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Properties

Label 507.2.t.b
Level $507$
Weight $2$
Character orbit 507.t
Analytic conductor $4.048$
Analytic rank $0$
Dimension $360$
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(4,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.t (of order \(78\), degree \(24\), 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: \(360\)
Relative dimension: \(15\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9} + 6 q^{11} - 28 q^{12} + 6 q^{13} + 4 q^{14} - 6 q^{15} + 20 q^{16} - 6 q^{19} - 12 q^{20} - 28 q^{22} + 46 q^{23} - 6 q^{25} - 39 q^{26} + 30 q^{27} - 6 q^{28} + 43 q^{29} + 26 q^{31} - 195 q^{32} + 19 q^{33} + 65 q^{34} + 84 q^{35} - 14 q^{36} - 65 q^{38} - 2 q^{39} + 12 q^{41} - 128 q^{42} + 83 q^{43} - 39 q^{44} - 6 q^{45} - 20 q^{48} + 72 q^{49} - 52 q^{50} - 55 q^{52} + 49 q^{53} - 49 q^{55} - 2 q^{56} + 26 q^{57} + 26 q^{58} - 202 q^{59} - 182 q^{60} - 3 q^{61} - 65 q^{62} - 3 q^{63} - 14 q^{64} - 58 q^{65} + 48 q^{66} - 41 q^{67} + 139 q^{68} + 6 q^{69} - 60 q^{71} - 52 q^{73} - 269 q^{74} + 23 q^{75} - 14 q^{76} + 70 q^{77} - 65 q^{78} + 18 q^{79} + 492 q^{80} + 15 q^{81} - 65 q^{82} + 78 q^{83} - 6 q^{84} - 91 q^{85} - 169 q^{86} + 48 q^{87} - 522 q^{88} - 12 q^{89} + 373 q^{91} + 72 q^{92} - 3 q^{93} - 13 q^{94} - 110 q^{95} + 65 q^{96} - 121 q^{97} - 104 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.60202 − 0.104858i −0.278217 + 0.960518i 4.76600 + 0.384751i −0.705070 − 0.267398i 0.824645 − 2.47011i 0.861409 + 2.02180i −7.19059 − 0.873096i −0.845190 − 0.534466i 1.80657 + 0.769706i
4.2 −2.53957 − 0.102341i −0.278217 + 0.960518i 4.44542 + 0.358871i 1.66050 + 0.629744i 0.804853 − 2.41083i −2.05060 − 4.81293i −6.20652 − 0.753608i −0.845190 − 0.534466i −4.15250 − 1.76922i
4.3 −1.85726 − 0.0748449i −0.278217 + 0.960518i 1.45029 + 0.117079i −1.06783 − 0.404975i 0.588611 − 1.76311i 0.332955 + 0.781475i 1.00563 + 0.122106i −0.845190 − 0.534466i 1.95292 + 0.832063i
4.4 −1.79660 − 0.0724007i −0.278217 + 0.960518i 1.22903 + 0.0992178i 1.73729 + 0.658868i 0.569389 − 1.70553i 1.36244 + 3.19777i 1.36900 + 0.166227i −0.845190 − 0.534466i −3.07353 − 1.30951i
4.5 −1.38591 − 0.0558504i −0.278217 + 0.960518i −0.0758757 − 0.00612532i −1.08360 − 0.410955i 0.439231 − 1.31566i −0.389218 − 0.913528i 2.85867 + 0.347105i −0.845190 − 0.534466i 1.47882 + 0.630068i
4.6 −0.772377 − 0.0311257i −0.278217 + 0.960518i −1.39792 − 0.112852i 3.22103 + 1.22158i 0.244786 − 0.733223i −1.25005 − 2.93398i 2.61094 + 0.317026i −0.845190 − 0.534466i −2.44983 − 1.04377i
4.7 −0.218325 − 0.00879818i −0.278217 + 0.960518i −1.94593 − 0.157091i −3.83328 − 1.45377i 0.0691925 − 0.207257i 1.70758 + 4.00784i 0.857279 + 0.104093i −0.845190 − 0.534466i 0.824109 + 0.351120i
4.8 −0.208145 − 0.00838797i −0.278217 + 0.960518i −1.95026 − 0.157441i −1.53730 − 0.583019i 0.0659665 − 0.197594i −0.815402 − 1.91382i 0.818208 + 0.0993485i −0.845190 − 0.534466i 0.315091 + 0.134248i
