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Properties

Label 90.14.c.b
Level $90$
Weight $14$
Character orbit 90.c
Analytic conductor $96.508$
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] = [90,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.19");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(96.5078360567\)
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: \( x^{6} - 2x^{5} + 2x^{4} + 160950x^{3} + 43599609x^{2} + 975553632x + 10914144768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2}\cdot 5^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 4096 q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 412) q^{5}+ \cdots + 4096 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 4096 q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 412) q^{5}+ \cdots + ( - 20304576 \beta_{5} + \cdots + 23463331893 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24576 q^{4} + 2470 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24576 q^{4} + 2470 q^{5} - 2269440 q^{10} - 673672 q^{11} + 1211648 q^{14} + 100663296 q^{16} + 142606200 q^{19} - 10117120 q^{20} + 1820907150 q^{25} + 3369156096 q^{26} - 1402368660 q^{29} - 22270466688 q^{31} + 10743816192 q^{34} - 40910703880 q^{35} + 9295626240 q^{40} + 159550828628 q^{41} + 2759360512 q^{44} - 48346742016 q^{46} - 142584010062 q^{49} - 33045516800 q^{50} - 465712133640 q^{55} - 4962910208 q^{56} + 129517581080 q^{59} + 2208324934212 q^{61} - 412316860416 q^{64} + 475107396240 q^{65} - 1324581354240 q^{70} - 1016718596592 q^{71} - 1548283182592 q^{74} - 584114995200 q^{76} - 23303633760 q^{79} + 41439723520 q^{80} + 9460560132480 q^{85} + 10908216246016 q^{86} + 1102941191140 q^{89} - 9640398296208 q^{91} + 20956004804352 q^{94} - 26900168949000 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 160950x^{3} + 43599609x^{2} + 975553632x + 10914144768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2902156 \nu^{5} + 38332616 \nu^{4} - 467184200 \nu^{3} - 252252560280 \nu^{2} + \cdots - 14\!\cdots\!56 ) / 21586271220765 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8857615 \nu^{5} + 124928126 \nu^{4} - 53829158030 \nu^{3} - 717491956170 \nu^{2} + \cdots - 82\!\cdots\!16 ) / 11512677984408 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 602533027 \nu^{5} - 401566369498 \nu^{4} + 18522916830250 \nu^{3} + \cdots - 10\!\cdots\!12 ) / 633197289142440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3016377017 \nu^{5} + 247785621010 \nu^{4} + 7563441998270 \nu^{3} + \cdots + 52\!\cdots\!00 ) / 379918373485464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51002602211 \nu^{5} - 671582849926 \nu^{4} - 5501876772650 \nu^{3} + \cdots + 24\!\cdots\!16 ) / 949795933713660 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{5} - 7\beta_{4} - 6\beta_{3} - 65\beta_{2} + 29\beta _1 + 1278 ) / 3840 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 198\beta_{5} - 628\beta_{2} + 82677\beta_1 ) / 1200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 69314\beta_{5} + 233095\beta_{4} + 195270\beta_{3} - 2091519\beta_{2} + 24506371\beta _1 - 1545067710 ) / 19200 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 29696\beta_{5} + 234715\beta_{4} - 172470\beta_{3} - 115931\beta_{2} - 115931\beta _1 - 7027499250 ) / 240 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 34367486 \beta_{5} + 340981753 \beta_{4} + 213450234 \beta_{3} + 2992855487 \beta_{2} + \cdots - 3401768055042 ) / 3840 \) 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
−11.7164 − 11.7164i
−50.1479 − 50.1479i
62.8643 + 62.8643i
−11.7164 + 11.7164i
−50.1479 + 50.1479i
62.8643 − 62.8643i
− 64.0000i 0 −4096.00 −33325.2 + 10494.5i 0 386506.i 262144.i 0 671650. + 2.13281e6i
19.2 − 64.0000i 0 −4096.00 272.592 − 34937.5i 0 − 454502.i 262144.i 0 −2.23600e6 − 17445.9i
19.3 − 64.0000i 0 −4096.00 34287.6 + 6712.97i 0 77461.9i 262144.i 0 429630. − 2.19441e6i
19.4 64.0000i 0 −4096.00 −33325.2 − 10494.5i 0 − 386506.i − 262144.i 0 671650. − 2.13281e6i
19.5 64.0000i 0 −4096.00 272.592 + 34937.5i 0 454502.i − 262144.i 0 −2.23600e6 + 17445.9i
19.6 64.0000i 0 −4096.00 34287.6 − 6712.97i 0 − 77461.9i − 262144.i 0 429630. + 2.19441e6i
\(n\): 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 90.14.c.b 6
3.b odd 2 1 10.14.b.a ✓ 6
5.b even 2 1 inner 90.14.c.b 6
12.b even 2 1 80.14.c.b 6
15.d odd 2 1 10.14.b.a ✓ 6
15.e even 4 1 50.14.a.i 3
15.e even 4 1 50.14.a.j 3
60.h even 2 1 80.14.c.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.14.b.a ✓ 6 3.b odd 2 1
10.14.b.a ✓ 6 15.d odd 2 1
50.14.a.i 3 15.e even 4 1
50.14.a.j 3 15.e even 4 1
80.14.c.b 6 12.b even 2 1
80.14.c.b 6 60.h even 2 1
90.14.c.b 6 1.a even 1 1 trivial
90.14.c.b 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 361959036252T_{7}^{4} + 32994989415706252067568T_{7}^{2} + 185165400716248404090131777994304 \) acting on \(S_{14}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4096)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T^{3} + \cdots + 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 40\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 92\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots + 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 22\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
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