Properties

Label 300.3.k.c
Level $300$
Weight $3$
Character orbit 300.k
Analytic conductor $8.174$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,3,Mod(157,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.157");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.k (of order \(4\), degree \(2\), 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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 3 \beta_{3} q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 3 \beta_{3} q^{7} + 3 \beta_{2} q^{9} + 6 q^{11} + 5 \beta_1 q^{13} - 18 \beta_{3} q^{17} + 25 \beta_{2} q^{19} + 9 q^{21} - 6 \beta_1 q^{23} + 3 \beta_{3} q^{27} + 42 \beta_{2} q^{29} + 49 q^{31} + 6 \beta_1 q^{33} - 4 \beta_{3} q^{37} + 15 \beta_{2} q^{39} - 60 q^{41} - \beta_1 q^{43} - 42 \beta_{3} q^{47} + 22 \beta_{2} q^{49} + 54 q^{51} - 12 \beta_1 q^{53} + 25 \beta_{3} q^{57} - 78 \beta_{2} q^{59} - 13 q^{61} + 9 \beta_1 q^{63} + 43 \beta_{3} q^{67} - 18 \beta_{2} q^{69} - 60 q^{71} + 52 \beta_1 q^{73} - 18 \beta_{3} q^{77} - 106 \beta_{2} q^{79} - 9 q^{81} - 66 \beta_1 q^{83} + 42 \beta_{3} q^{87} - 60 \beta_{2} q^{89} + 45 q^{91} + 49 \beta_1 q^{93} + 99 \beta_{3} q^{97} + 18 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{11} + 36 q^{21} + 196 q^{31} - 240 q^{41} + 216 q^{51} - 52 q^{61} - 240 q^{71} - 36 q^{81} + 180 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 −3.67423 3.67423i 0 3.00000i 0
157.2 0 1.22474 1.22474i 0 0 0 3.67423 + 3.67423i 0 3.00000i 0
193.1 0 −1.22474 1.22474i 0 0 0 −3.67423 + 3.67423i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 3.67423 3.67423i 0 3.00000i 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
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.k.c 4
3.b odd 2 1 900.3.l.d 4
4.b odd 2 1 1200.3.bg.i 4
5.b even 2 1 inner 300.3.k.c 4
5.c odd 4 2 inner 300.3.k.c 4
15.d odd 2 1 900.3.l.d 4
15.e even 4 2 900.3.l.d 4
20.d odd 2 1 1200.3.bg.i 4
20.e even 4 2 1200.3.bg.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.k.c 4 1.a even 1 1 trivial
300.3.k.c 4 5.b even 2 1 inner
300.3.k.c 4 5.c odd 4 2 inner
900.3.l.d 4 3.b odd 2 1
900.3.l.d 4 15.d odd 2 1
900.3.l.d 4 15.e even 4 2
1200.3.bg.i 4 4.b odd 2 1
1200.3.bg.i 4 20.d odd 2 1
1200.3.bg.i 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 729 \) acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 729 \) Copy content Toggle raw display
$11$ \( (T - 6)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 5625 \) Copy content Toggle raw display
$17$ \( T^{4} + 944784 \) Copy content Toggle raw display
$19$ \( (T^{2} + 625)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 11664 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1764)^{2} \) Copy content Toggle raw display
$31$ \( (T - 49)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 2304 \) Copy content Toggle raw display
$41$ \( (T + 60)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 9 \) Copy content Toggle raw display
$47$ \( T^{4} + 28005264 \) Copy content Toggle raw display
$53$ \( T^{4} + 186624 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6084)^{2} \) Copy content Toggle raw display
$61$ \( (T + 13)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 30769209 \) Copy content Toggle raw display
$71$ \( (T + 60)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 65804544 \) Copy content Toggle raw display
$79$ \( (T^{2} + 11236)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 170772624 \) Copy content Toggle raw display
$89$ \( (T^{2} + 3600)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 864536409 \) Copy content Toggle raw display
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