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Properties

Label 336.7.be.a
Level $336$
Weight $7$
Character orbit 336.be
Analytic conductor $77.298$
Analytic rank $0$
Dimension $16$
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] = [336,7,Mod(79,336)]
 
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, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.79");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.be (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: \(77.2981720963\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 173080 x^{14} - 1401372 x^{13} + 11696366794 x^{12} + 263620079492 x^{11} + \cdots + 44\!\cdots\!17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2}\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_1 - 18) q^{3} + ( - \beta_{8} + 14 \beta_1 - 14) q^{5} + ( - \beta_{7} - \beta_{4} - 58 \beta_1 + 49) q^{7} + ( - 243 \beta_1 + 243) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_1 - 18) q^{3} + ( - \beta_{8} + 14 \beta_1 - 14) q^{5} + ( - \beta_{7} - \beta_{4} - 58 \beta_1 + 49) q^{7} + ( - 243 \beta_1 + 243) q^{9} + ( - \beta_{11} - \beta_{7} - \beta_{4} + \cdots + 21) q^{11}+ \cdots + ( - 243 \beta_{11} + 243 \beta_{9} + \cdots + 2673) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 216 q^{3} - 112 q^{5} + 324 q^{7} + 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 216 q^{3} - 112 q^{5} + 324 q^{7} + 1944 q^{9} + 252 q^{11} + 1960 q^{13} + 1792 q^{17} - 3780 q^{19} + 1944 q^{21} + 6048 q^{23} - 49684 q^{25} + 9344 q^{29} - 138996 q^{31} - 2268 q^{33} + 97200 q^{35} + 68348 q^{37} - 26460 q^{39} + 98224 q^{41} + 27216 q^{45} - 192600 q^{47} - 113984 q^{49} - 48384 q^{51} + 99416 q^{53} + 68040 q^{57} - 325836 q^{59} + 103376 q^{61} - 131220 q^{63} - 181784 q^{65} + 399924 q^{67} - 108864 q^{69} + 213556 q^{73} + 1341468 q^{75} + 1620400 q^{77} - 2014740 q^{79} - 472392 q^{81} + 1124544 q^{85} - 126144 q^{87} + 338800 q^{89} - 272052 q^{91} + 1250964 q^{93} + 4604544 q^{95} + 2772616 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 173080 x^{14} - 1401372 x^{13} + 11696366794 x^{12} + 263620079492 x^{11} + \cdots + 44\!\cdots\!17 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 57\!\cdots\!88 \nu^{15} + \cdots - 15\!\cdots\!33 ) / 59\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57\!\cdots\!88 \nu^{15} + \cdots + 15\!\cdots\!33 ) / 59\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 69\!\cdots\!91 \nu^{15} + \cdots + 49\!\cdots\!25 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!29 \nu^{15} + \cdots - 11\!\cdots\!64 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!09 \nu^{15} + \cdots - 98\!\cdots\!94 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 95\!\cdots\!01 \nu^{15} + \cdots - 35\!\cdots\!48 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\!\cdots\!37 \nu^{15} + \cdots + 84\!\cdots\!30 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!59 \nu^{15} + \cdots + 23\!\cdots\!48 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27\!\cdots\!32 \nu^{15} + \cdots - 10\!\cdots\!51 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 54\!\cdots\!41 \nu^{15} + \cdots + 78\!\cdots\!71 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!97 \nu^{15} + \cdots + 83\!\cdots\!32 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 36\!\cdots\!82 \nu^{15} + \cdots - 15\!\cdots\!67 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 44\!\cdots\!39 \nu^{15} + \cdots - 73\!\cdots\!90 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 47\!\cdots\!69 \nu^{15} + \cdots + 20\!\cdots\!59 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 71\!\cdots\!41 \nu^{15} + \cdots - 52\!\cdots\!52 ) / 27\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4 \beta_{13} - 4 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - \beta_{8} + 3 \beta_{7} + \cdots + 21635 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 15 \beta_{15} + 150 \beta_{13} - 186 \beta_{12} + 63 \beta_{11} + 528 \beta_{10} + 90 \beta_{9} + \cdots + 489863 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 450 \beta_{15} - 1668 \beta_{14} + 232648 \beta_{13} - 234664 \beta_{12} + 174387 \beta_{11} + \cdots + 824958885 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 882325 \beta_{15} - 255405 \beta_{14} + 8277492 \beta_{13} - 11782512 \beta_{12} + 10700361 \beta_{11} + \cdots + 37056950680 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20632230 \beta_{15} - 193958163 \beta_{14} + 11979157384 \beta_{13} - 12159712540 \beta_{12} + \cdots + 36018672875464 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 46820043725 \beta_{15} - 36402928338 \beta_{14} + 491273324514 \beta_{13} - 744768261804 \beta_{12} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 673419423810 \beta_{15} - 15825396050622 \beta_{14} + 601403782553704 \beta_{13} - 616238809404040 \beta_{12} + \cdots + 16\!