Properties

Label 507.4.a.r
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} + 6) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} + \beta_{6} + 7 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} + 6) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} + \beta_{6} + 7 \beta_1) q^{8} + 9 q^{9} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 8) q^{10} + ( - \beta_{9} - \beta_{8} + 2 \beta_{6} - 4 \beta_{2} + 5 \beta_1) q^{11} + (3 \beta_{4} + 18) q^{12} + ( - 2 \beta_{5} - 5 \beta_{4} + \beta_{3} - 6) q^{14} + ( - 3 \beta_{8} + 3 \beta_{2} + 3 \beta_1) q^{15} + (2 \beta_{5} + 7 \beta_{4} - 6 \beta_{3} + 50) q^{16} + (\beta_{7} + 4 \beta_{3} + 21) q^{17} + 9 \beta_1 q^{18} + ( - \beta_{9} + 3 \beta_{8} - 4 \beta_{6} - 12 \beta_{2} + 9 \beta_1) q^{19} + (4 \beta_{9} - 2 \beta_{8} - 5 \beta_{6} + 34 \beta_{2} + 18 \beta_1) q^{20} + ( - 3 \beta_{9} + 6 \beta_{2}) q^{21} + (\beta_{7} - 3 \beta_{5} + 5 \beta_{4} - 21 \beta_{3} + 58) q^{22} + (3 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - 12) q^{23} + (3 \beta_{9} - 3 \beta_{8} + 3 \beta_{6} + 21 \beta_1) q^{24} + (\beta_{5} - \beta_{4} + 21 \beta_{3} + 96) q^{25} + 27 q^{27} + ( - \beta_{9} + 9 \beta_{8} - 4 \beta_{6} + 10 \beta_{2} - 45 \beta_1) q^{28} + ( - 3 \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + 99) q^{29} + ( - 3 \beta_{7} + 3 \beta_{5} + 6 \beta_{4} + 9 \beta_{3} + 24) q^{30} + ( - 2 \beta_{9} - 3 \beta_{8} - 6 \beta_{6} - 22 \beta_{2} - 9 \beta_1) q^{31} + (3 \beta_{9} - 3 \beta_{8} + 3 \beta_{6} - 96 \beta_{2} + 51 \beta_1) q^{32} + ( - 3 \beta_{9} - 3 \beta_{8} + 6 \beta_{6} - 12 \beta_{2} + 15 \beta_1) q^{33} + (6 \beta_{8} - \beta_{6} + 50 \beta_{2} + 18 \beta_1) q^{34} + ( - 3 \beta_{7} + 5 \beta_{5} - 13 \beta_{4} - 35 \beta_{3} - 12) q^{35} + (9 \beta_{4} + 54) q^{36} + ( - \beta_{9} + 12 \beta_{8} + 27 \beta_{2} - 36 \beta_1) q^{37} + ( - \beta_{7} - \beta_{5} - 7 \beta_{4} + 17 \beta_{3} + 138) q^{38} + (\beta_{7} + 7 \beta_{5} + 14 \beta_{4} + 63 \beta_{3} + 200) q^{40} + (3 \beta_{9} - 8 \beta_{8} + 6 \beta_{6} + 71 \beta_{2} - 32 \beta_1) q^{41} + ( - 6 \beta_{5} - 15 \beta_{4} + 3 \beta_{3} - 18) q^{42} + ( - \beta_{7} - 10 \beta_{5} - 4 \beta_{4} - 18 \beta_{3} - 74) q^{43} + (7 \beta_{9} + 15 \beta_{8} + 16 \beta_{6} - 250 \beta_{2} + 69 \beta_1) q^{44} + ( - 9 \beta_{8} + 9 \beta_{2} + 9 \beta_1) q^{45} + ( - 3 \beta_{9} + 21 \beta_{8} + 10 \beta_{6} - 26 \beta_{2} - 27 \beta_1) q^{46} + (\beta_{9} - 3 \beta_{8} + 4 \beta_{6} - 4 \beta_{2} + 47 \beta_1) q^{47} + (6 \beta_{5} + 21 \beta_{4} - 18 \beta_{3} + 150) q^{48} + ( - 2 \beta_{7} - 7 \beta_{5} - 19 \beta_{4} - 3 \beta_{3} + 155) q^{49} + (\beta_{9} - \beta_{8} - 23 \beta_{6} + 288 \beta_{2} + 84 \beta_1) q^{50} + (3 \beta_{7} + 12 \beta_{3} + 63) q^{51} + ( - 4 \beta_{7} + 9 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 33) q^{53} + 27 \beta_1 q^{54} + (\beta_{7} + 11 \beta_{5} - 