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Properties

Label 2592.2.c.b
Level $2592$
Weight $2$
Character orbit 2592.c
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(2591,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.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: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_{8} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + \beta_{8} q^{7} + \beta_{11} q^{11} + ( - \beta_{5} - 1) q^{13} + ( - \beta_{13} - \beta_{10} + \beta_{2}) q^{17} + (\beta_{9} + \beta_{8} + \beta_{6}) q^{19} + ( - \beta_{14} - \beta_{12}) q^{23} + ( - \beta_{4} + 2 \beta_1 - 3) q^{25} + ( - \beta_{10} - \beta_{3} + \beta_{2}) q^{29} + ( - \beta_{8} + \beta_{7} - \beta_{6}) q^{31} + (\beta_{15} + \beta_{14} + \cdots + \beta_{11}) q^{35}+ \cdots + (3 \beta_{4} + \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{13} - 48 q^{25} + 72 q^{37} - 48 q^{49} - 56 q^{61} + 96 q^{73} - 24 q^{85} - 32 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 6491 \nu^{14} - 58908 \nu^{12} + 105540 \nu^{10} + 538508 \nu^{8} - 2050314 \nu^{6} + \cdots + 250599968 ) / 168671414 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4911576 \nu^{15} - 154540353 \nu^{13} + 1087211884 \nu^{11} - 3393895680 \nu^{9} + \cdots + 175507889011 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5080251 \nu^{15} + 21986582 \nu^{13} + 260696896 \nu^{11} - 2899209176 \nu^{9} + \cdots - 1205684410658 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6521823 \nu^{14} + 83168284 \nu^{12} - 411692072 \nu^{10} + 522082916 \nu^{8} + \cdots - 42580169892 ) / 28505468966 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4056333 \nu^{14} - 56001301 \nu^{12} + 280032755 \nu^{10} - 321867369 \nu^{8} + \cdots + 36303001371 ) / 14252734483 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83 \nu^{14} + 1503 \nu^{12} - 10320 \nu^{10} + 33444 \nu^{8} - 14420 \nu^{6} - 116448 \nu^{4} + \cdots - 1185873 ) / 266006 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26615768 \nu^{14} - 377833303 \nu^{12} + 2192636052 \nu^{10} - 6160635588 \nu^{8} + \cdots + 286797162301 ) / 57010937932 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28468673 \nu^{14} - 411068542 \nu^{12} + 2198160292 \nu^{10} - 4361097768 \nu^{8} + \cdots + 430455083838 ) / 57010937932 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32378109 \nu^{14} + 370944239 \nu^{12} - 1761661872 \nu^{10} + 2452720884 \nu^{8} + \cdots - 259428206505 ) / 57010937932 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 54547925 \nu^{15} + 822019962 \nu^{13} - 4738102516 \nu^{11} + 9772987000 \nu^{9} + \cdots - 449210584186 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 86485643 \nu^{15} + 1208458904 \nu^{13} - 7402336548 \nu^{11} + 23588475888 \nu^{9} + \cdots - 863408348660 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 92342067 \nu^{15} - 1160236910 \nu^{13} + 5159374028 \nu^{11} - 968568264 \nu^{9} + \cdots + 799165681366 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 125576328 \nu^{15} - 1354271587 \nu^{13} + 4789011908 \nu^{11} + 2755526352 \nu^{9} + \cdots + 3392094955665 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 173392139 \nu^{15} - 2824088503 \nu^{13} + 19393996784 \nu^{11} - 67793496188 \nu^{9} + \cdots + 2730588590353 \nu ) / 741142193116 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 527130867 \nu^{15} + 7409119988 \nu^{13} - 40895142084 \nu^{11} + 90655347520 \nu^{9} + \cdots - 5739594309176 \nu ) / 741142193116 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{14} + 2\beta_{13} + \beta_{12} + 2\beta_{11} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} + 2\beta_{7} + 3\beta_{6} + 2\beta_{4} - 14\beta _1 + 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + 2\beta_{14} + 2\beta_{13} + \beta_{12} + 8\beta_{11} + 12\beta_{10} + 8\beta_{3} + 18\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{9} - 9\beta_{8} + 17\beta_{7} + 11\beta_{6} + 28\beta_{5} + 26\beta_{4} - 38\beta _1 + 60 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 15 \beta_{15} + 6 \beta_{14} + 26 \beta_{13} - 49 \beta_{12} + 84 \beta_{11} + 24 \beta_{10} + \cdots + 106 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -9\beta_{9} - 9\beta_{8} + 9\beta_{7} + 4\beta_{6} + 40\beta_{5} + 62\beta_{4} - 20\beta _1 + 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 130 \beta_{15} + 6 \beta_{14} + 26 \beta_{13} - 363 \beta_{12} + 461 \beta_{11} + \cdots - 462 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -407\beta_{9} - 201\beta_{8} - 319\beta_{7} - 193\beta_{6} + 932\beta_{5} + 1246\beta_{4} - 578\beta _1 + 756 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 190 \beta_{15} + 