Properties

Label 900.5.g.a
Level $900$
Weight $5$
Character orbit 900.g
Analytic conductor $93.033$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,5,Mod(701,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.701");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 900.g (of order \(2\), degree \(1\), 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: \(93.0329667755\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 27\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 68 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 68 q^{7} - 4 \beta q^{11} + 16 q^{13} - 13 \beta q^{17} - 208 q^{19} - 4 \beta q^{23} + 17 \beta q^{29} + 1652 q^{31} + 442 q^{37} - 13 \beta q^{41} - 1160 q^{43} + 12 \beta q^{47} + 2223 q^{49} - 65 \beta q^{53} + 88 \beta q^{59} - 3910 q^{61} - 6392 q^{67} + 172 \beta q^{71} + 2224 q^{73} + 272 \beta q^{77} - 7060 q^{79} + 156 \beta q^{83} - 221 \beta q^{89} - 1088 q^{91} - 4352 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 136 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 136 q^{7} + 32 q^{13} - 416 q^{19} + 3304 q^{31} + 884 q^{37} - 2320 q^{43} + 4446 q^{49} - 7820 q^{61} - 12784 q^{67} + 4448 q^{73} - 14120 q^{79} - 2176 q^{91} - 8704 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
1.41421i
1.41421i
0 0 0 0 0 −68.0000 0 0 0
701.2 0 0 0 0 0 −68.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.5.g.a 2
3.b odd 2 1 inner 900.5.g.a 2
5.b even 2 1 36.5.c.a 2
5.c odd 4 2 900.5.b.a 4
15.d odd 2 1 36.5.c.a 2
15.e even 4 2 900.5.b.a 4
20.d odd 2 1 144.5.e.a 2
40.e odd 2 1 576.5.e.a 2
40.f even 2 1 576.5.e.j 2
45.h odd 6 2 324.5.g.c 4
45.j even 6 2 324.5.g.c 4
60.h even 2 1 144.5.e.a 2
120.i odd 2 1 576.5.e.j 2
120.m even 2 1 576.5.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.c.a 2 5.b even 2 1
36.5.c.a 2 15.d odd 2 1
144.5.e.a 2 20.d odd 2 1
144.5.e.a 2 60.h even 2 1
324.5.g.c 4 45.h odd 6 2
324.5.g.c 4 45.j even 6 2
576.5.e.a 2 40.e odd 2 1
576.5.e.a 2 120.m even 2 1
576.5.e.j 2 40.f even 2 1
576.5.e.j 2 120.i odd 2 1
900.5.b.a 4 5.c odd 4 2
900.5.b.a 4 15.e even 4 2
900.5.g.a 2 1.a even 1 1 trivial
900.5.g.a 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 68 \) acting on \(S_{5}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 68)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 23328 \) Copy content Toggle raw display
$13$ \( (T - 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 246402 \) Copy content Toggle raw display
$19$ \( (T + 208)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 23328 \) Copy content Toggle raw display
$29$ \( T^{2} + 421362 \) Copy content Toggle raw display
$31$ \( (T - 1652)^{2} \) Copy content Toggle raw display
$37$ \( (T - 442)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 246402 \) Copy content Toggle raw display
$43$ \( (T + 1160)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 209952 \) Copy content Toggle raw display
$53$ \( T^{2} + 6160050 \) Copy content Toggle raw display
$59$ \( T^{2} + 11290752 \) Copy content Toggle raw display
$61$ \( (T + 3910)^{2} \) Copy content Toggle raw display
$67$ \( (T + 6392)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 43133472 \) Copy content Toggle raw display
$73$ \( (T - 2224)^{2} \) Copy content Toggle raw display
$79$ \( (T + 7060)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 35481888 \) Copy content Toggle raw display
$89$ \( T^{2} + 71210178 \) Copy content Toggle raw display
$97$ \( (T + 4352)^{2} \) Copy content Toggle raw display
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