Properties

Label 405.4.e.r
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: 6.0.95327307.1
Defining polynomial: \(x^{6} - 3 x^{5} + 20 x^{4} - 35 x^{3} + 85 x^{2} - 68 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 135)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( -\beta_{1} - 7 \beta_{3} - \beta_{4} + \beta_{5} ) q^{4} + 5 \beta_{3} q^{5} + ( -14 + 14 \beta_{3} + 2 \beta_{4} ) q^{7} + ( 9 + 8 \beta_{1} + \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{4} q^{2} + ( -\beta_{1} - 7 \beta_{3} - \beta_{4} + \beta_{5} ) q^{4} + 5 \beta_{3} q^{5} + ( -14 + 14 \beta_{3} + 2 \beta_{4} ) q^{7} + ( 9 + 8 \beta_{1} + \beta_{2} ) q^{8} -5 \beta_{1} q^{10} + ( 12 + 4 \beta_{2} - 12 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{11} + ( 10 \beta_{1} - 14 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} ) q^{13} + ( -16 \beta_{1} - 30 \beta_{3} - 16 \beta_{4} + 2 \beta_{5} ) q^{14} + ( -58 + 58 \beta_{3} + 17 \beta_{4} ) q^{16} + ( 3 + 2 \beta_{1} + 8 \beta_{2} ) q^{17} + ( 59 + 14 \beta_{1} - 4 \beta_{2} ) q^{19} + ( 35 + 5 \beta_{2} - 35 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} ) q^{20} + ( 42 \beta_{1} - 54 \beta_{3} + 42 \beta_{4} + 2 \beta_{5} ) q^{22} + ( 32 \beta_{1} - 39 \beta_{3} + 32 \beta_{4} - 4 \beta_{5} ) q^{23} + ( -25 + 25 \beta_{3} ) q^{25} + ( -126 - 28 \beta_{1} - 10 \beta_{2} ) q^{26} + ( 116 + 46 \beta_{1} + 16 \beta_{2} ) q^{28} + ( 42 - 8 \beta_{2} - 42 \beta_{3} - 42 \beta_{4} - 8 \beta_{5} ) q^{29} + ( 30 \beta_{1} - 83 \beta_{3} + 30 \beta_{4} + 8 \beta_{5} ) q^{31} + ( -11 \beta_{1} - 183 \beta_{3} - 11 \beta_{4} + 9 \beta_{5} ) q^{32} + ( 18 - 2 \beta_{2} - 18 \beta_{3} + 69 \beta_{4} - 2 \beta_{5} ) q^{34} + ( -70 - 10 \beta_{1} ) q^{35} + ( 50 - 64 \beta_{1} - 8 \beta_{2} ) q^{37} + ( -234 - 14 \beta_{2} + 234 \beta_{3} + 41 \beta_{4} - 14 \beta_{5} ) q^{38} + ( 40 \beta_{1} + 45 \beta_{3} + 40 \beta_{4} - 5 \beta_{5} ) q^{40} + ( 42 \beta_{1} - 132 \beta_{3} + 42 \beta_{4} - 16 \beta_{5} ) q^{41} + ( 16 + 8 \beta_{2} - 16 \beta_{3} + 78 \beta_{4} + 8 \beta_{5} ) q^{43} + ( -546 + 12 \beta_{1} - 10 \beta_{2} ) q^{44} + ( -456 - 25 \beta_{1} - 32 \beta_{2} ) q^{46} + ( -168 + 4 \beta_{2} + 168 \beta_{3} - 28 \beta_{4} + 4 \beta_{5} ) q^{47} + ( -60 \beta_{1} + 87 \beta_{3} - 60 \beta_{4} + 4 \beta_{5} ) q^{49} + ( -25 \beta_{1} - 25 \beta_{4} ) q^{50} + ( 472 - 4 \beta_{2} - 472 \beta_{3} - 154 \beta_{4} - 4 \beta_{5} ) q^{52} + ( -165 - 14 \beta_{1} - 12 \beta_{2} ) q^{53} + ( 60 - 10 \beta_{1} + 20 \beta_{2} ) q^{55} + ( -354 - 30 \beta_{2} + 354 \beta_{3} + 162 \beta_{4} - 30 \beta_{5} ) q^{56} + ( 20 \beta_{1} + 678 \beta_{3} + 20 \beta_{4} - 42 \beta_{5} ) q^{58} + ( -82 \beta_{1} + 78 \beta_{3} - 82 \beta_{4} + 12 \beta_{5} ) q^{59} + ( -173 - 16 \beta_{2} + 173 \beta_{3} + 60 \beta_{4} - 16 \beta_{5} ) q^{61} + ( -498 + 117 \beta_{1} - 30 \beta_{2} ) q^{62} + ( -353 + 130 \beta_{1} + 11 \beta_{2} ) q^{64} + ( 70 - 20 \beta_{2} - 70 \beta_{3} + 50 \beta_{4} - 20 \beta_{5} ) q^{65} + ( 52 \beta_{1} - 302 \beta_{3} + 52 \beta_{4} + 24 \beta_{5} ) q^{67} + ( -51 \beta_{1} - 999 \beta_{3} - 51 \beta_{4} + 5 \beta_{5} ) q^{68} + ( 150 + 10 \beta_{2} - 150 \beta_{3} - 80 \beta_{4} + 10 \beta_{5} ) q^{70} + ( 192 - 34 \beta_{1} + 60 \beta_{2} ) q^{71} + ( 404 + 22 \beta_{1} + 60 \beta_{2} ) q^{73} + ( 912 + 64 \beta_{2} - 912 \beta_{3} - 78 \beta_{4} + 64 \beta_{5} ) q^{74} + ( -275 \beta_{1} - 59 \beta_{3} - 275 \beta_{4} + 73 \beta_{5} ) q^{76} + ( 56 \beta_{1} + 60 \beta_{3} + 56 \beta_{4} - 52 \beta_{5} ) q^{77} + ( -221 + 12 \beta_{2} + 221 \beta_{3} - 22 \beta_{4} + 12 \beta_{5} ) q^{79} + ( -290 - 85 \beta_{1} ) q^{80} + ( -534 - 38 \beta_{1} - 42 \beta_{2} ) q^{82} + ( -489 + 60 \beta_{2} + 489 \beta_{3} - 120 \beta_{4} + 60 \beta_{5} ) q^{83} + ( 10 \beta_{1} + 15 \beta_{3} + 10 \beta_{4} - 40 \beta_{5} ) q^{85} + ( 2 \beta_{1} - 1218 \beta_{3} + 2 \beta_{4} + 78 \beta_{5} ) q^{86} + ( 192 + 4 \beta_{2} - 192 \beta_{3} - 278 \beta_{4} + 4 \beta_{5} ) q^{88} + ( -756 + 66 \beta_{1} + 48 \beta_{2} ) q^{89} + ( -56 - 196 \beta_{1} - 76 \beta_{2} ) q^{91} + ( 495 - 7 \beta_{2} - 495 \beta_{3} - 481 \beta_{4} - 7 \beta_{5} ) q^{92} + ( -108 \beta_{1} + 396 \beta_{3} - 108 \beta_{4} - 28 \beta_{5} ) q^{94} + ( 70 \beta_{1} + 295 \beta_{3} + 70 \beta_{4} + 20 \beta_{5} ) q^{95} + ( -440 - 8 \beta_{2} + 440 \beta_{3} + 64 \beta_{4} - 8 \beta_{5} ) q^{97} + ( 876 + 5 \beta_{1} + 60 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 23 q^{4} + 15 q^{5} - 44 q^{7} + 72 q^{8} + O(q^{10}) \) \( 6 q - q^{2} - 23 q^{4} + 15 q^{5} - 44 q^{7} + 72 q^{8} - 10 q^{10} + 38 q^{11} - 28 q^{13} - 108 q^{14} - 191 q^{16} + 38 q^{17} + 374 q^{19} + 115 q^{20} - 122 q^{22} - 81 q^{23} - 75 q^{25} - 832 q^{26} + 820 q^{28} + 160 q^{29} - 227 q^{31} - 569 q^{32} - 17 q^{34} - 440 q^{35} + 156 q^{37} - 757 q^{38} + 180 q^{40} - 338 q^{41} - 22 q^{43} - 3272 q^{44} - 2850 q^{46} - 472 q^{47} + 197 q^{49} - 25 q^{50} + 1566 q^{52} - 1042 q^{53} + 380 q^{55} - 1254 q^{56} + 2096 q^{58} + 140 q^{59} - 595 q^{61} - 2814 q^{62} - 1836 q^{64} + 140 q^{65} - 878 q^{67} - 3053 q^{68} + 540 q^{70} + 1204 q^{71} + 2588 q^{73} + 2878 q^{74} - 525 q^{76} + 288 q^{77} - 629 q^{79} - 1910 q^{80} - 3364 q^{82} - 1287 q^{83} + 95 q^{85} - 3730 