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Properties

Label 1470.2.n.l
Level $1470$
Weight $2$
Character orbit 1470.n
Analytic conductor $11.738$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1470,2,Mod(79,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.n (of order \(6\), degree \(2\), not 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: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{14} + 227x^{12} - 1394x^{10} + 6177x^{8} - 14768x^{6} + 24768x^{4} - 11264x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{5} q^{3} + (\beta_{3} + 1) q^{4} - \beta_{8} q^{5} - q^{6} + (\beta_{5} - \beta_{2}) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{5} q^{3} + (\beta_{3} + 1) q^{4} - \beta_{8} q^{5} - q^{6} + (\beta_{5} - \beta_{2}) q^{8} - \beta_{3} q^{9} + \beta_{13} q^{10} + (\beta_{14} - \beta_{13} + \cdots - \beta_{5}) q^{11}+ \cdots + ( - \beta_{15} + \beta_{12} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 4 q^{5} - 16 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 4 q^{5} - 16 q^{6} + 8 q^{9} - 8 q^{16} - 24 q^{19} + 8 q^{20} - 8 q^{24} - 4 q^{25} + 32 q^{29} - 4 q^{30} + 32 q^{31} + 16 q^{34} + 16 q^{36} - 48 q^{41} - 4 q^{45} - 8 q^{46} - 8 q^{50} - 8 q^{51} - 8 q^{54} - 40 q^{59} + 24 q^{61} - 16 q^{64} - 28 q^{65} + 16 q^{69} - 80 q^{71} - 16 q^{74} + 4 q^{75} - 48 q^{76} - 16 q^{79} + 4 q^{80} - 8 q^{81} + 56 q^{85} - 8 q^{86} - 88 q^{89} - 24 q^{94} - 24 q^{95} + 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 18x^{14} + 227x^{12} - 1394x^{10} + 6177x^{8} - 14768x^{6} + 24768x^{4} - 11264x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19 \nu^{14} - 326 \nu^{12} + 3705 \nu^{10} - 18886 \nu^{8} + 51859 \nu^{6} - 85120 \nu^{4} + \cdots - 716288 ) / 496640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3301 \nu^{15} - 53930 \nu^{13} + 643695 \nu^{11} - 3281194 \nu^{9} + 12013925 \nu^{7} + \cdots + 46691840 \nu ) / 33771520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 269 \nu^{14} - 4686 \nu^{12} + 58575 \nu^{10} - 344566 \nu^{8} + 1506549 \nu^{6} - 3327060 \nu^{4} + \cdots - 2710528 ) / 2110720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2031 \nu^{15} + 32894 \nu^{13} - 396045 \nu^{11} + 2018814 \nu^{9} - 7668591 \nu^{7} + \cdots - 33238528 \nu ) / 16885760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4893 \nu^{15} + 93154 \nu^{13} - 1194855 \nu^{11} + 7811442 \nu^{9} - 35273581 \nu^{7} + \cdots + 65518592 \nu ) / 33771520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 733 \nu^{14} + 12142 \nu^{12} - 151775 \nu^{10} + 854742 \nu^{8} - 3903653 \nu^{6} + \cdots + 1554176 ) / 4221440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1385 \nu^{15} + 5482 \nu^{14} - 21810 \nu^{13} - 111332 \nu^{12} + 270075 \nu^{11} + \cdots - 87668736 ) / 33771520 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1385 \nu^{15} - 5482 \nu^{14} - 21810 \nu^{13} + 111332 \nu^{12} + 270075 \nu^{11} + \cdots + 87668736 ) / 33771520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3715 \nu^{15} - 69038 \nu^{13} + 878105 \nu^{11} - 5601470 \nu^{9} + 25102547 \nu^{7} + \cdots - 46285824 \nu ) / 16885760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4339 \nu^{15} + 3301 \nu^{14} - 75250 \nu^{13} - 53930 \nu^{12} + 940625 \nu^{11} + \cdots + 29806080 ) / 16885760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4339 \nu^{15} - 3301 \nu^{14} - 75250 \nu^{13} + 53930 \nu^{12} + 940625 \nu^{11} + \cdots - 29806080 ) / 16885760 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6491 \nu^{15} - 761 \nu^{14} - 112738 \nu^{13} + 11858 \nu^{12} + 1409225 \nu^{11} + \cdots - 19785216 ) / 16885760 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11597 \nu^{15} + 6596 \nu^{14} + 203666 \nu^{13} - 113560 \nu^{12} - 2548375 \nu^{11} + \cdots - 14535680 ) / 33771520 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 485 \nu^{15} - 194 \nu^{14} + 8730 \nu^{13} + 3340 \nu^{12} - 110095 \nu^{11} - 41750 \nu^{10} + \cdots + 427520 ) / 993280 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10588 \nu^{15} + 761 \nu^{14} - 186280 \nu^{13} - 11858 \nu^{12} + 2328500 \nu^{11} + \cdots + 19785216 ) / 16885760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} - \beta_{7} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + 4\beta_{6} + \beta_{5} + 8\beta_{3} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{15} + 5\beta_{12} - 9\beta_{11} - 9\beta_{10} + 8\beta_{9} + 13\beta_{5} - 8\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 17 \beta_{15} - 17 \beta_{14} + 17 \beta_{13} + 17 \beta_{12} + 9 \beta_{8} - 9 \beta_{7} + \cdots - 44 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 37 \beta_{14} - 37 \beta_{13} - 81 \beta_{11} - 81 \beta_{10} + 112 \beta_{9} + 81 \beta_{8} + \cdots + 141 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -201\beta_{15} + 201\beta_{12} - 89\beta_{11} + 89\beta_{10} - 201\beta_{5} + 201\beta_{2} - 436\beta _1 - 472 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 325 \beta_{15} - 325 \beta_{14} - 325 \beta_{13} - 325 \beta_{12} + 761 \beta_{8} + \cdots + 1485 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2145 \beta_{14} - 2145 \beta_{13} - 905 \beta_{11} + 905 \beta_{10} - 905 \beta_{8} + 905 \beta_{7} + \cdots - 4344 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3077 \beta_{15} - 3077 \beta_{12} + 7361 \beta_{11} + 7361 \beta_{10} - 12864 \beta_{9} + \cdots + 15277 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 22041 \beta_{15} + 22041 \beta_{14} - 22041 \beta_{13} - 22041 \beta_{12} - 9177 \beta_{8} + \cdots + 42308 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 30053 \beta_{14} + 30053 \beta_{13} + 72361 \beta_{11} + 72361 \beta_{10} - 130472 \beta_{9} + \cdots - 154925 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 222961 \beta_{15} - 222961 \beta_{12} + 92489 \beta_{11} - 92489 \beta_{10} + 222961 \beta_{5} + \cdots + 409656 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 297317 \beta_{15} + 297317 \beta_{14} + 297317 \beta_{13} + 297317 \beta_{12} - 717233 \beta_{8} + \cdots - 1559117 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 2239977 \beta_{14} + 2239977 \beta_{13} + 928217 \beta_{11} - 928217 \beta_{10} + 928217 \beta_{8} + \cdots + 4058200 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2957317 \beta_{15} + 2957317 \beta_{12} - 7138009 \beta_{11} - 7138009 \beta_{10} + \cdots - 15630093 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.599547 + 0.346149i
−2.73710 − 1.58027i
−1.46557 − 0.846149i
1.87108 + 1.08027i
−1.87108 − 1.08027i
−0.599547 − 0.346149i
1.46557 + 0.846149i
2.73710 + 1.58027i
0.599547 − 0.346149i
−2.73710 + 1.58027i
−1.46557 + 0.846149i
1.87108 − 1.08027i
−1.87108 + 1.08027i
−0.599547 + 0.346149i
1.46557 − 0.846149i
2.73710 − 1.58027i
−0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −1.98078 + 1.03755i −1.00000 0 − 1.00000i 0.500000 − 0.866025i 2.23418 + 0.0918501i
79.2 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −0.506647 − 2.17791i −1.00000 0 − 1.00000i 0.500000 − 0.866025i −0.650187 + 2.13945i
79.3 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.51486 − 1.64475i −1.00000 0 − 1.00000i 0.500000 − 0.866025i −2.13428 + 0.666969i
79.4 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.97257 + 1.05307i −1.00000 0 − 1.00000i 0.500000 − 0.866025i −1.18176 − 1.89827i
79.5 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.89827 − 1.18176i −1.00000 0 1.00000i 0.500000 − 0.866025i −1.05307 − 1.97257i
79.6 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.0918501 + 2.23418i −1.00000 0 1.00000i 0.500000 − 0.866025i −1.03755 + 1.98078i
79.7 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.666969 − 2.13428i −1.00000 0 1.00000i 0.500000 − 0.866025i 1.64475 − 1.51486i
79.8 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 2.13945 − 0.650187i −1.00000 0 1.00000i 0.500000 − 0.866025i 2.17791 + 0.506647i
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i −1.98078 − 1.03755i −1.00000 0 1.00000i 0.500000 + 0.866025i 2.23418 − 0.0918501i
