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Properties

Label 272.2.bf.b
Level $272$
Weight $2$
Character orbit 272.bf
Analytic conductor $2.172$
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] = [272,2,Mod(31,272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(272, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 272.bf (of order \(16\), degree \(8\), 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: \(2.17193093498\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 224x^{12} + 1040x^{10} + 2564x^{8} + 3312x^{6} + 2016x^{4} + 416x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{11} - 3 \beta_{10} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots - \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{13} + 16 q^{17} + 48 q^{21} - 16 q^{25} - 16 q^{29} - 16 q^{41} - 16 q^{45} - 64 q^{49} - 16 q^{53} - 96 q^{57} - 64 q^{61} - 32 q^{65} - 32 q^{69} + 48 q^{73} + 32 q^{77} + 80 q^{81} + 16 q^{85} + 48 q^{89} + 64 q^{93} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 224x^{12} + 1040x^{10} + 2564x^{8} + 3312x^{6} + 2016x^{4} + 416x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4 \nu^{15} - 16 \nu^{14} + 145 \nu^{13} - 354 \nu^{12} + 1966 \nu^{11} - 2892 \nu^{10} + 12819 \nu^{9} + \cdots + 1352 ) / 1808 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4 \nu^{15} + 16 \nu^{14} + 145 \nu^{13} + 354 \nu^{12} + 1966 \nu^{11} + 2892 \nu^{10} + 12819 \nu^{9} + \cdots - 1352 ) / 1808 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} + 22\nu^{11} + 180\nu^{9} + 680\nu^{7} + 1202\nu^{5} + 884\nu^{3} + 168\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31 \nu^{15} + 98 \nu^{14} - 700 \nu^{13} + 2140 \nu^{12} - 5914 \nu^{11} + 17318 \nu^{10} + \cdots + 3584 ) / 3616 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31 \nu^{15} + 98 \nu^{14} + 700 \nu^{13} + 2140 \nu^{12} + 5914 \nu^{11} + 17318 \nu^{10} + \cdots + 3584 ) / 3616 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57 \nu^{15} + 82 \nu^{14} - 1360 \nu^{13} + 1786 \nu^{12} - 12478 \nu^{11} + 14426 \nu^{10} + \cdots - 7720 ) / 3616 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49 \nu^{15} + 144 \nu^{14} - 1070 \nu^{13} + 3186 \nu^{12} - 8546 \nu^{11} + 26254 \nu^{10} + \cdots - 6744 ) / 3616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49 \nu^{15} + 144 \nu^{14} + 1070 \nu^{13} + 3186 \nu^{12} + 8546 \nu^{11} + 26254 \nu^{10} + \cdots - 6744 ) / 3616 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 57 \nu^{15} + 82 \nu^{14} + 1360 \nu^{13} + 1786 \nu^{12} + 12478 \nu^{11} + 14426 \nu^{10} + \cdots - 7720 ) / 3616 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 145 \nu^{15} - 124 \nu^{14} + 3420 \nu^{13} - 2800 \nu^{12} + 31096 \nu^{11} - 23882 \nu^{10} + \cdots - 1952 ) / 3616 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 145 \nu^{15} + 124 \nu^{14} + 3420 \nu^{13} + 2800 \nu^{12} + 31096 \nu^{11} + 23882 \nu^{10} + \cdots + 1952 ) / 3616 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 303 \nu^{15} - 228 \nu^{14} + 7170 \nu^{13} - 5214 \nu^{12} + 65700 \nu^{11} - 45166 \nu^{10} + \cdots - 2656 ) / 3616 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 375 \nu^{15} - 30 \nu^{14} + 8650 \nu^{13} - 466 \nu^{12} + 76228 \nu^{11} - 1298 \nu^{10} + \cdots + 5360 ) / 3616 