Properties

Label 950.2.j.d
Level $950$
Weight $2$
Character orbit 950.j
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.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} + 5 q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( 1 - \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 3 \zeta_{12}^{3} q^{18} + ( 3 - 5 \zeta_{12}^{2} ) q^{19} -5 \zeta_{12} q^{22} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{23} + 2 q^{26} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{28} + 6 \zeta_{12}^{2} q^{29} + 4 q^{31} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{32} + ( 3 - 3 \zeta_{12}^{2} ) q^{36} -11 \zeta_{12}^{3} q^{37} + ( -3 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{38} + ( 9 - 9 \zeta_{12}^{2} ) q^{41} + 6 \zeta_{12} q^{43} + 5 \zeta_{12}^{2} q^{44} + q^{46} + 6 q^{49} -2 \zeta_{12} q^{52} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{53} + q^{56} -6 \zeta_{12}^{3} q^{58} -4 \zeta_{12} q^{62} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} - q^{64} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{67} + ( -6 + 6 \zeta_{12}^{2} ) q^{71} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{72} + 14 \zeta_{12} q^{73} + ( -11 + 11 \zeta_{12}^{2} ) q^{74} + ( 5 - 2 \zeta_{12}^{2} ) q^{76} + 5 \zeta_{12}^{3} q^{77} + ( 10 - 10 \zeta_{12}^{2} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{82} -14 \zeta_{12}^{3} q^{83} -6 \zeta_{12}^{2} q^{86} -5 \zeta_{12}^{3} q^{88} -7 \zeta_{12}^{2} q^{89} -2 \zeta_{12}^{2} q^{91} -\zeta_{12} q^{92} + 2 \zeta_{12} q^{97} -6 \zeta_{12} q^{98} -15 \zeta_{12}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 6q^{9} + 20q^{11} + 2q^{14} - 2q^{16} + 2q^{19} + 8q^{26} + 12q^{29} + 16q^{31} + 6q^{36} + 18q^{41} + 10q^{44} + 4q^{46} + 24q^{49} + 4q^{56} - 4q^{64} - 12q^{71} - 22q^{74} + 16q^{76} + 20q^{79} - 18q^{81} - 12q^{86} - 14q^{89} - 4q^{91} - 30q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i −1.50000 2.59808i 0
49.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i −1.50000 2.59808i 0
349.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i −1.50000 + 2.59808i 0
349.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i −1.50000 + 2.59808i 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
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.j.d 4
5.b even 2 1 inner 950.2.j.d 4
5.c odd 4 1 190.2.e.a 2
5.c odd 4 1 950.2.e.f 2
15.e even 4 1 1710.2.l.h 2
19.c even 3 1 inner 950.2.j.d 4
20.e even 4 1 1520.2.q.f 2
95.i even 6 1 inner 950.2.j.d 4
95.l even 12 1 3610.2.a.c 1
95.m odd 12 1 190.2.e.a 2
95.m odd 12 1 950.2.e.f 2
95.m odd 12 1 3610.2.a.g 1
285.v even 12 1 1710.2.l.h 2
380.v even 12 1 1520.2.q.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.a 2 5.c odd 4 1
190.2.e.a 2 95.m odd 12 1
950.2.e.f 2 5.c odd 4 1
950.2.e.f 2 95.m odd 12 1
950.2.j.d 4 1.a even 1 1 trivial
950.2.j.d 4 5.b even 2 1 inner
950.2.j.d 4 19.c even 3 1 inner
950.2.j.d 4 95.i even 6 1 inner
1520.2.q.f 2 20.e even 4 1
1520.2.q.f 2 380.v even 12 1
1710.2.l.h 2 15.e even 4 1
1710.2.l.h 2 285.v even 12 1
3610.2.a.c 1 95.l even 12 1
3610.2.a.g 1 95.m odd 12 1

Hecke kernels

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

\( T_{3} \)
\( T_{7}^{2} + 1 \)
\( T_{11} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -5 + T )^{4} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 19 - T + T^{2} )^{2} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 36 - 6 T + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( 121 + T^{2} )^{2} \)
$41$ \( ( 81 - 9 T + T^{2} )^{2} \)
$43$ \( 1296 - 36 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( 625 - 25 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( 20736 - 144 T^{2} + T^{4} \)
$71$ \( ( 36 + 6 T + T^{2} )^{2} \)
$73$ \( 38416 - 196 T^{2} + T^{4} \)
$79$ \( ( 100 - 10 T + T^{2} )^{2} \)
$83$ \( ( 196 + T^{2} )^{2} \)
$89$ \( ( 49 + 7 T + T^{2} )^{2} \)
$97$ \( 16 - 4 T^{2} + T^{4} \)
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