186 \beta_{14} - 186 \beta_{13} - 519 \beta_{12} + 1273 \beta_{11} + \cdots - 4150 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3212 \beta_{9} - 1518 \beta_{8} - 3014 \beta_{7} - 2267 \beta_{6} + 2640 \beta_{5} + 3586 \beta_{4} + \cdots + 12012 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1408 \beta_{15} + 6778 \beta_{14} - 1894 \beta_{13} + 4039 \beta_{12} + 12993 \beta_{11} + \cdots - 49454 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -2267\beta_{9} - 1371\beta_{8} - 1181\beta_{7} - 862\beta_{6} - 8328\beta _1 + 14507 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20751 \beta_{15} + 52566 \beta_{14} + 14150 \beta_{13} + 55971 \beta_{12} + 66826 \beta_{11} + \cdots - 125562 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 88188 \beta_{9} - 75206 \beta_{8} + 41794 \beta_{7} + 30537 \beta_{6} - 19792 \beta_{5} + \cdots + 680932 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 40077 \beta_{15} + 120086 \beta_{14} + 120086 \beta_{13} + 110107 \beta_{12} + 177664 \beta_{11} + \cdots + 175046 \beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
−2.39101 − 0.123030i
2.39101 − 0.123030i
−0.850627 + 1.24273i
0.850627 + 1.24273i
2.13219 − 1.08896i
−2.13219 − 1.08896i
−0.115299 + 1.50155i
0.115299 + 1.50155i
0.115299 − 1.50155i
−0.115299 − 1.50155i
−2.13219 + 1.08896i
2.13219 + 1.08896i
0.850627 − 1.24273i
−0.850627 − 1.24273i
2.39101 + 0.123030i
−2.39101 + 0.123030i
0 0 0 − 4.01832i 0 − 3.08003i 0 0 0
2591.2 0 0 0 − 4.01832i 0 3.08003i 0 0 0
2591.3 0 0 0 − 2.71606i 0 − 4.24703i 0 0 0
2591.4 0 0 0 − 2.71606i 0 4.24703i 0 0 0
2591.5 0 0 0 − 2.60410i 0 − 0.347982i 0 0 0
2591.6 0 0 0 − 2.60410i 0 0.347982i 0 0 0
2591.7 0 0 0 − 1.30185i 0 − 3.51498i 0 0 0
2591.8 0 0 0 − 1.30185i 0 3.51498i 0 0 0
2591.9 0 0 0 1.30185i 0 − 3.51498i 0 0 0
2591.10 0 0 0 1.30185i 0 3.51498i 0 0 0
2591.11 0 0 0 2.60410i 0 − 0.347982i 0 0 0
2591.12 0 0 0 2.60410i 0 0.347982i 0 0 0
2591.13 0 0 0 2.71606i 0 − 4.24703i 0 0 0
2591.14 0 0 0 2.71606i 0 4.24703i 0 0 0
2591.15 0 0 0 4.01832i 0 − 3.08003i 0 0 0
2591.16 0 0 0 4.01832i 0 3.08003i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.c.b ✓ 16
3.b odd 2 1 inner 2592.2.c.b ✓ 16
4.b odd 2 1 inner 2592.2.c.b ✓ 16
8.b even 2 1 5184.2.c.l 16
8.d odd 2 1 5184.2.c.l 16
9.c even 3 1 2592.2.s.i 16
9.c even 3 1 2592.2.s.j 16
9.d odd 6 1 2592.2.s.i 16
9.d odd 6 1 2592.2.s.j 16
12.b even 2 1 inner 2592.2.c.b ✓ 16
24.f even 2 1 5184.2.c.l 16
24.h odd 2 1 5184.2.c.l 16
36.f odd 6 1 2592.2.s.i 16
36.f odd 6 1 2592.2.s.j 16
36.h even 6 1 2592.2.s.i 16
36.h even 6 1 2592.2.s.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.c.b ✓ 16 1.a even 1 1 trivial
2592.2.c.b ✓ 16 3.b odd 2 1 inner
2592.2.c.b ✓ 16 4.b odd 2 1 inner
2592.2.c.b ✓ 16 12.b even 2 1 inner
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 36.f odd 6 1
2592.2.s.i 16 36.h even 6 1
2592.2.s.j 16 9.c even 3 1
2592.2.s.j 16 9.d odd 6 1
2592.2.s.j 16 36.f odd 6 1
2592.2.s.j 16 36.h even 6 1
5184.2.c.l 16 8.b even 2 1
5184.2.c.l 16 8.d odd 2 1
5184.2.c.l 16 24.f even 2 1
5184.2.c.l 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 32T_{5}^{6} + 330T_{5}^{4} + 1280T_{5}^{2} + 1369 \) acting on \(S_{2}^{\mathrm{new}}(2592, [\chi])\). 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
$5$ \( (T^{8} + 32 T^{6} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 40 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 44 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 18 T^{2} + \cdots + 37)^{4} \) 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
$23$ \( (T^{8} - 116 T^{6} + \cdots + 43264)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 56 T^{6} + 594 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 184 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 18 T^{3} + \cdots + 141)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 128 T^{6} + \cdots + 80656)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 232 T^{6} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 176 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 144 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 176 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{3} + \cdots - 3191)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 328 T^{6} + \cdots + 19642624)^{2} \) 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
$79$ \( (T^{8} + 232 T^{6} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 752 T^{6} + \cdots + 951599104)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 360 T^{6} + \cdots + 1418481)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 8 T^{3} + \cdots + 13312)^{4} \) Copy content Toggle raw display
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