Properties

Label 1078.2.e.q
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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu - 1 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} + 6 \nu - 1 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2\)

Character values

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

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

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
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
−0.500000 0.866025i −0.618034 + 1.07047i −0.500000 + 0.866025i 0.618034 + 1.07047i 1.23607 0 1.00000 0.736068 + 1.27491i 0.618034 1.07047i
67.2 −0.500000 0.866025i 1.61803 2.80252i −0.500000 + 0.866025i −1.61803 2.80252i −3.23607 0 1.00000 −3.73607 6.47106i −1.61803 + 2.80252i
177.1 −0.500000 + 0.866025i −0.618034 1.07047i −0.500000 0.866025i 0.618034 1.07047i 1.23607 0 1.00000 0.736068 1.27491i 0.618034 + 1.07047i
177.2 −0.500000 + 0.866025i 1.61803 + 2.80252i −0.500000 0.866025i −1.61803 + 2.80252i −3.23607 0 1.00000 −3.73607 + 6.47106i −1.61803 2.80252i
\(n\): 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.q 4
7.b odd 2 1 1078.2.e.n 4
7.c even 3 1 154.2.a.d 2
7.c even 3 1 inner 1078.2.e.q 4
7.d odd 6 1 1078.2.a.w 2
7.d odd 6 1 1078.2.e.n 4
21.g even 6 1 9702.2.a.cu 2
21.h odd 6 1 1386.2.a.m 2
28.f even 6 1 8624.2.a.bf 2
28.g odd 6 1 1232.2.a.p 2
35.j even 6 1 3850.2.a.bj 2
35.l odd 12 2 3850.2.c.q 4
56.k odd 6 1 4928.2.a.bk 2
56.p even 6 1 4928.2.a.bt 2
77.h odd 6 1 1694.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 7.c even 3 1
1078.2.a.w 2 7.d odd 6 1
1078.2.e.n 4 7.b odd 2 1
1078.2.e.n 4 7.d odd 6 1
1078.2.e.q 4 1.a even 1 1 trivial
1078.2.e.q 4 7.c even 3 1 inner
1232.2.a.p 2 28.g odd 6 1
1386.2.a.m 2 21.h odd 6 1
1694.2.a.l 2 77.h odd 6 1
3850.2.a.bj 2 35.j even 6 1
3850.2.c.q 4 35.l odd 12 2
4928.2.a.bk 2 56.k odd 6 1
4928.2.a.bt 2 56.p even 6 1
8624.2.a.bf 2 28.f even 6 1
9702.2.a.cu 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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

\( T_{3}^{4} - 2 T_{3}^{3} + 8 T_{3}^{2} + 8 T_{3} + 16 \)
\( T_{5}^{4} + 2 T_{5}^{3} + 8 T_{5}^{2} - 8 T_{5} + 16 \)
\( T_{13}^{2} + 2 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( 16 - 8 T + 8 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T + T^{2} )^{2} \)
$13$ \( ( -4 + 2 T + T^{2} )^{2} \)
$17$ \( 256 + 64 T + 32 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 400 - 200 T + 80 T^{2} - 10 T^{3} + T^{4} \)
$23$ \( ( 16 + 4 T + T^{2} )^{2} \)
$29$ \( ( -20 + T^{2} )^{2} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( 5776 + 304 T + 92 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( -16 - 4 T + T^{2} )^{2} \)
$43$ \( ( 16 + 12 T + T^{2} )^{2} \)
$47$ \( ( 4 - 2 T + T^{2} )^{2} \)
$53$ \( 16 - 32 T + 68 T^{2} + 8 T^{3} + T^{4} \)
$59$ \( 400 + 200 T + 80 T^{2} + 10 T^{3} + T^{4} \)
$61$ \( 16 - 24 T + 32 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( 30976 + 704 T + 192 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( ( -16 - 4 T + T^{2} )^{2} \)
$73$ \( 4096 - 512 T + 128 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( -124 + 2 T + T^{2} )^{2} \)
$89$ \( ( 100 + 10 T + T^{2} )^{2} \)
$97$ \( ( 44 - 16 T + T^{2} )^{2} \)
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