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Properties

Label 353.2.f.a
Level $353$
Weight $2$
Character orbit 353.f
Analytic conductor $2.819$
Analytic rank $0$
Dimension $232$
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] = [353,2,Mod(36,353)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(353, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("353.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 353.f (of order \(16\), 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: \(2.81871919135\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(29\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} + 24 q^{10} + 8 q^{11} + 72 q^{12} - 8 q^{13} - 8 q^{14} - 8 q^{15} - 248 q^{16} - 8 q^{17} + 24 q^{18} - 8 q^{19} + 8 q^{20} + 24 q^{21} - 48 q^{22} + 16 q^{23} + 32 q^{24} - 32 q^{25} + 16 q^{26} + 16 q^{27} - 40 q^{28} - 8 q^{29} - 8 q^{30} - 8 q^{31} + 8 q^{32} - 136 q^{33} + 80 q^{36} - 8 q^{37} - 48 q^{38} + 40 q^{39} + 40 q^{40} - 64 q^{41} + 96 q^{43} + 112 q^{44} - 160 q^{45} - 40 q^{46} + 104 q^{47} + 16 q^{48} - 24 q^{49} - 72 q^{50} - 144 q^{51} + 24 q^{52} - 8 q^{53} + 48 q^{54} - 72 q^{55} - 56 q^{56} + 24 q^{57} + 128 q^{58} - 8 q^{59} - 64 q^{60} - 8 q^{61} - 40 q^{62} - 8 q^{63} - 80 q^{65} - 32 q^{66} - 64 q^{67} - 16 q^{68} + 208 q^{69} + 8 q^{70} + 16 q^{71} + 24 q^{72} + 8 q^{73} - 8 q^{75} + 120 q^{76} - 48 q^{77} + 24 q^{78} + 16 q^{79} - 16 q^{80} - 24 q^{81} + 16 q^{82} - 104 q^{83} + 184 q^{84} + 80 q^{85} + 120 q^{86} - 104 q^{87} - 72 q^{89} - 208 q^{90} + 24 q^{91} + 24 q^{92} - 24 q^{93} + 56 q^{94} - 40 q^{95} - 120 q^{96} - 48 q^{97} + 40 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −1.83777 + 1.83777i 1.65514 − 0.329228i − 4.75476i −0.991536 + 1.48394i −2.43672 + 3.64681i −0.869971 + 0.581296i 5.06260 + 5.06260i −0.140537 + 0.0582122i −0.904920 − 4.54934i
36.2 −1.83165 + 1.83165i −3.10034 + 0.616696i − 4.70991i −0.939070 + 1.40542i 4.54918 − 6.80832i 1.69432 − 1.13211i 4.96362 + 4.96362i 6.46015 − 2.67588i −0.854187 − 4.29429i
36.3 −1.74016 + 1.74016i −1.19038 + 0.236781i − 4.05633i −0.0819079 + 0.122584i 1.65942 − 2.48349i −2.85021 + 1.90445i 3.57835 + 3.57835i −1.41070 + 0.584330i −0.0707827 − 0.355849i
36.4 −1.60440 + 1.60440i 2.28136 − 0.453790i − 3.14821i 1.28352 − 1.92092i −2.93215 + 4.38827i 1.74145 − 1.16360i 1.84218 + 1.84218i 2.22703 − 0.922464i 1.02265 + 5.14121i
36.5 −1.54451 + 1.54451i −1.53117 + 0.304569i − 2.77104i 2.16346 − 3.23785i 1.89451 − 2.83533i −0.326993 + 0.218490i 1.19088 + 1.19088i −0.519907 + 0.215353i 1.65940 + 8.34239i
36.6 −1.31564 + 1.31564i −0.357461 + 0.0711034i − 1.46182i −1.61496 + 2.41696i 0.376744 − 0.563837i 2.12278 − 1.41840i −0.708048 − 0.708048i −2.64892 + 1.09722i −1.05514 − 5.30455i
36.7 −1.12294 + 1.12294i 1.04641 − 0.208145i − 0.522007i 1.59156 − 2.38194i −0.941329 + 1.40880i −0.347660 + 0.232299i −1.65970 − 1.65970i −1.71998 + 0.712440i 0.887551 + 4.46202i
