Properties

Label 405.4.e.q
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: 6.0.84779568.3
Defining polynomial: \( x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - 2 \beta_{3} - \beta_1) q^{2} + ( - 3 \beta_{5} - \beta_{4} + 7 \beta_{3} - 7) q^{4} + (5 \beta_{3} - 5) q^{5} + (\beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1) q^{7} + (5 \beta_{2} + 7 \beta_1 + 29) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 2 \beta_{3} - \beta_1) q^{2} + ( - 3 \beta_{5} - \beta_{4} + 7 \beta_{3} - 7) q^{4} + (5 \beta_{3} - 5) q^{5} + (\beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1) q^{7} + (5 \beta_{2} + 7 \beta_1 + 29) q^{8} + (5 \beta_1 + 10) q^{10} + ( - 9 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} + 9 \beta_1) q^{11} + ( - 6 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 5) q^{13} + (16 \beta_{5} + 5 \beta_{4} + 13 \beta_{3} - 13) q^{14} + (37 \beta_{5} + 9 \beta_{4} - 69 \beta_{3} - 9 \beta_{2} - 37 \beta_1) q^{16} + ( - 5 \beta_{2} + 3 \beta_1 + 51) q^{17} + ( - \beta_{2} - 33 \beta_1 - 28) q^{19} + (15 \beta_{5} + 5 \beta_{4} - 35 \beta_{3} - 5 \beta_{2} - 15 \beta_1) q^{20} + (37 \beta_{5} + 19 \beta_{4} - 95 \beta_{3} + 95) q^{22} + (25 \beta_{5} + 5 \beta_{4} + 85 \beta_{3} - 85) q^{23} - 25 \beta_{3} q^{25} + (10 \beta_{2} + 21 \beta_1 + 72) q^{26} + ( - 2 \beta_{2} - 36 \beta_1 - 124) q^{28} + (\beta_{5} + 25 \beta_{4} - 47 \beta_{3} - 25 \beta_{2} - \beta_1) q^{29} + (19 \beta_{5} - 17 \beta_{4} - 39 \beta_{3} + 39) q^{31} + ( - 95 \beta_{5} - 15 \beta_{4} + 295 \beta_{3} - 295) q^{32} + (29 \beta_{5} - 7 \beta_{4} - 145 \beta_{3} + 7 \beta_{2} - 29 \beta_1) q^{34} + ( - 15 \beta_{2} + 5 \beta_1 - 10) q^{35} + (25 \beta_{2} - 55 \beta_1 - 138) q^{37} + ( - 66 \beta_{5} - 35 \beta_{4} + 417 \beta_{3} + 35 \beta_{2} + 66 \beta_1) q^{38} + ( - 35 \beta_{5} - 25 \beta_{4} + 145 \beta_{3} - 145) q^{40} + (26 \beta_{5} - 10 \beta_{4} + 188 \beta_{3} - 188) q^{41} + ( - 17 \beta_{5} - 29 \beta_{4} + 281 \beta_{3} + 29 \beta_{2} + 17 \beta_1) q^{43} + ( - 35 \beta_{2} - 155 \beta_1 - 535) q^{44} + ( - 35 \beta_{2} + 35 \beta_1 - 95) q^{46} + (123 \beta_{5} - 5 \beta_{4} + 9 \beta_{3} + 5 \beta_{2} - 123 \beta_1) q^{47} + (53 \beta_{5} + 41 \beta_{4} + 161 \beta_{3} - 161) q^{49} + ( - 25 \beta_{5} + 50 \beta_{3} - 50) q^{50} + (95 \beta_{5} + 25 \beta_{4} - 315 \beta_{3} - 25 \beta_{2} - 95 \beta_1) q^{52} + (30 \beta_{2} - 38 \beta_1 - 136) q^{53} + ( - 25 \beta_{2} - 45 \beta_1 - 15) q^{55} + ( - 42 \beta_{5} + 744 \beta_{3} + 42 \beta_1) q^{56} + ( - 173 \beta_{5} - 51 \beta_{4} + 55 \beta_{3} - 55) q^{58} + (68 \beta_{5} + 104 \beta_{3} - 104) q^{59} + (43 \beta_{5} + 31 \beta_{4} + 26 \beta_{3} - 31 \beta_{2} - 43 \beta_1) q^{61} + (15 \beta_{2} + 27 \beta_1 - 321) q^{62} + (53 \beta_{2} + 169 \beta_1 + 1053) q^{64} + (30 \beta_{5} + 10 \beta_{4} - 25 \beta_{3} - 10 \beta_{2} - 30 \beta_1) q^{65} + ( - 121 \beta_{5} + 3 \beta_{4} + 40 \beta_{3} - 40) q^{67} + ( - 115 \beta_{5} - 55 \beta_{4} + 215 \beta_{3} - 215) q^{68} + ( - 80 \beta_{5} - 25 \beta_{4} - 65 \beta_{3} + 25 \beta_{2} + 80 \beta_1) q^{70} + ( - 30 \beta_{2} + 182 \beta_1 + 64) q^{71} + (3 \beta_{2} + 59 \beta_1 - 306) q^{73} + ( - 68 \beta_{5} - 5 \beta_{4} + 931 \beta_{3} + 5 \beta_{2} + 68 \beta_1) q^{74} + (394 \beta_{5} + 128 \beta_{4} - 1266 \beta_{3} + 1266) q^{76} + ( - 