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Properties

Label 1116.4.i.e
Level $1116$
Weight $4$
Character orbit 1116.i
Analytic conductor $65.846$
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] = [1116,4,Mod(253,1116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1116, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1116.253");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1116.i (of order \(3\), degree \(2\), 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: \(65.8461315664\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 29x^{6} - 58x^{5} + 824x^{4} - 1198x^{3} + 1933x^{2} + 129x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 124)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{6} + \cdots + \beta_{3}) q^{5}+ \cdots + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{6} + \cdots + \beta_{3}) q^{5}+ \cdots + (37 \beta_{3} - 42 \beta_{2} + \cdots + 315) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{5} - 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{5} - 32 q^{7} + 80 q^{11} - 28 q^{13} + 8 q^{17} - 56 q^{19} + 624 q^{23} - 48 q^{25} + 216 q^{29} + 528 q^{31} - 592 q^{35} + 96 q^{37} - 552 q^{41} - 112 q^{43} + 304 q^{47} - 4 q^{49} - 1316 q^{53} - 208 q^{55} - 224 q^{59} + 2664 q^{61} - 504 q^{65} - 272 q^{67} - 1120 q^{71} + 248 q^{73} - 624 q^{77} + 824 q^{79} - 1616 q^{83} - 2232 q^{85} - 184 q^{89} - 6928 q^{91} - 544 q^{95} + 2520 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 29x^{6} - 58x^{5} + 824x^{4} - 1198x^{3} + 1933x^{2} + 129x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 117404 \nu^{7} + 31169 \nu^{6} + 3211838 \nu^{5} - 1568182 \nu^{4} + 88088321 \nu^{3} + \cdots + 588938652 ) / 49422123 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 42896 \nu^{7} + 27104 \nu^{6} + 1173512 \nu^{5} - 572968 \nu^{4} + 28919957 \nu^{3} + \cdots + 88248789 ) / 16474041 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75068 \nu^{7} - 47432 \nu^{6} - 2053646 \nu^{5} + 1002694 \nu^{4} - 54728435 \nu^{3} + \cdots - 109131768 ) / 16474041 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81647 \nu^{7} - 87009 \nu^{6} + 2364375 \nu^{5} - 4882215 \nu^{4} + 67348749 \nu^{3} + \cdots + 10506036 ) / 10982694 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 612323 \nu^{7} + 659755 \nu^{6} - 17928125 \nu^{5} + 36812338 \nu^{4} - 510679055 \nu^{3} + \cdots + 3614310 ) / 16474041 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1469587 \nu^{7} + 1580537 \nu^{6} - 42949375 \nu^{5} + 88271321 \nu^{4} - 1223404357 \nu^{3} + \cdots + 8658594 ) / 32948082 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4377592 \nu^{7} - 4517564 \nu^{6} + 126818713 \nu^{5} - 257729570 \nu^{4} + 3609005434 \nu^{3} + \cdots + 564019506 ) / 49422123 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{4} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{7} + 3\beta_{6} - \beta_{5} - 57\beta_{4} + 3\beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} - 7\beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -168\beta_{7} - 111\beta_{6} + 43\beta_{5} + 1497\beta_{4} + 168\beta _1 - 1497 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90\beta_{7} + 463\beta_{6} - 781\beta_{5} - 2189\beta_{4} + 463\beta_{3} + 781\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -831\beta_{3} - 406\beta_{2} - 1149\beta _1 + 10362 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5148\beta_{7} - 14365\beta_{6} + 22009\beta_{5} + 84083\beta_{4} + 5148\beta _1 - 84083 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times\).

\(n\) \(497\) \(559\) \(685\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
253.1
−0.0334318 + 0.0579056i
0.833026 − 1.44284i
−2.74884 + 4.76113i
2.44925 − 4.24222i
−0.0334318 − 0.0579056i
0.833026 + 1.44284i
−2.74884 − 4.76113i
2.44925 + 4.24222i
0 0 0 −7.27011 − 12.5922i 0 −4.03211 + 6.98382i 0 0 0
253.2 0 0 0 3.98613 + 6.90419i 0 3.19423 − 5.53257i 0 0 0
253.3 0 0 0 4.12668 + 7.14762i 0 2.48358 − 4.30169i 0 0 0
253.4 0 0 0 7.15729 + 12.3968i 0 −17.6457 + 30.5633i 0 0 0
397.1 0 0 0 −7.27011 + 12.5922i 0 −4.03211 − 6.98382i 0 0 0
397.2 0 0 0 3.98613 − 6.90419i 0 3.19423 + 5.53257i 0 0 0
397.3 0 0 0 4.12668 − 7.14762i 0 2.48358 + 4.30169i 0 0 0
397.4 0 0 0 7.15729 − 12.3968i 0 −17.6457 − 30.5633i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 253.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1116.4.i.e 8
3.b odd 2 1 124.4.e.c ✓ 8
12.b even 2 1 496.4.i.e 8
31.c even 3 1 inner 1116.4.i.e 8
93.h odd 6 1 124.4.e.c ✓ 8
372.p even 6 1 496.4.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.e.c ✓ 8 3.b odd 2 1
124.4.e.c ✓ 8 93.h odd 6 1
496.4.i.e 8 12.b even 2 1
496.4.i.e 8 372.p even 6 1
1116.4.i.e 8 1.a even 1 1 trivial
1116.4.i.e 8 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1116, [\chi])\):

\( T_{5}^{8} - 16 T_{5}^{7} + 402 T_{5}^{6} - 4448 T_{5}^{5} + 89283 T_{5}^{4} - 933472 T_{5}^{3} + \cdots + 187553025 \) Copy content Toggle raw display
\( T_{13}^{8} + 28 T_{13}^{7} + 7442 T_{13}^{6} + 269232 T_{13}^{5} + 52246619 T_{13}^{4} + \cdots + 2366893017841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{7} + \cdots + 187553025 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{7} + \cdots + 81558961 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 4788778401 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2366893017841 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 12137208561 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 77787196287025 \) Copy content Toggle raw display
$23$ \( (T^{4} - 312 T^{3} + \cdots - 279217152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 108 T^{3} + \cdots + 638715456)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 78\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 44\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 1137223889649 \) Copy content Toggle raw display
$47$ \( (T^{4} - 152 T^{3} + \cdots + 13923790848)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 74\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 49\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( (T^{4} - 1332 T^{3} + \cdots + 725031296)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 29\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 69\!\cdots\!29 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 35\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 32\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{4} + 92 T^{3} + \cdots + 75650082624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1260 T^{3} + \cdots - 319459930240)^{2} \) Copy content Toggle raw display
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