Properties

Label 1575.4.a.k
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 8q^{4} + 7q^{7} + O(q^{10}) \) \( q + 4q^{2} + 8q^{4} + 7q^{7} - 62q^{11} + 62q^{13} + 28q^{14} - 64q^{16} + 84q^{17} + 100q^{19} - 248q^{22} - 42q^{23} + 248q^{26} + 56q^{28} + 10q^{29} - 48q^{31} - 256q^{32} + 336q^{34} + 246q^{37} + 400q^{38} + 248q^{41} - 68q^{43} - 496q^{44} - 168q^{46} + 324q^{47} + 49q^{49} + 496q^{52} + 258q^{53} + 40q^{58} - 120q^{59} + 622q^{61} - 192q^{62} - 512q^{64} - 904q^{67} + 672q^{68} + 678q^{71} + 642q^{73} + 984q^{74} + 800q^{76} - 434q^{77} + 740q^{79} + 992q^{82} + 468q^{83} - 272q^{86} - 200q^{89} + 434q^{91} - 336q^{92} + 1296q^{94} + 1266q^{97} + 196q^{98} + O(q^{100}) \)

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
4.00000 0 8.00000 0 0 7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 1575.4.a.k 1
3.b odd 2 1 525.4.a.b 1
5.b even 2 1 63.4.a.a 1
15.d odd 2 1 21.4.a.b 1
15.e even 4 2 525.4.d.b 2
20.d odd 2 1 1008.4.a.m 1
35.c odd 2 1 441.4.a.b 1
35.i odd 6 2 441.4.e.n 2
35.j even 6 2 441.4.e.m 2
60.h even 2 1 336.4.a.h 1
105.g even 2 1 147.4.a.g 1
105.o odd 6 2 147.4.e.c 2
105.p even 6 2 147.4.e.b 2
120.i odd 2 1 1344.4.a.w 1
120.m even 2 1 1344.4.a.i 1
420.o odd 2 1 2352.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 15.d odd 2 1
63.4.a.a 1 5.b even 2 1
147.4.a.g 1 105.g even 2 1
147.4.e.b 2 105.p even 6 2
147.4.e.c 2 105.o odd 6 2
336.4.a.h 1 60.h even 2 1
441.4.a.b 1 35.c odd 2 1
441.4.e.m 2 35.j even 6 2
441.4.e.n 2 35.i odd 6 2
525.4.a.b 1 3.b odd 2 1
525.4.d.b 2 15.e even 4 2
1008.4.a.m 1 20.d odd 2 1
1344.4.a.i 1 120.m even 2 1
1344.4.a.w 1 120.i odd 2 1
1575.4.a.k 1 1.a even 1 1 trivial
2352.4.a.l 1 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2} - 4 \)
\( T_{11} + 62 \)
\( T_{13} - 62 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -7 + T \)
$11$ \( 62 + T \)
$13$ \( -62 + T \)
$17$ \( -84 + T \)
$19$ \( -100 + T \)
$23$ \( 42 + T \)
$29$ \( -10 + T \)
$31$ \( 48 + T \)
$37$ \( -246 + T \)
$41$ \( -248 + T \)
$43$ \( 68 + T \)
$47$ \( -324 + T \)
$53$ \( -258 + T \)
$59$ \( 120 + T \)
$61$ \( -622 + T \)
$67$ \( 904 + T \)
$71$ \( -678 + T \)
$73$ \( -642 + T \)
$79$ \( -740 + T \)
$83$ \( -468 + T \)
$89$ \( 200 + T \)
$97$ \( -1266 + T \)
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