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Properties

Label 18.5.d.a
Level $18$
Weight $5$
Character orbit 18.d
Analytic conductor $1.861$
Analytic rank $0$
Dimension $8$
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] = [18,5,Mod(5,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 18.d (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: \(1.86065933551\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.221456830464.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{5} - 3 \beta_{2} + \beta_1 + 2) q^{3} + ( - 8 \beta_{2} + 8) q^{4} + ( - \beta_{6} - 3 \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots - 23) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{5} - 3 \beta_{2} + \beta_1 + 2) q^{3} + ( - 8 \beta_{2} + 8) q^{4} + ( - \beta_{6} - 3 \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + (93 \beta_{7} - 372 \beta_{6} + \cdots - 2181) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 32 q^{4} + 18 q^{5} + 48 q^{6} - 26 q^{7} - 78 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 32 q^{4} + 18 q^{5} + 48 q^{6} - 26 q^{7} - 78 q^{9} - 720 q^{11} - 144 q^{12} + 10 q^{13} + 288 q^{14} + 1134 q^{15} - 256 q^{16} - 384 q^{18} + 100 q^{19} + 144 q^{20} + 438 q^{21} + 336 q^{22} + 1278 q^{23} + 384 q^{24} + 794 q^{25} - 1296 q^{27} - 416 q^{28} - 1854 q^{29} - 3456 q^{30} - 1478 q^{31} - 3384 q^{33} - 96 q^{34} + 1056 q^{36} - 32 q^{37} + 6768 q^{38} + 5274 q^{39} - 36 q^{41} + 2592 q^{42} - 68 q^{43} + 3402 q^{45} + 2112 q^{46} + 2214 q^{47} - 1536 q^{48} + 2442 q^{49} - 15552 q^{50} - 12006 q^{51} - 80 q^{52} + 7056 q^{54} - 3996 q^{55} + 2304 q^{56} + 10902 q^{57} - 2400 q^{58} + 9108 q^{59} + 6480 q^{60} - 4478 q^{61} - 6654 q^{63} - 4096 q^{64} - 22554 q^{65} - 19872 q^{66} + 7504 q^{67} - 11088 q^{68} - 5994 q^{69} + 6048 q^{70} + 5376 q^{72} + 20716 q^{73} + 15264 q^{74} + 16590 q^{75} + 400 q^{76} + 34434 q^{77} + 24096 q^{78} - 6050 q^{79} - 21150 q^{81} + 1152 q^{82} - 3834 q^{83} - 9600 q^{84} - 16092 q^{85} - 12528 q^{86} + 10170 q^{87} - 2688 q^{88} + 2592 q^{90} - 45868 q^{91} + 10224 q^{92} - 10926 q^{93} + 672 q^{94} + 20880 q^{95} + 31336 q^{97} - 22338 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -19\nu^{7} - 286\nu^{6} + 545\nu^{5} - 8755\nu^{4} + 14939\nu^{3} - 71959\nu^{2} + 65748\nu - 147099 ) / 4230 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 290\nu^{5} + 655\nu^{4} - 2912\nu^{3} + 3727\nu^{2} - 7344\nu + 3777 ) / 1410 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -19\nu^{7} + 419\nu^{6} - 1570\nu^{5} + 10985\nu^{4} - 21016\nu^{3} + 78911\nu^{2} - 67497\nu + 146886 ) / 4230 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 179\nu^{7} - 979\nu^{6} + 6665\nu^{5} - 23380\nu^{4} + 70091\nu^{3} - 149926\nu^{2} + 205212\nu - 201981 ) / 4230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -217\nu^{7} + 407\nu^{6} - 5575\nu^{5} + 5870\nu^{4} - 40213\nu^{3} + 18698\nu^{2} - 73716\nu + 843 ) / 4230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 467\nu^{7} - 1987\nu^{6} + 14990\nu^{5} - 36385\nu^{4} + 122048\nu^{3} - 129703\nu^{2} + 188301\nu + 78702 ) / 4230 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 532\nu^{7} - 1157\nu^{6} + 14350\nu^{5} - 13595\nu^{4} + 101998\nu^{3} + 17587\nu^{2} + 197916\nu + 235632 ) / 4230 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} - 5\beta_{3} - 5\beta _1 + 10 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 2\beta _1 - 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} + 11\beta_{6} - 