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 375 \nu^{15} + 30 \nu^{14} + 8650 \nu^{13} + 466 \nu^{12} + 76228 \nu^{11} + 1298 \nu^{10} + \cdots - 5360 ) / 3616 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 303 \nu^{15} + 228 \nu^{14} + 7170 \nu^{13} + 5214 \nu^{12} + 65700 \nu^{11} + 45166 \nu^{10} + \cdots + 2656 ) / 3616 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{13} - 3 \beta_{11} - 3 \beta_{10} - 6 \beta_{9} + 5 \beta_{8} - 5 \beta_{7} + \cdots + 8 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{15} + 5 \beta_{14} - 5 \beta_{13} - 5 \beta_{12} + 3 \beta_{9} - 5 \beta_{8} - 5 \beta_{7} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} - 11 \beta_{14} - 11 \beta_{13} + \beta_{12} + 30 \beta_{11} + 30 \beta_{10} + \cdots - 54 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 42 \beta_{15} - 42 \beta_{14} + 42 \beta_{13} + 42 \beta_{12} + \beta_{11} - \beta_{10} - 19 \beta_{9} + \cdots - 102 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9 \beta_{15} + 51 \beta_{14} + 51 \beta_{13} - 9 \beta_{12} - 126 \beta_{11} - 126 \beta_{10} + \cdots + 186 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 340 \beta_{15} + 336 \beta_{14} - 336 \beta_{13} - 340 \beta_{12} - 24 \beta_{11} + 24 \beta_{10} + \cdots + 702 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 104 \beta_{15} - 444 \beta_{14} - 444 \beta_{13} + 104 \beta_{12} + 1020 \beta_{11} + 1020 \beta_{10} + \cdots - 1319 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2723 \beta_{15} - 2643 \beta_{14} + 2643 \beta_{13} + 2723 \beta_{12} + 344 \beta_{11} - 344 \beta_{10} + \cdots - 5018 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1012 \beta_{15} + 3739 \beta_{14} + 3739 \beta_{13} - 1012 \beta_{12} - 8173 \beta_{11} + \cdots + 9564 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 21698 \beta_{15} + 20650 \beta_{14} - 20650 \beta_{13} - 21698 \beta_{12} - 3920 \beta_{11} + \cdots + 36664 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9063 \beta_{15} - 30849 \beta_{14} - 30849 \beta_{13} + 9063 \beta_{12} + 65182 \beta_{11} + \cdots - 70466 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 172260 \beta_{15} - 160884 \beta_{14} + 160884 \beta_{13} + 172260 \beta_{12} + 39438 \beta_{11} + \cdots - 271764 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 77562 \beta_{15} + 251046 \beta_{14} + 251046 \beta_{13} - 77562 \beta_{12} - 518004 \beta_{11} + \cdots + 525276 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
− 1.09993i
2.51414i
1.37802i
− 2.79223i
1.09993i
− 2.51414i
− 1.37802i
2.79223i
1.51472i
− 0.100508i
0.609948i
− 2.02416i
− 1.51472i
0.100508i
− 0.609948i
2.02416i
0 −2.78499 − 0.553969i 0 −1.03153 + 0.689246i 0 0.553969 − 0.829074i 0 4.67766 + 1.93755i 0
31.2 0 2.78499 + 0.553969i 0 −1.03153 + 0.689246i 0 −0.553969 + 0.829074i 0 4.67766 + 1.93755i 0
63.1 0 −0.639223 − 0.956665i 0 2.33809 + 0.465076i 0 0.956665 + 4.80948i 0 0.641449 − 1.54859i 0
63.2 0 0.639223 + 0.956665i 0 2.33809 + 0.465076i 0 −0.956665 − 4.80948i 0 0.641449 − 1.54859i 0
79.1 0 −2.78499 + 0.553969i 0 −1.03153 − 0.689246i 0 0.553969 + 0.829074i 0 4.67766 − 1.93755i 0
79.2 0 2.78499 − 0.553969i 0 −1.03153 − 0.689246i 0 −0.553969 − 0.829074i 0 4.67766 − 1.93755i 0
95.1 0 −0.639223 + 0.956665i 0 2.33809 − 0.465076i 0 0.956665 − 4.80948i 0 0.641449 + 1.54859i 0
