Newspace parameters
Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 405.g (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.0354507066\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{16} + 256x^{12} + 15630x^{8} + 235936x^{4} + 28561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{8} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 256x^{12} + 15630x^{8} + 235936x^{4} + 28561 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 2453\nu^{14} + 630165\nu^{10} + 38949635\nu^{6} + 634533683\nu^{2} ) / 220010960 \) |
\(\beta_{3}\) | \(=\) | \( ( 11049 \nu^{14} + 30589 \nu^{12} + 3051455 \nu^{10} + 6367075 \nu^{8} + 219702375 \nu^{6} + 175867315 \nu^{4} + 4061058769 \nu^{2} + \cdots - 1387973171 ) / 880043840 \) |
\(\beta_{4}\) | \(=\) | \( ( 11049 \nu^{14} - 30589 \nu^{12} + 3051455 \nu^{10} - 6367075 \nu^{8} + 219702375 \nu^{6} - 175867315 \nu^{4} + 4061058769 \nu^{2} + \cdots + 1387973171 ) / 880043840 \) |
\(\beta_{5}\) | \(=\) | \( ( - 17171 \nu^{14} - 2197 \nu^{12} - 4411155 \nu^{10} - 609245 \nu^{8} - 272647445 \nu^{6} - 55782675 \nu^{4} - 4221724821 \nu^{2} + \cdots - 1470016587 ) / 440021920 \) |
\(\beta_{6}\) | \(=\) | \( ( - 17171 \nu^{14} + 2197 \nu^{12} - 4411155 \nu^{10} + 609245 \nu^{8} - 272647445 \nu^{6} + 55782675 \nu^{4} - 4221724821 \nu^{2} + \cdots + 1470016587 ) / 440021920 \) |
\(\beta_{7}\) | \(=\) | \( ( 57891 \nu^{14} + 65741 \nu^{12} + 14614085 \nu^{10} + 16114995 \nu^{8} + 854054365 \nu^{6} + 848379155 \nu^{4} + 11330324731 \nu^{2} + \cdots + 6071492141 ) / 880043840 \) |
\(\beta_{8}\) | \(=\) | \( ( - 57891 \nu^{14} + 65741 \nu^{12} - 14614085 \nu^{10} + 16114995 \nu^{8} - 854054365 \nu^{6} + 848379155 \nu^{4} - 11330324731 \nu^{2} + \cdots + 6071492141 ) / 880043840 \) |
\(\beta_{9}\) | \(=\) | \( ( 2453\nu^{15} + 630165\nu^{11} + 38949635\nu^{7} + 634533683\nu^{3} ) / 220010960 \) |
\(\beta_{10}\) | \(=\) | \( ( - 305879 \nu^{15} - 175253 \nu^{13} - 77850245 \nu^{11} - 42252535 \nu^{9} - 4682276425 \nu^{7} - 2207321675 \nu^{5} - 68431243099 \nu^{3} + \cdots - 23975329833 \nu ) / 11440569920 \) |
\(\beta_{11}\) | \(=\) | \( ( 305879 \nu^{15} - 175253 \nu^{13} + 77850245 \nu^{11} - 42252535 \nu^{9} + 4682276425 \nu^{7} - 2207321675 \nu^{5} + 68431243099 \nu^{3} + \cdots - 23975329833 \nu ) / 11440569920 \) |
\(\beta_{12}\) | \(=\) | \( ( - 109903 \nu^{15} + 53573 \nu^{13} - 28146153 \nu^{11} + 14433107 \nu^{9} - 1715329841 \nu^{7} + 1020068283 \nu^{5} - 25218938263 \nu^{3} + \cdots + 21046244141 \nu ) / 2288113984 \) |
\(\beta_{13}\) | \(=\) | \( ( 427697 \nu^{15} + 46306 \nu^{13} + 109290505 \nu^{11} + 14956500 \nu^{9} + 6629462815 \nu^{7} + 1446509870 \nu^{5} + 97262967207 \nu^{3} + \cdots + 40627945436 \nu ) / 5720284960 \) |
\(\beta_{14}\) | \(=\) | \( ( 472597 \nu^{15} - 192998 \nu^{13} + 121553855 \nu^{11} - 49288850 \nu^{9} + 7516984195 \nu^{7} - 2928656770 \nu^{5} + 116709898297 \nu^{3} + \cdots - 34052489718 \nu ) / 5720284960 \) |
