Properties

Label 1078.2.i.a
Level $1078$
Weight $2$
Character orbit 1078.i
Analytic conductor $8.608$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{48}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{48}^{4} q^{2} + ( -\zeta_{48} - \zeta_{48}^{5} - \zeta_{48}^{7} + \zeta_{48}^{9} + \zeta_{48}^{11} + \zeta_{48}^{15} ) q^{3} + \zeta_{48}^{8} q^{4} + ( -\zeta_{48} - \zeta_{48}^{3} - \zeta_{48}^{7} + \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{5} + ( -\zeta_{48}^{3} - \zeta_{48}^{5} - \zeta_{48}^{9} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{6} + \zeta_{48}^{12} q^{8} + ( 1 + 2 \zeta_{48}^{6} - \zeta_{48}^{8} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{9} +O(q^{10})\) \( q + \zeta_{48}^{4} q^{2} + ( -\zeta_{48} - \zeta_{48}^{5} - \zeta_{48}^{7} + \zeta_{48}^{9} + \zeta_{48}^{11} + \zeta_{48}^{15} ) q^{3} + \zeta_{48}^{8} q^{4} + ( -\zeta_{48} - \zeta_{48}^{3} - \zeta_{48}^{7} + \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{5} + ( -\zeta_{48}^{3} - \zeta_{48}^{5} - \zeta_{48}^{9} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{6} + \zeta_{48}^{12} q^{8} + ( 1 + 2 \zeta_{48}^{6} - \zeta_{48}^{8} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{9} + ( \zeta_{48} - \zeta_{48}^{5} - \zeta_{48}^{7} - \zeta_{48}^{9} - \zeta_{48}^{11} + \zeta_{48}^{15} ) q^{10} + ( -\zeta_{48}^{2} + 3 \zeta_{48}^{4} - 3 \zeta_{48}^{12} - \zeta_{48}^{14} ) q^{11} + ( -\zeta_{48} - \zeta_{48}^{3} - \zeta_{48}^{7} + \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{12} + ( 3 \zeta_{48}^{3} + 3 \zeta_{48}^{5} - 3 \zeta_{48}^{13} ) q^{13} + ( 4 + 2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} ) q^{15} + ( -1 + \zeta_{48}^{8} ) q^{16} + ( 2 \zeta_{48}^{2} + \zeta_{48}^{4} - \zeta_{48}^{12} - 2 \zeta_{48}^{14} ) q^{18} + ( -3 \zeta_{48} + 3 \zeta_{48}^{7} ) q^{19} + ( -\zeta_{48}^{3} + \zeta_{48}^{5} - \zeta_{48}^{9} - \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{20} + ( 3 + \zeta_{48}^{2} - \zeta_{48}^{6} - \zeta_{48}^{10} ) q^{22} + ( -2 + 3 \zeta_{48}^{6} + 2 \zeta_{48}^{8} - 3 \zeta_{48}^{10} - 3 \zeta_{48}^{14} ) q^{23} + ( \zeta_{48} - \zeta_{48}^{5} - \zeta_{48}^{7} - \zeta_{48}^{9} - \zeta_{48}^{11} + \zeta_{48}^{15} ) q^{24} + ( 2 \zeta_{48}^{2} - \zeta_{48}^{8} + 2 \zeta_{48}^{14} ) q^{25} + ( 3 \zeta_{48} + 3 \zeta_{48}^{7} ) q^{26} + ( -2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} + 2 \zeta_{48}^{9} - 2 \zeta_{48}^{13} + 2 \zeta_{48}^{15} ) q^{27} + ( -3 \zeta_{48}^{2} + 3 \zeta_{48}^{6} + 3 \zeta_{48}^{10} + 6 \zeta_{48}^{12} ) q^{29} + ( 