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Properties

Label 342.2.h.f
Level $342$
Weight $2$
Character orbit 342.h
Analytic conductor $2.731$
Analytic rank $0$
Dimension $10$
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(121,342)]
 
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([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.h (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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.50911458676875.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 12x^{8} - 28x^{7} + 58x^{6} - 111x^{5} + 174x^{4} - 252x^{3} + 324x^{2} - 324x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} + 12x^{8} - 28x^{7} + 58x^{6} - 111x^{5} + 174x^{4} - 252x^{3} + 324x^{2} - 324x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13 \nu^{9} - 86 \nu^{8} + 135 \nu^{7} - 329 \nu^{6} + 1517 \nu^{5} - 2331 \nu^{4} + 4398 \nu^{3} + \cdots - 12798 ) / 2997 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + 4\nu^{8} - 12\nu^{7} + 28\nu^{6} - 58\nu^{5} + 111\nu^{4} - 174\nu^{3} + 252\nu^{2} - 324\nu + 324 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{9} - 8\nu^{8} + 16\nu^{7} - 22\nu^{6} - 37\nu^{5} + 74\nu^{4} - 324\nu^{3} + 384\nu^{2} - 576\nu + 1458 ) / 999 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6 \nu^{9} + 40 \nu^{8} - 80 \nu^{7} + 184 \nu^{6} - 370 \nu^{5} + 629 \nu^{4} - 970 \nu^{3} + \cdots + 1368 ) / 333 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{9} + 65 \nu^{8} - 130 \nu^{7} + 373 \nu^{6} - 740 \nu^{5} + 1147 \nu^{4} - 2085 \nu^{3} + \cdots + 2889 ) / 999 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 91 \nu^{9} - 508 \nu^{8} + 1164 \nu^{7} - 2692 \nu^{6} + 5476 \nu^{5} - 9768 \nu^{4} + 15168 \nu^{3} + \cdots - 21303 ) / 2997 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 56 \nu^{9} + 176 \nu^{8} - 426 \nu^{7} + 965 \nu^{6} - 1850 \nu^{5} + 3108 \nu^{4} - 4416 \nu^{3} + \cdots + 3888 ) / 999 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 170 \nu^{9} + 566 \nu^{8} - 1206 \nu^{7} + 3203 \nu^{6} - 6290 \nu^{5} + 10989 \nu^{4} + \cdots + 18225 ) / 2997 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 2\beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} - 2\beta_{8} - 4\beta_{6} + \beta_{5} - 2\beta_{4} + 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{8} + 6\beta_{5} - 7\beta_{4} + 8\beta_{3} + 4\beta_{2} - 2\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{9} + \beta_{8} + 4\beta_{7} + 4\beta_{6} + 16\beta_{4} - 2\beta_{3} + 10\beta_{2} + 3\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14\beta_{9} - 11\beta_{8} + 2\beta_{7} + 2\beta_{6} + 3\beta_{4} - 9\beta_{3} + \beta_{2} + 9\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{9} - 24 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} + 47 \beta_{5} - 38 \beta_{4} + 3 \beta_{3} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4\beta_{9} + 16\beta_{8} + 24\beta_{7} - 20\beta_{6} + 29\beta_{5} + 96\beta_{4} - 7\beta_{3} + 20\beta_{2} - 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(\beta_{4}\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.69141 + 0.373009i
0.927286 + 1.46292i
0.776413 − 1.54828i
−0.378732 − 1.69014i
−1.01638 + 1.40249i
1.69141 − 0.373009i
0.927286 − 1.46292i
0.776413 + 1.54828i
−0.378732 + 1.69014i
−1.01638 − 1.40249i
−0.500000 + 0.866025i −1.69141 + 0.373009i −0.500000 − 0.866025i 1.04534 0.522670 − 1.65131i 0.0934846 + 0.161920i 1.00000 2.72173 − 1.26182i −0.522670 + 0.905290i
121.2 −0.500000 + 0.866025i −0.927286 + 1.46292i −0.500000 − 0.866025i −1.60657 −0.803284 − 1.53451i −0.448388 − 0.776631i 1.00000 −1.28028 − 2.71309i 0.803284 − 1.39133i
121.3 −0.500000 + 0.866025i −0.776413 − 1.54828i −0.500000 − 0.866025i 3.45812 1.72906 + 0.101749i −2.30306 − 3.98902i 1.00000 −1.79437 + 2.40422i −1.72906 + 2.99482i
