Properties

Label 507.4.a.p
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + 3 q^{3} + (\beta_{5} + \beta_{3} + 5) q^{4} + (\beta_{8} - \beta_{3} + 3) q^{5} + (3 \beta_{3} + 3) q^{6} + ( - \beta_{8} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 9) q^{7} + ( - 2 \beta_{8} + 2 \beta_{5} + \beta_{4} + 6 \beta_{3} - \beta_{2} + \beta_1 + 13) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + 3 q^{3} + (\beta_{5} + \beta_{3} + 5) q^{4} + (\beta_{8} - \beta_{3} + 3) q^{5} + (3 \beta_{3} + 3) q^{6} + ( - \beta_{8} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 9) q^{7} + ( - 2 \beta_{8} + 2 \beta_{5} + \beta_{4} + 6 \beta_{3} - \beta_{2} + \beta_1 + 13) q^{8} + 9 q^{9} + (\beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 7 \beta_{3} - \beta_1 - 3) q^{10} + (2 \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - 6 \beta_{3} - \beta_{2} + 8) q^{11} + (3 \beta_{5} + 3 \beta_{3} + 15) q^{12} + ( - \beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 15 \beta_{3} + \cdots + 27) q^{14}+ \cdots + (18 \beta_{8} - 9 \beta_{6} - 9 \beta_{5} + 9 \beta_{4} - 54 \beta_{3} - 9 \beta_{2} + \cdots + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4276 \nu^{8} + 993 \nu^{7} - 231050 \nu^{6} - 177378 \nu^{5} + 3665268 \nu^{4} + 4377940 \nu^{3} - 13739936 \nu^{2} - 15644592 \nu - 1151901 ) / 1169766 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64999 \nu^{8} + 190713 \nu^{7} - 3890801 \nu^{6} - 11363973 \nu^{5} + 68463819 \nu^{4} + 197174581 \nu^{3} - 296579777 \nu^{2} - 794287746 \nu - 40902156 ) / 15206958 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24987 \nu^{8} - 7166 \nu^{7} - 1397001 \nu^{6} - 297443 \nu^{5} + 23628597 \nu^{4} + 13299657 \nu^{3} - 105271205 \nu^{2} - 45679772 \nu + 3317895 ) / 5068986 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 90230 \nu^{8} + 32184 \nu^{7} - 5333239 \nu^{6} - 3836655 \nu^{5} + 94118139 \nu^{4} + 86430869 \nu^{3} - 422915233 \nu^{2} - 219614865 \nu + 6113457 ) / 15206958 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 135002 \nu^{8} - 25263 \nu^{7} - 7390255 \nu^{6} - 2116755 \nu^{5} + 120654441 \nu^{4} + 76680635 \nu^{3} - 485797741 \nu^{2} - 246575877 \nu - 154170810 ) / 15206958 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 279719 \nu^{8} - 175299 \nu^{7} - 15606016 \nu^{6} + 2115444 \nu^{5} + 265872168 \nu^{4} + 57886724 \nu^{3} - 1250692846 \nu^{2} + \cdots + 313226181 ) / 15206958 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 450169 \nu^{8} - 76689 \nu^{7} - 24966176 \nu^{6} - 9175896 \nu^{5} + 417633228 \nu^{4} + 314702644 \nu^{3} - 1808976728 \nu^{2} - 1087356531 \nu + 15335469 ) / 15206958 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 735613 \nu^{8} - 134067 \nu^{7} - 41110133 \nu^{6} - 11744913 \nu^{5} + 695505123 \nu^{4} + 411567511 \nu^{3} - 3109582931 \nu^{2} + \cdots + 129805956 ) / 15206958 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 12\beta_{3} - \beta _1 - 5 ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} + 10\beta_{5} - 10\beta_{3} - 20\beta _1 + 147 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 13 \beta_{8} + 28 \beta_{7} + 36 \beta_{6} + 49 \beta_{5} + 19 \beta_{4} - 258 \beta_{3} + 13 \beta_{2} - 53 \beta _1 - 18 ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{8} + 104 \beta_{7} - 53 \beta_{6} + 233 \beta_{5} - 46 \beta_{4} - 494 \beta_{3} + 26 \beta_{2} - 603 \beta _1 + 3056 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 377 \beta_{8} + 633 \beta_{7} + 987 \beta_{6} + 1585 \beta_{5} + 407 \beta_{4} - 6283 \beta_{3} + 442 \beta_{2} - 1658 \beta _1 + 1824 ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 507 \beta_{8} + 3033 \beta_{7} - 872 \beta_{6} + 6044 \beta_{5} - 2007 \beta_{4} - 17455 \beta_{3} + 1027 \beta_{2} - 15967 \beta _1 + 71121 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8788 \beta_{8} + 14072 \beta_{7} + 25574 \beta_{6} + 46062 \beta_{5} + 8136 \beta_{4} - 162988 \beta_{3} + 13598 \beta_{2} - 47536 \beta _1 + 94223 ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 15964 \beta_{8} + 82361 \beta_{7} - 9523 \beta_{6} + 167537 \beta_{5} - 69817 \beta_{4} - 554392 \beta_{3} + 35074 \beta_{2} - 413731 \beta _1 + 1753215 ) / 13 \) 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
