Properties

Label 825.6.a.k
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 9 q^{3} + (\beta_{3} + \beta_1 + 25) q^{4} + 9 \beta_1 q^{6} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 - 24) q^{7} + (3 \beta_{3} + 5 \beta_{2} + 20 \beta_1 + 26) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 9 q^{3} + (\beta_{3} + \beta_1 + 25) q^{4} + 9 \beta_1 q^{6} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 - 24) q^{7} + (3 \beta_{3} + 5 \beta_{2} + 20 \beta_1 + 26) q^{8} + 81 q^{9} - 121 q^{11} + (9 \beta_{3} + 9 \beta_1 + 225) q^{12} + ( - \beta_{4} + 10 \beta_{3} + 2 \beta_{2} - 28 \beta_1 + 189) q^{13} + ( - 3 \beta_{4} - 10 \beta_{3} + 8 \beta_{2} - 4 \beta_1 + 77) q^{14} + (5 \beta_{4} + 19 \beta_{3} + 10 \beta_{2} + 107 \beta_1 + 352) q^{16} + (9 \beta_{3} + 37 \beta_{2} + 27 \beta_1 + 42) q^{17} + 81 \beta_1 q^{18} + ( - 7 \beta_{4} - 3 \beta_{3} + 25 \beta_{2} + 145 \beta_1 + 657) q^{19} + (9 \beta_{3} - 27 \beta_{2} + 27 \beta_1 - 216) q^{21} - 121 \beta_1 q^{22} + ( - 8 \beta_{4} - 16 \beta_{3} - 32 \beta_{2} - 352 \beta_1 + 924) q^{23} + (27 \beta_{3} + 45 \beta_{2} + 180 \beta_1 + 234) q^{24} + (3 \beta_{4} + 4 \beta_{3} + 16 \beta_{2} + 446 \beta_1 - 1863) q^{26} + 729 q^{27} + (11 \beta_{4} - 10 \beta_{3} - 58 \beta_{2} - 202 \beta_1 + 1021) q^{28} + (9 \beta_{4} + 85 \beta_{3} - 79 \beta_{2} + 169 \beta_1 - 637) q^{29} + ( - 24 \beta_{4} - 32 \beta_{2} + 512 \beta_1 - 1320) q^{31} + (5 \beta_{4} + 89 \beta_{3} + 85 \beta_{2} + 248 \beta_1 + 4883) q^{32} - 1089 q^{33} + (37 \beta_{4} + 230 \beta_{3} + 8 \beta_{2} + 414 \beta_1 + 2037) q^{34} + (81 \beta_{3} + 81 \beta_1 + 2025) q^{36} + ( - 6 \beta_{4} - 110 \beta_{3} + 170 \beta_{2} + 122 \beta_1 - 1332) q^{37} + (32 \beta_{4} + 278 \beta_{3} - 264 \beta_{2} + 932 \beta_1 + 8890) q^{38} + ( - 9 \beta_{4} + 90 \beta_{3} + 18 \beta_{2} - 252 \beta_1 + 1701) q^{39} + (7 \beta_{4} - 69 \beta_{3} - 161 \beta_{2} - 57 \beta_1 - 2963) q^{41} + ( - 27 \beta_{4} - 90 \beta_{3} + 72 \beta_{2} - 36 \beta_1 + 693) q^{42} + ( - 48 \beta_{4} - 25 \beta_{3} - 101 \beta_{2} - 91 \beta_1 - 2128) q^{43} + ( - 121 \beta_{3} - 121 \beta_1 - 3025) q^{44} + ( - 24 \beta_{4} - 528 \beta_{3} - 304 \beta_{2} + 212 \beta_1 - 20232) q^{46} + (16 \beta_{4} + 40 \beta_{3} - 280 \beta_{2} + 1304 \beta_1 + 8020) q^{47} + (45 \beta_{4} + 171 \beta_{3} + 90 \beta_{2} + 963 \beta_1 + 3168) q^{48} + ( - 22 \beta_{4} - 100 \beta_{3} + 108 \beta_{2} - 1032 \beta_1 - 2189) q^{49} + (81 \beta_{3} + 333 \beta_{2} + 243 \beta_1 + 378) q^{51} + (45 \beta_{4} + 208 \beta_{3} + 36 \beta_{2} - 426 \beta_1 + 19583) q^{52} + (2 \beta_{4} + 60 \beta_{3} - 324 \beta_{2} + 536 \beta_1 + 12360) q^{53} + 729 \beta_1 q^{54} + (27 \beta_{4} - 214 \beta_{3} + 104 \beta_{2} + 304 \beta_1 - 14897) q^{56} + ( - 63 \beta_{4} - 27 \beta_{3} + 225 \beta_{2} + 1305 \beta_1 + 5913) q^{57} + ( - 88 \beta_{4} - 74 \beta_{3} + 792 \beta_{2} + 1334 \beta_1 + 5330) q^{58} + (616 \beta_{3} + 40 \beta_{2} - 152 \beta_1 + 1612) q^{59} + (82 \beta_{4} + 258 \beta_{3} - 246 \beta_{2} + 1210 \beta_1 + 11492) q^{61} + ( - 8 \beta_{4} + 400 \beta_{3} - 736 \beta_{2} - 448 \beta_1 + 