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Properties

Label 403.2.bj.a
Level $403$
Weight $2$
Character orbit 403.bj
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(100,403)]
 
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([20, 28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.bj (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9} + 12 q^{10} + 3 q^{11} - 50 q^{12} - 24 q^{13} + 24 q^{15} + 23 q^{16} - 9 q^{17} - 15 q^{18} - 3 q^{19} - 75 q^{20} - 34 q^{21} - 22 q^{22} - 12 q^{23} + 43 q^{24} - 112 q^{25} - 21 q^{26} + 48 q^{27} + 33 q^{28} - 11 q^{29} + 24 q^{30} - 22 q^{31} + q^{32} + 53 q^{33} - 49 q^{34} - 34 q^{35} - 67 q^{36} - 34 q^{37} - 23 q^{38} - 25 q^{39} + 31 q^{40} + 7 q^{41} + 28 q^{42} + 41 q^{43} - 42 q^{44} - q^{45} - 39 q^{46} - 20 q^{47} + 84 q^{48} + 42 q^{49} + 24 q^{50} - 71 q^{51} + 109 q^{52} - 8 q^{53} + 31 q^{54} + 35 q^{55} + 63 q^{56} + 21 q^{57} - 191 q^{58} - 46 q^{59} + 2 q^{60} + 53 q^{61} + 18 q^{62} + 17 q^{63} - 91 q^{64} + 47 q^{65} - 62 q^{66} + 7 q^{67} - 166 q^{68} + 53 q^{69} + 134 q^{70} - 79 q^{71} + 65 q^{72} - 31 q^{73} + 12 q^{74} + 73 q^{75} - 46 q^{76} - 80 q^{77} + 114 q^{78} - 142 q^{79} + 87 q^{80} - 27 q^{81} - 9 q^{82} + 94 q^{83} + 85 q^{84} - 45 q^{85} - 34 q^{86} - 11 q^{87} + 61 q^{88} + 75 q^{89} + 43 q^{90} - 104 q^{91} - 294 q^{92} + 26 q^{93} - 57 q^{94} - 101 q^{95} + 24 q^{96} + 46 q^{97} - 146 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1 −2.71366 − 0.576807i 0.313803 − 0.965786i 5.20416 + 2.31704i −0.254988 + 0.441652i −1.40863 + 2.43981i 0.0372303 + 0.354222i −8.29696 − 6.02809i 1.59278 + 1.15722i 0.946699 − 1.05142i
100.2 −2.57899 − 0.548181i −0.805625 + 2.47946i 4.52359 + 2.01403i −1.73519 + 3.00543i 3.43689 − 5.95286i −0.217284 − 2.06732i −6.29611 − 4.57439i −3.07163 − 2.23167i 6.12254 − 6.79977i
100.3 −2.28137 − 0.484920i −0.170532 + 0.524843i 3.14241 + 1.39909i 0.529585 − 0.917267i 0.643553 − 1.11467i −0.0507599 − 0.482948i −2.71676 − 1.97384i 2.18067 + 1.58435i −1.65298 + 1.83582i
100.4 −2.24686 − 0.477586i −0.679969 + 2.09273i 2.99322 + 1.33267i 0.449810 − 0.779094i 2.52726 − 4.37734i 0.399570 + 3.80166i −2.37217 − 1.72348i −1.49011 − 1.08263i −1.38275 + 1.53570i
100.5 −2.23294 − 0.474625i 0.848869 − 2.61255i 2.93364 + 1.30614i 1.01379 − 1.75593i −3.13545 + 5.43076i −0.161425 − 1.53586i −2.23702 − 1.62529i −3.67779 − 2.67207i −3.09713 + 3.43971i
100.6 −2.14008 − 0.454887i −0.792588 + 2.43933i 2.54591 + 1.13351i 1.99812 − 3.46084i 2.80582 − 4.85982i −0.425169 − 4.04521i −1.39275 − 1.01189i −2.89510 − 2.10342i −5.85041 + 6.49754i
100.7 −2.08345 − 0.442850i 0.752630 − 2.31636i 2.31754 + 1.03184i −1.53370 + 2.65645i −2.59386 + 4.49271i 0.0462700 + 0.440229i −0.925131 − 0.672147i −2.37201 − 1.72337i 4.37179 − 4.85536i
100.8 −1.52609 − 0.324380i −0.102259 + 0.314720i 0.396636 + 0.176594i −0.0573847 + 0.0993931i 0.258145 − 0.447120i −0.116771 − 1.11100i 1.97641 + 1.43595i 2.33846 + 1.69899i 0.119815 − 0.133068i
