Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + \beta_1 q^{6} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + \beta_1 q^{6} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} + q^{9} + (\beta_{3} + 4) q^{10} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 1) q^{12} + (\beta_{3} + 4) q^{14} + (\beta_{2} - \beta_1) q^{15} + 3 q^{16} + (2 \beta_{3} - 2) q^{17} + \beta_1 q^{18} + (\beta_{2} - \beta_1) q^{19} + (3 \beta_{2} + 5 \beta_1) q^{20} + (\beta_{2} - \beta_1) q^{21} + (\beta_{3} + 2) q^{22} + 4 q^{23} + ( - \beta_{2} - 2 \beta_1) q^{24} - 3 q^{25} + q^{27} + (3 \beta_{2} + 5 \beta_1) q^{28} + 2 q^{29} + (\beta_{3} + 4) q^{30} + (\beta_{2} + 3 \beta_1) q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + ( - \beta_{2} - \beta_1) q^{33} + (2 \beta_{2} + 4 \beta_1) q^{34} - 8 q^{35} + ( - \beta_{3} - 1) q^{36} + ( - 3 \beta_{2} + \beta_1) q^{37} + (\beta_{3} + 4) q^{38} + ( - 3 \beta_{3} - 4) q^{40} + ( - 3 \beta_{2} - 5 \beta_1) q^{41} + (\beta_{3} + 4) q^{42} + ( - 2 \beta_{3} - 4) q^{43} + ( - \beta_{2} + 3 \beta_1) q^{44} + (\beta_{2} - \beta_1) q^{45} + 4 \beta_1 q^{46} + ( - 5 \beta_{2} - \beta_1) q^{47} + 3 q^{48} - q^{49} - 3 \beta_1 q^{50} + (2 \beta_{3} - 2) q^{51} - 2 q^{53} + \beta_1 q^{54} - 2 \beta_{3} q^{55} + ( - 3 \beta_{3} - 4) q^{56} + (\beta_{2} - \beta_1) q^{57} + 2 \beta_1 q^{58} + (3 \beta_{2} - \beta_1) q^{59} + (3 \beta_{2} + 5 \beta_1) q^{60} + ( - 4 \beta_{3} + 2) q^{61} + ( - 3 \beta_{3} - 8) q^{62} + (\beta_{2} - \beta_1) q^{63} + (\beta_{3} + 7) q^{64} + (\beta_{3} + 2) q^{66} + ( - 3 \beta_{2} - \beta_1) q^{67} - 14 q^{68} + 4 q^{69} - 8 \beta_1 q^{70} + ( - \beta_{2} - \beta_1) q^{71} + ( - \beta_{2} - 2 \beta_1) q^{72} + (\beta_{2} + 5 \beta_1) q^{73} + ( - \beta_{3} - 6) q^{74} - 3 q^{75} + (3 \beta_{2} + 5 \beta_1) q^{76} - 2 \beta_{3} q^{77} + 4 \beta_{3} q^{79} + (3 \beta_{2} - 3 \beta_1) q^{80} + q^{81} + (5 \beta_{3} + 12) q^{82} + ( - \beta_{2} + 3 \beta_1) q^{83} + (3 \beta_{2} + 5 \beta_1) q^{84} + ( - 10 \beta_{2} - 6 \beta_1) q^{85} + ( - 2 \beta_{2} - 10 \beta_1) q^{86} + 2 q^{87} + ( - \beta_{3} - 6) q^{88} + (7 \beta_{2} + 5 \beta_1) q^{89} + (\beta_{3} + 4) q^{90} + ( - 4 \beta_{3} - 4) q^{92} + (\beta_{2} + 3 \beta_1) q^{93} + (\beta_{3} - 2) q^{94} - 8 q^{95} + ( - 2 \beta_{2} - \beta_1) q^{96} + ( - \beta_{2} + 3 \beta_1) q^{97} - \beta_1 q^{98} + ( - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 16 q^{10} - 4 q^{12} + 16 q^{14} + 12 q^{16} - 8 q^{17} + 8 q^{22} + 16 q^{23} - 12 q^{25} + 4 q^{27} + 8 q^{29} + 16 q^{30} - 32 q^{35} - 4 q^{36} + 16 q^{38} - 16 q^{40} + 16 q^{42} - 16 q^{43} + 12 q^{48} - 4 q^{49} - 8 q^{51} - 8 q^{53} - 16 q^{56} + 8 q^{61} - 32 q^{62} + 28 q^{64} + 8 q^{66} - 56 q^{68} + 16 q^{69} - 24 q^{74} - 12 q^{75} + 4 q^{81} + 48 q^{82} + 8 q^{87} - 24 q^{88} + 16 q^{90} - 16 q^{92} - 8 q^{94} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 1.00000 −3.82843 2.82843i 2.41421i 2.82843i 4.41421i 1.00000 6.82843
