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Properties

Label 90.11.g.c
Level $90$
Weight $11$
Character orbit 90.g
Analytic conductor $57.182$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,11,Mod(37,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.37");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 10)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 16 \beta_1 + 16) q^{2} - 512 \beta_1 q^{4} + ( - \beta_{3} - \beta_{2} - 895 \beta_1 - 910) q^{5} + ( - 5 \beta_{5} + 3 \beta_{3} + \cdots + 2252) q^{7}+ \cdots + ( - 8192 \beta_1 - 8192) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \beta_1 + 16) q^{2} - 512 \beta_1 q^{4} + ( - \beta_{3} - \beta_{2} - 895 \beta_1 - 910) q^{5} + ( - 5 \beta_{5} + 3 \beta_{3} + \cdots + 2252) q^{7}+ \cdots + ( - 1982560 \beta_{5} + \cdots - 2145674544) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 96 q^{2} - 5460 q^{5} + 13512 q^{7} - 49152 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 96 q^{2} - 5460 q^{5} + 13512 q^{7} - 49152 q^{8} - 173280 q^{10} - 647832 q^{11} - 742902 q^{13} - 1572864 q^{16} + 755118 q^{17} - 2749440 q^{20} - 10365312 q^{22} + 15052992 q^{23} + 42644850 q^{25} - 23772864 q^{26} - 6918144 q^{28} + 153847152 q^{31} - 25165824 q^{32} + 208942440 q^{35} - 32574498 q^{37} + 17493120 q^{38} + 737280 q^{40} - 226153272 q^{41} - 63628752 q^{43} + 481695744 q^{46} + 19700448 q^{47} + 788800800 q^{50} - 380365824 q^{52} + 950001042 q^{53} - 135145080 q^{55} - 221380608 q^{56} - 310652160 q^{58} + 1603984392 q^{61} + 2461554432 q^{62} - 1398176070 q^{65} + 2502647712 q^{67} - 386620416 q^{68} + 174645120 q^{70} + 7137533808 q^{71} + 2304462438 q^{73} + 559779840 q^{76} - 597969864 q^{77} + 1431306240 q^{80} - 3618452352 q^{82} + 88180632 q^{83} + 1729841610 q^{85} - 2036120064 q^{86} + 5307039744 q^{88} - 43718253408 q^{91} + 7707131904 q^{92} + 34456200 q^{95} + 33281088582 q^{97} - 12874047264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68121\nu^{5} + 149814\nu^{4} + 329476\nu^{3} - 39101454\nu^{2} + 4554477405\nu - 10394598718 ) / 10016360270 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10648\nu^{5} + 290852\nu^{4} + 2779128\nu^{3} - 6111952\nu^{2} + 8954834136 ) / 3490021 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6061417 \nu^{5} - 5857957 \nu^{4} + 63870702 \nu^{3} - 8487433493 \nu^{2} + \cdots - 1816170252876 ) / 1001636027 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6091265 \nu^{5} - 32584545 \nu^{4} - 71661030 \nu^{3} + 8504566245 \nu^{2} + \cdots + 82266862440 ) / 1001636027 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21309084 \nu^{5} - 46863656 \nu^{4} + 510965616 \nu^{3} + 32264134756 \nu^{2} + \cdots + 2899104846792 ) / 1001636027 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 4\beta_{4} - \beta_{2} ) / 400 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 8\beta_{4} - 8\beta_{3} + 17400\beta_1 ) / 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 261\beta_{5} + 1044\beta_{3} + 261\beta_{2} - 229600\beta _1 + 229600 ) / 400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -1331\beta_{4} - 1331\beta_{3} - 13\beta_{2} - 2270700 ) / 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13165\beta_{5} + 61844\beta_{4} + 13165\beta_{2} + 19975200\beta _1 + 19975200 ) / 80 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−12.3957 + 12.3957i
2.29143 − 2.29143i
10.1043 − 10.1043i
−12.3957 − 12.3957i
2.29143 + 2.29143i
10.1043 + 10.1043i
16.0000 + 16.0000i 0 512.000i −3124.94 − 19.4459i 0 −1507.13 − 1507.13i −8192.00 + 8192.00i 0 −49687.9 − 50310.2i
37.2 16.0000 + 16.0000i 0 512.000i −2583.39 + 1758.32i 0 −13021.6 − 13021.6i −8192.00 + 8192.00i 0 −69467.5 − 13201.2i
37.3 16.0000 + 16.0000i 0 512.000i 2978.33 + 946.124i 0 21284.7 + 21284.7i −8192.00 + 8192.00i 0 32515.4 + 62791.3i
73.1 16.0000 − 16.0000i 0 − 512.000i −3124.94 + 19.4459i 0 −1507.13 + 1507.13i −8192.00 − 8192.00i 0 −49687.9 + 50310.2i
73.2 16.0000 − 16.0000i 0 − 512.000i −2583.39 − 1758.32i 0 −13021.6 + 13021.6i −8192.00 − 8192.00i 0 −69467.5 + 13201.2i
73.3 16.0000 − 16.0000i 0 − 512.000i 2978.33 − 946.124i 0 21284.7 − 21284.7i −8192.00 − 8192.00i 0 32515.4 − 62791.3i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.11.g.c 6
3.b odd 2 1 10.11.c.c ✓ 6
5.c odd 4 1 inner 90.11.g.c 6
12.b even 2 1 80.11.p.c 6
15.d odd 2 1 50.11.c.e 6
15.e even 4 1 10.11.c.c ✓ 6
15.e even 4 1 50.11.c.e 6
60.l odd 4 1 80.11.p.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.c ✓ 6 3.b odd 2 1
10.11.c.c ✓ 6 15.e even 4 1
50.11.c.e 6 15.d odd 2 1
50.11.c.e 6 15.e even 4 1
80.11.p.c 6 12.b even 2 1
80.11.p.c 6 60.l odd 4 1
90.11.g.c 6 1.a even 1 1 trivial
90.11.g.c 6 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(90, [\chi])\):

\( T_{7}^{6} - 13512 T_{7}^{5} + 91287072 T_{7}^{4} + 9497368375544 T_{7}^{3} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
\( T_{11}^{3} + 323916T_{11}^{2} + 9059018052T_{11} - 2666505072779552 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32 T + 512)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( (T^{3} + \cdots - 26\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 77\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 71\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 28\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 31\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 89\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 85\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 15\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 29\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 58\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
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