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Properties

Label 336.6.bl.f
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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Show commands: Magma / PariGP / SageMath

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: \( x^{14} - 2 x^{13} + 1803 x^{12} + 29002 x^{11} + 2626116 x^{10} + 28853720 x^{9} + 1155561735 x^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{6}\cdot 7^{3} \)
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.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + (\beta_{6} - 3 \beta_1 + 3) q^{5} + (\beta_{7} - \beta_{3} + 8 \beta_1 - 5) q^{7} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + (\beta_{6} - 3 \beta_1 + 3) q^{5} + (\beta_{7} - \beta_{3} + 8 \beta_1 - 5) q^{7} + ( - 81 \beta_1 - 81) q^{9} + ( - \beta_{11} - \beta_{7} + \beta_{5} + \cdots + 75) q^{11}+ \cdots + (81 \beta_{13} + 81 \beta_{11} + \cdots - 2997) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 63 q^{3} + 66 q^{5} - 113 q^{7} - 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 63 q^{3} + 66 q^{5} - 113 q^{7} - 567 q^{9} + 786 q^{11} - 2785 q^{19} - 180 q^{21} - 7260 q^{23} + 8653 q^{25} + 10206 q^{27} - 11712 q^{29} + 3133 q^{31} - 7074 q^{33} + 23502 q^{35} + 7901 q^{37} - 1107 q^{39} - 5346 q^{45} + 10326 q^{47} + 509 q^{49} + 564 q^{53} - 54456 q^{55} + 50130 q^{57} + 44328 q^{59} - 66120 q^{61} + 10773 q^{63} + 82398 q^{65} - 226371 q^{67} - 88659 q^{73} + 77877 q^{75} - 192276 q^{77} + 61131 q^{79} - 45927 q^{81} + 42732 q^{83} + 45312 q^{85} + 52704 q^{87} + 287412 q^{89} + 137853 q^{91} + 28197 q^{93} - 2166 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 1803 x^{12} + 29002 x^{11} + 2626116 x^{10} + 28853720 x^{9} + 1155561735 x^{8} + \cdots + 22\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 22\!\cdots\!92 \nu^{13} + \cdots + 58\!\cdots\!08 ) / 32\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 42\!\cdots\!16 \nu^{13} + \cdots + 33\!\cdots\!84 ) / 57\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38\!\cdots\!47 \nu^{13} + \cdots - 71\!\cdots\!52 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 70\!\cdots\!15 \nu^{13} + \cdots + 14\!\cdots\!16 ) / 40\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!54 \nu^{13} + \cdots - 43\!\cdots\!36 ) / 57\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\!\cdots\!00 \nu^{13} + \cdots + 12\!\cdots\!56 ) / 57\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!88 \nu^{13} + \cdots - 40\!\cdots\!96 ) / 63\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!68 \nu^{13} + \cdots - 23\!\cdots\!60 ) / 57\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!46 \nu^{13} + \cdots - 88\!\cdots\!08 ) / 57\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!65 \nu^{13} + \cdots + 19\!\cdots\!36 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 41\!\cdots\!26 \nu^{13} + \cdots - 10\!\cdots\!36 ) / 57\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21\!\cdots\!33 \nu^{13} + \cdots + 10\!\cdots\!08 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 85\!\cdots\!07 \nu^{13} + \cdots + 43\!\cdots\!76 ) / 66\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 4 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} - \beta_{9} - 14 \beta_{7} + 37 \beta_{6} + \cdots + 3 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{13} - 48 \beta_{12} + 4 \beta_{11} + 127 \beta_{10} - 96 \beta_{9} + 70 \beta_{8} + \cdots - 86301 ) / 168 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 766 \beta_{13} + 824 \beta_{12} - 766 \beta_{11} + 2088 \beta_{10} - 824 \beta_{9} + 567 \beta_{8} + \cdots - 579395 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 42596 \beta_{13} + 148992 \beta_{12} + 21298 \beta_{11} + 72939 \beta_{10} + 74496 \beta_{9} + \cdots + 34734 ) / 168 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1946862 \beta_{13} + 2906383 \beta_{12} + 3893724 \beta_{11} - 4061678 \beta_{10} + 5812766 \beta_{9} + \cdots + 2922622658 ) / 168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1967782 \beta_{13} - 4642301 \beta_{12} + 1967782 \beta_{11} - 14315969 \beta_{10} + \cdots + 5641356658 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6119446428 \beta_{13} - 10655462330 \beta_{12} - 3059723214 \beta_{11} - 7510091181 \beta_{10} + \cdots - 1420946937 ) / 168 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 111739921270 \beta_{13} - 233325569840 \beta_{12} - 223479842540 \beta_{11} + \cdots - 272067444349937 ) / 168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2630596057582 \beta_{13} + 4890517667642 \beta_{12} - 2630596057582 \beta_{11} + 15145905352314 \beta_{10} + \cdots - 55\!\cdots\!23 ) / 84 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 423343654181204 \beta_{13} + 845338334936232 \beta_{12} + 211671827090602 \beta_{11} + \cdots + 123383185319730 ) / 168 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 93\!\cdots\!70 \beta_{13} + \cdots + 20\!\cdots\!78 ) / 168 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 69\!\cdots\!23 \beta_{13} + \cdots + 15\!\cdots\!95 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 33\!\cdots\!20 \beta_{13} + \cdots - 96\!\cdots\!05 ) / 168 \) Copy content Toggle raw display

