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Properties

Label 407.2.l.a
Level $407$
Weight $2$
Character orbit 407.l
Analytic conductor $3.250$
Analytic rank $0$
Dimension $84$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [407,2,Mod(12,407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("407.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.l (of order \(9\), degree \(6\), 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: \(3.24991136227\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{5} + 3 q^{7} - 9 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{5} + 3 q^{7} - 9 q^{8} + 6 q^{9} + 18 q^{10} + 42 q^{11} - 15 q^{12} - 3 q^{13} + 21 q^{15} - 12 q^{17} - 45 q^{18} + 12 q^{19} + 27 q^{20} - 36 q^{21} - 36 q^{24} - 18 q^{25} + 9 q^{26} - 15 q^{27} - 66 q^{28} + 3 q^{29} - 3 q^{30} - 24 q^{31} + 21 q^{32} - 3 q^{34} - 27 q^{35} + 18 q^{36} + 6 q^{37} + 24 q^{38} + 27 q^{39} + 57 q^{40} - 30 q^{41} - 24 q^{42} - 39 q^{45} + 45 q^{46} - 9 q^{47} - 30 q^{48} - 3 q^{49} - 93 q^{50} + 54 q^{51} + 39 q^{52} + 3 q^{53} - 87 q^{54} - 12 q^{55} + 36 q^{56} - 18 q^{57} - 54 q^{58} - 51 q^{59} - 15 q^{60} + 84 q^{61} - 24 q^{62} + 54 q^{63} - 3 q^{64} - 24 q^{65} + 27 q^{67} - 84 q^{68} + 24 q^{69} + 84 q^{70} + 21 q^{71} + 42 q^{72} - 168 q^{73} + 81 q^{74} - 42 q^{75} + 75 q^{76} + 6 q^{77} - 102 q^{78} + 3 q^{79} + 60 q^{80} + 36 q^{81} - 108 q^{82} + 3 q^{83} + 63 q^{84} + 120 q^{85} + 45 q^{86} + 24 q^{87} + 9 q^{88} + 114 q^{89} + 87 q^{90} - 75 q^{91} - 135 q^{92} + 54 q^{93} + 69 q^{94} - 114 q^{95} + 12 q^{96} - 12 q^{97} + 6 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −1.99969 + 1.67794i 1.46916 + 1.23277i 0.835983 − 4.74109i −0.725927 + 0.264216i −5.00636 2.36043 − 0.859127i 3.67315 + 6.36208i 0.117757 + 0.667831i 1.00829 − 1.74641i
12.2 −1.69214 + 1.41988i −1.45088 − 1.21743i 0.500002 − 2.83565i −2.86773 + 1.04377i 4.18371 4.13181 − 1.50386i 0.971266 + 1.68228i 0.101967 + 0.578282i 3.37059 − 5.83803i
12.3 −1.46267 + 1.22732i −1.00523 − 0.843486i 0.285777 − 1.62072i 2.21537 − 0.806330i 2.50555 −3.52375 + 1.28254i −0.338222 − 0.585818i −0.221931 − 1.25863i −2.25073 + 3.89838i
12.4 −0.981481 + 0.823561i −0.235390 − 0.197516i −0.0622429 + 0.352997i 1.77539 − 0.646189i 0.393697 1.89503 − 0.689736i −1.51086 − 2.61688i −0.504548 − 2.86144i −1.21034 + 2.09636i
12.5 −0.781379 + 0.655655i −2.16731 − 1.81858i −0.166627 + 0.944986i −2.47806 + 0.901939i 2.88585 −2.58682 + 0.941526i −1.50940 − 2.61436i 0.869016 + 4.92843i 1.34494 − 2.32950i
12.6 −0.421600 + 0.353764i 2.41106 + 2.02312i −0.294699 + 1.67132i 0.885113 − 0.322155i −1.73221 1.31358 − 0.478106i −1.01737 − 1.76213i 1.19925 + 6.80129i −0.259197 + 0.448942i
12.7 −0.220542 + 0.185057i 0.897119 + 0.752772i −0.332904 + 1.88799i −2.50116 + 0.910348i −0.337158 −2.04507 + 0.744345i −0.563863 − 0.976640i −0.282788 − 1.60377i 0.383145 − 0.663627i
