LMFDB - Embedded newform 361.3.d.a.69.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [361,3,Mod(69,361)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(361, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("361.69");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 361 = 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	361.d (of order $6$, degree $2$, not 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:	$9.83653754341$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			69.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	361.69
Dual form			361.3.d.a.293.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -5.00000 q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	$q+(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -5.00000 q^{7} +(-4.50000 + 7.79423i) q^{9} +3.00000 q^{11} +(-8.00000 - 13.8564i) q^{16} +(-7.50000 - 12.9904i) q^{17} -36.0000 q^{20} +(15.0000 - 25.9808i) q^{23} +(-28.0000 + 48.4974i) q^{25} +(10.0000 - 17.3205i) q^{28} +(-22.5000 - 38.9711i) q^{35} +(-18.0000 - 31.1769i) q^{36} +(42.5000 + 73.6122i) q^{43} +(-6.00000 + 10.3923i) q^{44} -81.0000 q^{45} +(-37.5000 + 64.9519i) q^{47} -24.0000 q^{49} +(13.5000 + 23.3827i) q^{55} +(-51.5000 + 89.2006i) q^{61} +(22.5000 - 38.9711i) q^{63} +64.0000 q^{64} +60.0000 q^{68} +(12.5000 + 21.6506i) q^{73} -15.0000 q^{77} +(72.0000 - 124.708i) q^{80} +(-40.5000 - 70.1481i) q^{81} +90.0000 q^{83} +(67.5000 - 116.913i) q^{85} +(60.0000 + 103.923i) q^{92} +(-13.5000 + 23.3827i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 + 9 * q^5 - 10 * q^7 - 9 * q^9	$ 2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9} + 6 q^{11} - 16 q^{16} - 15 q^{17} - 72 q^{20} + 30 q^{23} - 56 q^{25} + 20 q^{28} - 45 q^{35} - 36 q^{36} + 85 q^{43} - 12 q^{44} - 162 q^{45} - 75 q^{47} - 48 q^{49} + 27 q^{55} - 103 q^{61} + 45 q^{63} + 128 q^{64} + 120 q^{68} + 25 q^{73} - 30 q^{77} + 144 q^{80} - 81 q^{81} + 180 q^{83} + 135 q^{85} + 120 q^{92} - 27 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 + 9 * q^5 - 10 * q^7 - 9 * q^9 + 6 * q^11 - 16 * q^16 - 15 * q^17 - 72 * q^20 + 30 * q^23 - 56 * q^25 + 20 * q^28 - 45 * q^35 - 36 * q^36 + 85 * q^43 - 12 * q^44 - 162 * q^45 - 75 * q^47 - 48 * q^49 + 27 * q^55 - 103 * q^61 + 45 * q^63 + 128 * q^64 + 120 * q^68 + 25 * q^73 - 30 * q^77 + 144 * q^80 - 81 * q^81 + 180 * q^83 + 135 * q^85 + 120 * q^92 - 27 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{5}{6}\right)$

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		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$	4.50000	+	7.79423i	0.900000	+	1.55885i	0.827492	+	0.561478i	$0.189767\pi$
$5$	4.50000	+	7.79423i	0.900000	+	1.55885i	0.0725083	+	0.997368i	$0.476900\pi$
$6$		0			0
$7$	−5.00000			−0.714286			−0.357143	−	0.934050i	$-0.616249\pi$
$7$	−5.00000			−0.714286			−0.357143	+	0.934050i	$0.616249\pi$
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$10$		0			0
$11$	3.00000			0.272727			0.136364	−	0.990659i	$-0.456458\pi$
$11$	3.00000			0.272727			0.136364	+	0.990659i	$0.456458\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$	−7.50000	−	12.9904i	−0.441176	−	0.764140i	0.556601	−	0.830780i	$-0.312105\pi$
$17$	−7.50000	−	12.9904i	−0.441176	−	0.764140i	−0.997777	+	0.0666402i	$0.978772\pi$
$18$		0			0
$19$		0			0
$19$		0			0
$20$	−36.0000			−1.80000
$21$		0			0
$22$		0			0
$23$	15.0000	−	25.9808i	0.652174	−	1.12960i	−0.330420	−	0.943834i	$-0.607191\pi$
$23$	15.0000	−	25.9808i	0.652174	−	1.12960i	0.982594	−	0.185764i	$-0.0594761\pi$
$24$		0			0
$25$	−28.0000	+	48.4974i	−1.12000	+	1.93990i
$26$		0			0
$27$		0			0
$28$	10.0000	−	17.3205i	0.357143	−	0.618590i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−22.5000	−	38.9711i	−0.642857	−	1.11346i
$36$	−18.0000	−	31.1769i	−0.500000	−	0.866025i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	42.5000	+	73.6122i	0.988372	+	1.71191i	0.625869	+	0.779928i	$0.284744\pi$
$43$	42.5000	+	73.6122i	0.988372	+	1.71191i	0.362503	+	0.931983i	$0.381922\pi$
$44$	−6.00000	+	10.3923i	−0.136364	+	0.236189i
$45$	−81.0000			−1.80000
$46$		0			0
$47$	−37.5000	+	64.9519i	−0.797872	+	1.38196i	0.123127	+	0.992391i	$0.460708\pi$
