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Properties

Label 241.2.m.a
Level $241$
Weight $2$
Character orbit 241.m
Analytic conductor $1.924$
Analytic rank $0$
Dimension $160$
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] = [241,2,Mod(2,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.m (of order \(24\), degree \(8\), 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: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} - 8 q^{3} - 36 q^{4} - 8 q^{5} + 8 q^{7} - 16 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} - 8 q^{3} - 36 q^{4} - 8 q^{5} + 8 q^{7} - 16 q^{8} - 36 q^{9} + 4 q^{10} - 8 q^{11} - 24 q^{12} - 8 q^{13} + 16 q^{14} - 20 q^{15} + 80 q^{16} - 24 q^{17} + 8 q^{18} + 4 q^{19} - 16 q^{20} + 16 q^{22} + 20 q^{23} - 12 q^{24} + 20 q^{26} + 4 q^{27} - 48 q^{28} - 24 q^{29} + 84 q^{30} - 16 q^{31} + 24 q^{32} + 20 q^{33} - 52 q^{34} + 8 q^{35} - 376 q^{36} - 36 q^{37} - 60 q^{38} - 32 q^{39} - 20 q^{41} - 64 q^{42} - 116 q^{44} + 32 q^{45} + 120 q^{46} + 8 q^{47} + 204 q^{48} - 36 q^{49} + 224 q^{50} + 56 q^{51} + 32 q^{52} - 32 q^{53} - 12 q^{54} + 36 q^{55} - 80 q^{56} + 72 q^{57} - 24 q^{58} + 20 q^{59} + 36 q^{60} + 48 q^{61} + 32 q^{62} - 112 q^{63} - 68 q^{65} + 12 q^{66} - 28 q^{67} - 72 q^{68} + 12 q^{69} + 28 q^{70} + 68 q^{72} - 36 q^{74} + 20 q^{75} - 40 q^{76} + 40 q^{77} - 100 q^{78} - 40 q^{79} + 152 q^{80} + 52 q^{81} + 96 q^{82} - 96 q^{83} + 56 q^{84} - 60 q^{85} + 92 q^{86} + 40 q^{87} + 44 q^{88} - 64 q^{89} - 12 q^{90} - 8 q^{91} - 188 q^{92} - 56 q^{93} - 52 q^{94} + 8 q^{95} - 168 q^{96} - 72 q^{97} + 144 q^{98} + 108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.704088 − 2.62769i 2.52559 + 0.676731i −4.67697 + 2.70025i 0.313348 + 0.313348i − 7.11296i 2.71787 − 2.08550i 6.54122 + 6.54122i 3.32258 + 1.91829i 0.602757 − 1.04401i
2.2 −0.667750 − 2.49208i −2.07790 − 0.556771i −4.03251 + 2.32817i 2.64848 + 2.64848i 5.55006i −0.592030 + 0.454281i 4.84603 + 4.84603i 1.40958 + 0.813823i 4.83169 − 8.36872i
2.3 −0.601890 − 2.24628i 0.155005 + 0.0415335i −2.95147 + 1.70403i −0.474194 − 0.474194i − 0.373184i −2.57684 + 1.97728i 2.31542 + 2.31542i −2.57577 − 1.48712i −0.779762 + 1.35059i
2.4 −0.542269 − 2.02378i −3.00395 − 0.804907i −2.06956 + 1.19486i −2.34995 − 2.34995i 6.51580i −1.24675 + 0.956667i 0.577378 + 0.577378i 5.77778 + 3.33581i −3.48147 + 6.03008i
2.5 −0.427399 − 1.59507i 1.86568 + 0.499906i −0.629538 + 0.363464i 1.72756 + 1.72756i − 3.18955i −0.628487 + 0.482255i −1.48653 − 1.48653i 0.632762 + 0.365326i 2.01722 − 3.49393i
2.6 −0.420927 − 1.57092i −1.37342 − 0.368007i −0.558561 + 0.322485i −0.0180347 − 0.0180347i 2.31244i 3.75903 − 2.88441i −1.55828 − 1.55828i −0.847224 − 0.489145i −0.0207398 + 0.0359224i
