Properties

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

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 8 \beta_1 + 8) q^{2} + (\beta_{3} + 11 \beta_1 - 7) q^{3} - 64 \beta_1 q^{4} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - 1) q^{5}+ \cdots + ( - 3 \beta_{6} + 6 \beta_{5} + \cdots - 144) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 8 \beta_1 + 8) q^{2} + (\beta_{3} + 11 \beta_1 - 7) q^{3} - 64 \beta_1 q^{4} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - 1) q^{5}+ \cdots + ( - 6801 \beta_{7} - 14976 \beta_{6} + \cdots + 1393263) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{2} - 12 q^{3} - 256 q^{4} + 54 q^{5} + 480 q^{6} - 44 q^{7} - 4096 q^{8} - 2238 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{2} - 12 q^{3} - 256 q^{4} + 54 q^{5} + 480 q^{6} - 44 q^{7} - 4096 q^{8} - 2238 q^{9} + 864 q^{10} + 2172 q^{11} + 4608 q^{12} - 6398 q^{13} + 352 q^{14} + 17604 q^{15} - 16384 q^{16} - 51972 q^{17} - 22416 q^{18} + 90712 q^{19} + 3456 q^{20} + 192390 q^{21} - 17376 q^{22} - 2028 q^{23} + 6144 q^{24} - 173446 q^{25} - 102368 q^{26} - 288360 q^{27} + 5632 q^{28} + 283098 q^{29} + 172800 q^{30} - 812 q^{31} + 131072 q^{32} + 342108 q^{33} - 207888 q^{34} - 1492992 q^{35} - 36096 q^{36} + 441688 q^{37} + 362848 q^{38} + 67428 q^{39} - 27648 q^{40} + 610704 q^{41} + 349200 q^{42} - 84380 q^{43} - 278016 q^{44} - 652698 q^{45} - 32448 q^{46} + 855708 q^{47} - 245760 q^{48} - 1273038 q^{49} + 1387568 q^{50} + 1748556 q^{51} - 409472 q^{52} - 1842600 q^{53} + 640800 q^{54} + 7016760 q^{55} + 22528 q^{56} - 4893306 q^{57} - 2264784 q^{58} + 380796 q^{59} + 255744 q^{60} - 3151130 q^{61} - 12992 q^{62} - 3149076 q^{63} + 2097152 q^{64} - 326538 q^{65} - 6234192 q^{66} - 9961580 q^{67} + 1663104 q^{68} + 15322554 q^{69} - 5971968 q^{70} + 8970384 q^{71} + 1145856 q^{72} + 29234932 q^{73} + 1766752 q^{74} - 26321244 q^{75} - 2902784 q^{76} - 5954826 q^{77} + 6127680 q^{78} - 17309396 q^{79} - 442368 q^{80} + 6907266 q^{81} + 9771264 q^{82} - 11520192 q^{83} - 9519360 q^{84} - 14675148 q^{85} + 675040 q^{86} + 37274652 q^{87} - 1112064 q^{88} + 26066064 q^{89} + 28333152 q^{90} + 26231936 q^{91} - 129792 q^{92} - 48684366 q^{93} - 6845664 q^{94} - 31016952 q^{95} - 2359296 q^{96} - 22003112 q^{97} - 20368608 q^{98} + 26558784 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 1336\nu^{5} + 539944\nu^{3} + 62785035\nu + 22961400 ) / 45922800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 167 \nu^{7} + 176252 \nu^{5} - 328020 \nu^{4} + 58868168 \nu^{3} - 184675260 \nu^{2} + \cdots - 19238373000 ) / 160729800 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 167 \nu^{7} + 176252 \nu^{5} - 328020 \nu^{4} + 58868168 \nu^{3} - 253559460 \nu^{2} + \cdots - 42245695800 ) / 160729800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 167 \nu^{7} + 176252 \nu^{5} - 328020 \nu^{4} + 58868168 \nu^{3} - 414289260 \nu^{2} + \cdots - 95929449000 ) / 53576600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{6} + 2081\nu^{4} + 677750\nu^{2} + 15435\nu + 67639185 ) / 5145 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 186 \nu^{7} + 3905 \nu^{6} + 189921 \nu^{5} + 4186160 \nu^{4} + 58431309 \nu^{3} + \cdots + 143583726000 ) / 20091225 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3841 \nu^{7} + 4053796 \nu^{5} + 10168620 \nu^{4} + 1353967864 \nu^{3} + 7171501260 \nu^{2} + \cdots + 1079543341800 ) / 160729800 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 10\beta_{3} + 7\beta_{2} ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{4} - 56\beta_{3} + 77\beta_{2} - 18036 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 91\beta_{7} - 378\beta_{6} + 189\beta_{5} - 327\beta_{4} + 4614\beta_{3} - 1946\beta_{2} - 19656\beta _1 + 9639 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 490\beta_{7} + 4921\beta_{4} + 32018\beta_{3} - 58051\beta_{2} + 6987168 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 64218 \beta_{7} + 252504 \beta_{6} - 126252 \beta_{5} + 123001 \beta_{4} - 2199832 \beta_{3} + \cdots - 10857672 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -509845\beta_{7} + 138915\beta_{5} - 2755893\beta_{4} - 14260554\beta_{3} + 34254668\beta_{2} - 2984456799 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 36660344 \beta_{7} - 133246512 \beta_{6} + 66623256 \beta_{5} - 50552683 \beta_{4} + 1075524286 \beta_{3} + \cdots + 8061413976 ) / 54 \) Copy content Toggle raw display

