Properties

Label 950.2.e.k
Level $950$
Weight $2$
Character orbit 950.e
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0.500000 0.866025i −1.32288 + 2.29129i −0.500000 0.866025i 0 1.32288 + 2.29129i 1.64575 −1.00000 −2.00000 3.46410i 0
201.2 0.500000 0.866025i 1.32288 2.29129i −0.500000 0.866025i 0 −1.32288 2.29129i −3.64575 −1.00000 −2.00000 3.46410i 0
501.1 0.500000 + 0.866025i −1.32288 2.29129i −0.500000 + 0.866025i 0 1.32288 2.29129i 1.64575 −1.00000 −2.00000 + 3.46410i 0
501.2 0.500000 + 0.866025i 1.32288 + 2.29129i −0.500000 + 0.866025i 0 −1.32288 + 2.29129i −3.64575 −1.00000 −2.00000 + 3.46410i 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.k 4
5.b even 2 1 38.2.c.b 4
5.c odd 4 2 950.2.j.g 8
15.d odd 2 1 342.2.g.f 4
19.c even 3 1 inner 950.2.e.k 4
20.d odd 2 1 304.2.i.e 4
40.e odd 2 1 1216.2.i.k 4
40.f even 2 1 1216.2.i.l 4
60.h even 2 1 2736.2.s.v 4
95.d odd 2 1 722.2.c.j 4
95.h odd 6 1 722.2.a.g 2
95.h odd 6 1 722.2.c.j 4
95.i even 6 1 38.2.c.b 4
95.i even 6 1 722.2.a.j 2
95.m odd 12 2 950.2.j.g 8
95.o odd 18 6 722.2.e.o 12
95.p even 18 6 722.2.e.n 12
285.n odd 6 1 342.2.g.f 4
285.n odd 6 1 6498.2.a.ba 2
285.q even 6 1 6498.2.a.bg 2
380.p odd 6 1 304.2.i.e 4
380.p odd 6 1 5776.2.a.ba 2
380.s even 6 1 5776.2.a.z 2
760.z even 6 1 1216.2.i.l 4
760.bm odd 6 1 1216.2.i.k 4
1140.bn even 6 1 2736.2.s.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 5.b even 2 1
38.2.c.b 4 95.i even 6 1
304.2.i.e 4 20.d odd 2 1
304.2.i.e 4 380.p odd 6 1
342.2.g.f 4 15.d odd 2 1
342.2.g.f 4 285.n odd 6 1
722.2.a.g 2 95.h odd 6 1
722.2.a.j 2 95.i even 6 1
722.2.c.j 4 95.d odd 2 1
722.2.c.j 4 95.h odd 6 1
722.2.e.n 12 95.p even 18 6
722.2.e.o 12 95.o odd 18 6
950.2.e.k 4 1.a even 1 1 trivial
950.2.e.k 4 19.c even 3 1 inner
950.2.j.g 8 5.c odd 4 2
950.2.j.g 8 95.m odd 12 2
1216.2.i.k 4 40.e odd 2 1
1216.2.i.k 4 760.bm odd 6 1
1216.2.i.l 4 40.f even 2 1
1216.2.i.l 4 760.z even 6 1
2736.2.s.v 4 60.h even 2 1
2736.2.s.v 4 1140.bn even 6 1
5776.2.a.z 2 380.s even 6 1
5776.2.a.ba 2 380.p odd 6 1
6498.2.a.ba 2 285.n odd 6 1
6498.2.a.bg 2 285.q even 6 1

Hecke kernels

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

\( T_{3}^{4} + 7 T_{3}^{2} + 49 \)
\( T_{7}^{2} + 2 T_{7} - 6 \)
\( T_{11}^{2} + 4 T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 49 + 7 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -6 + 2 T + T^{2} )^{2} \)
$11$ \( ( -3 + 4 T + T^{2} )^{2} \)
$13$ \( ( 4 - 2 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( 361 - 228 T + 67 T^{2} - 12 T^{3} + T^{4} \)
$23$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( ( 2 + 6 T + T^{2} )^{2} \)
$37$ \( ( 2 + 6 T + T^{2} )^{2} \)
$41$ \( 9 - 30 T + 103 T^{2} + 10 T^{3} + T^{4} \)
$43$ \( 64 - 96 T + 136 T^{2} - 12 T^{3} + T^{4} \)
$47$ \( 1764 - 588 T + 154 T^{2} - 14 T^{3} + T^{4} \)
$53$ \( 11664 - 432 T + 124 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 3969 + 63 T^{2} + T^{4} \)
$61$ \( 196 + 196 T + 210 T^{2} - 14 T^{3} + T^{4} \)
$67$ \( 9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4} \)
$71$ \( 1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4} \)
$73$ \( 441 - 294 T + 175 T^{2} - 14 T^{3} + T^{4} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( ( -63 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( 2809 + 954 T + 271 T^{2} + 18 T^{3} + T^{4} \)
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