Properties

Label 507.4.b.e
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-17})\)
Defining polynomial: \( x^{4} - 17x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} - 9 q^{4} + (3 \beta_{2} - 2 \beta_1) q^{5} - 3 \beta_1 q^{6} + (11 \beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} - 9 q^{4} + (3 \beta_{2} - 2 \beta_1) q^{5} - 3 \beta_1 q^{6} + (11 \beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8} + 9 q^{9} + ( - 3 \beta_{3} + 34) q^{10} + ( - 21 \beta_{2} + \beta_1) q^{11} + 27 q^{12} + ( - 11 \beta_{3} + 51) q^{14} + ( - 9 \beta_{2} + 6 \beta_1) q^{15} - 55 q^{16} + ( - \beta_{3} + 36) q^{17} + 9 \beta_1 q^{18} + ( - 37 \beta_{2} - 9 \beta_1) q^{19} + ( - 27 \beta_{2} + 18 \beta_1) q^{20} + ( - 33 \beta_{2} + 9 \beta_1) q^{21} + (21 \beta_{3} - 17) q^{22} + (7 \beta_{3} + 69) q^{23} + 3 \beta_1 q^{24} + (12 \beta_{3} + 30) q^{25} - 27 q^{27} + ( - 99 \beta_{2} + 27 \beta_1) q^{28} + (22 \beta_{3} + 3) q^{29} + (9 \beta_{3} - 102) q^{30} + (78 \beta_{2} - 42 \beta_1) q^{31} - 63 \beta_1 q^{32} + (63 \beta_{2} - 3 \beta_1) q^{33} + ( - 17 \beta_{2} + 36 \beta_1) q^{34} + (31 \beta_{3} - 201) q^{35} - 81 q^{36} + (82 \beta_{2} + 45 \beta_1) q^{37} + (37 \beta_{3} + 153) q^{38} + (3 \beta_{3} - 34) q^{40} + (30 \beta_{2} + \beta_1) q^{41} + (33 \beta_{3} - 153) q^{42} + (15 \beta_{3} - 235) q^{43} + (189 \beta_{2} - 9 \beta_1) q^{44} + (27 \beta_{2} - 18 \beta_1) q^{45} + (119 \beta_{2} + 69 \beta_1) q^{46} + (111 \beta_{2} + 79 \beta_1) q^{47} + 165 q^{48} + (66 \beta_{3} - 173) q^{49} + (204 \beta_{2} + 30 \beta_1) q^{50} + (3 \beta_{3} - 108) q^{51} + ( - 18 \beta_{3} - 567) q^{53} - 27 \beta_1 q^{54} + ( - 45 \beta_{3} + 223) q^{55} + (11 \beta_{3} - 51) q^{56} + (111 \beta_{2} + 27 \beta_1) q^{57} + (374 \beta_{2} + 3 \beta_1) q^{58} + ( - 360 \beta_{2} - 8 \beta_1) q^{59} + (81 \beta_{2} - 54 \beta_1) q^{60} + (87 \beta_{3} + 80) q^{61} + ( - 78 \beta_{3} + 714) q^{62} + (99 \beta_{2} - 27 \beta_1) q^{63} + 631 q^{64} + ( - 63 \beta_{3} + 51) q^{66} + ( - 83 \beta_{2} + 21 \beta_1) q^{67} + (9 \beta_{3} - 324) q^{68} + ( - 21 \beta_{3} - 207) q^{69} + (527 \beta_{2} - 201 \beta_1) q^{70} + (219 \beta_{2} - 17 \beta_1) q^{71} - 9 \beta_1 q^{72} + 225 \beta_{2} q^{73} + ( - 82 \beta_{3} - 765) q^{74} + ( - 36 \beta_{3} - 90) q^{75} + (333 \beta_{2} + 81 \beta_1) q^{76} + ( - 74 \beta_{3} + 744) q^{77} + (126 \beta_{3} + 2) q^{79} + ( - 165 \beta_{2} + 110 \beta_1) q^{80} + 81 q^{81} + ( - 30 \beta_{3} - 17) q^{82} + ( - 177 \beta_{2} + 241 \beta_1) q^{83} + (297 \beta_{2} - 81 \beta_1) q^{84} + (142 \beta_{2} - 81 \beta_1) q^{85} + (255 \beta_{2} - 235 \beta_1) q^{86} + ( - 66 \beta_{3} - 9) q^{87} + ( - 21 \beta_{3} + 17) q^{88} + ( - 42 \beta_{2} - 242 \beta_1) q^{89} + ( - 27 \beta_{3} + 306) q^{90} + ( - 63 \beta_{3} - 621) q^{92} + ( - 234 \beta_{2} + 126 \beta_1) q^{93} + ( - 111 \beta_{3} - 1343) q^{94} + ( - 47 \beta_{3} + 27) q^{95} + 189 \beta_1 q^{96} + (56 \beta_{2} + 402 \beta_1) q^{97} + (1122 \beta_{2} - 173 \beta_1) q^{98} + ( - 189 \beta_{2} + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 17 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 17 ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 34\nu ) / 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 17\beta_{2} + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 17\beta_1 \) 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
