Properties

Label 342.2.e.c
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: \( x^{12} + 8x^{10} + 55x^{8} - 2x^{7} + 70x^{6} - 32x^{5} + 73x^{4} - 18x^{3} + 13x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 8x^{10} + 55x^{8} - 2x^{7} + 70x^{6} - 32x^{5} + 73x^{4} - 18x^{3} + 13x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 411176 \nu^{11} + 3327904 \nu^{10} - 3541281 \nu^{9} + 25462287 \nu^{8} - 25192919 \nu^{7} + 174805454 \nu^{6} - 53789475 \nu^{5} + \cdots + 10915007 ) / 8057223 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 485726 \nu^{11} - 2471284 \nu^{10} + 2275851 \nu^{9} - 19554387 \nu^{8} + 14692937 \nu^{7} - 134803199 \nu^{6} - 43318125 \nu^{5} - 171073498 \nu^{4} + \cdots - 3803162 ) / 8057223 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 573380 \nu^{11} + 1123298 \nu^{10} + 5455884 \nu^{9} + 8537454 \nu^{8} + 37790027 \nu^{7} + 57094012 \nu^{6} + 80439177 \nu^{5} + 34029107 \nu^{4} + \cdots - 8741327 ) / 8057223 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 723523 \nu^{11} - 2374420 \nu^{10} - 5988987 \nu^{9} - 18519324 \nu^{8} - 41432173 \nu^{7} - 125252072 \nu^{6} - 57042816 \nu^{5} - 116563363 \nu^{4} + \cdots - 5341028 ) / 8057223 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1042276 \nu^{11} + 503822 \nu^{10} - 5877465 \nu^{9} + 5217513 \nu^{8} - 37931593 \nu^{7} + 38234422 \nu^{6} + 58653528 \nu^{5} + 120977024 \nu^{4} + \cdots + 5590234 ) / 8057223 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 475714 \nu^{11} - 219777 \nu^{10} + 3753182 \nu^{9} - 1727756 \nu^{8} + 25748182 \nu^{7} - 12842883 \nu^{6} + 30886544 \nu^{5} - 29151715 \nu^{4} + \cdots + 755148 ) / 2685741 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 491336 \nu^{11} - 66931 \nu^{10} + 3789188 \nu^{9} - 896846 \nu^{8} + 25810123 \nu^{7} - 7407675 \nu^{6} + 26207109 \nu^{5} - 38561249 \nu^{4} + \cdots - 5226000 ) / 2685741 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 496946 \nu^{11} - 348319 \nu^{10} + 4407278 \nu^{9} - 2829986 \nu^{8} + 30660580 \nu^{7} - 20565930 \nu^{6} + 58131969 \nu^{5} - 43917038 \nu^{4} + \cdots - 7544085 ) / 2685741 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 826298 \nu^{11} + 506575 \nu^{10} - 6411425 \nu^{9} + 3835658 \nu^{8} - 44063809 \nu^{7} + 27624861 \nu^{6} - 49514907 \nu^{5} + 48686057 \nu^{4} + \cdots - 4239471 ) / 2685741 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5325568 \nu^{11} - 291295 \nu^{10} - 42073479 \nu^{9} - 1655457 \nu^{8} - 288320839 \nu^{7} - 261647 \nu^{6} - 340849530 \nu^{5} + 184170155 \nu^{4} + \cdots + 3040609 ) / 8057223 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5636270 \nu^{11} - 3730295 \nu^{10} - 45709146 \nu^{9} - 29404869 \nu^{8} - 313840082 \nu^{7} - 190033528 \nu^{6} - 412733472 \nu^{5} + \cdots - 28561402 ) / 8057223 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - 2\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 8\beta_{6} + \beta_{5} + 2\beta_{4} - 5\beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{11} + 11\beta_{10} + 8\beta_{7} - 5\beta_{6} - 3\beta_{5} - 2\beta_{4} + 2\beta_{2} - 8\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21 \beta_{10} - 16 \beta_{9} - 6 \beta_{8} + 17 \beta_{7} + 46 \beta_{6} + \beta_{5} + \beta_{3} - 12 \beta_{2} + 8 \beta _1 - 46 