 44 q^{9} - 16 q^{17} + 36 q^{25} - 32 q^{45} - 28 q^{49} + 24 q^{57} - 48 q^{61} - 24 q^{73} + 52 q^{81} + 24 q^{85} - 64 q^{93}+O(q^{100}) $ 36 * q + 44 * q^9 - 16 * q^17 + 36 * q^25 - 32 * q^45 - 28 * q^49 + 24 * q^57 - 48 * q^61 - 24 * q^73 + 52 * q^81 + 24 * q^85 - 64 * 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
$3$	−0.963498			−0.556276			−0.278138	−	0.960541i	$-0.589717\pi$
$3$	−0.963498			−0.556276			−0.278138	+	0.960541i	$0.589717\pi$
$4$		0			0
$5$	3.73336			1.66961			0.834804	−	0.550547i	$-0.185581\pi$
$5$	3.73336			1.66961			0.834804	+	0.550547i	$0.185581\pi$
$6$		0			0
$7$		−	0.145369i		−	0.0549444i	−0.999623	−	0.0274722i	$-0.991254\pi$
$7$		−	0.145369i		−	0.0549444i	0.999623	−	0.0274722i	$-0.00874578\pi$
$8$		0			0
$9$	−2.07167			−0.690557
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			5.37224i			1.48999i	0.667070	+	0.744995i	$0.267548\pi$
$13$			5.37224i			1.48999i	−0.667070	+	0.744995i	$0.732452\pi$
$14$		0			0
$15$	−3.59709			−0.928763
$16$		0			0
$17$	1.20440			0.292111			0.146055	−	0.989276i	$-0.453342\pi$
$17$	1.20440			0.292111			0.146055	+	0.989276i	$0.453342\pi$
$18$		0			0
$19$	−3.41798	−	2.70507i	−0.784139	−	0.620585i
$19$	−3.41798	−	2.70507i	−0.784139	−	0.620585i
$20$		0			0
$21$			0.140063i			0.0305643i
$22$		0			0
$23$		−	6.87893i		−	1.43436i	−0.696890	−	0.717178i	$-0.745433\pi$
$23$		−	6.87893i		−	1.43436i	0.696890	−	0.717178i	$-0.254567\pi$
$24$		0			0
$25$	8.93797			1.78759
$26$		0			0
$27$	4.88655			0.940416
$28$		0			0
$29$		−	1.91582i		−	0.355758i	−0.984052	−	0.177879i	$-0.943076\pi$
$29$		−	1.91582i		−	0.355758i	0.984052	−	0.177879i	$-0.0569236\pi$
$30$		0			0
$31$	6.32456			1.13592			0.567962	−	0.823055i	$-0.307732\pi$
$31$	6.32456			1.13592			0.567962	+	0.823055i	$0.307732\pi$
$32$		0			0
$33$		−	0.963498i		−	0.167724i
$34$		0			0
$35$		−	0.542716i		−	0.0917357i
$36$		0			0
$37$			8.26609i			1.35894i	0.733705	+	0.679469i	$0.237790\pi$
$37$			8.26609i			1.35894i	−0.733705	+	0.679469i	$0.762210\pi$
$38$		0			0
$39$		−	5.17614i		−	0.828846i
$40$		0			0
$41$			7.55484i			1.17987i	0.807451	+	0.589934i	$0.200846\pi$
$41$			7.55484i			1.17987i	−0.807451	+	0.589934i	$0.799154\pi$
$42$		0			0
$43$			3.30043i			0.503311i	0.967817	+	0.251655i	$0.0809750\pi$
$43$			3.30043i			0.503311i	−0.967817	+	0.251655i	$0.919025\pi$
$44$		0			0
$45$	−7.73429			−1.15296
$46$		0			0
$47$			9.21781i			1.34456i	0.740299	+	0.672278i	$0.234684\pi$
$47$			9.21781i			1.34456i	−0.740299	+	0.672278i	$0.765316\pi$
$48$		0			0
$49$	6.97887			0.996981
$50$		0			0
$51$	−1.16044			−0.162494
$52$		0			0
$53$			0.682633i			0.0937668i	0.998900	+	0.0468834i	$0.0149289\pi$
$53$			0.682633i			0.0937668i	−0.998900	+	0.0468834i	$0.985071\pi$
$54$		0			0
$55$			3.73336i			0.503406i
$56$		0			0
$57$	3.29322	+	2.60633i	0.436198	+	0.345217i
$58$		0			0
$59$	−10.4693			−1.36299			−0.681493	−	0.731825i	$-0.738669\pi$
$59$	−10.4693			−1.36299			−0.681493	+	0.731825i	$0.738669\pi$
$60$		0			0
$61$	2.20573			0.282415			0.141208	−	0.989980i	$-0.454901\pi$
$61$	2.20573			0.282415			0.141208	+	0.989980i	$0.454901\pi$
$62$		0			0
$63$			0.301157i			0.0379423i
$64$		0			0
$65$			20.0565i			2.48770i
$66$		0			0
$67$	8.98497			1.09769			0.548844	−	0.835925i	$-0.315068\pi$
$67$	8.98497			1.09769			0.548844	+	0.835925i	$0.315068\pi$
$68$		0			0
$69$			6.62784i			0.797898i
$70$		0			0
$71$	10.6511			1.26405			0.632026	−	0.774947i	$-0.282223\pi$
$71$	10.6511			1.26405			0.632026	+	0.774947i	$0.282223\pi$
$72$		0			0
$73$	9.94951			1.16450			0.582251	−	0.813009i	$-0.302172\pi$
$73$	9.94951			1.16450			0.582251	+	0.813009i	$0.302172\pi$
$74$		0			0
$75$	−8.61172			−0.994396
$76$		0			0
$77$	0.145369			0.0165664
$78$		0			0
$79$	−16.0101			−1.80127			−0.900637	−	0.434572i	$-0.856900\pi$
$79$	−16.0101			−1.80127			−0.900637	+	0.434572i	$0.856900\pi$
$80$		0			0
$81$	1.50683			0.167426
$82$		0			0
$83$			1.70000i			0.186599i	0.995638	+	0.0932994i	$0.0297414\pi$
$83$			1.70000i			0.186599i	−0.995638	+	0.0932994i	$0.970259\pi$
$84$		0			0
$85$	4.49647			0.487711
$86$		0			0
$87$			1.84589i			0.197900i
$88$		0			0
$89$			15.4603i			1.63878i	0.573233	+	0.819392i	$0.305689\pi$
$89$			15.4603i			1.63878i	−0.573233	+	0.819392i	$0.694311\pi$
$90$		0			0
$91$	0.780958			0.0818667
$92$		0			0
$93$	−6.09370			−0.631887
$94$		0			0
$95$	−12.7606	−	10.0990i	−1.30921	−	1.03613i
$96$		0			0
$97$		−	8.95894i		−	0.909642i	−0.890583	−	0.454821i	$-0.849703\pi$
$97$		−	8.95894i		−	0.909642i	0.890583	−	0.454821i	$-0.150297\pi$
$98$		0			0
$99$		−	2.07167i		−	0.208211i
$100$		0			0
$101$	−1.54717			−0.153949			−0.0769747	−	0.997033i	$-0.524526\pi$
$101$	−1.54717			−0.153949			−0.0769747	+	0.997033i	$0.524526\pi$
$102$		0			0
$103$	−2.01938			−0.198976			−0.0994878	−	0.995039i	$-0.531720\pi$
$103$	−2.01938			−0.198976			−0.0994878	+	0.995039i	$0.531720\pi$
$104$		0			0
$105$			0.522906i			0.0510304i
$106$		0			0
$107$	8.48284			0.820067			0.410033	−	0.912071i	$-0.365517\pi$
$107$	8.48284			0.820067			0.410033	+	0.912071i	$0.365517\pi$
