Properties

Label 700.2.r.d
Level $700$
Weight $2$
Character orbit 700.r
Analytic conductor $5.590$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{6} + \beta_{11} ) q^{3} + ( \beta_{4} - \beta_{10} ) q^{7} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{6} + \beta_{11} ) q^{3} + ( \beta_{4} - \beta_{10} ) q^{7} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{9} + ( -1 + \beta_{2} + \beta_{5} - \beta_{8} ) q^{11} + ( 3 \beta_{10} + \beta_{11} ) q^{13} + ( -\beta_{4} - 4 \beta_{7} - 2 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{17} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{8} ) q^{19} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{8} ) q^{21} + ( \beta_{4} + 2 \beta_{6} - \beta_{9} ) q^{23} + ( 2 \beta_{4} + \beta_{6} + \beta_{9} - 3 \beta_{10} + 4 \beta_{11} ) q^{27} + ( -\beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{8} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{8} ) q^{31} + ( \beta_{4} - \beta_{6} - 6 \beta_{7} - \beta_{9} ) q^{33} + ( 3 \beta_{6} + \beta_{7} ) q^{37} + ( -6 - \beta_{1} + 6 \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{8} ) q^{39} + ( 2 + \beta_{1} + 2 \beta_{3} - 4 \beta_{5} - \beta_{8} ) q^{41} + ( 2 \beta_{4} + \beta_{6} + \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{43} + ( \beta_{4} + \beta_{7} - \beta_{9} ) q^{47} + ( -1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{5} + 2 \beta_{8} ) q^{49} + ( 3 \beta_{2} + 6 \beta_{8} ) q^{51} + ( -\beta_{4} - 3 \beta_{6} + 5 \beta_{7} - 2 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} ) q^{53} + ( 3 \beta_{10} + 2 \beta_{11} ) q^{57} + ( -2 + 2 \beta_{2} - \beta_{5} + \beta_{8} ) q^{59} + ( 2 \beta_{1} - 7 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{8} ) q^{61} + ( -2 \beta_{6} - 12 \beta_{7} - \beta_{9} + 6 \beta_{10} - 4 \beta_{11} ) q^{63} + ( -2 \beta_{6} - 6 \beta_{7} + 6 \beta_{10} - 2 \beta_{11} ) q^{67} + ( 9 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{8} ) q^{69} + ( -7 - 2 \beta_{5} ) q^{71} + ( 4 \beta_{7} - 4 \beta_{10} ) q^{73} + ( -2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{10} - 4 \beta_{11} ) q^{77} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 4 \beta_{8} ) q^{79} + ( -6 + 6 \beta_{2} + 5 \beta_{5} - 5 \beta_{8} ) q^{81} + ( 2 \beta_{4} + \beta_{6} + \beta_{9} + 10 \beta_{10} + 5 \beta_{11} ) q^{83} + ( 2 \beta_{4} + 4 \beta_{6} + 9 \beta_{7} + 4 \beta_{9} - 9 \beta_{10} + 6 \beta_{11} ) q^{87} + ( 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{5} ) q^{89} + ( 6 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{5} + \beta_{8} ) q^{91} + ( -2 \beta_{4} + 3 \beta_{6} + 9 \beta_{7} + 2 \beta_{9} ) q^{93} + ( -4 \beta_{4} - 2 \beta_{6} - 2 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} ) q^{97} + ( -6 + 2 \beta_{1} + 4 \beta_{3} + 5 \beta_{5} - 2 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{9} + O(q^{10}) \) \( 12 q + 16 q^{9} - 8 q^{11} - 4 q^{19} - 22 q^{21} + 6 q^{31} - 28 q^{39} + 44 q^{41} - 24 q^{49} + 6 q^{51} - 10 q^{59} - 34 q^{61} + 96 q^{69} - 76 q^{71} - 2 q^{79} - 46 q^{81} - 28 q^{89} + 34 q^{91} - 84 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 117 \nu^{10} - 767 \nu^{8} + 7472 \nu^{6} - 27234 \nu^{4} + 90554 \nu^{2} - 60864 \)\()/21995\)
\(\beta_{2}\)\(=\)\((\)\( -1298 \nu^{10} + 10953 \nu^{8} - 71803 \nu^{6} + 202311 \nu^{4} - 490451 \nu^{2} + 240966 \)\()/197955\)
