LMFDB - Embedded newform 3344.2.o.a.1519.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3344.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.o (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31411 q^{3} +1.36581 q^{5} -4.52438i q^{7} -1.27312 q^{9} +O(q^{10})$	$q-1.31411 q^{3} +1.36581 q^{5} -4.52438i q^{7} -1.27312 q^{9} -1.00000i q^{11} -6.15181i q^{13} -1.79482 q^{15} -7.08674 q^{17} +(2.02768 - 3.85856i) q^{19} +5.94552i q^{21} -1.57220i q^{23} -3.13457 q^{25} +5.61534 q^{27} -6.00931i q^{29} -1.17161 q^{31} +1.31411i q^{33} -6.17943i q^{35} +10.6012i q^{37} +8.08415i q^{39} +6.26175i q^{41} -0.714490i q^{43} -1.73884 q^{45} -1.81799i q^{47} -13.4700 q^{49} +9.31275 q^{51} +10.0904i q^{53} -1.36581i q^{55} +(-2.66460 + 5.07057i) q^{57} -5.62345 q^{59} -2.63038 q^{61} +5.76007i q^{63} -8.40220i q^{65} -12.8075 q^{67} +2.06604i q^{69} +11.4751 q^{71} +7.28533 q^{73} +4.11916 q^{75} -4.52438 q^{77} +16.1356 q^{79} -3.55981 q^{81} +4.65498i q^{83} -9.67913 q^{85} +7.89689i q^{87} +7.01151i q^{89} -27.8331 q^{91} +1.53962 q^{93} +(2.76943 - 5.27006i) q^{95} -3.74462i q^{97} +1.27312i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.31411			−0.758701			−0.379350	−	0.925253i	$-0.623853\pi$
$3$	−1.31411			−0.758701			−0.379350	+	0.925253i	$0.623853\pi$
$4$		0			0
$5$	1.36581			0.610808			0.305404	−	0.952223i	$-0.401208\pi$
$5$	1.36581			0.610808			0.305404	+	0.952223i	$0.401208\pi$
$6$		0			0
$7$		−	4.52438i		−	1.71005i	−0.518584	−	0.855027i	$-0.673541\pi$
$7$		−	4.52438i		−	1.71005i	0.518584	−	0.855027i	$-0.326459\pi$
$8$		0			0
$9$	−1.27312			−0.424373
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$		−	6.15181i		−	1.70621i	−0.521742	−	0.853103i	$-0.674718\pi$
$13$		−	6.15181i		−	1.70621i	0.521742	−	0.853103i	$-0.325282\pi$
$14$		0			0
$15$	−1.79482			−0.463421
$16$		0			0
$17$	−7.08674			−1.71879			−0.859394	−	0.511315i	$-0.829159\pi$
$17$	−7.08674			−1.71879			−0.859394	+	0.511315i	$0.829159\pi$
$18$		0			0
$19$	2.02768	−	3.85856i	0.465183	−	0.885215i
$19$	2.02768	−	3.85856i	0.465183	−	0.885215i
$20$		0			0
$21$			5.94552i			1.29742i
$22$		0			0
$23$		−	1.57220i		−	0.327826i	−0.986475	−	0.163913i	$-0.947588\pi$
$23$		−	1.57220i		−	0.327826i	0.986475	−	0.163913i	$-0.0524117\pi$
$24$		0			0
$25$	−3.13457			−0.626913
$26$		0			0
$27$	5.61534			1.08067
$28$		0			0
$29$		−	6.00931i		−	1.11590i	−0.829874	−	0.557951i	$-0.811588\pi$
$29$		−	6.00931i		−	1.11590i	0.829874	−	0.557951i	$-0.188412\pi$
$30$		0			0
$31$	−1.17161			−0.210427			−0.105213	−	0.994450i	$-0.533553\pi$
$31$	−1.17161			−0.210427			−0.105213	+	0.994450i	$0.533553\pi$
$32$		0			0
$33$			1.31411i			0.228757i
$34$		0			0
$35$		−	6.17943i		−	1.04451i
$36$		0			0
$37$			10.6012i			1.74282i	0.490557	+	0.871409i	$0.336793\pi$
$37$			10.6012i			1.74282i	−0.490557	+	0.871409i	$0.663207\pi$
$38$		0			0
$39$			8.08415i			1.29450i
$40$		0			0
$41$			6.26175i			0.977922i	0.872306	+	0.488961i	$0.162624\pi$
$41$			6.26175i			0.977922i	−0.872306	+	0.488961i	$0.837376\pi$
$42$		0			0
$43$		−	0.714490i		−	0.108959i	−0.998515	−	0.0544794i	$-0.982650\pi$
$43$		−	0.714490i		−	0.108959i	0.998515	−	0.0544794i	$-0.0173499\pi$
$44$		0			0
$45$	−1.73884			−0.259211
$46$		0			0
$47$		−	1.81799i		−	0.265181i	−0.991171	−	0.132591i	$-0.957670\pi$
$47$		−	1.81799i		−	0.265181i	0.991171	−	0.132591i	$-0.0423295\pi$
$48$		0			0
$49$	−13.4700			−1.92428
$50$		0			0
$51$	9.31275			1.30405
$52$		0			0
$53$			10.0904i			1.38602i	0.720928	+	0.693010i	$0.243716\pi$
$53$			10.0904i			1.38602i	−0.720928	+	0.693010i	$0.756284\pi$
$54$		0			0
$55$		−	1.36581i		−	0.184166i
$56$		0			0
$57$	−2.66460	+	5.07057i	−0.352934	+	0.671613i
$58$		0			0
$59$	−5.62345			−0.732111			−0.366055	−	0.930593i	$-0.619292\pi$
$59$	−5.62345			−0.732111			−0.366055	+	0.930593i	$0.619292\pi$
$60$		0			0
$61$	−2.63038			−0.336786			−0.168393	−	0.985720i	$-0.553858\pi$
$61$	−2.63038			−0.336786			−0.168393	+	0.985720i	$0.553858\pi$
$62$		0			0
$63$			5.76007i			0.725701i
$64$		0			0
$65$		−	8.40220i		−	1.04216i
$66$		0			0
$67$	−12.8075			−1.56469			−0.782345	−	0.622845i	$-0.785977\pi$
$67$	−12.8075			−1.56469			−0.782345	+	0.622845i	$0.785977\pi$
$68$		0			0
$69$			2.06604i			0.248722i
$70$		0			0
$71$	11.4751			1.36185			0.680924	−	0.732354i	$-0.261578\pi$
$71$	11.4751			1.36185			0.680924	+	0.732354i	$0.261578\pi$
$72$		0			0
$73$	7.28533			0.852683			0.426342	−	0.904562i	$-0.359802\pi$
$73$	7.28533			0.852683			0.426342	+	0.904562i	$0.359802\pi$
$74$		0			0
$75$	4.11916			0.475640
$76$		0			0
$77$	−4.52438			−0.515600
$78$		0			0
$79$	16.1356			1.81540			0.907698	−	0.419625i	$-0.137838\pi$
$79$	16.1356			1.81540			0.907698	+	0.419625i	$0.137838\pi$
$80$		0			0
$81$	−3.55981			−0.395534
$82$		0			0
$83$			4.65498i			0.510951i	0.966816	+	0.255475i	$0.0822320\pi$
$83$			4.65498i			0.510951i	−0.966816	+	0.255475i	$0.917768\pi$
$84$		0			0
$85$	−9.67913			−1.04985
$86$		0			0
$87$			7.89689i			0.846635i
$88$		0			0
$89$			7.01151i			0.743219i	0.928389	+	0.371609i	$0.121194\pi$
$89$			7.01151i			0.743219i	−0.928389	+	0.371609i	$0.878806\pi$
