Properties

Label 90.6.a.b
Level $90$
Weight $6$
Character orbit 90.a
Self dual yes
Analytic conductor $14.435$
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] = [90,6,Mod(1,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 90.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: \(14.4345437832\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} + 25 q^{5} - 118 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} + 25 q^{5} - 118 q^{7} - 64 q^{8} - 100 q^{10} - 192 q^{11} + 1106 q^{13} + 472 q^{14} + 256 q^{16} - 762 q^{17} - 2740 q^{19} + 400 q^{20} + 768 q^{22} - 1566 q^{23} + 625 q^{25} - 4424 q^{26} - 1888 q^{28} - 5910 q^{29} - 6868 q^{31} - 1024 q^{32} + 3048 q^{34} - 2950 q^{35} - 5518 q^{37} + 10960 q^{38} - 1600 q^{40} + 378 q^{41} - 2434 q^{43} - 3072 q^{44} + 6264 q^{46} - 13122 q^{47} - 2883 q^{49} - 2500 q^{50} + 17696 q^{52} + 9174 q^{53} - 4800 q^{55} + 7552 q^{56} + 23640 q^{58} + 34980 q^{59} - 9838 q^{61} + 27472 q^{62} + 4096 q^{64} + 27650 q^{65} + 33722 q^{67} - 12192 q^{68} + 11800 q^{70} - 70212 q^{71} + 21986 q^{73} + 22072 q^{74} - 43840 q^{76} + 22656 q^{77} + 4520 q^{79} + 6400 q^{80} - 1512 q^{82} + 109074 q^{83} - 19050 q^{85} + 9736 q^{86} + 12288 q^{88} - 38490 q^{89} - 130508 q^{91} - 25056 q^{92} + 52488 q^{94} - 68500 q^{95} - 1918 q^{97} + 11532 q^{98}+O(q^{100}) \) 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
−4.00000 0 16.0000 25.0000 0 −118.000 −64.0000 0 −100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 90.6.a.b 1
3.b odd 2 1 10.6.a.c 1
4.b odd 2 1 720.6.a.v 1
5.b even 2 1 450.6.a.u 1
5.c odd 4 2 450.6.c.f 2
12.b even 2 1 80.6.a.c 1
15.d odd 2 1 50.6.a.b 1
15.e even 4 2 50.6.b.b 2
21.c even 2 1 490.6.a.k 1
24.f even 2 1 320.6.a.k 1
24.h odd 2 1 320.6.a.f 1
60.h even 2 1 400.6.a.i 1
60.l odd 4 2 400.6.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 3.b odd 2 1
50.6.a.b 1 15.d odd 2 1
50.6.b.b 2 15.e even 4 2
80.6.a.c 1 12.b even 2 1
90.6.a.b 1 1.a even 1 1 trivial
320.6.a.f 1 24.h odd 2 1
320.6.a.k 1 24.f even 2 1
400.6.a.i 1 60.h even 2 1
400.6.c.i 2 60.l odd 4 2
450.6.a.u 1 5.b even 2 1
450.6.c.f 2 5.c odd 4 2
490.6.a.k 1 21.c even 2 1
720.6.a.v 1 4.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(90))\):

\( T_{7} + 118 \) Copy content Toggle raw display
\( T_{11} + 192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T + 118 \) Copy content Toggle raw display
$11$ \( T + 192 \) Copy content Toggle raw display
$13$ \( T - 1106 \) Copy content Toggle raw display
$17$ \( T + 762 \) Copy content Toggle raw display
$19$ \( T + 2740 \) Copy content Toggle raw display
$23$ \( T + 1566 \) Copy content Toggle raw display
$29$ \( T + 5910 \) Copy content Toggle raw display
$31$ \( T + 6868 \) Copy content Toggle raw display
$37$ \( T + 5518 \) Copy content Toggle raw display
$41$ \( T - 378 \) Copy content Toggle raw display
$43$ \( T + 2434 \) Copy content Toggle raw display
$47$ \( T + 13122 \) Copy content Toggle raw display
$53$ \( T - 9174 \) Copy content Toggle raw display
$59$ \( T - 34980 \) Copy content Toggle raw display
$61$ \( T + 9838 \) Copy content Toggle raw display
$67$ \( T - 33722 \) Copy content Toggle raw display
$71$ \( T + 70212 \) Copy content Toggle raw display
$73$ \( T - 21986 \) Copy content Toggle raw display
$79$ \( T - 4520 \) Copy content Toggle raw display
$83$ \( T - 109074 \) Copy content Toggle raw display
$89$ \( T + 38490 \) Copy content Toggle raw display
$97$ \( T + 1918 \) Copy content Toggle raw display
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