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Properties

Label 15.9.d.c
Level $15$
Weight $9$
Character orbit 15.d
Analytic conductor $6.111$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [15,9,Mod(14,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.14");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 15.d (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: \(6.11067915092\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2096 x^{10} + 1966565 x^{8} + 942732880 x^{6} + 407378811520 x^{4} + 58185108489824 x^{2} + 32\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{18}\cdot 5^{7} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} + \beta_{2}) q^{3} + ( - \beta_1 + 142) q^{4} + ( - \beta_{5} - \beta_{4} + \cdots + 7 \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_1 - 251) q^{6} + ( - \beta_{8} - 2 \beta_{4} + \cdots + 2 \beta_{2}) q^{7}+ \cdots + (16119 \beta_{11} - 6372 \beta_{9} + \cdots - 53363340) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 1704 q^{4} - 3012 q^{6} - 22824 q^{9} - 35280 q^{10} - 95520 q^{15} + 413328 q^{16} + 276192 q^{19} - 604044 q^{21} + 1173336 q^{24} - 2036220 q^{25} + 3563460 q^{30} - 279216 q^{31} - 1225344 q^{34}+ \cdots - 640360080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2096 x^{10} + 1966565 x^{8} + 942732880 x^{6} + 407378811520 x^{4} + 58185108489824 x^{2} + 32\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 27046273969224 \nu^{10} + \cdots + 20\!\cdots\!06 ) / 53\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18\!\cdots\!08 \nu^{11} + \cdots - 55\!\cdots\!98 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18\!\cdots\!47 \nu^{11} + \cdots - 82\!\cdots\!32 \nu ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!58 \nu^{11} + \cdots + 46\!\cdots\!98 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!70 \nu^{11} + \cdots + 52\!\cdots\!56 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45\!\cdots\!50 \nu^{11} + \cdots + 98\!\cdots\!52 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45\!\cdots\!50 \nu^{11} + \cdots + 15\!\cdots\!48 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 40\!\cdots\!82 \nu^{11} + \cdots + 44\!\cdots\!42 \nu ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16\!\cdots\!61 \nu^{11} + \cdots - 47\!\cdots\!36 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!35 \nu^{11} + \cdots + 60\!\cdots\!32 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27\!\cdots\!27 \nu^{11} + \cdots + 86\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{10} - 4\beta_{8} - \beta_{7} - \beta_{6} - 6\beta_{4} - 20\beta_{3} + 546\beta_{2} - \beta_1 ) / 1080 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 36 \beta_{11} + 16 \beta_{9} - 55 \beta_{7} + 43 \beta_{6} + 368 \beta_{5} + 72 \beta_{4} + \cdots - 188640 ) / 540 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 405 \beta_{11} - 146 \beta_{10} + 1210 \beta_{8} + 73 \beta_{7} + 73 \beta_{6} + 405 \beta_{5} + \cdots + 73 \beta_1 ) / 270 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7308 \beta_{11} - 6100 \beta_{9} - 11710 \beta_{7} - 8630 \beta_{6} - 57632 \beta_{5} + \cdots + 6901290 ) / 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1838250 \beta_{11} + 8006 \beta_{10} - 786322 \beta_{8} - 4003 \beta_{7} - 4003 \beta_{6} + \cdots - 4003 \beta_1 ) / 540 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22828914 \beta_{11} + 21596458 \beta_{9} + 76448465 \beta_{7} + 12420259 \beta_{6} + \cdots + 14822160480 ) / 270 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 908939475 \beta_{11} + 106657574 \beta_{10} - 673007455 \beta_{8} - 53328787 \beta_{7} + \cdots - 53328787 \beta_1 ) / 270 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6416158896 \beta_{11} - 5896584400 \beta_{9} - 25482970880 \beta_{7} + 2881127600 \beta_{6} + \cdots - 18730791768210 ) / 90 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 545120161605 \beta_{11} - 240444715972 \beta_{10} + 1519724010869 \beta_{8} + \cdots + 120222357986 \beta_1 ) / 270 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9248043682710 \beta_{11} + 8079802159058 \beta_{9} + 53791826718790 \beta_{7} - 26480028306886 \beta_{6} + \cdots + 94\!