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Properties

Label 240.9.c.d
Level $240$
Weight $9$
Character orbit 240.c
Analytic conductor $97.771$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.209");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(97.7708664147\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 7378 x^{14} + 23156928 x^{12} + 101588726286 x^{10} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{20}\cdot 5^{15} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{6} q^{7} + ( - \beta_{3} + \beta_{2} - 922) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{6} q^{7} + ( - \beta_{3} + \beta_{2} - 922) q^{9} + (\beta_{4} + 3 \beta_{3}) q^{11} + ( - \beta_{9} + \beta_{5} - 27 \beta_1) q^{13} + ( - \beta_{12} + \beta_{5} - 226) q^{15} + ( - \beta_{14} - 2 \beta_{5} + \cdots - 47 \beta_1) q^{17}+ \cdots + ( - 72 \beta_{15} + 72 \beta_{14} + \cdots - 1060492) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 14756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 14756 q^{9} - 3620 q^{15} + 48024 q^{19} + 4724 q^{21} + 224520 q^{25} - 470104 q^{31} + 2849664 q^{39} + 3895180 q^{45} - 19816920 q^{49} - 5026040 q^{51} - 20570480 q^{55} + 18849944 q^{61} - 38669180 q^{69} - 24090640 q^{75} + 90778632 q^{79} + 16242056 q^{81} + 63385880 q^{85} - 22375296 q^{91} - 16968560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 7378 x^{14} + 23156928 x^{12} + 101588726286 x^{10} + \cdots + 34\!\cdots\!81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24860517443 \nu^{15} - 1917610715646 \nu^{14} + 45246948231071 \nu^{13} + \cdots + 41\!\cdots\!74 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24860517443 \nu^{15} - 1917610715646 \nu^{14} + 45246948231071 \nu^{13} + \cdots - 10\!\cdots\!26 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24860517443 \nu^{15} + 2779591272516 \nu^{14} - 45246948231071 \nu^{13} + \cdots + 10\!\cdots\!96 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} - 7378 \nu^{13} - 23156928 \nu^{11} - 101588726286 \nu^{9} + \cdots - 51\!\cdots\!66 \nu ) / 19\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9990143 \nu^{15} + 13366675591 \nu^{13} - 104908177452681 \nu^{11} + \cdots - 22\!\cdots\!41 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49721034886 \nu^{15} - 4154823217233 \nu^{14} - 90493896462142 \nu^{13} + \cdots - 22\!\cdots\!23 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 269662534991 \nu^{15} - 63316590089076 \nu^{14} + \cdots - 65\!\cdots\!80 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 515192467 \nu^{15} - 615081057479 \nu^{13} + \cdots + 14\!\cdots\!29 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 786028665971 \nu^{15} - 483201141673257 \nu^{14} + \cdots - 27\!\cdots\!47 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 327171200152 \nu^{15} - 130384486268541 \nu^{14} - 837569098937492 \nu^{13} + \cdots - 36\!\cdots\!01 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 62428779098 \nu^{15} + 5117553683829 \nu^{14} - 151753762897234 \nu^{13} + \cdots + 31\!\cdots\!29 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4049141812412 \nu^{15} - 214843604579271 \nu^{14} + \cdots + 55\!\cdots\!89 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 84271574038 \nu^{15} - 319601785941 \nu^{14} - 781672829197186 \nu^{13} + \cdots - 17\!\cdots\!71 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3465320720078 \nu^{15} + 460308043912221 \nu^{14} + \cdots + 26\!\cdots\!81 ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 922 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - 2 \beta_{14} + \beta_{13} + 2 \beta_{12} - \beta_{10} - 11 \beta_{9} - \beta_{8} + \cdots - 916 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 63 \beta_{15} + 63 \beta_{14} + 18 \beta_{13} + 60 \beta_{12} - 357 \beta_{11} + 12 \beta_{10} + \cdots + 1015138 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4624 \beta_{15} - 7519 \beta_{14} + 6959 \beta_{13} - 20831 \beta_{12} + 4624 \beta_{10} + 22595 \beta_{9} + \cdots + 11583 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 609156 \beta_{15} - 609156 \beta_{14} + 774072 \beta_{13} + 2920197 \beta_{12} + 899655 \beta_{11} + \cdots - 24229375936 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14029492 \beta_{15} + 69784480 \beta_{14} + 41799856 \beta_{13} + 288620 \beta_{12} - 14029492 \beta_{10} + \cdots + 27770364 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 455274180 \beta_{15} - 455274180 \beta_{14} - 6127927920 \beta_{13} - 6214353672 \beta_{12} + \cdots - 493412860983155 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 372515274100 \beta_{15} - 181942319680 \beta_{14} + 191909050640 \beta_{13} - 1309454872940 \beta_{12} + \cdots + 564424324740 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 17864334358740 \beta_{15} + 17864334358740 \beta_{14} - 13579912603800 \beta_{13} + \cdots + 27\!