Properties

Label 950.2.b.h
Level $950$
Weight $2$
Character orbit 950.b
Analytic conductor $7.586$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.63107136.1
Defining polynomial: \(x^{6} + 13 x^{4} + 42 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} - q^{4} + ( 1 - \beta_{2} ) q^{6} + ( \beta_{3} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} - q^{4} + ( 1 - \beta_{2} ) q^{6} + ( \beta_{3} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{9} + 2 \beta_{2} q^{11} + ( -\beta_{1} + \beta_{3} ) q^{12} + ( -\beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{13} + ( -1 - \beta_{4} ) q^{14} + q^{16} + ( -\beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{17} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{18} - q^{19} + ( 2 + \beta_{2} ) q^{21} + 2 \beta_{1} q^{22} + ( \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{23} + ( -1 + \beta_{2} ) q^{24} + ( -2 + \beta_{2} + \beta_{4} ) q^{26} + ( -3 \beta_{1} + 6 \beta_{3} + 2 \beta_{5} ) q^{27} + ( -\beta_{3} + \beta_{5} ) q^{28} + ( -2 - \beta_{2} - \beta_{4} ) q^{29} + ( 2 + 2 \beta_{2} ) q^{31} + \beta_{3} q^{32} + ( -2 \beta_{1} + 8 \beta_{3} + 2 \beta_{5} ) q^{33} + ( -4 + \beta_{2} + \beta_{4} ) q^{34} + ( 2 - 2 \beta_{2} - \beta_{4} ) q^{36} + ( 2 \beta_{1} - \beta_{3} ) q^{37} -\beta_{3} q^{38} + ( 5 - 5 \beta_{2} - \beta_{4} ) q^{39} + ( \beta_{1} + 2 \beta_{3} ) q^{42} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{43} -2 \beta_{2} q^{44} + ( -1 - \beta_{2} - 2 \beta_{4} ) q^{46} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{47} + ( \beta_{1} - \beta_{3} ) q^{48} + ( -3 - 3 \beta_{2} - 3 \beta_{4} ) q^{49} + ( 7 - 7 \beta_{2} - \beta_{4} ) q^{51} + ( \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{52} + ( -4 \beta_{1} + 5 \beta_{3} - \beta_{5} ) q^{53} + ( -6 + 3 \beta_{2} + 2 \beta_{4} ) q^{54} + ( 1 + \beta_{4} ) q^{56} + ( -\beta_{1} + \beta_{3} ) q^{57} + ( -\beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{58} + ( -1 + \beta_{2} + 4 \beta_{4} ) q^{59} + 2 \beta_{4} q^{61} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{62} + ( \beta_{1} + 5 \beta_{3} - 2 \beta_{5} ) q^{63} - q^{64} + ( -8 + 2 \beta_{2} + 2 \beta_{4} ) q^{66} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{67} + ( \beta_{1} - 4 \beta_{3} - \beta_{5} ) q^{68} + ( -1 + 4 \beta_{2} + \beta_{4} ) q^{69} + ( -4 + 4 \beta_{2} - 2 \beta_{4} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{72} + ( -\beta_{1} + 5 \beta_{3} - 2 \beta_{5} ) q^{73} + ( 1 - 2 \beta_{2} ) q^{74} + q^{76} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{77} + ( -5 \beta_{1} + 5 \beta_{3} + \beta_{5} ) q^{78} + ( -8 + 4 \beta_{2} ) q^{79} + ( 10 - 7 \beta_{2} ) q^{81} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{83} + ( -2 - \beta_{2} ) q^{84} + ( -4 - 2 \beta_{2} ) q^{86} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{87} -2 \beta_{1} q^{88} + ( -6 + 4 \beta_{2} ) q^{89} + ( 6 + 2 \beta_{2} + \beta_{4} ) q^{91} + ( -\beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{92} + ( 6 \beta_{3} + 2 \beta_{5} ) q^{93} + ( -1 - 2 \beta_{2} - 2 \beta_{4} ) q^{94} + ( 1 - \beta_{2} ) q^{96} + ( -6 \beta_{1} - 8 \beta_{3} + 2 \beta_{5} ) q^{97} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} ) q^{98} + ( 14 - 8 \beta_{2} - 2 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 4q^{6} - 10q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 4q^{6} - 10q^{9} + 4q^{11} - 4q^{14} + 6q^{16} - 6q^{19} + 14q^{21} - 4q^{24} - 12q^{26} - 12q^{29} + 16q^{31} - 24q^{34} + 10q^{36} + 22q^{39} - 4q^{44} - 4q^{46} - 18q^{49} + 30q^{51} - 34q^{54} + 4q^{56} - 12q^{59} - 4q^{61} - 6q^{64} - 48q^{66} - 12q^{71} + 2q^{74} + 6q^{76} - 40q^{79} + 46q^{81} - 14q^{84} - 28q^{86} - 28q^{89} + 38q^{91} - 6q^{94} + 4q^{96} + 72q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 13 x^{4} + 42 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 7 \nu^{2} + 3 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 10 \nu^{3} + 21 \nu \)\()/9\)
\(\beta_{4}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} - 19 \nu^{3} - 75 \nu \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 4\)
\(\nu^{3}\)\(=\)\(-\beta_{5} - \beta_{3} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-7 \beta_{4} + 3 \beta_{2} + 25\)
\(\nu^{5}\)\(=\)\(10 \beta_{5} + 19 \beta_{3} + 39 \beta_{1}\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
2.77339i
0.480031i
2.25342i
2.25342i
0.480031i
2.77339i
1.00000i 1.77339i −1.00000 0 −1.77339 2.69168i 1.00000i −0.144903 0
799.2 1.00000i 0.519969i −1.00000 0 0.519969 4.76957i 1.00000i 2.72963 0
799.3 1.00000i 3.25342i −1.00000 0 3.25342 0.0778929i 1.00000i −7.58473 0
799.4 1.00000i 3.25342i −1.00000 0 3.25342 0.0778929i 1.00000i −7.58473 0
799.5 1.00000i 0.519969i −1.00000 0 0.519969 4.76957i 1.00000i 2.72963 0
799.6 1.00000i 1.77339i −1.00000 0 −1.77339 2.69168i 1.00000i −0.144903 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.b.h 6
5.b even 2 1 inner 950.2.b.h 6
5.c odd 4 1 950.2.a.j 3
5.c odd 4 1 950.2.a.l yes 3
15.e even 4 1 8550.2.a.ci 3
15.e even 4 1 8550.2.a.cp 3
20.e even 4 1 7600.2.a.bk 3
20.e even 4 1 7600.2.a.bz 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 5.c odd 4 1
950.2.a.l yes 3 5.c odd 4 1
950.2.b.h 6 1.a even 1 1 trivial
950.2.b.h 6 5.b even 2 1 inner
7600.2.a.bk 3 20.e even 4 1
7600.2.a.bz 3 20.e even 4 1
8550.2.a.ci 3 15.e even 4 1
8550.2.a.cp 3 15.e even 4 1

