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Properties

Label 847.6.a.m
Level $847$
Weight $6$
Character orbit 847.a
Self dual yes
Analytic conductor $135.845$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,6,Mod(1,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

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: yes
Analytic conductor: \(135.845095382\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 316 x^{11} + 1264 x^{10} + 38184 x^{9} - 118956 x^{8} - 2174258 x^{7} + \cdots - 544885000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{5} + \beta_1) q^{3} + (\beta_{2} - \beta_1 + 19) q^{4} + ( - \beta_{7} - \beta_1 - 5) q^{5} + (\beta_{10} - \beta_{7} + \beta_{5} + \cdots - 26) q^{6}+ \cdots + ( - \beta_{10} + 2 \beta_{7} + \cdots + 93) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{5} + \beta_1) q^{3} + (\beta_{2} - \beta_1 + 19) q^{4} + ( - \beta_{7} - \beta_1 - 5) q^{5} + (\beta_{10} - \beta_{7} + \beta_{5} + \cdots - 26) q^{6}+ \cdots + ( - 2401 \beta_1 + 2401) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 8 q^{2} - q^{3} + 244 q^{4} - 69 q^{5} - 338 q^{6} + 637 q^{7} + 384 q^{8} + 1184 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 8 q^{2} - q^{3} + 244 q^{4} - 69 q^{5} - 338 q^{6} + 637 q^{7} + 384 q^{8} + 1184 q^{9} + 566 q^{10} - 850 q^{12} + 120 q^{13} + 392 q^{14} + 1191 q^{15} + 3516 q^{16} + 4272 q^{17} + 3660 q^{18} - 756 q^{19} - 1414 q^{20} - 49 q^{21} - 3565 q^{23} - 9718 q^{24} + 12842 q^{25} - 3906 q^{26} + 4925 q^{27} + 11956 q^{28} + 8848 q^{29} + 29872 q^{30} + 11559 q^{31} - 6764 q^{32} - 13940 q^{34} - 3381 q^{35} + 55420 q^{36} + 1873 q^{37} - 9678 q^{38} + 35436 q^{39} - 2958 q^{40} + 39720 q^{41} - 16562 q^{42} - 5820 q^{43} - 80134 q^{45} + 10304 q^{46} + 4388 q^{47} - 17406 q^{48} + 31213 q^{49} + 75328 q^{50} + 21732 q^{51} + 14658 q^{52} + 1890 q^{53} - 178272 q^{54} + 18816 q^{56} + 104848 q^{57} + 21272 q^{58} + 2633 q^{59} + 23396 q^{60} + 95472 q^{61} + 28676 q^{62} + 58016 q^{63} - 89872 q^{64} - 69848 q^{65} + 20197 q^{67} + 226352 q^{68} - 27001 q^{69} + 27734 q^{70} + 30033 q^{71} + 308036 q^{72} + 74792 q^{73} + 136716 q^{74} - 116660 q^{75} - 188590 q^{76} + 150280 q^{78} + 119924 q^{79} - 123770 q^{80} + 105697 q^{81} - 235316 q^{82} + 173804 q^{83} - 41650 q^{84} - 183320 q^{85} + 40160 q^{86} + 27688 q^{87} + 270017 q^{89} + 135570 q^{90} + 5880 q^{91} - 273212 q^{92} + 196247 q^{93} + 160284 q^{94} + 8672 q^{95} + 368714 q^{96} + 143183 q^{97} + 19208 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 316 x^{11} + 1264 x^{10} + 38184 x^{9} - 118956 x^{8} - 2174258 x^{7} + \cdots - 544885000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 46466130699 \nu^{12} + 4132847165210 \nu^{11} - 29757585544726 \nu^{10} + \cdots + 25\!\cdots\!80 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2816870547 \nu^{12} + 34354326860 \nu^{11} + 466470191532 \nu^{10} + \cdots + 54\!\cdots\!80 ) / 70\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 129791624517 \nu^{12} + 1709202074910 \nu^{11} + 25958580669262 \nu^{10} + \cdots + 72\!\cdots\!00 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 129791624517 \nu^{12} - 1709202074910 \nu^{11} - 25958580669262 \nu^{10} + \cdots + 12\!\cdots\!40 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 93033448049 \nu^{12} - 1307385698750 \nu^{11} - 17440003539694 \nu^{10} + \cdots - 10\!\cdots\!20 ) / 65\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 81206042228 \nu^{12} + 999867245305 \nu^{11} + 16589657756673 \nu^{10} + \cdots - 21\!\cdots\!20 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 710937463433 \nu^{12} - 9625892035750 \nu^{11} - 142370749989718 \nu^{10} + \cdots + 22\!\cdots\!00 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 390571804089 \nu^{12} + 5566707790930 \nu^{11} + 74080191621314 \nu^{10} + \cdots + 17\!\cdots\!00 ) / 98\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 520593436441 \nu^{12} - 7130618498210 \nu^{11} - 97745635977106 \nu^{10} + \cdots + 53\!\cdots\!80 ) / 98\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 244800880711 \nu^{12} - 3616388097610 \nu^{11} - 44257251017626 \nu^{10} + \cdots - 13\!