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Properties

Label 90.12.c.b
Level $90$
Weight $12$
Character orbit 90.c
Analytic conductor $69.151$
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] = [90,12,Mod(19,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.19");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\), degree \(1\), 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: \(69.1508862504\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 198x^{3} + 3568321x^{2} - 6762620x + 6408200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{2}\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 1024 q^{4} + ( - \beta_{5} + 63 \beta_{2} - 88) q^{5} + (3 \beta_{5} - \beta_{3} - 338 \beta_{2} + \cdots - 1) q^{7}+ \cdots - 1024 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - 1024 q^{4} + ( - \beta_{5} + 63 \beta_{2} - 88) q^{5} + (3 \beta_{5} - \beta_{3} - 338 \beta_{2} + \cdots - 1) q^{7}+ \cdots + (6471648 \beta_{5} - 2157216 \beta_{3} + \cdots - 2157216) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6144 q^{4} - 530 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6144 q^{4} - 530 q^{5} - 385920 q^{10} + 642728 q^{11} + 2125952 q^{14} + 6291456 q^{16} - 24109080 q^{19} + 542720 q^{20} - 181718850 q^{25} - 89251584 q^{26} + 256409820 q^{29} + 458481792 q^{31} - 225288192 q^{34} + 697136360 q^{35} + 395182080 q^{40} + 164768948 q^{41} - 658153472 q^{44} - 2956208256 q^{46} - 675514158 q^{49} - 3912262400 q^{50} + 3688644360 q^{55} - 2176974848 q^{56} - 17663962360 q^{59} - 5020792428 q^{61} - 6442450944 q^{64} + 19996916880 q^{65} + 24956826240 q^{70} - 56788418832 q^{71} + 64135292672 q^{74} + 24687697920 q^{76} + 2602550880 q^{79} - 555745280 q^{80} - 85024210560 q^{85} - 111995790464 q^{86} - 249448412540 q^{89} - 184446766128 q^{91} - 337749482112 q^{94} - 104896380600 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 198x^{3} + 3568321x^{2} - 6762620x + 6408200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 658267 \nu^{5} + 1708186 \nu^{4} - 1920228396 \nu^{3} + 610277066 \nu^{2} + \cdots + 5659254908590 ) / 120904744321 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5715200 \nu^{5} - 6014576 \nu^{4} + 315216 \nu^{3} + 11372932784 \nu^{2} + \cdots - 19324355382560 ) / 604523721605 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4302597700 \nu^{5} + 23516657224 \nu^{4} + 363593908716 \nu^{3} + 3164711100384 \nu^{2} + \cdots + 60\!\cdots\!40 ) / 5440713494445 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5055815380 \nu^{5} - 88999461104 \nu^{4} + 1060740724164 \nu^{3} - 3090142550064 \nu^{2} + \cdots - 21\!\cdots\!55 ) / 5440713494445 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13587225885 \nu^{5} - 7991399102 \nu^{4} + 163630622832 \nu^{3} - 9396474407982 \nu^{2} + \cdots - 30\!\cdots\!55 ) / 5440713494445 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} - 4\beta_{3} + 10\beta_{2} - 31\beta _1 + 640 ) / 1920 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -54\beta_{5} + 18\beta_{3} + 15755\beta_{2} - 26\beta _1 + 18 ) / 400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8371\beta_{5} + 9235\beta_{4} + 38668\beta_{3} - 64430\beta_{2} - 312531\beta _1 - 956512 ) / 9600 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17533\beta_{5} - 16505\beta_{4} + 36094\beta_{3} - 17533\beta _1 - 951669746 ) / 400 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4172683 \beta_{5} + 3484939 \beta_{4} + 14283628 \beta_{3} + 16916450 \beta_{2} + 110784085 \beta _1 - 599300608 ) / 1920 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.947541 − 0.947541i
−30.7073 + 30.7073i
30.7598 − 30.7598i
0.947541 + 0.947541i
−30.7073 − 30.7073i
30.7598 + 30.7598i
− 32.0000i 0 −1024.00 −2827.02 + 6390.31i 0 − 15908.8i 32768.0i 0 204490. + 90464.5i
19.2 − 32.0000i 0 −1024.00 −1594.62 − 6803.33i 0 − 24477.0i 32768.0i 0 −217707. + 51027.9i
19.3 − 32.0000i 0 −1024.00 4156.64 − 5616.98i 0 73603.8i 32768.0i 0 −179743. − 133012.i
19.4 32.0000i 0 −1024.00 −2827.02 − 6390.31i 0 15908.8i − 32768.0i 0 204490. − 90464.5i
19.5 32.0000i 0 −1024.00 −1594.62 + 6803.33i 0 24477.0i − 32768.0i 0 −217707. − 51027.9i
19.6 32.0000i 0 −1024.00 4156.64 + 5616.98i 0 − 73603.8i − 32768.0i 0 −179743. + 133012.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.12.c.b 6
3.b odd 2 1 10.12.b.a ✓ 6
5.b even 2 1 inner 90.12.c.b 6
12.b even 2 1 80.12.c.c 6
15.d odd 2 1 10.12.b.a ✓ 6
15.e even 4 1 50.12.a.i 3
15.e even 4 1 50.12.a.j 3
60.h even 2 1 80.12.c.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.b.a ✓ 6 3.b odd 2 1
10.12.b.a ✓ 6 15.d odd 2 1
50.12.a.i 3 15.e even 4 1
50.12.a.j 3 15.e even 4 1
80.12.c.c 6 12.b even 2 1
80.12.c.c 6 60.h even 2 1
90.12.c.b 6 1.a even 1 1 trivial
90.12.c.b 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 6269737308T_{7}^{4} + 4768524644880488688T_{7}^{2} + 821471635536743699888328256 \) acting on \(S_{12}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1024)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 82\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T^{3} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 42\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 25\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 54\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 54\!\cdots\!72)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 73\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 87\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
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