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Properties

Label 68.3.f.a
Level $68$
Weight $3$
Character orbit 68.f
Analytic conductor $1.853$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [68,3,Mod(47,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 68.f (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: \(1.85286579765\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{4} - 8 q^{5} - 22 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{4} - 8 q^{5} - 22 q^{6} - 14 q^{10} + 26 q^{12} - 8 q^{13} + 28 q^{14} - 60 q^{16} + 16 q^{17} + 12 q^{18} + 6 q^{20} - 8 q^{21} + 2 q^{22} - 22 q^{24} - 44 q^{28} + 16 q^{29} - 136 q^{30} + 72 q^{33} - 50 q^{34} - 24 q^{37} + 80 q^{38} + 282 q^{40} - 224 q^{41} + 118 q^{44} - 176 q^{46} + 86 q^{48} + 388 q^{50} - 332 q^{52} + 224 q^{54} + 100 q^{56} - 184 q^{57} + 238 q^{58} - 88 q^{61} + 28 q^{62} + 92 q^{64} - 296 q^{65} - 102 q^{68} + 408 q^{69} + 252 q^{72} + 176 q^{73} - 402 q^{74} - 524 q^{78} + 210 q^{80} + 552 q^{81} - 168 q^{82} - 1016 q^{84} + 260 q^{86} - 566 q^{88} - 8 q^{89} - 326 q^{90} - 328 q^{92} + 38 q^{96} + 56 q^{97} - 560 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.94997 + 0.444526i −0.571697 + 0.571697i 3.60479 − 1.73363i 0.237302 + 0.237302i 0.860659 − 1.36893i 5.04525 + 5.04525i −6.25861 + 4.98295i 8.34633i −0.568220 − 0.357246i
47.2 −1.85239 − 0.754082i 3.43488 − 3.43488i 2.86272 + 2.79371i 5.06370 + 5.06370i −8.95293 + 3.77256i −2.03562 − 2.03562i −3.19619 − 7.33378i − 14.5968i −5.56151 − 13.1984i
47.3 −1.79166 − 0.888799i −0.489773 + 0.489773i 2.42007 + 3.18485i −4.70828 − 4.70828i 1.31282 − 0.442196i −4.96891 − 4.96891i −1.50525 − 7.85711i 8.52024i 4.25091 + 12.6203i
47.4 −1.47284 + 1.35305i 2.80972 − 2.80972i 0.338524 − 3.98565i −3.83971 − 3.83971i −0.336587 + 7.93994i −7.94331 − 7.94331i 4.89418 + 6.32827i − 6.78900i 10.8506 + 0.459974i
47.5 −1.15440 + 1.63321i −3.27297 + 3.27297i −1.33474 − 3.77074i −1.26728 − 1.26728i −1.56714 − 9.12373i −1.40415 − 1.40415i 7.69922 + 2.17301i − 12.4246i 3.53268 − 0.606791i
47.6 −0.977713 − 1.74473i −1.61477 + 1.61477i −2.08815 + 3.41169i 3.28978 + 3.28978i 4.39611 + 1.23855i 2.43706 + 2.43706i 7.99408 + 0.307608i 3.78506i 2.52331 − 8.95624i
47.7 −0.618260 + 1.90204i 1.53396 − 1.53396i −3.23551 − 2.35191i 3.77530 + 3.77530i 1.96926 + 3.86603i 6.38155 + 6.38155i 6.47381 − 4.69998i 4.29396i −9.51488 + 4.84665i
47.8 −0.464918 − 1.94521i 3.01125 − 3.01125i −3.56770 + 1.80873i −4.55081 − 4.55081i −7.25749 − 4.45753i 6.08115 + 6.08115i 5.17705 + 6.09903i − 9.13519i −6.73654 + 10.9680i