$108$		0			0
$109$		−	9.93472i		−	0.951574i	−0.879561	−	0.475787i	$-0.842163\pi$
$109$		−	9.93472i		−	0.951574i	0.879561	−	0.475787i	$-0.157837\pi$
$110$		0			0
$111$		−	7.96437i		−	0.755944i
$112$		0			0
$113$		−	3.07754i		−	0.289511i	−0.989467	−	0.144755i	$-0.953761\pi$
$113$		−	3.07754i		−	0.289511i	0.989467	−	0.144755i	$-0.0462395\pi$
$114$		0			0
$115$		−	25.6815i		−	2.39482i
$116$		0			0
$117$		−	11.1295i		−	1.02892i
$118$		0			0
$119$		−	0.175083i		−	0.0160499i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	7.27908i		−	0.656332i
$124$		0			0
$125$	14.7019			1.31497
$126$		0			0
$127$	−3.41389			−0.302933			−0.151467	−	0.988462i	$-0.548400\pi$
$127$	−3.41389			−0.302933			−0.151467	+	0.988462i	$0.548400\pi$
$128$		0			0
$129$		−	3.17996i		−	0.279980i
$130$		0			0
$131$			19.5730i			1.71010i	0.518543	+	0.855052i	$0.326475\pi$
$131$			19.5730i			1.71010i	−0.518543	+	0.855052i	$0.673525\pi$
$132$		0			0
$133$	−0.393234	+	0.496870i	−0.0340977	+	0.0430841i
$134$		0			0
$135$	18.2432			1.57013
$136$		0			0
$137$	10.7217			0.916017			0.458008	−	0.888948i	$-0.348563\pi$
$137$	10.7217			0.916017			0.458008	+	0.888948i	$0.348563\pi$
$138$		0			0
$139$			17.0391i			1.44523i	0.691249	+	0.722617i	$0.257061\pi$
$139$			17.0391i			1.44523i	−0.691249	+	0.722617i	$0.742939\pi$
$140$		0			0
$141$		−	8.88135i		−	0.747944i
$142$		0			0
$143$	−5.37224			−0.449249
$144$		0			0
$145$		−	7.15243i		−	0.593977i
$146$		0			0
$147$	−6.72413			−0.554597
$148$		0			0
$149$	−6.53937			−0.535726			−0.267863	−	0.963457i	$-0.586318\pi$
$149$	−6.53937			−0.535726			−0.267863	+	0.963457i	$0.586318\pi$
$150$		0			0
$151$	−6.17015			−0.502119			−0.251060	−	0.967972i	$-0.580779\pi$
$151$	−6.17015			−0.502119			−0.251060	+	0.967972i	$0.580779\pi$
$152$		0			0
$153$	−2.49513			−0.201719
$154$		0			0
$155$	23.6118			1.89655
$156$		0			0
$157$	2.09106			0.166885			0.0834426	−	0.996513i	$-0.473408\pi$
$157$	2.09106			0.166885			0.0834426	+	0.996513i	$0.473408\pi$
$158$		0			0
$159$		−	0.657716i		−	0.0521602i
$160$		0			0
$161$	−0.999986			−0.0788099
$162$		0			0
$163$			12.5631i			0.984021i	0.870589	+	0.492011i	$0.163738\pi$
$163$			12.5631i			0.984021i	−0.870589	+	0.492011i	$0.836262\pi$
$164$		0			0
$165$		−	3.59709i		−	0.280033i
$166$		0			0
$167$	7.26947			0.562529			0.281264	−	0.959630i	$-0.409246\pi$
$167$	7.26947			0.562529			0.281264	+	0.959630i	$0.409246\pi$
$168$		0			0
$169$	−15.8609			−1.22007
$170$		0			0
$171$	7.08093	+	5.60401i	0.541493	+	0.428550i
$172$		0			0
$173$		−	6.12611i		−	0.465760i	−0.972505	−	0.232880i	$-0.925185\pi$
$173$		−	6.12611i		−	0.465760i	0.972505	−	0.232880i	$-0.0748149\pi$
$174$		0			0
$175$		−	1.29931i		−	0.0982183i
$176$		0			0
$177$	10.0871			0.758197
$178$		0			0
$179$	8.73434			0.652836			0.326418	−	0.945226i	$-0.394158\pi$
$179$	8.73434			0.652836			0.326418	+	0.945226i	$0.394158\pi$
$180$		0			0
$181$			14.1398i			1.05101i	0.850792	+	0.525503i	$0.176123\pi$
$181$			14.1398i			1.05101i	−0.850792	+	0.525503i	$0.823877\pi$
$182$		0			0
$183$	−2.12522			−0.157101
$184$		0			0
$185$			30.8603i			2.26889i
$186$		0			0
$187$			1.20440i			0.0880747i
$188$		0			0
$189$		−	0.710354i		−	0.0516706i
$190$		0			0
$191$		−	4.94417i		−	0.357748i	−0.983872	−	0.178874i	$-0.942755\pi$
$191$		−	4.94417i		−	0.357748i	0.983872	−	0.178874i	$-0.0572454\pi$
$192$		0			0
$193$		−	11.3723i		−	0.818599i	−0.912400	−	0.409299i	$-0.865773\pi$
$193$		−	11.3723i		−	0.818599i	0.912400	−	0.409299i	$-0.134227\pi$
$194$		0			0
$195$		−	19.3244i		−	1.38385i
$196$		0			0
$197$	4.31103			0.307148			0.153574	−	0.988137i	$-0.450922\pi$
$197$	4.31103			0.307148			0.153574	+	0.988137i	$0.450922\pi$
$198$		0			0
$199$			8.41752i			0.596702i	0.954456	+	0.298351i	$0.0964367\pi$
$199$			8.41752i			0.596702i	−0.954456	+	0.298351i	$0.903563\pi$
$200$		0			0
$201$	−8.65700			−0.610618
$202$		0			0
$203$	−0.278501			−0.0195469
$204$		0			0
$205$			28.2049i			1.96992i
$206$		0			0
$207$			14.2509i			0.990505i
$208$		0			0
$209$	2.70507	−	3.41798i	0.187114	−	0.236427i
$210$		0			0
$211$	13.7484			0.946481			0.473241	−	0.880933i	$-0.343084\pi$
$211$	13.7484			0.946481			0.473241	+	0.880933i	$0.343084\pi$
$212$		0			0
$213$	−10.2623			−0.703162
$214$		0			0
$215$			12.3217i			0.840332i
$216$		0			0
$217$		−	0.919396i		−	0.0624127i
$218$		0			0
$219$	−9.58634			−0.647785
$220$		0			0
$221$			6.47034i			0.435242i
$222$		0			0
$223$	10.6474			0.713005			0.356502	−	0.934294i	$-0.383969\pi$
$223$	10.6474			0.713005			0.356502	+	0.934294i	$0.383969\pi$
$224$		0			0
$225$	−18.5165			−1.23444
$226$		0			0
$227$	−25.6493			−1.70241			−0.851203	−	0.524837i	$-0.824126\pi$
$227$	−25.6493			−1.70241			−0.851203	+	0.524837i	$0.824126\pi$
$228$		0			0
$229$	−6.70881			−0.443331			−0.221665	−	0.975123i	$-0.571149\pi$
$229$	−6.70881			−0.443331			−0.221665	+	0.975123i	$0.571149\pi$
$230$		0			0
$231$	−0.140063			−0.00921548
$232$		0			0
$233$	5.41335			0.354640			0.177320	−	0.984153i	$-0.443257\pi$
$233$	5.41335			0.354640			0.177320	+	0.984153i	$0.443257\pi$
