Properties

Label 1950.2.bc.a
Level $1950$
Weight $2$
Character orbit 1950.bc
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.bc (of order \(6\), degree \(2\), 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: \(\Z[a_1, a_2]\)
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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i −2.59808 1.50000i 1.00000i −0.500000 + 0.866025i 0
751.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 2.59808 + 1.50000i 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i −2.59808 + 1.50000i 1.00000i −0.500000 0.866025i 0
901.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 2.59808 1.50000i 1.00000i −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.e 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.bc.a 4
5.b even 2 1 390.2.bb.a 4
5.c odd 4 1 1950.2.y.d 4
5.c odd 4 1 1950.2.y.e 4
13.e even 6 1 inner 1950.2.bc.a 4
15.d odd 2 1 1170.2.bs.d 4
65.l even 6 1 390.2.bb.a 4
65.l even 6 1 5070.2.b.p 4
65.n even 6 1 5070.2.b.p 4
65.r odd 12 1 1950.2.y.d 4
65.r odd 12 1 1950.2.y.e 4
65.s odd 12 1 5070.2.a.ba 2
65.s odd 12 1 5070.2.a.be 2
195.y odd 6 1 1170.2.bs.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 5.b even 2 1
390.2.bb.a 4 65.l even 6 1
1170.2.bs.d 4 15.d odd 2 1
1170.2.bs.d 4 195.y odd 6 1
1950.2.y.d 4 5.c odd 4 1
1950.2.y.d 4 65.r odd 12 1
1950.2.y.e 4 5.c odd 4 1
1950.2.y.e 4 65.r odd 12 1
1950.2.bc.a 4 1.a even 1 1 trivial
1950.2.bc.a 4 13.e even 6 1 inner
5070.2.a.ba 2 65.s odd 12 1
5070.2.a.be 2 65.s odd 12 1
5070.2.b.p 4 65.l even 6 1
5070.2.b.p 4 65.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 9 T_{7}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 81 - 9 T^{2} + T^{4} \)
$11$ \( 1 - 6 T + 11 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( 169 + 23 T^{2} + T^{4} \)
$17$ \( ( 16 - 4 T + T^{2} )^{2} \)
$19$ \( 169 + 78 T - T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 144 + 12 T^{2} + T^{4} \)
$29$ \( 64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( 1936 + 104 T^{2} + T^{4} \)
$37$ \( 2209 - 1128 T + 239 T^{2} - 24 T^{3} + T^{4} \)
$41$ \( 256 - 16 T^{2} + T^{4} \)
$43$ \( ( 36 - 6 T + T^{2} )^{2} \)
$47$ \( 9 + 42 T^{2} + T^{4} \)
$53$ \( ( 1 - 4 T + T^{2} )^{2} \)
$59$ \( 2704 - 624 T - 4 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 64 + 96 T + 56 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 52 - 20 T + T^{2} )^{2} \)
$83$ \( 576 + 96 T^{2} + T^{4} \)
$89$ \( 20449 + 6006 T + 731 T^{2} + 42 T^{3} + T^{4} \)
$97$ \( 10816 + 3744 T + 536 T^{2} + 36 T^{3} + T^{4} \)
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