Properties

Label 405.4.a.d
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + \beta q^{4} + 5 q^{5} + ( - \beta + 5) q^{7} + (7 \beta - 8) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + \beta q^{4} + 5 q^{5} + ( - \beta + 5) q^{7} + (7 \beta - 8) q^{8} - 5 \beta q^{10} + ( - 5 \beta - 16) q^{11} + (8 \beta - 60) q^{13} + ( - 4 \beta + 8) q^{14} + ( - 7 \beta - 56) q^{16} + (25 \beta + 26) q^{17} + ( - 25 \beta + 30) q^{19} + 5 \beta q^{20} + (21 \beta + 40) q^{22} + (23 \beta - 145) q^{23} + 25 q^{25} + (52 \beta - 64) q^{26} + (4 \beta - 8) q^{28} + (39 \beta + 143) q^{29} - 6 q^{31} + (7 \beta + 120) q^{32} + ( - 51 \beta - 200) q^{34} + ( - 5 \beta + 25) q^{35} + ( - 10 \beta - 314) q^{37} + ( - 5 \beta + 200) q^{38} + (35 \beta - 40) q^{40} + (60 \beta + 89) q^{41} + (87 \beta - 92) q^{43} + ( - 21 \beta - 40) q^{44} + (122 \beta - 184) q^{46} + ( - 11 \beta - 445) q^{47} + ( - 9 \beta - 310) q^{49} - 25 \beta q^{50} + ( - 52 \beta + 64) q^{52} + ( - 100 \beta + 162) q^{53} + ( - 25 \beta - 80) q^{55} + (36 \beta - 96) q^{56} + ( - 182 \beta - 312) q^{58} + ( - 49 \beta - 18) q^{59} + (195 \beta - 221) q^{61} + 6 \beta q^{62} + ( - 71 \beta + 392) q^{64} + (40 \beta - 300) q^{65} + ( - 184 \beta - 211) q^{67} + (51 \beta + 200) q^{68} + ( - 20 \beta + 40) q^{70} + ( - 110 \beta + 252) q^{71} + (205 \beta - 508) q^{73} + (324 \beta + 80) q^{74} + (5 \beta - 200) q^{76} + ( - 4 \beta - 40) q^{77} + ( - 76 \beta - 382) q^{79} + ( - 35 \beta - 280) q^{80} + ( - 149 \beta - 480) q^{82} + ( - 453 \beta + 33) q^{83} + (125 \beta + 130) q^{85} + (5 \beta - 696) q^{86} + ( - 107 \beta - 152) q^{88} + ( - 315 \beta - 375) q^{89} + (92 \beta - 364) q^{91} + ( - 122 \beta + 184) q^{92} + (456 \beta + 88) q^{94} + ( - 125 \beta + 150) q^{95} + ( - 131 \beta - 450) q^{97} + (319 \beta + 72) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} + 10 q^{5} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{4} + 10 q^{5} + 9 q^{7} - 9 q^{8} - 5 q^{10} - 37 q^{11} - 112 q^{13} + 12 q^{14} - 119 q^{16} + 77 q^{17} + 35 q^{19} + 5 q^{20} + 101 q^{22} - 267 q^{23} + 50 q^{25} - 76 q^{26} - 12 q^{28} + 325 q^{29} - 12 q^{31} + 247 q^{32} - 451 q^{34} + 45 q^{35} - 638 q^{37} + 395 q^{38} - 45 q^{40} + 238 q^{41} - 97 q^{43} - 101 q^{44} - 246 q^{46} - 901 q^{47} - 629 q^{49} - 25 q^{50} + 76 q^{52} + 224 q^{53} - 185 q^{55} - 156 q^{56} - 806 q^{58} - 85 q^{59} - 247 q^{61} + 6 q^{62} + 713 q^{64} - 560 q^{65} - 606 q^{67} + 451 q^{68} + 60 q^{70} + 394 q^{71} - 811 q^{73} + 484 q^{74} - 395 q^{76} - 84 q^{77} - 840 q^{79} - 595 q^{80} - 1109 q^{82} - 387 q^{83} + 385 q^{85} - 1387 q^{86} - 411 q^{88} - 1065 q^{89} - 636 q^{91} + 246 q^{92} + 632 q^{94} + 175 q^{95} - 1031 q^{97} + 463 q^{98}+O(q^{100}) \) 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
3.37228
−2.37228
−3.37228 0 3.37228 5.00000 0 1.62772 15.6060 0 −16.8614
1.2 2.37228 0 −2.37228 5.00000 0 7.37228 −24.6060 0 11.8614
\(n\): 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.d 2
3.b odd 2 1 405.4.a.e 2
5.b even 2 1 2025.4.a.l 2
9.c even 3 2 45.4.e.a 4
9.d odd 6 2 135.4.e.a 4
15.d odd 2 1 2025.4.a.j 2
45.j even 6 2 225.4.e.a 4
45.k odd 12 4 225.4.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.a 4 9.c even 3 2
135.4.e.a 4 9.d odd 6 2
225.4.e.a 4 45.j even 6 2
225.4.k.a 8 45.k odd 12 4
405.4.a.d 2 1.a even 1 1 trivial
405.4.a.e 2 3.b odd 2 1
2025.4.a.j 2 15.d odd 2 1
2025.4.a.l 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 37T + 136 \) Copy content Toggle raw display
$13$ \( T^{2} + 112T + 2608 \) Copy content Toggle raw display
$17$ \( T^{2} - 77T - 3674 \) Copy content Toggle raw display
$19$ \( T^{2} - 35T - 4850 \) Copy content Toggle raw display
$23$ \( T^{2} + 267T + 13458 \) Copy content Toggle raw display
$29$ \( T^{2} - 325T + 13858 \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 638T + 100936 \) Copy content Toggle raw display
$41$ \( T^{2} - 238T - 15539 \) Copy content Toggle raw display
$43$ \( T^{2} + 97T - 60092 \) Copy content Toggle raw display
$47$ \( T^{2} + 901T + 201952 \) Copy content Toggle raw display
$53$ \( T^{2} - 224T - 69956 \) Copy content Toggle raw display
$59$ \( T^{2} + 85T - 18002 \) Copy content Toggle raw display
$61$ \( T^{2} + 247T - 298454 \) Copy content Toggle raw display
$67$ \( T^{2} + 606T - 187503 \) Copy content Toggle raw display
$71$ \( T^{2} - 394T - 61016 \) Copy content Toggle raw display
$73$ \( T^{2} + 811T - 182276 \) Copy content Toggle raw display
$79$ \( T^{2} + 840T + 128748 \) Copy content Toggle raw display
$83$ \( T^{2} + 387 T - 1655532 \) Copy content Toggle raw display
$89$ \( T^{2} + 1065 T - 535050 \) Copy content Toggle raw display
$97$ \( T^{2} + 1031 T + 124162 \) Copy content Toggle raw display
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