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Properties

Label 403.2.bs.a
Level $403$
Weight $2$
Character orbit 403.bs
Analytic conductor $3.218$
Analytic rank $0$
Dimension $288$
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] = [403,2,Mod(4,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.bs (of order \(30\), degree \(8\), 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: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9} + 3 q^{10} - 18 q^{11} - 46 q^{12} - q^{13} - 32 q^{14} + 18 q^{15} + 21 q^{16} - 15 q^{17} - 15 q^{19} + 51 q^{20} - 10 q^{22} - 4 q^{23} - 51 q^{24} - 296 q^{25} - 6 q^{26} - 52 q^{27} + 21 q^{28} + q^{29} + 60 q^{30} + 138 q^{32} + 69 q^{33} - 10 q^{35} + 128 q^{36} - 18 q^{37} + 32 q^{38} - 14 q^{39} + 60 q^{40} - 15 q^{41} - 49 q^{42} - 36 q^{43} + 6 q^{45} - 69 q^{46} + 21 q^{48} - 23 q^{49} + 117 q^{50} + 8 q^{51} + 26 q^{52} - 48 q^{53} + 75 q^{54} + 46 q^{55} - 98 q^{56} - 21 q^{58} - 105 q^{59} - 74 q^{61} - 3 q^{62} - 90 q^{63} + 90 q^{64} + 89 q^{65} - 8 q^{66} + 6 q^{67} - 182 q^{68} + 29 q^{69} + 3 q^{71} - 183 q^{72} - 53 q^{74} - 38 q^{75} + 144 q^{76} - 128 q^{78} - 72 q^{79} - 72 q^{80} + 11 q^{81} - 11 q^{82} - 33 q^{84} + 72 q^{85} - 18 q^{87} - 14 q^{88} + 81 q^{89} - 34 q^{90} - 48 q^{91} + 8 q^{92} + 72 q^{93} - 6 q^{94} + 141 q^{97} + 96 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.552779 + 2.60062i −0.533122 − 0.592092i −4.63058 − 2.06167i 1.27621i 1.83451 − 1.05915i 1.08933 − 2.44667i 4.79578 − 6.60083i 0.247231 − 2.35225i −3.31895 − 0.705465i
4.2 −0.525326 + 2.47147i 0.803900 + 0.892821i −4.00509 − 1.78318i 0.793316i −2.62889 + 1.51779i −0.479028 + 1.07591i 3.54075 − 4.87343i 0.162711 − 1.54809i −1.96065 − 0.416750i
4.3 −0.518746 + 2.44051i −1.86014 − 2.06590i −3.85989 − 1.71854i 2.00004i 6.00678 − 3.46802i −1.59632 + 3.58539i 3.26332 − 4.49158i −0.494216 + 4.70215i −4.88112 − 1.03751i
4.4 −0.502235 + 2.36283i 0.181860 + 0.201976i −3.50363 − 1.55992i − 2.70808i −0.568571 + 0.328265i −0.369305 + 0.829472i 2.60573 − 3.58648i 0.305864 − 2.91010i 6.39874 + 1.36009i
4.5 −0.490829 + 2.30917i 2.17363 + 2.41407i −3.26425 − 1.45334i − 1.85570i −6.64136 + 3.83439i −1.05002 + 2.35838i 2.18296 − 3.00459i −0.789440 + 7.51102i 4.28511 + 0.910829i
4.6 −0.431958 + 2.03220i 1.72231 + 1.91282i −2.11617 − 0.942181i 3.56131i −4.63120 + 2.67382i 1.47626 − 3.31574i 0.386433 − 0.531879i −0.378938 + 3.60536i −7.23732 − 1.53834i
4.7 −0.364510 + 1.71488i −0.797644 − 0.885874i −0.980870 − 0.436711i 3.57659i 1.80992 − 1.04496i 0.0329582 − 0.0740253i −0.954559 + 1.31384i 0.165050 − 1.57034i −6.13344 − 1.30370i
4.8 −0.360928 + 1.69803i 0.922989 + 1.02508i −0.925954 − 0.412261i − 3.21206i −2.07376 + 1.19728i 2.07463 − 4.65970i −1.00652 + 1.38535i 0.114699 − 1.09128i 5.45418 + 1.15932i
