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Properties

Label 41.6.c.a
Level $41$
Weight $6$
Character orbit 41.c
Analytic conductor $6.576$
Analytic rank $0$
Dimension $34$
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] = [41,6,Mod(9,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.9");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 41.c (of order \(4\), degree \(2\), 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: \(6.57573661233\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 20 q^{3} - 548 q^{4} - 124 q^{6} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 20 q^{3} - 548 q^{4} - 124 q^{6} - 2 q^{7} + 1660 q^{10} + 654 q^{11} + 98 q^{12} + 854 q^{13} + 2236 q^{14} - 998 q^{15} + 8828 q^{16} + 1730 q^{17} + 2068 q^{18} + 716 q^{19} - 7750 q^{22} - 8156 q^{23} + 5496 q^{24} - 28870 q^{25} + 11470 q^{26} - 26806 q^{27} - 5942 q^{28} - 19190 q^{29} + 21430 q^{30} - 6376 q^{31} + 4228 q^{34} + 20934 q^{35} + 768 q^{37} - 6552 q^{38} - 47024 q^{40} - 9618 q^{41} + 126236 q^{42} - 57632 q^{44} + 81112 q^{45} + 2372 q^{47} + 88202 q^{48} + 24936 q^{51} + 151982 q^{52} - 94170 q^{53} - 153650 q^{54} - 105662 q^{55} + 8144 q^{56} - 69412 q^{57} - 63094 q^{58} + 87684 q^{59} + 58746 q^{60} - 76328 q^{63} - 109780 q^{64} + 88904 q^{65} - 427468 q^{66} + 212150 q^{67} - 199904 q^{68} + 125672 q^{69} + 113566 q^{70} + 189392 q^{71} + 579216 q^{72} + 199072 q^{75} + 128522 q^{76} - 327232 q^{78} - 28880 q^{79} - 557850 q^{81} - 241122 q^{82} + 384572 q^{83} - 114880 q^{85} - 453980 q^{86} + 338338 q^{88} - 391266 q^{89} + 955316 q^{92} + 394684 q^{93} - 42650 q^{94} - 404378 q^{95} - 254372 q^{96} - 324538 q^{97} - 167276 q^{98} - 685716 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 − 10.9013i −5.43752 − 5.43752i −86.8384 − 24.5024i −59.2761 + 59.2761i 59.9844 + 59.9844i 597.810i − 183.867i −267.108
9.2 − 8.99972i 17.9662 + 17.9662i −48.9949 59.9700i 161.691 − 161.691i 107.188 + 107.188i 152.949i 402.568i 539.713
9.3 − 7.92145i 7.97023 + 7.97023i −30.7494 − 9.82957i 63.1358 − 63.1358i −151.212 − 151.212i − 9.90681i − 115.951i −77.8644
9.4 − 7.89466i −15.6318 − 15.6318i −30.3257 75.0683i −123.408 + 123.408i −100.122 − 100.122i − 13.2179i 245.709i 592.639
9.5 − 5.81352i 3.20738 + 3.20738i −1.79697 − 42.7691i 18.6461 − 18.6461i 22.3392 + 22.3392i − 175.586i − 222.425i −248.639
9.6 − 5.68402i −19.1932 − 19.1932i −0.308102 − 102.985i −109.094 + 109.094i 57.7111 + 57.7111i − 180.137i 493.754i −585.370
9.7 − 3.95940i −4.71456 − 4.71456i 16.3231 87.8737i −18.6668 + 18.6668i 158.504 + 158.504i − 191.331i − 198.546i 347.927
9.8 − 1.51674i 21.1352 + 21.1352i 29.6995 − 77.0125i 32.0565 − 32.0565i 44.5407 + 44.5407i − 93.5820i 650.390i −116.808
9.9 0.0172797i −10.8720 − 10.8720i 31.9997 3.25644i 0.187866 − 0.187866i −62.9759 − 62.9759i 1.10590i − 6.59809i −0.0562704
9.10 0.127223i 14.0147 + 14.0147i 31.9838 90.6024i −1.78299 + 1.78299i −136.937 − 136.937i 8.14022i 149.822i −11.5267
9.11 0.668322i 4.32517 + 4.32517i 31.5533 − 28.4441i −2.89060 + 2.89060i 39.7770 + 39.7770i 42.4741i − 205.586i 19.0098
9.12 5.24792i −17.0066 − 17.0066i 4.45931 13.4881i 89.2494 − 89.2494i 50.5983 + 50.5983i 191.336i 335.450i −70.7843
9.13 6.27235i 9.76614 + 9.76614i −7.34234 22.9635i −61.2566 + 61.2566i 80.8932 + 80.8932i 154.661i − 52.2451i −144.035
9.14 6.29735i −0.450543 − 0.450543i −7.65657 − 106.704i 2.83723 − 2.83723i −126.086 − 126.086i 153.299i − 242.594i 671.954
9.15 9.30165i −5.19740 − 5.19740i −54.5206 78.5132i 48.3444 − 48.3444i −115.965 − 115.965i − 209.479i − 188.974i −730.302
9.16 10.2709i 20.0569 + 20.0569i −73.4913 − 29.1061i −206.002 + 206.002i −68.6428 − 68.6428i − 426.153i 561.559i 298.946
9.17 10.4878i −9.93818 − 9.93818i −77.9945 − 58.3823i 104.230 − 104.230i 139.405 + 139.405i − 482.383i − 45.4650i 612.304
32.1 − 10.4878i −9.93818 + 9.93818i −77.9945 58.3823i 104.230 + 104.230i 139.405 − 139.405i 482.383i 45.4650i 612.304
32.2 − 10.2709i 20.0569 − 20.0569i −73.4913 29.1061i −206.002 − 206.002i −68.6428 + 68.6428i 426.153i − 561.559i 298.946
32.3 − 9.30165i −5.19740 + 5.19740i −54.5206 − 78.5132i 48.3444 + 48.3444i −115.965 + 115.965i 209.479i 188.974i −730.302
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.6.c.a ✓ 34
41.c even 4 1 inner 41.6.c.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.6.c.a ✓ 34 1.a even 1 1 trivial
41.6.c.a ✓ 34 41.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(41, [\chi])\).