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Properties

Label 338.8.b.h
Level $338$
Weight $8$
Character orbit 338.b
Analytic conductor $105.586$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,8,Mod(337,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.b (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: \(105.586138614\)
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: \( x^{8} + 13962x^{6} + 63111321x^{4} + 101620417536x^{2} + 51674244710400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 26)
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.

\(f(q)\) \(=\) \( q - 8 \beta_{2} q^{2} + \beta_{3} q^{3} - 64 q^{4} + (\beta_{7} + 70 \beta_{2} - \beta_1) q^{5} - 8 \beta_1 q^{6} + ( - 2 \beta_{5} - 138 \beta_{2} - 7 \beta_1) q^{7} + 512 \beta_{2} q^{8} + ( - 3 \beta_{6} + 6 \beta_{4} + \cdots + 1302) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 \beta_{2} q^{2} + \beta_{3} q^{3} - 64 q^{4} + (\beta_{7} + 70 \beta_{2} - \beta_1) q^{5} - 8 \beta_1 q^{6} + ( - 2 \beta_{5} - 138 \beta_{2} - 7 \beta_1) q^{7} + 512 \beta_{2} q^{8} + ( - 3 \beta_{6} + 6 \beta_{4} + \cdots + 1302) q^{9}+ \cdots + ( - 7152 \beta_{7} + \cdots + 136674 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 512 q^{4} + 10428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 512 q^{4} + 10428 q^{9} + 4448 q^{10} - 8768 q^{14} + 32768 q^{16} + 56632 q^{17} - 118272 q^{22} - 66776 q^{23} - 173756 q^{25} + 212544 q^{27} - 186280 q^{29} - 248448 q^{30} - 283088 q^{35} - 667392 q^{36} + 1598208 q^{38} - 284672 q^{40} - 1457184 q^{42} - 1138528 q^{43} - 1435596 q^{49} + 5459856 q^{51} + 2470700 q^{53} + 2185024 q^{55} + 561152 q^{56} - 1371368 q^{61} + 4978560 q^{62} - 2097152 q^{64} + 3818208 q^{66} - 3624448 q^{68} + 11200068 q^{69} - 154176 q^{74} + 44401920 q^{75} + 27830412 q^{77} - 14107904 q^{79} - 7516008 q^{81} - 1326272 q^{82} + 14365800 q^{87} + 7569408 q^{88} + 36854304 q^{90} + 4273664 q^{92} + 9187200 q^{94} - 46668176 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 13962x^{6} + 63111321x^{4} + 101620417536x^{2} + 51674244710400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 13962\nu^{5} + 55922841\nu^{3} + 51437638656\nu ) / 254644715520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 6981\nu^{2} + 7188480 ) / 35424 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 11298\nu^{4} + 34774929\nu^{2} + 23080121856 ) / 17853696 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -127\nu^{7} - 1613430\nu^{5} - 6458592231\nu^{3} - 7776509193216\nu ) / 118834200576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 12054\nu^{4} + 43028181\nu^{2} + 38896536960 ) / 8926848 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -625\nu^{7} - 7288554\nu^{5} - 24207873417\nu^{3} - 18760060670976\nu ) / 356502601728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{6} - 6\beta_{4} - 9\beta_{3} - 3489 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 72\beta_{7} - 216\beta_{5} - 26640\beta_{2} - 4965\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -20943\beta_{6} + 41886\beta_{4} + 98253\beta_{3} + 17168229 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -290088\beta_{7} + 1614168\beta_{5} + 309780720\beta_{2} + 27791001\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 132289227\beta_{6} - 246724758\beta_{4} - 797088033\beta_{3} - 95717045817 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23764104\beta_{7} - 10457679960\beta_{5} - 2580729212880\beta_{2} - 161798689053\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 71.1584i
− 39.9844i
31.8716i
79.2712i
71.1584i
39.9844i
− 31.8716i
− 79.2712i
− 8.00000i −71.1584 −64.0000 523.489i 569.267i 416.801i 512.000i 2876.52 4187.91
337.2 − 8.00000i −39.9844 −64.0000 − 323.700i 319.875i − 568.591i 512.000i −588.246 −2589.60
337.3 − 8.00000i 31.8716 −64.0000 − 54.4265i − 254.973i 1112.71i 512.000i −1171.20 −435.412
337.4 − 8.00000i 79.2712 −64.0000 132.638i − 634.170i − 1508.92i 512.000i 4096.92 1061.10
337.5 8.00000i −71.1584 −64.0000 − 523.489i − 569.267i − 416.801i − 512.000i 2876.52 4187.91
337.6 8.00000i −39.9844 −64.0000 323.700i − 319.875i 568.591i − 512.000i −588.246 −2589.60
337.7 8.00000i 31.8716 −64.0000 54.4265i 254.973i − 1112.71i − 512.000i −1171.20 −435.412
337.8 8.00000i 79.2712 −64.0000 − 132.638i 634.170i 1508.92i − 512.000i 4096.92 1061.10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.8.b.h 8
13.b even 2 1 inner 338.8.b.h 8
13.d odd 4 1 338.8.a.i 4
13.d odd 4 1 338.8.a.j 4
13.f odd 12 2 26.8.c.b ✓ 8
39.k even 12 2 234.8.h.b 8
52.l even 12 2 208.8.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.b ✓ 8 13.f odd 12 2
208.8.i.b 8 52.l even 12 2
234.8.h.b 8 39.k even 12 2
338.8.a.i 4 13.d odd 4 1
338.8.a.j 4 13.d odd 4 1
338.8.b.h 8 1.a even 1 1 trivial
338.8.b.h 8 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6981T_{3}^{2} - 35424T_{3} + 7188480 \) acting on \(S_{8}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 6981 T^{2} + \cdots + 7188480)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots - 59\!\cdots\!71)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 80\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 25\!\cdots\!13)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 75\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 44\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 77\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 60\!\cdots\!45)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 46\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
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