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Properties

Label 693.2.j.f
Level $693$
Weight $2$
Character orbit 693.j
Analytic conductor $5.534$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [693,2,Mod(232,693)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("693.232"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(693, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-1] 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 11 x^{16} - 12 x^{15} + 83 x^{14} - 88 x^{13} + 337 x^{12} - 336 x^{11} + 966 x^{10} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{13} q^{3} + ( - \beta_{11} + \beta_{9} + \cdots - \beta_1) q^{4} + (\beta_{8} - \beta_{7} - \beta_{4} + \cdots - \beta_1) q^{5} + (\beta_{17} - \beta_{16} + \beta_{11} + \cdots + 1) q^{6}+ \cdots + (\beta_{16} - \beta_{15} + \beta_{14} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} + q^{3} - 3 q^{4} - 6 q^{5} + 2 q^{6} - 9 q^{7} - 12 q^{8} - 5 q^{9} - 26 q^{10} - 9 q^{11} - 5 q^{12} + 8 q^{13} - q^{14} + 16 q^{15} + 5 q^{16} - 28 q^{17} - 16 q^{18} - 24 q^{19} - 19 q^{20}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 11 x^{16} - 12 x^{15} + 83 x^{14} - 88 x^{13} + 337 x^{12} - 336 x^{11} + 966 x^{10} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 67230361684167 \nu^{17} + \cdots + 85\!\cdots\!36 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 74007602581948 \nu^{17} - 920317722200969 \nu^{16} + \cdots - 54\!\cdots\!56 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 145071587986955 \nu^{17} - 814627707675891 \nu^{16} + \cdots - 30\!\cdots\!98 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 126966058373428 \nu^{17} + 735762095983319 \nu^{16} + \cdots + 24\!\cdots\!72 ) / 13\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 548078964059175 \nu^{17} - 672045980755002 \nu^{16} + \cdots - 27\!\cdots\!24 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 595548517106988 \nu^{17} + 219079190568903 \nu^{16} + \cdots + 33\!\cdots\!02 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 846310119619021 \nu^{17} + 867493501935613 \nu^{16} + \cdots - 296030410327792 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 902967963605217 \nu^{17} - 491605673477205 \nu^{16} + \cdots - 37\!\cdots\!48 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 488078246213362 \nu^{17} - 482156850072706 \nu^{16} + \cdots + 29\!\cdots\!54 ) / 13\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 546343411958140 \nu^{17} + \cdots + 54\!\cdots\!02 ) / 13\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!89 \nu^{17} + \cdots + 88\!\cdots\!22 ) / 13\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26\!\cdots\!29 \nu^{17} + \cdots - 10\!\cdots\!94 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!24 \nu^{17} + \cdots - 15\!\cdots\!38 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 27\!\cdots\!42 \nu^{17} + \cdots - 30\!\cdots\!18 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 48\!\cdots\!62 \nu^{17} + \cdots + 18\!\cdots\!54 ) / 27\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 70\!\cdots\!97 \nu^{17} + \cdots + 28\!\cdots\!48 ) / 27\!\cdots\!26 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - 3\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - 4\beta_{8} - \beta_{6} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + 5\beta_{7} - 6\beta_{5} + 13\beta_{3} + 6\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} + 6 \beta_{16} + \beta_{15} - \beta_{14} - 6 \beta_{13} - \beta_{12} - 7 \beta_{11} + \cdots - 19 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} - \beta_{15} + \beta_{14} - \beta_{10} - 31 \beta_{9} - 31 \beta_{8} - \beta_{7} + 23 \beta_{4} + \cdots + 60 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 