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Properties

Label 3744.2.m.h
Level $3744$
Weight $2$
Character orbit 3744.m
Analytic conductor $29.896$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,2,Mod(1585,3744)]
 
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([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 936)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{14} q^{5} + \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{5} + \beta_{5} q^{7} + (\beta_{14} - \beta_{11}) q^{11} + ( - \beta_{15} - \beta_{6}) q^{13} - \beta_{3} q^{17} + ( - \beta_{15} + \beta_{12} - \beta_{6}) q^{19} - \beta_{2} q^{23} + \beta_1 q^{25} + \beta_{7} q^{29} + \beta_{10} q^{31} + (\beta_{9} - 2 \beta_{7}) q^{35} + (2 \beta_{15} - 2 \beta_{13}) q^{37} + ( - 6 \beta_{8} - 3 \beta_{4}) q^{41} + (\beta_{13} + \beta_{12}) q^{43} + ( - \beta_{8} - 4 \beta_{4}) q^{47} + ( - \beta_1 - 8) q^{49} + ( - \beta_{9} - 2 \beta_{7}) q^{53} + (3 \beta_1 + 5) q^{55} + ( - \beta_{14} - 3 \beta_{11}) q^{59} - 3 \beta_{6} q^{61} + (2 \beta_{8} - \beta_{4} + \cdots + \beta_{2}) q^{65}+ \cdots + ( - \beta_{10} + \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 396 \nu^{14} + 1812 \nu^{12} - 7496 \nu^{10} - 25482 \nu^{8} - 34944 \nu^{6} - 37770 \nu^{4} + \cdots + 1991875 ) / 881375 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7468 \nu^{14} - 15136 \nu^{12} + 146738 \nu^{10} + 531446 \nu^{8} - 573668 \nu^{6} + \cdots - 4522500 ) / 881375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2458 \nu^{14} + 5148 \nu^{12} - 42734 \nu^{10} - 183828 \nu^{8} + 123374 \nu^{6} + 1088952 \nu^{4} + \cdots + 1368050 ) / 176275 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7797 \nu^{15} - 1879 \nu^{13} + 194307 \nu^{11} + 471569 \nu^{9} - 1255952 \nu^{7} + \cdots - 4720750 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10661 \nu^{15} + 14207 \nu^{13} - 225956 \nu^{11} - 888177 \nu^{9} + 521116 \nu^{7} + \cdots + 26244000 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23588 \nu^{14} + 17416 \nu^{12} - 373728 \nu^{10} - 1083526 \nu^{8} + 2038808 \nu^{6} + \cdots + 8274750 ) / 881375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28806 \nu^{14} + 29832 \nu^{12} - 448156 \nu^{10} - 1496702 \nu^{8} + 2047066 \nu^{6} + \cdots + 13465750 ) / 881375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} + \cdots - 598500 \nu ) / 400625 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} + \cdots - 22476000 ) / 881375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8289 \nu^{15} + 23943 \nu^{13} - 103294 \nu^{11} - 572573 \nu^{9} - 243116 \nu^{7} + \cdots + 11379000 \nu ) / 881375 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 68641 \nu^{15} - 53217 \nu^{13} + 1141536 \nu^{11} + 3533287 \nu^{9} - 5705496 \nu^{7} + \cdots - 30934000 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 78923 \nu^{15} - 26760 \nu^{14} - 34611 \nu^{13} + 11580 \nu^{12} + 1339113 \nu^{11} + \cdots - 13844375 ) / 4406875 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 78923 \nu^{15} - 144700 \nu^{14} + 34611 \nu^{13} - 75500 \nu^{12} - 1339113 \nu^{11} + \cdots - 55218125 ) / 4406875 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 108813 \nu^{15} + 85331 \nu^{13} - 1829673 \nu^{11} - 5646091 \nu^{9} + 9386828 \nu^{7} + \cdots + 50144500 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 128244 \nu^{15} - 144700 \nu^{14} + 47208 \nu^{13} - 75500 \nu^{12} - 2180589 \nu^{11} + \cdots - 55218125 ) / 4406875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} + \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} - \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{14} - \beta_{13} + \beta_{12} + 12\beta_{11} + \beta_{10} + 8\beta_{8} - \beta_{6} - 3\beta_{5} - 2\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{13} + 3\beta_{12} - 2\beta_{9} - 4\beta_{7} + 5\beta_{6} - 2\beta_{2} - 8\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14\beta_{15} - 19\beta_{13} + 5\beta_{12} - 6\beta_{8} - 5\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -11\beta_{13} - 11\beta_{12} + 11\beta_{9} + 17\beta_{7} - 15\beta_{6} - 5\beta_{3} - 6\beta_{2} - 26\beta _1 + 56 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10 \beta_{15} + 110 \beta_{14} - 9 \beta_{13} - \beta_{12} + 182 \beta_{11} - 9 \beta_{10} + \cdots + 72 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 35 \beta_{13} + 35 \beta_{12} - 20 \beta_{9} - 32 \beta_{7} + 53 \beta_{6} - 32 \beta_{3} - 52 \beta_{2} + \cdots + 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 274 \beta_{15} + 114 \beta_{14} - 361 \beta_{13} + 87 \beta_{12} + 190 \beta_{11} + 87 \beta_{10} + \cdots - 304 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21\beta_{9} + 36\beta_{7} - 