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Properties

Label 206.2.g.b
Level $206$
Weight $2$
Character orbit 206.g
Analytic conductor $1.645$
Analytic rank $0$
Dimension $128$
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] = [206,2,Mod(7,206)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(206, base_ring=CyclotomicField(102))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("206.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 206 = 2 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 206.g (of order \(51\), degree \(32\), 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: \(1.64491828164\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(4\) over \(\Q(\zeta_{51})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{51}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{5} - q^{6} - 4 q^{7} - 8 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{5} - q^{6} - 4 q^{7} - 8 q^{8} - 10 q^{9} + 22 q^{10} - 5 q^{11} - q^{12} - 12 q^{13} - 9 q^{14} + 33 q^{15} + 4 q^{16} + q^{17} + 5 q^{18} - 11 q^{19} - 11 q^{20} + 29 q^{21} - 7 q^{22} - 9 q^{23} - 15 q^{24} - 28 q^{25} - 11 q^{26} - 16 q^{27} - 4 q^{28} - 17 q^{29} - 32 q^{30} - 4 q^{31} + 4 q^{32} - 4 q^{33} - 2 q^{34} - 81 q^{35} - 29 q^{36} + 3 q^{37} - 11 q^{38} - 32 q^{39} + 6 q^{40} - 37 q^{41} + 44 q^{42} + 13 q^{43} - 5 q^{44} - 19 q^{45} - 4 q^{46} - 68 q^{47} + 33 q^{48} - 24 q^{49} - 45 q^{50} - 17 q^{51} + 6 q^{52} + 95 q^{53} + 8 q^{54} - 168 q^{55} - 4 q^{56} - 82 q^{57} - 34 q^{58} + 10 q^{59} - q^{60} - 83 q^{61} - 49 q^{62} - 86 q^{63} - 8 q^{64} - 42 q^{65} - 60 q^{66} - 54 q^{67} + q^{68} + 64 q^{69} + 21 q^{70} - 95 q^{71} - 10 q^{72} - 22 q^{73} - 10 q^{74} + 260 q^{75} - 12 q^{76} - 115 q^{77} - q^{78} - 24 q^{79} - 12 q^{80} - 32 q^{81} - 3 q^{82} - 50 q^{83} - 22 q^{84} + 87 q^{85} + 98 q^{86} + 47 q^{87} - 22 q^{88} + 7 q^{89} + 55 q^{90} + 327 q^{91} + 30 q^{92} - 22 q^{93} + 51 q^{94} - 34 q^{95} - q^{96} + 175 q^{97} + 129 q^{98} + 152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 0.650618 − 0.759405i −0.0570115 + 0.615253i −0.153392 − 0.988165i 2.84128 + 0.351827i 0.430133 + 0.443589i −2.63974 − 1.41751i −0.850217 − 0.526432i 2.57363 + 0.481096i 2.11577 − 1.92878i
7.2 0.650618 − 0.759405i −0.0511717 + 0.552231i −0.153392 − 0.988165i 0.780378 + 0.0966318i 0.386073 + 0.398151i 2.05313 + 1.10251i −0.850217 − 0.526432i 2.64658 + 0.494731i 0.581111 − 0.529753i
7.3 0.650618 − 0.759405i 0.164466 − 1.77488i −0.153392 − 0.988165i −3.25087 − 0.402545i −1.24084 − 1.27966i 0.654025 + 0.351204i −0.850217 − 0.526432i −0.174214 − 0.0325662i −2.42077 + 2.20682i
7.4 0.650618 − 0.759405i 0.258602 − 2.79076i −0.153392 − 0.988165i 1.39332 + 0.172531i −1.95106 − 2.01210i 0.743654 + 0.399334i −0.850217 − 0.526432i −4.77253 − 0.892141i 1.03754 − 0.945844i
15.1 −0.696134 − 0.717912i −1.93030 − 2.55613i −0.0307951 + 0.999526i 1.39661 + 3.96330i −0.491330 + 3.16520i −0.478437 + 2.18461i 0.739009 − 0.673696i −1.98676 + 6.98275i 1.87307 − 3.76163i
15.2 −0.696134 − 0.717912i −0.685235 − 0.907398i −0.0307951 + 0.999526i −0.325600 − 0.923987i −0.174416 + 1.12361i 0.204101 − 0.931954i 0.739009 − 0.673696i 0.467165 − 1.64191i −0.436680 + 0.876971i
