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Properties

Label 176.11.h.e
Level $176$
Weight $11$
Character orbit 176.h
Analytic conductor $111.823$
Analytic rank $0$
Dimension $10$
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] = [176,11,Mod(65,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(2))
 
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("176.65");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 176.h (of order \(2\), degree \(1\), 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: \(111.822876471\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 135903 x^{8} - 6427236 x^{7} + 6935435151 x^{6} + 631292713590 x^{5} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{5}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 11) q^{3} + ( - \beta_{6} - 2 \beta_1 + 113) q^{5} + ( - \beta_{5} + \beta_{4}) q^{7} + (3 \beta_{6} - \beta_{2} - 74 \beta_1 + 7775) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 11) q^{3} + ( - \beta_{6} - 2 \beta_1 + 113) q^{5} + ( - \beta_{5} + \beta_{4}) q^{7} + (3 \beta_{6} - \beta_{2} - 74 \beta_1 + 7775) q^{9} + ( - \beta_{9} + 2 \beta_{8} + \cdots - 9550) q^{11}+ \cdots + (1569 \beta_{9} + 21326 \beta_{8} + \cdots - 4881790889) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 106 q^{3} + 1138 q^{5} + 78044 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 106 q^{3} + 1138 q^{5} + 78044 q^{9} - 95414 q^{11} - 1441618 q^{15} - 17496838 q^{23} - 1494468 q^{25} - 54656930 q^{27} + 91050970 q^{31} - 12170158 q^{33} - 82676974 q^{37} - 124619384 q^{45} - 352507996 q^{47} - 374605478 q^{49} + 571129876 q^{53} - 1363103126 q^{55} + 1508647610 q^{59} - 3146811782 q^{67} + 5332296166 q^{69} + 328577450 q^{71} + 18684358968 q^{75} + 4256837904 q^{77} - 16957790722 q^{81} + 17791426978 q^{89} - 40311734544 q^{91} - 11674310138 q^{93} - 62585189614 q^{97} - 48880194572 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 135903 x^{8} - 6427236 x^{7} + 6935435151 x^{6} + 631292713590 x^{5} + \cdots + 88\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33\!\cdots\!54 \nu^{9} + \cdots + 23\!\cdots\!76 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 80\!\cdots\!66 \nu^{9} + \cdots + 57\!\cdots\!44 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!57 \nu^{9} + \cdots - 27\!\cdots\!48 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!08 \nu^{9} + \cdots - 13\!\cdots\!92 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 41\!\cdots\!49 \nu^{9} + \cdots - 21\!\cdots\!76 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 80\!\cdots\!14 \nu^{9} + \cdots - 51\!\cdots\!16 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81\!\cdots\!62 \nu^{9} + \cdots + 52\!\cdots\!84 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!99 \nu^{9} + \cdots - 66\!\cdots\!16 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!87 \nu^{9} + \cdots + 34\!\cdots\!08 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -11\beta_{4} + 32\beta_{2} - 1408\beta _1 + 1120 ) / 8448 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 32 \beta_{9} + 16 \beta_{8} + 976 \beta_{7} + 3664 \beta_{6} - 9 \beta_{4} + 1104 \beta_{2} + \cdots + 114786288 ) / 4224 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6624 \beta_{9} + 3216 \beta_{8} + 456432 \beta_{7} + 2164464 \beta_{6} - 11568 \beta_{5} + \cdots + 16951237728 ) / 8448 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1031840 \beta_{9} + 597072 \beta_{8} + 39744112 \beta_{7} + 163731760 \beta_{6} + \cdots + 1952753761080 ) / 2112 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 703337760 \beta_{9} + 401646000 \beta_{8} + 37592826768 \beta_{7} + 171769116048 \beta_{6} + \cdots + 11\!\cdots\!56 ) / 8448 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 121573226400 \beta_{9} + 77044485008 \beta_{8} + 5043567262992 \beta_{7} + 21653256530064 \beta_{6} + \cdots + 16\!\cdots\!88 ) / 4224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 52027027557792 \beta_{9} + 32928506633904 \beta_{8} + \cdots + 60\!\cdots\!16 ) / 8448 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 18\!\cdots\!12 \beta_{9} + \cdots + 18\!\cdots\!36 ) / 1056 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 34\!\cdots\!04 \beta_{9} + \cdots + 31\!\cdots\!44 ) / 8448 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−126.950 + 1.41421i
−126.950 − 1.41421i
230.003 − 1.41421i
230.003 + 1.41421i
171.602 + 1.41421i
171.602 − 1.41421i
−138.673 − 1.41421i
−138.673 + 1.41421i
−134.982 + 1.41421i
−134.982 − 1.41421i
0 −380.412 0 1511.55 0 − 14680.5i 0 85664.2 0
65.2 0 −380.412 0 1511.55 0 14680.5i 0 85664.2 0
65.3 0 −189.131 0 −1492.29 0 − 21897.3i 0 −23278.6 0
65.4 0 −189.131 0 −1492.29 0 21897.3i 0 −23278.6 0
65.5 0 81.3085 0 2188.57 0 − 24594.3i 0 −52437.9 0
65.6 0 81.3085 0 2188.57 0 24594.3i 0 −52437.9 0
65.7 0 251.838 0 3507.04 0 − 17033.0i 0 4373.51 0
65.8 0 251.838 0 3507.04 0 17033.0i 0 4373.51 0
65.9 0 289.396 0 −5145.87 0 − 3108.94i 0 24700.9 0
65.10 0 289.396 0 −5145.87 0 3108.94i 0 24700.9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.11.h.e 10
4.b odd 2 1 22.11.b.a ✓ 10
11.b odd 2 1 inner 176.11.h.e 10
12.b even 2 1 198.11.d.a 10
44.c even 2 1 22.11.b.a ✓ 10
132.d odd 2 1 198.11.d.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.11.b.a ✓ 10 4.b odd 2 1
22.11.b.a ✓ 10 44.c even 2 1
176.11.h.e 10 1.a even 1 1 trivial
176.11.h.e 10 11.b odd 2 1 inner
198.11.d.a 10 12.b even 2 1
198.11.d.a 10 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 53T_{3}^{4} - 165729T_{3}^{3} + 15856353T_{3}^{2} + 5034774672T_{3} - 426349422396 \) acting on \(S_{11}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{5} - 53 T^{4} + \cdots - 426349422396)^{2} \) Copy content Toggle raw display
$5$ \( (T^{5} + \cdots - 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 17\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 26\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 23\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 33\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 37\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 44\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 36\!\cdots\!92)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 28\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots - 22\!\cdots\!32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 22\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 11\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 21\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 57\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 67\!\cdots\!44)^{2} \) Copy content Toggle raw display
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