\(\beta_{3}\)\(=\)\((\)\( 461 \nu^{10} - 5466 \nu^{8} + 28501 \nu^{6} - 98847 \nu^{4} + 142112 \nu^{2} - 186237 \)\()/65985\)
\(\beta_{4}\)\(=\)\((\)\( 569 \nu^{11} - 6174 \nu^{9} + 40474 \nu^{7} - 169668 \nu^{5} + 474413 \nu^{3} - 729693 \nu \)\()/197955\)
\(\beta_{5}\)\(=\)\((\)\( -288 \nu^{10} + 1888 \nu^{8} - 9933 \nu^{6} + 12896 \nu^{4} - 6336 \nu^{2} - 68439 \)\()/21995\)
\(\beta_{6}\)\(=\)\((\)\( 81 \nu^{11} - 531 \nu^{9} + 3481 \nu^{7} - 3627 \nu^{5} + 1782 \nu^{3} + 98293 \nu \)\()/21995\)
\(\beta_{7}\)\(=\)\((\)\( 288 \nu^{11} - 1888 \nu^{9} + 9933 \nu^{7} - 12896 \nu^{5} + 6336 \nu^{3} + 90434 \nu \)\()/65985\)
\(\beta_{8}\)\(=\)\((\)\( 1138 \nu^{10} - 12348 \nu^{8} + 80948 \nu^{6} - 273351 \nu^{4} + 552916 \nu^{2} - 271656 \)\()/65985\)
\(\beta_{9}\)\(=\)\((\)\( 2027 \nu^{11} - 15732 \nu^{9} + 103132 \nu^{7} - 234954 \nu^{5} + 506489 \nu^{3} + 445716 \nu \)\()/197955\)
\(\beta_{10}\)\(=\)\((\)\( 88 \nu^{11} - 783 \nu^{9} + 4868 \nu^{7} - 13716 \nu^{5} + 26581 \nu^{3} - 2916 \nu \)\()/7155\)
\(\beta_{11}\)\(=\)\((\)\( -1298 \nu^{11} + 10953 \nu^{9} - 71803 \nu^{7} + 202311 \nu^{5} - 450860 \nu^{3} + 43011 \nu \)\()/39591\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} + 2 \beta_{6} + \beta_{4}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{8} + 2 \beta_{5} + \beta_{3} - 9 \beta_{2} - \beta_{1} + 9\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{11} + 5 \beta_{9} + 5 \beta_{6} + 10 \beta_{4}\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{8} + 2 \beta_{5} - 2 \beta_{3} - 12 \beta_{2} - 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(18 \beta_{11} + 9 \beta_{10} + 46 \beta_{9} - 9 \beta_{7} - 5 \beta_{6} + 23 \beta_{4}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(32 \beta_{8} - 5 \beta_{5} - 64 \beta_{3} - 32 \beta_{1} - 153\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(106 \beta_{9} - 81 \beta_{7} - 116 \beta_{6} - 106 \beta_{4}\)\()/3\)
\(\nu^{8}\)\(=\)\(51 \beta_{8} - 51 \beta_{5} - 55 \beta_{3} + 222 \beta_{2} + 55 \beta_{1} - 222\)
\(\nu^{9}\)\(=\)\((\)\(-495 \beta_{11} - 531 \beta_{10} - 493 \beta_{9} - 493 \beta_{6} - 986 \beta_{4}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-191 \beta_{8} - 835 \beta_{5} + 835 \beta_{3} + 2952 \beta_{2} + 1670 \beta_{1}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-2505 \beta_{11} - 3078 \beta_{10} - 4624 \beta_{9} + 3078 \beta_{7} - 193 \beta_{6} - 2312 \beta_{4}\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.617942 0.356769i
−1.65604 0.956115i
−1.90412 1.09935i
1.90412 + 1.09935i
1.65604 + 0.956115i
0.617942 + 0.356769i
−0.617942 + 0.356769i
−1.65604 + 0.956115i
−1.90412 + 1.09935i
1.90412 1.09935i
1.65604 0.956115i
0.617942 0.356769i
0 −2.77509 + 1.60220i 0 0 0 2.15715 + 1.53189i 0 3.63409 6.29444i 0
149.2 0 −2.22469 + 1.28442i 0 0 0 0.568650 2.58392i 0 1.79949 3.11682i 0
149.3 0 −0.315621 + 0.182224i 0 0 0 −1.58850 2.11581i 0 −1.43359 + 2.48305i 0
149.4 0 0.315621 0.182224i 0 0 0 1.58850 + 2.11581i 0 −1.43359 + 2.48305i 0
149.5 0 2.22469 1.28442i 0 0 0 −0.568650 + 2.58392i 0 1.79949 3.11682i 0
149.6 0 2.77509 1.60220i 0 0 0 −2.15715 1.53189i 0 3.63409 6.29444i 0
249.1 0 −2.77509 1.60220i 0 0 0 2.15715 1.53189i 0 3.63409 + 6.29444i 0
249.2 0 −2.22469 1.28442i 0 0 0 0.568650 + 2.58392i 0 1.79949 + 3.11682i 0
249.3 0 −0.315621 0.182224i 0 0 0 −1.58850 + 2.11581i 0 −1.43359 2.48305i 0
249.4 0 0.315621 + 0.182224i 0 0 0 1.58850 2.11581i 0 −1.43359 2.48305i 0
