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Properties

Label 465.2.br.a
Level $465$
Weight $2$
Character orbit 465.br
Analytic conductor $3.713$
Analytic rank $0$
Dimension $336$
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] = [465,2,Mod(11,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 0, 23]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.br (of order \(30\), degree \(8\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(42\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 76 q^{4} - 18 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q + 76 q^{4} - 18 q^{6} - 2 q^{10} - 30 q^{12} + 18 q^{13} - 80 q^{16} - 10 q^{18} - 30 q^{19} + 42 q^{21} + 36 q^{24} + 168 q^{25} + 30 q^{27} - 132 q^{28} - 12 q^{31} + 34 q^{33} - 22 q^{34} - 6 q^{36} - 18 q^{37} - 32 q^{39} - 64 q^{40} - 20 q^{42} + 68 q^{43} - 4 q^{45} + 50 q^{46} - 108 q^{48} + 62 q^{49} - 32 q^{51} - 144 q^{52} - 220 q^{54} - 8 q^{55} - 168 q^{57} - 140 q^{58} - 88 q^{63} + 20 q^{64} + 116 q^{66} + 32 q^{67} - 264 q^{69} - 136 q^{72} + 10 q^{73} + 56 q^{76} - 56 q^{78} + 30 q^{79} + 172 q^{81} + 236 q^{82} + 100 q^{84} - 34 q^{87} + 72 q^{88} - 48 q^{90} + 30 q^{91} + 150 q^{93} - 160 q^{94} + 142 q^{96} - 22 q^{97} + 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.59594 + 0.843473i −1.42902 − 0.978719i 4.40943 − 3.20364i 0.866025 − 0.500000i 4.53519 + 1.33536i −0.237334 − 2.25808i −5.53569 + 7.61922i 1.08422 + 2.79723i −1.82642 + 2.02844i
11.2 −2.46266 + 0.800167i −1.68173 + 0.414488i 3.80640 − 2.76551i −0.866025 + 0.500000i 3.80986 − 2.36641i 0.278026 + 2.64524i −4.11699 + 5.66655i 2.65640 − 1.39411i 1.73264 − 1.92430i
11.3 −2.30067 + 0.747532i 1.73130 − 0.0509297i 3.11623 − 2.26408i 0.866025 − 0.500000i −3.94508 + 1.41138i −0.120716 − 1.14854i −2.63317 + 3.62424i 2.99481 − 0.176349i −1.61867 + 1.79772i
11.4 −2.21287 + 0.719003i −0.758090 + 1.55734i 2.76177 − 2.00654i −0.866025 + 0.500000i 0.557820 − 3.99125i −0.495842 − 4.71762i −1.93346 + 2.66118i −1.85060 − 2.36120i 1.55690 − 1.72911i
11.5 −2.17782 + 0.707617i 1.29095 + 1.15475i 2.62415 − 1.90655i −0.866025 + 0.500000i −3.62858 − 1.60134i −0.222601 − 2.11791i −1.67388 + 2.30390i 0.333112 + 2.98145i 1.53224 − 1.70172i
11.6 −2.14287 + 0.696259i 1.60365 − 0.654463i 2.48906 − 1.80841i −0.866025 + 0.500000i −2.98072 + 2.51898i 0.239268 + 2.27648i −1.42588 + 1.96255i 2.14336 − 2.09905i 1.50765 − 1.67441i
11.7 −1.98234 + 0.644100i −0.459134 − 1.67009i 1.89676 − 1.37807i −0.866025 + 0.500000i 1.98586 + 3.01495i −0.0338045 − 0.321628i −0.422090 + 0.580956i −2.57839 + 1.53359i 1.39470 − 1.54897i
11.8 −1.91559 + 0.622413i −1.26550 + 1.18259i 1.66405 − 1.20900i 0.866025 − 0.500000i 1.68811 − 3.05302i 0.189742 + 1.80527i −0.0673404 + 0.0926861i 0.202965 − 2.99313i −1.34774 + 1.49682i
