Properties

Label 700.2.i.d
Level $700$
Weight $2$
Character orbit 700.i
Analytic conductor $5.590$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{3} + \beta_{5} ) q^{3} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} + \beta_{5} ) q^{3} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 3 - \beta_{3} ) q^{13} + ( -4 - 2 \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{5} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{19} + ( \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{21} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{27} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{31} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{33} + ( \beta_{4} - 3 \beta_{5} ) q^{37} + ( 6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{39} + ( 2 - \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{41} + ( -4 - \beta_{1} + \beta_{2} ) q^{43} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{49} + ( -3 \beta_{4} - 6 \beta_{5} ) q^{51} + ( -5 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} ) q^{53} + ( -3 + 2 \beta_{3} ) q^{57} + ( 2 - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{59} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{61} + ( 6 + \beta_{1} + \beta_{2} + 4 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} ) q^{63} + ( -6 - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -9 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{69} + ( -7 + 2 \beta_{3} ) q^{71} + ( -4 - 4 \beta_{4} ) q^{73} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{77} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{79} + ( -6 - 5 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} ) q^{81} + ( 10 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{83} + ( 9 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 6 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{91} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} + 7 \beta_{5} ) q^{93} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{97} + ( 6 + 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 4 q^{7} - 8 q^{9} + O(q^{10}) \) \( 6 q - q^{3} + 4 q^{7} - 8 q^{9} - 4 q^{11} + 16 q^{13} - 10 q^{17} + 2 q^{19} - 11 q^{21} + 3 q^{23} + 20 q^{27} + 3 q^{31} - 18 q^{33} - 6 q^{37} + 14 q^{39} + 22 q^{41} - 22 q^{43} - 4 q^{47} + 12 q^{49} + 3 q^{51} - 14 q^{53} - 14 q^{57} + 5 q^{59} - 17 q^{61} + 5 q^{63} - 20 q^{67} - 48 q^{69} - 38 q^{71} - 12 q^{73} - 13 q^{77} + q^{79} - 23 q^{81} + 56 q^{83} + 27 q^{87} + 14 q^{89} + 17 q^{91} + 28 q^{93} + 30 q^{97} + 42 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{5} + 15 \nu^{4} + 8 \nu^{3} + 57 \nu^{2} + 47 \nu + 180 \)\()/83\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 131 \nu - 240 \)\()/83\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{5} + 25 \nu^{4} - 42 \nu^{3} + 95 \nu^{2} - 60 \nu + 300 \)\()/83\)
\(\beta_{4}\)\(=\)\((\)\( -20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu - 45 \)\()/249\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{5} + 3 \nu^{4} + 68 \nu^{3} + 28 \nu^{2} + 358 \nu + 36 \)\()/83\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 9\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{3} - 5 \beta_{2} + 5 \beta_{1}\)\()/3\)
\(\nu^{4}\)\(=\)\(5 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(18 \beta_{5} + 9 \beta_{4} + 5 \beta_{3} - 23 \beta_{2} - 46 \beta_{1} + 9\)\()/3\)

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0.356769 0.617942i
1.09935 1.90412i
−0.956115 + 1.65604i
0.356769 + 0.617942i
1.09935 + 1.90412i
−0.956115 1.65604i
0 −1.60220 2.77509i 0 0 0 1.53189 2.15715i 0 −3.63409 + 6.29444i 0
401.2 0 −0.182224 0.315621i 0 0 0 −2.11581 + 1.58850i 0 1.43359 2.48305i 0
401.3 0 1.28442 + 2.22469i 0 0 0 2.58392 + 0.568650i 0 −1.79949 + 3.11682i 0
501.1 0 −1.60220 + 2.77509i 0 0 0 1.53189 + 2.15715i 0 −3.63409 6.29444i 0
501.2 0 −0.182224 + 0.315621i 0 0 0 −2.11581 1.58850i 0 1.43359 + 2.48305i 0
501.3 0 1.28442 2.22469i 0 0 0 2.58392 0.568650i 0 −1.79949 3.11682i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.i.d 6
5.b even 2 1 700.2.i.e yes 6
5.c odd 4 2 700.2.r.d 12
7.c even 3 1 inner 700.2.i.d 6
7.c even 3 1 4900.2.a.bc 3
7.d odd 6 1 4900.2.a.bb 3
35.i odd 6 1 4900.2.a.bd 3
35.j even 6 1 700.2.i.e yes 6
35.j even 6 1 4900.2.a.ba 3
35.k even 12 2 4900.2.e.s 6
35.l odd 12 2 700.2.r.d 12
35.l odd 12 2 4900.2.e.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 1.a even 1 1 trivial
700.2.i.d 6 7.c even 3 1 inner
700.2.i.e yes 6 5.b even 2 1
700.2.i.e yes 6 35.j even 6 1
700.2.r.d 12 5.c odd 4 2
700.2.r.d 12 35.l odd 12 2
4900.2.a.ba 3 35.j even 6 1
4900.2.a.bb 3 7.d odd 6 1
4900.2.a.bc 3 7.c even 3 1
4900.2.a.bd 3 35.i odd 6 1
4900.2.e.s 6 35.k even 12 2
4900.2.e.t 6 35.l odd 12 2