4.9 −0.350844 + 1.65059i −1.46122 − 1.62285i −0.774264 − 0.344725i − 3.64318i 3.19131 − 1.84250i −1.03720 + 2.32958i −1.14309 + 1.57333i −0.184888 + 1.75909i 6.01339 + 1.27819i
4.10 −0.323679 + 1.52279i 0.859963 + 0.955086i −0.387033 − 0.172318i 2.26589i −1.73275 + 1.00040i −1.51411 + 3.40074i −1.44246 + 1.98538i 0.140933 − 1.34089i −3.45048 − 0.733422i
4.11 −0.319792 + 1.50450i −0.741178 − 0.823162i −0.334172 − 0.148783i − 1.09342i 1.47547 − 0.851865i 1.19871 − 2.69235i −1.47745 + 2.03354i 0.185335 − 1.76335i 1.64506 + 0.349668i
4.12 −0.254380 + 1.19677i −1.60706 − 1.78482i 0.459552 + 0.204606i 0.244018i 2.54482 − 1.46925i −0.0246816 + 0.0554357i −1.80008 + 2.47760i −0.289361 + 2.75309i −0.292033 − 0.0620735i
4.13 −0.230096 + 1.08252i 0.418891 + 0.465225i 0.708195 + 0.315309i − 1.86982i −0.599999 + 0.346409i −2.06505 + 4.63818i −1.80528 + 2.48476i 0.272620 − 2.59381i 2.02411 + 0.430238i
4.14 −0.129772 + 0.610527i −0.324888 − 0.360825i 1.47119 + 0.655015i − 0.658726i 0.262455 − 0.151528i 0.662294 − 1.48754i −1.32458 + 1.82312i 0.288943 − 2.74911i 0.402170 + 0.0854839i
4.15 −0.117135 + 0.551075i 1.94238 + 2.15723i 1.53713 + 0.684373i − 0.526119i −1.41631 + 0.817710i 0.000659214 − 0.00148062i −1.21949 + 1.67849i −0.567221 + 5.39675i 0.289931 + 0.0616267i
4.16 −0.0758408 + 0.356803i 1.47267 + 1.63557i 1.70553 + 0.759353i − 1.67783i −0.695264 + 0.401411i 0.425932 − 0.956658i −0.829106 + 1.14117i −0.192734 + 1.83374i 0.598654 + 0.127248i
4.17 −0.0525799 + 0.247369i 0.690263 + 0.766614i 1.76866 + 0.787460i 3.00880i −0.225931 + 0.130441i −0.278018 + 0.624439i −0.585086 + 0.805302i 0.202350 − 1.92523i −0.744285 − 0.158203i
4.18 −0.00361138 + 0.0169902i −2.04909 − 2.27575i 1.82682 + 0.813351i 1.58318i 0.0460655 − 0.0265959i −0.684412 + 1.53721i −0.0408357 + 0.0562056i −0.666663 + 6.34287i −0.0268985 − 0.00571745i
4.19 0.00797121 − 0.0375016i −0.129846 − 0.144209i 1.82575 + 0.812875i 2.93327i −0.00644308 + 0.00371992i 1.61684 − 3.63149i 0.0901082 − 0.124023i 0.309649 − 2.94612i 0.110002 + 0.0233817i
4.20 0.0550517 − 0.258998i −1.89412 − 2.10363i 1.76304 + 0.784957i − 4.39411i −0.649111 + 0.374764i 0.553765 − 1.24378i 0.611633 − 0.841841i −0.523995 + 4.98548i −1.13807 − 0.241903i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner
31.d even 5 1 inner
403.bs even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bs.a ✓ 288
13.e even 6 1 inner 403.2.bs.a ✓ 288
31.d even 5 1 inner 403.2.bs.a ✓ 288
403.bs even 30 1 inner 403.2.bs.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bs.a ✓ 288 1.a even 1 1 trivial
403.2.bs.a ✓ 288 13.e even 6 1 inner
403.2.bs.a ✓ 288 31.d even 5 1 inner
403.2.bs.a ✓ 288 403.bs even 30 1 inner

Hecke kernels

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