Properties

Label 405.4.a.l
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} - \beta_1 + 6) q^{4} + 5 q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 7) q^{7} + ( - \beta_{4} + 2 \beta_{3} - 4 \beta_1 + 12) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} - \beta_1 + 6) q^{4} + 5 q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 7) q^{7} + ( - \beta_{4} + 2 \beta_{3} - 4 \beta_1 + 12) q^{8} + ( - 5 \beta_1 + 5) q^{10} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 14) q^{11} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 4) q^{13} + (3 \beta_{5} - \beta_{4} + 4 \beta_{3} - \beta_{2} - 7 \beta_1 + 32) q^{14} + (2 \beta_{5} - 3 \beta_{4} - 16 \beta_1 + 14) q^{16} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 19) q^{17} + ( - 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 7) q^{19} + (5 \beta_{3} - 5 \beta_1 + 30) q^{20} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 10 \beta_{2} - 20 \beta_1 - 5) q^{22} + (6 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 34) q^{23} + 25 q^{25} + ( - 7 \beta_{5} + 6 \beta_{4} - 5 \beta_{3} + 15 \beta_{2} + 2 \beta_1 + 2) q^{26} + (3 \beta_{5} - 6 \beta_{4} + 13 \beta_{3} - 29 \beta_{2} - 40 \beta_1 + 70) q^{28} + ( - 3 \beta_{5} - 4 \beta_{4} + \beta_{3} + 8 \beta_{2} - 12 \beta_1 + 52) q^{29} + ( - 4 \beta_{5} - 3 \beta_{4} - 10 \beta_{3} + 6 \beta_{2} - 16 \beta_1 - 13) q^{31} + (6 \beta_{5} - \beta_{4} + 6 \beta_{3} - 24 \beta_{2} + 18 \beta_1 + 114) q^{32} + ( - 4 \beta_{5} + 9 \beta_{4} - 7 \beta_{3} + 38 \beta_{2} - 25 \beta_1 - 60) q^{34} + ( - 5 \beta_{4} - 5 \beta_{2} - 10 \beta_1 + 35) q^{35} + (3 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 28 \beta_{2} - 10 \beta_1 - 29) q^{37} + ( - 4 \beta_{5} + 14 \beta_{4} - 14 \beta_{3} + 20 \beta_{2} + 31 \beta_1 - 9) q^{38} + ( - 5 \beta_{4} + 10 \beta_{3} - 20 \beta_1 + 60) q^{40} + (10 \beta_{5} - 2 \beta_{4} - 26 \beta_{3} - 7 \beta_{2} - 16 \beta_1 + 62) q^{41} + (\beta_{5} - \beta_{4} + 25 \beta_{3} + 38 \beta_{2} - 42 \beta_1 + 99) q^{43} + ( - 6 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 120) q^{44} + ( - 4 \beta_{5} - 13 \beta_{4} - 14 \beta_{3} - 74 \beta_{2} - 22 \beta_1 - 28) q^{46} + ( - 9 \beta_{5} - 6 \beta_{4} - 29 \beta_{3} + 14 \beta_{2} + 44 \beta_1 + 23) q^{47} + (12 \beta_{5} - 15 \beta_{4} + 16 \beta_{3} - 12 \beta_{2} - 26 \beta_1 + 15) q^{49} + ( - 25 \beta_1 + 25) q^{50} + ( - 19 \beta_{5} + 16 \beta_{4} - 11 \beta_{3} + 75 \beta_{2} + 28 \beta_1 - 82) q^{52} + (5 \beta_{5} - 24 \beta_{4} - 27 \beta_{3} - 58 \beta_{2} + 20 \beta_1 + 127) q^{53} + ( - 5 \beta_{5} + 5 \beta_{3} - 10 \beta_{2} + 10 \beta_1 + 70) q^{55} + (17 \beta_{5} - 20 \beta_{4} + 33 \beta_{3} - 57 \beta_{2} - 92 \beta_1 + 410) q^{56} + (4 \beta_{4} + 21 \beta_{3} + 44 \beta_{2} - 58 \beta_1 + 169) q^{58} + ( - 8 \beta_{4} + 6 \beta_{3} + 15 \beta_{2} + 30 \beta_1 + 294) q^{59} + (22 \beta_{5} + 2 \beta_{4} - 38 \beta_{3} + 30 \beta_{2} + 72 \beta_1 - 84) q^{61} + (19 \beta_{4} + 12 \beta_{3} + 54 \beta_{2} + 73 \beta_1 + 155) q^{62} + (10 \beta_{5} - \beta_{4} - 10 \beta_{3} - 96 \beta_{2} - 22 \beta_1 - 158) q^{64} + ( - 5 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} + 