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Properties

Label 241.2.e.a
Level $241$
Weight $2$
Character orbit 241.e
Analytic conductor $1.924$
Analytic rank $0$
Dimension $72$
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(87,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.87");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.e (of order \(5\), degree \(4\), 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: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 16 q^{2} + q^{3} + 52 q^{4} + 3 q^{5} - 7 q^{6} + 7 q^{7} - 48 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 16 q^{2} + q^{3} + 52 q^{4} + 3 q^{5} - 7 q^{6} + 7 q^{7} - 48 q^{8} - 5 q^{9} - 3 q^{10} - 2 q^{11} + 4 q^{12} - 10 q^{13} - 11 q^{14} - 28 q^{15} + 12 q^{16} - q^{17} + 17 q^{18} - 8 q^{19} - 20 q^{20} - 12 q^{21} + 8 q^{22} + 9 q^{23} - 17 q^{24} + 17 q^{25} - 14 q^{26} + 7 q^{27} + 23 q^{28} + 25 q^{29} + 12 q^{30} + 7 q^{31} - 72 q^{32} - 12 q^{33} - 38 q^{34} - 14 q^{35} - 19 q^{36} + 28 q^{37} + 14 q^{38} + 29 q^{39} - 2 q^{40} + 8 q^{41} + 46 q^{42} - 33 q^{43} - 24 q^{44} + 11 q^{46} - q^{47} - 3 q^{48} - 5 q^{49} + 19 q^{50} + 32 q^{51} + 7 q^{52} - 11 q^{53} + 19 q^{54} - 4 q^{55} - 59 q^{56} - 12 q^{57} - 24 q^{58} + 17 q^{59} + 40 q^{60} + 29 q^{61} - 25 q^{62} - 66 q^{63} - 68 q^{64} - 94 q^{65} - 45 q^{66} + 4 q^{67} + 71 q^{68} + 2 q^{69} + 45 q^{70} + 3 q^{71} - 52 q^{72} + 31 q^{73} - 2 q^{74} - 57 q^{75} - 60 q^{76} - 29 q^{77} - 64 q^{78} - 34 q^{79} - 47 q^{80} + 26 q^{81} - 29 q^{82} + 26 q^{83} - 89 q^{84} + 26 q^{85} + 61 q^{86} - 11 q^{87} + 16 q^{88} + 76 q^{89} + 87 q^{90} + 49 q^{91} + 52 q^{92} + 77 q^{93} - 42 q^{94} - 67 q^{95} + 30 q^{96} - 54 q^{97} - 16 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1 −2.64454 −1.15953 + 0.842447i 4.99359 −1.19823 − 0.870566i 3.06642 − 2.22788i 0.0629932 + 0.193873i −7.91667 −0.292261 + 0.899487i 3.16877 + 2.30225i
87.2 −2.52090 2.54215 − 1.84698i 4.35493 −1.33961 − 0.973283i −6.40849 + 4.65604i −0.858993 − 2.64371i −5.93653 2.12413 − 6.53740i 3.37702 + 2.45355i
87.3 −2.17338 0.886115 − 0.643800i 2.72358 2.93664 + 2.13360i −1.92586 + 1.39922i 0.108070 + 0.332605i −1.57262 −0.556330 + 1.71221i −6.38244 − 4.63712i
87.4 −1.94649 −1.92573 + 1.39912i 1.78884 1.54347 + 1.12139i 3.74841 − 2.72338i 0.621613 + 1.91313i 0.411030 0.823830 − 2.53549i −3.00435 − 2.18279i
87.5 −1.79904 1.53379 − 1.11436i 1.23656 −1.36592 − 0.992403i −2.75936 + 2.00479i 1.44935 + 4.46065i 1.37347 0.183656 − 0.565236i 2.45736 + 1.78538i
