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Properties

Label 525.3.s.d
Level $525$
Weight $3$
Character orbit 525.s
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(124,525)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.124");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.s (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: \(14.3052138789\)
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 21)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (4 \zeta_{12}^{2} - 2) q^{6} - 7 \zeta_{12} q^{7} - 8 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (4 \zeta_{12}^{2} - 2) q^{6} - 7 \zeta_{12} q^{7} - 8 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} - 10 \zeta_{12}^{2} q^{11} + (7 \zeta_{12}^{3} - 14 \zeta_{12}) q^{13} - 14 \zeta_{12}^{2} q^{14} + ( - 16 \zeta_{12}^{2} + 16) q^{16} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{17} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{18} + ( - 19 \zeta_{12}^{2} - 19) q^{19} + ( - 14 \zeta_{12}^{2} + 7) q^{21} - 20 \zeta_{12}^{3} q^{22} + 40 \zeta_{12} q^{23} + ( - 8 \zeta_{12}^{2} + 16) q^{24} + ( - 14 \zeta_{12}^{2} - 14) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} - 16 q^{29} + ( - 3 \zeta_{12}^{2} + 6) q^{31} + ( - 20 \zeta_{12}^{3} + 10 \zeta_{12}) q^{33} + ( - 16 \zeta_{12}^{2} + 8) q^{34} - 5 \zeta_{12} q^{37} + ( - 38 \zeta_{12}^{3} - 38 \zeta_{12}) q^{38} - 21 \zeta_{12}^{2} q^{39} + ( - 28 \zeta_{12}^{2} + 14) q^{41} + ( - 28 \zeta_{12}^{3} + 14 \zeta_{12}) q^{42} + 19 \zeta_{12}^{3} q^{43} + 80 \zeta_{12}^{2} q^{46} + ( - 60 \zeta_{12}^{3} + 30 \zeta_{12}) q^{47} + ( - 16 \zeta_{12}^{3} + 32 \zeta_{12}) q^{48} + 49 \zeta_{12}^{2} q^{49} + ( - 12 \zeta_{12}^{2} + 12) q^{51} + ( - 32 \zeta_{12}^{3} + 32 \zeta_{12}) q^{53} + ( - 6 \zeta_{12}^{2} - 6) q^{54} + (56 \zeta_{12}^{2} - 56) q^{56} - 57 \zeta_{12}^{3} q^{57} - 32 \zeta_{12} q^{58} + (24 \zeta_{12}^{2} - 48) q^{59} + (12 \zeta_{12}^{2} + 12) q^{61} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{62} + ( - 21 \zeta_{12}^{3} + 21 \zeta_{12}) q^{63} - 64 q^{64} + ( - 20 \zeta_{12}^{2} + 40) q^{66} + ( - 59 \zeta_{12}^{3} + 59 \zeta_{12}) q^{67} + (80 \zeta_{12}^{2} - 40) q^{69} - 26 q^{71} + 24 \zeta_{12} q^{72} + (11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{73} - 10 \zeta_{12}^{2} q^{74} + 70 \zeta_{12}^{3} q^{77} - 42 \zeta_{12}^{3} q^{78} + ( - 47 \zeta_{12}^{2} + 47) q^{79} - 9 \zeta_{12}^{2} q^{81} + ( - 56 \zeta_{12}^{3} + 28 \zeta_{12}) q^{82} + ( - 14 \zeta_{12}^{3} + 28 \zeta_{12}) q^{83} + (38 \zeta_{12}^{2} - 38) q^{86} + ( - 16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{87} + (80 \zeta_{12}^{3} - 80 \zeta_{12}) q^{88} + ( - 68 \zeta_{12}^{2} - 68) q^{89} + (49 \zeta_{12}^{2} + 49) q^{91} + 9 \zeta_{12} q^{93} + ( - 60 \zeta_{12}^{2} + 120) q^{94} + (28 \zeta_{12}^{3} - 56 \zeta_{12}) q^{97} + 98 \zeta_{12}^{3} q^{98} + 30 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} - 20 q^{11} - 28 q^{14} + 32 q^{16} - 114 q^{19} + 48 q^{24} - 84 q^{26} - 64 q^{29} + 18 q^{31} - 42 q^{39} + 160 q^{46} + 98 q^{49} + 24 q^{51} - 36 q^{54} - 112 q^{56} - 144 q^{59} + 72 q^{61} - 256 q^{64} + 120 q^{66} - 104 q^{71} - 20 q^{74} + 94 q^{79} - 18 q^{81} - 76 q^{86} - 408 q^{89} + 294 q^{91} + 360 q^{94} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−1.73205 − 1.00000i −0.866025 − 1.50000i 0 0 3.46410i 6.06218 + 3.50000i 8.00000i −1.50000 + 2.59808i 0