$47$	−37.5000	+	64.9519i	−0.797872	+	1.38196i	−0.920999	+	0.389564i	$0.872626\pi$
$48$		0			0
$49$	−24.0000			−0.489796
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$	13.5000	+	23.3827i	0.245455	+	0.425140i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−51.5000	+	89.2006i	−0.844262	+	1.46231i	0.0419980	+	0.999118i	$0.486628\pi$
$61$	−51.5000	+	89.2006i	−0.844262	+	1.46231i	−0.886260	+	0.463187i	$0.846706\pi$
$62$		0			0
$63$	22.5000	−	38.9711i	0.357143	−	0.618590i
$64$	64.0000			1.00000
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	60.0000			0.882353
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	12.5000	+	21.6506i	0.171233	+	0.296584i	0.938851	−	0.344323i	$-0.111892\pi$
$73$	12.5000	+	21.6506i	0.171233	+	0.296584i	−0.767618	+	0.640907i	$0.778558\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−15.0000			−0.194805
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	72.0000	−	124.708i	0.900000	−	1.55885i
$81$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$82$		0			0
$83$	90.0000			1.08434			0.542169	−	0.840270i	$-0.317603\pi$
$83$	90.0000			1.08434			0.542169	+	0.840270i	$0.317603\pi$
$84$		0			0
$85$	67.5000	−	116.913i	0.794118	−	1.37545i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$	60.0000	+	103.923i	0.652174	+	1.12960i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$		0			0
$99$	−13.5000	+	23.3827i	−0.136364	+	0.236189i
$100$	−112.000	−	193.990i	−1.12000	−	1.93990i
$101$	51.0000	−	88.3346i	0.504950	−	0.874600i	−0.495033	−	0.868874i	$-0.664844\pi$
$101$	51.0000	−	88.3346i	0.504950	−	0.874600i	0.999984	−	0.00572580i	$-0.00182259\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$	40.0000	+	69.2820i	0.357143	+	0.618590i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	270.000			2.34783
$116$		0			0
$117$		0			0
$118$		0			0
$119$	37.5000	+	64.9519i	0.315126	+	0.545814i
$120$		0			0
$121$	−112.000			−0.925620
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−279.000			−2.23200
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	106.500	+	184.463i	0.812977	+	1.40812i	0.910771	+	0.412911i	$0.135488\pi$
$131$	106.500	+	184.463i	0.812977	+	1.40812i	−0.0977943	+	0.995207i	$0.531179\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−127.500	+	220.836i	−0.930657	+	1.61195i	−0.148456	+	0.988919i	$0.547430\pi$
$137$	−127.500	+	220.836i	−0.930657	+	1.61195i	−0.782201	+	0.623026i	$0.785903\pi$
$138$		0			0
$139$	98.5000	−	170.607i	0.708633	−	1.22739i	−0.256731	−	0.966483i	$-0.582646\pi$
$139$	98.5000	−	170.607i	0.708633	−	1.22739i	0.965364	−	0.260906i	$-0.0840212\pi$
$140$	180.000			1.28571
$141$		0			0
$142$		0			0
$143$		0			0
$144$	144.000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	88.5000	+	153.286i	0.593960	+	1.02877i	0.993693	+	0.112137i	$0.0357695\pi$
$149$	88.5000	+	153.286i	0.593960	+	1.02877i	−0.399733	+	0.916632i	$0.630897\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	135.000			0.882353
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−5.00000	−	8.66025i	−0.0318471	−	0.0551609i	0.849663	−	0.527327i	$-0.176806\pi$
$157$	−5.00000	−	8.66025i	−0.0318471	−	0.0551609i	−0.881510	+	0.472166i	$0.843472\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−75.0000	+	129.904i	−0.465839	+	0.806856i
$162$		0			0
$163$	250.000			1.53374			0.766871	−	0.641801i	$-0.221813\pi$
$163$	250.000			1.53374			0.766871	+	0.641801i	$0.221813\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−84.5000	−	146.358i	−0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$	−340.000			−1.97674
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	140.000	−	242.487i	0.800000	−	1.38564i
$176$	−24.0000	−	41.5692i	−0.136364	−	0.236189i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	162.000	−	280.592i	0.900000	−	1.55885i
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−22.5000	−	38.9711i	−0.120321	−	0.208402i
$188$	−150.000	−	259.808i	−0.797872	−	1.38196i
$189$		0			0
$190$		0			0
$191$	−93.0000			−0.486911			−0.243455	−	0.969912i	$-0.578281\pi$
$191$	−93.0000			−0.486911			−0.243455	+	0.969912i	$0.578281\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	48.0000	−	83.1384i	0.244898	−	0.424176i
