Properties

Label 1078.2.e.m
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
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.

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

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 −1.20711 + 2.09077i −0.500000 + 0.866025i −0.292893 0.507306i 2.41421 0 1.00000 −1.41421 2.44949i −0.292893 + 0.507306i
67.2 −0.500000 0.866025i 0.207107 0.358719i −0.500000 + 0.866025i −1.70711 2.95680i −0.414214 0 1.00000 1.41421 + 2.44949i −1.70711 + 2.95680i
177.1 −0.500000 + 0.866025i −1.20711 2.09077i −0.500000 0.866025i −0.292893 + 0.507306i 2.41421 0 1.00000 −1.41421 + 2.44949i −0.292893 0.507306i
177.2 −0.500000 + 0.866025i 0.207107 + 0.358719i −0.500000 0.866025i −1.70711 + 2.95680i −0.414214 0 1.00000 1.41421 2.44949i −1.70711 2.95680i
\(n\): 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.m 4
7.b odd 2 1 154.2.e.e 4
7.c even 3 1 1078.2.a.x 2
7.c even 3 1 inner 1078.2.e.m 4
7.d odd 6 1 154.2.e.e 4
7.d odd 6 1 1078.2.a.t 2
21.c even 2 1 1386.2.k.t 4
21.g even 6 1 1386.2.k.t 4
21.g even 6 1 9702.2.a.cx 2
21.h odd 6 1 9702.2.a.ch 2
28.d even 2 1 1232.2.q.f 4
28.f even 6 1 1232.2.q.f 4
28.f even 6 1 8624.2.a.cc 2
28.g odd 6 1 8624.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 7.b odd 2 1
154.2.e.e 4 7.d odd 6 1
1078.2.a.t 2 7.d odd 6 1
1078.2.a.x 2 7.c even 3 1
1078.2.e.m 4 1.a even 1 1 trivial
1078.2.e.m 4 7.c even 3 1 inner
1232.2.q.f 4 28.d even 2 1
1232.2.q.f 4 28.f even 6 1
1386.2.k.t 4 21.c even 2 1
1386.2.k.t 4 21.g even 6 1
8624.2.a.bh 2 28.g odd 6 1
8624.2.a.cc 2 28.f even 6 1
9702.2.a.ch 2 21.h odd 6 1
9702.2.a.cx 2 21.g even 6 1

Hecke kernels

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

\( T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1 \)
\( T_{5}^{4} + 4 T_{5}^{3} + 14 T_{5}^{2} + 8 T_{5} + 4 \)
\( T_{13}^{2} - 2 T_{13} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T + T^{2} )^{2} \)
$13$ \( ( -7 - 2 T + T^{2} )^{2} \)
$17$ \( 784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( 196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( ( -23 + 6 T + T^{2} )^{2} \)
$31$ \( ( 16 + 4 T + T^{2} )^{2} \)
$37$ \( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( ( 14 - 8 T + T^{2} )^{2} \)
$43$ \( ( -32 + T^{2} )^{2} \)
$47$ \( 4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 8836 + 376 T + 110 T^{2} - 4 T^{3} + T^{4} \)
$59$ \( 2209 + 658 T + 149 T^{2} + 14 T^{3} + T^{4} \)
$61$ \( 5329 - 1314 T + 251 T^{2} - 18 T^{3} + T^{4} \)
$67$ \( 961 + 434 T + 165 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( ( -34 + 8 T + T^{2} )^{2} \)
$73$ \( 3844 + 992 T + 194 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( 3969 - 1134 T + 261 T^{2} - 18 T^{3} + T^{4} \)
$83$ \( ( -196 + 4 T + T^{2} )^{2} \)
$89$ \( 3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( ( -7 - 2 T + T^{2} )^{2} \)
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