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Properties

Label 975.2.bt.l
Level $975$
Weight $2$
Character orbit 975.bt
Analytic conductor $7.785$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(68,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bt (of order \(12\), degree \(4\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{6} + 128 q^{16} - 336 q^{31} - 4 q^{36} + 168 q^{46} - 160 q^{51} + 64 q^{61} - 280 q^{66} + 16 q^{76} - 20 q^{81} + 152 q^{91} + 488 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −0.720951 + 2.69062i −1.64930 + 0.528962i −4.98764 − 2.87962i 0 −0.234174 − 4.81901i 0.853080 + 3.18374i 7.40446 − 7.40446i 2.44040 − 1.74484i 0
68.2 −0.720951 + 2.69062i −1.16386 − 1.28275i −4.98764 − 2.87962i 0 4.29047 − 2.20670i −0.853080 − 3.18374i 7.40446 − 7.40446i −0.290875 + 2.98587i 0
68.3 −0.649446 + 2.42377i 0.840051 + 1.51470i −3.72081 − 2.14821i 0 −4.21685 + 1.05237i 0.535168 + 1.99727i 4.07459 − 4.07459i −1.58863 + 2.54485i 0
68.4 −0.649446 + 2.42377i 1.48486 − 0.891742i −3.72081 − 2.14821i 0 1.19704 + 4.17808i −0.535168 − 1.99727i 4.07459 − 4.07459i 1.40959 − 2.64822i 0
68.5 −0.518588 + 1.93540i −0.840505 + 1.51445i −1.74477 − 1.00734i 0 −2.49518 − 2.41208i −0.951092 − 3.54952i 0.0208099 − 0.0208099i −1.58710 − 2.54580i 0
68.6 −0.518588 + 1.93540i 0.0293248 − 1.73180i −1.74477 − 1.00734i 0 3.33651 + 0.954846i 0.951092 + 3.54952i 0.0208099 − 0.0208099i −2.99828 − 0.101569i 0
68.7 −0.404333 + 1.50899i −0.722483 + 1.57417i −0.381523 − 0.220272i 0 −2.08329 − 1.72671i −0.257836 − 0.962257i −1.72267 + 1.72267i −1.95604 − 2.27463i 0
68.8 −0.404333 + 1.50899i 0.161398 − 1.72451i −0.381523 − 0.220272i 0 2.53702 + 0.940827i 0.257836 + 0.962257i −1.72267 + 1.72267i −2.94790 − 0.556665i 0
68.9 −0.310559 + 1.15902i 1.58277 + 0.703457i 0.485165 + 0.280110i 0 −1.30687 + 1.61600i 0.435316 + 1.62462i −2.17225 + 2.17225i 2.01030 + 2.22682i 0
68.10 −0.310559 + 1.15902i 1.72244 + 0.182171i 0.485165 + 0.280110i 0 −0.746061 + 1.93978i −0.435316 − 1.62462i −2.17225 + 2.17225i 2.93363 + 0.627559i 0
68.11 −0.0574860 + 0.214541i −1.32933 + 1.11036i 1.68933 + 0.975334i 0 −0.161799 − 0.349025i 0.320723 + 1.19696i −0.620471 + 0.620471i 0.534223 − 2.95205i 0
68.12 −0.0574860 + 0.214541i −0.596054 − 1.62626i 1.68933 + 0.975334i 0 0.383163 − 0.0343906i −0.320723 − 1.19696i −0.620471 + 0.620471i −2.28944 + 1.93868i 0
68.13 0.0574860 − 0.214541i 0.596054 + 1.62626i 1.68933 + 0.975334i 0 0.383163 − 0.0343906i 0.320723 + 1.19696i 0.620471 − 0.620471i −2.28944 + 1.93868i 0
68.14 0.0574860 − 0.214541i 1.32933 − 1.11036i 1.68933 + 0.975334i 0 −0.161799 − 0.349025i −0.320723 − 1.19696i 0.620471 − 0.620471i 0.534223 − 2.95205i 0
