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Properties

Label 1734.2.f.k
Level $1734$
Weight $2$
Character orbit 1734.f
Analytic conductor $13.846$
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] = [1734,2,Mod(829,1734)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1734, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1734.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.f (of order \(4\), 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: \(13.8460597105\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 102)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{5} + \beta_{5} q^{6} + (2 \beta_{7} - \beta_{3} + 1) q^{7} - \beta_{3} q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{5} + \beta_{5} q^{6} + (2 \beta_{7} - \beta_{3} + 1) q^{7} - \beta_{3} q^{8} + \beta_{3} q^{9} + (\beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots + 1) q^{10}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{7} + 8 q^{10} - 16 q^{11} + 8 q^{14} + 8 q^{16} - 8 q^{18} + 8 q^{20} - 16 q^{22} + 8 q^{23} - 8 q^{28} - 24 q^{29} - 16 q^{30} - 8 q^{31} - 16 q^{33} - 16 q^{35} + 8 q^{37} + 16 q^{38} - 8 q^{39} - 8 q^{40} + 16 q^{41} + 16 q^{44} + 8 q^{45} + 8 q^{46} + 32 q^{47} - 24 q^{50} - 8 q^{56} + 16 q^{57} + 24 q^{58} + 24 q^{61} + 8 q^{62} + 8 q^{63} - 8 q^{64} - 32 q^{65} + 16 q^{67} + 40 q^{71} + 8 q^{72} - 8 q^{74} + 24 q^{75} + 8 q^{78} - 8 q^{79} - 8 q^{80} - 8 q^{81} + 16 q^{82} + 16 q^{88} + 32 q^{89} + 8 q^{90} + 64 q^{91} - 8 q^{92} + 16 q^{97} + 24 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1159\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
−0.382683 + 0.923880i
0.382683 − 0.923880i
0.923880 + 0.382683i
−0.923880 − 0.382683i
−0.382683 − 0.923880i
0.382683 + 0.923880i
0.923880 − 0.382683i
−0.923880 + 0.382683i
1.00000i −0.707107 − 0.707107i −1.00000 −2.95541 − 2.95541i 0.707107 − 0.707107i −1.61313 + 1.61313i − 1.00000i 1.00000i 2.95541 − 2.95541i
829.2 1.00000i −0.707107 − 0.707107i −1.00000 −1.87302 − 1.87302i 0.707107 − 0.707107i 3.61313 − 3.61313i − 1.00000i 1.00000i 1.87302 − 1.87302i
829.3 1.00000i 0.707107 + 0.707107i −1.00000 −0.892349 − 0.892349i −0.707107 + 0.707107i 2.08239 − 2.08239i − 1.00000i 1.00000i 0.892349 − 0.892349i
829.4 1.00000i 0.707107 + 0.707107i −1.00000 1.72078 + 1.72078i −0.707107 + 0.707107i −0.0823922 + 0.0823922i − 1.00000i 1.00000i −1.72078 + 1.72078i
1483.1 − 1.00000i −0.707107 + 0.707107i −1.00000 −2.95541 + 2.95541i 0.707107 + 0.707107i −1.61313 − 1.61313i 1.00000i − 1.00000i 2.95541 + 2.95541i
1483.2 − 1.00000i −0.707107 + 0.707107i −1.00000 −1.87302 + 1.87302i 0.707107 + 0.707107i 3.61313 + 3.61313i 1.00000i − 1.00000i 1.87302 + 1.87302i
1483.3 − 1.00000i 0.707107 − 0.707107i −1.00000 −0.892349 + 0.892349i −0.707107 − 0.707107i 2.08239 + 2.08239i 1.00000i − 1.00000i 0.892349 + 0.892349i
1483.4 − 1.00000i 0.707107 − 0.707107i −1.00000 1.72078 − 1.72078i −0.707107 − 0.707107i −0.0823922 − 0.0823922i 1.00000i − 1.00000i −1.72078 − 1.72078i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1734.2.f.k 8
17.b even 2 1 1734.2.f.l 8
17.c even 4 1 inner 1734.2.f.k 8
17.c even 4 1 1734.2.f.l 8
17.d even 8 1 1734.2.a.t 4
17.d even 8 1 1734.2.a.u 4
17.d even 8 2 1734.2.b.l 8
17.e odd 16 2 102.2.h.b ✓ 8
51.g odd 8 1 5202.2.a.bu 4
51.g odd 8 1 5202.2.a.bx 4
51.i even 16 2 306.2.l.e 8
68.i even 16 2 816.2.bq.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.h.b ✓ 8 17.e odd 16 2
306.2.l.e 8 51.i even 16 2
816.2.bq.c 8 68.i even 16 2
1734.2.a.t 4 17.d even 8 1
1734.2.a.u 4 17.d even 8 1
1734.2.b.l 8 17.d even 8 2
1734.2.f.k 8 1.a even 1 1 trivial
1734.2.f.k 8 17.c even 4 1 inner
1734.2.f.l 8 17.b even 2 1
1734.2.f.l 8 17.c even 4 1
5202.2.a.bu 4 51.g odd 8 1
5202.2.a.bx 4 51.g odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1734, [\chi])\):

\( T_{5}^{8} + 8T_{5}^{7} + 32T_{5}^{6} + 48T_{5}^{5} + 68T_{5}^{4} + 272T_{5}^{3} + 1152T_{5}^{2} + 1632T_{5} + 1156 \) Copy content Toggle raw display
\( T_{7}^{8} - 8T_{7}^{7} + 32T_{7}^{6} - 16T_{7}^{5} + 8T_{7}^{4} - 160T_{7}^{3} + 1152T_{7}^{2} + 192T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} + \cdots + 1156 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 16 T^{7} + \cdots + 73984 \) Copy content Toggle raw display
$13$ \( (T^{4} - 44 T^{2} + \cdots + 316)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 64 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 24 T^{7} + \cdots + 1156 \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + \cdots + 9604 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots + 3844 \) Copy content Toggle raw display
$43$ \( T^{8} + 64 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( (T^{4} - 16 T^{3} + \cdots - 376)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 280 T^{6} + \cdots + 99856 \) Copy content Toggle raw display
$59$ \( T^{8} + 256 T^{6} + \cdots + 153664 \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + \cdots + 8836 \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots - 136)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 40 T^{7} + \cdots + 107412496 \) Copy content Toggle raw display
$73$ \( T^{8} + 800 T^{5} + \cdots + 8836 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 6574096 \) Copy content Toggle raw display
$83$ \( T^{8} + 320 T^{6} + \cdots + 7268416 \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{3} + \cdots + 6596)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + \cdots + 81468676 \) Copy content Toggle raw display
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