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: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - 2 \beta ) q^{2} + ( -3 + 8 \beta ) q^{4} + 7 q^{7} + ( -5 - 2 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 - 2 \beta ) q^{2} + ( -3 + 8 \beta ) q^{4} + 7 q^{7} + ( -5 - 2 \beta ) q^{8} + ( 48 - 4 \beta ) q^{11} + ( 34 - 76 \beta ) q^{13} + ( -7 - 14 \beta ) q^{14} + ( 33 - 48 \beta ) q^{16} + ( 22 - 88 \beta ) q^{17} + ( -28 - 52 \beta ) q^{19} + ( -40 - 84 \beta ) q^{22} + ( -180 + 40 \beta ) q^{23} + ( 118 + 160 \beta ) q^{26} + ( -21 + 56 \beta ) q^{28} + ( 106 + 24 \beta ) q^{29} + ( 72 - 204 \beta ) q^{31} + ( 103 + 94 \beta ) q^{32} + ( 154 + 220 \beta ) q^{34} + ( -126 + 48 \beta ) q^{37} + ( 132 + 212 \beta ) q^{38} + ( 58 - 160 \beta ) q^{41} + ( -196 + 256 \beta ) q^{43} + ( -176 + 364 \beta ) q^{44} + ( 100 + 240 \beta ) q^{46} + ( 32 + 336 \beta ) q^{47} + 49 q^{49} + ( -710 - 108 \beta ) q^{52} + ( 94 - 172 \beta ) q^{53} + ( -35 - 14 \beta ) q^{56} + ( -154 - 284 \beta ) q^{58} + ( 268 - 72 \beta ) q^{59} + ( -258 - 168 \beta ) q^{61} + ( 336 + 468 \beta ) q^{62} + ( -555 - 104 \beta ) q^{64} + ( -532 + 328 \beta ) q^{67} + ( -770 - 264 \beta ) q^{68} + ( 508 - 276 \beta ) q^{71} + ( -90 - 244 \beta ) q^{73} + ( 30 + 108 \beta ) q^{74} + ( -332 - 484 \beta ) q^{76} + ( 336 - 28 \beta ) q^{77} + ( -688 + 968 \beta ) q^{79} + ( 262 + 364 \beta ) q^{82} + ( 220 + 168 \beta ) q^{83} + ( -316 - 376 \beta ) q^{86} + ( -232 - 68 \beta ) q^{88} + ( 778 - 224 \beta ) q^{89} + ( 238 - 532 \beta ) q^{91} + ( 860 - 1240 \beta ) q^{92} + ( -704 - 1072 \beta ) q^{94} + ( 1310 - 172 \beta ) q^{97} + ( -49 - 98 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 2q^{4} + 14q^{7} - 12q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 2q^{4} + 14q^{7} - 12q^{8} + 92q^{11} - 8q^{13} - 28q^{14} + 18q^{16} - 44q^{17} - 108q^{19} - 164q^{22} - 320q^{23} + 396q^{26} + 14q^{28} + 236q^{29} - 60q^{31} + 300q^{32} + 528q^{34} - 204q^{37} + 476q^{38} - 44q^{41} - 136q^{43} + 12q^{44} + 440q^{46} + 400q^{47} + 98q^{49} - 1528q^{52} + 16q^{53} - 84q^{56} - 592q^{58} + 464q^{59} - 684q^{61} + 1140q^{62} - 1214q^{64} - 736q^{67} - 1804q^{68} + 740q^{71} - 424q^{73} + 168q^{74} - 1148q^{76} + 644q^{77} - 408q^{79} + 888q^{82} + 608q^{83} - 1008q^{86} - 532q^{88} + 1332q^{89} - 56q^{91} + 480q^{92} - 2480q^{94} + 2448q^{97} - 196q^{98} + O(q^{100}) \)

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
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:

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 15 T^{2} + 32 T^{3} + 64 T^{4} \)
$3$ 1
$5$ 1
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 - 92 T + 4758 T^{2} - 122452 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 8 T - 2810 T^{2} + 17576 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 44 T + 630 T^{2} + 216172 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 108 T + 13254 T^{2} + 740772 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 320 T + 47934 T^{2} + 3893440 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 236 T + 61982 T^{2} - 5755804 T^{3} + 594823321 T^{4} \)
$31$ \( 1 + 60 T + 8462 T^{2} + 1787460 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 204 T + 108830 T^{2} + 10333212 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 44 T + 106326 T^{2} + 3032524 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 136 T + 81718 T^{2} + 10812952 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 400 T + 106526 T^{2} - 41529200 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 16 T + 260838 T^{2} - 2382032 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 464 T + 458102 T^{2} - 95295856 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 684 T + 535646 T^{2} + 155255004 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 736 T + 602470 T^{2} + 221361568 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 740 T + 757502 T^{2} - 264854140 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 424 T + 748558 T^{2} + 164943208 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 408 T - 143586 T^{2} + 201159912 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 608 T + 1200710 T^{2} - 347646496 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 1332 T + 1790774 T^{2} - 939018708 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 2448 T + 3286542 T^{2} - 2234223504 T^{3} + 832972004929 T^{4} \)
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