Properties

Label 750.2.h.b
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: \(\Q(\zeta_{20})\)
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.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} -\zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{6} + ( -\zeta_{20}^{3} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} +O(q^{10})\) \( 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} + ( -\zeta_{20}^{3} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{7} + \zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} + ( -1 + \zeta_{20}^{2} - 2 \zeta_{20}^{6} ) q^{11} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{12} + ( 4 \zeta_{20}^{3} - 4 \zeta_{20}^{5} ) q^{13} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{14} + \zeta_{20}^{4} q^{16} + ( 2 \zeta_{20} + 2 \zeta_{20}^{5} ) q^{17} -\zeta_{20}^{5} q^{18} + ( 4 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{19} + ( -1 + 2 \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{21} + ( -\zeta_{20} + \zeta_{20}^{3} - 2 \zeta_{20}^{7} ) q^{22} + ( 2 \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{23} + q^{24} + ( 4 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{26} -\zeta_{20} q^{27} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{28} + ( -4 + 6 \zeta_{20}^{2} - 4 \zeta_{20}^{4} ) q^{29} + ( -5 + 6 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 5 \zeta_{20}^{6} ) q^{31} + \zeta_{20}^{5} q^{32} + ( \zeta_{20} - 3 \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{33} + ( 2 \zeta_{20}^{2} + 2 \zeta_{20}^{6} ) q^{34} -\zeta_{20}^{6} q^{36} + ( -8 \zeta_{20} + 8 \zeta_{20}^{3} - 8 \zeta_{20}^{5} + 8 \zeta_{20}^{7} ) q^{37} + ( 4 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{38} + ( 4 - 4 \zeta_{20}^{2} ) q^{39} + ( -6 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{41} + ( -\zeta_{20} + 2 \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{42} + ( -6 \zeta_{20}^{3} + 4 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{43} + ( 2 - 3 \zeta_{20}^{2} + 3 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{44} + ( 4 - 2 \zeta_{20}^{2} + 4 \zeta_{20}^{4} ) q^{46} + ( -6 \zeta_{20} + 6 \zeta_{20}^{3} + 8 \zeta_{20}^{7} ) q^{47} + \zeta_{20} q^{48} + ( 5 + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{49} + ( 2 + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{51} + ( 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{52} + ( -5 \zeta_{20} + 5 \zeta_{20}^{3} + 6 \zeta_{20}^{7} ) q^{53} -\zeta_{20}^{2} q^{54} + ( -1 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{56} + ( -6 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{57} + ( -4 \zeta_{20} + 6 \zeta_{20}^{3} - 4 \zeta_{20}^{5} ) q^{58} + ( -\zeta_{20}^{2} + 3 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{59} + ( 2 - 2 \zeta_{20}^{2} - 6 \zeta_{20}^{6} ) q^{61} + ( -5 \zeta_{20} + 6 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 5 \zeta_{20}^{7} ) q^{62} + ( \zeta_{20} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{63} + \zeta_{20}^{6} q^{64} + ( \zeta_{20}^{2} - 3 \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{66} + ( 4 \zeta_{20} + 4 \zeta_{20}^{3} + 4 \zeta_{20}^{5} ) q^{67} + ( 2 \zeta_{20}^{3} + 2 \zeta_{20}^{7} ) q^{68} + ( 2 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{69} + ( -4 - 8 \zeta_{20}^{2} - 4 \zeta_{20}^{4} ) q^{71} -\zeta_{20}^{7} q^{72} + ( -10 \zeta_{20} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{73} -8 q^{74} + ( 4 + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{76} + ( 3 \zeta_{20} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{77} + ( 4 \zeta_{20} - 4 \zeta_{20}^{3} ) q^{78} + ( 1 - 5 \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{79} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{81} + ( -6 \zeta_{20}^{3} + 2 \zeta_{20}^{5} - 6 \zeta_{20}^{7} ) q^{82} + ( 3 \zeta_{20} + 4 \zeta_{20}^{3} + 3 \zeta_{20}^{5} ) q^{83} + ( -\zeta_{20}^{2} + 2 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{84} + ( 6 - 6 \zeta_{20}^{2} - 2 \zeta_{20}^{6} ) q^{86} + ( 2 \zeta_{20} - 6 \zeta_{20}^{3} + 6 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{87} + ( 2 \zeta_{20} - 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{88} + ( -4 + 4 \zeta_{20}^{2} + 10 \zeta_{20}^{6} ) q^{89} + ( 8 \zeta_{20}^{2} - 12 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{91} + ( 4 \zeta_{20} - 2 \zeta_{20}^{3} + 4 \zeta_{20}^{5} ) q^{92} + ( -\zeta_{20}^{3} + 6 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{93} + ( -8 + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{94} + \zeta_{20}^{2} q^{96} + ( -\zeta_{20} + \zeta_{20}^{3} + 5 \zeta_{20}^{7} ) q^{97} + ( 5 \zeta_{20} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{98} + ( -2 + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{4} + 2q^{6} + 2q^{9} + O(q^{10}) \) \( 8q + 2q^{4} + 2q^{6} + 2q^{9} - 10q^{11} + 8q^{14} - 2q^{16} + 32q^{19} - 2q^{21} + 8q^{24} - 16q^{26} - 12q^{29} - 6q^{31} + 8q^{34} - 2q^{36} + 24q^{39} - 28q^{41} + 20q^{46} + 28q^{49} + 8q^{51} - 2q^{54} + 2q^{56} - 10q^{59} + 2q^{64} + 10q^{66} + 20q^{69} - 40q^{71} - 64q^{74} + 8q^{76} - 4q^{79} - 2q^{81} - 8q^{84} + 32q^{86} - 4q^{89} + 56q^{91} - 40q^{94} + 2q^{96} - 20q^{99} + O(q^{100}) \)

