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Properties

Label 538.3.c.a
Level $538$
Weight $3$
Character orbit 538.c
Analytic conductor $14.659$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,3,Mod(187,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(4))
 
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("538.187");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 538.c (of order \(4\), degree \(2\), 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: \(14.6594382226\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 44 q^{2} + 2 q^{3} + 4 q^{7} - 88 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 44 q^{2} + 2 q^{3} + 4 q^{7} - 88 q^{8} - 4 q^{12} + 8 q^{14} + 38 q^{15} - 176 q^{16} - 120 q^{18} + 18 q^{19} - 16 q^{21} + 68 q^{23} - 8 q^{24} + 196 q^{25} + 16 q^{26} - 22 q^{27} + 8 q^{28} + 18 q^{29} + 88 q^{31} - 176 q^{32} + 96 q^{33} + 10 q^{35} - 240 q^{36} + 180 q^{37} + 36 q^{38} - 54 q^{39} + 144 q^{41} - 16 q^{42} + 68 q^{46} - 156 q^{47} - 8 q^{48} + 196 q^{50} + 32 q^{52} - 188 q^{53} - 44 q^{54} - 88 q^{57} + 36 q^{58} - 98 q^{59} - 76 q^{60} - 244 q^{61} + 176 q^{62} - 10 q^{63} + 192 q^{66} + 220 q^{67} - 70 q^{69} + 20 q^{70} + 156 q^{71} - 240 q^{72} + 180 q^{74} - 102 q^{75} + 36 q^{76} - 28 q^{77} - 108 q^{78} + 392 q^{81} + 144 q^{82} - 250 q^{83} - 342 q^{85} - 44 q^{86} - 60 q^{87} - 192 q^{90} + 66 q^{91} - 536 q^{93} - 156 q^{94} - 472 q^{95} + 404 q^{98} - 188 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
187.1 1.00000 + 1.00000i −3.89974 − 3.89974i 2.00000i −6.52679 − 7.79948i −0.905272 + 0.905272i −2.00000 + 2.00000i 21.4159i −6.52679 − 6.52679i
187.2 1.00000 + 1.00000i −3.11799 − 3.11799i 2.00000i −4.51105 − 6.23598i 6.48641 − 6.48641i −2.00000 + 2.00000i 10.4437i −4.51105 − 4.51105i
187.3 1.00000 + 1.00000i −2.92326 − 2.92326i 2.00000i 5.47933 − 5.84652i −7.43471 + 7.43471i −2.00000 + 2.00000i 8.09090i 5.47933 + 5.47933i
187.4 1.00000 + 1.00000i −2.74968 − 2.74968i 2.00000i 6.97749 − 5.49937i 1.80329 − 1.80329i −2.00000 + 2.00000i 6.12153i 6.97749 + 6.97749i
187.5 1.00000 + 1.00000i −2.64193 − 2.64193i 2.00000i −4.20274 − 5.28386i −7.77918 + 7.77918i −2.00000 + 2.00000i 4.95961i −4.20274 − 4.20274i
187.6 1.00000 + 1.00000i −2.63017 − 2.63017i 2.00000i 7.51373 − 5.26034i 7.42431 − 7.42431i −2.00000 + 2.00000i 4.83558i 7.51373 + 7.51373i
187.7 1.00000 + 1.00000i −2.30188 − 2.30188i 2.00000i −2.14972 − 4.60375i 2.52965 − 2.52965i −2.00000 + 2.00000i 1.59726i −2.14972 − 2.14972i
187.8 1.00000 + 1.00000i −1.45158 − 1.45158i 2.00000i 1.48345 − 2.90315i −2.38989 + 2.38989i −2.00000 + 2.00000i − 4.78586i 1.48345 + 1.48345i
187.9 1.00000 + 1.00000i −0.447549 − 0.447549i 2.00000i 4.15921 − 0.895099i −0.142659 + 0.142659i −2.00000 + 2.00000i − 8.59940i 4.15921 + 4.15921i
187.10 1.00000 + 1.00000i −0.382952 − 0.382952i 2.00000i −9.21654 − 0.765903i 4.16457 − 4.16457i −2.00000 + 2.00000i − 8.70670i −9.21654 − 9.21654i
187.11 1.00000 + 1.00000i −0.283316 − 0.283316i 2.00000i −1.59047 − 0.566631i 4.29967 − 4.29967i −2.00000 + 2.00000i − 8.83946i −1.59047 − 1.59047i
187.12 1.00000 + 1.00000i −0.209469 − 0.209469i 2.00000i −0.0245183 − 0.418938i −4.25464 + 4.25464i −2.00000 + 2.00000i − 8.91225i −0.0245183 − 0.0245183i
187.13 1.00000 + 1.00000i 0.991340 + 0.991340i 2.00000i −6.41200 1.98268i −6.33449 + 6.33449i −2.00000 + 2.00000i − 7.03449i −6.41200 − 6.41200i
187.14 1.00000 + 1.00000i 1.65205 + 1.65205i 2.00000i 6.81237 3.30411i 5.20547 − 5.20547i −2.00000 + 2.00000i − 3.54144i 6.81237 + 6.81237i
187.15 1.00000 + 1.00000i 1.79446 + 1.79446i 2.00000i −4.75177 3.58893i 9.51668 − 9.51668i −2.00000 + 2.00000i − 2.55979i −4.75177 − 4.75177i
187.16 1.00000 + 1.00000i 1.87375 + 1.87375i 2.00000i −6.42972 3.74750i −4.44577 + 4.44577i −2.00000 + 2.00000i − 1.97811i −6.42972 − 6.42972i
187.17 1.00000 + 1.00000i 1.93508 + 1.93508i 2.00000i 9.46172 3.87015i −8.10752 + 8.10752i −2.00000 + 2.00000i − 1.51096i 9.46172 + 9.46172i
187.18 1.00000 + 1.00000i 2.05500 + 2.05500i 2.00000i 2.94721 4.11001i 4.83908 − 4.83908i −2.00000 + 2.00000i − 0.553923i 2.94721 + 2.94721i
187.19 1.00000 + 1.00000i 2.93671 + 2.93671i 2.00000i −5.34414 5.87341i 2.19065 − 2.19065i −2.00000 + 2.00000i 8.24850i −5.34414 − 5.34414i
187.20 1.00000 + 1.00000i 3.20740 + 3.20740i 2.00000i −1.33372 6.41480i −6.57572 + 6.57572i −2.00000 + 2.00000i 11.5748i −1.33372 − 1.33372i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 187.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
269.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.3.c.a ✓ 44
269.c odd 4 1 inner 538.3.c.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.3.c.a ✓ 44 1.a even 1 1 trivial
538.3.c.a ✓ 44 269.c odd 4 1 inner