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Properties

Label 403.2.ch.a
Level $403$
Weight $2$
Character orbit 403.ch
Analytic conductor $3.218$
Analytic rank $0$
Dimension $560$
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(11,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([35, 46]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.ch (of order \(60\), degree \(16\), 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: \(560\)
Relative dimension: \(35\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9} - 24 q^{10} - 18 q^{11} - 26 q^{12} + 6 q^{13} + 4 q^{14} - 8 q^{15} - 58 q^{16} - 30 q^{17} + 44 q^{18} - 30 q^{19} + 10 q^{20} - 42 q^{21} - 60 q^{22} - 30 q^{23} + 32 q^{24} + 24 q^{26} - 100 q^{27} - 66 q^{28} - 10 q^{29} - 12 q^{31} + 20 q^{32} + 70 q^{33} + 28 q^{34} - 2 q^{35} - 78 q^{36} - 38 q^{37} + 108 q^{38} - 22 q^{39} - 28 q^{40} - 10 q^{41} - 40 q^{42} - 126 q^{43} + 60 q^{44} - 120 q^{45} - 4 q^{46} - 32 q^{47} - 64 q^{48} + 78 q^{49} - 150 q^{50} - 36 q^{51} - 26 q^{52} - 84 q^{53} - 62 q^{54} - 10 q^{55} + 216 q^{56} + 22 q^{57} - 158 q^{58} - 50 q^{59} - 152 q^{60} - 36 q^{61} - 42 q^{62} + 78 q^{63} - 270 q^{64} + 54 q^{65} - 56 q^{66} - 46 q^{67} + 162 q^{69} + 90 q^{70} + 34 q^{71} - 42 q^{72} - 40 q^{73} + 80 q^{74} - 38 q^{75} + 10 q^{76} - 72 q^{77} + 6 q^{78} - 32 q^{79} + 78 q^{80} - 62 q^{81} - 30 q^{82} + 16 q^{83} + 126 q^{84} - 42 q^{85} + 154 q^{86} - 56 q^{87} - 54 q^{88} - 46 q^{89} + 90 q^{90} + 134 q^{91} - 12 q^{93} - 24 q^{94} - 78 q^{95} - 136 q^{96} + 204 q^{97} - 130 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.57671 + 0.135040i −2.14204 − 0.695992i 4.63216 − 0.486860i −0.765094 + 2.85537i 5.61341 + 1.50411i 2.68803 − 1.03184i −6.77303 + 1.07274i 1.67689 + 1.21833i 1.58584 − 7.46078i
11.2 −2.52766 + 0.132469i 2.14081 + 0.695592i 4.38247 − 0.460616i −0.253940 + 0.947716i −5.50339 − 1.47463i 0.297869 − 0.114341i −6.01643 + 0.952909i 1.67218 + 1.21491i 0.516330 − 2.42914i
11.3 −2.38428 + 0.124955i −0.874415 − 0.284115i 3.68013 − 0.386797i 0.207198 − 0.773272i 2.12035 + 0.568146i −0.517083 + 0.198489i −4.00980 + 0.635090i −1.74317 − 1.26649i −0.397393 + 1.86959i
11.4 −2.34602 + 0.122950i −2.92390 − 0.950033i 3.49967 − 0.367830i 0.583120 − 2.17623i 6.97635 + 1.86931i −4.00906 + 1.53893i −3.52444 + 0.558217i 5.21959 + 3.79225i −1.10045 + 5.17719i
11.5 −2.15166 + 0.112764i 2.54241 + 0.826080i 2.62788 − 0.276201i 0.835228 − 3.11711i −5.56356 − 1.49075i −3.32428 + 1.27607i −1.36698 + 0.216509i 3.35441 + 2.43712i −1.44563 + 6.80115i
11.6 −1.98259 + 0.103903i 1.93799 + 0.629693i 1.93081 − 0.202936i −0.925407 + 3.45367i −3.90767 − 1.04706i 3.57658 − 1.37292i 0.114823 − 0.0181862i 0.932260 + 0.677327i 1.47585 − 6.94334i
11.7 −1.96385 + 0.102921i 1.22501 + 0.398029i 1.85706 − 0.195185i 0.861644 − 3.21570i −2.44669 − 0.655589i 4.08147 − 1.56673i 0.257774 − 0.0408275i −1.08484 − 0.788180i −1.36117 + 6.40382i