\cdots\!80 ) / 270 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 60068117290725 \beta_{11} + 324679499790154 \beta_{10} + \cdots - 162339749895077 \beta_1 ) / 270 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
14.6004 + 17.1593i
14.6004 − 17.1593i
8.79222 − 30.8359i
8.79222 + 30.8359i
2.83292 − 10.0603i
2.83292 + 10.0603i
−2.83292 − 10.0603i
−2.83292 + 10.0603i
−8.79222 − 30.8359i
−8.79222 + 30.8359i
−14.6004 + 17.1593i
−14.6004 − 17.1593i
−29.2008 −12.3596 − 80.0515i 596.686 218.046 − 585.731i 360.909 + 2337.57i − 2074.09i −9948.30 −6255.48 + 1978.80i −6367.13 + 17103.8i
14.2 −29.2008 −12.3596 + 80.0515i 596.686 218.046 + 585.731i 360.909 − 2337.57i 2074.09i −9948.30 −6255.48 − 1978.80i −6367.13 − 17103.8i
14.3 −17.5844 74.7380 − 31.2287i 53.2123 −31.8709 + 624.187i −1314.22 + 549.140i − 3426.92i 3565.91 4610.53 − 4667.95i 560.432 − 10976.0i
14.4 −17.5844 74.7380 + 31.2287i 53.2123 −31.8709 − 624.187i −1314.22 − 549.140i 3426.92i 3565.91 4610.53 + 4667.95i 560.432 + 10976.0i
14.5 −5.66585 −35.3550 − 72.8768i −223.898 531.836 + 328.291i 200.316 + 412.909i 1674.62i 2719.03 −4061.05 + 5153.12i −3013.30 − 1860.05i
14.6 −5.66585 −35.3550 + 72.8768i −223.898 531.836 − 328.291i 200.316 − 412.909i − 1674.62i 2719.03 −4061.05 − 5153.12i −3013.30 + 1860.05i
14.7 5.66585 35.3550 − 72.8768i −223.898 −531.836 − 328.291i 200.316 − 412.909i 1674.62i −2719.03 −4061.05 − 5153.12i −3013.30 − 1860.05i
14.8 5.66585 35.3550 + 72.8768i −223.898 −531.836 + 328.291i 200.316 + 412.909i − 1674.62i −2719.03 −4061.05 + 5153.12i −3013.30 + 1860.05i
14.9 17.5844 −74.7380 − 31.2287i 53.2123 31.8709 − 624.187i −1314.22 − 549.140i − 3426.92i −3565.91 4610.53 + 4667.95i 560.432 − 10976.0i
14.10 17.5844 −74.7380 + 31.2287i 53.2123 31.8709 + 624.187i −1314.22 + 549.140i 3426.92i −3565.91 4610.53 − 4667.95i 560.432 + 10976.0i
14.11 29.2008 12.3596 − 80.0515i 596.686 −218.046 + 585.731i 360.909 − 2337.57i − 2074.09i 9948.30 −6255.48 − 1978.80i −6367.13 + 17103.8i
14.12 29.2008 12.3596 + 80.0515i 596.686 −218.046 − 585.731i 360.909 + 2337.57i 2074.09i 9948.30 −6255.48 + 1978.80i −6367.13 − 17103.8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.9.d.c ✓ 12
3.b odd 2 1 inner 15.9.d.c ✓ 12
4.b odd 2 1 240.9.c.c 12
5.b even 2 1 inner 15.9.d.c ✓ 12
5.c odd 4 2 75.9.c.h 12
12.b even 2 1 240.9.c.c 12
15.d odd 2 1 inner 15.9.d.c ✓ 12
15.e even 4 2 75.9.c.h 12
20.d odd 2 1 240.9.c.c 12
60.h even 2 1 240.9.c.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.d.c ✓ 12 1.a even 1 1 trivial
15.9.d.c ✓ 12 3.b odd 2 1 inner
15.9.d.c ✓ 12 5.b even 2 1 inner
15.9.d.c ✓ 12 15.d odd 2 1 inner
75.9.c.h 12 5.c odd 4 2
75.9.c.h 12 15.e even 4 2
240.9.c.c 12 4.b odd 2 1
240.9.c.c 12 12.b even 2 1
240.9.c.c 12 20.d odd 2 1
240.9.c.c 12 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 1194T_{2}^{4} + 300960T_{2}^{2} - 8464000 \) acting on \(S_{9}^{\mathrm{new}}(15, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 1194 T^{4} + \cdots - 8464000)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 79\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 35\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 466181174169488)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 37\!\cdots\!32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 34\!\cdots\!28)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 37\!\cdots\!68)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
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