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Properties

Label 9.8.c.a
Level $9$
Weight $8$
Character orbit 9.c
Analytic conductor $2.811$
Analytic rank $0$
Dimension $12$
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] = [9,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(2.81146522936\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{15} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{2} + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{3} + ( - \beta_{10} - \beta_{6} + \beta_{5} + \cdots - 51) q^{4}+ \cdots + (3 \beta_{11} + 12 \beta_{10} + \cdots + 393) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{2} + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{3} + ( - \beta_{10} - \beta_{6} + \beta_{5} + \cdots - 51) q^{4}+ \cdots + (4125 \beta_{11} - 27738 \beta_{10} + \cdots - 3933651) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{2} + 24 q^{3} - 321 q^{4} - 180 q^{5} - 1233 q^{6} - 84 q^{7} + 5922 q^{8} + 990 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 9 q^{2} + 24 q^{3} - 321 q^{4} - 180 q^{5} - 1233 q^{6} - 84 q^{7} + 5922 q^{8} + 990 q^{9} + 252 q^{10} - 8460 q^{11} + 8052 q^{12} - 1848 q^{13} - 16272 q^{14} - 1188 q^{15} - 12417 q^{16} + 30564 q^{17} + 42876 q^{18} + 24432 q^{19} - 40788 q^{20} - 187224 q^{21} - 35001 q^{22} - 51588 q^{23} + 215469 q^{24} + 4746 q^{25} + 536472 q^{26} + 322272 q^{27} + 75516 q^{28} - 414648 q^{29} - 1112112 q^{30} + 8196 q^{31} - 1048977 q^{32} - 148518 q^{33} - 106623 q^{34} + 2210616 q^{35} + 2501811 q^{36} + 139344 q^{37} - 1952685 q^{38} - 2057316 q^{39} + 305496 q^{40} - 1731582 q^{41} + 538866 q^{42} + 408372 q^{43} + 5169114 q^{44} + 2687580 q^{45} - 1684008 q^{46} - 1631484 q^{47} - 7434525 q^{48} - 179010 q^{49} - 1654461 q^{50} + 2525688 q^{51} + 681594 q^{52} + 2835648 q^{53} + 5816529 q^{54} - 16056 q^{55} - 1784466 q^{56} - 3071850 q^{57} - 948384 q^{58} - 2055636 q^{59} + 371484 q^{60} - 2723196 q^{61} - 1026828 q^{62} - 2238804 q^{63} + 7178178 q^{64} - 1387620 q^{65} + 5754762 q^{66} + 3806556 q^{67} + 2142639 q^{68} - 3002292 q^{69} + 953442 q^{70} + 2408400 q^{71} - 9638325 q^{72} - 10670052 q^{73} + 9846504 q^{74} + 19174632 q^{75} - 6727827 q^{76} + 3478824 q^{77} - 13339962 q^{78} + 6020916 q^{79} - 38072448 q^{80} - 28538730 q^{81} + 9403002 q^{82} + 9605052 q^{83} + 30090090 q^{84} - 1698624 q^{85} + 34278561 q^{86} + 10290708 q^{87} - 16459029 q^{88} - 24630264 q^{89} - 13660596 q^{90} + 13570104 q^{91} + 39143394 q^{92} + 27331212 q^{93} + 12602808 q^{94} + 10422072 q^{95} - 3404376 q^{96} + 9977226 q^{97} - 95833314 q^{98} - 49382676 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + \cdots + 3098815785 ) / 479986560 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1958197 \nu^{11} - 11583479 \nu^{10} + 732599113 \nu^{9} - 3710871544 \nu^{8} + \cdots - 21\!\cdots\!74 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3916394 \nu^{11} - 21540167 \nu^{10} + 1457064271 \nu^{9} - 6395237967 \nu^{8} + \cdots - 87187467749373 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8715547 \nu^{11} + 179001836 \nu^{10} + 3102424856 \nu^{9} + 63919669287 \nu^{8} + \cdots + 43\!