Properties

Label 825.6.a.n
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 28) q^{4} + 9 \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} - 6 \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} - 24 \beta_1 - 18) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 28) q^{4} + 9 \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} - 6 \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} - 24 \beta_1 - 18) q^{8} + 81 q^{9} + 121 q^{11} + ( - 9 \beta_{2} - 252) q^{12} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 7 \beta_{2} + 22 \beta_1 - 207) q^{13} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 8 \beta_{2} - 29 \beta_1 + 370) q^{14} + (\beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 28 \beta_{2} - 17 \beta_1 + 536) q^{16} + ( - \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + 14 \beta_{2} - 56 \beta_1 - 79) q^{17} - 81 \beta_1 q^{18} + ( - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 9 \beta_{2} - 42 \beta_1 + 372) q^{19} + (9 \beta_{4} - 9 \beta_{2} + 54 \beta_1 - 9) q^{21} - 121 \beta_1 q^{22} + ( - 3 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} - \beta_{3} + 5 \beta_{2} - 74 \beta_1 - 66) q^{23} + (9 \beta_{3} - 9 \beta_{2} + 216 \beta_1 + 162) q^{24} + (2 \beta_{6} + 7 \beta_{5} - 23 \beta_{4} + 9 \beta_{3} - 74 \beta_{2} + \cdots - 1162) q^{26}+ \cdots + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} - 63 q^{3} + 195 q^{4} + 9 q^{6} - 153 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} - 63 q^{3} + 195 q^{4} + 9 q^{6} - 153 q^{8} + 567 q^{9} + 847 q^{11} - 1755 q^{12} - 1418 q^{13} + 2548 q^{14} + 3699 q^{16} - 630 q^{17} - 81 q^{18} + 2572 q^{19} - 121 q^{22} - 536 q^{23} + 1377 q^{24} - 7626 q^{26} - 5103 q^{27} + 11368 q^{28} - 1038 q^{29} + 1872 q^{31} + 7523 q^{32} - 7623 q^{33} + 20790 q^{34} + 15795 q^{36} - 24298 q^{37} + 18952 q^{38} + 12762 q^{39} - 17658 q^{41} - 22932 q^{42} - 7244 q^{43} + 23595 q^{44} + 31016 q^{46} - 34560 q^{47} - 33291 q^{48} + 78735 q^{49} + 5670 q^{51} - 110222 q^{52} + 10214 q^{53} + 729 q^{54} + 81124 q^{56} - 23148 q^{57} + 5718 q^{58} + 94676 q^{59} + 69538 q^{61} + 4208 q^{62} + 112339 q^{64} + 1089 q^{66} - 64908 q^{67} + 136010 q^{68} + 4824 q^{69} + 61816 q^{71} - 12393 q^{72} + 11890 q^{73} - 124050 q^{74} - 47216 q^{76} + 68634 q^{78} + 18928 q^{79} + 45927 q^{81} - 36398 q^{82} - 17492 q^{83} - 102312 q^{84} - 216688 q^{86} + 9342 q^{87} - 18513 q^{88} + 25302 q^{89} + 3392 q^{91} + 27408 q^{92} - 16848 q^{93} - 30800 q^{94} - 67707 q^{96} + 172546 q^{97} + 615271 q^{98} + 68607 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 88\nu - 78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -15\nu^{6} + 71\nu^{5} + 2323\nu^{4} - 11587\nu^{3} - 70328\nu^{2} + 346212\nu - 34776 ) / 2344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{6} + 45\nu^{5} - 1631\nu^{4} - 7641\nu^{3} + 104468\nu^{2} + 291764\nu - 1252344 ) / 2344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} + 277\nu^{5} + 2097\nu^{4} - 41409\nu^{3} - 113220\nu^{2} + 1195020\nu + 889544 ) / 2344 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 88\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 3\beta_{5} - 2\beta_{4} - 2\beta_{3} + 124\beta_{2} - 17\beta _1 + 5272 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{6} + 9\beta_{5} + 153\beta_{3} - 216\beta_{2} + 8775\beta _1 + 1126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 188\beta_{6} - 422\beta_{5} - 466\beta_{4} - 358\beta_{3} + 14265\beta_{2} - 5994\beta _1 + 524252 \) 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
10.4220
9.12102
5.17374
−0.0402667
−5.88025
−6.96291
−10.8333
−10.4220 −9.00000 76.6176 0 93.7978 101.242 −465.003 81.0000 0
1.2 −9.12102 −9.00000 51.1931 0 82.0892 −202.409 −175.061 81.0000 0
1.3 −5.17374 −9.00000 −5.23241 0 46.5637 −24.5405 192.631 81.0000 0
1.4 0.0402667 −9.00000 −31.9984 0 −0.362400 −37.9248 −2.57700 81.0000 0
1.5 5.88025 −9.00000 2.57733 0 −52.9222 219.194 −173.013 81.0000 0
1.6 6.96291 −9.00000 16.4822 0 −62.6662 −244.038 −108.049 81.0000 0
1.7 10.8333 −9.00000 85.3606 0 −97.4998 188.476 578.072 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.n 7
5.b even 2 1 165.6.a.h 7
15.d odd 2 1 495.6.a.n 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.h 7 5.b even 2 1
495.6.a.n 7 15.d odd 2 1
825.6.a.n 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + T_{2}^{6} - 209T_{2}^{5} - 137T_{2}^{4} + 12724T_{2}^{3} + 1040T_{2}^{2} - 218208T_{2} + 8784 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + T^{6} - 209 T^{5} - 137 T^{4} + \cdots + 8784 \) Copy content Toggle raw display
$3$ \( (T + 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots - 192283411073024 \) Copy content Toggle raw display
$11$ \( (T - 121)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + 1418 T^{6} + \cdots - 62\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{7} + 630 T^{6} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{7} - 2572 T^{6} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} + 536 T^{6} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{7} + 1038 T^{6} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} - 1872 T^{6} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{7} + 24298 T^{6} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{7} + 17658 T^{6} + \cdots + 71\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{7} + 7244 T^{6} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} + 34560 T^{6} + \cdots - 24\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{7} - 10214 T^{6} + \cdots - 47\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{7} - 94676 T^{6} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} - 69538 T^{6} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{7} + 64908 T^{6} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{7} - 61816 T^{6} + \cdots + 83\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{7} - 11890 T^{6} + \cdots - 45\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{7} - 18928 T^{6} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + 17492 T^{6} + \cdots + 39\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{7} - 25302 T^{6} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} - 172546 T^{6} + \cdots + 29\!\cdots\!68 \) Copy content Toggle raw display
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