Properties

Label 1575.4.a.p
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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} + ( 7 + 3 \beta ) q^{4} -7 q^{7} + ( -41 - 5 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} + ( 7 + 3 \beta ) q^{4} -7 q^{7} + ( -41 - 5 \beta ) q^{8} + ( 8 - 10 \beta ) q^{11} + ( -14 + 12 \beta ) q^{13} + ( 7 + 7 \beta ) q^{14} + ( 55 + 27 \beta ) q^{16} + ( -2 - 2 \beta ) q^{17} + ( 44 - 24 \beta ) q^{19} + ( 132 + 12 \beta ) q^{22} + ( 20 - 34 \beta ) q^{23} + ( -154 - 10 \beta ) q^{26} + ( -49 - 21 \beta ) q^{28} + ( 138 - 24 \beta ) q^{29} + ( -16 + 72 \beta ) q^{31} + ( -105 - 69 \beta ) q^{32} + ( 30 + 6 \beta ) q^{34} + ( 106 + 36 \beta ) q^{37} + ( 292 + 4 \beta ) q^{38} + ( 210 + 30 \beta ) q^{41} + ( -212 + 48 \beta ) q^{43} + ( -364 - 76 \beta ) q^{44} + ( 456 + 48 \beta ) q^{46} + ( -40 + 68 \beta ) q^{47} + 49 q^{49} + ( 406 + 78 \beta ) q^{52} + ( -554 + 4 \beta ) q^{53} + ( 287 + 35 \beta ) q^{56} + ( 198 - 90 \beta ) q^{58} + ( -460 + 116 \beta ) q^{59} + ( -250 + 72 \beta ) q^{61} + ( -992 - 128 \beta ) q^{62} + ( 631 + 27 \beta ) q^{64} + ( -20 - 108 \beta ) q^{67} + ( -98 - 26 \beta ) q^{68} + ( -492 + 30 \beta ) q^{71} + ( -530 - 12 \beta ) q^{73} + ( -610 - 178 \beta ) q^{74} + ( -700 - 108 \beta ) q^{76} + ( -56 + 70 \beta ) q^{77} + ( -232 - 108 \beta ) q^{79} + ( -630 - 270 \beta ) q^{82} + ( 924 + 96 \beta ) q^{83} + ( -460 + 116 \beta ) q^{86} + ( 372 + 420 \beta ) q^{88} + ( -254 + 142 \beta ) q^{89} + ( 98 - 84 \beta ) q^{91} + ( -1288 - 280 \beta ) q^{92} + ( -912 - 96 \beta ) q^{94} + ( -266 - 276 \beta ) q^{97} + ( -49 - 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 17q^{4} - 14q^{7} - 87q^{8} + O(q^{10}) \) \( 2q - 3q^{2} + 17q^{4} - 14q^{7} - 87q^{8} + 6q^{11} - 16q^{13} + 21q^{14} + 137q^{16} - 6q^{17} + 64q^{19} + 276q^{22} + 6q^{23} - 318q^{26} - 119q^{28} + 252q^{29} + 40q^{31} - 279q^{32} + 66q^{34} + 248q^{37} + 588q^{38} + 450q^{41} - 376q^{43} - 804q^{44} + 960q^{46} - 12q^{47} + 98q^{49} + 890q^{52} - 1104q^{53} + 609q^{56} + 306q^{58} - 804q^{59} - 428q^{61} - 2112q^{62} + 1289q^{64} - 148q^{67} - 222q^{68} - 954q^{71} - 1072q^{73} - 1398q^{74} - 1508q^{76} - 42q^{77} - 572q^{79} - 1530q^{82} + 1944q^{83} - 804q^{86} + 1164q^{88} - 366q^{89} + 112q^{91} - 2856q^{92} - 1920q^{94} - 808q^{97} - 147q^{98} + O(q^{100}) \)

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
4.27492
−3.27492
−5.27492 0 19.8248 0 0 −7.00000 −62.3746 0 0
1.2 2.27492 0 −2.82475 0 0 −7.00000 −24.6254 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.p 2
3.b odd 2 1 525.4.a.n 2
5.b even 2 1 63.4.a.e 2
15.d odd 2 1 21.4.a.c 2
15.e even 4 2 525.4.d.g 4
20.d odd 2 1 1008.4.a.ba 2
35.c odd 2 1 441.4.a.r 2
35.i odd 6 2 441.4.e.p 4
35.j even 6 2 441.4.e.q 4
60.h even 2 1 336.4.a.m 2
105.g even 2 1 147.4.a.i 2
105.o odd 6 2 147.4.e.l 4
105.p even 6 2 147.4.e.m 4
120.i odd 2 1 1344.4.a.bg 2
120.m even 2 1 1344.4.a.bo 2
420.o odd 2 1 2352.4.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 15.d odd 2 1
63.4.a.e 2 5.b even 2 1
147.4.a.i 2 105.g even 2 1
147.4.e.l 4 105.o odd 6 2
147.4.e.m 4 105.p even 6 2
336.4.a.m 2 60.h even 2 1
441.4.a.r 2 35.c odd 2 1
441.4.e.p 4 35.i odd 6 2
441.4.e.q 4 35.j even 6 2
525.4.a.n 2 3.b odd 2 1
525.4.d.g 4 15.e even 4 2
1008.4.a.ba 2 20.d odd 2 1
1344.4.a.bg 2 120.i odd 2 1
1344.4.a.bo 2 120.m even 2 1
1575.4.a.p 2 1.a even 1 1 trivial
2352.4.a.bz 2 420.o odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 3 T_{2} - 12 \)
\( T_{11}^{2} - 6 T_{11} - 1416 \)
\( T_{13}^{2} + 16 T_{13} - 1988 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -12 + 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 7 + T )^{2} \)
$11$ \( -1416 - 6 T + T^{2} \)
$13$ \( -1988 + 16 T + T^{2} \)
$17$ \( -48 + 6 T + T^{2} \)
$19$ \( -7184 - 64 T + T^{2} \)
$23$ \( -16464 - 6 T + T^{2} \)
$29$ \( 7668 - 252 T + T^{2} \)
$31$ \( -73472 - 40 T + T^{2} \)
$37$ \( -3092 - 248 T + T^{2} \)
$41$ \( 37800 - 450 T + T^{2} \)
$43$ \( 2512 + 376 T + T^{2} \)
$47$ \( -65856 + 12 T + T^{2} \)
$53$ \( 304476 + 1104 T + T^{2} \)
$59$ \( -30144 + 804 T + T^{2} \)
$61$ \( -28076 + 428 T + T^{2} \)
$67$ \( -160736 + 148 T + T^{2} \)
$71$ \( 214704 + 954 T + T^{2} \)
$73$ \( 285244 + 1072 T + T^{2} \)
$79$ \( -84416 + 572 T + T^{2} \)
$83$ \( 813456 - 1944 T + T^{2} \)
$89$ \( -253848 + 366 T + T^{2} \)
$97$ \( -922292 + 808 T + T^{2} \)
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