Properties

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

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Show commands: Magma / PariGP / SageMath

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 26)
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.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + i q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + i q^{7} + i q^{8} + 2 q^{11} + i q^{13} + q^{14} + q^{16} + 3 i q^{17} - 6 q^{19} - 2 i q^{22} - 4 i q^{23} + q^{26} - i q^{28} + 2 q^{29} + 4 q^{31} - i q^{32} + 3 q^{34} + 3 i q^{37} + 6 i q^{38} + 5 i q^{43} - 2 q^{44} - 4 q^{46} - 13 i q^{47} + 6 q^{49} - i q^{52} + 12 i q^{53} - q^{56} - 2 i q^{58} - 10 q^{59} - 8 q^{61} - 4 i q^{62} - q^{64} - 2 i q^{67} - 3 i q^{68} + 5 q^{71} + 10 i q^{73} + 3 q^{74} + 6 q^{76} + 2 i q^{77} + 4 q^{79} + 5 q^{86} + 2 i q^{88} + 6 q^{89} - q^{91} + 4 i q^{92} - 13 q^{94} + 14 i q^{97} - 6 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 2 q^{26} + 4 q^{29} + 8 q^{31} + 6 q^{34} - 4 q^{44} - 8 q^{46} + 12 q^{49} - 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} + 10 q^{71} + 6 q^{74} + 12 q^{76} + 8 q^{79} + 10 q^{86} + 12 q^{89} - 2 q^{91} - 26 q^{94}+O(q^{100}) \) 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 1.00000i 1.00000i 0 0
5149.2 1.00000i 0 −1.00000 0 0 1.00000i 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.v 2
3.b odd 2 1 650.2.b.a 2
5.b even 2 1 inner 5850.2.e.v 2
5.c odd 4 1 234.2.a.b 1
5.c odd 4 1 5850.2.a.bn 1
15.d odd 2 1 650.2.b.a 2
15.e even 4 1 26.2.a.b 1
15.e even 4 1 650.2.a.g 1
20.e even 4 1 1872.2.a.m 1
40.i odd 4 1 7488.2.a.w 1
40.k even 4 1 7488.2.a.v 1
45.k odd 12 2 2106.2.e.t 2
45.l even 12 2 2106.2.e.h 2
60.l odd 4 1 208.2.a.d 1
60.l odd 4 1 5200.2.a.c 1
65.f even 4 1 3042.2.b.f 2
65.h odd 4 1 3042.2.a.l 1
65.k even 4 1 3042.2.b.f 2
105.k odd 4 1 1274.2.a.o 1
105.w odd 12 2 1274.2.f.a 2
105.x even 12 2 1274.2.f.l 2
120.q odd 4 1 832.2.a.a 1
120.w even 4 1 832.2.a.j 1
165.l odd 4 1 3146.2.a.a 1
195.j odd 4 1 338.2.b.a 2
195.s even 4 1 338.2.a.a 1
195.s even 4 1 8450.2.a.y 1
195.u odd 4 1 338.2.b.a 2
195.bc odd 12 2 338.2.e.d 4
195.bf even 12 2 338.2.c.g 2
195.bl even 12 2 338.2.c.c 2
195.bn odd 12 2 338.2.e.d 4
240.z odd 4 1 3328.2.b.k 2
240.bb even 4 1 3328.2.b.g 2
240.bd odd 4 1 3328.2.b.k 2
240.bf even 4 1 3328.2.b.g 2
255.o even 4 1 7514.2.a.i 1
285.j odd 4 1 9386.2.a.f 1
780.u even 4 1 2704.2.f.j 2
780.w odd 4 1 2704.2.a.n 1
780.bn even 4 1 2704.2.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 15.e even 4 1
208.2.a.d 1 60.l odd 4 1
234.2.a.b 1 5.c odd 4 1
338.2.a.a 1 195.s even 4 1
338.2.b.a 2 195.j odd 4 1
338.2.b.a 2 195.u odd 4 1
338.2.c.c 2 195.bl even 12 2
338.2.c.g 2 195.bf even 12 2
338.2.e.d 4 195.bc odd 12 2
338.2.e.d 4 195.bn odd 12 2
650.2.a.g 1 15.e even 4 1
650.2.b.a 2 3.b odd 2 1
650.2.b.a 2 15.d odd 2 1
832.2.a.a 1 120.q odd 4 1
832.2.a.j 1 120.w even 4 1
1274.2.a.o 1 105.k odd 4 1
1274.2.f.a 2 105.w odd 12 2
1274.2.f.l 2 105.x even 12 2
1872.2.a.m 1 20.e even 4 1
2106.2.e.h 2 45.l even 12 2
2106.2.e.t 2 45.k odd 12 2
2704.2.a.n 1 780.w odd 4 1
2704.2.f.j 2 780.u even 4 1
2704.2.f.j 2 780.bn even 4 1
3042.2.a.l 1 65.h odd 4 1
3042.2.b.f 2 65.f even 4 1
3042.2.b.f 2 65.k even 4 1
3146.2.a.a 1 165.l odd 4 1
3328.2.b.g 2 240.bb even 4 1
3328.2.b.g 2 240.bf even 4 1
3328.2.b.k 2 240.z odd 4 1
3328.2.b.k 2 240.bd odd 4 1
5200.2.a.c 1 60.l odd 4 1
5850.2.a.bn 1 5.c odd 4 1
5850.2.e.v 2 1.a even 1 1 trivial
5850.2.e.v 2 5.b even 2 1 inner
7488.2.a.v 1 40.k even 4 1
7488.2.a.w 1 40.i odd 4 1
7514.2.a.i 1 255.o even 4 1
8450.2.a.y 1 195.s even 4 1
9386.2.a.f 1 285.j odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 9 \) Copy content Toggle raw display
\( T_{19} + 6 \) Copy content Toggle raw display
\( T_{29} - 2 \) 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
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 25 \) Copy content Toggle raw display
$47$ \( T^{2} + 169 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T - 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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