Properties

Label 950.2.e.d
Level $950$
Weight $2$
Character orbit 950.e
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(201,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.e (of order \(3\), degree \(2\), 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: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + 4 q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + 4 q^{7} + q^{8} + 2 \zeta_{6} q^{9} + 3 q^{11} - q^{12} + 2 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{14} + (\zeta_{6} - 1) q^{16} + (6 \zeta_{6} - 6) q^{17} - 2 q^{18} + ( - 3 \zeta_{6} - 2) q^{19} + ( - 4 \zeta_{6} + 4) q^{21} + (3 \zeta_{6} - 3) q^{22} - 6 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} - 2 q^{26} + 5 q^{27} - 4 \zeta_{6} q^{28} + 2 q^{31} - \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{33} - 6 \zeta_{6} q^{34} + ( - 2 \zeta_{6} + 2) q^{36} + 10 q^{37} + ( - 2 \zeta_{6} + 5) q^{38} + 2 q^{39} + (9 \zeta_{6} - 9) q^{41} + 4 \zeta_{6} q^{42} + (4 \zeta_{6} - 4) q^{43} - 3 \zeta_{6} q^{44} + 6 q^{46} + \zeta_{6} q^{48} + 9 q^{49} + 6 \zeta_{6} q^{51} + ( - 2 \zeta_{6} + 2) q^{52} + 6 \zeta_{6} q^{53} + (5 \zeta_{6} - 5) q^{54} + 4 q^{56} + (2 \zeta_{6} - 5) q^{57} + ( - 9 \zeta_{6} + 9) q^{59} + 4 \zeta_{6} q^{61} + (2 \zeta_{6} - 2) q^{62} + 8 \zeta_{6} q^{63} + q^{64} + 3 \zeta_{6} q^{66} - 7 \zeta_{6} q^{67} + 6 q^{68} - 6 q^{69} + ( - 6 \zeta_{6} + 6) q^{71} + 2 \zeta_{6} q^{72} + (\zeta_{6} - 1) q^{73} + (10 \zeta_{6} - 10) q^{74} + (5 \zeta_{6} - 3) q^{76} + 12 q^{77} + (2 \zeta_{6} - 2) q^{78} + ( - 4 \zeta_{6} + 4) q^{79} + (\zeta_{6} - 1) q^{81} - 9 \zeta_{6} q^{82} - 3 q^{83} - 4 q^{84} - 4 \zeta_{6} q^{86} + 3 q^{88} - 6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + (6 \zeta_{6} - 6) q^{92} + ( - 2 \zeta_{6} + 2) q^{93} - q^{96} + ( - 17 \zeta_{6} + 17) q^{97} + (9 \zeta_{6} - 9) q^{98} + 6 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} + 8 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} + 8 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 4 q^{14} - q^{16} - 6 q^{17} - 4 q^{18} - 7 q^{19} + 4 q^{21} - 3 q^{22} - 6 q^{23} + q^{24} - 4 q^{26} + 10 q^{27} - 4 q^{28} + 4 q^{31} - q^{32} + 3 q^{33} - 6 q^{34} + 2 q^{36} + 20 q^{37} + 8 q^{38} + 4 q^{39} - 9 q^{41} + 4 q^{42} - 4 q^{43} - 3 q^{44} + 12 q^{46} + q^{48} + 18 q^{49} + 6 q^{51} + 2 q^{52} + 6 q^{53} - 5 q^{54} + 8 q^{56} - 8 q^{57} + 9 q^{59} + 4 q^{61} - 2 q^{62} + 8 q^{63} + 2 q^{64} + 3 q^{66} - 7 q^{67} + 12 q^{68} - 12 q^{69} + 6 q^{71} + 2 q^{72} - q^{73} - 10 q^{74} - q^{76} + 24 q^{77} - 2 q^{78} + 4 q^{79} - q^{81} - 9 q^{82} - 6 q^{83} - 8 q^{84} - 4 q^{86} + 6 q^{88} - 6 q^{89} + 8 q^{91} - 6 q^{92} + 2 q^{93} - 2 q^{96} + 17 q^{97} - 9 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 4.00000 1.00000 1.00000 + 1.73205i 0
501.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 4.00000 1.00000 1.00000 1.73205i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.e.d 2
5.b even 2 1 38.2.c.a 2
5.c odd 4 2 950.2.j.e 4
15.d odd 2 1 342.2.g.b 2
19.c even 3 1 inner 950.2.e.d 2
20.d odd 2 1 304.2.i.c 2
40.e odd 2 1 1216.2.i.d 2
40.f even 2 1 1216.2.i.h 2
60.h even 2 1 2736.2.s.m 2
95.d odd 2 1 722.2.c.b 2
95.h odd 6 1 722.2.a.d 1
95.h odd 6 1 722.2.c.b 2
95.i even 6 1 38.2.c.a 2
95.i even 6 1 722.2.a.c 1
95.m odd 12 2 950.2.j.e 4
95.o odd 18 6 722.2.e.i 6
95.p even 18 6 722.2.e.j 6
285.n odd 6 1 342.2.g.b 2
285.n odd 6 1 6498.2.a.s 1
285.q even 6 1 6498.2.a.e 1
380.p odd 6 1 304.2.i.c 2
380.p odd 6 1 5776.2.a.g 1
380.s even 6 1 5776.2.a.n 1
760.z even 6 1 1216.2.i.h 2
760.bm odd 6 1 1216.2.i.d 2
1140.bn even 6 1 2736.2.s.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 5.b even 2 1
38.2.c.a 2 95.i even 6 1
304.2.i.c 2 20.d odd 2 1
304.2.i.c 2 380.p odd 6 1
342.2.g.b 2 15.d odd 2 1
342.2.g.b 2 285.n odd 6 1
722.2.a.c 1 95.i even 6 1
722.2.a.d 1 95.h odd 6 1
722.2.c.b 2 95.d odd 2 1
722.2.c.b 2 95.h odd 6 1
722.2.e.i 6 95.o odd 18 6
722.2.e.j 6 95.p even 18 6
950.2.e.d 2 1.a even 1 1 trivial
950.2.e.d 2 19.c even 3 1 inner
950.2.j.e 4 5.c odd 4 2
950.2.j.e 4 95.m odd 12 2
1216.2.i.d 2 40.e odd 2 1
1216.2.i.d 2 760.bm odd 6 1
1216.2.i.h 2 40.f even 2 1
1216.2.i.h 2 760.z even 6 1
2736.2.s.m 2 60.h even 2 1
2736.2.s.m 2 1140.bn even 6 1
5776.2.a.g 1 380.p odd 6 1
5776.2.a.n 1 380.s even 6 1
6498.2.a.e 1 285.q even 6 1
6498.2.a.s 1 285.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
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