LMFDB - Embedded newform 3344.2.o.a.1519.10

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			1519.10
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.a.1519.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71785 q^{3} -3.66778 q^{5} -2.65082i q^{7} -0.0489799 q^{9} +O(q^{10})$	$q-1.71785 q^{3} -3.66778 q^{5} -2.65082i q^{7} -0.0489799 q^{9} -1.00000i q^{11} -4.00362i q^{13} +6.30071 q^{15} +5.00587 q^{17} +(-1.66960 - 4.02647i) q^{19} +4.55372i q^{21} -0.204072i q^{23} +8.45262 q^{25} +5.23770 q^{27} +6.62744i q^{29} +8.91321 q^{31} +1.71785i q^{33} +9.72263i q^{35} -11.2007i q^{37} +6.87764i q^{39} -1.16260i q^{41} -8.59761i q^{43} +0.179648 q^{45} +7.02526i q^{47} -0.0268544 q^{49} -8.59936 q^{51} -5.64742i q^{53} +3.66778i q^{55} +(2.86813 + 6.91688i) q^{57} +10.7855 q^{59} -15.0588 q^{61} +0.129837i q^{63} +14.6844i q^{65} +9.02412 q^{67} +0.350565i q^{69} -3.63267 q^{71} -7.45373 q^{73} -14.5204 q^{75} -2.65082 q^{77} +11.2460 q^{79} -8.85066 q^{81} -3.09050i q^{83} -18.3604 q^{85} -11.3850i q^{87} +1.40070i q^{89} -10.6129 q^{91} -15.3116 q^{93} +(6.12373 + 14.7682i) q^{95} -16.0507i q^{97} +0.0489799i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.71785			−0.991803			−0.495902	−	0.868379i	$-0.665162\pi$
$3$	−1.71785			−0.991803			−0.495902	+	0.868379i	$0.665162\pi$
$4$		0			0
$5$	−3.66778			−1.64028			−0.820141	−	0.572162i	$-0.806105\pi$
$5$	−3.66778			−1.64028			−0.820141	+	0.572162i	$0.806105\pi$
$6$		0			0
$7$		−	2.65082i		−	1.00192i	−0.865471	−	0.500958i	$-0.832981\pi$
$7$		−	2.65082i		−	1.00192i	0.865471	−	0.500958i	$-0.167019\pi$
$8$		0			0
$9$	−0.0489799			−0.0163266
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$		−	4.00362i		−	1.11040i	−0.831715	−	0.555202i	$-0.812641\pi$
$13$		−	4.00362i		−	1.11040i	0.831715	−	0.555202i	$-0.187359\pi$
$14$		0			0
$15$	6.30071			1.62684
$16$		0			0
$17$	5.00587			1.21410			0.607051	−	0.794663i	$-0.292352\pi$
$17$	5.00587			1.21410			0.607051	+	0.794663i	$0.292352\pi$
$18$		0			0
$19$	−1.66960	−	4.02647i	−0.383033	−	0.923735i
$19$	−1.66960	−	4.02647i	−0.383033	−	0.923735i
$20$		0			0
$21$			4.55372i			0.993704i
$22$		0			0
$23$		−	0.204072i		−	0.0425519i	−0.999774	−	0.0212759i	$-0.993227\pi$
$23$		−	0.204072i		−	0.0425519i	0.999774	−	0.0212759i	$-0.00677285\pi$
$24$		0			0
$25$	8.45262			1.69052
$26$		0			0
$27$	5.23770			1.00800
$28$		0			0
$29$			6.62744i			1.23069i	0.788260	+	0.615343i	$0.210982\pi$
$29$			6.62744i			1.23069i	−0.788260	+	0.615343i	$0.789018\pi$
$30$		0			0
$31$	8.91321			1.60086			0.800430	−	0.599426i	$-0.204605\pi$
$31$	8.91321			1.60086			0.800430	+	0.599426i	$0.204605\pi$
$32$		0			0
$33$			1.71785i			0.299040i
$34$		0			0
$35$			9.72263i			1.64343i
$36$		0			0
$37$		−	11.2007i		−	1.84139i	−0.390286	−	0.920694i	$-0.627624\pi$
$37$		−	11.2007i		−	1.84139i	0.390286	−	0.920694i	$-0.372376\pi$
$38$		0			0
$39$			6.87764i			1.10130i
$40$		0			0
$41$		−	1.16260i		−	0.181568i	−0.995871	−	0.0907840i	$-0.971063\pi$
$41$		−	1.16260i		−	0.181568i	0.995871	−	0.0907840i	$-0.0289373\pi$
$42$		0			0
$43$		−	8.59761i		−	1.31112i	−0.755142	−	0.655562i	$-0.772432\pi$
$43$		−	8.59761i		−	1.31112i	0.755142	−	0.655562i	$-0.227568\pi$
$44$		0			0
$45$	0.179648			0.0267803
$46$		0			0
$47$			7.02526i			1.02474i	0.858765	+	0.512370i	$0.171232\pi$
$47$			7.02526i			1.02474i	−0.858765	+	0.512370i	$0.828768\pi$
$48$		0			0
$49$	−0.0268544			−0.00383635
$50$		0			0
$51$	−8.59936			−1.20415
$52$		0			0
$53$		−	5.64742i		−	0.775733i	−0.921715	−	0.387867i	$-0.873212\pi$
$53$		−	5.64742i		−	0.775733i	0.921715	−	0.387867i	$-0.126788\pi$
$54$		0			0
$55$			3.66778i			0.494564i
$56$		0			0
$57$	2.86813	+	6.91688i	0.379893	+	0.916163i
$58$		0			0
$59$	10.7855			1.40416			0.702078	−	0.712100i	$-0.252256\pi$
$59$	10.7855			1.40416			0.702078	+	0.712100i	$0.252256\pi$
$60$		0			0
$61$	−15.0588			−1.92808			−0.964041	−	0.265753i	$-0.914380\pi$
$61$	−15.0588			−1.92808			−0.964041	+	0.265753i	$0.914380\pi$
$62$		0			0
$63$			0.129837i			0.0163579i
$64$		0			0
$65$			14.6844i			1.82138i
$66$		0			0
$67$	9.02412			1.10247			0.551236	−	0.834349i	$-0.314156\pi$
$67$	9.02412			1.10247			0.551236	+	0.834349i	$0.314156\pi$
$68$		0			0
$69$			0.350565i			0.0422031i
$70$		0			0
$71$	−3.63267			−0.431118			−0.215559	−	0.976491i	$-0.569157\pi$
$71$	−3.63267			−0.431118			−0.215559	+	0.976491i	$0.569157\pi$
$72$		0			0
$73$	−7.45373			−0.872393			−0.436197	−	0.899851i	$-0.643675\pi$
$73$	−7.45373			−0.872393			−0.436197	+	0.899851i	$0.643675\pi$
$74$		0			0
$75$	−14.5204			−1.67667
$76$		0			0
$77$	−2.65082			−0.302089
$78$		0			0
$79$	11.2460			1.26527			0.632636	−	0.774449i	$-0.281973\pi$
$79$	11.2460			1.26527			0.632636	+	0.774449i	$0.281973\pi$
$80$		0			0
$81$	−8.85066			−0.983407
$82$		0			0
$83$		−	3.09050i		−	0.339226i	−0.985511	−	0.169613i	$-0.945748\pi$
$83$		−	3.09050i		−	0.339226i	0.985511	−	0.169613i	$-0.0542518\pi$
$84$		0			0
$85$	−18.3604			−1.99147
$86$		0			0
$87$		−	11.3850i		−	1.22060i
$88$		0			0
$89$			1.40070i			0.148474i	0.997241	+	0.0742370i	$0.0236521\pi$
$89$			1.40070i			0.148474i	−0.997241	+	0.0742370i	$0.976348\pi$
