Properties

Label 1950.2.y.l
Level $1950$
Weight $2$
Character orbit 1950.y
Analytic conductor $15.571$
Analytic rank $0$
Dimension $8$
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.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{4} q^{2} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{3} + ( -1 + \zeta_{24}^{4} ) q^{4} + \zeta_{24}^{2} q^{6} + ( 1 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{4} ) q^{7} - q^{8} + ( 1 - \zeta_{24}^{4} ) q^{9} +O(q^{10})\) \( q + \zeta_{24}^{4} q^{2} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{3} + ( -1 + \zeta_{24}^{4} ) q^{4} + \zeta_{24}^{2} q^{6} + ( 1 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{4} ) q^{7} - q^{8} + ( 1 - \zeta_{24}^{4} ) q^{9} + ( -2 - \zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{4} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{11} + \zeta_{24}^{6} q^{12} + ( 3 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{13} + ( 1 + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{14} -\zeta_{24}^{4} q^{16} + ( 3 \zeta_{24} - \zeta_{24}^{2} + 3 \zeta_{24}^{3} ) q^{17} + q^{18} + ( -1 + \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{19} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{21} + ( -1 - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{22} + ( 2 - 4 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{23} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{24} + ( -\zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{26} -\zeta_{24}^{6} q^{27} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{28} + ( \zeta_{24} + \zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} + 4 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{29} + ( 2 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} - 5 \zeta_{24}^{7} ) q^{31} + ( 1 - \zeta_{24}^{4} ) q^{32} + ( -2 - \zeta_{24}^{2} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{33} + ( 3 \zeta_{24}^{5} - \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{34} + \zeta_{24}^{4} q^{36} + ( 5 \zeta_{24} + 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{37} + ( 1 - \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{38} + ( 3 \zeta_{24} + \zeta_{24}^{5} ) q^{39} + ( -3 \zeta_{24} + 5 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} - 5 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{41} + ( \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{42} + ( -2 + 5 \zeta_{24} - \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{43} + ( 1 + \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{44} + ( 1 - \zeta_{24} - \zeta_{24}^{2} - 4 \zeta_{24}^{3} + \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{46} + ( -4 - 3 \zeta_{24} + 4 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{47} -\zeta_{24}^{2} q^{48} + ( 2 \zeta_{24} + \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{49} + ( -1 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{51} + ( -4 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{52} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{53} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{54} + ( -1 - \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{4} ) q^{56} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{57} + ( 1 - 4 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{4} + 5 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{58} + ( 1 + 4 \zeta_{24} - 3 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{59} + ( -4 - 3 \zeta_{24} - 4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{61} + ( 4 + 2 \zeta_{24} + \zeta_{24}^{2} + 5 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 5 \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{62} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{63} + q^{64} + ( -2 - \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{66} + ( 4 \zeta_{24} - 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{67} + ( -3 \zeta_{24} + \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{68} + ( -1 - \zeta_{24} + \zeta_{24}^{2} - 3 \zeta_{24}^{3} + \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{69} + ( 5 + \zeta_{24} + \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{71} + ( -1 + \zeta_{24}^{4} ) q^{72} + ( 3 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{73} + ( -4 + 3 \zeta_{24} - 2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{74} + ( 2 - 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{76} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{77} + ( -\zeta_{24} + 4 \zeta_{24}^{5} ) q^{78} + ( 1 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{79} -\zeta_{24}^{4} q^{81} + ( \zeta_{24} + 5 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{82} + ( 1 - \zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{83} + ( \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{84} + ( 2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{86} + ( 1 + 4 \zeta_{24} - \zeta_{24}^{2} + 5 \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{87} + ( 2 + \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{4} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{88} + ( -4 + 5 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{6} - 5 \zeta_{24}^{7} ) q^{89} + ( -1 + \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{91} + ( -1 + 3 \zeta_{24} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{92} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - \zeta_{24}^{4} - 5 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{93} + ( \zeta_{24} + 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{94} -\zeta_{24}^{6} q^{96} + ( 2 \zeta_{24} + \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{97} + ( -4 + 2 \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{98} + ( -1 - \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 4q^{4} + 4q^{7} - 8q^{8} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{2} - 4q^{4} + 4q^{7} - 8q^{8} + 4q^{9} - 12q^{11} + 8q^{14} - 4q^{16} + 8q^{18} - 12q^{19} - 12q^{22} + 12q^{23} + 4q^{28} - 4q^{29} + 4q^{32} - 8q^{33} + 4q^{36} + 16q^{37} - 24q^{43} + 12q^{46} - 32q^{47} + 16q^{49} - 8q^{51} - 4q^{56} + 4q^{58} + 12q^{59} - 16q^{61} + 24q^{62} - 4q^{63} + 8q^{64} - 16q^{66} - 24q^{67} - 4q^{69} + 60q^{71} - 4q^{72} + 24q^{73} - 16q^{74} + 12q^{76} + 8q^{79} - 4q^{81} + 8q^{83} + 12q^{87} + 12q^{88} - 24q^{89} + 8q^{91} - 4q^{93} - 16q^{94} - 16q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1 - \zeta_{24}^{4}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −0.465926 + 0.807007i −1.00000 0.500000 0.866025i 0
49.2 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i 1.46593 2.53906i −1.00000 0.500000 0.866025i 0
49.3 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 0.241181 0.417738i −1.00000 0.500000 0.866025i 0
49.4 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 0.758819 1.31431i −1.00000 0.500000 0.866025i 0
199.1 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i −0.465926 0.807007i −1.00000 0.500000 + 0.866025i 0
199.2 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i 1.46593 + 2.53906i −1.00000 0.500000 + 0.866025i 0
199.3 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 0.241181 + 0.417738i −1.00000 0.500000 + 0.866025i 0
199.4 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 0.758819 + 1.31431i −1.00000 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.y.l 8
5.b even 2 1 1950.2.y.i 8
5.c odd 4 1 1950.2.bc.e 8
5.c odd 4 1 1950.2.bc.f yes 8
13.e even 6 1 1950.2.y.i 8
65.l even 6 1 inner 1950.2.y.l 8
65.r odd 12 1 1950.2.bc.e 8
65.r odd 12 1 1950.2.bc.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.y.i 8 5.b even 2 1
1950.2.y.i 8 13.e even 6 1
1950.2.y.l 8 1.a even 1 1 trivial
1950.2.y.l 8 65.l even 6 1 inner
1950.2.bc.e 8 5.c odd 4 1
1950.2.bc.e 8 65.r odd 12 1
1950.2.bc.f yes 8 5.c odd 4 1
1950.2.bc.f yes 8 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 4 - 8 T + 20 T^{2} - 8 T^{3} + 22 T^{4} - 16 T^{5} + 14 T^{6} - 4 T^{7} + T^{8} \)
$11$ \( 529 + 828 T + 294 T^{2} - 216 T^{3} - 85 T^{4} + 72 T^{5} + 54 T^{6} + 12 T^{7} + T^{8} \)
$13$ \( 28561 + 191 T^{4} + T^{8} \)
$17$ \( 2116 + 4968 T + 2140 T^{2} - 4104 T^{3} + 1398 T^{4} - 38 T^{6} + T^{8} \)
$19$ \( 324 + 648 T + 324 T^{2} - 216 T^{3} - 90 T^{4} + 72 T^{5} + 54 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( 58081 + 89652 T + 37452 T^{2} - 13392 T^{3} - 433 T^{4} + 432 T^{5} + 12 T^{6} - 12 T^{7} + T^{8} \)
$29$ \( 341056 + 270976 T + 166240 T^{2} + 43648 T^{3} + 9496 T^{4} + 592 T^{5} + 100 T^{6} + 4 T^{7} + T^{8} \)
$31$ \( 145924 + 171672 T^{2} + 12128 T^{4} + 204 T^{6} + T^{8} \)
$37$ \( 16056049 - 4199336 T + 1082276 T^{2} - 132416 T^{3} + 20791 T^{4} - 2032 T^{5} + 260 T^{6} - 16 T^{7} + T^{8} \)
$41$ \( 21316 - 61320 T + 73692 T^{2} - 42840 T^{3} + 10550 T^{4} - 102 T^{6} + T^{8} \)
$43$ \( 386884 - 365736 T + 111516 T^{2} + 3528 T^{3} - 5290 T^{4} - 144 T^{5} + 186 T^{6} + 24 T^{7} + T^{8} \)
$47$ \( ( -1724 - 544 T + 20 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$53$ \( 9604 + 13720 T^{2} + 1824 T^{4} + 76 T^{6} + T^{8} \)
$59$ \( 5740816 - 4542816 T + 1466624 T^{2} - 212352 T^{3} + 7356 T^{4} + 1344 T^{5} - 64 T^{6} - 12 T^{7} + T^{8} \)
$61$ \( 8567329 + 3606064 T + 1412452 T^{2} + 138016 T^{3} + 23935 T^{4} + 1888 T^{5} + 292 T^{6} + 16 T^{7} + T^{8} \)
$67$ \( 20647936 + 2617344 T + 913408 T^{2} + 144384 T^{3} + 34752 T^{4} + 4224 T^{5} + 448 T^{6} + 24 T^{7} + T^{8} \)
$71$ \( 6115729 - 7329972 T + 3907740 T^{2} - 1173744 T^{3} + 213623 T^{4} - 23760 T^{5} + 1596 T^{6} - 60 T^{7} + T^{8} \)
$73$ \( ( -2231 + 804 T - 34 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$79$ \( ( 622 + 212 T - 102 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$83$ \( ( -263 + 340 T - 70 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$89$ \( 5080516 - 2461368 T + 275772 T^{2} + 58968 T^{3} - 8074 T^{4} - 1296 T^{5} + 138 T^{6} + 24 T^{7} + T^{8} \)
$97$ \( 61606801 + 2637264 T + 1792582 T^{2} - 71904 T^{3} + 37947 T^{4} - 672 T^{5} + 214 T^{6} + T^{8} \)
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