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Properties

Label 4050.2.c.ba
Level $4050$
Weight $2$
Character orbit 4050.c
Analytic conductor $32.339$
Analytic rank $0$
Dimension $4$
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] = [4050,2,Mod(649,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, 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("4050.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.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: \(32.3394128186\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 17x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 9\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times\).

\(n\) \(2351\) \(3727\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
− 3.37228i
2.37228i
− 2.37228i
3.37228i
− 1.00000i 0 −1.00000 0 0 − 3.37228i 1.00000i 0 0
649.2 − 1.00000i 0 −1.00000 0 0 2.37228i 1.00000i 0 0
649.3 1.00000i 0 −1.00000 0 0 − 2.37228i − 1.00000i 0 0
649.4 1.00000i 0 −1.00000 0 0 3.37228i − 1.00000i 0 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 4050.2.c.ba 4
3.b odd 2 1 4050.2.c.v 4
5.b even 2 1 inner 4050.2.c.ba 4
5.c odd 4 1 810.2.a.k 2
5.c odd 4 1 4050.2.a.bo 2
9.c even 3 2 1350.2.j.f 8
9.d odd 6 2 450.2.j.g 8
15.d odd 2 1 4050.2.c.v 4
15.e even 4 1 810.2.a.i 2
15.e even 4 1 4050.2.a.bw 2
20.e even 4 1 6480.2.a.bn 2
45.h odd 6 2 450.2.j.g 8
45.j even 6 2 1350.2.j.f 8
45.k odd 12 2 270.2.e.c 4
45.k odd 12 2 1350.2.e.l 4
45.l even 12 2 90.2.e.c ✓ 4
45.l even 12 2 450.2.e.j 4
60.l odd 4 1 6480.2.a.be 2
180.v odd 12 2 720.2.q.f 4
180.x even 12 2 2160.2.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c ✓ 4 45.l even 12 2
270.2.e.c 4 45.k odd 12 2
450.2.e.j 4 45.l even 12 2
450.2.j.g 8 9.d odd 6 2
450.2.j.g 8 45.h odd 6 2
720.2.q.f 4 180.v odd 12 2
810.2.a.i 2 15.e even 4 1
810.2.a.k 2 5.c odd 4 1
1350.2.e.l 4 45.k odd 12 2
1350.2.j.f 8 9.c even 3 2
1350.2.j.f 8 45.j even 6 2
2160.2.q.f 4 180.x even 12 2
4050.2.a.bo 2 5.c odd 4 1
4050.2.a.bw 2 15.e even 4 1
4050.2.c.v 4 3.b odd 2 1
4050.2.c.v 4 15.d odd 2 1
4050.2.c.ba 4 1.a even 1 1 trivial
4050.2.c.ba 4 5.b even 2 1 inner
6480.2.a.be 2 60.l odd 4 1
6480.2.a.bn 2 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 \(S_{2}^{\mathrm{new}}(4050, [\chi])\):

\( T_{7}^{4} + 17T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{4} + 68T_{13}^{2} + 1024 \) Copy content Toggle raw display
\( T_{17}^{4} + 57T_{17}^{2} + 144 \) Copy content Toggle raw display
\( T_{19}^{2} + T_{19} - 8 \) Copy content Toggle raw display
\( T_{29}^{2} + 3T_{29} - 6 \) Copy content Toggle raw display
\( T_{41} - 3 \) Copy content Toggle raw display
\( T_{71} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 17T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 68T^{2} + 1024 \) Copy content Toggle raw display
$17$ \( T^{4} + 57T^{2} + 144 \) Copy content Toggle raw display
$19$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 21T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T - 3)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 161T^{2} + 4096 \) Copy content Toggle raw display
$47$ \( T^{4} + 57T^{2} + 144 \) Copy content Toggle raw display
$53$ \( (T^{2} + 132)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 209T^{2} + 1936 \) Copy content Toggle raw display
$79$ \( (T + 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 57T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T^{2} - 15 T - 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 77T^{2} + 484 \) Copy content Toggle raw display
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