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Properties

Label 5184.2.f.d
Level $5184$
Weight $2$
Character orbit 5184.f
Analytic conductor $41.394$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5184,2,Mod(2591,5184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("5184.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.f (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: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.33418400425706520576.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} - \beta_1 q^{7} + \beta_{11} q^{11} + \beta_{7} q^{13} + \beta_{4} q^{17} + \beta_{2} q^{19} - \beta_{9} q^{23} + ( - \beta_{6} + 1) q^{25} - \beta_{8} q^{29} - 2 \beta_1 q^{31} + \beta_{11} q^{35}+ \cdots + ( - 4 \beta_{6} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{25} + 48 q^{49} - 32 q^{73} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 116\nu^{14} - 791\nu^{12} + 4704\nu^{10} - 6208\nu^{8} + 9544\nu^{6} + 480\nu^{4} + 7316\nu^{2} - 599 ) / 1584 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -20\nu^{14} + 147\nu^{12} - 896\nu^{10} + 1568\nu^{8} - 2568\nu^{6} + 512\nu^{4} - 84\nu^{2} - 1197 ) / 176 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 179 \nu^{15} - 1853 \nu^{13} + 12032 \nu^{11} - 38392 \nu^{9} + 67208 \nu^{7} - 74912 \nu^{5} + \cdots - 10029 \nu ) / 1056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 127\nu^{15} - 945\nu^{13} + 5760\nu^{11} - 10520\nu^{9} + 17640\nu^{7} - 10080\nu^{5} + 8911\nu^{3} - 225\nu ) / 352 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 13\nu^{13} + 88\nu^{11} - 340\nu^{9} + 624\nu^{7} - 760\nu^{5} + 357\nu^{3} - 101\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -15\nu^{14} + 112\nu^{12} - 672\nu^{10} + 1176\nu^{8} - 1616\nu^{6} + 384\nu^{4} - 63\nu^{2} - 80 ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{14} - 64\nu^{12} + 384\nu^{10} - 600\nu^{8} + 944\nu^{6} - 336\nu^{4} + 441\nu^{2} - 40 ) / 24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 887 \nu^{15} - 7257 \nu^{13} + 44736 \nu^{11} - 100024 \nu^{9} + 157992 \nu^{7} - 116832 \nu^{5} + \cdots - 16713 \nu ) / 1056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -41\nu^{15} + 329\nu^{13} - 2016\nu^{11} + 4312\nu^{9} - 6664\nu^{7} + 4608\nu^{5} - 2345\nu^{3} + 665\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1060 \nu^{14} + 8407 \nu^{12} - 51360 \nu^{10} + 106688 \nu^{8} - 162152 \nu^{6} + 98592 \nu^{4} + \cdots + 4759 ) / 1584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -45\nu^{15} + 357\nu^{13} - 2176\nu^{11} + 4496\nu^{9} - 6664\nu^{7} + 3808\nu^{5} - 1301\nu^{3} + 85\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -80\nu^{14} + 599\nu^{12} - 3584\nu^{10} + 6272\nu^{8} - 8336\nu^{6} + 2048\nu^{4} - 336\nu^{2} - 201 ) / 44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 93\nu^{14} - 752\nu^{12} + 4608\nu^{10} - 9960\nu^{8} + 15088\nu^{6} - 9696\nu^{4} + 3789\nu^{2} - 416 ) / 48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -147\nu^{15} + 1155\nu^{13} - 7040\nu^{11} + 14296\nu^{9} - 21560\nu^{7} + 12320\nu^{5} - 5635\nu^{3} + 275\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 549 \nu^{15} - 4347 \nu^{13} + 26496 \nu^{11} - 54552 \nu^{9} + 81144 \nu^{7} - 46368 \nu^{5} + \cdots - 1035 \nu ) / 176 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} - 3\beta_{14} - 3\beta_{9} - 6\beta_{8} + \beta_{5} - 2\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{13} + \beta_{12} - 21\beta_{10} - 6\beta_{7} - 6\beta_{6} + 4\beta_{2} + 3\beta _1 + 24 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{15} - 6\beta_{14} - 3\beta_{11} - 6\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -36\beta_{13} - 7\beta_{12} - 123\beta_{10} - 30\beta_{7} + 48\beta_{6} - 22\beta_{2} - 21\beta _1 - 102 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 80 \beta_{15} - 63 \beta_{14} - 42 \beta_{11} + 93 \beta_{9} + 84 \beta_{8} - 49 \beta_{5} + \cdots + 126 \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{12} + 48\beta_{6} - 20\beta_{2} - 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 448 \beta_{15} + 345 \beta_{14} + 246 \beta_{11} + 501 \beta_{9} + 444 \beta_{8} - 281 \beta_{5} + \cdots + 738 \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1140 \beta_{13} - 239 \beta_{12} + 3885 \beta_{10} + 846 \beta_{7} + 1632 \beta_{6} - 662 \beta_{2} + \cdots - 2874 ) / 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1247\beta_{15} + 