\cdots\!31 ) / 390418907964480 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39397984 \nu^{11} - 61492349 \nu^{10} - 12116718047 \nu^{9} - 12538416999 \nu^{8} + \cdots + 18\!\cdots\!03 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19698992 \nu^{11} - 29932779 \nu^{10} - 6062426001 \nu^{9} - 5755955939 \nu^{8} + \cdots + 24\!\cdots\!99 ) / 390418907964480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39397984 \nu^{11} - 1379844477 \nu^{10} + 19323402177 \nu^{9} - 585440672363 \nu^{8} + \cdots - 10\!\cdots\!05 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 40909916 \nu^{11} - 196334331 \nu^{10} - 10253989353 \nu^{9} - 119631669557 \nu^{8} + \cdots - 12\!\cdots\!79 ) / 260279271976320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 207307619 \nu^{11} + 5152671906 \nu^{10} - 103684920984 \nu^{9} + 1740136007971 \nu^{8} + \cdots + 58\!\cdots\!79 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 717004532 \nu^{11} - 3489650237 \nu^{10} + 265837762141 \nu^{9} - 1011651363039 \nu^{8} + \cdots + 75\!\cdots\!23 ) / 780837815928960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1388736754 \nu^{11} - 8080539299 \nu^{10} + 492206445007 \nu^{9} - 2374979635447 \nu^{8} + \cdots - 16\!\cdots\!29 ) / 780837815928960 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{3} - 2\beta_{2} - 178 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{11} - 9 \beta_{10} - 6 \beta_{9} - 3 \beta_{8} - \beta_{7} - 32 \beta_{6} + 9 \beta_{5} + \cdots - 702 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{11} - 18 \beta_{10} - 8 \beta_{9} - 25 \beta_{8} + 31 \beta_{7} + 414 \beta_{6} - 363 \beta_{5} + \cdots + 51168 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 369 \beta_{11} + 1851 \beta_{10} + 839 \beta_{9} + 262 \beta_{8} + 267 \beta_{7} + 5027 \beta_{6} + \cdots + 159606 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1117 \beta_{11} + 5598 \beta_{10} + 1815 \beta_{9} + 4413 \beta_{8} - 5278 \beta_{7} - 50291 \beta_{6} + \cdots - 5293926 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 51457 \beta_{11} - 295056 \beta_{10} - 106151 \beta_{9} - 21340 \beta_{8} - 46445 \beta_{7} + \cdots - 28191453 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 211050 \beta_{11} - 1206390 \beta_{10} - 335266 \beta_{9} - 665999 \beta_{8} + 700323 \beta_{7} + \cdots + 561011751 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6852619 \beta_{11} + 42782841 \beta_{10} + 13099101 \beta_{9} + 1508520 \beta_{8} + 7177271 \beta_{7} + \cdots + 4403976597 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 35853809 \beta_{11} + 223001406 \beta_{10} + 56945803 \beta_{9} + 96246053 \beta_{8} + \cdots - 59944805034 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 905631615 \beta_{11} - 5885979663 \beta_{10} - 1596910777 \beta_{9} - 50624621 \beta_{8} + \cdots - 645741529713 ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 − 11.4952i
0.500000 − 6.17443i
0.500000 − 2.70685i
0.500000 + 1.48508i
0.500000 + 9.08282i
0.500000 + 9.80854i
0.500000 + 11.4952i
0.500000 + 6.17443i
0.500000 + 2.70685i
0.500000 − 1.48508i
0.500000 − 9.08282i
