Properties

Label 405.3.g.f
Level $405$
Weight $3$
Character orbit 405.g
Analytic conductor $11.035$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,3,Mod(82,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.g (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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: \( x^{16} + 256x^{12} + 15630x^{8} + 235936x^{4} + 28561 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + ( - \beta_{6} - \beta_{5} - 3 \beta_{2}) q^{4} + (\beta_{15} + \beta_{11} + \beta_{9}) q^{5} + (2 \beta_{6} + \beta_{3} - 2 \beta_{2} + 2) q^{7} + ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + ( - \beta_{6} - \beta_{5} - 3 \beta_{2}) q^{4} + (\beta_{15} + \beta_{11} + \beta_{9}) q^{5} + (2 \beta_{6} + \beta_{3} - 2 \beta_{2} + 2) q^{7} + ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + 3 \beta_1) q^{8} + (\beta_{8} - 2 \beta_{6} - \beta_{5} + \beta_{4} - 5 \beta_{2} - 3) q^{10} + ( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10}) q^{11} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 4) q^{13} + (\beta_{15} + 2 \beta_{14} - \beta_{13} + 2 \beta_{12} + \beta_{11} - 7 \beta_{10} - \beta_{9} - \beta_1) q^{14} + ( - 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \cdots - 5) q^{16}+ \cdots + ( - \beta_{15} - \beta_{14} - 4 \beta_{13} - 4 \beta_{12} - 23 \beta_{11} - 26 \beta_{10} + \cdots + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{7} - 56 q^{10} - 44 q^{13} - 32 q^{16} + 32 q^{22} - 92 q^{25} + 176 q^{28} + 320 q^{31} + 4 q^{37} - 528 q^{40} + 256 q^{43} - 16 q^{46} - 308 q^{52} - 364 q^{55} + 492 q^{58} + 8 q^{61} + 88 q^{67} - 36 q^{70} + 364 q^{73} - 912 q^{76} + 32 q^{82} + 64 q^{85} + 840 q^{88} + 224 q^{91} + 304 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 256x^{12} + 15630x^{8} + 235936x^{4} + 28561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2453\nu^{14} + 630165\nu^{10} + 38949635\nu^{6} + 634533683\nu^{2} ) / 220010960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11049 \nu^{14} + 30589 \nu^{12} + 3051455 \nu^{10} + 6367075 \nu^{8} + 219702375 \nu^{6} + 175867315 \nu^{4} + 4061058769 \nu^{2} + \cdots - 1387973171 ) / 880043840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11049 \nu^{14} - 30589 \nu^{12} + 3051455 \nu^{10} - 6367075 \nu^{8} + 219702375 \nu^{6} - 175867315 \nu^{4} + 4061058769 \nu^{2} + \cdots + 1387973171 ) / 880043840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17171 \nu^{14} - 2197 \nu^{12} - 4411155 \nu^{10} - 609245 \nu^{8} - 272647445 \nu^{6} - 55782675 \nu^{4} - 4221724821 \nu^{2} + \cdots - 1470016587 ) / 440021920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17171 \nu^{14} + 2197 \nu^{12} - 4411155 \nu^{10} + 609245 \nu^{8} - 272647445 \nu^{6} + 55782675 \nu^{4} - 4221724821 \nu^{2} + \cdots + 1470016587 ) / 440021920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 57891 \nu^{14} + 65741 \nu^{12} + 14614085 \nu^{10} + 16114995 \nu^{8} + 854054365 \nu^{6} + 848379155 \nu^{4} + 11330324731 \nu^{2} + \cdots + 6071492141 ) / 880043840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 57891 \nu^{14} + 65741 \nu^{12} - 14614085 \nu^{10} + 16114995 \nu^{8} - 854054365 \nu^{6} + 848379155 \nu^{4} - 11330324731 \nu^{2} + \cdots + 6071492141 ) / 880043840 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2453\nu^{15} + 630165\nu^{11} + 38949635\nu^{7} + 634533683\nu^{3} ) / 220010960 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 305879 \nu^{15} - 175253 \nu^{13} - 77850245 \nu^{11} - 42252535 \nu^{9} - 4682276425 \nu^{7} - 2207321675 \nu^{5} - 68431243099 \nu^{3} + \cdots - 23975329833 \nu ) / 11440569920 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 305879 \nu^{15} - 175253 \nu^{13} + 77850245 \nu^{11} - 42252535 \nu^{9} + 4682276425 \nu^{7} - 2207321675 \nu^{5} + 68431243099 \nu^{3} + \cdots - 23975329833 \nu ) / 11440569920 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 109903 \nu^{15} + 53573 \nu^{13} - 28146153 \nu^{11} + 14433107 \nu^{9} - 1715329841 \nu^{7} + 1020068283 \nu^{5} - 25218938263 \nu^{3} + \cdots + 21046244141 \nu ) / 2288113984 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 427697 \nu^{15} + 46306 \nu^{13} + 109290505 \nu^{11} + 14956500 \nu^{9} + 6629462815 \nu^{7} + 1446509870 \nu^{5} + 97262967207 \nu^{3} + \cdots + 40627945436 \nu ) / 5720284960 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 472597 \nu^{15} - 192998 \nu^{13} + 121553855 \nu^{11} - 49288850 \nu^{9} + 7516984195 \nu^{7} - 2928656770 \nu^{5} + 116709898297 \nu^{3} + \cdots - 34052489718 \nu ) / 5720284960 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1251073 \nu^{15} - 210743 \nu^{13} - 320957955 \nu^{11} - 56325165 \nu^{9} - 19716244815 \nu^{7} - 3649991865 \nu^{5} + \cdots - 44129649603 \nu ) / 11440569920 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + 11\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} - 2\beta_{7} + 16\beta_{6} - 16\beta_{5} - 2\beta_{4} + 2\beta_{3} - 73 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 22\beta_{15} + 22\beta_{14} + 16\beta_{13} + 16\beta_{12} - 18\beta_{11} - 24\beta_{10} - 133\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 38\beta_{8} - 38\beta_{7} - 189\beta_{6} - 189\beta_{5} - 46\beta_{4} - 46\beta_{3} - 771\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -219\beta_{15} + 219\beta_{14} - 349\beta_{13} + 349\beta_{12} + 427\beta_{11} - 297\beta_{10} - 1663\beta_{9} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 568\beta_{8} + 568\beta_{7} - 3096\beta_{6} + 3096\beta_{5} + 776\beta_{4} - 776\beta_{3} + 10401 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5008 \beta_{15} - 5008 \beta_{14} - 2888 \beta_{13} - 2888 \beta_{12} + 4496 \beta_{11} + 6616 \beta_{10} + 21233 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7896 \beta_{8} + 7896 \beta_{7} + 33625 \beta_{6} + 33625 \beta_{5} + 11624 \beta_{4} + 11624 \beta_{3} + 116023 \beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 37793 \beta_{15} - 37793 \beta_{14} + 68937 \beta_{13} - 68937 \beta_{12} - 95473 \beta_{11} + 64329 \beta_{10} + 274651 \beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 106730 \beta_{8} - 106730 \beta_{7} + 552440 \beta_{6} - 552440 \beta_{5} - 164410 \beta_{4} + 164410 \beta_{3} - 1699881 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 930310 \beta_{15} + 930310 \beta_{14} + 494760 \beta_{13} + 494760 \beta_{12} - 889890 \beta_{11} - 1325440 \beta_{10} - 3580821 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1425070 \beta_{8} - 1425070 \beta_{7} - 5895781 \beta_{6} - 5895781 \beta_{5} - 2255750 \beta_{4} - 2255750 \beta_{3} - 19284627 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 6490171 \beta_{15} + 6490171 \beta_{14} - 12426741 \beta_{13} + 12426741 \beta_{12} + 18005211 \beta_{11} - 12068641 \beta_{10} - 46906671 \beta_{9} \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
1.96165 1.96165i
1.54374 1.54374i
0.417936 0.417936i
−0.417936 + 0.417936i
−1.54374 + 1.54374i
−1.96165 + 1.96165i
−2.56790 + 2.56790i
2.56790 + 2.56790i
1.96165 + 1.96165i
1.54374 + 1.54374i
0.417936 + 0.417936i
−0.417936 0.417936i
−1.54374 1.54374i
−1.96165 1.96165i
−2.56790 2.56790i
−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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.8
Significant digits:
Format:

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 \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 256 T^{12} + 15630 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 46 T^{14} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( (T^{8} - 20 T^{7} + 200 T^{6} + \cdots + 1249924)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 448 T^{6} + 58740 T^{4} + \cdots + 399424)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 163814401)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 1245766 T^{12} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 2010 T^{6} + 1076085 T^{4} + \cdots + 50552100)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 2218954 T^{12} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{8} + 3822 T^{6} + \cdots + 411193867536)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 80 T^{3} + 1782 T^{2} + \cdots - 18944)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 23253810064)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 4582 T^{6} + \cdots + 4015503424)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 128 T^{7} + \cdots + 27685065955600)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 68774722 T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + 30966538 T^{12} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + 16374 T^{6} + \cdots + 1000583424)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} - 11109 T^{2} + \cdots + 270238)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 44 T^{7} + \cdots + 136557921640000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 16084 T^{6} + \cdots + 243517148561296)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 182 T^{7} + \cdots + 75176865856)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 25308 T^{6} + \cdots + 11811016864656)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 52607176 T^{12} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + 25626 T^{6} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 152 T^{7} + \cdots + 500450545045504)^{2} \) Copy content Toggle raw display
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