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Properties

Label 847.2.e.f
Level $847$
Weight $2$
Character orbit 847.e
Analytic conductor $6.763$
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] = [847,2,Mod(485,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.485");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.e (of order \(3\), degree \(2\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 11 x^{12} - 2 x^{11} + 88 x^{10} - 10 x^{9} + 310 x^{8} + 46 x^{7} + 791 x^{6} + 186 x^{5} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{10} + \beta_{4}) q^{3} + ( - \beta_{12} - \beta_{11} - \beta_{9} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{13} - \beta_{12} + \cdots - 2 \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{10} + \beta_{4}) q^{3} + ( - \beta_{12} - \beta_{11} - \beta_{9} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{13} - 4 \beta_{12} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{3} - 8 q^{4} - 4 q^{5} - 4 q^{6} + 2 q^{7} - 6 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{3} - 8 q^{4} - 4 q^{5} - 4 q^{6} + 2 q^{7} - 6 q^{8} - 10 q^{9} - 6 q^{10} - 13 q^{12} + 12 q^{13} + 12 q^{14} - 2 q^{15} - 6 q^{16} + 3 q^{17} - 14 q^{18} + 9 q^{19} + 4 q^{20} - 6 q^{21} + 6 q^{23} - 4 q^{24} - 7 q^{25} - 25 q^{26} + 24 q^{27} - 36 q^{28} + 12 q^{29} - 16 q^{30} - 14 q^{31} + 36 q^{32} + 16 q^{34} + 18 q^{35} - 4 q^{36} + 8 q^{37} + 10 q^{38} + 9 q^{39} - 18 q^{40} - 20 q^{41} - 8 q^{42} + 34 q^{43} - 12 q^{45} - 16 q^{46} - 30 q^{47} + 112 q^{48} + 14 q^{49} + 42 q^{50} + 20 q^{51} + 37 q^{52} + 10 q^{53} + 7 q^{54} - 33 q^{56} - 62 q^{57} + 13 q^{58} - 20 q^{59} - 42 q^{60} - 7 q^{61} + 52 q^{62} + 37 q^{63} + 42 q^{64} - 12 q^{65} - 7 q^{67} - 7 q^{68} + 18 q^{69} - 39 q^{70} - 22 q^{71} + q^{72} + 6 q^{73} - 8 q^{74} + q^{75} - 112 q^{76} + 30 q^{78} + 14 q^{79} - 12 q^{80} - 35 q^{81} + 7 q^{82} + 34 q^{83} + 40 q^{84} - 4 q^{85} + 15 q^{87} - 40 q^{89} + 136 q^{90} - 32 q^{91} - 52 q^{92} - 19 q^{94} - 20 q^{95} + 63 q^{96} - 44 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 11 x^{12} - 2 x^{11} + 88 x^{10} - 10 x^{9} + 310 x^{8} + 46 x^{7} + 791 x^{6} + 186 x^{5} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 264377324 \nu^{13} - 1552789332 \nu^{12} + 2434085922 \nu^{11} - 15976163615 \nu^{10} + \cdots - 2769021545193 ) / 909375762219 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1514602168 \nu^{13} - 793131972 \nu^{12} - 12002255852 \nu^{11} - 4273053430 \nu^{10} + \cdots + 891314822619 ) / 2728127286657 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5120621638 \nu^{13} - 87439272219 \nu^{12} + 60937104272 \nu^{11} - 989644306820 \nu^{10} + \cdots - 38669157038421 ) / 8184381859971 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15219014966 \nu^{13} + 113311355787 \nu^{12} + 97189960210 \nu^{11} + 1197525580805 \nu^{10} + \cdots + 42036246745902 ) / 8184381859971 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11003886699 \nu^{13} - 1514602168 \nu^{12} - 121835885661 \nu^{11} + 10005517546 \nu^{10} + \cdots - 108394416522 ) / 2728127286657 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43887916724 \nu^{13} - 73110782037 \nu^{12} + 516218986960 \nu^{11} - 908095812328 \nu^{10} + \cdots - 31356178550721 ) / 8184381859971 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18302830254 \nu^{13} - 17909964458 \nu^{12} - 191773032494 \nu^{11} + \cdots - 9689220683829 ) / 2728127286657 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 157825292764 \nu^{13} + 39995780802 \nu^{12} - 1779899351258 \nu^{11} + \cdots - 29545999845552 ) / 8184381859971 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 60247680609 \nu^{13} + 34411700773 \nu^{12} - 656430346233 \nu^{11} + 487669814708 \nu^{10} + \cdots + 573975068229 ) / 2728127286657 