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Properties

Label 324.8.e.m
Level $324$
Weight $8$
Character orbit 324.e
Analytic conductor $101.213$
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] = [324,8,Mod(109,324)]
 
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, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.109");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), 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: \(101.212748257\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3205 x^{14} + 7140274 x^{12} + 8220484645 x^{10} + 6820694102626 x^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{44} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{8}) q^{5} + (\beta_{3} - \beta_{2} - 70 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{8}) q^{5} + (\beta_{3} - \beta_{2} - 70 \beta_1) q^{7} + (\beta_{14} - \beta_{9} + \beta_{8}) q^{11} + ( - \beta_{6} + 2 \beta_{2} + \cdots - 185) q^{13}+ \cdots + (197 \beta_{7} - 197 \beta_{6} + \cdots + 4208030 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 560 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 560 q^{7} - 1480 q^{13} - 55264 q^{19} - 77936 q^{25} + 247424 q^{31} + 220400 q^{37} - 897040 q^{43} - 1329672 q^{49} + 1910880 q^{55} - 494968 q^{61} - 4698160 q^{67} + 21452240 q^{73} - 8887312 q^{79} - 15973992 q^{85} + 30743008 q^{91} - 33664240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3205 x^{14} + 7140274 x^{12} + 8220484645 x^{10} + 6820694102626 x^{8} + \cdots + 57\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 81\!\cdots\!29 \nu^{14} + \cdots + 15\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39\!\cdots\!87 \nu^{14} + \cdots + 51\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!27 \nu^{14} + \cdots + 14\!\cdots\!00 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\!\cdots\!24 \nu^{14} + \cdots + 68\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!29 \nu^{14} + \cdots - 48\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 28\!\cdots\!02 \nu^{14} + \cdots - 27\!\cdots\!00 ) / 77\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 69\!\cdots\!72 \nu^{14} + \cdots - 14\!\cdots\!00 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!63 \nu^{15} + \cdots - 19\!\cdots\!00 \nu ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 60\!\cdots\!59 \nu^{15} + \cdots - 55\!\cdots\!00 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 58\!\cdots\!98 \nu^{15} + \cdots - 11\!\cdots\!00 \nu ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13\!\cdots\!39 \nu^{15} + \cdots + 29\!\cdots\!00 \nu ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!14 \nu^{15} + \cdots - 21\!\cdots\!00 \nu ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 64\!\cdots\!87 \nu^{15} + \cdots + 32\!\cdots\!00 \nu ) / 46\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 62\!\cdots\!02 \nu^{15} + \cdots - 46\!\cdots\!00 \nu ) / 35\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 47\!\cdots\!59 \nu^{15} + \cdots - 21\!\cdots\!00 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{15} - 3\beta_{13} + 3\beta_{12} + 5\beta_{11} + 102\beta_{10} - 102\beta_{8} ) / 1944 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -18\beta_{7} + 18\beta_{6} - 25\beta_{4} - 193\beta_{3} + 193\beta_{2} + 519210\beta_1 ) / 648 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2389\beta_{14} + 2389\beta_{13} - 1057\beta_{12} - 5983\beta_{11} - 47818\beta_{10} - 5983\beta_{9} ) / 648 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -27486\beta_{6} - 38231\beta_{5} - 360911\beta_{2} - 649380726\beta _1 - 649380726 ) / 648 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4321281\beta_{15} - 10956693\beta_{14} + 37841743\beta_{9} + 194605338\beta_{8} ) / 1944 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13609134\beta_{7} + 17422055\beta_{5} + 17422055\beta_{4} + 193573919\beta_{3} + 298901892150 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6212159139 \beta_{15} - 16108833423 \beta_{13} + 6212159139 \beta_{12} + \cdots - 260356185246 \beta_{8} ) / 1944 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 61123511214 \beta_{7} + 61123511214 \beta_{6} - 70492802719 \beta_{4} - 906248856439 \beta_{3} + \cdots + 12\!