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Properties

Label 240.9.bg.c
Level $240$
Weight $9$
Character orbit 240.bg
Analytic conductor $97.771$
Analytic rank $0$
Dimension $16$
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] = [240,9,Mod(97,240)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.97");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 240.bg (of order \(4\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 30946474 x^{14} - 8341419336 x^{13} + 380689791299777 x^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{20}\cdot 5^{13} \)
Twist minimal: no (minimal twist has level 60)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{7} + 2 \beta_{4} + \cdots - 112) q^{5}+ \cdots + 2187 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{7} + 2 \beta_{4} + \cdots - 112) q^{5}+ \cdots + (2187 \beta_{15} + 4374 \beta_{14} + \cdots - 2187) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 1788 q^{5} - 4220 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 1788 q^{5} - 4220 q^{7} - 23616 q^{11} - 18900 q^{13} - 59616 q^{15} - 44940 q^{17} + 163944 q^{21} - 196440 q^{23} + 247724 q^{25} - 3742624 q^{31} - 134460 q^{33} + 1574664 q^{35} - 2141100 q^{37} + 16347000 q^{41} - 12080280 q^{43} - 734832 q^{45} + 14942400 q^{47} - 7693704 q^{51} + 23760300 q^{53} + 27374868 q^{55} - 27530280 q^{57} + 85401912 q^{61} - 9229140 q^{63} - 94222620 q^{65} + 99451240 q^{67} - 73302480 q^{71} + 124097320 q^{73} + 13129128 q^{75} - 185945400 q^{77} - 76527504 q^{81} + 22058160 q^{83} + 18420356 q^{85} + 110300940 q^{87} - 170997360 q^{91} + 9969480 q^{93} + 184494000 q^{95} + 185269800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 30946474 x^{14} - 8341419336 x^{13} + 380689791299777 x^{12} + \cdots + 57\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!83 \nu^{15} + \cdots + 37\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!01 \nu^{15} + \cdots + 33\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!89 \nu^{15} + \cdots + 66\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!11 \nu^{15} + \cdots - 65\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!63 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 69\!\cdots\!03 \nu^{15} + \cdots - 40\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!67 \nu^{15} + \cdots - 11\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 61\!\cdots\!34 \nu^{15} + \cdots + 23\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!54 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!63 \nu^{15} + \cdots - 94\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!11 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!57 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!89 \nu^{15} + \cdots + 89\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!13 \nu^{15} + \cdots + 40\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28\!\cdots\!43 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - 5\beta_{4} - 5\beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 109 \beta_{15} + 459 \beta_{12} - 1085 \beta_{11} + 319 \beta_{10} - 482 \beta_{9} + 801 \beta_{8} + \cdots + 7737302 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 373215 \beta_{15} + 1446 \beta_{14} + 1248 \beta_{13} - 194739 \beta_{12} - 1263267 \beta_{11} + \cdots + 3179294864 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 79690539 \beta_{15} + 4438176 \beta_{14} + 2744088 \beta_{13} + 2950348785 \beta_{12} + \cdots + 49101145584572 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4119745164515 \beta_{15} + 27879150070 \beta_{14} + 22922588410 \beta_{13} + 187428056005 \beta_{12} + \cdots + 55\!\cdots\!48 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 44\!\cdots\!83 \beta_{15} + 101481755389920 \beta_{14} + 57505561221960 \beta_{13} + \cdots + 38\!