Hecke kernels

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

\( T_{3}^{6} + 14 T_{3}^{4} + 37 T_{3}^{2} + 9 \)
\( T_{7}^{6} + 30 T_{7}^{4} + 165 T_{7}^{2} + 1 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} + 24 \)
\( T_{13}^{6} + 42 T_{13}^{4} + 429 T_{13}^{2} + 1225 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( 9 + 37 T^{2} + 14 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 + 165 T^{2} + 30 T^{4} + T^{6} \)
$11$ \( ( 24 - 24 T - 2 T^{2} + T^{3} )^{2} \)
$13$ \( 1225 + 429 T^{2} + 42 T^{4} + T^{6} \)
$17$ \( 81 + 1305 T^{2} + 78 T^{4} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 12321 + 2157 T^{2} + 106 T^{4} + T^{6} \)
$29$ \( ( -9 - 3 T + 6 T^{2} + T^{3} )^{2} \)
$31$ \( ( 56 - 4 T - 8 T^{2} + T^{3} )^{2} \)
$37$ \( 1 + 627 T^{2} + 51 T^{4} + T^{6} \)
$41$ \( T^{6} \)
$43$ \( 576 + 2272 T^{2} + 116 T^{4} + T^{6} \)
$47$ \( 2025 + 2979 T^{2} + 123 T^{4} + T^{6} \)
$53$ \( 751689 + 28365 T^{2} + 310 T^{4} + T^{6} \)
$59$ \( ( -1431 - 201 T + 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( -120 - 56 T + 2 T^{2} + T^{3} )^{2} \)
$67$ \( 5625 + 1129 T^{2} + 62 T^{4} + T^{6} \)
$71$ \( ( -1512 - 192 T + 6 T^{2} + T^{3} )^{2} \)
$73$ \( 100489 + 8337 T^{2} + 198 T^{4} + T^{6} \)
$79$ \( ( -320 + 32 T + 20 T^{2} + T^{3} )^{2} \)
$83$ \( 28224 + 10320 T^{2} + 232 T^{4} + T^{6} \)
$89$ \( ( -312 - 36 T + 14 T^{2} + T^{3} )^{2} \)
$97$ \( 5953600 + 138576 T^{2} + 696 T^{4} + T^{6} \)
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