Properties

Label 300.9.b.b
Level $300$
Weight $9$
Character orbit 300.b
Analytic conductor $122.214$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,9,Mod(149,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.149");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 300.b (of order \(2\), degree \(1\), not 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: \(122.213583018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 81 i q^{3} + 4034 i q^{7} - 6561 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 81 i q^{3} + 4034 i q^{7} - 6561 q^{9} + 35806 i q^{13} + 258526 q^{19} + 326754 q^{21} + 531441 i q^{27} - 1809406 q^{31} + 503522 i q^{37} + 2900286 q^{39} - 3492194 i q^{43} - 10508355 q^{49} - 20940606 i q^{57} - 23826526 q^{61} - 26467074 i q^{63} - 5421406 i q^{67} - 16169282 i q^{73} + 18887038 q^{79} + 43046721 q^{81} - 144441404 q^{91} + 146561886 i q^{93} + 176908034 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 13122 q^{9} + 517052 q^{19} + 653508 q^{21} - 3618812 q^{31} + 5800572 q^{39} - 21016710 q^{49} - 47653052 q^{61} + 37774076 q^{79} + 86093442 q^{81} - 288882808 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.00000i
1.00000i
0 81.0000i 0 0 0 4034.00i 0 −6561.00 0
149.2 0 81.0000i 0 0 0 4034.00i 0 −6561.00 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 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.9.b.b 2
3.b odd 2 1 CM 300.9.b.b 2
5.b even 2 1 inner 300.9.b.b 2
5.c odd 4 1 12.9.c.a 1
5.c odd 4 1 300.9.g.a 1
15.d odd 2 1 inner 300.9.b.b 2
15.e even 4 1 12.9.c.a 1
15.e even 4 1 300.9.g.a 1
20.e even 4 1 48.9.e.a 1
40.i odd 4 1 192.9.e.a 1
40.k even 4 1 192.9.e.b 1
45.k odd 12 2 324.9.g.a 2
45.l even 12 2 324.9.g.a 2
60.l odd 4 1 48.9.e.a 1
120.q odd 4 1 192.9.e.b 1
120.w even 4 1 192.9.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.a 1 5.c odd 4 1
12.9.c.a 1 15.e even 4 1
48.9.e.a 1 20.e even 4 1
48.9.e.a 1 60.l odd 4 1
192.9.e.a 1 40.i odd 4 1
192.9.e.a 1 120.w even 4 1
192.9.e.b 1 40.k even 4 1
192.9.e.b 1 120.q odd 4 1
300.9.b.b 2 1.a even 1 1 trivial
300.9.b.b 2 3.b odd 2 1 CM
300.9.b.b 2 5.b even 2 1 inner
300.9.b.b 2 15.d odd 2 1 inner
300.9.g.a 1 5.c odd 4 1
300.9.g.a 1 15.e even 4 1
324.9.g.a 2 45.k odd 12 2
324.9.g.a 2 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 16273156 \) acting on \(S_{9}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16273156 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1282069636 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 258526)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 1809406)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 253534404484 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 12195418933636 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 23826526)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 29391643016836 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 261445680395524 \) Copy content Toggle raw display
$79$ \( (T - 18887038)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 31\!\cdots\!56 \) Copy content Toggle raw display
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