337.2 0.414214i 1.00000 1.82843 2.82843i 0.414214i 2.82843i 1.58579i 1.00000 1.17157
337.3 0.414214i 1.00000 1.82843 2.82843i 0.414214i 2.82843i 1.58579i 1.00000 1.17157
337.4 2.41421i 1.00000 −3.82843 2.82843i 2.41421i 2.82843i 4.41421i 1.00000 6.82843
\(n\): 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 507.2.b.e 4
3.b odd 2 1 1521.2.b.j 4
13.b even 2 1 inner 507.2.b.e 4
13.c even 3 2 507.2.j.f 8
13.d odd 4 1 39.2.a.b 2
13.d odd 4 1 507.2.a.h 2
13.e even 6 2 507.2.j.f 8
13.f odd 12 2 507.2.e.d 4
13.f odd 12 2 507.2.e.h 4
39.d odd 2 1 1521.2.b.j 4
39.f even 4 1 117.2.a.c 2
39.f even 4 1 1521.2.a.f 2
52.f even 4 1 624.2.a.k 2
52.f even 4 1 8112.2.a.bm 2
65.f even 4 1 975.2.c.h 4
65.g odd 4 1 975.2.a.l 2
65.k even 4 1 975.2.c.h 4
91.i even 4 1 1911.2.a.h 2
104.j odd 4 1 2496.2.a.bf 2
104.m even 4 1 2496.2.a.bi 2
117.y odd 12 2 1053.2.e.m 4
117.z even 12 2 1053.2.e.e 4
143.g even 4 1 4719.2.a.p 2
156.l odd 4 1 1872.2.a.w 2
195.j odd 4 1 2925.2.c.u 4
195.n even 4 1 2925.2.a.v 2
195.u odd 4 1 2925.2.c.u 4
273.o odd 4 1 5733.2.a.u 2
312.w odd 4 1 7488.2.a.co 2
312.y even 4 1 7488.2.a.cl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.d odd 4 1
117.2.a.c 2 39.f even 4 1
507.2.a.h 2 13.d odd 4 1
507.2.b.e 4 1.a even 1 1 trivial
507.2.b.e 4 13.b even 2 1 inner
507.2.e.d 4 13.f odd 12 2
507.2.e.h 4 13.f odd 12 2
507.2.j.f 8 13.c even 3 2
507.2.j.f 8 13.e even 6 2
624.2.a.k 2 52.f even 4 1
975.2.a.l 2 65.g odd 4 1
975.2.c.h 4 65.f even 4 1
975.2.c.h 4 65.k even 4 1
1053.2.e.e 4 117.z even 12 2
1053.2.e.m 4 117.y odd 12 2
1521.2.a.f 2 39.f even 4 1
1521.2.b.j 4 3.b odd 2 1
1521.2.b.j 4 39.d odd 2 1
1872.2.a.w 2 156.l odd 4 1
1911.2.a.h 2 91.i even 4 1
2496.2.a.bf 2 104.j odd 4 1
2496.2.a.bi 2 104.m even 4 1
2925.2.a.v 2 195.n even 4 1
2925.2.c.u 4 195.j odd 4 1
2925.2.c.u 4 195.u odd 4 1
4719.2.a.p 2 143.g even 4 1
5733.2.a.u 2 273.o odd 4 1
7488.2.a.cl 2 312.y even 4 1
7488.2.a.co 2 312.w odd 4 1
8112.2.a.bm 2 52.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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$41$ \( T^{4} + 144T^{2} + 3136 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T + 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$89$ \( T^{4} + 304 T^{2} + 18496 \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
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