Properties

Label 405.4.a.n
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 44 x^{5} + 74 x^{4} + 479 x^{3} - 460 x^{2} - 1200 x + 288\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 5 + \beta_{2} ) q^{4} + 5 q^{5} + ( 3 - \beta_{5} ) q^{7} + ( 1 + 7 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 5 + \beta_{2} ) q^{4} + 5 q^{5} + ( 3 - \beta_{5} ) q^{7} + ( 1 + 7 \beta_{1} + \beta_{3} ) q^{8} + 5 \beta_{1} q^{10} + ( 2 + 3 \beta_{1} + \beta_{4} ) q^{11} + ( 15 - 4 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} + ( -4 + 7 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{14} + ( 46 - 6 \beta_{1} + 7 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{16} + ( 22 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{17} + ( 40 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{19} + ( 25 + 5 \beta_{2} ) q^{20} + ( 44 - \beta_{1} + 5 \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{22} + ( 11 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{23} + 25 q^{25} + ( -59 + 32 \beta_{1} - 10 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{26} + ( 52 - 14 \beta_{1} - \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{28} + ( -46 + 15 \beta_{1} - 5 \beta_{2} - \beta_{5} + 3 \beta_{6} ) q^{29} + ( 37 - 8 \beta_{1} - 9 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{31} + ( -58 + 53 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} ) q^{32} + ( 10 + 45 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{34} + ( 15 - 5 \beta_{5} ) q^{35} + ( 61 - 14 \beta_{1} - 11 \beta_{2} + 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{37} + ( 26 + 49 \beta_{1} + 17 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 5 \beta_{6} ) q^{38} + ( 5 + 35 \beta_{1} + 5 \beta_{3} ) q^{40} + ( -8 + 4 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + \beta_{4} - 9 \beta_{5} - \beta_{6} ) q^{41} + ( 82 - 29 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} ) q^{43} + ( -6 + 45 \beta_{1} + 15 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} ) q^{44} + ( 94 + 41 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{46} + ( 30 - 4 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + 8 \beta_{5} + \beta_{6} ) q^{47} + ( 172 + 19 \beta_{1} - 20 \beta_{2} - \beta_{4} - 6 \beta_{5} - 10 \beta_{6} ) q^{49} + 25 \beta_{1} q^{50} + ( 265 - 98 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{52} + ( 74 - 4 \beta_{1} + 24 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 10 \beta_{6} ) q^{53} + ( 10 + 15 \beta_{1} + 5 \beta_{4} ) q^{55} + ( -143 + 29 \beta_{1} - 42 \beta_{2} + \beta_{3} - 2 \beta_{4} - 13 \beta_{5} - 3 \beta_{6} ) q^{56} + ( 189 - 92 \beta_{1} + 7 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} ) q^{58} + ( 1 - 53 \beta_{1} - \beta_{2} - 10 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 13 \beta_{6} ) q^{59} + ( 95 + 101 \beta_{1} - 20 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} ) q^{61} + ( -101 - 18 \beta_{1} - 46 \beta_{2} - 7 \beta_{3} - 10 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} ) q^{62} + ( 335 - 88 \beta_{1} + 41 \beta_{2} + 2 \beta_{3} + 12 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{64} + ( 75 - 20 \beta_{1} + 5 \beta_{2} - 5 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} ) q^{65} + ( 171 - 21 \beta_{1} - 12 \beta_{2} + 8 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} ) q^{67} + ( 364 - 99 \beta_{1} + 77 \beta_{2} - 3 \beta_{3} + 10 \beta_{4} - 7 \beta_{5} - \beta_{6} ) q^{68} + ( -20 + 35 \beta_{1} - 10 \beta_{2} - 10 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} ) q^{70} + ( 89 - 144 \beta_{1} - 3 \beta_{2} + \beta_{4} + 7 \beta_{5} + 7 \beta_{6} ) q^{71} + ( 293 + 29 \beta_{1} - 3 \beta_{2} + 10 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{73} + ( -173 - 60 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} + 14 \beta_{5} + 6 \beta_{6} ) q^{74} + ( 348 + 169 \beta_{1} + 41 \beta_{2} + 9 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 3 \beta_{6} ) q^{76} + ( 37 - 104 \beta_{1} - 37 \beta_{2} + 4 \beta_{3} + 9 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} ) q^{77} + ( 322 - 58 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} + 16 \beta_{6} ) q^{79} + ( 230 - 30 \beta_{1} + 35 \beta_{2} - 5 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} - 10 \beta_{6} ) q^{80} + ( 5 - 39 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} - 14 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} ) q^{82} + ( 223 - 138 \beta_{1} + 14 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 8 \beta_{6} ) q^{83} + ( 110 - 5 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} + 10 \beta_{5} - 10 \beta_{6} ) q^{85} + ( -354 - 39 \beta_{1} - 35 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} + 11 \beta_{5} + 9 \beta_{6} ) q^{86} + ( 294 + 55 \beta_{1} + 85 \beta_{2} + 11 \beta_{3} + 22 \beta_{4} + 15 \beta_{5} - 23 \beta_{6} ) q^{88} + ( -406 - 133 \beta_{1} - 13 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} - 19 \beta_{5} - 13 \beta_{6} ) q^{89} + ( 443 + 98 \beta_{1} + 13 \beta_{2} - 22 \beta_{3} - 9 \beta_{4} + 11 \beta_{5} - 15 \beta_{6} ) q^{91} + ( 398 - 44 \beta_{1} + 41 \beta_{2} - \beta_{3} + 4 \beta_{4} - 9 \beta_{5} - 7 \beta_{6} ) q^{92} + ( -63 - 91 \beta_{1} + 16 \beta_{2} - 17 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} - 17 \beta_{6} ) q^{94} + ( 200 + 15 \beta_{1} + 10 \beta_{2} + 10 \beta_{3} - 5 \beta_{4} ) q^{95} + ( 307 - 175 \beta_{1} + 37 \beta_{2} + 22 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} ) q^{97} + ( 188 - \beta_{1} + 25 \beta_{2} - 11 \beta_{3} - 34 \beta_{4} - 9 \beta_{5} + 5 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + O(q^{10}) \) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 44 x^{5} + 74 x^{4} + 479 x^{3} - 460 x^{2} - 1200 x + 288\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 13 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 23 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 39 \nu^{4} + 35 \nu^{3} + 310 \nu^{2} - 114 \nu - 240 \)\()/12\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{5} + 42 \nu^{4} - 140 \nu^{3} - 397 \nu^{2} + 756 \nu + 672 \)\()/24\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{5} - 48 \nu^{4} - 82 \nu^{3} + 595 \nu^{2} + 732 \nu - 1104 \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 23 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 31 \beta_{2} - 6 \beta_{1} + 294\)
\(\nu^{5}\)\(=\)\(2 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 36 \beta_{3} - 2 \beta_{2} + 597 \beta_{1} - 26\)
\(\nu^{6}\)\(=\)\(-76 \beta_{6} + 84 \beta_{5} + 92 \beta_{4} - 38 \beta_{3} + 897 \beta_{2} - 328 \beta_{1} + 7615\)

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
−5.38503
−3.04174
−1.57021
0.225250
2.19444
4.26178
5.31551
−5.38503 0 20.9986 5.00000 0 12.5702 −69.9976 0 −26.9252
1.2 −3.04174 0 1.25219 5.00000 0 −13.7122 20.5251 0 −15.2087
1.3 −1.57021 0 −5.53444 5.00000 0 34.2398 21.2519 0 −7.85104
1.4 0.225250 0 −7.94926 5.00000 0 −31.1940 −3.59257 0 1.12625
1.5 2.19444 0 −3.18442 5.00000 0 2.76605 −24.5436 0 10.9722
1.6 4.26178 0 10.1628 5.00000 0 30.7639 9.21718 0 21.3089
1.7 5.31551 0 20.2546 5.00000 0 −13.4337 65.1396 0 26.5775
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.n 7
3.b odd 2 1 405.4.a.m 7
5.b even 2 1 2025.4.a.ba 7
9.c even 3 2 135.4.e.c 14
9.d odd 6 2 45.4.e.c 14
15.d odd 2 1 2025.4.a.bb 7
45.h odd 6 2 225.4.e.d 14
