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Properties

Label 403.2.bt.a
Level $403$
Weight $2$
Character orbit 403.bt
Analytic conductor $3.218$
Analytic rank $0$
Dimension $280$
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(82,403)]
 
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, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.bt (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: \(280\)
Relative dimension: \(35\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9} + 3 q^{10} - 8 q^{12} - 6 q^{13} + 4 q^{14} - 45 q^{15} + 23 q^{16} + 27 q^{17} + 45 q^{18} - 15 q^{19} - 12 q^{20} - 76 q^{21} + 41 q^{22} - 10 q^{23} - 33 q^{24} + 96 q^{25} + 9 q^{26} - 24 q^{27} - 32 q^{28} + 13 q^{29} + 36 q^{30} + 2 q^{31} - 141 q^{32} - 93 q^{33} - 9 q^{34} - 43 q^{35} - 194 q^{36} + 3 q^{37} - 49 q^{38} + 50 q^{39} - 75 q^{40} - 15 q^{41} + 17 q^{42} + 33 q^{43} + 18 q^{44} - 15 q^{45} - 9 q^{46} - 59 q^{48} + 3 q^{49} + 36 q^{50} + 47 q^{51} - 56 q^{52} + 12 q^{53} - 33 q^{54} - 5 q^{55} - 50 q^{56} - 105 q^{57} - 3 q^{58} - 15 q^{59} + 90 q^{60} - 57 q^{61} - 72 q^{62} + 201 q^{63} + 13 q^{64} - 43 q^{65} + 22 q^{66} - 71 q^{68} - 7 q^{69} - 42 q^{71} + 90 q^{72} + 9 q^{73} - 113 q^{74} + 45 q^{75} + 14 q^{76} - 24 q^{77} + 61 q^{78} + 54 q^{79} + 30 q^{80} + 106 q^{81} + 16 q^{82} + 54 q^{83} + 60 q^{84} + 18 q^{85} + 84 q^{86} + 42 q^{87} - 98 q^{88} - 99 q^{89} + 11 q^{90} + 60 q^{91} + 266 q^{92} - 104 q^{93} + 33 q^{94} - 120 q^{95} + 204 q^{96} - 50 q^{97} - 15 q^{98} - 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −0.558017 + 2.62526i −2.44380 + 0.519446i −4.75353 − 2.11641i −1.49782 + 0.864767i − 6.70549i −0.203294 − 0.279810i 5.05356 − 6.95562i 2.96171 − 1.31864i −1.43443 − 4.41473i
82.2 −0.554356 + 2.60804i 0.790982 − 0.168128i −4.66747 − 2.07809i 2.97528 − 1.71778i 2.15612i −2.45564 − 3.37990i 4.87275 − 6.70677i −2.14325 + 0.954237i 2.83067 + 8.71192i
82.3 −0.526303 + 2.47606i 1.87677 − 0.398919i −4.02678 − 1.79284i −3.24110 + 1.87125i 4.85693i −1.04985 − 1.44500i 3.58267 − 4.93112i 0.622476 − 0.277144i −2.92753 − 9.01000i
82.4 −0.510705 + 2.40268i −1.82246 + 0.387376i −3.68496 − 1.64065i 3.06150 − 1.76756i − 4.57663i 2.12591 + 2.92607i 2.93626 − 4.04141i 0.430673 − 0.191748i 2.68335 + 8.25849i
82.5 −0.475411 + 2.23663i 0.254115 − 0.0540139i −2.94943 − 1.31317i −0.204817 + 0.118251i 0.594042i 1.78231 + 2.45313i 1.65121 − 2.27270i −2.67898 + 1.19276i −0.167112 − 0.514319i
82.6 −0.448118 + 2.10823i 3.13762 − 0.666922i −2.41674 − 1.07600i 0.330706 − 0.190933i 6.91369i 0.412693 + 0.568023i 0.817700 − 1.12547i 6.65924 − 2.96488i 0.254336 + 0.782765i
82.7 −0.387542 + 1.82324i −0.271363 + 0.0576799i −1.34692 − 0.599689i 0.212568 − 0.122726i − 0.517113i −2.14262 − 2.94906i −0.575865 + 0.792611i −2.67033 + 1.18891i 0.141380 + 0.435123i
