Embedded newform 75.9.j.a.11.8

• Label: 75.9.j.a.11.8
• Level: $75$
• Weight: $9$
• Character: 75.11
• Analytic conductor: $30.553$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(30.5533957546\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(78\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			11.8
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.3948 - 22.5655i) q^{2} +(-80.8710 - 4.56896i) q^{3} +(-161.305 + 496.445i) q^{4} +(295.172 - 550.907i) q^{5} +(1222.76 + 1899.80i) q^{6} +1770.00 q^{7} +(7056.09 - 2292.66i) q^{8} +(6519.25 + 738.994i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.3948	−	22.5655i	−1.02468	−	1.41035i	−0.908873	−	0.417073i	\(-0.863056\pi\)
							−0.115803	−	0.993272i	\(-0.536944\pi\)
\(3\)	−80.8710	−	4.56896i	−0.998408	−	0.0564070i
							\(5\)	295.172	−	550.907i	0.472275	−	0.881451i
\(7\)	1770.00			0.737194										0.368597	−	0.929589i	\(-0.379838\pi\)
							0.368597	+	0.929589i	\(0.379838\pi\)
\(11\)	−8190.57	−	11273.4i	−0.559427	−	0.769985i	0.431827	−	0.901957i	\(-0.357869\pi\)
							−0.991254	+	0.131972i	\(0.957869\pi\)
\(13\)	−6812.21	−	4949.36i	−0.238514	−	0.173291i	0.462107	−	0.886824i	\(-0.347094\pi\)
							−0.700621	+	0.713533i	\(0.747094\pi\)
\(17\)	−62200.7	+	20210.2i	−0.744731	+	0.241978i	−0.656713	−	0.754141i	\(-0.728054\pi\)
							−0.0880187	+	0.996119i	\(0.528054\pi\)
\(19\)	−49835.0	−	153376.i	−0.382402	−	1.17691i	−0.938348	−	0.345693i	\(-0.887644\pi\)
							0.555946	−	0.831219i	\(-0.312356\pi\)
\(23\)	184978.	+	254600.i	0.661010	+	0.909802i	0.999514	−	0.0311631i	\(-0.00992114\pi\)
							−0.338505	+	0.940965i	\(0.609921\pi\)
\(29\)	−955076.	−	310323.i	−1.35035	−	0.438755i	−0.457540	−	0.889189i	\(-0.651269\pi\)
							−0.892810	+	0.450434i	\(0.851269\pi\)
\(31\)	−403731.	−	1.24256e6i	−0.437165	−	1.34546i	−0.890851	−	0.454295i	\(-0.849891\pi\)
							0.453686	−	0.891162i	\(-0.350109\pi\)
\(37\)	1.96112e6	+	1.42484e6i	1.04640	+	0.760254i	0.971525	−	0.236939i	\(-0.0761442\pi\)
							0.0748752	+	0.997193i	\(0.476144\pi\)
\(41\)	1.32291e6	−	1.82084e6i	0.468162	−	0.644370i	−0.508014	−	0.861349i	\(-0.669620\pi\)
							0.976176	+	0.216979i	\(0.0696202\pi\)
\(43\)	1.57016e6			0.459272			0.229636	−	0.973277i	\(-0.426246\pi\)
							0.229636	+	0.973277i	\(0.426246\pi\)
\(47\)	598557.	+	194483.i	0.122663	+	0.0398557i	0.369705	−	0.929149i	\(-0.379459\pi\)
							−0.247042	+	0.969005i	\(0.579459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.8	✓	312
3.2	odd	2	inner	75.9.j.a.11.71	yes	312
25.16	even	5	inner	75.9.j.a.41.71	yes	312
75.41	odd	10	inner	75.9.j.a.41.8	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.8	✓	312	1.1	even	1	trivial
75.9.j.a.11.71	yes	312	3.2	odd	2	inner
75.9.j.a.41.8	yes	312	75.41	odd	10	inner
75.9.j.a.41.71	yes	312	25.16	even	5	inner