95.2 0 0.639223 − 0.956665i 0 2.33809 − 0.465076i 0 −0.956665 + 4.80948i 0 0.641449 + 1.54859i 0
143.1 0 −1.86281 − 1.24469i 0 0.490334 + 2.46508i 0 −1.24469 − 0.247585i 0 0.772765 + 1.86562i 0
143.2 0 1.86281 + 1.24469i 0 0.490334 + 2.46508i 0 1.24469 + 0.247585i 0 0.772765 + 1.86562i 0
159.1 0 −0.604270 − 3.03787i 0 −1.79690 + 2.68925i 0 −3.03787 + 2.02984i 0 −6.09187 + 2.52334i 0
159.2 0 0.604270 + 3.03787i 0 −1.79690 + 2.68925i 0 3.03787 − 2.02984i 0 −6.09187 + 2.52334i 0
175.1 0 −1.86281 + 1.24469i 0 0.490334 − 2.46508i 0 −1.24469 + 0.247585i 0 0.772765 − 1.86562i 0
175.2 0 1.86281 − 1.24469i 0 0.490334 − 2.46508i 0 1.24469 − 0.247585i 0 0.772765 − 1.86562i 0
207.1 0 −0.604270 + 3.03787i 0 −1.79690 − 2.68925i 0 −3.03787 − 2.02984i 0 −6.09187 − 2.52334i 0
207.2 0 0.604270 − 3.03787i 0 −1.79690 − 2.68925i 0 3.03787 + 2.02984i 0 −6.09187 − 2.52334i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.e odd 16 1 inner
68.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.2.bf.b ✓ 16
4.b odd 2 1 inner 272.2.bf.b ✓ 16
17.e odd 16 1 inner 272.2.bf.b ✓ 16
68.i even 16 1 inner 272.2.bf.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
272.2.bf.b ✓ 16 1.a even 1 1 trivial
272.2.bf.b ✓ 16 4.b odd 2 1 inner
272.2.bf.b ✓ 16 17.e odd 16 1 inner
272.2.bf.b ✓ 16 68.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 92T_{3}^{12} + 168T_{3}^{10} + 4232T_{3}^{8} - 23856T_{3}^{6} + 129112T_{3}^{4} + 102800T_{3}^{2} + 264196 \) acting on \(S_{2}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 92 T^{12} + \cdots + 264196 \) Copy content Toggle raw display
$5$ \( (T^{8} + 4 T^{6} + \cdots + 578)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 32 T^{14} + \cdots + 264196 \) Copy content Toggle raw display
$11$ \( T^{16} + 16 T^{14} + \cdots + 264196 \) Copy content Toggle raw display
$13$ \( (T^{8} - 8 T^{7} + \cdots + 4624)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 8 T^{7} + \cdots + 83521)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 88263656464 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 22065914116 \) Copy content Toggle raw display
$29$ \( (T^{8} + 8 T^{7} + \cdots + 167042)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 243990554116 \) Copy content Toggle raw display
$37$ \( (T^{8} - 12 T^{6} + \cdots + 578)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{7} + \cdots + 578)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 88263656464 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1412218503424 \) Copy content Toggle raw display
$53$ \( (T^{8} + 8 T^{7} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 73\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{8} + 32 T^{7} + \cdots + 555458)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 400 T^{6} + \cdots + 19013888)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 653349691218436 \) Copy content Toggle raw display
$73$ \( (T^{8} - 24 T^{7} + \cdots + 555458)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 243990554116 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 88263656464 \) Copy content Toggle raw display
$89$ \( (T^{8} - 24 T^{7} + \cdots + 4624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 32 T^{7} + \cdots + 498837698)^{2} \) Copy content Toggle raw display
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