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Properties

Label 800.6.d.d
Level $800$
Weight $6$
Character orbit 800.d
Analytic conductor $128.307$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,6,Mod(401,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.401");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.d (of order \(2\), degree \(1\), not 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: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 130 x^{17} + 144 x^{16} + 1560 x^{15} - 12320 x^{14} - 56128 x^{13} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{99}\cdot 5^{4}\cdot 31 \)
Twist minimal: no (minimal twist has level 200)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_1 + 10) q^{7} + ( - \beta_{5} - 81) q^{9} + \beta_{3} q^{11} + (\beta_{9} - 3 \beta_{2}) q^{13} + ( - \beta_{4} + \beta_1) q^{17} + (\beta_{13} + 7 \beta_{2}) q^{19} + (\beta_{17} - 7 \beta_{2}) q^{21}+ \cdots + (6 \beta_{19} - \beta_{18} + \cdots + 52 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 196 q^{7} - 1620 q^{9} + 7184 q^{23} - 7160 q^{31} - 2836 q^{33} - 22452 q^{39} - 5804 q^{41} - 44180 q^{47} + 62652 q^{49} + 43696 q^{57} + 1240 q^{63} + 7724 q^{71} - 105136 q^{73} + 7780 q^{79}+ \cdots - 73688 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 130 x^{17} + 144 x^{16} + 1560 x^{15} - 12320 x^{14} - 56128 x^{13} + \cdots + 11\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 483 \nu^{19} + 764979 \nu^{18} - 4445008 \nu^{17} + 8154694 \nu^{16} + \cdots + 22\!\cdots\!72 ) / 64\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4229911 \nu^{19} + 16221639 \nu^{18} + 26166864 \nu^{17} - 1019195218 \nu^{16} + \cdots + 73\!\cdots\!20 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12041203 \nu^{19} - 49677539 \nu^{18} + 956079088 \nu^{17} - 1939600486 \nu^{16} + \cdots - 31\!\cdots\!72 ) / 18\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 82295 \nu^{19} + 8750759 \nu^{18} - 61367344 \nu^{17} - 28062354 \nu^{16} + \cdots + 33\!\cdots\!24 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 310971 \nu^{19} + 3945067 \nu^{18} - 9206448 \nu^{17} - 50179594 \nu^{16} + \cdots + 67\!\cdots\!60 ) / 38\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 130059 \nu^{19} - 5166133 \nu^{18} + 48198720 \nu^{17} - 85837418 \nu^{16} + \cdots - 24\!\cdots\!56 ) / 48\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 335778113 \nu^{19} + 6524261487 \nu^{18} - 9233781936 \nu^{17} - 58219258882 \nu^{16} + \cdots + 23\!\cdots\!92 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1417891 \nu^{19} + 3636973 \nu^{18} - 140074640 \nu^{17} - 48761286 \nu^{16} + \cdots + 56\!\cdots\!04 ) / 19\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 156057097 \nu^{19} - 478425817 \nu^{18} - 816890416 \nu^{17} + 9692817006 \nu^{16} + \cdots + 11\!\cdots\!32 ) / 18\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 591517 \nu^{19} + 2610515 \nu^{18} + 6359312 \nu^{17} - 123801786 \nu^{16} + \cdots - 81\!\cdots\!32 ) / 64\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1573257 \nu^{19} + 2243561 \nu^{18} + 2064288 \nu^{17} + 147082642 \nu^{16} + \cdots + 45\!\cdots\!88 ) / 97\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1615251 \nu^{19} + 3132829 \nu^{18} - 15438384 \nu^{17} + 4307546 \nu^{16} + \cdots + 23\!\cdots\!92 ) / 97\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 271939405 \nu^{19} - 1012567773 \nu^{18} - 7212198768 \nu^{17} + 16706379238 \nu^{16} + \cdots + 44\!\cdots\!68 ) / 14\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1114623673 \nu^{19} - 1333531767 \nu^{18} + 9944640048 \nu^{17} - 144274865422 \nu^{16} + \cdots + 12\!\cdots\!48 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1239299791 \nu^{19} - 2033538303 \nu^{18} - 5188074192 \nu^{17} - 77890939934 \nu^{16} + \cdots + 18\!