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 56 \beta_{11} - 77 \beta_{10} + 2 \beta_{9} - 56 \beta_{8} - \beta_{7} + 36 \beta_{6} + 54 \beta_{5} + 8 \beta_{4} + 33 \beta_{3} + 31 \beta _1 + 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 41 \beta_{11} - 41 \beta_{10} + 31 \beta_{9} - 151 \beta_{7} + 10 \beta_{6} - 55 \beta_{5} - 76 \beta_{4} + 227 \beta_{3} + 76 \beta_{2} - 45 \beta _1 + 244 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6 \beta_{10} - 4 \beta_{9} + 379 \beta_{8} - 365 \beta_{7} - 56 \beta_{6} - 204 \beta_{5} - 204 \beta_{3} - 43 \beta_{2} + 155 \beta _1 + 56 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 97 \beta_{11} - 215 \beta_{10} + 178 \beta_{9} + 97 \beta_{8} + 68 \beta_{7} - 689 \beta_{6} + 94 \beta_{5} + 166 \beta_{4} - 532 \beta_{3} - 27 \beta _1 + 121 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2548 \beta_{11} + 3490 \beta_{10} - 81 \beta_{9} + 2557 \beta_{7} - 1360 \beta_{6} - 1020 \beta_{5} - 250 \beta_{4} - 258 \beta_{3} + 250 \beta_{2} - 2380 \beta _1 - 1515 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6288 \beta_{10} - 4928 \beta_{9} - 2077 \beta_{8} + 5645 \beta_{7} + 13253 \beta_{6} + 621 \beta_{5} + 621 \beta_{3} - 3296 \beta_{2} + 2371 \beta _1 - 13253 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 17111 \beta_{11} - 22451 \beta_{10} + 294 \beta_{9} - 17111 \beta_{8} - 621 \beta_{7} + 13154 \beta_{6} + 15773 \beta_{5} + 1469 \beta_{4} + 11381 \beta_{3} + 9095 \beta _1 + 6678 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\beta_{6}\) \(1\)

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
1.30509 2.26048i
0.425485 0.736962i
−0.122364 + 0.211941i
−1.27710 + 2.21200i
0.289933 0.502178i
−0.621041 + 1.07567i
1.30509 + 2.26048i
0.425485 + 0.736962i
−0.122364 0.211941i
−1.27710 2.21200i
0.289933 + 0.502178i
−0.621041 1.07567i
−0.500000 + 0.866025i −1.70951 + 0.278542i −0.500000 0.866025i −1.71495 2.97037i 0.613529 1.61975i −0.0414111 + 0.0717262i 1.00000 2.84483 0.952340i 3.42989
115.2 −0.500000 + 0.866025i −1.05505 + 1.37363i −0.500000 0.866025i 1.72549 + 2.98863i −0.662079 1.60052i −1.90190 + 3.29418i 1.00000 −0.773746 2.89850i −3.45098
115.3 −0.500000 + 0.866025i 0.147638 1.72575i −0.500000 0.866025i −0.573032 0.992520i 1.42072 + 0.990732i 0.817447 1.41586i 1.00000 −2.95641 0.509572i 1.14606
115.4 −0.500000 + 0.866025i 0.178697 + 1.72281i −0.500000 0.866025i −1.95772 3.39088i −1.58134 0.706648i −1.65131 + 2.86015i 1.00000 −2.93613 + 0.615721i 3.91545
115.5 −0.500000 + 0.866025i 1.71412 + 0.248580i −0.500000 0.866025i 2.19415 + 3.80037i −1.07234 + 1.36018i 1.88373 3.26272i 1.00000 2.87642 + 0.852192i −4.38829
115.6 −0.500000 + 0.866025i 1.72410 0.165767i −0.500000 0.866025i 0.326066 + 0.564763i −0.718491 + 1.57600i −2.10656 + 3.64868i 1.00000 2.94504 0.571599i −0.652132
229.1 −0.500000 0.866025i −1.70951 0.278542i −0.500000 + 0.866025i −1.71495 + 2.97037i 0.613529 + 1.61975i −0.0414111 0.0717262i 1.00000 2.84483 + 0.952340i 3.42989
229.2 −0.500000 0.866025i −1.05505 1.37363i −0.500000 + 0.866025i 1.72549 2.98863i −0.662079 + 1.60052i −1.90190 3.29418i 1.00000 −0.773746 + 2.89850i −3.45098
229.3 −0.500000 0.866025i 0.147638 + 1.72575i −0.500000 + 0.866025i −0.573032 + 0.992520i 1.42072 0.990732i 0.817447 + 1.41586i 1.00000 −2.95641 + 0.509572i 1.14606
