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Properties

Label 56.4.b.b
Level $56$
Weight $4$
Character orbit 56.b
Analytic conductor $3.304$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,4,Mod(29,56)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.29");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 56.b (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: \(3.30410696032\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} + 4x^{8} + 2x^{7} + 12x^{6} + 32x^{5} + 96x^{4} + 128x^{3} + 2048x^{2} - 12288x + 32768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + \beta_{2} q^{4} + ( - \beta_{6} - \beta_1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4} - 2) q^{6} + 7 q^{7} + (\beta_{4} + \beta_{3} - 2) q^{8} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \cdots - 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{4} q^{3} + \beta_{2} q^{4} + ( - \beta_{6} - \beta_1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4} - 2) q^{6} + 7 q^{7} + (\beta_{4} + \beta_{3} - 2) q^{8} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \cdots - 11) q^{9}+ \cdots + ( - 25 \beta_{8} - 27 \beta_{7} + \cdots - 38) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{2} + q^{4} - 26 q^{6} + 70 q^{7} - 15 q^{8} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{2} + q^{4} - 26 q^{6} + 70 q^{7} - 15 q^{8} - 126 q^{9} + 48 q^{10} + 62 q^{12} + 21 q^{14} - 60 q^{15} - 103 q^{16} + 196 q^{17} - 219 q^{18} + 324 q^{20} + 466 q^{22} + 180 q^{23} - 498 q^{24} - 378 q^{25} - 736 q^{26} + 7 q^{28} + 864 q^{30} - 264 q^{31} - 447 q^{32} + 280 q^{33} - 106 q^{34} + 519 q^{36} + 630 q^{38} + 524 q^{39} - 484 q^{40} - 436 q^{41} - 182 q^{42} + 406 q^{44} + 400 q^{46} - 2666 q^{48} + 490 q^{49} - 1601 q^{50} + 1932 q^{52} + 2196 q^{54} + 360 q^{55} - 105 q^{56} - 1084 q^{57} - 1716 q^{58} + 3824 q^{60} + 1392 q^{62} - 882 q^{63} - 1583 q^{64} + 428 q^{65} - 2340 q^{66} + 914 q^{68} + 336 q^{70} - 64 q^{71} - 2913 q^{72} - 652 q^{73} - 1196 q^{74} + 2478 q^{76} + 3904 q^{78} + 2160 q^{79} - 1228 q^{80} + 2566 q^{81} - 3650 q^{82} + 434 q^{84} + 1842 q^{86} - 1256 q^{87} - 3538 q^{88} - 2068 q^{89} - 2776 q^{90} - 1408 q^{92} + 432 q^{94} - 2156 q^{95} + 614 q^{96} + 1292 q^{97} + 147 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} + 4x^{8} + 2x^{7} + 12x^{6} + 32x^{5} + 96x^{4} + 128x^{3} + 2048x^{2} - 12288x + 32768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9 \nu^{9} - 29 \nu^{8} - 70 \nu^{7} - 38 \nu^{6} - 88 \nu^{5} - 1880 \nu^{4} + 20704 \nu^{3} + \cdots - 133120 ) / 23552 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9 \nu^{9} + 29 \nu^{8} + 70 \nu^{7} + 38 \nu^{6} + 88 \nu^{5} + 1880 \nu^{4} + 2848 \nu^{3} + \cdots + 180224 ) / 23552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - 3\nu^{8} + 4\nu^{7} + 2\nu^{6} + 12\nu^{5} + 32\nu^{4} + 96\nu^{3} + 128\nu^{2} + 2048\nu - 11264 ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45 \nu^{9} + 99 \nu^{8} + 120 \nu^{7} + 1110 \nu^{6} + 1820 \nu^{5} - 720 \nu^{4} - 3424 \nu^{3} + \cdots + 806912 ) / 47104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 67 \nu^{9} + 55 \nu^{8} - 148 \nu^{7} - 794 \nu^{6} + 2340 \nu^{5} + 4384 \nu^{4} + 1696 \nu^{3} + \cdots - 483328 ) / 47104 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20 \nu^{9} - 21 \nu^{8} - 61 \nu^{7} + 396 \nu^{6} - 354 \nu^{5} + 3172 \nu^{4} - 2240 \nu^{3} + \cdots - 147968 ) / 11776 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28 \nu^{9} - 35 \nu^{8} - 163 \nu^{7} - 76 \nu^{6} - 222 \nu^{5} + 380 \nu^{4} - 10112 \nu^{3} + \cdots + 75264 ) / 11776 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 7\beta_{4} + 2\beta_{2} - 4\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{9} - 3\beta_{8} + 9\beta_{7} + 11\beta_{6} + 9\beta_{5} - \beta_{4} + 2\beta_{3} - 8\beta_{2} - 4\beta _1 - 35 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{9} + 11 \beta_{8} - 9 \beta_{7} + 21 \beta_{6} - \beta_{5} - 17 \beta_{4} + 4 \beta_{3} + \cdots - 153 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 23 \beta_{9} - 21 \beta_{8} - 33 \beta_{7} - 11 \beta_{6} + 55 \beta_{5} + 55 \beta_{4} - 40 \beta_{3} + \cdots - 309 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 75 \beta_{9} + 15 \beta_{8} + 35 \beta_{7} + \beta_{6} - 245 \beta_{5} - 9 \beta_{4} - 44 \beta_{3} + \cdots - 1865 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 195 \beta_{9} + 111 \beta_{8} + 115 \beta_{7} - 95 \beta_{6} - 69 \beta_{5} - 521 \beta_{4} + \cdots + 8111 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−2.63061 − 1.03917i
