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Properties

Label 59.18.a.a
Level $59$
Weight $18$
Character orbit 59.a
Self dual yes
Analytic conductor $108.101$
Analytic rank $1$
Dimension $38$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,18,Mod(1,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 59.a (trivial)

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: yes
Analytic conductor: \(108.101031533\)
Analytic rank: \(1\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 15 q^{2} + 2080511 q^{4} - 1062222 q^{5} - 3454906 q^{6} - 43925040 q^{7} + 17944905 q^{8} + 1407964240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 15 q^{2} + 2080511 q^{4} - 1062222 q^{5} - 3454906 q^{6} - 43925040 q^{7} + 17944905 q^{8} + 1407964240 q^{9} - 753380082 q^{10} - 1247087592 q^{11} - 12806174710 q^{13} - 9920461605 q^{14} - 26267835313 q^{15} + 150793103155 q^{16} - 32714582925 q^{17} - 321256617405 q^{18} - 275283244180 q^{19} - 219435764020 q^{20} - 301109113003 q^{21} + 141637789665 q^{22} + 101540066360 q^{23} + 2412933601890 q^{24} + 4991161115804 q^{25} + 990445756111 q^{26} - 1179423443835 q^{27} - 10869216693165 q^{28} - 12862783749830 q^{29} - 27886581579060 q^{30} - 7513791056524 q^{31} - 31047860166815 q^{32} - 16499870737070 q^{33} - 62883745276639 q^{34} - 41836456706373 q^{35} - 74757290464031 q^{36} - 113711477231480 q^{37} - 86782161105620 q^{38} - 63680125347522 q^{39} - 53946909479582 q^{40} - 40841718887048 q^{41} - 8891162142170 q^{42} - 81961546008910 q^{43} - 24535596549333 q^{44} - 43121875478149 q^{45} - 432155860006696 q^{46} + 48939724777580 q^{47} + 513518246048640 q^{48} + 888582051628886 q^{49} + 20\!\cdots\!21 q^{50}+ \cdots - 51\!\cdots\!14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −685.279 2207.48 338535. 777125. −1.51274e6 −1.15256e7 −1.42170e8 −1.24267e8 −5.32547e8
1.2 −684.581 15529.2 337580. 464435. −1.06310e7 7.56088e6 −1.41371e8 1.12016e8 −3.17944e8
1.3 −673.813 −18107.9 322952. −791552. 1.22013e7 −620489. −1.29291e8 1.98755e8 5.33358e8
1.4 −654.278 −632.542 297008. −1.03634e6 413858. −2.84362e7 −1.08568e8 −1.28740e8 6.78055e8
1.5 −517.700 7661.35 136942. −102200. −3.96628e6 2.07408e7 −3.03866e6 −7.04438e7 5.29091e7
1.6 −498.464 10126.9 117395. 1.14682e6 −5.04791e6 −2.46174e7 6.81766e6 −2.65857e7 −5.71648e8
1.7 −474.769 −20635.2 94334.0 876020. 9.79694e6 8.48652e6 1.74421e7 2.96669e8 −4.15907e8
1.8 −441.923 −16267.1 64223.7 −1.10399e6 7.18882e6 2.16741e7 2.95418e7 1.35480e8 4.87879e8
1.9 −438.751 −5455.29 61430.4 378259. 2.39351e6 −6.36839e6 3.05553e7 −9.93800e7 −1.65962e8
1.10 −352.028 20528.2 −7148.36 −641568. −7.22651e6 2.69602e7 4.86574e7 2.92268e8 2.25850e8
1.11 −345.599 12699.6 −11633.2 −1.32893e6 −4.38898e6 1.45947e6 4.93188e7 3.21402e7 4.59278e8
1.12 −311.440 −13591.0 −34077.3 797906. 4.23276e6 2.23561e7 5.14341e7 5.55740e7 −2.48500e8
1.13 −252.042 17232.6 −67547.0 1.07007e6 −4.34333e6 −1.20393e7 5.00603e7 1.67822e8 −2.69702e8
1.14 −249.983 −14283.8 −68580.6 271188. 3.57071e6 −2.24320e7 4.99097e7 7.48875e7 −6.77922e7
1.15 −186.227 586.247 −96391.7 −1.36112e6 −109175. −1.02808e7 4.23598e7 −1.28796e8 2.53477e8
1.16 −177.299 2894.60 −99637.2 589485. −513208. 1.99287e7 4.09044e7 −1.20761e8 −1.04515e8
1.17 −174.045 22053.8 −100780. 230534. −3.83836e6 −1.27047e7 4.03528e7 3.57231e8 −4.01233e7
1.18 −38.7395 −2676.81 −129571. 119311. 103698. −2.40565e7 1.00972e7 −1.21975e8 −4.62206e6
1.19 15.6332 9151.12 −130828. 1.36327e6 143061. 970140. −4.09432e6 −4.53971e7 2.13122e7
1.20 47.3291 −20318.3 −128832. 1.59162e6 −961646. 4.91038e6 −1.23010e7 2.83692e8 7.53300e7
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(59\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.18.a.a ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.18.a.a ✓ 38 1.a even 1 1 trivial