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Properties

Label 1152.2.i.g
Level $1152$
Weight $2$
Character orbit 1152.i
Analytic conductor $9.199$
Analytic rank $0$
Dimension $10$
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] = [1152,2,Mod(385,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, 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("1152.385");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.i (of order \(3\), degree \(2\), 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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.8528759163648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{5}) q^{3} + \beta_{8} q^{5} + (\beta_{8} + \beta_{5} + \beta_{4} + \cdots - 1) q^{7}+ \cdots + (\beta_{9} - \beta_{6} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{5}) q^{3} + \beta_{8} q^{5} + (\beta_{8} + \beta_{5} + \beta_{4} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{8} - \beta_{7} + \cdots + 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} - 4 q^{7} - q^{9} - q^{11} - 6 q^{13} + 12 q^{15} - 6 q^{17} + 18 q^{19} - 16 q^{21} + 4 q^{23} + q^{25} - 2 q^{27} + 4 q^{29} - 8 q^{31} - 13 q^{33} - 24 q^{35} + 20 q^{37} - 18 q^{39} - 5 q^{41} - 13 q^{43} + 12 q^{45} - 6 q^{47} + 3 q^{49} + 3 q^{51} + 12 q^{55} + 27 q^{57} - 13 q^{59} - 10 q^{61} - 20 q^{63} - 17 q^{67} + 10 q^{69} + 8 q^{71} - 34 q^{73} - 29 q^{75} - 8 q^{77} - 6 q^{79} - q^{81} + 12 q^{83} - 18 q^{85} + 10 q^{87} + 44 q^{89} + 36 q^{91} - 26 q^{93} - 6 q^{95} + 27 q^{97} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + \nu^{8} + 4\nu^{7} + 12\nu^{6} + 45\nu^{5} - 63\nu^{4} + 27\nu - 567 ) / 486 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 4\nu^{7} + 15\nu^{6} - 18\nu^{5} + 9\nu^{4} + 81\nu^{3} + 162\nu^{2} - 270\nu + 81 ) / 243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 4\nu^{7} - 12\nu^{6} + 9\nu^{5} + 9\nu^{4} + 297\nu + 81 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - \nu^{7} - 9\nu^{5} + 36\nu^{4} - 27\nu^{3} - 27\nu + 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{9} + 7\nu^{8} + 28\nu^{7} - 24\nu^{6} + 45\nu^{5} - 63\nu^{4} + 162\nu^{3} - 648\nu^{2} - 297\nu - 567 ) / 486 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{9} + 11\nu^{8} - 10\nu^{7} - 12\nu^{6} - 99\nu^{5} + 117\nu^{4} - 189\nu + 1053 ) / 486 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{9} - 4\nu^{8} + 11\nu^{7} + 6\nu^{6} - 45\nu^{5} + 36\nu^{4} + 81\nu^{3} - 81\nu^{2} - 108\nu + 324 ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17 \nu^{9} + 19 \nu^{8} + 22 \nu^{7} - 60 \nu^{6} - 171 \nu^{5} + 369 \nu^{4} + 162 \nu^{3} + \cdots + 2349 ) / 486 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} + \beta_{6} + 3\beta_{5} + 4\beta_{4} + 2\beta_{3} + 6\beta_{2} - 5\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{9} + 6\beta_{8} + 3\beta_{7} + 4\beta_{6} + 9\beta_{5} + \beta_{4} + 5\beta_{3} + 6\beta_{2} + 7\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{9} - 6 \beta_{8} - 18 \beta_{7} - 11 \beta_{6} - 12 \beta_{5} - 20 \beta_{4} - 10 \beta_{3} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9 \beta_{9} - 18 \beta_{8} + 15 \beta_{7} + \beta_{6} - 6 \beta_{5} - 14 \beta_{4} + 2 \beta_{3} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3\beta_{9} - 30\beta_{8} - 33\beta_{7} + 13\beta_{6} - 35\beta_{4} + 14\beta_{3} - 102\beta_{2} + 70\beta _1 + 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
−1.41743 + 0.995434i
0.756905 + 1.55791i
−1.13593 − 1.30754i
1.72806 + 0.117480i
1.06839 − 1.36328i
−1.41743 − 0.995434i
0.756905 − 1.55791i
−1.13593 + 1.30754i
1.72806 − 0.117480i
1.06839 + 1.36328i
0 −1.57079 + 0.729814i 0 −0.115851 − 0.200661i 0 −0.230793 + 0.399745i 0 1.93474 − 2.29277i 0
385.2 0 −0.970741 − 1.43446i 0 1.07447 + 1.86104i 0 0.153174 − 0.265305i 0 −1.11533 + 2.78497i 0