\cdots\!74 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 718952827444355 \beta_{15} + \cdots + 192632758655220 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 25\!\cdots\!45 \beta_{15} + \cdots - 35\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12\!\cdots\!16 \beta_{15} + \cdots - 61\!\cdots\!25 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 44\!\cdots\!12 \beta_{15} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 19\!\cdots\!44 \beta_{15} + \cdots - 10\!\cdots\!28 \) 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
−77.5456 − 23.4026i
−77.5456 + 23.4026i
−62.9079 − 51.0255i
−62.9079 + 51.0255i
−32.2702 − 74.2942i
−32.2702 + 74.2942i
−16.2916 − 79.3447i
−16.2916 + 79.3447i
16.2916 − 79.3447i
16.2916 + 79.3447i
32.2702 − 74.2942i
32.2702 + 74.2942i
62.9079 − 51.0255i
62.9079 + 51.0255i
77.5456 − 23.4026i
77.5456 + 23.4026i
0 −77.5456 − 23.4026i 0 220.972 + 584.633i 0 256.799i 0 5465.63 + 3629.54i 0
209.2 0 −77.5456 + 23.4026i 0 220.972 − 584.633i 0 − 256.799i 0 5465.63 − 3629.54i 0
209.3 0 −62.9079 − 51.0255i 0 −605.408 − 155.260i 0 459.019i 0 1353.80 + 6419.81i 0
209.4 0 −62.9079 + 51.0255i 0 −605.408 + 155.260i 0 − 459.019i 0 1353.80 − 6419.81i 0
209.5 0 −32.2702 − 74.2942i 0 615.732 − 107.234i 0 − 4017.33i 0 −4478.26 + 4794.99i 0
209.6 0 −32.2702 + 74.2942i 0 615.732 + 107.234i 0 4017.33i 0 −4478.26 − 4794.99i 0
209.7 0 −16.2916 − 79.3447i 0 121.823 − 613.012i 0 3405.57i 0 −6030.17 + 2585.30i 0
209.8 0 −16.2916 + 79.3447i 0 121.823 + 613.012i 0 − 3405.57i 0 −6030.17 − 2585.30i 0
209.9 0 16.2916 − 79.3447i 0 −121.823 + 613.012i 0 3405.57i 0 −6030.17 − 2585.30i 0
209.10 0 16.2916 + 79.3447i 0 −121.823 − 613.012i 0 − 3405.57i 0 −6030.17 + 2585.30i 0
209.11 0 32.2702 − 74.2942i 0 −615.732 + 107.234i 0 − 4017.33i 0 −4478.26 − 4794.99i 0
209.12 0 32.2702 + 74.2942i 0 −615.732 − 107.234i 0 4017.33i 0 −4478.26 + 4794.99i 0
209.13 0 62.9079 − 51.0255i 0 605.408 + 155.260i 0 459.019i 0 1353.80 − 6419.81i 0
209.14 0 62.9079 + 51.0255i 0 605.408 − 155.260i 0 − 459.019i 0 1353.80 + 6419.81i 0
209.15 0 77.5456 − 23.4026i 0 −220.972 − 584.633i 0 256.799i 0 5465.63 − 3629.54i 0
209.16 0 77.5456 + 23.4026i 0 −220.972 + 584.633i 0 − 256.799i 0 5465.63 + 3629.54i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.16
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 240.9.c.d 16
3.b odd 2 1 inner 240.9.c.d 16
4.b odd 2 1 60.9.b.a ✓ 16
5.b even 2 1 inner 240.9.c.d 16
12.b even 2 1 60.9.b.a ✓ 16
15.d odd 2 1 inner 240.9.c.d 16
20.d odd 2 1 60.9.b.a ✓ 16
20.e even 4 2 300.9.g.h 16
60.h even 2 1 60.9.b.a ✓ 16
60.l odd 4 2 300.9.g.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.9.b.a ✓ 16 4.b odd 2 1
60.9.b.a ✓ 16 12.b even 2 1
60.9.b.a ✓ 16 20.d odd 2 1
60.9.b.a ✓ 16 60.h even 2 1
240.9.c.d 16 1.a even 1 1 trivial
240.9.c.d 16 3.b odd 2 1 inner
240.9.c.d 16 5.b even 2 1 inner
240.9.c.d 16 15.d odd 2 1 inner
300.9.g.h 16 20.e even 4 2
300.9.g.h 16 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 \(S_{9}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{8} + 28013434T_{7}^{6} + 194864255690496T_{7}^{4} + 52166784840953810944T_{7}^{2} + 2600757082907060397408256 \) Copy content Toggle raw display
\( T_{17}^{8} - 36442812140 T_{17}^{6} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 26\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 36\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 14\!\cdots\!16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 11\!\cdots\!96)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 49\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 21\!\cdots\!64)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 89\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 62\!\cdots\!16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 93\!\cdots\!76)^{2} \) Copy content Toggle raw display
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