Properties

Label 1078.2.c.a
Level $1078$
Weight $2$
Character orbit 1078.c
Analytic conductor $8.608$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(199\) \(981\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1077.1
0.923880 + 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
−0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
1.00000i 2.61313i −1.00000 2.61313i −2.61313 0 1.00000i −3.82843 2.61313
1077.2 1.00000i 1.08239i −1.00000 1.08239i −1.08239 0 1.00000i 1.82843 1.08239
1077.3 1.00000i 1.08239i −1.00000 1.08239i 1.08239 0 1.00000i 1.82843 −1.08239
1077.4 1.00000i 2.61313i −1.00000 2.61313i 2.61313 0 1.00000i −3.82843 −2.61313
1077.5 1.00000i 2.61313i −1.00000 2.61313i 2.61313 0 1.00000i −3.82843 −2.61313
1077.6 1.00000i 1.08239i −1.00000 1.08239i 1.08239 0 1.00000i 1.82843 −1.08239
1077.7 1.00000i 1.08239i −1.00000 1.08239i −1.08239 0 1.00000i 1.82843 1.08239
1077.8 1.00000i 2.61313i −1.00000 2.61313i −2.61313 0 1.00000i −3.82843 2.61313
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1077.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 8 + 8 T^{2} + T^{4} )^{2} \)
$5$ \( ( 8 + 8 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 121 + 14 T^{2} + T^{4} )^{2} \)
$13$ \( ( 162 - 36 T^{2} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( 162 - 36 T^{2} + T^{4} )^{2} \)
$23$ \( ( -14 - 4 T + T^{2} )^{4} \)
$29$ \( ( 324 + 108 T^{2} + T^{4} )^{2} \)
$31$ \( ( 98 + 52 T^{2} + T^{4} )^{2} \)
$37$ \( ( -6 + T )^{8} \)
$41$ \( ( 2592 - 144 T^{2} + T^{4} )^{2} \)
$43$ \( ( 18 + T^{2} )^{4} \)
$47$ \( ( 1058 + 68 T^{2} + T^{4} )^{2} \)
$53$ \( ( -56 + 8 T + T^{2} )^{4} \)
$59$ \( ( 7688 + 200 T^{2} + T^{4} )^{2} \)
$61$ \( ( 162 - 36 T^{2} + T^{4} )^{2} \)
$67$ \( ( 4 - 12 T + T^{2} )^{4} \)
$71$ \( ( -2 + T )^{8} \)
$73$ \( ( 10368 - 288 T^{2} + T^{4} )^{2} \)
$79$ \( ( 72 + T^{2} )^{4} \)
$83$ \( ( 162 - 180 T^{2} + T^{4} )^{2} \)
$89$ \( ( 1922 + 164 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1922 + 100 T^{2} + T^{4} )^{2} \)
show more
show less