45.l even 12 4 225.4.k.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.c 14 9.d odd 6 2
135.4.e.c 14 9.c even 3 2
225.4.e.d 14 45.h odd 6 2
225.4.k.d 28 45.l even 12 4
405.4.a.m 7 3.b odd 2 1
405.4.a.n 7 1.a even 1 1 trivial
2025.4.a.ba 7 5.b even 2 1
2025.4.a.bb 7 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 2 T_{2}^{6} - 44 T_{2}^{5} + 74 T_{2}^{4} + 479 T_{2}^{3} - 460 T_{2}^{2} - 1200 T_{2} + 288 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 288 - 1200 T - 460 T^{2} + 479 T^{3} + 74 T^{4} - 44 T^{5} - 2 T^{6} + T^{7} \)
$3$ \( T^{7} \)
$5$ \( ( -5 + T )^{7} \)
$7$ \( 210450744 - 68052735 T - 4867848 T^{2} + 650337 T^{3} + 26142 T^{4} - 1571 T^{5} - 22 T^{6} + T^{7} \)
$11$ \( -20019444768 - 2234576400 T + 4473656 T^{2} + 5647316 T^{3} + 46184 T^{4} - 4166 T^{5} - 23 T^{6} + T^{7} \)
$13$ \( 159234392576 + 54441888 T - 558177424 T^{2} + 1597000 T^{3} + 436772 T^{4} - 3188 T^{5} - 96 T^{6} + T^{7} \)
$17$ \( 60588009792 + 7871650176 T - 2499369808 T^{2} - 27251020 T^{3} + 2125940 T^{4} - 9644 T^{5} - 161 T^{6} + T^{7} \)
$19$ \( -1377989598400 + 1417080000 T + 6594215888 T^{2} - 224819660 T^{3} + 1854320 T^{4} + 15442 T^{5} - 279 T^{6} + T^{7} \)
$23$ \( -4061042235738 + 319448690109 T - 7434227790 T^{2} + 899433 T^{3} + 1879776 T^{4} - 14439 T^{5} - 96 T^{6} + T^{7} \)
$29$ \( -12128150971026 - 2048542254411 T - 92897749316 T^{2} - 1753720489 T^{3} - 13795250 T^{4} - 16301 T^{5} + 296 T^{6} + T^{7} \)
$31$ \( 29481260805504 + 2691313729344 T - 21117961440 T^{2} - 1452653784 T^{3} + 22032132 T^{4} - 69584 T^{5} - 244 T^{6} + T^{7} \)
$37$ \( -83646911884544 - 8022049775936 T - 196405787520 T^{2} + 585288704 T^{3} + 34349204 T^{4} - 86580 T^{5} - 404 T^{6} + T^{7} \)
$41$ \( 1524782419390161 + 58383942911499 T + 692598630925 T^{2} + 1021803995 T^{3} - 31456877 T^{4} - 161351 T^{5} + 47 T^{6} + T^{7} \)
$43$ \( -1328908142680384 + 146770517416224 T - 1523916256048 T^{2} - 3470994956 T^{3} + 73393160 T^{4} - 94898 T^{5} - 525 T^{6} + T^{7} \)
$47$ \( 6505162038618372 + 292893869594349 T + 4217212042898 T^{2} + 20820619937 T^{3} - 8230474 T^{4} - 267791 T^{5} - 164 T^{6} + T^{7} \)
$53$ \( 111107669935704192 + 3313074698297664 T - 34008897116512 T^{2} + 39458567216 T^{3} + 279547832 T^{4} - 500516 T^{5} - 506 T^{6} + T^{7} \)
$59$ \( -595651050751670208 - 38178523175987136 T + 14741411894096 T^{2} + 375330469568 T^{3} - 71749996 T^{4} - 1106312 T^{5} + 85 T^{6} + T^{7} \)
$61$ \( 1223316036378446 - 192599375971827 T + 7109002299836 T^{2} - 97105034237 T^{3} + 471979010 T^{4} - 451073 T^{5} - 828 T^{6} + T^{7} \)
$67$ \( 29225224702431603 + 686210632919667 T - 11724435478647 T^{2} - 9567655617 T^{3} + 273237249 T^{4} - 88235 T^{5} - 1093 T^{6} + T^{7} \)
$71$ \( 329554038570193728 - 2740127166757248 T - 99927513780416 T^{2} + 322939201712 T^{3} + 436205092 T^{4} - 1229012 T^{5} - 328 T^{6} + T^{7} \)
$73$ \( 82873748372160512 + 2981349592535040 T + 20542100801024 T^{2} - 128439358016 T^{3} - 117820816 T^{4} + 1327444 T^{5} - 2085 T^{6} + T^{7} \)
$79$ \( 16163225966939188608 + 91608055797302208 T - 217501872207264 T^{2} - 941918607696 T^{3} + 2167211400 T^{4} + 15124 T^{5} - 2110 T^{6} + T^{7} \)
$83$ \( -6624473796314288172 + 68135962294721277 T - 175355495435376 T^{2} - 206497270215 T^{3} + 1234542222 T^{4} - 720135 T^{5} - 1290 T^{6} + T^{7} \)
$89$ \( \)\(32\!\cdots\!50\)\( + 277911272642123745 T - 3417793050944628 T^{2} - 8680332870945 T^{3} - 6568870986 T^{4} + 788319 T^{5} + 3048 T^{6} + T^{7} \)
$97$ \( \)\(47\!\cdots\!76\)\( + 331725638028814336 T - 3031047860438448 T^{2} + 320540186240 T^{3} + 4996509932 T^{4} - 2304468 T^{5} - 1787 T^{6} + T^{7} \)
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