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Properties

Label 6724.2.a.j
Level $6724$
Weight $2$
Character orbit 6724.a
Self dual yes
Analytic conductor $53.691$
Analytic rank $0$
Dimension $16$
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] = [6724,2,Mod(1,6724)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6724, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6724.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6724 = 2^{2} \cdot 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6724.a (trivial)

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: yes
Analytic conductor: \(53.6914103191\)
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: \( x^{16} - 39x^{14} + 594x^{12} - 4428x^{10} + 16529x^{8} - 28236x^{6} + 17856x^{4} - 4032x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} + ( - \beta_{11} + \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} + ( - \beta_{11} + \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} + 2) q^{9} + (\beta_{15} - \beta_{11} + \beta_1) q^{11} + (\beta_{15} + \beta_{9} + \beta_{7}) q^{13} + (\beta_{14} - \beta_{11} + \cdots + \beta_1) q^{15}+ \cdots + ( - 2 \beta_{15} - 3 \beta_{14} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 39x^{14} + 594x^{12} - 4428x^{10} + 16529x^{8} - 28236x^{6} + 17856x^{4} - 4032x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} - 31\nu^{12} + 354\nu^{10} - 1780\nu^{8} + 3361\nu^{6} + 1148\nu^{4} - 7896\nu^{2} + 2048 ) / 288 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{14} - 437\nu^{12} + 4594\nu^{10} - 21148\nu^{8} + 43311\nu^{6} - 47456\nu^{4} + 51712\nu^{2} - 22336 ) / 2304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{14} - 437\nu^{12} + 4594\nu^{10} - 21148\nu^{8} + 43311\nu^{6} - 47456\nu^{4} + 49408\nu^{2} - 10816 ) / 2304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{14} - 87\nu^{12} - 202\nu^{10} + 11372\nu^{8} - 69979\nu^{6} + 141936\nu^{4} - 76096\nu^{2} + 6464 ) / 2304 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} - 107\nu^{12} + 2558\nu^{10} - 24708\nu^{8} + 105505\nu^{6} - 180752\nu^{4} + 87104\nu^{2} - 10944 ) / 2304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 107\nu^{13} + 2558\nu^{11} - 24708\nu^{9} + 105505\nu^{7} - 180752\nu^{5} + 87104\nu^{3} - 8640\nu ) / 2304 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -17\nu^{14} + 523\nu^{12} - 5998\nu^{10} + 31908\nu^{8} - 81137\nu^{6} + 92416\nu^{4} - 35968\nu^{2} + 2112 ) / 1152 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17 \nu^{15} - 651 \nu^{13} + 9710 \nu^{11} - 70564 \nu^{9} + 254321 \nu^{7} - 408960 \nu^{5} + \cdots - 46528 \nu ) / 4608 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17 \nu^{15} - 731 \nu^{13} + 11934 \nu^{11} - 91844 \nu^{9} + 331313 \nu^{7} - 460496 \nu^{5} + \cdots + 24640 \nu ) / 4608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19 \nu^{15} + 753 \nu^{13} - 11674 \nu^{11} + 88844 \nu^{9} - 340723 \nu^{7} + 607536 \nu^{5} + \cdots + 98624 \nu ) / 4608 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17 \nu^{15} - 811 \nu^{13} + 14542 \nu^{11} - 124260 \nu^{9} + 523505 \nu^{7} - 1014304 \nu^{5} + \cdots - 139968 \nu ) / 4608 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 49 \nu^{14} + 1707 \nu^{12} - 23086 \nu^{10} + 152420 \nu^{8} - 504337 \nu^{6} + 753984 \nu^{4} + \cdots + 39872 ) / 2304 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 51 \nu^{15} + 1921 \nu^{13} - 28202 \nu^{11} + 201836 \nu^{9} - 715347 \nu^{7} + 1115488 \nu^{5} + \cdots + 61760 \nu ) / 4608 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21 \nu^{15} + 835 \nu^{13} - 12890 \nu^{11} + 96428 \nu^{9} - 354309 \nu^{7} + 568588 \nu^{5} + \cdots + 33536 \nu ) / 1152 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{13} + 2\beta_{8} - 5\beta_{6} + \beta_{5} - 11\beta_{4} + 9\beta_{3} + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{15} + 16\beta_{14} + \beta_{12} - 10\beta_{11} + 13\beta_{10} + 10\beta_{9} - 17\beta_{7} + 85\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{13} + 33\beta_{8} - 74\beta_{6} + 14\beta_{5} - 108\beta_{4} + 84\beta_{3} + 2\beta_{2} + 361 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 54 \beta_{15} + 201 \beta_{14} + 12 \beta_{12} - 101 \beta_{11} + 141 \beta_{10} + 95 \beta_{9} + \cdots + 822 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -351\beta_{13} + 408\beta_{8} - 867\beta_{6} + 147\beta_{5} - 1047\beta_{4} + 810\beta_{3} + 