Properties

Label 950.2.bc.a
Level $950$
Weight $2$
Character orbit 950.bc
Analytic conductor $7.586$
Analytic rank $0$
Dimension $600$
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(61,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([72, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.bc (of order \(45\), degree \(24\), 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: \(600\)
Relative dimension: \(25\) over \(\Q(\zeta_{45})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{45}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 600 q - 42 q^{7} + 75 q^{8} - 9 q^{11} - 24 q^{15} + 18 q^{17} + 624 q^{18} + 36 q^{19} - 6 q^{20} - 18 q^{22} - 6 q^{23} + 120 q^{25} + 48 q^{26} - 18 q^{29} - 90 q^{33} + 18 q^{34} - 51 q^{35} + 12 q^{38} + 36 q^{39} + 36 q^{41} + 108 q^{43} - 18 q^{44} + 24 q^{45} + 36 q^{46} - 66 q^{47} - 282 q^{49} - 9 q^{50} - 48 q^{51} + 18 q^{53} - 18 q^{54} - 87 q^{55} - 36 q^{56} - 18 q^{57} - 36 q^{58} - 30 q^{59} + 6 q^{60} + 174 q^{61} + 12 q^{62} + 18 q^{63} + 75 q^{64} + 54 q^{65} - 18 q^{66} + 18 q^{67} - 42 q^{68} + 24 q^{69} + 69 q^{70} + 48 q^{71} + 18 q^{73} + 12 q^{74} - 120 q^{75} - 72 q^{77} + 12 q^{78} - 36 q^{79} - 60 q^{81} - 24 q^{82} + 3 q^{83} + 75 q^{84} - 18 q^{85} - 12 q^{86} + 6 q^{87} - 9 q^{88} - 90 q^{89} - 36 q^{90} - 30 q^{91} - 42 q^{92} - 102 q^{93} - 150 q^{94} - 102 q^{95} + 42 q^{97} + 108 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 0.961262 0.275637i −3.36425 0.472815i 0.848048 0.529919i −1.20385 1.88434i −3.36425 + 0.472815i 0.735407 1.27376i 0.669131 0.743145i 8.21086 + 2.35443i −1.67661 1.47952i
61.2 0.961262 0.275637i −3.10593 0.436510i 0.848048 0.529919i −0.194755 + 2.22757i −3.10593 + 0.436510i 1.10411 1.91237i 0.669131 0.743145i 6.57247 + 1.88463i 0.426791 + 2.19496i
61.3 0.961262 0.275637i −2.52861 0.355373i 0.848048 0.529919i 2.12329 + 0.701180i −2.52861 + 0.355373i 1.00675 1.74375i 0.669131 0.743145i 3.38381 + 0.970291i 2.23431 + 0.0887607i
61.4 0.961262 0.275637i −2.47903 0.348405i 0.848048 0.529919i 1.80399 1.32121i −2.47903 + 0.348405i −0.861925 + 1.49290i 0.669131 0.743145i 3.14043 + 0.900503i 1.36993 1.76728i
61.5 0.961262 0.275637i −2.26586 0.318446i 0.848048 0.529919i 0.868613 2.06046i −2.26586 + 0.318446i −1.77884 + 3.08104i 0.669131 0.743145i 2.14894 + 0.616198i 0.267024 2.22007i
61.6 0.961262 0.275637i −1.99159 0.279900i 0.848048 0.529919i −2.16545 0.557508i −1.99159 + 0.279900i 0.493923 0.855500i 0.669131 0.743145i 1.00432 + 0.287984i −2.23524 + 0.0609689i
61.7 0.961262 0.275637i −1.97569 0.277665i 0.848048 0.529919i −0.0284411 + 2.23589i −1.97569 + 0.277665i −1.68657 + 2.92123i 0.669131 0.743145i 0.942462 + 0.270247i 0.588955 + 2.15711i
61.8 0.961262 0.275637i −1.54503 0.217140i 0.848048 0.529919i 1.55455 1.60729i −1.54503 + 0.217140i 2.43412 4.21602i 0.669131 0.743145i −0.543813 0.155936i 1.05130 1.97352i
61.9 0.961262 0.275637i −1.18489 0.166526i 0.848048 0.529919i −2.08057 + 0.819293i −1.18489 + 0.166526i −0.801519 + 1.38827i 0.669131 0.743145i −1.50754 0.432281i −1.77414 + 1.36104i
