Properties

Label 1216.4.a.bb
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, 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("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(71.7463225670\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 106x^{3} - 401x^{2} + 356x + 2112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 4) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_1 + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 4) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_1 + 16) q^{9} + (2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1) q^{11} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 18) q^{13} + (5 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 - 39) q^{15} + (\beta_{4} + \beta_{3} - 7 \beta_{2} - 2 \beta_1 + 49) q^{17} + 19 q^{19} + (\beta_{4} + 4 \beta_{3} + 7 \beta_{2} - 2 \beta_1 - 65) q^{21} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 7 \beta_1 + 8) q^{23} + ( - 4 \beta_{4} - 6 \beta_{3} - 11 \beta_{2} - \beta_1 + 113) q^{25} + (5 \beta_{4} + 7 \beta_{3} + 10 \beta_{2} + 5 \beta_1 - 39) q^{27} + ( - 3 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 43) q^{29} + ( - 11 \beta_{4} + \beta_{3} + 11 \beta_{2} - 3 \beta_1 + 37) q^{31} + ( - \beta_{4} + 15 \beta_{3} + 5 \beta_{2} - 19 \beta_1 + 131) q^{33} + ( - 18 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} + 5 \beta_1 + 4) q^{35} + (11 \beta_{4} - \beta_{3} - 11 \beta_{2} + 3 \beta_1 - 115) q^{37} + (4 \beta_{4} + 9 \beta_{3} + 12 \beta_{2} + 9 \beta_1 + 26) q^{39} + (9 \beta_{4} - 9 \beta_{3} + 11 \beta_{2} + 7 \beta_1 + 39) q^{41} + ( - 4 \beta_{4} - 4 \beta_{3} - 9 \beta_{2} - 23 \beta_1 - 14) q^{43} + ( - 18 \beta_{4} - 49 \beta_{2} - 13 \beta_1 + 114) q^{45} + ( - 2 \beta_{4} - 4 \beta_{3} + 15 \beta_{2} - 7 \beta_1 + 204) q^{47} + (3 \beta_{4} - 17 \beta_{3} + 13 \beta_{2} + 2 \beta_1 + 94) q^{49} + (14 \beta_{4} + 2 \beta_{3} + 53 \beta_{2} + 2 \beta_1 - 294) q^{51} + (11 \beta_{4} - 8 \beta_{3} - 9 \beta_{2} + 2 \beta_1 + 15) q^{53} + (2 \beta_{4} - 32 \beta_{3} + 55 \beta_{2} + 25 \beta_1 + 112) q^{55} + 19 \beta_{2} q^{57} + (3 \beta_{4} - 15 \beta_{3} - 32 \beta_{2} + 9 \beta_1 - 201) q^{59} + (16 \beta_{4} + 4 \beta_{3} - 15 \beta_{2} - 21 \beta_1 + 56) q^{61} + (4 \beta_{3} - \beta_{2} - 21 \beta_1 + 350) q^{63} + ( - 14 \beta_{4} + 12 \beta_{3} - 40 \beta_{2} + 24 \beta_1 - 14) q^{65} + ( - 12 \beta_{4} + 28 \beta_{3} + 7 \beta_{2} - 10 \beta_1 - 72) q^{67} + (28 \beta_{4} + 21 \beta_{3} + 44 \beta_{2} + 21 \beta_1 + 50) q^{69} + ( - 10 \beta_{4} + 4 \beta_{3} + 40 \beta_{2} + 2 \beta_1 + 238) q^{71} + (43 \beta_{4} + 15 \beta_{3} - 35 \beta_{2} + 4 \beta_1 + 3) q^{73} + (19 \beta_{4} - 39 \beta_{3} + 84 \beta_{2} + 55 \beta_1 - 421) q^{75} + ( - 32 \beta_{4} - 6 \beta_{3} + 97 \beta_{2} + 45 \beta_1 + 218) q^{77} + (15 \beta_{4} + 15 \beta_{3} + 99 \beta_{2} - 5 \beta_1 - 49) q^{79} + ( - 8 \beta_{4} + 8 \beta_{3} - 12 \beta_{2} - 44 \beta_1 - 71) q^{81} + ( - 8 \beta_{4} - 10 \beta_{3} + 52 \beta_{2} + 6 \beta_1 - 136) q^{83} + ( - 40 \beta_{4} - 32 \beta_{3} - 45 \beta_{2} - 59 \beta_1 + 686) q^{85} + (32 \beta_{4} - 23 \beta_{3} - 76 \beta_{2} + 53 \beta_1 - 122) q^{87} + ( - 51 \beta_{4} - 39 \beta_{3} + 27 \beta_{2} - 25 \beta_1 + 131) q^{89} + (15 \beta_{4} - 23 \beta_{3} + 32 \beta_{2} - 49 \beta_1 + 475) q^{91} + (12 \beta_{4} + 22 \beta_{3} + 150 \beta_{2} + 58 \beta_1 + 232) q^{93} + ( - 19 \beta_{2} - 19 \beta_1) q^{95} + ( - 6 \beta_{4} + 10 \beta_{3} - 84 \beta_{2} + 32 \beta_1 + 584) q^{97} + (18 \beta_{4} + 64 \beta_{3} + 243 \beta_{2} + 21 \beta_1 - 60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} - 22 q^{7} + 83 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} - 22 q^{7} + 83 q^{9} + 6 q^{11} - 82 q^{13} - 204 q^{15} + 234 q^{17} + 95 q^{19} - 308 q^{21} + 60 q^{23} + 549 q^{25} - 190 q^{27} - 210 q^{29} + 224 q^{31} + 704 q^{33} + 42 q^{35} - 614 q^{37} + 132 q^{39} + 194 q^{41} - 38 q^{43} + 516 q^{45} + 1066 q^{47} + 489 q^{49} - 1382 q^{51} + 42 q^{53} + 618 q^{55} + 38 q^{57} - 1090 q^{59} + 276 q^{61} + 