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Properties

Label 2300.2.c.j
Level $2300$
Weight $2$
Character orbit 2300.c
Analytic conductor $18.366$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,2,Mod(1749,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2300.1749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (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: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 53x^{4} + 29x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{5} - \beta_1) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{5} - \beta_1) q^{7} + (\beta_{2} - 1) q^{9} + (\beta_{6} - \beta_{3}) q^{11} + (\beta_{7} + \beta_{5} + \beta_1) q^{13} + ( - 2 \beta_{4} - 2 \beta_1) q^{17} + (\beta_{6} - \beta_{3} - 2) q^{19} + 2 q^{21} - \beta_{4} q^{23} + (\beta_{5} - \beta_{4}) q^{27} + ( - \beta_{6} - \beta_{2} + 2) q^{29} + ( - \beta_{3} - \beta_{2} + 4) q^{31} + ( - 2 \beta_{7} - 6 \beta_{4}) q^{33} + (2 \beta_{7} - 2 \beta_1) q^{37} + (\beta_{6} - 2 \beta_{3} - 3) q^{39} + ( - \beta_{6} - 2 \beta_{3} + 2 \beta_{2}) q^{41} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{43} + ( - \beta_{7} + 2 \beta_{5} - 4 \beta_{4}) q^{47} + ( - \beta_{6} - \beta_{3} - 2 \beta_{2} + 1) q^{49} + (2 \beta_{3} - 2 \beta_{2} + 8) q^{51} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{53} + ( - 2 \beta_{7} - 6 \beta_{4} - 2 \beta_1) q^{57} + (\beta_{6} - \beta_{3} - 3) q^{59} + ( - 2 \beta_{6} + 6) q^{61} + ( - 3 \beta_{5} - \beta_1) q^{63} + ( - 2 \beta_{7} - 4 \beta_{4}) q^{67} + \beta_{3} q^{69} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{71} + (\beta_{7} + \beta_{5} + \cdots + 2 \beta_1) q^{73}+ \cdots + (\beta_{6} + 7 \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 2 q^{11} - 14 q^{19} + 16 q^{21} + 10 q^{29} + 28 q^{31} - 22 q^{39} + 6 q^{41} - 2 q^{49} + 56 q^{51} - 22 q^{59} + 44 q^{61} + 36 q^{71} - 50 q^{79} + 50 q^{91} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 53x^{4} + 29x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 7\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 51\nu^{3} + 15\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 53\nu^{3} + 27\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 13\nu^{4} + 44\nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\nu^{7} - 27\nu^{5} - 95\nu^{3} - 28\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} - 7\beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - 7\beta_{5} + 11\beta_{4} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 13\beta_{3} + 47\beta_{2} - 175 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{7} + 47\beta_{5} - 101\beta_{4} - 269\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
− 2.66337i
− 2.50653i
− 0.631352i
− 0.474520i
0.474520i
0.631352i
2.50653i
2.66337i
0 − 2.66337i 0 0 0 0.750930i 0 −4.09352 0
1749.2 0 − 2.50653i 0 0 0 0.797915i 0 −3.28271 0
1749.3 0 − 0.631352i 0 0 0 3.16780i 0 2.60139 0
1749.4 0 − 0.474520i 0 0 0 4.21479i 0 2.77483 0
1749.5 0 0.474520i 0 0 0 − 4.21479i 0 2.77483 0
1749.6 0 0.631352i 0 0 0 − 3.16780i 0 2.60139 0
1749.7 0 2.50653i 0 0 0 − 0.797915i 0 −3.28271 0
1749.8 0 2.66337i 0 0 0 − 0.750930i 0 −4.09352 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1749.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.2.c.j 8
5.b even 2 1 inner 2300.2.c.j 8
5.c odd 4 1 2300.2.a.l ✓ 4
5.c odd 4 1 2300.2.a.m yes 4
20.e even 4 1 9200.2.a.cm 4
20.e even 4 1 9200.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.a.l ✓ 4 5.c odd 4 1
2300.2.a.m yes 4 5.c odd 4 1
2300.2.c.j 8 1.a even 1 1 trivial
2300.2.c.j 8 5.b even 2 1 inner
9200.2.a.cm 4 20.e even 4 1
9200.2.a.co 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2300, [\chi])\):

\( T_{3}^{8} + 14T_{3}^{6} + 53T_{3}^{4} + 29T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 29T_{7}^{6} + 212T_{7}^{4} + 224T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} - 26T_{11}^{2} + 24T_{11} + 120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 14 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 29 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} - 26 T^{2} + \cdots + 120)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 63 T^{6} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( T^{8} + 72 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( (T^{4} + 7 T^{3} - 8 T^{2} + \cdots + 72)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 5 T^{3} - 31 T^{2} + \cdots - 93)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 14 T^{3} + \cdots - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 200 T^{6} + \cdots + 173056 \) Copy content Toggle raw display
$41$ \( (T^{4} - 3 T^{3} + \cdots + 381)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 65 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$47$ \( T^{8} + 210 T^{6} + \cdots + 1542564 \) Copy content Toggle raw display
$53$ \( T^{8} + 132 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$59$ \( (T^{4} + 11 T^{3} + \cdots + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 22 T^{3} + \cdots - 416)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 200 T^{6} + \cdots + 984064 \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + \cdots - 1800)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 119 T^{6} + \cdots + 19321 \) Copy content Toggle raw display
$79$ \( (T^{4} + 25 T^{3} + \cdots + 40)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 201 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( (T^{4} - 236 T^{2} + \cdots - 3840)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 504 T^{6} + \cdots + 1081600 \) Copy content Toggle raw display
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