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Properties

Label 336.3.z.f
Level $336$
Weight $3$
Character orbit 336.z
Analytic conductor $9.155$
Analytic rank $0$
Dimension $20$
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] = [336,3,Mod(47,336)]
 
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, 3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.z (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: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 83 x^{18} + 2702 x^{16} + 44346 x^{14} + 396449 x^{12} + 1961403 x^{10} + 5268164 x^{8} + \cdots + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{9} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{13} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{11} - \beta_{4}) q^{7} + ( - \beta_{6} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{13} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{11} - \beta_{4}) q^{7} + ( - \beta_{6} + \beta_{4} + 1) q^{9} + (\beta_{19} + \beta_{17} + \cdots + \beta_{4}) q^{11}+ \cdots + ( - 2 \beta_{18} + 10 \beta_{17} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 3 q^{3} + 16 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 3 q^{3} + 16 q^{7} + 11 q^{9} + 44 q^{19} - 5 q^{21} - 98 q^{25} + 144 q^{27} + 70 q^{31} + 15 q^{33} - 64 q^{37} - 6 q^{39} + 21 q^{45} - 184 q^{49} + 33 q^{51} - 120 q^{55} + 218 q^{57} - 102 q^{61} - 27 q^{63} + 144 q^{67} + 372 q^{73} + 246 q^{75} - 258 q^{79} + 287 q^{81} - 108 q^{85} - 318 q^{87} - 510 q^{91} - 173 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 83 x^{18} + 2702 x^{16} + 44346 x^{14} + 396449 x^{12} + 1961403 x^{10} + 5268164 x^{8} + \cdots + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13\!\cdots\!97 \nu^{19} + \cdots + 18\!\cdots\!72 ) / 11\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 87\!\cdots\!39 \nu^{18} + \cdots + 17\!\cdots\!48 ) / 17\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!97 \nu^{19} + \cdots - 21\!\cdots\!64 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38906132571 \nu^{19} - 3222931741433 \nu^{17} - 104620580034354 \nu^{15} + \cdots - 10\!\cdots\!84 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\!\cdots\!73 \nu^{19} + \cdots + 28\!\cdots\!96 ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!85 \nu^{19} + \cdots - 37\!\cdots\!64 ) / 46\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!65 \nu^{19} + \cdots - 69\!\cdots\!68 ) / 46\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!65 \nu^{19} + \cdots - 16\!\cdots\!64 ) / 46\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 50\!\cdots\!73 \nu^{19} + \cdots - 92\!\cdots\!12 ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50\!\cdots\!73 \nu^{19} + \cdots - 92\!\cdots\!12 ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76\!\cdots\!11 \nu^{19} + \cdots + 39\!\cdots\!96 ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 76\!\cdots\!11 \nu^{19} + \cdots - 39\!\cdots\!96 ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 77\!\cdots\!85 \nu^{19} + \cdots + 34\!\cdots\!96 ) / 69\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!74 \nu^{19} + \cdots - 47\!\cdots\!32 ) / 11\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\!\cdots\!13 \nu^{19} + \cdots + 50\!\cdots\!88 ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 15\!\cdots\!55 \nu^{19} + \cdots + 51\!\cdots\!64 ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{19} + \cdots + 93\!\cdots\!16 ) / 77\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 13\!\cdots\!07 \nu^{19} + \cdots - 23\!\cdots\!80 ) / 46\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 33\!\cdots\!59 \nu^{19} + \cdots + 20\!