LMFDB - Embedded newform 3344.2.o.a.1519.17

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			1519.17
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.a.1519.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.711963 q^{3} +0.887508 q^{5} -2.32992i q^{7} -2.49311 q^{9} +O(q^{10})$	$q-0.711963 q^{3} +0.887508 q^{5} -2.32992i q^{7} -2.49311 q^{9} +1.00000i q^{11} +0.0550862i q^{13} -0.631873 q^{15} -0.537779 q^{17} +(3.30891 + 2.83745i) q^{19} +1.65882i q^{21} +2.34463i q^{23} -4.21233 q^{25} +3.91089 q^{27} -5.27761i q^{29} +4.62073 q^{31} -0.711963i q^{33} -2.06782i q^{35} +2.51560i q^{37} -0.0392193i q^{39} -12.4472i q^{41} -9.53266i q^{43} -2.21265 q^{45} +4.22413i q^{47} +1.57148 q^{49} +0.382879 q^{51} -4.95065i q^{53} +0.887508i q^{55} +(-2.35582 - 2.02016i) q^{57} +6.86465 q^{59} -7.13712 q^{61} +5.80874i q^{63} +0.0488895i q^{65} -3.31376 q^{67} -1.66929i q^{69} -8.62601 q^{71} -13.5549 q^{73} +2.99902 q^{75} +2.32992 q^{77} -16.6133 q^{79} +4.69492 q^{81} +3.87134i q^{83} -0.477284 q^{85} +3.75746i q^{87} +3.77934i q^{89} +0.128346 q^{91} -3.28979 q^{93} +(2.93668 + 2.51826i) q^{95} -17.5949i q^{97} -2.49311i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.711963			−0.411052			−0.205526	−	0.978652i	$-0.565891\pi$
$3$	−0.711963			−0.411052			−0.205526	+	0.978652i	$0.565891\pi$
$4$		0			0
$5$	0.887508			0.396906			0.198453	−	0.980110i	$-0.436408\pi$
$5$	0.887508			0.396906			0.198453	+	0.980110i	$0.436408\pi$
$6$		0			0
$7$		−	2.32992i		−	0.880627i	−0.897844	−	0.440313i	$-0.854867\pi$
$7$		−	2.32992i		−	0.880627i	0.897844	−	0.440313i	$-0.145133\pi$
$8$		0			0
$9$	−2.49311			−0.831036
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			0.0550862i			0.0152782i	0.999971	+	0.00763908i	$0.00243162\pi$
$13$			0.0550862i			0.0152782i	−0.999971	+	0.00763908i	$0.997568\pi$
$14$		0			0
$15$	−0.631873			−0.163149
$16$		0			0
$17$	−0.537779			−0.130431			−0.0652153	−	0.997871i	$-0.520773\pi$
$17$	−0.537779			−0.130431			−0.0652153	+	0.997871i	$0.520773\pi$
$18$		0			0
$19$	3.30891	+	2.83745i	0.759116	+	0.650956i
$19$	3.30891	+	2.83745i	0.759116	+	0.650956i
$20$		0			0
$21$			1.65882i			0.361983i
$22$		0			0
$23$			2.34463i			0.488889i	0.969663	+	0.244445i	$0.0786057\pi$
$23$			2.34463i			0.488889i	−0.969663	+	0.244445i	$0.921394\pi$
$24$		0			0
$25$	−4.21233			−0.842466
$26$		0			0
$27$	3.91089			0.752651
$28$		0			0
$29$		−	5.27761i		−	0.980028i	−0.871715	−	0.490014i	$-0.836992\pi$
$29$		−	5.27761i		−	0.980028i	0.871715	−	0.490014i	$-0.163008\pi$
$30$		0			0
$31$	4.62073			0.829908			0.414954	−	0.909842i	$-0.363798\pi$
$31$	4.62073			0.829908			0.414954	+	0.909842i	$0.363798\pi$
$32$		0			0
$33$		−	0.711963i		−	0.123937i
$34$		0			0
$35$		−	2.06782i		−	0.349526i
$36$		0			0
$37$			2.51560i			0.413562i	0.978387	+	0.206781i	$0.0662987\pi$
$37$			2.51560i			0.413562i	−0.978387	+	0.206781i	$0.933701\pi$
$38$		0			0
$39$		−	0.0392193i		−	0.00628012i
$40$		0			0
$41$		−	12.4472i		−	1.94392i	−0.235144	−	0.971961i	$-0.575556\pi$
$41$		−	12.4472i		−	1.94392i	0.235144	−	0.971961i	$-0.424444\pi$
$42$		0			0
$43$		−	9.53266i		−	1.45372i	−0.686787	−	0.726858i	$-0.740980\pi$
$43$		−	9.53266i		−	1.45372i	0.686787	−	0.726858i	$-0.259020\pi$
$44$		0			0
$45$	−2.21265			−0.329843
$46$		0			0
$47$			4.22413i			0.616153i	0.951362	+	0.308077i	$0.0996853\pi$
$47$			4.22413i			0.616153i	−0.951362	+	0.308077i	$0.900315\pi$
$48$		0			0
$49$	1.57148			0.224497
$50$		0			0
$51$	0.382879			0.0536138
$52$		0			0
$53$		−	4.95065i		−	0.680023i	−0.940421	−	0.340012i	$-0.889569\pi$
$53$		−	4.95065i		−	0.680023i	0.940421	−	0.340012i	$-0.110431\pi$
$54$		0			0
$55$			0.887508i			0.119672i
$56$		0			0
$57$	−2.35582	−	2.02016i	−0.312036	−	0.267577i
$58$		0			0
$59$	6.86465			0.893701			0.446851	−	0.894609i	$-0.352546\pi$
$59$	6.86465			0.893701			0.446851	+	0.894609i	$0.352546\pi$
$60$		0			0
$61$	−7.13712			−0.913815			−0.456907	−	0.889514i	$-0.651043\pi$
$61$	−7.13712			−0.913815			−0.456907	+	0.889514i	$0.651043\pi$
$62$		0			0
$63$			5.80874i			0.731833i
$64$		0			0
$65$			0.0488895i			0.00606399i
$66$		0			0
$67$	−3.31376			−0.404840			−0.202420	−	0.979299i	$-0.564881\pi$
$67$	−3.31376			−0.404840			−0.202420	+	0.979299i	$0.564881\pi$
$68$		0			0
$69$		−	1.66929i		−	0.200959i
$70$		0			0
$71$	−8.62601			−1.02372			−0.511860	−	0.859069i	$-0.671043\pi$
$71$	−8.62601			−1.02372			−0.511860	+	0.859069i	$0.671043\pi$
$72$		0			0
$73$	−13.5549			−1.58648			−0.793241	−	0.608908i	$-0.791608\pi$
$73$	−13.5549			−1.58648			−0.793241	+	0.608908i	$0.791608\pi$
$74$		0			0
$75$	2.99902			0.346297
$76$		0			0
$77$	2.32992			0.265519
$78$		0			0
$79$	−16.6133			−1.86914			−0.934569	−	0.355781i	$-0.884215\pi$
$79$	−16.6133			−1.86914			−0.934569	+	0.355781i	$0.884215\pi$
$80$		0			0
$81$	4.69492			0.521658
$82$		0			0
$83$			3.87134i			0.424935i	0.977168	+	0.212468i	$0.0681500\pi$
$83$			3.87134i			0.424935i	−0.977168	+	0.212468i	$0.931850\pi$
$84$		0			0
$85$	−0.477284			−0.0517687
$86$		0			0
$87$			3.75746i			0.402842i
$88$		0			0
$89$			3.77934i			0.400609i	0.979734	+	0.200305i	$0.0641932\pi$
