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Properties

Label 1134.3.q.c
Level $1134$
Weight $3$
Character orbit 1134.q
Analytic conductor $30.899$
Analytic rank $0$
Dimension $8$
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] = [1134,3,Mod(701,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1134.q (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: \(30.8992619785\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 126)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - 2 \beta_{2} + 2) q^{4} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4}) q^{5} + \beta_{5} q^{7} + 2 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - 2 \beta_{2} + 2) q^{4} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4}) q^{5} + \beta_{5} q^{7} + 2 \beta_{6} q^{8} + (4 \beta_1 + 2) q^{10} + ( - 2 \beta_{4} - 4 \beta_{3}) q^{11} + ( - 4 \beta_{5} - 8 \beta_{2} + \cdots + 8) q^{13}+ \cdots - 7 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 16 q^{10} + 32 q^{13} - 16 q^{16} + 160 q^{19} - 16 q^{22} + 132 q^{25} - 16 q^{31} + 104 q^{34} + 304 q^{37} + 16 q^{40} - 80 q^{43} + 32 q^{46} - 28 q^{49} - 64 q^{52} - 928 q^{55} - 152 q^{58} - 232 q^{61} - 64 q^{64} + 192 q^{67} - 112 q^{70} - 192 q^{73} + 160 q^{76} - 304 q^{79} - 432 q^{82} - 328 q^{85} + 32 q^{88} + 224 q^{91} + 96 q^{94} + 288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 533\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 203\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 341\nu^{3} + 81\nu ) / 297 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 79\nu^{7} - 605\nu^{5} + 4345\nu^{3} - 5688\nu ) / 1485 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 4\beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 11\beta_{6} + 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} - 23\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31\beta_{7} + 79\beta_{6} - 79\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta _1 - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -533\beta_{4} - 203\beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
−2.23256 + 1.28897i
1.00781 − 0.581861i
−1.00781 + 0.581861i
2.23256 − 1.28897i
−2.23256 − 1.28897i
1.00781 + 0.581861i
−1.00781 − 0.581861i
2.23256 + 1.28897i
−1.22474 + 0.707107i 0 1.00000 − 1.73205i −7.70549 − 4.44876i 0 −1.32288 − 2.29129i 2.82843i 0 12.5830
701.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 5.25600 + 3.03455i 0 1.32288 + 2.29129i 2.82843i 0 −8.58301
701.3 1.22474 − 0.707107i 0 1.00000 − 1.73205i −5.25600 − 3.03455i 0 1.32288 + 2.29129i − 2.82843i 0 −8.58301
701.4 1.22474 − 0.707107i 0 1.00000 − 1.73205i 7.70549 + 4.44876i 0 −1.32288 − 2.29129i − 2.82843i 0 12.5830
1079.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −7.70549 + 4.44876i 0 −1.32288 + 2.29129i − 2.82843i 0 12.5830
1079.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 5.25600 − 3.03455i 0 1.32288 − 2.29129i − 2.82843i 0 −8.58301
1079.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −5.25600 + 3.03455i 0 1.32288 − 2.29129i 2.82843i 0 −8.58301
1079.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 7.70549 − 4.44876i 0 −1.32288 + 2.29129i 2.82843i 0 12.5830
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.3.q.c 8
3.b odd 2 1 inner 1134.3.q.c 8
9.c even 3 1 126.3.b.a ✓ 4
9.c even 3 1 inner 1134.3.q.c 8
9.d odd 6 1 126.3.b.a ✓ 4
9.d odd 6 1 inner 1134.3.q.c 8
36.f odd 6 1 1008.3.d.a 4
36.h even 6 1 1008.3.d.a 4
45.h odd 6 1 3150.3.e.e 4
45.j even 6 1 3150.3.e.e 4
45.k odd 12 2 3150.3.c.b 8
45.l even 12 2 3150.3.c.b 8
63.g even 3 1 882.3.s.e 8
63.h even 3 1 882.3.s.e 8
63.i even 6 1 882.3.s.i 8
63.j odd 6 1 882.3.s.e 8
63.k odd 6 1 882.3.s.i 8
63.l odd 6 1 882.3.b.f 4
63.n odd 6 1 882.3.s.e 8
63.o even 6 1 882.3.b.f 4
63.s even 6 1 882.3.s.i 8
63.t odd 6 1 882.3.s.i 8
72.j odd 6 1 4032.3.d.i 4
72.l even 6 1 4032.3.d.j 4
72.n even 6 1 4032.3.d.i 4
72.p odd 6 1 4032.3.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a ✓ 4 9.c even 3 1
126.3.b.a ✓ 4 9.d odd 6 1
882.3.b.f 4 63.l odd 6 1
882.3.b.f 4 63.o even 6 1
882.3.s.e 8 63.g even 3 1
882.3.s.e 8 63.h even 3 1
882.3.s.e 8 63.j odd 6 1
882.3.s.e 8 63.n odd 6 1
882.3.s.i 8 63.i even 6 1
882.3.s.i 8 63.k odd 6 1
882.3.s.i 8 63.s even 6 1
882.3.s.i 8 63.t odd 6 1
1008.3.d.a 4 36.f odd 6 1
1008.3.d.a 4 36.h even 6 1
1134.3.q.c 8 1.a even 1 1 trivial
1134.3.q.c 8 3.b odd 2 1 inner
1134.3.q.c 8 9.c even 3 1 inner
1134.3.q.c 8 9.d odd 6 1 inner
3150.3.c.b 8 45.k odd 12 2
3150.3.c.b 8 45.l even 12 2
3150.3.e.e 4 45.h odd 6 1
3150.3.e.e 4 45.j even 6 1
4032.3.d.i 4 72.j odd 6 1
4032.3.d.i 4 72.n even 6 1
4032.3.d.j 4 72.l even 6 1
4032.3.d.j 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 116T_{5}^{6} + 10540T_{5}^{4} - 338256T_{5}^{2} + 8503056 \) acting on \(S_{3}^{\mathrm{new}}(1134, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 116 T^{6} + \cdots + 8503056 \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 2176782336 \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{3} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 788 T^{2} + 79524)^{2} \) Copy content Toggle raw display
$19$ \( (T - 20)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 2176782336 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 61505984016 \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$37$ \( (T - 38)^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 828311133456 \) Copy content Toggle raw display
$43$ \( (T^{4} + 40 T^{3} + \cdots + 13191424)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 288 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 16164 T^{2} + 64738116)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 3392 T^{6} + \cdots + 84934656 \) Copy content Toggle raw display
$61$ \( (T^{4} + 116 T^{3} + \cdots + 2471184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 96 T^{3} + \cdots + 23658496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 464 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 48 T - 2224)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 152 T^{3} + \cdots + 15872256)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{4} + 19476 T^{2} + 443556)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 144 T^{3} + \cdots + 70023424)^{2} \) Copy content Toggle raw display
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