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: \( x^{2} - x - 44 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{2} + 9 q^{3} + ( - 5 \beta + 21) q^{4} + ( - 9 \beta + 27) q^{6} + ( - 10 \beta + 148) q^{7} + (\beta + 187) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 3) q^{2} + 9 q^{3} + ( - 5 \beta + 21) q^{4} + ( - 9 \beta + 27) q^{6} + ( - 10 \beta + 148) q^{7} + (\beta + 187) q^{8} + 81 q^{9} - 121 q^{11} + ( - 45 \beta + 189) q^{12} + ( - 38 \beta + 102) q^{13} + ( - 168 \beta + 884) q^{14} + ( - 25 \beta - 155) q^{16} + ( - 60 \beta + 430) q^{17} + ( - 81 \beta + 243) q^{18} + ( - 12 \beta - 732) q^{19} + ( - 90 \beta + 1332) q^{21} + (121 \beta - 363) q^{22} + ( - 366 \beta + 1868) q^{23} + (9 \beta + 1683) q^{24} + ( - 178 \beta + 1978) q^{26} + 729 q^{27} + ( - 900 \beta + 5308) q^{28} + (412 \beta + 3094) q^{29} + (344 \beta - 3936) q^{31} + (73 \beta - 5349) q^{32} - 1089 q^{33} + ( - 550 \beta + 3930) q^{34} + ( - 405 \beta + 1701) q^{36} + (136 \beta + 14890) q^{37} + (708 \beta - 1668) q^{38} + ( - 342 \beta + 918) q^{39} + ( - 712 \beta - 2534) q^{41} + ( - 1512 \beta + 7956) q^{42} + (1496 \beta + 7580) q^{43} + (605 \beta - 2541) q^{44} + ( - 2600 \beta + 21708) q^{46} + (2526 \beta - 5188) q^{47} + ( - 225 \beta - 1395) q^{48} + ( - 2860 \beta + 9497) q^{49} + ( - 540 \beta + 3870) q^{51} + ( - 1118 \beta + 10502) q^{52} + ( - 2206 \beta - 5986) q^{53} + ( - 729 \beta + 2187) q^{54} + ( - 1732 \beta + 27236) q^{56} + ( - 108 \beta - 6588) q^{57} + ( - 2270 \beta - 8846) q^{58} + ( - 1476 \beta + 9388) q^{59} + (2330 \beta - 2638) q^{61} + (4624 \beta - 26944) q^{62} + ( - 810 \beta + 11988) q^{63} + (6295 \beta - 14299) q^{64} + (1089 \beta - 3267) q^{66} + ( - 3200 \beta - 14068) q^{67} + ( - 3110 \beta + 22230) q^{68} + ( - 3294 \beta + 16812) q^{69} + ( - 3098 \beta - 15356) q^{71} + (81 \beta + 15147) q^{72} + ( - 7536 \beta - 26554) q^{73} + ( - 14618 \beta + 38686) q^{74} + (3468 \beta - 12732) q^{76} + (1210 \beta - 17908) q^{77} + ( - 1602 \beta + 17802) q^{78} + ( - 9482 \beta + 5676) q^{79} + 6561 q^{81} + (1110 \beta + 23726) q^{82} + ( - 2592 \beta + 30444) q^{83} + ( - 8100 \beta + 47772) q^{84} + ( - 4588 \beta - 43084) q^{86} + (3708 \beta + 27846) q^{87} + ( - 121 \beta - 22627) q^{88} + (15056 \beta + 38666) q^{89} + ( - 6264 \beta + 31816) q^{91} + ( - 15196 \beta + 119748) q^{92} + (3096 \beta - 35424) q^{93} + (10240 \beta - 126708) q^{94} + (657 \beta - 48141) q^{96} + (21300 \beta - 14210) q^{97} + ( - 15217 \beta + 154331) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 18 q^{3} + 37 q^{4} + 45 q^{6} + 286 q^{7} + 375 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} + 18 q^{3} + 37 q^{4} + 45 q^{6} + 286 q^{7} + 375 q^{8} + 162 q^{9} - 242 q^{11} + 333 q^{12} + 166 q^{13} + 1600 q^{14} - 335 q^{16} + 800 q^{17} + 405 q^{18} - 1476 q^{19} + 2574 q^{21} - 605 q^{22} + 3370 q^{23} + 3375 q^{24} + 3778 q^{26} + 1458 q^{27} + 9716 q^{28} + 6600 q^{29} - 7528 q^{31} - 10625 q^{32} - 2178 q^{33} + 7310 q^{34} + 2997 q^{36} + 29916 q^{37} - 2628 q^{38} + 1494 q^{39} - 5780 q^{41} + 14400 q^{42} + 16656 q^{43} - 4477 q^{44} + 40816 q^{46} - 7850 q^{47} - 3015 q^{48} + 16134 q^{49} + 7200 q^{51} + 19886 q^{52} - 14178 q^{53} + 3645 q^{54} + 52740 q^{56} - 13284 q^{57} - 19962 q^{58} + 17300 q^{59} - 2946 q^{61} - 49264 q^{62} + 23166 q^{63} - 22303 q^{64} - 5445 q^{66} - 31336 q^{67} + 41350 q^{68} + 30330 q^{69} - 33810 q^{71} + 30375 q^{72} - 60644 q^{73} + 62754 q^{74} - 21996 q^{76} - 34606 q^{77} + 34002 q^{78} + 1870 q^{79} + 13122 q^{81} + 48562 q^{82} + 58296 q^{83} + 87444 q^{84} - 90756 q^{86} + 59400 q^{87} - 45375 q^{88} + 92388 q^{89} + 57368 q^{91} + 224300 q^{92} - 67752 q^{93} - 243176 q^{94} - 95625 q^{96} - 7120 q^{97} + 293445 q^{98} - 19602 q^{99}+O(q^{100}) \) 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
7.15207
−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 \( T_{2}^{2} - 5T_{2} - 38 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

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