Properties

Label 9.6.c.a
Level $9$
Weight $6$
Character orbit 9.c
Analytic conductor $1.443$
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] = [9,6,Mod(4,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(1.44345437832\)
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: \( x^{8} + 40x^{6} + 568x^{4} + 3363x^{2} + 7056 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} + \beta_{5} - \beta_{3} + 3 \beta_1 - 3) q^{3} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4} - \beta_{3} - \beta_{2} - 13 \beta_1) q^{4} + (\beta_{7} - \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 20 \beta_1) q^{5} + (3 \beta_{7} - \beta_{6} - 4 \beta_{5} - 12 \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 24) q^{6} + ( - 3 \beta_{7} - \beta_{6} - 10 \beta_{5} - 11 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{7}+ \cdots + (3 \beta_{7} - 3 \beta_{6} - 24 \beta_{5} + 12 \beta_{4} - 3 \beta_{2} + 27 \beta_1 - 72) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} + \beta_{5} - \beta_{3} + 3 \beta_1 - 3) q^{3} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4} - \beta_{3} - \beta_{2} - 13 \beta_1) q^{4} + (\beta_{7} - \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 20 \beta_1) q^{5} + (3 \beta_{7} - \beta_{6} - 4 \beta_{5} - 12 \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 24) q^{6} + ( - 3 \beta_{7} - \beta_{6} - 10 \beta_{5} - 11 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{7}+ \cdots + ( - 1143 \beta_{7} - 858 \beta_{6} - 1011 \beta_{5} - 666 \beta_{4} + \cdots - 6381) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} - 12 q^{3} - 49 q^{4} + 78 q^{5} + 171 q^{6} + 28 q^{7} - 750 q^{8} - 414 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{2} - 12 q^{3} - 49 q^{4} + 78 q^{5} + 171 q^{6} + 28 q^{7} - 750 q^{8} - 414 q^{9} + 60 q^{10} + 444 q^{11} + 2724 q^{12} - 182 q^{13} + 1392 q^{14} - 2052 q^{15} - 289 q^{16} - 4356 q^{17} - 8100 q^{18} + 952 q^{19} + 6684 q^{20} + 8670 q^{21} + 1011 q^{22} + 8844 q^{23} + 549 q^{24} - 1654 q^{25} - 24888 q^{26} - 10152 q^{27} - 1604 q^{28} + 12018 q^{29} + 22104 q^{30} + 1132 q^{31} + 8703 q^{32} - 8820 q^{33} + 10125 q^{34} - 16224 q^{35} - 14589 q^{36} - 15176 q^{37} - 11145 q^{38} + 8220 q^{39} - 8736 q^{40} + 1248 q^{41} + 10098 q^{42} - 6092 q^{43} + 49530 q^{44} + 43038 q^{45} + 45960 q^{46} - 60 q^{47} - 57405 q^{48} + 9090 q^{49} - 57057 q^{50} - 9396 q^{51} - 32510 q^{52} + 20952 q^{53} + 8181 q^{54} - 36120 q^{55} - 61170 q^{56} - 104298 q^{57} + 8328 q^{58} + 2076 q^{59} + 88308 q^{60} + 48142 q^{61} + 241764 q^{62} + 133524 q^{63} - 20926 q^{64} - 13146 q^{65} - 100998 q^{66} - 7148 q^{67} - 123129 q^{68} - 125982 q^{69} - 654 q^{70} - 71856 q^{71} + 35451 q^{72} + 122452 q^{73} - 160320 q^{74} - 18732 q^{75} - 49571 q^{76} + 39534 q^{77} + 181422 q^{78} - 59516 q^{79} + 124512 q^{80} + 194562 q^{81} - 233598 q^{82} + 117696 q^{83} + 108354 q^{84} + 28836 q^{85} - 15915 q^{86} - 142956 q^{87} + 104523 q^{88} - 451728 q^{89} - 539892 q^{90} + 111392 q^{91} + 134034 q^{92} + 113898 q^{93} + 169464 q^{94} + 294888 q^{95} + 42768 q^{96} + 33976 q^{97} + 57654 q^{98} + 33696 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 40x^{6} + 568x^{4} + 3363x^{2} + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 40\nu^{5} + 484\nu^{3} + 1683\nu + 84 ) / 168 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} - 4\nu^{6} - 96\nu^{5} - 88\nu^{4} - 984\nu^{3} - 592\nu^{2} - 3273\nu - 1644 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 158\nu^{5} - 42\nu^{4} - 1580\nu^{3} - 798\nu^{2} - 4929\nu - 3108 ) / 28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} + 28\nu^{6} - 276\nu^{5} + 868\nu^{4} - 2592\nu^{3} + 8428\nu^{2} - 7167\nu + 25116 ) / 168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 31\nu^{4} - 301\nu^{2} - 897 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 31\nu^{4} + 310\nu^{2} - 9\nu + 987 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 51\nu^{7} - 28\nu^{6} + 1620\nu^{5} - 1372\nu^{4} + 16368\nu^{3} - 18508\nu^{2} + 51477\nu - 67452 ) / 168 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - 3\beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + 4\beta_{5} - \beta_{4} - 2\beta_{2} - 180 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 18\beta_{6} + 20\beta_{5} + 22\beta_{4} - 3\beta_{3} + 11\beta_{2} - 18\beta _1 + 9 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17\beta_{7} - 57\beta_{6} - 74\beta_{5} + 23\beta_{4} - 6\beta_{3} + 40\beta_{2} + 2088 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -199\beta_{7} - 471\beta_{6} - 632\beta_{5} - 793\beta_{4} + 126\beta_{3} - 272\beta_{2} + 1080\beta _1 - 540 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -113\beta_{7} + 432\beta_{6} + 518\beta_{5} - 206\beta_{4} + 93\beta_{3} - 319\beta_{2} - 13347 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2867 \beta_{7} + 6465 \beta_{6} + 9286 \beta_{5} + 12107 \beta_{4} - 2136 \beta_{3} + 3598 \beta_{2} - 22752 \beta _1 + 11376 ) / 18 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
