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Properties

Label 336.7.f.c
Level $336$
Weight $7$
Character orbit 336.f
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(97,336)]
 
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, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.97");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.f (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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 313x^{6} + 386x^{5} + 76605x^{4} + 40588x^{3} + 5917828x^{2} + 2216528x + 365115664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{9}\cdot 7 \)
Twist minimal: no (minimal twist has level 42)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{4} - 2 \beta_1) q^{5} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 26) q^{7}+ \cdots - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{4} - 2 \beta_1) q^{5} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 26) q^{7}+ \cdots + (243 \beta_{7} - 2187 \beta_{6} + \cdots + 233523) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 212 q^{7} - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 212 q^{7} - 1944 q^{9} - 7728 q^{11} - 4536 q^{15} + 972 q^{21} + 96 q^{23} + 27968 q^{25} - 3072 q^{29} + 96024 q^{35} - 116584 q^{37} + 81648 q^{39} - 58280 q^{43} + 410288 q^{49} + 248184 q^{51} - 289872 q^{53} + 227448 q^{57} - 51516 q^{63} - 482640 q^{65} - 1172440 q^{67} - 1720848 q^{71} - 362400 q^{77} + 1736696 q^{79} + 472392 q^{81} - 1277544 q^{85} - 1779072 q^{91} + 400464 q^{93} + 4246320 q^{95} + 1877904 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 313x^{6} + 386x^{5} + 76605x^{4} + 40588x^{3} + 5917828x^{2} + 2216528x + 365115664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9483289533 \nu^{7} - 1046907411615 \nu^{6} + 4034112096339 \nu^{5} - 253747145386359 \nu^{4} + \cdots - 36\!\cdots\!66 ) / 27\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1707816653181 \nu^{7} + 822121139497260 \nu^{6} + \cdots + 30\!\cdots\!72 ) / 21\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 566397 \nu^{7} + 11829030 \nu^{6} - 141832041 \nu^{5} - 355231734 \nu^{4} + \cdots - 64281213965736 ) / 1593280420795 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 107668872535283 \nu^{7} + \cdots - 56\!\cdots\!24 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 163542898290077 \nu^{7} + \cdots - 82\!\cdots\!84 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 153271985117117 \nu^{7} + \cdots + 20\!\cdots\!96 ) / 85\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 368357042034276 \nu^{7} + 305841901306225 \nu^{6} + \cdots - 33\!\cdots\!28 ) / 42\!\cdots\!10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -28\beta_{6} + 20\beta_{5} - 60\beta_{4} + 19\beta_{3} + 3\beta_{2} + 8\beta _1 + 112 ) / 504 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 14\beta_{7} + 28\beta_{6} - 24\beta_{5} - 40\beta_{4} - 363\beta_{3} - 187\beta_{2} - 2174\beta _1 - 19586 ) / 252 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2800 \beta_{7} + 11592 \beta_{6} + 172 \beta_{5} + 2284 \beta_{4} - 12543 \beta_{3} - 2112 \beta_{2} + \cdots - 186368 ) / 504 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5726 \beta_{7} + 12824 \beta_{6} + 11290 \beta_{5} + 23390 \beta_{4} - 58347 \beta_{3} + \cdots - 3731154 ) / 252 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 452256 \beta_{7} - 1300740 \beta_{6} + 548880 \beta_{5} + 1971408 \beta_{4} + 2434340 \beta_{3} + \cdots + 37034424 ) / 504 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 535108 \beta_{7} - 1254904 \beta_{6} - 101474 \beta_{5} - 230686 \beta_{4} + 6761849 \beta_{3} + \cdots + 231516300 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 119158984 \beta_{7} - 305772180 \beta_{6} - 207159100 \beta_{5} - 570112540 \beta_{4} + \cdots + 12038677616 ) / 504 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
8.07582 − 13.9877i
4.53758 − 7.85933i
−4.74469 + 8.21805i
−6.86872 + 11.8970i
−6.86872 − 11.8970i
−4.74469 − 8.21805i
4.53758 + 7.85933i
8.07582 + 13.9877i
0 − 15.5885i 0 − 165.496i 0 −170.635 + 297.545i 0 −243.000 0
97.2 0 − 15.5885i 0 − 88.9684i 0 297.538 − 170.646i 0 −243.000 0
97.3 0 − 15.5885i 0 − 5.82312i 0 −342.119 + 24.5679i 0 −243.000 0
97.4 0 − 15.5885i 0 114.795i 0 321.215 − 120.290i 0 −243.000 0
97.5 0 15.5885i 0 − 114.795i 0 321.215 + 120.290i 0 −243.000 0
97.6 0 15.5885i 0 5.82312i 0 −342.119 − 24.5679i 0 −243.000 0
97.7 0 15.5885i 0 88.9684i 0 297.538 + 170.646i 0 −243.000 0
97.8 0 15.5885i 0 165.496i 0 −170.635 − 297.545i 0 −243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.f.c 8
4.b odd 2 1 42.7.c.a ✓ 8
7.b odd 2 1 inner 336.7.f.c 8
12.b even 2 1 126.7.c.b 8
28.d even 2 1 42.7.c.a ✓ 8
28.f even 6 1 294.7.g.a 8
28.f even 6 1 294.7.g.f 8
28.g odd 6 1 294.7.g.a 8
28.g odd 6 1 294.7.g.f 8
84.h odd 2 1 126.7.c.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.c.a ✓ 8 4.b odd 2 1
42.7.c.a ✓ 8 28.d even 2 1
126.7.c.b 8 12.b even 2 1
126.7.c.b 8 84.h odd 2 1
294.7.g.a 8 28.f even 6 1
294.7.g.a 8 28.g odd 6 1
294.7.g.f 8 28.f even 6 1
294.7.g.f 8 28.g odd 6 1
336.7.f.c 8 1.a even 1 1 trivial
336.7.f.c 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 48516T_{5}^{6} + 683671716T_{5}^{4} + 2879999625600T_{5}^{2} + 96872837760000 \) 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
$3$ \( (T^{2} + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 96872837760000 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + 3864 T^{3} + \cdots - 1155384)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 58\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 31\!\cdots\!48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 95\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 56\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 33\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 19\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 75\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 97\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 86\!\cdots\!84 \) Copy content Toggle raw display
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