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Properties

Label 459.3.c.g
Level $459$
Weight $3$
Character orbit 459.c
Analytic conductor $12.507$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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(458,459)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("459.458"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(459, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 459 = 3^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 459.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,-36,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.5068441341\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 68 x^{18} + 1808 x^{16} + 24602 x^{14} + 187648 x^{12} + 817824 x^{10} + 1959913 x^{8} + \cdots + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + (\beta_1 - 2) q^{4} + \beta_{4} q^{5} - \beta_{11} q^{7} + (\beta_{17} - 3 \beta_{10}) q^{8} + ( - \beta_{16} + 2 \beta_{13}) q^{10} - \beta_{5} q^{11} + ( - \beta_{7} - \beta_1 + 5) q^{13}+ \cdots + ( - \beta_{19} - 4 \beta_{17} + \cdots + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 36 q^{4} + 100 q^{13} + 228 q^{16} + 172 q^{19} + 272 q^{25} - 80 q^{34} + 124 q^{43} + 4 q^{49} - 888 q^{52} + 268 q^{55} - 1408 q^{64} + 80 q^{67} - 300 q^{70} - 252 q^{76} - 484 q^{85} + 172 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 68 x^{18} + 1808 x^{16} + 24602 x^{14} + 187648 x^{12} + 817824 x^{10} + 1959913 x^{8} + \cdots + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 767620267 \nu^{18} - 50855917898 \nu^{16} - 1300953678904 \nu^{14} + \cdots + 21294627749000 ) / 6496079363320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2302860801 \nu^{18} - 152567753694 \nu^{16} - 3902861036712 \nu^{14} + \cdots - 72533783382720 ) / 12992158726640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30876386417 \nu^{18} - 2220164159358 \nu^{16} - 63542709309384 \nu^{14} + \cdots - 41\!\cdots\!20 ) / 90945111086480 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 63815218589 \nu^{18} + 4320174900301 \nu^{16} + 114173390927428 \nu^{14} + \cdots + 23\!\cdots\!80 ) / 136417666629720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 100568244519 \nu^{18} - 6828318336271 \nu^{16} - 181121211517568 \nu^{14} + \cdots - 13\!\cdots\!00 ) / 136417666629720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 323154142767 \nu^{18} + 22016619677078 \nu^{16} + 586930174546464 \nu^{14} + \cdots + 11\!\cdots\!00 ) / 272835333259440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28749334305 \nu^{18} - 1950923736014 \nu^{16} - 51708327412936 \nu^{14} + \cdots - 10\!\cdots\!16 ) / 18189022217296 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 214961023007 \nu^{18} - 14605351654338 \nu^{16} - 387791581873464 \nu^{14} + \cdots - 60\!\cdots\!40 ) / 90945111086480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 721117070257 \nu^{18} - 49377703155698 \nu^{16} + \cdots - 19\!\cdots\!80 ) / 272835333259440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27964422669 \nu^{19} + 1898510260424 \nu^{17} + 50356252513960 \nu^{15} + \cdots + 10\!\cdots\!48 \nu ) / 51968634906560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 333586096465 \nu^{19} - 22569753138618 \nu^{17} - 595681618898992 \nu^{15} + \cdots + 51\!\cdots\!72 \nu ) / 545670666518880 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1100897208789 \nu^{19} - 743028439176 \nu^{18} + 74582505204804 \nu^{17} + \cdots - 26\!\cdots\!40 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 83893268007 \nu^{19} - 5695530781272 \nu^{17} - 151068757541880 \nu^{15} + \cdots - 28\!\cdots\!64 \nu ) / 51968634906560 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 733324141323 \nu^{19} + 49839391818860 \nu^{17} + \cdots + 88\!