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Properties

Label 206.2.g.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 4 q^{2} + 2 q^{3} + 4 q^{4} + q^{6} + 8 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 4 q^{2} + 2 q^{3} + 4 q^{4} + q^{6} + 8 q^{8} - 10 q^{9} - 34 q^{10} - q^{11} - q^{12} - 8 q^{13} - 17 q^{14} - 29 q^{15} + 4 q^{16} + q^{17} - 5 q^{18} - 7 q^{19} + 17 q^{20} - 41 q^{21} + 15 q^{22} - 45 q^{23} + 15 q^{24} - 28 q^{25} + 13 q^{26} + 20 q^{27} + 11 q^{29} - 24 q^{30} - 4 q^{31} - 4 q^{32} - 8 q^{33} + 2 q^{34} + 79 q^{35} - 29 q^{36} - 77 q^{37} + 7 q^{38} - 28 q^{39} + 43 q^{41} - 48 q^{42} - 5 q^{43} - q^{44} - 71 q^{45} - 14 q^{46} + 68 q^{47} - 35 q^{48} - 52 q^{49} + 45 q^{50} + 29 q^{51} + 4 q^{52} - 105 q^{53} + 10 q^{54} + 66 q^{55} + 18 q^{57} - 28 q^{58} - 74 q^{59} + 5 q^{60} + 75 q^{61} + 49 q^{62} + 40 q^{63} - 8 q^{64} + 58 q^{65} - 84 q^{66} - 50 q^{67} + q^{68} - 112 q^{69} - 45 q^{70} - 29 q^{71} + 10 q^{72} - 38 q^{73} + 38 q^{74} - 282 q^{75} - 20 q^{76} - 9 q^{77} + 3 q^{78} + 12 q^{79} + 48 q^{81} - 9 q^{82} + 16 q^{83} - 24 q^{84} + 43 q^{85} - 12 q^{86} + 71 q^{87} - 16 q^{88} + 71 q^{89} + 113 q^{90} - 129 q^{91} + 48 q^{92} - 34 q^{93} + 85 q^{94} - 10 q^{95} + q^{96} + 27 q^{97} + 171 q^{98} + 170 q^{99}+O(q^{100}) \) 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 −0.188135 + 2.03030i −0.153392 − 0.988165i 3.65821 + 0.452985i −1.41942 − 1.46382i 1.40760 + 0.755866i 0.850217 + 0.526432i −1.13781 − 0.212694i −2.72410 + 2.48335i
7.2 −0.650618 + 0.759405i −0.0165711 + 0.178831i −0.153392 − 0.988165i −1.97150 − 0.244124i −0.125024 − 0.128935i −4.38427 − 2.35431i 0.850217 + 0.526432i 2.91721 + 0.545322i 1.46808 − 1.33833i
7.3 −0.650618 + 0.759405i 0.00288780 − 0.0311643i −0.153392 − 0.988165i −0.458869 − 0.0568203i 0.0217875 + 0.0224691i 0.790516 + 0.424498i 0.850217 + 0.526432i 2.94796 + 0.551069i 0.341698 − 0.311499i
7.4 −0.650618 + 0.759405i 0.253210 − 2.73257i −0.153392 − 0.988165i 1.92651 + 0.238554i 1.91038 + 1.97015i 0.120173 + 0.0645316i 0.850217 + 0.526432i −4.45389 − 0.832576i −1.43458 + 1.30780i
15.1 0.696134 + 0.717912i −1.18039 − 1.56308i −0.0307951 + 0.999526i −0.468040 − 1.32820i 0.300450 − 1.93553i 0.504687 − 2.30447i −0.739009 + 0.673696i −0.228932 + 0.804612i 0.627713 − 1.26062i
15.2 0.696134 + 0.717912i −0.641243 − 0.849143i −0.0307951 + 0.999526i 0.735324 + 2.08670i 0.163219 − 1.05147i −0.781854 + 3.57005i −0.739009 + 0.673696i 0.511138 − 1.79646i −0.986182 + 1.98052i
15.3 0.696134 + 0.717912i 0.990432 + 1.31154i −0.0307951 + 0.999526i 0.593749 + 1.68494i −0.252100 + 1.62405i 0.588630 − 2.68777i −0.739009 + 0.673696i 0.0817964 − 0.287484i −0.796307 + 1.59920i
