Properties

Label 700.2.i.f
Level $700$
Weight $2$
Character orbit 700.i
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{4} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{4} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{4} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{13} + ( 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{7} ) q^{19} + ( -1 + 3 \beta_{2} - \beta_{7} ) q^{21} + ( 2 \beta_{3} - 3 \beta_{6} ) q^{23} + 3 \beta_{4} q^{27} + \beta_{7} q^{29} + ( 1 - \beta_{1} - \beta_{2} ) q^{31} + ( 3 \beta_{3} - 3 \beta_{6} ) q^{33} + ( \beta_{3} - 5 \beta_{6} ) q^{37} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{39} + ( -8 + \beta_{7} ) q^{41} + ( \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( -\beta_{3} + 3 \beta_{6} ) q^{47} + ( -3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{7} ) q^{49} + ( -\beta_{1} - 5 \beta_{2} + \beta_{7} ) q^{51} + ( 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{53} + ( 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{57} + ( -1 + \beta_{1} + \beta_{2} ) q^{59} + ( -2 \beta_{1} + 5 \beta_{2} + 2 \beta_{7} ) q^{61} + ( -5 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{67} + ( -5 - 2 \beta_{7} ) q^{69} + ( 4 - 2 \beta_{7} ) q^{71} + ( 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 \beta_{4} - 3 \beta_{5} ) q^{77} + ( \beta_{1} - 3 \beta_{2} - \beta_{7} ) q^{79} + ( 9 - 9 \beta_{2} ) q^{81} + ( -3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{83} + ( -\beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{87} + 7 \beta_{2} q^{89} + ( -4 + 2 \beta_{1} + 12 \beta_{2} ) q^{91} + ( -3 \beta_{3} + \beta_{6} ) q^{93} + 4 \beta_{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 6 q^{11} + 2 q^{19} + 4 q^{29} + 2 q^{31} + 12 q^{39} - 60 q^{41} - 10 q^{49} - 18 q^{51} - 2 q^{59} + 24 q^{61} - 48 q^{69} + 24 q^{71} - 14 q^{79} + 36 q^{81} + 28 q^{89} + 20 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} - 56 \nu^{4} - 224 \nu^{2} - 895 \)\()/280\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{6} + 56 \nu^{4} + 504 \nu^{2} + 2025 \)\()/1400\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{7} + 224 \nu^{5} + 616 \nu^{3} + 9475 \nu \)\()/7000\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 16 \nu^{5} + 44 \nu^{3} + 175 \nu \)\()/500\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 14 \nu^{5} + 126 \nu^{3} + 505 \nu \)\()/350\)
\(\beta_{6}\)\(=\)\((\)\( 47 \nu^{7} + 448 \nu^{5} + 1232 \nu^{3} + 4975 \nu \)\()/7000\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} - 9 \nu^{4} - 31 \nu^{2} - 100 \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} - \beta_{4} + 3 \beta_{3}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + 13 \beta_{2} - \beta_{1} - 14\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{6} + 12 \beta_{5} + \beta_{4} - 12 \beta_{3}\)\()/3\)
\(\nu^{4}\)\(=\)\(-3 \beta_{7} - 17 \beta_{2} - 3 \beta_{1} + 3\)
\(\nu^{5}\)\(=\)\((\)\(34 \beta_{6} - 33 \beta_{5} + 67 \beta_{4}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-56 \beta_{7} + 56 \beta_{2} + 112 \beta_{1} + 53\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(281 \beta_{6} - 559 \beta_{4} - 3 \beta_{3}\)\()/3\)

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−0.656712 + 2.13746i
1.52274 1.63746i
−1.52274 + 1.63746i
0.656712 2.13746i
−0.656712 2.13746i
1.52274 + 1.63746i
−1.52274 1.63746i
0.656712 + 2.13746i