4 \zeta_{48}^{4} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{30} + ( -3 \zeta_{48} - 2 \zeta_{48}^{5} - 3 \zeta_{48}^{7} + 3 \zeta_{48}^{9} + 2 \zeta_{48}^{11} + 3 \zeta_{48}^{15} ) q^{31} + ( -\zeta_{48}^{4} + \zeta_{48}^{12} ) q^{32} + ( -\zeta_{48} - 3 \zeta_{48}^{3} + 5 \zeta_{48}^{7} + 3 \zeta_{48}^{11} + 3 \zeta_{48}^{13} ) q^{33} + ( 1 + 2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} ) q^{36} + ( -6 + 6 \zeta_{48}^{8} ) q^{37} + ( -3 \zeta_{48}^{5} + 3 \zeta_{48}^{11} ) q^{38} + ( -6 \zeta_{48}^{2} - 6 \zeta_{48}^{4} + 6 \zeta_{48}^{12} + 6 \zeta_{48}^{14} ) q^{39} + ( \zeta_{48} + \zeta_{48}^{3} - \zeta_{48}^{7} - \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{40} + ( 6 \zeta_{48}^{3} + 6 \zeta_{48}^{5} - 6 \zeta_{48}^{13} ) q^{41} + ( 3 \zeta_{48}^{2} - 3 \zeta_{48}^{6} - 3 \zeta_{48}^{10} ) q^{43} + ( 3 \zeta_{48}^{4} + \zeta_{48}^{6} - \zeta_{48}^{10} - \zeta_{48}^{14} ) q^{44} + ( -5 \zeta_{48} - \zeta_{48}^{5} - 5 \zeta_{48}^{7} + 5 \zeta_{48}^{9} + \zeta_{48}^{11} + 5 \zeta_{48}^{15} ) q^{45} + ( 3 \zeta_{48}^{2} - 2 \zeta_{48}^{4} + 2 \zeta_{48}^{12} - 3 \zeta_{48}^{14} ) q^{46} + ( 4 \zeta_{48} - \zeta_{48}^{3} + 4 \zeta_{48}^{7} + \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{47} + ( -\zeta_{48}^{3} + \zeta_{48}^{5} - \zeta_{48}^{9} - \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{48} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} - \zeta_{48}^{12} ) q^{50} + ( 3 \zeta_{48}^{5} + 3 \zeta_{48}^{11} ) q^{52} + ( 6 \zeta_{48}^{2} + 4 \zeta_{48}^{8} + 6 \zeta_{48}^{14} ) q^{53} + ( 2 \zeta_{48} - 2 \zeta_{48}^{3} - 2 \zeta_{48}^{7} + 2 \zeta_{48}^{11} + 2 \zeta_{48}^{13} ) q^{54} + ( -3 \zeta_{48}^{3} - 3 \zeta_{48}^{5} - \zeta_{48}^{9} + 3 \zeta_{48}^{13} + 5 \zeta_{48}^{15} ) q^{55} -6 \zeta_{48}^{12} q^{57} + ( -6 - 3 \zeta_{48}^{6} + 6 \zeta_{48}^{8} + 3 \zeta_{48}^{10} + 3 \zeta_{48}^{14} ) q^{58} + ( 7 \zeta_{48} - \zeta_{48}^{5} + 7 \zeta_{48}^{7} - 7 \zeta_{48}^{9} + \zeta_{48}^{11} - 7 \zeta_{48}^{15} ) q^{59} + ( 2 \zeta_{48}^{2} + 4 \zeta_{48}^{8} + 2 \zeta_{48}^{14} ) q^{60} + ( -3 \zeta_{48} + 3 \zeta_{48}^{7} ) q^{61} + ( -3 \zeta_{48}^{3} - 3 \zeta_{48}^{5} - 2 \zeta_{48}^{9} + 3 \zeta_{48}^{13} + 2 \zeta_{48}^{15} ) q^{62} - q^{64} + ( -6 \zeta_{48}^{4} - 6 \zeta_{48}^{6} - 6 \zeta_{48}^{10} + 6 \zeta_{48}^{14} ) q^{65} + ( -3 \zeta_{48} - \zeta_{48}^{5} - 3 \zeta_{48}^{7} + 3 \zeta_{48}^{9} + 5 \zeta_{48}^{11} + 3 \zeta_{48}^{15} ) q^{66} + ( 4 \zeta_{48}^{2} - 6 \zeta_{48}^{8} + 4 \zeta_{48}^{14} ) q^{67} + ( -2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} + 4 \zeta_{48}^{9} - 2 \zeta_{48}^{13} + 4 \zeta_{48}^{15} ) q^{69} + 2 q^{71} + ( \zeta_{48}^{4} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{72} + ( -6 \zeta_{48} - 6 \zeta_{48}^{5} + 6 \zeta_{48}^{7} + 6 \zeta_{48}^{9} - 6 \zeta_{48}^{11} - 6 \zeta_{48}^{15} ) q^{73} + ( -6 \zeta_{48}^{4} + 6 \zeta_{48}^{12} ) q^{74} + ( -3 \zeta_{48} + \zeta_{48}^{3} - 3 \zeta_{48}^{7} - \zeta_{48}^{11} + \zeta_{48}^{13} ) q^{75} + ( -3 \zeta_{48}^{9} + 3 \zeta_{48}^{15} ) q^{76} + ( -6 - 6 \zeta_{48}^{2} - 6 \zeta_{48}^{6} + 6 \zeta_{48}^{10} ) q^{78} + ( -6 \zeta_{48}^{6} - 6 \zeta_{48}^{10} + 6 \zeta_{48}^{14} ) q^{79} + ( \zeta_{48} + \zeta_{48}^{5} + \zeta_{48}^{7} - \zeta_{48}^{9} - \zeta_{48}^{11} - \zeta_{48}^{15} ) q^{80} + ( 2 \zeta_{48}^{2} + 3 \zeta_{48}^{8} + 2 \zeta_{48}^{14} ) q^{81} + ( 6 \zeta_{48} + 6 \zeta_{48}^{7} ) q^{82} + ( -3 \zeta_{48}^{3} - 3 \zeta_{48}^{5} + 6 \zeta_{48}^{9} + 3 \zeta_{48}^{13} - 6 \zeta_{48}^{15} ) q^{83} + ( 3 \zeta_{48}^{6} - 3 \zeta_{48}^{10} - 3 \zeta_{48}^{14} ) q^{86} + ( 6 \zeta_{48} - 12 \zeta_{48}^{5} - 6 \zeta_{48}^{7} - 6 \zeta_{48}^{9} - 12 \zeta_{48}^{11} + 6 \zeta_{48}^{15} ) q^{87} + ( \zeta_{48}^{2} + 3 \zeta_{48}^{8} - \zeta_{48}^{14} ) q^{88} + ( 5 \zeta_{48} + 4 \zeta_{48}^{3} + 5 \zeta_{48}^{7} - 4 \zeta_{48}^{11} + 4 \zeta_{48}^{13} ) q^{89} + ( -5 \zeta_{48}^{3} - 5 \zeta_{48}^{5} - \zeta_{48}^{9} + 5 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{90} + ( -2 + 3 \zeta_{48}^{2} + 3 \zeta_{48}^{6} - 3 \zeta_{48}^{10} ) q^{92} + ( 10 + 6 \zeta_{48}^{6} - 10 \zeta_{48}^{8} - 6 \zeta_{48}^{10} - 6 \zeta_{48}^{14} ) q^{93} + ( \zeta_{48} + 4 \zeta_{48}^{5} - \zeta_{48}^{7} - \zeta_{48}^{9} + 4 \zeta_{48}^{11} + \zeta_{48}^{15} ) q^{94} + ( 6 \zeta_{48}^{4} - 6 \zeta_{48}^{12} ) q^{95} + ( \zeta_{48} + \zeta_{48}^{3} - \zeta_{48}^{7} - \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{96} + ( -3 \zeta_{48}^{3} + 3 \zeta_{48}^{5} + 4 \zeta_{48}^{9} - 3 \zeta_{48}^{13} + 4 \zeta_{48}^{15} ) q^{97} + ( -4 + 5 \zeta_{48}^{2} - 7 \zeta_{48}^{6} - 5 \zeta_{48}^{10} - 3 \zeta_{48}^{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 8 q^{9} + O(q^{10}) \) \( 16 q + 8 q^{4} + 8 q^{9} + 64 q^{15} - 8 q^{16} + 48 q^{22} - 16 q^{23} - 8 q^{25} + 16 q^{36} - 48 q^{37} + 32 q^{53} - 48 q^{58} + 32 q^{60} - 16 q^{64} - 48 q^{67} + 32 q^{71} - 96 q^{78} + 24 q^{81} + 24 q^{88} - 32 q^{92} + 80 q^{93} - 64 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(981\)
