Properties

Label 112.7.s.b
Level $112$
Weight $7$
Character orbit 112.s
Analytic conductor $25.766$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 7)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 13 \beta_{3} - 3 \beta_1 - 6) q^{3} + ( - 50 \beta_{3} + 50 \beta_{2} + 25 \beta_1 - 25) q^{5} + (126 \beta_{3} + 7 \beta_{2} + 126 \beta_1 - 7) q^{7} + (78 \beta_{3} + 78 \beta_{2} + 312 \beta_1 + 312) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 13 \beta_{3} - 3 \beta_1 - 6) q^{3} + ( - 50 \beta_{3} + 50 \beta_{2} + 25 \beta_1 - 25) q^{5} + (126 \beta_{3} + 7 \beta_{2} + 126 \beta_1 - 7) q^{7} + (78 \beta_{3} + 78 \beta_{2} + 312 \beta_1 + 312) q^{9} + (181 \beta_{3} - 362 \beta_{2} + 941 \beta_1) q^{11} + ( - 238 \beta_{2} + 2940 \beta_1 + 1470) q^{13} + (950 \beta_{3} - 475 \beta_{2} + 4125) q^{15} + ( - 78 \beta_{3} + 2243 \beta_1 + 4486) q^{17} + (267 \beta_{3} - 267 \beta_{2} + 3013 \beta_1 - 3013) q^{19} + (1372 \beta_{3} - 2058 \beta_{2} - 10731 \beta_1 - 9408) q^{21} + ( - 3431 \beta_{3} - 3431 \beta_{2} - 1235 \beta_1 - 1235) q^{23} + (2500 \beta_{3} - 5000 \beta_{2} - 1250 \beta_1) q^{25} + (4719 \beta_{2} - 9666 \beta_1 - 4833) q^{27} + ( - 7140 \beta_{3} + 3570 \beta_{2} - 8636) q^{29} + ( - 2869 \beta_{3} + 2867 \beta_1 + 5734) q^{31} + (10604 \beta_{3} - 10604 \beta_{2} + 11295 \beta_1 - 11295) q^{33} + ( - 6125 \beta_{3} - 3675 \beta_{2} - 8575 \beta_1 - 42875) q^{35} + ( - 6264 \beta_{3} - 6264 \beta_{2} - 21935 \beta_1 - 21935) q^{37} + (18396 \beta_{3} - 36792 \beta_{2} + 5334 \beta_1) q^{39} + ( - 9338 \beta_{2} - 4116 \beta_1 - 2058) q^{41} + (5460 \beta_{3} - 2730 \beta_{2} - 80054) q^{43} + ( - 21450 \beta_{3} - 31200 \beta_1 - 62400) q^{45} + (15499 \beta_{3} - 15499 \beta_{2} - 59119 \beta_1 + 59119) q^{47} + ( - 35280 \beta_{3} + 31654 \beta_{2} + 88200 \beta_1 + 79135) q^{49} + ( - 28925 \beta_{3} - 28925 \beta_{2} - 14103 \beta_1 - 14103) q^{51} + (47580 \beta_{3} - 95160 \beta_{2} - 52265 \beta_1) q^{53} + ( - 33475 \beta_{2} + 61550 \beta_1 + 30775) q^{55} + (76736 \beta_{3} - 38368 \beta_{2} + 6291) q^{57} + (32299 \beta_{3} + 10441 \beta_1 + 20882) q^{59} + (23680 \beta_{3} - 23680 \beta_{2} + 23447 \beta_1 - 23447) q^{61} + ( - 22386 \beta_{3} + 50778 \beta_{2} + 119028 \beta_1 + 14196) q^{63} + ( - 67550 \beta_{3} - 67550 \beta_{2} - 38850 \beta_1 - 38850) q^{65} + (22525 \beta_{3} - 45050 \beta_{2} + 216247 \beta_1) q^{67} + (46934 \beta_{2} + 542646 \beta_1 + 271323) q^{69} + ( - 14532 \beta_{3} + 7266 \beta_{2} + 370382) q^{71} + (159984 \beta_{3} - 161963 \beta_1 - 323926) q^{73} + ( - 38750 \beta_{3} + 38750 \beta_{2} + 198750 \beta_1 - 198750) q^{75} + ( - 103614 \beta_{3} + 143906 \beta_{2} - 254387 \beta_1 + 33474) q^{77} + ( - 34847 \beta_{3} - 34847 \beta_{2} - 467995 \beta_1 - 467995) q^{79} + (8190 \beta_{3} - 16380 \beta_{2} - 552033 \beta_1) q^{81} + (91588 \beta_{2} - 702072 \beta_1 - 351036) q^{83} + ( - 220400 \beta_{3} + 110200 \beta_{2} - 144825) q^{85} + (144398 \beta_{3} + 304368 \beta_1 + 608736) q^{87} + ( - 234984 \beta_{3} + 234984 \beta_{2} + 113215 \beta_1 - 113215) q^{89} + ( - 175812 \beta_{3} + 382396 \beta_{2} - 385728 \beta_1 - 370734) q^{91} + ( - 28664 \beta_{3} - 28664 \beta_{2} + 197979 \beta_1 + 197979) q^{93} + (143975 \beta_{3} - 287950 \beta_{2} - 145875 \beta_1) q^{95} + (155722 \beta_{2} - 777924 \beta_1 - 388962) q^{97} + ( - 33852 \beta_{3} + 16926 \beta_{2} - 39468) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} - 150 q^{5} - 280 q^{7} + 624 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{3} - 150 q^{5} - 280 q^{7} + 624 q^{9} - 1882 q^{11} + 16500 q^{15} + 13458 q^{17} - 18078 q^{19} - 16170 q^{21} - 2470 q^{23} + 2500 q^{25} - 34544 q^{29} + 17202 q^{31} - 67770 q^{33} - 154350 q^{35} - 43870 q^{37} - 10668 q^{39} - 320216 q^{43} - 187200 q^{45} + 354714 q^{47} + 140140 q^{49} - 28206 q^{51} + 104530 q^{53} + 25164 q^{57} + 62646 q^{59} - 140682 q^{61} - 181272 q^{63} - 77700 q^{65} - 432494 q^{67} + 1481528 q^{71} - 971778 q^{73} - 1192500 q^{75} + 642670 q^{77} - 935990 q^{79} + 1104066 q^{81} - 579300 q^{85} + 1826208 q^{87} - 679290 q^{89} - 711480 q^{91} + 395958 q^{93} + 291750 q^{95} - 157872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
