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Properties

Label 975.2.bt.m
Level $975$
Weight $2$
Character orbit 975.bt
Analytic conductor $7.785$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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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: no (minimal twist has level 195)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{3} - 4 q^{6} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{3} - 4 q^{6} + 4 q^{7} + 28 q^{12} + 24 q^{13} + 16 q^{16} - 16 q^{18} - 8 q^{21} + 12 q^{22} + 32 q^{27} + 44 q^{28} + 16 q^{31} + 46 q^{33} - 36 q^{36} - 20 q^{37} - 8 q^{42} + 16 q^{43} - 64 q^{48} + 48 q^{51} + 76 q^{52} - 68 q^{57} + 20 q^{58} + 18 q^{63} + 32 q^{66} + 52 q^{67} - 6 q^{72} - 64 q^{73} - 104 q^{76} - 144 q^{78} + 52 q^{82} - 14 q^{87} - 84 q^{88} + 64 q^{91} - 32 q^{93} - 240 q^{96} - 48 q^{97}+O(q^{100}) \) 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 −0.683455 + 2.55069i 1.50518 + 0.856995i −4.30685 − 2.48656i 0 −3.21465 + 3.25352i −0.572219 − 2.13555i 5.55152 − 5.55152i 1.53112 + 2.57986i 0
68.2 −0.617961 + 2.30626i −0.192722 + 1.72130i −3.20491 − 1.85036i 0 −3.85066 − 1.50816i 0.570618 + 2.12957i 2.87132 − 2.87132i −2.92572 − 0.663463i 0
68.3 −0.604143 + 2.25469i −1.42796 − 0.980278i −2.98660 − 1.72431i 0 3.07291 − 2.62737i −0.0109257 − 0.0407754i 2.39103 − 2.39103i 1.07811 + 2.79959i 0
68.4 −0.576369 + 2.15104i −1.57479 + 0.721142i −2.56272 − 1.47959i 0 −0.643546 − 3.80307i −0.598395 − 2.23324i 1.51038 − 1.51038i 1.95991 − 2.27129i 0
68.5 −0.472349 + 1.76283i 1.28238 − 1.16426i −1.15241 − 0.665344i 0 1.44666 + 2.81056i 0.0579742 + 0.216363i −0.863735 + 0.863735i 0.289000 − 2.98605i 0
68.6 −0.402483 + 1.50209i 1.52467 − 0.821812i −0.362218 − 0.209127i 0 0.620778 + 2.62095i 0.727252 + 2.71414i −1.73929 + 1.73929i 1.64925 − 2.50599i 0
68.7 −0.395659 + 1.47662i −0.249228 − 1.71403i −0.291809 − 0.168476i 0 2.62957 + 0.310154i −1.22324 − 4.56520i −1.79769 + 1.79769i −2.87577 + 0.854368i 0
68.8 −0.283199 + 1.05691i −0.136195 + 1.72669i 0.695185 + 0.401365i 0 −1.78639 − 0.632943i −0.767714 − 2.86515i −2.16851 + 2.16851i −2.96290 − 0.470332i 0
68.9 −0.200533 + 0.748401i −1.65787 − 0.501470i 1.21216 + 0.699841i 0 0.707759 − 1.14019i 0.994250 + 3.71059i −1.86258 + 1.86258i 2.49706 + 1.66274i 0
68.10 −0.181708 + 0.678144i 0.369290 + 1.69222i 1.30519 + 0.753551i 0 −1.21468 − 0.0570595i 0.708309 + 2.64344i −1.74105 + 1.74105i −2.72725 + 1.24984i 0
68.11 −0.151334 + 0.564785i 1.44147 + 0.960299i 1.43597 + 0.829058i 0 −0.760505 + 0.668794i −0.448316 − 1.67314i −1.51245 + 1.51245i 1.15565 + 2.76848i 0
68.12 −0.115776 + 0.432084i −0.493867 − 1.66015i 1.55876 + 0.899950i 0 0.774502 − 0.0211855i 0.196382 + 0.732909i −1.20194 + 1.20194i −2.51219 + 1.63978i 0
68.13 0.115776 − 0.432084i 1.25778 − 1.19080i 1.55876 + 0.899950i 0 −0.368904 − 0.681331i 0.196382 + 0.732909i 1.20194 − 1.20194i 0.163999 − 2.99551i 0
