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Properties

Label 240.10.f.c
Level $240$
Weight $10$
Character orbit 240.f
Analytic conductor $123.609$
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] = [240,10,Mod(49,240)]
 
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, 0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 240.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: \(123.608600679\)
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: \( x^{8} + 939x^{6} + 217699x^{4} + 14559561x^{2} + 31136400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{12}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{7} - \beta_{5} - \beta_{4} + \cdots - 86) q^{5}+ \cdots - 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{7} - \beta_{5} - \beta_{4} + \cdots - 86) q^{5}+ \cdots + (6561 \beta_{7} + 13122 \beta_{6} + \cdots - 59101488) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 690 q^{5} - 52488 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 690 q^{5} - 52488 q^{9} + 71988 q^{11} - 80190 q^{15} - 851584 q^{19} - 1593108 q^{21} + 1695500 q^{25} - 73572 q^{29} - 474088 q^{31} + 36357180 q^{35} - 12959676 q^{39} + 93320088 q^{41} + 4527090 q^{45} + 51329600 q^{49} + 108196236 q^{51} - 64428480 q^{55} - 236526036 q^{59} - 357427760 q^{61} + 19848300 q^{65} + 167059584 q^{69} + 156890664 q^{71} + 528573600 q^{75} - 863922280 q^{79} + 344373768 q^{81} - 2223350420 q^{85} + 357382224 q^{89} - 214754328 q^{91} - 1698584640 q^{95} - 472313268 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 939x^{6} + 217699x^{4} + 14559561x^{2} + 31136400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 111\nu^{6} + 52926\nu^{4} - 12161985\nu^{2} - 1591447600 ) / 590048 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 597\nu^{7} + 559746\nu^{5} + 132904173\nu^{3} + 10255365504\nu ) / 182914880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 676 \nu^{7} - 69255 \nu^{6} - 160080 \nu^{5} - 55626750 \nu^{4} - 468247556 \nu^{3} + \cdots + 52856476560 ) / 159312960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 676 \nu^{7} + 69255 \nu^{6} - 160080 \nu^{5} + 55626750 \nu^{4} - 468247556 \nu^{3} + \cdots - 52856476560 ) / 159312960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -20285\nu^{7} - 18191922\nu^{5} - 3715480325\nu^{3} - 205760824848\nu ) / 1234675440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42361 \nu^{7} + 1720035 \nu^{6} - 38063082 \nu^{5} + 1370729790 \nu^{4} + \cdots + 5687751607920 ) / 2469350880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10441\nu^{7} + 9375834\nu^{5} + 1924192249\nu^{3} + 95661340776\nu ) / 205779240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -9\beta_{7} - 27\beta_{5} + 4\beta_{2} ) / 540 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{7} + 14\beta_{6} + 7\beta_{5} - 11\beta_{4} + 11\beta_{3} - \beta _1 - 42243 ) / 180 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 389\beta_{7} + 1643\beta_{5} + 32\beta_{4} + 32\beta_{3} + 2140\beta_{2} ) / 60 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4567\beta_{7} - 9134\beta_{6} - 4567\beta_{5} + 7811\beta_{4} - 7811\beta_{3} - 2279\beta _1 + 20074923 ) / 180 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -611583\beta_{7} - 2947689\beta_{5} - 92880\beta_{4} - 92880\beta_{3} - 5089012\beta_{2} ) / 180 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2944567 \beta_{7} + 5889134 \beta_{6} + 2944567 \beta_{5} - 4929611 \beta_{4} + 4929611 \beta_{3} + \cdots - 11619671523 ) / 180 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 121718629\beta_{7} + 607018843\beta_{5} + 21904192\beta_{4} + 21904192\beta_{3} + 1124824220\beta_{2} ) / 60 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 25.1000i
13.6993i
1.48693i
10.9137i
25.1000i
− 13.6993i
− 1.48693i
− 10.9137i
0 − 81.0000i 0 −1324.50 − 445.892i 0 176.118i 0 −6561.00 0
49.2 0 − 81.0000i 0 −743.946 − 1183.08i 0 3575.78i 0 −6561.00 0
49.3 0 − 81.0000i 0 343.445 + 1354.68i 0 − 10707.3i 0 −6561.00 0
49.4 0 − 81.0000i 0 1380.00 − 220.717i 0 − 2878.61i 0 −6561.00 0
49.5 0 81.0000i 0 −1324.50 + 445.892i 0 − 176.118i 0 −6561.00 0
49.6 0 81.0000i 0 −743.946 + 1183.08i 0 − 3575.78i 0 −6561.00 0
49.7 0 81.0000i 0 343.445 − 1354.68i 0 10707.3i 0 −6561.00 0
49.8 0 81.0000i 0 1380.00 + 220.717i 0 2878.61i 0 −6561.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
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 240.10.f.c 8
4.b odd 2 1 15.10.b.a ✓ 8
5.b even 2 1 inner 240.10.f.c 8
12.b even 2 1 45.10.b.c 8
20.d odd 2 1 15.10.b.a ✓ 8
20.e even 4 1 75.10.a.i 4
20.e even 4 1 75.10.a.l 4
60.h even 2 1 45.10.b.c 8
60.l odd 4 1 225.10.a.q 4
60.l odd 4 1 225.10.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.b.a ✓ 8 4.b odd 2 1
15.10.b.a ✓ 8 20.d odd 2 1
45.10.b.c 8 12.b even 2 1
45.10.b.c 8 60.h even 2 1
75.10.a.i 4 20.e even 4 1
75.10.a.l 4 20.e even 4 1
225.10.a.q 4 60.l odd 4 1
225.10.a.u 4 60.l odd 4 1
240.10.f.c 8 1.a even 1 1 trivial
240.10.f.c 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 135749628 T_{7}^{6} + \cdots + 37\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6561)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 69\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 30\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 86\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 70\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
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