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Properties

Label 1856.4.a.bi
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
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] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

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: yes
Analytic conductor: \(109.507544971\)
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: \( x^{10} - 4 x^{9} - 135 x^{8} + 788 x^{7} + 3323 x^{6} - 26136 x^{5} + 2315 x^{4} + 188664 x^{3} + \cdots + 128592 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{2} q^{5} - \beta_{6} q^{7} + (\beta_{9} - \beta_{5} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{2} q^{5} - \beta_{6} q^{7} + (\beta_{9} - \beta_{5} + 5) q^{9} + (\beta_{7} - \beta_{6} + \cdots + 4 \beta_1) q^{11}+ \cdots + (11 \beta_{8} + 28 \beta_{7} + \cdots + 21 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{5} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{5} + 50 q^{9} + 220 q^{13} + 76 q^{17} + 104 q^{21} + 446 q^{25} + 290 q^{29} - 1120 q^{33} + 1708 q^{37} - 980 q^{41} - 348 q^{45} + 1146 q^{49} + 44 q^{53} - 40 q^{57} + 1492 q^{61} - 2016 q^{65} + 2328 q^{69} - 3100 q^{73} + 3016 q^{77} - 2174 q^{81} + 9440 q^{85} + 636 q^{89} + 2536 q^{93} + 620 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 135 x^{8} + 788 x^{7} + 3323 x^{6} - 26136 x^{5} + 2315 x^{4} + 188664 x^{3} + \cdots + 128592 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10854023209 \nu^{9} - 1714044196 \nu^{8} + 1354984318315 \nu^{7} + \cdots + 836484917813280 ) / 52871845454856 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24688532 \nu^{9} + 182635072 \nu^{8} + 3815648153 \nu^{7} - 28379917644 \nu^{6} + \cdots + 3934134509796 ) / 71642067012 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5620429 \nu^{9} + 6830280 \nu^{8} - 731642547 \nu^{7} + 691738306 \nu^{6} + 23855482859 \nu^{5} + \cdots - 137905539504 ) / 7960229668 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4877782454 \nu^{9} + 17799628726 \nu^{8} + 703860283880 \nu^{7} - 3453513745104 \nu^{6} + \cdots + 778464445245516 ) / 6608980681857 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30036802 \nu^{9} + 95458676 \nu^{8} + 4237603342 \nu^{7} - 19853351823 \nu^{6} + \cdots + 3779470536804 ) / 35821033506 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6205797484 \nu^{9} + 17424622718 \nu^{8} + 859815349318 \nu^{7} - 3872373077583 \nu^{6} + \cdots + 589724260107732 ) / 6608980681857 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9575261599 \nu^{9} - 32238786260 \nu^{8} - 1348953020665 \nu^{7} + 6542711940150 \nu^{6} + \cdots - 870803080538256 ) / 5874649494984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42392072837 \nu^{9} + 105910314400 \nu^{8} + 5862590273195 \nu^{7} + \cdots + 36\!\cdots\!68 ) / 17623948484952 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 261879455 \nu^{9} + 525655384 \nu^{8} + 35733837842 \nu^{7} - 137637367368 \nu^{6} + \cdots + 21816689635044 ) / 71642067012 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 2\beta_{5} + \beta_{4} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} - 2\beta_{7} + 5\beta_{6} + 6\beta_{5} - 7\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 228 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18 \beta_{9} + 16 \beta_{8} + 48 \beta_{7} + 35 \beta_{6} - 152 \beta_{5} + 75 \beta_{4} + 10 \beta_{3} + \cdots - 512 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 292 \beta_{9} - 120 \beta_{8} - 466 \beta_{7} + 277 \beta_{6} + 1080 \beta_{5} - 723 \beta_{4} + \cdots + 15424 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3002 \beta_{9} + 1704 \beta_{8} + 6120 \beta_{7} + 889 \beta_{6} - 15044 \beta_{5} + 7813 \beta_{4} + \cdots - 95352 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 35558 \beta_{9} - 17168 \beta_{8} - 62882 \beta_{7} + 16525 \beta_{6} + 139914 \beta_{5} + \cdots + 1433244 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 379714 \beta_{9} + 192288 \beta_{8} + 720456 \beta_{7} - 18773 \beta_{6} - 1650272 \beta_{5} + \cdots - 12690928 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4194712 \beta_{9} - 2075960 \beta_{8} - 7633538 \beta_{7} + 1253921 \beta_{6} + 16909396 \beta_{5} + \cdots + 153648712 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 45223890 \beta_{9} + 22357240 \beta_{8} + 84047344 \beta_{7} - 7015343 \beta_{6} - 188470276 \beta_{5} + \cdots - 1548739832 ) / 8 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.40120
1.25115
−3.21629
1.47788
−10.8075
−0.612154
5.97060
6.55010
−5.42259
2.40763
0 −8.07880 0 −14.4117 0 −28.4619 0 38.2670 0
1.2 0 −7.53335 0 7.78444 0 −5.94345 0 29.7513 0
1.3 0 −4.75556 0 22.0670 0 32.1228 0 −4.38468 0
1.4 0 −3.91971 0 −7.82384 0 17.9594 0 −11.6359 0
1.5 0 −0.0474689 0 −5.61587 0 −9.39051 0 −26.9977 0
1.6 0 0.0474689 0 −5.61587 0 9.39051 0 −26.9977 0
1.7 0 3.91971 0 −7.82384 0 −17.9594 0 −11.6359 0
1.8 0 4.75556 0 22.0670 0 −32.1228 0 −4.38468 0
1.9 0 7.53335 0 7.78444 0 5.94345 0 29.7513 0
1.10 0 8.07880 0 −14.4117 0 28.4619 0 38.2670 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.4.a.bi 10
4.b odd 2 1 inner 1856.4.a.bi 10
8.b even 2 1 928.4.a.f ✓ 10
8.d odd 2 1 928.4.a.f ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.f ✓ 10 8.b even 2 1
928.4.a.f ✓ 10 8.d odd 2 1
1856.4.a.bi 10 1.a even 1 1 trivial
1856.4.a.bi 10 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{10} - 160T_{3}^{8} + 8686T_{3}^{6} - 183092T_{3}^{4} + 1287417T_{3}^{2} - 2900 \) Copy content Toggle raw display
\( T_{5}^{5} - 2T_{5}^{4} - 422T_{5}^{3} - 1676T_{5}^{2} + 21917T_{5} + 108774 \) Copy content Toggle raw display
\( T_{7}^{10} - 2288T_{7}^{8} + 1700448T_{7}^{6} - 452968704T_{7}^{4} + 37753198848T_{7}^{2} - 839836258304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 160 T^{8} + \cdots - 2900 \) Copy content Toggle raw display
$5$ \( (T^{5} - 2 T^{4} + \cdots + 108774)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 839836258304 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 162994861358676 \) Copy content Toggle raw display
$13$ \( (T^{5} - 110 T^{4} + \cdots + 28673602)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 38 T^{4} + \cdots - 4092920640)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T - 29)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 22\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{5} - 854 T^{4} + \cdots + 293417725952)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 5366289891808)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 19\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{5} - 22 T^{4} + \cdots - 2494384982)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 40\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 53613741397184)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 16\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 34\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{5} + 1550 T^{4} + \cdots - 801523248128)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 32\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 33585949543200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 2383191738336)^{2} \) Copy content Toggle raw display
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