Properties

Label 3120.2.r.l
Level $3120$
Weight $2$
Character orbit 3120.r
Analytic conductor $24.913$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3120,2,Mod(2209,3120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3120.2209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.r (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: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 8x^{8} + 6x^{7} + 25x^{6} - 68x^{5} + 90x^{4} - 10x^{3} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1560)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{8} + 1) q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{8} + 1) q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 1) q^{7} - q^{9} + (\beta_{8} + \beta_{4} + 2 \beta_{3}) q^{11} + (\beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{13} + (\beta_{6} - \beta_{3}) q^{15} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1) q^{17} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{19} + ( - \beta_{8} + \beta_{3} + \beta_{2}) q^{21} + ( - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{23} + (\beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{25} + \beta_{3} q^{27} + (\beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_1 - 1) q^{29} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{31} + ( - \beta_{7} - \beta_{6} + 2) q^{33} + ( - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_1 - 1) q^{35} + (\beta_{9} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_1 - 2) q^{37} + (\beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{39} + ( - \beta_{9} - \beta_{7} + \beta_{6} + \beta_1) q^{41} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{43} + (\beta_{8} - 1) q^{45} + (\beta_{9} + \beta_{8} - \beta_{4} + \beta_1 - 2) q^{47} + (\beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + \beta_1 - 1) q^{49} + ( - \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_1 + 3) q^{51} + (2 \beta_{8} - 2 \beta_{2}) q^{53} + ( - \beta_{9} - 3 \beta_{6} + \beta_{5} + 3 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{55} + ( - \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_1 - 1) q^{57} + ( - 2 \beta_{9} + \beta_{8} + 4 \beta_{7} - 4 \beta_{6} - \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{59} + (3 \beta_{8} - \beta_{7} - \beta_{6} - 3 \beta_{4} - 6) q^{61} + (\beta_{7} + \beta_{6} - \beta_{5} + 1) q^{63} + (2 \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{65} + (\beta_{9} + 4 \beta_{8} - 4 \beta_{4} + \beta_1) q^{67} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{69} + ( - \beta_{7} + \beta_{6} - 8 \beta_{3}) q^{71} + ( - 2 \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_1 + 5) q^{73} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{75} + (2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2}) q^{77} + (\beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} - 3 \beta_{4} + \beta_1 - 2) q^{79} + q^{81} + ( - 2 \beta_{9} - \beta_{8} + 2 \beta_{5} + \beta_{4} - 2 \beta_1 + 6) q^{83} + ( - 2 \beta_{9} + \beta_{7} - 3 \beta_{6} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2}) q^{85} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \beta_{3} - \beta_{2} + \beta_1) q^{87} + ( - 2 \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{4} - 6 \beta_{3} + 4 \beta_{2}) q^{89} + ( - \beta_{9} + 3 \beta_{8} + 2 \beta_{6} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1) q^{91} + ( - 2 \beta_{8} + 2 \beta_{4} + 2) q^{93} + ( - 2 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 3 \beta_{2}) q^{95} + ( - 2 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + 9) q^{97} + ( - \beta_{8} - \beta_{4} - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6 q^{5} - 12 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 6 