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: \( x^{3} - x^{2} - 13x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{4} + 5 q^{5} + ( - \beta_{2} + 2 \beta_1 - 14) q^{7} + (2 \beta_{2} + \beta_1 - 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{4} + 5 q^{5} + ( - \beta_{2} + 2 \beta_1 - 14) q^{7} + (2 \beta_{2} + \beta_1 - 8) q^{8} + 5 \beta_{2} q^{10} + ( - 11 \beta_{2} - 3 \beta_1) q^{11} + (13 \beta_{2} + \beta_1 + 18) q^{13} + ( - 25 \beta_{2} + \beta_1 - 17) q^{14} + ( - 8 \beta_{2} + 6 \beta_1 - 5) q^{16} + ( - 3 \beta_{2} + \beta_1 - 56) q^{17} + ( - 7 \beta_{2} - 3 \beta_1 - 58) q^{19} + ( - 5 \beta_{2} - 5 \beta_1 + 15) q^{20} + (29 \beta_{2} + 11 \beta_1 - 112) q^{22} + ( - 3 \beta_{2} + 12 \beta_1 + 60) q^{23} + 25 q^{25} + ( - \beta_{2} - 13 \beta_1 + 140) q^{26} + (10 \beta_{2} + 9 \beta_1 - 166) q^{28} + (4 \beta_{2} + 10 \beta_1 - 107) q^{29} + ( - 51 \beta_{2} - 11 \beta_1 - 138) q^{31} + ( - 49 \beta_{2} - 42) q^{32} + ( - 59 \beta_{2} + 3 \beta_1 - 36) q^{34} + ( - 5 \beta_{2} + 10 \beta_1 - 70) q^{35} + ( - 7 \beta_{2} - 17 \beta_1 + 126) q^{37} + ( - 33 \beta_{2} + 7 \beta_1 - 68) q^{38} + (10 \beta_{2} + 5 \beta_1 - 40) q^{40} + ( - 17 \beta_{2} - 23 \beta_1 + 49) q^{41} + (37 \beta_{2} - 29 \beta_1 - 198) q^{43} + ( - 119 \beta_{2} - 5 \beta_1 + 286) q^{44} + ( - 9 \beta_{2} + 3 \beta_1 - 69) q^{46} + ( - 54 \beta_{2} - 5 \beta_1 + 202) q^{47} + (63 \beta_{2} - 37 \beta_1 + 152) q^{49} + 25 \beta_{2} q^{50} + (115 \beta_{2} - 7 \beta_1 - 116) q^{52} + (62 \beta_{2} + 8 \beta_1 - 220) q^{53} + ( - 55 \beta_{2} - 15 \beta_1) q^{55} + ( - 30 \beta_{2} - 18 \beta_1 + 219) q^{56} + ( - 171 \beta_{2} - 4 \beta_1 + 14) q^{58} + (118 \beta_{2} - 36 \beta_1 - 72) q^{59} + (45 \beta_{2} - 25 \beta_1 - 473) q^{61} + ( - 21 \beta_{2} + 51 \beta_1 - 528) q^{62} + (71 \beta_{2} + \beta_1 - 499) q^{64} + (65 \beta_{2} + 5 \beta_1 + 90) q^{65} + ( - 48 \beta_{2} + 7 \beta_1 - 630) q^{67} + (29 \beta_{2} + 51 \beta_1 - 210) q^{68} + ( - 125 \beta_{2} + 5 \beta_1 - 85) q^{70} + (7 \beta_{2} - 83 \beta_1 - 2) q^{71} + ( - 20 \beta_{2} + 64 \beta_1 - 108) q^{73} + (235 \beta_{2} + 7 \beta_1 - 26) q^{74} + ( - 21 \beta_{2} + 57 \beta_1 + 80) q^{76} + (239 \beta_{2} + \beta_1 - 236) q^{77} + (48 \beta_{2} + 64 \beta_1 - 90) q^{79} + ( - 40 \beta_{2} + 30 \beta_1 - 25) q^{80} + (204 \beta_{2} + 17 \beta_1 - 118) q^{82} + ( - 13 \beta_{2} + 118 \beta_1 - 242) q^{83} + ( - 15 \beta_{2} + 5 \beta_1 - 280) q^{85} + ( - 61 \beta_{2} - 37 \beta_1 + 494) q^{86} + (203 \beta_{2} + 31 \beta_1 - 398) q^{88} + (88 \beta_{2} + 80 \beta_1 + 629) q^{89} + ( - 331 \beta_{2} + 45 \beta_1 - 332) q^{91} + ( - 54 \beta_{2} - 87 \beta_1 - 588) q^{92} + (286 \beta_{2} + 54 \beta_1 - 579) q^{94} + ( - 35 \beta_{2} - 15 \beta_1 - 290) q^{95} + ( - 58 \beta_{2} - 20 \beta_1 - 120) q^{97} + (311 \beta_{2} - 63 \beta_1 + 804) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 11 q^{4} + 15 q^{5} - 43 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 11 q^{4} + 15 q^{5} - 43 q^{7} - 27 q^{8} - 5 q^{10} + 14 q^{11} + 40 q^{13} - 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 55 q^{20} - 376 q^{22} + 171 q^{23} + 75 q^{25} + 434 q^{26} - 517 q^{28} - 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 215 q^{35} + 402 q^{37} - 178 q^{38} - 135 q^{40} + 187 q^{41} - 602 q^{43} + 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} - 25 q^{50} - 456 q^{52} - 730 q^{53} + 70 q^{55} + 705 q^{56} + 217 q^{58} - 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 200 q^{65} - 1849 q^{67} - 710 q^{68} - 135 q^{70} + 70 q^{71} - 368 q^{73} - 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} - 65 q^{80} - 575 q^{82} - 831 q^{83} - 830 q^{85} + 1580 q^{86} - 1428 q^{88} + 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} - 820 q^{95} - 282 q^{97} + 2164 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 4\nu - 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 2\nu - 9 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{2} + 2\beta _1 + 29 ) / 3 \) 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
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 \( T_{2}^{3} + T_{2}^{2} - 17T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\). Copy content Toggle raw display

Hecke characteristic polynomials

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