$90$		0			0
$91$	−10.6129			−1.11253
$92$		0			0
$93$	−15.3116			−1.58774
$94$		0			0
$95$	6.12373	+	14.7682i	0.628282	+	1.51519i
$96$		0			0
$97$		−	16.0507i		−	1.62970i	−0.579669	−	0.814852i	$-0.696818\pi$
$97$		−	16.0507i		−	1.62970i	0.579669	−	0.814852i	$-0.303182\pi$
$98$		0			0
$99$			0.0489799i			0.00492267i
$100$		0			0
$101$	5.68888			0.566065			0.283032	−	0.959110i	$-0.408660\pi$
$101$	5.68888			0.566065			0.283032	+	0.959110i	$0.408660\pi$
$102$		0			0
$103$	−1.86903			−0.184161			−0.0920807	−	0.995752i	$-0.529352\pi$
$103$	−1.86903			−0.184161			−0.0920807	+	0.995752i	$0.529352\pi$
$104$		0			0
$105$		−	16.7021i		−	1.62995i
$106$		0			0
$107$	15.8824			1.53541			0.767703	−	0.640806i	$-0.221400\pi$
$107$	15.8824			1.53541			0.767703	+	0.640806i	$0.221400\pi$
$108$		0			0
$109$		−	5.05519i		−	0.484200i	−0.970251	−	0.242100i	$-0.922164\pi$
$109$		−	5.05519i		−	0.484200i	0.970251	−	0.242100i	$-0.0778361\pi$
$110$		0			0
$111$			19.2412i			1.82629i
$112$		0			0
$113$		−	20.9592i		−	1.97168i	−0.167693	−	0.985839i	$-0.553632\pi$
$113$		−	20.9592i		−	1.97168i	0.167693	−	0.985839i	$-0.446368\pi$
$114$		0			0
$115$			0.748490i			0.0697971i
$116$		0			0
$117$			0.196097i			0.0181292i
$118$		0			0
$119$		−	13.2697i		−	1.21643i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			1.99718i			0.180080i
$124$		0			0
$125$	−12.6635			−1.13265
$126$		0			0
$127$	−2.61496			−0.232040			−0.116020	−	0.993247i	$-0.537014\pi$
$127$	−2.61496			−0.232040			−0.116020	+	0.993247i	$0.537014\pi$
$128$		0			0
$129$			14.7694i			1.30038i
$130$		0			0
$131$		−	12.6760i		−	1.10751i	−0.832680	−	0.553755i	$-0.813195\pi$
$131$		−	12.6760i		−	1.10751i	0.832680	−	0.553755i	$-0.186805\pi$
$132$		0			0
$133$	−10.6734	+	4.42581i	−0.925505	+	0.383767i
$134$		0			0
$135$	−19.2107			−1.65340
$136$		0			0
$137$	3.63104			0.310221			0.155110	−	0.987897i	$-0.450427\pi$
$137$	3.63104			0.310221			0.155110	+	0.987897i	$0.450427\pi$
$138$		0			0
$139$			5.78076i			0.490318i	0.969483	+	0.245159i	$0.0788402\pi$
$139$			5.78076i			0.490318i	−0.969483	+	0.245159i	$0.921160\pi$
$140$		0			0
$141$		−	12.0684i		−	1.01634i
$142$		0			0
$143$	−4.00362			−0.334800
$144$		0			0
$145$		−	24.3080i		−	2.01867i
$146$		0			0
$147$	0.0461320			0.00380490
$148$		0			0
$149$	−2.59920			−0.212935			−0.106467	−	0.994316i	$-0.533954\pi$
$149$	−2.59920			−0.212935			−0.106467	+	0.994316i	$0.533954\pi$
$150$		0			0
$151$	3.20707			0.260987			0.130494	−	0.991449i	$-0.458344\pi$
$151$	3.20707			0.260987			0.130494	+	0.991449i	$0.458344\pi$
$152$		0			0
$153$	−0.245187			−0.0198222
$154$		0			0
$155$	−32.6917			−2.62586
$156$		0			0
$157$	−20.4857			−1.63494			−0.817468	−	0.575974i	$-0.804623\pi$
$157$	−20.4857			−1.63494			−0.817468	+	0.575974i	$0.804623\pi$
$158$		0			0
$159$			9.70145i			0.769375i
$160$		0			0
$161$	−0.540957			−0.0426334
$162$		0			0
$163$			19.1956i			1.50351i	0.659440	+	0.751757i	$0.270794\pi$
$163$			19.1956i			1.50351i	−0.659440	+	0.751757i	$0.729206\pi$
$164$		0			0
$165$		−	6.30071i		−	0.490510i
$166$		0			0
$167$	−24.4044			−1.88847			−0.944233	−	0.329278i	$-0.893195\pi$
$167$	−24.4044			−1.88847			−0.944233	+	0.329278i	$0.893195\pi$
$168$		0			0
$169$	−3.02899			−0.232999
$170$		0			0
$171$	0.0817769	+	0.197216i	0.00625364	+	0.0150815i
$172$		0			0
$173$		−	22.7675i		−	1.73098i	−0.500929	−	0.865489i	$-0.667008\pi$
$173$		−	22.7675i		−	1.73098i	0.500929	−	0.865489i	$-0.332992\pi$
$174$		0			0
$175$		−	22.4064i		−	1.69376i
$176$		0			0
$177$	−18.5280			−1.39265
$178$		0			0
$179$	2.38421			0.178204			0.0891022	−	0.996022i	$-0.471600\pi$
$179$	2.38421			0.178204			0.0891022	+	0.996022i	$0.471600\pi$
$180$		0			0
$181$			23.3640i			1.73663i	0.496014	+	0.868315i	$0.334797\pi$
$181$			23.3640i			1.73663i	−0.496014	+	0.868315i	$0.665203\pi$
$182$		0			0
$183$	25.8688			1.91228
$184$		0			0
$185$			41.0818i			3.02039i
$186$		0			0
$187$		−	5.00587i		−	0.366066i
$188$		0			0
$189$		−	13.8842i		−	1.00993i
$190$		0			0
$191$		−	25.3871i		−	1.83695i	−0.395483	−	0.918473i	$-0.629423\pi$
$191$		−	25.3871i		−	1.83695i	0.395483	−	0.918473i	$-0.370577\pi$
$192$		0			0
$193$			7.82933i			0.563568i	0.959478	+	0.281784i	$0.0909261\pi$
$193$			7.82933i			0.563568i	−0.959478	+	0.281784i	$0.909074\pi$
$194$		0			0
$195$		−	25.2257i		−	1.80645i
$196$		0			0
$197$	−7.25592			−0.516963			−0.258482	−	0.966016i	$-0.583222\pi$
$197$	−7.25592			−0.516963			−0.258482	+	0.966016i	$0.583222\pi$
$198$		0			0
$199$			11.4278i			0.810093i	0.914296	+	0.405047i	$0.132745\pi$
$199$			11.4278i			0.810093i	−0.914296	+	0.405047i	$0.867255\pi$
$200$		0			0
$201$	−15.5021			−1.09344
$202$		0			0
$203$	17.5682			1.23304
$204$		0			0
$205$			4.26417i			0.297823i
$206$		0			0
$207$			0.00999541i			0.000694729i
$208$		0			0
$209$	−4.02647	+	1.66960i	−0.278517	+	0.115489i
$210$		0			0
$211$	−3.37087			−0.232060			−0.116030	−	0.993246i	$-0.537017\pi$
$211$	−3.37087			−0.232060			−0.116030	+	0.993246i	$0.537017\pi$
$212$		0			0
$213$	6.24039			0.427584
$214$		0			0
$215$			31.5342i			2.15061i
$216$		0			0
