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: \( x^{2} - x + 1 \) 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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - 2 \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - 2 \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + ( - \zeta_{6} + 1) q^{11} + 2 q^{13} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + ( - 3 \zeta_{6} + 3) q^{18} + 2 q^{20} - q^{22} + 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 2 \zeta_{6} q^{26} - 2 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + (\zeta_{6} - 1) q^{32} + 2 q^{34} - 3 q^{36} + 2 \zeta_{6} q^{37} - 2 \zeta_{6} q^{40} + 10 q^{41} + 4 q^{43} + \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{45} + ( - 8 \zeta_{6} + 8) q^{46} - 8 \zeta_{6} q^{47} - q^{50} + (2 \zeta_{6} - 2) q^{52} + (6 \zeta_{6} - 6) q^{53} - 2 q^{55} + 2 \zeta_{6} q^{58} - 10 \zeta_{6} q^{61} - 8 q^{62} + q^{64} - 4 \zeta_{6} q^{65} + ( - 12 \zeta_{6} + 12) q^{67} - 2 \zeta_{6} q^{68} + 16 q^{71} + 3 \zeta_{6} q^{72} + ( - 14 \zeta_{6} + 14) q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + (2 \zeta_{6} - 2) q^{80} + (9 \zeta_{6} - 9) q^{81} - 10 \zeta_{6} q^{82} + 4 q^{85} - 4 \zeta_{6} q^{86} + ( - \zeta_{6} + 1) q^{88} + 6 \zeta_{6} q^{89} - 6 q^{90} - 8 q^{92} + (8 \zeta_{6} - 8) q^{94} + 10 q^{97} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} + 3 q^{9} - 2 q^{10} + q^{11} + 4 q^{13} - q^{16} - 2 q^{17} + 3 q^{18} + 4 q^{20} - 2 q^{22} + 8 q^{23} + q^{25} - 2 q^{26} - 4 q^{29} + 8 q^{31} - q^{32} + 4 q^{34} - 6 q^{36} + 2 q^{37} - 2 q^{40} + 20 q^{41} + 8 q^{43} + q^{44} + 6 q^{45} + 8 q^{46} - 8 q^{47} - 2 q^{50} - 2 q^{52} - 6 q^{53} - 4 q^{55} + 2 q^{58} - 10 q^{61} - 16 q^{62} + 2 q^{64} - 4 q^{65} + 12 q^{67} - 2 q^{68} + 32 q^{71} + 3 q^{72} + 14 q^{73} + 2 q^{74} - 2 q^{80} - 9 q^{81} - 10 q^{82} + 8 q^{85} - 4 q^{86} + q^{88} + 6 q^{89} - 12 q^{90} - 16 q^{92} - 8 q^{94} + 20 q^{97} + 6 q^{99}+O(q^{100}) \) 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 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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

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

Hecke characteristic polynomials

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