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Properties

Label 3240.2.f.j
Level $3240$
Weight $2$
Character orbit 3240.f
Analytic conductor $25.872$
Analytic rank $0$
Dimension $16$
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] = [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: \( x^{16} + 3x^{14} + 85x^{12} + 198x^{10} + 3006x^{8} + 4950x^{6} + 53125x^{4} + 46875x^{2} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - \beta_1 q^{7} + \beta_{10} q^{11} - \beta_{2} q^{13} + (\beta_{12} - \beta_{7} + \beta_{5}) q^{17} + (\beta_{4} - 1) q^{19} + ( - \beta_{14} - \beta_{12} + \cdots - \beta_{5}) q^{23}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{9} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{19} + 6 q^{25} - 12 q^{49} - 32 q^{55} - 52 q^{61} + 32 q^{79} + 42 q^{85} + 44 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3x^{14} + 85x^{12} + 198x^{10} + 3006x^{8} + 4950x^{6} + 53125x^{4} + 46875x^{2} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4 \nu^{14} + 445 \nu^{12} + 931 \nu^{10} + 24003 \nu^{8} + 26212 \nu^{6} + 443517 \nu^{4} + \cdots + 2929375 ) / 1258750 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2371 \nu^{14} + 93963 \nu^{12} + 445835 \nu^{10} + 5568583 \nu^{8} + 15832901 \nu^{6} + \cdots + 1286343750 ) / 62937500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6581 \nu^{14} + 170482 \nu^{12} + 48165 \nu^{10} + 12179837 \nu^{8} + 13547689 \nu^{6} + \cdots + 3195609375 ) / 125875000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} + 3\nu^{12} + 85\nu^{10} + 198\nu^{8} + 3006\nu^{6} + 4950\nu^{4} + 37500\nu^{2} + 31250 ) / 15625 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14237 \nu^{15} - 101814 \nu^{13} + 729695 \nu^{11} - 8184449 \nu^{9} + 13289847 \nu^{7} + \cdots - 2525046875 \nu ) / 629375000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 212 \nu^{15} - 2236 \nu^{13} - 17195 \nu^{11} - 129851 \nu^{9} - 538447 \nu^{7} + \cdots - 32296875 \nu ) / 5937500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5781 \nu^{15} + 19218 \nu^{13} + 440135 \nu^{11} + 1180263 \nu^{9} + 11633311 \nu^{7} + \cdots + 96578125 \nu ) / 125875000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16229 \nu^{14} + 73688 \nu^{12} - 864215 \nu^{10} + 4398533 \nu^{8} - 17963499 \nu^{6} + \cdots + 1612390625 ) / 62937500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8351 \nu^{14} - 53028 \nu^{12} - 549385 \nu^{10} - 2738873 \nu^{8} - 14670281 \nu^{6} + \cdots - 381078125 ) / 25175000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 193 \nu^{15} + 1704 \nu^{13} + 14155 \nu^{11} + 85714 \nu^{9} + 387283 \nu^{7} + \cdots + 12109375 \nu ) / 2968750 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 46861 \nu^{14} + 101408 \nu^{12} + 2570035 \nu^{10} + 4858603 \nu^{8} + 61916891 \nu^{6} + \cdots + 675046875 ) / 125875000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1223 \nu^{15} + 5544 \nu^{13} + 85830 \nu^{11} + 377154 \nu^{9} + 2403838 \nu^{7} + \cdots + 87468750 \nu ) / 16562500 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 641 \nu^{15} + 2898 \nu^{13} + 43035 \nu^{11} + 119793 \nu^{9} + 991771 \nu^{7} + \cdots + 9703125 \nu ) / 5937500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 996 \nu^{15} + 512 \nu^{13} - 56660 \nu^{11} + 74667 \nu^{9} - 1438476 \nu^{7} + \cdots + 54625000 \nu ) / 8281250 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25 \nu^{15} - 12 \nu^{13} + 1164 \nu^{11} - 1420 \nu^{9} + 23424 \nu^{7} - 51372 \nu^{5} + \cdots - 727500 \nu ) / 165625 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + 2\beta_{13} + 4\beta_{12} - 2\beta_{10} - 6\beta_{7} + 2\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{11} - 3\beta_{9} - \beta_{8} - 4\beta_{4} + 2\beta_{2} - 14\beta _1 - 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + 2\beta_{12} + 2\beta_{10} - \beta_{7} + 2\beta_{6} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 23\beta_{11} + 23\beta_{9} + 9\beta_{8} + 8\beta_{4} - 8\beta_{3} + 14\beta_{2} + 22\beta _1 - 156 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 33 \beta_{15} + 24 \beta_{14} - 34 \beta_{13} - 100 \beta_{12} - 6 \beta_{10} + 126 \beta_{7} + \cdots + 6 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} - 4\beta_{9} + 6\beta_{8} + 28\beta_{4} + 2\beta_{3} - 18\beta_{2} + 14\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 77 \beta_{15} - 344 \beta_{14} + 30 \beta_{13} - 252 \beta_{12} - 262 \beta_{10} + \cdots - 418 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -463\beta_{11} - 415\beta_{9} - 449\beta_{8} - 560\beta_{4} + 632\beta_{3} - 366\beta_{2} - 502\beta _1 + 1428 