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Properties

Label 867.2.i.d
Level $867$
Weight $2$
Character orbit 867.i
Analytic conductor $6.923$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [867,2,Mod(65,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.i (of order \(16\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{4} + 8 q^{9} + 32 q^{10} + 24 q^{12} - 16 q^{13} - 16 q^{15} + 16 q^{18} + 16 q^{19} + 16 q^{21} - 48 q^{22} + 8 q^{24} - 16 q^{25} - 48 q^{27} - 64 q^{28} - 8 q^{30} + 16 q^{31} - 8 q^{36}+ \cdots + 96 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 −0.762502 − 1.84084i 0.275439 + 1.71001i −1.39308 + 1.39308i −0.240611 − 0.360100i 2.93784 − 1.81093i 2.28935 + 1.52969i −0.0550167 − 0.0227887i −2.84827 + 0.942008i −0.479421 + 0.717504i
65.2 −0.295520 − 0.713449i −1.72548 − 0.150682i 0.992537 − 0.992537i 1.44335 + 2.16012i 0.402411 + 1.27557i −3.21323 − 2.14701i −2.42834 − 1.00585i 2.95459 + 0.520000i 1.11460 − 1.66811i
65.3 0.295520 + 0.713449i 1.65180 + 0.521102i 0.992537 − 0.992537i −1.44335 − 2.16012i 0.116362 + 1.33247i −3.21323 − 2.14701i 2.42834 + 1.00585i 2.45691 + 1.72151i 1.11460 − 1.66811i
65.4 0.762502 + 1.84084i −0.908865 + 1.47444i −1.39308 + 1.39308i 0.240611 + 0.360100i −3.40722 − 0.548817i 2.28935 + 1.52969i 0.0550167 + 0.0227887i −1.34793 − 2.68013i −0.479421 + 0.717504i
131.1 −1.88095 + 0.779115i 0.508244 + 1.65580i 1.51674 − 1.51674i −0.405238 + 2.03727i −2.24604 − 2.71850i −1.95490 + 0.388855i −0.112963 + 0.272718i −2.48338 + 1.68311i −0.825033 − 4.14772i
131.2 −0.419490 + 0.173759i −0.0473558 − 1.73140i −1.26843 + 1.26843i −0.596910 + 3.00087i 0.320711 + 0.718078i 2.33759 − 0.464975i 0.659211 − 1.59148i −2.99551 + 0.163984i −0.271029 − 1.36255i
131.3 0.419490 − 0.173759i 1.58149 + 0.706330i −1.26843 + 1.26843i 0.596910 − 3.00087i 0.786149 + 0.0215020i 2.33759 − 0.464975i −0.659211 + 1.59148i 2.00219 + 2.23410i −0.271029 − 1.36255i
131.4 1.88095 − 0.779115i −1.33527 − 1.10321i 1.51674 − 1.51674i 0.405238 − 2.03727i −3.37109 − 1.03475i −1.95490 + 0.388855i 0.112963 − 0.272718i 0.565877 + 2.94615i −0.825033 − 4.14772i
158.1 −1.34436 + 0.556851i 1.44078 + 0.961322i 0.0830021 − 0.0830021i −2.99276 − 0.595296i −2.47224 − 0.490059i −0.420765 − 2.11533i 1.04834 − 2.53091i 1.15172 + 2.77012i 4.35483 − 0.866229i
158.2 −0.774648 + 0.320870i −1.73205 − 0.00133765i −0.917091 + 0.917091i 0.802626 + 0.159652i 1.34216 − 0.554726i 0.0380817 + 0.191449i 1.05790 − 2.55399i 3.00000 + 0.00463376i −0.672980 + 0.133864i
158.3 0.774648 − 0.320870i 0.661591 − 1.60072i −0.917091 + 0.917091i −0.802626 − 0.159652i −0.00112159 − 1.45228i 0.0380817 + 0.191449i −1.05790 + 2.55399i −2.12459 − 2.11804i −0.672980 + 0.133864i
158.4 1.34436 − 0.556851i 0.336782 + 1.69899i 0.0830021 − 0.0830021i 2.99276 + 0.595296i 1.39884 + 2.09652i −0.420765 − 2.11533i −1.04834 + 2.53091i −2.77316 + 1.14438i 4.35483 − 0.866229i
224.1 −0.978223 − 2.36164i −1.73097 − 0.0611151i −3.20621 + 3.20621i −1.92973 + 1.28940i 1.54895 + 4.14772i −0.0590250 + 0.0883372i 5.98501 + 2.47907i 2.99253 + 0.211577i 4.93281 + 3.29600i
224.2 −0.503008 − 1.21437i 1.11133 + 1.32851i 0.192538 − 0.192538i 0.116989 − 0.0781694i 1.05430 − 2.01782i 0.982905 − 1.47102i −2.75940 − 1.14298i −0.529896 + 2.95283i −0.153773 − 0.102748i
224.3 0.503008 + 1.21437i 1.53513 − 0.802099i 0.192538 − 0.192538i −0.116989 + 0.0781694i 1.74623 + 1.46076i 0.982905 − 1.47102i 2.75940 + 1.14298i 1.71327 − 2.46266i −0.153773 − 0.102748i