$217$		−	23.6273i		−	1.60393i
$218$		0			0
$219$	12.8044			0.865242
$220$		0			0
$221$		−	20.0416i		−	1.34815i
$222$		0			0
$223$	11.5024			0.770259			0.385130	−	0.922862i	$-0.374157\pi$
$223$	11.5024			0.770259			0.385130	+	0.922862i	$0.374157\pi$
$224$		0			0
$225$	−0.414009			−0.0276006
$226$		0			0
$227$	−7.63696			−0.506883			−0.253441	−	0.967351i	$-0.581562\pi$
$227$	−7.63696			−0.506883			−0.253441	+	0.967351i	$0.581562\pi$
$228$		0			0
$229$	−19.4568			−1.28574			−0.642871	−	0.765974i	$-0.722257\pi$
$229$	−19.4568			−1.28574			−0.642871	+	0.765974i	$0.722257\pi$
$230$		0			0
$231$	4.55372			0.299613
$232$		0			0
$233$	16.6632			1.09164			0.545821	−	0.837902i	$-0.316218\pi$
$233$	16.6632			1.09164			0.545821	+	0.837902i	$0.316218\pi$
$234$		0			0
$235$		−	25.7671i		−	1.68086i
$236$		0			0
$237$	−19.3189			−1.25490
$238$		0			0
$239$		−	14.8632i		−	0.961420i	−0.876880	−	0.480710i	$-0.840379\pi$
$239$		−	14.8632i		−	0.961420i	0.876880	−	0.480710i	$-0.159621\pi$
$240$		0			0
$241$			15.6223i			1.00632i	0.864194	+	0.503159i	$0.167829\pi$
$241$			15.6223i			1.00632i	−0.864194	+	0.503159i	$0.832171\pi$
$242$		0			0
$243$	−0.508963			−0.0326500
$244$		0			0
$245$	0.0984962			0.00629269
$246$		0			0
$247$	−16.1205	+	6.68445i	−1.02572	+	0.425321i
$248$		0			0
$249$			5.30902i			0.336445i
$250$		0			0
$251$		−	9.32092i		−	0.588331i	−0.955754	−	0.294166i	$-0.904958\pi$
$251$		−	9.32092i		−	0.588331i	0.955754	−	0.294166i	$-0.0950417\pi$
$252$		0			0
$253$	−0.204072			−0.0128299
$254$		0			0
$255$	31.5406			1.97515
$256$		0			0
$257$		−	2.94776i		−	0.183876i	−0.995765	−	0.0919382i	$-0.970694\pi$
$257$		−	2.94776i		−	0.183876i	0.995765	−	0.0919382i	$-0.0293062\pi$
$258$		0			0
$259$	−29.6911			−1.84492
$260$		0			0
$261$		−	0.324612i		−	0.0200930i
$262$		0			0
$263$			16.3181i			1.00622i	0.864223	+	0.503109i	$0.167811\pi$
$263$			16.3181i			1.00622i	−0.864223	+	0.503109i	$0.832189\pi$
$264$		0			0
$265$			20.7135i			1.27242i
$266$		0			0
$267$		−	2.40620i		−	0.147257i
$268$		0			0
$269$		−	2.63414i		−	0.160606i	−0.996770	−	0.0803032i	$-0.974411\pi$
$269$		−	2.63414i		−	0.160606i	0.996770	−	0.0803032i	$-0.0255889\pi$
$270$		0			0
$271$		−	10.7783i		−	0.654738i	−0.944897	−	0.327369i	$-0.893838\pi$
$271$		−	10.7783i		−	0.654738i	0.944897	−	0.327369i	$-0.106162\pi$
$272$		0			0
$273$	18.2314			1.10341
$274$		0			0
$275$		−	8.45262i		−	0.509712i
$276$		0			0
$277$	29.6339			1.78053			0.890265	−	0.455443i	$-0.150519\pi$
$277$	29.6339			1.78053			0.890265	+	0.455443i	$0.150519\pi$
$278$		0			0
$279$	−0.436568			−0.0261367
$280$		0			0
$281$			19.6765i			1.17380i	0.809659	+	0.586901i	$0.199652\pi$
$281$			19.6765i			1.17380i	−0.809659	+	0.586901i	$0.800348\pi$
$282$		0			0
$283$		−	4.89175i		−	0.290784i	−0.989374	−	0.145392i	$-0.953556\pi$
$283$		−	4.89175i		−	0.290784i	0.989374	−	0.145392i	$-0.0464444\pi$
$284$		0			0
$285$	−10.5197	−	25.3696i	−0.623132	−	1.50277i
$286$		0			0
$287$	−3.08185			−0.181916
$288$		0			0
$289$	8.05876			0.474045
$290$		0			0
$291$			27.5728i			1.61634i
$292$		0			0
$293$			24.3037i			1.41984i	0.704283	+	0.709919i	$0.251269\pi$
$293$			24.3037i			1.41984i	−0.704283	+	0.709919i	$0.748731\pi$
$294$		0			0
$295$	−39.5590			−2.30321
$296$		0			0
$297$		−	5.23770i		−	0.303922i
$298$		0			0
$299$	−0.817025			−0.0472498
$300$		0			0
$301$	−22.7907			−1.31364
$302$		0			0
$303$	−9.77266			−0.561425
$304$		0			0
$305$	55.2324			3.16260
$306$		0			0
$307$	14.2075			0.810867			0.405433	−	0.914125i	$-0.367121\pi$
$307$	14.2075			0.810867			0.405433	+	0.914125i	$0.367121\pi$
$308$		0			0
$309$	3.21073			0.182652
$310$		0			0
$311$		−	29.9799i		−	1.70001i	−0.526778	−	0.850003i	$-0.676600\pi$
$311$		−	29.9799i		−	1.70001i	0.526778	−	0.850003i	$-0.323400\pi$
$312$		0			0
$313$	−3.33727			−0.188634			−0.0943169	−	0.995542i	$-0.530067\pi$
$313$	−3.33727			−0.188634			−0.0943169	+	0.995542i	$0.530067\pi$
$314$		0			0
$315$		−	0.476214i		−	0.0268316i
$316$		0			0
$317$			4.21066i			0.236494i	0.992984	+	0.118247i	$0.0377275\pi$
$317$			4.21066i			0.236494i	−0.992984	+	0.118247i	$0.962272\pi$
$318$		0			0
$319$	6.62744			0.371066
$320$		0			0
$321$	−27.2836			−1.52282
$322$		0			0
$323$	−8.35781	−	20.1560i	−0.465041	−	1.12151i
$324$		0			0
$325$		−	33.8411i		−	1.87717i
$326$		0			0
$327$			8.68408i			0.480231i
$328$		0			0
$329$	18.6227			1.02670
$330$		0			0
$331$	−16.5725			−0.910906			−0.455453	−	0.890260i	$-0.650523\pi$
$331$	−16.5725			−0.910906			−0.455453	+	0.890260i	$0.650523\pi$
$332$		0			0
$333$			0.548610i			0.0300637i
$334$		0			0
$335$	−33.0985			−1.80836
$336$		0			0
$337$		−	5.18321i		−	0.282347i	−0.989985	−	0.141174i	$-0.954912\pi$
$337$		−	5.18321i		−	0.282347i	0.989985	−	0.141174i	$-0.0450876\pi$
$338$		0			0
$339$			36.0049i			1.95552i
$340$		0			0
$341$		−	8.91321i		−	0.482677i
$342$		0			0
$343$		−	18.4846i		−	0.998073i
$344$		0			0
$345$		−	1.28580i		−	0.0692249i
$346$		0			0
$347$			18.0317i			0.967992i	0.875070	+	0.483996i	$0.160815\pi$
$347$			18.0317i			0.967992i	−0.875070	+	0.483996i	$0.839185\pi$
$348$		0			0
$349$	−8.10853			−0.434040			−0.217020	−	0.976167i	$-0.569634\pi$
