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Properties

Label 272.10.b.f
Level $272$
Weight $10$
Character orbit 272.b
Analytic conductor $140.090$
Analytic rank $0$
Dimension $20$
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] = [272,10,Mod(33,272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(272, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.33");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 272.b (of order \(2\), degree \(1\), 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: \(140.089747437\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 269112 x^{18} + 29775318564 x^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{8}\cdot 43^{2} \)
Twist minimal: no (minimal twist has level 136)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{10} - \beta_1) q^{5} + (\beta_{11} + 5 \beta_1) q^{7} + (\beta_{2} - 7228) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{10} - \beta_1) q^{5} + (\beta_{11} + 5 \beta_1) q^{7} + (\beta_{2} - 7228) q^{9} + ( - \beta_{12} + \beta_{10} + 24 \beta_1) q^{11} + (\beta_{5} - \beta_{2} + 8387) q^{13} + (\beta_{4} - 4 \beta_{2} + 26822) q^{15} + ( - \beta_{14} + 35 \beta_{10} + \cdots + 1513) q^{17}+ \cdots + ( - 6 \beta_{19} + \cdots - 1796509 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 144564 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 144564 q^{9} + 167736 q^{13} + 536448 q^{15} + 30260 q^{17} + 928928 q^{19} - 2602048 q^{21} - 10286604 q^{25} - 12939328 q^{33} + 5838544 q^{35} + 37016784 q^{43} - 37133280 q^{47} - 13774548 q^{49} - 146859120 q^{51} + 35272008 q^{53} + 43617536 q^{55} + 260532080 q^{59} + 332639008 q^{67} + 524679408 q^{69} + 339280912 q^{77} + 1708083636 q^{81} + 821368592 q^{83} + 1847368816 q^{85} + 370537120 q^{87} + 168151576 q^{89} - 1691367952 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 269112 x^{18} + 29775318564 x^{16} + \cdots + 27\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 26911 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!27 \nu^{18} + \cdots + 80\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48\!\cdots\!84 \nu^{18} + \cdots + 11\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!33 \nu^{18} + \cdots + 42\!\cdots\!20 ) / 32\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 78\!\cdots\!17 \nu^{18} + \cdots + 28\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!11 \nu^{18} + \cdots - 20\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!03 \nu^{18} + \cdots + 18\!\cdots\!00 ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35\!\cdots\!91 \nu^{18} + \cdots + 15\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30\!\cdots\!87 \nu^{19} + \cdots - 47\!\cdots\!00 \nu ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51\!\cdots\!07 \nu^{19} + \cdots - 53\!\cdots\!00 \nu ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 79\!\cdots\!13 \nu^{19} + \cdots + 32\!\cdots\!00 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 39\!\cdots\!39 \nu^{19} + \cdots + 44\!\cdots\!20 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25\!\cdots\!59 \nu^{19} + \cdots - 70\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25\!\cdots\!59 \nu^{19} + \cdots + 70\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!97 \nu^{19} + \cdots + 16\!\cdots\!00 \nu ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 44\!\cdots\!83 \nu^{19} + \cdots + 98\!\cdots\!00 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 28\!\cdots\!21 \nu^{19} + \cdots + 48\!\cdots\!00 \nu ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 59\!\cdots\!29 \nu^{19} + \cdots + 11\!