35 \beta_{4} - 33 \beta_{3} + 52) q^{55} + (5 \beta_{7} + 9 \beta_{5} - 27 \beta_{4} + 19 \beta_{3} - 534) q^{56} + ( - 3 \beta_{9} + 9 \beta_{8} - 12 \beta_{6} - 36 \beta_{2} + 27 \beta_1) q^{57} + ( - 3 \beta_{9} + 3 \beta_{8} + 17 \beta_{6} - 136 \beta_{2} + 135 \beta_1) q^{58} + ( - 4 \beta_{9} - 6 \beta_{8} - 16 \beta_{6} + 52 \beta_{2} + 26 \beta_1) q^{59} + (12 \beta_{9} - 6 \beta_{8} - 15 \beta_{6} + 102 \beta_{2} + 54 \beta_1) q^{60} + (\beta_{7} - 3 \beta_{5} - 7 \beta_{4} + 9 \beta_{3} + 275) q^{61} + ( - 9 \beta_{7} + 5 \beta_{5} - 28 \beta_{4} + 36 \beta_{3} - 156) q^{62} + ( - 9 \beta_{9} + 18 \beta_{2}) q^{63} + ( - 10 \beta_{5} + 19 \beta_{4} - 66 \beta_{3} + 314) q^{64} + (3 \beta_{7} - 9 \beta_{5} + 15 \beta_{4} - 63 \beta_{3} + 174) q^{66} + ( - 3 \beta_{9} + 12 \beta_{8} + 14 \beta_{6} + 106 \beta_{2} - 36 \beta_1) q^{67} + ( - 3 \beta_{7} - 5 \beta_{5} + 10 \beta_{4} + 15 \beta_{3} + 120) q^{68} + (9 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 36) q^{69} + ( - 3 \beta_{9} - 15 \beta_{8} + 8 \beta_{6} - 502 \beta_{2} - 135 \beta_1) q^{70} + ( - 5 \beta_{9} - \beta_{8} + 4 \beta_{6} - 108 \beta_{2} - 95 \beta_1) q^{71} + (9 \beta_{9} - 9 \beta_{8} + 9 \beta_{6} + 63 \beta_1) q^{72} + ( - 10 \beta_{6} + 57 \beta_{2} - 72 \beta_1) q^{73} + (12 \beta_{7} - 14 \beta_{5} - 53 \beta_{4} + 2 \beta_{3} - 438) q^{74} + (3 \beta_{5} - 3 \beta_{4} + 63 \beta_{3} + 288) q^{75} + ( - \beta_{9} - 21 \beta_{8} + 6 \beta_{6} + 346 \beta_{2} + 9 \beta_1) q^{76} + ( - 2 \beta_{7} + 8 \beta_{5} - 58 \beta_{4} + 56 \beta_{3} + 432) q^{77} + (2 \beta_{7} - 13 \beta_{5} - 11 \beta_{4} + 63 \beta_{3} + 110) q^{79} + ( - 4 \beta_{9} - 6 \beta_{8} - 13 \beta_{6} + 562 \beta_{2} + 158 \beta_1) q^{80} + 81 q^{81} + ( - 2 \beta_{7} + 8 \beta_{5} + 3 \beta_{4} + 36 \beta_{3} - 478) q^{82} + ( - 19 \beta_{9} - 37 \beta_{8} - 10 \beta_{6} - 172 \beta_{2} - 35 \beta_1) q^{83} + ( - 3 \beta_{9} + 27 \beta_{8} - 12 \beta_{6} + 30 \beta_{2} - 135 \beta_1) q^{84} + (21 \beta_{9} - 42 \beta_{8} + 14 \beta_{6} + 35 \beta_{2} - 54 \beta_1) q^{85} + ( - 24 \beta_{9} + 18 \beta_{8} + 21 \beta_{6} - 186 \beta_{2} - 77 \beta_1) q^{86} + ( - 9 \beta_{5} + 9 \beta_{4} - 33 \beta_{3} + 297) q^{87} + (23 \beta_{7} + 7 \beta_{5} + 81 \beta_{4} - 249 \beta_{3} + 634) q^{88} + (10 \beta_{9} - 6 \beta_{8} + 4 \beta_{6} - 136 \beta_{2} - 118 \beta_1) q^{89} + ( - 9 \beta_{7} + 9 \beta_{5} + 18 \beta_{4} + 27 \beta_{3} + 72) q^{90} + (7 \beta_{7} - 29 \beta_{5} - 35 \beta_{4} - 153 \beta_{3} - 174) q^{92} + ( - 6 \beta_{9} - 9 \beta_{8} - 18 \beta_{6} - 66 \beta_{2} - 27 \beta_1) q^{93} + (\beta_{7} + \beta_{5} + 63 \beta_{4} - 33 \beta_{3} + 646) q^{94} + ( - 31 \beta_{7} - 13 \beta_{5} + 47 \beta_{4} - 25 \beta_{3} - 276) q^{95} + (9 \beta_{9} - 9 \beta_{8} + 9 \beta_{6} - 288 \beta_{2} + 153 \beta_1) q^{96} + (22 \beta_{9} - 3 \beta_{8} - 32 \beta_{6} + 250 \beta_{2} - 81 \beta_1) q^{97} + ( - 33 \beta_{9} + 21 \beta_{8} - 15 \beta_{6} + 12 \beta_{2} + 11 \beta_1) q^{98} + ( - 9 \beta_{9} - 9 \beta_{8} + 18 \beta_{6} - 36 \beta_{2} + 45 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36} + 1380 q^{38} + 2000 q^{40} - 180 q^{42} - 740 q^{43} + 1500 q^{48} + 1550 q^{49} + 630 q^{51} + 330 q^{53} + 520 q^{55} - 5340 q^{56} + 2750 q^{61} - 1560 q^{62} + 3140 q^{64} + 1740 q^{66} + 1200 q^{68} - 360 q^{69} - 4380 q^{74} + 2880 q^{75} + 4320 q^{77} + 1100 q^{79} + 810 q^{81} - 4780 q^{82} + 2970 q^{87} + 6340 q^{88} + 720 q^{90} - 1740 q^{92} + 6460 q^{94} - 2760 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 35\nu^{3} - 210\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 35\nu^{4} - 210\nu^{2} ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 51\nu^{4} - 706\nu^{2} + 1792 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 49\nu^{5} + 700\nu^{3} - 2940\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - 61\nu^{6} + 1168\nu^{4} - 7140\nu^{2} + 8064 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 64\nu^{7} + 1309\nu^{5} - 9030\nu^{3} + 15912\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} - 70\nu^{7} + 1603\nu^{5} - 12654\nu^{3} + 20304\nu ) / 576 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} + 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 31\beta_{4} - 6\beta_{3} + 322 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35\beta_{9} - 35\beta_{8} + 35\beta_{6} - 96\beta_{2} + 595\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 70\beta_{5} + 875\beta_{4} - 306\beta_{3} + 8330 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1015\beta_{9} - 1015\beta_{8} + 1111\beta_{6} - 4704\beta_{2} + 15995\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96\beta_{7} + 1934\beta_{5} + 24307\beta_{4} - 11658\beta_{3} + 223930 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28175\beta_{9} - 27599\beta_{8} + 34319\beta_{6} - 175392\beta_{2} + 436603\beta_1 \) 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
−5.36472
−5.04537
−3.27897
−2.04224
−0.917374
0.917374
2.04224
3.27897
5.04537
5.36472
−5.36472 3.00000 20.7803 2.69631 −16.0942 15.2025 −68.5626 9.00000 −14.4650
1.2 −5.04537 3.00000 17.4557 −20.1174 −15.1361 15.4279 −47.7076 9.00000 101.500
1.3 −3.27897 3.00000 2.75167 17.5414 −9.83692 −26.6999 17.2091 9.00000 −57.5178
1.4 −2.04224 3.00000 −3.82924 −12.0825 −6.12673 −29.7373 24.1582 9.00000 24.6753
1.5 −0.917374 3.00000 −7.15843 15.4704 −2.75212 20.5833 13.9059 9.00000 −14.1922
1.6 0.917374 3.00000 −7.15843 −15.4704 2.75212 −20.5833 −13.9059 9.00000 −14.1922
1.7 2.04224 3.00000 −3.82924 12.0825 6.12673 29.7373 −24.1582 9.00000 24.6753
1.8 3.27897 3.00000 2.75167 −17.5414 9.83692 26.6999 −17.2091 9.00000 −57.5178
1.9 5.04537 3.00000 17.4557 20.1174 15.1361 −15.4279 47.7076 9.00000 101.500
1.10 5.36472 3.00000 20.7803 −2.69631 16.0942 −15.2025 68.5626 9.00000 −14.4650
