Properties

Label 950.2.e.j
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: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} -\beta_{3} q^{3} + \beta_{1} q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} + q^{7} - q^{8} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} -\beta_{3} q^{3} + \beta_{1} q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} + q^{7} - q^{8} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -1 - \beta_{2} ) q^{11} + \beta_{2} q^{12} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + ( -1 - \beta_{1} ) q^{16} + ( 2 + 2 \beta_{1} - \beta_{3} ) q^{17} + ( -2 - \beta_{2} ) q^{18} + ( -2 + 3 \beta_{1} ) q^{19} -\beta_{3} q^{21} + ( -1 - \beta_{1} - \beta_{3} ) q^{22} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{23} + \beta_{3} q^{24} + ( -3 + \beta_{2} ) q^{26} + 5 q^{27} + \beta_{1} q^{28} + ( 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( -2 - \beta_{2} ) q^{31} -\beta_{1} q^{32} + ( 5 + 5 \beta_{1} + 2 \beta_{3} ) q^{33} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{36} + ( 2 - 2 \beta_{2} ) q^{37} + ( -5 - 2 \beta_{1} ) q^{38} + ( -5 + 2 \beta_{2} ) q^{39} + ( 3 + 3 \beta_{1} + 3 \beta_{3} ) q^{41} + ( \beta_{2} - \beta_{3} ) q^{42} + ( 3 + 3 \beta_{1} + \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{44} + ( -1 + 2 \beta_{2} ) q^{46} + ( \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{47} + ( -\beta_{2} + \beta_{3} ) q^{48} -6 q^{49} + ( 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -3 - 3 \beta_{1} + \beta_{3} ) q^{52} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 5 + 5 \beta_{1} ) q^{54} - q^{56} + ( 3 \beta_{2} + 2 \beta_{3} ) q^{57} + ( -4 - \beta_{2} ) q^{58} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{59} + ( -7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{61} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{62} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{63} + q^{64} + ( 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{66} + ( -10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -2 + \beta_{2} ) q^{68} + ( -10 - \beta_{2} ) q^{69} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{71} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{72} + \beta_{3} q^{73} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{74} + ( -3 - 5 \beta_{1} ) q^{76} + ( -1 - \beta_{2} ) q^{77} + ( -5 - 5 \beta_{1} + 2 \beta_{3} ) q^{78} + ( -1 - \beta_{1} - 5 \beta_{3} ) q^{79} + ( 6 + 6 \beta_{1} - 2 \beta_{3} ) q^{81} + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{82} + ( -1 - \beta_{2} ) q^{83} + \beta_{2} q^{84} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{86} + ( 5 + 5 \beta_{2} ) q^{87} + ( 1 + \beta_{2} ) q^{88} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{92} + ( 5 + 5 \beta_{1} + 3 \beta_{3} ) q^{93} + ( -1 - 4 \beta_{2} ) q^{94} -\beta_{2} q^{96} + ( -4 - 4 \beta_{1} - 3 \beta_{3} ) q^{97} + ( -6 - 6 \beta_{1} ) q^{98} + ( -7 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - q^{3} - 2q^{4} + q^{6} + 4q^{7} - 4q^{8} - 5q^{9} + O(q^{10}) \) \( 4q + 2q^{2} - q^{3} - 2q^{4} + q^{6} + 4q^{7} - 4q^{8} - 5q^{9} - 6q^{11} + 2q^{12} - 5q^{13} + 2q^{14} - 2q^{16} + 3q^{17} - 10q^{18} - 14q^{19} - q^{21} - 3q^{22} + q^{24} - 10q^{26} + 20q^{27} - 2q^{28} - 9q^{29} - 10q^{31} + 2q^{32} + 12q^{33} - 3q^{34} - 5q^{36} + 4q^{37} - 16q^{38} - 16q^{39} + 9q^{41} + q^{42} + 7q^{43} + 3q^{44} - 6q^{47} - q^{48} - 24q^{49} - 9q^{51} - 5q^{52} + 10q^{54} - 4q^{56} + 8q^{57} - 18q^{58} + 11q^{61} - 5q^{62} - 5q^{63} + 4q^{64} - 12q^{66} + 22q^{67} - 6q^{68} - 42q^{69} + 5q^{72} + q^{73} + 2q^{74} - 2q^{76} - 6q^{77} - 8q^{78} - 7q^{79} + 10q^{81} - 9q^{82} - 6q^{83} + 2q^{84} - 7q^{86} + 30q^{87} + 6q^{88} - 5q^{91} + 13q^{93} - 12q^{94} - 2q^{96} - 11q^{97} - 12q^{98} + 18q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - \nu - 4 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 3 \nu + 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} + \nu^{2} + \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{3} - \beta_{2} - \beta_{1} + 7\)\()/3\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
1.39564 0.228425i
−0.895644 + 1.09445i
1.39564 + 0.228425i
−0.895644 1.09445i
0.500000 0.866025i −1.39564 + 2.41733i −0.500000 0.866025i 0 1.39564 + 2.41733i 1.00000 −1.00000 −2.39564 4.14938i 0
201.2 0.500000 0.866025i 0.895644 1.55130i −0.500000 0.866025i 0 −0.895644 1.55130i 1.00000 −1.00000 −0.104356 0.180750i 0
501.1 0.500000 + 0.866025i −1.39564 2.41733i −0.500000 + 0.866025i 0 1.39564 2.41733i 1.00000 −1.00000 −2.39564 + 4.14938i 0
501.2 0.500000 + 0.866025i 0.895644 + 1.55130i −0.500000 + 0.866025i 0 −0.895644 + 1.55130i 1.00000 −1.00000 −0.104356 + 0.180750i 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.j yes 4
5.b even 2 1 950.2.e.i 4
5.c odd 4 2 950.2.j.h 8
19.c even 3 1 inner 950.2.e.j yes 4
95.i even 6 1 950.2.e.i 4
95.m odd 12 2 950.2.j.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.i 4 5.b even 2 1
950.2.e.i 4 95.i even 6 1
950.2.e.j yes 4 1.a even 1 1 trivial
950.2.e.j yes 4 19.c even 3 1 inner
950.2.j.h 8 5.c odd 4 2
950.2.j.h 8 95.m odd 12 2

