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Properties

Label 330.3.k.b
Level $330$
Weight $3$
Character orbit 330.k
Analytic conductor $8.992$
Analytic rank $0$
Dimension $48$
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] = [330,3,Mod(197,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("330.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 330.k (of order \(4\), degree \(2\), 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: \(8.99184872389\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} + 2 q^{3} + 4 q^{6} - 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} + 2 q^{3} + 4 q^{6} - 96 q^{8} - 8 q^{11} + 4 q^{12} + 8 q^{15} - 192 q^{16} + 16 q^{17} - 8 q^{18} - 32 q^{21} - 8 q^{22} + 44 q^{25} - 16 q^{27} - 6 q^{30} - 16 q^{31} - 192 q^{32} - 26 q^{33} - 104 q^{35} - 16 q^{36} + 148 q^{37} + 64 q^{41} - 32 q^{42} + 64 q^{45} - 8 q^{48} - 20 q^{50} + 56 q^{51} - 20 q^{55} - 104 q^{57} - 112 q^{58} - 28 q^{60} - 16 q^{62} + 48 q^{63} + 40 q^{65} - 52 q^{66} - 124 q^{67} - 32 q^{68} + 16 q^{70} - 16 q^{72} - 64 q^{75} + 144 q^{77} - 248 q^{78} + 404 q^{81} + 64 q^{82} + 168 q^{83} + 96 q^{87} + 16 q^{88} + 50 q^{90} + 608 q^{91} + 226 q^{93} + 208 q^{95} - 16 q^{96} + 12 q^{97} + 320 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 1.00000 + 1.00000i −2.99432 + 0.184455i 2.00000i −4.99066 − 0.305423i −3.17878 − 2.80987i 9.01053 − 9.01053i −2.00000 + 2.00000i 8.93195 − 1.10464i −4.68524 − 5.29609i
197.2 1.00000 + 1.00000i −2.98462 − 0.303406i 2.00000i −3.09674 + 3.92559i −2.68121 − 3.28802i −5.06758 + 5.06758i −2.00000 + 2.00000i 8.81589 + 1.81110i −7.02232 + 0.828849i
197.3 1.00000 + 1.00000i −2.86700 − 0.883351i 2.00000i 1.54612 − 4.75495i −1.98365 − 3.75035i −2.18643 + 2.18643i −2.00000 + 2.00000i 7.43938 + 5.06514i 6.30107 − 3.20883i
197.4 1.00000 + 1.00000i −2.86673 + 0.884213i 2.00000i 4.81692 − 1.34062i −3.75095 − 1.98252i 2.08440 − 2.08440i −2.00000 + 2.00000i 7.43633 − 5.06961i 6.15754 + 3.47630i
197.5 1.00000 + 1.00000i −2.67100 + 1.36593i 2.00000i 2.49377 + 4.33372i −4.03693 − 1.30507i −2.25330 + 2.25330i −2.00000 + 2.00000i 5.26847 − 7.29680i −1.83994 + 6.82749i
197.6 1.00000 + 1.00000i −2.04846 − 2.19176i 2.00000i 4.36435 + 2.43976i 0.143308 − 4.24022i 3.38397 − 3.38397i −2.00000 + 2.00000i −0.607659 + 8.97946i 1.92460 + 6.80411i
197.7 1.00000 + 1.00000i −1.80049 − 2.39963i 2.00000i −3.73198 − 3.32752i 0.599136 − 4.20012i −3.72619 + 3.72619i −2.00000 + 2.00000i −2.51645 + 8.64104i −0.404460 − 7.05949i
197.8 1.00000 + 1.00000i −1.36593 + 2.67100i 2.00000i −2.49377 − 4.33372i −4.03693 + 1.30507i 2.25330 − 2.25330i −2.00000 + 2.00000i −5.26847 − 7.29680i 1.83994 − 6.82749i
197.9 1.00000 + 1.00000i −0.884213 + 2.86673i 2.00000i −4.81692 + 1.34062i −3.75095 + 1.98252i −2.08440 + 2.08440i −2.00000 + 2.00000i −7.43633 − 5.06961i −6.15754 − 3.47630i