\(\chi(n)\) \(1 - \zeta_{48}^{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0.608761 + 0.793353i
−0.793353 + 0.608761i
0.793353 0.608761i
−0.608761 0.793353i
0.991445 + 0.130526i
0.130526 0.991445i
−0.130526 + 0.991445i
−0.991445 0.130526i
0.608761 0.793353i
−0.793353 0.608761i
0.793353 + 0.608761i
−0.608761 + 0.793353i
0.991445 0.130526i
0.130526 + 0.991445i
−0.130526 0.991445i
−0.991445 + 0.130526i
−0.866025 0.500000i −2.26303 + 1.30656i 0.500000 + 0.866025i −2.26303 1.30656i 2.61313 0 1.00000i 1.91421 3.31552i 1.30656 + 2.26303i
901.2 −0.866025 0.500000i −0.937379 + 0.541196i 0.500000 + 0.866025i −0.937379 0.541196i 1.08239 0 1.00000i −0.914214 + 1.58346i 0.541196 + 0.937379i
901.3 −0.866025 0.500000i 0.937379 0.541196i 0.500000 + 0.866025i 0.937379 + 0.541196i −1.08239 0 1.00000i −0.914214 + 1.58346i −0.541196 0.937379i
901.4 −0.866025 0.500000i 2.26303 1.30656i 0.500000 + 0.866025i 2.26303 + 1.30656i −2.61313 0 1.00000i 1.91421 3.31552i −1.30656 2.26303i
901.5 0.866025 + 0.500000i −2.26303 + 1.30656i 0.500000 + 0.866025i −2.26303 1.30656i −2.61313 0 1.00000i 1.91421 3.31552i −1.30656 2.26303i
901.6 0.866025 + 0.500000i −0.937379 + 0.541196i 0.500000 + 0.866025i −0.937379 0.541196i −1.08239 0 1.00000i −0.914214 + 1.58346i −0.541196 0.937379i
901.7 0.866025 + 0.500000i 0.937379 0.541196i 0.500000 + 0.866025i 0.937379 + 0.541196i 1.08239 0 1.00000i −0.914214 + 1.58346i 0.541196 + 0.937379i
901.8 0.866025 + 0.500000i 2.26303 1.30656i 0.500000 + 0.866025i 2.26303 + 1.30656i 2.61313 0 1.00000i 1.91421 3.31552i 1.30656 + 2.26303i
1011.1 −0.866025 + 0.500000i −2.26303 1.30656i 0.500000 0.866025i −2.26303 + 1.30656i 2.61313 0 1.00000i 1.91421 + 3.31552i 1.30656 2.26303i
1011.2 −0.866025 + 0.500000i −0.937379 0.541196i 0.500000 0.866025i −0.937379 + 0.541196i 1.08239 0 1.00000i −0.914214 1.58346i 0.541196 0.937379i
1011.3 −0.866025 + 0.500000i 0.937379 + 0.541196i 0.500000 0.866025i 0.937379 0.541196i −1.08239 0 1.00000i −0.914214 1.58346i −0.541196 + 0.937379i
1011.4 −0.866025 + 0.500000i 2.26303 + 1.30656i 0.500000 0.866025i 2.26303 1.30656i −2.61313 0 1.00000i 1.91421 + 3.31552i −1.30656 + 2.26303i
1011.5 0.866025 0.500000i −2.26303 1.30656i 0.500000 0.866025i −2.26303 + 1.30656i −2.61313 0 1.00000i 1.91421 + 3.31552i −1.30656 + 2.26303i
1011.6 0.866025 0.500000i −0.937379 0.541196i 0.500000 0.866025i −0.937379 + 0.541196i −1.08239 0 1.00000i −0.914214 1.58346i −0.541196 + 0.937379i
1011.7 0.866025 0.500000i 0.937379 + 0.541196i 0.500000 0.866025i 0.937379 0.541196i 1.08239 0 1.00000i −0.914214 1.58346i 0.541196 0.937379i