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 199685537987 \nu^{13} + 51679398804 \nu^{12} - 2119933619446 \nu^{11} + \cdots + 2599860888081 ) / 8184381859971 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 22354218198 \nu^{13} - 16117932056 \nu^{12} + 245120458775 \nu^{11} - 216874387178 \nu^{10} + \cdots - 303224946210 ) / 909375762219 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 78390217983 \nu^{13} + 37463058359 \nu^{12} - 861062148420 \nu^{11} + 558485888017 \nu^{10} + \cdots + 544112700525 ) / 2728127286657 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} + 3\beta_{6} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 5\beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + 6\beta_{12} + 7\beta_{11} - \beta_{10} + 7\beta_{9} - 7\beta_{8} - 15\beta_{6} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{13} - 9 \beta_{12} - 10 \beta_{11} - 8 \beta_{10} - 9 \beta_{9} + 8 \beta_{8} + 13 \beta_{6} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{8} + 11\beta_{7} + 12\beta_{5} - 7\beta_{4} + 25\beta_{3} + 37\beta_{2} + 91 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 67 \beta_{13} + 73 \beta_{12} + 74 \beta_{11} + 54 \beta_{10} + 85 \beta_{9} - 74 \beta_{8} + \cdots + 183 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 109 \beta_{13} - 246 \beta_{12} - 335 \beta_{11} + 35 \beta_{10} - 334 \beta_{9} + 245 \beta_{8} + \cdots - 601 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 88 \beta_{11} - 88 \beta_{9} + 111 \beta_{8} + 111 \beta_{7} + 480 \beta_{5} + 349 \beta_{4} + \cdots + 986 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 897 \beta_{13} + 1720 \beta_{12} + 2353 \beta_{11} - 128 \beta_{10} + 2376 \beta_{9} + \cdots + 1970 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3424 \beta_{13} - 4300 \beta_{12} - 5243 \beta_{11} - 2240 \beta_{10} - 4610 \beta_{9} + 3667 \beta_{8} + \cdots - 7827 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 310 \beta_{11} - 310 \beta_{9} + 4690 \beta_{8} + 4690 \beta_{7} + 7060 \beta_{5} - 111 \beta_{4} + \cdots + 29108 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 24531 \beta_{13} + 32193 \beta_{12} + 35493 \beta_{11} + 14481 \beta_{10} + 39873 \beta_{9} + \cdots + 58939 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(-\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
1.17952 − 2.04300i
0.944230 − 1.63545i
0.680247 − 1.17822i
−0.200568 + 0.347394i
−0.406024 + 0.703253i
−0.838875 + 1.45297i
−1.35854 + 2.35305i
1.17952 + 2.04300i
0.944230 + 1.63545i
0.680247 + 1.17822i
−0.200568 − 0.347394i
−0.406024 − 0.703253i
−0.838875 − 1.45297i
−1.35854 − 2.35305i
−1.17952 + 2.04300i −1.32623 − 2.29710i −1.78256 − 3.08748i −1.66306 + 2.88050i 6.25730 0.348611 − 2.62268i 3.69218 −2.01779 + 3.49491i −3.92324 − 6.79525i
485.2 −0.944230 + 1.63545i 0.0669258 + 0.115919i −0.783142 − 1.35644i 1.17111 − 2.02842i −0.252774 1.25670 + 2.32824i −0.819057 1.49104 − 2.58256i 2.21159 + 3.83059i
485.3 −0.680247 + 1.17822i 1.49682 + 2.59257i 0.0745274 + 0.129085i −0.872163 + 1.51063i −4.07283 −2.38576 − 1.14375i −2.92378 −2.98094 + 5.16315i −1.18657 − 2.05520i
485.4 0.200568 − 0.347394i −1.61927 − 2.80466i 0.919545 + 1.59270i 0.638896 − 1.10660i −1.29909 −0.695454 + 2.55271i 1.54000 −3.74408 + 6.48494i −0.256284 − 0.443897i
485.5 0.406024 − 0.703253i 0.229130 + 0.396865i 0.670290 + 1.16098i −2.14775 + 3.72001i 0.372128 2.49240 + 0.887667i 2.71271 1.39500 − 2.41621i 1.74407 + 3.02082i
485.6 0.838875 − 1.45297i 0.537398 + 0.930801i −0.407422 − 0.705675i 0.752739 − 1.30378i 1.80324 −2.64562 + 0.0259872i 1.98840 0.922406 − 1.59765i −1.26291 − 2.18742i
485.7 1.35854 − 2.35305i −0.884770 − 1.53247i −2.69124 − 4.66137i 0.120225 − 0.208235i −4.80797 2.62913 − 0.296126i −9.19045 −0.0656351 + 0.113683i −0.326659 − 0.565790i