15.4 0.696134 + 0.717912i 1.69900 + 2.24984i −0.0307951 + 0.999526i −1.05597 − 2.99663i −0.432456 + 2.78593i −0.520422 + 2.37632i −0.739009 + 0.673696i −1.35420 + 4.75951i 1.41622 − 2.84415i
17.1 −0.816197 − 0.577774i −1.71328 + 1.56186i 0.332355 + 0.943154i 0.494012 − 0.745540i 2.30077 − 0.284898i −0.728853 + 0.751654i 0.273663 − 0.961826i 0.219112 − 2.36460i −0.833965 + 0.323080i
17.2 −0.816197 − 0.577774i 0.104496 − 0.0952611i 0.332355 + 0.943154i 0.838829 − 1.26592i −0.140329 + 0.0173765i 0.949990 − 0.979709i 0.273663 − 0.961826i −0.274960 + 2.96729i −1.41606 + 0.548586i
17.3 −0.816197 − 0.577774i 0.134333 − 0.122461i 0.332355 + 0.943154i −1.95246 + 2.94656i −0.180397 + 0.0223380i 2.10181 − 2.16756i 0.273663 − 0.961826i −0.273756 + 2.95430i 3.29604 − 1.27689i
17.4 −0.816197 − 0.577774i 1.77856 − 1.62137i 0.332355 + 0.943154i 1.95207 − 2.94597i −2.38844 + 0.295753i −1.03328 + 1.06560i 0.273663 − 0.961826i 0.257625 − 2.78022i −3.29538 + 1.27664i
19.1 0.0307951 − 0.999526i −0.441469 + 1.55160i −0.998103 − 0.0615609i −1.88101 + 1.51364i 1.53727 + 0.489041i −2.58746 − 1.19042i −0.0922684 + 0.995734i 0.338080 + 0.209330i 1.45500 + 1.92673i
19.2 0.0307951 − 0.999526i −0.317047 + 1.11430i −0.998103 − 0.0615609i 1.00587 − 0.809423i 1.10401 + 0.351212i 2.01960 + 0.929163i −0.0922684 + 0.995734i 1.40949 + 0.872722i −0.778063 − 1.03032i
19.3 0.0307951 − 0.999526i 0.559183 − 1.96533i −0.998103 − 0.0615609i 0.131391 − 0.105730i −1.94717 − 0.619440i −4.37686 − 2.01367i −0.0922684 + 0.995734i −0.999168 − 0.618659i −0.101633 − 0.134584i
19.4 0.0307951 − 0.999526i 0.686066 − 2.41127i −0.998103 − 0.0615609i 1.82652 − 1.46979i −2.38900 − 0.759996i 3.60263 + 1.65747i −0.0922684 + 0.995734i −2.79290 − 1.72929i −1.41285 − 1.87091i
25.1 0.908465 − 0.417960i −2.14905 + 1.95911i 0.650618 − 0.759405i −3.02475 − 0.186560i −1.13350 + 2.67800i −2.11755 − 0.532584i 0.273663 − 0.961826i 0.503465 − 5.43326i −2.82585 + 1.09474i
25.2 0.908465 − 0.417960i −1.08084 + 0.985320i 0.650618 − 0.759405i −0.0593355 − 0.00365969i −0.570085 + 1.34688i 3.96201 + 0.996485i 0.273663 − 0.961826i −0.0794356 + 0.857247i −0.0554339 + 0.0214752i
25.3 0.908465 − 0.417960i 0.460314 − 0.419632i 0.650618 − 0.759405i 2.64045 + 0.162857i 0.242790 − 0.573614i −1.90075 − 0.478059i 0.273663 − 0.961826i −0.241007 + 2.60088i 2.46683 − 0.955653i
25.4 0.908465 − 0.417960i 1.70685 − 1.55600i 0.650618 − 0.759405i −1.96406 − 0.121139i 0.900267 − 2.12697i −1.00679 − 0.253217i 0.273663 − 0.961826i 0.215397 − 2.32450i −1.83492 + 0.710850i
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:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{128} - 2 T_{3}^{127} + 19 T_{3}^{126} - 46 T_{3}^{125} + 247 T_{3}^{124} - 568 T_{3}^{123} + \cdots + 174380572921 \) acting on \(S_{2}^{\mathrm{new}}(206, [\chi])\). Copy content Toggle raw display