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Properties

Label 3744.2.d.f
Level $3744$
Weight $2$
Character orbit 3744.d
Analytic conductor $29.896$
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] = [3744,2,Mod(287,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.d (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: \(29.8959905168\)
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: \( x^{12} - 12x^{10} + 47x^{8} - 44x^{6} + 81x^{4} + 848x^{2} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + ( - \beta_{7} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} + ( - \beta_{7} - \beta_1) q^{7} - \beta_{5} q^{11} + q^{13} + ( - \beta_{10} + 2 \beta_{3}) q^{17} + ( - \beta_{7} + \beta_{6} - 2 \beta_1) q^{19} + (\beta_{11} - \beta_{5} - \beta_{2}) q^{23} + ( - \beta_{9} + \beta_{4} - 4) q^{25} + \beta_{3} q^{29} + \beta_{7} q^{31} + ( - \beta_{5} + \beta_{2}) q^{35} + ( - \beta_{9} + \beta_{4} - 3) q^{37} + ( - \beta_{8} - 2 \beta_{3}) q^{41} + (2 \beta_{7} + \beta_{6} - 2 \beta_1) q^{43} + (\beta_{11} - \beta_{5}) q^{47} + (2 \beta_{9} - \beta_{4} - 3) q^{49} + (\beta_{10} + 2 \beta_{3}) q^{53} - \beta_{6} q^{55} + (2 \beta_{5} - 3 \beta_{2}) q^{59} + \beta_{4} q^{61} - \beta_{8} q^{65} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{67} + ( - 6 \beta_{5} + \beta_{2}) q^{71} + (\beta_{9} + \beta_{4} - 9) q^{73} + (\beta_{10} + \beta_{3}) q^{77} + ( - 2 \beta_{7} - 2 \beta_{6}) q^{79} + (6 \beta_{5} + \beta_{2}) q^{83} + ( - \beta_{9} - 3 \beta_{4} - 1) q^{85} + ( - \beta_{8} + 4 \beta_{3}) q^{89} + ( - \beta_{7} - \beta_1) q^{91} + ( - 2 \beta_{11} - 11 \beta_{5} + 3 \beta_{2}) q^{95} + ( - \beta_{9} - \beta_{4} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{13} - 52 q^{25} - 40 q^{37} - 28 q^{49} - 104 q^{73} - 16 q^{85} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 12x^{10} + 47x^{8} - 44x^{6} + 81x^{4} + 848x^{2} + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -191\nu^{10} + 3596\nu^{8} - 23761\nu^{6} + 69052\nu^{4} - 53263\nu^{2} - 122584 ) / 186544 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 70637 \nu^{11} + 1693220 \nu^{9} - 16520739 \nu^{7} + 75097460 \nu^{5} - 123092317 \nu^{3} - 129121480 \nu ) / 152966080 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 583\nu^{11} - 8156\nu^{9} + 40681\nu^{7} - 43212\nu^{5} - 118857\nu^{3} + 694024\nu ) / 859360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5808\nu^{10} - 56120\nu^{8} + 154966\nu^{6} + 21080\nu^{4} - 1192192\nu^{2} + 8722760 ) / 2390095 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1177\nu^{11} - 16404\nu^{9} + 79159\nu^{7} - 142628\nu^{5} + 362697\nu^{3} + 181416\nu ) / 1167680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -34469\nu^{10} + 474620\nu^{8} - 2736123\nu^{6} + 6567820\nu^{4} - 10577109\nu^{2} - 18309320 ) / 9560380 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37191\nu^{10} - 468000\nu^{8} + 1997237\nu^{6} - 3667440\nu^{4} + 12138511\nu^{2} + 17505240 ) / 9560380 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -31521\nu^{11} + 417164\nu^{9} - 1752127\nu^{7} + 1740388\nu^{5} + 3013479\nu^{3} - 39405176\nu ) / 19120760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11544\nu^{10} - 151050\nu^{8} + 564798\nu^{6} + 397450\nu^{4} - 3001696\nu^{2} + 8389345 ) / 2390095 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -156073\nu^{11} + 1583780\nu^{9} - 4357671\nu^{7} + 345460\nu^{5} - 28097593\nu^{3} - 229397880\nu ) / 76483040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -80093\nu^{11} + 1057684\nu^{9} - 5355891\nu^{7} + 12770548\nu^{5} - 25705533\nu^{3} - 15419336\nu ) / 19120760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{5} + \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - \beta_{6} - 3\beta_{4} + 4\beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} - 2\beta_{10} + 4\beta_{8} + 9\beta_{5} + 2\beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{9} - 4\beta_{7} - 13\beta_{6} - 