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Properties

Label 847.4.a.m
Level $847$
Weight $4$
Character orbit 847.a
Self dual yes
Analytic conductor $49.975$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [847,4,Mod(1,847)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("847.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(49.9746177749\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 67 x^{12} + 380 x^{11} + 1774 x^{10} - 8872 x^{9} - 23849 x^{8} + 93880 x^{7} + \cdots - 1952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + \beta_{5} q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{10} - \beta_{6}) q^{5} + (\beta_{12} + \beta_{10} + \beta_{9} + \cdots - 2) q^{6} + 7 q^{7} + (\beta_{10} + \beta_{8} + \beta_{6} + \cdots - 9) q^{8}+ \cdots + (49 \beta_1 - 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 8 q^{2} + 4 q^{3} + 60 q^{4} + 6 q^{5} - 34 q^{6} + 98 q^{7} - 96 q^{8} + 58 q^{9} - 4 q^{10} - 14 q^{12} - 164 q^{13} - 56 q^{14} - 240 q^{15} + 356 q^{16} - 276 q^{17} - 516 q^{18} + 12 q^{19}+ \cdots - 392 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 67 x^{12} + 380 x^{11} + 1774 x^{10} - 8872 x^{9} - 23849 x^{8} + 93880 x^{7} + \cdots - 1952 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 207518521 \nu^{13} - 102387013460 \nu^{12} + 714379018579 \nu^{11} + \cdots + 76\!\cdots\!56 ) / 228191618192384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 176241 \nu^{13} + 4768704 \nu^{12} - 582621 \nu^{11} - 360437770 \nu^{10} + \cdots + 110342724816 ) / 76114615808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1412398839 \nu^{13} + 12633753668 \nu^{12} - 191397229933 \nu^{11} - 963537634950 \nu^{10} + \cdots + 16\!\cdots\!48 ) / 228191618192384 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1855629 \nu^{13} + 6195384 \nu^{12} + 146733095 \nu^{11} - 362342442 \nu^{10} + \cdots + 172011216 ) / 38057307904 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2668284117 \nu^{13} + 1648551052 \nu^{12} - 262371039527 \nu^{11} - 219440401986 \nu^{10} + \cdots + 213918763787792 ) / 20744692562944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39696286779 \nu^{13} + 224446809540 \nu^{12} + 2779858306041 \nu^{11} + \cdots + 46\!\cdots\!92 ) / 228191618192384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 46973032437 \nu^{13} - 264496584324 \nu^{12} - 3136525559239 \nu^{11} + \cdots - 982332880695920 ) / 228191618192384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 54175806459 \nu^{13} - 250623399260 \nu^{12} - 4040717705769 \nu^{11} + \cdots - 15\!\cdots\!00 ) / 228191618192384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14378731329 \nu^{13} + 64091431896 \nu^{12} + 1045181396911 \nu^{11} + \cdots - 828151703725200 ) / 57047904548096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38872132517 \nu^{13} + 228958778896 \nu^{12} + 2646226456807 \nu^{11} + \cdots + 19\!\cdots\!36 ) / 114095809096192 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 294401036709 \nu^{13} - 1348194208532 \nu^{12} - 21237803013751 \nu^{11} + \cdots - 11\!\cdots\!04 ) / 228191618192384 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{2} + 21\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} + \cdots + 254 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 37 \beta_{10} - 14 \beta_{9} + 34 \beta_{8} + 7 \beta_{7} + \cdots + 546 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{13} - 81 \beta_{12} + 61 \beta_{11} + 59 \beta_{10} - 107 \beta_{9} + 101 \beta_{8} + \cdots + 6526 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 119 \beta_{13} - 177 \beta_{12} + 242 \beta_{11} + 1076 \beta_{10} - 753 \beta_{9} + 1066 \beta_{8} + \cdots + 20626 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 586 \beta_{13} - 2794 \beta_{12} + 2332 \beta_{11} + 2222 \beta_{10} - 4500 \beta_{9} + 4068 \beta_{8} + \cdots + 185156 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5216 \beta_{13} - 9664 \beta_{12} + 9864 \beta_{11} + 28245 \beta_{10} - 30708 \beta_{9} + 33883 \beta_{8} + \cdots + 734508 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 27570 \beta_{13} - 97904 \beta_{12} + 76905 \beta_{11} + 65223 \beta_{10} - 174924 \beta_{9} + \cdots + 5577966 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 206880 \beta_{13} - 441936 \beta_{12} + 344202 \beta_{11} + 673679 \beta_{10} - 1153580 \beta_{9} + \cdots + 25497414 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1146651 \beta_{13} - 3554831 \beta_{12} + 2377717 \beta_{11} + 1495409 \beta_{10} - 6567473 \beta_{9} + \cdots + 174811338 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7859267 \beta_{13} - 18548169 \beta_{12} + 11099276 \beta_{11} + 13752770 \beta_{10} - 42068925 \beta_{9} + \cdots + 874205358 \) Copy content Toggle raw display

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.61440
−4.41395
−3.06738
−2.77106
−1.89225
−1.17862
−0.0148533
0.136463
2.16256
2.21504
3.66175
4.40341
5.47367
5.89963
−5.61440 2.60717 23.5215 15.0474 −14.6377 7.00000 −87.1440 −20.2027 −84.4819
1.2 −5.41395 8.77003 21.3108 −13.8023 −47.4805 7.00000 −72.0641 49.9135 74.7248
1.3 −4.06738 −6.15905 8.54358 −5.80582 25.0512 7.00000 −2.21096 10.9339 23.6145
1.4 −3.77106 −10.1713 6.22092 7.94861 38.3564 7.00000 6.70904 76.4544 −29.9747
1.5 −2.89225 −0.970502 0.365134 −11.4090 2.80694 7.00000 22.0820 −26.0581 32.9978
1.6 −2.17862 3.68217 −3.25361 13.1297 −8.02205 7.00000 24.5174 −13.4417 −28.6047
1.7 −1.01485 7.71197 −6.97007 9.89975 −7.82652 7.00000 15.1924 32.4745 −10.0468
1.8 −0.863537 3.33740 −7.25430 −21.5184 −2.88197 7.00000 13.1727 −15.8618 18.5820
1.9 1.16256 −4.41996 −6.64845 8.78946 −5.13848 7.00000 −17.0297 −7.46391 10.2183
1.10 1.21504 1.52687 −6.52367 2.62996 1.85521 7.00000 −17.6469 −24.6687 3.19552
1.11 2.66175 6.90109 −0.915080 −3.45427 18.3690 7.00000 −23.7297 20.6250 −9.19441
1.12 3.40341 −6.23211 3.58318 21.0508 −21.2104 7.00000 −15.0322 11.8392 71.6443
1.13 4.47367 1.36499 12.0137 −9.85366 6.10653 7.00000 17.9559 −25.1368 −44.0820
1.14 4.89963 −3.94881 16.0064 −6.65208 −19.3477 7.00000 39.2282 −11.4069 −32.5927
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.4.a.m ✓ 14
11.b odd 2 1 847.4.a.n yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.4.a.m ✓ 14 1.a even 1 1 trivial
847.4.a.n yes 14 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 8 T_{2}^{13} - 54 T_{2}^{12} - 528 T_{2}^{11} + 817 T_{2}^{10} + 12740 T_{2}^{9} + \cdots + 722128 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 8 T^{13} + \cdots + 722128 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 206111908 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 33697424911244 \) Copy content Toggle raw display
$7$ \( (T - 7)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 70\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 81\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 36\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 70\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 11\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 21\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 16\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 13\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 49\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 53\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 21\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 17\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
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