Properties

Label 825.6.a.g
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(3\)
Coefficient field: 3.3.18257.1
Defining polynomial: \( x^{3} - x^{2} - 26x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + (9 \beta_1 - 9) q^{6} + (6 \beta_{2} + 20 \beta_1 + 14) q^{7} + ( - 2 \beta_{2} - 37 \beta_1 + 21) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + (9 \beta_1 - 9) q^{6} + (6 \beta_{2} + 20 \beta_1 + 14) q^{7} + ( - 2 \beta_{2} - 37 \beta_1 + 21) q^{8} + 81 q^{9} + 121 q^{11} + (9 \beta_{2} - 9 \beta_1 - 126) q^{12} + ( - 44 \beta_{2} + 24 \beta_1 - 90) q^{13} + (14 \beta_{2} + 68 \beta_1 + 278) q^{14} + ( - 67 \beta_{2} + 35 \beta_1 - 186) q^{16} + (102 \beta_{2} - 236 \beta_1 - 100) q^{17} + (81 \beta_1 - 81) q^{18} + ( - 46 \beta_{2} + 172 \beta_1 - 994) q^{19} + (54 \beta_{2} + 180 \beta_1 + 126) q^{21} + (121 \beta_1 - 121) q^{22} + ( - 64 \beta_{2} - 688 \beta_1 + 464) q^{23} + ( - 18 \beta_{2} - 333 \beta_1 + 189) q^{24} + (68 \beta_{2} - 486 \beta_1 + 850) q^{26} + 729 q^{27} + ( - 138 \beta_{2} - 236 \beta_1 + 318) q^{28} + (330 \beta_{2} + 652 \beta_1 - 1840) q^{29} + ( - 44 \beta_{2} - 56 \beta_1 - 4956) q^{31} + (166 \beta_{2} + 395 \beta_1 + 645) q^{32} + 1089 q^{33} + ( - 338 \beta_{2} + 818 \beta_1 - 4728) q^{34} + (81 \beta_{2} - 81 \beta_1 - 1134) q^{36} + ( - 648 \beta_{2} + 448 \beta_1 + 2130) q^{37} + (218 \beta_{2} - 1408 \beta_1 + 4286) q^{38} + ( - 396 \beta_{2} + 216 \beta_1 - 810) q^{39} + ( - 330 \beta_{2} - 1804 \beta_1 - 2264) q^{41} + (126 \beta_{2} + 612 \beta_1 + 2502) q^{42} + ( - 618 \beta_{2} - 1340 \beta_1 + 11850) q^{43} + (121 \beta_{2} - 121 \beta_1 - 1694) q^{44} + ( - 624 \beta_{2} - 112 \beta_1 - 11648) q^{46} + ( - 364 \beta_{2} + 1544 \beta_1 + 7820) q^{47} + ( - 603 \beta_{2} + 315 \beta_1 - 1674) q^{48} + (280 \beta_{2} + 2832 \beta_1 - 5935) q^{49} + (918 \beta_{2} - 2124 \beta_1 - 900) q^{51} + (854 \beta_{2} + 694 \beta_1 - 6776) q^{52} + (1196 \beta_{2} + 360 \beta_1 + 7126) q^{53} + (729 \beta_1 - 729) q^{54} + ( - 546 \beta_{2} - 3100 \beta_1 - 12122) q^{56} + ( - 414 \beta_{2} + 1548 \beta_1 - 8946) q^{57} + (322 \beta_{2} + 1130 \beta_1 + 10284) q^{58} + ( - 2864 \beta_{2} - 4928 \beta_1 - 1988) q^{59} + ( - 616 \beta_{2} + 432 \beta_1 + 8262) q^{61} + ( - 12 \beta_{2} - 5352 \beta_1 + 4356) q^{62} + (486 \beta_{2} + 1620 \beta_1 + 1134) q^{63} + (2373 \beta_{2} + 1019 \beta_1 + 10694) q^{64} + (1089 \beta_1 - 1089) q^{66} + (64 \beta_{2} - 3360 \beta_1 - 4524) q^{67} + ( - 2108 \beta_{2} - 218 \beta_1 + 24538) q^{68} + ( - 576 \beta_{2} - 6192 \beta_1 + 4176) q^{69} + (3576 \beta_{2} - 3376 \beta_1 + 1552) q^{71} + ( - 162 \beta_{2} - 2997 \beta_1 + 1701) q^{72} + ( - 1028 \beta_{2} + 1656 \beta_1 - 846) q^{73} + (1096 \beta_{2} - 3702 \beta_1 + 10670) q^{74} + ( - 154 \beta_{2} + 744 \beta_1 + 1842) q^{76} + (726 \beta_{2} + 2420 \beta_1 + 1694) q^{77} + (612 \beta_{2} - 4374 \beta_1 + 7650) q^{78} + ( - 4202 \beta_{2} - 8252 \beta_1 - 8130) q^{79} + 6561 q^{81} + ( - 1474 \beta_{2} - 5234 \beta_1 - 25764) q^{82} + ( - 2800 \beta_{2} + 5744 \beta_1 + 15284) q^{83} + ( - 1242 \beta_{2} - 2124 \beta_1 + 2862) q^{84} + ( - 722 \beta_{2} + 6288 \beta_1 - 29686) q^{86} + (2970 \beta_{2} + 5868 \beta_1 - 16560) q^{87} + ( - 242 \beta_{2} - 4477 \beta_1 + 2541) q^{88} + (2640 \beta_{2} + 8896 \beta_1 - 66838) q^{89} + (1436 \beta_{2} - 5496 \beta_1 - 29716) q^{91} + (2560 \beta_{2} + 4752 \beta_1 - 112) q^{92} + ( - 396 \beta_{2} - 504 \beta_1 - 44604) q^{93} + (1908 \beta_{2} + 4544 \beta_1 + 21340) q^{94} + (1494 \beta_{2} + 3555 \beta_1 + 5805) q^{96} + (48 \beta_{2} + 13696 \beta_1 - 87090) q^{97} + (2552 \beta_{2} - 3415 \beta_1 + 51839) q^{98} + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9} + 