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Properties

Label 241.4.l.a
Level $241$
Weight $4$
Character orbit 241.l
Analytic conductor $14.219$
Analytic rank $0$
Dimension $480$
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] = [241,4,Mod(6,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.6");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 241.l (of order \(20\), 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: \(14.2194603114\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{3} - 1960 q^{4} - 10 q^{5} + 106 q^{6} - 40 q^{7} + 1190 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{3} - 1960 q^{4} - 10 q^{5} + 106 q^{6} - 40 q^{7} + 1190 q^{9} + 10 q^{10} - 30 q^{11} - 200 q^{12} - 88 q^{13} + 62 q^{14} + 356 q^{15} + 8248 q^{16} - 26 q^{17} + 70 q^{18} - 596 q^{19} - 570 q^{20} + 446 q^{21} - 198 q^{22} - 948 q^{24} + 1846 q^{25} + 234 q^{26} - 10 q^{27} - 172 q^{28} - 10 q^{29} - 108 q^{31} + 414 q^{33} - 1584 q^{34} + 362 q^{35} - 7442 q^{36} - 912 q^{37} + 942 q^{38} + 190 q^{39} - 76 q^{40} - 10 q^{41} + 3802 q^{42} + 382 q^{43} - 458 q^{44} - 280 q^{45} + 1392 q^{46} - 10 q^{47} + 1320 q^{48} + 1190 q^{49} + 1850 q^{50} + 2150 q^{51} + 2056 q^{52} + 470 q^{53} + 3818 q^{54} - 474 q^{55} + 746 q^{56} + 586 q^{57} + 1174 q^{58} - 1110 q^{59} - 3168 q^{60} - 3530 q^{61} - 3540 q^{62} - 572 q^{63} - 30936 q^{64} + 7722 q^{65} - 268 q^{66} + 3860 q^{67} + 1774 q^{68} - 7390 q^{69} + 6346 q^{70} - 212 q^{71} - 8600 q^{72} + 802 q^{73} + 3624 q^{74} + 4660 q^{75} + 10816 q^{76} - 4250 q^{77} - 7726 q^{78} - 1250 q^{79} + 7700 q^{80} - 6228 q^{81} + 2822 q^{82} + 6850 q^{83} - 6626 q^{84} + 746 q^{85} + 14 q^{86} - 2566 q^{87} + 5850 q^{88} - 1350 q^{89} - 480 q^{90} - 1078 q^{91} + 5410 q^{92} - 1270 q^{93} - 2830 q^{94} - 2226 q^{95} + 14492 q^{96} + 4716 q^{97} - 682 q^{98} + 8202 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 − 5.43744i −8.71742 − 2.83246i −21.5657 8.02439 − 2.60728i −15.4013 + 47.4004i 0.140911 − 0.889677i 73.7627i 46.1271 + 33.5133i −14.1769 − 43.6321i
6.2 − 5.39315i 3.26132 + 1.05967i −21.0861 3.81423 − 1.23932i 5.71494 − 17.5888i 4.44762 − 28.0812i 70.5750i −12.3302 − 8.95838i −6.68382 − 20.5707i
6.3 − 5.24721i 8.70976 + 2.82997i −19.5332 16.8575 − 5.47733i 14.8495 − 45.7019i −5.00286 + 31.5868i 60.5171i 46.0078 + 33.4266i −28.7407 − 88.4547i
6.4 − 5.12174i −1.39929 − 0.454658i −18.2323 −1.79124 + 0.582010i −2.32864 + 7.16682i −3.59462 + 22.6955i 52.4070i −20.0921 − 14.5978i 2.98091 + 9.17429i
6.5 − 5.10495i −3.74663 − 1.21736i −18.0605 −18.6948 + 6.07432i −6.21454 + 19.1264i 1.63685 − 10.3347i 51.3585i −9.28815 − 6.74823i 31.0091 + 95.4363i
