Properties

Label 206.2.e.a
Level $206$
Weight $2$
Character orbit 206.e
Analytic conductor $1.645$
Analytic rank $0$
Dimension $16$
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(9,206)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(206, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("206.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 206 = 2 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 206.e (of order \(17\), degree \(16\), 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: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{34}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{34}^{10} q^{2} + ( - \zeta_{34}^{15} + \zeta_{34}^{12} - \zeta_{34}^{5}) q^{3} - \zeta_{34}^{3} q^{4} + ( - \zeta_{34}^{14} + \zeta_{34}^{12} - \zeta_{34}^{7} + \zeta_{34}^{5} + \zeta_{34}^{2} + 1) q^{5} + (\zeta_{34}^{15} - \zeta_{34}^{8} + \zeta_{34}^{5}) q^{6} + ( - \zeta_{34}^{14} + \zeta_{34}^{13} - \zeta_{34}^{12} + \zeta_{34}^{11} - \zeta_{34}^{8} + 2 \zeta_{34}^{5} + 2 \zeta_{34}^{3} - 1) q^{7} + \zeta_{34}^{13} q^{8} + ( - \zeta_{34}^{13} - \zeta_{34}^{7} - 2 \zeta_{34}^{3} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{34}^{10} q^{2} + ( - \zeta_{34}^{15} + \zeta_{34}^{12} - \zeta_{34}^{5}) q^{3} - \zeta_{34}^{3} q^{4} + ( - \zeta_{34}^{14} + \zeta_{34}^{12} - \zeta_{34}^{7} + \zeta_{34}^{5} + \zeta_{34}^{2} + 1) q^{5} + (\zeta_{34}^{15} - \zeta_{34}^{8} + \zeta_{34}^{5}) q^{6} + ( - \zeta_{34}^{14} + \zeta_{34}^{13} - \zeta_{34}^{12} + \zeta_{34}^{11} - \zeta_{34}^{8} + 2 \zeta_{34}^{5} + 2 \zeta_{34}^{3} - 1) q^{7} + \zeta_{34}^{13} q^{8} + ( - \zeta_{34}^{13} - \zeta_{34}^{7} - 2 \zeta_{34}^{3} + 2) q^{9} + ( - \zeta_{34}^{15} - \zeta_{34}^{12} - \zeta_{34}^{10} - \zeta_{34}^{7} + \zeta_{34}^{5} - 1) q^{10} + ( - \zeta_{34}^{15} + \zeta_{34}^{14} + \zeta_{34}^{13} - \zeta_{34}^{12} - \zeta_{34}^{8} + \zeta_{34}^{7} + \zeta_{34}^{6} - \zeta_{34}^{5}) q^{11} + ( - \zeta_{34}^{15} + \zeta_{34}^{8} - \zeta_{34}) q^{12} + (2 \zeta_{34}^{14} - \zeta_{34}^{13} + \zeta_{34}^{12} - 2 \zeta_{34}^{11} + 2 \zeta_{34}^{8} - \zeta_{34}^{7} - \zeta_{34}^{6} - \zeta_{34}^{5} + \cdots + 2) q^{13} + \cdots + ( - \zeta_{34}^{15} + 3 \zeta_{34}^{14} - \zeta_{34}^{13} + \zeta_{34}^{12} - \zeta_{34}^{11} + \zeta_{34}^{10} - 3 \zeta_{34}^{9} + \zeta_{34}^{8} + \cdots + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - 3 q^{3} - q^{4} + 15 q^{5} + 3 q^{6} - 7 q^{7} + q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{2} - 3 q^{3} - q^{4} + 15 q^{5} + 3 q^{6} - 7 q^{7} + q^{8} + 28 q^{9} - 15 q^{10} - 3 q^{12} + 22 q^{13} - 10 q^{14} + 11 q^{15} - q^{16} - 15 q^{17} - 11 q^{18} + 32 q^{19} - 19 q^{20} - 21 q^{21} + 17 q^{22} + 5 q^{23} + 3 q^{24} - 16 q^{25} - 5 q^{26} - 9 q^{27} - 7 q^{28} - 6 q^{29} - 28 q^{30} + 4 q^{31} + q^{32} - 2 q^{34} + 20 q^{35} - 6 q^{36} - 4 q^{37} - 32 q^{38} - 19 q^{39} + 2 q^{40} - 35 q^{41} - 13 q^{42} - q^{43} + 17 q^{44} - 12 q^{45} + 12 q^{46} - 18 q^{47} - 3 q^{48} + 60 q^{49} - 18 q^{50} - 45 q^{51} + 22 q^{52} - 84 q^{53} + 9 q^{54} + 51 q^{55} - 10 q^{56} - 6 q^{57} + 40 q^{58} + 13 q^{59} + 11 q^{60} + 28 q^{61} + 13 q^{62} - 8 q^{63} - q^{64} - 7 q^{65} + 17 q^{66} - 18 q^{67} + 2 q^{68} - 2 q^{69} + 48 q^{70} - 23 q^{71} + 6 q^{72} + 16 q^{73} + 4 q^{74} - 14 q^{75} - 19 q^{76} - 34 q^{77} + 36 q^{78} + 44 q^{79} - 2 q^{80} + 8 q^{81} + 35 q^{82} - 77 q^{83} + 30 q^{84} + 4 q^{85} - 16 q^{86} - 69 q^{87} - 16 q^{89} - 39 q^{90} - 33 q^{91} + 5 q^{92} - 22 q^{93} - 50 q^{94} - 4 q^{95} + 3 q^{96} + 37 q^{97} + 42 q^{98} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/206\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(\zeta_{34}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.739009 + 0.673696i
