Properties

Label 2240.2.g.p
Level $2240$
Weight $2$
Character orbit 2240.g
Analytic conductor $17.886$
Analytic rank $0$
Dimension $12$
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] = [2240,2,Mod(449,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 148x^{8} + 494x^{6} + 708x^{4} + 304x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1120)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} - \beta_{10} q^{5} - \beta_{5} q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} - \beta_{10} q^{5} - \beta_{5} q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{3} - 1) q^{9} - \beta_{2} q^{11} + (\beta_{11} + \beta_{10} + \beta_{4}) q^{13} + ( - \beta_{9} + \beta_{5} + \beta_{2} + \beta_1) q^{15} + ( - 2 \beta_{6} + \beta_{4}) q^{17} + (\beta_{8} + \beta_{7} - 2 \beta_1) q^{19} - \beta_{3} q^{21} + (2 \beta_{9} - 2 \beta_{5}) q^{23} + ( - 2 \beta_{11} + \beta_{6} - \beta_{3}) q^{25} + (\beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{5}) q^{27} + (\beta_{11} - \beta_{10} - \beta_{3} + 4) q^{29} + (2 \beta_{8} + 2 \beta_{7}) q^{31} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{4}) q^{33} + \beta_{7} q^{35} + (2 \beta_{11} + 2 \beta_{10} - 2 \beta_{6}) q^{37} + (2 \beta_{8} + 2 \beta_{7} - \beta_{2} - 2 \beta_1) q^{39} + (2 \beta_{3} + 2) q^{41} + ( - 2 \beta_{9} - 4 \beta_{5}) q^{43} + (2 \beta_{11} + \beta_{10} - \beta_{4} - 4) q^{45} + (\beta_{9} + 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{5}) q^{47} - q^{49} + (4 \beta_{8} + 4 \beta_{7} + \beta_{2}) q^{51} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{53} + ( - 3 \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_1) q^{55} + ( - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{6} - 2 \beta_{4}) q^{57} + ( - \beta_{8} - \beta_{7} + 2 \beta_1) q^{59} + (\beta_{11} - \beta_{10} + 2 \beta_{3} - 4) q^{61} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5}) q^{63} + (\beta_{11} + \beta_{10} - 3 \beta_{6} - 2 \beta_{3} + 5) q^{65} + (2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{5}) q^{67} + (2 \beta_{11} - 2 \beta_{10} + 8) q^{69} + 2 \beta_1 q^{71} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{73} + (3 \beta_{9} - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{75} + \beta_{4} q^{77} + ( - 2 \beta_{8} - 2 \beta_{7} + \beta_{2} + 2 \beta_1) q^{79} + ( - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{3} - 1) q^{81} + ( - 2 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 2 \beta_{5}) q^{83} + (\beta_{11} + \beta_{10} + 2 \beta_{4} - 5 \beta_{3} - 2) q^{85} + ( - 5 \beta_{9} + \beta_{8} - \beta_{7} + 6 \beta_{5}) q^{87} - 6 q^{89} + ( - \beta_{8} - \beta_{7} + \beta_{2}) q^{91} + (4 \beta_{6} - 4 \beta_{4}) q^{93} + ( - \beta_{9} + 4 \beta_{8} + 3 \beta_{5} - 2 \beta_{2} + \beta_1) q^{95} + (2 \beta_{6} + \beta_{4}) q^{97} + (2 \beta_{8} + 2 \beta_{7} + 2 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{9} - 4 q^{21} - 4 q^{25} + 44 q^{29} + 32 q^{41} - 48 q^{45} - 12 q^{49} - 40 q^{61} + 52 q^{65} + 96 q^{69} - 4 q^{81} - 44 q^{85} - 72 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 148x^{8} + 494x^{6} + 708x^{4} + 304x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{10} + 67\nu^{8} + 382\nu^{6} + 874\nu^{4} + 751\nu^{2} + 116 ) / 62 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{10} - 143\nu^{8} - 689\nu^{6} - 773\nu^{4} + 1077\nu^{2} + 607 ) / 124 