68.14 0.151334 − 0.564785i −1.72850 + 0.110909i 1.43597 + 0.829058i 0 −0.198940 + 0.993014i −0.448316 − 1.67314i 1.51245 − 1.51245i 2.97540 − 0.383413i 0
68.15 0.181708 − 0.678144i −1.16593 + 1.28086i 1.30519 + 0.753551i 0 0.656753 + 1.02341i 0.708309 + 2.64344i 1.74105 − 1.74105i −0.281230 − 2.98679i 0
68.16 0.200533 − 0.748401i 1.68649 + 0.394648i 1.21216 + 0.699841i 0 0.633553 − 1.18303i 0.994250 + 3.71059i 1.86258 − 1.86258i 2.68851 + 1.33114i 0
68.17 0.283199 − 1.05691i −0.745396 + 1.56345i 0.695185 + 0.401365i 0 1.44134 + 1.23059i −0.767714 − 2.86515i 2.16851 − 2.16851i −1.88877 − 2.33078i 0
68.18 0.395659 − 1.47662i 1.07285 − 1.35978i −0.291809 − 0.168476i 0 −1.58339 − 2.12220i −1.22324 − 4.56520i 1.79769 − 1.79769i −0.697981 − 2.91767i 0
68.19 0.402483 − 1.50209i −0.909499 − 1.47405i −0.362218 − 0.209127i 0 −2.58020 + 0.772867i 0.727252 + 2.71414i 1.73929 − 1.73929i −1.34562 + 2.68129i 0
68.20 0.472349 − 1.76283i −0.528445 − 1.64947i −1.15241 − 0.665344i 0 −3.15735 + 0.152433i 0.0579742 + 0.216363i 0.863735 − 0.863735i −2.44149 + 1.74331i 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.c odd 4 1 inner
13.c even 3 1 inner
15.e even 4 1 inner
39.i odd 6 1 inner
65.q odd 12 1 inner
195.bl even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bt.m 96
3.b odd 2 1 inner 975.2.bt.m 96
5.b even 2 1 195.2.bl.a ✓ 96
5.c odd 4 1 195.2.bl.a ✓ 96
5.c odd 4 1 inner 975.2.bt.m 96
13.c even 3 1 inner 975.2.bt.m 96
15.d odd 2 1 195.2.bl.a ✓ 96
15.e even 4 1 195.2.bl.a ✓ 96
15.e even 4 1 inner 975.2.bt.m 96
39.i odd 6 1 inner 975.2.bt.m 96
65.n even 6 1 195.2.bl.a ✓ 96
65.q odd 12 1 195.2.bl.a ✓ 96
65.q odd 12 1 inner 975.2.bt.m 96
195.x odd 6 1 195.2.bl.a ✓ 96
195.bl even 12 1 195.2.bl.a ✓ 96
195.bl even 12 1 inner 975.2.bt.m 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bl.a ✓ 96 5.b even 2 1
195.2.bl.a ✓ 96 5.c odd 4 1
195.2.bl.a ✓ 96 15.d odd 2 1
195.2.bl.a ✓ 96 15.e even 4 1
195.2.bl.a ✓ 96 65.n even 6 1
195.2.bl.a ✓ 96 65.q odd 12 1
195.2.bl.a ✓ 96 195.x odd 6 1
195.2.bl.a ✓ 96 195.bl even 12 1
975.2.bt.m 96 1.a even 1 1 trivial
975.2.bt.m 96 3.b odd 2 1 inner
975.2.bt.m 96 5.c odd 4 1 inner
975.2.bt.m 96 13.c even 3 1 inner
975.2.bt.m 96 15.e even 4 1 inner
975.2.bt.m 96 39.i odd 6 1 inner
975.2.bt.m 96 65.q odd 12 1 inner
975.2.bt.m 96 195.bl even 12 1 inner

Hecke kernels

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

\( T_{2}^{96} - 160 T_{2}^{92} + 15348 T_{2}^{88} - 962308 T_{2}^{84} + 44647360 T_{2}^{80} + \cdots + 57648010000 \) Copy content Toggle raw display
\( T_{7}^{48} - 2 T_{7}^{47} + 2 T_{7}^{46} - 48 T_{7}^{45} - 405 T_{7}^{44} + 1330 T_{7}^{43} + \cdots + 11316496 \) Copy content Toggle raw display
\( T_{59}^{48} + 730 T_{59}^{46} + 301572 T_{59}^{44} + 85362180 T_{59}^{42} + 18299038235 T_{59}^{40} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display