Properties

Label 336.6.bl.e
Level $336$
Weight $6$
Character orbit 336.bl
Analytic conductor $53.889$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,6,Mod(31,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.31");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.bl (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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 434 x^{12} - 1773 x^{11} + 161210 x^{10} - 411219 x^{9} + 13011143 x^{8} + 40309634 x^{7} + 721029148 x^{6} + 394325472 x^{5} + \cdots + 122943744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + ( - \beta_{2} + 2 \beta_1 - 1) q^{5} + (\beta_{4} + 18 \beta_1 + 10) q^{7} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + ( - \beta_{2} + 2 \beta_1 - 1) q^{5} + (\beta_{4} + 18 \beta_1 + 10) q^{7} + ( - 81 \beta_1 - 81) q^{9} + (\beta_{11} - \beta_{9} - \beta_{3} + \beta_{2} + 26 \beta_1 + 52) q^{11} + ( - \beta_{9} - \beta_{6} - \beta_{5} + \beta_{2} - 25 \beta_1 - 13) q^{13} + (9 \beta_{3} + 9 \beta_{2} - 36 \beta_1 - 18) q^{15} + (\beta_{13} - 2 \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + \cdots - 82) q^{17}+ \cdots + ( - 81 \beta_{7} + 81 \beta_{4} + 162 \beta_{3} + 81 \beta_{2} + \cdots - 2187) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 63 q^{3} - 33 q^{5} + 13 q^{7} - 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 63 q^{3} - 33 q^{5} + 13 q^{7} - 567 q^{9} + 549 q^{11} - 870 q^{17} + 146 q^{19} - 1773 q^{21} + 6396 q^{23} + 5134 q^{25} + 10206 q^{27} + 16458 q^{29} - 3137 q^{31} - 4941 q^{33} - 20910 q^{35} + 2162 q^{37} + 2376 q^{39} + 2673 q^{45} - 720 q^{47} + 23477 q^{49} + 7830 q^{51} - 44835 q^{53} - 72666 q^{55} - 2628 q^{57} - 44841 q^{59} + 93060 q^{61} + 14904 q^{63} - 4884 q^{65} + 68484 q^{67} - 163632 q^{73} + 46206 q^{75} + 99681 q^{77} + 134205 q^{79} - 45927 q^{81} + 188634 q^{83} + 276972 q^{85} - 74061 q^{87} - 49302 q^{89} + 110976 q^{91} - 28233 q^{93} + 60222 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 434 x^{12} - 1773 x^{11} + 161210 x^{10} - 411219 x^{9} + 13011143 x^{8} + 40309634 x^{7} + 721029148 x^{6} + 394325472 x^{5} + \cdots + 122943744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 71\!\cdots\!37 \nu^{13} + \cdots - 31\!\cdots\!64 ) / 29\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22\!\cdots\!35 \nu^{13} + \cdots + 15\!\cdots\!08 ) / 22\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!78 \nu^{13} + \cdots - 14\!\cdots\!92 ) / 56\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54\!\cdots\!91 \nu^{13} + \cdots - 65\!\cdots\!48 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41\!\cdots\!57 \nu^{13} + \cdots - 57\!\cdots\!16 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!59 \nu^{13} + \cdots + 21\!\cdots\!08 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\!\cdots\!85 \nu^{13} + \cdots - 36\!\cdots\!84 ) / 91\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35\!\cdots\!52 \nu^{13} + \cdots - 37\!\cdots\!04 ) / 45\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!31 \nu^{13} + \cdots - 36\!\cdots\!80 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!73 \nu^{13} + \cdots + 38\!\cdots\!92 ) / 91\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!95 \nu^{13} + \cdots - 36\!\cdots\!48 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!85 \nu^{13} + \cdots - 87\!\cdots\!04 ) / 39\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!77 \nu^{13} + \cdots - 25\!