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 56 \beta_{15} - 69 \beta_{14} + 19 \beta_{13} + 258 \beta_{12} + 90 \beta_{10} - 113 \beta_{7} + \cdots - 245 \beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1963 \beta_{11} + 3579 \beta_{9} - 1759 \beta_{8} - 4620 \beta_{4} - 1472 \beta_{3} + 6254 \beta_{2} + \cdots - 2772 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 195 \beta_{15} + 8096 \beta_{14} + 3874 \beta_{13} + 3132 \beta_{12} - 898 \beta_{10} + \cdots + 9178 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -488\beta_{11} - 1208\beta_{9} + 1948\beta_{8} + 1920\beta_{4} - 2928\beta_{3} + 172\beta_{2} - 260\beta _1 + 2009 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1537 \beta_{15} + 4000 \beta_{14} + 11322 \beta_{13} - 38540 \beta_{12} + 7206 \beta_{10} + \cdots + 59994 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 16723 \beta_{11} - 98211 \beta_{9} + 40327 \beta_{8} + 19556 \beta_{4} + 61760 \beta_{3} + \cdots - 24340 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 18344 \beta_{15} - 12575 \beta_{14} - 17529 \beta_{13} - 3970 \beta_{12} + 16178 \beta_{10} + \cdots - 30653 \beta_{5} \) 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
−1.43446 − 1.71532i
−1.43446 + 1.71532i
−1.59693 − 1.56518i
−1.59693 + 1.56518i
1.13055 + 1.92921i
1.13055 − 1.92921i
1.83411 + 1.27908i
1.83411 − 1.27908i
−1.83411 + 1.27908i
−1.83411 − 1.27908i
−1.13055 + 1.92921i
−1.13055 − 1.92921i
1.59693 − 1.56518i
1.59693 + 1.56518i
1.43446 − 1.71532i
1.43446 + 1.71532i
0 0 0 −2.20274 − 0.384624i 0 2.36064i 0 0 0
649.2 0 0 0 −2.20274 + 0.384624i 0 − 2.36064i 0 0 0
649.3 0 0 0 −2.15396 − 0.600392i 0 1.54653i 0 0 0
649.4 0 0 0 −2.15396 + 0.600392i 0 − 1.54653i 0 0 0
649.5 0 0 0 −1.10547 − 1.94369i 0 0.930801i 0 0 0
649.6 0 0 0 −1.10547 + 1.94369i 0 − 0.930801i 0 0 0
649.7 0 0 0 −0.190658 − 2.22792i 0 4.70842i 0 0 0
649.8 0 0 0 −0.190658 + 2.22792i 0 − 4.70842i 0 0 0
649.9 0 0 0 0.190658 − 2.22792i 0 − 4.70842i 0 0 0
649.10 0 0 0 0.190658 + 2.22792i 0 4.70842i 0 0 0
649.11 0 0 0 1.10547 − 1.94369i 0 − 0.930801i 0 0 0
649.12 0 0 0 1.10547 + 1.94369i 0 0.930801i 0 0 0
649.13 0 0 0 2.15396 − 0.600392i 0 − 1.54653i 0 0 0
649.14 0 0 0 2.15396 + 0.600392i 0 1.54653i 0 0 0
649.15 0 0 0 2.20274 − 0.384624i 0 − 2.36064i 0 0 0
649.16 0 0 0 2.20274 + 0.384624i 0 2.36064i 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
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 3240.2.f.j ✓ 16
3.b odd 2 1 inner 3240.2.f.j ✓ 16
5.b even 2 1 inner 3240.2.f.j ✓ 16
15.d odd 2 1 inner 3240.2.f.j ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.f.j ✓ 16 1.a even 1 1 trivial
3240.2.f.j ✓ 16 3.b odd 2 1 inner
3240.2.f.j ✓ 16 5.b even 2 1 inner
3240.2.f.j ✓ 16 15.d odd 2 1 inner

Hecke kernels

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

\( T_{7}^{8} + 31T_{7}^{6} + 216T_{7}^{4} + 460T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} - 34T_{11}^{6} + 273T_{11}^{4} - 316T_{11}^{2} + 64 \) 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
$5$ \( T^{16} - 3 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 31 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 34 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 46 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 45 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 5 T^{3} - 9 T^{2} + \cdots - 8)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 97 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 97 T^{6} + \cdots + 4624)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 51 T^{2} + \cdots - 72)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 55 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 121 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 120 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 189 T^{6} + \cdots + 576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 265 T^{6} + \cdots + 8133904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 177 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 13 T^{3} + \cdots + 1696)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 208 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 406 T^{6} + \cdots + 16516096)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 403 T^{6} + \cdots + 45751696)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots - 1664)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 532 T^{6} + \cdots + 96510976)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 561 T^{6} + \cdots + 12194064)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 460 T^{6} + \cdots + 38539264)^{2} \) Copy content Toggle raw display
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