q^{86} + 858 q^{88} - 4308 q^{89} - 880 q^{91} + 1959 q^{92} + 1108 q^{94} + 935 q^{95} - 1392 q^{97} + 5386 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 20 x^{4} - 35 x^{3} + 85 x^{2} - 68 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} + 14 \nu^{2} - 13 \nu + 24 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} - 8 \nu^{2} + 7 \nu + 12 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 38 \nu^{3} - 52 \nu^{2} + 149 \nu - 64 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{5} - 28 \nu^{4} + 206 \nu^{3} - 287 \nu^{2} + 786 \nu - 356 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{5} + 18 \nu^{4} - 132 \nu^{3} + 183 \nu^{2} - 514 \nu + 220 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_{1} + 6\)\()/9\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 5 \beta_{1} - 48\)\()/9\)
\(\nu^{3}\)\(=\)\((\)\(17 \beta_{5} + 8 \beta_{4} + 75 \beta_{3} + 13 \beta_{2} + 13 \beta_{1} - 114\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{5} + 6 \beta_{4} + 51 \beta_{3} - 5 \beta_{2} - 7 \beta_{1} + 102\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-136 \beta_{5} - 10 \beta_{4} - 879 \beta_{3} - 158 \beta_{2} - 95 \beta_{1} + 1524\)\()/9\)

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
0.500000 + 0.0280269i
0.500000 3.26212i
0.500000 + 2.36807i
0.500000 0.0280269i
0.500000 + 3.26212i
0.500000 2.36807i
−2.60034 + 4.50391i 0 −9.52349 16.4952i 2.50000 + 4.33013i 0 −12.2007 + 21.1322i 57.4517 0 −26.0034
136.2 −0.129356 + 0.224051i 0 3.96653 + 6.87024i 2.50000 + 4.33013i 0 −7.25871 + 12.5725i −4.12208 0 −1.29356
136.3 2.22969 3.86194i 0 −5.94305 10.2937i 2.50000 + 4.33013i 0 −2.54062 + 4.40048i −17.3296 0 22.2969
271.1 −2.60034 4.50391i 0 −9.52349 + 16.4952i 2.50000 4.33013i 0 −12.2007 21.1322i 57.4517 0 −26.0034
271.2 −0.129356 0.224051i 0 3.96653 6.87024i 2.50000 4.33013i 0 −7.25871 12.5725i −4.12208 0 −1.29356
271.3 2.22969 + 3.86194i 0 −5.94305 + 10.2937i 2.50000 4.33013i 0 −2.54062 4.40048i −17.3296 0 22.2969
\(n\): 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.r 6
3.b odd 2 1 405.4.e.t 6
9.c even 3 1 135.4.a.g yes 3
9.c even 3 1 inner 405.4.e.r 6
9.d odd 6 1 135.4.a.f 3
9.d odd 6 1 405.4.e.t 6
36.f odd 6 1 2160.4.a.be 3
36.h even 6 1 2160.4.a.bm 3
45.h odd 6 1 675.4.a.r 3
45.j even 6 1 675.4.a.q 3
45.k odd 12 2 675.4.b.k 6
45.l even 12 2 675.4.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 9.d odd 6 1
135.4.a.g yes 3 9.c even 3 1
405.4.e.r 6 1.a even 1 1 trivial
405.4.e.r 6 9.c even 3 1 inner
405.4.e.t 6 3.b odd 2 1
405.4.e.t 6 9.d odd 6 1
675.4.a.q 3 45.j even 6 1
675.4.a.r 3 45.h odd 6 1
675.4.b.k 6 45.k odd 12 2