197.10 1.00000 + 1.00000i −0.669003 − 2.92445i 2.00000i 3.40288 + 3.66339i 2.25545 − 3.59346i −6.65454 + 6.65454i −2.00000 + 2.00000i −8.10487 + 3.91294i −0.260516 + 7.06627i
197.11 1.00000 + 1.00000i −0.456925 − 2.96500i 2.00000i 1.12684 − 4.87137i 2.50807 − 3.42192i 6.36468 − 6.36468i −2.00000 + 2.00000i −8.58244 + 2.70957i 5.99821 − 3.74452i
197.12 1.00000 + 1.00000i −0.243722 − 2.99008i 2.00000i −4.46945 + 2.24143i 2.74636 − 3.23381i 4.99009 − 4.99009i −2.00000 + 2.00000i −8.88120 + 1.45750i −6.71088 − 2.22802i
197.13 1.00000 + 1.00000i −0.184455 + 2.99432i 2.00000i 4.99066 + 0.305423i −3.17878 + 2.80987i −9.01053 + 9.01053i −2.00000 + 2.00000i −8.93195 − 1.10464i 4.68524 + 5.29609i
197.14 1.00000 + 1.00000i 0.303406 + 2.98462i 2.00000i 3.09674 − 3.92559i −2.68121 + 3.28802i 5.06758 − 5.06758i −2.00000 + 2.00000i −8.81589 + 1.81110i 7.02232 − 0.828849i
197.15 1.00000 + 1.00000i 0.883351 + 2.86700i 2.00000i −1.54612 + 4.75495i −1.98365 + 3.75035i 2.18643 − 2.18643i −2.00000 + 2.00000i −7.43938 + 5.06514i −6.30107 + 3.20883i
197.16 1.00000 + 1.00000i 1.71697 − 2.46008i 2.00000i −4.52746 − 2.12183i 4.17705 − 0.743114i −8.81914 + 8.81914i −2.00000 + 2.00000i −3.10403 − 8.44778i −2.40563 − 6.64928i
197.17 1.00000 + 1.00000i 1.80862 − 2.39351i 2.00000i −1.70087 + 4.70181i 4.20213 − 0.584882i 0.348525 − 0.348525i −2.00000 + 2.00000i −2.45775 − 8.65791i −6.40268 + 3.00094i
197.18 1.00000 + 1.00000i 2.19176 + 2.04846i 2.00000i −4.36435 − 2.43976i 0.143308 + 4.24022i −3.38397 + 3.38397i −2.00000 + 2.00000i 0.607659 + 8.97946i −1.92460 − 6.80411i
197.19 1.00000 + 1.00000i 2.39351 − 1.80862i 2.00000i 1.70087 − 4.70181i 4.20213 + 0.584882i −0.348525 + 0.348525i −2.00000 + 2.00000i 2.45775 − 8.65791i 6.40268 − 3.00094i
197.20 1.00000 + 1.00000i 2.39963 + 1.80049i 2.00000i 3.73198 + 3.32752i 0.599136 + 4.20012i 3.72619 − 3.72619i −2.00000 + 2.00000i 2.51645 + 8.64104i 0.404460 + 7.05949i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
33.d even 2 1 inner
165.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.3.k.b yes 48
3.b odd 2 1 330.3.k.a ✓ 48
5.c odd 4 1 inner 330.3.k.b yes 48
11.b odd 2 1 330.3.k.a ✓ 48
15.e even 4 1 330.3.k.a ✓ 48
33.d even 2 1 inner 330.3.k.b yes 48
55.e even 4 1 330.3.k.a ✓ 48
165.l odd 4 1 inner 330.3.k.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.3.k.a ✓ 48 3.b odd 2 1
330.3.k.a ✓ 48 11.b odd 2 1
330.3.k.a ✓ 48 15.e even 4 1
330.3.k.a ✓ 48 55.e even 4 1
330.3.k.b yes 48 1.a even 1 1 trivial
330.3.k.b yes 48 5.c odd 4 1 inner
330.3.k.b yes 48 33.d even 2 1 inner
330.3.k.b yes 48 165.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{24} - 8 T_{17}^{23} + 32 T_{17}^{22} + 8712 T_{17}^{21} + 721440 T_{17}^{20} + \cdots + 19\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(330, [\chi])\). Copy content Toggle raw display