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Properties

Label 960.3.l.j
Level $960$
Weight $3$
Character orbit 960.l
Analytic conductor $26.158$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(641,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.641");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.l (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: \(26.1581053786\)
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: \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 1094 x^{12} - 3652 x^{11} + 5452 x^{10} + 1164 x^{9} + \cdots + 20736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 480)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{4} q^{5} + (\beta_{5} + \beta_1) q^{7} + (\beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + \beta_{4} q^{5} + (\beta_{5} + \beta_1) q^{7} + (\beta_{6} + 1) q^{9} + ( - \beta_{13} + \beta_{7} + \cdots + \beta_1) q^{11}+ \cdots + (4 \beta_{13} - 4 \beta_{11} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} + 16 q^{13} + 104 q^{21} - 80 q^{25} + 192 q^{33} - 144 q^{37} - 40 q^{45} - 128 q^{49} - 80 q^{57} - 144 q^{61} + 280 q^{69} + 192 q^{81} + 96 q^{93} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 1094 x^{12} - 3652 x^{11} + 5452 x^{10} + 1164 x^{9} + \cdots + 20736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 265120597 \nu^{15} + 942387 \nu^{14} + 1401383345 \nu^{13} + 23583208255 \nu^{12} + \cdots - 219855865256640 ) / 24362570035008 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1388522339 \nu^{15} - 15286379448 \nu^{14} + 109021438216 \nu^{13} - 566220766030 \nu^{12} + \cdots + 75251029162752 ) / 24362570035008 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 505236429 \nu^{15} + 4979134639 \nu^{14} - 34906315777 \nu^{13} + 175567630407 \nu^{12} + \cdots + 20673942144192 ) / 8120856678336 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 815120 \nu^{15} - 6113400 \nu^{14} + 43546246 \nu^{13} - 190330699 \nu^{12} + \cdots - 1809374112 ) / 11502629856 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 798796512 \nu^{15} + 5835652835 \nu^{14} - 40881356273 \nu^{13} + 174012166641 \nu^{12} + \cdots - 13495915113408 ) / 8120856678336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1237882115 \nu^{15} - 11984663208 \nu^{14} + 82872054007 \nu^{13} - 408200868067 \nu^{12} + \cdots + 66289229426880 ) / 12181285017504 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 364118865 \nu^{15} - 2121631763 \nu^{14} + 14855155525 \nu^{13} - 52668235763 \nu^{12} + \cdots + 3345747297984 ) / 2706952226112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 289826899 \nu^{15} - 1960713902 \nu^{14} + 13581507538 \nu^{13} - 55261425500 \nu^{12} + \cdots - 11609626651776 ) / 2030214169584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1800067744 \nu^{15} - 10199009661 \nu^{14} + 70944100568 \nu^{13} - 244437213062 \nu^{12} + \cdots - 24853781473920 ) / 12181285017504 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 562185629 \nu^{15} + 4214896266 \nu^{14} - 30075895252 \nu^{13} + 131275899592 \nu^{12} + \cdots - 37610128370880 ) / 3045321254376 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2262094677 \nu^{15} + 18447021626 \nu^{14} - 129698062940 \nu^{13} + 591401105904 \nu^{12} + \cdots - 8951407210368 ) / 8120856678336 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3877174285 \nu^{15} + 29080303089 \nu^{14} - 200800728677 \nu^{13} + \cdots - 53395813114272 ) / 12181285017504 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 8508808957 \nu^{15} - 59838095547 \nu^{14} + 422583847199 \nu^{13} - 1764447726707 \nu^{12} + \cdots - 83338409891520 ) / 24362570035008 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4423480295 \nu^{15} - 28375958595 \nu^{14} + 198651799903 \nu^{13} - 768982727755 \nu^{12} + \cdots - 3906025371936 ) / 12181285017504 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 15402346 \nu^{15} - 115517595 \nu^{14} + 824284025 \nu^{13} - 3605829305 \nu^{12} + \cdots - 32776379136 ) / 41716729512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{15} + 3 \beta_{13} - 6 \beta_{11} - 3 \beta_{8} - 5 \beta_{7} + 4 \beta_{5} - 24 \beta_{4} + \cdots + 24 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{14} - 9 \beta_{13} - 6 \beta_{11} - 6 \beta_{10} + 8 \beta_{9} - 3 \beta_{8} + 23 \beta_{7} + \cdots - 68 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 15 \beta_{15} - 12 \beta_{14} - 9 \beta_{13} - 18 \beta_{12} + 3 \beta_{11} - 9 \beta_{10} + \cdots - 72 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 30 \beta_{15} + 19 \beta_{14} + 96 \beta_{13} - 18 \beta_{12} + 30 \beta_{11} - 12 \beta_{10} + \cdots - 151 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 417 \beta_{15} + 310 \beta_{14} + 405 \beta_{13} + 360 \beta_{12} + 435 \beta_{11} + 60 \beta_{10} + \cdots - 244 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 279 \beta_{15} - 70 \beta_{14} - 1167 \beta_{13} + 585 \beta_{12} + 201 \beta_{11} + 1362 \beta_{10} + \cdots + 16243 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 20175 \beta_{15} - 3122 \beta_{14} - 4965 \beta_{13} - 504 \beta_{12} - 15225 \beta_{11} + \cdots + 111404 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6646 \beta_{15} - 2565 \beta_{14} - 3364 \beta_{13} - 