Properties

Label 192.7.e.f
Level $192$
Weight $7$
Character orbit 192.e
Analytic conductor $44.170$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,7,Mod(65,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 192.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: \(44.1703840550\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 6)
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 \(\beta = 12\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 21) q^{3} + 10 \beta q^{5} - 2 q^{7} + (42 \beta + 153) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 21) q^{3} + 10 \beta q^{5} - 2 q^{7} + (42 \beta + 153) q^{9} - 2 \beta q^{11} + 2950 q^{13} + (210 \beta - 2880) q^{15} + 264 \beta q^{17} + 5258 q^{19} + ( - 2 \beta - 42) q^{21} + 604 \beta q^{23} - 13175 q^{25} + (1035 \beta - 8883) q^{27} - 130 \beta q^{29} - 22898 q^{31} + ( - 42 \beta + 576) q^{33} - 20 \beta q^{35} - 34058 q^{37} + (2950 \beta + 61950) q^{39} - 988 \beta q^{41} - 6406 q^{43} + (1530 \beta - 120960) q^{45} - 10600 \beta q^{47} - 117645 q^{49} + (5544 \beta - 76032) q^{51} + 11346 \beta q^{53} + 5760 q^{55} + (5258 \beta + 110418) q^{57} - 19258 \beta q^{59} + 62566 q^{61} + ( - 84 \beta - 306) q^{63} + 29500 \beta q^{65} + 438698 q^{67} + (12684 \beta - 173952) q^{69} + 4020 \beta q^{71} - 730510 q^{73} + ( - 13175 \beta - 276675) q^{75} + 4 \beta q^{77} - 340562 q^{79} + (12852 \beta - 484623) q^{81} + 29242 \beta q^{83} - 760320 q^{85} + ( - 2730 \beta + 37440) q^{87} - 22788 \beta q^{89} - 5900 q^{91} + ( - 22898 \beta - 480858) q^{93} + 52580 \beta q^{95} - 281086 q^{97} + ( - 306 \beta + 24192) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 42 q^{3} - 4 q^{7} + 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 42 q^{3} - 4 q^{7} + 306 q^{9} + 5900 q^{13} - 5760 q^{15} + 10516 q^{19} - 84 q^{21} - 26350 q^{25} - 17766 q^{27} - 45796 q^{31} + 1152 q^{33} - 68116 q^{37} + 123900 q^{39} - 12812 q^{43} - 241920 q^{45} - 235290 q^{49} - 152064 q^{51} + 11520 q^{55} + 220836 q^{57} + 125132 q^{61} - 612 q^{63} + 877396 q^{67} - 347904 q^{69} - 1461020 q^{73} - 553350 q^{75} - 681124 q^{79} - 969246 q^{81} - 1520640 q^{85} + 74880 q^{87} - 11800 q^{91} - 961716 q^{93} - 562172 q^{97} + 48384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.41421i
1.41421i
0 21.0000 16.9706i 0 169.706i 0 −2.00000 0 153.000 712.764i 0
65.2 0 21.0000 + 16.9706i 0 169.706i 0 −2.00000 0 153.000 + 712.764i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.7.e.f 2
3.b odd 2 1 inner 192.7.e.f 2
4.b odd 2 1 192.7.e.c 2
8.b even 2 1 48.7.e.b 2
8.d odd 2 1 6.7.b.a 2
12.b even 2 1 192.7.e.c 2
24.f even 2 1 6.7.b.a 2
24.h odd 2 1 48.7.e.b 2
40.e odd 2 1 150.7.d.a 2
40.k even 4 2 150.7.b.a 4
56.e even 2 1 294.7.b.a 2
72.l even 6 2 162.7.d.b 4
72.p odd 6 2 162.7.d.b 4
120.m even 2 1 150.7.d.a 2
120.q odd 4 2 150.7.b.a 4
168.e odd 2 1 294.7.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 8.d odd 2 1
6.7.b.a 2 24.f even 2 1
48.7.e.b 2 8.b even 2 1
48.7.e.b 2 24.h odd 2 1
150.7.b.a 4 40.k even 4 2
150.7.b.a 4 120.q odd 4 2
150.7.d.a 2 40.e odd 2 1
150.7.d.a 2 120.m even 2 1
162.7.d.b 4 72.l even 6 2
162.7.d.b 4 72.p odd 6 2
192.7.e.c 2 4.b odd 2 1
192.7.e.c 2 12.b even 2 1
192.7.e.f 2 1.a even 1 1 trivial
192.7.e.f 2 3.b odd 2 1 inner
294.7.b.a 2 56.e even 2 1
294.7.b.a 2 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{2} + 28800 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 42T + 729 \) Copy content Toggle raw display
$5$ \( T^{2} + 28800 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1152 \) Copy content Toggle raw display
$13$ \( (T - 2950)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 20072448 \) Copy content Toggle raw display
$19$ \( (T - 5258)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 105067008 \) Copy content Toggle raw display
$29$ \( T^{2} + 4867200 \) Copy content Toggle raw display
$31$ \( (T + 22898)^{2} \) Copy content Toggle raw display
$37$ \( (T + 34058)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 281129472 \) Copy content Toggle raw display
$43$ \( (T + 6406)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 32359680000 \) Copy content Toggle raw display
$53$ \( T^{2} + 37074734208 \) Copy content Toggle raw display
$59$ \( T^{2} + 106810722432 \) Copy content Toggle raw display
$61$ \( (T - 62566)^{2} \) Copy content Toggle raw display
$67$ \( (T - 438698)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4654195200 \) Copy content Toggle raw display
$73$ \( (T + 730510)^{2} \) Copy content Toggle raw display
$79$ \( (T + 340562)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 246267234432 \) Copy content Toggle raw display
$89$ \( T^{2} + 149556367872 \) Copy content Toggle raw display
$97$ \( (T + 281086)^{2} \) Copy content Toggle raw display
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