Properties

Label 1078.2.e.o
Level $1078$
Weight $2$
Character orbit 1078.e
Analytic conductor $8.608$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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

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

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.500000 0.866025i −1.41421 + 2.44949i −0.500000 + 0.866025i 0 2.82843 0 1.00000 −2.50000 4.33013i 0
67.2 −0.500000 0.866025i 1.41421 2.44949i −0.500000 + 0.866025i 0 −2.82843 0 1.00000 −2.50000 4.33013i 0
177.1 −0.500000 + 0.866025i −1.41421 2.44949i −0.500000 0.866025i 0 2.82843 0 1.00000 −2.50000 + 4.33013i 0
177.2 −0.500000 + 0.866025i 1.41421 + 2.44949i −0.500000 0.866025i 0 −2.82843 0 1.00000 −2.50000 + 4.33013i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.o 4
7.b odd 2 1 inner 1078.2.e.o 4
7.c even 3 1 1078.2.a.v 2
7.c even 3 1 inner 1078.2.e.o 4
7.d odd 6 1 1078.2.a.v 2
7.d odd 6 1 inner 1078.2.e.o 4
21.g even 6 1 9702.2.a.co 2
21.h odd 6 1 9702.2.a.co 2
28.f even 6 1 8624.2.a.bz 2
28.g odd 6 1 8624.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.v 2 7.c even 3 1
1078.2.a.v 2 7.d odd 6 1
1078.2.e.o 4 1.a even 1 1 trivial
1078.2.e.o 4 7.b odd 2 1 inner
1078.2.e.o 4 7.c even 3 1 inner
1078.2.e.o 4 7.d odd 6 1 inner
8624.2.a.bz 2 28.f even 6 1
8624.2.a.bz 2 28.g odd 6 1
9702.2.a.co 2 21.g even 6 1
9702.2.a.co 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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

\( T_{3}^{4} + 8 T_{3}^{2} + 64 \)
\( T_{5} \)
\( T_{13}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 64 + 8 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( -18 + T^{2} )^{2} \)
$17$ \( 64 + 8 T^{2} + T^{4} \)
$19$ \( 324 + 18 T^{2} + T^{4} \)
$23$ \( ( 36 + 6 T + T^{2} )^{2} \)
$29$ \( ( 4 + T )^{4} \)
$31$ \( 2500 + 50 T^{2} + T^{4} \)
$37$ \( ( 4 + 2 T + T^{2} )^{2} \)
$41$ \( ( -8 + T^{2} )^{2} \)
$43$ \( ( -10 + T )^{4} \)
$47$ \( 26244 + 162 T^{2} + T^{4} \)
$53$ \( ( 4 + 2 T + T^{2} )^{2} \)
$59$ \( 16384 + 128 T^{2} + T^{4} \)
$61$ \( 9604 + 98 T^{2} + T^{4} \)
$67$ \( ( 64 + 8 T + T^{2} )^{2} \)
$71$ \( ( -16 + T )^{4} \)
$73$ \( 5184 + 72 T^{2} + T^{4} \)
$79$ \( ( 64 - 8 T + T^{2} )^{2} \)
$83$ \( ( -162 + T^{2} )^{2} \)
$89$ \( 2500 + 50 T^{2} + T^{4} \)
$97$ \( ( -50 + T^{2} )^{2} \)
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