Properties

Label 750.2.h.a
Level $750$
Weight $2$
Character orbit 750.h
Analytic conductor $5.989$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{6} -2 \zeta_{20}^{5} q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{6} -2 \zeta_{20}^{5} q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} + ( -2 + 2 \zeta_{20}^{2} + 4 \zeta_{20}^{6} ) q^{11} + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{12} + ( 3 \zeta_{20} - 3 \zeta_{20}^{7} ) q^{13} -2 \zeta_{20}^{6} q^{14} + \zeta_{20}^{4} q^{16} + ( 3 \zeta_{20} + 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} ) q^{17} -\zeta_{20}^{5} q^{18} + ( -4 + 6 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{19} + 2 \zeta_{20}^{2} q^{21} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} + 4 \zeta_{20}^{7} ) q^{22} + 6 \zeta_{20} q^{23} - q^{24} + ( 3 + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{26} + \zeta_{20} q^{27} -2 \zeta_{20}^{7} q^{28} + ( 1 - 3 \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{29} + ( -6 \zeta_{20}^{2} + 6 \zeta_{20}^{4} ) q^{31} + \zeta_{20}^{5} q^{32} + ( -2 \zeta_{20} - 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} ) q^{33} + ( 3 \zeta_{20}^{2} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{6} ) q^{34} -\zeta_{20}^{6} q^{36} + ( -4 \zeta_{20} + 7 \zeta_{20}^{3} - 7 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{37} + ( -4 \zeta_{20} + 6 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{38} + ( -3 + 3 \zeta_{20}^{2} + 3 \zeta_{20}^{6} ) q^{39} + ( -5 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 5 \zeta_{20}^{6} ) q^{41} + 2 \zeta_{20}^{3} q^{42} + ( -2 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{43} + ( -4 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{44} + 6 \zeta_{20}^{2} q^{46} + ( 2 \zeta_{20} - 2 \zeta_{20}^{3} + 6 \zeta_{20}^{7} ) q^{47} -\zeta_{20} q^{48} + 3 q^{49} + ( -6 - 3 \zeta_{20}^{4} + 3 \zeta_{20}^{6} ) q^{51} + ( 3 \zeta_{20} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{52} + ( 9 \zeta_{20} - 9 \zeta_{20}^{3} - 3 \zeta_{20}^{7} ) q^{53} + \zeta_{20}^{2} q^{54} + ( 2 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{56} + ( 2 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{57} + ( \zeta_{20} - 3 \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{58} + ( -8 \zeta_{20}^{2} + 4 \zeta_{20}^{4} - 8 \zeta_{20}^{6} ) q^{59} + ( -3 + 3 \zeta_{20}^{2} + 7 \zeta_{20}^{6} ) q^{61} + ( -6 \zeta_{20}^{3} + 6 \zeta_{20}^{5} ) q^{62} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{63} + \zeta_{20}^{6} q^{64} + ( -2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{66} + ( -6 \zeta_{20} - 6 \zeta_{20}^{5} ) q^{67} + ( 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{68} + ( -6 + 6 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{69} + ( 10 - 2 \zeta_{20}^{2} + 10 \zeta_{20}^{4} ) q^{71} -\zeta_{20}^{7} q^{72} + ( -5 \zeta_{20} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{73} + ( -4 + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{74} + ( -4 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{76} + ( 8 \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{77} + ( -3 \zeta_{20} + 3 \zeta_{20}^{3} + 3 \zeta_{20}^{7} ) q^{78} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{81} + ( -5 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 5 \zeta_{20}^{7} ) q^{82} -6 \zeta_{20}^{3} q^{83} + 2 \zeta_{20}^{4} q^{84} + ( 2 - 2 \zeta_{20}^{2} ) q^{86} + ( 2 \zeta_{20} - 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{87} + ( -4 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{88} + ( 1 - \zeta_{20}^{2} + 2 \zeta_{20}^{6} ) q^{89} + ( -6 \zeta_{20}^{2} - 6 \zeta_{20}^{6} ) q^{91} + 6 \zeta_{20}^{3} q^{92} + ( -6 \zeta_{20}^{3} + 6 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{93} + ( -6 + 8 \zeta_{20}^{2} - 8 \zeta_{20}^{4} + 6 \zeta_{20}^{6} ) q^{94} -\zeta_{20}^{2} q^{96} + ( 