2.7 −0.362641 − 1.35340i 2.40372 + 0.644075i 0.0318802 − 0.0184060i −2.75079 − 2.75079i − 3.48675i 1.31419 − 1.00841i −2.01798 − 2.01798i 2.76496 + 1.59635i −2.72536 + 4.72046i
2.8 −0.222155 − 0.829092i 0.311636 + 0.0835027i 1.09401 − 0.631627i −1.02800 − 1.02800i − 0.276926i −0.302572 + 0.232172i −1.98059 − 1.98059i −2.50793 − 1.44796i −0.623932 + 1.08068i
2.9 −0.104391 − 0.389594i −2.83228 − 0.758906i 1.59117 − 0.918660i 2.04617 + 2.04617i 1.18266i 0.0933977 − 0.0716666i −1.09441 − 1.09441i 4.84778 + 2.79887i 0.583573 − 1.01078i
2.10 −0.0692757 − 0.258540i −1.49309 − 0.400072i 1.67001 − 0.964179i −2.09042 − 2.09042i 0.413739i −2.26261 + 1.73616i −0.743500 − 0.743500i −0.528818 − 0.305313i −0.395643 + 0.685274i
2.11 0.0371148 + 0.138514i 0.407739 + 0.109253i 1.71424 − 0.989718i 2.06635 + 2.06635i 0.0605326i 0.858916 − 0.659069i 0.403513 + 0.403513i −2.44376 − 1.41091i −0.209527 + 0.362912i
2.12 0.103325 + 0.385613i 2.49259 + 0.667887i 1.59403 − 0.920313i −0.673429 − 0.673429i 1.03018i −2.81988 + 2.16377i 1.08416 + 1.08416i 3.16885 + 1.82954i 0.190101 − 0.329265i
2.13 0.312380 + 1.16582i −1.80359 − 0.483270i 0.470498 − 0.271642i 0.138268 + 0.138268i − 2.25362i 1.25476 − 0.962814i 2.17054 + 2.17054i 0.421309 + 0.243243i −0.118003 + 0.204388i
2.14 0.365174 + 1.36285i 1.64953 + 0.441989i 0.00804676 − 0.00464580i −1.42518 − 1.42518i 2.40946i 3.20408 − 2.45858i 2.00462 + 2.00462i −0.0724922 − 0.0418534i 1.42187 − 2.46275i
2.15 0.393808 + 1.46971i −2.34258 − 0.627694i −0.272916 + 0.157568i 0.386161 + 0.386161i − 3.69011i −3.60979 + 2.76989i 1.81275 + 1.81275i 2.49563 + 1.44085i −0.415472 + 0.719618i
2.16 0.456332 + 1.70305i 1.41788 + 0.379919i −0.960099 + 0.554313i 0.286711 + 0.286711i 2.58809i −1.54437 + 1.18504i 1.11129 + 1.11129i −0.732035 − 0.422641i −0.357449 + 0.619120i
2.17 0.590754 + 2.20472i 0.415803 + 0.111414i −2.77976 + 1.60490i 2.55729 + 2.55729i 0.982547i 1.45041 − 1.11294i −1.95257 − 1.95257i −2.43760 − 1.40735i −4.12738 + 7.14883i
2.18 0.610972 + 2.28018i −2.28675 − 0.612733i −3.09388 + 1.78626i −2.05274 − 2.05274i − 5.58857i 2.61515 − 2.00668i −2.62485 − 2.62485i 2.25571 + 1.30234i 3.42644 − 5.93477i
2.19 0.655534 + 2.44649i −0.436959 − 0.117083i −3.82352 + 2.20751i 0.510985 + 0.510985i − 1.14577i −1.23402 + 0.946899i −4.32520 − 4.32520i −2.42085 − 1.39768i −0.915150 + 1.58509i
2.20 0.722236 + 2.69542i 3.10525 + 0.832050i −5.01162 + 2.89346i −1.52568 − 1.52568i 8.97090i 0.201148 − 0.154346i −7.47229 − 7.47229i 6.35222 + 3.66746i 3.01045 − 5.21425i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.m even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.m.a ✓ 160
241.m even 24 1 inner 241.2.m.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.m.a ✓ 160 1.a even 1 1 trivial
241.2.m.a ✓ 160 241.m even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(241, [\chi])\).