Newform orbit 2275.2.a.i

• Label: 2275.2.a.i
• Level: $2275$
• Weight: $2$
• Character orbit: 2275.a
• Self dual: yes
• Analytic conductor: $18.166$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2275.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.1659664598\)
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:	yes
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 + (b - 1) * q^2 - b * q^3 + (-2*b + 2) * q^4 + (b - 3) * q^6 + q^7 + (2*b - 6) * q^8 + (-b + 3) * q^11 + (-2*b + 6) * q^12 + q^13 + (b - 1) * q^14 + (-4*b + 8) * q^16 + (2*b - 4) * q^17 + (3*b - 1) * q^19 - b * q^21 + (4*b - 6) * q^22 - 3 * q^23 + (6*b - 6) * q^24 + (b - 1) * q^26 + 3*b * q^27 + (-2*b + 2) * q^28 + (2*b - 5) * q^29 + (-b + 3) * q^31 + (8*b - 8) * q^32 + (-3*b + 3) * q^33 + (-6*b + 10) * q^34 + (2*b + 4) * q^37 + (-4*b + 10) * q^38 - b * q^39 + (-b - 3) * q^41 + (b - 3) * q^42 - 11 * q^43 + (-8*b + 12) * q^44 + (-3*b + 3) * q^46 + (-4*b - 4) * q^47 + (-8*b + 12) * q^48 + q^49 + (4*b - 6) * q^51 + (-2*b + 2) * q^52 + (2*b - 9) * q^53 + (-3*b + 9) * q^54 + (2*b - 6) * q^56 + (b - 9) * q^57 + (-7*b + 11) * q^58 + (2*b - 4) * q^59 + (b + 2) * q^61 + (4*b - 6) * q^62 + (-8*b + 16) * q^64 + (6*b - 12) * q^66 - 6 * q^67 + (12*b - 20) * q^68 + 3*b * q^69 + (2*b + 4) * q^71 + 4 * q^73 + (2*b + 2) * q^74 + (8*b - 20) * q^76 + (-b + 3) * q^77 + (b - 3) * q^78 + (2*b + 3) * q^79 - 9 * q^81 - 2*b * q^82 + (-3*b - 9) * q^83 + (-2*b + 6) * q^84 + (-11*b + 11) * q^86 + (5*b - 6) * q^87 + (12*b - 24) * q^88 + (-6*b - 6) * q^89 + q^91 + (6*b - 6) * q^92 + (-3*b + 3) * q^93 - 8 * q^94 + (8*b - 24) * q^96 + (-2*b - 16) * q^97 + (b - 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^2 + 4 * q^4 - 6 * q^6 + 2 * q^7 - 12 * q^8 + 6 * q^11 + 12 * q^12 + 2 * q^13 - 2 * q^14 + 16 * q^16 - 8 * q^17 - 2 * q^19 - 12 * q^22 - 6 * q^23 - 12 * q^24 - 2 * q^26 + 4 * q^28 - 10 * q^29 + 6 * q^31 - 16 * q^32 + 6 * q^33 + 20 * q^34 + 8 * q^37 + 20 * q^38 - 6 * q^41 - 6 * q^42 - 22 * q^43 + 24 * q^44 + 6 * q^46 - 8 * q^47 + 24 * q^48 + 2 * q^49 - 12 * q^51 + 4 * q^52 - 18 * q^53 + 18 * q^54 - 12 * q^56 - 18 * q^57 + 22 * q^58 - 8 * q^59 + 4 * q^61 - 12 * q^62 + 32 * q^64 - 24 * q^66 - 12 * q^67 - 40 * q^68 + 8 * q^71 + 8 * q^73 + 4 * q^74 - 40 * q^76 + 6 * q^77 - 6 * q^78 + 6 * q^79 - 18 * q^81 - 18 * q^83 + 12 * q^84 + 22 * q^86 - 12 * q^87 - 48 * q^88 - 12 * q^89 + 2 * q^91 - 12 * q^92 + 6 * q^93 - 16 * q^94 - 48 * q^96 - 32 * q^97 - 2 * q^98

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

1.73205

5.46410

0

−4.73205

1.00000

−9.46410

0

0

1.2

0.732051

−1.73205

−1.46410

0

−1.26795

1.00000

−2.53590

0

0

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\( +1 \)
\(7\)	\( -1 \)
\(13\)	\( -1 \)

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	2275.2.a.i	✓	2
5.b	even	2	1		2275.2.a.k	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2275.2.a.i	✓	2	1.a	even	1	1	trivial
2275.2.a.k	yes	2	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2275))\):

\( T_{2}^{2} + 2T_{2} - 2 \)

\( T_{3}^{2} - 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T - 2 \)
$3$	\( T^{2} - 3 \)
$5$	\( T^{2} \)
$7$	\( (T - 1)^{2} \)
$11$	\( T^{2} - 6T + 6 \)