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: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
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.

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

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

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display

Character values

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

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

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

\( T_{3}^{4} - 2T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} + 8T_{5}^{2} - 8T_{5} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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