$197$	90.0000			0.456853			0.228426	−	0.973561i	$-0.426642\pi$
$197$	90.0000			0.456853			0.228426	+	0.973561i	$0.426642\pi$
$198$		0			0
$199$	−113.500	+	196.588i	−0.570352	+	0.987878i	0.426178	+	0.904639i	$0.359860\pi$
$199$	−113.500	+	196.588i	−0.570352	+	0.987878i	−0.996530	+	0.0832388i	$0.973474\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	135.000	+	233.827i	0.652174	+	1.12960i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−382.500	+	662.509i	−1.77907	+	3.08144i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−108.000			−0.490909
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$	−252.000	−	436.477i	−1.12000	−	1.93990i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−17.0000			−0.0742358			−0.0371179	−	0.999311i	$-0.511818\pi$
$229$	−17.0000			−0.0742358			−0.0371179	+	0.999311i	$0.511818\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	232.500	+	402.702i	0.997854	+	1.72833i	0.555632	+	0.831428i	$0.312476\pi$
$233$	232.500	+	402.702i	0.997854	+	1.72833i	0.442222	+	0.896905i	$0.354190\pi$
$234$		0			0
$235$	−675.000			−2.87234
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−453.000			−1.89540			−0.947699	−	0.319166i	$-0.896597\pi$
$239$	−453.000			−1.89540			−0.947699	+	0.319166i	$0.896597\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$		0			0
$244$	−206.000	−	356.802i	−0.844262	−	1.46231i
$245$	−108.000	−	187.061i	−0.440816	−	0.763516i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.5000	+	23.3827i	−0.0537849	+	0.0931581i	−0.891664	−	0.452697i	$-0.850462\pi$
$251$	−13.5000	+	23.3827i	−0.0537849	+	0.0931581i	0.837879	+	0.545855i	$0.183795\pi$
$252$	90.0000	+	155.885i	0.357143	+	0.618590i
$253$	45.0000	−	77.9423i	0.177866	−	0.308072i
$254$		0			0
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	202.500	+	350.740i	0.769962	+	1.33361i	0.937583	+	0.347762i	$0.113058\pi$
$263$	202.500	+	350.740i	0.769962	+	1.33361i	−0.167621	+	0.985852i	$0.553608\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	71.0000	+	122.976i	0.261993	+	0.453785i	0.966771	−	0.255643i	$-0.0822871\pi$
$271$	71.0000	+	122.976i	0.261993	+	0.453785i	−0.704779	+	0.709427i	$0.748954\pi$
$272$	−120.000	+	207.846i	−0.441176	+	0.764140i
$273$		0			0
$274$		0			0
$275$	−84.0000	+	145.492i	−0.305455	+	0.529063i
$276$		0			0
$277$	535.000			1.93141			0.965704	−	0.259646i	$-0.0836057\pi$
$277$	535.000			1.93141			0.965704	+	0.259646i	$0.0836057\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	−197.500	−	342.080i	−0.697880	−	1.20876i	−0.969200	−	0.246274i	$-0.920794\pi$
$283$	−197.500	−	342.080i	−0.697880	−	1.20876i	0.271320	−	0.962489i	$-0.412540\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	32.0000	−	55.4256i	0.110727	−	0.191784i
$290$		0			0
$291$		0			0
$292$	−100.000			−0.342466
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−212.500	−	368.061i	−0.705980	−	1.22279i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−927.000			−3.03934
$306$		0			0
$307$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$307$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$308$	30.0000	−	51.9615i	0.0974026	−	0.168706i
$309$		0			0
$310$		0			0
$311$	603.000			1.93891			0.969453	−	0.245276i	$-0.0788785\pi$
$311$	603.000			1.93891			0.969453	+	0.245276i	$0.0788785\pi$
$312$		0			0
$313$	295.000	−	510.955i	0.942492	−	1.63244i	0.181795	−	0.983336i	$-0.441809\pi$
$313$	295.000	−	510.955i	0.942492	−	1.63244i	0.760697	−	0.649108i	$-0.224858\pi$
$314$		0			0
$315$	405.000			1.28571
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	288.000	+	498.831i	0.900000	+	1.55885i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	324.000			1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	187.500	−	324.760i	0.569909	−	0.987111i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−180.000	+	311.769i	−0.542169	+	0.939064i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$		0			0
$339$		0			0
$340$	270.000	+	467.654i	0.794118	+	1.37545i
$341$		0			0
$342$		0			0
$343$	365.000			1.06414
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−337.500	−	584.567i	−0.972622	−	1.68463i	−0.687568	−	0.726120i	$-0.741322\pi$