11.9 −1.87211 + 0.608286i 0.401263 − 1.68493i 1.51676 − 1.10199i 0.866025 − 0.500000i 0.273709 + 3.39846i −0.297852 − 2.83387i 0.144837 − 0.199350i −2.67798 − 1.35220i −1.31715 + 1.46285i
11.10 −1.81428 + 0.589495i −1.28383 − 1.16266i 1.32607 − 0.963448i 0.866025 − 0.500000i 3.01461 + 1.35259i 0.311823 + 2.96680i 0.404656 − 0.556961i 0.296421 + 2.98532i −1.27646 + 1.41766i
11.11 −1.44318 + 0.468916i −0.564342 + 1.63753i 0.244839 − 0.177886i 0.866025 − 0.500000i 0.0465783 − 2.62788i −0.00638236 − 0.0607241i 1.51393 − 2.08375i −2.36304 − 1.84826i −1.01537 + 1.12768i
11.12 −1.12702 + 0.366191i 1.55095 + 0.771077i −0.481957 + 0.350162i 0.866025 − 0.500000i −2.03031 − 0.301076i 0.451880 + 4.29935i 1.80802 − 2.48853i 1.81088 + 2.39180i −0.792932 + 0.880640i
11.13 −1.08138 + 0.351363i −1.72293 − 0.177517i −0.572100 + 0.415655i −0.866025 + 0.500000i 1.92552 − 0.413409i 0.285217 + 2.71366i 1.80928 − 2.49026i 2.93698 + 0.611699i 0.760824 − 0.844981i
11.14 −0.894407 + 0.290610i 1.72906 − 0.101721i −0.902525 + 0.655723i −0.866025 + 0.500000i −1.51692 + 0.593463i −0.268864 − 2.55807i 1.72221 − 2.37042i 2.97931 − 0.351762i 0.629274 − 0.698879i
11.15 −0.859594 + 0.279299i −1.15408 − 1.29155i −0.957139 + 0.695402i −0.866025 + 0.500000i 1.35277 + 0.787877i −0.222156 − 2.11368i 1.69104 − 2.32752i −0.336207 + 2.98110i 0.604781 − 0.671677i
11.16 −0.794622 + 0.258188i 0.879799 − 1.49196i −1.05327 + 0.765247i −0.866025 + 0.500000i −0.313900 + 1.41270i 0.390821 + 3.71841i 1.62158 − 2.23191i −1.45191 − 2.62526i 0.559068 − 0.620908i
11.17 −0.717669 + 0.233185i 0.262684 + 1.71202i −1.15736 + 0.840871i 0.866025 − 0.500000i −0.587736 − 1.16741i −0.279891 − 2.66299i 1.52161 − 2.09432i −2.86199 + 0.899437i −0.504927 + 0.560779i
11.18 −0.515520 + 0.167503i −0.107795 − 1.72869i −1.38033 + 1.00287i 0.866025 − 0.500000i 0.345131 + 0.873119i 0.229375 + 2.18236i 1.18082 − 1.62526i −2.97676 + 0.372690i −0.362702 + 0.402821i
11.19 −0.249885 + 0.0811926i 1.69653 − 0.348958i −1.56218 + 1.13499i 0.866025 − 0.500000i −0.395606 + 0.224945i −0.414216 − 3.94100i 0.607088 − 0.835586i 2.75646 − 1.18404i −0.175811 + 0.195257i
11.20 −0.221949 + 0.0721157i −1.57712 + 0.716031i −1.57397 + 1.14356i 0.866025 − 0.500000i 0.298403 − 0.272658i −0.274533 − 2.61200i 0.541218 − 0.744923i 1.97460 − 2.25853i −0.156156 + 0.173429i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.h odd 30 1 inner
93.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.br.a ✓ 336
3.b odd 2 1 inner 465.2.br.a ✓ 336
31.h odd 30 1 inner 465.2.br.a ✓ 336
93.p even 30 1 inner 465.2.br.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.br.a ✓ 336 1.a even 1 1 trivial
465.2.br.a ✓ 336 3.b odd 2 1 inner
465.2.br.a ✓ 336 31.h odd 30 1 inner
465.2.br.a ✓ 336 93.p even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).