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Properties

Label 112.11.s.a
Level $112$
Weight $11$
Character orbit 112.s
Analytic conductor $71.160$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.17");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(71.1600122995\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 7168 x^{10} - 191104 x^{9} + 39872585 x^{8} - 837614684 x^{7} + 83400850488 x^{6} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{7}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3} + 27 \beta_1 - 54) q^{3} + ( - \beta_{9} - \beta_{7} + \beta_{6} + \cdots - 371) q^{5}+ \cdots + (6 \beta_{11} - 3 \beta_{10} + \cdots - 7594) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{3} + 27 \beta_1 - 54) q^{3} + ( - \beta_{9} - \beta_{7} + \beta_{6} + \cdots - 371) q^{5}+ \cdots + ( - 816957 \beta_{11} + \cdots + 1101162600) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 486 q^{3} - 6666 q^{5} - 30576 q^{7} - 45528 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 486 q^{3} - 6666 q^{5} - 30576 q^{7} - 45528 q^{9} + 111210 q^{11} - 1065684 q^{15} + 1439502 q^{17} + 452814 q^{19} - 2940462 q^{21} - 853074 q^{23} + 16905804 q^{25} + 60157248 q^{29} - 87231186 q^{31} - 303597198 q^{33} - 153795138 q^{35} - 7666506 q^{37} + 71628084 q^{39} - 1066803336 q^{43} - 948611736 q^{45} + 985909398 q^{47} + 456183924 q^{49} + 319431390 q^{51} - 600022554 q^{53} + 3357095652 q^{57} + 2101762050 q^{59} - 2201391150 q^{61} - 2691345552 q^{63} + 1536128076 q^{65} + 1590058326 q^{67} + 7739561160 q^{71} + 2008593834 q^{73} - 12301086492 q^{75} + 464655282 q^{77} - 2310562242 q^{79} + 7562341422 q^{81} - 39581743596 q^{85} - 24592790952 q^{87} + 2541648690 q^{89} + 4866473640 q^{91} - 7836264270 q^{93} - 29024266590 q^{95} + 13213951200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 7168 x^{10} - 191104 x^{9} + 39872585 x^{8} - 837614684 x^{7} + 83400850488 x^{6} + \cdots + 16\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 59\!\cdots\!91 \nu^{11} + \cdots + 20\!\cdots\!92 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!80 \nu^{11} + \cdots - 53\!\cdots\!16 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 84\!\cdots\!23 \nu^{11} + \cdots + 99\!\cdots\!36 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50\!\cdots\!44 \nu^{11} + \cdots - 50\!\cdots\!08 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!99 \nu^{11} + \cdots + 12\!\cdots\!04 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 47\!\cdots\!35 \nu^{11} + \cdots - 41\!\cdots\!56 ) / 51\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 65\!\cdots\!12 \nu^{11} + \cdots - 10\!\cdots\!88 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 76\!\cdots\!68 \nu^{11} + \cdots - 16\!\cdots\!16 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!65 \nu^{11} + \cdots + 97\!\cdots\!84 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50\!\cdots\!28 \nu^{11} + \cdots + 53\!\cdots\!40 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21\!\cdots\!09 \nu^{11} + \cdots + 13\!\cdots\!08 ) / 40\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -16\beta_{11} + 8\beta_{10} - 27\beta_{5} + 109\beta_{4} + 245\beta_{3} - 27\beta_{2} ) / 1344 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 64 \beta_{11} + 64 \beta_{10} - 20 \beta_{9} - 10 \beta_{8} + 477 \beta_{7} + 430 \beta_{5} + \cdots - 20 ) / 336 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 35432 \beta_{11} - 70864 \beta_{10} + 6848 \beta_{9} - 6848 \beta_{8} + 59776 \beta_{6} + \cdots + 64210944 ) / 1344 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1990928 \beta_{11} + 995464 \beta_{10} + 381860 \beta_{9} + 763720 \beta_{8} - 5364178 \beta_{7} + \cdots - 6352984184 ) / 672 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 191169672 \beta_{11} + 191169672 \beta_{10} - 125806880 \beta_{9} - 62903440 \beta_{8} + \cdots - 125806880 ) / 1344 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3372257800 \beta_{11} - 6744515600 \beta_{10} + 1417094874 \beta_{9} - 1417094874 \beta_{8} + \cdots + 16447490022208 ) / 336 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2293152767376 \beta_{11} + 1146576383688 \beta_{10} + 438954601568 \beta_{9} + \cdots - 42\!\cdots\!72 ) / 1344 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 43843202670360 \beta_{11} + 43843202670360 \beta_{10} - 36971525932584 \beta_{9} + \cdots - 36971525932584 ) / 672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 71\!\cdots\!20 \beta_{11} + \cdots + 28\!\cdots\!80 ) / 1344 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 28\!\cdots\!16 \beta_{11} + \cdots - 59\!\cdots\!24 ) / 336 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 45\!\cdots\!16 \beta_{11} + \cdots - 36\!\cdots\!44 ) / 1344 \) Copy content Toggle raw display

