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Properties

Label 507.2.p.a
Level $507$
Weight $2$
Character orbit 507.p
Analytic conductor $4.048$
Analytic rank $0$
Dimension $168$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(25,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.p (of order \(26\), degree \(12\), 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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(14\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 14 q^{3} + 12 q^{4} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 14 q^{3} + 12 q^{4} - 14 q^{9} - 4 q^{10} + 12 q^{12} + 13 q^{13} + 2 q^{14} - 8 q^{16} - 4 q^{17} - 72 q^{22} + 48 q^{23} - 44 q^{25} - 39 q^{26} - 14 q^{27} + 45 q^{29} - 4 q^{30} - 26 q^{31} + 130 q^{32} + 13 q^{33} - 65 q^{34} - 35 q^{35} + 12 q^{36} + 61 q^{38} + 12 q^{40} - 63 q^{42} + 72 q^{43} - 39 q^{44} - 8 q^{48} - 68 q^{49} - 52 q^{50} - 4 q^{51} + 65 q^{52} - q^{53} + 53 q^{55} - 14 q^{56} - 13 q^{57} - 26 q^{58} - 104 q^{59} + 117 q^{60} + 12 q^{61} + 49 q^{62} - 32 q^{64} - 52 q^{65} - 46 q^{66} + 26 q^{67} - 84 q^{68} - 4 q^{69} - 39 q^{71} - 52 q^{73} + 29 q^{74} + 8 q^{75} - 130 q^{76} + 60 q^{77} + 65 q^{78} + 14 q^{79} - 14 q^{81} + 45 q^{82} + 78 q^{83} - 13 q^{85} - 13 q^{86} - 46 q^{87} - 26 q^{88} - 4 q^{90} - 208 q^{91} + 82 q^{92} - 39 q^{93} + 25 q^{94} - 66 q^{95} + 65 q^{96} + 26 q^{97} - 104 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −2.41243 − 0.914913i 0.568065 + 0.822984i 3.48572 + 3.08808i −3.08975 − 0.375164i −0.617456 − 2.50512i −0.149094 − 0.284076i −3.18567 − 6.06979i −0.354605 + 0.935016i 7.11057 + 3.73191i
25.2 −2.23311 − 0.846906i 0.568065 + 0.822984i 2.77250 + 2.45622i 0.980994 + 0.119114i −0.571560 − 2.31891i 1.47491 + 2.81021i −1.89129 − 3.60355i −0.354605 + 0.935016i −2.08979 − 1.09680i
25.3 −1.83220 − 0.694863i 0.568065 + 0.822984i 1.37712 + 1.22002i −0.128032 − 0.0155459i −0.468949 − 1.90260i −2.04557 − 3.89750i 0.145878 + 0.277948i −0.354605 + 0.935016i 0.223779 + 0.117448i
25.4 −1.24255 − 0.471239i 0.568065 + 0.822984i −0.175147 − 0.155167i −1.73849 − 0.211091i −0.318029 − 1.29030i −0.0913225 − 0.174001i 1.37966 + 2.62872i −0.354605 + 0.935016i 2.06070 + 1.08154i
25.5 −0.911757 − 0.345784i 0.568065 + 0.822984i −0.785288 − 0.695704i 2.27932 + 0.276760i −0.233362 − 0.946788i 1.78194 + 3.39521i 1.38175 + 2.63271i −0.354605 + 0.935016i −1.98249 − 1.04049i
25.6 −0.642861 − 0.243805i 0.568065 + 0.822984i −1.14319 − 1.01278i 4.00656 + 0.486484i −0.164539 − 0.667561i −0.644348 − 1.22770i 1.12702 + 2.14737i −0.354605 + 0.935016i −2.45705 − 1.28956i
25.7 −0.239472 − 0.0908198i 0.568065 + 0.822984i −1.44792 − 1.28275i −3.85869 − 0.468530i −0.0612925 − 0.248673i 0.285060 + 0.543137i 0.468284 + 0.892241i −0.354605 + 0.935016i 0.881497 + 0.462645i