$349$	−8.10853			−0.434040			−0.217020	+	0.976167i	$0.569634\pi$
$350$		0			0
$351$		−	20.9698i		−	1.11928i
$352$		0			0
$353$	−0.417540			−0.0222234			−0.0111117	−	0.999938i	$-0.503537\pi$
$353$	−0.417540			−0.0222234			−0.0111117	+	0.999938i	$0.503537\pi$
$354$		0			0
$355$	13.3238			0.707155
$356$		0			0
$357$			22.7954i			1.20646i
$358$		0			0
$359$			18.3062i			0.966163i	0.875575	+	0.483082i	$0.160483\pi$
$359$			18.3062i			0.966163i	−0.875575	+	0.483082i	$0.839517\pi$
$360$		0			0
$361$	−13.4249	+	13.4452i	−0.706572	+	0.707641i
$362$		0			0
$363$	1.71785			0.0901639
$364$		0			0
$365$	27.3387			1.43097
$366$		0			0
$367$		−	12.0040i		−	0.626605i	−0.949653	−	0.313303i	$-0.898565\pi$
$367$		−	12.0040i		−	0.626605i	0.949653	−	0.313303i	$-0.101435\pi$
$368$		0			0
$369$			0.0569442i			0.00296440i
$370$		0			0
$371$	−14.9703			−0.777220
$372$		0			0
$373$			22.5730i			1.16878i	0.811471	+	0.584392i	$0.198667\pi$
$373$			22.5730i			1.16878i	−0.811471	+	0.584392i	$0.801333\pi$
$374$		0			0
$375$	21.7540			1.12337
$376$		0			0
$377$	26.5338			1.36656
$378$		0			0
$379$	−9.87561			−0.507276			−0.253638	−	0.967299i	$-0.581627\pi$
$379$	−9.87561			−0.507276			−0.253638	+	0.967299i	$0.581627\pi$
$380$		0			0
$381$	4.49211			0.230138
$382$		0			0
$383$	24.5253			1.25319			0.626593	−	0.779347i	$-0.284449\pi$
$383$	24.5253			1.25319			0.626593	+	0.779347i	$0.284449\pi$
$384$		0			0
$385$	9.72263			0.495511
$386$		0			0
$387$			0.421110i			0.0214062i
$388$		0			0
$389$	10.2454			0.519462			0.259731	−	0.965681i	$-0.416366\pi$
$389$	10.2454			0.519462			0.259731	+	0.965681i	$0.416366\pi$
$390$		0			0
$391$		−	1.02156i		−	0.0516623i
$392$		0			0
$393$			21.7755i			1.09843i
$394$		0			0
$395$	−41.2478			−2.07540
$396$		0			0
$397$	−27.8533			−1.39791			−0.698957	−	0.715163i	$-0.746352\pi$
$397$	−27.8533			−1.39791			−0.698957	+	0.715163i	$0.746352\pi$
$398$		0			0
$399$	18.3354	−	7.60290i	0.917919	−	0.380621i
$400$		0			0
$401$			0.250244i			0.0124966i	0.999980	+	0.00624830i	$0.00198891\pi$
$401$			0.250244i			0.0124966i	−0.999980	+	0.00624830i	$0.998011\pi$
$402$		0			0
$403$		−	35.6851i		−	1.77760i
$404$		0			0
$405$	32.4623			1.61306
$406$		0			0
$407$	−11.2007			−0.555199
$408$		0			0
$409$		−	4.25022i		−	0.210160i	−0.994464	−	0.105080i	$-0.966490\pi$
$409$		−	4.25022i		−	0.210160i	0.994464	−	0.105080i	$-0.0335098\pi$
$410$		0			0
$411$	−6.23760			−0.307678
$412$		0			0
$413$		−	28.5905i		−	1.40685i
$414$		0			0
$415$			11.3353i			0.556426i
$416$		0			0
$417$		−	9.93050i		−	0.486299i
$418$		0			0
$419$		−	4.73744i		−	0.231439i	−0.993282	−	0.115719i	$-0.963083\pi$
$419$		−	4.73744i		−	0.231439i	0.993282	−	0.115719i	$-0.0369174\pi$
$420$		0			0
$421$		−	12.6268i		−	0.615390i	−0.951485	−	0.307695i	$-0.900442\pi$
$421$		−	12.6268i		−	0.615390i	0.951485	−	0.307695i	$-0.0995577\pi$
$422$		0			0
$423$		−	0.344097i		−	0.0167306i
$424$		0			0
$425$	42.3128			2.05247
$426$		0			0
$427$			39.9182i			1.93178i
$428$		0			0
$429$	6.87764			0.332055
$430$		0			0
$431$	−16.4612			−0.792909			−0.396455	−	0.918054i	$-0.629760\pi$
$431$	−16.4612			−0.792909			−0.396455	+	0.918054i	$0.629760\pi$
$432$		0			0
$433$		−	31.8234i		−	1.52933i	−0.644425	−	0.764667i	$-0.722903\pi$
$433$		−	31.8234i		−	1.52933i	0.644425	−	0.764667i	$-0.277097\pi$
$434$		0			0
$435$			41.7576i			2.00212i
$436$		0			0
$437$	−0.821687	+	0.340718i	−0.0393066	+	0.0162988i
$438$		0			0
$439$	35.2196			1.68094			0.840470	−	0.541858i	$-0.182279\pi$
$439$	35.2196			1.68094			0.840470	+	0.541858i	$0.182279\pi$
$440$		0			0
$441$	0.00131533			6.26346e−5
$442$		0			0
$443$			23.8869i			1.13490i	0.823407	+	0.567451i	$0.192070\pi$
$443$			23.8869i			1.13490i	−0.823407	+	0.567451i	$0.807930\pi$
$444$		0			0
$445$		−	5.13746i		−	0.243539i
$446$		0			0
$447$	4.46505			0.211189
$448$		0			0
$449$			36.2880i			1.71254i	0.516530	+	0.856269i	$0.327224\pi$
$449$			36.2880i			1.71254i	−0.516530	+	0.856269i	$0.672776\pi$
$450$		0			0
$451$	−1.16260			−0.0547448
$452$		0			0
$453$	−5.50927			−0.258848
$454$		0			0
$455$	38.9258			1.82487
$456$		0			0
$457$	−3.71654			−0.173852			−0.0869262	−	0.996215i	$-0.527704\pi$
$457$	−3.71654			−0.173852			−0.0869262	+	0.996215i	$0.527704\pi$
$458$		0			0
$459$	26.2193			1.22381
$460$		0			0
$461$	−2.36558			−0.110176			−0.0550880	−	0.998482i	$-0.517544\pi$
$461$	−2.36558			−0.110176			−0.0550880	+	0.998482i	$0.517544\pi$
$462$		0			0
$463$			16.6355i			0.773118i	0.922265	+	0.386559i	$0.126337\pi$
$463$			16.6355i			0.773118i	−0.922265	+	0.386559i	$0.873663\pi$
$464$		0			0
$465$	56.1596			2.60434
$466$		0			0
$467$			26.8430i			1.24215i	0.783752	+	0.621074i	$0.213303\pi$
$467$			26.8430i			1.24215i	−0.783752	+	0.621074i	$0.786697\pi$
$468$		0			0
$469$		−	23.9213i		−	1.10458i
$470$		0			0
$471$	35.1914			1.62153
$472$		0			0
$473$	−8.59761			−0.395319
$474$		0			0
$475$	−14.1125	−	34.0342i	−0.647526	−	1.56160i
$476$		0			0
$477$			0.276610i			0.0126651i
$478$		0			0
$479$			32.2789i			1.47486i	0.675422	+	0.737431i	$0.263961\pi$
$479$			32.2789i			1.47486i	−0.675422	+	0.737431i	$0.736039\pi$
