Properties

Label 507.4.a.n
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 \) 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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{4} + ( - \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_1 - 5) q^{5} + ( - 3 \beta_1 + 3) q^{6} + ( - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 - 1) q^{7} + ( - \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 10) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{4} + ( - \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_1 - 5) q^{5} + ( - 3 \beta_1 + 3) q^{6} + ( - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 - 1) q^{7} + ( - \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 10) q^{8} + 9 q^{9} + (5 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - \beta_{2} - 8 \beta_1 + 21) q^{10} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2) q^{11} + ( - 3 \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 9) q^{12} + (6 \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} - 8 \beta_{3} + 3 \beta_{2} + \cdots + 11) q^{14}+ \cdots + ( - 18 \beta_{7} - 9 \beta_{6} + 27 \beta_{5} + 9 \beta_{4} - 36 \beta_{3} + \cdots - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9} + 198 q^{10} - 37 q^{11} - 96 q^{12} + 98 q^{14} + 123 q^{15} + 32 q^{16} - 134 q^{17} - 72 q^{18} + 72 q^{19} - 356 q^{20} + 3 q^{21} + 274 q^{22} + 226 q^{23} + 333 q^{24} + 612 q^{25} - 243 q^{27} - 132 q^{28} - 547 q^{29} - 594 q^{30} + 521 q^{31} - 721 q^{32} + 111 q^{33} + 100 q^{34} + 138 q^{35} + 288 q^{36} - 584 q^{37} - 416 q^{38} + 1342 q^{40} - 482 q^{41} - 294 q^{42} + 158 q^{43} - 1453 q^{44} - 369 q^{45} - 1537 q^{46} - 1500 q^{47} - 96 q^{48} + 642 q^{49} - 2777 q^{50} + 402 q^{51} + 1399 q^{53} + 216 q^{54} - 1408 q^{55} - 616 q^{56} - 216 q^{57} - 1455 q^{58} - 1541 q^{59} + 1068 q^{60} + 2092 q^{61} - 293 q^{62} - 9 q^{63} + 2481 q^{64} - 822 q^{66} - 252 q^{67} - 1579 q^{68} - 678 q^{69} - 2492 q^{70} - 2352 q^{71} - 999 q^{72} - 903 q^{73} + 1037 q^{74} - 1836 q^{75} + 485 q^{76} - 1686 q^{77} - 115 q^{79} - 5701 q^{80} + 729 q^{81} - 5147 q^{82} - 1207 q^{83} + 396 q^{84} - 4308 q^{85} - 5691 q^{86} + 1641 q^{87} - 484 q^{88} - 2336 q^{89} + 1782 q^{90} + 2087 q^{92} - 1563 q^{93} - 468 q^{94} - 222 q^{95} + 2163 q^{96} - 2155 q^{97} - 5593 q^{98} - 333 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{8} - 719 \nu^{7} + 1734 \nu^{6} + 27181 \nu^{5} - 27838 \nu^{4} - 334818 \nu^{3} - 58625 \nu^{2} + 1425628 \nu + 513836 ) / 198848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{8} + 719 \nu^{7} - 1734 \nu^{6} - 27181 \nu^{5} + 27838 \nu^{4} + 334818 \nu^{3} + 257473 \nu^{2} - 1425628 \nu - 2502316 ) / 198848 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 127 \nu^{8} + 483 \nu^{7} - 10310 \nu^{6} - 16209 \nu^{5} + 248606 \nu^{4} + 170474 \nu^{3} - 2016507 \nu^{2} - 623892 \nu + 3265492 ) / 198848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 171 \nu^{8} + 831 \nu^{7} - 8974 \nu^{6} - 35013 \nu^{5} + 139382 \nu^{4} + 389090 \nu^{3} - 558855 \nu^{2} - 606900 \nu - 168892 ) / 198848 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 383 \nu^{8} - 1291 \nu^{7} + 14878 \nu^{6} + 61649 \nu^{5} - 141702 \nu^{4} - 860826 \nu^{3} - 82469 \nu^{2} + 3151596 \nu + 2264092 ) / 198848 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 743 \nu^{8} - 2293 \nu^{7} - 33670 \nu^{6} + 79991 \nu^{5} + 493806 \nu^{4} - 772630 \nu^{3} - 2400467 \nu^{2} + 1774556 \nu + 2379924 ) / 198848 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 591 \nu^{8} + 1757 \nu^{7} + 24206 \nu^{6} - 67775 \nu^{5} - 300654 \nu^{4} + 771566 \nu^{3} + 1205827 \nu^{2} - 2318484 \nu - 1800516 ) / 99424 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - 3\beta_{5} + \beta_{4} + \beta_{3} + 17\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 2\beta_{6} - 3\beta_{5} + 4\beta_{4} + 21\beta_{3} + 27\beta_{2} + \beta _1 + 161 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{8} - 15\beta_{7} - 21\beta_{6} - 76\beta_{5} + 33\beta_{4} + 44\beta_{3} + 15\beta_{2} + 313\beta _1 + 146 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{8} - 56 \beta_{7} - 87 \beta_{6} - 125 \beta_{5} + 127 \beta_{4} + 434 \beta_{3} + 651 \beta_{2} + 60 \beta _1 + 3031 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 404 \beta_{8} - 655 \beta_{7} - 449 \beta_{6} - 1646 \beta_{5} + 919 \beta_{4} + 1311 \beta_{3} + 670 \beta_{2} + 5996 \beta _1 + 3912 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 714 \beta_{8} - 2012 \beta_{7} - 2778 \beta_{6} - 3688 \beta_{5} + 3420 \beta_{4} + 9290 \beta_{3} + 15240 \beta_{2} + 2151 \beta _1 + 61010 \) 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