q^{5} - 12 q^{7} - 10 q^{9} - 8 q^{13} - 2 q^{25} + 12 q^{29} + 20 q^{33} - 16 q^{35} - 24 q^{37} + 2 q^{39} - 6 q^{45} - 8 q^{47} + 18 q^{49} + 16 q^{51} + 52 q^{55} - 24 q^{57} - 36 q^{61} + 12 q^{63} + 32 q^{65} + 36 q^{67} - 20 q^{69} + 40 q^{73} - 4 q^{75} + 8 q^{79} + 10 q^{81} + 40 q^{83} - 20 q^{85} + 4 q^{91} + 4 q^{93} - 20 q^{95} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} + 8x^{8} + 6x^{7} + 25x^{6} - 68x^{5} + 90x^{4} - 10x^{3} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 20289 \nu^{9} - 27913 \nu^{8} + 301505 \nu^{7} - 843151 \nu^{6} + 3442683 \nu^{5} + 2255638 \nu^{4} + 7091017 \nu^{3} - 19441646 \nu^{2} + 17822350 \nu - 2847572 ) / 4427135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 44413 \nu^{9} + 600458 \nu^{8} - 1891760 \nu^{7} + 2417367 \nu^{6} + 2815661 \nu^{5} + 14459192 \nu^{4} - 29717829 \nu^{3} + 22592138 \nu^{2} + \cdots + 2033447 ) / 4427135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23486 \nu^{9} + 28951 \nu^{8} + 69457 \nu^{7} - 654191 \nu^{6} - 994108 \nu^{5} - 46474 \nu^{4} + 2199641 \nu^{3} - 5823214 \nu^{2} + 331634 \nu + 17151 ) / 885427 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 251948 \nu^{9} - 1742428 \nu^{8} + 4805420 \nu^{7} - 3653532 \nu^{6} + 431409 \nu^{5} - 36258467 \nu^{4} + 71006404 \nu^{3} - 53366378 \nu^{2} + \cdots - 2182552 ) / 4427135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64993 \nu^{9} + 257345 \nu^{8} - 513275 \nu^{7} - 406958 \nu^{6} - 1643522 \nu^{5} + 4313381 \nu^{4} - 6058074 \nu^{3} + 331634 \nu^{2} + 902578 \nu - 838455 ) / 885427 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 637662 \nu^{9} + 2237951 \nu^{8} - 4150070 \nu^{7} - 5409667 \nu^{6} - 19316427 \nu^{5} + 32262199 \nu^{4} - 46694286 \nu^{3} - 10027296 \nu^{2} + \cdots - 7344086 ) / 4427135 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1059601 \nu^{9} - 4076561 \nu^{8} + 7731595 \nu^{7} + 7902081 \nu^{6} + 27343613 \nu^{5} - 69897989 \nu^{4} + 81137268 \nu^{3} + 8123614 \nu^{2} + \cdots - 8585234 ) / 4427135 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1262419 \nu^{9} + 4347082 \nu^{8} - 7328505 \nu^{7} - 13031214 \nu^{6} - 36014989 \nu^{5} + 67988193 \nu^{4} - 66731682 \nu^{3} - 45813532 \nu^{2} + \cdots + 2931208 ) / 4427135 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1332935 \nu^{9} + 5449273 \nu^{8} - 11054205 \nu^{7} - 7144335 \nu^{6} - 32542477 \nu^{5} + 94405962 \nu^{4} - 121874305 \nu^{3} + \cdots - 1842018 ) / 4427135 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 3\beta_{4} - 8\beta_{3} + 2\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{9} + 3\beta_{8} - 3\beta_{7} + 3\beta_{6} - 10\beta_{5} + 7\beta_{4} - 14\beta_{3} + 10\beta_{2} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -10\beta_{9} + 15\beta_{8} - 5\beta_{7} - 5\beta_{6} - 36\beta_{5} - 15\beta_{4} - 10\beta _1 - 80 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 35 \beta_{8} + 35 \beta_{7} - 41 \beta_{6} - 108 \beta_{5} - 149 \beta_{4} + 176 \beta_{3} - 108 \beta_{2} - 36 \beta _1 - 176 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 108 \beta_{9} - 37 \beta_{8} + 185 \beta_{7} - 185 \beta_{6} - 511 \beta_{4} + 910 \beta_{3} - 474 \beta_{2} - 108 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 474 \beta_{9} - 511 \beta_{8} + 511 \beta_{7} - 401 \beta_{6} + 1238 \beta_{5} - 837 \beta_{4} + 2148 \beta_{3} - 1238 \beta_{2} + 2148 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1238 \beta_{9} - 2223 \beta_{8} + 363 \beta_{7} + 363 \beta_{6} + 5830 \beta_{5} + 2223 \beta_{4} + 1238 \beta _1 + 10706 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4699 \beta_{8} - 4699 \beta_{7} + 6193 \beta_{6} + 14568 \beta_{5} + 20761 \beta_{4} - 25862 \beta_{3} + 14568 \beta_{2} + 5830 \beta _1 + 25862 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2209.1
