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Properties

Label 403.2.bk.a
Level $403$
Weight $2$
Character orbit 403.bk
Analytic conductor $3.218$
Analytic rank $0$
Dimension $280$
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] = [403,2,Mod(9,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.bk (of order \(15\), degree \(8\), 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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(35\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9} - 9 q^{10} - 6 q^{11} - 50 q^{12} - 6 q^{13} - 21 q^{15} + 23 q^{16} + 3 q^{17} - 15 q^{18} - 9 q^{19} + 60 q^{20} - 34 q^{21} + 41 q^{22} - 12 q^{23} - 89 q^{24} - 112 q^{25} + 39 q^{26} + 48 q^{27} + 66 q^{28} - 11 q^{29} + 24 q^{30} - 22 q^{31} + q^{32} - 37 q^{33} - 49 q^{34} + 41 q^{35} + 134 q^{36} + 17 q^{37} - 23 q^{38} + 10 q^{39} + 31 q^{40} + q^{41} - 71 q^{42} - 7 q^{43} - 42 q^{44} - 13 q^{45} + 21 q^{46} - 20 q^{47} - 3 q^{48} - 69 q^{49} - 42 q^{50} - 71 q^{51} - 140 q^{52} - 8 q^{53} - 149 q^{54} + 5 q^{55} - 126 q^{56} + 21 q^{57} + 79 q^{58} + 77 q^{59} + 2 q^{60} + 53 q^{61} + 72 q^{62} + 17 q^{63} - 91 q^{64} - 25 q^{65} - 62 q^{66} - 14 q^{67} + 83 q^{68} - 31 q^{69} + 134 q^{70} + 32 q^{71} - 70 q^{72} - 31 q^{73} + 51 q^{74} + 79 q^{75} + 98 q^{76} - 80 q^{77} - 141 q^{78} - 142 q^{79} + 6 q^{80} + 66 q^{81} - 48 q^{82} + 94 q^{83} + 130 q^{84} + 90 q^{85} - 34 q^{86} + 22 q^{87} - 122 q^{88} - 15 q^{89} + 43 q^{90} + 56 q^{91} - 294 q^{92} - 52 q^{93} - 57 q^{94} + 94 q^{95} + 24 q^{96} + 46 q^{97} + 73 q^{98} + 12 q^{99}+O(q^{100}) \) 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 −1.83126 − 2.03382i −0.500677 + 0.556058i −0.573858 + 5.45990i −0.330308 − 0.572109i 2.04780 −3.67674 − 2.67131i 7.72715 − 5.61410i 0.255062 + 2.42676i −0.558690 + 1.71947i
9.2 −1.65225 − 1.83501i 2.14947 − 2.38723i −0.428272 + 4.07474i −1.22323 − 2.11870i −7.93205 −1.37387 − 0.998173i 4.18946 − 3.04382i −0.765054 − 7.27901i −1.86675 + 5.74525i
9.3 −1.59232 − 1.76846i −1.58986 + 1.76572i −0.382880 + 3.64286i 1.32032 + 2.28686i 5.65416 2.08439 + 1.51440i 3.20149 − 2.32602i −0.276520 − 2.63091i 1.94183 − 5.97634i
9.4 −1.55327 − 1.72508i 1.03676 − 1.15144i −0.354201 + 3.36999i 0.252831 + 0.437917i −3.59669 1.69707 + 1.23299i 2.60770 − 1.89460i 0.0626468 + 0.596045i 0.362727 − 1.11636i
9.5 −1.46781 − 1.63017i −2.12808 + 2.36348i −0.293925 + 2.79651i −0.951364 − 1.64781i 6.97649 −1.54320 − 1.12120i 1.44087 − 1.04685i −0.743696 − 7.07579i −1.28979 + 3.96956i
9.6 −1.44960 − 1.60995i 0.384158 − 0.426651i −0.281525 + 2.67853i −1.83417 − 3.17687i −1.24376 0.633388 + 0.460183i 1.21509 − 0.882816i 0.279132 + 2.65576i −2.45578 + 7.55812i
9.7 −1.32661 − 1.47335i 0.650439 − 0.722386i −0.201810 + 1.92010i 1.17248 + 2.03079i −1.92721 0.184244 + 0.133861i −0.111198 + 0.0807903i 0.214815 + 2.04383i 1.43665 − 4.42155i
