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Properties

Label 819.2.bm.g
Level $819$
Weight $2$
Character orbit 819.bm
Analytic conductor $6.540$
Analytic rank $0$
Dimension $20$
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] = [819,2,Mod(478,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.478");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.bm (of order \(6\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 33 x^{18} + 455 x^{16} + 3403 x^{14} + 15006 x^{12} + 39799 x^{10} + 62505 x^{8} + 55993 x^{6} + \cdots + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2}) q^{2} + (\beta_1 - 1) q^{4} + \beta_{7} q^{5} + (\beta_{15} - \beta_{13}) q^{7} + (\beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2}) q^{2} + (\beta_1 - 1) q^{4} + \beta_{7} q^{5} + (\beta_{15} - \beta_{13}) q^{7} + (\beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{8}+ \cdots + ( - \beta_{19} + 2 \beta_{18} + \cdots - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 26 q^{4} - 6 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 26 q^{4} - 6 q^{5} + 2 q^{7} + 2 q^{10} + 12 q^{11} + 8 q^{13} - 2 q^{14} + 42 q^{16} - 16 q^{17} - 9 q^{19} - 9 q^{22} + 36 q^{23} + 12 q^{25} + 16 q^{26} - 2 q^{28} + 3 q^{29} - 18 q^{31} - 18 q^{35} - 9 q^{38} + 5 q^{40} - 21 q^{41} + 16 q^{43} + 6 q^{44} - 21 q^{47} - 24 q^{49} + 54 q^{50} - 41 q^{52} + 26 q^{53} + 17 q^{55} + 6 q^{56} + 42 q^{58} - 4 q^{62} - 46 q^{64} + 50 q^{65} - 3 q^{67} - 6 q^{68} - 15 q^{71} - 9 q^{73} - 12 q^{74} + 75 q^{76} - 20 q^{77} + 3 q^{79} + 24 q^{80} + 15 q^{82} - 78 q^{85} - 3 q^{86} - 22 q^{88} + 4 q^{91} - 142 q^{92} + 36 q^{94} + 84 q^{95} - 15 q^{97} - 81 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 33 x^{18} + 455 x^{16} + 3403 x^{14} + 15006 x^{12} + 39799 x^{10} + 62505 x^{8} + 55993 x^{6} + \cdots + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2116 \nu^{18} - 67386 \nu^{16} - 887545 \nu^{14} - 6247121 \nu^{12} - 25326750 \nu^{10} + \cdots - 1430232 ) / 78814 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2116 \nu^{18} + 67386 \nu^{16} + 887545 \nu^{14} + 6247121 \nu^{12} + 25326750 \nu^{10} + \cdots + 1430232 ) / 78814 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 43788 \nu^{18} - 1443263 \nu^{16} - 19879084 \nu^{14} - 148563367 \nu^{12} - 654599666 \nu^{10} + \cdots - 138397894 ) / 1339838 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43788 \nu^{18} + 1443263 \nu^{16} + 19879084 \nu^{14} + 148563367 \nu^{12} + 654599666 \nu^{10} + \cdots + 138397894 ) / 1339838 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39625 \nu^{19} - 283077 \nu^{18} - 1345812 \nu^{17} - 9180339 \nu^{16} - 19234190 \nu^{15} + \cdots - 394130712 ) / 4019514 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43633 \nu^{19} - 161830 \nu^{18} + 1428531 \nu^{17} - 5224322 \nu^{16} + 19439089 \nu^{15} + \cdots - 163312344 ) / 2679676 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43633 \nu^{19} - 161830 \nu^{18} - 1428531 \nu^{17} - 5224322 \nu^{16} - 19439089 \nu^{15} + \cdots - 163312344 ) / 2679676 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 50615 \nu^{19} + 177050 \nu^{18} - 1547557 \nu^{17} + 5676836 \nu^{16} - 19031503 \nu^{15} + \cdots + 132712408 ) / 2679676 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43633 \nu^{19} - 379746 \nu^{18} - 1428531 \nu^{17} - 12287124 \nu^{16} - 19439089 \nu^{15} + \cdots - 360716608 ) / 2679676 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 59593 \nu^{19} + 1915785 \nu^{17} + 25497551 \nu^{15} + 181493899 \nu^{13} + 744321654 \nu^{11} + \cdots + 945768 ) / 1891536 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 347073 \nu^{19} - 375400 \nu^{18} + 11381641 \nu^{17} - 12340840 \nu^{16} + 155349919 \nu^{15} + \cdots - 749416136 ) / 10718704 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 347073 \nu^{19} + 375400 \nu^{18} + 11381641 \nu^{17} + 12340840 \nu^{16} + 155349919 \nu^{15} + \cdots + 749416136 ) / 10718704 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1741799 \nu^{19} - 1134696 \nu^{18} + 56733111 \nu^{17} - 36066624 \nu^{16} + 