$234$		0			0
$235$			34.4134i			2.24488i
$236$		0			0
$237$	15.4257			1.00201
$238$		0			0
$239$		−	6.65412i		−	0.430420i	−0.976568	−	0.215210i	$-0.930956\pi$
$239$		−	6.65412i		−	0.430420i	0.976568	−	0.215210i	$-0.0690435\pi$
$240$		0			0
$241$		−	0.0157840i		−	0.00101674i	−1.00000		0.000508368i	$-0.999838\pi$
$241$		−	0.0157840i		−	0.00101674i	1.00000		0.000508368i	$-0.000161818\pi$
$242$		0			0
$243$	−16.1115			−1.03355
$244$		0			0
$245$	26.0546			1.66457
$246$		0			0
$247$	14.5323	−	18.3622i	0.924666	−	1.16836i
$248$		0			0
$249$		−	1.63794i		−	0.103800i
$250$		0			0
$251$			5.79427i			0.365731i	0.983138	+	0.182865i	$0.0585373\pi$
$251$			5.79427i			0.365731i	−0.983138	+	0.182865i	$0.941463\pi$
$252$		0			0
$253$	6.87893			0.432475
$254$		0			0
$255$	−4.33234			−0.271302
$256$		0			0
$257$		−	1.94936i		−	0.121597i	−0.998150	−	0.0607987i	$-0.980635\pi$
$257$		−	1.94936i		−	0.121597i	0.998150	−	0.0607987i	$-0.0193648\pi$
$258$		0			0
$259$	1.20164			0.0746660
$260$		0			0
$261$			3.96894i			0.245671i
$262$		0			0
$263$			17.0147i			1.04917i	0.851358	+	0.524585i	$0.175780\pi$
$263$			17.0147i			1.04917i	−0.851358	+	0.524585i	$0.824220\pi$
$264$		0			0
$265$			2.54851i			0.156554i
$266$		0			0
$267$		−	14.8959i		−	0.911617i
$268$		0			0
$269$		−	20.4050i		−	1.24412i	−0.782971	−	0.622058i	$-0.786297\pi$
$269$		−	20.4050i		−	1.24412i	0.782971	−	0.622058i	$-0.213703\pi$
$270$		0			0
$271$		−	24.0656i		−	1.46188i	−0.682442	−	0.730940i	$-0.739082\pi$
$271$		−	24.0656i		−	1.46188i	0.682442	−	0.730940i	$-0.260918\pi$
$272$		0			0
$273$	−0.752452			−0.0455405
$274$		0			0
$275$			8.93797i			0.538980i
$276$		0			0
$277$	10.3572			0.622303			0.311152	−	0.950360i	$-0.399285\pi$
$277$	10.3572			0.622303			0.311152	+	0.950360i	$0.399285\pi$
$278$		0			0
$279$	−13.1024			−0.784420
$280$		0			0
$281$			3.27046i			0.195100i	0.995231	+	0.0975498i	$0.0311005\pi$
$281$			3.27046i			0.195100i	−0.995231	+	0.0975498i	$0.968899\pi$
$282$		0			0
$283$			4.27629i			0.254199i	0.991890	+	0.127100i	$0.0405668\pi$
$283$			4.27629i			0.254199i	−0.991890	+	0.127100i	$0.959433\pi$
$284$		0			0
$285$	12.2948	+	9.73036i	0.728280	+	0.576377i
$286$		0			0
$287$	1.09824			0.0648272
$288$		0			0
$289$	−15.5494			−0.914671
$290$		0			0
$291$			8.63192i			0.506012i
$292$		0			0
$293$			29.7810i			1.73982i	0.493207	+	0.869912i	$0.335824\pi$
$293$			29.7810i			1.73982i	−0.493207	+	0.869912i	$0.664176\pi$
$294$		0			0
$295$	−39.0856			−2.27565
$296$		0			0
$297$			4.88655i			0.283546i
$298$		0			0
$299$	36.9553			2.13718
$300$		0			0
$301$	0.479781			0.0276541
$302$		0			0
$303$	1.49070			0.0856383
$304$		0			0
$305$	8.23479			0.471523
$306$		0			0
$307$	−25.0519			−1.42979			−0.714894	−	0.699232i	$-0.753525\pi$
$307$	−25.0519			−1.42979			−0.714894	+	0.699232i	$0.753525\pi$
$308$		0			0
$309$	1.94567			0.110685
$310$		0			0
$311$		−	30.9983i		−	1.75775i	−0.477051	−	0.878876i	$-0.658294\pi$
$311$		−	30.9983i		−	1.75775i	0.477051	−	0.878876i	$-0.341706\pi$
$312$		0			0
$313$	−1.14168			−0.0645318			−0.0322659	−	0.999479i	$-0.510272\pi$
$313$	−1.14168			−0.0645318			−0.0322659	+	0.999479i	$0.510272\pi$
$314$		0			0
$315$			1.12433i			0.0633487i
$316$		0			0
$317$			30.8739i			1.73405i	0.498263	+	0.867026i	$0.333971\pi$
$317$			30.8739i			1.73405i	−0.498263	+	0.867026i	$0.666029\pi$
$318$		0			0
$319$	1.91582			0.107265
$320$		0			0
$321$	−8.17320			−0.456183
$322$		0			0
$323$	−4.11663	−	3.25800i	−0.229055	−	0.181280i
$324$		0			0
$325$			48.0169i			2.66350i
$326$		0			0
$327$			9.57209i			0.529338i
$328$		0			0
$329$	1.33999			0.0738759
$330$		0			0
$331$	28.8624			1.58642			0.793212	−	0.608946i	$-0.208407\pi$
$331$	28.8624			1.58642			0.793212	+	0.608946i	$0.208407\pi$
$332$		0			0
$333$		−	17.1246i		−	0.938423i
$334$		0			0
$335$	33.5441			1.83271
$336$		0			0
$337$		−	30.6362i		−	1.66886i	−0.551115	−	0.834429i	$-0.685798\pi$
$337$		−	30.6362i		−	1.66886i	0.551115	−	0.834429i	$-0.314202\pi$
$338$		0			0
$339$			2.96520i			0.161048i
$340$		0			0
$341$			6.32456i			0.342494i
$342$		0			0
$343$		−	2.03210i		−	0.109723i
$344$		0			0
$345$			24.7441i			1.33218i
$346$		0			0
$347$		−	31.6230i		−	1.69761i	−0.528703	−	0.848807i	$-0.677322\pi$
$347$		−	31.6230i		−	1.69761i	0.528703	−	0.848807i	$-0.322678\pi$
$348$		0			0
$349$	27.6352			1.47928			0.739638	−	0.673005i	$-0.234997\pi$
$349$	27.6352			1.47928			0.739638	+	0.673005i	$0.234997\pi$
$350$		0			0
$351$			26.2517i			1.40121i
$352$		0			0
$353$	−16.5724			−0.882062			−0.441031	−	0.897492i	$-0.645387\pi$
$353$	−16.5724			−0.882062			−0.441031	+	0.897492i	$0.645387\pi$
$354$		0			0
$355$	39.7644			2.11047
$356$		0			0
$357$			0.168693i			0.00892816i
$358$		0			0
$359$		−	0.201482i		−	0.0106338i	−0.999986	−	0.00531690i	$-0.998308\pi$
$359$		−	0.201482i		−	0.0106338i	0.999986	−	0.00531690i	$-0.00169243\pi$
$360$		0			0
$361$	4.36521	+	18.4918i	0.229748	+	0.973250i
$362$		0			0
$363$	0.963498			0.0505706
$364$		0			0
$365$	37.1451			1.94426
$366$		0			0
$367$			27.9007i			1.45640i	0.685363	+	0.728202i	$0.259644\pi$