45\beta_{2} + 3413 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 714 \beta_{15} + 2283 \beta_{14} + 102 \beta_{12} - 1059 \beta_{11} + 1455 \beta_{10} + \cdots + 8051 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3789 \beta_{13} + 4554 \beta_{8} - 9465 \beta_{6} + 1353 \beta_{5} - 10184 \beta_{4} + 7949 \beta_{3} + \cdots + 32920 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8430 \beta_{15} + 24698 \beta_{14} + 678 \beta_{12} - 11354 \beta_{11} + 14738 \beta_{10} + \cdots + 79440 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 39523 \beta_{13} + 48544 \beta_{8} - 100444 \beta_{6} + 11312 \beta_{5} - 99547 \beta_{4} + \cdots + 320558 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 94467 \beta_{15} + 260600 \beta_{14} + 2621 \beta_{12} - 122846 \beta_{11} + 148091 \beta_{10} + \cdots + 787337 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 405561 \beta_{13} + 505779 \beta_{8} - 1051960 \beta_{6} + 85168 \beta_{5} - 976761 \beta_{4} + \cdots + 3136949 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1030527 \beta_{15} + 2713503 \beta_{14} - 18969 \beta_{12} - 1331173 \beta_{11} + \cdots + 7825161 \beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.19728
−3.12433
−3.11423
−2.36013
−1.66505
−0.770134
−0.527140
−0.322385
0.322385
0.527140
0.770134
1.66505
2.36013
3.11423
3.12433
3.19728
0 −3.19728 0 2.17635 0 −4.47386 0 7.22259 0
1.2 0 −3.12433 0 4.24501 0 −1.30863 0 6.76146 0
1.3 0 −3.11423 0 −0.759223 0 −0.569900 0 6.69841 0
1.4 0 −2.36013 0 1.80249 0 −2.68829 0 2.57023 0
1.5 0 −1.66505 0 −3.40359 0 0.711682 0 −0.227602 0
1.6 0 −0.770134 0 −3.70327 0 −0.0752960 0 −2.40689 0
1.7 0 −0.527140 0 −0.0258714 0 2.09135 0 −2.72212 0
1.8 0 −0.322385 0 2.66810 0 3.97926 0 −2.89607 0
1.9 0 0.322385 0 2.66810 0 −3.97926 0 −2.89607 0
1.10 0 0.527140 0 −0.0258714 0 −2.09135 0 −2.72212 0
1.11 0 0.770134 0 −3.70327 0 0.0752960 0 −2.40689 0
1.12 0 1.66505 0 −3.40359 0 −0.711682 0 −0.227602 0
1.13 0 2.36013 0 1.80249 0 2.68829 0 2.57023 0
1.14 0 3.11423 0 −0.759223 0 0.569900 0 6.69841 0
1.15 0 3.12433 0 4.24501 0 1.30863 0 6.76146 0
1.16 0 3.19728 0 2.17635 0 4.47386 0 7.22259 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(41\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6724.2.a.j 16
41.b even 2 1 inner 6724.2.a.j 16
41.g even 20 2 164.2.k.a ✓ 16
123.m odd 20 2 1476.2.bb.b 16
164.n odd 20 2 656.2.be.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.k.a ✓ 16 41.g even 20 2
656.2.be.e 16 164.n odd 20 2
1476.2.bb.b 16 123.m odd 20 2
6724.2.a.j 16 1.a even 1 1 trivial
6724.2.a.j 16 41.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 39T_{3}^{14} + 594T_{3}^{12} - 4428T_{3}^{10} + 16529T_{3}^{8} - 28236T_{3}^{6} + 17856T_{3}^{4} - 4032T_{3}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 39 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{7} - 25 T^{6} + \cdots + 11)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 50 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{16} - 94 T^{14} + \cdots + 633616 \) Copy content Toggle raw display
$13$ \( T^{16} - 70 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{16} - 85 T^{14} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 12204946576 \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} + \cdots - 116524)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 203233536 \) Copy content Toggle raw display
$31$ \( (T^{8} + T^{7} + \cdots + 569536)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 6 T^{7} + \cdots - 61259)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} + 24 T^{7} + \cdots + 1396)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 208925783056 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1151855721 \) Copy content Toggle raw display
$59$ \( (T^{8} + 16 T^{7} + \cdots + 2321216)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 3 T^{7} + \cdots + 2982501)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 796242120976 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5235328391056 \) Copy content Toggle raw display
$73$ \( (T^{8} + 17 T^{7} + \cdots - 1437444)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 34930114816 \) Copy content Toggle raw display
$83$ \( (T^{8} - 6 T^{7} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 11764439296 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 733165775001 \) Copy content Toggle raw display
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