Embedded newform 23.15.d.a.5.17

• Label: 23.15.d.a.5.17
• Level: $23$
• Weight: $15$
• Character: 23.5
• Analytic conductor: $28.596$
• Analytic rank: $0$
• Dimension: $270$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.5956626749\)
Analytic rank:	\(0\)
Dimension:	\(270\)
Relative dimension:	\(27\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.17
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(67.1845 + 19.7271i) q^{2} +(-2168.41 + 2502.47i) q^{3} +(-9658.50 - 6207.14i) q^{4} +(-59403.5 - 8540.93i) q^{5} +(-195050. + 125351. i) q^{6} +(1.32939e6 - 607114. i) q^{7} +(-1.27772e6 - 1.47457e6i) q^{8} +(-879702. - 6.11846e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	67.1845	+	19.7271i	0.524879	+	0.154118i	0.533428	−	0.845845i	\(-0.320903\pi\)
							−0.00854982	+	0.999963i	\(0.502722\pi\)
\(3\)	−2168.41	+	2502.47i	−0.991498	+	1.14425i	−0.00195641	+	0.999998i	\(0.500623\pi\)
							−0.989541	+	0.144251i	\(0.953923\pi\)
\(5\)	−59403.5	−	8540.93i	−0.760364	−	0.109324i	−0.248784	−	0.968559i	\(-0.580031\pi\)
							−0.511581	+	0.859235i	\(0.670940\pi\)
\(7\)	1.32939e6	−	607114.i	1.61424	−	0.737198i	0.615535	−	0.788110i	\(-0.288940\pi\)
							0.998703	+	0.0509118i	\(0.0162128\pi\)
\(11\)	−3.33921e6	−	1.13723e7i	−0.171354	−	0.583579i	−0.999726	−	0.0234004i	\(-0.992551\pi\)
							0.828372	−	0.560178i	\(-0.189267\pi\)
\(13\)	−2.53664e7	+	5.55448e7i	−0.404256	+	0.885196i	0.592565	+	0.805522i	\(0.298115\pi\)
							−0.996821	+	0.0796740i	\(0.974612\pi\)
\(17\)	1.08899e7	+	1.69450e7i	0.0265388	+	0.0412952i	0.854270	−	0.519830i	\(-0.174005\pi\)
							−0.827731	+	0.561126i	\(0.810368\pi\)
\(19\)	−1.94804e8	+	3.03122e8i	−0.217933	+	0.339111i	−0.932956	−	0.359990i	\(-0.882780\pi\)
							0.715023	+	0.699101i	\(0.246416\pi\)
\(23\)	−1.73436e9	+	2.92999e9i	−0.509384	+	0.860539i
							\(29\)	2.42002e10	−	1.55525e10i	1.40292	−	0.901601i	0.403010	−	0.915195i	\(-0.367964\pi\)
0.999908	+	0.0135948i	\(0.00432748\pi\)
\(31\)	−2.11237e9	−	2.43780e9i	−0.0767782	−	0.0886068i	0.716061	−	0.698038i	\(-0.245943\pi\)
							−0.792839	+	0.609431i	\(0.791398\pi\)
\(37\)	1.59566e11	−	2.29421e10i	1.68084	−	0.241669i	0.765238	−	0.643748i	\(-0.222621\pi\)
							0.915605	+	0.402079i	\(0.131712\pi\)
\(41\)	−4.02988e10	+	2.80284e11i	−0.206921	+	1.43917i	0.576205	+	0.817305i	\(0.304533\pi\)
							−0.783126	+	0.621863i	\(0.786376\pi\)
\(43\)	3.52109e11	+	3.05104e11i	1.29538	+	1.12245i	0.985132	+	0.171800i	\(0.0549583\pi\)
							0.310250	+	0.950655i	\(0.399587\pi\)
\(47\)	3.79146e11			0.748379			0.374190	−	0.927352i	\(-0.377921\pi\)
							0.374190	+	0.927352i	\(0.377921\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.17	✓	270
23.14	odd	22	inner	23.15.d.a.14.17	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.17	✓	270	1.1	even	1	trivial
23.15.d.a.14.17	yes	270	23.14	odd	22	inner