Properties

Label 90.3.j.b
Level $90$
Weight $3$
Character orbit 90.j
Analytic conductor $2.452$
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] = [90,3,Mod(29,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 90.j (of order \(6\), 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: \(2.45232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{13} q^{3} + 2 \beta_{6} q^{4} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 2) q^{5}+ \cdots + ( - \beta_{15} + \beta_{14} - \beta_{12} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{13} q^{3} + 2 \beta_{6} q^{4} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 2) q^{5}+ \cdots + (2 \beta_{15} - 2 \beta_{14} + \cdots + 34) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 30 q^{5} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 30 q^{5} - 8 q^{6} - 16 q^{9} - 16 q^{10} + 60 q^{11} + 12 q^{14} - 98 q^{15} - 32 q^{16} + 144 q^{19} - 60 q^{20} - 56 q^{21} + 8 q^{24} + 6 q^{25} + 72 q^{29} + 52 q^{30} + 28 q^{31} - 136 q^{34} + 40 q^{36} - 276 q^{39} + 16 q^{40} - 180 q^{41} + 242 q^{45} - 56 q^{46} + 12 q^{49} + 144 q^{50} - 8 q^{51} + 260 q^{54} + 20 q^{55} - 24 q^{56} - 228 q^{59} + 20 q^{60} + 68 q^{61} + 128 q^{64} - 102 q^{65} + 440 q^{66} - 16 q^{69} - 112 q^{70} + 72 q^{74} - 274 q^{75} - 144 q^{76} + 420 q^{79} - 500 q^{81} + 176 q^{84} - 136 q^{85} - 48 q^{86} + 40 q^{90} - 168 q^{91} - 164 q^{94} + 276 q^{95} + 16 q^{96} + 268 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} + 98 \nu^{14} - 1694 \nu^{13} - 11468 \nu^{12} + 16043 \nu^{11} - 2128 \nu^{10} + \cdots + 3632930676 ) / 1084139640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 98 \nu^{14} + 1694 \nu^{13} - 11468 \nu^{12} - 16043 \nu^{11} - 2128 \nu^{10} + \cdots + 3632930676 ) / 1084139640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{14} + 55 \nu^{12} - 2032 \nu^{10} + 15657 \nu^{8} + 213588 \nu^{6} - 1070253 \nu^{4} + \cdots + 65426292 ) / 20076660 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 256 \nu^{15} + 703 \nu^{13} + 72245 \nu^{11} - 469200 \nu^{9} - 2736405 \nu^{7} + \cdots - 1856323413 \nu ) / 4878628380 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 212 \nu^{14} + 1034 \nu^{12} + 27619 \nu^{10} - 245004 \nu^{8} - 1736091 \nu^{6} + \cdots - 865185948 ) / 271034910 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1225 \nu^{15} - 16666 \nu^{13} - 256559 \nu^{11} + 3638289 \nu^{9} + 19138311 \nu^{7} + \cdots + 12751926795 \nu ) / 4878628380 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 644 \nu^{15} + 5394 \nu^{14} + 13265 \nu^{13} - 7680 \nu^{12} + 167749 \nu^{11} + \cdots - 376260228 ) / 3252418920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 644 \nu^{15} + 5394 \nu^{14} - 13265 \nu^{13} - 7680 \nu^{12} - 167749 \nu^{11} + \cdots - 376260228 ) / 3252418920 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4495 \nu^{15} - 18027 \nu^{14} - 29809 \nu^{13} + 278145 \nu^{12} - 779024 \nu^{11} + \cdots - 179743975020 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4495 \nu^{15} + 18027 \nu^{14} - 29809 \nu^{13} - 278145 \nu^{12} - 779024 \nu^{11} + \cdots + 179743975020 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2413 \nu^{14} + 6352 \nu^{12} + 417695 \nu^{10} - 1418565 \nu^{8} - 33448095 \nu^{6} + \cdots - 5822999037 ) / 542069820 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 212 \nu^{15} + 1034 \nu^{13} + 27619 \nu^{11} - 245004 \nu^{9} - 1736091 \nu^{7} + \cdots - 594151038 \nu ) / 271034910 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1561 \nu^{15} - 578 \nu^{14} - 8533 \nu^{13} + 16745 \nu^{12} - 279308 \nu^{11} + \cdots - 7814839905 ) / 1084139640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1561 \nu^{15} + 1426 \nu^{14} - 8533 \nu^{13} - 20881 \nu^{12} - 