$367$			27.9007i			1.45640i	−0.685363	+	0.728202i	$0.740356\pi$
$368$		0			0
$369$		−	15.6511i		−	0.814766i
$370$		0			0
$371$	0.0992339			0.00515197
$372$		0			0
$373$		−	13.7701i		−	0.712989i	−0.934297	−	0.356495i	$-0.883972\pi$
$373$		−	13.7701i		−	0.712989i	0.934297	−	0.356495i	$-0.116028\pi$
$374$		0			0
$375$	−14.1652			−0.731488
$376$		0			0
$377$	10.2922			0.530076
$378$		0			0
$379$	18.1643			0.933039			0.466520	−	0.884511i	$-0.345508\pi$
$379$	18.1643			0.933039			0.466520	+	0.884511i	$0.345508\pi$
$380$		0			0
$381$	3.28927			0.168515
$382$		0			0
$383$	6.52440			0.333381			0.166691	−	0.986009i	$-0.446692\pi$
$383$	6.52440			0.333381			0.166691	+	0.986009i	$0.446692\pi$
$384$		0			0
$385$	0.542716			0.0276594
$386$		0			0
$387$		−	6.83741i		−	0.347565i
$388$		0			0
$389$	25.3246			1.28401			0.642005	−	0.766700i	$-0.278103\pi$
$389$	25.3246			1.28401			0.642005	+	0.766700i	$0.278103\pi$
$390$		0			0
$391$		−	8.28502i		−	0.418991i
$392$		0			0
$393$		−	18.8586i		−	0.951290i
$394$		0			0
$395$	−59.7714			−3.00742
$396$		0			0
$397$	4.97493			0.249685			0.124842	−	0.992177i	$-0.460157\pi$
$397$	4.97493			0.249685			0.124842	+	0.992177i	$0.460157\pi$
$398$		0			0
$399$	0.378880	−	0.478733i	0.0189677	−	0.0239666i
$400$		0			0
$401$			7.29806i			0.364448i	0.983257	+	0.182224i	$0.0583296\pi$
$401$			7.29806i			0.364448i	−0.983257	+	0.182224i	$0.941670\pi$
$402$		0			0
$403$			33.9770i			1.69252i
$404$		0			0
$405$	5.62555			0.279536
$406$		0			0
$407$	−8.26609			−0.409735
$408$		0			0
$409$		−	10.0334i		−	0.496122i	−0.968744	−	0.248061i	$-0.920207\pi$
$409$		−	10.0334i		−	0.496122i	0.968744	−	0.248061i	$-0.0797934\pi$
$410$		0			0
$411$	−10.3303			−0.509558
$412$		0			0
$413$			1.52191i			0.0748885i
$414$		0			0
$415$			6.34669i			0.311547i
$416$		0			0
$417$		−	16.4171i		−	0.803949i
$418$		0			0
$419$		−	28.0694i		−	1.37128i	−0.727940	−	0.685641i	$-0.759522\pi$
$419$		−	28.0694i		−	1.37128i	0.727940	−	0.685641i	$-0.240478\pi$
$420$		0			0
$421$			12.8790i			0.627684i	0.949475	+	0.313842i	$0.101616\pi$
$421$			12.8790i			0.627684i	−0.949475	+	0.313842i	$0.898384\pi$
$422$		0			0
$423$		−	19.0963i		−	0.928492i
$424$		0			0
$425$	10.7649			0.522176
$426$		0			0
$427$		−	0.320646i		−	0.0155171i
$428$		0			0
$429$	5.17614			0.249906
$430$		0			0
$431$	0.561458			0.0270445			0.0135222	−	0.999909i	$-0.495696\pi$
$431$	0.561458			0.0270445			0.0135222	+	0.999909i	$0.495696\pi$
$432$		0			0
$433$		−	37.3926i		−	1.79697i	−0.439001	−	0.898487i	$-0.644668\pi$
$433$		−	37.3926i		−	1.79697i	0.439001	−	0.898487i	$-0.355332\pi$
$434$		0			0
$435$			6.89136i			0.330415i
$436$		0			0
$437$	−18.6080	+	23.5121i	−0.890141	+	1.12474i
$438$		0			0
$439$	−4.92623			−0.235116			−0.117558	−	0.993066i	$-0.537507\pi$
$439$	−4.92623			−0.235116			−0.117558	+	0.993066i	$0.537507\pi$
$440$		0			0
$441$	−14.4579			−0.688472
$442$		0			0
$443$		−	12.4916i		−	0.593493i	−0.954956	−	0.296747i	$-0.904098\pi$
$443$		−	12.4916i		−	0.593493i	0.954956	−	0.296747i	$-0.0959017\pi$
$444$		0			0
$445$			57.7187i			2.73613i
$446$		0			0
$447$	6.30068			0.298012
$448$		0			0
$449$		−	34.5228i		−	1.62923i	−0.580002	−	0.814615i	$-0.696948\pi$
$449$		−	34.5228i		−	1.62923i	0.580002	−	0.814615i	$-0.303052\pi$
$450$		0			0
$451$	−7.55484			−0.355744
$452$		0			0
$453$	5.94493			0.279317
$454$		0			0
$455$	2.91560			0.136685
$456$		0			0
$457$	39.9843			1.87039			0.935194	−	0.354136i	$-0.115225\pi$
$457$	39.9843			1.87039			0.935194	+	0.354136i	$0.115225\pi$
$458$		0			0
$459$	5.88538			0.274706
$460$		0			0
$461$	−29.6211			−1.37959			−0.689796	−	0.724003i	$-0.742300\pi$
$461$	−29.6211			−1.37959			−0.689796	+	0.724003i	$0.742300\pi$
$462$		0			0
$463$		−	13.3489i		−	0.620375i	−0.950675	−	0.310187i	$-0.899608\pi$
$463$		−	13.3489i		−	0.620375i	0.950675	−	0.310187i	$-0.100392\pi$
$464$		0			0
$465$	−22.7500			−1.05500
$466$		0			0
$467$			9.11342i			0.421719i	0.977516	+	0.210859i	$0.0676262\pi$
$467$			9.11342i			0.421719i	−0.977516	+	0.210859i	$0.932374\pi$
$468$		0			0
$469$		−	1.30614i		−	0.0603119i
$470$		0			0
$471$	−2.01474			−0.0928342
$472$		0			0
$473$	−3.30043			−0.151754
$474$		0			0
$475$	−30.5498	−	24.1778i	−1.40172	−	1.10935i
$476$		0			0
$477$		−	1.41419i		−	0.0647513i
$478$		0			0
$479$			12.5441i			0.573153i	0.958057	+	0.286576i	$0.0925172\pi$
$479$			12.5441i			0.573153i	−0.958057	+	0.286576i	$0.907483\pi$
$480$		0			0
$481$	−44.4074			−2.02480
$482$		0			0
$483$	0.963485			0.0438401
$484$		0			0
$485$		−	33.4469i		−	1.51875i
$486$		0			0
$487$	−13.9556			−0.632388			−0.316194	−	0.948695i	$-0.602405\pi$
$487$	−13.9556			−0.632388			−0.316194	+	0.948695i	$0.602405\pi$
$488$		0			0
$489$		−	12.1046i		−	0.547387i
$490$		0			0
$491$			42.3144i			1.90962i	0.297214	+	0.954811i	$0.403942\pi$
$491$			42.3144i			1.90962i	−0.297214	+	0.954811i	$0.596058\pi$
$492$		0			0
$493$		−	2.30742i		−	0.103921i
$494$		0			0
$495$		−	7.73429i		−	0.347631i
$496$		0			0
$497$		−	1.54834i		−	0.0694526i
$498$		0			0
