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Properties

Label 9.7.d.a
Level $9$
Weight $7$
Character orbit 9.d
Analytic conductor $2.070$
Analytic rank $0$
Dimension $10$
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] = [9,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(2.07048675258\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{6} + \beta_{3} - \beta_{2} + \cdots + 2) q^{3}+ \cdots + ( - 6 \beta_{9} + 6 \beta_{8} + \cdots - 174) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{6} + \beta_{3} - \beta_{2} + \cdots + 2) q^{3}+ \cdots + (7770 \beta_{9} + 6696 \beta_{8} + \cdots + 187050) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} + 24 q^{3} + 127 q^{4} - 219 q^{5} + 333 q^{6} - 121 q^{7} - 1980 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} + 24 q^{3} + 127 q^{4} - 219 q^{5} + 333 q^{6} - 121 q^{7} - 1980 q^{9} - 132 q^{10} + 483 q^{11} - 1830 q^{12} - 841 q^{13} + 12174 q^{14} + 7965 q^{15} - 1985 q^{16} - 7884 q^{18} + 6176 q^{19} - 63186 q^{20} - 25845 q^{21} + 3471 q^{22} + 53565 q^{23} + 111519 q^{24} + 8452 q^{25} - 101034 q^{27} - 22660 q^{28} - 80679 q^{29} - 37782 q^{30} - 24601 q^{31} + 218295 q^{32} + 229995 q^{33} + 7425 q^{34} - 274977 q^{36} + 12764 q^{37} - 371877 q^{38} - 112749 q^{39} + 54150 q^{40} + 232251 q^{41} + 270540 q^{42} - 93271 q^{43} + 63801 q^{45} + 112512 q^{46} - 142887 q^{47} - 143283 q^{48} + 86238 q^{49} + 318459 q^{50} + 57078 q^{51} + 186920 q^{52} + 13851 q^{54} - 419982 q^{55} + 342546 q^{56} - 1086 q^{57} - 380658 q^{58} - 995061 q^{59} - 1011402 q^{60} - 59305 q^{61} + 526455 q^{63} + 403066 q^{64} + 1642029 q^{65} + 1610586 q^{66} + 158513 q^{67} - 1693791 q^{68} - 851715 q^{69} - 304788 q^{70} + 1469907 q^{72} + 933896 q^{73} + 595182 q^{74} - 757524 q^{75} + 666641 q^{76} - 2198883 q^{77} - 3481884 q^{78} + 468707 q^{79} + 1774548 q^{81} - 2038470 q^{82} + 3008337 q^{83} + 1543746 q^{84} - 1189944 q^{85} + 1905549 q^{86} - 615591 q^{87} - 349773 q^{88} + 84294 q^{90} - 211778 q^{91} - 973788 q^{92} - 2954553 q^{93} + 809124 q^{94} - 2562954 q^{95} - 1022112 q^{96} + 336029 q^{97} - 432567 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5928919 \nu^{9} + 4480309 \nu^{8} - 11157967 \nu^{7} + 1123354358 \nu^{6} + \cdots + 38990423194624 ) / 25215567395904 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5928919 \nu^{9} + 4480309 \nu^{8} - 11157967 \nu^{7} + 1123354358 \nu^{6} + \cdots + 38990423194624 ) / 12607783697952 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3461507741 \nu^{9} - 4504997485 \nu^{8} + 258824546191 \nu^{7} - 4959213290 \nu^{6} + \cdots + 112526238150656 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 324925231 \nu^{9} - 340790038123 \nu^{8} - 479648728859 \nu^{7} - 24727965197840 \nu^{6} + \cdots - 12\!\cdots\!20 ) / 184914160903296 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21277637089 \nu^{9} - 1722250368305 \nu^{8} - 1807267633093 \nu^{7} - 131725210021426 \nu^{6} + \cdots - 91\!\cdots\!52 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 53802043985 \nu^{9} + 313884513313 \nu^{8} - 3438448123435 \nu^{7} + \cdots + 15\!\cdots\!24 ) / 739656643613184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 694109900449 \nu^{9} + 1669624327615 \nu^{8} + 50589384866123 \nu^{7} + \cdots + 34\!\cdots\!44 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1021374746999 \nu^{9} + 455147885623 \nu^{8} - 71821429448413 \nu^{7} + \cdots + 96\!\cdots\!28 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 274880637151 \nu^{9} + 534542556065 \nu^{8} + 20460229640149 \nu^{7} + \cdots + 47\!\cdots\!04 ) / 493104429075456 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{5} + 89\beta_{3} - \beta_{2} - 89 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{9} + 6\beta_{8} + 27\beta_{7} - 16\beta_{6} - 2\beta_{4} - 150\beta_{2} - 156\beta _1 - 104 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{9} - 6\beta_{7} - 57\beta_{6} - 68\beta_{5} + 3\beta_{4} - 4423\beta_{3} + 128\beta_{2} - 188\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 142 \beta_{9} - 534 \beta_{8} - 1635 \beta_{7} - 935 \beta_{6} - 534 \beta_{5} - 367 \beta_{4} + \cdots + 23867 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -47\beta_{9} - 1410\beta_{8} - 1269\beta_{7} + 1400\beta_{6} - 42\beta_{4} + 2268\beta_{2} + 3678\beta _1 + 84528 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7569 \beta_{9} - 8550 \beta_{7} + 48525 \beta_{6} + 14418 \beta_{5} + 10419 \beta_{4} + 769455 \beta_{3} + \cdots - 7569 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5526 \beta_{9} + 262018 \beta_{8} + 277201 \beta_{7} - 17043 \beta_{6} + 262018 \beta_{5} + \cdots - 15279353 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1854889 \beta_{9} + 3309126 \beta_{8} + 8873793 \beta_{7} - 6031564 \beta_{6} - 493670 \beta_{4} + \cdots - 182871716 ) / 9 \) Copy content Toggle raw display

