Properties

Label 18.8.c.a
Level $18$
Weight $8$
Character orbit 18.c
Analytic conductor $5.623$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,8,Mod(7,18)]
 
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([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.7");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), 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: \(5.62293045871\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.14601465675.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 52x^{4} - 99x^{3} + 709x^{2} - 660x + 1872 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - 8 \beta_1 q^{2} + ( - \beta_{4} - \beta_{2} + 19 \beta_1 - 14) q^{3} + (64 \beta_1 - 64) q^{4} + (2 \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 18) q^{5}+ \cdots + (3 \beta_{5} + 15 \beta_{4} + 54 \beta_{3} + \cdots - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 \beta_1 q^{2} + ( - \beta_{4} - \beta_{2} + 19 \beta_1 - 14) q^{3} + (64 \beta_1 - 64) q^{4} + (2 \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 18) q^{5}+ \cdots + (111600 \beta_{5} + 44541 \beta_{4} + \cdots - 5915781) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{2} - 27 q^{3} - 192 q^{4} + 54 q^{5} + 792 q^{6} + 210 q^{7} + 3072 q^{8} + 2295 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{2} - 27 q^{3} - 192 q^{4} + 54 q^{5} + 792 q^{6} + 210 q^{7} + 3072 q^{8} + 2295 q^{9} - 864 q^{10} + 6579 q^{11} - 4608 q^{12} + 10092 q^{13} + 1680 q^{14} + 756 q^{15} - 12288 q^{16} - 29790 q^{17} + 19008 q^{18} - 137490 q^{19} + 3456 q^{20} + 85464 q^{21} + 52632 q^{22} - 39654 q^{23} - 13824 q^{24} - 7923 q^{25} - 161472 q^{26} + 87480 q^{27} - 26880 q^{28} + 239832 q^{29} + 283392 q^{30} + 145704 q^{31} - 98304 q^{32} + 175365 q^{33} + 119160 q^{34} - 1087776 q^{35} - 298944 q^{36} - 720384 q^{37} + 549960 q^{38} + 2386548 q^{39} + 27648 q^{40} + 1086993 q^{41} - 752400 q^{42} - 299967 q^{43} - 842112 q^{44} - 2798874 q^{45} + 634464 q^{46} + 131634 q^{47} + 405504 q^{48} + 482175 q^{49} - 63384 q^{50} - 3719223 q^{51} + 645888 q^{52} + 1909152 q^{53} - 1222776 q^{54} - 5494824 q^{55} + 107520 q^{56} + 6076107 q^{57} + 1918656 q^{58} + 2504853 q^{59} - 2315520 q^{60} + 7309038 q^{61} - 2331264 q^{62} - 1254960 q^{63} + 1572864 q^{64} - 1786698 q^{65} + 2211408 q^{66} + 3433035 q^{67} + 953280 q^{68} - 5815206 q^{69} + 4351104 q^{70} + 2269368 q^{71} + 1175040 q^{72} - 15901746 q^{73} + 2881536 q^{74} + 4230945 q^{75} + 4399680 q^{76} - 6147432 q^{77} - 12143088 q^{78} + 7076928 q^{79} - 442368 q^{80} + 4749435 q^{81} - 17391888 q^{82} - 10914444 q^{83} + 549504 q^{84} + 17613396 q^{85} - 2399736 q^{86} - 11533266 q^{87} + 3368448 q^{88} + 23800356 q^{89} + 18592416 q^{90} - 37170552 q^{91} - 2537856 q^{92} + 30872466 q^{93} + 1053072 q^{94} - 17342424 q^{95} - 2359296 q^{96} + 519357 q^{97} - 7714800 q^{98} - 11453238 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 52x^{4} - 99x^{3} + 709x^{2} - 660x + 1872 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 6\nu^{3} + 4\nu^{2} + 1067\nu + 612 ) / 2292 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 93\nu^{4} - 188\nu^{3} + 2481\nu^{2} + 517\nu + 3132 ) / 382 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 98\nu^{4} + 194\nu^{3} - 2485\nu^{2} + 5292\nu - 6036 ) / 382 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 184\nu^{5} + 113\nu^{4} + 6282\nu^{3} + 14530\nu^{2} + 25031\nu + 172896 ) / 2292 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -187\nu^{5} + 181\nu^{4} - 