$499$		−	36.1909i		−	1.62013i	−0.586341	−	0.810065i	$-0.699432\pi$
$499$		−	36.1909i		−	1.62013i	0.586341	−	0.810065i	$-0.300568\pi$
$500$		0			0
$501$	−7.00412			−0.312921
$502$		0			0
$503$			35.8043i			1.59644i	0.602368	+	0.798218i	$0.294224\pi$
$503$			35.8043i			1.59644i	−0.602368	+	0.798218i	$0.705776\pi$
$504$		0			0
$505$	−5.77615			−0.257035
$506$		0			0
$507$	15.2820			0.678696
$508$		0			0
$509$		−	13.3072i		−	0.589832i	−0.955523	−	0.294916i	$-0.904708\pi$
$509$		−	13.3072i		−	0.589832i	0.955523	−	0.294916i	$-0.0952917\pi$
$510$		0			0
$511$		−	1.44635i		−	0.0639829i
$512$		0			0
$513$	−16.7021	−	13.2184i	−0.737417	−	0.583609i
$514$		0			0
$515$	−7.53908			−0.332211
$516$		0			0
$517$	−9.21781			−0.405399
$518$		0			0
$519$			5.90250i			0.259091i
$520$		0			0
$521$		−	32.6853i		−	1.43197i	−0.698116	−	0.715984i	$-0.745978\pi$
$521$		−	32.6853i		−	1.43197i	0.698116	−	0.715984i	$-0.254022\pi$
$522$		0			0
$523$	6.17135			0.269854			0.134927	−	0.990856i	$-0.456920\pi$
$523$	6.17135			0.269854			0.134927	+	0.990856i	$0.456920\pi$
$524$		0			0
$525$			1.25188i			0.0546365i
$526$		0			0
$527$	7.61732			0.331816
$528$		0			0
$529$	−24.3197			−1.05738
$530$		0			0
$531$	21.6889			0.941220
$532$		0			0
$533$	−40.5864			−1.75799
$534$		0			0
$535$	31.6695			1.36919
$536$		0			0
$537$	−8.41553			−0.363157
$538$		0			0
$539$			6.97887i			0.300601i
$540$		0			0
$541$	3.51513			0.151127			0.0755636	−	0.997141i	$-0.475924\pi$
$541$	3.51513			0.151127			0.0755636	+	0.997141i	$0.475924\pi$
$542$		0			0
$543$		−	13.6237i		−	0.584649i
$544$		0			0
$545$		−	37.0899i		−	1.58876i
$546$		0			0
$547$	32.9843			1.41031			0.705154	−	0.709054i	$-0.250878\pi$
$547$	32.9843			1.41031			0.705154	+	0.709054i	$0.250878\pi$
$548$		0			0
$549$	−4.56955			−0.195024
$550$		0			0
$551$	−5.18242	+	6.54823i	−0.220778	+	0.278964i
$552$		0			0
$553$			2.32737i			0.0989700i
$554$		0			0
$555$		−	29.7338i		−	1.26213i
$556$		0			0
$557$	−41.8208			−1.77200			−0.886002	−	0.463682i	$-0.846528\pi$
$557$	−41.8208			−1.77200			−0.886002	+	0.463682i	$0.846528\pi$
$558$		0			0
$559$	−17.7307			−0.749929
$560$		0			0
$561$		−	1.16044i		−	0.0489939i
$562$		0			0
$563$	−12.2816			−0.517608			−0.258804	−	0.965930i	$-0.583328\pi$
$563$	−12.2816			−0.517608			−0.258804	+	0.965930i	$0.583328\pi$
$564$		0			0
$565$		−	11.4896i		−	0.483369i
$566$		0			0
$567$		−	0.219047i		−	0.00919912i
$568$		0			0
$569$		−	0.515104i		−	0.0215943i	−0.999942	−	0.0107972i	$-0.996563\pi$
$569$		−	0.515104i		−	0.0215943i	0.999942	−	0.0107972i	$-0.00343691\pi$
$570$		0			0
$571$		−	11.3733i		−	0.475959i	−0.971270	−	0.237979i	$-0.923515\pi$
$571$		−	11.3733i		−	0.475959i	0.971270	−	0.237979i	$-0.0764851\pi$
$572$		0			0
$573$			4.76370i			0.199007i
$574$		0			0
$575$		−	61.4837i		−	2.56405i
$576$		0			0
$577$	−35.4134			−1.47428			−0.737139	−	0.675741i	$-0.763824\pi$
$577$	−35.4134			−1.47428			−0.737139	+	0.675741i	$0.763824\pi$
$578$		0			0
$579$			10.9572i			0.455367i
$580$		0			0
$581$	0.247127			0.0102526
$582$		0			0
$583$	−0.682633			−0.0282718
$584$		0			0
$585$		−	41.5504i		−	1.71790i
$586$		0			0
$587$			15.3147i			0.632106i	0.948742	+	0.316053i	$0.102358\pi$
$587$			15.3147i			0.632106i	−0.948742	+	0.316053i	$0.897642\pi$
$588$		0			0
$589$	−21.6172	−	17.1084i	−0.890722	−	0.704938i
$590$		0			0
$591$	−4.15367			−0.170859
$592$		0			0
$593$	−24.4454			−1.00385			−0.501926	−	0.864911i	$-0.667375\pi$
$593$	−24.4454			−1.00385			−0.501926	+	0.864911i	$0.667375\pi$
$594$		0			0
$595$		−	0.653649i		−	0.0267970i
$596$		0			0
$597$		−	8.11027i		−	0.331931i
$598$		0			0
$599$	14.2309			0.581460			0.290730	−	0.956805i	$-0.406102\pi$
$599$	14.2309			0.581460			0.290730	+	0.956805i	$0.406102\pi$
$600$		0			0
$601$			25.4717i			1.03901i	0.854467	+	0.519506i	$0.173884\pi$
$601$			25.4717i			1.03901i	−0.854467	+	0.519506i	$0.826116\pi$
$602$		0			0
$603$	−18.6139			−0.758017
$604$		0			0
$605$	−3.73336			−0.151783
$606$		0			0
$607$	−13.3172			−0.540530			−0.270265	−	0.962786i	$-0.587111\pi$
$607$	−13.3172			−0.540530			−0.270265	+	0.962786i	$0.587111\pi$
$608$		0			0
$609$	0.268335			0.0108735
$610$		0			0
$611$	−49.5203			−2.00338
$612$		0			0
$613$	16.5548			0.668644			0.334322	−	0.942459i	$-0.391493\pi$
$613$	16.5548			0.668644			0.334322	+	0.942459i	$0.391493\pi$
$614$		0			0
$615$		−	27.1754i		−	1.09582i
$616$		0			0
$617$	−3.59490			−0.144725			−0.0723626	−	0.997378i	$-0.523054\pi$
$617$	−3.59490			−0.144725			−0.0723626	+	0.997378i	$0.523054\pi$
$618$		0			0
$619$		−	14.0240i		−	0.563671i	−0.959463	−	0.281835i	$-0.909057\pi$
$619$		−	14.0240i		−	0.563671i	0.959463	−	0.281835i	$-0.0909432\pi$
$620$		0			0
$621$		−	33.6142i		−	1.34889i
$622$		0			0
$623$	2.24745			0.0900421
$624$		0			0
$625$	10.1974			0.407898
$626$		0			0
$627$	−2.60633	+	3.29322i	−0.104087	+	0.131519i
$628$		0			0
$629$			9.95571i			0.396960i
$630$		0			0
$631$			8.20683i			0.326709i	0.986567	+	0.163354i	$0.0522314\pi$
$631$			8.20683i			0.326709i	−0.986567	+	0.163354i	$0.947769\pi$
$632$		0			0
$633$	−13.2466			−0.526505
