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Properties

Label 68.6.h.a
Level $68$
Weight $6$
Character orbit 68.h
Analytic conductor $10.906$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [68,6,Mod(9,68)]
 
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([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.9");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 68.h (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: \(10.9060997473\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 44 q^{5} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 44 q^{5} + 44 q^{9} - 600 q^{11} - 4372 q^{15} + 1212 q^{17} + 284 q^{19} + 1672 q^{23} - 3676 q^{25} - 20796 q^{27} - 11316 q^{29} + 16632 q^{31} + 32888 q^{33} - 6464 q^{35} + 19564 q^{37} - 15880 q^{39} - 14620 q^{41} - 17628 q^{43} + 64172 q^{45} - 64812 q^{49} + 3256 q^{51} + 5396 q^{53} + 35236 q^{57} + 57012 q^{59} + 70364 q^{61} + 93844 q^{63} - 56064 q^{65} + 62528 q^{67} - 159656 q^{69} - 56144 q^{71} + 61828 q^{73} + 92856 q^{75} + 5236 q^{77} - 181064 q^{79} + 129980 q^{83} - 250260 q^{85} - 498484 q^{87} - 20472 q^{91} - 336876 q^{93} - 3044 q^{95} + 317004 q^{97} + 268076 q^{99}+O(q^{100}) \) 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 0 −26.9790 − 11.1751i 0 28.5599 − 68.9498i 0 −85.1002 − 205.450i 0 431.157 + 431.157i 0
9.2 0 −16.8745 − 6.98965i 0 −20.7374 + 50.0645i 0 16.1796 + 39.0610i 0 64.0668 + 64.0668i 0
9.3 0 −8.12020 − 3.36350i 0 4.98604 − 12.0374i 0 17.5239 + 42.3064i 0 −117.202 − 117.202i 0
9.4 0 1.09028 + 0.451609i 0 37.3851 − 90.2555i 0 63.2435 + 152.683i 0 −170.842 − 170.842i 0
9.5 0 8.19879 + 3.39605i 0 −35.1418 + 84.8399i 0 −19.3921 − 46.8167i 0 −116.140 − 116.140i 0
9.6 0 12.2250 + 5.06374i 0 11.5618 − 27.9127i 0 −82.3932 − 198.915i 0 −48.0189 − 48.0189i 0
9.7 0 22.6815 + 9.39498i 0 −2.17857 + 5.25954i 0 48.2192 + 116.411i 0 254.358 + 254.358i 0
25.1 0 −7.82515 + 18.8916i 0 −27.5210 − 11.3996i 0 −77.1634 + 31.9621i 0 −123.832 − 123.832i 0
25.2 0 −7.17240 + 17.3157i 0 77.4309 + 32.0729i 0 91.1642 − 37.7614i 0 −76.5634 − 76.5634i 0
25.3 0 −1.40601 + 3.39440i 0 −69.4607 − 28.7716i 0 120.687 − 49.9901i 0 162.282 + 162.282i 0
25.4 0 2.54560 − 6.14561i 0 8.67300 + 3.59247i 0 −214.730 + 88.9439i 0 140.538 + 140.538i 0
25.5 0 3.99622 − 9.64772i 0 −5.75150 − 2.38235i 0 94.7274 − 39.2374i 0 94.7182 + 94.7182i 0
25.6 0 6.22649 − 15.0321i 0 76.2008 + 31.5634i 0 32.9168 − 13.6346i 0 −15.3674 − 15.3674i 0
25.7 0 11.4134 − 27.5544i 0 −62.0065 − 25.6839i 0 −5.88291 + 2.43678i 0 −457.154 − 457.154i 0
49.1 0 −7.82515 − 18.8916i 0 −27.5210 + 11.3996i 0 −77.1634 − 31.9621i 0 −123.832 + 123.832i 0
49.2 0 −7.17240 − 17.3157i 0 77.4309 − 32.0729i 0 91.1642 + 37.7614i 0 −76.5634 + 76.5634i 0
49.3 0 −1.40601 − 3.39440i 0 −69.4607 + 28.7716i 0 120.687 + 49.9901i 0 162.282 − 162.282i 0
49.4 0 2.54560 + 6.14561i 0 8.67300 − 3.59247i 0 −214.730 − 88.9439i 0 140.538 − 140.538i 0
49.5 0 3.99622 + 9.64772i 0 −5.75150 + 2.38235i 0 94.7274 + 39.2374i 0 94.7182 − 94.7182i 0
49.6 0 6.22649 + 15.0321i 0 76.2008 − 31.5634i 0 32.9168 + 13.6346i 0 −15.3674 + 15.3674i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.6.h.a ✓ 28
17.d even 8 1 inner 68.6.h.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.6.h.a ✓ 28 1.a even 1 1 trivial
68.6.h.a ✓ 28 17.d even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(68, [\chi])\).