$90$		0			0
$91$	−27.8331			−2.91770
$92$		0			0
$93$	1.53962			0.159651
$94$		0			0
$95$	2.76943	−	5.27006i	0.284137	−	0.540696i
$96$		0			0
$97$		−	3.74462i		−	0.380209i	−0.981764	−	0.190104i	$-0.939117\pi$
$97$		−	3.74462i		−	0.380209i	0.981764	−	0.190104i	$-0.0608826\pi$
$98$		0			0
$99$			1.27312i			0.127953i
$100$		0			0
$101$	17.2823			1.71966			0.859828	−	0.510584i	$-0.170571\pi$
$101$	17.2823			1.71966			0.859828	+	0.510584i	$0.170571\pi$
$102$		0			0
$103$	17.4647			1.72084			0.860422	−	0.509583i	$-0.170200\pi$
$103$	17.4647			1.72084			0.860422	+	0.509583i	$0.170200\pi$
$104$		0			0
$105$			8.12044i			0.792474i
$106$		0			0
$107$	10.4639			1.01158			0.505791	−	0.862656i	$-0.331201\pi$
$107$	10.4639			1.01158			0.505791	+	0.862656i	$0.331201\pi$
$108$		0			0
$109$		−	2.48940i		−	0.238441i	−0.992868	−	0.119220i	$-0.961960\pi$
$109$		−	2.48940i		−	0.238441i	0.992868	−	0.119220i	$-0.0380395\pi$
$110$		0			0
$111$		−	13.9311i		−	1.32228i
$112$		0			0
$113$		−	10.7800i		−	1.01410i	−0.861917	−	0.507050i	$-0.830736\pi$
$113$		−	10.7800i		−	1.01410i	0.861917	−	0.507050i	$-0.169264\pi$
$114$		0			0
$115$		−	2.14733i		−	0.200239i
$116$		0			0
$117$			7.83199i			0.724068i
$118$		0			0
$119$			32.0631i			2.93922i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	8.22862i		−	0.741950i
$124$		0			0
$125$	−11.1103			−0.993732
$126$		0			0
$127$	−3.41764			−0.303267			−0.151633	−	0.988437i	$-0.548453\pi$
$127$	−3.41764			−0.303267			−0.151633	+	0.988437i	$0.548453\pi$
$128$		0			0
$129$			0.938917i			0.0826671i
$130$		0			0
$131$			18.4803i			1.61463i	0.590122	+	0.807314i	$0.299080\pi$
$131$			18.4803i			1.61463i	−0.590122	+	0.807314i	$0.700920\pi$
$132$		0			0
$133$	−17.4576	−	9.17400i	−1.51376	−	0.795487i
$134$		0			0
$135$	7.66948			0.660084
$136$		0			0
$137$	2.18594			0.186757			0.0933786	−	0.995631i	$-0.470233\pi$
$137$	2.18594			0.186757			0.0933786	+	0.995631i	$0.470233\pi$
$138$		0			0
$139$		−	0.120434i		−	0.0102151i	−0.999987	−	0.00510756i	$-0.998374\pi$
$139$		−	0.120434i		−	0.0102151i	0.999987	−	0.00510756i	$-0.00162579\pi$
$140$		0			0
$141$			2.38904i			0.201193i
$142$		0			0
$143$	−6.15181			−0.514441
$144$		0			0
$145$		−	8.20757i		−	0.681602i
$146$		0			0
$147$	17.7010			1.45995
$148$		0			0
$149$	−17.0025			−1.39290			−0.696448	−	0.717608i	$-0.745237\pi$
$149$	−17.0025			−1.39290			−0.696448	+	0.717608i	$0.745237\pi$
$150$		0			0
$151$	−18.1987			−1.48099			−0.740496	−	0.672061i	$-0.765409\pi$
$151$	−18.1987			−1.48099			−0.740496	+	0.672061i	$0.765409\pi$
$152$		0			0
$153$	9.02227			0.729407
$154$		0			0
$155$	−1.60019			−0.128530
$156$		0			0
$157$	−17.2570			−1.37725			−0.688627	−	0.725115i	$-0.741786\pi$
$157$	−17.2570			−1.37725			−0.688627	+	0.725115i	$0.741786\pi$
$158$		0			0
$159$		−	13.2599i		−	1.05158i
$160$		0			0
$161$	−7.11323			−0.560601
$162$		0			0
$163$		−	21.8774i		−	1.71357i	−0.515674	−	0.856785i	$-0.672458\pi$
$163$		−	21.8774i		−	1.71357i	0.515674	−	0.856785i	$-0.327542\pi$
$164$		0			0
$165$			1.79482i			0.139727i
$166$		0			0
$167$	2.56662			0.198611			0.0993055	−	0.995057i	$-0.468338\pi$
$167$	2.56662			0.198611			0.0993055	+	0.995057i	$0.468338\pi$
$168$		0			0
$169$	−24.8448			−1.91114
$170$		0			0
$171$	−2.58148	+	4.91241i	−0.197411	+	0.375661i
$172$		0			0
$173$		−	7.19048i		−	0.546682i	−0.961917	−	0.273341i	$-0.911871\pi$
$173$		−	7.19048i		−	0.546682i	0.961917	−	0.273341i	$-0.0881288\pi$
$174$		0			0
$175$			14.1820i			1.07206i
$176$		0			0
$177$	7.38982			0.555453
$178$		0			0
$179$	−7.17814			−0.536519			−0.268260	−	0.963347i	$-0.586448\pi$
$179$	−7.17814			−0.536519			−0.268260	+	0.963347i	$0.586448\pi$
$180$		0			0
$181$			11.4286i			0.849481i	0.905315	+	0.424740i	$0.139635\pi$
$181$			11.4286i			0.849481i	−0.905315	+	0.424740i	$0.860365\pi$
$182$		0			0
$183$	3.45661			0.255520
$184$		0			0
$185$			14.4791i			1.06453i
$186$		0			0
$187$			7.08674i			0.518234i
$188$		0			0
$189$		−	25.4059i		−	1.84801i
$190$		0			0
$191$		−	2.60966i		−	0.188829i	−0.995533	−	0.0944144i	$-0.969902\pi$
$191$		−	2.60966i		−	0.188829i	0.995533	−	0.0944144i	$-0.0300979\pi$
$192$		0			0
$193$		−	6.51514i		−	0.468970i	−0.972120	−	0.234485i	$-0.924660\pi$
$193$		−	6.51514i		−	0.468970i	0.972120	−	0.234485i	$-0.0753404\pi$
$194$		0			0
$195$			11.0414i			0.790691i
$196$		0			0
$197$	20.8746			1.48725			0.743626	−	0.668596i	$-0.233104\pi$
$197$	20.8746			1.48725			0.743626	+	0.668596i	$0.233104\pi$
$198$		0			0
$199$		−	15.1308i		−	1.07260i	−0.844029	−	0.536298i	$-0.819822\pi$
$199$		−	15.1308i		−	1.07260i	0.844029	−	0.536298i	$-0.180178\pi$
$200$		0			0
$201$	16.8305			1.18713
$202$		0			0
$203$	−27.1884			−1.90825
$204$		0			0
$205$			8.55236i			0.597323i
$206$		0			0
$207$			2.00160i			0.139121i
$208$		0			0
$209$	−3.85856	−	2.02768i	−0.266902	−	0.140258i
$210$		0			0
$211$	−17.9753			−1.23747			−0.618736	−	0.785599i	$-0.712355\pi$
$211$	−17.9753			−1.23747			−0.618736	+	0.785599i	$0.712355\pi$
$212$		0			0
$213$	−15.0796			−1.03324
$214$		0			0
$215$		−	0.975857i		−	0.0665529i
$216$		0			0
$217$			5.30079i			0.359841i
