Properties

Label 825.6.a.p
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + ( - \beta_{6} - \beta_{2} - 3 \beta_1 + 9) q^{7} + ( - \beta_{3} - \beta_{2} - 8 \beta_1 - 25) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + ( - \beta_{6} - \beta_{2} - 3 \beta_1 + 9) q^{7} + ( - \beta_{3} - \beta_{2} - 8 \beta_1 - 25) q^{8} + 81 q^{9} - 121 q^{11} + (9 \beta_{2} + 9 \beta_1 + 117) q^{12} + (\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 42 \beta_1 - 52) q^{13} + (\beta_{7} + 5 \beta_{6} - 2 \beta_{5} - \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 128) q^{14} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 31 \beta_1 - 43) q^{16} + (10 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 35) q^{17} + ( - 81 \beta_1 - 81) q^{18} + ( - 3 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \cdots - 125) q^{19}+ \cdots - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9} - 968 q^{11} + 963 q^{12} - 382 q^{13} + 1048 q^{14} - 325 q^{16} - 288 q^{17} - 729 q^{18} - 988 q^{19} + 594 q^{21} + 1089 q^{22} - 5972 q^{23} - 1863 q^{24} - 14579 q^{26} + 5832 q^{27} - 5942 q^{28} + 1032 q^{29} - 4682 q^{31} - 3863 q^{32} - 8712 q^{33} + 2206 q^{34} + 8667 q^{36} + 17200 q^{37} - 11011 q^{38} - 3438 q^{39} - 13220 q^{41} + 9432 q^{42} - 22872 q^{43} - 12947 q^{44} + 9101 q^{46} - 6700 q^{47} - 2925 q^{48} - 43466 q^{49} - 2592 q^{51} - 5009 q^{52} - 6224 q^{53} - 6561 q^{54} + 20992 q^{56} - 8892 q^{57} - 33015 q^{58} - 77556 q^{59} + 11554 q^{61} - 12135 q^{62} + 5346 q^{63} - 149917 q^{64} + 9801 q^{66} - 20894 q^{67} - 91776 q^{68} - 53748 q^{69} - 21648 q^{71} - 16767 q^{72} - 64660 q^{73} - 179522 q^{74} + 24401 q^{76} - 7986 q^{77} - 131211 q^{78} - 22660 q^{79} + 52488 q^{81} + 56080 q^{82} - 100390 q^{83} - 53478 q^{84} + 47271 q^{86} + 9288 q^{87} + 25047 q^{88} - 25578 q^{89} + 73250 q^{91} - 95311 q^{92} - 42138 q^{93} - 170120 q^{94} - 34767 q^{96} - 142828 q^{97} - 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu^{2} - 70\nu - 44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 14\nu^{5} + 164\nu^{4} + 1600\nu^{3} - 7927\nu^{2} - 36334\nu + 55536 ) / 160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 160\nu^{5} + 176\nu^{4} + 7407\nu^{3} - 13084\nu^{2} - 85572\nu + 99104 ) / 320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 188\nu^{5} + 344\nu^{4} + 10287\nu^{3} - 15018\nu^{2} - 146720\nu + 89856 ) / 320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} - 4\nu^{6} + 504\nu^{5} + 208\nu^{4} - 24861\nu^{3} + 6664\nu^{2} + 314100\nu - 129568 ) / 320 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_{2} + 72\beta _1 - 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 91\beta_{2} - 159\beta _1 + 3164 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} + 2\beta_{6} + 10\beta_{5} - 10\beta_{4} + 115\beta_{3} - 306\beta_{2} + 5750\beta _1 - 6972 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -56\beta_{7} - 356\beta_{6} + 188\beta_{5} + 308\beta_{4} - 338\beta_{3} + 8081\beta_{2} - 19783\beta _1 + 252852 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 640 \beta_{7} + 672 \beta_{6} + 1568 \beta_{5} - 1952 \beta_{4} + 11345 \beta_{3} - 37078 \beta_{2} + 487168 \beta _1 - 869884 \) 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
8.70245
7.96197
5.43413
0.894159
0.292443
−4.15857
−8.09383
−10.0327
−9.70245 9.00000 62.1375 0 −87.3220 79.4231 −292.408 81.0000 0
1.2 −8.96197 9.00000 48.3168 0 −80.6577 −191.454 −146.231 81.0000 0
1.3 −6.43413 9.00000 9.39797 0 −57.9071 9.96708 145.424 81.0000 0
1.4 −1.89416 9.00000 −28.4122 0 −17.0474 191.987 114.430 81.0000 0
1.5 −1.29244 9.00000 −30.3296 0 −11.6320 −91.7671 80.5574 81.0000 0
1.6 3.15857 9.00000 −22.0234 0 28.4272 35.7175 −170.637 81.0000 0
1.7 7.09383 9.00000 18.3224 0 63.8444 −4.62438 −97.0268 81.0000 0
1.8 9.03274 9.00000 49.5905 0 81.2947 36.7509 158.890 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.p 8
5.b even 2 1 825.6.a.q yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.p 8 1.a even 1 1 trivial
825.6.a.q yes 8 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 9T_{2}^{7} - 141T_{2}^{6} - 1251T_{2}^{5} + 5160T_{2}^{4} + 45922T_{2}^{3} - 22268T_{2}^{2} - 305880T_{2} - 277200 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 9 T^{7} - 141 T^{6} + \cdots - 277200 \) Copy content Toggle raw display
$3$ \( (T - 9)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 66 T^{7} + \cdots - 16208419192764 \) Copy content Toggle raw display
$11$ \( (T + 121)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 382 T^{7} + \cdots + 29\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{8} + 288 T^{7} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + 988 T^{7} + \cdots - 27\!\cdots\!23 \) Copy content Toggle raw display
$23$ \( T^{8} + 5972 T^{7} + \cdots - 31\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{8} - 1032 T^{7} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{8} + 4682 T^{7} + \cdots - 95\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} - 17200 T^{7} + \cdots - 29\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + 13220 T^{7} + \cdots - 33\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{8} + 22872 T^{7} + \cdots + 68\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{8} + 6700 T^{7} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + 6224 T^{7} + \cdots - 55\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{8} + 77556 T^{7} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} - 11554 T^{7} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + 20894 T^{7} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + 21648 T^{7} + \cdots + 39\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{8} + 64660 T^{7} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + 22660 T^{7} + \cdots + 21\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{8} + 100390 T^{7} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + 25578 T^{7} + \cdots + 21\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{8} + 142828 T^{7} + \cdots - 13\!\cdots\!75 \) Copy content Toggle raw display
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