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Properties

Label 800.6.d.c
Level $800$
Weight $6$
Character orbit 800.d
Analytic conductor $128.307$
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] = [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: \( x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{93}\cdot 3^{4}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{4} - 10) q^{7} + (\beta_{5} - \beta_{4} - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{4} - 10) q^{7} + (\beta_{5} - \beta_{4} - 81) q^{9} + (\beta_{3} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{8} + 7 \beta_1) q^{13} + \beta_{7} q^{17} + ( - \beta_{15} + \beta_{3} + \cdots - 3 \beta_1) q^{19}+ \cdots + ( - 14 \beta_{19} - 14 \beta_{17} + \cdots + 101 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 196 q^{7} - 1620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 196 q^{7} - 1620 q^{9} - 4676 q^{23} - 7160 q^{31} - 5672 q^{33} + 44904 q^{39} + 11608 q^{41} + 44180 q^{47} + 18756 q^{49} - 5032 q^{57} + 240620 q^{63} + 200312 q^{71} + 105136 q^{73} - 282080 q^{79} + 65172 q^{81} - 332592 q^{87} - 3160 q^{89} - 147376 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 79348087 \nu^{19} + 164467438 \nu^{18} + 2750967783 \nu^{17} + 2347892670 \nu^{16} + \cdots + 38\!\cdots\!76 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6673115 \nu^{19} - 30164170 \nu^{18} + 153706635 \nu^{17} + 266710470 \nu^{16} + \cdots + 15\!\cdots\!20 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 392347319 \nu^{19} + 3248676206 \nu^{18} + 37138350375 \nu^{17} + 56596678206 \nu^{16} + \cdots + 46\!\cdots\!32 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13756221 \nu^{19} - 7338982 \nu^{18} + 200878797 \nu^{17} - 336652854 \nu^{16} + \cdots + 16\!\cdots\!56 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16634889 \nu^{19} + 110715086 \nu^{18} - 463943001 \nu^{17} - 1145810466 \nu^{16} + \cdots - 79\!\cdots\!76 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 916605 \nu^{19} - 557722 \nu^{18} - 72289293 \nu^{17} + 76715958 \nu^{16} + \cdots - 96\!\cdots\!52 ) / 269036751421440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 836343 \nu^{19} + 4825138 \nu^{18} + 26902617 \nu^{17} - 67997406 \nu^{16} + \cdots + 32\!\cdots\!04 ) / 241205363343360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2596081553 \nu^{19} + 2571847202 \nu^{18} + 86300109729 \nu^{17} + 14967625938 \nu^{16} + \cdots + 80\!\cdots\!24 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 93629767 \nu^{19} + 702882478 \nu^{18} + 981182583 \nu^{17} - 7979058114 \nu^{16} + \cdots + 40\!\cdots\!60 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1017903 \nu^{19} - 5922366 \nu^{18} - 40945759 \nu^{17} - 18273230 \nu^{16} + \cdots - 10\!\cdots\!80 ) / 160803575562240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 139676643 \nu^{19} + 2240750630 \nu^{18} + 9107502675 \nu^{17} - 41092516746 \nu^{16} + \cdots + 52\!\cdots\!64 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1494615 \nu^{19} - 139442 \nu^{18} + 12535687 \nu^{17} - 26796322 \nu^{16} + \cdots + 93\!\cdots\!08 ) / 145728240353280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 55930047 \nu^{19} + 298875042 \nu^{18} - 471238127 \nu^{17} - 811773934 \nu^{16} + \cdots - 48\!\cdots\!04 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3023485621 \nu^{19} - 3513130646 \nu^{18} + 23790813957 \nu^{17} - 48592005414 \nu^{16} + \cdots + 26\!\cdots\!04 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13747776289 \nu^{19} - 4153513666 \nu^{18} - 348265483569 \nu^{17} + 147182976654 \nu^{16} + \cdots - 60\!