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Properties

Label 663.2.bp.c
Level $663$
Weight $2$
Character orbit 663.bp
Analytic conductor $5.294$
Analytic rank $0$
Dimension $304$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [663,2,Mod(50,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 7, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.50");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.bp (of order \(12\), degree \(4\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(76\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 24 q^{4} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q - 24 q^{4} + 20 q^{9} - 32 q^{13} - 12 q^{15} + 96 q^{16} - 8 q^{18} - 12 q^{21} - 12 q^{30} - 16 q^{33} + 4 q^{34} + 84 q^{36} + 32 q^{42} - 24 q^{43} - 24 q^{49} - 48 q^{52} + 8 q^{55} + 120 q^{60} - 160 q^{66} - 32 q^{67} - 24 q^{69} + 8 q^{70} - 184 q^{72} - 104 q^{76} + 108 q^{81} - 204 q^{84} - 108 q^{85} + 20 q^{87} - 72 q^{93} + 16 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −2.64709 − 0.709284i −1.60517 + 0.650706i 4.77193 + 2.75507i 0.343905 + 0.343905i 4.71057 − 0.583951i 3.10677 − 0.832458i −6.80197 − 6.80197i 2.15316 − 2.08899i −0.666420 − 1.15427i
50.2 −2.64709 − 0.709284i 1.60517 − 0.650706i 4.77193 + 2.75507i −0.343905 − 0.343905i −4.71057 + 0.583951i −3.10677 + 0.832458i −6.80197 − 6.80197i 2.15316 − 2.08899i 0.666420 + 1.15427i
50.3 −2.48285 − 0.665277i −1.44191 − 0.959628i 3.98989 + 2.30356i 0.365207 + 0.365207i 2.94163 + 3.34188i −1.06279 + 0.284774i −4.73863 − 4.73863i 1.15823 + 2.76740i −0.663789 − 1.14972i
50.4 −2.48285 − 0.665277i 1.44191 + 0.959628i 3.98989 + 2.30356i −0.365207 − 0.365207i −2.94163 − 3.34188i 1.06279 − 0.284774i −4.73863 − 4.73863i 1.15823 + 2.76740i 0.663789 + 1.14972i
50.5 −2.37283 − 0.635798i −0.171459 − 1.72354i 3.49404 + 2.01728i 1.12219 + 1.12219i −0.688982 + 4.19869i −2.80851 + 0.752539i −3.53411 − 3.53411i −2.94120 + 0.591035i −1.94927 − 3.37624i
50.6 −2.37283 − 0.635798i 0.171459 + 1.72354i 3.49404 + 2.01728i −1.12219 − 1.12219i 0.688982 − 4.19869i 2.80851 − 0.752539i −3.53411 − 3.53411i −2.94120 + 0.591035i 1.94927 + 3.37624i
50.7 −2.36858 − 0.634660i −0.270859 + 1.71074i 3.47534 + 2.00649i −2.55357 − 2.55357i 1.72729 − 3.88013i −3.70993 + 0.994073i −3.49034 − 3.49034i −2.85327 − 0.926739i 4.42769 + 7.66898i
50.8 −2.36858 − 0.634660i 0.270859 − 1.71074i 3.47534 + 2.00649i 2.55357 + 2.55357i −1.72729 + 3.88013i 3.70993 − 0.994073i −3.49034 − 3.49034i −2.85327 − 0.926739i −4.42769 − 7.66898i
50.9 −2.16844 − 0.581032i −1.66125 − 0.490151i 2.63249 + 1.51987i −2.28380 − 2.28380i 3.31753 + 2.02810i −0.618949 + 0.165847i −1.65049 − 1.65049i 2.51950 + 1.62853i 3.62532 + 6.27924i
50.10 −2.16844 − 0.581032i 1.66125 + 0.490151i 2.63249 + 1.51987i 2.28380 + 2.28380i −3.31753 − 2.02810i 0.618949 − 0.165847i −1.65049 − 1.65049i 2.51950 + 1.62853i −3.62532 − 6.27924i
