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Properties

Label 342.3.c.b
Level $342$
Weight $3$
Character orbit 342.c
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(305,342)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.c (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: 8.0.691798081536.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} - 12x^{5} + 127x^{4} - 144x^{3} - 282x^{2} + 900x + 1350 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - 2 q^{4} + \beta_{7} q^{5} + (\beta_{5} + \beta_{2} + 1) q^{7} - 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - 2 q^{4} + \beta_{7} q^{5} + (\beta_{5} + \beta_{2} + 1) q^{7} - 2 \beta_{3} q^{8} + ( - \beta_{5} + \beta_1 - 1) q^{10} + ( - \beta_{7} - 2 \beta_{6} - \beta_{3}) q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{7} + \beta_{4} + \beta_{3}) q^{14} + 4 q^{16} + ( - 4 \beta_{6} - \beta_{4} + 5 \beta_{3}) q^{17} - \beta_{5} q^{19} - 2 \beta_{7} q^{20} + ( - 3 \beta_{5} - \beta_1 + 3) q^{22} + ( - 5 \beta_{7} - 2 \beta_{6} + \cdots - 5 \beta_{3}) q^{23}+ \cdots + (9 \beta_{7} + 4 \beta_{6} + \cdots + 12 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 8 q^{7} - 4 q^{10} - 8 q^{13} + 32 q^{16} + 20 q^{22} - 24 q^{25} - 16 q^{28} - 24 q^{31} - 84 q^{34} + 72 q^{37} + 8 q^{40} - 180 q^{43} + 104 q^{46} + 116 q^{49} + 16 q^{52} + 76 q^{55} + 160 q^{58} + 40 q^{61} - 64 q^{64} + 128 q^{67} - 228 q^{70} - 316 q^{73} + 400 q^{79} - 56 q^{82} - 324 q^{85} - 40 q^{88} - 560 q^{91} + 116 q^{94} - 288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} - 12x^{5} + 127x^{4} - 144x^{3} - 282x^{2} + 900x + 1350 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 48316 \nu^{7} - 85773 \nu^{6} + 1779500 \nu^{5} - 522333 \nu^{4} - 14944036 \nu^{3} + \cdots - 237742020 ) / 41176665 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 67412 \nu^{7} + 428751 \nu^{6} - 587440 \nu^{5} - 3792624 \nu^{4} - 7184788 \nu^{3} + \cdots + 179989965 ) / 41176665 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 200\nu^{7} - 233\nu^{6} - 1969\nu^{5} + 2560\nu^{4} + 21101\nu^{3} - 71453\nu^{2} + 13578\nu + 147000 ) / 98745 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36339 \nu^{7} - 385198 \nu^{6} - 52923 \nu^{5} + 3967853 \nu^{4} + 2427573 \nu^{3} + \cdots + 84082725 ) / 13725555 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7454 \nu^{7} + 2968 \nu^{6} - 115450 \nu^{5} - 137627 \nu^{4} + 1137374 \nu^{3} - 447413 \nu^{2} + \cdots + 5419443 ) / 2745111 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -6\nu^{7} + 11\nu^{6} + 87\nu^{5} - 18\nu^{4} - 729\nu^{3} + 1575\nu^{2} - 1404\nu - 5050 ) / 695 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 510358 \nu^{7} - 579492 \nu^{6} - 6288608 \nu^{5} + 740859 \nu^{4} + 62007385 \nu^{3} + \cdots + 385086375 ) / 41176665 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{7} + \beta_{6} - 14\beta_{5} - 2\beta_{4} - 17\beta_{3} - 2\beta_{2} - 4\beta _1 + 2 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{7} - 4\beta_{6} - \beta_{5} - 2\beta_{4} + 8\beta_{3} - 3\beta_{2} - \beta _1 + 18 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 36\beta_{7} + 31\beta_{6} - 10\beta_{5} - 2\beta_{4} - 67\beta_{3} - 20\beta_{2} - 10\beta _1 + 50 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -44\beta_{7} - 24\beta_{6} - 34\beta_{5} - 12\beta_{4} + 188\beta_{3} - 17\beta_{2} - 39\beta _1 - 53 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 152\beta_{7} + 217\beta_{6} + 48\beta_{5} - 34\beta_{4} + 121\beta_{3} - 156\beta_{2} + 188\beta _1 + 1856 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 228\beta_{7} + 308\beta_{6} - 510\beta_{5} - 96\beta_{4} + 144\beta_{3} + 45\beta_{2} - 510\beta _1 - 1655 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3638 \beta_{7} - 2683 \beta_{6} + 812 \beta_{5} - 1424 \beta_{4} + 14821 \beta_{3} - 984 \beta_{2} + \cdots + 15964 ) / 10 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
−1.31342 − 0.517638i
3.04547 − 0.517638i
1.31342 + 1.93185i
−3.04547 + 1.93185i
−3.04547 − 1.93185i
1.31342 − 1.93185i
3.04547 + 0.517638i
−1.31342 + 0.517638i
− 1.41421i 0 −2.00000 − 8.85053i 0 9.86326 2.82843i 0 −12.5165
305.2 − 1.41421i 0 −2.00000 − 0.429787i 0 −9.59531 2.82843i 0 −0.607811
305.3 − 1.41421i 0 −2.00000 2.80489i 0 7.23641 2.82843i 0 3.96671
305.4 − 1.41421i 0 −2.00000 5.06122i 0 −3.50436 2.82843i 0 7.15765
305.5 1.41421i 0 −2.00000 − 5.06122i 0 −3.50436 − 2.82843i 0 7.15765
305.6 1.41421i 0 −2.00000 − 2.80489i 0 7.23641 − 2.82843i 0 3.96671
305.7 1.41421i 0 −2.00000 0.429787i 0 −9.59531 − 2.82843i 0 −0.607811
305.8 1.41421i 0 −2.00000 8.85053i 0 9.86326 − 2.82843i 0 −12.5165
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.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.c.b ✓ 8
3.b odd 2 1 inner 342.3.c.b ✓ 8
4.b odd 2 1 2736.3.h.b 8
12.b even 2 1 2736.3.h.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.3.c.b ✓ 8 1.a even 1 1 trivial
342.3.c.b ✓ 8 3.b odd 2 1 inner
2736.3.h.b 8 4.b odd 2 1
2736.3.h.b 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 112T_{5}^{6} + 2845T_{5}^{4} + 16308T_{5}^{2} + 2916 \) 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
$5$ \( T^{8} + 112 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} + \cdots + 2400)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 580 T^{6} + \cdots + 26132544 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + \cdots + 17728)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 3458145636 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 56933777664 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 1494904896 \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + \cdots + 840064)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 36 T^{3} + \cdots + 1291648)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 7364840702976 \) Copy content Toggle raw display
$43$ \( (T^{4} + 90 T^{3} + \cdots - 9680000)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 122595218496 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 3771925056 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 247669456896 \) Copy content Toggle raw display
$61$ \( (T^{4} - 20 T^{3} + \cdots - 226412)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 64 T^{3} + \cdots + 1132800)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 2721391312896 \) Copy content Toggle raw display
$73$ \( (T^{4} + 158 T^{3} + \cdots - 16900608)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 200 T^{3} + \cdots + 1561600)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 79\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + 144 T^{3} + \cdots - 21980096)^{2} \) Copy content Toggle raw display
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