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Properties

Label 261.3.n.a
Level $261$
Weight $3$
Character orbit 261.n
Analytic conductor $7.112$
Analytic rank $0$
Dimension $120$
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] = [261,3,Mod(35,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.n (of order \(14\), degree \(6\), 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: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 40 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 40 q^{4} + 16 q^{7} + 36 q^{13} - 72 q^{16} - 116 q^{22} - 44 q^{25} - 288 q^{28} + 80 q^{34} + 168 q^{40} - 308 q^{43} + 420 q^{49} + 300 q^{52} + 616 q^{55} + 240 q^{58} - 504 q^{61} - 204 q^{64} - 88 q^{67} - 588 q^{73} - 980 q^{76} - 644 q^{79} - 368 q^{82} + 308 q^{85} - 784 q^{88} + 428 q^{91} - 124 q^{94} + 196 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −2.44238 + 3.06265i 0 −2.52451 − 11.0606i −7.28523 − 5.80977i 0 1.25078 − 5.48001i 25.9231 + 12.4839i 0 35.5866 − 8.12240i
35.2 −2.14266 + 2.68681i 0 −1.73788 − 7.61417i 3.12825 + 2.49470i 0 1.06438 − 4.66334i 11.7966 + 5.68094i 0 −13.4056 + 3.05973i
35.3 −2.01945 + 2.53231i 0 −1.44433 − 6.32803i 5.40353 + 4.30917i 0 0.595397 − 2.60860i 7.26854 + 3.50035i 0 −21.8243 + 4.98126i
35.4 −1.69878 + 2.13021i 0 −0.761832 − 3.33780i −4.59177 − 3.66181i 0 −0.752227 + 3.29572i −1.41486 − 0.681359i 0 15.6008 − 3.56079i
35.5 −1.48458 + 1.86161i 0 −0.371514 − 1.62771i −1.61309 − 1.28640i 0 −1.89233 + 8.29085i −4.99943 − 2.40760i 0 4.78953 − 1.09318i
35.6 −1.13494 + 1.42317i 0 0.152759 + 0.669279i 6.00177 + 4.78625i 0 −2.92808 + 12.8288i −7.68602 − 3.70139i 0 −13.6233 + 3.10943i
35.7 −1.09606 + 1.37442i 0 0.202410 + 0.886818i 0.734527 + 0.585766i 0 1.66645 − 7.30118i −7.77613 − 3.74479i 0 −1.61017 + 0.367512i
35.8 −0.881091 + 1.10485i 0 0.445704 + 1.95276i 1.05806 + 0.843774i 0 2.66834 − 11.6908i −7.64307 − 3.68071i 0 −1.86449 + 0.425558i
35.9 −0.576678 + 0.723131i 0 0.699722 + 3.06568i −5.64609 − 4.50260i 0 0.147568 − 0.646538i −5.95370 − 2.86715i 0 6.51195 − 1.48631i
35.10 −0.0423928 + 0.0531589i 0 0.889055 + 3.89520i −4.04085 − 3.22247i 0 −0.710351 + 3.11225i −0.489792 − 0.235872i 0 0.342606 − 0.0781976i
35.11 0.0423928 − 0.0531589i 0 0.889055 + 3.89520i 4.04085 + 3.22247i 0 −0.710351 + 3.11225i 0.489792 + 0.235872i 0 0.342606 − 0.0781976i
35.12 0.576678 − 0.723131i 0 0.699722 + 3.06568i 5.64609 + 4.50260i 0 0.147568 − 0.646538i 5.95370 + 2.86715i 0 6.51195 − 1.48631i
35.13 0.881091 − 1.10485i 0 0.445704 + 1.95276i −1.05806 − 0.843774i 0 2.66834 − 11.6908i 7.64307 + 3.68071i 0 −1.86449 + 0.425558i
35.14 1.09606 − 1.37442i 0 0.202410 + 0.886818i −0.734527 − 0.585766i 0 1.66645 − 7.30118i 7.77613 + 3.74479i 0 −1.61017 + 0.367512i
35.15 1.13494 − 1.42317i 0 0.152759 + 0.669279i −6.00177 − 4.78625i 0 −2.92808 + 12.8288i 7.68602 + 3.70139i 0 −13.6233 + 3.10943i
35.16 1.48458 − 1.86161i 0 −0.371514 − 1.62771i 1.61309 + 1.28640i 0 −1.89233 + 8.29085i 4.99943 + 2.40760i 0 4.78953 − 1.09318i
35.17 1.69878 − 2.13021i 0 −0.761832 − 3.33780i 4.59177 + 3.66181i 0 −0.752227 + 3.29572i 1.41486 + 0.681359i 0 15.6008 − 3.56079i
35.18 2.01945 − 2.53231i 0 −1.44433 − 6.32803i −5.40353 − 4.30917i 0 0.595397 − 2.60860i −7.26854 − 3.50035i 0 −21.8243 + 4.98126i
35.19 2.14266 − 2.68681i 0 −1.73788 − 7.61417i −3.12825 − 2.49470i 0 1.06438 − 4.66334i −11.7966 − 5.68094i 0 −13.4056 + 3.05973i
35.20 2.44238 − 3.06265i 0 −2.52451 − 11.0606i 7.28523 + 5.80977i 0 1.25078 − 5.48001i −25.9231 − 12.4839i 0 35.5866 − 8.12240i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.e even 14 1 inner
87.h odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.n.a ✓ 120
3.b odd 2 1 inner 261.3.n.a ✓ 120
29.e even 14 1 inner 261.3.n.a ✓ 120
87.h odd 14 1 inner 261.3.n.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.3.n.a ✓ 120 1.a even 1 1 trivial
261.3.n.a ✓ 120 3.b odd 2 1 inner
261.3.n.a ✓ 120 29.e even 14 1 inner
261.3.n.a ✓ 120 87.h odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).