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

Character values

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

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

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
4.07727 + 7.06203i
1.22025 + 2.11353i
1.07323 + 1.85889i
−2.32209 − 4.02197i
−3.54866 − 6.14646i
4.07727 − 7.06203i
1.22025 − 2.11353i
1.07323 − 1.85889i
−2.32209 + 4.02197i
−3.54866 + 6.14646i
−12.2318 − 7.06203i 9.56099 + 25.2505i 67.7447 + 117.337i −103.839 + 59.9512i 61.3717 − 376.379i −128.891 + 223.245i − 1009.72i −546.175 + 482.840i 1693.51
2.2 −3.66074 − 2.11353i −26.3977 − 5.67123i −23.0660 − 39.9514i −64.9866 + 37.5201i 84.6488 + 76.5532i 181.066 − 313.616i 465.535i 664.674 + 299.415i 317.199
2.3 −3.21969 − 1.85889i 22.3857 − 15.0957i −25.0891 − 43.4555i 136.563 − 78.8448i −100.136 + 6.99111i −256.037 + 443.470i 424.489i 273.237 − 675.857i −586.255
2.4 6.96626 + 4.02197i 0.282380 + 26.9985i 0.352523 + 0.610587i 80.3236 − 46.3749i −106.620 + 189.214i 60.0074 − 103.936i − 509.141i −728.841 + 15.2477i 746.074
2.5 10.6460 + 6.14646i 6.16862 − 26.2859i 43.5579 + 75.4444i −157.562 + 90.9682i 227.236 − 241.924i 83.3541 − 144.373i 284.159i −652.896 − 324.295i −2236.53
5.1 −12.2318 + 7.06203i 9.56099 − 25.2505i 67.7447 − 117.337i −103.839 − 59.9512i 61.3717 + 376.379i −128.891 − 223.245i 1009.72i −546.175 − 482.840i 1693.51
5.2 −3.66074 + 2.11353i −26.3977 + 5.67123i −23.0660 + 39.9514i −64.9866 − 37.5201i 84.6488 − 76.5532i 181.066 + 313.616i − 465.535i 664.674 − 299.415i 317.199
5.3 −3.21969 + 1.85889i 22.3857 + 15.0957i −25.0891 + 43.4555i 136.563 + 78.8448i −100.136 − 6.99111i −256.037 − 443.470i − 424.489i 273.237 + 675.857i −586.255
5.4 6.96626 − 4.02197i 0.282380 − 26.9985i 0.352523 − 0.610587i 80.3236 + 46.3749i −106.620 − 189.214i 60.0074 + 103.936i 509.141i −728.841 − 15.2477i 746.074
5.5 10.6460 − 6.14646i 6.16862 + 26.2859i 43.5579 − 75.4444i −157.562 − 90.9682i 227.236 + 241.924i 83.3541 + 144.373i − 284.159i −652.896 + 324.295i −2236.53
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.5
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 9.7.d.a ✓ 10
3.b odd 2 1 27.7.d.a 10
4.b odd 2 1 144.7.q.a 10
9.c even 3 1 27.7.d.a 10
9.c even 3 1 81.7.b.a 10
9.d odd 6 1 inner 9.7.d.a ✓ 10
9.d odd 6 1 81.7.b.a 10
12.b even 2 1 432.7.q.a 10
36.f odd 6 1 432.7.q.a 10
36.h even 6 1 144.7.q.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.7.d.a ✓ 10 1.a even 1 1 trivial
9.7.d.a ✓ 10 9.d odd 6 1 inner
27.7.d.a 10 3.b odd 2 1
27.7.d.a 10 9.c even 3 1
81.7.b.a 10 9.c even 3 1
81.7.b.a 10 9.d odd 6 1
144.7.q.a 10 4.b odd 2 1
144.7.q.a 10 36.h even 6 1
432.7.q.a 10 12.b even 2 1
432.7.q.a 10 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 481738752 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 205891132094649 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 91\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 96\!\cdots\!27 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 44\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 48\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 73\!\cdots\!43 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 52\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 73\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
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