121.4 −0.500000 + 0.866025i 0.378732 − 1.69014i −0.500000 − 0.866025i 2.54867 1.27434 + 1.17306i 2.28394 + 3.95589i 1.00000 −2.71312 − 1.28022i −1.27434 + 2.20721i
121.5 −0.500000 + 0.866025i 1.01638 + 1.40249i −0.500000 − 0.866025i −3.44556 −1.72278 + 0.178963i −2.12597 − 3.68229i 1.00000 −0.933957 + 2.85092i 1.72278 − 2.98394i
277.1 −0.500000 − 0.866025i −1.69141 − 0.373009i −0.500000 + 0.866025i 1.04534 0.522670 + 1.65131i 0.0934846 − 0.161920i 1.00000 2.72173 + 1.26182i −0.522670 − 0.905290i
277.2 −0.500000 − 0.866025i −0.927286 − 1.46292i −0.500000 + 0.866025i −1.60657 −0.803284 + 1.53451i −0.448388 + 0.776631i 1.00000 −1.28028 + 2.71309i 0.803284 + 1.39133i
277.3 −0.500000 − 0.866025i −0.776413 + 1.54828i −0.500000 + 0.866025i 3.45812 1.72906 − 0.101749i −2.30306 + 3.98902i 1.00000 −1.79437 − 2.40422i −1.72906 − 2.99482i
277.4 −0.500000 − 0.866025i 0.378732 + 1.69014i −0.500000 + 0.866025i 2.54867 1.27434 − 1.17306i 2.28394 − 3.95589i 1.00000 −2.71312 + 1.28022i −1.27434 − 2.20721i
277.5 −0.500000 − 0.866025i 1.01638 − 1.40249i −0.500000 + 0.866025i −3.44556 −1.72278 − 0.178963i −2.12597 + 3.68229i 1.00000 −0.933957 − 2.85092i 1.72278 + 2.98394i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g 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.h.f yes 10
3.b odd 2 1 1026.2.h.f 10
9.c even 3 1 342.2.f.f ✓ 10
9.d odd 6 1 1026.2.f.f 10
19.c even 3 1 342.2.f.f ✓ 10
57.h odd 6 1 1026.2.f.f 10
171.g even 3 1 inner 342.2.h.f yes 10
171.n odd 6 1 1026.2.h.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.f.f ✓ 10 9.c even 3 1
342.2.f.f ✓ 10 19.c even 3 1
342.2.h.f yes 10 1.a even 1 1 trivial
342.2.h.f yes 10 171.g even 3 1 inner
1026.2.f.f 10 9.d odd 6 1
1026.2.f.f 10 57.h odd 6 1
1026.2.h.f 10 3.b odd 2 1
1026.2.h.f 10 171.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 2T_{5}^{4} - 15T_{5}^{3} + 28T_{5}^{2} + 37T_{5} - 51 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 4 T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( (T^{5} - 2 T^{4} - 15 T^{3} + \cdots - 51)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 5 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$11$ \( T^{10} - T^{9} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{10} + 7 T^{9} + \cdots + 1156 \) Copy content Toggle raw display
$17$ \( T^{10} + 9 T^{9} + \cdots + 1896129 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + \cdots + 553536 \) Copy content Toggle raw display
$29$ \( (T^{5} - 13 T^{4} + \cdots + 15)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 18 T^{9} + \cdots + 45091225 \) Copy content Toggle raw display
$37$ \( (T^{5} + 24 T^{4} + 191 T^{3} + \cdots - 2)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + 5 T^{4} + \cdots + 21456)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} - 15 T^{9} + \cdots + 4946176 \) Copy content Toggle raw display
$47$ \( (T^{5} + 7 T^{4} - 18 T^{3} + \cdots - 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 14 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$59$ \( (T^{5} - 15 T^{4} + \cdots + 13491)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 14 T^{4} + \cdots + 26869)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 118461456 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 118919025 \) Copy content Toggle raw display
$73$ \( T^{10} - 4 T^{9} + \cdots + 114921 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 118810000 \) Copy content Toggle raw display
$83$ \( T^{10} - 21 T^{9} + \cdots + 12131289 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1936088001 \) Copy content Toggle raw display
$97$ \( T^{10} + 24 T^{9} + \cdots + 1653796 \) Copy content Toggle raw display
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