↑

Properties

Label 950.2.r.a
Level $950$
Weight $2$
Character orbit 950.r
Analytic conductor $7.586$
Analytic rank $0$
Dimension $200$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(11,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([24, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.r (of order \(15\), degree \(8\), 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: \(200\)
Relative dimension: \(25\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9} + 4 q^{10} - 2 q^{11} + 8 q^{13} + 7 q^{14} + 10 q^{15} + 25 q^{16} - 13 q^{17} - 216 q^{18} - 36 q^{19} - 2 q^{20} + 4 q^{22} - 13 q^{23} + 33 q^{25} - 64 q^{26} - 2 q^{28} + 12 q^{29} - 20 q^{30} + 12 q^{31} + 100 q^{32} - 12 q^{33} + 8 q^{34} - 22 q^{35} + 37 q^{36} + 8 q^{37} + 18 q^{38} + 28 q^{39} - q^{40} + 2 q^{41} + 70 q^{43} - 4 q^{44} + 60 q^{45} + 24 q^{46} - 22 q^{47} + 156 q^{49} + 26 q^{50} - 42 q^{51} + 8 q^{52} - 18 q^{53} - 10 q^{55} - 4 q^{56} + 74 q^{57} + 24 q^{58} + 50 q^{59} + 12 q^{61} - 4 q^{62} - 45 q^{63} - 50 q^{64} + 120 q^{65} + 2 q^{66} - 18 q^{67} - 84 q^{68} + 4 q^{69} - 13 q^{70} + 19 q^{71} - 17 q^{72} - 24 q^{73} + 4 q^{74} - 136 q^{75} - 6 q^{76} - 12 q^{77} - 26 q^{78} - 8 q^{79} + q^{80} - 63 q^{81} + 28 q^{82} + 16 q^{83} - 20 q^{84} - 5 q^{85} + 20 q^{86} - 64 q^{87} + 2 q^{88} + 16 q^{89} - 65 q^{90} + 72 q^{91} + 12 q^{92} - 8 q^{93} - 44 q^{94} - 73 q^{95} - 11 q^{97} - 27 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 0.104528 + 0.994522i −2.23670 − 2.48410i −0.978148 + 0.207912i 2.20697 − 0.359579i 2.23670 − 2.48410i −4.60253 −0.309017 − 0.951057i −0.854373 + 8.12881i 0.588300 + 2.15729i
11.2 0.104528 + 0.994522i −2.02768 − 2.25196i −0.978148 + 0.207912i −0.209287 − 2.22625i 2.02768 − 2.25196i 1.71696 −0.309017 − 0.951057i −0.646281 + 6.14895i 2.19218 − 0.440848i
11.3 0.104528 + 0.994522i −1.82987 − 2.03228i −0.978148 + 0.207912i −1.78813 + 1.34260i 1.82987 − 2.03228i 1.23976 −0.309017 − 0.951057i −0.468141 + 4.45407i −1.52216 − 1.63800i
11.4 0.104528 + 0.994522i −1.76134 − 1.95617i −0.978148 + 0.207912i −1.69383 + 1.45977i 1.76134 − 1.95617i −4.49296 −0.309017 − 0.951057i −0.410684 + 3.90739i −1.62883 − 1.53197i
11.5 0.104528 + 0.994522i −1.74362 − 1.93649i −0.978148 + 0.207912i −1.31558 − 1.80811i 1.74362 − 1.93649i −0.103099 −0.309017 − 0.951057i −0.396182 + 3.76942i 1.66069 − 1.49737i
11.6 0.104528 + 0.994522i −1.36974 − 1.52125i −0.978148 + 0.207912i 2.23379 − 0.100934i 1.36974 − 1.52125i 5.15541 −0.309017 − 0.951057i −0.124432 + 1.18389i 0.333876 + 2.21100i
11.7 0.104528 + 0.994522i −1.32901 − 1.47601i −0.978148 + 0.207912i 1.09596 + 1.94907i 1.32901 − 1.47601i −0.0211674 −0.309017 − 0.951057i −0.0987671 + 0.939706i −1.82383 + 1.29369i
11.8 0.104528 + 0.994522i −0.955473 − 1.06116i −0.978148 + 0.207912i 0.663784 − 2.13527i 0.955473 − 1.06116i −1.22437 −0.309017 − 0.951057i 0.100453 − 0.955746i 2.19296 + 0.436951i
