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Properties

Label 459.3.b.d
Level $459$
Weight $3$
Character orbit 459.b
Analytic conductor $12.507$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [459,3,Mod(188,459)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(459, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("459.188"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 459 = 3^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 459.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-20,0,0,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.5068441341\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 34x^{10} + 395x^{8} + 1888x^{6} + 3523x^{4} + 1566x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{9} q^{5} + ( - \beta_{7} + \beta_{2} - 5) q^{7} + ( - \beta_{10} + \beta_{9} + \cdots - 3 \beta_1) q^{8} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots - 1) q^{10}+ \cdots + ( - 2 \beta_{11} - 9 \beta_{10} + \cdots + 40 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{4} - 54 q^{7} - 12 q^{10} - 50 q^{13} + 108 q^{16} - 50 q^{19} - 106 q^{22} - 48 q^{25} + 382 q^{28} + 2 q^{31} - 244 q^{37} + 208 q^{40} - 78 q^{43} - 18 q^{46} + 470 q^{49} + 42 q^{52} + 290 q^{55}+ \cdots + 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 34x^{10} + 395x^{8} + 1888x^{6} + 3523x^{4} + 1566x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} + 11\nu^{8} + 886\nu^{6} + 8858\nu^{4} + 22331\nu^{2} + 4083 ) / 888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{10} - 323\nu^{8} - 2686\nu^{6} - 3794\nu^{4} + 17425\nu^{2} + 13389 ) / 1776 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\nu^{10} + 835\nu^{8} + 9374\nu^{6} + 41398\nu^{4} + 61993\nu^{2} + 9591 ) / 1776 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{11} + 523\nu^{9} + 7574\nu^{7} + 47350\nu^{5} + 116845\nu^{3} + 72351\nu ) / 5328 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -17\nu^{10} - 553\nu^{8} - 5954\nu^{6} - 24942\nu^{4} - 37733\nu^{2} - 8881 ) / 592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 91\nu^{11} + 2995\nu^{9} + 33038\nu^{7} + 147634\nu^{5} + 286447\nu^{3} + 283347\nu ) / 15984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 101\nu^{11} + 3551\nu^{9} + 43270\nu^{7} + 218894\nu^{5} + 414341\nu^{3} + 125523\nu ) / 5328 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\nu^{11} + 365\nu^{9} + 4066\nu^{7} + 18050\nu^{5} + 29699\nu^{3} + 9009\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -299\nu^{11} - 10031\nu^{9} - 114262\nu^{7} - 532274\nu^{5} - 975815\nu^{3} - 445959\nu ) / 5328 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + 2\beta_{8} - 2\beta_{6} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 4\beta_{5} + \beta_{3} - 13\beta_{2} + 65 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 22\beta_{10} - 22\beta_{9} - 35\beta_{8} + 46\beta_{6} + 139\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 44\beta_{7} + 92\beta_{5} + 10\beta_{4} - 27\beta_{3} + 174\beta_{2} - 832 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -37\beta_{11} - 401\beta_{10} + 375\beta_{9} + 529\beta_{8} - 814\beta_{6} - 1893\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -776\beta_{7} - 1664\beta_{5} - 284\beta_{4} + 550\beta_{3} - 2422\beta_{2} + 11489 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 834\beta_{11} + 6756\beta_{10} - 5976\beta_{9} - 7830\beta_{8} + 13376\beta_{6} + 26869\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 12732\beta_{7} + 27776\beta_{5} + 5736\beta_{4} - 9902\beta_{3} + 34699\beta_{2} - 164906 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -15638\beta_{11} - 109313\beta_{10} + 93081\beta_{9} + 116308\beta_{8} - 213038\beta_{6} - 391047\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/459\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(190\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
− 3.90664i
− 3.15039i
− 2.15341i
− 1.89037i
− 0.736967i
− 0.243755i
0.243755i
0.736967i
1.89037i
2.15341i
3.15039i
3.90664i
− 3.90664i 0 −11.2618 − 3.88566i 0 −9.05048 28.3693i 0 −15.1799
188.2 − 3.15039i 0 −5.92497 0.888739i 0 −10.5667 6.06443i 0 2.79988
188.3 − 2.15341i 0 −0.637159 8.83083i 0 −10.7410 − 7.24156i 0 19.0164
188.4 − 1.89037i 0 0.426490 − 5.04259i 0 −0.681340 − 8.36772i 0 −9.53237
188.5 − 0.736967i 0 3.45688 − 5.75031i 0 −8.26227 − 5.49547i 0 −4.23779
188.6 − 0.243755i 0 3.94058 4.65129i 0 12.3018 − 1.93556i 0 1.13378
188.7 0.243755i 0 3.94058 − 4.65129i 0 12.3018 1.93556i 0 1.13378
188.8 0.736967i 0 3.45688 5.75031i 0 −8.26227 5.49547i 0 −4.23779
188.9 1.89037i 0 0.426490 5.04259i 0 −0.681340 8.36772i 0 −9.53237
188.10 2.15341i 0 −0.637159 − 8.83083i 0 −10.7410 7.24156i 0 19.0164
188.11 3.15039i 0 −5.92497 − 0.888739i 0 −10.5667 − 6.06443i 0 2.79988
188.12 3.90664i 0 −11.2618 3.88566i 0 −9.05048 − 28.3693i 0 −15.1799
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 459.3.b.d ✓ 12
3.b odd 2 1 inner 459.3.b.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
459.3.b.d ✓ 12 1.a even 1 1 trivial
459.3.b.d ✓ 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 34T_{2}^{10} + 395T_{2}^{8} + 1888T_{2}^{6} + 3523T_{2}^{4} + 1566T_{2}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(459, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 34 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 174 T^{10} + \cdots + 16916769 \) Copy content Toggle raw display
$7$ \( (T^{6} + 27 T^{5} + \cdots - 71136)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 191886050304 \) Copy content Toggle raw display
$13$ \( (T^{6} + 25 T^{5} + \cdots - 209014)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 17)^{6} \) Copy content Toggle raw display
$19$ \( (T^{6} + 25 T^{5} + \cdots - 1329948)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 13661569176336 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 605806956017316 \) Copy content Toggle raw display
$31$ \( (T^{6} - T^{5} + \cdots - 1507775392)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 122 T^{5} + \cdots + 1736384)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{6} + 39 T^{5} + \cdots - 335565954)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 215001926747136 \) Copy content Toggle raw display
$61$ \( (T^{6} + 36 T^{5} + \cdots + 7959296)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 301 T^{5} + \cdots + 267266912854)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{6} + 151 T^{5} + \cdots + 1842291936)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 14 T^{5} + \cdots - 441485318592)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 67\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{6} - 33 T^{5} + \cdots + 12301504112)^{2} \) Copy content Toggle raw display
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