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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1078,2,Mod(901,1078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1078.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{10} + \beta_{7} - \beta_{2}) q^{3} + \beta_{3} q^{4} + ( - \beta_{13} - \beta_{2}) q^{5} + ( - \beta_{15} - \beta_{11}) q^{6} + \beta_{5} q^{8} + (2 \beta_{14} - 2 \beta_{6} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{10} + \beta_{7} - \beta_{2}) q^{3} + \beta_{3} q^{4} + ( - \beta_{13} - \beta_{2}) q^{5} + ( - \beta_{15} - \beta_{11}) q^{6} + \beta_{5} q^{8} + (2 \beta_{14} - 2 \beta_{6} - \beta_{3} + 1) q^{9} + ( - \beta_{11} + \beta_{8} - \beta_{4}) q^{10} + ( - \beta_{6} - 3 \beta_{5} + 3 \beta_1) q^{11} + ( - \beta_{13} - \beta_{2}) q^{12} + 3 \beta_{15} q^{13} + (2 \beta_{14} + 4) q^{15} + (\beta_{3} - 1) q^{16} + (2 \beta_{9} - \beta_{5} + \beta_1) q^{18} - 3 \beta_{8} q^{19} + ( - \beta_{13} + \beta_{10} - \beta_{7}) q^{20} + ( - \beta_{12} + \beta_{9} + 3) q^{22} + (3 \beta_{14} - 3 \beta_{6} + 2 \beta_{3} - 2) q^{23} + ( - \beta_{11} + \beta_{8} - \beta_{4}) q^{24} + (2 \beta_{6} - \beta_{3}) q^{25} + 3 \beta_{2} q^{26} + ( - 2 \beta_{13} + 2 \beta_{10} + 2 \beta_{7}) q^{27} + (3 \beta_{12} - 3 \beta_{9} + 6 \beta_{5}) q^{29} + (2 \beta_{12} + 4 \beta_1) q^{30} + ( - 2 \beta_{10} + 3 \beta_{7} - 3 \beta_{2}) q^{31} + (\beta_{5} - \beta_1) q^{32} + ( - 3 \beta_{15} - 3 \beta_{8} + 3 \beta_{4} + 2 \beta_{2}) q^{33} + (2 \beta_{14} + 1) q^{36} + (6 \beta_{3} - 6) q^{37} - 3 \beta_{10} q^{38} + ( - 6 \beta_{9} + 6 \beta_{5} - 6 \beta_1) q^{39} + (\beta_{15} + \beta_{8} - \beta_{4}) q^{40} + 6 \beta_{15} q^{41} + ( - 3 \beta_{12} + 3 \beta_{9}) q^{43} + (\beta_{14} - \beta_{6} + 3 \beta_1) q^{44} + ( - \beta_{10} + 5 \beta_{7} - 5 \beta_{2}) q^{45} + (3 \beta_{9} + 2 \beta_{5} - 2 \beta_1) q^{46} + ( - \beta_{13} + 4 \beta_{2}) q^{47} + ( - \beta_{13} + \beta_{10} - \beta_{7}) q^{48} + (2 \beta_{12} - 2 \beta_{9} - \beta_{5}) q^{50} + 3 \beta_{4} q^{52} + (6 \beta_{6} + 4 \beta_{3}) q^{53} + ( - 2 \beta_{15} + 2 \beta_{8} + 2 \beta_{4}) q^{54} + ( - 3 \beta_{15} - 3 \beta_{11} + 2 \beta_{7}) q^{55} - 6 \beta_{5} q^{57} + ( - 3 \beta_{14} + 3 \beta_{6} + 6 \beta_{3} - 6) q^{58} + ( - \beta_{10} - 7 \beta_{7} + 7 \beta_{2}) q^{59} + (2 \beta_{6} + 4 \beta_{3}) q^{60} - 3 \beta_{8} q^{61} + ( - 3 \beta_{15} - 2 \beta_{11}) q^{62} - q^{64} + ( - 6 \beta_{12} - 6 \beta_1) q^{65} + ( - 3 \beta_{10} + 3 \beta_{7} + 2 \beta_{4} - 3 \beta_{2}) q^{66} + (4 \beta_{6} - 6 \beta_{3}) q^{67} + ( - 2 \beta_{13} + 2 \beta_{10} + 4 \beta_{7}) q^{69} + 2 q^{71} + (2 \beta_{12} + \beta_1) q^{72} + (6 \beta_{11} - 6 \beta_{8} - 6 \beta_{4}) q^{73} + (6 \beta_{5} - 6 \beta_1) q^{74} + (\beta_{13} - 3 \beta_{2}) q^{75} - 3 \beta_{11} q^{76} + ( - 6 \beta_{14} - 6) q^{78} - 6 \beta_{12} q^{79} + (\beta_{10} - \beta_{7} + \beta_{2}) q^{80} + (2 \beta_{6} + 3 \beta_{3}) q^{81} + 6 \beta_{2} q^{82} + ( - 3 \beta_{15} + 6 \beta_{11}) q^{83} + (3 \beta_{14} - 3 \beta_{6}) q^{86} + ( - 6 \beta_{11} + 6 \beta_{8} - 12 \beta_{4}) q^{87} + (\beta_{9} + 3 \beta_{3}) q^{88} + (4 \beta_{13} + 5 \beta_{2}) q^{89} + ( - 5 \beta_{15} - \beta_{11}) q^{90} + (3 \beta_{14} - 2) q^{92} + (6 \beta_{14} - 6 \beta_{6} - 10 \beta_{3} + 10) q^{93} + ( - \beta_{11} + \beta_{8} + 4 \beta_{4}) q^{94} + ( - 6 \beta_{5} + 6 \beta_1) q^{95} + (\beta_{15} + \beta_{8} - \beta_{4}) q^{96} + ( - 3 \beta_{13} + 3 \beta_{10} + 4 \beta_{7}) q^{97} + ( - \beta_{14} - 6 \beta_{12} + 6 \beta_{9} - 3 \beta_{5} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 8 q^{9} + 64 q^{15} - 8 q^{16} + 48 q^{22} - 16 q^{23} - 8 q^{25} + 16 q^{36} - 48 q^{37} + 32 q^{53} - 48 q^{58} + 32 q^{60} - 16 q^{64} - 48 q^{67} + 32 q^{71} - 96 q^{78} + 24 q^{81} + 24 q^{88} - 32 q^{92} + 80 q^{93} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{48}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{48}^{7} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{48}^{8} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{48}^{11} + \zeta_{48}^{5} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{48}^{12} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{48}^{7} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{48}^{11} + \zeta_{48}^{5} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( \zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( -\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{5} + \zeta_{48}^{3} \) Copy content Toggle raw display
\(\zeta_{48}\)\(=\) \( ( \beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{2}\)\(=\) \( ( \beta_{9} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{3}\)\(=\) \( ( \beta_{15} + \beta_{13} - \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{48}^{5}\)\(=\) \( ( \beta_{10} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{6}\)\(=\) \( ( \beta_{14} + \beta_{12} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{7}\)\(=\) \( ( -\beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{8}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{48}^{9}\)\(=\) \( ( \beta_{11} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{10}\)\(=\) \( ( -\beta_{14} + \beta_{12} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{11}\)\(=\) \( ( -\beta_{10} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{12}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\zeta_{48}^{13}\)\(=\) \( ( -\beta_{15} + \beta_{13} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{14}\)\(=\) \( ( -\beta_{9} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{15}\)\(=\) \( ( -\beta_{11} + \beta_{7} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(981\)
\(\chi(n)\) \(1 - \beta_{3}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 901.8
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 \( T_{3}^{8} - 8T_{3}^{6} + 56T_{3}^{4} - 64T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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