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Properties

Label 208.4.w.c
Level $208$
Weight $4$
Character orbit 208.w
Analytic conductor $12.272$
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] = [208,4,Mod(17,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.w (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: \(12.2723972812\)
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: \( x^{8} + 51x^{6} - 224x^{5} + 2520x^{4} - 5712x^{3} + 16675x^{2} + 9072x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 52)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{4} + \beta_{2} - 3 \beta_1 + 2) q^{5} + ( - \beta_{7} - \beta_{3} - 3 \beta_1 + 6) q^{7} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{4} + \beta_{2} - 3 \beta_1 + 2) q^{5} + ( - \beta_{7} - \beta_{3} - 3 \beta_1 + 6) q^{7} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \cdots + 1) q^{9}+ \cdots + (16 \beta_{7} - 16 \beta_{6} + \cdots - 536) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{7} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{7} - 70 q^{9} - 72 q^{11} + 62 q^{13} - 96 q^{15} + 88 q^{17} + 144 q^{19} + 20 q^{23} - 84 q^{25} + 432 q^{27} - 484 q^{29} + 1038 q^{33} - 40 q^{35} + 996 q^{37} + 236 q^{39} + 156 q^{41} - 504 q^{43} - 1530 q^{45} + 922 q^{49} + 1808 q^{51} - 1164 q^{53} + 1128 q^{55} - 600 q^{59} - 1224 q^{61} - 6480 q^{63} + 670 q^{65} - 960 q^{67} + 1738 q^{69} + 2964 q^{71} - 1448 q^{75} - 3972 q^{77} + 3968 q^{79} - 4132 q^{81} + 3870 q^{85} + 1660 q^{87} + 5430 q^{89} + 1720 q^{91} + 3324 q^{93} - 2400 q^{95} - 3042 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 51x^{6} - 224x^{5} + 2520x^{4} - 5712x^{3} + 16675x^{2} + 9072x + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9826 \nu^{7} - 6885 \nu^{6} + 485520 \nu^{5} - 2541224 \nu^{4} + 25532640 \nu^{3} + \cdots + 88255737 ) / 116697672 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64117 \nu^{7} + 352863 \nu^{6} + 3930864 \nu^{5} + 22456 \nu^{4} + 83154960 \nu^{3} + \cdots + 57304989 ) / 233395344 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5032 \nu^{7} + 13121 \nu^{6} + 248640 \nu^{5} - 563584 \nu^{4} + 9050020 \nu^{3} + \cdots + 125518095 ) / 12966408 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 99661 \nu^{7} + 145287 \nu^{6} + 4161696 \nu^{5} - 21247016 \nu^{4} + 157075128 \nu^{3} + \cdots + 16692237 ) / 233395344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 114845 \nu^{7} - 91287 \nu^{6} + 6437424 \nu^{5} - 28710472 \nu^{4} + 338532768 \nu^{3} + \cdots - 377106597 ) / 233395344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 167831 \nu^{7} + 351945 \nu^{6} - 7530096 \nu^{5} + 51933448 \nu^{4} - 462351960 \nu^{3} + \cdots - 5197882509 ) / 233395344 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 168443 \nu^{7} - 134451 \nu^{6} + 7560336 \nu^{5} - 52001992 \nu^{4} + 400635648 \nu^{3} + \cdots - 4523137929 ) / 233395344 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + \beta_{6} + 3\beta_{5} + 4\beta_{4} - \beta_{3} - 2\beta_{2} - \beta _1 - 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{7} - 14\beta_{6} + 15\beta_{5} - 2\beta_{4} + 8\beta_{3} + 4\beta_{2} - 307\beta _1 + 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{7} + 29\beta_{6} - 46\beta_{4} + 73\beta_{3} - 46\beta_{2} + 46\beta _1 + 481 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 938 \beta_{7} + 469 \beta_{6} - 429 \beta_{5} + 652 \beta_{4} - 469 \beta_{3} - 326 \beta_{2} + \cdots - 14797 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3661 \beta_{7} - 7322 \beta_{6} + 393 \beta_{5} - 4754 \beta_{4} - 3268 \beta_{3} + 9508 \beta_{2} + \cdots + 4754 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7462\beta_{7} + 7462\beta_{6} - 6692\beta_{4} + 10598\beta_{3} - 6692\beta_{2} + 6692\beta _1 + 205265 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 469082 \beta_{7} + 234541 \beta_{6} - 70521 \beta_{5} + 543028 \beta_{4} - 234541 \beta_{3} + \cdots - 6062797 ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.287051 + 0.497187i
−3.99238 + 6.91500i
2.54083 − 4.40084i
1.73860 − 3.01134i
−0.287051 − 0.497187i
−3.99238 − 6.91500i
2.54083 + 4.40084i
1.73860 + 3.01134i
0 −4.89420 + 8.47701i 0 − 2.49640i 0 15.5619 − 8.98467i 0 −34.4065 − 59.5937i 0
17.2 0 −0.230301 + 0.398893i 0 − 11.9830i 0 −26.4624 + 15.2781i 0 13.3939 + 23.1990i 0
17.3 0 0.643469 − 1.11452i 0 15.8968i 0 1.02552 − 0.592083i 0 12.6719 + 21.9484i 0
17.4 0 4.48104 − 7.76138i 0 − 11.8097i 0 27.8750 − 16.0936i 0 −26.6594 − 46.1754i 0
49.1 0 −4.89420 − 8.47701i 0 2.49640i 0 15.5619 + 8.98467i 0 −34.4065 + 59.5937i 0
49.2 0 −0.230301 − 0.398893i 0 11.9830i 0 −26.4624 − 15.2781i 0 13.3939 − 23.1990i 0
49.3 0 0.643469 + 1.11452i 0 − 15.8968i 0 1.02552 + 0.592083i 0 12.6719 − 21.9484i 0
49.4 0 4.48104 + 7.76138i 0 11.8097i 0 27.8750 + 16.0936i 0 −26.6594 + 46.1754i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.w.c 8
4.b odd 2 1 52.4.h.a ✓ 8
12.b even 2 1 468.4.t.g 8
13.e even 6 1 inner 208.4.w.c 8
52.b odd 2 1 676.4.h.e 8
52.f even 4 2 676.4.e.h 16
52.i odd 6 1 52.4.h.a ✓ 8
52.i odd 6 1 676.4.d.d 8
52.j odd 6 1 676.4.d.d 8
52.j odd 6 1 676.4.h.e 8
52.l even 12 2 676.4.a.g 8
52.l even 12 2 676.4.e.h 16
156.r even 6 1 468.4.t.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.4.h.a ✓ 8 4.b odd 2 1
52.4.h.a ✓ 8 52.i odd 6 1
208.4.w.c 8 1.a even 1 1 trivial
208.4.w.c 8 13.e even 6 1 inner
468.4.t.g 8 12.b even 2 1
468.4.t.g 8 156.r even 6 1
676.4.a.g 8 52.l even 12 2
676.4.d.d 8 52.i odd 6 1
676.4.d.d 8 52.j odd 6 1
676.4.e.h 16 52.f even 4 2
676.4.e.h 16 52.l even 12 2
676.4.h.e 8 52.b odd 2 1
676.4.h.e 8 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 89T_{3}^{6} - 144T_{3}^{5} + 7869T_{3}^{4} - 6408T_{3}^{3} + 9812T_{3}^{2} + 3744T_{3} + 2704 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 89 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$5$ \( T^{8} + 542 T^{6} + \cdots + 31539456 \) Copy content Toggle raw display
$7$ \( T^{8} - 36 T^{7} + \cdots + 437981184 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 1312546000896 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 9335475270801 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 5522368400784 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10\!\cdots\!29 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 534450801348864 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 34\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 47\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{4} + 582 T^{3} + \cdots + 492936516)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 19\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 43\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} - 1984 T^{3} + \cdots - 112716401664)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 79\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
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