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 2.61803i 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 2.61803i −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 2.61803i 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 2.61803i −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 0.381966i −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 0.381966i 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 0.381966i −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 0.381966i 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.b 8
5.b even 2 1 inner 750.2.h.b 8
5.c odd 4 1 150.2.g.a 4
5.c odd 4 1 750.2.g.b 4
15.e even 4 1 450.2.h.c 4
25.d even 5 1 inner 750.2.h.b 8
25.d even 5 1 3750.2.c.b 4
25.e even 10 1 inner 750.2.h.b 8
25.e even 10 1 3750.2.c.b 4
25.f odd 20 1 150.2.g.a 4
25.f odd 20 1 750.2.g.b 4
25.f odd 20 1 3750.2.a.d 2
25.f odd 20 1 3750.2.a.f 2
75.l even 20 1 450.2.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.a 4 5.c odd 4 1
150.2.g.a 4 25.f odd 20 1
450.2.h.c 4 15.e even 4 1
450.2.h.c 4 75.l even 20 1
750.2.g.b 4 5.c odd 4 1
750.2.g.b 4 25.f odd 20 1
750.2.h.b 8 1.a even 1 1 trivial
750.2.h.b 8 5.b even 2 1 inner
750.2.h.b 8 25.d even 5 1 inner
750.2.h.b 8 25.e even 10 1 inner
3750.2.a.d 2 25.f odd 20 1
3750.2.a.f 2 25.f odd 20 1
3750.2.c.b 4 25.d even 5 1
3750.2.c.b 4 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 7 T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$5$ \( \)
$7$ \( ( 1 - 21 T^{2} + 197 T^{4} - 1029 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 + 5 T - T^{2} - 55 T^{3} - 184 T^{4} - 605 T^{5} - 121 T^{6} + 6655 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( 1 + 42 T^{2} + 795 T^{4} + 11332 T^{6} + 153429 T^{8} + 1915108 T^{10} + 22705995 T^{12} + 202725978 T^{14} + 815730721 T^{16} \)
$17$ \( 1 + 38 T^{2} + 555 T^{4} - 2432 T^{6} - 168571 T^{8} - 702848 T^{10} + 46354155 T^{12} + 917227622 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 - 16 T + 117 T^{2} - 578 T^{3} + 2525 T^{4} - 10982 T^{5} + 42237 T^{6} - 109744 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( 1 + 26 T^{2} + 147 T^{4} - 9932 T^{6} - 335995 T^{8} - 5254028 T^{10} + 41136627 T^{12} + 3848933114 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 + 6 T + 47 T^{2} + 288 T^{3} + 2365 T^{4} + 8352 T^{5} + 39527 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} )^{2}( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 10 T^{2} - 1269 T^{4} - 26380 T^{6} + 1473461 T^{8} - 36114220 T^{10} - 2378310309 T^{12} + 25657264090 T^{14} + 3512479453921 T^{16} \)
$41$ \( ( 1 + 14 T + 95 T^{2} + 786 T^{3} + 6569 T^{4} + 32226 T^{5} + 159695 T^{6} + 964894 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 80 T^{2} + 5118 T^{4} - 147920 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 174 T^{2} + 11867 T^{4} + 388392 T^{6} + 10496005 T^{8} + 857957928 T^{10} + 57907174427 T^{12} + 1875583467246 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 + 150 T^{2} + 5691 T^{4} - 410500 T^{6} - 45809019 T^{8} - 1153094500 T^{10} + 44904727371 T^{12} + 3324654169350 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 + 5 T - 49 T^{2} - 295 T^{3} + 1736 T^{4} - 17405 T^{5} - 170569 T^{6} + 1026895 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 25010 T^{5} - 78141 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( 1 + 198 T^{2} + 10235 T^{4} - 834852 T^{6} - 122622251 T^{8} - 3747650628 T^{10} + 206246723435 T^{12} + 17910759669462 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 + 20 T + 89 T^{2} - 1420 T^{3} - 22639 T^{4} - 100820 T^{5} + 448649 T^{6} + 7158220 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( 1 - 98 T^{2} + 13875 T^{4} - 944068 T^{6} + 101584229 T^{8} - 5030938372 T^{10} + 394025593875 T^{12} - 14830754176322 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 + 2 T - 55 T^{2} - 578 T^{3} + 3679 T^{4} - 45662 T^{5} - 343255 T^{6} + 986078 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( 1 + 207 T^{2} + 12695 T^{4} - 1121523 T^{6} - 205113656 T^{8} - 7726171947 T^{10} + 602483385095 T^{12} + 67676657287383 T^{14} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 2 T + 35 T^{2} + 632 T^{3} + 8789 T^{4} + 56248 T^{5} + 277235 T^{6} + 1409938 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 + 195 T^{2} + 20651 T^{4} + 1202145 T^{6} + 55933936 T^{8} + 11310982305 T^{10} + 1828218181931 T^{12} + 162429540961155 T^{14} + 7837433594376961 T^{16} \)
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