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Properties

Label 9.11.d.a
Level $9$
Weight $11$
Character orbit 9.d
Analytic conductor $5.718$
Analytic rank $0$
Dimension $18$
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] = [9,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(5.71821527406\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 2219 x^{16} + 4286 x^{15} + 3372866 x^{14} + 7237076 x^{13} + 2694115412 x^{12} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{36} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \cdots + 3) q^{3}+ \cdots + ( - 2 \beta_{17} + \beta_{16} + \cdots - 2688) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \cdots + 3) q^{3}+ \cdots + (17511 \beta_{17} + 64225 \beta_{16} + \cdots - 2142641798) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} + 51 q^{3} + 4095 q^{4} + 4956 q^{5} - 5283 q^{6} - 6120 q^{7} - 3951 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{2} + 51 q^{3} + 4095 q^{4} + 4956 q^{5} - 5283 q^{6} - 6120 q^{7} - 3951 q^{9} - 2052 q^{10} + 969 q^{11} + 1031514 q^{12} + 140274 q^{13} - 2134578 q^{14} + 1174770 q^{15} - 1571841 q^{16} - 10430748 q^{18} + 2771370 q^{19} + 14542734 q^{20} + 7893876 q^{21} - 3475521 q^{22} - 9944382 q^{23} + 9173583 q^{24} + 14726277 q^{25} - 10369242 q^{27} - 22968324 q^{28} - 41383956 q^{29} - 67505382 q^{30} + 28428390 q^{31} + 165205143 q^{32} + 47221821 q^{33} - 66263103 q^{34} + 107351775 q^{36} + 63695196 q^{37} - 621662181 q^{38} - 161939274 q^{39} - 75566250 q^{40} + 452543883 q^{41} + 733154220 q^{42} + 102086037 q^{43} - 772291584 q^{45} + 44251488 q^{46} - 289445970 q^{47} - 411656979 q^{48} + 6735231 q^{49} + 1915252779 q^{50} + 1858869675 q^{51} + 19035576 q^{52} - 4488556437 q^{54} - 289455552 q^{55} - 4272894318 q^{56} - 1034132379 q^{57} - 83900178 q^{58} + 3065773551 q^{59} + 11567968038 q^{60} + 141349086 q^{61} - 4692569886 q^{63} - 28980486 q^{64} - 11094862926 q^{65} - 5862645990 q^{66} + 954074115 q^{67} + 21469701105 q^{68} + 11337524370 q^{69} - 1440686628 q^{70} - 12325971501 q^{72} + 772009902 q^{73} - 30101814978 q^{74} - 12481839489 q^{75} - 40906287 q^{76} + 25001072268 q^{77} + 31785497316 q^{78} + 2079634212 q^{79} - 14104914183 q^{81} - 1423292742 q^{82} - 23396438442 q^{83} - 15587763342 q^{84} - 5975304714 q^{85} + 39111097821 q^{86} + 39360802212 q^{87} + 2974531059 q^{88} - 56325384666 q^{90} + 2180413260 q^{91} - 40885098684 q^{92} + 709626342 q^{93} + 6395997924 q^{94} + 22335455946 q^{95} + 61717992336 q^{96} - 14510723337 q^{97} - 23929366734 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 2219 x^{16} + 4286 x^{15} + 3372866 x^{14} + 7237076 x^{13} + 2694115412 x^{12} + \cdots + 64\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\!\cdots\!65 \nu^{17} + \cdots + 52\!\cdots\!16 ) / 53\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!37 \nu^{17} + \cdots + 71\!\cdots\!80 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!37 \nu^{17} + \cdots - 71\!\cdots\!80 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50\!\cdots\!01 \nu^{17} + \cdots + 11\!\cdots\!56 ) / 97\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 76\!\cdots\!15 \nu^{17} + \cdots + 25\!\cdots\!84 ) / 78\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!29 \nu^{17} + \cdots - 14\!\cdots\!00 ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!77 \nu^{17} + \cdots + 41\!\cdots\!28 ) / 78\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 52\!\cdots\!43 \nu^{17} + \cdots - 18\!\cdots\!48 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\!\cdots\!59 \nu^{17} + \cdots - 10\!\cdots\!88 ) / 87\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!61 \nu^{17} + \cdots + 94\!