1260 \beta_{12} - 10916 \beta_{11} - 9684 \beta_{10} + \cdots - 117649 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 348639 \beta_{15} - 110442 \beta_{14} - 163941 \beta_{13} - 161856 \beta_{12} + 140007 \beta_{11} + \cdots - 3968316 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 858231 \beta_{15} + 132980 \beta_{14} + 775377 \beta_{13} - 372105 \beta_{12} + 870879 \beta_{11} + \cdots + 361867 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1968045 \beta_{15} + 5502750 \beta_{14} + 10839051 \beta_{13} + 4009104 \beta_{12} + 5295855 \beta_{11} + \cdots + 47715300 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 9733056 \beta_{15} + 3633163 \beta_{14} - 12022230 \beta_{13} + 16044930 \beta_{12} - 3025842 \beta_{11} + \cdots + 170848733 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 247038333 \beta_{15} - 98372378 \beta_{14} - 267689277 \beta_{13} - 2123784 \beta_{12} + \cdots + 1358442572 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 376491231 \beta_{15} - 206037562 \beta_{14} - 129411177 \beta_{13} - 262867059 \beta_{12} + \cdots - 3804271997 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 5022518529 \beta_{15} - 416959010 \beta_{14} + 1509604131 \beta_{13} - 2213887320 \beta_{12} + \cdots - 76790358052 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
2.45733 − 4.45426i
2.45733 + 4.45426i
−0.317839 + 0.355703i
−0.317839 − 0.355703i
−0.317839 + 1.17756i
−0.317839 − 1.17756i
−1.45733 − 4.45426i
−1.45733 + 4.45426i
2.45733 − 0.803980i
2.45733 + 0.803980i
1.31784 + 0.355703i
1.31784 − 0.355703i
1.31784 + 1.17756i
1.31784 − 1.17756i
−1.45733 − 0.803980i
−1.45733 + 0.803980i
0 −2.99655 − 0.143843i 0 2.23607i 0 2.24307 0 8.95862 + 0.862068i 0
641.2 0 −2.99655 + 0.143843i 0 − 2.23607i 0 2.24307 0 8.95862 − 0.862068i 0
641.3 0 −2.65305 − 1.40048i 0 2.23607i 0 −12.5386 0 5.07731 + 7.43108i 0
641.4 0 −2.65305 + 1.40048i 0 − 2.23607i 0 −12.5386 0 5.07731 − 7.43108i 0
641.5 0 −1.56887 − 2.55708i 0 − 2.23607i 0 0.823421 0 −4.07731 + 8.02344i 0
641.6 0 −1.56887 + 2.55708i 0 2.23607i 0 0.823421 0 −4.07731 − 8.02344i 0
641.7 0 −0.721589 − 2.91193i 0 2.23607i 0 −1.03633 0 −7.95862 + 4.20243i 0
641.8 0 −0.721589 + 2.91193i 0 − 2.23607i 0 −1.03633 0 −7.95862 − 4.20243i 0
641.9 0 0.721589 − 2.91193i 0 − 2.23607i 0 1.03633 0 −7.95862 − 4.20243i 0
641.10 0 0.721589 + 2.91193i 0 2.23607i 0 1.03633 0 −7.95862 + 4.20243i 0
641.11 0 1.56887 − 2.55708i 0 2.23607i 0 −0.823421 0 −4.07731 − 8.02344i 0
641.12 0 1.56887 + 2.55708i 0 − 2.23607i 0 −0.823421 0 −4.07731 + 8.02344i 0
641.13 0 2.65305 − 1.40048i 0 − 2.23607i 0 12.5386 0 5.07731 − 7.43108i 0
641.14 0 2.65305 + 1.40048i 0 2.23607i 0 12.5386 0 5.07731 + 7.43108i 0
641.15 0 2.99655 − 0.143843i 0 − 2.23607i 0 −2.24307 0 8.95862 − 0.862068i 0
641.16 0 2.99655 + 0.143843i 0 2.23607i 0 −2.24307 0 8.95862 + 0.862068i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.16
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 960.3.l.j 16
3.b odd 2 1 inner 960.3.l.j 16
4.b odd 2 1 inner 960.3.l.j 16
8.b even 2 1 480.3.l.b ✓ 16
8.d odd 2 1 480.3.l.b ✓ 16
12.b even 2 1 inner 960.3.l.j 16
24.f even 2 1 480.3.l.b ✓ 16
24.h odd 2 1 480.3.l.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.l.b ✓ 16 8.b even 2 1
480.3.l.b ✓ 16 8.d odd 2 1
480.3.l.b ✓ 16 24.f even 2 1
480.3.l.b ✓ 16 24.h odd 2 1
960.3.l.j 16 1.a even 1 1 trivial
960.3.l.j 16 3.b odd 2 1 inner
960.3.l.j 16 4.b odd 2 1 inner
960.3.l.j 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 164T_{7}^{6} + 1076T_{7}^{4} - 1504T_{7}^{2} + 576 \) acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 164 T^{6} + \cdots + 576)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 472 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots + 20576)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1632 T^{6} + \cdots + 2005606656)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 1480 T^{6} + \cdots + 13604889600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2580 T^{6} + \cdots + 28237441600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 4808 T^{6} + \cdots + 115883053056)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 5144 T^{6} + \cdots + 243104219136)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 36 T^{3} + \cdots + 19296)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 6280 T^{6} + \cdots + 15336345600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 2103149650176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 2975293809216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 82199318835456)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 12328 T^{6} + \cdots + 435726729216)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 36 T^{3} + \cdots - 975104)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 15340 T^{6} + \cdots + 189747360000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 32895088459776)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16840 T^{2} + \cdots + 13055760)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 44014414160896)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 4467034423296)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 40 T^{3} + \cdots + 143760)^{4} \) Copy content Toggle raw display
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