Properties

Label 825.6.a.e
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta ) q^{2} + 9 q^{3} + ( 21 - 5 \beta ) q^{4} + ( 27 - 9 \beta ) q^{6} + ( 148 - 10 \beta ) q^{7} + ( 187 + \beta ) q^{8} + 81 q^{9} +O(q^{10})\) \( q + ( 3 - \beta ) q^{2} + 9 q^{3} + ( 21 - 5 \beta ) q^{4} + ( 27 - 9 \beta ) q^{6} + ( 148 - 10 \beta ) q^{7} + ( 187 + \beta ) q^{8} + 81 q^{9} -121 q^{11} + ( 189 - 45 \beta ) q^{12} + ( 102 - 38 \beta ) q^{13} + ( 884 - 168 \beta ) q^{14} + ( -155 - 25 \beta ) q^{16} + ( 430 - 60 \beta ) q^{17} + ( 243 - 81 \beta ) q^{18} + ( -732 - 12 \beta ) q^{19} + ( 1332 - 90 \beta ) q^{21} + ( -363 + 121 \beta ) q^{22} + ( 1868 - 366 \beta ) q^{23} + ( 1683 + 9 \beta ) q^{24} + ( 1978 - 178 \beta ) q^{26} + 729 q^{27} + ( 5308 - 900 \beta ) q^{28} + ( 3094 + 412 \beta ) q^{29} + ( -3936 + 344 \beta ) q^{31} + ( -5349 + 73 \beta ) q^{32} -1089 q^{33} + ( 3930 - 550 \beta ) q^{34} + ( 1701 - 405 \beta ) q^{36} + ( 14890 + 136 \beta ) q^{37} + ( -1668 + 708 \beta ) q^{38} + ( 918 - 342 \beta ) q^{39} + ( -2534 - 712 \beta ) q^{41} + ( 7956 - 1512 \beta ) q^{42} + ( 7580 + 1496 \beta ) q^{43} + ( -2541 + 605 \beta ) q^{44} + ( 21708 - 2600 \beta ) q^{46} + ( -5188 + 2526 \beta ) q^{47} + ( -1395 - 225 \beta ) q^{48} + ( 9497 - 2860 \beta ) q^{49} + ( 3870 - 540 \beta ) q^{51} + ( 10502 - 1118 \beta ) q^{52} + ( -5986 - 2206 \beta ) q^{53} + ( 2187 - 729 \beta ) q^{54} + ( 27236 - 1732 \beta ) q^{56} + ( -6588 - 108 \beta ) q^{57} + ( -8846 - 2270 \beta ) q^{58} + ( 9388 - 1476 \beta ) q^{59} + ( -2638 + 2330 \beta ) q^{61} + ( -26944 + 4624 \beta ) q^{62} + ( 11988 - 810 \beta ) q^{63} + ( -14299 + 6295 \beta ) q^{64} + ( -3267 + 1089 \beta ) q^{66} + ( -14068 - 3200 \beta ) q^{67} + ( 22230 - 3110 \beta ) q^{68} + ( 16812 - 3294 \beta ) q^{69} + ( -15356 - 3098 \beta ) q^{71} + ( 15147 + 81 \beta ) q^{72} + ( -26554 - 7536 \beta ) q^{73} + ( 38686 - 14618 \beta ) q^{74} + ( -12732 + 3468 \beta ) q^{76} + ( -17908 + 1210 \beta ) q^{77} + ( 17802 - 1602 \beta ) q^{78} + ( 5676 - 9482 \beta ) q^{79} + 6561 q^{81} + ( 23726 + 1110 \beta ) q^{82} + ( 30444 - 2592 \beta ) q^{83} + ( 47772 - 8100 \beta ) q^{84} + ( -43084 - 4588 \beta ) q^{86} + ( 27846 + 3708 \beta ) q^{87} + ( -22627 - 121 \beta ) q^{88} + ( 38666 + 15056 \beta ) q^{89} + ( 31816 - 6264 \beta ) q^{91} + ( 119748 - 15196 \beta ) q^{92} + ( -35424 + 3096 \beta ) q^{93} + ( -126708 + 10240 \beta ) q^{94} + ( -48141 + 657 \beta ) q^{96} + ( -14210 + 21300 \beta ) q^{97} + ( 154331 - 15217 \beta ) q^{98} -9801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{2} + 18q^{3} + 37q^{4} + 45q^{6} + 286q^{7} + 375q^{8} + 162q^{9} + O(q^{10}) \) \( 2q + 5q^{2} + 18q^{3} + 