\cdots\!76 ) / 19\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 77\!\cdots\!45 \nu^{17} + \cdots - 16\!\cdots\!68 ) / 97\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!47 \nu^{17} + \cdots + 10\!\cdots\!32 ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!39 \nu^{17} + \cdots + 28\!\cdots\!08 ) / 87\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!27 \nu^{17} + \cdots + 25\!\cdots\!64 ) / 96\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 46\!\cdots\!69 \nu^{17} + \cdots + 71\!\cdots\!44 ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 17\!\cdots\!75 \nu^{17} + \cdots + 33\!\cdots\!24 ) / 65\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 84\!\cdots\!47 \nu^{17} + \cdots - 14\!\cdots\!40 ) / 19\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} + 4\beta_{2} - 1478\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{17} - \beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \cdots - 8927 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{17} + 50 \beta_{16} - 21 \beta_{15} - 126 \beta_{14} - 67 \beta_{13} + 12 \beta_{12} + \cdots - 3663791 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1609 \beta_{17} + 1689 \beta_{16} - 5191 \beta_{15} + 5508 \beta_{14} + 789 \beta_{13} + \cdots + 40546 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11733 \beta_{17} + 44851 \beta_{16} + 30762 \beta_{15} - 168720 \beta_{14} + 70990 \beta_{13} + \cdots + 3510498061 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2433188 \beta_{17} - 1661638 \beta_{16} + 3565787 \beta_{15} + 1281690 \beta_{14} + 1425149 \beta_{13} + \cdots + 29850933861 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 27274509 \beta_{17} - 157092317 \beta_{16} + 23790249 \beta_{15} + 466176966 \beta_{14} + \cdots + 234050764 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2958325829 \beta_{17} + 28811165 \beta_{16} - 1276353050 \beta_{15} - 1695996176 \beta_{14} + \cdots - 43535758364645 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3307673636 \beta_{17} + 94911826662 \beta_{16} - 91066033443 \beta_{15} - 300993268086 \beta_{14} + \cdots - 38\!\cdots\!17 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2211126449939 \beta_{17} + 10822394681859 \beta_{16} - 10719896840407 \beta_{15} + \cdots + 69821973575596 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 56045972711217 \beta_{17} + 128145728277215 \beta_{16} + 64137121028586 \beta_{15} + \cdots + 43\!\cdots\!97 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 30\!\cdots\!20 \beta_{17} + \cdots + 79\!\cdots\!09 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 70\!\cdots\!69 \beta_{17} + \cdots + 41\!\cdots\!92 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 50\!\cdots\!29 \beta_{17} + \cdots - 10\!\cdots\!01 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 15\!\cdots\!60 \beta_{17} + \cdots - 56\!\cdots\!25 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 63\!\cdots\!31 \beta_{17} + \cdots + 85\!\cdots\!48 ) / 27 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
17.4033 + 30.1435i
13.2461 + 22.9429i
8.49892 + 14.7206i
5.37603 + 9.31155i
−2.54168 − 4.40231i
−2.95332 − 5.11529i
−9.48684 − 16.4317i
−13.3248 − 23.0793i
−15.7177 − 27.2239i
17.4033 − 30.1435i
13.2461 − 22.9429i
8.49892 − 14.7206i
5.37603 − 9.31155i
−2.54168 + 4.40231i
−2.95332 + 5.11529i
−9.48684 + 16.4317i
−13.3248 + 23.0793i
−15.7177 + 27.2239i
−52.2100 − 30.1435i 234.393 − 64.1019i 1305.26 + 2260.78i 3011.61 − 1738.75i −14169.9 − 3718.65i −696.306 + 1206.04i − 95646.4i 50830.9 − 30050.1i −209648.
2.2 −39.7383 − 22.9429i −164.813 + 178.565i 540.755 + 936.616i −2605.51 + 1504.29i 10646.2 − 3314.59i 13406.9 − 23221.4i − 2638.94i −4722.16 − 58859.9i 138051.
