Properties

Label 300.2.j.b
Level $300$
Weight $2$
Character orbit 300.j
Analytic conductor $2.396$
Analytic rank $0$
Dimension $8$
CM no
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(7,300)]
 
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([2, 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("300.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.j (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: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{2} + \beta_{6} q^{3} + (\beta_{5} + \beta_{3}) q^{4} + \beta_1 q^{6} + 2 \beta_{4} q^{7} + (\beta_{7} + 2 \beta_{6}) q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{2} + \beta_{6} q^{3} + (\beta_{5} + \beta_{3}) q^{4} + \beta_1 q^{6} + 2 \beta_{4} q^{7} + (\beta_{7} + 2 \beta_{6}) q^{8} + \beta_{3} q^{9} + ( - 4 \beta_1 + 2) q^{11} + ( - \beta_{4} + \beta_{2}) q^{12} + (4 \beta_{7} - 2 \beta_{6}) q^{13} + (2 \beta_{5} - 2 \beta_{3}) q^{14} + (3 \beta_1 - 2) q^{16} + \beta_{7} q^{18} + ( - 4 \beta_{5} + 2 \beta_{3}) q^{19} + 2 q^{21} + (6 \beta_{4} - 2 \beta_{2}) q^{22} + 4 \beta_{6} q^{23} + ( - \beta_{5} + 3 \beta_{3}) q^{24} + (2 \beta_1 - 8) q^{26} - \beta_{4} q^{27} + ( - 2 \beta_{7} + 4 \beta_{6}) q^{28} - 8 \beta_{3} q^{29} + ( - 4 \beta_1 + 2) q^{31} + ( - 5 \beta_{4} + \beta_{2}) q^{32} + ( - 4 \beta_{7} + 2 \beta_{6}) q^{33} + (\beta_1 - 2) q^{36} + (2 \beta_{4} + 4 \beta_{2}) q^{37} + (2 \beta_{7} - 8 \beta_{6}) q^{38} + ( - 4 \beta_{5} + 2 \beta_{3}) q^{39} - 2 q^{41} + (2 \beta_{4} + 2 \beta_{2}) q^{42} - 8 \beta_{6} q^{43} + (6 \beta_{5} - 10 \beta_{3}) q^{44} + 4 \beta_1 q^{46} + (3 \beta_{7} - 2 \beta_{6}) q^{48} + 3 \beta_{3} q^{49} + ( - 10 \beta_{4} - 6 \beta_{2}) q^{52} + ( - 8 \beta_{7} + 4 \beta_{6}) q^{53} + ( - \beta_{5} + \beta_{3}) q^{54} + (2 \beta_1 + 4) q^{56} + ( - 2 \beta_{4} - 4 \beta_{2}) q^{57} - 8 \beta_{7} q^{58} + (4 \beta_{5} - 2 \beta_{3}) q^{59} + 6 q^{61} + (6 \beta_{4} - 2 \beta_{2}) q^{62} + 2 \beta_{6} q^{63} + ( - 5 \beta_{5} + 7 \beta_{3}) q^{64} + ( - 2 \beta_1 + 8) q^{66} - 12 \beta_{4} q^{67} + 4 \beta_{3} q^{69} + ( - 3 \beta_{4} - \beta_{2}) q^{72} + (8 \beta_{7} - 4 \beta_{6}) q^{73} + (2 \beta_{5} + 6 \beta_{3}) q^{74} + ( - 6 \beta_1 - 4) q^{76} + ( - 4 \beta_{4} - 8 \beta_{2}) q^{77} + (2 \beta_{7} - 8 \beta_{6}) q^{78} + (4 \beta_{5} - 2 \beta_{3}) q^{79} - q^{81} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{82} - 12 \beta_{6} q^{83} + (2 \beta_{5} + 2 \beta_{3}) q^{84} - 8 \beta_1 q^{86} + 8 \beta_{4} q^{87} + ( - 10 \beta_{7} + 12 \beta_{6}) q^{88} + 6 \beta_{3} q^{89} + (8 \beta_1 - 4) q^{91} + ( - 4 \beta_{4} + 4 \beta_{2}) q^{92} + ( - 4 \beta_{7} + 2 \beta_{6}) q^{93} + (\beta_1 - 6) q^{96} + (4 \beta_{4} + 8 \beta_{2}) q^{97} + 3 \beta_{7} q^{98} + (4 \beta_{5} - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{6} - 4 q^{16} + 16 q^{21} - 56 q^{26} - 12 q^{36} - 16 q^{41} + 16 q^{46} + 40 q^{56} + 48 q^{61} + 56 q^{66} - 56 q^{76} - 8 q^{81} - 32 q^{86} - 44 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 7\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 7\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 5\nu^{3} ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7\beta_{7} - 10\beta_{6} \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.28897 0.581861i
