Properties

Label 1575.4.a.u
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -4 + \beta ) q^{4} + 7 q^{7} + ( 4 - 11 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -4 + \beta ) q^{4} + 7 q^{7} + ( 4 - 11 \beta ) q^{8} + ( 4 - 4 \beta ) q^{11} + 22 \beta q^{13} + 7 \beta q^{14} + ( -12 - 15 \beta ) q^{16} + ( -42 + 26 \beta ) q^{17} + ( 22 - 44 \beta ) q^{19} -16 q^{22} + ( -34 - 14 \beta ) q^{23} + ( 88 + 22 \beta ) q^{26} + ( -28 + 7 \beta ) q^{28} + ( 152 + 30 \beta ) q^{29} + ( -114 + 18 \beta ) q^{31} + ( -92 + 61 \beta ) q^{32} + ( 104 - 16 \beta ) q^{34} + ( -18 + 30 \beta ) q^{37} + ( -176 - 22 \beta ) q^{38} + ( 114 - 52 \beta ) q^{41} + ( 60 - 166 \beta ) q^{43} + ( -32 + 16 \beta ) q^{44} + ( -56 - 48 \beta ) q^{46} + ( -272 + 30 \beta ) q^{47} + 49 q^{49} + ( 88 - 66 \beta ) q^{52} + ( -378 - 52 \beta ) q^{53} + ( 28 - 77 \beta ) q^{56} + ( 120 + 182 \beta ) q^{58} + 284 q^{59} + ( -354 + 90 \beta ) q^{61} + ( 72 - 96 \beta ) q^{62} + ( 340 + 89 \beta ) q^{64} + ( -396 + 98 \beta ) q^{67} + ( 272 - 120 \beta ) q^{68} + ( 488 - 162 \beta ) q^{71} + ( 104 - 290 \beta ) q^{73} + ( 120 + 12 \beta ) q^{74} + ( -264 + 154 \beta ) q^{76} + ( 28 - 28 \beta ) q^{77} + ( 136 + 328 \beta ) q^{79} + ( -208 + 62 \beta ) q^{82} + ( 284 - 300 \beta ) q^{83} + ( -664 - 106 \beta ) q^{86} + ( 192 - 16 \beta ) q^{88} + ( 202 - 476 \beta ) q^{89} + 154 \beta q^{91} + ( 80 + 8 \beta ) q^{92} + ( 120 - 242 \beta ) q^{94} + ( -572 - 482 \beta ) q^{97} + 49 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8} + O(q^{10}) \) \( 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8} + 4 q^{11} + 22 q^{13} + 7 q^{14} - 39 q^{16} - 58 q^{17} - 32 q^{22} - 82 q^{23} + 198 q^{26} - 49 q^{28} + 334 q^{29} - 210 q^{31} - 123 q^{32} + 192 q^{34} - 6 q^{37} - 374 q^{38} + 176 q^{41} - 46 q^{43} - 48 q^{44} - 160 q^{46} - 514 q^{47} + 98 q^{49} + 110 q^{52} - 808 q^{53} - 21 q^{56} + 422 q^{58} + 568 q^{59} - 618 q^{61} + 48 q^{62} + 769 q^{64} - 694 q^{67} + 424 q^{68} + 814 q^{71} - 82 q^{73} + 252 q^{74} - 374 q^{76} + 28 q^{77} + 600 q^{79} - 354 q^{82} + 268 q^{83} - 1434 q^{86} + 368 q^{88} - 72 q^{89} + 154 q^{91} + 168 q^{92} - 2 q^{94} - 1626 q^{97} + 49 q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
−1.56155 0 −5.56155 0 0 7.00000 21.1771 0 0
1.2 2.56155 0 −1.43845 0 0 7.00000 −24.1771 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.u 2
3.b odd 2 1 1575.4.a.r 2
5.b even 2 1 315.4.a.h 2
15.d odd 2 1 315.4.a.j yes 2
35.c odd 2 1 2205.4.a.y 2
105.g even 2 1 2205.4.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.h 2 5.b even 2 1
315.4.a.j yes 2 15.d odd 2 1
1575.4.a.r 2 3.b odd 2 1
1575.4.a.u 2 1.a even 1 1 trivial
2205.4.a.y 2 35.c odd 2 1
2205.4.a.ba 2 105.g even 2 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 4 \)
\( T_{11}^{2} - 4 T_{11} - 64 \)
\( T_{13}^{2} - 22 T_{13} - 1936 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -64 - 4 T + T^{2} \)
$13$ \( -1936 - 22 T + T^{2} \)
$17$ \( -2032 + 58 T + T^{2} \)
$19$ \( -8228 + T^{2} \)
$23$ \( 848 + 82 T + T^{2} \)
$29$ \( 24064 - 334 T + T^{2} \)
$31$ \( 9648 + 210 T + T^{2} \)
$37$ \( -3816 + 6 T + T^{2} \)
$41$ \( -3748 - 176 T + T^{2} \)
$43$ \( -116584 + 46 T + T^{2} \)
$47$ \( 62224 + 514 T + T^{2} \)
$53$ \( 151724 + 808 T + T^{2} \)
$59$ \( ( -284 + T )^{2} \)
$61$ \( 61056 + 618 T + T^{2} \)
$67$ \( 79592 + 694 T + T^{2} \)
$71$ \( 54112 - 814 T + T^{2} \)
$73$ \( -355744 + 82 T + T^{2} \)
$79$ \( -367232 - 600 T + T^{2} \)
$83$ \( -364544 - 268 T + T^{2} \)
$89$ \( -961652 + 72 T + T^{2} \)
$97$ \( -326408 + 1626 T + T^{2} \)
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