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Properties

Label 88.4.k.b
Level $88$
Weight $4$
Character orbit 88.k
Analytic conductor $5.192$
Analytic rank $0$
Dimension $128$
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] = [88,4,Mod(19,88)]
 
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([5, 5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 88.k (of order \(10\), 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: \(5.19216808051\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 5 q^{2} - 26 q^{3} - 9 q^{4} - 45 q^{6} - 5 q^{8} - 130 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 5 q^{2} - 26 q^{3} - 9 q^{4} - 45 q^{6} - 5 q^{8} - 130 q^{9} - 46 q^{11} + 230 q^{12} + 66 q^{14} - 105 q^{16} - 10 q^{17} - 630 q^{18} - 540 q^{19} + 134 q^{20} - 803 q^{22} + 125 q^{24} + 894 q^{25} + 178 q^{26} + 532 q^{27} - 530 q^{28} + 1150 q^{30} + 392 q^{33} + 906 q^{34} - 10 q^{35} + 1128 q^{36} + 734 q^{38} - 1090 q^{40} - 10 q^{41} - 596 q^{42} - 1914 q^{44} - 1260 q^{46} - 1374 q^{48} - 1770 q^{49} - 665 q^{50} + 3000 q^{51} + 3060 q^{52} + 2268 q^{56} - 1760 q^{57} - 18 q^{58} - 3232 q^{59} - 14 q^{60} + 3970 q^{62} - 2265 q^{64} + 5180 q^{66} + 1884 q^{67} - 3840 q^{68} - 1892 q^{70} - 5555 q^{72} - 10 q^{73} - 2870 q^{74} + 6664 q^{75} - 4428 q^{78} + 3764 q^{80} - 1388 q^{81} - 1527 q^{82} - 13460 q^{83} + 5890 q^{84} - 3277 q^{86} + 5267 q^{88} + 2036 q^{89} + 10160 q^{90} - 4882 q^{91} - 6610 q^{92} + 7280 q^{94} - 4230 q^{96} + 7212 q^{97} + 1134 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.82137 − 0.199704i 1.71363 + 5.27403i 7.92024 + 1.12688i 6.72897 − 9.26163i −3.78155 − 15.2222i 4.18093 − 12.8676i −22.1209 − 4.76104i −3.03535 + 2.20531i −20.8345 + 24.7867i
19.2 −2.75875 − 0.623929i 1.37382 + 4.22818i 7.22143 + 3.44253i −5.58036 + 7.68071i −1.15194 − 12.5217i −3.98666 + 12.2697i −17.7742 − 14.0027i 5.85332 − 4.25269i 20.1870 − 17.7074i
19.3 −2.73544 − 0.719293i −2.31383 − 7.12124i 6.96524 + 3.93516i −7.45856 + 10.2658i 1.20708 + 21.1440i −3.81271 + 11.7343i −16.2224 − 15.7744i −23.5148 + 17.0845i 27.7866 − 22.7166i
19.4 −2.69422 + 0.860909i −2.22291 − 6.84142i 6.51767 − 4.63896i 11.7141 − 16.1230i 11.8789 + 16.5186i −2.63231 + 8.10140i −13.5663 + 18.1095i −20.0202 + 14.5455i −17.6798 + 53.5238i
19.5 −2.66454 + 0.948814i −0.462608 − 1.42376i 6.19950 − 5.05630i −5.51936 + 7.59675i 2.58352 + 3.35474i 2.25660 − 6.94509i −11.7213 + 19.3549i 20.0304 − 14.5529i 7.49864 − 25.4787i
19.6 −2.36830 − 1.54634i −1.47481 − 4.53901i 3.21764 + 7.32440i 4.99068 − 6.86908i −3.52607 + 13.0303i 5.86854 − 18.0615i 3.70570 − 22.3219i 3.41596 − 2.48184i −22.4413 + 8.55070i
19.7 −2.21992 + 1.75270i 0.342626 + 1.05449i 1.85611 − 7.78170i 1.34278 − 1.84818i −2.60881 − 1.74038i −6.97098 + 21.4545i 9.51855 + 20.5280i 20.8489 − 15.1476i 0.258430 + 6.45630i
19.8 −2.18124 + 1.80060i 2.96196 + 9.11598i 1.51565 − 7.85511i −10.8751 + 14.9683i −22.8750 − 14.5509i 3.10904 − 9.56863i 10.8379 + 19.8630i −52.4843 + 38.1321i −3.23072 − 52.2314i
