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Properties

Label 1134.2.h.v
Level $1134$
Weight $2$
Character orbit 1134.h
Analytic conductor $9.055$
Analytic rank $0$
Dimension $8$
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] = [1134,2,Mod(109,1134)]
 
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([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.h (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: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.454201344.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{4} - 1) q^{4} + (\beta_{5} + 1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{4} - 1) q^{4} + (\beta_{5} + 1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{7} - q^{8} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{10} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - 3 \beta_{6} - \beta_{5} + \cdots + \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} + 8 q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} + 8 q^{5} - 8 q^{8} + 4 q^{10} + 4 q^{11} + 2 q^{13} + 6 q^{14} - 4 q^{16} - 8 q^{17} + 10 q^{19} - 4 q^{20} + 2 q^{22} + 4 q^{23} + 24 q^{25} - 2 q^{26} + 6 q^{28} + 14 q^{29} + 2 q^{31} + 4 q^{32} + 8 q^{34} + 4 q^{35} + 4 q^{37} + 20 q^{38} - 8 q^{40} - 12 q^{41} + 16 q^{43} - 2 q^{44} + 2 q^{46} - 18 q^{47} + 8 q^{49} + 12 q^{50} - 4 q^{52} - 6 q^{53} + 56 q^{55} + 28 q^{58} - 16 q^{59} + 14 q^{61} + 4 q^{62} + 8 q^{64} - 24 q^{65} - 18 q^{67} + 16 q^{68} + 8 q^{70} - 20 q^{71} + 24 q^{73} + 8 q^{74} + 10 q^{76} - 28 q^{77} - 10 q^{79} - 4 q^{80} + 12 q^{82} - 14 q^{83} + 20 q^{85} + 32 q^{86} - 4 q^{88} - 12 q^{89} + 26 q^{91} - 2 q^{92} + 18 q^{94} - 20 q^{95} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 20\nu^{7} + 402\nu^{6} + 43\nu^{5} - 760\nu^{4} + 4607\nu^{3} - 5750\nu^{2} + 6903\nu + 37211 ) / 21903 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1429 \nu^{7} + 249 \nu^{6} - 28687 \nu^{5} + 55213 \nu^{4} + 130948 \nu^{3} - 312274 \nu^{2} + \cdots + 257062 ) / 284739 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 503\nu^{7} + 619\nu^{6} - 16806\nu^{5} + 39294\nu^{4} + 20588\nu^{3} - 177467\nu^{2} + 403957\nu - 361166 ) / 94913 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -62\nu^{7} - 84\nu^{6} + 716\nu^{5} - 773\nu^{4} - 2168\nu^{3} + 3968\nu^{2} - 6246\nu - 1235 ) / 5811 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4786 \nu^{7} + 17697 \nu^{6} + 40087 \nu^{5} - 168580 \nu^{4} + 95639 \nu^{3} + 434146 \nu^{2} + \cdots + 1041560 ) / 284739 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6835 \nu^{7} + 4986 \nu^{6} + 23635 \nu^{5} - 68815 \nu^{4} - 68611 \nu^{3} - 39062 \nu^{2} + \cdots - 500461 ) / 284739 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8923 \nu^{7} + 19965 \nu^{6} + 49810 \nu^{5} - 252307 \nu^{4} + 142553 \nu^{3} + 495529 \nu^{2} + \cdots + 888953 ) / 284739 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - 8\beta_{6} - 2\beta_{5} + 11\beta_{4} + 5\beta_{3} + 4\beta_{2} + 2\beta _1 - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} - 2\beta_{6} + 8\beta_{5} + 7\beta_{4} + 4\beta_{3} + 4\beta_{2} - 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{7} - 13\beta_{6} - \beta_{5} + 46\beta_{4} + 4\beta_{3} + 5\beta_{2} + 58\beta _1 - 91 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -32\beta_{7} + 18\beta_{6} + 45\beta_{5} - 18\beta_{4} - 7\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -19\beta_{7} + 40\beta_{6} - 131\beta_{5} - 232\beta_{4} - 187\beta_{3} - 131\beta_{2} + 368\beta _1 - 659 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
2.02641 + 1.27503i
0.338876 + 1.46735i
1.02715 − 1.10132i
−2.39244 + 0.0909984i
2.02641 − 1.27503i
0.338876 − 1.46735i
1.02715 + 1.10132i
