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Properties

Label 272.10.b.d
Level $272$
Weight $10$
Character orbit 272.b
Analytic conductor $140.090$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 196356 x^{12} + 14438324660 x^{10} + 509584678872768 x^{8} + \cdots + 89\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 68)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{7} - \beta_1) q^{5} + (\beta_{8} - \beta_1) q^{7} + (\beta_{2} - 8368) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{7} - \beta_1) q^{5} + (\beta_{8} - \beta_1) q^{7} + (\beta_{2} - 8368) q^{9} + ( - \beta_{9} - 2 \beta_{8} + 15 \beta_1) q^{11} + (\beta_{3} + 2525) q^{13} + ( - \beta_{4} - \beta_{3} + 34667) q^{15} + (\beta_{11} - \beta_{9} - 3 \beta_{8} + \cdots - 23820) q^{17}+ \cdots + ( - 152 \beta_{13} + \cdots + 1893841 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 117150 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 117150 q^{9} + 35348 q^{13} + 485344 q^{15} - 333474 q^{17} - 635720 q^{19} + 300584 q^{21} - 2207882 q^{25} - 5711576 q^{33} + 1299504 q^{35} - 16229784 q^{43} - 67994544 q^{47} - 136889654 q^{49} + 89536928 q^{51} - 24892188 q^{53} + 12622496 q^{55} + 481509672 q^{59} - 328705096 q^{67} - 234819080 q^{69} + 1420601208 q^{77} + 1592694790 q^{81} + 27071832 q^{83} - 1499524848 q^{85} + 2611633760 q^{87} + 2440879068 q^{89} - 2523196424 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 196356 x^{12} + 14438324660 x^{10} + 509584678872768 x^{8} + \cdots + 89\!\cdots\!08 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 28051 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 48\!\cdots\!60 \nu^{12} + \cdots - 38\!\cdots\!48 ) / 87\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 56\!\cdots\!83 \nu^{12} + \cdots + 41\!\cdots\!32 ) / 69\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19\!\cdots\!53 \nu^{12} + \cdots - 16\!\cdots\!32 ) / 69\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29\!\cdots\!49 \nu^{12} + \cdots - 27\!\cdots\!32 ) / 87\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 72\!\cdots\!63 \nu^{13} + \cdots + 33\!\cdots\!72 \nu ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!81 \nu^{13} + \cdots + 13\!\cdots\!08 \nu ) / 80\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!83 \nu^{13} + \cdots - 95\!\cdots\!80 \nu ) / 75\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 36\!\cdots\!39 \nu^{13} + \cdots + 52\!\cdots\!00 ) / 60\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 36\!\cdots\!39 \nu^{13} + \cdots + 52\!\cdots\!00 ) / 60\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!41 \nu^{13} + \cdots - 12\!\cdots\!04 \nu ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21\!\cdots\!37 \nu^{13} + \cdots - 19\!\cdots\!08 \nu ) / 13\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 28051 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} + 3\beta_{11} - 3\beta_{10} - 22\beta_{9} - 116\beta_{8} - 364\beta_{7} - 49353\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 24 \beta_{11} + 24 \beta_{10} + 120 \beta_{6} - 343 \beta_{5} - 373 \beta_{4} + 507 \beta_{3} + \cdots + 1382729620 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 80499 \beta_{13} + 78523 \beta_{12} - 275725 \beta_{11} + 275725 \beta_{10} + 2723427 \beta_{9} + \cdots + 3027851317 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7865424 \beta_{11} - 7865424 \beta_{10} - 19044792 \beta_{6} + 41549898 \beta_{5} + \cdots - 84891899733664 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5840430970 \beta_{13} - 5270655922 \beta_{12} + 22993157598 \beta_{11} - 22993157598 \beta_{10} + \cdots - 201107060646318 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 928255684848 \beta_{11} + 928255684848 \beta_{10} + 1982820921456 \beta_{6} - 3784426583332 \beta_{5} + \cdots + 56\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 421328001195348 \beta_{13} + 346360329812884 \beta_{12} + \cdots + 13\!\cdots\!60 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 85\!\cdots\!04 \beta_{11} + \cdots - 39\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 30\!\cdots\!48 \beta_{13} + \cdots - 98\!\cdots\!80 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 70\!\cdots\!48 \beta_{11} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 22\!\cdots\!32 \beta_{13} + \cdots + 70\!\cdots\!36 \beta_1 \) 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
− 273.007i
− 229.350i
− 177.098i
− 107.663i
− 102.660i
− 96.4665i
− 80.1348i
80.1348i
96.4665i
102.660i
107.663i
177.098i
229.350i
273.007i
0 − 273.007i 0 686.912i 0 − 7157.59i 0 −54850.0 0
33.2 0 − 229.350i 0 − 1619.14i 0 5417.26i 0 −32918.4 0
33.3 0 − 177.098i 0 2131.96i 0 9240.78i 0 −11680.6 0
33.4 0 − 107.663i 0 1557.83i 0 4865.08i 0 8091.58 0
33.5 0 − 102.660i 0 − 873.124i 0 − 2814.66i 0 9143.85 0
33.6 0 − 96.4665i 0 − 1439.77i 0 − 177.975i 0 10377.2 0
33.7 0 − 80.1348i 0 1369.31i 0 − 12382.5i 0 13261.4 0
33.8 0 80.1348i 0 − 1369.31i 0 12382.5i 0 13261.4 0
33.9 0 96.4665i 0 1439.77i 0 177.975i 0 10377.2 0
33.10 0 102.660i 0 873.124i 0 2814.66i 0 9143.85 0
33.11 0 107.663i 0 − 1557.83i 0 − 4865.08i 0 8091.58 0
33.12 0 177.098i 0 − 2131.96i 0 − 9240.78i 0 −11680.6 0
33.13 0 229.350i 0 1619.14i 0 − 5417.26i 0 −32918.4 0
33.14 0 273.007i 0 − 686.912i 0 7157.59i 0 −54850.0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.14
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.d 14
4.b odd 2 1 68.10.b.a ✓ 14
17.b even 2 1 inner 272.10.b.d 14
68.d odd 2 1 68.10.b.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.10.b.a ✓ 14 4.b odd 2 1
68.10.b.a ✓ 14 68.d odd 2 1
272.10.b.d 14 1.a even 1 1 trivial
272.10.b.d 14 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 196356 T_{3}^{12} + 14438324660 T_{3}^{10} + 509584678872768 T_{3}^{8} + \cdots + 89\!\cdots\!08 \) acting on \(S_{10}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 89\!\cdots\!08 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots - 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 32\!\cdots\!13 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots - 22\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 62\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 95\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 50\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 41\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 35\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
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