Properties

Label 700.2.c.c
Level $700$
Weight $2$
Character orbit 700.c
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
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] = [700,2,Mod(699,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.699");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.c (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: \(5.58952814149\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + (\beta_{3} - \beta_{2}) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_1 + 2) q^{6} + ( - \beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{3} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{2} + (\beta_{3} - \beta_{2}) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_1 + 2) q^{6} + ( - \beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{3} - \beta_1 - 1) q^{8} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{12} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{13} + (2 \beta_{3} + \beta_1) q^{14} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{16} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{17} + 6 q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} + 3) q^{21} + ( - 2 \beta_{2} - \beta_1 + 2) q^{22} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{23} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 + 1) q^{24} + ( - 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{26} + (3 \beta_{3} - 3 \beta_{2}) q^{27} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{28} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 3) q^{29} + 6 q^{31} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{32} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{33} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 5) q^{34} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{37} + ( - 6 \beta_{3} - 6) q^{38} + (6 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{39} + (2 \beta_{3} - 2 \beta_{2}) q^{41} + ( - 3 \beta_{3} + 2 \beta_1 - 7) q^{42} + 2 q^{43} + ( - 4 \beta_{3} + \beta_{2} + \beta_1) q^{44} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{46} + (\beta_{3} - \beta_{2}) q^{47} + (2 \beta_{3} - 2 \beta_{2} - 6) q^{48} + (4 \beta_{3} - 4 \beta_{2} + 1) q^{49} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{51} + ( - 2 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 6) q^{52} + ( - \beta_{3} + \beta_1 - 1) q^{53} + ( - 3 \beta_1 + 6) q^{54} + ( - \beta_{3} - 4 \beta_{2} + \beta_1 + 1) q^{56} + (6 \beta_{3} - 6 \beta_{2}) q^{57} + ( - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 5) q^{58} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{59} + (\beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{61} + ( - 6 \beta_{3} - 6) q^{62} + ( - 4 \beta_{3} + 4 \beta_1 - 4) q^{64} + (5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{66} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{67} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 6) q^{68} + ( - 5 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{69} + (2 \beta_{2} - 2 \beta_1 + 2) q^{71} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 8) q^{73} + (6 \beta_{2} - 2 \beta_1 + 4) q^{74} + (6 \beta_{2} + 6 \beta_1) q^{76} + ( - 5 \beta_{3} + \beta_1) q^{77} + ( - 6 \beta_{2} - 3 \beta_1 + 6) q^{78} + (2 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 3) q^{79} - 9 q^{81} + ( - 2 \beta_1 + 4) q^{82} + (4 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 6) q^{83} + (4 \beta_{3} + 5 \beta_{2} + \beta_1 + 4) q^{84} + ( - 2 \beta_{3} - 2) q^{86} + (5 \beta_{3} + \beta_{2} - 6 \beta_1 + 6) q^{87} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 5) q^{88} + ( - 5 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{89} + ( - 4 \beta_{3} + 7 \beta_{2} + 5 \beta_1 - 7) q^{91} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 6) q^{92} + (6 \beta_{3} - 6 \beta_{2}) q^{93} + ( - \beta_1 + 2) q^{94} + (6 \beta_{3} - 2 \beta_1 + 