Properties

Label 1078.2.a.w
Level $1078$
Weight $2$
Character orbit 1078.a
Self dual yes
Analytic conductor $8.608$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( -1 - \beta ) q^{5} + ( 1 + \beta ) q^{6} + q^{8} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( -1 - \beta ) q^{5} + ( 1 + \beta ) q^{6} + q^{8} + ( 3 + 2 \beta ) q^{9} + ( -1 - \beta ) q^{10} + q^{11} + ( 1 + \beta ) q^{12} + ( 1 - \beta ) q^{13} + ( -6 - 2 \beta ) q^{15} + q^{16} + ( 2 + 2 \beta ) q^{17} + ( 3 + 2 \beta ) q^{18} + ( 5 - \beta ) q^{19} + ( -1 - \beta ) q^{20} + q^{22} + 4 q^{23} + ( 1 + \beta ) q^{24} + ( 1 + 2 \beta ) q^{25} + ( 1 - \beta ) q^{26} + ( 10 + 2 \beta ) q^{27} -2 \beta q^{29} + ( -6 - 2 \beta ) q^{30} -2 q^{31} + q^{32} + ( 1 + \beta ) q^{33} + ( 2 + 2 \beta ) q^{34} + ( 3 + 2 \beta ) q^{36} + ( -2 - 4 \beta ) q^{37} + ( 5 - \beta ) q^{38} -4 q^{39} + ( -1 - \beta ) q^{40} + ( -2 - 2 \beta ) q^{41} + ( -6 + 2 \beta ) q^{43} + q^{44} + ( -13 - 5 \beta ) q^{45} + 4 q^{46} + 2 q^{47} + ( 1 + \beta ) q^{48} + ( 1 + 2 \beta ) q^{50} + ( 12 + 4 \beta ) q^{51} + ( 1 - \beta ) q^{52} + ( 4 - 2 \beta ) q^{53} + ( 10 + 2 \beta ) q^{54} + ( -1 - \beta ) q^{55} + 4 \beta q^{57} -2 \beta q^{58} + ( -5 - \beta ) q^{59} + ( -6 - 2 \beta ) q^{60} + ( 3 + \beta ) q^{61} -2 q^{62} + q^{64} + 4 q^{65} + ( 1 + \beta ) q^{66} + ( -2 - 6 \beta ) q^{67} + ( 2 + 2 \beta ) q^{68} + ( 4 + 4 \beta ) q^{69} + ( 2 - 2 \beta ) q^{71} + ( 3 + 2 \beta ) q^{72} + ( -4 + 4 \beta ) q^{73} + ( -2 - 4 \beta ) q^{74} + ( 11 + 3 \beta ) q^{75} + ( 5 - \beta ) q^{76} -4 q^{78} + ( -1 - \beta ) q^{80} + ( 11 + 6 \beta ) q^{81} + ( -2 - 2 \beta ) q^{82} + ( 1 - 5 \beta ) q^{83} + ( -12 - 4 \beta ) q^{85} + ( -6 + 2 \beta ) q^{86} + ( -10 - 2 \beta ) q^{87} + q^{88} -10 q^{89} + ( -13 - 5 \beta ) q^{90} + 4 q^{92} + ( -2 - 2 \beta ) q^{93} + 2 q^{94} -4 \beta q^{95} + ( 1 + \beta ) q^{96} + ( -8 + 2 \beta ) q^{97} + ( 3 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 6 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 6 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} - 12 q^{15} + 2 q^{16} + 4 q^{17} + 6 q^{18} + 10 q^{19} - 2 q^{20} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 20 q^{27} - 12 q^{30} - 4 q^{31} + 2 q^{32} + 2 q^{33} + 4 q^{34} + 6 q^{36} - 4 q^{37} + 10 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 12 q^{43} + 2 q^{44} - 26 q^{45} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{50} + 24 q^{51} + 2 q^{52} + 8 q^{53} + 20 q^{54} - 2 q^{55} - 10 q^{59} - 12 q^{60} + 6 q^{61} - 4 q^{62} + 2 q^{64} + 8 q^{65} + 2 q^{66} - 4 q^{67} + 4 q^{68} + 8 q^{69} + 4 q^{71} + 6 q^{72} - 8 q^{73} - 4 q^{74} + 22 q^{75} + 10 q^{76} - 8 q^{78} - 2 q^{80} + 22 q^{81} - 4 q^{82} + 2 q^{83} - 24 q^{85} - 12 q^{86} - 20 q^{87} + 2 q^{88} - 20 q^{89} - 26 q^{90} + 8 q^{92} - 4 q^{93} + 4 q^{94} + 2 q^{96} - 16 q^{97} + 6 q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
1.00000 −1.23607 1.00000 1.23607 −1.23607 0 1.00000 −1.47214 1.23607
1.2 1.00000 3.23607 1.00000 −3.23607 3.23607 0 1.00000 7.47214 −3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1078.2.a.w 2
3.b odd 2 1 9702.2.a.cu 2
4.b odd 2 1 8624.2.a.bf 2
7.b odd 2 1 154.2.a.d 2
7.c even 3 2 1078.2.e.n 4
7.d odd 6 2 1078.2.e.q 4
21.c even 2 1 1386.2.a.m 2
28.d even 2 1 1232.2.a.p 2
35.c odd 2 1 3850.2.a.bj 2
35.f even 4 2 3850.2.c.q 4
56.e even 2 1 4928.2.a.bk 2
56.h odd 2 1 4928.2.a.bt 2
77.b even 2 1 1694.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 7.b odd 2 1
1078.2.a.w 2 1.a even 1 1 trivial
1078.2.e.n 4 7.c even 3 2
1078.2.e.q 4 7.d odd 6 2
1232.2.a.p 2 28.d even 2 1
1386.2.a.m 2 21.c even 2 1
1694.2.a.l 2 77.b even 2 1
3850.2.a.bj 2 35.c odd 2 1
3850.2.c.q 4 35.f even 4 2
4928.2.a.bk 2 56.e even 2 1
4928.2.a.bt 2 56.h odd 2 1
8624.2.a.bf 2 4.b odd 2 1
9702.2.a.cu 2 3.b odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2 T_{3} - 4 \)
\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{13}^{2} - 2 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -4 - 2 T + T^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( -16 - 4 T + T^{2} \)
$19$ \( 20 - 10 T + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( -76 + 4 T + T^{2} \)
$41$ \( -16 + 4 T + T^{2} \)
$43$ \( 16 + 12 T + T^{2} \)
$47$ \( ( -2 + T )^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( 20 + 10 T + T^{2} \)
$61$ \( 4 - 6 T + T^{2} \)
$67$ \( -176 + 4 T + T^{2} \)
$71$ \( -16 - 4 T + T^{2} \)
$73$ \( -64 + 8 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -124 - 2 T + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 44 + 16 T + T^{2} \)
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