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Properties

Label 975.2.bu.h
Level $975$
Weight $2$
Character orbit 975.bu
Analytic conductor $7.785$
Analytic rank $0$
Dimension $32$
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] = [975,2,Mod(7,975)]
 
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([0, 3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bu (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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} - 4 q^{6} - 16 q^{11} - 48 q^{16} + 24 q^{19} - 8 q^{21} + 16 q^{24} + 24 q^{26} - 12 q^{29} - 8 q^{31} - 56 q^{34} - 16 q^{39} - 12 q^{41} + 88 q^{44} - 32 q^{46} + 32 q^{49} - 4 q^{54} - 96 q^{56} - 92 q^{59} - 56 q^{61} - 128 q^{64} - 12 q^{71} + 120 q^{74} - 200 q^{76} + 16 q^{81} + 48 q^{84} + 96 q^{86} - 68 q^{89} + 216 q^{94} + 136 q^{96} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.40866 − 1.39064i 0.258819 + 0.965926i 2.86775 + 4.96709i 0 0.719847 − 2.68651i −1.64946 − 2.85694i − 10.3895i −0.866025 + 0.500000i 0
7.2 −1.95100 − 1.12641i −0.258819 − 0.965926i 1.53760 + 2.66321i 0 −0.583073 + 2.17606i −0.867592 − 1.50271i − 2.42225i −0.866025 + 0.500000i 0
7.3 −0.930243 − 0.537076i 0.258819 + 0.965926i −0.423099 − 0.732829i 0 0.278011 − 1.03755i −0.795895 − 1.37853i 3.05725i −0.866025 + 0.500000i 0
7.4 −0.163153 − 0.0941966i −0.258819 − 0.965926i −0.982254 − 1.70131i 0 −0.0487598 + 0.181974i 0.164622 + 0.285134i 0.746886i −0.866025 + 0.500000i 0
7.5 0.163153 + 0.0941966i 0.258819 + 0.965926i −0.982254 − 1.70131i 0 −0.0487598 + 0.181974i −0.164622 − 0.285134i − 0.746886i −0.866025 + 0.500000i 0
7.6 0.930243 + 0.537076i −0.258819 − 0.965926i −0.423099 − 0.732829i 0 0.278011 − 1.03755i 0.795895 + 1.37853i − 3.05725i −0.866025 + 0.500000i 0
7.7 1.95100 + 1.12641i 0.258819 + 0.965926i 1.53760 + 2.66321i 0 −0.583073 + 2.17606i 0.867592 + 1.50271i 2.42225i −0.866025 + 0.500000i 0
7.8 2.40866 + 1.39064i −0.258819 − 0.965926i 2.86775 + 4.96709i 0 0.719847 − 2.68651i 1.64946 + 2.85694i 10.3895i −0.866025 + 0.500000i 0
232.1 −2.26631 + 1.30846i 0.965926 + 0.258819i 2.42412 − 4.19869i 0 −2.52774 + 0.677307i 0.342531 − 0.593281i 7.45358i 0.866025 + 0.500000i 0
232.2 −1.85389 + 1.07034i −0.965926 − 0.258819i 1.29127 − 2.23654i 0 2.06774 − 0.554050i 2.03627 − 3.52692i 1.24701i 0.866025 + 0.500000i 0
232.3 −1.29975 + 0.750409i 0.965926 + 0.258819i 0.126227 − 0.218632i 0 −1.44968 + 0.388440i −0.215517 + 0.373286i − 2.62275i 0.866025 + 0.500000i 0
232.4 −0.487427 + 0.281416i −0.965926 − 0.258819i −0.841610 + 1.45771i 0 0.543655 − 0.145672i 1.24734 − 2.16046i − 2.07304i 0.866025 + 0.500000i 0
232.5 0.487427 − 0.281416i 0.965926 + 0.258819i −0.841610 + 1.45771i 0 0.543655 − 0.145672i −1.24734 + 2.16046i 2.07304i 0.866025 + 0.500000i 0
232.6 1.29975 − 0.750409i −0.965926 − 0.258819i 0.126227 − 0.218632i 0 −1.44968 + 0.388440i 0.215517 − 0.373286i 2.62275i 0.866025 + 0.500000i 0
232.7 1.85389 − 1.07034i 0.965926 + 0.258819i 1.29127 − 2.23654i 0 2.06774 − 0.554050i −2.03627 + 3.52692i − 1.24701i 0.866025 + 0.500000i 0
232.8 2.26631 − 1.30846i −0.965926 − 0.258819i 2.42412 − 4.19869i 0 −2.52774 + 0.677307i −0.342531 + 0.593281i − 7.45358i 0.866025 + 0.500000i 0
418.1 −2.40866 + 1.39064i 0.258819 − 0.965926i 2.86775 − 4.96709i 0 0.719847 + 2.68651i −1.64946 + 2.85694i 10.3895i −0.866025 − 0.500000i 0
418.2 −1.95100 + 1.12641i −0.258819 + 0.965926i 1.53760 − 2.66321i 0 −0.583073 − 2.17606i −0.867592 + 1.50271i 2.42225i −0.866025 − 0.500000i 0
418.3 −0.930243 + 0.537076i 0.258819 − 0.965926i −0.423099 + 0.732829i 0 0.278011 + 1.03755i −0.795895 + 1.37853i − 3.05725i −0.866025 − 0.500000i 0
418.4 −0.163153 + 0.0941966i −0.258819 + 0.965926i −0.982254 + 1.70131i 0 −0.0487598 − 0.181974i 0.164622 − 0.285134i − 0.746886i −0.866025 − 0.500000i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.o even 12 1 inner
65.t 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.bu.h yes 32
5.b even 2 1 inner 975.2.bu.h yes 32
5.c odd 4 2 975.2.bl.h ✓ 32
13.f odd 12 1 975.2.bl.h ✓ 32
65.o even 12 1 inner 975.2.bu.h yes 32
65.s odd 12 1 975.2.bl.h ✓ 32
65.t even 12 1 inner 975.2.bu.h yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bl.h ✓ 32 5.c odd 4 2
975.2.bl.h ✓ 32 13.f odd 12 1
975.2.bl.h ✓ 32 65.s odd 12 1
975.2.bu.h yes 32 1.a even 1 1 trivial
975.2.bu.h yes 32 5.b even 2 1 inner
975.2.bu.h yes 32 65.o even 12 1 inner
975.2.bu.h yes 32 65.t 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 \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{32} - 28 T_{2}^{30} + 472 T_{2}^{28} - 5216 T_{2}^{26} + 42784 T_{2}^{24} - 261520 T_{2}^{22} + \cdots + 1296 \) Copy content Toggle raw display
\( T_{7}^{32} + 40 T_{7}^{30} + 1024 T_{7}^{28} + 15536 T_{7}^{26} + 170158 T_{7}^{24} + 1254016 T_{7}^{22} + \cdots + 6561 \) Copy content Toggle raw display