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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,2,Mod(115,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.115");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.e (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: \(2.73088374913\)
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: \( x^{12} - 2x^{10} - 6x^{9} + 10x^{8} + 15x^{6} + 90x^{4} - 162x^{3} - 162x^{2} + 729 \) 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

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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} - 6x^{9} + 10x^{8} + 15x^{6} + 90x^{4} - 162x^{3} - 162x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11 \nu^{11} + 15 \nu^{10} - 679 \nu^{9} + 363 \nu^{8} + 605 \nu^{7} + 2688 \nu^{6} - 4299 \nu^{5} + \cdots + 74601 ) / 16767 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52 \nu^{11} + 42 \nu^{10} - 742 \nu^{9} + 768 \nu^{8} - 376 \nu^{7} + 4173 \nu^{6} - 8187 \nu^{5} + \cdots + 84807 ) / 16767 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{11} - 18 \nu^{10} - 119 \nu^{9} - 132 \nu^{8} + 217 \nu^{7} + 252 \nu^{6} - 444 \nu^{5} + \cdots + 9963 ) / 5589 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 82 \nu^{11} - 321 \nu^{10} - 263 \nu^{9} + 15 \nu^{8} + 1405 \nu^{7} - 888 \nu^{6} - 489 \nu^{5} + \cdots + 66825 ) / 16767 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 100 \nu^{11} - 654 \nu^{10} + 997 \nu^{9} - 1047 \nu^{8} + 3016 \nu^{7} - 8025 \nu^{6} + \cdots - 70227 ) / 16767 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112 \nu^{11} - 186 \nu^{10} - 26 \nu^{9} - 30 \nu^{8} + 1192 \nu^{7} - 915 \nu^{6} + 1392 \nu^{5} + \cdots + 50787 ) / 16767 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7 \nu^{11} - 3 \nu^{10} + 5 \nu^{9} + 21 \nu^{8} - 34 \nu^{7} + 78 \nu^{6} - 33 \nu^{5} + \cdots + 243 ) / 729 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62 \nu^{11} - 66 \nu^{10} - 214 \nu^{9} - 24 \nu^{8} + 305 \nu^{7} + 96 \nu^{6} - 294 \nu^{5} + \cdots + 27216 ) / 5589 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 64 \nu^{11} + 12 \nu^{10} - 74 \nu^{9} - 165 \nu^{8} + \nu^{7} + 39 \nu^{6} - 417 \nu^{5} + \cdots + 8262 ) / 5589 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 194 \nu^{11} + 507 \nu^{10} + 289 \nu^{9} + 15 \nu^{8} - 2597 \nu^{7} + 1803 \nu^{6} + \cdots - 117612 ) / 16767 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{7} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 3\beta_{5} - \beta_{4} - 3\beta_{3} + 2\beta_{2} + 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{11} - \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + 8 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{11} - 13 \beta_{9} - 2 \beta_{8} + 12 \beta_{7} - 2 \beta_{6} + 8 \beta_{5} + 2 \beta_{4} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5 \beta_{11} - 10 \beta_{10} + 12 \beta_{9} - 9 \beta_{8} + 10 \beta_{7} + 5 \beta_{6} - 24 \beta_{5} + \cdots - 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 24 \beta_{11} + 22 \beta_{10} - 51 \beta_{9} + 24 \beta_{8} + 26 \beta_{7} - 26 \beta_{6} + 23 \beta_{5} + \cdots - 61 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 27 \beta_{11} - 41 \beta_{10} - 19 \beta_{9} - 44 \beta_{8} + 8 \beta_{7} - 34 \beta_{6} - 21 \beta_{5} + \cdots + 43 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4 \beta_{11} - 10 \beta_{10} + 40 \beta_{9} - 16 \beta_{8} + 184 \beta_{7} - 26 \beta_{6} - 73 \beta_{5} + \cdots - 284 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
−1.35650 + 1.07698i
0.476591 + 1.66519i
−1.41330 − 1.00129i
1.65241 + 0.519186i
−0.904252 − 1.47727i
1.54506 − 0.782805i
−1.35650 − 1.07698i
0.476591 − 1.66519i
−1.41330 + 1.00129i
1.65241 − 0.519186i
−0.904252 + 1.47727i
1.54506 + 0.782805i
0.500000 − 0.866025i −1.61095 − 0.636275i −0.500000 − 0.866025i 1.22474 + 2.12132i −1.35650 + 1.07698i −1.71112 + 2.96375i −1.00000 2.19031 + 2.05001i 2.44949