$634$		0			0
$635$	−12.7453			−0.505780
$636$		0			0
$637$			37.4921i			1.48549i
$638$		0			0
$639$	−22.0656			−0.872900
$640$		0			0
$641$		−	39.3008i		−	1.55229i	−0.630557	−	0.776143i	$-0.717173\pi$
$641$		−	39.3008i		−	1.55229i	0.630557	−	0.776143i	$-0.282827\pi$
$642$		0			0
$643$		−	18.1453i		−	0.715579i	−0.933802	−	0.357790i	$-0.883530\pi$
$643$		−	18.1453i		−	0.715579i	0.933802	−	0.357790i	$-0.116470\pi$
$644$		0			0
$645$		−	11.8719i		−	0.467457i
$646$		0			0
$647$			8.03455i			0.315871i	0.987449	+	0.157935i	$0.0504838\pi$
$647$			8.03455i			0.315871i	−0.987449	+	0.157935i	$0.949516\pi$
$648$		0			0
$649$		−	10.4693i		−	0.410956i
$650$		0			0
$651$			0.885837i			0.0347187i
$652$		0			0
$653$	−16.6020			−0.649687			−0.324844	−	0.945768i	$-0.605312\pi$
$653$	−16.6020			−0.649687			−0.324844	+	0.945768i	$0.605312\pi$
$654$		0			0
$655$			73.0731i			2.85520i
$656$		0			0
$657$	−20.6121			−0.804155
$658$		0			0
$659$	−7.59636			−0.295912			−0.147956	−	0.988994i	$-0.547269\pi$
$659$	−7.59636			−0.295912			−0.147956	+	0.988994i	$0.547269\pi$
$660$		0			0
$661$			20.0110i			0.778339i	0.921166	+	0.389169i	$0.127238\pi$
$661$			20.0110i			0.778339i	−0.921166	+	0.389169i	$0.872762\pi$
$662$		0			0
$663$		−	6.23416i		−	0.242115i
$664$		0			0
$665$	−1.46808	+	1.85499i	−0.0569298	+	0.0719335i
$666$		0			0
$667$	−13.1788			−0.510284
$668$		0			0
$669$	−10.2588			−0.396627
$670$		0			0
$671$			2.20573i			0.0851514i
$672$		0			0
$673$		−	45.2052i		−	1.74253i	−0.490811	−	0.871266i	$-0.663300\pi$
$673$		−	45.2052i		−	1.74253i	0.490811	−	0.871266i	$-0.336700\pi$
$674$		0			0
$675$	43.6758			1.68108
$676$		0			0
$677$		−	33.3188i		−	1.28055i	−0.768148	−	0.640273i	$-0.778821\pi$
$677$		−	33.3188i		−	1.28055i	0.768148	−	0.640273i	$-0.221179\pi$
$678$		0			0
$679$	−1.30235			−0.0499798
$680$		0			0
$681$	24.7131			0.947008
$682$		0			0
$683$	16.3600			0.625999			0.313000	−	0.949753i	$-0.398666\pi$
$683$	16.3600			0.625999			0.313000	+	0.949753i	$0.398666\pi$
$684$		0			0
$685$	40.0279			1.52939
$686$		0			0
$687$	6.46393			0.246614
$688$		0			0
$689$	−3.66727			−0.139712
$690$		0			0
$691$		−	1.68975i		−	0.0642809i	−0.999483	−	0.0321405i	$-0.989768\pi$
$691$		−	1.68975i		−	0.0642809i	0.999483	−	0.0321405i	$-0.0102324\pi$
$692$		0			0
$693$	−0.301157			−0.0114400
$694$		0			0
$695$			63.6129i			2.41298i
$696$		0			0
$697$			9.09908i			0.344652i
$698$		0			0
$699$	−5.21575			−0.197278
$700$		0			0
$701$	22.8124			0.861613			0.430806	−	0.902444i	$-0.358229\pi$
$701$	22.8124			0.861613			0.430806	+	0.902444i	$0.358229\pi$
$702$		0			0
$703$	22.3603	−	28.2534i	0.843336	−	1.06560i
$704$		0			0
$705$		−	33.1572i		−	1.24877i
$706$		0			0
$707$			0.224911i			0.00845866i
$708$		0			0
$709$	41.3520			1.55301			0.776504	−	0.630112i	$-0.216991\pi$
$709$	41.3520			1.55301			0.776504	+	0.630112i	$0.216991\pi$
$710$		0			0
$711$	33.1676			1.24388
$712$		0			0
$713$		−	43.5062i		−	1.62932i
$714$		0			0
$715$	−20.0565			−0.750070
$716$		0			0
$717$			6.41124i			0.239432i
$718$		0			0
$719$		−	26.9432i		−	1.00481i	−0.864632	−	0.502406i	$-0.832448\pi$
$719$		−	26.9432i		−	1.00481i	0.864632	−	0.502406i	$-0.167552\pi$
$720$		0			0
$721$			0.293556i			0.0109326i
$722$		0			0
$723$			0.0152078i			0.000565585i
$724$		0			0
$725$		−	17.1235i		−	0.635951i
$726$		0			0
$727$		−	50.0523i		−	1.85634i	−0.372160	−	0.928168i	$-0.621383\pi$
$727$		−	50.0523i		−	1.85634i	0.372160	−	0.928168i	$-0.378617\pi$
$728$		0			0
$729$	11.0029			0.407514
$730$		0			0
$731$			3.97505i			0.147023i
$732$		0			0
$733$	−39.2384			−1.44930			−0.724652	−	0.689115i	$-0.758000\pi$
$733$	−39.2384			−1.44930			−0.724652	+	0.689115i	$0.758000\pi$
$734$		0			0
$735$	−25.1036			−0.925960
$736$		0			0
$737$			8.98497i			0.330966i
$738$		0			0
$739$			29.9501i			1.10173i	0.834593	+	0.550867i	$0.185703\pi$
$739$			29.9501i			1.10173i	−0.834593	+	0.550867i	$0.814297\pi$
$740$		0			0
$741$	−14.0018	+	17.6920i	−0.514370	+	0.649930i
$742$		0			0
$743$	−26.6019			−0.975928			−0.487964	−	0.872864i	$-0.662260\pi$
$743$	−26.6019			−0.975928			−0.487964	+	0.872864i	$0.662260\pi$
$744$		0			0
$745$	−24.4138			−0.894453
$746$		0			0
$747$		−	3.52183i		−	0.128857i
$748$		0			0
$749$		−	1.23314i		−	0.0450581i
$750$		0			0
$751$	−30.6706			−1.11919			−0.559594	−	0.828767i	$-0.689043\pi$
$751$	−30.6706			−1.11919			−0.559594	+	0.828767i	$0.689043\pi$
$752$		0			0
$753$		−	5.58277i		−	0.203447i
$754$		0			0
$755$	−23.0354			−0.838343
$756$		0			0
$757$	11.6467			0.423308			0.211654	−	0.977345i	$-0.432115\pi$
$757$	11.6467			0.423308			0.211654	+	0.977345i	$0.432115\pi$
$758$		0			0
$759$	−6.62784			−0.240575
$760$		0			0
$761$	−52.6468			−1.90845			−0.954223	−	0.299096i	$-0.903315\pi$
$761$	−52.6468			−1.90845			−0.954223	+	0.299096i	$0.903315\pi$
$762$		0			0
$763$	−1.44420			−0.0522837
$764$		0			0
$765$	−9.31521			−0.336792
$766$		0			0
$767$		−	56.2435i		−	2.03084i
$768$		0			0
$769$	−48.3390			−1.74315			−0.871575	−	0.490262i	$-0.836901\pi$
$769$	−48.3390			−1.74315			−0.871575	+	0.490262i	$0.836901\pi$
