Properties

Label 1575.4.a.n
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -2 - \beta ) q^{2} + ( 1 + 4 \beta ) q^{4} + 7 q^{7} + ( -6 - \beta ) q^{8} +O(q^{10})\) \( q + ( -2 - \beta ) q^{2} + ( 1 + 4 \beta ) q^{4} + 7 q^{7} + ( -6 - \beta ) q^{8} + ( 46 - 2 \beta ) q^{11} + ( -4 - 38 \beta ) q^{13} + ( -14 - 7 \beta ) q^{14} + ( 9 - 24 \beta ) q^{16} + ( -22 - 44 \beta ) q^{17} + ( -54 - 26 \beta ) q^{19} + ( -82 - 42 \beta ) q^{22} + ( -160 + 20 \beta ) q^{23} + ( 198 + 80 \beta ) q^{26} + ( 7 + 28 \beta ) q^{28} + ( 118 + 12 \beta ) q^{29} + ( -30 - 102 \beta ) q^{31} + ( 150 + 47 \beta ) q^{32} + ( 264 + 110 \beta ) q^{34} + ( -102 + 24 \beta ) q^{37} + ( 238 + 106 \beta ) q^{38} + ( -22 - 80 \beta ) q^{41} + ( -68 + 128 \beta ) q^{43} + ( 6 + 182 \beta ) q^{44} + ( 220 + 120 \beta ) q^{46} + ( 200 + 168 \beta ) q^{47} + 49 q^{49} + ( -764 - 54 \beta ) q^{52} + ( 8 - 86 \beta ) q^{53} + ( -42 - 7 \beta ) q^{56} + ( -296 - 142 \beta ) q^{58} + ( 232 - 36 \beta ) q^{59} + ( -342 - 84 \beta ) q^{61} + ( 570 + 234 \beta ) q^{62} + ( -607 - 52 \beta ) q^{64} + ( -368 + 164 \beta ) q^{67} + ( -902 - 132 \beta ) q^{68} + ( 370 - 138 \beta ) q^{71} + ( -212 - 122 \beta ) q^{73} + ( 84 + 54 \beta ) q^{74} + ( -574 - 242 \beta ) q^{76} + ( 322 - 14 \beta ) q^{77} + ( -204 + 484 \beta ) q^{79} + ( 444 + 182 \beta ) q^{82} + ( 304 + 84 \beta ) q^{83} + ( -504 - 188 \beta ) q^{86} + ( -266 - 34 \beta ) q^{88} + ( 666 - 112 \beta ) q^{89} + ( -28 - 266 \beta ) q^{91} + ( 240 - 620 \beta ) q^{92} + ( -1240 - 536 \beta ) q^{94} + ( 1224 - 86 \beta ) q^{97} + ( -98 - 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 2 q^{4} + 14 q^{7} - 12 q^{8} + O(q^{10}) \) \( 2 q - 4 q^{2} + 2 q^{4} + 14 q^{7} - 12 q^{8} + 92 q^{11} - 8 q^{13} - 28 q^{14} + 18 q^{16} - 44 q^{17} - 108 q^{19} - 164 q^{22} - 320 q^{23} + 396 q^{26} + 14 q^{28} + 236 q^{29} - 60 q^{31} + 300 q^{32} + 528 q^{34} - 204 q^{37} + 476 q^{38} - 44 q^{41} - 136 q^{43} + 12 q^{44} + 440 q^{46} + 400 q^{47} + 98 q^{49} - 1528 q^{52} + 16 q^{53} - 84 q^{56} - 592 q^{58} + 464 q^{59} - 684 q^{61} + 1140 q^{62} - 1214 q^{64} - 736 q^{67} - 1804 q^{68} + 740 q^{71} - 424 q^{73} + 168 q^{74} - 1148 q^{76} + 644 q^{77} - 408 q^{79} + 888 q^{82} + 608 q^{83} - 1008 q^{86} - 532 q^{88} + 1332 q^{89} - 56 q^{91} + 480 q^{92} - 2480 q^{94} + 2448 q^{97} - 196 q^{98} + O(q^{100}) \)

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
−4.23607 0 9.94427 0 0 7.00000 −8.23607 0 0
1.2 0.236068 0 −7.94427 0 0 7.00000 −3.76393 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.n 2
3.b odd 2 1 525.4.a.o 2
5.b even 2 1 315.4.a.l 2
15.d odd 2 1 105.4.a.d 2
15.e even 4 2 525.4.d.k 4
35.c odd 2 1 2205.4.a.be 2
60.h even 2 1 1680.4.a.bd 2
105.g even 2 1 735.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 15.d odd 2 1
315.4.a.l 2 5.b even 2 1
525.4.a.o 2 3.b odd 2 1
525.4.d.k 4 15.e even 4 2
735.4.a.m 2 105.g even 2 1
1575.4.a.n 2 1.a even 1 1 trivial
1680.4.a.bd 2 60.h even 2 1
2205.4.a.be 2 35.c odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 4 T_{2} - 1 \)
\( T_{11}^{2} - 92 T_{11} + 2096 \)
\( T_{13}^{2} + 8 T_{13} - 7204 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 4 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( 2096 - 92 T + T^{2} \)
$13$ \( -7204 + 8 T + T^{2} \)
$17$ \( -9196 + 44 T + T^{2} \)
$19$ \( -464 + 108 T + T^{2} \)
$23$ \( 23600 + 320 T + T^{2} \)
$29$ \( 13204 - 236 T + T^{2} \)
$31$ \( -51120 + 60 T + T^{2} \)
$37$ \( 7524 + 204 T + T^{2} \)
$41$ \( -31516 + 44 T + T^{2} \)
$43$ \( -77296 + 136 T + T^{2} \)
$47$ \( -101120 - 400 T + T^{2} \)
$53$ \( -36916 - 16 T + T^{2} \)
$59$ \( 47344 - 464 T + T^{2} \)
$61$ \( 81684 + 684 T + T^{2} \)
$67$ \( 944 + 736 T + T^{2} \)
$71$ \( 41680 - 740 T + T^{2} \)
$73$ \( -29476 + 424 T + T^{2} \)
$79$ \( -1129664 + 408 T + T^{2} \)
$83$ \( 57136 - 608 T + T^{2} \)
$89$ \( 380836 - 1332 T + T^{2} \)
$97$ \( 1461196 - 2448 T + T^{2} \)
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