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Properties

Label 1080.4.f.a
Level $1080$
Weight $4$
Character orbit 1080.f
Analytic conductor $63.722$
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] = [1080,4,Mod(649,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1080.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.f (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: \(63.7220628062\)
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: \( x^{16} - 216 x^{14} + 19078 x^{12} - 888840 x^{10} + 23532927 x^{8} - 353885448 x^{6} + \cdots + 15738957025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{9}) q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{9}) q^{5} + \beta_{4} q^{7} + (\beta_{13} - \beta_{9} + \beta_{8}) q^{11} + (\beta_{7} + \beta_{5} + \beta_1) q^{13} + ( - \beta_{15} + \beta_{14} - 23 \beta_{10}) q^{17} + (\beta_{5} - \beta_{3} - \beta_1 + 7) q^{19} + ( - \beta_{15} + 2 \beta_{14} - 27 \beta_{10}) q^{23} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 11) q^{25}+ \cdots + ( - 3 \beta_{7} + 26 \beta_{5} + \cdots + 26 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 100 q^{19} - 186 q^{25} - 288 q^{31} - 2124 q^{49} + 1222 q^{55} + 188 q^{61} + 4964 q^{79} + 636 q^{85} - 4144 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 216 x^{14} + 19078 x^{12} - 888840 x^{10} + 23532927 x^{8} - 353885448 x^{6} + \cdots + 15738957025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15258859194448 \nu^{14} + \cdots - 12\!\cdots\!65 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 42983520316 \nu^{14} - 11582164234825 \nu^{12} + \cdots - 11\!\cdots\!65 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1842194879 \nu^{14} + 308270786452 \nu^{12} - 17820900631131 \nu^{10} + \cdots + 38\!\cdots\!45 ) / 72\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35268877925747 \nu^{14} + \cdots - 23\!\cdots\!25 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43396483984573 \nu^{14} + \cdots - 20\!\cdots\!30 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44386280649 \nu^{14} + 8165222543552 \nu^{12} - 575869875487621 \nu^{10} + \cdots + 19\!\cdots\!25 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 218487966787189 \nu^{14} + \cdots + 36\!\cdots\!45 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!73 \nu^{15} + \cdots - 40\!\cdots\!06 \nu ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25\!\cdots\!43 \nu^{15} + \cdots - 33\!\cdots\!53 \nu ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 52586385563 \nu^{15} - 10532528622210 \nu^{13} + 843444805720725 \nu^{11} + \cdots - 96\!\cdots\!57 \nu ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!90 \nu^{15} + \cdots + 12\!\cdots\!07 \nu ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 91\!\cdots\!70 \nu^{15} + \cdots - 18\!\cdots\!55 \nu ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 20\!\cdots\!77 \nu^{15} + \cdots + 62\!\cdots\!56 \nu ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3537238187730 \nu^{15} - 760112487320573 \nu^{13} + \cdots - 18\!\cdots\!71 \nu ) / 13\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 345583816839806 \nu^{15} + \cdots - 70\!\cdots\!65 \nu ) / 72\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{13} - \beta_{12} + \beta_{11} + 12\beta_{10} - 6\beta_{9} + 6\beta_{8} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 4\beta_{6} + 2\beta_{5} + \beta_{4} - 8\beta_{3} - 2\beta_{2} + 10\beta _1 + 644 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{15} - 12 \beta_{14} + 93 \beta_{13} - 88 \beta_{12} + 5 \beta_{11} + 972 \beta_{10} + \cdots + 237 \beta_{8} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{7} + 272\beta_{6} + 700\beta_{5} + 177\beta_{4} - 496\beta_{3} - 188\beta_{2} + 284\beta _1 + 25588 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 690 \beta_{15} - 1260 \beta_{14} + 4139 \beta_{13} - 6227 \beta_{12} - 384 \beta_{11} + \cdots + 11139 \beta_{8} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 281 \beta_{7} + 3675 \beta_{6} + 14361 \beta_{5} + 4781 \beta_{4} - 6840 \beta_{3} - 3222 \beta_{2} + \cdots + 289896 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 53872 \beta_{15} - 100184 \beta_{14} + 187848 \beta_{13} - 366131 \beta_{12} - 33923 \beta_{11} + \cdots + 594008 \beta_{8} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 140911 \beta_{7} + 746192 \beta_{6} + 3811792 \beta_{5} + 1554183 \beta_{4} - 1444960 \beta_{3} + \cdots + 55441876 ) / 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3684006 \beta_{15} - 7108740 \beta_{14} + 8729415 \beta_{13} - 19779052 \beta_{12} + \cdots + 32812299 \beta_{8} ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11405349 \beta_{7} + 36919592 \beta_{6} + 233711458 \beta_{5} + 110686387 \beta_{4} - 73833496 \beta_{3} + \cdots + 2691779788 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 237428026 \beta_{15} - 472363628 \beta_{14} + 410173911 \beta_{13} - 1013532925 \beta_{12} + \cdots + 1805613119 \beta_{8} ) / 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 33157430 \beta_{7} + 74432541 \beta_{6} + 572538300 \beta_{5} + 305200645 \beta_{4} - 152037724 \beta_{3} + \cdots + 5398524635 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14788073664 \beta_{15} - 30032184624 \beta_{14} + 19135441318 \beta_{13} - 49636080809 \beta_{12} + \cdots + 97681982034 \beta_{8} ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 51497131363 \beta_{7} + 83786380940 \beta_{6} + 785651181598 \beta_{5} + 462351888065 \beta_{4} + \cdots + 6070248020788 ) / 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 898442845278 \beta_{15} - 1847741873484 \beta_{14} + 868005228621 \beta_{13} + \cdots + 5170786891221 \beta_{8} ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−4.34334 − 0.500000i
−4.34334 + 0.500000i
−7.44320 + 0.500000i
−7.44320 − 0.500000i
1.86958 + 0.500000i
1.86958 − 0.500000i
5.58918 − 0.500000i
5.58918 + 0.500000i
−5.58918 − 0.500000i
−5.58918 + 0.500000i
−1.86958 + 0.500000i
−1.86958 − 0.500000i
7.44320 + 0.500000i
7.44320 − 0.500000i
4.34334 − 0.500000i
4.34334 + 0.500000i
0 0 0 −10.9468 − 2.27337i 0 − 9.40838i 0 0 0
649.2 0 0 0 −10.9468 + 2.27337i 0 9.40838i 0 0 0
649.3 0 0 0 −9.92541 − 5.14648i 0 − 35.8405i 0 0 0
649.4 0 0 0 −9.92541 + 5.14648i 0 35.8405i 0 0 0
649.5 0 0 0 −2.88139 − 10.8027i 0 14.8724i 0 0 0
649.6 0 0 0 −2.88139 + 10.8027i 0 − 14.8724i 0 0 0
649.7 0 0 0 −0.319421 − 11.1758i 0 17.5714i 0 0 0
649.8 0 0 0 −0.319421 + 11.1758i 0 − 17.5714i 0 0 0
649.9 0 0 0 0.319421 − 11.1758i 0 − 17.5714i 0 0 0
649.10 0 0 0 0.319421 + 11.1758i 0 17.5714i 0 0 0
649.11 0 0 0 2.88139 − 10.8027i 0 − 14.8724i 0 0 0
649.12 0 0 0 2.88139 + 10.8027i 0 14.8724i 0 0 0
649.13 0 0 0 9.92541 − 5.14648i 0 35.8405i 0 0 0
649.14 0 0 0 9.92541 + 5.14648i 0 − 35.8405i 0 0 0
649.15 0 0 0 10.9468 − 2.27337i 0 9.40838i 0 0 0
649.16 0 0 0 10.9468 + 2.27337i 0 − 9.40838i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.4.f.a ✓ 16
3.b odd 2 1 inner 1080.4.f.a ✓ 16
5.b even 2 1 inner 1080.4.f.a ✓ 16
15.d odd 2 1 inner 1080.4.f.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.f.a ✓ 16 1.a even 1 1 trivial
1080.4.f.a ✓ 16 3.b odd 2 1 inner
1080.4.f.a ✓ 16 5.b even 2 1 inner
1080.4.f.a ✓ 16 15.d 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 \(S_{4}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7}^{8} + 1903T_{7}^{6} + 909636T_{7}^{4} + 154025824T_{7}^{2} + 7765134400 \) Copy content Toggle raw display
\( T_{11}^{8} - 5287T_{11}^{6} + 6322020T_{11}^{4} - 705522208T_{11}^{2} + 20116316224 \) 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
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 1903 T^{6} + \cdots + 7765134400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 5287 T^{6} + \cdots + 20116316224)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 3874283622400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 72822670304400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 25 T^{3} + \cdots + 6363850)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 240387721731364)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 385546105667584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 72 T^{3} + \cdots + 110592945)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 47\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 88\!\cdots\!61)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 47 T^{3} + \cdots + 691349980)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 28\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1241 T^{3} + \cdots - 133941569024)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 95\!\cdots\!29)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
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