$89$			3.77934i			0.400609i	−0.979734	+	0.200305i	$0.935807\pi$
$90$		0			0
$91$	0.128346			0.0134544
$92$		0			0
$93$	−3.28979			−0.341135
$94$		0			0
$95$	2.93668	+	2.51826i	0.301297	+	0.258368i
$96$		0			0
$97$		−	17.5949i		−	1.78649i	−0.449566	−	0.893247i	$-0.648421\pi$
$97$		−	17.5949i		−	1.78649i	0.449566	−	0.893247i	$-0.351579\pi$
$98$		0			0
$99$		−	2.49311i		−	0.250567i
$100$		0			0
$101$	−12.4792			−1.24172			−0.620862	−	0.783920i	$-0.713217\pi$
$101$	−12.4792			−1.24172			−0.620862	+	0.783920i	$0.713217\pi$
$102$		0			0
$103$	4.55342			0.448661			0.224331	−	0.974513i	$-0.427980\pi$
$103$	4.55342			0.448661			0.224331	+	0.974513i	$0.427980\pi$
$104$		0			0
$105$			1.47221i			0.143673i
$106$		0			0
$107$	−8.51152			−0.822840			−0.411420	−	0.911446i	$-0.634967\pi$
$107$	−8.51152			−0.822840			−0.411420	+	0.911446i	$0.634967\pi$
$108$		0			0
$109$		−	13.9816i		−	1.33919i	−0.742726	−	0.669595i	$-0.766468\pi$
$109$		−	13.9816i		−	1.33919i	0.742726	−	0.669595i	$-0.233532\pi$
$110$		0			0
$111$		−	1.79101i		−	0.169995i
$112$		0			0
$113$			0.515718i			0.0485146i	0.999706	+	0.0242573i	$0.00772210\pi$
$113$			0.515718i			0.0485146i	−0.999706	+	0.0242573i	$0.992278\pi$
$114$		0			0
$115$			2.08088i			0.194043i
$116$		0			0
$117$		−	0.137336i		−	0.0126967i
$118$		0			0
$119$			1.25298i			0.114861i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			8.86192i			0.799053i
$124$		0			0
$125$	−8.17602			−0.731285
$126$		0			0
$127$	14.8566			1.31831			0.659157	−	0.752006i	$-0.270913\pi$
$127$	14.8566			1.31831			0.659157	+	0.752006i	$0.270913\pi$
$128$		0			0
$129$			6.78690i			0.597553i
$130$		0			0
$131$			3.47723i			0.303807i	0.988395	+	0.151904i	$0.0485404\pi$
$131$			3.47723i			0.303807i	−0.988395	+	0.151904i	$0.951460\pi$
$132$		0			0
$133$	6.61103	−	7.70949i	0.573249	−	0.668497i
$134$		0			0
$135$	3.47095			0.298731
$136$		0			0
$137$	−12.6343			−1.07942			−0.539709	−	0.841852i	$-0.681466\pi$
$137$	−12.6343			−1.07942			−0.539709	+	0.841852i	$0.681466\pi$
$138$		0			0
$139$		−	4.67797i		−	0.396780i	−0.980123	−	0.198390i	$-0.936429\pi$
$139$		−	4.67797i		−	0.396780i	0.980123	−	0.198390i	$-0.0635713\pi$
$140$		0			0
$141$		−	3.00743i		−	0.253271i
$142$		0			0
$143$	−0.0550862			−0.00460654
$144$		0			0
$145$		−	4.68392i		−	0.388979i
$146$		0			0
$147$	−1.11883			−0.0922799
$148$		0			0
$149$	5.88000			0.481709			0.240854	−	0.970561i	$-0.422572\pi$
$149$	5.88000			0.481709			0.240854	+	0.970561i	$0.422572\pi$
$150$		0			0
$151$	−15.3497			−1.24914			−0.624570	−	0.780969i	$-0.714726\pi$
$151$	−15.3497			−1.24914			−0.624570	+	0.780969i	$0.714726\pi$
$152$		0			0
$153$	1.34074			0.108393
$154$		0			0
$155$	4.10094			0.329395
$156$		0			0
$157$	−4.93139			−0.393568			−0.196784	−	0.980447i	$-0.563050\pi$
$157$	−4.93139			−0.393568			−0.196784	+	0.980447i	$0.563050\pi$
$158$		0			0
$159$			3.52468i			0.279525i
$160$		0			0
$161$	5.46280			0.430529
$162$		0			0
$163$		−	21.5985i		−	1.69173i	−0.533398	−	0.845864i	$-0.679085\pi$
$163$		−	21.5985i		−	1.69173i	0.533398	−	0.845864i	$-0.320915\pi$
$164$		0			0
$165$		−	0.631873i		−	0.0491912i
$166$		0			0
$167$	14.0998			1.09107			0.545537	−	0.838087i	$-0.316326\pi$
$167$	14.0998			1.09107			0.545537	+	0.838087i	$0.316326\pi$
$168$		0			0
$169$	12.9970			0.999767
$170$		0			0
$171$	−8.24947	−	7.07407i	−0.630853	−	0.540968i
$172$		0			0
$173$			4.33072i			0.329258i	0.986356	+	0.164629i	$0.0526427\pi$
$173$			4.33072i			0.329258i	−0.986356	+	0.164629i	$0.947357\pi$
$174$		0			0
$175$			9.81439i			0.741898i
$176$		0			0
$177$	−4.88738			−0.367358
$178$		0			0
$179$	−25.1073			−1.87661			−0.938304	−	0.345812i	$-0.887604\pi$
$179$	−25.1073			−1.87661			−0.938304	+	0.345812i	$0.887604\pi$
$180$		0			0
$181$		−	10.2246i		−	0.759991i	−0.924988	−	0.379995i	$-0.875926\pi$
$181$		−	10.2246i		−	0.759991i	0.924988	−	0.379995i	$-0.124074\pi$
$182$		0			0
$183$	5.08137			0.375625
$184$		0			0
$185$			2.23261i			0.164145i
$186$		0			0
$187$		−	0.537779i		−	0.0393263i
$188$		0			0
$189$		−	9.11205i		−	0.662804i
$190$		0			0
$191$		−	14.8356i		−	1.07347i	−0.843751	−	0.536735i	$-0.819658\pi$
$191$		−	14.8356i		−	1.07347i	0.843751	−	0.536735i	$-0.180342\pi$
$192$		0			0
$193$			21.0892i			1.51803i	0.651073	+	0.759015i	$0.274319\pi$
$193$			21.0892i			1.51803i	−0.651073	+	0.759015i	$0.725681\pi$
$194$		0			0
$195$		−	0.0348075i		−	0.00249262i
$196$		0			0
$197$	−22.6632			−1.61468			−0.807342	−	0.590083i	$-0.799095\pi$
$197$	−22.6632			−1.61468			−0.807342	+	0.590083i	$0.799095\pi$
$198$		0			0
$199$			17.2587i			1.22344i	0.791075	+	0.611719i	$0.209522\pi$
$199$			17.2587i			1.22344i	−0.791075	+	0.611719i	$0.790478\pi$
$200$		0			0
$201$	2.35927			0.166410
$202$		0			0
$203$	−12.2964			−0.863038
$204$		0			0
$205$		−	11.0470i		−	0.771553i
$206$		0			0
$207$		−	5.84542i		−	0.406285i
$208$		0			0
$209$	−2.83745	+	3.30891i	−0.196271	+	0.228882i
$210$		0			0
$211$	7.99157			0.550162			0.275081	−	0.961421i	$-0.411295\pi$
$211$	7.99157			0.550162			0.275081	+	0.961421i	$0.411295\pi$
$212$		0			0
$213$	6.14140			0.420802
$214$		0			0
$215$		−	8.46031i		−	0.576988i
$216$		0			0