\cdots\!26 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 24\!\cdots\!25 \beta_{15} + \cdots + 13\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11\!\cdots\!90 \beta_{15} + \cdots + 80\!\cdots\!99 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 12\!\cdots\!70 \beta_{15} + \cdots + 75\!\cdots\!45 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 79\!\cdots\!20 \beta_{15} + \cdots + 39\!\cdots\!27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 69\!\cdots\!50 \beta_{15} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 13\!\cdots\!80 \beta_{15} + \cdots + 19\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 37\!\cdots\!75 \beta_{15} + \cdots + 23\!\cdots\!09 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
230.550 + 0.866025i
179.351 + 0.866025i
81.2666 + 0.866025i
17.4157 + 0.866025i
−32.8215 + 0.866025i
−106.134 + 0.866025i
−153.605 + 0.866025i
−212.023 + 0.866025i
230.550 − 0.866025i
179.351 − 0.866025i
81.2666 − 0.866025i
17.4157 − 0.866025i
−32.8215 − 0.866025i
−106.134 − 0.866025i
−153.605 − 0.866025i
−212.023 − 0.866025i
0 −13.5000 + 7.79423i 0 −122.025 + 211.354i 0 −292.124 − 179.757i 0 121.500 − 210.444i 0
79.2 0 −13.5000 + 7.79423i 0 −96.4257 + 167.014i 0 297.005 − 171.572i 0 121.500 − 210.444i 0
79.3 0 −13.5000 + 7.79423i 0 −47.3833 + 82.0703i 0 −118.618 + 321.836i 0 121.500 − 210.444i 0
79.4 0 −13.5000 + 7.79423i 0 −15.4579 + 26.7738i 0 337.064 + 63.5355i 0 121.500 − 210.444i 0
79.5 0 −13.5000 + 7.79423i 0 9.66076 − 16.7329i 0 −318.080 − 128.351i 0 121.500 − 210.444i 0
79.6 0 −13.5000 + 7.79423i 0 46.3170 − 80.2234i 0 172.181 − 296.653i 0 121.500 − 210.444i 0
79.7 0 −13.5000 + 7.79423i 0 70.0525 − 121.334i 0 −14.5983 − 342.689i 0 121.500 − 210.444i 0
79.8 0 −13.5000 + 7.79423i 0 99.2617 − 171.926i 0 99.1709 + 328.351i 0 121.500 − 210.444i 0
319.1 0 −13.5000 − 7.79423i 0 −122.025 − 211.354i 0 −292.124 + 179.757i 0 121.500 + 210.444i 0
319.2 0 −13.5000 − 7.79423i 0 −96.4257 − 167.014i 0 297.005 + 171.572i 0 121.500 + 210.444i 0
319.3 0 −13.5000 − 7.79423i 0 −47.3833 − 82.0703i 0 −118.618 − 321.836i 0 121.500 + 210.444i 0
319.4 0 −13.5000 − 7.79423i 0 −15.4579 − 26.7738i 0 337.064 − 63.5355i 0 121.500 + 210.444i 0
319.5 0 −13.5000 − 7.79423i 0 9.66076 + 16.7329i 0 −318.080 + 128.351i 0 121.500 + 210.444i 0
319.6 0 −13.5000 − 7.79423i 0 46.3170 + 80.2234i 0 172.181 + 296.653i 0 121.500 + 210.444i 0
319.7 0 −13.5000 − 7.79423i 0 70.0525 + 121.334i 0 −14.5983 + 342.689i 0 121.500 + 210.444i 0
319.8 0 −13.5000 − 7.79423i 0 99.2617 + 171.926i 0 99.1709 − 328.351i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.be.a ✓ 16
4.b odd 2 1 336.7.be.d yes 16
7.c even 3 1 336.7.be.d yes 16
28.g odd 6 1 inner 336.7.be.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.7.be.a ✓ 16 1.a even 1 1 trivial
336.7.be.a ✓ 16 28.g odd 6 1 inner
336.7.be.d yes 16 4.b odd 2 1
336.7.be.d yes 16 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{16} + 112 T_{5}^{15} + 93614 T_{5}^{14} + 2541504 T_{5}^{13} + 5287633159 T_{5}^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
\( T_{11}^{16} - 252 T_{11}^{15} - 9893886 T_{11}^{14} + 2498593608 T_{11}^{13} + 69963978470163 T_{11}^{12} + \cdots + 66\!\cdots\!56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 243)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 66\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 30\!\cdots\!92)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 26\!\cdots\!12)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 90\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 61\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 56\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 72\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 73\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 61\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 13\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 65\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
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