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Properties

Label 768.4.j.f
Level $768$
Weight $4$
Character orbit 768.j
Analytic conductor $45.313$
Analytic rank $0$
Dimension $16$
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(193,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.j (of order \(4\), degree \(2\), 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: \(16\)
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: \( x^{16} + 2042x^{12} + 1366657x^{8} + 357692760x^{4} + 29376588816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 3 \beta_{10} q^{3} + \beta_{5} q^{5} + ( - \beta_{15} + \beta_{12} + \beta_{10}) q^{7} + 9 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_{10} q^{3} + \beta_{5} q^{5} + ( - \beta_{15} + \beta_{12} + \beta_{10}) q^{7} + 9 \beta_1 q^{9} + (\beta_{15} + \beta_{14} + \cdots + 13 \beta_{3}) q^{11}+ \cdots + ( - 9 \beta_{15} + 9 \beta_{14} + \cdots - 9 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{5} - 256 q^{13} + 336 q^{17} + 24 q^{21} + 536 q^{29} + 624 q^{33} - 176 q^{37} - 72 q^{45} + 1968 q^{49} + 8 q^{53} + 1744 q^{61} + 6672 q^{65} + 1248 q^{69} + 3984 q^{77} - 1296 q^{81} + 5136 q^{85} + 552 q^{93} + 7456 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2042x^{12} + 1366657x^{8} + 357692760x^{4} + 29376588816 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -97255\nu^{14} - 205964738\nu^{10} - 151024270687\nu^{6} - 58038765690852\nu^{2} ) / 486652915860192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{12} + 17548\nu^{8} + 7082861\nu^{4} + 656982252 ) / 3669462 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -393985\nu^{13} - 637965170\nu^{9} - 272700140749\nu^{5} - 30022640411868\nu ) / 7052940809568 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79453\nu^{13} + 162220670\nu^{9} + 94861574269\nu^{5} + 14393704732668\nu ) / 881617601196 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8127869 \nu^{14} - 141633402 \nu^{12} - 14062447318 \nu^{10} - 243237744300 \nu^{8} + \cdots - 14\!\cdots\!20 ) / 162217638620064 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8127869 \nu^{14} + 141633402 \nu^{12} - 14062447318 \nu^{10} + 243237744300 \nu^{8} + \cdots + 14\!\cdots\!56 ) / 162217638620064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37705445 \nu^{14} - 744163344 \nu^{12} - 61782266710 \nu^{10} - 1199849166288 \nu^{8} + \cdots - 58\!\cdots\!12 ) / 486652915860192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37705445 \nu^{14} + 744163344 \nu^{12} - 61782266710 \nu^{10} + 1199849166288 \nu^{8} + \cdots + 58\!\cdots\!12 ) / 486652915860192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -185\nu^{14} - 292072\nu^{10} - 115114859\nu^{6} - 9855348732\nu^{2} ) / 1342801962 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62692309 \nu^{15} - 102513570254 \nu^{11} - 44700617719573 \nu^{7} - 51\!\cdots\!12 \nu^{3} ) / 22\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 338032045 \nu^{15} + 5264628102 \nu^{13} - 601205274062 \nu^{11} + \cdots + 69\!\cdots\!80 \nu ) / 67\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 338032045 \nu^{15} + 5264628102 \nu^{13} + 601205274062 \nu^{11} + \cdots + 69\!\cdots\!80 \nu ) / 67\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 61556369\nu^{15} + 118409147806\nu^{11} + 66215608922105\nu^{7} + 10215693190408428\nu^{3} ) / 8394762798588312 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1081582501 \nu^{15} - 24408551970 \nu^{13} + 1679107231022 \nu^{11} + \cdots - 12\!\cdots\!60 \nu ) / 67\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1081582501 \nu^{15} - 20657026800 \nu^{13} - 1679107231022 \nu^{11} + \cdots - 10\!\cdots\!64 \nu ) / 67\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{12} + 2\beta_{11} - \beta_{4} + 4\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{7} - 86\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{15} + 12\beta_{14} - 19\beta_{13} + 50\beta_{12} - 50\beta_{11} + 268\beta_{10} - 12\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 49\beta_{8} - 49\beta_{7} - 12\beta_{6} + 12\beta_{5} - 43\beta_{2} - 2042 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 540\beta_{15} + 540\beta_{14} - 1334\beta_{12} - 1334\beta_{11} + 277\beta_{4} - 9784\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1471\beta_{9} - 1879\beta_{8} - 1879\beta_{7} + 780\beta_{6} + 780\beta_{5} + 51746\beta _1 + 780 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19284 \beta_{15} - 19284 \beta_{14} + 1219 \beta_{13} - 37394 \beta_{12} + 37394 \beta_{11} + \cdots + 19284 \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -66133\beta_{8} + 66133\beta_{7} + 34956\beta_{6} - 34956\beta_{5} + 