47.9 0.464918 − 1.94521i −3.01125 + 3.01125i −3.56770 − 1.80873i −4.55081 − 4.55081i 4.45753 + 7.25749i −6.08115 − 6.08115i −5.17705 + 6.09903i − 9.13519i −10.9680 + 6.73654i
47.10 0.618260 + 1.90204i −1.53396 + 1.53396i −3.23551 + 2.35191i 3.77530 + 3.77530i −3.86603 − 1.96926i −6.38155 − 6.38155i −6.47381 − 4.69998i 4.29396i −4.84665 + 9.51488i
47.11 0.977713 − 1.74473i 1.61477 − 1.61477i −2.08815 − 3.41169i 3.28978 + 3.28978i −1.23855 − 4.39611i −2.43706 − 2.43706i −7.99408 + 0.307608i 3.78506i 8.95624 − 2.52331i
47.12 1.15440 + 1.63321i 3.27297 − 3.27297i −1.33474 + 3.77074i −1.26728 − 1.26728i 9.12373 + 1.56714i 1.40415 + 1.40415i −7.69922 + 2.17301i − 12.4246i 0.606791 − 3.53268i
47.13 1.47284 + 1.35305i −2.80972 + 2.80972i 0.338524 + 3.98565i −3.83971 − 3.83971i −7.93994 + 0.336587i 7.94331 + 7.94331i −4.89418 + 6.32827i − 6.78900i −0.459974 − 10.8506i
47.14 1.79166 − 0.888799i 0.489773 − 0.489773i 2.42007 − 3.18485i −4.70828 − 4.70828i 0.442196 − 1.31282i 4.96891 + 4.96891i 1.50525 − 7.85711i 8.52024i −12.6203 − 4.25091i
47.15 1.85239 − 0.754082i −3.43488 + 3.43488i 2.86272 − 2.79371i 5.06370 + 5.06370i −3.77256 + 8.95293i 2.03562 + 2.03562i 3.19619 − 7.33378i − 14.5968i 13.1984 + 5.56151i
47.16 1.94997 + 0.444526i 0.571697 − 0.571697i 3.60479 + 1.73363i 0.237302 + 0.237302i 1.36893 − 0.860659i −5.04525 − 5.04525i 6.25861 + 4.98295i 8.34633i 0.357246 + 0.568220i
55.1 −1.94997 − 0.444526i −0.571697 − 0.571697i 3.60479 + 1.73363i 0.237302 − 0.237302i 0.860659 + 1.36893i 5.04525 − 5.04525i −6.25861 − 4.98295i − 8.34633i −0.568220 + 0.357246i
55.2 −1.85239 + 0.754082i 3.43488 + 3.43488i 2.86272 − 2.79371i 5.06370 − 5.06370i −8.95293 − 3.77256i −2.03562 + 2.03562i −3.19619 + 7.33378i 14.5968i −5.56151 + 13.1984i
55.3 −1.79166 + 0.888799i −0.489773 − 0.489773i 2.42007 − 3.18485i −4.70828 + 4.70828i 1.31282 + 0.442196i −4.96891 + 4.96891i −1.50525 + 7.85711i − 8.52024i 4.25091 − 12.6203i
55.4 −1.47284 − 1.35305i 2.80972 + 2.80972i 0.338524 + 3.98565i −3.83971 + 3.83971i −0.336587 − 7.93994i −7.94331 + 7.94331i 4.89418 − 6.32827i 6.78900i 10.8506 − 0.459974i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.c even 4 1 inner
68.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.3.f.a ✓ 32
4.b odd 2 1 inner 68.3.f.a ✓ 32
17.c even 4 1 inner 68.3.f.a ✓ 32
68.f odd 4 1 inner 68.3.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.3.f.a ✓ 32 1.a even 1 1 trivial
68.3.f.a ✓ 32 4.b odd 2 1 inner
68.3.f.a ✓ 32 17.c even 4 1 inner
68.3.f.a ✓ 32 68.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(68, [\chi])\).