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Properties

Label 241.8.a.a
Level $241$
Weight $8$
Character orbit 241.a
Self dual yes
Analytic conductor $75.285$
Analytic rank $1$
Dimension $67$
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] = [241,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(75.2847911417\)
Analytic rank: \(1\)
Dimension: \(67\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 67 q - 33 q^{2} - 95 q^{3} + 3813 q^{4} - 1320 q^{5} - 1815 q^{6} - 3233 q^{7} - 6273 q^{8} + 38442 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 67 q - 33 q^{2} - 95 q^{3} + 3813 q^{4} - 1320 q^{5} - 1815 q^{6} - 3233 q^{7} - 6273 q^{8} + 38442 q^{9} - 2559 q^{10} - 36302 q^{11} - 12293 q^{12} - 13283 q^{13} - 59568 q^{14} - 59503 q^{15} + 191485 q^{16} - 44242 q^{17} - 89271 q^{18} - 75226 q^{19} - 232370 q^{20} - 98094 q^{21} - 72922 q^{22} - 490672 q^{23} - 334401 q^{24} + 860001 q^{25} - 439156 q^{26} - 315425 q^{27} - 612655 q^{28} - 836508 q^{29} - 112535 q^{30} - 834568 q^{31} - 778757 q^{32} - 710048 q^{33} - 454625 q^{34} - 999597 q^{35} + 661496 q^{36} - 229370 q^{37} - 1276180 q^{38} - 3336349 q^{39} - 468916 q^{40} - 1641475 q^{41} - 1718091 q^{42} - 1172854 q^{43} - 6462300 q^{44} - 2620433 q^{45} - 3061509 q^{46} - 3784046 q^{47} - 1521275 q^{48} + 5799118 q^{49} - 4451043 q^{50} - 3027559 q^{51} - 3808521 q^{52} - 3253453 q^{53} - 6395658 q^{54} - 2853847 q^{55} - 10653521 q^{56} - 6375334 q^{57} - 5299639 q^{58} - 11095546 q^{59} - 6310703 q^{60} + 3781012 q^{61} + 8303743 q^{62} + 2582820 q^{63} + 33373431 q^{64} + 193115 q^{65} + 21003910 q^{66} - 1392022 q^{67} + 5452483 q^{68} - 3142142 q^{69} + 3315024 q^{70} - 35280386 q^{71} - 9035071 q^{72} - 3069604 q^{73} - 20796004 q^{74} - 17223212 q^{75} - 13953054 q^{76} - 12611287 q^{77} - 36061904 q^{78} - 22272291 q^{79} - 64114985 q^{80} + 8334163 q^{81} - 22943793 q^{82} - 29501732 q^{83} - 78746254 q^{84} - 45256860 q^{85} - 62121171 q^{86} - 44916852 q^{87} - 94290852 q^{88} - 47825385 q^{89} - 136862131 q^{90} - 45821860 q^{91} - 82615152 q^{92} - 75530800 q^{93} - 121422914 q^{94} - 105643731 q^{95} - 144157853 q^{96} - 51660607 q^{97} - 108206468 q^{98} - 122297768 q^{99}+O(q^{100}) \) 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 −22.4377 17.3992 375.450 −488.813 −390.398 −949.862 −5552.21 −1884.27 10967.8
1.2 −22.3447 22.8424 371.286 222.131 −510.407 −200.661 −5436.15 −1665.23 −4963.45
1.3 −21.4781 −59.8355 333.310 −327.575 1285.16 1024.08 −4409.68 1393.29 7035.70
1.4 −20.9047 58.8813 309.006 −69.2114 −1230.89 1605.36 −3783.87 1280.01 1446.84
1.5 −20.4270 68.0690 289.262 351.046 −1390.44 −904.040 −3294.10 2446.39 −7170.81
1.6 −19.1744 −7.99779 239.658 −72.5808 153.353 −743.124 −2140.99 −2123.04 1391.69
1.7 −18.8548 −36.5589 227.503 −78.1521 689.310 −1668.13 −1876.11 −850.449 1473.54
1.8 −18.3746 41.6123 209.627 257.225 −764.609 −1125.57 −1499.86 −455.420 −4726.41
1.9 −17.9165 −75.4804 193.000 340.322 1352.34 −43.7137 −1164.56 3510.29 −6097.37
1.10 −17.3313 −91.2051 172.374 78.2512 1580.70 1065.05 −769.060 6131.37 −1356.19
1.11 −16.1774 0.768286 133.710 −550.633 −12.4289 1472.67 −92.3679 −2186.41 8907.83
1.12 −15.6011 52.9447 115.395 −196.354 −825.997 390.770 196.649 616.143 3063.34
1.13 −14.9597 −58.7283 95.7912 143.592 878.556 793.622 481.832 1262.02 −2148.09
1.14 −14.3440 23.8967 77.7496 166.552 −342.774 825.267 720.790 −1615.95 −2389.02
1.15 −13.9256 −57.3689 65.9215 −258.078 798.895 71.5078 864.479 1104.19 3593.89
1.16 −12.7741 92.1274 35.1768 36.9431 −1176.84 −72.7889 1185.73 6300.46 −471.913
1.17 −12.7230 −7.62066 33.8745 256.459 96.9576 627.231 1197.56 −2128.93 −3262.93
1.18 −12.3096 81.6901 23.5268 −476.132 −1005.57 549.430 1286.03 4486.27 5861.01
1.19 −12.2788 −46.0825 22.7692 157.495 565.838 1768.91 1292.11 −63.4064 −1933.85
1.20 −12.1574 −77.0365 19.8018 217.721 936.562 −1494.73 1315.41 3747.63 −2646.91
See all 67 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.67
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 241.8.a.a ✓ 67
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.8.a.a ✓ 67 1.a even 1 1 trivial