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Properties

Label 671.2.e.b
Level $671$
Weight $2$
Character orbit 671.e
Analytic conductor $5.358$
Analytic rank $0$
Dimension $52$
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] = [671,2,Mod(474,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.474");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.e (of order \(3\), 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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 2 q^{3} - 30 q^{4} + 2 q^{6} + 5 q^{7} + 12 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 2 q^{3} - 30 q^{4} + 2 q^{6} + 5 q^{7} + 12 q^{8} + 50 q^{9} - 52 q^{11} - 5 q^{12} - 5 q^{13} - 3 q^{14} - 12 q^{15} - 42 q^{16} - 8 q^{17} + 20 q^{18} - q^{19} + 26 q^{20} + 10 q^{21} - 8 q^{23} + 6 q^{24} - 24 q^{25} + 7 q^{26} + 8 q^{27} - 64 q^{28} + 10 q^{29} + 32 q^{30} + 8 q^{31} - 12 q^{32} - 2 q^{33} + 28 q^{34} - 10 q^{35} + 4 q^{36} - 36 q^{37} - 20 q^{38} - 17 q^{39} + 8 q^{40} - 30 q^{41} + 48 q^{42} + 18 q^{43} + 30 q^{44} - 5 q^{46} - 16 q^{47} + 38 q^{48} - 61 q^{49} + 64 q^{50} + 68 q^{51} + 12 q^{52} - 18 q^{53} - 2 q^{54} - 16 q^{56} + 24 q^{57} + 64 q^{58} + 7 q^{59} + 40 q^{60} - 8 q^{61} - 94 q^{62} + 5 q^{63} + 52 q^{64} + 55 q^{65} - 2 q^{66} + 23 q^{67} + 35 q^{68} - 60 q^{69} - 118 q^{70} - 4 q^{71} + 46 q^{72} + 23 q^{73} - 21 q^{74} - 28 q^{75} - 4 q^{76} - 5 q^{77} - 32 q^{78} + 17 q^{79} + 28 q^{81} + 22 q^{82} + 32 q^{83} - 56 q^{84} - 124 q^{85} + 13 q^{86} + 7 q^{87} - 12 q^{88} - 156 q^{89} - 31 q^{90} + 38 q^{91} - 21 q^{92} - 30 q^{93} - 38 q^{94} + 124 q^{95} - 45 q^{96} + 9 q^{97} + 58 q^{98} - 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1 −1.38749 − 2.40320i −0.290631 −2.85026 + 4.93679i 1.20549 + 2.08797i 0.403247 + 0.698444i 1.78220 + 3.08686i 10.2689 −2.91553 3.34521 − 5.79408i
474.2 −1.29820 − 2.24854i −2.20150 −2.37062 + 4.10604i −1.99654 − 3.45810i 2.85797 + 4.95015i −0.139079 − 0.240891i 7.11734 1.84659 −5.18379 + 8.97858i
474.3 −1.25349 − 2.17111i 2.71514 −2.14247 + 3.71087i −1.24474 − 2.15595i −3.40340 − 5.89487i 2.18763 + 3.78909i 5.72832 4.37199 −3.12053 + 5.40492i
474.4 −1.15795 − 2.00563i −1.02649 −1.68170 + 2.91279i 1.61946 + 2.80498i 1.18862 + 2.05876i −2.13295 − 3.69438i 3.15752 −1.94632 3.75050 − 6.49606i
474.5 −1.10060 − 1.90630i 1.94702 −1.42266 + 2.46412i 0.944317 + 1.63561i −2.14290 − 3.71161i −1.57831 − 2.73372i 1.86074 0.790879 2.07864 − 3.60031i
474.6 −0.992734 − 1.71947i 0.874220 −0.971041 + 1.68189i −0.834468 − 1.44534i −0.867868 − 1.50319i −0.386123 − 0.668785i −0.114996 −2.23574 −1.65681 + 2.86968i