37q^{4} + 45q^{6} + 286q^{7} + 375q^{8} + 162q^{9} - 242q^{11} + 333q^{12} + 166q^{13} + 1600q^{14} - 335q^{16} + 800q^{17} + 405q^{18} - 1476q^{19} + 2574q^{21} - 605q^{22} + 3370q^{23} + 3375q^{24} + 3778q^{26} + 1458q^{27} + 9716q^{28} + 6600q^{29} - 7528q^{31} - 10625q^{32} - 2178q^{33} + 7310q^{34} + 2997q^{36} + 29916q^{37} - 2628q^{38} + 1494q^{39} - 5780q^{41} + 14400q^{42} + 16656q^{43} - 4477q^{44} + 40816q^{46} - 7850q^{47} - 3015q^{48} + 16134q^{49} + 7200q^{51} + 19886q^{52} - 14178q^{53} + 3645q^{54} + 52740q^{56} - 13284q^{57} - 19962q^{58} + 17300q^{59} - 2946q^{61} - 49264q^{62} + 23166q^{63} - 22303q^{64} - 5445q^{66} - 31336q^{67} + 41350q^{68} + 30330q^{69} - 33810q^{71} + 30375q^{72} - 60644q^{73} + 62754q^{74} - 21996q^{76} - 34606q^{77} + 34002q^{78} + 1870q^{79} + 13122q^{81} + 48562q^{82} + 58296q^{83} + 87444q^{84} - 90756q^{86} + 59400q^{87} - 45375q^{88} + 92388q^{89} + 57368q^{91} + 224300q^{92} - 67752q^{93} - 243176q^{94} - 95625q^{96} - 7120q^{97} + 293445q^{98} - 19602q^{99} + O(q^{100}) \)

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
7.15207
−6.15207
−4.15207 9.00000 −14.7603 0 −37.3686 76.4793 194.152 81.0000 0
1.2 9.15207 9.00000 51.7603 0 82.3686 209.521 180.848 81.0000 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 825.6.a.e 2
5.b even 2 1 33.6.a.c 2
15.d odd 2 1 99.6.a.f 2
20.d odd 2 1 528.6.a.s 2
55.d odd 2 1 363.6.a.j 2
165.d even 2 1 1089.6.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.c 2 5.b even 2 1
99.6.a.f 2 15.d odd 2 1
363.6.a.j 2 55.d odd 2 1
528.6.a.s 2 20.d odd 2 1
825.6.a.e 2 1.a even 1 1 trivial
1089.6.a.j 2 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 T_{2} - 38 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -38 - 5 T + T^{2} \)
$3$ \( ( -9 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 16024 - 286 T + T^{2} \)
$11$ \( ( 121 + T )^{2} \)
$13$ \( -57008 - 166 T + T^{2} \)
$17$ \( 700 - 800 T + T^{2} \)
$19$ \( 538272 + 1476 T + T^{2} \)
$23$ \( -3088328 - 3370 T + T^{2} \)
$29$ \( 3378828 - 6600 T + T^{2} \)
$31$ \( 8931328 + 7528 T + T^{2} \)
$37$ \( 222923316 - 29916 T + T^{2} \)
$41$ \( -14080172 + 5780 T + T^{2} \)
$43$ \( -29676624 - 16656 T + T^{2} \)
$47$ \( -266939288 + 7850 T + T^{2} \)
$53$ \( -165085872 + 14178 T + T^{2} \)
$59$ \( -21579488 - 17300 T + T^{2} \)
$61$ \( -238059096 + 2946 T + T^{2} \)
$67$ \( -207633776 + 31336 T + T^{2} \)
$71$ \( -138914952 + 33810 T + T^{2} \)
$73$ \( -1593591164 + 60644 T + T^{2} \)
$79$ \( -3977569112 - 1870 T + T^{2} \)
$83$ \( 552313872 - 58296 T + T^{2} \)
$89$ \( -7896843132 - 92388 T + T^{2} \)
$97$ \( -20063108900 + 7120 T + T^{2} \)
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