0 −32.0772 18.5198i 0 −143.566 + 82.8879i 0 197.286 + 280.583i 0 321.463 + 556.790i 0
17.2 0 23.0772 + 13.3236i 0 68.5660 39.5866i 0 −337.286 62.3451i 0 −9.46299 16.3904i 0
33.1 0 −32.0772 + 18.5198i 0 −143.566 82.8879i 0 197.286 280.583i 0 321.463 556.790i 0
33.2 0 23.0772 13.3236i 0 68.5660 + 39.5866i 0 −337.286 + 62.3451i 0 −9.46299 + 16.3904i 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
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.b 4
4.b odd 2 1 7.7.d.b 4
7.d odd 6 1 inner 112.7.s.b 4
12.b even 2 1 63.7.m.b 4
28.d even 2 1 49.7.d.c 4
28.f even 6 1 7.7.d.b 4
28.f even 6 1 49.7.b.b 4
28.g odd 6 1 49.7.b.b 4
28.g odd 6 1 49.7.d.c 4
84.j odd 6 1 63.7.m.b 4
84.j odd 6 1 441.7.d.b 4
84.n even 6 1 441.7.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.d.b 4 4.b odd 2 1
7.7.d.b 4 28.f even 6 1
49.7.b.b 4 28.f even 6 1
49.7.b.b 4 28.g odd 6 1
49.7.d.c 4 28.d even 2 1
49.7.d.c 4 28.g odd 6 1
63.7.m.b 4 12.b even 2 1
63.7.m.b 4 84.j odd 6 1
112.7.s.b 4 1.a even 1 1 trivial
112.7.s.b 4 7.d odd 6 1 inner
441.7.d.b 4 84.j odd 6 1
441.7.d.b 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 18T_{3}^{3} - 879T_{3}^{2} - 17766T_{3} + 974169 \) acting on \(S_{7}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 18 T^{3} - 879 T^{2} + \cdots + 974169 \) Copy content Toggle raw display
$5$ \( T^{4} + 150 T^{3} + \cdots + 172265625 \) Copy content Toggle raw display
$7$ \( T^{4} + 280 T^{3} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{4} + 1882 T^{3} + \cdots + 87487583089 \) Copy content Toggle raw display
$13$ \( T^{4} + 13645128 T^{2} + \cdots + 37734434122896 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 226702498429449 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 718603078673529 \) Copy content Toggle raw display
$23$ \( T^{4} + 2470 T^{3} + \cdots + 44\!\cdots\!29 \) Copy content Toggle raw display
$29$ \( (T^{2} + 17272 T - 154827704)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 611468988954201 \) Copy content Toggle raw display
$37$ \( T^{4} + 43870 T^{3} + \cdots + 50\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( T^{4} + 1071791112 T^{2} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{2} + 160108 T + 6274490716)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 354714 T^{3} + \cdots + 81\!\cdots\!29 \) Copy content Toggle raw display
$53$ \( T^{4} - 104530 T^{3} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{4} - 62646 T^{3} + \cdots + 35\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{4} + 140682 T^{3} + \cdots + 29\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{4} + 432494 T^{3} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{2} - 740764 T + 136232520316)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 971778 T^{3} + \cdots + 56\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{4} + 935990 T^{3} + \cdots + 38\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( T^{4} + 840017980704 T^{2} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{4} + 679290 T^{3} + \cdots + 85\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{4} + 1198740720072 T^{2} + \cdots + 95\!\cdots\!84 \) Copy content Toggle raw display
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