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Properties

Label 68.3.g.e
Level $68$
Weight $3$
Character orbit 68.g
Analytic conductor $1.853$
Analytic rank $0$
Dimension $48$
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(15,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.15");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 68.g (of order \(8\), degree \(4\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} + 8 q^{5} + 36 q^{6} - 36 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} + 8 q^{5} + 36 q^{6} - 36 q^{8} + 40 q^{9} - 44 q^{10} - 40 q^{12} - 4 q^{14} - 8 q^{16} - 8 q^{17} + 56 q^{18} + 156 q^{20} + 20 q^{22} - 280 q^{24} - 240 q^{25} - 260 q^{26} - 188 q^{28} + 112 q^{29} - 52 q^{32} - 16 q^{33} + 164 q^{34} + 156 q^{36} + 72 q^{37} + 168 q^{40} + 176 q^{41} + 448 q^{42} + 300 q^{44} - 144 q^{45} + 500 q^{46} + 84 q^{48} + 24 q^{49} - 480 q^{50} + 120 q^{52} - 240 q^{53} + 36 q^{54} + 180 q^{56} - 936 q^{57} + 224 q^{58} + 240 q^{60} - 440 q^{61} + 244 q^{62} + 584 q^{65} - 268 q^{66} - 204 q^{68} + 432 q^{69} - 600 q^{70} + 864 q^{73} + 84 q^{74} - 480 q^{76} + 152 q^{77} - 136 q^{78} - 520 q^{80} + 360 q^{82} - 128 q^{84} - 416 q^{85} - 1440 q^{86} + 168 q^{88} - 452 q^{90} - 532 q^{92} - 40 q^{93} - 264 q^{94} - 700 q^{96} - 88 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −1.98899 + 0.209613i 3.81369 − 1.57968i 3.91212 − 0.833835i −1.70415 + 0.705880i −7.25425 + 3.94136i 2.44274 − 5.89729i −7.60638 + 2.47852i 5.68488 − 5.68488i 3.24156 − 1.76120i
15.2 −1.97514 + 0.314369i −1.56664 + 0.648923i 3.80234 − 1.24185i 1.13809 − 0.471410i 2.89033 − 1.77422i −4.44731 + 10.7367i −7.11976 + 3.64816i −4.33070 + 4.33070i −2.09968 + 1.28888i
15.3 −1.58451 − 1.22038i −3.85971 + 1.59874i 1.02136 + 3.86740i 5.55446 − 2.30073i 8.06683 + 2.17707i 2.53795 − 6.12716i 3.10133 − 7.37440i 5.97742 − 5.97742i −11.6089 − 3.13299i
15.4 −1.50795 + 1.31380i −3.47219 + 1.43823i 0.547845 − 3.96231i −2.32297 + 0.962204i 3.34635 − 6.73055i 4.98490 − 12.0346i 4.37956 + 6.69473i 3.62364 − 3.62364i 2.23878 − 4.50288i
15.5 −1.22038 − 1.58451i 3.85971 − 1.59874i −1.02136 + 3.86740i 5.55446 − 2.30073i −7.24353 − 4.16469i −2.53795 + 6.12716i 7.37440 − 3.10133i 5.97742 − 5.97742i −10.4241 − 5.99336i
15.6 −0.952847 + 1.75843i 2.29433 − 0.950341i −2.18417 − 3.35103i 6.60775 − 2.73702i −0.515031 + 4.93995i 0.112326 − 0.271178i 7.97374 − 0.647689i −2.00318 + 2.00318i −1.48331 + 14.2272i
15.7 0.209613 − 1.98899i −3.81369 + 1.57968i −3.91212 − 0.833835i −1.70415 + 0.705880i 2.34257 + 7.91650i −2.44274 + 5.89729i −2.47852 + 7.60638i 5.68488 − 5.68488i 1.04677 + 3.53748i
15.8 0.314369 − 1.97514i 1.56664 − 0.648923i −3.80234 − 1.24185i 1.13809 − 0.471410i −0.789210 − 3.29833i 4.44731 − 10.7367i −3.64816 + 7.11976i −4.33070 + 4.33070i −0.573322 − 2.39607i
