Properties

Label 1078.2.e.b
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.

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

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 −1.00000 1.73205i 0 0 1.00000 1.50000 + 2.59808i −1.00000 + 1.73205i
177.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 + 1.73205i 0 0 1.00000 1.50000 2.59808i −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.b 2
7.b odd 2 1 1078.2.e.c 2
7.c even 3 1 154.2.a.c 1
7.c even 3 1 inner 1078.2.e.b 2
7.d odd 6 1 1078.2.a.j 1
7.d odd 6 1 1078.2.e.c 2
21.g even 6 1 9702.2.a.v 1
21.h odd 6 1 1386.2.a.b 1
28.f even 6 1 8624.2.a.o 1
28.g odd 6 1 1232.2.a.h 1
35.j even 6 1 3850.2.a.f 1
35.l odd 12 2 3850.2.c.l 2
56.k odd 6 1 4928.2.a.o 1
56.p even 6 1 4928.2.a.n 1
77.h odd 6 1 1694.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.c 1 7.c even 3 1
1078.2.a.j 1 7.d odd 6 1
1078.2.e.b 2 1.a even 1 1 trivial
1078.2.e.b 2 7.c even 3 1 inner
1078.2.e.c 2 7.b odd 2 1
1078.2.e.c 2 7.d odd 6 1
1232.2.a.h 1 28.g odd 6 1
1386.2.a.b 1 21.h odd 6 1
1694.2.a.c 1 77.h odd 6 1
3850.2.a.f 1 35.j even 6 1
3850.2.c.l 2 35.l odd 12 2
4928.2.a.n 1 56.p even 6 1
4928.2.a.o 1 56.k odd 6 1
8624.2.a.o 1 28.f even 6 1
9702.2.a.v 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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 64 - 8 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 64 - 8 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 64 + 8 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 100 + 10 T + T^{2} \)
$67$ \( 144 - 12 T + T^{2} \)
$71$ \( ( -16 + T )^{2} \)
$73$ \( 196 - 14 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( -10 + T )^{2} \)
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