Properties

Label 825.6.a.f
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.307532.1
Defining polynomial: \( x^{3} - x^{2} - 76x + 168 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + 9 q^{3} + (\beta_{2} + 2 \beta_1 + 24) q^{4} + ( - 9 \beta_1 - 18) q^{6} + ( - 4 \beta_{2} + 6 \beta_1 - 34) q^{7} + ( - 7 \beta_{2} - 10 \beta_1 - 76) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + 9 q^{3} + (\beta_{2} + 2 \beta_1 + 24) q^{4} + ( - 9 \beta_1 - 18) q^{6} + ( - 4 \beta_{2} + 6 \beta_1 - 34) q^{7} + ( - 7 \beta_{2} - 10 \beta_1 - 76) q^{8} + 81 q^{9} + 121 q^{11} + (9 \beta_{2} + 18 \beta_1 + 216) q^{12} + (\beta_{2} - 113 \beta_1 + 68) q^{13} + (14 \beta_{2} + 106 \beta_1 - 292) q^{14} + (13 \beta_{2} + 138 \beta_1 - 180) q^{16} + (12 \beta_{2} + 58 \beta_1 - 660) q^{17} + ( - 81 \beta_1 - 162) q^{18} + ( - 11 \beta_{2} + 57 \beta_1 + 672) q^{19} + ( - 36 \beta_{2} + 54 \beta_1 - 306) q^{21} + ( - 121 \beta_1 - 242) q^{22} + ( - 14 \beta_{2} - 286 \beta_1 - 316) q^{23} + ( - 63 \beta_{2} - 90 \beta_1 - 684) q^{24} + (108 \beta_{2} - 86 \beta_1 + 5752) q^{26} + 729 q^{27} + ( - 48 \beta_{2} - 152 \beta_1 - 3672) q^{28} + (145 \beta_{2} + 53 \beta_1 + 1498) q^{29} + ( - 34 \beta_{2} + 582 \beta_1 - 3768) q^{31} + (21 \beta_{2} + 266 \beta_1 - 4228) q^{32} + 1089 q^{33} + ( - 118 \beta_{2} + 444 \beta_1 - 1552) q^{34} + (81 \beta_{2} + 162 \beta_1 + 1944) q^{36} + (202 \beta_{2} + 274 \beta_1 + 2706) q^{37} + ( - 2 \beta_{2} - 474 \beta_1 - 4440) q^{38} + (9 \beta_{2} - 1017 \beta_1 + 612) q^{39} + (35 \beta_{2} + 879 \beta_1 + 1710) q^{41} + (126 \beta_{2} + 954 \beta_1 - 2628) q^{42} + (322 \beta_{2} + 1992 \beta_1 - 6846) q^{43} + (121 \beta_{2} + 242 \beta_1 + 2904) q^{44} + (356 \beta_{2} + 568 \beta_1 + 15336) q^{46} + ( - 134 \beta_{2} + 442 \beta_1 - 18692) q^{47} + (117 \beta_{2} + 1242 \beta_1 - 1620) q^{48} + ( - 140 \beta_{2} - 672 \beta_1 + 813) q^{49} + (108 \beta_{2} + 522 \beta_1 - 5940) q^{51} + ( - 486 \beta_{2} - 4080 \beta_1 - 7912) q^{52} + ( - 74 \beta_{2} + 1946 \beta_1 - 3742) q^{53} + ( - 729 \beta_1 - 1458) q^{54} + ( - 56 \beta_{2} + 1144 \beta_1 + 24016) q^{56} + ( - 99 \beta_{2} + 513 \beta_1 + 6048) q^{57} + ( - 778 \beta_{2} - 4108 \beta_1 - 4012) q^{58} + ( - 14 \beta_{2} - 566 \beta_1 - 19548) q^{59} + (544 \beta_{2} - 968 \beta_1 + 27138) q^{61} + ( - 412 \beta_{2} + 4380 \beta_1 - 23136) q^{62} + ( - 324 \beta_{2} + 486 \beta_1 - 2754) q^{63} + ( - 787 \beta_{2} - 566 \beta_1 + 636) q^{64} + ( - 1089 \beta_1 - 2178) q^{66} + (844 \beta_{2} - 2136 \beta_1 - 496) q^{67} + ( - 238 \beta_{2} + 1820 \beta_1 - 280) q^{68} + ( - 126 \beta_{2} - 2574 \beta_1 - 2844) q^{69} + ( - 1064 \beta_{2} - 1228 \beta_1 - 25000) q^{71} + ( - 567 \beta_{2} - 810 \beta_1 - 6156) q^{72} + ( - 563 \beta_{2} + 6451 \beta_1 + 18092) q^{73} + ( - 1284 \beta_{2} - 6342 \beta_1 - 17236) q^{74} + (836 \beta_{2} + 2652 \beta_1 + 12000) q^{76} + ( - 484 \beta_{2} + 726 \beta_1 - 4114) q^{77} + (972 \beta_{2} - 774 \beta_1 + 51768) q^{78} + ( - 1927 \beta_{2} + 2985 \beta_1 - 29492) q^{79} + 6561 q^{81} + ( - 1054 \beta_{2} - 2340 \beta_1 - 48708) q^{82} + ( - 1715 \beta_{2} + 2811 \beta_1 - 26430) q^{83} + ( - 432 \beta_{2} - 1368 \beta_1 - 33048) q^{84} + ( - 3602 \beta_{2} + 1050 \beta_1 - 86028) q^{86} + (1305 \beta_{2} + 477 \beta_1 + 13482) q^{87} + ( - 847 \beta_{2} - 1210 \beta_1 - 9196) q^{88} + ( - 1410 \beta_{2} - 8246 \beta_1 + 14726) q^{89} + (466 \beta_{2} + 13682 \beta_1 - 46568) q^{91} + ( - 1900 \beta_{2} - 12592 \beta_1 - 45824) q^{92} + ( - 306 \beta_{2} + 5238 \beta_1 - 33912) q^{93} + (228 \beta_{2} + 21104 \beta_1 + 12792) q^{94} + (189 \beta_{2} + 2394 \beta_1 - 38052) q^{96} + (2108 \beta_{2} - 88 \beta_1 - 10430) q^{97} + (1372 \beta_{2} + 1707 \beta_1 + 