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Properties

Label 403.2.r.a
Level $403$
Weight $2$
Character orbit 403.r
Analytic conductor $3.218$
Analytic rank $0$
Dimension $68$
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(218,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.218");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.r (of order \(6\), 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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 32 q^{4} - 34 q^{9} + 8 q^{10} - 12 q^{11} - 16 q^{12} + 6 q^{13} - 8 q^{14} - 36 q^{16} - 6 q^{17} + 12 q^{19} - 12 q^{20} - 20 q^{22} - 8 q^{23} + 48 q^{24} - 72 q^{25} - 12 q^{27} - 6 q^{28} + 32 q^{30} + 6 q^{33} + 30 q^{35} + 40 q^{36} - 42 q^{37} - 36 q^{38} - 14 q^{39} + 8 q^{40} + 18 q^{41} - 16 q^{42} + 12 q^{43} + 60 q^{45} + 30 q^{46} - 46 q^{48} + 22 q^{49} + 56 q^{51} + 20 q^{53} - 114 q^{54} - 6 q^{55} - 2 q^{56} - 12 q^{58} + 6 q^{59} + 6 q^{61} - 8 q^{62} - 30 q^{63} + 24 q^{64} + 24 q^{65} + 8 q^{66} - 48 q^{67} + 58 q^{68} - 28 q^{69} - 30 q^{71} + 72 q^{72} + 8 q^{74} - 4 q^{75} - 12 q^{76} - 20 q^{77} + 26 q^{78} + 16 q^{79} + 42 q^{80} - 58 q^{81} - 42 q^{82} - 72 q^{84} + 30 q^{85} - 20 q^{87} + 64 q^{88} + 18 q^{89} + 52 q^{90} - 22 q^{91} + 48 q^{92} + 8 q^{94} - 32 q^{95} - 168 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
218.1 −2.26368 + 1.30694i −0.194798 − 0.337400i 2.41618 − 4.18495i − 3.31493i 0.881922 + 0.509178i −1.89144 − 1.09203i 7.40344i 1.42411 − 2.46663i 4.33241 + 7.50395i
218.2 −2.19517 + 1.26738i −1.69135 − 2.92950i 2.21251 − 3.83219i − 0.452856i 7.42559 + 4.28717i 1.73117 + 0.999489i 6.14687i −4.22131 + 7.31153i 0.573941 + 0.994095i
218.3 −2.18149 + 1.25949i −0.435889 − 0.754983i 2.17261 − 3.76307i 3.48615i 1.90178 + 1.09799i −2.02718 − 1.17039i 5.90753i 1.12000 − 1.93990i −4.39076 − 7.60502i
218.4 −2.14952 + 1.24103i −0.331917 − 0.574896i 2.08030 − 3.60319i − 0.685218i 1.42693 + 0.823836i 3.86933 + 2.23396i 5.36274i 1.27966 − 2.21644i 0.850375 + 1.47289i
218.5 −1.98176 + 1.14417i 0.827863 + 1.43390i 1.61825 − 2.80289i − 1.16564i −3.28125 − 1.89443i −2.58921 − 1.49488i 2.82952i 0.129286 − 0.223931i 1.33368 + 2.31001i
218.6 −1.86490 + 1.07670i 1.32944 + 2.30267i 1.31857 − 2.28383i 3.07578i −4.95857 − 2.86283i −0.437880 − 0.252810i 1.37202i −2.03485 + 3.52446i −3.31169 − 5.73602i
218.7 −1.60943 + 0.929207i 1.27215 + 2.20343i 0.726853 − 1.25895i − 3.22618i −4.09489 − 2.36418i 2.41260 + 1.39291i − 1.01524i −1.73673 + 3.00811i 2.99779 + 5.19232i
218.8 −1.56202 + 0.901835i 0.553992 + 0.959542i 0.626612 − 1.08532i 0.771735i −1.73070 − 0.999218i 3.80672 + 2.19781i − 1.34694i 0.886187 − 1.53492i −0.695977 − 1.20547i