606.1 −1.17952 − 2.04300i −1.32623 + 2.29710i −1.78256 + 3.08748i −1.66306 − 2.88050i 6.25730 0.348611 + 2.62268i 3.69218 −2.01779 − 3.49491i −3.92324 + 6.79525i
606.2 −0.944230 − 1.63545i 0.0669258 − 0.115919i −0.783142 + 1.35644i 1.17111 + 2.02842i −0.252774 1.25670 − 2.32824i −0.819057 1.49104 + 2.58256i 2.21159 − 3.83059i
606.3 −0.680247 − 1.17822i 1.49682 − 2.59257i 0.0745274 − 0.129085i −0.872163 − 1.51063i −4.07283 −2.38576 + 1.14375i −2.92378 −2.98094 − 5.16315i −1.18657 + 2.05520i
606.4 0.200568 + 0.347394i −1.61927 + 2.80466i 0.919545 − 1.59270i 0.638896 + 1.10660i −1.29909 −0.695454 − 2.55271i 1.54000 −3.74408 − 6.48494i −0.256284 + 0.443897i
606.5 0.406024 + 0.703253i 0.229130 − 0.396865i 0.670290 − 1.16098i −2.14775 − 3.72001i 0.372128 2.49240 − 0.887667i 2.71271 1.39500 + 2.41621i 1.74407 − 3.02082i
606.6 0.838875 + 1.45297i 0.537398 − 0.930801i −0.407422 + 0.705675i 0.752739 + 1.30378i 1.80324 −2.64562 − 0.0259872i 1.98840 0.922406 + 1.59765i −1.26291 + 2.18742i
606.7 1.35854 + 2.35305i −0.884770 + 1.53247i −2.69124 + 4.66137i 0.120225 + 0.208235i −4.80797 2.62913 + 0.296126i −9.19045 −0.0656351 − 0.113683i −0.326659 + 0.565790i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.e.f ✓ 14
7.c even 3 1 inner 847.2.e.f ✓ 14
7.c even 3 1 5929.2.a.bp 7
7.d odd 6 1 5929.2.a.bo 7
11.b odd 2 1 847.2.e.g yes 14
11.c even 5 4 847.2.n.m 56
11.d odd 10 4 847.2.n.l 56
77.h odd 6 1 847.2.e.g yes 14
77.h odd 6 1 5929.2.a.bq 7
77.i even 6 1 5929.2.a.bn 7
77.m even 15 4 847.2.n.m 56
77.o odd 30 4 847.2.n.l 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.e.f ✓ 14 1.a even 1 1 trivial
847.2.e.f ✓ 14 7.c even 3 1 inner
847.2.e.g yes 14 11.b odd 2 1
847.2.e.g yes 14 77.h odd 6 1
847.2.n.l 56 11.d odd 10 4
847.2.n.l 56 77.o odd 30 4
847.2.n.m 56 11.c even 5 4
847.2.n.m 56 77.m even 15 4
5929.2.a.bn 7 77.i even 6 1
5929.2.a.bo 7 7.d odd 6 1
5929.2.a.bp 7 7.c even 3 1
5929.2.a.bq 7 77.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 11 T_{2}^{12} + 2 T_{2}^{11} + 88 T_{2}^{10} + 10 T_{2}^{9} + 310 T_{2}^{8} - 46 T_{2}^{7} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 11 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{14} + 3 T^{13} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{14} + 4 T^{13} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{14} - 2 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( (T^{7} - 6 T^{6} + \cdots - 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} - 3 T^{13} + \cdots + 400689 \) Copy content Toggle raw display
$19$ \( T^{14} - 9 T^{13} + \cdots + 34963569 \) Copy content Toggle raw display
$23$ \( T^{14} - 6 T^{13} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( (T^{7} - 6 T^{6} - 29 T^{5} + \cdots + 45)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 4970955025 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 11249784225 \) Copy content Toggle raw display
$41$ \( (T^{7} + 10 T^{6} + \cdots - 49431)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 17 T^{6} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 2795131161 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 31381059609 \) Copy content Toggle raw display
$59$ \( T^{14} + 20 T^{13} + \cdots + 2712609 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 2469593025 \) Copy content Toggle raw display
$67$ \( T^{14} + 7 T^{13} + \cdots + 19580625 \) Copy content Toggle raw display
$71$ \( (T^{7} + 11 T^{6} + \cdots - 19845)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 158482921 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 83160140625 \) Copy content Toggle raw display
$83$ \( (T^{7} - 17 T^{6} + \cdots - 3915)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + 40 T^{13} + \cdots + 27720225 \) Copy content Toggle raw display
$97$ \( (T^{7} + 22 T^{6} + \cdots - 134047)^{2} \) Copy content Toggle raw display
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