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Properties

Label 547.3.b.b
Level $547$
Weight $3$
Character orbit 547.b
Analytic conductor $14.905$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [547,3,Mod(546,547)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("547.546");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 547.b (of order \(2\), degree \(1\), 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: \(14.9046704605\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 192 q^{4} - 306 q^{9} - 4 q^{10} - 32 q^{11} + 26 q^{13} - 26 q^{14} + 22 q^{15} + 236 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} - 544 q^{25} - 96 q^{29} + 26 q^{34} + 10 q^{35} + 364 q^{36} + 44 q^{40} + 124 q^{44} - 288 q^{46} - 310 q^{47} - 694 q^{49} + 86 q^{51} - 316 q^{52} + 24 q^{53} - 266 q^{54} + 158 q^{56} - 80 q^{60} + 40 q^{62} - 652 q^{64} + 528 q^{66} + 28 q^{67} + 16 q^{69} + 94 q^{73} - 614 q^{74} - 28 q^{76} - 98 q^{78} + 928 q^{81} - 772 q^{82} + 358 q^{84} + 74 q^{85} - 410 q^{86} - 214 q^{90} + 656 q^{93} - 724 q^{96} + 346 q^{97} + 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
546.1 − 3.90365i 5.00487i −11.2385 − 3.32024i 19.5373 0.969581i 28.2567i −16.0487 −12.9611
546.2 − 3.88566i − 2.17695i −11.0983 − 9.52039i −8.45889 − 6.36562i 27.5816i 4.26087 −36.9930
546.3 − 3.79607i − 0.894948i −10.4101 3.75049i −3.39728 0.715898i 24.3333i 8.19907 14.2371
546.4 − 3.76304i − 4.52828i −10.1605 3.85159i −17.0401 − 7.12637i 23.1820i −11.5053 14.4937
546.5 − 3.58531i − 0.541221i −8.85444 − 2.75520i −1.94044 8.99411i 17.4047i 8.70708 −9.87823
546.6 − 3.54864i 2.90313i −8.59283 1.02412i 10.3021 − 3.84607i 16.2983i 0.571853 3.63422
546.7 − 3.54186i − 4.64741i −8.54479 0.255244i −16.4605 11.6148i 16.0970i −12.5984 0.904038
546.8 − 3.52162i 0.549187i −8.40183 7.97338i 1.93403 − 12.0038i 15.5016i 8.69839 28.0792
546.9 − 3.42438i 4.48969i −7.72636 7.63002i 15.3744 10.8574i 12.7605i −11.1573 26.1281
546.10 − 3.26612i 1.63931i −6.66753 − 7.05581i 5.35418 7.78718i 8.71245i 6.31266 −23.0451
546.11 − 3.08562i 1.35400i −5.52104 − 5.29090i 4.17794 − 11.3849i 4.69333i 7.16667 −16.3257
546.12 − 3.05754i − 3.93936i −5.34857 − 2.27125i −12.0448 − 6.60828i 4.12330i −6.51856 −6.94444
546.13 − 3.03308i − 3.99539i −5.19958 9.10946i −12.1184 2.89551i 3.63844i −6.96316 27.6297
546.14 − 2.90201i 2.26440i −4.42164 8.06772i 6.57132 4.29626i 1.22362i 3.87247 23.4126
546.15 − 2.87053i − 4.81682i −4.23994 − 8.18119i −13.8268 5.20188i 0.688763i −14.2018 −23.4844
546.16 − 2.83323i − 2.06299i −4.02719 − 2.31552i −5.84492 − 4.45862i 0.0770388i 4.74408 −6.56039
546.17 − 2.78422i 5.61272i −3.75190 5.19773i 15.6271 − 12.7126i − 0.690758i −22.5026 14.4716
546.18 − 2.76044i − 1.30646i −3.62000 3.69969i −3.60639 4.19330i − 1.04896i 7.29317 10.2127
546.19 − 2.71290i 2.08874i −3.35982 − 3.69381i 5.66654 8.32703i − 1.73675i 4.63716 −10.0209
546.20 − 2.64774i 5.22856i −3.01055 − 9.23817i 13.8439 − 6.29191i − 2.61981i −18.3379 −24.4603
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 546.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
547.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.3.b.b ✓ 88
547.b odd 2 1 inner 547.3.b.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.3.b.b ✓ 88 1.a even 1 1 trivial
547.3.b.b ✓ 88 547.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 272 T_{2}^{86} + 35653 T_{2}^{84} + 2999930 T_{2}^{82} + 182111318 T_{2}^{80} + \cdots + 93\!\cdots\!48 \) acting on \(S_{3}^{\mathrm{new}}(547, [\chi])\). Copy content Toggle raw display