Properties

Label 112.7.s.c
Level $112$
Weight $7$
Character orbit 112.s
Analytic conductor $25.766$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,7,Mod(17,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
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("112.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), 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: \(25.7660573654\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{3}) q^{3} + ( - 4 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 28 \beta_1 - 56) q^{5} + (8 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} - 5 \beta_{4} + 13 \beta_{3} - 8 \beta_{2} + \cdots - 89) q^{7}+ \cdots + (7 \beta_{6} + 7 \beta_{4} + 7 \beta_{3} + \beta_{2} + 189 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{3}) q^{3} + ( - 4 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 28 \beta_1 - 56) q^{5} + (8 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} - 5 \beta_{4} + 13 \beta_{3} - 8 \beta_{2} + \cdots - 89) q^{7}+ \cdots + (4200 \beta_{7} - 4851 \beta_{6} + 5883 \beta_{5} + 9702 \beta_{4} + \cdots + 578241) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9} + 1356 q^{11} - 27144 q^{15} - 17304 q^{17} + 32004 q^{19} + 9756 q^{21} + 4128 q^{23} + 4664 q^{25} - 30312 q^{29} + 3108 q^{31} + 3276 q^{33} - 98028 q^{35} - 6124 q^{37} - 100764 q^{39} + 297376 q^{43} - 172116 q^{45} - 313908 q^{47} + 32432 q^{49} - 253692 q^{51} + 278484 q^{53} - 81288 q^{57} + 835464 q^{59} - 995316 q^{61} - 1216188 q^{63} + 8316 q^{65} - 648808 q^{67} - 190128 q^{71} - 1617084 q^{73} + 2042208 q^{75} + 1456224 q^{77} - 70096 q^{79} + 1177920 q^{81} + 2190984 q^{85} + 2057076 q^{87} + 739116 q^{89} - 2233752 q^{91} - 23364 q^{93} - 725640 q^{95} + 4625928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10737582 \nu^{7} + 884171859 \nu^{6} - 3763944460 \nu^{5} + 189328804101 \nu^{4} - 195625686972 \nu^{3} + \cdots + 41\!\cdots\!50 ) / 33\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1538745699 \nu^{7} - 106190192862 \nu^{6} + 452054636280 \nu^{5} - 43422948065793 \nu^{4} + 23494900020696 \nu^{3} + \cdots - 50\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13703534741 \nu^{7} - 783676536967 \nu^{6} + 17549027818005 \nu^{5} - 218960513207588 \nu^{4} + \cdots - 15\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 585607110 \nu^{7} - 16432330321 \nu^{6} + 225553521577 \nu^{5} - 4727688326325 \nu^{4} + 42323775633178 \nu^{3} + \cdots - 43\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4000422684 \nu^{7} + 194241509733 \nu^{6} - 1687131486945 \nu^{5} + 40398246731187 \nu^{4} - 146692237678314 \nu^{3} + \cdots + 18\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58886751167 \nu^{7} + 585831373629 \nu^{6} - 24486826868135 \nu^{5} + 30203231462756 \nu^{4} + \cdots - 11\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 111050436319 \nu^{7} + 841446413228 \nu^{6} - 32007850574370 \nu^{5} + 52581850337617 \nu^{4} + \cdots - 37\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -4\beta_{6} - 4\beta_{4} - 4\beta_{3} + 5\beta_{2} + 12\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 9\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} + 849\beta _1 - 849 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 296\beta_{7} - 286\beta_{6} + 53\beta_{5} + 572\beta_{4} - 138\beta_{3} - 233\beta_{2} - 3822 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 710\beta_{7} + 427\beta_{6} + 427\beta_{4} + 1847\beta_{3} - 3435\beta_{2} - 139071\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 21934 \beta_{7} + 47494 \beta_{6} + 9318 \beta_{5} - 23747 \beta_{4} + 69428 \beta_{3} - 23747 \beta_{2} - 644823 \beta _1 + 644823 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 397436 \beta_{7} + 130311 \beta_{6} + 593257 \beta_{5} - 260622 \beta_{4} - 68407 \beta_{3} + 723568 \beta_{2} + 25540707 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5192390 \beta_{7} - 4421299 \beta_{6} - 4421299 \beta_{4} - 14806079 \beta_{3} + 12740605 \beta_{2} + 180033087 \beta_1 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−4.86132 8.42006i
