Properties

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

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{10}) \)
Defining polynomial: \(x^{2} - 10\)
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 \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + ( 2 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta q^{5} + ( 2 - \beta ) q^{7} + q^{9} + 4 q^{13} -\beta q^{15} + ( 4 - \beta ) q^{17} + ( 2 + \beta ) q^{19} + ( -2 + \beta ) q^{21} - q^{23} + 5 q^{25} - q^{27} + ( 2 - 2 \beta ) q^{29} + 2 \beta q^{31} + ( -10 + 2 \beta ) q^{35} + ( -2 + 2 \beta ) q^{37} -4 q^{39} -2 q^{41} + ( -2 - \beta ) q^{43} + \beta q^{45} + ( -6 - 2 \beta ) q^{47} + ( 7 - 4 \beta ) q^{49} + ( -4 + \beta ) q^{51} + ( -4 - 3 \beta ) q^{53} + ( -2 - \beta ) q^{57} + ( -2 + 2 \beta ) q^{59} + ( -6 - 2 \beta ) q^{61} + ( 2 - \beta ) q^{63} + 4 \beta q^{65} + ( -2 - \beta ) q^{67} + q^{69} + ( -2 + 4 \beta ) q^{73} -5 q^{75} + ( 10 - \beta ) q^{79} + q^{81} -4 q^{83} + ( -10 + 4 \beta ) q^{85} + ( -2 + 2 \beta ) q^{87} + 3 \beta q^{89} + ( 8 - 4 \beta ) q^{91} -2 \beta q^{93} + ( 10 + 2 \beta ) q^{95} + ( -10 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{7} + 2q^{9} + 8q^{13} + 8q^{17} + 4q^{19} - 4q^{21} - 2q^{23} + 10q^{25} - 2q^{27} + 4q^{29} - 20q^{35} - 4q^{37} - 8q^{39} - 4q^{41} - 4q^{43} - 12q^{47} + 14q^{49} - 8q^{51} - 8q^{53} - 4q^{57} - 4q^{59} - 12q^{61} + 4q^{63} - 4q^{67} + 2q^{69} - 4q^{73} - 10q^{75} + 20q^{79} + 2q^{81} - 8q^{83} - 20q^{85} - 4q^{87} + 16q^{91} + 20q^{95} - 20q^{97} + O(q^{100}) \)

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
−3.16228
3.16228
0 −1.00000 0 −3.16228 0 5.16228 0 1.00000 0
1.2 0 −1.00000 0 3.16228 0 −1.16228 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.a 2
3.b odd 2 1 828.2.a.f 2
4.b odd 2 1 1104.2.a.n 2
5.b even 2 1 6900.2.a.p 2
5.c odd 4 2 6900.2.f.k 4
8.b even 2 1 4416.2.a.bk 2
8.d odd 2 1 4416.2.a.be 2
12.b even 2 1 3312.2.a.y 2
23.b odd 2 1 6348.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.a.a 2 1.a even 1 1 trivial
828.2.a.f 2 3.b odd 2 1
1104.2.a.n 2 4.b odd 2 1
3312.2.a.y 2 12.b even 2 1
4416.2.a.be 2 8.d odd 2 1
4416.2.a.bk 2 8.b even 2 1
6348.2.a.e 2 23.b odd 2 1
6900.2.a.p 2 5.b even 2 1
6900.2.f.k 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 10 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(276))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -10 + T^{2} \)
$7$ \( -6 - 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -4 + T )^{2} \)
$17$ \( 6 - 8 T + T^{2} \)
$19$ \( -6 - 4 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -36 - 4 T + T^{2} \)
$31$ \( -40 + T^{2} \)
$37$ \( -36 + 4 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( -6 + 4 T + T^{2} \)
$47$ \( -4 + 12 T + T^{2} \)
$53$ \( -74 + 8 T + T^{2} \)
$59$ \( -36 + 4 T + T^{2} \)
$61$ \( -4 + 12 T + T^{2} \)
$67$ \( -6 + 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -156 + 4 T + T^{2} \)
$79$ \( 90 - 20 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( -90 + T^{2} \)
$97$ \( 60 + 20 T + T^{2} \)
show more
show less