6864\nu^{3} + 3239\nu^{2} - 51221\nu + 13674 ) / 1146 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 6\beta _1 + 6 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 4\beta_{4} + 3\beta_{3} + \beta_{2} + 6\beta _1 - 288 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{4} - 17\beta_{3} - 23\beta_{2} + 432\beta _1 - 648 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -52\beta_{5} - 92\beta_{4} - 123\beta_{3} - 11\beta_{2} + 858\beta _1 + 5478 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -126\beta_{5} - 240\beta_{4} + 283\beta_{3} + 577\beta_{2} - 16566\beta _1 + 23772 ) / 18 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 1.89756i
0.500000 + 5.21243i
0.500000 4.18090i
0.500000 + 1.89756i
0.500000 5.21243i
0.500000 + 4.18090i
−4.00000 + 6.92820i −43.9518 + 15.9763i −32.0000 55.4256i −1.70819 2.95867i 65.1201 368.412i 407.819 706.363i 512.000 1676.52 1404.37i 27.3311
7.2 −4.00000 + 6.92820i −11.7819 45.2569i −32.0000 55.4256i 187.789 + 325.261i 360.677 + 99.4005i −528.867 + 916.024i 512.000 −1909.38 + 1066.42i −3004.63
7.3 −4.00000 + 6.92820i 42.2336 20.0828i −32.0000 55.4256i −159.081 275.536i −29.7966 + 372.935i 226.048 391.526i 512.000 1380.36 1696.34i 2545.30
13.1 −4.00000 6.92820i −43.9518 15.9763i −32.0000 + 55.4256i −1.70819 + 2.95867i 65.1201 + 368.412i 407.819 + 706.363i 512.000 1676.52 + 1404.37i 27.3311
13.2 −4.00000 6.92820i −11.7819 + 45.2569i −32.0000 + 55.4256i 187.789 325.261i 360.677 99.4005i −528.867 916.024i 512.000 −1909.38 1066.42i −3004.63
13.3 −4.00000 6.92820i 42.2336 + 20.0828i −32.0000 + 55.4256i −159.081 + 275.536i −29.7966 372.935i 226.048 + 391.526i 512.000 1380.36 + 1696.34i 2545.30
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.8.c.a 6
3.b odd 2 1 54.8.c.a 6
4.b odd 2 1 144.8.i.a 6
9.c even 3 1 inner 18.8.c.a 6
9.c even 3 1 162.8.a.f 3
9.d odd 6 1 54.8.c.a 6
9.d odd 6 1 162.8.a.e 3
12.b even 2 1 432.8.i.a 6
36.f odd 6 1 144.8.i.a 6
36.h even 6 1 432.8.i.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.8.c.a 6 1.a even 1 1 trivial
18.8.c.a 6 9.c even 3 1 inner
54.8.c.a 6 3.b odd 2 1
54.8.c.a 6 9.d odd 6 1
144.8.i.a 6 4.b odd 2 1
144.8.i.a 6 36.f odd 6 1
162.8.a.e 3 9.d odd 6 1
162.8.a.f 3 9.c even 3 1
432.8.i.a 6 12.b even 2 1
432.8.i.a 6 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 54T_{5}^{5} + 122607T_{5}^{4} + 7279794T_{5}^{3} + 14303890521T_{5}^{2} + 48862653840T_{5} + 166659897600 \) acting on \(S_{8}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8 T + 64)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 10460353203 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 166659897600 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 35\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots + 2214690411708)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 22074972070832)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots - 81\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 14\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots + 33\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 94\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 65\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 53\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 47\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 84\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 87\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 88\!\cdots\!61 \) Copy content Toggle raw display
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