Properties

Label 700.2.t.b
Level $700$
Weight $2$
Character orbit 700.t
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
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 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} + 1) q^{3} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{6} + (2 \zeta_{12}^{2} + 1) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} + 1) q^{3} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{6} + (2 \zeta_{12}^{2} + 1) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} - \zeta_{12} q^{11} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{12} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{13} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{17} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{19} + (5 \zeta_{12}^{2} - 1) q^{21} + ( - \zeta_{12}^{3} - 1) q^{22} + \zeta_{12}^{2} q^{23} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{24} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{26} + ( - 6 \zeta_{12}^{2} + 3) q^{27} + (4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} - 4 q^{29} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{33} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{34} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{37} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{38} - 6 \zeta_{12} q^{39} + (4 \zeta_{12}^{2} - 2) q^{41} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 5) q^{42} - 2 q^{43} - 2 \zeta_{12}^{2} q^{44} + (\zeta_{12}^{2} + \zeta_{12} - 1) q^{46} + ( - 5 \zeta_{12}^{2} + 10) q^{47} + (8 \zeta_{12}^{2} - 4) q^{48} + (8 \zeta_{12}^{2} - 3) q^{49} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{51} + ( - 4 \zeta_{12}^{2} - 4) q^{52} - \zeta_{12} q^{53} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{54} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{56} - 9 \zeta_{12}^{3} q^{57} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{58} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{59} + (3 \zeta_{12}^{2} - 6) q^{61} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{62} + 8 \zeta_{12}^{3} q^{64} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{66} + ( - 3 \zeta_{12}^{2} + 3) q^{67} + ( - 2 \zeta_{12}^{2} + 4) q^{68} + (2 \zeta_{12}^{2} - 1) q^{69} + 14 \zeta_{12}^{3} q^{71} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{73} + ( - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{74} + ( - 12 \zeta_{12}^{2} + 6) q^{76} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{77} + ( - 6 \zeta_{12}^{3} - 6) q^{78} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{79} + ( - 9 \zeta_{12}^{2} + 9) q^{81} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{82} + ( - 16 \zeta_{12}^{2} + 8) q^{83} + (10 \zeta_{12}^{3} - 2 \zeta_{12}) q^{84} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{86} + ( - 4 \zeta_{12}^{2} - 4) q^{87} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{88} + (9 \zeta_{12}^{2} - 18) q^{89} + ( - 2 \zeta_{12}^{3} - 8 \zeta_{12}) q^{91} + 2 \zeta_{12}^{3} q^{92} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{93} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{94} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{96} + (10 \zeta_{12}^{3} - 20 \zeta_{12}) q^{97} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 6 q^{3} + 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 6 q^{3} + 8 q^{7} + 8 q^{8} - 2 q^{14} + 8 q^{16} + 6 q^{21} - 4 q^{22} + 2 q^{23} + 12 q^{24} - 12 q^{26} - 16 q^{29} - 8 q^{32} - 18 q^{38} - 12 q^{42} - 8 q^{43} - 4 q^{44} - 2 q^{46} + 30 q^{47} + 4 q^{49} - 24 q^{52} + 18 q^{54} + 16 q^{56} - 8 q^{58} - 18 q^{61} - 6 q^{66} + 6 q^{67} + 12 q^{68} + 6 q^{74} - 24 q^{78} + 18 q^{81} - 12 q^{82} - 4 q^{86} - 24 q^{87} + 4 q^{88} - 54 q^{89} + 30 q^{94} - 24 q^{96} - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.366025 + 1.36603i 1.50000 + 0.866025i −1.73205 1.00000i 0 −1.73205 + 1.73205i 2.00000 + 1.73205i 2.00000 2.00000i 0 0