15\beta_{4} + 28\beta _1 + 31 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{11} - 17\beta_{10} + 64\beta_{8} + 79\beta_{5} + 105\beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12\beta_{9} - 40\beta_{7} - 68\beta_{6} - 27\beta_{4} + 80\beta _1 + 52 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 38\beta_{11} - 101\beta_{10} + 502\beta_{8} + 163\beta_{5} + 901\beta_{3} - 11\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -23\beta_{9} - 588\beta_{7} - 985\beta_{6} + 43\beta_{4} + 1228\beta _1 - 79 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -207\beta_{11} - 410\beta_{10} + 1736\beta_{8} - 787\beta_{5} + 2922\beta_{3} + 67\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -888\beta_{9} - 3122\beta_{7} - 6047\beta_{6} + 2941\beta_{4} + 8316\beta _1 - 8016 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -7410\beta_{11} - 5309\beta_{10} + 19918\beta_{8} - 25059\beta_{5} + 32189\beta_{3} + 3499\beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
2.57272 + 0.525570i
−2.57272 + 0.525570i
−0.366740 − 1.25423i
0.366740 − 1.25423i
1.49887 + 1.07270i
−1.49887 + 1.07270i
−1.49887 − 1.07270i
1.49887 − 1.07270i
0.366740 + 1.25423i
−0.366740 + 1.25423i
−2.57272 − 0.525570i
2.57272 − 0.525570i
0 0 0 − 4.09430i 0 − 0.513465i 0 0 0
287.2 0 0 0 − 4.09430i 0 0.513465i 0 0 0
287.3 0 0 0 − 3.24195i 0 − 1.54751i 0 0 0
287.4 0 0 0 − 3.24195i 0 1.54751i 0 0 0
287.5 0 0 0 − 0.852353i 0 − 5.03404i 0 0 0
287.6 0 0 0 − 0.852353i 0 5.03404i 0 0 0
287.7 0 0 0 0.852353i 0 − 5.03404i 0 0 0
287.8 0 0 0 0.852353i 0 5.03404i 0 0 0
287.9 0 0 0 3.24195i 0 − 1.54751i 0 0 0
287.10 0 0 0 3.24195i 0 1.54751i 0 0 0
287.11 0 0 0 4.09430i 0 − 0.513465i 0 0 0
287.12 0 0 0 4.09430i 0 0.513465i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.d.f ✓ 12
3.b odd 2 1 inner 3744.2.d.f ✓ 12
4.b odd 2 1 inner 3744.2.d.f ✓ 12
8.b even 2 1 7488.2.d.k 12
8.d odd 2 1 7488.2.d.k 12
12.b even 2 1 inner 3744.2.d.f ✓ 12
24.f even 2 1 7488.2.d.k 12
24.h odd 2 1 7488.2.d.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.2.d.f ✓ 12 1.a even 1 1 trivial
3744.2.d.f ✓ 12 3.b odd 2 1 inner
3744.2.d.f ✓ 12 4.b odd 2 1 inner
3744.2.d.f ✓ 12 12.b even 2 1 inner
7488.2.d.k 12 8.b even 2 1
7488.2.d.k 12 8.d odd 2 1
7488.2.d.k 12 24.f even 2 1
7488.2.d.k 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 28T_{5}^{4} + 196T_{5}^{2} + 128 \) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 28 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 28 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2)^{6} \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 62 T^{4} + \cdots + 4232)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 120 T^{4} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 152 T^{4} + \cdots - 80000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} + 24 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 10 T^{2} + \cdots - 256)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 52 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 216 T^{4} + \cdots + 73984)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 118 T^{4} + \cdots - 27848)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 78 T^{4} + \cdots + 200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 414 T^{4} + \cdots - 908552)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 28 T + 32)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 92 T^{4} + \cdots + 21904)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 238 T^{4} + \cdots - 95048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 26 T^{2} + \cdots - 320)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 288 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 286 T^{4} + \cdots - 291848)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 124 T^{4} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 2 T^{2} + \cdots + 320)^{4} \) Copy content Toggle raw display
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