Properties

Label 1575.2.a.t
Level 1575
Weight 2
Character orbit 1575.a
Self dual yes
Analytic conductor 12.576
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} - q^{7} + 3 q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} - q^{7} + 3 q^{8} + 3 q^{11} + ( -2 + 2 \beta ) q^{13} -\beta q^{14} + ( -2 + \beta ) q^{16} + 2 \beta q^{17} + ( 2 + 2 \beta ) q^{19} + 3 \beta q^{22} + ( 3 - 4 \beta ) q^{23} + 6 q^{26} + ( -1 - \beta ) q^{28} + ( 3 + 2 \beta ) q^{29} + ( 2 - 4 \beta ) q^{31} + ( -3 - \beta ) q^{32} + ( 6 + 2 \beta ) q^{34} + ( -5 + 4 \beta ) q^{37} + ( 6 + 4 \beta ) q^{38} + ( -5 - 2 \beta ) q^{43} + ( 3 + 3 \beta ) q^{44} + ( -12 - \beta ) q^{46} + ( 6 + 2 \beta ) q^{47} + q^{49} + ( 4 + 2 \beta ) q^{52} + ( -6 + 4 \beta ) q^{53} -3 q^{56} + ( 6 + 5 \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( 8 - 4 \beta ) q^{61} + ( -12 - 2 \beta ) q^{62} + ( 1 - 6 \beta ) q^{64} + ( -11 - 2 \beta ) q^{67} + ( 6 + 4 \beta ) q^{68} + 3 q^{71} + ( 4 - 2 \beta ) q^{73} + ( 12 - \beta ) q^{74} + ( 8 + 6 \beta ) q^{76} -3 q^{77} + ( -1 - 6 \beta ) q^{79} + ( 6 - 4 \beta ) q^{83} + ( -6 - 7 \beta ) q^{86} + 9 q^{88} + ( 6 - 6 \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( -9 - 5 \beta ) q^{92} + ( 6 + 8 \beta ) q^{94} + ( 10 - 4 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 3q^{4} - 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + q^{2} + 3q^{4} - 2q^{7} + 6q^{8} + 6q^{11} - 2q^{13} - q^{14} - 3q^{16} + 2q^{17} + 6q^{19} + 3q^{22} + 2q^{23} + 12q^{26} - 3q^{28} + 8q^{29} - 7q^{32} + 14q^{34} - 6q^{37} + 16q^{38} - 12q^{43} + 9q^{44} - 25q^{46} + 14q^{47} + 2q^{49} + 10q^{52} - 8q^{53} - 6q^{56} + 17q^{58} + 14q^{59} + 12q^{61} - 26q^{62} - 4q^{64} - 24q^{67} + 16q^{68} + 6q^{71} + 6q^{73} + 23q^{74} + 22q^{76} - 6q^{77} - 8q^{79} + 8q^{83} - 19q^{86} + 18q^{88} + 6q^{89} + 2q^{91} - 23q^{92} + 20q^{94} + 16q^{97} + q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
−1.30278 0 −0.302776 0 0 −1.00000 3.00000 0 0
1.2 2.30278 0 3.30278 0 0 −1.00000 3.00000 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.t 2
3.b odd 2 1 525.2.a.f 2
5.b even 2 1 1575.2.a.o 2
5.c odd 4 2 1575.2.d.g 4
12.b even 2 1 8400.2.a.df 2
15.d odd 2 1 525.2.a.h yes 2
15.e even 4 2 525.2.d.d 4
21.c even 2 1 3675.2.a.w 2
60.h even 2 1 8400.2.a.cw 2
105.g even 2 1 3675.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.f 2 3.b odd 2 1
525.2.a.h yes 2 15.d odd 2 1
525.2.d.d 4 15.e even 4 2
1575.2.a.o 2 5.b even 2 1
1575.2.a.t 2 1.a even 1 1 trivial
1575.2.d.g 4 5.c odd 4 2
3675.2.a.w 2 21.c even 2 1
3675.2.a.bb 2 105.g even 2 1
8400.2.a.cw 2 60.h even 2 1
8400.2.a.df 2 12.b even 2 1

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} - T_{2} - 3 \)
\( T_{11} - 3 \)
\( T_{13}^{2} + 2 T_{13} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ 1
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 + 2 T + 14 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 2 T + 22 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 6 T + 34 T^{2} - 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 2 T - 5 T^{2} - 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 + 10 T^{2} + 961 T^{4} \)
$37$ \( 1 + 6 T + 31 T^{2} + 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( 1 + 12 T + 109 T^{2} + 516 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 14 T + 130 T^{2} - 658 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 8 T + 70 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 14 T + 154 T^{2} - 826 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 24 T + 265 T^{2} + 1608 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 3 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 6 T + 142 T^{2} - 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T + 57 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 8 T + 130 T^{2} - 664 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 16 T + 206 T^{2} - 1552 T^{3} + 9409 T^{4} \)
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