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Properties

Label 405.3.l.j
Level $405$
Weight $3$
Character orbit 405.l
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(28,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.28");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.l (of order \(12\), degree \(4\), not 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: \(4\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{15} - \beta_{11} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{15} - 2 \beta_{13} + \cdots + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{15} - \beta_{11} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{15} - 8 \beta_{13} + \cdots - 19) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} + 12 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} + 12 q^{5} - 20 q^{7} - 56 q^{10} + 22 q^{13} - 168 q^{14} + 16 q^{16} - 96 q^{20} - 16 q^{22} + 36 q^{23} + 46 q^{25} + 176 q^{28} - 252 q^{29} - 160 q^{31} - 114 q^{32} + 4 q^{37} + 192 q^{38} + 264 q^{40} - 128 q^{43} - 16 q^{46} + 204 q^{47} + 540 q^{50} + 154 q^{52} - 364 q^{55} - 246 q^{58} + 36 q^{59} - 4 q^{61} + 24 q^{65} - 44 q^{67} + 966 q^{68} + 18 q^{70} + 364 q^{73} + 288 q^{74} + 456 q^{76} - 204 q^{77} + 32 q^{82} + 12 q^{83} - 32 q^{85} - 420 q^{88} + 224 q^{91} - 960 q^{92} + 114 q^{95} - 152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
\(\beta_{2}\)\(=\) \( ( - 1037 \nu^{14} + 2691 \nu^{12} - 250015 \nu^{10} + 583505 \nu^{8} - 13061075 \nu^{6} + \cdots + 51693551 ) / 135391360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2453\nu^{14} + 630165\nu^{10} + 38949635\nu^{6} + 634533683\nu^{2} ) / 220010960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12265 \nu^{14} - 16211 \nu^{13} - 2197 \nu^{12} + 3150825 \nu^{10} - 4332705 \nu^{9} + \cdots - 1029994667 ) / 880043840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12265 \nu^{14} - 16211 \nu^{13} + 2197 \nu^{12} - 3150825 \nu^{10} - 4332705 \nu^{9} + \cdots + 1029994667 ) / 880043840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17171 \nu^{14} - 7124 \nu^{13} - 2197 \nu^{12} - 4411155 \nu^{10} - 2301000 \nu^{9} + \cdots - 1470016587 ) / 880043840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9812 \nu^{15} + 13481 \nu^{14} + 34983 \nu^{12} - 2520660 \nu^{11} + 3250195 \nu^{10} + \cdots + 672016163 ) / 1760087680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9812 \nu^{15} - 13481 \nu^{14} - 34983 \nu^{12} - 2520660 \nu^{11} - 3250195 \nu^{10} + \cdots - 672016163 ) / 1760087680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9812 \nu^{15} - 35579 \nu^{14} + 26195 \nu^{12} + 2520660 \nu^{11} - 9353105 \nu^{10} + \cdots - 3447962505 ) / 1760087680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 305879 \nu^{15} - 175253 \nu^{13} - 77850245 \nu^{11} - 42252535 \nu^{9} + \cdots + 11440569920 ) / 22881139840 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 305879 \nu^{15} + 127556 \nu^{14} - 175253 \nu^{13} + 77850245 \nu^{11} + \cdots - 23975329833 \nu ) / 22881139840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 945194 \nu^{15} - 625027 \nu^{14} - 385996 \nu^{13} - 854633 \nu^{12} + 243107710 \nu^{11} + \cdots - 78929397833 ) / 22881139840 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 945194 \nu^{15} - 625027 \nu^{14} + 385996 \nu^{13} - 854633 \nu^{12} - 243107710 \nu^{11} + \cdots - 78929397833 ) / 22881139840 