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Properties

Label 430.3.f.b
Level $430$
Weight $3$
Character orbit 430.f
Analytic conductor $11.717$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [430,3,Mod(87,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.87");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 430.f (of order \(4\), degree \(2\), 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: \(11.7166513675\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 44 q^{2} + 4 q^{3} + 4 q^{5} + 8 q^{6} - 88 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 44 q^{2} + 4 q^{3} + 4 q^{5} + 8 q^{6} - 88 q^{8} + 4 q^{10} + 8 q^{12} - 8 q^{13} - 12 q^{15} - 176 q^{16} - 12 q^{17} + 116 q^{18} + 24 q^{21} - 12 q^{23} - 84 q^{25} - 16 q^{26} + 16 q^{27} - 24 q^{30} - 176 q^{32} + 68 q^{33} + 216 q^{35} + 232 q^{36} - 32 q^{37} - 24 q^{38} - 8 q^{40} - 40 q^{41} + 24 q^{42} + 120 q^{45} - 24 q^{46} + 92 q^{47} - 16 q^{48} - 100 q^{50} + 240 q^{51} - 16 q^{52} + 144 q^{53} - 200 q^{55} - 268 q^{57} + 32 q^{58} - 24 q^{60} - 352 q^{61} - 60 q^{63} - 184 q^{65} + 136 q^{66} - 76 q^{67} + 24 q^{68} + 208 q^{70} + 8 q^{71} + 232 q^{72} - 208 q^{73} + 180 q^{75} - 48 q^{76} + 132 q^{77} - 8 q^{78} - 16 q^{80} - 644 q^{81} - 40 q^{82} - 36 q^{83} - 16 q^{85} + 496 q^{87} + 188 q^{90} + 40 q^{91} - 24 q^{92} + 48 q^{93} - 288 q^{95} - 32 q^{96} + 160 q^{97} - 460 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1 1.00000 + 1.00000i −3.97439 + 3.97439i 2.00000i 1.38808 + 4.80346i −7.94879 −6.92505 − 6.92505i −2.00000 + 2.00000i − 22.5916i −3.41538 + 6.19154i
87.2 1.00000 + 1.00000i −3.59694 + 3.59694i 2.00000i 4.03848 − 2.94800i −7.19388 2.61762 + 2.61762i −2.00000 + 2.00000i − 16.8759i 6.98648 + 1.09048i
87.3 1.00000 + 1.00000i −3.32097 + 3.32097i 2.00000i −3.99148 − 3.01133i −6.64193 −2.36128 − 2.36128i −2.00000 + 2.00000i − 13.0576i −0.980144 − 7.00281i
87.4 1.00000 + 1.00000i −2.65967 + 2.65967i 2.00000i −0.473312 − 4.97755i −5.31935 8.69401 + 8.69401i −2.00000 + 2.00000i − 5.14773i 4.50424 − 5.45086i
87.5 1.00000 + 1.00000i −2.47666 + 2.47666i 2.00000i 3.84322 + 3.19839i −4.95333 2.42313 + 2.42313i −2.00000 + 2.00000i − 3.26772i 0.644829 + 7.04160i
87.6 1.00000 + 1.00000i −2.13117 + 2.13117i 2.00000i −4.30411 + 2.54453i −4.26234 3.93083 + 3.93083i −2.00000 + 2.00000i − 0.0837693i −6.84864 − 1.75957i
87.7 1.00000 + 1.00000i −1.56932 + 1.56932i 2.00000i −2.87512 + 4.09068i −3.13863 −8.02886 − 8.02886i −2.00000 + 2.00000i 4.07449i −6.96580 + 1.21556i
87.8 1.00000 + 1.00000i −1.03257 + 1.03257i 2.00000i 1.69544 + 4.70377i −2.06514 5.76808 + 5.76808i −2.00000 + 2.00000i 6.86761i −3.00833 + 6.39921i
87.9 1.00000 + 1.00000i −0.744097 + 0.744097i 2.00000i 1.05249 − 4.88797i −1.48819 −4.98039 − 4.98039i −2.00000 + 2.00000i 7.89264i 5.94046 − 3.83549i
87.10 1.00000 + 1.00000i −0.416400 + 0.416400i 2.00000i −0.917062 − 4.91518i −0.832800 −4.02965 − 4.02965i −2.00000 + 2.00000i 8.65322i 3.99812 − 5.83224i
87.11 1.00000 + 1.00000i 0.148527 − 0.148527i 2.00000i 3.67287 + 3.39264i 0.297053 −7.25854 − 7.25854i −2.00000 + 2.00000i 8.95588i 0.280231 + 7.06551i
87.12 1.00000 + 1.00000i 0.413367 − 0.413367i 2.00000i 3.79133 − 3.25973i 0.826735 7.88446 + 7.88446i −2.00000 + 2.00000i 8.65825i 7.05106 + 0.531606i
87.13 1.00000 + 1.00000i 0.599129 − 0.599129i 2.00000i −4.20095 − 2.71146i 1.19826 0.431885 + 0.431885i −2.00000 + 2.00000i 8.28209i −1.48949 − 6.91241i
87.14 1.00000 + 1.00000i 1.40694 − 1.40694i 2.00000i −5.00000 + 0.00652978i 2.81388 −2.89728 − 2.89728i −2.00000 + 2.00000i 5.04105i −5.00653 − 4.99347i
87.15 1.00000 + 1.00000i 1.45062 − 1.45062i 2.00000i −0.946027 + 4.90969i 2.90124 1.02774 + 1.02774i −2.00000 + 2.00000i 4.79140i −5.85571 + 3.96366i
87.16 1.00000 + 1.00000i 1.60684 − 1.60684i 2.00000i 4.86329 − 1.16122i 3.21368 5.52381 + 5.52381i −2.00000 + 2.00000i 3.83614i 6.02451 + 3.70207i
87.17 1.00000 + 1.00000i 1.73699 − 1.73699i 2.00000i 4.96336 + 0.604171i 3.47398 −1.78292 − 1.78292i −2.00000 + 2.00000i 2.96572i 4.35919 + 5.56753i
87.18 1.00000 + 1.00000i 2.73249 − 2.73249i 2.00000i −3.21472 + 3.82957i 5.46499 6.32285 + 6.32285i −2.00000 + 2.00000i − 5.93303i −7.04429 + 0.614850i
87.19 1.00000 + 1.00000i 3.12245 − 3.12245i 2.00000i 1.89062 − 4.62877i 6.24490 −7.46970 − 7.46970i −2.00000 + 2.00000i − 10.4994i 6.51939 − 2.73816i
87.20 1.00000 + 1.00000i 3.20581 − 3.20581i 2.00000i −4.78260 + 1.45834i 6.41161 −7.39870 − 7.39870i −2.00000 + 2.00000i − 11.5544i −6.24094 − 3.32426i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.3.f.b ✓ 44
5.c odd 4 1 inner 430.3.f.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.3.f.b ✓ 44 1.a even 1 1 trivial
430.3.f.b ✓ 44 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{44} - 4 T_{3}^{43} + 8 T_{3}^{42} + 8 T_{3}^{41} + 2662 T_{3}^{40} - 10388 T_{3}^{39} + \cdots + 29\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display