Properties

Label 507.2.f.a
Level $507$
Weight $2$
Character orbit 507.f
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} -\zeta_{8}^{2} q^{4} -2 \zeta_{8} q^{5} + ( 1 - \zeta_{8} - \zeta_{8}^{2} ) q^{6} + ( -1 - \zeta_{8}^{2} ) q^{7} -3 \zeta_{8}^{3} q^{8} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + \zeta_{8} q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} -\zeta_{8}^{2} q^{4} -2 \zeta_{8} q^{5} + ( 1 - \zeta_{8} - \zeta_{8}^{2} ) q^{6} + ( -1 - \zeta_{8}^{2} ) q^{7} -3 \zeta_{8}^{3} q^{8} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} -2 \zeta_{8}^{2} q^{10} + 4 \zeta_{8}^{3} q^{11} + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{12} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{14} + ( -2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{15} + q^{16} + ( -2 - \zeta_{8} + 2 \zeta_{8}^{2} ) q^{18} + ( -1 + \zeta_{8}^{2} ) q^{19} + 2 \zeta_{8}^{3} q^{20} + ( 1 + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{21} -4 q^{22} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{23} + ( -3 - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{24} -\zeta_{8}^{2} q^{25} + ( 5 - \zeta_{8} - \zeta_{8}^{3} ) q^{27} + ( -1 + \zeta_{8}^{2} ) q^{28} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{30} + ( 5 - 5 \zeta_{8}^{2} ) q^{31} -5 \zeta_{8} q^{32} + ( 4 + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{33} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{36} + ( -1 - \zeta_{8}^{2} ) q^{37} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{38} -6 q^{40} -2 \zeta_{8} q^{41} + ( -2 + \zeta_{8} + \zeta_{8}^{3} ) q^{42} -6 \zeta_{8}^{2} q^{43} + 4 \zeta_{8} q^{44} + ( 4 + 2 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{45} + ( -6 - 6 \zeta_{8}^{2} ) q^{46} + 4 \zeta_{8}^{3} q^{47} + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{48} -5 \zeta_{8}^{2} q^{49} -\zeta_{8}^{3} q^{50} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{53} + ( 1 + 5 \zeta_{8} - \zeta_{8}^{2} ) q^{54} + 8 q^{55} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{56} + ( 1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{57} + ( -2 + 2 \zeta_{8}^{2} ) q^{58} + 4 \zeta_{8}^{3} q^{59} + ( 2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{60} + 8 q^{61} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{62} + ( 1 + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} -7 \zeta_{8}^{2} q^{64} + ( 4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{66} + ( 5 - 5 \zeta_{8}^{2} ) q^{67} + ( 6 \zeta_{8} + 12 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{69} + ( -2 + 2 \zeta_{8}^{2} ) q^{70} + 4 \zeta_{8} q^{71} + ( 6 + 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{72} + ( -1 - \zeta_{8}^{2} ) q^{73} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{74} + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{75} + ( 1 + \zeta_{8}^{2} ) q^{76} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{77} -10 q^{79} -2 \zeta_{8} q^{80} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} -2 \zeta_{8}^{2} q^{82} -8 \zeta_{8} q^{83} + ( 1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{84} -6 \zeta_{8}^{3} q^{86} + ( 4 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{87} + 12 \zeta_{8}^{2} q^{88} -14 \zeta_{8}^{3} q^{89} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{90} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{92} + ( -5 - 10 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{93} -4 q^{94} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{95} + ( -5 + 5 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{96} + ( -7 + 7 \zeta_{8}^{2} ) q^{97} -5 \zeta_{8}^{3} q^{98} + ( -8 - 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{6} - 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{6} - 