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Properties

Label 430.2.x.a
Level $430$
Weight $2$
Character orbit 430.x
Analytic conductor $3.434$
Analytic rank $0$
Dimension $528$
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] = [430,2,Mod(3,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(84))
 
chi = DirichletCharacter(H, H._module([63, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.x (of order \(84\), degree \(24\), 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: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(528\)
Relative dimension: \(22\) over \(\Q(\zeta_{84})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 528 q - 4 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 528 q - 4 q^{6} - 12 q^{7} + 40 q^{13} + 88 q^{16} - 4 q^{17} + 16 q^{21} - 12 q^{23} + 8 q^{25} - 12 q^{28} + 36 q^{30} - 72 q^{31} - 124 q^{33} + 40 q^{35} - 268 q^{36} - 44 q^{38} + 56 q^{41} - 168 q^{43} - 24 q^{46} - 72 q^{47} - 24 q^{50} + 280 q^{51} + 16 q^{52} - 132 q^{53} - 24 q^{55} + 8 q^{56} - 20 q^{57} - 8 q^{60} - 72 q^{61} + 48 q^{62} - 84 q^{65} - 72 q^{66} - 40 q^{67} - 32 q^{68} - 24 q^{71} + 192 q^{73} + 48 q^{76} - 148 q^{77} - 40 q^{78} + 56 q^{82} - 28 q^{83} - 20 q^{86} - 216 q^{87} + 8 q^{90} - 48 q^{91} + 12 q^{92} + 108 q^{93} - 4 q^{95} + 4 q^{96} - 120 q^{97} + 96 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.330279 − 0.943883i −2.12618 − 2.47066i −0.781831 + 0.623490i −1.43957 + 1.71103i −1.62979 + 2.82287i 0.781116 − 2.91516i 0.846724 + 0.532032i −1.13642 + 7.53964i 2.09047 + 0.793673i
3.2 −0.330279 − 0.943883i −1.48270 − 1.72293i −0.781831 + 0.623490i 1.46328 + 1.69080i −1.13654 + 1.96854i −0.664636 + 2.48045i 0.846724 + 0.532032i −0.322955 + 2.14267i 1.11263 − 1.93960i
3.3 −0.330279 − 0.943883i −1.40744 − 1.63547i −0.781831 + 0.623490i 1.98609 − 1.02735i −1.07885 + 1.86862i 0.821541 − 3.06603i 0.846724 + 0.532032i −0.246762 + 1.63716i −1.62566 − 1.53533i
3.4 −0.330279 − 0.943883i −1.23630 − 1.43661i −0.781831 + 0.623490i 1.28939 − 1.82688i −0.947664 + 1.64140i −1.29317 + 4.82619i 0.846724 + 0.532032i −0.0882734 + 0.585656i −2.15022 − 0.613651i
3.5 −0.330279 − 0.943883i −0.777489 − 0.903458i −0.781831 + 0.623490i −2.03981 − 0.916062i −0.595971 + 1.03225i 0.237699 − 0.887105i 0.846724 + 0.532032i 0.235379 − 1.56164i −0.190949 + 2.22790i
3.6 −0.330279 − 0.943883i −0.0131640 − 0.0152969i −0.781831 + 0.623490i −1.62810 + 1.53273i −0.0100907 + 0.0174775i 0.140005 − 0.522506i 0.846724 + 0.532032i 0.447066 − 2.96609i 1.98445 + 1.03051i
3.7 −0.330279 − 0.943883i 0.658429 + 0.765108i −0.781831 + 0.623490i −0.824716 − 2.07842i 0.504708 − 0.874179i −1.09920 + 4.10225i 0.846724 + 0.532032i 0.295265 − 1.95895i −1.68940 + 1.46490i
3.8 −0.330279 − 0.943883i 0.671823 + 0.780672i −0.781831 + 0.623490i 0.417921 − 2.19667i 0.514974 − 0.891962i 0.647385 − 2.41607i 0.846724 + 0.532032i 0.289024 − 1.91755i −2.21143 + 0.331044i