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{10} + 58\nu^{8} + 395\nu^{6} + 1105\nu^{4} + 1098\nu^{2} + 211 ) / 62 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 71\nu^{9} + 979\nu^{7} + 5039\nu^{5} + 9945\nu^{3} + 5113\nu ) / 186 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{11} + 223\nu^{9} + 1655\nu^{7} + 5395\nu^{5} + 7095\nu^{3} + 2303\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{11} - 125\nu^{9} - 808\nu^{7} - 2351\nu^{5} - 3213\nu^{3} - 1660\nu ) / 186 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16 \nu^{11} - 12 \nu^{10} - 299 \nu^{9} - 201 \nu^{8} - 1993 \nu^{7} - 1146 \nu^{6} - 5666 \nu^{5} - 2436 \nu^{4} - 6321 \nu^{3} - 951 \nu^{2} - 1549 \nu + 768 ) / 372 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16 \nu^{11} - 12 \nu^{10} + 299 \nu^{9} - 201 \nu^{8} + 1993 \nu^{7} - 1146 \nu^{6} + 5666 \nu^{5} - 2436 \nu^{4} + 6321 \nu^{3} - 951 \nu^{2} + 1549 \nu + 768 ) / 372 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\nu^{11} + 473\nu^{9} + 3271\nu^{7} + 10097\nu^{5} + 13521\nu^{3} + 6367\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20 \nu^{11} - 15 \nu^{10} - 397 \nu^{9} - 228 \nu^{8} - 2933 \nu^{7} - 1107 \nu^{6} - 9826 \nu^{5} - 1743 \nu^{4} - 13923 \nu^{3} - 282 \nu^{2} - 4889 \nu - 249 ) / 372 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20 \nu^{11} + 15 \nu^{10} - 397 \nu^{9} + 228 \nu^{8} - 2933 \nu^{7} + 1107 \nu^{6} - 9826 \nu^{5} + 1743 \nu^{4} - 13923 \nu^{3} + 282 \nu^{2} - 4889 \nu + 249 ) / 372 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} - \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} - 4\beta_{9} - \beta_{8} + \beta_{7} - 6\beta_{6} + 8\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{11} + 7\beta_{10} - 5\beta_{8} - 5\beta_{7} - 7\beta_{3} + 9\beta _1 + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{11} - 7\beta_{10} + 21\beta_{9} + 11\beta_{8} - 11\beta_{7} + 35\beta_{6} - 61\beta_{5} + 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{11} - 24\beta_{10} + 12\beta_{8} + 12\beta_{7} + 23\beta_{3} + 2\beta_{2} - 33\beta _1 - 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 40 \beta_{11} + 40 \beta_{10} - 121 \beta_{9} - 96 \beta_{8} + 96 \beta_{7} - 213 \beta_{6} + 433 \beta_{5} - 24 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -335\beta_{11} + 335\beta_{10} - 115\beta_{8} - 115\beta_{7} - 291\beta_{3} - 56\beta_{2} + 459\beta _1 + 1329 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 209 \beta_{11} - 209 \beta_{10} + 730 \beta_{9} + 765 \beta_{8} - 765 \beta_{7} + 1344 \beta_{6} - 2974 \beta_{5} + 220 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2369 \beta_{11} - 2369 \beta_{10} + 539 \beta_{8} + 539 \beta_{7} + 1823 \beta_{3} + 556 \beta_{2} - 3133 \beta _1 - 8461 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1007 \beta_{11} + 1007 \beta_{10} - 4523 \beta_{9} - 5815 \beta_{8} + 5815 \beta_{7} - 8705 \beta_{6} + 20179 \beta_{5} - 1830 \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.67709i
0.178681i
0.810591i
2.36682i
1.97020i
2.64856i
2.64856i
1.97020i
2.36682i
0.810591i
0.178681i
1.67709i
0 2.85577i 0 0.649832 2.13956i 0 1.00000i 0 −5.15544 0
449.2 0 2.85577i 0 0.649832 + 2.13956i 0 1.00000i 0 −5.15544 0
449.3 0 2.17741i 0 1.45926 1.69427i 0 1.00000i 0 −1.74111 0
449.4 0 2.17741i 0 1.45926 + 1.69427i 0 1.00000i 0 −1.74111 0
