Properties

Label 9.5.d.a
Level $9$
Weight $5$
Character orbit 9.d
Analytic conductor $0.930$
Analytic rank $0$
Dimension $6$
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] = [9,5,Mod(2,9)]
 
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([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(0.930329667755\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.39400128.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{4} - 3 \beta_{3} + \beta_1 - 3) q^{3} + (\beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{4} + ( - 3 \beta_{5} - \beta_1 - 3) q^{5} + (\beta_{5} + \beta_{4} + 6 \beta_{3} - 8 \beta_{2} - 7 \beta_1 - 14) q^{6} + (4 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 7 \beta_{2} + 14 \beta_1 + 4) q^{7} + ( - 3 \beta_{5} + 3 \beta_{4} - 54 \beta_{3} + \beta_{2} + \beta_1 - 30) q^{8} + (3 \beta_{4} + 45 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 45) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{4} - 3 \beta_{3} + \beta_1 - 3) q^{3} + (\beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{4} + ( - 3 \beta_{5} - \beta_1 - 3) q^{5} + (\beta_{5} + \beta_{4} + 6 \beta_{3} - 8 \beta_{2} - 7 \beta_1 - 14) q^{6} + (4 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 7 \beta_{2} + 14 \beta_1 + 4) q^{7} + ( - 3 \beta_{5} + 3 \beta_{4} - 54 \beta_{3} + \beta_{2} + \beta_1 - 30) q^{8} + (3 \beta_{4} + 45 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 45) q^{9} + ( - \beta_{5} - \beta_{4} + 8 \beta_{2} - 8 \beta_1 + 2) q^{10} + ( - 3 \beta_{4} + 54 \beta_{3} - 2 \beta_{2} + 108) q^{11} + (\beta_{5} - 7 \beta_{4} - 114 \beta_{3} + \beta_{2} + 21 \beta_1 - 8) q^{12} + ( - 5 \beta_{5} + 10 \beta_{4} - 20 \beta_{3} - 26 \beta_{2} - 13 \beta_1 - 25) q^{13} + (21 \beta_{5} + 135 \beta_{3} + 2 \beta_1 - 114) q^{14} + ( - 10 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 13 \beta_1 - 175) q^{15} + ( - 30 \beta_{5} + 15 \beta_{4} - 41 \beta_{3} - 21 \beta_{2} - 42 \beta_1 - 30) q^{16} + (21 \beta_{5} - 21 \beta_{4} - 162 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 60) q^{17} + (3 \beta_{5} - 24 \beta_{4} + 207 \beta_{3} + 3 \beta_{2} + 33 \beta_1 + 336) q^{18} + (9 \beta_{5} + 9 \beta_{4} - 9 \beta_{2} + 9 \beta_1 - 52) q^{19} + (24 \beta_{4} + 162 \beta_{3} - 28 \beta_{2} + 324) q^{20} + ( - 8 \beta_{5} + 15 \beta_{4} - 342 \beta_{3} + 19 \beta_{2} - 74 \beta_1 - 56) q^{21} + ( - 2 \beta_{5} + 4 \beta_{4} - 60 \beta_{3} + 130 \beta_{2} + 65 \beta_1 - 62) q^{22} + ( - 42 \beta_{5} + 27 \beta_{3} + 29 \beta_1 - 69) q^{23} + (38 \beta_{5} - 13 \beta_{4} + 219 \beta_{3} + 65 \beta_{2} + 16 \beta_1 - 82) q^{24} + (70 \beta_{5} - 35 \beta_{4} + 127 \beta_{3} + \beta_{2} + 2 \beta_1 + 70) q^{25} + ( - 39 \beta_{5} + 39 \beta_{4} - 486 \beta_{3} - 26 \beta_{2} + \cdots - 282) q^{26}+ \cdots + (3 \beta_{5} + 165 \beta_{4} + 3771 \beta_{3} + 1542 \beta_{2} + \cdots + 984) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{3} + 15 q^{4} - 12 q^{5} - 99 q^{6} + 12 q^{7} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{3} + 15 q^{4} - 12 q^{5} - 99 