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Properties

Label 102.5.g.d
Level $102$
Weight $5$
Character orbit 102.g
Analytic conductor $10.544$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [102,5,Mod(53,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 102.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: \(10.5437362346\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 88 q^{2} - 4 q^{3} + 72 q^{5} - 40 q^{6} + 132 q^{7} - 704 q^{8} + 324 q^{9} + 384 q^{10} - 616 q^{11} - 128 q^{12} + 416 q^{14} - 908 q^{15} - 2816 q^{16} + 424 q^{17} + 1016 q^{18} + 568 q^{19}+ \cdots + 91704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 2.00000 − 2.00000i −8.98767 − 0.470907i − 8.00000i 44.5723 − 18.4624i −18.9172 + 17.0335i −18.9619 + 45.7780i −16.0000 − 16.0000i 80.5565 + 8.46471i 52.2197 − 126.069i
53.2 2.00000 − 2.00000i −8.64365 + 2.50744i − 8.00000i −4.86643 + 2.01574i −12.2724 + 22.3022i 5.00624 − 12.0861i −16.0000 − 16.0000i 68.4255 − 43.3468i −5.70137 + 13.7643i
53.3 2.00000 − 2.00000i −6.49064 − 6.23471i − 8.00000i −13.4598 + 5.57524i −25.4507 + 0.511855i −17.8254 + 43.0343i −16.0000 − 16.0000i 3.25676 + 80.9345i −15.7692 + 38.0701i
53.4 2.00000 − 2.00000i −5.22697 − 7.32658i − 8.00000i 12.7918 − 5.29855i −25.1071 − 4.19923i 35.6639 − 86.1003i −16.0000 − 16.0000i −26.3576 + 76.5916i 14.9866 − 36.1808i
53.5 2.00000 − 2.00000i −4.70328 + 7.67327i − 8.00000i −10.0719 + 4.17192i 5.93998 + 24.7531i 7.83473 − 18.9147i −16.0000 − 16.0000i −36.7583 − 72.1792i −11.8000 + 28.4876i
53.6 2.00000 − 2.00000i −0.0925879 + 8.99952i − 8.00000i 7.71818 − 3.19697i 17.8139 + 18.1842i −32.7174 + 78.9869i −16.0000 − 16.0000i −80.9829 − 1.66649i 9.04241 − 21.8303i
53.7 2.00000 − 2.00000i 3.73212 − 8.18971i − 8.00000i −36.5016 + 15.1195i −8.91517 − 23.8437i −20.7919 + 50.1961i −16.0000 − 16.0000i −53.1426 − 61.1299i −42.7643 + 103.242i
53.8 2.00000 − 2.00000i 5.34583 + 7.24031i − 8.00000i −40.4335 + 16.7481i 25.1723 + 3.78896i 9.74985 − 23.5382i −16.0000 − 16.0000i −23.8442 + 77.4110i −47.3708 + 114.363i
53.9 2.00000 − 2.00000i 5.68748 + 6.97514i − 8.00000i 24.3911 − 10.1031i 25.3252 + 2.57533i 18.4989 − 44.6604i −16.0000 − 16.0000i −16.3052 + 79.3419i 28.5759 − 68.9884i
53.10 2.00000 − 2.00000i 8.05326 − 4.01809i − 8.00000i 40.0560 − 16.5917i 8.07035 − 24.1427i −2.16514 + 5.22710i −16.0000 − 16.0000i 48.7100 − 64.7174i 46.9285 − 113.295i
53.11 2.00000 − 2.00000i 8.91190 − 1.25620i − 8.00000i −14.6814 + 6.08124i 15.3114 − 20.3362i 11.9385 − 28.8221i −16.0000 − 16.0000i 77.8439 − 22.3902i −17.2004 + 41.5253i
59.1 2.00000 + 2.00000i −8.81431 − 1.81879i 8.00000i 5.39248 − 13.0186i −13.9910 − 21.2662i −32.8806 + 13.6196i −16.0000 + 16.0000i 74.3840 + 32.0627i 36.8221 − 15.2522i
59.2 2.00000 + 2.00000i −7.69652 + 4.66514i 8.00000i −13.6292 + 32.9039i −24.7233 − 6.06275i 1.24493 − 0.515667i −16.0000 + 16.0000i 37.4728 − 71.8108i −93.0663 + 38.5493i
59.3 2.00000 + 2.00000i −7.34085 + 5.20691i 8.00000i 8.97188 − 21.6600i −25.0955 − 4.26787i 85.5528 − 35.4371i −16.0000 + 16.0000i 26.7761 − 76.4463i 61.2638 − 25.3763i
59.4 2.00000 + 2.00000i −6.83333 − 5.85710i 8.00000i 2.47614 − 5.97792i −1.95247 − 25.3809i −1.01811 + 0.421715i −16.0000 + 16.0000i 12.3889 + 80.0470i 16.9081 − 7.00357i
59.5 2.00000 + 2.00000i −3.67062 + 8.21745i 8.00000i 16.1337 − 38.9501i −23.7761 + 9.09367i −77.2612 + 32.0026i −16.0000 + 16.0000i −54.0531 − 60.3263i 110.167 − 45.6329i
59.6 2.00000 + 2.00000i 1.58351 − 8.85960i 8.00000i 11.1111 − 26.8246i 20.8862 − 14.5522i 27.8275 − 11.5265i −16.0000 + 16.0000i −75.9850 − 28.0586i 75.8714 − 31.4270i
59.7 2.00000 + 2.00000i 1.73792 + 8.83061i 8.00000i −7.65773 + 18.4874i −14.1854 + 21.1371i 3.64521 − 1.50990i −16.0000 + 16.0000i −74.9592 + 30.6938i −52.2903 + 21.6593i
59.8 2.00000 + 2.00000i 7.10130 − 5.52916i 8.00000i 3.88053 − 9.36843i 25.2609 + 3.14427i 15.3463 − 6.35666i −16.0000 + 16.0000i 19.8568 − 78.5284i 26.4979 − 10.9758i
59.9 2.00000 + 2.00000i 7.44631 + 5.05494i 8.00000i 14.7889 − 35.7036i 4.78274 + 25.0025i 33.4426 − 13.8524i −16.0000 + 16.0000i 29.8951 + 75.2814i 100.985 − 41.8294i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.5.g.d yes 44
3.b odd 2 1 102.5.g.c ✓ 44
17.d even 8 1 102.5.g.c ✓ 44
51.g odd 8 1 inner 102.5.g.d yes 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.5.g.c ✓ 44 3.b odd 2 1
102.5.g.c ✓ 44 17.d even 8 1
102.5.g.d yes 44 1.a even 1 1 trivial
102.5.g.d yes 44 51.g odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{44} - 72 T_{5}^{43} + 1648 T_{5}^{42} + 38112 T_{5}^{41} - 3210880 T_{5}^{40} + \cdots + 24\!\cdots\!12 \) acting on \(S_{5}^{\mathrm{new}}(102, [\chi])\). Copy content Toggle raw display