Properties

Label 1575.4.a.s
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
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.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + 7 q^{7} + ( -10 + 5 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + 7 q^{7} + ( -10 + 5 \beta ) q^{8} + ( 23 + 10 \beta ) q^{11} + ( 30 - 8 \beta ) q^{13} -7 \beta q^{14} + ( -66 - 3 \beta ) q^{16} + ( 54 - 5 \beta ) q^{17} + ( -32 + 7 \beta ) q^{19} + ( -100 - 33 \beta ) q^{22} + ( 13 + 5 \beta ) q^{23} + ( 80 - 22 \beta ) q^{26} + ( 14 + 7 \beta ) q^{28} + ( 193 + 27 \beta ) q^{29} + ( -114 + 66 \beta ) q^{31} + ( 110 + 29 \beta ) q^{32} + ( 50 - 49 \beta ) q^{34} + ( 9 + 57 \beta ) q^{37} + ( -70 + 25 \beta ) q^{38} + ( 222 + 61 \beta ) q^{41} + ( -75 + 77 \beta ) q^{43} + ( 146 + 53 \beta ) q^{44} + ( -50 - 18 \beta ) q^{46} + ( -58 - 108 \beta ) q^{47} + 49 q^{49} + ( -20 + 6 \beta ) q^{52} + ( 174 - 86 \beta ) q^{53} + ( -70 + 35 \beta ) q^{56} + ( -270 - 220 \beta ) q^{58} + ( 10 - 210 \beta ) q^{59} + ( 468 + 54 \beta ) q^{61} + ( -660 + 48 \beta ) q^{62} + ( 238 - 115 \beta ) q^{64} + ( 537 - 166 \beta ) q^{67} + ( 58 + 39 \beta ) q^{68} + ( -215 + 303 \beta ) q^{71} + ( -34 - 269 \beta ) q^{73} + ( -570 - 66 \beta ) q^{74} + ( 6 - 11 \beta ) q^{76} + ( 161 + 70 \beta ) q^{77} + ( -539 - 41 \beta ) q^{79} + ( -610 - 283 \beta ) q^{82} + ( 724 + 69 \beta ) q^{83} + ( -770 - 2 \beta ) q^{86} + ( 270 + 65 \beta ) q^{88} + ( 848 + 17 \beta ) q^{89} + ( 210 - 56 \beta ) q^{91} + ( 76 + 28 \beta ) q^{92} + ( 1080 + 166 \beta ) q^{94} + ( 1018 - 272 \beta ) q^{97} -49 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8} + O(q^{10}) \) \( 2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8} + 56 q^{11} + 52 q^{13} - 7 q^{14} - 135 q^{16} + 103 q^{17} - 57 q^{19} - 233 q^{22} + 31 q^{23} + 138 q^{26} + 35 q^{28} + 413 q^{29} - 162 q^{31} + 249 q^{32} + 51 q^{34} + 75 q^{37} - 115 q^{38} + 505 q^{41} - 73 q^{43} + 345 q^{44} - 118 q^{46} - 224 q^{47} + 98 q^{49} - 34 q^{52} + 262 q^{53} - 105 q^{56} - 760 q^{58} - 190 q^{59} + 990 q^{61} - 1272 q^{62} + 361 q^{64} + 908 q^{67} + 155 q^{68} - 127 q^{71} - 337 q^{73} - 1206 q^{74} + q^{76} + 392 q^{77} - 1119 q^{79} - 1503 q^{82} + 1517 q^{83} - 1542 q^{86} + 605 q^{88} + 1713 q^{89} + 364 q^{91} + 180 q^{92} + 2326 q^{94} + 1764 q^{97} - 49 q^{98} + O(q^{100}) \)

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
3.70156
−2.70156
−3.70156 0 5.70156 0 0 7.00000 8.50781 0 0
1.2 2.70156 0 −0.701562 0 0 7.00000 −23.5078 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.s 2
3.b odd 2 1 175.4.a.e yes 2
5.b even 2 1 1575.4.a.v 2
15.d odd 2 1 175.4.a.d 2
15.e even 4 2 175.4.b.d 4
21.c even 2 1 1225.4.a.t 2
105.g even 2 1 1225.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.d 2 15.d odd 2 1
175.4.a.e yes 2 3.b odd 2 1
175.4.b.d 4 15.e even 4 2
1225.4.a.r 2 105.g even 2 1
1225.4.a.t 2 21.c even 2 1
1575.4.a.s 2 1.a even 1 1 trivial
1575.4.a.v 2 5.b even 2 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} - 10 \)
\( T_{11}^{2} - 56 T_{11} - 241 \)
\( T_{13}^{2} - 52 T_{13} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -10 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -241 - 56 T + T^{2} \)
$13$ \( 20 - 52 T + T^{2} \)
$17$ \( 2396 - 103 T + T^{2} \)
$19$ \( 310 + 57 T + T^{2} \)
$23$ \( -16 - 31 T + T^{2} \)
$29$ \( 35170 - 413 T + T^{2} \)
$31$ \( -38088 + 162 T + T^{2} \)
$37$ \( -31896 - 75 T + T^{2} \)
$41$ \( 25616 - 505 T + T^{2} \)
$43$ \( -59440 + 73 T + T^{2} \)
$47$ \( -107012 + 224 T + T^{2} \)
$53$ \( -58648 - 262 T + T^{2} \)
$59$ \( -443000 + 190 T + T^{2} \)
$61$ \( 215136 - 990 T + T^{2} \)
$67$ \( -76333 - 908 T + T^{2} \)
$71$ \( -937010 + 127 T + T^{2} \)
$73$ \( -713308 + 337 T + T^{2} \)
$79$ \( 295810 + 1119 T + T^{2} \)
$83$ \( 526522 - 1517 T + T^{2} \)
$89$ \( 730630 - 1713 T + T^{2} \)
$97$ \( 19588 - 1764 T + T^{2} \)
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