Properties

Label 825.6.a.q
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
−10.0327
−8.09383
−4.15857
0.292443
0.894159
5.43413
7.96197
8.70245
−9.03274 −9.00000 49.5905 0 81.2947 −36.7509 −158.890 81.0000 0
1.2 −7.09383 −9.00000 18.3224 0 63.8444 4.62438 97.0268 81.0000 0
1.3 −3.15857 −9.00000 −22.0234 0 28.4272 −35.7175 170.637 81.0000 0
1.4 1.29244 −9.00000 −30.3296 0 −11.6320 91.7671 −80.5574 81.0000 0
1.5 1.89416 −9.00000 −28.4122 0 −17.0474 −191.987 −114.430 81.0000 0
1.6 6.43413 −9.00000 9.39797 0 −57.9071 −9.96708 −145.424 81.0000 0
1.7 8.96197 −9.00000 48.3168 0 −80.6577 191.454 146.231 81.0000 0
1.8 9.70245 −9.00000 62.1375 0 −87.3220 −79.4231 292.408 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.q yes 8
5.b even 2 1 825.6.a.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.p 8 5.b even 2 1
825.6.a.q yes 8 1.a even 1 1 trivial

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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