279308 \nu^{11} + \cdots + 10191444057 ) / 1084139640 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{12} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} + 2 \beta_{14} + 7 \beta_{13} + \beta_{11} + \beta_{10} - 4 \beta_{9} + 4 \beta_{8} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 7 \beta_{9} + 7 \beta_{8} + \beta_{6} + 6 \beta_{4} + \cdots + 55 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{15} + 7 \beta_{14} - \beta_{13} - 19 \beta_{11} - 19 \beta_{10} + 10 \beta_{9} - 10 \beta_{8} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 114 \beta_{15} - 114 \beta_{14} + 38 \beta_{12} - 31 \beta_{11} + 31 \beta_{10} + 47 \beta_{9} + \cdots - 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 116 \beta_{15} + 116 \beta_{14} + 376 \beta_{13} + 49 \beta_{11} + 49 \beta_{10} - 199 \beta_{9} + \cdots + 116 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 108 \beta_{15} + 108 \beta_{14} + 340 \beta_{12} - 122 \beta_{11} + 122 \beta_{10} + 265 \beta_{9} + \cdots - 518 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 571 \beta_{15} + 571 \beta_{14} - 553 \beta_{13} - 2401 \beta_{11} - 2401 \beta_{10} + 424 \beta_{9} + \cdots + 571 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8337 \beta_{15} - 8337 \beta_{14} + 1685 \beta_{12} - 4243 \beta_{11} + 4243 \beta_{10} - 1396 \beta_{9} + \cdots - 27157 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4480 \beta_{15} - 4480 \beta_{14} - 18467 \beta_{13} + 1090 \beta_{11} + 1090 \beta_{10} + \cdots - 4480 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18420 \beta_{15} + 18420 \beta_{14} + 5920 \beta_{12} - 6440 \beta_{11} + 6440 \beta_{10} + \cdots - 319289 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 47170 \beta_{15} + 47170 \beta_{14} + 6140 \beta_{13} - 115510 \beta_{11} - 115510 \beta_{10} + \cdots + 47170 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 363051 \beta_{15} - 363051 \beta_{14} + 91811 \beta_{12} - 375340 \beta_{11} + 375340 \beta_{10} + \cdots - 2664409 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 961348 \beta_{15} - 961348 \beta_{14} - 4958843 \beta_{13} + 360511 \beta_{11} + 360511 \beta_{10} + \cdots - 961348 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.173499 2.99498i
−2.87096 + 0.870383i
2.97088 + 0.416995i
−0.633522 + 2.93235i
0.633522 2.93235i
−2.97088 0.416995i
2.87096 0.870383i
0.173499 + 2.99498i
−0.173499 + 2.99498i
−2.87096 0.870383i
2.97088 0.416995i
−0.633522 2.93235i
0.633522 + 2.93235i
−2.97088 + 0.416995i
2.87096 + 0.870383i
0.173499 2.99498i
−0.707107 + 1.22474i −2.68048 1.34723i −1.00000 1.73205i 4.82012 + 1.32909i 3.54540 2.33026i 2.75375 + 1.58988i 2.82843 5.36992 + 7.22246i −5.03613 + 4.96360i
29.2 −0.707107 + 1.22474i −0.681708 + 2.92152i −1.00000 1.73205i −0.137425 + 4.99811i −3.09608 2.90074i −4.16400 2.40409i 2.82843 −8.07055 3.98325i −6.02424 3.70251i
29.3 −0.707107 + 1.22474i 1.84657 2.36436i −1.00000 1.73205i 0.410361 4.98313i 1.59002 + 3.93343i −8.30302 4.79375i 2.82843 −2.18038 8.73189i 5.81290 + 4.02619i
29.4 −0.707107 + 1.22474i 2.22272 + 2.01482i −1.00000 1.73205i 3.82116 3.22471i −4.03934 + 1.29758i 7.59195 + 4.38322i 2.82843 0.881011 + 8.95678i 1.24747 + 6.96016i
29.5 0.707107 1.22474i −2.22272 2.01482i −1.00000 1.73205i 4.70326 1.69687i −4.03934 + 1.29758i −7.59195 4.38322i −2.82843 0.881011 + 8.95678i 1.24747 6.96016i
29.6 0.707107 1.22474i −1.84657 + 2.36436i −1.00000 1.73205i 4.52070 + 2.13618i 1.59002 + 3.93343i 8.30302 + 4.79375i −2.82843 −2.18038 8.73189i 5.81290 4.02619i
29.7 0.707107 1.22474i 0.681708 2.92152i −1.00000 1.73205i −4.39720 2.38004i −3.09608 2.90074i 4.16400 + 2.40409i −2.82843 −8.07055 3.98325i −6.02424 + 3.70251i
29.8 0.707107 1.22474i 2.68048 + 1.34723i −1.00000 1.73205i 1.25903 4.83889i 3.54540 2.33026i −2.75375 1.58988i −2.82843 5.36992 + 7.22246i −5.03613 4.96360i
