Properties

Label 112.6.a.h
Level $112$
Weight $6$
Character orbit 112.a
Self dual yes
Analytic conductor $17.963$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,6,Mod(1,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

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: yes
Analytic conductor: \(17.9629878191\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta + 3) q^{3} + (5 \beta - 9) q^{5} - 49 q^{7} + (18 \beta + 279) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta + 3) q^{3} + (5 \beta - 9) q^{5} - 49 q^{7} + (18 \beta + 279) q^{9} + (62 \beta - 198) q^{11} + (63 \beta - 175) q^{13} + ( - 12 \beta + 828) q^{15} + ( - 38 \beta + 900) q^{17} + (9 \beta + 1633) q^{19} + ( - 147 \beta - 147) q^{21} + ( - 284 \beta - 1044) q^{23} + ( - 90 \beta - 1619) q^{25} + (162 \beta + 3186) q^{27} + (126 \beta + 3348) q^{29} + ( - 270 \beta + 10) q^{31} + ( - 408 \beta + 10008) q^{33} + ( - 245 \beta + 441) q^{35} + (270 \beta + 3116) q^{37} + ( - 336 \beta + 10248) q^{39} + ( - 546 \beta - 3024) q^{41} + (2394 \beta + 1510) q^{43} + (1233 \beta + 2619) q^{45} + ( - 1874 \beta - 5850) q^{47} + 2401 q^{49} + (2586 \beta - 3798) q^{51} + ( - 104 \beta + 4734) q^{53} + ( - 1548 \beta + 19452) q^{55} + (4926 \beta + 6438) q^{57} + (1025 \beta + 21969) q^{59} + ( - 2403 \beta - 32377) q^{61} + ( - 882 \beta - 13671) q^{63} + ( - 1442 \beta + 19530) q^{65} + (972 \beta - 12392) q^{67} + ( - 3984 \beta - 51696) q^{69} + (2100 \beta - 48708) q^{71} + ( - 2628 \beta + 8726) q^{73} + ( - 5127 \beta - 20247) q^{75} + ( - 3038 \beta + 9702) q^{77} + ( - 7452 \beta - 25628) q^{79} + (5670 \beta - 30537) q^{81} + ( - 7875 \beta - 58779) q^{83} + (4842 \beta - 18930) q^{85} + (10422 \beta + 31590) q^{87} + ( - 11104 \beta + 42138) q^{89} + ( - 3087 \beta + 8575) q^{91} + ( - 780 \beta - 46140) q^{93} + (8084 \beta - 12132) q^{95} + (4410 \beta + 10388) q^{97} + (13734 \beta + 8370) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9} - 396 q^{11} - 350 q^{13} + 1656 q^{15} + 1800 q^{17} + 3266 q^{19} - 294 q^{21} - 2088 q^{23} - 3238 q^{25} + 6372 q^{27} + 6696 q^{29} + 20 q^{31} + 20016 q^{33} + 882 q^{35} + 6232 q^{37} + 20496 q^{39} - 6048 q^{41} + 3020 q^{43} + 5238 q^{45} - 11700 q^{47} + 4802 q^{49} - 7596 q^{51} + 9468 q^{53} + 38904 q^{55} + 12876 q^{57} + 43938 q^{59} - 64754 q^{61} - 27342 q^{63} + 39060 q^{65} - 24784 q^{67} - 103392 q^{69} - 97416 q^{71} + 17452 q^{73} - 40494 q^{75} + 19404 q^{77} - 51256 q^{79} - 61074 q^{81} - 117558 q^{83} - 37860 q^{85} + 63180 q^{87} + 84276 q^{89} + 17150 q^{91} - 92280 q^{93} - 24264 q^{95} + 20776 q^{97} + 16740 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.27492
4.27492
0 −19.6495 0 −46.7492 0 −49.0000 0 143.103 0
1.2 0 25.6495 0 28.7492 0 −49.0000 0 414.897 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.6.a.h 2
3.b odd 2 1 1008.6.a.bq 2
4.b odd 2 1 7.6.a.b 2
7.b odd 2 1 784.6.a.v 2
8.b even 2 1 448.6.a.u 2
8.d odd 2 1 448.6.a.w 2
12.b even 2 1 63.6.a.f 2
20.d odd 2 1 175.6.a.c 2
20.e even 4 2 175.6.b.c 4
28.d even 2 1 49.6.a.f 2
28.f even 6 2 49.6.c.d 4
28.g odd 6 2 49.6.c.e 4
44.c even 2 1 847.6.a.c 2
84.h odd 2 1 441.6.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 4.b odd 2 1
49.6.a.f 2 28.d even 2 1
49.6.c.d 4 28.f even 6 2
49.6.c.e 4 28.g odd 6 2
63.6.a.f 2 12.b even 2 1
112.6.a.h 2 1.a even 1 1 trivial
175.6.a.c 2 20.d odd 2 1
175.6.b.c 4 20.e even 4 2
441.6.a.l 2 84.h odd 2 1
448.6.a.u 2 8.b even 2 1
448.6.a.w 2 8.d odd 2 1
784.6.a.v 2 7.b odd 2 1
847.6.a.c 2 44.c even 2 1
1008.6.a.bq 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6T_{3} - 504 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(112))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 6T - 504 \) Copy content Toggle raw display
$5$ \( T^{2} + 18T - 1344 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 396T - 179904 \) Copy content Toggle raw display
$13$ \( T^{2} + 350T - 195608 \) Copy content Toggle raw display
$17$ \( T^{2} - 1800 T + 727692 \) Copy content Toggle raw display
$19$ \( T^{2} - 3266 T + 2662072 \) Copy content Toggle raw display
$23$ \( T^{2} + 2088 T - 3507456 \) Copy content Toggle raw display
$29$ \( T^{2} - 6696 T + 10304172 \) Copy content Toggle raw display
$31$ \( T^{2} - 20T - 4155200 \) Copy content Toggle raw display
$37$ \( T^{2} - 6232 T + 5554156 \) Copy content Toggle raw display
$41$ \( T^{2} + 6048 T - 7848036 \) Copy content Toggle raw display
$43$ \( T^{2} - 3020 T - 324400352 \) Copy content Toggle raw display
$47$ \( T^{2} + 11700 T - 165954432 \) Copy content Toggle raw display
$53$ \( T^{2} - 9468 T + 21794244 \) Copy content Toggle raw display
$59$ \( T^{2} - 43938 T + 422751336 \) Copy content Toggle raw display
$61$ \( T^{2} + 64754 T + 719128816 \) Copy content Toggle raw display
$67$ \( T^{2} + 24784 T + 99708976 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 2121099264 \) Copy content Toggle raw display
$73$ \( T^{2} - 17452 T - 317520812 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2508546944 \) Copy content Toggle raw display
$83$ \( T^{2} + 117558 T - 79919784 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 5252421468 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 1000631156 \) Copy content Toggle raw display
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