Properties

Label 1472.4.a.s
Level $1472$
Weight $4$
Character orbit 1472.a
Self dual yes
Analytic conductor $86.851$
Analytic rank $0$
Dimension $3$
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] = [1472,4,Mod(1,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1472.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1472.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: \(86.8508115285\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.761.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 184)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} - 2 \beta_1 - 5) q^{5} + (3 \beta_1 + 5) q^{7} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} - 2 \beta_1 - 5) q^{5} + (3 \beta_1 + 5) q^{7} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{9} + ( - 4 \beta_{2} + 7 \beta_1 - 17) q^{11} + ( - 4 \beta_{2} + 4 \beta_1 + 29) q^{13} + ( - 8 \beta_{2} + 7 \beta_1 + 75) q^{15} + ( - 5 \beta_{2} - 2 \beta_1 - 15) q^{17} + ( - 8 \beta_{2} + 20 \beta_1 - 18) q^{19} + (9 \beta_{2} - 14 \beta_1 - 93) q^{21} - 23 q^{23} + ( - 7 \beta_{2} + 9 \beta_1 + 139) q^{25} + (3 \beta_{2} + 2 \beta_1 - 54) q^{27} + ( - 25 \beta_{2} + 9 \beta_1 + 49) q^{29} + (7 \beta_{2} - 14 \beta_1 + 242) q^{31} + (13 \beta_{2} - 20 \beta_1 - 165) q^{33} + (19 \beta_{2} - 31 \beta_1 - 250) q^{35} + ( - 39 \beta_{2} + 14 \beta_1 + 9) q^{37} + (4 \beta_{2} - 57 \beta_1 - 72) q^{39} + ( - 21 \beta_{2} - 25 \beta_1 + 95) q^{41} + (26 \beta_{2} - 42 \beta_1 - 60) q^{43} + (32 \beta_{2} - 74 \beta_1 + 22) q^{45} + ( - 19 \beta_{2} + 48 \beta_1 + 150) q^{47} + ( - 27 \beta_{2} + 57 \beta_1 - 39) q^{49} + ( - 16 \beta_{2} + \beta_1 + 127) q^{51} + (40 \beta_{2} + 14 \beta_1 + 72) q^{53} + (89 \beta_{2} - 75 \beta_1 - 84) q^{55} + (44 \beta_{2} - 74 \beta_1 - 516) q^{57} + ( - 18 \beta_{2} - 36 \beta_1 - 326) q^{59} + (18 \beta_{2} - 46 \beta_1 + 426) q^{61} + ( - 24 \beta_{2} + 90 \beta_1 + 182) q^{63} + (19 \beta_{2} - 146 \beta_1 - 89) q^{65} + ( - 6 \beta_{2} - 49 \beta_1 + 273) q^{67} + 23 \beta_1 q^{69} + ( - 38 \beta_{2} - 19 \beta_1 + 70) q^{71} + ( - 46 \beta_{2} + 32 \beta_1 - 137) q^{73} + (13 \beta_{2} - 194 \beta_1 - 188) q^{75} + ( - 59 \beta_{2} + 95 \beta_1 + 410) q^{77} + ( - 66 \beta_{2} - 14 \beta_1 - 26) q^{79} + (93 \beta_{2} - 21 \beta_1 - 209) q^{81} + ( - 26 \beta_{2} - 53 \beta_1 + 167) q^{83} + (19 \beta_{2} - 31 \beta_1 + 670) q^{85} + ( - 23 \beta_{2} - 176 \beta_1 + 46) q^{87} + ( - 94 \beta_{2} - 52 \beta_1 - 454) q^{89} + ( - 32 \beta_{2} + 191 \beta_1 + 361) q^{91} + ( - 28 \beta_{2} - 172 \beta_1 + 343) q^{93} + (210 \beta_{2} - 224 \beta_1 - 698) q^{95} + (77 \beta_{2} + 60 \beta_1 + 801) q^{97} + (74 \beta_{2} + 88 \beta_1 + 910) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 16 q^{5} + 18 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 16 q^{5} + 18 q^{7} + 18 q^{9} - 40 q^{11} + 95 q^{13} + 240 q^{15} - 42 q^{17} - 26 q^{19} - 302 q^{21} - 69 q^{23} + 433 q^{25} - 163 q^{27} + 181 q^{29} + 705 q^{31} - 528 q^{33} - 800 q^{35} + 80 q^{37} - 277 q^{39} + 281 q^{41} - 248 q^{43} - 40 q^{45} + 517 q^{47} - 33 q^{49} + 398 q^{51} + 190 q^{53} - 416 