Properties

Label 1456.2.s.q
Level $1456$
Weight $2$
Character orbit 1456.s
Analytic conductor $11.626$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(113,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.s (of order \(3\), 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{5}) q^{3} + ( - \beta_{3} - 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{5}) q^{3} + ( - \beta_{3} - 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4}) q^{9} + ( - \beta_{6} + \beta_{5}) q^{11} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{13} + ( - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{15} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{5} + 3 \beta_1) q^{19} - \beta_{6} q^{21} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{23} + (3 \beta_{3} - \beta_{2} + 2) q^{25} + ( - 3 \beta_{6} - \beta_{3} + 7) q^{27} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{2}) q^{29} + ( - \beta_{2} + 1) q^{31} + (\beta_{7} - \beta_{5} - 5 \beta_{4}) q^{33} + ( - 2 \beta_{4} - \beta_1) q^{35} + (\beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{37} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 4) q^{39} + ( - \beta_{7} + 6 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{41} + ( - \beta_{4} - \beta_1) q^{43} + (2 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{45} + ( - \beta_{6} - 3 \beta_{3} - 1) q^{47} + ( - \beta_{4} - 1) q^{49} + ( - \beta_{6} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{51} + ( - 4 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 1) q^{53} + (2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{55} + ( - \beta_{6} - 4 \beta_{3} - 3 \beta_{2} - 5) q^{57} + (\beta_{7} - 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{59} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{61} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - 2) q^{63} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{65} + ( - \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - \beta_{4} + 6 \beta_{3} + \beta_{2} + 6 \beta_1 - 1) q^{67} + (2 \beta_{7} + 2 \beta_{5} - 4 \beta_{4}) q^{69} + (2 \beta_{7} + 2 \beta_{5} + 4 \beta_{4}) q^{71} + ( - 2 \beta_{6} - 2 \beta_{3} - 3 \beta_{2} - 2) q^{73} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{75} + \beta_{6} q^{77} + (3 \beta_{6} + \beta_{3} - \beta_{2} + 6) q^{79} + (7 \beta_{6} - 7 \beta_{5} - 8 \beta_{4} - \beta_{3} - \beta_1 - 8) q^{81} + (4 \beta_{3} + \beta_{2} + 1) q^{83} + (\beta_{7} + \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{85} + (2 \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - \beta_1) q^{87} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{89} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{91} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{7} + 3 \beta_{5} - 5 \beta_{4} - 2 \beta_1) q^{95} + ( - \beta_{7} - 5 \beta_{5} - \beta_{4} - 2 \beta_1) q^{97} + (6 \beta_{6} + \beta_{3} - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 14 q^{5} - 4 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 14 q^{5} - 4 q^{7} - 7 q^{9} - q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{17} + q^{19} - 2 q^{21} - 2 q^{23} + 10 q^{25} + 52 q^{27} - q^{29} + 8 q^{31} + 19 q^{33} + 7 q^{35} + 10 q^{37} - 20 q^{39} + 22 q^{41} + 3 q^{43} + 11 q^{45} - 4 q^{47} - 4 q^{49} - 14 q^{51} + 4 q^{53} - 3 q^{55} - 34 q^{57} - 8 q^{59} - 8 q^{61} - 7 q^{63} + 7 q^{65} - 6 q^{67} + 18 q^{69} - 14 q^{71} - 16 q^{73} - 7 q^{75} + 2 q^{77} + 52 q^{79} - 24 q^{81} - 5 q^{85} + 13 q^{87} + q^{89} + 4 q^{91} + 7 q^{93} + 21 q^{95} - 3 q^{97} - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -12\nu^{7} - 7\nu^{6} - 76\nu^{5} - 44\nu^{4} - 602\nu^{3} - 36\nu^{2} - 8\nu + 1249 ) / 458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\nu^{7} - 24\nu^{6} + 361\nu^{5} + 209\nu^{4} + 2287\nu^{3} + 171\nu^{2} + 38\nu + 193 ) / 916 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 193\nu^{7} - 250\nu^{6} + 1375\nu^{5} - 361\nu^{4} + 7125\nu^{3} - 5375\nu^{2} + 2724\nu - 375 ) / 916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 174\nu^{7} - 242\nu^{6} + 1331\nu^{5} - 507\nu^{4} + 6897\nu^{3} - 5203\nu^{2} + 5383\nu - 363 ) / 458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -375\nu^{7} + 182\nu^{6} - 2375\nu^{5} - 1375\nu^{4} - 13889\nu^{3} - 1125\nu^{2} - 250\nu - 2933 ) / 916 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -273\nu^{7} + 356\nu^{6} - 1958\nu^{5} + 602\nu^{4} - 10146\nu^{3} + 7654\nu^{2} - 3617\nu + 534 ) / 458 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - 3\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 7\beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + \beta_{5} + 18\beta_{4} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{7} - 7\beta_{6} + 7\beta_{5} + 15\beta_{4} - 48\beta_{3} - 10\beta_{2} - 48\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{6} - 86\beta_{3} - 48\beta_{2} + 117 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -86\beta_{7} - 48\beta_{5} - 152\beta_{4} + 337\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