$217$		−	10.7659i		−	0.730839i
$218$		0			0
$219$	9.65059			0.652126
$220$		0			0
$221$		−	0.0296242i		−	0.00199274i
$222$		0			0
$223$	16.6863			1.11740			0.558699	−	0.829371i	$-0.311301\pi$
$223$	16.6863			1.11740			0.558699	+	0.829371i	$0.311301\pi$
$224$		0			0
$225$	10.5018			0.700120
$226$		0			0
$227$	15.2658			1.01322			0.506612	−	0.862174i	$-0.330898\pi$
$227$	15.2658			1.01322			0.506612	+	0.862174i	$0.330898\pi$
$228$		0			0
$229$	−11.7379			−0.775664			−0.387832	−	0.921730i	$-0.626776\pi$
$229$	−11.7379			−0.775664			−0.387832	+	0.921730i	$0.626776\pi$
$230$		0			0
$231$	−1.65882			−0.109142
$232$		0			0
$233$	−7.44556			−0.487775			−0.243887	−	0.969804i	$-0.578423\pi$
$233$	−7.44556			−0.487775			−0.243887	+	0.969804i	$0.578423\pi$
$234$		0			0
$235$			3.74895i			0.244555i
$236$		0			0
$237$	11.8280			0.768313
$238$		0			0
$239$			0.348169i			0.0225212i	0.999937	+	0.0112606i	$0.00358443\pi$
$239$			0.348169i			0.0225212i	−0.999937	+	0.0112606i	$0.996416\pi$
$240$		0			0
$241$			5.85132i			0.376917i	0.982081	+	0.188458i	$0.0603490\pi$
$241$			5.85132i			0.376917i	−0.982081	+	0.188458i	$0.939651\pi$
$242$		0			0
$243$	−15.0753			−0.967079
$244$		0			0
$245$	1.39470			0.0891041
$246$		0			0
$247$	−0.156304	+	0.182275i	−0.00994541	+	0.0115979i
$248$		0			0
$249$		−	2.75625i		−	0.174670i
$250$		0			0
$251$		−	0.830707i		−	0.0524338i	−0.999656	−	0.0262169i	$-0.991654\pi$
$251$		−	0.830707i		−	0.0524338i	0.999656	−	0.0262169i	$-0.00834605\pi$
$252$		0			0
$253$	−2.34463			−0.147406
$254$		0			0
$255$	0.339808			0.0212796
$256$		0			0
$257$			6.28344i			0.391950i	0.980609	+	0.195975i	$0.0627872\pi$
$257$			6.28344i			0.391950i	−0.980609	+	0.195975i	$0.937213\pi$
$258$		0			0
$259$	5.86114			0.364193
$260$		0			0
$261$			13.1577i			0.814439i
$262$		0			0
$263$		−	10.3376i		−	0.637444i	−0.947848	−	0.318722i	$-0.896746\pi$
$263$		−	10.3376i		−	0.637444i	0.947848	−	0.318722i	$-0.103254\pi$
$264$		0			0
$265$		−	4.39374i		−	0.269905i
$266$		0			0
$267$		−	2.69075i		−	0.164671i
$268$		0			0
$269$		−	21.8304i		−	1.33102i	−0.746387	−	0.665512i	$-0.768213\pi$
$269$		−	21.8304i		−	1.33102i	0.746387	−	0.665512i	$-0.231787\pi$
$270$		0			0
$271$		−	26.9412i		−	1.63656i	−0.574817	−	0.818282i	$-0.694927\pi$
$271$		−	26.9412i		−	1.63656i	0.574817	−	0.818282i	$-0.305073\pi$
$272$		0			0
$273$	−0.0913779			−0.00553044
$274$		0			0
$275$		−	4.21233i		−	0.254013i
$276$		0			0
$277$	−18.5568			−1.11497			−0.557485	−	0.830187i	$-0.688233\pi$
$277$	−18.5568			−1.11497			−0.557485	+	0.830187i	$0.688233\pi$
$278$		0			0
$279$	−11.5200			−0.689684
$280$		0			0
$281$		−	24.5952i		−	1.46723i	−0.679568	−	0.733613i	$-0.737833\pi$
$281$		−	24.5952i		−	1.46723i	0.679568	−	0.733613i	$-0.262167\pi$
$282$		0			0
$283$		−	4.20154i		−	0.249756i	−0.992172	−	0.124878i	$-0.960146\pi$
$283$		−	4.20154i		−	0.249756i	0.992172	−	0.124878i	$-0.0398539\pi$
$284$		0			0
$285$	−2.09081	−	1.79291i	−0.123849	−	0.106203i
$286$		0			0
$287$	−29.0009			−1.71187
$288$		0			0
$289$	−16.7108			−0.982988
$290$		0			0
$291$			12.5269i			0.734342i
$292$		0			0
$293$			12.4922i			0.729802i	0.931046	+	0.364901i	$0.118897\pi$
$293$			12.4922i			0.729802i	−0.931046	+	0.364901i	$0.881103\pi$
$294$		0			0
$295$	6.09243			0.354715
$296$		0			0
$297$			3.91089i			0.226933i
$298$		0			0
$299$	−0.129157			−0.00746934
$300$		0			0
$301$	−22.2103			−1.28018
$302$		0			0
$303$	8.88471			0.510413
$304$		0			0
$305$	−6.33425			−0.362698
$306$		0			0
$307$	16.5197			0.942830			0.471415	−	0.881911i	$-0.343743\pi$
$307$	16.5197			0.942830			0.471415	+	0.881911i	$0.343743\pi$
$308$		0			0
$309$	−3.24186			−0.184423
$310$		0			0
$311$			7.94014i			0.450244i	0.974331	+	0.225122i	$0.0722780\pi$
$311$			7.94014i			0.450244i	−0.974331	+	0.225122i	$0.927722\pi$
$312$		0			0
$313$	13.6803			0.773258			0.386629	−	0.922235i	$-0.373639\pi$
$313$	13.6803			0.773258			0.386629	+	0.922235i	$0.373639\pi$
$314$		0			0
$315$			5.15530i			0.290468i
$316$		0			0
$317$		−	20.8148i		−	1.16907i	−0.811367	−	0.584537i	$-0.801276\pi$
$317$		−	20.8148i		−	1.16907i	0.811367	−	0.584537i	$-0.198724\pi$
$318$		0			0
$319$	5.27761			0.295489
$320$		0			0
$321$	6.05989			0.338230
$322$		0			0
$323$	−1.77946	−	1.52592i	−0.0990120	−	0.0849046i
$324$		0			0
$325$		−	0.232041i		−	0.0128713i
$326$		0			0
$327$			9.95435i			0.550477i
$328$		0			0
$329$	9.84189			0.542601
$330$		0			0
$331$	10.5353			0.579072			0.289536	−	0.957167i	$-0.406499\pi$
$331$	10.5353			0.579072			0.289536	+	0.957167i	$0.406499\pi$
$332$		0			0
$333$		−	6.27166i		−	0.343685i
$334$		0			0
$335$	−2.94098			−0.160683
$336$		0			0
$337$		−	24.7712i		−	1.34937i	−0.738105	−	0.674686i	$-0.764279\pi$
$337$		−	24.7712i		−	1.34937i	0.738105	−	0.674686i	$-0.235721\pi$
$338$		0			0
$339$		−	0.367172i		−	0.0199420i
$340$		0			0
$341$			4.62073i			0.250227i
$342$		0			0
$343$		−	19.9708i		−	1.07832i
$344$		0			0
$345$		−	1.48151i		−	0.0797617i
$346$		0			0
$347$		−	22.4906i		−	1.20736i	−0.797226	−	0.603681i	$-0.793700\pi$
$347$		−	22.4906i		−	1.20736i	0.797226	−	0.603681i	$-0.206300\pi$
$348$		0			0