0 −0.866025 1.50000i 0 0 0 0.209313 + 2.63746i 0 0 0
401.2 0 −0.866025 1.50000i 0 0 0 2.38876 1.13746i 0 0 0
401.3 0 0.866025 + 1.50000i 0 0 0 −2.38876 + 1.13746i 0 0 0
401.4 0 0.866025 + 1.50000i 0 0 0 −0.209313 2.63746i 0 0 0
501.1 0 −0.866025 + 1.50000i 0 0 0 0.209313 2.63746i 0 0 0
501.2 0 −0.866025 + 1.50000i 0 0 0 2.38876 + 1.13746i 0 0 0
501.3 0 0.866025 1.50000i 0 0 0 −2.38876 1.13746i 0 0 0
501.4 0 0.866025 1.50000i 0 0 0 −0.209313 + 2.63746i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.4
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.i.f 8
5.b even 2 1 inner 700.2.i.f 8
5.c odd 4 1 140.2.q.a 4
5.c odd 4 1 140.2.q.b yes 4
7.c even 3 1 inner 700.2.i.f 8
7.c even 3 1 4900.2.a.be 4
7.d odd 6 1 4900.2.a.bf 4
15.e even 4 1 1260.2.bm.a 4
15.e even 4 1 1260.2.bm.b 4
20.e even 4 1 560.2.bw.a 4
20.e even 4 1 560.2.bw.e 4
35.f even 4 1 980.2.q.b 4
35.f even 4 1 980.2.q.g 4
35.i odd 6 1 4900.2.a.bf 4
35.j even 6 1 inner 700.2.i.f 8
35.j even 6 1 4900.2.a.be 4
35.k even 12 2 980.2.e.c 4
35.k even 12 1 980.2.q.b 4
35.k even 12 1 980.2.q.g 4
35.l odd 12 1 140.2.q.a 4
35.l odd 12 1 140.2.q.b yes 4
35.l odd 12 2 980.2.e.f 4
105.x even 12 1 1260.2.bm.a 4
105.x even 12 1 1260.2.bm.b 4
140.w even 12 1 560.2.bw.a 4
140.w even 12 1 560.2.bw.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 5.c odd 4 1
140.2.q.a 4 35.l odd 12 1
140.2.q.b yes 4 5.c odd 4 1
140.2.q.b yes 4 35.l odd 12 1
560.2.bw.a 4 20.e even 4 1
560.2.bw.a 4 140.w even 12 1
560.2.bw.e 4 20.e even 4 1
560.2.bw.e 4 140.w even 12 1
700.2.i.f 8 1.a even 1 1 trivial
700.2.i.f 8 5.b even 2 1 inner
700.2.i.f 8 7.c even 3 1 inner
700.2.i.f 8 35.j even 6 1 inner
980.2.e.c 4 35.k even 12 2
980.2.e.f 4 35.l odd 12 2
980.2.q.b 4 35.f even 4 1
980.2.q.b 4 35.k even 12 1
980.2.q.g 4 35.f even 4 1
980.2.q.g 4 35.k even 12 1
1260.2.bm.a 4 15.e even 4 1
1260.2.bm.a 4 105.x even 12 1
1260.2.bm.b 4 15.e even 4 1
1260.2.bm.b 4 105.x even 12 1
4900.2.a.be 4 7.c even 3 1
4900.2.a.be 4 35.j even 6 1
4900.2.a.bf 4 7.d odd 6 1
4900.2.a.bf 4 35.i odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 3 T_{3}^{2} + 9 \)
\( T_{11}^{4} - 3 T_{11}^{3} + 21 T_{11}^{2} + 36 T_{11} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 2401 + 245 T^{2} - 24 T^{4} + 5 T^{6} + T^{8} \)
$11$ \( ( 144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$13$ \( ( 256 - 44 T^{2} + T^{4} )^{2} \)
$17$ \( 16 + 92 T^{2} + 525 T^{4} + 23 T^{6} + T^{8} \)
$19$ \( ( 196 + 14 T + 15 T^{2} - T^{3} + T^{4} )^{2} \)
$23$ \( 2401 + 3038 T^{2} + 3795 T^{4} + 62 T^{6} + T^{8} \)
$29$ \( ( -14 - T + T^{2} )^{4} \)
$31$ \( ( 196 + 14 T + 15 T^{2} - T^{3} + T^{4} )^{2} \)
$37$ \( 9834496 + 410816 T^{2} + 14025 T^{4} + 131 T^{6} + T^{8} \)
$41$ \( ( 42 + 15 T + T^{2} )^{4} \)
$43$ \( ( 196 - 47 T^{2} + T^{4} )^{2} \)
$47$ \( 38416 + 9212 T^{2} + 2013 T^{4} + 47 T^{6} + T^{8} \)
$53$ \( 3111696 + 153468 T^{2} + 5805 T^{4} + 87 T^{6} + T^{8} \)
$59$ \( ( 196 - 14 T + 15 T^{2} + T^{3} + T^{4} )^{2} \)
$61$ \( ( 441 + 252 T + 165 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$67$ \( 5764801 + 494606 T^{2} + 40035 T^{4} + 206 T^{6} + T^{8} \)
$71$ \( ( -48 - 6 T + T^{2} )^{4} \)
$73$ \( 38416 + 9212 T^{2} + 2013 T^{4} + 47 T^{6} + T^{8} \)
$79$ \( ( 4 - 14 T + 51 T^{2} + 7 T^{3} + T^{4} )^{2} \)
$83$ \( ( 1764 - 87 T^{2} + T^{4} )^{2} \)
$89$ \( ( 49 - 7 T + T^{2} )^{4} \)
$97$ \( ( -48 + T^{2} )^{4} \)
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