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Properties

Label 3240.2.f.i
Level $3240$
Weight $2$
Character orbit 3240.f
Analytic conductor $25.872$
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] = [3240,2,Mod(649,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.f (of order \(2\), degree \(1\), 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: \(25.8715302549\)
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: \( x^{16} - 2 x^{15} - x^{14} + 26 x^{13} - 44 x^{12} + 6 x^{11} + 225 x^{10} - 174 x^{9} + 102 x^{8} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 360)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{10} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{10} q^{7} + ( - \beta_{7} + 1) q^{11} - \beta_{5} q^{13} + (\beta_{13} + \beta_{11} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{14} - \beta_{13} + \cdots + \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{5} + 16 q^{11} + 4 q^{19} + 6 q^{25} + 20 q^{29} + 12 q^{31} - 2 q^{35} - 8 q^{41} - 36 q^{49} + 10 q^{55} + 20 q^{61} + 10 q^{65} - 8 q^{71} - 4 q^{79} - 36 q^{85} + 48 q^{89} - 4 q^{91} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - x^{14} + 26 x^{13} - 44 x^{12} + 6 x^{11} + 225 x^{10} - 174 x^{9} + 102 x^{8} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 35 \nu^{15} - 56 \nu^{14} + 42 \nu^{13} - 794 \nu^{12} + 134 \nu^{11} + 1964 \nu^{10} + \cdots + 5625000 ) / 2000000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} + \nu^{13} - 26 \nu^{12} + 44 \nu^{11} - 6 \nu^{10} - 225 \nu^{9} + \cdots + 156250 ) / 78125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 133 \nu^{15} + 104 \nu^{14} - 1398 \nu^{13} + 3638 \nu^{12} + 918 \nu^{11} - 26132 \nu^{10} + \cdots - 51875000 ) / 5000000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 58 \nu^{15} - 761 \nu^{14} + 4082 \nu^{13} - 4672 \nu^{12} - 13672 \nu^{11} + 63328 \nu^{10} + \cdots + 78828125 ) / 5000000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{15} + 500 \nu^{14} - 1620 \nu^{13} + 432 \nu^{12} + 9204 \nu^{11} - 21126 \nu^{10} + \cdots - 22531250 ) / 2000000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 95 \nu^{15} + 84 \nu^{14} + 122 \nu^{13} - 1094 \nu^{12} + 934 \nu^{11} - 2616 \nu^{10} + \cdots - 5125000 ) / 2000000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 378 \nu^{15} - 1361 \nu^{14} + 482 \nu^{13} + 6008 \nu^{12} - 15512 \nu^{11} + 9288 \nu^{10} + \cdots + 18828125 ) / 10000000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 477 \nu^{15} + 604 \nu^{14} - 3948 \nu^{13} + 4448 \nu^{12} + 1388 \nu^{11} - 89462 \nu^{10} + \cdots - 207656250 ) / 10000000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 713 \nu^{15} + 3821 \nu^{14} - 8802 \nu^{13} + 3392 \nu^{12} + 62992 \nu^{11} + \cdots - 146171875 ) / 20000000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 506 \nu^{15} - 1617 \nu^{14} + 354 \nu^{13} + 9336 \nu^{12} - 21144 \nu^{11} + 10056 \nu^{10} + \cdots - 1171875 ) / 10000000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 762 \nu^{15} + 1489 \nu^{14} + 1182 \nu^{13} - 15352 \nu^{12} + 15768 \nu^{11} + \cdots + 51171875 ) / 10000000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 596 \nu^{15} + 3007 \nu^{14} - 3534 \nu^{13} - 8936 \nu^{12} + 36664 \nu^{11} + \cdots - 76484375 ) / 10000000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1327 \nu^{15} + 5969 \nu^{14} - 9978 \nu^{13} - 12592 \nu^{12} + 68128 \nu^{11} + \cdots - 256796875 ) / 20000000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 180 \nu^{15} - 313 \nu^{14} - 234 \nu^{13} + 3128 \nu^{12} - 4388 \nu^{11} - 3148 \nu^{10} + \cdots - 9734375 ) / 2000000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{8} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 4 \beta_{14} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + \beta_{9} + \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} + 6 \beta_{14} - 8 \beta_{13} - 3 \beta_{12} - 10 \beta_{11} - 2 \beta_{10} - 4 \beta_{9} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} - 8 \beta_{14} - 19 \beta_{13} + 8 \beta_{12} - 29 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{15} + 8 \beta_{14} - 30 \beta_{13} + 25 \beta_{12} + 20 \beta_{11} - 16 \beta_{10} - 18 \beta_{9} + \cdots - 96 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17 \beta_{15} - 28 \beta_{14} + 5 \beta_{13} - 24 \beta_{12} - 46 \beta_{11} - 48 \beta_{10} + 43 \beta_{9} + \cdots - 31 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 61 \beta_{15} + 18 \beta_{14} + 129 \beta_{13} - 169 \beta_{12} + 230 \beta_{11} - 126 \beta_{10} + \cdots + 572 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 440 \beta_{15} - 88 \beta_{14} + 254 \beta_{13} - 367 \beta_{12} + 155 \beta_{11} - 48 \beta_{10} + \cdots + 192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 950 \beta_{15} + 674 \beta_{14} - 354 \beta_{13} - 1128 \beta_{12} - 178 \beta_{11} + 546 \beta_{10} + \cdots - 1072 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 894 \beta_{15} + 264 \beta_{14} + 664 \beta_{13} + 16 \beta_{12} + 1664 \beta_{11} + 2856 \beta_{10} + \cdots - 4657 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 394 \beta_{15} - 2834 \beta_{14} + 4386 \beta_{13} + 1222 \beta_{12} + 6250 \beta_{11} + 5358 \beta_{10} + \cdots - 10552 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6680 \beta_{15} + 9320 \beta_{14} + 50 \beta_{13} + 1728 \beta_{12} + 445 \beta_{11} + 4704 \beta_{10} + \cdots + 12240 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 13215 \beta_{15} - 8322 \beta_{14} - 1253 \beta_{13} + 11216 \beta_{12} + 7090 \beta_{11} - 5778 \beta_{10} + \cdots + 75268 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
2.08944 + 0.796405i
2.08944 − 0.796405i
1.76404 + 1.37410i
1.76404 − 1.37410i
1.51358 + 1.64593i
1.51358 − 1.64593i
0.659117 + 2.13672i
0.659117 − 2.13672i
0.535737 + 2.17094i
0.535737 − 2.17094i
−1.17326 + 1.90354i
−1.17326 − 1.90354i
−2.15319 + 0.603133i
−2.15319 − 0.603133i
−2.23546 + 0.0521238i
−2.23546 − 0.0521238i
0 0 0 −2.08944 − 0.796405i 0 − 2.53599i 0 0 0
649.2 0 0 0 −2.08944 + 0.796405i 0 2.53599i 0 0 0
649.3 0 0 0 −1.76404 − 1.37410i 0 3.79229i 0 0 0
649.4 0 0 0 −1.76404 + 1.37410i 0 − 3.79229i 0 0 0
649.5 0 0 0 −1.51358 − 1.64593i 0 − 4.50582i 0 0 0
649.6 0 0 0 −1.51358 + 1.64593i 0 4.50582i 0 0 0
649.7 0 0 0 −0.659117 − 2.13672i 0 2.90572i 0 0 0
649.8 0 0 0 −0.659117 + 2.13672i 0 − 2.90572i 0 0 0
649.9 0 0 0 −0.535737 − 2.17094i 0 − 2.28645i 0 0 0
649.10 0 0 0 −0.535737 + 2.17094i 0 2.28645i 0 0 0
649.11 0 0 0 1.17326 − 1.90354i 0 1.37527i 0 0 0
649.12 0 0 0 1.17326 + 1.90354i 0 − 1.37527i 0 0 0
649.13 0 0 0 2.15319 − 0.603133i 0 − 0.703111i 0 0 0
649.14 0 0 0 2.15319 + 0.603133i 0 0.703111i 0 0 0
649.15 0 0 0 2.23546 − 0.0521238i 0 − 4.10221i 0 0 0
649.16 0 0 0 2.23546 + 0.0521238i 0 4.10221i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.16
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 3240.2.f.i 16
3.b odd 2 1 3240.2.f.k 16
5.b even 2 1 inner 3240.2.f.i 16
9.c even 3 2 1080.2.bi.b 32
9.d odd 6 2 360.2.bi.b ✓ 32
15.d odd 2 1 3240.2.f.k 16
36.f odd 6 2 2160.2.by.f 32
36.h even 6 2 720.2.by.f 32
45.h odd 6 2 360.2.bi.b ✓ 32
45.j even 6 2 1080.2.bi.b 32
180.n even 6 2 720.2.by.f 32
180.p odd 6 2 2160.2.by.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bi.b ✓ 32 9.d odd 6 2
360.2.bi.b ✓ 32 45.h odd 6 2
720.2.by.f 32 36.h even 6 2
720.2.by.f 32 180.n even 6 2
1080.2.bi.b 32 9.c even 3 2
1080.2.bi.b 32 45.j even 6 2
2160.2.by.f 32 36.f odd 6 2
2160.2.by.f 32 180.p odd 6 2
3240.2.f.i 16 1.a even 1 1 trivial
3240.2.f.i 16 5.b even 2 1 inner
3240.2.f.k 16 3.b odd 2 1
3240.2.f.k 16 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7}^{16} + 74 T_{7}^{14} + 2215 T_{7}^{12} + 34544 T_{7}^{10} + 301567 T_{7}^{8} + 1471538 T_{7}^{6} + \cdots + 1304164 \) Copy content Toggle raw display
\( T_{11}^{8} - 8T_{11}^{7} - 20T_{11}^{6} + 216T_{11}^{5} + 27T_{11}^{4} - 1260T_{11}^{3} + 1448T_{11}^{2} - 328T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 74 T^{14} + \cdots + 1304164 \) Copy content Toggle raw display
$11$ \( (T^{8} - 8 T^{7} - 20 T^{6} + \cdots - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 122 T^{14} + \cdots + 24049216 \) Copy content Toggle raw display
$17$ \( T^{16} + 138 T^{14} + \cdots + 2985984 \) Copy content Toggle raw display
$19$ \( (T^{8} - 2 T^{7} + \cdots + 10432)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 2161506064 \) Copy content Toggle raw display
$29$ \( (T^{8} - 10 T^{7} + \cdots - 21724)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 6 T^{7} + \cdots + 114696)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1073741824 \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} + \cdots - 500887)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 132 T^{14} + \cdots + 8503056 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3196545444 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 145887330304 \) Copy content Toggle raw display
$59$ \( (T^{8} - 156 T^{6} + \cdots + 1266624)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} + \cdots + 17627854)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4512901160881 \) Copy content Toggle raw display
$71$ \( (T^{8} + 4 T^{7} + \cdots - 5345152)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1007681536 \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} + \cdots + 5679136)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2960216598784 \) Copy content Toggle raw display
$89$ \( (T^{8} - 24 T^{7} + \cdots + 396576)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 244359424 \) Copy content Toggle raw display
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