$349$	−18.2505			−0.976925			−0.488462	−	0.872585i	$-0.662442\pi$
$349$	−18.2505			−0.976925			−0.488462	+	0.872585i	$0.662442\pi$
$350$		0			0
$351$			0.215436i			0.0114991i
$352$		0			0
$353$	33.3802			1.77665			0.888324	−	0.459217i	$-0.151870\pi$
$353$	33.3802			1.77665			0.888324	+	0.459217i	$0.151870\pi$
$354$		0			0
$355$	−7.65565			−0.406320
$356$		0			0
$357$		−	0.892077i		−	0.0472137i
$358$		0			0
$359$			23.5969i			1.24540i	0.782462	+	0.622698i	$0.213964\pi$
$359$			23.5969i			1.24540i	−0.782462	+	0.622698i	$0.786036\pi$
$360$		0			0
$361$	2.89775	+	18.7777i	0.152513	+	0.988301i
$362$		0			0
$363$	0.711963			0.0373684
$364$		0			0
$365$	−12.0301			−0.629683
$366$		0			0
$367$		−	7.74050i		−	0.404051i	−0.979380	−	0.202025i	$-0.935248\pi$
$367$		−	7.74050i		−	0.404051i	0.979380	−	0.202025i	$-0.0647523\pi$
$368$		0			0
$369$			31.0321i			1.61547i
$370$		0			0
$371$	−11.5346			−0.598847
$372$		0			0
$373$		−	35.3793i		−	1.83187i	−0.401323	−	0.915936i	$-0.631450\pi$
$373$		−	35.3793i		−	1.83187i	0.401323	−	0.915936i	$-0.368550\pi$
$374$		0			0
$375$	5.82102			0.300596
$376$		0			0
$377$	0.290724			0.0149730
$378$		0			0
$379$	−19.5036			−1.00183			−0.500917	−	0.865496i	$-0.667004\pi$
$379$	−19.5036			−1.00183			−0.500917	+	0.865496i	$0.667004\pi$
$380$		0			0
$381$	−10.5774			−0.541895
$382$		0			0
$383$	30.8221			1.57493			0.787467	−	0.616357i	$-0.211392\pi$
$383$	30.8221			1.57493			0.787467	+	0.616357i	$0.211392\pi$
$384$		0			0
$385$	2.06782			0.105386
$386$		0			0
$387$			23.7660i			1.20809i
$388$		0			0
$389$	24.3756			1.23589			0.617946	−	0.786220i	$-0.287965\pi$
$389$	24.3756			1.23589			0.617946	+	0.786220i	$0.287965\pi$
$390$		0			0
$391$		−	1.26089i		−	0.0637662i
$392$		0			0
$393$		−	2.47566i		−	0.124881i
$394$		0			0
$395$	−14.7444			−0.741872
$396$		0			0
$397$	−9.60268			−0.481945			−0.240972	−	0.970532i	$-0.577466\pi$
$397$	−9.60268			−0.481945			−0.240972	+	0.970532i	$0.577466\pi$
$398$		0			0
$399$	−4.70681	+	5.48887i	−0.235635	+	0.274787i
$400$		0			0
$401$			28.4337i			1.41991i	0.704246	+	0.709956i	$0.251285\pi$
$401$			28.4337i			1.41991i	−0.704246	+	0.709956i	$0.748715\pi$
$402$		0			0
$403$			0.254539i			0.0126795i
$404$		0			0
$405$	4.16678			0.207049
$406$		0			0
$407$	−2.51560			−0.124694
$408$		0			0
$409$		−	4.08938i		−	0.202207i	−0.994876	−	0.101103i	$-0.967763\pi$
$409$		−	4.08938i		−	0.202207i	0.994876	−	0.101103i	$-0.0322373\pi$
$410$		0			0
$411$	8.99512			0.443696
$412$		0			0
$413$		−	15.9941i		−	0.787017i
$414$		0			0
$415$			3.43585i			0.168659i
$416$		0			0
$417$			3.33054i			0.163097i
$418$		0			0
$419$			3.79345i			0.185322i	0.995698	+	0.0926611i	$0.0295373\pi$
$419$			3.79345i			0.185322i	−0.995698	+	0.0926611i	$0.970463\pi$
$420$		0			0
$421$		−	30.3197i		−	1.47769i	−0.673875	−	0.738846i	$-0.735371\pi$
$421$		−	30.3197i		−	1.47769i	0.673875	−	0.738846i	$-0.264629\pi$
$422$		0			0
$423$		−	10.5312i		−	0.512046i
$424$		0			0
$425$	2.26530			0.109883
$426$		0			0
$427$			16.6289i			0.804729i
$428$		0			0
$429$	0.0392193			0.00189353
$430$		0			0
$431$	15.0264			0.723797			0.361899	−	0.932217i	$-0.382129\pi$
$431$	15.0264			0.723797			0.361899	+	0.932217i	$0.382129\pi$
$432$		0			0
$433$			25.3689i			1.21915i	0.792727	+	0.609577i	$0.208661\pi$
$433$			25.3689i			1.21915i	−0.792727	+	0.609577i	$0.791339\pi$
$434$		0			0
$435$			3.33478i			0.159890i
$436$		0			0
$437$	−6.65277	+	7.75817i	−0.318245	+	0.371124i
$438$		0			0
$439$	−22.5729			−1.07735			−0.538673	−	0.842515i	$-0.681074\pi$
$439$	−22.5729			−1.07735			−0.538673	+	0.842515i	$0.681074\pi$
$440$		0			0
$441$	−3.91787			−0.186565
$442$		0			0
$443$			15.5170i			0.737233i	0.929582	+	0.368617i	$0.120168\pi$
$443$			15.5170i			0.737233i	−0.929582	+	0.368617i	$0.879832\pi$
$444$		0			0
$445$			3.35420i			0.159004i
$446$		0			0
$447$	−4.18634			−0.198007
$448$		0			0
$449$			30.5368i			1.44112i	0.693391	+	0.720561i	$0.256116\pi$
$449$			30.5368i			1.44112i	−0.693391	+	0.720561i	$0.743884\pi$
$450$		0			0
$451$	12.4472			0.586114
$452$		0			0
$453$	10.9284			0.513461
$454$		0			0
$455$	0.113908			0.00534011
$456$		0			0
$457$	−1.49042			−0.0697188			−0.0348594	−	0.999392i	$-0.511098\pi$
$457$	−1.49042			−0.0697188			−0.0348594	+	0.999392i	$0.511098\pi$
$458$		0			0
$459$	−2.10320			−0.0981688
$460$		0			0
$461$	13.3751			0.622939			0.311470	−	0.950256i	$-0.399179\pi$
$461$	13.3751			0.622939			0.311470	+	0.950256i	$0.399179\pi$
$462$		0			0
$463$		−	3.12253i		−	0.145116i	−0.997364	−	0.0725581i	$-0.976884\pi$
$463$		−	3.12253i		−	0.145116i	0.997364	−	0.0725581i	$-0.0231163\pi$
$464$		0			0
$465$	−2.91972			−0.135399
$466$		0			0
$467$			40.7800i			1.88707i	0.331268	+	0.943537i	$0.392523\pi$
$467$			40.7800i			1.88707i	−0.331268	+	0.943537i	$0.607477\pi$
$468$		0			0
$469$			7.72078i			0.356513i
$470$		0			0
$471$	3.51097			0.161777
$472$		0			0
$473$	9.53266			0.438312
$474$		0			0
$475$	−13.9382	−	11.9523i	−0.639529	−	0.548408i
$476$		0			0
$477$			12.3425i			0.565124i
$478$		0			0
$479$		−	13.7350i		−	0.627568i	−0.949494	−	0.313784i	$-0.898403\pi$
$479$		−	13.7350i		−	0.627568i	0.949494	−	0.313784i	$-0.101597\pi$
