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Properties

Label 504.2.bk.a
Level $504$
Weight $2$
Character orbit 504.bk
Analytic conductor $4.024$
Analytic rank $0$
Dimension $12$
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] = [504,2,Mod(19,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bk (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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.144054149089536.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{11} - \beta_{10} + \cdots - \beta_{3}) q^{4}+ \cdots + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{11} - \beta_{10} + \cdots - \beta_{3}) q^{4}+ \cdots + (2 \beta_{11} - 2 \beta_{10} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{8} + 6 q^{10} + 6 q^{11} - 6 q^{14} + 6 q^{17} - 6 q^{19} + 24 q^{22} - 6 q^{26} + 6 q^{28} - 18 q^{35} + 24 q^{38} + 42 q^{40} - 6 q^{44} - 18 q^{46} - 12 q^{49} + 48 q^{50} - 24 q^{52} + 18 q^{58} - 42 q^{59} - 72 q^{64} + 12 q^{65} + 30 q^{67} + 36 q^{68} + 30 q^{70} + 18 q^{73} - 12 q^{74} - 36 q^{80} + 54 q^{82} + 6 q^{88} - 18 q^{89} - 72 q^{91} - 60 q^{92} - 12 q^{94} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3057 \nu^{11} + 45996 \nu^{10} - 86008 \nu^{9} - 97831 \nu^{8} - 142271 \nu^{7} + \cdots + 2179472 ) / 1375087 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1900 \nu^{11} - 5021 \nu^{10} + 17481 \nu^{9} + 21324 \nu^{8} - 69565 \nu^{7} - 118672 \nu^{6} + \cdots - 210768 ) / 289492 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 89065 \nu^{11} + 12805 \nu^{10} + 165954 \nu^{9} + 1037153 \nu^{8} - 3089444 \nu^{7} + \cdots + 1209724 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17437 \nu^{11} + 7066 \nu^{10} - 100969 \nu^{9} - 163381 \nu^{8} + 761375 \nu^{7} - 420189 \nu^{6} + \cdots - 1691468 ) / 785764 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 246051 \nu^{11} + 871028 \nu^{10} + 6163 \nu^{9} - 885875 \nu^{8} - 12628835 \nu^{7} + \cdots + 3141260 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 146932 \nu^{11} + 319768 \nu^{10} + 291913 \nu^{9} + 229501 \nu^{8} - 7300724 \nu^{7} + \cdots + 7497160 ) / 2750174 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2236 \nu^{11} + 4785 \nu^{10} + 6307 \nu^{9} - 1316 \nu^{8} - 113135 \nu^{7} + 323728 \nu^{6} + \cdots - 27316 ) / 41356 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 228352 \nu^{11} - 272019 \nu^{10} - 940304 \nu^{9} - 525407 \nu^{8} + 10909881 \nu^{7} + \cdots + 2200946 ) / 2750174 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 475721 \nu^{11} - 1135999 \nu^{10} - 834872 \nu^{9} + 354939 \nu^{8} + 23253862 \nu^{7} + \cdots + 5254820 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26339 \nu^{11} + 54578 \nu^{10} + 59599 \nu^{9} + 15779 \nu^{8} - 1269817 \nu^{7} + \cdots - 105412 ) / 289492 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 191973 \nu^{11} + 499534 \nu^{10} + 212817 \nu^{9} - 108091 \nu^{8} - 9329027 \nu^{7} + \cdots + 1072580 ) / 785764 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{10} - 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 5 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots + 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 22 \beta_{7} + 8 \beta_{6} + \beta_{5} + 8 \beta_{4} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{11} - 24 \beta_{10} - 16 \beta_{8} + 23 \beta_{7} - 5 \beta_{6} + 13 \beta_{5} + 25 \beta_{3} + \cdots - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23 \beta_{11} + 93 \beta_{10} - 63 \beta_{9} + 73 \beta_{8} - 17 \beta_{7} + 9 \beta_{6} - 40 \beta_{5} + \cdots + 78 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 43 \beta_{11} - 247 \beta_{10} + 69 \beta_{9} - 293 \beta_{8} - 342 \beta_{7} + 134 \beta_{6} + \cdots - 153 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 212 \beta_{11} + 474 \beta_{10} - 164 \beta_{9} + 666 \beta_{8} + 1329 \beta_{7} - 483 \beta_{6} + \cdots - 181 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 291 \beta_{11} + 501 \beta_{10} - 291 \beta_{9} - 467 \beta_{8} - 4069 \beta_{7} + 1347 \beta_{6} + \cdots + 1638 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 47 \beta_{11} - 4907 \beta_{10} + 2599 \beta_{9} - 2699 \beta_{8} + 6710 \beta_{7} - 2216 \beta_{6} + \cdots - 7147 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3036 \beta_{11} + 22620 \beta_{10} - 8568 \beta_{9} + 20124 \beta_{8} + 4207 \beta_{7} - 2529 \beta_{6} + \cdots + 18491 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15225 \beta_{11} - 56695 \beta_{10} + 24185 \beta_{9} - 63533 \beta_{8} - 73831 \beta_{7} + 26995 \beta_{6} + \cdots - 24794 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1 + \beta_{7}\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.186445 − 1.54034i
2.00233 + 0.854000i
−2.37165 + 1.78079i
1.09935 − 0.468876i
0.609850 + 0.457915i
−0.0263223 − 0.217464i
2.00233 − 0.854000i
0.186445 + 1.54034i
1.09935 + 0.468876i
−2.37165 − 1.78079i
−0.0263223 + 0.217464i
0.609850 − 0.457915i
−1.13090 − 0.849154i 0 0.557875 + 1.92062i −1.03926 + 1.80005i 0 1.25203 − 2.33076i 1.00000 − 2.64575i 0 2.70382 − 1.15319i
19.2 −1.13090 + 0.849154i 0 0.557875 − 1.92062i 1.59713 − 2.76632i 0 −0.694153 − 2.55307i 1.00000 + 2.64575i 0 0.542829 + 4.48465i
19.3 −0.169938 − 1.40397i 0 −1.94224 + 0.477176i −0.345107 + 0.597743i 0 −2.63639 + 0.222310i 1.00000 + 2.64575i 0 0.897858 + 0.382939i
19.4 −0.169938 + 1.40397i 0 −1.94224 − 0.477176i −1.59713 + 2.76632i 0 0.694153 + 2.55307i 1.00000 − 2.64575i 0 −3.61240 − 2.71243i
19.5 1.30084 − 0.554812i 0 1.38437 − 1.44344i 0.345107 − 0.597743i 0 2.63639 − 0.222310i 1.00000 − 2.64575i 0 0.117294 − 0.969037i
19.6 1.30084 + 0.554812i 0 1.38437 + 1.44344i 1.03926 − 1.80005i 0 −1.25203 + 2.33076i 1.00000 + 2.64575i 0 2.35060 − 1.76498i
451.1 −1.13090 − 0.849154i 0 0.557875 + 1.92062i 1.59713 + 2.76632i 0 −0.694153 + 2.55307i 1.00000 − 2.64575i 0 0.542829 − 4.48465i
451.2 −1.13090 + 0.849154i 0 0.557875 − 1.92062i −1.03926 − 1.80005i 0 1.25203 + 2.33076i 1.00000 + 2.64575i 0 2.70382 + 1.15319i
451.3 −0.169938 − 1.40397i 0 −1.94224 + 0.477176i −1.59713 − 2.76632i 0 0.694153 − 2.55307i 1.00000 + 2.64575i 0 −3.61240 + 2.71243i
451.4 −0.169938 + 1.40397i 0 −1.94224 − 0.477176i −0.345107 − 0.597743i 0 −2.63639 − 0.222310i 1.00000 − 2.64575i 0 0.897858 − 0.382939i
451.5 1.30084 − 0.554812i 0 1.38437 − 1.44344i 1.03926 + 1.80005i 0 −1.25203 − 2.33076i 1.00000 − 2.64575i 0 2.35060 + 1.76498i
451.6 1.30084 + 0.554812i 0 1.38437 + 1.44344i 0.345107 + 0.597743i 0 2.63639 + 0.222310i 1.00000 + 2.64575i 0 0.117294 + 0.969037i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
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 504.2.bk.a 12
3.b odd 2 1 56.2.m.a ✓ 12
4.b odd 2 1 2016.2.bs.a 12
7.d odd 6 1 inner 504.2.bk.a 12
8.b even 2 1 2016.2.bs.a 12
8.d odd 2 1 inner 504.2.bk.a 12
12.b even 2 1 224.2.q.a 12