36.8 −0.940706 + 0.940706i −2.07728 + 0.413196i 0.230146i 0.351153 − 0.525537i 1.56541 − 2.34280i 2.02247 − 1.35137i −2.09791 − 2.09791i 1.37270 − 0.568593i 0.164044 + 0.824707i
36.9 −0.874350 + 0.874350i −2.02567 + 0.402931i 0.471023i −2.25919 + 3.38111i 1.41884 − 2.12345i −3.49957 + 2.33834i −2.16054 − 2.16054i 1.16934 − 0.484358i −0.980957 − 4.93160i
36.10 −0.871993 + 0.871993i 3.26786 − 0.650017i 0.479258i −0.866765 + 1.29721i −2.28274 + 3.41636i −0.726107 + 0.485169i −2.16189 − 2.16189i 7.48472 − 3.10027i −0.375341 − 1.88697i
36.11 −0.647366 + 0.647366i 0.0843190 − 0.0167721i 1.16183i 0.0823853 − 0.123298i −0.0437276 + 0.0654430i −1.44393 + 0.964802i −2.04686 − 2.04686i −2.76481 + 1.14522i 0.0264857 + 0.133153i
36.12 −0.437732 + 0.437732i 2.45538 − 0.488405i 1.61678i −0.0560046 + 0.0838168i −0.861006 + 1.28859i 3.51011 − 2.34538i −1.58318 − 1.58318i 3.01870 − 1.25039i −0.0121743 − 0.0612043i
36.13 −0.275582 + 0.275582i −2.92249 + 0.581319i 1.84811i 1.51305 − 2.26445i 0.645185 − 0.965587i −4.01242 + 2.68102i −1.06047 − 1.06047i 5.43136 − 2.24974i 0.207070 + 1.04101i
36.14 −0.180049 + 0.180049i −2.30248 + 0.457992i 1.93516i 0.547244 − 0.819009i 0.332098 − 0.497020i 2.90843 − 1.94335i −0.708521 − 0.708521i 2.32003 − 0.960988i 0.0489309 + 0.245992i
36.15 −0.178912 + 0.178912i 1.02428 − 0.203741i 1.93598i −1.80359 + 2.69927i −0.146804 + 0.219707i 0.254371 − 0.169965i −0.704196 − 0.704196i −1.76401 + 0.730677i −0.160247 − 0.805618i
36.16 0.166074 − 0.166074i 2.69712 − 0.536491i 1.94484i 2.33567 − 3.49558i 0.358824 − 0.537018i −2.03978 + 1.36294i 0.655134 + 0.655134i 4.21501 − 1.74592i −0.192630 − 0.968417i
36.17 0.250180 − 0.250180i 1.03619 − 0.206111i 1.87482i 0.717988 − 1.07455i 0.207669 − 0.310799i 3.23793 − 2.16352i 0.969401 + 0.969401i −1.74043 + 0.720910i −0.0892033 − 0.448455i
36.18 0.412197 − 0.412197i −0.876254 + 0.174298i 1.66019i −0.207113 + 0.309967i −0.289344 + 0.433034i −1.71425 + 1.14542i 1.50872 + 1.50872i −2.03420 + 0.842592i 0.0423959 + 0.213138i
36.19 0.477143 − 0.477143i 1.96598 − 0.391057i 1.54467i −0.883144 + 1.32172i 0.751462 − 1.12464i −3.34389 + 2.23432i 1.69132 + 1.69132i 0.940495 − 0.389566i 0.209263 + 1.05204i
36.20 0.778358 − 0.778358i −1.28403 + 0.255410i 0.788317i 1.89695 − 2.83899i −0.800637 + 1.19824i 1.01666 − 0.679308i 2.17031 + 2.17031i −1.18813 + 0.492142i −0.733242 − 3.68626i
See next 80 embeddings (of 232 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
353.f even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 353.2.f.a ✓ 232
353.f even 16 1 inner 353.2.f.a ✓ 232
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
353.2.f.a ✓ 232 1.a even 1 1 trivial
353.2.f.a ✓ 232 353.f even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(353, [\chi])\).