11.8 −1.76182 + 0.0923331i −1.13628 − 0.369201i 1.10645 − 0.116292i −0.276623 + 1.03237i 2.03602 + 0.545550i 1.09450 − 0.420140i 1.54642 − 0.244928i −1.27222 − 0.924320i 0.392039 − 1.84440i
11.9 −1.62361 + 0.0850899i 1.10108 + 0.357762i 0.639831 − 0.0672490i −0.314520 + 1.17380i −1.81817 − 0.487176i −2.76717 + 1.06222i 2.17853 − 0.345045i −1.34267 − 0.975508i 0.410779 − 1.93256i
11.10 −1.39534 + 0.0731266i −2.17674 − 0.707266i −0.0474228 + 0.00498434i 0.992216 − 3.70300i 3.08901 + 0.827698i 2.67826 − 1.02809i 2.82591 − 0.447580i 1.81093 + 1.31572i −1.11369 + 5.23949i
11.11 −1.34882 + 0.0706888i −0.497934 − 0.161789i −0.174716 + 0.0183634i 0.0990127 − 0.369520i 0.683062 + 0.183026i −1.81118 + 0.695246i 2.90245 − 0.459704i −2.20529 − 1.60224i −0.107430 + 0.505417i
11.12 −0.995730 + 0.0521840i −2.77166 − 0.900568i −1.00029 + 0.105134i −0.373706 + 1.39469i 2.80682 + 0.752086i 2.77787 − 1.06632i 2.96017 − 0.468845i 4.44404 + 3.22878i 0.299330 − 1.40824i
11.13 −0.754685 + 0.0395514i 2.67759 + 0.870001i −1.42106 + 0.149359i −0.552551 + 2.06215i −2.05515 − 0.550675i −3.74165 + 1.43629i 2.55938 − 0.405366i 3.98552 + 2.89565i 0.335441 − 1.57813i
11.14 −0.636159 + 0.0333397i 3.08296 + 1.00171i −1.58546 + 0.166638i 0.0795633 − 0.296934i −1.99465 − 0.534465i 2.99900 − 1.15121i 2.26143 − 0.358175i 6.07415 + 4.41312i −0.0407153 + 0.191550i
11.15 −0.596380 + 0.0312549i 0.657945 + 0.213779i −1.63435 + 0.171777i 0.974547 − 3.63706i −0.399067 − 0.106930i −2.39734 + 0.920252i 2.14902 − 0.340371i −2.03986 − 1.48205i −0.467524 + 2.19953i
11.16 −0.284729 + 0.0149220i −2.30585 − 0.749216i −1.90820 + 0.200559i 0.150829 − 0.562901i 0.667721 + 0.178915i −3.98575 + 1.52998i 1.10354 − 0.174784i 2.32856 + 1.69180i −0.0345457 + 0.162525i
11.17 −0.268345 + 0.0140634i −1.39144 − 0.452107i −1.91723 + 0.201509i −1.07119 + 3.99775i 0.379744 + 0.101752i −1.20967 + 0.464347i 1.04246 − 0.165109i −0.695342 − 0.505196i 0.231227 − 1.08784i
11.18 −0.173208 + 0.00907747i −0.409936 − 0.133196i −1.95913 + 0.205912i −0.381329 + 1.42314i 0.0722135 + 0.0193495i 4.38084 − 1.68165i 0.680089 − 0.107716i −2.27674 − 1.65415i 0.0531309 − 0.249961i
11.19 −0.124958 + 0.00654878i 0.895096 + 0.290834i −1.97347 + 0.207420i 0.216048 − 0.806301i −0.113754 − 0.0304803i 1.43092 − 0.549278i 0.492421 − 0.0779919i −1.71044 − 1.24271i −0.0217166 + 0.102169i
11.20 0.0361928 − 0.00189679i 0.910576 + 0.295864i −1.98774 + 0.208920i 0.464170 − 1.73230i 0.0335175 + 0.00898099i 0.685669 − 0.263204i −0.143138 + 0.0226709i −1.68544 − 1.22454i 0.0135138 − 0.0635775i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.ch even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.ch.a yes 560
13.f odd 12 1 403.2.cc.a ✓ 560
31.h odd 30 1 403.2.cc.a ✓ 560
403.ch even 60 1 inner 403.2.ch.a yes 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.cc.a ✓ 560 13.f odd 12 1
403.2.cc.a ✓ 560 31.h odd 30 1
403.2.ch.a yes 560 1.a even 1 1 trivial
403.2.ch.a yes 560 403.ch even 60 1 inner

Hecke kernels

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