Properties

Label 960.3.e.c
Level $960$
Weight $3$
Character orbit 960.e
Analytic conductor $26.158$
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] = [960,3,Mod(511,960)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.511");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.e (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: \(8\)
Coefficient field: 8.0.85100625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 60)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{5} + 2 \beta_{2}) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{5} + 2 \beta_{2}) q^{7} - 3 q^{9} + ( - \beta_{7} - \beta_{5}) q^{11} + ( - \beta_{6} - 2 \beta_1 - 2) q^{13} + \beta_{4} q^{15} + ( - \beta_{3} - 4 \beta_1) q^{17} + (\beta_{7} - 4 \beta_{4} + 6 \beta_{2}) q^{19} + ( - \beta_{3} + 6) q^{21} + (2 \beta_{7} - 3 \beta_{5} - 2 \beta_{4} + 4 \beta_{2}) q^{23} + 5 q^{25} + 3 \beta_{2} q^{27} + (2 \beta_{6} + 2 \beta_{3} - 2 \beta_1 - 8) q^{29} + ( - \beta_{7} + 2 \beta_{5} - 8 \beta_{4} - 6 \beta_{2}) q^{31} + (\beta_{6} + \beta_{3}) q^{33} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{4}) q^{35} + (\beta_{6} - 2 \beta_{3} - 10 \beta_1 + 14) q^{37} + ( - 3 \beta_{7} - 2 \beta_{4} + 2 \beta_{2}) q^{39} + ( - 2 \beta_{6} - 2) q^{41} + (2 \beta_{7} + 6 \beta_{5} - 4 \beta_{4} - 16 \beta_{2}) q^{43} - 3 \beta_1 q^{45} + ( - 2 \beta_{7} - \beta_{5} - 10 \beta_{4} - 8 \beta_{2}) q^{47} + (4 \beta_{3} - 16 \beta_1 - 7) q^{49} + ( - 3 \beta_{5} - 4 \beta_{4}) q^{51} + ( - 2 \beta_{6} - 2 \beta_1 - 44) q^{53} + (\beta_{7} - 3 \beta_{5}) q^{55} + ( - \beta_{6} + 12 \beta_1 + 18) q^{57} + ( - 3 \beta_{7} - 3 \beta_{5} - 4 \beta_{4} + 4 \beta_{2}) q^{59} + (2 \beta_{6} + 4 \beta_{3} - 4 \beta_1 + 22) q^{61} + ( - 3 \beta_{5} - 6 \beta_{2}) q^{63} + (2 \beta_{6} - \beta_{3} - 2 \beta_1 - 10) q^{65} + ( - 4 \beta_{7} - 4 \beta_{5} - 12 \beta_{4} + 8 \beta_{2}) q^{67} + ( - 2 \beta_{6} + 3 \beta_{3} + 6 \beta_1 + 12) q^{69} + (8 \beta_{5} - 4 \beta_{4} - 12 \beta_{2}) q^{71} + (2 \beta_{6} - 4 \beta_{3} - 4 \beta_1 - 30) q^{73} - 5 \beta_{2} q^{75} + ( - 2 \beta_{6} - 2 \beta_{3} + 28 \beta_1 + 36) q^{77} + (9 \beta_{7} + 2 \beta_{5} - 16 \beta_{4} + 14 \beta_{2}) q^{79} + 9 q^{81} + ( - 2 \beta_{7} + 2 \beta_{5} - 24 \beta_{4} - 4 \beta_{2}) q^{83} + ( - \beta_{6} - 2 \beta_{3} - 20) q^{85} + (6 \beta_{7} + 6 \beta_{5} - 2 \beta_{4} + 8 \beta_{2}) q^{87} + (2 \beta_{6} + 44 \beta_1 + 10) q^{89} + (4 \beta_{7} - 6 \beta_{5} - 8 \beta_{4} - 12 \beta_{2}) q^{91} + (\beta_{6} - 2 \beta_{3} + 24 \beta_1 - 18) q^{93} + ( - 2 \beta_{7} + \beta_{5} - 6 \beta_{4} + 20 \beta_{2}) q^{95} + ( - 2 \beta_{6} - 6 \beta_{3} - 12 \beta_1 + 54) q^{97} + (3 \beta_{7} + 3 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} - 16 q^{13} + 48 q^{21} + 40 q^{25} - 64 q^{29} + 112 q^{37} - 16 q^{41} - 56 q^{49} - 352 q^{53} + 144 q^{57} + 176 q^{61} - 80 q^{65} + 96 q^{69} - 240 q^{73} + 288 q^{77} + 72 q^{81} - 160 q^{85} + 80 q^{89} - 144 q^{93} + 432 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} - \nu^{6} - 4\nu^{5} - 5\nu^{4} - 5\nu^{3} + 16\nu^{2} - 8\nu - 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{7} + \nu^{6} - 4\nu^{5} - 3\nu^{4} + 5\nu^{3} + 8\nu^{2} - 8\nu - 40 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} + 7\nu^{6} - 12\nu^{5} - 21\nu^{4} + 3\nu^{3} + 8\nu^{2} + 40\nu - 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + \nu^{6} - 6\nu^{5} - \nu^{4} + 5\nu^{3} + 14\nu^{2} - 4\nu - 36 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - \nu^{6} - 4\nu^{5} - 5\nu^{4} + 11\nu^{3} + 16\nu^{2} - 8\nu - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{7} + 3\nu^{5} + \nu^{4} - 4\nu^{3} - \nu^{2} + 10\nu + 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -15\nu^{7} - 11\nu^{6} + 20\nu^{5} + 25\nu^{4} - 23\nu^{3} - 80\nu^{2} + 40\nu + 184 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} - 10\beta_{2} + 2\beta _1 + 10 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - 4\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + \beta_{6} - 3\beta_{5} + 10\beta_{4} - 3\beta_{3} + 2\beta_{2} - 10\beta _1 + 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} + 3\beta_{5} - 10\beta_{4} - 3\beta_{3} + 18\beta_{2} - 10\beta _1 - 18 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} - 3\beta_{5} - 8\beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} - \beta_{5} - 2\beta_{4} - \beta_{3} + 86\beta_{2} + 2\beta _1 + 86 ) / 16 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
−0.600040 + 1.28061i
1.40906 + 0.120653i
1.04064 0.957636i
−1.34966 + 0.422403i
1.40906 0.120653i
−0.600040 1.28061i
−1.34966 0.422403i
1.04064 + 0.957636i
0 1.73205i 0 −2.23607 0 0.596540i 0 −3.00000 0
511.2 0 1.73205i 0 −2.23607 0 6.33166i 0 −3.00000 0
511.3 0 1.73205i 0 2.23607 0 5.46770i 0 −3.00000 0
511.4 0 1.73205i 0 2.23607 0 12.3959i 0 −3.00000 0
511.5 0 1.73205i 0 −2.23607 0 6.33166i 0 −3.00000 0
511.6 0 1.73205i 0 −2.23607 0 0.596540i 0 −3.00000 0
511.7 0 1.73205i 0 2.23607 0 12.3959i 0 −3.00000 0
511.8 0 1.73205i 0 2.23607 0 5.46770i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.e.c 8
3.b odd 2 1 2880.3.e.j 8
4.b odd 2 1 inner 960.3.e.c 8
8.b even 2 1 60.3.c.a 8
8.d odd 2 1 60.3.c.a 8
12.b even 2 1 2880.3.e.j 8
24.f even 2 1 180.3.c.b 8
24.h odd 2 1 180.3.c.b 8
40.e odd 2 1 300.3.c.d 8
40.f even 2 1 300.3.c.d 8
40.i odd 4 2 300.3.f.b 16
40.k even 4 2 300.3.f.b 16
120.i odd 2 1 900.3.c.u 8
120.m even 2 1 900.3.c.u 8
120.q odd 4 2 900.3.f.f 16
120.w even 4 2 900.3.f.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 8.b even 2 1
60.3.c.a 8 8.d odd 2 1
180.3.c.b 8 24.f even 2 1
180.3.c.b 8 24.h odd 2 1
300.3.c.d 8 40.e odd 2 1
300.3.c.d 8 40.f even 2 1
300.3.f.b 16 40.i odd 4 2
300.3.f.b 16 40.k even 4 2
900.3.c.u 8 120.i odd 2 1
900.3.c.u 8 120.m even 2 1
900.3.f.f 16 120.q odd 4 2
900.3.f.f 16 120.w even 4 2
960.3.e.c 8 1.a even 1 1 trivial
960.3.e.c 8 4.b odd 2 1 inner
2880.3.e.j 8 3.b odd 2 1
2880.3.e.j 8 12.b even 2 1

