Properties

Label 112.10.a.i
Level $112$
Weight $10$
Character orbit 112.a
Self dual yes
Analytic conductor $57.684$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(57.6840136504\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 823x - 4578 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 56)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 31) q^{3} + ( - \beta_{2} + \beta_1 + 92) q^{5} + 2401 q^{7} + (31 \beta_{2} + 90 \beta_1 + 17310) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 31) q^{3} + ( - \beta_{2} + \beta_1 + 92) q^{5} + 2401 q^{7} + (31 \beta_{2} + 90 \beta_1 + 17310) q^{9} + ( - \beta_{2} - 86 \beta_1 + 15093) q^{11} + ( - 57 \beta_{2} + 427 \beta_1 - 3558) q^{13} + (29 \beta_{2} - 404 \beta_1 + 31717) q^{15} + ( - 298 \beta_{2} + 1394 \beta_1 - 18058) q^{17} + ( - 378 \beta_{2} - 501 \beta_1 - 162965) q^{19} + (2401 \beta_1 + 74431) q^{21} + ( - 953 \beta_{2} + 4044 \beta_1 + 86295) q^{23} + ( - 649 \beta_{2} - 1186 \beta_1 - 1291052) q^{25} + (2852 \beta_{2} + 20142 \beta_1 + 3391494) q^{27} + (4550 \beta_{2} + 1526 \beta_1 - 256114) q^{29} + (366 \beta_{2} + 23942 \beta_1 + 2737752) q^{31} + ( - 2668 \beta_{2} + 9464 \beta_1 - 2638036) q^{33} + ( - 2401 \beta_{2} + 2401 \beta_1 + 220892) q^{35} + (8132 \beta_{2} + 44278 \beta_1 + 2414780) q^{37} + (13123 \beta_{2} - 10000 \beta_1 + 14866847) q^{39} + ( - 25092 \beta_{2} - 1594 \beta_1 - 19289496) q^{41} + ( - 18675 \beta_{2} - 52946 \beta_1 + 12423479) q^{43} + (7217 \beta_{2} + 4293 \beta_1 - 15176694) q^{45} + ( - 21540 \beta_{2} - 115606 \beta_1 + 9705642) q^{47} + 5764801 q^{49} + (42618 \beta_{2} - 101202 \beta_1 + 47533044) q^{51} + (67042 \beta_{2} + 34936 \beta_1 + 15035992) q^{53} + ( - 20870 \beta_{2} + 51660 \beta_1 - 469090) q^{55} + ( - 16287 \beta_{2} - 402314 \beta_1 - 25813073) q^{57} + ( - 23532 \beta_{2} + 199363 \beta_1 + 68519233) q^{59} + ( - 160805 \beta_{2} - 353767 \beta_1 - 36485944) q^{61} + (74431 \beta_{2} + 216090 \beta_1 + 41561310) q^{63} + ( - 5991 \beta_{2} - 237354 \beta_1 + 47608427) q^{65} + (13481 \beta_{2} - 839484 \beta_1 + 17156461) q^{67} + (123458 \beta_{2} - 204024 \beta_1 + 141558402) q^{69} + ( - 176974 \beta_{2} + 654892 \beta_1 + 50293334) q^{71} + (134624 \beta_{2} - 175700 \beta_1 + 253706558) q^{73} + ( - 38064 \beta_{2} - 1721221 \beta_1 - 87407947) q^{75} + ( - 2401 \beta_{2} - 206486 \beta_1 + 36238293) q^{77} + (312160 \beta_{2} + 453212 \beta_1 + 71510068) q^{79} + (19933 \beta_{2} + 4391262 \beta_1 + 510620412) q^{81} + (30906 \beta_{2} + 1058511 \beta_1 + 12379499) q^{83} + ( - 82168 \beta_{2} - 875662 \beta_1 + 224750186) q^{85} + (56406 \beta_{2} + 2359170 \beta_1 + 79655148) q^{87} + ( - 281998 \beta_{2} + 2554448 \beta_1 + 440894948) q^{89} + ( - 136857 \beta_{2} + 1025227 \beta_1 - 8542758) q^{91} + (742934 \beta_{2} + 4353460 \beta_1 + 950171578) q^{93} + ( - 100321 \beta_{2} - 263684 \beta_1 + 206699087) q^{95} + ( - 562272 \beta_{2} - 5009758 \beta_1 + 32439608) q^{97} + (307731 \beta_{2} - 1867662 \beta_1 - 56969343) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9} + 45364 q^{11} - 11158 q^{13} + 95584 q^{15} - 55866 q^{17} - 488772 q^{19} + 220892 q^{21} + 253888 q^{23} - 3872619 q^{25} + 10157192 q^{27} - 765318 q^{29} + 8189680 q^{31} - 7926240 q^{33} + 657874 q^{35} + 7208194 q^{37} + 44623664 q^{39} - 57891986 q^{41} + 37304708 q^{43} - 45527158 