$218$		0			0
$219$	−9.57371			−0.646932
$220$		0			0
$221$			43.5963i			2.93261i
$222$		0			0
$223$	−15.7739			−1.05630			−0.528151	−	0.849151i	$-0.677114\pi$
$223$	−15.7739			−1.05630			−0.528151	+	0.849151i	$0.677114\pi$
$224$		0			0
$225$	3.99068			0.266045
$226$		0			0
$227$	−10.6481			−0.706742			−0.353371	−	0.935483i	$-0.614965\pi$
$227$	−10.6481			−0.706742			−0.353371	+	0.935483i	$0.614965\pi$
$228$		0			0
$229$	10.6593			0.704384			0.352192	−	0.935928i	$-0.385436\pi$
$229$	10.6593			0.704384			0.352192	+	0.935928i	$0.385436\pi$
$230$		0			0
$231$	5.94552			0.391186
$232$		0			0
$233$	4.53532			0.297119			0.148559	−	0.988903i	$-0.452536\pi$
$233$	4.53532			0.297119			0.148559	+	0.988903i	$0.452536\pi$
$234$		0			0
$235$		−	2.48303i		−	0.161975i
$236$		0			0
$237$	−21.2039			−1.37734
$238$		0			0
$239$		−	16.1688i		−	1.04587i	−0.852372	−	0.522935i	$-0.824837\pi$
$239$		−	16.1688i		−	1.04587i	0.852372	−	0.522935i	$-0.175163\pi$
$240$		0			0
$241$			6.96214i			0.448471i	0.974535	+	0.224236i	$0.0719885\pi$
$241$			6.96214i			0.448471i	−0.974535	+	0.224236i	$0.928012\pi$
$242$		0			0
$243$	−12.1680			−0.780581
$244$		0			0
$245$	−18.3974			−1.17537
$246$		0			0
$247$	−23.7372	−	12.4739i	−1.51036	−	0.793697i
$248$		0			0
$249$		−	6.11715i		−	0.387659i
$250$		0			0
$251$			7.65949i			0.483463i	0.970343	+	0.241731i	$0.0777153\pi$
$251$			7.65949i			0.483463i	−0.970343	+	0.241731i	$0.922285\pi$
$252$		0			0
$253$	−1.57220			−0.0988434
$254$		0			0
$255$	12.7194			0.796522
$256$		0			0
$257$			19.9033i			1.24153i	0.783996	+	0.620766i	$0.213178\pi$
$257$			19.9033i			1.24153i	−0.783996	+	0.620766i	$0.786822\pi$
$258$		0			0
$259$	47.9636			2.98031
$260$		0			0
$261$			7.65057i			0.473558i
$262$		0			0
$263$		−	6.84632i		−	0.422162i	−0.977469	−	0.211081i	$-0.932302\pi$
$263$		−	6.84632i		−	0.422162i	0.977469	−	0.211081i	$-0.0676984\pi$
$264$		0			0
$265$			13.7815i			0.846593i
$266$		0			0
$267$		−	9.21389i		−	0.563881i
$268$		0			0
$269$			24.0554i			1.46668i	0.679860	+	0.733342i	$0.262041\pi$
$269$			24.0554i			1.46668i	−0.679860	+	0.733342i	$0.737959\pi$
$270$		0			0
$271$			10.5993i			0.643863i	0.946763	+	0.321931i	$0.104332\pi$
$271$			10.5993i			0.643863i	−0.946763	+	0.321931i	$0.895668\pi$
$272$		0			0
$273$	36.5757			2.21366
$274$		0			0
$275$			3.13457i			0.189021i
$276$		0			0
$277$	−7.85391			−0.471896			−0.235948	−	0.971766i	$-0.575820\pi$
$277$	−7.85391			−0.471896			−0.235948	+	0.971766i	$0.575820\pi$
$278$		0			0
$279$	1.49160			0.0892995
$280$		0			0
$281$		−	4.22570i		−	0.252084i	−0.992025	−	0.126042i	$-0.959773\pi$
$281$		−	4.22570i		−	0.252084i	0.992025	−	0.126042i	$-0.0402274\pi$
$282$		0			0
$283$		−	31.4215i		−	1.86781i	−0.357519	−	0.933906i	$-0.616377\pi$
$283$		−	31.4215i		−	1.86781i	0.357519	−	0.933906i	$-0.383623\pi$
$284$		0			0
$285$	−3.63933	+	6.92543i	−0.215575	+	0.410227i
$286$		0			0
$287$	28.3305			1.67230
$288$		0			0
$289$	33.2219			1.95423
$290$		0			0
$291$			4.92084i			0.288465i
$292$		0			0
$293$		−	21.5099i		−	1.25662i	−0.777962	−	0.628312i	$-0.783746\pi$
$293$		−	21.5099i		−	1.25662i	0.777962	−	0.628312i	$-0.216254\pi$
$294$		0			0
$295$	−7.68056			−0.447179
$296$		0			0
$297$		−	5.61534i		−	0.325835i
$298$		0			0
$299$	−9.67189			−0.559340
$300$		0			0
$301$	−3.23262			−0.186325
$302$		0			0
$303$	−22.7109			−1.30470
$304$		0			0
$305$	−3.59260			−0.205712
$306$		0			0
$307$	−16.7379			−0.955281			−0.477640	−	0.878555i	$-0.658508\pi$
$307$	−16.7379			−0.955281			−0.477640	+	0.878555i	$0.658508\pi$
$308$		0			0
$309$	−22.9504			−1.30561
$310$		0			0
$311$		−	18.9708i		−	1.07574i	−0.843029	−	0.537869i	$-0.819230\pi$
$311$		−	18.9708i		−	1.07574i	0.843029	−	0.537869i	$-0.180770\pi$
$312$		0			0
$313$	−6.64777			−0.375754			−0.187877	−	0.982193i	$-0.560161\pi$
$313$	−6.64777			−0.375754			−0.187877	+	0.982193i	$0.560161\pi$
$314$		0			0
$315$			7.86715i			0.443264i
$316$		0			0
$317$		−	32.4401i		−	1.82202i	−0.412389	−	0.911008i	$-0.635306\pi$
$317$		−	32.4401i		−	1.82202i	0.412389	−	0.911008i	$-0.364694\pi$
$318$		0			0
$319$	−6.00931			−0.336457
$320$		0			0
$321$	−13.7507			−0.767488
$322$		0			0
$323$	−14.3697	+	27.3446i	−0.799550	+	1.52150i
$324$		0			0
$325$			19.2833i			1.06964i
$326$		0			0
$327$			3.27134i			0.180905i
$328$		0			0
$329$	−8.22527			−0.453474
$330$		0			0
$331$	−16.4186			−0.902446			−0.451223	−	0.892411i	$-0.649012\pi$
$331$	−16.4186			−0.902446			−0.451223	+	0.892411i	$0.649012\pi$
$332$		0			0
$333$		−	13.4965i		−	0.739605i
$334$		0			0
$335$	−17.4926			−0.955725
$336$		0			0
$337$		−	13.7459i		−	0.748787i	−0.927270	−	0.374393i	$-0.877851\pi$
$337$		−	13.7459i		−	0.748787i	0.927270	−	0.374393i	$-0.122149\pi$
$338$		0			0
$339$			14.1661i			0.769398i
$340$		0			0
$341$			1.17161i			0.0634461i
$342$		0			0
$343$			29.2726i			1.58057i
$344$		0			0
$345$			2.82182i			0.151922i
$346$		0			0
$347$		−	4.55648i		−	0.244605i	−0.992493	−	0.122302i	$-0.960972\pi$
$347$		−	4.55648i		−	0.244605i	0.992493	−	0.122302i	$-0.0390278\pi$
$348$		0			0
$349$	−16.0704			−0.860229			−0.430114	−	0.902774i	$-0.641527\pi$
