Properties

Label 575.6.a.b
Level $575$
Weight $6$
Character orbit 575.a
Self dual yes
Analytic conductor $92.221$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.2206963925\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7925.1
Defining polynomial: \( x^{3} - x^{2} - 13x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 10 \beta_{2} + \beta_1 - 73) q^{6} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 10 \beta_{2} + \beta_1 - 73) q^{6} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9} + (28 \beta_{2} + 3 \beta_1 + 37) q^{11} + ( - 16 \beta_{2} + 34 \beta_1 - 190) q^{12} + (99 \beta_{2} - 45 \beta_1 + 324) q^{13} + (60 \beta_{2} - 180 \beta_1 + 592) q^{14} + ( - 128 \beta_{2} + 8 \beta_1 + 88) q^{16} + (146 \beta_{2} + 57 \beta_1 + 269) q^{17} + ( - 182 \beta_{2} + 8 \beta_1 - 1364) q^{18} + (200 \beta_{2} + 40 \beta_1 + 498) q^{19} + ( - 52 \beta_{2} + 193 \beta_1 - 475) q^{21} + ( - 68 \beta_{2} - 78 \beta_1 - 454) q^{22} + 529 q^{23} + (216 \beta_{2} + 360 \beta_1 + 1248) q^{24} + ( - 18 \beta_{2} - 747 \beta_1 + 423) q^{26} + (559 \beta_{2} - 22 \beta_1 + 3373) q^{27} + (472 \beta_{2} - 1068 \beta_1 + 2908) q^{28} + (523 \beta_{2} - 745 \beta_1 - 704) q^{29} + (479 \beta_{2} - 914 \beta_1 - 1471) q^{31} + ( - 288 \beta_{2} + 464 \beta_1 - 1968) q^{32} + (406 \beta_{2} + 329 \beta_1 + 2431) q^{33} + ( - 520 \beta_{2} - 276 \beta_1 - 3656) q^{34} + ( - 980 \beta_{2} + 968 \beta_1 - 200) q^{36} + ( - 1662 \beta_{2} + 1117 \beta_1 - 2053) q^{37} + ( - 560 \beta_{2} - 698 \beta_1 - 3622) q^{38} + (1017 \beta_{2} + 1494 \beta_1 + 5499) q^{39} + ( - 1847 \beta_{2} + 1171 \beta_1 - 3258) q^{41} + ( - 668 \beta_{2} + 1544 \beta_1 - 6116) q^{42} + (1150 \beta_{2} - 2050 \beta_1 + 4510) q^{43} + ( - 448 \beta_{2} + 104 \beta_1 + 1888) q^{44} + ( - 529 \beta_1 + 529) q^{46} + (1021 \beta_{2} + 682 \beta_1 - 7613) q^{47} + ( - 1360 \beta_{2} - 968 \beta_1 - 7576) q^{48} + (1428 \beta_{2} - 4306 \beta_1 - 949) q^{49} + (2648 \beta_{2} + 1909 \beta_1 + 16517) q^{51} + ( - 144 \beta_{2} - 2682 \beta_1 + 14958) q^{52} + ( - 534 \beta_{2} - 4444 \beta_1 - 7006) q^{53} + ( - 1030 \beta_{2} - 4601 \beta_1 - 3727) q^{54} + (1408 \beta_{2} - 3432 \beta_1 + 12600) q^{56} + (3338 \beta_{2} + 2836 \beta_1 + 20486) q^{57} + (1934 \beta_{2} - 4067 \beta_1 + 16559) q^{58} + (1980 \beta_{2} + 934 \beta_1 + 17710) q^{59} + ( - 42 \beta_{2} + 1322 \beta_1 - 23320) q^{61} + (2698 \beta_{2} - 4057 \beta_1 + 21985) q^{62} + (52 \beta_{2} + 2906 \beta_1 - 12614) q^{63} + (2816 \beta_{2} + 4608 \beta_1 - 16064) q^{64} + ( - 2128 \beta_{2} - 1598 \beta_1 - 14110) q^{66} + (1436 \beta_{2} + 4323 \beta_1 + 21949) q^{67} + ( - 2528 \beta_{2} + 1492 \beta_1 + 4124) q^{68} + (529 \beta_{2} + 1058 \beta_1 + 3703) q^{69} + (2227 \beta_{2} - 1628 \beta_1 + 31515) q^{71} + (3912 \beta_{2} + 6744 \beta_1 + 25224) q^{72} + (21 \beta_{2} - 3241 \beta_1 + 26170) q^{73} + ( - 1144 \beta_{2} + 10962 \beta_1 - 15646) q^{74} + ( - 2488 \beta_{2} - 28 \beta_1 + 11316) q^{76} + (876 \beta_{2} - 532 \beta_1 - 368) q^{77} + ( - 8010 \beta_{2} - 63 \beta_1 - 58041) q^{78} + ( - 6662 \beta_{2} - 4778 \beta_1 + 20036) q^{79} + ( - 124 \beta_{2} + 5680 \beta_1 + 51917) q^{81} + ( - 990 \beta_{2} + 12807 \beta_1 - 16043) q^{82} + (13758 \beta_{2} - 5867 \beta_1 - 9839) q^{83} + ( - 3176 \beta_{2} + 8996 \beta_1 - 32516) q^{84} + (5900 \beta_{2} - 17060 \beta_1 + 56060) q^{86} + ( - 2623 \beta_{2} + 2554 \beta_1 - 28441) q^{87} + (2656 \beta_{2} + 2024 \beta_1 + 19256) q^{88} + (8666 \beta_{2} - 5964 \beta_1 + 4304) q^{89} + (6984 \beta_{2} - 14229 \beta_1 + 43587) q^{91} + (2116 \beta_{2} - 3174 \beta_1 + 1058) q^{92} + ( - 5777 \beta_{2} + 455 \beta_1 - 50366) q^{93} + ( - 4770 \beta_{2} + 8981 \beta_1 - 44413) q^{94} + ( - 320 \beta_{2} - 6064 \beta_1 + 3472) q^{96} + ( - 618 \beta_{2} - 3351 \beta_1 + 90313) q^{97} + (14368 \beta_{2} - 23437 \beta_1 + 121157) q^{98} + (4118 \beta_{2} + 8116 \beta_1 + 62360) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 20 q^{3} + 16 q^{4} - 230 q^{6} + 282 q^{7} + 360 q^{8} + 