0.602635 0.798017i
−0.739009 0.673696i
−0.445738 0.895163i
0.850217 + 0.526432i
−0.0922684 + 0.995734i
0.982973 0.183750i
0.602635 + 0.798017i
−0.0922684 0.995734i
0.850217 0.526432i
−0.932472 0.361242i
0.273663 0.961826i
0.273663 + 0.961826i
0.982973 + 0.183750i
−0.445738 + 0.895163i
−0.932472 + 0.361242i
−0.445738 + 0.895163i −1.60817 + 0.995734i −0.602635 0.798017i 2.14064 1.95145i −0.174523 1.88341i 3.68272 + 0.688420i 0.982973 0.183750i 0.257496 0.517122i 0.792703 + 2.78606i
13.1 0.982973 + 0.183750i −0.0891263 + 0.961826i 0.932472 + 0.361242i −1.18911 + 1.57463i −0.264344 + 0.929072i −4.40875 + 2.72978i 0.850217 + 0.526432i 2.03175 + 0.379801i −1.45820 + 1.32932i
23.1 −0.445738 0.895163i −1.60817 0.995734i −0.602635 + 0.798017i 2.14064 + 1.95145i −0.174523 + 1.88341i 3.68272 0.688420i 0.982973 + 0.183750i 0.257496 + 0.517122i 0.792703 2.78606i
61.1 −0.0922684 + 0.995734i 0.875383 0.798017i −0.982973 0.183750i 1.47261 + 2.95739i 0.713843 + 0.945281i −0.0766996 0.269571i 0.273663 0.961826i −0.147341 + 1.59006i −3.08065 + 1.19345i
79.1 −0.739009 + 0.673696i 2.31068 0.895163i 0.0922684 0.995734i 2.09248 + 1.29561i −1.10455 + 2.21823i −2.21495 + 2.93307i 0.602635 + 0.798017i 2.32091 2.11579i −2.41921 + 0.452228i
81.1 0.602635 + 0.798017i −0.0914964 + 0.183750i −0.273663 + 0.961826i −0.311945 + 3.36642i −0.201774 + 0.0377181i −2.33749 0.905548i −0.932472 + 0.361242i 1.78251 + 2.36043i −2.87445 + 1.77978i
93.1 0.273663 + 0.961826i −0.272797 + 0.361242i −0.850217 + 0.526432i 2.50903 0.469018i −0.422106 0.163525i 2.08154 1.89757i −0.739009 0.673696i 0.764912 + 2.68839i 1.13774 + 2.28489i
111.1 0.982973 0.183750i −0.0891263 0.961826i 0.932472 0.361242i −1.18911 1.57463i −0.264344 0.929072i −4.40875 2.72978i 0.850217 0.526432i 2.03175 0.379801i −1.45820 1.32932i
117.1 0.602635 0.798017i −0.0914964 0.183750i −0.273663 0.961826i −0.311945 3.36642i −0.201774 0.0377181i −2.33749 + 0.905548i −0.932472 0.361242i 1.78251 2.36043i −2.87445 1.77978i
133.1 −0.739009 0.673696i 2.31068 + 0.895163i 0.0922684 + 0.995734i 2.09248 1.29561i −1.10455 2.21823i −2.21495 2.93307i 0.602635 0.798017i 2.32091 + 2.11579i −2.41921 0.452228i
137.1 0.850217 + 0.526432i 0.191683 0.673696i 0.445738 + 0.895163i 0.443053 + 0.171640i 0.517627 0.471880i −0.0228867 0.246987i −0.0922684 + 0.995734i 2.13353 + 1.32102i 0.286335 + 0.379169i
167.1 −0.932472 + 0.361242i −2.81616 + 0.526432i 0.739009 0.673696i 0.343246 1.20639i 2.43582 1.50820i −0.203480 0.408642i −0.445738 + 0.895163i 4.85623 1.88131i 0.115729 + 1.24892i
169.1 −0.932472 0.361242i −2.81616 0.526432i 0.739009 + 0.673696i 0.343246 + 1.20639i 2.43582 + 1.50820i −0.203480 + 0.408642i −0.445738 0.895163i 4.85623 + 1.88131i 0.115729 1.24892i
