Properties

Label 18.9.d.a
Level $18$
Weight $9$
Character orbit 18.d
Analytic conductor $7.333$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.5");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(7.33281498110\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 5476 x^{14} - 38192 x^{13} + 11414542 x^{12} - 67991120 x^{11} + \cdots + 19\!\cdots\!29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{22} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 7) q^{3} + 128 \beta_1 q^{4} + ( - \beta_{10} - \beta_{6} + \beta_{5} + \cdots - 73) q^{5}+ \cdots + (2 \beta_{15} - \beta_{13} + \cdots - 2382) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 7) q^{3} + 128 \beta_1 q^{4} + ( - \beta_{10} - \beta_{6} + \beta_{5} + \cdots - 73) q^{5}+ \cdots + (16401 \beta_{15} - 13995 \beta_{14} + \cdots + 1076400) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 126 q^{3} + 1024 q^{4} - 882 q^{5} + 384 q^{6} - 1846 q^{7} - 28662 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 126 q^{3} + 1024 q^{4} - 882 q^{5} + 384 q^{6} - 1846 q^{7} - 28662 q^{9} + 45756 q^{11} + 5376 q^{12} - 3370 q^{13} - 94464 q^{14} + 128754 q^{15} - 131072 q^{16} - 236544 q^{18} + 362180 q^{19} - 112896 q^{20} - 299166 q^{21} - 61824 q^{22} + 1311138 q^{23} - 147456 q^{24} + 963394 q^{25} - 208656 q^{27} - 472576 q^{28} - 2851290 q^{29} - 1253376 q^{30} + 542438 q^{31} + 3875796 q^{33} + 220416 q^{34} - 3655680 q^{36} + 3343328 q^{37} - 1314432 q^{38} - 5896002 q^{39} + 9218592 q^{41} + 14237952 q^{42} + 339512 q^{43} - 32740578 q^{45} + 7417344 q^{46} - 34980606 q^{47} - 1376256 q^{48} - 2364654 q^{49} + 27744768 q^{50} + 50877810 q^{51} + 431360 q^{52} - 5648256 q^{54} - 4584276 q^{55} - 12091392 q^{56} - 34049898 q^{57} - 7852800 q^{58} + 93924216 q^{59} + 18604800 q^{60} - 841954 q^{61} - 14043234 q^{63} - 33554432 q^{64} - 126568134 q^{65} - 35179776 q^{66} + 29946644 q^{67} - 5476608 q^{68} + 70499610 q^{69} - 34359552 q^{70} - 11894784 q^{72} - 7547764 q^{73} + 35124480 q^{74} + 114494910 q^{75} + 23179520 q^{76} + 9309294 q^{77} + 23014656 q^{78} + 33813002 q^{79} - 46018134 q^{81} - 137346048 q^{82} + 114200226 q^{83} - 15040512 q^{84} - 125696772 q^{85} - 171379584 q^{86} - 159599970 q^{87} + 7913472 q^{88} + 129745152 q^{90} + 268578316 q^{91} + 167825664 q^{92} + 120711534 q^{93} - 11832576 q^{94} - 143949240 q^{95} - 25165824 q^{96} - 89415484 q^{97} + 366888330 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 5476 x^{14} - 38192 x^{13} + 11414542 x^{12} - 67991120 x^{11} + \cdots + 19\!\cdots\!29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 34\!\cdots\!96 \nu^{15} + \cdots + 63\!\cdots\!32 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 81\!\cdots\!41 \nu^{15} + \cdots - 11\!\cdots\!72 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38\!\cdots\!49 \nu^{15} + \cdots - 62\!\cdots\!12 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\!\cdots\!81 \nu^{15} + \cdots + 18\!\cdots\!72 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47\!\cdots\!81 \nu^{15} + \cdots + 19\!\cdots\!43 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42\!\cdots\!02 \nu^{15} + \cdots - 27\!\cdots\!59 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!09 \nu^{15} + \cdots + 62\!\cdots\!98 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 54\!