Properties

Label 405.4.e.s
Level $405$
Weight $4$
Character orbit 405.e
Analytic conductor $23.896$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.148347072.2
Defining polynomial: \(x^{6} + 29 x^{4} + 223 x^{2} + 243\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{3} ) q^{2} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{4} + ( 5 + 5 \beta_{2} ) q^{5} + ( -5 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} ) q^{7} + ( -8 + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{3} ) q^{2} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{4} + ( 5 + 5 \beta_{2} ) q^{5} + ( -5 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} ) q^{7} + ( -8 + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} + 5 \beta_{3} q^{10} + ( -7 \beta_{1} + 17 \beta_{2} + 7 \beta_{3} + 5 \beta_{4} ) q^{11} + ( 20 + 13 \beta_{1} + 20 \beta_{2} + 13 \beta_{5} ) q^{13} + ( -56 - 9 \beta_{1} - 56 \beta_{2} - 11 \beta_{5} ) q^{14} + ( -5 \beta_{1} - 44 \beta_{2} + 5 \beta_{3} - 9 \beta_{4} ) q^{16} + ( 14 + 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{17} + ( -5 - 10 \beta_{3} - 14 \beta_{4} + 14 \beta_{5} ) q^{19} + ( -5 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} ) q^{20} + ( -80 + 20 \beta_{1} - 80 \beta_{2} - 17 \beta_{5} ) q^{22} + ( 22 + 15 \beta_{1} + 22 \beta_{2} + 19 \beta_{5} ) q^{23} + 25 \beta_{2} q^{25} + ( 104 + 59 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} ) q^{26} + ( 12 - 47 \beta_{3} + 11 \beta_{4} - 11 \beta_{5} ) q^{28} + ( -65 \beta_{1} + 95 \beta_{2} + 65 \beta_{3} - 13 \beta_{4} ) q^{29} + ( -211 + 5 \beta_{1} - 211 \beta_{2} + \beta_{5} ) q^{31} + ( -96 - 43 \beta_{1} - 96 \beta_{2} + 21 \beta_{5} ) q^{32} + ( -38 \beta_{1} - 64 \beta_{2} + 38 \beta_{3} + 8 \beta_{4} ) q^{34} + ( 50 + 25 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} ) q^{35} + ( -156 - 54 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{37} + ( -13 \beta_{1} + 128 \beta_{2} + 13 \beta_{3} + 38 \beta_{4} ) q^{38} + ( -40 + 15 \beta_{1} - 40 \beta_{2} + 5 \beta_{5} ) q^{40} + ( -59 + 2 \beta_{1} - 59 \beta_{2} - 62 \beta_{5} ) q^{41} + ( -28 \beta_{1} - 288 \beta_{2} + 28 \beta_{3} - 8 \beta_{4} ) q^{43} + ( 98 - 38 \beta_{3} - 14 \beta_{4} + 14 \beta_{5} ) q^{44} + ( 112 + 75 \beta_{3} + 23 \beta_{4} - 23 \beta_{5} ) q^{46} + ( -67 \beta_{1} + 56 \beta_{2} + 67 \beta_{3} - 31 \beta_{4} ) q^{47} + ( -301 - 11 \beta_{1} - 301 \beta_{2} - 7 \beta_{5} ) q^{49} + 25 \beta_{1} q^{50} + ( -33 \beta_{1} - 456 \beta_{2} + 33 \beta_{3} + 19 \beta_{4} ) q^{52} + ( 186 + 53 \beta_{3} - 21 \beta_{4} + 21 \beta_{5} ) q^{53} + ( -85 + 35 \beta_{3} + 25 \beta_{4} - 25 \beta_{5} ) q^{55} + ( -15 \beta_{1} + 15 \beta_{3} - 63 \beta_{4} ) q^{56} + ( -624 + 4 \beta_{1} - 624 \beta_{2} - 39 \beta_{5} ) q^{58} + ( -159 + 58 \beta_{1} - 159 \beta_{2} + 90 \beta_{5} ) q^{59} + ( 122 \beta_{1} + 6 \beta_{2} - 122 \beta_{3} + 58 \beta_{4} ) q^{61} + ( 48 - 204 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{62} + ( -120 - 57 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} ) q^{64} + ( 65 \beta_{1} + 100 \beta_{2} - 65 \beta_{3} + 65 \beta_{4} ) q^{65} + ( -38 - 154 \beta_{1} - 38 \beta_{2} + 18 \beta_{5} ) q^{67} + ( -284 - 22 \beta_{1} - 284 \beta_{2} + 10 \beta_{5} ) q^{68} + ( -45 \beta_{1} - 280 \beta_{2} + 45 \beta_{3} - 55 \beta_{4} ) q^{70} + ( 177 + 79 \beta_{3} + 69 \beta_{4} - 69 \beta_{5} ) q^{71} + ( -226 + 32 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} ) q^{73} + ( 214 \beta_{1} + 536 \beta_{2} - 214 \beta_{3} + 50 \beta_{4} ) q^{74} + ( -246 + 111 \beta_{1} - 246 \beta_{2} + 23 \beta_{5} ) q^{76} + ( -662 + 162 \beta_{1} - 662 \beta_{2} - 98 \beta_{5} ) q^{77} + ( 82 \beta_{1} - 184 \beta_{2} - 82 \beta_{3} - 54 \beta_{4} ) q^{79} + ( 220 + 25 \beta_{3} - 45 \beta_{4} + 45 \beta_{5} ) q^{80} + ( 144 - 181 \beta_{3} - 126 \beta_{4} + 126 \beta_{5} ) q^{82} + ( 210 \beta_{1} + 648 \beta_{2} - 210 \beta_{3} - 30 \beta_{4} ) q^{83} + ( 70 + 40 \beta_{1} + 70 \beta_{2} + 40 \beta_{5} ) q^{85} + ( -264 - 332 \beta_{1} - 264 \beta_{2} - 12 \beta_{5} ) q^{86} + ( 72 \beta_{1} - 232 \beta_{2} - 72 \beta_{3} - 70 \beta_{4} ) q^{88} + ( -297 - 39 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{89} + ( -216 + 581 \beta_{3} - 161 \beta_{4} + 161 \beta_{5} ) q^{91} + ( -21 \beta_{1} - 620 \beta_{2} + 21 \beta_{3} + 31 \beta_{4} ) q^{92} + ( -608 - 73 \beta_{1} - 608 \beta_{2} - 5 \beta_{5} ) q^{94} + ( -25 - 50 \beta_{1} - 25 \beta_{2} + 70 \beta_{5} ) q^{95} + ( 336 \beta_{1} + 112 \beta_{2} - 336 \beta_{3} + 32 \beta_{4} ) q^{97} + ( -96 - 326 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8} + O(q^{10}) \) \( 6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8} - 10 q^{10} - 58 q^{11} + 47 q^{13} - 159 q^{14} + 127 q^{16} + 68 q^{17} - 10 q^{19} + 25 q^{20} - 260 q^{22} + 51 q^{23} - 75 q^{25} + 506 q^{26} + 166 q^{28} - 350 q^{29} - 638 q^{31} - 245 q^{32} + 154 q^{34} + 250 q^{35} - 828 q^{37} - 397 q^{38} - 135 q^{40} - 179 q^{41} + 836 q^{43} + 664 q^{44} + 522 q^{46} - 235 q^{47} - 892 q^{49} - 25 q^{50} + 1335 q^{52} + 1010 q^{53} - 580 q^{55} - 15 q^{56} - 1876 q^{58} - 535 q^{59} + 104 q^{61} + 696 q^{62} - 606 q^{64} - 235 q^{65} + 40 q^{67} - 830 q^{68} + 795 q^{70} + 904 q^{71} - 1420 q^{73} - 1394 q^{74} - 849 q^{76} - 2148 q^{77} + 634 q^{79} + 1270 q^{80} + 1226 q^{82} - 1734 q^{83} + 170 q^{85} - 460 q^{86} + 768 q^{88} - 1704 q^{89} - 2458 q^{91} + 1839 q^{92} - 1751 q^{94} - 25 q^{95} + 76 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 29 x^{4} + 223 x^{2} + 243\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu^{2} + 13 \nu + 9 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 20 \nu^{3} + 79 \nu - 36 \)\()/72\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{2} + 9 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 9 \nu^{4} - 20 \nu^{3} - 144 \nu^{2} + 29 \nu - 207 \)\()/72\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 9 \nu^{4} - 20 \nu^{3} + 144 \nu^{2} + 29 \nu + 207 \)\()/72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 2 \beta_{2} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{3} - 9\)
\(\nu^{3}\)\(=\)\((\)\(-13 \beta_{5} - 13 \beta_{4} - 6 \beta_{3} - 26 \beta_{2} + 12 \beta_{1} - 13\)\()/3\)
\(\nu^{4}\)\(=\)\(4 \beta_{5} - 4 \beta_{4} - 32 \beta_{3} + 121\)
\(\nu^{5}\)\(=\)\((\)\(181 \beta_{5} + 181 \beta_{4} + 120 \beta_{3} + 578 \beta_{2} - 240 \beta_{1} + 289\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
4.00586i
3.41374i
1.13993i
4.00586i
3.41374i
1.13993i
−1.76174 + 3.05142i 0 −2.20744 3.82340i 2.50000 + 4.33013i 0 −12.7162 + 22.0252i −12.6321 0 −17.6174
136.2 −0.663404 + 1.14905i 0 3.11979 + 5.40363i 2.50000 + 4.33013i 0 12.0521 20.8749i −18.8932 0 −6.63404
136.3 1.92514 3.33444i 0 −3.41235 5.91036i 2.50000 + 4.33013i 0 13.1641 22.8009i 4.52526 0 19.2514