124.2 1.73205 + 1.00000i 0.866025 + 1.50000i 0 0 3.46410i −6.06218 − 3.50000i − 8.00000i −1.50000 + 2.59808i 0
199.1 −1.73205 + 1.00000i −0.866025 + 1.50000i 0 0 − 3.46410i 6.06218 − 3.50000i − 8.00000i −1.50000 − 2.59808i 0
199.2 1.73205 − 1.00000i 0.866025 − 1.50000i 0 0 − 3.46410i −6.06218 + 3.50000i 8.00000i −1.50000 − 2.59808i 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.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.s.d 4
5.b even 2 1 inner 525.3.s.d 4
5.c odd 4 1 21.3.f.c ✓ 2
5.c odd 4 1 525.3.o.b 2
7.d odd 6 1 inner 525.3.s.d 4
15.e even 4 1 63.3.m.a 2
20.e even 4 1 336.3.bh.c 2
35.f even 4 1 147.3.f.e 2
35.i odd 6 1 inner 525.3.s.d 4
35.k even 12 1 21.3.f.c ✓ 2
35.k even 12 1 147.3.d.a 2
35.k even 12 1 525.3.o.b 2
35.l odd 12 1 147.3.d.a 2
35.l odd 12 1 147.3.f.e 2
60.l odd 4 1 1008.3.cg.f 2
105.k odd 4 1 441.3.m.b 2
105.w odd 12 1 63.3.m.a 2
105.w odd 12 1 441.3.d.d 2
105.x even 12 1 441.3.d.d 2
105.x even 12 1 441.3.m.b 2
140.w even 12 1 2352.3.f.b 2
140.x odd 12 1 336.3.bh.c 2
140.x odd 12 1 2352.3.f.b 2
420.br even 12 1 1008.3.cg.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.c ✓ 2 5.c odd 4 1
21.3.f.c ✓ 2 35.k even 12 1
63.3.m.a 2 15.e even 4 1
63.3.m.a 2 105.w odd 12 1
147.3.d.a 2 35.k even 12 1
147.3.d.a 2 35.l odd 12 1
147.3.f.e 2 35.f even 4 1
147.3.f.e 2 35.l odd 12 1
336.3.bh.c 2 20.e even 4 1
336.3.bh.c 2 140.x odd 12 1
441.3.d.d 2 105.w odd 12 1
441.3.d.d 2 105.x even 12 1
441.3.m.b 2 105.k odd 4 1
441.3.m.b 2 105.x even 12 1
525.3.o.b 2 5.c odd 4 1
525.3.o.b 2 35.k even 12 1
525.3.s.d 4 1.a even 1 1 trivial
525.3.s.d 4 5.b even 2 1 inner
525.3.s.d 4 7.d odd 6 1 inner
525.3.s.d 4 35.i odd 6 1 inner
1008.3.cg.f 2 60.l odd 4 1
1008.3.cg.f 2 420.br even 12 1
2352.3.f.b 2 140.w even 12 1
2352.3.f.b 2 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} - 4T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 10T_{11} + 100 \) Copy content Toggle raw display
\( T_{13}^{2} - 147 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$19$ \( (T^{2} + 57 T + 1083)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 1600 T^{2} + 2560000 \) Copy content Toggle raw display
$29$ \( (T + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$41$ \( (T^{2} + 588)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 361)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2700 T^{2} + 7290000 \) Copy content Toggle raw display
$53$ \( T^{4} - 1024 T^{2} + 1048576 \) Copy content Toggle raw display
$59$ \( (T^{2} + 72 T + 1728)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 36 T + 432)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 3481 T^{2} + 12117361 \) Copy content Toggle raw display
$71$ \( (T + 26)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 363 T^{2} + 131769 \) Copy content Toggle raw display
$79$ \( (T^{2} - 47 T + 2209)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 588)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 204 T + 13872)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2352)^{2} \) Copy content Toggle raw display
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