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Properties

Label 960.3.j.f
Level $960$
Weight $3$
Character orbit 960.j
Analytic conductor $26.158$
Analytic rank $0$
Dimension $12$
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] = [960,3,Mod(319,960)]
 
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([1, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.j (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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{6} + \beta_1) q^{5} + ( - \beta_{8} + \beta_{5}) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{6} + \beta_1) q^{5} + ( - \beta_{8} + \beta_{5}) q^{7} + 3 q^{9} + (\beta_{11} + \beta_{9} + \cdots - \beta_{6}) q^{11}+ \cdots + (3 \beta_{11} + 3 \beta_{9} + \cdots - 3 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{5} + 36 q^{9} - 24 q^{15} - 80 q^{23} + 28 q^{25} + 40 q^{29} + 144 q^{35} + 136 q^{41} + 224 q^{43} - 12 q^{45} + 208 q^{47} + 212 q^{49} - 192 q^{55} - 40 q^{61} + 96 q^{65} - 352 q^{67} + 192 q^{75} + 108 q^{81} + 64 q^{85} + 48 q^{87} - 8 q^{89} - 176 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} - 41 \nu^{8} + 134 \nu^{7} - 533 \nu^{6} + 1151 \nu^{5} - 2707 \nu^{4} + \cdots - 1776 ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 154 \nu^{11} + 847 \nu^{10} - 5377 \nu^{9} + 17844 \nu^{8} - 43226 \nu^{7} + 73948 \nu^{6} + \cdots - 332836 ) / 34910 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} - 41 \nu^{8} + 134 \nu^{7} - 533 \nu^{6} + 1151 \nu^{5} - 2707 \nu^{4} + \cdots - 1592 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 354 \nu^{11} + 1947 \nu^{10} - 14627 \nu^{9} + 51219 \nu^{8} - 186684 \nu^{7} + 428001 \nu^{6} + \cdots + 155264 ) / 6982 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10480 \nu^{11} + 89059 \nu^{10} - 599903 \nu^{9} + 2799641 \nu^{8} - 9872926 \nu^{7} + \cdots + 48840464 ) / 139640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10480 \nu^{11} - 82077 \nu^{10} + 564993 \nu^{9} - 2527343 \nu^{8} + 8993194 \nu^{7} + \cdots - 37669264 ) / 139640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10836 \nu^{11} - 70071 \nu^{10} + 508147 \nu^{9} - 2019463 \nu^{8} + 7336738 \nu^{7} + \cdots - 20056728 ) / 139640 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10480 \nu^{11} + 33203 \nu^{10} - 320623 \nu^{9} + 593329 \nu^{8} - 2723358 \nu^{7} + \cdots - 23437200 ) / 139640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10836 \nu^{11} + 49125 \nu^{10} - 403417 \nu^{9} + 1188605 \nu^{8} - 4641686 \nu^{7} + \cdots - 4631624 ) / 139640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3292 \nu^{11} - 18106 \nu^{10} + 141782 \nu^{9} - 502224 \nu^{8} + 1930250 \nu^{7} - 4538905 \nu^{6} + \cdots - 2623076 ) / 34910 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2540 \nu^{11} - 13970 \nu^{10} + 107002 \nu^{9} - 376734 \nu^{8} + 1407884 \nu^{7} - 3267292 \nu^{6} + \cdots - 1513948 ) / 17455 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} + 2\beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{9} + 2\beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} + 2\beta_{3} - 10\beta _1 - 42 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{11} - 5 \beta_{10} + 7 \beta_{9} - 20 \beta_{8} - 7 \beta_{7} + 20 \beta_{6} + 3 \beta_{4} + \cdots - 65 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 18 \beta_{11} - 11 \beta_{10} + 11 \beta_{9} - 38 \beta_{8} - 19 \beta_{7} + 58 \beta_{6} + \cdots + 284 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 117 \beta_{11} + 26 \beta_{10} - 80 \beta_{9} + 206 \beta_{8} + 60 \beta_{7} - 156 \beta_{6} + \cdots + 819 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 198 \beta_{11} + 53 \beta_{10} - 83 \beta_{9} + 314 \beta_{8} + 165 \beta_{7} - 463 \beta_{6} + \cdots - 1090 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1195 \beta_{11} - 146 \beta_{10} + 976 \beta_{9} - 2062 \beta_{8} - 332 \beta_{7} + 844 \beta_{6} + \cdots - 10573 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6670 \beta_{11} - 1105 \beta_{10} + 2845 \beta_{9} - 9814 \beta_{8} - 4773 \beta_{7} + 12762 \beta_{6} + \cdots + 16140 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9135 \beta_{11} + 731 \beta_{10} - 11065 \beta_{9} + 17732 \beta_{8} - 1559 \beta_{7} + 3052 \beta_{6} + \cdots + 139639 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 98562 \beta_{11} + 12741 \beta_{10} - 47421 \beta_{9} + 143834 \beta_{8} + 60869 \beta_{7} + \cdots - 74712 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 20611 \beta_{11} + 1310 \beta_{10} + 107068 \beta_{9} - 95370 \beta_{8} + 89920 \beta_{7} + \cdots - 1816613 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.500000 − 3.59014i
