Properties

Label 342.3.m.d
Level $342$
Weight $3$
Character orbit 342.m
Analytic conductor $9.319$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,3,Mod(145,342)]
 
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([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.m (of order \(6\), 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: \(9.31882504112\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 80 x^{14} + 4582 x^{12} - 116048 x^{10} + 2090635 x^{8} - 20507888 x^{6} + 145417654 x^{4} - 570337064 x^{2} + 1506138481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + (2 \beta_1 + 2) q^{4} + ( - \beta_{12} - \beta_{10}) q^{5} + ( - \beta_{7} + 1) q^{7} + 2 \beta_{9} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + (2 \beta_1 + 2) q^{4} + ( - \beta_{12} - \beta_{10}) q^{5} + ( - \beta_{7} + 1) q^{7} + 2 \beta_{9} q^{8} + ( - \beta_{6} + \beta_{5}) q^{10} + (\beta_{11} + 2 \beta_{9} + 4 \beta_{8}) q^{11} + ( - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2}) q^{13} + ( - \beta_{13} + \beta_{10} - 2 \beta_{8}) q^{14} + 4 \beta_1 q^{16} + (\beta_{15} + 2 \beta_{14} + \beta_{13} - 3 \beta_{12} - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8}) q^{17} + (\beta_{7} - 2 \beta_{5} - \beta_{3} + \beta_{2} + 7 \beta_1 + 4) q^{19} - 2 \beta_{10} q^{20} + (\beta_{5} - 2 \beta_{2} - 4 \beta_1 - 8) q^{22} + (\beta_{15} - \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{10} + 4 \beta_{9} - 6 \beta_{8}) q^{23} + ( - \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{2} - 8 \beta_1 - 8) q^{25} + (\beta_{11} - \beta_{10}) q^{26} + (2 \beta_{4} + 2 \beta_1 + 2) q^{28} + ( - 2 \beta_{12} - 4 \beta_{10}) q^{29} + (\beta_{7} + 2 \beta_{4} + 3 \beta_{3} - 8 \beta_1 - 4) q^{31} + (4 \beta_{9} + 4 \beta_{8}) q^{32} + ( - 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{34}+ \cdots + ( - 3 \beta_{15} - 7 \beta_{12} - 3 \beta_{11} + 7 \beta_{10} - 27 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} + 24 q^{7} - 32 q^{16} - 96 q^{22} - 80 q^{25} + 24 q^{28} - 24 q^{34} + 212 q^{43} + 432 q^{49} + 80 q^{55} - 192 q^{61} - 128 q^{64} + 108 q^{67} - 408 q^{70} - 40 q^{73} - 168 q^{76} - 732 q^{79} + 72 q^{82} - 352 q^{85} - 204 q^{91} + 360 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 80 x^{14} + 4582 x^{12} - 116048 x^{10} + 2090635 x^{8} - 20507888 x^{6} + 145417654 x^{4} - 570337064 x^{2} + 1506138481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 163455856112 \nu^{14} + 12380288973470 \nu^{12} - 699476967224640 \nu^{10} + \cdots + 29\!\cdots\!60 ) / 46\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 625586030420 \nu^{14} + 49550082836104 \nu^{12} + \cdots + 65\!\cdots\!92 ) / 46\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1947351576330 \nu^{14} + 148577931152752 \nu^{12} + \cdots + 62\!\cdots\!96 ) / 46\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16507449391907 \nu^{14} + \cdots - 18\!\cdots\!07 ) / 18\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14098651228589 \nu^{14} + 891087694938548 \nu^{12} + \cdots - 40\!\cdots\!16 ) / 93\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6279628042228 \nu^{14} - 537478001105760 \nu^{12} + \cdots - 23\!\cdots\!50 ) / 31\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1505410050 \nu^{14} - 108378542575 \nu^{12} + 5956393343625 \nu^{10} - 121511691139200 \nu^{8} + \cdots + 99\!\cdots\!77 ) / 47\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 285440682588860 \nu^{15} + \cdots - 39\!\cdots\!67 \nu ) / 54\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3737417132 \nu^{15} + 295169480981 \nu^{13} - 16676852520672 \nu^{11} + 405931558747333 \nu^{9} + \cdots + 71\!\cdots\!38 \nu ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 203457785384199 \nu^{15} + \cdots - 63\!\cdots\!48 \nu ) / 52\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19\!