Properties

Label 102.4.a.d
Level $102$
Weight $4$
Character orbit 102.a
Self dual yes
Analytic conductor $6.018$
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] = [102,4,Mod(1,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("102.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 102.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: \(6.01819482059\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 5 q^{5} - 6 q^{6} + 12 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 5 q^{5} - 6 q^{6} + 12 q^{7} + 8 q^{8} + 9 q^{9} + 10 q^{10} + 37 q^{11} - 12 q^{12} + 19 q^{13} + 24 q^{14} - 15 q^{15} + 16 q^{16} + 17 q^{17} + 18 q^{18} + 37 q^{19} + 20 q^{20} - 36 q^{21} + 74 q^{22} - 3 q^{23} - 24 q^{24} - 100 q^{25} + 38 q^{26} - 27 q^{27} + 48 q^{28} - 86 q^{29} - 30 q^{30} - 142 q^{31} + 32 q^{32} - 111 q^{33} + 34 q^{34} + 60 q^{35} + 36 q^{36} - 296 q^{37} + 74 q^{38} - 57 q^{39} + 40 q^{40} - 121 q^{41} - 72 q^{42} + 3 q^{43} + 148 q^{44} + 45 q^{45} - 6 q^{46} + 402 q^{47} - 48 q^{48} - 199 q^{49} - 200 q^{50} - 51 q^{51} + 76 q^{52} + 174 q^{53} - 54 q^{54} + 185 q^{55} + 96 q^{56} - 111 q^{57} - 172 q^{58} + 270 q^{59} - 60 q^{60} - 520 q^{61} - 284 q^{62} + 108 q^{63} + 64 q^{64} + 95 q^{65} - 222 q^{66} - 780 q^{67} + 68 q^{68} + 9 q^{69} + 120 q^{70} + 84 q^{71} + 72 q^{72} - 302 q^{73} - 592 q^{74} + 300 q^{75} + 148 q^{76} + 444 q^{77} - 114 q^{78} + 178 q^{79} + 80 q^{80} + 81 q^{81} - 242 q^{82} + 698 q^{83} - 144 q^{84} + 85 q^{85} + 6 q^{86} + 258 q^{87} + 296 q^{88} + 1512 q^{89} + 90 q^{90} + 228 q^{91} - 12 q^{92} + 426 q^{93} + 804 q^{94} + 185 q^{95} - 96 q^{96} - 500 q^{97} - 398 q^{98} + 333 q^{99}+O(q^{100}) \) 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
2.00000 −3.00000 4.00000 5.00000 −6.00000 12.0000 8.00000 9.00000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(17\) \(-1\)

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 102.4.a.d 1
3.b odd 2 1 306.4.a.a 1
4.b odd 2 1 816.4.a.h 1
12.b even 2 1 2448.4.a.g 1
17.b even 2 1 1734.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.4.a.d 1 1.a even 1 1 trivial
306.4.a.a 1 3.b odd 2 1
816.4.a.h 1 4.b odd 2 1
1734.4.a.f 1 17.b even 2 1
2448.4.a.g 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(102))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 12 \) Copy content Toggle raw display
$11$ \( T - 37 \) Copy content Toggle raw display
$13$ \( T - 19 \) Copy content Toggle raw display
$17$ \( T - 17 \) Copy content Toggle raw display
$19$ \( T - 37 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T + 86 \) Copy content Toggle raw display
$31$ \( T + 142 \) Copy content Toggle raw display
$37$ \( T + 296 \) Copy content Toggle raw display
$41$ \( T + 121 \) Copy content Toggle raw display
$43$ \( T - 3 \) Copy content Toggle raw display
$47$ \( T - 402 \) Copy content Toggle raw display
$53$ \( T - 174 \) Copy content Toggle raw display
$59$ \( T - 270 \) Copy content Toggle raw display
$61$ \( T + 520 \) Copy content Toggle raw display
$67$ \( T + 780 \) Copy content Toggle raw display
$71$ \( T - 84 \) Copy content Toggle raw display
$73$ \( T + 302 \) Copy content Toggle raw display
$79$ \( T - 178 \) Copy content Toggle raw display
$83$ \( T - 698 \) Copy content Toggle raw display
$89$ \( T - 1512 \) Copy content Toggle raw display
$97$ \( T + 500 \) Copy content Toggle raw display
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