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 982950 \nu^{15} - 1071473 \nu^{14} - 92612 \nu^{13} + 797511 \nu^{12} - 251349590 \nu^{11} + \cdots + 40708966571 ) / 22881139840 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 549515 \nu^{15} - 159445 \nu^{14} - 28561 \nu^{12} + 140730765 \nu^{11} - 40960725 \nu^{10} + \cdots - 13389930671 ) / 11440569920 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} + 2 \beta_{13} - 2 \beta_{12} + 4 \beta_{10} - \beta_{9} - 12 \beta_{8} - 11 \beta_{7} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 12 \beta_{9} + \cdots - 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 36 \beta_{11} - 36 \beta_{10} + 16 \beta_{9} + 16 \beta_{8} - 32 \beta_{6} + 22 \beta_{5} + 22 \beta_{4} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 38 \beta_{15} + 76 \beta_{14} + 38 \beta_{13} + 38 \beta_{12} + 38 \beta_{11} - 38 \beta_{10} + \cdots + 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 698 \beta_{15} - 438 \beta_{13} + 438 \beta_{12} + 156 \beta_{11} - 1032 \beta_{10} + 349 \beta_{9} + \cdots + 516 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 568 \beta_{15} + 1136 \beta_{14} - 568 \beta_{13} - 568 \beta_{12} + 568 \beta_{11} - 568 \beta_{10} + \cdots + 4777 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8992 \beta_{11} + 8992 \beta_{10} - 2888 \beta_{9} - 2888 \beta_{8} + 5776 \beta_{6} - 5008 \beta_{5} + \cdots - 1608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7896 \beta_{15} - 15792 \beta_{14} - 7896 \beta_{13} - 7896 \beta_{12} - 7896 \beta_{11} + \cdots - 7896 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 137874 \beta_{15} + 75586 \beta_{13} - 75586 \beta_{12} - 53072 \beta_{11} + 204244 \beta_{10} + \cdots - 102122 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 106730 \beta_{15} - 213460 \beta_{14} + 106730 \beta_{13} + 106730 \beta_{12} - 106730 \beta_{11} + \cdots - 701731 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1779780 \beta_{11} - 1779780 \beta_{10} + 494760 \beta_{9} + 494760 \beta_{8} - 989520 \beta_{6} + \cdots + 395130 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1425070 \beta_{15} + 2850140 \beta_{14} + 1425070 \beta_{13} + 1425070 \beta_{12} + 1425070 \beta_{11} + \cdots + 1425070 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 24853482 \beta_{15} - 12980342 \beta_{13} + 12980342 \beta_{12} + 11156940 \beta_{11} + \cdots + 18558812 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(\beta_{3}\) \(-1 + \beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−2.56790 + 2.56790i
−1.54374 + 1.54374i
0.417936 − 0.417936i
1.96165 − 1.96165i
−2.56790 − 2.56790i
−1.54374 − 1.54374i
0.417936 + 0.417936i
1.96165 + 1.96165i
2.56790 − 2.56790i
1.54374 − 1.54374i
−0.417936 + 0.417936i
−1.96165 + 1.96165i
2.56790 + 2.56790i
1.54374 + 1.54374i
−0.417936 − 0.417936i
−1.96165 − 1.96165i
−3.50782 − 0.939918i 0 7.95726 + 4.59412i 1.14088 + 4.86810i 0 6.20430 + 1.66244i −13.3229 − 13.3229i 0 0.573606 − 18.1488i
28.2 −2.10878 − 0.565047i 0 0.663588 + 0.383123i 2.51904 − 4.31908i 0 −1.97124 − 0.528192i 4.99206 + 4.99206i 0 −7.75259 + 7.68463i
28.3 0.570911 + 0.152975i 0 −3.16156 − 1.82533i 4.29180 + 2.56524i 0 −9.62941 − 2.58019i −3.19748 − 3.19748i 0 2.05782 + 2.12106i
28.4 2.67967 + 0.718015i 0 3.20097 + 1.84808i −4.08569 − 2.88220i 0 −8.26391 − 2.21431i −0.596019 − 0.596019i 0 −8.87884 − 10.6569i
