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Properties

Label 800.6.f.b
Level $800$
Weight $6$
Character orbit 800.f
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(49,800)]
 
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, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.f (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: \( 2^{93}\cdot 3^{4}\cdot 5^{8} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - 2) q^{3} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{7} - 3 \beta_{4} + 82) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - 2) q^{3} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{7} - 3 \beta_{4} + 82) q^{9} + (\beta_{5} - 5 \beta_1) q^{11} + (\beta_{11} - 7 \beta_{4} + 1) q^{13} + ( - \beta_{10} + 2 \beta_{6} + \beta_{5}) q^{17} + (\beta_{15} - \beta_{10} + \cdots - 28 \beta_1) q^{19}+ \cdots + ( - 36 \beta_{19} + 35 \beta_{17} + \cdots + 759 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 36 q^{3} + 1620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 36 q^{3} + 1620 q^{9} - 11664 q^{27} - 7160 q^{31} - 3608 q^{37} - 44904 q^{39} + 11608 q^{41} + 51772 q^{43} - 18756 q^{49} + 928 q^{53} + 161604 q^{67} + 200312 q^{71} + 26008 q^{77} + 282080 q^{79} + 65172 q^{81} + 99092 q^{83} + 3160 q^{89} + 293472 q^{93}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 1334623 \nu^{19} + 6032834 \nu^{18} - 30741327 \nu^{17} - 53342094 \nu^{16} + \cdots - 31\!\cdots\!84 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19644053 \nu^{19} + 2191429078 \nu^{18} - 12822196965 \nu^{17} - 59620134810 \nu^{16} + \cdots - 23\!\cdots\!16 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 169540969 \nu^{19} - 1318276306 \nu^{18} - 7653071097 \nu^{17} + 12155775486 \nu^{16} + \cdots - 46\!\cdots\!52 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1159161 \nu^{19} + 876270 \nu^{18} - 12454985 \nu^{17} + 11755262 \nu^{16} + \cdots - 12\!\cdots\!68 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4890613 \nu^{19} + 48190762 \nu^{18} + 537302853 \nu^{17} + 847637274 \nu^{16} + \cdots + 66\!\cdots\!04 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 272744459 \nu^{19} - 336548630 \nu^{18} - 9052177851 \nu^{17} - 8430572454 \nu^{16} + \cdots - 12\!\cdots\!12 ) / 42\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6652887 \nu^{19} + 36722546 \nu^{18} - 184242311 \nu^{17} - 304984990 \nu^{16} + \cdots - 28\!\cdots\!80 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1367865 \nu^{19} + 29312786 \nu^{18} + 135387273 \nu^{17} + 168905730 \nu^{16} + \cdots + 34\!\cdots\!84 ) / 699495553695744 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31899645 \nu^{19} + 809516954 \nu^{18} + 3244972941 \nu^{17} - 17640677046 \nu^{16} + \cdots + 17\!\cdots\!84 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3694915601 \nu^{19} + 11110737314 \nu^{18} + 93222909729 \nu^{17} + \cdots + 34\!\cdots\!28 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55721043 \nu^{19} + 219860218 \nu^{18} - 898998723 \nu^{17} - 1638138966 \nu^{16} + \cdots - 20\!\cdots\!16 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 78280929 \nu^{19} - 160368670 \nu^{18} + 1667183025 \nu^{17} + 8566089042 \nu^{16} + \cdots + 54\!\cdots\!92 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 50108067 \nu^{19} - 24960934 \nu^{18} - 634618131 \nu^{17} + 2052577290 \nu^{16} + \cdots - 32\!\cdots\!40 ) / 69\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 9200216753 \nu^{19} + 21998014562 \nu^{18} + 285245726913 \nu^{17} + \cdots + 26\!