249.5 0 2.22469 + 1.28442i 0 0 0 −0.568650 2.58392i 0 1.79949 + 3.11682i 0
249.6 0 2.77509 + 1.60220i 0 0 0 −2.15715 + 1.53189i 0 3.63409 + 6.29444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.6
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.d 12
5.b even 2 1 inner 700.2.r.d 12
5.c odd 4 1 700.2.i.d 6
5.c odd 4 1 700.2.i.e yes 6
7.c even 3 1 inner 700.2.r.d 12
7.c even 3 1 4900.2.e.t 6
7.d odd 6 1 4900.2.e.s 6
35.i odd 6 1 4900.2.e.s 6
35.j even 6 1 inner 700.2.r.d 12
35.j even 6 1 4900.2.e.t 6
35.k even 12 1 4900.2.a.bb 3
35.k even 12 1 4900.2.a.bd 3
35.l odd 12 1 700.2.i.d 6
35.l odd 12 1 700.2.i.e yes 6
35.l odd 12 1 4900.2.a.ba 3
35.l odd 12 1 4900.2.a.bc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 5.c odd 4 1
700.2.i.d 6 35.l odd 12 1
700.2.i.e yes 6 5.c odd 4 1
700.2.i.e yes 6 35.l odd 12 1
700.2.r.d 12 1.a even 1 1 trivial
700.2.r.d 12 5.b even 2 1 inner
700.2.r.d 12 7.c even 3 1 inner
700.2.r.d 12 35.j even 6 1 inner
4900.2.a.ba 3 35.l odd 12 1
4900.2.a.bb 3 35.k even 12 1
4900.2.a.bc 3 35.l odd 12 1
4900.2.a.bd 3 35.k even 12 1
4900.2.e.s 6 7.d odd 6 1
4900.2.e.s 6 35.i odd 6 1
4900.2.e.t 6 7.c even 3 1
4900.2.e.t 6 35.j even 6 1

Hecke kernels

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

\( T_{3}^{12} - 17 T_{3}^{10} + 219 T_{3}^{8} - 1172 T_{3}^{6} + 4747 T_{3}^{4} - 630 T_{3}^{2} + 81 \)
\( T_{11}^{6} + 4 T_{11}^{5} + 19 T_{11}^{4} + 6 T_{11}^{3} + 45 T_{11}^{2} + 27 T_{11} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 81 - 630 T^{2} + 4747 T^{4} - 1172 T^{6} + 219 T^{8} - 17 T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( 117649 + 28812 T^{2} + 5880 T^{4} + 947 T^{6} + 120 T^{8} + 12 T^{10} + T^{12} \)
$11$ \( ( 81 + 27 T + 45 T^{2} + 6 T^{3} + 19 T^{4} + 4 T^{5} + T^{6} )^{2} \)
$13$ \( ( 9 + 217 T^{2} + 38 T^{4} + T^{6} )^{2} \)
$17$ \( 43046721 - 11160261 T^{2} + 2355399 T^{4} - 126360 T^{6} + 5023 T^{8} - 82 T^{10} + T^{12} \)
$19$ \( ( 441 - 483 T + 487 T^{2} - 88 T^{3} + 27 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$23$ \( 43046721 - 11691702 T^{2} + 2644083 T^{4} - 131220 T^{6} + 4779 T^{8} - 81 T^{10} + T^{12} \)
$29$ \( ( -81 - 45 T + T^{3} )^{4} \)
$31$ \( ( 1369 + 1554 T + 1653 T^{2} + 200 T^{3} + 51 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$37$ \( 492884401 - 127811157 T^{2} + 29546487 T^{4} - 888232 T^{6} + 20487 T^{8} - 162 T^{10} + T^{12} \)
$41$ \( ( 873 - 78 T - 11 T^{2} + T^{3} )^{4} \)
$43$ \( ( 5041 + 1758 T^{2} + 93 T^{4} + T^{6} )^{2} \)
$47$ \( 6561 - 41553 T^{2} + 258471 T^{4} - 29592 T^{6} + 2851 T^{8} - 58 T^{10} + T^{12} \)
$53$ \( 90842562801 - 4700951397 T^{2} + 175149783 T^{4} - 2922120 T^{6} + 35479 T^{8} - 226 T^{10} + T^{12} \)
$59$ \( ( 81 + 45 T^{2} + 18 T^{3} + 25 T^{4} + 5 T^{5} + T^{6} )^{2} \)
$61$ \( ( 1 + 26 T + 659 T^{2} + 440 T^{3} + 263 T^{4} + 17 T^{5} + T^{6} )^{2} \)
$67$ \( 26873856 - 36910080 T^{2} + 49657600 T^{4} - 1413632 T^{6} + 32880 T^{8} - 200 T^{10} + T^{12} \)
$71$ \( ( 45 + 87 T + 19 T^{2} + T^{3} )^{4} \)
$73$ \( ( 256 - 16 T^{2} + T^{4} )^{3} \)
$79$ \( ( 201601 - 52982 T + 13475 T^{2} - 1016 T^{3} + 119 T^{4} + T^{5} + T^{6} )^{2} \)
$83$ \( ( 962361 + 71577 T^{2} + 526 T^{4} + T^{6} )^{2} \)
$89$ \( ( 2025 - 1755 T + 2151 T^{2} + 636 T^{3} + 157 T^{4} + 14 T^{5} + T^{6} )^{2} \)
$97$ \( ( 25 + 474 T^{2} + 261 T^{4} + T^{6} )^{2} \)
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