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Properties

Label 1200.5.j.d
Level $1200$
Weight $5$
Character orbit 1200.j
Analytic conductor $124.044$
Analytic rank $0$
Dimension $8$
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] = [1200,5,Mod(799,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.799");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1200.j (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: \(124.043955701\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.70892257536.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 21x^{6} + 320x^{4} + 2541x^{2} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta_{2} q^{3} + (\beta_{7} - 13 \beta_{2}) q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_{2} q^{3} + (\beta_{7} - 13 \beta_{2}) q^{7} + 27 q^{9} + (3 \beta_{4} + 20 \beta_1) q^{11} + 37 \beta_{3} q^{13} + ( - \beta_{6} + 96 \beta_{3}) q^{17} + ( - 13 \beta_{4} + 49 \beta_1) q^{19} + (3 \beta_{5} + 117) q^{21} + ( - 9 \beta_{7} - 16 \beta_{2}) q^{23} - 81 \beta_{2} q^{27} + ( - 13 \beta_{5} - 204) q^{29} + ( - 23 \beta_{4} + 405 \beta_1) q^{31} + (9 \beta_{6} + 180 \beta_{3}) q^{33} + (2 \beta_{6} + 526 \beta_{3}) q^{37} + 111 \beta_1 q^{39} + ( - 29 \beta_{5} - 666) q^{41} + ( - 11 \beta_{7} - 243 \beta_{2}) q^{43} + (36 \beta_{7} - 118 \beta_{2}) q^{47} + (26 \beta_{5} - 346) q^{49} + ( - 9 \beta_{4} + 288 \beta_1) q^{51} + ( - 7 \beta_{6} + 1170 \beta_{3}) q^{53} + ( - 39 \beta_{6} + 441 \beta_{3}) q^{57} + (12 \beta_{4} + 1990 \beta_1) q^{59} + (46 \beta_{5} - 1897) q^{61} + (27 \beta_{7} - 351 \beta_{2}) q^{63} + (57 \beta_{7} + 1579 \beta_{2}) q^{67} + ( - 27 \beta_{5} + 144) q^{69} + (63 \beta_{4} - 226 \beta_1) q^{71} + ( - 22 \beta_{6} + 2710 \beta_{3}) q^{73} + (19 \beta_{6} - 3864 \beta_{3}) q^{77} + (52 \beta_{4} + 1612 \beta_1) q^{79} + 729 q^{81} + (120 \beta_{7} - 2766 \beta_{2}) q^{83} + ( - 117 \beta_{7} + 612 \beta_{2}) q^{87} + (80 \beta_{5} + 5904) q^{89} + ( - 37 \beta_{4} + 481 \beta_1) q^{91} + ( - 69 \beta_{6} + 3645 \beta_{3}) q^{93} + (116 \beta_{6} + 5195 \beta_{3}) q^{97} + (81 \beta_{4} + 540 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 216 q^{9} + 936 q^{21} - 1632 q^{29} - 5328 q^{41} - 2768 q^{49} - 15176 q^{61} + 1152 q^{69} + 5832 q^{81} + 47232 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 21x^{6} + 320x^{4} + 2541x^{2} + 14641 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21\nu^{6} + 320\nu^{4} + 6720\nu^{2} + 34001 ) / 19360 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 100\nu^{5} - 890\nu^{3} - 9559\nu ) / 13310 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{7} + 320\nu^{5} + 2848\nu^{3} + 14641\nu ) / 42592 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} - 2457 ) / 80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -12\nu^{6} - 252\nu^{4} - 2388\nu^{2} - 15246 ) / 121 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 129\nu^{7} + 3072\nu^{5} + 52896\nu^{3} + 699501\nu ) / 21296 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 63\nu^{7} + 960\nu^{5} + 16530\nu^{3} + 43923\nu ) / 6655 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + 6\beta_{3} + 6\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + 126\beta _1 - 126 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - 96\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{5} - 7\beta_{4} - 398\beta _1 - 398 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -89\beta_{7} - 89\beta_{6} + 3306\beta_{3} - 3306\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 80\beta_{4} + 2457 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -659\beta_{7} + 659\beta_{6} + 46194\beta_{3} + 46194\beta_{2} ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
2.07237 + 2.58945i
2.07237 − 2.58945i
−1.20635 + 3.08945i
−1.20635 − 3.08945i
1.20635 − 3.08945i
1.20635 + 3.08945i
−2.07237 − 2.58945i
−2.07237 + 2.58945i
0 −5.19615 0 0 0 −61.8613 0 27.0000 0
799.2 0 −5.19615 0 0 0 −61.8613 0 27.0000 0
799.3 0 −5.19615 0 0 0 16.8280 0 27.0000 0
799.4 0 −5.19615 0 0 0 16.8280 0 27.0000 0
799.5 0 5.19615 0 0 0 −16.8280 0 27.0000 0
799.6 0 5.19615 0 0 0 −16.8280 0 27.0000 0
799.7 0 5.19615 0 0 0 61.8613 0 27.0000 0
799.8 0 5.19615 0 0 0 61.8613 0 27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.5.j.d 8
4.b odd 2 1 inner 1200.5.j.d 8
5.b even 2 1 inner 1200.5.j.d 8
5.c odd 4 1 1200.5.e.e ✓ 4
5.c odd 4 1 1200.5.e.f yes 4
20.d odd 2 1 inner 1200.5.j.d 8
20.e even 4 1 1200.5.e.e ✓ 4
20.e even 4 1 1200.5.e.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.5.e.e ✓ 4 5.c odd 4 1
1200.5.e.e ✓ 4 20.e even 4 1
1200.5.e.f yes 4 5.c odd 4 1
1200.5.e.f yes 4 20.e even 4 1
1200.5.j.d 8 1.a even 1 1 trivial
1200.5.j.d 8 4.b odd 2 1 inner
1200.5.j.d 8 5.b even 2 1 inner
1200.5.j.d 8 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 4110T_{7}^{2} + 1083681 \) acting on \(S_{5}^{\mathrm{new}}(1200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4110 T^{2} + 1083681)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 30264 T^{2} + 162103824)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1369)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 27720 T^{2} + 20903184)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 537630 T^{2} + 64723939281)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 252312 T^{2} + 15530144400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 408 T - 743220)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2621934 T^{2} + 106809351489)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 590504 T^{2} + 66615610000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1332 T - 3462048)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 728910 T^{2} + 103245921)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 3859008798096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 1302666126336)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 135894694982544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3794 T - 6228095)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 6003827973441)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 35889492934656)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 25973333731216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 13030944825600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 90486936 T^{2} + 437010900624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 11808 T + 5135616)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 12\!\cdots\!21)^{2} \) Copy content Toggle raw display
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