0.500000 − 9.80854i
−10.7051 − 18.5418i 21.6493 − 41.4525i −165.199 + 286.133i 32.6274 − 56.5123i −1000.36 + 42.3372i −118.194 − 204.717i 4333.37 −1249.62 − 1794.83i −1397.12
4.2 −6.09721 − 10.5607i −33.3118 + 32.8226i −10.3519 + 17.9301i −246.026 + 426.130i 549.738 + 151.669i −382.311 − 662.182i −1308.41 32.3523 − 2186.76i 6000.29
4.3 −3.09420 − 5.35931i 36.5784 + 29.1379i 44.8519 − 77.6857i 167.952 − 290.901i 42.9781 − 286.194i 442.025 + 765.610i −1347.24 488.965 + 2131.64i −2078.70
4.4 0.536120 + 0.928588i −22.8679 − 40.7929i 63.4251 − 109.856i 47.9866 − 83.1153i 25.6198 − 43.1048i −189.000 − 327.358i 273.261 −1141.12 + 1865.70i 102.906
4.5 7.11595 + 12.3252i 45.7172 − 9.84574i −37.2735 + 64.5595i −145.304 + 251.673i 446.672 + 493.411i −555.940 − 962.916i 760.739 1993.12 − 900.239i −4135.89
4.6 7.74445 + 13.4138i −35.7652 + 30.1306i −55.9529 + 96.9133i 52.7641 − 91.3900i −681.146 − 246.401i 761.419 + 1318.82i 249.280 371.296 − 2155.25i 1634.51
7.1 −10.7051 + 18.5418i 21.6493 + 41.4525i −165.199 − 286.133i 32.6274 + 56.5123i −1000.36 − 42.3372i −118.194 + 204.717i 4333.37 −1249.62 + 1794.83i −1397.12
7.2 −6.09721 + 10.5607i −33.3118 − 32.8226i −10.3519 − 17.9301i −246.026 − 426.130i 549.738 − 151.669i −382.311 + 662.182i −1308.41 32.3523 + 2186.76i 6000.29
7.3 −3.09420 + 5.35931i 36.5784 − 29.1379i 44.8519 + 77.6857i 167.952 + 290.901i 42.9781 + 286.194i 442.025 − 765.610i −1347.24 488.965 − 2131.64i −2078.70
7.4 0.536120 − 0.928588i −22.8679 + 40.7929i 63.4251 + 109.856i 47.9866 + 83.1153i 25.6198 + 43.1048i −189.000 + 327.358i 273.261 −1141.12 − 1865.70i 102.906
7.5 7.11595 − 12.3252i 45.7172 + 9.84574i −37.2735 − 64.5595i −145.304 − 251.673i 446.672 − 493.411i −555.940 + 962.916i 760.739 1993.12 + 900.239i −4135.89
7.6 7.74445 − 13.4138i −35.7652 − 30.1306i −55.9529 − 96.9133i 52.7641 + 91.3900i −681.146 + 246.401i 761.419 − 1318.82i 249.280 371.296 + 2155.25i 1634.51
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
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 9.8.c.a ✓ 12
3.b odd 2 1 27.8.c.a 12
4.b odd 2 1 144.8.i.c 12
9.c even 3 1 inner 9.8.c.a ✓ 12
9.c even 3 1 81.8.a.e 6
9.d odd 6 1 27.8.c.a 12
9.d odd 6 1 81.8.a.c 6
12.b even 2 1 432.8.i.c 12
36.f odd 6 1 144.8.i.c 12
36.h even 6 1 432.8.i.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.8.c.a ✓ 12 1.a even 1 1 trivial
9.8.c.a ✓ 12 9.c even 3 1 inner
27.8.c.a 12 3.b odd 2 1
27.8.c.a 12 9.d odd 6 1
81.8.a.c 6 9.d odd 6 1
81.8.a.e 6 9.c even 3 1
144.8.i.c 12 4.b odd 2 1
144.8.i.c 12 36.f odd 6 1
432.8.i.c 12 12.b even 2 1
432.8.i.c 12 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 145838444544 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 83\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 27\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 54\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 74\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 21\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 81\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 83\!\cdots\!69 \) Copy content Toggle raw display
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