Properties

Label 950.2.e.h
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: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} ) q^{6} + ( -3 - \beta_{3} ) q^{7} + q^{8} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} ) q^{6} + ( -3 - \beta_{3} ) q^{7} + q^{8} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + q^{11} -\beta_{3} q^{12} + ( -2 + 2 \beta_{2} ) q^{13} + ( -\beta_{1} + 3 \beta_{2} ) q^{14} -\beta_{2} q^{16} + 2 \beta_{1} q^{17} + ( 1 - \beta_{3} ) q^{18} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{21} -\beta_{2} q^{22} + ( 3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{23} -\beta_{1} q^{24} + 2 q^{26} + ( 4 + \beta_{3} ) q^{27} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{28} + ( -2 + 2 \beta_{2} ) q^{29} -4 \beta_{3} q^{31} + ( -1 + \beta_{2} ) q^{32} -\beta_{1} q^{33} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{34} + ( -\beta_{1} - \beta_{2} ) q^{36} + ( -3 - 3 \beta_{3} ) q^{37} + ( -1 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{38} -2 \beta_{3} q^{39} + ( 2 \beta_{1} + \beta_{2} ) q^{41} + ( -4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{42} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 + \beta_{2} ) q^{44} + ( -3 - 3 \beta_{3} ) q^{46} + ( 8 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{47} + ( \beta_{1} + \beta_{3} ) q^{48} + ( 6 + 5 \beta_{3} ) q^{49} + ( 8 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{51} -2 \beta_{2} q^{52} + ( -5 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{53} + ( \beta_{1} - 4 \beta_{2} ) q^{54} + ( -3 - \beta_{3} ) q^{56} + ( 8 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{57} + 2 q^{58} + ( -\beta_{1} - 12 \beta_{2} ) q^{59} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{61} -4 \beta_{1} q^{62} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( \beta_{1} + \beta_{3} ) q^{66} + ( 12 + \beta_{1} - 12 \beta_{2} + \beta_{3} ) q^{67} + 2 \beta_{3} q^{68} + 12 q^{69} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{71} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{73} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{74} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{76} + ( -3 - \beta_{3} ) q^{77} -2 \beta_{1} q^{78} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{79} + 7 \beta_{2} q^{81} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{82} + ( -2 - 5 \beta_{3} ) q^{83} + ( 4 + 2 \beta_{3} ) q^{84} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{86} -2 \beta_{3} q^{87} + q^{88} + ( -5 - 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{89} + ( 6 + 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{92} + ( -4 \beta_{1} - 16 \beta_{2} ) q^{93} + ( -8 - 2 \beta_{3} ) q^{94} -\beta_{3} q^{96} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{97} + ( 5 \beta_{1} - 6 \beta_{2} ) q^{98} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - q^{3} - 2q^{4} - q^{6} - 10q^{7} + 4q^{8} - 3q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - q^{3} - 2q^{4} - q^{6} - 10q^{7} + 4q^{8} - 3q^{9} + 4q^{11} + 2q^{12} - 4q^{13} + 5q^{14} - 2q^{16} + 2q^{17} + 6q^{18} - 10q^{19} - 6q^{21} - 2q^{22} + 3q^{23} - q^{24} + 8q^{26} + 14q^{27} + 5q^{28} - 4q^{29} + 8q^{31} - 2q^{32} - q^{33} + 2q^{34} - 3q^{36} - 6q^{37} + 5q^{38} + 4q^{39} + 4q^{41} - 6q^{42} - 2q^{43} - 2q^{44} - 6q^{46} + 14q^{47} - q^{48} + 14q^{49} + 18q^{51} - 4q^{52} - 11q^{53} - 7q^{54} - 10q^{56} + 28q^{57} + 8q^{58} - 25q^{59} - 2q^{61} - 4q^{62} - q^{63} + 4q^{64} - q^{66} + 23q^{67} - 4q^{68} + 48q^{69} - 16q^{71} - 3q^{72} + 9q^{73} + 3q^{74} + 5q^{76} - 10q^{77} - 2q^{78} + 14q^{79} + 14q^{81} + 4q^{82} + 2q^{83} + 12q^{84} - 2q^{86} + 4q^{87} + 4q^{88} - 7q^{89} + 10q^{91} + 3q^{92} - 36q^{93} - 28q^{94} + 2q^{96} - 9q^{97} - 7q^{98} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

