Properties

Label 950.2.b.f
Level $950$
Weight $2$
Character orbit 950.b
Analytic conductor $7.586$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} + ( 1 - \beta_{3} ) q^{6} -\beta_{1} q^{7} -\beta_{2} q^{8} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} + ( 1 - \beta_{3} ) q^{6} -\beta_{1} q^{7} -\beta_{2} q^{8} + ( -2 + \beta_{3} ) q^{9} + 4 q^{11} -\beta_{1} q^{12} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + q^{16} + ( -\beta_{1} - 6 \beta_{2} ) q^{17} + ( \beta_{1} - \beta_{2} ) q^{18} + q^{19} + ( 5 - \beta_{3} ) q^{21} + 4 \beta_{2} q^{22} -3 \beta_{1} q^{23} + ( -1 + \beta_{3} ) q^{24} + ( -1 + 3 \beta_{3} ) q^{26} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + \beta_{1} q^{28} + ( 1 - 3 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{3} ) q^{31} + \beta_{2} q^{32} + 4 \beta_{1} q^{33} + ( 5 + \beta_{3} ) q^{34} + ( 2 - \beta_{3} ) q^{36} + 6 \beta_{2} q^{37} + \beta_{2} q^{38} + ( 13 - \beta_{3} ) q^{39} + ( 6 - 4 \beta_{3} ) q^{41} + ( -\beta_{1} + 4 \beta_{2} ) q^{42} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{43} -4 q^{44} + ( -3 + 3 \beta_{3} ) q^{46} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{47} + \beta_{1} q^{48} + ( 2 + \beta_{3} ) q^{49} + ( -1 + 5 \beta_{3} ) q^{51} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{52} + ( \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 - \beta_{3} ) q^{54} + ( 1 - \beta_{3} ) q^{56} + \beta_{1} q^{57} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -1 + \beta_{3} ) q^{59} + ( 8 - 2 \beta_{3} ) q^{61} + 2 \beta_{1} q^{62} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{63} - q^{64} + ( 4 - 4 \beta_{3} ) q^{66} -\beta_{1} q^{67} + ( \beta_{1} + 6 \beta_{2} ) q^{68} + ( 15 - 3 \beta_{3} ) q^{69} + ( 4 - 4 \beta_{3} ) q^{71} + ( -\beta_{1} + \beta_{2} ) q^{72} + ( 3 \beta_{1} + 6 \beta_{2} ) q^{73} -6 q^{74} - q^{76} -4 \beta_{1} q^{77} + ( -\beta_{1} + 12 \beta_{2} ) q^{78} + ( 2 - 2 \beta_{3} ) q^{79} -7 q^{81} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 2 \beta_{1} + 8 \beta_{2} ) q^{83} + ( -5 + \beta_{3} ) q^{84} + ( 6 + 2 \beta_{3} ) q^{86} + ( \beta_{1} - 12 \beta_{2} ) q^{87} -4 \beta_{2} q^{88} -2 q^{89} + ( -13 + \beta_{3} ) q^{91} + 3 \beta_{1} q^{92} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{93} -4 \beta_{3} q^{94} + ( 1 - \beta_{3} ) q^{96} -6 \beta_{2} q^{97} + ( \beta_{1} + 3 \beta_{2} ) q^{98} + ( -8 + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 2q^{6} - 6q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 2q^{6} - 6q^{9} + 16q^{11} - 2q^{14} + 4q^{16} + 4q^{19} + 18q^{21} - 2q^{24} + 2q^{26} - 2q^{29} - 4q^{31} + 22q^{34} + 6q^{36} + 50q^{39} + 16q^{41} - 16q^{44} - 6q^{46} + 10q^{49} + 6q^{51} - 14q^{54} + 2q^{56} - 2q^{59} + 28q^{61} - 4q^{64} + 8q^{66} + 54q^{69} + 8q^{71} - 24q^{74} - 4q^{76} + 4q^{79} - 28q^{81} - 18q^{84} + 28q^{86} - 8q^{89} - 50q^{91} - 8q^{94} + 2q^{96} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

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

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
1.56155i
2.56155i
2.56155i
1.56155i
1.00000i 1.56155i −1.00000 0 −1.56155 1.56155i 1.00000i 0.561553 0
799.2 1.00000i 2.56155i −1.00000 0 2.56155 2.56155i 1.00000i −3.56155 0
799.3 1.00000i 2.56155i −1.00000 0 2.56155 2.56155i 1.00000i −3.56155 0
799.4 1.00000i 1.56155i −1.00000 0 −1.56155 1.56155i 1.00000i 0.561553 0
\(n\): e.g. 2-40 or 990-1000
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.f 4
5.b even 2 1 inner 950.2.b.f 4
5.c odd 4 1 190.2.a.d 2
5.c odd 4 1 950.2.a.h 2
15.e even 4 1 1710.2.a.w 2
15.e even 4 1 8550.2.a.br 2
20.e even 4 1 1520.2.a.n 2
20.e even 4 1 7600.2.a.y 2
35.f even 4 1 9310.2.a.bc 2
40.i odd 4 1 6080.2.a.bh 2
40.k even 4 1 6080.2.a.bb 2
95.g even 4 1 3610.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 5.c odd 4 1
950.2.a.h 2 5.c odd 4 1
950.2.b.f 4 1.a even 1 1 trivial
950.2.b.f 4 5.b even 2 1 inner
1520.2.a.n 2 20.e even 4 1
1710.2.a.w 2 15.e even 4 1
3610.2.a.t 2 95.g even 4 1
6080.2.a.bb 2 40.k even 4 1
6080.2.a.bh 2 40.i odd 4 1
7600.2.a.y 2 20.e even 4 1
8550.2.a.br 2 15.e even 4 1
9310.2.a.bc 2 35.f even 4 1

Hecke kernels

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

\( T_{3}^{4} + 9 T_{3}^{2} + 16 \)
\( T_{7}^{4} + 9 T_{7}^{2} + 16 \)
\( T_{11} - 4 \)
\( T_{13}^{4} + 77 T_{13}^{2} + 1444 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 16 + 9 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 + 9 T^{2} + T^{4} \)
$11$ \( ( -4 + T )^{4} \)
$13$ \( 1444 + 77 T^{2} + T^{4} \)
$17$ \( 676 + 69 T^{2} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 1296 + 81 T^{2} + T^{4} \)
$29$ \( ( -38 + T + T^{2} )^{2} \)
$31$ \( ( -16 + 2 T + T^{2} )^{2} \)
$37$ \( ( 36 + T^{2} )^{2} \)
$41$ \( ( -52 - 8 T + T^{2} )^{2} \)
$43$ \( 1024 + 132 T^{2} + T^{4} \)
$47$ \( 4096 + 144 T^{2} + T^{4} \)
$53$ \( 4 + 21 T^{2} + T^{4} \)
$59$ \( ( -4 + T + T^{2} )^{2} \)
$61$ \( ( 32 - 14 T + T^{2} )^{2} \)
$67$ \( 16 + 9 T^{2} + T^{4} \)
$71$ \( ( -64 - 4 T + T^{2} )^{2} \)
$73$ \( 324 + 117 T^{2} + T^{4} \)
$79$ \( ( -16 - 2 T + T^{2} )^{2} \)
$83$ \( 1024 + 132 T^{2} + T^{4} \)
$89$ \( ( 2 + T )^{4} \)
$97$ \( ( 36 + T^{2} )^{2} \)
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