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Properties

Label 56.3.k.d
Level $56$
Weight $3$
Character orbit 56.k
Analytic conductor $1.526$
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] = [56,3,Mod(11,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 56.k (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: \(1.52588948042\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{10} - \beta_{9} + \beta_{8}) q^{3} + (\beta_{9} + \beta_{7} + \beta_{2}) q^{4} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{11} + 2 \beta_{10} + \cdots - 5) q^{6}+ \cdots + (4 \beta_{9} - 6 \beta_{7} - 4 \beta_{6} + \cdots - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{10} - \beta_{9} + \beta_{8}) q^{3} + (\beta_{9} + \beta_{7} + \beta_{2}) q^{4} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{11} + 2 \beta_{10} + \cdots - 5) q^{6}+ \cdots + (21 \beta_{10} - 21 \beta_{8} + \cdots + 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 6 q^{3} - 4 q^{4} - 56 q^{6} + 8 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 6 q^{3} - 4 q^{4} - 56 q^{6} + 8 q^{8} - 40 q^{9} - 6 q^{10} + 30 q^{11} + 32 q^{12} + 52 q^{14} + 16 q^{16} + 30 q^{17} - 16 q^{18} + 78 q^{19} + 48 q^{20} + 24 q^{22} - 76 q^{24} - 92 q^{25} - 128 q^{26} + 156 q^{27} - 4 q^{28} - 16 q^{30} + 112 q^{32} - 78 q^{33} + 76 q^{34} - 222 q^{35} - 248 q^{36} + 80 q^{38} + 44 q^{40} - 232 q^{41} + 132 q^{42} - 200 q^{43} + 132 q^{44} - 156 q^{46} + 176 q^{48} + 372 q^{49} + 48 q^{50} + 10 q^{51} + 132 q^{52} - 36 q^{54} - 112 q^{56} + 332 q^{57} + 4 q^{58} - 110 q^{59} + 84 q^{60} - 96 q^{62} - 160 q^{64} - 32 q^{65} - 138 q^{66} + 434 q^{67} + 96 q^{68} - 188 q^{70} - 328 q^{72} + 102 q^{73} - 34 q^{74} - 60 q^{75} - 168 q^{76} + 720 q^{78} - 256 q^{80} - 82 q^{81} - 24 q^{82} - 536 q^{83} - 624 q^{84} + 240 q^{86} - 204 q^{88} + 214 q^{89} + 440 q^{90} - 8 q^{91} + 160 q^{92} - 16 q^{94} + 48 q^{96} - 152 q^{97} + 382 q^{98} + 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2848753 \nu^{11} - 128409 \nu^{10} + 36949178 \nu^{9} + 33927641 \nu^{8} - 280693408 \nu^{7} + \cdots - 114476732 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 908144 \nu^{11} - 6494043 \nu^{10} + 3252536 \nu^{9} + 28087343 \nu^{8} + 91567337 \nu^{7} + \cdots - 26684466 ) / 5128417 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2016555 \nu^{11} + 4608465 \nu^{10} + 11867940 \nu^{9} + 1459025 \nu^{8} - 175954938 \nu^{7} + \cdots + 13204764 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 894552 \nu^{11} - 880921 \nu^{10} - 8410783 \nu^{9} - 6680758 \nu^{8} + 80688819 \nu^{7} + \cdots + 16276324 ) / 2930524 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3736366 \nu^{11} + 5070151 \nu^{10} + 28320593 \nu^{9} + 24171184 \nu^{8} - 310515677 \nu^{7} + \cdots - 32192540 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8673537 \nu^{11} + 20159299 \nu^{10} + 48705482 \nu^{9} + 5607853 \nu^{8} - 742433540 \nu^{7} + \cdots + 29235872 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8864783 \nu^{11} + 20455402 \nu^{10} + 48834261 \nu^{9} + 8785933 \nu^{8} - 751559539 \nu^{7} + \cdots + 49862132 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2781796 \nu^{11} + 6284192 \nu^{10} + 15510516 \nu^{9} + 3410783 \nu^{8} - 234558069 \nu^{7} + \cdots + 15160294 ) / 5128417 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15221487 \nu^{11} - 28755150 \nu^{10} - 95322325 \nu^{9} - 55362465 \nu^{8} + 1263522623 \nu^{7} + \cdots - 55637788 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8729167 \nu^{11} + 23461665 \nu^{10} + 41996941 \nu^{9} - 12733522 \nu^{8} - 752734228 \nu^{7} + \cdots + 65016998 