Properties

Label 900.2.bj.a
Level $900$
Weight $2$
Character orbit 900.bj
Analytic conductor $7.187$
Analytic rank $0$
Dimension $8$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(127,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bj (of order \(20\), degree \(8\), minimal)

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: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{2} + 2 \zeta_{20} q^{4} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} + 2 \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}) q^{5} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{2} + 2 \zeta_{20} q^{4} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} + 2 \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}) q^{5} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{8} + ( - \zeta_{20}^{7} - 3 \zeta_{20}^{2}) q^{10} + (5 \zeta_{20}^{6} - 2 \zeta_{20}^{4} - 3 \zeta_{20}^{3} + 2 \zeta_{20}^{2} - 2 \zeta_{20} - 2) q^{13} + 4 \zeta_{20}^{2} q^{16} + (4 \zeta_{20}^{7} + 4 \zeta_{20}^{6} - \zeta_{20}^{2} + \zeta_{20}) q^{17} + (4 \zeta_{20}^{5} + 2) q^{20} + (3 \zeta_{20}^{6} - 3 \zeta_{20}^{4} + 4 \zeta_{20}^{3} + 3 \zeta_{20}^{2} - 3) q^{25} + ( - 5 \zeta_{20}^{7} + \zeta_{20}^{6} + 5 \zeta_{20}^{5} - \zeta_{20}^{4} - 5 \zeta_{20}^{3} + 10 \zeta_{20}) q^{26} + ( - 2 \zeta_{20}^{7} + 5 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - 10 \zeta_{20}^{4} + 5 \zeta_{20}^{2} + 2 \zeta_{20} - 5) q^{29} + ( - 4 \zeta_{20}^{5} - 4) q^{32} + ( - 3 \zeta_{20}^{7} - 5 \zeta_{20}^{4} - 3 \zeta_{20}^{3} + 3 \zeta_{20} + 5) q^{34} + ( - 6 \zeta_{20}^{7} + 5 \zeta_{20}^{5} - \zeta_{20}^{4} - 6 \zeta_{20}^{3} + 6 \zeta_{20} - 6) q^{37} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{3} - 2 \zeta_{20}^{2} + 2) q^{40} + ( - 4 \zeta_{20}^{4} - 5 \zeta_{20}^{3} + 4 \zeta_{20}^{2} - 5 \zeta_{20} - 4) q^{41} + 7 \zeta_{20}^{5} q^{49} + ( - 7 \zeta_{20}^{6} - \zeta_{20}) q^{50} + (10 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - 6 \zeta_{20}^{4} + 4 \zeta_{20}^{3} - 4 \zeta_{20}^{2} - 4 \zeta_{20}) q^{52} + ( - 2 \zeta_{20}^{7} + 7 \zeta_{20}^{5} + 7 \zeta_{20}^{2} - 2) q^{53} + (7 \zeta_{20}^{7} - 7 \zeta_{20}^{6} + 3 \zeta_{20}^{2} + 3 \zeta_{20}) q^{58} + ( - 10 \zeta_{20}^{7} + 5 \zeta_{20}^{5} + 6 \zeta_{20}^{4} - 5 \zeta_{20}^{3} - 6 \zeta_{20}^{2} + 5 \zeta_{20}) q^{61} + 8 \zeta_{20}^{3} q^{64} + ( - 4 \zeta_{20}^{7} - \zeta_{20}^{5} - 7 \zeta_{20}^{2} - 8) q^{65} + (8 \zeta_{20}^{7} + 8 \zeta_{20}^{6} - 8 \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 10 \zeta_{20}^{2} - 8) q^{68} + (5 \zeta_{20}^{7} - 8 \zeta_{20}^{5} - 3 \zeta_{20}^{4} + 8 \zeta_{20}^{3} + 8 \zeta_{20}^{2} - 8 \zeta_{20}) q^{73} + (7 \zeta_{20}^{7} - 5 \zeta_{20}^{6} + 5 \zeta_{20}^{4} + 7 \zeta_{20}^{3} - 10 \zeta_{20}^{2} + 5) q^{74} + (8 \zeta_{20}^{6} + 4 \zeta_{20}) q^{80} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{5} + 9 \zeta_{20}^{4} - \zeta_{20}^{3} + 10 \zeta_{20}) q^{82} + (2 \zeta_{20}^{6} + 6 \zeta_{20}^{5} - 9 \zeta_{20} - 7) q^{85} + (5 \zeta_{20}^{6} + 8 \zeta_{20}^{5} + 8 \zeta_{20} + 5) q^{89} + (9 \zeta_{20}^{7} + 9 \zeta_{20}^{6} - 9 \zeta_{20}^{5} - 5 \zeta_{20}^{4} + 5 \zeta_{20}^{3} + 9 \zeta_{20}^{2} + \cdots - 9) q^{97}+ \cdots + ( - 7 \zeta_{20}^{6} + 7 \zeta_{20}^{4} - 7 \zeta_{20}^{3} - 7 \zeta_{20}^{2} + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 4 q^{5} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 4 q^{5} + 4 q^{8} - 6 q^{10} + 2 q^{13} + 8 q^{16} + 6 q^{17} + 16 q^{20} - 6 q^{25} + 4 q^{26} - 32 q^{32} + 50 q^{34} - 46 q^{37} + 4 q^{40} - 16 q^{41} - 14 q^{50} + 4 q^{52} - 2 q^{53} - 8 q^{58} - 24 q^{61} - 78 q^{65} - 12 q^{68} + 22 q^{73} + 16 q^{80} - 16 q^{82} - 52 q^{85} + 50 q^{89} - 26 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{20}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 + 0.309017i
