Properties

Label 261.3.s.c
Level $261$
Weight $3$
Character orbit 261.s
Analytic conductor $7.112$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [261,3,Mod(10,261)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("261.10"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(261, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([0, 23])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.s (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 24 q^{10} + 72 q^{16} - 32 q^{19} + 336 q^{22} + 52 q^{25} - 212 q^{31} - 420 q^{34} - 316 q^{37} - 332 q^{40} + 208 q^{43} - 184 q^{46} - 844 q^{49} - 260 q^{52} - 40 q^{55} - 260 q^{58} + 240 q^{61}+ \cdots + 1988 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −3.23541 + 2.03294i 0 4.59948 − 9.55092i 8.30581 + 1.89575i 0 −0.117247 + 0.0564633i 2.82396 + 25.0634i 0 −30.7266 + 10.7517i
10.2 −2.77904 + 1.74618i 0 2.93835 − 6.10155i −7.25275 − 1.65539i 0 3.23708 − 1.55889i 1.01873 + 9.04147i 0 23.0463 − 8.06424i
10.3 −2.05173 + 1.28919i 0 0.812066 − 1.68627i −1.21789 − 0.277976i 0 3.74303 − 1.80255i −0.577442 − 5.12494i 0 2.85716 − 0.999762i
10.4 −1.56943 + 0.986140i 0 −0.244887 + 0.508512i 1.95162 + 0.445444i 0 −8.32237 + 4.00784i −0.947252 − 8.40709i 0 −3.50220 + 1.22547i
10.5 −0.0881801 + 0.0554072i 0 −1.73083 + 3.59410i 5.73523 + 1.30903i 0 4.97303 − 2.39488i −0.0931557 − 0.826780i 0 −0.578262 + 0.202343i
10.6 0.0881801 − 0.0554072i 0 −1.73083 + 3.59410i −5.73523 − 1.30903i 0 4.97303 − 2.39488i 0.0931557 + 0.826780i 0 −0.578262 + 0.202343i
10.7 1.56943 − 0.986140i 0 −0.244887 + 0.508512i −1.95162 − 0.445444i 0 −8.32237 + 4.00784i 0.947252 + 8.40709i 0 −3.50220 + 1.22547i
10.8 2.05173 − 1.28919i 0 0.812066 − 1.68627i 1.21789 + 0.277976i 0 3.74303 − 1.80255i 0.577442 + 5.12494i 0 2.85716 − 0.999762i
10.9 2.77904 − 1.74618i 0 2.93835 − 6.10155i 7.25275 + 1.65539i 0 3.23708 − 1.55889i −1.01873 − 9.04147i 0 23.0463 − 8.06424i
10.10 3.23541 − 2.03294i 0 4.59948 − 9.55092i −8.30581 − 1.89575i 0 −0.117247 + 0.0564633i −2.82396 − 25.0634i 0 −30.7266 + 10.7517i
19.1 −1.83277 − 2.91684i 0 −3.41338 + 7.08796i −6.77901 − 1.54727i 0 −2.18838 + 1.05387i 13.2376 − 1.49152i 0 7.91127 + 22.6091i
19.2 −1.46336 − 2.32892i 0 −1.54691 + 3.21220i 5.07856 + 1.15915i 0 −8.02635 + 3.86529i −1.18820 + 0.133878i 0 −4.73218 − 13.5238i
19.3 −1.21264 − 1.92991i 0 −0.518510 + 1.07670i 1.07114 + 0.244480i 0 7.68731 − 3.70202i −6.35304 + 0.715815i 0 −0.827080 − 2.36366i
19.4 −0.402987 − 0.641350i 0 1.48660 − 3.08696i −0.142387 − 0.0324990i 0 −6.58114 + 3.16931i −5.58965 + 0.629802i 0 0.0365370 + 0.104417i
19.5 −0.174786 − 0.278170i 0 1.68871 − 3.50664i −9.52764 − 2.17462i 0 5.59504 − 2.69443i −2.57644 + 0.290295i 0 1.06038 + 3.03040i
19.6 0.174786 + 0.278170i 0 1.68871 − 3.50664i 9.52764 + 2.17462i 0 5.59504 − 2.69443i 2.57644 − 0.290295i 0 1.06038 + 3.03040i
19.7 0.402987 + 0.641350i 0 1.48660 − 3.08696i 0.142387 + 0.0324990i 0 −6.58114 + 3.16931i 5.58965 − 0.629802i 0 0.0365370 + 0.104417i
19.8 1.21264 + 1.92991i 0 −0.518510 + 1.07670i −1.07114 − 0.244480i 0 7.68731 − 3.70202i 6.35304 − 0.715815i 0 −0.827080 − 2.36366i
19.9 1.46336 + 2.32892i 0 −1.54691 + 3.21220i −5.07856 − 1.15915i 0 −8.02635 + 3.86529i 1.18820 − 0.133878i 0 −4.73218 − 13.5238i
19.10 1.83277 + 2.91684i 0 −3.41338 + 7.08796i 6.77901 + 1.54727i 0 −2.18838 + 1.05387i −13.2376 + 1.49152i 0 7.91127 + 22.6091i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 10.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.f odd 28 1 inner
87.k even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.s.c ✓ 120
3.b odd 2 1 inner 261.3.s.c ✓ 120
29.f odd 28 1 inner 261.3.s.c ✓ 120
87.k even 28 1 inner 261.3.s.c ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.3.s.c ✓ 120 1.a even 1 1 trivial
261.3.s.c ✓ 120 3.b odd 2 1 inner
261.3.s.c ✓ 120 29.f odd 28 1 inner
261.3.s.c ✓ 120 87.k even 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} - 298 T_{2}^{116} + 64847 T_{2}^{112} - 449470 T_{2}^{110} - 13161037 T_{2}^{108} + \cdots + 11\!\cdots\!21 \) acting on \(S_{3}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display