Properties

Label 950.2.j.g
Level $950$
Weight $2$
Character orbit 950.j
Analytic conductor $7.586$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} -\beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{4} + \beta_{7} q^{6} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} + ( 4 + 4 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{2} -\beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{4} + \beta_{7} q^{6} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} + ( 4 + 4 \beta_{3} ) q^{9} + ( -2 - \beta_{5} + \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{6} ) q^{12} -2 \beta_{2} q^{13} + ( -\beta_{3} + \beta_{5} ) q^{14} + \beta_{3} q^{16} + ( 4 \beta_{2} + 4 \beta_{4} ) q^{18} + ( -2 + 2 \beta_{3} - \beta_{7} ) q^{19} + ( 7 \beta_{3} - \beta_{5} ) q^{21} + ( -\beta_{1} - 2 \beta_{4} ) q^{22} + ( -\beta_{2} + \beta_{6} ) q^{23} + \beta_{5} q^{24} + 2 q^{26} + ( -\beta_{1} - \beta_{6} ) q^{27} + ( -\beta_{2} - \beta_{6} ) q^{28} + ( -1 - \beta_{3} - \beta_{7} ) q^{29} + ( -3 - \beta_{5} + \beta_{7} ) q^{31} + \beta_{2} q^{32} + ( 2 \beta_{1} + 7 \beta_{4} ) q^{33} + 4 \beta_{3} q^{36} + ( \beta_{1} - 3 \beta_{2} - 3 \beta_{4} + \beta_{6} ) q^{37} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{38} + ( -2 \beta_{5} + 2 \beta_{7} ) q^{39} + ( 5 \beta_{3} + 2 \beta_{5} ) q^{41} + ( 7 \beta_{2} + \beta_{6} ) q^{42} + ( -2 \beta_{1} - 6 \beta_{4} ) q^{43} + ( -2 - 2 \beta_{3} + \beta_{7} ) q^{44} + ( 1 - \beta_{5} + \beta_{7} ) q^{46} + ( 7 \beta_{2} - \beta_{6} ) q^{47} -\beta_{6} q^{48} + ( -1 - 2 \beta_{5} + 2 \beta_{7} ) q^{49} + 2 \beta_{4} q^{52} + ( 2 \beta_{2} + 4 \beta_{6} ) q^{53} + \beta_{5} q^{54} + ( 1 + \beta_{5} - \beta_{7} ) q^{56} + ( 2 \beta_{1} - 7 \beta_{2} - 7 \beta_{4} - 2 \beta_{6} ) q^{57} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} ) q^{58} -3 \beta_{5} q^{59} + ( 7 + 7 \beta_{3} + 3 \beta_{7} ) q^{61} + ( -\beta_{1} - 3 \beta_{4} ) q^{62} + ( -4 \beta_{2} - 4 \beta_{6} ) q^{63} - q^{64} + ( 7 + 7 \beta_{3} - 2 \beta_{7} ) q^{66} + ( -2 \beta_{2} + \beta_{6} ) q^{67} + ( 7 - \beta_{5} + \beta_{7} ) q^{69} + ( -8 \beta_{3} - 2 \beta_{5} ) q^{71} + 4 \beta_{2} q^{72} + ( 2 \beta_{1} - 7 \beta_{4} ) q^{73} + ( -3 \beta_{3} - \beta_{5} ) q^{74} + ( -4 - 2 \beta_{3} - \beta_{5} ) q^{76} + ( 3 \beta_{1} + 9 \beta_{2} + 9 \beta_{4} + 3 \beta_{6} ) q^{77} -2 \beta_{1} q^{78} + 4 \beta_{3} q^{79} -5 \beta_{3} q^{81} + ( 5 \beta_{2} - 2 \beta_{6} ) q^{82} + ( 3 \beta_{1} + 3 \beta_{6} ) q^{83} + ( -7 - \beta_{5} + \beta_{7} ) q^{84} + ( -6 - 6 \beta_{3} + 2 \beta_{7} ) q^{86} + ( \beta_{1} - 7 \beta_{2} - 7 \beta_{4} + \beta_{6} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{88} + ( -2 - 2 \beta_{3} + 2 \beta_{7} ) q^{91} + ( -\beta_{1} + \beta_{4} ) q^{92} + ( 3 \beta_{1} + 7 \beta_{4} ) q^{93} + ( -7 + \beta_{5} - \beta_{7} ) q^{94} + ( \beta_{5} - \beta_{7} ) q^{96} + ( 2 \beta_{1} - 9 \beta_{4} ) q^{97} + ( -2 \beta_{1} - \beta_{4} ) q^{98} + ( -8 - 8 \beta_{3} + 4 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + 16q^{9} + O(q^{10}) \) \( 8q + 4q^{4} + 16q^{9} - 16q^{11} + 4q^{14} - 4q^{16} - 24q^{19} - 28q^{21} + 16q^{26} - 4q^{29} - 24q^{31} - 16q^{36} - 20q^{41} - 8q^{44} + 8q^{46} - 8q^{49} + 8q^{56} + 28q^{61} - 8q^{64} + 28q^{66} + 56q^{69} + 32q^{71} + 12q^{74} - 24q^{76} - 16q^{79} + 20q^{81} - 56q^{84} - 24q^{86} - 8q^{91} - 56q^{94} - 32q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 15 \nu^{4} + 5 \nu^{2} + 12 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{5} + 5 \nu^{3} + 12 \nu \)\()/40\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{6} - 5 \nu^{4} - 15 \nu^{2} - 36 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 7 \nu \)\()/10\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 13 \nu \)\()/10\)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{6} - 15 \nu^{4} - 5 \nu^{2} - 48 \)\()/20\)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 132 \nu \)\()/40\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 3 \beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - \beta_{5} + 5 \beta_{4} + 5 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} - 11 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} - 5 \beta_{1} - 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{5} - 13 \beta_{4}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.228425 + 1.39564i
1.09445 0.895644i
0.228425 1.39564i
−1.09445 + 0.895644i
−0.228425 1.39564i
1.09445 + 0.895644i
0.228425 + 1.39564i
−1.09445 0.895644i
−0.866025 0.500000i −2.29129 1.32288i 0.500000 + 0.866025i 0 1.32288 + 2.29129i 1.64575i 1.00000i 2.00000 + 3.46410i 0
49.2 −0.866025 0.500000i 2.29129 + 1.32288i 0.500000 + 0.866025i 0 −1.32288 2.29129i 3.64575i 1.00000i 2.00000 + 3.46410i 0
49.3 0.866025 + 0.500000i −2.29129 1.32288i 0.500000 + 0.866025i 0 −1.32288 2.29129i 3.64575i 1.00000i 2.00000 + 3.46410i 0
49.4 0.866025 + 0.500000i 2.29129 + 1.32288i 0.500000 + 0.866025i 0 1.32288 + 2.29129i 1.64575i 1.00000i 2.00000 + 3.46410i 0
349.1 −0.866025 + 0.500000i −2.29129 + 1.32288i 0.500000 0.866025i 0 1.32288 2.29129i 1.64575i 1.00000i 2.00000 3.46410i 0
349.2 −0.866025 + 0.500000i 2.29129 1.32288i 0.500000 0.866025i 0 −1.32288 + 2.29129i 3.64575i 1.00000i 2.00000 3.46410i 0
349.3 0.866025 0.500000i −2.29129 + 1.32288i 0.500000 0.866025i 0 −1.32288 + 2.29129i 3.64575i 1.00000i 2.00000 3.46410i 0
349.4 0.866025 0.500000i 2.29129 1.32288i 0.500000 0.866025i 0 1.32288 2.29129i 1.64575i 1.00000i 2.00000 3.46410i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.j.g 8
5.b even 2 1 inner 950.2.j.g 8
5.c odd 4 1 38.2.c.b 4
5.c odd 4 1 950.2.e.k 4
15.e even 4 1 342.2.g.f 4
19.c even 3 1 inner 950.2.j.g 8
20.e even 4 1 304.2.i.e 4