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

Character values

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

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

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
−13.1137 + 22.7136i
−12.0227 + 20.8239i
8.80516 − 15.2510i
1.72535 − 2.98840i
1.06930 − 1.85209i
21.3516 − 36.9820i
−6.81497 + 11.8039i
−13.1137 − 22.7136i
−12.0227 − 20.8239i
8.80516 + 15.2510i
1.72535 + 2.98840i
1.06930 + 1.85209i
21.3516 + 36.9820i
−6.81497 − 11.8039i
0 −4.50000 + 7.79423i 0 −85.4360 + 49.3265i 0 −124.613 − 35.7563i 0 −40.5000 − 70.1481i 0
31.2 0 −4.50000 + 7.79423i 0 −49.9818 + 28.8570i 0 118.297 − 53.0350i 0 −40.5000 − 70.1481i 0
31.3 0 −4.50000 + 7.79423i 0 −18.2166 + 10.5173i 0 11.9043 + 129.094i 0 −40.5000 − 70.1481i 0
31.4 0 −4.50000 + 7.79423i 0 −0.920368 + 0.531375i 0 −119.208 − 50.9556i 0 −40.5000 − 70.1481i 0
31.5 0 −4.50000 + 7.79423i 0 36.3441 − 20.9833i 0 −45.6015 + 121.357i 0 −40.5000 − 70.1481i 0
31.6 0 −4.50000 + 7.79423i 0 73.0945 − 42.2011i 0 113.495 + 62.6565i 0 −40.5000 − 70.1481i 0
31.7 0 −4.50000 + 7.79423i 0 78.1160 − 45.1003i 0 −10.7742 − 129.193i 0 −40.5000 − 70.1481i 0
271.1 0 −4.50000 − 7.79423i 0 −85.4360 − 49.3265i 0 −124.613 + 35.7563i 0 −40.5000 + 70.1481i 0
271.2 0 −4.50000 − 7.79423i 0 −49.9818 − 28.8570i 0 118.297 + 53.0350i 0 −40.5000 + 70.1481i 0
271.3 0 −4.50000 − 7.79423i 0 −18.2166 − 10.5173i 0 11.9043 − 129.094i 0 −40.5000 + 70.1481i 0
271.4 0 −4.50000 − 7.79423i 0 −0.920368 − 0.531375i 0 −119.208 + 50.9556i 0 −40.5000 + 70.1481i 0
271.5 0 −4.50000 − 7.79423i 0 36.3441 + 20.9833i 0 −45.6015 − 121.357i 0 −40.5000 + 70.1481i 0
271.6 0 −4.50000 − 7.79423i 0 73.0945 + 42.2011i 0 113.495 − 62.6565i 0 −40.5000 + 70.1481i 0
271.7 0 −4.50000 − 7.79423i 0 78.1160 + 45.1003i 0 −10.7742 + 129.193i 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.f ✓ 14
4.b odd 2 1 336.6.bl.h yes 14
7.d odd 6 1 336.6.bl.h yes 14
28.f even 6 1 inner 336.6.bl.f ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.6.bl.f ✓ 14 1.a even 1 1 trivial
336.6.bl.f ✓ 14 28.f even 6 1 inner
336.6.bl.h yes 14 4.b odd 2 1
336.6.bl.h 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 \(S_{6}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{14} - 66 T_{5}^{13} - 13086 T_{5}^{12} + 959508 T_{5}^{11} + 140554755 T_{5}^{10} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
\( T_{11}^{14} - 786 T_{11}^{13} - 515934 T_{11}^{12} + 567386676 T_{11}^{11} + 255201496227 T_{11}^{10} + \cdots + 12\!\cdots\!08 \) 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
$5$ \( T^{14} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 37\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 45\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 92\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 46\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 38\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 42\!\cdots\!03 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 80\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 28\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 51\!\cdots\!12 \) Copy content Toggle raw display
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