$480$		0			0
$481$	−44.8435			−2.04469
$482$		0			0
$483$	0.929285			0.0422839
$484$		0			0
$485$			58.8705i			2.67317i
$486$		0			0
$487$	29.3882			1.33171			0.665854	−	0.746082i	$-0.268067\pi$
$487$	29.3882			1.33171			0.665854	+	0.746082i	$0.268067\pi$
$488$		0			0
$489$		−	32.9752i		−	1.49119i
$490$		0			0
$491$			21.9234i			0.989388i	0.869067	+	0.494694i	$0.164720\pi$
$491$			21.9234i			0.989388i	−0.869067	+	0.494694i	$0.835280\pi$
$492$		0			0
$493$			33.1761i			1.49418i
$494$		0			0
$495$		−	0.179648i		−	0.00807456i
$496$		0			0
$497$			9.62955i			0.431944i
$498$		0			0
$499$			3.41686i			0.152959i	0.997071	+	0.0764797i	$0.0243680\pi$
$499$			3.41686i			0.152959i	−0.997071	+	0.0764797i	$0.975632\pi$
$500$		0			0
$501$	41.9231			1.87299
$502$		0			0
$503$		−	38.3118i		−	1.70824i	−0.520078	−	0.854118i	$-0.674097\pi$
$503$		−	38.3118i		−	1.70824i	0.520078	−	0.854118i	$-0.325903\pi$
$504$		0			0
$505$	−20.8656			−0.928506
$506$		0			0
$507$	5.20336			0.231089
$508$		0			0
$509$			7.07903i			0.313772i	0.987617	+	0.156886i	$0.0501456\pi$
$509$			7.07903i			0.313772i	−0.987617	+	0.156886i	$0.949854\pi$
$510$		0			0
$511$			19.7585i			0.874065i
$512$		0			0
$513$	−8.74487	−	21.0894i	−0.386095	−	0.931121i
$514$		0			0
$515$	6.85521			0.302077
$516$		0			0
$517$	7.02526			0.308971
$518$		0			0
$519$			39.1111i			1.71679i
$520$		0			0
$521$		−	6.25070i		−	0.273848i	−0.990582	−	0.136924i	$-0.956278\pi$
$521$		−	6.25070i		−	0.273848i	0.990582	−	0.136924i	$-0.0437216\pi$
$522$		0			0
$523$	−1.45340			−0.0635529			−0.0317764	−	0.999495i	$-0.510116\pi$
$523$	−1.45340			−0.0635529			−0.0317764	+	0.999495i	$0.510116\pi$
$524$		0			0
$525$			38.4909i			1.67988i
$526$		0			0
$527$	44.6184			1.94361
$528$		0			0
$529$	22.9584			0.998189
$530$		0			0
$531$	−0.528275			−0.0229252
$532$		0			0
$533$	−4.65462			−0.201614
$534$		0			0
$535$	−58.2530			−2.51850
$536$		0			0
$537$	−4.09573			−0.176744
$538$		0			0
$539$			0.0268544i			0.00115670i
$540$		0			0
$541$	−28.8041			−1.23839			−0.619193	−	0.785239i	$-0.712540\pi$
$541$	−28.8041			−1.23839			−0.619193	+	0.785239i	$0.712540\pi$
$542$		0			0
$543$		−	40.1359i		−	1.72239i
$544$		0			0
$545$			18.5413i			0.794224i
$546$		0			0
$547$	29.3286			1.25400			0.627001	−	0.779019i	$-0.284282\pi$
$547$	29.3286			1.25400			0.627001	+	0.779019i	$0.284282\pi$
$548$		0			0
$549$	0.737579			0.0314791
$550$		0			0
$551$	26.6852	−	11.0652i	1.13683	−	0.471393i
$552$		0			0
$553$		−	29.8111i		−	1.26770i
$554$		0			0
$555$		−	70.5725i		−	2.99564i
$556$		0			0
$557$	7.45427			0.315847			0.157924	−	0.987451i	$-0.449520\pi$
$557$	7.45427			0.315847			0.157924	+	0.987451i	$0.449520\pi$
$558$		0			0
$559$	−34.4216			−1.45588
$560$		0			0
$561$			8.59936i			0.363065i
$562$		0			0
$563$	−3.27872			−0.138181			−0.0690907	−	0.997610i	$-0.522010\pi$
$563$	−3.27872			−0.138181			−0.0690907	+	0.997610i	$0.522010\pi$
$564$		0			0
$565$			76.8739i			3.23411i
$566$		0			0
$567$			23.4615i			0.985291i
$568$		0			0
$569$			6.12088i			0.256601i	0.991735	+	0.128300i	$0.0409522\pi$
$569$			6.12088i			0.256601i	−0.991735	+	0.128300i	$0.959048\pi$
$570$		0			0
$571$		−	43.6973i		−	1.82867i	−0.404953	−	0.914337i	$-0.632712\pi$
$571$		−	43.6973i		−	1.82867i	0.404953	−	0.914337i	$-0.367288\pi$
$572$		0			0
$573$			43.6113i			1.82189i
$574$		0			0
$575$		−	1.72494i		−	0.0719350i
$576$		0			0
$577$	−27.0802			−1.12737			−0.563683	−	0.825991i	$-0.690616\pi$
$577$	−27.0802			−1.12737			−0.563683	+	0.825991i	$0.690616\pi$
$578$		0			0
$579$		−	13.4496i		−	0.558948i
$580$		0			0
$581$	−8.19235			−0.339876
$582$		0			0
$583$	−5.64742			−0.233892
$584$		0			0
$585$		−	0.719241i		−	0.0297370i
$586$		0			0
$587$			19.8786i			0.820478i	0.911978	+	0.410239i	$0.134555\pi$
$587$			19.8786i			0.820478i	−0.911978	+	0.410239i	$0.865445\pi$
$588$		0			0
$589$	−14.8815	−	35.8887i	−0.613182	−	1.47877i
$590$		0			0
$591$	12.4646			0.512726
$592$		0			0
$593$	21.6987			0.891061			0.445530	−	0.895267i	$-0.353015\pi$
$593$	21.6987			0.891061			0.445530	+	0.895267i	$0.353015\pi$
$594$		0			0
$595$			48.6703i			1.99529i
$596$		0			0
$597$		−	19.6312i		−	0.803453i
$598$		0			0
$599$	−42.7503			−1.74673			−0.873366	−	0.487065i	$-0.838068\pi$
$599$	−42.7503			−1.74673			−0.873366	+	0.487065i	$0.838068\pi$
$600$		0			0
$601$		−	10.3839i		−	0.423568i	−0.977316	−	0.211784i	$-0.932073\pi$
$601$		−	10.3839i		−	0.423568i	0.977316	−	0.211784i	$-0.0679274\pi$
$602$		0			0
$603$	−0.442001			−0.0179997
$604$		0			0
$605$	3.66778			0.149117
$606$		0			0
$607$	−12.1604			−0.493575			−0.246787	−	0.969070i	$-0.579375\pi$
$607$	−12.1604			−0.493575			−0.246787	+	0.969070i	$0.579375\pi$
$608$		0			0
$609$	−30.1795			−1.22294
$610$		0			0
$611$	28.1265			1.13788
$612$		0			0
$613$	11.8015			0.476656			0.238328	−	0.971185i	$-0.423401\pi$
$613$	11.8015			0.476656			0.238328	+	0.971185i	$0.423401\pi$
$614$		0			0
$615$		−	7.32522i		−	0.295382i
$616$		0			0
$617$	−23.1184			−0.930714			−0.465357	−	0.885123i	$-0.654074\pi$
$617$	−23.1184			−0.930714			−0.465357	+	0.885123i	$0.654074\pi$
$618$		0			0