9.8 −1.24715 − 1.38510i −0.607940 + 0.675186i −0.154065 + 1.46583i 1.19707 + 2.07339i 1.69340 −2.18489 − 1.58741i −0.793286 + 0.576356i 0.227301 + 2.16262i 1.37893 − 4.24390i
9.9 −0.918709 − 1.02033i −0.334947 + 0.371997i 0.0120106 − 0.114273i −0.585861 − 1.01474i 0.687278 3.13175 + 2.27535i −2.34917 + 1.70677i 0.287394 + 2.73437i −0.497134 + 1.53002i
9.10 −0.769629 − 0.854759i 1.59325 − 1.76948i 0.0707719 − 0.673349i −0.220512 − 0.381938i −2.73870 −1.17517 − 0.853812i −2.49107 + 1.80987i −0.279042 − 2.65490i −0.156753 + 0.482435i
9.11 −0.736014 − 0.817427i 2.06902 − 2.29788i 0.0825875 − 0.785768i 1.51346 + 2.62138i −3.40117 2.69694 + 1.95944i −2.48286 + 1.80390i −0.685820 − 6.52514i 1.02886 − 3.16652i
9.12 −0.718707 − 0.798205i −0.717726 + 0.797115i 0.0884655 − 0.841693i 1.01665 + 1.76088i 1.15210 −3.07573 − 2.23465i −2.47334 + 1.79699i 0.193323 + 1.83935i 0.674874 − 2.07705i
9.13 −0.669055 − 0.743061i −1.14502 + 1.27167i 0.104552 − 0.994745i −1.77181 − 3.06886i 1.71101 −2.25690 − 1.63973i −2.42696 + 1.76329i 0.00750493 + 0.0714046i −1.09491 + 3.36979i
9.14 −0.628504 − 0.698025i −1.76109 + 1.95588i 0.116836 − 1.11162i −0.604953 − 1.04781i 2.47210 2.24460 + 1.63080i −2.36917 + 1.72130i −0.410474 − 3.90540i −0.351181 + 1.08082i
9.15 −0.165338 − 0.183626i 1.65099 − 1.83361i 0.202675 − 1.92832i −1.68082 − 2.91126i −0.609670 0.0202552 + 0.0147163i −0.787407 + 0.572085i −0.322771 − 3.07096i −0.256681 + 0.789984i
9.16 −0.142826 − 0.158624i 0.434282 − 0.482319i 0.204295 − 1.94373i 1.97676 + 3.42385i −0.138534 0.326262 + 0.237043i −0.682872 + 0.496136i 0.269555 + 2.56464i 0.260774 − 0.802579i
9.17 −0.133023 − 0.147737i 0.0833490 − 0.0925684i 0.204926 − 1.94974i −0.998333 − 1.72916i −0.0247631 −1.37062 − 0.995817i −0.636973 + 0.462788i 0.311964 + 2.96813i −0.122660 + 0.377509i
9.18 −0.0525666 − 0.0583811i −1.70784 + 1.89675i 0.208412 − 1.98291i 1.96469 + 3.40295i 0.200510 2.32100 + 1.68631i −0.253832 + 0.184420i −0.367355 − 3.49515i 0.0953908 − 0.293583i
9.19 0.117605 + 0.130614i 0.508986 − 0.565287i 0.205828 − 1.95832i 0.00796452 + 0.0137949i 0.133693 3.75325 + 2.72689i 0.564372 − 0.410040i 0.253104 + 2.40812i −0.000865141 0.00266263i
9.20 0.169063 + 0.187763i −1.38560 + 1.53887i 0.202384 − 1.92556i 0.388050 + 0.672122i −0.523196 −2.21450 − 1.60893i 0.804576 − 0.584559i −0.134633 − 1.28095i −0.0605950 + 0.186492i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bk even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bk.a yes 280
13.c even 3 1 403.2.bj.a ✓ 280
31.g even 15 1 403.2.bj.a ✓ 280
403.bk even 15 1 inner 403.2.bk.a yes 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bj.a ✓ 280 13.c even 3 1
403.2.bj.a ✓ 280 31.g even 15 1
403.2.bk.a yes 280 1.a even 1 1 trivial
403.2.bk.a yes 280 403.bk even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).