Properties

Label 1617.4.a.o
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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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: \( x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{3} + 2 \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{4} - 2 \beta_{3} + \cdots - 7 \beta_1) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{3} + 2 \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{4} - 2 \beta_{3} + \cdots - 7 \beta_1) q^{8}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 7 q^{5} - 3 q^{6} - 12 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 7 q^{5} - 3 q^{6} - 12 q^{8} + 45 q^{9} - 113 q^{10} - 55 q^{11} + 63 q^{12} - 23 q^{13} - 21 q^{15} + 281 q^{16} + 102 q^{17} - 9 q^{18} + 155 q^{19} - 291 q^{20} + 11 q^{22} + 192 q^{23} - 36 q^{24} + 394 q^{25} - 41 q^{26} + 135 q^{27} + 591 q^{29} - 339 q^{30} - 366 q^{31} - 87 q^{32} - 165 q^{33} - 594 q^{34} + 189 q^{36} + 259 q^{37} - 175 q^{38} - 69 q^{39} - 1890 q^{40} + 104 q^{41} + 224 q^{43} - 231 q^{44} - 63 q^{45} + 1416 q^{46} + 453 q^{47} + 843 q^{48} + 1350 q^{50} + 306 q^{51} - 815 q^{52} + 1032 q^{53} - 27 q^{54} + 77 q^{55} + 465 q^{57} + 293 q^{58} + 517 q^{59} - 873 q^{60} - 958 q^{61} + 3030 q^{62} + 516 q^{64} - 197 q^{65} + 33 q^{66} + 361 q^{67} + 1680 q^{68} + 576 q^{69} + 548 q^{71} - 108 q^{72} + 1035 q^{73} - 2555 q^{74} + 1182 q^{75} + 5201 q^{76} - 123 q^{78} + 776 q^{79} + 1273 q^{80} + 405 q^{81} - 376 q^{82} + 1974 q^{83} + 1838 q^{85} + 1224 q^{86} + 1773 q^{87} + 132 q^{88} - 2710 q^{89} - 1017 q^{90} - 2834 q^{92} - 1098 q^{93} + 563 q^{94} - 355 q^{95} - 261 q^{96} - 1988 q^{97} - 495 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 30\nu^{2} - \nu + 82 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} + 28\nu^{2} - 45\nu - 58 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{3} + \beta_{2} + 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} + 30\beta_{2} + \beta _1 + 278 \) 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
5.22587
2.48454
−0.183399
−1.50528
−5.02173
−5.22587 3.00000 19.3097 9.27763 −15.6776 0 −59.1032 9.00000 −48.4837
1.2 −2.48454 3.00000 −1.82705 −13.9229 −7.45363 0 24.4157 9.00000 34.5919
1.3 0.183399 3.00000 −7.96636 17.9271 0.550196 0 −2.92821 9.00000 3.28780
1.4 1.50528 3.00000 −5.73413 0.155265 4.51584 0 −20.6737 9.00000 0.233718
1.5 5.02173 3.00000 17.2178 −20.4371 15.0652 0 46.2894 9.00000 −102.630
\(n\): 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.o 5
7.b odd 2 1 231.4.a.j 5
21.c even 2 1 693.4.a.q 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.j 5 7.b odd 2 1
693.4.a.q 5 21.c even 2 1
1617.4.a.o 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{5} + T_{2}^{4} - 30T_{2}^{3} - 21T_{2}^{2} + 103T_{2} - 18 \) Copy content Toggle raw display
\( T_{5}^{5} + 7T_{5}^{4} - 485T_{5}^{3} - 1951T_{5}^{2} + 47640T_{5} - 7348 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} + \cdots - 18 \) Copy content Toggle raw display
$3$ \( (T - 3)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 7 T^{4} + \cdots - 7348 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( (T + 11)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 23 T^{4} + \cdots - 122236 \) Copy content Toggle raw display
$17$ \( T^{5} - 102 T^{4} + \cdots + 159061888 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 1688336424 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 2862756736 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 144277546716 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 131921372032 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 244887003636 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 1412698624 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 7075944192 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 384442814216 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 299252301984 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 896870843568 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 10265429183328 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 1110831992688 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 27412560709632 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 25877686218916 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 264536335362048 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 11601305339904 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 7305960217248 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 2636714723104 \) Copy content Toggle raw display
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