q^{6} + 12 q^{7} + 99 q^{9} - 36 q^{10} + 483 q^{11} + 330 q^{12} - 6 q^{13} - 1146 q^{14} - 1026 q^{15} + 15 q^{16} + 1404 q^{18} - 258 q^{19} + 1614 q^{20} + 480 q^{21} - 369 q^{22} - 282 q^{23} - 1449 q^{24} - 273 q^{25} + 54 q^{27} + 1308 q^{28} - 1056 q^{29} - 1278 q^{30} + 1290 q^{31} - 1161 q^{32} + 279 q^{33} + 513 q^{34} - 2385 q^{36} + 12 q^{37} - 789 q^{38} + 1974 q^{39} - 1314 q^{40} + 7629 q^{41} + 9612 q^{42} - 285 q^{43} - 4212 q^{45} - 5760 q^{46} - 9642 q^{47} - 6771 q^{48} - 1863 q^{49} + 3027 q^{50} + 2457 q^{51} - 240 q^{52} - 405 q^{54} + 2016 q^{55} - 462 q^{56} + 5367 q^{57} + 6462 q^{58} + 6225 q^{59} + 7470 q^{60} + 3630 q^{61} - 7578 q^{63} + 15450 q^{64} - 7158 q^{65} - 13734 q^{66} - 5055 q^{67} - 10503 q^{68} - 13878 q^{69} - 9684 q^{70} + 8451 q^{72} - 14622 q^{73} + 26454 q^{74} + 21021 q^{75} - 4047 q^{76} + 2580 q^{77} - 12060 q^{78} + 4764 q^{79} + 18387 q^{81} - 9702 q^{82} - 1866 q^{83} - 6486 q^{84} + 12366 q^{85} - 37731 q^{86} - 21564 q^{87} + 14787 q^{88} + 20790 q^{90} + 34836 q^{91} + 33636 q^{92} + 19254 q^{93} - 12708 q^{94} - 13362 q^{95} - 3672 q^{96} - 28959 q^{97} - 9126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 11\nu^{4} - 121\nu^{3} + 98\nu^{2} + 1118\nu - 220 ) / 1098 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 11\nu^{4} + 121\nu^{3} - 98\nu^{2} + 529\nu + 220 ) / 549 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55\nu^{5} - 56\nu^{4} + 616\nu^{3} + 649\nu^{2} + 5488\nu + 22 ) / 1098 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 373\nu^{5} - 260\nu^{4} + 3958\nu^{3} + 6817\nu^{2} + 37558\nu + 15082 ) / 1098 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -406\nu^{5} + 623\nu^{4} - 4657\nu^{3} - 3583\nu^{2} - 36898\nu + 6206 ) / 1098 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} - 21\beta_{3} + 2\beta_{2} + \beta _1 - 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} + 11\beta_{2} - 11\beta _1 - 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 22\beta_{5} - 11\beta_{4} + 237\beta_{3} - 23\beta_{2} - 46\beta _1 + 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23\beta_{5} - 46\beta_{4} + 549\beta_{3} - 270\beta_{2} - 135\beta _1 + 572 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
1.89154 3.27625i
−0.102534 + 0.177594i
−1.28901 + 2.23263i
1.89154 + 3.27625i
−0.102534 0.177594i
−1.28901 2.23263i
−5.67463 3.27625i −1.11837 8.93024i 13.4676 + 23.3266i 10.2044 5.89150i −22.9114 + 54.3399i 26.6364 46.1356i 71.6534i −78.4985 + 19.9746i −77.2081
2.2 0.307601 + 0.177594i 8.32172 + 3.42768i −7.93692 13.7472i −30.0804 + 17.3669i 1.95104 + 2.53225i 15.6054 27.0294i 11.3212i 57.5020 + 57.0484i −12.3370
2.3 3.86703 + 2.23263i −8.70335 2.29167i 1.96929 + 3.41090i 13.8760 8.01130i −28.5397 28.2933i −36.2418 + 62.7727i 53.8574i 70.4965 + 39.8904i 71.5451
5.1 −5.67463 + 3.27625i −1.11837 + 8.93024i 13.4676 23.3266i 10.2044 + 5.89150i −22.9114 54.3399i 26.6364 + 46.1356i 71.6534i −78.4985 19.9746i −77.2081
5.2 0.307601 0.177594i 8.32172 3.42768i −7.93692 + 13.7472i −30.0804 17.3669i 1.95104 2.53225i 15.6054 + 27.0294i 11.3212i 57.5020 57.0484i −12.3370