$480$		0			0
$481$	−0.138575			−0.00631846
$482$		0			0
$483$	−3.88931			−0.176970
$484$		0			0
$485$		−	15.6156i		−	0.709070i
$486$		0			0
$487$	−26.9267			−1.22017			−0.610083	−	0.792338i	$-0.708864\pi$
$487$	−26.9267			−1.22017			−0.610083	+	0.792338i	$0.708864\pi$
$488$		0			0
$489$			15.3774i			0.695388i
$490$		0			0
$491$		−	27.7745i		−	1.25345i	−0.779242	−	0.626723i	$-0.784396\pi$
$491$		−	27.7745i		−	1.25345i	0.779242	−	0.626723i	$-0.215604\pi$
$492$		0			0
$493$			2.83819i			0.127826i
$494$		0			0
$495$		−	2.21265i		−	0.0994514i
$496$		0			0
$497$			20.0979i			0.901514i
$498$		0			0
$499$			13.0780i			0.585450i	0.956197	+	0.292725i	$0.0945621\pi$
$499$			13.0780i			0.585450i	−0.956197	+	0.292725i	$0.905438\pi$
$500$		0			0
$501$	−10.0385			−0.448488
$502$		0			0
$503$			24.8338i			1.10728i	0.832755	+	0.553642i	$0.186762\pi$
$503$			24.8338i			1.10728i	−0.832755	+	0.553642i	$0.813238\pi$
$504$		0			0
$505$	−11.0754			−0.492847
$506$		0			0
$507$	−9.25336			−0.410956
$508$		0			0
$509$			28.0888i			1.24501i	0.782614	+	0.622507i	$0.213886\pi$
$509$			28.0888i			1.24501i	−0.782614	+	0.622507i	$0.786114\pi$
$510$		0			0
$511$			31.5818i			1.39710i
$512$		0			0
$513$	12.9408	+	11.0970i	0.571349	+	0.489943i
$514$		0			0
$515$	4.04119			0.178076
$516$		0			0
$517$	−4.22413			−0.185777
$518$		0			0
$519$		−	3.08331i		−	0.135342i
$520$		0			0
$521$		−	23.8899i		−	1.04663i	−0.852138	−	0.523317i	$-0.824694\pi$
$521$		−	23.8899i		−	1.04663i	0.852138	−	0.523317i	$-0.175306\pi$
$522$		0			0
$523$	23.7408			1.03811			0.519057	−	0.854739i	$-0.326283\pi$
$523$	23.7408			1.03811			0.519057	+	0.854739i	$0.326283\pi$
$524$		0			0
$525$		−	6.98748i		−	0.304959i
$526$		0			0
$527$	−2.48494			−0.108245
$528$		0			0
$529$	17.5027			0.760987
$530$		0			0
$531$	−17.1143			−0.742698
$532$		0			0
$533$	0.685668			0.0296996
$534$		0			0
$535$	−7.55404			−0.326590
$536$		0			0
$537$	17.8755			0.771383
$538$		0			0
$539$			1.57148i			0.0676884i
$540$		0			0
$541$	−2.19316			−0.0942913			−0.0471457	−	0.998888i	$-0.515012\pi$
$541$	−2.19316			−0.0942913			−0.0471457	+	0.998888i	$0.515012\pi$
$542$		0			0
$543$			7.27956i			0.312396i
$544$		0			0
$545$		−	12.4087i		−	0.531532i
$546$		0			0
$547$	10.4939			0.448688			0.224344	−	0.974510i	$-0.427976\pi$
$547$	10.4939			0.448688			0.224344	+	0.974510i	$0.427976\pi$
$548$		0			0
$549$	17.7936			0.759413
$550$		0			0
$551$	14.9750	−	17.4631i	0.637955	−	0.743954i
$552$		0			0
$553$			38.7076i			1.64601i
$554$		0			0
$555$		−	1.58954i		−	0.0674721i
$556$		0			0
$557$	−3.87419			−0.164155			−0.0820773	−	0.996626i	$-0.526155\pi$
$557$	−3.87419			−0.164155			−0.0820773	+	0.996626i	$0.526155\pi$
$558$		0			0
$559$	0.525118			0.0222101
$560$		0			0
$561$			0.382879i			0.0161652i
$562$		0			0
$563$	−26.7084			−1.12562			−0.562812	−	0.826585i	$-0.690281\pi$
$563$	−26.7084			−1.12562			−0.562812	+	0.826585i	$0.690281\pi$
$564$		0			0
$565$			0.457703i			0.0192557i
$566$		0			0
$567$		−	10.9388i		−	0.459386i
$568$		0			0
$569$			18.8604i			0.790669i	0.918537	+	0.395335i	$0.129371\pi$
$569$			18.8604i			0.790669i	−0.918537	+	0.395335i	$0.870629\pi$
$570$		0			0
$571$		−	22.4837i		−	0.940914i	−0.882423	−	0.470457i	$-0.844089\pi$
$571$		−	22.4837i		−	0.940914i	0.882423	−	0.470457i	$-0.155911\pi$
$572$		0			0
$573$			10.5624i			0.441252i
$574$		0			0
$575$		−	9.87636i		−	0.411873i
$576$		0			0
$577$	24.3471			1.01358			0.506791	−	0.862069i	$-0.330832\pi$
$577$	24.3471			1.01358			0.506791	+	0.862069i	$0.330832\pi$
$578$		0			0
$579$		−	15.0147i		−	0.623990i
$580$		0			0
$581$	9.01991			0.374209
$582$		0			0
$583$	4.95065			0.205035
$584$		0			0
$585$		−	0.121887i		−	0.00503940i
$586$		0			0
$587$		−	9.65715i		−	0.398593i	−0.979939	−	0.199297i	$-0.936134\pi$
$587$		−	9.65715i		−	0.398593i	0.979939	−	0.199297i	$-0.0638657\pi$
$588$		0			0
$589$	15.2896	+	13.1111i	0.629996	+	0.540234i
$590$		0			0
$591$	16.1353			0.663719
$592$		0			0
$593$	36.6414			1.50468			0.752341	−	0.658774i	$-0.228925\pi$
$593$	36.6414			1.50468			0.752341	+	0.658774i	$0.228925\pi$
$594$		0			0
$595$			1.11203i			0.0455889i
$596$		0			0
$597$		−	12.2876i		−	0.502897i
$598$		0			0
$599$	−9.63445			−0.393653			−0.196827	−	0.980438i	$-0.563064\pi$
$599$	−9.63445			−0.393653			−0.196827	+	0.980438i	$0.563064\pi$
$600$		0			0
$601$			7.62962i			0.311219i	0.987819	+	0.155609i	$0.0497341\pi$
$601$			7.62962i			0.311219i	−0.987819	+	0.155609i	$0.950266\pi$
$602$		0			0
$603$	8.26155			0.336436
$604$		0			0
$605$	−0.887508			−0.0360823
$606$		0			0
$607$	22.8697			0.928251			0.464126	−	0.885769i	$-0.346369\pi$
$607$	22.8697			0.928251			0.464126	+	0.885769i	$0.346369\pi$
$608$		0			0
$609$	8.75458			0.354754
$610$		0			0
$611$	−0.232692			−0.00941369
$612$		0			0
$613$	38.2107			1.54331			0.771657	−	0.636039i	$-0.219428\pi$
$613$	38.2107			1.54331			0.771657	+	0.636039i	$0.219428\pi$
$614$		0			0
$615$			7.86503i			0.317148i
$616$		0			0
$617$	12.6009			0.507294			0.253647	−	0.967297i	$-0.418370\pi$
$617$	12.6009			0.507294			0.253647	+	0.967297i	$0.418370\pi$
$618$		0			0
