Properties

Label 1078.2.e.j
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: \(x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 4 \zeta_{6} q^{5} - q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 4 \zeta_{6} q^{5} - q^{8} + 3 \zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{11} + 2 q^{13} -\zeta_{6} q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + ( -3 + 3 \zeta_{6} ) q^{18} + 6 \zeta_{6} q^{19} -4 q^{20} + q^{22} -4 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} -2 q^{29} + ( 2 - 2 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 4 q^{34} -3 q^{36} -10 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} -4 \zeta_{6} q^{40} + 4 q^{41} -8 q^{43} + \zeta_{6} q^{44} + ( -12 + 12 \zeta_{6} ) q^{45} + ( 4 - 4 \zeta_{6} ) q^{46} -2 \zeta_{6} q^{47} -11 q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + 4 q^{55} -2 \zeta_{6} q^{58} + ( 12 - 12 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} + 2 q^{62} + q^{64} + 8 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{67} + 4 \zeta_{6} q^{68} -8 q^{71} -3 \zeta_{6} q^{72} + ( -4 + 4 \zeta_{6} ) q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} -6 q^{76} + ( 4 - 4 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + 4 \zeta_{6} q^{82} -6 q^{83} + 16 q^{85} -8 \zeta_{6} q^{86} + ( -1 + \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} -12 q^{90} + 4 q^{92} + ( 2 - 2 \zeta_{6} ) q^{94} + ( -24 + 24 \zeta_{6} ) q^{95} -14 q^{97} + 3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 4 q^{5} - 2 q^{8} + 3 q^{9} + O(q^{10}) \) \( 2 q + q^{2} - q^{4} + 4 q^{5} - 2 q^{8} + 3 q^{9} - 4 q^{10} + q^{11} + 4 q^{13} - q^{16} + 4 q^{17} - 3 q^{18} + 6 q^{19} - 8 q^{20} + 2 q^{22} - 4 q^{23} - 11 q^{25} + 2 q^{26} - 4 q^{29} + 2 q^{31} + q^{32} + 8 q^{34} - 6 q^{36} - 10 q^{37} - 6 q^{38} - 4 q^{40} + 8 q^{41} - 16 q^{43} + q^{44} - 12 q^{45} + 4 q^{46} - 2 q^{47} - 22 q^{50} - 2 q^{52} - 6 q^{53} + 8 q^{55} - 2 q^{58} + 12 q^{59} + 14 q^{61} + 4 q^{62} + 2 q^{64} + 8 q^{65} + 12 q^{67} + 4 q^{68} - 16 q^{71} - 3 q^{72} - 4 q^{73} + 10 q^{74} - 12 q^{76} + 4 q^{80} - 9 q^{81} + 4 q^{82} - 12 q^{83} + 32 q^{85} - 8 q^{86} - q^{88} + 6 q^{89} - 24 q^{90} + 8 q^{92} + 2 q^{94} - 24 q^{95} - 28 q^{97} + 6 q^{99} + O(q^{100}) \)

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 0 −0.500000 + 0.866025i 2.00000 + 3.46410i 0 0 −1.00000 1.50000 + 2.59808i −2.00000 + 3.46410i
177.1 0.500000 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i 0 0 −1.00000 1.50000 2.59808i −2.00000 3.46410i
\(n\): 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.j 2
7.b odd 2 1 1078.2.e.i 2
7.c even 3 1 154.2.a.a 1
7.c even 3 1 inner 1078.2.e.j 2
7.d odd 6 1 1078.2.a.d 1
7.d odd 6 1 1078.2.e.i 2
21.g even 6 1 9702.2.a.ba 1
21.h odd 6 1 1386.2.a.l 1
28.f even 6 1 8624.2.a.r 1
28.g odd 6 1 1232.2.a.e 1
35.j even 6 1 3850.2.a.u 1
35.l odd 12 2 3850.2.c.j 2
56.k odd 6 1 4928.2.a.w 1
56.p even 6 1 4928.2.a.v 1
77.h odd 6 1 1694.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.a 1 7.c even 3 1
1078.2.a.d 1 7.d odd 6 1
1078.2.e.i 2 7.b odd 2 1
1078.2.e.i 2 7.d odd 6 1
1078.2.e.j 2 1.a even 1 1 trivial
1078.2.e.j 2 7.c even 3 1 inner
1232.2.a.e 1 28.g odd 6 1
1386.2.a.l 1 21.h odd 6 1
1694.2.a.g 1 77.h odd 6 1
3850.2.a.u 1 35.j even 6 1
3850.2.c.j 2 35.l odd 12 2
4928.2.a.v 1 56.p even 6 1
4928.2.a.w 1 56.k odd 6 1
8624.2.a.r 1 28.f even 6 1
9702.2.a.ba 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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 4 - 2 T + T^{2} \)
$37$ \( 100 + 10 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( 4 + 2 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 196 - 14 T + T^{2} \)
$67$ \( 144 - 12 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 16 + 4 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( 14 + T )^{2} \)
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