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Properties

Label 1008.5.m.g
Level $1008$
Weight $5$
Character orbit 1008.m
Analytic conductor $104.197$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,5,Mod(127,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.m (of order \(2\), degree \(1\), 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: \(104.196922789\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 198 x^{14} + 26419 x^{12} + 1982934 x^{10} + 108625521 x^{8} + 3296539080 x^{6} + \cdots + 280476160000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{52}\cdot 3^{8}\cdot 7^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + \beta_1 q^{7} + \beta_{5} q^{11} + (\beta_{6} + 8) q^{13} + (\beta_{11} - \beta_{3}) q^{17} + (\beta_{8} + 5 \beta_{2}) q^{19} + ( - \beta_{12} - \beta_{9} + \beta_{5}) q^{23} + (\beta_{7} + 141) q^{25} + (\beta_{15} - \beta_{3}) q^{29} + (2 \beta_{13} - 3 \beta_{8} + \cdots - 6 \beta_1) q^{31}+ \cdots + ( - 8 \beta_{7} - 22 \beta_{6} + \cdots - 2446) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 128 q^{13} + 2256 q^{25} + 3584 q^{37} - 5488 q^{49} + 11904 q^{61} - 24032 q^{73} - 9184 q^{85} - 39136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 198 x^{14} + 26419 x^{12} + 1982934 x^{10} + 108625521 x^{8} + 3296539080 x^{6} + \cdots + 280476160000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4519521 \nu^{14} + 856469158 \nu^{12} + 112999420499 \nu^{10} + 8207737622614 \nu^{8} + \cdots + 32\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36\!\cdots\!99 \nu^{14} + \cdots - 27\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!47 \nu^{15} + \cdots + 19\!\cdots\!00 \nu ) / 96\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16352015 \nu^{14} - 3201106210 \nu^{12} - 363230130669 \nu^{10} - 23292270705762 \nu^{8} + \cdots - 22\!\cdots\!08 ) / 44\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!37 \nu^{15} + \cdots + 56\!\cdots\!00 \nu ) / 96\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 174650989406155 \nu^{14} + \cdots - 76\!\cdots\!68 ) / 32\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!55 \nu^{14} + \cdots - 72\!\cdots\!76 ) / 29\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33\!\cdots\!77 \nu^{14} + \cdots - 30\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!53 \nu^{15} + \cdots + 48\!\cdots\!00 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!81 \nu^{15} + \cdots + 31\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60\!\cdots\!09 \nu^{15} + \cdots - 79\!\cdots\!00 \nu ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 63\!\cdots\!61 \nu^{15} + \cdots + 19\!\cdots\!00 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!47 \nu^{14} + \cdots - 10\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 51\!\cdots\!41 \nu^{15} + \cdots + 64\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!71 \nu^{15} + \cdots + 28\!\cdots\!00 \nu ) / 32\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 12 \beta_{15} - 2 \beta_{14} + 36 \beta_{12} - 38 \beta_{11} - 21 \beta_{10} + \cdots + 144 \beta_{3} ) / 4704 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{13} + 14\beta_{8} - 14\beta_{6} - 9\beta_{4} - 518\beta_{2} - 425\beta _1 - 8316 ) / 336 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 183\beta_{15} + 55\beta_{14} + 604\beta_{11} - 4401\beta_{3} ) / 588 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 63 \beta_{13} - 294 \beta_{8} + 28 \beta_{7} - 210 \beta_{6} - 275 \beta_{4} + 11872 \beta_{2} + \cdots - 190876 ) / 112 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 52548 \beta_{15} - 20812 \beta_{14} - 157644 \beta_{12} - 157288 \beta_{11} + \cdots + 1517868 \beta_{3} ) / 4704 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -12180\beta_{7} + 27874\beta_{6} + 68631\beta_{4} + 41615532 