Properties

Label 825.6.a.h
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: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + 9 q^{3} + ( - 4 \beta_1 - 8) q^{4} + ( - 9 \beta_{2} - 9) q^{6} + (7 \beta_{2} + 31 \beta_1 - 38) q^{7} + (28 \beta_{2} + 8 \beta_1 + 20) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + 9 q^{3} + ( - 4 \beta_1 - 8) q^{4} + ( - 9 \beta_{2} - 9) q^{6} + (7 \beta_{2} + 31 \beta_1 - 38) q^{7} + (28 \beta_{2} + 8 \beta_1 + 20) q^{8} + 81 q^{9} - 121 q^{11} + ( - 36 \beta_1 - 72) q^{12} + ( - 2 \beta_{2} + 77 \beta_1 + 207) q^{13} + (138 \beta_{2} - 34 \beta_1 + 32) q^{14} + (32 \beta_{2} + 224 \beta_1 - 368) q^{16} + ( - 119 \beta_{2} + 219 \beta_1 + 138) q^{17} + ( - 81 \beta_{2} - 81) q^{18} + (149 \beta_{2} + 206 \beta_1 + 721) q^{19} + (63 \beta_{2} + 279 \beta_1 - 342) q^{21} + (121 \beta_{2} + 121) q^{22} + ( - 150 \beta_{2} + 190 \beta_1 - 1416) q^{23} + (252 \beta_{2} + 72 \beta_1 + 180) q^{24} + (22 \beta_{2} - 162 \beta_1 + 224) q^{26} + 729 q^{27} + ( - 220 \beta_{2} - 372 \beta_1 - 2160) q^{28} + (129 \beta_{2} - 308 \beta_1 + 1801) q^{29} + ( - 32 \beta_{2} - 242 \beta_1 + 2018) q^{31} + (176 \beta_{2} - 576 \beta_1 + 112) q^{32} - 1089 q^{33} + (400 \beta_{2} - 914 \beta_1 + 3694) q^{34} + ( - 324 \beta_1 - 648) q^{36} + ( - 1076 \beta_{2} - 928 \beta_1 - 6290) q^{37} + (46 \beta_{2} + 184 \beta_1 - 3118) q^{38} + ( - 18 \beta_{2} + 693 \beta_1 + 1863) q^{39} + ( - 781 \beta_{2} + 1892 \beta_1 + 8131) q^{41} + (1242 \beta_{2} - 306 \beta_1 + 288) q^{42} + ( - 1903 \beta_{2} - 949 \beta_1 - 7808) q^{43} + (484 \beta_1 + 968) q^{44} + (1836 \beta_{2} - 980 \beta_1 + 5816) q^{46} + ( - 520 \beta_{2} + 598 \beta_1 - 1118) q^{47} + (288 \beta_{2} + 2016 \beta_1 - 3312) q^{48} + ( - 10 \beta_{2} - 196 \beta_1 + 3775) q^{49} + ( - 1071 \beta_{2} + 1971 \beta_1 + 1242) q^{51} + ( - 624 \beta_{2} - 2052 \beta_1 - 8164) q^{52} + (3174 \beta_{2} + 2146 \beta_1 - 3606) q^{53} + ( - 729 \beta_{2} - 729) q^{54} + ( - 3592 \beta_{2} + 952 \beta_1 + 4336) q^{56} + (1341 \beta_{2} + 1854 \beta_1 + 6489) q^{57} + ( - 2596 \beta_{2} + 1132 \beta_1 - 6308) q^{58} + ( - 1822 \beta_{2} - 2954 \beta_1 + 5732) q^{59} + ( - 2786 \beta_{2} + 1694 \beta_1 + 2410) q^{61} + ( - 2776 \beta_{2} + 356 \beta_1 - 2492) q^{62} + (567 \beta_{2} + 2511 \beta_1 - 3078) q^{63} + ( - 2688 \beta_{2} - 5312 \beta_1 + 4736) q^{64} + (1089 \beta_{2} + 1089) q^{66} + (5936 \beta_{2} + 4430 \beta_1 + 31566) q^{67} + ( - 2228 \beta_{2} - 3580 \beta_1 - 21880) q^{68} + ( - 1350 \beta_{2} + 1710 \beta_1 - 12744) q^{69} + ( - 9528 \beta_{2} + 3010 \beta_1 + 14670) q^{71} + (2268 \beta_{2} + 648 \beta_1 + 1620) q^{72} + ( - 1244 \beta_{2} - 12157 \beta_1 + 13479) q^{73} + (2430 \beta_{2} - 2448 \beta_1 + 26398) q^{74} + ( - 1052 \beta_{2} - 6776 \beta_1 - 20092) q^{76} + ( - 847 \beta_{2} - 3751 \beta_1 + 4598) q^{77} + (198 \beta_{2} - 1458 \beta_1 + 2016) q^{78} + (3973 \beta_{2} - 770 \beta_1 + 3189) q^{79} + 6561 q^{81} + ( - 3236 \beta_{2} - 6908 \beta_1 + 19292) q^{82} + ( - 1382 \beta_{2} - 10151 \beta_1 - 35935) q^{83} + ( - 1980 \beta_{2} - 3348 \beta_1 - 19440) q^{84} + (3058 \beta_{2} - 