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2679889 \nu^{11} + 7514003 \nu^{10} + 13528514 \nu^{9} - 8114019 \nu^{8} - 240378660 \nu^{7} + \cdots + 6095104 ) / 2930524 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{7} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} + 5\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} - 2\beta_{3} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{11} - 6\beta_{8} - 3\beta_{6} - 6\beta_{3} + 8\beta_{2} - 8\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{11} - 20\beta_{10} + 16\beta_{9} + 8\beta_{8} + 37\beta_{7} + 19\beta_{4} + 17\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 45\beta_{9} + 106\beta_{7} - 45\beta_{6} + 20\beta_{5} - 61\beta_{4} - 60\beta_{3} - 54\beta _1 + 106 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 231 \beta_{11} - 160 \beta_{10} + 10 \beta_{8} + 96 \beta_{6} - 160 \beta_{5} + 10 \beta_{3} + \cdots - 225 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -311\beta_{11} - 340\beta_{10} + 497\beta_{9} + 542\beta_{8} + 1168\beta_{7} - 311\beta_{4} - 276\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -356\beta_{9} - 837\beta_{7} + 356\beta_{6} - 1084\beta_{5} - 2367\beta_{4} - 452\beta_{3} - 2101\beta _1 - 837 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 401 \beta_{11} - 4112 \beta_{10} + 4372 \beta_{8} + 4779 \beta_{6} - 4112 \beta_{5} + 4372 \beta_{3} + \cdots - 11236 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 21471 \beta_{11} + 5536 \beta_{10} + 1610 \beta_{9} + 8394 \beta_{8} + 3785 \beta_{7} + \cdots - 19059 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 40931 \beta_{9} - 96238 \beta_{7} + 40931 \beta_{6} - 42140 \beta_{5} - 17901 \beta_{4} + \cdots - 96238 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.121721 − 0.507075i
−0.407369 − 0.812545i
0.378279 − 0.358951i
−2.29733 − 1.90372i
2.79733 + 1.03769i
0.907369 − 0.0534805i
0.121721 + 0.507075i
−0.407369 + 0.812545i
0.378279 + 0.358951i
−2.29733 + 1.90372i
2.79733 − 1.03769i
0.907369 + 0.0534805i
−1.78207 − 0.907869i 1.99052 − 3.44767i 2.35155 + 3.23577i 1.63031 − 0.941260i −6.67727 + 4.33687i −5.14749 − 4.74377i −1.25297 − 7.90127i −3.42430 − 5.93106i −3.75987 + 0.197284i
11.2 −1.19654 + 1.60259i −2.66613 + 4.61787i −1.13656 − 3.83513i 1.86796 − 1.07847i −4.21039 − 9.79818i −6.91861 + 1.06433i 7.50608 + 2.76746i −9.71647 − 16.8294i −0.506759 + 4.28400i
11.3 0.104798 − 1.99725i 1.99052 − 3.44767i −3.97803 − 0.418616i −1.63031 + 0.941260i −6.67727 − 4.33687i 5.14749 + 4.74377i −1.25297 + 7.90127i −3.42430 − 5.93106i 1.70908 + 3.35478i
11.4 0.371518 + 1.96519i −0.824388 + 1.42788i −3.72395 + 1.46021i −3.95004 + 2.28056i −3.11234 − 1.08960i 6.75545 + 1.83408i −4.25310 − 6.77577i 3.14077 + 5.43997i −5.94924 − 6.91531i
11.5 1.51615 + 1.30434i −0.824388 + 1.42788i 0.597396 + 3.95514i 3.95004 − 2.28056i −3.11234 + 1.08960i −6.75545 − 1.83408i −4.25310 + 6.77577i 3.14077 + 5.43997i 8.96346 + 1.69454i
11.6 1.98615 − 0.234945i −2.66613 + 4.61787i 3.88960 − 0.933271i −1.86796 + 1.07847i −4.21039 + 9.79818i 6.91861 − 1.06433i 7.50608 − 2.76746i −9.71647 − 16.8294i −3.45667 + 2.58086i
51.1 −1.78207 + 0.907869i 1.99052 + 3.44767i 2.35155 − 3.23577i 1.63031 + 0.941260i −6.67727 − 4.33687i −5.14749 + 4.74377i −1.25297 + 7.90127i −3.42430 + 5.93106i −3.75987 − 0.197284i
51.2 −1.19654 − 1.60259i −2.66613 − 4.61787i −1.13656 + 3.83513i 1.86796 + 1.07847i −4.21039 + 9.79818i −6.91861 − 1.06433i 7.50608 − 2.76746i −9.71647 + 16.8294i −0.506759 − 4.28400i
51.3 0.104798 + 1.99725i 1.99052 + 3.44767i −3.97803 + 0.418616i −1.63031 − 0.941260i −6.67727 + 4.33687i 5.14749 − 4.74377i −1.25297 − 7.90127i −3.42430 + 5.93106i 1.70908 − 3.35478i
