Embedded newform 684.2.bs.a.119.11

• Label: 684.2.bs.a.119.11
• Level: $684$
• Weight: $2$
• Character: 684.119
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $696$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$684 = 2^{2} \cdot 3^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	684.bs (of order $18$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.46176749826$
Analytic rank:	$0$
Dimension:	$696$
Relative dimension:	$116$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			119.11
Character	$\chi$	$=$	684.119
Dual form			684.2.bs.a.23.11

$q$-expansion

$f(q)$	$=$	$q+(-1.36753 - 0.360368i) q^{2} +(0.356216 - 1.69503i) q^{3} +(1.74027 + 0.985628i) q^{4} +(-1.53473 - 0.270614i) q^{5} +(-1.09797 + 2.18963i) q^{6} -0.923751i q^{7} +(-2.02468 - 1.97501i) q^{8} +(-2.74622 - 1.20759i) q^{9} +(2.00127 + 0.923141i) q^{10} +(1.38916 + 2.40610i) q^{11} +(2.29058 - 2.59870i) q^{12} +(5.07753 - 4.26056i) q^{13} +(-0.332890 + 1.26326i) q^{14} +(-1.00539 + 2.50501i) q^{15} +(2.05708 + 3.43052i) q^{16} +(-0.788778 - 2.16715i) q^{17} +(3.32036 + 2.64107i) q^{18} +(-3.66424 - 2.36079i) q^{19} +(-2.40412 - 1.98361i) q^{20} +(-1.56578 - 0.329055i) q^{21} +(-1.03264 - 3.79102i) q^{22} +(-8.65925 - 3.15171i) q^{23} +(-4.06892 + 2.72835i) q^{24} +(-2.41630 - 0.879461i) q^{25} +(-8.47904 + 3.99665i) q^{26} +(-3.02515 + 4.22475i) q^{27} +(0.910474 - 1.60758i) q^{28} +(3.10017 + 3.69463i) q^{29} +(2.27763 - 3.06336i) q^{30} +(3.56920 + 2.06068i) q^{31} +(-1.57686 - 5.43263i) q^{32} +(4.57324 - 1.49757i) q^{33} +(0.297705 + 3.24789i) q^{34} +(-0.249980 + 1.41771i) q^{35} +(-3.58893 - 4.80828i) q^{36} -5.89377 q^{37} +(4.16021 + 4.54892i) q^{38} +(-5.41305 - 10.1242i) q^{39} +(2.57287 + 3.57902i) q^{40} +(0.179922 + 0.494332i) q^{41} +(2.02267 + 1.01425i) q^{42} +(-2.04848 - 5.62815i) q^{43} +(0.0459988 + 5.55646i) q^{44} +(3.88791 + 2.59649i) q^{45} +(10.7060 + 7.43057i) q^{46} +(-2.45486 + 2.05987i) q^{47} +(6.54757 - 2.26479i) q^{48} +6.14668 q^{49} +(2.98743 + 2.07345i) q^{50} +(-3.95435 + 0.565024i) q^{51} +(13.0356 - 2.40996i) q^{52} +(-3.83515 - 4.57056i) q^{53} +(5.65944 - 4.68730i) q^{54} +(-1.48086 - 4.06864i) q^{55} +(-1.82442 + 1.87030i) q^{56} +(-5.30686 + 5.37004i) q^{57} +(-2.90814 - 6.16972i) q^{58} +(-2.01639 - 1.69196i) q^{59} +(-4.21866 + 3.36845i) q^{60} +(2.41706 + 13.7078i) q^{61} +(-4.13839 - 4.10427i) q^{62} +(-1.11551 + 2.53682i) q^{63} +(0.198657 + 7.99753i) q^{64} +(-8.94561 + 5.16475i) q^{65} +(-6.79372 + 0.399924i) q^{66} +(-14.1422 + 2.49365i) q^{67} +(0.763316 - 4.54887i) q^{68} +(-8.42679 + 13.5550i) q^{69} +(0.852752 - 1.84867i) q^{70} +(-7.01828 - 5.88904i) q^{71} +(3.17521 + 7.86880i) q^{72} +(1.56692 - 8.88643i) q^{73} +(8.05990 + 2.12393i) q^{74} +(-2.35143 + 3.78241i) q^{75} +(-4.04992 - 7.71999i) q^{76} +(2.22264 - 1.28324i) q^{77} +(3.75405 + 15.7959i) q^{78} +(-6.78987 + 8.09186i) q^{79} +(-2.22871 - 5.82159i) q^{80} +(6.08345 + 6.63262i) q^{81} +(-0.0679072 - 0.740852i) q^{82} +4.95989 q^{83} +(-2.40056 - 2.11592i) q^{84} +(0.624099 + 3.53944i) q^{85} +(0.773148 + 8.43487i) q^{86} +(7.36683 - 3.93877i) q^{87} +(1.93947 - 7.61520i) q^{88} +(4.70122 - 0.828952i) q^{89} +(-4.38114 - 4.95186i) q^{90} +(-3.93569 - 4.69038i) q^{91} +(-11.9630 - 14.0196i) q^{92} +(4.76432 - 5.31584i) q^{93} +(4.09940 - 1.93228i) q^{94} +(4.98476 + 4.61477i) q^{95} +(-9.77015 + 0.737629i) q^{96} +(0.862927 - 4.89390i) q^{97} +(-8.40577 - 2.21507i) q^{98} +(-0.909361 - 8.28522i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	696 * q - 9 * q^2 - 3 * q^4 - 18 * q^5 - 6 * q^6 - 18 * q^9 - 12 * q^10 - 3 * q^12 - 6 * q^13 - 9 * q^14 - 3 * q^16 - 12 * q^18 - 18 * q^20 + 24 * q^21 + 9 * q^22 + 45 * q^24 - 6 * q^25 + 72 * q^26 - 18 * q^29 - 9 * q^32 + 12 * q^33 + 15 * q^34 - 66 * q^36 - 48 * q^37 - 9 * q^38 + 12 * q^40 - 108 * q^41 + 126 * q^42 + 135 * q^44 - 6 * q^45 - 6 * q^46 - 18 * q^48 - 564 * q^49 - 9 * q^50 - 3 * q^52 - 66 * q^54 + 45 * q^56 - 12 * q^57 - 6 * q^58 + 9 * q^60 - 6 * q^61 - 18 * q^62 + 12 * q^64 - 18 * q^65 - 18 * q^66 - 6 * q^69 + 18 * q^70 + 12 * q^72 + 12 * q^73 - 90 * q^74 - 3 * q^76 - 36 * q^77 - 21 * q^78 + 18 * q^81 - 12 * q^82 + 81 * q^84 - 6 * q^85 - 9 * q^86 + 90 * q^88 - 117 * q^90 - 9 * q^92 - 30 * q^93 - 6 * q^94 - 87 * q^96 + 12 * q^97 + 45 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{8}{9}\right)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−1.36753	−	0.360368i	−0.966989	−	0.254819i
							$3$	0.356216	−	1.69503i	0.205662	−	0.978623i
$5$	−1.53473	−	0.270614i	−0.686352	−	0.121022i								−0.180412	−	0.983591i	$-0.557743\pi$
							−0.505940	+	0.862569i	$0.668854\pi$
$7$		−	0.923751i		−	0.349145i	−0.984644	−	0.174573i	$-0.944146\pi$
							0.984644	−	0.174573i	$-0.0558544\pi$
$11$	1.38916	+	2.40610i	0.418848	+	0.725467i	0.995824	−	0.0912948i	$-0.0291006\pi$
							−0.576976	+	0.816761i	$0.695767\pi$
$13$	5.07753	−	4.26056i	1.40825	−	1.18167i	0.450962	−	0.892543i	$-0.351081\pi$
							0.957292	−	0.289122i	$-0.0933634\pi$
$17$	−0.788778	−	2.16715i	−0.191307	−	0.525611i	0.806541	−	0.591178i	$-0.201337\pi$
							−0.997848	+	0.0655666i	$0.979115\pi$
$19$	−3.66424	−	2.36079i	−0.840635	−	0.541602i
							$23$	−8.65925	−	3.15171i	−1.80558	−	0.657177i	−0.997697	−	0.0678308i	$-0.978392\pi$
−0.807881	−	0.589346i	$-0.799386\pi$
$29$	3.10017	+	3.69463i	0.575686	+	0.686076i	0.972788	−	0.231698i	$-0.0744283\pi$
							−0.397101	+	0.917775i	$0.629984\pi$
$31$	3.56920	+	2.06068i	0.641048	+	0.370109i	0.785018	−	0.619473i	$-0.212654\pi$
							−0.143970	+	0.989582i	$0.545987\pi$
$37$	−5.89377			−0.968930			−0.484465	−	0.874811i	$-0.660986\pi$
							−0.484465	+	0.874811i	$0.660986\pi$
$41$	0.179922	+	0.494332i	0.0280991	+	0.0772018i	0.952949	−	0.303131i	$-0.0980318\pi$
							−0.924850	+	0.380332i	$0.875810\pi$
$43$	−2.04848	−	5.62815i	−0.312390	−	0.858285i	−0.992173	−	0.124872i	$-0.960148\pi$
							0.679783	−	0.733414i	$-0.262074\pi$
$47$	−2.45486	+	2.05987i	−0.358078	+	0.300463i	−0.804024	−	0.594597i	$-0.797312\pi$
							0.445946	+	0.895060i	$0.352867\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.2.bs.a.119.11	yes	696
4.3	odd	2	inner	684.2.bs.a.119.4	yes	696
9.5	odd	6		684.2.ch.a.347.30	yes	696
19.4	even	9		684.2.ch.a.479.37	yes	696
36.23	even	6		684.2.ch.a.347.37	yes	696
76.23	odd	18		684.2.ch.a.479.30	yes	696
171.23	odd	18	inner	684.2.bs.a.23.4	✓	696
684.23	even	18	inner	684.2.bs.a.23.11	yes	696

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.bs.a.23.4	✓	696	171.23	odd	18	inner
684.2.bs.a.23.11	yes	696	684.23	even	18	inner
684.2.bs.a.119.4	yes	696	4.3	odd	2	inner
684.2.bs.a.119.11	yes	696	1.1	even	1	trivial
684.2.ch.a.347.30	yes	696	9.5	odd	6
684.2.ch.a.347.37	yes	696	36.23	even	6
684.2.ch.a.479.30	yes	696	76.23	odd	18
684.2.ch.a.479.37	yes	696	19.4	even	9