Properties

Label 272.10.a.d
Level $272$
Weight $10$
Character orbit 272.a
Self dual yes
Analytic conductor $140.090$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [272,10,Mod(1,272)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 272.a (trivial)

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: yes
Analytic conductor: \(140.089747437\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 24887x^{2} - 1623822x - 15343848 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 34)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 57) q^{3} + ( - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 412) q^{5} + (8 \beta_{3} + \beta_{2} - 15 \beta_1 - 671) q^{7} + (63 \beta_{3} + 31 \beta_{2} - 24 \beta_1 + 23605) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 57) q^{3} + ( - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 412) q^{5} + (8 \beta_{3} + \beta_{2} - 15 \beta_1 - 671) q^{7} + (63 \beta_{3} + 31 \beta_{2} - 24 \beta_1 + 23605) q^{9} + (70 \beta_{3} - 70 \beta_{2} + 27 \beta_1 - 32013) q^{11} + ( - 29 \beta_{3} + 87 \beta_{2} + 368 \beta_1 - 4026) q^{13} + ( - 258 \beta_{3} + 492 \beta_{2} + 1710 \beta_1 - 125586) q^{15} - 83521 q^{17} + (1384 \beta_{3} - 302 \beta_{2} + 2290 \beta_1 - 125346) q^{19} + (579 \beta_{3} + 183 \beta_{2} - 2356 \beta_1 + 509664) q^{21} + (970 \beta_{3} + 755 \beta_{2} - 2973 \beta_1 + 99723) q^{23} + (5478 \beta_{3} - 1174 \beta_{2} - 4676 \beta_1 + 1624911) q^{25} + ( - 954 \beta_{3} - 6148 \beta_{2} - 26148 \beta_1 - 239116) q^{27} + (301 \beta_{3} - 6693 \beta_{2} - 11212 \beta_1 + 1142092) q^{29} + (14514 \beta_{3} + 4003 \beta_{2} + 113 \beta_1 + 1238201) q^{31} + ( - 6321 \beta_{3} + 12603 \beta_{2} + 38820 \beta_1 - 525132) q^{33} + ( - 4786 \beta_{3} + 1484 \beta_{2} + 27050 \beta_1 - 3733982) q^{35} + (21377 \beta_{3} - 14241 \beta_{2} - 38888 \beta_1 + 2803204) q^{37} + ( - 20226 \beta_{3} - 28692 \beta_{2} + 16840 \beta_1 - 13877136) q^{39} + ( - 33186 \beta_{3} + 21758 \beta_{2} - 10964 \beta_1 - 2879710) q^{41} + ( - 37122 \beta_{3} + 25006 \beta_{2} - 22222 \beta_1 + 13410614) q^{43} + ( - 66807 \beta_{3} - 90765 \beta_{2} + 107496 \beta_1 - 48110544) q^{45} + ( - 30898 \beta_{3} - 34386 \beta_{2} - 115376 \beta_1 + 14342696) q^{47} + ( - 22727 \beta_{3} + 3541 \beta_{2} - 48212 \beta_1 - 24568279) q^{49} + (83521 \beta_1 + 4760697) q^{51} + (110456 \beta_{3} + 35632 \beta_{2} - 163500 \beta_1 - 24607866) q^{53} + (135858 \beta_{3} - 8528 \beta_{2} - 42898 \beta_1 + 72637726) q^{55} + ( - 216138 \beta_{3} - 23826 \beta_{2} + \cdots - 107726652) q^{57}+ \cdots + ( - 3293208 \beta_{3} - 2308206 \beta_{2} + \cdots - 768638811) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 226 q^{3} - 1656 q^{5} - 2654 q^{7} + 94468 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 226 q^{3} - 1656 q^{5} - 2654 q^{7} + 94468 q^{9} - 128106 q^{11} - 16840 q^{13} - 505764 q^{15} - 334084 q^{17} - 505964 q^{19} + 2043368 q^{21} + 404838 q^{23} + 6508996 q^{25} - 904168 q^{27} + 4590792 q^{29} + 4952578 q^{31} - 2178168 q^{33} - 14990028 q^{35} + 11290592 q^{37} - 55542224 q^{39} - 11496912 q^{41} + 53686900 q^{43} - 192657168 q^{45} + 57601536 q^{47} - 98176692 q^{49} + 18875746 q^{51} - 98104464 q^{53} + 290636700 q^{55} - 431148328 q^{57} - 1116948 q^{59} - 62897560 q^{61} + 278468302 q^{63} - 73031304 q^{65} - 69022280 q^{67} + 396672096 q^{69} - 202453914 q^{71} + 312667256 q^{73} + 17055314 q^{75} + 203294832 q^{77} - 886063442 q^{79} + 2402225260 q^{81} - 1760033004 q^{83} + 138310776 q^{85} + 1469185668 q^{87} + 100343712 q^{89} - 667416592 q^{91} - 1212199744 q^{93} + 2064031944 q^{95} + 63193280 q^{97} - 3076730454 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 24887x^{2} - 1623822x - 15343848 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{3} + 947\nu^{2} + 208140\nu + 1911753 ) / 12849 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\nu^{3} - 1894\nu^{2} - 339186\nu - 3874902 ) / 12849 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{3} + 90\nu^{2} - 212894\nu - 11055150 ) / 12849 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 4 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 99\beta_{3} + 79\beta_{2} + 230\beta _1 + 74782 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8523\beta_{3} + 25723\beta_{2} + 50636\beta _1 + 7556512 ) / 6 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
185.613
−99.7587
−11.4440
−72.4099
0 −277.187 0 −1826.15 0 −1008.55 0 57149.5 0
1.2 0 −173.189 0 2313.45 0 −1296.04 0 10311.3 0
1.3 0 −31.3410 0 220.193 0 −5673.79 0 −18700.7 0
1.4 0 255.716 0 −2363.49 0 5324.38 0 45707.9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.10.a.d 4
4.b odd 2 1 34.10.a.d 4
12.b even 2 1 306.10.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.10.a.d 4 4.b odd 2 1
272.10.a.d 4 1.a even 1 1 trivial
306.10.a.l 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 226T_{3}^{3} - 61062T_{3}^{2} - 14380776T_{3} - 384736824 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(272))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 226 T^{3} + \cdots - 384736824 \) Copy content Toggle raw display
$5$ \( T^{4} + 1656 T^{3} + \cdots + 2198652595200 \) Copy content Toggle raw display
$7$ \( T^{4} + 2654 T^{3} + \cdots - 39487188537440 \) Copy content Toggle raw display
$11$ \( T^{4} + 128106 T^{3} + \cdots - 84\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{4} + 16840 T^{3} + \cdots + 22\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( (T + 83521)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 505964 T^{3} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( T^{4} - 404838 T^{3} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} - 4590792 T^{3} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} - 4952578 T^{3} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} - 11290592 T^{3} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{4} + 11496912 T^{3} + \cdots - 28\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{4} - 53686900 T^{3} + \cdots - 21\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{4} - 57601536 T^{3} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + 98104464 T^{3} + \cdots - 10\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{4} + 1116948 T^{3} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + 62897560 T^{3} + \cdots + 85\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{4} + 69022280 T^{3} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{4} + 202453914 T^{3} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} - 312667256 T^{3} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} + 886063442 T^{3} + \cdots + 89\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{4} + 1760033004 T^{3} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} - 100343712 T^{3} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} - 63193280 T^{3} + \cdots + 38\!\cdots\!52 \) Copy content Toggle raw display
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