$770$		0			0
$771$			1.87820i			0.0676418i
$772$		0			0
$773$		−	39.0056i		−	1.40293i	−0.712703	−	0.701466i	$-0.752529\pi$
$773$		−	39.0056i		−	1.40293i	0.712703	−	0.701466i	$-0.247471\pi$
$774$		0			0
$775$	56.5287			2.03057
$776$		0			0
$777$	−1.15777			−0.0415349
$778$		0			0
$779$	20.4364	−	25.8223i	0.732209	−	0.925181i
$780$		0			0
$781$			10.6511i			0.381126i
$782$		0			0
$783$		−	9.36173i		−	0.334561i
$784$		0			0
$785$	7.80669			0.278633
$786$		0			0
$787$	−53.9078			−1.92161			−0.960803	−	0.277231i	$-0.910583\pi$
$787$	−53.9078			−1.92161			−0.960803	+	0.277231i	$0.910583\pi$
$788$		0			0
$789$		−	16.3936i		−	0.583629i
$790$		0			0
$791$	−0.447380			−0.0159070
$792$		0			0
$793$			11.8497i			0.420796i
$794$		0			0
$795$		−	2.45549i		−	0.0870872i
$796$		0			0
$797$		−	11.0651i		−	0.391945i	−0.980609	−	0.195973i	$-0.937214\pi$
$797$		−	11.0651i		−	0.391945i	0.980609	−	0.195973i	$-0.0627864\pi$
$798$		0			0
$799$			11.1020i			0.392759i
$800$		0			0
$801$		−	32.0286i		−	1.13167i
$802$		0			0
$803$			9.94951i			0.351111i
$804$		0			0
$805$	−3.73331			−0.131582
$806$		0			0
$807$			19.6602i			0.692072i
$808$		0			0
$809$	−33.2375			−1.16857			−0.584283	−	0.811550i	$-0.698624\pi$
$809$	−33.2375			−1.16857			−0.584283	+	0.811550i	$0.698624\pi$
$810$		0			0
$811$	−28.7994			−1.01128			−0.505642	−	0.862744i	$-0.668744\pi$
$811$	−28.7994			−1.01128			−0.505642	+	0.862744i	$0.668744\pi$
$812$		0			0
$813$			23.1871i			0.813209i
$814$		0			0
$815$			46.9027i			1.64293i
$816$		0			0
$817$	8.92789	−	11.2808i	0.312347	−	0.394666i
$818$		0			0
$819$	−1.61789			−0.0565336
$820$		0			0
$821$	37.5363			1.31002			0.655012	−	0.755618i	$-0.272663\pi$
$821$	37.5363			1.31002			0.655012	+	0.755618i	$0.272663\pi$
$822$		0			0
$823$			20.0662i			0.699463i	0.936850	+	0.349732i	$0.113727\pi$
$823$			20.0662i			0.699463i	−0.936850	+	0.349732i	$0.886273\pi$
$824$		0			0
$825$		−	8.61172i		−	0.299822i
$826$		0			0
$827$	31.4445			1.09343			0.546716	−	0.837318i	$-0.315878\pi$
$827$	31.4445			1.09343			0.546716	+	0.837318i	$0.315878\pi$
$828$		0			0
$829$			23.8694i			0.829020i	0.910045	+	0.414510i	$0.136047\pi$
$829$			23.8694i			0.829020i	−0.910045	+	0.414510i	$0.863953\pi$
$830$		0			0
$831$	−9.97914			−0.346172
$832$		0			0
$833$	8.40538			0.291229
$834$		0			0
$835$	27.1395			0.939203
$836$		0			0
$837$	30.9052			1.06824
$838$		0			0
$839$	46.5425			1.60683			0.803414	−	0.595421i	$-0.203015\pi$
$839$	46.5425			1.60683			0.803414	+	0.595421i	$0.203015\pi$
$840$		0			0
$841$	25.3296			0.873436
$842$		0			0
$843$		−	3.15109i		−	0.108529i
$844$		0			0
$845$	−59.2145			−2.03704
$846$		0			0
$847$			0.145369i			0.00499495i
$848$		0			0
$849$		−	4.12020i		−	0.141405i
$850$		0			0
$851$	56.8619			1.94920
$852$		0			0
$853$	35.5962			1.21879			0.609395	−	0.792867i	$-0.291412\pi$
$853$	35.5962			1.21879			0.609395	+	0.792867i	$0.291412\pi$
$854$		0			0
$855$	26.4357	+	20.9218i	0.904081	+	0.715510i
$856$		0			0
$857$		−	7.17112i		−	0.244961i	−0.992471	−	0.122480i	$-0.960915\pi$
$857$		−	7.17112i		−	0.244961i	0.992471	−	0.122480i	$-0.0390849\pi$
$858$		0			0
$859$			16.5060i			0.563179i	0.959535	+	0.281589i	$0.0908616\pi$
$859$			16.5060i			0.563179i	−0.959535	+	0.281589i	$0.909138\pi$
$860$		0			0
$861$	−1.05815			−0.0360618
$862$		0			0
$863$	−9.32157			−0.317310			−0.158655	−	0.987334i	$-0.550716\pi$
$863$	−9.32157			−0.317310			−0.158655	+	0.987334i	$0.550716\pi$
$864$		0			0
$865$		−	22.8710i		−	0.777637i
$866$		0			0
$867$	14.9818			0.508810
$868$		0			0
$869$		−	16.0101i		−	0.543105i
$870$		0			0
$871$			48.2694i			1.63555i
$872$		0			0
$873$			18.5600i			0.628160i
$874$		0			0
$875$		−	2.13720i		−	0.0722505i
$876$		0			0
$877$		−	6.12442i		−	0.206807i	−0.994639	−	0.103403i	$-0.967027\pi$
$877$		−	6.12442i		−	0.206807i	0.994639	−	0.103403i	$-0.0329733\pi$
$878$		0			0
$879$		−	28.6939i		−	0.967822i
$880$		0			0
$881$	1.21874			0.0410606			0.0205303	−	0.999789i	$-0.493465\pi$
$881$	1.21874			0.0410606			0.0205303	+	0.999789i	$0.493465\pi$
$882$		0			0
$883$		−	32.5115i		−	1.09410i	−0.837100	−	0.547050i	$-0.815751\pi$
$883$		−	32.5115i		−	1.09410i	0.837100	−	0.547050i	$-0.184249\pi$
$884$		0			0
$885$	37.6589			1.26589
$886$		0			0
$887$	−40.2357			−1.35098			−0.675491	−	0.737368i	$-0.736068\pi$
$887$	−40.2357			−1.35098			−0.675491	+	0.737368i	$0.736068\pi$
$888$		0			0
$889$			0.496274i			0.0166445i
$890$		0			0
$891$			1.50683i			0.0504808i
$892$		0			0
$893$	24.9348	−	31.5063i	0.834412	−	1.05432i
$894$		0			0
$895$	32.6084			1.08998
$896$		0			0
$897$	−35.6063			−1.18886
$898$		0			0
$899$		−	12.1167i		−	0.404114i
$900$		0			0
$901$			0.822166i			0.0273903i
$902$		0			0
$903$	−0.462269			−0.0153833
$904$		0			0
$905$			52.7891i			1.75477i
$906$		0			0
$907$	−7.46363			−0.247826			−0.123913	−	0.992293i	$-0.539544\pi$
$907$	−7.46363			−0.247826			−0.123913	+	0.992293i	$0.539544\pi$
$908$		0			0
$909$	3.20523			0.106311
$910$		0			0
$911$	−45.3310			−1.50188			−0.750942	−	0.660369i	$-0.770400\pi$
