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Properties

Label 471.2.w.b
Level $471$
Weight $2$
Character orbit 471.w
Analytic conductor $3.761$
Analytic rank $0$
Dimension $2400$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [471,2,Mod(5,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(156))
 
chi = DirichletCharacter(H, H._module([78, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.w (of order \(156\), degree \(48\), 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: \(2400\)
Relative dimension: \(50\) over \(\Q(\zeta_{156})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{156}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2400 q - 46 q^{3} - 104 q^{4} - 52 q^{6} - 104 q^{7} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2400 q - 46 q^{3} - 104 q^{4} - 52 q^{6} - 104 q^{7} - 42 q^{9} - 68 q^{10} - 10 q^{12} - 132 q^{13} - 60 q^{15} + 104 q^{16} - 60 q^{18} - 92 q^{19} - 52 q^{21} - 80 q^{22} - 84 q^{24} - 236 q^{25} - 52 q^{27} - 168 q^{28} - 38 q^{30} - 56 q^{31} - 70 q^{33} - 136 q^{34} - 130 q^{36} + 164 q^{37} - 58 q^{39} + 148 q^{40} - 94 q^{42} - 80 q^{43} - 20 q^{45} - 72 q^{46} + 154 q^{48} - 104 q^{49} + 2 q^{51} - 348 q^{52} + 186 q^{54} - 84 q^{55} - 46 q^{57} - 104 q^{58} - 182 q^{60} - 200 q^{61} - 44 q^{63} - 104 q^{64} - 20 q^{66} - 88 q^{67} - 164 q^{70} - 10 q^{72} + 12 q^{73} - 20 q^{75} - 236 q^{76} - 314 q^{78} - 92 q^{79} - 118 q^{81} - 104 q^{82} - 42 q^{84} + 56 q^{85} + 36 q^{87} - 112 q^{88} - 52 q^{90} - 552 q^{91} + 40 q^{93} - 336 q^{94} - 152 q^{96} - 224 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.806761 − 2.58899i 1.51358 − 0.842071i −4.40603 + 3.04126i 0.0497706 − 2.47109i −3.40121 − 3.23928i 1.47996 − 0.271214i 7.15908 + 5.60878i 1.58183 − 2.54908i −6.43778 + 1.86473i
5.2 −0.796307 − 2.55544i −0.0192181 + 1.73194i −4.25020 + 2.93370i 0.0413163 − 2.05134i 4.44118 − 1.33005i 1.94698 − 0.356797i 6.66735 + 5.22354i −2.99926 − 0.0665694i −5.27498 + 1.52792i
5.3 −0.770409 − 2.47233i 0.318267 − 1.70256i −3.87291 + 2.67328i −0.0579684 + 2.87811i −4.45448 + 0.524806i −1.14668 + 0.210137i 5.51600 + 4.32151i −2.79741 − 1.08374i 7.16029 − 2.07400i
5.4 −0.762829 − 2.44801i −0.942528 + 1.45315i −3.76485 + 2.59869i −0.0699377 + 3.47238i 4.27630 + 1.19881i −3.90943 + 0.716429i 5.19671 + 4.07136i −1.22328 − 2.73927i 8.55375 − 2.47763i
5.5 −0.740299 − 2.37570i −1.51136 − 0.846042i −3.44995 + 2.38133i −0.0286084 + 1.42040i −0.891085 + 4.21687i 3.23007 − 0.591933i 4.29371 + 3.36391i 1.56843 + 2.55735i 3.39562 − 0.983553i
5.6 −0.729154 − 2.33994i 1.42576 + 0.983464i −3.29768 + 2.27622i −0.0104872 + 0.520685i 1.26165 − 4.05329i −3.08574 + 0.565483i 3.87210 + 3.03360i 1.06560 + 2.80437i 1.22602 − 0.355120i
5.7 −0.729004 − 2.33946i −1.71279 − 0.257563i −3.29564 + 2.27481i 0.0533293 − 2.64778i 0.646077 + 4.19477i −3.19688 + 0.585849i 3.86652 + 3.02922i 2.86732 + 0.882303i −6.23324 + 1.80548i
5.8 −0.628483 − 2.01687i 1.73109 + 0.0575727i −2.02682 + 1.39901i −0.0388136 + 1.92708i −0.971847 − 3.52758i 1.68312 − 0.308444i 0.769563 + 0.602914i 2.99337 + 0.199327i 3.91107 − 1.13286i