19.9 −1.81840 − 2.16643i 0.327347 + 1.00747i −1.38687 + 7.87887i 7.82667 − 10.7725i 1.58737 − 2.54116i −9.82364 + 30.2341i 19.5909 − 11.3223i 20.9356 − 15.2106i −37.5699 + 2.63269i
19.10 −1.70111 + 2.25969i −2.85280 − 8.78001i −2.21243 − 7.68799i −4.86887 + 6.70142i 24.6931 + 8.48933i 6.17777 − 19.0132i 21.1361 + 8.07872i −47.1066 + 34.2249i −6.86066 − 22.4020i
19.11 −1.56324 + 2.35718i 1.55292 + 4.77939i −3.11258 − 7.36966i 9.32679 − 12.8372i −13.6934 − 3.81081i 9.39487 − 28.9144i 22.2373 + 4.18361i 1.41249 − 1.02623i 15.6796 + 42.0525i
19.12 −1.49849 − 2.39886i 0.327347 + 1.00747i −3.50908 + 7.18932i −7.82667 + 10.7725i 1.92626 − 2.29494i 9.82364 − 30.2341i 22.5045 − 2.35529i 20.9356 − 15.2106i 37.5699 + 2.63269i
19.13 −0.905250 + 2.67965i −0.861739 − 2.65216i −6.36104 − 4.85151i −1.28778 + 1.77248i 7.88695 + 0.0917104i −1.19904 + 3.69026i 18.7587 − 12.6535i 15.5521 − 11.2993i −3.58387 − 5.05535i
19.14 −0.738817 − 2.73023i −1.47481 − 4.53901i −6.90830 + 4.03428i −4.99068 + 6.86908i −11.3029 + 7.38007i −5.86854 + 18.0615i 16.1185 + 15.8807i 3.41596 − 2.48184i 22.4413 + 8.55070i
19.15 −0.270313 + 2.81548i 2.24775 + 6.91786i −7.85386 − 1.52212i 2.71861 − 3.74185i −20.0847 + 4.45851i −8.05725 + 24.7977i 6.40851 − 21.7009i −20.9610 + 15.2291i 9.80023 + 8.66567i
19.16 0.161209 − 2.82383i −2.31383 − 7.12124i −7.94802 − 0.910452i 7.45856 − 10.2658i −20.4822 + 5.38586i 3.81271 − 11.7343i −3.85225 + 22.2971i −23.5148 + 17.0845i −27.7866 − 22.7166i
19.17 0.259110 − 2.81653i 1.37382 + 4.22818i −7.86572 − 1.45958i 5.58036 − 7.68071i 12.2648 − 2.77385i 3.98666 − 12.2697i −6.14905 + 21.7759i 5.85332 − 4.25269i −20.1870 − 17.7074i
19.18 0.313041 + 2.81105i 0.430245 + 1.32416i −7.80401 + 1.75995i −12.0755 + 16.6205i −3.58759 + 1.62396i 5.57362 − 17.1538i −7.39028 − 21.3865i 20.2752 − 14.7308i −50.5013 − 28.7420i
19.19 0.681921 − 2.74499i 1.71363 + 5.27403i −7.06997 − 3.74374i −6.72897 + 9.26163i 15.6457 − 1.10745i −4.18093 + 12.8676i −15.0977 + 16.8541i −3.03535 + 2.20531i 20.8345 + 24.7867i
19.20 0.789761 + 2.71593i −2.55284 − 7.85684i −6.75256 + 4.28987i 0.247062 − 0.340051i 19.3225 − 13.1384i −10.4173 + 32.0612i −16.9839 − 14.9515i −33.3695 + 24.2444i 1.11868 + 0.402443i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.d odd 10 1 inner
88.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.4.k.b ✓ 128
8.d odd 2 1 inner 88.4.k.b ✓ 128
11.d odd 10 1 inner 88.4.k.b ✓ 128
88.k even 10 1 inner 88.4.k.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.k.b ✓ 128 1.a even 1 1 trivial
88.4.k.b ✓ 128 8.d odd 2 1 inner
88.4.k.b ✓ 128 11.d odd 10 1 inner
88.4.k.b ✓ 128 88.k even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 13 T_{3}^{63} + 333 T_{3}^{62} + 3202 T_{3}^{61} + 57483 T_{3}^{60} + 498538 T_{3}^{59} + \cdots + 38\!\cdots\!25 \) acting on \(S_{4}^{\mathrm{new}}(88, [\chi])\). Copy content Toggle raw display