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Properties

Label 5.18.b.a
Level $5$
Weight $18$
Character orbit 5.b
Analytic conductor $9.161$
Analytic rank $0$
Dimension $8$
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] = [5,18,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), 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: \(9.16110436723\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 203459x^{6} + 12362849196x^{4} + 237701205446144x^{2} + 1320400799499206656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{8}\cdot 5^{12}\cdot 11 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 72387) q^{4} + ( - \beta_{4} + 24 \beta_{2} + \cdots + 47400) q^{5}+ \cdots + ( - 3 \beta_{7} - 12 \beta_{6} + \cdots - 29364543) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 72387) q^{4} + ( - \beta_{4} + 24 \beta_{2} + \cdots + 47400) q^{5}+ \cdots + ( - 1702446174 \beta_{7} + \cdots - 70\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 579096 q^{4} + 379200 q^{5} + 357816 q^{6} - 234916344 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 579096 q^{4} + 379200 q^{5} + 357816 q^{6} - 234916344 q^{9} + 329570200 q^{10} + 463296576 q^{11} - 29937907992 q^{14} + 30646226400 q^{15} + 30848001568 q^{16} - 20615713280 q^{19} - 47558579400 q^{20} - 75039699024 q^{21} + 1768741136160 q^{24} - 1789249435000 q^{25} - 838901194224 q^{26} - 4079017824720 q^{29} + 2416984007400 q^{30} + 11329328658496 q^{31} - 36406243632832 q^{34} + 4019663899200 q^{35} + 59729752432728 q^{36} + 40318460422272 q^{39} - 209747532172000 q^{40} + 97217252847456 q^{41} - 116357853210912 q^{44} - 366841998003600 q^{45} + 10\!\cdots\!36 q^{46}+ \cdots - 56\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 203459x^{6} + 12362849196x^{4} + 237701205446144x^{2} + 1320400799499206656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{7} - 3990585\nu^{5} - 233779723524\nu^{3} - 2952850120132352\nu ) / 7999594960896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 203459 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19613 \nu^{7} - 2309984 \nu^{6} - 3663672279 \nu^{5} - 423805335072 \nu^{4} + \cdots - 20\!\cdots\!04 ) / 66663291340800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 699511 \nu^{7} + 6929952 \nu^{6} - 125502752613 \nu^{5} + 1271416005216 \nu^{4} + \cdots + 60\!\cdots\!12 ) / 199989874022400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 8925\nu^{4} - 13674601908\nu^{2} - 261009316476832 ) / 32033232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 235831 \nu^{7} + 41579712 \nu^{6} - 44063831973 \nu^{5} + 7628496031296 \nu^{4} + \cdots + 36\!\cdots\!32 ) / 39997974804480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 203459 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 3\beta_{5} - 27\beta_{4} + 3\beta_{3} + 1409\beta_{2} - 328045\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} - 666\beta_{6} - 240\beta_{4} - 117523\beta_{3} - 60\beta_{2} - 48\beta _1 + 16669866289 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 143227 \beta_{7} - 359457 \beta_{5} + 3937353 \beta_{4} - 429681 \beta_{3} - 313921307 \beta_{2} + 31317568479 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 107100 \beta_{7} + 122188878 \beta_{6} - 2142000 \beta_{4} + 12625709133 \beta_{3} + \cdots - 15\!\cdots\!19 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 17777883909 \beta_{7} + 38584449567 \beta_{5} - 494752067703 \beta_{4} + 53333651727 \beta_{3} + \cdots - 31\!\cdots\!97 \beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 330.281i
− 256.320i
− 137.667i
− 98.5951i
98.5951i
137.667i
256.320i
330.281i
− 660.562i − 11309.3i −305271. −21187.7 + 873207.i −7.47047e6 − 2.37065e7i 1.15069e8i 1.24087e6 5.76808e8 + 1.39958e7i
4.2 − 512.640i 18245.5i −131728. 346646. − 801733.i 9.35337e6 − 1.34806e7i 336370.i −2.03757e8 −4.11001e8 − 1.77705e8i
4.3 − 275.335i 2990.60i 55262.8 −756669. + 436339.i 823415. 2.43909e7i − 5.13044e7i 1.20196e8 1.20139e8 + 2.08337e8i
4.4 − 197.190i − 12817.1i 92188.0 620811. − 614437.i −2.52741e6 4.49140e6i − 4.40247e7i −3.51381e7 −1.21161e8 − 1.22418e8i
4.5 197.190i 12817.1i 92188.0 620811. + 614437.i −2.52741e6 − 4.49140e6i 4.40247e7i −3.51381e7 −1.21161e8 + 1.22418e8i
4.6 275.335i − 2990.60i 55262.8 −756669. − 436339.i 823415. − 2.43909e7i 5.13044e7i 1.20196e8 1.20139e8 − 2.08337e8i
4.7 512.640i − 18245.5i −131728. 346646. + 801733.i 9.35337e6 1.34806e7i − 336370.i −2.03757e8 −4.11001e8 + 1.77705e8i
4.8 660.562i 11309.3i −305271. −21187.7 − 873207.i −7.47047e6 2.37065e7i − 1.15069e8i 1.24087e6 5.76808e8 − 1.39958e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.18.b.a ✓ 8
3.b odd 2 1 45.18.b.b 8
4.b odd 2 1 80.18.c.b 8
5.b even 2 1 inner 5.18.b.a ✓ 8
5.c odd 4 2 25.18.a.f 8
15.d odd 2 1 45.18.b.b 8
20.d odd 2 1 80.18.c.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.18.b.a ✓ 8 1.a even 1 1 trivial
5.18.b.a ✓ 8 5.b even 2 1 inner
25.18.a.f 8 5.c odd 4 2
45.18.b.b 8 3.b odd 2 1
45.18.b.b 8 15.d odd 2 1
80.18.c.b 8 4.b odd 2 1
80.18.c.b 8 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 78\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 51\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 56\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 23\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 76\!\cdots\!56 \) Copy content Toggle raw display
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