Properties

Label 1950.2.y.d
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.

\(f(q)\) \(=\) \( q -\zeta_{12}^{2} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} -\zeta_{12} q^{6} + ( 3 - 3 \zeta_{12}^{2} ) q^{7} + q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q -\zeta_{12}^{2} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} -\zeta_{12} q^{6} + ( 3 - 3 \zeta_{12}^{2} ) q^{7} + q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + \zeta_{12}^{3} q^{12} + ( 3 + \zeta_{12}^{2} ) q^{13} -3 q^{14} -\zeta_{12}^{2} q^{16} -4 \zeta_{12} q^{17} - q^{18} + ( -1 + 4 \zeta_{12} - \zeta_{12}^{2} ) q^{19} -3 \zeta_{12}^{3} q^{21} + ( 1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{22} + ( 4 - 2 \zeta_{12}^{2} ) q^{23} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{24} + ( 1 - 4 \zeta_{12}^{2} ) q^{26} -\zeta_{12}^{3} q^{27} + 3 \zeta_{12}^{2} q^{28} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{29} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{31} + ( -1 + \zeta_{12}^{2} ) q^{32} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{33} + 4 \zeta_{12}^{3} q^{34} + \zeta_{12}^{2} q^{36} + ( -4 \zeta_{12} - \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{37} + ( -1 + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{38} + ( 4 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{39} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{42} + 6 \zeta_{12} q^{43} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{44} + ( -2 - 2 \zeta_{12}^{2} ) q^{46} + ( 3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} -\zeta_{12} q^{48} -2 \zeta_{12}^{2} q^{49} -4 q^{51} + ( -4 + 3 \zeta_{12}^{2} ) q^{52} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{53} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{54} + ( 3 - 3 \zeta_{12}^{2} ) q^{56} + ( 4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{57} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{58} + ( 2 - 8 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{59} + ( 4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{61} + ( -8 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{62} -3 \zeta_{12}^{2} q^{63} + q^{64} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{66} + ( 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{68} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{69} + ( -4 + 6 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{71} + ( 1 - \zeta_{12}^{2} ) q^{72} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + ( -1 - 4 \zeta_{12} + \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{74} + ( 2 - 4 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + ( -3 + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{77} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{78} + ( -10 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} -4 \zeta_{12} q^{82} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} + 3 \zeta_{12} q^{84} -6 \zeta_{12}^{3} q^{86} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{87} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{88} + ( 14 - 2 \zeta_{12} - 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{89} + ( 12 - 9 \zeta_{12}^{2} ) q^{91} + ( -2 + 4 \zeta_{12}^{2} ) q^{92} + ( -4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{93} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{94} + \zeta_{12}^{3} q^{96} + ( -2 - 6 \zeta_{12} + 2 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{97} + ( -2 + 2 \zeta_{12}^{2} ) q^{98} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} + 6q^{7} + 4q^{8} + 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} + 6q^{7} + 4q^{8} + 2q^{9} - 6q^{11} + 14q^{13} - 12q^{14} - 2q^{16} - 4q^{18} - 6q^{19} + 6q^{22} + 12q^{23} - 4q^{26} + 6q^{28} - 4q^{29} - 2q^{32} - 4q^{33} + 2q^{36} - 2q^{37} - 12q^{46} + 12q^{47} - 4q^{49} - 16q^{51} - 10q^{52} + 6q^{56} + 16q^{57} - 4q^{58} + 12q^{59} + 8q^{61} - 24q^{62} - 6q^{63} + 4q^{64} + 8q^{66} + 4q^{67} - 24q^{71} + 2q^{72} - 2q^{74} + 6q^{76} - 40q^{79} - 2q^{81} + 24q^{83} - 12q^{87} - 6q^{88} + 42q^{89} + 30q^{91} - 4q^{93} - 6q^{94} - 4q^{97} - 4q^{98} + O(q^{100}) \)

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 1.50000 2.59808i 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 1.50000 2.59808i 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 1.50000 + 2.59808i 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 1.50000 + 2.59808i 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.d 4
5.b even 2 1 1950.2.y.e 4
5.c odd 4 1 390.2.bb.a 4
5.c odd 4 1 1950.2.bc.a 4
13.e even 6 1 1950.2.y.e 4
15.e even 4 1 1170.2.bs.d 4
65.l even 6 1 inner 1950.2.y.d 4
65.o even 12 1 5070.2.a.be 2
65.q odd 12 1 5070.2.b.p 4
65.r odd 12 1 390.2.bb.a 4
65.r odd 12 1 1950.2.bc.a 4
65.r odd 12 1 5070.2.b.p 4
65.t even 12 1 5070.2.a.ba 2
195.bf even 12 1 1170.2.bs.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 5.c odd 4 1
390.2.bb.a 4 65.r odd 12 1
1170.2.bs.d 4 15.e even 4 1
1170.2.bs.d 4 195.bf even 12 1
1950.2.y.d 4 1.a even 1 1 trivial
1950.2.y.d 4 65.l even 6 1 inner
1950.2.y.e 4 5.b even 2 1
1950.2.y.e 4 13.e even 6 1
1950.2.bc.a 4 5.c odd 4 1
1950.2.bc.a 4 65.r odd 12 1
5070.2.a.ba 2 65.t even 12 1
5070.2.a.be 2 65.o even 12 1
5070.2.b.p 4 65.q odd 12 1
5070.2.b.p 4 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 3 T_{7} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 9 - 3 T + T^{2} )^{2} \)
$11$ \( 1 - 6 T + 11 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( ( 13 - 7 T + T^{2} )^{2} \)
$17$ \( 256 - 16 T^{2} + T^{4} \)
$19$ \( 169 - 78 T - T^{2} + 6 T^{3} + T^{4} \)
$23$ \( ( 12 - 6 T + T^{2} )^{2} \)
$29$ \( 64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4} \)
$31$ \( 1936 + 104 T^{2} + T^{4} \)
$37$ \( 2209 - 94 T + 51 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 256 - 16 T^{2} + T^{4} \)
$43$ \( 1296 - 36 T^{2} + T^{4} \)
$47$ \( ( -3 - 6 T + T^{2} )^{2} \)
$53$ \( 1 + 14 T^{2} + T^{4} \)
$59$ \( 2704 + 624 T - 4 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( 16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( 144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4} \)
$73$ \( ( -48 + T^{2} )^{2} \)
$79$ \( ( 52 + 20 T + T^{2} )^{2} \)
$83$ \( ( 24 - 12 T + T^{2} )^{2} \)
$89$ \( 20449 - 6006 T + 731 T^{2} - 42 T^{3} + T^{4} \)
$97$ \( 10816 - 416 T + 120 T^{2} + 4 T^{3} + T^{4} \)
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