\cdots\!04 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18774007967 \nu^{19} + 17287544002 \nu^{18} - 462627382671 \nu^{17} + 48908230770 \nu^{16} + \cdots - 48\!\cdots\!88 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 45543489 \nu^{19} - 74343774 \nu^{18} + 1184585489 \nu^{17} - 3074485742 \nu^{16} + \cdots + 68\!\cdots\!28 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 21649789733 \nu^{19} + 29698586038 \nu^{18} - 215095951989 \nu^{17} + 51227734854 \nu^{16} + \cdots - 22\!\cdots\!16 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 43230699 \nu^{19} - 28761802 \nu^{18} + 422450907 \nu^{17} + 517520646 \nu^{16} + \cdots + 83\!\cdots\!52 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 940503239 \nu^{19} - 698250446 \nu^{18} - 17248345527 \nu^{17} + 18650325474 \nu^{16} + \cdots - 17\!\cdots\!80 ) / 59\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{19} + 2 \beta_{18} + \beta_{17} - 4 \beta_{16} - \beta_{15} - 9 \beta_{14} + 4 \beta_{13} + \cdots + 556 ) / 5120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 30 \beta_{19} - 6 \beta_{18} + 23 \beta_{17} + 4 \beta_{16} + 17 \beta_{15} - 15 \beta_{14} + \cdots + 9572 ) / 5120 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{19} + 9 \beta_{18} + 38 \beta_{17} + 12 \beta_{16} - 38 \beta_{15} - 42 \beta_{14} + \cdots - 7822 ) / 1280 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 34 \beta_{19} - 86 \beta_{18} + 21 \beta_{17} - 4 \beta_{16} + 51 \beta_{15} - 45 \beta_{14} + \cdots - 38652 ) / 1024 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 934 \beta_{19} - 38 \beta_{18} + 1647 \beta_{17} + 516 \beta_{16} - 1327 \beta_{15} + \cdots - 685204 ) / 5120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 670 \beta_{19} - 19 \beta_{18} - 1696 \beta_{17} + 972 \beta_{16} + 856 \beta_{15} - 220 \beta_{14} + \cdots - 2004002 ) / 1280 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 18 \beta_{19} - 1790 \beta_{18} + 2729 \beta_{17} - 14580 \beta_{16} - 34489 \beta_{15} + \cdots - 31790052 ) / 5120 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5642 \beta_{19} + 14610 \beta_{18} - 15429 \beta_{17} - 364 \beta_{16} + 13949 \beta_{15} + \cdots + 13371108 ) / 1024 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5410 \beta_{19} - 17391 \beta_{18} - 127190 \beta_{17} - 62388 \beta_{16} - 27930 \beta_{15} + \cdots + 95486538 ) / 1280 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 868090 \beta_{19} + 1098354 \beta_{18} - 702647 \beta_{17} - 750548 \beta_{16} + 1921647 \beta_{15} + \cdots - 1197052188 ) / 5120 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 683274 \beta_{19} + 1416090 \beta_{18} + 22463 \beta_{17} + 3301700 \beta_{16} - 3028623 \beta_{15} + \cdots + 5245909948 ) / 5120 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 797486 \beta_{19} - 27087 \beta_{18} + 602036 \beta_{17} - 365604 \beta_{16} + 299764 \beta_{15} + \cdots - 1223945154 ) / 256 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2685154 \beta_{19} - 39712990 \beta_{18} + 43194057 \beta_{17} + 35995340 \beta_{16} + \cdots + 110353849708 ) / 5120 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 462442210 \beta_{19} + 130246554 \beta_{18} + 311292087 \beta_{17} + 241839652 \beta_{16} + \cdots - 131995518908 ) / 5120 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 70086250 \beta_{19} - 207186151 \beta_{18} - 94712530 \beta_{17} + 90720972 \beta_{16} + \cdots - 326790446302 ) / 1280 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 562015806 \beta_{19} + 758694250 \beta_{18} + 1588946133 \beta_{17} - 335778052 \beta_{16} + \cdots + 655882842148 ) / 1024 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 