$619$		−	29.7739i		−	1.19671i	−0.801230	−	0.598357i	$-0.795821\pi$
$619$		−	29.7739i		−	1.19671i	0.801230	−	0.598357i	$-0.204179\pi$
$620$		0			0
$621$		−	1.06887i		−	0.0428921i
$622$		0			0
$623$	3.71301			0.148758
$624$		0			0
$625$	4.18372			0.167349
$626$		0			0
$627$	6.91688	−	2.86813i	0.276234	−	0.114542i
$628$		0			0
$629$		−	56.0694i		−	2.23563i
$630$		0			0
$631$		−	27.8991i		−	1.11065i	−0.831635	−	0.555323i	$-0.812595\pi$
$631$		−	27.8991i		−	1.11065i	0.831635	−	0.555323i	$-0.187405\pi$
$632$		0			0
$633$	5.79066			0.230158
$634$		0			0
$635$	9.59108			0.380611
$636$		0			0
$637$			0.107515i			0.00425990i
$638$		0			0
$639$	0.177928			0.00703871
$640$		0			0
$641$			3.51931i			0.139005i	0.997582	+	0.0695023i	$0.0221411\pi$
$641$			3.51931i			0.139005i	−0.997582	+	0.0695023i	$0.977859\pi$
$642$		0			0
$643$			14.0262i			0.553139i	0.960994	+	0.276569i	$0.0891976\pi$
$643$			14.0262i			0.553139i	−0.960994	+	0.276569i	$0.910802\pi$
$644$		0			0
$645$		−	54.1711i		−	2.13298i
$646$		0			0
$647$		−	24.8561i		−	0.977194i	−0.872510	−	0.488597i	$-0.837509\pi$
$647$		−	24.8561i		−	0.977194i	0.872510	−	0.488597i	$-0.162491\pi$
$648$		0			0
$649$		−	10.7855i		−	0.423369i
$650$		0			0
$651$			40.5883i			1.59078i
$652$		0			0
$653$	−13.0746			−0.511647			−0.255824	−	0.966723i	$-0.582347\pi$
$653$	−13.0746			−0.511647			−0.255824	+	0.966723i	$0.582347\pi$
$654$		0			0
$655$			46.4929i			1.81663i
$656$		0			0
$657$	0.365083			0.0142433
$658$		0			0
$659$	20.7481			0.808233			0.404116	−	0.914708i	$-0.367579\pi$
$659$	20.7481			0.808233			0.404116	+	0.914708i	$0.367579\pi$
$660$		0			0
$661$		−	14.4434i		−	0.561784i	−0.959739	−	0.280892i	$-0.909370\pi$
$661$		−	14.4434i		−	0.561784i	0.959739	−	0.280892i	$-0.0906303\pi$
$662$		0			0
$663$			34.4286i			1.33709i
$664$		0			0
$665$	39.1479	−	16.2329i	1.51809	−	0.629486i
$666$		0			0
$667$	1.35247			0.0523679
$668$		0			0
$669$	−19.7595			−0.763945
$670$		0			0
$671$			15.0588i			0.581339i
$672$		0			0
$673$		−	23.3974i		−	0.901904i	−0.892548	−	0.450952i	$-0.851085\pi$
$673$		−	23.3974i		−	0.901904i	0.892548	−	0.450952i	$-0.148915\pi$
$674$		0			0
$675$	44.2723			1.70404
$676$		0			0
$677$			4.15282i			0.159606i	0.996811	+	0.0798029i	$0.0254291\pi$
$677$			4.15282i			0.159606i	−0.996811	+	0.0798029i	$0.974571\pi$
$678$		0			0
$679$	−42.5476			−1.63283
$680$		0			0
$681$	13.1192			0.502728
$682$		0			0
$683$	−0.280657			−0.0107390			−0.00536952	−	0.999986i	$-0.501709\pi$
$683$	−0.280657			−0.0107390			−0.00536952	+	0.999986i	$0.501709\pi$
$684$		0			0
$685$	−13.3179			−0.508850
$686$		0			0
$687$	33.4239			1.27520
$688$		0			0
$689$	−22.6102			−0.861378
$690$		0			0
$691$			21.6198i			0.822455i	0.911533	+	0.411228i	$0.134900\pi$
$691$			21.6198i			0.822455i	−0.911533	+	0.411228i	$0.865100\pi$
$692$		0			0
$693$	0.129837			0.00493210
$694$		0			0
$695$		−	21.2026i		−	0.804260i
$696$		0			0
$697$		−	5.81984i		−	0.220442i
$698$		0			0
$699$	−28.6249			−1.08269
$700$		0			0
$701$	17.3334			0.654673			0.327337	−	0.944908i	$-0.393849\pi$
$701$	17.3334			0.654673			0.327337	+	0.944908i	$0.393849\pi$
$702$		0			0
$703$	−45.0993	+	18.7007i	−1.70095	+	0.705312i
$704$		0			0
$705$			44.2642i			1.66708i
$706$		0			0
$707$		−	15.0802i		−	0.567150i
$708$		0			0
$709$	41.4277			1.55585			0.777925	−	0.628357i	$-0.216272\pi$
$709$	41.4277			1.55585			0.777925	+	0.628357i	$0.216272\pi$
$710$		0			0
$711$	−0.550827			−0.0206576
$712$		0			0
$713$		−	1.81893i		−	0.0681196i
$714$		0			0
$715$	14.6844			0.549166
$716$		0			0
$717$			25.5328i			0.953540i
$718$		0			0
$719$		−	1.29720i		−	0.0483775i	−0.999707	−	0.0241888i	$-0.992300\pi$
$719$		−	1.29720i		−	0.0483775i	0.999707	−	0.0241888i	$-0.00770028\pi$
$720$		0			0
$721$			4.95448i			0.184514i
$722$		0			0
$723$		−	26.8367i		−	0.998069i
$724$		0			0
$725$			56.0193i			2.08050i
$726$		0			0
$727$		−	31.4537i		−	1.16655i	−0.812274	−	0.583276i	$-0.801770\pi$
$727$		−	31.4537i		−	1.16655i	0.812274	−	0.583276i	$-0.198230\pi$
$728$		0			0
$729$	27.4263			1.01579
$730$		0			0
$731$		−	43.0385i		−	1.59184i
$732$		0			0
$733$	−20.3465			−0.751516			−0.375758	−	0.926718i	$-0.622618\pi$
$733$	−20.3465			−0.751516			−0.375758	+	0.926718i	$0.622618\pi$
$734$		0			0
$735$	−0.169202			−0.00624111
$736$		0			0
$737$		−	9.02412i		−	0.332408i
$738$		0			0
$739$			35.2610i			1.29710i	0.761173	+	0.648549i	$0.224624\pi$
$739$			35.2610i			1.29710i	−0.761173	+	0.648549i	$0.775376\pi$
$740$		0			0
$741$	27.6926	−	11.4829i	1.01731	−	0.421835i
$742$		0			0
$743$	−37.0758			−1.36018			−0.680090	−	0.733129i	$-0.738059\pi$
$743$	−37.0758			−1.36018			−0.680090	+	0.733129i	$0.738059\pi$
$744$		0			0
$745$	9.53330			0.349273
$746$		0			0
$747$			0.151372i			0.00553842i
$748$		0			0
$749$		−	42.1013i		−	1.53835i
$750$		0			0
$751$	15.7816			0.575880			0.287940	−	0.957648i	$-0.407030\pi$
$751$	15.7816			0.575880			0.287940	+	0.957648i	$0.407030\pi$
$752$		0			0
$753$			16.0120i			0.583509i
$754$		0			0
$755$	−11.7628			−0.428093
$756$		0			0
$757$	−3.48836			−0.126787			−0.0633933	−	0.997989i	$-0.520192\pi$
$757$	−3.48836			−0.126787			−0.0633933	+	0.997989i	$0.520192\pi$