−0.951057 0.309017i
−0.587785 0.809017i
−0.587785 + 0.809017i
0.587785 0.809017i
0.587785 + 0.809017i
−1.39680 0.221232i 0 1.90211 + 0.618034i 1.56909 + 1.59310i 0 0 −2.52015 1.28408i 0 −1.83927 2.57237i
163.1 −1.39680 + 0.221232i 0 1.90211 0.618034i 1.56909 1.59310i 0 0 −2.52015 + 1.28408i 0 −1.83927 + 2.57237i
487.1 −0.221232 1.39680i 0 −1.90211 + 0.618034i −0.333023 2.21113i 0 0 1.28408 + 2.52015i 0 −3.01484 + 0.954339i
523.1 −0.221232 + 1.39680i 0 −1.90211 0.618034i −0.333023 + 2.21113i 0 0 1.28408 2.52015i 0 −3.01484 0.954339i
667.1 −0.642040 + 1.26007i 0 −1.17557 1.61803i −2.20582 0.366554i 0 0 2.79360 0.442463i 0 1.87811 2.54415i
703.1 −0.642040 1.26007i 0 −1.17557 + 1.61803i −2.20582 + 0.366554i 0 0 2.79360 + 0.442463i 0 1.87811 + 2.54415i
847.1 1.26007 0.642040i 0 1.17557 1.61803i −1.03025 + 1.98459i 0 0 0.442463 2.79360i 0 −0.0240055 + 3.16219i
883.1 1.26007 + 0.642040i 0 1.17557 + 1.61803i −1.03025 1.98459i 0 0 0.442463 + 2.79360i 0 −0.0240055 3.16219i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bj.a 8
3.b odd 2 1 100.2.l.a 8
4.b odd 2 1 CM 900.2.bj.a 8
12.b even 2 1 100.2.l.a 8
15.d odd 2 1 500.2.l.b 8
15.e even 4 1 500.2.l.a 8
15.e even 4 1 500.2.l.c 8
25.f odd 20 1 inner 900.2.bj.a 8
60.h even 2 1 500.2.l.b 8
60.l odd 4 1 500.2.l.a 8
60.l odd 4 1 500.2.l.c 8
75.h odd 10 1 500.2.l.a 8
75.j odd 10 1 500.2.l.c 8
75.l even 20 1 100.2.l.a 8
75.l even 20 1 500.2.l.b 8
100.l even 20 1 inner 900.2.bj.a 8
300.n even 10 1 500.2.l.c 8
300.r even 10 1 500.2.l.a 8
300.u odd 20 1 100.2.l.a 8
300.u odd 20 1 500.2.l.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.l.a 8 3.b odd 2 1
100.2.l.a 8 12.b even 2 1
100.2.l.a 8 75.l even 20 1
100.2.l.a 8 300.u odd 20 1
500.2.l.a 8 15.e even 4 1
500.2.l.a 8 60.l odd 4 1
500.2.l.a 8 75.h odd 10 1
500.2.l.a 8 300.r even 10 1
500.2.l.b 8 15.d odd 2 1
500.2.l.b 8 60.h even 2 1
500.2.l.b 8 75.l even 20 1
500.2.l.b 8 300.u odd 20 1
500.2.l.c 8 15.e even 4 1
500.2.l.c 8 60.l odd 4 1
500.2.l.c 8 75.j odd 10 1
500.2.l.c 8 300.n even 10 1
900.2.bj.a 8 1.a even 1 1 trivial
900.2.bj.a 8 4.b odd 2 1 CM
900.2.bj.a 8 25.f odd 20 1 inner
900.2.bj.a 8 100.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{13}^{8} - 2T_{13}^{7} + 2T_{13}^{6} + 260T_{13}^{5} - 2084T_{13}^{4} + 2210T_{13}^{3} + 54673T_{13}^{2} - 189816T_{13} + 516961 \) Copy content Toggle raw display
\( T_{17}^{8} - 6T_{17}^{7} + 18T_{17}^{6} - 680T_{17}^{5} + 2396T_{17}^{4} - 3570T_{17}^{3} + 101117T_{17}^{2} + 136708T_{17} + 57121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + 11 T^{6} + 24 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 516961 \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 57121 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{6} + 1450 T^{5} + \cdots + 4583881 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 46 T^{7} + 883 T^{6} + \cdots + 2825761 \) Copy content Toggle raw display
$41$ \( T^{8} + 16 T^{7} + 192 T^{6} + \cdots + 383161 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 2 T^{7} + 227 T^{6} + \cdots + 6974881 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 24 T^{7} + 432 T^{6} + \cdots + 20967241 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 22 T^{7} + 242 T^{6} + \cdots + 9740641 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} - 50 T^{7} + 1189 T^{6} + \cdots + 77070841 \) Copy content Toggle raw display
$97$ \( T^{8} + 26 T^{7} + 338 T^{6} + \cdots + 6974881 \) Copy content Toggle raw display
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