Properties

Label 405.4.a.g
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7032.1
Defining polynomial: \( x^{3} - x^{2} - 14x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + 5 q^{5} + (3 \beta_{2} + 5 \beta_1 - 10) q^{7} + ( - \beta_{2} + 3 \beta_1 + 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + 5 q^{5} + (3 \beta_{2} + 5 \beta_1 - 10) q^{7} + ( - \beta_{2} + 3 \beta_1 + 8) q^{8} - 5 \beta_1 q^{10} + ( - 5 \beta_{2} - 7 \beta_1 - 17) q^{11} + ( - 13 \beta_{2} + 13 \beta_1 - 20) q^{13} + ( - 11 \beta_{2} + 9 \beta_1 - 56) q^{14} + ( - 9 \beta_{2} + 5 \beta_1 - 44) q^{16} + ( - 8 \beta_{2} + 8 \beta_1 - 14) q^{17} + (14 \beta_{2} + 10 \beta_1 - 5) q^{19} + (5 \beta_{2} - 5 \beta_1 + 10) q^{20} + (17 \beta_{2} + 20 \beta_1 + 80) q^{22} + (19 \beta_{2} - 15 \beta_1 + 22) q^{23} + 25 q^{25} + (13 \beta_{2} + 59 \beta_1 - 104) q^{26} + ( - 11 \beta_{2} + 47 \beta_1 + 12) q^{28} + (13 \beta_{2} - 65 \beta_1 - 95) q^{29} + ( - \beta_{2} + 5 \beta_1 + 211) q^{31} + (21 \beta_{2} + 43 \beta_1 - 96) q^{32} + (8 \beta_{2} + 38 \beta_1 - 64) q^{34} + (15 \beta_{2} + 25 \beta_1 - 50) q^{35} + ( - 2 \beta_{2} + 54 \beta_1 - 156) q^{37} + ( - 38 \beta_{2} - 13 \beta_1 - 128) q^{38} + ( - 5 \beta_{2} + 15 \beta_1 + 40) q^{40} + ( - 62 \beta_{2} - 2 \beta_1 - 59) q^{41} + ( - 8 \beta_{2} + 28 \beta_1 - 288) q^{43} + ( - 14 \beta_{2} - 38 \beta_1 - 98) q^{44} + ( - 23 \beta_{2} - 75 \beta_1 + 112) q^{46} + (31 \beta_{2} - 67 \beta_1 - 56) q^{47} + (7 \beta_{2} - 11 \beta_1 + 301) q^{49} - 25 \beta_1 q^{50} + (19 \beta_{2} + 33 \beta_1 - 456) q^{52} + ( - 21 \beta_{2} + 53 \beta_1 - 186) q^{53} + ( - 25 \beta_{2} - 35 \beta_1 - 85) q^{55} + (63 \beta_{2} - 15 \beta_1) q^{56} + (39 \beta_{2} + 4 \beta_1 + 624) q^{58} + (90 \beta_{2} - 58 \beta_1 - 159) q^{59} + (58 \beta_{2} - 122 \beta_1 + 6) q^{61} + ( - 3 \beta_{2} - 204 \beta_1 - 48) q^{62} + ( - 13 \beta_{2} + 57 \beta_1 - 120) q^{64} + ( - 65 \beta_{2} + 65 \beta_1 - 100) q^{65} + ( - 18 \beta_{2} - 154 \beta_1 + 38) q^{67} + (10 \beta_{2} + 22 \beta_1 - 284) q^{68} + ( - 55 \beta_{2} + 45 \beta_1 - 280) q^{70} + (69 \beta_{2} + 79 \beta_1 - 177) q^{71} + ( - 12 \beta_{2} - 32 \beta_1 - 226) q^{73} + ( - 50 \beta_{2} + 214 \beta_1 - 536) q^{74} + ( - 23 \beta_{2} + 111 \beta_1 + 246) q^{76} + ( - 98 \beta_{2} - 162 \beta_1 - 662) q^{77} + ( - 54 \beta_{2} - 82 \beta_1 - 184) q^{79} + ( - 45 \beta_{2} + 25 \beta_1 - 220) q^{80} + (126 \beta_{2} + 181 \beta_1 + 144) q^{82} + (30 \beta_{2} + 210 \beta_1 - 648) q^{83} + ( - 40 \beta_{2} + 40 \beta_1 - 70) q^{85} + ( - 12 \beta_{2} + 332 \beta_1 - 264) q^{86} + ( - 70 \beta_{2} - 72 \beta_1 - 232) q^{88} + (3 \beta_{2} - 39 \beta_1 + 297) q^{89} + (161 \beta_{2} - 581 \beta_1 - 216) q^{91} + ( - 31 \beta_{2} - 21 \beta_1 + 620) q^{92} + (5 \beta_{2} - 73 \beta_1 + 608) q^{94} + (70 \beta_{2} + 50 \beta_1 - 25) q^{95} + (32 \beta_{2} - 336 \beta_1 + 112) q^{97} + ( - 3 \beta_{2} - 326 \beta_1 + 96) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} + 15 q^{5} - 25 q^{7} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} + 15 q^{5} - 25 q^{7} + 27 q^{8} - 5 q^{10} - 58 q^{11} - 47 q^{13} - 159 q^{14} - 127 q^{16} - 34 q^{17} - 5 q^{19} + 25 q^{20} + 260 q^{22} + 51 q^{23} + 75 q^{25} - 253 q^{26} + 83 q^{28} - 350 q^{29} + 638 q^{31} - 245 q^{32} - 154 