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 $2$

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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: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} + 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} + 4 \beta_{2} q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{10} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{3} - \beta_1) q^{12} - 5 q^{13} + ( - \beta_{3} - 7) q^{15} + \beta_{2} q^{16} + (6 \beta_{2} + 6) q^{17} + (4 \beta_{2} + 4) q^{18} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 1) q^{20} - q^{22} + (\beta_{3} - \beta_{2} + \beta_1) q^{23} - \beta_1 q^{24} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{25} + 5 \beta_{2} q^{26} + \beta_{3} q^{27} + ( - 2 \beta_{3} + 1) q^{29} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{30} + ( - 4 \beta_{2} - 4) q^{31} + (\beta_{2} + 1) q^{32} + ( - \beta_{3} - \beta_1) q^{33} + 6 q^{34} + 4 q^{36} + (\beta_{3} + \beta_{2} + \beta_1) q^{37} + (3 \beta_{2} + \beta_1 + 3) q^{38} - 5 \beta_1 q^{39} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{40} + (3 \beta_{3} - 3) q^{41} - 4 q^{43} + \beta_{2} q^{44} + (4 \beta_{2} - 4 \beta_1 + 4) q^{45} + ( - \beta_{2} + \beta_1 - 1) q^{46} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{47} + \beta_{3} q^{48} + ( - 2 \beta_{3} - 3) q^{50} + (6 \beta_{3} + 6 \beta_1) q^{51} + (5 \beta_{2} + 5) q^{52} + (\beta_{2} - \beta_1 + 1) q^{53} + (\beta_{3} + \beta_1) q^{54} + ( - \beta_{3} - 1) q^{55} + (3 \beta_{3} - 7) q^{57} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{58} + (2 \beta_{2} + \beta_1 + 2) q^{59} + (7 \beta_{2} - \beta_1 + 7) q^{60} + ( - 2 \beta_{3} - 9 \beta_{2} - 2 \beta_1) q^{61} - 4 q^{62} + q^{64} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{65} - \beta_1 q^{66} + (4 \beta_{2} + 3 \beta_1 + 4) q^{67} - 6 \beta_{2} q^{68} + ( - \beta_{3} - 7) q^{69} + (\beta_{3} + 7) q^{71} - 4 \beta_{2} q^{72} + (3 \beta_{2} - \beta_1 + 3) q^{73} + (\beta_{2} + \beta_1 + 1) q^{74} + ( - 3 \beta_{3} + 14 \beta_{2} - 3 \beta_1) q^{75} + ( - \beta_{3} + 3) q^{76} + 5 \beta_{3} q^{78} + ( - \beta_{3} - \beta_1) q^{79} + (\beta_{2} - \beta_1 + 1) q^{80} + (5 \beta_{2} + 5) q^{81} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{82} + ( - 2 \beta_{3} - 8) q^{83} + (6 \beta_{3} + 6) q^{85} + 4 \beta_{2} q^{86} + (14 \beta_{2} + \beta_1 + 14) q^{87} + (\beta_{2} + 1) q^{88} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{89} + (4 \beta_{3} + 4) q^{90} + ( - \beta_{3} - 1) q^{92} + ( - 4 \beta_{3} - 4 \beta_1) q^{93} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{94} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{95} + (\beta_{3} + \beta_1) q^{96} + ( - 2 \beta_{3} + 11) q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{8} - 8 q^{9} - 2 q^{10} - 2 q^{11} - 20 q^{13} - 28 q^{15} - 2 q^{16} + 12 q^{17} + 8 q^{18} - 6 q^{19} - 4 q^{20} - 4 q^{22} + 2 q^{23} - 6 q^{25} - 10 q^{26} + 4 q^{29} - 14 q^{30} - 8 q^{31} + 2 q^{32} + 24 q^{34} + 16 q^{36} - 2 q^{37} + 6 q^{38} - 2 q^{40} - 12 q^{41} - 16 q^{43} - 2 q^{44} + 8 q^{45} - 2 q^{46} + 16 q^{47} - 12 q^{50} + 10 q^{52} + 2 q^{53} - 4 q^{55} - 28 q^{57} + 2 q^{58} + 4 q^{59} + 14 q^{60} + 18 q^{61} - 16 q^{62} + 4 q^{64} - 10 q^{65} + 8 q^{67} + 12 q^{68} - 28 q^{69} + 28 q^{71} + 8 q^{72} + 6 q^{73} + 2 q^{74} - 28 q^{75} + 12 q^{76} + 2 q^{80} + 10 q^{81} - 6 q^{82} - 32 q^{83} + 24 q^{85} - 8 q^{86} + 28 q^{87} + 2 q^{88} - 8 q^{89} + 16 q^{90} - 4 q^{92} - 16 q^{94} - 8 q^{95} + 44 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{5}^{4} - 2T_{5}^{3} + 10T_{5}^{2} + 12T_{5} + 36 \) Copy content Toggle raw display
\( T_{13} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

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