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3845340 \beta_{15} - 1833452 \beta_{14} + 11536020 \beta_{12} - 10965728 \beta_{11} + \cdots + 122164836 \beta_{3} ) / 4704 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 451521 \beta_{13} + 1772358 \beta_{8} + 445900 \beta_{7} - 434658 \beta_{6} - 1836515 \beta_{4} + \cdots - 1036833308 ) / 112 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 70986369\beta_{15} + 39054359\beta_{14} + 200686358\beta_{11} - 2383573635\beta_{3} ) / 588 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 164680397 \beta_{13} - 463719998 \beta_{8} + 132180132 \beta_{7} - 67179602 \beta_{6} + \cdots - 236462338476 ) / 336 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 21082049052 \beta_{15} - 13081505884 \beta_{14} - 63246147156 \beta_{12} + \cdots + 732862075524 \beta_{3} ) / 4704 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -4121675292\beta_{7} + 1313664786\beta_{6} + 11353918371\beta_{4} + 6053838040348 ) / 56 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1570750185156 \beta_{15} - 1084696754668 \beta_{14} + 4712250555468 \beta_{12} + \cdots + 55910334310380 \beta_{3} ) / 4704 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 220858114987 \beta_{13} + 517223619250 \beta_{8} + 159823308204 \beta_{7} - 37753694638 \beta_{6} + \cdots - 200545370851572 ) / 48 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 29327381592279 \beta_{15} + 22348556976563 \beta_{14} + 90570730113152 \beta_{11} - 10\!\cdots\!05 \beta_{3} ) / 588 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−4.29732 − 7.44318i
−4.29732 + 7.44318i
3.23543 − 5.60393i
3.23543 + 5.60393i
0.730929 − 1.26601i
0.730929 + 1.26601i
−4.47557 − 7.75192i
−4.47557 + 7.75192i
4.47557 + 7.75192i
4.47557 − 7.75192i
−0.730929 + 1.26601i
−0.730929 − 1.26601i
−3.23543 + 5.60393i
−3.23543 − 5.60393i
4.29732 + 7.44318i
4.29732 − 7.44318i
0 0 0 −37.8078 0 − 18.5203i 0 0 0
127.2 0 0 0 −37.8078 0 18.5203i 0 0 0
127.3 0 0 0 −28.7002 0 − 18.5203i 0 0 0
127.4 0 0 0 −28.7002 0 18.5203i 0 0 0
127.5 0 0 0 −27.2939 0 − 18.5203i 0 0 0
127.6 0 0 0 −27.2939 0 18.5203i 0 0 0
127.7 0 0 0 −8.11861 0 − 18.5203i 0 0 0
127.8 0 0 0 −8.11861 0 18.5203i 0 0 0
127.9 0 0 0 8.11861 0 − 18.5203i 0 0 0
127.10 0 0 0 8.11861 0 18.5203i 0 0 0
127.11 0 0 0 27.2939 0 − 18.5203i 0 0 0
127.12 0 0 0 27.2939 0 18.5203i 0 0 0
127.13 0 0 0 28.7002 0 − 18.5203i 0 0 0
127.14 0 0 0 28.7002 0 18.5203i 0 0 0
127.15 0 0 0 37.8078 0 − 18.5203i 0 0 0
127.16 0 0 0 37.8078 0 18.5203i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.5.m.g ✓ 16
3.b odd 2 1 inner 1008.5.m.g ✓ 16
4.b odd 2 1 inner 1008.5.m.g ✓ 16
12.b even 2 1 inner 1008.5.m.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.5.m.g ✓ 16 1.a even 1 1 trivial
1008.5.m.g ✓ 16 3.b odd 2 1 inner
1008.5.m.g ✓ 16 4.b odd 2 1 inner
1008.5.m.g ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 3064T_{5}^{6} + 3053520T_{5}^{4} - 1065369088T_{5}^{2} + 57813339136 \) acting on \(S_{5}^{\mathrm{new}}(1008, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3064 T^{6} + \cdots + 57813339136)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 343)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 32 T^{3} + \cdots + 76339216)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 78\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 2416277508880)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 47\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 43434561115152)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 84\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 116672058374000)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 61\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 911961951068528)^{4} \) Copy content Toggle raw display
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