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Properties

Label 592.2.bq.b
Level $592$
Weight $2$
Character orbit 592.bq
Analytic conductor $4.727$
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] = [592,2,Mod(65,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bq (of order \(18\), degree \(6\), 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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3} + \beta_1) q^{3} + ( - \beta_{11} - \beta_{8} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{6} + \beta_{3} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3} + \beta_1) q^{3} + ( - \beta_{11} - \beta_{8} + \cdots - \beta_{2}) q^{5}+ \cdots + (4 \beta_{11} - 2 \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 12 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 12 q^{7} - 6 q^{9} + 6 q^{11} + 6 q^{13} + 18 q^{19} - 6 q^{21} - 18 q^{25} + 6 q^{27} + 18 q^{29} - 6 q^{33} - 18 q^{35} + 30 q^{37} - 30 q^{39} + 24 q^{41} - 18 q^{45} - 6 q^{47} + 12 q^{49} - 12 q^{53} + 18 q^{55} - 36 q^{57} - 36 q^{61} + 6 q^{63} + 36 q^{65} + 30 q^{67} - 18 q^{69} - 12 q^{71} + 36 q^{75} + 12 q^{77} - 6 q^{79} + 24 q^{81} + 48 q^{83} + 18 q^{85} - 36 q^{87} - 18 q^{89} + 6 q^{91} - 12 q^{93} + 36 q^{97} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{36}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{36}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{36}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{36}^{7} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{36}^{8} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{36}^{10} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{36}^{9} + \zeta_{36}^{5} + \zeta_{36}^{3} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{36}^{11} + \zeta_{36}^{9} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{36}^{11} + \zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}^{3} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( \zeta_{36}^{11} + \zeta_{36}^{9} - \zeta_{36}^{7} - \zeta_{36}^{5} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \zeta_{36}^{11} - \zeta_{36}^{9} - \zeta_{36}^{7} - \zeta_{36}^{5} + \zeta_{36}^{3} \) Copy content Toggle raw display

Display \(\zeta_{36}^j\) in terms of \(\beta_i\)

\(\zeta_{36}\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{36}^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} + 2\beta_{9} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{36}^{5}\)\(=\) \( ( -2\beta_{11} + \beta_{10} - \beta_{8} + 2\beta_{7} - \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{36}^{7}\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{7} + 2\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{8}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\zeta_{36}^{9}\)\(=\) \( ( -\beta_{11} + 2\beta_{10} + \beta_{9} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{10}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{36}^{11}\)\(=\) \( ( -\beta_{11} + 2\beta_{10} - 2\beta_{8} + \beta_{7} + \beta_{4} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{36}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(\beta_{2} - \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.984808 + 0.173648i
−0.984808 − 0.173648i
−0.642788 − 0.766044i
0.642788 + 0.766044i
−0.342020 + 0.939693i
0.342020 − 0.939693i
−0.642788 + 0.766044i
0.642788 − 0.766044i
0.984808 − 0.173648i
−0.984808 + 0.173648i
−0.342020 − 0.939693i
0.342020 + 0.939693i
0 0.266044 − 1.50881i 0 −1.97937 + 2.35892i 0 −0.153180 − 0.128533i 0 0.613341 + 0.223238i 0
65.2 0 0.266044 − 1.50881i 0 −0.247315 + 0.294739i 0 2.50048 + 2.09815i 0 0.613341 + 0.223238i 0
225.1 0 −1.43969 + 1.20805i 0 −1.45842 + 4.00698i 0 3.39364 + 1.23518i 0 0.0923963 − 0.524005i 0
225.2 0 −1.43969 + 1.20805i 0 0.273629 − 0.751790i 0 0.138449 + 0.0503913i 0 0.0923963 − 0.524005i 0
289.1 0 −0.326352 − 0.118782i 0 0.839712 − 0.148064i 0 −0.240460 − 1.36372i 0 −2.20574 − 1.85083i 0
289.2 0 −0.326352 − 0.118782i 0 2.57176 − 0.453471i 0 0.361075 + 2.04776i 0 −2.20574 − 1.85083i 0
321.1 0 −1.43969 − 1.20805i 0 −1.45842 − 4.00698i 0 3.39364 − 1.23518i 0 0.0923963 + 0.524005i 0
321.2 0 −1.43969 − 1.20805i 0 0.273629 + 0.751790i 0 0.138449 − 0.0503913i 0 0.0923963 + 0.524005i 0
337.1 0 0.266044 + 1.50881i 0 −1.97937 − 2.35892i 0 −0.153180 + 0.128533i 0 0.613341 − 0.223238i 0
337.2 0 0.266044 + 1.50881i 0 −0.247315 − 0.294739i 0 2.50048 − 2.09815i 0 0.613341 − 0.223238i 0
465.1 0 −0.326352 + 0.118782i 0 0.839712 + 0.148064i 0 −0.240460 + 1.36372i 0 −2.20574 + 1.85083i 0
465.2 0 −0.326352 + 0.118782i 0 2.57176 + 0.453471i 0 0.361075 − 2.04776i 0 −2.20574 + 1.85083i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.bq.b 12
4.b odd 2 1 74.2.h.a ✓ 12
12.b even 2 1 666.2.bj.c 12
37.h even 18 1 inner 592.2.bq.b 12
148.o odd 18 1 74.2.h.a ✓ 12
148.q even 36 1 2738.2.a.r 6
148.q even 36 1 2738.2.a.s 6
444.ba even 18 1 666.2.bj.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.h.a ✓ 12 4.b odd 2 1
74.2.h.a ✓ 12 148.o odd 18 1
592.2.bq.b 12 1.a even 1 1 trivial
592.2.bq.b 12 37.h even 18 1 inner
666.2.bj.c 12 12.b even 2 1
666.2.bj.c 12 444.ba even 18 1
2738.2.a.r 6 148.q even 36 1
2738.2.a.s 6 148.q even 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3T_{3}^{5} + 6T_{3}^{4} + 8T_{3}^{3} + 12T_{3}^{2} + 6T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 408321 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 288369 \) Copy content Toggle raw display
$17$ \( T^{12} + 36 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{12} - 18 T^{11} + \cdots + 10439361 \) Copy content Toggle raw display
$23$ \( T^{12} - 63 T^{10} + \cdots + 431649 \) Copy content Toggle raw display
$29$ \( T^{12} - 18 T^{11} + \cdots + 110889 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 317445489 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} - 24 T^{11} + \cdots + 331776 \) Copy content Toggle raw display
$43$ \( T^{12} + 156 T^{10} + \cdots + 2277081 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 2027430729 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 45041148441 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17477104401 \) Copy content Toggle raw display
$61$ \( T^{12} + 36 T^{11} + \cdots + 331776 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 59697637561 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 4131885551616 \) Copy content Toggle raw display
$73$ \( (T^{6} - 366 T^{4} + \cdots + 94609)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 6 T^{11} + \cdots + 47961 \) Copy content Toggle raw display
$83$ \( T^{12} - 48 T^{11} + \cdots + 110889 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 687331089 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 27455164416 \) Copy content Toggle raw display
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