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Properties

Label 430.2.t.a
Level $430$
Weight $2$
Character orbit 430.t
Analytic conductor $3.434$
Analytic rank $0$
Dimension $264$
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(9,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.t (of order \(42\), degree \(12\), 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: \(264\)
Relative dimension: \(22\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9} - 8 q^{11} + 10 q^{14} + 32 q^{15} - 44 q^{16} + 4 q^{19} - 4 q^{20} + 24 q^{21} + 2 q^{24} + 28 q^{25} - 12 q^{26} - 46 q^{29} - 36 q^{31} + 12 q^{34} - 68 q^{35} - 134 q^{36} - 64 q^{39} - 20 q^{41} + 8 q^{44} - 70 q^{45} + 112 q^{49} - 28 q^{50} - 28 q^{51} + 68 q^{54} - 30 q^{55} + 4 q^{56} - 40 q^{59} - 4 q^{60} + 20 q^{61} + 44 q^{64} + 18 q^{65} - 44 q^{66} + 32 q^{69} - 48 q^{70} + 20 q^{71} + 40 q^{74} + 122 q^{75} + 52 q^{76} + 16 q^{79} + 4 q^{80} - 16 q^{81} - 24 q^{84} + 120 q^{85} - 14 q^{86} - 142 q^{89} - 68 q^{90} - 4 q^{94} - 22 q^{95} - 2 q^{96} - 268 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.781831 + 0.623490i −0.511545 + 3.39388i 0.222521 − 0.974928i 1.50535 + 1.65346i −1.71611 − 2.97238i 1.77402 + 1.02423i 0.433884 + 0.900969i −8.39001 − 2.58798i −2.20784 − 0.354160i
9.2 −0.781831 + 0.623490i −0.362457 + 2.40475i 0.222521 − 0.974928i −2.23136 + 0.145098i −1.21595 − 2.10610i −1.90024 − 1.09710i 0.433884 + 0.900969i −2.78472 − 0.858971i 1.65408 − 1.50467i
9.3 −0.781831 + 0.623490i −0.299194 + 1.98502i 0.222521 − 0.974928i 0.934065 − 2.03163i −1.00372 − 1.73850i 1.47449 + 0.851296i 0.433884 + 0.900969i −0.984076 − 0.303547i 0.536420 + 2.17077i
9.4 −0.781831 + 0.623490i −0.192290 + 1.27576i 0.222521 − 0.974928i 2.15300 − 0.603819i −0.645086 − 1.11732i −4.41993 − 2.55185i 0.433884 + 0.900969i 1.27612 + 0.393632i −1.30681 + 1.81446i
9.5 −0.781831 + 0.623490i −0.115718 + 0.767737i 0.222521 − 0.974928i −0.503717 + 2.17859i −0.388204 − 0.672390i 0.338318 + 0.195328i 0.433884 + 0.900969i 2.29069 + 0.706584i −0.964509 − 2.01736i
9.6 −0.781831 + 0.623490i 0.0612820 − 0.406579i 0.222521 − 0.974928i 2.16843 + 0.545829i 0.205586 + 0.356085i 1.79116 + 1.03412i 0.433884 + 0.900969i 2.70517 + 0.834434i −2.03566 + 0.925245i
9.7 −0.781831 + 0.623490i 0.0684663 − 0.454244i 0.222521 − 0.974928i −0.0829298 − 2.23453i 0.229688 + 0.397830i −1.76000 − 1.01613i 0.433884 + 0.900969i 2.66507 + 0.822065i 1.45804 + 1.69532i
9.8 −0.781831 + 0.623490i 0.225928 − 1.49894i 0.222521 − 0.974928i −2.22164 + 0.253613i 0.757933 + 1.31278i 3.55836 + 2.05442i 0.433884 + 0.900969i 0.670955 + 0.206962i 1.57882 − 1.58345i
