Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.4
Defining polynomial: \(x^{8} + 17 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{5} q^{3} -\beta_{2} q^{4} + q^{6} + \beta_{3} q^{7} + \beta_{5} q^{8} -2 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{5} q^{3} -\beta_{2} q^{4} + q^{6} + \beta_{3} q^{7} + \beta_{5} q^{8} -2 \beta_{2} q^{9} + \beta_{1} q^{12} + \beta_{5} q^{13} -\beta_{6} q^{14} - q^{16} + \beta_{3} q^{17} + 2 \beta_{5} q^{18} + ( 2 \beta_{2} - \beta_{6} ) q^{19} -\beta_{4} q^{21} + \beta_{7} q^{23} -\beta_{2} q^{24} - q^{26} + 5 \beta_{1} q^{27} + \beta_{7} q^{28} + \beta_{6} q^{29} -\beta_{1} q^{32} -\beta_{6} q^{34} -2 q^{36} + 2 \beta_{1} q^{37} + ( -2 \beta_{5} + \beta_{7} ) q^{38} -\beta_{2} q^{39} -2 \beta_{4} q^{41} + \beta_{3} q^{42} + \beta_{4} q^{46} + 2 \beta_{3} q^{47} + \beta_{5} q^{48} + 8 \beta_{2} q^{49} -\beta_{4} q^{51} -\beta_{1} q^{52} + 9 \beta_{5} q^{53} -5 \beta_{2} q^{54} + \beta_{4} q^{56} + ( -2 \beta_{1} + \beta_{3} ) q^{57} -\beta_{7} q^{58} + 3 \beta_{6} q^{59} + 10 q^{61} + 2 \beta_{7} q^{63} + \beta_{2} q^{64} + 13 \beta_{1} q^{67} + \beta_{7} q^{68} -\beta_{6} q^{69} + 2 \beta_{4} q^{71} -2 \beta_{1} q^{72} -\beta_{7} q^{73} -2 \beta_{2} q^{74} + ( 2 + \beta_{4} ) q^{76} + \beta_{5} q^{78} + 4 \beta_{6} q^{79} - q^{81} + 2 \beta_{3} q^{82} + 2 \beta_{7} q^{83} -\beta_{6} q^{84} -\beta_{3} q^{87} + 4 \beta_{6} q^{89} + \beta_{4} q^{91} -\beta_{3} q^{92} -2 \beta_{6} q^{94} - q^{96} -8 \beta_{1} q^{97} -8 \beta_{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{6} + O(q^{10}) \) \( 8q + 8q^{6} - 8q^{16} - 8q^{26} - 16q^{36} + 80q^{61} + 16q^{76} - 8q^{81} - 8q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 17 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 5 \nu \)\()/28\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 33 \nu^{2} \)\()/112\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 61 \nu \)\()/28\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} + 17 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 13 \nu^{3} \)\()/448\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - \nu^{2} \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -11 \nu^{7} - 251 \nu^{3} \)\()/448\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 7 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + 11 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(7 \beta_{4} - 17\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{3} + 61 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-33 \beta_{6} - 7 \beta_{2}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} - 251 \beta_{5}\)\()/2\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1
−1.01575 + 1.72286i
1.72286 1.01575i
−1.72286 + 1.01575i
1.01575 1.72286i
−1.01575 1.72286i
1.72286 + 1.01575i
−1.72286 1.01575i
1.01575 + 1.72286i
−0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 −2.73861 + 2.73861i 0.707107 0.707107i 2.00000i 0
493.2 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 2.73861 2.73861i 0.707107 0.707107i 2.00000i 0
493.3 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 −2.73861 + 2.73861i −0.707107 + 0.707107i 2.00000i 0
493.4 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 2.73861 2.73861i −0.707107 + 0.707107i 2.00000i 0
607.1 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 −2.73861 2.73861i 0.707107 + 0.707107i 2.00000i 0
607.2 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 2.73861 + 2.73861i 0.707107 + 0.707107i 2.00000i 0
607.3 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 −2.73861 2.73861i −0.707107 0.707107i 2.00000i 0
607.4 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 2.73861 + 2.73861i −0.707107 0.707107i 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.d odd 2 1 inner
95.g even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.b 8
5.b even 2 1 inner 950.2.f.b 8
5.c odd 4 2 inner 950.2.f.b 8
19.b odd 2 1 inner 950.2.f.b 8
95.d odd 2 1 inner 950.2.f.b 8
95.g even 4 2 inner 950.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.f.b 8 1.a even 1 1 trivial
950.2.f.b 8 5.b even 2 1 inner
950.2.f.b 8 5.c odd 4 2 inner
950.2.f.b 8 19.b odd 2 1 inner
950.2.f.b 8 95.d odd 2 1 inner
950.2.f.b 8 95.g even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 225 + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( ( 1 + T^{4} )^{2} \)
$17$ \( ( 225 + T^{4} )^{2} \)
$19$ \( ( 361 - 22 T^{2} + T^{4} )^{2} \)
$23$ \( ( 225 + T^{4} )^{2} \)
$29$ \( ( -15 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( ( 16 + T^{4} )^{2} \)
$41$ \( ( 60 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( ( 3600 + T^{4} )^{2} \)
$53$ \( ( 6561 + T^{4} )^{2} \)
$59$ \( ( -135 + T^{2} )^{4} \)
$61$ \( ( -10 + T )^{8} \)
$67$ \( ( 28561 + T^{4} )^{2} \)
$71$ \( ( 60 + T^{2} )^{4} \)
$73$ \( ( 225 + T^{4} )^{2} \)
$79$ \( ( -240 + T^{2} )^{4} \)
$83$ \( ( 3600 + T^{4} )^{2} \)
$89$ \( ( -240 + T^{2} )^{4} \)
$97$ \( ( 4096 + T^{4} )^{2} \)
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