Properties

Label 103.9.b.b
Level $103$
Weight $9$
Character orbit 103.b
Analytic conductor $41.960$
Analytic rank $0$
Dimension $64$
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] = [103,9,Mod(102,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.102");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 103.b (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: \(41.9599968363\)
Analytic rank: \(0\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{2} + 7678 q^{4} - 5152 q^{7} + 8216 q^{8} - 177928 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{2} + 7678 q^{4} - 5152 q^{7} + 8216 q^{8} - 177928 q^{9} - 82538 q^{13} + 31646 q^{14} + 3390 q^{15} + 721814 q^{16} - 11006 q^{17} + 585188 q^{18} - 79586 q^{19} - 46496 q^{23} - 8775454 q^{25} - 1390328 q^{26} - 1165662 q^{28} + 383944 q^{29} + 518752 q^{30} + 13195274 q^{32} + 2070786 q^{33} + 1044152 q^{34} - 28755300 q^{36} - 2690534 q^{38} + 5231488 q^{41} - 53939576 q^{46} + 54368824 q^{49} - 3248516 q^{50} - 5642618 q^{52} + 50989976 q^{55} - 58033248 q^{56} - 35954808 q^{58} - 15930038 q^{59} - 85873038 q^{60} - 1272590 q^{61} + 13979120 q^{63} + 94446812 q^{64} + 53421702 q^{66} + 106176242 q^{68} + 79481976 q^{72} - 9704364 q^{76} + 253556574 q^{79} - 125427648 q^{81} + 64418768 q^{82} - 25696694 q^{83} - 196090314 q^{91} - 264251796 q^{92} + 458960072 q^{93} + 3061336 q^{97} - 353849536 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
102.1 −29.3312 87.6804i 604.317 634.155i 2571.77i 972.142 −10216.6 −1126.85 18600.5i
102.2 −29.3312 87.6804i 604.317 634.155i 2571.77i 972.142 −10216.6 −1126.85 18600.5i
102.3 −28.8717 63.7101i 577.575 899.927i 1839.42i 2825.20 −9284.41 2502.02 25982.4i
102.4 −28.8717 63.7101i 577.575 899.927i 1839.42i 2825.20 −9284.41 2502.02 25982.4i
102.5 −27.9433 148.875i 524.827 866.759i 4160.05i −4361.90 −7511.92 −15602.7 24220.1i
102.6 −27.9433 148.875i 524.827 866.759i 4160.05i −4361.90 −7511.92 −15602.7 24220.1i
102.7 −25.5002 139.257i 394.260 85.7228i 3551.08i 3044.06 −3525.66 −12831.4 2185.95i
102.8 −25.5002 139.257i 394.260 85.7228i 3551.08i 3044.06 −3525.66 −12831.4 2185.95i
102.9 −24.6420 51.4776i 351.226 30.8704i 1268.51i −941.911 −2346.56 3911.05 760.707i
102.10 −24.6420 51.4776i 351.226 30.8704i 1268.51i −941.911 −2346.56 3911.05 760.707i
102.11 −22.6597 138.463i 257.460 943.340i 3137.52i −3425.78 −33.0865 −12611.0 21375.8i
102.12 −22.6597 138.463i 257.460 943.340i 3137.52i −3425.78 −33.0865 −12611.0 21375.8i
102.13 −21.3916 22.4279i 201.601 876.654i 479.770i −2429.16 1163.68 6057.99 18753.0i
102.14 −21.3916 22.4279i 201.601 876.654i 479.770i −2429.16 1163.68 6057.99 18753.0i
102.15 −18.9906 9.68308i 104.644 820.259i 183.888i 3261.63 2874.34 6467.24 15577.2i
102.16 −18.9906 9.68308i 104.644 820.259i 183.888i 3261.63 2874.34 6467.24 15577.2i
102.17 −17.6874 104.607i 56.8444 374.832i 1850.22i 878.650 3522.55 −4381.57 6629.81i
102.18 −17.6874 104.607i 56.8444 374.832i 1850.22i 878.650 3522.55 −4381.57 6629.81i
102.19 −15.1473 80.0038i −26.5585 1018.10i 1211.84i −1821.42 4280.01 160.387 15421.5i
102.20 −15.1473 80.0038i −26.5585 1018.10i 1211.84i −1821.42 4280.01 160.387 15421.5i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 102.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.9.b.b 64
103.b odd 2 1 inner 103.9.b.b 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.9.b.b 64 1.a even 1 1 trivial
103.9.b.b 64 103.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + T_{2}^{31} - 6015 T_{2}^{30} - 7214 T_{2}^{29} + 16213023 T_{2}^{28} + 21330897 T_{2}^{27} - 25891004273 T_{2}^{26} - 34998546836 T_{2}^{25} + 27290668796296 T_{2}^{24} + \cdots + 12\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(103, [\chi])\). Copy content Toggle raw display