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Properties

Label 176.15.h.e
Level $176$
Weight $15$
Character orbit 176.h
Analytic conductor $218.819$
Analytic rank $0$
Dimension $14$
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] = [176,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.65");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(218.818983947\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 38299509 x^{12} + 1255603312 x^{11} + 548839279225666 x^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{64}\cdot 3^{10}\cdot 11^{7} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 314) q^{3} + ( - \beta_{2} + 4983) q^{5} + ( - \beta_{5} - 11 \beta_{4}) q^{7} + (\beta_{3} - 19 \beta_{2} + \cdots + 787001) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 314) q^{3} + ( - \beta_{2} + 4983) q^{5} + ( - \beta_{5} - 11 \beta_{4}) q^{7} + (\beta_{3} - 19 \beta_{2} + \cdots + 787001) q^{9}+ \cdots + ( - 1063506 \beta_{13} + \cdots - 8489273865276) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4394 q^{3} + 69758 q^{5} + 11016572 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4394 q^{3} + 69758 q^{5} + 11016572 q^{9} - 20143042 q^{11} + 1359602 q^{15} + 7305755542 q^{23} + 19291879452 q^{25} - 34093422830 q^{27} + 33569873942 q^{31} + 2885838062 q^{33} + 73167823966 q^{37} + 2000205168616 q^{45} + 1612717386124 q^{47} + 3424602524990 q^{49} - 3530064068164 q^{53} + 3715439610854 q^{55} + 818496564070 q^{59} - 16485465276922 q^{67} - 11394452631206 q^{69} + 19380879179878 q^{71} - 23016770893992 q^{75} + 60534793808304 q^{77} - 10394309810662 q^{81} - 117770741987650 q^{89} - 150621364097712 q^{91} + 27345122803162 q^{93} + 123398138843566 q^{97} - 118861332531788 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 38299509 x^{12} + 1255603312 x^{11} + 548839279225666 x^{10} + \cdots + 61\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 55\!\cdots\!75 \nu^{13} + \cdots + 60\!\cdots\!40 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!14 \nu^{13} + \cdots + 14\!\cdots\!20 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!50 \nu^{13} + \cdots - 57\!\cdots\!00 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!50 \nu^{13} + \cdots - 12\!\cdots\!80 ) / 73\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!89 \nu^{13} + \cdots - 12\!\cdots\!00 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!52 \nu^{13} + \cdots - 16\!\cdots\!60 ) / 73\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!47 \nu^{13} + \cdots - 46\!\cdots\!20 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21\!\cdots\!97 \nu^{13} + \cdots + 16\!\cdots\!40 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!47 \nu^{13} + \cdots + 12\!\cdots\!20 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 67\!\cdots\!23 \nu^{13} + \cdots + 84\!\cdots\!80 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37\!\cdots\!27 \nu^{13} + \cdots + 63\!\cdots\!00 ) / 69\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17\!\cdots\!53 \nu^{13} + \cdots - 17\!\cdots\!80 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 77\!\cdots\!33 \nu^{13} + \cdots - 14\!\cdots\!00 ) / 11\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 3072\beta_1 ) / 3072 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{6} + \beta_{4} + 1536\beta_{3} - 29184\beta_{2} - 79872\beta _1 + 8404027392 ) / 1536 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 26040 \beta_{13} - 25344 \beta_{12} + 114360 \beta_{11} - 445440 \beta_{10} + 516864 \beta_{9} + \cdots - 780383455560 ) / 3072 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5569472 \beta_{13} + 5876992 \beta_{12} - 188982208 \beta_{11} - 14635776 \beta_{10} + \cdots + 40\!\cdots\!96 ) / 768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 828534961192 \beta_{13} - 804757068032 \beta_{12} + 1752015481640 \beta_{11} + \cdots + 17\!\cdots\!96 ) / 3072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 26530745839232 \beta_{13} + 36533233598464 \beta_{12} + \cdots + 89\!\cdots\!28 ) / 1536 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13\!\cdots\!04 \beta_{13} + \cdots + 56\!\cdots\!92 ) / 3072 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 13\!\cdots\!56 \beta_{13} + \cdots + 25\!\cdots\!00 ) / 384 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 18\!\cdots\!60 \beta_{13} + \cdots + 10\!\cdots\!68 ) / 3072 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16\!\cdots\!20 \beta_{13} + \cdots + 12\!\cdots\!64 ) / 1536 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 24\!\cdots\!12 \beta_{13} + \cdots + 17\!\cdots\!12 ) / 3072 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 16\!\cdots\!60 \beta_{13} + \cdots + 74\!\cdots\!32 ) / 768 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 30\!\cdots\!76 \beta_{13} + \cdots + 26\!\cdots\!64 ) / 3072 \) 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
−3417.25 + 1.41421i
−3417.25 − 1.41421i
−2599.55 − 1.41421i
−2599.55 + 1.41421i
−1087.11 − 1.41421i
−1087.11 + 1.41421i
320.191 + 1.41421i
320.191 − 1.41421i
1021.72 − 1.41421i
1021.72 + 1.41421i
2199.62 + 1.41421i
2199.62 − 1.41421i
3563.38 + 1.41421i
3563.38 − 1.41421i
0 −3731.25 0 97060.9 0 − 769857.i 0 9.13929e6 0
65.2 0 −3731.25 0 97060.9 0 769857.i 0 9.13929e6 0
65.3 0 −2913.55 0 −75323.6 0 − 867659.i 0 3.70579e6 0
65.4 0 −2913.55 0 −75323.6 0 867659.i 0 3.70579e6 0
65.5 0 −1401.11 0 44552.9 0 − 500199.i 0 −2.81985e6 0
65.6 0 −1401.11 0 44552.9 0 500199.i 0 −2.81985e6 0
65.7 0 6.19137 0 −140247. 0 − 1.02360e6i 0 −4.78293e6 0
65.8 0 6.19137 0 −140247. 0 1.02360e6i 0 −4.78293e6 0
65.9 0 707.723 0 −8968.27 0 − 531246.i 0 −4.28210e6 0
65.10 0 707.723 0 −8968.27 0 531246.i 0 −4.28210e6 0
65.11 0 1885.62 0 124487. 0 − 166636.i 0 −1.22739e6 0
65.12 0 1885.62 0 124487. 0 166636.i 0 −1.22739e6 0
65.13 0 3249.38 0 −6683.13 0 − 286020.i 0 5.77548e6 0
65.14 0 3249.38 0 −6683.13 0 286020.i 0 5.77548e6 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.14
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 176.15.h.e 14
4.b odd 2 1 22.15.b.a ✓ 14
11.b odd 2 1 inner 176.15.h.e 14
44.c even 2 1 22.15.b.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.15.b.a ✓ 14 4.b odd 2 1
22.15.b.a ✓ 14 44.c even 2 1
176.15.h.e 14 1.a even 1 1 trivial
176.15.h.e 14 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 2197 T_{3}^{6} - 17081130 T_{3}^{5} - 28374383334 T_{3}^{4} + 73286898396561 T_{3}^{3} + \cdots + 40\!\cdots\!16 \) acting on \(S_{15}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{7} + \cdots + 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{7} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 74\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 25\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 28\!\cdots\!92)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 27\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 20\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 63\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 47\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 59\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 29\!\cdots\!08)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 77\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots + 72\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 96\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 62\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 23\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 43\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 97\!\cdots\!16)^{2} \) Copy content Toggle raw display
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