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Properties

Label 336.7.d.b
Level $336$
Weight $7$
Character orbit 336.d
Analytic conductor $77.298$
Analytic rank $0$
Dimension $12$
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(113,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.d (of order \(2\), degree \(1\), not 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 592x^{10} + 135745x^{8} + 15239528x^{6} + 859756808x^{4} + 21444428576x^{2} + 133666284816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{10}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{3} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{6} + \beta_{2} + 100) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{3} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{6} + \beta_{2} + 100) q^{9} + (\beta_{7} - \beta_{4} + \cdots - 13 \beta_1) q^{11}+ \cdots + (356 \beta_{11} - 328 \beta_{10} + \cdots + 6418) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 1196 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 1196 q^{9} - 4032 q^{13} - 7976 q^{15} + 22920 q^{19} - 5488 q^{21} - 5388 q^{25} + 35388 q^{27} + 25008 q^{31} - 118400 q^{33} + 67248 q^{37} - 3800 q^{39} + 251376 q^{43} - 5960 q^{45} + 201684 q^{49} - 258376 q^{51} + 525936 q^{55} + 110208 q^{57} + 45360 q^{61} - 107016 q^{63} - 251904 q^{67} - 209336 q^{69} + 731640 q^{73} - 266612 q^{75} - 336768 q^{79} - 323716 q^{81} + 3152472 q^{85} + 489160 q^{87} + 337512 q^{91} + 948200 q^{93} - 1063416 q^{97} + 76480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 592x^{10} + 135745x^{8} + 15239528x^{6} + 859756808x^{4} + 21444428576x^{2} + 133666284816 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1235205 \nu^{11} + 94173497 \nu^{10} + 610135035 \nu^{9} + 47427582829 \nu^{8} + \cdots + 13\!\cdots\!00 ) / 71334730865376 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11116845 \nu^{11} - 12582413995 \nu^{10} + 5491215315 \nu^{9} - 6335872929581 \nu^{8} + \cdots - 18\!\cdots\!96 ) / 642012577788384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 931426637 \nu^{11} - 1695122946 \nu^{10} - 474562407856 \nu^{9} - 853696490922 \nu^{8} + \cdots - 24\!\cdots\!00 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1598437337 \nu^{11} - 12127876822 \nu^{10} + 804035326756 \nu^{9} - 6109576941122 \nu^{8} + \cdots - 17\!\cdots\!08 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3054868981 \nu^{11} - 308799679982 \nu^{10} + 1545388192424 \nu^{9} - 156103454272702 \nu^{8} + \cdots - 47\!\cdots\!72 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18965284911 \nu^{11} + 92412138796 \nu^{10} - 9547384401666 \nu^{9} + \cdots + 13\!\cdots\!48 ) / 25\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1992714980 \nu^{11} - 1412602455 \nu^{10} + 1009021340605 \nu^{9} - 711413742435 \nu^{8} + \cdots - 20\!\cdots\!00 ) / 214004192596128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27596940673 \nu^{11} - 610985461736 \nu^{10} + 13923442828850 \nu^{9} + \cdots - 94\!\cdots\!96 ) / 25\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7814763307 \nu^{11} - 32379992463 \nu^{10} - 3940726687721 \nu^{9} + \cdots - 49\!\cdots\!48 ) / 642012577788384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20115075361 \nu^{11} + 18646352406 \nu^{10} + 10135113269684 \nu^{9} + \cdots + 27\!\cdots\!00 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2926220197 \nu^{11} + 376693988 \nu^{10} + 1475644645298 \nu^{9} + 189710331316 \nu^{8} + \cdots + 55\!