Properties

Label 1764.2.b.m
Level $1764$
Weight $2$
Character orbit 1764.b
Analytic conductor $14.086$
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] = [1764,2,Mod(1567,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1764.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.15911316233388032.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} - 160 x^{3} + 160 x^{2} - 128 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 588)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{11} - \beta_{2}) q^{4} + (\beta_{11} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{11} - \beta_{2}) q^{4} + (\beta_{11} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} - 1) q^{8} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \cdots + 1) q^{10}+ \cdots + (2 \beta_{10} - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 4 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} - 4 q^{4} - 4 q^{8} - 4 q^{16} + 24 q^{20} - 12 q^{25} - 24 q^{26} - 32 q^{29} + 16 q^{31} - 4 q^{32} + 32 q^{34} + 32 q^{37} + 24 q^{38} - 32 q^{40} + 24 q^{44} + 24 q^{46} + 28 q^{50} + 32 q^{52} + 32 q^{53} + 16 q^{55} + 16 q^{58} + 16 q^{59} - 8 q^{62} - 4 q^{64} + 8 q^{68} + 32 q^{74} - 32 q^{76} + 16 q^{80} + 32 q^{82} + 16 q^{83} + 16 q^{85} + 24 q^{86} + 24 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} - 160 x^{3} + 160 x^{2} - 128 x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 2\nu^{9} + 3\nu^{7} + 4\nu^{6} - 4\nu^{5} + 4\nu^{4} - 4\nu^{3} + 40\nu^{2} + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 8 \nu^{9} - 16 \nu^{8} + 23 \nu^{7} - 37 \nu^{6} + 58 \nu^{5} - 70 \nu^{4} + 88 \nu^{3} - 80 \nu^{2} + 80 \nu - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 31 \nu^{7} - 51 \nu^{6} + 72 \nu^{5} - 102 \nu^{4} + 116 \nu^{3} - 136 \nu^{2} + 128 \nu - 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 4 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 35 \nu^{7} - 56 \nu^{6} + 84 \nu^{5} - 112 \nu^{4} + 140 \nu^{3} - 160 \nu^{2} + 160 \nu - 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 34 \nu^{7} - 55 \nu^{6} + 78 \nu^{5} - 90 \nu^{4} + 132 \nu^{3} - 112 \nu^{2} + 112 \nu - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{11} + 5 \nu^{10} - 12 \nu^{9} + 24 \nu^{8} - 34 \nu^{7} + 63 \nu^{6} - 84 \nu^{5} + 110 \nu^{4} - 128 \nu^{3} + 136 \nu^{2} - 128 \nu + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 7 \nu^{9} - 12 \nu^{8} + 19 \nu^{7} - 33 \nu^{6} + 47 \nu^{5} - 54 \nu^{4} + 70 \nu^{3} - 72 \nu^{2} + 80 \nu - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} - 8 \nu^{9} + 14 \nu^{8} - 23 \nu^{7} + 37 \nu^{6} - 50 \nu^{5} + 64 \nu^{4} - 80 \nu^{3} + 84 \nu^{2} - 80 \nu + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - 8 \nu^{10} + 18 \nu^{9} - 28 \nu^{8} + 49 \nu^{7} - 76 \nu^{6} + 104 \nu^{5} - 136 \nu^{4} + 164 \nu^{3} - 160 \nu^{2} + 144 \nu - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4 \nu^{11} + 9 \nu^{10} - 20 \nu^{9} + 36 \nu^{8} - 56 \nu^{7} + 83 \nu^{6} - 120 \nu^{5} + 146 \nu^{4} - 192 \nu^{3} + 192 \nu^{2} - 160 \nu + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{11} - 11 \nu^{10} + 24 \nu^{9} - 44 \nu^{8} + 77 \nu^{7} - 117 \nu^{6} + 166 \nu^{5} - 210 \nu^{4} + 248 \nu^{3} - 