Character values

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

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

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
13.4379i
15.0676i
22.3446i
20.7149i
13.4379i
15.0676i
22.3446i
20.7149i
4.00000 6.92820i −46.0138 + 8.35049i −32.0000 55.4256i 103.252 + 178.837i −126.201 + 352.195i −808.604 + 1400.54i −512.000 2047.54 768.476i 1652.03
7.2 4.00000 6.92820i −11.3727 + 45.3615i −32.0000 55.4256i −247.107 428.002i 268.783 + 260.238i 648.241 1122.79i −512.000 −1928.33 1031.76i −3953.72
7.3 4.00000 6.92820i 14.5665 44.4389i −32.0000 55.4256i −47.1766 81.7123i −249.616 278.675i −101.366 + 175.572i −512.000 −1762.64 1294.64i −754.826
7.4 4.00000 6.92820i 36.8200 + 28.8321i −32.0000 55.4256i 218.032 + 377.643i 347.034 139.768i 239.730 415.224i −512.000 524.422 + 2123.19i 3488.51
13.1 4.00000 + 6.92820i −46.0138 8.35049i −32.0000 + 55.4256i 103.252 178.837i −126.201 352.195i −808.604 1400.54i −512.000 2047.54 + 768.476i 1652.03
13.2 4.00000 + 6.92820i −11.3727 45.3615i −32.0000 + 55.4256i −247.107 + 428.002i 268.783 260.238i 648.241 + 1122.79i −512.000 −1928.33 + 1031.76i −3953.72
13.3 4.00000 + 6.92820i 14.5665 + 44.4389i −32.0000 + 55.4256i −47.1766 + 81.7123i −249.616 + 278.675i −101.366 175.572i −512.000 −1762.64 + 1294.64i −754.826
13.4 4.00000 + 6.92820i 36.8200 28.8321i −32.0000 + 55.4256i 218.032 377.643i 347.034 + 139.768i 239.730 + 415.224i −512.000 524.422 2123.19i 3488.51
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
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.b 8
3.b odd 2 1 54.8.c.b 8
4.b odd 2 1 144.8.i.b 8
9.c even 3 1 inner 18.8.c.b 8
9.c even 3 1 162.8.a.h 4
9.d odd 6 1 54.8.c.b 8
9.d odd 6 1 162.8.a.i 4
12.b even 2 1 432.8.i.b 8
36.f odd 6 1 144.8.i.b 8
36.h even 6 1 432.8.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.8.c.b 8 1.a even 1 1 trivial
18.8.c.b 8 9.c even 3 1 inner
54.8.c.b 8 3.b odd 2 1
54.8.c.b 8 9.d odd 6 1
144.8.i.b 8 4.b odd 2 1
144.8.i.b 8 36.f odd 6 1
162.8.a.h 4 9.c even 3 1
162.8.a.i 4 9.d odd 6 1
432.8.i.b 8 12.b even 2 1
432.8.i.b 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 54 T_{5}^{7} + 244431 T_{5}^{6} - 33030990 T_{5}^{5} + 55374420825 T_{5}^{4} + \cdots + 17\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 22876792454961 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots - 21\!\cdots\!28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 57\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 14\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 32\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 73\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 72\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 25\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
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