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Properties

Label 1764.2.w.a
Level $1764$
Weight $2$
Character orbit 1764.w
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
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] = [1764,2,Mod(509,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.509");
 
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.w (of order \(6\), degree \(2\), 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + \beta_{15} q^{5} - \beta_{11} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} + \beta_{15} q^{5} - \beta_{11} q^{9} + ( - \beta_{13} - \beta_{11} - \beta_{7}) q^{11} + ( - \beta_{14} + \beta_{10} + \cdots - \beta_{2}) q^{13}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{11} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{9} - 6 q^{11} - 12 q^{15} - 6 q^{23} - 8 q^{25} - 12 q^{29} - 2 q^{37} - 36 q^{39} + 4 q^{43} + 12 q^{51} + 36 q^{53} - 42 q^{57} - 28 q^{67} - 40 q^{79} - 18 q^{81} + 6 q^{85} - 6 q^{93} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 15\nu^{13} + 72\nu^{11} + 153\nu^{9} - 423\nu^{7} - 891\nu^{5} + 1944\nu^{3} + 17496\nu ) / 15309 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + 15\nu^{12} + 72\nu^{10} + 153\nu^{8} - 423\nu^{6} - 891\nu^{4} + 1944\nu^{2} + 12393 ) / 5103 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{14} + 12\nu^{12} - 18\nu^{10} - 369\nu^{8} + 153\nu^{6} + 1782\nu^{4} + 4617\nu^{2} - 9477 ) / 5103 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} + 3\nu^{13} + 9\nu^{11} + 9\nu^{9} - 225\nu^{7} + 81\nu^{5} + 729\nu^{3} + 2187\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{15} + 3\nu^{13} - 18\nu^{11} + 90\nu^{9} + 18\nu^{7} + 324\nu^{5} - 3159\nu^{3} + 2187\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{14} - 3\nu^{12} - 9\nu^{10} - 9\nu^{8} + 225\nu^{6} - 81\nu^{4} - 729\nu^{2} - 2187 ) / 729 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\nu^{15} + 6\nu^{13} - 9\nu^{11} - 279\nu^{9} - 396\nu^{7} + 324\nu^{5} + 6561\nu^{3} + 13122\nu ) / 15309 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{15} + 12\nu^{13} - 18\nu^{11} - 369\nu^{9} + 153\nu^{7} + 1782\nu^{5} + 4617\nu^{3} - 4374\nu ) / 5103 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -20\nu^{15} + 78\nu^{13} + 72\nu^{11} + 909\nu^{9} - 3447\nu^{7} - 2592\nu^{5} - 9963\nu^{3} + 63423\nu ) / 15309 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -20\nu^{14} + 15\nu^{12} + 72\nu^{10} + 342\nu^{8} - 1179\nu^{6} + 243\nu^{4} - 1458\nu^{2} + 2187 ) / 5103 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23\nu^{14} - 33\nu^{12} - 234\nu^{10} - 1017\nu^{8} + 3879\nu^{6} + 8424\nu^{4} + 2187\nu^{2} - 82377 ) / 5103 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{14} + \nu^{12} - 12\nu^{10} - 36\nu^{8} + 81\nu^{6} + 306\nu^{4} + 54\nu^{2} - 1215 ) / 189 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\nu^{15} - 3\nu^{13} - 153\nu^{11} - 396\nu^{9} + 1395\nu^{7} + 2862\nu^{5} - 162\nu^{3} - 16767\nu ) / 5103 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 62\nu^{15} - 15\nu^{13} - 639\nu^{11} - 2421\nu^{9} + 7794\nu^{7} + 17901\nu^{5} + 3159\nu^{3} - 139968\nu ) / 15309 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{14} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{9} - \beta_{8} - \beta_{6} + 2\beta_{5} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 2\beta_{7} - 2\beta_{4} + \beta_{3} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{10} + 3\beta_{6} + 