q^{45} + 29210992 q^{47} + 17294403 q^{49} + 142742952 q^{51} + 45140082 q^{53} - 1479800 q^{55} - 77053192 q^{57} + 205334804 q^{59} - 109264870 q^{61} + 124542271 q^{63} + 143056644 q^{65} + 52322348 q^{67} + 425002688 q^{69} + 150048136 q^{71} + 761429998 q^{73} - 260540684 q^{75} + 108918964 q^{77} + 214389152 q^{79} + 1527489907 q^{81} + 36110892 q^{83} + 675044052 q^{85} + 236662680 q^{87} + 1319848398 q^{89} - 26790358 q^{91} + 2846904208 q^{93} + 620260624 q^{95} + 101766310 q^{97} - 168732636 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 823x - 4578 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} + 12\nu - 1099 ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{2} + 192\nu + 4391 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 4\beta _1 + 1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + 16\beta _1 + 4395 ) / 8 \) 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
−5.79960
−25.3451
31.1447
0 −189.265 0 −729.944 0 2401.00 0 16138.2 0
1.2 0 7.32110 0 1191.17 0 2401.00 0 −19629.4 0
1.3 0 273.944 0 −187.226 0 2401.00 0 55362.2 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 112.10.a.i 3
4.b odd 2 1 56.10.a.a 3
28.d even 2 1 392.10.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.10.a.a 3 4.b odd 2 1
112.10.a.i 3 1.a even 1 1 trivial
392.10.a.d 3 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 92T_{3}^{2} - 51228T_{3} + 379584 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(112))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 92 T^{2} - 51228 T + 379584 \) Copy content Toggle raw display
$5$ \( T^{3} - 274 T^{2} + \cdots - 162790400 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 45364 T^{2} + \cdots + 3858186796800 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 467510359287776 \) Copy content Toggle raw display
$17$ \( T^{3} + 55866 T^{2} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{3} + 488772 T^{2} + \cdots - 36\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{3} - 253888 T^{2} + \cdots + 83\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{3} + 765318 T^{2} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{3} - 8189680 T^{2} + \cdots + 35\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{3} - 7208194 T^{2} + \cdots - 33\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{3} + 57891986 T^{2} + \cdots - 99\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{3} - 37304708 T^{2} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{3} - 29210992 T^{2} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{3} - 45140082 T^{2} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{3} - 205334804 T^{2} + \cdots - 98\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{3} + 109264870 T^{2} + \cdots - 17\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{3} - 52322348 T^{2} + \cdots + 82\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{3} - 150048136 T^{2} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{3} - 761429998 T^{2} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{3} - 214389152 T^{2} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{3} - 36110892 T^{2} + \cdots - 27\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{3} - 1319848398 T^{2} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{3} - 101766310 T^{2} + \cdots + 92\!\cdots\!28 \) Copy content Toggle raw display
show more
show less