31638) q^{98} + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9} + 363 q^{11} + 657 q^{12} + 90 q^{13} - 784 q^{14} - 415 q^{16} - 1934 q^{17} - 567 q^{18} + 2084 q^{19} - 828 q^{21} - 847 q^{22} - 1220 q^{23} - 2079 q^{24} + 17062 q^{26} + 2187 q^{27} - 11120 q^{28} + 4402 q^{29} - 10688 q^{31} - 12439 q^{32} + 3267 q^{33} - 4094 q^{34} + 5913 q^{36} + 8190 q^{37} - 13792 q^{38} + 810 q^{39} + 5974 q^{41} - 7056 q^{42} - 18868 q^{43} + 8833 q^{44} + 46220 q^{46} - 55500 q^{47} - 3735 q^{48} + 1907 q^{49} - 17406 q^{51} - 27330 q^{52} - 9206 q^{53} - 5103 q^{54} + 73248 q^{56} + 18756 q^{57} - 15366 q^{58} - 59196 q^{59} + 79902 q^{61} - 64616 q^{62} - 7452 q^{63} + 2129 q^{64} - 7623 q^{66} - 4468 q^{67} + 1218 q^{68} - 10980 q^{69} - 75164 q^{71} - 18711 q^{72} + 61290 q^{73} - 56766 q^{74} + 37816 q^{76} - 11132 q^{77} + 153558 q^{78} - 83564 q^{79} + 19683 q^{81} - 147410 q^{82} - 74764 q^{83} - 100080 q^{84} - 253432 q^{86} + 39618 q^{87} - 27951 q^{88} + 37342 q^{89} - 126488 q^{91} - 148164 q^{92} - 96192 q^{93} + 59252 q^{94} - 111951 q^{96} - 33486 q^{97} + 95249 q^{98} + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 76x + 168 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2\nu - 52 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 + 52 \) 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
7.91848
2.30119
−9.21967
−9.91848 9.00000 66.3762 0 −89.2663 −92.6461 −340.959 81.0000 0
1.2 −4.30119 9.00000 −13.4998 0 −38.7107 148.216 195.703 81.0000 0
1.3 7.21967 9.00000 20.1236 0 64.9770 −147.570 −85.7438 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.f 3
5.b even 2 1 165.6.a.e 3
15.d odd 2 1 495.6.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.e 3 5.b even 2 1
495.6.a.a 3 15.d odd 2 1
825.6.a.f 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 7T_{2}^{2} - 60T_{2} - 308 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 7 T^{2} - 60 T - 308 \) Copy content Toggle raw display
$3$ \( (T - 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 92 T^{2} - 21932 T - 2026368 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 90 T^{2} + \cdots - 210673232 \) Copy content Toggle raw display
$17$ \( T^{3} + 1934 T^{2} + \cdots - 123909408 \) Copy content Toggle raw display
$19$ \( T^{3} - 2084 T^{2} + \cdots + 14437440 \) Copy content Toggle raw display
$23$ \( T^{3} + 1220 T^{2} + \cdots - 2404328128 \) Copy content Toggle raw display
$29$ \( T^{3} - 4402 T^{2} + \cdots + 80705567064 \) Copy content Toggle raw display
$31$ \( T^{3} + 10688 T^{2} + \cdots + 593236224 \) Copy content Toggle raw display
$37$ \( T^{3} - 8190 T^{2} + \cdots + 165140256344 \) Copy content Toggle raw display
$41$ \( T^{3} - 5974 T^{2} + \cdots + 127610311752 \) Copy content Toggle raw display
$43$ \( T^{3} + 18868 T^{2} + \cdots - 5672527691040 \) Copy content Toggle raw display
$47$ \( T^{3} + 55500 T^{2} + \cdots + 5576540180928 \) Copy content Toggle raw display
$53$ \( T^{3} + 9206 T^{2} + \cdots + 850686588776 \) Copy content Toggle raw display
$59$ \( T^{3} + 59196 T^{2} + \cdots + 7185357358784 \) Copy content Toggle raw display
$61$ \( T^{3} - 79902 T^{2} + \cdots - 2993338614376 \) Copy content Toggle raw display
$67$ \( T^{3} + 4468 T^{2} + \cdots + 6432661987328 \) Copy content Toggle raw display
$71$ \( T^{3} + 75164 T^{2} + \cdots - 31171869026560 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 152306824713328 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 283704612543488 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 200710881230832 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 340522035911288 \) Copy content Toggle raw display
$97$ \( T^{3} + 33486 T^{2} + \cdots + 93893760682568 \) Copy content Toggle raw display
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