Properties

Label 1078.2.e.v
Level $1078$
Weight $2$
Character orbit 1078.e
Analytic conductor $8.608$
Analytic rank $0$
Dimension $4$
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.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} -\beta_{3} q^{6} - q^{8} + 4 \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} -\beta_{3} q^{6} - q^{8} + 4 \beta_{2} q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{10} + ( -1 - \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{3} ) q^{12} -5 q^{13} + ( -7 - \beta_{3} ) q^{15} + \beta_{2} q^{16} + ( 6 + 6 \beta_{2} ) q^{17} + ( 4 + 4 \beta_{2} ) q^{18} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{19} + ( -1 - \beta_{3} ) q^{20} - q^{22} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} -\beta_{1} q^{24} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{25} + 5 \beta_{2} q^{26} + \beta_{3} q^{27} + ( 1 - 2 \beta_{3} ) q^{29} + ( -\beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{30} + ( -4 - 4 \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{32} + ( -\beta_{1} - \beta_{3} ) q^{33} + 6 q^{34} + 4 q^{36} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{38} -5 \beta_{1} q^{39} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{40} + ( -3 + 3 \beta_{3} ) q^{41} -4 q^{43} + \beta_{2} q^{44} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{45} + ( -1 + \beta_{1} - \beta_{2} ) q^{46} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{47} + \beta_{3} q^{48} + ( -3 - 2 \beta_{3} ) q^{50} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{51} + ( 5 + 5 \beta_{2} ) q^{52} + ( 1 - \beta_{1} + \beta_{2} ) q^{53} + ( \beta_{1} + \beta_{3} ) q^{54} + ( -1 - \beta_{3} ) q^{55} + ( -7 + 3 \beta_{3} ) q^{57} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{58} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{59} + ( 7 - \beta_{1} + 7 \beta_{2} ) q^{60} + ( -2 \beta_{1} - 9 \beta_{2} - 2 \beta_{3} ) q^{61} -4 q^{62} + q^{64} + ( -5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{65} -\beta_{1} q^{66} + ( 4 + 3 \beta_{1} + 4 \beta_{2} ) q^{67} -6 \beta_{2} q^{68} + ( -7 - \beta_{3} ) q^{69} + ( 7 + \beta_{3} ) q^{71} -4 \beta_{2} q^{72} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{73} + ( 1 + \beta_{1} + \beta_{2} ) q^{74} + ( -3 \beta_{1} + 14 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 3 - \beta_{3} ) q^{76} + 5 \beta_{3} q^{78} + ( -\beta_{1} - \beta_{3} ) q^{79} + ( 1 - \beta_{1} + \beta_{2} ) q^{80} + ( 5 + 5 \beta_{2} ) q^{81} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{82} + ( -8 - 2 \beta_{3} ) q^{83} + ( 6 + 6 \beta_{3} ) q^{85} + 4 \beta_{2} q^{86} + ( 14 + \beta_{1} + 14 \beta_{2} ) q^{87} + ( 1 + \beta_{2} ) q^{88} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 4 + 4 \beta_{3} ) q^{90} + ( -1 - \beta_{3} ) q^{92} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{93} + ( -8 + 2 \beta_{1} - 8 \beta_{2} ) q^{94} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{95} + ( \beta_{1} + \beta_{3} ) q^{96} + ( 11 - 2 \beta_{3} ) q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{8} - 8 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{8} - 8 q^{9} - 2 q^{10} - 2 q^{11} - 20 q^{13} - 28 q^{15} - 2 q^{16} + 12 q^{17} + 8 q^{18} - 6 q^{19} - 4 q^{20} - 4 q^{22} + 2 q^{23} - 6 