100.9 −1.41577 − 0.300932i −0.586565 + 1.80526i 0.0867606 + 0.0386283i −2.18816 + 3.79001i 1.37370 − 2.37932i 0.150521 + 1.43211i 2.23074 + 1.62072i −0.487856 − 0.354448i 4.23848 − 4.70731i
100.10 −1.36969 − 0.291136i 0.233287 − 0.717984i −0.0358094 − 0.0159434i 1.52418 − 2.63996i −0.528561 + 0.915495i 0.0830158 + 0.789842i 2.31012 + 1.67840i 1.96597 + 1.42836i −2.85624 + 3.17218i
100.11 −1.34661 − 0.286231i 0.272316 − 0.838103i −0.0956586 − 0.0425900i −1.29583 + 2.24445i −0.606595 + 1.05065i −0.453578 − 4.31551i 2.34416 + 1.70313i 1.79879 + 1.30690i 2.38741 − 2.65149i
100.12 −1.10979 − 0.235894i −0.532074 + 1.63756i −0.651094 − 0.289886i 0.103550 − 0.179354i 0.976783 − 1.69184i 0.239412 + 2.27785i 2.49000 + 1.80909i 0.0285618 + 0.0207514i −0.157228 + 0.174619i
100.13 −0.881283 − 0.187322i 0.643229 − 1.97966i −1.08552 − 0.483305i −0.866187 + 1.50028i −0.937701 + 1.62414i 0.293789 + 2.79521i 2.32392 + 1.68843i −1.07824 − 0.783389i 1.04439 − 1.15991i
100.14 −0.661543 − 0.140615i −1.01562 + 3.12577i −1.40922 − 0.627427i −0.398234 + 0.689762i 1.11141 − 1.92502i −0.202627 − 1.92787i 1.93835 + 1.40829i −6.31190 − 4.58586i 0.360441 − 0.400310i
100.15 −0.528168 − 0.112266i 0.698235 − 2.14895i −1.56073 − 0.694883i 1.10173 − 1.90824i −0.610039 + 1.05662i −0.519865 − 4.94618i 1.62000 + 1.17700i −1.70339 − 1.23759i −0.796127 + 0.884188i
100.16 −0.247139 − 0.0525310i −0.639897 + 1.96940i −1.76877 − 0.787508i 0.388050 − 0.672122i 0.261598 − 0.453101i −0.286123 − 2.72228i 0.804576 + 0.584559i −1.04202 − 0.757070i −0.131209 + 0.145723i
100.17 −0.171917 − 0.0365421i 0.235059 − 0.723438i −1.79887 − 0.800909i 0.00796452 − 0.0137949i −0.0668467 + 0.115782i 0.484936 + 4.61385i 0.564372 + 0.410040i 1.95894 + 1.42325i −0.00187333 + 0.00208055i
100.18 0.0768429 + 0.0163335i −0.788714 + 2.42741i −1.82145 − 0.810963i 1.96469 − 3.40295i −0.100255 + 0.173647i 0.299883 + 2.85320i −0.253832 − 0.184420i −2.84321 − 2.06571i 0.206555 − 0.229402i
100.19 0.194455 + 0.0413328i 0.0384921 − 0.118467i −1.79099 − 0.797399i −0.998333 + 1.72916i 0.0123816 − 0.0214455i −0.177091 − 1.68490i −0.636973 − 0.462788i 2.41450 + 1.75424i −0.265602 + 0.294981i
100.20 0.208786 + 0.0443788i 0.200559 − 0.617258i −1.78547 − 0.794942i 1.97676 − 3.42385i 0.0692671 − 0.119974i 0.0421544 + 0.401072i −0.682872 − 0.496136i 2.08627 + 1.51576i 0.564667 − 0.627126i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bj even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bj.a ✓ 280
13.c even 3 1 403.2.bk.a yes 280
31.g even 15 1 403.2.bk.a yes 280
403.bj even 15 1 inner 403.2.bj.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bj.a ✓ 280 1.a even 1 1 trivial
403.2.bj.a ✓ 280 403.bj even 15 1 inner
403.2.bk.a yes 280 13.c even 3 1
403.2.bk.a yes 280 31.g even 15 1

Hecke kernels

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