10) q^{96} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 6) q^{97} + ( - \beta_{3} - 4 \beta_1 + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 6 q^{6} - 8 q^{7} - 8 q^{8} + 12 q^{13} - 2 q^{14} + 8 q^{16} + 12 q^{17} + 24 q^{19} + 12 q^{21} + 10 q^{22} - 8 q^{23} + 6 q^{26} - 4 q^{29} + 24 q^{31} + 8 q^{32} - 12 q^{33} - 18 q^{34} - 12 q^{38} - 18 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{46} - 24 q^{48} + 4 q^{49} - 24 q^{52} + 18 q^{54} + 16 q^{56} + 26 q^{58} - 12 q^{62} - 6 q^{66} + 24 q^{68} + 24 q^{73} + 12 q^{77} + 30 q^{78} - 36 q^{81} + 12 q^{82} - 4 q^{86} - 16 q^{88} - 24 q^{91} + 24 q^{92} + 6 q^{94} + 24 q^{96} + 12 q^{97} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
699.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 1.73205i 1.73205 + 1.00000i 0 0.633975 2.36603i −2.00000 1.73205i −2.00000 2.00000i 0 0
699.2 −1.36603 + 0.366025i 1.73205i 1.73205 1.00000i 0 0.633975 + 2.36603i −2.00000 + 1.73205i −2.00000 + 2.00000i 0 0
699.3 0.366025 1.36603i 1.73205i −1.73205 1.00000i 0 2.36603 + 0.633975i −2.00000 1.73205i −2.00000 + 2.00000i 0 0
699.4 0.366025 + 1.36603i 1.73205i −1.73205 + 1.00000i 0 2.36603 0.633975i −2.00000 + 1.73205i −2.00000 2.00000i 0 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
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.c.c 4
4.b odd 2 1 700.2.c.e 4
5.b even 2 1 700.2.c.f 4
5.c odd 4 1 140.2.g.b yes 4
5.c odd 4 1 700.2.g.g 4
7.b odd 2 1 700.2.c.b 4
15.e even 4 1 1260.2.c.b 4
20.d odd 2 1 700.2.c.b 4
20.e even 4 1 140.2.g.a 4
20.e even 4 1 700.2.g.f 4
28.d even 2 1 700.2.c.f 4
35.c odd 2 1 700.2.c.e 4
35.f even 4 1 140.2.g.a 4
35.f even 4 1 700.2.g.f 4
35.k even 12 1 980.2.o.a 4
35.k even 12 1 980.2.o.c 4
35.l odd 12 1 980.2.o.b 4
35.l odd 12 1 980.2.o.d 4
40.i odd 4 1 2240.2.k.b 4
40.k even 4 1 2240.2.k.a 4
60.l odd 4 1 1260.2.c.a 4
105.k odd 4 1 1260.2.c.a 4
140.c even 2 1 inner 700.2.c.c 4
140.j odd 4 1 140.2.g.b yes 4
140.j odd 4 1 700.2.g.g 4
140.w even 12 1 980.2.o.a 4
140.w even 12 1 980.2.o.c 4
140.x odd 12 1 980.2.o.b 4
140.x odd 12 1 980.2.o.d 4
280.s even 4 1 2240.2.k.a 4
280.y odd 4 1 2240.2.k.b 4
420.w even 4 1 1260.2.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 20.e even 4 1
140.2.g.a 4 35.f even 4 1
140.2.g.b yes 4 5.c odd 4 1
140.2.g.b yes 4 140.j odd 4 1
700.2.c.b 4 7.b odd 2 1
700.2.c.b 4 20.d odd 2 1
700.2.c.c 4 1.a even 1 1 trivial
700.2.c.c 4 140.c even 2 1 inner
700.2.c.e 4 4.b odd 2 1
700.2.c.e 4 35.c odd 2 1
700.2.c.f 4 5.b even 2 1
700.2.c.f 4 28.d even 2 1
700.2.g.f 4 20.e even 4 1
700.2.g.f 4 35.f even 4 1
700.2.g.g 4 5.c odd 4 1
700.2.g.g 4 140.j odd 4 1
980.2.o.a 4 35.k even 12 1
980.2.o.a 4 140.w even 12 1
980.2.o.b 4 35.l odd 12 1
980.2.o.b 4 140.x odd 12 1
980.2.o.c 4 35.k even 12 1
980.2.o.c 4 140.w even 12 1
980.2.o.d 4 35.l odd 12 1
980.2.o.d 4 140.x odd 12 1
1260.2.c.a 4 60.l odd 4 1
1260.2.c.a 4 105.k odd 4 1
1260.2.c.b 4 15.e even 4 1
1260.2.c.b 4 420.w even 4 1
2240.2.k.a 4 40.k even 4 1
2240.2.k.a 4 280.s even 4 1
2240.2.k.b 4 40.i odd 4 1
2240.2.k.b 4 280.y odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{4} + 14T_{11}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} - 3 \) Copy content Toggle raw display
\( T_{19} - 6 \) Copy content Toggle raw display
\( T_{23}^{2} + 4T_{23} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T - 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T - 6)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 47)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$67$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T - 12)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 222T^{2} + 1521 \) Copy content Toggle raw display
$83$ \( T^{4} + 312 T^{2} + 17424 \) Copy content Toggle raw display
$89$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T - 99)^{2} \) Copy content Toggle raw display
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