Properties

Label 48.15.g.b
Level $48$
Weight $15$
Character orbit 48.g
Analytic conductor $59.678$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \( x^{4} - x^{3} + 3669553x^{2} + 3669552x + 13465611880704 \) 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.

\(f(q)\) \(=\) \( q + 729 \beta_1 q^{3} + (\beta_{2} + 25278) q^{5} + ( - 5 \beta_{3} - 414484 \beta_1) q^{7} - 1594323 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 729 \beta_1 q^{3} + (\beta_{2} + 25278) q^{5} + ( - 5 \beta_{3} - 414484 \beta_1) q^{7} - 1594323 q^{9} + ( - 110 \beta_{3} - 4984188 \beta_1) q^{11} + ( - 390 \beta_{2} - 28411214) q^{13} + ( - 729 \beta_{3} + 18427662 \beta_1) q^{15} + ( - 4330 \beta_{2} - 55535766) q^{17} + ( - 230 \beta_{3} + 419025484 \beta_1) q^{19} + ( - 10935 \beta_{2} + 906476508) q^{21} + (10710 \beta_{3} + 1191600720 \beta_1) q^{23} + (50556 \beta_{2} + 2990110043) q^{25} - 1162261467 \beta_1 q^{27} + (209615 \beta_{2} - 4383176730) q^{29} + ( - 88645 \beta_{3} + 3525228468 \beta_1) q^{31} + ( - 240570 \beta_{2} + 10900419156) q^{33} + (288094 \beta_{3} + 31795915368 \beta_1) q^{35} + ( - 929340 \beta_{2} + 53126740106) q^{37} + (284310 \beta_{3} - 20711775006 \beta_1) q^{39} + (85770 \beta_{2} - 268267911174) q^{41} + ( - 3102410 \beta_{3} - 10594113948 \beta_1) q^{43} + ( - 1594323 \beta_{2} - 40301296794) q^{45} + (2829370 \beta_{3} - 229834692024 \beta_1) q^{47} + (12434520 \beta_{2} - 471266514719) q^{49} + (3156570 \beta_{3} - 40485573414 \beta_1) q^{51} + ( - 8100705 \beta_{2} + 317712640662) q^{53} + (2203608 \beta_{3} + 804021017976 \beta_1) q^{55} + ( - 503010 \beta_{2} - 916408733508) q^{57} + ( - 8795320 \beta_{3} + 1027208495148 \beta_1) q^{59} + ( - 2053560 \beta_{2} - 2672800762390) q^{61} + (7971615 \beta_{3} + 660821374332 \beta_1) q^{63} + ( - 38269634 \beta_{2} - 4015491537252) q^{65} + ( - 32229240 \beta_{3} + 2656900077412 \beta_1) q^{67} + (23422770 \beta_{2} - 2606030774640) q^{69} + (9841870 \beta_{3} + 2890058549088 \beta_1) q^{71} + ( - 46820040 \beta_{2} - 10233190147598) q^{73} + ( - 36855324 \beta_{3} + 2179790221347 \beta_1) q^{75} + (211542540 \beta_{2} - 20147768370576) q^{77} + (192261295 \beta_{3} + 2150637439156 \beta_1) q^{79} + 2541865828329 q^{81} + (8978690 \beta_{3} + 15341394888348 \beta_1) q^{83} + ( - 164989506 \beta_{2} - 38012460595668) q^{85} + ( - 152809335 \beta_{3} - 3195335836170 \beta_1) q^{87} + (63654580 \beta_{2} - 75235880686926) q^{89} + ( - 19592690 \beta_{3} - 4710570725224 \beta_1) q^{91} + ( - 193866615 \beta_{2} - 7709674659516) q^{93} + ( - 424839424 \beta_{3} + 12536695312872 \beta_1) q^{95} + ( - 101383260 \beta_{2} - 85085855560766) q^{97} + (175375530 \beta_{3} + 7946405564724 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 101112 q^{5} - 6377292 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 101112 q^{5} - 6377292 q^{9} - 113644856 q^{13} - 222143064 q^{17} + 3625906032 q^{21} + 11960440172 q^{25} - 17532706920 q^{29} + 43601676624 q^{33} + 212506960424 q^{37} - 1073071644696 q^{41} - 161205187176 q^{45} - 1885066058876 q^{49} + 1270850562648 q^{53} - 3665634934032 q^{57} - 10691203049560 q^{61} - 16061966149008 q^{65} - 10424123098560 q^{69} - 40932760590392 q^{73} - 80591073482304 q^{77} + 10167463313316 q^{81} - 152049842382672 q^{85} - 300943522747704 q^{89} - 30838698638064 q^{93} - 340343422243064 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 3669553x^{2} + 3669552x + 13465611880704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 3669553\nu^{2} - 3669553\nu + 6732804105576 ) / 6732807775128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\nu^{3} + 264207768 ) / 3669553 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} + 7339105\nu + 1834776 ) / 76449 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 176138520\beta _1 - 176138520 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3669553\beta_{2} - 264207768 ) / 48 \) 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
958.054 1659.40i
−957.554 + 1658.53i
958.054 + 1659.40i
−957.554 1658.53i
0 1262.67i 0 −66671.2 0 1.51421e6i 0 −1.59432e6 0
31.2 0 1262.67i 0 117227. 0 78395.8i 0 −1.59432e6 0
31.3 0 1262.67i 0 −66671.2 0 1.51421e6i 0 −1.59432e6 0
31.4 0 1262.67i 0 117227. 0 78395.8i 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.b 4
3.b odd 2 1 144.15.g.d 4
4.b odd 2 1 inner 48.15.g.b 4
12.b even 2 1 144.15.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.15.g.b 4 1.a even 1 1 trivial
48.15.g.b 4 4.b odd 2 1 inner
144.15.g.d 4 3.b odd 2 1
144.15.g.d 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 50556T_{5} - 7815671100 \) 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
$5$ \( (T^{2} - 50556 T - 7815671100)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 53\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 478754938252604)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + \cdots - 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 44\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots + 71\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 59\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots - 45\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 86\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 85\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 49\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 56\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 71\!\cdots\!56)^{2} \) Copy content Toggle raw display
show more
show less