280 \nu^{2} + 240 \nu - 128 ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{7} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + \beta_{9} - 2\beta_{8} - \beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{11} + \beta_{9} + 6\beta_{8} + 3\beta_{7} - 2\beta_{5} + 2\beta_{4} - 5\beta_{3} + 5\beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{10} + 4\beta_{8} + 4\beta_{6} - 3\beta_{5} + 3\beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6\beta_{11} - 5\beta_{9} - 2\beta_{8} - 3\beta_{7} + 2\beta_{6} + 2\beta_{5} - \beta_{3} - 5\beta_{2} + 5\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5 \beta_{10} + 14 \beta_{9} + 6 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 9 \beta_{4} - 11 \beta_{3} - 3 \beta_{2} - 3 \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10 \beta_{11} + 15 \beta_{9} - 14 \beta_{8} + 5 \beta_{7} + 12 \beta_{6} - 14 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 11 \beta_{2} - 7 \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - \beta_{10} - 4 \beta_{9} - 16 \beta_{8} - 12 \beta_{7} - 16 \beta_{6} - 13 \beta_{5} + 21 \beta_{4} - 37 \beta_{3} + 7 \beta_{2} + 3 \beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2 \beta_{11} - 24 \beta_{10} + \beta_{9} + 42 \beta_{8} - \beta_{7} - 46 \beta_{6} - 10 \beta_{5} + 36 \beta_{4} - 39 \beta_{3} - 39 \beta_{2} + 15 \beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.34902 0.424442i
1.34902 + 0.424442i
1.22594 0.705031i
1.22594 + 0.705031i
0.639847 1.26119i
0.639847 + 1.26119i
0.250649 1.39182i
0.250649 + 1.39182i
−0.476589 1.33149i
−0.476589 + 1.33149i
−0.988865 1.01101i
−0.988865 + 1.01101i
−1.34902 0.424442i 0 1.63970 + 1.14516i 0.127929i 0 0 −1.72593 2.24080i 0 −0.0542984 + 0.172579i
1567.2 −1.34902 + 0.424442i 0 1.63970 1.14516i 0.127929i 0 0 −1.72593 + 2.24080i 0 −0.0542984 0.172579i
1567.3 −1.22594 0.705031i 0 1.00586 + 1.72865i 3.64758i 0 0 −0.0143727 2.82839i 0 −2.57166 + 4.47172i
1567.4 −1.22594 + 0.705031i 0 1.00586 1.72865i 3.64758i 0 0 −0.0143727 + 2.82839i 0 −2.57166 4.47172i
1567.5 −0.639847 1.26119i 0 −1.18119 + 1.61393i 3.10455i 0 0 2.79126 + 0.457034i 0 −3.91542 + 1.98644i
1567.6 −0.639847 + 1.26119i 0 −1.18119 1.61393i 3.10455i 0 0 2.79126 0.457034i 0 −3.91542 1.98644i
1567.7 −0.250649 1.39182i 0 −1.87435 + 0.697718i 3.39209i 0 0 1.44090 + 2.43388i 0 4.72119 0.850222i
1567.8 −0.250649 + 1.39182i 0 −1.87435 0.697718i 3.39209i 0 0 1.44090 2.43388i 0 4.72119 + 0.850222i
1567.9 0.476589 1.33149i 0 −1.54572 1.26915i 0.509876i 0 0 −2.42653 + 1.45325i 0 0.678894 + 0.243001i
1567.10 0.476589 + 1.33149i 0 −1.54572 + 1.26915i 0.509876i 0 0 −2.42653 1.45325i 0 0.678894 0.243001i
1567.11 0.988865 1.01101i 0 −0.0442929 1.99951i 1.12886i 0 0 −2.06533 1.93246i 0 1.14129 + 1.11629i
1567.12 0.988865 + 1.01101i 0 −0.0442929 + 1.99951i 1.12886i 0 0 −2.06533 + 1.93246i 0 1.14129 1.11629i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.m 12
3.b odd 2 1 588.2.b.d yes 12
4.b odd 2 1 1764.2.b.l 12
7.b odd 2 1 1764.2.b.l 12
12.b even 2 1 588.2.b.c 12
21.c even 2 1 588.2.b.c 12
21.g even 6 2 588.2.o.f 24
21.h odd 6 2 588.2.o.e 24
28.d even 2 1 inner 1764.2.b.m 12
84.h odd 2 1 588.2.b.d yes 12
84.j odd 6 2 588.2.o.e 24