Hecke kernels

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

\( T_{7}^{8} + 224T_{7}^{6} + 12032T_{7}^{4} + 188416T_{7}^{2} + 65536 \) Copy content Toggle raw display
\( T_{13}^{4} + 8T_{13}^{3} - 472T_{13}^{2} - 5792T_{13} - 12464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 224 T^{6} + 12032 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$11$ \( (T^{4} + 208 T^{2} + 10496)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{3} - 472 T^{2} - 5792 T - 12464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 424 T^{2} + 3840 T - 8816)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1696 T^{6} + \cdots + 6544162816 \) Copy content Toggle raw display
$23$ \( T^{8} + 3616 T^{6} + \cdots + 101419319296 \) Copy content Toggle raw display
$29$ \( (T^{4} + 32 T^{3} - 2152 T^{2} + \cdots + 1334416)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 5408 T^{6} + \cdots + 59895709696 \) Copy content Toggle raw display
$37$ \( (T^{4} - 56 T^{3} - 1528 T^{2} + \cdots - 244784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} - 1800 T^{2} + \cdots + 87184)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 10816 T^{6} + \cdots + 33624411406336 \) Copy content Toggle raw display
$47$ \( T^{8} + 8032 T^{6} + \cdots + 1056981385216 \) Copy content Toggle raw display
$53$ \( (T^{4} + 176 T^{3} + 9752 T^{2} + \cdots - 478064)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 4896 T^{6} + \cdots + 173909016576 \) Copy content Toggle raw display
$61$ \( (T^{4} - 88 T^{3} - 2536 T^{2} + \cdots - 2142704)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 16064 T^{6} + \cdots + 281086590976 \) Copy content Toggle raw display
$71$ \( T^{8} + 13952 T^{6} + \cdots + 16079971680256 \) Copy content Toggle raw display
$73$ \( (T^{4} + 120 T^{3} - 1576 T^{2} + \cdots + 4962064)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 41888 T^{6} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + 36928 T^{6} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} - 40 T^{3} - 20584 T^{2} + \cdots + 70652944)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 216 T^{3} + 5880 T^{2} + \cdots - 59281776)^{2} \) Copy content Toggle raw display
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