104 \beta_{5} + 80 \beta_{4} + 658 \beta_{3} - 658) q^{77} + ( - 38 \beta_{5} + 54 \beta_{4} - 343 \beta_{3} - 54 \beta_{2} + 38 \beta_1) q^{79} + (45 \beta_{2} + 185 \beta_1 + 345) q^{80} + ( - 6 \beta_{2} + 212 \beta_1 + 70) q^{82} + ( - 24 \beta_{5} + 60 \beta_{4} - 102 \beta_{3} - 60 \beta_{2} + 24 \beta_1) q^{83} + ( - 15 \beta_{5} + 25 \beta_{4} + 255 \beta_{3} - 255) q^{85} + (443 \beta_{5} + 75 \beta_{4} - 691 \beta_{3} + 691) q^{86} + ( - 569 \beta_{5} - 73 \beta_{4} + 1945 \beta_{3} + 73 \beta_{2} + 569 \beta_1) q^{88} + (60 \beta_{2} + 360) q^{89} + (23 \beta_{2} - 81 \beta_1 - 230) q^{91} + ( - 35 \beta_{5} + 5 \beta_{4} + 415 \beta_{3} - 5 \beta_{2} + 35 \beta_1) q^{92} + ( - 89 \beta_{5} - 113 \beta_{4} + 1345 \beta_{3} - 1345) q^{94} + (165 \beta_{5} + 5 \beta_{4} - 140 \beta_{3} + 140) q^{95} + ( - 113 \beta_{5} - \beta_{4} - 202 \beta_{3} + \beta_{2} + 113 \beta_1) q^{97} + ( - 135 \beta_{2} - 97 \beta_1 - 179) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} - 17 q^{4} - 15 q^{5} + 4 q^{7} + 150 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 5 q^{2} - 17 q^{4} - 15 q^{5} + 4 q^{7} + 150 q^{8} + 50 q^{10} - 5 q^{11} - 7 q^{13} - 60 q^{14} - 161 q^{16} + 310 q^{17} - 100 q^{19} - 85 q^{20} + 229 q^{22} - 285 q^{23} - 75 q^{25} + 370 q^{26} - 668 q^{28} - 115 q^{29} + 115 q^{31} - 775 q^{32} - 413 q^{34} - 40 q^{35} - 768 q^{37} + 1150 q^{38} - 375 q^{40} - 580 q^{41} + 797 q^{43} - 2830 q^{44} - 570 q^{46} + 145 q^{47} - 577 q^{49} - 125 q^{50} - 825 q^{52} - 800 q^{53} + 50 q^{55} + 2190 q^{56} + 59 q^{58} - 380 q^{59} + 152 q^{61} - 2010 q^{62} + 5874 q^{64} - 35 q^{65} - 2 q^{67} - 475 q^{68} - 300 q^{70} + 80 q^{71} - 1960 q^{73} + 2720 q^{74} + 3276 q^{76} - 1950 q^{77} - 1013 q^{79} + 1610 q^{80} + 8 q^{82} - 270 q^{83} - 775 q^{85} + 1555 q^{86} + 5193 q^{88} + 2040 q^{89} - 1264 q^{91} + 1215 q^{92} - 3833 q^{94} + 250 q^{95} - 720 q^{97} - 610 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{5} + 143\nu^{4} + 21\nu^{3} + 1672\nu^{2} - 1056\nu + 13728 ) / 3760 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -17\nu^{5} + 221\nu^{4} - 993\nu^{3} + 2584\nu^{2} - 1632\nu + 17456 ) / 3760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39\nu^{5} - 37\nu^{4} + 481\nu^{3} + 182\nu^{2} + 5624\nu + 208 ) / 3760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -61\nu^{5} + 88\nu^{4} - 1144\nu^{3} + 1047\nu^{2} - 13376\nu + 8448 ) / 1880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -31\nu^{5} + 27\nu^{4} - 351\nu^{3} + 200\nu^{2} - 4104\nu + 2592 ) / 752 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{5} + \beta_{4} + 23\beta_{3} - 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{2} + 17\beta _1 - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -23\beta_{5} - 3\beta_{4} - 93\beta_{3} + 3\beta_{2} + 23\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -233\beta_{5} + 131\beta_{4} - 227\beta_{3} + 227 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
0.327167 0.566669i
−1.66402 + 2.88216i
1.83685 3.18152i
0.327167 + 0.566669i
−1.66402 2.88216i
1.83685 + 3.18152i
−2.72938 + 4.72742i 0 −10.8990 18.8776i −2.50000 4.33013i 0 5.90326 10.2247i 75.3201 0 27.2938
136.2 −1.06306 + 1.84127i 0 1.73981 + 3.01344i −2.50000 4.33013i 0 −15.3500 + 26.5870i −24.4070 0 10.6306
136.3 1.29244 2.23857i 0 0.659207 + 1.14178i −2.50000 4.33013i 0 11.4468 19.8264i 24.0869 0 −12.9244