385.3 0 0.564403 + 1.63751i 0 −1.59327 − 2.75962i 0 −0.607060 + 1.05146i 0 −2.36290 + 1.84844i 0
385.4 0 0.762291 − 1.55529i 0 −0.705463 − 1.22190i 0 1.17123 − 2.02864i 0 −1.83783 − 2.37116i 0
385.5 0 1.71483 − 0.243611i 0 1.34011 + 2.32114i 0 −2.48656 + 4.30684i 0 2.88131 − 0.835506i 0
769.1 0 −1.57079 − 0.729814i 0 −0.115851 + 0.200661i 0 −0.230793 − 0.399745i 0 1.93474 + 2.29277i 0
769.2 0 −0.970741 + 1.43446i 0 1.07447 − 1.86104i 0 0.153174 + 0.265305i 0 −1.11533 − 2.78497i 0
769.3 0 0.564403 − 1.63751i 0 −1.59327 + 2.75962i 0 −0.607060 − 1.05146i 0 −2.36290 − 1.84844i 0
769.4 0 0.762291 + 1.55529i 0 −0.705463 + 1.22190i 0 1.17123 + 2.02864i 0 −1.83783 + 2.37116i 0
769.5 0 1.71483 + 0.243611i 0 1.34011 − 2.32114i 0 −2.48656 − 4.30684i 0 2.88131 + 0.835506i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 385.5
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 1152.2.i.g yes 10
3.b odd 2 1 3456.2.i.f 10
4.b odd 2 1 1152.2.i.f yes 10
8.b even 2 1 1152.2.i.e ✓ 10
8.d odd 2 1 1152.2.i.h yes 10
9.c even 3 1 inner 1152.2.i.g yes 10
9.d odd 6 1 3456.2.i.f 10
12.b even 2 1 3456.2.i.g 10
24.f even 2 1 3456.2.i.h 10
24.h odd 2 1 3456.2.i.e 10
36.f odd 6 1 1152.2.i.f yes 10
36.h even 6 1 3456.2.i.g 10
72.j odd 6 1 3456.2.i.e 10
72.l even 6 1 3456.2.i.h 10
72.n even 6 1 1152.2.i.e ✓ 10
72.p odd 6 1 1152.2.i.h yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.e ✓ 10 8.b even 2 1
1152.2.i.e ✓ 10 72.n even 6 1
1152.2.i.f yes 10 4.b odd 2 1
1152.2.i.f yes 10 36.f odd 6 1
1152.2.i.g yes 10 1.a even 1 1 trivial
1152.2.i.g yes 10 9.c even 3 1 inner
1152.2.i.h yes 10 8.d odd 2 1
1152.2.i.h yes 10 72.p odd 6 1
3456.2.i.e 10 24.h odd 2 1
3456.2.i.e 10 72.j odd 6 1
3456.2.i.f 10 3.b odd 2 1
3456.2.i.f 10 9.d odd 6 1
3456.2.i.g 10 12.b even 2 1
3456.2.i.g 10 36.h even 6 1
3456.2.i.h 10 24.f even 2 1
3456.2.i.h 10 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{10} + 12T_{5}^{8} - 4T_{5}^{7} + 117T_{5}^{6} - 18T_{5}^{5} + 328T_{5}^{4} + 198T_{5}^{3} + 717T_{5}^{2} + 162T_{5} + 36 \) Copy content Toggle raw display
\( T_{7}^{10} + 4T_{7}^{9} + 24T_{7}^{8} + 129T_{7}^{6} + 138T_{7}^{5} + 240T_{7}^{4} + 48T_{7}^{3} + 33T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + 12 T^{8} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{10} + 4 T^{9} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + \cdots + 961 \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 2304 \) Copy content Toggle raw display
$17$ \( (T^{5} + 3 T^{4} + \cdots + 108)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 9 T^{4} + \cdots - 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 4 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( T^{10} - 4 T^{9} + \cdots + 34596 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots + 22810176 \) Copy content Toggle raw display
$37$ \( (T^{5} - 10 T^{4} + \cdots - 344)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 319587129 \) Copy content Toggle raw display
$43$ \( T^{10} + 13 T^{9} + \cdots + 720801 \) Copy content Toggle raw display
$47$ \( T^{10} + 6 T^{9} + \cdots + 12194064 \) Copy content Toggle raw display
$53$ \( (T^{5} - 144 T^{3} + \cdots - 9648)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 2802325969 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 2249415184 \) Copy content Toggle raw display
$67$ \( T^{10} + 17 T^{9} + \cdots + 63001 \) Copy content Toggle raw display
$71$ \( (T^{5} - 4 T^{4} + \cdots - 14472)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 17 T^{4} + \cdots - 2972)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 6986619396 \) Copy content Toggle raw display
$83$ \( T^{10} - 12 T^{9} + \cdots + 8088336 \) Copy content Toggle raw display
$89$ \( (T^{5} - 22 T^{4} + \cdots - 27464)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 10710387081 \) Copy content Toggle raw display
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