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Properties

Label 56.3.h.d
Level $56$
Weight $3$
Character orbit 56.h
Analytic conductor $1.526$
Analytic rank $0$
Dimension $8$
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(13,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 56.h (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.976966189056.51
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + \beta_{6} q^{3} + (\beta_{5} - \beta_1 - 2) q^{4} + (\beta_{6} - \beta_{4}) q^{5} + (\beta_{4} + \beta_{3}) q^{6} + (\beta_{7} - \beta_1 - 2) q^{7} + ( - 2 \beta_{5} + 2 \beta_{2} + \cdots + 2) q^{8}+ \cdots + (4 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + \beta_{6} q^{3} + (\beta_{5} - \beta_1 - 2) q^{4} + (\beta_{6} - \beta_{4}) q^{5} + (\beta_{4} + \beta_{3}) q^{6} + (\beta_{7} - \beta_1 - 2) q^{7} + ( - 2 \beta_{5} + 2 \beta_{2} + \cdots + 2) q^{8}+ \cdots + (10 \beta_{5} - 36 \beta_{2} + \cdots - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 16 q^{4} - 16 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 16 q^{4} - 16 q^{7} + 8 q^{8} + 8 q^{9} - 4 q^{14} + 64 q^{15} - 16 q^{16} + 44 q^{18} - 72 q^{22} - 32 q^{23} - 24 q^{25} + 56 q^{28} - 56 q^{30} + 176 q^{32} - 112 q^{36} - 128 q^{39} + 120 q^{42} - 96 q^{44} + 160 q^{46} - 280 q^{49} - 132 q^{50} + 56 q^{56} + 304 q^{57} + 288 q^{58} - 80 q^{60} - 112 q^{63} - 64 q^{64} + 272 q^{65} - 216 q^{70} + 544 q^{71} - 280 q^{72} - 240 q^{74} - 200 q^{78} - 32 q^{79} - 328 q^{81} + 336 q^{84} - 264 q^{86} + 336 q^{88} - 224 q^{92} - 256 q^{95} + 236 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 434 \nu^{7} - 4500 \nu^{6} - 13152 \nu^{5} + 45925 \nu^{4} + 160296 \nu^{3} - 212975 \nu^{2} + \cdots + 2383525 ) / 1582309 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1923912 \nu^{7} + 1015375 \nu^{6} + 14571202 \nu^{5} - 37305672 \nu^{4} - 246088672 \nu^{3} + \cdots - 1128588579 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2149232 \nu^{7} + 17601190 \nu^{6} + 42539286 \nu^{5} - 118886838 \nu^{4} - 311444344 \nu^{3} + \cdots - 1083690552 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2679764 \nu^{7} - 208186 \nu^{6} + 31814108 \nu^{5} + 10053780 \nu^{4} - 411714584 \nu^{3} + \cdots - 982876622 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2867686 \nu^{7} + 2156686 \nu^{6} - 26119292 \nu^{5} - 29939305 \nu^{4} + 342306416 \nu^{3} + \cdots - 49189703 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3872412 \nu^{7} - 6558661 \nu^{6} + 34456727 \nu^{5} + 55216902 \nu^{4} - 394683447 \nu^{3} + \cdots - 1714881404 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12558 \nu^{7} - 22917 \nu^{6} - 180905 \nu^{5} + 49278 \nu^{4} + 1537867 \nu^{3} - 5066755 \nu^{2} + \cdots + 1827118 ) / 1582309 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta_{2} - \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 4\beta_{6} + 14\beta_{5} - 4\beta_{4} - 3\beta_{3} + 4\beta_{2} + 8\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{7} - 8\beta_{6} - 33\beta_{5} + 24\beta_{4} + 16\beta_{3} - 42\beta_{2} + 29\beta _1 - 94 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -50\beta_{7} + 94\beta_{6} + 56\beta_{5} - 138\beta_{4} - 25\beta_{3} - 6\beta_{2} - 214\beta _1 + 372 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -82\beta_{7} - 402\beta_{6} + 93\beta_{5} + 518\beta_{4} - 32\beta_{3} - 48\beta_{2} + 801\beta _1 - 1498 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 861\beta_{7} + 969\beta_{6} - 1432\beta_{5} - 1067\beta_{4} + 693\beta_{3} - 1306\beta_{2} - 3538\beta _1 + 4072 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.52814 + 1.99551i
−3.26020 + 1.99551i
1.52814 − 1.99551i
−3.26020 − 1.99551i
−0.639946 − 0.719687i
2.37200 − 0.719687i
−0.639946 + 0.719687i
2.37200 + 0.719687i
−1.36603 − 1.46081i −1.75265 −0.267949 + 3.99102i −6.54099 2.39417 + 2.56030i −0.267949 + 6.99487i 6.19615 − 5.06040i −5.92820 8.93516 + 9.55517i
13.2 −1.36603 − 1.46081i 1.75265 −0.267949 + 3.99102i 6.54099 −2.39417 − 2.56030i −0.267949 − 6.99487i 6.19615 − 5.06040i −5.92820 −8.93516 − 9.55517i
