Properties

Label 1950.2.y.f
Level $1950$
Weight $2$
Character orbit 1950.y
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.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −1.00000 + 1.73205i −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 −1.00000 + 1.73205i −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 −1.00000 1.73205i −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 −1.00000 1.73205i −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
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.f 4
5.b even 2 1 1950.2.y.c 4
5.c odd 4 1 390.2.bb.b 4
5.c odd 4 1 1950.2.bc.b 4
13.e even 6 1 1950.2.y.c 4
15.e even 4 1 1170.2.bs.e 4
65.l even 6 1 inner 1950.2.y.f 4
65.o even 12 1 5070.2.a.y 2
65.q odd 12 1 5070.2.b.o 4
65.r odd 12 1 390.2.bb.b 4
65.r odd 12 1 1950.2.bc.b 4
65.r odd 12 1 5070.2.b.o 4
65.t even 12 1 5070.2.a.bg 2
195.bf even 12 1 1170.2.bs.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 5.c odd 4 1
390.2.bb.b 4 65.r odd 12 1
1170.2.bs.e 4 15.e even 4 1
1170.2.bs.e 4 195.bf even 12 1
1950.2.y.c 4 5.b even 2 1
1950.2.y.c 4 13.e even 6 1
1950.2.y.f 4 1.a even 1 1 trivial
1950.2.y.f 4 65.l even 6 1 inner
1950.2.bc.b 4 5.c odd 4 1
1950.2.bc.b 4 65.r odd 12 1
5070.2.a.y 2 65.o even 12 1
5070.2.a.bg 2 65.t even 12 1
5070.2.b.o 4 65.q odd 12 1
5070.2.b.o 4 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 2 T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 4 + 2 T + T^{2} )^{2} \)
$11$ \( 9 - 36 T + 51 T^{2} - 12 T^{3} + T^{4} \)
$13$ \( 169 - 22 T^{2} + T^{4} \)
$17$ \( 256 - 16 T^{2} + T^{4} \)
$19$ \( 16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 1 + 6 T + 11 T^{2} - 6 T^{3} + T^{4} \)
$29$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( 121 - 88 T + 75 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 16 - 4 T^{2} + T^{4} \)
$43$ \( 529 + 552 T + 215 T^{2} + 24 T^{3} + T^{4} \)
$47$ \( ( 37 - 14 T + T^{2} )^{2} \)
$53$ \( 144 + 168 T^{2} + T^{4} \)
$59$ \( 169 - 156 T + 35 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 11664 + 108 T^{2} + T^{4} \)
$67$ \( 2704 - 832 T + 204 T^{2} - 16 T^{3} + T^{4} \)
$71$ \( 10816 + 3744 T + 536 T^{2} + 36 T^{3} + T^{4} \)
$73$ \( ( -2 + T )^{4} \)
$79$ \( ( 1 - 14 T + T^{2} )^{2} \)
$83$ \( ( -44 - 4 T + T^{2} )^{2} \)
$89$ \( 16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4} \)
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