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Properties

Label 2592.2.s.j
Level $2592$
Weight $2$
Character orbit 2592.s
Analytic conductor $20.697$
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] = [2592,2,Mod(863,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.863");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.s (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: \(20.6972242039\)
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: \( x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{15} + \beta_{14} + \cdots - \beta_{5}) q^{5}+ \cdots + (\beta_{10} + 2 \beta_{8} + \cdots - \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} + \beta_{14} + \cdots - \beta_{5}) q^{5}+ \cdots + ( - 2 \beta_{8} + 3 \beta_{7} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{7} + 4 q^{13} + 24 q^{25} + 72 q^{31} + 72 q^{37} + 84 q^{43} + 24 q^{49} + 28 q^{61} + 36 q^{67} + 96 q^{73} + 12 q^{79} + 12 q^{85} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4506447 \nu^{14} - 121241175 \nu^{12} + 1034556480 \nu^{10} - 3947923892 \nu^{8} + \cdots - 42326241815 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 206016 \nu^{14} - 375731 \nu^{12} + 17700672 \nu^{10} - 197309312 \nu^{8} + \cdots + 8568605045 ) / 8770913528 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13282279 \nu^{14} + 200884791 \nu^{12} - 1177246560 \nu^{10} + 3219861076 \nu^{8} + \cdots - 296484914921 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19681995 \nu^{15} - 459692017 \nu^{13} + 3195349744 \nu^{11} - 7267451924 \nu^{9} + \cdots + 67134153359 \nu ) / 2964568772464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5080251 \nu^{15} + 21986582 \nu^{13} + 260696896 \nu^{11} - 2899209176 \nu^{9} + \cdots - 1205684410658 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4209 \nu^{14} + 71399 \nu^{12} - 338828 \nu^{10} + 257052 \nu^{8} + 4901436 \nu^{6} + \cdots - 83054881 ) / 43420364 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11703162 \nu^{14} - 120180421 \nu^{12} + 499243320 \nu^{10} - 345490376 \nu^{8} + \cdots + 202336233411 ) / 114021875864 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30695 \nu^{14} - 419209 \nu^{12} + 2520928 \nu^{10} - 7255428 \nu^{8} + 1602740 \nu^{6} + \cdots + 379906423 ) / 289763344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16691371 \nu^{15} + 405805691 \nu^{13} - 5421908520 \nu^{11} + 32843046644 \nu^{9} + \cdots + 175474228267 \nu ) / 1482284386232 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25286987 \nu^{14} - 353399906 \nu^{12} + 1901413416 \nu^{10} - 4117794124 \nu^{8} + \cdots + 313392480206 ) / 114021875864 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5281529 \nu^{15} - 16310755 \nu^{13} + 607738864 \nu^{11} - 4165926004 \nu^{9} + \cdots - 198374633795 \nu ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 599257 \nu^{15} + 8739462 \nu^{13} - 46755552 \nu^{11} + 102847788 \nu^{9} + \cdots - 9359839554 \nu ) / 14676083032 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 144082388 \nu^{15} - 2115255563 \nu^{13} + 13246128560 \nu^{11} - 45286056288 \nu^{9} + \cdots + 1840873477261 \nu ) / 1482284386232 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 319559619 \nu^{15} - 4593759155 \nu^{13} + 25201681824 \nu^{11} - 52599749252 \nu^{9} + \cdots + 3620395773293 \nu ) / 2964568772464 