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Properties

Label 240.5.c.e
Level $240$
Weight $5$
Character orbit 240.c
Analytic conductor $24.809$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,5,Mod(209,240)]
 
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([0, 0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.209");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 240.c (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: \(24.8087911401\)
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: \( x^{8} + 110x^{6} + 2705x^{4} + 17000x^{2} + 25600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} - \beta_{6} q^{7} + (\beta_{5} - \beta_{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{2} q^{5} - \beta_{6} q^{7} + (\beta_{5} - \beta_{2} + 6) q^{9} + ( - \beta_{7} - \beta_{5} - \beta_{2}) q^{11} + (3 \beta_{6} + \beta_{3} + 3 \beta_1) q^{13} + ( - \beta_{7} - 3 \beta_{6} + \cdots + 12) q^{15}+ \cdots + (39 \beta_{7} - 29 \beta_{5} + \cdots + 3084) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 52 q^{9} + 100 q^{15} - 408 q^{19} + 212 q^{21} + 1528 q^{31} + 1152 q^{39} - 2900 q^{45} + 480 q^{49} - 7400 q^{51} - 7600 q^{55} - 10808 q^{61} - 8300 q^{69} + 14000 q^{75} - 15144 q^{79} + 34688 q^{81} + 22600 q^{85} + 54912 q^{91} + 24400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 110x^{6} + 2705x^{4} + 17000x^{2} + 25600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -49\nu^{7} + 80\nu^{6} - 5190\nu^{5} + 8200\nu^{4} - 113385\nu^{3} + 148200\nu^{2} - 536200\nu + 148000 ) / 53600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\nu^{7} + 136\nu^{6} + 3010\nu^{5} + 15280\nu^{4} + 45535\nu^{3} + 379240\nu^{2} - 120200\nu + 1511200 ) / 32160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 147 \nu^{7} - 240 \nu^{6} - 15570 \nu^{5} - 24600 \nu^{4} - 340155 \nu^{3} - 444600 \nu^{2} + \cdots - 444000 ) / 53600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 157 \nu^{7} - 790 \nu^{6} + 14920 \nu^{5} - 89350 \nu^{4} + 199555 \nu^{3} - 2259100 \nu^{2} + \cdots - 9334000 ) / 40200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 133 \nu^{7} - 824 \nu^{6} + 14470 \nu^{5} - 83120 \nu^{4} + 338005 \nu^{3} - 1559960 \nu^{2} + \cdots - 4670720 ) / 32160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 89\nu^{7} + 9290\nu^{5} + 187485\nu^{3} + 476200\nu ) / 13400 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 379 \nu^{7} + 344 \nu^{6} + 40690 \nu^{5} + 33920 \nu^{4} + 913315 \nu^{3} + 590360 \nu^{2} + \cdots + 1579760 ) / 16080 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{7} + 15\beta_{6} - 5\beta_{5} - 4\beta_{4} - 15\beta_{3} - 25\beta_{2} - 41\beta_1 ) / 300 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 5\beta_{5} - \beta_{4} + 20\beta_{3} + 2\beta_{2} - 59\beta _1 - 822 ) / 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 25\beta_{7} - 150\beta_{6} + 25\beta_{5} + 29\beta_{4} + 170\beta_{2} - 29\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -95\beta_{7} + 475\beta_{5} + 23\beta_{4} - 1560\beta_{3} + 170\beta_{2} + 4657\beta _1 + 49890 ) / 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1555\beta_{7} + 11850\beta_{6} - 1555\beta_{5} - 2459\beta_{4} + 1900\beta_{3} - 13850\beta_{2} + 8159\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1577\beta_{7} - 7885\beta_{5} - 101\beta_{4} + 23900\beta_{3} - 4226\beta_{2} - 71599\beta _1 - 729294 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22465 \beta_{7} - 184890 \beta_{6} + 22465 \beta_{5} + 39545 \beta_{4} - 38060 \beta_{3} + \cdots - 153725 \beta_1 ) / 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
8.83726i
− 8.83726i
− 4.83397i
4.83397i
− 1.49058i
1.49058i
− 2.51271i
2.51271i
0 −8.97294 − 0.697422i 0 −9.58741 + 23.0886i 0 − 67.4301i 0 80.0272 + 12.5158i 0
209.2 0 −8.97294 + 0.697422i 0 −9.58741 − 23.0886i 0 67.4301i 0 80.0272 − 12.5158i 0
209.3 0 −2.64318 − 8.60312i 0 23.0886 + 9.58741i 0 11.6269i 0 −67.0272 + 45.4792i 0
209.4 0 −2.64318 + 8.60312i 0 23.0886 − 9.58741i 0 − 11.6269i 0 −67.0272 − 45.4792i 0
209.5 0 2.64318 − 8.60312i 0 −23.0886 − 9.58741i 0 11.6269i 0 −67.0272 − 45.4792i 0
209.6 0 2.64318 + 8.60312i 0 −23.0886 + 9.58741i 0 − 11.6269i 0 −67.0272 + 45.4792i 0
209.7 0 8.97294 − 0.697422i 0 9.58741 − 23.0886i 0 − 67.4301i 0 80.0272 − 12.5158i 0
209.8 0 8.97294 + 0.697422i 0 9.58741 + 23.0886i 0 67.4301i 0 80.0272 + 12.5158i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d 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.c.e 8
3.b odd 2 1 inner 240.5.c.e 8
4.b odd 2 1 60.5.b.a ✓ 8
5.b even 2 1 inner 240.5.c.e 8
12.b even 2 1 60.5.b.a ✓ 8
15.d odd 2 1 inner 240.5.c.e 8
20.d odd 2 1 60.5.b.a ✓ 8
20.e even 4 2 300.5.g.h 8
60.h even 2 1 60.5.b.a ✓ 8
60.l odd 4 2 300.5.g.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.b.a ✓ 8 4.b odd 2 1
60.5.b.a ✓ 8 12.b even 2 1
60.5.b.a ✓ 8 20.d odd 2 1
60.5.b.a ✓ 8 60.h even 2 1
240.5.c.e 8 1.a even 1 1 trivial
240.5.c.e 8 3.b odd 2 1 inner
240.5.c.e 8 5.b even 2 1 inner
240.5.c.e 8 15.d odd 2 1 inner
300.5.g.h 8 20.e even 4 2
300.5.g.h 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{4} + 4682T_{7}^{2} + 614656 \) Copy content Toggle raw display
\( T_{17}^{4} - 100900T_{17}^{2} + 1236544000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 26 T^{6} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 4682 T^{2} + 614656)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 55600 T^{2} + 760384000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 42912 T^{2} + 11943936)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 100900 T^{2} + 1236544000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 102 T - 192024)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 296350 T^{2} + 21409129000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 1031180544000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 382 T - 158144)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 15774750401536)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 11048532544000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1375562 T^{2} + 386456182336)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 137109500089000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 21069103104000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 40800768064000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2702 T - 7711424)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 27\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 114709561344000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 226681208836096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 3786 T + 1831824)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 63525673849000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 62\!\cdots\!56)^{2} \) Copy content Toggle raw display
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