$349$	−16.0704			−0.860229			−0.430114	+	0.902774i	$0.641527\pi$
$350$		0			0
$351$		−	34.5445i		−	1.84385i
$352$		0			0
$353$	−6.00621			−0.319679			−0.159839	−	0.987143i	$-0.551098\pi$
$353$	−6.00621			−0.319679			−0.159839	+	0.987143i	$0.551098\pi$
$354$		0			0
$355$	15.6728			0.831828
$356$		0			0
$357$		−	42.1344i		−	2.22999i
$358$		0			0
$359$			25.6100i			1.35164i	0.737065	+	0.675822i	$0.236211\pi$
$359$			25.6100i			1.35164i	−0.737065	+	0.675822i	$0.763789\pi$
$360$		0			0
$361$	−10.7770	−	15.6479i	−0.567210	−	0.823573i
$362$		0			0
$363$	1.31411			0.0689728
$364$		0			0
$365$	9.95037			0.520826
$366$		0			0
$367$		−	18.2903i		−	0.954745i	−0.878701	−	0.477372i	$-0.841589\pi$
$367$		−	18.2903i		−	0.954745i	0.878701	−	0.477372i	$-0.158411\pi$
$368$		0			0
$369$		−	7.97196i		−	0.415004i
$370$		0			0
$371$	45.6527			2.37017
$372$		0			0
$373$		−	9.86117i		−	0.510592i	−0.966863	−	0.255296i	$-0.917827\pi$
$373$		−	9.86117i		−	0.510592i	0.966863	−	0.255296i	$-0.0821729\pi$
$374$		0			0
$375$	14.6001			0.753945
$376$		0			0
$377$	−36.9682			−1.90396
$378$		0			0
$379$	21.1325			1.08550			0.542752	−	0.839893i	$-0.317382\pi$
$379$	21.1325			1.08550			0.542752	+	0.839893i	$0.317382\pi$
$380$		0			0
$381$	4.49115			0.230089
$382$		0			0
$383$	−2.62725			−0.134246			−0.0671230	−	0.997745i	$-0.521382\pi$
$383$	−2.62725			−0.134246			−0.0671230	+	0.997745i	$0.521382\pi$
$384$		0			0
$385$	−6.17943			−0.314933
$386$		0			0
$387$			0.909631i			0.0462391i
$388$		0			0
$389$	−16.1936			−0.821048			−0.410524	−	0.911850i	$-0.634654\pi$
$389$	−16.1936			−0.821048			−0.410524	+	0.911850i	$0.634654\pi$
$390$		0			0
$391$			11.1418i			0.563464i
$392$		0			0
$393$		−	24.2851i		−	1.22502i
$394$		0			0
$395$	22.0381			1.10886
$396$		0			0
$397$	18.2529			0.916085			0.458042	−	0.888930i	$-0.348551\pi$
$397$	18.2529			0.916085			0.458042	+	0.888930i	$0.348551\pi$
$398$		0			0
$399$	22.9412	+	12.0556i	1.14849	+	0.603537i
$400$		0			0
$401$			23.1837i			1.15774i	0.815420	+	0.578870i	$0.196506\pi$
$401$			23.1837i			1.15774i	−0.815420	+	0.578870i	$0.803494\pi$
$402$		0			0
$403$			7.20751i			0.359032i
$404$		0			0
$405$	−4.86202			−0.241596
$406$		0			0
$407$	10.6012			0.525480
$408$		0			0
$409$			5.30810i			0.262469i	0.991351	+	0.131234i	$0.0418940\pi$
$409$			5.30810i			0.262469i	−0.991351	+	0.131234i	$0.958106\pi$
$410$		0			0
$411$	−2.87256			−0.141693
$412$		0			0
$413$			25.4426i			1.25195i
$414$		0			0
$415$			6.35782i			0.312093i
$416$		0			0
$417$			0.158264i			0.00775021i
$418$		0			0
$419$			27.9520i			1.36555i	0.730630	+	0.682773i	$0.239226\pi$
$419$			27.9520i			1.36555i	−0.730630	+	0.682773i	$0.760774\pi$
$420$		0			0
$421$			11.6942i			0.569939i	0.958537	+	0.284970i	$0.0919835\pi$
$421$			11.6942i			0.569939i	−0.958537	+	0.284970i	$0.908017\pi$
$422$		0			0
$423$			2.31452i			0.112536i
$424$		0			0
$425$	22.2139			1.07753
$426$		0			0
$427$			11.9008i			0.575922i
$428$		0			0
$429$	8.08415			0.390306
$430$		0			0
$431$	−32.6237			−1.57143			−0.785713	−	0.618591i	$-0.787704\pi$
$431$	−32.6237			−1.57143			−0.785713	+	0.618591i	$0.787704\pi$
$432$		0			0
$433$			22.5834i			1.08529i	0.839963	+	0.542644i	$0.182577\pi$
$433$			22.5834i			1.08529i	−0.839963	+	0.542644i	$0.817423\pi$
$434$		0			0
$435$			10.7856i			0.517132i
$436$		0			0
$437$	−6.06643	−	3.18793i	−0.290197	−	0.152499i
$438$		0			0
$439$	−3.91705			−0.186951			−0.0934754	−	0.995622i	$-0.529798\pi$
$439$	−3.91705			−0.186951			−0.0934754	+	0.995622i	$0.529798\pi$
$440$		0			0
$441$	17.1489			0.816613
$442$		0			0
$443$		−	21.0543i		−	1.00032i	−0.865933	−	0.500160i	$-0.833275\pi$
$443$		−	21.0543i		−	1.00032i	0.865933	−	0.500160i	$-0.166725\pi$
$444$		0			0
$445$			9.57638i			0.453964i
$446$		0			0
$447$	22.3431			1.05679
$448$		0			0
$449$		−	9.02038i		−	0.425698i	−0.977085	−	0.212849i	$-0.931726\pi$
$449$		−	9.02038i		−	0.425698i	0.977085	−	0.212849i	$-0.0682742\pi$
$450$		0			0
$451$	6.26175			0.294854
$452$		0			0
$453$	23.9151			1.12363
$454$		0			0
$455$	−38.0147			−1.78216
$456$		0			0
$457$	−11.9244			−0.557798			−0.278899	−	0.960320i	$-0.589969\pi$
$457$	−11.9244			−0.557798			−0.278899	+	0.960320i	$0.589969\pi$
$458$		0			0
$459$	−39.7945			−1.85745
$460$		0			0
$461$	−35.7802			−1.66645			−0.833226	−	0.552933i	$-0.813509\pi$
$461$	−35.7802			−1.66645			−0.833226	+	0.552933i	$0.813509\pi$
$462$		0			0
$463$			6.29863i			0.292722i	0.989231	+	0.146361i	$0.0467562\pi$
$463$			6.29863i			0.292722i	−0.989231	+	0.146361i	$0.953244\pi$
$464$		0			0
$465$	2.10283			0.0975162
$466$		0			0
$467$		−	10.9998i		−	0.509008i	−0.967072	−	0.254504i	$-0.918088\pi$
$467$		−	10.9998i		−	0.509008i	0.967072	−	0.254504i	$-0.0819122\pi$
$468$		0			0
$469$			57.9461i			2.67570i
$470$		0			0
$471$	22.6775			1.04492
$472$		0			0
$473$	−0.714490			−0.0328523
$474$		0			0
$475$	−6.35591	+	12.0949i	−0.291629	+	0.554953i
$476$		0			0
$477$		−	12.8463i		−	0.588190i
$478$		0			0
$479$		−	26.8446i		−	1.22656i	−0.789866	−	0.613280i	$-0.789850\pi$
$479$		−	26.8446i		−	1.22656i	0.789866	−	0.613280i	$-0.210150\pi$
