Properties

Label 825.6.a.d
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{313}) \)
Defining polynomial: \( x^{2} - x - 78 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{313})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 9 q^{3} + (\beta + 46) q^{4} - 9 \beta q^{6} + ( - 2 \beta + 10) q^{7} + ( - 15 \beta - 78) q^{8} + 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 9 q^{3} + (\beta + 46) q^{4} - 9 \beta q^{6} + ( - 2 \beta + 10) q^{7} + ( - 15 \beta - 78) q^{8} + 81 q^{9} + 121 q^{11} + (9 \beta + 414) q^{12} + (106 \beta - 20) q^{13} + ( - 8 \beta + 156) q^{14} + (61 \beta - 302) q^{16} + ( - 4 \beta + 462) q^{17} - 81 \beta q^{18} + (4 \beta - 1468) q^{19} + ( - 18 \beta + 90) q^{21} - 121 \beta q^{22} + ( - 26 \beta - 2610) q^{23} + ( - 135 \beta - 702) q^{24} + ( - 86 \beta - 8268) q^{26} + 729 q^{27} + ( - 84 \beta + 304) q^{28} + ( - 132 \beta - 6234) q^{29} + (608 \beta + 4664) q^{31} + (721 \beta - 2262) q^{32} + 1089 q^{33} + ( - 458 \beta + 312) q^{34} + (81 \beta + 3726) q^{36} + (320 \beta - 3158) q^{37} + (1464 \beta - 312) q^{38} + (954 \beta - 180) q^{39} + ( - 728 \beta + 12486) q^{41} + ( - 72 \beta + 1404) q^{42} + ( - 1240 \beta - 9560) q^{43} + (121 \beta + 5566) q^{44} + (2636 \beta + 2028) q^{46} + (778 \beta + 2514) q^{47} + (549 \beta - 2718) q^{48} + ( - 36 \beta - 16395) q^{49} + ( - 36 \beta + 4158) q^{51} + (4962 \beta + 7348) q^{52} + ( - 594 \beta - 20088) q^{53} - 729 \beta q^{54} + (36 \beta + 1560) q^{56} + (36 \beta - 13212) q^{57} + (6366 \beta + 10296) q^{58} + ( - 3676 \beta + 10944) q^{59} + (2746 \beta - 7072) q^{61} + ( - 5272 \beta - 47424) q^{62} + ( - 162 \beta + 810) q^{63} + ( - 411 \beta - 46574) q^{64} - 1089 \beta q^{66} + ( - 768 \beta - 32300) q^{67} + (274 \beta + 20940) q^{68} + ( - 234 \beta - 23490) q^{69} + ( - 3102 \beta + 32274) q^{71} + ( - 1215 \beta - 6318) q^{72} + ( - 320 \beta - 26546) q^{73} + (2838 \beta - 24960) q^{74} + ( - 1280 \beta - 67216) q^{76} + ( - 242 \beta + 1210) q^{77} + ( - 774 \beta - 74412) q^{78} + ( - 2130 \beta + 9626) q^{79} + 6561 q^{81} + ( - 11758 \beta + 56784) q^{82} + (3528 \beta + 5388) q^{83} + ( - 756 \beta + 2736) q^{84} + (10800 \beta + 96720) q^{86} + ( - 1188 \beta - 56106) q^{87} + ( - 1815 \beta - 9438) q^{88} + (3024 \beta - 30582) q^{89} + (888 \beta - 16736) q^{91} + ( - 3832 \beta - 122088) q^{92} + (5472 \beta + 41976) q^{93} + ( - 3292 \beta - 60684) q^{94} + (6489 \beta - 20358) q^{96} + ( - 1092 \beta + 92074) q^{97} + (16431 \beta + 2808) q^{98} + 9801 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9} + 242 q^{11} + 837 q^{12} + 66 q^{13} + 304 q^{14} - 543 q^{16} + 920 q^{17} - 81 q^{18} - 2932 q^{19} + 162 q^{21} - 121 q^{22} - 5246 q^{23} - 1539 q^{24} - 16622 q^{26} + 1458 q^{27} + 524 q^{28} - 12600 q^{29} + 9936 q^{31} - 3803 q^{32} + 2178 q^{33} + 166 q^{34} + 7533 q^{36} - 5996 q^{37} + 840 q^{38} + 594 q^{39} + 24244 q^{41} + 2736 q^{42} - 20360 q^{43} + 