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Properties

Label 403.2.k.d
Level $403$
Weight $2$
Character orbit 403.k
Analytic conductor $3.218$
Analytic rank $0$
Dimension $48$
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] = [403,2,Mod(66,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.66");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.k (of order \(5\), degree \(4\), 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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 7 q^{2} - 2 q^{3} - 7 q^{4} - 12 q^{5} - 10 q^{6} + 25 q^{7} - 14 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 7 q^{2} - 2 q^{3} - 7 q^{4} - 12 q^{5} - 10 q^{6} + 25 q^{7} - 14 q^{8} - 8 q^{9} - 19 q^{10} - 9 q^{11} + 15 q^{12} + 12 q^{13} - 25 q^{14} - 30 q^{15} - 21 q^{16} + 11 q^{17} + 17 q^{18} + 36 q^{19} + 30 q^{20} + 11 q^{21} + 15 q^{22} - 7 q^{23} - 20 q^{24} - 16 q^{25} + 8 q^{26} - 5 q^{27} - 9 q^{28} + 12 q^{29} + 18 q^{30} + 22 q^{31} - 76 q^{32} - 49 q^{33} - 26 q^{34} + 8 q^{35} + 2 q^{36} + 64 q^{37} - 27 q^{38} - 3 q^{39} - 24 q^{40} + 46 q^{41} + 20 q^{42} - 28 q^{43} - 23 q^{45} + 34 q^{46} + 5 q^{47} - 20 q^{48} - 11 q^{49} + 9 q^{50} + 59 q^{51} + 17 q^{52} + 23 q^{53} + 41 q^{54} - 10 q^{55} - 60 q^{56} + 24 q^{57} - 37 q^{58} + 71 q^{59} - 72 q^{60} + 22 q^{61} + 43 q^{62} - 106 q^{63} - 52 q^{64} + 2 q^{65} - 21 q^{66} - 56 q^{67} - 104 q^{68} - 12 q^{69} - 32 q^{70} - 36 q^{71} + 147 q^{72} - 12 q^{73} + 10 q^{74} + 34 q^{75} - 49 q^{76} - 30 q^{77} + 5 q^{78} - 70 q^{79} + q^{81} + 130 q^{82} + 11 q^{83} + 77 q^{84} + 8 q^{85} + 11 q^{86} - 88 q^{87} + 96 q^{88} - 40 q^{89} - 48 q^{90} + 10 q^{91} + 112 q^{92} + 50 q^{93} + 78 q^{94} + 41 q^{95} - 75 q^{96} - 47 q^{97} - 46 q^{98} + 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
66.1 −0.743865 − 2.28938i 0.00536078 − 0.0164988i −3.06990 + 2.23041i 0.430402 −0.0417597 3.00261 − 2.18153i 3.49493 + 2.53921i 2.42681 + 1.76318i −0.320161 − 0.985354i
66.2 −0.604780 − 1.86132i 0.854563 − 2.63007i −1.48072 + 1.07581i −1.67501 −5.41223 1.22914 − 0.893026i −0.268732 − 0.195245i −3.75996 − 2.73177i 1.01301 + 3.11773i
66.3 −0.599354 − 1.84462i −0.251540 + 0.774160i −1.42537 + 1.03559i −3.24431 1.57879 1.80822 − 1.31375i −0.373685 − 0.271498i 1.89100 + 1.37389i 1.94449 + 5.98453i
66.4 −0.365564 − 1.12509i −0.709225 + 2.18277i 0.485840 − 0.352983i 3.83735 2.71508 3.69636 − 2.68556i −2.48886 − 1.80827i −1.83443 − 1.33279i −1.40280 − 4.31737i
66.5 −0.288777 − 0.888765i 0.813772 − 2.50453i 0.911523 − 0.662261i 1.99869 −2.46094 0.592448 − 0.430438i −2.36388 − 1.71746i −3.18340 − 2.31288i −0.577176 − 1.77636i
66.6 −0.0757307 − 0.233075i 0.263551 − 0.811126i 1.56945 − 1.14027i −2.48983 −0.209012 2.14015 − 1.55491i −0.781154 − 0.567542i 1.83859 + 1.33581i 0.188556 + 0.580316i