13.3 −1.36603 + 1.46081i −1.75265 −0.267949 − 3.99102i −6.54099 2.39417 − 2.56030i −0.267949 − 6.99487i 6.19615 + 5.06040i −5.92820 8.93516 − 9.55517i
13.4 −1.36603 + 1.46081i 1.75265 −0.267949 − 3.99102i 6.54099 −2.39417 + 2.56030i −0.267949 + 6.99487i 6.19615 + 5.06040i −5.92820 −8.93516 + 9.55517i
13.5 0.366025 − 1.96622i −4.11439 −3.73205 − 1.43937i −1.10245 −1.50597 + 8.08980i −3.73205 − 5.92214i −4.19615 + 6.81119i 7.92820 −0.403524 + 2.16766i
13.6 0.366025 − 1.96622i 4.11439 −3.73205 − 1.43937i 1.10245 1.50597 − 8.08980i −3.73205 + 5.92214i −4.19615 + 6.81119i 7.92820 0.403524 − 2.16766i
13.7 0.366025 + 1.96622i −4.11439 −3.73205 + 1.43937i −1.10245 −1.50597 − 8.08980i −3.73205 + 5.92214i −4.19615 − 6.81119i 7.92820 −0.403524 − 2.16766i
13.8 0.366025 + 1.96622i 4.11439 −3.73205 + 1.43937i 1.10245 1.50597 + 8.08980i −3.73205 − 5.92214i −4.19615 − 6.81119i 7.92820 0.403524 + 2.16766i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.3.h.d ✓ 8
3.b odd 2 1 504.3.l.f 8
4.b odd 2 1 224.3.h.d 8
7.b odd 2 1 inner 56.3.h.d ✓ 8
7.c even 3 2 392.3.j.d 16
7.d odd 6 2 392.3.j.d 16
8.b even 2 1 inner 56.3.h.d ✓ 8
8.d odd 2 1 224.3.h.d 8
12.b even 2 1 2016.3.l.f 8
21.c even 2 1 504.3.l.f 8
24.f even 2 1 2016.3.l.f 8
24.h odd 2 1 504.3.l.f 8
28.d even 2 1 224.3.h.d 8
56.e even 2 1 224.3.h.d 8
56.h odd 2 1 inner 56.3.h.d ✓ 8
56.j odd 6 2 392.3.j.d 16
56.p even 6 2 392.3.j.d 16
84.h odd 2 1 2016.3.l.f 8
168.e odd 2 1 2016.3.l.f 8
168.i even 2 1 504.3.l.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.d ✓ 8 1.a even 1 1 trivial
56.3.h.d ✓ 8 7.b odd 2 1 inner
56.3.h.d ✓ 8 8.b even 2 1 inner
56.3.h.d ✓ 8 56.h odd 2 1 inner
224.3.h.d 8 4.b odd 2 1
224.3.h.d 8 8.d odd 2 1
224.3.h.d 8 28.d even 2 1
224.3.h.d 8 56.e even 2 1
392.3.j.d 16 7.c even 3 2
392.3.j.d 16 7.d odd 6 2
392.3.j.d 16 56.j odd 6 2
392.3.j.d 16 56.p even 6 2
504.3.l.f 8 3.b odd 2 1
504.3.l.f 8 21.c even 2 1
504.3.l.f 8 24.h odd 2 1
504.3.l.f 8 168.i even 2 1
2016.3.l.f 8 12.b even 2 1
2016.3.l.f 8 24.f even 2 1
2016.3.l.f 8 84.h odd 2 1
2016.3.l.f 8 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 20T_{3}^{2} + 52 \) acting on \(S_{3}^{\mathrm{new}}(56, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 6 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 20 T^{2} + 52)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 44 T^{2} + 52)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 120 T^{2} + 528)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 332 T^{2} + 27508)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1008 T^{2} + 247104)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 788 T^{2} + 114868)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T - 416)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1920 T^{2} + 33792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3984 T^{2} + 3322176)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 864 T^{2} + 76032)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1344 T^{2} + 439296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6072 T^{2} + 7730448)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3984 T^{2} + 3322176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3456 T^{2} + 2737152)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4724 T^{2} + 5029492)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 44 T^{2} + 52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6072 T^{2} + 7730448)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 136 T + 2596)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11952 T^{2} + 29899584)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 956)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 20948 T^{2} + 114868)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 14256 T^{2} + 29899584)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 24048 T^{2} + 102163776)^{2} \) Copy content Toggle raw display
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