$347$	−337.500	−	584.567i	−0.972622	−	1.68463i	−0.285055	−	0.958511i	$-0.592012\pi$
$348$		0			0
$349$	527.000			1.51003			0.755014	−	0.655708i	$-0.227630\pi$
$349$	527.000			1.51003			0.755014	+	0.655708i	$0.227630\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−510.000			−1.44476			−0.722380	−	0.691497i	$-0.756952\pi$
$353$	−510.000			−1.44476			−0.722380	+	0.691497i	$0.756952\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−121.500	−	210.444i	−0.338440	−	0.586195i	0.645699	−	0.763592i	$-0.276566\pi$
$359$	−121.500	−	210.444i	−0.338440	−	0.586195i	−0.984140	+	0.177396i	$0.943233\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−112.500	+	194.856i	−0.308219	+	0.533851i
$366$		0			0
$367$	−25.0000	+	43.3013i	−0.0681199	+	0.117987i	−0.898074	−	0.439845i	$-0.855033\pi$
$367$	−25.0000	+	43.3013i	−0.0681199	+	0.117987i	0.829954	+	0.557832i	$0.188367\pi$
$368$	−480.000			−1.30435
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$	−67.5000	−	116.913i	−0.175325	−	0.303671i
$386$		0			0
$387$	−765.000			−1.97674
$388$		0			0
$389$	76.5000	−	132.502i	0.196658	−	0.340622i	−0.750785	−	0.660547i	$-0.770324\pi$
$389$	76.5000	−	132.502i	0.196658	−	0.340622i	0.947443	+	0.319925i	$0.103658\pi$
$390$		0			0
$391$	−450.000			−1.15090
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−54.0000	−	93.5307i	−0.136364	−	0.236189i
$397$	372.500	+	645.189i	0.938287	+	1.62516i	0.768665	+	0.639652i	$0.220921\pi$
$397$	372.500	+	645.189i	0.938287	+	1.62516i	0.169622	+	0.985509i	$0.445745\pi$
$398$		0			0
$399$		0			0
$400$	896.000			2.24000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$	204.000	+	353.338i	0.504950	+	0.874600i
$405$	364.500	−	631.333i	0.900000	−	1.55885i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	405.000	+	701.481i	0.975904	+	1.69031i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	762.000			1.81862			0.909308	−	0.416124i	$-0.136612\pi$
$419$	762.000			1.81862			0.909308	+	0.416124i	$0.136612\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$	−337.500	−	584.567i	−0.797872	−	1.38196i
$424$		0			0
$425$	840.000			1.97647
$426$		0			0
$427$	257.500	−	446.003i	0.603044	−	1.04450i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$433$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	108.000	−	187.061i	0.244898	−	0.424176i
$442$		0			0
$443$	22.5000	−	38.9711i	0.0507901	−	0.0879710i	−0.839513	−	0.543340i	$-0.817159\pi$
$443$	22.5000	−	38.9711i	0.0507901	−	0.0879710i	0.890303	+	0.455369i	$0.150493\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−320.000			−0.714286
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−625.000			−1.36761			−0.683807	−	0.729663i	$-0.739677\pi$
$457$	−625.000			−1.36761			−0.683807	+	0.729663i	$0.739677\pi$
$458$		0			0
$459$		0			0
$460$	−540.000	+	935.307i	−1.17391	+	2.03328i
$461$	−223.500	−	387.113i	−0.484816	−	0.839725i	0.515032	−	0.857171i	$-0.327780\pi$
$461$	−223.500	−	387.113i	−0.484816	−	0.839725i	−0.999848	+	0.0174455i	$0.994447\pi$
$462$		0			0
$463$	755.000			1.63067			0.815335	−	0.578990i	$-0.196553\pi$
$463$	755.000			1.63067			0.815335	+	0.578990i	$0.196553\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	915.000			1.95931			0.979657	−	0.200677i	$-0.0643143\pi$
$467$	915.000			1.95931			0.979657	+	0.200677i	$0.0643143\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	127.500	+	220.836i	0.269556	+	0.466885i
$474$		0			0
$475$		0			0
$476$	−300.000			−0.630252
$477$		0			0
$478$		0			0
$479$	471.000	−	815.796i	0.983299	−	1.70312i	0.334033	−	0.942561i	$-0.391590\pi$
$479$	471.000	−	815.796i	0.983299	−	1.70312i	0.649266	−	0.760562i	$-0.275076\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	224.000	−	387.979i	0.462810	−	0.801610i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	459.000	+	795.011i	0.934827	+	1.61917i	0.774942	+	0.632032i	$0.217779\pi$
$491$	459.000	+	795.011i	0.934827	+	1.61917i	0.159885	+	0.987136i	$0.448888\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−243.000			−0.490909
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−261.500	−	452.931i	−0.524048	−	0.907678i	−0.999608	−	0.0279946i	$-0.991088\pi$