−0.581861 1.28897i
0.581861 + 1.28897i
1.28897 + 0.581861i
−1.28897 + 0.581861i
−0.581861 + 1.28897i
0.581861 1.28897i
1.28897 0.581861i
−1.28897 0.581861i −0.707107 0.707107i 1.32288 + 1.50000i 0 0.500000 + 1.32288i −1.41421 + 1.41421i −0.832353 2.70318i 1.00000i 0
7.2 −0.581861 1.28897i 0.707107 + 0.707107i −1.32288 + 1.50000i 0 0.500000 1.32288i 1.41421 1.41421i 2.70318 + 0.832353i 1.00000i 0
7.3 0.581861 + 1.28897i −0.707107 0.707107i −1.32288 + 1.50000i 0 0.500000 1.32288i −1.41421 + 1.41421i −2.70318 0.832353i 1.00000i 0
7.4 1.28897 + 0.581861i 0.707107 + 0.707107i 1.32288 + 1.50000i 0 0.500000 + 1.32288i 1.41421 1.41421i 0.832353 + 2.70318i 1.00000i 0
43.1 −1.28897 + 0.581861i −0.707107 + 0.707107i 1.32288 1.50000i 0 0.500000 1.32288i −1.41421 1.41421i −0.832353 + 2.70318i 1.00000i 0
43.2 −0.581861 + 1.28897i 0.707107 0.707107i −1.32288 1.50000i 0 0.500000 + 1.32288i 1.41421 + 1.41421i 2.70318 0.832353i 1.00000i 0
43.3 0.581861 1.28897i −0.707107 + 0.707107i −1.32288 1.50000i 0 0.500000 + 1.32288i −1.41421 1.41421i −2.70318 + 0.832353i 1.00000i 0
43.4 1.28897 0.581861i 0.707107 0.707107i 1.32288 1.50000i 0 0.500000 1.32288i 1.41421 + 1.41421i 0.832353 2.70318i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.j.b 8
3.b odd 2 1 900.2.k.h 8
4.b odd 2 1 inner 300.2.j.b 8
5.b even 2 1 inner 300.2.j.b 8
5.c odd 4 2 inner 300.2.j.b 8
12.b even 2 1 900.2.k.h 8
15.d odd 2 1 900.2.k.h 8
15.e even 4 2 900.2.k.h 8
20.d odd 2 1 inner 300.2.j.b 8
20.e even 4 2 inner 300.2.j.b 8
60.h even 2 1 900.2.k.h 8
60.l odd 4 2 900.2.k.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.j.b 8 1.a even 1 1 trivial
300.2.j.b 8 4.b odd 2 1 inner
300.2.j.b 8 5.b even 2 1 inner
300.2.j.b 8 5.c odd 4 2 inner
300.2.j.b 8 20.d odd 2 1 inner
300.2.j.b 8 20.e even 4 2 inner
900.2.k.h 8 3.b odd 2 1
900.2.k.h 8 12.b even 2 1
900.2.k.h 8 15.d odd 2 1
900.2.k.h 8 15.e even 4 2
900.2.k.h 8 60.h even 2 1
900.2.k.h 8 60.l odd 4 2

Hecke kernels

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

\( T_{7}^{4} + 16 \) Copy content Toggle raw display
\( T_{19}^{2} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{4} + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 784)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 12544)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$61$ \( (T - 6)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 12544)^{2} \) Copy content Toggle raw display
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