Properties

Label 4598.2.a.cc
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $10$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 118x^{6} - 220x^{5} - 270x^{4} + 512x^{3} + 176x^{2} - 392x + 44 \) 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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 118x^{6} - 220x^{5} - 270x^{4} + 512x^{3} + 176x^{2} - 392x + 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59 \nu^{9} - 21 \nu^{8} - 1216 \nu^{7} + 276 \nu^{6} + 8338 \nu^{5} - 1532 \nu^{4} - 22372 \nu^{3} + 3632 \nu^{2} + 19636 \nu - 2940 ) / 892 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53 \nu^{9} + 155 \nu^{8} - 858 \nu^{7} - 2738 \nu^{6} + 3340 \nu^{5} + 12412 \nu^{4} - 2272 \nu^{3} - 13640 \nu^{2} - 2000 \nu - 1492 ) / 892 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97 \nu^{9} - 95 \nu^{8} - 1848 \nu^{7} + 1376 \nu^{6} + 11448 \nu^{5} - 6442 \nu^{4} - 26576 \nu^{3} + 9252 \nu^{2} + 20188 \nu - 2596 ) / 892 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 159 \nu^{9} - 19 \nu^{8} + 3020 \nu^{7} + 632 \nu^{6} - 18048 \nu^{5} - 2894 \nu^{4} + 39820 \nu^{3} + 3456 \nu^{2} - 27004 \nu + 16 ) / 892 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 120 \nu^{9} - 48 \nu^{8} + 2254 \nu^{7} + 1045 \nu^{6} - 13213 \nu^{5} - 4776 \nu^{4} + 29144 \nu^{3} + 6008 \nu^{2} - 21508 \nu - 1368 ) / 446 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 120 \nu^{9} + 48 \nu^{8} - 2254 \nu^{7} - 1045 \nu^{6} + 13213 \nu^{5} + 4776 \nu^{4} - 29144 \nu^{3} - 5562 \nu^{2} + 21508 \nu - 862 ) / 446 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 199 \nu^{9} + 35 \nu^{8} - 3697 \nu^{7} - 906 \nu^{6} + 21263 \nu^{5} + 3148 \nu^{4} - 44926 \nu^{3} + 42 \nu^{2} + 31200 \nu - 4912 ) / 446 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 535 \nu^{9} + 9 \nu^{8} + 10142 \nu^{7} + 710 \nu^{6} - 60980 \nu^{5} - 554 \nu^{4} + 138928 \nu^{3} - 11496 \nu^{2} - 104624 \nu + 17316 ) / 892 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} + 11\beta_{7} + 10\beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 15 \beta_{9} - 15 \beta_{8} + 5 \beta_{7} + 14 \beta_{6} - 21 \beta_{5} - 14 \beta_{4} - 2 \beta_{3} - 30 \beta_{2} + 61 \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 21 \beta_{9} - 17 \beta_{8} + 103 \beta_{7} + 96 \beta_{6} - 19 \beta_{5} - 38 \beta_{4} - 30 \beta_{3} - 66 \beta_{2} + 59 \beta _1 + 336 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 175 \beta_{9} - 173 \beta_{8} + 101 \beta_{7} + 174 \beta_{6} - 199 \beta_{5} - 160 \beta_{4} - 38 \beta_{3} - 362 \beta_{2} + 577 \beta _1 + 502 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 301 \beta_{9} - 235 \beta_{8} + 977 \beta_{7} + 950 \beta_{6} - 275 \beta_{5} - 510 \beta_{4} - 348 \beta_{3} - 844 \beta_{2} + 849 \beta _1 + 3214 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1901 \beta_{9} - 1855 \beta_{8} + 1465 \beta_{7} + 2074 \beta_{6} - 1927 \beta_{5} - 1754 \beta_{4} - 536 \beta_{3} - 4060 \beta_{2} + 5697 \beta _1 + 6318 \) 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
−2.84834
−2.53955
−1.58152
−1.28398
0.120966
1.20026
1.39462
2.10092
2.12868
3.30795
−1.00000 −2.84834 1.00000 −3.07094 2.84834 −4.67138 −1.00000 5.11302 3.07094
1.2 −1.00000 −2.53955 1.00000 1.81466 2.53955 −0.280173 −1.00000 3.44929 −1.81466
1.3 −1.00000 −1.58152 1.00000 2.60241 1.58152 −1.55057 −1.00000 −0.498782 −2.60241
1.4 −1.00000 −1.28398 1.00000 −1.35936 1.28398 3.16828 −1.00000 −1.35139 1.35936
1.5 −1.00000 0.120966 1.00000 −2.16907 −0.120966 −3.98760 −1.00000 −2.98537 2.16907