0.500000 + 3.59014i
0.500000 − 1.80359i
0.500000 + 1.80359i
0.500000 − 2.65258i
0.500000 + 2.65258i
0.500000 + 0.723587i
0.500000 − 0.723587i
0.500000 − 2.16741i
0.500000 + 2.16741i
0.500000 + 2.02498i
0.500000 − 2.02498i
0 −1.73205 0 −1.77322 − 4.67501i 0 2.37003 0 3.00000 0
319.2 0 −1.73205 0 −1.77322 + 4.67501i 0 2.37003 0 3.00000 0
319.3 0 −1.73205 0 −0.621940 − 4.96117i 0 −9.43574 0 3.00000 0
319.4 0 −1.73205 0 −0.621940 + 4.96117i 0 −9.43574 0 3.00000 0
319.5 0 −1.73205 0 4.85927 − 1.17794i 0 7.06571 0 3.00000 0
319.6 0 −1.73205 0 4.85927 + 1.17794i 0 7.06571 0 3.00000 0
319.7 0 1.73205 0 −4.94155 − 0.762320i 0 7.21051 0 3.00000 0
319.8 0 1.73205 0 −4.94155 + 0.762320i 0 7.21051 0 3.00000 0
319.9 0 1.73205 0 −3.65509 − 3.41179i 0 −13.0243 0 3.00000 0
319.10 0 1.73205 0 −3.65509 + 3.41179i 0 −13.0243 0 3.00000 0
319.11 0 1.73205 0 4.13253 − 2.81463i 0 5.81384 0 3.00000 0
319.12 0 1.73205 0 4.13253 + 2.81463i 0 5.81384 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.j.f 12
4.b odd 2 1 960.3.j.g 12
5.b even 2 1 960.3.j.g 12
8.b even 2 1 480.3.j.a ✓ 12
8.d odd 2 1 480.3.j.b yes 12
20.d odd 2 1 inner 960.3.j.f 12
24.f even 2 1 1440.3.j.c 12
24.h odd 2 1 1440.3.j.d 12
40.e odd 2 1 480.3.j.a ✓ 12
40.f even 2 1 480.3.j.b yes 12
40.i odd 4 1 2400.3.e.h 12
40.i odd 4 1 2400.3.e.i 12
40.k even 4 1 2400.3.e.h 12
40.k even 4 1 2400.3.e.i 12
120.i odd 2 1 1440.3.j.c 12
120.m even 2 1 1440.3.j.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.j.a ✓ 12 8.b even 2 1
480.3.j.a ✓ 12 40.e odd 2 1
480.3.j.b yes 12 8.d odd 2 1
480.3.j.b yes 12 40.f even 2 1
960.3.j.f 12 1.a even 1 1 trivial
960.3.j.f 12 20.d odd 2 1 inner
960.3.j.g 12 4.b odd 2 1
960.3.j.g 12 5.b even 2 1
1440.3.j.c 12 24.f even 2 1
1440.3.j.c 12 120.i odd 2 1
1440.3.j.d 12 24.h odd 2 1
1440.3.j.d 12 120.m even 2 1
2400.3.e.h 12 40.i odd 4 1
2400.3.e.h 12 40.k even 4 1
2400.3.e.i 12 40.i odd 4 1
2400.3.e.i 12 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{6} - 200T_{7}^{4} + 704T_{7}^{3} + 9232T_{7}^{2} - 59648T_{7} + 86272 \) Copy content Toggle raw display
\( T_{11}^{12} + 592T_{11}^{10} + 77920T_{11}^{8} + 3053312T_{11}^{6} + 38007040T_{11}^{4} + 136323072T_{11}^{2} + 144769024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 200 T^{4} + \cdots + 86272)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 144769024 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1071241560064 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 1963047583744 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 5879810228224 \) Copy content Toggle raw display
$23$ \( (T^{6} + 40 T^{5} + \cdots + 67145728)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 20 T^{5} + \cdots + 174739456)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 652427275534336 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 62\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{6} - 68 T^{5} + \cdots + 997804096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 112 T^{5} + \cdots + 49254400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 104 T^{5} + \cdots + 818692096)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{6} + 20 T^{5} + \cdots - 24763044800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 176 T^{5} + \cdots - 85655552)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{6} - 11408 T^{4} + \cdots + 243552256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 4 T^{5} + \cdots + 34204460608)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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