\(\beta_{15}\) | \(=\) | \( ( - 1251073 \nu^{15} - 210743 \nu^{13} - 320957955 \nu^{11} - 56325165 \nu^{9} - 19716244815 \nu^{7} - 3649991865 \nu^{5} + \cdots - 44129649603 \nu ) / 11440569920 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} + \beta_{5} + 7\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + 11\beta_{9} \) |
\(\nu^{4}\) | \(=\) | \( -2\beta_{8} - 2\beta_{7} + 16\beta_{6} - 16\beta_{5} - 2\beta_{4} + 2\beta_{3} - 73 \) |
\(\nu^{5}\) | \(=\) | \( 22\beta_{15} + 22\beta_{14} + 16\beta_{13} + 16\beta_{12} - 18\beta_{11} - 24\beta_{10} - 133\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 38\beta_{8} - 38\beta_{7} - 189\beta_{6} - 189\beta_{5} - 46\beta_{4} - 46\beta_{3} - 771\beta_{2} \) |
\(\nu^{7}\) | \(=\) | \( -219\beta_{15} + 219\beta_{14} - 349\beta_{13} + 349\beta_{12} + 427\beta_{11} - 297\beta_{10} - 1663\beta_{9} \) |
\(\nu^{8}\) | \(=\) | \( 568\beta_{8} + 568\beta_{7} - 3096\beta_{6} + 3096\beta_{5} + 776\beta_{4} - 776\beta_{3} + 10401 \) |
\(\nu^{9}\) | \(=\) | \( - 5008 \beta_{15} - 5008 \beta_{14} - 2888 \beta_{13} - 2888 \beta_{12} + 4496 \beta_{11} + 6616 \beta_{10} + 21233 \beta_1 \) |
\(\nu^{10}\) | \(=\) | \( - 7896 \beta_{8} + 7896 \beta_{7} + 33625 \beta_{6} + 33625 \beta_{5} + 11624 \beta_{4} + 11624 \beta_{3} + 116023 \beta_{2} \) |
\(\nu^{11}\) | \(=\) | \( 37793 \beta_{15} - 37793 \beta_{14} + 68937 \beta_{13} - 68937 \beta_{12} - 95473 \beta_{11} + 64329 \beta_{10} + 274651 \beta_{9} \) |
\(\nu^{12}\) | \(=\) | \( - 106730 \beta_{8} - 106730 \beta_{7} + 552440 \beta_{6} - 552440 \beta_{5} - 164410 \beta_{4} + 164410 \beta_{3} - 1699881 \) |
\(\nu^{13}\) | \(=\) | \( 930310 \beta_{15} + 930310 \beta_{14} + 494760 \beta_{13} + 494760 \beta_{12} - 889890 \beta_{11} - 1325440 \beta_{10} - 3580821 \beta_1 \) |
\(\nu^{14}\) | \(=\) | \( 1425070 \beta_{8} - 1425070 \beta_{7} - 5895781 \beta_{6} - 5895781 \beta_{5} - 2255750 \beta_{4} - 2255750 \beta_{3} - 19284627 \beta_{2} \) |
\(\nu^{15}\) | \(=\) | \( - 6490171 \beta_{15} + 6490171 \beta_{14} - 12426741 \beta_{13} + 12426741 \beta_{12} + 18005211 \beta_{11} - 12068641 \beta_{10} - 46906671 \beta_{9} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
\(n\) | \(82\) | \(326\) |
\(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 |
|
−2.56790 | − | 2.56790i | 0 | 9.18825i | −3.64546 | − | 3.42208i | 0 | −4.54186 | − | 4.54186i | 13.3229 | − | 13.3229i | 0 | 0.573606 | + | 18.1488i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.2 | −1.96165 | − | 1.96165i | 0 | 3.69616i | −0.453215 | − | 4.97942i | 0 | 6.04960 | + | 6.04960i | −0.596019 | + | 0.596019i | 0 | −8.87884 | + | 10.6569i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.3 | −1.54374 | − | 1.54374i | 0 | 0.766246i | 4.99995 | − | 0.0220096i | 0 | 1.44305 | + | 1.44305i | −4.99206 | + | 4.99206i | 0 | −7.75259 | − | 7.68463i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.4 | −0.417936 | − | 0.417936i | 0 | − | 3.65066i | 0.0756586 | + | 4.99943i | 0 | 7.04922 | + | 7.04922i | −3.19748 | + | 3.19748i | 0 | 2.05782 | − | 2.12106i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.5 | 0.417936 | + | 0.417936i | 0 | − | 3.65066i | −0.0756586 | − | 4.99943i | 0 | 7.04922 | + | 7.04922i | 3.19748 | − | 3.19748i | 0 | 2.05782 | − | 2.12106i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.6 | 1.54374 | + | 1.54374i | 0 | 0.766246i | −4.99995 | + | 0.0220096i | 0 | 1.44305 | + | 1.44305i | 4.99206 | − | 4.99206i | 0 | −7.75259 | − | 7.68463i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.7 | 1.96165 | + | 1.96165i | 0 | 3.69616i | 0.453215 | + | 4.97942i | 0 | 6.04960 | + | 6.04960i | 0.596019 | − | 0.596019i | 0 | −8.87884 | + | 10.6569i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
82.8 | 2.56790 | + | 2.56790i | 0 | 9.18825i | 3.64546 | + | 3.42208i | 0 | −4.54186 | − | 4.54186i | −13.3229 | + | 13.3229i | 0 | 0.573606 | + | 18.1488i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.1 | −2.56790 | + | 2.56790i | 0 | − | 9.18825i | −3.64546 | + | 3.42208i | 0 | −4.54186 | + | 4.54186i | 13.3229 | + | 13.3229i | 0 | 0.573606 | − | 18.1488i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.2 | −1.96165 | + | 1.96165i | 0 | − | 3.69616i | −0.453215 | + | 4.97942i | 0 | 6.04960 | − | 6.04960i | −0.596019 | − | 0.596019i | 0 | −8.87884 | − | 10.6569i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.3 | −1.54374 | + | 1.54374i | 0 | − | 0.766246i | 4.99995 | + | 0.0220096i | 0 | 1.44305 | − | 1.44305i | −4.99206 | − | 4.99206i | 0 | −7.75259 | + | 7.68463i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.4 | −0.417936 | + | 0.417936i | 0 | 3.65066i | 0.0756586 | − | 4.99943i | 0 | 7.04922 | − | 7.04922i | −3.19748 | − | 3.19748i | 0 | 2.05782 | + | 2.12106i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.5 | 0.417936 | − | 0.417936i | 0 | 3.65066i | −0.0756586 | + | 4.99943i | 0 | 7.04922 | − | 7.04922i | 3.19748 | + | 3.19748i | 0 | 2.05782 | + | 2.12106i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.6 | 1.54374 | − | 1.54374i | 0 | − | 0.766246i | −4.99995 | − | 0.0220096i | 0 | 1.44305 | − | 1.44305i | 4.99206 | + | 4.99206i | 0 | −7.75259 | + | 7.68463i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.7 | 1.96165 | − | 1.96165i | 0 | − | 3.69616i | 0.453215 | − | 4.97942i | 0 | 6.04960 | − | 6.04960i | 0.596019 | + | 0.596019i | 0 | −8.87884 | − | 10.6569i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.8 | 2.56790 | − | 2.56790i | 0 | − | 9.18825i | 3.64546 | − | 3.42208i | 0 | −4.54186 | + | 4.54186i | −13.3229 | − | 13.3229i | 0 | 0.573606 | − | 18.1488i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 405.3.g.f | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 405.3.g.f | ✓ | 16 |
