Properties

Label 324.5.c.a
Level $324$
Weight $5$
Character orbit 324.c
Analytic conductor $33.492$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,5,Mod(161,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.161");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.c (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: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{20} \)
Twist minimal: no (minimal twist has level 36)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{3} - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{3} - 3) q^{7} - \beta_{2} q^{11} + (\beta_{5} + \beta_{3} + 2) q^{13} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{17} + (\beta_{6} + 70) q^{19} + (\beta_{7} + 2 \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{6} - 2 \beta_{5} + 4 \beta_{3} - 88) q^{25} + (\beta_{7} + \beta_{4} - 5 \beta_{2} - 8 \beta_1) q^{29} + ( - \beta_{6} + 3 \beta_{5} - 4 \beta_{3} - 46) q^{31} + (2 \beta_{7} - \beta_{4} + 6 \beta_{2} - 15 \beta_1) q^{35} + ( - \beta_{6} - 5 \beta_{5} - 19 \beta_{3} - 5) q^{37} + (2 \beta_{7} - 3 \beta_{4} - \beta_{2} + 26 \beta_1) q^{41} + (\beta_{6} - 2 \beta_{5} - 3 \beta_{3} + 15) q^{43} + (4 \beta_{7} - 3 \beta_{4} + 4 \beta_{2} + 11 \beta_1) q^{47} + (2 \beta_{6} - 11 \beta_{5} - 19 \beta_{3} + 71) q^{49} + (5 \beta_{7} + 5 \beta_{4} - 25 \beta_{2} + 61 \beta_1) q^{53} + ( - \beta_{6} + 13 \beta_{5} + 34 \beta_{3} - 212) q^{55} + (7 \beta_{7} - \beta_{4} - 20 \beta_{2} - 63 \beta_1) q^{59} + (2 \beta_{6} + 19 \beta_{5} - 53 \beta_{3} + 480) q^{61} + (9 \beta_{7} + 3 \beta_{4} + 41 \beta_{2} + 106 \beta_1) q^{65} + ( - 3 \beta_{5} + 96 \beta_{3} - 16) q^{67} + (13 \beta_{7} + 9 \beta_{4} + 17 \beta_{2} + 35 \beta_1) q^{71} + ( - 8 \beta_{6} - \beta_{5} + 83 \beta_{3} - 953) q^{73} + (6 \beta_{7} - 14 \beta_{4} + 18 \beta_{2} - 105 \beta_1) q^{77} + (7 \beta_{6} - 3 \beta_{5} + 90 \beta_{3} + 568) q^{79} + (12 \beta_{7} - 7 \beta_{4} - 12 \beta_{2} - 229 \beta_1) q^{83} + (7 \beta_{6} - 7 \beta_{5} - 97 \beta_{3} + 731) q^{85} + (7 \beta_{7} + 3 \beta_{4} - 19 \beta_{2} - 241 \beta_1) q^{89} + (7 \beta_{6} + 7 \beta_{5} - 162 \beta_{3} + 1940) q^{91} + (16 \beta_{7} + 20 \beta_{4} - 14 \beta_{2} + 384 \beta_1) q^{95} + ( - 9 \beta_{6} + \beta_{5} - 17 \beta_{3} - 1822) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 26 q^{7} + 10 q^{13} + 562 q^{19} - 706 q^{25} - 374 q^{31} + 16 q^{37} + 136 q^{43} + 654 q^{49} - 1818 q^{55} + 3874 q^{61} - 308 q^{67} - 7802 q^{73} + 4390 q^{79} + 6084 q^{85} + 15830 q^{91} - 14564 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\nu^{7} - 142\nu^{6} + 635\nu^{5} - 1115\nu^{4} + 17923\nu^{3} - 159143\nu^{2} + 464580\nu - 386178 ) / 104247 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53 \nu^{7} - 367 \nu^{6} + 5765 \nu^{5} - 70703 \nu^{4} + 113944 \nu^{3} + 106240 \nu^{2} - 1050579 \nu - 42441945 ) / 104247 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 5\nu^{6} + 34\nu^{5} + 116\nu^{4} + 197\nu^{3} - 223\nu^{2} + 25974\nu + 60657 ) / 729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 118\nu^{7} + 49\nu^{6} - 1775\nu^{5} + 11717\nu^{4} + 118754\nu^{3} + 166664\nu^{2} + 172422\nu + 2470650 ) / 34749 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} - 38\nu^{6} + 73\nu^{5} + 410\nu^{4} + 428\nu^{3} - 18433\nu^{2} + 18954\nu + 178665 ) / 729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 2\nu^{6} + 143\nu^{5} + 661\nu^{4} + 4441\nu^{3} + 15841\nu^{2} + 96174\nu + 445521 ) / 729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1052 \nu^{7} - 1433 \nu^{6} + 40423 \nu^{5} - 237610 \nu^{4} - 608323 \nu^{3} + 281948 \nu^{2} + 16720785 \nu - 71046885 ) / 104247 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - 3\beta_{6} - 3\beta_{5} + 6\beta_{4} + 18\beta_{3} - 3\beta_{2} + 22\beta _1 + 186 ) / 486 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 3\beta_{5} + 11\beta_{4} - 3\beta_{3} - \beta_{2} - 116\beta _1 - 60 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 19\beta_{7} + 24\beta_{6} + 27\beta_{5} + 56\beta_{4} - 228\beta_{3} - 37\beta_{2} + 125\beta _1 - 7947 ) / 162 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14\beta_{7} + 111\beta_{6} + 96\beta_{5} - 64\beta_{4} - 570\beta_{3} - 220\beta_{2} - 219\beta _1 - 120225 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 217 \beta_{7} + 930 \beta_{6} + 285 \beta_{5} - 2155 \beta_{4} - 2505 \beta_{3} + 515 \beta_{2} + 2208 \beta _1 - 206598 ) / 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4159 \beta_{7} + 651 \beta_{6} - 6591 \beta_{5} - 13543 \beta_{4} + 15891 \beta_{3} + 5291 \beta_{2} + 68865 \beta _1 + 432933 ) / 162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 26773 \beta_{7} - 19995 \beta_{6} - 32457 \beta_{5} - 47548 \beta_{4} + 237246 \beta_{3} + 67505 \beta_{2} + 53688 \beta _1 + 13250454 ) / 162 \) Copy content Toggle raw display

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−3.41053 2.74723i
−3.05006 3.25531i
4.23522 + 4.06612i
3.72537 + 4.42407i
3.72537 4.42407i
4.23522 4.06612i
−3.05006 + 3.25531i
−3.41053 + 2.74723i
0 0 0 40.2664i 0 14.7738 0 0 0
161.2 0 0 0 31.6564i 0 −75.3660 0 0 0
161.3 0 0 0 12.2819i 0 −14.2840 0 0 0
161.4 0 0 0 8.86801i 0 61.8763 0 0 0
161.5 0 0 0 8.86801i 0 61.8763 0 0 0
161.6 0 0 0 12.2819i 0 −14.2840 0 0 0
161.7 0 0 0 31.6564i 0 −75.3660 0 0 0
161.8 0 0 0 40.2664i 0 14.7738 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
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 324.5.c.a 8
3.b odd 2 1 inner 324.5.c.a 8
4.b odd 2 1 1296.5.e.g 8
9.c even 3 1 36.5.g.a 8
9.c even 3 1 108.5.g.a 8
9.d odd 6 1 36.5.g.a 8
9.d odd 6 1 108.5.g.a 8
12.b even 2 1 1296.5.e.g 8
36.f odd 6 1 144.5.q.c 8
36.f odd 6 1 432.5.q.c 8
36.h even 6 1 144.5.q.c 8
36.h even 6 1 432.5.q.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a 8 9.c even 3 1
36.5.g.a 8 9.d odd 6 1
108.5.g.a 8 9.c even 3 1
108.5.g.a 8 9.d odd 6 1
144.5.q.c 8 36.f odd 6 1
144.5.q.c 8 36.h even 6 1
324.5.c.a 8 1.a even 1 1 trivial
324.5.c.a 8 3.b odd 2 1 inner
432.5.q.c 8 36.f odd 6 1
432.5.q.c 8 36.h even 6 1
1296.5.e.g 8 4.b odd 2 1
1296.5.e.g 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2853T_{5}^{6} + 2238759T_{5}^{4} + 403999407T_{5}^{2} + 19274879556 \) acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2853 T^{6} + \cdots + 19274879556 \) Copy content Toggle raw display
$7$ \( (T^{4} + 13 T^{3} - 4881 T^{2} + \cdots + 984106)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 49752 T^{6} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} - 72165 T^{2} + \cdots + 627597820)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 380043 T^{6} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{4} - 281 T^{3} + \cdots - 16594049624)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2134773 T^{6} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + 1949445 T^{6} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + 187 T^{3} + \cdots + 76986598576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} - 3885276 T^{2} + \cdots + 400007987296)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 9956700 T^{6} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{4} - 68 T^{3} - 719976 T^{2} + \cdots + 20007524971)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 19055493 T^{6} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + 57977928 T^{6} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + 65598192 T^{6} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( (T^{4} - 1937 T^{3} + \cdots - 106249943015384)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 154 T^{3} + \cdots + 56193518999761)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 134421732 T^{6} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + 3901 T^{3} + \cdots + 523952945824816)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2195 T^{3} + \cdots + 832106966500150)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 298478277 T^{6} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + 207178632 T^{6} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + 7282 T^{3} + \cdots - 361232872000949)^{2} \) Copy content Toggle raw display
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