Hecke kernels

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

\( T_{3}^{4} + T_{3}^{3} + 6 T_{3}^{2} - 5 T_{3} + 25 \)
\( T_{7} - 1 \)
\( T_{11}^{2} + 3 T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 25 - 5 T + 6 T^{2} + T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( ( -3 + 3 T + T^{2} )^{2} \)
$13$ \( 1 + 5 T + 24 T^{2} + 5 T^{3} + T^{4} \)
$17$ \( 9 + 9 T + 12 T^{2} - 3 T^{3} + T^{4} \)
$19$ \( ( 19 + 7 T + T^{2} )^{2} \)
$23$ \( 441 + 21 T^{2} + T^{4} \)
$29$ \( 225 + 135 T + 66 T^{2} + 9 T^{3} + T^{4} \)
$31$ \( ( 1 + 5 T + T^{2} )^{2} \)
$37$ \( ( -20 - 2 T + T^{2} )^{2} \)
$41$ \( 729 + 243 T + 108 T^{2} - 9 T^{3} + T^{4} \)
$43$ \( 49 - 49 T + 42 T^{2} - 7 T^{3} + T^{4} \)
$47$ \( 5625 - 450 T + 111 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 441 + 21 T^{2} + T^{4} \)
$59$ \( 441 + 21 T^{2} + T^{4} \)
$61$ \( 289 + 187 T + 138 T^{2} - 11 T^{3} + T^{4} \)
$67$ \( 10000 - 2200 T + 384 T^{2} - 22 T^{3} + T^{4} \)
$71$ \( 441 + 21 T^{2} + T^{4} \)
$73$ \( 25 + 5 T + 6 T^{2} - T^{3} + T^{4} \)
$79$ \( 14161 - 833 T + 168 T^{2} + 7 T^{3} + T^{4} \)
$83$ \( ( -3 + 3 T + T^{2} )^{2} \)
$89$ \( 441 + 21 T^{2} + T^{4} \)
$97$ \( 289 - 187 T + 138 T^{2} + 11 T^{3} + T^{4} \)
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