Properties

Label 700.2.r.b
Level 700
Weight 2
Character orbit 700.r
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.r (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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -2 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -2 \zeta_{12}^{2} q^{9} + ( 3 - 3 \zeta_{12}^{2} ) q^{11} -2 \zeta_{12}^{3} q^{13} -3 \zeta_{12} q^{17} -\zeta_{12}^{2} q^{19} + ( 1 - 3 \zeta_{12}^{2} ) q^{21} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} -5 \zeta_{12}^{3} q^{27} + 6 q^{29} + ( 7 - 7 \zeta_{12}^{2} ) q^{31} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{33} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{37} + ( 2 - 2 \zeta_{12}^{2} ) q^{39} + 6 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -3 \zeta_{12}^{2} q^{51} + 3 \zeta_{12} q^{53} -\zeta_{12}^{3} q^{57} + ( 9 - 9 \zeta_{12}^{2} ) q^{59} + \zeta_{12}^{2} q^{61} + ( -2 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} + 7 \zeta_{12} q^{67} -3 q^{69} -\zeta_{12} q^{73} + ( -9 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} -13 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} -12 \zeta_{12}^{3} q^{83} + 6 \zeta_{12} q^{87} + 15 \zeta_{12}^{2} q^{89} + ( -6 + 4 \zeta_{12}^{2} ) q^{91} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{93} -10 \zeta_{12}^{3} q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 6q^{11} - 2q^{19} - 2q^{21} + 24q^{29} + 14q^{31} + 4q^{39} + 24q^{41} - 4q^{49} - 6q^{51} + 18q^{59} + 2q^{61} - 12q^{69} - 26q^{79} - 2q^{81} + 30q^{89} - 16q^{91} - 24q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 0 0 1.73205 2.00000i 0 −1.00000 + 1.73205i 0
149.2 0 0.866025 0.500000i 0 0 0 −1.73205 + 2.00000i 0 −1.00000 + 1.73205i 0
249.1 0 −0.866025 0.500000i 0 0 0 1.73205 + 2.00000i 0 −1.00000 1.73205i 0
249.2 0 0.866025 + 0.500000i 0 0 0 −1.73205 2.00000i 0 −1.00000 1.73205i 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.c even 3 1 inner
35.j 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.r.b 4
5.b even 2 1 inner 700.2.r.b 4
5.c odd 4 1 28.2.e.a 2
5.c odd 4 1 700.2.i.c 2
7.c even 3 1 inner 700.2.r.b 4
7.c even 3 1 4900.2.e.i 2
7.d odd 6 1 4900.2.e.h 2
15.e even 4 1 252.2.k.c 2
20.e even 4 1 112.2.i.b 2
35.f even 4 1 196.2.e.a 2
35.i odd 6 1 4900.2.e.h 2
35.j even 6 1 inner 700.2.r.b 4
35.j even 6 1 4900.2.e.i 2
35.k even 12 1 196.2.a.a 1
35.k even 12 1 196.2.e.a 2
35.k even 12 1 4900.2.a.n 1
35.l odd 12 1 28.2.e.a 2
35.l odd 12 1 196.2.a.b 1
35.l odd 12 1 700.2.i.c 2
35.l odd 12 1 4900.2.a.g 1
40.i odd 4 1 448.2.i.e 2
40.k even 4 1 448.2.i.c 2
45.k odd 12 1 2268.2.i.a 2
45.k odd 12 1 2268.2.l.h 2
45.l even 12 1 2268.2.i.h 2
45.l even 12 1 2268.2.l.a 2
60.l odd 4 1 1008.2.s.p 2
105.k odd 4 1 1764.2.k.b 2
105.w odd 12 1 1764.2.a.j 1
105.w odd 12 1 1764.2.k.b 2
105.x even 12 1 252.2.k.c 2
105.x even 12 1 1764.2.a.a 1
140.j odd 4 1 784.2.i.d 2
140.w even 12 1 112.2.i.b 2
140.w even 12 1 784.2.a.d 1
140.x odd 12 1 784.2.a.g 1
140.x odd 12 1 784.2.i.d 2
280.bp odd 12 1 3136.2.a.k 1
280.br even 12 1 448.2.i.c 2
280.br even 12 1 3136.2.a.s 1
280.bt odd 12 1 448.2.i.e 2
280.bt odd 12 1 3136.2.a.h 1
280.bv even 12 1 3136.2.a.v 1
315.bt odd 12 1 2268.2.l.h 2
315.bv even 12 1 2268.2.l.a 2
315.bx even 12 1 2268.2.i.h 2
315.ch odd 12 1 2268.2.i.a 2
420.bp odd 12 1 1008.2.s.p 2
420.bp odd 12 1 7056.2.a.f 1
420.br even 12 1 7056.2.a.bw 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 5.c odd 4 1
28.2.e.a 2 35.l odd 12 1
112.2.i.b 2 20.e even 4 1
112.2.i.b 2 140.w even 12 1
196.2.a.a 1 35.k even 12 1
196.2.a.b 1 35.l odd 12 1
196.2.e.a 2 35.f even 4 1
196.2.e.a 2 35.k even 12 1
252.2.k.c 2 15.e even 4 1
252.2.k.c 2 105.x even 12 1
448.2.i.c 2 40.k even 4 1
448.2.i.c 2 280.br even 12 1
448.2.i.e 2 40.i odd 4 1
448.2.i.e 2 280.bt odd 12 1
700.2.i.c 2 5.c odd 4 1
700.2.i.c 2 35.l odd 12 1
700.2.r.b 4 1.a even 1 1 trivial
700.2.r.b 4 5.b even 2 1 inner
700.2.r.b 4 7.c even 3 1 inner
700.2.r.b 4 35.j even 6 1 inner
784.2.a.d 1 140.w even 12 1
784.2.a.g 1 140.x odd 12 1
784.2.i.d 2 140.j odd 4 1
784.2.i.d 2 140.x odd 12 1
1008.2.s.p 2 60.l odd 4 1
1008.2.s.p 2 420.bp odd 12 1
1764.2.a.a 1 105.x even 12 1
1764.2.a.j 1 105.w odd 12 1
1764.2.k.b 2 105.k odd 4 1
1764.2.k.b 2 105.w odd 12 1
2268.2.i.a 2 45.k odd 12 1
2268.2.i.a 2 315.ch odd 12 1
2268.2.i.h 2 45.l even 12 1
2268.2.i.h 2 315.bx even 12 1
2268.2.l.a 2 45.l even 12 1
2268.2.l.a 2 315.bv even 12 1
2268.2.l.h 2 45.k odd 12 1
2268.2.l.h 2 315.bt odd 12 1
3136.2.a.h 1 280.bt odd 12 1
3136.2.a.k 1 280.bp odd 12 1
3136.2.a.s 1 280.br even 12 1
3136.2.a.v 1 280.bv even 12 1
4900.2.a.g 1 35.l odd 12 1
4900.2.a.n 1 35.k even 12 1
4900.2.e.h 2 7.d odd 6 1
4900.2.e.h 2 35.i odd 6 1
4900.2.e.i 2 7.c even 3 1
4900.2.e.i 2 35.j even 6 1
7056.2.a.f 1 420.bp odd 12 1
7056.2.a.bw 1 420.br even 12 1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{2} + 1 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 97 T^{2} + 6600 T^{4} + 272473 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( 1 + 145 T^{2} + 15696 T^{4} + 772705 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \)
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