$758$		0			0
$759$	0.350565			0.0127247
$760$		0			0
$761$	22.6483			0.821002			0.410501	−	0.911860i	$-0.365354\pi$
$761$	22.6483			0.821002			0.410501	+	0.911860i	$0.365354\pi$
$762$		0			0
$763$	−13.4004			−0.485127
$764$		0			0
$765$	0.899293			0.0325140
$766$		0			0
$767$		−	43.1812i		−	1.55918i
$768$		0			0
$769$	−10.2418			−0.369328			−0.184664	−	0.982802i	$-0.559120\pi$
$769$	−10.2418			−0.369328			−0.184664	+	0.982802i	$0.559120\pi$
$770$		0			0
$771$			5.06383i			0.182369i
$772$		0			0
$773$		−	13.4205i		−	0.482701i	−0.970438	−	0.241350i	$-0.922410\pi$
$773$		−	13.4205i		−	0.482701i	0.970438	−	0.241350i	$-0.0775903\pi$
$774$		0			0
$775$	75.3400			2.70629
$776$		0			0
$777$	51.0050			1.82979
$778$		0			0
$779$	−4.68118	+	1.94108i	−0.167721	+	0.0695465i
$780$		0			0
$781$			3.63267i			0.129987i
$782$		0			0
$783$			34.7126i			1.24053i
$784$		0			0
$785$	75.1371			2.68176
$786$		0			0
$787$	40.1920			1.43269			0.716345	−	0.697746i	$-0.245814\pi$
$787$	40.1920			1.43269			0.716345	+	0.697746i	$0.245814\pi$
$788$		0			0
$789$		−	28.0321i		−	0.997970i
$790$		0			0
$791$	−55.5592			−1.97546
$792$		0			0
$793$			60.2898i			2.14095i
$794$		0			0
$795$		−	35.5828i		−	1.26199i
$796$		0			0
$797$			32.8581i			1.16389i	0.813227	+	0.581947i	$0.197709\pi$
$797$			32.8581i			1.16389i	−0.813227	+	0.581947i	$0.802291\pi$
$798$		0			0
$799$			35.1676i			1.24414i
$800$		0			0
$801$		−	0.0686062i		−	0.00242408i
$802$		0			0
$803$			7.45373i			0.263037i
$804$		0			0
$805$	1.98411			0.0699308
$806$		0			0
$807$			4.52507i			0.159290i
$808$		0			0
$809$	3.82725			0.134559			0.0672795	−	0.997734i	$-0.478568\pi$
$809$	3.82725			0.134559			0.0672795	+	0.997734i	$0.478568\pi$
$810$		0			0
$811$	−22.7176			−0.797723			−0.398861	−	0.917011i	$-0.630595\pi$
$811$	−22.7176			−0.797723			−0.398861	+	0.917011i	$0.630595\pi$
$812$		0			0
$813$			18.5156i			0.649371i
$814$		0			0
$815$		−	70.4052i		−	2.46619i
$816$		0			0
$817$	−34.6180	+	14.3546i	−1.21113	+	0.502203i
$818$		0			0
$819$	0.519818			0.0181639
$820$		0			0
$821$	−19.6944			−0.687339			−0.343669	−	0.939091i	$-0.611670\pi$
$821$	−19.6944			−0.687339			−0.343669	+	0.939091i	$0.611670\pi$
$822$		0			0
$823$			22.6781i			0.790509i	0.918572	+	0.395255i	$0.129344\pi$
$823$			22.6781i			0.790509i	−0.918572	+	0.395255i	$0.870656\pi$
$824$		0			0
$825$			14.5204i			0.505534i
$826$		0			0
$827$	35.5689			1.23685			0.618425	−	0.785844i	$-0.287771\pi$
$827$	35.5689			1.23685			0.618425	+	0.785844i	$0.287771\pi$
$828$		0			0
$829$			29.0453i			1.00878i	0.863475	+	0.504392i	$0.168283\pi$
$829$			29.0453i			1.00878i	−0.863475	+	0.504392i	$0.831717\pi$
$830$		0			0
$831$	−50.9067			−1.76593
$832$		0			0
$833$	−0.134430			−0.00465772
$834$		0			0
$835$	89.5099			3.09762
$836$		0			0
$837$	46.6847			1.61366
$838$		0			0
$839$	6.10711			0.210841			0.105420	−	0.994428i	$-0.466381\pi$
$839$	6.10711			0.210841			0.105420	+	0.994428i	$0.466381\pi$
$840$		0			0
$841$	−14.9230			−0.514586
$842$		0			0
$843$		−	33.8013i		−	1.16418i
$844$		0			0
$845$	11.1097			0.382184
$846$		0			0
$847$			2.65082i			0.0910833i
$848$		0			0
$849$			8.40331i			0.288401i
$850$		0			0
$851$	−2.28575			−0.0783545
$852$		0			0
$853$	37.3492			1.27881			0.639406	−	0.768870i	$-0.279180\pi$
$853$	37.3492			1.27881			0.639406	+	0.768870i	$0.279180\pi$
$854$		0			0
$855$	−0.299940	−	0.723345i	−0.0102577	−	0.0247379i
$856$		0			0
$857$		−	0.574397i		−	0.0196210i	−0.999952	−	0.00981052i	$-0.996877\pi$
$857$		−	0.574397i		−	0.0196210i	0.999952	−	0.00981052i	$-0.00312284\pi$
$858$		0			0
$859$		−	20.8875i		−	0.712672i	−0.934358	−	0.356336i	$-0.884026\pi$
$859$		−	20.8875i		−	0.712672i	0.934358	−	0.356336i	$-0.115974\pi$
$860$		0			0
$861$	5.29417			0.180425
$862$		0			0
$863$	31.0812			1.05802			0.529009	−	0.848616i	$-0.322564\pi$
$863$	31.0812			1.05802			0.529009	+	0.848616i	$0.322564\pi$
$864$		0			0
$865$			83.5060i			2.83929i
$866$		0			0
$867$	−13.8438			−0.470159
$868$		0			0
$869$		−	11.2460i		−	0.381494i
$870$		0			0
$871$		−	36.1292i		−	1.22419i
$872$		0			0
$873$			0.786163i			0.0266076i
$874$		0			0
$875$			33.5686i			1.13483i
$876$		0			0
$877$		−	28.0821i		−	0.948264i	−0.880454	−	0.474132i	$-0.842762\pi$
$877$		−	28.0821i		−	0.948264i	0.880454	−	0.474132i	$-0.157238\pi$
$878$		0			0
$879$		−	41.7502i		−	1.40820i
$880$		0			0
$881$	−4.38646			−0.147784			−0.0738918	−	0.997266i	$-0.523542\pi$
$881$	−4.38646			−0.147784			−0.0738918	+	0.997266i	$0.523542\pi$
$882$		0			0
$883$		−	29.9628i		−	1.00833i	−0.863608	−	0.504164i	$-0.831801\pi$
$883$		−	29.9628i		−	1.00833i	0.863608	−	0.504164i	$-0.168199\pi$
$884$		0			0
$885$	67.9565			2.28433
$886$		0			0
$887$	−16.2704			−0.546307			−0.273154	−	0.961970i	$-0.588067\pi$
$887$	−16.2704			−0.546307			−0.273154	+	0.961970i	$0.588067\pi$
$888$		0			0
$889$			6.93178i			0.232484i
$890$		0			0
$891$			8.85066i			0.296508i
$892$		0			0
$893$	28.2870	−	11.7294i	0.946588	−	0.392509i
$894$		0			0
$895$	−8.74477			−0.292306
$896$		0			0
$897$	1.40353			0.0468625
$898$		0			0
$899$			59.0718i			1.97015i