3.62198i
2.34949i
3.84183i
2.56934i
3.62198i
2.34949i
3.84183i
2.56934i
−3.77467 6.53793i −15.5883 + 0.0716805i −12.4963 + 21.6443i 43.7155 75.7175i 59.3094 + 101.645i 20.6033 + 35.6859i −52.9007 242.990 2.23475i −660.048
4.2 −1.47673 2.55778i 11.3240 10.7129i 11.6385 20.1585i −32.4255 + 56.1625i −44.1239 13.1441i 80.0952 + 138.729i −163.259 13.4657 242.627i 191.535
4.3 1.78978 + 3.09998i 1.34189 + 15.5306i 9.59340 16.6162i 4.05388 7.02152i −45.7429 + 31.9561i −87.7139 151.925i 183.226 −239.399 + 41.6808i 29.0221
4.4 4.96163 + 8.59380i −3.07760 15.2816i −33.2356 + 57.5657i 23.6560 40.9735i 116.057 102.270i 1.01546 + 1.75882i −342.066 −224.057 + 94.0614i 469.490
7.1 −3.77467 + 6.53793i −15.5883 0.0716805i −12.4963 21.6443i 43.7155 + 75.7175i 59.3094 101.645i 20.6033 35.6859i −52.9007 242.990 + 2.23475i −660.048
7.2 −1.47673 + 2.55778i 11.3240 + 10.7129i 11.6385 + 20.1585i −32.4255 56.1625i −44.1239 + 13.1441i 80.0952 138.729i −163.259 13.4657 + 242.627i 191.535
7.3 1.78978 3.09998i 1.34189 15.5306i 9.59340 + 16.6162i 4.05388 + 7.02152i −45.7429 31.9561i −87.7139 + 151.925i 183.226 −239.399 41.6808i 29.0221
7.4 4.96163 8.59380i −3.07760 + 15.2816i −33.2356 57.5657i 23.6560 + 40.9735i 116.057 + 102.270i 1.01546 1.75882i −342.066 −224.057 94.0614i 469.490
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
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 9.6.c.a 8
3.b odd 2 1 27.6.c.a 8
4.b odd 2 1 144.6.i.c 8
9.c even 3 1 inner 9.6.c.a 8
9.c even 3 1 81.6.a.c 4
9.d odd 6 1 27.6.c.a 8
9.d odd 6 1 81.6.a.d 4
12.b even 2 1 432.6.i.c 8
36.f odd 6 1 144.6.i.c 8
36.h even 6 1 432.6.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.6.c.a 8 1.a even 1 1 trivial
9.6.c.a 8 9.c even 3 1 inner
27.6.c.a 8 3.b odd 2 1
27.6.c.a 8 9.d odd 6 1
81.6.a.c 4 9.c even 3 1
81.6.a.d 4 9.d odd 6 1
144.6.i.c 8 4.b odd 2 1
144.6.i.c 8 36.f odd 6 1
432.6.i.c 8 12.b even 2 1
432.6.i.c 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} + 93 T^{6} + \cdots + 627264 \) Copy content Toggle raw display
$3$ \( T^{8} + 12 T^{7} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{8} - 78 T^{7} + \cdots + 4730520600576 \) Copy content Toggle raw display
$7$ \( T^{8} - 28 T^{7} + \cdots + 5530756283536 \) Copy content Toggle raw display
$11$ \( T^{8} - 444 T^{7} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{8} + 182 T^{7} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( (T^{4} + 2178 T^{3} + \cdots - 917747509932)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 476 T^{3} + \cdots + 1740240514672)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 8844 T^{7} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} - 12018 T^{7} + \cdots + 87\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{8} - 1132 T^{7} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + 7588 T^{3} + \cdots - 24412884987584)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 1248 T^{7} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + 6092 T^{7} + \cdots + 72\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{8} + 60 T^{7} + \cdots + 99\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} - 10476 T^{3} + \cdots + 92\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 2076 T^{7} + \cdots + 83\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{8} - 48142 T^{7} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{8} + 7148 T^{7} + \cdots + 53\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{4} + 35928 T^{3} + \cdots - 82\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 61226 T^{3} + \cdots - 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 59516 T^{7} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{8} - 117696 T^{7} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + 225864 T^{3} + \cdots + 30\!\cdots\!60)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 33976 T^{7} + \cdots + 26\!\cdots\!09 \) Copy content Toggle raw display
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