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Properties

Label 1944.2.i.q
Level $1944$
Weight $2$
Character orbit 1944.i
Analytic conductor $15.523$
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] = [1944,2,Mod(649,1944)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1944, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1944.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1944.i (of order \(3\), 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: \(15.5229181529\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 18x^{10} + 153x^{8} - 647x^{6} + 1512x^{4} - 1989x^{2} + 1369 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + (\beta_{9} - \beta_{8} - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + (\beta_{9} - \beta_{8} - \beta_{2} - \beta_1) q^{7} + \beta_{10} q^{11} + ( - \beta_{6} - \beta_{4} - \beta_{3}) q^{13} + ( - \beta_{9} + \beta_{4} + \beta_{3} + \cdots - 2) q^{17}+ \cdots + (\beta_{10} - 3 \beta_{8} + \cdots - 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{7} - 3 q^{11} - 3 q^{13} - 18 q^{17} + 18 q^{19} - 6 q^{23} - 18 q^{25} - 3 q^{29} - 24 q^{31} + 30 q^{35} + 30 q^{37} + 21 q^{41} - 18 q^{43} - 6 q^{47} - 21 q^{49} - 24 q^{53} + 36 q^{55} - 12 q^{59} - 18 q^{61} + 33 q^{65} - 24 q^{67} + 30 q^{71} + 12 q^{73} + 3 q^{77} - 21 q^{79} - 18 q^{83} - 24 q^{85} - 36 q^{89} + 60 q^{91} - 39 q^{95} - 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 18x^{10} + 153x^{8} - 647x^{6} + 1512x^{4} - 1989x^{2} + 1369 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{10} + 182\nu^{8} - 1494\nu^{6} + 5474\nu^{4} - 8897\nu^{2} - 528 ) / 4165 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 67\nu^{10} - 1141\nu^{8} + 8970\nu^{6} - 32011\nu^{4} + 51562\nu^{2} - 35697 ) / 4165 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 229 \nu^{11} + 5772 \nu^{10} - 1939 \nu^{9} - 89096 \nu^{8} - 1482 \nu^{7} + 655455 \nu^{6} + \cdots - 2677357 ) / 308210 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 330 \nu^{11} - 2479 \nu^{10} - 609 \nu^{9} + 42217 \nu^{8} + 45858 \nu^{7} - 331890 \nu^{6} + \cdots + 1320789 ) / 308210 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{11} - 287\nu^{9} + 2199\nu^{7} - 7650\nu^{5} + 14636\nu^{3} - 11937\nu + 3145 ) / 6290 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1191 \nu^{11} - 1998 \nu^{10} + 18256 \nu^{9} + 27454 \nu^{8} - 127574 \nu^{7} - 170496 \nu^{6} + \cdots - 743145 ) / 308210 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41 \nu^{11} - 28 \nu^{10} - 700 \nu^{9} + 301 \nu^{8} + 5482 \nu^{7} - 1120 \nu^{6} - 19516 \nu^{5} + \cdots - 31808 ) / 8330 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 234 \nu^{11} - 296 \nu^{10} + 3102 \nu^{9} + 5069 \nu^{8} - 19152 \nu^{7} - 39516 \nu^{6} + \cdots + 191475 ) / 44030 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2093 \nu^{11} + 6327 \nu^{10} + 30681 \nu^{9} - 98679 \nu^{8} - 210931 \nu^{7} + 729233 \nu^{6} + \cdots - 3284749 ) / 308210 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3461 \nu^{11} + 4070 \nu^{10} + 56896 \nu^{9} - 54131 \nu^{8} - 436552 \nu^{7} + 328227 \nu^{6} + \cdots + 560809 ) / 308210 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3461 \nu^{11} + 4070 \nu^{10} - 56896 \nu^{9} - 54131 \nu^{8} + 436552 \nu^{7} + 328227 \nu^{6} + \cdots + 560809 ) / 308210 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{11} + 4 \beta_{10} + \beta_{9} - 19 \beta_{8} + 5 \beta_{7} - 5 \beta_{6} - 11 \beta_{5} + \cdots + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{11} - 10 \beta_{10} + 6 \beta_{9} - 9 \beta_{8} - 15 \beta_{7} - 15 \beta_{6} + 6 \beta_{5} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{11} - 11 \beta_{10} + 7 \beta_{9} - 100 \beta_{8} - 22 \beta_{7} + 22 \beta_{6} - 161 \beta_{5} + \cdots + 84 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 59 \beta_{11} - 59 \beta_{10} + 28 \beta_{9} - 50 \beta_{8} - 94 \beta_{7} - 94 \beta_{6} + \cdots - 288 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 311 \beta_{11} - 311 \beta_{10} + 109 \beta_{9} - 130 \beta_{8} - 484 \beta_{7} + 484 \beta_{6} + \cdots + 525 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 93 \beta_{11} - 93 \beta_{10} + 53 \beta_{9} - 88 \beta_{8} - 158 \beta_{7} - 158 \beta_{6} + \cdots - 3052 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2690 \beta_{11} - 2690 \beta_{10} + 916 \beta_{9} + 3752 \beta_{8} - 4114 \beta_{7} + 4114 \beta_{6} + \cdots + 858 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2307 \beta_{11} + 2307 \beta_{10} - 749 \beta_{9} + 1665 \beta_{8} + 3497 \beta_{7} + 3497 \beta_{6} + \cdots - 17498 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12161 \beta_{11} - 12161 \beta_{10} + 4114 \beta_{9} + 47328 \beta_{8} - 18556 \beta_{7} + 18556 \beta_{6} + \cdots - 20160 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.16625 − 0.642788i
2.53804 + 0.984808i
−1.64263 − 0.342020i
1.64263 − 0.342020i
−1.16625 − 0.642788i
−2.53804 + 0.984808i
1.16625 + 0.642788i
2.53804 − 0.984808i
−1.64263 + 0.342020i
1.64263 + 0.342020i
−1.16625 + 0.642788i
−2.53804 − 0.984808i
0 0 0 −1.94637 − 3.37122i 0 0.709291 − 1.22853i 0 0 0
649.2 0 0 0 −1.32672 − 2.29795i 0 −2.30565 + 3.99350i 0 0 0
649.3 0 0 0 −0.635428 − 1.10059i 0 −1.56360 + 2.70823i 0 0 0
649.4 0 0 0 0.809076 + 1.40136i 0 2.59569 − 4.49586i 0 0 0
649.5 0 0 0 1.00668 + 1.74362i 0 −0.861995 + 1.49302i 0 0 0
649.6 0 0 0 2.09277 + 3.62478i 0 −0.0737368 + 0.127716i 0 0 0
1297.1 0 0 0 −1.94637 + 3.37122i 0 0.709291 + 1.22853i 0 0 0
1297.2 0 0 0 −1.32672 + 2.29795i 0 −2.30565 − 3.99350i 0 0 0
1297.3 0 0 0 −0.635428 + 1.10059i 0 −1.56360 − 2.70823i 0 0 0
1297.4 0 0 0 0.809076 − 1.40136i 0 2.59569 + 4.49586i 0 0 0
1297.5 0 0 0 1.00668 − 1.74362i 0 −0.861995 − 1.49302i 0 0 0
1297.6 0 0 0 2.09277 − 3.62478i 0 −0.0737368 − 0.127716i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.2.i.q 12
3.b odd 2 1 1944.2.i.r 12
9.c even 3 1 1944.2.a.r yes 6
9.c even 3 1 inner 1944.2.i.q 12
9.d odd 6 1 1944.2.a.q ✓ 6
9.d odd 6 1 1944.2.i.r 12
36.f odd 6 1 3888.2.a.bl 6
36.h even 6 1 3888.2.a.bm 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.2.a.q ✓ 6 9.d odd 6 1
1944.2.a.r yes 6 9.c even 3 1
1944.2.i.q 12 1.a even 1 1 trivial
1944.2.i.q 12 9.c even 3 1 inner
1944.2.i.r 12 3.b odd 2 1
1944.2.i.r 12 9.d odd 6 1
3888.2.a.bl 6 36.f odd 6 1
3888.2.a.bm 6 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1944, [\chi])\):

\( T_{5}^{12} + 24 T_{5}^{10} + 4 T_{5}^{9} + 441 T_{5}^{8} + 36 T_{5}^{7} + 2886 T_{5}^{6} - 846 T_{5}^{5} + \cdots + 32041 \) Copy content Toggle raw display
\( T_{7}^{12} + 3 T_{7}^{11} + 36 T_{7}^{10} + 105 T_{7}^{9} + 981 T_{7}^{8} + 2538 T_{7}^{7} + 8865 T_{7}^{6} + \cdots + 729 \) 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
$5$ \( T^{12} + 24 T^{10} + \cdots + 32041 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} + \cdots + 1495729 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( (T^{6} + 9 T^{5} + \cdots + 136)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 9 T^{5} + \cdots + 568)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 6 T^{11} + \cdots + 506944 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 45927729 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 2565523801 \) Copy content Toggle raw display
$37$ \( (T^{6} - 15 T^{5} + \cdots - 23976)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 21 T^{11} + \cdots + 419904 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 558093376 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 29934536256 \) Copy content Toggle raw display
$53$ \( (T^{6} + 12 T^{5} + \cdots - 15263)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 105247081 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 58940186176 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 81001713664 \) Copy content Toggle raw display
$71$ \( (T^{6} - 15 T^{5} + \cdots + 303832)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 6 T^{5} + \cdots - 11177)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 21 T^{11} + \cdots + 128881 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 19405325809 \) Copy content Toggle raw display
$89$ \( (T^{6} + 18 T^{5} + \cdots - 174312)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 13333551841 \) Copy content Toggle raw display
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