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Properties

Label 272.10.b.e
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: \( x^{20} + 270012 x^{18} + 30304010244 x^{16} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{9} + \beta_1) q^{5} + (\beta_{10} - 4 \beta_1) q^{7} + (\beta_{2} - 7318) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{9} + \beta_1) q^{5} + (\beta_{10} - 4 \beta_1) q^{7} + (\beta_{2} - 7318) q^{9} + ( - \beta_{11} - \beta_{10} + \cdots + 21 \beta_1) q^{11}+ \cdots + ( - 24 \beta_{19} + \cdots - 701079 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 146364 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 146364 q^{9} - 1616 q^{13} - 536448 q^{15} - 397764 q^{17} + 246928 q^{19} + 2345784 q^{21} - 4614140 q^{25} - 11447880 q^{33} + 471744 q^{35} - 3134992 q^{43} + 21441600 q^{47} - 110754508 q^{49} + 47604864 q^{51} + 49259368 q^{53} + 62421280 q^{55} - 307011760 q^{59} - 148917712 q^{67} - 491348856 q^{69} + 745516136 q^{77} + 457451916 q^{81} + 1437262352 q^{83} + 300591488 q^{85} + 866290560 q^{87} - 2926970112 q^{89} + 422064264 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 270012 x^{18} + 30304010244 x^{16} + \cdots + 40\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 27001 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36\!\cdots\!23 \nu^{18} + \cdots + 24\!\cdots\!24 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!77 \nu^{18} + \cdots + 10\!\cdots\!28 ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 58\!\cdots\!63 \nu^{18} + \cdots + 13\!\cdots\!12 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!81 \nu^{18} + \cdots + 24\!\cdots\!24 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 66\!\cdots\!06 \nu^{18} + \cdots + 16\!\cdots\!76 ) / 72\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 68\!\cdots\!43 \nu^{18} + \cdots - 11\!\cdots\!32 ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63\!\cdots\!09 \nu^{19} + \cdots + 86\!\cdots\!96 \nu ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!45 \nu^{19} + \cdots + 88\!\cdots\!24 \nu ) / 12\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18\!\cdots\!61 \nu^{19} + \cdots - 21\!\cdots\!04 \nu ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 51\!\cdots\!41 \nu^{19} + \cdots + 32\!\cdots\!40 ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 51\!\cdots\!41 \nu^{19} + \cdots - 32\!\cdots\!40 ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 51\!\cdots\!41 \nu^{19} + \cdots - 41\!\cdots\!68 ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 31\!\cdots\!67 \nu^{19} + \cdots - 13\!\cdots\!52 \nu ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 35\!\cdots\!67 \nu^{19} + \cdots + 17\!\cdots\!28 \nu ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 15\!\cdots\!89 \nu^{19} + \cdots - 13\!\cdots\!44 \nu ) / 63\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 64\!\cdots\!41 \nu^{19} + \cdots - 23\!\cdots\!96 \nu ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 27\!\cdots\!05 \nu^{19} + \cdots - 51\!\cdots\!20 \nu ) / 42\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 27001 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{13} - \beta_{12} + 2\beta_{11} + 10\beta_{10} + 39\beta_{9} - 45545\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 199 \beta_{14} + 466 \beta_{13} - 267 \beta_{12} + 11 \beta_{8} - 8 \beta_{7} + 91 \beta_{6} + \cdots + 1229833133 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 716 \beta_{19} - 12444 \beta_{18} - 16334 \beta_{17} - 86365 \beta_{16} + 2634 \beta_{15} + \cdots + 2486103475 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26709260 \beta_{14} - 56477612 \beta_{13} + 29768352 \beta_{12} - 1362466 \beta_{8} + \cdots - 67186879074966 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 116455636 \beta_{19} + 1539886248 \beta_{18} + 1914175018 \beta_{17} + 6459790708 \beta_{16} + \cdots - 150087853622232 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2504282715750 \beta_{14} + 5169903328182 \beta_{13} - 2665620612432 \beta_{12} + \cdots + 40\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12732453941064 \beta_{19} - 142648201984896 \beta_{18} - 165735914432892 \beta_{17} + \cdots + 96\!