15.3 −0.696134 − 0.717912i −0.243807 − 0.322852i −0.0307951 + 0.999526i 0.412164 + 1.16964i −0.0620573 + 0.399780i 0.334185 − 1.52594i 0.739009 − 0.673696i 0.776197 − 2.72805i 0.552774 − 1.11012i
15.4 −0.696134 − 0.717912i 1.19038 + 1.57632i −0.0307951 + 0.999526i −0.0309529 − 0.0878379i 0.302995 − 1.95192i −0.961845 + 4.39192i 0.739009 − 0.673696i −0.246793 + 0.867389i −0.0415125 + 0.0833684i
17.1 0.816197 + 0.577774i −0.921743 + 0.840279i 0.332355 + 0.943154i −0.663156 + 1.00080i −1.23781 + 0.153275i −0.879650 + 0.907169i −0.273663 + 0.961826i −0.133265 + 1.43816i −1.11950 + 0.433698i
17.2 0.816197 + 0.577774i −0.793565 + 0.723430i 0.332355 + 0.943154i 2.27017 − 3.42603i −1.06568 + 0.131960i 3.00468 − 3.09867i −0.273663 + 0.961826i −0.170411 + 1.83903i 3.83238 − 1.48467i
17.3 0.816197 + 0.577774i 1.43475 − 1.30795i 0.332355 + 0.943154i 1.32756 − 2.00349i 1.92674 − 0.238582i −2.62871 + 2.71095i −0.273663 + 0.961826i 0.0709768 − 0.765962i 2.24112 − 0.868214i
17.4 0.816197 + 0.577774i 1.60925 − 1.46703i 0.332355 + 0.943154i −1.10204 + 1.66315i 2.16108 − 0.267600i 1.95327 − 2.01438i −0.273663 + 0.961826i 0.160720 − 1.73444i −1.86041 + 0.720726i
19.1 −0.0307951 + 0.999526i −0.671191 + 2.35899i −0.998103 − 0.0615609i −1.25697 + 1.01148i −2.33720 − 0.743518i −1.74064 − 0.800822i 0.0922684 − 0.995734i −2.56370 − 1.58737i −0.972288 − 1.28752i
19.2 −0.0307951 + 0.999526i 0.127696 − 0.448803i −0.998103 − 0.0615609i −1.74281 + 1.40244i 0.444658 + 0.141456i 3.30796 + 1.52190i 0.0922684 − 0.995734i 2.36553 + 1.46468i −1.34810 − 1.78517i
19.3 −0.0307951 + 0.999526i 0.139902 − 0.491704i −0.998103 − 0.0615609i 1.74465 − 1.40391i 0.487163 + 0.154978i −0.989152 − 0.455082i 0.0922684 − 0.995734i 2.32845 + 1.44172i 1.34952 + 1.78705i
19.4 −0.0307951 + 0.999526i 0.769682 − 2.70515i −0.998103 − 0.0615609i 0.492201 − 0.396073i 2.68017 + 0.852622i 0.929320 + 0.427555i 0.0922684 − 0.995734i −4.17478 − 2.58491i 0.380727 + 0.504165i
25.1 −0.908465 + 0.417960i −2.24067 + 2.04264i 0.650618 − 0.759405i −1.63816 − 0.101038i 1.18183 − 2.79218i −2.08076 − 0.523331i −0.273663 + 0.961826i 0.571417 − 6.16657i 1.53044 − 0.592896i
25.2 −0.908465 + 0.417960i −1.27707 + 1.16420i 0.650618 − 0.759405i 1.06987 + 0.0659870i 0.673584 − 1.59140i 2.88569 + 0.725780i −0.273663 + 0.961826i −0.00126556 + 0.0136575i −0.999515 + 0.387214i
25.3 −0.908465 + 0.417960i 0.132447 − 0.120741i 0.650618 − 0.759405i −3.86068 − 0.238118i −0.0698582 + 0.165046i 3.41155 + 0.858040i −0.273663 + 0.961826i −0.273841 + 2.95522i 3.60682 − 1.39729i
25.4 −0.908465 + 0.417960i 1.49759 − 1.36524i 0.650618 − 0.759405i 1.11765 + 0.0689340i −0.789897 + 1.86621i 1.09409 + 0.275176i −0.273663 + 0.961826i 0.102110 − 1.10194i −1.04415 + 0.404507i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.g even 51 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 206.2.g.b ✓ 128
103.g even 51 1 inner 206.2.g.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
206.2.g.b ✓ 128 1.a even 1 1 trivial
206.2.g.b ✓ 128 103.g even 51 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{128} - 2 T_{3}^{127} + 19 T_{3}^{126} - 34 T_{3}^{125} + 243 T_{3}^{124} - 440 T_{3}^{123} + \cdots + 40\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(206, [\chi])\). Copy content Toggle raw display