\cdots\!80 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 38\!\cdots\!25 \beta_{15} + \cdots + 71\!\cdots\!76 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 72\!\cdots\!77 \beta_{15} + \cdots + 34\!\cdots\!00 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 36\!\cdots\!47 \beta_{15} + \cdots + 83\!\cdots\!80 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 88\!\cdots\!55 \beta_{15} + \cdots + 33\!\cdots\!08 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 35\!\cdots\!45 \beta_{15} + \cdots + 92\!\cdots\!08 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 98\!\cdots\!73 \beta_{15} + \cdots + 33\!\cdots\!48 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 35\!\cdots\!19 \beta_{15} + \cdots + 98\!\cdots\!12 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 10\!\cdots\!47 \beta_{15} + \cdots + 33\!\cdots\!96 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 35\!\cdots\!25 \beta_{15} + \cdots + 10\!\cdots\!60 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−1675.46 + 1.00000i
1773.40 + 1.00000i
1685.41 + 1.00000i
−2402.08 + 1.00000i
3225.69 + 1.00000i
−681.621 + 1.00000i
−2315.27 + 1.00000i
391.920 + 1.00000i
−1675.46 − 1.00000i
1773.40 − 1.00000i
1685.41 − 1.00000i
−2402.08 − 1.00000i
3225.69 − 1.00000i
−681.621 − 1.00000i
−2315.27 − 1.00000i
391.920 − 1.00000i
0 −33.0681 + 33.0681i 0 −615.939 − 106.040i 0 −1939.46 − 1939.46i 0 − 2187.00i 0
97.2 0 −33.0681 + 33.0681i 0 −179.099 + 598.789i 0 1509.40 + 1509.40i 0 − 2187.00i 0
97.3 0 −33.0681 + 33.0681i 0 15.8453 − 624.799i 0 1421.41 + 1421.41i 0 − 2187.00i 0
97.4 0 −33.0681 + 33.0681i 0 566.118 + 264.830i 0 −2666.08 − 2666.08i 0 − 2187.00i 0
97.5 0 33.0681 − 33.0681i 0 −599.429 − 176.946i 0 2961.69 + 2961.69i 0 − 2187.00i 0
97.6 0 33.0681 − 33.0681i 0 −468.797 − 413.345i 0 −945.621 − 945.621i 0 − 2187.00i 0
97.7 0 33.0681 − 33.0681i 0 −151.420 + 606.380i 0 −2579.27 − 2579.27i 0 − 2187.00i 0
97.8 0 33.0681 − 33.0681i 0 538.720 − 316.869i 0 127.920 + 127.920i 0 − 2187.00i 0
193.1 0 −33.0681 − 33.0681i 0 −615.939 + 106.040i 0 −1939.46 + 1939.46i 0 2187.00i 0
193.2 0 −33.0681 − 33.0681i 0 −179.099 − 598.789i 0 1509.40 − 1509.40i 0 2187.00i 0
193.3 0 −33.0681 − 33.0681i 0 15.8453 + 624.799i 0 1421.41 − 1421.41i 0 2187.00i 0
193.4 0 −33.0681 − 33.0681i 0 566.118 − 264.830i 0 −2666.08 + 2666.08i 0 2187.00i 0
193.5 0 33.0681 + 33.0681i 0 −599.429 + 176.946i 0 2961.69 − 2961.69i 0 2187.00i 0
193.6 0 33.0681 + 33.0681i 0 −468.797 + 413.345i 0 −945.621 + 945.621i 0 2187.00i 0
193.7 0 33.0681 + 33.0681i 0 −151.420 − 606.380i 0 −2579.27 + 2579.27i 0 2187.00i 0
193.8 0 33.0681 + 33.0681i 0 538.720 + 316.869i 0 127.920 − 127.920i 0 2187.00i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.9.bg.c 16
4.b odd 2 1 60.9.k.a ✓ 16
5.c odd 4 1 inner 240.9.bg.c 16
12.b even 2 1 180.9.l.c 16
20.d odd 2 1 300.9.k.e 16
20.e even 4 1 60.9.k.a ✓ 16
20.e even 4 1 300.9.k.e 16
60.l odd 4 1 180.9.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.9.k.a ✓ 16 4.b odd 2 1
60.9.k.a ✓ 16 20.e even 4 1
180.9.l.c 16 12.b even 2 1
180.9.l.c 16 60.l odd 4 1
240.9.bg.c 16 1.a even 1 1 trivial
240.9.bg.c 16 5.c odd 4 1 inner
300.9.k.e 16 20.d odd 2 1
300.9.k.e 16 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 4220 T_{7}^{15} + 8904200 T_{7}^{14} - 3419547960 T_{7}^{13} + 337814088968876 T_{7}^{12} + \cdots + 26\!\cdots\!96 \) acting on \(S_{9}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 4782969)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 28\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 35\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 82\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
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