−2.63061 + 1.03917i
−1.26262 − 2.53097i
−1.26262 + 2.53097i
0.642469 − 2.75449i
0.642469 + 2.75449i
2.23748 − 1.73022i
2.23748 + 1.73022i
2.51328 − 1.29747i
2.51328 + 1.29747i
−2.63061 − 1.03917i − 0.152232i 5.84027 + 5.46729i − 4.08829i −0.158195 + 0.400465i 7.00000 −9.68207 − 20.4513i 26.9768 −4.24841 + 10.7547i
29.2 −2.63061 + 1.03917i 0.152232i 5.84027 − 5.46729i 4.08829i −0.158195 − 0.400465i 7.00000 −9.68207 + 20.4513i 26.9768 −4.24841 − 10.7547i
29.3 −1.26262 − 2.53097i − 8.47096i −4.81160 + 6.39128i − 9.14067i −21.4397 + 10.6956i 7.00000 22.2513 + 4.10827i −44.7572 −23.1347 + 11.5412i
29.4 −1.26262 + 2.53097i 8.47096i −4.81160 − 6.39128i 9.14067i −21.4397 − 10.6956i 7.00000 22.2513 − 4.10827i −44.7572 −23.1347 − 11.5412i
29.5 0.642469 − 2.75449i 6.36522i −7.17447 − 3.53935i 18.2234i 17.5330 + 4.08946i 7.00000 −14.3585 + 17.4881i −13.5161 50.1962 + 11.7080i
29.6 0.642469 + 2.75449i − 6.36522i −7.17447 + 3.53935i − 18.2234i 17.5330 − 4.08946i 7.00000 −14.3585 − 17.4881i −13.5161 50.1962 − 11.7080i
29.7 2.23748 − 1.73022i 1.66488i 2.01266 − 7.74269i − 11.2765i 2.88062 + 3.72515i 7.00000 −8.89328 − 20.8065i 24.2282 −19.5109 − 25.2311i
29.8 2.23748 + 1.73022i − 1.66488i 2.01266 + 7.74269i 11.2765i 2.88062 − 3.72515i 7.00000 −8.89328 + 20.8065i 24.2282 −19.5109 + 25.2311i
29.9 2.51328 − 1.29747i − 9.10668i 4.63314 − 6.52181i 15.9525i −11.8157 − 22.8876i 7.00000 3.18252 − 22.4025i −55.9317 20.6979 + 40.0931i
29.10 2.51328 + 1.29747i 9.10668i 4.63314 + 6.52181i − 15.9525i −11.8157 + 22.8876i 7.00000 3.18252 + 22.4025i −55.9317 20.6979 − 40.0931i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.4.b.b ✓ 10
3.b odd 2 1 504.4.c.b 10
4.b odd 2 1 224.4.b.b 10
8.b even 2 1 inner 56.4.b.b ✓ 10
8.d odd 2 1 224.4.b.b 10
12.b even 2 1 2016.4.c.b 10
16.e even 4 2 1792.4.a.y 10
16.f odd 4 2 1792.4.a.ba 10
24.f even 2 1 2016.4.c.b 10
24.h odd 2 1 504.4.c.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.b.b ✓ 10 1.a even 1 1 trivial
56.4.b.b ✓ 10 8.b even 2 1 inner
224.4.b.b 10 4.b odd 2 1
224.4.b.b 10 8.d odd 2 1
504.4.c.b 10 3.b odd 2 1
504.4.c.b 10 24.h odd 2 1
1792.4.a.y 10 16.e even 4 2
1792.4.a.ba 10 16.f odd 4 2
2016.4.c.b 10 12.b even 2 1
2016.4.c.b 10 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 198T_{3}^{8} + 12764T_{3}^{6} + 275272T_{3}^{4} + 674688T_{3}^{2} + 15488 \) acting on \(S_{4}^{\mathrm{new}}(56, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 3 T^{9} + \cdots + 32768 \) Copy content Toggle raw display
$3$ \( T^{10} + 198 T^{8} + \cdots + 15488 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 15007434752 \) Copy content Toggle raw display
$7$ \( (T - 7)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( (T^{5} - 98 T^{4} + \cdots - 1639487904)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{5} - 90 T^{4} + \cdots + 590836224)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( (T^{5} + 132 T^{4} + \cdots + 14256308224)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 75\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( (T^{5} + 218 T^{4} + \cdots + 174769110432)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 51\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{5} - 155520 T^{3} + \cdots + 266971643904)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 28\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 41\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 46\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( (T^{5} + 32 T^{4} + \cdots - 499619856384)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 1244847071008)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 150313487564800)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 6440532062880)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 601386864409184)^{2} \) Copy content Toggle raw display
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