$499$	−261.500	−	452.931i	−0.524048	−	0.907678i	0.475560	−	0.879683i	$-0.342245\pi$
$500$	558.000	−	966.484i	1.11600	−	1.93297i
$501$		0			0
$502$		0			0
$503$	−465.000	+	805.404i	−0.924453	+	1.60120i	−0.132015	+	0.991248i	$0.542145\pi$
$503$	−465.000	+	805.404i	−0.924453	+	1.60120i	−0.792438	+	0.609952i	$0.791189\pi$
$504$		0			0
$505$	918.000			1.81782
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	−62.5000	−	108.253i	−0.122309	−	0.211846i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−112.500	+	194.856i	−0.217602	+	0.376897i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$	−852.000			−1.62595
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−185.500	−	321.295i	−0.350662	−	0.607364i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−72.0000			−0.133581
$540$		0			0
$541$	228.500	−	395.774i	0.422366	−	0.731559i	−0.573804	−	0.818992i	$-0.694533\pi$
$541$	228.500	−	395.774i	0.422366	−	0.731559i	0.996170	+	0.0874330i	$0.0278664\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	−510.000	−	883.346i	−0.930657	−	1.61195i
$549$	−463.500	−	802.806i	−0.844262	−	1.46231i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	394.000	+	682.428i	0.708633	+	1.22739i
$557$	−547.500	+	948.298i	−0.982944	+	1.70251i	−0.332207	+	0.943206i	$0.607793\pi$
$557$	−547.500	+	948.298i	−0.982944	+	1.70251i	−0.650737	+	0.759303i	$0.725540\pi$
$558$		0			0
$559$		0			0
$560$	−360.000	+	623.538i	−0.642857	+	1.11346i
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	202.500	+	350.740i	0.357143	+	0.618590i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	458.000			0.802102			0.401051	−	0.916056i	$-0.368645\pi$
$571$	458.000			0.802102			0.401051	+	0.916056i	$0.368645\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	840.000	+	1454.92i	1.46087	+	2.53030i
$576$	−288.000	+	498.831i	−0.500000	+	0.866025i
$577$	−1145.00			−1.98440			−0.992201	−	0.124648i	$-0.960220\pi$
$577$	−1145.00			−1.98440			−0.992201	+	0.124648i	$0.960220\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−450.000			−0.774527
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	562.500	+	974.279i	0.958262	+	1.65976i	0.726719	+	0.686934i	$0.241044\pi$
$587$	562.500	+	974.279i	0.958262	+	1.65976i	0.231543	+	0.972825i	$0.425623\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.0000	−	25.9808i	0.0252951	−	0.0438124i	−0.853101	−	0.521746i	$-0.825281\pi$
$593$	15.0000	−	25.9808i	0.0252951	−	0.0438124i	0.878396	+	0.477934i	$0.158614\pi$
$594$		0			0
$595$	−337.500	+	584.567i	−0.567227	+	0.982466i
$596$	−708.000			−1.18792
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−504.000	−	872.954i	−0.833058	−	1.44290i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−270.000	+	467.654i	−0.441176	+	0.764140i
$613$	−147.500	−	255.477i	−0.240620	−	0.416766i	0.720271	−	0.693693i	$-0.244017\pi$
$613$	−147.500	−	255.477i	−0.240620	−	0.416766i	−0.960891	+	0.276927i	$0.910684\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	532.500	−	922.317i	0.863047	−	1.49484i	−0.00592639	−	0.999982i	$-0.501886\pi$
$617$	532.500	−	922.317i	0.863047	−	1.49484i	0.868973	−	0.494859i	$-0.164780\pi$
$618$		0			0
$619$	−662.000			−1.06947			−0.534733	−	0.845021i	$-0.679588\pi$
$619$	−662.000			−1.06947			−0.534733	+	0.845021i	$0.679588\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−555.500	−	962.154i	−0.888800	−	1.53945i
$626$		0			0
$627$		0			0
$628$	40.0000			0.0636943
$629$		0			0
$630$		0			0
$631$	518.500	−	898.068i	0.821712	−	1.42325i	−0.0826952	−	0.996575i	$-0.526353\pi$
$631$	518.500	−	898.068i	0.821712	−	1.42325i	0.904407	−	0.426671i	$-0.140314\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−557.500	−	965.618i	−0.867030	−	1.50174i	−0.865018	−	0.501741i	$-0.832693\pi$
$643$	−557.500	−	965.618i	−0.867030	−	1.50174i	−0.00201180	−	0.999998i	$-0.500640\pi$
$644$	−300.000	−	519.615i	−0.465839	−	0.806856i
$645$		0			0
$646$		0			0
$647$	−1005.00			−1.55332			−0.776662	−	0.629918i	$-0.783088\pi$
$647$	−1005.00			−1.55332			−0.776662	+	0.629918i	$0.783088\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−500.000	+	866.025i	−0.766871	+	1.32826i
$653$	375.000			0.574273			0.287136	−	0.957890i	$-0.407297\pi$
