Properties

Label 650.2.c.a
Level $650$
Weight $2$
Character orbit 650.c
Analytic conductor $5.190$
Analytic rank $0$
Dimension $2$
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] = [650,2,Mod(649,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.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: \(5.19027613138\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + i q^{3} + q^{4} - i q^{6} - 3 q^{7} - q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + i q^{3} + q^{4} - i q^{6} - 3 q^{7} - q^{8} + 2 q^{9} + i q^{12} + ( - 2 i + 3) q^{13} + 3 q^{14} + q^{16} + 3 i q^{17} - 2 q^{18} + 6 i q^{19} - 3 i q^{21} + 6 i q^{23} - i q^{24} + (2 i - 3) q^{26} + 5 i q^{27} - 3 q^{28} - q^{32} - 3 i q^{34} + 2 q^{36} - 3 q^{37} - 6 i q^{38} + (3 i + 2) q^{39} + 3 i q^{42} + i q^{43} - 6 i q^{46} - 3 q^{47} + i q^{48} + 2 q^{49} - 3 q^{51} + ( - 2 i + 3) q^{52} + 6 i q^{53} - 5 i q^{54} + 3 q^{56} - 6 q^{57} + 6 i q^{59} - 8 q^{61} - 6 q^{63} + q^{64} + 12 q^{67} + 3 i q^{68} - 6 q^{69} + 15 i q^{71} - 2 q^{72} + 6 q^{73} + 3 q^{74} + 6 i q^{76} + ( - 3 i - 2) q^{78} - 10 q^{79} + q^{81} + 6 q^{83} - 3 i q^{84} - i q^{86} + 6 i q^{89} + (6 i - 9) q^{91} + 6 i q^{92} + 3 q^{94} - i q^{96} + 12 q^{97} - 2 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} + 4 q^{9} + 6 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{18} - 6 q^{26} - 6 q^{28} - 2 q^{32} + 4 q^{36} - 6 q^{37} + 4 q^{39} - 6 q^{47} + 4 q^{49} - 6 q^{51} + 6 q^{52} + 6 q^{56} - 12 q^{57} - 16 q^{61} - 12 q^{63} + 2 q^{64} + 24 q^{67} - 12 q^{69} - 4 q^{72} + 12 q^{73} + 6 q^{74} - 4 q^{78} - 20 q^{79} + 2 q^{81} + 12 q^{83} - 18 q^{91} + 6 q^{94} + 24 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-1\)

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
1.00000i
1.00000i
−1.00000 1.00000i 1.00000 0 1.00000i −3.00000 −1.00000 2.00000 0
649.2 −1.00000 1.00000i 1.00000 0 1.00000i −3.00000 −1.00000 2.00000 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
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.c.a 2
5.b even 2 1 650.2.c.d 2
5.c odd 4 1 26.2.b.a 2
5.c odd 4 1 650.2.d.b 2
13.b even 2 1 650.2.c.d 2
15.e even 4 1 234.2.b.b 2
20.e even 4 1 208.2.f.a 2
35.f even 4 1 1274.2.d.c 2
35.k even 12 2 1274.2.n.c 4
35.l odd 12 2 1274.2.n.d 4
40.i odd 4 1 832.2.f.d 2
40.k even 4 1 832.2.f.b 2
60.l odd 4 1 1872.2.c.f 2
65.d even 2 1 inner 650.2.c.a 2
65.f even 4 1 338.2.a.d 1
65.f even 4 1 8450.2.a.u 1
65.h odd 4 1 26.2.b.a 2
65.h odd 4 1 650.2.d.b 2
65.k even 4 1 338.2.a.b 1
65.k even 4 1 8450.2.a.h 1
65.o even 12 2 338.2.c.f 2
65.q odd 12 2 338.2.e.c 4
65.r odd 12 2 338.2.e.c 4
65.t even 12 2 338.2.c.b 2
195.j odd 4 1 3042.2.a.j 1
195.s even 4 1 234.2.b.b 2
195.u odd 4 1 3042.2.a.g 1
260.l odd 4 1 2704.2.a.j 1
260.p even 4 1 208.2.f.a 2
260.s odd 4 1 2704.2.a.k 1
455.s even 4 1 1274.2.d.c 2
455.cv odd 12 2 1274.2.n.d 4
455.df even 12 2 1274.2.n.c 4
520.bc even 4 1 832.2.f.b 2
520.bg odd 4 1 832.2.f.d 2
780.w odd 4 1 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 5.c odd 4 1
26.2.b.a 2 65.h odd 4 1
208.2.f.a 2 20.e even 4 1
208.2.f.a 2 260.p even 4 1
234.2.b.b 2 15.e even 4 1
234.2.b.b 2 195.s even 4 1
338.2.a.b 1 65.k even 4 1
338.2.a.d 1 65.f even 4 1
338.2.c.b 2 65.t even 12 2
338.2.c.f 2 65.o even 12 2
338.2.e.c 4 65.q odd 12 2
338.2.e.c 4 65.r odd 12 2
650.2.c.a 2 1.a even 1 1 trivial
650.2.c.a 2 65.d even 2 1 inner
650.2.c.d 2 5.b even 2 1
650.2.c.d 2 13.b even 2 1
650.2.d.b 2 5.c odd 4 1
650.2.d.b 2 65.h odd 4 1
832.2.f.b 2 40.k even 4 1
832.2.f.b 2 520.bc even 4 1
832.2.f.d 2 40.i odd 4 1
832.2.f.d 2 520.bg odd 4 1
1274.2.d.c 2 35.f even 4 1
1274.2.d.c 2 455.s even 4 1
1274.2.n.c 4 35.k even 12 2
1274.2.n.c 4 455.df even 12 2
1274.2.n.d 4 35.l odd 12 2
1274.2.n.d 4 455.cv odd 12 2
1872.2.c.f 2 60.l odd 4 1
1872.2.c.f 2 780.w odd 4 1
2704.2.a.j 1 260.l odd 4 1
2704.2.a.k 1 260.s odd 4 1
3042.2.a.g 1 195.u odd 4 1
3042.2.a.j 1 195.j odd 4 1
8450.2.a.h 1 65.k even 4 1
8450.2.a.u 1 65.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display
\( T_{37} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 225 \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( (T - 12)^{2} \) Copy content Toggle raw display
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