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Properties

Label 1200.5.e.m
Level $1200$
Weight $5$
Character orbit 1200.e
Analytic conductor $124.044$
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] = [1200,5,Mod(751,1200)]
 
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, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.751");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1200.e (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: \(124.043955701\)
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: \( x^{8} + 44x^{6} - 78x^{5} + 1654x^{4} - 1716x^{3} + 13929x^{2} + 10998x + 79524 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 240)
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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + ( - \beta_{2} - 6 \beta_1) q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + ( - \beta_{2} - 6 \beta_1) q^{7} - 27 q^{9} + (\beta_{4} - \beta_{2} - 8 \beta_1) q^{11} + (\beta_{7} + \beta_{5} + 3 \beta_{3} + 6) q^{13} + ( - \beta_{7} + \beta_{5} + 3 \beta_{3} + 48) q^{17} + (\beta_{6} - 6 \beta_{2} + 11 \beta_1) q^{19} + ( - 9 \beta_{3} + 45) q^{21} + (\beta_{6} + 6 \beta_{4} + \cdots - 45 \beta_1) q^{23}+ \cdots + ( - 27 \beta_{4} + 27 \beta_{2} + 216 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 216 q^{9} + 36 q^{13} + 372 q^{17} + 396 q^{21} - 588 q^{29} + 540 q^{33} + 972 q^{37} + 6312 q^{41} - 6112 q^{49} - 4980 q^{53} - 1008 q^{57} + 16784 q^{61} + 3168 q^{69} - 21240 q^{73} - 29688 q^{77} + 5832 q^{81} + 9168 q^{89} - 9000 q^{93} - 6960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 44x^{6} - 78x^{5} + 1654x^{4} - 1716x^{3} + 13929x^{2} + 10998x + 79524 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 138424 \nu^{7} + 855118 \nu^{6} - 5203484 \nu^{5} + 42941735 \nu^{4} - 262302898 \nu^{3} + \cdots + 5316713085 ) / 5293591059 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9704572 \nu^{7} - 196805121 \nu^{6} - 116431867 \nu^{5} - 9839364456 \nu^{4} + \cdots - 1107972994500 ) / 26467955295 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 504658 \nu^{7} - 1476909 \nu^{6} - 18970553 \nu^{5} + 19681662 \nu^{4} - 551845687 \nu^{3} + \cdots + 9494023587 ) / 563147985 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36706288 \nu^{7} - 403992354 \nu^{6} - 545119468 \nu^{5} - 22380650274 \nu^{4} + \cdots - 2058913317510 ) / 26467955295 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1115752 \nu^{7} + 3609006 \nu^{6} + 41942132 \nu^{5} - 43514328 \nu^{4} + 803665798 \nu^{3} + \cdots - 18066768678 ) / 563147985 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24766636 \nu^{7} + 59016302 \nu^{6} + 1893471464 \nu^{5} + 1249149937 \nu^{4} + \cdots + 419952402735 ) / 8822651765 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1016136 \nu^{7} - 1598988 \nu^{6} - 38197476 \nu^{5} + 39629304 \nu^{4} - 1167810024 \nu^{3} + \cdots - 13195096881 ) / 187715995 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 3\beta_{5} + 3\beta_{4} + 6\beta_{3} - 6\beta_{2} - 3\beta _1 + 3 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} + 8\beta_{3} + 8\beta_{2} + 444\beta _1 - 436 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} - 84\beta_{5} - 228\beta_{3} + 1641 ) / 60 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{7} - 15\beta_{6} - \beta_{5} - \beta_{4} + 86\beta_{3} - 86\beta_{2} - 2787\beta _1 - 2701 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 824 \beta_{7} + 824 \beta_{6} + 2733 \beta_{5} - 2733 \beta_{4} + 9276 \beta_{3} + 9276 \beta_{2} + \cdots - 124062 ) / 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2827\beta_{7} - 1154\beta_{5} - 19628\beta_{3} + 492601 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 43057 \beta_{7} - 43057 \beta_{6} + 95979 \beta_{5} + 95979 \beta_{4} + 394158 \beta_{3} + \cdots - 6576051 ) / 120 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
1.92268 − 3.33017i
−1.11749 + 1.93555i
2.48967 − 4.31224i
−3.29485 + 5.70685i
−3.29485 − 5.70685i
2.48967 + 4.31224i
−1.11749 − 1.93555i
1.92268 + 3.33017i
0 − 5.19615i 0 0 0 − 59.0422i 0 −27.0000 0
751.2 0 − 5.19615i 0 0 0 − 6.21699i 0 −27.0000 0
751.3 0 − 5.19615i 0 0 0 8.13170i 0 −27.0000 0
751.4 0 − 5.19615i 0 0 0 95.2326i 0 −27.0000 0
751.5 0 5.19615i 0 0 0 − 95.2326i 0 −27.0000 0
751.6 0 5.19615i 0 0 0 − 8.13170i 0 −27.0000 0
751.7 0 5.19615i 0 0 0 6.21699i 0 −27.0000 0
751.8 0 5.19615i 0 0 0 59.0422i 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.e.m 8
4.b odd 2 1 inner 1200.5.e.m 8
5.b even 2 1 1200.5.e.k 8
5.c odd 4 2 240.5.j.b ✓ 16
15.e even 4 2 720.5.j.f 16
20.d odd 2 1 1200.5.e.k 8
20.e even 4 2 240.5.j.b ✓ 16
40.i odd 4 2 960.5.j.b 16
40.k even 4 2 960.5.j.b 16
60.l odd 4 2 720.5.j.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.5.j.b ✓ 16 5.c odd 4 2
240.5.j.b ✓ 16 20.e even 4 2
720.5.j.f 16 15.e even 4 2
720.5.j.f 16 60.l odd 4 2
960.5.j.b 16 40.i odd 4 2
960.5.j.b 16 40.k even 4 2
1200.5.e.k 8 5.b even 2 1
1200.5.e.k 8 20.d odd 2 1
1200.5.e.m 8 1.a even 1 1 trivial
1200.5.e.m 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{8} + 12660T_{7}^{6} + 32933232T_{7}^{4} + 3344587200T_{7}^{2} + 80801473536 \) Copy content Toggle raw display
\( T_{13}^{4} - 18T_{13}^{3} - 39876T_{13}^{2} - 228312T_{13} + 294152256 \) 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
$7$ \( T^{8} + \cdots + 80801473536 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 90610904088576 \) Copy content Toggle raw display
$13$ \( (T^{4} - 18 T^{3} + \cdots + 294152256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 186 T^{3} + \cdots - 133059456)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + 294 T^{3} + \cdots + 567049792000)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{4} - 486 T^{3} + \cdots + 140040048960)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 3359786346896)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 50752186801344)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 184329485874800)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 22961323431936)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 28833102007696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 57\!\cdots\!64)^{2} \) Copy content Toggle raw display
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