Properties

Label 241.6.a.a
Level $241$
Weight $6$
Character orbit 241.a
Self dual yes
Analytic conductor $38.653$
Analytic rank $1$
Dimension $47$
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] = [241,6,Mod(1,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 241.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: \(38.6525005749\)
Analytic rank: \(1\)
Dimension: \(47\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 47 q - 21 q^{2} - 35 q^{3} + 645 q^{4} - 268 q^{5} - 231 q^{6} - 451 q^{7} - 993 q^{8} + 2976 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 47 q - 21 q^{2} - 35 q^{3} + 645 q^{4} - 268 q^{5} - 231 q^{6} - 451 q^{7} - 993 q^{8} + 2976 q^{9} - 383 q^{10} - 3386 q^{11} - 2165 q^{12} - 515 q^{13} - 2576 q^{14} - 6439 q^{15} + 6461 q^{16} - 2446 q^{17} - 3771 q^{18} - 3766 q^{19} - 15882 q^{20} - 5262 q^{21} - 5554 q^{22} - 20950 q^{23} - 12153 q^{24} + 22065 q^{25} - 12932 q^{26} - 12197 q^{27} - 13199 q^{28} - 18872 q^{29} - 31247 q^{30} - 28986 q^{31} - 31957 q^{32} - 5348 q^{33} - 15241 q^{34} - 33681 q^{35} + 6896 q^{36} - 29828 q^{37} - 36468 q^{38} - 67477 q^{39} - 26804 q^{40} - 47709 q^{41} - 25587 q^{42} - 3306 q^{43} - 133196 q^{44} - 44951 q^{45} - 48573 q^{46} - 108634 q^{47} - 86915 q^{48} + 61808 q^{49} - 57631 q^{50} - 97171 q^{51} + 15975 q^{52} - 90955 q^{53} - 106218 q^{54} - 91839 q^{55} - 129681 q^{56} - 29254 q^{57} - 89511 q^{58} - 212922 q^{59} - 211919 q^{60} - 96388 q^{61} - 317095 q^{62} - 413072 q^{63} - 371097 q^{64} - 213225 q^{65} - 528956 q^{66} - 233030 q^{67} - 481975 q^{68} - 286062 q^{69} - 594432 q^{70} - 617818 q^{71} - 684723 q^{72} - 236152 q^{73} - 467428 q^{74} - 379416 q^{75} - 542552 q^{76} - 215189 q^{77} - 741318 q^{78} - 386211 q^{79} - 676197 q^{80} - 193633 q^{81} - 287713 q^{82} - 289856 q^{83} - 842126 q^{84} - 236394 q^{85} - 446489 q^{86} - 551988 q^{87} - 151532 q^{88} - 245283 q^{89} - 240527 q^{90} - 548076 q^{91} - 766850 q^{92} - 247594 q^{93} - 215802 q^{94} - 490461 q^{95} - 359681 q^{96} - 82501 q^{97} - 142488 q^{98} - 813124 q^{99}+O(q^{100}) \) 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 −10.7206 11.5622 82.9310 −65.4773 −123.953 98.2220 −546.010 −109.317 701.956
1.2 −10.2895 26.3385 73.8735 −43.0235 −271.010 −187.960 −430.856 450.718 442.690
1.3 −10.2411 −23.2816 72.8806 −9.01991 238.430 168.636 −418.664 299.033 92.3741
1.4 −9.93843 −19.3294 66.7723 −104.407 192.103 −127.795 −345.582 130.624 1037.64
1.5 −9.92645 −12.6407 66.5344 62.3940 125.477 188.856 −342.804 −83.2134 −619.351
1.6 −9.63170 21.5246 60.7697 21.9387 −207.319 −64.7970 −277.101 220.310 −211.307
1.7 −9.03131 −24.7487 49.5645 88.0875 223.513 −61.1400 −158.630 369.498 −795.546
1.8 −8.56012 −10.4236 41.2757 −41.0951 89.2271 −137.755 −79.4012 −134.349 351.779
1.9 −8.38800 3.30188 38.3585 −47.3447 −27.6961 224.163 −53.3350 −232.098 397.127
1.10 −7.94385 22.5005 31.1048 46.4850 −178.741 51.0662 7.11130 263.273 −369.270
1.11 −7.40336 13.2418 22.8098 34.5473 −98.0338 −151.594 68.0385 −67.6549 −255.767
1.12 −7.15052 −11.7753 19.1300 36.8177 84.1996 −10.3585 92.0272 −104.342 −263.266
1.13 −6.39218 3.09349 8.85997 −97.8228 −19.7741 −190.960 147.915 −233.430 625.301
1.14 −6.09017 5.61699 5.09012 70.8471 −34.2084 179.955 163.886 −211.449 −431.471
1.15 −5.45931 −11.8068 −2.19591 83.2099 64.4571 −5.76248 186.686 −103.599 −454.269
1.16 −3.90261 16.1434 −16.7696 −47.7344 −63.0012 130.876 190.329 17.6081 186.289
1.17 −3.85750 −23.7750 −17.1197 −53.9863 91.7120 −208.479 189.479 322.250 208.252
1.18 −3.57364 −12.1250 −19.2291 −91.1682 43.3305 66.0046 183.074 −95.9836 325.802
1.19 −3.36464 21.4437 −20.6792 −7.37461 −72.1504 −58.5773 177.247 216.832 24.8130
1.20 −2.85211 16.8003 −23.8655 −2.03317 −47.9163 −89.3811 159.334 39.2509 5.79881
See all 47 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.47
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(241\) \(1\)

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 241.6.a.a 47
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.6.a.a 47 1.a even 1 1 trivial