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Properties

Label 350.3.s.b
Level $350$
Weight $3$
Character orbit 350.s
Analytic conductor $9.537$
Analytic rank $0$
Dimension $128$
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] = [350,3,Mod(113,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([19, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.113");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.s (of order \(20\), 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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 32 q^{2} - 4 q^{3} + 4 q^{5} - 8 q^{6} + 64 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 32 q^{2} - 4 q^{3} + 4 q^{5} - 8 q^{6} + 64 q^{8} + 40 q^{9} + 12 q^{10} - 8 q^{11} - 8 q^{12} + 16 q^{13} - 8 q^{15} + 128 q^{16} - 24 q^{17} + 496 q^{18} + 20 q^{19} + 56 q^{20} + 32 q^{22} + 28 q^{23} - 56 q^{25} + 112 q^{26} + 188 q^{27} - 100 q^{29} - 124 q^{30} - 96 q^{31} - 512 q^{32} - 284 q^{33} - 200 q^{34} - 28 q^{35} - 288 q^{36} - 144 q^{37} - 72 q^{38} + 200 q^{39} + 8 q^{40} + 48 q^{41} - 52 q^{43} - 40 q^{44} + 28 q^{45} - 44 q^{46} + 76 q^{47} + 16 q^{48} - 28 q^{50} + 232 q^{51} + 32 q^{52} - 372 q^{53} - 160 q^{54} - 332 q^{55} - 148 q^{57} + 124 q^{58} + 460 q^{59} + 288 q^{60} - 312 q^{61} - 236 q^{62} - 224 q^{63} + 1192 q^{65} + 372 q^{66} + 692 q^{67} + 48 q^{68} - 280 q^{69} - 84 q^{70} + 72 q^{71} - 8 q^{72} + 596 q^{73} + 812 q^{75} - 144 q^{76} - 56 q^{77} - 216 q^{78} - 640 q^{79} - 16 q^{80} + 680 q^{81} - 192 q^{82} + 136 q^{83} - 404 q^{85} - 284 q^{86} - 1396 q^{87} + 16 q^{88} - 80 q^{89} + 584 q^{90} - 544 q^{92} - 216 q^{93} - 380 q^{94} + 20 q^{95} + 32 q^{96} + 64 q^{97} + 224 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 −1.39680 + 0.221232i −2.57603 + 5.05575i 1.90211 − 0.618034i −2.12569 − 4.52564i 2.47972 − 7.63179i 1.87083 − 1.87083i −2.52015 + 1.28408i −13.6346 − 18.7664i 3.97039 + 5.85116i
113.2 −1.39680 + 0.221232i −2.54181 + 4.98857i 1.90211 − 0.618034i 2.39562 + 4.38873i 2.44677 − 7.53038i −1.87083 + 1.87083i −2.52015 + 1.28408i −13.1350 − 18.0788i −4.31714 − 5.60020i
113.3 −1.39680 + 0.221232i −1.81921 + 3.57041i 1.90211 − 0.618034i −3.95517 + 3.05886i 1.75119 − 5.38962i 1.87083 − 1.87083i −2.52015 + 1.28408i −4.14819 − 5.70950i 4.84787 − 5.14763i
113.4 −1.39680 + 0.221232i −1.76370 + 3.46145i 1.90211 − 0.618034i 3.23952 − 3.80861i 1.69775 − 5.22515i −1.87083 + 1.87083i −2.52015 + 1.28408i −3.58094 − 4.92875i −3.68238 + 6.03656i
113.5 −1.39680 + 0.221232i −1.26616 + 2.48498i 1.90211 − 0.618034i 2.61589 + 4.26111i 1.21882 − 3.75114i 1.87083 − 1.87083i −2.52015 + 1.28408i 0.718103 + 0.988384i −4.59658 − 5.37322i
113.6 −1.39680 + 0.221232i −0.771657 + 1.51446i 1.90211 − 0.618034i −2.28068 − 4.44955i 0.742805 − 2.28612i −1.87083 + 1.87083i −2.52015 + 1.28408i 3.59193 + 4.94386i 4.17004 + 5.71059i
