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Properties

Label 507.2.q.a
Level $507$
Weight $2$
Character orbit 507.q
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(16,507)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.q (of order \(39\), 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 \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q + q^{2} - 15 q^{3} + 15 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 6 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 360 q + q^{2} - 15 q^{3} + 15 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 6 q^{8} + 15 q^{9} - q^{10} - 2 q^{11} + 30 q^{12} - 19 q^{13} + q^{15} + 15 q^{16} - 7 q^{17} - 2 q^{18} - 6 q^{19} + q^{20} + 4 q^{21} + 22 q^{22} + 46 q^{23} + 3 q^{24} + 28 q^{25} + 34 q^{26} + 30 q^{27} - 2 q^{28} + 36 q^{29} + q^{30} - 34 q^{31} - 190 q^{32} + 15 q^{33} - 51 q^{34} - 80 q^{35} + 15 q^{36} + q^{37} - 53 q^{38} + 5 q^{39} - 45 q^{40} + 9 q^{41} + 130 q^{42} + 78 q^{43} + 35 q^{44} - q^{45} - 318 q^{46} - 12 q^{47} - 15 q^{48} - 71 q^{49} + 48 q^{50} - 14 q^{51} - 63 q^{52} - 26 q^{53} - q^{54} - 61 q^{55} - 4 q^{56} - 38 q^{57} - 27 q^{58} + 208 q^{59} - 180 q^{60} - 3 q^{61} - 61 q^{62} + 2 q^{63} + 140 q^{64} + 45 q^{65} - 60 q^{66} - 80 q^{67} - 140 q^{68} + 6 q^{69} - 100 q^{70} + 84 q^{71} - 3 q^{72} + 134 q^{73} - 151 q^{74} - 12 q^{75} + 32 q^{76} + 82 q^{77} - 63 q^{78} + 4 q^{79} - 155 q^{80} + 15 q^{81} - 56 q^{82} + 106 q^{83} + 2 q^{84} - 19 q^{85} + q^{86} - 49 q^{87} + 8 q^{88} - 14 q^{89} + 2 q^{90} - 368 q^{91} + 36 q^{92} - 4 q^{93} - 59 q^{94} + 104 q^{95} + 75 q^{96} - 28 q^{97} + 101 q^{98} + 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.74572 − 0.221657i 0.845190 + 0.534466i 5.51573 + 0.896393i 0.154238 + 0.136643i −2.20218 − 1.65483i −1.20996 + 1.25970i −9.59673 − 2.36538i 0.428693 + 0.903450i −0.393207 − 0.409372i
16.2 −2.25005 − 0.181643i 0.845190 + 0.534466i 3.05563 + 0.496588i −2.62979 − 2.32979i −1.80464 − 1.35610i 0.735447 − 0.765681i −2.40157 − 0.591933i 0.428693 + 0.903450i 5.49396 + 5.71982i
16.3 −2.21324 − 0.178671i 0.845190 + 0.534466i 2.89239 + 0.470059i −0.0344706 − 0.0305383i −1.77511 − 1.33391i 3.31537 − 3.45167i −2.00573 − 0.494367i 0.428693 + 0.903450i 0.0708354 + 0.0737474i
16.4 −1.75204 − 0.141439i 0.845190 + 0.534466i 1.07555 + 0.174793i 0.0960286 + 0.0850739i −1.40521 − 1.05595i −1.50147 + 1.56320i 1.55365 + 0.382940i 0.428693 + 0.903450i −0.156213 − 0.162635i
16.5 −1.46442 − 0.118220i 0.845190 + 0.534466i 0.156450 + 0.0254257i 3.08910 + 2.73671i −1.17453 − 0.882601i −0.144485 + 0.150425i 2.62688 + 0.647468i 0.428693 + 0.903450i −4.20021 − 4.37289i
16.6 −0.779114 − 0.0628966i 0.845190 + 0.534466i −1.37104 − 0.222815i −2.45133 − 2.17169i −0.624884 − 0.469570i −1.20443 + 1.25394i 2.57205 + 0.633954i 0.428693 + 0.903450i 1.77327 + 1.84617i
16.7 −0.676006 − 0.0545728i 0.845190 + 0.534466i −1.52010 − 0.247040i −1.01786 − 0.901745i −0.542186 − 0.407426i −0.418694 + 0.435907i 2.33111 + 0.574566i 0.428693 + 0.903450i 0.638868 + 0.665132i
