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Properties

Label 240.4.w.a
Level $240$
Weight $4$
Character orbit 240.w
Analytic conductor $14.160$
Analytic rank $0$
Dimension $12$
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] = [240,4,Mod(127,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.127");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.w (of order \(4\), degree \(2\), 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: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} - 624 x^{9} - 2192 x^{8} + 5348 x^{7} + 190832 x^{6} + 74508 x^{5} + \cdots + 8925147729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{8} + 3) q^{5} + ( - \beta_{11} + \beta_{10} + \cdots + 2 \beta_{3}) q^{7}+ \cdots + 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{8} + 3) q^{5} + ( - \beta_{11} + \beta_{10} + \cdots + 2 \beta_{3}) q^{7}+ \cdots + (9 \beta_{5} - 36 \beta_{3} - 36 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 32 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 32 q^{5} - 4 q^{13} - 308 q^{17} + 192 q^{21} + 492 q^{25} - 408 q^{33} - 540 q^{37} + 720 q^{41} + 36 q^{45} - 2908 q^{53} - 768 q^{57} + 2368 q^{61} + 2868 q^{65} - 596 q^{73} + 1552 q^{77} - 972 q^{81} + 3692 q^{85} - 720 q^{93} - 3764 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} - 624 x^{9} - 2192 x^{8} + 5348 x^{7} + 190832 x^{6} + 74508 x^{5} + \cdots + 8925147729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 43\!\cdots\!20 \nu^{11} + \cdots + 79\!\cdots\!66 ) / 30\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43\!\cdots\!22 \nu^{11} + \cdots - 90\!\cdots\!36 ) / 22\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 89\!\cdots\!50 \nu^{11} + \cdots - 17\!\cdots\!46 ) / 30\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!25 \nu^{11} + \cdots - 23\!\cdots\!92 ) / 30\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65\!\cdots\!45 \nu^{11} + \cdots + 15\!\cdots\!48 ) / 93\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!86 \nu^{11} + \cdots - 38\!\cdots\!53 ) / 15\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!03 \nu^{11} + \cdots + 31\!\cdots\!67 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!29 \nu^{11} + \cdots - 26\!\cdots\!53 ) / 10\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63\!\cdots\!24 \nu^{11} + \cdots - 17\!\cdots\!27 ) / 51\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20\!\cdots\!35 \nu^{11} + \cdots - 35\!\cdots\!08 ) / 86\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!10 \nu^{11} + \cdots + 40\!\cdots\!16 ) / 91\!\cdots\!65 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - \beta_{5} + \cdots + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 11 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} + 21 \beta_{5} + \cdots + 94 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 80 \beta_{11} - 85 \beta_{10} + 39 \beta_{9} + 69 \beta_{8} + 69 \beta_{7} + 39 \beta_{6} + \cdots + 894 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 334 \beta_{11} + 1635 \beta_{10} - 1623 \beta_{9} + 1623 \beta_{8} + 1641 \beta_{7} + 1641 \beta_{6} + \cdots + 18612 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2074 \beta_{11} - 10505 \beta_{10} - 17961 \beta_{9} + 4137 \beta_{8} - 4137 \beta_{7} + 17961 \beta_{6} + \cdots + 44562 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 17530 \beta_{10} - 11067 \beta_{9} - 11067 \beta_{8} + 17157 \beta_{7} - 17157 \beta_{6} + \cdots - 1480 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 414530 \beta_{11} + 166721 \beta_{10} + 983433 \beta_{9} + 207267 \beta_{8} + 207267 \beta_{7} + \cdots + 13379718 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2040466 \beta_{11} - 454389 \beta_{10} - 17232441 \beta_{9} + 17232441 \beta_{8} + 5220447 \beta_{7} + \cdots - 45663300 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7300360 \beta_{11} - 13264925 \beta_{10} - 18560031 \beta_{9} - 40314441 \beta_{8} + 40314441 \beta_{7} + \cdots + 602081562 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 324989645 \beta_{10} - 1437339 \beta_{9} - 1437339 \beta_{8} - 742952793 \beta_{7} + \cdots - 181665422 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1222354138 \beta_{11} + 3449185009 \beta_{10} - 2722105245 \beta_{9} + 4351336113 \beta_{8} + \cdots - 13939747230 ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−3.42194 − 8.06099i
