Embedded newform 115.4.l.a.102.4

• Label: 115.4.l.a.102.4
• Level: $115$
• Weight: $4$
• Character: 115.102
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $680$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.l (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.78521965066\)
Analytic rank:	\(0\)
Dimension:	\(680\)
Relative dimension:	\(34\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			102.4
Character	\(\chi\)	\(=\)	115.102
Dual form			115.4.l.a.53.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.341249 + 4.77128i) q^{2} +(6.15772 - 1.33953i) q^{3} +(-14.7301 - 2.11787i) q^{4} +(11.1479 - 0.850466i) q^{5} +(4.28996 + 29.8374i) q^{6} +(31.1635 - 17.0166i) q^{7} +(6.99722 - 32.1657i) q^{8} +(11.5631 - 5.28070i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 18 q^{2} - 26 q^{3} - 22 q^{5} - 76 q^{6} - 22 q^{7} - 50 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.341249	+	4.77128i	−0.120650	+	1.68690i	0.472122	+	0.881533i	\(0.343488\pi\)
							−0.592772	+	0.805371i	\(0.701966\pi\)
\(3\)	6.15772	−	1.33953i	1.18505	−	0.257793i	0.423501	−	0.905896i	\(-0.360801\pi\)
							0.761553	+	0.648103i	\(0.224437\pi\)
\(5\)	11.1479	−	0.850466i	0.997103	−	0.0760679i
							\(7\)	31.1635	−	17.0166i	1.68267	−	0.918809i	0.704855	−	0.709352i	\(-0.251012\pi\)
0.977818	−	0.209457i	\(-0.0671696\pi\)
\(11\)	2.59952	+	2.25250i	0.0712533	+	0.0617413i	0.689759	−	0.724039i	\(-0.257716\pi\)
							−0.618506	+	0.785780i	\(0.712262\pi\)
\(13\)	−3.35434	+	6.14302i	−0.0715637	+	0.131059i	−0.911011	−	0.412382i	\(-0.864697\pi\)
							0.839447	+	0.543441i	\(0.182879\pi\)
\(17\)	−50.4864	−	37.7936i	−0.720279	−	0.539194i	0.175072	−	0.984556i	\(-0.443984\pi\)
							−0.895351	+	0.445361i	\(0.853075\pi\)
\(19\)	−10.4871	+	72.9396i	−0.126627	+	0.880710i	0.823159	+	0.567811i	\(0.192209\pi\)
							−0.949786	+	0.312900i	\(0.898700\pi\)
\(23\)	−80.9598	−	74.9167i	−0.733969	−	0.679183i
							\(29\)	−248.606	+	35.7442i	−1.59190	+	0.228880i	−0.880547	−	0.473959i	\(-0.842825\pi\)
−0.711349	+	0.702839i	\(0.751915\pi\)
\(31\)	−200.632	+	128.938i	−1.16240	+	0.747031i	−0.972071	−	0.234688i	\(-0.924593\pi\)
							−0.190333	+	0.981720i	\(0.560957\pi\)
\(37\)	293.809	+	109.585i	1.30546	+	0.486911i	0.903437	−	0.428720i	\(-0.141035\pi\)
							0.402021	+	0.915631i	\(0.368308\pi\)
\(41\)	12.0984	−	26.4918i	0.0460842	−	0.100910i	−0.885189	−	0.465231i	\(-0.845971\pi\)
							0.931274	+	0.364320i	\(0.118699\pi\)
\(43\)	62.2309	+	286.071i	0.220701	+	1.01454i	0.945510	+	0.325594i	\(0.105564\pi\)
							−0.724809	+	0.688950i	\(0.758072\pi\)
\(47\)	−141.204	−	141.204i	−0.438229	−	0.438229i	0.453187	−	0.891416i	\(-0.350287\pi\)
							−0.891416	+	0.453187i	\(0.850287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.l.a.102.4	yes	680
5.3	odd	4	inner	115.4.l.a.33.31	yes	680
23.7	odd	22	inner	115.4.l.a.7.31	✓	680
115.53	even	44	inner	115.4.l.a.53.4	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.l.a.7.31	✓	680	23.7	odd	22	inner
115.4.l.a.33.31	yes	680	5.3	odd	4	inner
115.4.l.a.53.4	yes	680	115.53	even	44	inner
115.4.l.a.102.4	yes	680	1.1	even	1	trivial