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Properties

Label 342.2.bf.a
Level $342$
Weight $2$
Character orbit 342.bf
Analytic conductor $2.731$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,2,Mod(155,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([3, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.bf (of order \(18\), degree \(6\), 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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.1952986685049.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3 \nu^{11} - 15 \nu^{10} + 66 \nu^{9} - 175 \nu^{8} + 387 \nu^{7} - 619 \nu^{6} + 804 \nu^{5} + \cdots - 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3 \nu^{11} - 18 \nu^{10} + 81 \nu^{9} - 239 \nu^{8} + 553 \nu^{7} - 970 \nu^{6} + 1339 \nu^{5} + \cdots - 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 7 \nu^{11} - 38 \nu^{10} + 167 \nu^{9} - 463 \nu^{8} + 1031 \nu^{7} - 1704 \nu^{6} + 2234 \nu^{5} + \cdots - 38 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( - 7 \nu^{11} + 39 \nu^{10} - 172 \nu^{9} + 485 \nu^{8} - 1089 \nu^{7} + 1831 \nu^{6} - 2433 \nu^{5} + \cdots + 46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( - 8 \nu^{11} + 46 \nu^{10} - 204 \nu^{9} + 586 \nu^{8} - 1328 \nu^{7} + 2269 \nu^{6} - 3048 \nu^{5} + \cdots + 62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 16 \nu^{11} - 87 \nu^{10} + 383 \nu^{9} - 1064 \nu^{8} + 2375 \nu^{7} - 3936 \nu^{6} + 5176 \nu^{5} + \cdots - 85 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + \cdots + 110 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( - 24 \nu^{11} + 131 \nu^{10} - 577 \nu^{9} + 1608 \nu^{8} - 3595 \nu^{7} + 5982 \nu^{6} - 7891 \nu^{5} + \cdots + 133 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2443 \nu^{8} - 5472 \nu^{7} + 9134 \nu^{6} - 12076 \nu^{5} + \cdots + 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + \cdots + 217 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 42 \nu^{11} - 231 \nu^{10} + 1019 \nu^{9} - 2853 \nu^{8} + 6396 \nu^{7} - 10689 \nu^{6} + 14157 \nu^{5} + \cdots - 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{5} - 2\beta_{3} + \beta_{2} - \beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \cdots - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11 \beta_{11} + 17 \beta_{10} + 5 \beta_{9} - 7 \beta_{8} - 5 \beta_{7} - 9 \beta_{5} + 2 \beta_{4} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 19 \beta_{11} - \beta_{10} - 31 \beta_{9} + 15 \beta_{8} - 21 \beta_{7} - 24 \beta_{6} + 5 \beta_{5} + \cdots + 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 85 \beta_{11} - 97 \beta_{10} - 55 \beta_{9} + 55 \beta_{8} + 11 \beta_{7} - 18 \beta_{6} + 46 \beta_{5} + \cdots - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 20 \beta_{11} - 118 \beta_{10} + 134 \beta_{9} - 4 \beta_{8} + 142 \beta_{7} + 135 \beta_{6} + \cdots - 120 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 503 \beta_{11} + 386 \beta_{10} + 440 \beta_{9} - 279 \beta_{8} + 99 \beta_{7} + 228 \beta_{6} + \cdots - 73 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 425 \beta_{11} + 1076 \beta_{10} - 319 \beta_{9} - 305 \beta_{8} - 679 \beta_{7} - 564 \beta_{6} + \cdots + 472 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2299 \beta_{11} - 862 \beta_{10} - 2725 \beta_{9} + 1061 \beta_{8} - 1277 \beta_{7} - 1824 \beta_{6} + \cdots + 825 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4708 \beta_{11} - 6628 \beta_{10} - 985 \beta_{9} + 2697 \beta_{8} + 2277 \beta_{7} + 1329 \beta_{6} + \cdots - 1414 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1 - \beta_{11}\) \(\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
0.500000 − 0.258654i
0.500000 + 2.22827i
0.500000 + 0.0126039i
0.500000 + 1.27297i
0.500000 + 1.00210i
0.500000 − 1.68614i
0.500000 + 0.258654i
0.500000 − 2.22827i
0.500000 − 1.00210i
0.500000 + 1.68614i
0.500000 − 0.0126039i
0.500000 − 1.27297i
−0.939693 + 0.342020i −0.986166 − 1.42389i 0.766044 − 0.642788i 0.312518 + 0.858637i 1.41369 + 1.00073i 3.82739 −0.500000 + 0.866025i −1.05495 + 2.80839i −0.587342 − 0.699967i
155.2 −0.939693 + 0.342020i 0.159815 + 1.72466i 0.766044 − 0.642788i −0.833463 − 2.28992i −0.740046 − 1.56599i −0.480091 −0.500000 + 0.866025i −2.94892 + 0.551252i 1.56640 + 1.86676i