3 \zeta_{20} - 3 \zeta_{20}^{3} - 12 \zeta_{20}^{7} ) q^{97} + 3 \zeta_{20} q^{98} + ( 4 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{4} - 2q^{6} + 2q^{9} + O(q^{10}) \) \( 8q + 2q^{4} - 2q^{6} + 2q^{9} - 4q^{11} - 4q^{14} - 2q^{16} + 4q^{21} - 8q^{24} + 12q^{26} - 24q^{31} + 6q^{34} - 2q^{36} - 12q^{39} - 24q^{41} - 16q^{44} + 12q^{46} + 24q^{49} - 36q^{51} + 2q^{54} + 4q^{56} - 40q^{59} - 4q^{61} + 2q^{64} - 4q^{66} - 12q^{69} + 56q^{71} - 44q^{74} - 40q^{76} - 2q^{81} - 4q^{84} + 12q^{86} + 10q^{89} - 24q^{91} - 4q^{94} - 2q^{96} + 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(\zeta_{20}^{2}\) \(1\)

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
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 2.00000i 0.951057 + 0.309017i 0.809017 0.587785i 0
49.2 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 2.00000i −0.951057 0.309017i 0.809017 0.587785i 0
199.1 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 2.00000i 0.951057 0.309017i 0.809017 + 0.587785i 0
199.2 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 2.00000i −0.951057 + 0.309017i 0.809017 + 0.587785i 0
349.1 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 2.00000i −0.587785 + 0.809017i −0.309017 + 0.951057i 0
349.2 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 2.00000i 0.587785 0.809017i −0.309017 + 0.951057i 0
649.1 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 2.00000i −0.587785 0.809017i −0.309017 0.951057i 0
649.2 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 2.00000i 0.587785 + 0.809017i −0.309017 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.h.a 8
5.b even 2 1 inner 750.2.h.a 8
5.c odd 4 1 150.2.g.b 4
5.c odd 4 1 750.2.g.a 4
15.e even 4 1 450.2.h.b 4
25.d even 5 1 inner 750.2.h.a 8
25.d even 5 1 3750.2.c.c 4
25.e even 10 1 inner 750.2.h.a 8
25.e even 10 1 3750.2.c.c 4
25.f odd 20 1 150.2.g.b 4
25.f odd 20 1 750.2.g.a 4
25.f odd 20 1 3750.2.a.b 2
25.f odd 20 1 3750.2.a.g 2
75.l even 20 1 450.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.b 4 5.c odd 4 1
150.2.g.b 4 25.f odd 20 1
450.2.h.b 4 15.e even 4 1
450.2.h.b 4 75.l even 20 1
750.2.g.a 4 5.c odd 4 1
750.2.g.a 4 25.f odd 20 1
750.2.h.a 8 1.a even 1 1 trivial
750.2.h.a 8 5.b even 2 1 inner
750.2.h.a 8 25.d even 5 1 inner
750.2.h.a 8 25.e even 10 1 inner
3750.2.a.b 2 25.f odd 20 1
3750.2.a.g 2 25.f odd 20 1
3750.2.c.c 4 25.d even 5 1
3750.2.c.c 4 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 4 + T^{2} )^{4} \)
$11$ \( ( 16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$13$ \( 6561 + 729 T^{2} + 486 T^{4} - 36 T^{6} + T^{8} \)
$17$ \( 6561 - 8019 T^{2} + 3726 T^{4} + 36 T^{6} + T^{8} \)
$19$ \( ( 400 + 200 T + 40 T^{2} + T^{4} )^{2} \)
$23$ \( 1679616 - 46656 T^{2} + 1296 T^{4} - 36 T^{6} + T^{8} \)
$29$ \( ( 25 + 25 T + 10 T^{2} + T^{4} )^{2} \)
$31$ \( ( 1296 + 648 T + 144 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$37$ \( 130321 - 44764 T^{2} + 5806 T^{4} + 41 T^{6} + T^{8} \)
$41$ \( ( 961 + 403 T + 94 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$43$ \( ( 16 + 12 T^{2} + T^{4} )^{2} \)
$47$ \( 3748096 + 30976 T^{2} + 5856 T^{4} - 124 T^{6} + T^{8} \)
$53$ \( 96059601 - 352836 T^{2} + 13446 T^{4} - 171 T^{6} + T^{8} \)
$59$ \( ( 6400 + 1600 T + 240 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$61$ \( ( 361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$67$ \( 1679616 - 186624 T^{2} + 7776 T^{4} + 36 T^{6} + T^{8} \)
$71$ \( ( 13456 - 2552 T + 384 T^{2} - 28 T^{3} + T^{4} )^{2} \)
$73$ \( 923521 - 145111 T^{2} + 8566 T^{4} + 44 T^{6} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( 1679616 - 46656 T^{2} + 1296 T^{4} - 36 T^{6} + T^{8} \)
$89$ \( ( 25 + 10 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$97$ \( 96059601 - 2734479 T^{2} + 29646 T^{4} + 36 T^{6} + T^{8} \)
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