2.3 −25.4968 − 14.7206i −111.624 − 215.845i −78.6099 − 136.156i 205.899 − 118.876i −331.312 + 7146.52i −3544.69 + 6139.58i 34776.4i −34129.2 + 48187.0i −6999.68
2.4 −16.1281 − 9.31155i 148.124 + 192.635i −338.590 − 586.455i −748.776 + 432.306i −595.225 − 4486.10i −13727.6 + 23777.0i 31681.3i −15167.5 + 57067.8i 16101.8
2.5 7.62503 + 4.40231i 209.019 − 123.935i −473.239 − 819.674i −620.145 + 358.041i 2139.38 − 24.8403i 9365.12 − 16220.9i − 17349.3i 28329.2 − 51809.7i −6304.84
2.6 8.85995 + 5.11529i −203.520 + 132.773i −459.668 − 796.168i 5050.65 − 2916.00i −2482.35 + 135.303i 1337.45 − 2316.53i − 19881.5i 23791.5 − 54044.0i 59664.7
2.7 28.4605 + 16.4317i −228.812 − 81.8178i 28.0010 + 48.4991i −4600.38 + 2656.03i −5167.70 − 6088.34i −1786.25 + 3093.87i − 31811.7i 45660.7 + 37441.8i −174572.
2.8 39.9745 + 23.0793i 88.3708 + 226.362i 553.305 + 958.352i −794.787 + 458.870i −1691.69 + 11088.2i 4431.20 − 7675.06i 3813.13i −43430.2 + 40007.5i −42361.6
2.9 47.1532 + 27.2239i 54.3618 − 236.841i 970.286 + 1680.58i 3579.43 − 2066.59i 9011.09 − 9687.89i −11845.8 + 20517.5i 49905.4i −53138.6 − 25750.2i 225043.
5.1 −52.2100 + 30.1435i 234.393 + 64.1019i 1305.26 − 2260.78i 3011.61 + 1738.75i −14169.9 + 3718.65i −696.306 − 1206.04i 95646.4i 50830.9 + 30050.1i −209648.
5.2 −39.7383 + 22.9429i −164.813 − 178.565i 540.755 − 936.616i −2605.51 − 1504.29i 10646.2 + 3314.59i 13406.9 + 23221.4i 2638.94i −4722.16 + 58859.9i 138051.
5.3 −25.4968 + 14.7206i −111.624 + 215.845i −78.6099 + 136.156i 205.899 + 118.876i −331.312 − 7146.52i −3544.69 − 6139.58i − 34776.4i −34129.2 − 48187.0i −6999.68
5.4 −16.1281 + 9.31155i 148.124 − 192.635i −338.590 + 586.455i −748.776 − 432.306i −595.225 + 4486.10i −13727.6 − 23777.0i − 31681.3i −15167.5 − 57067.8i 16101.8
5.5 7.62503 − 4.40231i 209.019 + 123.935i −473.239 + 819.674i −620.145 − 358.041i 2139.38 + 24.8403i 9365.12 + 16220.9i 17349.3i 28329.2 + 51809.7i −6304.84
5.6 8.85995 − 5.11529i −203.520 − 132.773i −459.668 + 796.168i 5050.65 + 2916.00i −2482.35 − 135.303i 1337.45 + 2316.53i 19881.5i 23791.5 + 54044.0i 59664.7
5.7 28.4605 − 16.4317i −228.812 + 81.8178i 28.0010 − 48.4991i −4600.38 − 2656.03i −5167.70 + 6088.34i −1786.25 − 3093.87i 31811.7i 45660.7 − 37441.8i −174572.
5.8 39.9745 − 23.0793i 88.3708 − 226.362i 553.305 − 958.352i −794.787 − 458.870i −1691.69 − 11088.2i 4431.20 + 7675.06i − 3813.13i −43430.2 − 40007.5i −42361.6
5.9 47.1532 − 27.2239i 54.3618 + 236.841i 970.286 − 1680.58i 3579.43 + 2066.59i 9011.09 + 9687.89i −11845.8 − 20517.5i − 49905.4i −53138.6 + 25750.2i 225043.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.9
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 9.11.d.a ✓ 18
3.b odd 2 1 27.11.d.a 18
9.c even 3 1 27.11.d.a 18
9.c even 3 1 81.11.b.a 18
9.d odd 6 1 inner 9.11.d.a ✓ 18
9.d odd 6 1 81.11.b.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.11.d.a ✓ 18 1.a even 1 1 trivial
9.11.d.a ✓ 18 9.d odd 6 1 inner
27.11.d.a 18 3.b odd 2 1
27.11.d.a 18 9.c even 3 1
81.11.b.a 18 9.c even 3 1
81.11.b.a 18 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 87\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 74\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 76\!\cdots\!43 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 56\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 35\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 92\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 24\!\cdots\!87 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 28\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 15\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 65\!\cdots\!23 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 58\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 56\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 33\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
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