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

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

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
1.28078 2.21837i
−0.780776 + 1.35234i
1.28078 + 2.21837i
−0.780776 1.35234i
−0.500000 + 0.866025i −1.28078 + 2.21837i −0.500000 0.866025i 0 −1.28078 2.21837i −0.438447 1.00000 −1.78078 3.08440i 0
201.2 −0.500000 + 0.866025i 0.780776 1.35234i −0.500000 0.866025i 0 0.780776 + 1.35234i −4.56155 1.00000 0.280776 + 0.486319i 0
501.1 −0.500000 0.866025i −1.28078 2.21837i −0.500000 + 0.866025i 0 −1.28078 + 2.21837i −0.438447 1.00000 −1.78078 + 3.08440i 0
501.2 −0.500000 0.866025i 0.780776 + 1.35234i −0.500000 + 0.866025i 0 0.780776 1.35234i −4.56155 1.00000 0.280776 0.486319i 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.h 4
5.b even 2 1 190.2.e.c 4
5.c odd 4 2 950.2.j.f 8
15.d odd 2 1 1710.2.l.m 4
19.c even 3 1 inner 950.2.e.h 4
20.d odd 2 1 1520.2.q.h 4
95.h odd 6 1 3610.2.a.u 2
95.i even 6 1 190.2.e.c 4
95.i even 6 1 3610.2.a.k 2
95.m odd 12 2 950.2.j.f 8
285.n odd 6 1 1710.2.l.m 4
380.p odd 6 1 1520.2.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.c 4 5.b even 2 1
190.2.e.c 4 95.i even 6 1
950.2.e.h 4 1.a even 1 1 trivial
950.2.e.h 4 19.c even 3 1 inner
950.2.j.f 8 5.c odd 4 2
950.2.j.f 8 95.m odd 12 2
1520.2.q.h 4 20.d odd 2 1
1520.2.q.h 4 380.p odd 6 1
1710.2.l.m 4 15.d odd 2 1
1710.2.l.m 4 285.n odd 6 1
3610.2.a.k 2 95.i even 6 1
3610.2.a.u 2 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 \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{4} + T_{3}^{3} + 5 T_{3}^{2} - 4 T_{3} + 16 \)
\( T_{7}^{2} + 5 T_{7} + 2 \)
\( T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 16 - 4 T + 5 T^{2} + T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2 + 5 T + T^{2} )^{2} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( 4 + 2 T + T^{2} )^{2} \)
$17$ \( 256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( ( 19 + 5 T + T^{2} )^{2} \)
$23$ \( 1296 + 108 T + 45 T^{2} - 3 T^{3} + T^{4} \)
$29$ \( ( 4 + 2 T + T^{2} )^{2} \)
$31$ \( ( -64 - 4 T + T^{2} )^{2} \)
$37$ \( ( -36 + 3 T + T^{2} )^{2} \)
$41$ \( 169 + 52 T + 29 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( 256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( 1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4} \)
$53$ \( 676 + 286 T + 95 T^{2} + 11 T^{3} + T^{4} \)
$59$ \( 23104 + 3800 T + 473 T^{2} + 25 T^{3} + T^{4} \)
$61$ \( 256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4} \)
$67$ \( 16384 - 2944 T + 401 T^{2} - 23 T^{3} + T^{4} \)
$71$ \( 16 - 64 T + 260 T^{2} + 16 T^{3} + T^{4} \)
$73$ \( 324 + 162 T + 99 T^{2} - 9 T^{3} + T^{4} \)
$79$ \( 1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4} \)
$83$ \( ( -106 - T + T^{2} )^{2} \)
$89$ \( 676 - 182 T + 75 T^{2} + 7 T^{3} + T^{4} \)
$97$ \( 324 - 162 T + 99 T^{2} + 9 T^{3} + T^{4} \)
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