Properties

Label 324.5.d.c
Level $324$
Weight $5$
Character orbit 324.d
Analytic conductor $33.492$
Analytic rank $0$
Dimension $10$
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] = [324,5,Mod(163,324)]
 
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([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.163");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.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: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 24 x^{8} - 64 x^{7} + 418 x^{6} - 1866 x^{5} + 9456 x^{4} - 36840 x^{3} + 125205 x^{2} - 309645 x + 565056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{6} + \beta_1 - 4) q^{4} - \beta_{3} q^{5} + \beta_{9} q^{7} + (\beta_{8} - 2 \beta_{6} + \beta_{2} + 5 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{6} + \beta_1 - 4) q^{4} - \beta_{3} q^{5} + \beta_{9} q^{7} + (\beta_{8} - 2 \beta_{6} + \beta_{2} + 5 \beta_1 + 1) q^{8} + ( - \beta_{4} + 2 \beta_{3} + \beta_1 - 4) q^{10} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1 - 1) q^{11} + ( - 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{3} - 2 \beta_1 - 19) q^{13} + ( - 2 \beta_{9} + \beta_{8} - 3 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 2) q^{14}+ \cdots + ( - 72 \beta_{9} + 88 \beta_{8} + 36 \beta_{7} + 148 \beta_{6} + \cdots + 1489) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 11 q^{2} - 35 q^{4} + 2 q^{5} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 11 q^{2} - 35 q^{4} + 2 q^{5} + 7 q^{8} - 43 q^{10} - 178 q^{13} + 24 q^{14} - 143 q^{16} - 22 q^{17} + 341 q^{20} - 144 q^{22} - 384 q^{25} + 227 q^{26} - 792 q^{28} + 2066 q^{29} - 1601 q^{32} - 835 q^{34} - 82 q^{37} - 720 q^{38} + 647 q^{40} - 3652 q^{41} - 1608 q^{44} - 5496 q^{46} - 7430 q^{49} - 6024 q^{50} + 827 q^{52} + 6188 q^{53} + 6912 q^{56} - 3283 q^{58} - 9682 q^{61} + 5232 q^{62} - 4031 q^{64} + 18214 q^{65} - 12355 q^{68} - 10032 q^{70} + 10022 q^{73} - 8005 q^{74} + 13944 q^{76} - 3984 q^{77} - 22975 q^{80} - 26098 q^{82} - 37502 q^{85} - 9984 q^{86} + 11280 q^{88} + 19274 q^{89} + 31416 q^{92} - 6576 q^{94} - 15772 q^{97} + 14173 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 24 x^{8} - 64 x^{7} + 418 x^{6} - 1866 x^{5} + 9456 x^{4} - 36840 x^{3} + 125205 x^{2} - 309645 x + 565056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{9} + 2 \nu^{8} - 26 \nu^{7} + 90 \nu^{6} - 508 \nu^{5} + 2374 \nu^{4} - 11830 \nu^{3} + 48670 \nu^{2} - 173875 \nu + 417984 ) / 65536 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{9} + 2 \nu^{8} - 26 \nu^{7} + 90 \nu^{6} - 508 \nu^{5} + 2374 \nu^{4} - 11830 \nu^{3} + 48670 \nu^{2} + 88269 \nu + 417984 ) / 65536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 6 \nu^{8} - 26 \nu^{7} - 206 \nu^{6} + 152 \nu^{5} - 786 \nu^{4} + 10330 \nu^{3} - 19930 \nu^{2} + 38823 \nu - 77248 ) / 16384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{9} - 46 \nu^{8} - 10 \nu^{7} + 234 \nu^{6} + 4052 \nu^{5} - 2122 \nu^{4} + 18170 \nu^{3} - 202930 \nu^{2} + 290237 \nu - 301888 ) / 32768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{9} - 2 \nu^{8} - 54 \nu^{7} + 38 \nu^{6} - 572 \nu^{5} + 2106 \nu^{4} - 11482 \nu^{3} + 158434 \nu^{2} - 13617 \nu + 469056 ) / 32768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{9} + 2 \nu^{8} - 106 \nu^{7} + 218 \nu^{6} - 1588 \nu^{5} + 6854 \nu^{4} - 35142 \nu^{3} + 124702 \nu^{2} - 361367 \nu + 682432 ) / 32768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{9} - 14 \nu^{8} + 22 \nu^{7} - 118 \nu^{6} + 116 \nu^{5} - 2026 \nu^{4} + 32602 \nu^{3} - 34770 \nu^{2} + 335005 \nu - 594752 ) / 16384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29 \nu^{9} + 86 \nu^{8} + 578 \nu^{7} + 542 \nu^{6} + 6044 \nu^{5} - 17470 \nu^{4} + 110350 \nu^{3} - 243446 \nu^{2} + 400503 \nu + 122432 ) / 65536 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{9} + 22 \nu^{8} - 38 \nu^{7} + 542 \nu^{6} - 1024 \nu^{5} + 13762 \nu^{4} - 39994 \nu^{3} + 181450 \nu^{2} - 469019 \nu + 735936 ) / 16384 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 4\beta _1 - 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{8} + 2\beta_{7} + 8\beta_{6} + 2\beta_{3} - 7\beta_{2} - 15\beta _1 + 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{9} - 6\beta_{8} + 2\beta_{7} - 37\beta_{6} - 11\beta_{5} + 10\beta_{3} + 36\beta_{2} + 98\beta _1 - 155 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16 \beta_{9} + 4 \beta_{8} - 28 \beta_{7} + 24 \beta_{6} + 56 \beta_{5} + 16 \beta_{4} + 20 \beta_{3} - 135 \beta_{2} - 485 \beta _1 + 1152 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 96 \beta_{9} + 180 \beta_{8} + 100 \beta_{7} + 623 \beta_{6} - 79 \beta_{5} + 80 \beta_{4} - 92 \beta_{3} + 416 \beta_{2} + 40 \beta _1 - 6843 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 304 \beta_{9} - 858 \beta_{8} + 70 \beta_{7} - 4720 \beta_{6} - 56 \beta_{5} - 208 \beta_{4} - 1066 \beta_{3} - 543 \beta_{2} + 12989 \beta _1 + 3268 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 584 \beta_{9} + 2446 \beta_{8} - 1274 \beta_{7} + 16283 \beta_{6} + 853 \beta_{5} - 1040 \beta_{4} + 1582 \beta_{3} - 9836 \beta_{2} - 85354 \beta _1 + 176621 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4512 \beta_{9} + 3464 \beta_{8} - 56 \beta_{7} - 31984 \beta_{6} - 9840 \beta_{5} + 2400 \beta_{4} + 12520 \beta_{3} + 94865 \beta_{2} + 258071 \beta _1 - 1168736 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
2.82663 1.16485i
2.82663 + 1.16485i
1.89260 2.76276i
1.89260 + 2.76276i
0.260513 3.79620i
0.260513 + 3.79620i
−0.902661 3.99882i
−0.902661 + 3.99882i
−3.57709 3.05919i
−3.57709 + 3.05919i
−3.82663 1.16485i 0 13.2862 + 8.91492i −12.2326 0 52.9747i −40.4570 49.5906i 0 46.8098 + 14.2492i
163.2 −3.82663 + 1.16485i 0 13.2862 8.91492i −12.2326 0 52.9747i −40.4570 + 49.5906i 0 46.8098 14.2492i
163.3 −2.89260 2.76276i 0 0.734275 + 15.9831i 38.7998 0 44.6730i 42.0337 48.2615i 0 −112.232 107.195i
163.4 −2.89260 + 2.76276i 0 0.734275 15.9831i 38.7998 0 44.6730i 42.0337 + 48.2615i 0 −112.232 + 107.195i
163.5 −1.26051 3.79620i 0 −12.8222 + 9.57031i −34.6728 0 58.2241i 52.4934 + 36.6121i 0 43.7055 + 131.625i
163.6 −1.26051 + 3.79620i 0 −12.8222 9.57031i −34.6728 0 58.2241i 52.4934 36.6121i 0 43.7055 131.625i
163.7 −0.0973389 3.99882i 0 −15.9811 + 0.778481i 8.69308 0 84.0818i 4.66858 + 63.8295i 0 −0.846174 34.7620i
163.8 −0.0973389 + 3.99882i 0 −15.9811 0.778481i 8.69308 0 84.0818i 4.66858 63.8295i 0 −0.846174 + 34.7620i
163.9 2.57709 3.05919i 0 −2.71726 15.7676i 0.412540 0 21.4057i −55.2386 32.3218i 0 1.06315 1.26204i
163.10 2.57709 + 3.05919i 0 −2.71726 + 15.7676i 0.412540 0 21.4057i −55.2386 + 32.3218i 0 1.06315 + 1.26204i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.d.c 10
3.b odd 2 1 324.5.d.d yes 10
4.b odd 2 1 inner 324.5.d.c 10
12.b even 2 1 324.5.d.d yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.5.d.c 10 1.a even 1 1 trivial
324.5.d.c 10 4.b odd 2 1 inner
324.5.d.d yes 10 3.b odd 2 1
324.5.d.d yes 10 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - T_{5}^{4} - 1466T_{5}^{3} - 3718T_{5}^{2} + 144841T_{5} - 59017 \) acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 11 T^{9} + 78 T^{8} + \cdots + 1048576 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} - T^{4} - 1466 T^{3} - 3718 T^{2} + \cdots - 59017)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 15720 T^{8} + \cdots + 61\!\cdots\!88 \) Copy content Toggle raw display
$11$ \( T^{10} + 91848 T^{8} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T^{5} + 89 T^{4} - 78122 T^{3} + \cdots + 18547259857)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 11 T^{4} + \cdots - 667567300561)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 737352 T^{8} + \cdots + 42\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( T^{10} + 1100520 T^{8} + \cdots + 59\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( (T^{5} - 1033 T^{4} + \cdots - 189507124642945)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 2953344 T^{8} + \cdots + 71\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( (T^{5} + 41 T^{4} + \cdots - 852313726745807)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + 1826 T^{4} + \cdots + 920805364605728)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 18102408 T^{8} + \cdots + 73\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{10} + 25575072 T^{8} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( (T^{5} - 3094 T^{4} + \cdots + 302749170293792)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 112922016 T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + 4841 T^{4} + \cdots + 815431223481217)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 125329992 T^{8} + \cdots + 21\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{10} + 132091560 T^{8} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} - 5011 T^{4} + \cdots - 29\!\cdots\!47)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 257560488 T^{8} + \cdots + 67\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{10} + 335564832 T^{8} + \cdots + 33\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( (T^{5} - 9637 T^{4} + \cdots + 21\!\cdots\!15)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 7886 T^{4} + \cdots - 43\!\cdots\!16)^{2} \) Copy content Toggle raw display
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