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Properties

Label 735.2.y.b
Level $735$
Weight $2$
Character orbit 735.y
Analytic conductor $5.869$
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] = [735,2,Mod(128,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.128");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.y (of order \(12\), degree \(4\), 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{24}^{7} q^{2} + ( - \zeta_{24}^{7} + \cdots - \zeta_{24}^{2}) q^{3}+ \cdots + (2 \zeta_{24}^{7} + \cdots + \zeta_{24}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{24}^{7} q^{2} + ( - \zeta_{24}^{7} + \cdots - \zeta_{24}^{2}) q^{3}+ \cdots + (4 \zeta_{24}^{7} + \cdots - 7 \zeta_{24}^{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 4 q^{5} - 24 q^{10} + 12 q^{11} + 16 q^{12} + 12 q^{15} - 16 q^{16} - 4 q^{17} - 32 q^{18} - 24 q^{19} + 16 q^{20} - 32 q^{22} + 12 q^{23} + 8 q^{27} - 16 q^{30} + 16 q^{33} - 8 q^{36} - 4 q^{37} - 24 q^{38} - 24 q^{43} - 8 q^{45} + 16 q^{46} - 4 q^{47} - 16 q^{48} + 16 q^{51} + 24 q^{52} - 24 q^{53} + 40 q^{54} + 36 q^{57} + 40 q^{59} - 16 q^{60} - 12 q^{65} + 16 q^{67} + 8 q^{68} - 32 q^{69} + 8 q^{75} - 16 q^{78} - 16 q^{80} - 28 q^{81} + 48 q^{82} - 16 q^{83} + 16 q^{85} + 96 q^{86} + 60 q^{87} - 48 q^{89} + 32 q^{90} + 72 q^{94} + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(-\zeta_{24}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−1.93185 + 0.517638i 1.33195 + 1.10721i 1.73205 − 1.00000i 2.03906 + 0.917738i −3.14626 − 1.44949i 0 0 0.548188 + 2.94949i −4.41421 − 0.717439i
128.2 1.93185 − 0.517638i −0.599900 + 1.62484i 1.73205 − 1.00000i −1.30701 + 1.81431i −0.317837 + 3.44949i 0 0 −2.28024 − 1.94949i −1.58579 + 4.18154i
263.1 −0.517638 + 1.93185i −1.10721 − 1.33195i −1.73205 − 1.00000i −1.81431 + 1.30701i 3.14626 − 1.44949i 0 0 −0.548188 + 2.94949i −1.58579 − 4.18154i
263.2 0.517638 − 1.93185i −1.62484 + 0.599900i −1.73205 − 1.00000i −0.917738 − 2.03906i 0.317837 + 3.44949i 0 0 2.28024 − 1.94949i −4.41421 + 0.717439i
422.1 −0.517638 − 1.93185i −1.10721 + 1.33195i −1.73205 + 1.00000i −1.81431 − 1.30701i 3.14626 + 1.44949i 0 0 −0.548188 − 2.94949i −1.58579 + 4.18154i
422.2 0.517638 + 1.93185i −1.62484 − 0.599900i −1.73205 + 1.00000i −0.917738 + 2.03906i 0.317837 − 3.44949i 0 0 2.28024 + 1.94949i −4.41421 − 0.717439i
557.1 −1.93185 − 0.517638i 1.33195 − 1.10721i 1.73205 + 1.00000i 2.03906 − 0.917738i −3.14626 + 1.44949i 0 0 0.548188 − 2.94949i −4.41421 + 0.717439i
557.2 1.93185 + 0.517638i −0.599900 − 1.62484i 1.73205 + 1.00000i −1.30701 − 1.81431i −0.317837 − 3.44949i 0 0 −2.28024 + 1.94949i −1.58579 − 4.18154i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner
35.f even 4 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.y.b 8
3.b odd 2 1 735.2.y.a 8
5.c odd 4 1 735.2.y.c 8
7.b odd 2 1 735.2.y.c 8
7.c even 3 1 735.2.j.a ✓ 8
7.c even 3 1 735.2.y.d 8
7.d odd 6 1 735.2.j.b yes 8
7.d odd 6 1 735.2.y.a 8
15.e even 4 1 735.2.y.d 8
21.c even 2 1 735.2.y.d 8
21.g even 6 1 735.2.j.a ✓ 8
21.g even 6 1 inner 735.2.y.b 8
21.h odd 6 1 735.2.j.b yes 8
21.h odd 6 1 735.2.y.c 8
35.f even 4 1 inner 735.2.y.b 8
35.k even 12 1 735.2.j.a ✓ 8
35.k even 12 1 735.2.y.d 8
35.l odd 12 1 735.2.j.b yes 8
35.l odd 12 1 735.2.y.a 8
105.k odd 4 1 735.2.y.a 8
105.w odd 12 1 735.2.j.b yes 8
105.w odd 12 1 735.2.y.c 8
105.x even 12 1 735.2.j.a ✓ 8
105.x even 12 1 inner 735.2.y.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.j.a ✓ 8 7.c even 3 1
735.2.j.a ✓ 8 21.g even 6 1
735.2.j.a ✓ 8 35.k even 12 1
735.2.j.a ✓ 8 105.x even 12 1
735.2.j.b yes 8 7.d odd 6 1
735.2.j.b yes 8 21.h odd 6 1
735.2.j.b yes 8 35.l odd 12 1
735.2.j.b yes 8 105.w odd 12 1
735.2.y.a 8 3.b odd 2 1
735.2.y.a 8 7.d odd 6 1
735.2.y.a 8 35.l odd 12 1
735.2.y.a 8 105.k odd 4 1
735.2.y.b 8 1.a even 1 1 trivial
735.2.y.b 8 21.g even 6 1 inner
735.2.y.b 8 35.f even 4 1 inner
735.2.y.b 8 105.x even 12 1 inner
735.2.y.c 8 5.c odd 4 1
735.2.y.c 8 7.b odd 2 1
735.2.y.c 8 21.h odd 6 1
735.2.y.c 8 105.w odd 12 1
735.2.y.d 8 7.c even 3 1
735.2.y.d 8 15.e even 4 1
735.2.y.d 8 21.c even 2 1
735.2.y.d 8 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{8} - 16T_{2}^{4} + 256 \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{3} + 7T_{11}^{2} + 30T_{11} + 25 \) Copy content Toggle raw display
\( T_{13}^{8} + 146T_{13}^{4} + 625 \) Copy content Toggle raw display
\( T_{17}^{8} + 4T_{17}^{7} + 8T_{17}^{6} + 232T_{17}^{5} - 161T_{17}^{4} - 5800T_{17}^{3} + 5000T_{17}^{2} - 62500T_{17} + 390625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 6 T^{3} + 7 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 146T^{4} + 625 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} + \cdots + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 12 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} - 75)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$41$ \( (T^{4} + 56 T^{2} + 400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 12 T^{3} + \cdots + 900)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T + 100)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$71$ \( (T^{4} + 112 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 10000 T^{4} + 100000000 \) Copy content Toggle raw display
$79$ \( T^{8} - 146 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{3} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + \cdots + 14400)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 146T^{4} + 625 \) Copy content Toggle raw display
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