Embedded newform 702.2.bb.a.89.10

• Label: 702.2.bb.a.89.10
• Level: $702$
• Weight: $2$
• Character: 702.89
• Analytic conductor: $5.605$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	702.bb (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.60549822189\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			89.10
Character	\(\chi\)	\(=\)	702.89
Dual form			702.2.bb.a.71.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(-0.994452 - 0.266463i) q^{5} +(0.339163 + 0.0908784i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-0.891601 + 0.514766i) q^{10} +(-2.56893 - 2.56893i) q^{11} +(0.0241746 - 3.60547i) q^{13} +(0.304085 - 0.175564i) q^{14} -1.00000 q^{16} +(1.67236 - 2.89660i) q^{17} +(-0.969723 + 0.259837i) q^{19} +(-0.266463 + 0.994452i) q^{20} -3.63301 q^{22} +(-0.735260 + 1.27351i) q^{23} +(-3.41219 - 1.97003i) q^{25} +(-2.53236 - 2.56655i) q^{26} +(0.0908784 - 0.339163i) q^{28} -5.82075i q^{29} +(1.48558 - 5.54424i) q^{31} +(-0.707107 + 0.707107i) q^{32} +(-0.865675 - 3.23074i) q^{34} +(-0.313065 - 0.180748i) q^{35} +(4.96682 + 1.33085i) q^{37} +(-0.501966 + 0.869430i) q^{38} +(0.514766 + 0.891601i) q^{40} +(2.41936 + 9.02916i) q^{41} +(2.37192 - 1.36943i) q^{43} +(-2.56893 + 2.56893i) q^{44} +(0.380599 + 1.42041i) q^{46} +(-8.18256 + 2.19251i) q^{47} +(-5.95541 - 3.43835i) q^{49} +(-3.80581 + 1.01976i) q^{50} +(-3.60547 - 0.0241746i) q^{52} +2.67501i q^{53} +(1.87015 + 3.23920i) q^{55} +(-0.175564 - 0.304085i) q^{56} +(-4.11589 - 4.11589i) q^{58} +(3.42622 + 3.42622i) q^{59} +(-2.86325 - 4.95929i) q^{61} +(-2.86991 - 4.97083i) q^{62} +1.00000i q^{64} +(-0.984764 + 3.57903i) q^{65} +(14.5603 - 3.90143i) q^{67} +(-2.89660 - 1.67236i) q^{68} +(-0.349179 + 0.0935622i) q^{70} +(1.53985 + 5.74681i) q^{71} +(2.47165 - 2.47165i) q^{73} +(4.45313 - 2.57101i) q^{74} +(0.259837 + 0.969723i) q^{76} +(-0.637824 - 1.10474i) q^{77} +(-1.76544 + 3.05783i) q^{79} +(0.994452 + 0.266463i) q^{80} +(8.09532 + 4.67384i) q^{82} +(2.23652 + 8.34680i) q^{83} +(-2.43491 + 2.43491i) q^{85} +(0.708868 - 2.64553i) q^{86} +3.63301i q^{88} +(2.68520 - 10.0213i) q^{89} +(0.335858 - 1.22064i) q^{91} +(1.27351 + 0.735260i) q^{92} +(-4.23561 + 7.33629i) q^{94} +1.03358 q^{95} +(-3.82280 + 14.2669i) q^{97} +(-6.64239 + 1.77982i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	56 * q + 4 * q^7 - 56 * q^16 - 8 * q^19 - 4 * q^28 + 8 * q^31 - 24 * q^35 - 4 * q^37 + 36 * q^38 + 48 * q^41 + 12 * q^43 - 60 * q^47 + 24 * q^50 - 4 * q^52 + 120 * q^65 - 56 * q^67 - 24 * q^71 + 28 * q^73 - 48 * q^74 + 4 * q^76 + 24 * q^77 - 24 * q^79 + 60 * q^83 - 48 * q^86 + 4 * q^91 + 24 * q^92 + 20 * q^97 + 48 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/702\mathbb{Z}\right)^\times\).

