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Properties

Label 950.2.u.h
Level $950$
Weight $2$
Character orbit 950.u
Analytic conductor $7.586$
Analytic rank $0$
Dimension $48$
CM no
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(99,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.u (of order \(18\), degree \(6\), 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: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{11} + 30 q^{14} + 30 q^{19} - 36 q^{21} - 18 q^{26} + 24 q^{29} + 18 q^{31} + 18 q^{34} - 132 q^{39} + 36 q^{41} - 6 q^{46} + 54 q^{49} - 6 q^{51} - 54 q^{54} - 12 q^{56} - 72 q^{59} + 24 q^{61} + 24 q^{64} + 96 q^{66} - 42 q^{69} - 78 q^{71} - 36 q^{74} + 12 q^{76} + 84 q^{79} - 72 q^{81} - 18 q^{84} - 78 q^{86} + 72 q^{89} + 24 q^{91} - 24 q^{94} + 12 q^{96} - 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −0.342020 + 0.939693i −2.84564 + 0.501764i −0.766044 − 0.642788i 0 0.501764 − 2.84564i −3.49580 − 2.01830i 0.866025 − 0.500000i 5.02684 − 1.82962i 0
99.2 −0.342020 + 0.939693i −1.28901 + 0.227288i −0.766044 − 0.642788i 0 0.227288 − 1.28901i 1.93191 + 1.11539i 0.866025 − 0.500000i −1.20918 + 0.440106i 0
99.3 −0.342020 + 0.939693i 0.634880 − 0.111946i −0.766044 − 0.642788i 0 −0.111946 + 0.634880i 0.369892 + 0.213557i 0.866025 − 0.500000i −2.42854 + 0.883915i 0
99.4 −0.342020 + 0.939693i 2.51497 − 0.443457i −0.766044 − 0.642788i 0 −0.443457 + 2.51497i −3.95328 − 2.28243i 0.866025 − 0.500000i 3.30934 − 1.20450i 0
99.5 0.342020 − 0.939693i −2.51497 + 0.443457i −0.766044 − 0.642788i 0 −0.443457 + 2.51497i 3.95328 + 2.28243i −0.866025 + 0.500000i 3.30934 − 1.20450i 0
99.6 0.342020 − 0.939693i −0.634880 + 0.111946i −0.766044 − 0.642788i 0 −0.111946 + 0.634880i −0.369892 − 0.213557i −0.866025 + 0.500000i −2.42854 + 0.883915i 0
99.7 0.342020 − 0.939693i 1.28901 − 0.227288i −0.766044 − 0.642788i 0 0.227288 − 1.28901i −1.93191 − 1.11539i −0.866025 + 0.500000i −1.20918 + 0.440106i 0
99.8 0.342020 − 0.939693i 2.84564 − 0.501764i −0.766044 − 0.642788i 0 0.501764 − 2.84564i 3.49580 + 2.01830i −0.866025 + 0.500000i 5.02684 − 1.82962i 0
149.1 −0.642788 + 0.766044i −0.934611 − 2.56782i −0.173648 − 0.984808i 0 2.56782 + 0.934611i 1.85059 − 1.06844i 0.866025 + 0.500000i −3.42208 + 2.87147i 0
149.2 −0.642788 + 0.766044i −0.0291678 − 0.0801377i −0.173648 − 0.984808i 0 0.0801377 + 0.0291678i 1.59412 − 0.920368i 0.866025 + 0.500000i 2.29256 − 1.92369i 0
149.3 −0.642788 + 0.766044i 0.291155 + 0.799943i −0.173648 − 0.984808i 0 −0.799943 − 0.291155i −4.37330 + 2.52492i 0.866025 + 0.500000i 1.74300 − 1.46255i 0
149.4 −0.642788 + 0.766044i 1.01464 + 2.78771i −0.173648 − 0.984808i 0 −2.78771 − 1.01464i 0.787850 − 0.454865i 0.866025 + 0.500000i −4.44370 + 3.72870i 0
149.5 0.642788 − 0.766044i −1.01464 − 2.78771i −0.173648 − 0.984808i 0 −2.78771 − 1.01464i −0.787850 + 0.454865i −0.866025 − 0.500000i −4.44370 + 3.72870i 0
149.6 0.642788 − 0.766044i −0.291155 − 0.799943i −0.173648 − 0.984808i 0 −0.799943 − 0.291155i 4.37330 − 2.52492i −0.866025 − 0.500000i 1.74300 − 1.46255i 0
149.7 0.642788 − 0.766044i 0.0291678 + 0.0801377i −0.173648 − 0.984808i 0 0.0801377 + 0.0291678i −1.59412 + 0.920368i −0.866025 − 0.500000i 2.29256 − 1.92369i 0
149.8 0.642788 − 0.766044i 0.934611 + 2.56782i −0.173648 − 0.984808i 0 2.56782 + 0.934611i −1.85059 + 1.06844i −0.866025 − 0.500000i −3.42208 + 2.87147i 0
199.1 −0.984808 + 0.173648i −2.09100 − 2.49196i 0.939693 − 0.342020i 0 2.49196 + 2.09100i 0.964479 + 0.556842i −0.866025 + 0.500000i −1.31663 + 7.46696i 0
199.2 −0.984808 + 0.173648i −0.536848 − 0.639791i 0.939693 − 0.342020i 0 0.639791 + 0.536848i −3.62467 − 2.09271i −0.866025 + 0.500000i 0.399818 − 2.26748i 0
199.3 −0.984808 + 0.173648i 0.549102 + 0.654394i 0.939693 − 0.342020i 0 −0.654394 − 0.549102i 2.40903 + 1.39086i −0.866025 + 0.500000i 0.394226 − 2.23577i 0
199.4 −0.984808 + 0.173648i 1.43596 + 1.71131i 0.939693 − 0.342020i 0 −1.71131 − 1.43596i −2.43877 − 1.40802i −0.866025 + 0.500000i −0.345659 + 1.96033i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.u.h 48
5.b even 2 1 inner 950.2.u.h 48
5.c odd 4 1 950.2.l.j ✓ 24
5.c odd 4 1 950.2.l.k yes 24
19.e even 9 1 inner 950.2.u.h 48
95.p even 18 1 inner 950.2.u.h 48
95.q odd 36 1 950.2.l.j ✓ 24
95.q odd 36 1 950.2.l.k yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.l.j ✓ 24 5.c odd 4 1
950.2.l.j ✓ 24 95.q odd 36 1
950.2.l.k yes 24 5.c odd 4 1
950.2.l.k yes 24 95.q odd 36 1
950.2.u.h 48 1.a even 1 1 trivial
950.2.u.h 48 5.b even 2 1 inner
950.2.u.h 48 19.e even 9 1 inner
950.2.u.h 48 95.p even 18 1 inner

Hecke kernels

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

\( T_{3}^{48} + 18 T_{3}^{44} - 1091 T_{3}^{42} - 1179 T_{3}^{40} - 2961 T_{3}^{38} + 984224 T_{3}^{36} + \cdots + 130321 \) Copy content Toggle raw display
\( T_{7}^{48} - 111 T_{7}^{46} + 7080 T_{7}^{44} - 305633 T_{7}^{42} + 9903747 T_{7}^{40} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display