\cdots\!68 ) / 92\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{17} + \beta_{15} + 2\beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + 2\beta_{4} - \beta_{3} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{18} + \beta_{17} + \beta_{15} + \beta_{14} - \beta_{11} + \beta_{10} - \beta_{8} - \beta_{6} + \cdots - 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 19 \beta_{17} - 6 \beta_{16} - 16 \beta_{15} + \beta_{14} - 18 \beta_{13} - 50 \beta_{12} - 34 \beta_{11} + \cdots + 9 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21 \beta_{18} - 26 \beta_{17} - 32 \beta_{15} - 21 \beta_{14} - 12 \beta_{12} + 32 \beta_{11} + \cdots + 487 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 48 \beta_{19} + 42 \beta_{18} - 329 \beta_{17} + 174 \beta_{16} + 308 \beta_{15} - 35 \beta_{14} + \cdots - 189 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 488 \beta_{18} + 549 \beta_{17} + 834 \beta_{15} + 452 \beta_{14} + 528 \beta_{12} - 792 \beta_{11} + \cdots - 10825 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2112 \beta_{19} - 1980 \beta_{18} + 5693 \beta_{17} - 4094 \beta_{16} - 6682 \beta_{15} + 1409 \beta_{14} + \cdots + 3153 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11955 \beta_{18} - 10727 \beta_{17} - 20231 \beta_{15} - 9939 \beta_{14} - 17442 \beta_{12} + \cdots + 255103 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 69768 \beta_{19} + 67374 \beta_{18} - 97363 \beta_{17} + 90402 \beta_{16} + 154420 \beta_{15} + \cdots - 28463 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 297202 \beta_{18} + 197504 \beta_{17} + 477842 \beta_{15} + 217987 \beta_{14} + 517386 \beta_{12} + \cdots - 6169146 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2069544 \beta_{19} - 2041764 \beta_{18} + 1582999 \beta_{17} - 1940138 \beta_{16} - 3689466 \beta_{15} + \cdots - 687853 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 7412481 \beta_{18} - 3343002 \beta_{17} - 11166747 \beta_{15} - 4726590 \beta_{14} - 14574060 \beta_{12} + \cdots + 151218224 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 58296240 \beta_{19} + 58634862 \beta_{18} - 22433225 \beta_{17} + 40821054 \beta_{16} + 89805680 \beta_{15} + \cdots + 53256663 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 185089352 \beta_{18} + 47656511 \beta_{17} + 259910690 \beta_{15} + 100981490 \beta_{14} + \cdots - 3737205231 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1596946176 \beta_{19} - 1635056796 \beta_{18} + 198078473 \beta_{17} - 841741854 \beta_{16} + \cdots - 2224093583 ) / 6 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 4627889247 \beta_{18} - 375768709 \beta_{17} - 6044378731 \beta_{15} - 2120069061 \beta_{14} + \cdots + 92885863545 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 43036469976 \beta_{19} + 44774298630 \beta_{18} + 3004567049 \beta_{17} + 16921544922 \beta_{16} + \cdots + 77233127965 ) / 6 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 115920769138 \beta_{18} - 9294414474 \beta_{17} + 140685962130 \beta_{15} + 43562752969 \beta_{14} + \cdots - 2318597508184 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 1147888980120 \beta_{19} - 1211138880060 \beta_{18} - 254012019461 \beta_{17} - 328197915154 \beta_{16} + \cdots - 2449292337953 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
− 0.836090i
− 1.24117i
− 3.70117i
− 5.07301i
− 2.61521i
− 1.01894i
2.86395i
4.75207i
1.63179i
0.0416279i
0.836090i
1.24117i
3.70117i
5.07301i
2.61521i
1.01894i
− 2.86395i
− 4.75207i
− 1.63179i
− 0.0416279i
0 −2.78299 − 1.12025i 0 1.01058 − 1.75037i 0 −0.0684533 + 6.99967i 0 6.49009 + 6.23528i 0
47.2 0 −2.66635 + 1.37497i 0 1.28887 − 2.23238i 0 −6.74427 − 1.87480i 0 5.21890 − 7.33233i 0
47.3 0 −2.50321 − 1.65346i 0 −3.20326 + 5.54821i 0 2.53573 − 6.52457i 0 3.53214 + 8.27792i 0
47.4 0 −1.43046 + 2.63700i 0 −4.89508 + 8.47853i 0 1.55722 + 6.82459i 0 −4.90756 − 7.54426i 0