\cdots\!84 \nu^{15} + \cdots + 26\!\cdots\!27 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 99\!\cdots\!79 \nu^{15} + \cdots - 36\!\cdots\!53 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20\!\cdots\!20 \nu^{15} + \cdots - 76\!\cdots\!19 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!21 \nu^{15} + \cdots - 56\!\cdots\!48 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!19 \nu^{15} + \cdots + 54\!\cdots\!65 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21\!\cdots\!81 \nu^{15} + \cdots - 33\!\cdots\!92 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!96 \nu^{15} + \cdots - 30\!\cdots\!27 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 64\!\cdots\!87 \nu^{15} + \cdots - 59\!\cdots\!74 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 2 \beta_{15} - 3 \beta_{14} - 8 \beta_{11} + \beta_{7} + 34 \beta_{6} + 10 \beta_{5} + 24 \beta_{4} + \cdots + 206 ) / 432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 20 \beta_{15} - 59 \beta_{14} + 92 \beta_{13} - 38 \beta_{12} - 58 \beta_{11} + 32 \beta_{8} + \cdots - 146924 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3672 \beta_{15} + 5635 \beta_{14} + 1302 \beta_{13} + 262 \beta_{12} + 9944 \beta_{11} + \cdots - 1593186 ) / 432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 22378 \beta_{15} + 34620 \beta_{14} - 71740 \beta_{13} + 28764 \beta_{12} + 47418 \beta_{11} + \cdots + 94798366 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6032470 \beta_{15} - 9352216 \beta_{14} - 1843886 \beta_{13} - 600646 \beta_{12} + \cdots + 2860174232 ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 81622314 \beta_{15} - 86641573 \beta_{14} + 215082324 \beta_{13} - 81614740 \beta_{12} + \cdots - 279461384412 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10020101254 \beta_{15} + 15541382265 \beta_{14} + 1962435236 \beta_{13} + 1432177632 \beta_{12} + \cdots - 4875666715538 ) / 432 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 35633514154 \beta_{15} + 29773148095 \beta_{14} - 82097391760 \beta_{13} + 29468779843 \beta_{12} + \cdots + 106744244912947 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16589997986976 \beta_{15} - 25883898471449 \beta_{14} - 1650338438634 \beta_{13} + \cdots + 83\!\cdots\!86 ) / 432 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 244807683855094 \beta_{15} - 177684355620726 \beta_{14} + 511525743001360 \beta_{13} + \cdots - 66\!\cdots\!96 ) / 216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27\!\cdots\!38 \beta_{15} + \cdots - 14\!\cdots\!12 ) / 432 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 20\!\cdots\!82 \beta_{15} + \cdots + 51\!\cdots\!56 ) / 108 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 44\!\cdots\!86 \beta_{15} + \cdots + 25\!\cdots\!90 ) / 432 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 70\!\cdots\!80 \beta_{15} + \cdots - 16\!\cdots\!04 ) / 216 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 71\!\cdots\!24 \beta_{15} + \cdots - 43\!\cdots\!94 ) / 432 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 14.2466i
0.500000 39.6902i
0.500000 + 35.1177i
0.500000 9.67409i
0.500000 + 40.7320i
0.500000 11.5235i
0.500000 22.4281i
0.500000 6.78035i
0.500000 14.2466i
0.500000 + 39.6902i
0.500000 35.1177i
0.500000 + 9.67409i
0.500000 40.7320i
0.500000 + 11.5235i
0.500000 + 22.4281i
