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Properties

Label 525.3.e.a
Level $525$
Weight $3$
Character orbit 525.e
Analytic conductor $14.305$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,3,Mod(349,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.349");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.e (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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + 3 q^{4} + \beta_{2} q^{6} + (4 \beta_{3} - \beta_1) q^{7} - 7 \beta_1 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + 3 q^{4} + \beta_{2} q^{6} + (4 \beta_{3} - \beta_1) q^{7} - 7 \beta_1 q^{8} + 3 q^{9} + 10 q^{11} - 3 \beta_{3} q^{12} - 4 \beta_{3} q^{13} + ( - 4 \beta_{2} - 1) q^{14} + 5 q^{16} - 3 \beta_1 q^{18} + 12 \beta_{2} q^{19} + (\beta_{2} - 12) q^{21} - 10 \beta_1 q^{22} - 14 \beta_1 q^{23} + 7 \beta_{2} q^{24} + 4 \beta_{2} q^{26} - 3 \beta_{3} q^{27} + (12 \beta_{3} - 3 \beta_1) q^{28} + 38 q^{29} + 16 \beta_{2} q^{31} - 33 \beta_1 q^{32} - 10 \beta_{3} q^{33} + 9 q^{36} - 26 \beta_1 q^{37} + 12 \beta_{3} q^{38} + 12 q^{39} - 40 \beta_{2} q^{41} + (\beta_{3} + 12 \beta_1) q^{42} + 26 \beta_1 q^{43} + 30 q^{44} - 14 q^{46} + 16 \beta_{3} q^{47} - 5 \beta_{3} q^{48} + ( - 8 \beta_{2} + 47) q^{49} - 12 \beta_{3} q^{52} + 10 \beta_1 q^{53} + 3 \beta_{2} q^{54} + ( - 28 \beta_{2} - 7) q^{56} - 36 \beta_1 q^{57} - 38 \beta_1 q^{58} - 44 \beta_{2} q^{59} + 20 \beta_{2} q^{61} + 16 \beta_{3} q^{62} + (12 \beta_{3} - 3 \beta_1) q^{63} - 13 q^{64} + 10 \beta_{2} q^{66} - 74 \beta_1 q^{67} + 14 \beta_{2} q^{69} - 62 q^{71} - 21 \beta_1 q^{72} + 24 \beta_{3} q^{73} - 26 q^{74} + 36 \beta_{2} q^{76} + (40 \beta_{3} - 10 \beta_1) q^{77} - 12 \beta_1 q^{78} + 46 q^{79} + 9 q^{81} - 40 \beta_{3} q^{82} - 52 \beta_{3} q^{83} + (3 \beta_{2} - 36) q^{84} + 26 q^{86} - 38 \beta_{3} q^{87} - 70 \beta_1 q^{88} + 24 \beta_{2} q^{89} + (4 \beta_{2} - 48) q^{91} - 42 \beta_1 q^{92} - 48 \beta_1 q^{93} - 16 \beta_{2} q^{94} + 33 \beta_{2} q^{96} + 32 \beta_{3} q^{97} + ( - 8 \beta_{3} - 47 \beta_1) q^{98} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{4} + 12 q^{9} + 40 q^{11} - 4 q^{14} + 20 q^{16} - 48 q^{21} + 152 q^{29} + 36 q^{36} + 48 q^{39} + 120 q^{44} - 56 q^{46} + 188 q^{49} - 28 q^{56} - 52 q^{64} - 248 q^{71} - 104 q^{74} + 184 q^{79} + 36 q^{81} - 144 q^{84} + 104 q^{86} - 192 q^{91} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
− 1.00000i −1.73205 3.00000 0 1.73205i 6.92820 − 1.00000i − 7.00000i 3.00000 0
349.2 − 1.00000i 1.73205 3.00000 0 − 1.73205i −6.92820 − 1.00000i − 7.00000i 3.00000 0
349.3 1.00000i −1.73205 3.00000 0 − 1.73205i 6.92820 + 1.00000i 7.00000i 3.00000 0
349.4 1.00000i 1.73205 3.00000 0 1.73205i −6.92820 + 1.00000i 7.00000i 3.00000 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
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.e.a 4
5.b even 2 1 inner 525.3.e.a 4
5.c odd 4 1 21.3.d.a ✓ 2
5.c odd 4 1 525.3.h.a 2
7.b odd 2 1 inner 525.3.e.a 4
15.e even 4 1 63.3.d.b 2
20.e even 4 1 336.3.f.a 2
35.c odd 2 1 inner 525.3.e.a 4
35.f even 4 1 21.3.d.a ✓ 2
35.f even 4 1 525.3.h.a 2
35.k even 12 1 147.3.f.b 2
35.k even 12 1 147.3.f.d 2
35.l odd 12 1 147.3.f.b 2
35.l odd 12 1 147.3.f.d 2
40.i odd 4 1 1344.3.f.c 2
40.k even 4 1 1344.3.f.b 2
60.l odd 4 1 1008.3.f.d 2
105.k odd 4 1 63.3.d.b 2
105.w odd 12 1 441.3.m.d 2
105.w odd 12 1 441.3.m.f 2
105.x even 12 1 441.3.m.d 2
105.x even 12 1 441.3.m.f 2
140.j odd 4 1 336.3.f.a 2
280.s even 4 1 1344.3.f.c 2
280.y odd 4 1 1344.3.f.b 2
420.w even 4 1 1008.3.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a ✓ 2 5.c odd 4 1
21.3.d.a ✓ 2 35.f even 4 1
63.3.d.b 2 15.e even 4 1
63.3.d.b 2 105.k odd 4 1
147.3.f.b 2 35.k even 12 1
147.3.f.b 2 35.l odd 12 1
147.3.f.d 2 35.k even 12 1
147.3.f.d 2 35.l odd 12 1
336.3.f.a 2 20.e even 4 1
336.3.f.a 2 140.j odd 4 1
441.3.m.d 2 105.w odd 12 1
441.3.m.d 2 105.x even 12 1
441.3.m.f 2 105.w odd 12 1
441.3.m.f 2 105.x even 12 1
525.3.e.a 4 1.a even 1 1 trivial
525.3.e.a 4 5.b even 2 1 inner
525.3.e.a 4 7.b odd 2 1 inner
525.3.e.a 4 35.c odd 2 1 inner
525.3.h.a 2 5.c odd 4 1
525.3.h.a 2 35.f even 4 1
1008.3.f.d 2 60.l odd 4 1
1008.3.f.d 2 420.w even 4 1
1344.3.f.b 2 40.k even 4 1
1344.3.f.b 2 280.y odd 4 1
1344.3.f.c 2 40.i odd 4 1
1344.3.f.c 2 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 94T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 10)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 432)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$29$ \( (T - 38)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 768)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 768)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 5808)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 5476)^{2} \) Copy content Toggle raw display
$71$ \( (T + 62)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 1728)^{2} \) Copy content Toggle raw display
$79$ \( (T - 46)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8112)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1728)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 3072)^{2} \) Copy content Toggle raw display
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