\cdots\!76 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20\!\cdots\!64 \beta_{14} + \cdots - 26\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 11\!\cdots\!76 \beta_{19} + \cdots - 64\!\cdots\!32 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16\!\cdots\!48 \beta_{14} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 99\!\cdots\!68 \beta_{19} + \cdots + 44\!\cdots\!92 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12\!\cdots\!40 \beta_{14} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 79\!\cdots\!48 \beta_{19} + \cdots - 31\!\cdots\!56 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 90\!\cdots\!76 \beta_{14} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 61\!\cdots\!00 \beta_{19} + \cdots + 22\!\cdots\!24 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 67\!\cdots\!00 \beta_{14} + \cdots - 60\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 46\!\cdots\!16 \beta_{19} + \cdots - 15\!\cdots\!64 \beta_1 \) 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
− 269.855i
− 224.314i
− 211.875i
− 184.649i
− 161.663i
− 142.179i
− 111.088i
− 77.0132i
− 50.1594i
− 27.3945i
27.3945i
50.1594i
77.0132i
111.088i
142.179i
161.663i
184.649i
211.875i
224.314i
269.855i
0 − 269.855i 0 186.530i 0 7416.24i 0 −53138.6 0
33.2 0 − 224.314i 0 − 2037.30i 0 − 1139.97i 0 −30633.7 0
33.3 0 − 211.875i 0 − 860.161i 0 − 8572.48i 0 −25208.0 0
33.4 0 − 184.649i 0 601.324i 0 − 2058.96i 0 −14412.1 0
33.5 0 − 161.663i 0 2158.77i 0 − 5500.15i 0 −6451.86 0
33.6 0 − 142.179i 0 1772.40i 0 9293.02i 0 −531.738 0
33.7 0 − 111.088i 0 − 2148.27i 0 10178.1i 0 7342.49 0
33.8 0 − 77.0132i 0 − 771.037i 0 6588.65i 0 13752.0 0
33.9 0 − 50.1594i 0 − 1881.16i 0 − 6607.82i 0 17167.0 0
33.10 0 − 27.3945i 0 38.0573i 0 − 4196.17i 0 18932.5 0
33.11 0 27.3945i 0 − 38.0573i 0 4196.17i 0 18932.5 0
33.12 0 50.1594i 0 1881.16i 0 6607.82i 0 17167.0 0
33.13 0 77.0132i 0 771.037i 0 − 6588.65i 0 13752.0 0
33.14 0 111.088i 0 2148.27i 0 − 10178.1i 0 7342.49 0
33.15 0 142.179i 0 − 1772.40i 0 − 9293.02i 0 −531.738 0
33.16 0 161.663i 0 − 2158.77i 0 5500.15i 0 −6451.86 0
33.17 0 184.649i 0 − 601.324i 0 2058.96i 0 −14412.1 0
33.18 0 211.875i 0 860.161i 0 8572.48i 0 −25208.0 0
33.19 0 224.314i 0 2037.30i 0 1139.97i 0 −30633.7 0
33.20 0 269.855i 0 − 186.530i 0 − 7416.24i 0 −53138.6 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.e 20
4.b odd 2 1 136.10.b.a ✓ 20
17.b even 2 1 inner 272.10.b.e 20
68.d odd 2 1 136.10.b.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.10.b.a ✓ 20 4.b odd 2 1
136.10.b.a ✓ 20 68.d odd 2 1
272.10.b.e 20 1.a even 1 1 trivial
272.10.b.e 20 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 270012 T_{3}^{18} + 30304010244 T_{3}^{16} + \cdots + 40\!\cdots\!76 \) 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
$3$ \( T^{20} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 76\!\cdots\!48)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 55\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 48\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 66\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 94\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 19\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 55\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 81\!\cdots\!04)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 37\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 31\!\cdots\!88)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
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