47107\beta_{2} + 1436450 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 641388 \beta_{15} - 641388 \beta_{14} + 1087574 \beta_{12} + 1087574 \beta_{11} + \cdots + 10966216 \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1477555 \beta_{9} + 2235847 \beta_{8} + 2235847 \beta_{7} - 1346700 \beta_{6} - 1346700 \beta_{5} + \cdots - 1346700 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 20763828 \beta_{15} + 20763828 \beta_{14} + 7769981 \beta_{13} + 32497634 \beta_{12} + \cdots - 20763828 \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 73949245 \beta_{8} - 73949245 \beta_{7} - 48037596 \beta_{6} + 48037596 \beta_{5} - 46126615 \beta_{2} - 1215595586 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 664809900 \beta_{15} + 664809900 \beta_{14} - 990133190 \beta_{12} - 990133190 \beta_{11} + \cdots - 11433125032 \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1441635823 \beta_{9} - 2413961263 \beta_{8} - 2413961263 \beta_{7} + 1640777148 \beta_{6} + \cdots + 1640777148 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 21188931636 \beta_{15} - 21188931636 \beta_{14} - 12013336637 \beta_{13} - 30585548786 \beta_{12} + \cdots + 21188931636 \beta_{3} ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
2.47963 + 2.47963i
−3.98991 − 3.98991i
3.28280 + 3.28280i
−3.18674 − 3.18674i
−2.47963 − 2.47963i
3.98991 + 3.98991i
−3.28280 − 3.28280i
3.18674 + 3.18674i
2.47963 − 2.47963i
−3.98991 + 3.98991i
3.28280 − 3.28280i
−3.18674 + 3.18674i
−2.47963 + 2.47963i
3.98991 − 3.98991i
−3.28280 + 3.28280i
3.18674 − 3.18674i
0 −2.12132 + 2.12132i 0 −12.3126 − 12.3126i 0 − 1.18094i 0 − 9.00000i 0
193.2 0 −2.12132 + 2.12132i 0 −3.39861 − 3.39861i 0 − 24.2508i 0 − 9.00000i 0
193.3 0 −2.12132 + 2.12132i 0 4.13066 + 4.13066i 0 15.4881i 0 − 9.00000i 0
193.4 0 −2.12132 + 2.12132i 0 9.58055 + 9.58055i 0 7.11520i 0 − 9.00000i 0
193.5 0 2.12132 − 2.12132i 0 −12.3126 − 12.3126i 0 1.18094i 0 − 9.00000i 0
193.6 0 2.12132 − 2.12132i 0 −3.39861 − 3.39861i 0 24.2508i 0 − 9.00000i 0
193.7 0 2.12132 − 2.12132i 0 4.13066 + 4.13066i 0 − 15.4881i 0 − 9.00000i 0
193.8 0 2.12132 − 2.12132i 0 9.58055 + 9.58055i 0 − 7.11520i 0 − 9.00000i 0
577.1 0 −2.12132 − 2.12132i 0 −12.3126 + 12.3126i 0 1.18094i 0 9.00000i 0
577.2 0 −2.12132 − 2.12132i 0 −3.39861 + 3.39861i 0 24.2508i 0 9.00000i 0
577.3 0 −2.12132 − 2.12132i 0 4.13066 − 4.13066i 0 − 15.4881i 0 9.00000i 0
577.4 0 −2.12132 − 2.12132i 0 9.58055 − 9.58055i 0 − 7.11520i 0 9.00000i 0
577.5 0 2.12132 + 2.12132i 0 −12.3126 + 12.3126i 0 − 1.18094i 0 9.00000i 0
577.6 0 2.12132 + 2.12132i 0 −3.39861 + 3.39861i 0 − 24.2508i 0 9.00000i 0
577.7 0 2.12132 + 2.12132i 0 4.13066 − 4.13066i 0 15.4881i 0 9.00000i 0
577.8 0 2.12132 + 2.12132i 0 9.58055 − 9.58055i 0 7.11520i 0 9.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.j.f ✓ 16
4.b odd 2 1 inner 768.4.j.f ✓ 16
8.b even 2 1 768.4.j.g yes 16
8.d odd 2 1 768.4.j.g yes 16
16.e even 4 1 inner 768.4.j.f ✓ 16
16.e even 4 1 768.4.j.g yes 16
16.f odd 4 1 inner 768.4.j.f ✓ 16
16.f odd 4 1 768.4.j.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.4.j.f ✓ 16 1.a even 1 1 trivial
768.4.j.f ✓ 16 4.b odd 2 1 inner
768.4.j.f ✓ 16 16.e even 4 1 inner
768.4.j.f ✓ 16 16.f odd 4 1 inner
768.4.j.g yes 16 8.b even 2 1
768.4.j.g yes 16 8.d odd 2 1
768.4.j.g yes 16 16.e even 4 1
768.4.j.g yes 16 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4T_{5}^{7} + 8T_{5}^{6} - 1264T_{5}^{5} + 58576T_{5}^{4} - 77952T_{5}^{3} + 18432T_{5}^{2} + 1271808T_{5} + 43877376 \) acting on \(S_{4}^{\mathrm{new}}(768, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 81)^{4} \) Copy content Toggle raw display
$5$ \( (T^{8} + 4 T^{7} + \cdots + 43877376)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 880 T^{6} + \cdots + 9960336)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 1210906569744)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 84 T^{3} + \cdots + 35624016)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 141654937964544)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 26\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 729699359650704)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 71\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 31\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 26\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 26\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 39\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 65\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 52\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1864 T^{3} + \cdots - 510246098688)^{4} \) Copy content Toggle raw display
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