6.6 − 4.88246i 7.17944 + 2.33274i −15.8384 −12.9862 + 4.21946i 11.3895 − 35.0534i −0.200118 + 1.26350i 38.2709i 24.2593 + 17.6254i 20.6014 + 63.4045i
6.7 − 4.79602i −2.45247 − 0.796855i −15.0018 18.7999 − 6.10844i −3.82173 + 11.7621i −0.959050 + 6.05520i 33.5806i −16.4638 − 11.9617i −29.2962 − 90.1644i
6.8 − 4.38477i 5.62014 + 1.82610i −11.2262 5.05605 − 1.64281i 8.00700 − 24.6430i 2.16461 − 13.6668i 14.1461i 6.40792 + 4.65563i −7.20334 − 22.1696i
6.9 − 4.00444i −8.72794 − 2.83588i −8.03551 −9.94745 + 3.23212i −11.3561 + 34.9505i −1.84489 + 11.6482i 0.142209i 46.2912 + 33.6326i 12.9428 + 39.8339i
6.10 − 4.00161i −5.16474 − 1.67813i −8.01286 7.05912 − 2.29365i −6.71520 + 20.6673i 4.28106 − 27.0295i 0.0514781i 2.01497 + 1.46396i −9.17828 − 28.2478i
6.11 − 3.91882i 0.586333 + 0.190511i −7.35714 −0.392752 + 0.127613i 0.746578 − 2.29773i −1.26854 + 8.00924i − 2.51926i −21.5360 − 15.6468i 0.500091 + 1.53912i
6.12 − 3.79472i −5.87698 − 1.90955i −6.39992 2.07643 − 0.674674i −7.24620 + 22.3015i 3.48733 − 22.0181i − 6.07186i 9.04911 + 6.57456i −2.56020 − 7.87949i
6.13 − 3.68420i 6.98766 + 2.27043i −5.57332 −17.1608 + 5.57587i 8.36471 − 25.7439i −1.27701 + 8.06271i − 8.94036i 21.8291 + 15.8598i 20.5426 + 63.2237i
6.14 − 3.54891i 3.03932 + 0.987536i −4.59474 −3.89545 + 1.26571i 3.50467 − 10.7863i −4.50462 + 28.4411i − 12.0850i −13.5812 − 9.86732i 4.49188 + 13.8246i
6.15 − 3.25737i −1.57299 − 0.511095i −2.61046 −12.4034 + 4.03011i −1.66483 + 5.12381i 2.48904 − 15.7152i − 17.5557i −19.6304 − 14.2623i 13.1275 + 40.4024i
6.16 − 3.24273i −6.30302 − 2.04798i −2.51533 15.8065 − 5.13585i −6.64104 + 20.4390i −3.45395 + 21.8074i − 17.7853i 13.6904 + 9.94665i −16.6542 − 51.2564i
6.17 − 3.07869i 4.02143 + 1.30664i −1.47831 16.8403 − 5.47173i 4.02274 − 12.3807i 0.828297 − 5.22966i − 20.0782i −7.37887 − 5.36106i −16.8458 − 51.8459i
6.18 − 2.80316i 9.26647 + 3.01086i 0.142309 6.15659 − 2.00040i 8.43991 − 25.9754i 2.85184 − 18.0058i − 22.8242i 54.9588 + 39.9299i −5.60743 − 17.2579i
6.19 − 2.00769i −6.37017 − 2.06979i 3.96917 6.20933 − 2.01753i −4.15551 + 12.7893i −2.94910 + 18.6199i − 24.0304i 14.4515 + 10.4996i −4.05059 − 12.4664i
6.20 − 1.99525i 4.72936 + 1.53666i 4.01896 −11.2852 + 3.66677i 3.06603 − 9.43628i 4.97729 − 31.4253i − 23.9809i −1.83794 − 1.33534i 7.31614 + 22.5168i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.4.l.a ✓ 480
241.l even 20 1 inner 241.4.l.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.4.l.a ✓ 480 1.a even 1 1 trivial
241.4.l.a ✓ 480 241.l even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(241, [\chi])\).