271.1 −2.72938 4.72742i 0 −10.8990 + 18.8776i −2.50000 + 4.33013i 0 5.90326 + 10.2247i 75.3201 0 27.2938
271.2 −1.06306 1.84127i 0 1.73981 3.01344i −2.50000 + 4.33013i 0 −15.3500 26.5870i −24.4070 0 10.6306
271.3 1.29244 + 2.23857i 0 0.659207 1.14178i −2.50000 + 4.33013i 0 11.4468 + 19.8264i 24.0869 0 −12.9244
\(n\): 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.q 6
3.b odd 2 1 405.4.e.v 6
9.c even 3 1 135.4.a.h yes 3
9.c even 3 1 inner 405.4.e.q 6
9.d odd 6 1 135.4.a.e 3
9.d odd 6 1 405.4.e.v 6
36.f odd 6 1 2160.4.a.bq 3
36.h even 6 1 2160.4.a.bi 3
45.h odd 6 1 675.4.a.s 3
45.j even 6 1 675.4.a.p 3
45.k odd 12 2 675.4.b.n 6
45.l even 12 2 675.4.b.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.e 3 9.d odd 6 1
135.4.a.h yes 3 9.c even 3 1
405.4.e.q 6 1.a even 1 1 trivial
405.4.e.q 6 9.c even 3 1 inner
405.4.e.v 6 3.b odd 2 1
405.4.e.v 6 9.d odd 6 1
675.4.a.p 3 45.j even 6 1
675.4.a.s 3 45.h odd 6 1
675.4.b.m 6 45.l even 12 2
675.4.b.n 6 45.k odd 12 2
2160.4.a.bi 3 36.h even 6 1
2160.4.a.bq 3 36.f odd 6 1

Hecke kernels

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

\( T_{2}^{6} + 5T_{2}^{5} + 33T_{2}^{4} + 20T_{2}^{3} + 214T_{2}^{2} + 240T_{2} + 900 \) Copy content Toggle raw display
\( T_{7}^{6} - 4T_{7}^{5} + 811T_{7}^{4} - 13416T_{7}^{3} + 665217T_{7}^{2} - 6596910T_{7} + 68856804 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{5} + 33 T^{4} + 20 T^{3} + \cdots + 900 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} + 811 T^{4} + \cdots + 68856804 \) Copy content Toggle raw display
$11$ \( T^{6} + 5 T^{5} + 2913 T^{4} + \cdots + 977187600 \) Copy content Toggle raw display
$13$ \( T^{6} + 7 T^{5} + 818 T^{4} + \cdots + 41280625 \) Copy content Toggle raw display
$17$ \( (T^{3} - 155 T^{2} + 5608 T - 41760)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 50 T^{2} - 16663 T - 368012)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 285 T^{5} + \cdots + 306362250000 \) Copy content Toggle raw display
$29$ \( T^{6} + 115 T^{5} + \cdots + 41477979315600 \) Copy content Toggle raw display
$31$ \( T^{6} - 115 T^{5} + \cdots + 880414396416 \) Copy content Toggle raw display
$37$ \( (T^{3} + 384 T^{2} - 67923 T - 22667198)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 580 T^{5} + \cdots + 15345082598400 \) Copy content Toggle raw display
$43$ \( T^{6} - 797 T^{5} + \cdots + 28707478180096 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 207021450297600 \) Copy content Toggle raw display
$53$ \( (T^{3} + 400 T^{2} - 58172 T - 12658320)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 380 T^{5} + \cdots + 27093274214400 \) Copy content Toggle raw display
$61$ \( T^{6} - 152 T^{5} + \cdots + 25695430112356 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 431363822490000 \) Copy content Toggle raw display
$71$ \( (T^{3} - 40 T^{2} - 677372 T + 216071280)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 980 T^{2} + 264533 T + 16447954)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 1013 T^{5} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{6} + 270 T^{5} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} - 1020 T^{2} + 46800 T + 125064000)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 752435512191364 \) Copy content Toggle raw display
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