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Properties

Label 324.5.g.f
Level $324$
Weight $5$
Character orbit 324.g
Analytic conductor $33.492$
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] = [324,5,Mod(53,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.g (of order \(6\), degree \(2\), not 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: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 6242x^{12} + 11954017x^{8} + 7856017920x^{4} + 1485512441856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{44} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_{5}) q^{5} + (\beta_{7} - 7 \beta_{4} - 7) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_{5}) q^{5} + (\beta_{7} - 7 \beta_{4} - 7) q^{7} + \beta_{11} q^{11} + ( - \beta_{8} + 28 \beta_{4} + \cdots + 1) q^{13}+ \cdots + (\beta_{8} - 120 \beta_{7} + \cdots + 383) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 52 q^{7} - 220 q^{13} - 616 q^{19} + 328 q^{25} + 1472 q^{31} + 2456 q^{37} - 388 q^{43} - 3432 q^{49} - 72 q^{55} - 1804 q^{61} + 2180 q^{67} - 9088 q^{73} + 5012 q^{79} + 5364 q^{85} - 3512 q^{91} + 2528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 6242x^{12} + 11954017x^{8} + 7856017920x^{4} + 1485512441856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -242119\nu^{12} + 377715634\nu^{8} + 3899090215289\nu^{4} + 2069290573299968 ) / 8736247696384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1640615\nu^{12} + 9039954286\nu^{8} + 13667354391239\nu^{4} + 4900500625833728 ) / 17472495392768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -789857\nu^{12} - 4080368050\nu^{8} - 5021469016913\nu^{4} - 952265633413376 ) / 4368123848192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 445\nu^{14} + 1558874\nu^{10} - 898861667\nu^{6} - 3117636999936\nu^{2} - 27750626721792 ) / 55501253443584 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4505253991 \nu^{15} + 15098400600 \nu^{13} - 9928164553838 \nu^{11} + \cdots - 27\!\cdots\!04 \nu ) / 36\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 305295704 \nu^{14} - 11281527531 \nu^{12} + 1726916798848 \nu^{10} - 58279896858150 \nu^{8} + \cdots - 13\!\cdots\!08 ) / 12\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6367013039 \nu^{14} - 93731616180 \nu^{12} + 37164139122430 \nu^{10} + \cdots - 27\!\cdots\!96 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51666206939 \nu^{14} + 110661941664 \nu^{12} + 266248425707734 \nu^{10} + \cdots - 94\!\cdots\!04 ) / 79\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49328274079 \nu^{15} - 768671626224 \nu^{13} - 293896505584094 \nu^{11} + \cdots - 47\!\cdots\!56 \nu ) / 73\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5067618563 \nu^{15} - 768671626224 \nu^{13} + 36544397170726 \nu^{11} + \cdots - 25\!\cdots\!48 \nu ) / 27\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 141213009923 \nu^{15} - 3309606661584 \nu^{13} + 864721736095270 \nu^{11} + \cdots - 46\!\cdots\!20 \nu ) / 73\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 134875765081 \nu^{15} - 674944523472 \nu^{13} - 876446173670162 \nu^{11} + \cdots - 16\!\cdots\!88 \nu ) / 36\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7698005279 \nu^{15} - 191906514960 \nu^{13} - 42681748798174 \nu^{11} + \cdots - 50\!\cdots\!72 \nu ) / 68\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 123930808595 \nu^{15} + 6487922866746 \nu^{13} - 622594814537542 \nu^{11} + \cdots + 17\!\cdots\!68 \nu ) / 91\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1143205125577 \nu^{15} - 19273251240336 \nu^{13} + \cdots - 88\!\cdots\!24 \nu ) / 73\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} + 9\beta_{12} - 18\beta_{11} + 6\beta_{10} - 97\beta_{9} + 36\beta_{5} ) / 486 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{8} + 44\beta_{7} - 50\beta_{6} + 2598\beta_{4} + 25\beta_{3} + 22\beta_{2} - 3\beta _1 + 1302 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 37 \beta_{15} + 32 \beta_{14} + 185 \beta_{13} - 117 \beta_{12} + 666 \beta_{11} + \cdots - 2692 \beta_{5} ) / 486 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 381\beta_{3} + 698\beta_{2} - 121\beta _1 - 84048 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2131 \beta_{15} + 2816 \beta_{14} - 4555 \beta_{13} - 30573 \beta_{12} + 28746 \beta_{11} + \cdots - 160324 \beta_{5} ) / 486 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 38214 \beta_{8} - 126700 \beta_{7} + 101554 \beta_{6} - 11429478 \beta_{4} - 50777 \beta_{3} + \cdots - 5733846 ) / 162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 