q^{25} - 10 q^{26} + 4 q^{29} - 14 q^{30} - 8 q^{31} + 2 q^{32} + 24 q^{34} + 16 q^{36} - 2 q^{37} + 6 q^{38} - 2 q^{40} - 12 q^{41} - 16 q^{43} - 2 q^{44} + 8 q^{45} - 2 q^{46} + 16 q^{47} - 12 q^{50} + 10 q^{52} + 2 q^{53} - 4 q^{55} - 28 q^{57} + 2 q^{58} + 4 q^{59} + 14 q^{60} + 18 q^{61} - 16 q^{62} + 4 q^{64} - 10 q^{65} + 8 q^{67} + 12 q^{68} - 28 q^{69} + 28 q^{71} + 8 q^{72} + 6 q^{73} + 2 q^{74} - 28 q^{75} + 12 q^{76} + 2 q^{80} + 10 q^{81} - 6 q^{82} - 32 q^{83} + 24 q^{85} - 8 q^{86} + 28 q^{87} + 2 q^{88} - 8 q^{89} + 16 q^{90} - 4 q^{92} - 16 q^{94} - 8 q^{95} + 44 q^{97} + 16 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0.500000 + 0.866025i −1.32288 + 2.29129i −0.500000 + 0.866025i 1.82288 + 3.15731i −2.64575 0 −1.00000 −2.00000 3.46410i −1.82288 + 3.15731i
67.2 0.500000 + 0.866025i 1.32288 2.29129i −0.500000 + 0.866025i −0.822876 1.42526i 2.64575 0 −1.00000 −2.00000 3.46410i 0.822876 1.42526i
177.1 0.500000 0.866025i −1.32288 2.29129i −0.500000 0.866025i 1.82288 3.15731i −2.64575 0 −1.00000 −2.00000 + 3.46410i −1.82288 3.15731i
177.2 0.500000 0.866025i 1.32288 + 2.29129i −0.500000 0.866025i −0.822876 + 1.42526i 2.64575 0 −1.00000 −2.00000 + 3.46410i 0.822876 + 1.42526i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.v 4
7.b odd 2 1 154.2.e.f 4
7.c even 3 1 1078.2.a.n 2
7.c even 3 1 inner 1078.2.e.v 4
7.d odd 6 1 154.2.e.f 4
7.d odd 6 1 1078.2.a.s 2
21.c even 2 1 1386.2.k.s 4
21.g even 6 1 1386.2.k.s 4
21.g even 6 1 9702.2.a.cz 2
21.h odd 6 1 9702.2.a.dr 2
28.d even 2 1 1232.2.q.g 4
28.f even 6 1 1232.2.q.g 4
28.f even 6 1 8624.2.a.ca 2
28.g odd 6 1 8624.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 7.b odd 2 1
154.2.e.f 4 7.d odd 6 1
1078.2.a.n 2 7.c even 3 1
1078.2.a.s 2 7.d odd 6 1
1078.2.e.v 4 1.a even 1 1 trivial
1078.2.e.v 4 7.c even 3 1 inner
1232.2.q.g 4 28.d even 2 1
1232.2.q.g 4 28.f even 6 1
1386.2.k.s 4 21.c even 2 1
1386.2.k.s 4 21.g even 6 1
8624.2.a.bk 2 28.g odd 6 1
8624.2.a.ca 2 28.f even 6 1
9702.2.a.cz 2 21.g even 6 1
9702.2.a.dr 2 21.h odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 7 T_{3}^{2} + 49 \)
\( T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36 \)
\( T_{13} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 49 + 7 T^{2} + T^{4} \)
$5$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T + T^{2} )^{2} \)
$13$ \( ( 5 + T )^{4} \)
$17$ \( ( 36 - 6 T + T^{2} )^{2} \)
$19$ \( 4 + 12 T + 34 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( ( -27 - 2 T + T^{2} )^{2} \)
$31$ \( ( 16 + 4 T + T^{2} )^{2} \)
$37$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( ( -54 + 6 T + T^{2} )^{2} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( 1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4} \)
$53$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$59$ \( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( 2809 - 954 T + 271 T^{2} - 18 T^{3} + T^{4} \)
$67$ \( 2209 + 376 T + 111 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( ( 42 - 14 T + T^{2} )^{2} \)
$73$ \( 4 - 12 T + 34 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( 49 + 7 T^{2} + T^{4} \)
$83$ \( ( 36 + 16 T + T^{2} )^{2} \)
$89$ \( 9216 - 768 T + 160 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( ( 93 - 22 T + T^{2} )^{2} \)
show more
show less