$619$			37.7476i			1.51720i	0.651554	+	0.758602i	$0.274117\pi$
$619$			37.7476i			1.51720i	−0.651554	+	0.758602i	$0.725883\pi$
$620$		0			0
$621$			9.16959i			0.367963i
$622$		0			0
$623$	8.80556			0.352787
$624$		0			0
$625$	13.8054			0.552215
$626$		0			0
$627$	2.02016	−	2.35582i	0.0806774	−	0.0940824i
$628$		0			0
$629$		−	1.35284i		−	0.0539411i
$630$		0			0
$631$			35.1749i			1.40029i	0.714000	+	0.700146i	$0.246882\pi$
$631$			35.1749i			1.40029i	−0.714000	+	0.700146i	$0.753118\pi$
$632$		0			0
$633$	−5.68970			−0.226145
$634$		0			0
$635$	13.1854			0.523246
$636$		0			0
$637$			0.0865668i			0.00342990i
$638$		0			0
$639$	21.5056			0.850748
$640$		0			0
$641$			37.8964i			1.49682i	0.663238	+	0.748409i	$0.269182\pi$
$641$			37.8964i			1.49682i	−0.663238	+	0.748409i	$0.730818\pi$
$642$		0			0
$643$		−	36.8841i		−	1.45457i	−0.686337	−	0.727284i	$-0.740783\pi$
$643$		−	36.8841i		−	1.45457i	0.686337	−	0.727284i	$-0.259217\pi$
$644$		0			0
$645$			6.02343i			0.237172i
$646$		0			0
$647$		−	38.5863i		−	1.51698i	−0.651682	−	0.758492i	$-0.725937\pi$
$647$		−	38.5863i		−	1.51698i	0.651682	−	0.758492i	$-0.274063\pi$
$648$		0			0
$649$			6.86465i			0.269461i
$650$		0			0
$651$			7.66495i			0.300413i
$652$		0			0
$653$	−8.82383			−0.345303			−0.172652	−	0.984983i	$-0.555233\pi$
$653$	−8.82383			−0.345303			−0.172652	+	0.984983i	$0.555233\pi$
$654$		0			0
$655$			3.08607i			0.120583i
$656$		0			0
$657$	33.7939			1.31842
$658$		0			0
$659$	34.3489			1.33804			0.669021	−	0.743243i	$-0.266713\pi$
$659$	34.3489			1.33804			0.669021	+	0.743243i	$0.266713\pi$
$660$		0			0
$661$		−	12.1440i		−	0.472345i	−0.971711	−	0.236173i	$-0.924107\pi$
$661$		−	12.1440i		−	0.472345i	0.971711	−	0.236173i	$-0.0758931\pi$
$662$		0			0
$663$			0.0210914i			0.000819120i
$664$		0			0
$665$	5.86734	−	6.84223i	0.227526	−	0.265330i
$666$		0			0
$667$	12.3741			0.479125
$668$		0			0
$669$	−11.8800			−0.459309
$670$		0			0
$671$		−	7.13712i		−	0.275526i
$672$		0			0
$673$			24.5180i			0.945099i	0.881304	+	0.472550i	$0.156666\pi$
$673$			24.5180i			0.945099i	−0.881304	+	0.472550i	$0.843334\pi$
$674$		0			0
$675$	−16.4740			−0.634083
$676$		0			0
$677$			36.9924i			1.42173i	0.703327	+	0.710866i	$0.251697\pi$
$677$			36.9924i			1.42173i	−0.703327	+	0.710866i	$0.748303\pi$
$678$		0			0
$679$	−40.9948			−1.57323
$680$		0			0
$681$	−10.8687			−0.416488
$682$		0			0
$683$	23.5821			0.902345			0.451173	−	0.892437i	$-0.351006\pi$
$683$	23.5821			0.902345			0.451173	+	0.892437i	$0.351006\pi$
$684$		0			0
$685$	−11.2130			−0.428427
$686$		0			0
$687$	8.35697			0.318838
$688$		0			0
$689$	0.272712			0.0103895
$690$		0			0
$691$			8.28297i			0.315099i	0.987511	+	0.157550i	$0.0503594\pi$
$691$			8.28297i			0.315099i	−0.987511	+	0.157550i	$0.949641\pi$
$692$		0			0
$693$	−5.80874			−0.220656
$694$		0			0
$695$		−	4.15173i		−	0.157484i
$696$		0			0
$697$			6.69383i			0.253547i
$698$		0			0
$699$	5.30096			0.200501
$700$		0			0
$701$	−6.40604			−0.241953			−0.120976	−	0.992655i	$-0.538603\pi$
$701$	−6.40604			−0.241953			−0.120976	+	0.992655i	$0.538603\pi$
$702$		0			0
$703$	−7.13788	+	8.32388i	−0.269210	+	0.313941i
$704$		0			0
$705$		−	2.66911i		−	0.100525i
$706$		0			0
$707$			29.0755i			1.09350i
$708$		0			0
$709$	−29.8898			−1.12254			−0.561268	−	0.827634i	$-0.689686\pi$
$709$	−29.8898			−1.12254			−0.561268	+	0.827634i	$0.689686\pi$
$710$		0			0
$711$	41.4187			1.55332
$712$		0			0
$713$			10.8339i			0.405733i
$714$		0			0
$715$	−0.0488895			−0.00182836
$716$		0			0
$717$		−	0.247884i		−	0.00925738i
$718$		0			0
$719$			43.9435i			1.63881i	0.573212	+	0.819407i	$0.305697\pi$
$719$			43.9435i			1.63881i	−0.573212	+	0.819407i	$0.694303\pi$
$720$		0			0
$721$		−	10.6091i		−	0.395103i
$722$		0			0
$723$		−	4.16592i		−	0.154932i
$724$		0			0
$725$			22.2310i			0.825640i
$726$		0			0
$727$		−	0.462251i		−	0.0171439i	−0.999963	−	0.00857197i	$-0.997271\pi$
$727$		−	0.462251i		−	0.0171439i	0.999963	−	0.00857197i	$-0.00272858\pi$
$728$		0			0
$729$	−3.35172			−0.124138
$730$		0			0
$731$			5.12647i			0.189609i
$732$		0			0
$733$	7.69709			0.284299			0.142149	−	0.989845i	$-0.454599\pi$
$733$	7.69709			0.284299			0.142149	+	0.989845i	$0.454599\pi$
$734$		0			0
$735$	−0.992975			−0.0366264
$736$		0			0
$737$		−	3.31376i		−	0.122064i
$738$		0			0
$739$		−	13.6855i		−	0.503428i	−0.967802	−	0.251714i	$-0.919006\pi$
$739$		−	13.6855i		−	0.503428i	0.967802	−	0.251714i	$-0.0809943\pi$
$740$		0			0
$741$	0.111283	−	0.129773i	0.00408808	−	0.00476734i
$742$		0			0
$743$	36.9287			1.35478			0.677392	−	0.735622i	$-0.263110\pi$
$743$	36.9287			1.35478			0.677392	+	0.735622i	$0.263110\pi$
$744$		0			0
$745$	5.21855			0.191193
$746$		0			0
$747$		−	9.65168i		−	0.353136i
$748$		0			0
$749$			19.8312i			0.724615i
$750$		0			0
$751$	−36.9950			−1.34997			−0.674983	−	0.737834i	$-0.735849\pi$
$751$	−36.9950			−1.34997			−0.674983	+	0.737834i	$0.735849\pi$
$752$		0			0
$753$			0.591433i			0.0215530i
$754$		0			0
$755$	−13.6230			−0.495791
$756$		0			0
$757$	−15.2818			−0.555426			−0.277713	−	0.960664i	$-0.589576\pi$
$757$	−15.2818			−0.555426			−0.277713	+	0.960664i	$0.589576\pi$