3.9 −0.330279 − 0.943883i 1.23231 + 1.43197i −0.781831 + 0.623490i 1.04170 + 1.97860i 0.944603 − 1.63610i 1.31062 − 4.89129i 0.846724 + 0.532032i −0.0848203 + 0.562746i 1.52352 − 1.63673i
3.10 −0.330279 − 0.943883i 1.64699 + 1.91384i −0.781831 + 0.623490i 2.22442 + 0.227923i 1.26247 − 2.18667i −0.541490 + 2.02087i 0.846724 + 0.532032i −0.503069 + 3.33764i −0.519547 − 2.17487i
3.11 −0.330279 − 0.943883i 1.75583 + 2.04031i −0.781831 + 0.623490i −2.23481 + 0.0750623i 1.34590 − 2.33117i −0.285871 + 1.06689i 0.846724 + 0.532032i −0.632803 + 4.19837i 0.808960 + 2.08461i
3.12 0.330279 + 0.943883i −1.92406 − 2.23579i −0.781831 + 0.623490i −2.01247 − 0.974668i 1.47485 − 2.55452i −0.743303 + 2.77404i −0.846724 − 0.532032i −0.849656 + 5.63710i 0.255297 − 2.22145i
3.13 0.330279 + 0.943883i −1.35577 − 1.57544i −0.781831 + 0.623490i 1.79734 − 1.33025i 1.03925 − 1.80003i −0.398389 + 1.48681i −0.846724 − 0.532032i −0.196755 + 1.30538i 1.84922 + 1.25713i
3.14 0.330279 + 0.943883i −1.26939 − 1.47505i −0.781831 + 0.623490i −0.345539 + 2.20921i 0.973028 − 1.68533i −0.0286434 + 0.106899i −0.846724 − 0.532032i −0.117316 + 0.778339i −2.19936 + 0.403507i
3.15 0.330279 + 0.943883i −0.667495 − 0.775643i −0.781831 + 0.623490i −1.70213 + 1.45008i 0.511657 − 0.886215i 0.854652 − 3.18961i −0.846724 − 0.532032i 0.291054 − 1.93102i −1.93089 − 1.12768i
3.16 0.330279 + 0.943883i −0.515215 − 0.598691i −0.781831 + 0.623490i 2.22238 + 0.247005i 0.394930 − 0.684038i 0.301636 − 1.12572i −0.846724 − 0.532032i 0.354143 − 2.34958i 0.500863 + 2.17925i
3.17 0.330279 + 0.943883i 0.0217181 + 0.0252369i −0.781831 + 0.623490i −2.02196 − 0.954820i −0.0166477 + 0.0288346i −0.751874 + 2.80603i −0.846724 − 0.532032i 0.446962 − 2.96540i 0.233427 − 2.22385i
3.18 0.330279 + 0.943883i 0.447979 + 0.520562i −0.781831 + 0.623490i −0.398933 − 2.20019i −0.343391 + 0.594771i 0.739536 − 2.75998i −0.846724 − 0.532032i 0.376828 − 2.50009i 1.94497 − 1.10322i
3.19 0.330279 + 0.943883i 0.993378 + 1.15433i −0.781831 + 0.623490i 1.09418 + 1.95007i −0.761458 + 1.31888i −0.436975 + 1.63081i −0.846724 − 0.532032i 0.101458 − 0.673127i −1.47925 + 1.67685i
3.20 0.330279 + 0.943883i 1.27566 + 1.48234i −0.781831 + 0.623490i 1.46555 − 1.68884i −0.977833 + 1.69366i −1.09982 + 4.10459i −0.846724 − 0.532032i −0.122904 + 0.815417i 2.07811 + 0.825518i
See next 80 embeddings (of 528 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.h odd 42 1 inner
215.x even 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.x.a ✓ 528
5.c odd 4 1 inner 430.2.x.a ✓ 528
43.h odd 42 1 inner 430.2.x.a ✓ 528
215.x even 84 1 inner 430.2.x.a ✓ 528
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.x.a ✓ 528 1.a even 1 1 trivial
430.2.x.a ✓ 528 5.c odd 4 1 inner
430.2.x.a ✓ 528 43.h odd 42 1 inner
430.2.x.a ✓ 528 215.x even 84 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(430, [\chi])\).