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Properties

Label 1856.4.a.bk
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 245x^{10} + 21294x^{8} - 755514x^{6} + 8955005x^{4} - 27099393x^{2} + 23710340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{19} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} - \beta_{7} q^{7} + (\beta_{3} - \beta_{2} + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} - \beta_{7} q^{7} + (\beta_{3} - \beta_{2} + 14) q^{9} + (\beta_{10} + \beta_{7}) q^{11} + (\beta_{4} + 2 \beta_{2} - 3) q^{13} + ( - \beta_{11} + \beta_{9} + \cdots - 4 \beta_1) q^{15}+ \cdots + ( - 23 \beta_{11} + 18 \beta_{10} + \cdots - 15 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 245x^{10} + 21294x^{8} - 755514x^{6} + 8955005x^{4} - 27099393x^{2} + 23710340 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -491\nu^{10} + 125541\nu^{8} - 11255035\nu^{6} + 402225787\nu^{4} - 4412660266\nu^{2} + 8041961952 ) / 103502808 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -491\nu^{10} + 125541\nu^{8} - 11255035\nu^{6} + 402225787\nu^{4} - 4309157458\nu^{2} + 3798346824 ) / 103502808 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6265 \nu^{10} - 1235890 \nu^{8} + 79290224 \nu^{6} - 1758043882 \nu^{4} + 8400204463 \nu^{2} + 7986096484 ) / 621016848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 787\nu^{10} - 169018\nu^{8} + 12655964\nu^{6} - 385175254\nu^{4} + 4137831661\nu^{2} - 7570448972 ) / 51751404 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10809 \nu^{10} - 2602664 \nu^{8} + 220850310 \nu^{6} - 7505285884 \nu^{4} + 78171035497 \nu^{2} - 125081524228 ) / 207005616 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 46274 \nu^{11} + 11740775 \nu^{9} - 1029868159 \nu^{7} + 34119647501 \nu^{5} + \cdots - 616650760628 \nu ) / 55581007896 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 309941 \nu^{11} - 79156292 \nu^{9} + 7073821954 \nu^{7} - 250114895120 \nu^{5} + \cdots - 7870458399796 \nu ) / 111162015792 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 263196 \nu^{11} - 64165295 \nu^{9} + 5516969457 \nu^{7} - 190373414485 \nu^{5} + \cdots - 3601582924456 \nu ) / 9263501316 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 549710 \nu^{11} - 132445209 \nu^{9} + 11239206649 \nu^{7} - 381914057707 \nu^{5} + \cdots - 6042365174700 \nu ) / 18527002632 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3468293 \nu^{11} - 849139832 \nu^{9} + 73277455438 \nu^{7} - 2534595868940 \nu^{5} + \cdots - 43085788356244 \nu ) / 111162015792 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + 41 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{8} + 72\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} - 5\beta_{5} + 83\beta_{3} - 55\beta_{2} + 2913 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -109\beta_{11} - 11\beta_{10} + 123\beta_{9} + 61\beta_{8} - 75\beta_{7} + 5473\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 542\beta_{6} - 490\beta_{5} - 198\beta_{4} + 6607\beta_{3} - 2633\beta_{2} + 220481 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10487\beta_{11} - 2116\beta_{10} + 13197\beta_{9} + 2279\beta_{8} - 10416\beta_{7} + 421882\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 58378\beta_{6} - 40019\beta_{5} - 33102\beta_{4} + 526837\beta_{3} - 82947\beta_{2} + 16956905 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -974343\beta_{11} - 260291\beta_{10} + 1311371\beta_{9} - 27801\beta_{8} - 1130871\beta_{7} + 32795819\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5779036\beta_{6} - 3096104\beta_{5} - 3924972\beta_{4} + 42260097\beta_{3} + 2867611\beta_{2} + 1315829017 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 88828697 \beta_{11} - 27059108 \beta_{10} + 124349773 \beta_{9} - 18154327 \beta_{8} + \cdots + 2568117924 \beta_1 \) 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
−9.17689
−8.55609
−8.30801
−3.82226
−1.50033
−1.30165
1.30165
1.50033
3.82226
8.30801
8.55609
9.17689
0 −9.17689 0 14.0091 0 −13.6574 0 57.2153 0
1.2 0 −8.55609 0 −17.9229 0 −6.79699 0 46.2068 0
1.3 0 −8.30801 0 −11.7163 0 23.1998 0 42.0230 0
1.4 0 −3.82226 0 −3.68395 0 23.1652 0 −12.3903 0
1.5 0 −1.50033 0 −0.787262 0 −27.3506 0 −24.7490 0
1.6 0 −1.30165 0 15.1013 0 −22.0989 0 −25.3057 0
1.7 0 1.30165 0 15.1013 0 22.0989 0 −25.3057 0
1.8 0 1.50033 0 −0.787262 0 27.3506 0 −24.7490 0
1.9 0 3.82226 0 −3.68395 0 −23.1652 0 −12.3903 0
1.10 0 8.30801 0 −11.7163 0 −23.1998 0 42.0230 0
1.11 0 8.55609 0 −17.9229 0 6.79699 0 46.2068 0
1.12 0 9.17689 0 14.0091 0 13.6574 0 57.2153 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bk 12
4.b odd 2 1 inner 1856.4.a.bk 12
8.b even 2 1 928.4.a.i ✓ 12
8.d odd 2 1 928.4.a.i ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.i ✓ 12 8.b even 2 1
928.4.a.i ✓ 12 8.d odd 2 1
1856.4.a.bk 12 1.a even 1 1 trivial
1856.4.a.bk 12 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{12} - 245T_{3}^{10} + 21294T_{3}^{8} - 755514T_{3}^{6} + 8955005T_{3}^{4} - 27099393T_{3}^{2} + 23710340 \) Copy content Toggle raw display
\( T_{5}^{6} + 5T_{5}^{5} - 436T_{5}^{4} - 1814T_{5}^{3} + 43849T_{5}^{2} + 199089T_{5} + 128842 \) Copy content Toggle raw display
\( T_{7}^{12} - 2544 T_{7}^{10} + 2529632 T_{7}^{8} - 1231220992 T_{7}^{6} + 297097423104 T_{7}^{4} + \cdots + 909258653696000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 245 T^{10} + \cdots + 23710340 \) Copy content Toggle raw display
$5$ \( (T^{6} + 5 T^{5} + \cdots + 128842)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 909258653696000 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 73\!\cdots\!40 \) Copy content Toggle raw display
$13$ \( (T^{6} + 15 T^{5} + \cdots - 3382882886)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 168 T^{5} + \cdots + 25740107392)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 21\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 36\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( (T - 29)^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 40279997186048)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 4616524334080)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 23\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 16\!\cdots\!86)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 88\!\cdots\!60)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 57\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 338596891021312)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 44\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 36\!\cdots\!32)^{2} \) Copy content Toggle raw display
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