Properties

Label 405.4.a.h
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: 3.3.2292.1
Defining polynomial: \(x^{3} - x^{2} - 13 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 3 - \beta_{1} - \beta_{2} ) q^{4} + 5 q^{5} + ( -14 + 2 \beta_{1} - \beta_{2} ) q^{7} + ( -8 + \beta_{1} + 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 3 - \beta_{1} - \beta_{2} ) q^{4} + 5 q^{5} + ( -14 + 2 \beta_{1} - \beta_{2} ) q^{7} + ( -8 + \beta_{1} + 2 \beta_{2} ) q^{8} + 5 \beta_{2} q^{10} + ( -3 \beta_{1} - 11 \beta_{2} ) q^{11} + ( 18 + \beta_{1} + 13 \beta_{2} ) q^{13} + ( -17 + \beta_{1} - 25 \beta_{2} ) q^{14} + ( -5 + 6 \beta_{1} - 8 \beta_{2} ) q^{16} + ( -56 + \beta_{1} - 3 \beta_{2} ) q^{17} + ( -58 - 3 \beta_{1} - 7 \beta_{2} ) q^{19} + ( 15 - 5 \beta_{1} - 5 \beta_{2} ) q^{20} + ( -112 + 11 \beta_{1} + 29 \beta_{2} ) q^{22} + ( 60 + 12 \beta_{1} - 3 \beta_{2} ) q^{23} + 25 q^{25} + ( 140 - 13 \beta_{1} - \beta_{2} ) q^{26} + ( -166 + 9 \beta_{1} + 10 \beta_{2} ) q^{28} + ( -107 + 10 \beta_{1} + 4 \beta_{2} ) q^{29} + ( -138 - 11 \beta_{1} - 51 \beta_{2} ) q^{31} + ( -42 - 49 \beta_{2} ) q^{32} + ( -36 + 3 \beta_{1} - 59 \beta_{2} ) q^{34} + ( -70 + 10 \beta_{1} - 5 \beta_{2} ) q^{35} + ( 126 - 17 \beta_{1} - 7 \beta_{2} ) q^{37} + ( -68 + 7 \beta_{1} - 33 \beta_{2} ) q^{38} + ( -40 + 5 \beta_{1} + 10 \beta_{2} ) q^{40} + ( 49 - 23 \beta_{1} - 17 \beta_{2} ) q^{41} + ( -198 - 29 \beta_{1} + 37 \beta_{2} ) q^{43} + ( 286 - 5 \beta_{1} - 119 \beta_{2} ) q^{44} + ( -69 + 3 \beta_{1} - 9 \beta_{2} ) q^{46} + ( 202 - 5 \beta_{1} - 54 \beta_{2} ) q^{47} + ( 152 - 37 \beta_{1} + 63 \beta_{2} ) q^{49} + 25 \beta_{2} q^{50} + ( -116 - 7 \beta_{1} + 115 \beta_{2} ) q^{52} + ( -220 + 8 \beta_{1} + 62 \beta_{2} ) q^{53} + ( -15 \beta_{1} - 55 \beta_{2} ) q^{55} + ( 219 - 18 \beta_{1} - 30 \beta_{2} ) q^{56} + ( 14 - 4 \beta_{1} - 171 \beta_{2} ) q^{58} + ( -72 - 36 \beta_{1} + 118 \beta_{2} ) q^{59} + ( -473 - 25 \beta_{1} + 45 \beta_{2} ) q^{61} + ( -528 + 51 \beta_{1} - 21 \beta_{2} ) q^{62} + ( -499 + \beta_{1} + 71 \beta_{2} ) q^{64} + ( 90 + 5 \beta_{1} + 65 \beta_{2} ) q^{65} + ( -630 + 7 \beta_{1} - 48 \beta_{2} ) q^{67} + ( -210 + 51 \beta_{1} + 29 \beta_{2} ) q^{68} + ( -85 + 5 \beta_{1} - 125 \beta_{2} ) q^{70} + ( -2 - 83 \beta_{1} + 7 \beta_{2} ) q^{71} + ( -108 + 64 \beta_{1} - 20 \beta_{2} ) q^{73} + ( -26 + 7 \beta_{1} + 235 \beta_{2} ) q^{74} + ( 80 + 57 \beta_{1} - 21 \beta_{2} ) q^{76} + ( -236 + \beta_{1} + 239 \beta_{2} ) q^{77} + ( -90 + 64 \beta_{1} + 48 \beta_{2} ) q^{79} + ( -25 + 30 \beta_{1} - 40 \beta_{2} ) q^{80} + ( -118 + 17 \beta_{1} + 204 \beta_{2} ) q^{82} + ( -242 + 118 \beta_{1} - 13 \beta_{2} ) q^{83} + ( -280 + 5 \beta_{1} - 15 \beta_{2} ) q^{85} + ( 494 - 37 \beta_{1} - 61 \beta_{2} ) q^{86} + ( -398 + 31 \beta_{1} + 203 \beta_{2} ) q^{88} + ( 629 + 80 \beta_{1} + 88 \beta_{2} ) q^{89} + ( -332 + 45 \beta_{1} - 331 \beta_{2} ) q^{91} + ( -588 - 87 \beta_{1} - 54 \beta_{2} ) q^{92} + ( -579 + 54 \beta_{1} + 286 \beta_{2} ) q^{94} + ( -290 - 15 \beta_{1} - 35 \beta_{2} ) q^{95} + ( -120 - 20 \beta_{1} - 58 \beta_{2} ) q^{97} + ( 804 - 63 \beta_{1} + 311 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 11 q^{4} + 15 q^{5} - 43 q^{7} - 