Embedded newform 75.9.j.a.11.1

• Label: 75.9.j.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.2305 - 25.0921i) q^{2} +(-80.9595 - 2.56149i) q^{3} +(-218.155 + 671.412i) q^{4} +(-34.5911 + 624.042i) q^{5} +(1411.66 + 2078.14i) q^{6} -3717.22 q^{7} +(13272.8 - 4312.60i) q^{8} +(6547.88 + 414.754i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.2305	−	25.0921i	−1.13940	−	1.56826i	−0.768829	−	0.639454i	\(-0.779160\pi\)
							−0.370576	−	0.928802i	\(-0.620840\pi\)
\(3\)	−80.9595	−	2.56149i	−0.999500	−	0.0316233i
							\(5\)	−34.5911	+	624.042i	−0.0553458	+	0.998467i
\(7\)	−3717.22			−1.54820										−0.774098	−	0.633066i	\(-0.781796\pi\)
							−0.774098	+	0.633066i	\(0.781796\pi\)
\(11\)	5239.78	+	7211.94i	0.357884	+	0.492585i	0.949558	−	0.313593i	\(-0.101533\pi\)
							−0.591674	+	0.806178i	\(0.701533\pi\)
\(13\)	7440.50	+	5405.84i	0.260512	+	0.189273i	0.710373	−	0.703825i	\(-0.248526\pi\)
							−0.449860	+	0.893099i	\(0.648526\pi\)
\(17\)	7615.31	−	2474.36i	0.0911783	−	0.0296256i	−0.263073	−	0.964776i	\(-0.584736\pi\)
							0.354251	+	0.935150i	\(0.384736\pi\)
\(19\)	49121.1	+	151179.i	0.376924	+	1.16005i	0.942171	+	0.335131i	\(0.108781\pi\)
							−0.565247	+	0.824921i	\(0.691219\pi\)
\(23\)	45264.3	+	62301.0i	0.161750	+	0.222630i	0.882197	−	0.470880i	\(-0.156063\pi\)
							−0.720447	+	0.693510i	\(0.756063\pi\)
\(29\)	571929.	+	185831.i	0.808631	+	0.262740i	0.684018	−	0.729465i	\(-0.260231\pi\)
							0.124613	+	0.992205i	\(0.460231\pi\)
\(31\)	405771.	+	1.24883e6i	0.439374	+	1.35225i	0.888538	+	0.458804i	\(0.151722\pi\)
							−0.449164	+	0.893449i	\(0.648278\pi\)
\(37\)	1.68333e6	+	1.22301e6i	0.898176	+	0.652563i	0.937997	−	0.346643i	\(-0.112679\pi\)
							−0.0398205	+	0.999207i	\(0.512679\pi\)
\(41\)	2.08087e6	−	2.86407e6i	0.736391	−	1.01356i	−0.262427	−	0.964952i	\(-0.584523\pi\)
							0.998818	−	0.0486038i	\(-0.0154772\pi\)
\(43\)	1.02474e6			0.299736			0.149868	−	0.988706i	\(-0.452115\pi\)
							0.149868	+	0.988706i	\(0.452115\pi\)
\(47\)	3.44613e6	+	1.11971e6i	0.706220	+	0.229465i	0.640038	−	0.768343i	\(-0.278918\pi\)
							0.0661814	+	0.997808i	\(0.478918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.1	✓	312
3.2	odd	2	inner	75.9.j.a.11.78	yes	312
25.16	even	5	inner	75.9.j.a.41.78	yes	312
75.41	odd	10	inner	75.9.j.a.41.1	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.1	✓	312	1.1	even	1	trivial
75.9.j.a.11.78	yes	312	3.2	odd	2	inner
75.9.j.a.41.1	yes	312	75.41	odd	10	inner
75.9.j.a.41.78	yes	312	25.16	even	5	inner