50.11 −2.06193 − 0.552492i −1.02783 + 1.39412i 2.21425 + 1.27840i 1.35122 + 1.35122i 2.88955 − 2.30671i 0.0545286 − 0.0146109i −0.840456 − 0.840456i −0.887142 − 2.86583i −2.03959 − 3.53267i
50.12 −2.06193 − 0.552492i 1.02783 − 1.39412i 2.21425 + 1.27840i −1.35122 − 1.35122i −2.88955 + 2.30671i −0.0545286 + 0.0146109i −0.840456 − 0.840456i −0.887142 − 2.86583i 2.03959 + 3.53267i
50.13 −1.95380 − 0.523519i −0.711011 − 1.57939i 1.81121 + 1.04570i −1.50678 − 1.50678i 0.562333 + 3.45803i 2.65440 − 0.711244i −0.130734 − 0.130734i −1.98893 + 2.24592i 2.15512 + 3.73277i
50.14 −1.95380 − 0.523519i 0.711011 + 1.57939i 1.81121 + 1.04570i 1.50678 + 1.50678i −0.562333 − 3.45803i −2.65440 + 0.711244i −0.130734 − 0.130734i −1.98893 + 2.24592i −2.15512 − 3.73277i
50.15 −1.79017 − 0.479674i −1.72145 − 0.191303i 1.24257 + 0.717397i 2.93851 + 2.93851i 2.98993 + 1.16820i 1.90254 − 0.509785i 0.740700 + 0.740700i 2.92681 + 0.658637i −3.85090 − 6.66996i
50.16 −1.79017 − 0.479674i 1.72145 + 0.191303i 1.24257 + 0.717397i −2.93851 − 2.93851i −2.98993 − 1.16820i −1.90254 + 0.509785i 0.740700 + 0.740700i 2.92681 + 0.658637i 3.85090 + 6.66996i
50.17 −1.76331 − 0.472477i −1.44371 + 0.956921i 1.15397 + 0.666242i 0.552271 + 0.552271i 2.99783 − 1.00523i −0.943563 + 0.252827i 0.861649 + 0.861649i 1.16860 − 2.76304i −0.712888 − 1.23476i
50.18 −1.76331 − 0.472477i 1.44371 − 0.956921i 1.15397 + 0.666242i −0.552271 − 0.552271i −2.99783 + 1.00523i 0.943563 − 0.252827i 0.861649 + 0.861649i 1.16860 − 2.76304i 0.712888 + 1.23476i
50.19 −1.51731 − 0.406562i −1.40066 + 1.01890i 0.404885 + 0.233761i −2.73832 − 2.73832i 2.53948 − 0.976524i 4.47071 − 1.19792i 1.70220 + 1.70220i 0.923705 − 2.85425i 3.04158 + 5.26817i
50.20 −1.51731 − 0.406562i 1.40066 − 1.01890i 0.404885 + 0.233761i 2.73832 + 2.73832i −2.53948 + 0.976524i −4.47071 + 1.19792i 1.70220 + 1.70220i 0.923705 − 2.85425i −3.04158 − 5.26817i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
17.b even 2 1 inner
39.k even 12 1 inner
51.c odd 2 1 inner
221.w odd 12 1 inner
663.bp even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.bp.c ✓ 304
3.b odd 2 1 inner 663.2.bp.c ✓ 304
13.f odd 12 1 inner 663.2.bp.c ✓ 304
17.b even 2 1 inner 663.2.bp.c ✓ 304
39.k even 12 1 inner 663.2.bp.c ✓ 304
51.c odd 2 1 inner 663.2.bp.c ✓ 304
221.w odd 12 1 inner 663.2.bp.c ✓ 304
663.bp even 12 1 inner 663.2.bp.c ✓ 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.bp.c ✓ 304 1.a even 1 1 trivial
663.2.bp.c ✓ 304 3.b odd 2 1 inner
663.2.bp.c ✓ 304 13.f odd 12 1 inner
663.2.bp.c ✓ 304 17.b even 2 1 inner
663.2.bp.c ✓ 304 39.k even 12 1 inner
663.2.bp.c ✓ 304 51.c odd 2 1 inner
663.2.bp.c ✓ 304 221.w odd 12 1 inner
663.2.bp.c ✓ 304 663.bp even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\):

\( T_{2}^{152} + 6 T_{2}^{150} - 272 T_{2}^{148} - 1704 T_{2}^{146} + 41982 T_{2}^{144} + \cdots + 830702150185216 \) Copy content Toggle raw display
\( T_{5}^{152} + 2372 T_{5}^{148} + 2591202 T_{5}^{144} + 1731386970 T_{5}^{140} + 792688364039 T_{5}^{136} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display