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Properties

Label 208.8.f.c
Level $208$
Weight $8$
Character orbit 208.f
Analytic conductor $64.976$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [208,8,Mod(129,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.129"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(64.9760853007\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 17770x^{8} + 98320641x^{6} + 176057788072x^{4} + 109845194658832x^{2} + 14762086704451584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 5) q^{3} - \beta_{4} q^{5} + ( - \beta_{9} - \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{2} + 1396) q^{9} + ( - \beta_{8} - \beta_{7} + \cdots - 3 \beta_{3}) q^{11} + (2 \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 361) q^{13}+ \cdots + (399 \beta_{9} + \cdots + 171024 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 54 q^{3} + 13960 q^{9} + 3432 q^{13} - 6918 q^{17} - 94164 q^{23} - 330788 q^{25} + 53550 q^{27} - 131304 q^{29} - 873450 q^{35} + 1569460 q^{39} - 2740774 q^{43} - 3083904 q^{49} - 1103186 q^{51}+ \cdots - 16547484 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 17770x^{8} + 98320641x^{6} + 176057788072x^{4} + 109845194658832x^{2} + 14762086704451584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5867 \nu^{8} - 94017378 \nu^{6} - 429800329035 \nu^{4} - 434217718330916 \nu^{2} - 51\!\cdots\!64 ) / 999398686918320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5867 \nu^{8} + 94017378 \nu^{6} + 429800329035 \nu^{4} + 534157587022748 \nu^{2} + 40\!\cdots\!20 ) / 99939868691832 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35664299 \nu^{9} + 574351546782 \nu^{7} + \cdots - 47\!\cdots\!36 \nu ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 498043218541 \nu^{9} + \cdots - 82\!\cdots\!84 \nu ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 498043218541 \nu^{9} - 218544744439512 \nu^{8} + \cdots - 97\!\cdots\!48 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 498043218541 \nu^{9} + \cdots + 26\!\cdots\!84 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 210213627636 \nu^{9} + \cdots + 10\!\cdots\!72 \nu ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3933011561659 \nu^{9} + \cdots - 18\!\cdots\!40 \nu ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1759324984357 \nu^{9} + \cdots + 60\!\cdots\!36 \nu ) / 24\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{7} - 3\beta_{4} ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10\beta _1 - 3558 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7963\beta_{9} + 720\beta_{8} - 5659\beta_{7} + 36273\beta_{4} - 42774\beta_{3} ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 255 \beta_{9} - 255 \beta_{7} - 117 \beta_{6} - 627 \beta_{5} + 255 \beta_{4} - 8374 \beta_{2} + \cdots + 23873625 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -60815635\beta_{9} - 9249552\beta_{8} + 40634899\beta_{7} - 287064345\beta_{4} + 481907598\beta_{3} ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2090175 \beta_{9} + 2090175 \beta_{7} + 1567809 \beta_{6} + 5748159 \beta_{5} - 2090175 \beta_{4} + \cdots - 180094740693 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 472018270723 \beta_{9} + 94209298704 \beta_{8} - 306654898435 \beta_{7} + \cdots - 4900691001102 \beta_{3} ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14813991675 \beta_{9} - 14813991675 \beta_{7} - 16552693521 \beta_{6} - 46180676871 \beta_{5} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 36\!\cdots\!79 \beta_{9} - 893550136102416 \beta_{8} + \cdots + 46\!\cdots\!90 \beta_{3} ) / 48 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
− 83.5063i
83.5063i
32.3570i
− 32.3570i
13.5100i
− 13.5100i
36.7123i
− 36.7123i
− 90.6609i
90.6609i
0 −88.5063 0 − 433.113i 0 − 805.670i 0 5646.37 0
129.2 0 −88.5063 0 433.113i 0 805.670i 0 5646.37 0
129.3 0 −37.3570 0 − 477.073i 0 855.664i 0 −791.454 0
129.4 0 −37.3570 0 477.073i 0 − 855.664i 0 −791.454 0
129.5 0 −18.5100 0 − 6.61448i 0 − 1604.67i 0 −1844.38 0
129.6 0 −18.5100 0 6.61448i 0 1604.67i 0 −1844.38 0
129.7 0 31.7123 0 − 319.050i 0 − 1008.64i 0 −1181.33 0
129.8 0 31.7123 0 319.050i 0 1008.64i 0 −1181.33 0
129.9 0 85.6609 0 − 197.476i 0 − 828.309i 0 5150.80 0
129.10 0 85.6609 0 197.476i 0 828.309i 0 5150.80 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.10
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 208.8.f.c 10
4.b odd 2 1 26.8.b.a ✓ 10
12.b even 2 1 234.8.b.c 10
13.b even 2 1 inner 208.8.f.c 10
52.b odd 2 1 26.8.b.a ✓ 10
52.f even 4 1 338.8.a.k 5
52.f even 4 1 338.8.a.l 5
156.h even 2 1 234.8.b.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.b.a ✓ 10 4.b odd 2 1
26.8.b.a ✓ 10 52.b odd 2 1
208.8.f.c 10 1.a even 1 1 trivial
208.8.f.c 10 13.b even 2 1 inner
234.8.b.c 10 12.b even 2 1
234.8.b.c 10 156.h even 2 1
338.8.a.k 5 52.f even 4 1
338.8.a.l 5 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 27T_{3}^{4} - 8593T_{3}^{3} - 208131T_{3}^{2} + 8127144T_{3} + 166250448 \) acting on \(S_{8}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{5} + 27 T^{4} + \cdots + 166250448)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 85\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 97\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 84\!\cdots\!72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 44\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 40\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 25\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 37\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
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