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

Character values

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

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

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
−7.35708 − 12.7428i
−39.8637 − 69.0460i
−20.0351 − 34.7018i
21.7454 + 37.6641i
26.6851 + 46.2199i
18.8254 + 32.6066i
−7.35708 + 12.7428i
−39.8637 + 69.0460i
−20.0351 + 34.7018i
21.7454 − 37.6641i
26.6851 − 46.2199i
18.8254 − 32.6066i
0 −251.837 − 145.398i 0 −4757.26 + 2746.61i 0 −15953.8 + 5286.95i 0 12756.6 + 22095.0i 0
17.2 0 −181.381 − 104.720i 0 −1221.89 + 705.456i 0 16732.6 + 1579.95i 0 −7591.90 − 13149.6i 0
17.3 0 −173.391 − 100.107i 0 4505.30 − 2601.14i 0 −7983.44 − 14789.9i 0 −9481.60 − 16422.6i 0
17.4 0 −43.3342 − 25.0190i 0 1336.07 − 771.378i 0 −14974.8 − 7630.84i 0 −28272.6 − 48969.6i 0
17.5 0 102.202 + 59.0063i 0 32.1973 − 18.5891i 0 11036.6 + 12675.5i 0 −22561.0 − 39076.8i 0
17.6 0 304.740 + 175.942i 0 −3227.42 + 1863.35i 0 −4145.11 + 16287.8i 0 32386.5 + 56095.1i 0
33.1 0 −251.837 + 145.398i 0 −4757.26 − 2746.61i 0 −15953.8 − 5286.95i 0 12756.6 − 22095.0i 0
33.2 0 −181.381 + 104.720i 0 −1221.89 − 705.456i 0 16732.6 − 1579.95i 0 −7591.90 + 13149.6i 0
33.3 0 −173.391 + 100.107i 0 4505.30 + 2601.14i 0 −7983.44 + 14789.9i 0 −9481.60 + 16422.6i 0
33.4 0 −43.3342 + 25.0190i 0 1336.07 + 771.378i 0 −14974.8 + 7630.84i 0 −28272.6 + 48969.6i 0
33.5 0 102.202 − 59.0063i 0 32.1973 + 18.5891i 0 11036.6 − 12675.5i 0 −22561.0 + 39076.8i 0
33.6 0 304.740 − 175.942i 0 −3227.42 − 1863.35i 0 −4145.11 − 16287.8i 0 32386.5 − 56095.1i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
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 112.11.s.a 12
4.b odd 2 1 14.11.d.a ✓ 12
7.d odd 6 1 inner 112.11.s.a 12
12.b even 2 1 126.11.n.b 12
28.d even 2 1 98.11.d.a 12
28.f even 6 1 14.11.d.a ✓ 12
28.f even 6 1 98.11.b.c 12
28.g odd 6 1 98.11.b.c 12
28.g odd 6 1 98.11.d.a 12
84.j odd 6 1 126.11.n.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.11.d.a ✓ 12 4.b odd 2 1
14.11.d.a ✓ 12 28.f even 6 1
98.11.b.c 12 28.f even 6 1
98.11.b.c 12 28.g odd 6 1
98.11.d.a 12 28.d even 2 1
98.11.d.a 12 28.g odd 6 1
112.11.s.a 12 1.a even 1 1 trivial
112.11.s.a 12 7.d odd 6 1 inner
126.11.n.b 12 12.b even 2 1
126.11.n.b 12 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 486 T_{3}^{11} - 36285 T_{3}^{10} - 55898262 T_{3}^{9} + 2562735438 T_{3}^{8} + \cdots + 64\!\cdots\!29 \) acting on \(S_{11}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 64\!\cdots\!29 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 50\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 29\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 63\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 17\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 34\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 38\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 82\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 36\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
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