$480$		0			0
$481$	65.2163			2.97361
$482$		0			0
$483$	9.34755			0.425328
$484$		0			0
$485$		−	5.11444i		−	0.232235i
$486$		0			0
$487$	11.6553			0.528151			0.264075	−	0.964502i	$-0.414933\pi$
$487$	11.6553			0.528151			0.264075	+	0.964502i	$0.414933\pi$
$488$		0			0
$489$			28.7493i			1.30009i
$490$		0			0
$491$			11.7621i			0.530815i	0.964136	+	0.265407i	$0.0855064\pi$
$491$			11.7621i			0.530815i	−0.964136	+	0.265407i	$0.914494\pi$
$492$		0			0
$493$			42.5864i			1.91800i
$494$		0			0
$495$			1.73884i			0.0781549i
$496$		0			0
$497$		−	51.9179i		−	2.32883i
$498$		0			0
$499$			17.2939i			0.774180i	0.922042	+	0.387090i	$0.126520\pi$
$499$			17.2939i			0.774180i	−0.922042	+	0.387090i	$0.873480\pi$
$500$		0			0
$501$	−3.37282			−0.150686
$502$		0			0
$503$		−	29.9943i		−	1.33738i	−0.743542	−	0.668689i	$-0.766856\pi$
$503$		−	29.9943i		−	1.33738i	0.743542	−	0.668689i	$-0.233144\pi$
$504$		0			0
$505$	23.6044			1.05038
$506$		0			0
$507$	32.6488			1.44998
$508$		0			0
$509$		−	4.12809i		−	0.182974i	−0.995806	−	0.0914872i	$-0.970838\pi$
$509$		−	4.12809i		−	0.182974i	0.995806	−	0.0914872i	$-0.0291620\pi$
$510$		0			0
$511$		−	32.9616i		−	1.45813i
$512$		0			0
$513$	11.3861	−	21.6671i	0.502710	−	0.956628i
$514$		0			0
$515$	23.8534			1.05111
$516$		0			0
$517$	−1.81799			−0.0799551
$518$		0			0
$519$			9.44907i			0.414768i
$520$		0			0
$521$		−	21.3573i		−	0.935680i	−0.883813	−	0.467840i	$-0.845032\pi$
$521$		−	21.3573i		−	0.935680i	0.883813	−	0.467840i	$-0.154968\pi$
$522$		0			0
$523$	9.35716			0.409160			0.204580	−	0.978850i	$-0.434417\pi$
$523$	9.35716			0.409160			0.204580	+	0.978850i	$0.434417\pi$
$524$		0			0
$525$		−	18.6366i		−	0.813369i
$526$		0			0
$527$	8.30288			0.361679
$528$		0			0
$529$	20.5282			0.892530
$530$		0			0
$531$	7.15932			0.310688
$532$		0			0
$533$	38.5211			1.66854
$534$		0			0
$535$	14.2917			0.617883
$536$		0			0
$537$	9.43285			0.407057
$538$		0			0
$539$			13.4700i			0.580193i
$540$		0			0
$541$	−26.2569			−1.12887			−0.564436	−	0.825477i	$-0.690906\pi$
$541$	−26.2569			−1.12887			−0.564436	+	0.825477i	$0.690906\pi$
$542$		0			0
$543$		−	15.0184i		−	0.644502i
$544$		0			0
$545$		−	3.40004i		−	0.145642i
$546$		0			0
$547$	−9.81380			−0.419608			−0.209804	−	0.977744i	$-0.567283\pi$
$547$	−9.81380			−0.419608			−0.209804	+	0.977744i	$0.567283\pi$
$548$		0			0
$549$	3.34879			0.142923
$550$		0			0
$551$	−23.1873	−	12.1850i	−0.987812	−	0.519098i
$552$		0			0
$553$		−	73.0035i		−	3.10442i
$554$		0			0
$555$		−	19.0272i		−	0.807658i
$556$		0			0
$557$	33.8288			1.43337			0.716685	−	0.697397i	$-0.245658\pi$
$557$	33.8288			1.43337			0.716685	+	0.697397i	$0.245658\pi$
$558$		0			0
$559$	−4.39541			−0.185906
$560$		0			0
$561$		−	9.31275i		−	0.393184i
$562$		0			0
$563$	−10.5530			−0.444755			−0.222377	−	0.974961i	$-0.571382\pi$
$563$	−10.5530			−0.444755			−0.222377	+	0.974961i	$0.571382\pi$
$564$		0			0
$565$		−	14.7235i		−	0.619420i
$566$		0			0
$567$			16.1059i			0.676385i
$568$		0			0
$569$			2.27960i			0.0955657i	0.998858	+	0.0477829i	$0.0152156\pi$
$569$			2.27960i			0.0955657i	−0.998858	+	0.0477829i	$0.984784\pi$
$570$		0			0
$571$		−	4.45694i		−	0.186517i	−0.995642	−	0.0932585i	$-0.970272\pi$
$571$		−	4.45694i		−	0.186517i	0.995642	−	0.0932585i	$-0.0297283\pi$
$572$		0			0
$573$			3.42938i			0.143264i
$574$		0			0
$575$			4.92817i			0.205519i
$576$		0			0
$577$	11.7707			0.490022			0.245011	−	0.969520i	$-0.421208\pi$
$577$	11.7707			0.490022			0.245011	+	0.969520i	$0.421208\pi$
$578$		0			0
$579$			8.56160i			0.355808i
$580$		0			0
$581$	21.0609			0.873753
$582$		0			0
$583$	10.0904			0.417901
$584$		0			0
$585$			10.6970i			0.442267i
$586$		0			0
$587$			2.49167i			0.102842i	0.998677	+	0.0514211i	$0.0163751\pi$
$587$			2.49167i			0.102842i	−0.998677	+	0.0514211i	$0.983625\pi$
$588$		0			0
$589$	−2.37565	+	4.52072i	−0.0978869	+	0.186273i
$590$		0			0
$591$	−27.4315			−1.12838
$592$		0			0
$593$	32.5858			1.33814			0.669068	−	0.743201i	$-0.266693\pi$
$593$	32.5858			1.33814			0.669068	+	0.743201i	$0.266693\pi$
$594$		0			0
$595$			43.7920i			1.79530i
$596$		0			0
$597$			19.8836i			0.813780i
$598$		0			0
$599$	−20.9674			−0.856704			−0.428352	−	0.903612i	$-0.640906\pi$
$599$	−20.9674			−0.856704			−0.428352	+	0.903612i	$0.640906\pi$
$600$		0			0
$601$		−	6.88018i		−	0.280648i	−0.990106	−	0.140324i	$-0.955186\pi$
$601$		−	6.88018i		−	0.280648i	0.990106	−	0.140324i	$-0.0448145\pi$
$602$		0			0
$603$	16.3055			0.664012
$604$		0			0
$605$	−1.36581			−0.0555280
$606$		0			0
$607$	36.6400			1.48717			0.743585	−	0.668641i	$-0.233124\pi$
$607$	36.6400			1.48717			0.743585	+	0.668641i	$0.233124\pi$
$608$		0			0
$609$	35.7285			1.44779
$610$		0			0
$611$	−11.1839			−0.452454
$612$		0			0
$613$	−25.7969			−1.04193			−0.520964	−	0.853579i	$-0.674427\pi$
$613$	−25.7969			−1.04193			−0.520964	+	0.853579i	$0.674427\pi$
$614$		0			0
$615$		−	11.2387i		−	0.453189i
$616$		0			0
$617$	14.3504			0.577727			0.288864	−	0.957370i	$-0.406723\pi$
$617$	14.3504			0.577727			0.288864	+	0.957370i	$0.406723\pi$
$618$		0			0