\cdots\!60 ) / 32\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 3\beta_{2} + 81\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2 \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 5 \beta_{6} + 3 \beta_{5} + \beta_{4} + \cdots + 4047 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + 12 \beta_{11} + 26 \beta_{10} + 2 \beta_{9} + 7 \beta_{8} - \beta_{7} + 145 \beta_{6} + \cdots + 10985 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41 \beta_{12} + 205 \beta_{11} + 508 \beta_{10} + 157 \beta_{9} + 214 \beta_{8} + 48 \beta_{7} + \cdots + 380163 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 636 \beta_{12} + 2582 \beta_{11} + 6268 \beta_{10} + 522 \beta_{9} + 1774 \beta_{8} - 2382 \beta_{7} + \cdots + 1780107 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11998 \beta_{12} + 33715 \beta_{11} + 92922 \beta_{10} + 20051 \beta_{9} + 34113 \beta_{8} + \cdots + 39530069 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 156309 \beta_{12} + 421052 \beta_{11} + 1120986 \beta_{10} + 98282 \beta_{9} + 319283 \beta_{8} + \cdots + 260104777 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2353821 \beta_{12} + 5100941 \beta_{11} + 14981996 \beta_{10} + 2458693 \beta_{9} + 4957266 \beta_{8} + \cdots + 4450477223 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 28960736 \beta_{12} + 62567866 \beta_{11} + 178651940 \beta_{10} + 15999238 \beta_{9} + \cdots + 36324110907 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 392445686 \beta_{12} + 739305279 \beta_{11} + 2264316202 \beta_{10} + 303197343 \beta_{9} + \cdots + 531537224709 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
11.6158
9.71908
8.05013
7.07634
3.73109
2.88343
0.257717
−2.17393
−3.04182
−7.28721
−7.81786
−9.00303
−9.00976
−10.6158 −3.27030 80.6957 −3.96535 34.7169 49.0000 −516.945 −232.305 42.0954
1.2 −8.71908 28.6420 44.0224 −77.7806 −249.732 49.0000 −104.824 577.362 678.175
1.3 −7.05013 −20.0366 17.7043 13.3130 141.261 49.0000 100.786 158.467 −93.8583
1.4 −6.07634 14.0390 4.92195 81.3007 −85.3058 49.0000 164.536 −45.9064 −494.011
1.5 −2.73109 −18.3701 −24.5412 −86.2855 50.1703 49.0000 154.419 94.4598 235.653
1.6 −1.88343 12.9168 −28.4527 −68.8885 −24.3279 49.0000 113.858 −76.1557 129.747
1.7 0.742283 −2.45671 −31.4490 62.8840 −1.82357 49.0000 −47.0972 −236.965 46.6777
1.8 3.17393 25.5757 −21.9261 18.6089 81.1757 49.0000 −171.158 411.119 59.0633
1.9 4.04182 −13.6331 −15.6637 −7.36582 −55.1024 49.0000 −192.648 −57.1395 −29.7713
1.10 8.28721 2.56348 36.6778 −71.2607 21.2441 49.0000 38.7658 −236.429 −590.552
1.11 8.81786 −16.6294 45.7547 102.346 −146.636 49.0000 121.287 33.5373 902.475
1.12 10.0030 19.5934 68.0605 47.3194 195.993 49.0000 360.715 140.900 473.337
1.13 10.0098 −29.9342 68.1953 −79.2257 −299.634 49.0000 362.307 653.055 −793.031
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 847.6.a.m yes 13
11.b odd 2 1 847.6.a.l ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.6.a.l ✓ 13 11.b odd 2 1
847.6.a.m yes 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 8 T_{2}^{12} - 298 T_{2}^{11} + 2256 T_{2}^{10} + 33059 T_{2}^{9} - 228252 T_{2}^{8} + \cdots - 1421084672 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(847))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + \cdots - 1421084672 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots + 133899082581600 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( (T - 49)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 45\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots - 20\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 75\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 63\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 18\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 22\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 34\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 19\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 10\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 35\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 19\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 23\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 70\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 83\!\cdots\!28 \) Copy content Toggle raw display
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