Properties

Label 75.12.a.a
Level $75$
Weight $12$
Character orbit 75.a
Self dual yes
Analytic conductor $57.626$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.6257385420\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 78 q^{2} + 243 q^{3} + 4036 q^{4} - 18954 q^{6} + 27760 q^{7} - 155064 q^{8} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 78 q^{2} + 243 q^{3} + 4036 q^{4} - 18954 q^{6} + 27760 q^{7} - 155064 q^{8} + 59049 q^{9} + 637836 q^{11} + 980748 q^{12} - 766214 q^{13} - 2165280 q^{14} + 3829264 q^{16} - 3084354 q^{17} - 4605822 q^{18} - 19511404 q^{19} + 6745680 q^{21} - 49751208 q^{22} - 15312360 q^{23} - 37680552 q^{24} + 59764692 q^{26} + 14348907 q^{27} + 112039360 q^{28} + 10751262 q^{29} - 50937400 q^{31} + 18888480 q^{32} + 154994148 q^{33} + 240579612 q^{34} + 238321764 q^{36} - 664740830 q^{37} + 1521889512 q^{38} - 186190002 q^{39} + 898833450 q^{41} - 526163040 q^{42} + 957947188 q^{43} + 2574306096 q^{44} + 1194364080 q^{46} + 1555741344 q^{47} + 930511152 q^{48} - 1206709143 q^{49} - 749498022 q^{51} - 3092439704 q^{52} - 3792417030 q^{53} - 1119214746 q^{54} - 4304576640 q^{56} - 4741271172 q^{57} - 838598436 q^{58} + 555306924 q^{59} + 4950420998 q^{61} + 3973117200 q^{62} + 1639200240 q^{63} - 9315634112 q^{64} - 12089543544 q^{66} - 5292399284 q^{67} - 12448452744 q^{68} - 3720903480 q^{69} - 14831086248 q^{71} - 9156374136 q^{72} - 13971005210 q^{73} + 51849784740 q^{74} - 78748026544 q^{76} + 17706327360 q^{77} + 14522820156 q^{78} + 3720542360 q^{79} + 3486784401 q^{81} - 70109009100 q^{82} - 8768454036 q^{83} + 27225564480 q^{84} - 74719880664 q^{86} + 2612556666 q^{87} - 98905401504 q^{88} - 25472769174 q^{89} - 21270100640 q^{91} - 61800684960 q^{92} - 12377788200 q^{93} - 121347824832 q^{94} + 4589900640 q^{96} + 39092494846 q^{97} + 94123313154 q^{98} + 37663577964 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−78.0000 243.000 4036.00 0 −18954.0 27760.0 −155064. 59049.0 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 75.12.a.a 1
3.b odd 2 1 225.12.a.f 1
5.b even 2 1 3.12.a.a 1
5.c odd 4 2 75.12.b.a 2
15.d odd 2 1 9.12.a.a 1
15.e even 4 2 225.12.b.a 2
20.d odd 2 1 48.12.a.f 1
35.c odd 2 1 147.12.a.c 1
40.e odd 2 1 192.12.a.g 1
40.f even 2 1 192.12.a.q 1
45.h odd 6 2 81.12.c.e 2
45.j even 6 2 81.12.c.a 2
60.h even 2 1 144.12.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 5.b even 2 1
9.12.a.a 1 15.d odd 2 1
48.12.a.f 1 20.d odd 2 1
75.12.a.a 1 1.a even 1 1 trivial
75.12.b.a 2 5.c odd 4 2
81.12.c.a 2 45.j even 6 2
81.12.c.e 2 45.h odd 6 2
144.12.a.l 1 60.h even 2 1
147.12.a.c 1 35.c odd 2 1
192.12.a.g 1 40.e odd 2 1
192.12.a.q 1 40.f even 2 1
225.12.a.f 1 3.b odd 2 1
225.12.b.a 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 78 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 78 \) Copy content Toggle raw display
$3$ \( T - 243 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 27760 \) Copy content Toggle raw display
$11$ \( T - 637836 \) Copy content Toggle raw display
$13$ \( T + 766214 \) Copy content Toggle raw display
$17$ \( T + 3084354 \) Copy content Toggle raw display
$19$ \( T + 19511404 \) Copy content Toggle raw display
$23$ \( T + 15312360 \) Copy content Toggle raw display
$29$ \( T - 10751262 \) Copy content Toggle raw display
$31$ \( T + 50937400 \) Copy content Toggle raw display
$37$ \( T + 664740830 \) Copy content Toggle raw display
$41$ \( T - 898833450 \) Copy content Toggle raw display
$43$ \( T - 957947188 \) Copy content Toggle raw display
$47$ \( T - 1555741344 \) Copy content Toggle raw display
$53$ \( T + 3792417030 \) Copy content Toggle raw display
$59$ \( T - 555306924 \) Copy content Toggle raw display
$61$ \( T - 4950420998 \) Copy content Toggle raw display
$67$ \( T + 5292399284 \) Copy content Toggle raw display
$71$ \( T + 14831086248 \) Copy content Toggle raw display
$73$ \( T + 13971005210 \) Copy content Toggle raw display
$79$ \( T - 3720542360 \) Copy content Toggle raw display
$83$ \( T + 8768454036 \) Copy content Toggle raw display
$89$ \( T + 25472769174 \) Copy content Toggle raw display
$97$ \( T - 39092494846 \) Copy content Toggle raw display
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