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Properties

Label 430.2.p.a
Level $430$
Weight $2$
Character orbit 430.p
Analytic conductor $3.434$
Analytic rank $0$
Dimension $132$
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(59,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.p (of order \(14\), degree \(6\), 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: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q + 22 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q + 22 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{9} + 8 q^{11} - 10 q^{14} - 20 q^{15} - 22 q^{16} - 4 q^{19} + 4 q^{20} - 24 q^{21} + 4 q^{24} - 16 q^{25} + 12 q^{26} + 40 q^{29} + 24 q^{31} - 4 q^{35} + 128 q^{36} - 56 q^{39} - 28 q^{41} - 8 q^{44} - 80 q^{45} - 12 q^{46} - 136 q^{49} - 56 q^{50} + 172 q^{51} + 16 q^{54} - 4 q^{56} + 16 q^{59} - 8 q^{60} - 44 q^{61} + 22 q^{64} + 42 q^{65} + 8 q^{66} - 62 q^{69} + 12 q^{70} - 8 q^{71} - 4 q^{74} - 122 q^{75} - 52 q^{76} - 64 q^{79} - 4 q^{80} - 56 q^{81} + 24 q^{84} - 72 q^{85} - 10 q^{86} + 16 q^{89} + 2 q^{90} - 24 q^{91} - 20 q^{94} + 106 q^{95} - 4 q^{96} - 104 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −0.974928 + 0.222521i −2.62517 − 0.599178i 0.900969 − 0.433884i −2.23603 + 0.0136065i 2.69268 − 3.67204i −0.781831 + 0.623490i 3.82959 + 1.84424i 2.17694 − 0.510828i
59.2 −0.974928 + 0.222521i −2.59733 − 0.592824i 0.900969 − 0.433884i −0.425312 − 2.19525i 2.66413 4.04951i −0.781831 + 0.623490i 3.69178 + 1.77787i 0.903137 + 2.04557i
59.3 −0.974928 + 0.222521i −2.18942 − 0.499720i 0.900969 − 0.433884i 2.05139 + 0.889832i 2.24572 1.14095i −0.781831 + 0.623490i 1.84092 + 0.886539i −2.19796 − 0.411045i
59.4 −0.974928 + 0.222521i −1.09644 − 0.250255i 0.900969 − 0.433884i 1.67031 − 1.48663i 1.12463 − 4.66029i −0.781831 + 0.623490i −1.56336 − 0.752874i −1.29763 + 1.82103i
59.5 −0.974928 + 0.222521i −0.468512 − 0.106935i 0.900969 − 0.433884i −2.22867 − 0.181698i 0.480560 − 0.219529i −0.781831 + 0.623490i −2.49484 − 1.20145i 2.21323 − 0.318784i
59.6 −0.974928 + 0.222521i 0.271317 + 0.0619263i 0.900969 − 0.433884i −0.882134 + 2.05471i −0.278294 4.13300i −0.781831 + 0.623490i −2.63313 − 1.26805i 0.402800 − 2.19949i
59.7 −0.974928 + 0.222521i 0.607117 + 0.138570i 0.900969 − 0.433884i −0.987643 − 2.00613i −0.622730 0.713157i −0.781831 + 0.623490i −2.35352 − 1.13339i 1.40929 + 1.73606i
59.8 −0.974928 + 0.222521i 1.43560 + 0.327666i 0.900969 − 0.433884i 0.699215 + 2.12393i −1.47252 − 4.43745i −0.781831 + 0.623490i −0.749332 − 0.360859i −1.15430 − 1.91509i
59.9 −0.974928 + 0.222521i 1.83835 + 0.419592i 0.900969 − 0.433884i 1.44701 − 1.70474i −1.88563 0.473994i −0.781831 + 0.623490i 0.500573 + 0.241063i −1.03139 + 1.98399i
59.10 −0.974928 + 0.222521i 2.10581 + 0.480636i 0.900969 − 0.433884i 2.17028 + 0.538429i −2.15996 1.96131i −0.781831 + 0.623490i 1.50050 + 0.722602i −2.23567 − 0.0419976i
59.11 −0.974928 + 0.222521i 3.15256 + 0.719552i 0.900969 − 0.433884i −1.55589 + 1.60599i −3.23364 0.962433i −0.781831 + 0.623490i 6.71798 + 3.23521i 1.15952 − 1.91194i
59.12 0.974928 − 0.222521i −3.15256 − 0.719552i 0.900969 − 0.433884i 1.91194 − 1.15952i −3.23364 − 0.962433i 0.781831 − 0.623490i 6.71798 + 3.23521i 1.60599 − 1.55589i
59.13 0.974928 − 0.222521i −2.10581 − 0.480636i 0.900969 − 0.433884i 0.0419976 + 2.23567i −2.15996 − 1.96131i 0.781831 − 0.623490i 1.50050 + 0.722602i 0.538429 + 2.17028i
59.14 0.974928 − 0.222521i −1.83835 − 0.419592i 0.900969 − 0.433884i −1.98399 + 1.03139i −1.88563 − 0.473994i 0.781831 − 0.623490i 0.500573 + 0.241063i −1.70474 + 1.44701i
59.15 0.974928 − 0.222521i −1.43560 − 0.327666i 0.900969 − 0.433884i 1.91509 + 1.15430i −1.47252 4.43745i 0.781831 − 0.623490i −0.749332 − 0.360859i 2.12393 + 0.699215i
59.16 0.974928 − 0.222521i −0.607117 − 0.138570i 0.900969 − 0.433884i −1.73606 − 1.40929i −0.622730 − 0.713157i 0.781831 − 0.623490i −2.35352 − 1.13339i −2.00613 − 0.987643i
59.17 0.974928 − 0.222521i −0.271317 − 0.0619263i 0.900969 − 0.433884i 2.19949 − 0.402800i −0.278294 − 4.13300i 0.781831 − 0.623490i −2.63313 − 1.26805i 2.05471 − 0.882134i
59.18 0.974928 − 0.222521i 0.468512 + 0.106935i 0.900969 − 0.433884i 0.318784 − 2.21323i 0.480560 0.219529i 0.781831 − 0.623490i −2.49484 − 1.20145i −0.181698 − 2.22867i
59.19 0.974928 − 0.222521i 1.09644 + 0.250255i 0.900969 − 0.433884i −1.82103 + 1.29763i 1.12463 4.66029i 0.781831 − 0.623490i −1.56336 − 0.752874i −1.48663 + 1.67031i
59.20 0.974928 − 0.222521i 2.18942 + 0.499720i 0.900969 − 0.433884i 0.411045 + 2.19796i 2.24572 − 1.14095i 0.781831 − 0.623490i 1.84092 + 0.886539i 0.889832 + 2.05139i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
43.e even 7 1 inner
215.p even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.p.a ✓ 132
5.b even 2 1 inner 430.2.p.a ✓ 132
43.e even 7 1 inner 430.2.p.a ✓ 132
215.p even 14 1 inner 430.2.p.a ✓ 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.p.a ✓ 132 1.a even 1 1 trivial
430.2.p.a ✓ 132 5.b even 2 1 inner
430.2.p.a ✓ 132 43.e even 7 1 inner
430.2.p.a ✓ 132 215.p even 14 1 inner

Hecke kernels

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