Properties

Label 525.2.r.a
Level $525$
Weight $2$
Character orbit 525.r
Analytic conductor $4.192$
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] = [525,2,Mod(424,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.r (of order \(6\), degree \(2\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{2} + 3) q^{7} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{8} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{2} + 3) q^{7} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{11} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{12} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{13} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} - 4) q^{14} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{16} + ( - 5 \zeta_{12}^{3} - \zeta_{12}^{2} + 5 \zeta_{12} + 2) q^{17} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{18} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{19} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{21} - 2 \zeta_{12}^{3} q^{22} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{23} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{24} + (6 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{26} - \zeta_{12}^{3} q^{27} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12} + 2) q^{28} + (3 \zeta_{12}^{3} - 6 \zeta_{12} - 1) q^{29} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12}) q^{31} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 16) q^{32} + (\zeta_{12}^{2} + \zeta_{12} + 1) q^{33} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 2) q^{34} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 2) q^{36} + ( - 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{37} + ( - 7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 7 \zeta_{12} - 6) q^{38} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} - \zeta_{12}) q^{39} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 1) q^{41} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 5 \zeta_{12} + 3) q^{42} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{43} + ( - 4 \zeta_{12}^{2} + 4) q^{44} + (4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12}) q^{46} - 2 \zeta_{12} q^{47} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{48} + ( - 5 \zeta_{12}^{2} + 8) q^{49} + ( - 2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + \zeta_{12} + 5) q^{51} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12} - 12) q^{52} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12} + 12) q^{53} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 1) q^{54} + (12 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{56} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{57} + ( - 2 \zeta_{12}^{2} + 8 \zeta_{12} - 2) q^{58} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 3 \zeta_{12}) q^{59} + (4 \zeta_{12}^{2} - 4) q^{61} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{62} + ( - 3 \zeta_{12}^{2} + 2) q^{63} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12} - 16) q^{64} - 2 \zeta_{12}^{2} q^{66} + (6 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 6 \zeta_{12} + 10) q^{67} + ( - 8 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{68} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{69} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 1) q^{71} + ( - 2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{72} + (4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 4 \zeta_{12} + 10) q^{73} + ( - \zeta_{12}^{3} + 7 \zeta_{12}^{2} - \zeta_{12}) q^{74} + (6 \zeta_{12}^{3} - 12 \zeta_{12} + 14) q^{76} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{77} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{78} + ( - 12 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 6 \zeta_{12} + 3) q^{79} - \zeta_{12}^{2} q^{81} - 2 \zeta_{12} q^{82} + ( - 3 \zeta_{12}^{3} + 14 \zeta_{12}^{2} - 7) q^{83} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} - 8) q^{84} + (2 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - \zeta_{12} + 7) q^{86} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} - 6) q^{87} + (4 \zeta_{12}^{2} - 