Properties

Label 405.4.e.w
Level $405$
Weight $4$
Character orbit 405.e
Analytic conductor $23.896$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} - 20304 x^{3} + 10368 x^{2} + 124416 x + 746496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{2} + (\beta_{9} + \beta_{5} + 6 \beta_{2} - 6) q^{4} + (5 \beta_{2} - 5) q^{5} + (\beta_{10} - 2 \beta_{3} - 7 \beta_{2} + \beta_1) q^{7} + ( - \beta_{10} - 4 \beta_{9} + \beta_{8} - 2 \beta_{6} + 4 \beta_{3} + 12) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{2} + (\beta_{9} + \beta_{5} + 6 \beta_{2} - 6) q^{4} + (5 \beta_{2} - 5) q^{5} + (\beta_{10} - 2 \beta_{3} - 7 \beta_{2} + \beta_1) q^{7} + ( - \beta_{10} - 4 \beta_{9} + \beta_{8} - 2 \beta_{6} + 4 \beta_{3} + 12) q^{8} + ( - 5 \beta_{9} + 5 \beta_{3} + 5) q^{10} + ( - \beta_{11} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 14 \beta_{2} + 2 \beta_1) q^{11} + (\beta_{11} + 2 \beta_{8} - 3 \beta_{7} - \beta_{5} - \beta_{4} + 4 \beta_{2} - 4) q^{13} + ( - 3 \beta_{11} + 7 \beta_{9} - \beta_{8} + \beta_{7} + 4 \beta_{5} + 3 \beta_{4} + \cdots - 32) q^{14}+ \cdots + ( - 67 \beta_{10} - 111 \beta_{9} + 67 \beta_{8} - 156 \beta_{7} + \cdots + 345) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8} + 40 q^{10} - 88 q^{11} - 20 q^{13} - 180 q^{14} - 58 q^{16} + 248 q^{17} - 92 q^{19} - 170 q^{20} + 74 q^{22} - 210 q^{23} - 150 q^{25} + 8 q^{26} + 704 q^{28} - 296 q^{29} + 104 q^{31} - 722 q^{32} + 428 q^{34} + 400 q^{35} - 408 q^{37} + 20 q^{38} - 330 q^{40} - 344 q^{41} - 512 q^{43} + 1432 q^{44} - 372 q^{46} - 238 q^{47} - 68 q^{49} - 100 q^{50} + 468 q^{52} + 1700 q^{53} + 880 q^{55} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} + 364 q^{61} + 2076 q^{62} - 1980 q^{64} - 100 q^{65} - 88 q^{67} - 236 q^{68} - 900 q^{70} + 2728 q^{71} + 1672 q^{73} - 1316 q^{74} + 2106 q^{76} - 840 q^{77} + 680 q^{79} + 580 q^{80} + 3484 q^{82} - 2148 q^{83} - 620 q^{85} - 2872 q^{86} - 1296 q^{88} + 6000 q^{89} - 6116 q^{91} - 1002 q^{92} + 3662 q^{94} + 230 q^{95} + 612 q^{97} + 3964 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} - 20304 x^{3} + 10368 x^{2} + 124416 x + 746496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 39168669767185 \nu^{11} + 622492575406102 \nu^{10} - 116378803247278 \nu^{9} + \cdots + 57\!\cdots\!56 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21833561 \nu^{11} - 6933902 \nu^{10} + 191663726 \nu^{9} - 824768992 \nu^{8} - 19024412783 \nu^{7} + 8759970096 \nu^{6} + \cdots + 11197076020992 ) / 28100111293440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8315698756077 \nu^{11} + 101245330415030 \nu^{10} - 394560515037286 \nu^{9} + \cdots + 15\!\cdots\!84 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4200645557245 \nu^{11} + 6160897283854 \nu^{10} - 314135929010806 \nu^{9} + \cdots + 62\!\cdots\!12 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 335658953887093 \nu^{11} + \cdots - 17\!