Properties

Label 1575.4.a.m
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - \beta ) q^{2} + ( 5 + 7 \beta ) q^{4} + 7 q^{7} + ( -19 - 25 \beta ) q^{8} +O(q^{10})\) \( q + ( -3 - \beta ) q^{2} + ( 5 + 7 \beta ) q^{4} + 7 q^{7} + ( -19 - 25 \beta ) q^{8} + ( 8 + 10 \beta ) q^{11} + ( -18 + 22 \beta ) q^{13} + ( -21 - 7 \beta ) q^{14} + ( 117 + 63 \beta ) q^{16} + ( -6 + 28 \beta ) q^{17} + ( 100 - 26 \beta ) q^{19} + ( -64 - 48 \beta ) q^{22} + ( 64 + 56 \beta ) q^{23} + ( -34 - 70 \beta ) q^{26} + ( 35 + 49 \beta ) q^{28} + ( -26 + 84 \beta ) q^{29} + ( 156 + 18 \beta ) q^{31} + ( -451 - 169 \beta ) q^{32} + ( -94 - 106 \beta ) q^{34} + ( 78 - 24 \beta ) q^{37} + ( -196 + 4 \beta ) q^{38} + ( -30 - 140 \beta ) q^{41} + ( -216 + 68 \beta ) q^{43} + ( 320 + 176 \beta ) q^{44} + ( -416 - 288 \beta ) q^{46} + ( 92 + 108 \beta ) q^{47} + 49 q^{49} + ( 526 + 138 \beta ) q^{52} + ( -90 + 214 \beta ) q^{53} + ( -133 - 175 \beta ) q^{56} + ( -258 - 310 \beta ) q^{58} + ( -164 - 36 \beta ) q^{59} + ( 522 - 252 \beta ) q^{61} + ( -540 - 228 \beta ) q^{62} + ( 1093 + 623 \beta ) q^{64} + ( 408 - 28 \beta ) q^{67} + ( 754 + 294 \beta ) q^{68} + ( -392 + 330 \beta ) q^{71} + ( -478 + 178 \beta ) q^{73} + ( -138 + 18 \beta ) q^{74} + ( -228 + 388 \beta ) q^{76} + ( 56 + 70 \beta ) q^{77} + ( 160 + 88 \beta ) q^{79} + ( 650 + 590 \beta ) q^{82} + ( 700 - 264 \beta ) q^{83} + ( 376 - 56 \beta ) q^{86} + ( -1152 - 640 \beta ) q^{88} + ( -382 + 728 \beta ) q^{89} + ( -126 + 154 \beta ) q^{91} + ( 1888 + 1120 \beta ) q^{92} + ( -708 - 524 \beta ) q^{94} + ( 322 - 146 \beta ) q^{97} + ( -147 - 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + O(q^{10}) \) \( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32} - 294 q^{34} + 132 q^{37} - 388 q^{38} - 200 q^{41} - 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 98 q^{49} + 1190 q^{52} + 34 q^{53} - 441 q^{56} - 826 q^{58} - 364 q^{59} + 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} - 68 q^{76} + 182 q^{77} + 408 q^{79} + 1890 q^{82} + 1136 q^{83} + 696 q^{86} - 2944 q^{88} - 36 q^{89} - 98 q^{91} + 4896 q^{92} - 1940 q^{94} + 498 q^{97} - 343 q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
−5.56155 0 22.9309 0 0 7.00000 −83.0388 0 0
1.2 −1.43845 0 −5.93087 0 0 7.00000 20.0388 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.m 2
3.b odd 2 1 525.4.a.p 2
5.b even 2 1 315.4.a.m 2
15.d odd 2 1 105.4.a.c 2
15.e even 4 2 525.4.d.i 4
35.c odd 2 1 2205.4.a.bh 2
60.h even 2 1 1680.4.a.bk 2
105.g even 2 1 735.4.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 15.d odd 2 1
315.4.a.m 2 5.b even 2 1
525.4.a.p 2 3.b odd 2 1
525.4.d.i 4 15.e even 4 2
735.4.a.k 2 105.g even 2 1
1575.4.a.m 2 1.a even 1 1 trivial
1680.4.a.bk 2 60.h even 2 1
2205.4.a.bh 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + 7 T_{2} + 8 \)
\( T_{11}^{2} - 26 T_{11} - 256 \)
\( T_{13}^{2} + 14 T_{13} - 2008 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + 7 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -256 - 26 T + T^{2} \)
$13$ \( -2008 + 14 T + T^{2} \)
$17$ \( -3268 - 16 T + T^{2} \)
$19$ \( 4696 - 174 T + T^{2} \)
$23$ \( -4864 - 184 T + T^{2} \)
$29$ \( -29732 - 32 T + T^{2} \)
$31$ \( 25848 - 330 T + T^{2} \)
$37$ \( 1908 - 132 T + T^{2} \)
$41$ \( -73300 + 200 T + T^{2} \)
$43$ \( 13472 + 364 T + T^{2} \)
$47$ \( -28256 - 292 T + T^{2} \)
$53$ \( -194344 - 34 T + T^{2} \)
$59$ \( 27616 + 364 T + T^{2} \)
$61$ \( -113076 - 792 T + T^{2} \)
$67$ \( 151904 - 788 T + T^{2} \)
$71$ \( -411296 + 454 T + T^{2} \)
$73$ \( 16664 + 778 T + T^{2} \)
$79$ \( 8704 - 408 T + T^{2} \)
$83$ \( 26416 - 1136 T + T^{2} \)
$89$ \( -2252108 + 36 T + T^{2} \)
$97$ \( -28592 - 498 T + T^{2} \)
show more
show less