8) q^{88} + (14 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 7 \zeta_{12} + 3) q^{89} + ( - 8 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 4 \zeta_{12} + 1) q^{91} + ( - 12 \zeta_{12}^{3} + 16 \zeta_{12}^{2} - 8) q^{92} + ( - 2 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{93} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{94} + ( - 16 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{96} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{97} + ( - 5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 8 \zeta_{12} - 13) q^{98} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} + 4 q^{4} + 4 q^{6} + 10 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{2} + 4 q^{4} + 4 q^{6} + 10 q^{7} + 2 q^{9} + 2 q^{11} - 12 q^{12} - 18 q^{14} - 16 q^{16} + 6 q^{17} - 6 q^{18} + 2 q^{19} + 6 q^{23} + 12 q^{24} + 2 q^{26} + 16 q^{28} - 4 q^{29} - 6 q^{31} + 48 q^{32} + 6 q^{33} + 8 q^{34} + 8 q^{36} - 18 q^{37} - 18 q^{38} - 8 q^{39} + 4 q^{41} + 10 q^{42} + 8 q^{44} - 12 q^{46} + 22 q^{49} + 10 q^{51} - 36 q^{52} + 36 q^{53} + 2 q^{54} - 12 q^{56} - 12 q^{58} + 10 q^{59} - 8 q^{61} + 2 q^{63} - 64 q^{64} - 4 q^{66} + 30 q^{67} - 48 q^{68} - 12 q^{69} + 4 q^{71} - 12 q^{72} + 30 q^{73} + 14 q^{74} + 56 q^{76} + 8 q^{77} + 6 q^{79} - 2 q^{81} - 36 q^{84} + 14 q^{86} - 18 q^{87} - 24 q^{88} + 6 q^{89} - 6 q^{91} - 12 q^{93} - 4 q^{94} - 16 q^{96} - 48 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \zeta_{12}^{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−2.36603 + 1.36603i −0.866025 0.500000i 2.73205 4.73205i 0 2.73205 2.50000 + 0.866025i 9.46410i 0.500000 + 0.866025i 0
424.2 −0.633975 + 0.366025i 0.866025 + 0.500000i −0.732051 + 1.26795i 0 −0.732051 2.50000 + 0.866025i 2.53590i 0.500000 + 0.866025i 0
499.1 −2.36603 1.36603i −0.866025 + 0.500000i 2.73205 + 4.73205i 0 2.73205 2.50000 0.866025i 9.46410i 0.500000 0.866025i 0
499.2 −0.633975 0.366025i 0.866025 0.500000i −0.732051 1.26795i 0 −0.732051 2.50000 0.866025i 2.53590i 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.r.a 4
5.b even 2 1 525.2.r.f 4
5.c odd 4 1 105.2.i.d 4
5.c odd 4 1 525.2.i.f 4
7.c even 3 1 525.2.r.f 4
15.e even 4 1 315.2.j.c 4
20.e even 4 1 1680.2.bg.o 4
35.f even 4 1 735.2.i.l 4
35.j even 6 1 inner 525.2.r.a 4
35.k even 12 1 735.2.a.h 2
35.k even 12 1 735.2.i.l 4
35.k even 12 1 3675.2.a.be 2
35.l odd 12 1 105.2.i.d 4
35.l odd 12 1 525.2.i.f 4
35.l odd 12 1 735.2.a.g 2
35.l odd 12 1 3675.2.a.bg 2
105.w odd 12 1 2205.2.a.ba 2
105.x even 12 1 315.2.j.c 4
105.x even 12 1 2205.2.a.z 2
140.w even 12 1 1680.2.bg.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 5.c odd 4 1
105.2.i.d 4 35.l odd 12 1
315.2.j.c 4 15.e even 4 1
315.2.j.c 4 105.x even 12 1
525.2.i.f 4 5.c odd 4 1
525.2.i.f 4 35.l odd 12 1
525.2.r.a 4 1.a even 1 1 trivial
525.2.r.a 4 35.j even 6 1 inner
525.2.r.f 4 5.b even 2 1
525.2.r.f 4 7.c even 3 1
735.2.a.g 2 35.l odd 12 1
735.2.a.h 2 35.k even 12 1
735.2.i.l 4 35.f even 4 1
735.2.i.l 4 35.k even 12 1
1680.2.bg.o 4 20.e even 4 1
1680.2.bg.o 4 140.w even 12 1
2205.2.a.z 2 105.x even 12 1
2205.2.a.ba 2 105.w odd 12 1
3675.2.a.be 2 35.k even 12 1
3675.2.a.bg 2 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{3} + 14T_{2}^{2} + 12T_{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} + 38T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} - 10 T^{2} + 132 T + 484 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$37$ \( T^{4} + 18 T^{3} + 131 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 62T^{2} + 529 \) Copy content Toggle raw display
$47$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 30 T^{3} + 339 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 30 T^{3} + 359 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + 135 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( T^{4} + 312 T^{2} + 19044 \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + 174 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
$97$ \( T^{4} + 224T^{2} + 256 \) Copy content Toggle raw display
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