\cdots\!33 \nu^{15} + \cdots - 94\!\cdots\!40 \nu ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 108973994552152 \nu^{15} + \cdots + 39\!\cdots\!11 \nu ) / 19\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 96\!\cdots\!55 \nu^{15} + \cdots + 41\!\cdots\!28 \nu ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 33\!\cdots\!83 \nu^{15} + \cdots + 60\!\cdots\!22 \nu ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 39\!\cdots\!84 \nu^{15} + \cdots - 13\!\cdots\!83 \nu ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + \beta_{12} - 2\beta_{11} + 2\beta_{10} + 3\beta_{9} + 3\beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} - \beta_{2} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 82 \beta_{15} + 18 \beta_{14} + 18 \beta_{13} - 14 \beta_{12} - 41 \beta_{11} - 7 \beta_{10} + 285 \beta_{9} + 18 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} - 6\beta_{5} + 12\beta_{4} + 58\beta_{3} + 58\beta_{2} - 697\beta _1 - 697 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1879\beta_{15} + 792\beta_{13} - 905\beta_{12} + 1879\beta_{11} + 113\beta_{10} - 15321\beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -864\beta_{7} + 450\beta_{6} - 900\beta_{5} - 2925\beta_{3} + 5850\beta_{2} - 30374 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 89099 \beta_{15} - 34182 \beta_{14} + 51085 \beta_{12} + 178198 \beta_{11} + 67988 \beta_{10} - 821499 \beta_{9} - 821499 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 47304 \beta_{7} + 50184 \beta_{6} - 25092 \beta_{5} - 47304 \beta_{4} - 286504 \beta_{3} + 143252 \beta_{2} + 1417885 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8551250 \beta_{15} - 1559232 \beta_{14} - 1559232 \beta_{13} + 5166382 \beta_{12} + 4275625 \beta_{11} + 2583191 \beta_{10} - 40498419 \beta_{9} - 1559232 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1277436 \beta_{6} + 1277436 \beta_{5} - 2389920 \beta_{4} - 6956069 \beta_{3} - 6956069 \beta_{2} + 67677800 \beta _1 + 67677800 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 206090633 \beta_{15} - 73536354 \beta_{13} + 126810967 \beta_{12} - 206090633 \beta_{11} - 53274613 \beta_{10} + 1898588067 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 117426564 \beta_{7} - 62957970 \beta_{6} + 125915940 \beta_{5} + 336788262 \beta_{3} - 673576524 \beta_{2} + 3255835501 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 9950641183 \beta_{15} + 3519826488 \beta_{14} - 6164259137 \beta_{12} - 19901282366 \beta_{11} - 8808691786 \beta_{10} + 95591247825 \beta_{9} + 95591247825 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5710899168 \beta_{7} - 6131093868 \beta_{6} + 3065546934 \beta_{5} + 5710899168 \beta_{4} + 32577285170 \beta_{3} - 16288642585 \beta_{2} - 157087053074 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 961510531510 \beta_{15} + 169484931702 \beta_{14} + 169484931702 \beta_{13} - 597156820250 \beta_{12} - 480755265755 \beta_{11} - 298578410125 \beta_{10} + \cdots + 169484931702 \beta_{8} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
2.29967 + 1.32771i
−2.15207 1.24250i
6.02079 + 3.47611i
−3.71890 2.14711i
3.71890 + 2.14711i
−6.02079 3.47611i
2.15207 + 1.24250i
−2.29967 1.32771i
2.29967 1.32771i
−2.15207 + 1.24250i
6.02079 3.47611i
−3.71890 + 2.14711i
3.71890 2.14711i
−6.02079 + 3.47611i
2.15207 1.24250i
−2.29967 + 1.32771i
−1.22474 + 0.707107i 0 1.00000 1.73205i −4.33044 7.50054i 0 2.96184 2.82843i 0 10.6074 + 6.12417i
145.2 −1.22474 + 0.707107i 0 1.00000 1.73205i −0.0575331 0.0996502i 0 −10.7734 2.82843i 0 0.140927 + 0.0813641i
145.3 −1.22474 + 0.707107i 0 1.00000 1.73205i 0.375080 + 0.649658i 0 0.433375 2.82843i 0 −0.918755 0.530444i
145.4 −1.22474 + 0.707107i 0 1.00000 1.73205i 4.01289 + 6.95053i 0 13.3781 2.82843i 0 −9.82954 5.67509i
