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Properties

Label 342.3.d.b
Level $342$
Weight $3$
Character orbit 342.d
Analytic conductor $9.319$
Analytic rank $0$
Dimension $8$
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] = [342,3,Mod(37,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.d (of order \(2\), degree \(1\), 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: \(9.31882504112\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.184143974400.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 2 q^{4} - \beta_{5} q^{5} + (\beta_{5} - \beta_1 - 2) q^{7} - 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - 2 q^{4} - \beta_{5} q^{5} + (\beta_{5} - \beta_1 - 2) q^{7} - 2 \beta_{2} q^{8} + ( - \beta_{3} - \beta_{2}) q^{10} - \beta_{6} q^{11} + (\beta_{7} - 6 \beta_{2}) q^{13} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{14} + 4 q^{16} + ( - \beta_{5} + 2 \beta_1) q^{17} + ( - \beta_{6} - \beta_{4} + 3 \beta_{2} + \cdots - 4) q^{19}+ \cdots + ( - 7 \beta_{7} + 4 \beta_{4} + \cdots - 6 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 4 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 4 q^{5} - 12 q^{7} - 4 q^{11} + 32 q^{16} - 4 q^{17} - 36 q^{19} + 8 q^{20} + 56 q^{23} + 140 q^{25} + 96 q^{26} + 24 q^{28} - 236 q^{35} - 48 q^{38} + 100 q^{43} + 8 q^{44} + 188 q^{47} - 36 q^{49} + 28 q^{55} + 168 q^{58} - 180 q^{61} + 96 q^{62} - 64 q^{64} + 8 q^{68} - 356 q^{73} + 192 q^{74} + 72 q^{76} - 68 q^{77} - 16 q^{80} + 72 q^{82} - 136 q^{83} + 148 q^{85} - 112 q^{92} + 140 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 17\nu^{4} + 39\nu^{3} + 88\nu^{2} - 108\nu - 36 ) / 45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 82\nu^{7} - 287\nu^{6} - 1511\nu^{5} + 4495\nu^{4} + 12457\nu^{3} - 23324\nu^{2} - 30864\nu + 19476 ) / 26190 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 119\nu^{7} - 1144\nu^{6} - 862\nu^{5} + 29750\nu^{4} - 20941\nu^{3} - 249898\nu^{2} + 163452\nu + 694512 ) / 26190 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{7} + 125\nu^{6} - 101\nu^{5} - 2485\nu^{4} + 2809\nu^{3} + 20450\nu^{2} - 15840\nu - 57474 ) / 1746 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -85\nu^{7} + 443\nu^{6} + 491\nu^{5} - 5536\nu^{4} + 6602\nu^{3} + 14624\nu^{2} - 58884\nu + 22482 ) / 13095 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -205\nu^{7} + 572\nu^{6} + 4214\nu^{5} - 8764\nu^{4} - 36817\nu^{3} + 45506\nu^{2} + 223824\nu - 95832 ) / 26190 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -32\nu^{7} + 112\nu^{6} + 760\nu^{5} - 2180\nu^{4} - 6224\nu^{3} + 11572\nu^{2} + 15792\nu - 9900 ) / 2619 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{6} + 10\beta_{2} + \beta _1 + 8 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 4\beta_{6} + 8\beta_{4} + 4\beta_{3} + 12\beta_{2} - 14\beta _1 + 148 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 21\beta_{7} + 34\beta_{6} + 30\beta_{5} + 12\beta_{4} + 6\beta_{3} + 238\beta_{2} - 29\beta _1 + 188 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14\beta_{7} + 16\beta_{6} + 15\beta_{5} + 38\beta_{4} + 34\beta_{3} + 132\beta_{2} - 56\beta _1 + 292 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 630\beta_{7} + 286\beta_{6} + 390\beta_{5} + 360\beta_{4} + 330\beta_{3} + 4030\beta_{2} - 611\beta _1 + 2372 ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2111 \beta_{7} + 700 \beta_{6} + 1020 \beta_{5} + 2492 \beta_{4} + 2716 \beta_{3} + 11808 \beta_{2} + \cdots + 8200 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12453 \beta_{7} + 1690 \beta_{6} + 2910 \beta_{5} + 7476 \beta_{4} + 8358 \beta_{3} + 64054 \beta_{2} + \cdots + 20180 ) / 20 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.51885 − 0.707107i
3.87998 − 0.707107i
2.51885 − 0.707107i
−2.87998 − 0.707107i
−1.51885 + 0.707107i
3.87998 + 0.707107i
2.51885 + 0.707107i
−2.87998 + 0.707107i
− 1.41421i 0 −2.00000 −9.91350 0 3.01452 2.82843i 0 14.0198i
37.2 − 1.41421i 0 −2.00000 −2.94975 0 5.84873 2.82843i 0 4.17158i
37.3 − 1.41421i 0 −2.00000 4.01452 0 −10.9135 2.82843i 0 − 5.67739i
37.4 − 1.41421i 0 −2.00000 6.84873 0 −3.94975 2.82843i 0 − 9.68557i
37.5 1.41421i 0 −2.00000 −9.91350 0 3.01452 − 2.82843i 0 − 14.0198i
37.6 1.41421i 0 −2.00000 −2.94975 0 5.84873 − 2.82843i 0 − 4.17158i
37.7 1.41421i 0 −2.00000 4.01452 0 −10.9135 − 2.82843i 0 5.67739i
37.8 1.41421i 0 −2.00000 6.84873 0 −3.94975 − 2.82843i 0 9.68557i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.3.d.b 8
3.b odd 2 1 114.3.d.a ✓ 8
4.b odd 2 1 2736.3.o.n 8
12.b even 2 1 912.3.o.d 8
19.b odd 2 1 inner 342.3.d.b 8
57.d even 2 1 114.3.d.a ✓ 8
76.d even 2 1 2736.3.o.n 8
228.b odd 2 1 912.3.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.d.a ✓ 8 3.b odd 2 1
114.3.d.a ✓ 8 57.d even 2 1
342.3.d.b 8 1.a even 1 1 trivial
342.3.d.b 8 19.b odd 2 1 inner
912.3.o.d 8 12.b even 2 1
912.3.o.d 8 228.b odd 2 1
2736.3.o.n 8 4.b odd 2 1
2736.3.o.n 8 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{3} - 83T_{5}^{2} + 36T_{5} + 804 \) acting on \(S_{3}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} + \cdots + 804)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{3} + \cdots + 760)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + \cdots + 28650)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 240 T^{2} + 576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{3} - 179 T^{2} + \cdots - 60)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{4} - 28 T^{3} + \cdots - 302040)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 30611001600 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 6046617600 \) Copy content Toggle raw display
$37$ \( (T^{4} + 2976 T^{2} + 831744)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 4246237209600 \) Copy content Toggle raw display
$43$ \( (T^{4} - 50 T^{3} + \cdots - 1616000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 94 T^{3} + \cdots + 89850)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 492631164527616 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 652283427225600 \) Copy content Toggle raw display
$61$ \( (T^{4} + 90 T^{3} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 292222411776 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 110381078937600 \) Copy content Toggle raw display
$73$ \( (T^{4} + 178 T^{3} + \cdots + 577600)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{4} + 68 T^{3} + \cdots - 57240)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 109746576000000 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
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