0.500000 + 6.78035i
−9.79796 + 5.65685i −69.4086 + 41.7547i 64.0000 110.851i −719.139 415.195i 443.862 801.745i 1343.41 + 2326.85i 1448.15i 3074.10 5796.26i 9394.80
5.2 −9.79796 + 5.65685i −24.1444 77.3178i 64.0000 110.851i −69.6596 40.2180i 673.941 + 620.976i 370.849 + 642.329i 1448.15i −5395.10 + 3733.58i 910.029
5.3 −9.79796 + 5.65685i 22.8219 + 77.7185i 64.0000 110.851i 683.343 + 394.528i −663.250 632.382i 89.7132 + 155.388i 1448.15i −5519.32 + 3547.36i −8927.15
5.4 −9.79796 + 5.65685i 77.7361 22.7616i 64.0000 110.851i −115.044 66.4206i −632.896 + 662.760i −1060.32 1836.53i 1448.15i 5524.82 3538.80i 1502.93
5.5 9.79796 5.65685i −47.1686 + 65.8492i 64.0000 110.851i −476.096 274.874i −89.6569 + 912.014i −1631.36 2825.60i 1448.15i −2111.24 6212.04i −6219.69
5.6 9.79796 5.65685i 7.38176 80.6629i 64.0000 110.851i 935.250 + 539.967i −383.972 832.090i −2056.09 3561.25i 1448.15i −6452.02 1190.87i 12218.1
5.7 9.79796 5.65685i 36.7596 72.1785i 64.0000 110.851i −944.709 545.428i −48.1341 915.145i 1250.70 + 2166.27i 1448.15i −3858.46 5306.50i −12341.6
5.8 9.79796 5.65685i 59.0222 + 55.4742i 64.0000 110.851i 265.055 + 153.030i 892.106 + 209.654i 770.107 + 1333.86i 1448.15i 406.233 + 6548.41i 3462.67
11.1 −9.79796 5.65685i −69.4086 41.7547i 64.0000 + 110.851i −719.139 + 415.195i 443.862 + 801.745i 1343.41 2326.85i 1448.15i 3074.10 + 5796.26i 9394.80
11.2 −9.79796 5.65685i −24.1444 + 77.3178i 64.0000 + 110.851i −69.6596 + 40.2180i 673.941 620.976i 370.849 642.329i 1448.15i −5395.10 3733.58i 910.029
11.3 −9.79796 5.65685i 22.8219 77.7185i 64.0000 + 110.851i 683.343 394.528i −663.250 + 632.382i 89.7132 155.388i 1448.15i −5519.32 3547.36i −8927.15
11.4 −9.79796 5.65685i 77.7361 + 22.7616i 64.0000 + 110.851i −115.044 + 66.4206i −632.896 662.760i −1060.32 + 1836.53i 1448.15i 5524.82 + 3538.80i 1502.93
11.5 9.79796 + 5.65685i −47.1686 65.8492i 64.0000 + 110.851i −476.096 + 274.874i −89.6569 912.014i −1631.36 + 2825.60i 1448.15i −2111.24 + 6212.04i −6219.69
11.6 9.79796 + 5.65685i 7.38176 + 80.6629i 64.0000 + 110.851i 935.250 539.967i −383.972 + 832.090i −2056.09 + 3561.25i 1448.15i −6452.02 + 1190.87i 12218.1
11.7 9.79796 + 5.65685i 36.7596 + 72.1785i 64.0000 + 110.851i −944.709 + 545.428i −48.1341 + 915.145i 1250.70 2166.27i 1448.15i −3858.46 + 5306.50i −12341.6
11.8 9.79796 + 5.65685i 59.0222 55.4742i 64.0000 + 110.851i 265.055 153.030i 892.106 209.654i 770.107 1333.86i 1448.15i 406.233 6548.41i 3462.67
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
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 18.9.d.a 16
3.b odd 2 1 54.9.d.a 16
4.b odd 2 1 144.9.q.b 16
9.c even 3 1 54.9.d.a 16
9.c even 3 1 162.9.b.c 16
9.d odd 6 1 inner 18.9.d.a 16
9.d odd 6 1 162.9.b.c 16
12.b even 2 1 432.9.q.c 16
36.f odd 6 1 432.9.q.c 16
36.h even 6 1 144.9.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.d.a 16 1.a even 1 1 trivial
18.9.d.a 16 9.d odd 6 1 inner
54.9.d.a 16 3.b odd 2 1
54.9.d.a 16 9.c even 3 1
144.9.q.b 16 4.b odd 2 1
144.9.q.b 16 36.h even 6 1
162.9.b.c 16 9.c even 3 1
162.9.b.c 16 9.d odd 6 1
432.9.q.c 16 12.b even 2 1
432.9.q.c 16 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 128 T^{2} + 16384)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 52\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
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