Properties

Label 507.4.b.c
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} + 3 q^{3} - q^{4} - 9 i q^{5} + 9 i q^{6} - 2 i q^{7} + 21 i q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{2} + 3 q^{3} - q^{4} - 9 i q^{5} + 9 i q^{6} - 2 i q^{7} + 21 i q^{8} + 9 q^{9} + 27 q^{10} - 30 i q^{11} - 3 q^{12} + 6 q^{14} - 27 i q^{15} - 71 q^{16} + 111 q^{17} + 27 i q^{18} - 46 i q^{19} + 9 i q^{20} - 6 i q^{21} + 90 q^{22} + 6 q^{23} + 63 i q^{24} + 44 q^{25} + 27 q^{27} + 2 i q^{28} - 105 q^{29} + 81 q^{30} - 100 i q^{31} - 45 i q^{32} - 90 i q^{33} + 333 i q^{34} - 18 q^{35} - 9 q^{36} - 17 i q^{37} + 138 q^{38} + 189 q^{40} - 231 i q^{41} + 18 q^{42} + 514 q^{43} + 30 i q^{44} - 81 i q^{45} + 18 i q^{46} + 162 i q^{47} - 213 q^{48} + 339 q^{49} + 132 i q^{50} + 333 q^{51} + 639 q^{53} + 81 i q^{54} - 270 q^{55} + 42 q^{56} - 138 i q^{57} - 315 i q^{58} - 600 i q^{59} + 27 i q^{60} + 233 q^{61} + 300 q^{62} - 18 i q^{63} - 433 q^{64} + 270 q^{66} + 926 i q^{67} - 111 q^{68} + 18 q^{69} - 54 i q^{70} - 930 i q^{71} + 189 i q^{72} + 253 i q^{73} + 51 q^{74} + 132 q^{75} + 46 i q^{76} - 60 q^{77} - 1324 q^{79} + 639 i q^{80} + 81 q^{81} + 693 q^{82} + 810 i q^{83} + 6 i q^{84} - 999 i q^{85} + 1542 i q^{86} - 315 q^{87} + 630 q^{88} - 498 i q^{89} + 243 q^{90} - 6 q^{92} - 300 i q^{93} - 486 q^{94} - 414 q^{95} - 135 i q^{96} + 1358 i q^{97} + 1017 i q^{98} - 270 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9} + 54 q^{10} - 6 q^{12} + 12 q^{14} - 142 q^{16} + 222 q^{17} + 180 q^{22} + 12 q^{23} + 88 q^{25} + 54 q^{27} - 210 q^{29} + 162 q^{30} - 36 q^{35} - 18 q^{36} + 276 q^{38} + 378 q^{40} + 36 q^{42} + 1028 q^{43} - 426 q^{48} + 678 q^{49} + 666 q^{51} + 1278 q^{53} - 540 q^{55} + 84 q^{56} + 466 q^{61} + 600 q^{62} - 866 q^{64} + 540 q^{66} - 222 q^{68} + 36 q^{69} + 102 q^{74} + 264 q^{75} - 120 q^{77} - 2648 q^{79} + 162 q^{81} + 1386 q^{82} - 630 q^{87} + 1260 q^{88} + 486 q^{90} - 12 q^{92} - 972 q^{94} - 828 q^{95}+O(q^{100}) \) 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
3.00000i 3.00000 −1.00000 9.00000i 9.00000i 2.00000i 21.0000i 9.00000 27.0000
337.2 3.00000i 3.00000 −1.00000 9.00000i 9.00000i 2.00000i 21.0000i 9.00000 27.0000
\(n\): 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.c 2
13.b even 2 1 inner 507.4.b.c 2
13.d odd 4 1 507.4.a.a 1
13.d odd 4 1 507.4.a.e 1
13.f odd 12 2 39.4.e.a 2
39.f even 4 1 1521.4.a.c 1
39.f even 4 1 1521.4.a.j 1
39.k even 12 2 117.4.g.b 2
52.l even 12 2 624.4.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 13.f odd 12 2
117.4.g.b 2 39.k even 12 2
507.4.a.a 1 13.d odd 4 1
507.4.a.e 1 13.d odd 4 1
507.4.b.c 2 1.a even 1 1 trivial
507.4.b.c 2 13.b even 2 1 inner
624.4.q.b 2 52.l even 12 2
1521.4.a.c 1 39.f even 4 1
1521.4.a.j 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 \(S_{4}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 900 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 111)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2116 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 105)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 10000 \) Copy content Toggle raw display
$37$ \( T^{2} + 289 \) Copy content Toggle raw display
$41$ \( T^{2} + 53361 \) Copy content Toggle raw display
$43$ \( (T - 514)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 26244 \) Copy content Toggle raw display
$53$ \( (T - 639)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 360000 \) Copy content Toggle raw display
$61$ \( (T - 233)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 857476 \) Copy content Toggle raw display
$71$ \( T^{2} + 864900 \) Copy content Toggle raw display
$73$ \( T^{2} + 64009 \) Copy content Toggle raw display
$79$ \( (T + 1324)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 656100 \) Copy content Toggle raw display
$89$ \( T^{2} + 248004 \) Copy content Toggle raw display
$97$ \( T^{2} + 1844164 \) Copy content Toggle raw display
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