59.1 −0.707107 1.22474i −2.68048 + 1.34723i −1.00000 + 1.73205i 4.82012 1.32909i 3.54540 + 2.33026i 2.75375 1.58988i 2.82843 5.36992 7.22246i −5.03613 4.96360i
59.2 −0.707107 1.22474i −0.681708 2.92152i −1.00000 + 1.73205i −0.137425 4.99811i −3.09608 + 2.90074i −4.16400 + 2.40409i 2.82843 −8.07055 + 3.98325i −6.02424 + 3.70251i
59.3 −0.707107 1.22474i 1.84657 + 2.36436i −1.00000 + 1.73205i 0.410361 + 4.98313i 1.59002 3.93343i −8.30302 + 4.79375i 2.82843 −2.18038 + 8.73189i 5.81290 4.02619i
59.4 −0.707107 1.22474i 2.22272 2.01482i −1.00000 + 1.73205i 3.82116 + 3.22471i −4.03934 1.29758i 7.59195 4.38322i 2.82843 0.881011 8.95678i 1.24747 6.96016i
59.5 0.707107 + 1.22474i −2.22272 + 2.01482i −1.00000 + 1.73205i 4.70326 + 1.69687i −4.03934 1.29758i −7.59195 + 4.38322i −2.82843 0.881011 8.95678i 1.24747 + 6.96016i
59.6 0.707107 + 1.22474i −1.84657 2.36436i −1.00000 + 1.73205i 4.52070 2.13618i 1.59002 3.93343i 8.30302 4.79375i −2.82843 −2.18038 + 8.73189i 5.81290 + 4.02619i
59.7 0.707107 + 1.22474i 0.681708 + 2.92152i −1.00000 + 1.73205i −4.39720 + 2.38004i −3.09608 + 2.90074i 4.16400 2.40409i −2.82843 −8.07055 + 3.98325i −6.02424 3.70251i
59.8 0.707107 + 1.22474i 2.68048 1.34723i −1.00000 + 1.73205i 1.25903 + 4.83889i 3.54540 + 2.33026i −2.75375 + 1.58988i −2.82843 5.36992 7.22246i −5.03613 + 4.96360i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.3.j.b 16
3.b odd 2 1 270.3.j.b 16
5.b even 2 1 inner 90.3.j.b 16
5.c odd 4 2 450.3.i.e 16
9.c even 3 1 270.3.j.b 16
9.c even 3 1 810.3.b.b 16
9.d odd 6 1 inner 90.3.j.b 16
9.d odd 6 1 810.3.b.b 16
15.d odd 2 1 270.3.j.b 16
15.e even 4 2 1350.3.i.e 16
45.h odd 6 1 inner 90.3.j.b 16
45.h odd 6 1 810.3.b.b 16
45.j even 6 1 270.3.j.b 16
45.j even 6 1 810.3.b.b 16
45.k odd 12 2 1350.3.i.e 16
45.l even 12 2 450.3.i.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.j.b 16 1.a even 1 1 trivial
90.3.j.b 16 5.b even 2 1 inner
90.3.j.b 16 9.d odd 6 1 inner
90.3.j.b 16 45.h odd 6 1 inner
270.3.j.b 16 3.b odd 2 1
270.3.j.b 16 9.c even 3 1
270.3.j.b 16 15.d odd 2 1
270.3.j.b 16 45.j even 6 1
450.3.i.e 16 5.c odd 4 2
450.3.i.e 16 45.l even 12 2
810.3.b.b 16 9.c even 3 1
810.3.b.b 16 9.d odd 6 1
810.3.b.b 16 45.h odd 6 1
810.3.b.b 16 45.j even 6 1
1350.3.i.e 16 15.e even 4 2
1350.3.i.e 16 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 202 T_{7}^{14} + 27898 T_{7}^{12} - 2058640 T_{7}^{10} + 109528039 T_{7}^{8} + \cdots + 2726544000625 \) acting on \(S_{3}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 2726544000625 \) Copy content Toggle raw display
$11$ \( (T^{8} - 30 T^{7} + \cdots + 110923024)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 1991891886336 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1520 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 36 T^{3} + \cdots - 118800)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 26\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} - 36 T^{7} + \cdots + 334196141409)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 27544049296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 3016 T^{6} + \cdots + 1414963456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 90 T^{7} + \cdots + 15399072649)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$53$ \( (T^{8} - 3264 T^{6} + \cdots + 81293414400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 28666857972736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 34 T^{7} + \cdots + 116562836569)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{8} + 11328 T^{6} + \cdots + 39661519104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 12985356390400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
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