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Properties

Label 240.5.e.c
Level $240$
Weight $5$
Character orbit 240.e
Analytic conductor $24.809$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,5,Mod(31,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 240.e (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: \(24.8087911401\)
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: \( x^{8} - x^{7} + 113x^{6} + 106x^{5} + 10018x^{4} + 4722x^{3} + 283257x^{2} + 7587x + 6395841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q - 3 \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{4} + \beta_{2} - 2 \beta_1) q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{4} + \beta_{2} - 2 \beta_1) q^{7} - 27 q^{9} + (\beta_{5} - 2 \beta_{4} + \cdots - 20 \beta_1) q^{11}+ \cdots + ( - 27 \beta_{5} + 54 \beta_{4} + \cdots + 540 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 216 q^{9} - 304 q^{13} + 192 q^{17} - 144 q^{21} + 1000 q^{25} + 2112 q^{29} - 1440 q^{33} + 2512 q^{37} - 2448 q^{41} - 4792 q^{49} - 1440 q^{53} + 6192 q^{57} - 14576 q^{61} - 3600 q^{65} - 4032 q^{69} + 8720 q^{73} + 36192 q^{77} + 5832 q^{81} - 2400 q^{85} - 17712 q^{89} - 2160 q^{93} + 60080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 113x^{6} + 106x^{5} + 10018x^{4} + 4722x^{3} + 283257x^{2} + 7587x + 6395841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 222155 \nu^{7} + 59963879 \nu^{6} - 67921171 \nu^{5} + 5376676693 \nu^{4} + \cdots + 10189240320510 ) / 6775284487014 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 280 \nu^{7} - 139375 \nu^{6} + 461570 \nu^{5} - 15827150 \nu^{4} + 4499470 \nu^{3} + \cdots - 19154481615 ) / 661980924 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14000 \nu^{7} - 142145 \nu^{6} + 1265500 \nu^{5} + 1842500 \nu^{4} + 50650220 \nu^{3} + \cdots + 33526547865 ) / 2407485564 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 940239048 \nu^{7} + 11899574281 \nu^{6} - 288002361814 \nu^{5} + 1236017920922 \nu^{4} + \cdots + 12\!\cdots\!53 ) / 49685419571436 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3965074 \nu^{7} - 4649543 \nu^{6} - 330284576 \nu^{5} - 1422709144 \nu^{4} + \cdots - 1060433191881 ) / 112750573914 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26212928 \nu^{7} + 1322063 \nu^{6} - 2369461456 \nu^{5} - 3449808560 \nu^{4} + \cdots + 2173407635889 ) / 265224659634 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 72460024 \nu^{7} - 121793332 \nu^{6} + 6549868598 \nu^{5} + 9536256730 \nu^{4} + \cdots + 5170988691252 ) / 397836989451 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5\beta_{6} - 15\beta_{5} - 6\beta_{3} + 6\beta_{2} + 30\beta _1 + 30 ) / 240 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -10\beta_{7} - 5\beta_{6} - 15\beta_{5} + 30\beta_{4} + 254\beta_{3} + 264\beta_{2} + 6750\beta _1 - 6750 ) / 240 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -55\beta_{7} - 85\beta_{6} + 278\beta_{3} - 2460 ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1340 \beta_{7} - 885 \beta_{6} + 2655 \beta_{5} - 4020 \beta_{4} + 29542 \beta_{3} - 30882 \beta_{2} + \cdots - 462270 ) / 240 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5190 \beta_{7} + 5261 \beta_{6} + 15783 \beta_{5} - 15570 \beta_{4} - 27630 \beta_{3} - 32820 \beta_{2} + \cdots + 293646 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18760\beta_{7} + 13970\beta_{6} - 350144\beta_{3} + 4517775 ) / 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2496080 \beta_{7} + 2195405 \beta_{6} - 6586215 \beta_{5} + 7488240 \beta_{4} - 15373078 \beta_{3} + \cdots + 174311790 ) / 240 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−4.05736 − 7.02756i
4.86638 + 8.42882i
2.67907 + 4.64029i
−2.98809 − 5.17553i
4.86638 − 8.42882i
−4.05736 + 7.02756i
−2.98809 + 5.17553i
2.67907 − 4.64029i
0 − 5.19615i 0 −11.1803 0 − 26.1824i 0 −27.0000 0
31.2 0 − 5.19615i 0 −11.1803 0 50.2381i 0 −27.0000 0
31.3 0 − 5.19615i 0 11.1803 0 − 82.4854i 0 −27.0000 0
31.4 0 − 5.19615i 0 11.1803 0 44.5733i 0 −27.0000 0
31.5 0 5.19615i 0 −11.1803 0 − 50.2381i 0 −27.0000 0
31.6 0 5.19615i 0 −11.1803 0 26.1824i 0 −27.0000 0
31.7 0 5.19615i 0 11.1803 0 − 44.5733i 0 −27.0000 0
31.8 0 5.19615i 0 11.1803 0 82.4854i 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.5.e.c ✓ 8
3.b odd 2 1 720.5.e.f 8
4.b odd 2 1 inner 240.5.e.c ✓ 8
5.b even 2 1 1200.5.e.n 8
5.c odd 4 2 1200.5.j.h 16
8.b even 2 1 960.5.e.c 8
8.d odd 2 1 960.5.e.c 8
12.b even 2 1 720.5.e.f 8
20.d odd 2 1 1200.5.e.n 8
20.e even 4 2 1200.5.j.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.5.e.c ✓ 8 1.a even 1 1 trivial
240.5.e.c ✓ 8 4.b odd 2 1 inner
720.5.e.f 8 3.b odd 2 1
720.5.e.f 8 12.b even 2 1
960.5.e.c 8 8.b even 2 1
960.5.e.c 8 8.d odd 2 1
1200.5.e.n 8 5.b even 2 1
1200.5.e.n 8 20.d odd 2 1
1200.5.j.h 16 5.c odd 4 2
1200.5.j.h 16 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 12000T_{7}^{6} + 43460352T_{7}^{4} + 58592747520T_{7}^{2} + 23387824521216 \) acting on \(S_{5}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 23387824521216 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} + 152 T^{3} + \cdots + 1191470416)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 96 T^{3} + \cdots + 18766921104)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} - 1056 T^{3} + \cdots - 288363596400)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 2218247408080)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 7502818014576)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 67718177333136)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 476401204598000)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 89\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 353799306666224)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 25\!\cdots\!24)^{2} \) Copy content Toggle raw display
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