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Properties

Label 209.2.g.a
Level $209$
Weight $2$
Character orbit 209.g
Analytic conductor $1.669$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,2,Mod(65,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.g (of order \(6\), degree \(2\), 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: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.1132927402587890625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 4 x^{14} - 9 x^{13} + 12 x^{12} + 12 x^{11} + 8 x^{10} - 54 x^{9} - 19 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_1) q^{2} + (\beta_{6} - 1) q^{3} + (\beta_{15} + \beta_{13} + \beta_{9} + \cdots - 1) q^{4}+ \cdots + (\beta_{15} - \beta_{7} - 2 \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_1) q^{2} + (\beta_{6} - 1) q^{3} + (\beta_{15} + \beta_{13} + \beta_{9} + \cdots - 1) q^{4}+ \cdots + (\beta_{15} - 2 \beta_{14} + 2 \beta_{13} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{3} - 10 q^{4} + 8 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{3} - 10 q^{4} + 8 q^{5} + 12 q^{9} + 16 q^{11} + 24 q^{14} - 12 q^{15} - 38 q^{16} - 20 q^{20} - 30 q^{22} - 4 q^{23} + 32 q^{25} + 20 q^{26} - 12 q^{33} - 6 q^{34} + 20 q^{36} + 14 q^{38} + 8 q^{42} - 20 q^{44} + 24 q^{45} - 4 q^{47} - 18 q^{48} + 8 q^{49} + 8 q^{55} - 156 q^{58} + 12 q^{59} + 30 q^{60} + 8 q^{64} + 50 q^{66} - 96 q^{67} + 24 q^{70} + 48 q^{71} + 40 q^{77} - 30 q^{78} + 38 q^{80} - 40 q^{81} + 42 q^{82} + 114 q^{86} + 12 q^{89} - 60 q^{91} + 10 q^{92} - 56 q^{93} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 4 x^{14} - 9 x^{13} + 12 x^{12} + 12 x^{11} + 8 x^{10} - 54 x^{9} - 19 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 19847 \nu^{15} + 76377 \nu^{14} - 108454 \nu^{13} + 171765 \nu^{12} - 283119 \nu^{11} + \cdots + 135736 ) / 171100 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21208 \nu^{15} + 79218 \nu^{14} - 115771 \nu^{13} + 202015 \nu^{12} - 332721 \nu^{11} + \cdots - 69666 ) / 171100 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22643 \nu^{15} + 11713 \nu^{14} + 129284 \nu^{13} - 207225 \nu^{12} + 553139 \nu^{11} + \cdots - 205276 ) / 171100 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34192 \nu^{15} + 82317 \nu^{14} - 98349 \nu^{13} + 322290 \nu^{12} - 430499 \nu^{11} + \cdots - 19679 ) / 171100 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 55054 \nu^{15} - 141874 \nu^{14} + 148803 \nu^{13} - 408215 \nu^{12} + 469003 \nu^{11} + \cdots + 191488 ) / 171100 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 56068 \nu^{15} - 170743 \nu^{14} + 181331 \nu^{13} - 331410 \nu^{12} + 391521 \nu^{11} + \cdots + 192341 ) / 171100 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 83098 \nu^{15} - 379863 \nu^{14} + 830371 \nu^{13} - 1635885 \nu^{12} + 2762636 \nu^{11} + \cdots + 11676 ) / 171100 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 103957 \nu^{15} - 363917 \nu^{14} + 596044 \nu^{13} - 1244025 \nu^{12} + 1924749 \nu^{11} + \cdots - 33776 ) / 171100 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26616 \nu^{15} + 85866 \nu^{14} - 126177 \nu^{13} + 266925 \nu^{12} - 373527 \nu^{11} + \cdots - 83737 ) / 42775 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 478 \nu^{15} - 1926 \nu^{14} + 3618 \nu^{13} - 7137 \nu^{12} + 11727 \nu^{11} - 3321 \nu^{10} + \cdots - 531 ) / 590 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 147732 \nu^{15} + 457072 \nu^{14} - 598939 \nu^{13} + 1248325 \nu^{12} - 1629909 \nu^{11} + \cdots - 308024 ) / 171100 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 178467 \nu^{15} - 607507 \nu^{14} + 928474 \nu^{13} - 1893075 \nu^{12} + 2797479 \nu^{11} + \cdots + 87394 ) / 171100 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 182146 \nu^{15} - 631401 \nu^{14} + 1001137 \nu^{13} - 2035935 \nu^{12} + 3031272 \nu^{11} + \cdots + 224612 ) / 171100 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 184371 \nu^{15} - 675361 \nu^{14} + 1153017 \nu^{13} - 2304400 \nu^{12} + 3548042 \nu^{11} + \cdots + 464422 ) / 171100 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19914 \nu^{15} - 61635 \nu^{14} + 85953 \nu^{13} - 189891 \nu^{12} + 261750 \nu^{11} + 206304 \nu^{10} + \cdots + 57702 ) / 17110 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + \beta_{9} + \beta_{8} + 2 \beta_{5} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 3 \beta_{9} + \cdots - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 7 \beta_{14} + 4 \beta_{11} - 7 \beta_{9} + \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + \cdots - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{15} + 9 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} - 2 \beta_{11} - 8 \beta_{10} + 11 \beta_{9} + \cdots + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{15} + 6\beta_{14} - 6\beta_{12} - 6\beta_{11} + 9\beta_{9} + 3\beta_{8} + 12\beta_{5} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7 \beta_{15} - 28 \beta_{14} - 23 \beta_{12} + 35 \beta_{11} + 35 \beta_{10} - 39 \beta_{9} - 16 \beta_{8} + \cdots - 93 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 36 \beta_{15} - 29 \beta_{14} + 58 \beta_{13} - 7 \beta_{12} + 141 \beta_{11} + 7 \beta_{10} + \cdots - 127 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 89 \beta_{15} + 286 \beta_{14} + 134 \beta_{13} - 67 \beta_{12} - 67 \beta_{11} - 67 \beta_{10} + \cdots + 67 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 89 \beta_{15} + 324 \beta_{14} - 101 \beta_{12} - 235 \beta_{11} + 235 \beta_{10} + 353 \beta_{9} + \cdots - 369 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 180 \beta_{15} - 228 \beta_{14} + 114 \beta_{12} + 258 \beta_{11} + 180 \beta_{10} - 413 \beta_{9} + \cdots - 413 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1044 \beta_{15} - 509 \beta_{14} + 754 \beta_{13} + 1553 \beta_{12} + 1553 \beta_{11} - 377 \beta_{10} + \cdots + 377 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1253 \beta_{15} + 4059 \beta_{14} + 1630 \beta_{12} - 4318 \beta_{11} - 1630 \beta_{10} + 5271 \beta_{9} + \cdots + 6070 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 756 \beta_{15} - 2597 \beta_{14} - 5194 \beta_{13} + 3353 \beta_{12} - 5859 \beta_{11} + 3353 \beta_{10} + \cdots + 847 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 13060 \beta_{15} - 27373 \beta_{14} - 6706 \beta_{13} + 14313 \beta_{12} + 14313 \beta_{11} + \cdots - 3353 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1080 \beta_{14} + 4635 \beta_{12} - 1080 \beta_{11} - 7375 \beta_{10} + 4635 \beta_{8} + 10350 \beta_{3} + \cdots + 16443 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(134\)
\(\chi(n)\) \(-\beta_{9}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.32605 + 0.281861i
0.0498873 + 0.474646i
−0.721521 − 0.153364i
−0.435999 + 0.194119i
−0.219018 − 2.08382i
−0.907123 − 1.00746i
1.91415 − 0.852233i
0.493578 + 0.548174i
1.32605 − 0.281861i
0.0498873 − 0.474646i
−0.721521 + 0.153364i
−0.435999 − 0.194119i
−0.219018 + 2.08382i
−0.907123 + 1.00746i
1.91415 + 0.852233i
0.493578 − 0.548174i
−1.34697 + 2.33302i 0.315575 + 0.182197i −2.62864 − 4.55295i 0.500000 − 0.866025i −0.850137 + 0.490827i − 3.29655i 8.77493 −1.43361 − 2.48308i 1.34697 + 2.33302i
65.2 −1.15218 + 1.99563i −2.86886 − 1.65633i −1.65502 − 2.86657i 0.500000 − 0.866025i 6.61085 − 3.81678i − 0.161926i 3.01878 3.98689 + 6.90550i 1.15218 + 1.99563i
65.3 −0.300310 + 0.520153i −1.81557 − 1.04822i 0.819627 + 1.41964i 0.500000 − 0.866025i 1.09047 − 0.629584i 2.57000i −2.18581 0.697541 + 1.20818i 0.300310 + 0.520153i
65.4 −0.134102 + 0.232271i 1.36886 + 0.790309i 0.964034 + 1.66976i 0.500000 − 0.866025i −0.367131 + 0.211963i 2.91576i −1.05352 −0.250822 − 0.434437i 0.134102 + 0.232271i
65.5 0.134102 − 0.232271i 1.36886 + 0.790309i 0.964034 + 1.66976i 0.500000 − 0.866025i 0.367131 − 0.211963i − 2.91576i 1.05352 −0.250822 − 0.434437i −0.134102 − 0.232271i