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1582155 \nu^{15} - 21956880 \nu^{13} + 119050416 \nu^{11} - 229166428 \nu^{9} + \cdots + 14779694904 \nu ) / 14676083032 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{5} + 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} + \beta_{7} - \beta_{6} + 7\beta_{3} + \beta_{2} + \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{15} + \beta_{14} - \beta_{13} - 6\beta_{12} - \beta_{11} - \beta_{9} + \beta_{5} + 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{10} + 21\beta_{8} + 4\beta_{7} - 9\beta_{6} + 24\beta_{3} + 4\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -10\beta_{15} - 27\beta_{14} + 3\beta_{13} - 67\beta_{12} - 13\beta_{11} - 3\beta_{9} + 10\beta_{5} + 84\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -31\beta_{10} + 25\beta_{8} + 31\beta_{7} - 20\beta_{6} + 16\beta_{3} + 11\beta_{2} + 15\beta _1 - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 247 \beta_{15} - 453 \beta_{14} + 16 \beta_{13} - 453 \beta_{12} - 13 \beta_{11} - 3 \beta_{9} + \cdots + 260 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -260\beta_{10} - 172\beta_{8} + 363\beta_{7} - 103\beta_{6} - 289\beta_{3} - 103\beta_{2} + 475\beta _1 - 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1274\beta_{15} - 1413\beta_{14} - 141\beta_{13} - 753\beta_{12} + 93\beta_{11} - 93\beta_{9} - 141\beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 473 \beta_{10} - 1595 \beta_{8} + 1320 \beta_{7} + 473 \beta_{6} - 1320 \beta_{3} - 1320 \beta_{2} + \cdots + 2244 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12650 \beta_{15} - 9635 \beta_{14} - 3389 \beta_{13} - 947 \beta_{12} + 947 \beta_{11} + \cdots - 5104 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 896\beta_{10} + 896\beta_{8} + 896\beta_{7} + 2552\beta_{6} + 6156\beta_{3} - 3448\beta_{2} + 7052\beta _1 + 12335 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 23781 \beta_{15} - 2821 \beta_{14} - 19208 \beta_{13} + 16971 \beta_{12} - 7075 \beta_{11} + \cdots - 16706 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 9896 \beta_{10} + 120086 \beta_{8} - 3405 \beta_{7} + 13301 \beta_{6} + 194461 \beta_{3} - 13301 \beta_{2} + \cdots + 110190 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 53817 \beta_{15} + 60043 \beta_{14} - 21919 \beta_{13} + 4418 \beta_{12} - 60043 \beta_{11} + \cdots + 63796 \beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(\beta_{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
−2.13219 + 1.08896i
0.115299 − 1.50155i
−2.39101 + 0.123030i
−0.850627 − 1.24273i
0.850627 + 1.24273i
2.39101 − 0.123030i
−0.115299 + 1.50155i
2.13219 − 1.08896i
−2.13219 − 1.08896i
0.115299 + 1.50155i
−2.39101 − 0.123030i
−0.850627 + 1.24273i
0.850627 − 1.24273i
2.39101 + 0.123030i
−0.115299 − 1.50155i
2.13219 + 1.08896i
0 0 0 −3.47996 − 2.00916i 0 2.66739 − 1.54002i 0 0 0
863.2 0 0 0 −2.35218 − 1.35803i 0 3.67803 − 2.12351i 0 0 0
863.3 0 0 0 −2.25522 − 1.30205i 0 −0.301361 + 0.173991i 0 0 0
863.4 0 0 0 −1.12743 − 0.650924i 0 −3.04406 + 1.75749i 0 0 0
863.5 0 0 0 1.12743 + 0.650924i 0 −3.04406 + 1.75749i 0 0 0
863.6 0 0 0 2.25522 + 1.30205i 0 −0.301361 + 0.173991i 0 0 0
863.7 0 0 0 2.35218 + 1.35803i 0 3.67803 − 2.12351i 0 0 0
863.8 0 0 0 3.47996 + 2.00916i 0 2.66739 − 1.54002i 0 0 0
1727.1 0 0 0 −3.47996 + 2.00916i 0 2.66739 + 1.54002i 0 0 0
