Properties

Label 825.4.a.s
Level $825$
Weight $4$
Character orbit 825.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.23612.1
Defining polynomial: \(x^{3} - x^{2} - 20 x + 26\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{1} + 4 \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{1} + 4 \beta_{2} ) q^{8} + 9 q^{9} + 11 q^{11} + ( 21 + 3 \beta_{1} + 3 \beta_{2} ) q^{12} + ( 4 - 12 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -22 + 10 \beta_{1} + 4 \beta_{2} ) q^{14} + ( 9 + 23 \beta_{1} + 7 \beta_{2} ) q^{16} + ( 76 - 10 \beta_{1} - 6 \beta_{2} ) q^{17} + ( 9 + 9 \beta_{1} ) q^{18} + ( 48 + 2 \beta_{1} + 4 \beta_{2} ) q^{19} + ( 6 - 6 \beta_{1} + 6 \beta_{2} ) q^{21} + ( 11 + 11 \beta_{1} ) q^{22} + ( 60 + 20 \beta_{1} + 8 \beta_{2} ) q^{23} + ( 45 + 9 \beta_{1} + 12 \beta_{2} ) q^{24} + ( -168 - 4 \beta_{1} - 18 \beta_{2} ) q^{26} + 27 q^{27} + ( 110 + 10 \beta_{1} + 6 \beta_{2} ) q^{28} + ( 34 - 34 \beta_{1} + 20 \beta_{2} ) q^{29} + ( -24 + 4 \beta_{1} - 16 \beta_{2} ) q^{31} + ( 225 + 13 \beta_{1} + 12 \beta_{2} ) q^{32} + 33 q^{33} + ( -76 + 52 \beta_{1} - 28 \beta_{2} ) q^{34} + ( 63 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( 122 + 24 \beta_{1} + 4 \beta_{2} ) q^{37} + ( 84 + 64 \beta_{1} + 14 \beta_{2} ) q^{38} + ( 12 - 36 \beta_{1} - 6 \beta_{2} ) q^{39} + ( -66 + 2 \beta_{1} - 20 \beta_{2} ) q^{41} + ( -66 + 30 \beta_{1} + 12 \beta_{2} ) q^{42} + ( 150 + 74 \beta_{1} - 14 \beta_{2} ) q^{43} + ( 77 + 11 \beta_{1} + 11 \beta_{2} ) q^{44} + ( 356 + 92 \beta_{1} + 44 \beta_{2} ) q^{46} + ( 36 - 48 \beta_{1} - 28 \beta_{2} ) q^{47} + ( 27 + 69 \beta_{1} + 21 \beta_{2} ) q^{48} + ( -51 - 4 \beta_{1} - 20 \beta_{2} ) q^{49} + ( 228 - 30 \beta_{1} - 18 \beta_{2} ) q^{51} + ( -292 - 144 \beta_{1} - 42 \beta_{2} ) q^{52} + ( 42 + 32 \beta_{1} - 52 \beta_{2} ) q^{53} + ( 27 + 27 \beta_{1} ) q^{54} + ( 438 + 54 \beta_{1} - 4 \beta_{2} ) q^{56} + ( 144 + 6 \beta_{1} + 12 \beta_{2} ) q^{57} + ( -402 + 114 \beta_{1} + 26 \beta_{2} ) q^{58} + ( -308 - 120 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 218 - 12 \beta_{1} - 44 \beta_{2} ) q^{61} + ( -88 \beta_{1} - 44 \beta_{2} ) q^{62} + ( 18 - 18 \beta_{1} + 18 \beta_{2} ) q^{63} + ( 359 + 89 \beta_{1} - 7 \beta_{2} ) q^{64} + ( 33 + 33 \beta_{1} ) q^{66} + ( 128 - 148 \beta_{1} + 28 \beta_{2} ) q^{67} + ( -12 - 108 \beta_{1} + 16 \beta_{2} ) q^{68} + ( 180 + 60 \beta_{1} + 24 \beta_{2} ) q^{69} + ( -216 + 104 \beta_{1} - 16 \beta_{2} ) q^{71} + ( 135 + 27 \beta_{1} + 36 \beta_{2} ) q^{72} + ( -356 + 168 \beta_{1} - 14 \beta_{2} ) q^{73} + ( 466 + 138 \beta_{1} + 36 \beta_{2} ) q^{74} + ( 624 + 124 \beta_{1} + 74 \beta_{2} ) q^{76} + ( 22 - 22 \beta_{1} + 22 \beta_{2} ) q^{77} + ( -504 - 12 \beta_{1} - 54 \beta_{2} ) q^{78} + ( -524 - 14 \beta_{1} - 52 \beta_{2} ) q^{79} + 81 q^{81} + ( -78 - 146 \beta_{1} - 58 \beta_{2} ) q^{82} + ( 582 - 164 \beta_{1} - 70 \beta_{2} ) q^{83} + ( 330 + 30 \beta_{1} + 18 \beta_{2} ) q^{84} + ( 1158 + 94 \beta_{1} + 32 \beta_{2} ) q^{86} + ( 102 - 102 \beta_{1} + 60 \beta_{2} ) q^{87} + ( 165 + 33 \beta_{1} + 44 \beta_{2} ) q^{88} + ( -730 + 68 \beta_{1} ) q^{89} + ( 56 - 176 \beta_{1} + 4 \beta_{2} ) q^{91} + ( 1252 + 372 \beta_{1} + 160 \beta_{2} ) q^{92} + ( -72 + 12 \beta_{1} - 48 \beta_{2} ) q^{93} + ( -692 - 76 \beta_{1} - 132 \beta_{2} ) q^{94} + ( 675 + 39 \beta_{1} + 36 \beta_{2} ) q^{96} + ( -286 + 240 \beta_{1} - 64 \beta_{2} ) q^{97} + ( -147 - 131 \beta_{1} - 64 \beta_{2} ) q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 4q^{2} + 9q^{3} + 22q^{4} + 12q^{6} + 4q^{7} + 48q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + 4q^{2} + 9q^{3} + 22q^{4} + 12q^{6} + 4q^{7} + 48q^{8} + 27q^{9} + 33q^{11} + 66q^{12} - 56q^{14} + 50q^{16} + 218q^{17} + 36q^{18} + 146q^{19} + 12q^{21} + 44q^{22} + 200q^{23} + 144q^{24} - 508q^{26} + 81q^{27} + 340q^{28} + 68q^{29} - 68q^{31} + 688q^{32} + 99q^{33} - 176q^{34} + 198q^{36} + 390q^{37} + 316q^{38} - 196q^{41} - 168q^{42} + 524q^{43} + 242q^{44} + 1160q^{46} + 60q^{47} + 150q^{48} - 157q^{49} + 654q^{51} - 1020q^{52} + 158q^{53} + 108q^{54} + 1368q^{56} + 438q^{57} - 1092q^{58} - 1044q^{59} + 642q^{61} - 88q^{62} + 36q^{63} + 1166q^{64} + 132q^{66} + 236q^{67} - 144q^{68} + 600q^{69} - 544q^{71} + 432q^{72} - 900q^{73} + 1536q^{74} + 1996q^{76} + 44q^{77} - 1524q^{78} - 1586q^{79} + 243q^{81} - 380q^{82} + 1582q^{83} + 1020q^{84} + 3568q^{86} + 204q^{87} + 528q^{88} - 2122q^{89} - 8q^{91} + 4128q^{92} - 204q^{93} - 2152q^{94} + 2064q^{96} - 618q^{97} - 572q^{98} + 297q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 20 x + 26\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 14\)

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
−4.59056
1.32906
4.26150
−3.59056 3.00000 4.89212 0 −10.7717 16.1465 11.1590 9.00000 0
1.2 2.32906 3.00000 −2.57547 0 6.98719 −22.4672 −24.6309 9.00000 0
1.3 5.26150 3.00000 19.6833 0 15.7845 10.3207 61.4719 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 825.4.a.s 3
3.b odd 2 1 2475.4.a.s 3
5.b even 2 1 165.4.a.d 3
5.c odd 4 2 825.4.c.l 6
15.d odd 2 1 495.4.a.l 3
55.d odd 2 1 1815.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 5.b even 2 1
495.4.a.l 3 15.d odd 2 1
825.4.a.s 3 1.a even 1 1 trivial
825.4.c.l 6 5.c odd 4 2
1815.4.a.s 3 55.d odd 2 1
2475.4.a.s 3 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2}^{3} - 4 T_{2}^{2} - 15 T_{2} + 44 \)
\( T_{7}^{3} - 4 T_{7}^{2} - 428 T_{7} + 3744 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 44 - 15 T - 4 T^{2} + T^{3} \)
$3$ \( ( -3 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( 3744 - 428 T - 4 T^{2} + T^{3} \)
$11$ \( ( -11 + T )^{3} \)
$13$ \( 34144 - 3560 T + T^{3} \)
$17$ \( 235104 + 9680 T - 218 T^{2} + T^{3} \)
$19$ \( -30960 + 5376 T - 146 T^{2} + T^{3} \)
$23$ \( -1664 - 2672 T - 200 T^{2} + T^{3} \)
$29$ \( 3163056 - 54364 T - 68 T^{2} + T^{3} \)
$31$ \( -1812096 - 23232 T + 68 T^{2} + T^{3} \)
$37$ \( -618952 + 36460 T - 390 T^{2} + T^{3} \)
$41$ \( -4364208 - 26076 T + 196 T^{2} + T^{3} \)
$43$ \( 31273920 - 28668 T - 524 T^{2} + T^{3} \)
$47$ \( 20966976 - 135920 T - 60 T^{2} + T^{3} \)
$53$ \( -39574952 - 260852 T - 158 T^{2} + T^{3} \)
$59$ \( -84227264 + 64144 T + 1044 T^{2} + T^{3} \)
$61$ \( 22757384 - 60548 T - 642 T^{2} + T^{3} \)
$67$ \( -87537664 - 462208 T - 236 T^{2} + T^{3} \)
$71$ \( 6553600 - 129728 T + 544 T^{2} + T^{3} \)
$73$ \( 5609344 - 299576 T + 900 T^{2} + T^{3} \)
$79$ \( -14694992 + 562208 T + 1586 T^{2} + T^{3} \)
$83$ \( 924645384 - 307644 T - 1582 T^{2} + T^{3} \)
$89$ \( 293444632 + 1406940 T + 2122 T^{2} + T^{3} \)
$97$ \( 223543736 - 1291700 T + 618 T^{2} + T^{3} \)
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