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Properties

Label 570.4.f.b
Level $570$
Weight $4$
Character orbit 570.f
Analytic conductor $33.631$
Analytic rank $0$
Dimension $40$
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] = [570,4,Mod(341,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.341");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 570.f (of order \(2\), degree \(1\), 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: \(33.6310887033\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 80 q^{2} + 10 q^{3} + 160 q^{4} + 20 q^{6} - 20 q^{7} + 320 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 80 q^{2} + 10 q^{3} + 160 q^{4} + 20 q^{6} - 20 q^{7} + 320 q^{8} + 10 q^{9} + 40 q^{12} - 40 q^{14} + 640 q^{16} + 20 q^{18} + 68 q^{19} - 78 q^{21} + 80 q^{24} - 1000 q^{25} + 292 q^{27} - 80 q^{28} - 60 q^{29} + 1280 q^{32} + 440 q^{33} + 40 q^{36} + 136 q^{38} - 634 q^{39} + 888 q^{41} - 156 q^{42} + 488 q^{43} + 160 q^{45} + 160 q^{48} + 924 q^{49} - 2000 q^{50} + 2098 q^{51} - 108 q^{53} + 584 q^{54} - 160 q^{56} + 1562 q^{57} - 120 q^{58} + 132 q^{59} - 1496 q^{61} + 762 q^{63} + 2560 q^{64} + 120 q^{65} + 880 q^{66} - 1222 q^{69} - 1128 q^{71} + 80 q^{72} - 316 q^{73} - 250 q^{75} + 272 q^{76} - 1268 q^{78} - 942 q^{81} + 1776 q^{82} - 312 q^{84} + 976 q^{86} + 1830 q^{87} + 1776 q^{89} + 320 q^{90} - 1568 q^{93} - 660 q^{95} + 320 q^{96} + 1848 q^{98} - 4016 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 2.00000 −5.18941 − 0.264627i 4.00000 − 5.00000i −10.3788 − 0.529255i −11.4208 8.00000 26.8599 + 2.74652i − 10.0000i
341.2 2.00000 −5.18941 + 0.264627i 4.00000 5.00000i −10.3788 + 0.529255i −11.4208 8.00000 26.8599 − 2.74652i 10.0000i
341.3 2.00000 −4.96315 − 1.53855i 4.00000 5.00000i −9.92630 − 3.07711i 25.5559 8.00000 22.2657 + 15.2721i 10.0000i
341.4 2.00000 −4.96315 + 1.53855i 4.00000 − 5.00000i −9.92630 + 3.07711i 25.5559 8.00000 22.2657 − 15.2721i − 10.0000i
341.5 2.00000 −4.80682 − 1.97346i 4.00000 − 5.00000i −9.61363 − 3.94691i −16.4640 8.00000 19.2109 + 18.9721i − 10.0000i
341.6 2.00000 −4.80682 + 1.97346i 4.00000 5.00000i −9.61363 + 3.94691i −16.4640 8.00000 19.2109 − 18.9721i 10.0000i
341.7 2.00000 −3.88445 − 3.45124i 4.00000 − 5.00000i −7.76890 − 6.90247i −0.609196 8.00000 3.17792 + 26.8123i − 10.0000i
341.8 2.00000 −3.88445 + 3.45124i 4.00000 5.00000i −7.76890 + 6.90247i −0.609196 8.00000 3.17792 − 26.8123i 10.0000i
341.9 2.00000 −3.72490 − 3.62286i 4.00000 5.00000i −7.44980 − 7.24572i −15.6708 8.00000 0.749737 + 26.9896i 10.0000i
341.10 2.00000 −3.72490 + 3.62286i 4.00000 − 5.00000i −7.44980 + 7.24572i −15.6708 8.00000 0.749737 − 26.9896i − 10.0000i
341.11 2.00000 −3.00767 − 4.23721i 4.00000 − 5.00000i −6.01534 − 8.47441i 25.9641 8.00000 −8.90783 + 25.4882i − 10.0000i
341.12 2.00000 −3.00767 + 4.23721i 4.00000 5.00000i −6.01534 + 8.47441i 25.9641 8.00000 −8.90783 − 25.4882i 10.0000i
341.13 2.00000 −2.37233 − 4.62299i 4.00000 5.00000i −4.74466 − 9.24598i 12.6718 8.00000 −15.7441 + 21.9345i 10.0000i
341.14 2.00000 −2.37233 + 4.62299i 4.00000 − 5.00000i −4.74466 + 9.24598i 12.6718 8.00000 −15.7441 − 21.9345i − 10.0000i
341.15 2.00000 −1.49896 − 4.97525i 4.00000 − 5.00000i −2.99792 − 9.95050i 9.34413 8.00000 −22.5062 + 14.9154i − 10.0000i
341.16 2.00000 −1.49896 + 4.97525i 4.00000 5.00000i −2.99792 + 9.95050i 9.34413 8.00000 −22.5062 − 14.9154i 10.0000i
341.17 2.00000 −1.06997 − 5.08480i 4.00000 5.00000i −2.13994 − 10.1696i 2.56764 8.00000 −24.7103 + 10.8812i 10.0000i
341.18 2.00000 −1.06997 + 5.08480i 4.00000 − 5.00000i −2.13994 + 10.1696i 2.56764 8.00000 −24.7103 − 10.8812i − 10.0000i
341.19 2.00000 −0.670363 − 5.15273i 4.00000 5.00000i −1.34073 − 10.3055i −23.2517 8.00000 −26.1012 + 6.90839i 10.0000i
341.20 2.00000 −0.670363 + 5.15273i 4.00000 − 5.00000i −1.34073 + 10.3055i −23.2517 8.00000 −26.1012 − 6.90839i − 10.0000i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.4.f.b yes 40
3.b odd 2 1 570.4.f.a ✓ 40
19.b odd 2 1 570.4.f.a ✓ 40
57.d even 2 1 inner 570.4.f.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.f.a ✓ 40 3.b odd 2 1
570.4.f.a ✓ 40 19.b odd 2 1
570.4.f.b yes 40 1.a even 1 1 trivial
570.4.f.b yes 40 57.d even 2 1 inner