1.6 −1.00000 1.20026 1.00000 −4.16480 −1.20026 −0.346148 −1.00000 −1.55937 4.16480
1.7 −1.00000 1.39462 1.00000 2.16074 −1.39462 1.18224 −1.00000 −1.05504 −2.16074
1.8 −1.00000 2.10092 1.00000 2.46005 −2.10092 −3.09995 −1.00000 1.41386 −2.46005
1.9 −1.00000 2.12868 1.00000 −0.444747 −2.12868 0.813941 −1.00000 1.53127 0.444747
1.10 −1.00000 3.30795 1.00000 −0.828943 −3.30795 −2.22864 −1.00000 7.94251 0.828943
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.cc 10
11.b odd 2 1 4598.2.a.cd 10
11.c even 5 2 418.2.f.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.h 20 11.c even 5 2
4598.2.a.cc 10 1.a even 1 1 trivial
4598.2.a.cd 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{10} - 2 T_{3}^{9} - 19 T_{3}^{8} + 36 T_{3}^{7} + 118 T_{3}^{6} - 220 T_{3}^{5} - 270 T_{3}^{4} + 512 T_{3}^{3} + 176 T_{3}^{2} - 392 T_{3} + 44 \) Copy content Toggle raw display
\( T_{5}^{10} + 3 T_{5}^{9} - 23 T_{5}^{8} - 56 T_{5}^{7} + 191 T_{5}^{6} + 369 T_{5}^{5} - 635 T_{5}^{4} - 1062 T_{5}^{3} + 591 T_{5}^{2} + 1191 T_{5} + 349 \) Copy content Toggle raw display
\( T_{7}^{10} + 11 T_{7}^{9} + 27 T_{7}^{8} - 88 T_{7}^{7} - 449 T_{7}^{6} - 307 T_{7}^{5} + 809 T_{7}^{4} + 856 T_{7}^{3} - 329 T_{7}^{2} - 351 T_{7} - 59 \) Copy content Toggle raw display
\( T_{13}^{10} + 11 T_{13}^{9} - T_{13}^{8} - 356 T_{13}^{7} - 940 T_{13}^{6} + 1480 T_{13}^{5} + 6152 T_{13}^{4} + 576 T_{13}^{3} - 6912 T_{13}^{2} - 1216 T_{13} + 1984 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 2 T^{9} - 19 T^{8} + 36 T^{7} + \cdots + 44 \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} - 23 T^{8} - 56 T^{7} + \cdots + 349 \) Copy content Toggle raw display
$7$ \( T^{10} + 11 T^{9} + 27 T^{8} - 88 T^{7} + \cdots - 59 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 11 T^{9} - T^{8} - 356 T^{7} + \cdots + 1984 \) Copy content Toggle raw display
$17$ \( T^{10} + 12 T^{9} - 11 T^{8} + \cdots - 1381 \) Copy content Toggle raw display
$19$ \( (T + 1)^{10} \) Copy content Toggle raw display
$23$ \( T^{10} - 14 T^{9} - 3 T^{8} + \cdots - 377581 \) Copy content Toggle raw display
$29$ \( T^{10} + 16 T^{9} - 38 T^{8} + \cdots - 19407424 \) Copy content Toggle raw display
$31$ \( T^{10} - 12 T^{9} - 127 T^{8} + \cdots - 5575484 \) Copy content Toggle raw display
$37$ \( T^{10} + T^{9} - 117 T^{8} + \cdots + 260516 \) Copy content Toggle raw display
$41$ \( T^{10} - 5 T^{9} - 309 T^{8} + \cdots + 301928884 \) Copy content Toggle raw display
$43$ \( T^{10} + 22 T^{9} + 52 T^{8} + \cdots + 350900 \) Copy content Toggle raw display
$47$ \( T^{10} - 8 T^{9} - 185 T^{8} + \cdots - 347771 \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} - 369 T^{8} + \cdots - 210485900 \) Copy content Toggle raw display
$59$ \( T^{10} + 7 T^{9} - 121 T^{8} + \cdots - 23104 \) Copy content Toggle raw display
$61$ \( T^{10} + 35 T^{9} + 297 T^{8} + \cdots - 342661 \) Copy content Toggle raw display
$67$ \( T^{10} - 9 T^{9} - 335 T^{8} + \cdots + 6908404 \) Copy content Toggle raw display
$71$ \( T^{10} + 4 T^{9} - 429 T^{8} + \cdots - 23572844 \) Copy content Toggle raw display
$73$ \( T^{10} + 5 T^{9} - 387 T^{8} + \cdots + 36593104 \) Copy content Toggle raw display
$79$ \( T^{10} + 18 T^{9} - 145 T^{8} + \cdots + 8477116 \) Copy content Toggle raw display
$83$ \( T^{10} + 7 T^{9} + \cdots - 3201446201 \) Copy content Toggle raw display
$89$ \( T^{10} - 22 T^{9} - 135 T^{8} + \cdots - 64913216 \) Copy content Toggle raw display
$97$ \( T^{10} - 32 T^{9} + 69 T^{8} + \cdots + 86549804 \) Copy content Toggle raw display
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