\cdots\!00 \nu ) / 58\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 26911 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + 13\beta_{12} + 3\beta_{11} - 740\beta_{10} - 47828\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 381 \beta_{15} + 381 \beta_{14} + 8 \beta_{9} - 19 \beta_{8} + 131 \beta_{7} - 44 \beta_{6} + \cdots + 1287050070 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2016 \beta_{19} - 1452 \beta_{18} + 8142 \beta_{17} - 18666 \beta_{16} - 137965 \beta_{15} + \cdots + 2764761803 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 38452416 \beta_{15} - 38452416 \beta_{14} - 1064432 \beta_{9} + 2225626 \beta_{8} + \cdots - 74385417393318 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 292197852 \beta_{19} + 299392188 \beta_{18} - 856264638 \beta_{17} + 2455835202 \beta_{16} + \cdots - 177231392816684 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2931504785136 \beta_{15} + 2931504785136 \beta_{14} + 102709760576 \beta_{9} - 187519477198 \beta_{8} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 31093824444552 \beta_{19} - 36140382753048 \beta_{18} + 67413068523252 \beta_{17} + \cdots + 12\!\cdots\!40 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20\!\cdots\!80 \beta_{15} + \cdots - 32\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 29\!\cdots\!44 \beta_{19} + \cdots - 85\!\cdots\!32 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14\!\cdots\!72 \beta_{15} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25\!\cdots\!96 \beta_{19} + \cdots + 62\!\cdots\!68 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 95\!\cdots\!36 \beta_{15} + \cdots - 16\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 21\!\cdots\!68 \beta_{19} + \cdots - 45\!\cdots\!80 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 65\!\cdots\!68 \beta_{15} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 17\!\cdots\!68 \beta_{19} + \cdots + 33\!\cdots\!52 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 45\!\cdots\!24 \beta_{15} + \cdots - 91\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 13\!\cdots\!40 \beta_{19} + \cdots - 25\!\cdots\!64 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
− 275.918i
− 236.127i
− 194.375i
− 184.543i
− 177.253i
− 117.659i
− 110.684i
− 83.8369i
− 25.1892i
− 14.5255i
14.5255i
25.1892i
83.8369i
110.684i
117.659i
177.253i
184.543i
194.375i
236.127i
275.918i
0 − 275.918i 0 2640.74i 0 − 2732.73i 0 −56448.0 0
33.2 0 − 236.127i 0 − 1060.31i 0 813.037i 0 −36072.8 0
33.3 0 − 194.375i 0 − 868.674i 0 − 9920.16i 0 −18098.8 0
33.4 0 − 184.543i 0 − 2021.55i 0 4948.74i 0 −14373.2 0
33.5 0 − 177.253i 0 966.196i 0 8468.59i 0 −11735.7 0
33.6 0 − 117.659i 0 877.655i 0 − 10815.5i 0 5839.41 0
33.7 0 − 110.684i 0 966.814i 0 − 2205.83i 0 7432.13 0
33.8 0 − 83.8369i 0 442.321i 0 4446.19i 0 12654.4 0
33.9 0 − 25.1892i 0 − 2564.31i 0 − 6231.81i 0 19048.5 0
33.10 0 − 14.5255i 0 − 1525.00i 0 5218.62i 0 19472.0 0
33.11 0 14.5255i 0 1525.00i 0 − 5218.62i 0 19472.0 0
33.12 0 25.1892i 0 2564.31i 0 6231.81i 0 19048.5 0
33.13 0 83.8369i 0 − 442.321i 0 − 4446.19i 0 12654.4 0
33.14 0 110.684i 0 − 966.814i 0 2205.83i 0 7432.13 0
33.15 0 117.659i 0 − 877.655i 0 10815.5i 0 5839.41 0
33.16 0 177.253i 0 − 966.196i 0 − 8468.59i 0 −11735.7 0
33.17 0 184.543i 0 2021.55i 0 − 4948.74i 0 −14373.2 0
33.18 0 194.375i 0 868.674i 0 9920.16i 0 −18098.8 0
33.19 0 236.127i 0 1060.31i 0 − 813.037i 0 −36072.8 0
33.20 0 275.918i 0 − 2640.74i 0 2732.73i 0 −56448.0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.10.b.f 20
4.b odd 2 1 136.10.b.b ✓ 20
17.b even 2 1 inner 272.10.b.f 20
68.d odd 2 1 136.10.b.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.10.b.b ✓ 20 4.b odd 2 1
136.10.b.b ✓ 20 68.d odd 2 1
272.10.b.f 20 1.a even 1 1 trivial
272.10.b.f 20 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 269112 T_{3}^{18} + 29775318564 T_{3}^{16} + \cdots + 27\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 90\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 14\!\cdots\!12)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 55\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 44\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 56\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 86\!\cdots\!84)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 71\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 41\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 62\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 42\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 52\!\cdots\!92 \) Copy content Toggle raw display
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