25.8 0.00155211 0.000588639i 0.568065 + 0.822984i −1.49702 − 1.32624i −0.897149 − 0.108934i 0.000397260 0.00161175i −0.178306 − 0.339734i −0.00308573 − 0.00587937i −0.354605 + 0.935016i −0.00132835 0.000697174i
25.9 0.725144 + 0.275011i 0.568065 + 0.822984i −1.04682 − 0.927401i 1.67905 + 0.203873i 0.185599 + 0.753005i −1.40603 − 2.67897i −1.22487 − 2.33380i −0.354605 + 0.935016i 1.16148 + 0.609593i
25.10 0.759337 + 0.287979i 0.568065 + 0.822984i −1.00336 − 0.888900i 0.883771 + 0.107309i 0.194351 + 0.788513i 2.17066 + 4.13585i −1.26072 − 2.40210i −0.354605 + 0.935016i 0.640177 + 0.335991i
25.11 1.35653 + 0.514462i 0.568065 + 0.822984i 0.0784691 + 0.0695176i −3.71157 − 0.450666i 0.347200 + 1.40865i −1.65187 − 3.14737i −1.27776 − 2.43458i −0.354605 + 0.935016i −4.80299 − 2.52080i
25.12 1.96188 + 0.744043i 0.568065 + 0.822984i 1.79835 + 1.59320i 1.22751 + 0.149046i 0.502139 + 2.03726i −0.253099 − 0.482239i 0.392545 + 0.747932i −0.354605 + 0.935016i 2.29732 + 1.20573i
25.13 2.15817 + 0.818487i 0.568065 + 0.822984i 2.49077 + 2.20663i −1.10490 − 0.134159i 0.552380 + 2.24110i 1.27631 + 2.43180i 1.42410 + 2.71339i −0.354605 + 0.935016i −2.27476 − 1.19389i
25.14 2.55177 + 0.967759i 0.568065 + 0.822984i 4.07796 + 3.61275i 1.13985 + 0.138403i 0.653121 + 2.64982i −2.09908 − 3.99947i 4.37317 + 8.33237i −0.354605 + 0.935016i 2.77471 + 1.45628i
64.1 −2.32286 − 0.282046i −0.748511 + 0.663123i 3.37425 + 0.831678i −0.0918625 − 0.175029i 1.92572 − 1.32923i −0.0642964 − 0.0243844i −3.22762 − 1.22407i 0.120537 − 0.992709i 0.164018 + 0.432479i
64.2 −2.29315 − 0.278439i −0.748511 + 0.663123i 3.23914 + 0.798376i 1.23177 + 2.34695i 1.90109 − 1.31223i −4.14543 − 1.57215i −2.88578 − 1.09443i 0.120537 − 0.992709i −2.17116 − 5.72488i
64.3 −2.27015 − 0.275646i −0.748511 + 0.663123i 3.13571 + 0.772883i −1.62219 − 3.09083i 1.88202 − 1.29906i 2.36774 + 0.897966i −2.62905 − 0.997067i 0.120537 − 0.992709i 2.83064 + 7.46378i
64.4 −1.39250 − 0.169080i −0.748511 + 0.663123i −0.0314082 − 0.00774143i −1.23718 − 2.35725i 1.15442 − 0.796842i −2.90606 − 1.10212i 2.66558 + 1.01092i 0.120537 − 0.992709i 1.32421 + 3.49165i
64.5 −1.16970 − 0.142027i −0.748511 + 0.663123i −0.593855 − 0.146372i 1.79511 + 3.42030i 0.969716 − 0.669346i 2.92200 + 1.10817i 2.87729 + 1.09121i 0.120537 − 0.992709i −1.61397 − 4.25568i
64.6 −0.915602 − 0.111174i −0.748511 + 0.663123i −1.11592 − 0.275048i 0.391488 + 0.745917i 0.759060 − 0.523942i −1.12887 − 0.428124i 2.71594 + 1.03002i 0.120537 − 0.992709i −0.275520 − 0.726487i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.h even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.p.a ✓ 168
169.h even 26 1 inner 507.2.p.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.p.a ✓ 168 1.a even 1 1 trivial
507.2.p.a ✓ 168 169.h even 26 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{168} - 20 T_{2}^{166} + 248 T_{2}^{164} - 26 T_{2}^{163} - 2380 T_{2}^{162} + 650 T_{2}^{161} + \cdots + 455625 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display