65.6 0.300310 − 0.520153i −1.81557 − 1.04822i 0.819627 + 1.41964i 0.500000 − 0.866025i −1.09047 + 0.629584i − 2.57000i 2.18581 0.697541 + 1.20818i −0.300310 − 0.520153i
65.7 1.15218 − 1.99563i −2.86886 − 1.65633i −1.65502 − 2.86657i 0.500000 − 0.866025i −6.61085 + 3.81678i 0.161926i −3.01878 3.98689 + 6.90550i −1.15218 − 1.99563i
65.8 1.34697 − 2.33302i 0.315575 + 0.182197i −2.62864 − 4.55295i 0.500000 − 0.866025i 0.850137 − 0.490827i 3.29655i −8.77493 −1.43361 − 2.48308i −1.34697 − 2.33302i
164.1 −1.34697 − 2.33302i 0.315575 − 0.182197i −2.62864 + 4.55295i 0.500000 + 0.866025i −0.850137 − 0.490827i 3.29655i 8.77493 −1.43361 + 2.48308i 1.34697 − 2.33302i
164.2 −1.15218 − 1.99563i −2.86886 + 1.65633i −1.65502 + 2.86657i 0.500000 + 0.866025i 6.61085 + 3.81678i 0.161926i 3.01878 3.98689 − 6.90550i 1.15218 − 1.99563i
164.3 −0.300310 − 0.520153i −1.81557 + 1.04822i 0.819627 − 1.41964i 0.500000 + 0.866025i 1.09047 + 0.629584i − 2.57000i −2.18581 0.697541 − 1.20818i 0.300310 − 0.520153i
164.4 −0.134102 − 0.232271i 1.36886 − 0.790309i 0.964034 − 1.66976i 0.500000 + 0.866025i −0.367131 − 0.211963i − 2.91576i −1.05352 −0.250822 + 0.434437i 0.134102 − 0.232271i
164.5 0.134102 + 0.232271i 1.36886 − 0.790309i 0.964034 − 1.66976i 0.500000 + 0.866025i 0.367131 + 0.211963i 2.91576i 1.05352 −0.250822 + 0.434437i −0.134102 + 0.232271i
164.6 0.300310 + 0.520153i −1.81557 + 1.04822i 0.819627 − 1.41964i 0.500000 + 0.866025i −1.09047 − 0.629584i 2.57000i 2.18581 0.697541 − 1.20818i −0.300310 + 0.520153i
164.7 1.15218 + 1.99563i −2.86886 + 1.65633i −1.65502 + 2.86657i 0.500000 + 0.866025i −6.61085 − 3.81678i − 0.161926i −3.01878 3.98689 − 6.90550i −1.15218 + 1.99563i
164.8 1.34697 + 2.33302i 0.315575 − 0.182197i −2.62864 + 4.55295i 0.500000 + 0.866025i 0.850137 + 0.490827i − 3.29655i −8.77493 −1.43361 + 2.48308i −1.34697 + 2.33302i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.d odd 6 1 inner
209.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.g.a ✓ 16
11.b odd 2 1 inner 209.2.g.a ✓ 16
19.d odd 6 1 inner 209.2.g.a ✓ 16
209.g even 6 1 inner 209.2.g.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.g.a ✓ 16 1.a even 1 1 trivial
209.2.g.a ✓ 16 11.b odd 2 1 inner
209.2.g.a ✓ 16 19.d odd 6 1 inner
209.2.g.a ✓ 16 209.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 13T_{2}^{14} + 125T_{2}^{12} + 538T_{2}^{10} + 1714T_{2}^{8} + 722T_{2}^{6} + 245T_{2}^{4} + 17T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 13 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{8} + 6 T^{7} + 9 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 26 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 50 T^{14} + \cdots + 160000 \) Copy content Toggle raw display
$17$ \( T^{16} - 26 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{8} + 2 T^{7} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 1016096256256 \) Copy content Toggle raw display
$31$ \( (T^{8} + 108 T^{6} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 102 T^{6} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 43218738217216 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 236421376 \) Copy content Toggle raw display
$47$ \( (T^{8} + 2 T^{7} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 130 T^{6} + \cdots + 1092025)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 6 T^{7} + \cdots + 9024016)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{8} + 48 T^{7} + \cdots + 1597696)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 24 T^{7} + \cdots + 3444736)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 507422576896 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 49\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{8} + 466 T^{6} + \cdots + 2085136)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 6 T^{7} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 130 T^{6} + \cdots + 1092025)^{2} \) Copy content Toggle raw display
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