↑

Properties

Label 403.2.k.e
Level $403$
Weight $2$
Character orbit 403.k
Analytic conductor $3.218$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(68\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 3 q^{2} - 2 q^{3} - 23 q^{4} + 12 q^{5} + 4 q^{6} + 2 q^{7} - 3 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 3 q^{2} - 2 q^{3} - 23 q^{4} + 12 q^{5} + 4 q^{6} + 2 q^{7} - 3 q^{8} - 23 q^{9} - 13 q^{10} - 5 q^{11} - 28 q^{12} - 17 q^{13} - 3 q^{14} - 14 q^{15} + 9 q^{16} + 12 q^{17} - 19 q^{18} - 4 q^{19} - 53 q^{20} - 13 q^{21} - 14 q^{22} - 9 q^{23} + 2 q^{24} + 96 q^{25} + 12 q^{26} + 25 q^{27} - 25 q^{28} - 78 q^{30} - 2 q^{31} + 76 q^{32} + 29 q^{33} - 15 q^{34} - 36 q^{35} + 52 q^{36} + 24 q^{37} - 19 q^{38} + 3 q^{39} - 12 q^{40} - 40 q^{41} + 11 q^{42} - 22 q^{43} + 4 q^{44} + 63 q^{45} - 24 q^{46} + 3 q^{47} + 68 q^{48} + 33 q^{49} - 76 q^{50} - 59 q^{51} - 13 q^{52} - q^{53} + 18 q^{54} - 22 q^{55} + 78 q^{56} - 16 q^{57} + 5 q^{58} - 18 q^{59} + 43 q^{60} - 32 q^{61} - 39 q^{62} + 20 q^{63} + 23 q^{64} + 2 q^{65} + 11 q^{66} + 114 q^{67} + 98 q^{68} - 46 q^{69} + 32 q^{70} - 2 q^{71} + 28 q^{72} + 10 q^{73} - 43 q^{74} - 12 q^{75} - 35 q^{76} - 3 q^{77} - 6 q^{78} - 10 q^{79} + 68 q^{80} - 54 q^{81} - 80 q^{82} - 22 q^{83} - 14 q^{84} - 50 q^{85} - 66 q^{86} + 76 q^{87} - 34 q^{88} - 10 q^{89} - 63 q^{90} - 8 q^{91} - 64 q^{92} - 16 q^{93} + 30 q^{94} + 15 q^{95} + 34 q^{96} - 7 q^{97} + 138 q^{98} - 48 q^{99}+O(q^{100}) \) 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 −0.831221 − 2.55823i −0.796905 + 2.45262i −4.23560 + 3.07735i −0.866460 6.93679 1.08832 − 0.790714i 7.04097 + 5.11556i −2.95324 − 2.14566i 0.720220 + 2.21661i
66.2 −0.717286 − 2.20758i 0.693769 − 2.13520i −2.74087 + 1.99136i 3.88389 −5.21126 1.00783 − 0.732228i 2.60631 + 1.89360i −1.65072 − 1.19932i −2.78586 − 8.57400i
66.3 −0.651553 − 2.00527i −0.602423 + 1.85407i −1.97856 + 1.43751i 2.75944 4.11042 −1.61436 + 1.17290i 0.760164 + 0.552291i −0.647600 − 0.470509i −1.79792 − 5.53344i
66.4 −0.537701 − 1.65487i 0.587091 − 1.80688i −0.831453 + 0.604086i −2.54418 −3.30584 −1.97022 + 1.43145i −1.36868 − 0.994405i −0.493085 − 0.358248i 1.36801 + 4.21030i
66.5 −0.471037 − 1.44970i 0.0410411 − 0.126311i −0.261732 + 0.190160i 1.10606 −0.202446 0.346937 − 0.252065i −2.06742 − 1.50207i 2.41278 + 1.75299i −0.520995 − 1.60346i
66.6 −0.285917 − 0.879963i −0.982047 + 3.02243i 0.925448 − 0.672377i −3.20777 2.94041 3.53175 − 2.56597i −2.35335 − 1.70981i −5.74362 − 4.17298i 0.917156 + 2.82271i
66.7 −0.100028 − 0.307853i 0.126453 − 0.389181i 1.53327 − 1.11398i 0.747578 −0.132459 2.14953 − 1.56172i −1.02006 − 0.741120i 2.29158 + 1.66493i −0.0747784 − 0.230144i