−0.115680 0.200364i
−1.11000 1.92258i
1.37054 + 2.37385i
0.355143 + 0.615126i
−0.115680 + 0.200364i
−1.11000 + 1.92258i
1.37054 2.37385i
0.355143 0.615126i
0 −1.66113 + 2.87716i 0 −2.23136 0 −0.500000 0.866025i 0 −4.01868 6.96056i 0
113.2 0 0.274776 0.475925i 0 −4.22001 0 −0.500000 0.866025i 0 1.34900 + 2.33653i 0
113.3 0 0.682410 1.18197i 0 0.741082 0 −0.500000 0.866025i 0 0.568634 + 0.984903i 0
113.4 0 1.20394 2.08529i 0 −1.28971 0 −0.500000 0.866025i 0 −1.39895 2.42305i 0
1121.1 0 −1.66113 2.87716i 0 −2.23136 0 −0.500000 + 0.866025i 0 −4.01868 + 6.96056i 0
1121.2 0 0.274776 + 0.475925i 0 −4.22001 0 −0.500000 + 0.866025i 0 1.34900 2.33653i 0
1121.3 0 0.682410 + 1.18197i 0 0.741082 0 −0.500000 + 0.866025i 0 0.568634 0.984903i 0
1121.4 0 1.20394 + 2.08529i 0 −1.28971 0 −0.500000 + 0.866025i 0 −1.39895 + 2.42305i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.s.q 8
4.b odd 2 1 91.2.f.c 8
12.b even 2 1 819.2.o.h 8
13.c even 3 1 inner 1456.2.s.q 8
28.d even 2 1 637.2.f.i 8
28.f even 6 1 637.2.g.j 8
28.f even 6 1 637.2.h.i 8
28.g odd 6 1 637.2.g.k 8
28.g odd 6 1 637.2.h.h 8
52.i odd 6 1 1183.2.a.l 4
52.j odd 6 1 91.2.f.c 8
52.j odd 6 1 1183.2.a.k 4
52.l even 12 2 1183.2.c.g 8
156.p even 6 1 819.2.o.h 8
364.q odd 6 1 637.2.h.h 8
364.v even 6 1 637.2.f.i 8
364.v even 6 1 8281.2.a.bp 4
364.ba even 6 1 637.2.g.j 8
364.bc even 6 1 8281.2.a.bt 4
364.bi odd 6 1 637.2.g.k 8
364.br even 6 1 637.2.h.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 4.b odd 2 1
91.2.f.c 8 52.j odd 6 1
637.2.f.i 8 28.d even 2 1
637.2.f.i 8 364.v even 6 1
637.2.g.j 8 28.f even 6 1
637.2.g.j 8 364.ba even 6 1
637.2.g.k 8 28.g odd 6 1
637.2.g.k 8 364.bi odd 6 1
637.2.h.h 8 28.g odd 6 1
637.2.h.h 8 364.q odd 6 1
637.2.h.i 8 28.f even 6 1
637.2.h.i 8 364.br even 6 1
819.2.o.h 8 12.b even 2 1
819.2.o.h 8 156.p even 6 1
1183.2.a.k 4 52.j odd 6 1
1183.2.a.l 4 52.i odd 6 1
1183.2.c.g 8 52.l even 12 2
1456.2.s.q 8 1.a even 1 1 trivial
1456.2.s.q 8 13.c even 3 1 inner
8281.2.a.bp 4 364.v even 6 1
8281.2.a.bt 4 364.bc even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\):

\( T_{3}^{8} - T_{3}^{7} + 10T_{3}^{6} - 23T_{3}^{5} + 103T_{3}^{4} - 156T_{3}^{3} + 202T_{3}^{2} - 96T_{3} + 36 \) Copy content Toggle raw display
\( T_{5}^{4} + 7T_{5}^{3} + 12T_{5}^{2} - T_{5} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + 10 T^{6} - 23 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( (T^{4} + 7 T^{3} + 12 T^{2} - T - 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + 10 T^{6} + 23 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + 16 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + 28 T^{6} + \cdots + 2809 \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} + 56 T^{6} + \cdots + 250000 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + 42 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + 23 T^{6} - 64 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 13 T^{2} + 26 T + 54)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 10 T^{7} + 83 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} - 22 T^{7} + 335 T^{6} + \cdots + 318096 \) Copy content Toggle raw display
$43$ \( T^{8} - 3 T^{7} + 12 T^{6} - 7 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( (T^{4} + 2 T^{3} - 51 T^{2} + 12 T + 100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} - 140 T^{2} + 890 T - 1389)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + 141 T^{6} + \cdots + 498436 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + 79 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} + 247 T^{6} + \cdots + 121220100 \) Copy content Toggle raw display
$71$ \( T^{8} + 14 T^{7} + 212 T^{6} + \cdots + 4129024 \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} - 119 T^{2} - 88 T + 1772)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 26 T^{3} + 134 T^{2} + 1024 T - 7680)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 97 T^{2} + 442 T - 426)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} + 72 T^{6} - 297 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{8} + 3 T^{7} + 314 T^{6} + \cdots + 246238864 \) Copy content Toggle raw display
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