\cdots\!00 ) / 142669461730752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 675 \beta_{11} + 474 \beta_{10} - 192 \beta_{9} - 144 \beta_{8} + 417 \beta_{7} - 336 \beta_{6} + \cdots - 96 ) / 74088 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 42 \beta_{10} - 42 \beta_{9} + 63 \beta_{8} - 21 \beta_{7} + 21 \beta_{6} - 63 \beta_{5} + \cdots - 261030 ) / 2646 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 42201 \beta_{11} - 27852 \beta_{10} + 8832 \beta_{9} + 6624 \beta_{8} - 37515 \beta_{7} + \cdots + 4416 ) / 37044 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1323 \beta_{11} + 12369 \beta_{10} + 5754 \beta_{9} - 16569 \beta_{8} + 2877 \beta_{7} + \cdots + 34821780 ) / 2646 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3354021 \beta_{11} + 2337009 \beta_{10} - 385608 \beta_{9} - 289206 \beta_{8} + 3385047 \beta_{7} + \cdots - 192804 ) / 18522 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 386316 \beta_{11} - 2701167 \beta_{10} - 757680 \beta_{9} + 3466323 \beta_{8} - 378840 \beta_{7} + \cdots - 5339819352 ) / 2646 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 581650362 \beta_{11} - 436115865 \beta_{10} + 19909392 \beta_{9} + 14932044 \beta_{8} + \cdots + 9954696 ) / 18522 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 89354097 \beta_{11} + 538731039 \beta_{10} + 103819926 \beta_{9} - 679995099 \beta_{8} + \cdots + 895857224892 ) / 2646 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 105595807302 \beta_{11} + 84357949749 \beta_{10} + 3107368104 \beta_{9} + 2330526078 \beta_{8} + \cdots + 1553684052 ) / 18522 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 19246755276 \beta_{11} - 104158889709 \beta_{10} - 14982975672 \beta_{9} + 130897132821 \beta_{8} + \cdots - 159019041543600 ) / 2646 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19679405869080 \beta_{11} - 16504182405651 \beta_{10} - 1537465965456 \beta_{9} + \cdots - 768732982728 ) / 18522 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
14.0035i
− 14.0035i
− 8.32514i
8.32514i
− 9.31880i
9.31880i
− 9.39419i
9.39419i
11.9372i
− 11.9372i
3.00096i
− 3.00096i
0 −24.4388 − 11.4781i 0 − 89.7454i 0 −129.642 0 465.508 + 561.020i 0
113.2 0 −24.4388 + 11.4781i 0 89.7454i 0 −129.642 0 465.508 − 561.020i 0
113.3 0 −22.9026 − 14.2993i 0 163.744i 0 129.642 0 320.059 + 654.984i 0
113.4 0 −22.9026 + 14.2993i 0 − 163.744i 0 129.642 0 320.059 − 654.984i 0
113.5 0 −12.2497 − 24.0613i 0 − 132.148i 0 129.642 0 −428.892 + 589.485i 0
113.6 0 −12.2497 + 24.0613i 0 132.148i 0 129.642 0 −428.892 − 589.485i 0
113.7 0 5.14276 − 26.5057i 0 − 71.4773i 0 −129.642 0 −676.104 − 272.625i 0
113.8 0 5.14276 + 26.5057i 0 71.4773i 0 −129.642 0 −676.104 + 272.625i 0
113.9 0 21.5692 − 16.2409i 0 − 44.1221i 0 129.642 0 201.465 − 700.609i 0
113.10 0 21.5692 + 16.2409i 0 44.1221i 0 129.642 0 201.465 + 700.609i 0
113.11 0 26.8790 − 2.55298i 0 192.507i 0 −129.642 0 715.965 − 137.243i 0
113.12 0 26.8790 + 2.55298i 0 − 192.507i 0 −129.642 0 715.965 + 137.243i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.d.b 12
3.b odd 2 1 inner 336.7.d.b 12
4.b odd 2 1 84.7.c.a ✓ 12
12.b even 2 1 84.7.c.a ✓ 12
28.d even 2 1 588.7.c.f 12
84.h odd 2 1 588.7.c.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.c.a ✓ 12 4.b odd 2 1
84.7.c.a ✓ 12 12.b even 2 1
336.7.d.b 12 1.a even 1 1 trivial
336.7.d.b 12 3.b odd 2 1 inner
588.7.c.f 12 28.d even 2 1
588.7.c.f 12 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 96444 T_{5}^{10} + 3404737932 T_{5}^{8} + 54729980328800 T_{5}^{6} + \cdots + 13\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 16807)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 96\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 36\!\cdots\!48)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 41\!\cdots\!08)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 15\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 33\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 91\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 58\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 27\!\cdots\!52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 48\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 77\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 29\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 79\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 53\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
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