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Properties

Label 348.6.a.a
Level $348$
Weight $6$
Character orbit 348.a
Self dual yes
Analytic conductor $55.814$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,6,Mod(1,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 348.a (trivial)

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: yes
Analytic conductor: \(55.8135692949\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 7568x^{4} - 107678x^{3} + 10875535x^{2} + 149445472x - 2970578880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 q^{3} + ( - \beta_{2} - 6) q^{5} + ( - \beta_{5} + 20) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + ( - \beta_{2} - 6) q^{5} + ( - \beta_{5} + 20) q^{7} + 81 q^{9} + ( - \beta_{5} - 2 \beta_{4} + \cdots + 42) q^{11}+ \cdots + ( - 81 \beta_{5} - 162 \beta_{4} + \cdots + 3402) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 54 q^{3} - 36 q^{5} + 117 q^{7} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 54 q^{3} - 36 q^{5} + 117 q^{7} + 486 q^{9} + 247 q^{11} - 183 q^{13} + 324 q^{15} - 1105 q^{17} - 1178 q^{19} - 1053 q^{21} - 2434 q^{23} - 3024 q^{25} - 4374 q^{27} - 5046 q^{29} - 5726 q^{31} - 2223 q^{33} - 8076 q^{35} + 988 q^{37} + 1647 q^{39} - 148 q^{41} + 18674 q^{43} - 2916 q^{45} + 13397 q^{47} + 17435 q^{49} + 9945 q^{51} + 23170 q^{53} + 44948 q^{55} + 10602 q^{57} + 57790 q^{59} + 66368 q^{61} + 9477 q^{63} + 15544 q^{65} + 135667 q^{67} + 21906 q^{69} + 119566 q^{71} + 109352 q^{73} + 27216 q^{75} + 38413 q^{77} + 145584 q^{79} + 39366 q^{81} + 167942 q^{83} + 216184 q^{85} + 45414 q^{87} + 163977 q^{89} + 245777 q^{91} + 51534 q^{93} + 220730 q^{95} + 122458 q^{97} + 20007 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 7568x^{4} - 107678x^{3} + 10875535x^{2} + 149445472x - 2970578880 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12946 \nu^{5} - 1484045 \nu^{4} - 10149244 \nu^{3} + 5048575637 \nu^{2} - 83782289214 \nu - 2449601393472 ) / 24754505976 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 84901 \nu^{5} - 2269857 \nu^{4} - 542309561 \nu^{3} + 5382584481 \nu^{2} + \cdots - 6893364090528 ) / 99018023904 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28729 \nu^{5} - 1885761 \nu^{4} - 167313752 \nu^{3} + 8264416401 \nu^{2} + \cdots - 6186658029048 ) / 24754505976 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 199817 \nu^{5} + 9812901 \nu^{4} + 1211564569 \nu^{3} - 38440250085 \nu^{2} + \cdots + 31540978182816 ) / 99018023904 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 163693 \nu^{5} - 6627949 \nu^{4} - 968129283 \nu^{3} + 16540877269 \nu^{2} + \cdots - 6548207662080 ) / 49509011952 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -40\beta_{5} + 15\beta_{4} + 39\beta_{3} + 127\beta_{2} + 16\beta _1 + 10103 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1664\beta_{5} + 4649\beta_{4} + 3849\beta_{3} + 10665\beta_{2} + 2432\beta _1 + 244105 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -254696\beta_{5} + 205775\beta_{4} + 243943\beta_{3} + 1058495\beta_{2} + 127472\beta _1 + 48036023 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 14902336 \beta_{5} + 27855465 \beta_{4} + 22244297 \beta_{3} + 86645225 \beta_{2} + 17928160 \beta _1 + 2521360201 ) / 4 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
85.6180
−58.7935
−43.1401
11.8455
37.3713
−30.9012
0 −9.00000 0 −79.6117 0 130.624 0 81.0000 0
1.2 0 −9.00000 0 −38.9358 0 200.640 0 81.0000 0
1.3 0 −9.00000 0 −33.3645 0 −160.894 0 81.0000 0
1.4 0 −9.00000 0 0.435537 0 −113.109 0 81.0000 0
1.5 0 −9.00000 0 50.9851 0 −71.3560 0 81.0000 0
1.6 0 −9.00000 0 64.4913 0 131.095 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.6.a.a ✓ 6
3.b odd 2 1 1044.6.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.6.a.a ✓ 6 1.a even 1 1 trivial
1044.6.a.g 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 36T_{5}^{5} - 7215T_{5}^{4} - 208630T_{5}^{3} + 11347080T_{5}^{2} + 335156904T_{5} - 148108176 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(348))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 36 T^{5} + \cdots - 148108176 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 4461620729520 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 32\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 99\!\cdots\!50 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 20\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( (T + 841)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 31\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 94\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 49\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 17\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 28\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 21\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 69\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 90\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 34\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
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