↑

Properties

Label 2548.2.j.r
Level $2548$
Weight $2$
Character orbit 2548.j
Analytic conductor $20.346$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,2,Mod(1145,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.1145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2548.j (of order \(3\), degree \(2\), not 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: \(20.3458824350\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 10 x^{10} - 8 x^{9} + 80 x^{8} - 56 x^{7} + 220 x^{6} - 240 x^{5} + 484 x^{4} - 336 x^{3} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{2} + \beta_1) q^{3} + ( - \beta_{10} + 2 \beta_{6} - 2) q^{5} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \cdots - 1) q^{9} + ( - \beta_{10} + \beta_{6} + \cdots + 2 \beta_1) q^{11}+ \cdots + (\beta_{9} + 8 \beta_{5} - 4 \beta_{4} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{5} - 6 q^{9} + 4 q^{11} - 12 q^{13} + 8 q^{15} - 16 q^{17} - 10 q^{19} + 2 q^{23} - 12 q^{25} - 24 q^{27} + 4 q^{29} - 6 q^{31} - 28 q^{33} - 16 q^{37} + 16 q^{41} + 12 q^{43} - 34 q^{45} - 30 q^{47}+ \cdots - 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 10 x^{10} - 8 x^{9} + 80 x^{8} - 56 x^{7} + 220 x^{6} - 240 x^{5} + 484 x^{4} - 336 x^{3} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 379000 \nu^{11} + 322300 \nu^{10} + 3778645 \nu^{9} + 152800 \nu^{8} + 28166440 \nu^{7} + \cdots - 12684696 ) / 86544668 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3905120 \nu^{11} - 7830811 \nu^{10} - 41548809 \nu^{9} - 37653720 \nu^{8} - 264964348 \nu^{7} + \cdots + 1118396056 ) / 432723340 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3705810 \nu^{11} + 4692768 \nu^{10} + 38183017 \nu^{9} + 13824960 \nu^{8} + 266775954 \nu^{7} + \cdots + 270746912 ) / 216361670 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11205 \nu^{11} + 10949 \nu^{10} + 114461 \nu^{9} + 15880 \nu^{8} + 824777 \nu^{7} + 187358 \nu^{6} + \cdots + 510976 ) / 347290 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3171174 \nu^{11} - 379000 \nu^{10} - 32034040 \nu^{9} + 21590747 \nu^{8} - 253846720 \nu^{7} + \cdots + 103206288 ) / 86544668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27192621 \nu^{11} - 7411620 \nu^{10} + 262540674 \nu^{9} - 293907002 \nu^{8} + 2147759760 \nu^{7} + \cdots - 826492776 ) / 432723340 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29498324 \nu^{11} - 4899106 \nu^{10} - 291648120 \nu^{9} + 197195993 \nu^{8} + \cdots + 139153820 ) / 432723340 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15940620 \nu^{11} - 13784186 \nu^{10} - 161115064 \nu^{9} - 8253520 \nu^{8} + \cdots - 608382084 ) / 216361670 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35536656 \nu^{11} - 2624119 \nu^{10} - 364956680 \nu^{9} + 245386747 \nu^{8} + \cdots + 201658580 ) / 432723340 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52697053 \nu^{11} - 16024810 \nu^{10} + 514656112 \nu^{9} - 582294921 \nu^{8} + \cdots - 1617413908 ) / 432723340 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} - 4\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{5} - \beta_{4} + 6\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{10} - 9\beta_{8} - 2\beta_{7} + 25\beta_{6} - 2\beta_{5} + 8\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{11} + 2 \beta_{10} + 13 \beta_{8} + 21 \beta_{7} - 22 \beta_{6} + 13 \beta_{4} + \cdots - 42 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{9} + 28\beta_{5} - 74\beta_{4} - 40\beta_{3} + 68\beta_{2} + 176 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 70 \beta_{11} - 28 \beta_{10} - 70 \beta_{9} - 130 \beta_{8} - 178 \beta_{7} + 216 \beta_{6} + \cdots - 216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 60 \beta_{11} + 286 \beta_{10} + 594 \beta_{8} + 292 \beta_{7} - 1308 \beta_{6} + 594 \beta_{4} + \cdots - 590 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 534\beta_{9} + 1436\beta_{5} - 1190\beta_{4} - 304\beta_{3} + 2432\beta_{2} + 2020 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 656 \beta_{11} - 2128 \beta_{10} - 656 \beta_{9} - 4754 \beta_{8} - 2732 \beta_{7} + 10022 \beta_{6} + \cdots - 10022 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4098 \beta_{11} + 2980 \beta_{10} + 10466 \beta_{8} + 11478 \beta_{7} - 18252 \beta_{6} + \cdots - 19228 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(1\) \(-1 + \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1145.1
0.652846 + 1.13076i
0.0779999 + 0.135100i
1.23667 + 2.14198i
0.379586 + 0.657462i
−1.43795 − 2.49061i
−0.909151 − 1.57470i
0.652846 − 1.13076i
0.0779999 − 0.135100i
1.23667 − 2.14198i
0.379586 − 0.657462i
−1.43795 + 2.49061i
−0.909151 + 1.57470i
0 −1.35995 + 2.35551i 0 −1.92853 − 3.34030i 0 0 0 −2.19894 − 3.80868i 0
1145.2 0 −0.785107 + 1.35984i 0 0.299524 + 0.518790i 0 0 0 0.267215 + 0.462830i 0
1145.3 0 −0.529566 + 0.917235i 0 −2.04647 − 3.54459i 0 0 0 0.939121 + 1.62660i 0
1145.4 0 0.327521 − 0.567283i 0 0.369059 + 0.639229i 0 0 0 1.28546 + 2.22648i 0
1145.5 0 0.730846 − 1.26586i 0 −0.163891 − 0.283868i 0 0 0 0.431728 + 0.747776i 0
1145.6 0 1.61626 − 2.79944i 0 −1.52970 − 2.64951i 0 0 0 −3.72458 − 6.45116i 0
1353.1 0 −1.35995 − 2.35551i 0 −1.92853 + 3.34030i 0 0 0 −2.19894 + 3.80868i 0
1353.2 0 −0.785107 − 1.35984i 0 0.299524 − 0.518790i 0 0 0 0.267215 − 0.462830i 0
1353.3 0 −0.529566 − 0.917235i 0 −2.04647 + 3.54459i 0 0 0 0.939121 − 1.62660i 0
1353.4 0 0.327521 + 0.567283i 0 0.369059 − 0.639229i 0 0 0 1.28546 − 2.22648i 0
1353.5 0 0.730846 + 1.26586i 0 −0.163891 + 0.283868i 0 0 0 0.431728 − 0.747776i 0
1353.6 0 1.61626 + 2.79944i 0 −1.52970 + 2.64951i 0 0 0 −3.72458 + 6.45116i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1145.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.j.r 12
7.b odd 2 1 2548.2.j.s 12
7.c even 3 1 2548.2.a.s yes 6
7.c even 3 1 inner 2548.2.j.r 12
7.d odd 6 1 2548.2.a.r ✓ 6
7.d odd 6 1 2548.2.j.s 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2548.2.a.r ✓ 6 7.d odd 6 1
2548.2.a.s yes 6 7.c even 3 1
2548.2.j.r 12 1.a even 1 1 trivial
2548.2.j.r 12 7.c even 3 1 inner
2548.2.j.s 12 7.b odd 2 1
2548.2.j.s 12 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2548, [\chi])\):

