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Properties

Label 6480.2.h.e
Level $6480$
Weight $2$
Character orbit 6480.h
Analytic conductor $51.743$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6480,2,Mod(2591,6480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6480.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.h (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: \(51.7430605098\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 796 x^{12} - 2228 x^{11} + 5254 x^{10} - 10232 x^{9} + \cdots + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} - \beta_{7} q^{7} + \beta_{8} q^{11} + (\beta_{10} + 1) q^{13} + \beta_{12} q^{17} + \beta_{14} q^{19} + (\beta_{9} - \beta_{6}) q^{23} - q^{25} + (\beta_{12} - \beta_1) q^{29} + ( - \beta_{15} - \beta_{5}) q^{31} - \beta_{9} q^{35} + ( - \beta_{10} - \beta_{2} - 1) q^{37} + ( - \beta_{13} - \beta_1) q^{41} + (\beta_{15} + 2 \beta_{5}) q^{43} + ( - \beta_{11} - \beta_{9} + \cdots - \beta_{6}) q^{47}+ \cdots + (3 \beta_{2} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{13} - 16 q^{25} - 16 q^{37} - 32 q^{49} - 16 q^{61} + 16 q^{73} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 796 x^{12} - 2228 x^{11} + 5254 x^{10} - 10232 x^{9} + \cdots + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 422 \nu^{15} + 3165 \nu^{14} - 24897 \nu^{13} + 113828 \nu^{12} - 471033 \nu^{11} + 1444179 \nu^{10} + \cdots + 563355 ) / 20265 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8 \nu^{14} - 56 \nu^{13} + 405 \nu^{12} - 1702 \nu^{11} + 6511 \nu^{10} - 18288 \nu^{9} + 44999 \nu^{8} + \cdots + 6120 ) / 105 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29 \nu^{14} - 203 \nu^{13} + 1245 \nu^{12} - 4831 \nu^{11} + 15688 \nu^{10} - 38994 \nu^{9} + \cdots + 4230 ) / 105 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 100 \nu^{15} + 750 \nu^{14} - 4632 \nu^{13} + 18733 \nu^{12} - 61838 \nu^{11} + 159071 \nu^{10} + \cdots - 8847 ) / 579 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 104 \nu^{15} - 780 \nu^{14} + 7334 \nu^{13} - 35841 \nu^{12} + 163776 \nu^{11} - 532543 \nu^{10} + \cdots - 446310 ) / 2895 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19 \nu^{14} - 133 \nu^{13} + 828 \nu^{12} - 3239 \nu^{11} + 10715 \nu^{10} - 27054 \nu^{9} + \cdots + 3132 ) / 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6858 \nu^{15} - 51435 \nu^{14} + 326363 \nu^{13} - 1341262 \nu^{12} + 4589547 \nu^{11} + \cdots - 1557540 ) / 20265 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17 \nu^{14} + 119 \nu^{13} - 735 \nu^{12} + 2863 \nu^{11} - 9374 \nu^{10} + 23462 \nu^{9} + \cdots - 2160 ) / 15 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 124 \nu^{14} - 868 \nu^{13} + 5385 \nu^{12} - 21026 \nu^{11} + 69193 \nu^{10} - 173914 \nu^{9} + \cdots + 16740 ) / 105 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 191 \nu^{14} + 1337 \nu^{13} - 8265 \nu^{12} + 32209 \nu^{11} - 105562 \nu^{10} + 264426 \nu^{9} + \cdots - 25155 ) / 105 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 288 \nu^{14} + 2016 \nu^{13} - 12480 \nu^{12} + 48672 \nu^{11} - 159776 \nu^{10} + 400768 \nu^{9} + \cdots - 40770 ) / 105 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20152 \nu^{15} + 151140 \nu^{14} - 954192 \nu^{13} + 3909958 \nu^{12} - 13297608 \nu^{11} + \cdots + 4048470 ) / 20265 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25586 \nu^{15} + 191895 \nu^{14} - 1197951 \nu^{13} + 4876274 \nu^{12} - 16345719 \nu^{11} + \cdots + 1456470 ) / 20265 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25024 \nu^{15} + 187680 \nu^{14} - 1173054 \nu^{13} + 4778371 \nu^{12} - 16062636 \nu^{11} + \cdots + 4903380 ) / 20265 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10288 \nu^{15} + 77160 \nu^{14} - 484430 \nu^{13} + 1978535 \nu^{12} - 6680328 \nu^{11} + \cdots + 1131282 ) / 4053 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + 3\beta_{7} + \beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \cdots - 15 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9 \beta_{15} - 3 \beta_{14} - 8 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} + 6 \beta_{10} + 6 \beta_{9} + \cdots - 48 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9 \beta_{15} - 3 \beta_{14} - 8 \beta_{13} - 12 \beta_{12} + 7 \beta_{11} - 23 \beta_{10} - 20 \beta_{9} + \cdots + 66 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 84 \beta_{15} + 30 \beta_{14} + 83 \beta_{13} + 60 \beta_{12} + 40 \beta_{11} - 125 \beta_{10} + \cdots + 411 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 297 \beta_{15} + 105 \beta_{14} + 289 \beta_{13} + 241 \beta_{12} - 87 \beta_{11} + 345 \beta_{10} + \cdots - 723 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 258 \beta_{15} - 99 \beta_{14} - 260 \beta_{13} - 147 \beta_{12} - 224 \beta_{11} + 826 \beta_{10} + \cdots - 2013 