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Properties

Label 103.11.b.c
Level $103$
Weight $11$
Character orbit 103.b
Analytic conductor $65.442$
Analytic rank $0$
Dimension $80$
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] = [103,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.102");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(65.4417970254\)
Analytic rank: \(0\)
Dimension: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{2} + 36862 q^{4} + 56700 q^{7} - 71176 q^{8} - 1908760 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{2} + 36862 q^{4} + 56700 q^{7} - 71176 q^{8} - 1908760 q^{9} + 621454 q^{13} + 2591230 q^{14} - 2799822 q^{15} + 13465814 q^{16} - 849110 q^{17} - 5592316 q^{18} + 2716630 q^{19} + 3255084 q^{23} - 164471282 q^{25} + 36372824 q^{26} - 53645934 q^{28} - 27510948 q^{29} + 6162496 q^{30} - 126628726 q^{32} - 82386042 q^{33} + 59481176 q^{34} - 1233908724 q^{36} + 525233114 q^{38} - 228689068 q^{41} + 1201449960 q^{46} + 507708148 q^{49} - 672841524 q^{50} - 1449748714 q^{52} - 1863464912 q^{55} + 1257693440 q^{56} + 1861485688 q^{58} - 1044583318 q^{59} - 3500235390 q^{60} - 1704257534 q^{61} - 3155272876 q^{63} + 1214108476 q^{64} + 4906249542 q^{66} - 18621400382 q^{68} - 2003729496 q^{72} + 5279540724 q^{76} - 8890763554 q^{79} + 34283672808 q^{81} - 50187013312 q^{82} - 17308259190 q^{83} + 12715876426 q^{91} - 9922117972 q^{92} + 29079204200 q^{93} + 10334863084 q^{97} - 13991159184 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
102.1 −62.5598 425.999i 2889.73 1391.96i − 26650.4i 8083.15 −116720. −122426. − 87080.9i
102.2 −62.5598 − 425.999i 2889.73 − 1391.96i 26650.4i 8083.15 −116720. −122426. 87080.9i
102.3 −58.3970 − 204.919i 2386.21 4754.56i 11966.6i −23025.5 −79548.6 17057.3 − 277652.i
102.4 −58.3970 204.919i 2386.21 − 4754.56i − 11966.6i −23025.5 −79548.6 17057.3 277652.i
102.5 −58.3816 − 136.357i 2384.41 − 2936.37i 7960.71i −2005.03 −79422.7 40455.9 171430.i
102.6 −58.3816 136.357i 2384.41 2936.37i − 7960.71i −2005.03 −79422.7 40455.9 − 171430.i
102.7 −51.2095 127.567i 1598.41 − 4183.55i − 6532.61i 22524.8 −29415.2 42775.8 214237.i
102.8 −51.2095 − 127.567i 1598.41 4183.55i 6532.61i 22524.8 −29415.2 42775.8 − 214237.i
102.9 −49.9698 286.148i 1472.98 2761.66i − 14298.8i −12599.8 −22435.6 −22831.7 − 138000.i
102.10 −49.9698 − 286.148i 1472.98 − 2761.66i 14298.8i −12599.8 −22435.6 −22831.7 138000.i
102.11 −49.4994 − 220.310i 1426.19 − 1955.58i 10905.2i −24436.4 −19908.1 10512.6 96800.2i
102.12 −49.4994 220.310i 1426.19 1955.58i − 10905.2i −24436.4 −19908.1 10512.6 − 96800.2i
102.13 −49.1976 − 420.166i 1396.41 3897.01i 20671.1i 1570.10 −18321.5 −117490. − 191724.i
102.14 −49.1976 420.166i 1396.41 − 3897.01i − 20671.1i 1570.10 −18321.5 −117490. 191724.i
102.15 −43.3209 − 305.466i 852.697 693.547i 13233.0i 24770.7 7421.01 −34260.2 − 30045.0i
102.16 −43.3209 305.466i 852.697 − 693.547i − 13233.0i 24770.7 7421.01 −34260.2 30045.0i
102.17 −42.8957 324.570i 816.042 5955.42i − 13922.6i 18670.8 8920.51 −46296.4 − 255462.i
102.18 −42.8957 − 324.570i 816.042 − 5955.42i 13922.6i 18670.8 8920.51 −46296.4 255462.i
102.19 −36.1111 − 68.2815i 280.012 1913.75i 2465.72i −2725.82 26866.2 54386.6 − 69107.7i
102.20 −36.1111 68.2815i 280.012 − 1913.75i − 2465.72i −2725.82 26866.2 54386.6 69107.7i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 102.80
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 103.11.b.c ✓ 80
103.b odd 2 1 inner 103.11.b.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.11.b.c ✓ 80 1.a even 1 1 trivial
103.11.b.c ✓ 80 103.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + T_{2}^{39} - 29695 T_{2}^{38} - 17150 T_{2}^{37} + 404099303 T_{2}^{36} + \cdots + 62\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(103, [\chi])\). Copy content Toggle raw display