\( T_{3}^{12} + 12 T_{3}^{10} + 8 T_{3}^{9} + 116 T_{3}^{8} + 56 T_{3}^{7} + 324 T_{3}^{6} + 80 T_{3}^{5} + \cdots + 196 \) Copy content Toggle raw display
\( T_{5}^{12} + 10 T_{5}^{11} + 71 T_{5}^{10} + 274 T_{5}^{9} + 808 T_{5}^{8} + 1178 T_{5}^{7} + 1381 T_{5}^{6} + \cdots + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 12 T^{10} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( T^{12} + 10 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 6724 \) Copy content Toggle raw display
$13$ \( (T + 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 16 T^{11} + \cdots + 64516 \) Copy content Toggle raw display
$19$ \( T^{12} + 10 T^{11} + \cdots + 13300609 \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} + \cdots + 23804641 \) Copy content Toggle raw display
$29$ \( (T^{6} - 2 T^{5} - 57 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + \cdots + 7921 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2907150724 \) Copy content Toggle raw display
$41$ \( (T^{6} - 8 T^{5} + \cdots - 76816)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 6 T^{5} - 105 T^{4} + \cdots + 73)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 209699361 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1070532961 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 368793616 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1193840704 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 397444096 \) Copy content Toggle raw display
$71$ \( (T^{6} - 212 T^{4} + \cdots - 15842)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 14 T^{11} + \cdots + 1896129 \) Copy content Toggle raw display
$79$ \( T^{12} + 14 T^{11} + \cdots + 703921 \) Copy content Toggle raw display
$83$ \( (T^{6} + 14 T^{5} + \cdots - 38089)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 73501174321 \) Copy content Toggle raw display
$97$ \( (T^{6} - 22 T^{5} + \cdots - 233)^{2} \) Copy content Toggle raw display
show more
show less