271.1 −1.76174 3.05142i 0 −2.20744 + 3.82340i 2.50000 4.33013i 0 −12.7162 22.0252i −12.6321 0 −17.6174
271.2 −0.663404 1.14905i 0 3.11979 5.40363i 2.50000 4.33013i 0 12.0521 + 20.8749i −18.8932 0 −6.63404
271.3 1.92514 + 3.33444i 0 −3.41235 + 5.91036i 2.50000 4.33013i 0 13.1641 + 22.8009i 4.52526 0 19.2514
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.e.s 6
3.b odd 2 1 405.4.e.u 6
9.c even 3 1 405.4.a.i yes 3
9.c even 3 1 inner 405.4.e.s 6
9.d odd 6 1 405.4.a.g 3
9.d odd 6 1 405.4.e.u 6
45.h odd 6 1 2025.4.a.r 3
45.j even 6 1 2025.4.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.g 3 9.d odd 6 1
405.4.a.i yes 3 9.c even 3 1
405.4.e.s 6 1.a even 1 1 trivial
405.4.e.s 6 9.c even 3 1 inner
405.4.e.u 6 3.b odd 2 1
405.4.e.u 6 9.d odd 6 1
2025.4.a.p 3 45.j even 6 1
2025.4.a.r 3 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{6} + T_{2}^{5} + 15 T_{2}^{4} + 22 T_{2}^{3} + 214 T_{2}^{2} + 252 T_{2} + 324 \)
\( T_{7}^{6} - 25 T_{7}^{5} + 1273 T_{7}^{4} - 16080 T_{7}^{3} + 823404 T_{7}^{2} - 10458720 T_{7} + 260499600 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 324 + 252 T + 214 T^{2} + 22 T^{3} + 15 T^{4} + T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( 25 - 5 T + T^{2} )^{3} \)
$7$ \( 260499600 - 10458720 T + 823404 T^{2} - 16080 T^{3} + 1273 T^{4} - 25 T^{5} + T^{6} \)
$11$ \( 9000000 - 2733000 T + 655921 T^{2} - 58838 T^{3} + 4275 T^{4} + 58 T^{5} + T^{6} \)
$13$ \( 137160603904 - 2752456064 T + 72641168 T^{2} - 391400 T^{3} + 9641 T^{4} - 47 T^{5} + T^{6} \)
$17$ \( ( 90984 - 2708 T - 34 T^{2} + T^{3} )^{2} \)
$19$ \( ( -299645 - 10777 T + 5 T^{2} + T^{3} )^{2} \)
$23$ \( 1084005816336 - 15867217440 T + 285356556 T^{2} - 1305072 T^{3} + 17841 T^{4} - 51 T^{5} + T^{6} \)
$29$ \( 126287249817600 + 225463178880 T + 4335739969 T^{2} + 15453470 T^{3} + 142563 T^{4} + 350 T^{5} + T^{6} \)
$31$ \( 90993741996096 + 1290835679544 T + 12225850209 T^{2} + 67256670 T^{3} + 271723 T^{4} + 638 T^{5} + T^{6} \)
$37$ \( ( -577760 + 16032 T + 414 T^{2} + T^{3} )^{2} \)
$41$ \( 316826721339129 + 2702285972259 T + 26234534722 T^{2} + 8424011 T^{3} + 183858 T^{4} + 179 T^{5} + T^{6} \)
$43$ \( 349427949912064 - 4128010809344 T + 33139430912 T^{2} - 147229568 T^{3} + 478064 T^{4} - 836 T^{5} + T^{6} \)
$47$ \( 81096796901376 + 627494599680 T + 6971565760 T^{2} + 1635952 T^{3} + 124905 T^{4} + 235 T^{5} + T^{6} \)
$53$ \( ( 1500684 + 35128 T - 505 T^{2} + T^{3} )^{2} \)
$59$ \( 498077813969649 - 5607289003593 T + 51186113506 T^{2} - 179053529 T^{3} + 537474 T^{4} + 535 T^{5} + T^{6} \)
$61$ \( 554264938580224 - 6931292254784 T + 89126880272 T^{2} - 16466816 T^{3} + 305228 T^{4} - 104 T^{5} + T^{6} \)
$67$ \( 246999193899264 - 5896406917440 T + 141388680720 T^{2} - 16425216 T^{3} + 376780 T^{4} - 40 T^{5} + T^{6} \)
$71$ \( ( 116183454 - 264923 T - 452 T^{2} + T^{3} )^{2} \)
$73$ \( ( 8707528 + 144236 T + 710 T^{2} + T^{3} )^{2} \)
$79$ \( 25533374939136 + 607822000128 T + 11265565440 T^{2} + 86368704 T^{3} + 522244 T^{4} - 634 T^{5} + T^{6} \)
$83$ \( 49432784635183104 - 62687755063296 T + 465025556736 T^{2} + 933574464 T^{3} + 2724804 T^{4} + 1734 T^{5} + T^{6} \)
$89$ \( ( 17926434 + 220725 T + 852 T^{2} + T^{3} )^{2} \)
$97$ \( 494595736877400064 - 1107776287801344 T + 2481154228224 T^{2} - 1406550016 T^{3} + 1575168 T^{4} + T^{6} \)
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