9.9 −0.781831 + 0.623490i 0.287268 − 1.90590i 0.222521 − 0.974928i 0.339942 + 2.21008i 0.963713 + 1.66920i −2.79107 − 1.61143i 0.433884 + 0.900969i −0.683210 − 0.210742i −1.64374 − 1.51596i
9.10 −0.781831 + 0.623490i 0.320497 − 2.12636i 0.222521 − 0.974928i 1.14747 − 1.91919i 1.07519 + 1.86228i 3.36198 + 1.94104i 0.433884 + 0.900969i −1.55196 − 0.478717i 0.299466 + 2.21592i
9.11 −0.781831 + 0.623490i 0.408860 − 2.71261i 0.222521 − 0.974928i −2.17831 + 0.504950i 1.37162 + 2.37572i −2.50700 − 1.44741i 0.433884 + 0.900969i −4.32436 − 1.33389i 1.38824 − 1.75294i
9.12 0.781831 − 0.623490i −0.408860 + 2.71261i 0.222521 − 0.974928i −0.325780 − 2.21221i 1.37162 + 2.37572i 2.50700 + 1.44741i −0.433884 − 0.900969i −4.32436 − 1.33389i −1.63399 − 1.52645i
9.13 0.781831 − 0.623490i −0.320497 + 2.12636i 0.222521 − 0.974928i −1.36731 + 1.76931i 1.07519 + 1.86228i −3.36198 − 1.94104i −0.433884 − 0.900969i −1.55196 − 0.478717i 0.0341455 + 2.23581i
9.14 0.781831 − 0.623490i −0.287268 + 1.90590i 0.222521 − 0.974928i 2.18150 − 0.490989i 0.963713 + 1.66920i 2.79107 + 1.61143i −0.433884 − 0.900969i −0.683210 − 0.210742i 1.39944 − 1.74401i
9.15 0.781831 − 0.623490i −0.225928 + 1.49894i 0.222521 − 0.974928i −0.575574 − 2.16072i 0.757933 + 1.31278i −3.55836 − 2.05442i −0.433884 − 0.900969i 0.670955 + 0.206962i −1.79719 − 1.33046i
9.16 0.781831 − 0.623490i −0.0684663 + 0.454244i 0.222521 − 0.974928i −2.11036 + 0.739168i 0.229688 + 0.397830i 1.76000 + 1.01613i −0.433884 − 0.900969i 2.66507 + 0.822065i −1.18908 + 1.89369i
9.17 0.781831 − 0.623490i −0.0612820 + 0.406579i 0.222521 − 0.974928i 1.30031 + 1.81912i 0.205586 + 0.356085i −1.79116 − 1.03412i −0.433884 − 0.900969i 2.70517 + 0.834434i 2.15083 + 0.611511i
9.18 0.781831 − 0.623490i 0.115718 − 0.767737i 0.222521 − 0.974928i 1.84397 − 1.26483i −0.388204 − 0.672390i −0.338318 − 0.195328i −0.433884 − 0.900969i 2.29069 + 0.706584i 0.653065 − 2.13858i
9.19 0.781831 − 0.623490i 0.192290 − 1.27576i 0.222521 − 0.974928i 0.224499 + 2.22477i −0.645086 − 1.11732i 4.41993 + 2.55185i −0.433884 − 0.900969i 1.27612 + 0.393632i 1.56264 + 1.59942i
9.20 0.781831 − 0.623490i 0.299194 − 1.98502i 0.222521 − 0.974928i −1.54994 + 1.61173i −1.00372 − 1.73850i −1.47449 − 0.851296i −0.433884 − 0.900969i −0.984076 − 0.303547i −0.206891 + 2.22648i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
43.g even 21 1 inner
215.u even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.t.a ✓ 264
5.b even 2 1 inner 430.2.t.a ✓ 264
43.g even 21 1 inner 430.2.t.a ✓ 264
215.u even 42 1 inner 430.2.t.a ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.t.a ✓ 264 1.a even 1 1 trivial
430.2.t.a ✓ 264 5.b even 2 1 inner
430.2.t.a ✓ 264 43.g even 21 1 inner
430.2.t.a ✓ 264 215.u even 42 1 inner

Hecke kernels

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