Properties

Label 1575.4.a.y
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 3 + 3 \beta ) q^{4} -7 q^{7} + ( 25 + \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 3 + 3 \beta ) q^{4} -7 q^{7} + ( 25 + \beta ) q^{8} + ( -32 + 2 \beta ) q^{11} + ( -2 + 10 \beta ) q^{13} + ( -7 - 7 \beta ) q^{14} + ( 11 + 3 \beta ) q^{16} + ( 26 - 12 \beta ) q^{17} + ( -60 - 2 \beta ) q^{19} + ( -12 - 28 \beta ) q^{22} + ( 32 - 48 \beta ) q^{23} + ( 98 + 18 \beta ) q^{26} + ( -21 - 21 \beta ) q^{28} + ( -170 - 12 \beta ) q^{29} + ( 52 - 38 \beta ) q^{31} + ( -159 + 9 \beta ) q^{32} + ( -94 + 2 \beta ) q^{34} + ( 134 - 80 \beta ) q^{37} + ( -80 - 64 \beta ) q^{38} + ( -62 + 108 \beta ) q^{41} + ( 248 - 100 \beta ) q^{43} + ( -36 - 84 \beta ) q^{44} + ( -448 - 64 \beta ) q^{46} + ( -164 + 140 \beta ) q^{47} + 49 q^{49} + ( 294 + 54 \beta ) q^{52} + ( 462 + 58 \beta ) q^{53} + ( -175 - 7 \beta ) q^{56} + ( -290 - 194 \beta ) q^{58} + ( -220 - 76 \beta ) q^{59} + ( -398 - 84 \beta ) q^{61} + ( -328 - 24 \beta ) q^{62} + ( -157 - 165 \beta ) q^{64} + ( 64 + 228 \beta ) q^{67} + ( -282 + 6 \beta ) q^{68} + ( -112 - 86 \beta ) q^{71} + ( -182 + 38 \beta ) q^{73} + ( -666 - 26 \beta ) q^{74} + ( -240 - 192 \beta ) q^{76} + ( 224 - 14 \beta ) q^{77} + ( 920 - 8 \beta ) q^{79} + ( 1018 + 154 \beta ) q^{82} + ( -228 - 224 \beta ) q^{83} + ( -752 + 48 \beta ) q^{86} + ( -780 + 20 \beta ) q^{88} + ( -230 - 336 \beta ) q^{89} + ( 14 - 70 \beta ) q^{91} + ( -1344 - 192 \beta ) q^{92} + ( 1236 + 116 \beta ) q^{94} + ( 474 - 278 \beta ) q^{97} + ( 49 + 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8} - 62 q^{11} + 6 q^{13} - 21 q^{14} + 25 q^{16} + 40 q^{17} - 122 q^{19} - 52 q^{22} + 16 q^{23} + 214 q^{26} - 63 q^{28} - 352 q^{29} + 66 q^{31} - 309 q^{32} - 186 q^{34} + 188 q^{37} - 224 q^{38} - 16 q^{41} + 396 q^{43} - 156 q^{44} - 960 q^{46} - 188 q^{47} + 98 q^{49} + 642 q^{52} + 982 q^{53} - 357 q^{56} - 774 q^{58} - 516 q^{59} - 880 q^{61} - 680 q^{62} - 479 q^{64} + 356 q^{67} - 558 q^{68} - 310 q^{71} - 326 q^{73} - 1358 q^{74} - 672 q^{76} + 434 q^{77} + 1832 q^{79} + 2190 q^{82} - 680 q^{83} - 1456 q^{86} - 1540 q^{88} - 796 q^{89} - 42 q^{91} - 2880 q^{92} + 2588 q^{94} + 670 q^{97} + 147 q^{98} + O(q^{100}) \)

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
−2.70156
3.70156
−1.70156 0 −5.10469 0 0 −7.00000 22.2984 0 0
1.2 4.70156 0 14.1047 0 0 −7.00000 28.7016 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.y 2
3.b odd 2 1 525.4.a.i 2
5.b even 2 1 315.4.a.g 2
15.d odd 2 1 105.4.a.g 2
15.e even 4 2 525.4.d.j 4
35.c odd 2 1 2205.4.a.v 2
60.h even 2 1 1680.4.a.y 2
105.g even 2 1 735.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 15.d odd 2 1
315.4.a.g 2 5.b even 2 1
525.4.a.i 2 3.b odd 2 1
525.4.d.j 4 15.e even 4 2
735.4.a.q 2 105.g even 2 1
1575.4.a.y 2 1.a even 1 1 trivial
1680.4.a.y 2 60.h even 2 1
2205.4.a.v 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} - 3 T_{2} - 8 \)
\( T_{11}^{2} + 62 T_{11} + 920 \)
\( T_{13}^{2} - 6 T_{13} - 1016 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -8 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 7 + T )^{2} \)
$11$ \( 920 + 62 T + T^{2} \)
$13$ \( -1016 - 6 T + T^{2} \)
$17$ \( -1076 - 40 T + T^{2} \)
$19$ \( 3680 + 122 T + T^{2} \)
$23$ \( -23552 - 16 T + T^{2} \)
$29$ \( 29500 + 352 T + T^{2} \)
$31$ \( -13712 - 66 T + T^{2} \)
$37$ \( -56764 - 188 T + T^{2} \)
$41$ \( -119492 + 16 T + T^{2} \)
$43$ \( -63296 - 396 T + T^{2} \)
$47$ \( -192064 + 188 T + T^{2} \)
$53$ \( 206600 - 982 T + T^{2} \)
$59$ \( 7360 + 516 T + T^{2} \)
$61$ \( 121276 + 880 T + T^{2} \)
$67$ \( -501152 - 356 T + T^{2} \)
$71$ \( -51784 + 310 T + T^{2} \)
$73$ \( 11768 + 326 T + T^{2} \)
$79$ \( 838400 - 1832 T + T^{2} \)
$83$ \( -398704 + 680 T + T^{2} \)
$89$ \( -998780 + 796 T + T^{2} \)
$97$ \( -679936 - 670 T + T^{2} \)
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