$619$		−	6.61957i		−	0.266063i	−0.991112	−	0.133031i	$-0.957529\pi$
$619$		−	6.61957i		−	0.266063i	0.991112	−	0.133031i	$-0.0424711\pi$
$620$		0			0
$621$		−	8.82844i		−	0.354273i
$622$		0			0
$623$	31.7227			1.27094
$624$		0			0
$625$	0.498343			0.0199337
$626$		0			0
$627$	5.07057	+	2.66460i	0.202499	+	0.106414i
$628$		0			0
$629$		−	75.1276i		−	2.99553i
$630$		0			0
$631$		−	16.8663i		−	0.671436i	−0.941963	−	0.335718i	$-0.891021\pi$
$631$		−	16.8663i		−	0.671436i	0.941963	−	0.335718i	$-0.108979\pi$
$632$		0			0
$633$	23.6215			0.938871
$634$		0			0
$635$	−4.66784			−0.185238
$636$		0			0
$637$			82.8648i			3.28322i
$638$		0			0
$639$	−14.6092			−0.577932
$640$		0			0
$641$		−	14.7897i		−	0.584157i	−0.956394	−	0.292078i	$-0.905653\pi$
$641$		−	14.7897i		−	0.584157i	0.956394	−	0.292078i	$-0.0943468\pi$
$642$		0			0
$643$		−	44.0804i		−	1.73836i	−0.494496	−	0.869180i	$-0.664647\pi$
$643$		−	44.0804i		−	1.73836i	0.494496	−	0.869180i	$-0.335353\pi$
$644$		0			0
$645$			1.28238i			0.0504937i
$646$		0			0
$647$			25.4850i			1.00192i	0.865471	+	0.500960i	$0.167020\pi$
$647$			25.4850i			1.00192i	−0.865471	+	0.500960i	$0.832980\pi$
$648$		0			0
$649$			5.62345i			0.220740i
$650$		0			0
$651$		−	6.96582i		−	0.273012i
$652$		0			0
$653$	28.9603			1.13330			0.566651	−	0.823958i	$-0.308239\pi$
$653$	28.9603			1.13330			0.566651	+	0.823958i	$0.308239\pi$
$654$		0			0
$655$			25.2405i			0.986228i
$656$		0			0
$657$	−9.27509			−0.361856
$658$		0			0
$659$	48.6991			1.89705			0.948523	−	0.316709i	$-0.102578\pi$
$659$	48.6991			1.89705			0.948523	+	0.316709i	$0.102578\pi$
$660$		0			0
$661$			21.3401i			0.830034i	0.909814	+	0.415017i	$0.136224\pi$
$661$			21.3401i			0.830034i	−0.909814	+	0.415017i	$0.863776\pi$
$662$		0			0
$663$		−	57.2903i		−	2.22497i
$664$		0			0
$665$	−23.8437	−	12.5299i	−0.924620	−	0.485890i
$666$		0			0
$667$	−9.44785			−0.365822
$668$		0			0
$669$	20.7287			0.801417
$670$		0			0
$671$			2.63038i			0.101545i
$672$		0			0
$673$			17.6568i			0.680621i	0.940313	+	0.340310i	$0.110532\pi$
$673$			17.6568i			0.680621i	−0.940313	+	0.340310i	$0.889468\pi$
$674$		0			0
$675$	−17.6017			−0.677488
$676$		0			0
$677$		−	36.7772i		−	1.41346i	−0.707482	−	0.706732i	$-0.750169\pi$
$677$		−	36.7772i		−	1.41346i	0.707482	−	0.706732i	$-0.249831\pi$
$678$		0			0
$679$	−16.9421			−0.650177
$680$		0			0
$681$	13.9928			0.536206
$682$		0			0
$683$	1.20194			0.0459911			0.0229955	−	0.999736i	$-0.492680\pi$
$683$	1.20194			0.0459911			0.0229955	+	0.999736i	$0.492680\pi$
$684$		0			0
$685$	2.98557			0.114073
$686$		0			0
$687$	−14.0074			−0.534417
$688$		0			0
$689$	62.0742			2.36484
$690$		0			0
$691$		−	13.4619i		−	0.512116i	−0.966661	−	0.256058i	$-0.917576\pi$
$691$		−	13.4619i		−	0.512116i	0.966661	−	0.256058i	$-0.0824239\pi$
$692$		0			0
$693$	5.76007			0.218807
$694$		0			0
$695$		−	0.164490i		−	0.00623947i
$696$		0			0
$697$		−	44.3754i		−	1.68084i
$698$		0			0
$699$	−5.95990			−0.225424
$700$		0			0
$701$	−13.1083			−0.495094			−0.247547	−	0.968876i	$-0.579624\pi$
$701$	−13.1083			−0.495094			−0.247547	+	0.968876i	$0.579624\pi$
$702$		0			0
$703$	40.9052	+	21.4958i	1.54277	+	0.810729i
$704$		0			0
$705$			3.26297i			0.122890i
$706$		0			0
$707$		−	78.1917i		−	2.94070i
$708$		0			0
$709$	12.7076			0.477243			0.238621	−	0.971113i	$-0.423305\pi$
$709$	12.7076			0.477243			0.238621	+	0.971113i	$0.423305\pi$
$710$		0			0
$711$	−20.5425			−0.770405
$712$		0			0
$713$			1.84200i			0.0689835i
$714$		0			0
$715$	−8.40220			−0.314224
$716$		0			0
$717$			21.2475i			0.793503i
$718$		0			0
$719$			24.0774i			0.897934i	0.893548	+	0.448967i	$0.148208\pi$
$719$			24.0774i			0.897934i	−0.893548	+	0.448967i	$0.851792\pi$
$720$		0			0
$721$		−	79.0166i		−	2.94273i
$722$		0			0
$723$		−	9.14901i		−	0.340255i
$724$		0			0
$725$			18.8366i			0.699573i
$726$		0			0
$727$			23.5200i			0.872310i	0.899871	+	0.436155i	$0.143660\pi$
$727$			23.5200i			0.872310i	−0.899871	+	0.436155i	$0.856340\pi$
$728$		0			0
$729$	26.6696			0.987762
$730$		0			0
$731$			5.06341i			0.187277i
$732$		0			0
$733$	−45.1693			−1.66837			−0.834183	−	0.551487i	$-0.814060\pi$
$733$	−45.1693			−1.66837			−0.834183	+	0.551487i	$0.814060\pi$
$734$		0			0
$735$	24.1762			0.891752
$736$		0			0
$737$			12.8075i			0.471772i
$738$		0			0
$739$		−	45.3574i		−	1.66850i	−0.551387	−	0.834249i	$-0.685901\pi$
$739$		−	45.3574i		−	1.66850i	0.551387	−	0.834249i	$-0.314099\pi$
$740$		0			0
$741$	31.1932	+	16.3921i	1.14591	+	0.602179i
$742$		0			0
$743$	28.2802			1.03750			0.518750	−	0.854926i	$-0.326398\pi$
$743$	28.2802			1.03750			0.518750	+	0.854926i	$0.326398\pi$
$744$		0			0
$745$	−23.2221			−0.850792
$746$		0			0
$747$		−	5.92635i		−	0.216834i
$748$		0			0
$749$		−	47.3426i		−	1.72986i
$750$		0			0
$751$	−36.0025			−1.31375			−0.656875	−	0.753999i	$-0.728122\pi$
$751$	−36.0025			−1.31375			−0.656875	+	0.753999i	$0.728122\pi$
$752$		0			0
$753$		−	10.0654i		−	0.366804i
$754$		0			0
$755$	−24.8560			−0.904602
$756$		0			0
$757$	−31.0167			−1.12732			−0.563661	−	0.826006i	$-0.690608\pi$
$757$	−31.0167			−1.12732			−0.563661	+	0.826006i	$0.690608\pi$
