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Properties

Label 960.2.y.e
Level $960$
Weight $2$
Character orbit 960.y
Analytic conductor $7.666$
Analytic rank $0$
Dimension $16$
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] = [960,2,Mod(847,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.847");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.y (of order \(4\), degree \(2\), 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: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 14 x^{14} - 10 x^{13} - 26 x^{12} + 78 x^{11} - 66 x^{10} - 74 x^{9} + 233 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} + (\beta_{7} + \beta_{5}) q^{5} + (\beta_{15} + \beta_{12} + \cdots + \beta_{3}) q^{7}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta_{7} + \beta_{5}) q^{5} + (\beta_{15} + \beta_{12} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - \beta_{14} - \beta_{10} + \cdots - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} - 4 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} - 4 q^{5} + 4 q^{7} + 16 q^{9} + 4 q^{15} - 8 q^{17} - 8 q^{19} - 4 q^{21} + 32 q^{25} - 16 q^{27} + 12 q^{29} + 20 q^{35} - 4 q^{45} + 32 q^{47} + 8 q^{51} + 16 q^{53} + 4 q^{55} + 8 q^{57} + 24 q^{59} + 40 q^{61} + 4 q^{63} - 4 q^{65} + 8 q^{73} - 32 q^{75} - 72 q^{77} + 48 q^{79} + 16 q^{81} + 8 q^{83} - 8 q^{85} - 12 q^{87} + 40 q^{91} - 8 q^{95} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 14 x^{14} - 10 x^{13} - 26 x^{12} + 78 x^{11} - 66 x^{10} - 74 x^{9} + 233 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13 \nu^{15} - 28 \nu^{14} + 6 \nu^{13} + 74 \nu^{12} - 158 \nu^{11} + 26 \nu^{10} + 314 \nu^{9} + \cdots + 640 ) / 256 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{15} + 36 \nu^{14} - 58 \nu^{13} + 6 \nu^{12} + 182 \nu^{11} - 322 \nu^{10} + 30 \nu^{9} + \cdots + 1024 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4 \nu^{15} - 35 \nu^{14} + 76 \nu^{13} - 30 \nu^{12} - 190 \nu^{11} + 434 \nu^{10} - 214 \nu^{9} + \cdots - 2304 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6 \nu^{15} - 31 \nu^{14} + 57 \nu^{13} - 10 \nu^{12} - 168 \nu^{11} + 328 \nu^{10} - 112 \nu^{9} + \cdots - 1632 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33 \nu^{15} - 168 \nu^{14} + 290 \nu^{13} - 6 \nu^{12} - 902 \nu^{11} + 1618 \nu^{10} - 366 \nu^{9} + \cdots - 7680 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24 \nu^{15} - 117 \nu^{14} + 206 \nu^{13} - 18 \nu^{12} - 630 \nu^{11} + 1178 \nu^{10} - 318 \nu^{9} + \cdots - 5568 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 87 \nu^{15} - 378 \nu^{14} + 582 \nu^{13} + 130 \nu^{12} - 2094 \nu^{11} + 3290 \nu^{10} + \cdots - 12544 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33 \nu^{15} + 138 \nu^{14} - 203 \nu^{13} - 66 \nu^{12} + 768 \nu^{11} - 1152 \nu^{10} - 48 \nu^{9} + \cdots + 4128 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 109 \nu^{15} - 578 \nu^{14} + 1062 \nu^{13} - 146 \nu^{12} - 3130 \nu^{11} + 6014 \nu^{10} + \cdots - 29056 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 67 \nu^{15} + 316 \nu^{14} - 530 \nu^{13} - 14 \nu^{12} + 1714 \nu^{11} - 2998 \nu^{10} + \cdots + 13184 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40 \nu^{15} + 182 \nu^{14} - 297 \nu^{13} - 28 \nu^{12} + 1002 \nu^{11} - 1678 \nu^{10} + \cdots + 7008 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 75 \nu^{15} - 372 \nu^{14} + 652 \nu^{13} - 34 \nu^{12} - 2014 \nu^{11} + 3674 \nu^{10} - 902 \nu^{9} + \cdots - 17216 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 185 \nu^{15} - 840 \nu^{14} + 1370 \nu^{13} + 122 \nu^{12} - 4582 \nu^{11} + 