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Properties

Label 525.3.s.c
Level $525$
Weight $3$
Character orbit 525.s
Analytic conductor $14.305$
Analytic rank $0$
Dimension $4$
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 3 \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} - 1) q^{6} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{7} - 7 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 3 \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} - 1) q^{6} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{7} - 7 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + 11 \zeta_{12}^{2} q^{11} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{12} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{13} - 7 q^{14} + (5 \zeta_{12}^{2} - 5) q^{16} + ( - 14 \zeta_{12}^{3} - 14 \zeta_{12}) q^{17} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{18} + (2 \zeta_{12}^{2} + 2) q^{19} + ( - 7 \zeta_{12}^{2} - 7) q^{21} + 11 \zeta_{12}^{3} q^{22} - 28 \zeta_{12} q^{23} + ( - 7 \zeta_{12}^{2} + 14) q^{24} + ( - 4 \zeta_{12}^{2} - 4) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + 21 \zeta_{12} q^{28} - 25 q^{29} + (19 \zeta_{12}^{2} - 38) q^{31} + (33 \zeta_{12}^{3} - 33 \zeta_{12}) q^{32} + (22 \zeta_{12}^{3} - 11 \zeta_{12}) q^{33} + ( - 28 \zeta_{12}^{2} + 14) q^{34} + 9 q^{36} - 58 \zeta_{12} q^{37} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{38} - 12 \zeta_{12}^{2} q^{39} + (4 \zeta_{12}^{2} - 2) q^{41} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{42} + 26 \zeta_{12}^{3} q^{43} + ( - 33 \zeta_{12}^{2} + 33) q^{44} - 28 \zeta_{12}^{2} q^{46} + (88 \zeta_{12}^{3} - 44 \zeta_{12}) q^{47} + (5 \zeta_{12}^{3} - 10 \zeta_{12}) q^{48} + ( - 49 \zeta_{12}^{2} + 49) q^{49} + ( - 42 \zeta_{12}^{2} + 42) q^{51} + (12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{52} + ( - 31 \zeta_{12}^{3} + 31 \zeta_{12}) q^{53} + ( - 3 \zeta_{12}^{2} - 3) q^{54} + 49 \zeta_{12}^{2} q^{56} + 6 \zeta_{12}^{3} q^{57} - 25 \zeta_{12} q^{58} + ( - 5 \zeta_{12}^{2} + 10) q^{59} + (8 \zeta_{12}^{2} + 8) q^{61} + (19 \zeta_{12}^{3} - 38 \zeta_{12}) q^{62} - 21 \zeta_{12}^{3} q^{63} - 13 q^{64} + (11 \zeta_{12}^{2} - 22) q^{66} + ( - 52 \zeta_{12}^{3} + 52 \zeta_{12}) q^{67} + (84 \zeta_{12}^{3} - 42 \zeta_{12}) q^{68} + ( - 56 \zeta_{12}^{2} + 28) q^{69} + 64 q^{71} + 21 \zeta_{12} q^{72} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{73} - 58 \zeta_{12}^{2} q^{74} + ( - 12 \zeta_{12}^{2} + 6) q^{76} - 77 \zeta_{12} q^{77} - 12 \zeta_{12}^{3} q^{78} + ( - 17 \zeta_{12}^{2} + 17) q^{79} - 9 \zeta_{12}^{2} q^{81} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{82} + (31 \zeta_{12}^{3} - 62 \zeta_{12}) q^{83} + (42 \zeta_{12}^{2} - 21) q^{84} + (26 \zeta_{12}^{2} - 26) q^{86} + ( - 25 \zeta_{12}^{3} - 25 \zeta_{12}) q^{87} + ( - 77 \zeta_{12}^{3} + 77 \zeta_{12}) q^{88} + (46 \zeta_{12}^{2} + 46) q^{89} + ( - 28 \zeta_{12}^{2} + 56) q^{91} + 84 \zeta_{12}^{3} q^{92} - 57 \zeta_{12} q^{93} + (44 \zeta_{12}^{2} - 88) q^{94} + ( - 33 \zeta_{12}^{2} - 33) q^{96} + ( - 53 \zeta_{12}^{3} + 106 \zeta_{12}) q^{97} + ( - 49 \zeta_{12}^{3} + 49 \zeta_{12}) q^{98} - 33 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} - 6 q^{9} + 22 q^{11} - 28 q^{14} - 10 q^{16} + 12 q^{19} - 42 q^{21} + 42 q^{24} - 24 q^{26} - 100 q^{29} - 114 q^{31} + 36 q^{36} - 24 q^{39} + 66 q^{44} - 56 q^{46} + 98 q^{49} + 84 q^{51} - 18 q^{54} + 98 q^{56} + 30 q^{59} + 48 q^{61} - 52 q^{64} - 66 q^{66} + 256 q^{71} - 116 q^{74} + 34 q^{79} - 18 q^{81} - 52 q^{86} + 276 q^{89} + 168 q^{91} - 264 q^{94} - 198 q^{96} - 132 q^{99}+O(q^{100}) \) 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