\cdots\!08 \nu ) / 272835333259440 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 228136740797 \nu^{19} - 15497171354284 \nu^{17} - 411426387001848 \nu^{15} + \cdots - 13\!\cdots\!64 \nu ) / 72756088869184 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2293734916471 \nu^{19} - 155695416659174 \nu^{17} + \cdots - 93\!\cdots\!20 \nu ) / 545670666518880 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 135216391743 \nu^{19} + 9187415794732 \nu^{17} + 243975540496376 \nu^{15} + \cdots + 49\!\cdots\!80 \nu ) / 25984317453280 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 1466507678598 \nu^{19} - 99729706275325 \nu^{17} + \cdots - 82\!\cdots\!68 \nu ) / 272835333259440 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 3788302540113 \nu^{19} - 257637716576022 \nu^{17} + \cdots - 12\!\cdots\!20 \nu ) / 181890222172960 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + 3\beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{2} + 3\beta _1 - 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{17} + 3\beta_{16} - 3\beta_{14} - 16\beta_{13} - 3\beta_{11} - 42\beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{9} + 3\beta_{7} + 8\beta_{6} - 4\beta_{5} + 4\beta_{4} - 3\beta_{3} + 44\beta_{2} - 66\beta _1 + 315 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{19} + 10 \beta_{18} - 84 \beta_{17} - 95 \beta_{16} + 3 \beta_{15} + 70 \beta_{14} + \cdots + 3 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 134 \beta_{9} + 9 \beta_{8} - 120 \beta_{7} - 214 \beta_{6} + 188 \beta_{5} - 188 \beta_{4} + \cdots - 6096 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 135 \beta_{19} - 399 \beta_{18} + 2103 \beta_{17} + 2513 \beta_{16} - 147 \beta_{15} + \cdots - 144 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3664 \beta_{9} - 498 \beta_{8} + 3732 \beta_{7} + 4880 \beta_{6} - 5896 \beta_{5} + 6184 \beta_{4} + \cdots + 132069 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4296 \beta_{19} + 11868 \beta_{18} - 51693 \beta_{17} - 63327 \beta_{16} + 5070 \beta_{15} + \cdots + 4866 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 94238 \beta_{9} + 17772 \beta_{8} - 104703 \beta_{7} - 109888 \beta_{6} + 162206 \beta_{5} + \cdots - 3015381 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 120399 \beta_{19} - 319286 \beta_{18} + 1265640 \beta_{17} + 1568193 \beta_{16} + \cdots - 141711 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 787900 \beta_{9} - 177031 \beta_{8} + 925922 \beta_{7} + 837812 \beta_{6} - 1406896 \beta_{5} + \cdots + 23575346 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3177843 \beta_{19} + 8217313 \beta_{18} - 30943899 \beta_{17} - 38547119 \beta_{16} + \cdots + 3827928 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 58571736 \beta_{9} + 14542992 \beta_{8} - 71321496 \beta_{7} - 58547064 \beta_{6} + 106839870 \beta_{5} + \cdots - 1684109253 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 81228660 \beta_{19} - 206777056 \beta_{18} + 755913093 \beta_{17} + 944158553 \beta_{16} + \cdots - 99214410 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 1441973360 \beta_{9} - 379840176 \beta_{8} + 1795516347 \beta_{7} + 1384026880 \beta_{6} + \cdots + 40462340235 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 2038454451 \beta_{19} + 5138716650 \beta_{18} - 18454773048 \beta_{17} - 23082908703 \beta_{16} + \cdots + 2510857443 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 35367636514 \beta_{9} + 9656109345 \beta_{8} - 44660887824 \beta_{7} - 33065769122 \beta_{6} + \cdots - 977500581288 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 50597750907 \beta_{19} - 126775433275 \beta_{18} + 450363055359 \beta_{17} + \cdots - 62645675580 \beta_{2} ) / 3 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
458.1
− 2.93798i
− 4.93798i
− 3.92773i
− 1.92773i
− 0.719204i
− 2.71920i
− 2.18928i
− 0.189282i
0.257931i
− 1.74207i
