Properties

Label 75.10.b.d
Level $75$
Weight $10$
Character orbit 75.b
Analytic conductor $38.628$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 81 i q^{3} + 496 q^{4} -324 q^{6} + 7680 i q^{7} + 4032 i q^{8} -6561 q^{9} +O(q^{10})\) \( q + 4 i q^{2} + 81 i q^{3} + 496 q^{4} -324 q^{6} + 7680 i q^{7} + 4032 i q^{8} -6561 q^{9} -86404 q^{11} + 40176 i q^{12} -149978 i q^{13} -30720 q^{14} + 237824 q^{16} + 207622 i q^{17} -26244 i q^{18} -716284 q^{19} -622080 q^{21} -345616 i q^{22} + 1369920 i q^{23} -326592 q^{24} + 599912 q^{26} -531441 i q^{27} + 3809280 i q^{28} + 3194402 q^{29} -2349000 q^{31} + 3015680 i q^{32} -6998724 i q^{33} -830488 q^{34} -3254256 q^{36} -18735710 i q^{37} -2865136 i q^{38} + 12148218 q^{39} -29282630 q^{41} -2488320 i q^{42} -1516724 i q^{43} -42856384 q^{44} -5479680 q^{46} -615752 i q^{47} + 19263744 i q^{48} -18628793 q^{49} -16817382 q^{51} -74389088 i q^{52} + 4747430 i q^{53} + 2125764 q^{54} -30965760 q^{56} -58019004 i q^{57} + 12777608 i q^{58} -60616076 q^{59} -126745682 q^{61} -9396000 i q^{62} -50388480 i q^{63} + 109703168 q^{64} + 27994896 q^{66} + 111182652 i q^{67} + 102980512 i q^{68} -110963520 q^{69} -175551608 q^{71} -26453952 i q^{72} -61233350 i q^{73} + 74942840 q^{74} -355276864 q^{76} -663582720 i q^{77} + 48592872 i q^{78} -234431160 q^{79} + 43046721 q^{81} -117130520 i q^{82} + 118910388 i q^{83} -308551680 q^{84} + 6066896 q^{86} + 258746562 i q^{87} -348380928 i q^{88} + 316534326 q^{89} + 1151831040 q^{91} + 679480320 i q^{92} -190269000 i q^{93} + 2463008 q^{94} -244270080 q^{96} -242912258 i q^{97} -74515172 i q^{98} + 566896644 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 992q^{4} - 648q^{6} - 13122q^{9} + O(q^{10}) \) \( 2q + 992q^{4} - 648q^{6} - 13122q^{9} - 172808q^{11} - 61440q^{14} + 475648q^{16} - 1432568q^{19} - 1244160q^{21} - 653184q^{24} + 1199824q^{26} + 6388804q^{29} - 4698000q^{31} - 1660976q^{34} - 6508512q^{36} + 24296436q^{39} - 58565260q^{41} - 85712768q^{44} - 10959360q^{46} - 37257586q^{49} - 33634764q^{51} + 4251528q^{54} - 61931520q^{56} - 121232152q^{59} - 253491364q^{61} + 219406336q^{64} + 55989792q^{66} - 221927040q^{69} - 351103216q^{71} + 149885680q^{74} - 710553728q^{76} - 468862320q^{79} + 86093442q^{81} - 617103360q^{84} + 12133792q^{86} + 633068652q^{89} + 2303662080q^{91} + 4926016q^{94} - 488540160q^{96} + 1133793288q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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
1.00000i
1.00000i
4.00000i 81.0000i 496.000 0 −324.000 7680.00i 4032.00i −6561.00 0
49.2 4.00000i 81.0000i 496.000 0 −324.000 7680.00i 4032.00i −6561.00 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.10.b.d 2
3.b odd 2 1 225.10.b.e 2
5.b even 2 1 inner 75.10.b.d 2
5.c odd 4 1 15.10.a.a 1
5.c odd 4 1 75.10.a.c 1
15.d odd 2 1 225.10.b.e 2
15.e even 4 1 45.10.a.b 1
15.e even 4 1 225.10.a.c 1
20.e even 4 1 240.10.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.a 1 5.c odd 4 1
45.10.a.b 1 15.e even 4 1
75.10.a.c 1 5.c odd 4 1
75.10.b.d 2 1.a even 1 1 trivial
75.10.b.d 2 5.b even 2 1 inner
225.10.a.c 1 15.e even 4 1
225.10.b.e 2 3.b odd 2 1
225.10.b.e 2 15.d odd 2 1
240.10.a.c 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 16 \) acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 1008 T^{2} + 262144 T^{4} \)
$3$ \( 1 + 6561 T^{2} \)
$5$ 1
$7$ \( 1 - 21724814 T^{2} + 1628413597910449 T^{4} \)
$11$ \( ( 1 + 86404 T + 2357947691 T^{2} )^{2} \)
$13$ \( 1 + 1284401738 T^{2} + \)\(11\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 - 194068858110 T^{2} + \)\(14\!\cdots\!09\)\( T^{4} \)
$19$ \( ( 1 + 716284 T + 322687697779 T^{2} )^{2} \)
$23$ \( 1 - 1725624516526 T^{2} + \)\(32\!\cdots\!69\)\( T^{4} \)
$29$ \( ( 1 - 3194402 T + 14507145975869 T^{2} )^{2} \)
$31$ \( ( 1 + 2349000 T + 26439622160671 T^{2} )^{2} \)
$37$ \( 1 + 91103349613946 T^{2} + \)\(16\!\cdots\!29\)\( T^{4} \)
$41$ \( ( 1 + 29282630 T + 327381934393961 T^{2} )^{2} \)
$43$ \( 1 - 1002884772181510 T^{2} + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 2237881795680030 T^{2} + \)\(12\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 - 6576989091999366 T^{2} + \)\(10\!\cdots\!89\)\( T^{4} \)
$59$ \( ( 1 + 60616076 T + 8662995818654939 T^{2} )^{2} \)
$61$ \( ( 1 + 126745682 T + 11694146092834141 T^{2} )^{2} \)
$67$ \( 1 - 42051486686836790 T^{2} + \)\(74\!\cdots\!09\)\( T^{4} \)
$71$ \( ( 1 + 175551608 T + 45848500718449031 T^{2} )^{2} \)
$73$ \( 1 - 113993650264313326 T^{2} + \)\(34\!\cdots\!69\)\( T^{4} \)
$79$ \( ( 1 + 234431160 T + 119851595982618319 T^{2} )^{2} \)
$83$ \( 1 - 359740830160770262 T^{2} + \)\(34\!\cdots\!09\)\( T^{4} \)
$89$ \( ( 1 - 316534326 T + 350356403707485209 T^{2} )^{2} \)
$97$ \( 1 - 1461455752222471870 T^{2} + \)\(57\!\cdots\!89\)\( T^{4} \)
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