Properties

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

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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: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 11 q^{4} + 7 q^{7} + 3 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 11 q^{4} + 7 q^{7} + 3 \beta q^{8} + 10 \beta q^{11} -82 q^{13} + 7 \beta q^{14} -31 q^{16} -18 \beta q^{17} -20 q^{19} + 190 q^{22} -30 \beta q^{23} -82 \beta q^{26} + 77 q^{28} -56 \beta q^{29} + 156 q^{31} -55 \beta q^{32} -342 q^{34} -186 q^{37} -20 \beta q^{38} + 38 \beta q^{41} -164 q^{43} + 110 \beta q^{44} -570 q^{46} + 108 \beta q^{47} + 49 q^{49} -902 q^{52} -36 \beta q^{53} + 21 \beta q^{56} -1064 q^{58} -36 \beta q^{59} + 790 q^{61} + 156 \beta q^{62} -797 q^{64} + 44 q^{67} -198 \beta q^{68} -102 \beta q^{71} -126 q^{73} -186 \beta q^{74} -220 q^{76} + 70 \beta q^{77} -712 q^{79} + 722 q^{82} -336 \beta q^{83} -164 \beta q^{86} + 570 q^{88} + 334 \beta q^{89} -574 q^{91} -330 \beta q^{92} + 2052 q^{94} -798 q^{97} + 49 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 22q^{4} + 14q^{7} + O(q^{10}) \) \( 2q + 22q^{4} + 14q^{7} - 164q^{13} - 62q^{16} - 40q^{19} + 380q^{22} + 154q^{28} + 312q^{31} - 684q^{34} - 372q^{37} - 328q^{43} - 1140q^{46} + 98q^{49} - 1804q^{52} - 2128q^{58} + 1580q^{61} - 1594q^{64} + 88q^{67} - 252q^{73} - 440q^{76} - 1424q^{79} + 1444q^{82} + 1140q^{88} - 1148q^{91} + 4104q^{94} - 1596q^{97} + O(q^{100}) \)

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
−4.35890
4.35890
−4.35890 0 11.0000 0 0 7.00000 −13.0767 0 0
1.2 4.35890 0 11.0000 0 0 7.00000 13.0767 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.4.a.t 2
3.b odd 2 1 inner 1575.4.a.t 2
5.b even 2 1 63.4.a.d 2
15.d odd 2 1 63.4.a.d 2
20.d odd 2 1 1008.4.a.be 2
35.c odd 2 1 441.4.a.q 2
35.i odd 6 2 441.4.e.s 4
35.j even 6 2 441.4.e.r 4
60.h even 2 1 1008.4.a.be 2
105.g even 2 1 441.4.a.q 2
105.o odd 6 2 441.4.e.r 4
105.p even 6 2 441.4.e.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 5.b even 2 1
63.4.a.d 2 15.d odd 2 1
441.4.a.q 2 35.c odd 2 1
441.4.a.q 2 105.g even 2 1
441.4.e.r 4 35.j even 6 2
441.4.e.r 4 105.o odd 6 2
441.4.e.s 4 35.i odd 6 2
441.4.e.s 4 105.p even 6 2
1008.4.a.be 2 20.d odd 2 1
1008.4.a.be 2 60.h even 2 1
1575.4.a.t 2 1.a even 1 1 trivial
1575.4.a.t 2 3.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{2} - 19 \)
\( T_{11}^{2} - 1900 \)
\( T_{13} + 82 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -19 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -1900 + T^{2} \)
$13$ \( ( 82 + T )^{2} \)
$17$ \( -6156 + T^{2} \)
$19$ \( ( 20 + T )^{2} \)
$23$ \( -17100 + T^{2} \)
$29$ \( -59584 + T^{2} \)
$31$ \( ( -156 + T )^{2} \)
$37$ \( ( 186 + T )^{2} \)
$41$ \( -27436 + T^{2} \)
$43$ \( ( 164 + T )^{2} \)
$47$ \( -221616 + T^{2} \)
$53$ \( -24624 + T^{2} \)
$59$ \( -24624 + T^{2} \)
$61$ \( ( -790 + T )^{2} \)
$67$ \( ( -44 + T )^{2} \)
$71$ \( -197676 + T^{2} \)
$73$ \( ( 126 + T )^{2} \)
$79$ \( ( 712 + T )^{2} \)
$83$ \( -2145024 + T^{2} \)
$89$ \( -2119564 + T^{2} \)
$97$ \( ( 798 + T )^{2} \)
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