16\beta_{5} + 4\beta_{4} + 103\beta_{3} + 60\beta_{2} + 88\beta _1 - 242 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - 15\beta_{5} - 16\beta_{4} + 22\beta_{3} + 18\beta_{2} + 53\beta _1 + 177 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 97\beta_{7} - 118\beta_{6} + 122\beta_{5} - 239\beta_{4} - 1250\beta_{3} - 1392\beta_{2} - 821\beta _1 + 3940 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{7} - 53\beta_{6} + 194\beta_{5} + 180\beta_{4} - 505\beta_{3} - 690\beta_{2} - 894\beta _1 - 1271 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1367 \beta_{7} + 1061 \beta_{6} - 934 \beta_{5} + 5005 \beta_{4} + 12613 \beta_{3} + 19800 \beta_{2} + \cdots - 56654 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 − 1.74753i
0.500000 + 3.16175i
0.500000 + 2.20403i
0.500000 − 3.61825i
0.500000 + 1.74753i
0.500000 − 3.16175i
0.500000 − 2.20403i
0.500000 + 3.61825i
−2.44949 + 1.41421i −7.80760 − 4.47676i 4.00000 − 6.92820i −32.5033 − 18.7658i 25.4557 − 0.0758456i −1.35458 − 2.34620i 22.6274i 40.9173 + 69.9055i 106.155
5.2 −2.44949 + 1.41421i 6.85811 − 5.82806i 4.00000 − 6.92820i 37.0033 + 21.3638i −8.55675 + 23.9746i −19.8424 − 34.3680i 22.6274i 13.0674 − 79.9390i −120.852
5.3 2.44949 − 1.41421i −1.67960 − 8.84189i 4.00000 − 6.92820i 6.41371 + 3.70296i −16.6185 − 19.2828i 30.1882 + 52.2875i − 22.6274i −75.3579 + 29.7016i 20.9471
5.4 2.44949 − 1.41421i 5.62909 + 7.02235i 4.00000 − 6.92820i −1.91371 − 1.10488i 23.7195 + 9.24044i −21.9913 − 38.0900i − 22.6274i −17.6268 + 79.0588i −6.25016
11.1 −2.44949 − 1.41421i −7.80760 + 4.47676i 4.00000 + 6.92820i −32.5033 + 18.7658i 25.4557 + 0.0758456i −1.35458 + 2.34620i − 22.6274i 40.9173 − 69.9055i 106.155
11.2 −2.44949 − 1.41421i 6.85811 + 5.82806i 4.00000 + 6.92820i 37.0033 − 21.3638i −8.55675 − 23.9746i −19.8424 + 34.3680i − 22.6274i 13.0674 + 79.9390i −120.852
11.3 2.44949 + 1.41421i −1.67960 + 8.84189i 4.00000 + 6.92820i 6.41371 − 3.70296i −16.6185 + 19.2828i 30.1882 − 52.2875i 22.6274i −75.3579 − 29.7016i 20.9471
11.4 2.44949 + 1.41421i 5.62909 − 7.02235i 4.00000 + 6.92820i −1.91371 + 1.10488i 23.7195 − 9.24044i −21.9913 + 38.0900i 22.6274i −17.6268 − 79.0588i −6.25016
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.5.d.a ✓ 8
3.b odd 2 1 54.5.d.a 8
4.b odd 2 1 144.5.q.b 8
9.c even 3 1 54.5.d.a 8
9.c even 3 1 162.5.b.c 8
9.d odd 6 1 inner 18.5.d.a ✓ 8
9.d odd 6 1 162.5.b.c 8
12.b even 2 1 432.5.q.b 8
36.f odd 6 1 432.5.q.b 8
36.f odd 6 1 1296.5.e.e 8
36.h even 6 1 144.5.q.b 8
36.h even 6 1 1296.5.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.5.d.a ✓ 8 1.a even 1 1 trivial
18.5.d.a ✓ 8 9.d odd 6 1 inner
54.5.d.a 8 3.b odd 2 1
54.5.d.a 8 9.c even 3 1
144.5.q.b 8 4.b odd 2 1
144.5.q.b 8 36.h even 6 1
162.5.b.c 8 9.c even 3 1
162.5.b.c 8 9.d odd 6 1
432.5.q.b 8 12.b even 2 1
432.5.q.b 8 36.f odd 6 1
1296.5.e.e 8 36.f odd 6 1
1296.5.e.e 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{8} - 18 T^{7} + \cdots + 688747536 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 81510250000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 53\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} - 50 T^{3} + \cdots + 8154234820)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots - 305304165104)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 80\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 78\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 13267734129308)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
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