\cdots\!64 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 2945865 \nu^{19} + 4679527 \nu^{18} - 67695024 \nu^{17} - 198444050 \nu^{16} + \cdots + 47\!\cdots\!80 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1315816601 \nu^{19} + 8802739305 \nu^{18} + 3729213232 \nu^{17} - 46737146190 \nu^{16} + \cdots - 12\!\cdots\!28 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 1431644519 \nu^{19} + 373231977 \nu^{18} - 25407984592 \nu^{17} + 123379016370 \nu^{16} + \cdots + 10\!\cdots\!44 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 524680729 \nu^{19} - 674529431 \nu^{18} - 13564174928 \nu^{17} + 116266793394 \nu^{16} + \cdots - 54\!\cdots\!84 ) / 18\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{11} - \beta_{9} - 2\beta_{5} + \beta_{3} - 5\beta_{2} + \beta _1 + 51 ) / 1024 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{11} - 4 \beta_{10} + \beta_{9} + 4 \beta_{7} - 10 \beta_{5} + 4 \beta_{4} + \cdots + 57 ) / 1024 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{19} - 24 \beta_{18} + \beta_{15} - 24 \beta_{14} - 8 \beta_{13} - 8 \beta_{12} + \cdots + 19987 ) / 1024 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24 \beta_{19} + 136 \beta_{18} - 128 \beta_{17} - 64 \beta_{16} - \beta_{15} - 56 \beta_{14} + \cdots - 2683 ) / 1024 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 200 \beta_{19} - 152 \beta_{18} + 640 \beta_{17} - 448 \beta_{16} - 19 \beta_{15} + 40 \beta_{14} + \cdots - 400601 ) / 1024 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 40 \beta_{19} + 3640 \beta_{18} - 2432 \beta_{17} - 448 \beta_{16} - 761 \beta_{15} + 3832 \beta_{14} + \cdots + 5905813 ) / 1024 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7208 \beta_{19} + 3528 \beta_{18} + 3456 \beta_{17} - 7744 \beta_{16} + 11821 \beta_{15} + \cdots + 23368823 ) / 1024 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 48760 \beta_{19} + 46680 \beta_{18} + 64128 \beta_{17} + 29248 \beta_{16} - 102881 \beta_{15} + \cdots - 331287411 ) / 1024 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 148248 \beta_{19} - 215928 \beta_{18} + 379776 \beta_{17} + 233664 \beta_{16} - 14507 \beta_{15} + \cdots + 2642052975 ) / 1024 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 871432 \beta_{19} - 1485352 \beta_{18} + 176768 \beta_{17} - 798144 \beta_{16} + 4266791 \beta_{15} + \cdots - 6472793403 ) / 1024 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9337560 \beta_{19} + 12296136 \beta_{18} + 29287296 \beta_{17} - 7439168 \beta_{16} + \cdots - 6360254825 ) / 1024 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 38621368 \beta_{19} - 137369192 \beta_{18} + 114097792 \beta_{17} - 44914624 \beta_{16} + \cdots + 66445988829 ) / 1024 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 106292760 \beta_{19} + 468732808 \beta_{18} - 339050624 \beta_{17} - 348597568 \beta_{16} + \cdots + 919435035487 ) / 1024 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1252382712 \beta_{19} + 839552472 \beta_{18} + 8223046272 \beta_{17} - 1563537856 \beta_{16} + \cdots + 534647590709 ) / 1024 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1481624920 \beta_{19} + 25320812360 \beta_{18} + 6491591552 \beta_{17} + 18626462912 \beta_{16} + \cdots - 144287881689 ) / 1024 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 3302070216 \beta_{19} - 40276926696 \beta_{18} - 5861947776 \beta_{17} + 6958525504 \beta_{16} + \cdots + 212772075877197 ) / 1024 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 259586292584 \beta_{19} + 240640883464 \beta_{18} + 227080894336 \beta_{17} + \cdots + 10\!\cdots\!71 ) / 1024 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 654964349832 \beta_{19} + 1270779714648 \beta_{18} + 4943214228096 \beta_{17} + 684029549120 \beta_{16} + \cdots + 21\!\cdots\!29 ) / 1024 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 8579465733672 \beta_{19} - 14630041116472 \beta_{18} - 19388279776384 \beta_{17} + \cdots - 50\!\cdots\!09 ) / 1024 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−1.78549 + 5.36768i