28536) q^{62} + (81 \beta_{3} - 243 \beta_{2} + 243 \beta_1 - 1944) q^{63} + ( - 80 \beta_{4} + 233 \beta_{3} + 200 \beta_{2} + 4181 \beta_1 + 1893) q^{64} - 1089 \beta_1 q^{66} + (144 \beta_{4} - 322 \beta_{3} - 234 \beta_{2} + 650 \beta_1 + 3060) q^{67} + ( - 29 \beta_{4} + 552 \beta_{3} + 1142 \beta_{2} + 6888 \beta_1 + 15255) q^{68} + ( - 72 \beta_{4} - 144 \beta_{3} - 288 \beta_{2} - 3168 \beta_1 + 8316) q^{69} + ( - 8 \beta_{4} + 74 \beta_{3} + 274 \beta_{2} - 3602 \beta_1 + 18332) q^{71} + (243 \beta_{3} + 405 \beta_{2} + 1620 \beta_1 + 2106) q^{72} + (145 \beta_{4} + 1088 \beta_{3} - 176 \beta_{2} + 810 \beta_1 + 23263) q^{73} + (176 \beta_{4} + 764 \beta_{3} - 912 \beta_{2} - 3446 \beta_1 + 13940) q^{74} + ( - 72 \beta_{4} + 200 \beta_{3} + 1878 \beta_{2} + 11010 \beta_1 + 17906) q^{76} + ( - 121 \beta_{3} + 363 \beta_{2} - 363 \beta_1 + 2904) q^{77} + (27 \beta_{4} + 36 \beta_{3} + 144 \beta_{2} + 4014 \beta_1 - 16767) q^{78} + (39 \beta_{4} - 487 \beta_{3} + 53 \beta_{2} - 6223 \beta_1 + 12799) q^{79} + 6561 q^{81} + ( - 168 \beta_{4} - 1014 \beta_{3} + 40 \beta_{2} - 5430 \beta_1 - 4498) q^{82} + ( - 97 \beta_{4} - 820 \beta_{3} + 348 \beta_{2} + 6170 \beta_1 + 403) q^{83} + (99 \beta_{4} - 90 \beta_{3} - 522 \beta_{2} - 1818 \beta_1 + 9189) q^{84} + ( - 53 \beta_{4} - 550 \beta_{3} - 1560 \beta_{2} - 2260 \beta_1 - 6485) q^{86} + (81 \beta_{4} + 765 \beta_{3} - 711 \beta_{2} + 1521 \beta_1 - 5733) q^{87} + ( - 363 \beta_{3} - 605 \beta_{2} - 2420 \beta_1 - 3146) q^{88} + ( - 160 \beta_{4} + 1638 \beta_{3} - 130 \beta_{2} + 3090 \beta_1 + 13682) q^{89} + (304 \beta_{4} - 124 \beta_{3} - 1020 \beta_{2} + 716 \beta_1 - 472) q^{91} + ( - 24 \beta_{4} - 1804 \beta_{3} - 2080 \beta_{2} - 22940 \beta_1 - 7476) q^{92} + ( - 216 \beta_{4} - 288 \beta_{2} + 4608 \beta_1 - 11880) q^{93} + ( - 296 \beta_{4} - 48 \beta_{3} + 992 \beta_{2} + 9220 \beta_1 + 67192) q^{94} + (45 \beta_{4} + 801 \beta_{3} + 765 \beta_{2} + 2232 \beta_1 + 43947) q^{96} + ( - 360 \beta_{4} - 374 \beta_{3} + 114 \beta_{2} - 7922 \beta_1 + 13566) q^{97} + (130 \beta_{4} - 648 \beta_{3} - 1312 \beta_{2} - 5079 \beta_1 - 53434) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} + 45 q^{3} + 127 q^{4} + 9 q^{6} - 116 q^{7} + 153 q^{8} + 405 q^{9} - 605 q^{11} + 1143 q^{12} + 926 q^{13} + 368 q^{14} + 1891 q^{16} + 246 q^{17} + 81 q^{18} + 3420 q^{19} - 1044 q^{21} - 121 q^{22} + 4244 q^{23} + 1377 q^{24} - 8862 q^{26} + 3645 q^{27} + 4904 q^{28} - 2922 q^{29} - 6112 q^{31} + 24757 q^{32} - 5445 q^{33} + 10866 q^{34} + 10287 q^{36} - 6654 q^{37} + 45692 q^{38} + 8334 q^{39} - 14934 q^{41} + 3312 q^{42} - 10804 q^{43} - 15367 q^{44} - 101500 q^{46} + 41460 q^{47} + 17019 q^{48} - 12099 q^{49} + 2214 q^{51} + 97742 q^{52} + 62398 q^{53} + 729 q^{54} - 74368 q^{56} + 30780 q^{57} + 27822 q^{58} + 8524 q^{59} + 59010 q^{61} + 142624 q^{62} - 9396 q^{63} + 13799 q^{64} - 1089 q^{66} + 15772 q^{67} + 83686 q^{68} + 38196 q^{69} + 88124 q^{71} + 12393 q^{72} + 118358 q^{73} + 67194 q^{74} + 100668 q^{76} + 14036 q^{77} - 79758 q^{78} + 57324 q^{79} + 32805 q^{81} - 29102 q^{82} + 7268 q^{83} + 44136 q^{84} - 35288 q^{86} - 