−4.48584
−3.76649
−2.37739
−1.73419
−0.614643
2.05129
2.86460
4.23649
4.82618
−5.48584 −3.00000 22.0945 −13.3185 16.4575 −21.4234 −77.3200 9.00000 73.0635
1.2 −4.76649 −3.00000 14.7194 −18.8390 14.2995 23.8593 −32.0282 9.00000 89.7961
1.3 −3.37739 −3.00000 3.40677 15.7127 10.1322 17.1681 15.5131 9.00000 −53.0679
1.4 −2.73419 −3.00000 −0.524213 −21.1246 8.20257 −25.8618 23.3068 9.00000 57.7586
1.5 −1.61464 −3.00000 −5.39293 −1.20859 4.84393 −28.2769 21.6248 9.00000 1.95145
1.6 1.05129 −3.00000 −6.89480 −17.8886 −3.15386 30.1975 −15.6587 9.00000 −18.8061
1.7 1.86460 −3.00000 −4.52327 2.36060 −5.59380 4.86461 −23.3509 9.00000 4.40158
1.8 3.23649 −3.00000 2.47490 13.5815 −9.70948 −1.42933 −17.8820 9.00000 43.9566
1.9 3.82618 −3.00000 6.63963 −0.275426 −11.4785 −0.0981245 −5.20502 9.00000 −1.05383
\(n\): 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.n 9
3.b odd 2 1 1521.4.a.bj 9
13.b even 2 1 507.4.a.q yes 9
13.d odd 4 2 507.4.b.j 18
39.d odd 2 1 1521.4.a.be 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.n 9 1.a even 1 1 trivial
507.4.a.q yes 9 13.b even 2 1
507.4.b.j 18 13.d odd 4 2
1521.4.a.be 9 39.d odd 2 1
1521.4.a.bj 9 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{9} + 8T_{2}^{8} - 20T_{2}^{7} - 251T_{2}^{6} + 8T_{2}^{5} + 2521T_{2}^{4} + 1303T_{2}^{3} - 8946T_{2}^{2} - 3640T_{2} + 9464 \) Copy content Toggle raw display
\( T_{5}^{9} + 41 T_{5}^{8} - 28 T_{5}^{7} - 18187 T_{5}^{6} - 130261 T_{5}^{5} + 2089839 T_{5}^{4} + 19041401 T_{5}^{3} - 23660364 T_{5}^{2} - 65644332 T_{5} - 15899689 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 8 T^{8} - 20 T^{7} + \cdots + 9464 \) Copy content Toggle raw display
$3$ \( (T + 3)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 41 T^{8} - 28 T^{7} + \cdots - 15899689 \) Copy content Toggle raw display
$7$ \( T^{9} + T^{8} - 1864 T^{7} + \cdots + 132217379 \) Copy content Toggle raw display
$11$ \( T^{9} + 37 T^{8} + \cdots - 982063511267 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 134 T^{8} + \cdots + 7231752990904 \) Copy content Toggle raw display
$19$ \( T^{9} - 72 T^{8} + \cdots - 82\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{9} - 226 T^{8} + \cdots + 91\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{9} + 547 T^{8} + \cdots - 12\!\cdots\!83 \) Copy content Toggle raw display
$31$ \( T^{9} - 521 T^{8} + \cdots - 68\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( T^{9} + 584 T^{8} + \cdots - 52\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{9} + 482 T^{8} + \cdots - 52\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{9} - 158 T^{8} + \cdots - 14\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{9} + 1500 T^{8} + \cdots + 26\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{9} - 1399 T^{8} + \cdots + 26\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{9} + 1541 T^{8} + \cdots - 64\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{9} - 2092 T^{8} + \cdots - 18\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{9} + 252 T^{8} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{9} + 2352 T^{8} + \cdots + 48\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{9} + 903 T^{8} + \cdots + 15\!\cdots\!77 \) Copy content Toggle raw display
$79$ \( T^{9} + 115 T^{8} + \cdots - 63\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{9} + 1207 T^{8} + \cdots - 17\!\cdots\!51 \) Copy content Toggle raw display
$89$ \( T^{9} + 2336 T^{8} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{9} + 2155 T^{8} + \cdots - 25\!\cdots\!69 \) Copy content Toggle raw display
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