675.4.b.l 6 45.l even 12 2
2160.4.a.be 3 36.f odd 6 1
2160.4.a.bm 3 36.h even 6 1

Hecke kernels

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

\( T_{2}^{6} + T_{2}^{5} + 24 T_{2}^{4} - 11 T_{2}^{3} + 535 T_{2}^{2} + 138 T_{2} + 36 \)
\( T_{7}^{6} + 44 T_{7}^{5} + 1384 T_{7}^{4} + 20688 T_{7}^{3} + 225504 T_{7}^{2} + 993600 T_{7} + 3240000 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 36 + 138 T + 535 T^{2} - 11 T^{3} + 24 T^{4} + T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( 25 - 5 T + T^{2} )^{3} \)
$7$ \( 3240000 + 993600 T + 225504 T^{2} + 20688 T^{3} + 1384 T^{4} + 44 T^{5} + T^{6} \)
$11$ \( 6935558400 - 217527360 T + 9987184 T^{2} - 67304 T^{3} + 4056 T^{4} - 38 T^{5} + T^{6} \)
$13$ \( 10024014400 + 458149120 T + 23743136 T^{2} + 72112 T^{3} + 5360 T^{4} + 28 T^{5} + T^{6} \)
$17$ \( ( 553887 - 11477 T - 19 T^{2} + T^{3} )^{2} \)
$19$ \( ( 525871 + 3587 T - 187 T^{2} + T^{3} )^{2} \)
$23$ \( 4177858328361 + 47626801281 T + 708499062 T^{2} + 2200581 T^{3} + 29862 T^{4} + 81 T^{5} + T^{6} \)
$29$ \( 62295660417600 - 377021359680 T + 3544623424 T^{2} - 8142640 T^{3} + 73368 T^{4} - 160 T^{5} + T^{6} \)
$31$ \( 60674035041 - 4427127333 T + 267113862 T^{2} - 4572513 T^{3} + 69502 T^{4} + 227 T^{5} + T^{6} \)
$37$ \( ( 13637080 - 99924 T - 78 T^{2} + T^{3} )^{2} \)
$41$ \( 146812964889600 + 518737591680 T + 5928291664 T^{2} + 9762824 T^{3} + 157056 T^{4} + 338 T^{5} + T^{6} \)
$43$ \( 340939975993600 + 2952778576960 T + 25979347376 T^{2} + 33410968 T^{3} + 160400 T^{4} + 22 T^{5} + T^{6} \)
$47$ \( 80202240000 - 15351705600 T + 3072177664 T^{2} + 26152576 T^{3} + 168576 T^{4} + 472 T^{5} + T^{6} \)
$53$ \( ( -939789 + 61387 T + 521 T^{2} + T^{3} )^{2} \)
$59$ \( 1164957926990400 - 5681116583040 T + 32483343904 T^{2} - 44960240 T^{3} + 186048 T^{4} - 140 T^{5} + T^{6} \)
$61$ \( 3177687716449 + 4900386643 T + 1068208166 T^{2} + 1929559 T^{3} + 356774 T^{4} + 595 T^{5} + T^{6} \)
$67$ \( 127577025000000 - 857832660000 T + 15685108704 T^{2} + 89272344 T^{3} + 694936 T^{4} + 878 T^{5} + T^{6} \)
$71$ \( ( 280550880 - 583652 T - 602 T^{2} + T^{3} )^{2} \)
$73$ \( ( 404091280 - 95908 T - 1294 T^{2} + T^{3} )^{2} \)
$79$ \( 4041318151809 + 195117998877 T + 8155968894 T^{2} + 57029505 T^{3} + 298582 T^{4} + 629 T^{5} + T^{6} \)
$83$ \( 119996015960256921 + 126741406683447 T + 579688456086 T^{2} + 221925123 T^{3} + 2022246 T^{4} + 1287 T^{5} + T^{6} \)
$89$ \( ( 74325600 + 1057572 T + 2154 T^{2} + T^{3} )^{2} \)
$97$ \( 4044390164070400 + 34591892766720 T + 207341408256 T^{2} + 629967872 T^{3} + 1393728 T^{4} + 1392 T^{5} + T^{6} \)
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