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Properties

Label 768.4.f.d
Level $768$
Weight $4$
Character orbit 768.f
Analytic conductor $45.313$
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] = [768,4,Mod(383,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (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: \(45.3134668844\)
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: \( x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 96)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{10} q^{5} + (\beta_{2} + \beta_1) q^{7} + (\beta_{6} - \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{10} q^{5} + (\beta_{2} + \beta_1) q^{7} + (\beta_{6} - \beta_{4} + 2) q^{9} + (2 \beta_{11} + \beta_{9} + \beta_{4} + \cdots - 1) q^{11}+ \cdots + (39 \beta_{11} - 3 \beta_{9} + \cdots + 511) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 20 q^{9} - 360 q^{19} + 132 q^{25} - 844 q^{27} + 80 q^{33} + 1992 q^{43} - 540 q^{49} + 3072 q^{51} + 888 q^{57} - 5640 q^{67} - 2424 q^{73} - 6444 q^{75} + 2924 q^{81} + 7536 q^{91} - 2952 q^{97} + 6176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -34\nu^{10} + 531\nu^{8} - 3334\nu^{6} + 8476\nu^{4} - 11310\nu^{2} - 5219 ) / 3573 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -14314\nu^{10} + 227124\nu^{8} - 1482220\nu^{6} + 3882820\nu^{4} - 5290314\nu^{2} - 2443736 ) / 217953 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4609 \nu^{11} - 14976 \nu^{10} - 59990 \nu^{9} + 230854 \nu^{8} + 287852 \nu^{7} - 1327664 \nu^{6} + \cdots - 8719100 ) / 1888926 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4609 \nu^{11} - 14976 \nu^{10} + 59990 \nu^{9} + 230854 \nu^{8} - 287852 \nu^{7} + \cdots - 8089458 ) / 629642 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2707 \nu^{11} - 10504 \nu^{10} - 41518 \nu^{9} + 183781 \nu^{8} + 254446 \nu^{7} - 1315990 \nu^{6} + \cdots - 2381353 ) / 314821 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9871 \nu^{11} - 624 \nu^{10} + 174654 \nu^{9} + 35854 \nu^{8} - 1216484 \nu^{7} - 265200 \nu^{6} + \cdots + 6510296 ) / 629642 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -90923\nu^{11} + 1469640\nu^{9} - 9170186\nu^{7} + 21145592\nu^{5} - 19445841\nu^{3} - 3859978\nu ) / 2833389 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 156764 \nu^{11} + 256646 \nu^{10} + 2706942 \nu^{9} - 4541004 \nu^{8} - 19353116 \nu^{7} + \cdots + 60157864 ) / 2833389 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -21656\nu^{11} + 332144\nu^{9} - 2035568\nu^{7} + 4568096\nu^{5} - 4003144\nu^{3} - 10837024\nu ) / 314821 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17681\nu^{11} - 294456\nu^{9} + 2017166\nu^{7} - 5925944\nu^{5} + 8242515\nu^{3} + 2046478\nu ) / 217953 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 193771 \nu^{11} + 14976 \nu^{10} - 3100778 \nu^{9} - 230854 \nu^{8} + 19800164 \nu^{7} + \cdots + 8719100 ) / 1888926 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -4\beta_{10} - 3\beta_{9} - 4\beta_{8} - 12\beta_{5} + 4\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{8} - 6 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \cdots + 130 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 18 \beta_{11} - 11 \beta_{10} - 30 \beta_{9} - 14 \beta_{8} - 15 \beta_{7} - 42 \beta_{5} + \cdots - 12 ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 15 \beta_{11} - 6 \beta_{10} - 15 \beta_{9} - 6 \beta_{8} - 30 \beta_{6} + 18 \beta_{5} - 2 \beta_{4} + \cdots + 398 ) / 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 120 \beta_{11} - 29 \beta_{10} - 222 \beta_{9} - 20 \beta_{8} - 63 \beta_{7} - 60 \beta_{5} + \cdots - 200 ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{11} - 20 \beta_{10} - 3 \beta_{9} - 20 \beta_{8} - 6 \beta_{6} + 60 \beta_{5} - 22 \beta_{4} + \cdots - 566 ) / 48 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 714 \beta_{11} + 278 \beta_{10} - 1383 \beta_{9} + 314 \beta_{8} + 288 \beta_{7} + 942 \beta_{5} + \cdots - 1652 ) / 96 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 135 \beta_{11} - 10 \beta_{10} + 135 \beta_{9} - 10 \beta_{8} + 270 \beta_{6} + 30 \beta_{5} + \cdots - 4154 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3420 \beta_{11} + 4898 \beta_{10} - 6753 \beta_{9} + 4856 \beta_{8} + 7134 \beta_{7} + 14568 \beta_{5} + \cdots - 8904 ) / 96 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10203 \beta_{11} - 139 \beta_{10} + 10203 \beta_{9} - 139 \beta_{8} + 20406 \beta_{6} + 417 \beta_{5} + \cdots - 285146 ) / 48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7326 \beta_{11} + 46909 \beta_{10} - 14754 \beta_{9} + 45334 \beta_{8} + 71829 \beta_{7} + \cdots - 21252 ) / 96 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