229.4 −0.500000 0.866025i 0.178697 1.72281i −0.500000 + 0.866025i −1.95772 + 3.39088i −1.58134 + 0.706648i −1.65131 2.86015i 1.00000 −2.93613 0.615721i 3.91545
229.5 −0.500000 0.866025i 1.71412 0.248580i −0.500000 + 0.866025i 2.19415 3.80037i −1.07234 1.36018i 1.88373 + 3.26272i 1.00000 2.87642 0.852192i −4.38829
229.6 −0.500000 0.866025i 1.72410 + 0.165767i −0.500000 + 0.866025i 0.326066 0.564763i −0.718491 1.57600i −2.10656 3.64868i 1.00000 2.94504 + 0.571599i −0.652132
\(n\): 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.c 12
3.b odd 2 1 1026.2.e.d 12
9.c even 3 1 inner 342.2.e.c 12
9.c even 3 1 3078.2.a.x 6
9.d odd 6 1 1026.2.e.d 12
9.d odd 6 1 3078.2.a.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.e.c 12 1.a even 1 1 trivial
342.2.e.c 12 9.c even 3 1 inner
1026.2.e.d 12 3.b odd 2 1
1026.2.e.d 12 9.d odd 6 1
3078.2.a.v 6 9.d odd 6 1
3078.2.a.x 6 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 30 T_{5}^{10} + 16 T_{5}^{9} + 672 T_{5}^{8} + 336 T_{5}^{7} + 6600 T_{5}^{6} + 3936 T_{5}^{5} + 48192 T_{5}^{4} + 19456 T_{5}^{3} + 43872 T_{5}^{2} - 14592 T_{5} + 23104 \) 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
$3$ \( T^{12} - 2 T^{11} - 2 T^{9} - 8 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 30 T^{10} + 16 T^{9} + \cdots + 23104 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + 48 T^{10} + 144 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + 60 T^{10} + \cdots + 760384 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + 84 T^{10} + \cdots + 19035769 \) Copy content Toggle raw display
$17$ \( (T^{6} - 66 T^{4} - 18 T^{3} + 549 T^{2} + \cdots + 405)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + 138 T^{10} + \cdots + 66049 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + 66 T^{10} + \cdots + 72361 \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + 138 T^{10} + \cdots + 202094656 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} - 135 T^{4} + 628 T^{3} + \cdots - 90329)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 6 T^{11} + 66 T^{10} + \cdots + 1600 \) Copy content Toggle raw display
$43$ \( T^{12} + 12 T^{11} + 210 T^{10} + \cdots + 72863296 \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} + \cdots + 2953161649 \) Copy content Toggle raw display
$53$ \( (T^{6} + 18 T^{5} + 36 T^{4} - 664 T^{3} + \cdots + 1927)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 12 T^{11} + 234 T^{10} + \cdots + 20331081 \) Copy content Toggle raw display
$61$ \( T^{12} + 12 T^{11} + 174 T^{10} + \cdots + 4494400 \) Copy content Toggle raw display
$67$ \( T^{12} + 18 T^{11} + \cdots + 275017385241 \) Copy content Toggle raw display
$71$ \( (T^{6} - 18 T^{5} - 30 T^{4} + 1232 T^{3} + \cdots - 50840)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 24 T^{5} + 54 T^{4} + 2622 T^{3} + \cdots - 59967)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 18 T^{11} + \cdots + 1126273600 \) Copy content Toggle raw display
$83$ \( T^{12} - 12 T^{11} + 138 T^{10} + \cdots + 10497600 \) Copy content Toggle raw display
$89$ \( (T^{6} - 6 T^{5} - 102 T^{4} + 252 T^{3} + \cdots - 5400)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 18 T^{11} + \cdots + 5833925283904 \) Copy content Toggle raw display
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