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{5} + 9 T_{3}^{4} - 2 T_{3}^{3} + 67 T_{3}^{2} + 24 T_{3} + 9 \)
\( T_{11}^{6} + 4 T_{11}^{5} + 19 T_{11}^{4} + 6 T_{11}^{3} + 45 T_{11}^{2} + 27 T_{11} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 9 + 24 T + 67 T^{2} - 2 T^{3} + 9 T^{4} + T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 343 - 196 T + 14 T^{2} + 11 T^{3} + 2 T^{4} - 4 T^{5} + T^{6} \)
$11$ \( 81 + 27 T + 45 T^{2} + 6 T^{3} + 19 T^{4} + 4 T^{5} + T^{6} \)
$13$ \( ( 3 + 13 T - 8 T^{2} + T^{3} )^{2} \)
$17$ \( 6561 - 729 T + 891 T^{2} + 252 T^{3} + 91 T^{4} + 10 T^{5} + T^{6} \)
$19$ \( 441 + 483 T + 487 T^{2} + 88 T^{3} + 27 T^{4} - 2 T^{5} + T^{6} \)
$23$ \( 6561 - 2916 T + 1539 T^{2} - 54 T^{3} + 45 T^{4} - 3 T^{5} + T^{6} \)
$29$ \( ( 81 - 45 T + T^{3} )^{2} \)
$31$ \( 1369 + 1554 T + 1653 T^{2} + 200 T^{3} + 51 T^{4} - 3 T^{5} + T^{6} \)
$37$ \( 22201 + 9387 T + 4863 T^{2} - 80 T^{3} + 99 T^{4} + 6 T^{5} + T^{6} \)
$41$ \( ( 873 - 78 T - 11 T^{2} + T^{3} )^{2} \)
$43$ \( ( -71 + 14 T + 11 T^{2} + T^{3} )^{2} \)
$47$ \( 81 + 189 T + 477 T^{2} - 66 T^{3} + 37 T^{4} + 4 T^{5} + T^{6} \)
$53$ \( 301401 + 8235 T + 7911 T^{2} + 888 T^{3} + 211 T^{4} + 14 T^{5} + T^{6} \)
$59$ \( 81 + 45 T^{2} - 18 T^{3} + 25 T^{4} - 5 T^{5} + T^{6} \)
$61$ \( 1 + 26 T + 659 T^{2} + 440 T^{3} + 263 T^{4} + 17 T^{5} + T^{6} \)
$67$ \( 5184 + 7200 T + 8560 T^{2} + 1856 T^{3} + 300 T^{4} + 20 T^{5} + T^{6} \)
$71$ \( ( 45 + 87 T + 19 T^{2} + T^{3} )^{2} \)
$73$ \( ( 16 + 4 T + T^{2} )^{3} \)
$79$ \( 201601 + 52982 T + 13475 T^{2} + 1016 T^{3} + 119 T^{4} - T^{5} + T^{6} \)
$83$ \( ( 981 + 129 T - 28 T^{2} + T^{3} )^{2} \)
$89$ \( 2025 + 1755 T + 2151 T^{2} - 636 T^{3} + 157 T^{4} - 14 T^{5} + T^{6} \)
$97$ \( ( 5 - 18 T - 15 T^{2} + T^{3} )^{2} \)
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