0.248859 − 0.707107i
−0.248859 − 0.707107i
0.248859 + 0.707107i
−0.248859 + 0.707107i
2.59708 + 0.707107i
−2.59708 + 0.707107i
2.59708 − 0.707107i
−2.59708 − 0.707107i
−1.64111 − 0.707107i
1.64111 − 0.707107i
−1.64111 + 0.707107i
1.64111 + 0.707107i
0 −5.17226 − 0.497717i 0 −18.4532 0 17.1600i 0 26.5046 + 5.14865i 0
383.2 0 −5.17226 − 0.497717i 0 18.4532 0 − 17.1600i 0 26.5046 + 5.14865i 0
383.3 0 −5.17226 + 0.497717i 0 −18.4532 0 − 17.1600i 0 26.5046 − 5.14865i 0
383.4 0 −5.17226 + 0.497717i 0 18.4532 0 17.1600i 0 26.5046 − 5.14865i 0
383.5 0 0.143987 − 5.19416i 0 −7.96714 0 − 25.6706i 0 −26.9585 − 1.49578i 0
383.6 0 0.143987 − 5.19416i 0 7.96714 0 25.6706i 0 −26.9585 − 1.49578i 0
383.7 0 0.143987 + 5.19416i 0 −7.96714 0 25.6706i 0 −26.9585 + 1.49578i 0
383.8 0 0.143987 + 5.19416i 0 7.96714 0 − 25.6706i 0 −26.9585 + 1.49578i 0
383.9 0 4.02827 − 3.28223i 0 −2.00080 0 − 14.5105i 0 5.45398 − 26.4434i 0
383.10 0 4.02827 − 3.28223i 0 2.00080 0 14.5105i 0 5.45398 − 26.4434i 0
383.11 0 4.02827 + 3.28223i 0 −2.00080 0 14.5105i 0 5.45398 + 26.4434i 0
383.12 0 4.02827 + 3.28223i 0 2.00080 0 − 14.5105i 0 5.45398 + 26.4434i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.f.d 12
3.b odd 2 1 inner 768.4.f.d 12
4.b odd 2 1 768.4.f.i 12
8.b even 2 1 768.4.f.i 12
8.d odd 2 1 inner 768.4.f.d 12
12.b even 2 1 768.4.f.i 12
16.e even 4 1 96.4.c.a ✓ 12
16.e even 4 1 192.4.c.d 12
16.f odd 4 1 96.4.c.a ✓ 12
16.f odd 4 1 192.4.c.d 12
24.f even 2 1 inner 768.4.f.d 12
24.h odd 2 1 768.4.f.i 12
48.i odd 4 1 96.4.c.a ✓ 12
48.i odd 4 1 192.4.c.d 12
48.k even 4 1 96.4.c.a ✓ 12
48.k even 4 1 192.4.c.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.4.c.a ✓ 12 16.e even 4 1
96.4.c.a ✓ 12 16.f odd 4 1
96.4.c.a ✓ 12 48.i odd 4 1
96.4.c.a ✓ 12 48.k even 4 1
192.4.c.d 12 16.e even 4 1
192.4.c.d 12 16.f odd 4 1
192.4.c.d 12 48.i odd 4 1
192.4.c.d 12 48.k even 4 1
768.4.f.d 12 1.a even 1 1 trivial
768.4.f.d 12 3.b odd 2 1 inner
768.4.f.d 12 8.d odd 2 1 inner
768.4.f.d 12 24.f even 2 1 inner
768.4.f.i 12 4.b odd 2 1
768.4.f.i 12 8.b even 2 1
768.4.f.i 12 12.b even 2 1
768.4.f.i 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{6} - 408T_{5}^{4} + 23232T_{5}^{2} - 86528 \) Copy content Toggle raw display
\( T_{19}^{3} + 90T_{19}^{2} + 12T_{19} - 50184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 2 T^{5} + \cdots + 19683)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 408 T^{4} + \cdots - 86528)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 1164 T^{4} + \cdots + 40857664)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 5400 T^{4} + \cdots + 3202560512)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 9324 T^{4} + \cdots + 17936709184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 24576 T^{4} + \cdots + 391378894848)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 90 T^{2} + \cdots - 50184)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 26976 T^{4} + \cdots - 124728147968)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 19385223967232)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 58956 T^{4} + \cdots + 708600302656)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 10556936732736)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 359281246896128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 498 T^{2} + \cdots + 4917544)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 165630483365888)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 152455972372992)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 61\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 321912780357184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 1410 T^{2} + \cdots - 73008616)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 61\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 606 T^{2} + \cdots - 145079064)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 76\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 22\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 761571160915968)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 738 T^{2} + \cdots - 38714344)^{4} \) Copy content Toggle raw display
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