\cdots\!48 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 445476863 \nu^{19} + 755206978 \nu^{18} - 2379736431 \nu^{17} + 690187890 \nu^{16} + \cdots - 55\!\cdots\!20 ) / 31\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 279607999 \nu^{19} + 207980926 \nu^{18} + 4651875759 \nu^{17} - 5763844146 \nu^{16} + \cdots + 21\!\cdots\!60 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 244379043 \nu^{19} - 882778202 \nu^{18} + 3542156307 \nu^{17} - 8059440138 \nu^{16} + \cdots + 22\!\cdots\!32 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 57874591541 \nu^{19} - 28605234794 \nu^{18} - 940175038341 \nu^{17} + 1158073375782 \nu^{16} + \cdots - 96\!\cdots\!68 ) / 21\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 28 \beta_{19} + 18 \beta_{18} - 2 \beta_{17} - 12 \beta_{16} - 3 \beta_{15} + 23 \beta_{14} + \cdots + 2538 ) / 25600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 28 \beta_{19} - 38 \beta_{18} - 78 \beta_{17} + 12 \beta_{16} + 83 \beta_{15} + 17 \beta_{14} + \cdots + 48802 ) / 25600 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 18 \beta_{19} - 39 \beta_{18} - 108 \beta_{17} + 112 \beta_{16} - 32 \beta_{15} + 142 \beta_{14} + \cdots - 39819 ) / 6400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20 \beta_{19} + 34 \beta_{18} - 2 \beta_{17} + 20 \beta_{16} - 19 \beta_{15} + 23 \beta_{14} + \cdots - 39478 ) / 1024 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1308 \beta_{19} - 694 \beta_{18} - 1822 \beta_{17} + 268 \beta_{16} - 5733 \beta_{15} + \cdots - 3461694 ) / 25600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10606 \beta_{19} - 1491 \beta_{18} + 7856 \beta_{17} - 5024 \beta_{16} - 9566 \beta_{15} + \cdots - 9927991 ) / 6400 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 99876 \beta_{19} - 28942 \beta_{18} - 28626 \beta_{17} + 38324 \beta_{16} - 106699 \beta_{15} + \cdots - 157676822 ) / 25600 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8172 \beta_{19} - 2390 \beta_{18} + 6946 \beta_{17} - 20 \beta_{16} - 677 \beta_{15} + \cdots + 13514578 ) / 1024 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 91758 \beta_{19} + 145665 \beta_{18} + 136748 \beta_{17} - 210672 \beta_{16} + 11092 \beta_{15} + \cdots + 480581725 ) / 6400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1462268 \beta_{19} + 1872850 \beta_{18} - 2420018 \beta_{17} - 7964428 \beta_{16} + \cdots - 5939140390 ) / 25600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 11180916 \beta_{19} - 27219414 \beta_{18} - 9920766 \beta_{17} + 11878284 \beta_{16} + \cdots + 26439427586 ) / 25600 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 377730 \beta_{19} + 227613 \beta_{18} - 609080 \beta_{17} + 191424 \beta_{16} + 976198 \beta_{15} + \cdots - 1250502055 ) / 256 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 206557204 \beta_{19} + 27478386 \beta_{18} + 91894446 \beta_{17} + 293929396 \beta_{16} + \cdots + 545833861386 ) / 25600 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2382349028 \beta_{19} + 536563450 \beta_{18} + 872846578 \beta_{17} - 190007412 \beta_{16} + \cdots - 682288380990 ) / 25600 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 160118542 \beta_{19} + 506109225 \beta_{18} + 880534852 \beta_{17} - 86269328 \beta_{16} + \cdots - 1625292214715 ) / 6400 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 894641740 \beta_{19} - 747940190 \beta_{18} - 240157186 \beta_{17} + 1194117652 \beta_{16} + \cdots + 652260593290 ) / 1024 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 28884382716 \beta_{19} + 79191908618 \beta_{18} + 92698079266 \beta_{17} + 5046178316 \beta_{16} + \cdots + 109761795011138 ) / 25600 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 72314538066 \beta_{19} + 46613619069 \beta_{18} - 46315267616 \beta_{17} - 50923610336 \beta_{16} + \cdots + 31192842076569 ) / 6400 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 800902948092 \beta_{19} - 238043720974 \beta_{18} - 362774695378 \beta_{17} - 1025926382668 \beta_{16} + \cdots + 510561238570026 ) / 25600 \) 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
