Properties

Label 150.2.e.b
Level $150$
Weight $2$
Character orbit 150.e
Analytic conductor $1.198$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2} + ( -2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3} -\zeta_{24}^{6} q^{4} + ( 2 - \zeta_{24}^{4} ) q^{6} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{3} q^{8} + 3 \zeta_{24}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2} + ( -2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3} -\zeta_{24}^{6} q^{4} + ( 2 - \zeta_{24}^{4} ) q^{6} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{3} q^{8} + 3 \zeta_{24}^{2} q^{9} + ( -3 + 6 \zeta_{24}^{4} ) q^{11} + ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{12} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{14} - q^{16} + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{17} + ( -3 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{18} -\zeta_{24}^{6} q^{19} -6 \zeta_{24}^{4} q^{21} + ( -3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{22} + 6 \zeta_{24}^{3} q^{23} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{24} + ( -3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{27} + ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{28} -2 q^{31} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{32} -9 \zeta_{24}^{7} q^{33} + 3 \zeta_{24}^{6} q^{34} + ( 3 - 3 \zeta_{24}^{4} ) q^{36} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{37} + \zeta_{24}^{3} q^{38} + ( 3 - 6 \zeta_{24}^{4} ) q^{41} + 6 \zeta_{24} q^{42} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{43} + ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{44} -6 q^{46} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{48} + 5 \zeta_{24}^{6} q^{49} + ( -6 + 3 \zeta_{24}^{4} ) q^{51} -6 \zeta_{24}^{3} q^{53} + ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{54} + ( -2 + 4 \zeta_{24}^{4} ) q^{56} + ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{57} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{59} + 14 q^{61} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{62} + ( 6 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{63} + \zeta_{24}^{6} q^{64} + 9 \zeta_{24}^{4} q^{66} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{67} -3 \zeta_{24}^{3} q^{68} + ( -6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{69} + 3 \zeta_{24}^{5} q^{72} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{73} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{74} - q^{76} + ( -18 \zeta_{24} + 18 \zeta_{24}^{5} ) q^{77} -14 \zeta_{24}^{6} q^{79} + 9 \zeta_{24}^{4} q^{81} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{82} -3 \zeta_{24}^{3} q^{83} + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{84} + ( 2 - 4 \zeta_{24}^{4} ) q^{86} + ( -3 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{88} + ( 18 \zeta_{24}^{2} - 9 \zeta_{24}^{6} ) q^{89} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{92} + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{93} + ( -2 + \zeta_{24}^{4} ) q^{96} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{97} -5 \zeta_{24}^{3} q^{98} + ( -9 \zeta_{24}^{2} + 18 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{6} + O(q^{10}) \) \( 8q + 12q^{6} - 8q^{16} - 24q^{21} - 16q^{31} + 12q^{36} - 48q^{46} - 36q^{51} + 112q^{61} + 36q^{66} - 8q^{76} + 36q^{81} - 12q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.965926 + 0.258819i
−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
−0.707107 + 0.707107i −1.67303 0.448288i 1.00000i 0 1.50000 0.866025i 2.44949 + 2.44949i 0.707107 + 0.707107i 2.59808 + 1.50000i 0
107.2 −0.707107 + 0.707107i −0.448288 1.67303i 1.00000i 0 1.50000 + 0.866025i −2.44949 2.44949i 0.707107 + 0.707107i −2.59808 + 1.50000i 0
107.3 0.707107 0.707107i 0.448288 + 1.67303i 1.00000i 0 1.50000 + 0.866025i 2.44949 + 2.44949i −0.707107 0.707107i −2.59808 + 1.50000i 0
107.4 0.707107 0.707107i 1.67303 + 0.448288i 1.00000i 0 1.50000 0.866025i −2.44949 2.44949i −0.707107 0.707107i 2.59808 + 1.50000i 0
143.1 −0.707107 0.707107i −1.67303 + 0.448288i 1.00000i 0 1.50000 + 0.866025i 2.44949 2.44949i 0.707107 0.707107i 2.59808 1.50000i 0
143.2 −0.707107 0.707107i −0.448288 + 1.67303i 1.00000i 0 1.50000 0.866025i −2.44949 + 2.44949i 0.707107 0.707107i −2.59808 1.50000i 0
143.3 0.707107 + 0.707107i 0.448288 1.67303i 1.00000i 0 1.50000 0.866025i 2.44949 2.44949i −0.707107 + 0.707107i −2.59808 1.50000i 0
143.4 0.707107 + 0.707107i 1.67303 0.448288i 1.00000i 0 1.50000 + 0.866025i −2.44949 + 2.44949i −0.707107 + 0.707107i 2.59808 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.e.b 8
3.b odd 2 1 inner 150.2.e.b 8
4.b odd 2 1 1200.2.v.l 8
5.b even 2 1 inner 150.2.e.b 8
5.c odd 4 2 inner 150.2.e.b 8
12.b even 2 1 1200.2.v.l 8
15.d odd 2 1 inner 150.2.e.b 8
15.e even 4 2 inner 150.2.e.b 8
20.d odd 2 1 1200.2.v.l 8
20.e even 4 2 1200.2.v.l 8
60.h even 2 1 1200.2.v.l 8
60.l odd 4 2 1200.2.v.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.e.b 8 1.a even 1 1 trivial
150.2.e.b 8 3.b odd 2 1 inner
150.2.e.b 8 5.b even 2 1 inner
150.2.e.b 8 5.c odd 4 2 inner
150.2.e.b 8 15.d odd 2 1 inner
150.2.e.b 8 15.e even 4 2 inner
1200.2.v.l 8 4.b odd 2 1
1200.2.v.l 8 12.b even 2 1
1200.2.v.l 8 20.d odd 2 1
1200.2.v.l 8 20.e even 4 2
1200.2.v.l 8 60.h even 2 1
1200.2.v.l 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 144 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( 81 - 9 T^{4} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 144 + T^{4} )^{2} \)
$11$ \( ( 27 + T^{2} )^{4} \)
$13$ \( T^{8} \)
$17$ \( ( 81 + T^{4} )^{2} \)
$19$ \( ( 1 + T^{2} )^{4} \)
$23$ \( ( 1296 + T^{4} )^{2} \)
$29$ \( T^{8} \)
$31$ \( ( 2 + T )^{8} \)
$37$ \( ( 144 + T^{4} )^{2} \)
$41$ \( ( 27 + T^{2} )^{4} \)
$43$ \( ( 144 + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( ( 1296 + T^{4} )^{2} \)
$59$ \( ( -108 + T^{2} )^{4} \)
$61$ \( ( -14 + T )^{8} \)
$67$ \( ( 729 + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( ( 5625 + T^{4} )^{2} \)
$79$ \( ( 196 + T^{2} )^{4} \)
$83$ \( ( 81 + T^{4} )^{2} \)
$89$ \( ( -243 + T^{2} )^{4} \)
$97$ \( ( 2304 + T^{4} )^{2} \)
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