Properties

Label 507.4.b.d
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + 3 q^{3} + 7 q^{4} - 7 i q^{5} + 3 i q^{6} - 10 i q^{7} + 15 i q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + 3 q^{3} + 7 q^{4} - 7 i q^{5} + 3 i q^{6} - 10 i q^{7} + 15 i q^{8} + 9 q^{9} + 7 q^{10} - 22 i q^{11} + 21 q^{12} + 10 q^{14} - 21 i q^{15} + 41 q^{16} - 37 q^{17} + 9 i q^{18} - 30 i q^{19} - 49 i q^{20} - 30 i q^{21} + 22 q^{22} + 162 q^{23} + 45 i q^{24} + 76 q^{25} + 27 q^{27} - 70 i q^{28} - 113 q^{29} + 21 q^{30} - 196 i q^{31} + 161 i q^{32} - 66 i q^{33} - 37 i q^{34} - 70 q^{35} + 63 q^{36} + 13 i q^{37} + 30 q^{38} + 105 q^{40} - 285 i q^{41} + 30 q^{42} + 246 q^{43} - 154 i q^{44} - 63 i q^{45} + 162 i q^{46} - 462 i q^{47} + 123 q^{48} + 243 q^{49} + 76 i q^{50} - 111 q^{51} - 537 q^{53} + 27 i q^{54} - 154 q^{55} + 150 q^{56} - 90 i q^{57} - 113 i q^{58} + 576 i q^{59} - 147 i q^{60} - 635 q^{61} + 196 q^{62} - 90 i q^{63} + 167 q^{64} + 66 q^{66} - 202 i q^{67} - 259 q^{68} + 486 q^{69} - 70 i q^{70} + 1086 i q^{71} + 135 i q^{72} - 805 i q^{73} - 13 q^{74} + 228 q^{75} - 210 i q^{76} - 220 q^{77} + 884 q^{79} - 287 i q^{80} + 81 q^{81} + 285 q^{82} - 518 i q^{83} - 210 i q^{84} + 259 i q^{85} + 246 i q^{86} - 339 q^{87} + 330 q^{88} + 194 i q^{89} + 63 q^{90} + 1134 q^{92} - 588 i q^{93} + 462 q^{94} - 210 q^{95} + 483 i q^{96} + 1202 i q^{97} + 243 i q^{98} - 198 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9} + 14 q^{10} + 42 q^{12} + 20 q^{14} + 82 q^{16} - 74 q^{17} + 44 q^{22} + 324 q^{23} + 152 q^{25} + 54 q^{27} - 226 q^{29} + 42 q^{30} - 140 q^{35} + 126 q^{36} + 60 q^{38} + 210 q^{40} + 60 q^{42} + 492 q^{43} + 246 q^{48} + 486 q^{49} - 222 q^{51} - 1074 q^{53} - 308 q^{55} + 300 q^{56} - 1270 q^{61} + 392 q^{62} + 334 q^{64} + 132 q^{66} - 518 q^{68} + 972 q^{69} - 26 q^{74} + 456 q^{75} - 440 q^{77} + 1768 q^{79} + 162 q^{81} + 570 q^{82} - 678 q^{87} + 660 q^{88} + 126 q^{90} + 2268 q^{92} + 924 q^{94} - 420 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 3.00000 7.00000 7.00000i 3.00000i 10.0000i 15.0000i 9.00000 7.00000
337.2 1.00000i 3.00000 7.00000 7.00000i 3.00000i 10.0000i 15.0000i 9.00000 7.00000
\(n\): 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.d 2
13.b even 2 1 inner 507.4.b.d 2
13.d odd 4 1 507.4.a.b 1
13.d odd 4 1 507.4.a.d 1
13.f odd 12 2 39.4.e.b 2
39.f even 4 1 1521.4.a.e 1
39.f even 4 1 1521.4.a.h 1
39.k even 12 2 117.4.g.a 2
52.l even 12 2 624.4.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 13.f odd 12 2
117.4.g.a 2 39.k even 12 2
507.4.a.b 1 13.d odd 4 1
507.4.a.d 1 13.d odd 4 1
507.4.b.d 2 1.a even 1 1 trivial
507.4.b.d 2 13.b even 2 1 inner
624.4.q.c 2 52.l even 12 2
1521.4.a.e 1 39.f even 4 1
1521.4.a.h 1 39.f even 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 49 \) Copy content Toggle raw display
$7$ \( T^{2} + 100 \) Copy content Toggle raw display
$11$ \( T^{2} + 484 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 37)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 900 \) Copy content Toggle raw display
$23$ \( (T - 162)^{2} \) Copy content Toggle raw display
$29$ \( (T + 113)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 38416 \) Copy content Toggle raw display
$37$ \( T^{2} + 169 \) Copy content Toggle raw display
$41$ \( T^{2} + 81225 \) Copy content Toggle raw display
$43$ \( (T - 246)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 213444 \) Copy content Toggle raw display
$53$ \( (T + 537)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 331776 \) Copy content Toggle raw display
$61$ \( (T + 635)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 40804 \) Copy content Toggle raw display
$71$ \( T^{2} + 1179396 \) Copy content Toggle raw display
$73$ \( T^{2} + 648025 \) Copy content Toggle raw display
$79$ \( (T - 884)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 268324 \) Copy content Toggle raw display
$89$ \( T^{2} + 37636 \) Copy content Toggle raw display
$97$ \( T^{2} + 1444804 \) Copy content Toggle raw display
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