−2.39244 − 0.0909984i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.23483 0 1.18403 + 2.36603i −1.00000 0 −1.11741 + 1.93542i
109.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.880398 0 −2.56867 − 0.633975i −1.00000 0 −0.440199 + 0.762447i
109.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.88040 0 2.56867 − 0.633975i −1.00000 0 1.44020 − 2.49450i
109.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 4.23483 0 −1.18403 + 2.36603i −1.00000 0 2.11741 − 3.66747i
541.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.23483 0 1.18403 − 2.36603i −1.00000 0 −1.11741 − 1.93542i
541.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.880398 0 −2.56867 + 0.633975i −1.00000 0 −0.440199 − 0.762447i
541.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.88040 0 2.56867 + 0.633975i −1.00000 0 1.44020 + 2.49450i
541.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 4.23483 0 −1.18403 − 2.36603i −1.00000 0 2.11741 + 3.66747i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.h.v 8
3.b odd 2 1 1134.2.h.u 8
7.c even 3 1 1134.2.e.u 8
9.c even 3 1 1134.2.e.u 8
9.c even 3 1 1134.2.g.p yes 8
9.d odd 6 1 1134.2.e.v 8
9.d odd 6 1 1134.2.g.o ✓ 8
21.h odd 6 1 1134.2.e.v 8
63.g even 3 1 inner 1134.2.h.v 8
63.g even 3 1 7938.2.a.cl 4
63.h even 3 1 1134.2.g.p yes 8
63.j odd 6 1 1134.2.g.o ✓ 8
63.k odd 6 1 7938.2.a.cc 4
63.n odd 6 1 1134.2.h.u 8
63.n odd 6 1 7938.2.a.cm 4
63.s even 6 1 7938.2.a.cv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.u 8 7.c even 3 1
1134.2.e.u 8 9.c even 3 1
1134.2.e.v 8 9.d odd 6 1
1134.2.e.v 8 21.h odd 6 1
1134.2.g.o ✓ 8 9.d odd 6 1
1134.2.g.o ✓ 8 63.j odd 6 1
1134.2.g.p yes 8 9.c even 3 1
1134.2.g.p yes 8 63.h even 3 1
1134.2.h.u 8 3.b odd 2 1
1134.2.h.u 8 63.n odd 6 1
1134.2.h.v 8 1.a even 1 1 trivial
1134.2.h.v 8 63.g even 3 1 inner
7938.2.a.cc 4 63.k odd 6 1
7938.2.a.cl 4 63.g even 3 1
7938.2.a.cm 4 63.n odd 6 1
7938.2.a.cv 4 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{4} - 4T_{5}^{3} - 8T_{5}^{2} + 24T_{5} + 24 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 20T_{11}^{2} - 12T_{11} + 6 \) Copy content Toggle raw display
\( T_{17}^{8} + 8T_{17}^{7} + 60T_{17}^{6} + 80T_{17}^{5} + 220T_{17}^{4} + 96T_{17}^{3} + 624T_{17}^{2} + 288T_{17} + 144 \) Copy content Toggle raw display
\( T_{23}^{4} - 2T_{23}^{3} - 80T_{23}^{2} + 102T_{23} + 1473 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{3} - 8 T^{2} + \cdots + 24)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 20 T^{2} + \cdots + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots + 14884 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} + \cdots + 1473)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 14 T^{7} + \cdots + 125316 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 541696 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 2259009 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} + \cdots + 341056 \) Copy content Toggle raw display
$47$ \( T^{8} + 18 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$53$ \( T^{8} + 6 T^{7} + \cdots + 171396 \) Copy content Toggle raw display
$59$ \( T^{8} + 16 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$61$ \( T^{8} - 14 T^{7} + \cdots + 86436 \) Copy content Toggle raw display
$67$ \( T^{8} + 18 T^{7} + \cdots + 1119364 \) Copy content Toggle raw display
$71$ \( (T^{4} + 10 T^{3} + \cdots - 747)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 24 T^{7} + \cdots + 52441 \) Copy content Toggle raw display
$79$ \( T^{8} + 10 T^{7} + \cdots + 43441281 \) Copy content Toggle raw display
$83$ \( T^{8} + 14 T^{7} + \cdots + 248004 \) Copy content Toggle raw display
$89$ \( T^{8} + 12 T^{7} + \cdots + 15896169 \) Copy content Toggle raw display
$97$ \( T^{8} + 164 T^{6} + \cdots + 21270544 \) Copy content Toggle raw display
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