Properties

Label 1078.2.a.q
Level $1078$
Weight $2$
Character orbit 1078.a
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} -\beta q^{5} -\beta q^{6} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + \beta q^{3} + q^{4} -\beta q^{5} -\beta q^{6} - q^{8} - q^{9} + \beta q^{10} - q^{11} + \beta q^{12} -2 \beta q^{13} -2 q^{15} + q^{16} + 4 \beta q^{17} + q^{18} -2 \beta q^{19} -\beta q^{20} + q^{22} -2 q^{23} -\beta q^{24} -3 q^{25} + 2 \beta q^{26} -4 \beta q^{27} -6 q^{29} + 2 q^{30} + \beta q^{31} - q^{32} -\beta q^{33} -4 \beta q^{34} - q^{36} -2 q^{37} + 2 \beta q^{38} -4 q^{39} + \beta q^{40} -4 \beta q^{41} + 4 q^{43} - q^{44} + \beta q^{45} + 2 q^{46} -3 \beta q^{47} + \beta q^{48} + 3 q^{50} + 8 q^{51} -2 \beta q^{52} -12 q^{53} + 4 \beta q^{54} + \beta q^{55} -4 q^{57} + 6 q^{58} -3 \beta q^{59} -2 q^{60} + 4 \beta q^{61} -\beta q^{62} + q^{64} + 4 q^{65} + \beta q^{66} -2 q^{67} + 4 \beta q^{68} -2 \beta q^{69} -10 q^{71} + q^{72} + 10 \beta q^{73} + 2 q^{74} -3 \beta q^{75} -2 \beta q^{76} + 4 q^{78} -8 q^{79} -\beta q^{80} -5 q^{81} + 4 \beta q^{82} + 2 \beta q^{83} -8 q^{85} -4 q^{86} -6 \beta q^{87} + q^{88} + 5 \beta q^{89} -\beta q^{90} -2 q^{92} + 2 q^{93} + 3 \beta q^{94} + 4 q^{95} -\beta q^{96} + 11 \beta q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 2 q^{32} - 2 q^{36} - 4 q^{37} - 8 q^{39} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 6 q^{50} + 16 q^{51} - 24 q^{53} - 8 q^{57} + 12 q^{58} - 4 q^{60} + 2 q^{64} + 8 q^{65} - 4 q^{67} - 20 q^{71} + 2 q^{72} + 4 q^{74} + 8 q^{78} - 16 q^{79} - 10 q^{81} - 16 q^{85} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{93} + 8 q^{95} + 2 q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.a.q 2
3.b odd 2 1 9702.2.a.dq 2
4.b odd 2 1 8624.2.a.bw 2
7.b odd 2 1 inner 1078.2.a.q 2
7.c even 3 2 1078.2.e.s 4
7.d odd 6 2 1078.2.e.s 4
21.c even 2 1 9702.2.a.dq 2
28.d even 2 1 8624.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.q 2 1.a even 1 1 trivial
1078.2.a.q 2 7.b odd 2 1 inner
1078.2.e.s 4 7.c even 3 2
1078.2.e.s 4 7.d odd 6 2
8624.2.a.bw 2 4.b odd 2 1
8624.2.a.bw 2 28.d even 2 1
9702.2.a.dq 2 3.b odd 2 1
9702.2.a.dq 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\):

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} - 2 \)
\( T_{13}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( -2 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( -32 + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -2 + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( -32 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -18 + T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( -18 + T^{2} \)
$61$ \( -32 + T^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( ( 10 + T )^{2} \)
$73$ \( -200 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( -8 + T^{2} \)
$89$ \( -50 + T^{2} \)
$97$ \( -242 + T^{2} \)
show more
show less