$911$	−45.3310			−1.50188			−0.750942	+	0.660369i	$0.770400\pi$
$912$		0			0
$913$	−1.70000			−0.0562616
$914$		0			0
$915$	−7.93421			−0.262297
$916$		0			0
$917$	2.84532			0.0939607
$918$		0			0
$919$			35.5044i			1.17118i	0.810607	+	0.585591i	$0.199138\pi$
$919$			35.5044i			1.17118i	−0.810607	+	0.585591i	$0.800862\pi$
$920$		0			0
$921$	24.1375			0.795357
$922$		0			0
$923$			57.2202i			1.88343i
$924$		0			0
$925$			73.8821i			2.42923i
$926$		0			0
$927$	4.18349			0.137404
$928$		0			0
$929$	−44.5001			−1.46000			−0.730001	−	0.683446i	$-0.760480\pi$
$929$	−44.5001			−1.46000			−0.730001	+	0.683446i	$0.760480\pi$
$930$		0			0
$931$	−23.8536	−	18.8783i	−0.781772	−	0.618712i
$932$		0			0
$933$			29.8668i			0.977795i
$934$		0			0
$935$			4.49647i			0.147050i
$936$		0			0
$937$	31.5129			1.02948			0.514741	−	0.857345i	$-0.327888\pi$
$937$	31.5129			1.02948			0.514741	+	0.857345i	$0.327888\pi$
$938$		0			0
$939$	1.10001			0.0358975
$940$		0			0
$941$			7.82317i			0.255028i	0.991837	+	0.127514i	$0.0406998\pi$
$941$			7.82317i			0.255028i	−0.991837	+	0.127514i	$0.959300\pi$
$942$		0			0
$943$	51.9693			1.69235
$944$		0			0
$945$		−	2.65201i		−	0.0862698i
$946$		0			0
$947$			1.53142i			0.0497644i	0.999690	+	0.0248822i	$0.00792107\pi$
$947$			1.53142i			0.0497644i	−0.999690	+	0.0248822i	$0.992079\pi$
$948$		0			0
$949$			53.4511i			1.73510i
$950$		0			0
$951$		−	29.7470i		−	0.964611i
$952$		0			0
$953$		−	28.1295i		−	0.911204i	−0.890183	−	0.455602i	$-0.849424\pi$
$953$		−	28.1295i		−	0.911204i	0.890183	−	0.455602i	$-0.150576\pi$
$954$		0			0
$955$		−	18.4584i		−	0.597299i
$956$		0			0
$957$	−1.84589			−0.0596690
$958$		0			0
$959$		−	1.55861i		−	0.0503300i
$960$		0			0
$961$	9.00000			0.290323
$962$		0			0
$963$	−17.5736			−0.566303
$964$		0			0
$965$		−	42.4570i		−	1.36674i
$966$		0			0
$967$		−	39.1415i		−	1.25871i	−0.777120	−	0.629353i	$-0.783320\pi$
$967$		−	39.1415i		−	1.25871i	0.777120	−	0.629353i	$-0.216680\pi$
$968$		0			0
$969$	3.96637	+	3.13907i	0.127418	+	0.100842i
$970$		0			0
$971$	22.7425			0.729843			0.364922	−	0.931038i	$-0.381096\pi$
$971$	22.7425			0.729843			0.364922	+	0.931038i	$0.381096\pi$
$972$		0			0
$973$	2.47696			0.0794076
$974$		0			0
$975$		−	46.2642i		−	1.48164i
$976$		0			0
$977$			22.6631i			0.725057i	0.931973	+	0.362528i	$0.118086\pi$
$977$			22.6631i			0.725057i	−0.931973	+	0.362528i	$0.881914\pi$
$978$		0			0
$979$	−15.4603			−0.494112
$980$		0			0
$981$			20.5815i			0.657116i
$982$		0			0
$983$	7.04992			0.224858			0.112429	−	0.993660i	$-0.464137\pi$
$983$	7.04992			0.224858			0.112429	+	0.993660i	$0.464137\pi$
$984$		0			0
$985$	16.0946			0.512817
$986$		0			0
$987$	−1.29107			−0.0410954
$988$		0			0
$989$	22.7034			0.721928
$990$		0			0
$991$	1.52858			0.0485570			0.0242785	−	0.999705i	$-0.492271\pi$
$991$	1.52858			0.0485570			0.0242785	+	0.999705i	$0.492271\pi$
$992$		0			0
$993$	−27.8089			−0.882490
$994$		0			0
$995$			31.4256i			0.996260i
$996$		0			0
$997$	40.5835			1.28529			0.642647	−	0.766163i	$-0.277836\pi$
$997$	40.5835			1.28529			0.642647	+	0.766163i	$0.277836\pi$
$998$		0			0
$999$			40.3926i			1.27797i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.a.1519.14	yes	36
4.3	odd	2	inner	3344.2.o.a.1519.24	yes	36
19.18	odd	2	inner	3344.2.o.a.1519.23	yes	36
76.75	even	2	inner	3344.2.o.a.1519.13	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.a.1519.13	✓	36	76.75	even	2	inner
3344.2.o.a.1519.14	yes	36	1.1	even	1	trivial
3344.2.o.a.1519.23	yes	36	19.18	odd	2	inner
3344.2.o.a.1519.24	yes	36	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)

\(f(q)\)	\(=\)	\(q-0.963498 q^{3} +3.73336 q^{5} -0.145369i q^{7} -2.07167 q^{9} +O(q^{10})\)	\(q-0.963498 q^{3} +3.73336 q^{5} -0.145369i q^{7} -2.07167 q^{9} +1.00000i q^{11} +5.37224i q^{13} -3.59709 q^{15} +1.20440 q^{17} +(-3.41798 - 2.70507i) q^{19} +0.140063i q^{21} -6.87893i q^{23} +8.93797 q^{25} +4.88655 q^{27} -1.91582i q^{29} +6.32456 q^{31} -0.963498i q^{33} -0.542716i q^{35} +8.26609i q^{37} -5.17614i q^{39} +7.55484i q^{41} +3.30043i q^{43} -7.73429 q^{45} +9.21781i q^{47} +6.97887 q^{49} -1.16044 q^{51} +0.682633i q^{53} +3.73336i q^{55} +(3.29322 + 2.60633i) q^{57} -10.4693 q^{59} +2.20573 q^{61} +0.301157i q^{63} +20.0565i q^{65} +8.98497 q^{67} +6.62784i q^{69} +10.6511 q^{71} +9.94951 q^{73} -8.61172 q^{75} +0.145369 q^{77} -16.0101 q^{79} +1.50683 q^{81} +1.70000i q^{83} +4.49647 q^{85} +1.84589i q^{87} +15.4603i q^{89} +0.780958 q^{91} -6.09370 q^{93} +(-12.7606 - 10.0990i) q^{95} -8.95894i q^{97} -2.07167i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 44 q^{9}+O(q^{10}) \) 36 * q + 44 * q^9	\( 36 q + 44 q^{9} - 16 q^{17} + 36 q^{25} - 32 q^{45} - 28 q^{49} + 24 q^{57} - 48 q^{61} - 24 q^{73} + 52 q^{81} + 24 q^{85} - 64 q^{93}+O(q^{100}) \) 36 * q + 44 * q^9 - 16 * q^17 + 36 * q^25 - 32 * q^45 - 28 * q^49 + 24 * q^57 - 48 * q^61 - 24 * q^73 + 52 * q^81 + 24 * q^85 - 64 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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