Properties

Label 1521.4.a.bl
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 82x^{10} + 2137x^{8} - 22488x^{6} + 89784x^{4} - 67392x^{2} + 11664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{2} + 3) q^{4} + (\beta_{8} - \beta_{6}) q^{5} + (\beta_{9} - \beta_{7}) q^{7} + ( - \beta_{11} + 3 \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{2} + 3) q^{4} + (\beta_{8} - \beta_{6}) q^{5} + (\beta_{9} - \beta_{7}) q^{7} + ( - \beta_{11} + 3 \beta_{6}) q^{8} + (\beta_{5} - \beta_{2} - 10) q^{10} + (\beta_{11} - \beta_{8} - 4 \beta_{6}) q^{11} + (\beta_{4} + 2 \beta_1) q^{14} + ( - 2 \beta_{5} + 3 \beta_{2} + 17) q^{16} + ( - 2 \beta_{4} + \beta_{3} - 3 \beta_1) q^{17} + ( - \beta_{10} - 4 \beta_{9} + 3 \beta_{7}) q^{19} + (2 \beta_{11} - 5 \beta_{8} - 9 \beta_{6}) q^{20} + (\beta_{5} - 12 \beta_{2} - 53) q^{22} + ( - \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{23} + ( - 4 \beta_{5} - 9 \beta_{2} + 6) q^{25} + ( - \beta_{10} - 4 \beta_{9} + 31 \beta_{7}) q^{28} + (9 \beta_{4} - 17 \beta_1) q^{29} + (9 \beta_{10} - 13 \beta_{9} - 10 \beta_{7}) q^{31} + (3 \beta_{11} - 6 \beta_{8} + 15 \beta_{6}) q^{32} + (11 \beta_{10} - 11 \beta_{9} - 43 \beta_{7}) q^{34} + ( - 7 \beta_{4} + 3 \beta_{3} + 12 \beta_1) q^{35} + ( - 12 \beta_{10} + 13 \beta_{9} + 24 \beta_{7}) q^{37} + ( - 5 \beta_{4} - \beta_{3} - 4 \beta_1) q^{38} + ( - 9 \beta_{5} - 17 \beta_{2} - 40) q^{40} + (3 \beta_{11} - 28 \beta_{8} - 15 \beta_{6}) q^{41} + (9 \beta_{5} - 15 \beta_{2} - 143) q^{43} + (5 \beta_{11} + 11 \beta_{8} - 116 \beta_{6}) q^{44} + ( - 35 \beta_{10} + 14 \beta_{9} + 67 \beta_{7}) q^{46} + ( - \beta_{11} - 19 \beta_{8} - 24 \beta_{6}) q^{47} + ( - 4 \beta_{5} - 13 \beta_{2} - 181) q^{49} + (5 \beta_{11} - 12 \beta_{8} - 70 \beta_{6}) q^{50} + (11 \beta_{4} + 2 \beta_{3} + 79 \beta_1) q^{53} + (11 \beta_{5} + 28 \beta_{2} - 71) q^{55} + ( - 13 \beta_{4} - \beta_{3} + 8 \beta_1) q^{56} + (26 \beta_{10} + \beta_{9} - 178 \beta_{7}) q^{58} + ( - 8 \beta_{11} - 8 \beta_{8} - 20 \beta_{6}) q^{59} + ( - 11 \beta_{5} - 42 \beta_{2} - 226) q^{61} + ( - 4 \beta_{4} + 9 \beta_{3} - 94 \beta_1) q^{62} + (16 \beta_{5} - 33 \beta_{2} - 1) q^{64} + (26 \beta_{10} - 17 \beta_{9} + 139 \beta_{7}) q^{67} + (16 \beta_{4} + 3 \beta_{3} - 107 \beta_1) q^{68} + (11 \beta_{10} - 14 \beta_{9} + 101 \beta_{7}) q^{70} + ( - 7 \beta_{11} - 9 \beta_{8} + 168 \beta_{6}) q^{71} + (8 \beta_{10} - 14 \beta_{9} - 39 \beta_{7}) q^{73} + (\beta_{4} - 12 \beta_{3} + 123 \beta_1) q^{74} + ( - 3 \beta_{10} + 22 \beta_{9} - 65 \beta_{7}) q^{76} + (18 \beta_{4} - 2 \beta_{3} - 24 \beta_1) q^{77} + (36 \beta_{5} - 19 \beta_{2} - 550) q^{79} + ( - 8 \beta_{11} + 13 \beta_{8} - 113 \beta_{6}) q^{80} + ( - 22 \beta_{5} - 39 \beta_{2} - 217) q^{82} + ( - 23 \beta_{11} - \beta_{8} - 200 \beta_{6}) q^{83} + ( - 22 \beta_{10} + 67 \beta_{9} - 202 \beta_{7}) q^{85} + (24 \beta_{11} + 27 \beta_{8} - 254 \beta_{6}) q^{86} + (13 \beta_{5} - 60 \beta_{2} - 881) q^{88} + ( - 34 \beta_{11} - 4 \beta_{8} - 126 \beta_{6}) q^{89} + ( - 13 \beta_{4} - 11 \beta_{3} + 252 \beta_1) q^{92} + ( - 21 \beta_{5} - 16 \beta_{2} - 275) q^{94} + (25 \beta_{4} - 13 \beta_{3} - 64 \beta_1) q^{95} + ( - 59 \beta_{10} - 5 \beta_{9} - 74 \beta_{7}) q^{97} + (9 \beta_{11} - 12 \beta_{8} - 289 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 82x^{10} + 2137x^{8} - 22488x^{6} + 89784x^{4} - 67392x^{2} + 