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Properties

Label 336.7.bh.e
Level $336$
Weight $7$
Character orbit 336.bh
Analytic conductor $77.298$
Analytic rank $0$
Dimension $8$
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(145,336)]
 
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([0, 0, 0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.145");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 82x^{6} - 165x^{5} + 5606x^{4} - 7807x^{3} + 102447x^{2} + 132594x + 1162084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 84)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{4} - 3 \beta_1 - 7) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 54) q^{7}+ \cdots - 243 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{4} - 3 \beta_1 - 7) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 54) q^{7}+ \cdots + (972 \beta_{7} - 1215 \beta_{6} + \cdots - 8019) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9} - 126 q^{11} - 756 q^{15} - 2532 q^{17} - 19998 q^{19} - 5886 q^{21} + 15648 q^{23} + 42698 q^{25} + 129468 q^{29} + 42096 q^{31} - 3402 q^{33} - 73524 q^{35} + 9866 q^{37} - 29106 q^{39} + 71764 q^{43} - 10206 q^{45} - 86988 q^{47} + 314588 q^{49} - 22788 q^{51} + 391710 q^{53} - 359964 q^{57} + 553434 q^{59} - 1009104 q^{61} - 181278 q^{63} + 589452 q^{65} - 229762 q^{67} - 208488 q^{71} - 1249290 q^{73} + 1152846 q^{75} + 1009566 q^{77} - 693808 q^{79} - 236196 q^{81} - 1302600 q^{85} + 1747818 q^{87} - 1414692 q^{89} + 766410 q^{91} + 378864 q^{93} - 3047568 q^{95} - 61236 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 82x^{6} - 165x^{5} + 5606x^{4} - 7807x^{3} + 102447x^{2} + 132594x + 1162084 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 9659521 \nu^{7} + 77530555 \nu^{6} - 720692020 \nu^{5} + 6152911039 \nu^{4} + \cdots - 195965705782 ) / 6272450648002 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3595888 \nu^{7} - 332009022 \nu^{6} + 8071513719 \nu^{5} - 71932276648 \nu^{4} + \cdots - 126708615949482 ) / 570222786182 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1501215361 \nu^{7} + 19919106314 \nu^{6} - 75483296654 \nu^{5} + 1097694255988 \nu^{4} + \cdots - 647680862469410 ) / 6272450648002 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1550256892 \nu^{7} - 2575341491 \nu^{6} + 78777555545 \nu^{5} - 1099026853639 \nu^{4} + \cdots - 13\!\cdots\!96 ) / 6272450648002 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1165235138 \nu^{7} - 24628162010 \nu^{6} - 47869345266 \nu^{5} - 2232961192814 \nu^{4} + \cdots - 974144033632308 ) / 3136225324001 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4035826132 \nu^{7} + 13794693635 \nu^{6} - 245740299185 \nu^{5} + 1166566827679 \nu^{4} + \cdots - 971154652898584 ) / 6272450648002 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4725027148 \nu^{7} + 2418333230 \nu^{6} + 254322741186 \nu^{5} - 1229041627066 \nu^{4} + \cdots - 766559230402812 ) / 3136225324001 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} + 46\beta _1 + 42 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 16\beta_{6} + 2\beta_{5} - 4\beta_{4} - 8\beta_{3} + 16\beta_{2} + 6856\beta _1 + 12 ) / 168 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 62\beta_{7} + 202\beta_{6} - 31\beta_{5} - 34\beta_{4} - 236\beta_{3} + 5368 ) / 168 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -173\beta_{7} - 173\beta_{5} - 180\beta_{4} + 760\beta_{3} - 1340\beta_{2} - 363476\beta _1 - 364056 ) / 168 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1383 \beta_{7} - 14834 \beta_{6} + 2766 \beta_{5} - 14716 \beta_{4} + 14598 \beta_{3} + 14834 \beta_{2} + \cdots + 118 ) / 168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4390\beta_{7} + 14472\beta_{6} - 2195\beta_{5} + 6332\beta_{4} - 8140\beta_{3} + 3106880 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 84519 \beta_{7} - 84519 \beta_{5} + 856582 \beta_{4} + 95366 \beta_{3} - 1047314 \beta_{2} + \cdots - 94757726 ) / 168 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−1.59227 − 2.75789i
−4.24382 − 7.35052i
2.91150 + 5.04287i
3.42459 + 5.93157i
−1.59227 + 2.75789i
−4.24382 + 7.35052i
2.91150 − 5.04287i
3.42459 − 5.93157i
0 13.5000 − 7.79423i 0 −181.505 − 104.792i 0 223.761 − 259.961i 0 121.500 − 210.444i 0
145.2 0 13.5000 − 7.79423i 0 −68.1069 − 39.3216i 0 −335.720 + 70.2945i 0 121.500 − 210.444i 0
145.3 0 13.5000 − 7.79423i 0 27.1744 + 15.6892i 0 342.316 + 21.6478i 0 121.500 − 210.444i 0
145.4 0 13.5000 − 7.79423i 0 201.438 + 116.300i 0 −184.358 − 289.242i 0 121.500 − 210.444i 0
241.1 0 13.5000 + 7.79423i 0 −181.505 + 104.792i 0 223.761 + 259.961i 0 121.500 + 210.444i 0
241.2 0 13.5000 + 7.79423i 0 −68.1069 + 39.3216i 0 −335.720 − 70.2945i 0 121.500 + 210.444i 0
241.3 0 13.5000 + 7.79423i 0 27.1744 − 15.6892i 0 342.316 − 21.6478i 0 121.500 + 210.444i 0
241.4 0 13.5000 + 7.79423i 0 201.438 − 116.300i 0 −184.358 + 289.242i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d 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.bh.e 8
4.b odd 2 1 84.7.m.a ✓ 8
7.d odd 6 1 inner 336.7.bh.e 8
12.b even 2 1 252.7.z.d 8
28.d even 2 1 588.7.m.c 8
28.f even 6 1 84.7.m.a ✓ 8
28.f even 6 1 588.7.d.b 8
28.g odd 6 1 588.7.d.b 8
28.g odd 6 1 588.7.m.c 8
84.j odd 6 1 252.7.z.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.m.a ✓ 8 4.b odd 2 1
84.7.m.a ✓ 8 28.f even 6 1
252.7.z.d 8 12.b even 2 1
252.7.z.d 8 84.j odd 6 1
336.7.bh.e 8 1.a even 1 1 trivial
336.7.bh.e 8 7.d odd 6 1 inner
588.7.d.b 8 28.f even 6 1
588.7.d.b 8 28.g odd 6 1
588.7.m.c 8 28.d even 2 1
588.7.m.c 8 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 42 T_{5}^{7} - 51717 T_{5}^{6} - 2196810 T_{5}^{5} + 2561019525 T_{5}^{4} + \cdots + 14\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 86\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 94\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 95\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 76\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
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