↑

Properties

Label 41.10.b.a
Level $41$
Weight $10$
Character orbit 41.b
Analytic conductor $21.116$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [41,10,Mod(40,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.40");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 41.b (of order \(2\), degree \(1\), 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: \(21.1164692827\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{2} + 7166 q^{4} - 1028 q^{5} + 7338 q^{8} - 162774 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{2} + 7166 q^{4} - 1028 q^{5} + 7338 q^{8} - 162774 q^{9} - 93712 q^{10} + 2229126 q^{16} - 556354 q^{18} - 769120 q^{20} + 1442016 q^{21} - 3101352 q^{23} + 5184442 q^{25} - 2041304 q^{31} + 33907754 q^{32} - 25082912 q^{33} - 21434034 q^{36} + 55804068 q^{37} + 29457720 q^{39} - 70642512 q^{40} - 22046202 q^{41} - 3550388 q^{42} - 31648768 q^{43} + 112258332 q^{45} + 102840944 q^{46} + 34766618 q^{49} + 71493050 q^{50} - 204235488 q^{51} - 181255672 q^{57} - 281334496 q^{59} - 171694980 q^{61} - 45594368 q^{62} + 918376598 q^{64} - 852066260 q^{66} - 966302838 q^{72} + 153880612 q^{73} + 865501760 q^{74} + 360157864 q^{77} + 1902337040 q^{78} - 2801006464 q^{80} + 81029782 q^{81} - 97561770 q^{82} - 647433336 q^{83} - 880161700 q^{84} + 147000760 q^{86} + 4159077048 q^{87} + 3598608000 q^{90} + 231429256 q^{91} - 2434987536 q^{92} + 58556054 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −43.1975 − 116.439i 1354.03 −141.297 5029.86i 4885.90i −36373.5 6125.04 6103.70
40.2 −43.1975 116.439i 1354.03 −141.297 − 5029.86i − 4885.90i −36373.5 6125.04 6103.70
40.3 −33.9187 − 258.262i 638.476 749.627 8759.90i − 2779.26i −4289.89 −47016.2 −25426.3
40.4 −33.9187 258.262i 638.476 749.627 − 8759.90i 2779.26i −4289.89 −47016.2 −25426.3
40.5 −32.3364 − 130.772i 533.644 −1694.14 4228.70i − 3134.67i −699.897 2581.71 54782.3
40.6 −32.3364 130.772i 533.644 −1694.14 − 4228.70i 3134.67i −699.897 2581.71 54782.3
40.7 −31.8453 − 7.91734i 502.122 2058.26 252.130i 10173.2i 314.578 19620.3 −65545.8
40.8 −31.8453 7.91734i 502.122 2058.26 − 252.130i − 10173.2i 314.578 19620.3 −65545.8
40.9 −18.4707 − 112.559i −170.833 −1394.73 2079.05i 9202.83i 12612.4 7013.41 25761.7
40.10 −18.4707 112.559i −170.833 −1394.73 − 2079.05i − 9202.83i 12612.4 7013.41 25761.7
40.11 −12.2311 − 109.628i −362.400 468.830 1340.87i − 5083.77i 10694.9 7664.69 −5734.31
40.12 −12.2311 109.628i −362.400 468.830 − 1340.87i 5083.77i 10694.9 7664.69 −5734.31
40.13 −9.46833 − 217.334i −422.351 1510.83 2057.79i 3131.89i 8846.74 −27551.0 −14305.0
40.14 −9.46833 217.334i −422.351 1510.83 − 2057.79i − 3131.89i 8846.74 −27551.0 −14305.0
40.15 2.02233 − 228.190i −507.910 −2606.28 − 461.475i − 4605.63i −2062.59 −32387.8 −5270.74
40.16 2.02233 228.190i −507.910 −2606.28 461.475i 4605.63i −2062.59 −32387.8 −5270.74
40.17 8.63628 − 11.8710i −437.415 −717.809 − 102.521i − 6784.62i −8199.41 19542.1 −6199.20
40.18 8.63628 11.8710i −437.415 −717.809 102.521i 6784.62i −8199.41 19542.1 −6199.20
40.19 12.2276 − 96.3878i −362.486 2276.10 − 1178.59i − 2948.33i −10692.9 10392.4 27831.2
40.20 12.2276 96.3878i −362.486 2276.10 1178.59i 2948.33i −10692.9 10392.4 27831.2
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.10.b.a ✓ 30
41.b even 2 1 inner 41.10.b.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.10.b.a ✓ 30 1.a even 1 1 trivial
41.10.b.a ✓ 30 41.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(41, [\chi])\).