Embedded newform 100.10.c.a.49.2

• Label: 100.10.c.a.49.2
• Level: $100$
• Weight: $10$
• Character: 100.49
• Analytic conductor: $51.504$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(51.5035836164\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	100.49
Dual form			100.10.c.a.49.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+228.000i q^{3} +6328.00i q^{7} -32301.0 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64602 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(1\)	\(-1\)

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	0
			\(3\)	228.000i	1.62513i	0.582868	+	0.812567i	\(0.301931\pi\)
−0.582868	+	0.812567i				\(0.698069\pi\)
\(5\)	0	0
			\(7\)	6328.00i	0.996151i	0.867134	+	0.498076i	\(0.165960\pi\)
−0.867134	+	0.498076i				\(0.834040\pi\)
\(11\)	−30420.0	−0.626458	−0.313229	−	0.949678i	\(-0.601411\pi\)
			−0.313229	+	0.949678i	\(0.601411\pi\)
\(13\)	− 32338.0i	− 0.314028i	−0.987596	−	0.157014i	\(-0.949813\pi\)
			0.987596	−	0.157014i	\(-0.0501867\pi\)
\(17\)	− 590994.i	− 1.71618i	−0.513499	−	0.858090i	\(-0.671651\pi\)
			0.513499	−	0.858090i	\(-0.328349\pi\)
\(19\)	−34676.0	−0.0610433	−0.0305216	−	0.999534i	\(-0.509717\pi\)
			−0.0305216	+	0.999534i	\(0.509717\pi\)
\(23\)	1.04854e6i	0.781282i	0.920543	+	0.390641i	\(0.127747\pi\)
			−0.920543	+	0.390641i	\(0.872253\pi\)
\(29\)	−4.40941e6	−1.15768	−0.578841	−	0.815441i	\(-0.696495\pi\)
			−0.578841	+	0.815441i	\(0.696495\pi\)
\(31\)	−7.40118e6	−1.43937	−0.719687	−	0.694299i	\(-0.755715\pi\)
			−0.719687	+	0.694299i	\(0.755715\pi\)
\(37\)	− 1.02345e7i	− 0.897757i	−0.893593	−	0.448879i	\(-0.851824\pi\)
			0.893593	−	0.448879i	\(-0.148176\pi\)
\(41\)	1.83527e7	1.01432	0.507158	−	0.861853i	\(-0.330696\pi\)
			0.507158	+	0.861853i	\(0.330696\pi\)
\(43\)	− 252340.i	− 0.0112558i	−0.999984	−	0.00562792i	\(-0.998209\pi\)
			0.999984	−	0.00562792i	\(-0.00179143\pi\)
\(47\)	4.95171e7i	1.48018i	0.672507	+	0.740091i	\(0.265218\pi\)
			−0.672507	+	0.740091i	\(0.734782\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.10.c.a.49.2		2
4.3	odd	2		400.10.c.a.49.1		2
5.2	odd	4		4.10.a.a.1.1	✓	1
5.3	odd	4		100.10.a.a.1.1		1
5.4	even	2	inner	100.10.c.a.49.1		2
15.2	even	4		36.10.a.b.1.1		1
20.3	even	4		400.10.a.k.1.1		1
20.7	even	4		16.10.a.a.1.1		1
20.19	odd	2		400.10.c.a.49.2		2
35.2	odd	12		196.10.e.a.165.1		2
35.12	even	12		196.10.e.b.165.1		2
35.17	even	12		196.10.e.b.177.1		2
35.27	even	4		196.10.a.a.1.1		1
35.32	odd	12		196.10.e.a.177.1		2
40.27	even	4		64.10.a.i.1.1		1
40.37	odd	4		64.10.a.a.1.1		1
45.2	even	12		324.10.e.b.109.1		2
45.7	odd	12		324.10.e.e.109.1		2
45.22	odd	12		324.10.e.e.217.1		2
45.32	even	12		324.10.e.b.217.1		2
60.47	odd	4		144.10.a.j.1.1		1
80.27	even	4		256.10.b.b.129.2		2
80.37	odd	4		256.10.b.j.129.1		2
80.67	even	4		256.10.b.b.129.1		2
80.77	odd	4		256.10.b.j.129.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.10.a.a.1.1	✓	1	5.2	odd	4
16.10.a.a.1.1		1	20.7	even	4
36.10.a.b.1.1		1	15.2	even	4
64.10.a.a.1.1		1	40.37	odd	4
64.10.a.i.1.1		1	40.27	even	4
100.10.a.a.1.1		1	5.3	odd	4
100.10.c.a.49.1		2	5.4	even	2	inner
100.10.c.a.49.2		2	1.1	even	1	trivial
144.10.a.j.1.1		1	60.47	odd	4
196.10.a.a.1.1		1	35.27	even	4
196.10.e.a.165.1		2	35.2	odd	12
196.10.e.a.177.1		2	35.32	odd	12
196.10.e.b.165.1		2	35.12	even	12
196.10.e.b.177.1		2	35.17	even	12
256.10.b.b.129.1		2	80.67	even	4
256.10.b.b.129.2		2	80.27	even	4
256.10.b.j.129.1		2	80.37	odd	4
256.10.b.j.129.2		2	80.77	odd	4
324.10.e.b.109.1		2	45.2	even	12
324.10.e.b.217.1		2	45.32	even	12
324.10.e.e.109.1		2	45.7	odd	12
324.10.e.e.217.1		2	45.22	odd	12
400.10.a.k.1.1		1	20.3	even	4
400.10.c.a.49.1		2	4.3	odd	2
400.10.c.a.49.2		2	20.19	odd	2