Properties

Label 700.1.u.a
Level $700$
Weight $1$
Character orbit 700.u
Analytic conductor $0.349$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 700.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.349345508843\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.980.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{12} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} - q^{6} -\zeta_{12}^{5} q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{12} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} - q^{6} -\zeta_{12}^{5} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12} q^{12} - q^{14} + \zeta_{12}^{4} q^{16} -\zeta_{12}^{4} q^{21} -\zeta_{12} q^{23} -\zeta_{12}^{2} q^{24} + \zeta_{12}^{3} q^{27} + \zeta_{12} q^{28} + q^{29} -\zeta_{12}^{5} q^{32} - q^{41} + \zeta_{12}^{5} q^{42} + \zeta_{12}^{3} q^{43} + \zeta_{12}^{2} q^{46} -2 \zeta_{12} q^{47} + \zeta_{12}^{3} q^{48} -\zeta_{12}^{4} q^{49} -\zeta_{12}^{4} q^{54} -\zeta_{12}^{2} q^{56} -\zeta_{12} q^{58} -\zeta_{12}^{4} q^{61} - q^{64} + \zeta_{12}^{5} q^{67} - q^{69} + \zeta_{12}^{2} q^{81} + \zeta_{12} q^{82} + \zeta_{12}^{3} q^{83} + q^{84} -\zeta_{12}^{4} q^{86} -\zeta_{12}^{5} q^{87} + \zeta_{12}^{4} q^{89} -\zeta_{12}^{3} q^{92} + 2 \zeta_{12}^{2} q^{94} -\zeta_{12}^{4} q^{96} + \zeta_{12}^{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 4q^{6} + O(q^{10}) \) \( 4q + 2q^{4} - 4q^{6} - 4q^{14} - 2q^{16} + 2q^{21} - 2q^{24} + 4q^{29} - 4q^{41} + 2q^{46} + 2q^{49} + 2q^{54} - 2q^{56} + 2q^{61} - 4q^{64} - 4q^{69} + 2q^{81} + 4q^{84} + 2q^{86} - 2q^{89} + 4q^{94} + 2q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 −1.00000 0.866025 0.500000i 1.00000i 0 0
51.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.00000 −0.866025 + 0.500000i 1.00000i 0 0
151.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.00000 0.866025 + 0.500000i 1.00000i 0 0
151.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 −1.00000 −0.866025 0.500000i 1.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.u.a 4
4.b odd 2 1 inner 700.1.u.a 4
5.b even 2 1 inner 700.1.u.a 4
5.c odd 4 1 140.1.p.a 2
5.c odd 4 1 140.1.p.b yes 2
7.c even 3 1 inner 700.1.u.a 4
15.e even 4 1 1260.1.ci.a 2
15.e even 4 1 1260.1.ci.b 2
20.d odd 2 1 CM 700.1.u.a 4
20.e even 4 1 140.1.p.a 2
20.e even 4 1 140.1.p.b yes 2
28.g odd 6 1 inner 700.1.u.a 4
35.f even 4 1 980.1.p.a 2
35.f even 4 1 980.1.p.b 2
35.j even 6 1 inner 700.1.u.a 4
35.k even 12 1 980.1.f.a 1
35.k even 12 1 980.1.f.d 1
35.k even 12 1 980.1.p.a 2
35.k even 12 1 980.1.p.b 2
35.l odd 12 1 140.1.p.a 2
35.l odd 12 1 140.1.p.b yes 2
35.l odd 12 1 980.1.f.b 1
35.l odd 12 1 980.1.f.c 1
40.i odd 4 1 2240.1.bt.a 2
40.i odd 4 1 2240.1.bt.b 2
40.k even 4 1 2240.1.bt.a 2
40.k even 4 1 2240.1.bt.b 2
60.l odd 4 1 1260.1.ci.a 2
60.l odd 4 1 1260.1.ci.b 2
105.x even 12 1 1260.1.ci.a 2
105.x even 12 1 1260.1.ci.b 2
140.j odd 4 1 980.1.p.a 2
140.j odd 4 1 980.1.p.b 2
140.p odd 6 1 inner 700.1.u.a 4
140.w even 12 1 140.1.p.a 2
140.w even 12 1 140.1.p.b yes 2
140.w even 12 1 980.1.f.b 1
140.w even 12 1 980.1.f.c 1
140.x odd 12 1 980.1.f.a 1
140.x odd 12 1 980.1.f.d 1
140.x odd 12 1 980.1.p.a 2
140.x odd 12 1 980.1.p.b 2
280.br even 12 1 2240.1.bt.a 2
280.br even 12 1 2240.1.bt.b 2
280.bt odd 12 1 2240.1.bt.a 2
280.bt odd 12 1 2240.1.bt.b 2
420.bp odd 12 1 1260.1.ci.a 2
420.bp odd 12 1 1260.1.ci.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 5.c odd 4 1
140.1.p.a 2 20.e even 4 1
140.1.p.a 2 35.l odd 12 1
140.1.p.a 2 140.w even 12 1
140.1.p.b yes 2 5.c odd 4 1
140.1.p.b yes 2 20.e even 4 1
140.1.p.b yes 2 35.l odd 12 1
140.1.p.b yes 2 140.w even 12 1
700.1.u.a 4 1.a even 1 1 trivial
700.1.u.a 4 4.b odd 2 1 inner
700.1.u.a 4 5.b even 2 1 inner
700.1.u.a 4 7.c even 3 1 inner
700.1.u.a 4 20.d odd 2 1 CM
700.1.u.a 4 28.g odd 6 1 inner
700.1.u.a 4 35.j even 6 1 inner
700.1.u.a 4 140.p odd 6 1 inner
980.1.f.a 1 35.k even 12 1
980.1.f.a 1 140.x odd 12 1
980.1.f.b 1 35.l odd 12 1
980.1.f.b 1 140.w even 12 1
980.1.f.c 1 35.l odd 12 1
980.1.f.c 1 140.w even 12 1
980.1.f.d 1 35.k even 12 1
980.1.f.d 1 140.x odd 12 1
980.1.p.a 2 35.f even 4 1
980.1.p.a 2 35.k even 12 1
980.1.p.a 2 140.j odd 4 1
980.1.p.a 2 140.x odd 12 1
980.1.p.b 2 35.f even 4 1
980.1.p.b 2 35.k even 12 1
980.1.p.b 2 140.j odd 4 1
980.1.p.b 2 140.x odd 12 1
1260.1.ci.a 2 15.e even 4 1
1260.1.ci.a 2 60.l odd 4 1
1260.1.ci.a 2 105.x even 12 1
1260.1.ci.a 2 420.bp odd 12 1
1260.1.ci.b 2 15.e even 4 1
1260.1.ci.b 2 60.l odd 4 1
1260.1.ci.b 2 105.x even 12 1
1260.1.ci.b 2 420.bp odd 12 1
2240.1.bt.a 2 40.i odd 4 1
2240.1.bt.a 2 40.k even 4 1
2240.1.bt.a 2 280.br even 12 1
2240.1.bt.a 2 280.bt odd 12 1
2240.1.bt.b 2 40.i odd 4 1
2240.1.bt.b 2 40.k even 4 1
2240.1.bt.b 2 280.br even 12 1
2240.1.bt.b 2 280.bt odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( -1 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 1 + T )^{4} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( 16 - 4 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 1 + T + T^{2} )^{2} \)
$97$ \( T^{4} \)
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