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Properties

Label 1170.2.bj.c
Level $1170$
Weight $2$
Character orbit 1170.bj
Analytic conductor $9.342$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1170,2,Mod(199,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{2} + \beta_{4} q^{4} + (\beta_{11} - \beta_{9} - \beta_{3} - \beta_1) q^{5} + (\beta_{7} + \beta_{3}) q^{7} + q^{8} - \beta_{11} q^{10} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{11}+ \cdots + ( - 2 \beta_{10} + 2 \beta_{9} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 6 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8} + 4 q^{10} - 6 q^{11} + 8 q^{13} - 4 q^{14} - 6 q^{16} + 18 q^{17} - 6 q^{19} - 2 q^{20} + 6 q^{22} + 6 q^{23} - 10 q^{25} + 2 q^{26} + 2 q^{28} - 14 q^{29}+ \cdots + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 203419 \nu^{11} + 163110633 \nu^{10} - 591783880 \nu^{9} - 97338749 \nu^{8} + \cdots - 81183629852 ) / 63907274600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 99774878 \nu^{11} - 1446867854 \nu^{10} + 1099642740 \nu^{9} + 2834573637 \nu^{8} + \cdots - 399707750949 ) / 415397284900 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 120243408 \nu^{11} - 454030419 \nu^{10} - 2051209685 \nu^{9} + 7036618932 \nu^{8} + \cdots - 618192439839 ) / 415397284900 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 164152294 \nu^{11} + 1358027358 \nu^{10} - 5882189980 \nu^{9} + 1843354751 \nu^{8} + \cdots - 1308690012227 ) / 415397284900 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 480376508 \nu^{11} - 958108569 \nu^{10} - 1722573835 \nu^{9} + 8639610832 \nu^{8} + \cdots - 997979945989 ) / 415397284900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23159569 \nu^{11} + 44665642 \nu^{10} + 118225633 \nu^{9} - 603932751 \nu^{8} + \cdots + 21026586399 ) / 16615891396 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 236542471 \nu^{11} + 346656978 \nu^{10} + 1242595595 \nu^{9} - 6207967149 \nu^{8} + \cdots + 460186493 ) / 166158913960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 632187142 \nu^{11} + 553303131 \nu^{10} + 5382274615 \nu^{9} - 11841029268 \nu^{8} + \cdots + 880210105411 ) / 415397284900 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 320232543 \nu^{11} - 1004615314 \nu^{10} - 1143072025 \nu^{9} + 9975044447 \nu^{8} + \cdots - 144016360059 ) / 166158913960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1027654074 \nu^{11} + 2460612632 \nu^{10} + 3624161555 \nu^{9} - 25210679196 \nu^{8} + \cdots + 27977691092 ) / 415397284900 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{5} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 2\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 10 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{11} + 2 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{11} + 2 \beta_{10} - \beta_{9} - 10 \beta_{8} - 12 \beta_{7} - 34 \beta_{6} - 9 \beta_{5} + \cdots - 82 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 46 \beta_{11} - 40 \beta_{10} + 46 \beta_{9} + 42 \beta_{8} - 90 \beta_{7} - 13 \beta_{6} - 30 \beta_{5} + \cdots - 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 33 \beta_{11} - 67 \beta_{10} + 79 \beta_{9} - 57 \beta_{8} - 234 \beta_{7} - 76 \beta_{6} + 5 \beta_{5} + \cdots - 51 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 93 \beta_{11} - 174 \beta_{10} + 231 \beta_{9} + 181 \beta_{8} - 364 \beta_{7} + 130 \beta_{6} + \cdots + 362 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1124 \beta_{11} + 316 \beta_{10} - 358 \beta_{9} - 776 \beta_{8} + 646 \beta_{7} + 682 \beta_{6} + \cdots + 1700 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 246 \beta_{11} - 81 \beta_{10} - 196 \beta_{9} + 482 \beta_{8} + 630 \beta_{7} + 1314 \beta_{6} + \cdots + 2556 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7310 \beta_{11} + 5070 \beta_{10} - 9673 \beta_{9} - 6494 \beta_{8} + 12576 \beta_{7} + 2066 \beta_{6} + \cdots + 2574 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{4}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−2.39378 − 0.0429626i
1.75374 − 1.62986i
−0.330925 − 1.46916i
−1.44229 − 0.433312i
2.00607 + 1.30680i
1.40719 + 0.536449i
−2.39378 + 0.0429626i
1.75374 + 1.62986i
