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Properties

Label 525.2.i.e
Level $525$
Weight $2$
Character orbit 525.i
Analytic conductor $4.192$
Analytic rank $0$
Dimension $2$
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(151,525)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.i (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + 2 q^{6} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + 2 q^{6} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + 2 \zeta_{6} q^{12} - q^{13} + (6 \zeta_{6} - 2) q^{14} + 4 \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{18} - \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 3) q^{21} + 4 q^{22} - 2 \zeta_{6} q^{26} - q^{27} + (4 \zeta_{6} - 6) q^{28} + 4 q^{29} + (9 \zeta_{6} - 9) q^{31} + (8 \zeta_{6} - 8) q^{32} - 2 \zeta_{6} q^{33} + 2 q^{36} + 3 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{38} + (\zeta_{6} - 1) q^{39} - 10 q^{41} + (2 \zeta_{6} + 4) q^{42} - 5 q^{43} + 4 \zeta_{6} q^{44} - 6 \zeta_{6} q^{47} + 4 q^{48} + (5 \zeta_{6} + 3) q^{49} + ( - 2 \zeta_{6} + 2) q^{52} + ( - 12 \zeta_{6} + 12) q^{53} - 2 \zeta_{6} q^{54} - q^{57} + 8 \zeta_{6} q^{58} + ( - 12 \zeta_{6} + 12) q^{59} - 10 \zeta_{6} q^{61} - 18 q^{62} + ( - 3 \zeta_{6} + 1) q^{63} - 8 q^{64} + ( - 4 \zeta_{6} + 4) q^{66} + (5 \zeta_{6} - 5) q^{67} - 6 q^{71} + (3 \zeta_{6} - 3) q^{73} + (6 \zeta_{6} - 6) q^{74} + 2 q^{76} + ( - 4 \zeta_{6} + 6) q^{77} - 2 q^{78} + \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 20 \zeta_{6} q^{82} - 6 q^{83} + (6 \zeta_{6} - 2) q^{84} - 10 \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 4) q^{87} - 16 \zeta_{6} q^{89} + ( - \zeta_{6} - 2) q^{91} + 9 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{94} + 8 \zeta_{6} q^{96} + 6 q^{97} + (16 \zeta_{6} - 10) q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} - 2 q^{4} + 4 q^{6} + 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} - 2 q^{4} + 4 q^{6} + 5 q^{7} - q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{14} + 4 q^{16} + 2 q^{18} - q^{19} + 4 q^{21} + 8 q^{22} - 2 q^{26} - 2 q^{27} - 8 q^{28} + 8 q^{29} - 9 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{36} + 3 q^{37} + 2 q^{38} - q^{39} - 20 q^{41} + 10 q^{42} - 10 q^{43} + 4 q^{44} - 6 q^{47} + 8 q^{48} + 11 q^{49} + 2 q^{52} + 12 q^{53} - 2 q^{54} - 2 q^{57} + 8 q^{58} + 12 q^{59} - 10 q^{61} - 36 q^{62} - q^{63} - 16 q^{64} + 4 q^{66} - 5 q^{67} - 12 q^{71} - 3 q^{73} - 6 q^{74} + 4 q^{76} + 8 q^{77} - 4 q^{78} + q^{79} - q^{81} - 20 q^{82} - 12 q^{83} + 2 q^{84} - 10 q^{86} + 4 q^{87} - 16 q^{89} - 5 q^{91} + 9 q^{93} + 12 q^{94} + 8 q^{96} + 12 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.500000 + 0.866025i
0.500000 − 0.866025i
1.00000 + 1.73205i 0.500000 − 0.866025i −1.00000 + 1.73205i 0 2.00000 2.50000 + 0.866025i 0 −0.500000 − 0.866025i 0
