↑

Properties

Label 51.6.e.a
Level $51$
Weight $6$
Character orbit 51.e
Analytic conductor $8.180$
Analytic rank $0$
Dimension $32$
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] = [51,6,Mod(4,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 51.e (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: \(8.17957481046\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 576 q^{4} + 76 q^{5} - 180 q^{6} + 236 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 576 q^{4} + 76 q^{5} - 180 q^{6} + 236 q^{7} + 724 q^{10} - 672 q^{11} - 2684 q^{13} + 2832 q^{14} + 8824 q^{16} - 4868 q^{17} + 1296 q^{18} + 4464 q^{20} - 3528 q^{21} - 1628 q^{22} + 5584 q^{23} + 10872 q^{24} - 36792 q^{28} + 4820 q^{29} - 1728 q^{30} + 13880 q^{31} + 7236 q^{33} + 5068 q^{34} - 12152 q^{35} - 33328 q^{37} - 29256 q^{38} - 11160 q^{39} - 43288 q^{40} - 50932 q^{41} + 116000 q^{44} - 6156 q^{45} + 47596 q^{46} - 16552 q^{47} + 27360 q^{48} + 139080 q^{50} + 12924 q^{51} + 186888 q^{52} - 14580 q^{54} - 129428 q^{55} - 117024 q^{56} + 19440 q^{57} - 212912 q^{58} + 118952 q^{61} - 219736 q^{62} + 19116 q^{63} - 143848 q^{64} + 220976 q^{65} + 98632 q^{67} + 367680 q^{68} + 156060 q^{69} - 205512 q^{71} - 62208 q^{72} - 40924 q^{73} - 178760 q^{74} - 58896 q^{75} + 87012 q^{78} - 309916 q^{79} - 664992 q^{80} - 209952 q^{81} + 350116 q^{82} + 6264 q^{84} + 335336 q^{85} + 1139576 q^{86} - 307696 q^{88} - 62680 q^{89} + 58644 q^{90} + 174460 q^{91} - 94256 q^{92} + 367240 q^{95} - 508824 q^{96} - 299020 q^{97} - 1191536 q^{98} - 54432 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 − 10.8134i 6.36396 − 6.36396i −84.9307 −46.9107 + 46.9107i −68.8164 − 68.8164i 105.836 + 105.836i 572.363i − 81.0000i 507.266 + 507.266i
4.2 − 8.79415i −6.36396 + 6.36396i −45.3370 −31.3579 + 31.3579i 55.9656 + 55.9656i 72.5068 + 72.5068i 117.288i − 81.0000i 275.766 + 275.766i
4.3 − 8.22028i 6.36396 − 6.36396i −35.5730 66.1764 − 66.1764i −52.3135 − 52.3135i 13.8481 + 13.8481i 29.3711i − 81.0000i −543.989 − 543.989i
4.4 − 6.59084i −6.36396 + 6.36396i −11.4392 50.2012 − 50.2012i 41.9439 + 41.9439i 116.537 + 116.537i − 135.513i − 81.0000i −330.868 − 330.868i
4.5 − 5.57214i 6.36396 − 6.36396i 0.951217 −32.5739 + 32.5739i −35.4609 − 35.4609i −135.342 − 135.342i − 183.609i − 81.0000i 181.507 + 181.507i
4.6 − 4.47439i −6.36396 + 6.36396i 11.9798 −16.4382 + 16.4382i 28.4749 + 28.4749i −104.696 − 104.696i − 196.783i − 81.0000i 73.5510 + 73.5510i
4.7 − 1.96981i 6.36396 − 6.36396i 28.1198 6.47739 − 6.47739i −12.5358 − 12.5358i 125.155 + 125.155i − 118.425i − 81.0000i −12.7593 − 12.7593i
4.8 0.447157i −6.36396 + 6.36396i 31.8001 24.1504 − 24.1504i −2.84569 − 2.84569i −32.5034 − 32.5034i 28.5286i − 81.0000i 10.7990 + 10.7990i
4.9 2.45426i 6.36396 − 6.36396i 25.9766 43.6768 − 43.6768i 15.6188 + 15.6188i −148.036 − 148.036i 142.290i − 81.0000i 107.194 + 107.194i
4.10 2.86766i 6.36396 − 6.36396i 23.7765 −61.2889 + 61.2889i 18.2497 + 18.2497i 0.538399 + 0.538399i 159.948i − 81.0000i −175.756 − 175.756i
4.11 4.92532i −6.36396 + 6.36396i 7.74124 32.1773 − 32.1773i −31.3445 − 31.3445i 151.888 + 151.888i 195.738i − 81.0000i 158.484 + 158.484i
4.12 5.19230i −6.36396 + 6.36396i 5.04003 −41.1274 + 41.1274i −33.0436 − 33.0436i −81.2277 − 81.2277i 192.323i − 81.0000i −213.546 − 213.546i
4.13 8.37886i 6.36396 − 6.36396i −38.2052 50.4918 − 50.4918i 53.3227 + 53.3227i 101.302 + 101.302i − 51.9926i − 81.0000i 423.064 + 423.064i
4.14 9.73313i −6.36396 + 6.36396i −62.7338 74.8108 − 74.8108i −61.9412 − 61.9412i −147.938 − 147.938i − 299.136i − 81.0000i 728.143 + 728.143i
4.15 9.80384i 6.36396 − 6.36396i −64.1153 −34.6261 + 34.6261i 62.3913 + 62.3913i −73.5970 − 73.5970i − 314.853i − 81.0000i −339.469 − 339.469i
4.16 10.6326i −6.36396 + 6.36396i −81.0511 −45.8392 + 45.8392i −67.6651 − 67.6651i 153.729 + 153.729i − 521.539i − 81.0000i −487.388 − 487.388i
13.1 − 10.6326i −6.36396 − 6.36396i −81.0511 −45.8392 − 45.8392i −67.6651 + 67.6651i 153.729 − 153.729i 521.539i 81.0000i −487.388 + 487.388i
13.2 − 9.80384i 6.36396 + 6.36396i −64.1153 −34.6261 − 34.6261i 62.3913 − 62.3913i −73.5970 + 73.5970i 314.853i 81.0000i −339.469 + 339.469i
13.3 − 9.73313i −6.36396 − 6.36396i −62.7338 74.8108 + 74.8108i −61.9412 + 61.9412i −147.938 + 147.938i 299.136i 81.0000i 728.143 − 728.143i
13.4 − 8.37886i 6.36396 + 6.36396i −38.2052 50.4918 + 50.4918i 53.3227 − 53.3227i 101.302 − 101.302i 51.9926i 81.0000i 423.064 − 423.064i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.6.e.a ✓ 32
3.b odd 2 1 153.6.f.c 32
17.c even 4 1 inner 51.6.e.a ✓ 32
17.d even 8 1 867.6.a.n 16
17.d even 8 1 867.6.a.o 16
51.f odd 4 1 153.6.f.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.6.e.a ✓ 32 1.a even 1 1 trivial
51.6.e.a ✓ 32 17.c even 4 1 inner
153.6.f.c 32 3.b odd 2 1
153.6.f.c 32 51.f odd 4 1
867.6.a.n 16 17.d even 8 1
867.6.a.o 16 17.d even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(51, [\chi])\).