Properties

Label 1950.2.y.l
Level $1950$
Weight $2$
Character orbit 1950.y
Analytic conductor $15.571$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{4} q^{2} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{3} + (\zeta_{24}^{4} - 1) q^{4} + \zeta_{24}^{2} q^{6} + ( - \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} + 1) q^{7} - q^{8} + ( - \zeta_{24}^{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{4} q^{2} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{3} + (\zeta_{24}^{4} - 1) q^{4} + \zeta_{24}^{2} q^{6} + ( - \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} + 1) q^{7} - q^{8} + ( - \zeta_{24}^{4} + 1) q^{9} + (\zeta_{24}^{7} + 2 \zeta_{24}^{6} + \zeta_{24}^{4} - 2 \zeta_{24}^{2} - \zeta_{24} - 2) q^{11} + \zeta_{24}^{6} q^{12} + (\zeta_{24}^{7} + 3 \zeta_{24}^{3}) q^{13} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} + 1) q^{14} - \zeta_{24}^{4} q^{16} + (3 \zeta_{24}^{3} - \zeta_{24}^{2} + 3 \zeta_{24}) q^{17} + q^{18} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{4} + 2 \zeta_{24}^{3} + \zeta_{24} - 1) q^{19} + ( - \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{3} - \zeta_{24}) q^{21} + (\zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 1) q^{22} + (4 \zeta_{24}^{7} + \zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3} - \zeta_{24}^{2} - 4 \zeta_{24} + 2) q^{23} + (\zeta_{24}^{6} - \zeta_{24}^{2}) q^{24} + (4 \zeta_{24}^{7} - \zeta_{24}^{3}) q^{26} - \zeta_{24}^{6} q^{27} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24}) q^{28} + (\zeta_{24}^{7} + \zeta_{24}^{6} + 4 \zeta_{24}^{5} - \zeta_{24}^{4} + 4 \zeta_{24}^{3} + \zeta_{24}^{2} + \zeta_{24}) q^{29} + ( - 5 \zeta_{24}^{7} - \zeta_{24}^{6} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{4} + 3 \zeta_{24}^{3} - 3 \zeta_{24} + 2) q^{31} + ( - \zeta_{24}^{4} + 1) q^{32} + (\zeta_{24}^{7} + 2 \zeta_{24}^{6} + \zeta_{24}^{5} + 2 \zeta_{24}^{4} - \zeta_{24}^{3} - \zeta_{24}^{2} - 2) q^{33} + (3 \zeta_{24}^{7} - \zeta_{24}^{6} + 3 \zeta_{24}^{5}) q^{34} + \zeta_{24}^{4} q^{36} + (5 \zeta_{24}^{7} + 2 \zeta_{24}^{6} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + 5 \zeta_{24}) q^{37} + (\zeta_{24}^{7} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24} + 1) q^{38} + (\zeta_{24}^{5} + 3 \zeta_{24}) q^{39} + (3 \zeta_{24}^{7} - 5 \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + 5 \zeta_{24}^{2} - 3 \zeta_{24}) q^{41} + ( - \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2}) q^{42} + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{3} - \zeta_{24}^{2} + 5 \zeta_{24} - 2) q^{43} + ( - 2 \zeta_{24}^{6} - \zeta_{24}^{5} - 2 \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} + 1) q^{44} + (3 \zeta_{24}^{7} - 3 \zeta_{24}^{5} + \zeta_{24}^{4} - 4 \zeta_{24}^{3} - \zeta_{24}^{2} - \zeta_{24} + 1) q^{46} + (4 \zeta_{24}^{7} - 2 \zeta_{24}^{6} - \zeta_{24}^{5} - 3 \zeta_{24}^{3} + 4 \zeta_{24}^{2} - 3 \zeta_{24} - 4) q^{47} - \zeta_{24}^{2} q^{48} + (2 \zeta_{24}^{7} + \zeta_{24}^{6} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + \zeta_{24}^{2} + 2 \zeta_{24}) q^{49} + ( - 3 \zeta_{24}^{7} + 3 \zeta_{24}^{3} + 3 \zeta_{24} - 1) q^{51} + (3 \zeta_{24}^{7} - 4 \zeta_{24}^{3}) q^{52} + ( - \zeta_{24}^{7} - \zeta_{24}^{6} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} - 2) q^{53} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{54} + (\zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24} - 1) q^{56} + ( - \zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{5} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2} + 2 \zeta_{24}) q^{57} + (5 \zeta_{24}^{7} + 2 \zeta_{24}^{6} + 5 \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3} - \zeta_{24}^{2} - 4 \zeta_{24} + 1) q^{58} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + \zeta_{24}^{4} + 6 \zeta_{24}^{3} - 3 \zeta_{24}^{2} + 4 \zeta_{24} + 1) q^{59} + (8 \zeta_{24}^{6} + 4 \zeta_{24}^{4} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{2} - 3 \zeta_{24} - 4) q^{61} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{6} - 5 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + 5 \zeta_{24}^{3} + \zeta_{24}^{2} + 2 \zeta_{24} + 4) q^{62} + (\zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24}) q^{63} + q^{64} + (\zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} - \zeta_{24} - 2) q^{66} + (4 \zeta_{24}^{7} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{5} - 6 \zeta_{24}^{4} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{2} + 4 \zeta_{24}) q^{67} + (3 \zeta_{24}^{7} - \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{3} + \zeta_{24}^{2} - 3 \zeta_{24}) q^{68} + (4 \zeta_{24}^{7} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{5} + \zeta_{24}^{4} - 3 \zeta_{24}^{3} + \zeta_{24}^{2} - \zeta_{24} - 1) q^{69} + ( - 3 \zeta_{24}^{7} + 3 \zeta_{24}^{5} + 5 \zeta_{24}^{4} + 4 \zeta_{24}^{3} + \zeta_{24}^{2} + \zeta_{24} + 5) q^{71} + (\zeta_{24}^{4} - 1) q^{72} + (4 \zeta_{24}^{7} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{2} + 3) q^{73} + (2 \zeta_{24}^{7} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - 5 \zeta_{24}^{3} - 2 \zeta_{24}^{2} + \cdots - 4) q^{74} + \cdots + (2 \zeta_{24}^{6} + \zeta_{24}^{5} + 2 \zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{7} - 8 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{7} - 8 q^{8} + 4 q^{9} - 12 q^{11} + 8 q^{14} - 4 q^{16} + 8 q^{18} - 12 q^{19} - 12 q^{22} + 12 q^{23} + 4 q^{28} - 4 q^{29} + 4 q^{32} - 8 q^{33} + 4 q^{36} + 16 q^{37} - 24 q^{43} + 12 q^{46} - 32 q^{47} + 16 q^{49} - 8 q^{51} - 4 q^{56} + 4 q^{58} + 12 q^{59} - 16 q^{61} + 24 q^{62} - 4 q^{63} + 8 q^{64} - 16 q^{66} - 24 q^{67} - 4 q^{69} + 60 q^{71} - 4 q^{72} + 24 q^{73} - 16 q^{74} + 12 q^{76} + 8 q^{79} - 4 q^{81} + 8 q^{83} + 12 q^{87} + 12 q^{88} - 24 q^{89} + 8 q^{91} - 4 q^{93} - 16 q^{94} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1 - \zeta_{24}^{4}\) \(1\) \(-1\)

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
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −0.465926 + 0.807007i −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 1.46593 2.53906i −1.00000 0.500000 0.866025i 0
49.3 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 0.241181 0.417738i −1.00000 0.500000 0.866025i 0
49.4 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 0.758819 1.31431i −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 −0.465926 0.807007i −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 1.46593 + 2.53906i −1.00000 0.500000 + 0.866025i 0
199.3 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 0.241181 + 0.417738i −1.00000 0.500000 + 0.866025i 0
199.4 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 0.758819 + 1.31431i −1.00000 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
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.l 8
5.b even 2 1 1950.2.y.i 8
5.c odd 4 1 1950.2.bc.e 8
5.c odd 4 1 1950.2.bc.f yes 8
13.e even 6 1 1950.2.y.i 8
65.l even 6 1 inner 1950.2.y.l 8
65.r odd 12 1 1950.2.bc.e 8
65.r odd 12 1 1950.2.bc.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.y.i 8 5.b even 2 1
1950.2.y.i 8 13.e even 6 1
1950.2.y.l 8 1.a even 1 1 trivial
1950.2.y.l 8 65.l even 6 1 inner
1950.2.bc.e 8 5.c odd 4 1
1950.2.bc.e 8 65.r odd 12 1
1950.2.bc.f yes 8 5.c odd 4 1
1950.2.bc.f yes 8 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} + 14T_{7}^{6} - 16T_{7}^{5} + 22T_{7}^{4} - 8T_{7}^{3} + 20T_{7}^{2} - 8T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + 14 T^{6} - 16 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 529 \) Copy content Toggle raw display
$13$ \( T^{8} + 191 T^{4} + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 38 T^{6} + 1398 T^{4} + \cdots + 2116 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 324 \) Copy content Toggle raw display
$23$ \( T^{8} - 12 T^{7} + 12 T^{6} + \cdots + 58081 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + 100 T^{6} + \cdots + 341056 \) Copy content Toggle raw display
$31$ \( T^{8} + 204 T^{6} + 12128 T^{4} + \cdots + 145924 \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + 260 T^{6} + \cdots + 16056049 \) Copy content Toggle raw display
$41$ \( T^{8} - 102 T^{6} + 10550 T^{4} + \cdots + 21316 \) Copy content Toggle raw display
$43$ \( T^{8} + 24 T^{7} + 186 T^{6} + \cdots + 386884 \) Copy content Toggle raw display
$47$ \( (T^{4} + 16 T^{3} + 20 T^{2} - 544 T - 1724)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 76 T^{6} + 1824 T^{4} + \cdots + 9604 \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} - 64 T^{6} + \cdots + 5740816 \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + 292 T^{6} + \cdots + 8567329 \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + 448 T^{6} + \cdots + 20647936 \) Copy content Toggle raw display
$71$ \( T^{8} - 60 T^{7} + 1596 T^{6} + \cdots + 6115729 \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} - 34 T^{2} + 804 T - 2231)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} - 102 T^{2} + 212 T + 622)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 4 T^{3} - 70 T^{2} + 340 T - 263)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 24 T^{7} + 138 T^{6} + \cdots + 5080516 \) Copy content Toggle raw display
$97$ \( T^{8} + 214 T^{6} + \cdots + 61606801 \) Copy content Toggle raw display
show more
show less