11253 q^{44} + 6692 q^{46} + 5806 q^{47} - 4887 q^{48} - 32826 q^{49} + 8280 q^{51} + 19658 q^{52} - 40770 q^{53} - 729 q^{54} + 3156 q^{56} - 26388 q^{57} + 26958 q^{58} + 18212 q^{59} - 11398 q^{61} - 100120 q^{62} + 1458 q^{63} - 93559 q^{64} - 1089 q^{66} - 65368 q^{67} + 42154 q^{68} - 47214 q^{69} + 61446 q^{71} - 13851 q^{72} - 53412 q^{73} - 47082 q^{74} - 135712 q^{76} + 2178 q^{77} - 149598 q^{78} + 17122 q^{79} + 13122 q^{81} + 101810 q^{82} + 14304 q^{83} + 4716 q^{84} + 204240 q^{86} - 113400 q^{87} - 20691 q^{88} - 58140 q^{89} - 32584 q^{91} - 248008 q^{92} + 89424 q^{93} - 124660 q^{94} - 34227 q^{96} + 183056 q^{97} + 22047 q^{98} + 19602 q^{99}+O(q^{100}) \) 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
9.34590
−8.34590
−9.34590 9.00000 55.3459 0 −84.1131 −8.69181 −218.189 81.0000 0
1.2 8.34590 9.00000 37.6541 0 75.1131 26.6918 47.1885 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.d 2
5.b even 2 1 33.6.a.d 2
15.d odd 2 1 99.6.a.e 2
20.d odd 2 1 528.6.a.q 2
55.d odd 2 1 363.6.a.g 2
165.d even 2 1 1089.6.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.d 2 5.b even 2 1
99.6.a.e 2 15.d odd 2 1
363.6.a.g 2 55.d odd 2 1
528.6.a.q 2 20.d odd 2 1
825.6.a.d 2 1.a even 1 1 trivial
1089.6.a.o 2 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 78 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 78 \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 18T - 232 \) Copy content Toggle raw display
$11$ \( (T - 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 66T - 878128 \) Copy content Toggle raw display
$17$ \( T^{2} - 920T + 210348 \) Copy content Toggle raw display
$19$ \( T^{2} + 2932 T + 2147904 \) Copy content Toggle raw display
$23$ \( T^{2} + 5246 T + 6827232 \) Copy content Toggle raw display
$29$ \( T^{2} + 12600 T + 38326572 \) Copy content Toggle raw display
$31$ \( T^{2} - 9936 T - 4245184 \) Copy content Toggle raw display
$37$ \( T^{2} + 5996 T + 975204 \) Copy content Toggle raw display
$41$ \( T^{2} - 24244 T + 105471636 \) Copy content Toggle raw display
$43$ \( T^{2} + 20360 T - 16684800 \) Copy content Toggle raw display
$47$ \( T^{2} - 5806 T - 38936064 \) Copy content Toggle raw display
$53$ \( T^{2} + 40770 T + 387938808 \) Copy content Toggle raw display
$59$ \( T^{2} - 18212 T - 974471136 \) Copy content Toggle raw display
$61$ \( T^{2} + 11398 T - 557566776 \) Copy content Toggle raw display
$67$ \( T^{2} + 65368 T + 1022090128 \) Copy content Toggle raw display
$71$ \( T^{2} - 61446 T + 190949616 \) Copy content Toggle raw display
$73$ \( T^{2} + 53412 T + 705197636 \) Copy content Toggle raw display
$79$ \( T^{2} - 17122 T - 281721704 \) Copy content Toggle raw display
$83$ \( T^{2} - 14304 T - 922809744 \) Copy content Toggle raw display
$89$ \( T^{2} + 58140 T + 129501828 \) Copy content Toggle raw display
$97$ \( T^{2} - 183056 T + 8284064476 \) Copy content Toggle raw display
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