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Properties

Label 896.2.q.a
Level $896$
Weight $2$
Character orbit 896.q
Analytic conductor $7.155$
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] = [896,2,Mod(703,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.q (of order \(6\), degree \(2\), 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24x^{14} + 226x^{12} - 972x^{10} + 1575x^{8} + 252x^{6} + 550x^{4} + 156x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{3} + \beta_{2} - 2) q^{3} + ( - \beta_{13} + \beta_{10} + \beta_{5}) q^{5} + (\beta_{10} + \beta_{8} - \beta_{4}) q^{7} + (\beta_{12} + \beta_{11} - \beta_{9} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{3} + \beta_{2} - 2) q^{3} + ( - \beta_{13} + \beta_{10} + \beta_{5}) q^{5} + (\beta_{10} + \beta_{8} - \beta_{4}) q^{7} + (\beta_{12} + \beta_{11} - \beta_{9} + \cdots + 2) q^{9}+ \cdots + (\beta_{12} - \beta_{11} + 2 \beta_{9} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{3} + 8 q^{9} + 4 q^{11} + 12 q^{19} - 16 q^{25} + 24 q^{33} - 20 q^{35} + 16 q^{49} + 52 q^{51} + 48 q^{57} - 60 q^{59} + 24 q^{65} - 12 q^{67} - 24 q^{73} + 120 q^{75} - 32 q^{81} + 24 q^{89} - 72 q^{91} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24x^{14} + 226x^{12} - 972x^{10} + 1575x^{8} + 252x^{6} + 550x^{4} + 156x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13543 \nu^{14} + 429249 \nu^{12} - 5601614 \nu^{10} + 36966313 \nu^{8} - 121375292 \nu^{6} + \cdots + 28089012 ) / 39788476 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2742 \nu^{14} - 63413 \nu^{12} + 569287 \nu^{10} - 2259390 \nu^{8} + 2967075 \nu^{6} + \cdots - 889595 ) / 3060652 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31987 \nu^{15} - 825179 \nu^{13} + 8757356 \nu^{11} - 47162623 \nu^{9} + 130204306 \nu^{7} + \cdots - 7068028 \nu ) / 39788476 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45956 \nu^{15} + 1339385 \nu^{13} - 16099727 \nu^{11} + 98925010 \nu^{9} - 309182623 \nu^{7} + \cdots + 77606203 \nu ) / 39788476 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60741 \nu^{14} - 1736125 \nu^{12} + 20612094 \nu^{10} - 126911873 \nu^{8} + 409643968 \nu^{6} + \cdots - 52088512 ) / 39788476 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8928 \nu^{14} - 219390 \nu^{12} + 2137764 \nu^{10} - 9777437 \nu^{8} + 18618924 \nu^{6} + \cdots + 498261 ) / 1421017 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 281971 \nu^{15} - 6968099 \nu^{13} + 68614748 \nu^{11} - 320930859 \nu^{9} + 651534178 \nu^{7} + \cdots + 6883280 \nu ) / 39788476 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 394087 \nu^{14} - 9092864 \nu^{12} + 80172959 \nu^{10} - 297924065 \nu^{8} + 245316181 \nu^{6} + \cdots + 174076245 ) / 39788476 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 317762 \nu^{15} - 7371063 \nu^{13} + 65516719 \nu^{11} - 246474572 \nu^{9} + 203226999 \nu^{7} + \cdots + 132348297 \nu ) / 39788476 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 432725 \nu^{14} - 10433869 \nu^{12} + 98985936 \nu^{10} - 432430181 \nu^{8} + 737763466 \nu^{6} + \cdots - 13901972 ) / 39788476 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 582531 \nu^{14} - 13910723 \nu^{12} + 129886478 \nu^{10} - 548229671 \nu^{8} + 825018216 \nu^{6} + \cdots + 132028820 ) / 39788476 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 528296 \nu^{15} + 13111829 \nu^{13} - 129828765 \nu^{11} + 612489648 \nu^{9} + \cdots + 213399561 \nu ) / 39788476 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 682235 \nu^{15} + 16582331 \nu^{13} - 159108005 \nu^{11} + 708348501 \nu^{9} + \cdots - 83407358 \nu ) / 19894238 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 718355 \nu^{15} - 17401158 \nu^{13} + 166244059 \nu^{11} - 735570487 \nu^{9} + \cdots + 48167315 \nu ) / 19894238 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{6} + \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - 2\beta_{13} + 2\beta_{8} + 4\beta_{5} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{11} - 6\beta_{9} + 3\beta_{7} - 6\beta_{6} + 10\beta_{3} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{15} - 2\beta_{14} - 18\beta_{13} - 2\beta_{10} + 39\beta_{8} + 52\beta_{5} - 3\beta_{4} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{12} + 6\beta_{11} - 24\beta_{9} + 42\beta_{7} - 29\beta_{6} + 80\beta_{3} + 48\beta_{2} + 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -122\beta_{15} - 12\beta_{14} - 128\beta_{13} - 19\beta_{10} + 427\beta_{8} + 457\beta_{5} - 65\beta_{4} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -96\beta_{12} - 104\beta_{11} - 24\beta_{9} + 397\beta_{7} - 108\beta_{6} + 568\beta_{3} + 188\beta_{2} + 139 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 860 \beta_{15} + 16 \beta_{14} - 756 \beta_{13} - 80 \beta_{10} + 3757 \beta_{8} + 3268 \beta_{5} + \cdots - 697 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -869\beta_{12} - 1557\beta_{11} + 801\beta_{9} + 2930\beta_{7} - 84\beta_{6} + 3743\beta_{3} - 261\beta_{2} - 1030 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5039 \beta_{15} + 1214 \beta_{14} - 3482 \beta_{13} + 345 \beta_{10} + 28675 \beta_{8} + \cdots - 9573 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5980 \beta_{12} - 14612 \beta_{11} + 11130 \beta_{9} + 16606 \beta_{7} + 3936 \beta_{6} + \cdots - 17303 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 22828 \beta_{15} + 17030 \beta_{14} - 8216 \beta_{13} + 11050 \beta_{10} + 192318 \beta_{8} + \cdots - 88053 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 29406 \beta_{12} - 110881 \beta_{11} + 102665 \beta_{9} + 55380 \beta_{7} + 56229 \beta_{6} + \cdots - 158729 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 48729 \beta_{15} + 169312 \beta_{14} + 62152 \beta_{13} + 139906 \beta_{10} + 1103468 \beta_{8} + \cdots - 663575 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(1 - \beta_{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
−2.37177 − 0.500000i
2.37177 + 0.500000i
0.526379 + 0.500000i
−0.526379 − 0.500000i
−0.245327 + 0.500000i
0.245327 − 0.500000i
2.65282 − 0.500000i
−2.65282 + 0.500000i
−2.37177 + 0.500000i
2.37177 − 0.500000i
0.526379 − 0.500000i
−0.526379 + 0.500000i
−0.245327 − 0.500000i
0.245327 + 0.500000i
2.65282 + 0.500000i
−2.65282 − 0.500000i
0 −2.80401 − 1.61890i 0 −1.53015 − 2.65030i 0 −2.57882 + 0.591357i 0 3.74165 + 6.48073i 0
703.2 0 −2.80401 − 1.61890i 0 1.53015 + 2.65030i 0 2.57882 − 0.591357i 0 3.74165 + 6.48073i 0
703.3 0 −1.20586 − 0.696202i 0 −2.07821 − 3.59957i 0 0.632797 − 2.56896i 0 −0.530605 − 0.919035i 0
703.4 0 −1.20586 − 0.696202i 0 2.07821 + 3.59957i 0 −0.632797 + 2.56896i 0 −0.530605 − 0.919035i 0
703.5 0 −0.537541 − 0.310349i 0 −0.0337794 − 0.0585076i 0 −1.98532 + 1.74886i 0 −1.30737 − 2.26443i 0
703.6 0 −0.537541 − 0.310349i 0 0.0337794 + 0.0585076i 0 1.98532 − 1.74886i 0 −1.30737 − 2.26443i 0
