Properties

Label 950.2.e.a
Level $950$
Weight $2$
Character orbit 950.e
Analytic conductor $7.586$
Analytic rank $0$
Dimension $2$
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i −2.00000 1.00000 1.00000 + 1.73205i 0
501.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.00000 1.00000 1.00000 1.73205i 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
19.c even 3 1 inner

Twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 36 - 6 T + T^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 19 - 7 T + T^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( 4 - 2 T + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( 25 + 5 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 25 + 5 T + T^{2} \)
$61$ \( 196 + 14 T + T^{2} \)
$67$ \( 25 + 5 T + T^{2} \)
$71$ \( 36 - 6 T + T^{2} \)
$73$ \( 81 + 9 T + T^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( ( -11 + T )^{2} \)
$89$ \( 196 + 14 T + T^{2} \)
$97$ \( 225 + 15 T + T^{2} \)
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