q^{34} - 125 q^{35} - 414 q^{37} - 397 q^{38} + 135 q^{40} - 179 q^{41} - 836 q^{43} - 332 q^{44} + 261 q^{46} - 235 q^{47} + 892 q^{49} - 25 q^{50} - 1335 q^{52} - 505 q^{53} - 290 q^{55} - 15 q^{56} + 1876 q^{58} - 535 q^{59} - 104 q^{61} - 348 q^{62} - 303 q^{64} - 235 q^{65} - 40 q^{67} - 830 q^{68} - 795 q^{70} - 452 q^{71} - 710 q^{73} - 1394 q^{74} + 849 q^{76} - 2148 q^{77} - 634 q^{79} - 635 q^{80} + 613 q^{82} - 1734 q^{83} - 170 q^{85} - 460 q^{86} - 768 q^{88} + 852 q^{89} - 1229 q^{91} + 1839 q^{92} + 1751 q^{94} - 25 q^{95} - 38 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 14x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 10 \) 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
3.52348
1.32681
−3.85028
−3.52348 0 4.41489 5.00000 0 25.4325 12.6321 0 −17.6174
1.2 −1.32681 0 −6.23958 5.00000 0 −24.1043 18.8932 0 −6.63404
1.3 3.85028 0 6.82469 5.00000 0 −26.3282 −4.52526 0 19.2514
\(n\): e.g. 2-40 or 990-1000
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.g 3
3.b odd 2 1 405.4.a.i yes 3
5.b even 2 1 2025.4.a.r 3
9.c even 3 2 405.4.e.u 6
9.d odd 6 2 405.4.e.s 6
15.d odd 2 1 2025.4.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.g 3 1.a even 1 1 trivial
405.4.a.i yes 3 3.b odd 2 1
405.4.e.s 6 9.d odd 6 2
405.4.e.u 6 9.c even 3 2
2025.4.a.p 3 15.d odd 2 1
2025.4.a.r 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + T_{2}^{2} - 14T_{2} - 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 14 T - 18 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 25 T^{2} - 648 T - 16140 \) Copy content Toggle raw display
$11$ \( T^{3} + 58 T^{2} - 911 T + 3000 \) Copy content Toggle raw display
$13$ \( T^{3} + 47 T^{2} - 7432 T - 370352 \) Copy content Toggle raw display
$17$ \( T^{3} + 34 T^{2} - 2708 T - 90984 \) Copy content Toggle raw display
$19$ \( T^{3} + 5 T^{2} - 10777 T - 299645 \) Copy content Toggle raw display
$23$ \( T^{3} - 51 T^{2} - 15240 T + 1041156 \) Copy content Toggle raw display
$29$ \( T^{3} + 350 T^{2} + \cdots - 11237760 \) Copy content Toggle raw display
$31$ \( T^{3} - 638 T^{2} + 135321 T - 9539064 \) Copy content Toggle raw display
$37$ \( T^{3} + 414 T^{2} + 16032 T - 577760 \) Copy content Toggle raw display
$41$ \( T^{3} + 179 T^{2} + \cdots - 17799627 \) Copy content Toggle raw display
$43$ \( T^{3} + 836 T^{2} + \cdots + 18692992 \) Copy content Toggle raw display
$47$ \( T^{3} + 235 T^{2} - 69680 T - 9005376 \) Copy content Toggle raw display
$53$ \( T^{3} + 505 T^{2} + 35128 T - 1500684 \) Copy content Toggle raw display
$59$ \( T^{3} + 535 T^{2} + \cdots + 22317657 \) Copy content Toggle raw display
$61$ \( T^{3} + 104 T^{2} + \cdots - 23542832 \) Copy content Toggle raw display
$67$ \( T^{3} + 40 T^{2} - 375180 T - 15716208 \) Copy content Toggle raw display
$71$ \( T^{3} + 452 T^{2} + \cdots - 116183454 \) Copy content Toggle raw display
$73$ \( T^{3} + 710 T^{2} + 144236 T + 8707528 \) Copy content Toggle raw display
$79$ \( T^{3} + 634 T^{2} - 120288 T + 5053056 \) Copy content Toggle raw display
$83$ \( T^{3} + 1734 T^{2} + \cdots - 222334848 \) Copy content Toggle raw display
$89$ \( T^{3} - 852 T^{2} + \cdots - 17926434 \) Copy content Toggle raw display
$97$ \( T^{3} - 1575168 T - 703275008 \) Copy content Toggle raw display
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