21.c even 2 1 392.2.m.g 12
21.g even 6 1 56.2.m.a ✓ 12
21.g even 6 1 392.2.e.e 12
21.h odd 6 1 392.2.e.e 12
21.h odd 6 1 392.2.m.g 12
24.f even 2 1 56.2.m.a ✓ 12
24.h odd 2 1 224.2.q.a 12
28.f even 6 1 2016.2.bs.a 12
56.j odd 6 1 2016.2.bs.a 12
56.m even 6 1 inner 504.2.bk.a 12
84.h odd 2 1 1568.2.q.g 12
84.j odd 6 1 224.2.q.a 12
84.j odd 6 1 1568.2.e.e 12
84.n even 6 1 1568.2.e.e 12
84.n even 6 1 1568.2.q.g 12
168.e odd 2 1 392.2.m.g 12
168.i even 2 1 1568.2.q.g 12
168.s odd 6 1 1568.2.e.e 12
168.s odd 6 1 1568.2.q.g 12
168.v even 6 1 392.2.e.e 12
168.v even 6 1 392.2.m.g 12
168.ba even 6 1 224.2.q.a 12
168.ba even 6 1 1568.2.e.e 12
168.be odd 6 1 56.2.m.a ✓ 12
168.be odd 6 1 392.2.e.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.m.a ✓ 12 3.b odd 2 1
56.2.m.a ✓ 12 21.g even 6 1
56.2.m.a ✓ 12 24.f even 2 1
56.2.m.a ✓ 12 168.be odd 6 1
224.2.q.a 12 12.b even 2 1
224.2.q.a 12 24.h odd 2 1
224.2.q.a 12 84.j odd 6 1
224.2.q.a 12 168.ba even 6 1
392.2.e.e 12 21.g even 6 1
392.2.e.e 12 21.h odd 6 1
392.2.e.e 12 168.v even 6 1
392.2.e.e 12 168.be odd 6 1
392.2.m.g 12 21.c even 2 1
392.2.m.g 12 21.h odd 6 1
392.2.m.g 12 168.e odd 2 1
392.2.m.g 12 168.v even 6 1
504.2.bk.a 12 1.a even 1 1 trivial
504.2.bk.a 12 7.d odd 6 1 inner
504.2.bk.a 12 8.d odd 2 1 inner
504.2.bk.a 12 56.m even 6 1 inner
1568.2.e.e 12 84.j odd 6 1
1568.2.e.e 12 84.n even 6 1
1568.2.e.e 12 168.s odd 6 1
1568.2.e.e 12 168.ba even 6 1
1568.2.q.g 12 84.h odd 2 1
1568.2.q.g 12 84.n even 6 1
1568.2.q.g 12 168.i even 2 1
1568.2.q.g 12 168.s odd 6 1
2016.2.bs.a 12 4.b odd 2 1
2016.2.bs.a 12 8.b even 2 1
2016.2.bs.a 12 28.f even 6 1
2016.2.bs.a 12 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 15T_{5}^{10} + 174T_{5}^{8} + 723T_{5}^{6} + 2286T_{5}^{4} + 1071T_{5}^{2} + 441 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{3} + 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{10} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} + 12 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 36 T^{4} + \cdots - 336)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 3 T^{5} + \cdots + 1083)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 3 T^{5} + \cdots + 147)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 69 T^{10} + \cdots + 45252529 \) Copy content Toggle raw display
$29$ \( (T^{6} + 48 T^{4} + \cdots + 112)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 99 T^{10} + \cdots + 441 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 3074369809 \) Copy content Toggle raw display
$41$ \( (T^{6} + 144 T^{4} + \cdots + 52272)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} + 123 T^{10} + \cdots + 6456681 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 108651322129 \) Copy content Toggle raw display
$59$ \( (T^{6} + 21 T^{5} + \cdots + 133563)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 42362284041 \) Copy content Toggle raw display
$67$ \( (T^{6} - 15 T^{5} + \cdots + 52441)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 28)^{6} \) Copy content Toggle raw display
$73$ \( (T^{6} - 9 T^{5} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 31317319089 \) Copy content Toggle raw display
$83$ \( (T^{6} + 228 T^{4} + \cdots + 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 3)^{6} \) Copy content Toggle raw display
$97$ \( (T^{6} + 360 T^{4} + \cdots + 134832)^{2} \) Copy content Toggle raw display
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