Properties

Label 48.17.g.a
Level $48$
Weight $17$
Character orbit 48.g
Analytic conductor $77.916$
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] = [48,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 17 \)
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: \(77.9157810512\)
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: \( x^{4} - x^{3} + 379501x^{2} + 379500x + 144020250000 \) 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.

\(f(q)\) \(=\) \( q - 2187 \beta_1 q^{3} + (11 \beta_{2} - 35106) q^{5} + ( - 91 \beta_{3} - 323708 \beta_1) q^{7} - 14348907 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2187 \beta_1 q^{3} + (11 \beta_{2} - 35106) q^{5} + ( - 91 \beta_{3} - 323708 \beta_1) q^{7} - 14348907 q^{9} + (1942 \beta_{3} - 6709020 \beta_1) q^{11} + (15990 \beta_{2} - 146671382) q^{13} + (24057 \beta_{3} + 76776822 \beta_1) q^{15} + (219082 \beta_{2} - 142751334) q^{17} + ( - 120658 \beta_{3} + 1529491436 \beta_1) q^{19} + (597051 \beta_{2} - 2123848188) q^{21} + ( - 1376694 \beta_{3} + 6301259136 \beta_1) q^{23} + ( - 772332 \beta_{2} - 45556861693) q^{25} + 31381059609 \beta_1 q^{27} + (2334541 \beta_{2} - 146405354106) q^{29} + (11038093 \beta_{3} + 300555197100 \beta_1) q^{31} + ( - 12741462 \beta_{2} - 44017880220) q^{33} + (6755434 \beta_{3} + 886607037624 \beta_1) q^{35} + ( - 109846044 \beta_{2} - 692617627774) q^{37} + (34970130 \beta_{3} + 320770312434 \beta_1) q^{39} + (284088582 \beta_{2} - 2436174278406) q^{41} + ( - 205269646 \beta_{3} + 1065445224228 \beta_1) q^{43} + ( - 157837977 \beta_{2} + 503732729142) q^{45} + ( - 132108746 \beta_{3} - 2288615374248 \beta_1) q^{47} + (176744568 \beta_{2} + 11196631428241) q^{49} + (479132334 \beta_{3} + 312197167458 \beta_1) q^{51} + (802278741 \beta_{2} + 23487743348550) q^{53} + (5623368 \beta_{3} - 18442734664392 \beta_1) q^{55} + (791637138 \beta_{2} + 10034993311596) q^{57} + (1058399672 \beta_{3} - 16132594649700 \beta_1) q^{59} + ( - 8295469968 \beta_{2} + 93540586578098) q^{61} + (1305750537 \beta_{3} + 4644855987156 \beta_1) q^{63} + ( - 2174730142 \beta_{2} + 158941734369132) q^{65} + ( - 11248193352 \beta_{3} + 48630251654804 \beta_1) q^{67} + (9032489334 \beta_{2} + 41342561191296) q^{69} + (11191437202 \beta_{3} + 291444192678864 \beta_1) q^{71} + (19699004856 \beta_{2} + 344987429136130) q^{73} + ( - 1689090084 \beta_{3} + 99632856522591 \beta_1) q^{75} + ( - 54360348 \beta_{2} + 457045200125136) q^{77} + (7611186377 \beta_{3} + 715644094048268 \beta_1) q^{79} + 205891132094649 q^{81} + ( - 20433438250 \beta_{3} + 792787966468476 \beta_1) q^{83} + ( - 9261357366 \beta_{2} + 21\!\cdots\!56) q^{85}+ \cdots + ( - 27865577394 \beta_{3} + 96267104041140 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 140424 q^{5} - 57395628 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 140424 q^{5} - 57395628 q^{9} - 586685528 q^{13} - 571005336 q^{17} - 8495392752 q^{21} - 182227446772 q^{25} - 585621416424 q^{29} - 176071520880 q^{33} - 2770470511096 q^{37} - 9744697113624 q^{41} + 2014930916568 q^{45} + 44786525712964 q^{49} + 93950973394200 q^{53} + 40139973246384 q^{57} + 374162346312392 q^{61} + 635766937476528 q^{65} + 165370244765184 q^{69} + 13\!\cdots\!20 q^{73}+ \cdots + 45\!\cdots\!40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 379501x^{2} + 379500x + 144020250000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 379501\nu^{2} - 379501\nu + 72009935250 ) / 72010314750 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\nu^{3} + 27324024 ) / 379501 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{3} - 4\nu^{2} + 3036004\nu + 759000 ) / 31625 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 18216024\beta _1 - 18216024 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 379501\beta_{2} - 27324024 ) / 48 \) 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
308.268 + 533.936i
−307.768 533.070i
308.268 533.936i
−307.768 + 533.070i
0 3788.00i 0 −360373. 0 5.22136e6i 0 −1.43489e7 0
31.2 0 3788.00i 0 290161. 0 4.10000e6i 0 −1.43489e7 0
31.3 0 3788.00i 0 −360373. 0 5.22136e6i 0 −1.43489e7 0
31.4 0 3788.00i 0 290161. 0 4.10000e6i 0 −1.43489e7 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 48.17.g.a 4
4.b odd 2 1 inner 48.17.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.17.g.a 4 1.a even 1 1 trivial
48.17.g.a 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 70212T_{5} - 104566166460 \) acting on \(S_{17}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 14348907)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 70212 T - 104566166460)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + \cdots - 41\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 97\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots + 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 10\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 64\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 90\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots - 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 51\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 62\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 21\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
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