115.2 0.500000 − 0.866025i −1.20380 + 1.24534i −0.500000 − 0.866025i −1.22474 − 2.12132i 0.476591 + 1.66519i −0.442633 + 0.766662i −1.00000 −0.101719 − 2.99828i −2.44949
115.3 0.500000 − 0.866025i 0.160489 − 1.72460i −0.500000 − 0.866025i 1.22474 + 2.12132i −1.41330 − 1.00129i 1.59944 − 2.77031i −1.00000 −2.94849 − 0.553557i 2.44949
115.4 0.500000 − 0.866025i 0.376574 + 1.69062i −0.500000 − 0.866025i −1.22474 − 2.12132i 1.65241 + 0.519186i 1.76747 − 3.06135i −1.00000 −2.71638 + 1.27329i −2.44949
115.5 0.500000 − 0.866025i 0.827228 − 1.52174i −0.500000 − 0.866025i −1.22474 − 2.12132i −0.904252 − 1.47727i −2.32484 + 4.02674i −1.00000 −1.63139 − 2.51765i −2.44949
115.6 0.500000 − 0.866025i 1.45046 + 0.946661i −0.500000 − 0.866025i 1.22474 + 2.12132i 1.54506 − 0.782805i −0.888320 + 1.53862i −1.00000 1.20767 + 2.74619i 2.44949
229.1 0.500000 + 0.866025i −1.61095 + 0.636275i −0.500000 + 0.866025i 1.22474 − 2.12132i −1.35650 − 1.07698i −1.71112 − 2.96375i −1.00000 2.19031 − 2.05001i 2.44949
229.2 0.500000 + 0.866025i −1.20380 − 1.24534i −0.500000 + 0.866025i −1.22474 + 2.12132i 0.476591 − 1.66519i −0.442633 − 0.766662i −1.00000 −0.101719 + 2.99828i −2.44949
229.3 0.500000 + 0.866025i 0.160489 + 1.72460i −0.500000 + 0.866025i 1.22474 − 2.12132i −1.41330 + 1.00129i 1.59944 + 2.77031i −1.00000 −2.94849 + 0.553557i 2.44949
229.4 0.500000 + 0.866025i 0.376574 − 1.69062i −0.500000 + 0.866025i −1.22474 + 2.12132i 1.65241 − 0.519186i 1.76747 + 3.06135i −1.00000 −2.71638 − 1.27329i −2.44949
229.5 0.500000 + 0.866025i 0.827228 + 1.52174i −0.500000 + 0.866025i −1.22474 + 2.12132i −0.904252 + 1.47727i −2.32484 − 4.02674i −1.00000 −1.63139 + 2.51765i −2.44949
229.6 0.500000 + 0.866025i 1.45046 − 0.946661i −0.500000 + 0.866025i 1.22474 − 2.12132i 1.54506 + 0.782805i −0.888320 − 1.53862i −1.00000 1.20767 − 2.74619i 2.44949
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.6
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 342.2.e.d ✓ 12
3.b odd 2 1 1026.2.e.c 12
9.c even 3 1 inner 342.2.e.d ✓ 12
9.c even 3 1 3078.2.a.u 6
9.d odd 6 1 1026.2.e.c 12
9.d odd 6 1 3078.2.a.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.e.d ✓ 12 1.a even 1 1 trivial
342.2.e.d ✓ 12 9.c even 3 1 inner
1026.2.e.c 12 3.b odd 2 1
1026.2.e.c 12 9.d odd 6 1
3078.2.a.u 6 9.c even 3 1
3078.2.a.w 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 6T_{5}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{4} + 6 T^{2} + 36)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + \cdots + 80089 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 129600 \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 5625 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} - 38 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots + 1750329 \) Copy content Toggle raw display
$29$ \( T^{12} + 54 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( T^{12} + 8 T^{11} + \cdots + 16064064 \) Copy content Toggle raw display
$37$ \( (T^{6} - 22 T^{5} + \cdots + 55291)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 4 T^{11} + \cdots + 419904 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 120297024 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 51515834841 \) Copy content Toggle raw display
$53$ \( (T^{6} - 4 T^{5} + \cdots + 41211)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 16 T^{11} + \cdots + 97121025 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 3966984256 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 243640881 \) Copy content Toggle raw display
$71$ \( (T^{6} - 4 T^{5} + \cdots + 41256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 4 T^{5} + \cdots - 1497359)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 217385536 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 5769921600 \) Copy content Toggle raw display
$89$ \( (T^{6} + 4 T^{5} + \cdots - 172440)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 313431616 \) Copy content Toggle raw display
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