Embedded newform 75.9.j.a.11.5

• Label: 75.9.j.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-17.0444 - 23.4596i) q^{2} +(-29.5485 - 75.4181i) q^{3} +(-180.733 + 556.239i) q^{4} +(-601.366 - 170.248i) q^{5} +(-1265.64 + 1978.65i) q^{6} +1578.44 q^{7} +(9069.54 - 2946.87i) q^{8} +(-4814.77 + 4456.98i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−17.0444	−	23.4596i	−1.06527	−	1.46622i	−0.874770	−	0.484538i	\(-0.838988\pi\)
							−0.190505	−	0.981686i	\(-0.561012\pi\)
\(3\)	−29.5485	−	75.4181i	−0.364796	−	0.931087i
							\(5\)	−601.366	−	170.248i	−0.962185	−	0.272396i
\(7\)	1578.44			0.657408										0.328704	−	0.944433i	\(-0.393388\pi\)
							0.328704	+	0.944433i	\(0.393388\pi\)
\(11\)	−1149.69	−	1582.41i	−0.0785251	−	0.108081i	0.767948	−	0.640513i	\(-0.221278\pi\)
							−0.846473	+	0.532432i	\(0.821278\pi\)
\(13\)	27847.4	+	20232.3i	0.975014	+	0.708389i	0.956589	−	0.291441i	\(-0.0941346\pi\)
							0.0184255	+	0.999830i	\(0.494135\pi\)
\(17\)	49045.1	−	15935.7i	0.587219	−	0.190799i	−0.000312986	−	1.00000i	\(-0.500100\pi\)
							0.587532	+	0.809201i	\(0.300100\pi\)
\(19\)	−52053.7	−	160205.i	−0.399427	−	1.22931i	−0.925460	−	0.378845i	\(-0.876321\pi\)
							0.526034	−	0.850464i	\(-0.323679\pi\)
\(23\)	−129535.	−	178290.i	−0.462889	−	0.637112i	0.512216	−	0.858857i	\(-0.328825\pi\)
							−0.975105	+	0.221745i	\(0.928825\pi\)
\(29\)	1.14751e6	+	372849.i	1.62243	+	0.527159i	0.972513	−	0.232849i	\(-0.0748048\pi\)
							0.649914	+	0.760007i	\(0.274805\pi\)
\(31\)	342721.	+	1.05479e6i	0.371102	+	1.14213i	0.946071	+	0.323960i	\(0.105014\pi\)
							−0.574969	+	0.818175i	\(0.694986\pi\)
\(37\)	5494.47	+	3991.97i	0.00293170	+	0.00213000i	0.589250	−	0.807951i	\(-0.299423\pi\)
							−0.586318	+	0.810081i	\(0.699423\pi\)
\(41\)	−1.12723e6	+	1.55150e6i	−0.398913	+	0.549056i	−0.960471	−	0.278381i	\(-0.910202\pi\)
							0.561558	+	0.827437i	\(0.310202\pi\)
\(43\)	−3.14991e6			−0.921351			−0.460675	−	0.887569i	\(-0.652393\pi\)
							−0.460675	+	0.887569i	\(0.652393\pi\)
\(47\)	4.80769e6	+	1.56211e6i	0.985246	+	0.320126i	0.756955	−	0.653467i	\(-0.226686\pi\)
							0.228291	+	0.973593i	\(0.426686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.5	✓	312
3.2	odd	2	inner	75.9.j.a.11.74	yes	312
25.16	even	5	inner	75.9.j.a.41.74	yes	312
75.41	odd	10	inner	75.9.j.a.41.5	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.5	✓	312	1.1	even	1	trivial
75.9.j.a.11.74	yes	312	3.2	odd	2	inner
75.9.j.a.41.5	yes	312	75.41	odd	10	inner
75.9.j.a.41.74	yes	312	25.16	even	5	inner