226.1 1.00000 − 1.73205i 0.500000 + 0.866025i −1.00000 − 1.73205i 0 2.00000 2.50000 − 0.866025i 0 −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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.e 2
5.b even 2 1 21.2.e.a ✓ 2
5.c odd 4 2 525.2.r.e 4
7.c even 3 1 inner 525.2.i.e 2
7.c even 3 1 3675.2.a.a 1
7.d odd 6 1 3675.2.a.c 1
15.d odd 2 1 63.2.e.b 2
20.d odd 2 1 336.2.q.f 2
35.c odd 2 1 147.2.e.a 2
35.i odd 6 1 147.2.a.b 1
35.i odd 6 1 147.2.e.a 2
35.j even 6 1 21.2.e.a ✓ 2
35.j even 6 1 147.2.a.c 1
35.l odd 12 2 525.2.r.e 4
40.e odd 2 1 1344.2.q.c 2
40.f even 2 1 1344.2.q.m 2
45.h odd 6 1 567.2.g.f 2
45.h odd 6 1 567.2.h.a 2
45.j even 6 1 567.2.g.a 2
45.j even 6 1 567.2.h.f 2
60.h even 2 1 1008.2.s.d 2
105.g even 2 1 441.2.e.e 2
105.o odd 6 1 63.2.e.b 2
105.o odd 6 1 441.2.a.b 1
105.p even 6 1 441.2.a.a 1
105.p even 6 1 441.2.e.e 2
140.c even 2 1 2352.2.q.c 2
140.p odd 6 1 336.2.q.f 2
140.p odd 6 1 2352.2.a.d 1
140.s even 6 1 2352.2.a.w 1
140.s even 6 1 2352.2.q.c 2
280.ba even 6 1 9408.2.a.k 1
280.bf even 6 1 1344.2.q.m 2
280.bf even 6 1 9408.2.a.bg 1
280.bi odd 6 1 1344.2.q.c 2
280.bi odd 6 1 9408.2.a.cv 1
280.bk odd 6 1 9408.2.a.bz 1
315.r even 6 1 567.2.g.a 2
315.v odd 6 1 567.2.h.a 2
315.bo even 6 1 567.2.h.f 2
315.br odd 6 1 567.2.g.f 2
420.ba even 6 1 1008.2.s.d 2
420.ba even 6 1 7056.2.a.bp 1
420.be odd 6 1 7056.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a ✓ 2 5.b even 2 1
21.2.e.a ✓ 2 35.j even 6 1
63.2.e.b 2 15.d odd 2 1
63.2.e.b 2 105.o odd 6 1
147.2.a.b 1 35.i odd 6 1
147.2.a.c 1 35.j even 6 1
147.2.e.a 2 35.c odd 2 1
147.2.e.a 2 35.i odd 6 1
336.2.q.f 2 20.d odd 2 1
336.2.q.f 2 140.p odd 6 1
441.2.a.a 1 105.p even 6 1
441.2.a.b 1 105.o odd 6 1
441.2.e.e 2 105.g even 2 1
441.2.e.e 2 105.p even 6 1
525.2.i.e 2 1.a even 1 1 trivial
525.2.i.e 2 7.c even 3 1 inner
525.2.r.e 4 5.c odd 4 2
525.2.r.e 4 35.l odd 12 2
567.2.g.a 2 45.j even 6 1
567.2.g.a 2 315.r even 6 1
567.2.g.f 2 45.h odd 6 1
567.2.g.f 2 315.br odd 6 1
567.2.h.a 2 45.h odd 6 1
567.2.h.a 2 315.v odd 6 1
567.2.h.f 2 45.j even 6 1
567.2.h.f 2 315.bo even 6 1
1008.2.s.d 2 60.h even 2 1
1008.2.s.d 2 420.ba even 6 1
1344.2.q.c 2 40.e odd 2 1
1344.2.q.c 2 280.bi odd 6 1
1344.2.q.m 2 40.f even 2 1
1344.2.q.m 2 280.bf even 6 1
2352.2.a.d 1 140.p odd 6 1
2352.2.a.w 1 140.s even 6 1
2352.2.q.c 2 140.c even 2 1
2352.2.q.c 2 140.s even 6 1
3675.2.a.a 1 7.c even 3 1
3675.2.a.c 1 7.d odd 6 1
7056.2.a.m 1 420.be odd 6 1
7056.2.a.bp 1 420.ba even 6 1
9408.2.a.k 1 280.ba even 6 1
9408.2.a.bg 1 280.bf even 6 1
9408.2.a.bz 1 280.bk odd 6 1
9408.2.a.cv 1 280.bi odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( (T + 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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