$653$	375.000			0.574273			0.287136	+	0.957890i	$0.407297\pi$
$654$		0			0
$655$	−958.500	+	1660.17i	−1.46336	+	2.53461i
$656$		0			0
$657$	−225.000			−0.342466
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−154.500	+	267.602i	−0.230253	+	0.398811i
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	676.000			1.00000
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−2295.00			−3.35036
$686$		0			0
$687$		0			0
$688$	680.000	−	1177.79i	0.988372	−	1.71191i
$689$		0			0
$690$		0			0
$691$	−157.000			−0.227207			−0.113603	−	0.993526i	$-0.536239\pi$
$691$	−157.000			−0.227207			−0.113603	+	0.993526i	$0.536239\pi$
$692$		0			0
$693$	67.5000	−	116.913i	0.0974026	−	0.168706i
$694$		0			0
$695$	1773.00			2.55108
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	560.000	+	969.948i	0.800000	+	1.38564i
$701$	−549.000	−	950.896i	−0.783167	−	1.35648i	−0.930088	−	0.367337i	$-0.880270\pi$
$701$	−549.000	−	950.896i	−0.783167	−	1.35648i	0.146921	−	0.989148i	$-0.453064\pi$
$702$		0			0
$703$		0			0
$704$	192.000			0.272727
$705$		0			0
$706$		0			0
$707$	−255.000	+	441.673i	−0.360679	+	0.624714i
$708$		0			0
$709$	659.000	−	1141.42i	0.929478	−	1.60990i	0.145282	−	0.989390i	$-0.453591\pi$
$709$	659.000	−	1141.42i	0.929478	−	1.60990i	0.784196	−	0.620513i	$-0.213076\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−481.500	−	833.982i	−0.669680	−	1.15992i	−0.977994	−	0.208635i	$-0.933098\pi$
$719$	−481.500	−	833.982i	−0.669680	−	1.15992i	0.308313	−	0.951285i	$-0.400235\pi$
$720$	648.000	+	1122.37i	0.900000	+	1.55885i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	42.5000	+	73.6122i	0.0584594	+	0.101255i	0.893774	−	0.448518i	$-0.148048\pi$
$727$	42.5000	+	73.6122i	0.0584594	+	0.101255i	−0.835315	+	0.549772i	$0.814715\pi$
$728$		0			0
$729$	729.000			1.00000
$730$		0			0
$731$	637.500	−	1104.18i	0.872093	−	1.51051i
$732$		0			0
$733$	−1270.00			−1.73261			−0.866303	−	0.499519i	$-0.833510\pi$
$733$	−1270.00			−1.73261			−0.866303	+	0.499519i	$0.833510\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−273.500	−	473.716i	−0.370095	−	0.641023i	0.619485	−	0.785008i	$-0.287341\pi$
$739$	−273.500	−	473.716i	−0.370095	−	0.641023i	−0.989580	+	0.143986i	$0.954008\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$	−796.500	+	1379.58i	−1.06913	+	1.85178i
$746$		0			0
$747$	−405.000	+	701.481i	−0.542169	+	0.939064i
$748$	180.000			0.240642
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	1200.00			1.59574
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	392.500	+	679.830i	0.518494	+	0.898058i	0.999769	+	0.0214884i	$0.00684050\pi$
$757$	392.500	+	679.830i	0.518494	+	0.898058i	−0.481275	+	0.876570i	$0.659826\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1503.00			1.97503			0.987516	−	0.157516i	$-0.0503486\pi$
$761$	1503.00			1.97503			0.987516	+	0.157516i	$0.0503486\pi$
$762$		0			0
$763$		0			0
$764$	186.000	−	322.161i	0.243455	−	0.421677i
$765$	607.500	+	1052.22i	0.794118	+	1.37545i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−531.500	+	920.585i	−0.691157	+	1.19712i	0.280302	+	0.959912i	$0.409566\pi$
$769$	−531.500	+	920.585i	−0.691157	+	1.19712i	−0.971459	+	0.237208i	$0.923768\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	192.000	+	332.554i	0.244898	+	0.424176i
$785$	45.0000	−	77.9423i	0.0573248	−	0.0992895i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−180.000	+	311.769i	−0.228426	+	0.395646i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−454.000	−	786.351i	−0.570352	−	0.987878i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	1125.00			1.40801
$800$		0			0
$801$		0			0
$802$		0			0
$803$	37.5000	+	64.9519i	0.0466999	+	0.0808866i
$804$		0			0
$805$	−1350.00			−1.67702
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1593.00			−1.96910			−0.984549	−	0.175110i	$-0.943972\pi$
$809$	−1593.00			−1.96910			−0.984549	+	0.175110i	$0.943972\pi$
$810$		0			0
$811$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$811$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1125.00	+	1948.56i	1.38037	+	2.39087i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−583.500	+	1010.65i	−0.710719	+	1.23100i	0.253869	+	0.967239i	$0.418297\pi$
