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Properties

Label 471.2.q.b
Level $471$
Weight $2$
Character orbit 471.q
Analytic conductor $3.761$
Analytic rank $0$
Dimension $336$
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] = [471,2,Mod(19,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([0, 62]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.q (of order \(39\), degree \(24\), 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: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(14\) over \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 2 q^{2} + 14 q^{3} - 26 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 2 q^{2} + 14 q^{3} - 26 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 14 q^{9} + q^{10} - 2 q^{11} - 182 q^{12} - 78 q^{13} - 38 q^{14} - 24 q^{15} - 14 q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 2 q^{22} - 7 q^{23} + 124 q^{25} + 118 q^{26} - 28 q^{27} - 80 q^{28} + 49 q^{29} + q^{30} - 12 q^{31} - 12 q^{32} + 11 q^{33} - 63 q^{34} + 65 q^{35} + 13 q^{36} - 25 q^{37} - 5 q^{38} + 73 q^{40} - 94 q^{41} - 7 q^{42} - 10 q^{43} - 12 q^{44} - 17 q^{45} - 16 q^{46} + 29 q^{47} + 7 q^{48} - 70 q^{49} - 36 q^{50} + 101 q^{51} - 27 q^{52} - 32 q^{53} - 2 q^{54} - 23 q^{55} - 130 q^{56} - 2 q^{57} + 302 q^{58} - 18 q^{59} - q^{60} - 40 q^{61} - 130 q^{62} + 28 q^{63} - 16 q^{64} + 52 q^{65} - 24 q^{66} - 114 q^{67} + 150 q^{68} - 3 q^{69} - 15 q^{70} + 27 q^{71} - 3 q^{73} - 125 q^{74} - 40 q^{75} + 368 q^{76} - 24 q^{77} - 41 q^{78} + 55 q^{79} + 78 q^{80} + 14 q^{81} + 24 q^{82} + 40 q^{83} + 105 q^{84} - 142 q^{85} - 48 q^{86} - 5 q^{87} - 255 q^{88} - 52 q^{89} - 2 q^{90} + 130 q^{91} - 366 q^{92} - 54 q^{93} - 64 q^{94} - 58 q^{95} + 136 q^{96} - 110 q^{97} - 6 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.67728 − 0.659891i −0.845190 − 0.534466i 4.96148 + 2.60399i 2.70271 − 2.03095i 1.91013 + 1.98865i 0.750173 − 0.393721i −7.43705 − 6.58866i 0.428693 + 0.903450i −8.57612 + 3.65394i
19.2 −2.54821 − 0.628078i −0.845190 − 0.534466i 4.32800 + 2.27151i −2.61874 + 1.96786i 1.81804 + 1.89278i 1.20275 − 0.631251i −5.67309 − 5.02592i 0.428693 + 0.903450i 7.90908 − 3.36974i
19.3 −2.06747 − 0.509586i −0.845190 − 0.534466i 2.24385 + 1.17766i −0.929494 + 0.698469i 1.47505 + 1.53569i −0.0122606 + 0.00643484i −0.851290 − 0.754178i 0.428693 + 0.903450i 2.27763 − 0.970408i
19.4 −1.84576 − 0.454940i −0.845190 − 0.534466i 1.42896 + 0.749975i 1.98261 − 1.48983i 1.31687 + 1.37101i −1.31241 + 0.688808i 0.549517 + 0.486830i 0.428693 + 0.903450i −4.33720 + 1.84791i
19.5 −1.32814 − 0.327356i −0.845190 − 0.534466i −0.114131 − 0.0599004i −1.19669 + 0.899253i 0.947566 + 0.986521i −3.33061 + 1.74804i 2.17972 + 1.93107i 0.428693 + 0.903450i 1.88374 − 0.802587i
19.6 −0.670127 − 0.165171i −0.845190 − 0.534466i −1.34912 − 0.708075i 2.10094 − 1.57875i 0.478106 + 0.497761i 2.63145 − 1.38109i 1.82035 + 1.61269i 0.428693 + 0.903450i −1.66866 + 0.710950i
19.7 −0.252108 − 0.0621390i −0.845190 − 0.534466i −1.71121 − 0.898115i 0.591924 − 0.444802i 0.179868 + 0.187262i −1.74335 + 0.914980i 0.764309 + 0.677118i 0.428693 + 0.903450i −0.176868 + 0.0753565i