−5.37943 − 1.74979i
2.74854 − 4.94424i
−2.73024 − 4.95437i
4.12326 + 3.87282i
5.63982 − 0.438664i
−5.25771 + 2.08722i
5.22286 − 2.17296i
0.858917 + 5.59127i
−2.94053 − 4.83252i
−2.94053 + 4.83252i
0.858917 − 5.59127i
5.22286 + 2.17296i
−5.25771 − 2.08722i
5.63982 + 0.438664i
4.12326 − 3.87282i
−2.73024 + 4.95437i
2.74854 + 4.94424i
−5.37943 + 1.74979i
−1.78549 − 5.36768i
0 − 29.6034i 0 0 0 −90.3364 0 −633.360 0
401.2 0 − 25.2673i 0 0 0 185.199 0 −395.434 0
401.3 0 − 23.5470i 0 0 0 39.5751 0 −311.462 0
401.4 0 − 18.7068i 0 0 0 −207.924 0 −106.946 0
401.5 0 − 18.2848i 0 0 0 2.25573 0 −91.3327 0
401.6 0 − 17.0419i 0 0 0 191.207 0 −47.4256 0
401.7 0 − 10.4354i 0 0 0 −66.0164 0 134.101 0
401.8 0 − 8.95730i 0 0 0 −179.198 0 162.767 0
401.9 0 − 2.58812i 0 0 0 27.4015 0 236.302 0
401.10 0 − 0.457817i 0 0 0 195.837 0 242.790 0
401.11 0 0.457817i 0 0 0 195.837 0 242.790 0
401.12 0 2.58812i 0 0 0 27.4015 0 236.302 0
401.13 0 8.95730i 0 0 0 −179.198 0 162.767 0
401.14 0 10.4354i 0 0 0 −66.0164 0 134.101 0
401.15 0 17.0419i 0 0 0 191.207 0 −47.4256 0
401.16 0 18.2848i 0 0 0 2.25573 0 −91.3327 0
401.17 0 18.7068i 0 0 0 −207.924 0 −106.946 0
401.18 0 23.5470i 0 0 0 39.5751 0 −311.462 0
401.19 0 25.2673i 0 0 0 185.199 0 −395.434 0
401.20 0 29.6034i 0 0 0 −90.3364 0 −633.360 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.d.d 20
4.b odd 2 1 200.6.d.c ✓ 20
5.b even 2 1 800.6.d.b 20
5.c odd 4 2 800.6.f.d 40
8.b even 2 1 inner 800.6.d.d 20
8.d odd 2 1 200.6.d.c ✓ 20
20.d odd 2 1 200.6.d.d yes 20
20.e even 4 2 200.6.f.d 40
40.e odd 2 1 200.6.d.d yes 20
40.f even 2 1 800.6.d.b 20
40.i odd 4 2 800.6.f.d 40
40.k even 4 2 200.6.f.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.d.c ✓ 20 4.b odd 2 1
200.6.d.c ✓ 20 8.d odd 2 1
200.6.d.d yes 20 20.d odd 2 1
200.6.d.d yes 20 40.e odd 2 1
200.6.f.d 40 20.e even 4 2
200.6.f.d 40 40.k even 4 2
800.6.d.b 20 5.b even 2 1
800.6.d.b 20 40.f even 2 1
800.6.d.d 20 1.a even 1 1 trivial
800.6.d.d 20 8.b even 2 1 inner
800.6.f.d 40 5.c odd 4 2
800.6.f.d 40 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{20} + 3240 T_{3}^{18} + 4338819 T_{3}^{16} + 3123786848 T_{3}^{14} + 1315141752042 T_{3}^{12} + \cdots + 12\!\cdots\!83 \) Copy content Toggle raw display
\( T_{7}^{10} - 98 T_{7}^{9} - 94896 T_{7}^{8} + 7891024 T_{7}^{7} + 2920561232 T_{7}^{6} + \cdots + 37\!\cdots\!52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 12\!\cdots\!83 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 37\!\cdots\!52)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 32\!\cdots\!75 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 29\!\cdots\!99)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 38\!\cdots\!23 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 20\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 85\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 49\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 13\!\cdots\!75)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 51\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 56\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 17\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 30\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 67\!\cdots\!63 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 32\!\cdots\!17)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 50\!\cdots\!47 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 15\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 76\!\cdots\!96)^{2} \) Copy content Toggle raw display
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