61.10 0.961262 0.275637i −1.08040 0.151840i 0.848048 0.529919i 1.81971 + 1.29948i −1.08040 + 0.151840i −1.07705 + 1.86551i 0.669131 0.743145i −1.73958 0.498818i 2.10741 + 0.747557i
61.11 0.961262 0.275637i −0.363561 0.0510952i 0.848048 0.529919i −1.52899 + 1.63162i −0.363561 + 0.0510952i 1.89241 3.27775i 0.669131 0.743145i −2.75422 0.789760i −1.02003 + 1.98986i
61.12 0.961262 0.275637i 0.0711673 + 0.0100019i 0.848048 0.529919i 2.19008 + 0.451146i 0.0711673 0.0100019i 0.840373 1.45557i 0.669131 0.743145i −2.87882 0.825488i 2.22960 0.170000i
61.13 0.961262 0.275637i 0.181877 + 0.0255611i 0.848048 0.529919i −1.97721 1.04434i 0.181877 0.0255611i −1.51682 + 2.62720i 0.669131 0.743145i −2.85136 0.817614i −2.18847 0.458892i
61.14 0.961262 0.275637i 0.198824 + 0.0279429i 0.848048 0.529919i −0.662616 2.13564i 0.198824 0.0279429i 0.864339 1.49708i 0.669131 0.743145i −2.84503 0.815801i −1.22561 1.87026i
61.15 0.961262 0.275637i 0.287928 + 0.0404657i 0.848048 0.529919i −0.0993678 2.23386i 0.287928 0.0404657i −0.701041 + 1.21424i 0.669131 0.743145i −2.80252 0.803610i −0.711253 2.11993i
61.16 0.961262 0.275637i 0.897293 + 0.126106i 0.848048 0.529919i 0.788589 + 2.09240i 0.897293 0.126106i −2.61399 + 4.52757i 0.669131 0.743145i −2.09455 0.600603i 1.33478 + 1.79398i
61.17 0.961262 0.275637i 1.28122 + 0.180063i 0.848048 0.529919i 0.0954887 + 2.23403i 1.28122 0.180063i 0.778156 1.34781i 0.669131 0.743145i −1.27469 0.365513i 0.707571 + 2.12117i
61.18 0.961262 0.275637i 1.47109 + 0.206748i 0.848048 0.529919i −2.22682 + 0.203157i 1.47109 0.206748i 2.40262 4.16146i 0.669131 0.743145i −0.762436 0.218625i −2.08456 + 0.809082i
61.19 0.961262 0.275637i 2.08096 + 0.292460i 0.848048 0.529919i 1.28691 1.82862i 2.08096 0.292460i 0.875211 1.51591i 0.669131 0.743145i 1.36107 + 0.390281i 0.733018 2.11251i
61.20 0.961262 0.275637i 2.20161 + 0.309416i 0.848048 0.529919i 1.57433 + 1.58792i 2.20161 0.309416i 1.15550 2.00139i 0.669131 0.743145i 1.86756 + 0.535515i 1.95103 + 1.09246i
See next 80 embeddings (of 600 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.25
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner
25.d even 5 1 inner
475.bc even 45 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bc.a 600
19.e even 9 1 inner 950.2.bc.a 600
25.d even 5 1 inner 950.2.bc.a 600
475.bc even 45 1 inner 950.2.bc.a 600
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bc.a 600 1.a even 1 1 trivial
950.2.bc.a 600 19.e even 9 1 inner
950.2.bc.a 600 25.d even 5 1 inner
950.2.bc.a 600 475.bc even 45 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{600} + 15 T_{3}^{596} + 42 T_{3}^{595} - 2705 T_{3}^{594} + 2271 T_{3}^{593} - 5376 T_{3}^{592} + 13890 T_{3}^{591} + 12024 T_{3}^{590} - 333228 T_{3}^{589} + 2090375 T_{3}^{588} - 7105839 T_{3}^{587} + \cdots + 14\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\). Copy content Toggle raw display