7.51287 + 13.0127i
4.65421 + 8.06134i
−6.30576 10.9219i
−4.86132 + 8.42006i
7.51287 13.0127i
4.65421 8.06134i
−6.30576 + 10.9219i
0 −27.6101 15.9407i 0 111.836 64.5687i 0 −298.743 + 168.528i 0 143.713 + 248.917i 0
17.2 0 −21.8891 12.6377i 0 −10.7486 + 6.20573i 0 −195.705 281.689i 0 −45.0775 78.0766i 0
17.3 0 12.7609 + 7.36750i 0 −106.741 + 61.6269i 0 309.691 147.446i 0 −255.940 443.301i 0
17.4 0 36.7384 + 21.2109i 0 −162.347 + 93.7310i 0 −141.244 + 312.569i 0 535.305 + 927.175i 0
33.1 0 −27.6101 + 15.9407i 0 111.836 + 64.5687i 0 −298.743 168.528i 0 143.713 248.917i 0
33.2 0 −21.8891 + 12.6377i 0 −10.7486 6.20573i 0 −195.705 + 281.689i 0 −45.0775 + 78.0766i 0
33.3 0 12.7609 7.36750i 0 −106.741 61.6269i 0 309.691 + 147.446i 0 −255.940 + 443.301i 0
33.4 0 36.7384 21.2109i 0 −162.347 93.7310i 0 −141.244 312.569i 0 535.305 927.175i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.7.s.c 8
4.b odd 2 1 14.7.d.a 8
7.d odd 6 1 inner 112.7.s.c 8
12.b even 2 1 126.7.n.c 8
28.d even 2 1 98.7.d.c 8
28.f even 6 1 14.7.d.a 8
28.f even 6 1 98.7.b.c 8
28.g odd 6 1 98.7.b.c 8
28.g odd 6 1 98.7.d.c 8
84.j odd 6 1 126.7.n.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.d.a 8 4.b odd 2 1
14.7.d.a 8 28.f even 6 1
98.7.b.c 8 28.f even 6 1
98.7.b.c 8 28.g odd 6 1
98.7.d.c 8 28.d even 2 1
98.7.d.c 8 28.g odd 6 1
112.7.s.c 8 1.a even 1 1 trivial
112.7.s.c 8 7.d odd 6 1 inner
126.7.n.c 8 12.b even 2 1
126.7.n.c 8 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 1836T_{3}^{6} + 2867193T_{3}^{4} + 32849712T_{3}^{3} - 818090820T_{3}^{2} - 9012254076T_{3} + 253716712209 \) acting on \(S_{7}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 1836 T^{6} + \cdots + 253716712209 \) Copy content Toggle raw display
$5$ \( T^{8} + 336 T^{7} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + 652 T^{7} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} - 1356 T^{7} + \cdots + 57\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{8} + 11814792 T^{6} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + 17304 T^{7} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} - 32004 T^{7} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} - 4128 T^{7} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{4} + 15156 T^{3} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 3108 T^{7} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{8} + 6124 T^{7} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{8} + 25142408520 T^{6} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{4} - 148688 T^{3} + \cdots - 56\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 313908 T^{7} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( T^{8} - 278484 T^{7} + \cdots + 50\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{8} - 835464 T^{7} + \cdots + 27\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{8} + 995316 T^{7} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{8} + 648808 T^{7} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T^{4} + 95064 T^{3} + \cdots + 56\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 1617084 T^{7} + \cdots + 79\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{8} + 70096 T^{7} + \cdots + 83\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + 1103755415712 T^{6} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} - 739116 T^{7} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{8} + 4399256051592 T^{6} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
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