↑

Properties

Label 41.10.c.a
Level $41$
Weight $10$
Character orbit 41.c
Analytic conductor $21.116$
Analytic rank $0$
Dimension $62$
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(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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.9");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(21.1164692827\)
Analytic rank: \(0\)
Dimension: \(62\)
Relative dimension: \(31\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 62 q - 148 q^{3} - 15876 q^{4} - 4372 q^{6} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 62 q - 148 q^{3} - 15876 q^{4} - 4372 q^{6} - 2 q^{7} + 64940 q^{10} - 101136 q^{11} + 88610 q^{12} - 212164 q^{13} - 486724 q^{14} - 280448 q^{15} + 4065276 q^{16} + 989130 q^{17} + 631900 q^{18} - 557304 q^{19} + 878738 q^{22} - 2585064 q^{23} + 8160792 q^{24} - 13556490 q^{25} - 4638930 q^{26} - 4560520 q^{27} - 15062678 q^{28} - 645332 q^{29} + 24289990 q^{30} - 10831104 q^{31} + 3975452 q^{34} + 29170524 q^{35} - 3116208 q^{37} + 29714800 q^{38} - 35676784 q^{40} + 38183180 q^{41} - 88501252 q^{42} + 88194656 q^{44} - 44608688 q^{45} - 37901682 q^{47} - 65214646 q^{48} + 22786536 q^{51} + 67233198 q^{52} + 98223480 q^{53} - 222847538 q^{54} + 341535988 q^{55} + 37845552 q^{56} + 543416000 q^{57} - 39712038 q^{58} + 476227372 q^{59} - 73813254 q^{60} - 329401706 q^{63} - 1473830612 q^{64} - 283361396 q^{65} + 1055141156 q^{66} + 137841616 q^{67} - 1702495104 q^{68} - 643435264 q^{69} - 1945980994 q^{70} + 43024650 q^{71} + 298006608 q^{72} + 511623892 q^{75} + 2630680074 q^{76} + 2932953920 q^{78} - 559013342 q^{79} - 3103395126 q^{81} - 1636160042 q^{82} + 4760276320 q^{83} + 234551440 q^{85} + 4427686276 q^{86} - 1446984286 q^{88} - 373768662 q^{89} - 546085324 q^{92} + 246761008 q^{93} - 931400058 q^{94} - 2067661068 q^{95} - 7929456644 q^{96} - 949806418 q^{97} + 5988587276 q^{98} + 1756047684 q^{99}+O(q^{100}) \) 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 − 44.0476i −107.424 − 107.424i −1428.19 1964.96i −4731.77 + 4731.77i −6990.74 − 6990.74i 40355.9i 3396.85i 86551.8
9.2 − 43.5939i 116.221 + 116.221i −1388.43 − 347.886i 5066.51 − 5066.51i 4266.58 + 4266.58i 38206.9i 7331.44i −15165.7
9.3 − 40.6130i −147.297 − 147.297i −1137.42 − 1955.60i −5982.18 + 5982.18i 6867.80 + 6867.80i 25400.1i 23709.9i −79422.8
9.4 − 35.0737i 75.3048 + 75.3048i −718.166 217.529i 2641.22 − 2641.22i −4539.45 − 4539.45i 7231.03i − 8341.37i 7629.54
9.5 − 34.0948i −25.2088 − 25.2088i −650.456 − 1459.50i −859.489 + 859.489i −3092.91 − 3092.91i 4720.64i − 18412.0i −49761.4
9.6 − 32.5005i 7.26533 + 7.26533i −544.284 1433.15i 236.127 − 236.127i 5548.45 + 5548.45i 1049.26i − 19577.4i 46578.0
9.7 − 28.6799i 172.163 + 172.163i −310.538 2765.16i 4937.61 − 4937.61i −1759.10 − 1759.10i − 5777.92i 39597.0i 79304.4
9.8 − 28.6591i −133.710 − 133.710i −309.346 1241.77i −3832.03 + 3832.03i 1725.41 + 1725.41i − 5807.88i 16074.0i 35588.0
9.9 − 25.7433i 159.796 + 159.796i −150.720 − 1915.98i 4113.68 − 4113.68i −929.116 − 929.116i − 9300.56i 31386.3i −49323.6
9.10 − 18.8163i −187.426 − 187.426i 157.945 − 836.977i −3526.68 + 3526.68i −7855.38 − 7855.38i − 12605.9i 50574.3i −15748.8
9.11 − 18.0564i −83.0950 − 83.0950i 185.966 − 415.379i −1500.40 + 1500.40i 2030.90 + 2030.90i − 12602.8i − 5873.43i −7500.27
9.12 − 12.4598i 32.3176 + 32.3176i 356.754 1203.69i 402.670 − 402.670i −5019.01 − 5019.01i − 10824.5i − 17594.1i 14997.8
9.13 − 12.1812i 95.7860 + 95.7860i 363.618 − 100.810i 1166.79 − 1166.79i 7261.36 + 7261.36i − 10666.1i − 1333.07i −1227.99
9.14 − 7.65294i −33.9566 − 33.9566i 453.432 − 2680.21i −259.868 + 259.868i 1109.50 + 1109.50i − 7388.40i − 17376.9i −20511.5
9.15 − 3.11710i −103.776 − 103.776i 502.284 2428.58i −323.482 + 323.482i −43.4878 − 43.4878i − 3161.63i 1856.11i 7570.15
9.16 − 2.91537i 83.6684 + 83.6684i 503.501 − 244.198i 243.925 − 243.925i −6830.06 − 6830.06i − 2960.56i − 5682.18i −711.928
9.17 5.80705i −164.374 − 164.374i 478.278 − 651.280i 954.527 − 954.527i 5590.28 + 5590.28i 5750.59i 34354.6i 3782.01
9.18 6.02559i 182.688 + 182.688i 475.692 583.218i −1100.80 + 1100.80i −581.237 − 581.237i 5951.43i 47066.7i −3514.23
9.19 11.0747i −54.4217 − 54.4217i 389.352 201.486i 602.701 − 602.701i 523.559 + 523.559i 9982.16i − 13759.6i −2231.39
9.20 13.9877i 64.9677 + 64.9677i 316.344 2181.64i −908.750 + 908.750i 4287.58 + 4287.58i 11586.6i − 11241.4i −30516.2
See all 62 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.31
Significant digits:
Format:

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 41.10.c.a ✓ 62
41.c even 4 1 inner 41.10.c.a ✓ 62
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.10.c.a ✓ 62 1.a even 1 1 trivial
41.10.c.a ✓ 62 41.c even 4 1 inner

Hecke kernels

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