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{2} + 20 q^{3} + 16 q^{4} - 230 q^{6} + 282 q^{7} + 360 q^{8} + 121 q^{9} + 136 q^{11} - 620 q^{12} + 1116 q^{13} + 2016 q^{14} + 128 q^{16} + 896 q^{17} - 4282 q^{18} + 1654 q^{19} - 1670 q^{21} - 1352 q^{22} + 1587 q^{23} + 3600 q^{24} + 1998 q^{26} + 10700 q^{27} + 10264 q^{28} - 844 q^{29} - 3020 q^{31} - 6656 q^{32} + 7370 q^{33} - 11212 q^{34} - 2548 q^{36} - 8938 q^{37} - 10728 q^{38} + 16020 q^{39} - 12792 q^{41} - 20560 q^{42} + 16730 q^{43} + 5112 q^{44} + 2116 q^{46} - 22500 q^{47} - 23120 q^{48} + 2887 q^{49} + 50290 q^{51} + 47412 q^{52} - 17108 q^{53} - 7610 q^{54} + 42640 q^{56} + 61960 q^{57} + 55678 q^{58} + 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 40696 q^{63} - 49984 q^{64} - 42860 q^{66} + 62960 q^{67} + 8352 q^{68} + 10580 q^{69} + 98400 q^{71} + 72840 q^{72} + 81772 q^{73} - 59044 q^{74} + 31488 q^{76} + 304 q^{77} - 182070 q^{78} + 58224 q^{79} + 149947 q^{81} - 61926 q^{82} - 9892 q^{83} - 109720 q^{84} + 191140 q^{86} - 90500 q^{87} + 58400 q^{88} + 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 157330 q^{93} - 146990 q^{94} + 16160 q^{96} + 273672 q^{97} + 401276 q^{98} + 183082 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 13x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} - \beta _1 + 17 ) / 2 \) 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
3.65736
0.917748
−3.57511
−5.31473 27.6631 −3.75366 0 −147.022 11.7843 190.021 522.249 0
1.2 0.164504 1.43100 −31.9729 0 0.235406 43.8366 −10.5238 −240.952 0
1.3 9.15022 −9.09413 51.7266 0 −83.2134 226.379 180.503 −160.297 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 575.6.a.b 3
5.b even 2 1 23.6.a.a 3
15.d odd 2 1 207.6.a.b 3
20.d odd 2 1 368.6.a.e 3
115.c odd 2 1 529.6.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.a 3 5.b even 2 1
207.6.a.b 3 15.d odd 2 1
368.6.a.e 3 20.d odd 2 1
529.6.a.a 3 115.c odd 2 1
575.6.a.b 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4T_{2}^{2} - 48T_{2} + 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(575))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 4 T^{2} - 48 T + 8 \) Copy content Toggle raw display
$3$ \( T^{3} - 20 T^{2} - 225 T + 360 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 282 T^{2} + 13108 T - 116944 \) Copy content Toggle raw display
$11$ \( T^{3} - 136 T^{2} - 43688 T + 840152 \) Copy content Toggle raw display
$13$ \( T^{3} - 1116 T^{2} + \cdots + 255615102 \) Copy content Toggle raw display
$17$ \( T^{3} - 896 T^{2} + \cdots - 220718408 \) Copy content Toggle raw display
$19$ \( T^{3} - 1654 T^{2} + \cdots + 460771768 \) Copy content Toggle raw display
$23$ \( (T - 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 844 T^{2} + \cdots - 33789223458 \) Copy content Toggle raw display
$31$ \( T^{3} + 3020 T^{2} + \cdots - 117638912880 \) Copy content Toggle raw display
$37$ \( T^{3} + 8938 T^{2} + \cdots - 1048082031344 \) Copy content Toggle raw display
$41$ \( T^{3} + 12792 T^{2} + \cdots - 1564944049486 \) Copy content Toggle raw display
$43$ \( T^{3} - 16730 T^{2} + \cdots + 95315904000 \) Copy content Toggle raw display
$47$ \( T^{3} + 22500 T^{2} + \cdots - 916008439440 \) Copy content Toggle raw display
$53$ \( T^{3} + 17108 T^{2} + \cdots - 7849670295504 \) Copy content Toggle raw display
$59$ \( T^{3} - 54176 T^{2} + \cdots - 1725012447168 \) Copy content Toggle raw display
$61$ \( T^{3} + 71324 T^{2} + \cdots + 11439907465152 \) Copy content Toggle raw display
$67$ \( T^{3} - 62960 T^{2} + \cdots + 11971711378840 \) Copy content Toggle raw display
$71$ \( T^{3} - 98400 T^{2} + \cdots - 24837760695040 \) Copy content Toggle raw display
$73$ \( T^{3} - 81772 T^{2} + \cdots - 7199078503954 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 235690469012368 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 297029282761704 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 125799322340896 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 693755159518744 \) Copy content Toggle raw display
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