Embedded newform 315.2.r.b.184.10

• Label: 315.2.r.b.184.10
• Level: $315$
• Weight: $2$
• Character: 315.184
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			184.10
Character	\(\chi\)	\(=\)	315.184
Dual form			315.2.r.b.214.33

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.80560i q^{2} +(1.73197 - 0.0166637i) q^{3} -1.26019 q^{4} +(-1.93209 - 1.12562i) q^{5} +(-0.0300880 - 3.12725i) q^{6} +(2.01307 + 1.71685i) q^{7} -1.33580i q^{8} +(2.99944 - 0.0577222i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 88 q^{4} + 3 q^{5} + 6 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.80560i		−	1.27675i	−0.769725	−	0.638376i	\(-0.779607\pi\)
							0.769725	−	0.638376i	\(-0.220393\pi\)
\(3\)	1.73197	−	0.0166637i	0.999954	−	0.00962081i
							\(5\)	−1.93209	−	1.12562i	−0.864057	−	0.503394i
\(7\)	2.01307	+	1.71685i	0.760868	+	0.648907i
							\(11\)	1.98491	−	3.43796i	0.598473	−	1.03659i	−0.394574	−	0.918864i	\(-0.629108\pi\)
0.993047	−	0.117721i	\(-0.0375588\pi\)
\(13\)	−0.246741	−	0.142456i	−0.0684338	−	0.0395103i	0.465393	−	0.885104i	\(-0.345913\pi\)
							−0.533826	+	0.845594i	\(0.679246\pi\)
\(17\)	−7.00194	+	4.04257i	−1.69822	+	0.980468i	−0.750771	+	0.660563i	\(0.770318\pi\)
							−0.947450	+	0.319905i	\(0.896349\pi\)
\(19\)	1.62907	−	2.82164i	0.373735	−	0.647329i	−0.616402	−	0.787432i	\(-0.711410\pi\)
							0.990137	+	0.140103i	\(0.0447435\pi\)
\(23\)	0.453376	−	0.261757i	0.0945355	−	0.0545801i	−0.451987	−	0.892025i	\(-0.649285\pi\)
							0.546522	+	0.837444i	\(0.315951\pi\)
\(29\)	3.35657	+	5.81375i	0.623299	+	1.07959i	0.988867	+	0.148801i	\(0.0475414\pi\)
							−0.365568	+	0.930785i	\(0.619125\pi\)
\(31\)	−4.27971			−0.768659			−0.384330	−	0.923196i	\(-0.625567\pi\)
							−0.384330	+	0.923196i	\(0.625567\pi\)
\(37\)	5.74650	+	3.31774i	0.944719	+	0.545433i	0.891436	−	0.453146i	\(-0.149698\pi\)
							0.0532822	+	0.998579i	\(0.483032\pi\)
\(41\)	3.90766	−	6.76827i	0.610274	−	1.05703i	−0.380920	−	0.924608i	\(-0.624393\pi\)
							0.991194	−	0.132418i	\(-0.0422741\pi\)
\(43\)	−4.26784	+	2.46404i	−0.650840	+	0.375763i	−0.788778	−	0.614678i	\(-0.789286\pi\)
							0.137938	+	0.990441i	\(0.455953\pi\)
\(47\)			11.4474i			1.66978i	0.550417	+	0.834890i	\(0.314469\pi\)
							−0.550417	+	0.834890i	\(0.685531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.r.b.184.10	✓	84
3.2	odd	2		945.2.r.b.604.33		84
5.4	even	2	inner	315.2.r.b.184.33	yes	84
7.4	even	3		315.2.bo.b.4.33	yes	84
9.2	odd	6		945.2.bo.b.289.33		84
9.7	even	3		315.2.bo.b.79.10	yes	84
15.14	odd	2		945.2.r.b.604.10		84
21.11	odd	6		945.2.bo.b.739.10		84
35.4	even	6		315.2.bo.b.4.10	yes	84
45.29	odd	6		945.2.bo.b.289.10		84
45.34	even	6		315.2.bo.b.79.33	yes	84
63.11	odd	6		945.2.r.b.424.33		84
63.25	even	3	inner	315.2.r.b.214.10	yes	84
105.74	odd	6		945.2.bo.b.739.33		84
315.74	odd	6		945.2.r.b.424.10		84
315.214	even	6	inner	315.2.r.b.214.33	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.r.b.184.10	✓	84	1.1	even	1	trivial
315.2.r.b.184.33	yes	84	5.4	even	2	inner
315.2.r.b.214.10	yes	84	63.25	even	3	inner
315.2.r.b.214.33	yes	84	315.214	even	6	inner
315.2.bo.b.4.10	yes	84	35.4	even	6
315.2.bo.b.4.33	yes	84	7.4	even	3
315.2.bo.b.79.10	yes	84	9.7	even	3
315.2.bo.b.79.33	yes	84	45.34	even	6
945.2.r.b.424.10		84	315.74	odd	6
945.2.r.b.424.33		84	63.11	odd	6
945.2.r.b.604.10		84	15.14	odd	2
945.2.r.b.604.33		84	3.2	odd	2
945.2.bo.b.289.10		84	45.29	odd	6
945.2.bo.b.289.33		84	9.2	odd	6
945.2.bo.b.739.10		84	21.11	odd	6
945.2.bo.b.739.33		84	105.74	odd	6