11664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -107\nu^{10} + 7946\nu^{8} - 168863\nu^{6} + 1219116\nu^{4} - 2374020\nu^{2} + 11664 ) / 241488 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 107\nu^{10} - 7946\nu^{8} + 168863\nu^{6} - 1219116\nu^{4} + 2494764\nu^{2} - 1702080 ) / 120744 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -121\nu^{10} + 9832\nu^{8} - 250765\nu^{6} + 2479146\nu^{4} - 7545996\nu^{2} - 2739096 ) / 120744 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -112\nu^{10} + 10057\nu^{8} - 303046\nu^{6} + 3796521\nu^{4} - 17347752\nu^{2} + 7515828 ) / 120744 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 197\nu^{10} - 15758\nu^{8} + 390641\nu^{6} - 3752148\nu^{4} + 12663828\nu^{2} - 4642848 ) / 80496 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 568\nu^{11} - 45613\nu^{9} + 1142302\nu^{7} - 11253417\nu^{5} + 40025268\nu^{3} - 14739084\nu ) / 2173392 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -568\nu^{11} + 45613\nu^{9} - 1142302\nu^{7} + 11253417\nu^{5} - 40025268\nu^{3} + 16912476\nu ) / 2173392 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2075\nu^{11} + 176756\nu^{9} - 4923155\nu^{7} + 56847558\nu^{5} - 252243036\nu^{3} + 199166040\nu ) / 4346784 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 269\nu^{11} - 21698\nu^{9} + 548249\nu^{7} - 5499624\nu^{5} + 20767500\nu^{3} - 16517520\nu ) / 334368 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3665\nu^{11} + 304706\nu^{9} - 8110061\nu^{7} + 86615472\nu^{5} - 327694428\nu^{3} + 68702256\nu ) / 4346784 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6199\nu^{11} - 515266\nu^{9} + 13717531\nu^{7} - 147138540\nu^{5} + 574132356\nu^{3} - 228990240\nu ) / 2173392 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{7} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 3\beta_{10} + 3\beta_{9} + 36\beta_{7} + 28\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} - 4\beta_{3} + 45\beta_{2} + 88\beta _1 + 424 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -59\beta_{11} - 175\beta_{10} + 145\beta_{9} - 6\beta_{8} + 1424\beta_{7} + 1046\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -154\beta_{5} - 36\beta_{4} - 234\beta_{3} + 2013\beta_{2} + 3780\beta _1 + 16244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2869\beta_{11} - 8323\beta_{10} + 6349\beta_{9} - 570\beta_{8} + 59660\beta_{7} + 43534\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -8282\beta_{5} - 2544\beta_{4} - 11192\beta_{3} + 89481\beta_{2} + 163856\beta _1 + 680236 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -131339\beta_{11} - 376083\beta_{10} + 276453\beta_{9} - 32478\beta_{8} + 2569644\beta_{7} + 1879370\beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -394786\beta_{5} - 132108\beta_{4} - 507422\beta_{3} + 3958701\beta_{2} + 7158788\beta _1 + 29400236 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5875753 \beta_{11} - 16718603 \beta_{10} + 12093389 \beta_{9} - 1580682 \beta_{8} + 112074172 \beta_{7} + 82151422 \beta_{6} \) 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
−6.63568
−3.17158
−4.27062
−0.806515
0.503193
−2.96091
2.96091
−0.503193
0.806515
4.27062
3.17158
6.63568
−4.90363 0 16.0456 5.61325 0 −3.21345 −39.4526 0 −27.5253
1.2 −4.90363 0 16.0456 5.61325 0 3.21345 −39.4526 0 −27.5253
1.3 −2.53857 0 −1.55568 −8.18941 0 −10.7837 24.2577 0 20.7894
