Properties

Label 1950.2.i.y
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i −1.36603 2.36603i 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 0.366025 + 0.633975i 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 −1.36603 + 2.36603i 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 0.366025 0.633975i 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.y 4
5.b even 2 1 1950.2.i.bh 4
5.c odd 4 1 390.2.y.b 4
5.c odd 4 1 390.2.y.c yes 4
13.c even 3 1 inner 1950.2.i.y 4
15.e even 4 1 1170.2.bp.d 4
15.e even 4 1 1170.2.bp.e 4
65.n even 6 1 1950.2.i.bh 4
65.q odd 12 1 390.2.y.b 4
65.q odd 12 1 390.2.y.c yes 4
195.bl even 12 1 1170.2.bp.d 4
195.bl even 12 1 1170.2.bp.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.b 4 5.c odd 4 1
390.2.y.b 4 65.q odd 12 1
390.2.y.c yes 4 5.c odd 4 1
390.2.y.c yes 4 65.q odd 12 1
1170.2.bp.d 4 15.e even 4 1
1170.2.bp.d 4 195.bl even 12 1
1170.2.bp.e 4 15.e even 4 1
1170.2.bp.e 4 195.bl even 12 1
1950.2.i.y 4 1.a even 1 1 trivial
1950.2.i.y 4 13.c even 3 1 inner
1950.2.i.bh 4 5.b even 2 1
1950.2.i.bh 4 65.n even 6 1

Hecke kernels

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

\( T_{7}^{4} + 2 T_{7}^{3} + 6 T_{7}^{2} - 4 T_{7} + 4 \)
\( T_{11}^{4} - 2 T_{11}^{3} + 6 T_{11}^{2} + 4 T_{11} + 4 \)
\( T_{17}^{4} + 2 T_{17}^{3} + 15 T_{17}^{2} - 22 T_{17} + 121 \)
\( T_{19}^{4} + 6 T_{19}^{3} + 30 T_{19}^{2} + 36 T_{19} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 169 + 78 T + 23 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 121 - 22 T + 15 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 676 - 52 T + 30 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( 9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( 2209 - 94 T + 51 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( 484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4} \)
$47$ \( ( -74 + 2 T + T^{2} )^{2} \)
$53$ \( ( 33 - 12 T + T^{2} )^{2} \)
$59$ \( 10816 + 416 T + 120 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( 20449 - 572 T + 159 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( 5476 + 148 T + 78 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( ( -59 - 8 T + T^{2} )^{2} \)
$79$ \( ( -12 + T )^{4} \)
$83$ \( ( 46 - 14 T + T^{2} )^{2} \)
$89$ \( 1936 - 176 T + 60 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( ( 100 - 10 T + T^{2} )^{2} \)
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