5.39246
2.37150
5.06791
−0.588238
0.107680
−0.100291
−4.83218
−4.14324
−3.27560
−4.83750 3.00000 15.4014 21.1983 −14.5125 16.2806 −35.8043 9.00000 −102.547
1.2 −3.17344 3.00000 2.07074 6.74147 −9.52033 −14.1726 18.8162 9.00000 −21.3937
1.3 −2.82093 3.00000 −0.0423641 −3.41089 −8.46278 13.3442 22.6869 9.00000 9.62187
1.4 −0.213700 3.00000 −7.95433 −15.3391 −0.641100 32.3928 3.40944 9.00000 3.27797
1.5 0.447278 3.00000 −7.79994 1.93073 1.34183 −8.14537 −7.06697 9.00000 0.863573
1.6 2.34727 3.00000 −2.49032 15.3991 7.04181 −10.1317 −24.6236 9.00000 36.1458
1.7 4.03025 3.00000 8.24289 8.08864 12.0907 5.95078 0.978887 9.00000 32.5992
1.8 4.69820 3.00000 14.0731 4.47249 14.0946 27.2096 28.5326 9.00000 21.0127
1.9 5.52257 3.00000 22.4988 −6.08065 16.5677 20.2718 80.0709 9.00000 −33.5808
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)

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 507.4.a.p yes 9
3.b odd 2 1 1521.4.a.bf 9
13.b even 2 1 507.4.a.o 9
13.d odd 4 2 507.4.b.k 18
39.d odd 2 1 1521.4.a.bi 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.o 9 13.b even 2 1
507.4.a.p yes 9 1.a even 1 1 trivial
507.4.b.k 18 13.d odd 4 2
1521.4.a.bf 9 3.b odd 2 1
1521.4.a.bi 9 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{9} - 6T_{2}^{8} - 40T_{2}^{7} + 251T_{2}^{6} + 452T_{2}^{5} - 3075T_{2}^{4} - 1401T_{2}^{3} + 11386T_{2}^{2} - 2288T_{2} - 1016 \) Copy content Toggle raw display
\( T_{5}^{9} - 33 T_{5}^{8} - 8 T_{5}^{7} + 8831 T_{5}^{6} - 66645 T_{5}^{5} - 239411 T_{5}^{4} + 3228609 T_{5}^{3} - 2833568 T_{5}^{2} - 29476244 T_{5} + 48900601 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 6 T^{8} - 40 T^{7} + \cdots - 1016 \) Copy content Toggle raw display
$3$ \( (T - 3)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 33 T^{8} - 8 T^{7} + \cdots + 48900601 \) Copy content Toggle raw display
$7$ \( T^{9} - 83 T^{8} + \cdots + 27017466139 \) Copy content Toggle raw display
$11$ \( T^{9} - 85 T^{8} + \cdots + 1288234570811 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 178 T^{8} + \cdots - 25\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{9} - 352 T^{8} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{9} - 150 T^{8} + \cdots + 60\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{9} + 97 T^{8} + \cdots + 34\!\cdots\!27 \) Copy content Toggle raw display
$31$ \( T^{9} - 717 T^{8} + \cdots - 40\!\cdots\!53 \) Copy content Toggle raw display
$37$ \( T^{9} - 1108 T^{8} + \cdots - 80\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{9} - 334 T^{8} + \cdots + 36\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{9} - 242 T^{8} + \cdots + 31\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{9} + 184 T^{8} + \cdots + 38\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{9} + 151 T^{8} + \cdots + 65\!\cdots\!51 \) Copy content Toggle raw display
$59$ \( T^{9} - 537 T^{8} + \cdots - 14\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{9} + 1340 T^{8} + \cdots - 61\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{9} - 2308 T^{8} + \cdots + 81\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{9} - 96 T^{8} + \cdots - 71\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{9} - 2505 T^{8} + \cdots + 19\!\cdots\!77 \) Copy content Toggle raw display
$79$ \( T^{9} + 1591 T^{8} + \cdots + 11\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( T^{9} - 1539 T^{8} + \cdots - 20\!\cdots\!97 \) Copy content Toggle raw display
$89$ \( T^{9} + 592 T^{8} + \cdots + 55\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{9} - 1445 T^{8} + \cdots + 31\!\cdots\!39 \) Copy content Toggle raw display
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