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Properties

Label 112.7.s.d
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: \( x^{8} - 2x^{7} + 59x^{6} + 246x^{5} + 2877x^{4} + 3788x^{3} + 5284x^{2} - 816x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{5} - 14 \beta_1 + 28) q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 121) q^{7}+ \cdots + (3 \beta_{7} - \beta_{6} + \cdots + 501 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{5} - 14 \beta_1 + 28) q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 121) q^{7}+ \cdots + ( - 561 \beta_{7} - 1122 \beta_{6} + \cdots - 245475) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 168 q^{5} + 452 q^{7} + 2004 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 168 q^{5} + 452 q^{7} + 2004 q^{9} + 540 q^{11} - 4056 q^{15} + 9912 q^{17} + 10836 q^{19} - 9612 q^{21} + 17832 q^{23} - 5272 q^{25} - 10632 q^{29} - 13524 q^{31} + 144396 q^{33} + 145956 q^{35} + 9164 q^{37} + 205524 q^{39} - 131456 q^{43} + 756252 q^{45} - 30156 q^{47} - 438352 q^{49} + 241860 q^{51} - 537060 q^{53} + 413496 q^{57} - 985992 q^{59} + 773220 q^{61} + 2228964 q^{63} - 266244 q^{65} + 988568 q^{67} - 74256 q^{71} + 2571156 q^{73} - 2230704 q^{75} - 2549880 q^{77} + 458984 q^{79} - 2816928 q^{81} + 1695192 q^{85} - 6079500 q^{87} - 439236 q^{89} + 2774472 q^{91} - 1808604 q^{93} + 3252768 q^{95} - 1963800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 59x^{6} + 246x^{5} + 2877x^{4} + 3788x^{3} + 5284x^{2} - 816x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3026387 \nu^{7} - 5496889 \nu^{6} + 176787043 \nu^{5} + 778541685 \nu^{4} + 8803356393 \nu^{3} + \cdots + 351091620 ) / 2835662628 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8177437 \nu^{7} + 31901728 \nu^{6} + 277489343 \nu^{5} + 5173815732 \nu^{4} + 28650445353 \nu^{3} + \cdots + 9396529260 ) / 2835662628 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6233889 \nu^{7} - 21530430 \nu^{6} + 364668203 \nu^{5} + 1115894738 \nu^{4} + 14197195821 \nu^{3} + \cdots - 6491507568 ) / 1890441752 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8803197 \nu^{7} - 28027038 \nu^{6} + 573607499 \nu^{5} + 1458531314 \nu^{4} + 24601618317 \nu^{3} + \cdots - 6076562928 ) / 1890441752 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39491329 \nu^{7} + 159748301 \nu^{6} - 2530731329 \nu^{5} - 4973834193 \nu^{4} + \cdots + 12664968420 ) / 2835662628 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21222949 \nu^{7} - 48708533 \nu^{6} + 1263861296 \nu^{5} + 4852394565 \nu^{4} + \cdots - 42517817580 ) / 83401842 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1697250521 \nu^{7} + 2898997426 \nu^{6} - 99341319811 \nu^{5} - 446431866870 \nu^{4} + \cdots - 1227770888304 ) / 5671325256 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + 21\beta_1 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 3\beta_{6} - 7\beta_{5} + \beta_{4} - \beta_{3} - 5\beta_{2} + 1197\beta _1 - 1197 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} - 8\beta_{6} - 134\beta_{5} - 106\beta_{4} - 224\beta_{3} + 71\beta_{2} - 5691 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -27\beta_{7} + 9\beta_{6} - 67\beta_{5} + 61\beta_{4} - 34\beta_{3} + 152\beta_{2} - 11163\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 828 \beta_{7} + 1242 \beta_{6} + 5559 \beta_{5} + 13534 \beta_{4} + 7388 \beta_{3} + 4731 \beta_{2} + \cdots + 550431 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4553\beta_{7} + 9106\beta_{6} + 88068\beta_{5} + 32614\beta_{4} + 78887\beta_{3} - 48587\beta_{2} + 5771241 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 108336 \beta_{7} - 36112 \beta_{6} + 370303 \beta_{5} - 526506 \beta_{4} + 418170 \beta_{3} + \cdots + 47061483 \beta_1 ) / 42 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−0.764537 − 1.32422i
4.48234 + 7.76363i
0.0782726 + 0.135572i
−2.79607 − 4.84294i
−0.764537 + 1.32422i
4.48234 − 7.76363i
0.0782726 − 0.135572i
−2.79607 + 4.84294i
0 −39.6171 − 22.8730i 0 142.845 − 82.4715i 0 313.113 − 140.034i 0 681.845 + 1180.99i 0
17.2 0 −10.1301 − 5.84860i 0 −136.498 + 78.8071i 0 −133.428 − 315.984i 0 −296.088 − 512.839i 0
17.3 0 5.11923 + 2.95559i 0 17.9741 − 10.3774i 0 −43.1560 + 340.274i 0 −347.029 − 601.072i 0
17.4 0 44.6280 + 25.7660i 0 59.6790 − 34.4557i 0 89.4711 − 331.125i 0 963.272 + 1668.44i 0
33.1 0 −39.6171 + 22.8730i 0 142.845 + 82.4715i 0 313.113 + 140.034i 0 681.845 − 1180.99i 0
33.2 0 −10.1301 + 5.84860i 0 −136.498 − 78.8071i 0 −133.428 + 315.984i 0 −296.088 + 512.839i 0
33.3 0 5.11923 − 2.95559i 0 17.9741 + 10.3774i 0 −43.1560 − 340.274i 0 −347.029 + 601.072i 0
33.4 0 44.6280 − 25.7660i 0 59.6790 + 34.4557i 0 89.4711 + 331.125i 0 963.272 − 1668.44i 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.d 8
4.b odd 2 1 28.7.h.a ✓ 8
7.d odd 6 1 inner 112.7.s.d 8
12.b even 2 1 252.7.z.c 8
28.d even 2 1 196.7.h.a 8
28.f even 6 1 28.7.h.a ✓ 8
28.f even 6 1 196.7.b.a 8
28.g odd 6 1 196.7.b.a 8
28.g odd 6 1 196.7.h.a 8
84.j odd 6 1 252.7.z.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.7.h.a ✓ 8 4.b odd 2 1
28.7.h.a ✓ 8 28.f even 6 1
112.7.s.d 8 1.a even 1 1 trivial
112.7.s.d 8 7.d odd 6 1 inner
196.7.b.a 8 28.f even 6 1
196.7.b.a 8 28.g odd 6 1
196.7.h.a 8 28.d even 2 1
196.7.h.a 8 28.g odd 6 1
252.7.z.c 8 12.b even 2 1
252.7.z.c 8 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2460T_{3}^{6} + 5888601T_{3}^{4} + 56412720T_{3}^{3} - 225685332T_{3}^{2} - 3737893068T_{3} + 26568674001 \) 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
$3$ \( T^{8} + \cdots + 26568674001 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 81\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 24\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 50\!\cdots\!49 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 15\!\cdots\!72)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 77\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 13\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 38\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 54\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 62\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 84\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 44\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 33\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 23\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
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