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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\):

\( T_{3}^{2} - 2T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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