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Properties

Label 403.2.e.a
Level $403$
Weight $2$
Character orbit 403.e
Analytic conductor $3.218$
Analytic rank $0$
Dimension $70$
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(191,403)]
 
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([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.e (of order \(3\), 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: \(70\)
Relative dimension: \(35\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9} - 6 q^{10} - 4 q^{11} + 5 q^{12} + q^{13} - 10 q^{14} + q^{15} - 28 q^{16} - 28 q^{17} - 20 q^{18} + 4 q^{19} + 25 q^{20} - 21 q^{21} + 4 q^{22} + 2 q^{23} + 4 q^{24} - 23 q^{25} - 24 q^{26} - 38 q^{27} - 21 q^{28} + 6 q^{29} + 31 q^{30} + 22 q^{31} + 14 q^{32} + 2 q^{33} - 11 q^{34} + 4 q^{35} + 56 q^{36} - 12 q^{37} - 7 q^{38} + 10 q^{39} - q^{40} + 4 q^{41} - 54 q^{42} + 2 q^{43} + 2 q^{44} + 58 q^{45} + 14 q^{46} - 2 q^{48} + 74 q^{49} + 7 q^{50} - 9 q^{51} + 5 q^{52} - 2 q^{53} + 24 q^{54} + 5 q^{55} + 26 q^{56} - q^{57} + 6 q^{58} - 42 q^{59} + 18 q^{60} - 3 q^{61} + 13 q^{62} - 32 q^{63} - 14 q^{64} + 20 q^{65} - 28 q^{66} + 4 q^{67} + 42 q^{68} - 64 q^{69} - 14 q^{70} + 43 q^{71} - 5 q^{72} + 11 q^{73} + 14 q^{74} - 74 q^{75} - 28 q^{76} - 10 q^{77} - 64 q^{78} + 2 q^{79} - 76 q^{80} - 11 q^{81} - 17 q^{82} + 56 q^{83} - 45 q^{84} - 5 q^{85} + 54 q^{86} + 48 q^{87} - 8 q^{88} + 30 q^{89} - 23 q^{90} - 36 q^{91} + 74 q^{92} + 22 q^{93} + 47 q^{94} - 9 q^{95} + 26 q^{96} + 29 q^{97} + 12 q^{98} + 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 −1.34742 + 2.33379i −0.887440 − 1.53709i −2.63106 − 4.55713i 0.115559 + 0.200154i 4.78300 1.93397 8.79087 −0.0750997 + 0.130077i −0.622824
191.2 −1.27542 + 2.20909i 0.569293 + 0.986044i −2.25340 − 3.90300i −0.675176 − 1.16944i −2.90435 1.90483 6.39445 0.851812 − 1.47538i 3.44453
191.3 −1.25452 + 2.17290i 1.68571 + 2.91973i −2.14766 − 3.71985i 0.669409 + 1.15945i −8.45904 1.15243 5.75904 −4.18323 + 7.24556i −3.35916
191.4 −1.18452 + 2.05166i −0.300552 − 0.520571i −1.80620 − 3.12843i 2.14607 + 3.71711i 1.42404 1.99772 3.81984 1.31934 − 2.28516i −10.1683
191.5 −1.06665 + 1.84749i 0.605846 + 1.04936i −1.27547 − 2.20918i −1.78766 − 3.09632i −2.58490 0.911932 1.17532 0.765900 − 1.32658i 7.62720
191.6 −1.05012 + 1.81887i −1.37200 − 2.37637i −1.20552 − 2.08802i 0.498066 + 0.862675i 5.76307 −3.52844 0.863293 −2.26475 + 3.92266i −2.09212
191.7 −0.948089 + 1.64214i −0.203682 − 0.352788i −0.797747 − 1.38174i 0.840173 + 1.45522i 0.772435 −3.74155 −0.767016 1.41703 − 2.45436i −3.18624
191.8 −0.937381 + 1.62359i 1.15361 + 1.99811i −0.757366 − 1.31180i −1.23736 − 2.14316i −4.32548 −4.49839 −0.909763 −1.16162 + 2.01199i 4.63950
191.9 −0.925450 + 1.60293i −1.55919 − 2.70060i −0.712914 − 1.23480i −0.0505045 − 0.0874763i 5.77182 4.42694 −1.06274 −3.36217 + 5.82344i 0.186957
191.10 −0.852641 + 1.47682i 0.937886 + 1.62447i −0.453992 − 0.786337i 1.48433 + 2.57094i −3.19872 0.0782312 −1.86219 −0.259260 + 0.449051i −5.06240
191.11 −0.770644 + 1.33479i −0.329492 − 0.570697i −0.187783 − 0.325250i −0.614094 − 1.06364i 1.01568 0.583300 −2.50372 1.28287 − 2.22200i 1.89299
191.12 −0.447963 + 0.775894i 0.0864423 + 0.149722i 0.598659 + 1.03691i −0.364951 − 0.632113i −0.154892 3.22911 −2.86456 1.48506 − 2.57219i 0.653937
191.13 −0.381984 + 0.661616i 1.18414 + 2.05098i 0.708176 + 1.22660i 0.878528 + 1.52165i −1.80929 −0.793025 −2.60998 −1.30436 + 2.25922i −1.34233
191.14 −0.356919 + 0.618202i −0.621239 − 1.07602i 0.745217 + 1.29075i −0.797665 − 1.38160i 0.886928 −1.56596 −2.49161 0.728125 − 1.26115i 1.13881
191.15 −0.351546 + 0.608895i 1.62249 + 2.81024i 0.752831 + 1.30394i −1.57412 − 2.72646i −2.28152 5.06668 −2.46480 −3.76495 + 6.52109i 2.21350
191.16 −0.333798 + 0.578156i −1.40074 − 2.42616i 0.777157 + 1.34608i −1.97182 − 3.41529i 1.87026 −1.53640 −2.37285 −2.42416 + 4.19877i 2.63276
191.17 −0.254102 + 0.440118i −0.777658 − 1.34694i 0.870864 + 1.50838i 1.96757 + 3.40793i 0.790418 3.56171 −1.90156 0.290495 − 0.503151i −1.99985
191.18 −0.0861040 + 0.149137i 1.23907 + 2.14613i 0.985172 + 1.70637i −0.342759 − 0.593675i −0.426756 −3.89977 −0.683725 −1.57059 + 2.72035i 0.118052
191.19 0.0840732 − 0.145619i 0.173155 + 0.299914i 0.985863 + 1.70757i 1.49511 + 2.58961i 0.0582309 −3.36152 0.667832 1.44003 − 2.49421i 0.502795
191.20 0.170223 − 0.294835i −1.12750 − 1.95288i 0.942048 + 1.63168i 0.624468 + 1.08161i −0.767702 −0.903522 1.32232 −1.04250 + 1.80566i 0.425195
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.35
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.e even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.e.a ✓ 70
13.c even 3 1 403.2.g.a yes 70
31.c even 3 1 403.2.g.a yes 70
403.e even 3 1 inner 403.2.e.a ✓ 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.e.a ✓ 70 1.a even 1 1 trivial
403.2.e.a ✓ 70 403.e even 3 1 inner
403.2.g.a yes 70 13.c even 3 1
403.2.g.a yes 70 31.c even 3 1

Hecke kernels

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