4.17154 + 0.802529i
0.250406 + 8.25846i
−8.06099 − 3.42194i
0.802529 + 4.17154i
8.25846 + 0.250406i
−3.42194 + 8.06099i
4.17154 − 0.802529i
0.250406 − 8.25846i
−8.06099 + 3.42194i
0.802529 − 4.17154i
8.25846 − 0.250406i
0 −2.12132 − 2.12132i 0 −9.48294 − 5.92233i 0 −5.14531 + 5.14531i 0 9.00000i 0
127.2 0 −2.12132 − 2.12132i 0 6.97406 + 8.73856i 0 −23.3388 + 23.3388i 0 9.00000i 0
127.3 0 −2.12132 − 2.12132i 0 10.5089 − 3.81623i 0 17.1704 − 17.1704i 0 9.00000i 0
127.4 0 2.12132 + 2.12132i 0 −9.48294 − 5.92233i 0 5.14531 − 5.14531i 0 9.00000i 0
127.5 0 2.12132 + 2.12132i 0 6.97406 + 8.73856i 0 23.3388 − 23.3388i 0 9.00000i 0
127.6 0 2.12132 + 2.12132i 0 10.5089 − 3.81623i 0 −17.1704 + 17.1704i 0 9.00000i 0
223.1 0 −2.12132 + 2.12132i 0 −9.48294 + 5.92233i 0 −5.14531 − 5.14531i 0 − 9.00000i 0
223.2 0 −2.12132 + 2.12132i 0 6.97406 − 8.73856i 0 −23.3388 − 23.3388i 0 − 9.00000i 0
223.3 0 −2.12132 + 2.12132i 0 10.5089 + 3.81623i 0 17.1704 + 17.1704i 0 − 9.00000i 0
223.4 0 2.12132 − 2.12132i 0 −9.48294 + 5.92233i 0 5.14531 + 5.14531i 0 − 9.00000i 0
223.5 0 2.12132 − 2.12132i 0 6.97406 − 8.73856i 0 23.3388 + 23.3388i 0 − 9.00000i 0
223.6 0 2.12132 − 2.12132i 0 10.5089 + 3.81623i 0 −17.1704 − 17.1704i 0 − 9.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.w.a ✓ 12
3.b odd 2 1 720.4.x.f 12
4.b odd 2 1 inner 240.4.w.a ✓ 12
5.c odd 4 1 inner 240.4.w.a ✓ 12
12.b even 2 1 720.4.x.f 12
15.e even 4 1 720.4.x.f 12
20.e even 4 1 inner 240.4.w.a ✓ 12
60.l odd 4 1 720.4.x.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.w.a ✓ 12 1.a even 1 1 trivial
240.4.w.a ✓ 12 4.b odd 2 1 inner
240.4.w.a ✓ 12 5.c odd 4 1 inner
240.4.w.a ✓ 12 20.e even 4 1 inner
720.4.x.f 12 3.b odd 2 1
720.4.x.f 12 12.b even 2 1
720.4.x.f 12 15.e even 4 1
720.4.x.f 12 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 1537288T_{7}^{8} + 416934543888T_{7}^{4} + 1156831381426176 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 81)^{3} \) Copy content Toggle raw display
$5$ \( (T^{6} - 16 T^{5} + \cdots + 1953125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{6} + 1796 T^{4} + \cdots + 49362048)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots + 2259533088)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 154 T^{5} + \cdots + 154697018912)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 25736 T^{4} + \cdots - 262737105408)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 11926634622016)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 13436928000000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 270 T^{5} + \cdots + 375497780000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 180 T^{2} + \cdots + 720000)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 675215395605632)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 680443723141632)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 592 T^{2} + \cdots - 1942704)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 45\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 17\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 47\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 88\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 48\!\cdots\!28)^{2} \) Copy content Toggle raw display
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