Properties

Label 4598.2.a.cd
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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

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.

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.cd 10
11.b odd 2 1 4598.2.a.cc 10
11.d odd 10 2 418.2.f.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.h 20 11.d odd 10 2
4598.2.a.cc 10 11.b odd 2 1
4598.2.a.cd 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{10} - 2 T_{3}^{9} - 19 T_{3}^{8} + 36 T_{3}^{7} + 118 T_{3}^{6} - 220 T_{3}^{5} - 270 T_{3}^{4} + 512 T_{3}^{3} + 176 T_{3}^{2} - 392 T_{3} + 44 \) 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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