22 \beta_{16} + 22 \beta_{15} + 9 \beta_{14} + 32 \beta_{13} + 32 \beta_{11} - \beta_{7} + 32 \beta_{6} + \cdots - 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11 \beta_{17} + 11 \beta_{16} + \beta_{15} - \beta_{14} - \beta_{13} + 9 \beta_{12} - 117 \beta_{11} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 62 \beta_{17} - 71 \beta_{16} - 167 \beta_{15} + \beta_{14} - 10 \beta_{13} + 61 \beta_{12} + 61 \beta_{11} + \cdots + 214 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 69 \beta_{16} + 69 \beta_{15} - 57 \beta_{14} + 13 \beta_{13} - 71 \beta_{12} + 566 \beta_{11} + \cdots - 1293 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 385 \beta_{17} + 867 \beta_{16} + 441 \beta_{15} - 370 \beta_{14} - 781 \beta_{13} - 355 \beta_{12} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 569 \beta_{17} - 143 \beta_{16} - 553 \beta_{15} + 441 \beta_{14} - 15 \beta_{13} + 128 \beta_{12} + \cdots + 6615 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1932 \beta_{16} + 1932 \beta_{15} + 1976 \beta_{14} + 4349 \beta_{13} - 143 \beta_{12} + 4335 \beta_{11} + \cdots - 5552 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3527 \beta_{17} + 3370 \beta_{16} + 1110 \beta_{15} - 967 \beta_{14} - 667 \beta_{13} + 1593 \beta_{12} + \cdots - 2560 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 12841 \beta_{17} - 14291 \beta_{16} - 23235 \beta_{15} + 1110 \beta_{14} - 2560 \beta_{13} + \cdots + 28952 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 11951 \beta_{16} + 11951 \beta_{15} - 7734 \beta_{14} + 5327 \beta_{13} - 14291 \beta_{12} + \cdots - 144298 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 71215 \beta_{17} + 119971 \beta_{16} + 77839 \beta_{15} - 63548 \beta_{14} - 98013 \beta_{13} + \cdots - 7667 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
232.1
1.10861 − 1.92017i
0.914478 − 1.58392i
0.640562 − 1.10949i
0.370913 − 0.642440i
0.217248 − 0.376285i
−0.151585 + 0.262553i
−0.411609 + 0.712928i
−1.06106 + 1.83781i
−1.12755 + 1.95298i
1.10861 + 1.92017i
0.914478 + 1.58392i
0.640562 + 1.10949i
0.370913 + 0.642440i
0.217248 + 0.376285i
−0.151585 − 0.262553i
−0.411609 − 0.712928i
−1.06106 − 1.83781i
−1.12755 − 1.95298i
−1.10861 + 1.92017i 0.386186 − 1.68845i −1.45802 − 2.52537i 0.197834 + 0.342659i 2.81397 + 2.61337i −0.500000 + 0.866025i 2.03107 −2.70172 − 1.30411i −0.877282
232.2 −0.914478 + 1.58392i −1.66791 − 0.466990i −0.672539 − 1.16487i −0.188691 − 0.326823i 2.26494 − 2.21478i −0.500000 + 0.866025i −1.19782 2.56384 + 1.55779i 0.690216
232.3 −0.640562 + 1.10949i 0.631775 + 1.61272i 0.179361 + 0.310662i −1.63801 − 2.83712i −2.19398 − 0.332101i −0.500000 + 0.866025i −3.02181 −2.20172 + 2.03775i 4.19699
232.4 −0.370913 + 0.642440i 1.41553 − 0.998130i 0.724847 + 1.25547i 1.80759 + 3.13084i 0.116198 + 1.27961i −0.500000 + 0.866025i −2.55907 1.00747 − 2.82577i −2.68183
232.5 −0.217248 + 0.376285i −0.191137 + 1.72147i 0.905606 + 1.56856i 0.601416 + 1.04168i −0.606240 − 0.445909i −0.500000 + 0.866025i −1.65596 −2.92693 − 0.658074i −0.522626
232.6 0.151585 − 0.262553i 1.72728 − 0.128452i 0.954044 + 1.65245i −0.267906 − 0.464026i 0.228105 − 0.472975i −0.500000 + 0.866025i 1.18482 2.96700 − 0.443744i −0.162442
232.7 0.411609 − 0.712928i −1.54939 − 0.774196i 0.661156 + 1.14515i −0.637369 − 1.10396i −1.18969 + 0.785940i −0.500000 + 0.866025i 2.73499 1.80124 + 2.39907i −1.04939
232.8 1.06106 − 1.83781i 0.729277 − 1.57104i −1.25170 − 2.16800i −1.40091 − 2.42645i −2.11346 − 3.00724i −0.500000 + 0.866025i −1.06826 −1.93631 − 2.29144i −5.94581
232.9 1.12755 − 1.95298i −0.981614 + 1.42704i −1.54276 − 2.67213i −1.47395 − 2.55295i 1.68015 + 3.52613i −0.500000 + 0.866025i −2.44794 −1.07287 − 2.80160i −6.64783
