Properties

Label 950.2.b.g
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: 6.0.4227136.2
Defining polynomial: \(x^{6} + 9 x^{4} + 22 x^{2} + 9\)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{3} - q^{4} + ( 1 + \beta_{2} - \beta_{5} ) q^{6} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{7} -\beta_{4} q^{8} + ( -4 - \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{2} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{3} - q^{4} + ( 1 + \beta_{2} - \beta_{5} ) q^{6} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{7} -\beta_{4} q^{8} + ( -4 - \beta_{5} ) q^{9} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{11} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{12} + ( 2 \beta_{1} - \beta_{3} ) q^{13} + ( -1 + 2 \beta_{2} + \beta_{5} ) q^{14} + q^{16} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{17} + ( -\beta_{1} - 4 \beta_{4} ) q^{18} + q^{19} + ( -2 - 3 \beta_{2} - 5 \beta_{5} ) q^{21} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{22} + ( \beta_{1} + \beta_{3} - 5 \beta_{4} ) q^{23} + ( -1 - \beta_{2} + \beta_{5} ) q^{24} + ( \beta_{2} - 2 \beta_{5} ) q^{26} + ( -\beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{27} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{28} + ( -4 + \beta_{2} - 2 \beta_{5} ) q^{29} + ( -2 - 2 \beta_{2} + 2 \beta_{5} ) q^{31} + \beta_{4} q^{32} + ( -4 \beta_{1} - 6 \beta_{3} ) q^{33} + ( 2 - \beta_{2} - 2 \beta_{5} ) q^{34} + ( 4 + \beta_{5} ) q^{36} + ( -2 \beta_{1} - 2 \beta_{3} - 5 \beta_{4} ) q^{37} + \beta_{4} q^{38} + ( -9 + 3 \beta_{2} - 2 \beta_{5} ) q^{39} + ( -4 + 4 \beta_{5} ) q^{41} + ( -5 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} ) q^{42} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{44} + ( 5 - \beta_{2} - \beta_{5} ) q^{46} + ( 4 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{47} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{48} + ( -9 - 3 \beta_{2} + 2 \beta_{5} ) q^{49} + ( -5 + 3 \beta_{2} + 4 \beta_{5} ) q^{51} + ( -2 \beta_{1} + \beta_{3} ) q^{52} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{53} + ( 2 - 3 \beta_{2} + \beta_{5} ) q^{54} + ( 1 - 2 \beta_{2} - \beta_{5} ) q^{56} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{57} + ( -2 \beta_{1} + \beta_{3} - 4 \beta_{4} ) q^{58} + ( 1 + \beta_{2} + 3 \beta_{5} ) q^{59} + ( 8 - 4 \beta_{2} - 2 \beta_{5} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{62} + ( \beta_{1} + 9 \beta_{3} - \beta_{4} ) q^{63} - q^{64} + ( 6 \beta_{2} + 4 \beta_{5} ) q^{66} + ( 3 \beta_{1} - 4 \beta_{3} - \beta_{4} ) q^{67} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{68} + ( -5 - 2 \beta_{2} + 7 \beta_{5} ) q^{69} -2 \beta_{5} q^{71} + ( \beta_{1} + 4 \beta_{4} ) q^{72} + ( 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} ) q^{73} + ( 5 + 2 \beta_{2} + 2 \beta_{5} ) q^{74} - q^{76} + ( 4 \beta_{1} - 6 \beta_{3} - 18 \beta_{4} ) q^{77} + ( -2 \beta_{1} + 3 \beta_{3} - 9 \beta_{4} ) q^{78} + ( -8 + 4 \beta_{2} ) q^{79} + ( -2 - \beta_{2} + 5 \beta_{5} ) q^{81} + ( 4 \beta_{1} - 4 \beta_{4} ) q^{82} + ( -6 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 2 + 3 \beta_{2} + 5 \beta_{5} ) q^{84} + ( -2 \beta_{2} + 2 \beta_{5} ) q^{86} + ( -6 \beta_{1} + 7 \beta_{3} - 5 \beta_{4} ) q^{87} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{88} + 10 q^{89} -7 \beta_{5} q^{91} + ( -\beta_{1} - \beta_{3} + 5 \beta_{4} ) q^{92} + ( 2 \beta_{1} + 14 \beta_{4} ) q^{93} + ( -3 - 2 \beta_{2} - 4 \beta_{5} ) q^{94} + ( 1 + \beta_{2} - \beta_{5} ) q^{96} -6 \beta_{3} q^{97} + ( 2 \beta_{1} - 3 \beta_{3} - 9 \beta_{4} ) q^{98} + ( -6 - 8 \beta_{2} - 6 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 4q^{6} - 26q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 4q^{6} - 26q^{9} + 4q^{11} - 4q^{14} + 6q^{16} + 6q^{19} - 22q^{21} - 4q^{24} - 4q^{26} - 28q^{29} - 8q^{31} + 8q^{34} + 26q^{36} - 58q^{39} - 16q^{41} - 4q^{44} + 28q^{46} - 50q^{49} - 22q^{51} + 14q^{54} + 4q^{56} + 12q^{59} + 44q^{61} - 6q^{64} + 8q^{66} - 16q^{69} - 4q^{71} + 34q^{74} - 6q^{76} - 48q^{79} - 2q^{81} + 22q^{84} + 4q^{86} + 60q^{89} - 14q^{91} - 26q^{94} + 4q^{96} - 48q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 22 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 6 \nu^{3} + 7 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\( \nu^{4} + 5 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} - 5 \beta_{2} + 12\)
\(\nu^{5}\)\(=\)\(3 \beta_{4} - 6 \beta_{3} + 17 \beta_{1}\)

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
2.19869i
1.91223i
0.713538i
0.713538i
1.91223i
2.19869i
1.00000i 3.03293i −1.00000 0 −3.03293 2.46980i 1.00000i −6.19869 0
799.2 1.00000i 2.25561i −1.00000 0 2.25561 4.22547i 1.00000i −2.08777 0
799.3 1.00000i 2.77733i −1.00000 0 2.77733 4.69527i 1.00000i −4.71354 0
799.4 1.00000i 2.77733i −1.00000 0 2.77733 4.69527i 1.00000i −4.71354 0
799.5 1.00000i 2.25561i −1.00000 0 2.25561 4.22547i 1.00000i −2.08777 0
799.6 1.00000i 3.03293i −1.00000 0 −3.03293 2.46980i 1.00000i −6.19869 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.g 6
5.b even 2 1 inner 950.2.b.g 6
5.c odd 4 1 950.2.a.k 3
5.c odd 4 1 950.2.a.m yes 3
15.e even 4 1 8550.2.a.cj 3
15.e even 4 1 8550.2.a.co 3
20.e even 4 1 7600.2.a.bm 3
20.e even 4 1 7600.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.k 3 5.c odd 4 1
950.2.a.m yes 3 5.c odd 4 1
950.2.b.g 6 1.a even 1 1 trivial
950.2.b.g 6 5.b even 2 1 inner
7600.2.a.bm 3 20.e even 4 1
7600.2.a.cb 3 20.e even 4 1
8550.2.a.cj 3 15.e even 4 1
8550.2.a.co 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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{6} + 22 T_{3}^{4} + 157 T_{3}^{2} + 361 \)
\( T_{7}^{6} + 46 T_{7}^{4} + 637 T_{7}^{2} + 2401 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 32 T_{11} + 24 \)
\( T_{13}^{6} + 50 T_{13}^{4} + 445 T_{13}^{2} + 441 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( 361 + 157 T^{2} + 22 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 2401 + 637 T^{2} + 46 T^{4} + T^{6} \)
$11$ \( ( 24 - 32 T - 2 T^{2} + T^{3} )^{2} \)
$13$ \( 441 + 445 T^{2} + 50 T^{4} + T^{6} \)
$17$ \( 49 + 169 T^{2} + 46 T^{4} + T^{6} \)
$19$ \( ( -1 + T )^{6} \)
$23$ \( 3969 + 1485 T^{2} + 82 T^{4} + T^{6} \)
$29$ \( ( 25 + 41 T + 14 T^{2} + T^{3} )^{2} \)
$31$ \( ( -152 - 36 T + 4 T^{2} + T^{3} )^{2} \)
$37$ \( 81 + 4275 T^{2} + 163 T^{4} + T^{6} \)
$41$ \( ( -64 - 48 T + 8 T^{2} + T^{3} )^{2} \)
$43$ \( 5184 + 1312 T^{2} + 84 T^{4} + T^{6} \)
$47$ \( 275625 + 14275 T^{2} + 219 T^{4} + T^{6} \)
$53$ \( 9 + 181 T^{2} + 78 T^{4} + T^{6} \)
$59$ \( ( 175 - 29 T - 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( 536 + 72 T - 22 T^{2} + T^{3} )^{2} \)
$67$ \( 219961 + 17161 T^{2} + 262 T^{4} + T^{6} \)
$71$ \( ( -24 - 16 T + 2 T^{2} + T^{3} )^{2} \)
$73$ \( 59049 + 6561 T^{2} + 198 T^{4} + T^{6} \)
$79$ \( ( -320 + 112 T + 24 T^{2} + T^{3} )^{2} \)
$83$ \( 2383936 + 63952 T^{2} + 472 T^{4} + T^{6} \)
$89$ \( ( -10 + T )^{6} \)
$97$ \( 419904 + 32400 T^{2} + 360 T^{4} + T^{6} \)
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