768476209 \nu^{15} + \cdots + 354108216 ) / 32156112 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1741799 \nu^{19} - 1134696 \nu^{18} - 56733111 \nu^{17} - 36066624 \nu^{16} - 768476209 \nu^{15} + \cdots + 354108216 ) / 32156112 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 310569 \nu^{19} - 344756 \nu^{18} - 10038565 \nu^{17} - 11208964 \nu^{16} - 134537471 \nu^{15} + \cdots - 567210144 ) / 5359352 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 42665 \nu^{19} - 1376697 \nu^{17} - 18397191 \nu^{15} - 131516931 \nu^{13} - 541707654 \nu^{11} + \cdots - 945768 ) / 630512 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 2792711 \nu^{19} + 2185608 \nu^{18} - 91371423 \nu^{17} + 70704936 \nu^{16} + \cdots + 2870972904 ) / 32156112 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 2792711 \nu^{19} - 2185608 \nu^{18} - 91371423 \nu^{17} - 70704936 \nu^{16} + \cdots - 2870972904 ) / 32156112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 5\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{15} - \beta_{14} + \beta_{5} - \beta_{4} - 7\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + 2 \beta_{17} - \beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{19} + 13 \beta_{18} - \beta_{16} + 12 \beta_{15} + 12 \beta_{14} - \beta_{13} + \beta_{12} + \cdots - 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{19} - 28 \beta_{17} + 10 \beta_{16} - 12 \beta_{15} + 12 \beta_{14} - 15 \beta_{13} - 15 \beta_{12} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 114 \beta_{19} - 128 \beta_{18} + 14 \beta_{16} - 111 \beta_{15} - 111 \beta_{14} + 18 \beta_{13} + \cdots + 570 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 75 \beta_{19} + 4 \beta_{18} + 294 \beta_{17} - 79 \beta_{16} + 103 \beta_{15} - 103 \beta_{14} + \cdots - 85 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1003 \beta_{19} + 1143 \beta_{18} - 140 \beta_{16} + 934 \beta_{15} + 934 \beta_{14} - 213 \beta_{13} + \cdots - 3823 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 502 \beta_{19} - 85 \beta_{18} - 2756 \beta_{17} + 587 \beta_{16} - 776 \beta_{15} + 776 \beta_{14} + \cdots + 997 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8493 \beta_{19} - 9733 \beta_{18} + 1240 \beta_{16} - 7520 \beta_{15} - 7520 \beta_{14} + 2123 \beta_{13} + \cdots + 26481 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3141 \beta_{19} + 1151 \beta_{18} + 24322 \beta_{17} - 4292 \beta_{16} + 5486 \beta_{15} + \cdots - 10034 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 70218 \beta_{19} + 80627 \beta_{18} - 10409 \beta_{16} + 59217 \beta_{15} + 59217 \beta_{14} + \cdots - 187868 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 18608 \beta_{19} - 12749 \beta_{18} - 206846 \beta_{17} + 31357 \beta_{16} - 37505 \beta_{15} + \cdots + 93048 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 571047 \beta_{19} - 656179 \beta_{18} + 85132 \beta_{16} - 461045 \beta_{15} - 461045 \beta_{14} + \cdots + 1357352 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 103921 \beta_{19} + 126294 \beta_{18} + 1716384 \beta_{17} - 230215 \beta_{16} + 251956 \beta_{15} + \cdots - 820761 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 4588963 \beta_{19} + 5275835 \beta_{18} - 686872 \beta_{16} + 3569231 \beta_{15} + 3569231 \beta_{14} + \cdots - 9946338 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 534336 \beta_{19} - 1167446 \beta_{18} - 13999628 \beta_{17} + 1701782 \beta_{16} - 1679181 \beta_{15} + \cdots + 7003479 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(1 - \beta_{11}\) \(-1 + \beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
478.1
− 2.50900i
− 2.44406i
− 2.04830i
− 0.915396i
− 0.560998i
− 0.493650i
0.871638i
1.46676i
2.11978i
2.78118i
− 2.78118i
− 2.11978i
− 1.46676i
− 0.871638i
0.493650i
0.560998i
0.915396i
2.04830i
2.44406i
2.50900i
− 2.50900i 0 −4.29509 2.97008 + 1.71478i 0 −0.0473512 − 2.64533i 5.75840i 0 4.30238 − 7.45195i
478.2 − 2.44406i 0 −3.97343 −3.36767 − 1.94432i 0 1.82552 − 1.91507i 4.82318i 0 −4.75204 + 8.23078i