15 \beta_{2} + 20) q^{65} + (10 \beta_{5} + 12 \beta_{4} - 30 \beta_{3} - 8 \beta_{2} + 164 \beta_1 - 36) q^{67} + ( - 32 \beta_{5} + 20 \beta_{4} - 8 \beta_{3} + 70 \beta_{2} + 54 \beta_1 + 28) q^{68} + (15 \beta_{5} - 5 \beta_{4} + 20 \beta_{3} - 5 \beta_{2} - 35 \beta_1 + 160) q^{70} + (9 \beta_{5} + 40 \beta_{4} - 9 \beta_{3} + 52 \beta_{2} - 82 \beta_1 + 268) q^{71} + ( - 7 \beta_{5} + 23 \beta_{4} + 33 \beta_{3} - 50 \beta_{2} + 138 \beta_1 + 101) q^{73} + (18 \beta_{5} + \beta_{4} - 5 \beta_{3} - 64 \beta_{2} + 59 \beta_1 + 200) q^{74} + ( - 32 \beta_{5} + 8 \beta_{4} - 41 \beta_{3} + 100 \beta_{2} + 77 \beta_1 - 374) q^{76} + ( - 9 \beta_{5} - 5 \beta_{4} + 19 \beta_{3} - 40 \beta_{2} + 46 \beta_1 + 123) q^{77} + ( - 12 \beta_{5} - 32 \beta_{4} + 12 \beta_{3} + 10 \beta_{2} + 144 \beta_1 - 172) q^{79} + (10 \beta_{5} - 15 \beta_{4} - 80 \beta_1 + 70) q^{80} + (11 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 127 \beta_{2} + 94 \beta_1 + 257) q^{82} + ( - 19 \beta_{5} + 27 \beta_{4} + 45 \beta_{3} + 34 \beta_{2} - 6 \beta_1 + 369) q^{83} + ( - 15 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} + 10 \beta_{2} + 30 \beta_1 + 95) q^{85} + ( - 36 \beta_{5} - 29 \beta_{4} + 69 \beta_{3} + 26 \beta_{2} - 249 \beta_1 + 552) q^{86} + ( - 12 \beta_{5} - 8 \beta_{4} + 26 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 214) q^{88} + (21 \beta_{5} + 18 \beta_{4} + 33 \beta_{3} - 76 \beta_{2} + 96 \beta_1 + 474) q^{89} + ( - 18 \beta_{5} + 21 \beta_{4} - 34 \beta_{3} + 44 \beta_{2} + 106 \beta_1 - 538) q^{91} + (52 \beta_{5} - 11 \beta_{4} + 50 \beta_{3} - 10 \beta_{2} + 64 \beta_1 + 142) q^{92} + ( - 2 \beta_{5} + 50 \beta_{4} - 61 \beta_{3} + 122 \beta_{2} + 151 \beta_1 - 644) q^{94} + ( - 10 \beta_{5} + 20 \beta_{4} - 20 \beta_{3} - 20 \beta_{2} + 10 \beta_1 - 35) q^{95} + ( - 41 \beta_{5} - 39 \beta_{4} - 13 \beta_{3} + 72 \beta_{2} + 30 \beta_1 - 125) q^{97} + (42 \beta_{5} - 67 \beta_{4} + 72 \beta_{3} - 156 \beta_{2} - 111 \beta_1 + 345) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8} + 20 q^{10} + 88 q^{11} + 20 q^{13} + 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 170 q^{20} - 74 q^{22} + 210 q^{23} + 150 q^{25} + 4 q^{26} + 352 q^{28} + 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 200 q^{35} - 204 q^{37} - 20 q^{38} + 330 q^{40} + 344 q^{41} + 512 q^{43} + 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 100 q^{50} - 468 q^{52} + 850 q^{53} + 440 q^{55} + 2316 q^{56} + 890 q^{58} + 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} + 100 q^{65} + 88 q^{67} + 236 q^{68} + 900 q^{70} + 1364 q^{71} + 836 q^{73} + 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} + 290 q^{80} + 1742 q^{82} + 2148 q^{83} + 620 q^{85} + 2872 q^{86} + 1296 q^{88} + 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} - 230 q^{95} - 612 q^{97} + 1982 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 26\nu^{3} + 30\nu^{2} + 141\nu - 36 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{3} - \nu^{2} - 19\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} - \nu^{3} - 21\nu^{2} + 3\nu + 30 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 20\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + \beta_{4} + 22\beta_{3} + 38\beta _1 + 255 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} + 28\beta_{4} + 40\beta_{3} + 24\beta_{2} + 425\beta _1 + 468 \) 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
