Embedded newform 702.2.bb.a.71.10

• Label: 702.2.bb.a.71.10
• Level: $702$
• Weight: $2$
• Character: 702.71
• 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			71.10
Character	\(\chi\)	\(=\)	702.71
Dual form			702.2.bb.a.89.10

$q$-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 \(\left(\mathbb{Z}/702\mathbb{Z}\right)^\times\).

\(n\)	\(379\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

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

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.71.10		56
3.2	odd	2		234.2.y.a.149.5	yes	56
9.2	odd	6		702.2.bc.a.305.3		56
9.7	even	3		234.2.z.a.227.12	yes	56
13.11	odd	12		702.2.bc.a.557.3		56
39.11	even	12		234.2.z.a.167.12	yes	56
117.11	even	12	inner	702.2.bb.a.89.10		56
117.115	odd	12		234.2.y.a.11.5	✓	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.y.a.11.5	✓	56	117.115	odd	12
234.2.y.a.149.5	yes	56	3.2	odd	2
234.2.z.a.167.12	yes	56	39.11	even	12
234.2.z.a.227.12	yes	56	9.7	even	3
702.2.bb.a.71.10		56	1.1	even	1	trivial
702.2.bb.a.89.10		56	117.11	even	12	inner
702.2.bc.a.305.3		56	9.2	odd	6
702.2.bc.a.557.3		56	13.11	odd	12