Properties

Label 1617.4.a.p
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
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: \( x^{5} - 28x^{3} - 11x^{2} + 108x - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{4} + \beta_1 - 1) q^{5} + (3 \beta_1 - 3) q^{6} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 12) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{4} + \beta_1 - 1) q^{5} + (3 \beta_1 - 3) q^{6} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 12) q^{8}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9} - 55 q^{10} + 55 q^{11} - 63 q^{12} - 111 q^{13} + 21 q^{15} + 201 q^{16} - 136 q^{17} + 45 q^{18} - 111 q^{19} - 219 q^{20} + 55 q^{22} - 28 q^{23} - 180 q^{24} + 190 q^{25} + q^{26} - 135 q^{27} + 61 q^{29} + 165 q^{30} + 280 q^{31} + 535 q^{32} - 165 q^{33} + 572 q^{34} + 189 q^{36} - 41 q^{37} - 267 q^{38} + 333 q^{39} + 336 q^{40} - 426 q^{41} + 424 q^{43} + 231 q^{44} - 63 q^{45} + 140 q^{46} - 75 q^{47} - 603 q^{48} + 490 q^{50} + 408 q^{51} + 269 q^{52} + 1500 q^{53} - 135 q^{54} - 77 q^{55} + 333 q^{57} - 1767 q^{58} - 757 q^{59} + 657 q^{60} - 658 q^{61} + 568 q^{62} - 748 q^{64} + 537 q^{65} - 165 q^{66} - 583 q^{67} + 1650 q^{68} + 84 q^{69} - 764 q^{71} + 540 q^{72} - 875 q^{73} - 825 q^{74} - 570 q^{75} - 213 q^{76} - 3 q^{78} - 244 q^{79} + 2577 q^{80} + 405 q^{81} + 2006 q^{82} - 924 q^{83} - 1402 q^{85} + 1272 q^{86} - 183 q^{87} + 660 q^{88} + 1110 q^{89} - 495 q^{90} - 2046 q^{92} - 840 q^{93} + 3349 q^{94} + 1923 q^{95} - 1605 q^{96} + 852 q^{97} + 495 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 28x^{3} - 11x^{2} + 108x - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 26\nu^{2} - 17\nu + 64 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 2\nu^{3} - 28\nu^{2} - 57\nu + 74 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 21\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} + 26\beta_{2} + 43\beta _1 + 222 \) 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
5.15106
1.30014
0.767088
−2.85551
−4.36278
−4.15106 −3.00000 9.23129 −3.26204 12.4532 0 −5.11115 9.00000 13.5409
1.2 −0.300143 −3.00000 −7.90991 20.3930 0.900430 0 4.77526 9.00000 −6.12084
1.3 0.232912 −3.00000 −7.94575 −7.75746 −0.698736 0 −3.71396 9.00000 −1.80680
1.4 3.85551 −3.00000 6.86494 −18.0422 −11.5665 0 −4.37623 9.00000 −69.5618
1.5 5.36278 −3.00000 20.7594 1.66863 −16.0883 0 68.4261 9.00000 8.94849
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( -1 \)
\(11\) \( -1 \)

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 1617.4.a.p 5
7.b odd 2 1 231.4.a.l 5
21.c even 2 1 693.4.a.n 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.l 5 7.b odd 2 1
693.4.a.n 5 21.c even 2 1
1617.4.a.p 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{5} - 5T_{2}^{4} - 18T_{2}^{3} + 85T_{2}^{2} + 7T_{2} - 6 \) Copy content Toggle raw display
\( T_{5}^{5} + 7T_{5}^{4} - 383T_{5}^{3} - 3499T_{5}^{2} - 2446T_{5} + 15536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 5 T^{4} + \cdots - 6 \) Copy content Toggle raw display
$3$ \( (T + 3)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 7 T^{4} + \cdots + 15536 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( (T - 11)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 111 T^{4} + \cdots + 276332 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots + 3184406528 \) Copy content Toggle raw display
$19$ \( T^{5} + 111 T^{4} + \cdots + 2596968 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 9848447488 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 1678137108 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 97533466624 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 82159135548 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 2830976555008 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 1905935989632 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 417603397024 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 3927875616864 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 211529924045472 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 14480622579168 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 8392173156048 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 2460667275264 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 1051447598248 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 93230064402432 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 15123167361792 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 1811669774112 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 346762905716192 \) Copy content Toggle raw display
show more
show less