175.1 0.273663 0.961826i −0.272797 0.361242i −0.850217 0.526432i 2.50903 + 0.469018i −0.422106 + 0.163525i 2.08154 + 1.89757i −0.739009 + 0.673696i 0.764912 2.68839i 1.13774 2.28489i
179.1 −0.0922684 0.995734i 0.875383 + 0.798017i −0.982973 + 0.183750i 1.47261 2.95739i 0.713843 0.945281i −0.0766996 + 0.269571i 0.273663 + 0.961826i −0.147341 1.59006i −3.08065 1.19345i
203.1 0.850217 0.526432i 0.191683 + 0.673696i 0.445738 0.895163i 0.443053 0.171640i 0.517627 + 0.471880i −0.0228867 + 0.246987i −0.0922684 0.995734i 2.13353 1.32102i 0.286335 0.379169i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.e even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 206.2.e.a 16
103.e even 17 1 inner 206.2.e.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
206.2.e.a 16 1.a even 1 1 trivial
206.2.e.a 16 103.e even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 3 T_{3}^{15} - 8 T_{3}^{14} - 24 T_{3}^{13} + 30 T_{3}^{12} + 90 T_{3}^{11} + 32 T_{3}^{10} + 28 T_{3}^{9} + 169 T_{3}^{8} - 3 T_{3}^{7} + 297 T_{3}^{6} + 92 T_{3}^{5} + 140 T_{3}^{4} + 63 T_{3}^{3} + 36 T_{3}^{2} + 6 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(206, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{15} + T^{14} - T^{13} + T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} - 8 T^{14} - 24 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} - 15 T^{15} + 123 T^{14} + \cdots + 57121 \) Copy content Toggle raw display
$7$ \( T^{16} + 7 T^{15} - 2 T^{14} - 184 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{16} + 17 T^{14} - 102 T^{13} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( T^{16} - 22 T^{15} + 263 T^{14} + \cdots + 5031049 \) Copy content Toggle raw display
$17$ \( T^{16} + 15 T^{15} + 157 T^{14} + \cdots + 21077281 \) Copy content Toggle raw display
$19$ \( T^{16} - 32 T^{15} + 531 T^{14} + \cdots + 33790969 \) Copy content Toggle raw display
$23$ \( T^{16} - 5 T^{15} + 42 T^{14} + \cdots + 5987809 \) Copy content Toggle raw display
$29$ \( T^{16} + 6 T^{15} + 87 T^{14} + \cdots + 329676649 \) Copy content Toggle raw display
$31$ \( T^{16} - 4 T^{15} + 16 T^{14} + \cdots + 47623801 \) Copy content Toggle raw display
$37$ \( T^{16} + 4 T^{15} - 35 T^{14} + \cdots + 26020201 \) Copy content Toggle raw display
$41$ \( T^{16} + 35 T^{15} + \cdots + 49807973329 \) Copy content Toggle raw display
$43$ \( T^{16} + T^{15} + \cdots + 1928284945129 \) Copy content Toggle raw display
$47$ \( (T^{8} + 9 T^{7} - 227 T^{6} + \cdots + 806957)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 132986631912289 \) Copy content Toggle raw display
$59$ \( T^{16} - 13 T^{15} + \cdots + 298232855449 \) Copy content Toggle raw display
$61$ \( T^{16} - 28 T^{15} + \cdots + 15851565409 \) Copy content Toggle raw display
$67$ \( T^{16} + 18 T^{15} + \cdots + 1248835305169 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 844940426843881 \) Copy content Toggle raw display
$73$ \( T^{16} - 16 T^{15} + \cdots + 82029049746121 \) Copy content Toggle raw display
$79$ \( T^{16} - 44 T^{15} + \cdots + 108222409 \) Copy content Toggle raw display
$83$ \( T^{16} + 77 T^{15} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 365135192637121 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 205995892438249 \) Copy content Toggle raw display
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