Properties

Label 1575.4.a.x
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( -3 + 3 \beta ) q^{4} + 7 q^{7} + ( 1 - 5 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( -3 + 3 \beta ) q^{4} + 7 q^{7} + ( 1 - 5 \beta ) q^{8} + ( 18 - 5 \beta ) q^{11} + ( -11 - 17 \beta ) q^{13} + ( 7 + 7 \beta ) q^{14} + ( 5 - 33 \beta ) q^{16} + ( -53 + 27 \beta ) q^{17} + ( -56 + 56 \beta ) q^{19} + ( -2 + 8 \beta ) q^{22} + ( 115 + 24 \beta ) q^{23} + ( -79 - 45 \beta ) q^{26} + ( -21 + 21 \beta ) q^{28} + ( 73 - 84 \beta ) q^{29} + ( -61 - 13 \beta ) q^{31} + ( -135 - 21 \beta ) q^{32} + ( 55 + \beta ) q^{34} + ( -66 + 19 \beta ) q^{37} + ( 168 + 56 \beta ) q^{38} + ( -95 - 45 \beta ) q^{41} + ( -373 - 58 \beta ) q^{43} + ( -114 + 54 \beta ) q^{44} + ( 211 + 163 \beta ) q^{46} + ( 120 - 88 \beta ) q^{47} + 49 q^{49} + ( -171 - 33 \beta ) q^{52} + ( 119 - 89 \beta ) q^{53} + ( 7 - 35 \beta ) q^{56} + ( -263 - 95 \beta ) q^{58} + ( 325 - 209 \beta ) q^{59} + ( 25 - 273 \beta ) q^{61} + ( -113 - 87 \beta ) q^{62} + ( -259 + 87 \beta ) q^{64} + ( -640 + 123 \beta ) q^{67} + ( 483 - 159 \beta ) q^{68} + ( -192 - 235 \beta ) q^{71} + ( -90 - 88 \beta ) q^{73} + ( 10 - 28 \beta ) q^{74} + ( 840 - 168 \beta ) q^{76} + ( 126 - 35 \beta ) q^{77} + ( -162 - 103 \beta ) q^{79} + ( -275 - 185 \beta ) q^{82} + ( 821 - 431 \beta ) q^{83} + ( -605 - 489 \beta ) q^{86} + ( 118 - 70 \beta ) q^{88} + ( -494 + 522 \beta ) q^{89} + ( -77 - 119 \beta ) q^{91} + ( -57 + 345 \beta ) q^{92} + ( -232 - 56 \beta ) q^{94} + ( 266 - 704 \beta ) q^{97} + ( 49 + 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + 31 q^{11} - 39 q^{13} + 21 q^{14} - 23 q^{16} - 79 q^{17} - 56 q^{19} + 4 q^{22} + 254 q^{23} - 203 q^{26} - 21 q^{28} + 62 q^{29} - 135 q^{31} - 291 q^{32} + 111 q^{34} - 113 q^{37} + 392 q^{38} - 235 q^{41} - 804 q^{43} - 174 q^{44} + 585 q^{46} + 152 q^{47} + 98 q^{49} - 375 q^{52} + 149 q^{53} - 21 q^{56} - 621 q^{58} + 441 q^{59} - 223 q^{61} - 313 q^{62} - 431 q^{64} - 1157 q^{67} + 807 q^{68} - 619 q^{71} - 268 q^{73} - 8 q^{74} + 1512 q^{76} + 217 q^{77} - 427 q^{79} - 735 q^{82} + 1211 q^{83} - 1699 q^{86} + 166 q^{88} - 466 q^{89} - 273 q^{91} + 231 q^{92} - 520 q^{94} - 172 q^{97} + 147 q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
−0.561553 0 −7.68466 0 0 7.00000 8.80776 0 0
1.2 3.56155 0 4.68466 0 0 7.00000 −11.8078 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.x 2
3.b odd 2 1 525.4.a.j 2
5.b even 2 1 1575.4.a.o 2
15.d odd 2 1 525.4.a.m yes 2
15.e even 4 2 525.4.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.j 2 3.b odd 2 1
525.4.a.m yes 2 15.d odd 2 1
525.4.d.m 4 15.e even 4 2
1575.4.a.o 2 5.b even 2 1
1575.4.a.x 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{2} - 3 T_{2} - 2 \)
\( T_{11}^{2} - 31 T_{11} + 134 \)
\( T_{13}^{2} + 39 T_{13} - 848 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( 134 - 31 T + T^{2} \)
$13$ \( -848 + 39 T + T^{2} \)
$17$ \( -1538 + 79 T + T^{2} \)
$19$ \( -12544 + 56 T + T^{2} \)
$23$ \( 13681 - 254 T + T^{2} \)
$29$ \( -29027 - 62 T + T^{2} \)
$31$ \( 3838 + 135 T + T^{2} \)
$37$ \( 1658 + 113 T + T^{2} \)
$41$ \( 5200 + 235 T + T^{2} \)
$43$ \( 147307 + 804 T + T^{2} \)
$47$ \( -27136 - 152 T + T^{2} \)
$53$ \( -28114 - 149 T + T^{2} \)
$59$ \( -137024 - 441 T + T^{2} \)
$61$ \( -304316 + 223 T + T^{2} \)
$67$ \( 270364 + 1157 T + T^{2} \)
$71$ \( -138916 + 619 T + T^{2} \)
$73$ \( -14956 + 268 T + T^{2} \)
$79$ \( 494 + 427 T + T^{2} \)
$83$ \( -422854 - 1211 T + T^{2} \)
$89$ \( -1103768 + 466 T + T^{2} \)
$97$ \( -2098972 + 172 T + T^{2} \)
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