3.57071 2.06155i
−3.57071 2.06155i
−3.57071 + 2.06155i
3.57071 + 2.06155i
4.12311i −3.00000 −9.00000 3.05006i 12.3693i 6.68324i 4.12311i 9.00000 12.5757
337.2 4.12311i −3.00000 −9.00000 13.4424i 12.3693i 31.4219i 4.12311i 9.00000 55.4243
337.3 4.12311i −3.00000 −9.00000 13.4424i 12.3693i 31.4219i 4.12311i 9.00000 55.4243
337.4 4.12311i −3.00000 −9.00000 3.05006i 12.3693i 6.68324i 4.12311i 9.00000 12.5757
\(n\): e.g. 2-40 or 990-1000
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.e 4
13.b even 2 1 inner 507.4.b.e 4
13.c even 3 1 39.4.j.b 4
13.d odd 4 2 507.4.a.k 4
13.e even 6 1 39.4.j.b 4
39.f even 4 2 1521.4.a.z 4
39.h odd 6 1 117.4.q.d 4
39.i odd 6 1 117.4.q.d 4
52.i odd 6 1 624.4.bv.c 4
52.j odd 6 1 624.4.bv.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.b 4 13.c even 3 1
39.4.j.b 4 13.e even 6 1
117.4.q.d 4 39.h odd 6 1
117.4.q.d 4 39.i odd 6 1
507.4.a.k 4 13.d odd 4 2
507.4.b.e 4 1.a even 1 1 trivial
507.4.b.e 4 13.b even 2 1 inner
624.4.bv.c 4 52.i odd 6 1
624.4.bv.c 4 52.j odd 6 1
1521.4.a.z 4 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 \(S_{4}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{2} + 17 \) Copy content Toggle raw display
\( T_{5}^{4} + 190T_{5}^{2} + 1681 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 17)^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 190T^{2} + 1681 \) Copy content Toggle raw display
$7$ \( T^{4} + 1032 T^{2} + 44100 \) Copy content Toggle raw display
$11$ \( T^{4} + 2680 T^{2} + \cdots + 1705636 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 72 T + 1245)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 10968 T^{2} + \cdots + 7452900 \) Copy content Toggle raw display
$23$ \( (T^{2} - 138 T + 2262)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 24675)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 96480 T^{2} + \cdots + 137733696 \) Copy content Toggle raw display
$37$ \( T^{4} + 109194 T^{2} + \cdots + 203148009 \) Copy content Toggle raw display
$41$ \( T^{4} + 5434 T^{2} + \cdots + 7198489 \) Copy content Toggle raw display
$43$ \( (T^{2} + 470 T + 43750)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 286120 T^{2} + \cdots + 4779509956 \) Copy content Toggle raw display
$53$ \( (T^{2} + 1134 T + 304965)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 779776 T^{2} + \cdots + 150320594944 \) Copy content Toggle raw display
$61$ \( (T^{2} - 160 T - 379619)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 56328 T^{2} + \cdots + 173448900 \) Copy content Toggle raw display
$71$ \( T^{4} + 297592 T^{2} + \cdots + 19312660900 \) Copy content Toggle raw display
$73$ \( (T^{2} + 151875)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 809672)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 2162728 T^{2} + \cdots + 798145692100 \) Copy content Toggle raw display
$89$ \( T^{4} + 2001760 T^{2} + \cdots + 980686167616 \) Copy content Toggle raw display
$97$ \( T^{4} + 5513352 T^{2} + \cdots + 7495877379600 \) Copy content Toggle raw display
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