\cdots\!56 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11005996995395 \nu^{11} - 8278688448046 \nu^{10} - 261584063656106 \nu^{9} + \cdots - 48\!\cdots\!08 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 238036479862585 \nu^{11} + \cdots + 17\!\cdots\!36 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!07 \nu^{11} - 357550815547730 \nu^{10} + \cdots - 22\!\cdots\!44 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 96943826623309 \nu^{11} + 480126841023914 \nu^{10} - 441998532012314 \nu^{9} + \cdots - 73\!\cdots\!36 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!33 \nu^{11} + \cdots + 22\!\cdots\!92 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!15 \nu^{11} + \cdots + 54\!\cdots\!64 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} + \beta_{3} - 2\beta_{2} + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - 5\beta_{9} - \beta_{8} - 8\beta_{7} + \beta_{6} - 2\beta_{5} - 5\beta_{3} + 8\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + 8 \beta_{10} + 12 \beta_{9} - 9 \beta_{8} - 8 \beta_{7} + 7 \beta_{6} - 17 \beta_{5} - 14 \beta_{3} + 17 \beta_{2} - 4 \beta _1 - 35 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 35 \beta_{10} + 35 \beta_{9} - 35 \beta_{8} - 24 \beta_{7} - 81 \beta_{6} + 40 \beta_{4} - 35 \beta_{3} - 24 \beta _1 - 1298 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 26 \beta_{11} + 365 \beta_{10} + 501 \beta_{9} - 279 \beta_{8} - 184 \beta_{7} - 451 \beta_{6} + 644 \beta_{5} + 48 \beta_{4} - 377 \beta_{3} - 1308 \beta_{2} - 464 \beta _1 - 904 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 16 \beta_{11} + 325 \beta_{10} + 758 \beta_{9} + 325 \beta_{8} + 1288 \beta_{7} - 488 \beta_{6} + 976 \beta_{5} - 8 \beta_{4} + 758 \beta_{3} - 2818 \beta_{2} - 1288 \beta _1 + 1409 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 144 \beta_{11} - 5057 \beta_{10} - 5627 \beta_{9} + 7763 \beta_{8} + 11704 \beta_{7} - 2351 \beta_{6} + 12820 \beta_{5} - 856 \beta_{4} + 8489 \beta_{3} - 36254 \beta_{2} + 3896 \beta _1 + 58398 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 24965 \beta_{10} - 17315 \beta_{9} + 24965 \beta_{8} + 32472 \beta_{7} + 46041 \beta_{6} - 13252 \beta_{4} + 17315 \beta_{3} + 32472 \beta _1 + 480200 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2095 \beta_{11} - 84984 \beta_{10} - 71512 \beta_{9} + 47317 \beta_{8} + 42152 \beta_{7} + 122651 \beta_{6} - 132301 \beta_{5} - 10160 \beta_{4} + 41934 \beta_{3} + 464143 \beta_{2} + \cdots + 260571 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 61008 \beta_{11} - 360417 \beta_{10} - 395679 \beta_{9} - 360417 \beta_{8} - 1058408 \beta_{7} + 612127 \beta_{6} - 1224254 \beta_{5} + 30504 \beta_{4} - 395679 \beta_{3} + \cdots - 2551826 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 216770 \beta_{11} + 1828317 \beta_{10} + 1259071 \beta_{9} - 3796381 \beta_{8} - 6773512 \beta_{7} - 139747 \beta_{6} - 5624698 \beta_{5} + 629000 \beta_{4} - 2414697 \beta_{3} + \cdots - 34930520 ) / 6 \) Copy content Toggle raw display