949.2 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i −0.506647 + 2.17791i −1.00000 0 1.00000i 0.500000 + 0.866025i −0.650187 − 2.13945i
949.3 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i 1.51486 + 1.64475i −1.00000 0 1.00000i 0.500000 + 0.866025i −2.13428 − 0.666969i
949.4 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i 1.97257 − 1.05307i −1.00000 0 1.00000i 0.500000 + 0.866025i −1.18176 + 1.89827i
949.5 0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i −1.89827 + 1.18176i −1.00000 0 − 1.00000i 0.500000 + 0.866025i −1.05307 + 1.97257i
949.6 0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 0.0918501 − 2.23418i −1.00000 0 − 1.00000i 0.500000 + 0.866025i −1.03755 − 1.98078i
949.7 0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 0.666969 + 2.13428i −1.00000 0 − 1.00000i 0.500000 + 0.866025i 1.64475 + 1.51486i
949.8 0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 2.13945 + 0.650187i −1.00000 0 − 1.00000i 0.500000 + 0.866025i 2.17791 − 0.506647i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.l 16
5.b even 2 1 inner 1470.2.n.l 16
7.b odd 2 1 1470.2.n.k 16
7.c even 3 1 1470.2.g.j ✓ 8
7.c even 3 1 inner 1470.2.n.l 16
7.d odd 6 1 1470.2.g.k yes 8
7.d odd 6 1 1470.2.n.k 16
35.c odd 2 1 1470.2.n.k 16
35.i odd 6 1 1470.2.g.k yes 8
35.i odd 6 1 1470.2.n.k 16
35.j even 6 1 1470.2.g.j ✓ 8
35.j even 6 1 inner 1470.2.n.l 16
35.k even 12 1 7350.2.a.dr 4
35.k even 12 1 7350.2.a.du 4
35.l odd 12 1 7350.2.a.ds 4
35.l odd 12 1 7350.2.a.dt 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j ✓ 8 7.c even 3 1
1470.2.g.j ✓ 8 35.j even 6 1
1470.2.g.k yes 8 7.d odd 6 1
1470.2.g.k yes 8 35.i odd 6 1
1470.2.n.k 16 7.b odd 2 1
1470.2.n.k 16 7.d odd 6 1
1470.2.n.k 16 35.c odd 2 1
1470.2.n.k 16 35.i odd 6 1
1470.2.n.l 16 1.a even 1 1 trivial
1470.2.n.l 16 5.b even 2 1 inner
1470.2.n.l 16 7.c even 3 1 inner
1470.2.n.l 16 35.j even 6 1 inner
7350.2.a.dr 4 35.k even 12 1
7350.2.a.ds 4 35.l odd 12 1
7350.2.a.dt 4 35.l odd 12 1
7350.2.a.du 4 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{8} + 18T_{11}^{6} - 32T_{11}^{5} + 292T_{11}^{4} - 288T_{11}^{3} + 832T_{11}^{2} + 512T_{11} + 1024 \) Copy content Toggle raw display
\( T_{17}^{16} - 80 T_{17}^{14} + 5064 T_{17}^{12} - 90880 T_{17}^{10} + 1129520 T_{17}^{8} + \cdots + 236421376 \) Copy content Toggle raw display
\( T_{19}^{8} + 12 T_{19}^{7} + 132 T_{19}^{6} + 592 T_{19}^{5} + 3280 T_{19}^{4} + 8064 T_{19}^{3} + \cdots + 200704 \) Copy content Toggle raw display
\( T_{31}^{8} - 16T_{31}^{7} + 178T_{31}^{6} - 992T_{31}^{5} + 3972T_{31}^{4} - 7936T_{31}^{3} + 11392T_{31}^{2} - 8192T_{31} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 18 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 52 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 236421376 \) Copy content Toggle raw display
$19$ \( (T^{8} + 12 T^{7} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 268435456 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 1568)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 16 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1600000000 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 376)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 236 T^{6} + \cdots + 541696)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1099511627776 \) Copy content Toggle raw display
$53$ \( T^{16} - 184 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$59$ \( (T^{8} + 20 T^{7} + \cdots + 9048064)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 12 T^{7} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 44 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + \cdots + 2944)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 3782742016 \) Copy content Toggle raw display
$79$ \( (T^{8} + 8 T^{7} + \cdots + 12845056)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 240 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 44 T^{7} + \cdots + 25482304)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 428 T^{6} + \cdots + 107661376)^{2} \) Copy content Toggle raw display
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