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Properties

Label 4050.2.c.r
Level $4050$
Weight $2$
Character orbit 4050.c
Analytic conductor $32.339$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4050,2,Mod(649,4050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,0,0,0,0,0,6,0,0,4,0,2,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.3394128186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + 2 i q^{7} + i q^{8} + 3 q^{11} - 2 i q^{13} + 2 q^{14} + q^{16} + 3 i q^{17} + q^{19} - 3 i q^{22} - 6 i q^{23} - 2 q^{26} - 2 i q^{28} + 6 q^{29} - 4 q^{31} - i q^{32} + 3 q^{34} + \cdots - 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 6 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{26} + 12 q^{29} - 8 q^{31} + 6 q^{34} - 18 q^{41} - 6 q^{44} - 12 q^{46} + 6 q^{49} - 4 q^{56} + 6 q^{59} + 16 q^{61} - 2 q^{64} + 24 q^{71}+ \cdots + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
1.00000i
− 1.00000i
− 1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 0
649.2 1.00000i 0 −1.00000 0 0 − 2.00000i − 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.r 2
3.b odd 2 1 4050.2.c.c 2
5.b even 2 1 inner 4050.2.c.r 2
5.c odd 4 1 162.2.a.b 1
5.c odd 4 1 4050.2.a.v 1
9.c even 3 2 1350.2.j.a 4
9.d odd 6 2 450.2.j.e 4
15.d odd 2 1 4050.2.c.c 2
15.e even 4 1 162.2.a.c 1
15.e even 4 1 4050.2.a.c 1
20.e even 4 1 1296.2.a.f 1
35.f even 4 1 7938.2.a.i 1
40.i odd 4 1 5184.2.a.q 1
40.k even 4 1 5184.2.a.p 1
45.h odd 6 2 450.2.j.e 4
45.j even 6 2 1350.2.j.a 4
45.k odd 12 2 54.2.c.a 2
45.k odd 12 2 1350.2.e.c 2
45.l even 12 2 18.2.c.a ✓ 2
45.l even 12 2 450.2.e.i 2
60.l odd 4 1 1296.2.a.g 1
105.k odd 4 1 7938.2.a.x 1
120.q odd 4 1 5184.2.a.o 1
120.w even 4 1 5184.2.a.r 1
180.v odd 12 2 144.2.i.c 2
180.x even 12 2 432.2.i.b 2
315.bs even 12 2 2646.2.e.c 2
315.bt odd 12 2 2646.2.e.b 2
315.bu odd 12 2 882.2.e.g 2
315.bv even 12 2 882.2.e.i 2
315.bw odd 12 2 882.2.h.b 2
315.bx even 12 2 882.2.h.c 2
315.cb even 12 2 2646.2.f.g 2
315.cf odd 12 2 882.2.f.d 2
315.cg even 12 2 2646.2.h.i 2
315.ch odd 12 2 2646.2.h.h 2
360.bo even 12 2 1728.2.i.f 2
360.br even 12 2 576.2.i.g 2
360.bt odd 12 2 576.2.i.a 2
360.bu odd 12 2 1728.2.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a ✓ 2 45.l even 12 2
54.2.c.a 2 45.k odd 12 2
144.2.i.c 2 180.v odd 12 2
162.2.a.b 1 5.c odd 4 1
162.2.a.c 1 15.e even 4 1
432.2.i.b 2 180.x even 12 2
450.2.e.i 2 45.l even 12 2
450.2.j.e 4 9.d odd 6 2
450.2.j.e 4 45.h odd 6 2
576.2.i.a 2 360.bt odd 12 2
576.2.i.g 2 360.br even 12 2
882.2.e.g 2 315.bu odd 12 2
882.2.e.i 2 315.bv even 12 2
882.2.f.d 2 315.cf odd 12 2
882.2.h.b 2 315.bw odd 12 2
882.2.h.c 2 315.bx even 12 2
1296.2.a.f 1 20.e even 4 1
1296.2.a.g 1 60.l odd 4 1
1350.2.e.c 2 45.k odd 12 2
1350.2.j.a 4 9.c even 3 2
1350.2.j.a 4 45.j even 6 2
1728.2.i.e 2 360.bu odd 12 2
1728.2.i.f 2 360.bo even 12 2
2646.2.e.b 2 315.bt odd 12 2
2646.2.e.c 2 315.bs even 12 2
2646.2.f.g 2 315.cb even 12 2
2646.2.h.h 2 315.ch odd 12 2
2646.2.h.i 2 315.cg even 12 2
4050.2.a.c 1 15.e even 4 1
4050.2.a.v 1 5.c odd 4 1
4050.2.c.c 2 3.b odd 2 1
4050.2.c.c 2 15.d odd 2 1
4050.2.c.r 2 1.a even 1 1 trivial
4050.2.c.r 2 5.b even 2 1 inner
5184.2.a.o 1 120.q odd 4 1
5184.2.a.p 1 40.k even 4 1
5184.2.a.q 1 40.i odd 4 1
5184.2.a.r 1 120.w even 4 1
7938.2.a.i 1 35.f even 4 1
7938.2.a.x 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4050, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 9 \) Copy content Toggle raw display
\( T_{19} - 1 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display
\( T_{41} + 9 \) Copy content Toggle raw display
\( T_{71} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 25 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 25 \) Copy content Toggle raw display
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