21323 \beta_{15} - 205280 \beta_{14} - 277961 \beta_{13} - 287163 \beta_{12} + \cdots + 9018628 \beta_{5} ) / 486 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1316749\beta_{3} - 2330970\beta_{2} + 693769\beta _1 + 202382160 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 17431747 \beta_{15} - 13929152 \beta_{14} + 32177659 \beta_{13} + 90484029 \beta_{12} + \cdots + 500455108 \beta_{5} ) / 486 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 159975462 \beta_{8} + 441959596 \beta_{7} - 269444818 \beta_{6} + 35516906310 \beta_{4} + \cdots + 17838440886 ) / 162 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 284826491 \beta_{15} + 903515552 \beta_{14} + 721748249 \beta_{13} + 1219932459 \beta_{12} + \cdots - 27776786308 \beta_{5} ) / 486 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 4081450397\beta_{3} + 7604199418\beta_{2} - 2814731161\beta _1 - 576270914640 ) / 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 76216373299 \beta_{15} + 56742572672 \beta_{14} - 142258577707 \beta_{13} + \cdots - 1548660037444 \beta_{5} ) / 486 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 595561496838 \beta_{8} - 1495885547116 \beta_{7} + 798722851570 \beta_{6} - 109099050497574 \beta_{4} + \cdots - 54847305997206 ) / 162 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1237701080747 \beta_{15} - 3480274370912 \beta_{14} - 2479678913897 \beta_{13} + \cdots + 86808384609796 \beta_{5} ) / 486 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−5.35887 − 5.35887i
−4.65176 − 4.65176i
2.99261 − 2.99261i
3.69971 − 3.69971i
−3.69971 + 3.69971i
−2.99261 + 2.99261i
4.65176 + 4.65176i
5.35887 + 5.35887i
−5.35887 + 5.35887i
−4.65176 + 4.65176i
2.99261 + 2.99261i
3.69971 + 3.69971i
−3.69971 − 3.69971i
−2.99261 − 2.99261i
4.65176 − 4.65176i
5.35887 − 5.35887i
0 0 0 −29.1904 − 16.8531i 0 16.7081 + 28.9393i 0 0 0
53.2 0 0 0 −22.8264 − 13.1788i 0 −10.2177 − 17.6977i 0 0 0
53.3 0 0 0 −20.5691 − 11.8756i 0 −43.3346 − 75.0578i 0 0 0
53.4 0 0 0 −14.2052 − 8.20137i 0 23.8442 + 41.2994i 0 0 0
53.5 0 0 0 14.2052 + 8.20137i 0 23.8442 + 41.2994i 0 0 0
53.6 0 0 0 20.5691 + 11.8756i 0 −43.3346 − 75.0578i 0 0 0
53.7 0 0 0 22.8264 + 13.1788i 0 −10.2177 − 17.6977i 0 0 0
53.8 0 0 0 29.1904 + 16.8531i 0 16.7081 + 28.9393i 0 0 0
269.1 0 0 0 −29.1904 + 16.8531i 0 16.7081 − 28.9393i 0 0 0
269.2 0 0 0 −22.8264 + 13.1788i 0 −10.2177 + 17.6977i 0 0 0
269.3 0 0 0 −20.5691 + 11.8756i 0 −43.3346 + 75.0578i 0 0 0
269.4 0 0 0 −14.2052 + 8.20137i 0 23.8442 − 41.2994i 0 0 0
269.5 0 0 0 14.2052 − 8.20137i 0 23.8442 − 41.2994i 0 0 0
269.6 0 0 0 20.5691 − 11.8756i 0 −43.3346 + 75.0578i 0 0 0
269.7 0 0 0 22.8264 − 13.1788i 0 −10.2177 + 17.6977i 0 0 0
269.8 0 0 0 29.1904 − 16.8531i 0 16.7081 − 28.9393i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.5.g.f 16
3.b odd 2 1 inner 324.5.g.f 16
9.c even 3 1 324.5.c.b ✓ 8
9.c even 3 1 inner 324.5.g.f 16
9.d odd 6 1 324.5.c.b ✓ 8
9.d odd 6 1 inner 324.5.g.f 16
36.f odd 6 1 1296.5.e.f 8
36.h even 6 1 1296.5.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.5.c.b ✓ 8 9.c even 3 1
324.5.c.b ✓ 8 9.d odd 6 1
324.5.g.f 16 1.a even 1 1 trivial
324.5.g.f 16 3.b odd 2 1 inner
324.5.g.f 16 9.c even 3 1 inner
324.5.g.f 16 9.d odd 6 1 inner
1296.5.e.f 8 36.f odd 6 1
1296.5.e.f 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{16} - 2664 T_{5}^{14} + 4630446 T_{5}^{12} - 4699658880 T_{5}^{10} + 3471457452291 T_{5}^{8} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
\( T_{7}^{8} + 26 T_{7}^{7} + 5998 T_{7}^{6} - 192436 T_{7}^{5} + 24798436 T_{7}^{4} - 290629936 T_{7}^{3} + \cdots + 7966032077056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 7966032077056)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 71\!\cdots\!09)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 88\!\cdots\!21)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 154 T^{3} + \cdots + 830765392)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 55\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 80\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 614 T^{3} + \cdots - 258318998879)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 49\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 40\!\cdots\!09)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 522346227455231)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 46\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 86\!\cdots\!29)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
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