7730 \nu^{10} + \cdots - 33536 ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 191 \nu^{15} - 886 \nu^{14} + 1454 \nu^{13} + 106 \nu^{12} - 4838 \nu^{11} + 8226 \nu^{10} + \cdots - 35072 ) / 256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 146 \nu^{15} - 657 \nu^{14} + 1056 \nu^{13} + 130 \nu^{12} - 3614 \nu^{11} + 5970 \nu^{10} + \cdots - 24192 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{11} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} - \beta_{13} + \beta_{12} - \beta_{8} + \beta_{6} + \beta_{5} + \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 2 \beta_{14} - 5 \beta_{11} - 2 \beta_{10} - 2 \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{8} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{15} + 6 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + \beta_{11} + 8 \beta_{10} + 2 \beta_{9} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{15} + 4 \beta_{14} - 4 \beta_{13} + 6 \beta_{12} + 4 \beta_{11} - 3 \beta_{9} - 2 \beta_{6} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3 \beta_{15} - 4 \beta_{14} - 6 \beta_{13} + 6 \beta_{12} + 3 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + \cdots - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4 \beta_{15} + 4 \beta_{14} - 9 \beta_{13} + 8 \beta_{12} + \beta_{11} - 9 \beta_{10} - 3 \beta_{9} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7 \beta_{15} + 32 \beta_{14} - 20 \beta_{13} - 8 \beta_{12} - 3 \beta_{11} + 8 \beta_{10} + 20 \beta_{9} + \cdots - 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11 \beta_{14} - 9 \beta_{13} + 9 \beta_{12} + 20 \beta_{11} + 8 \beta_{10} + 4 \beta_{9} - \beta_{8} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9 \beta_{15} - 2 \beta_{14} - 16 \beta_{13} - 16 \beta_{12} + 7 \beta_{11} - 58 \beta_{10} - 8 \beta_{9} + \cdots - 105 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 11 \beta_{15} - 29 \beta_{14} - 10 \beta_{13} + 21 \beta_{12} + 31 \beta_{11} - 27 \beta_{10} + \cdots - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3 \beta_{15} - 58 \beta_{14} + 10 \beta_{13} + 26 \beta_{12} - 43 \beta_{11} - 104 \beta_{10} + \cdots - 73 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 33 \beta_{15} + 16 \beta_{14} - 44 \beta_{13} + 46 \beta_{12} + 52 \beta_{11} + 80 \beta_{10} + \cdots + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 111 \beta_{15} - 156 \beta_{14} + 66 \beta_{13} - 2 \beta_{12} + 55 \beta_{11} + 2 \beta_{10} + \cdots - 83 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
847.1
1.40838 − 0.128355i
0.424183 + 1.34910i
0.885279 − 1.10285i
−1.20803 − 0.735291i
1.38194 + 0.300388i
1.28040 + 0.600471i
−1.40988 + 0.110627i
0.237728 − 1.39409i
1.40838 + 0.128355i
0.424183 − 1.34910i
0.885279 + 1.10285i
−1.20803 + 0.735291i
1.38194 − 0.300388i
1.28040 − 0.600471i
−1.40988 − 0.110627i
0.237728 + 1.39409i
0 −1.00000 0 −2.17005 + 0.539352i 0 −3.00806 − 3.00806i 0 1.00000 0
847.2 0 −1.00000 0 −2.15140 − 0.609492i 0 −0.566689 − 0.566689i 0 1.00000 0
847.3 0 −1.00000 0 −2.13688 − 0.658594i 0 3.54781 + 3.54781i 0 1.00000 0
847.4 0 −1.00000 0 −1.61356 + 1.54804i 0 −0.143894 − 0.143894i 0 1.00000 0
847.5 0 −1.00000 0 0.311968 − 2.21420i 0 1.96597 + 1.96597i 0 1.00000 0
847.6 0 −1.00000 0 1.45639 − 1.69674i 0 −1.12791 − 1.12791i 0 1.00000 0
847.7 0 −1.00000 0 2.06823 − 0.849960i 0 2.08016 + 2.08016i 0 1.00000 0
847.8 0 −1.00000 0 2.23531 − 0.0583995i 0 −0.747384 − 0.747384i 0 1.00000 0
943.1 0 −1.00000 0 −2.17005 − 0.539352i 0 −3.00806 + 3.00806i 0 1.00000 0
943.2 0 −1.00000 0 −2.15140 + 0.609492i 0 −0.566689 + 0.566689i 0 1.00000 0
943.3 0 −1.00000 0 −2.13688 + 0.658594i 0 3.54781 − 3.54781i 0 1.00000 0
