Properties

Label 4598.2.a.bw
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
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] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.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: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 11x^{6} + 22x^{5} + 34x^{4} - 68x^{3} - 28x^{2} + 60x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{6} - \beta_{3} - \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} + 2) q^{7} - q^{8} + (\beta_{6} - \beta_{5} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{6} - \beta_{3} - \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} + 2) q^{7} - q^{8} + (\beta_{6} - \beta_{5} + \beta_1) q^{9} + (\beta_{6} + \beta_{3} + \beta_1) q^{10} - \beta_1 q^{12} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{13} + (\beta_{4} - \beta_{2} - 2) q^{14} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{15} + q^{16} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{17} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{18} + q^{19} + ( - \beta_{6} - \beta_{3} - \beta_1) q^{20} + (\beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{21} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 2) q^{23} + \beta_1 q^{24} + ( - \beta_{7} + 3 \beta_{4} + \beta_1 - 1) q^{25} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{26} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{27} + ( - \beta_{4} + \beta_{2} + 2) q^{28} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{29} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1) q^{30} + (2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{31} - q^{32} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{34} + (\beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_1 - 1) q^{35} + (\beta_{6} - \beta_{5} + \beta_1) q^{36} + ( - \beta_{7} - \beta_{5} + \beta_1 + 2) q^{37} - q^{38} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} + 2) q^{39} + (\beta_{6} + \beta_{3} + \beta_1) q^{40} + ( - 3 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{41} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{42} + ( - 2 \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{2} - \beta_1 + 5) q^{43} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{45} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{46} + (2 \beta_{7} - \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{47} - \beta_1 q^{48} + (\beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2) q^{49} + (\beta_{7} - 3 \beta_{4} - \beta_1 + 1) q^{50} + ( - \beta_{7} + 3 \beta_{6} - \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 4) q^{51} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{52} + ( - 4 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{53} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{54} + (\beta_{4} - \beta_{2} - 2) q^{56} - \beta_1 q^{57} + (\beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{58} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{59} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{60} + ( - \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{61} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{62} + (3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{63} + q^{64} + ( - \beta_{5} + 5 \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{65} + ( - 2 \beta_{7} - 4 \beta_{6} + 6 \beta_{5} - 5 \beta_{4} + \beta_{3} + 5 \beta_{2} + \cdots - 1) q^{67}+ \cdots + ( - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} + 2 q^{6} + 14 q^{7} - 8 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} + 2 q^{6} + 14 q^{7} - 8 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{12} + 9 q^{13} - 14 q^{14} + 5 q^{15} + 8 q^{16} + 3 q^{17} - 2 q^{18} + 8 q^{19} - 2 q^{20} + 7 q^{21} - 13 q^{23} + 2 q^{24} + 4 q^{25} - 9 q^{26} + 4 q^{27} + 14 q^{28} + 10 q^{29} - 5 q^{30} - 2 q^{31} - 8 q^{32} - 3 q^{34} - 7 q^{35} + 2 q^{36} + 15 q^{37} - 8 q^{38} + 11 q^{39} + 2 q^{40} + 9 q^{41} - 7 q^{42} + 33 q^{43} - 21 q^{45} + 13 q^{46} - 19 q^{47} - 2 q^{48} + 12 q^{49} - 4 q^{50} + 22 q^{51} + 9 q^{52} + 10 q^{53} - 4 q^{54} - 14 q^{56} - 2 q^{57} - 10 q^{58} + q^{59} + 5 q^{60} + 26 q^{61} + 2 q^{62} + 19 q^{63} + 8 q^{64} + 26 q^{65} - 25 q^{67} + 3 q^{68} - 12 q^{69} + 7 q^{70} - 10 q^{71} - 2 q^{72} + 11 q^{73} - 15 q^{74} - 33 q^{75} + 8 q^{76} - 11 q^{78} + 10 q^{79} - 2 q^{80} - 40 q^{81} - 9 q^{82} + 16 q^{83} + 7 q^{84} + 33 q^{85} - 33 q^{86} + 2 q^{87} - 14 q^{89} + 21 q^{90} + 25 q^{91} - 13 q^{92} + 20 q^{93} + 19 q^{94} - 2 q^{95} + 2 q^{96} + 22 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 11x^{6} + 22x^{5} + 34x^{4} - 68x^{3} - 28x^{2} + 60x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 9\nu^{3} - 