$758$		0			0
$759$	1.66929			0.0605914
$760$		0			0
$761$	30.3286			1.09941			0.549706	−	0.835358i	$-0.314740\pi$
$761$	30.3286			1.09941			0.549706	+	0.835358i	$0.314740\pi$
$762$		0			0
$763$	−32.5759			−1.17933
$764$		0			0
$765$	1.18992			0.0430216
$766$		0			0
$767$			0.378148i			0.0136541i
$768$		0			0
$769$	−3.48948			−0.125834			−0.0629170	−	0.998019i	$-0.520040\pi$
$769$	−3.48948			−0.125834			−0.0629170	+	0.998019i	$0.520040\pi$
$770$		0			0
$771$		−	4.47358i		−	0.161112i
$772$		0			0
$773$			43.1010i			1.55024i	0.631817	+	0.775118i	$0.282309\pi$
$773$			43.1010i			1.55024i	−0.631817	+	0.775118i	$0.717691\pi$
$774$		0			0
$775$	−19.4641			−0.699169
$776$		0			0
$777$	−4.17291			−0.149702
$778$		0			0
$779$	35.3182	−	41.1865i	1.26541	−	1.47566i
$780$		0			0
$781$		−	8.62601i		−	0.308663i
$782$		0			0
$783$		−	20.6402i		−	0.737619i
$784$		0			0
$785$	−4.37665			−0.156209
$786$		0			0
$787$	32.4792			1.15776			0.578879	−	0.815413i	$-0.303490\pi$
$787$	32.4792			1.15776			0.578879	+	0.815413i	$0.303490\pi$
$788$		0			0
$789$			7.35999i			0.262023i
$790$		0			0
$791$	1.20158			0.0427233
$792$		0			0
$793$		−	0.393157i		−	0.0139614i
$794$		0			0
$795$			3.12818i			0.110945i
$796$		0			0
$797$		−	33.2354i		−	1.17726i	−0.808404	−	0.588628i	$-0.799668\pi$
$797$		−	33.2354i		−	1.17726i	0.808404	−	0.588628i	$-0.200332\pi$
$798$		0			0
$799$		−	2.27165i		−	0.0803653i
$800$		0			0
$801$		−	9.42231i		−	0.332921i
$802$		0			0
$803$		−	13.5549i		−	0.478342i
$804$		0			0
$805$	4.84828			0.170879
$806$		0			0
$807$			15.5425i			0.547120i
$808$		0			0
$809$	8.09578			0.284633			0.142316	−	0.989821i	$-0.454545\pi$
$809$	8.09578			0.284633			0.142316	+	0.989821i	$0.454545\pi$
$810$		0			0
$811$	−34.1041			−1.19756			−0.598779	−	0.800915i	$-0.704347\pi$
$811$	−34.1041			−1.19756			−0.598779	+	0.800915i	$0.704347\pi$
$812$		0			0
$813$			19.1812i			0.672713i
$814$		0			0
$815$		−	19.1689i		−	0.671457i
$816$		0			0
$817$	27.0484	−	31.5427i	0.946305	−	1.10354i
$818$		0			0
$819$	−0.319982			−0.0111811
$820$		0			0
$821$	28.2388			0.985539			0.492770	−	0.870160i	$-0.335985\pi$
$821$	28.2388			0.985539			0.492770	+	0.870160i	$0.335985\pi$
$822$		0			0
$823$		−	39.9408i		−	1.39225i	−0.717922	−	0.696124i	$-0.754906\pi$
$823$		−	39.9408i		−	1.39225i	0.717922	−	0.696124i	$-0.245094\pi$
$824$		0			0
$825$			2.99902i			0.104413i
$826$		0			0
$827$	6.23108			0.216676			0.108338	−	0.994114i	$-0.465447\pi$
$827$	6.23108			0.216676			0.108338	+	0.994114i	$0.465447\pi$
$828$		0			0
$829$		−	43.8585i		−	1.52327i	−0.648008	−	0.761634i	$-0.724398\pi$
$829$		−	43.8585i		−	1.52327i	0.648008	−	0.761634i	$-0.275602\pi$
$830$		0			0
$831$	13.2117			0.458310
$832$		0			0
$833$	−0.845109			−0.0292813
$834$		0			0
$835$	12.5137			0.433054
$836$		0			0
$837$	18.0712			0.624631
$838$		0			0
$839$	−9.61542			−0.331961			−0.165981	−	0.986129i	$-0.553079\pi$
$839$	−9.61542			−0.331961			−0.165981	+	0.986129i	$0.553079\pi$
$840$		0			0
$841$	1.14683			0.0395457
$842$		0			0
$843$			17.5109i			0.603106i
$844$		0			0
$845$	11.5349			0.396813
$846$		0			0
$847$			2.32992i			0.0800570i
$848$		0			0
$849$			2.99134i			0.102663i
$850$		0			0
$851$	−5.89815			−0.202186
$852$		0			0
$853$	−31.8247			−1.08966			−0.544829	−	0.838547i	$-0.683405\pi$
$853$	−31.8247			−1.08966			−0.544829	+	0.838547i	$0.683405\pi$
$854$		0			0
$855$	−7.32147	−	6.27830i	−0.250389	−	0.214713i
$856$		0			0
$857$			16.0170i			0.547132i	0.961853	+	0.273566i	$0.0882032\pi$
$857$			16.0170i			0.547132i	−0.961853	+	0.273566i	$0.911797\pi$
$858$		0			0
$859$			13.6556i			0.465924i	0.972486	+	0.232962i	$0.0748418\pi$
$859$			13.6556i			0.465924i	−0.972486	+	0.232962i	$0.925158\pi$
$860$		0			0
$861$	20.6476			0.703667
$862$		0			0
$863$	−38.0815			−1.29631			−0.648155	−	0.761508i	$-0.724459\pi$
$863$	−38.0815			−1.29631			−0.648155	+	0.761508i	$0.724459\pi$
$864$		0			0
$865$			3.84355i			0.130684i
$866$		0			0
$867$	11.8975			0.404059
$868$		0			0
$869$		−	16.6133i		−	0.563566i
$870$		0			0
$871$		−	0.182542i		−	0.00618521i
$872$		0			0
$873$			43.8661i			1.48464i
$874$		0			0
$875$			19.0495i			0.643989i
$876$		0			0
$877$		−	20.8231i		−	0.703146i	−0.936161	−	0.351573i	$-0.885647\pi$
$877$		−	20.8231i		−	0.703146i	0.936161	−	0.351573i	$-0.114353\pi$
$878$		0			0
$879$		−	8.89399i		−	0.299987i
$880$		0			0
$881$	−34.0286			−1.14645			−0.573226	−	0.819397i	$-0.694308\pi$
$881$	−34.0286			−1.14645			−0.573226	+	0.819397i	$0.694308\pi$
$882$		0			0
$883$			45.1054i			1.51792i	0.651138	+	0.758959i	$0.274292\pi$
$883$			45.1054i			1.51792i	−0.651138	+	0.758959i	$0.725708\pi$
$884$		0			0
$885$	−4.33759			−0.145806
$886$		0			0
$887$	−38.7218			−1.30015			−0.650075	−	0.759870i	$-0.725263\pi$
$887$	−38.7218			−1.30015			−0.650075	+	0.759870i	$0.725263\pi$
$888$		0			0
$889$		−	34.6147i		−	1.16094i
$890$		0			0
$891$			4.69492i			0.157286i
$892$		0			0
$893$	−11.9858	+	13.9773i	−0.401088	+	0.467731i
$894$		0			0
$895$	−22.2829			−0.744836
$896$		0			0
$897$	0.0919549			0.00307028
$898$		0			0
$899$		−	24.3864i		−	0.813333i
$900$		0			0
$901$			2.66236i			0.0886959i
