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Properties

Label 241.12.a.b
Level $241$
Weight $12$
Character orbit 241.a
Self dual yes
Analytic conductor $185.171$
Analytic rank $0$
Dimension $113$
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] = [241,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(185.170706515\)
Analytic rank: \(0\)
Dimension: \(113\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 113 q + 183 q^{2} + 1225 q^{3} + 123397 q^{4} + 34760 q^{5} + 77385 q^{6} + 115273 q^{7} + 404991 q^{8} + 7430154 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 113 q + 183 q^{2} + 1225 q^{3} + 123397 q^{4} + 34760 q^{5} + 77385 q^{6} + 115273 q^{7} + 404991 q^{8} + 7430154 q^{9} + 245121 q^{10} + 3697942 q^{11} + 4206235 q^{12} + 1451845 q^{13} + 8791216 q^{14} + 16407437 q^{15} + 139387709 q^{16} + 4508018 q^{17} + 36053217 q^{18} + 14328594 q^{19} + 128529150 q^{20} + 40620666 q^{21} + 43226342 q^{22} + 225936530 q^{23} + 134662191 q^{24} + 1210650331 q^{25} + 214226716 q^{26} + 441735991 q^{27} + 209176593 q^{28} + 618976096 q^{29} + 505356985 q^{30} + 726126822 q^{31} + 917513051 q^{32} + 427863436 q^{33} + 949657903 q^{34} + 1377714243 q^{35} + 9080249384 q^{36} + 667909384 q^{37} + 1437129276 q^{38} + 4617222143 q^{39} - 528854516 q^{40} + 3559758807 q^{41} + 1359885861 q^{42} + 1635216502 q^{43} + 11086165252 q^{44} + 5592511417 q^{45} + 7459952203 q^{46} + 9045396422 q^{47} + 12080896405 q^{48} + 35424514914 q^{49} + 7719115877 q^{50} + 5132074673 q^{51} - 179510505 q^{52} + 8882383325 q^{53} + 10972215702 q^{54} + 7670932333 q^{55} + 35802566703 q^{56} + 13776489842 q^{57} + 25496978857 q^{58} + 48708939654 q^{59} + 42622577617 q^{60} + 3792970484 q^{61} - 41262070855 q^{62} - 14824784456 q^{63} + 96086036551 q^{64} + 22261303815 q^{65} + 46966343428 q^{66} + 47152989450 q^{67} + 105685434305 q^{68} + 53911088742 q^{69} + 222311116528 q^{70} + 199796274614 q^{71} + 298241434869 q^{72} + 107310911904 q^{73} + 202614247364 q^{74} + 265381089216 q^{75} + 274496710064 q^{76} + 129381932347 q^{77} + 542958089106 q^{78} + 151780593705 q^{79} + 399990109419 q^{80} + 532958458529 q^{81} + 141564344559 q^{82} + 204656892112 q^{83} + 305065972498 q^{84} + 87942032162 q^{85} + 146834700511 q^{86} + 64940359644 q^{87} - 41627026796 q^{88} + 122850400509 q^{89} - 554630029535 q^{90} + 141982382196 q^{91} + 313151927950 q^{92} + 30268671458 q^{93} - 691191090322 q^{94} + 517011446775 q^{95} - 666595503329 q^{96} - 43162160137 q^{97} - 310094765604 q^{98} + 351924947972 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −88.9119 −140.256 5857.32 −402.606 12470.4 21934.7 −338694. −157475. 35796.5
1.2 −88.7455 601.343 5827.76 13613.9 −53366.5 −64649.4 −335437. 184466. −1.20817e6
1.3 −87.0571 361.418 5530.94 −3188.17 −31464.0 31017.8 −303215. −46523.8 277553.
1.4 −85.8694 345.195 5325.55 4242.24 −29641.7 −44162.3 −281441. −57987.4 −364279.
1.5 −84.7492 −96.3305 5134.42 −4479.43 8163.93 −67946.0 −261571. −167867. 379628.
1.6 −81.7690 −694.852 4638.16 5240.06 56817.3 −88348.3 −211795. 305672. −428475.
1.7 −80.0275 293.610 4356.40 7788.78 −23496.9 1849.12 −184735. −90940.1 −623316.
1.8 −79.5984 647.372 4287.91 −3130.80 −51529.8 −20093.5 −178293. 241943. 249206.
1.9 −79.4922 −783.128 4271.00 −38.7288 62252.5 21137.5 −176711. 436142. 3078.63
1.10 −78.7001 −493.459 4145.70 51.9372 38835.3 73400.3 −165090. 66355.1 −4087.46
1.11 −78.6352 130.049 4135.49 −11725.3 −10226.4 51789.5 −164150. −160234. 922024.
1.12 −75.7781 190.224 3694.32 4307.14 −14414.8 −59046.4 −124755. −140962. −326387.
1.13 −75.1131 −508.987 3593.97 −9757.34 38231.6 −64238.9 −116123. 81920.6 732904.
1.14 −70.6764 −479.573 2947.15 −2807.83 33894.5 81231.4 −63548.6 52843.2 198447.
1.15 −69.4474 683.032 2774.94 9551.68 −47434.8 31490.9 −50484.3 289386. −663340.
1.16 −68.2644 −230.593 2612.03 −11800.5 15741.3 19110.3 −38502.8 −123974. 805556.
1.17 −68.0042 −647.263 2576.58 11486.8 44016.6 29211.2 −35945.5 241803. −781151.
1.18 −67.3124 591.123 2482.96 7568.22 −39789.9 57572.8 −29278.5 172279. −509435.
1.19 −67.1666 −371.911 2463.35 −925.774 24980.0 −12741.9 −27897.5 −38829.2 62181.0
1.20 −64.3490 757.114 2092.80 −9052.99 −48719.6 72325.4 −2882.53 396075. 582551.
See next 80 embeddings (of 113 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.113
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 241.12.a.b ✓ 113
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.12.a.b ✓ 113 1.a even 1 1 trivial