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Properties

Label 1600.3.g.f
Level $1600$
Weight $3$
Character orbit 1600.g
Analytic conductor $43.597$
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] = [1600,3,Mod(351,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.351");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.g (of order \(2\), degree \(1\), 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: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{6} q^{7} + ( - \beta_{3} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{6} q^{7} + ( - \beta_{3} - 4) q^{9} + ( - 6 \beta_{2} - 5 \beta_1) q^{11} - \beta_{5} q^{13} + (3 \beta_{3} - 5) q^{17} + ( - 7 \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{21} + (\beta_{7} + \beta_{6}) q^{23} + (8 \beta_{2} - \beta_1) q^{27} - 2 \beta_{5} q^{29} + ( - 2 \beta_{7} + \beta_{6}) q^{31} + ( - 6 \beta_{3} + 25) q^{33} + (\beta_{5} - 5 \beta_{4}) q^{37} + ( - \beta_{7} - 2 \beta_{6}) q^{39} + ( - 2 \beta_{3} + 7) q^{41} + (9 \beta_{2} - 9 \beta_1) q^{43} + ( - 8 \beta_{7} + 2 \beta_{6}) q^{47} + (16 \beta_{3} - 63) q^{49} + (20 \beta_{2} + 3 \beta_1) q^{51} + (5 \beta_{5} - 3 \beta_{4}) q^{53} + ( - 7 \beta_{3} + 37) q^{57} + ( - 23 \beta_{2} + 7 \beta_1) q^{59} + (6 \beta_{5} - \beta_{4}) q^{61} + ( - 5 \beta_{7} + 2 \beta_{6}) q^{63} + (2 \beta_{2} - 37 \beta_1) q^{67} + (3 \beta_{5} + 5 \beta_{4}) q^{69} + ( - 4 \beta_{7} - 6 \beta_{6}) q^{71} + (13 \beta_{3} + 19) q^{73} + ( - 7 \beta_{5} - 23 \beta_{4}) q^{77} + (3 \beta_{7} - 5 \beta_{6}) q^{79} + (17 \beta_{3} - 5) q^{81} + (44 \beta_{2} - 3 \beta_1) q^{83} + ( - 2 \beta_{7} - 4 \beta_{6}) q^{87} + ( - 23 \beta_{3} - 13) q^{89} + (48 \beta_{2} - 16 \beta_1) q^{91} - \beta_{4} q^{93} + (16 \beta_{3} + 58) q^{97} + ( - \beta_{2} + 39 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{9} - 40 q^{17} + 200 q^{33} + 56 q^{41} - 504 q^{49} + 296 q^{57} + 152 q^{73} - 40 q^{81} - 104 q^{89} + 464 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} - 2\zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 8\zeta_{24}^{4} - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -8\zeta_{24}^{5} - 8\zeta_{24}^{3} + 8\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -8\zeta_{24}^{7} + 4\zeta_{24}^{6} - 4\zeta_{24}^{5} + 4\zeta_{24}^{3} - 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 8\zeta_{24}^{6} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + 4\beta_{3} + 4\beta_{2} + 4\beta_1 ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} - 4\beta_{2} + 4\beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{5} + 4\beta_{2} + 4\beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - 2\beta_{6} - \beta_{5} + 4\beta_{3} - 4\beta_{2} - 4\beta_1 ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} - \beta_{5} - 4\beta_{3} + 4\beta_{2} + 4\beta_1 ) / 32 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
−0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 − 0.965926i
0.258819 + 0.965926i
0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.965926 + 0.258819i
0 −3.14626 0 0 0 − 13.7980i 0 0.898979 0
351.2 0 −3.14626 0 0 0 13.7980i 0 0.898979 0
351.3 0 −0.317837 0 0 0 − 5.79796i 0 −8.89898 0
351.4 0 −0.317837 0 0 0 5.79796i 0 −8.89898 0
351.5 0 0.317837 0 0 0 − 5.79796i 0 −8.89898 0
351.6 0 0.317837 0 0 0 5.79796i 0 −8.89898 0
351.7 0 3.14626 0 0 0 − 13.7980i 0 0.898979 0
351.8 0 3.14626 0 0 0 13.7980i 0 0.898979 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.g.f ✓ 8
4.b odd 2 1 inner 1600.3.g.f ✓ 8
5.b even 2 1 1600.3.g.g yes 8
5.c odd 4 1 1600.3.e.h 8
5.c odd 4 1 1600.3.e.k 8
8.b even 2 1 inner 1600.3.g.f ✓ 8
8.d odd 2 1 inner 1600.3.g.f ✓ 8
20.d odd 2 1 1600.3.g.g yes 8
20.e even 4 1 1600.3.e.h 8
20.e even 4 1 1600.3.e.k 8
40.e odd 2 1 1600.3.g.g yes 8
40.f even 2 1 1600.3.g.g yes 8
40.i odd 4 1 1600.3.e.h 8
40.i odd 4 1 1600.3.e.k 8
40.k even 4 1 1600.3.e.h 8
40.k even 4 1 1600.3.e.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.3.e.h 8 5.c odd 4 1
1600.3.e.h 8 20.e even 4 1
1600.3.e.h 8 40.i odd 4 1
1600.3.e.h 8 40.k even 4 1
1600.3.e.k 8 5.c odd 4 1
1600.3.e.k 8 20.e even 4 1
1600.3.e.k 8 40.i odd 4 1
1600.3.e.k 8 40.k even 4 1
1600.3.g.f ✓ 8 1.a even 1 1 trivial
1600.3.g.f ✓ 8 4.b odd 2 1 inner
1600.3.g.f ✓ 8 8.b even 2 1 inner
1600.3.g.f ✓ 8 8.d odd 2 1 inner
1600.3.g.g yes 8 5.b even 2 1
1600.3.g.g yes 8 20.d odd 2 1
1600.3.g.g yes 8 40.e odd 2 1
1600.3.g.g yes 8 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} - 10T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 10T_{17} - 191 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 224 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 490 T^{2} + 57121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 10 T - 191)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 586 T^{2} + 37249)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 480 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 512)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 480 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2656 T^{2} + 1149184)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 14 T - 47)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 972)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 7040 T^{2} + 7573504)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 7264 T^{2} + 7661824)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6424 T^{2} + 4787344)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 9312 T^{2} + 20793600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 14026 T^{2} + 4464769)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 13184 T^{2} + 102400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 38 T - 3695)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4832 T^{2} + 5683456)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 19978 T^{2} + 10660225)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 26 T - 12527)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 116 T - 2780)^{4} \) Copy content Toggle raw display
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