Properties

Label 880.2.m.g
Level $880$
Weight $2$
Character orbit 880.m
Analytic conductor $7.027$
Analytic rank $0$
Dimension $8$
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] = [880,2,Mod(879,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.879");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.m (of order \(2\), degree \(1\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + ( - \beta_{2} + 1) q^{5} - \beta_{4} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} + ( - \beta_{2} + 1) q^{5} - \beta_{4} q^{7} + 3 q^{9} + \beta_{6} q^{11} - \beta_1 q^{13} + ( - \beta_{7} - \beta_{6} - \beta_{5}) q^{15} + \beta_1 q^{17} + ( - \beta_{6} + \beta_{5}) q^{19} + \beta_{3} q^{21} - \beta_{7} q^{23} + ( - 2 \beta_{2} - 3) q^{25} + ( - \beta_{6} - \beta_{5}) q^{31} + (3 \beta_{2} - \beta_1) q^{33} + (2 \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{35} + 4 \beta_{2} q^{37} + (3 \beta_{6} - 3 \beta_{5}) q^{39} - \beta_{3} q^{41} - \beta_{4} q^{43} + ( - 3 \beta_{2} + 3) q^{45} - 3 \beta_{7} q^{47} - 13 q^{49} + ( - 3 \beta_{6} + 3 \beta_{5}) q^{51} - 2 \beta_{2} q^{53} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4}) q^{55} + 2 \beta_1 q^{57} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{59} - \beta_{3} q^{61} - 3 \beta_{4} q^{63} + ( - \beta_{3} - \beta_1) q^{65} - 3 \beta_{7} q^{67} + 6 q^{69} + \beta_1 q^{73} + (3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5}) q^{75} + (5 \beta_{2} + 2 \beta_1) q^{77} + ( - \beta_{6} + \beta_{5}) q^{79} - 9 q^{81} + \beta_{4} q^{83} + (\beta_{3} + \beta_1) q^{85} + 4 q^{89} + (5 \beta_{6} + 5 \beta_{5}) q^{91} - 6 \beta_{2} q^{93} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4}) q^{95} + 6 \beta_{2} q^{97} + 3 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 24 q^{9} - 24 q^{25} + 24 q^{45} - 104 q^{49} + 48 q^{69} - 72 q^{81} + 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 3\nu^{5} + 3\nu^{4} - \nu^{3} - 9\nu^{2} + 9\nu + 170 ) / 33 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{6} - 12\nu^{5} - 10\nu^{4} + 40\nu^{3} + 118\nu^{2} - 140\nu + 328 ) / 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 6\nu^{4} - 22\nu^{3} - 54\nu^{2} + 66\nu - 160 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} + 4\nu^{5} + 130\nu^{4} + 104\nu^{3} - 314\nu^{2} + 2636\nu - 1260 ) / 507 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -518\nu^{7} + 1813\nu^{6} + 969\nu^{5} - 6955\nu^{4} - 12493\nu^{3} + 26601\nu^{2} - 42499\nu + 16541 ) / 5577 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 62\nu^{7} - 217\nu^{6} - 69\nu^{5} + 715\nu^{4} + 1417\nu^{3} - 2949\nu^{2} + 6919\nu - 2939 ) / 507 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62\nu^{7} - 217\nu^{6} - 69\nu^{5} + 715\nu^{4} + 1417\nu^{3} - 2949\nu^{2} + 5905\nu - 2432 ) / 507 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + \beta_{3} + 6\beta_{2} - 2\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 19\beta_{6} + 11\beta_{5} - 18\beta_{4} + 3\beta_{3} + 18\beta_{2} - 6\beta _1 + 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} + 17\beta_{6} + 11\beta_{5} - 18\beta_{4} + 10\beta_{3} + 54\beta_{2} + 4\beta _1 - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 146\beta_{7} + 27\beta_{6} + 187\beta_{5} - 120\beta_{4} + 45\beta_{3} + 240\beta_{2} + 30\beta _1 - 136 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 228\beta_{7} - \beta_{6} + 253\beta_{5} - 135\beta_{4} + 48\beta_{3} + 261\beta_{2} + 144\beta _1 - 774 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1122\beta_{7} - 497\beta_{6} + 737\beta_{5} - 126\beta_{4} + 91\beta_{3} + 504\beta_{2} + 448\beta _1 - 2462 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
879.1
−1.84278 1.22474i
0.393289 1.22474i
0.393289 + 1.22474i
−1.84278 + 1.22474i
0.606711 + 1.22474i
2.84278 + 1.22474i
2.84278 1.22474i
0.606711 1.22474i
0 −2.44949 0 1.00000 2.00000i 0 4.47214i 0 3.00000 0
879.2 0 −2.44949 0 1.00000 2.00000i 0 4.47214i 0 3.00000 0
879.3 0 −2.44949 0 1.00000 + 2.00000i 0 4.47214i 0 3.00000 0
879.4 0 −2.44949 0 1.00000 + 2.00000i 0 4.47214i 0 3.00000 0
879.5 0 2.44949 0 1.00000 2.00000i 0 4.47214i 0 3.00000 0
879.6 0 2.44949 0 1.00000 2.00000i 0 4.47214i 0 3.00000 0
879.7 0 2.44949 0 1.00000 + 2.00000i 0 4.47214i 0 3.00000 0
879.8 0 2.44949 0 1.00000 + 2.00000i 0 4.47214i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 879.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
20.d odd 2 1 inner
44.c even 2 1 inner
55.d odd 2 1 inner
220.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.m.g 8
4.b odd 2 1 inner 880.2.m.g 8
5.b even 2 1 inner 880.2.m.g 8
11.b odd 2 1 inner 880.2.m.g 8
20.d odd 2 1 inner 880.2.m.g 8
44.c even 2 1 inner 880.2.m.g 8
55.d odd 2 1 inner 880.2.m.g 8
220.g even 2 1 inner 880.2.m.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.m.g 8 1.a even 1 1 trivial
880.2.m.g 8 4.b odd 2 1 inner
880.2.m.g 8 5.b even 2 1 inner
880.2.m.g 8 11.b odd 2 1 inner
880.2.m.g 8 20.d odd 2 1 inner
880.2.m.g 8 44.c even 2 1 inner
880.2.m.g 8 55.d odd 2 1 inner
880.2.m.g 8 220.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{2} + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 120)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 120)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$89$ \( (T - 4)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
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