703.7 0 1.54741 + 0.893397i 0 −0.581841 − 1.00778i 0 −2.23781 − 1.41146i 0 0.0963180 + 0.166828i 0
703.8 0 1.54741 + 0.893397i 0 0.581841 + 1.00778i 0 2.23781 + 1.41146i 0 0.0963180 + 0.166828i 0
831.1 0 −2.80401 + 1.61890i 0 −1.53015 + 2.65030i 0 −2.57882 − 0.591357i 0 3.74165 − 6.48073i 0
831.2 0 −2.80401 + 1.61890i 0 1.53015 − 2.65030i 0 2.57882 + 0.591357i 0 3.74165 − 6.48073i 0
831.3 0 −1.20586 + 0.696202i 0 −2.07821 + 3.59957i 0 0.632797 + 2.56896i 0 −0.530605 + 0.919035i 0
831.4 0 −1.20586 + 0.696202i 0 2.07821 − 3.59957i 0 −0.632797 − 2.56896i 0 −0.530605 + 0.919035i 0
831.5 0 −0.537541 + 0.310349i 0 −0.0337794 + 0.0585076i 0 −1.98532 − 1.74886i 0 −1.30737 + 2.26443i 0
831.6 0 −0.537541 + 0.310349i 0 0.0337794 − 0.0585076i 0 1.98532 + 1.74886i 0 −1.30737 + 2.26443i 0
831.7 0 1.54741 − 0.893397i 0 −0.581841 + 1.00778i 0 −2.23781 + 1.41146i 0 0.0963180 − 0.166828i 0
831.8 0 1.54741 − 0.893397i 0 0.581841 − 1.00778i 0 2.23781 − 1.41146i 0 0.0963180 − 0.166828i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.q.a ✓ 16
4.b odd 2 1 896.2.q.d yes 16
7.d odd 6 1 inner 896.2.q.a ✓ 16
8.b even 2 1 896.2.q.d yes 16
8.d odd 2 1 inner 896.2.q.a ✓ 16
28.f even 6 1 896.2.q.d yes 16
56.j odd 6 1 896.2.q.d yes 16
56.m even 6 1 inner 896.2.q.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.q.a ✓ 16 1.a even 1 1 trivial
896.2.q.a ✓ 16 7.d odd 6 1 inner
896.2.q.a ✓ 16 8.d odd 2 1 inner
896.2.q.a ✓ 16 56.m even 6 1 inner
896.2.q.d yes 16 4.b odd 2 1
896.2.q.d yes 16 8.b even 2 1
896.2.q.d yes 16 28.f even 6 1
896.2.q.d yes 16 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 6T_{3}^{7} + 10T_{3}^{6} - 12T_{3}^{5} - 27T_{3}^{4} + 36T_{3}^{3} + 118T_{3}^{2} + 90T_{3} + 25 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 6 T^{7} + 10 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 28 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} - 2 T^{7} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 48 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 50 T^{6} + \cdots + 167281)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + \cdots + 441)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 64 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$29$ \( (T^{8} + 96 T^{6} + \cdots + 24336)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 72 T^{14} + \cdots + 194481 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 64013554081 \) Copy content Toggle raw display
$41$ \( (T^{8} + 112 T^{6} + \cdots + 150544)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 96 T^{2} + \cdots + 960)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3154956561 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 151807041 \) Copy content Toggle raw display
$59$ \( (T^{8} + 30 T^{7} + \cdots + 227529)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1534548635361 \) Copy content Toggle raw display
$67$ \( (T^{8} + 6 T^{7} + \cdots + 4149369)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 448 T^{6} + \cdots + 114233344)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 12 T^{7} + \cdots + 378225)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 863609933592081 \) Copy content Toggle raw display
$83$ \( (T^{8} + 192 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 12 T^{7} + \cdots + 1274641)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 192 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
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