Properties

Label 105.4.a.b
Level $105$
Weight $4$
Character orbit 105.a
Self dual yes
Analytic conductor $6.195$
Analytic rank $0$
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] = [105,4,Mod(1,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, 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("105.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.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: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + 5 q^{2} - 3 q^{3} + 17 q^{4} + 5 q^{5} - 15 q^{6} + 7 q^{7} + 45 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{2} - 3 q^{3} + 17 q^{4} + 5 q^{5} - 15 q^{6} + 7 q^{7} + 45 q^{8} + 9 q^{9} + 25 q^{10} + 12 q^{11} - 51 q^{12} + 30 q^{13} + 35 q^{14} - 15 q^{15} + 89 q^{16} - 134 q^{17} + 45 q^{18} - 92 q^{19} + 85 q^{20} - 21 q^{21} + 60 q^{22} + 112 q^{23} - 135 q^{24} + 25 q^{25} + 150 q^{26} - 27 q^{27} + 119 q^{28} - 58 q^{29} - 75 q^{30} - 224 q^{31} + 85 q^{32} - 36 q^{33} - 670 q^{34} + 35 q^{35} + 153 q^{36} - 146 q^{37} - 460 q^{38} - 90 q^{39} + 225 q^{40} + 18 q^{41} - 105 q^{42} + 340 q^{43} + 204 q^{44} + 45 q^{45} + 560 q^{46} + 208 q^{47} - 267 q^{48} + 49 q^{49} + 125 q^{50} + 402 q^{51} + 510 q^{52} - 754 q^{53} - 135 q^{54} + 60 q^{55} + 315 q^{56} + 276 q^{57} - 290 q^{58} + 380 q^{59} - 255 q^{60} + 718 q^{61} - 1120 q^{62} + 63 q^{63} - 287 q^{64} + 150 q^{65} - 180 q^{66} + 412 q^{67} - 2278 q^{68} - 336 q^{69} + 175 q^{70} - 960 q^{71} + 405 q^{72} + 1066 q^{73} - 730 q^{74} - 75 q^{75} - 1564 q^{76} + 84 q^{77} - 450 q^{78} + 896 q^{79} + 445 q^{80} + 81 q^{81} + 90 q^{82} + 436 q^{83} - 357 q^{84} - 670 q^{85} + 1700 q^{86} + 174 q^{87} + 540 q^{88} - 1038 q^{89} + 225 q^{90} + 210 q^{91} + 1904 q^{92} + 672 q^{93} + 1040 q^{94} - 460 q^{95} - 255 q^{96} - 702 q^{97} + 245 q^{98} + 108 q^{99}+O(q^{100}) \) 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
5.00000 −3.00000 17.0000 5.00000 −15.0000 7.00000 45.0000 9.00000 25.0000
\(n\): 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 105.4.a.b 1
3.b odd 2 1 315.4.a.a 1
4.b odd 2 1 1680.4.a.u 1
5.b even 2 1 525.4.a.a 1
5.c odd 4 2 525.4.d.a 2
7.b odd 2 1 735.4.a.j 1
15.d odd 2 1 1575.4.a.l 1
21.c even 2 1 2205.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 1.a even 1 1 trivial
315.4.a.a 1 3.b odd 2 1
525.4.a.a 1 5.b even 2 1
525.4.d.a 2 5.c odd 4 2
735.4.a.j 1 7.b odd 2 1
1575.4.a.l 1 15.d odd 2 1
1680.4.a.u 1 4.b odd 2 1
2205.4.a.b 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 5 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T - 30 \) Copy content Toggle raw display
$17$ \( T + 134 \) Copy content Toggle raw display
$19$ \( T + 92 \) Copy content Toggle raw display
$23$ \( T - 112 \) Copy content Toggle raw display
$29$ \( T + 58 \) Copy content Toggle raw display
$31$ \( T + 224 \) Copy content Toggle raw display
$37$ \( T + 146 \) Copy content Toggle raw display
$41$ \( T - 18 \) Copy content Toggle raw display
$43$ \( T - 340 \) Copy content Toggle raw display
$47$ \( T - 208 \) Copy content Toggle raw display
$53$ \( T + 754 \) Copy content Toggle raw display
$59$ \( T - 380 \) Copy content Toggle raw display
$61$ \( T - 718 \) Copy content Toggle raw display
$67$ \( T - 412 \) Copy content Toggle raw display
$71$ \( T + 960 \) Copy content Toggle raw display
$73$ \( T - 1066 \) Copy content Toggle raw display
$79$ \( T - 896 \) Copy content Toggle raw display
$83$ \( T - 436 \) Copy content Toggle raw display
$89$ \( T + 1038 \) Copy content Toggle raw display
$97$ \( T + 702 \) Copy content Toggle raw display
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