Properties

Label 750.2.c.b
Level $750$
Weight $2$
Character orbit 750.c
Analytic conductor $5.989$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [750,2,Mod(499,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("750.499");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{3} q^{3} - q^{4} - q^{6} + 2 \beta_1 q^{7} - \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{3} q^{3} - q^{4} - q^{6} + 2 \beta_1 q^{7} - \beta_{3} q^{8} - q^{9} + ( - \beta_{2} + 2) q^{11} - \beta_{3} q^{12} + (\beta_{3} + 3 \beta_1) q^{13} - 2 \beta_{2} q^{14} + q^{16} + (3 \beta_{3} + 3 \beta_1) q^{17} - \beta_{3} q^{18} - 6 q^{19} - 2 \beta_{2} q^{21} + (2 \beta_{3} - \beta_1) q^{22} + 5 \beta_1 q^{23} + q^{24} + ( - 3 \beta_{2} - 1) q^{26} - \beta_{3} q^{27} - 2 \beta_1 q^{28} + ( - 7 \beta_{2} - 5) q^{29} + ( - 3 \beta_{2} + 4) q^{31} + \beta_{3} q^{32} + (2 \beta_{3} - \beta_1) q^{33} + ( - 3 \beta_{2} - 3) q^{34} + q^{36} + ( - 6 \beta_{3} - 3 \beta_1) q^{37} - 6 \beta_{3} q^{38} + ( - 3 \beta_{2} - 1) q^{39} + ( - 2 \beta_{2} - 2) q^{41} - 2 \beta_1 q^{42} + (2 \beta_{3} - 3 \beta_1) q^{43} + (\beta_{2} - 2) q^{44} - 5 \beta_{2} q^{46} + (9 \beta_{3} + 3 \beta_1) q^{47} + \beta_{3} q^{48} + (4 \beta_{2} + 3) q^{49} + ( - 3 \beta_{2} - 3) q^{51} + ( - \beta_{3} - 3 \beta_1) q^{52} + ( - 2 \beta_{3} - 2 \beta_1) q^{53} + q^{54} + 2 \beta_{2} q^{56} - 6 \beta_{3} q^{57} + ( - 5 \beta_{3} - 7 \beta_1) q^{58} + (\beta_{2} - 7) q^{59} + (12 \beta_{2} + 6) q^{61} + (4 \beta_{3} - 3 \beta_1) q^{62} - 2 \beta_1 q^{63} - q^{64} + (\beta_{2} - 2) q^{66} + ( - 3 \beta_{3} + \beta_1) q^{67} + ( - 3 \beta_{3} - 3 \beta_1) q^{68} - 5 \beta_{2} q^{69} + ( - 8 \beta_{2} - 4) q^{71} + \beta_{3} q^{72} + 12 \beta_{3} q^{73} + (3 \beta_{2} + 6) q^{74} + 6 q^{76} + ( - 2 \beta_{3} + 6 \beta_1) q^{77} + ( - \beta_{3} - 3 \beta_1) q^{78} + ( - 3 \beta_{2} + 3) q^{79} + q^{81} + ( - 2 \beta_{3} - 2 \beta_1) q^{82} + (2 \beta_{3} + 10 \beta_1) q^{83} + 2 \beta_{2} q^{84} + (3 \beta_{2} - 2) q^{86} + ( - 5 \beta_{3} - 7 \beta_1) q^{87} + ( - 2 \beta_{3} + \beta_1) q^{88} + (2 \beta_{2} + 8) q^{89} + (4 \beta_{2} - 6) q^{91} - 5 \beta_1 q^{92} + (4 \beta_{3} - 3 \beta_1) q^{93} + ( - 3 \beta_{2} - 9) q^{94} - q^{96} + ( - 12 \beta_{3} - 10 \beta_1) q^{97} + (3 \beta_{3} + 4 \beta_1) q^{98} + (\beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 10 q^{11} + 4 q^{14} + 4 q^{16} - 24 q^{19} + 4 q^{21} + 4 q^{24} + 2 q^{26} - 6 q^{29} + 22 q^{31} - 6 q^{34} + 4 q^{36} + 2 q^{39} - 4 q^{41} - 10 q^{44} + 10 q^{46} + 4 q^{49} - 6 q^{51} + 4 q^{54} - 4 q^{56} - 30 q^{59} - 4 q^{64} - 10 q^{66} + 10 q^{69} + 18 q^{74} + 24 q^{76} + 18 q^{79} + 4 q^{81} - 4 q^{84} - 14 q^{86} + 28 q^{89} - 32 q^{91} - 30 q^{94} - 4 q^{96} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
0.618034i
1.61803i
1.61803i
0.618034i
1.00000i 1.00000i −1.00000 0 −1.00000 1.23607i 1.00000i −1.00000 0
499.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.23607i 1.00000i −1.00000 0
499.3 1.00000i 1.00000i −1.00000 0 −1.00000 3.23607i 1.00000i −1.00000 0
499.4 1.00000i 1.00000i −1.00000 0 −1.00000 1.23607i 1.00000i −1.00000 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
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 750.2.c.b 4
3.b odd 2 1 2250.2.c.b 4
4.b odd 2 1 6000.2.f.a 4
5.b even 2 1 inner 750.2.c.b 4
5.c odd 4 1 750.2.a.c 2
5.c odd 4 1 750.2.a.f yes 2
15.d odd 2 1 2250.2.c.b 4
15.e even 4 1 2250.2.a.c 2
15.e even 4 1 2250.2.a.n 2
20.d odd 2 1 6000.2.f.a 4
20.e even 4 1 6000.2.a.d 2
20.e even 4 1 6000.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.2.a.c 2 5.c odd 4 1
750.2.a.f yes 2 5.c odd 4 1
750.2.c.b 4 1.a even 1 1 trivial
750.2.c.b 4 5.b even 2 1 inner
2250.2.a.c 2 15.e even 4 1
2250.2.a.n 2 15.e even 4 1
2250.2.c.b 4 3.b odd 2 1
2250.2.c.b 4 15.d odd 2 1
6000.2.a.d 2 20.e even 4 1
6000.2.a.y 2 20.e even 4 1
6000.2.f.a 4 4.b odd 2 1
6000.2.f.a 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 12T_{7}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 121 \) Copy content Toggle raw display
$17$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$19$ \( (T + 6)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 75T^{2} + 625 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T - 59)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 11 T + 19)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 63T^{2} + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 47T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{4} + 135T^{2} + 2025 \) Copy content Toggle raw display
$53$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 15 T + 55)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 27T^{2} + 121 \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 9 T + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 268 T^{2} + 13456 \) Copy content Toggle raw display
$89$ \( (T^{2} - 14 T + 44)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 348T^{2} + 5776 \) Copy content Toggle raw display
show more
show less