− 0.257931i
1.74207i
2.18928i
0.189282i
0.719204i
2.71920i
3.92773i
1.92773i
2.93798i
4.93798i
− 3.93798i 0 −11.5077 −7.90288 0 − 3.69602i 29.5650i 0 31.1213i
458.2 − 3.93798i 0 −11.5077 7.90288 0 3.69602i 29.5650i 0 − 31.1213i
458.3 − 2.92773i 0 −4.57161 −4.30653 0 7.91861i 1.67353i 0 12.6084i
458.4 − 2.92773i 0 −4.57161 4.30653 0 − 7.91861i 1.67353i 0 − 12.6084i
458.5 − 1.71920i 0 1.04434 −8.53267 0 9.34999i − 8.67225i 0 14.6694i
458.6 − 1.71920i 0 1.04434 8.53267 0 − 9.34999i − 8.67225i 0 − 14.6694i
458.7 − 1.18928i 0 2.58561 −5.01447 0 − 4.18362i − 7.83215i 0 5.96362i
458.8 − 1.18928i 0 2.58561 5.01447 0 4.18362i − 7.83215i 0 − 5.96362i
458.9 − 0.742069i 0 3.44933 −3.74795 0 − 7.91897i − 5.52792i 0 2.78123i
458.10 − 0.742069i 0 3.44933 3.74795 0 7.91897i − 5.52792i 0 − 2.78123i
458.11 0.742069i 0 3.44933 −3.74795 0 7.91897i 5.52792i 0 − 2.78123i
458.12 0.742069i 0 3.44933 3.74795 0 − 7.91897i 5.52792i 0 2.78123i
458.13 1.18928i 0 2.58561 −5.01447 0 4.18362i 7.83215i 0 − 5.96362i
458.14 1.18928i 0 2.58561 5.01447 0 − 4.18362i 7.83215i 0 5.96362i
458.15 1.71920i 0 1.04434 −8.53267 0 − 9.34999i 8.67225i 0 − 14.6694i
458.16 1.71920i 0 1.04434 8.53267 0 9.34999i 8.67225i 0 14.6694i
458.17 2.92773i 0 −4.57161 −4.30653 0 − 7.91861i − 1.67353i 0 − 12.6084i
458.18 2.92773i 0 −4.57161 4.30653 0 7.91861i − 1.67353i 0 12.6084i
458.19 3.93798i 0 −11.5077 −7.90288 0 3.69602i − 29.5650i 0 − 31.1213i
458.20 3.93798i 0 −11.5077 7.90288 0 − 3.69602i − 29.5650i 0 31.1213i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 458.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.b even 2 1 inner
51.c 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.c.g ✓ 20
3.b odd 2 1 inner 459.3.c.g ✓ 20
17.b even 2 1 inner 459.3.c.g ✓ 20
51.c odd 2 1 inner 459.3.c.g ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
459.3.c.g ✓ 20 1.a even 1 1 trivial
459.3.c.g ✓ 20 3.b odd 2 1 inner
459.3.c.g ✓ 20 17.b even 2 1 inner
459.3.c.g ✓ 20 51.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(459, [\chi])\):

\( T_{2}^{10} + 29T_{2}^{8} + 258T_{2}^{6} + 815T_{2}^{4} + 931T_{2}^{2} + 306 \) Copy content Toggle raw display
\( T_{5}^{10} - 193T_{5}^{8} + 13437T_{5}^{6} - 415188T_{5}^{4} + 5797332T_{5}^{2} - 29787264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 29 T^{8} + \cdots + 306)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 193 T^{8} + \cdots - 29787264)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 244 T^{8} + \cdots + 82192356)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 792 T^{8} + \cdots - 38682238600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} - 25 T^{4} + \cdots - 353000)^{4} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 40\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{5} - 43 T^{4} + \cdots - 102170)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} - 2127 T^{8} + \cdots - 29588500000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 2440 T^{8} + \cdots - 32400534600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 957778704000000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 12094953728400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 1581841990144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 31 T^{4} + \cdots - 15650000)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 635537240653824)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 66\!\cdots\!50)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 20 T^{4} + \cdots - 187591500)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 89\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 14\!\cdots\!54)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 250760134125000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 91055451120804)^{2} \) Copy content Toggle raw display
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