\cdots\!74 \beta_1 ) / 648 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7866081533119 \beta_{14} + 7866081533119 \beta_{13} - 3009980206723 \beta_{12} + \cdots - 37797535067469 \beta_{9} ) / 648 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 91643444457498 \beta_{6} - 95078810276285 \beta_{5} + \cdots - 17\!\cdots\!50 ) / 648 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13\!\cdots\!11 \beta_{15} + \cdots + 46\!\cdots\!54 \beta_{8} ) / 1944 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 45\!\cdots\!62 \beta_{7} + \cdots + 84\!\cdots\!42 ) / 216 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 19\!\cdots\!81 \beta_{15} + \cdots - 62\!\cdots\!38 \beta_{8} ) / 1944 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 20\!\cdots\!02 \beta_{7} + \cdots + 36\!\cdots\!50 \beta_1 ) / 648 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24\!\cdots\!41 \beta_{14} + \cdots - 15\!\cdots\!07 \beta_{9} ) / 648 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−18.0990 + 31.3483i
8.11082 − 14.0484i
−6.08630 + 10.5418i
−19.2574 + 33.3549i
19.2574 − 33.3549i
6.08630 − 10.5418i
−8.11082 + 14.0484i
18.0990 − 31.3483i
−18.0990 − 31.3483i
8.11082 + 14.0484i
−6.08630 − 10.5418i
−19.2574 − 33.3549i
19.2574 + 33.3549i
6.08630 + 10.5418i
−8.11082 − 14.0484i
18.0990 + 31.3483i
0 0 0 −265.209 + 459.356i 0 −280.503 − 485.846i 0 0 0
109.2 0 0 0 −103.256 + 178.845i 0 −503.506 − 872.097i 0 0 0
109.3 0 0 0 −82.6178 + 143.098i 0 122.405 + 212.011i 0 0 0
109.4 0 0 0 −6.58929 + 11.4130i 0 801.604 + 1388.42i 0 0 0
109.5 0 0 0 6.58929 − 11.4130i 0 801.604 + 1388.42i 0 0 0
109.6 0 0 0 82.6178 − 143.098i 0 122.405 + 212.011i 0 0 0
109.7 0 0 0 103.256 − 178.845i 0 −503.506 − 872.097i 0 0 0
109.8 0 0 0 265.209 − 459.356i 0 −280.503 − 485.846i 0 0 0
217.1 0 0 0 −265.209 − 459.356i 0 −280.503 + 485.846i 0 0 0
217.2 0 0 0 −103.256 − 178.845i 0 −503.506 + 872.097i 0 0 0
217.3 0 0 0 −82.6178 − 143.098i 0 122.405 − 212.011i 0 0 0
217.4 0 0 0 −6.58929 − 11.4130i 0 801.604 − 1388.42i 0 0 0
217.5 0 0 0 6.58929 + 11.4130i 0 801.604 − 1388.42i 0 0 0
217.6 0 0 0 82.6178 + 143.098i 0 122.405 − 212.011i 0 0 0
217.7 0 0 0 103.256 + 178.845i 0 −503.506 + 872.097i 0 0 0
217.8 0 0 0 265.209 + 459.356i 0 −280.503 + 485.846i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
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 324.8.e.m 16
3.b odd 2 1 inner 324.8.e.m 16
9.c even 3 1 324.8.a.e ✓ 8
9.c even 3 1 inner 324.8.e.m 16
9.d odd 6 1 324.8.a.e ✓ 8
9.d odd 6 1 inner 324.8.e.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.8.a.e ✓ 8 9.c even 3 1
324.8.a.e ✓ 8 9.d odd 6 1
324.8.e.m 16 1.a even 1 1 trivial
324.8.e.m 16 3.b odd 2 1 inner
324.8.e.m 16 9.c even 3 1 inner
324.8.e.m 16 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{16} + 351468 T_{5}^{14} + 102624277914 T_{5}^{12} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
\( T_{7}^{8} - 280 T_{7}^{7} + 2018704 T_{7}^{6} + 1400490560 T_{7}^{5} + 3423043183696 T_{7}^{4} + \cdots + 49\!\cdots\!00 \) 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
$5$ \( T^{16} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 75\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 91\!\cdots\!25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 94\!\cdots\!00)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 78\!\cdots\!45)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 80\!\cdots\!25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 83\!\cdots\!75)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 69\!\cdots\!81)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
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