3.46430 + 1.99965i
−3.90102 − 0.884346i
0.236693 − 3.99299i
2.93366 + 2.71913i
3.18502 − 2.41984i
−3.80026 − 1.24819i
3.72553 − 1.45618i
−2.80358 + 2.85306i
0.593959 + 3.95566i
−2.63430 + 3.01006i
−2.63430 − 3.01006i
0.593959 − 3.95566i
−2.80358 − 2.85306i
3.72553 + 1.45618i
−3.80026 + 1.24819i
3.18502 + 2.41984i
2.93366 − 2.71913i
0.236693 + 3.99299i
−3.90102 + 0.884346i
3.46430 − 1.99965i
0 − 29.2080i 0 0 0 −168.173 0 −610.110 0
401.2 0 − 25.4343i 0 0 0 −56.4938 0 −403.904 0
401.3 0 − 25.0521i 0 0 0 103.624 0 −384.607 0
401.4 0 − 18.7876i 0 0 0 −107.536 0 −109.975 0
401.5 0 − 17.3148i 0 0 0 −9.19080 0 −56.8021 0
401.6 0 − 11.5927i 0 0 0 −231.529 0 108.609 0
401.7 0 − 10.8240i 0 0 0 163.706 0 125.841 0
401.8 0 − 10.7455i 0 0 0 198.733 0 127.535 0
401.9 0 − 6.93089i 0 0 0 47.1406 0 194.963 0
401.10 0 − 6.67450i 0 0 0 −38.2812 0 198.451 0
401.11 0 6.67450i 0 0 0 −38.2812 0 198.451 0
401.12 0 6.93089i 0 0 0 47.1406 0 194.963 0
401.13 0 10.7455i 0 0 0 198.733 0 127.535 0
401.14 0 10.8240i 0 0 0 163.706 0 125.841 0
401.15 0 11.5927i 0 0 0 −231.529 0 108.609 0
401.16 0 17.3148i 0 0 0 −9.19080 0 −56.8021 0
401.17 0 18.7876i 0 0 0 −107.536 0 −109.975 0
401.18 0 25.0521i 0 0 0 103.624 0 −384.607 0
401.19 0 25.4343i 0 0 0 −56.4938 0 −403.904 0
401.20 0 29.2080i 0 0 0 −168.173 0 −610.110 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.c 20
4.b odd 2 1 200.6.d.b 20
5.b even 2 1 160.6.d.a 20
5.c odd 4 1 800.6.f.b 20
5.c odd 4 1 800.6.f.c 20
8.b even 2 1 inner 800.6.d.c 20
8.d odd 2 1 200.6.d.b 20
20.d odd 2 1 40.6.d.a ✓ 20
20.e even 4 1 200.6.f.b 20
20.e even 4 1 200.6.f.c 20
40.e odd 2 1 40.6.d.a ✓ 20
40.f even 2 1 160.6.d.a 20
40.i odd 4 1 800.6.f.b 20
40.i odd 4 1 800.6.f.c 20
40.k even 4 1 200.6.f.b 20
40.k even 4 1 200.6.f.c 20
60.h even 2 1 360.6.k.b 20
120.m even 2 1 360.6.k.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a ✓ 20 20.d odd 2 1
40.6.d.a ✓ 20 40.e odd 2 1
160.6.d.a 20 5.b even 2 1
160.6.d.a 20 40.f even 2 1
200.6.d.b 20 4.b odd 2 1
200.6.d.b 20 8.d odd 2 1
200.6.f.b 20 20.e even 4 1
200.6.f.b 20 40.k even 4 1
200.6.f.c 20 20.e even 4 1
200.6.f.c 20 40.k even 4 1
360.6.k.b 20 60.h even 2 1
360.6.k.b 20 120.m even 2 1
800.6.d.c 20 1.a even 1 1 trivial
800.6.d.c 20 8.b even 2 1 inner
800.6.f.b 20 5.c odd 4 1
800.6.f.b 20 40.i odd 4 1
800.6.f.c 20 5.c odd 4 1
800.6.f.c 20 40.i odd 4 1

Hecke kernels

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

\( T_{3}^{20} + 3240 T_{3}^{18} + 4346772 T_{3}^{16} + 3151305344 T_{3}^{14} + 1355603009184 T_{3}^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
\( T_{7}^{10} + 98 T_{7}^{9} - 83922 T_{7}^{8} - 6806560 T_{7}^{7} + 2129001128 T_{7}^{6} + \cdots + 13\!\cdots\!08 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 29\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 88\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 42\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 16\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 82\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 50\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
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