363 q^{11} - 378 q^{12} - 290 q^{13} + 916 q^{14} - 590 q^{16} - 434 q^{17} - 162 q^{18} - 2856 q^{19} + 612 q^{21} - 242 q^{22} + 640 q^{23} + 216 q^{24} + 2132 q^{26} + 2187 q^{27} + 580 q^{28} - 4538 q^{29} - 14968 q^{31} + 2496 q^{32} + 3267 q^{33} - 13704 q^{34} - 3402 q^{36} + 6190 q^{37} + 11668 q^{38} - 2610 q^{39} - 8926 q^{41} + 8244 q^{42} + 33592 q^{43} - 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 5310 q^{48} - 14693 q^{49} - 3906 q^{51} - 18780 q^{52} + 22934 q^{53} - 1458 q^{54} - 40012 q^{56} - 25704 q^{57} + 32304 q^{58} - 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 5508 q^{63} + 35474 q^{64} - 2178 q^{66} - 16868 q^{67} + 71288 q^{68} + 5760 q^{69} + 4856 q^{71} + 1944 q^{72} - 1910 q^{73} + 29404 q^{74} + 6116 q^{76} + 8228 q^{77} + 19188 q^{78} - 36844 q^{79} + 19683 q^{81} - 84000 q^{82} + 48796 q^{83} + 5220 q^{84} - 83492 q^{86} - 40842 q^{87} + 2904 q^{88} - 188978 q^{89} - 93208 q^{91} + 6976 q^{92} - 134712 q^{93} + 70472 q^{94} + 22464 q^{96} - 247526 q^{97} + 154654 q^{98} + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 26x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 17 \) 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
−4.78415
0.305203
5.47894
−5.78415 9.00000 1.45634 0 −52.0573 −17.6498 176.669 81.0000 0
1.2 −0.694797 9.00000 −31.5173 0 −6.25317 −83.1683 44.1316 81.0000 0
1.3 4.47894 9.00000 −11.9391 0 40.3105 168.818 −196.801 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.g 3
5.b even 2 1 165.6.a.c 3
15.d odd 2 1 495.6.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.c 3 5.b even 2 1
495.6.a.b 3 15.d odd 2 1
825.6.a.g 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2T_{2}^{2} - 25T_{2} - 18 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 25 T - 18 \) Copy content Toggle raw display
$3$ \( (T - 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 68 T^{2} - 15552 T - 247808 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 290 T^{2} + \cdots - 132063592 \) Copy content Toggle raw display
$17$ \( T^{3} + 434 T^{2} + \cdots - 2547052488 \) Copy content Toggle raw display
$19$ \( T^{3} + 2856 T^{2} + \cdots + 137703680 \) Copy content Toggle raw display
$23$ \( T^{3} - 640 T^{2} + \cdots + 15777349632 \) Copy content Toggle raw display
$29$ \( T^{3} + 4538 T^{2} + \cdots - 44413548456 \) Copy content Toggle raw display
$31$ \( T^{3} + 14968 T^{2} + \cdots + 121645522944 \) Copy content Toggle raw display
$37$ \( T^{3} - 6190 T^{2} + \cdots - 28013661736 \) Copy content Toggle raw display
$41$ \( T^{3} + 8926 T^{2} + \cdots + 119305168392 \) Copy content Toggle raw display
$43$ \( T^{3} - 33592 T^{2} + \cdots + 38997547520 \) Copy content Toggle raw display
$47$ \( T^{3} - 24640 T^{2} + \cdots + 679997104128 \) Copy content Toggle raw display
$53$ \( T^{3} - 22934 T^{2} + \cdots + 4393759072056 \) Copy content Toggle raw display
$59$ \( T^{3} + 13756 T^{2} + \cdots - 20798004639936 \) Copy content Toggle raw display
$61$ \( T^{3} - 24602 T^{2} + \cdots + 43064794504 \) Copy content Toggle raw display
$67$ \( T^{3} + 16868 T^{2} + \cdots - 1826752720192 \) Copy content Toggle raw display
$71$ \( T^{3} - 4856 T^{2} + \cdots + 34155066048000 \) Copy content Toggle raw display
$73$ \( T^{3} + 1910 T^{2} + \cdots + 163103734088 \) Copy content Toggle raw display
$79$ \( T^{3} + 36844 T^{2} + \cdots - 70772253539328 \) Copy content Toggle raw display
$83$ \( T^{3} - 48796 T^{2} + \cdots + 70386077185728 \) Copy content Toggle raw display
$89$ \( T^{3} + 188978 T^{2} + \cdots - 16088649675432 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 148869121092488 \) Copy content Toggle raw display
show more
show less