16.8 −0.164145 − 0.0132511i 0.845190 + 0.534466i −1.94733 − 0.316472i 2.22303 + 1.96943i −0.131651 − 0.0989295i −0.158039 + 0.164536i 0.635238 + 0.156572i 0.428693 + 0.903450i −0.338802 − 0.352730i
16.9 0.350694 + 0.0283109i 0.845190 + 0.534466i −1.85192 − 0.300966i 0.194020 + 0.171887i 0.281272 + 0.211362i 2.30124 − 2.39584i −1.32416 − 0.326375i 0.428693 + 0.903450i 0.0631753 + 0.0657724i
16.10 1.03364 + 0.0834440i 0.845190 + 0.534466i −0.912654 − 0.148321i −1.98811 − 1.76131i 0.829023 + 0.622971i −2.70082 + 2.81186i −2.94471 − 0.725807i 0.428693 + 0.903450i −1.90802 − 1.98646i
16.11 1.26947 + 0.102482i 0.845190 + 0.534466i −0.373055 − 0.0606273i −2.62216 − 2.32303i 1.01817 + 0.765104i 1.77636 − 1.84939i −2.94055 − 0.724780i 0.428693 + 0.903450i −3.09068 − 3.21774i
16.12 1.66608 + 0.134500i 0.845190 + 0.534466i 0.783635 + 0.127353i 2.48154 + 2.19845i 1.33627 + 1.00414i 2.05375 − 2.13818i −1.95739 − 0.482453i 0.428693 + 0.903450i 3.83875 + 3.99656i
16.13 1.84825 + 0.149206i 0.845190 + 0.534466i 1.41967 + 0.230719i 1.63990 + 1.45283i 1.48238 + 1.11393i −1.92076 + 1.99973i −1.01128 − 0.249259i 0.428693 + 0.903450i 2.81418 + 2.92987i
16.14 2.40692 + 0.194306i 0.845190 + 0.534466i 3.78139 + 0.614535i −1.78205 − 1.57876i 1.93045 + 1.45064i 1.67721 − 1.74616i 4.29291 + 1.05811i 0.428693 + 0.903450i −3.98249 − 4.14621i
16.15 2.47293 + 0.199635i 0.845190 + 0.534466i 4.10141 + 0.666543i −0.0128422 − 0.0113772i 1.98339 + 1.49042i −1.37396 + 1.43044i 5.19165 + 1.27963i 0.428693 + 0.903450i −0.0294866 − 0.0306988i
55.1 −1.73581 + 2.12599i 0.996757 + 0.0804666i −1.10672 − 5.42106i −0.134154 − 1.10485i −1.90126 + 1.97942i −3.44326 + 2.17739i 8.58570 + 4.50612i 0.987050 + 0.160411i 2.58177 + 1.63261i
55.2 −1.58863 + 1.94572i 0.996757 + 0.0804666i −0.862027 − 4.22249i 0.458750 + 3.77814i −1.74005 + 1.81158i 2.22525 − 1.40716i 5.13689 + 2.69605i 0.987050 + 0.160411i −8.07999 − 5.10948i
55.3 −1.28348 + 1.57198i 0.996757 + 0.0804666i −0.423741 − 2.07562i −0.317374 − 2.61381i −1.40581 + 1.46360i 0.313349 − 0.198150i 0.212820 + 0.111697i 0.987050 + 0.160411i 4.51619 + 2.85586i
55.4 −1.18444 + 1.45068i 0.996757 + 0.0804666i −0.301514 − 1.47691i 0.266003 + 2.19074i −1.29733 + 1.35067i −0.690655 + 0.436744i −0.816908 − 0.428746i 0.987050 + 0.160411i −3.49312 − 2.20892i
55.5 −0.826374 + 1.01212i 0.996757 + 0.0804666i 0.0585493 + 0.286793i −0.0644718 − 0.530973i −0.905137 + 0.942347i 0.386707 − 0.244538i −2.65259 − 1.39219i 0.987050 + 0.160411i 0.590688 + 0.373529i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.i even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.q.a ✓ 360
169.i even 39 1 inner 507.2.q.a ✓ 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.q.a ✓ 360 1.a even 1 1 trivial
507.2.q.a ✓ 360 169.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{360} - T_{2}^{359} - 22 T_{2}^{358} + 23 T_{2}^{357} + 185 T_{2}^{356} - 169 T_{2}^{355} + \cdots + 90\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display