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Properties

Label 693.2.j.e
Level $693$
Weight $2$
Character orbit 693.j
Analytic conductor $5.534$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(232,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + x^{10} + 20 x^{9} - 34 x^{8} - 24 x^{7} + 133 x^{6} - 72 x^{5} - 306 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{4}) q^{2} + ( - \beta_{8} + \beta_{2}) q^{3} + ( - \beta_{10} + \beta_{6} - 2 \beta_{2} - 2) q^{4} + (2 \beta_{2} + 2) q^{5} + (\beta_{10} - \beta_{3} + 2 \beta_{2} + 3) q^{6} + \beta_{2} q^{7} + (\beta_{10} - \beta_{8} + \beta_{6} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{10} + \beta_{3} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{4}) q^{2} + ( - \beta_{8} + \beta_{2}) q^{3} + ( - \beta_{10} + \beta_{6} - 2 \beta_{2} - 2) q^{4} + (2 \beta_{2} + 2) q^{5} + (\beta_{10} - \beta_{3} + 2 \beta_{2} + 3) q^{6} + \beta_{2} q^{7} + (\beta_{10} - \beta_{8} + \beta_{6} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{8} + \beta_{7} + \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} - q^{3} - 7 q^{4} + 12 q^{5} + 15 q^{6} - 6 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} - q^{3} - 7 q^{4} + 12 q^{5} + 15 q^{6} - 6 q^{7} + 6 q^{8} - q^{9} - 4 q^{10} + 6 q^{11} + 2 q^{12} + 5 q^{13} - q^{14} - 10 q^{15} - 9 q^{16} + 12 q^{17} - 36 q^{18} - 4 q^{19} + 14 q^{20} - 4 q^{21} + q^{22} - q^{23} + q^{24} + 6 q^{25} + 10 q^{26} + 8 q^{27} + 14 q^{28} + 12 q^{29} - 6 q^{30} + 4 q^{31} - 17 q^{32} + 4 q^{33} - 24 q^{35} + q^{36} + 28 q^{37} + 10 q^{38} - 10 q^{39} + 6 q^{40} + 13 q^{41} - 18 q^{42} - 27 q^{43} - 14 q^{44} - 28 q^{45} + 68 q^{46} - 4 q^{47} + 55 q^{48} - 6 q^{49} + q^{50} - 42 q^{51} - 15 q^{52} + 8 q^{53} - 17 q^{54} + 24 q^{55} - 3 q^{56} + 14 q^{57} - 32 q^{58} + 21 q^{59} - 22 q^{60} - 26 q^{61} - 22 q^{62} - 13 q^{63} + 2 q^{64} - 10 q^{65} + 18 q^{66} - 22 q^{67} - 20 q^{68} - 33 q^{69} + 2 q^{70} + 30 q^{71} - 49 q^{72} + 26 q^{73} - 11 q^{74} + 4 q^{75} - 22 q^{76} + 6 q^{77} - 39 q^{78} - 8 q^{79} - 36 q^{80} - 29 q^{81} + 10 q^{82} + 50 q^{83} - 13 q^{84} + 12 q^{85} - 17 q^{86} + 34 q^{87} + 3 q^{88} - 26 q^{89} - 30 q^{90} - 10 q^{91} + 30 q^{92} + 7 q^{93} - 17 q^{94} - 4 q^{95} - 2 q^{96} - 5 q^{97} + 2 q^{98} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + x^{10} + 20 x^{9} - 34 x^{8} - 24 x^{7} + 133 x^{6} - 72 x^{5} - 306 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7 \nu^{11} + \nu^{10} + 47 \nu^{9} - 86 \nu^{8} - 59 \nu^{7} + 357 \nu^{6} - 175 \nu^{5} + \cdots + 1458 ) / 972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} + 50 \nu^{9} - 107 \nu^{8} - 5 \nu^{7} + 390 \nu^{6} - 448 \nu^{5} + \cdots + 7047 ) / 972 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 4 \nu^{10} + \nu^{9} + 20 \nu^{8} - 34 \nu^{7} - 24 \nu^{6} + 133 \nu^{5} - 72 \nu^{4} + \cdots - 972 ) / 243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} + 20 \nu^{10} - 14 \nu^{9} - 73 \nu^{8} + 139 \nu^{7} + 94 \nu^{6} - 504 \nu^{5} + \cdots + 2997 ) / 324 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{11} + 18 \nu^{10} - 6 \nu^{9} - 75 \nu^{8} + 103 \nu^{7} + 134 \nu^{6} - 464 \nu^{5} + \cdots + 2187 ) / 324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10 \nu^{11} + 15 \nu^{10} + 45 \nu^{9} - 135 \nu^{8} + 20 \nu^{7} + 397 \nu^{6} - 487 \nu^{5} + \cdots + 3483 ) / 324 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20 \nu^{11} - 53 \nu^{10} - 7 \nu^{9} + 211 \nu^{8} - 248 \nu^{7} - 399 \nu^{6} + 1175 \nu^{5} + \cdots - 3645 ) / 972 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37 \nu^{11} - 73 \nu^{10} - 155 \nu^{9} + 572 \nu^{8} - 163 \nu^{7} - 1575 \nu^{6} + 2257 \nu^{5} + \cdots - 15066 ) / 972 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23 \nu^{11} - 41 \nu^{10} - 73 \nu^{9} + 268 \nu^{8} - 86 \nu^{7} - 747 \nu^{6} + 1052 \nu^{5} + \cdots - 5589 ) / 486 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27 \nu^{11} - 65 \nu^{10} - 79 \nu^{9} + 400 \nu^{8} - 235 \nu^{7} - 1033 \nu^{6} + 1773 \nu^{5} + \cdots - 10530 ) / 324 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{8} + \beta_{7} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} + 2\beta_{10} + \beta_{9} + \beta_{8} + 2\beta_{6} - 4\beta_{5} - 2\beta_{4} - \beta_{3} - 2\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + 6 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{11} + 5 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} - 7 \beta_{7} + \beta_{6} - 6 \beta_{5} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{10} + 9 \beta_{9} - 19 \beta_{8} + 16 \beta_{7} - 21 \beta_{6} + 18 \beta_{5} + 4 \beta_{4} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{11} - 6 \beta_{10} + 21 \beta_{9} - 29 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \cdots - 20 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12 \beta_{11} - 9 \beta_{10} + 15 \beta_{9} - 14 \beta_{8} - 3 \beta_{7} - 36 \beta_{6} - 12 \beta_{5} + \cdots - 75 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 30 \beta_{11} - 36 \beta_{10} + 6 \beta_{9} + 28 \beta_{8} - 46 \beta_{7} - 15 \beta_{6} - 24 \beta_{5} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 50 \beta_{11} - 16 \beta_{10} - 30 \beta_{9} + 108 \beta_{8} - 92 \beta_{7} - 38 \beta_{6} + \cdots - 292 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
232.1
−1.73142 + 0.0468052i
1.26240 − 1.18590i
−1.41915 + 0.992983i
1.70323 − 0.314645i
1.64319 + 0.547660i
0.541751 + 1.64515i
−1.73142 − 0.0468052i
1.26240 + 1.18590i
−1.41915 − 0.992983i
1.70323 + 0.314645i
1.64319 − 0.547660i
0.541751 − 1.64515i
−1.32517 + 2.29527i −0.906244 − 1.47605i −2.51218 − 4.35122i 1.00000 + 1.73205i 4.58886 − 0.124050i −0.500000 + 0.866025i 8.01559 −1.35745 + 2.67532i −5.30070
232.2 −0.895820 + 1.55161i 1.65822 + 0.500319i −0.604985 − 1.04787i 1.00000 + 1.73205i −2.26176 + 2.12470i −0.500000 + 0.866025i −1.41545 2.49936 + 1.65927i −3.58328
232.3 −0.349626 + 0.605570i −1.56952 − 0.732528i 0.755523 + 1.30860i 1.00000 + 1.73205i 0.992343 − 0.694346i −0.500000 + 0.866025i −2.45511 1.92681 + 2.29944i −1.39850
232.4 0.0791249 − 0.137048i 1.12411 + 1.31772i 0.987479 + 1.71036i 1.00000 + 1.73205i 0.269536 − 0.0497926i −0.500000 + 0.866025i 0.629036 −0.472768 + 2.96251i 0.316499
232.5 0.795882 − 1.37851i 0.347307 + 1.69687i −0.266856 − 0.462207i 1.00000 + 1.73205i 2.61557 + 0.871745i −0.500000 + 0.866025i 2.33399 −2.75876 + 1.17867i 3.18353
