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Properties

Label 1456.2.r.n
Level $1456$
Weight $2$
Character orbit 1456.r
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(417,1456)]
 
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, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (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: 8.0.8681953329.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 82x^{4} - 102x^{3} + 102x^{2} - 57x + 21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 182)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_1) q^{3} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{6} - \beta_{5} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_1) q^{3} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_{6} - \beta_{5} + 3 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 7 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 7 q^{7} - 6 q^{9} + q^{11} - 8 q^{13} - 18 q^{15} - 3 q^{17} + 3 q^{19} + 9 q^{21} + 8 q^{23} - 14 q^{25} + 18 q^{27} + 34 q^{29} + 7 q^{31} + 6 q^{33} - 8 q^{35} - 24 q^{41} - 20 q^{43} - 27 q^{45} - 5 q^{47} + 23 q^{49} - 15 q^{51} + 3 q^{53} + 44 q^{55} + 48 q^{57} + 17 q^{59} + 5 q^{61} + 2 q^{65} + q^{67} - 72 q^{69} - 14 q^{71} - 14 q^{73} + 12 q^{75} + 16 q^{77} + 14 q^{79} + 12 q^{81} + 6 q^{85} - 6 q^{87} - 6 q^{89} + 7 q^{91} + 12 q^{93} - 24 q^{95} - 8 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 82x^{4} - 102x^{3} + 102x^{2} - 57x + 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\nu^{7} - 35\nu^{6} + 138\nu^{5} - 233\nu^{4} + 336\nu^{3} - 166\nu^{2} - 92\nu + 119 ) / 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} + 130\nu^{5} - 255\nu^{4} + 504\nu^{3} - 515\nu^{2} + 387\nu - 140 ) / 49 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10\nu^{7} + 35\nu^{6} - 138\nu^{5} + 282\nu^{4} - 434\nu^{3} + 509\nu^{2} - 202\nu + 77 ) / 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 14\nu^{6} + 37\nu^{5} - 278\nu^{4} + 497\nu^{3} - 1038\nu^{2} + 841\nu - 511 ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 35\nu^{6} - 110\nu^{5} + 408\nu^{4} - 630\nu^{3} + 1118\nu^{2} - 776\nu + 469 ) / 49 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\nu^{7} - 77\nu^{6} + 333\nu^{5} - 640\nu^{4} + 1190\nu^{3} - 1208\nu^{2} + 905\nu - 336 ) / 49 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -2\beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3\beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 7\beta_{6} + 7\beta_{5} - 2\beta_{4} - 8\beta_{3} + 2\beta_{2} + 6\beta _1 - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} + 16\beta_{6} + 16\beta_{5} - 2\beta_{4} - 17\beta_{3} + 8\beta_{2} - 9\beta _1 + 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -15\beta_{7} - 20\beta_{6} - 20\beta_{5} + 13\beta_{4} + 40\beta_{3} + 2\beta_{2} - 45\beta _1 + 104 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -60\beta_{7} - 98\beta_{6} - 104\beta_{5} + 10\beta_{4} + 163\beta_{3} - 49\beta_{2} - 3\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 36\beta_{7} + 19\beta_{6} - 2\beta_{5} - 98\beta_{4} - 83\beta_{3} - 91\beta_{2} + 249\beta _1 - 565 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
0.500000 − 2.38492i
0.500000 + 1.95148i
0.500000 + 1.05635i
0.500000 − 0.622917i
0.500000 + 2.38492i
0.500000 − 1.95148i
0.500000 − 1.05635i
0.500000 + 0.622917i
0 −1.46891 + 2.54423i 0 −0.846487 − 1.46616i 0 −2.26639 − 1.36509i 0 −2.81540 − 4.87641i 0
417.2 0 −0.529136 + 0.916490i 0 1.96917 + 3.41070i 0 −1.20215 + 2.35687i 0 0.940031 + 1.62818i 0
417.3 0 0.817059 − 1.41519i 0 −0.152231 − 0.263671i 0 2.55018 − 0.704672i 0 0.164829 + 0.285491i 0
417.4 0 1.18099 − 2.04553i 0 −1.97045 − 3.41292i 0 −2.58164 + 0.578917i 0 −1.28946 − 2.23341i 0
625.1 0 −1.46891 − 2.54423i 0 −0.846487 + 1.46616i 0 −2.26639 + 1.36509i 0 −2.81540 + 4.87641i 0
625.2 0 −0.529136 − 0.916490i 0 1.96917 − 3.41070i 0 −1.20215 − 2.35687i 0 0.940031 − 1.62818i 0
625.3 0 0.817059 + 1.41519i 0 −0.152231 + 0.263671i 0 2.55018 + 0.704672i 0 0.164829 − 0.285491i 0
625.4 0 1.18099 + 2.04553i 0 −1.97045 + 3.41292i 0 −2.58164 − 0.578917i 0 −1.28946 + 2.23341i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.r.n 8
4.b odd 2 1 182.2.f.c ✓ 8
7.c even 3 1 inner 1456.2.r.n 8
12.b even 2 1 1638.2.j.r 8
28.d even 2 1 1274.2.f.y 8
28.f even 6 1 1274.2.a.v 4
28.f even 6 1 1274.2.f.y 8
28.g odd 6 1 182.2.f.c ✓ 8
28.g odd 6 1 1274.2.a.w 4
84.n even 6 1 1638.2.j.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.f.c ✓ 8 4.b odd 2 1
182.2.f.c ✓ 8 28.g odd 6 1
1274.2.a.v 4 28.f even 6 1
1274.2.a.w 4 28.g odd 6 1
1274.2.f.y 8 28.d even 2 1
1274.2.f.y 8 28.f even 6 1
1456.2.r.n 8 1.a even 1 1 trivial
1456.2.r.n 8 7.c even 3 1 inner
1638.2.j.r 8 12.b even 2 1
1638.2.j.r 8 84.n even 6 1

Hecke kernels

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

\( T_{3}^{8} + 9T_{3}^{6} - 6T_{3}^{5} + 69T_{3}^{4} - 27T_{3}^{3} + 117T_{3}^{2} + 36T_{3} + 144 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} + 19T_{5}^{6} + 32T_{5}^{5} + 295T_{5}^{4} + 497T_{5}^{3} + 841T_{5}^{2} + 248T_{5} + 64 \) Copy content Toggle raw display
\( T_{11}^{8} - T_{11}^{7} + 31T_{11}^{6} - 34T_{11}^{5} + 940T_{11}^{4} - 976T_{11}^{3} + 784T_{11}^{2} - 256T_{11} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 9 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{8} + 7 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - T^{7} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( (T + 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots + 33489 \) Copy content Toggle raw display
$19$ \( T^{8} - 3 T^{7} + \cdots + 36864 \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + \cdots + 12544 \) Copy content Toggle raw display
$29$ \( (T^{4} - 17 T^{3} + \cdots - 296)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 7 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{8} + 9 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 4704)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} + \cdots + 526)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 5 T^{7} + \cdots + 26569 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{8} - 17 T^{7} + \cdots + 7268416 \) Copy content Toggle raw display
$61$ \( T^{8} - 5 T^{7} + \cdots + 1658944 \) Copy content Toggle raw display
$67$ \( T^{8} - T^{7} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{4} + 7 T^{3} + \cdots + 553)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 14 T^{7} + \cdots + 60964864 \) Copy content Toggle raw display
$79$ \( T^{8} - 14 T^{7} + \cdots + 60964864 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + 6 T^{7} + \cdots + 12616704 \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots + 112)^{2} \) Copy content Toggle raw display
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