$902$		0			0
$903$	15.8129			0.526221
$904$		0			0
$905$		−	9.07444i		−	0.301645i
$906$		0			0
$907$	−6.79264			−0.225546			−0.112773	−	0.993621i	$-0.535973\pi$
$907$	−6.79264			−0.225546			−0.112773	+	0.993621i	$0.535973\pi$
$908$		0			0
$909$	31.1119			1.03192
$910$		0			0
$911$	−19.6986			−0.652645			−0.326322	−	0.945259i	$-0.605810\pi$
$911$	−19.6986			−0.652645			−0.326322	+	0.945259i	$0.605810\pi$
$912$		0			0
$913$	−3.87134			−0.128123
$914$		0			0
$915$	4.50975			0.149088
$916$		0			0
$917$	8.10167			0.267541
$918$		0			0
$919$		−	48.6293i		−	1.60413i	−0.597235	−	0.802066i	$-0.703734\pi$
$919$		−	48.6293i		−	1.60413i	0.597235	−	0.802066i	$-0.296266\pi$
$920$		0			0
$921$	−11.7614			−0.387552
$922$		0			0
$923$		−	0.475174i		−	0.0156406i
$924$		0			0
$925$		−	10.5965i		−	0.348412i
$926$		0			0
$927$	−11.3522			−0.372854
$928$		0			0
$929$	33.7546			1.10745			0.553727	−	0.832698i	$-0.313205\pi$
$929$	33.7546			1.10745			0.553727	+	0.832698i	$0.313205\pi$
$930$		0			0
$931$	5.19988	+	4.45899i	0.170419	+	0.146138i
$932$		0			0
$933$		−	5.65308i		−	0.185074i
$934$		0			0
$935$		−	0.477284i		−	0.0156088i
$936$		0			0
$937$	−16.0164			−0.523233			−0.261616	−	0.965172i	$-0.584256\pi$
$937$	−16.0164			−0.523233			−0.261616	+	0.965172i	$0.584256\pi$
$938$		0			0
$939$	−9.73989			−0.317849
$940$		0			0
$941$			6.82145i			0.222373i	0.993800	+	0.111186i	$0.0354651\pi$
$941$			6.82145i			0.222373i	−0.993800	+	0.111186i	$0.964535\pi$
$942$		0			0
$943$	29.1840			0.950363
$944$		0			0
$945$		−	8.08702i		−	0.263071i
$946$		0			0
$947$		−	3.08216i		−	0.100157i	−0.998745	−	0.0500784i	$-0.984053\pi$
$947$		−	3.08216i		−	0.100157i	0.998745	−	0.0500784i	$-0.0159471\pi$
$948$		0			0
$949$		−	0.746688i		−	0.0242385i
$950$		0			0
$951$			14.8194i			0.480550i
$952$		0			0
$953$			30.3471i			0.983037i	0.870867	+	0.491519i	$0.163558\pi$
$953$			30.3471i			0.983037i	−0.870867	+	0.491519i	$0.836442\pi$
$954$		0			0
$955$		−	13.1668i		−	0.426066i
$956$		0			0
$957$	−3.75746			−0.121462
$958$		0			0
$959$			29.4368i			0.950563i
$960$		0			0
$961$	−9.64882			−0.311252
$962$		0			0
$963$	21.2202			0.683810
$964$		0			0
$965$			18.7168i			0.602515i
$966$		0			0
$967$			50.2143i			1.61478i	0.590016	+	0.807391i	$0.299121\pi$
$967$			50.2143i			1.61478i	−0.590016	+	0.807391i	$0.700879\pi$
$968$		0			0
$969$	1.26691	+	1.08640i	0.0406991	+	0.0349002i
$970$		0			0
$971$	−57.8802			−1.85746			−0.928732	−	0.370752i	$-0.879100\pi$
$971$	−57.8802			−1.85746			−0.928732	+	0.370752i	$0.879100\pi$
$972$		0			0
$973$	−10.8993			−0.349415
$974$		0			0
$975$			0.165205i			0.00529079i
$976$		0			0
$977$			26.5595i			0.849714i	0.905261	+	0.424857i	$0.139676\pi$
$977$			26.5595i			0.849714i	−0.905261	+	0.424857i	$0.860324\pi$
$978$		0			0
$979$	−3.77934			−0.120788
$980$		0			0
$981$			34.8575i			1.11292i
$982$		0			0
$983$	−23.3968			−0.746243			−0.373121	−	0.927783i	$-0.621712\pi$
$983$	−23.3968			−0.746243			−0.373121	+	0.927783i	$0.621712\pi$
$984$		0			0
$985$	−20.1138			−0.640877
$986$		0			0
$987$	−7.00706			−0.223037
$988$		0			0
$989$	22.3506			0.710707
$990$		0			0
$991$	−21.0983			−0.670210			−0.335105	−	0.942181i	$-0.608772\pi$
$991$	−21.0983			−0.670210			−0.335105	+	0.942181i	$0.608772\pi$
$992$		0			0
$993$	−7.50074			−0.238029
$994$		0			0
$995$			15.3172i			0.485589i
$996$		0			0
$997$	−9.87104			−0.312619			−0.156309	−	0.987708i	$-0.549960\pi$
$997$	−9.87104			−0.312619			−0.156309	+	0.987708i	$0.549960\pi$
$998$		0			0
$999$			9.83822i			0.311268i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.a.1519.17	✓	36
4.3	odd	2	inner	3344.2.o.a.1519.19	yes	36
19.18	odd	2	inner	3344.2.o.a.1519.20	yes	36
76.75	even	2	inner	3344.2.o.a.1519.18	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.a.1519.17	✓	36	1.1	even	1	trivial
3344.2.o.a.1519.18	yes	36	76.75	even	2	inner
3344.2.o.a.1519.19	yes	36	4.3	odd	2	inner
3344.2.o.a.1519.20	yes	36	19.18	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)

\(f(q)\)	\(=\)	\(q-0.711963 q^{3} +0.887508 q^{5} -2.32992i q^{7} -2.49311 q^{9} +O(q^{10})\)	\(q-0.711963 q^{3} +0.887508 q^{5} -2.32992i q^{7} -2.49311 q^{9} +1.00000i q^{11} +0.0550862i q^{13} -0.631873 q^{15} -0.537779 q^{17} +(3.30891 + 2.83745i) q^{19} +1.65882i q^{21} +2.34463i q^{23} -4.21233 q^{25} +3.91089 q^{27} -5.27761i q^{29} +4.62073 q^{31} -0.711963i q^{33} -2.06782i q^{35} +2.51560i q^{37} -0.0392193i q^{39} -12.4472i q^{41} -9.53266i q^{43} -2.21265 q^{45} +4.22413i q^{47} +1.57148 q^{49} +0.382879 q^{51} -4.95065i q^{53} +0.887508i q^{55} +(-2.35582 - 2.02016i) q^{57} +6.86465 q^{59} -7.13712 q^{61} +5.80874i q^{63} +0.0488895i q^{65} -3.31376 q^{67} -1.66929i q^{69} -8.62601 q^{71} -13.5549 q^{73} +2.99902 q^{75} +2.32992 q^{77} -16.6133 q^{79} +4.69492 q^{81} +3.87134i q^{83} -0.477284 q^{85} +3.75746i q^{87} +3.77934i q^{89} +0.128346 q^{91} -3.28979 q^{93} +(2.93668 + 2.51826i) q^{95} -17.5949i q^{97} -2.49311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 44 q^{9}+O(q^{10}) \) 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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