87.6 −1.43943 0.260638 − 0.189365i 0.0719673 −0.383543 − 0.278660i −0.375172 + 0.272578i −0.757716 − 2.33201i 2.77527 −0.894978 + 2.75446i 0.552085 + 0.401113i
87.7 −0.812326 −0.823920 + 0.598613i −1.34013 1.02654 + 0.745827i 0.669292 − 0.486269i −1.57651 − 4.85200i 2.71327 −0.606545 + 1.86675i −0.833888 − 0.605855i
87.8 −0.804360 −2.02094 + 1.46830i −1.35301 −2.15417 − 1.56509i 1.62556 − 1.18104i 0.525012 + 1.61582i 2.69702 1.00124 − 3.08151i 1.73273 + 1.25890i
87.9 −0.438844 2.37124 − 1.72281i −1.80742 1.58626 + 1.15249i −1.04061 + 0.756044i −0.348234 − 1.07176i 1.67086 1.72767 − 5.31723i −0.696121 − 0.505762i
87.10 0.000507158 0 −0.125671 + 0.0913051i −2.00000 −0.990883 − 0.719919i −6.37349e−5 0 4.63061e-5i 0.917685 + 2.82435i −0.00202863 −0.919594 + 2.83022i −0.000502534 0 0.000365112i
87.11 0.175527 −2.27191 + 1.65064i −1.96919 3.34187 + 2.42801i −0.398783 + 0.289733i −0.313612 − 0.965198i −0.696701 1.50992 − 4.64706i 0.586589 + 0.426182i
87.12 0.442188 1.61164 − 1.17093i −1.80447 −2.88270 − 2.09440i 0.712650 − 0.517770i −0.689816 − 2.12303i −1.68229 0.299274 − 0.921069i −1.27470 − 0.926120i
87.13 0.894076 −0.149113 + 0.108337i −1.20063 1.97578 + 1.43549i −0.133318 + 0.0968612i 0.715659 + 2.20257i −2.86160 −0.916553 + 2.82086i 1.76650 + 1.28344i
87.14 1.44214 −2.03533 + 1.47875i 0.0797711 −2.44817 − 1.77870i −2.93523 + 2.13257i −0.847826 − 2.60934i −2.76924 1.02880 − 3.16632i −3.53061 − 2.56514i
87.15 1.45762 2.33737 − 1.69820i 0.124670 0.552368 + 0.401318i 3.40700 − 2.47533i 1.05215 + 3.23818i −2.73353 1.65236 − 5.08545i 0.805144 + 0.584972i
87.16 1.72703 1.00013 − 0.726639i 0.982633 2.15655 + 1.56682i 1.72726 − 1.25493i −1.12430 − 3.46024i −1.75702 −0.454790 + 1.39970i 3.72443 + 2.70595i
87.17 2.10956 −1.99508 + 1.44951i 2.45024 0.896639 + 0.651446i −4.20873 + 3.05782i 0.839971 + 2.58517i 0.949809 0.952206 − 2.93059i 1.89151 + 1.37426i
87.18 2.33066 0.773154 − 0.561729i 3.43199 −1.94387 − 1.41230i 1.80196 − 1.30920i 0.297449 + 0.915452i 3.33750 −0.644824 + 1.98456i −4.53051 − 3.29160i
91.1 −2.74209 0.249649 + 0.768341i 5.51903 −0.0348161 + 0.107153i −0.684559 − 2.10686i 3.39018 + 2.46311i −9.64950 1.89903 − 1.37972i 0.0954686 − 0.293822i
91.2 −2.39058 −0.449714 − 1.38408i 3.71489 1.13621 − 3.49689i 1.07508 + 3.30875i −2.84013 − 2.06347i −4.09958 0.713624 − 0.518478i −2.71620 + 8.35961i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.e even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.e.a ✓ 72
241.e even 5 1 inner 241.2.e.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.e.a ✓ 72 1.a even 1 1 trivial
241.2.e.a ✓ 72 241.e even 5 1 inner

Hecke kernels

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