275\beta _1 + 607 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 768 \beta_{15} + 682 \beta_{14} - 1007 \beta_{13} + 239 \beta_{12} + 1080 \beta_{11} + \cdots + 1762 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 110 \beta_{13} - 110 \beta_{12} + 290 \beta_{9} + 462 \beta_{7} - 177 \beta_{6} - 462 \beta_{3} + \cdots + 153 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 790 \beta_{15} + 4630 \beta_{14} - 1037 \beta_{13} + 247 \beta_{12} + 7514 \beta_{11} + \cdots - 2884 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2071 \beta_{13} + 2071 \beta_{12} - 1039 \beta_{9} - 1678 \beta_{7} + 3360 \beta_{6} - 400 \beta_{3} + \cdots + 12151 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 4399\beta_{15} - 5754\beta_{13} + 1355\beta_{12} - 1521\beta_{8} - 1355\beta_{6} + 2475\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
0.752864 − 0.902863i
0.0783900 − 1.17295i
0.752864 + 0.902863i
0.0783900 + 1.17295i
0.556839 − 1.81878i
−1.90184 + 0.0324487i
0.556839 + 1.81878i
−1.90184 − 0.0324487i
1.90184 + 0.0324487i
−0.556839 − 1.81878i
1.90184 − 0.0324487i
−0.556839 + 1.81878i
−0.0783900 − 1.17295i
−0.752864 − 0.902863i
−0.0783900 + 1.17295i
−0.752864 + 0.902863i
0 0 0 −2.68999 0 − 4.15163i 0 0 0
1585.2 0 0 0 −2.68999 0 − 4.15163i 0 0 0
1585.3 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.4 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.5 0 0 0 −1.66251 0 − 3.57266i 0 0 0
1585.6 0 0 0 −1.66251 0 − 3.57266i 0 0 0
1585.7 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.8 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.9 0 0 0 1.66251 0 − 3.57266i 0 0 0
1585.10 0 0 0 1.66251 0 − 3.57266i 0 0 0
1585.11 0 0 0 1.66251 0 3.57266i 0 0 0
1585.12 0 0 0 1.66251 0 3.57266i 0 0 0
1585.13 0 0 0 2.68999 0 − 4.15163i 0 0 0
1585.14 0 0 0 2.68999 0 − 4.15163i 0 0 0
1585.15 0 0 0 2.68999 0 4.15163i 0 0 0
1585.16 0 0 0 2.68999 0 4.15163i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1585.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.m.h 16
3.b odd 2 1 inner 3744.2.m.h 16
4.b odd 2 1 936.2.m.h ✓ 16
8.b even 2 1 inner 3744.2.m.h 16
8.d odd 2 1 936.2.m.h ✓ 16
12.b even 2 1 936.2.m.h ✓ 16
13.b even 2 1 inner 3744.2.m.h 16
24.f even 2 1 936.2.m.h ✓ 16
24.h odd 2 1 inner 3744.2.m.h 16
39.d odd 2 1 inner 3744.2.m.h 16
52.b odd 2 1 936.2.m.h ✓ 16
104.e even 2 1 inner 3744.2.m.h 16
104.h odd 2 1 936.2.m.h ✓ 16
156.h even 2 1 936.2.m.h ✓ 16
312.b odd 2 1 inner 3744.2.m.h 16
312.h even 2 1 936.2.m.h ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.h ✓ 16 4.b odd 2 1
936.2.m.h ✓ 16 8.d odd 2 1
936.2.m.h ✓ 16 12.b even 2 1
936.2.m.h ✓ 16 24.f even 2 1
936.2.m.h ✓ 16 52.b odd 2 1
936.2.m.h ✓ 16 104.h odd 2 1
936.2.m.h ✓ 16 156.h even 2 1
936.2.m.h ✓ 16 312.h even 2 1
3744.2.m.h 16 1.a even 1 1 trivial
3744.2.m.h 16 3.b odd 2 1 inner
3744.2.m.h 16 8.b even 2 1 inner
3744.2.m.h 16 13.b even 2 1 inner
3744.2.m.h 16 24.h odd 2 1 inner
3744.2.m.h 16 39.d odd 2 1 inner
3744.2.m.h 16 104.e even 2 1 inner
3744.2.m.h 16 312.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 10T_{5}^{2} + 20 \) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 10 T^{2} + 20)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 30 T^{2} + 220)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 20)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 12 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 60 T^{2} + 880)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 34 T^{2} + 44)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 80 T^{2} + 880)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 28 T^{2} + 176)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 150 T^{2} + 5500)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 104 T^{2} + 704)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 126 T^{2} + 324)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{2} + 80)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 84 T^{2} + 1444)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 128 T^{2} + 176)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 100 T^{2} + 500)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 180 T^{2} + 6480)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 154 T^{2} + 5324)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 116 T^{2} + 484)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 200 T^{2} + 3520)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 76)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 100 T^{2} + 500)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 230 T^{2} + 12100)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 160 T^{2} + 3520)^{4} \) Copy content Toggle raw display
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