Properties

Label 575.2.a.h
Level $575$
Weight $2$
Character orbit 575.a
Self dual yes
Analytic conductor $4.591$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{8} + (\beta_{2} + 2) q^{9} + ( - 2 \beta_1 + 2) q^{11} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{12} + (2 \beta_{3} - \beta_{2} - 1) q^{13} + ( - 2 \beta_1 - 2) q^{14} + (2 \beta_{3} + 3 \beta_1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{3} - 3 \beta_1 + 1) q^{18} + (2 \beta_1 - 2) q^{19} + (2 \beta_{3} + 2) q^{21} + (2 \beta_{2} + 6) q^{22} + q^{23} + ( - 3 \beta_{2} - 4 \beta_1 - 7) q^{24} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{26} + ( - \beta_{2} + 3) q^{27} + (2 \beta_{3} + 2 \beta_1 + 4) q^{28} + (\beta_{3} + \beta_1 + 4) q^{29} + ( - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{31} + ( - 3 \beta_{2} - 3 \beta_1 - 5) q^{32} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{33} + ( - 2 \beta_{2} - 4) q^{34} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 + 5) q^{36} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{37} + ( - 2 \beta_{2} - 6) q^{38} + ( - 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 1) q^{39} + ( - \beta_{3} - \beta_1 + 4) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{42} + ( - 2 \beta_{3} + 2) q^{43} + ( - 2 \beta_{3} - 4 \beta_1 - 2) q^{44} - \beta_1 q^{46} + (\beta_{2} - 4 \beta_1 + 1) q^{47} + (\beta_{3} + 2 \beta_{2} + 10 \beta_1 + 1) q^{48} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{49} + (4 \beta_1 - 4) q^{51} + ( - \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{52} + (\beta_{3} + \beta_{2} + 5 \beta_1 - 7) q^{53} + (\beta_{3} - 2 \beta_1 - 1) q^{54} + ( - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 2) q^{56} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{57} + ( - \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 3) q^{58} + (3 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{61} + (\beta_{3} + 4 \beta_{2} + 5) q^{62} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{63} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 6) q^{64} + (6 \beta_{2} + 14) q^{66} + (3 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{67} + (2 \beta_{2} + 4 \beta_1) q^{68} + (\beta_{2} + 1) q^{69} + (3 \beta_{3} - 4 \beta_{2} + \beta_1 - 4) q^{71} + ( - \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 9) q^{72} + ( - 2 \beta_{3} + \beta_{2} + 9) q^{73} + (4 \beta_{3} + 4 \beta_1 - 6) q^{74} + (2 \beta_{3} + 4 \beta_1 + 2) q^{76} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 - 11) q^{78} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{79} - 7 q^{81} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{82} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{83} + (6 \beta_{2} + 8 \beta_1 + 6) q^{84} + (2 \beta_{3} + 2 \beta_{2}) q^{86} + (5 \beta_{2} + 4 \beta_1 + 5) q^{87} + (2 \beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{88} + ( - 4 \beta_{3} + 2 \beta_1) q^{89} + (6 \beta_{2} + 2 \beta_1 - 8) q^{91} + (\beta_{2} + \beta_1 + 1) q^{92} + (2 \beta_{3} - \beta_{2} - 8 \beta_1 + 5) q^{93} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 13) q^{94} + ( - 3 \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 14) q^{96} + ( - 2 \beta_{3} + 2 \beta_{2} + 6) q^{97} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{98} + ( - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} + 4 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} + 4 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 19 q^{12} - 12 q^{14} + 8 q^{16} + q^{17} - 3 q^{18} - 4 q^{19} + 10 q^{21} + 20 q^{22} + 4 q^{23} - 30 q^{24} - q^{26} + 14 q^{27} + 22 q^{28} + 19 q^{29} - q^{31} - 20 q^{32} + 2 q^{33} - 12 q^{34} + 23 q^{36} + 3 q^{37} - 20 q^{38} + 13 q^{41} - 6 q^{42} + 6 q^{43} - 18 q^{44} - 2 q^{46} - 6 q^{47} + 21 q^{48} + 9 q^{49} - 8 q^{51} + q^{52} - 19 q^{53} - 7 q^{54} - 10 q^{56} - 2 q^{57} - 21 q^{58} + 23 q^{59} + 13 q^{62} + 13 q^{63} + 27 q^{64} + 44 q^{66} + 3 q^{67} + 4 q^{68} + 2 q^{69} - 3 q^{71} - 39 q^{72} + 32 q^{73} - 12 q^{74} + 18 q^{76} - 18 q^{77} - 43 q^{78} + 2 q^{79} - 28 q^{81} + 5 q^{82} + 21 q^{83} + 28 q^{84} - 2 q^{86} + 18 q^{87} + 14 q^{88} - 40 q^{91} + 4 q^{92} + 8 q^{93} + 47 q^{94} - 61 q^{96} + 18 q^{97} - 16 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display