$821$	−583.500	+	1010.65i	−0.710719	+	1.23100i	−0.964588	+	0.263762i	$0.915037\pi$
$822$		0			0
$823$	782.500	−	1355.33i	0.950790	−	1.64682i	0.207068	−	0.978326i	$-0.433608\pi$
$823$	782.500	−	1355.33i	0.950790	−	1.64682i	0.743721	−	0.668490i	$-0.233059\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$827$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$828$	−1080.00			−1.30435
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	180.000	+	311.769i	0.216086	+	0.374273i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	760.500	−	1317.22i	0.900000	−	1.55885i
$846$		0			0
$847$	560.000			0.661157
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	515.000	+	892.006i	0.603751	+	1.04573i	0.992247	+	0.124278i	$0.0396614\pi$
$853$	515.000	+	892.006i	0.603751	+	1.04573i	−0.388496	+	0.921450i	$0.627005\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	746.500	−	1292.98i	0.869034	−	1.50521i	0.00604839	−	0.999982i	$-0.498075\pi$
$859$	746.500	−	1292.98i	0.869034	−	1.50521i	0.862985	−	0.505229i	$-0.168592\pi$
$860$	−1530.00	−	2650.04i	−1.77907	−	3.08144i
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1395.00			1.59429
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$	216.000	−	374.123i	0.245455	−	0.425140i
$881$	−537.000			−0.609535			−0.304767	−	0.952427i	$-0.598579\pi$
$881$	−537.000			−0.609535			−0.304767	+	0.952427i	$0.598579\pi$
$882$		0			0
$883$	−417.500	+	723.131i	−0.472820	+	0.818948i	−0.999516	−	0.0311055i	$-0.990097\pi$
$883$	−417.500	+	723.131i	−0.472820	+	0.818948i	0.526696	+	0.850054i	$0.323431\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−121.500	−	210.444i	−0.136364	−	0.236189i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2016.00			2.24000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$	459.000	+	795.011i	0.504950	+	0.874600i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	270.000			0.295728
$914$		0			0
$915$		0			0
$916$	34.0000	−	58.8897i	0.0371179	−	0.0642901i
$917$	−532.500	−	922.317i	−0.580698	−	1.00580i
$918$		0			0
$919$	1762.00			1.91730			0.958651	−	0.284585i	$-0.0918559\pi$
$919$	1762.00			1.91730			0.958651	+	0.284585i	$0.0918559\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−321.000	−	555.988i	−0.345533	−	0.598480i	0.639918	−	0.768444i	$-0.278968\pi$
$929$	−321.000	−	555.988i	−0.345533	−	0.598480i	−0.985450	+	0.169963i	$0.945635\pi$
$930$		0			0
$931$		0			0
$932$	−1860.00			−1.99571
$933$		0			0
$934$		0			0
$935$	202.500	−	350.740i	0.216578	−	0.375123i
$936$		0			0
$937$	−167.500	+	290.119i	−0.178762	+	0.309625i	−0.941457	−	0.337134i	$-0.890543\pi$
$937$	−167.500	+	290.119i	−0.178762	+	0.309625i	0.762695	+	0.646759i	$0.223876\pi$
$938$		0			0
$939$		0			0
$940$	1350.00	−	2338.27i	1.43617	−	2.48752i
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	915.000	+	1584.83i	0.966209	+	1.67352i	0.706331	+	0.707882i	$0.250349\pi$
$947$	915.000	+	1584.83i	0.966209	+	1.67352i	0.259878	+	0.965641i	$0.416318\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$	−418.500	−	724.863i	−0.438220	−	0.759019i
$956$	906.000	−	1569.24i	0.947699	−	1.64146i
$957$		0			0
$958$		0			0
$959$	637.500	−	1104.18i	0.664755	−	1.15139i
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	895.000	+	1550.19i	0.925543	+	1.60309i	0.790686	+	0.612222i	$0.209724\pi$
$967$	895.000	+	1550.19i	0.925543	+	1.60309i	0.134857	+	0.990865i	$0.456942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$	−492.500	+	853.035i	−0.506166	+	0.876706i
$974$		0			0
$975$		0			0
$976$	1648.00			1.68852
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	864.000			0.881633
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$	405.000	+	701.481i	0.411168	+	0.712163i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2550.00			2.57836
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2043.00			−2.05327
$996$		0			0
$997$	−987.500	+	1710.40i	−0.990471	+	1.71555i	−0.375968	+	0.926633i	$0.622690\pi$
$997$	−987.500	+	1710.40i	−0.990471	+	1.71555i	−0.614503	+	0.788914i	$0.710644\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	361.3.d.a.69.1		2
19.2	odd	18		361.3.f.a.333.1		6
