Properties

Label 735.4.a.h
Level $735$
Weight $4$
Character orbit 735.a
Self dual yes
Analytic conductor $43.366$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,4,Mod(1,735)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

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: yes
Analytic conductor: \(43.3664038542\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 3 q^{2} + 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} - 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} - 21 q^{8} + 9 q^{9} + 15 q^{10} - 45 q^{11} + 3 q^{12} + 31 q^{13} + 15 q^{15} - 71 q^{16} - 96 q^{17} + 27 q^{18} - 149 q^{19} + 5 q^{20} - 135 q^{22} - 141 q^{23} - 63 q^{24} + 25 q^{25} + 93 q^{26} + 27 q^{27} + 48 q^{29} + 45 q^{30} + 178 q^{31} - 45 q^{32} - 135 q^{33} - 288 q^{34} + 9 q^{36} + 371 q^{37} - 447 q^{38} + 93 q^{39} - 105 q^{40} - 225 q^{41} + 344 q^{43} - 45 q^{44} + 45 q^{45} - 423 q^{46} - 375 q^{47} - 213 q^{48} + 75 q^{50} - 288 q^{51} + 31 q^{52} - 663 q^{53} + 81 q^{54} - 225 q^{55} - 447 q^{57} + 144 q^{58} + 60 q^{59} + 15 q^{60} - 392 q^{61} + 534 q^{62} + 433 q^{64} + 155 q^{65} - 405 q^{66} - 280 q^{67} - 96 q^{68} - 423 q^{69} + 258 q^{71} - 189 q^{72} - 578 q^{73} + 1113 q^{74} + 75 q^{75} - 149 q^{76} + 279 q^{78} + 152 q^{79} - 355 q^{80} + 81 q^{81} - 675 q^{82} + 432 q^{83} - 480 q^{85} + 1032 q^{86} + 144 q^{87} + 945 q^{88} + 234 q^{89} + 135 q^{90} - 141 q^{92} + 534 q^{93} - 1125 q^{94} - 745 q^{95} - 135 q^{96} - 1352 q^{97} - 405 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
3.00000 3.00000 1.00000 5.00000 9.00000 0 −21.0000 9.00000 15.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.4.a.h 1
3.b odd 2 1 2205.4.a.d 1
7.b odd 2 1 735.4.a.g 1
7.d odd 6 2 105.4.i.a 2
21.c even 2 1 2205.4.a.h 1
21.g even 6 2 315.4.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.a 2 7.d odd 6 2
315.4.j.a 2 21.g even 6 2
735.4.a.g 1 7.b odd 2 1
735.4.a.h 1 1.a even 1 1 trivial
2205.4.a.d 1 3.b odd 2 1
2205.4.a.h 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{11} + 45 \) Copy content Toggle raw display
\( T_{13} - 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 45 \) Copy content Toggle raw display
$13$ \( T - 31 \) Copy content Toggle raw display
$17$ \( T + 96 \) Copy content Toggle raw display
$19$ \( T + 149 \) Copy content Toggle raw display
$23$ \( T + 141 \) Copy content Toggle raw display
$29$ \( T - 48 \) Copy content Toggle raw display
$31$ \( T - 178 \) Copy content Toggle raw display
$37$ \( T - 371 \) Copy content Toggle raw display
$41$ \( T + 225 \) Copy content Toggle raw display
$43$ \( T - 344 \) Copy content Toggle raw display
$47$ \( T + 375 \) Copy content Toggle raw display
$53$ \( T + 663 \) Copy content Toggle raw display
$59$ \( T - 60 \) Copy content Toggle raw display
$61$ \( T + 392 \) Copy content Toggle raw display
$67$ \( T + 280 \) Copy content Toggle raw display
$71$ \( T - 258 \) Copy content Toggle raw display
$73$ \( T + 578 \) Copy content Toggle raw display
$79$ \( T - 152 \) Copy content Toggle raw display
$83$ \( T - 432 \) Copy content Toggle raw display
$89$ \( T - 234 \) Copy content Toggle raw display
$97$ \( T + 1352 \) Copy content Toggle raw display
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