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Properties

Label 2169.2.d.c
Level $2169$
Weight $2$
Character orbit 2169.d
Analytic conductor $17.320$
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] = [2169,2,Mod(1927,2169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2169.1927");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.d (of order \(2\), degree \(1\), 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: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 41x^{14} + 652x^{12} + 5159x^{10} + 21933x^{8} + 50143x^{6} + 57208x^{4} + 26192x^{2} + 2048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 723)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + ( - \beta_{8} + \beta_{6} - \beta_{5} + 1) q^{4} - \beta_{5} q^{5} + \beta_1 q^{7} + ( - \beta_{11} + \beta_{9} + \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + ( - \beta_{8} + \beta_{6} - \beta_{5} + 1) q^{4} - \beta_{5} q^{5} + \beta_1 q^{7} + ( - \beta_{11} + \beta_{9} + \beta_{6}) q^{8} + (\beta_{9} - \beta_{5} - \beta_{3} + 1) q^{10} - \beta_{12} q^{11} + (\beta_{15} + \beta_{7} - \beta_1) q^{13} + (\beta_{14} - \beta_{10}) q^{14} + ( - \beta_{8} + \beta_{6} - \beta_{5} + \cdots + 2) q^{16}+ \cdots + ( - \beta_{13} - \beta_{11} + 2 \beta_{8} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 18 q^{4} - 4 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 18 q^{4} - 4 q^{5} + 12 q^{8} + 10 q^{10} + 38 q^{16} + 10 q^{20} + 24 q^{25} - 10 q^{29} + 16 q^{32} - 8 q^{40} + 24 q^{41} + 6 q^{47} + 30 q^{49} + 70 q^{50} - 18 q^{53} - 44 q^{58} + 38 q^{59} + 74 q^{61} - 4 q^{64} + 30 q^{67} - 36 q^{77} + 34 q^{79} + 50 q^{80} - 18 q^{82} - 10 q^{83} + 28 q^{91} - 26 q^{94} + 4 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 41x^{14} + 652x^{12} + 5159x^{10} + 21933x^{8} + 50143x^{6} + 57208x^{4} + 26192x^{2} + 2048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2565545 \nu^{15} - 127445673 \nu^{13} - 2542340516 \nu^{11} - 25974314943 \nu^{9} + \cdots - 183862718192 \nu ) / 11956147776 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 162427 \nu^{14} + 6125901 \nu^{12} + 85527598 \nu^{10} + 549147333 \nu^{8} + 1662915585 \nu^{6} + \cdots - 339820166 ) / 124543206 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 655357 \nu^{15} - 29800557 \nu^{13} - 534038884 \nu^{11} - 4797212235 \nu^{9} + \cdots - 30584658736 \nu ) / 1328460864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4731415 \nu^{14} - 182211459 \nu^{12} - 2618664712 \nu^{10} - 17479684785 \nu^{8} + \cdots + 9218176928 ) / 2989036944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 366365 \nu^{14} + 13343265 \nu^{12} + 177028184 \nu^{10} + 1052597139 \nu^{8} + 2874111045 \nu^{6} + \cdots + 164343824 ) / 166057608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8839579 \nu^{15} + 302907867 \nu^{13} + 3579805804 \nu^{11} + 16177662237 \nu^{9} + \cdots - 98708352176 \nu ) / 11956147776 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1216651 \nu^{14} + 47778117 \nu^{12} + 709145806 \nu^{10} + 5004701445 \nu^{8} + \cdots + 593606896 ) / 498172824 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7427737 \nu^{14} - 275368605 \nu^{12} - 3752509672 \nu^{10} - 23246897295 \nu^{8} + \cdots - 16840880608 ) / 2989036944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9940333 \nu^{15} + 367025253 \nu^{13} + 4923157612 \nu^{11} + 28799019003 \nu^{9} + \cdots - 87960515024 \nu ) / 11956147776 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1895674 \nu^{14} - 72449961 \nu^{12} - 1037300323 \nu^{10} - 7004419770 \nu^{8} + \cdots - 3017603644 ) / 747259236 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 732245 \nu^{15} - 27201821 \nu^{13} - 373588188 \nu^{11} - 2377354451 \nu^{9} + \cdots - 6195562416 \nu ) / 442820288 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7182833 \nu^{14} - 274897749 \nu^{12} - 3939666836 \nu^{10} - 26525477283 \nu^{8} + \cdots - 4718802368 ) / 747259236 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 40271117 \nu^{15} + 1526224629 \nu^{13} + 21519962780 \nu^{11} + 140869735323 \nu^{9} + \cdots - 39678856720 \nu ) / 11956147776 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 46956751 \nu^{15} - 1825894455 \nu^{13} - 26805524164 \nu^{11} - 187425858441 \nu^{9} + \cdots - 96737239792 \nu ) / 11956147776 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{6} + \beta_{5} + \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 2\beta_{14} + \beta_{12} + \beta_{10} + \beta_{7} - 2\beta_{4} - \beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} + 11\beta_{9} - 2\beta_{8} + 11\beta_{6} - 15\beta_{5} - 16\beta_{3} + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -32\beta_{15} - 33\beta_{14} - 16\beta_{12} - 19\beta_{10} - 11\beta_{7} + 25\beta_{4} + 22\beta_{2} + 91\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{13} + 37\beta_{11} - 132\beta_{9} + 43\beta_{8} - 126\beta_{6} + 202\beta_{5} + 213\beta_{3} - 550 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 430 \beta_{15} + 443 \beta_{14} + 207 \beta_{12} + 288 \beta_{10} + 112 \beta_{7} - 294 \beta_{4} + \cdots - 1016 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30 \beta_{13} - 585 \beta_{11} + 1635 \beta_{9} - 671 \beta_{8} + 1500 \beta_{6} - 2631 \beta_{5} + \cdots + 6673 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5496 \beta_{15} - 5609 \beta_{14} - 2583 \beta_{12} - 4022 \beta_{10} - 1229 \beta_{7} + \cdots + 11983 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 245 \beta_{13} + 8682 \beta_{11} - 20385 \beta_{9} + 9398 \beta_{8} - 18208 \beta_{6} + 33935 \beta_{5} + \cdots - 82647 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 68976 \beta_{15} + 69510 \beta_{14} + 32066 \beta_{12} + 54192 \beta_{10} + 14649 \beta_{7} + \cdots - 145252 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 135 \beta_{13} - 124463 \beta_{11} + 254305 \beta_{9} - 125801 \beta_{8} + 222894 \beta_{6} + \cdots + 1032076 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 859488 \beta_{15} - 853840 \beta_{14} - 398198 \beta_{12} - 718000 \beta_{10} - 184453 \beta_{7} + \cdots + 1784240 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 42965 \beta_{13} + 1746971 \beta_{11} - 3171627 \beta_{9} + 1649745 \beta_{8} - 2738297 \beta_{6} + \cdots - 12935289 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10679554 \beta_{15} + 10447799 \beta_{14} + 4949412 \beta_{12} + 9433247 \beta_{10} + \cdots - 22059982 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2169\mathbb{Z}\right)^\times\).

