Properties

Label 1575.4.a.w
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: \(\Q(\sqrt{65}) \)
Defining polynomial: \(x^{2} - x - 16\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 8 + \beta ) q^{4} + 7 q^{7} + ( 16 + \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 8 + \beta ) q^{4} + 7 q^{7} + ( 16 + \beta ) q^{8} + ( 10 + 2 \beta ) q^{11} + ( 12 - 2 \beta ) q^{13} + 7 \beta q^{14} + ( -48 + 9 \beta ) q^{16} + ( 66 - 16 \beta ) q^{17} + ( 58 - 14 \beta ) q^{19} + ( 32 + 12 \beta ) q^{22} + ( 140 - 20 \beta ) q^{23} + ( -32 + 10 \beta ) q^{26} + ( 56 + 7 \beta ) q^{28} + ( 74 + 48 \beta ) q^{29} + ( 54 + 42 \beta ) q^{31} + ( 16 - 47 \beta ) q^{32} + ( -256 + 50 \beta ) q^{34} + ( 30 + 36 \beta ) q^{37} + ( -224 + 44 \beta ) q^{38} + ( 138 - 100 \beta ) q^{41} + ( 156 + 32 \beta ) q^{43} + ( 112 + 28 \beta ) q^{44} + ( -320 + 120 \beta ) q^{46} + ( 304 - 48 \beta ) q^{47} + 49 q^{49} + ( 64 - 6 \beta ) q^{52} + ( 120 + 86 \beta ) q^{53} + ( 112 + 7 \beta ) q^{56} + ( 768 + 122 \beta ) q^{58} + ( 464 - 84 \beta ) q^{59} + ( -114 + 24 \beta ) q^{61} + ( 672 + 96 \beta ) q^{62} + ( -368 - 103 \beta ) q^{64} + ( 84 - 64 \beta ) q^{67} + ( 272 - 78 \beta ) q^{68} + ( -814 - 42 \beta ) q^{71} + ( 92 + 202 \beta ) q^{73} + ( 576 + 66 \beta ) q^{74} + ( 240 - 68 \beta ) q^{76} + ( 70 + 14 \beta ) q^{77} + ( -392 - 104 \beta ) q^{79} + ( -1600 + 38 \beta ) q^{82} + ( 356 + 216 \beta ) q^{83} + ( 512 + 188 \beta ) q^{86} + ( 192 + 44 \beta ) q^{88} + ( -290 - 8 \beta ) q^{89} + ( 84 - 14 \beta ) q^{91} + ( 800 - 40 \beta ) q^{92} + ( -768 + 256 \beta ) q^{94} + ( -104 - 314 \beta ) q^{97} + 49 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8} + O(q^{10}) \) \( 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8} + 22 q^{11} + 22 q^{13} + 7 q^{14} - 87 q^{16} + 116 q^{17} + 102 q^{19} + 76 q^{22} + 260 q^{23} - 54 q^{26} + 119 q^{28} + 196 q^{29} + 150 q^{31} - 15 q^{32} - 462 q^{34} + 96 q^{37} - 404 q^{38} + 176 q^{41} + 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} + 98 q^{49} + 122 q^{52} + 326 q^{53} + 231 q^{56} + 1658 q^{58} + 844 q^{59} - 204 q^{61} + 1440 q^{62} - 839 q^{64} + 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} + 412 q^{76} + 154 q^{77} - 888 q^{79} - 3162 q^{82} + 928 q^{83} + 1212 q^{86} + 428 q^{88} - 588 q^{89} + 154 q^{91} + 1560 q^{92} - 1280 q^{94} - 522 q^{97} + 49 q^{98} + O(q^{100}) \)

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
−3.53113
4.53113
−3.53113 0 4.46887 0 0 7.00000 12.4689 0 0
1.2 4.53113 0 12.5311 0 0 7.00000 20.5311 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.w 2
3.b odd 2 1 525.4.a.k 2
5.b even 2 1 315.4.a.i 2
15.d odd 2 1 105.4.a.f 2
15.e even 4 2 525.4.d.h 4
35.c odd 2 1 2205.4.a.z 2
60.h even 2 1 1680.4.a.bg 2
105.g even 2 1 735.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.f 2 15.d odd 2 1
315.4.a.i 2 5.b even 2 1
525.4.a.k 2 3.b odd 2 1
525.4.d.h 4 15.e even 4 2
735.4.a.p 2 105.g even 2 1
1575.4.a.w 2 1.a even 1 1 trivial
1680.4.a.bg 2 60.h even 2 1
2205.4.a.z 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} - T_{2} - 16 \)
\( T_{11}^{2} - 22 T_{11} + 56 \)
\( T_{13}^{2} - 22 T_{13} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -16 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( 56 - 22 T + T^{2} \)
$13$ \( 56 - 22 T + T^{2} \)
$17$ \( -796 - 116 T + T^{2} \)
$19$ \( -584 - 102 T + T^{2} \)
$23$ \( 10400 - 260 T + T^{2} \)
$29$ \( -27836 - 196 T + T^{2} \)
$31$ \( -23040 - 150 T + T^{2} \)
$37$ \( -18756 - 96 T + T^{2} \)
$41$ \( -154756 - 176 T + T^{2} \)
$43$ \( 12944 - 344 T + T^{2} \)
$47$ \( 40960 - 560 T + T^{2} \)
$53$ \( -93616 - 326 T + T^{2} \)
$59$ \( 63424 - 844 T + T^{2} \)
$61$ \( 1044 + 204 T + T^{2} \)
$67$ \( -63856 - 104 T + T^{2} \)
$71$ \( 668560 + 1670 T + T^{2} \)
$73$ \( -625816 - 386 T + T^{2} \)
$79$ \( 21376 + 888 T + T^{2} \)
$83$ \( -542864 - 928 T + T^{2} \)
$89$ \( 85396 + 588 T + T^{2} \)
$97$ \( -1534064 + 522 T + T^{2} \)
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