224.4 0.978223 + 2.36164i −1.62260 − 0.605951i −3.20621 + 3.20621i 1.92973 − 1.28940i −0.156224 − 4.42475i −0.0590250 + 0.0883372i −5.98501 − 2.47907i 2.26565 + 1.96643i 4.93281 + 3.29600i
329.1 −0.978223 + 2.36164i −1.73097 + 0.0611151i −3.20621 − 3.20621i −1.92973 − 1.28940i 1.54895 − 4.14772i −0.0590250 − 0.0883372i 5.98501 − 2.47907i 2.99253 − 0.211577i 4.93281 − 3.29600i
329.2 −0.503008 + 1.21437i 1.11133 − 1.32851i 0.192538 + 0.192538i 0.116989 + 0.0781694i 1.05430 + 2.01782i 0.982905 + 1.47102i −2.75940 + 1.14298i −0.529896 − 2.95283i −0.153773 + 0.102748i
329.3 0.503008 − 1.21437i 1.53513 + 0.802099i 0.192538 + 0.192538i −0.116989 − 0.0781694i 1.74623 − 1.46076i 0.982905 + 1.47102i 2.75940 − 1.14298i 1.71327 + 2.46266i −0.153773 + 0.102748i
329.4 0.978223 − 2.36164i −1.62260 + 0.605951i −3.20621 − 3.20621i 1.92973 + 1.28940i −0.156224 + 4.42475i −0.0590250 − 0.0883372i −5.98501 + 2.47907i 2.26565 − 1.96643i 4.93281 − 3.29600i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.e odd 16 1 inner
51.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.i.d 32
3.b odd 2 1 inner 867.2.i.d 32
17.b even 2 1 867.2.i.c 32
17.c even 4 1 51.2.i.a ✓ 32
17.c even 4 1 867.2.i.h 32
17.d even 8 1 867.2.i.b 32
17.d even 8 1 867.2.i.f 32
17.d even 8 1 867.2.i.g 32
17.d even 8 1 867.2.i.i 32
17.e odd 16 1 51.2.i.a ✓ 32
17.e odd 16 1 867.2.i.b 32
17.e odd 16 1 867.2.i.c 32
17.e odd 16 1 inner 867.2.i.d 32
17.e odd 16 1 867.2.i.f 32
17.e odd 16 1 867.2.i.g 32
17.e odd 16 1 867.2.i.h 32
17.e odd 16 1 867.2.i.i 32
51.c odd 2 1 867.2.i.c 32
51.f odd 4 1 51.2.i.a ✓ 32
51.f odd 4 1 867.2.i.h 32
51.g odd 8 1 867.2.i.b 32
51.g odd 8 1 867.2.i.f 32
51.g odd 8 1 867.2.i.g 32
51.g odd 8 1 867.2.i.i 32
51.i even 16 1 51.2.i.a ✓ 32
51.i even 16 1 867.2.i.b 32
51.i even 16 1 867.2.i.c 32
51.i even 16 1 inner 867.2.i.d 32
51.i even 16 1 867.2.i.f 32
51.i even 16 1 867.2.i.g 32
51.i even 16 1 867.2.i.h 32
51.i even 16 1 867.2.i.i 32
68.f odd 4 1 816.2.cj.c 32
68.i even 16 1 816.2.cj.c 32
204.l even 4 1 816.2.cj.c 32
204.t odd 16 1 816.2.cj.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.i.a ✓ 32 17.c even 4 1
51.2.i.a ✓ 32 17.e odd 16 1
51.2.i.a ✓ 32 51.f odd 4 1
51.2.i.a ✓ 32 51.i even 16 1
816.2.cj.c 32 68.f odd 4 1
816.2.cj.c 32 68.i even 16 1
816.2.cj.c 32 204.l even 4 1
816.2.cj.c 32 204.t odd 16 1
867.2.i.b 32 17.d even 8 1
867.2.i.b 32 17.e odd 16 1
867.2.i.b 32 51.g odd 8 1
867.2.i.b 32 51.i even 16 1
867.2.i.c 32 17.b even 2 1
867.2.i.c 32 17.e odd 16 1
867.2.i.c 32 51.c odd 2 1
867.2.i.c 32 51.i even 16 1
867.2.i.d 32 1.a even 1 1 trivial
867.2.i.d 32 3.b odd 2 1 inner
867.2.i.d 32 17.e odd 16 1 inner
867.2.i.d 32 51.i even 16 1 inner
867.2.i.f 32 17.d even 8 1
867.2.i.f 32 17.e odd 16 1
867.2.i.f 32 51.g odd 8 1
867.2.i.f 32 51.i even 16 1
867.2.i.g 32 17.d even 8 1
867.2.i.g 32 17.e odd 16 1
867.2.i.g 32 51.g odd 8 1
867.2.i.g 32 51.i even 16 1
867.2.i.h 32 17.c even 4 1
867.2.i.h 32 17.e odd 16 1
867.2.i.h 32 51.f odd 4 1
867.2.i.h 32 51.i even 16 1
867.2.i.i 32 17.d even 8 1
867.2.i.i 32 17.e odd 16 1
867.2.i.i 32 51.g odd 8 1
867.2.i.i 32 51.i even 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{32} + 8 T_{2}^{30} + 32 T_{2}^{28} - 40 T_{2}^{26} + 366 T_{2}^{24} + 2552 T_{2}^{22} + \cdots + 1156 \) Copy content Toggle raw display
\( T_{5}^{32} + 8 T_{5}^{30} - 52 T_{5}^{28} - 656 T_{5}^{26} + 456 T_{5}^{24} + 27912 T_{5}^{22} + \cdots + 1156 \) Copy content Toggle raw display
\( T_{7}^{16} - 12 T_{7}^{14} - 8 T_{7}^{13} + 150 T_{7}^{12} - 176 T_{7}^{11} - 60 T_{7}^{10} + 120 T_{7}^{9} + \cdots + 16 \) Copy content Toggle raw display