$900$		0			0
$901$		−	28.2703i		−	0.941820i
$902$		0			0
$903$	39.1511			1.30287
$904$		0			0
$905$		−	85.6939i		−	2.84856i
$906$		0			0
$907$	34.2924			1.13866			0.569331	−	0.822109i	$-0.307202\pi$
$907$	34.2924			1.13866			0.569331	+	0.822109i	$0.307202\pi$
$908$		0			0
$909$	−0.278641			−0.00924194
$910$		0			0
$911$	−21.2972			−0.705608			−0.352804	−	0.935697i	$-0.614772\pi$
$911$	−21.2972			−0.705608			−0.352804	+	0.935697i	$0.614772\pi$
$912$		0			0
$913$	−3.09050			−0.102280
$914$		0			0
$915$	−94.8812			−3.13668
$916$		0			0
$917$	−33.6019			−1.10963
$918$		0			0
$919$		−	32.3841i		−	1.06825i	−0.845404	−	0.534127i	$-0.820640\pi$
$919$		−	32.3841i		−	1.06825i	0.845404	−	0.534127i	$-0.179360\pi$
$920$		0			0
$921$	−24.4065			−0.804220
$922$		0			0
$923$			14.5438i			0.478716i
$924$		0			0
$925$		−	94.6755i		−	3.11291i
$926$		0			0
$927$	0.0915451			0.00300674
$928$		0			0
$929$	−35.7672			−1.17348			−0.586741	−	0.809774i	$-0.699589\pi$
$929$	−35.7672			−1.17348			−0.586741	+	0.809774i	$0.699589\pi$
$930$		0			0
$931$	0.0448362	+	0.108128i	0.00146945	+	0.00354377i
$932$		0			0
$933$			51.5011i			1.68607i
$934$		0			0
$935$			18.3604i			0.600451i
$936$		0			0
$937$	31.0421			1.01410			0.507051	−	0.861916i	$-0.330735\pi$
$937$	31.0421			1.01410			0.507051	+	0.861916i	$0.330735\pi$
$938$		0			0
$939$	5.73294			0.187088
$940$		0			0
$941$			40.4101i			1.31733i	0.752435	+	0.658667i	$0.228879\pi$
$941$			40.4101i			1.31733i	−0.752435	+	0.658667i	$0.771121\pi$
$942$		0			0
$943$	−0.237254			−0.00772606
$944$		0			0
$945$			50.9242i			1.65657i
$946$		0			0
$947$			34.2044i			1.11149i	0.831352	+	0.555747i	$0.187568\pi$
$947$			34.2044i			1.11149i	−0.831352	+	0.555747i	$0.812432\pi$
$948$		0			0
$949$			29.8419i			0.968710i
$950$		0			0
$951$		−	7.23330i		−	0.234556i
$952$		0			0
$953$			28.6127i			0.926857i	0.886134	+	0.463429i	$0.153381\pi$
$953$			28.6127i			0.926857i	−0.886134	+	0.463429i	$0.846619\pi$
$954$		0			0
$955$			93.1144i			3.01311i
$956$		0			0
$957$	−11.3850			−0.368024
$958$		0			0
$959$		−	9.62524i		−	0.310815i
$960$		0			0
$961$	48.4453			1.56275
$962$		0			0
$963$	−0.777916			−0.0250680
$964$		0			0
$965$		−	28.7163i		−	0.924410i
$966$		0			0
$967$			42.8423i			1.37771i	0.724897	+	0.688857i	$0.241887\pi$
$967$			42.8423i			1.37771i	−0.724897	+	0.688857i	$0.758113\pi$
$968$		0			0
$969$	14.3575	+	34.6250i	0.461229	+	1.11232i
$970$		0			0
$971$	22.5461			0.723537			0.361769	−	0.932268i	$-0.382173\pi$
$971$	22.5461			0.723537			0.361769	+	0.932268i	$0.382173\pi$
$972$		0			0
$973$	15.3238			0.491258
$974$		0			0
$975$			58.1341i			1.86178i
$976$		0			0
$977$		−	48.6562i		−	1.55665i	−0.627861	−	0.778325i	$-0.716069\pi$
$977$		−	48.6562i		−	1.55665i	0.627861	−	0.778325i	$-0.283931\pi$
$978$		0			0
$979$	1.40070			0.0447666
$980$		0			0
$981$			0.247603i			0.00790535i
$982$		0			0
$983$	−44.5854			−1.42205			−0.711027	−	0.703165i	$-0.751769\pi$
$983$	−44.5854			−1.42205			−0.711027	+	0.703165i	$0.751769\pi$
$984$		0			0
$985$	26.6131			0.847965
$986$		0			0
$987$	−31.9911			−1.01829
$988$		0			0
$989$	−1.75453			−0.0557907
$990$		0			0
$991$	−50.7244			−1.61131			−0.805656	−	0.592384i	$-0.798187\pi$
$991$	−50.7244			−1.61131			−0.805656	+	0.592384i	$0.798187\pi$
$992$		0			0
$993$	28.4691			0.903440
$994$		0			0
$995$		−	41.9146i		−	1.32878i
$996$		0			0
$997$	−24.4545			−0.774481			−0.387240	−	0.921979i	$-0.626572\pi$
$997$	−24.4545			−0.774481			−0.387240	+	0.921979i	$0.626572\pi$
$998$		0			0
$999$		−	58.6660i		−	1.85611i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.a.1519.10	yes	36
4.3	odd	2	inner	3344.2.o.a.1519.27	yes	36
19.18	odd	2	inner	3344.2.o.a.1519.28	yes	36
76.75	even	2	inner	3344.2.o.a.1519.9	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.a.1519.9	✓	36	76.75	even	2	inner
3344.2.o.a.1519.10	yes	36	1.1	even	1	trivial
3344.2.o.a.1519.27	yes	36	4.3	odd	2	inner
3344.2.o.a.1519.28	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)

\(f(q)\)	\(=\)	\(q-1.71785 q^{3} -3.66778 q^{5} -2.65082i q^{7} -0.0489799 q^{9} +O(q^{10})\)	\(q-1.71785 q^{3} -3.66778 q^{5} -2.65082i q^{7} -0.0489799 q^{9} -1.00000i q^{11} -4.00362i q^{13} +6.30071 q^{15} +5.00587 q^{17} +(-1.66960 - 4.02647i) q^{19} +4.55372i q^{21} -0.204072i q^{23} +8.45262 q^{25} +5.23770 q^{27} +6.62744i q^{29} +8.91321 q^{31} +1.71785i q^{33} +9.72263i q^{35} -11.2007i q^{37} +6.87764i q^{39} -1.16260i q^{41} -8.59761i q^{43} +0.179648 q^{45} +7.02526i q^{47} -0.0268544 q^{49} -8.59936 q^{51} -5.64742i q^{53} +3.66778i q^{55} +(2.86813 + 6.91688i) q^{57} +10.7855 q^{59} -15.0588 q^{61} +0.129837i q^{63} +14.6844i q^{65} +9.02412 q^{67} +0.350565i q^{69} -3.63267 q^{71} -7.45373 q^{73} -14.5204 q^{75} -2.65082 q^{77} +11.2460 q^{79} -8.85066 q^{81} -3.09050i q^{83} -18.3604 q^{85} -11.3850i q^{87} +1.40070i q^{89} -10.6129 q^{91} -15.3116 q^{93} +(6.12373 + 14.7682i) q^{95} -16.0507i q^{97} +0.0489799i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 44 q^{9}+O(q^{10}) \) 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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