5.3 3.86703 2.23263i −8.70335 + 2.29167i 1.96929 3.41090i 13.8760 + 8.01130i −28.5397 + 28.2933i −36.2418 62.7727i 53.8574i 70.4965 39.8904i 71.5451
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.3
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 9.5.d.a 6
3.b odd 2 1 27.5.d.a 6
4.b odd 2 1 144.5.q.a 6
9.c even 3 1 27.5.d.a 6
9.c even 3 1 81.5.b.a 6
9.d odd 6 1 inner 9.5.d.a 6
9.d odd 6 1 81.5.b.a 6
12.b even 2 1 432.5.q.a 6
36.f odd 6 1 432.5.q.a 6
36.f odd 6 1 1296.5.e.c 6
36.h even 6 1 144.5.q.a 6
36.h even 6 1 1296.5.e.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.5.d.a 6 1.a even 1 1 trivial
9.5.d.a 6 9.d odd 6 1 inner
27.5.d.a 6 3.b odd 2 1
27.5.d.a 6 9.c even 3 1
81.5.b.a 6 9.c even 3 1
81.5.b.a 6 9.d odd 6 1
144.5.q.a 6 4.b odd 2 1
144.5.q.a 6 36.h even 6 1
432.5.q.a 6 12.b even 2 1
432.5.q.a 6 36.f odd 6 1
1296.5.e.c 6 36.f odd 6 1
1296.5.e.c 6 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} - 27 T^{4} - 90 T^{3} + \cdots + 108 \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} - 45 T^{4} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{6} + 12 T^{5} - 729 T^{4} + \cdots + 43001388 \) Copy content Toggle raw display
$7$ \( T^{6} - 12 T^{5} + \cdots + 14524588324 \) Copy content Toggle raw display
$11$ \( T^{6} - 483 T^{5} + \cdots + 481294471563 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + \cdots + 708378089104 \) Copy content Toggle raw display
$17$ \( T^{6} + 155115 T^{4} + \cdots + 47166451632 \) Copy content Toggle raw display
$19$ \( (T^{3} + 129 T^{2} - 18024 T - 1195028)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 282 T^{5} + \cdots + 10049071819968 \) Copy content Toggle raw display
$29$ \( T^{6} + 1056 T^{5} + \cdots + 92\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{6} - 1290 T^{5} + \cdots + 63\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{3} - 6 T^{2} - 2222952 T + 1276743376)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 7629 T^{5} + \cdots + 32\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( T^{6} + 285 T^{5} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{6} + 9642 T^{5} + \cdots + 65\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{6} + 26693064 T^{4} + \cdots + 34\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{6} - 6225 T^{5} + \cdots + 74\!\cdots\!87 \) Copy content Toggle raw display
$61$ \( T^{6} - 3630 T^{5} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{6} + 5055 T^{5} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{6} + 64502244 T^{4} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( (T^{3} + 7311 T^{2} + \cdots - 15741832472)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 4764 T^{5} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{6} + 1866 T^{5} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{6} + 283928328 T^{4} + \cdots + 53\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{6} + 28959 T^{5} + \cdots + 52\!\cdots\!89 \) Copy content Toggle raw display
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