463.1 −1.10861 − 1.92017i 0.386186 + 1.68845i −1.45802 + 2.52537i 0.197834 − 0.342659i 2.81397 − 2.61337i −0.500000 − 0.866025i 2.03107 −2.70172 + 1.30411i −0.877282
463.2 −0.914478 − 1.58392i −1.66791 + 0.466990i −0.672539 + 1.16487i −0.188691 + 0.326823i 2.26494 + 2.21478i −0.500000 − 0.866025i −1.19782 2.56384 − 1.55779i 0.690216
463.3 −0.640562 − 1.10949i 0.631775 − 1.61272i 0.179361 − 0.310662i −1.63801 + 2.83712i −2.19398 + 0.332101i −0.500000 − 0.866025i −3.02181 −2.20172 − 2.03775i 4.19699
463.4 −0.370913 − 0.642440i 1.41553 + 0.998130i 0.724847 − 1.25547i 1.80759 − 3.13084i 0.116198 − 1.27961i −0.500000 − 0.866025i −2.55907 1.00747 + 2.82577i −2.68183
463.5 −0.217248 − 0.376285i −0.191137 − 1.72147i 0.905606 − 1.56856i 0.601416 − 1.04168i −0.606240 + 0.445909i −0.500000 − 0.866025i −1.65596 −2.92693 + 0.658074i −0.522626
463.6 0.151585 + 0.262553i 1.72728 + 0.128452i 0.954044 − 1.65245i −0.267906 + 0.464026i 0.228105 + 0.472975i −0.500000 − 0.866025i 1.18482 2.96700 + 0.443744i −0.162442
463.7 0.411609 + 0.712928i −1.54939 + 0.774196i 0.661156 − 1.14515i −0.637369 + 1.10396i −1.18969 − 0.785940i −0.500000 − 0.866025i 2.73499 1.80124 − 2.39907i −1.04939
463.8 1.06106 + 1.83781i 0.729277 + 1.57104i −1.25170 + 2.16800i −1.40091 + 2.42645i −2.11346 + 3.00724i −0.500000 − 0.866025i −1.06826 −1.93631 + 2.29144i −5.94581
463.9 1.12755 + 1.95298i −0.981614 − 1.42704i −1.54276 + 2.67213i −1.47395 + 2.55295i 1.68015 − 3.52613i −0.500000 − 0.866025i −2.44794 −1.07287 + 2.80160i −6.64783
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 232.9
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 693.2.j.f ✓ 18
9.c even 3 1 inner 693.2.j.f ✓ 18
9.c even 3 1 6237.2.a.bb 9
9.d odd 6 1 6237.2.a.ba 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.j.f ✓ 18 1.a even 1 1 trivial
693.2.j.f ✓ 18 9.c even 3 1 inner
6237.2.a.ba 9 9.d odd 6 1
6237.2.a.bb 9 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + T_{2}^{17} + 11 T_{2}^{16} + 12 T_{2}^{15} + 83 T_{2}^{14} + 88 T_{2}^{13} + 337 T_{2}^{12} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{18} - T^{17} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} + 6 T^{17} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{9} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{18} - 8 T^{17} + \cdots + 484 \) Copy content Toggle raw display
$17$ \( (T^{9} + 14 T^{8} + \cdots + 351144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} + 12 T^{8} + \cdots + 49158)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} - 19 T^{17} + \cdots + 16129 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 5920199457316 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 85745309329 \) Copy content Toggle raw display
$37$ \( (T^{9} + 19 T^{8} + \cdots + 98849)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 3636854958916 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 188526026108484 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 132833392963569 \) Copy content Toggle raw display
$53$ \( (T^{9} - 15 T^{8} + \cdots + 174637)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 8186087529 \) Copy content Toggle raw display
$61$ \( T^{18} - 2 T^{17} + \cdots + 50154724 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 41454694315024 \) Copy content Toggle raw display
$71$ \( (T^{9} + 9 T^{8} + \cdots + 2746847)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + 35 T^{8} + \cdots + 72704)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 74085218596 \) Copy content Toggle raw display
$89$ \( (T^{9} - 19 T^{8} + \cdots - 391191027)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 12\!\cdots\!29 \) Copy content Toggle raw display
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