Properties

Label 1575.2.a.n
Level 1575
Weight 2
Character orbit 1575.a
Self dual yes
Analytic conductor 12.576
Analytic rank 1
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(12.5764383184\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} + q^{7} + ( -1 + 2 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} + q^{7} + ( -1 + 2 \beta ) q^{8} + ( -1 - 2 \beta ) q^{11} -2 \beta q^{13} -\beta q^{14} -3 \beta q^{16} + 4 \beta q^{17} + ( -2 + 4 \beta ) q^{19} + ( 2 + 3 \beta ) q^{22} + ( -5 + 2 \beta ) q^{23} + ( 2 + 2 \beta ) q^{26} + ( -1 + \beta ) q^{28} -5 q^{29} -6 \beta q^{31} + ( 5 - \beta ) q^{32} + ( -4 - 4 \beta ) q^{34} + 3 q^{37} + ( -4 - 2 \beta ) q^{38} + ( -6 - 2 \beta ) q^{41} + ( 3 + 2 \beta ) q^{43} + ( -1 - \beta ) q^{44} + ( -2 + 3 \beta ) q^{46} + 2 q^{47} + q^{49} -2 q^{52} + ( -6 + 4 \beta ) q^{53} + ( -1 + 2 \beta ) q^{56} + 5 \beta q^{58} + ( -8 + 6 \beta ) q^{59} + ( -6 + 6 \beta ) q^{61} + ( 6 + 6 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( -3 + 2 \beta ) q^{67} + 4 q^{68} + ( -5 + 6 \beta ) q^{71} + ( -10 - 2 \beta ) q^{73} -3 \beta q^{74} + ( 6 - 2 \beta ) q^{76} + ( -1 - 2 \beta ) q^{77} + ( -5 + 10 \beta ) q^{79} + ( 2 + 8 \beta ) q^{82} + ( 4 - 6 \beta ) q^{83} + ( -2 - 5 \beta ) q^{86} + ( -3 - 4 \beta ) q^{88} + ( -16 + 2 \beta ) q^{89} -2 \beta q^{91} + ( 7 - 5 \beta ) q^{92} -2 \beta q^{94} + ( 4 - 2 \beta ) q^{97} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 2q^{7} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 2q^{7} - 4q^{11} - 2q^{13} - q^{14} - 3q^{16} + 4q^{17} + 7q^{22} - 8q^{23} + 6q^{26} - q^{28} - 10q^{29} - 6q^{31} + 9q^{32} - 12q^{34} + 6q^{37} - 10q^{38} - 14q^{41} + 8q^{43} - 3q^{44} - q^{46} + 4q^{47} + 2q^{49} - 4q^{52} - 8q^{53} + 5q^{58} - 10q^{59} - 6q^{61} + 18q^{62} + 4q^{64} - 4q^{67} + 8q^{68} - 4q^{71} - 22q^{73} - 3q^{74} + 10q^{76} - 4q^{77} + 12q^{82} + 2q^{83} - 9q^{86} - 10q^{88} - 30q^{89} - 2q^{91} + 9q^{92} - 2q^{94} + 6q^{97} - q^{98} + O(q^{100}) \)

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
1.61803
−0.618034
−1.61803 0 0.618034 0 0 1.00000 2.23607 0 0
1.2 0.618034 0 −1.61803 0 0 1.00000 −2.23607 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1575.2.a.n 2
3.b odd 2 1 175.2.a.e yes 2
5.b even 2 1 1575.2.a.s 2
5.c odd 4 2 1575.2.d.k 4
12.b even 2 1 2800.2.a.bp 2
15.d odd 2 1 175.2.a.d 2
15.e even 4 2 175.2.b.c 4
21.c even 2 1 1225.2.a.u 2
60.h even 2 1 2800.2.a.bh 2
60.l odd 4 2 2800.2.g.s 4
105.g even 2 1 1225.2.a.n 2
105.k odd 4 2 1225.2.b.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 15.d odd 2 1
175.2.a.e yes 2 3.b odd 2 1
175.2.b.c 4 15.e even 4 2
1225.2.a.n 2 105.g even 2 1
1225.2.a.u 2 21.c even 2 1
1225.2.b.k 4 105.k odd 4 2
1575.2.a.n 2 1.a even 1 1 trivial
1575.2.a.s 2 5.b even 2 1
1575.2.d.k 4 5.c odd 4 2
2800.2.a.bh 2 60.h even 2 1
2800.2.a.bp 2 12.b even 2 1
2800.2.g.s 4 60.l odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{11}^{2} + 4 T_{11} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ 1
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 + 4 T + 21 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( 1 + 18 T^{2} + 361 T^{4} \)
$23$ \( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 5 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 3 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 14 T + 126 T^{2} + 574 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 - 2 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 10 T + 98 T^{2} + 590 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 101 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 22 T + 262 T^{2} + 1606 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 33 T^{2} + 6241 T^{4} \)
$83$ \( 1 - 2 T + 122 T^{2} - 166 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 6 T + 198 T^{2} - 582 T^{3} + 9409 T^{4} \)
show more
show less