Properties

Label 507.4.a.q
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: \( x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 \) 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 \( q + ( - \beta_1 + 1) q^{2} - 3 q^{3} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{4} + (\beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_1 + 5) q^{5} + (3 \beta_1 - 3) q^{6} + (2 \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{6} + 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 10) q^{8} + 9 q^{9} + (5 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - \beta_{2} - 8 \beta_1 + 21) q^{10} + (2 \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 2) q^{11} + ( - 3 \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 9) q^{12} + (6 \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} - 8 \beta_{3} + 3 \beta_{2} + \cdots + 11) q^{14}+ \cdots + (18 \beta_{7} + 9 \beta_{6} - 27 \beta_{5} - 9 \beta_{4} + 36 \beta_{3} - 18 \beta_{2} + \cdots + 18) q^{99}+O(q^{100}) \) 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
4.82618
4.23649
2.86460
2.05129
−0.614643
−1.73419
−2.37739
−3.76649
−4.48584
−3.82618 −3.00000 6.63963 0.275426 11.4785 0.0981245 5.20502 9.00000 −1.05383
1.2 −3.23649 −3.00000 2.47490 −13.5815 9.70948 1.42933 17.8820 9.00000 43.9566
1.3 −1.86460 −3.00000 −4.52327 −2.36060 5.59380 −4.86461 23.3509 9.00000 4.40158
1.4 −1.05129 −3.00000 −6.89480 17.8886 3.15386 −30.1975 15.6587 9.00000 −18.8061
1.5 1.61464 −3.00000 −5.39293 1.20859 −4.84393 28.2769 −21.6248 9.00000 1.95145
1.6 2.73419 −3.00000 −0.524213 21.1246 −8.20257 25.8618 −23.3068 9.00000 57.7586
1.7 3.37739 −3.00000 3.40677 −15.7127 −10.1322 −17.1681 −15.5131 9.00000 −53.0679
1.8 4.76649 −3.00000 14.7194 18.8390 −14.2995 −23.8593 32.0282 9.00000 89.7961
1.9 5.48584 −3.00000 22.0945 13.3185 −16.4575 21.4234 77.3200 9.00000 73.0635
\(n\): 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.q yes 9
3.b odd 2 1 1521.4.a.be 9
13.b even 2 1 507.4.a.n 9
13.d odd 4 2 507.4.b.j 18
39.d odd 2 1 1521.4.a.bj 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.n 9 13.b even 2 1
507.4.a.q yes 9 1.a even 1 1 trivial
507.4.b.j 18 13.d odd 4 2
1521.4.a.be 9 3.b odd 2 1
1521.4.a.bj 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{9} - 8T_{2}^{8} - 20T_{2}^{7} + 251T_{2}^{6} + 8T_{2}^{5} - 2521T_{2}^{4} + 1303T_{2}^{3} + 8946T_{2}^{2} - 3640T_{2} - 9464 \) 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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