Properties

Label 1078.2.e.h
Level $1078$
Weight $2$
Character orbit 1078.e
Analytic conductor $8.608$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(199\) \(981\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 1.73205i −2.00000 0 −1.00000 −0.500000 0.866025i 1.00000 1.73205i
177.1 0.500000 0.866025i −1.00000 1.73205i −0.500000 0.866025i −1.00000 + 1.73205i −2.00000 0 −1.00000 −0.500000 + 0.866025i 1.00000 + 1.73205i
\(n\): 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.h 2
7.b odd 2 1 1078.2.e.l 2
7.c even 3 1 154.2.a.b 1
7.c even 3 1 inner 1078.2.e.h 2
7.d odd 6 1 1078.2.a.b 1
7.d odd 6 1 1078.2.e.l 2
21.g even 6 1 9702.2.a.bz 1
21.h odd 6 1 1386.2.a.f 1
28.f even 6 1 8624.2.a.z 1
28.g odd 6 1 1232.2.a.c 1
35.j even 6 1 3850.2.a.o 1
35.l odd 12 2 3850.2.c.d 2
56.k odd 6 1 4928.2.a.bf 1
56.p even 6 1 4928.2.a.d 1
77.h odd 6 1 1694.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 7.c even 3 1
1078.2.a.b 1 7.d odd 6 1
1078.2.e.h 2 1.a even 1 1 trivial
1078.2.e.h 2 7.c even 3 1 inner
1078.2.e.l 2 7.b odd 2 1
1078.2.e.l 2 7.d odd 6 1
1232.2.a.c 1 28.g odd 6 1
1386.2.a.f 1 21.h odd 6 1
1694.2.a.i 1 77.h odd 6 1
3850.2.a.o 1 35.j even 6 1
3850.2.c.d 2 35.l odd 12 2
4928.2.a.d 1 56.p even 6 1
4928.2.a.bf 1 56.k odd 6 1
8624.2.a.z 1 28.f even 6 1
9702.2.a.bz 1 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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$53$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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