6\beta_{5} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{11} + 6\beta_{7} + 3\beta_{4} + 3\beta_{3} + 3\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{15} + 6\beta_{14} - 3\beta_{10} + 3\beta_{9} - 12\beta_{8} - 3\beta_{5} + 18\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{13} + 3\beta_{12} - 6\beta_{7} - 24\beta_{4} + 12\beta_{3} + 9\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12\beta_{15} + 6\beta_{14} + 3\beta_{10} - 39\beta_{9} - 9\beta_{8} - 12\beta_{6} + 45\beta_{5} + 51\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -18\beta_{13} + 18\beta_{12} - 9\beta_{7} + 18\beta_{4} + 45\beta_{3} - 45\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9\beta_{15} - 72\beta_{14} - 45\beta_{10} + 54\beta_{9} + 9\beta_{8} + 54\beta_{6} - 45\beta_{5} + 54\beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 54\beta_{13} - 63\beta_{12} + 108\beta_{11} + 180\beta_{7} + 153\beta_{4} + 126\beta_{3} + 36\beta _1 - 198 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 108\beta_{15} - 27\beta_{14} + 297\beta_{10} - 243\beta_{8} - 216\beta_{6} - 216\beta_{5} + 270\beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 27\beta_{13} + 81\beta_{12} - 351\beta_{11} - 378\beta_{7} - 432\beta_{4} + 297\beta_{3} - 243\beta _1 - 1026 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 459 \beta_{15} - 567 \beta_{14} + 216 \beta_{10} - 540 \beta_{9} + 1242 \beta_{8} - 27 \beta_{6} + \cdots + 135 \beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
509.1
−0.604587 + 1.62311i
−1.71965 + 0.206851i
0.744857 + 1.56371i
−1.69483 − 0.357142i
1.69483 + 0.357142i
−0.744857 − 1.56371i
1.71965 − 0.206851i
0.604587 − 1.62311i
−0.604587 − 1.62311i
−1.71965 − 0.206851i
0.744857 − 1.56371i
−1.69483 + 0.357142i
1.69483 − 0.357142i
−0.744857 + 1.56371i
1.71965 + 0.206851i
0.604587 + 1.62311i
0 −1.70794 − 0.287965i 0 0.266780 + 0.462077i 0 0 0 2.83415 + 0.983658i 0
509.2 0 −1.03897 + 1.38584i 0 2.09336 + 3.62580i 0 0 0 −0.841101 − 2.87968i 0
509.3 0 −0.981784 − 1.42692i 0 0.276914 + 0.479629i 0 0 0 −1.07220 + 2.80185i 0
509.4 0 −0.538121 + 1.64634i 0 −1.21244 − 2.10001i 0 0 0 −2.42085 − 1.77186i 0
509.5 0 0.538121 − 1.64634i 0 1.21244 + 2.10001i 0 0 0 −2.42085 − 1.77186i 0
509.6 0 0.981784 + 1.42692i 0 −0.276914 − 0.479629i 0 0 0 −1.07220 + 2.80185i 0
509.7 0 1.03897 − 1.38584i 0 −2.09336 − 3.62580i 0 0 0 −0.841101 − 2.87968i 0
509.8 0 1.70794 + 0.287965i 0 −0.266780 − 0.462077i 0 0 0 2.83415 + 0.983658i 0
1109.1 0 −1.70794 + 0.287965i 0 0.266780 − 0.462077i 0 0 0 2.83415 − 0.983658i 0
1109.2 0 −1.03897 − 1.38584i 0 2.09336 − 3.62580i 0 0 0 −0.841101 + 2.87968i 0
1109.3 0 −0.981784 + 1.42692i 0 0.276914 − 0.479629i 0 0 0 −1.07220 − 2.80185i 0
1109.4 0 −0.538121 − 1.64634i 0 −1.21244 + 2.10001i 0 0 0 −2.42085 + 1.77186i 0
1109.5 0 0.538121 + 1.64634i 0 1.21244 − 2.10001i 0 0 0 −2.42085 + 1.77186i 0
1109.6 0 0.981784 − 1.42692i 0 −0.276914 + 0.479629i 0 0 0 −1.07220 − 2.80185i 0
1109.7 0 1.03897 + 1.38584i 0 −2.09336 + 3.62580i 0 0 0 −0.841101 + 2.87968i 0
1109.8 0 1.70794 − 0.287965i 0 −0.266780 + 0.462077i 0 0 0 2.83415 − 0.983658i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 509.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.w.a 16
3.b odd 2 1 5292.2.w.a 16
7.b odd 2 1 inner 1764.2.w.a 16
7.c even 3 1 252.2.x.a ✓ 16
7.c even 3 1 1764.2.bm.b 16
7.d odd 6 1 252.2.x.a ✓ 16
7.d odd 6 1 1764.2.bm.b 16
9.c even 3 1 5292.2.bm.b 16
9.d odd 6 1 1764.2.bm.b 16
21.c even 2 1 5292.2.w.a 16
21.g even 6 1 756.2.x.a 16
