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Properties

Label 92.9.c.a
Level $92$
Weight $9$
Character orbit 92.c
Analytic conductor $37.479$
Analytic rank $0$
Dimension $88$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [92,9,Mod(47,92)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(92, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("92.47");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 92.c (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: \(37.4788321256\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 336 q^{5} + 1617 q^{6} - 13707 q^{8} - 196264 q^{9} + 40114 q^{10} - 82415 q^{12} + 22832 q^{13} + 55992 q^{14} + 139024 q^{16} - 165648 q^{17} - 137887 q^{18} + 826404 q^{20} + 243648 q^{21} + 190556 q^{22}+ \cdots - 53187756 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −15.9956 − 0.376294i 130.796i 255.717 + 12.0381i −534.997 49.2176 − 2092.15i − 2047.49i −4085.81 − 288.781i −10546.5 8557.59 + 201.317i
47.2 −15.9956 + 0.376294i − 130.796i 255.717 − 12.0381i −534.997 49.2176 + 2092.15i 2047.49i −4085.81 + 288.781i −10546.5 8557.59 − 201.317i
47.3 −15.9695 − 0.988222i − 94.8206i 254.047 + 31.5627i 110.310 −93.7038 + 1514.23i − 4110.64i −4025.80 − 755.094i −2429.96 −1761.59 − 109.010i
47.4 −15.9695 + 0.988222i 94.8206i 254.047 − 31.5627i 110.310 −93.7038 − 1514.23i 4110.64i −4025.80 + 755.094i −2429.96 −1761.59 + 109.010i
47.5 −15.6955 − 3.10661i − 104.882i 236.698 + 97.5198i 1207.34 −325.828 + 1646.18i 1632.69i −3412.14 − 2265.95i −4439.27 −18949.9 − 3750.75i
47.6 −15.6955 + 3.10661i 104.882i 236.698 − 97.5198i 1207.34 −325.828 − 1646.18i − 1632.69i −3412.14 + 2265.95i −4439.27 −18949.9 + 3750.75i
47.7 −15.6747 − 3.20986i 49.0116i 235.394 + 100.627i −941.139 157.320 − 768.243i 885.862i −3366.73 − 2332.88i 4158.87 14752.1 + 3020.92i
47.8 −15.6747 + 3.20986i − 49.0116i 235.394 − 100.627i −941.139 157.320 + 768.243i − 885.862i −3366.73 + 2332.88i 4158.87 14752.1 − 3020.92i
47.9 −15.5166 − 3.90338i 43.7598i 225.527 + 121.134i 351.830 170.811 − 679.001i 1727.11i −3026.57 − 2759.90i 4646.08 −5459.20 − 1373.33i
47.10 −15.5166 + 3.90338i − 43.7598i 225.527 − 121.134i 351.830 170.811 + 679.001i − 1727.11i −3026.57 + 2759.90i 4646.08 −5459.20 + 1373.33i
47.11 −14.6855 − 6.35107i 28.7070i 175.328 + 186.537i 496.805 182.320 − 421.577i − 2901.60i −1390.06 − 3852.91i 5736.91 −7295.82 − 3155.24i
47.12 −14.6855 + 6.35107i − 28.7070i 175.328 − 186.537i 496.805 182.320 + 421.577i 2901.60i −1390.06 + 3852.91i 5736.91 −7295.82 + 3155.24i
47.13 −14.5818 − 6.58577i − 44.6750i 169.255 + 192.064i −184.175 −294.220 + 651.440i 1320.25i −1203.14 − 3915.31i 4565.14 2685.59 + 1212.93i
47.14 −14.5818 + 6.58577i 44.6750i 169.255 − 192.064i −184.175 −294.220 − 651.440i − 1320.25i −1203.14 + 3915.31i 4565.14 2685.59 − 1212.93i
47.15 −13.2258 − 9.00431i − 160.150i 93.8448 + 238.179i −225.940 −1442.04 + 2118.12i − 1273.33i 903.462 − 3995.12i −19087.0 2988.24 + 2034.43i
47.16 −13.2258 + 9.00431i 160.150i 93.8448 − 238.179i −225.940 −1442.04 − 2118.12i 1273.33i 903.462 + 3995.12i −19087.0 2988.24 − 2034.43i
47.17 −12.8088 − 9.58823i − 39.5308i 72.1315 + 245.628i −1109.36 −379.030 + 506.342i − 4026.66i 1431.22 − 3837.82i 4998.32 14209.6 + 10636.8i
47.18 −12.8088 + 9.58823i 39.5308i 72.1315 − 245.628i −1109.36 −379.030 − 506.342i 4026.66i 1431.22 + 3837.82i 4998.32 14209.6 − 10636.8i
47.19 −12.8034 − 9.59541i 137.683i 71.8563 + 245.709i 620.599 1321.13 − 1762.82i 4034.34i 1437.67 − 3835.41i −12395.6 −7945.80 − 5954.90i
47.20 −12.8034 + 9.59541i − 137.683i 71.8563 − 245.709i 620.599 1321.13 + 1762.82i − 4034.34i 1437.67 + 3835.41i −12395.6 −7945.80 + 5954.90i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.88
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.9.c.a ✓ 88
4.b odd 2 1 inner 92.9.c.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.9.c.a ✓ 88 1.a even 1 1 trivial
92.9.c.a ✓ 88 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(92, [\chi])\).