11.9 0.104528 + 0.994522i −0.588897 − 0.654036i −0.978148 + 0.207912i −2.19070 − 0.448160i 0.588897 − 0.654036i −3.02472 −0.309017 − 0.951057i 0.232622 − 2.21325i 0.216715 − 2.22554i
11.10 0.104528 + 0.994522i −0.286801 − 0.318525i −0.978148 + 0.207912i −1.03010 + 1.98466i 0.286801 − 0.318525i 3.29125 −0.309017 − 0.951057i 0.294382 − 2.80086i −2.08147 − 0.817006i
11.11 0.104528 + 0.994522i −0.278343 − 0.309131i −0.978148 + 0.207912i −1.78815 − 1.34258i 0.278343 − 0.309131i 2.38873 −0.309017 − 0.951057i 0.295498 − 2.81148i 1.14831 − 1.91869i
11.12 0.104528 + 0.994522i −0.261407 − 0.290322i −0.978148 + 0.207912i 1.28550 + 1.82962i 0.261407 − 0.290322i −0.691099 −0.309017 − 0.951057i 0.297632 − 2.83178i −1.68522 + 1.46970i
11.13 0.104528 + 0.994522i −0.175744 − 0.195183i −0.978148 + 0.207912i 1.83434 − 1.27875i 0.175744 − 0.195183i −2.84995 −0.309017 − 0.951057i 0.306375 − 2.91496i 1.46349 + 1.69062i
11.14 0.104528 + 0.994522i −0.0896612 − 0.0995788i −0.978148 + 0.207912i −1.17911 + 1.89992i 0.0896612 − 0.0995788i 2.27794 −0.309017 − 0.951057i 0.311709 − 2.96571i −2.01277 − 0.974050i
11.15 0.104528 + 0.994522i 0.388315 + 0.431268i −0.978148 + 0.207912i 1.69879 + 1.45400i −0.388315 + 0.431268i −3.61865 −0.309017 − 0.951057i 0.278382 − 2.64863i −1.26846 + 1.84147i
11.16 0.104528 + 0.994522i 0.831705 + 0.923702i −0.978148 + 0.207912i 2.14903 + 0.617800i −0.831705 + 0.923702i 0.754763 −0.309017 − 0.951057i 0.152093 − 1.44707i −0.389781 + 2.20183i
11.17 0.104528 + 0.994522i 0.856524 + 0.951266i −0.978148 + 0.207912i −1.58933 − 1.57290i −0.856524 + 0.951266i 1.37782 −0.309017 − 0.951057i 0.142311 − 1.35400i 1.39816 − 1.74504i
11.18 0.104528 + 0.994522i 0.895592 + 0.994656i −0.978148 + 0.207912i −2.22420 + 0.230050i −0.895592 + 0.994656i −2.57007 −0.309017 − 0.951057i 0.126330 − 1.20195i −0.461283 − 2.18797i
11.19 0.104528 + 0.994522i 1.14496 + 1.27160i −0.978148 + 0.207912i 1.79074 − 1.33913i −1.14496 + 1.27160i 1.05033 −0.309017 − 0.951057i 0.00753716 − 0.0717113i 1.51897 + 1.64095i
11.20 0.104528 + 0.994522i 1.32430 + 1.47079i −0.978148 + 0.207912i 0.202278 − 2.22690i −1.32430 + 1.47079i −4.37854 −0.309017 − 0.951057i −0.0958531 + 0.911982i 2.23584 − 0.0316046i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.25
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner
25.d even 5 1 inner
475.r even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.r.a ✓ 200
19.c even 3 1 inner 950.2.r.a ✓ 200
25.d even 5 1 inner 950.2.r.a ✓ 200
475.r even 15 1 inner 950.2.r.a ✓ 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.r.a ✓ 200 1.a even 1 1 trivial
950.2.r.a ✓ 200 19.c even 3 1 inner
950.2.r.a ✓ 200 25.d even 5 1 inner
950.2.r.a ✓ 200 475.r even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{200} - 56 T_{3}^{198} + 1433 T_{3}^{196} + 30 T_{3}^{195} - 20430 T_{3}^{194} + \cdots + 13\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\). Copy content Toggle raw display