113.7 −1.39680 + 0.221232i −0.286835 + 0.562944i 1.90211 − 0.618034i 4.98132 − 0.431784i 0.276110 − 0.849779i −1.87083 + 1.87083i −2.52015 + 1.28408i 5.05543 + 6.95821i −6.86240 + 1.70514i
113.8 −1.39680 + 0.221232i −0.201634 + 0.395729i 1.90211 − 0.618034i −3.71336 − 3.34828i 0.194095 − 0.597362i 1.87083 − 1.87083i −2.52015 + 1.28408i 5.17412 + 7.12157i 5.92757 + 3.85537i
113.9 −1.39680 + 0.221232i 0.561953 − 1.10290i 1.90211 − 0.618034i 3.18116 − 3.85749i −0.540942 + 1.66485i 1.87083 − 1.87083i −2.52015 + 1.28408i 4.38948 + 6.04160i −3.59005 + 6.09192i
113.10 −1.39680 + 0.221232i 1.06331 − 2.08686i 1.90211 − 0.618034i −1.73358 + 4.68985i −1.02355 + 3.15016i −1.87083 + 1.87083i −2.52015 + 1.28408i 2.06571 + 2.84321i 1.38393 − 6.93432i
113.11 −1.39680 + 0.221232i 1.13019 − 2.21813i 1.90211 − 0.618034i −4.96576 − 0.584162i −1.08794 + 3.34833i −1.87083 + 1.87083i −2.52015 + 1.28408i 1.64730 + 2.26731i 7.06542 − 0.282624i
113.12 −1.39680 + 0.221232i 1.42856 − 2.80371i 1.90211 − 0.618034i −1.79483 + 4.66675i −1.37515 + 4.23227i 1.87083 − 1.87083i −2.52015 + 1.28408i −0.529936 − 0.729395i 1.47459 − 6.91560i
113.13 −1.39680 + 0.221232i 1.83039 − 3.59234i 1.90211 − 0.618034i 1.21115 − 4.85109i −1.76195 + 5.42273i −1.87083 + 1.87083i −2.52015 + 1.28408i −4.26451 − 5.86960i −0.618525 + 7.04396i
113.14 −1.39680 + 0.221232i 2.23691 − 4.39017i 1.90211 − 0.618034i 3.71036 + 3.35161i −2.15327 + 6.62708i −1.87083 + 1.87083i −2.52015 + 1.28408i −8.97982 − 12.3597i −5.92411 − 3.86068i
113.15 −1.39680 + 0.221232i 2.36479 − 4.64117i 1.90211 − 0.618034i 4.99665 + 0.183108i −2.27637 + 7.00596i 1.87083 − 1.87083i −2.52015 + 1.28408i −10.6581 − 14.6696i −7.01984 + 0.849651i
113.16 −1.39680 + 0.221232i 2.40454 − 4.71917i 1.90211 − 0.618034i −4.47852 − 2.22326i −2.31463 + 7.12370i 1.87083 − 1.87083i −2.52015 + 1.28408i −11.1987 − 15.4137i 6.74746 + 2.11467i
127.1 −1.39680 − 0.221232i −2.57603 − 5.05575i 1.90211 + 0.618034i −2.12569 + 4.52564i 2.47972 + 7.63179i 1.87083 + 1.87083i −2.52015 − 1.28408i −13.6346 + 18.7664i 3.97039 − 5.85116i
127.2 −1.39680 − 0.221232i −2.54181 − 4.98857i 1.90211 + 0.618034i 2.39562 − 4.38873i 2.44677 + 7.53038i −1.87083 − 1.87083i −2.52015 − 1.28408i −13.1350 + 18.0788i −4.31714 + 5.60020i
127.3 −1.39680 − 0.221232i −1.81921 − 3.57041i 1.90211 + 0.618034i −3.95517 − 3.05886i 1.75119 + 5.38962i 1.87083 + 1.87083i −2.52015 − 1.28408i −4.14819 + 5.70950i 4.84787 + 5.14763i
127.4 −1.39680 − 0.221232i −1.76370 − 3.46145i 1.90211 + 0.618034i 3.23952 + 3.80861i 1.69775 + 5.22515i −1.87083 − 1.87083i −2.52015 − 1.28408i −3.58094 + 4.92875i −3.68238 − 6.03656i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.s.b ✓ 128
25.f odd 20 1 inner 350.3.s.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.s.b ✓ 128 1.a even 1 1 trivial
350.3.s.b ✓ 128 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{128} + 4 T_{3}^{127} - 12 T_{3}^{126} - 36 T_{3}^{125} - 1984 T_{3}^{124} - 8648 T_{3}^{123} + \cdots + 80\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display