1727.2 0 0 0 −2.35218 + 1.35803i 0 3.67803 + 2.12351i 0 0 0
1727.3 0 0 0 −2.25522 + 1.30205i 0 −0.301361 − 0.173991i 0 0 0
1727.4 0 0 0 −1.12743 + 0.650924i 0 −3.04406 − 1.75749i 0 0 0
1727.5 0 0 0 1.12743 − 0.650924i 0 −3.04406 − 1.75749i 0 0 0
1727.6 0 0 0 2.25522 − 1.30205i 0 −0.301361 − 0.173991i 0 0 0
1727.7 0 0 0 2.35218 − 1.35803i 0 3.67803 + 2.12351i 0 0 0
1727.8 0 0 0 3.47996 − 2.00916i 0 2.66739 + 1.54002i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.s.j 16
3.b odd 2 1 inner 2592.2.s.j 16
4.b odd 2 1 2592.2.s.i 16
9.c even 3 1 2592.2.c.b ✓ 16
9.c even 3 1 2592.2.s.i 16
9.d odd 6 1 2592.2.c.b ✓ 16
9.d odd 6 1 2592.2.s.i 16
12.b even 2 1 2592.2.s.i 16
36.f odd 6 1 2592.2.c.b ✓ 16
36.f odd 6 1 inner 2592.2.s.j 16
36.h even 6 1 2592.2.c.b ✓ 16
36.h even 6 1 inner 2592.2.s.j 16
72.j odd 6 1 5184.2.c.l 16
72.l even 6 1 5184.2.c.l 16
72.n even 6 1 5184.2.c.l 16
72.p odd 6 1 5184.2.c.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.c.b ✓ 16 9.c even 3 1
2592.2.c.b ✓ 16 9.d odd 6 1
2592.2.c.b ✓ 16 36.f odd 6 1
2592.2.c.b ✓ 16 36.h even 6 1
2592.2.s.i 16 4.b odd 2 1
2592.2.s.i 16 9.c even 3 1
2592.2.s.i 16 9.d odd 6 1
2592.2.s.i 16 12.b even 2 1
2592.2.s.j 16 1.a even 1 1 trivial
2592.2.s.j 16 3.b odd 2 1 inner
2592.2.s.j 16 36.f odd 6 1 inner
2592.2.s.j 16 36.h even 6 1 inner
5184.2.c.l 16 72.j odd 6 1
5184.2.c.l 16 72.l even 6 1
5184.2.c.l 16 72.n even 6 1
5184.2.c.l 16 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

\( T_{5}^{16} - 32 T_{5}^{14} + 694 T_{5}^{12} - 8000 T_{5}^{10} + 66571 T_{5}^{8} - 334784 T_{5}^{6} + \cdots + 1874161 \) Copy content Toggle raw display
\( T_{7}^{8} - 6T_{7}^{7} - 2T_{7}^{6} + 84T_{7}^{5} + 36T_{7}^{4} - 1008T_{7}^{3} + 1504T_{7}^{2} + 1152T_{7} + 256 \) 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
$5$ \( T^{16} - 32 T^{14} + \cdots + 1874161 \) Copy content Toggle raw display
$7$ \( (T^{8} - 6 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 44 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 96 T^{6} + 2298 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 120 T^{6} + \cdots + 389376)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1871773696 \) Copy content Toggle raw display
$29$ \( T^{16} - 56 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{8} - 36 T^{7} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 18 T^{3} + \cdots + 141)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 6505390336 \) Copy content Toggle raw display
$43$ \( (T^{8} - 42 T^{7} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 268435456 \) Copy content Toggle raw display
$53$ \( (T^{8} + 144 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 176 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$61$ \( (T^{8} - 14 T^{7} + \cdots + 10182481)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 18 T^{7} + \cdots + 19642624)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 164 T^{6} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 24 T^{3} + \cdots - 1263)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 6 T^{7} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 90\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{8} + 360 T^{6} + \cdots + 1418481)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 8 T^{7} + \cdots + 177209344)^{2} \) Copy content Toggle raw display
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