−0.330925 + 1.46916i
−1.44229 + 0.433312i
2.00607 − 1.30680i
1.40719 − 0.536449i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i −2.10012 − 0.767774i 0 0.823063 + 1.42559i 1.00000 0 1.71497 − 1.43487i
199.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.40066 + 1.74303i 0 −0.763837 − 1.32301i 1.00000 0 −0.809179 − 2.08452i
199.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.571769 − 2.16173i 0 0.603137 + 1.04466i 1.00000 0 2.15800 + 0.585699i
199.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.230377 + 2.22417i 0 −0.432713 − 0.749482i 1.00000 0 −1.81100 − 1.31160i
199.5 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.26873 − 1.84128i 0 2.17283 + 3.76344i 1.00000 0 0.960230 + 2.01940i
199.6 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.03420 − 0.928463i 0 −1.40247 − 2.42916i 1.00000 0 −0.213026 + 2.22590i
829.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.10012 + 0.767774i 0 0.823063 − 1.42559i 1.00000 0 1.71497 + 1.43487i
829.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.40066 − 1.74303i 0 −0.763837 + 1.32301i 1.00000 0 −0.809179 + 2.08452i
829.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.571769 + 2.16173i 0 0.603137 − 1.04466i 1.00000 0 2.15800 − 0.585699i
829.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.230377 − 2.22417i 0 −0.432713 + 0.749482i 1.00000 0 −1.81100 + 1.31160i
829.5 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.26873 + 1.84128i 0 2.17283 − 3.76344i 1.00000 0 0.960230 − 2.01940i
829.6 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.03420 + 0.928463i 0 −1.40247 + 2.42916i 1.00000 0 −0.213026 − 2.22590i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bj.c 12
3.b odd 2 1 390.2.x.b yes 12
5.b even 2 1 1170.2.bj.d 12
13.e even 6 1 1170.2.bj.d 12
15.d odd 2 1 390.2.x.a ✓ 12
15.e even 4 1 1950.2.bc.i 12
15.e even 4 1 1950.2.bc.j 12
39.h odd 6 1 390.2.x.a ✓ 12
65.l even 6 1 inner 1170.2.bj.c 12
195.y odd 6 1 390.2.x.b yes 12
195.bf even 12 1 1950.2.bc.i 12
195.bf even 12 1 1950.2.bc.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a ✓ 12 15.d odd 2 1
390.2.x.a ✓ 12 39.h odd 6 1
390.2.x.b yes 12 3.b odd 2 1
390.2.x.b yes 12 195.y odd 6 1
1170.2.bj.c 12 1.a even 1 1 trivial
1170.2.bj.c 12 65.l even 6 1 inner
1170.2.bj.d 12 5.b even 2 1
1170.2.bj.d 12 13.e even 6 1
1950.2.bc.i 12 15.e even 4 1
1950.2.bc.i 12 195.bf even 12 1
1950.2.bc.j 12 15.e even 4 1
1950.2.bc.j 12 195.bf even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{12} - 2 T_{7}^{11} + 19 T_{7}^{10} + 6 T_{7}^{9} + 205 T_{7}^{8} - 20 T_{7}^{7} + 708 T_{7}^{6} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{17}^{12} - 18 T_{17}^{11} + 95 T_{17}^{10} + 234 T_{17}^{9} - 2695 T_{17}^{8} - 4632 T_{17}^{7} + \cdots + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{12} - 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 18 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 1982464 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 190660864 \) Copy content Toggle raw display
$29$ \( T^{12} + 14 T^{11} + \cdots + 21904 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 177209344 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 227195329 \) Copy content Toggle raw display
$41$ \( T^{12} - 18 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 349241344 \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} + \cdots + 5956)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 2473271824 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 4983230464 \) Copy content Toggle raw display
$61$ \( T^{12} - 10 T^{11} + \cdots + 89718784 \) Copy content Toggle raw display
$67$ \( T^{12} + 4 T^{11} + \cdots + 83759104 \) Copy content Toggle raw display
$71$ \( T^{12} - 12 T^{11} + \cdots + 4194304 \) Copy content Toggle raw display
$73$ \( (T^{6} + 14 T^{5} + \cdots - 230528)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 2 T^{5} + \cdots - 29312)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 36 T^{5} + \cdots - 6912)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1341001056256 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 415519473664 \) Copy content Toggle raw display
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