Properties

Label 1950.2.i.bf
Level $1950$
Weight $2$
Character orbit 1950.i
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
Defining polynomial: \(x^{4} + 10 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 10 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/10\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(10 \beta_{2}\)
\(\nu^{3}\)\(=\)\(10 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−1.58114 2.73861i
1.58114 + 2.73861i
−1.58114 + 2.73861i
1.58114 2.73861i
0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −1.58114 2.73861i −1.00000 −0.500000 0.866025i 0
451.2 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 1.58114 + 2.73861i −1.00000 −0.500000 0.866025i 0
601.1 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i −1.58114 + 2.73861i −1.00000 −0.500000 + 0.866025i 0
601.2 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 1.58114 2.73861i −1.00000 −0.500000 + 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.i.bf yes 4
5.b even 2 1 1950.2.i.ba 4
5.c odd 4 2 1950.2.z.o 8
13.c even 3 1 inner 1950.2.i.bf yes 4
65.n even 6 1 1950.2.i.ba 4
65.q odd 12 2 1950.2.z.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.ba 4 5.b even 2 1
1950.2.i.ba 4 65.n even 6 1
1950.2.i.bf yes 4 1.a even 1 1 trivial
1950.2.i.bf yes 4 13.c even 3 1 inner
1950.2.z.o 8 5.c odd 4 2
1950.2.z.o 8 65.q odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7}^{4} + 10 T_{7}^{2} + 100 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{17}^{4} - 8 T_{17}^{3} + 58 T_{17}^{2} - 48 T_{17} + 36 \)
\( T_{19}^{4} + 10 T_{19}^{2} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 100 + 10 T^{2} + T^{4} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( 169 - 78 T + 25 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( 36 - 48 T + 58 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( 100 + 10 T^{2} + T^{4} \)
$23$ \( 81 - 18 T + 13 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( 1296 + 144 T + 52 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( ( 26 - 12 T + T^{2} )^{2} \)
$37$ \( 7921 + 178 T + 93 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( 8100 + 90 T^{2} + T^{4} \)
$43$ \( 36 + 24 T + 22 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( ( 6 + T )^{4} \)
$53$ \( ( 6 + 8 T + T^{2} )^{2} \)
$59$ \( ( 36 - 6 T + T^{2} )^{2} \)
$61$ \( 4225 - 650 T + 165 T^{2} + 10 T^{3} + T^{4} \)
$67$ \( ( 16 - 4 T + T^{2} )^{2} \)
$71$ \( 12321 - 2442 T + 373 T^{2} - 22 T^{3} + T^{4} \)
$73$ \( ( -15 + 10 T + T^{2} )^{2} \)
$79$ \( ( 26 - 12 T + T^{2} )^{2} \)
$83$ \( ( -159 - 2 T + T^{2} )^{2} \)
$89$ \( 36 - 48 T + 58 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( 6561 - 1782 T + 403 T^{2} - 22 T^{3} + T^{4} \)
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