Properties

Label 507.4.b.i
Level $507$
Weight $4$
Character orbit 507.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
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: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
5.36472i
5.04537i
3.27897i
2.04224i
0.917374i
0.917374i
2.04224i
3.27897i
5.04537i
5.36472i
5.36472i 3.00000 −20.7803 2.69631i 16.0942i 15.2025i 68.5626i 9.00000 14.4650
337.2 5.04537i 3.00000 −17.4557 20.1174i 15.1361i 15.4279i 47.7076i 9.00000 −101.500
337.3 3.27897i 3.00000 −2.75167 17.5414i 9.83692i 26.6999i 17.2091i 9.00000 57.5178
337.4 2.04224i 3.00000 3.82924 12.0825i 6.12673i 29.7373i 24.1582i 9.00000 −24.6753
337.5 0.917374i 3.00000 7.15843 15.4704i 2.75212i 20.5833i 13.9059i 9.00000 14.1922
337.6 0.917374i 3.00000 7.15843 15.4704i 2.75212i 20.5833i 13.9059i 9.00000 14.1922
337.7 2.04224i 3.00000 3.82924 12.0825i 6.12673i 29.7373i 24.1582i 9.00000 −24.6753
337.8 3.27897i 3.00000 −2.75167 17.5414i 9.83692i 26.6999i 17.2091i 9.00000 57.5178
337.9 5.04537i 3.00000 −17.4557 20.1174i 15.1361i 15.4279i 47.7076i 9.00000 −101.500
337.10 5.36472i 3.00000 −20.7803 2.69631i 16.0942i 15.2025i 68.5626i 9.00000 14.4650
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.10
Significant digits:
Format:

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.b.i 10
13.b even 2 1 inner 507.4.b.i 10
13.c even 3 1 39.4.j.c 10
13.d odd 4 2 507.4.a.r 10
13.e even 6 1 39.4.j.c 10
39.f even 4 2 1521.4.a.bk 10
39.h odd 6 1 117.4.q.e 10
39.i odd 6 1 117.4.q.e 10
52.i odd 6 1 624.4.bv.h 10
52.j odd 6 1 624.4.bv.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c 10 13.c even 3 1
39.4.j.c 10 13.e even 6 1
117.4.q.e 10 39.h odd 6 1
117.4.q.e 10 39.i odd 6 1
507.4.a.r 10 13.d odd 4 2
507.4.b.i 10 1.a even 1 1 trivial
507.4.b.i 10 13.b even 2 1 inner
624.4.bv.h 10 52.i odd 6 1
624.4.bv.h 10 52.j odd 6 1
1521.4.a.bk 10 39.f even 4 2

Hecke kernels

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

\( T_{2}^{10} + 70T_{2}^{8} + 1645T_{2}^{6} + 14700T_{2}^{4} + 44100T_{2}^{2} + 27648 \) 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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