12.8 0.378047 − 0.317219i −0.708023 − 0.594102i −0.305005 + 1.72977i −1.27318 + 0.463401i −0.456126 1.17841 − 0.428906i 0.926914 + 1.60546i −0.372605 − 2.11315i −0.334323 + 0.579065i
12.9 0.626141 − 0.525394i 1.24553 + 1.04513i −0.231283 + 1.31167i 3.14605 − 1.14507i 1.32898 −1.91073 + 0.695449i 1.36170 + 2.35853i −0.0618801 − 0.350939i 1.36826 − 2.36989i
12.10 0.971584 − 0.815256i −1.84836 − 1.55096i −0.0679627 + 0.385436i 2.95045 − 1.07388i −3.06027 3.28187 − 1.19450i 1.51651 + 2.62667i 0.490021 + 2.77905i 1.99113 − 3.44873i
12.11 1.11171 − 0.932838i 1.92999 + 1.61945i 0.0184226 − 0.104480i −2.79131 + 1.01596i 3.65628 0.459560 − 0.167266i 1.37426 + 2.38028i 0.581285 + 3.29663i −2.15542 + 3.73329i
12.12 1.36042 − 1.14153i −0.0212722 − 0.0178495i 0.200361 − 1.13631i −0.998861 + 0.363556i −0.0493149 2.90878 − 1.05871i 0.751354 + 1.30138i −0.520811 − 2.95366i −0.943862 + 1.63482i
12.13 1.91954 − 1.61068i 0.508174 + 0.426409i 0.743025 − 4.21390i 1.47127 − 0.535498i 1.66227 −1.86739 + 0.679673i −2.85522 − 4.94538i −0.444528 − 2.52104i 1.96163 − 3.39765i
12.14 1.95811 − 1.64305i −1.79061 − 1.50250i 0.787282 − 4.46490i −2.62648 + 0.955962i −5.97489 −0.397271 + 0.144595i −3.23833 − 5.60895i 0.427836 + 2.42638i −3.57225 + 6.18731i
34.1 −1.99969 − 1.67794i 1.46916 − 1.23277i 0.835983 + 4.74109i −0.725927 − 0.264216i −5.00636 2.36043 + 0.859127i 3.67315 − 6.36208i 0.117757 − 0.667831i 1.00829 + 1.74641i
34.2 −1.69214 − 1.41988i −1.45088 + 1.21743i 0.500002 + 2.83565i −2.86773 − 1.04377i 4.18371 4.13181 + 1.50386i 0.971266 − 1.68228i 0.101967 − 0.578282i 3.37059 + 5.83803i
34.3 −1.46267 − 1.22732i −1.00523 + 0.843486i 0.285777 + 1.62072i 2.21537 + 0.806330i 2.50555 −3.52375 − 1.28254i −0.338222 + 0.585818i −0.221931 + 1.25863i −2.25073 − 3.89838i
34.4 −0.981481 − 0.823561i −0.235390 + 0.197516i −0.0622429 − 0.352997i 1.77539 + 0.646189i 0.393697 1.89503 + 0.689736i −1.51086 + 2.61688i −0.504548 + 2.86144i −1.21034 − 2.09636i
34.5 −0.781379 − 0.655655i −2.16731 + 1.81858i −0.166627 − 0.944986i −2.47806 − 0.901939i 2.88585 −2.58682 − 0.941526i −1.50940 + 2.61436i 0.869016 − 4.92843i 1.34494 + 2.32950i
34.6 −0.421600 − 0.353764i 2.41106 − 2.02312i −0.294699 − 1.67132i 0.885113 + 0.322155i −1.73221 1.31358 + 0.478106i −1.01737 + 1.76213i 1.19925 − 6.80129i −0.259197 − 0.448942i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 407.2.l.a ✓ 84
37.f even 9 1 inner 407.2.l.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
407.2.l.a ✓ 84 1.a even 1 1 trivial
407.2.l.a ✓ 84 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} + 7 T_{2}^{81} - 33 T_{2}^{79} + 548 T_{2}^{78} + 360 T_{2}^{77} - 258 T_{2}^{76} + \cdots + 2996361 \) acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\). Copy content Toggle raw display