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Properties

Label 41.8.g.a
Level $41$
Weight $8$
Character orbit 41.g
Analytic conductor $12.808$
Analytic rank $0$
Dimension $184$
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] = [41,8,Mod(2,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([13]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.2");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 41.g (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: \(12.8077860448\)
Analytic rank: \(0\)
Dimension: \(184\)
Relative dimension: \(23\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 10 q^{2} + 18 q^{3} + 2682 q^{4} - 10 q^{5} - 4118 q^{6} - 8 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q - 10 q^{2} + 18 q^{3} + 2682 q^{4} - 10 q^{5} - 4118 q^{6} - 8 q^{7} - 10 q^{8} - 7294 q^{10} - 2080 q^{11} + 2436 q^{12} + 17036 q^{13} - 71638 q^{14} + 37888 q^{15} - 139782 q^{16} - 25576 q^{17} + 35754 q^{18} - 27184 q^{19} - 385540 q^{20} - 10 q^{21} + 88692 q^{22} - 236652 q^{23} + 84590 q^{24} + 220000 q^{25} + 190952 q^{26} - 289086 q^{27} + 306310 q^{28} + 711196 q^{29} - 106384 q^{30} - 53444 q^{31} - 1649750 q^{33} - 1732310 q^{34} - 173052 q^{35} - 2340490 q^{36} + 1159256 q^{37} + 2616700 q^{38} + 2015470 q^{39} + 1904956 q^{40} - 1734208 q^{41} + 3297840 q^{42} - 1237880 q^{43} - 3295194 q^{44} - 272752 q^{45} + 7027970 q^{46} - 1599036 q^{47} + 2982266 q^{48} + 535330 q^{49} + 3417792 q^{51} + 14272392 q^{52} + 2355178 q^{53} - 8647672 q^{54} + 4702156 q^{55} - 7882622 q^{56} - 3290334 q^{57} - 3730788 q^{58} - 5673738 q^{59} - 19610308 q^{60} + 10093470 q^{61} - 10 q^{62} + 220292 q^{63} + 15024004 q^{64} - 10603432 q^{65} - 18857354 q^{66} + 12462434 q^{67} - 17750084 q^{68} + 2608458 q^{69} + 29281548 q^{70} + 470884 q^{71} + 5778652 q^{72} - 55858910 q^{74} - 13537612 q^{75} - 434306 q^{76} - 19138580 q^{77} + 31064074 q^{78} + 24234934 q^{79} - 1949230 q^{80} - 64809164 q^{81} + 73078540 q^{82} - 27122608 q^{83} + 102875190 q^{84} + 20942564 q^{85} - 17444218 q^{86} - 9548010 q^{87} - 98556240 q^{88} - 21700344 q^{89} - 70158970 q^{90} - 55718376 q^{92} + 89891034 q^{93} - 21115306 q^{94} - 18169408 q^{95} + 143981332 q^{96} + 6835954 q^{97} + 40837212 q^{98} - 13274914 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −19.8127 + 6.43755i −12.2742 + 12.2742i 247.548 − 179.854i 108.500 + 149.338i 164.170 − 322.202i −281.314 − 552.109i −2179.43 + 2999.73i 1885.69i −3111.06 − 2260.31i
2.2 −17.7448 + 5.76564i −17.7013 + 17.7013i 178.082 − 129.384i −175.291 − 241.268i 212.048 − 416.167i 480.923 + 943.864i −1010.28 + 1390.54i 1560.32i 4501.58 + 3270.59i
2.3 −17.1738 + 5.58012i 49.6501 − 49.6501i 160.249 − 116.427i 100.832 + 138.783i −575.629 + 1129.74i 440.682 + 864.888i −743.811 + 1023.77i − 2743.26i −2506.10 − 1820.79i
2.4 −16.3016 + 5.29671i 39.2964 − 39.2964i 134.133 − 97.4531i −271.618 − 373.851i −432.452 + 848.736i −613.199 − 1203.47i −380.803 + 524.131i − 901.415i 6407.99 + 4655.68i
