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Properties

Label 6724.2.a.l
Level $6724$
Weight $2$
Character orbit 6724.a
Self dual yes
Analytic conductor $53.691$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 19 x^{16} + 125 x^{15} + 97 x^{14} - 1213 x^{13} + 139 x^{12} + 6021 x^{11} + \cdots - 419 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{17} + \beta_{12} + \beta_{7} + \cdots + 1) q^{5}+ \cdots + (\beta_{9} - \beta_{6} - \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{17} + \beta_{12} + \beta_{7} + \cdots + 1) q^{5}+ \cdots + (\beta_{16} - 3 \beta_{15} + 2 \beta_{14} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 5 q^{3} + 2 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 5 q^{3} + 2 q^{5} + 7 q^{7} + 9 q^{9} + 12 q^{11} + 12 q^{13} - 14 q^{15} + 21 q^{17} + 15 q^{19} + 5 q^{21} + 3 q^{23} + 8 q^{25} + 5 q^{27} + 38 q^{29} + 12 q^{31} + 25 q^{33} + 9 q^{35} + 12 q^{37} - 3 q^{39} + 21 q^{43} - 13 q^{45} - 29 q^{47} - 9 q^{49} + 9 q^{51} + 27 q^{53} + 17 q^{55} - 13 q^{57} + 25 q^{59} - 9 q^{61} + 68 q^{63} + 45 q^{65} - 5 q^{67} + 46 q^{69} + 12 q^{71} + 19 q^{73} + 24 q^{75} - 3 q^{77} - 27 q^{79} - 46 q^{81} - 46 q^{83} + 54 q^{85} - 11 q^{87} + 55 q^{89} - 31 q^{91} + 59 q^{93} + 27 q^{95} + 31 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} - 19 x^{16} + 125 x^{15} + 97 x^{14} - 1213 x^{13} + 139 x^{12} + 6021 x^{11} + \cdots - 419 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1991517643 \nu^{17} + 7257188382 \nu^{16} - 91033227445 \nu^{15} - 155505124518 \nu^{14} + \cdots + 3116004854587 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5117334346 \nu^{17} - 64254528752 \nu^{16} + 44758789498 \nu^{15} + 1472930210424 \nu^{14} + \cdots - 10083193492405 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5173972350 \nu^{17} + 24356871092 \nu^{16} + 92196221641 \nu^{15} - 576585972870 \nu^{14} + \cdots + 1444927175198 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7921568979 \nu^{17} + 54189470506 \nu^{16} + 83886922991 \nu^{15} - 1260964950102 \nu^{14} + \cdots + 5600600764393 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2655782727 \nu^{17} + 20960946416 \nu^{16} + 19321281480 \nu^{15} - 488949149590 \nu^{14} + \cdots + 2750677696801 ) / 20313809583 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9827313929 \nu^{17} - 83975285857 \nu^{16} - 49120926157 \nu^{15} + 1949876969196 \nu^{14} + \cdots - 11166522772634 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11803141429 \nu^{17} - 50603464797 \nu^{16} - 230371176244 \nu^{15} + 1210501821138 \nu^{14} + \cdots - 2648485890131 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12706462453 \nu^{17} + 85458250280 \nu^{16} + 140687773235 \nu^{15} - 1995721301484 \nu^{14} + \cdots + 9047936110015 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14581625611 \nu^{17} - 66622887610 \nu^{16} - 270768827727 \nu^{15} + 1584375684936 \nu^{14} + \cdots - 3319137402201 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15573324721 \nu^{17} - 83790223759 \nu^{16} - 246434536365 \nu^{15} + 1978326966189 \nu^{14} + \cdots - 6681662979711 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21496924875 \nu^{17} - 133248857093 \nu^{16} - 278095912429 \nu^{15} + 3123471135903 \nu^{14} + \cdots - 13206886037810 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24502450997 \nu^{17} - 127204175650 \nu^{16} - 404134485886 \nu^{15} + 3010800581655 \nu^{14} + \cdots - 9503983646651 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25546997768 \nu^{17} + 196973552528 \nu^{16} + 199773094002 \nu^{15} + \cdots + 24215425416438 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27849538169 \nu^{17} + 150193646706 \nu^{16} + 439074870923 \nu^{15} + \cdots + 11849230825906 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 37415229194 \nu^{17} - 164067200772 \nu^{16} - 718183719098 \nu^{15} + 3918845783796 \nu^{14} + \cdots - 8358561843844 ) / 60941428749 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 79656384008 \nu^{17} - 463557042919 \nu^{16} - 1138731763930 \nu^{15} + 10905779624805 \nu^{14} + \cdots - 41569411538363 ) / 60941428749 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{16} + \beta_{15} + \beta_{13} - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + 10 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{17} - 13 \beta_{16} + 13 \beta_{15} + 14 \beta_{13} + \beta_{12} - 9 \beta_{11} - 11 \beta_{10} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{17} + 16 \beta_{16} - 3 \beta_{15} - 10 \beta_{14} - 19 \beta_{13} - 14 \beta_{12} + \cdots + 197 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 104 \beta_{17} - 132 \beta_{16} + 135 \beta_{15} + 147 \beta_{13} + 18 \beta_{12} - 80 \beta_{11} + \cdots - 297 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 71 \beta_{17} + 191 \beta_{16} - 63 \beta_{15} - 87 \beta_{14} - 241 \beta_{13} - 150 \beta_{12} + \cdots + 1618 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 950 \beta_{17} - 1248 \beta_{16} + 1288 \beta_{15} + 2 \beta_{14} + 1407 \beta_{13} + 227 \beta_{12} + \cdots - 3253 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 917 \beta_{17} + 2068 \beta_{16} - 916 \beta_{15} - 753 \beta_{14} - 2665 \beta_{13} - 1474 \beta_{12} + \cdots + 13991 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8584 \beta_{17} - 11512 \beta_{16} + 11831 \beta_{15} + 71 \beta_{14} + 13052 \beta_{13} + 2512 \beta_{12} + \cdots - 33244 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 10499 \beta_{17} + 21403 \beta_{16} - 11376 \beta_{15} - 6565 \beta_{14} - 27678 \beta_{13} + \cdots + 125045 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 77360 \beta_{17} - 105399 \beta_{16} + 106933 \beta_{15} + 1437 \beta_{14} + 120056 \beta_{13} + \cdots - 328236 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 113091 \beta_{17} + 215889 \beta_{16} - 129556 \beta_{15} - 57520 \beta_{14} - 277921 \beta_{13} + \cdots + 1141037 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 698157 \beta_{17} - 964699 \beta_{16} + 961753 \beta_{15} + 22323 \beta_{14} + 1104623 \beta_{13} + \cdots - 3180929 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1174745 \beta_{17} + 2141314 \beta_{16} - 1398464 \beta_{15} - 505605 \beta_{14} - 2735268 \beta_{13} + \cdots + 10546614 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 6322786 \beta_{17} - 8853631 \beta_{16} + 8658163 \beta_{15} + 297935 \beta_{14} + 10197258 \beta_{13} + \cdots - 30500273 \) 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
−3.07173
−2.18470
−1.94679
−1.83914
−1.09341
−0.806218
−0.798145
−0.616613
−0.219101
0.651253
1.02637
1.45721
1.96499
2.33037
2.37969
2.44790
2.56056
2.75751
0 −3.07173 0 1.60569 0 4.23869 0 6.43553 0
1.2 0 −2.18470 0 −1.85770 0 1.59983 0 1.77293 0
1.3 0 −1.94679 0 −0.486957 0 −0.618115 0 0.789997 0
1.4 0 −1.83914 0 3.65876 0 0.00747803 0 0.382440 0
1.5 0 −1.09341 0 −1.38412 0 −2.79014 0 −1.80445 0
1.6 0 −0.806218 0 −1.77902 0 −0.188380 0 −2.35001 0
1.7 0 −0.798145 0 2.45730 0 0.750762 0 −2.36296 0
1.8 0 −0.616613 0 2.95448 0 −0.385921 0 −2.61979 0
1.9 0 −0.219101 0 −2.01513 0 0.320381 0 −2.95199 0
1.10 0 0.651253 0 −1.29601 0 −5.03218 0 −2.57587 0
1.11 0 1.02637 0 3.72268 0 −1.42593 0 −1.94656 0
1.12 0 1.45721 0 −0.302144 0 3.37813 0 −0.876550 0
1.13 0 1.96499 0 2.95734 0 2.48414 0 0.861202 0
1.14 0 2.33037 0 −4.04337 0 3.01023 0 2.43063 0
1.15 0 2.37969 0 1.28958 0 −0.144909 0 2.66290 0
1.16 0 2.44790 0 −2.79557 0 −3.83515 0 2.99222 0
1.17 0 2.56056 0 1.07147 0 1.61718 0 3.55649 0
1.18 0 2.75751 0 −1.75728 0 4.01390 0 4.60384 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6724.2.a.l yes 18
41.b even 2 1 6724.2.a.k ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6724.2.a.k ✓ 18 41.b even 2 1
6724.2.a.l yes 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 5 T_{3}^{17} - 19 T_{3}^{16} + 125 T_{3}^{15} + 97 T_{3}^{14} - 1213 T_{3}^{13} + \cdots - 419 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 5 T^{17} + \cdots - 419 \) Copy content Toggle raw display
$5$ \( T^{18} - 2 T^{17} + \cdots + 22653 \) Copy content Toggle raw display
$7$ \( T^{18} - 7 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{18} - 12 T^{17} + \cdots - 29511 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 114655507 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 2128858227 \) Copy content Toggle raw display
$19$ \( T^{18} - 15 T^{17} + \cdots + 196771 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 38210794551 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 777323664963 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 28513726123 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 656145023437 \) Copy content Toggle raw display
$41$ \( T^{18} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 13736188957 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 72741112263 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 2921564900367 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 5580460737837 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 119382632487 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 30147311431313 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 98\!\cdots\!53 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 281829445329643 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 923965602036121 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 410437034321931 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 756220759587 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 217185648279433 \) Copy content Toggle raw display
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