1790 q^{63} - 184 q^{65} - 314 q^{67} + 268 q^{69} + 1276 q^{71} - 106 q^{73} - 2066 q^{75} + 1226 q^{77} - 52 q^{79} - 283 q^{81} - 580 q^{83} + 3498 q^{85} - 900 q^{87} + 810 q^{89} + 2522 q^{91} + 1332 q^{93} + 2694 q^{97} + 126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 106x^{3} - 401x^{2} + 356x + 2112 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{4} + 16\nu^{3} + 1278\nu^{2} + 2011\nu - 11880 ) / 372 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\nu^{4} - 104\nu^{3} - 2262\nu^{2} - 749\nu + 15840 ) / 372 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{4} + 20\nu^{3} + 559\nu^{2} + 615\nu - 3256 ) / 31 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{4} - 30\nu^{3} - 854\nu^{2} - 721\nu + 5535 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{4} + \beta_{3} - 13\beta_{2} + 13\beta _1 + 181 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 52\beta_{4} + 33\beta_{3} - 101\beta_{2} + 65\beta _1 + 558 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 915\beta_{4} + 395\beta_{3} - 1987\beta_{2} + 1747\beta _1 + 18515 ) / 4 \) 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
−3.84548
11.6790
−3.22412
−6.76535
2.15594
0 −9.00153 0 19.8345 0 11.3235 0 54.0275 0
1.2 0 −5.36679 0 −12.8065 0 14.4426 0 1.80242 0
1.3 0 2.49563 0 15.7941 0 −30.5820 0 −20.7718 0
1.4 0 5.24575 0 −18.7154 0 −28.3947 0 0.517848 0
1.5 0 8.62694 0 −4.10677 0 11.2105 0 47.4241 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 1216.4.a.bb 5
4.b odd 2 1 1216.4.a.ba 5
8.b even 2 1 304.4.a.k 5
8.d odd 2 1 152.4.a.d 5
24.f even 2 1 1368.4.a.m 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.a.d 5 8.d odd 2 1
304.4.a.k 5 8.b even 2 1
1216.4.a.ba 5 4.b odd 2 1
1216.4.a.bb 5 1.a even 1 1 trivial
1368.4.a.m 5 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3}^{5} - 2T_{3}^{4} - 107T_{3}^{3} + 244T_{3}^{2} + 2236T_{3} - 5456 \) Copy content Toggle raw display
\( T_{5}^{5} - 587T_{5}^{3} - 1006T_{5}^{2} + 80568T_{5} + 308352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} - 107 T^{3} + \cdots - 5456 \) Copy content Toggle raw display
$5$ \( T^{5} - 587 T^{3} - 1006 T^{2} + \cdots + 308352 \) Copy content Toggle raw display
$7$ \( T^{5} + 22 T^{4} - 860 T^{3} + \cdots - 1592040 \) Copy content Toggle raw display
$11$ \( T^{5} - 6 T^{4} - 6899 T^{3} + \cdots - 91281200 \) Copy content Toggle raw display
$13$ \( T^{5} + 82 T^{4} - 2101 T^{3} + \cdots + 19005984 \) Copy content Toggle raw display
$17$ \( T^{5} - 234 T^{4} + \cdots + 607064382 \) Copy content Toggle raw display
$19$ \( (T - 19)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} - 60 T^{4} + \cdots - 4457888256 \) Copy content Toggle raw display
$29$ \( T^{5} + 210 T^{4} + \cdots - 7027069704 \) Copy content Toggle raw display
$31$ \( T^{5} - 224 T^{4} + \cdots - 376353439744 \) Copy content Toggle raw display
$37$ \( T^{5} + 614 T^{4} + \cdots + 345578559712 \) Copy content Toggle raw display
$41$ \( T^{5} - 194 T^{4} + \cdots - 344930807808 \) Copy content Toggle raw display
$43$ \( T^{5} + 38 T^{4} + \cdots + 1008111272576 \) Copy content Toggle raw display
$47$ \( T^{5} - 1066 T^{4} + \cdots + 399004393472 \) Copy content Toggle raw display
$53$ \( T^{5} - 42 T^{4} + \cdots + 115161670944 \) Copy content Toggle raw display
$59$ \( T^{5} + 1090 T^{4} + \cdots - 530364944 \) Copy content Toggle raw display
$61$ \( T^{5} - 276 T^{4} + \cdots - 185141518840 \) Copy content Toggle raw display
$67$ \( T^{5} + 314 T^{4} + \cdots + 4670019292608 \) Copy content Toggle raw display
$71$ \( T^{5} - 1276 T^{4} + \cdots + 79175200896 \) Copy content Toggle raw display
$73$ \( T^{5} + 106 T^{4} + \cdots + 38395259106486 \) Copy content Toggle raw display
$79$ \( T^{5} + 52 T^{4} + \cdots + 23640142664704 \) Copy content Toggle raw display
$83$ \( T^{5} + 580 T^{4} + \cdots + 1385360658176 \) Copy content Toggle raw display
$89$ \( T^{5} - 810 T^{4} + \cdots - 9913402008576 \) Copy content Toggle raw display
$97$ \( T^{5} - 2694 T^{4} + \cdots - 17678567904768 \) Copy content Toggle raw display
show more
show less