68.15 0.310559 − 1.15902i −1.72244 − 0.182171i 0.485165 + 0.280110i 0 −0.746061 + 1.93978i 0.435316 + 1.62462i 2.17225 − 2.17225i 2.93363 + 0.627559i 0
68.16 0.310559 − 1.15902i −1.58277 − 0.703457i 0.485165 + 0.280110i 0 −1.30687 + 1.61600i −0.435316 − 1.62462i 2.17225 − 2.17225i 2.01030 + 2.22682i 0
68.17 0.404333 − 1.50899i −0.161398 + 1.72451i −0.381523 − 0.220272i 0 2.53702 + 0.940827i −0.257836 − 0.962257i 1.72267 − 1.72267i −2.94790 − 0.556665i 0
68.18 0.404333 − 1.50899i 0.722483 − 1.57417i −0.381523 − 0.220272i 0 −2.08329 − 1.72671i 0.257836 + 0.962257i 1.72267 − 1.72267i −1.95604 − 2.27463i 0
68.19 0.518588 − 1.93540i −0.0293248 + 1.73180i −1.74477 − 1.00734i 0 3.33651 + 0.954846i −0.951092 − 3.54952i −0.0208099 + 0.0208099i −2.99828 − 0.101569i 0
68.20 0.518588 − 1.93540i 0.840505 − 1.51445i −1.74477 − 1.00734i 0 −2.49518 − 2.41208i 0.951092 + 3.54952i −0.0208099 + 0.0208099i −1.58710 − 2.54580i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
13.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
39.i odd 6 1 inner
65.n even 6 1 inner
65.q odd 12 2 inner
195.x odd 6 1 inner
195.bl even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bt.l ✓ 96
3.b odd 2 1 inner 975.2.bt.l ✓ 96
5.b even 2 1 inner 975.2.bt.l ✓ 96
5.c odd 4 2 inner 975.2.bt.l ✓ 96
13.c even 3 1 inner 975.2.bt.l ✓ 96
15.d odd 2 1 inner 975.2.bt.l ✓ 96
15.e even 4 2 inner 975.2.bt.l ✓ 96
39.i odd 6 1 inner 975.2.bt.l ✓ 96
65.n even 6 1 inner 975.2.bt.l ✓ 96
65.q odd 12 2 inner 975.2.bt.l ✓ 96
195.x odd 6 1 inner 975.2.bt.l ✓ 96
195.bl even 12 2 inner 975.2.bt.l ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bt.l ✓ 96 1.a even 1 1 trivial
975.2.bt.l ✓ 96 3.b odd 2 1 inner
975.2.bt.l ✓ 96 5.b even 2 1 inner
975.2.bt.l ✓ 96 5.c odd 4 2 inner
975.2.bt.l ✓ 96 13.c even 3 1 inner
975.2.bt.l ✓ 96 15.d odd 2 1 inner
975.2.bt.l ✓ 96 15.e even 4 2 inner
975.2.bt.l ✓ 96 39.i odd 6 1 inner
975.2.bt.l ✓ 96 65.n even 6 1 inner
975.2.bt.l ✓ 96 65.q odd 12 2 inner
975.2.bt.l ✓ 96 195.x odd 6 1 inner
975.2.bt.l ✓ 96 195.bl even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{48} - 124 T_{2}^{44} + 10436 T_{2}^{40} - 468558 T_{2}^{36} + 15117070 T_{2}^{32} + \cdots + 1336336 \) Copy content Toggle raw display
\( T_{7}^{48} - 330 T_{7}^{44} + 78243 T_{7}^{40} - 8698466 T_{7}^{36} + 700570089 T_{7}^{32} + \cdots + 53459728531456 \) Copy content Toggle raw display
\( T_{59}^{24} + 334 T_{59}^{22} + 70149 T_{59}^{20} + 9207448 T_{59}^{18} + 889389400 T_{59}^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display