199.2 1.36603 + 0.366025i 1.50000 + 0.866025i 1.73205 + 1.00000i 0 1.73205 + 1.73205i 2.00000 + 1.73205i 2.00000 + 2.00000i 0 0
299.1 −0.366025 1.36603i 1.50000 0.866025i −1.73205 + 1.00000i 0 −1.73205 1.73205i 2.00000 1.73205i 2.00000 + 2.00000i 0 0
299.2 1.36603 0.366025i 1.50000 0.866025i 1.73205 1.00000i 0 1.73205 1.73205i 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
7.d odd 6 1 inner
20.d odd 2 1 inner
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.t.b 4
4.b odd 2 1 700.2.t.a 4
5.b even 2 1 700.2.t.a 4
5.c odd 4 1 28.2.f.a 4
5.c odd 4 1 700.2.p.a 4
7.d odd 6 1 inner 700.2.t.b 4
15.e even 4 1 252.2.bf.e 4
20.d odd 2 1 inner 700.2.t.b 4
20.e even 4 1 28.2.f.a 4
20.e even 4 1 700.2.p.a 4
28.f even 6 1 700.2.t.a 4
35.f even 4 1 196.2.f.a 4
35.i odd 6 1 700.2.t.a 4
35.k even 12 1 28.2.f.a 4
35.k even 12 1 196.2.d.b 4
35.k even 12 1 700.2.p.a 4
35.l odd 12 1 196.2.d.b 4
35.l odd 12 1 196.2.f.a 4
40.i odd 4 1 448.2.p.d 4
40.k even 4 1 448.2.p.d 4
60.l odd 4 1 252.2.bf.e 4
105.w odd 12 1 252.2.bf.e 4
105.w odd 12 1 1764.2.b.a 4
105.x even 12 1 1764.2.b.a 4
140.j odd 4 1 196.2.f.a 4
140.s even 6 1 inner 700.2.t.b 4
140.w even 12 1 196.2.d.b 4
140.w even 12 1 196.2.f.a 4
140.x odd 12 1 28.2.f.a 4
140.x odd 12 1 196.2.d.b 4
140.x odd 12 1 700.2.p.a 4
280.bp odd 12 1 448.2.p.d 4
280.bp odd 12 1 3136.2.f.e 4
280.br even 12 1 3136.2.f.e 4
280.bt odd 12 1 3136.2.f.e 4
280.bv even 12 1 448.2.p.d 4
280.bv even 12 1 3136.2.f.e 4
420.bp odd 12 1 1764.2.b.a 4
420.br even 12 1 252.2.bf.e 4
420.br even 12 1 1764.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 5.c odd 4 1
28.2.f.a 4 20.e even 4 1
28.2.f.a 4 35.k even 12 1
28.2.f.a 4 140.x odd 12 1
196.2.d.b 4 35.k even 12 1
196.2.d.b 4 35.l odd 12 1
196.2.d.b 4 140.w even 12 1
196.2.d.b 4 140.x odd 12 1
196.2.f.a 4 35.f even 4 1
196.2.f.a 4 35.l odd 12 1
196.2.f.a 4 140.j odd 4 1
196.2.f.a 4 140.w even 12 1
252.2.bf.e 4 15.e even 4 1
252.2.bf.e 4 60.l odd 4 1
252.2.bf.e 4 105.w odd 12 1
252.2.bf.e 4 420.br even 12 1
448.2.p.d 4 40.i odd 4 1
448.2.p.d 4 40.k even 4 1
448.2.p.d 4 280.bp odd 12 1
448.2.p.d 4 280.bv even 12 1
700.2.p.a 4 5.c odd 4 1
700.2.p.a 4 20.e even 4 1
700.2.p.a 4 35.k even 12 1
700.2.p.a 4 140.x odd 12 1
700.2.t.a 4 4.b odd 2 1
700.2.t.a 4 5.b even 2 1
700.2.t.a 4 28.f even 6 1
700.2.t.a 4 35.i odd 6 1
700.2.t.b 4 1.a even 1 1 trivial
700.2.t.b 4 7.d odd 6 1 inner
700.2.t.b 4 20.d odd 2 1 inner
700.2.t.b 4 140.s even 6 1 inner
1764.2.b.a 4 105.w odd 12 1
1764.2.b.a 4 105.x even 12 1
1764.2.b.a 4 420.bp odd 12 1
1764.2.b.a 4 420.br even 12 1
3136.2.f.e 4 280.bp odd 12 1
3136.2.f.e 4 280.br even 12 1
3136.2.f.e 4 280.bt odd 12 1
3136.2.f.e 4 280.bv even 12 1

Hecke kernels

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

\( T_{3}^{2} - 3T_{3} + 3 \) Copy content Toggle raw display
\( T_{13}^{2} - 12 \) 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
$3$ \( (T^{2} - 3 T + 3)^{2} \) 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
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$19$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$37$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$59$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$61$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$79$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$83$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27 T + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
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