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Properties

Label 700.3.o.a
Level $700$
Weight $3$
Character orbit 700.o
Analytic conductor $19.074$
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] = [700,3,Mod(549,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.549");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 700.o (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: \(19.0736185052\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - 7 \zeta_{12}^{3} q^{7} + ( - 6 \zeta_{12}^{2} + 6) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - 7 \zeta_{12}^{3} q^{7} + ( - 6 \zeta_{12}^{2} + 6) q^{9} - 15 \zeta_{12}^{2} q^{11} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{13} + (17 \zeta_{12}^{3} + 17 \zeta_{12}) q^{17} + ( - 9 \zeta_{12}^{2} - 9) q^{19} + (7 \zeta_{12}^{2} - 14) q^{21} - 9 \zeta_{12} q^{23} + (15 \zeta_{12}^{3} - 30 \zeta_{12}) q^{27} + 6 q^{29} + (7 \zeta_{12}^{2} - 14) q^{31} + (30 \zeta_{12}^{3} - 15 \zeta_{12}) q^{33} - 31 \zeta_{12} q^{37} - 24 \zeta_{12}^{2} q^{39} + ( - 64 \zeta_{12}^{2} + 32) q^{41} - 10 \zeta_{12}^{3} q^{43} + (50 \zeta_{12}^{3} - 25 \zeta_{12}) q^{47} - 49 q^{49} + ( - 51 \zeta_{12}^{2} + 51) q^{51} + ( - 57 \zeta_{12}^{3} + 57 \zeta_{12}) q^{53} + 27 \zeta_{12}^{3} q^{57} + ( - 47 \zeta_{12}^{2} + 94) q^{59} + ( - 47 \zeta_{12}^{2} - 47) q^{61} - 42 \zeta_{12} q^{63} + (49 \zeta_{12}^{3} - 49 \zeta_{12}) q^{67} + (18 \zeta_{12}^{2} - 9) q^{69} - 126 q^{71} + (15 \zeta_{12}^{3} + 15 \zeta_{12}) q^{73} + (105 \zeta_{12}^{3} - 105 \zeta_{12}) q^{77} + (73 \zeta_{12}^{2} - 73) q^{79} - 9 \zeta_{12}^{2} q^{81} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{83} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{87} + ( - 33 \zeta_{12}^{2} - 33) q^{89} + ( - 112 \zeta_{12}^{2} + 56) q^{91} + 21 \zeta_{12} q^{93} + (16 \zeta_{12}^{3} - 32 \zeta_{12}) q^{97} - 90 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} - 30 q^{11} - 54 q^{19} - 42 q^{21} + 24 q^{29} - 42 q^{31} - 48 q^{39} - 196 q^{49} + 102 q^{51} + 282 q^{59} - 282 q^{61} - 504 q^{71} - 146 q^{79} - 18 q^{81} - 198 q^{89} - 360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 0 0 7.00000i 0 3.00000 + 5.19615i 0
549.2 0 0.866025 − 1.50000i 0 0 0 − 7.00000i 0 3.00000 + 5.19615i 0
649.1 0 −0.866025 − 1.50000i 0 0 0 − 7.00000i 0 3.00000 − 5.19615i 0
649.2 0 0.866025 + 1.50000i 0 0 0 7.00000i 0 3.00000 − 5.19615i 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 700.3.o.a 4
5.b even 2 1 inner 700.3.o.a 4
5.c odd 4 1 28.3.h.a ✓ 2
5.c odd 4 1 700.3.s.a 2
7.d odd 6 1 inner 700.3.o.a 4
15.e even 4 1 252.3.z.a 2
20.e even 4 1 112.3.s.a 2
35.f even 4 1 196.3.h.a 2
35.i odd 6 1 inner 700.3.o.a 4
35.k even 12 1 28.3.h.a ✓ 2
35.k even 12 1 196.3.b.a 2
35.k even 12 1 700.3.s.a 2
35.l odd 12 1 196.3.b.a 2
35.l odd 12 1 196.3.h.a 2
40.i odd 4 1 448.3.s.a 2
40.k even 4 1 448.3.s.b 2
60.l odd 4 1 1008.3.cg.c 2
105.k odd 4 1 1764.3.z.f 2
105.w odd 12 1 252.3.z.a 2
105.w odd 12 1 1764.3.d.a 2
105.x even 12 1 1764.3.d.a 2
105.x even 12 1 1764.3.z.f 2
140.j odd 4 1 784.3.s.b 2
140.w even 12 1 784.3.c.a 2
140.w even 12 1 784.3.s.b 2
140.x odd 12 1 112.3.s.a 2
140.x odd 12 1 784.3.c.a 2
280.bp odd 12 1 448.3.s.b 2
280.bv even 12 1 448.3.s.a 2
420.br even 12 1 1008.3.cg.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.h.a ✓ 2 5.c odd 4 1
28.3.h.a ✓ 2 35.k even 12 1
112.3.s.a 2 20.e even 4 1
112.3.s.a 2 140.x odd 12 1
196.3.b.a 2 35.k even 12 1
196.3.b.a 2 35.l odd 12 1
196.3.h.a 2 35.f even 4 1
196.3.h.a 2 35.l odd 12 1
252.3.z.a 2 15.e even 4 1
252.3.z.a 2 105.w odd 12 1
448.3.s.a 2 40.i odd 4 1
448.3.s.a 2 280.bv even 12 1
448.3.s.b 2 40.k even 4 1
448.3.s.b 2 280.bp odd 12 1
700.3.o.a 4 1.a even 1 1 trivial
700.3.o.a 4 5.b even 2 1 inner
700.3.o.a 4 7.d odd 6 1 inner
700.3.o.a 4 35.i odd 6 1 inner
700.3.s.a 2 5.c odd 4 1
700.3.s.a 2 35.k even 12 1
784.3.c.a 2 140.w even 12 1
784.3.c.a 2 140.x odd 12 1
784.3.s.b 2 140.j odd 4 1
784.3.s.b 2 140.w even 12 1
1008.3.cg.c 2 60.l odd 4 1
1008.3.cg.c 2 420.br even 12 1
1764.3.d.a 2 105.w odd 12 1
1764.3.d.a 2 105.x even 12 1
1764.3.z.f 2 105.k odd 4 1
1764.3.z.f 2 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 867 T^{2} + 751689 \) Copy content Toggle raw display
$19$ \( (T^{2} + 27 T + 243)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 961 T^{2} + 923521 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1875 T^{2} + 3515625 \) Copy content Toggle raw display
$53$ \( T^{4} - 3249 T^{2} + 10556001 \) Copy content Toggle raw display
$59$ \( (T^{2} - 141 T + 6627)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 141 T + 6627)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 2401 T^{2} + 5764801 \) Copy content Toggle raw display
$71$ \( (T + 126)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 675 T^{2} + 455625 \) Copy content Toggle raw display
$79$ \( (T^{2} + 73 T + 5329)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 99 T + 3267)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 768)^{2} \) Copy content Toggle raw display
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