943.4 0 −1.00000 0 −1.61356 − 1.54804i 0 −0.143894 + 0.143894i 0 1.00000 0
943.5 0 −1.00000 0 0.311968 + 2.21420i 0 1.96597 − 1.96597i 0 1.00000 0
943.6 0 −1.00000 0 1.45639 + 1.69674i 0 −1.12791 + 1.12791i 0 1.00000 0
943.7 0 −1.00000 0 2.06823 + 0.849960i 0 2.08016 − 2.08016i 0 1.00000 0
943.8 0 −1.00000 0 2.23531 + 0.0583995i 0 −0.747384 + 0.747384i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 847.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.y.e 16
4.b odd 2 1 240.2.y.e ✓ 16
5.c odd 4 1 960.2.bc.e 16
8.b even 2 1 1920.2.y.j 16
8.d odd 2 1 1920.2.y.i 16
12.b even 2 1 720.2.z.f 16
16.e even 4 1 240.2.bc.e yes 16
16.e even 4 1 1920.2.bc.i 16
16.f odd 4 1 960.2.bc.e 16
16.f odd 4 1 1920.2.bc.j 16
20.e even 4 1 240.2.bc.e yes 16
40.i odd 4 1 1920.2.bc.j 16
40.k even 4 1 1920.2.bc.i 16
48.i odd 4 1 720.2.bd.f 16
60.l odd 4 1 720.2.bd.f 16
80.i odd 4 1 240.2.y.e ✓ 16
80.j even 4 1 1920.2.y.j 16
80.s even 4 1 inner 960.2.y.e 16
80.t odd 4 1 1920.2.y.i 16
240.bb even 4 1 720.2.z.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.y.e ✓ 16 4.b odd 2 1
240.2.y.e ✓ 16 80.i odd 4 1
240.2.bc.e yes 16 16.e even 4 1
240.2.bc.e yes 16 20.e even 4 1
720.2.z.f 16 12.b even 2 1
720.2.z.f 16 240.bb even 4 1
720.2.bd.f 16 48.i odd 4 1
720.2.bd.f 16 60.l odd 4 1
960.2.y.e 16 1.a even 1 1 trivial
960.2.y.e 16 80.s even 4 1 inner
960.2.bc.e 16 5.c odd 4 1
960.2.bc.e 16 16.f odd 4 1
1920.2.y.i 16 8.d odd 2 1
1920.2.y.i 16 80.t odd 4 1
1920.2.y.j 16 8.b even 2 1
1920.2.y.j 16 80.j even 4 1
1920.2.bc.i 16 16.e even 4 1
1920.2.bc.i 16 40.k even 4 1
1920.2.bc.j 16 16.f odd 4 1
1920.2.bc.j 16 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{16} - 4 T_{7}^{15} + 8 T_{7}^{14} + 32 T_{7}^{13} + 392 T_{7}^{12} - 1264 T_{7}^{11} + \cdots + 2304 \) Copy content Toggle raw display
\( T_{11}^{16} + 8 T_{11}^{13} + 1288 T_{11}^{12} + 400 T_{11}^{11} + 32 T_{11}^{10} - 4864 T_{11}^{9} + \cdots + 952576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{15} + \cdots + 2304 \) Copy content Toggle raw display
$11$ \( T^{16} + 8 T^{13} + \cdots + 952576 \) Copy content Toggle raw display
$13$ \( T^{16} + 120 T^{14} + \cdots + 8386816 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} + \cdots + 9339136 \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{15} + \cdots + 5308416 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 330366976 \) Copy content Toggle raw display
$29$ \( T^{16} - 12 T^{15} + \cdots + 45373696 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1655277803776 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 793886976 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 177209344 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 37555339264 \) Copy content Toggle raw display
$47$ \( T^{16} - 32 T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( (T^{8} - 8 T^{7} + \cdots + 189328)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 8785580546304 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 55857327698176 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 330366976 \) Copy content Toggle raw display
$71$ \( (T^{8} - 392 T^{6} + \cdots + 3900672)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2179567984896 \) Copy content Toggle raw display
$79$ \( (T^{8} - 24 T^{7} + \cdots - 1507376)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{7} + \cdots - 5888)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 344 T^{6} + \cdots - 3899136)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16602430353664 \) Copy content Toggle raw display
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