40.i odd 4 1 1216.2.i.l 4
40.k even 4 1 1216.2.i.k 4
60.l odd 4 1 2736.2.s.v 4
95.g even 4 1 722.2.c.j 4
95.i even 6 1 inner 950.2.j.g 8
95.l even 12 1 722.2.a.g 2
95.l even 12 1 722.2.c.j 4
95.m odd 12 1 38.2.c.b 4
95.m odd 12 1 722.2.a.j 2
95.m odd 12 1 950.2.e.k 4
95.q odd 36 6 722.2.e.n 12
95.r even 36 6 722.2.e.o 12
285.v even 12 1 342.2.g.f 4
285.v even 12 1 6498.2.a.ba 2
285.w odd 12 1 6498.2.a.bg 2
380.v even 12 1 304.2.i.e 4
380.v even 12 1 5776.2.a.ba 2
380.w odd 12 1 5776.2.a.z 2
760.br odd 12 1 1216.2.i.l 4
760.bw even 12 1 1216.2.i.k 4
1140.bu odd 12 1 2736.2.s.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 5.c odd 4 1
38.2.c.b 4 95.m odd 12 1
304.2.i.e 4 20.e even 4 1
304.2.i.e 4 380.v even 12 1
342.2.g.f 4 15.e even 4 1
342.2.g.f 4 285.v even 12 1
722.2.a.g 2 95.l even 12 1
722.2.a.j 2 95.m odd 12 1
722.2.c.j 4 95.g even 4 1
722.2.c.j 4 95.l even 12 1
722.2.e.n 12 95.q odd 36 6
722.2.e.o 12 95.r even 36 6
950.2.e.k 4 5.c odd 4 1
950.2.e.k 4 95.m odd 12 1
950.2.j.g 8 1.a even 1 1 trivial
950.2.j.g 8 5.b even 2 1 inner
950.2.j.g 8 19.c even 3 1 inner
950.2.j.g 8 95.i even 6 1 inner
1216.2.i.k 4 40.k even 4 1
1216.2.i.k 4 760.bw even 12 1
1216.2.i.l 4 40.i odd 4 1
1216.2.i.l 4 760.br odd 12 1
2736.2.s.v 4 60.l odd 4 1
2736.2.s.v 4 1140.bu odd 12 1
5776.2.a.z 2 380.w odd 12 1
5776.2.a.ba 2 380.v even 12 1
6498.2.a.ba 2 285.v even 12 1
6498.2.a.bg 2 285.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{4} - 7 T_{3}^{2} + 49 \)
\( T_{7}^{4} + 16 T_{7}^{2} + 36 \)
\( T_{11}^{2} + 4 T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 49 - 7 T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 36 + 16 T^{2} + T^{4} )^{2} \)
$11$ \( ( -3 + 4 T + T^{2} )^{4} \)
$13$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( 361 + 228 T + 67 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$23$ \( 1296 - 576 T^{2} + 220 T^{4} - 16 T^{6} + T^{8} \)
$29$ \( ( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$31$ \( ( 2 + 6 T + T^{2} )^{4} \)
$37$ \( ( 4 + 32 T^{2} + T^{4} )^{2} \)
$41$ \( ( 9 - 30 T + 103 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$43$ \( 4096 - 8192 T^{2} + 16320 T^{4} - 128 T^{6} + T^{8} \)
$47$ \( 3111696 - 197568 T^{2} + 10780 T^{4} - 112 T^{6} + T^{8} \)
$53$ \( 136048896 - 2706048 T^{2} + 42160 T^{4} - 232 T^{6} + T^{8} \)
$59$ \( ( 3969 + 63 T^{2} + T^{4} )^{2} \)
$61$ \( ( 196 + 196 T + 210 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$67$ \( 81 - 198 T^{2} + 475 T^{4} - 22 T^{6} + T^{8} \)
$71$ \( ( 1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$73$ \( 194481 - 67914 T^{2} + 23275 T^{4} - 154 T^{6} + T^{8} \)
$79$ \( ( 16 + 4 T + T^{2} )^{4} \)
$83$ \( ( 63 + T^{2} )^{4} \)
$89$ \( T^{8} \)
$97$ \( 7890481 - 612362 T^{2} + 44715 T^{4} - 218 T^{6} + T^{8} \)
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