51.4 0.371518 − 1.96519i −0.824388 − 1.42788i −3.72395 − 1.46021i −3.95004 − 2.28056i −3.11234 + 1.08960i 6.75545 − 1.83408i −4.25310 + 6.77577i 3.14077 − 5.43997i −5.94924 + 6.91531i
51.5 1.51615 − 1.30434i −0.824388 − 1.42788i 0.597396 − 3.95514i 3.95004 + 2.28056i −3.11234 − 1.08960i −6.75545 + 1.83408i −4.25310 − 6.77577i 3.14077 − 5.43997i 8.96346 − 1.69454i
51.6 1.98615 + 0.234945i −2.66613 − 4.61787i 3.88960 + 0.933271i −1.86796 − 1.07847i −4.21039 − 9.79818i 6.91861 + 1.06433i 7.50608 + 2.76746i −9.71647 + 16.8294i −3.45667 − 2.58086i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.3.k.d ✓ 12
4.b odd 2 1 224.3.o.d 12
7.b odd 2 1 392.3.k.l 12
7.c even 3 1 inner 56.3.k.d ✓ 12
7.c even 3 1 392.3.g.j 6
7.d odd 6 1 392.3.g.i 6
7.d odd 6 1 392.3.k.l 12
8.b even 2 1 224.3.o.d 12
8.d odd 2 1 inner 56.3.k.d ✓ 12
28.f even 6 1 1568.3.g.l 6
28.g odd 6 1 224.3.o.d 12
28.g odd 6 1 1568.3.g.j 6
56.e even 2 1 392.3.k.l 12
56.j odd 6 1 1568.3.g.l 6
56.k odd 6 1 inner 56.3.k.d ✓ 12
56.k odd 6 1 392.3.g.j 6
56.m even 6 1 392.3.g.i 6
56.m even 6 1 392.3.k.l 12
56.p even 6 1 224.3.o.d 12
56.p even 6 1 1568.3.g.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d ✓ 12 1.a even 1 1 trivial
56.3.k.d ✓ 12 7.c even 3 1 inner
56.3.k.d ✓ 12 8.d odd 2 1 inner
56.3.k.d ✓ 12 56.k odd 6 1 inner
224.3.o.d 12 4.b odd 2 1
224.3.o.d 12 8.b even 2 1
224.3.o.d 12 28.g odd 6 1
224.3.o.d 12 56.p even 6 1
392.3.g.i 6 7.d odd 6 1
392.3.g.i 6 56.m even 6 1
392.3.g.j 6 7.c even 3 1
392.3.g.j 6 56.k odd 6 1
392.3.k.l 12 7.b odd 2 1
392.3.k.l 12 7.d odd 6 1
392.3.k.l 12 56.e even 2 1
392.3.k.l 12 56.m even 6 1
1568.3.g.j 6 28.g odd 6 1
1568.3.g.j 6 56.p even 6 1
1568.3.g.l 6 28.f even 6 1
1568.3.g.l 6 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(56, [\chi])\):

\( T_{3}^{6} + 3T_{3}^{5} + 28T_{3}^{4} + 13T_{3}^{3} + 466T_{3}^{2} + 665T_{3} + 1225 \) Copy content Toggle raw display
\( T_{5}^{12} - 29T_{5}^{10} + 654T_{5}^{8} - 4737T_{5}^{6} + 25022T_{5}^{4} - 64141T_{5}^{2} + 117649 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( (T^{6} + 3 T^{5} + \cdots + 1225)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 29 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} - 15 T^{5} + \cdots + 263169)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 928 T^{4} + \cdots + 20420848)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 15 T^{5} + \cdots + 1225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 39 T^{5} + \cdots + 290521)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{6} + 1384 T^{4} + \cdots + 35753200)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 41\!\cdots\!89 \) Copy content Toggle raw display
$41$ \( (T^{3} + 58 T^{2} + \cdots - 106036)^{4} \) Copy content Toggle raw display
$43$ \( (T^{3} + 50 T^{2} + \cdots - 77000)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 35\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 82\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( (T^{6} + 55 T^{5} + \cdots + 43020481)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( (T^{6} - 217 T^{5} + \cdots + 135936003025)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 7184 T^{4} + \cdots + 1372000000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 51 T^{5} + \cdots + 2144153025)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{3} + 134 T^{2} + \cdots - 185080)^{4} \) Copy content Toggle raw display
$89$ \( (T^{6} - 107 T^{5} + \cdots + 825668447569)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 38 T^{2} + \cdots - 117740)^{4} \) Copy content Toggle raw display
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