2.5 −15.0405 + 4.88696i −62.1240 + 62.1240i 98.7803 − 71.7681i 178.966 + 246.326i 630.779 − 1237.97i −33.7707 − 66.2787i 54.8511 − 75.4960i − 5531.78i −3895.53 − 2830.27i
2.6 −10.7883 + 3.50532i 14.3032 − 14.3032i 0.545073 − 0.396018i 256.143 + 352.551i −104.169 + 204.444i −479.838 − 941.735i 848.949 − 1168.48i 1777.84i −3999.14 − 2905.55i
2.7 −10.4933 + 3.40949i −40.7079 + 40.7079i −5.06884 + 3.68273i −160.368 − 220.728i 288.368 − 565.954i −299.588 − 587.975i 870.743 − 1198.47i − 1127.26i 2435.36 + 1769.40i
2.8 −8.80293 + 2.86025i 22.7681 − 22.7681i −34.2436 + 24.8794i 6.04615 + 8.32181i −135.304 + 265.548i 36.3335 + 71.3086i 926.668 − 1275.45i 1150.23i −77.0263 − 55.9629i
2.9 −7.72282 + 2.50930i −21.6491 + 21.6491i −50.2087 + 36.4788i 116.205 + 159.942i 112.868 − 221.516i 754.671 + 1481.13i 907.157 − 1248.59i 1249.64i −1298.77 − 943.613i
2.10 −2.69321 + 0.875078i 22.3769 − 22.3769i −97.0665 + 70.5230i −224.215 − 308.606i −40.6841 + 79.8471i 361.669 + 709.816i 412.763 − 568.120i 1185.55i 873.913 + 634.935i
2.11 −1.34366 + 0.436581i 62.7001 − 62.7001i −101.939 + 74.0633i −20.6782 − 28.4611i −56.8738 + 111.621i −258.167 − 506.682i 210.932 − 290.322i − 5675.59i 40.2099 + 29.2142i
2.12 1.02397 − 0.332708i −24.2757 + 24.2757i −102.616 + 74.5551i −59.3521 − 81.6912i −16.7809 + 32.9344i −559.158 − 1097.41i −161.276 + 221.977i 1008.38i −87.9541 − 63.9024i
2.13 1.79829 − 0.584300i −37.4505 + 37.4505i −100.662 + 73.1350i 194.908 + 268.268i −45.4646 + 89.2293i 104.846 + 205.772i −280.546 + 386.139i − 618.081i 507.251 + 368.539i
2.14 5.57990 − 1.81302i −62.8194 + 62.8194i −75.7060 + 55.0036i −257.320 − 354.171i −236.633 + 464.418i 608.861 + 1194.96i −764.125 + 1051.73i − 5705.55i −2077.94 − 1509.71i
2.15 5.91861 − 1.92307i 24.6212 − 24.6212i −72.2224 + 52.4726i 262.234 + 360.934i 98.3750 − 193.072i −210.393 − 412.919i −794.759 + 1093.89i 974.593i 2246.16 + 1631.93i
2.16 7.11485 − 2.31175i 39.6556 − 39.6556i −58.2773 + 42.3409i 160.467 + 220.864i 190.470 − 373.818i 624.096 + 1224.86i −879.596 + 1210.66i − 958.134i 1652.28 + 1200.45i
2.17 9.34960 − 3.03787i 6.07651 − 6.07651i −25.3678 + 18.4308i −173.143 − 238.312i 38.3533 − 75.2726i −84.5945 − 166.026i −920.820 + 1267.40i 2113.15i −2342.78 − 1702.13i
2.18 14.7986 − 4.80835i 41.7661 − 41.7661i 92.3233 − 67.0768i −22.4940 − 30.9603i 417.253 − 818.905i −670.238 − 1315.42i −126.967 + 174.754i − 1301.82i −481.746 − 350.009i
2.19 14.8256 − 4.81714i −59.2178 + 59.2178i 93.0408 − 67.5981i 129.880 + 178.764i −592.681 + 1163.20i −678.121 − 1330.89i −119.073 + 163.889i − 4826.48i 2786.68 + 2024.64i
2.20 15.2715 − 4.96200i −24.0933 + 24.0933i 105.042 − 76.3174i 57.3421 + 78.9247i −248.389 + 487.490i 311.247 + 610.856i 17.3556 − 23.8879i 1026.03i 1267.32 + 920.763i
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.8.g.a ✓ 184
41.g even 20 1 inner 41.8.g.a ✓ 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.8.g.a ✓ 184 1.a even 1 1 trivial
41.8.g.a ✓ 184 41.g even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(41, [\chi])\).