15.9 0.801004 + 1.83259i 5.17793 − 2.14477i −2.71679 + 2.93583i −4.03055 + 1.66951i 8.07802 + 7.77106i 0.784755 − 1.89457i −7.55633 − 2.62715i 15.8469 − 15.8469i −6.28802 − 6.04907i
15.10 1.31380 − 1.50795i 3.47219 − 1.43823i −0.547845 − 3.96231i −2.32297 + 0.962204i 2.39299 − 7.12545i −4.98490 + 12.0346i −6.69473 − 4.37956i 3.62364 − 3.62364i −1.60096 + 4.76707i
15.11 1.75843 − 0.952847i −2.29433 + 0.950341i 2.18417 − 3.35103i 6.60775 − 2.73702i −3.12889 + 3.85725i −0.112326 + 0.271178i 0.647689 − 7.97374i −2.00318 + 2.00318i 9.01132 − 11.1090i
15.12 1.83259 + 0.801004i −5.17793 + 2.14477i 2.71679 + 2.93583i −4.03055 + 1.66951i −11.2070 − 0.217057i −0.784755 + 1.89457i 2.62715 + 7.55633i 15.8469 − 15.8469i −8.72364 − 0.168959i
19.1 −1.99177 − 0.181232i −1.96587 − 4.74603i 3.93431 + 0.721946i 0.351391 + 0.848334i 3.05543 + 9.80929i −7.38074 − 3.05720i −7.70541 − 2.15097i −12.2962 + 12.2962i −0.546146 − 1.75337i
19.2 −1.91496 − 0.577001i 0.192601 + 0.464981i 3.33414 + 2.20987i −2.12349 − 5.12655i −0.100529 − 1.00155i 9.00459 + 3.72982i −5.10964 − 6.15561i 6.18485 − 6.18485i 1.10836 + 11.0424i
19.3 −1.89093 + 0.651435i 1.53980 + 3.71741i 3.15126 − 2.46364i 1.08393 + 2.61684i −5.33331 − 6.02629i −1.45805 − 0.603944i −4.35393 + 6.71143i −5.08418 + 5.08418i −3.75435 − 4.24216i
19.4 −1.44615 + 1.38154i −0.324616 − 0.783692i 0.182676 − 3.99583i −1.79167 − 4.32548i 1.55215 + 0.684862i −7.39353 − 3.06250i 5.25623 + 6.03092i 5.85516 − 5.85516i 8.56685 + 3.78000i
19.5 −1.10561 + 1.66662i −1.13913 − 2.75011i −1.55527 − 3.68526i 2.80733 + 6.77750i 5.84282 + 1.14203i 10.8207 + 4.48208i 7.86146 + 1.48240i 0.0984968 − 0.0984968i −14.3993 − 2.81448i
19.6 −0.577001 − 1.91496i −0.192601 − 0.464981i −3.33414 + 2.20987i −2.12349 − 5.12655i −0.779288 + 0.637118i −9.00459 − 3.72982i 6.15561 + 5.10964i 6.18485 − 6.18485i −8.59188 + 7.02441i
19.7 −0.181232 − 1.99177i 1.96587 + 4.74603i −3.93431 + 0.721946i 0.351391 + 0.848334i 9.09673 − 4.77570i 7.38074 + 3.05720i 2.15097 + 7.70541i −12.2962 + 12.2962i 1.62600 − 0.853637i
19.8 0.462186 + 1.94586i −1.81586 − 4.38389i −3.57277 + 1.79870i −3.57014 − 8.61908i 7.69117 − 5.55960i 5.82236 + 2.41170i −5.15131 − 6.12078i −9.55713 + 9.55713i 15.1215 − 10.9306i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.d even 8 1 inner
68.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.3.g.e ✓ 48
4.b odd 2 1 inner 68.3.g.e ✓ 48
17.d even 8 1 inner 68.3.g.e ✓ 48
68.g odd 8 1 inner 68.3.g.e ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.3.g.e ✓ 48 1.a even 1 1 trivial
68.3.g.e ✓ 48 4.b odd 2 1 inner
68.3.g.e ✓ 48 17.d even 8 1 inner
68.3.g.e ✓ 48 68.g odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(68, [\chi])\):

\( T_{3}^{48} - 20 T_{3}^{46} + 200 T_{3}^{44} + 3192 T_{3}^{42} + 920532 T_{3}^{40} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
\( T_{5}^{24} - 4 T_{5}^{23} + 68 T_{5}^{22} - 784 T_{5}^{21} + 4424 T_{5}^{20} - 19336 T_{5}^{19} + \cdots + 24562253185024 \) Copy content Toggle raw display