\(n\) \(730\) \(965\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1927.1
− 0.957617i
0.957617i
− 3.49643i
3.49643i
− 1.80262i
1.80262i
− 2.41919i
2.41919i
− 1.40217i
1.40217i
− 0.311622i
0.311622i
− 3.57003i
3.57003i
− 1.98688i
1.98688i
−2.53872 0 4.44511 1.00948 0 − 0.957617i −6.20747 0 −2.56278
1927.2 −2.53872 0 4.44511 1.00948 0 0.957617i −6.20747 0 −2.56278
1927.3 −1.89056 0 1.57423 −2.71501 0 − 3.49643i 0.804948 0 5.13289
1927.4 −1.89056 0 1.57423 −2.71501 0 3.49643i 0.804948 0 5.13289
1927.5 −0.639911 0 −1.59051 −2.38679 0 − 1.80262i 2.29761 0 1.52734
1927.6 −0.639911 0 −1.59051 −2.38679 0 1.80262i 2.29761 0 1.52734
1927.7 −0.583759 0 −1.65923 0.863321 0 − 2.41919i 2.13611 0 −0.503972
1927.8 −0.583759 0 −1.65923 0.863321 0 2.41919i 2.13611 0 −0.503972
1927.9 0.450369 0 −1.79717 2.74789 0 − 1.40217i −1.71013 0 1.23757
1927.10 0.450369 0 −1.79717 2.74789 0 1.40217i −1.71013 0 1.23757
1927.11 1.18100 0 −0.605236 −3.46309 0 − 0.311622i −3.07679 0 −4.08991
1927.12 1.18100 0 −0.605236 −3.46309 0 0.311622i −3.07679 0 −4.08991
1927.13 2.39980 0 3.75905 3.77689 0 − 3.57003i 4.22138 0 9.06380
1927.14 2.39980 0 3.75905 3.77689 0 3.57003i 4.22138 0 9.06380
1927.15 2.62178 0 4.87374 −1.83270 0 − 1.98688i 7.53433 0 −4.80493
1927.16 2.62178 0 4.87374 −1.83270 0 1.98688i 7.53433 0 −4.80493
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1927.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2169.2.d.c 16
3.b odd 2 1 723.2.d.b ✓ 16
241.b even 2 1 inner 2169.2.d.c 16
723.b odd 2 1 723.2.d.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
723.2.d.b ✓ 16 3.b odd 2 1
723.2.d.b ✓ 16 723.b odd 2 1
2169.2.d.c 16 1.a even 1 1 trivial
2169.2.d.c 16 241.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} - 12T_{2}^{6} + 9T_{2}^{5} + 41T_{2}^{4} - 17T_{2}^{3} - 35T_{2}^{2} + 2T_{2} + 6 \) acting on \(S_{2}^{\mathrm{new}}(2169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{7} - 12 T^{6} + \cdots + 6)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2 T^{7} + \cdots + 372)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 41 T^{14} + \cdots + 2048 \) Copy content Toggle raw display
$11$ \( T^{16} + 80 T^{14} + \cdots + 1089288 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 110469248 \) Copy content Toggle raw display
$17$ \( T^{16} + 80 T^{14} + \cdots + 1424672 \) Copy content Toggle raw display
$19$ \( T^{16} + 191 T^{14} + \cdots + 23887872 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 565690248 \) Copy content Toggle raw display
$29$ \( (T^{8} + 5 T^{7} - 94 T^{6} + \cdots - 92)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 455699313792 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 36230012928 \) Copy content Toggle raw display
$41$ \( (T^{8} - 12 T^{7} + \cdots + 24392)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 5323401773568 \) Copy content Toggle raw display
$47$ \( (T^{8} - 3 T^{7} + \cdots - 18992)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 9 T^{7} + \cdots - 19336)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 19 T^{7} + \cdots + 14712)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 37 T^{7} + \cdots - 1225894)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 15 T^{7} + \cdots - 5028)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 13190439578912 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2169560192 \) Copy content Toggle raw display
$79$ \( (T^{8} - 17 T^{7} + \cdots - 5334832)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 5 T^{7} + \cdots - 8011728)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1114446865352 \) Copy content Toggle raw display
$97$ \( (T^{8} - 2 T^{7} + \cdots - 9411126)^{2} \) Copy content Toggle raw display
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