84.n even 6 2 588.2.o.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.b.c 12 12.b even 2 1
588.2.b.c 12 21.c even 2 1
588.2.b.d yes 12 3.b odd 2 1
588.2.b.d yes 12 84.h odd 2 1
588.2.o.e 24 21.h odd 6 2
588.2.o.e 24 84.j odd 6 2
588.2.o.f 24 21.g even 6 2
588.2.o.f 24 84.n even 6 2
1764.2.b.l 12 4.b odd 2 1
1764.2.b.l 12 7.b odd 2 1
1764.2.b.m 12 1.a even 1 1 trivial
1764.2.b.m 12 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{12} + 36T_{5}^{10} + 446T_{5}^{8} + 2096T_{5}^{6} + 2428T_{5}^{4} + 528T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{12} + 104T_{11}^{10} + 4040T_{11}^{8} + 72672T_{11}^{6} + 603920T_{11}^{4} + 1962496T_{11}^{2} + 1968128 \) Copy content Toggle raw display
\( T_{19}^{6} - 72T_{19}^{4} - 32T_{19}^{3} + 1288T_{19}^{2} + 1280T_{19} - 256 \) Copy content Toggle raw display
\( T_{29}^{6} + 16T_{29}^{5} + 26T_{29}^{4} - 592T_{29}^{3} - 2268T_{29}^{2} + 1440T_{29} + 9544 \) Copy content Toggle raw display
\( T_{53}^{6} - 16T_{53}^{5} + 24T_{53}^{4} + 288T_{53}^{3} + 144T_{53}^{2} - 384T_{53} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + 10 T^{10} + 20 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 36 T^{10} + 446 T^{8} + 2096 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 104 T^{10} + 4040 T^{8} + \cdots + 1968128 \) Copy content Toggle raw display
$13$ \( T^{12} + 92 T^{10} + 3094 T^{8} + \cdots + 40328 \) Copy content Toggle raw display
$17$ \( T^{12} + 116 T^{10} + 5182 T^{8} + \cdots + 8405000 \) Copy content Toggle raw display
$19$ \( (T^{6} - 72 T^{4} - 32 T^{3} + 1288 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 216 T^{10} + \cdots + 518162432 \) Copy content Toggle raw display
$29$ \( (T^{6} + 16 T^{5} + 26 T^{4} - 592 T^{3} + \cdots + 9544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 8 T^{5} - 56 T^{4} + 416 T^{3} + \cdots - 6272)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 16 T^{5} - 22 T^{4} + 1280 T^{3} + \cdots + 10424)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 228 T^{10} + 14878 T^{8} + \cdots + 84872 \) Copy content Toggle raw display
$43$ \( T^{12} + 240 T^{10} + 17504 T^{8} + \cdots + 6422528 \) Copy content Toggle raw display
$47$ \( (T^{6} - 192 T^{4} + 160 T^{3} + \cdots - 62336)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 16 T^{5} + 24 T^{4} + 288 T^{3} + \cdots - 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 8 T^{5} - 144 T^{4} + 736 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 188 T^{10} + 11574 T^{8} + \cdots + 7688 \) Copy content Toggle raw display
$67$ \( T^{12} + 448 T^{10} + 59648 T^{8} + \cdots + 8388608 \) Copy content Toggle raw display
$71$ \( T^{12} + 296 T^{10} + 29000 T^{8} + \cdots + 4917248 \) Copy content Toggle raw display
$73$ \( T^{12} + 396 T^{10} + \cdots + 19046792 \) Copy content Toggle raw display
$79$ \( T^{12} + 432 T^{10} + \cdots + 1038221312 \) Copy content Toggle raw display
$83$ \( (T^{6} - 8 T^{5} - 208 T^{4} + 1472 T^{3} + \cdots + 82432)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 372 T^{10} + \cdots + 146068232 \) Copy content Toggle raw display
$97$ \( T^{12} + 492 T^{10} + \cdots + 1057448072 \) Copy content Toggle raw display
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