5.9 −0.578270 − 1.85573i 0.581756 − 1.63143i −1.46338 + 1.01010i 0.0312500 − 1.55155i −3.36391 − 0.136178i −2.85234 + 0.522710i −0.339461 − 0.265950i −2.32312 − 1.89819i −2.89734 + 0.839224i
5.10 −0.555464 − 1.78255i −0.542941 − 1.64475i −1.22297 + 0.844154i 0.0650492 − 3.22967i −2.63027 + 1.88142i 3.32658 − 0.609618i −0.755421 − 0.591835i −2.41043 + 1.78601i −5.79317 + 1.67801i
5.11 −0.514644 − 1.65155i −0.131041 + 1.72709i −0.816790 + 0.563789i −0.0847260 + 4.20662i 2.91981 − 0.672414i 4.05840 − 0.743729i −1.37198 − 1.07488i −2.96566 − 0.452637i 6.99104 − 2.02498i
5.12 −0.501835 − 1.61045i 0.737392 + 1.56724i −0.695730 + 0.480228i 0.00898903 − 0.446302i 2.15391 − 1.97403i −0.246423 + 0.0451587i −1.53316 − 1.20115i −1.91251 + 2.31135i −0.723256 + 0.209494i
5.13 −0.491284 − 1.57659i −1.50532 + 0.856739i −0.598295 + 0.412973i −0.0148874 + 0.739153i 2.09026 + 1.95237i −1.64538 + 0.301527i −1.65482 − 1.29647i 1.53200 − 2.57934i 1.17265 − 0.339663i
5.14 −0.478766 − 1.53642i −0.903218 − 1.47790i −0.485387 + 0.335039i −0.0365258 + 1.81349i −1.83824 + 2.09529i −0.111649 + 0.0204604i −1.78646 − 1.39960i −1.36840 + 2.66974i 2.80377 − 0.812121i
5.15 −0.436543 − 1.40091i 1.48969 + 0.883635i −0.126024 + 0.0869879i 0.0717448 − 3.56210i 0.587583 − 2.47268i 4.63283 − 0.848998i −2.13328 − 1.67132i 1.43838 + 2.63269i −5.02152 + 1.45450i
5.16 −0.406478 − 1.30443i −0.344988 + 1.69735i 0.109647 − 0.0756840i 0.0595518 − 2.95672i 2.35430 − 0.239919i −2.25008 + 0.412343i −2.29435 − 1.79751i −2.76197 − 1.17113i −3.88105 + 1.12416i
5.17 −0.362805 − 1.16428i −1.73205 − 0.00183740i 0.422041 − 0.291314i −0.0480498 + 2.38566i 0.626257 + 2.01726i −0.293905 + 0.0538602i −2.41223 − 1.88986i 2.99999 + 0.00636494i 2.79501 − 0.809585i
5.18 −0.289623 − 0.929432i 1.60894 − 0.641335i 0.866005 − 0.597760i 0.00859433 − 0.426705i −1.06206 − 1.30966i −0.297186 + 0.0544614i −2.33906 − 1.83253i 2.17738 − 2.06374i −0.399083 + 0.115596i
5.19 −0.264856 − 0.849952i 1.08949 + 1.34648i 0.993697 − 0.685900i −0.0538374 + 2.67301i 0.855886 − 1.28264i −4.66547 + 0.854979i −2.24777 − 1.76101i −0.626016 + 2.93396i 2.28619 − 0.662203i
5.20 −0.224383 − 0.720069i 0.915162 − 1.47054i 1.17782 − 0.812988i −0.0179039 + 0.888920i −1.26423 − 0.329017i 3.05277 − 0.559441i −2.03711 − 1.59597i −1.32496 − 2.69156i 0.644101 − 0.186566i
See next 80 embeddings (of 2400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
157.l odd 156 1 inner
471.w even 156 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.w.b ✓ 2400
3.b odd 2 1 inner 471.2.w.b ✓ 2400
157.l odd 156 1 inner 471.2.w.b ✓ 2400
471.w even 156 1 inner 471.2.w.b ✓ 2400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.w.b ✓ 2400 1.a even 1 1 trivial
471.2.w.b ✓ 2400 3.b odd 2 1 inner
471.2.w.b ✓ 2400 157.l odd 156 1 inner
471.2.w.b ✓ 2400 471.w even 156 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2400} + 52 T_{2}^{2398} + 614 T_{2}^{2396} - 14924 T_{2}^{2394} - 405939 T_{2}^{2392} + \cdots + 13\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\). Copy content Toggle raw display