4.9 0.805348 + 0.0324544i −0.278217 + 0.960518i −1.34598 − 0.108659i 3.20699 + 1.21625i −0.255235 + 0.764522i −0.483860 − 1.13566i −2.68071 − 0.325497i −0.845190 − 0.534466i 2.54327 + 1.08359i
4.10 0.989614 + 0.0398801i −0.278217 + 0.960518i −1.01577 − 0.0820013i −0.558145 − 0.211676i −0.313633 + 0.939447i −0.641263 − 1.50510i −2.96834 − 0.360422i −0.845190 − 0.534466i −0.543906 − 0.231737i
4.11 0.992569 + 0.0399992i −0.278217 + 0.960518i −1.00992 − 0.0815293i −0.620050 − 0.235154i −0.314570 + 0.942252i 1.02980 + 2.41704i −2.97142 − 0.360796i −0.845190 − 0.534466i −0.606036 − 0.258208i
4.12 1.81864 + 0.0732885i −0.278217 + 0.960518i 1.30855 + 0.105637i −3.76569 − 1.42814i −0.576371 + 1.72644i −1.29170 − 3.03172i −1.24165 − 0.150763i −0.845190 − 0.534466i −6.74376 − 2.87325i
4.13 1.92708 + 0.0776588i −0.278217 + 0.960518i 1.71410 + 0.138377i 3.62781 + 1.37585i −0.610741 + 1.82939i 1.17937 + 2.76808i −0.536697 − 0.0651668i −0.845190 − 0.534466i 6.88424 + 2.93310i
4.14 2.27767 + 0.0917871i −0.278217 + 0.960518i 3.18585 + 0.257189i −1.29798 − 0.492260i −0.721851 + 2.16221i 1.28355 + 3.01261i 2.70692 + 0.328679i −0.845190 − 0.534466i −2.91120 − 1.24035i
4.15 2.56929 + 0.103539i −0.278217 + 0.960518i 4.59701 + 0.371109i 1.84810 + 0.700893i −0.814272 + 2.43904i −1.21279 − 2.84653i 6.66737 + 0.809566i −0.845190 − 0.534466i 4.67574 + 1.99215i
10.1 −2.52917 − 0.516334i −0.987050 + 0.160411i 4.29016 + 1.82787i 0.764962 − 3.10357i 2.57925 + 0.103940i 4.06091 − 1.92693i −5.65796 − 3.90541i 0.948536 − 0.316668i −3.53720 + 7.45450i
10.2 −2.42455 − 0.494975i −0.987050 + 0.160411i 3.79347 + 1.61625i 0.0746454 − 0.302848i 2.47255 + 0.0996403i −4.01396 + 1.90465i −4.32440 − 2.98492i 0.948536 − 0.316668i −0.330883 + 0.697322i
10.3 −1.83778 − 0.375186i −0.987050 + 0.160411i 1.39672 + 0.595086i −0.959864 + 3.89432i 1.87417 + 0.0755263i 1.09680 − 0.520440i 0.743728 + 0.513358i 0.948536 − 0.316668i 3.22511 − 6.79678i
10.4 −1.49980 − 0.306186i −0.987050 + 0.160411i 0.315692 + 0.134504i −0.217058 + 0.880640i 1.52949 + 0.0616365i 0.786387 − 0.373145i 2.08725 + 1.44072i 0.948536 − 0.316668i 0.595184 − 1.25432i
10.5 −1.31845 − 0.269162i −0.987050 + 0.160411i −0.174110 − 0.0741814i 0.381569 − 1.54808i 1.34455 + 0.0541834i 1.87281 − 0.888658i 2.42447 + 1.67349i 0.948536 − 0.316668i −0.919764 + 1.93836i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.k even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.t.b ✓ 360
169.k even 78 1 inner 507.2.t.b ✓ 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.t.b ✓ 360 1.a even 1 1 trivial
507.2.t.b ✓ 360 169.k even 78 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{360} + 22 T_{2}^{358} + 186 T_{2}^{356} + 39 T_{2}^{355} + 197 T_{2}^{354} + \cdots + 20\!\cdots\!64 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display