\cdots\!72 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 2 \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} + 25 \beta_{3} + 14 \beta_{2} + 8 \beta _1 + 21 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - 2 \beta_{12} + 26 \beta_{11} - 12 \beta_{10} - 13 \beta_{9} + 5 \beta_{8} - 28 \beta_{7} + 26 \beta_{6} + \beta_{5} + 31 \beta_{4} + \beta_{3} - 49 \beta_{2} + 10402 \beta _1 + 33 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 160 \beta_{13} + 160 \beta_{12} - 120 \beta_{11} - 295 \beta_{10} + 778 \beta_{9} - 482 \beta_{8} + 140 \beta_{7} + 41 \beta_{6} - 201 \beta_{5} - 893 \beta_{4} - 1956 \beta_{3} + 1038 \beta_{2} + 643 \beta _1 + 23453 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 458 \beta_{13} + 229 \beta_{12} + 2770 \beta_{11} - 942 \beta_{10} - 2103 \beta_{9} + 511 \beta_{8} + 385 \beta_{7} + 229 \beta_{6} - 1927 \beta_{5} + 301 \beta_{4} + 7194 \beta_{3} + 6120 \beta_{2} + \cdots - 788412 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 56592 \beta_{13} - 113184 \beta_{12} - 30808 \beta_{11} + 228967 \beta_{10} + 186676 \beta_{9} + 89684 \beta_{8} - 87465 \beta_{7} - 35897 \beta_{6} + 56592 \beta_{5} + 353428 \beta_{4} + \cdots + 315040 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 35392 \beta_{13} + 35392 \beta_{12} - 494256 \beta_{11} + 121941 \beta_{10} + 305410 \beta_{9} - 149124 \beta_{8} + 264824 \beta_{7} - 180637 \beta_{6} + 145245 \beta_{5} - 366687 \beta_{4} + \cdots + 77457365 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 41336128 \beta_{13} + 20668064 \beta_{12} + 60331996 \beta_{11} - 55214356 \beta_{10} - 182542874 \beta_{9} + 39366814 \beta_{8} + 2644005 \beta_{7} + \cdots - 10445815093 ) / 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 58279538 \beta_{13} - 116559076 \beta_{12} + 228667001 \beta_{11} + 64316760 \beta_{10} + 57744381 \beta_{9} + 134323486 \beta_{8} - 409436104 \beta_{7} + \cdots + 598081116 ) / 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7729605064 \beta_{13} + 7729605064 \beta_{12} - 27772175880 \beta_{11} - 6276207827 \beta_{10} + 45998460754 \beta_{9} - 28180147806 \beta_{8} + \cdots + 4773557127293 ) / 84 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 103132781184 \beta_{13} + 51566390592 \beta_{12} + 313437460136 \beta_{11} - 163482921540 \beta_{10} - 450823416098 \beta_{9} + 85505305256 \beta_{8} + \cdots - 74734122914147 ) / 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2942827766400 \beta_{13} - 5885655532800 \beta_{12} + 1760003828540 \beta_{11} + 9847550119691 \beta_{10} + 8088118710784 \beta_{9} + \cdots + 19105401960776 ) / 84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1575539056014 \beta_{13} + 1575539056014 \beta_{12} - 12817309273608 \beta_{11} + 1708841987508 \beta_{10} + 11116792464900 \beta_{9} + \cdots + 20\!\cdots\!43 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22\!\cdots\!08 \beta_{13} + \cdots - 89\!\cdots\!21 ) / 84 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−3.37779 5.85050i
−10.0185 17.3526i
0.200900 + 0.347970i
8.48408 + 14.6949i
−0.796063 1.37882i
0.332438 + 0.575799i
5.67497 + 9.82934i
−3.37779 + 5.85050i
−10.0185 + 17.3526i
0.200900 0.347970i
8.48408 14.6949i
−0.796063 + 1.37882i
0.332438 0.575799i
5.67497 9.82934i
0 −4.50000 + 7.79423i 0 −88.4614 + 51.0732i 0 129.044 + 12.4399i 0 −40.5000 70.1481i 0
31.2 0 −4.50000 + 7.79423i 0 −54.0884 + 31.2279i 0 −39.2172 + 123.568i 0 −40.5000 70.1481i 0
31.3 0 −4.50000 + 7.79423i 0 −20.8640 + 12.0458i 0 −124.767 + 35.2169i 0 −40.5000 70.1481i 0
31.4 0 −4.50000 + 7.79423i 0 5.98712 3.45667i 0 77.5174 103.914i 0 −40.5000 70.1481i 0
