Properties

Label 950.2.b.b
Level $950$
Weight $2$
Character orbit 950.b
Analytic conductor $7.586$
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] = [950,2,Mod(799,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("950.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (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: \(7.58578819202\)
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 38)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + i q^{7} - i q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + i q^{7} - i q^{8} + 2 q^{9} - 6 q^{11} - i q^{12} + 5 i q^{13} - q^{14} + q^{16} - 3 i q^{17} + 2 i q^{18} - q^{19} - q^{21} - 6 i q^{22} + 3 i q^{23} + q^{24} - 5 q^{26} + 5 i q^{27} - i q^{28} - 9 q^{29} - 4 q^{31} + i q^{32} - 6 i q^{33} + 3 q^{34} - 2 q^{36} - 2 i q^{37} - i q^{38} - 5 q^{39} - i q^{42} + 8 i q^{43} + 6 q^{44} - 3 q^{46} + i q^{48} + 6 q^{49} + 3 q^{51} - 5 i q^{52} - 3 i q^{53} - 5 q^{54} + q^{56} - i q^{57} - 9 i q^{58} - 9 q^{59} - 10 q^{61} - 4 i q^{62} + 2 i q^{63} - q^{64} + 6 q^{66} - 5 i q^{67} + 3 i q^{68} - 3 q^{69} - 6 q^{71} - 2 i q^{72} - 7 i q^{73} + 2 q^{74} + q^{76} - 6 i q^{77} - 5 i q^{78} + 10 q^{79} + q^{81} - 6 i q^{83} + q^{84} - 8 q^{86} - 9 i q^{87} + 6 i q^{88} + 12 q^{89} - 5 q^{91} - 3 i q^{92} - 4 i q^{93} - q^{96} + 10 i q^{97} + 6 i q^{98} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 12 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{21} + 2 q^{24} - 10 q^{26} - 18 q^{29} - 8 q^{31} + 6 q^{34} - 4 q^{36} - 10 q^{39} + 12 q^{44} - 6 q^{46} + 12 q^{49} + 6 q^{51} - 10 q^{54} + 2 q^{56} - 18 q^{59} - 20 q^{61} - 2 q^{64} + 12 q^{66} - 6 q^{69} - 12 q^{71} + 4 q^{74} + 2 q^{76} + 20 q^{79} + 2 q^{81} + 2 q^{84} - 16 q^{86} + 24 q^{89} - 10 q^{91} - 2 q^{96} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 2.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 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
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 950.2.b.b 2
5.b even 2 1 inner 950.2.b.b 2
5.c odd 4 1 38.2.a.a 1
5.c odd 4 1 950.2.a.d 1
15.e even 4 1 342.2.a.e 1
15.e even 4 1 8550.2.a.m 1
20.e even 4 1 304.2.a.c 1
20.e even 4 1 7600.2.a.n 1
35.f even 4 1 1862.2.a.b 1
40.i odd 4 1 1216.2.a.e 1
40.k even 4 1 1216.2.a.m 1
55.e even 4 1 4598.2.a.p 1
60.l odd 4 1 2736.2.a.n 1
65.h odd 4 1 6422.2.a.h 1
95.g even 4 1 722.2.a.e 1
95.l even 12 2 722.2.c.c 2
95.m odd 12 2 722.2.c.e 2
95.q odd 36 6 722.2.e.f 6
95.r even 36 6 722.2.e.e 6
285.j odd 4 1 6498.2.a.f 1
380.j odd 4 1 5776.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 5.c odd 4 1
304.2.a.c 1 20.e even 4 1
342.2.a.e 1 15.e even 4 1
722.2.a.e 1 95.g even 4 1
722.2.c.c 2 95.l even 12 2
722.2.c.e 2 95.m odd 12 2
722.2.e.e 6 95.r even 36 6
722.2.e.f 6 95.q odd 36 6
950.2.a.d 1 5.c odd 4 1
950.2.b.b 2 1.a even 1 1 trivial
950.2.b.b 2 5.b even 2 1 inner
1216.2.a.e 1 40.i odd 4 1
1216.2.a.m 1 40.k even 4 1
1862.2.a.b 1 35.f even 4 1
2736.2.a.n 1 60.l odd 4 1
4598.2.a.p 1 55.e even 4 1
5776.2.a.m 1 380.j odd 4 1
6422.2.a.h 1 65.h odd 4 1
6498.2.a.f 1 285.j odd 4 1
7600.2.a.n 1 20.e even 4 1
8550.2.a.m 1 15.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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 9 \) Copy content Toggle raw display
$59$ \( (T + 9)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 25 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 49 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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