4181409978 \beta_{19} - 16438856870 \beta_{18} - 29917935729 \beta_{17} + 5950449540 \beta_{16} + \cdots + 21795997086988 ) / 5120 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 29813936850 \beta_{19} - 5198471939 \beta_{18} - 8594871064 \beta_{17} - 28009899828 \beta_{16} + \cdots + 6036352052878 ) / 1280 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 171922937266 \beta_{19} - 20126841278 \beta_{18} - 56544138647 \beta_{17} + 244253464716 \beta_{16} + \cdots + 94618048950716 ) / 5120 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
3.46430 + 1.99965i
3.46430 − 1.99965i
0.236693 + 3.99299i
0.236693 − 3.99299i
−3.80026 − 1.24819i
−3.80026 + 1.24819i
3.72553 + 1.45618i
3.72553 − 1.45618i
0.593959 − 3.95566i
0.593959 + 3.95566i
−2.63430 + 3.01006i
−2.63430 − 3.01006i
−2.80358 + 2.85306i
−2.80358 − 2.85306i
3.18502 + 2.41984i
3.18502 − 2.41984i
2.93366 − 2.71913i
2.93366 + 2.71913i
−3.90102 + 0.884346i
−3.90102 − 0.884346i
0 −29.2080 0 0 0 − 168.173i 0 610.110 0
49.2 0 −29.2080 0 0 0 168.173i 0 610.110 0
49.3 0 −25.0521 0 0 0 − 103.624i 0 384.607 0
49.4 0 −25.0521 0 0 0 103.624i 0 384.607 0
49.5 0 −11.5927 0 0 0 − 231.529i 0 −108.609 0
49.6 0 −11.5927 0 0 0 231.529i 0 −108.609 0
49.7 0 −10.8240 0 0 0 − 163.706i 0 −125.841 0
49.8 0 −10.8240 0 0 0 163.706i 0 −125.841 0
49.9 0 −6.93089 0 0 0 − 47.1406i 0 −194.963 0
49.10 0 −6.93089 0 0 0 47.1406i 0 −194.963 0
49.11 0 −6.67450 0 0 0 − 38.2812i 0 −198.451 0
49.12 0 −6.67450 0 0 0 38.2812i 0 −198.451 0
49.13 0 10.7455 0 0 0 − 198.733i 0 −127.535 0
49.14 0 10.7455 0 0 0 198.733i 0 −127.535 0
49.15 0 17.3148 0 0 0 − 9.19080i 0 56.8021 0
49.16 0 17.3148 0 0 0 9.19080i 0 56.8021 0
49.17 0 18.7876 0 0 0 − 107.536i 0 109.975 0
49.18 0 18.7876 0 0 0 107.536i 0 109.975 0
49.19 0 25.4343 0 0 0 − 56.4938i 0 403.904 0
49.20 0 25.4343 0 0 0 56.4938i 0 403.904 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f 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.f.b 20
4.b odd 2 1 200.6.f.c 20
5.b even 2 1 800.6.f.c 20
5.c odd 4 1 160.6.d.a 20
5.c odd 4 1 800.6.d.c 20
8.b even 2 1 800.6.f.c 20
8.d odd 2 1 200.6.f.b 20
20.d odd 2 1 200.6.f.b 20
20.e even 4 1 40.6.d.a ✓ 20
20.e even 4 1 200.6.d.b 20
40.e odd 2 1 200.6.f.c 20
40.f even 2 1 inner 800.6.f.b 20
40.i odd 4 1 160.6.d.a 20
40.i odd 4 1 800.6.d.c 20
40.k even 4 1 40.6.d.a ✓ 20
40.k even 4 1 200.6.d.b 20
60.l odd 4 1 360.6.k.b 20
120.q odd 4 1 360.6.k.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a ✓ 20 20.e even 4 1
40.6.d.a ✓ 20 40.k even 4 1
160.6.d.a 20 5.c odd 4 1
160.6.d.a 20 40.i odd 4 1
200.6.d.b 20 20.e even 4 1
200.6.d.b 20 40.k even 4 1
200.6.f.b 20 8.d odd 2 1
200.6.f.b 20 20.d odd 2 1
200.6.f.c 20 4.b odd 2 1
200.6.f.c 20 40.e odd 2 1
360.6.k.b 20 60.l odd 4 1
360.6.k.b 20 120.q odd 4 1
800.6.d.c 20 5.c odd 4 1
800.6.d.c 20 40.i odd 4 1
800.6.f.b 20 1.a even 1 1 trivial
800.6.f.b 20 40.f even 2 1 inner
800.6.f.c 20 5.b even 2 1
800.6.f.c 20 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 18 T_{3}^{9} - 1458 T_{3}^{8} - 23328 T_{3}^{7} + 690600 T_{3}^{6} + 10222224 T_{3}^{5} + \cdots + 377626901472 \) acting on \(S_{6}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots + 377626901472)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 78\!\cdots\!56 \) 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
$37$ \( (T^{10} + \cdots + 42\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 90\!\cdots\!88)^{2} \) 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
$67$ \( (T^{10} + \cdots + 71\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
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