954\beta_{14} + 693\beta_{11} + 918\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6342 \beta_{13} + 1337 \beta_{12} + 21603 \beta_{10} + 4662 \beta_{7} - 9114 \beta_{6} + 3668 \beta_{2} + \cdots + 15864 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13858 \beta_{15} + 10581 \beta_{14} + 7728 \beta_{11} - 15243 \beta_{9} - 13434 \beta_{8} + \cdots - 23184 \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 620\beta_{12} - 4224\beta_{6} + 1696\beta_{2} + 7327 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 76958 \beta_{15} - 58725 \beta_{14} - 42960 \beta_{11} - 84549 \beta_{9} - 74490 \beta_{8} + \cdots - 128880 \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 195642 \beta_{13} + 41335 \beta_{12} - 666291 \beta_{10} - 143274 \beta_{7} - 281562 \beta_{6} + \cdots + 487896 ) / 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -213649\beta_{15} - 163002\beta_{14} - 119301\beta_{11} - 156426\beta_{4} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
−2.04058 − 1.17813i
−0.367543 − 0.212201i
−0.367543 + 0.212201i
−2.04058 + 1.17813i
1.11871 + 0.645885i
0.670418 + 0.387066i
0.670418 − 0.387066i
1.11871 − 0.645885i
−1.11871 − 0.645885i
−0.670418 − 0.387066i
−0.670418 + 0.387066i
−1.11871 + 0.645885i
2.04058 + 1.17813i
0.367543 + 0.212201i
0.367543 − 0.212201i
2.04058 − 1.17813i
0 0 0 −2.78066 0 − 0.732051i 0 0 0
2591.2 0 0 0 −2.78066 0 − 0.732051i 0 0 0
2591.3 0 0 0 −2.78066 0 0.732051i 0 0 0
2591.4 0 0 0 −2.78066 0 0.732051i 0 0 0
2591.5 0 0 0 −2.06590 0 − 2.73205i 0 0 0
2591.6 0 0 0 −2.06590 0 − 2.73205i 0 0 0
2591.7 0 0 0 −2.06590 0 2.73205i 0 0 0
2591.8 0 0 0 −2.06590 0 2.73205i 0 0 0
2591.9 0 0 0 2.06590 0 − 2.73205i 0 0 0
2591.10 0 0 0 2.06590 0 − 2.73205i 0 0 0
2591.11 0 0 0 2.06590 0 2.73205i 0 0 0
2591.12 0 0 0 2.06590 0 2.73205i 0 0 0
2591.13 0 0 0 2.78066 0 − 0.732051i 0 0 0
2591.14 0 0 0 2.78066 0 − 0.732051i 0 0 0
2591.15 0 0 0 2.78066 0 0.732051i 0 0 0
2591.16 0 0 0 2.78066 0 0.732051i 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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.f.d ✓ 16
3.b odd 2 1 inner 5184.2.f.d ✓ 16
4.b odd 2 1 inner 5184.2.f.d ✓ 16
8.b even 2 1 inner 5184.2.f.d ✓ 16
8.d odd 2 1 inner 5184.2.f.d ✓ 16
12.b even 2 1 inner 5184.2.f.d ✓ 16
24.f even 2 1 inner 5184.2.f.d ✓ 16
24.h odd 2 1 inner 5184.2.f.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5184.2.f.d ✓ 16 1.a even 1 1 trivial
5184.2.f.d ✓ 16 3.b odd 2 1 inner
5184.2.f.d ✓ 16 4.b odd 2 1 inner
5184.2.f.d ✓ 16 8.b even 2 1 inner
5184.2.f.d ✓ 16 8.d odd 2 1 inner
5184.2.f.d ✓ 16 12.b even 2 1 inner
5184.2.f.d ✓ 16 24.f even 2 1 inner
5184.2.f.d ✓ 16 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5}^{4} - 12T_{5}^{2} + 33 \) Copy content Toggle raw display
\( T_{19}^{4} - 72T_{19}^{2} + 1188 \) 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
$5$ \( (T^{4} - 12 T^{2} + 33)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{2} + 4)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 36 T^{2} + 132)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 54 T^{2} + 297)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 36 T^{2} + 81)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 72 T^{2} + 1188)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 18)^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 36 T^{2} + 297)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 32 T^{2} + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 90 T^{2} + 297)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 54)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 72 T^{2} + 1188)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 72)^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} - 60 T^{2} + 132)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 144 T^{2} + 4752)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 162 T^{2} + 2673)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 216 T^{2} + 10692)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 54)^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 71)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 56 T^{2} + 676)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 240 T^{2} + 528)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 252 T^{2} + 13689)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 44)^{8} \) Copy content Toggle raw display
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