Properties

Label 1950.2.z.k
Level $1950$
Weight $2$
Character orbit 1950.z
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.z (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: \(\Z[a_1, a_2]\)
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1699.1
−0.866025 + 0.500000i
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 0.500000 + 0.866025i −1.73205 + 1.00000i 1.00000i 0.500000 + 0.866025i 0
1699.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.73205 1.00000i 1.00000i 0.500000 + 0.866025i 0
1849.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i −1.73205 1.00000i 1.00000i 0.500000 0.866025i 0
1849.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 1.73205 + 1.00000i 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
5.b even 2 1 inner
13.c even 3 1 inner
65.n 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.z.k 4
5.b even 2 1 inner 1950.2.z.k 4
5.c odd 4 1 390.2.i.c 2
5.c odd 4 1 1950.2.i.n 2
13.c even 3 1 inner 1950.2.z.k 4
15.e even 4 1 1170.2.i.d 2
65.n even 6 1 inner 1950.2.z.k 4
65.o even 12 1 5070.2.b.j 2
65.q odd 12 1 390.2.i.c 2
65.q odd 12 1 1950.2.i.n 2
65.q odd 12 1 5070.2.a.j 1
65.r odd 12 1 5070.2.a.v 1
65.t even 12 1 5070.2.b.j 2
195.bl even 12 1 1170.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.c 2 5.c odd 4 1
390.2.i.c 2 65.q odd 12 1
1170.2.i.d 2 15.e even 4 1
1170.2.i.d 2 195.bl even 12 1
1950.2.i.n 2 5.c odd 4 1
1950.2.i.n 2 65.q odd 12 1
1950.2.z.k 4 1.a even 1 1 trivial
1950.2.z.k 4 5.b even 2 1 inner
1950.2.z.k 4 13.c even 3 1 inner
1950.2.z.k 4 65.n even 6 1 inner
5070.2.a.j 1 65.q odd 12 1
5070.2.a.v 1 65.r odd 12 1
5070.2.b.j 2 65.o even 12 1
5070.2.b.j 2 65.t even 12 1

Hecke kernels

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

\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{17}^{4} - 36 T_{17}^{2} + 1296 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( ( 9 - 3 T + T^{2} )^{2} \)
$13$ \( 169 - 22 T^{2} + T^{4} \)
$17$ \( 1296 - 36 T^{2} + T^{4} \)
$19$ \( ( 4 - 2 T + T^{2} )^{2} \)
$23$ \( 81 - 9 T^{2} + T^{4} \)
$29$ \( ( 9 - 3 T + T^{2} )^{2} \)
$31$ \( ( -5 + T )^{4} \)
$37$ \( 2401 - 49 T^{2} + T^{4} \)
$41$ \( ( 36 + 6 T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( ( 9 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 81 + 9 T + T^{2} )^{2} \)
$61$ \( ( 4 + 2 T + T^{2} )^{2} \)
$67$ \( 4096 - 64 T^{2} + T^{4} \)
$71$ \( ( 144 - 12 T + T^{2} )^{2} \)
$73$ \( ( 196 + T^{2} )^{2} \)
$79$ \( ( 5 + T )^{4} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 324 + 18 T + T^{2} )^{2} \)
$97$ \( 38416 - 196 T^{2} + T^{4} \)
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