$758$		0			0
$759$	2.06604			0.0749926
$760$		0			0
$761$	−10.7742			−0.390563			−0.195282	−	0.980747i	$-0.562562\pi$
$761$	−10.7742			−0.390563			−0.195282	+	0.980747i	$0.562562\pi$
$762$		0			0
$763$	−11.2630			−0.407747
$764$		0			0
$765$	12.3227			0.445528
$766$		0			0
$767$			34.5944i			1.24913i
$768$		0			0
$769$	10.0744			0.363293			0.181647	−	0.983364i	$-0.441857\pi$
$769$	10.0744			0.363293			0.181647	+	0.983364i	$0.441857\pi$
$770$		0			0
$771$		−	26.1551i		−	0.941952i
$772$		0			0
$773$			26.1367i			0.940073i	0.882647	+	0.470037i	$0.155759\pi$
$773$			26.1367i			0.940073i	−0.882647	+	0.470037i	$0.844241\pi$
$774$		0			0
$775$	3.67248			0.131919
$776$		0			0
$777$	−63.0294			−2.26117
$778$		0			0
$779$	24.1614	+	12.6969i	0.865671	+	0.454912i
$780$		0			0
$781$		−	11.4751i		−	0.410613i
$782$		0			0
$783$		−	33.7443i		−	1.20592i
$784$		0			0
$785$	−23.5697			−0.841239
$786$		0			0
$787$	−3.63186			−0.129462			−0.0647310	−	0.997903i	$-0.520619\pi$
$787$	−3.63186			−0.129462			−0.0647310	+	0.997903i	$0.520619\pi$
$788$		0			0
$789$			8.99680i			0.320295i
$790$		0			0
$791$	−48.7729			−1.73416
$792$		0			0
$793$			16.1816i			0.574626i
$794$		0			0
$795$		−	18.1104i		−	0.642311i
$796$		0			0
$797$		−	31.0651i		−	1.10038i	−0.835039	−	0.550190i	$-0.814555\pi$
$797$		−	31.0651i		−	1.10038i	0.835039	−	0.550190i	$-0.185445\pi$
$798$		0			0
$799$			12.8836i			0.455790i
$800$		0			0
$801$		−	8.92649i		−	0.315402i
$802$		0			0
$803$		−	7.28533i		−	0.257094i
$804$		0			0
$805$	−9.71531			−0.342420
$806$		0			0
$807$		−	31.6114i		−	1.11277i
$808$		0			0
$809$	16.4773			0.579311			0.289656	−	0.957131i	$-0.406459\pi$
$809$	16.4773			0.579311			0.289656	+	0.957131i	$0.406459\pi$
$810$		0			0
$811$	−21.3659			−0.750260			−0.375130	−	0.926972i	$-0.622402\pi$
$811$	−21.3659			−0.750260			−0.375130	+	0.926972i	$0.622402\pi$
$812$		0			0
$813$		−	13.9287i		−	0.488499i
$814$		0			0
$815$		−	29.8803i		−	1.04666i
$816$		0			0
$817$	−2.75690	−	1.44876i	−0.0964519	−	0.0506857i
$818$		0			0
$819$	35.4349			1.23819
$820$		0			0
$821$	−10.8172			−0.377524			−0.188762	−	0.982023i	$-0.560447\pi$
$821$	−10.8172			−0.377524			−0.188762	+	0.982023i	$0.560447\pi$
$822$		0			0
$823$		−	0.350273i		−	0.0122097i	−0.999981	−	0.00610487i	$-0.998057\pi$
$823$		−	0.350273i		−	0.0122097i	0.999981	−	0.00610487i	$-0.00194325\pi$
$824$		0			0
$825$		−	4.11916i		−	0.143411i
$826$		0			0
$827$	7.69486			0.267577			0.133788	−	0.991010i	$-0.457286\pi$
$827$	7.69486			0.267577			0.133788	+	0.991010i	$0.457286\pi$
$828$		0			0
$829$			17.4792i			0.607077i	0.952819	+	0.303539i	$0.0981682\pi$
$829$			17.4792i			0.607077i	−0.952819	+	0.303539i	$0.901832\pi$
$830$		0			0
$831$	10.3209			0.358028
$832$		0			0
$833$	95.4582			3.30743
$834$		0			0
$835$	3.50551			0.121313
$836$		0			0
$837$	−6.57898			−0.227403
$838$		0			0
$839$	21.0651			0.727246			0.363623	−	0.931546i	$-0.381540\pi$
$839$	21.0651			0.727246			0.363623	+	0.931546i	$0.381540\pi$
$840$		0			0
$841$	−7.11184			−0.245236
$842$		0			0
$843$			5.55303i			0.191257i
$844$		0			0
$845$	−33.9333			−1.16734
$846$		0			0
$847$			4.52438i			0.155459i
$848$		0			0
$849$			41.2912i			1.41711i
$850$		0			0
$851$	16.6671			0.571342
$852$		0			0
$853$	2.49147			0.0853064			0.0426532	−	0.999090i	$-0.486419\pi$
$853$	2.49147			0.0853064			0.0426532	+	0.999090i	$0.486419\pi$
$854$		0			0
$855$	−3.52581	+	6.70941i	−0.120580	+	0.229457i
$856$		0			0
$857$			29.7866i			1.01749i	0.860917	+	0.508746i	$0.169891\pi$
$857$			29.7866i			1.01749i	−0.860917	+	0.508746i	$0.830109\pi$
$858$		0			0
$859$		−	8.98869i		−	0.306690i	−0.988173	−	0.153345i	$-0.950995\pi$
$859$		−	8.98869i		−	0.306690i	0.988173	−	0.153345i	$-0.0490046\pi$
$860$		0			0
$861$	−37.2294			−1.26877
$862$		0			0
$863$	−41.5449			−1.41421			−0.707103	−	0.707111i	$-0.749998\pi$
$863$	−41.5449			−1.41421			−0.707103	+	0.707111i	$0.749998\pi$
$864$		0			0
$865$		−	9.82082i		−	0.333918i
$866$		0			0
$867$	−43.6572			−1.48268
$868$		0			0
$869$		−	16.1356i		−	0.547362i
$870$		0			0
$871$			78.7896i			2.66968i
$872$		0			0
$873$			4.76735i			0.161350i
$874$		0			0
$875$			50.2670i			1.69933i
$876$		0			0
$877$			22.5568i			0.761691i	0.924639	+	0.380845i	$0.124367\pi$
$877$			22.5568i			0.761691i	−0.924639	+	0.380845i	$0.875633\pi$
$878$		0			0
$879$			28.2664i			0.953402i
$880$		0			0
$881$	−0.586038			−0.0197441			−0.00987205	−	0.999951i	$-0.503142\pi$
$881$	−0.586038			−0.0197441			−0.00987205	+	0.999951i	$0.503142\pi$
$882$		0			0
$883$			1.51276i			0.0509085i	0.999676	+	0.0254543i	$0.00810321\pi$
$883$			1.51276i			0.0509085i	−0.999676	+	0.0254543i	$0.991897\pi$
$884$		0			0
$885$	10.0931			0.339275
$886$		0			0
$887$	−47.2354			−1.58601			−0.793004	−	0.609216i	$-0.791484\pi$
$887$	−47.2354			−1.58601			−0.793004	+	0.609216i	$0.791484\pi$
$888$		0			0
$889$			15.4627i			0.518602i
$890$		0			0
$891$			3.55981i			0.119258i
$892$		0			0
$893$	−7.01483	−	3.68631i	−0.234742	−	0.123358i
$894$		0			0
$895$	−9.80396			−0.327710
$896$		0			0
$897$	12.7099			0.424371
$898$		0			0
$899$			7.04056i			0.234816i