478.3 − 2.04830i 0 −2.19554 −0.341406 − 0.197111i 0 −0.908042 + 2.48505i 0.400534i 0 −0.403742 + 0.699302i
478.4 − 0.915396i 0 1.16205 2.26729 + 1.30902i 0 2.57477 + 0.608732i − 2.89453i 0 1.19827 − 2.07547i
478.5 − 0.560998i 0 1.68528 −0.449963 − 0.259786i 0 0.680125 − 2.55684i − 2.06743i 0 −0.145740 + 0.252428i
478.6 − 0.493650i 0 1.75631 −2.40375 − 1.38781i 0 −0.134370 + 2.64234i − 1.85430i 0 −0.685090 + 1.18661i
478.7 0.871638i 0 1.24025 −1.34003 − 0.773665i 0 −2.02693 − 1.70046i 2.82432i 0 0.674356 − 1.16802i
478.8 1.46676i 0 −0.151375 1.88109 + 1.08605i 0 −2.04728 + 1.67590i 2.71148i 0 −1.59296 + 2.75910i
478.9 2.11978i 0 −2.49347 −3.01977 − 1.74346i 0 2.64461 + 0.0777267i − 1.04605i 0 3.69576 − 6.40125i
478.10 2.78118i 0 −5.73498 0.804122 + 0.464260i 0 −1.56105 − 2.13615i − 10.3876i 0 −1.29119 + 2.23641i
550.1 − 2.78118i 0 −5.73498 0.804122 − 0.464260i 0 −1.56105 + 2.13615i 10.3876i 0 −1.29119 − 2.23641i
550.2 − 2.11978i 0 −2.49347 −3.01977 + 1.74346i 0 2.64461 − 0.0777267i 1.04605i 0 3.69576 + 6.40125i
550.3 − 1.46676i 0 −0.151375 1.88109 − 1.08605i 0 −2.04728 − 1.67590i − 2.71148i 0 −1.59296 − 2.75910i
550.4 − 0.871638i 0 1.24025 −1.34003 + 0.773665i 0 −2.02693 + 1.70046i − 2.82432i 0 0.674356 + 1.16802i
550.5 0.493650i 0 1.75631 −2.40375 + 1.38781i 0 −0.134370 − 2.64234i 1.85430i 0 −0.685090 − 1.18661i
550.6 0.560998i 0 1.68528 −0.449963 + 0.259786i 0 0.680125 + 2.55684i 2.06743i 0 −0.145740 − 0.252428i
550.7 0.915396i 0 1.16205 2.26729 − 1.30902i 0 2.57477 − 0.608732i 2.89453i 0 1.19827 + 2.07547i
550.8 2.04830i 0 −2.19554 −0.341406 + 0.197111i 0 −0.908042 − 2.48505i − 0.400534i 0 −0.403742 − 0.699302i
550.9 2.44406i 0 −3.97343 −3.36767 + 1.94432i 0 1.82552 + 1.91507i − 4.82318i 0 −4.75204 − 8.23078i
550.10 2.50900i 0 −4.29509 2.97008 − 1.71478i 0 −0.0473512 + 2.64533i − 5.75840i 0 4.30238 + 7.45195i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 478.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.bm.g 20
3.b odd 2 1 273.2.t.d ✓ 20
7.c even 3 1 819.2.do.g 20
13.e even 6 1 819.2.do.g 20
21.h odd 6 1 273.2.bl.d yes 20
39.h odd 6 1 273.2.bl.d yes 20
91.k even 6 1 inner 819.2.bm.g 20
273.bp odd 6 1 273.2.t.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.d ✓ 20 3.b odd 2 1
273.2.t.d ✓ 20 273.bp odd 6 1
273.2.bl.d yes 20 21.h odd 6 1
273.2.bl.d yes 20 39.h odd 6 1
819.2.bm.g 20 1.a even 1 1 trivial
819.2.bm.g 20 91.k even 6 1 inner
819.2.do.g 20 7.c even 3 1
819.2.do.g 20 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{20} + 33 T_{2}^{18} + 455 T_{2}^{16} + 3403 T_{2}^{14} + 15006 T_{2}^{12} + 39799 T_{2}^{10} + \cdots + 576 \) Copy content Toggle raw display
\( T_{5}^{20} + 6 T_{5}^{19} - 13 T_{5}^{18} - 150 T_{5}^{17} + 139 T_{5}^{16} + 2484 T_{5}^{15} + \cdots + 46656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 33 T^{18} + \cdots + 576 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 6 T^{19} + \cdots + 46656 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 18257414400 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( (T^{10} + 8 T^{9} + \cdots + 49716)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 137768653584 \) Copy content Toggle raw display
$23$ \( (T^{10} - 18 T^{9} + \cdots - 651492)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 1191651690384 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 293380784803044 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 111696214861449 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 28328929344 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 13841465449744 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 14078614613904 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 38226506256 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 3923769600 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + 3 T^{19} + \cdots + 46656 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 29690641983744 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 548870434304256 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 5010940944 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
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