4q^{7} - 4q^{9} - 8q^{15} + 4q^{16} - 8q^{18} - 4q^{19} + 4q^{21} - 16q^{22} - 12q^{24} + 20q^{27} - 4q^{28} + 20q^{31} + 16q^{33} - 4q^{37} - 24q^{40} - 8q^{42} + 16q^{45} - 24q^{46} - 4q^{48} + 4q^{54} + 32q^{55} + 4q^{57} - 8q^{58} + 8q^{60} + 32q^{61} + 4q^{63} + 16q^{66} + 20q^{67} - 8q^{70} + 24q^{72} - 4q^{73} + 4q^{76} - 40q^{79} - 28q^{81} + 4q^{84} + 16q^{87} - 20q^{93} - 16q^{94} - 20q^{96} - 28q^{97} - 32q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i −1.00000 + 1.41421i 1.00000i 1.41421 + 1.41421i 1.70711 0.292893i −1.00000 1.00000i −2.12132 + 2.12132i −1.00000 2.82843i 2.00000i
239.2 0.707107 + 0.707107i −1.00000 1.41421i 1.00000i −1.41421 1.41421i 0.292893 1.70711i −1.00000 1.00000i 2.12132 2.12132i −1.00000 + 2.82843i 2.00000i
437.1 −0.707107 + 0.707107i −1.00000 1.41421i 1.00000i 1.41421 1.41421i 1.70711 + 0.292893i −1.00000 + 1.00000i −2.12132 2.12132i −1.00000 + 2.82843i 2.00000i
437.2 0.707107 0.707107i −1.00000 + 1.41421i 1.00000i −1.41421 + 1.41421i 0.292893 + 1.70711i −1.00000 + 1.00000i 2.12132 + 2.12132i −1.00000 2.82843i 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.a 4
3.b odd 2 1 inner 507.2.f.a 4
13.b even 2 1 39.2.f.a 4
13.c even 3 2 507.2.k.i 8
13.d odd 4 1 39.2.f.a 4
13.d odd 4 1 inner 507.2.f.a 4
13.e even 6 2 507.2.k.j 8
13.f odd 12 2 507.2.k.i 8
13.f odd 12 2 507.2.k.j 8
39.d odd 2 1 39.2.f.a 4
39.f even 4 1 39.2.f.a 4
39.f even 4 1 inner 507.2.f.a 4
39.h odd 6 2 507.2.k.j 8
39.i odd 6 2 507.2.k.i 8
39.k even 12 2 507.2.k.i 8
39.k even 12 2 507.2.k.j 8
52.b odd 2 1 624.2.bf.d 4
52.f even 4 1 624.2.bf.d 4
65.d even 2 1 975.2.o.j 4
65.f even 4 1 975.2.n.c 4
65.g odd 4 1 975.2.o.j 4
65.h odd 4 1 975.2.n.c 4
65.h odd 4 1 975.2.n.d 4
65.k even 4 1 975.2.n.d 4
156.h even 2 1 624.2.bf.d 4
156.l odd 4 1 624.2.bf.d 4
195.e odd 2 1 975.2.o.j 4
195.j odd 4 1 975.2.n.d 4
195.n even 4 1 975.2.o.j 4
195.s even 4 1 975.2.n.c 4
195.s even 4 1 975.2.n.d 4
195.u odd 4 1 975.2.n.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.f.a 4 13.b even 2 1
39.2.f.a 4 13.d odd 4 1
39.2.f.a 4 39.d odd 2 1
39.2.f.a 4 39.f even 4 1
507.2.f.a 4 1.a even 1 1 trivial
507.2.f.a 4 3.b odd 2 1 inner
507.2.f.a 4 13.d odd 4 1 inner
507.2.f.a 4 39.f even 4 1 inner
507.2.k.i 8 13.c even 3 2
507.2.k.i 8 13.f odd 12 2
507.2.k.i 8 39.i odd 6 2
507.2.k.i 8 39.k even 12 2
507.2.k.j 8 13.e even 6 2
507.2.k.j 8 13.f odd 12 2
507.2.k.j 8 39.h odd 6 2
507.2.k.j 8 39.k even 12 2
624.2.bf.d 4 52.b odd 2 1
624.2.bf.d 4 52.f even 4 1
624.2.bf.d 4 156.h even 2 1
624.2.bf.d 4 156.l odd 4 1
975.2.n.c 4 65.f even 4 1
975.2.n.c 4 65.h odd 4 1
975.2.n.c 4 195.s even 4 1
975.2.n.c 4 195.u odd 4 1
975.2.n.d 4 65.h odd 4 1
975.2.n.d 4 65.k even 4 1
975.2.n.d 4 195.j odd 4 1
975.2.n.d 4 195.s even 4 1
975.2.o.j 4 65.d even 2 1
975.2.o.j 4 65.g odd 4 1
975.2.o.j 4 195.e odd 2 1
975.2.o.j 4 195.n even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{4} + 1 \)
\( T_{5}^{4} + 16 \)
\( T_{7}^{2} + 2 T_{7} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( ( 3 + 2 T + T^{2} )^{2} \)
$5$ \( 16 + T^{4} \)
$7$ \( ( 2 + 2 T + T^{2} )^{2} \)
$11$ \( 256 + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 2 + 2 T + T^{2} )^{2} \)
$23$ \( ( -72 + T^{2} )^{2} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 50 - 10 T + T^{2} )^{2} \)
$37$ \( ( 2 + 2 T + T^{2} )^{2} \)
$41$ \( 16 + T^{4} \)
$43$ \( ( 36 + T^{2} )^{2} \)
$47$ \( 256 + T^{4} \)
$53$ \( ( 32 + T^{2} )^{2} \)
$59$ \( 256 + T^{4} \)
$61$ \( ( -8 + T )^{4} \)
$67$ \( ( 50 - 10 T + T^{2} )^{2} \)
$71$ \( 256 + T^{4} \)
$73$ \( ( 2 + 2 T + T^{2} )^{2} \)
$79$ \( ( 10 + T )^{4} \)
$83$ \( 4096 + T^{4} \)
$89$ \( 38416 + T^{4} \)
$97$ \( ( 98 + 14 T + T^{2} )^{2} \)
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