$900$		0			0
$901$		−	71.5079i		−	2.38227i
$902$		0			0
$903$	4.24802			0.141365
$904$		0			0
$905$			15.6093i			0.518870i
$906$		0			0
$907$	−20.9163			−0.694513			−0.347256	−	0.937770i	$-0.612887\pi$
$907$	−20.9163			−0.694513			−0.347256	+	0.937770i	$0.612887\pi$
$908$		0			0
$909$	−22.0025			−0.729776
$910$		0			0
$911$	32.4340			1.07459			0.537294	−	0.843395i	$-0.319447\pi$
$911$	32.4340			1.07459			0.537294	+	0.843395i	$0.319447\pi$
$912$		0			0
$913$	4.65498			0.154057
$914$		0			0
$915$	4.72106			0.156074
$916$		0			0
$917$	83.6116			2.76110
$918$		0			0
$919$			52.8102i			1.74205i	0.491240	+	0.871024i	$0.336544\pi$
$919$			52.8102i			1.74205i	−0.491240	+	0.871024i	$0.663456\pi$
$920$		0			0
$921$	21.9954			0.724772
$922$		0			0
$923$		−	70.5929i		−	2.32359i
$924$		0			0
$925$		−	33.2300i		−	1.09260i
$926$		0			0
$927$	−22.2346			−0.730280
$928$		0			0
$929$	36.2554			1.18950			0.594750	−	0.803911i	$-0.297251\pi$
$929$	36.2554			1.18950			0.594750	+	0.803911i	$0.297251\pi$
$930$		0			0
$931$	−27.3128	+	51.9747i	−0.895142	+	1.70340i
$932$		0			0
$933$			24.9297i			0.816163i
$934$		0			0
$935$			9.67913i			0.316541i
$936$		0			0
$937$	4.79474			0.156637			0.0783186	−	0.996928i	$-0.475045\pi$
$937$	4.79474			0.156637			0.0783186	+	0.996928i	$0.475045\pi$
$938$		0			0
$939$	8.73589			0.285085
$940$		0			0
$941$			18.7297i			0.610570i	0.952261	+	0.305285i	$0.0987517\pi$
$941$			18.7297i			0.610570i	−0.952261	+	0.305285i	$0.901248\pi$
$942$		0			0
$943$	9.84473			0.320589
$944$		0			0
$945$		−	34.6996i		−	1.12878i
$946$		0			0
$947$		−	6.58259i		−	0.213906i	−0.994264	−	0.106953i	$-0.965891\pi$
$947$		−	6.58259i		−	0.213906i	0.994264	−	0.106953i	$-0.0341094\pi$
$948$		0			0
$949$		−	44.8180i		−	1.45485i
$950$		0			0
$951$			42.6298i			1.38236i
$952$		0			0
$953$			48.3939i			1.56763i	0.620993	+	0.783816i	$0.286730\pi$
$953$			48.3939i			1.56763i	−0.620993	+	0.783816i	$0.713270\pi$
$954$		0			0
$955$		−	3.56430i		−	0.115338i
$956$		0			0
$957$	7.89689			0.255270
$958$		0			0
$959$		−	9.89000i		−	0.319365i
$960$		0			0
$961$	−29.6273			−0.955721
$962$		0			0
$963$	−13.3218			−0.429288
$964$		0			0
$965$		−	8.89844i		−	0.286451i
$966$		0			0
$967$		−	19.5354i		−	0.628217i	−0.949387	−	0.314108i	$-0.898294\pi$
$967$		−	19.5354i		−	0.628217i	0.949387	−	0.314108i	$-0.101706\pi$
$968$		0			0
$969$	18.8833	−	35.9338i	0.606619	−	1.15436i
$970$		0			0
$971$	−39.0656			−1.25368			−0.626838	−	0.779150i	$-0.715651\pi$
$971$	−39.0656			−1.25368			−0.626838	+	0.779150i	$0.715651\pi$
$972$		0			0
$973$	−0.544890			−0.0174684
$974$		0			0
$975$		−	25.3403i		−	0.811539i
$976$		0			0
$977$			4.11027i			0.131499i	0.997836	+	0.0657496i	$0.0209439\pi$
$977$			4.11027i			0.131499i	−0.997836	+	0.0657496i	$0.979056\pi$
$978$		0			0
$979$	7.01151			0.224089
$980$		0			0
$981$			3.16930i			0.101188i
$982$		0			0
$983$	−19.5833			−0.624609			−0.312305	−	0.949982i	$-0.601101\pi$
$983$	−19.5833			−0.624609			−0.312305	+	0.949982i	$0.601101\pi$
$984$		0			0
$985$	28.5107			0.908426
$986$		0			0
$987$	10.8089			0.344051
$988$		0			0
$989$	−1.12332			−0.0357196
$990$		0			0
$991$	38.3122			1.21703			0.608514	−	0.793543i	$-0.291766\pi$
$991$	38.3122			1.21703			0.608514	+	0.793543i	$0.291766\pi$
$992$		0			0
$993$	21.5758			0.684687
$994$		0			0
$995$		−	20.6658i		−	0.655151i
$996$		0			0
$997$	−39.7062			−1.25751			−0.628753	−	0.777605i	$-0.716434\pi$
$997$	−39.7062			−1.25751			−0.628753	+	0.777605i	$0.716434\pi$
$998$		0			0
$999$			59.5291i			1.88342i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.a.1519.12	yes	36
4.3	odd	2	inner	3344.2.o.a.1519.25	yes	36
19.18	odd	2	inner	3344.2.o.a.1519.26	yes	36
76.75	even	2	inner	3344.2.o.a.1519.11	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.a.1519.11	✓	36	76.75	even	2	inner
3344.2.o.a.1519.12	yes	36	1.1	even	1	trivial
3344.2.o.a.1519.25	yes	36	4.3	odd	2	inner
3344.2.o.a.1519.26	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)

\(f(q)\)	\(=\)	\(q-1.31411 q^{3} +1.36581 q^{5} -4.52438i q^{7} -1.27312 q^{9} +O(q^{10})\)	\(q-1.31411 q^{3} +1.36581 q^{5} -4.52438i q^{7} -1.27312 q^{9} -1.00000i q^{11} -6.15181i q^{13} -1.79482 q^{15} -7.08674 q^{17} +(2.02768 - 3.85856i) q^{19} +5.94552i q^{21} -1.57220i q^{23} -3.13457 q^{25} +5.61534 q^{27} -6.00931i q^{29} -1.17161 q^{31} +1.31411i q^{33} -6.17943i q^{35} +10.6012i q^{37} +8.08415i q^{39} +6.26175i q^{41} -0.714490i q^{43} -1.73884 q^{45} -1.81799i q^{47} -13.4700 q^{49} +9.31275 q^{51} +10.0904i q^{53} -1.36581i q^{55} +(-2.66460 + 5.07057i) q^{57} -5.62345 q^{59} -2.63038 q^{61} +5.76007i q^{63} -8.40220i q^{65} -12.8075 q^{67} +2.06604i q^{69} +11.4751 q^{71} +7.28533 q^{73} +4.11916 q^{75} -4.52438 q^{77} +16.1356 q^{79} -3.55981 q^{81} +4.65498i q^{83} -9.67913 q^{85} +7.89689i q^{87} +7.01151i q^{89} -27.8331 q^{91} +1.53962 q^{93} +(2.76943 - 5.27006i) q^{95} -3.74462i q^{97} +1.27312i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 44 q^{9}+O(q^{10}) \) 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

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