1011.8 0.866025 0.500000i 2.26303 + 1.30656i 0.500000 0.866025i 2.26303 1.30656i 2.61313 0 1.00000i 1.91421 + 3.31552i 1.30656 2.26303i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1011.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner
77.h odd 6 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.i.a 16
7.b odd 2 1 inner 1078.2.i.a 16
7.c even 3 1 1078.2.c.a 8
7.c even 3 1 inner 1078.2.i.a 16
7.d odd 6 1 1078.2.c.a 8
7.d odd 6 1 inner 1078.2.i.a 16
11.b odd 2 1 inner 1078.2.i.a 16
77.b even 2 1 inner 1078.2.i.a 16
77.h odd 6 1 1078.2.c.a 8
77.h odd 6 1 inner 1078.2.i.a 16
77.i even 6 1 1078.2.c.a 8
77.i even 6 1 inner 1078.2.i.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.c.a 8 7.c even 3 1
1078.2.c.a 8 7.d odd 6 1
1078.2.c.a 8 77.h odd 6 1
1078.2.c.a 8 77.i even 6 1
1078.2.i.a 16 1.a even 1 1 trivial
1078.2.i.a 16 7.b odd 2 1 inner
1078.2.i.a 16 7.c even 3 1 inner
1078.2.i.a 16 7.d odd 6 1 inner
1078.2.i.a 16 11.b odd 2 1 inner
1078.2.i.a 16 77.b even 2 1 inner
1078.2.i.a 16 77.h odd 6 1 inner
1078.2.i.a 16 77.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 8 T_{3}^{6} + 56 T_{3}^{4} - 64 T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( ( 64 - 64 T^{2} + 56 T^{4} - 8 T^{6} + T^{8} )^{2} \)
$5$ \( ( 64 - 64 T^{2} + 56 T^{4} - 8 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 14641 - 1694 T^{2} + 75 T^{4} - 14 T^{6} + T^{8} )^{2} \)
$13$ \( ( 162 - 36 T^{2} + T^{4} )^{4} \)
$17$ \( T^{16} \)
$19$ \( ( 26244 + 5832 T^{2} + 1134 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$23$ \( ( 196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$29$ \( ( 324 + 108 T^{2} + T^{4} )^{4} \)
$31$ \( ( 9604 - 5096 T^{2} + 2606 T^{4} - 52 T^{6} + T^{8} )^{2} \)
$37$ \( ( 36 + 6 T + T^{2} )^{8} \)
$41$ \( ( 2592 - 144 T^{2} + T^{4} )^{4} \)
$43$ \( ( 18 + T^{2} )^{8} \)
$47$ \( ( 1119364 - 71944 T^{2} + 3566 T^{4} - 68 T^{6} + T^{8} )^{2} \)
$53$ \( ( 3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$59$ \( ( 59105344 - 1537600 T^{2} + 32312 T^{4} - 200 T^{6} + T^{8} )^{2} \)
$61$ \( ( 26244 + 5832 T^{2} + 1134 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$67$ \( ( 16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4} )^{4} \)
$71$ \( ( -2 + T )^{16} \)
$73$ \( ( 107495424 + 2985984 T^{2} + 72576 T^{4} + 288 T^{6} + T^{8} )^{2} \)
$79$ \( ( 5184 - 72 T^{2} + T^{4} )^{4} \)
$83$ \( ( 162 - 180 T^{2} + T^{4} )^{4} \)
$89$ \( ( 3694084 - 315208 T^{2} + 24974 T^{4} - 164 T^{6} + T^{8} )^{2} \)
$97$ \( ( 1922 + 100 T^{2} + T^{4} )^{4} \)
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