474.7 −0.852247 − 1.47614i −2.43620 −0.452650 + 0.784013i −0.227714 − 0.394411i 2.07624 + 3.59616i −0.141463 − 0.245022i −1.86591 2.93505 −0.388136 + 0.672272i
474.8 −0.607552 − 1.05231i 1.43861 0.261761 − 0.453384i 1.53856 + 2.66486i −0.874029 − 1.51386i 1.94660 + 3.37161i −3.06634 −0.930407 1.86951 − 3.23808i
474.9 −0.545117 − 0.944170i −1.49207 0.405695 − 0.702684i −0.510299 − 0.883863i 0.813354 + 1.40877i −2.09294 − 3.62507i −3.06507 −0.773722 −0.556345 + 0.963618i
474.10 −0.473450 − 0.820039i 0.0815651 0.551690 − 0.955556i −2.02560 − 3.50844i −0.0386170 − 0.0668866i 1.68592 + 2.92010i −2.93859 −2.99335 −1.91804 + 3.32214i
474.11 −0.228810 − 0.396310i 3.12624 0.895292 − 1.55069i 0.274244 + 0.475004i −0.715314 − 1.23896i 0.670229 + 1.16087i −1.73465 6.77337 0.125499 − 0.217371i
474.12 −0.152344 − 0.263867i −0.931637 0.953583 − 1.65165i 0.808756 + 1.40081i 0.141929 + 0.245828i −0.258701 − 0.448083i −1.19046 −2.13205 0.246418 − 0.426808i
474.13 −0.0540138 − 0.0935546i 1.61653 0.994165 − 1.72194i −0.876082 − 1.51742i −0.0873151 − 0.151234i −1.77795 − 3.07950i −0.430849 −0.386818 −0.0946409 + 0.163923i
474.14 −0.0162350 − 0.0281199i −3.03811 0.999473 − 1.73114i 1.95212 + 3.38117i 0.0493239 + 0.0854315i 2.12263 + 3.67650i −0.129846 6.23014 0.0633855 − 0.109787i
474.15 0.296441 + 0.513451i −1.69870 0.824246 − 1.42764i −0.536117 − 0.928582i −0.503563 − 0.872197i 1.29043 + 2.23509i 2.16312 −0.114430 0.317854 − 0.550540i
474.16 0.386909 + 0.670147i −3.25763 0.700602 − 1.21348i −0.0296919 − 0.0514279i −1.26041 − 2.18309i −2.27586 − 3.94191i 2.63192 7.61213 0.0229762 − 0.0397959i
474.17 0.493598 + 0.854936i 1.52059 0.512723 − 0.888062i 0.571639 + 0.990108i 0.750561 + 1.30001i 1.94228 + 3.36413i 2.98671 −0.687796 −0.564319 + 0.977429i
474.18 0.536293 + 0.928887i 2.59874 0.424779 − 0.735739i 1.76038 + 3.04906i 1.39369 + 2.41394i −1.80235 − 3.12176i 3.05640 3.75346 −1.88816 + 3.27038i
474.19 0.539748 + 0.934872i 0.713258 0.417343 − 0.722860i −1.58975 − 2.75353i 0.384980 + 0.666804i −0.249244 − 0.431703i 3.06004 −2.49126 1.71613 − 2.97243i
474.20 0.910318 + 1.57672i −2.44727 −0.657357 + 1.13858i −0.0110391 − 0.0191203i −2.22779 − 3.85864i −0.468045 − 0.810677i 1.24766 2.98911 0.0200982 − 0.0348111i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 474.26
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.e.b ✓ 52
61.c even 3 1 inner 671.2.e.b ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.e.b ✓ 52 1.a even 1 1 trivial
671.2.e.b ✓ 52 61.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} + 41 T_{2}^{50} - 4 T_{2}^{49} + 948 T_{2}^{48} - 152 T_{2}^{47} + 15047 T_{2}^{46} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display