217.1 −3.50782 + 0.939918i 0 7.95726 − 4.59412i 1.14088 − 4.86810i 0 6.20430 − 1.66244i −13.3229 + 13.3229i 0 0.573606 + 18.1488i
217.2 −2.10878 + 0.565047i 0 0.663588 − 0.383123i 2.51904 + 4.31908i 0 −1.97124 + 0.528192i 4.99206 − 4.99206i 0 −7.75259 − 7.68463i
217.3 0.570911 − 0.152975i 0 −3.16156 + 1.82533i 4.29180 − 2.56524i 0 −9.62941 + 2.58019i −3.19748 + 3.19748i 0 2.05782 − 2.12106i
217.4 2.67967 − 0.718015i 0 3.20097 − 1.84808i −4.08569 + 2.88220i 0 −8.26391 + 2.21431i −0.596019 + 0.596019i 0 −8.87884 + 10.6569i
298.1 −0.939918 − 3.50782i 0 −7.95726 + 4.59412i 4.78634 + 1.44602i 0 −1.66244 − 6.20430i 13.3229 + 13.3229i 0 0.573606 − 18.1488i
298.2 −0.565047 − 2.10878i 0 −0.663588 + 0.383123i −2.48091 − 4.34109i 0 0.528192 + 1.97124i −4.99206 − 4.99206i 0 −7.75259 + 7.68463i
298.3 0.152975 + 0.570911i 0 3.16156 − 1.82533i 4.36746 − 2.43419i 0 2.58019 + 9.62941i 3.19748 + 3.19748i 0 2.05782 + 2.12106i
298.4 0.718015 + 2.67967i 0 −3.20097 + 1.84808i −4.53891 + 2.09721i 0 2.21431 + 8.26391i 0.596019 + 0.596019i 0 −8.87884 − 10.6569i
352.1 −0.939918 + 3.50782i 0 −7.95726 − 4.59412i 4.78634 − 1.44602i 0 −1.66244 + 6.20430i 13.3229 − 13.3229i 0 0.573606 + 18.1488i
352.2 −0.565047 + 2.10878i 0 −0.663588 − 0.383123i −2.48091 + 4.34109i 0 0.528192 − 1.97124i −4.99206 + 4.99206i 0 −7.75259 − 7.68463i
352.3 0.152975 − 0.570911i 0 3.16156 + 1.82533i 4.36746 + 2.43419i 0 2.58019 − 9.62941i 3.19748 − 3.19748i 0 2.05782 − 2.12106i
352.4 0.718015 − 2.67967i 0 −3.20097 − 1.84808i −4.53891 − 2.09721i 0 2.21431 − 8.26391i 0.596019 − 0.596019i 0 −8.87884 + 10.6569i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.l.j 16
3.b odd 2 1 405.3.l.m 16
5.c odd 4 1 405.3.l.m 16
9.c even 3 1 405.3.g.f ✓ 16
9.c even 3 1 405.3.l.m 16
9.d odd 6 1 405.3.g.f ✓ 16
9.d odd 6 1 inner 405.3.l.j 16
15.e even 4 1 inner 405.3.l.j 16
45.k odd 12 1 405.3.g.f ✓ 16
45.k odd 12 1 inner 405.3.l.j 16
45.l even 12 1 405.3.g.f ✓ 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 9.c even 3 1
405.3.g.f ✓ 16 9.d odd 6 1
405.3.g.f ✓ 16 45.k odd 12 1
405.3.g.f ✓ 16 45.l even 12 1
405.3.l.j 16 1.a even 1 1 trivial
405.3.l.j 16 9.d odd 6 1 inner
405.3.l.j 16 15.e even 4 1 inner
405.3.l.j 16 45.k odd 12 1 inner
405.3.l.m 16 3.b odd 2 1
405.3.l.m 16 5.c odd 4 1
405.3.l.m 16 9.c even 3 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 \( T_{2}^{16} + 6 T_{2}^{15} + 18 T_{2}^{14} + 36 T_{2}^{13} - 74 T_{2}^{12} - 450 T_{2}^{11} + \cdots + 28561 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{15} + \cdots + 28561 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 1562310005776 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 159539531776 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 2010 T^{6} + \cdots + 50552100)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{8} + 126 T^{7} + \cdots + 411193867536)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 80 T^{7} + \cdots + 358875136)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 2 T^{7} + \cdots + 23253810064)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} - 18 T^{7} + \cdots + 1000583424)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 2 T^{7} + \cdots + 73028576644)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 243517148561296)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 182 T^{7} + \cdots + 75176865856)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
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