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Properties

Label 336.7.f.a
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: \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{7}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{5} - 2 \beta_{3}) q^{5} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_1 - 3) q^{7} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{5} - 2 \beta_{3}) q^{5} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_1 - 3) q^{7} - 243 q^{9} + (3 \beta_{7} - 3 \beta_{4} + \cdots + 856) q^{11}+ \cdots + ( - 729 \beta_{7} + 729 \beta_{4} + \cdots - 208008) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 28 q^{7} - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 28 q^{7} - 1944 q^{9} + 6848 q^{11} + 4536 q^{15} + 6804 q^{21} + 24320 q^{23} - 77056 q^{25} + 60496 q^{29} - 103656 q^{35} - 39112 q^{37} - 108864 q^{39} + 29272 q^{43} - 94864 q^{49} - 103032 q^{51} + 232288 q^{53} + 180792 q^{57} + 6804 q^{63} - 836304 q^{65} + 2163848 q^{67} + 506288 q^{71} + 1536304 q^{77} + 93272 q^{79} + 472392 q^{81} + 3740760 q^{85} - 1890336 q^{91} - 2153952 q^{93} - 4154832 q^{95} - 1664064 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 72659960 \nu^{7} + 167751752 \nu^{6} + 14822976200 \nu^{5} - 32552006440 \nu^{4} + \cdots + 63618633670542 ) / 6112279547031 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 75463939 \nu^{7} - 32227930 \nu^{6} - 15395001205 \nu^{5} + 33808202321 \nu^{4} + \cdots - 676685780344617 ) / 6112279547031 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6298293453 \nu^{7} - 15562438353 \nu^{6} + 1313849863656 \nu^{5} - 8962913013219 \nu^{4} + \cdots + 10\!\cdots\!10 ) / 346362507665090 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45589983113 \nu^{7} - 330826219493 \nu^{6} + 9950286831806 \nu^{5} - 108683582885489 \nu^{4} + \cdots - 55\!\cdots\!30 ) / 445323224140830 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 236085965056 \nu^{7} - 195554084551 \nu^{6} + 52131292245397 \nu^{5} + \cdots + 84\!\cdots\!70 ) / 15\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 351490354093 \nu^{7} - 1157111077108 \nu^{6} + 72024347710666 \nu^{5} + \cdots + 32\!\cdots\!05 ) / 15\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1154674936661 \nu^{7} - 241903709621 \nu^{6} + 240107158945292 \nu^{5} + \cdots + 78\!\cdots\!60 ) / 31\!\cdots\!10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} - 5\beta _1 + 8 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 21\beta_{6} + 9\beta_{5} + 3\beta_{4} - 430\beta_{3} - 36\beta_{2} - 12\beta _1 - 3810 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} + 8\beta_{4} + 16\beta_{2} + 197\beta _1 + 2734 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 171 \beta_{7} - 4893 \beta_{6} - 2217 \beta_{5} - 99 \beta_{4} + 84166 \beta_{3} - 7524 \beta_{2} + \cdots - 748116 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 25141 \beta_{7} + 51349 \beta_{6} + 76405 \beta_{5} - 40405 \beta_{4} - 447334 \beta_{3} + \cdots - 4363358 ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1952\beta_{7} + 1952\beta_{4} + 348264\beta_{2} + 405717\beta _1 + 35068398 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8138989 \beta_{7} - 12792781 \beta_{6} - 16629229 \beta_{5} + 5057317 \beta_{4} + 131713054 \beta_{3} + \cdots - 1262908360 ) / 72 \) 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
−0.564972 − 0.978561i
6.44130 + 11.1567i
−7.66250 − 13.2718i
2.28617 + 3.95977i
2.28617 − 3.95977i
−7.66250 + 13.2718i
6.44130 − 11.1567i
−0.564972 + 0.978561i
0 − 15.5885i 0 − 126.447i 0 213.284 + 268.624i 0 −243.000 0
97.2 0 − 15.5885i 0 − 109.244i 0 −86.6767 − 331.868i 0 −243.000 0
97.3 0 − 15.5885i 0 175.511i 0 202.362 + 276.945i 0 −243.000 0
97.4 0 − 15.5885i 0 205.672i 0 −342.970 + 4.53729i 0 −243.000 0
97.5 0 15.5885i 0 − 205.672i 0 −342.970 − 4.53729i 0 −243.000 0
97.6 0 15.5885i 0 − 175.511i 0 202.362 − 276.945i 0 −243.000 0
97.7 0 15.5885i 0 109.244i 0 −86.6767 + 331.868i 0 −243.000 0
97.8 0 15.5885i 0 126.447i 0 213.284 − 268.624i 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.a 8
4.b odd 2 1 21.7.d.a ✓ 8
7.b odd 2 1 inner 336.7.f.a 8
12.b even 2 1 63.7.d.f 8
28.d even 2 1 21.7.d.a ✓ 8
28.f even 6 1 147.7.f.b 8
28.f even 6 1 147.7.f.c 8
28.g odd 6 1 147.7.f.b 8
28.g odd 6 1 147.7.f.c 8
84.h odd 2 1 63.7.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.d.a ✓ 8 4.b odd 2 1
21.7.d.a ✓ 8 28.d even 2 1
63.7.d.f 8 12.b even 2 1
63.7.d.f 8 84.h odd 2 1
147.7.f.b 8 28.f even 6 1
147.7.f.b 8 28.g odd 6 1
147.7.f.c 8 28.f even 6 1
147.7.f.c 8 28.g odd 6 1
336.7.f.a 8 1.a even 1 1 trivial
336.7.f.a 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 101028T_{5}^{6} + 3535164900T_{5}^{4} + 50334222960000T_{5}^{2} + 248637860496000000 \) 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
$5$ \( T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 3045338564760)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 14\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 66\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 64\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 36\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 16\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
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