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: \(x^{4} - x^{2} + 1\)
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.

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

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{2} - 3 T_{3} + 3 \)
\( T_{13}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )^{2}( 1 + 3 T^{2} )^{2} \)
$5$ 1
$7$ \( ( 1 - 4 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 - 31 T^{2} + 672 T^{4} - 8959 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 37 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \)
$23$ \( ( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 46 T^{2} + 961 T^{4} )( 1 - 13 T^{2} + 961 T^{4} ) \)
$37$ \( 1 + 65 T^{2} + 2856 T^{4} + 88985 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 2 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 - 15 T + 122 T^{2} - 705 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 + 105 T^{2} + 8216 T^{4} + 294945 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 54 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 71 T^{2} - 288 T^{4} - 378359 T^{6} + 28398241 T^{8} \)
$79$ \( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 + 26 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 27 T + 332 T^{2} + 2403 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 106 T^{2} + 9409 T^{4} )^{2} \)
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