232.6 1.19561 − 2.07086i −1.15386 + 1.29174i −1.85898 − 3.21986i 1.00000 + 1.73205i 1.29545 + 3.93392i −0.500000 + 0.866025i −4.10806 −0.337200 − 2.98099i 4.78246
463.1 −1.32517 − 2.29527i −0.906244 + 1.47605i −2.51218 + 4.35122i 1.00000 − 1.73205i 4.58886 + 0.124050i −0.500000 − 0.866025i 8.01559 −1.35745 − 2.67532i −5.30070
463.2 −0.895820 − 1.55161i 1.65822 − 0.500319i −0.604985 + 1.04787i 1.00000 − 1.73205i −2.26176 − 2.12470i −0.500000 − 0.866025i −1.41545 2.49936 − 1.65927i −3.58328
463.3 −0.349626 − 0.605570i −1.56952 + 0.732528i 0.755523 − 1.30860i 1.00000 − 1.73205i 0.992343 + 0.694346i −0.500000 − 0.866025i −2.45511 1.92681 − 2.29944i −1.39850
463.4 0.0791249 + 0.137048i 1.12411 − 1.31772i 0.987479 − 1.71036i 1.00000 − 1.73205i 0.269536 + 0.0497926i −0.500000 − 0.866025i 0.629036 −0.472768 − 2.96251i 0.316499
463.5 0.795882 + 1.37851i 0.347307 − 1.69687i −0.266856 + 0.462207i 1.00000 − 1.73205i 2.61557 − 0.871745i −0.500000 − 0.866025i 2.33399 −2.75876 − 1.17867i 3.18353
463.6 1.19561 + 2.07086i −1.15386 − 1.29174i −1.85898 + 3.21986i 1.00000 − 1.73205i 1.29545 − 3.93392i −0.500000 − 0.866025i −4.10806 −0.337200 + 2.98099i 4.78246
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 232.6
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.e ✓ 12
9.c even 3 1 inner 693.2.j.e ✓ 12
9.c even 3 1 6237.2.a.w 6
9.d odd 6 1 6237.2.a.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.j.e ✓ 12 1.a even 1 1 trivial
693.2.j.e ✓ 12 9.c even 3 1 inner
6237.2.a.v 6 9.d odd 6 1
6237.2.a.w 6 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + T_{2}^{11} + 10 T_{2}^{10} + 5 T_{2}^{9} + 70 T_{2}^{8} + 37 T_{2}^{7} + 197 T_{2}^{6} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 4)^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} - 5 T^{11} + \cdots + 1102500 \) Copy content Toggle raw display
$17$ \( (T^{6} - 6 T^{5} + \cdots + 6304)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 2 T^{5} - 62 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + T^{11} + \cdots + 831744 \) Copy content Toggle raw display
$29$ \( T^{12} - 12 T^{11} + \cdots + 13075456 \) Copy content Toggle raw display
$31$ \( T^{12} - 4 T^{11} + \cdots + 37662769 \) Copy content Toggle raw display
$37$ \( (T^{6} - 14 T^{5} + \cdots + 34833)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 13 T^{11} + \cdots + 19802500 \) Copy content Toggle raw display
$43$ \( T^{12} + 27 T^{11} + \cdots + 155236 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 129026881 \) Copy content Toggle raw display
$53$ \( (T^{6} - 4 T^{5} + \cdots + 1767)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1284218896 \) Copy content Toggle raw display
$61$ \( T^{12} + 26 T^{11} + \cdots + 24364096 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 4299162624 \) Copy content Toggle raw display
$71$ \( (T^{6} - 15 T^{5} + \cdots + 76392)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 13 T^{5} + \cdots - 29888)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 171700839424 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 108973932544 \) Copy content Toggle raw display
$89$ \( (T^{6} + 13 T^{5} + \cdots - 54096)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 5 T^{11} + \cdots + 6431296 \) Copy content Toggle raw display
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