5.c | odd | 4 | 1 | inner | 405.3.g.f | ✓ | 16 |
9.c | even | 3 | 1 | 405.3.l.j | 16 | ||
9.c | even | 3 | 1 | 405.3.l.m | 16 | ||
9.d | odd | 6 | 1 | 405.3.l.j | 16 | ||
9.d | odd | 6 | 1 | 405.3.l.m | 16 | ||
15.e | even | 4 | 1 | inner | 405.3.g.f | ✓ | 16 |
45.k | odd | 12 | 1 | 405.3.l.j | 16 | ||
45.k | odd | 12 | 1 | 405.3.l.m | 16 | ||
45.l | even | 12 | 1 | 405.3.l.j | 16 | ||
45.l | even | 12 | 1 | 405.3.l.m | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
405.3.g.f | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
405.3.g.f | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
405.3.g.f | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
405.3.g.f | ✓ | 16 | 15.e | even | 4 | 1 | inner |
405.3.l.j | 16 | 9.c | even | 3 | 1 | ||
405.3.l.j | 16 | 9.d | odd | 6 | 1 | ||
405.3.l.j | 16 | 45.k | odd | 12 | 1 | ||
405.3.l.j | 16 | 45.l | even | 12 | 1 | ||
405.3.l.m | 16 | 9.c | even | 3 | 1 | ||
405.3.l.m | 16 | 9.d | odd | 6 | 1 | ||
405.3.l.m | 16 | 45.k | odd | 12 | 1 | ||
405.3.l.m | 16 | 45.l | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 256T_{2}^{12} + 15630T_{2}^{8} + 235936T_{2}^{4} + 28561 \)
acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 256 T^{12} + 15630 T^{8} + \cdots + 28561 \)
$3$
\( T^{16} \)
$5$
\( T^{16} + 46 T^{14} + \cdots + 152587890625 \)
$7$
\( (T^{8} - 20 T^{7} + 200 T^{6} + \cdots + 1249924)^{2} \)
$11$
\( (T^{8} - 448 T^{6} + 58740 T^{4} + \cdots + 399424)^{2} \)
$13$
\( (T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 163814401)^{2} \)
$17$
\( T^{16} + 1245766 T^{12} + \cdots + 13\!\cdots\!16 \)
$19$
\( (T^{8} + 2010 T^{6} + 1076085 T^{4} + \cdots + 50552100)^{2} \)
$23$
\( T^{16} + 2218954 T^{12} + \cdots + 16\!\cdots\!76 \)
$29$
\( (T^{8} + 3822 T^{6} + \cdots + 411193867536)^{2} \)
$31$
\( (T^{4} - 80 T^{3} + 1782 T^{2} + \cdots - 18944)^{4} \)
$37$
\( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 23253810064)^{2} \)
$41$
\( (T^{8} - 4582 T^{6} + \cdots + 4015503424)^{2} \)
$43$
\( (T^{8} - 128 T^{7} + \cdots + 27685065955600)^{2} \)
$47$
\( T^{16} + 68774722 T^{12} + \cdots + 15\!\cdots\!00 \)
$53$
\( T^{16} + 30966538 T^{12} + \cdots + 76\!\cdots\!00 \)
$59$
\( (T^{8} + 16374 T^{6} + \cdots + 1000583424)^{2} \)
$61$
\( (T^{4} - 2 T^{3} - 11109 T^{2} + \cdots + 270238)^{4} \)
$67$
\( (T^{8} - 44 T^{7} + \cdots + 136557921640000)^{2} \)
$71$
\( (T^{8} - 16084 T^{6} + \cdots + 243517148561296)^{2} \)
$73$
\( (T^{8} - 182 T^{7} + \cdots + 75176865856)^{2} \)
$79$
\( (T^{8} + 25308 T^{6} + \cdots + 11811016864656)^{2} \)
$83$
\( T^{16} + 52607176 T^{12} + \cdots + 68\!\cdots\!76 \)
$89$
\( (T^{8} + 25626 T^{6} + \cdots + 11\!\cdots\!84)^{2} \)
$97$
\( (T^{8} - 152 T^{7} + \cdots + 500450545045504)^{2} \)
show more
show less