27 q^{8} + O(q^{10}) \) \( 3 q - q^{2} + 11 q^{4} + 15 q^{5} - 43 q^{7} - 27 q^{8} - 5 q^{10} + 14 q^{11} + 40 q^{13} - 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 55 q^{20} - 376 q^{22} + 171 q^{23} + 75 q^{25} + 434 q^{26} - 517 q^{28} - 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 215 q^{35} + 402 q^{37} - 178 q^{38} - 135 q^{40} + 187 q^{41} - 602 q^{43} + 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} - 25 q^{50} - 456 q^{52} - 730 q^{53} + 70 q^{55} + 705 q^{56} + 217 q^{58} - 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 200 q^{65} - 1849 q^{67} - 710 q^{68} - 135 q^{70} + 70 q^{71} - 368 q^{73} - 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} - 65 q^{80} - 575 q^{82} - 831 q^{83} - 830 q^{85} + 1580 q^{86} - 1428 q^{88} + 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} - 820 q^{95} - 282 q^{97} + 2164 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 13 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 4 \nu - 11 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 2 \nu - 9 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{2} + 2 \beta_{1} + 29\)\()/3\)

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
0.0765073
4.10645
−3.18296
−4.57358 0 12.9176 5.00000 0 −20.1145 −22.4912 0 −22.8679
1.2 −0.174985 0 −7.96938 5.00000 0 8.46371 2.79440 0 −0.874923
1.3 3.74857 0 6.05174 5.00000 0 −31.3492 −7.30318 0 18.7428
\(n\): 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.h 3
3.b odd 2 1 405.4.a.j 3
5.b even 2 1 2025.4.a.s 3
9.c even 3 2 45.4.e.b 6
9.d odd 6 2 135.4.e.b 6
15.d odd 2 1 2025.4.a.q 3
45.j even 6 2 225.4.e.c 6
45.k odd 12 4 225.4.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 9.c even 3 2
135.4.e.b 6 9.d odd 6 2
225.4.e.c 6 45.j even 6 2
225.4.k.c 12 45.k odd 12 4
405.4.a.h 3 1.a even 1 1 trivial
405.4.a.j 3 3.b odd 2 1
2025.4.a.q 3 15.d odd 2 1
2025.4.a.s 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + T_{2}^{2} - 17 T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 - 17 T + T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( -5 + T )^{3} \)
$7$ \( -5337 + 195 T + 43 T^{2} + T^{3} \)
$11$ \( -43548 - 2816 T - 14 T^{2} + T^{3} \)
$13$ \( 75364 - 2452 T - 40 T^{2} + T^{3} \)
$17$ \( 156324 + 8920 T + 166 T^{2} + T^{3} \)
$19$ \( 57316 + 7292 T + 164 T^{2} + T^{3} \)
$23$ \( 61209 - 4833 T - 171 T^{2} + T^{3} \)
$29$ \( -107067 + 27331 T + 335 T^{2} + T^{3} \)
$31$ \( -9860940 - 13932 T + 352 T^{2} + T^{3} \)
$37$ \( 3335284 + 24708 T - 402 T^{2} + T^{3} \)
$41$ \( 7228275 - 44597 T - 187 T^{2} + T^{3} \)
$43$ \( -15444524 + 9956 T + 602 T^{2} + T^{3} \)
$47$ \( -2487483 + 95281 T - 665 T^{2} + T^{3} \)
$53$ \( 3250536 + 106300 T + 730 T^{2} + T^{3} \)
$59$ \( -127375896 - 354644 T + 298 T^{2} + T^{3} \)
$61$ \( 55613497 + 589307 T + 1439 T^{2} + T^{3} \)
$67$ \( 208776159 + 1093677 T + 1849 T^{2} + T^{3} \)
$71$ \( 223775052 - 685460 T - 70 T^{2} + T^{3} \)
$73$ \( -134927744 - 372928 T + 368 T^{2} + T^{3} \)
$79$ \( -138322584 - 387924 T + 382 T^{2} + T^{3} \)
$83$ \( -955843821 - 1160973 T + 831 T^{2} + T^{3} \)
$89$ \( 125506395 + 238491 T - 1719 T^{2} + T^{3} \)
$97$ \( -16898264 - 67668 T + 282 T^{2} + T^{3} \)
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