167.1 0.173648 + 0.984808i −1.68842 − 0.386327i −0.939693 + 0.342020i 0.422372 − 0.0744757i 0.0872671 − 1.72985i 1.17453 −0.500000 − 0.866025i 2.70150 + 1.30456i 0.146688 + 0.403023i
167.2 0.173648 + 0.984808i 1.45446 − 0.940501i −0.939693 + 0.342020i −2.72051 + 0.479699i 1.17878 + 1.26905i 3.35755 −0.500000 − 0.866025i 1.23092 − 2.73584i −0.944822 − 2.59588i
173.1 0.766044 + 0.642788i −1.72962 + 0.0916693i 0.173648 + 0.984808i 2.16932 − 2.58529i −1.38389 − 1.04156i −1.76778 −0.500000 + 0.866025i 2.98319 − 0.317107i 3.32358 − 0.586038i
173.2 0.766044 + 0.642788i −0.210069 − 1.71926i 0.173648 + 0.984808i 0.649762 − 0.774356i 0.944200 − 1.45206i 2.88840 −0.500000 + 0.866025i −2.91174 + 0.722330i 0.995493 − 0.175532i
203.1 −0.939693 − 0.342020i −0.986166 + 1.42389i 0.766044 + 0.642788i 0.312518 − 0.858637i 1.41369 − 1.00073i 3.82739 −0.500000 − 0.866025i −1.05495 − 2.80839i −0.587342 + 0.699967i
203.2 −0.939693 − 0.342020i 0.159815 − 1.72466i 0.766044 + 0.642788i −0.833463 + 2.28992i −0.740046 + 1.56599i −0.480091 −0.500000 − 0.866025i −2.94892 − 0.551252i 1.56640 − 1.86676i
257.1 0.766044 − 0.642788i −1.72962 − 0.0916693i 0.173648 − 0.984808i 2.16932 + 2.58529i −1.38389 + 1.04156i −1.76778 −0.500000 − 0.866025i 2.98319 + 0.317107i 3.32358 + 0.586038i
257.2 0.766044 − 0.642788i −0.210069 + 1.71926i 0.173648 − 0.984808i 0.649762 + 0.774356i 0.944200 + 1.45206i 2.88840 −0.500000 − 0.866025i −2.91174 − 0.722330i 0.995493 + 0.175532i
299.1 0.173648 − 0.984808i −1.68842 + 0.386327i −0.939693 − 0.342020i 0.422372 + 0.0744757i 0.0872671 + 1.72985i 1.17453 −0.500000 + 0.866025i 2.70150 − 1.30456i 0.146688 − 0.403023i
299.2 0.173648 − 0.984808i 1.45446 + 0.940501i −0.939693 − 0.342020i −2.72051 − 0.479699i 1.17878 − 1.26905i 3.35755 −0.500000 + 0.866025i 1.23092 + 2.73584i −0.944822 + 2.59588i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.bd even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.bf.a yes 12
9.d odd 6 1 342.2.x.a ✓ 12
19.f odd 18 1 342.2.x.a ✓ 12
171.bd even 18 1 inner 342.2.bf.a yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.x.a ✓ 12 9.d odd 6 1
342.2.x.a ✓ 12 19.f odd 18 1
342.2.bf.a yes 12 1.a even 1 1 trivial
342.2.bf.a yes 12 171.bd even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 27 T_{5}^{9} + 36 T_{5}^{8} + 135 T_{5}^{7} + 9 T_{5}^{6} - 405 T_{5}^{5} + 1215 T_{5}^{4} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 27 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( (T^{6} - 9 T^{5} + 21 T^{4} + \cdots + 37)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 9 T^{11} + \cdots + 29241 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 567009 \) Copy content Toggle raw display
$17$ \( T^{12} + 27 T^{10} + \cdots + 3143529 \) Copy content Toggle raw display
$19$ \( T^{12} - 15 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 18 T^{11} + \cdots + 263169 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 2503100961 \) Copy content Toggle raw display
$31$ \( (T^{6} - 18 T^{5} + \cdots + 867)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 192 T^{10} + \cdots + 47961 \) Copy content Toggle raw display
$41$ \( T^{12} + 36 T^{11} + \cdots + 11758041 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 3825298801 \) Copy content Toggle raw display
$47$ \( T^{12} + 81 T^{10} + \cdots + 22155849 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 8176318929 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 114896961 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 3449565289 \) Copy content Toggle raw display
$67$ \( T^{12} + 33 T^{11} + \cdots + 45369 \) Copy content Toggle raw display
$71$ \( T^{12} - 9 T^{11} + \cdots + 33074001 \) Copy content Toggle raw display
$73$ \( T^{12} + 24 T^{11} + \cdots + 546121 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 110109157929 \) Copy content Toggle raw display
$83$ \( T^{12} + 315 T^{10} + \cdots + 51969681 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 9358821081 \) Copy content Toggle raw display
$97$ \( T^{12} + 18 T^{11} + \cdots + 2595321 \) Copy content Toggle raw display
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