18\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 9\nu^{5} + 2\nu^{4} + 20\nu^{3} - 10\nu^{2} - 12\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 11\nu^{5} + 36\nu^{3} + 2\nu^{2} - 36\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 11\nu^{5} - \nu^{4} - 35\nu^{3} + 4\nu^{2} + 32\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 11\nu^{5} - \nu^{4} - 35\nu^{3} + 6\nu^{2} + 30\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 11\nu^{5} - 9\nu^{4} - 34\nu^{3} + 16\nu^{2} + 28\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{6} - 7\beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} + 9\beta_{4} + 9\beta_{3} + 7\beta_{2} + 27\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 36\beta_{6} - 47\beta_{5} - 25\beta_{4} + 7\beta_{3} + 7\beta_{2} + 43\beta _1 + 83 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{6} + 65\beta_{5} + 67\beta_{4} + 63\beta_{3} + 41\beta_{2} + 151\beta _1 - 69 \) 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
2.40829
2.27943
1.59897
1.18171
0.0692689
−1.25705
−1.75021
−2.53041
−1.00000 −2.40829 1.00000 −3.43347 2.40829 1.05667 −1.00000 2.79988 3.43347
1.2 −1.00000 −2.27943 1.00000 −2.24748 2.27943 4.63064 −1.00000 2.19578 2.24748
1.3 −1.00000 −1.59897 1.00000 3.74775 1.59897 −0.838369 −1.00000 −0.443297 −3.74775
1.4 −1.00000 −1.18171 1.00000 −0.117838 1.18171 −1.74369 −1.00000 −1.60355 0.117838
1.5 −1.00000 −0.0692689 1.00000 1.10125 0.0692689 1.99611 −1.00000 −2.99520 −1.10125
1.6 −1.00000 1.25705 1.00000 1.68320 −1.25705 4.32620 −1.00000 −1.41983 −1.68320
1.7 −1.00000 1.75021 1.00000 −2.99749 −1.75021 0.219425 −1.00000 0.0632464 2.99749
1.8 −1.00000 2.53041 1.00000 0.264060 −2.53041 4.35301 −1.00000 3.40297 −0.264060
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

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 4598.2.a.bw 8
11.b odd 2 1 4598.2.a.bz 8
11.c even 5 2 418.2.f.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.g 16 11.c even 5 2
4598.2.a.bw 8 1.a even 1 1 trivial
4598.2.a.bz 8 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{8} + 2T_{3}^{7} - 11T_{3}^{6} - 22T_{3}^{5} + 34T_{3}^{4} + 68T_{3}^{3} - 28T_{3}^{2} - 60T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} - 20T_{5}^{6} - 36T_{5}^{5} + 99T_{5}^{4} + 100T_{5}^{3} - 180T_{5}^{2} + 20T_{5} + 5 \) Copy content Toggle raw display
\( T_{7}^{8} - 14T_{7}^{7} + 64T_{7}^{6} - 70T_{7}^{5} - 211T_{7}^{4} + 442T_{7}^{3} + 10T_{7}^{2} - 290T_{7} + 59 \) Copy content Toggle raw display
\( T_{13}^{8} - 9T_{13}^{7} - 23T_{13}^{6} + 394T_{13}^{5} - 744T_{13}^{4} - 1240T_{13}^{3} + 2080T_{13}^{2} + 1760T_{13} + 320 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} - 11 T^{6} - 22 T^{5} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} - 20 T^{6} - 36 T^{5} + \cdots + 5 \) Copy content Toggle raw display
$7$ \( T^{8} - 14 T^{7} + 64 T^{6} - 70 T^{5} + \cdots + 59 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 9 T^{7} - 23 T^{6} + 394 T^{5} + \cdots + 320 \) Copy content Toggle raw display
$17$ \( T^{8} - 3 T^{7} - 89 T^{6} + \cdots - 118571 \) Copy content Toggle raw display
$19$ \( (T - 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 13 T^{7} - 47 T^{6} + \cdots - 142921 \) Copy content Toggle raw display
$29$ \( T^{8} - 10 T^{7} - 28 T^{6} + \cdots + 33536 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} - 71 T^{6} + \cdots + 22996 \) Copy content Toggle raw display
$37$ \( T^{8} - 15 T^{7} + 35 T^{6} + \cdots + 2524 \) Copy content Toggle raw display
$41$ \( T^{8} - 9 T^{7} - 75 T^{6} + \cdots + 12100 \) Copy content Toggle raw display
$43$ \( T^{8} - 33 T^{7} + 336 T^{6} + \cdots - 1634476 \) Copy content Toggle raw display
$47$ \( T^{8} + 19 T^{7} - 129 T^{6} + \cdots - 2589455 \) Copy content Toggle raw display
$53$ \( T^{8} - 10 T^{7} - 237 T^{6} + \cdots + 104756 \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} - 149 T^{6} + 794 T^{5} + \cdots + 1600 \) Copy content Toggle raw display
$61$ \( T^{8} - 26 T^{7} + 152 T^{6} + \cdots - 223605 \) Copy content Toggle raw display
$67$ \( T^{8} + 25 T^{7} - 165 T^{6} + \cdots - 72329500 \) Copy content Toggle raw display
$71$ \( T^{8} + 10 T^{7} - 213 T^{6} + \cdots - 2243884 \) Copy content Toggle raw display
$73$ \( T^{8} - 11 T^{7} - 183 T^{6} + \cdots + 2788624 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} - 249 T^{6} + \cdots - 2701820 \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{7} - 98 T^{6} + \cdots - 1612721 \) Copy content Toggle raw display
$89$ \( T^{8} + 14 T^{7} - 233 T^{6} + \cdots - 481856 \) Copy content Toggle raw display
$97$ \( T^{8} - 22 T^{7} - 11 T^{6} + \cdots + 530500 \) Copy content Toggle raw display
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