Properties

Label 103.18.a.b
Level $103$
Weight $18$
Character orbit 103.a
Self dual yes
Analytic conductor $188.719$
Analytic rank $0$
Dimension $75$
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] = [103,18,Mod(1,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, 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("103.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(188.718749965\)
Analytic rank: \(0\)
Dimension: \(75\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 75 q + 1807 q^{2} + 15604 q^{3} + 5029631 q^{4} + 2453403 q^{5} - 1418177 q^{6} + 24063666 q^{7} + 311079204 q^{8} + 3821830473 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 75 q + 1807 q^{2} + 15604 q^{3} + 5029631 q^{4} + 2453403 q^{5} - 1418177 q^{6} + 24063666 q^{7} + 311079204 q^{8} + 3821830473 q^{9} + 46619918 q^{10} + 1854318275 q^{11} + 3846619095 q^{12} + 4087074053 q^{13} + 7006072263 q^{14} + 13076804947 q^{15} + 369598840507 q^{16} + 175498718319 q^{17} + 158633724936 q^{18} + 46304344061 q^{19} + 246340237869 q^{20} + 174178374934 q^{21} + 359633007585 q^{22} + 2000187252096 q^{23} + 306049148082 q^{24} + 13701438314894 q^{25} + 8401694936077 q^{26} + 224217466198 q^{27} - 9891975374384 q^{28} + 8007600657204 q^{29} + 23114400368558 q^{30} + 8982290233384 q^{31} + 88084390393860 q^{32} + 71411682241673 q^{33} + 52543316279238 q^{34} + 30057027650903 q^{35} + 301187806995744 q^{36} - 19693449079428 q^{37} - 60007141135838 q^{38} - 116946136532061 q^{39} - 325780683127237 q^{40} + 104672502851378 q^{41} - 720370049120813 q^{42} + 124411398015226 q^{43} - 152507311129672 q^{44} + 51191418104954 q^{45} - 439217262643903 q^{46} + 199444597200769 q^{47} + 12\!\cdots\!69 q^{48}+ \cdots + 15\!\cdots\!80 q^{99}+O(q^{100}) \) 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.

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 −684.133 15587.3 336966. 1.49942e6 −1.06638e7 −1.70692e6 −1.40859e8 1.13822e8 −1.02580e9
1.2 −683.258 6745.27 335769. −704392. −4.60876e6 −1.06552e7 −1.39861e8 −8.36414e7 4.81281e8
1.3 −681.827 15597.7 333816. 239807. −1.06349e7 −2.04143e7 −1.38236e8 1.14149e8 −1.63507e8
1.4 −660.506 899.492 305196. 178972. −594120. −2.07107e7 −1.15010e8 −1.28331e8 −1.18212e8
1.5 −655.732 −19150.8 298912. −43957.1 1.25578e7 −3.55696e6 −1.10058e8 2.37613e8 2.88241e7
1.6 −644.220 282.671 283947. −178170. −182102. 2.20767e7 −9.84850e7 −1.29060e8 1.14781e8
1.7 −619.709 12155.7 252967. −668724. −7.53303e6 1.48524e7 −7.55395e7 1.86221e7 4.14414e8
1.8 −607.179 −15256.1 237594. −1.13942e6 9.26320e6 −2.76514e7 −6.46780e7 1.03609e8 6.91832e8
1.9 −565.256 −19369.3 188442. −244340. 1.09486e7 2.06638e7 −3.24290e7 2.46030e8 1.38115e8
1.10 −560.848 −13739.1 183478. 964315. 7.70553e6 1.95070e7 −2.93921e7 5.96220e7 −5.40834e8
1.11 −524.816 −21497.2 144360. 1.49165e6 1.12821e7 −1.60270e7 −6.97353e6 3.32989e8 −7.82843e8
1.12 −505.425 12282.5 124383. −1.08481e6 −6.20786e6 1.56269e7 3.38103e6 2.17187e7 5.48290e8
1.13 −484.615 17274.0 103780. 1.05960e6 −8.37122e6 34849.8 1.32263e7 1.69250e8 −5.13500e8
1.14 −472.360 8120.08 92052.4 863260. −3.83561e6 −1.83518e7 1.84313e7 −6.32044e7 −4.07770e8
1.15 −457.450 −9015.89 78188.7 −815945. 4.12432e6 −2.21178e6 2.41915e7 −4.78539e7 3.73254e8
1.16 −429.129 −9834.29 53079.9 480699. 4.22018e6 2.34102e7 3.34687e7 −3.24269e7 −2.06282e8
1.17 −418.493 −3327.56 44064.0 777397. 1.39256e6 −1.89604e7 3.64122e7 −1.18068e8 −3.25335e8
1.18 −383.136 4061.48 15721.5 623786. −1.55610e6 6.34260e6 4.41950e7 −1.12645e8 −2.38995e8
1.19 −381.945 21120.0 14809.7 834842. −8.06668e6 2.86828e7 4.44058e7 3.16915e8 −3.18863e8
1.20 −379.273 2709.91 12775.7 −1.61373e6 −1.02779e6 2.70681e7 4.48665e7 −1.21797e8 6.12042e8
See all 75 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.75
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(103\) \(-1\)

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 103.18.a.b 75
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.18.a.b 75 1.a even 1 1 trivial