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Properties

Label 950.2.j.e
Level $950$
Weight $2$
Character orbit 950.j
Analytic conductor $7.586$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(49,950)]
 
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([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.j (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 4.00000i − 1.00000i −1.00000 − 1.73205i 0
49.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i − 4.00000i 1.00000i −1.00000 − 1.73205i 0
349.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i 0 0.500000 − 0.866025i − 4.00000i 1.00000i −1.00000 + 1.73205i 0
349.2 0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i 0 0.500000 − 0.866025i 4.00000i − 1.00000i −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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.j.e 4
5.b even 2 1 inner 950.2.j.e 4
5.c odd 4 1 38.2.c.a ✓ 2
5.c odd 4 1 950.2.e.d 2
15.e even 4 1 342.2.g.b 2
19.c even 3 1 inner 950.2.j.e 4
20.e even 4 1 304.2.i.c 2
40.i odd 4 1 1216.2.i.h 2
40.k even 4 1 1216.2.i.d 2
60.l odd 4 1 2736.2.s.m 2
95.g even 4 1 722.2.c.b 2
95.i even 6 1 inner 950.2.j.e 4
95.l even 12 1 722.2.a.d 1
95.l even 12 1 722.2.c.b 2
95.m odd 12 1 38.2.c.a ✓ 2
95.m odd 12 1 722.2.a.c 1
95.m odd 12 1 950.2.e.d 2
95.q odd 36 6 722.2.e.j 6
95.r even 36 6 722.2.e.i 6
285.v even 12 1 342.2.g.b 2
285.v even 12 1 6498.2.a.s 1
285.w odd 12 1 6498.2.a.e 1
380.v even 12 1 304.2.i.c 2
380.v even 12 1 5776.2.a.g 1
380.w odd 12 1 5776.2.a.n 1
760.br odd 12 1 1216.2.i.h 2
760.bw even 12 1 1216.2.i.d 2
1140.bu odd 12 1 2736.2.s.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a ✓ 2 5.c odd 4 1
38.2.c.a ✓ 2 95.m odd 12 1
304.2.i.c 2 20.e even 4 1
304.2.i.c 2 380.v even 12 1
342.2.g.b 2 15.e even 4 1
342.2.g.b 2 285.v even 12 1
722.2.a.c 1 95.m odd 12 1
722.2.a.d 1 95.l even 12 1
722.2.c.b 2 95.g even 4 1
722.2.c.b 2 95.l even 12 1
722.2.e.i 6 95.r even 36 6
722.2.e.j 6 95.q odd 36 6
950.2.e.d 2 5.c odd 4 1
950.2.e.d 2 95.m odd 12 1
950.2.j.e 4 1.a even 1 1 trivial
950.2.j.e 4 5.b even 2 1 inner
950.2.j.e 4 19.c even 3 1 inner
950.2.j.e 4 95.i even 6 1 inner
1216.2.i.d 2 40.k even 4 1
1216.2.i.d 2 760.bw even 12 1
1216.2.i.h 2 40.i odd 4 1
1216.2.i.h 2 760.br odd 12 1
2736.2.s.m 2 60.l odd 4 1
2736.2.s.m 2 1140.bu odd 12 1
5776.2.a.g 1 380.v even 12 1
5776.2.a.n 1 380.w odd 12 1
6498.2.a.e 1 285.w odd 12 1
6498.2.a.s 1 285.v even 12 1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 289 T^{2} + 83521 \) Copy content Toggle raw display
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