5714 \beta_1 + 46832) q^{86} + (1161 \beta_{2} - 2772 \beta_1 + 16209) q^{87} + ( - 3388 \beta_{2} - 968 \beta_1 - 2420) q^{88} + ( - 10502 \beta_{2} + 6468 \beta_1 - 14324) q^{89} + (4896 \beta_{2} + 8798 \beta_1 + 39554) q^{91} + ( - 2120 \beta_{2} + 3224 \beta_1 - 7632) q^{92} + ( - 288 \beta_{2} - 2178 \beta_1 + 18162) q^{93} + (2392 \beta_{2} - 3276 \beta_1 + 16068) q^{94} + (1584 \beta_{2} - 5184 \beta_1 + 1008) q^{96} + ( - 3184 \beta_{2} + 20686 \beta_1 + 40248) q^{97} + ( - 4373 \beta_{2} + 352 \beta_1 - 4525) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 11 ) / 2 \) 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
−1.87740
3.35386
−0.476452
−6.55890 9.00000 11.0192 0 −59.0301 −146.487 137.611 81.0000 0
1.2 −1.08127 9.00000 −30.8308 0 −9.73147 139.508 67.9374 81.0000 0
1.3 5.64018 9.00000 −0.188384 0 50.7616 −145.021 −181.548 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.h 3
5.b even 2 1 165.6.a.d 3
15.d odd 2 1 495.6.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.d 3 5.b even 2 1
495.6.a.c 3 15.d odd 2 1
825.6.a.h 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2T_{2}^{2} - 36T_{2} - 40 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 36 T - 40 \) Copy content Toggle raw display
$3$ \( (T - 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 152 T^{2} - 19424 T - 2963664 \) Copy content Toggle raw display
$11$ \( (T + 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 546 T^{2} - 76748 T + 7691848 \) Copy content Toggle raw display
$17$ \( T^{3} - 314 T^{2} + \cdots + 1079472216 \) Copy content Toggle raw display
$19$ \( T^{3} - 1808 T^{2} + \cdots + 729480096 \) Copy content Toggle raw display
$23$ \( T^{3} + 4288 T^{2} + \cdots + 857355136 \) Copy content Toggle raw display
$29$ \( T^{3} - 5582 T^{2} + \cdots - 329440872 \) Copy content Toggle raw display
$31$ \( T^{3} - 6328 T^{2} + \cdots - 5126546304 \) Copy content Toggle raw display
$37$ \( T^{3} + 16866 T^{2} + \cdots - 244700027368 \) Copy content Toggle raw display
$41$ \( T^{3} - 23282 T^{2} + \cdots + 945181300968 \) Copy content Toggle raw display
$43$ \( T^{3} + 20572 T^{2} + \cdots - 1240285492944 \) Copy content Toggle raw display
$47$ \( T^{3} + 3432 T^{2} + \cdots + 18016103040 \) Copy content Toggle raw display
$53$ \( T^{3} + 16138 T^{2} + \cdots + 985333601848 \) Copy content Toggle raw display
$59$ \( T^{3} - 21972 T^{2} + \cdots + 2567903224000 \) Copy content Toggle raw display
$61$ \( T^{3} - 8322 T^{2} + \cdots + 4408473611240 \) Copy content Toggle raw display
$67$ \( T^{3} - 84332 T^{2} + \cdots + 41154720036800 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 117803610062464 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 163971863295832 \) Copy content Toggle raw display
$79$ \( T^{3} - 6364 T^{2} + \cdots - 554174036768 \) Copy content Toggle raw display
$83$ \( T^{3} + 96272 T^{2} + \cdots - 3029676562224 \) Copy content Toggle raw display
$89$ \( T^{3} + 38938 T^{2} + \cdots + 96218735089528 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 251540556847112 \) Copy content Toggle raw display
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