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Properties

Label 88.6.i.a
Level $88$
Weight $6$
Character orbit 88.i
Analytic conductor $14.114$
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] = [88,6,Mod(9,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.9");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 88.i (of order \(5\), 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: \(14.1137761435\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{3} - 37 q^{5} + 105 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{3} - 37 q^{5} + 105 q^{7} - 26 q^{9} - 324 q^{11} + 1091 q^{13} + 3759 q^{15} + 1135 q^{17} - 3065 q^{19} - 7262 q^{21} - 2840 q^{23} - 7088 q^{25} + 3893 q^{27} + 1003 q^{29} + 335 q^{31} + 34023 q^{33} + 10255 q^{35} + 16423 q^{37} - 29735 q^{39} + 26467 q^{41} + 208 q^{43} - 38824 q^{45} + 4475 q^{47} + 2634 q^{49} - 91837 q^{51} - 6705 q^{53} - 2805 q^{55} + 53023 q^{57} + 48665 q^{59} - 4029 q^{61} - 72992 q^{63} + 10038 q^{65} - 15664 q^{67} + 66676 q^{69} + 59129 q^{71} - 83981 q^{73} + 162559 q^{75} - 115439 q^{77} - 123113 q^{79} - 182132 q^{81} - 219427 q^{83} + 196769 q^{85} + 221198 q^{87} + 55740 q^{89} + 301193 q^{91} + 170413 q^{93} - 135197 q^{95} + 183297 q^{97} - 126124 q^{99}+O(q^{100}) \) 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 −21.6063 + 15.6979i 0 −32.7922 − 100.924i 0 120.965 + 87.8864i 0 145.318 − 447.243i 0
9.2 0 −13.5733 + 9.86155i 0 15.3472 + 47.2339i 0 47.9836 + 34.8621i 0 11.8920 − 36.5999i 0
9.3 0 −10.9781 + 7.97603i 0 1.33662 + 4.11370i 0 −99.3118 − 72.1542i 0 −18.1903 + 55.9840i 0
9.4 0 7.10517 − 5.16221i 0 −22.6851 − 69.8175i 0 −109.326 − 79.4299i 0 −51.2561 + 157.750i 0
9.5 0 8.43564 − 6.12885i 0 19.3902 + 59.6769i 0 47.4953 + 34.5074i 0 −41.4939 + 127.705i 0
9.6 0 11.3023 − 8.21157i 0 −14.4845 − 44.5788i 0 180.903 + 131.434i 0 −14.7800 + 45.4881i 0
9.7 0 24.0957 − 17.5066i 0 10.6623 + 32.8153i 0 −131.714 − 95.6955i 0 199.033 − 612.562i 0
25.1 0 −8.17546 − 25.1615i 0 −2.05815 − 1.49533i 0 −61.3296 + 188.753i 0 −369.670 + 268.581i 0
25.2 0 −6.21406 − 19.1249i 0 34.6036 + 25.1410i 0 55.9296 − 172.134i 0 −130.557 + 94.8550i 0
25.3 0 −2.96101 − 9.11305i 0 −76.5778 − 55.6370i 0 16.4841 − 50.7327i 0 122.311 − 88.8642i 0
25.4 0 −0.949895 − 2.92347i 0 −10.7944 − 7.84258i 0 −37.9712 + 116.863i 0 188.947 − 137.278i 0
25.5 0 0.875667 + 2.69503i 0 82.1500 + 59.6855i 0 3.44034 − 10.5883i 0 190.095 − 138.112i 0
25.6 0 5.19082 + 15.9757i 0 −26.8188 − 19.4850i 0 48.6493 − 149.727i 0 −31.6872 + 23.0221i 0
25.7 0 6.95278 + 21.3985i 0 4.22095 + 3.06670i 0 −29.6985 + 91.4025i 0 −212.962 + 154.726i 0
49.1 0 −21.6063 − 15.6979i 0 −32.7922 + 100.924i 0 120.965 − 87.8864i 0 145.318 + 447.243i 0
49.2 0 −13.5733 − 9.86155i 0 15.3472 − 47.2339i 0 47.9836 − 34.8621i 0 11.8920 + 36.5999i 0
49.3 0 −10.9781 − 7.97603i 0 1.33662 − 4.11370i 0 −99.3118 + 72.1542i 0 −18.1903 − 55.9840i 0
49.4 0 7.10517 + 5.16221i 0 −22.6851 + 69.8175i 0 −109.326 + 79.4299i 0 −51.2561 − 157.750i 0
49.5 0 8.43564 + 6.12885i 0 19.3902 − 59.6769i 0 47.4953 − 34.5074i 0 −41.4939 − 127.705i 0
49.6 0 11.3023 + 8.21157i 0 −14.4845 + 44.5788i 0 180.903 − 131.434i 0 −14.7800 − 45.4881i 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
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.6.i.a ✓ 28
11.c even 5 1 inner 88.6.i.a ✓ 28
11.c even 5 1 968.6.a.n 14
11.d odd 10 1 968.6.a.o 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.6.i.a ✓ 28 1.a even 1 1 trivial
88.6.i.a ✓ 28 11.c even 5 1 inner
968.6.a.n 14 11.c even 5 1
968.6.a.o 14 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + T_{3}^{27} + 864 T_{3}^{26} - 4322 T_{3}^{25} + 806344 T_{3}^{24} + 3529195 T_{3}^{23} + \cdots + 15\!\cdots\!25 \) acting on \(S_{6}^{\mathrm{new}}(88, [\chi])\). Copy content Toggle raw display