Properties

Label 1950.2.i.i
Level $1950$
Weight $2$
Character orbit 1950.i
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} -5 \zeta_{6} q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} -5 \zeta_{6} q^{7} + q^{8} -\zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} - q^{12} + ( 4 - 3 \zeta_{6} ) q^{13} + 5 q^{14} + ( -1 + \zeta_{6} ) q^{16} -8 \zeta_{6} q^{17} + q^{18} + 5 \zeta_{6} q^{19} -5 q^{21} + 3 \zeta_{6} q^{22} + ( -4 + 4 \zeta_{6} ) q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( -1 + 4 \zeta_{6} ) q^{26} - q^{27} + ( -5 + 5 \zeta_{6} ) q^{28} + ( 4 - 4 \zeta_{6} ) q^{29} -2 q^{31} -\zeta_{6} q^{32} -3 \zeta_{6} q^{33} + 8 q^{34} + ( -1 + \zeta_{6} ) q^{36} + ( -7 + 7 \zeta_{6} ) q^{37} -5 q^{38} + ( 1 - 4 \zeta_{6} ) q^{39} + ( -6 + 6 \zeta_{6} ) q^{41} + ( 5 - 5 \zeta_{6} ) q^{42} + 6 \zeta_{6} q^{43} -3 q^{44} -4 \zeta_{6} q^{46} + 3 q^{47} + \zeta_{6} q^{48} + ( -18 + 18 \zeta_{6} ) q^{49} -8 q^{51} + ( -3 - \zeta_{6} ) q^{52} - q^{53} + ( 1 - \zeta_{6} ) q^{54} -5 \zeta_{6} q^{56} + 5 q^{57} + 4 \zeta_{6} q^{58} -12 \zeta_{6} q^{59} -2 \zeta_{6} q^{61} + ( 2 - 2 \zeta_{6} ) q^{62} + ( -5 + 5 \zeta_{6} ) q^{63} + q^{64} + 3 q^{66} + ( 8 - 8 \zeta_{6} ) q^{67} + ( -8 + 8 \zeta_{6} ) q^{68} + 4 \zeta_{6} q^{69} -2 \zeta_{6} q^{71} -\zeta_{6} q^{72} -7 \zeta_{6} q^{74} + ( 5 - 5 \zeta_{6} ) q^{76} -15 q^{77} + ( 3 + \zeta_{6} ) q^{78} -2 q^{79} + ( -1 + \zeta_{6} ) q^{81} -6 \zeta_{6} q^{82} -8 q^{83} + 5 \zeta_{6} q^{84} -6 q^{86} -4 \zeta_{6} q^{87} + ( 3 - 3 \zeta_{6} ) q^{88} + ( 11 - 11 \zeta_{6} ) q^{89} + ( -15 - 5 \zeta_{6} ) q^{91} + 4 q^{92} + ( -2 + 2 \zeta_{6} ) q^{93} + ( -3 + 3 \zeta_{6} ) q^{94} - q^{96} -18 \zeta_{6} q^{98} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} + q^{6} - 5q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} + q^{6} - 5q^{7} + 2q^{8} - q^{9} + 3q^{11} - 2q^{12} + 5q^{13} + 10q^{14} - q^{16} - 8q^{17} + 2q^{18} + 5q^{19} - 10q^{21} + 3q^{22} - 4q^{23} + q^{24} + 2q^{26} - 2q^{27} - 5q^{28} + 4q^{29} - 4q^{31} - q^{32} - 3q^{33} + 16q^{34} - q^{36} - 7q^{37} - 10q^{38} - 2q^{39} - 6q^{41} + 5q^{42} + 6q^{43} - 6q^{44} - 4q^{46} + 6q^{47} + q^{48} - 18q^{49} - 16q^{51} - 7q^{52} - 2q^{53} + q^{54} - 5q^{56} + 10q^{57} + 4q^{58} - 12q^{59} - 2q^{61} + 2q^{62} - 5q^{63} + 2q^{64} + 6q^{66} + 8q^{67} - 8q^{68} + 4q^{69} - 2q^{71} - q^{72} - 7q^{74} + 5q^{76} - 30q^{77} + 7q^{78} - 4q^{79} - q^{81} - 6q^{82} - 16q^{83} + 5q^{84} - 12q^{86} - 4q^{87} + 3q^{88} + 11q^{89} - 35q^{91} + 8q^{92} - 2q^{93} - 3q^{94} - 2q^{96} - 18q^{98} - 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −2.50000 4.33013i 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 −2.50000 + 4.33013i 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.i 2
5.b even 2 1 390.2.i.d 2
5.c odd 4 2 1950.2.z.j 4
13.c even 3 1 inner 1950.2.i.i 2
15.d odd 2 1 1170.2.i.g 2
65.l even 6 1 5070.2.a.x 1
65.n even 6 1 390.2.i.d 2
65.n even 6 1 5070.2.a.i 1
65.q odd 12 2 1950.2.z.j 4
65.s odd 12 2 5070.2.b.l 2
195.x odd 6 1 1170.2.i.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.d 2 5.b even 2 1
390.2.i.d 2 65.n even 6 1
1170.2.i.g 2 15.d odd 2 1
1170.2.i.g 2 195.x odd 6 1
1950.2.i.i 2 1.a even 1 1 trivial
1950.2.i.i 2 13.c even 3 1 inner
1950.2.z.j 4 5.c odd 4 2
1950.2.z.j 4 65.q odd 12 2
5070.2.a.i 1 65.n even 6 1
5070.2.a.x 1 65.l even 6 1
5070.2.b.l 2 65.s odd 12 2

Hecke kernels

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

\( T_{7}^{2} + 5 T_{7} + 25 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{17}^{2} + 8 T_{17} + 64 \)
\( T_{19}^{2} - 5 T_{19} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + 5 T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( 13 - 5 T + T^{2} \)
$17$ \( 64 + 8 T + T^{2} \)
$19$ \( 25 - 5 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( 16 - 4 T + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( 49 + 7 T + T^{2} \)
$41$ \( 36 + 6 T + T^{2} \)
$43$ \( 36 - 6 T + T^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( ( 1 + T )^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( 64 - 8 T + T^{2} \)
$71$ \( 4 + 2 T + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 121 - 11 T + T^{2} \)
$97$ \( T^{2} \)
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