Properties

Label 950.2.f.a
Level $950$
Weight $2$
Character orbit 950.f
Analytic conductor $7.586$
Analytic rank $0$
Dimension $4$
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.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( 1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8} q^{8} -3 \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( 1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8} q^{8} -3 \zeta_{8}^{2} q^{9} + 2 q^{11} -6 \zeta_{8} q^{13} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} - q^{16} + ( -3 - 3 \zeta_{8}^{2} ) q^{17} + 3 \zeta_{8} q^{18} + ( -3 \zeta_{8} - \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{19} + 2 \zeta_{8}^{3} q^{22} + ( -1 + \zeta_{8}^{2} ) q^{23} + 6 q^{26} + ( 1 - \zeta_{8}^{2} ) q^{28} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} -\zeta_{8}^{3} q^{32} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{34} -3 q^{36} + 6 \zeta_{8}^{3} q^{37} + ( 3 + \zeta_{8} - 3 \zeta_{8}^{2} ) q^{38} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{41} + ( 5 - 5 \zeta_{8}^{2} ) q^{43} -2 \zeta_{8}^{2} q^{44} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{46} + ( -7 - 7 \zeta_{8}^{2} ) q^{47} -5 \zeta_{8}^{2} q^{49} + 6 \zeta_{8}^{3} q^{52} -6 \zeta_{8} q^{53} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{56} + 8 q^{61} + ( -6 - 6 \zeta_{8}^{2} ) q^{62} + ( 3 - 3 \zeta_{8}^{2} ) q^{63} + \zeta_{8}^{2} q^{64} -12 \zeta_{8}^{3} q^{67} + ( -3 + 3 \zeta_{8}^{2} ) q^{68} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{71} -3 \zeta_{8}^{3} q^{72} + ( 1 - \zeta_{8}^{2} ) q^{73} -6 \zeta_{8}^{2} q^{74} + ( -1 + 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{76} + ( 2 + 2 \zeta_{8}^{2} ) q^{77} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{79} -9 q^{81} + ( 6 + 6 \zeta_{8}^{2} ) q^{82} + ( -9 + 9 \zeta_{8}^{2} ) q^{83} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{86} + 2 \zeta_{8} q^{88} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{89} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{91} + ( 1 + \zeta_{8}^{2} ) q^{92} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{94} -6 \zeta_{8}^{3} q^{97} + 5 \zeta_{8} q^{98} -6 \zeta_{8}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 8q^{11} - 4q^{16} - 12q^{17} - 4q^{23} + 24q^{26} + 4q^{28} - 12q^{36} + 12q^{38} + 20q^{43} - 28q^{47} + 32q^{61} - 24q^{62} + 12q^{63} - 12q^{68} + 4q^{73} - 4q^{76} + 8q^{77} - 36q^{81} + 24q^{82} - 36q^{83} + 4q^{92} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\)

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
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 1.00000 1.00000i 0.707107 0.707107i 3.00000i 0
493.2 0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 1.00000i −0.707107 + 0.707107i 3.00000i 0
607.1 −0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i 0.707107 + 0.707107i 3.00000i 0
607.2 0.707107 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i −0.707107 0.707107i 3.00000i 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.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.a 4
5.b even 2 1 190.2.f.a 4
5.c odd 4 1 190.2.f.a 4
5.c odd 4 1 inner 950.2.f.a 4
15.d odd 2 1 1710.2.p.a 4
15.e even 4 1 1710.2.p.a 4
19.b odd 2 1 inner 950.2.f.a 4
95.d odd 2 1 190.2.f.a 4
95.g even 4 1 190.2.f.a 4
95.g even 4 1 inner 950.2.f.a 4
285.b even 2 1 1710.2.p.a 4
285.j odd 4 1 1710.2.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.f.a 4 5.b even 2 1
190.2.f.a 4 5.c odd 4 1
190.2.f.a 4 95.d odd 2 1
190.2.f.a 4 95.g even 4 1
950.2.f.a 4 1.a even 1 1 trivial
950.2.f.a 4 5.c odd 4 1 inner
950.2.f.a 4 19.b odd 2 1 inner
950.2.f.a 4 95.g even 4 1 inner
1710.2.p.a 4 15.d odd 2 1
1710.2.p.a 4 15.e even 4 1
1710.2.p.a 4 285.b even 2 1
1710.2.p.a 4 285.j odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2 - 2 T + T^{2} )^{2} \)
$11$ \( ( -2 + T )^{4} \)
$13$ \( 1296 + T^{4} \)
$17$ \( ( 18 + 6 T + T^{2} )^{2} \)
$19$ \( 361 - 34 T^{2} + T^{4} \)
$23$ \( ( 2 + 2 T + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 72 + T^{2} )^{2} \)
$37$ \( 1296 + T^{4} \)
$41$ \( ( 72 + T^{2} )^{2} \)
$43$ \( ( 50 - 10 T + T^{2} )^{2} \)
$47$ \( ( 98 + 14 T + T^{2} )^{2} \)
$53$ \( 1296 + T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -8 + T )^{4} \)
$67$ \( 20736 + T^{4} \)
$71$ \( ( 72 + T^{2} )^{2} \)
$73$ \( ( 2 - 2 T + T^{2} )^{2} \)
$79$ \( ( -72 + T^{2} )^{2} \)
$83$ \( ( 162 + 18 T + T^{2} )^{2} \)
$89$ \( ( -72 + T^{2} )^{2} \)
$97$ \( 1296 + T^{4} \)
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