66.7 −0.00310087 − 0.00954350i −0.517074 + 1.59139i 1.61795 − 1.17551i 1.33726 0.0167908 −0.925697 + 0.672558i −0.0324719 − 0.0235922i 0.161891 + 0.117620i −0.00414667 − 0.0127621i
66.8 0.247442 + 0.761548i −0.990504 + 3.04846i 1.09931 − 0.798693i −0.800250 −2.56664 −0.623500 + 0.452999i 2.17588 + 1.58087i −5.88494 − 4.27566i −0.198015 − 0.609429i
66.9 0.376127 + 1.15760i 0.106335 − 0.327267i 0.419469 − 0.304762i −2.41222 0.418839 −3.46077 + 2.51440i 2.47999 + 1.80182i 2.33125 + 1.69376i −0.907299 − 2.79238i
66.10 0.655076 + 2.01612i −0.156598 + 0.481958i −2.01757 + 1.46585i 2.06047 −1.07427 0.413610 − 0.300506i −0.846966 − 0.615357i 2.21929 + 1.61241i 1.34976 + 4.15414i
66.11 0.700746 + 2.15667i −0.672668 + 2.07026i −2.54216 + 1.84699i −3.69852 −4.93624 0.888338 − 0.645416i −2.09561 − 1.52255i −1.40643 − 1.02183i −2.59172 − 7.97650i
66.12 0.774730 + 2.38437i 0.754027 − 2.32066i −3.46700 + 2.51892i −0.580097 6.11748 3.63826 − 2.64335i −4.63551 − 3.36789i −2.38983 − 1.73632i −0.449419 − 1.38317i
157.1 −1.95579 − 1.42096i −0.875508 + 0.636094i 1.18794 + 3.65610i 2.12456 2.61618 −0.805291 − 2.47843i 1.37774 − 4.24024i −0.565152 + 1.73936i −4.15519 − 3.01892i
157.2 −1.78093 − 1.29392i 1.61180 − 1.17104i 0.879438 + 2.70663i −1.55702 −4.38572 0.929842 + 2.86176i 0.575440 − 1.77102i 0.299507 − 0.921787i 2.77294 + 2.01466i
157.3 −1.43812 − 1.04485i −1.43056 + 1.03937i 0.358428 + 1.10313i 0.805848 3.14330 −0.358493 − 1.10333i −0.461478 + 1.42028i 0.0391825 − 0.120591i −1.15890 − 0.841992i
157.4 −0.804029 − 0.584162i 1.86138 − 1.35237i −0.312815 − 0.962747i 1.77944 −2.28660 −0.394812 − 1.21511i −0.925111 + 2.84720i 0.708770 − 2.18137i −1.43072 − 1.03948i
157.5 −0.0675977 − 0.0491126i −2.70745 + 1.96708i −0.615877 − 1.89547i 0.141769 0.279626 −0.415935 − 1.28012i −0.103100 + 0.317309i 2.53385 − 7.79838i −0.00958327 − 0.00696265i
157.6 0.607778 + 0.441577i 0.0305971 − 0.0222301i −0.443630 − 1.36535i 2.26489 0.0284125 1.52318 + 4.68785i 0.797580 − 2.45470i −0.926609 + 2.85181i 1.37655 + 1.00012i
157.7 0.715177 + 0.519606i 2.11502 − 1.53665i −0.376547 − 1.15889i −3.34815 2.31107 −0.00198090 − 0.00609658i 0.879216 − 2.70595i 1.18497 − 3.64695i −2.39452 − 1.73972i
157.8 1.04202 + 0.757072i −0.137474 + 0.0998807i −0.105385 − 0.324343i −3.49603 −0.218868 1.29282 + 3.97888i 0.931770 − 2.86769i −0.918128 + 2.82571i −3.64293 − 2.64675i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 66.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.k.d ✓ 48
31.d even 5 1 inner 403.2.k.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.k.d ✓ 48 1.a even 1 1 trivial
403.2.k.d ✓ 48 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 7 T_{2}^{47} + 40 T_{2}^{46} - 157 T_{2}^{45} + 594 T_{2}^{44} - 1866 T_{2}^{43} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display