21.g even 6 1 5292.2.bm.b 16
21.h odd 6 1 756.2.x.a 16
21.h odd 6 1 5292.2.bm.b 16
28.f even 6 1 1008.2.cc.c 16
28.g odd 6 1 1008.2.cc.c 16
63.g even 3 1 756.2.x.a 16
63.h even 3 1 2268.2.f.b 16
63.h even 3 1 5292.2.w.a 16
63.i even 6 1 inner 1764.2.w.a 16
63.i even 6 1 2268.2.f.b 16
63.j odd 6 1 inner 1764.2.w.a 16
63.j odd 6 1 2268.2.f.b 16
63.k odd 6 1 756.2.x.a 16
63.l odd 6 1 5292.2.bm.b 16
63.n odd 6 1 252.2.x.a ✓ 16
63.o even 6 1 1764.2.bm.b 16
63.s even 6 1 252.2.x.a ✓ 16
63.t odd 6 1 2268.2.f.b 16
63.t odd 6 1 5292.2.w.a 16
84.j odd 6 1 3024.2.cc.c 16
84.n even 6 1 3024.2.cc.c 16
252.n even 6 1 3024.2.cc.c 16
252.o even 6 1 1008.2.cc.c 16
252.bl odd 6 1 3024.2.cc.c 16
252.bn odd 6 1 1008.2.cc.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.x.a ✓ 16 7.c even 3 1
252.2.x.a ✓ 16 7.d odd 6 1
252.2.x.a ✓ 16 63.n odd 6 1
252.2.x.a ✓ 16 63.s even 6 1
756.2.x.a 16 21.g even 6 1
756.2.x.a 16 21.h odd 6 1
756.2.x.a 16 63.g even 3 1
756.2.x.a 16 63.k odd 6 1
1008.2.cc.c 16 28.f even 6 1
1008.2.cc.c 16 28.g odd 6 1
1008.2.cc.c 16 252.o even 6 1
1008.2.cc.c 16 252.bn odd 6 1
1764.2.w.a 16 1.a even 1 1 trivial
1764.2.w.a 16 7.b odd 2 1 inner
1764.2.w.a 16 63.i even 6 1 inner
1764.2.w.a 16 63.j odd 6 1 inner
1764.2.bm.b 16 7.c even 3 1
1764.2.bm.b 16 7.d odd 6 1
1764.2.bm.b 16 9.d odd 6 1
1764.2.bm.b 16 63.o even 6 1
2268.2.f.b 16 63.h even 3 1
2268.2.f.b 16 63.i even 6 1
2268.2.f.b 16 63.j odd 6 1
2268.2.f.b 16 63.t odd 6 1
3024.2.cc.c 16 84.j odd 6 1
3024.2.cc.c 16 84.n even 6 1
3024.2.cc.c 16 252.n even 6 1
3024.2.cc.c 16 252.bl odd 6 1
5292.2.w.a 16 3.b odd 2 1
5292.2.w.a 16 21.c even 2 1
5292.2.w.a 16 63.h even 3 1
5292.2.w.a 16 63.t odd 6 1
5292.2.bm.b 16 9.c even 3 1
5292.2.bm.b 16 21.g even 6 1
5292.2.bm.b 16 21.h odd 6 1
5292.2.bm.b 16 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 24T_{5}^{14} + 459T_{5}^{12} + 2682T_{5}^{10} + 12168T_{5}^{8} + 6939T_{5}^{6} + 2916T_{5}^{4} + 567T_{5}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 24 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 48 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{16} + 78 T^{14} + \cdots + 810000 \) Copy content Toggle raw display
$19$ \( T^{16} - 75 T^{14} + \cdots + 810000 \) Copy content Toggle raw display
$23$ \( (T^{8} + 3 T^{7} + \cdots + 50625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 6 T^{7} + \cdots + 245025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 72 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + T^{7} + \cdots + 372100)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 331869318561 \) Copy content Toggle raw display
$43$ \( (T^{8} - 2 T^{7} + \cdots + 461041)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 222 T^{6} + \cdots + 4968441)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 18 T^{7} + \cdots + 41990400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 96 T^{6} + \cdots + 441)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 351 T^{6} + \cdots + 35319249)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 7 T^{3} + \cdots - 1985)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 207 T^{6} + \cdots + 15876)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 5802782976 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots + 565)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 30237384321 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 83955602727441 \) Copy content Toggle raw display
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