5.11734
4.57457
1.14915
−1.07326
−3.53444
−4.23336
−4.11734 0 8.95250 5.00000 0 −20.0229 −3.92177 0 −20.5867
1.2 −3.57457 0 4.77759 5.00000 0 14.1597 11.5188 0 −17.8729
1.3 −0.149150 0 −7.97775 5.00000 0 20.1424 2.38308 0 −0.745751
1.4 2.07326 0 −3.70159 5.00000 0 −4.66112 −24.2604 0 10.3663
1.5 4.53444 0 12.5612 5.00000 0 −2.63618 20.6823 0 22.6722
1.6 5.23336 0 19.3881 5.00000 0 33.0180 59.5981 0 26.1668
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

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 405.4.a.l yes 6
3.b odd 2 1 405.4.a.k 6
5.b even 2 1 2025.4.a.y 6
9.c even 3 2 405.4.e.w 12
9.d odd 6 2 405.4.e.x 12
15.d odd 2 1 2025.4.a.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.k 6 3.b odd 2 1
405.4.a.l yes 6 1.a even 1 1 trivial
405.4.e.w 12 9.c even 3 2
405.4.e.x 12 9.d odd 6 2
2025.4.a.y 6 5.b even 2 1
2025.4.a.z 6 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 4T_{2}^{5} - 33T_{2}^{4} + 110T_{2}^{3} + 286T_{2}^{2} - 684T_{2} - 108 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 4 T^{5} - 33 T^{4} + 110 T^{3} + \cdots - 108 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 40 T^{5} - 263 T^{4} + \cdots - 2316924 \) Copy content Toggle raw display
$11$ \( T^{6} - 88 T^{5} + 1518 T^{4} + \cdots + 1197108 \) Copy content Toggle raw display
$13$ \( T^{6} - 20 T^{5} + \cdots - 185793728 \) Copy content Toggle raw display
$17$ \( T^{6} - 124 T^{5} + \cdots + 105966288 \) Copy content Toggle raw display
$19$ \( T^{6} + 46 T^{5} + \cdots - 36821611175 \) Copy content Toggle raw display
$23$ \( T^{6} - 210 T^{5} + \cdots + 332569842768 \) Copy content Toggle raw display
$29$ \( T^{6} - 296 T^{5} + \cdots + 635447099088 \) Copy content Toggle raw display
$31$ \( T^{6} + 104 T^{5} + \cdots - 69859854216 \) Copy content Toggle raw display
$37$ \( T^{6} + 204 T^{5} + \cdots + 12008297128192 \) Copy content Toggle raw display
$41$ \( T^{6} - 344 T^{5} + \cdots - 86213802377403 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 180465347194400 \) Copy content Toggle raw display
$47$ \( T^{6} - 238 T^{5} + \cdots - 49451750433900 \) Copy content Toggle raw display
$53$ \( T^{6} - 850 T^{5} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 168320389359483 \) Copy content Toggle raw display
$61$ \( T^{6} + 364 T^{5} + \cdots - 20\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{6} - 88 T^{5} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{6} - 1364 T^{5} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{6} - 836 T^{5} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{6} + 680 T^{5} + \cdots - 15\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{6} - 2148 T^{5} + \cdots + 41\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} - 3000 T^{5} + \cdots + 16\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{6} + 612 T^{5} + \cdots - 97\!\cdots\!24 \) Copy content Toggle raw display
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