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Properties

Label 160.6.c.d
Level $160$
Weight $6$
Character orbit 160.c
Analytic conductor $25.661$
Analytic rank $0$
Dimension $12$
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] = [160,6,Mod(129,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.c (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: \(25.6614111701\)
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: \( x^{12} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + \cdots + 1072562500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{46}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{5} - 5) q^{5} + ( - \beta_{6} - 3 \beta_{3} - 5 \beta_{2}) q^{7} + ( - 3 \beta_{5} + 3 \beta_{4} + \cdots - 22) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{5} - 5) q^{5} + ( - \beta_{6} - 3 \beta_{3} - 5 \beta_{2}) q^{7} + ( - 3 \beta_{5} + 3 \beta_{4} + \cdots - 22) q^{9}+ \cdots + ( - 51 \beta_{10} + 78 \beta_{8} + \cdots - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 60 q^{5} - 268 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 60 q^{5} - 268 q^{9} + 12080 q^{21} - 16420 q^{25} - 7864 q^{29} + 65560 q^{41} - 60420 q^{45} + 17956 q^{49} - 118232 q^{61} + 11360 q^{65} + 204464 q^{69} - 346436 q^{81} + 96320 q^{85} + 354936 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + \cdots + 1072562500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 307234436488338 \nu^{11} + \cdots - 84\!\cdots\!25 ) / 91\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\!\cdots\!39 \nu^{11} + \cdots + 50\!\cdots\!00 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35\!\cdots\!84 \nu^{11} + \cdots - 73\!\cdots\!00 ) / 10\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38\!\cdots\!07 \nu^{11} + \cdots + 37\!\cdots\!00 ) / 68\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\!\cdots\!97 \nu^{11} + \cdots - 20\!\cdots\!00 ) / 68\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!19 \nu^{11} + \cdots + 61\!\cdots\!00 ) / 32\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!48 \nu^{11} + \cdots - 18\!\cdots\!00 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!42 \nu^{11} + \cdots + 66\!\cdots\!00 ) / 19\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!62 \nu^{11} + \cdots + 31\!\cdots\!00 ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!66 \nu^{11} + \cdots - 18\!\cdots\!00 ) / 19\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!11 \nu^{11} + \cdots - 36\!\cdots\!00 ) / 34\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{11} + \beta_{10} - 8\beta_{9} + \beta_{8} + 8\beta_{7} + 28\beta_{5} + 26\beta_{4} + 16\beta_{3} + 640 ) / 1280 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - 4\beta_{9} + 240\beta_{6} + 14\beta_{5} + 13\beta_{4} + 590\beta_{3} - 800\beta_{2} ) / 320 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 94 \beta_{11} - 207 \beta_{10} - 976 \beta_{9} - 807 \beta_{8} - 976 \beta_{7} + 1440 \beta_{6} + \cdots - 122480 ) / 1280 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 103 \beta_{10} - 403 \beta_{8} - 484 \beta_{7} - 16680 \beta_{5} + 16680 \beta_{4} + 242 \beta_{3} + \cdots - 1245620 ) / 160 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8758 \beta_{11} - 36389 \beta_{10} + 135832 \beta_{9} - 137189 \beta_{8} - 135832 \beta_{7} + \cdots - 24504560 ) / 1280 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12679 \beta_{11} + 198916 \beta_{9} - 4843680 \beta_{6} - 1377546 \beta_{5} + \cdots + 24742400 \beta_{2} ) / 320 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1123186 \beta_{11} + 5765693 \beta_{10} + 20122144 \beta_{9} + 21395093 \beta_{8} + 20122144 \beta_{7} + \cdots + 4799527120 ) / 1280 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1379183 \beta_{10} + 5114333 \beta_{8} + 4799594 \beta_{7} + 91172610 \beta_{5} - 91172610 \beta_{4} + \cdots + 6655524745 ) / 40 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 163277722 \beta_{11} + 894814071 \beta_{10} - 3062582488 \beta_{9} + 3304285671 \beta_{8} + \cdots + 901210957840 ) / 1280 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 376612341 \beta_{11} - 7086035964 \beta_{9} + 111951031200 \beta_{6} + \cdots - 593572832000 \beta_{2} ) / 320 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 24877018814 \beta_{11} - 139055063647 \beta_{10} - 473195849456 \beta_{9} - 512742837847 \beta_{8} + \cdots - 163277454708080 ) / 1280 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−8.04538 − 8.04538i
9.04538 + 9.04538i
−3.92310 + 3.92310i
4.92310 − 4.92310i
−1.27837 − 1.27837i
2.27837 + 2.27837i
2.27837 − 2.27837i
−1.27837 + 1.27837i
4.92310 + 4.92310i
−3.92310 − 3.92310i
9.04538 − 9.04538i
−8.04538 + 8.04538i
0 − 21.9637i 0 18.7585 − 52.6604i 0 − 29.6983i 0 −239.406 0
129.2 0 − 21.9637i 0 18.7585 + 52.6604i 0 − 29.6983i 0 −239.406 0
129.3 0 − 16.9825i 0 12.3893 − 54.5115i 0 211.691i 0 −45.4059 0
129.4 0 − 16.9825i 0 12.3893 + 54.5115i 0 211.691i 0 −45.4059 0
129.5 0 − 5.01877i 0 −46.1478 − 31.5496i 0 15.3895i 0 217.812 0
129.6 0 − 5.01877i 0 −46.1478 + 31.5496i 0 15.3895i 0 217.812 0
129.7 0 5.01877i 0 −46.1478 − 31.5496i 0 − 15.3895i 0 217.812 0
129.8 0 5.01877i 0 −46.1478 + 31.5496i 0 − 15.3895i 0 217.812 0
129.9 0 16.9825i 0 12.3893 − 54.5115i 0 − 211.691i 0 −45.4059 0
129.10 0 16.9825i 0 12.3893 + 54.5115i 0 − 211.691i 0 −45.4059 0
129.11 0 21.9637i 0 18.7585 − 52.6604i 0 29.6983i 0 −239.406 0
129.12 0 21.9637i 0 18.7585 + 52.6604i 0 29.6983i 0 −239.406 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
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 160.6.c.d ✓ 12
4.b odd 2 1 inner 160.6.c.d ✓ 12
5.b even 2 1 inner 160.6.c.d ✓ 12
5.c odd 4 1 800.6.a.w 6
5.c odd 4 1 800.6.a.bb 6
8.b even 2 1 320.6.c.l 12
8.d odd 2 1 320.6.c.l 12
20.d odd 2 1 inner 160.6.c.d ✓ 12
20.e even 4 1 800.6.a.w 6
20.e even 4 1 800.6.a.bb 6
40.e odd 2 1 320.6.c.l 12
40.f even 2 1 320.6.c.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.d ✓ 12 1.a even 1 1 trivial
160.6.c.d ✓ 12 4.b odd 2 1 inner
160.6.c.d ✓ 12 5.b even 2 1 inner
160.6.c.d ✓ 12 20.d odd 2 1 inner
320.6.c.l 12 8.b even 2 1
320.6.c.l 12 8.d odd 2 1
320.6.c.l 12 40.e odd 2 1
320.6.c.l 12 40.f even 2 1
800.6.a.w 6 5.c odd 4 1
800.6.a.w 6 20.e even 4 1
800.6.a.bb 6 5.c odd 4 1
800.6.a.bb 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 796T_{3}^{4} + 158544T_{3}^{2} + 3504384 \) acting on \(S_{6}^{\mathrm{new}}(160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 796 T^{4} + \cdots + 3504384)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 30 T^{5} + \cdots + 30517578125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 45932 T^{4} + \cdots + 9360949504)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 1966 T^{2} + \cdots - 29310748632)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 16390 T^{2} + \cdots + 287527662200)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 30\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 1325305098360)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 408997402152984)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
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