66.8 −0.0877385 − 0.270031i 0.851349 − 2.62018i 1.55282 − 1.12819i 0.880734 −0.782228 −3.90316 + 2.83581i −0.900292 − 0.654101i −3.71351 − 2.69802i −0.0772743 − 0.237826i
66.9 0.139693 + 0.429932i −0.0999964 + 0.307757i 1.45271 − 1.05545i 4.04679 −0.146284 −3.32806 + 2.41798i 1.38815 + 1.00855i 2.34234 + 1.70181i 0.565310 + 1.73984i
66.10 0.165656 + 0.509838i 0.987660 − 3.03970i 1.38554 − 1.00665i −3.75225 1.71337 1.77021 − 1.28613i 1.61014 + 1.16984i −5.83728 − 4.24103i −0.621584 − 1.91304i
66.11 0.288452 + 0.887764i −0.712637 + 2.19327i 0.913113 − 0.663415i 1.71724 −2.15267 1.85539 − 1.34802i 2.36270 + 1.71660i −1.87554 − 1.36266i 0.495341 + 1.52450i
66.12 0.455627 + 1.40228i 0.322084 − 0.991272i −0.140749 + 0.102260i −2.15722 1.53679 0.0381524 − 0.0277193i 2.17817 + 1.58253i 1.54817 + 1.12481i −0.982888 − 3.02502i
66.13 0.545494 + 1.67886i −0.170573 + 0.524968i −0.902965 + 0.656042i −0.444421 −0.974393 3.36537 − 2.44508i 1.26228 + 0.917100i 2.18055 + 1.58427i −0.242429 − 0.746119i
66.14 0.614130 + 1.89010i −0.776144 + 2.38873i −1.57728 + 1.14596i 0.0289380 −4.99158 −3.33658 + 2.42417i 0.0810008 + 0.0588505i −2.67656 − 1.94463i 0.0177717 + 0.0546957i
66.15 0.767174 + 2.36112i 0.517972 − 1.59415i −3.36829 + 2.44720i 3.20234 4.16136 −2.60176 + 1.89029i −4.34523 − 3.15699i 0.154019 + 0.111901i 2.45675 + 7.56111i
66.16 0.773013 + 2.37909i 0.384456 − 1.18324i −3.44449 + 2.50257i −2.86335 3.11221 −1.19324 + 0.866941i −4.56892 − 3.31951i 1.17481 + 0.853551i −2.21341 − 6.81217i
66.17 0.860293 + 2.64771i −0.871148 + 2.68112i −4.65223 + 3.38004i 2.69869 −7.84827 2.17586 − 1.58085i −8.44709 − 6.13717i −4.00245 − 2.90795i 2.32167 + 7.14536i
157.1 −2.24707 − 1.63259i −0.470599 + 0.341910i 1.76594 + 5.43499i 0.224436 1.61567 0.946374 + 2.91264i 3.18833 − 9.81268i −0.822490 + 2.53136i −0.504323 − 0.366412i
157.2 −1.85614 − 1.34856i 0.0303627 − 0.0220598i 1.00860 + 3.10414i −4.26015 −0.0861064 −0.501866 − 1.54459i 0.896070 − 2.75782i −0.926616 + 2.85183i 7.90744 + 5.74509i
157.3 −1.84627 − 1.34139i 1.79347 − 1.30303i 0.991339 + 3.05103i 4.23714 −5.05910 −0.0173329 − 0.0533453i 0.851924 − 2.62195i 0.591588 − 1.82072i −7.82289 − 5.68366i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 66.17
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.e ✓ 68
31.d even 5 1 inner 403.2.k.e ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.k.e ✓ 68 1.a even 1 1 trivial
403.2.k.e ✓ 68 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{68} + 3 T_{2}^{67} + 33 T_{2}^{66} + 95 T_{2}^{65} + 604 T_{2}^{64} + 1537 T_{2}^{63} + \cdots + 13456 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display