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
2.69353
1.32973
−0.329727
−1.69353
−2.69353 2.56155 5.25508 0 −6.89961 2.74252 −8.76763 3.56155 0
1.2 −1.32973 −1.56155 −0.231826 0 2.07644 3.50407 2.96772 −0.561553 0
1.3 0.329727 −1.56155 −1.89128 0 −0.514886 −4.06562 −1.28306 −0.561553 0
1.4 1.69353 2.56155 0.868028 0 4.33805 0.819031 −1.91702 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(23\) \(-1\)

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 575.2.a.h 4
3.b odd 2 1 5175.2.a.bx 4
4.b odd 2 1 9200.2.a.cl 4
5.b even 2 1 115.2.a.c 4
5.c odd 4 2 575.2.b.e 8
15.d odd 2 1 1035.2.a.o 4
20.d odd 2 1 1840.2.a.u 4
35.c odd 2 1 5635.2.a.v 4
40.e odd 2 1 7360.2.a.cg 4
40.f even 2 1 7360.2.a.cj 4
115.c odd 2 1 2645.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.c 4 5.b even 2 1
575.2.a.h 4 1.a even 1 1 trivial
575.2.b.e 8 5.c odd 4 2
1035.2.a.o 4 15.d odd 2 1
1840.2.a.u 4 20.d odd 2 1
2645.2.a.m 4 115.c odd 2 1
5175.2.a.bx 4 3.b odd 2 1
5635.2.a.v 4 35.c odd 2 1
7360.2.a.cg 4 40.e odd 2 1
7360.2.a.cj 4 40.f even 2 1
9200.2.a.cl 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(575))\):

\( T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 5T_{2} + 2 \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} - 4 T^{2} - 5 T + 2 \) Copy content Toggle raw display
$3$ \( (T^{2} - T - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} - 14 T^{2} + 52 T - 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} - 16 T^{2} + 40 T + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 41T^{2} + 212 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} - 18 T^{2} + 24 T + 32 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} - 16 T^{2} - 40 T + 32 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 19 T^{3} + 117 T^{2} + \cdots + 202 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} - 101 T^{2} + 11 T + 2144 \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} - 116 T^{2} + \cdots + 2008 \) Copy content Toggle raw display
$41$ \( T^{4} - 13 T^{3} + 45 T^{2} - 3 T - 94 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} - 36 T^{2} + 16 T + 128 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} - 83 T^{2} - 548 T - 128 \) Copy content Toggle raw display
$53$ \( T^{4} + 19 T^{3} - 34 T^{2} + \cdots - 8776 \) Copy content Toggle raw display
$59$ \( T^{4} - 23 T^{3} + 100 T^{2} + \cdots - 3136 \) Copy content Toggle raw display
$61$ \( T^{4} - 56 T^{2} + 136 T - 32 \) Copy content Toggle raw display
$67$ \( T^{4} - 3 T^{3} - 98 T^{2} + \cdots + 2032 \) Copy content Toggle raw display
$71$ \( T^{4} + 3 T^{3} - 149 T^{2} - 535 T - 8 \) Copy content Toggle raw display
$73$ \( T^{4} - 32 T^{3} + 343 T^{2} + \cdots + 1684 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} - 140 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$83$ \( T^{4} - 21 T^{3} + 96 T^{2} + \cdots - 1216 \) Copy content Toggle raw display
$89$ \( T^{4} - 216 T^{2} - 1496 T - 2752 \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} + 72 T^{2} + \cdots - 1072 \) Copy content Toggle raw display
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