Character values

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

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

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
2.93142 + 2.93142i
−1.16241 + 1.16241i
−2.82176 2.82176i
3.41462 3.41462i
1.25636 + 1.25636i
−2.61824 + 2.61824i
2.93142 2.93142i
−1.16241 1.16241i
−2.82176 + 2.82176i
3.41462 + 3.41462i
1.25636 1.25636i
−2.61824 2.61824i
−2.61668 + 4.53223i 0 −9.69405 16.7906i −2.50000 4.33013i 0 −16.5090 + 28.5945i 59.5981 0 26.1668
136.2 −2.26722 + 3.92694i 0 −6.28058 10.8783i −2.50000 4.33013i 0 1.31809 2.28300i 20.6823 0 22.6722
136.3 −1.03663 + 1.79550i 0 1.85079 + 3.20567i −2.50000 4.33013i 0 2.33056 4.03665i −24.2604 0 10.3663
136.4 0.0745751 0.129168i 0 3.98888 + 6.90894i −2.50000 4.33013i 0 −10.0712 + 17.4439i 2.38308 0 −0.745751
136.5 1.78729 3.09567i 0 −2.38879 4.13751i −2.50000 4.33013i 0 −7.07987 + 12.2627i 11.5188 0 −17.8729
136.6 2.05867 3.56572i 0 −4.47625 7.75309i −2.50000 4.33013i 0 10.0115 17.3404i −3.92177 0 −20.5867
271.1 −2.61668 4.53223i 0 −9.69405 + 16.7906i −2.50000 + 4.33013i 0 −16.5090 28.5945i 59.5981 0 26.1668
271.2 −2.26722 3.92694i 0 −6.28058 + 10.8783i −2.50000 + 4.33013i 0 1.31809 + 2.28300i 20.6823 0 22.6722
271.3 −1.03663 1.79550i 0 1.85079 3.20567i −2.50000 + 4.33013i 0 2.33056 + 4.03665i −24.2604 0 10.3663
271.4 0.0745751 + 0.129168i 0 3.98888 6.90894i −2.50000 + 4.33013i 0 −10.0712 17.4439i 2.38308 0 −0.745751
271.5 1.78729 + 3.09567i 0 −2.38879 + 4.13751i −2.50000 + 4.33013i 0 −7.07987 12.2627i 11.5188 0 −17.8729
271.6 2.05867 + 3.56572i 0 −4.47625 + 7.75309i −2.50000 + 4.33013i 0 10.0115 + 17.3404i −3.92177 0 −20.5867
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.6
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.w 12
3.b odd 2 1 405.4.e.x 12
9.c even 3 1 405.4.a.l yes 6
9.c even 3 1 inner 405.4.e.w 12
9.d odd 6 1 405.4.a.k 6
9.d odd 6 1 405.4.e.x 12
45.h odd 6 1 2025.4.a.z 6
45.j even 6 1 2025.4.a.y 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.k 6 9.d odd 6 1
405.4.a.l yes 6 9.c even 3 1
405.4.e.w 12 1.a even 1 1 trivial
405.4.e.w 12 9.c even 3 1 inner
405.4.e.x 12 3.b odd 2 1
405.4.e.x 12 9.d odd 6 1
2025.4.a.y 6 45.j even 6 1
2025.4.a.z 6 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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{12} + 4 T_{2}^{11} + 49 T_{2}^{10} + 88 T_{2}^{9} + 1243 T_{2}^{8} + 2026 T_{2}^{7} + 18586 T_{2}^{6} + 13252 T_{2}^{5} + 153472 T_{2}^{4} + 171864 T_{2}^{3} + 498744 T_{2}^{2} - 73872 T_{2} + 11664 \) Copy content Toggle raw display
\( T_{7}^{12} + 40 T_{7}^{11} + 1863 T_{7}^{10} + 27280 T_{7}^{9} + 874429 T_{7}^{8} + 10054356 T_{7}^{7} + 293906532 T_{7}^{6} + 1439479296 T_{7}^{5} + 23417174988 T_{7}^{4} + \cdots + 5368136821776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + 49 T^{10} + \cdots + 11664 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 40 T^{11} + \cdots + 5368136821776 \) Copy content Toggle raw display
$11$ \( T^{12} + 88 T^{11} + \cdots + 1433067563664 \) Copy content Toggle raw display
$13$ \( T^{12} + 20 T^{11} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} - 124 T^{5} - 5004 T^{4} + \cdots + 105966288)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 46 T^{5} - 18169 T^{4} + \cdots - 36821611175)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 210 T^{11} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{12} + 296 T^{11} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} - 104 T^{11} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{6} + 204 T^{5} + \cdots + 12008297128192)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 344 T^{11} + \cdots + 74\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{12} + 512 T^{11} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + 238 T^{11} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} - 850 T^{5} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 1840 T^{11} + \cdots + 28\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{12} - 364 T^{11} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} + 88 T^{11} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} - 1364 T^{5} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 836 T^{5} + \cdots - 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 680 T^{11} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{12} + 2148 T^{11} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} - 3000 T^{5} + \cdots + 16\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 612 T^{11} + \cdots + 96\!\cdots\!76 \) Copy content Toggle raw display
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