Properties

Label 5850.2.e.bb
Level $5850$
Weight $2$
Character orbit 5850.e
Analytic conductor $46.712$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5850,2,Mod(5149,5850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5850, 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("5850.5149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.e (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: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
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 78)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + 4 i q^{7} - i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{4} + 4 i q^{7} - i q^{8} + 4 q^{11} - i q^{13} - 4 q^{14} + q^{16} - 2 i q^{17} + 8 q^{19} + 4 i q^{22} + q^{26} - 4 i q^{28} + 6 q^{29} - 4 q^{31} + i q^{32} + 2 q^{34} - 2 i q^{37} + 8 i q^{38} + 10 q^{41} - 4 i q^{43} - 4 q^{44} - 8 i q^{47} - 9 q^{49} + i q^{52} - 10 i q^{53} + 4 q^{56} + 6 i q^{58} + 4 q^{59} - 2 q^{61} - 4 i q^{62} - q^{64} - 16 i q^{67} + 2 i q^{68} + 8 q^{71} - 2 i q^{73} + 2 q^{74} - 8 q^{76} + 16 i q^{77} - 8 q^{79} + 10 i q^{82} + 12 i q^{83} + 4 q^{86} - 4 i q^{88} + 14 q^{89} + 4 q^{91} + 8 q^{94} + 10 i q^{97} - 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 8 q^{11} - 8 q^{14} + 2 q^{16} + 16 q^{19} + 2 q^{26} + 12 q^{29} - 8 q^{31} + 4 q^{34} + 20 q^{41} - 8 q^{44} - 18 q^{49} + 8 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} + 16 q^{71} + 4 q^{74} - 16 q^{76} - 16 q^{79} + 8 q^{86} + 28 q^{89} + 8 q^{91} + 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times\).

\(n\) \(2251\) \(3251\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5149.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 4.00000i 1.00000i 0 0
5149.2 1.00000i 0 −1.00000 0 0 4.00000i 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 5850.2.e.bb 2
3.b odd 2 1 1950.2.e.i 2
5.b even 2 1 inner 5850.2.e.bb 2
5.c odd 4 1 234.2.a.c 1
5.c odd 4 1 5850.2.a.d 1
15.d odd 2 1 1950.2.e.i 2
15.e even 4 1 78.2.a.a 1
15.e even 4 1 1950.2.a.w 1
20.e even 4 1 1872.2.a.c 1
40.i odd 4 1 7488.2.a.bz 1
40.k even 4 1 7488.2.a.bk 1
45.k odd 12 2 2106.2.e.j 2
45.l even 12 2 2106.2.e.q 2
60.l odd 4 1 624.2.a.h 1
65.f even 4 1 3042.2.b.g 2
65.h odd 4 1 3042.2.a.f 1
65.k even 4 1 3042.2.b.g 2
105.k odd 4 1 3822.2.a.j 1
120.q odd 4 1 2496.2.a.b 1
120.w even 4 1 2496.2.a.t 1
165.l odd 4 1 9438.2.a.t 1
195.j odd 4 1 1014.2.b.b 2
195.s even 4 1 1014.2.a.d 1
195.u odd 4 1 1014.2.b.b 2
195.bc odd 12 2 1014.2.i.d 4
195.bf even 12 2 1014.2.e.c 2
195.bl even 12 2 1014.2.e.f 2
195.bn odd 12 2 1014.2.i.d 4
780.w odd 4 1 8112.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 15.e even 4 1
234.2.a.c 1 5.c odd 4 1
624.2.a.h 1 60.l odd 4 1
1014.2.a.d 1 195.s even 4 1
1014.2.b.b 2 195.j odd 4 1
1014.2.b.b 2 195.u odd 4 1
1014.2.e.c 2 195.bf even 12 2
1014.2.e.f 2 195.bl even 12 2
1014.2.i.d 4 195.bc odd 12 2
1014.2.i.d 4 195.bn odd 12 2
1872.2.a.c 1 20.e even 4 1
1950.2.a.w 1 15.e even 4 1
1950.2.e.i 2 3.b odd 2 1
1950.2.e.i 2 15.d odd 2 1
2106.2.e.j 2 45.k odd 12 2
2106.2.e.q 2 45.l even 12 2
2496.2.a.b 1 120.q odd 4 1
2496.2.a.t 1 120.w even 4 1
3042.2.a.f 1 65.h odd 4 1
3042.2.b.g 2 65.f even 4 1
3042.2.b.g 2 65.k even 4 1
3822.2.a.j 1 105.k odd 4 1
5850.2.a.d 1 5.c odd 4 1
5850.2.e.bb 2 1.a even 1 1 trivial
5850.2.e.bb 2 5.b even 2 1 inner
7488.2.a.bk 1 40.k even 4 1
7488.2.a.bz 1 40.i odd 4 1
8112.2.a.v 1 780.w odd 4 1
9438.2.a.t 1 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5850, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} - 8 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display
\( T_{31} + 4 \) 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
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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