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Properties

Label 147.9.d.b
Level $147$
Weight $9$
Character orbit 147.d
Analytic conductor $59.885$
Analytic rank $0$
Dimension $12$
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] = [147,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.97");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(59.8846556790\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 1331 x^{10} + 1482 x^{9} + 1359381 x^{8} + 1222227 x^{7} + 474169529 x^{6} + \cdots + 556282980265104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{5}\cdot 7^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + 27 \beta_1 q^{3} + (\beta_{4} - \beta_{3} + 187) q^{4} + (\beta_{6} + \beta_{5} + \cdots - 76 \beta_1) q^{5}+ \cdots - 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + 27 \beta_1 q^{3} + (\beta_{4} - \beta_{3} + 187) q^{4} + (\beta_{6} + \beta_{5} + \cdots - 76 \beta_1) q^{5}+ \cdots + ( - 2187 \beta_{11} - 4374 \beta_{9} + \cdots + 6526008) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 2234 q^{4} + 3558 q^{8} - 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} + 2234 q^{4} + 3558 q^{8} - 26244 q^{9} - 36162 q^{11} + 75006 q^{15} + 597698 q^{16} + 13122 q^{18} + 419258 q^{22} + 132768 q^{23} - 1584750 q^{25} + 671106 q^{29} + 1065474 q^{30} - 1790298 q^{32} - 4885758 q^{36} - 4578886 q^{37} + 1861866 q^{39} - 10119958 q^{43} - 28247802 q^{44} + 39853088 q^{46} - 1162248 q^{50} - 3418524 q^{51} + 419022 q^{53} + 25718310 q^{57} - 3877870 q^{58} + 102929454 q^{60} + 272849458 q^{64} - 60881124 q^{65} + 115092114 q^{67} + 224566620 q^{71} - 7781346 q^{72} + 142030800 q^{74} + 102101148 q^{78} + 58628976 q^{79} + 57395628 q^{81} + 248647620 q^{85} - 254536536 q^{86} + 974141794 q^{88} + 364039200 q^{92} - 38730960 q^{93} + 191168976 q^{95} + 79086294 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 1331 x^{10} + 1482 x^{9} + 1359381 x^{8} + 1222227 x^{7} + 474169529 x^{6} + \cdots + 556282980265104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 26\!\cdots\!73 \nu^{11} + \cdots - 53\!\cdots\!06 ) / 37\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!96 \nu^{11} + \cdots - 18\!\cdots\!88 ) / 94\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!96 \nu^{11} + \cdots - 92\!\cdots\!89 ) / 94\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!82 \nu^{11} + \cdots - 11\!\cdots\!13 ) / 31\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!17 \nu^{11} + \cdots - 22\!\cdots\!70 ) / 41\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!24 \nu^{11} + \cdots + 40\!\cdots\!40 ) / 35\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!26 \nu^{11} + \cdots + 13\!\cdots\!76 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!94 \nu^{11} + \cdots - 39\!\cdots\!20 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35\!\cdots\!54 \nu^{11} + \cdots - 15\!\cdots\!40 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!80 \nu^{11} + \cdots + 66\!\cdots\!32 ) / 18\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!46 \nu^{11} + \cdots - 26\!\cdots\!72 ) / 54\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + 3\beta_{3} + 3\beta_{2} - 441\beta _1 - 444 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} - \beta_{4} + 739\beta_{3} - 1737 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{11} + \beta_{10} - \beta_{9} + 41 \beta_{8} - 24 \beta_{7} - 100 \beta_{6} - 1095 \beta_{5} + \cdots - 328904 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 144 \beta_{11} + 1142 \beta_{10} + 1142 \beta_{9} - 1286 \beta_{8} - 432 \beta_{7} - 4568 \beta_{6} + \cdots + 1938965 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11008\beta_{11} + 3506\beta_{9} - 56818\beta_{8} + 