449.5 0 0.321637i 0 −2.10909 0.742782i 0 1.00000i 0 2.89655 0
449.6 0 0.321637i 0 −2.10909 + 0.742782i 0 1.00000i 0 2.89655 0
449.7 0 0.321637i 0 −2.10909 0.742782i 0 1.00000i 0 2.89655 0
449.8 0 0.321637i 0 −2.10909 + 0.742782i 0 1.00000i 0 2.89655 0
449.9 0 2.17741i 0 1.45926 1.69427i 0 1.00000i 0 −1.74111 0
449.10 0 2.17741i 0 1.45926 + 1.69427i 0 1.00000i 0 −1.74111 0
449.11 0 2.85577i 0 0.649832 2.13956i 0 1.00000i 0 −5.15544 0
449.12 0 2.85577i 0 0.649832 + 2.13956i 0 1.00000i 0 −5.15544 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.g.p 12
4.b odd 2 1 inner 2240.2.g.p 12
5.b even 2 1 inner 2240.2.g.p 12
8.b even 2 1 1120.2.g.d 12
8.d odd 2 1 1120.2.g.d 12
20.d odd 2 1 inner 2240.2.g.p 12
40.e odd 2 1 1120.2.g.d 12
40.f even 2 1 1120.2.g.d 12
40.i odd 4 1 5600.2.a.by 6
40.i odd 4 1 5600.2.a.bz 6
40.k even 4 1 5600.2.a.by 6
40.k even 4 1 5600.2.a.bz 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.d 12 8.b even 2 1
1120.2.g.d 12 8.d odd 2 1
1120.2.g.d 12 40.e odd 2 1
1120.2.g.d 12 40.f even 2 1
2240.2.g.p 12 1.a even 1 1 trivial
2240.2.g.p 12 4.b odd 2 1 inner
2240.2.g.p 12 5.b even 2 1 inner
2240.2.g.p 12 20.d odd 2 1 inner
5600.2.a.by 6 40.i odd 4 1
5600.2.a.by 6 40.k even 4 1
5600.2.a.bz 6 40.i odd 4 1
5600.2.a.bz 6 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{6} + 13T_{3}^{4} + 40T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{6} - 45T_{11}^{4} + 592T_{11}^{2} - 1856 \) Copy content Toggle raw display
\( T_{19}^{6} - 104T_{19}^{4} + 2708T_{19}^{2} - 4176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 13 T^{4} + 40 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{4} + 16 T^{3} + 5 T^{2} + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} - 45 T^{4} + 592 T^{2} - 1856)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 65 T^{4} + 1056 T^{2} + 116)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 109 T^{4} + 3608 T^{2} + \cdots + 37584)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 104 T^{4} + 2708 T^{2} + \cdots - 4176)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 56 T^{4} + 656 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 11 T^{2} + 16 T + 12)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} - 128 T^{4} + 4416 T^{2} + \cdots - 29696)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 168 T^{4} + 8080 T^{2} + \cdots + 118784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} - 4 T + 48)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 116 T^{4} + 2048 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 257 T^{4} + 21680 T^{2} + \cdots + 602176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 204 T^{4} + 8176 T^{2} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 104 T^{4} + 2708 T^{2} + \cdots - 4176)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 10 T^{2} + 2 T - 124)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 196 T^{4} + 7168 T^{2} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 104 T^{4} + 1680 T^{2} + \cdots - 7424)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 204 T^{4} + 8176 T^{2} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 149 T^{4} + 648 T^{2} - 464)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 372 T^{4} + 34084 T^{2} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} + 189 T^{4} + 664 T^{2} + 464)^{2} \) Copy content Toggle raw display
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