145.5 1.22474 0.707107i 0 1.00000 1.73205i −4.01289 6.95053i 0 13.3781 2.82843i 0 −9.82954 5.67509i
145.6 1.22474 0.707107i 0 1.00000 1.73205i −0.375080 0.649658i 0 0.433375 2.82843i 0 −0.918755 0.530444i
145.7 1.22474 0.707107i 0 1.00000 1.73205i 0.0575331 + 0.0996502i 0 −10.7734 2.82843i 0 0.140927 + 0.0813641i
145.8 1.22474 0.707107i 0 1.00000 1.73205i 4.33044 + 7.50054i 0 2.96184 2.82843i 0 10.6074 + 6.12417i
217.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −4.33044 + 7.50054i 0 2.96184 2.82843i 0 10.6074 6.12417i
217.2 −1.22474 0.707107i 0 1.00000 + 1.73205i −0.0575331 + 0.0996502i 0 −10.7734 2.82843i 0 0.140927 0.0813641i
217.3 −1.22474 0.707107i 0 1.00000 + 1.73205i 0.375080 0.649658i 0 0.433375 2.82843i 0 −0.918755 + 0.530444i
217.4 −1.22474 0.707107i 0 1.00000 + 1.73205i 4.01289 6.95053i 0 13.3781 2.82843i 0 −9.82954 + 5.67509i
217.5 1.22474 + 0.707107i 0 1.00000 + 1.73205i −4.01289 + 6.95053i 0 13.3781 2.82843i 0 −9.82954 + 5.67509i
217.6 1.22474 + 0.707107i 0 1.00000 + 1.73205i −0.375080 + 0.649658i 0 0.433375 2.82843i 0 −0.918755 + 0.530444i
217.7 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0.0575331 0.0996502i 0 −10.7734 2.82843i 0 0.140927 0.0813641i
217.8 1.22474 + 0.707107i 0 1.00000 + 1.73205i 4.33044 7.50054i 0 2.96184 2.82843i 0 10.6074 6.12417i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.3.m.d 16
3.b odd 2 1 inner 342.3.m.d 16
19.d odd 6 1 inner 342.3.m.d 16
57.f even 6 1 inner 342.3.m.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.3.m.d 16 1.a even 1 1 trivial
342.3.m.d 16 3.b odd 2 1 inner
342.3.m.d 16 19.d odd 6 1 inner
342.3.m.d 16 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 140 T_{5}^{14} + 14688 T_{5}^{12} + 682112 T_{5}^{10} + 23737948 T_{5}^{8} + 13664928 T_{5}^{6} + 7573824 T_{5}^{4} + 100224 T_{5}^{2} + 1296 \) acting on \(S_{3}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 140 T^{14} + 14688 T^{12} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{3} - 134 T^{2} + 486 T - 185)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 812 T^{6} + 228184 T^{4} + \cdots + 738426276)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 234 T^{6} + 47763 T^{4} + \cdots + 48902049)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 2288 T^{14} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} - 196 T^{6} + \cdots + 16983563041)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 2876 T^{14} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} - 1680 T^{14} + \cdots + 557256278016 \) Copy content Toggle raw display
$31$ \( (T^{8} + 3912 T^{6} + \cdots + 3913628481)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 9900 T^{6} + \cdots + 372970368369)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 7056 T^{14} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} - 106 T^{7} + 9238 T^{6} + \cdots + 784056001)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 8816 T^{14} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} - 7620 T^{14} + \cdots + 97\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{16} - 23460 T^{14} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + 96 T^{7} + \cdots + 572481390625)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 54 T^{7} + \cdots + 15671836160289)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} - 13248 T^{14} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{8} + 20 T^{7} + \cdots + 20616240141121)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 366 T^{7} + \cdots + 34109606357649)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 21968 T^{6} + \cdots + 728195406129216)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 27924 T^{14} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} - 180 T^{7} + \cdots + 33546476292624)^{2} \) Copy content Toggle raw display
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