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Properties

Label 147.7.d.a
Level $147$
Weight $7$
Character orbit 147.d
Analytic conductor $33.818$
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] = [147,7,Mod(97,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("147.97");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.d (of order \(2\), degree \(1\), 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: \(33.8179502921\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} - 787x^{5} + 38792x^{4} - 92833x^{3} + 1563109x^{2} + 3107772x + 38787984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} - 9 \beta_1 q^{3} + ( - \beta_{5} + 43) q^{4} + ( - \beta_{6} - \beta_{4} - 3 \beta_1) q^{5} + (9 \beta_{2} - 9 \beta_1) q^{6} + ( - 2 \beta_{7} + 3 \beta_{5} + \cdots - 61) q^{8}+ \cdots - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} - 9 \beta_1 q^{3} + ( - \beta_{5} + 43) q^{4} + ( - \beta_{6} - \beta_{4} - 3 \beta_1) q^{5} + (9 \beta_{2} - 9 \beta_1) q^{6} + ( - 2 \beta_{7} + 3 \beta_{5} + \cdots - 61) q^{8}+ \cdots + (243 \beta_{7} + 2187 \beta_{5} + \cdots - 64395) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{2} + 346 q^{4} - 454 q^{8} - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{2} + 346 q^{4} - 454 q^{8} - 1944 q^{9} + 2140 q^{11} - 756 q^{15} - 7822 q^{16} - 2430 q^{18} - 78 q^{22} + 30448 q^{23} - 44548 q^{25} + 32524 q^{29} - 4698 q^{30} - 140406 q^{32} - 84078 q^{36} + 91340 q^{37} - 186732 q^{39} - 445660 q^{43} + 377658 q^{44} - 1051608 q^{46} - 1218884 q^{50} - 129816 q^{51} + 26068 q^{53} - 442908 q^{57} + 319002 q^{58} - 859410 q^{60} - 1410446 q^{64} + 778008 q^{65} - 768188 q^{67} + 225688 q^{71} + 110322 q^{72} - 2371060 q^{74} - 342792 q^{78} + 1119184 q^{79} + 472392 q^{81} + 1953576 q^{85} + 4604804 q^{86} - 609774 q^{88} - 113064 q^{92} - 723600 q^{93} - 2320224 q^{95} - 520020 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} - 787x^{5} + 38792x^{4} - 92833x^{3} + 1563109x^{2} + 3107772x + 38787984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1585013359 \nu^{7} - 18232571539 \nu^{6} + 303603349712 \nu^{5} - 4245938445433 \nu^{4} + \cdots - 10\!\cdots\!50 ) / 11\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8019055 \nu^{7} + 15616321 \nu^{6} + 1444380025 \nu^{5} - 2053828205 \nu^{4} + \cdots + 29614389599556 ) / 11465622241311 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8019055 \nu^{7} - 15616321 \nu^{6} - 1444380025 \nu^{5} + 2053828205 \nu^{4} + \cdots - 29614389599556 ) / 11465622241311 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27322708193 \nu^{7} + 1000103862583 \nu^{6} + 22563530233273 \nu^{5} + \cdots + 69\!\cdots\!80 ) / 11\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 39673486 \nu^{7} - 224426993 \nu^{6} + 7145928130 \nu^{5} - 10161113066 \nu^{4} + \cdots + 963541759742958 ) / 11465622241311 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 80062726973 \nu^{7} + 817429199804 \nu^{6} - 15380773767166 \nu^{5} + \cdots + 35\!\cdots\!42 ) / 59\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 313586723 \nu^{7} - 1214053715 \nu^{6} - 56482764965 \nu^{5} + 80315355913 \nu^{4} + \cdots - 487133839162452 ) / 7643748160874 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 2\beta_{3} - 2\beta_{2} - 106\beta _1 - 106 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 6\beta_{5} - 147\beta_{3} + 252 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} + 205\beta_{6} + 205\beta_{5} + 6\beta_{4} + 774\beta_{3} + 776\beta_{2} + 15886\beta _1 - 15886 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 424 \beta_{7} - 1560 \beta_{6} + 1560 \beta_{5} + 1272 \beta_{4} + 25013 \beta_{3} - 24589 \beta_{2} + \cdots - 90180 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 152\beta_{7} - 38461\beta_{5} - 199198\beta_{3} + 2727346 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 75858 \beta_{7} + 355626 \beta_{6} + 355626 \beta_{5} - 227574 \beta_{4} + 4549255 \beta_{3} + \cdots - 22658292 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−7.08935 + 12.2791i
−7.08935 − 12.2791i
−2.30325 + 3.98935i
−2.30325 − 3.98935i
4.15432 − 7.19549i
4.15432 + 7.19549i
5.73828 − 9.93899i
5.73828 + 9.93899i
−13.1787 − 15.5885i 109.678 − 79.5668i 205.435i 0 −601.976 −243.000 1048.59i
97.2 −13.1787 15.5885i 109.678 79.5668i − 205.435i 0 −601.976 −243.000 − 1048.59i
97.3 −3.60650 − 15.5885i −50.9932 83.0589i 56.2198i 0 414.723 −243.000 − 299.552i
97.4 −3.60650 15.5885i −50.9932 − 83.0589i − 56.2198i 0 414.723 −243.000 299.552i
97.5 9.30863 − 15.5885i 22.6506 174.756i − 145.107i 0 −384.906 −243.000 1626.74i
97.6 9.30863 15.5885i 22.6506 − 174.756i 145.107i 0 −384.906 −243.000 − 1626.74i
97.7 12.4766 − 15.5885i 91.6646 − 202.496i − 194.490i 0 345.159 −243.000 − 2526.46i
97.8 12.4766 15.5885i 91.6646 202.496i 194.490i 0 345.159 −243.000 2526.46i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.7.d.a 8
3.b odd 2 1 441.7.d.d 8
7.b odd 2 1 inner 147.7.d.a 8
7.c even 3 1 21.7.f.b ✓ 8
7.c even 3 1 147.7.f.a 8
7.d odd 6 1 21.7.f.b ✓ 8
7.d odd 6 1 147.7.f.a 8
21.c even 2 1 441.7.d.d 8
21.g even 6 1 63.7.m.c 8
21.h odd 6 1 63.7.m.c 8
28.f even 6 1 336.7.bh.b 8
28.g odd 6 1 336.7.bh.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b ✓ 8 7.c even 3 1
21.7.f.b ✓ 8 7.d odd 6 1
63.7.m.c 8 21.g even 6 1
63.7.m.c 8 21.h odd 6 1
147.7.d.a 8 1.a even 1 1 trivial
147.7.d.a 8 7.b odd 2 1 inner
147.7.f.a 8 7.c even 3 1
147.7.f.a 8 7.d odd 6 1
336.7.bh.b 8 28.f even 6 1
336.7.bh.b 8 28.g odd 6 1
441.7.d.d 8 3.b odd 2 1
441.7.d.d 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{3} - 202T_{2}^{2} + 914T_{2} + 5520 \) acting on \(S_{7}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 5 T^{3} + \cdots + 5520)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 1070 T^{3} + \cdots + 59090060580)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 42\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 96\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 25\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 76\!\cdots\!35)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
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