47.5 0 −0.779436 + 2.89698i 0 2.56804 − 4.44797i 0 6.71977 − 1.96078i 0 −7.78496 − 4.51602i 0
47.6 0 −0.180332 − 2.99458i 0 3.20326 − 5.54821i 0 2.53573 − 6.52457i 0 −8.93496 + 1.08003i 0
47.7 0 0.421334 − 2.97027i 0 −1.01058 + 1.75037i 0 −0.0684533 + 6.99967i 0 −8.64496 − 2.50294i 0
47.8 0 2.52394 − 1.62164i 0 −1.28887 + 2.23238i 0 −6.74427 − 1.87480i 0 3.74054 − 8.18586i 0
47.9 0 2.89857 + 0.773478i 0 −2.56804 + 4.44797i 0 6.71977 − 1.96078i 0 7.80346 + 4.48396i 0
47.10 0 2.99894 + 0.0796850i 0 4.89508 − 8.47853i 0 1.55722 + 6.82459i 0 8.98730 + 0.477941i 0
143.1 0 −2.78299 + 1.12025i 0 1.01058 + 1.75037i 0 −0.0684533 − 6.99967i 0 6.49009 − 6.23528i 0
143.2 0 −2.66635 − 1.37497i 0 1.28887 + 2.23238i 0 −6.74427 + 1.87480i 0 5.21890 + 7.33233i 0
143.3 0 −2.50321 + 1.65346i 0 −3.20326 − 5.54821i 0 2.53573 + 6.52457i 0 3.53214 − 8.27792i 0
143.4 0 −1.43046 − 2.63700i 0 −4.89508 − 8.47853i 0 1.55722 − 6.82459i 0 −4.90756 + 7.54426i 0
143.5 0 −0.779436 − 2.89698i 0 2.56804 + 4.44797i 0 6.71977 + 1.96078i 0 −7.78496 + 4.51602i 0
143.6 0 −0.180332 + 2.99458i 0 3.20326 + 5.54821i 0 2.53573 + 6.52457i 0 −8.93496 − 1.08003i 0
143.7 0 0.421334 + 2.97027i 0 −1.01058 − 1.75037i 0 −0.0684533 − 6.99967i 0 −8.64496 + 2.50294i 0
143.8 0 2.52394 + 1.62164i 0 −1.28887 − 2.23238i 0 −6.74427 + 1.87480i 0 3.74054 + 8.18586i 0
143.9 0 2.89857 − 0.773478i 0 −2.56804 − 4.44797i 0 6.71977 + 1.96078i 0 7.80346 − 4.48396i 0
143.10 0 2.99894 − 0.0796850i 0 4.89508 + 8.47853i 0 1.55722 − 6.82459i 0 8.98730 − 0.477941i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.z.f ✓ 20
3.b odd 2 1 inner 336.3.z.f ✓ 20
4.b odd 2 1 336.3.z.g yes 20
7.d odd 6 1 336.3.z.g yes 20
12.b even 2 1 336.3.z.g yes 20
21.g even 6 1 336.3.z.g yes 20
28.f even 6 1 inner 336.3.z.f ✓ 20
84.j odd 6 1 inner 336.3.z.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.z.f ✓ 20 1.a even 1 1 trivial
336.3.z.f ✓ 20 3.b odd 2 1 inner
336.3.z.f ✓ 20 28.f even 6 1 inner
336.3.z.f ✓ 20 84.j odd 6 1 inner
336.3.z.g yes 20 4.b odd 2 1
336.3.z.g yes 20 7.d odd 6 1
336.3.z.g yes 20 12.b even 2 1
336.3.z.g yes 20 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{20} + 174 T_{5}^{18} + 20952 T_{5}^{16} + 1244052 T_{5}^{14} + 52704513 T_{5}^{12} + \cdots + 7934677724736 \) Copy content Toggle raw display
\( T_{13}^{10} + 471T_{13}^{8} + 60084T_{13}^{6} + 1209600T_{13}^{4} + 5184000T_{13}^{2} + 248832 \) Copy content Toggle raw display
\( T_{19}^{10} - 22 T_{19}^{9} + 1338 T_{19}^{8} - 27356 T_{19}^{7} + 1270787 T_{19}^{6} + \cdots + 1766873660644 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 7934677724736 \) Copy content Toggle raw display
$7$ \( (T^{10} - 8 T^{9} + \cdots + 282475249)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 90\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{10} + 471 T^{8} + \cdots + 248832)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 50\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 1766873660644)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 1246086955008)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 954729105513481)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 3127882290724)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 39\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 587068342272)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 15\!\cdots\!72)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 4764296997612)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 82\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 14\!\cdots\!87)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 45\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 22\!\cdots\!92)^{2} \) Copy content Toggle raw display
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