Properties

Label 48.11.g.c
Level $48$
Weight $11$
Character orbit 48.g
Analytic conductor $30.497$
Analytic rank $0$
Dimension $4$
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] = [48,11,Mod(31,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 48.g (of order \(2\), degree \(1\), 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: \(30.4971481283\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{2545})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 637x^{2} + 636x + 404496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 81 \beta_1 q^{3} + ( - \beta_{2} + 750) q^{5} + (13 \beta_{3} + 428 \beta_1) q^{7} - 19683 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 \beta_1 q^{3} + ( - \beta_{2} + 750) q^{5} + (13 \beta_{3} + 428 \beta_1) q^{7} - 19683 q^{9} + ( - 82 \beta_{3} + 74148 \beta_1) q^{11} + (294 \beta_{2} + 216034) q^{13} + (81 \beta_{3} + 60750 \beta_1) q^{15} + ( - 1766 \beta_{2} - 13782) q^{17} + (646 \beta_{3} + 939628 \beta_1) q^{19} + (3159 \beta_{2} - 104004) q^{21} + ( - 1926 \beta_{3} - 325296 \beta_1) q^{23} + ( - 1500 \beta_{2} - 7737205) q^{25} - 1594323 \beta_1 q^{27} + (7297 \beta_{2} - 20197578) q^{29} + (3581 \beta_{3} + 4476852 \beta_1) q^{31} + ( - 19926 \beta_{2} - 18017964) q^{33} + (10178 \beta_{3} + 19377960 \beta_1) q^{35} + ( - 17604 \beta_{2} - 53127910) q^{37} + ( - 23814 \beta_{3} + 17498754 \beta_1) q^{39} + (63270 \beta_{2} - 136113318) q^{41} + ( - 28406 \beta_{3} + 93464964 \beta_1) q^{43} + (19683 \beta_{2} - 14762250) q^{45} + (25238 \beta_{3} + 145810248 \beta_1) q^{47} + (33384 \beta_{2} - 461295743) q^{49} + (143046 \beta_{3} - 1116342 \beta_1) q^{51} + ( - 366207 \beta_{2} - 222201018) q^{53} + (12648 \beta_{3} - 64594440 \beta_1) q^{55} + (156978 \beta_{2} - 228329604) q^{57} + ( - 323912 \beta_{3} + 38534412 \beta_1) q^{59} + (461112 \beta_{2} - 52083814) q^{61} + ( - 255879 \beta_{3} - 8424324 \beta_1) q^{63} + (4466 \beta_{2} - 268954980) q^{65} + (194808 \beta_{3} - 204681404 \beta_1) q^{67} + ( - 468018 \beta_{2} + 79046928) q^{69} + (1097858 \beta_{3} + 324580704 \beta_1) q^{71} + ( - 1409784 \beta_{2} + 1220417362) q^{73} + (121500 \beta_{3} - 626713605 \beta_1) q^{75} + (2786484 \beta_{2} + 4592806128) q^{77} + ( - 863351 \beta_{3} - 110548940 \beta_1) q^{79} + 387420489 q^{81} + ( - 1745378 \beta_{3} + 209882172 \beta_1) q^{83} + ( - 1310718 \beta_{2} + 2578478220) q^{85} + ( - 591057 \beta_{3} - 1636003818 \beta_1) q^{87} + ( - 2291668 \beta_{2} + 5385095634) q^{89} + (2682610 \beta_{3} - 5510283688 \beta_1) q^{91} + (870183 \beta_{2} - 1087875036) q^{93} + (1424128 \beta_{3} + 1651705320 \beta_1) q^{95} + (7560444 \beta_{2} + 3741442114) q^{97} + (1614006 \beta_{3} - 1459455084 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3000 q^{5} - 78732 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3000 q^{5} - 78732 q^{9} + 864136 q^{13} - 55128 q^{17} - 416016 q^{21} - 30948820 q^{25} - 80790312 q^{29} - 72071856 q^{33} - 212511640 q^{37} - 544453272 q^{41} - 59049000 q^{45} - 1845182972 q^{49} - 888804072 q^{53} - 913318416 q^{57} - 208335256 q^{61} - 1075819920 q^{65} + 316187712 q^{69} + 4881669448 q^{73} + 18371224512 q^{77} + 1549681956 q^{81} + 10313912880 q^{85} + 21540382536 q^{89} - 4351500144 q^{93} + 14965768456 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 637x^{2} + 636x + 404496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 637\nu^{2} - 637\nu + 201930 ) / 202566 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\nu^{3} + 45816 ) / 637 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{3} - 4\nu^{2} + 5092\nu + 1272 ) / 53 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 30552\beta _1 - 30552 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 637\beta_{2} - 45816 ) / 48 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−12.3620 + 21.4116i
12.8620 22.2776i
−12.3620 21.4116i
12.8620 + 22.2776i
0 140.296i 0 −460.752 0 26520.8i 0 −19683.0 0
31.2 0 140.296i 0 1960.75 0 28003.4i 0 −19683.0 0
31.3 0 140.296i 0 −460.752 0 26520.8i 0 −19683.0 0
31.4 0 140.296i 0 1960.75 0 28003.4i 0 −19683.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.11.g.c 4
3.b odd 2 1 144.11.g.e 4
4.b odd 2 1 inner 48.11.g.c 4
8.b even 2 1 192.11.g.b 4
8.d odd 2 1 192.11.g.b 4
12.b even 2 1 144.11.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.11.g.c 4 1.a even 1 1 trivial
48.11.g.c 4 4.b odd 2 1 inner
144.11.g.e 4 3.b odd 2 1
144.11.g.e 4 12.b even 2 1
192.11.g.b 4 8.b even 2 1
192.11.g.b 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 1500T_{5} - 903420 \) acting on \(S_{11}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 19683)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 1500 T - 903420)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 55\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{2} - 432068 T - 80037571964)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + \cdots - 4571656851996)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots + 329887474368804)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots + 23\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots + 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 69\!\cdots\!24)^{2} \) Copy content Toggle raw display
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