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Properties

Label 28.8.d.b
Level $28$
Weight $8$
Character orbit 28.d
Analytic conductor $8.747$
Analytic rank $0$
Dimension $24$
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] = [28,8,Mod(27,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.27");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 28.d (of order \(2\), degree \(1\), 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: \(8.74678071356\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 176 q^{4} + 1504 q^{8} + 20408 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 176 q^{4} + 1504 q^{8} + 20408 q^{9} + 29288 q^{14} + 70208 q^{16} + 70744 q^{18} - 19264 q^{21} + 114160 q^{22} - 457992 q^{25} + 551824 q^{28} - 686960 q^{29} - 228224 q^{30} - 892288 q^{32} - 168208 q^{36} - 775216 q^{37} - 1579648 q^{42} + 2436832 q^{44} + 4839504 q^{46} + 1385048 q^{49} + 750488 q^{50} + 5202128 q^{53} + 1999648 q^{56} + 5538240 q^{57} - 8383920 q^{58} - 3829376 q^{60} - 6125824 q^{64} - 1877952 q^{65} - 465920 q^{70} - 13537952 q^{72} - 13027440 q^{74} + 2665040 q^{77} + 18048 q^{78} - 16634728 q^{81} + 273280 q^{84} - 13883136 q^{85} + 26012912 q^{86} + 24483520 q^{88} + 30636704 q^{92} + 6153216 q^{93} + 33930680 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −11.2972 − 0.610357i −58.5163 127.255 + 13.7907i − 425.670i 661.072 + 35.7158i −322.049 − 848.427i −1429.21 − 233.467i 1237.16 −259.811 + 4808.90i
27.2 −11.2972 − 0.610357i 58.5163 127.255 + 13.7907i 425.670i −661.072 − 35.7158i 322.049 − 848.427i −1429.21 − 233.467i 1237.16 259.811 − 4808.90i
27.3 −11.2972 + 0.610357i −58.5163 127.255 − 13.7907i 425.670i 661.072 − 35.7158i −322.049 + 848.427i −1429.21 + 233.467i 1237.16 −259.811 − 4808.90i
27.4 −11.2972 + 0.610357i 58.5163 127.255 − 13.7907i − 425.670i −661.072 + 35.7158i 322.049 + 848.427i −1429.21 + 233.467i 1237.16 259.811 + 4808.90i
27.5 −8.50800 − 7.45747i −25.9662 16.7723 + 126.896i 67.6950i 220.920 + 193.642i 894.003 + 155.892i 803.627 − 1204.71i −1512.76 504.833 − 575.949i
27.6 −8.50800 − 7.45747i 25.9662 16.7723 + 126.896i − 67.6950i −220.920 − 193.642i −894.003 + 155.892i 803.627 − 1204.71i −1512.76 −504.833 + 575.949i
27.7 −8.50800 + 7.45747i −25.9662 16.7723 − 126.896i − 67.6950i 220.920 − 193.642i 894.003 − 155.892i 803.627 + 1204.71i −1512.76 504.833 + 575.949i
27.8 −8.50800 + 7.45747i 25.9662 16.7723 − 126.896i 67.6950i −220.920 + 193.642i −894.003 − 155.892i 803.627 + 1204.71i −1512.76 −504.833 − 575.949i
27.9 −2.92073 − 10.9302i −72.6453 −110.939 + 63.8484i 136.047i 212.178 + 794.028i −612.758 − 669.381i 1021.90 + 1026.10i 3090.34 1487.02 − 397.357i
27.10 −2.92073 − 10.9302i 72.6453 −110.939 + 63.8484i − 136.047i −212.178 − 794.028i 612.758 − 669.381i 1021.90 + 1026.10i 3090.34 −1487.02 + 397.357i
27.11 −2.92073 + 10.9302i −72.6453 −110.939 − 63.8484i − 136.047i 212.178 − 794.028i −612.758 + 669.381i 1021.90 − 1026.10i 3090.34 1487.02 + 397.357i
27.12 −2.92073 + 10.9302i 72.6453 −110.939 − 63.8484i 136.047i −212.178 + 794.028i 612.758 + 669.381i 1021.90 − 1026.10i 3090.34 −1487.02 − 397.357i
27.13 1.03691 − 11.2661i −29.9969 −125.850 − 23.3638i − 473.806i −31.1040 + 337.947i 334.749 + 843.496i −393.714 + 1393.61i −1287.19 −5337.94 − 491.293i
27.14 1.03691 − 11.2661i 29.9969 −125.850 − 23.3638i 473.806i 31.1040 − 337.947i −334.749 + 843.496i −393.714 + 1393.61i −1287.19 5337.94 + 491.293i
27.15 1.03691 + 11.2661i −29.9969 −125.850 + 23.3638i 473.806i −31.1040 − 337.947i 334.749 − 843.496i −393.714 − 1393.61i −1287.19 −5337.94 + 491.293i
27.16 1.03691 + 11.2661i 29.9969 −125.850 + 23.3638i − 473.806i 31.1040 + 337.947i −334.749 − 843.496i −393.714 − 1393.61i −1287.19 5337.94 − 491.293i
27.17 8.71517 − 7.21428i −81.5265 23.9082 − 125.747i 241.496i −710.517 + 588.155i 782.607 + 459.423i −698.813 − 1268.39i 4459.57 1742.22 + 2104.68i
27.18 8.71517 − 7.21428i 81.5265 23.9082 − 125.747i − 241.496i 710.517 − 588.155i −782.607 + 459.423i −698.813 − 1268.39i 4459.57 −1742.22 − 2104.68i
27.19 8.71517 + 7.21428i −81.5265 23.9082 + 125.747i − 241.496i −710.517 − 588.155i 782.607 − 459.423i −698.813 + 1268.39i 4459.57 1742.22 − 2104.68i
27.20 8.71517 + 7.21428i 81.5265 23.9082 + 125.747i 241.496i 710.517 + 588.155i −782.607 − 459.423i −698.813 + 1268.39i 4459.57 −1742.22 + 2104.68i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.8.d.b ✓ 24
4.b odd 2 1 inner 28.8.d.b ✓ 24
7.b odd 2 1 inner 28.8.d.b ✓ 24
28.d even 2 1 inner 28.8.d.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.8.d.b ✓ 24 1.a even 1 1 trivial
28.8.d.b ✓ 24 4.b odd 2 1 inner
28.8.d.b ✓ 24 7.b odd 2 1 inner
28.8.d.b ✓ 24 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 18224 T_{3}^{10} + 122701504 T_{3}^{8} - 379959023616 T_{3}^{6} + 559140361841664 T_{3}^{4} + \cdots + 94\!\cdots\!20 \) acting on \(S_{8}^{\mathrm{new}}(28, [\chi])\). Copy content Toggle raw display