q^{55} - 1666 q^{57} - 996 q^{59} + 1214 q^{61} + 660 q^{63} - 432 q^{65} + 776 q^{67} + 23 q^{69} + 229 q^{71} - 333 q^{73} - 771 q^{75} + 1384 q^{77} - 26 q^{79} - 741 q^{81} + 474 q^{83} + 1960 q^{85} - 15 q^{87} - 1320 q^{89} + 1306 q^{91} + 885 q^{93} - 2528 q^{95} + 2386 q^{97} + 2744 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} - 4\nu - 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 9 ) / 2 \) 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
−1.89195
3.06443
−0.172480
0 −7.72680 0 −18.6126 0 28.1804 0 32.7035 0
1.2 0 0.476220 0 −13.8291 0 3.57134 0 −26.7732 0
1.3 0 6.25058 0 16.4417 0 −13.7517 0 12.0698 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(23\) \( +1 \)

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 1472.4.a.s 3
4.b odd 2 1 1472.4.a.t 3
8.b even 2 1 184.4.a.d 3
8.d odd 2 1 368.4.a.i 3
24.h odd 2 1 1656.4.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.4.a.d 3 8.b even 2 1
368.4.a.i 3 8.d odd 2 1
1472.4.a.s 3 1.a even 1 1 trivial
1472.4.a.t 3 4.b odd 2 1
1656.4.a.i 3 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + T_{3}^{2} - 49T_{3} + 23 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1472))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} + \cdots + 23 \) Copy content Toggle raw display
$5$ \( T^{3} + 16 T^{2} + \cdots - 4232 \) Copy content Toggle raw display
$7$ \( T^{3} - 18 T^{2} + \cdots + 1384 \) Copy content Toggle raw display
$11$ \( T^{3} + 40 T^{2} + \cdots - 66056 \) Copy content Toggle raw display
$13$ \( T^{3} - 95 T^{2} + \cdots + 32179 \) Copy content Toggle raw display
$17$ \( T^{3} + 42 T^{2} + \cdots - 56456 \) Copy content Toggle raw display
$19$ \( T^{3} + 26 T^{2} + \cdots - 1143816 \) Copy content Toggle raw display
$23$ \( (T + 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 181 T^{2} + \cdots + 7111481 \) Copy content Toggle raw display
$31$ \( T^{3} - 705 T^{2} + \cdots - 10237599 \) Copy content Toggle raw display
$37$ \( T^{3} - 80 T^{2} + \cdots + 19357544 \) Copy content Toggle raw display
$41$ \( T^{3} - 281 T^{2} + \cdots - 2573499 \) Copy content Toggle raw display
$43$ \( T^{3} + 248 T^{2} + \cdots - 2773504 \) Copy content Toggle raw display
$47$ \( T^{3} - 517 T^{2} + \cdots + 647501 \) Copy content Toggle raw display
$53$ \( T^{3} - 190 T^{2} + \cdots + 18150072 \) Copy content Toggle raw display
$59$ \( T^{3} + 996 T^{2} + \cdots - 16600384 \) Copy content Toggle raw display
$61$ \( T^{3} - 1214 T^{2} + \cdots - 12905704 \) Copy content Toggle raw display
$67$ \( T^{3} - 776 T^{2} + \cdots + 14223944 \) Copy content Toggle raw display
$71$ \( T^{3} - 229 T^{2} + \cdots - 1059547 \) Copy content Toggle raw display
$73$ \( T^{3} + 333 T^{2} + \cdots + 8715807 \) Copy content Toggle raw display
$79$ \( T^{3} + 26 T^{2} + \cdots - 5488392 \) Copy content Toggle raw display
$83$ \( T^{3} - 474 T^{2} + \cdots - 8830248 \) Copy content Toggle raw display
$89$ \( T^{3} + 1320 T^{2} + \cdots - 655097408 \) Copy content Toggle raw display
$97$ \( T^{3} - 2386 T^{2} + \cdots + 449588984 \) Copy content Toggle raw display
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