−0.866025 − 0.500000i −0.866025 − 1.50000i −1.50000 − 2.59808i 0 1.73205i 6.06218 − 3.50000i 7.00000i −1.50000 + 2.59808i 0
124.2 0.866025 + 0.500000i 0.866025 + 1.50000i −1.50000 − 2.59808i 0 1.73205i −6.06218 + 3.50000i − 7.00000i −1.50000 + 2.59808i 0
199.1 −0.866025 + 0.500000i −0.866025 + 1.50000i −1.50000 + 2.59808i 0 − 1.73205i 6.06218 + 3.50000i − 7.00000i −1.50000 − 2.59808i 0
199.2 0.866025 − 0.500000i 0.866025 − 1.50000i −1.50000 + 2.59808i 0 − 1.73205i −6.06218 − 3.50000i 7.00000i −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.c 4
5.b even 2 1 inner 525.3.s.c 4
5.c odd 4 1 21.3.f.b ✓ 2
5.c odd 4 1 525.3.o.g 2
7.d odd 6 1 inner 525.3.s.c 4
15.e even 4 1 63.3.m.c 2
20.e even 4 1 336.3.bh.a 2
35.f even 4 1 147.3.f.c 2
35.i odd 6 1 inner 525.3.s.c 4
35.k even 12 1 21.3.f.b ✓ 2
35.k even 12 1 147.3.d.b 2
35.k even 12 1 525.3.o.g 2
35.l odd 12 1 147.3.d.b 2
35.l odd 12 1 147.3.f.c 2
60.l odd 4 1 1008.3.cg.g 2
105.k odd 4 1 441.3.m.e 2
105.w odd 12 1 63.3.m.c 2
105.w odd 12 1 441.3.d.b 2
105.x even 12 1 441.3.d.b 2
105.x even 12 1 441.3.m.e 2
140.w even 12 1 2352.3.f.d 2
140.x odd 12 1 336.3.bh.a 2
140.x odd 12 1 2352.3.f.d 2
420.br even 12 1 1008.3.cg.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.b ✓ 2 5.c odd 4 1
21.3.f.b ✓ 2 35.k even 12 1
63.3.m.c 2 15.e even 4 1
63.3.m.c 2 105.w odd 12 1
147.3.d.b 2 35.k even 12 1
147.3.d.b 2 35.l odd 12 1
147.3.f.c 2 35.f even 4 1
147.3.f.c 2 35.l odd 12 1
336.3.bh.a 2 20.e even 4 1
336.3.bh.a 2 140.x odd 12 1
441.3.d.b 2 105.w odd 12 1
441.3.d.b 2 105.x even 12 1
441.3.m.e 2 105.k odd 4 1
441.3.m.e 2 105.x even 12 1
525.3.o.g 2 5.c odd 4 1
525.3.o.g 2 35.k even 12 1
525.3.s.c 4 1.a even 1 1 trivial
525.3.s.c 4 5.b even 2 1 inner
525.3.s.c 4 7.d odd 6 1 inner
525.3.s.c 4 35.i odd 6 1 inner
1008.3.cg.g 2 60.l odd 4 1
1008.3.cg.g 2 420.br even 12 1
2352.3.f.d 2 140.w even 12 1
2352.3.f.d 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 \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 11T_{11} + 121 \) Copy content Toggle raw display
\( T_{13}^{2} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) 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
$11$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 588 T^{2} + 345744 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 784 T^{2} + 614656 \) Copy content Toggle raw display
$29$ \( (T + 25)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 57 T + 1083)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 3364 T^{2} + 11316496 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 5808 T^{2} + 33732864 \) Copy content Toggle raw display
$53$ \( T^{4} - 961 T^{2} + 923521 \) Copy content Toggle raw display
$59$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 24 T + 192)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 2704 T^{2} + 7311616 \) Copy content Toggle raw display
$71$ \( (T - 64)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$79$ \( (T^{2} - 17 T + 289)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2883)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 138 T + 6348)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8427)^{2} \) Copy content Toggle raw display
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