\(n\)	\(379\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

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

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.707107	−	0.707107i	0.500000	−	0.500000i
							\(3\)		0			0
\(5\)	−0.994452	−	0.266463i	−0.444732	−	0.119166i								0.0295011	−	0.999565i	\(-0.490608\pi\)
							−0.474234	+	0.880399i	\(0.657275\pi\)
\(7\)	0.339163	+	0.0908784i	0.128191	+	0.0343488i	0.322344	−	0.946622i	\(-0.395529\pi\)
							−0.194153	+	0.980971i	\(0.562196\pi\)
\(11\)	−2.56893	−	2.56893i	−0.774560	−	0.774560i	0.204340	−	0.978900i	\(-0.434495\pi\)
							−0.978900	+	0.204340i	\(0.934495\pi\)
\(13\)	0.0241746	−	3.60547i	0.00670484	−	0.999978i
							\(17\)	1.67236	−	2.89660i	0.405606	−	0.702530i	−0.588786	−	0.808289i	\(-0.700394\pi\)
0.994392	+	0.105759i	\(0.0337273\pi\)
\(19\)	−0.969723	+	0.259837i	−0.222470	+	0.0596106i	−0.368332	−	0.929694i	\(-0.620071\pi\)
							0.145862	+	0.989305i	\(0.453404\pi\)
\(23\)	−0.735260	+	1.27351i	−0.153312	+	0.265545i	−0.932443	−	0.361317i	\(-0.882327\pi\)
							0.779131	+	0.626861i	\(0.215661\pi\)
\(29\)		−	5.82075i		−	1.08089i	−0.841380	−	0.540443i	\(-0.818257\pi\)
							0.841380	−	0.540443i	\(-0.181743\pi\)
\(31\)	1.48558	−	5.54424i	0.266817	−	0.995775i	−0.694312	−	0.719675i	\(-0.744291\pi\)
							0.961129	−	0.276101i	\(-0.0890423\pi\)
\(37\)	4.96682	+	1.33085i	0.816540	+	0.218791i	0.642833	−	0.766006i	\(-0.277759\pi\)
							0.173707	+	0.984797i	\(0.444426\pi\)
\(41\)	2.41936	+	9.02916i	0.377840	+	1.41012i	0.849151	+	0.528151i	\(0.177114\pi\)
							−0.471311	+	0.881967i	\(0.656219\pi\)
\(43\)	2.37192	−	1.36943i	0.361714	−	0.208836i	−0.308118	−	0.951348i	\(-0.599699\pi\)
							0.669833	+	0.742512i	\(0.266366\pi\)
\(47\)	−8.18256	+	2.19251i	−1.19355	+	0.319811i	−0.800288	−	0.599616i	\(-0.795320\pi\)
							−0.393262	+	0.919427i	\(0.628653\pi\)
\(53\)			2.67501i			0.367442i	0.982978	+	0.183721i	\(0.0588142\pi\)
							−0.982978	+	0.183721i	\(0.941186\pi\)
\(59\)	3.42622	+	3.42622i	0.446055	+	0.446055i	0.894041	−	0.447985i	\(-0.147858\pi\)
							−0.447985	+	0.894041i	\(0.647858\pi\)
\(61\)	−2.86325	−	4.95929i	−0.366601	−	0.634972i	0.622431	−	0.782675i	\(-0.286145\pi\)
							−0.989032	+	0.147703i	\(0.952812\pi\)
\(67\)	14.5603	−	3.90143i	1.77883	−	0.476636i	0.788459	−	0.615087i	\(-0.210879\pi\)
							0.990369	+	0.138451i	\(0.0442125\pi\)
\(71\)	1.53985	+	5.74681i	0.182747	+	0.682021i	0.995102	+	0.0988571i	\(0.0315187\pi\)
							−0.812355	+	0.583164i	\(0.801815\pi\)
\(73\)	2.47165	−	2.47165i	0.289285	−	0.289285i	−0.547513	−	0.836797i	\(-0.684425\pi\)
							0.836797	+	0.547513i	\(0.184425\pi\)
\(79\)	−1.76544	+	3.05783i	−0.198627	+	0.344033i	−0.948084	−	0.318021i	\(-0.896982\pi\)
							0.749456	+	0.662054i	\(0.230315\pi\)
\(83\)	2.23652	+	8.34680i	0.245490	+	0.916181i	0.973137	+	0.230228i	\(0.0739474\pi\)
							−0.727647	+	0.685952i	\(0.759386\pi\)
\(89\)	2.68520	−	10.0213i	0.284630	−	1.06225i	−0.664479	−	0.747307i	\(-0.731347\pi\)
							0.949109	−	0.314947i	\(-0.101987\pi\)
\(97\)	−3.82280	+	14.2669i	−0.388147	+	1.44858i	0.444999	+	0.895531i	\(0.353204\pi\)
							−0.833146	+	0.553053i	\(0.813463\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	702.2.bb.a.89.10		56
3.2	odd	2		234.2.y.a.11.5	✓	56
9.4	even	3		234.2.z.a.167.12	yes	56
9.5	odd	6		702.2.bc.a.557.3		56
13.6	odd	12		702.2.bc.a.305.3		56
39.32	even	12		234.2.z.a.227.12	yes	56
117.32	even	12	inner	702.2.bb.a.71.10		56
117.58	odd	12		234.2.y.a.149.5	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.y.a.11.5	✓	56	3.2	odd	2
234.2.y.a.149.5	yes	56	117.58	odd	12
234.2.z.a.167.12	yes	56	9.4	even	3
234.2.z.a.227.12	yes	56	39.32	even	12
702.2.bb.a.71.10		56	117.32	even	12	inner
702.2.bb.a.89.10		56	1.1	even	1	trivial
702.2.bc.a.305.3		56	13.6	odd	12
702.2.bc.a.557.3		56	9.5	odd	6