Properties

Label 276.2.a.b
Level $276$
Weight $2$
Character orbit 276.a
Self dual yes
Analytic conductor $2.204$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( 2 + \beta ) q^{5} + \beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 2 + \beta ) q^{5} + \beta q^{7} + q^{9} -4 \beta q^{11} -4 \beta q^{13} + ( 2 + \beta ) q^{15} + ( 2 + 3 \beta ) q^{17} + ( -4 + 3 \beta ) q^{19} + \beta q^{21} + q^{23} + ( 1 + 4 \beta ) q^{25} + q^{27} + ( 6 - 2 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} -4 \beta q^{33} + ( 2 + 2 \beta ) q^{35} + ( -6 - 2 \beta ) q^{37} -4 \beta q^{39} -2 q^{41} + ( -4 + \beta ) q^{43} + ( 2 + \beta ) q^{45} + ( -2 + 6 \beta ) q^{47} -5 q^{49} + ( 2 + 3 \beta ) q^{51} + ( 6 + \beta ) q^{53} + ( -8 - 8 \beta ) q^{55} + ( -4 + 3 \beta ) q^{57} + ( -6 + 2 \beta ) q^{59} + ( 6 + 2 \beta ) q^{61} + \beta q^{63} + ( -8 - 8 \beta ) q^{65} + ( -12 - 3 \beta ) q^{67} + q^{69} -8 \beta q^{71} + ( 6 + 4 \beta ) q^{73} + ( 1 + 4 \beta ) q^{75} -8 q^{77} + 5 \beta q^{79} + q^{81} + ( -4 + 4 \beta ) q^{83} + ( 10 + 8 \beta ) q^{85} + ( 6 - 2 \beta ) q^{87} + ( 6 - 5 \beta ) q^{89} -8 q^{91} + ( -4 - 2 \beta ) q^{93} + ( -2 + 2 \beta ) q^{95} + ( 2 - 10 \beta ) q^{97} -4 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 4q^{5} + 2q^{9} + 4q^{15} + 4q^{17} - 8q^{19} + 2q^{23} + 2q^{25} + 2q^{27} + 12q^{29} - 8q^{31} + 4q^{35} - 12q^{37} - 4q^{41} - 8q^{43} + 4q^{45} - 4q^{47} - 10q^{49} + 4q^{51} + 12q^{53} - 16q^{55} - 8q^{57} - 12q^{59} + 12q^{61} - 16q^{65} - 24q^{67} + 2q^{69} + 12q^{73} + 2q^{75} - 16q^{77} + 2q^{81} - 8q^{83} + 20q^{85} + 12q^{87} + 12q^{89} - 16q^{91} - 8q^{93} - 4q^{95} + 4q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 1.00000 0 0.585786 0 −1.41421 0 1.00000 0
1.2 0 1.00000 0 3.41421 0 1.41421 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)

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 276.2.a.b 2
3.b odd 2 1 828.2.a.e 2
4.b odd 2 1 1104.2.a.l 2
5.b even 2 1 6900.2.a.m 2
5.c odd 4 2 6900.2.f.l 4
8.b even 2 1 4416.2.a.bc 2
8.d odd 2 1 4416.2.a.bi 2
12.b even 2 1 3312.2.a.s 2
23.b odd 2 1 6348.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.a.b 2 1.a even 1 1 trivial
828.2.a.e 2 3.b odd 2 1
1104.2.a.l 2 4.b odd 2 1
3312.2.a.s 2 12.b even 2 1
4416.2.a.bc 2 8.b even 2 1
4416.2.a.bi 2 8.d odd 2 1
6348.2.a.h 2 23.b odd 2 1
6900.2.a.m 2 5.b even 2 1
6900.2.f.l 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4 T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(276))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 2 - 4 T + T^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( -32 + T^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( -14 - 4 T + T^{2} \)
$19$ \( -2 + 8 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 28 - 12 T + T^{2} \)
$31$ \( 8 + 8 T + T^{2} \)
$37$ \( 28 + 12 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 14 + 8 T + T^{2} \)
$47$ \( -68 + 4 T + T^{2} \)
$53$ \( 34 - 12 T + T^{2} \)
$59$ \( 28 + 12 T + T^{2} \)
$61$ \( 28 - 12 T + T^{2} \)
$67$ \( 126 + 24 T + T^{2} \)
$71$ \( -128 + T^{2} \)
$73$ \( 4 - 12 T + T^{2} \)
$79$ \( -50 + T^{2} \)
$83$ \( -16 + 8 T + T^{2} \)
$89$ \( -14 - 12 T + T^{2} \)
$97$ \( -196 - 4 T + T^{2} \)
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