1018619\beta_{4} - 4270207\beta_{3} + 287483428 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 213456 \beta_{11} - 1126971 \beta_{10} + 1126971 \beta_{9} - 1427595 \beta_{8} + 640368 \beta_{7} + \cdots + 2132576737 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11877864 \beta_{11} - 5693475 \beta_{10} - 5693475 \beta_{9} + 61213803 \beta_{8} + 35633592 \beta_{7} + \cdots - 264685446720 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 243602496 \beta_{11} - 1077958192 \beta_{9} + 1498799344 \beta_{8} - 5786551156 \beta_{4} + \cdots - 2368714853229 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11967736640 \beta_{11} + 7425404788 \beta_{10} - 7425404788 \beta_{9} + 61682278196 \beta_{8} + \cdots - 248434513822364 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 257272951392 \beta_{11} + 1023600157301 \beta_{10} + 1023600157301 \beta_{9} - 1532015023061 \beta_{8} + \cdots + 26\!\cdots\!41 ) / 2 \) 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
−14.8255 − 25.6786i
−14.8255 + 25.6786i
−7.37103 − 12.7670i
−7.37103 + 12.7670i
−5.65790 − 9.79977i
−5.65790 + 9.79977i
3.88954 + 6.73687i
3.88954 − 6.73687i
9.75345 + 16.8935i
9.75345 − 16.8935i
15.7115 + 27.2131i
15.7115 − 27.2131i
−30.6511 − 46.7654i 683.489 651.152i 1433.41i 0 −13103.0 −2187.00 − 19958.5i
97.2 −30.6511 46.7654i 683.489 − 651.152i − 1433.41i 0 −13103.0 −2187.00 19958.5i
97.3 −15.7421 − 46.7654i −8.18726 − 1007.65i 736.184i 0 4158.85 −2187.00 15862.5i
97.4 −15.7421 46.7654i −8.18726 1007.65i − 736.184i 0 4158.85 −2187.00 − 15862.5i
97.5 −12.3158 − 46.7654i −104.321 692.574i 575.953i 0 4437.64 −2187.00 − 8529.61i
97.6 −12.3158 46.7654i −104.321 − 692.574i − 575.953i 0 4437.64 −2187.00 8529.61i
97.7 6.77907 − 46.7654i −210.044 − 267.490i − 317.026i 0 −3159.35 −2187.00 − 1813.34i
97.8 6.77907 46.7654i −210.044 267.490i 317.026i 0 −3159.35 −2187.00 1813.34i
97.9 18.5069 − 46.7654i 86.5052 − 295.384i − 865.482i 0 −3136.82 −2187.00 − 5466.65i
97.10 18.5069 46.7654i 86.5052 295.384i 865.482i 0 −3136.82 −2187.00 5466.65i
97.11 30.4230 − 46.7654i 669.558 1028.74i − 1422.74i 0 12581.7 −2187.00 31297.3i
97.12 30.4230 46.7654i 669.558 − 1028.74i 1422.74i 0 12581.7 −2187.00 − 31297.3i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
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 147.9.d.b 12
7.b odd 2 1 inner 147.9.d.b 12
7.c even 3 1 21.9.f.b ✓ 12
7.d odd 6 1 21.9.f.b ✓ 12
21.g even 6 1 63.9.m.d 12
21.h odd 6 1 63.9.m.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.f.b ✓ 12 7.c even 3 1
21.9.f.b ✓ 12 7.d odd 6 1
63.9.m.d 12 21.g even 6 1
63.9.m.d 12 21.h odd 6 1
147.9.d.b 12 1.a even 1 1 trivial
147.9.d.b 12 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{5} - 1322T_{2}^{4} - 4056T_{2}^{3} + 387808T_{2}^{2} + 1294464T_{2} - 22681728 \) acting on \(S_{9}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 3 T^{5} + \cdots - 22681728)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2187)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 95\!\cdots\!80)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 60\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 53\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 92\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 61\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 25\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 52\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 39\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 33\!\cdots\!12)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 15\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 88\!\cdots\!09)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
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