0.916837 0.916837i
0.312969 0.312969i
2.44396 2.44396i
−0.250470 + 0.250470i
−1.42330 + 1.42330i
0.916837 + 0.916837i
0.312969 + 0.312969i
2.44396 + 2.44396i
−0.250470 0.250470i
−1.42330 1.42330i
0 1.00000i 0 −1.34354 + 1.78743i 0 −3.74118 0 −1.00000 0
2209.2 0 1.00000i 0 −0.852717 2.06709i 0 2.76012 0 −1.00000 0
2209.3 0 1.00000i 0 1.26930 + 1.84089i 0 −0.793851 0 −1.00000 0
2209.4 0 1.00000i 0 1.69315 1.46056i 0 0.420177 0 −1.00000 0
2209.5 0 1.00000i 0 2.23380 0.100662i 0 −4.64527 0 −1.00000 0
2209.6 0 1.00000i 0 −1.34354 1.78743i 0 −3.74118 0 −1.00000 0
2209.7 0 1.00000i 0 −0.852717 + 2.06709i 0 2.76012 0 −1.00000 0
2209.8 0 1.00000i 0 1.26930 1.84089i 0 −0.793851 0 −1.00000 0
2209.9 0 1.00000i 0 1.69315 + 1.46056i 0 0.420177 0 −1.00000 0
2209.10 0 1.00000i 0 2.23380 + 0.100662i 0 −4.64527 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2209.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.2.r.l 10
4.b odd 2 1 1560.2.r.h yes 10
5.b even 2 1 3120.2.r.k 10
13.b even 2 1 3120.2.r.k 10
20.d odd 2 1 1560.2.r.g 10
52.b odd 2 1 1560.2.r.g 10
65.d even 2 1 inner 3120.2.r.l 10
260.g odd 2 1 1560.2.r.h yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.r.g 10 20.d odd 2 1
1560.2.r.g 10 52.b odd 2 1
1560.2.r.h yes 10 4.b odd 2 1
1560.2.r.h yes 10 260.g odd 2 1
3120.2.r.k 10 5.b even 2 1
3120.2.r.k 10 13.b even 2 1
3120.2.r.l 10 1.a even 1 1 trivial
3120.2.r.l 10 65.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{5} + 6T_{7}^{4} - 4T_{7}^{3} - 52T_{7}^{2} - 16T_{7} + 16 \) acting on \(S_{2}^{\mathrm{new}}(3120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - 6 T^{9} + 19 T^{8} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( (T^{5} + 6 T^{4} - 4 T^{3} - 52 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 72 T^{8} + 1572 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$13$ \( T^{10} + 8 T^{9} + 49 T^{8} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} + 88 T^{8} + 2800 T^{6} + \cdots + 565504 \) Copy content Toggle raw display
$19$ \( T^{10} + 104 T^{8} + 2768 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{10} + 124 T^{8} + 4640 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{5} - 6 T^{4} - 48 T^{3} + 460 T^{2} + \cdots + 944)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 164 T^{8} + \cdots + 16516096 \) Copy content Toggle raw display
$37$ \( (T^{5} + 12 T^{4} - 136 T^{3} + \cdots + 39296)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 192 T^{8} + 11396 T^{6} + \cdots + 3041536 \) Copy content Toggle raw display
$43$ \( T^{10} + 216 T^{8} + 11024 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( (T^{5} + 4 T^{4} - 22 T^{3} - 16 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 148 T^{8} + 7456 T^{6} + \cdots + 123904 \) Copy content Toggle raw display
$59$ \( T^{10} + 504 T^{8} + \cdots + 479785216 \) Copy content Toggle raw display
$61$ \( (T^{5} + 18 T^{4} - 68 T^{3} - 2088 T^{2} + \cdots + 44800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 18 T^{4} - 104 T^{3} + \cdots - 14176)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + 300 T^{8} + \cdots + 142468096 \) Copy content Toggle raw display
$73$ \( (T^{5} - 20 T^{4} - 36 T^{3} + 2212 T^{2} + \cdots - 27712)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 4 T^{4} - 156 T^{3} + 816 T^{2} + \cdots - 12352)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 20 T^{4} - 22 T^{3} + 2160 T^{2} + \cdots - 4000)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 656 T^{8} + \cdots + 6209440000 \) Copy content Toggle raw display
$97$ \( (T^{5} - 32 T^{4} + 148 T^{3} + \cdots - 94912)^{2} \) Copy content Toggle raw display
show more
show less