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Properties

Label 735.2.b.c
Level $735$
Weight $2$
Character orbit 735.b
Analytic conductor $5.869$
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] = [735,2,Mod(146,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("735.146");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.b (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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 1) q^{4} + q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} + \cdots - 1) q^{6}+ \cdots + (\beta_{7} + \beta_{6} + \cdots - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 1) q^{4} + q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + 6 \beta_{4} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 6 q^{4} + 8 q^{5} - 5 q^{6} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} - 6 q^{4} + 8 q^{5} - 5 q^{6} + q^{9} + 9 q^{12} - q^{15} - 2 q^{16} - 24 q^{17} - 7 q^{18} - 6 q^{20} - 40 q^{22} + 23 q^{24} + 8 q^{25} - 12 q^{26} - 4 q^{27} - 5 q^{30} - 2 q^{33} + 9 q^{36} - 14 q^{37} - 24 q^{38} - 12 q^{39} + 30 q^{41} + 16 q^{43} + q^{45} + 14 q^{46} - 12 q^{47} + 25 q^{48} - 6 q^{51} + 10 q^{54} + 6 q^{57} + 26 q^{58} - 24 q^{59} + 9 q^{60} - 24 q^{62} + 38 q^{64} + 38 q^{66} - 8 q^{67} + 13 q^{69} + q^{72} - q^{75} - 6 q^{78} + 58 q^{79} - 2 q^{80} + 13 q^{81} - 30 q^{83} - 24 q^{85} + 61 q^{87} + 4 q^{88} - 6 q^{89} - 7 q^{90} - 36 q^{93} + 39 q^{96} - 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + \nu^{3} + 6\nu^{2} + 4\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{4} + \nu^{3} - 6\nu^{2} + 4\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 22\nu^{2} + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 11\nu^{5} + 34\nu^{3} + 22\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{3} - 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} + 2\beta_{4} - 7\beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} + 9\beta_{5} - 9\beta_{3} + 32\beta_{2} - 70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} + 43\beta_{5} - 22\beta_{4} + 43\beta_{3} - 84\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
2.33086i
2.06288i
1.07834i
0.385731i
− 0.385731i
− 1.07834i
− 2.06288i
− 2.33086i
− 2.33086i 0.459555 − 1.66997i −3.43292 1.00000 −3.89248 − 1.07116i 0 3.33995i −2.57762 − 1.53489i − 2.33086i
146.2 − 2.06288i −1.71189 − 0.263509i −2.25548 1.00000 −0.543588 + 3.53142i 0 0.527019i 2.86113 + 0.902197i − 2.06288i
146.3 − 1.07834i −0.812371 + 1.52972i 0.837188 1.00000 1.64956 + 0.876010i 0 − 3.05945i −1.68011 − 2.48541i − 1.07834i
146.4 − 0.385731i 1.56470 + 0.742765i 1.85121 1.00000 0.286507 − 0.603555i 0 − 1.48553i 1.89660 + 2.32442i − 0.385731i
146.5 0.385731i 1.56470 − 0.742765i 1.85121 1.00000 0.286507 + 0.603555i 0 1.48553i 1.89660 − 2.32442i 0.385731i
146.6 1.07834i −0.812371 − 1.52972i 0.837188 1.00000 1.64956 − 0.876010i 0 3.05945i −1.68011 + 2.48541i 1.07834i
146.7 2.06288i −1.71189 + 0.263509i −2.25548 1.00000 −0.543588 − 3.53142i 0 − 0.527019i 2.86113 − 0.902197i 2.06288i
146.8 2.33086i 0.459555 + 1.66997i −3.43292 1.00000 −3.89248 + 1.07116i 0 − 3.33995i −2.57762 + 1.53489i 2.33086i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.b.c 8
3.b odd 2 1 735.2.b.d 8
7.b odd 2 1 735.2.b.d 8
7.c even 3 1 105.2.s.d yes 8
7.c even 3 1 735.2.s.k 8
7.d odd 6 1 105.2.s.c ✓ 8
7.d odd 6 1 735.2.s.l 8
21.c even 2 1 inner 735.2.b.c 8
21.g even 6 1 105.2.s.d yes 8
21.g even 6 1 735.2.s.k 8
21.h odd 6 1 105.2.s.c ✓ 8
21.h odd 6 1 735.2.s.l 8
35.i odd 6 1 525.2.t.g 8
35.j even 6 1 525.2.t.f 8
35.k even 12 2 525.2.q.f 16
35.l odd 12 2 525.2.q.e 16
105.o odd 6 1 525.2.t.g 8
105.p even 6 1 525.2.t.f 8
105.w odd 12 2 525.2.q.e 16
105.x even 12 2 525.2.q.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.c ✓ 8 7.d odd 6 1
105.2.s.c ✓ 8 21.h odd 6 1
105.2.s.d yes 8 7.c even 3 1
105.2.s.d yes 8 21.g even 6 1
525.2.q.e 16 35.l odd 12 2
525.2.q.e 16 105.w odd 12 2
525.2.q.f 16 35.k even 12 2
525.2.q.f 16 105.x even 12 2
525.2.t.f 8 35.j even 6 1
525.2.t.f 8 105.p even 6 1
525.2.t.g 8 35.i odd 6 1
525.2.t.g 8 105.o odd 6 1
735.2.b.c 8 1.a even 1 1 trivial
735.2.b.c 8 21.c even 2 1 inner
735.2.b.d 8 3.b odd 2 1
735.2.b.d 8 7.b odd 2 1
735.2.s.k 8 7.c even 3 1
735.2.s.k 8 21.g even 6 1
735.2.s.l 8 7.d odd 6 1
735.2.s.l 8 21.h odd 6 1

Hecke kernels

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

\( T_{2}^{8} + 11T_{2}^{6} + 36T_{2}^{4} + 32T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} + 12T_{17}^{3} + 42T_{17}^{2} + 42T_{17} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 11 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 56 T^{6} + \cdots + 21904 \) Copy content Toggle raw display
$13$ \( T^{8} + 21 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{3} + \cdots + 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 33 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$23$ \( T^{8} + 131 T^{6} + \cdots + 454276 \) Copy content Toggle raw display
$29$ \( T^{8} + 179 T^{6} + \cdots + 879844 \) Copy content Toggle raw display
$31$ \( T^{8} + 165 T^{6} + \cdots + 695556 \) Copy content Toggle raw display
$37$ \( (T^{4} + 7 T^{3} + \cdots + 124)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 15 T^{3} + \cdots - 378)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6 T^{3} + \cdots - 384)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 320 T^{6} + \cdots + 4194304 \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{3} + \cdots - 96)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 267 T^{6} + \cdots + 76176 \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots - 101)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 104 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$73$ \( T^{8} + 249 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$79$ \( (T^{4} - 29 T^{3} + \cdots - 3488)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 15 T^{3} + \cdots - 42)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{3} + \cdots - 1626)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 408 T^{6} + \cdots + 49449024 \) Copy content Toggle raw display
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