Newform orbit 1859.4.a.a

• Label: 1859.4.a.a
• Level: $1859$
• Weight: $4$
• Character orbit: 1859.a
• Self dual: yes
• Analytic conductor: $109.685$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1859.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(109.684550701\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{2} + (4 \beta - 1) q^{3} + ( - 2 \beta - 4) q^{4} + (8 \beta - 1) q^{5} + ( - 5 \beta + 13) q^{6} + ( - 4 \beta - 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 8 \beta + 22) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.73205
1.73205

−2.73205

−7.92820

−0.535898

−14.8564

21.6603

−3.07180

23.3205

35.8564

40.5885

1.2

0.732051

5.92820

−7.46410

12.8564

4.33975

−16.9282

−11.3205

8.14359

9.41154

Atkin-Lehner signs

\( p \)	Sign
\(11\)	\(-1\)
\(13\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1859.4.a.a		2
13.b	even	2	1		11.4.a.a	✓	2
39.d	odd	2	1		99.4.a.c		2
52.b	odd	2	1		176.4.a.i		2
65.d	even	2	1		275.4.a.b		2
65.h	odd	4	2		275.4.b.c		4
91.b	odd	2	1		539.4.a.e		2
104.e	even	2	1		704.4.a.p		2
104.h	odd	2	1		704.4.a.n		2
143.d	odd	2	1		121.4.a.c		2
143.l	odd	10	4		121.4.c.f		8
143.n	even	10	4		121.4.c.c		8
156.h	even	2	1		1584.4.a.bc		2
195.e	odd	2	1		2475.4.a.q		2
429.e	even	2	1		1089.4.a.v		2
572.b	even	2	1		1936.4.a.w		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.4.a.a	✓	2	13.b	even	2	1
99.4.a.c		2	39.d	odd	2	1
121.4.a.c		2	143.d	odd	2	1
121.4.c.c		8	143.n	even	10	4
121.4.c.f		8	143.l	odd	10	4
176.4.a.i		2	52.b	odd	2	1
275.4.a.b		2	65.d	even	2	1
275.4.b.c		4	65.h	odd	4	2
539.4.a.e		2	91.b	odd	2	1
704.4.a.n		2	104.h	odd	2	1
704.4.a.p		2	104.e	even	2	1
1089.4.a.v		2	429.e	even	2	1
1584.4.a.bc		2	156.h	even	2	1
1859.4.a.a		2	1.a	even	1	1	trivial
1936.4.a.w		2	572.b	even	2	1
2475.4.a.q		2	195.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T - 2 \)
$3$	\( T^{2} + 2T - 47 \)
$5$	\( T^{2} + 2T - 191 \)
$7$	\( T^{2} + 20T + 52 \)
$11$	\( (T - 11)^{2} \)