1.4 −2.53857 0 −1.55568 −8.18941 0 10.7837 24.2577 0 20.7894
1.5 −1.22886 0 −6.48991 17.3039 0 −19.1933 17.8060 0 −21.2640
1.6 −1.22886 0 −6.48991 17.3039 0 19.1933 17.8060 0 −21.2640
1.7 1.22886 0 −6.48991 −17.3039 0 −19.1933 −17.8060 0 −21.2640
1.8 1.22886 0 −6.48991 −17.3039 0 19.1933 −17.8060 0 −21.2640
1.9 2.53857 0 −1.55568 8.18941 0 −10.7837 −24.2577 0 20.7894
1.10 2.53857 0 −1.55568 8.18941 0 10.7837 −24.2577 0 20.7894
1.11 4.90363 0 16.0456 −5.61325 0 −3.21345 39.4526 0 −27.5253
1.12 4.90363 0 16.0456 −5.61325 0 3.21345 39.4526 0 −27.5253
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.bl 12
3.b odd 2 1 inner 1521.4.a.bl 12
13.b even 2 1 inner 1521.4.a.bl 12
13.f odd 12 2 117.4.q.f 12
39.d odd 2 1 inner 1521.4.a.bl 12
39.k even 12 2 117.4.q.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.q.f 12 13.f odd 12 2
117.4.q.f 12 39.k even 12 2
1521.4.a.bl 12 1.a even 1 1 trivial
1521.4.a.bl 12 3.b odd 2 1 inner
1521.4.a.bl 12 13.b even 2 1 inner
1521.4.a.bl 12 39.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{6} - 32T_{2}^{4} + 201T_{2}^{2} - 234 \) Copy content Toggle raw display
\( T_{5}^{6} - 398T_{5}^{4} + 31629T_{5}^{2} - 632736 \) Copy content Toggle raw display
\( T_{7}^{6} - 495T_{7}^{4} + 47844T_{7}^{2} - 442368 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 32 T^{4} + 201 T^{2} - 234)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 398 T^{4} + 31629 T^{2} + \cdots - 632736)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 495 T^{4} + 47844 T^{2} + \cdots - 442368)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3092 T^{4} + 2390016 T^{2} + \cdots - 245366784)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 11850 T^{4} + \cdots - 765135072)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 10056 T^{4} + \cdots - 231475968)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 60252 T^{4} + \cdots - 4223789826048)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 114954 T^{4} + \cdots - 16997248580832)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 98019 T^{4} + \cdots - 30656351232)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 132534 T^{4} + \cdots - 4986821583792)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 321494 T^{4} + \cdots - 237506188443264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 405 T^{2} - 20406 T - 11468392)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} - 159044 T^{4} + \cdots - 35969118142464)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 769410 T^{4} + \cdots - 581060481251328)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 169088 T^{4} + \cdots - 7878275997696)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 647 T^{2} - 185833 T - 93894047)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 664407 T^{4} + \cdots - 74\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 1058468 T^{4} + \cdots - 462569095931904)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 112065 T^{4} + \cdots - 13909107863187)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 1595 T^{2} + 72716 T - 515213504)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 2259428 T^{4} + \cdots - 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 2817944 T^{4} + \cdots - 72\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 2735019 T^{4} + \cdots - 587965056519168)^{2} \) Copy content Toggle raw display
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