Properties

Label 75.10.a.a
Level $75$
Weight $10$
Character orbit 75.a
Self dual yes
Analytic conductor $38.628$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 22q^{2} + 81q^{3} - 28q^{4} - 1782q^{6} + 5988q^{7} + 11880q^{8} + 6561q^{9} + O(q^{10}) \) \( q - 22q^{2} + 81q^{3} - 28q^{4} - 1782q^{6} + 5988q^{7} + 11880q^{8} + 6561q^{9} - 14648q^{11} - 2268q^{12} - 37906q^{13} - 131736q^{14} - 247024q^{16} + 441098q^{17} - 144342q^{18} + 441820q^{19} + 485028q^{21} + 322256q^{22} - 2264136q^{23} + 962280q^{24} + 833932q^{26} + 531441q^{27} - 167664q^{28} - 1049350q^{29} - 7910568q^{31} - 648032q^{32} - 1186488q^{33} - 9704156q^{34} - 183708q^{36} + 20992558q^{37} - 9720040q^{38} - 3070386q^{39} + 13285562q^{41} - 10670616q^{42} + 23130764q^{43} + 410144q^{44} + 49810992q^{46} + 13873688q^{47} - 20008944q^{48} - 4497463q^{49} + 35728938q^{51} + 1061368q^{52} + 57635174q^{53} - 11691702q^{54} + 71137440q^{56} + 35787420q^{57} + 23085700q^{58} - 32042120q^{59} + 110664022q^{61} + 174032496q^{62} + 39287268q^{63} + 140732992q^{64} + 26102736q^{66} + 118568268q^{67} - 12350744q^{68} - 183395016q^{69} + 276679712q^{71} + 77944680q^{72} + 264023294q^{73} - 461836276q^{74} - 12370960q^{76} - 87712224q^{77} + 67548492q^{78} + 448202760q^{79} + 43046721q^{81} - 292282364q^{82} - 851015796q^{83} - 13580784q^{84} - 508876808q^{86} - 84997350q^{87} - 174018240q^{88} + 189894930q^{89} - 226981128q^{91} + 63395808q^{92} - 640756008q^{93} - 305221136q^{94} - 52490592q^{96} + 1014149278q^{97} + 98944186q^{98} - 96105528q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−22.0000 81.0000 −28.0000 0 −1782.00 5988.00 11880.0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 22 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(75))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 22 T + 512 T^{2} \)
$3$ \( 1 - 81 T \)
$5$ 1
$7$ \( 1 - 5988 T + 40353607 T^{2} \)
$11$ \( 1 + 14648 T + 2357947691 T^{2} \)
$13$ \( 1 + 37906 T + 10604499373 T^{2} \)
$17$ \( 1 - 441098 T + 118587876497 T^{2} \)
$19$ \( 1 - 441820 T + 322687697779 T^{2} \)
$23$ \( 1 + 2264136 T + 1801152661463 T^{2} \)
$29$ \( 1 + 1049350 T + 14507145975869 T^{2} \)
$31$ \( 1 + 7910568 T + 26439622160671 T^{2} \)
$37$ \( 1 - 20992558 T + 129961739795077 T^{2} \)
$41$ \( 1 - 13285562 T + 327381934393961 T^{2} \)
$43$ \( 1 - 23130764 T + 502592611936843 T^{2} \)
$47$ \( 1 - 13873688 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 57635174 T + 3299763591802133 T^{2} \)
$59$ \( 1 + 32042120 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 110664022 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 118568268 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 276679712 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 264023294 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 448202760 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 851015796 T + 186940255267540403 T^{2} \)
$89$ \( 1 - 189894930 T + 350356403707485209 T^{2} \)
$97$ \( 1 - 1014149278 T + 760231058654565217 T^{2} \)
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