31.5 0 −4.50000 + 7.79423i 0 16.5738 9.56886i 0 −63.7331 112.894i 0 −40.5000 70.1481i 0
31.6 0 −4.50000 + 7.79423i 0 39.5034 22.8073i 0 115.029 + 59.7931i 0 −40.5000 70.1481i 0
31.7 0 −4.50000 + 7.79423i 0 84.8494 48.9878i 0 −87.3733 + 95.7753i 0 −40.5000 70.1481i 0
271.1 0 −4.50000 7.79423i 0 −88.4614 51.0732i 0 129.044 12.4399i 0 −40.5000 + 70.1481i 0
271.2 0 −4.50000 7.79423i 0 −54.0884 31.2279i 0 −39.2172 123.568i 0 −40.5000 + 70.1481i 0
271.3 0 −4.50000 7.79423i 0 −20.8640 12.0458i 0 −124.767 35.2169i 0 −40.5000 + 70.1481i 0
271.4 0 −4.50000 7.79423i 0 5.98712 + 3.45667i 0 77.5174 + 103.914i 0 −40.5000 + 70.1481i 0
271.5 0 −4.50000 7.79423i 0 16.5738 + 9.56886i 0 −63.7331 + 112.894i 0 −40.5000 + 70.1481i 0
271.6 0 −4.50000 7.79423i 0 39.5034 + 22.8073i 0 115.029 59.7931i 0 −40.5000 + 70.1481i 0
271.7 0 −4.50000 7.79423i 0 84.8494 + 48.9878i 0 −87.3733 95.7753i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.6.bl.e 14
4.b odd 2 1 336.6.bl.g yes 14
7.d odd 6 1 336.6.bl.g yes 14
28.f even 6 1 inner 336.6.bl.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.6.bl.e 14 1.a even 1 1 trivial
336.6.bl.e 14 28.f even 6 1 inner
336.6.bl.g yes 14 4.b odd 2 1
336.6.bl.g yes 14 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{14} + 33 T_{5}^{13} - 12960 T_{5}^{12} - 439659 T_{5}^{11} + 139139352 T_{5}^{10} + 3764574873 T_{5}^{9} - 413869138389 T_{5}^{8} - 9603746274096 T_{5}^{7} + \cdots + 82\!\cdots\!28 \) Copy content Toggle raw display
\( T_{11}^{14} - 549 T_{11}^{13} - 376572 T_{11}^{12} + 261894411 T_{11}^{11} + 146985837588 T_{11}^{10} - 60848112178125 T_{11}^{9} + \cdots + 87\!\cdots\!12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + 33 T^{13} + \cdots + 82\!\cdots\!28 \) Copy content Toggle raw display
$7$ \( T^{14} - 13 T^{13} + \cdots + 37\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{14} - 549 T^{13} + \cdots + 87\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{14} + 2417580 T^{12} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{14} + 870 T^{13} + \cdots + 37\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{14} - 146 T^{13} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{14} - 6396 T^{13} + \cdots + 24\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( (T^{7} - 8229 T^{6} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 3137 T^{13} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{14} - 2162 T^{13} + \cdots + 64\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{14} + 1111329108 T^{12} + \cdots + 22\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{14} + 1272023280 T^{12} + \cdots + 56\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{14} + 720 T^{13} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + 44835 T^{13} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{14} + 44841 T^{13} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{14} - 93060 T^{13} + \cdots + 50\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{14} - 68484 T^{13} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + 20486485152 T^{12} + \cdots + 10\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{14} + 163632 T^{13} + \cdots + 48\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{14} - 134205 T^{13} + \cdots + 16\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( (T^{7} - 94317 T^{6} + \cdots + 18\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + 49302 T^{13} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{14} + 42784338465 T^{12} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
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