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1746 \beta_{15} - 648 \beta_{14} - 1733 \beta_{13} - 1179 \beta_{12} + 182 \beta_{11} - 820 \beta_{10} + \cdots + 1515 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1350 \beta_{15} + 576 \beta_{14} + 1381 \beta_{13} + 526 \beta_{12} + 4500 \beta_{11} - 17829 \beta_{10} + \cdots + 39615 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 35109 \beta_{15} + 13365 \beta_{14} + 35003 \beta_{13} + 22125 \beta_{12} + 941 \beta_{11} + \cdots + 8475 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 22818 \beta_{15} + 8343 \beta_{14} + 22616 \beta_{13} + 15831 \beta_{12} - 41459 \beta_{11} + \cdots - 365601 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 156870 \beta_{15} - 60375 \beta_{14} - 156574 \beta_{13} - 96502 \beta_{12} - 24666 \beta_{11} + \cdots - 217158 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 534741 \beta_{15} - 203922 \beta_{14} - 533104 \beta_{13} - 336156 \beta_{12} + 345644 \beta_{11} + \cdots + 3057099 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1233723 \beta_{15} + 476748 \beta_{14} + 1231789 \beta_{13} + 753480 \beta_{12} + 405317 \beta_{11} + \cdots + 3579771 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 7593876 \beta_{15} + 2921307 \beta_{14} + 7578028 \beta_{13} + 4684201 \beta_{12} - 2497191 \beta_{11} + \cdots - 22121013 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6480\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(6401\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
0.500000 + 1.12119i
0.500000 + 1.76390i
0.500000 + 0.911095i
0.500000 + 1.97090i
0.500000 − 0.0390518i
0.500000 − 1.42873i
0.500000 − 1.24626i
0.500000 − 3.05304i
0.500000 + 3.05304i
0.500000 + 1.24626i
0.500000 + 1.42873i
0.500000 + 0.0390518i
0.500000 − 1.97090i
0.500000 − 0.911095i
0.500000 − 1.76390i
0.500000 − 1.12119i
0 0 0 − 1.00000i 0 − 4.17423i 0 0 0
2591.2 0 0 0 − 1.00000i 0 − 3.01017i 0 0 0
2591.3 0 0 0 − 1.00000i 0 − 2.33983i 0 0 0
2591.4 0 0 0 − 1.00000i 0 − 2.00996i 0 0 0
2591.5 0 0 0 − 1.00000i 0 2.00996i 0 0 0
2591.6 0 0 0 − 1.00000i 0 2.33983i 0 0 0
2591.7 0 0 0 − 1.00000i 0 3.01017i 0 0 0
2591.8 0 0 0 − 1.00000i 0 4.17423i 0 0 0
2591.9 0 0 0 1.00000i 0 − 4.17423i 0 0 0
2591.10 0 0 0 1.00000i 0 − 3.01017i 0 0 0
2591.11 0 0 0 1.00000i 0 − 2.33983i 0 0 0
2591.12 0 0 0 1.00000i 0 − 2.00996i 0 0 0
2591.13 0 0 0 1.00000i 0 2.00996i 0 0 0
2591.14 0 0 0 1.00000i 0 2.33983i 0 0 0
2591.15 0 0 0 1.00000i 0 3.01017i 0 0 0
2591.16 0 0 0 1.00000i 0 4.17423i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6480.2.h.e ✓ 16
3.b odd 2 1 inner 6480.2.h.e ✓ 16
4.b odd 2 1 inner 6480.2.h.e ✓ 16
12.b even 2 1 inner 6480.2.h.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6480.2.h.e ✓ 16 1.a even 1 1 trivial
6480.2.h.e ✓ 16 3.b odd 2 1 inner
6480.2.h.e ✓ 16 4.b odd 2 1 inner
6480.2.h.e ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6480, [\chi])\):

\( T_{7}^{8} + 36T_{7}^{6} + 432T_{7}^{4} + 2088T_{7}^{2} + 3492 \) Copy content Toggle raw display
\( T_{11}^{8} - 48T_{11}^{6} + 666T_{11}^{4} - 2736T_{11}^{2} + 873 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 36 T^{6} + \cdots + 3492)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 48 T^{6} + \cdots + 873)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} - 12 T^{2} + \cdots - 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 36 T^{6} + \cdots + 324)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 132 T^{6} + \cdots + 461817)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 120 T^{6} + \cdots + 55872)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 72 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 108 T^{6} + \cdots + 873)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} - 30 T^{2} + \cdots + 46)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 72 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 204 T^{6} + \cdots + 2182500)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 300 T^{6} + \cdots + 17603172)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 108 T^{6} + \cdots + 324)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 192 T^{6} + \cdots + 461817)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 1828)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 360 T^{6} + \cdots + 4525632)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 480 T^{6} + \cdots + 146036313)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4 T^{3} + \cdots - 242)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 288 T^{6} + \cdots + 13968)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 324 T^{6} + \cdots + 282852)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 504 T^{6} + \cdots + 179104689)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 16 T^{3} + \cdots - 7682)^{4} \) Copy content Toggle raw display
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