Properties

Label 825.6.a.i
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.3368.1
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} - 9 q^{3} + (4 \beta_{2} + 8) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (11 \beta_{2} - 14 \beta_1 + 69) q^{7} + (8 \beta_{2} - 4 \beta_1 - 12) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} - 9 q^{3} + (4 \beta_{2} + 8) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (11 \beta_{2} - 14 \beta_1 + 69) q^{7} + (8 \beta_{2} - 4 \beta_1 - 12) q^{8} + 81 q^{9} + 121 q^{11} + ( - 36 \beta_{2} - 72) q^{12} + (31 \beta_{2} + 25 \beta_1 - 152) q^{13} + ( - 34 \beta_{2} + 138 \beta_1 - 444) q^{14} + ( - 128 \beta_{2} + 32 \beta_1 - 400) q^{16} + (57 \beta_{2} - 34 \beta_1 + 81) q^{17} + (81 \beta_1 + 81) q^{18} + (136 \beta_{2} + 119 \beta_1 - 1351) q^{19} + ( - 99 \beta_{2} + 126 \beta_1 - 621) q^{21} + (121 \beta_1 + 121) q^{22} + ( - 164 \beta_{2} - 372 \beta_1 + 2284) q^{23} + ( - 72 \beta_{2} + 36 \beta_1 + 108) q^{24} + (162 \beta_{2} - 22 \beta_1 + 916) q^{26} - 729 q^{27} + (132 \beta_{2} - 304 \beta_1 + 2628) q^{28} + (580 \beta_{2} + 93 \beta_1 + 1185) q^{29} + (764 \beta_{2} - 920 \beta_1 - 764) q^{31} + ( - 384 \beta_{2} - 944 \beta_1 + 848) q^{32} - 1089 q^{33} + ( - 22 \beta_{2} + 400 \beta_1 - 1074) q^{34} + (324 \beta_{2} + 648) q^{36} + (1312 \beta_{2} - 410 \beta_1 - 1324) q^{37} + (748 \beta_{2} - 790 \beta_1 + 3698) q^{38} + ( - 279 \beta_{2} - 225 \beta_1 + 1368) q^{39} + (140 \beta_{2} - 521 \beta_1 + 3331) q^{41} + (306 \beta_{2} - 1242 \beta_1 + 3996) q^{42} + (37 \beta_{2} + 2 \beta_1 + 11831) q^{43} + (484 \beta_{2} + 968) q^{44} + ( - 1816 \beta_{2} + 1836 \beta_1 - 12716) q^{46} + ( - 1004 \beta_{2} + 640 \beta_1 + 1248) q^{47} + (1152 \beta_{2} - 288 \beta_1 + 3600) q^{48} + (1510 \beta_{2} - 3622 \beta_1 + 845) q^{49} + ( - 513 \beta_{2} + 306 \beta_1 - 729) q^{51} + ( - 756 \beta_{2} + 948 \beta_1 + 5408) q^{52} + (2746 \beta_{2} + 2226 \beta_1 + 4102) q^{53} + ( - 729 \beta_1 - 729) q^{54} + (136 \beta_{2} - 824 \beta_1 + 5376) q^{56} + ( - 1224 \beta_{2} - 1071 \beta_1 + 12159) q^{57} + (1532 \beta_{2} + 3992 \beta_1 + 6552) q^{58} + (844 \beta_{2} + 3292 \beta_1 - 26476) q^{59} + (916 \beta_{2} - 3326 \beta_1 - 22180) q^{61} + ( - 2152 \beta_{2} + 3976 \beta_1 - 34352) q^{62} + (891 \beta_{2} - 1134 \beta_1 + 5589) q^{63} + ( - 448 \beta_{2} - 1152 \beta_1 - 24320) q^{64} + ( - 1089 \beta_1 - 1089) q^{66} + (4474 \beta_{2} - 496 \beta_1 + 13726) q^{67} + ( - 268 \beta_{2} - 496 \beta_1 + 11868) q^{68} + (1476 \beta_{2} + 3348 \beta_1 - 20556) q^{69} + ( - 26 \beta_{2} + 1080 \beta_1 - 21206) q^{71} + (648 \beta_{2} - 324 \beta_1 - 972) q^{72} + ( - 3773 \beta_{2} + 8443 \beta_1 + 3802) q^{73} + (984 \beta_{2} + 5646 \beta_1 - 13378) q^{74} + ( - 6016 \beta_{2} + 4420 \beta_1 + 18364) q^{76} + (1331 \beta_{2} - 1694 \beta_1 + 8349) q^{77} + ( - 1458 \beta_{2} + 198 \beta_1 - 8244) q^{78} + (10742 \beta_{2} - 5831 \beta_1 + 9101) q^{79} + 6561 q^{81} + ( - 1804 \beta_{2} + 4552 \beta_1 - 16568) q^{82} + (2095 \beta_{2} + 4559 \beta_1 + 34496) q^{83} + ( - 1188 \beta_{2} + 2736 \beta_1 - 23652) q^{84} + (82 \beta_{2} + 12014 \beta_1 + 12020) q^{86} + ( - 5220 \beta_{2} - 837 \beta_1 - 10665) q^{87} + (968 \beta_{2} - 484 \beta_1 - 1452) q^{88} + ( - 12390 \beta_{2} + 2204 \beta_1 + 24872) q^{89} + ( - 2456 \beta_{2} + 4440 \beta_1 - 7224) q^{91} + (8960 \beta_{2} - 11728 \beta_1 - 19648) q^{92} + ( - 6876 \beta_{2} + 8280 \beta_1 + 6876) q^{93} + (552 \beta_{2} - 4412 \beta_1 + 23196) q^{94} + (3456 \beta_{2} + 8496 \beta_1 - 7632) q^{96} + ( - 3142 \beta_{2} - 10816 \beta_1 + 104024) q^{97} + ( - 11468 \beta_{2} + 12017 \beta_1 - 135883) q^{98} + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 15x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) 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
−3.76300
0.723686
4.03932
−7.52601 −9.00000 24.6408 0 67.7341 234.126 55.3856 81.0000 0
1.2 1.44737 −9.00000 −29.9051 0 −13.0263 −41.5023 −89.5997 81.0000 0
1.3 8.07863 −9.00000 33.2643 0 −72.7077 39.3760 10.2141 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.i 3
5.b even 2 1 165.6.a.b 3
15.d odd 2 1 495.6.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.b 3 5.b even 2 1
495.6.a.d 3 15.d odd 2 1
825.6.a.i 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2T_{2}^{2} - 60T_{2} + 88 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} - 60 T + 88 \) Copy content Toggle raw display
$3$ \( (T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 232 T^{2} - 2132 T + 382608 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 450 T^{2} + \cdots - 22659488 \) Copy content Toggle raw display
$17$ \( T^{3} - 334 T^{2} + \cdots + 57782448 \) Copy content Toggle raw display
$19$ \( T^{3} + 4036 T^{2} + \cdots - 1630951200 \) Copy content Toggle raw display
$23$ \( T^{3} - 7060 T^{2} + \cdots + 24275701568 \) Copy content Toggle raw display
$29$ \( T^{3} - 4042 T^{2} + \cdots + 65949214584 \) Copy content Toggle raw display
$31$ \( T^{3} + 608 T^{2} + \cdots - 211578448896 \) Copy content Toggle raw display
$37$ \( T^{3} + 2250 T^{2} + \cdots + 431879868536 \) Copy content Toggle raw display
$41$ \( T^{3} - 10654 T^{2} + \cdots - 7803557208 \) Copy content Toggle raw display
$43$ \( T^{3} - 35528 T^{2} + \cdots - 1659712050000 \) Copy content Toggle raw display
$47$ \( T^{3} - 2100 T^{2} + \cdots - 52162385088 \) Copy content Toggle raw display
$53$ \( T^{3} - 12826 T^{2} + \cdots - 2687939232856 \) Copy content Toggle raw display
$59$ \( T^{3} + 81876 T^{2} + \cdots - 3633753791296 \) Copy content Toggle raw display
$61$ \( T^{3} + 62298 T^{2} + \cdots - 12904038746056 \) Copy content Toggle raw display
$67$ \( T^{3} - 46148 T^{2} + \cdots + 40648408406912 \) Copy content Toggle raw display
$71$ \( T^{3} + 64724 T^{2} + \cdots + 8578136735360 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 144432126809632 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 351884592248992 \) Copy content Toggle raw display
$83$ \( T^{3} - 101024 T^{2} + \cdots - 5794291383408 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 246103360939432 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 179909862970168 \) Copy content Toggle raw display
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