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.a.r 10
3.b odd 2 1 1521.4.a.bk 10
13.b even 2 1 inner 507.4.a.r 10
13.d odd 4 2 507.4.b.i 10
13.f odd 12 2 39.4.j.c 10
39.d odd 2 1 1521.4.a.bk 10
39.k even 12 2 117.4.q.e 10
52.l even 12 2 624.4.bv.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c 10 13.f odd 12 2
117.4.q.e 10 39.k even 12 2
507.4.a.r 10 1.a even 1 1 trivial
507.4.a.r 10 13.b even 2 1 inner
507.4.b.i 10 13.d odd 4 2
624.4.bv.h 10 52.l even 12 2
1521.4.a.bk 10 3.b odd 2 1
1521.4.a.bk 10 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{10} - 70T_{2}^{8} + 1645T_{2}^{6} - 14700T_{2}^{4} + 44100T_{2}^{2} - 27648 \) Copy content Toggle raw display
\( T_{5}^{10} - 1105T_{5}^{8} + 441955T_{5}^{6} - 76029795T_{5}^{4} + 4880780280T_{5}^{2} - 31632011568 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 70 T^{8} + 1645 T^{6} + \cdots - 27648 \) Copy content Toggle raw display
$3$ \( (T - 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 1105 T^{8} + \cdots - 31632011568 \) Copy content Toggle raw display
$7$ \( T^{10} - 2490 T^{8} + \cdots - 14692478786352 \) Copy content Toggle raw display
$11$ \( T^{10} - 10780 T^{8} + \cdots - 50\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( (T^{5} - 105 T^{4} - 555 T^{3} + \cdots + 18224352)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 41340 T^{8} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T^{5} + 60 T^{4} - 33810 T^{3} + \cdots - 8153671248)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 495 T^{4} + 69915 T^{3} + \cdots + 427627836)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 116790 T^{8} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} - 237285 T^{8} + \cdots - 48\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{10} - 277405 T^{8} + \cdots - 36\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( (T^{5} + 370 T^{4} + \cdots - 227329236796)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 181660 T^{8} + \cdots - 21\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( (T^{5} - 165 T^{4} + \cdots - 46733997168)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 707800 T^{8} + \cdots - 41\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( (T^{5} - 1375 T^{4} + \cdots - 933851008945)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 935610 T^{8} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} - 930220 T^{8} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} - 600615 T^{8} + \cdots - 20\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( (T^{5} - 550 T^{4} + \cdots - 920208867136)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 3406900 T^{8} + \cdots - 16\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} - 1520560 T^{8} + \cdots - 28\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{10} - 4556430 T^{8} + \cdots - 15\!\cdots\!68 \) Copy content Toggle raw display
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