19.8 0.304362 + 0.0750185i −0.845190 − 0.534466i −1.68390 − 0.883781i −1.40241 + 1.05384i −0.217149 − 0.226076i 3.02560 − 1.58796i −0.915489 − 0.811052i 0.428693 + 0.903450i −0.505897 + 0.215543i
19.9 0.309649 + 0.0763217i −0.845190 − 0.534466i −1.68085 − 0.882180i −2.49078 + 1.87170i −0.220921 − 0.230003i 1.37070 − 0.719396i −0.930571 − 0.824414i 0.428693 + 0.903450i −0.914121 + 0.389471i
19.10 1.12662 + 0.277686i −0.845190 − 0.534466i −0.578755 − 0.303754i 2.59048 − 1.94662i −0.803792 − 0.836836i −2.02054 + 1.06046i −2.30473 − 2.04182i 0.428693 + 0.903450i 3.45903 − 1.47376i
19.11 1.92606 + 0.474730i −0.845190 − 0.534466i 1.71341 + 0.899266i −2.98759 + 2.24503i −1.37416 − 1.43065i −2.68643 + 1.40995i −0.0964276 − 0.0854274i 0.428693 + 0.903450i −6.82004 + 2.90575i
19.12 1.97256 + 0.486192i −0.845190 − 0.534466i 1.88370 + 0.988640i 2.48102 − 1.86436i −1.40733 − 1.46519i 1.36867 − 0.718334i 0.193693 + 0.171597i 0.428693 + 0.903450i 5.80039 − 2.47132i
19.13 2.10415 + 0.518627i −0.845190 − 0.534466i 2.38758 + 1.25310i −0.272381 + 0.204681i −1.50122 − 1.56294i 2.92878 − 1.53714i 1.12970 + 1.00083i 0.428693 + 0.903450i −0.679283 + 0.289416i
19.14 2.67476 + 0.659270i −0.845190 − 0.534466i 4.94881 + 2.59734i 0.358425 − 0.269339i −1.90833 − 1.98678i −1.72808 + 0.906968i 7.40054 + 6.55631i 0.428693 + 0.903450i 1.13627 − 0.484119i
37.1 −2.29227 + 1.20308i −0.996757 + 0.0804666i 2.67098 − 3.86958i −0.825574 + 0.859513i 2.18803 − 1.38363i −0.656511 − 0.951121i −0.843111 + 6.94364i 0.987050 − 0.160411i 0.858377 − 2.96346i
37.2 −1.69947 + 0.891951i −0.996757 + 0.0804666i 0.956496 − 1.38572i −2.86891 + 2.98685i 1.62219 − 1.02581i −1.57479 − 2.28148i 0.0731569 − 0.602501i 0.987050 − 0.160411i 2.21150 − 7.63499i
37.3 −1.68163 + 0.882587i −0.996757 + 0.0804666i 0.912786 − 1.32240i −1.29736 + 1.35069i 1.60516 − 1.01504i 1.91693 + 2.77716i 0.0900034 − 0.741244i 0.987050 − 0.160411i 0.989572 − 3.41640i
37.4 −1.44364 + 0.757679i −0.996757 + 0.0804666i 0.373879 − 0.541657i 1.60746 − 1.67354i 1.37799 − 0.871386i −2.41370 − 3.49684i 0.263700 − 2.17176i 0.987050 − 0.160411i −1.05258 + 3.63393i
37.5 −1.33614 + 0.701259i −0.996757 + 0.0804666i 0.157371 − 0.227991i 2.79334 − 2.90817i 1.27538 − 0.806500i 1.85954 + 2.69400i 0.313387 − 2.58098i 0.987050 − 0.160411i −1.69290 + 5.84458i
37.6 −0.369506 + 0.193932i −0.996757 + 0.0804666i −1.03720 + 1.50265i 0.841882 − 0.876493i 0.352702 − 0.223035i −0.419250 − 0.607388i 0.192443 − 1.58491i 0.987050 − 0.160411i −0.141101 + 0.487136i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.i even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.q.b ✓ 336
157.i even 39 1 inner 471.2.q.b ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.q.b ✓ 336 1.a even 1 1 trivial
471.2.q.b ✓ 336 157.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{336} + 2 T_{2}^{335} + 43 T_{2}^{334} + 86 T_{2}^{333} + 1027 T_{2}^{332} + 2058 T_{2}^{331} + \cdots + 33\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\). Copy content Toggle raw display