Properties

Label 690.4.a.b
Level $690$
Weight $4$
Character orbit 690.a
Self dual yes
Analytic conductor $40.711$
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] = [690,4,Mod(1,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, 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("690.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 690.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: \(40.7113179040\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 q^{6} - 18 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 q^{6} - 18 q^{7} - 8 q^{8} + 9 q^{9} - 10 q^{10} - 70 q^{11} - 12 q^{12} - 86 q^{13} + 36 q^{14} - 15 q^{15} + 16 q^{16} - 56 q^{17} - 18 q^{18} + 108 q^{19} + 20 q^{20} + 54 q^{21} + 140 q^{22} + 23 q^{23} + 24 q^{24} + 25 q^{25} + 172 q^{26} - 27 q^{27} - 72 q^{28} + 186 q^{29} + 30 q^{30} - 120 q^{31} - 32 q^{32} + 210 q^{33} + 112 q^{34} - 90 q^{35} + 36 q^{36} - 232 q^{37} - 216 q^{38} + 258 q^{39} - 40 q^{40} - 398 q^{41} - 108 q^{42} + 120 q^{43} - 280 q^{44} + 45 q^{45} - 46 q^{46} + 88 q^{47} - 48 q^{48} - 19 q^{49} - 50 q^{50} + 168 q^{51} - 344 q^{52} - 190 q^{53} + 54 q^{54} - 350 q^{55} + 144 q^{56} - 324 q^{57} - 372 q^{58} + 696 q^{59} - 60 q^{60} + 504 q^{61} + 240 q^{62} - 162 q^{63} + 64 q^{64} - 430 q^{65} - 420 q^{66} + 432 q^{67} - 224 q^{68} - 69 q^{69} + 180 q^{70} - 72 q^{71} - 72 q^{72} + 102 q^{73} + 464 q^{74} - 75 q^{75} + 432 q^{76} + 1260 q^{77} - 516 q^{78} - 218 q^{79} + 80 q^{80} + 81 q^{81} + 796 q^{82} + 82 q^{83} + 216 q^{84} - 280 q^{85} - 240 q^{86} - 558 q^{87} + 560 q^{88} - 828 q^{89} - 90 q^{90} + 1548 q^{91} + 92 q^{92} + 360 q^{93} - 176 q^{94} + 540 q^{95} + 96 q^{96} + 650 q^{97} + 38 q^{98} - 630 q^{99}+O(q^{100}) \) 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 −18.0000 −8.00000 9.00000 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 690.4.a.b 1
3.b odd 2 1 2070.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.4.a.b 1 1.a even 1 1 trivial
2070.4.a.h 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(690))\):

\( T_{7} + 18 \) Copy content Toggle raw display
\( T_{11} + 70 \) 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
$7$ \( T + 18 \) Copy content Toggle raw display
$11$ \( T + 70 \) Copy content Toggle raw display
$13$ \( T + 86 \) Copy content Toggle raw display
$17$ \( T + 56 \) Copy content Toggle raw display
$19$ \( T - 108 \) Copy content Toggle raw display
$23$ \( T - 23 \) Copy content Toggle raw display
$29$ \( T - 186 \) Copy content Toggle raw display
$31$ \( T + 120 \) Copy content Toggle raw display
$37$ \( T + 232 \) Copy content Toggle raw display
$41$ \( T + 398 \) Copy content Toggle raw display
$43$ \( T - 120 \) Copy content Toggle raw display
$47$ \( T - 88 \) Copy content Toggle raw display
$53$ \( T + 190 \) Copy content Toggle raw display
$59$ \( T - 696 \) Copy content Toggle raw display
$61$ \( T - 504 \) Copy content Toggle raw display
$67$ \( T - 432 \) Copy content Toggle raw display
$71$ \( T + 72 \) Copy content Toggle raw display
$73$ \( T - 102 \) Copy content Toggle raw display
$79$ \( T + 218 \) Copy content Toggle raw display
$83$ \( T - 82 \) Copy content Toggle raw display
$89$ \( T + 828 \) Copy content Toggle raw display
$97$ \( T - 650 \) Copy content Toggle raw display
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