Properties

Label 48.15.g.a
Level $48$
Weight $15$
Character orbit 48.g
Analytic conductor $59.678$
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] = [48,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(59.6779047129\)
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: \( x^{4} - x^{3} + 16897x^{2} + 16896x + 285474816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{9} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (7 \beta_{2} + 4590) q^{5} + ( - 17 \beta_{3} - 354 \beta_1) q^{7} - 1594323 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (7 \beta_{2} + 4590) q^{5} + ( - 17 \beta_{3} - 354 \beta_1) q^{7} - 1594323 q^{9} + (618 \beta_{3} + 2872 \beta_1) q^{11} + ( - 13338 \beta_{2} - 39597038) q^{13} + ( - 1701 \beta_{3} - 5388 \beta_1) q^{15} + (80378 \beta_{2} + 65624202) q^{17} + (36466 \beta_{3} + 222432 \beta_1) q^{19} + ( - 111537 \beta_{2} - 551675124) q^{21} + ( - 106818 \beta_{3} - 1129660 \beta_1) q^{23} + (64260 \beta_{2} - 4174928485) q^{25} + 1594323 \beta_1 q^{27} + ( - 1206967 \beta_{2} - 14985012042) q^{29} + (43631 \beta_{3} + 6621078 \beta_1) q^{31} + (4054698 \beta_{2} + 4116660084) q^{33} + ( - 666618 \beta_{3} - 20964964 \beta_1) q^{35} + ( - 7502004 \beta_{2} + 3681968330) q^{37} + (3241134 \beta_{3} + 41117570 \beta_1) q^{39} + (18965670 \beta_{2} + 111582587610) q^{41} + (102238 \beta_{3} + 83448528 \beta_1) q^{43} + ( - 11160261 \beta_{2} - 7317942570) q^{45} + ( - 2052654 \beta_{3} + 164683460 \beta_1) q^{47} + ( - 77189112 \beta_{2} + 183567183457) q^{49} + ( - 19531854 \beta_{3} - 74787294 \beta_1) q^{51} + (47863929 \beta_{2} + 209252792934) q^{53} + (7228728 \beta_{3} + 708274584 \beta_1) q^{55} + (239253426 \beta_{2} + 327353562972) q^{57} + (28471272 \beta_{3} + 2578572004 \beta_1) q^{59} + ( - 39111768 \beta_{2} - 957183231766) q^{61} + (27103491 \beta_{3} + 564390342 \beta_1) q^{63} + ( - 338400686 \beta_{2} - 3816391683780) q^{65} + ( - 72809688 \beta_{3} + 4833709164 \beta_1) q^{67} + ( - 700832898 \beta_{2} - 1721147969808) q^{69} + (120972054 \beta_{3} + 8842669124 \beta_1) q^{71} + (538384104 \beta_{2} - 2240021401742) q^{73} + ( - 15615180 \beta_{3} + 4167602845 \beta_1) q^{75} + (1691022420 \beta_{2} + 12467132686512) q^{77} + ( - 58878701 \beta_{3} + 22433757486 \beta_1) q^{79} + 2541865828329 q^{81} + ( - 896010918 \beta_{3} + 18369881600 \beta_1) q^{83} + (828304434 \beta_{2} + 22204438715340) q^{85} + (293292981 \beta_{3} + 15122606280 \beta_1) q^{87} + ( - 7693399124 \beta_{2} - 19824177873294) q^{89} + (1794662038 \beta_{3} + 50868538188 \beta_1) q^{91} + (286262991 \beta_{2} + 10523502959220) q^{93} + (516635904 \beta_{3} + 42078162392 \beta_1) q^{95} + (8416384380 \beta_{2} - 20423490292094) q^{97} + ( - 985291614 \beta_{3} - 4578895656 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18360 q^{5} - 6377292 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18360 q^{5} - 6377292 q^{9} - 158388152 q^{13} + 262496808 q^{17} - 2206700496 q^{21} - 16699713940 q^{25} - 59940048168 q^{29} + 16466640336 q^{33} + 14727873320 q^{37} + 446330350440 q^{41} - 29271770280 q^{45} + 734268733828 q^{49} + 837011171736 q^{53} + 1309414251888 q^{57} - 3828732927064 q^{61} - 15265566735120 q^{65} - 6884591879232 q^{69} - 8960085606968 q^{73} + 49868530746048 q^{77} + 10167463313316 q^{81} + 88817754861360 q^{85} - 79296711493176 q^{89} + 42094011836880 q^{93} - 81693961168376 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 16897x^{2} + 16896x + 285474816 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -243\nu^{3} + 4105971\nu^{2} - 4105971\nu + 34683137280 ) / 47581952 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -48\nu^{3} - 1216536 ) / 16897 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 202821\nu^{3} - 1165893\nu^{2} + 6852966981\nu - 6422600448 ) / 23790976 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 81\beta_{3} + 243\beta_{2} + 46\beta _1 + 5832 ) / 23328 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 81\beta_{3} - 243\beta_{2} + 270382\beta _1 - 197080776 ) / 23328 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -16897\beta_{2} - 1216536 ) / 48 \) 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
−64.7428 112.138i
65.2428 + 113.004i
−64.7428 + 112.138i
65.2428 113.004i
0 1262.67i 0 −39085.2 0 114233.i 0 −1.59432e6 0
31.2 0 1262.67i 0 48265.2 0 988060.i 0 −1.59432e6 0
31.3 0 1262.67i 0 −39085.2 0 114233.i 0 −1.59432e6 0
31.4 0 1262.67i 0 48265.2 0 988060.i 0 −1.59432e6 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 48.15.g.a 4
3.b odd 2 1 144.15.g.e 4
4.b odd 2 1 inner 48.15.g.a 4
12.b even 2 1 144.15.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.15.g.a 4 1.a even 1 1 trivial
48.15.g.a 4 4.b odd 2 1 inner
144.15.g.e 4 3.b odd 2 1
144.15.g.e 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9180T_{5} - 1886450940 \) acting on \(S_{15}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1594323)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 9180 T - 1886450940)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 53\!\cdots\!96)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + \cdots - 24\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots + 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 21\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots - 45\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 92\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 85\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 62\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 78\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 23\!\cdots\!64)^{2} \) Copy content Toggle raw display
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