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Properties

Label 300.2.e.a
Level $300$
Weight $2$
Character orbit 300.e
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,2,Mod(251,300)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.e (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: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{2} + 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{2} + 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9} + (2 \beta_{3} + \beta_1) q^{12} + ( - \beta_{2} - 4) q^{16} + (2 \beta_{3} + 2 \beta_1) q^{17} + 3 \beta_1 q^{18} + ( - 4 \beta_{2} - 2) q^{19} + (2 \beta_{3} - 2 \beta_1) q^{23} + (\beta_{2} - 4) q^{24} + ( - 3 \beta_{3} + 3 \beta_1) q^{27} + ( - 4 \beta_{2} - 2) q^{31} + ( - 2 \beta_{3} - 3 \beta_1) q^{32} + (2 \beta_{2} - 4) q^{34} + 3 \beta_{2} q^{36} + ( - 8 \beta_{3} + 2 \beta_1) q^{38} + ( - 2 \beta_{2} - 4) q^{46} + (6 \beta_{3} - 6 \beta_1) q^{47} + (2 \beta_{3} - 5 \beta_1) q^{48} + 7 q^{49} + (4 \beta_{2} + 2) q^{51} + (2 \beta_{3} + 2 \beta_1) q^{53} + (3 \beta_{2} + 6) q^{54} + ( - 6 \beta_{3} - 6 \beta_1) q^{57} - 2 q^{61} + ( - 8 \beta_{3} + 2 \beta_1) q^{62} + ( - 3 \beta_{2} + 4) q^{64} + (4 \beta_{3} - 6 \beta_1) q^{68} - 6 q^{69} + (6 \beta_{3} - 3 \beta_1) q^{72} + (2 \beta_{2} + 16) q^{76} + (4 \beta_{2} + 2) q^{79} + 9 q^{81} + ( - 2 \beta_{3} + 2 \beta_1) q^{83} + ( - 4 \beta_{3} - 2 \beta_1) q^{92} + ( - 6 \beta_{3} - 6 \beta_1) q^{93} + ( - 6 \beta_{2} - 12) q^{94} + ( - 5 \beta_{2} - 4) q^{96} + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 6 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 6 q^{6} + 12 q^{9} - 14 q^{16} - 18 q^{24} - 20 q^{34} - 6 q^{36} - 12 q^{46} + 28 q^{49} + 18 q^{54} - 8 q^{61} + 22 q^{64} - 24 q^{69} + 60 q^{76} + 36 q^{81} - 36 q^{94} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.866025 − 1.11803i
−0.866025 + 1.11803i
0.866025 − 1.11803i
0.866025 + 1.11803i
−0.866025 − 1.11803i −1.73205 −0.500000 + 1.93649i 0 1.50000 + 1.93649i 0 2.59808 − 1.11803i 3.00000 0
251.2 −0.866025 + 1.11803i −1.73205 −0.500000 − 1.93649i 0 1.50000 − 1.93649i 0 2.59808 + 1.11803i 3.00000 0
251.3 0.866025 − 1.11803i 1.73205 −0.500000 − 1.93649i 0 1.50000 − 1.93649i 0 −2.59808 − 1.11803i 3.00000 0
251.4 0.866025 + 1.11803i 1.73205 −0.500000 + 1.93649i 0 1.50000 + 1.93649i 0 −2.59808 + 1.11803i 3.00000 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.e.a 4
3.b odd 2 1 inner 300.2.e.a 4
4.b odd 2 1 inner 300.2.e.a 4
5.b even 2 1 inner 300.2.e.a 4
5.c odd 4 2 60.2.h.b ✓ 4
12.b even 2 1 inner 300.2.e.a 4
15.d odd 2 1 CM 300.2.e.a 4
15.e even 4 2 60.2.h.b ✓ 4
20.d odd 2 1 inner 300.2.e.a 4
20.e even 4 2 60.2.h.b ✓ 4
40.i odd 4 2 960.2.o.a 4
40.k even 4 2 960.2.o.a 4
60.h even 2 1 inner 300.2.e.a 4
60.l odd 4 2 60.2.h.b ✓ 4
120.q odd 4 2 960.2.o.a 4
120.w even 4 2 960.2.o.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.b ✓ 4 5.c odd 4 2
60.2.h.b ✓ 4 15.e even 4 2
60.2.h.b ✓ 4 20.e even 4 2
60.2.h.b ✓ 4 60.l odd 4 2
300.2.e.a 4 1.a even 1 1 trivial
300.2.e.a 4 3.b odd 2 1 inner
300.2.e.a 4 4.b odd 2 1 inner
300.2.e.a 4 5.b even 2 1 inner
300.2.e.a 4 12.b even 2 1 inner
300.2.e.a 4 15.d odd 2 1 CM
300.2.e.a 4 20.d odd 2 1 inner
300.2.e.a 4 60.h even 2 1 inner
960.2.o.a 4 40.i odd 4 2
960.2.o.a 4 40.k even 4 2
960.2.o.a 4 120.q odd 4 2
960.2.o.a 4 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(300, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T + 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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