218.9 −1.42642 + 0.823544i −1.30685 − 2.26354i 0.356451 − 0.617391i − 2.48483i 3.72825 + 2.15250i −4.08112 − 2.35624i − 2.11996i −1.91573 + 3.31814i 2.04637 + 3.54442i
218.10 −1.25179 + 0.722722i −0.0126133 − 0.0218468i 0.0446555 − 0.0773455i 1.27994i 0.0315784 + 0.0182318i −1.82792 − 1.05535i − 2.76180i 1.49968 − 2.59753i −0.925042 − 1.60222i
218.11 −0.697346 + 0.402613i 0.727281 + 1.25969i −0.675806 + 1.17053i − 3.57617i −1.01433 − 0.585626i −1.02623 − 0.592496i − 2.69880i 0.442123 − 0.765780i 1.43981 + 2.49383i
218.12 −0.624164 + 0.360361i −1.13079 − 1.95859i −0.740280 + 1.28220i − 4.09116i 1.41160 + 0.814986i 3.77469 + 2.17932i − 2.50852i −1.05738 + 1.83143i 1.47430 + 2.55356i
218.13 −0.565608 + 0.326554i −0.481134 − 0.833348i −0.786725 + 1.36265i − 0.172354i 0.544266 + 0.314232i 2.14068 + 1.23592i − 2.33385i 1.03702 − 1.79617i 0.0562829 + 0.0974848i
218.14 −0.544301 + 0.314252i 1.36076 + 2.35690i −0.802491 + 1.38996i 1.23905i −1.48132 − 0.855242i −3.53624 − 2.04165i − 2.26575i −2.20333 + 3.81627i −0.389373 − 0.674413i
218.15 −0.453928 + 0.262075i 0.409673 + 0.709575i −0.862633 + 1.49412i 4.38147i −0.371924 − 0.214730i 2.35938 + 1.36219i − 1.95260i 1.16434 − 2.01669i −1.14828 − 1.98887i
218.16 −0.441172 + 0.254711i −1.38443 − 2.39789i −0.870245 + 1.50731i 1.59074i 1.22154 + 0.705256i 0.253842 + 0.146556i − 1.90549i −2.33327 + 4.04134i −0.405179 − 0.701791i
218.17 −0.244810 + 0.141341i 1.61186 + 2.79182i −0.960045 + 1.66285i − 0.0582096i −0.789198 − 0.455643i 2.44759 + 1.41312i − 1.10814i −3.69618 + 6.40198i 0.00822740 + 0.0142503i
218.18 −0.117721 + 0.0679661i 0.0351399 + 0.0608641i −0.990761 + 1.71605i − 1.52124i −0.00827339 − 0.00477665i −2.01496 − 1.16334i − 0.541217i 1.49753 − 2.59380i 0.103393 + 0.179081i
218.19 0.228638 − 0.132004i −0.254003 − 0.439946i −0.965150 + 1.67169i 2.92638i −0.116149 − 0.0670588i −4.12569 − 2.38197i 1.03763i 1.37097 − 2.37458i 0.386294 + 0.669081i
218.20 0.514518 − 0.297057i −0.848877 − 1.47030i −0.823514 + 1.42637i − 2.73915i −0.873525 − 0.504330i −2.55168 − 1.47321i 2.16675i 0.0588173 − 0.101874i −0.813684 − 1.40934i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 218.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.r.a ✓ 68
13.e even 6 1 inner 403.2.r.a ✓ 68
13.f odd 12 1 5239.2.a.q 34
13.f odd 12 1 5239.2.a.r 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.r.a ✓ 68 1.a even 1 1 trivial
403.2.r.a ✓ 68 13.e even 6 1 inner
5239.2.a.q 34 13.f odd 12 1
5239.2.a.r 34 13.f odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).