26298 q^{87} - 18513 q^{88} + 72978 q^{89} - 1464 q^{91} - 62148 q^{92} - 55008 q^{93} + 344836 q^{94} + 222813 q^{96} + 59174 q^{97} - 272767 q^{98} - 49005 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 81\nu + 145 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 109\nu^{2} + 74\nu + 1465 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 5\beta_{2} + 84\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{4} + 115\beta_{3} + 10\beta_{2} + 203\beta _1 + 4800 \) 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
−9.46271
−6.98466
0.898099
5.93734
10.6119
−9.46271 9.00000 57.5429 0 −85.1644 112.299 −241.706 81.0000 0
1.2 −6.98466 9.00000 16.7854 0 −62.8619 −180.375 106.269 81.0000 0
1.3 0.898099 9.00000 −31.1934 0 8.08289 −120.732 −56.7540 81.0000 0
1.4 5.93734 9.00000 3.25206 0 53.4361 105.553 −170.686 81.0000 0
1.5 10.6119 9.00000 80.6130 0 95.5073 −32.7446 515.877 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.k 5
5.b even 2 1 165.6.a.g 5
15.d odd 2 1 495.6.a.i 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.g 5 5.b even 2 1
495.6.a.i 5 15.d odd 2 1
825.6.a.k 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - T_{2}^{4} - 143T_{2}^{3} + 71T_{2}^{2} + 4216T_{2} - 3740 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - T^{4} - 143 T^{3} + 71 T^{2} + \cdots - 3740 \) Copy content Toggle raw display
$3$ \( (T - 9)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 116 T^{4} + \cdots + 8452488256 \) Copy content Toggle raw display
$11$ \( (T + 121)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 926 T^{4} + \cdots - 13445413206016 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 475773584554112 \) Copy content Toggle raw display
$19$ \( T^{5} - 3420 T^{4} + \cdots - 25\!\cdots\!92 \) Copy content Toggle raw display
$23$ \( T^{5} - 4244 T^{4} + \cdots - 23\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{5} + 2922 T^{4} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{5} + 6112 T^{4} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} + 6654 T^{4} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{5} + 14934 T^{4} + \cdots - 86\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{5} + 10804 T^{4} + \cdots + 69\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{5} - 41460 T^{4} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{5} - 62398 T^{4} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{5} - 8524 T^{4} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{5} - 59010 T^{4} + \cdots + 27\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{5} - 15772 T^{4} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{5} - 88124 T^{4} + \cdots - 45\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{5} - 118358 T^{4} + \cdots - 70\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{5} - 57324 T^{4} + \cdots - 19\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{5} - 7268 T^{4} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{5} - 72978 T^{4} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{5} - 59174 T^{4} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
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