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Properties

Label 342.5.c.b
Level $342$
Weight $5$
Character orbit 342.c
Analytic conductor $35.353$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.305");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(35.3525273747\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 472 x^{10} + 2048 x^{9} + 80198 x^{8} - 427384 x^{7} - 5380292 x^{6} + \cdots + 7150951512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{6}\cdot 19^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} - 8 q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{6} - 7) q^{7} - 16 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} - 8 q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{6} - 7) q^{7} - 16 \beta_{2} q^{8} + (2 \beta_{3} + 6) q^{10} + (\beta_{10} + 2 \beta_{8} + \cdots - 5 \beta_{2}) q^{11}+ \cdots + (36 \beta_{11} + 36 \beta_{10} + \cdots + 510 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 96 q^{4} - 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 96 q^{4} - 80 q^{7} + 72 q^{10} - 400 q^{13} + 768 q^{16} + 184 q^{22} - 1836 q^{25} + 640 q^{28} - 2536 q^{31} + 1864 q^{34} - 896 q^{37} - 576 q^{40} + 6572 q^{43} + 16 q^{46} + 3744 q^{49} + 3200 q^{52} + 7356 q^{55} - 3888 q^{58} - 1776 q^{61} - 6144 q^{64} - 11224 q^{67} - 2136 q^{70} - 8012 q^{73} - 8600 q^{79} - 9152 q^{82} + 324 q^{85} - 1472 q^{88} + 18232 q^{91} - 21416 q^{94} + 41256 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 472 x^{10} + 2048 x^{9} + 80198 x^{8} - 427384 x^{7} - 5380292 x^{6} + \cdots + 7150951512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 30\!\cdots\!05 \nu^{11} + \cdots - 15\!\cdots\!72 ) / 16\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 47\!\cdots\!22 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 80\!\cdots\!13 \nu^{11} + \cdots + 23\!\cdots\!40 ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29\!\cdots\!24 \nu^{11} + \cdots - 41\!\cdots\!36 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!65 \nu^{11} + \cdots - 35\!\cdots\!16 ) / 18\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47\!\cdots\!89 \nu^{11} + \cdots + 69\!\cdots\!16 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 45\!\cdots\!20 \nu^{11} + \cdots + 52\!\cdots\!56 ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!49 \nu^{11} + \cdots - 28\!\cdots\!76 ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 56\!\cdots\!83 \nu^{11} + \cdots + 19\!\cdots\!32 ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 70\!\cdots\!65 \nu^{11} + \cdots + 47\!\cdots\!68 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 95\!\cdots\!84 \nu^{11} + \cdots + 31\!\cdots\!56 ) / 73\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 19\beta_{7} - 26\beta_{6} - 14\beta_{5} - 171\beta_{2} + 7\beta _1 + 64 ) / 171 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 14 \beta_{11} + 38 \beta_{10} - 52 \beta_{9} - 48 \beta_{8} - 9 \beta_{6} - 18 \beta_{5} + \cdots + 13689 ) / 171 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 540 \beta_{11} - 27 \beta_{9} - 459 \beta_{8} + 2185 \beta_{7} - 3728 \beta_{6} - 1547 \beta_{5} + \cdots - 4898 ) / 171 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6284 \beta_{11} + 9044 \beta_{10} - 15328 \beta_{9} - 17832 \beta_{8} + 342 \beta_{7} - 10395 \beta_{6} + \cdots + 1750149 ) / 171 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 152370 \beta_{11} + 1710 \beta_{10} - 52335 \beta_{9} - 157815 \beta_{8} + 279091 \beta_{7} + \cdots + 2197738 ) / 171 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1689686 \beta_{11} + 1856642 \beta_{10} - 3486820 \beta_{9} - 4323000 \beta_{8} + 510948 \beta_{7} + \cdots + 232443045 ) / 171 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 35583156 \beta_{11} + 3624516 \beta_{10} - 19992231 \beta_{9} - 41345703 \beta_{8} + 36232069 \beta_{7} + \cdots + 779210662 ) / 171 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 400406648 \beta_{11} + 360534728 \beta_{10} - 712196800 \beta_{9} - 933771792 \beta_{8} + \cdots + 29829710781 ) / 171 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7573182318 \beta_{11} + 1446275250 \beta_{10} - 5469373935 \beta_{9} - 9659238903 \beta_{8} + \cdots + 161442657466 ) / 171 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 87268036754 \beta_{11} + 67113934118 \beta_{10} - 137581760548 \beta_{9} - 188787051552 \beta_{8} + \cdots + 3510612907245 ) / 171 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1512783759168 \beta_{11} + 398758231080 \beta_{10} - 1271840372847 \beta_{9} - 2075400585423 \beta_{8} + \cdots + 25405260914422 ) / 171 \) 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
6.46285 + 1.41421i
−11.5274 + 1.41421i
1.31561 + 1.41421i
3.49316 + 1.41421i
−11.1374 + 1.41421i
13.3932 + 1.41421i
13.3932 − 1.41421i
−11.1374 − 1.41421i
3.49316 − 1.41421i
1.31561 − 1.41421i
−11.5274 − 1.41421i
6.46285 − 1.41421i
− 2.82843i 0 −8.00000 − 35.8577i 0 16.6454 22.6274i 0 −101.421
305.2 − 2.82843i 0 −8.00000 − 20.2535i 0 26.0682 22.6274i 0 −57.2855
305.3 − 2.82843i 0 −8.00000 − 16.3532i 0 −80.8606 22.6274i 0 −46.2538
305.4 − 2.82843i 0 −8.00000 15.4219i 0 −1.35080 22.6274i 0 43.6197
305.5 − 2.82843i 0 −8.00000 30.8350i 0 66.0097 22.6274i 0 87.2145
305.6 − 2.82843i 0 −8.00000 38.9354i 0 −66.5119 22.6274i 0 110.126
305.7 2.82843i 0 −8.00000 − 38.9354i 0 −66.5119 − 22.6274i 0 110.126
305.8 2.82843i 0 −8.00000 − 30.8350i 0 66.0097 − 22.6274i 0 87.2145
305.9 2.82843i 0 −8.00000 − 15.4219i 0 −1.35080 − 22.6274i 0 43.6197
305.10 2.82843i 0 −8.00000 16.3532i 0 −80.8606 − 22.6274i 0 −46.2538
305.11 2.82843i 0 −8.00000 20.2535i 0 26.0682 − 22.6274i 0 −57.2855
305.12 2.82843i 0 −8.00000 35.8577i 0 16.6454 − 22.6274i 0 −101.421
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.12
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 342.5.c.b ✓ 12
3.b odd 2 1 inner 342.5.c.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.5.c.b ✓ 12 1.a even 1 1 trivial
342.5.c.b ✓ 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 4668 T_{5}^{10} + 8319249 T_{5}^{8} + 7118901488 T_{5}^{6} + 3044031880824 T_{5}^{4} + \cdots + 48\!\cdots\!36 \) acting on \(S_{5}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( (T^{6} + 40 T^{5} + \cdots - 208084896)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 1103512500000)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6859)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 50\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 48\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 60\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 49\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 43\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
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