19.3	odd	18		361.3.f.a.127.1		6
19.4	even	9		361.3.f.a.116.1		6
19.5	even	9		361.3.f.a.262.1		6
19.6	even	9		361.3.f.a.299.1		6
19.7	even	3		19.3.b.a.18.1	✓	1
19.8	odd	6	inner	361.3.d.a.293.1		2
19.9	even	9		361.3.f.a.307.1		6
19.10	odd	18		361.3.f.a.307.1		6
19.11	even	3	inner	361.3.d.a.293.1		2
19.12	odd	6		19.3.b.a.18.1	✓	1
19.13	odd	18		361.3.f.a.299.1		6
19.14	odd	18		361.3.f.a.262.1		6
19.15	odd	18		361.3.f.a.116.1		6
19.16	even	9		361.3.f.a.127.1		6
19.17	even	9		361.3.f.a.333.1		6
19.18	odd	2	CM	361.3.d.a.69.1		2
57.26	odd	6		171.3.c.a.37.1		1
57.50	even	6		171.3.c.a.37.1		1
76.7	odd	6		304.3.e.a.113.1		1
76.31	even	6		304.3.e.a.113.1		1
95.7	odd	12		475.3.d.a.474.1		2
95.12	even	12		475.3.d.a.474.1		2
95.64	even	6		475.3.c.a.151.1		1
95.69	odd	6		475.3.c.a.151.1		1
95.83	odd	12		475.3.d.a.474.2		2
95.88	even	12		475.3.d.a.474.2		2
152.45	even	6		1216.3.e.a.1025.1		1
152.69	odd	6		1216.3.e.a.1025.1		1
152.83	odd	6		1216.3.e.b.1025.1		1
152.107	even	6		1216.3.e.b.1025.1		1
228.83	even	6		2736.3.o.a.721.1		1
228.107	odd	6		2736.3.o.a.721.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.b.a.18.1	✓	1	19.7	even	3
19.3.b.a.18.1	✓	1	19.12	odd	6
171.3.c.a.37.1		1	57.26	odd	6
171.3.c.a.37.1		1	57.50	even	6
304.3.e.a.113.1		1	76.7	odd	6
304.3.e.a.113.1		1	76.31	even	6
361.3.d.a.69.1		2	1.1	even	1	trivial
361.3.d.a.69.1		2	19.18	odd	2	CM
361.3.d.a.293.1		2	19.8	odd	6	inner
361.3.d.a.293.1		2	19.11	even	3	inner
361.3.f.a.116.1		6	19.4	even	9
361.3.f.a.116.1		6	19.15	odd	18
361.3.f.a.127.1		6	19.3	odd	18
361.3.f.a.127.1		6	19.16	even	9
361.3.f.a.262.1		6	19.5	even	9
361.3.f.a.262.1		6	19.14	odd	18
361.3.f.a.299.1		6	19.6	even	9
361.3.f.a.299.1		6	19.13	odd	18
361.3.f.a.307.1		6	19.9	even	9
361.3.f.a.307.1		6	19.10	odd	18
361.3.f.a.333.1		6	19.2	odd	18
361.3.f.a.333.1		6	19.17	even	9
475.3.c.a.151.1		1	95.64	even	6
475.3.c.a.151.1		1	95.69	odd	6
475.3.d.a.474.1		2	95.7	odd	12
475.3.d.a.474.1		2	95.12	even	12
475.3.d.a.474.2		2	95.83	odd	12
475.3.d.a.474.2		2	95.88	even	12
1216.3.e.a.1025.1		1	152.45	even	6
1216.3.e.a.1025.1		1	152.69	odd	6
1216.3.e.b.1025.1		1	152.83	odd	6
1216.3.e.b.1025.1		1	152.107	even	6
2736.3.o.a.721.1		1	228.83	even	6
2736.3.o.a.721.1		1	228.107	odd	6

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:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	361.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -5.00000 q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -5.00000 q^{7} +(-4.50000 + 7.79423i) q^{9} +3.00000 q^{11} +(-8.00000 - 13.8564i) q^{16} +(-7.50000 - 12.9904i) q^{17} -36.0000 q^{20} +(15.0000 - 25.9808i) q^{23} +(-28.0000 + 48.4974i) q^{25} +(10.0000 - 17.3205i) q^{28} +(-22.5000 - 38.9711i) q^{35} +(-18.0000 - 31.1769i) q^{36} +(42.5000 + 73.6122i) q^{43} +(-6.00000 + 10.3923i) q^{44} -81.0000 q^{45} +(-37.5000 + 64.9519i) q^{47} -24.0000 q^{49} +(13.5000 + 23.3827i) q^{55} +(-51.5000 + 89.2006i) q^{61} +(22.5000 - 38.9711i) q^{63} +64.0000 q^{64} +60.0000 q^{68} +(12.5000 + 21.6506i) q^{73} -15.0000 q^{77} +(72.0000 - 124.708i) q^{80} +(-40.5000 - 70.1481i) q^{81} +90.0000 q^{83} +(67.5000 - 116.913i) q^{85} +(60.0000 + 103.923i) q^{92} +(-13.5000 + 23.3827i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 + 9 * q^5 - 10 * q^7 - 9 * q^9	\( 2 q - 4 q^{4} + 9 q^{5} - 10 q^{7} - 9 q^{9} + 6 q^{11} - 16 q^{16} - 15 q^{17} - 72 q^{20} + 30 q^{23} - 56 q^{25} + 20 q^{28} - 45 q^{35} - 36 q^{36} + 85 q^{43} - 12 q^{44} - 162 q^{45} - 75 q^{47} - 48 q^{49} + 27 q^{55} - 103 q^{61} + 45 q^{63} + 128 q^{64} + 120 q^{68} + 25 q^{73} - 30 q^{77} + 144 q^{80} - 81 q^{81} + 180 q^{83} + 135 q^{85} + 120 q^{92} - 27 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 + 9 * q^5 - 10 * q^7 - 9 * q^9 + 6 * q^11 - 16 * q^16 - 15 * q^17 - 72 * q^20 + 30 * q^23 - 56 * q^25 + 20 * q^28 - 45 * q^35 - 36 * q^36 + 85 * q^43 - 12 * q^44 - 162 * q^45 - 75 * q^47 - 48 * q^49 + 27 * q^55 - 103 * q^61 + 45 * q^63 + 128 * q^64 + 120 * q^68 + 25 * q^73 - 30 * q^77 + 144 * q^80 - 81 * q^81 + 180 * q^83 + 135 * q^85 + 120 * q^92 - 27 * q^99

Introduction

L-functions

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