82.8 −0.384687 + 1.80981i −2.40054 + 0.510251i −1.30034 − 0.578951i −0.272473 + 0.157312i − 4.54082i −0.636632 − 0.876249i −0.627074 + 0.863094i 2.76162 − 1.22955i −0.179889 − 0.553641i
82.9 −0.309669 + 1.45688i 2.00373 − 0.425907i −0.199502 − 0.0888240i 3.03693 − 1.75337i 3.05108i 0.544135 + 0.748938i −1.55974 + 2.14679i 1.09292 − 0.486598i 1.61401 + 4.96740i
82.10 −0.285763 + 1.34441i 1.01586 − 0.215927i 0.101320 + 0.0451105i −1.34641 + 0.777351i 1.42743i 2.50350 + 3.44577i −1.70535 + 2.34722i −1.75529 + 0.781506i −0.660323 − 2.03226i
82.11 −0.226015 + 1.06331i −0.562471 + 0.119557i 0.747535 + 0.332824i 1.49545 − 0.863396i − 0.625106i −0.503379 − 0.692842i −1.80078 + 2.47856i −2.43856 + 1.08572i 0.580069 + 1.78527i
82.12 −0.219621 + 1.03324i 2.09766 − 0.445872i 0.807749 + 0.359633i −3.00000 + 1.73205i 2.26530i −0.0849215 − 0.116884i −1.79076 + 2.46477i 1.46075 − 0.650366i −1.13075 − 3.48010i
82.13 −0.190379 + 0.895663i −3.22061 + 0.684561i 1.06112 + 0.472442i 0.521441 − 0.301054i − 3.01491i 0.972860 + 1.33903i −1.70160 + 2.34205i 7.16305 − 3.18920i 0.170372 + 0.524350i
82.14 −0.136295 + 0.641215i −1.88071 + 0.399758i 1.43451 + 0.638685i −1.94312 + 1.12186i − 1.26043i 1.23396 + 1.69841i −1.37568 + 1.89347i 0.636641 − 0.283451i −0.454518 − 1.39886i
82.15 −0.107110 + 0.503915i 2.39868 − 0.509854i 1.58463 + 0.705524i 0.193800 − 0.111891i 1.26334i −2.62876 − 3.61818i −1.13088 + 1.55652i 2.75306 − 1.22574i 0.0356253 + 0.109644i
82.16 −0.0617274 + 0.290405i −0.640694 + 0.136184i 1.74657 + 0.777621i −2.50432 + 1.44587i − 0.194467i −1.38419 − 1.90518i −0.682654 + 0.939592i −2.34869 + 1.04571i −0.265302 − 0.816516i
82.17 −0.0166647 + 0.0784010i −2.19989 + 0.467602i 1.82122 + 0.810860i 3.63891 − 2.10092i − 0.180266i −2.71023 − 3.73031i −0.188147 + 0.258963i 1.88024 − 0.837136i 0.104073 + 0.320305i
82.18 0.0316136 − 0.148730i 1.83195 − 0.389394i 1.80597 + 0.804069i 1.52776 − 0.882054i − 0.284777i −0.353168 − 0.486094i 0.355432 − 0.489210i 0.463790 − 0.206493i −0.0828901 − 0.255110i
82.19 0.0633758 − 0.298160i −0.256430 + 0.0545059i 1.74221 + 0.775681i −2.38402 + 1.37641i 0.0799114i 0.229361 + 0.315689i 0.700029 − 0.963507i −2.67785 + 1.19226i 0.259302 + 0.798049i
82.20 0.0900306 − 0.423561i −1.34476 + 0.285838i 1.65579 + 0.737206i 1.67453 − 0.966790i 0.595322i 3.08797 + 4.25023i 0.970373 − 1.33560i −1.01396 + 0.451444i −0.258735 − 0.796306i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bt 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.bt.a yes 280
13.e even 6 1 403.2.bp.a ✓ 280
31.g even 15 1 403.2.bp.a ✓ 280
403.bt even 30 1 inner 403.2.bt.a yes 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bp.a ✓ 280 13.e even 6 1
403.2.bp.a ✓ 280 31.g even 15 1
403.2.bt.a yes 280 1.a even 1 1 trivial
403.2.bt.a yes 280 403.bt even 30 1 inner

Hecke kernels

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