↑

Properties

Label 2400.3.e.h
Level $2400$
Weight $3$
Character orbit 2400.e
Analytic conductor $65.395$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2400,3,Mod(1951,2400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.1951"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2400.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-36,0,0,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(65.3952634465\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} - \beta_{4} q^{7} - 3 q^{9} + ( - \beta_{11} - \beta_{4}) q^{11} + (\beta_{9} - \beta_{5} + \beta_{3}) q^{13} + ( - \beta_1 - 7) q^{17} + (\beta_{11} - \beta_{10} + \cdots - \beta_{4}) q^{19}+ \cdots + (3 \beta_{11} + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 36 q^{9} - 80 q^{17} + 40 q^{29} - 320 q^{37} + 136 q^{41} - 212 q^{49} + 176 q^{53} - 48 q^{57} + 40 q^{61} - 448 q^{73} - 448 q^{77} + 108 q^{81} + 8 q^{89} + 144 q^{93} - 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{8} + 4\nu^{7} - 35\nu^{6} + 91\nu^{5} - 362\nu^{4} + 577\nu^{3} - 1186\nu^{2} + 912\nu - 698 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 41 \nu^{8} - 134 \nu^{7} + 533 \nu^{6} - 1151 \nu^{5} + 2707 \nu^{4} + \cdots + 1666 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 41 \nu^{8} - 134 \nu^{7} + 533 \nu^{6} - 1151 \nu^{5} + 2707 \nu^{4} + \cdots + 1776 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 296 \nu^{11} - 1628 \nu^{10} + 13690 \nu^{9} - 49395 \nu^{8} + 204498 \nu^{7} - 496629 \nu^{6} + \cdots - 254594 ) / 17455 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{10} + 15 \nu^{9} - 118 \nu^{8} + 382 \nu^{7} - 1444 \nu^{6} + 3058 \nu^{5} - 6771 \nu^{4} + \cdots - 3498 ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1116 \nu^{11} + 6138 \nu^{10} - 48124 \nu^{9} + 170523 \nu^{8} - 660716 \nu^{7} + 1559698 \nu^{6} + \cdots + 1089810 ) / 34910 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1270 \nu^{11} - 6985 \nu^{10} + 53501 \nu^{9} - 188367 \nu^{8} + 703942 \nu^{7} - 1633646 \nu^{6} + \cdots - 756974 ) / 34910 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1270 \nu^{11} - 6985 \nu^{10} + 53501 \nu^{9} - 188367 \nu^{8} + 703942 \nu^{7} - 1633646 \nu^{6} + \cdots - 826794 ) / 34910 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{10} + 25 \nu^{9} - 197 \nu^{8} + 638 \nu^{7} - 2415 \nu^{6} + 5117 \nu^{5} - 11339 \nu^{4} + \cdots - 6166 ) / 10 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 354 \nu^{11} + 1947 \nu^{10} - 14627 \nu^{9} + 51219 \nu^{8} - 186684 \nu^{7} + 428001 \nu^{6} + \cdots + 155264 ) / 6982 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6170 \nu^{11} + 33935 \nu^{10} - 262671 \nu^{9} + 927507 \nu^{8} - 3516972 \nu^{7} + \cdots + 4895584 ) / 34910 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} + 2\beta_{3} - 2\beta_{2} - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} - 4\beta_{10} - 8\beta_{8} - 6\beta_{7} + 2\beta_{6} + \beta_{4} + 3\beta_{3} - 3\beta_{2} - 31 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{11} - 8 \beta_{10} - 4 \beta_{9} - 17 \beta_{8} - 11 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} + \cdots + 136 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25 \beta_{11} + 67 \beta_{10} - 10 \beta_{9} + 73 \beta_{8} + 131 \beta_{7} - 24 \beta_{6} + 15 \beta_{5} + \cdots + 392 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 85 \beta_{11} + 221 \beta_{10} + 62 \beta_{9} + 262 \beta_{8} + 420 \beta_{7} - 82 \beta_{6} + \cdots - 1065 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 249 \beta_{11} - 751 \beta_{10} + 252 \beta_{9} - 649 \beta_{8} - 1543 \beta_{7} + 208 \beta_{6} + \cdots - 5136 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1402 \beta_{11} - 4054 \beta_{10} - 624 \beta_{9} - 3859 \beta_{8} - 8157 \beta_{7} + 1224 \beta_{6} + \cdots + 8036 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1856 \beta_{11} + 6074 \beta_{10} - 4362 \beta_{9} + 4598 \beta_{8} + 12660 \beta_{7} - 1322 \beta_{6} + \cdots + 68687 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 20410 \beta_{11} + 62362 \beta_{10} + 3814 \beta_{9} + 53853 \beta_{8} + 127471 \beta_{7} - 16384 \beta_{6} + \cdots - 38162 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3761 \beta_{11} - 16523 \beta_{10} + 63844 \beta_{9} - 7393 \beta_{8} - 35447 \beta_{7} + \cdots - 900502 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1951.1
0.500000 − 1.80359i
0.500000 − 0.723587i
0.500000 + 2.02498i
0.500000 + 3.59014i
0.500000 − 2.65258i
0.500000 − 2.16741i
0.500000 + 2.16741i
0.500000 + 2.65258i
0.500000 − 3.59014i
0.500000 − 2.02498i
0.500000 + 0.723587i
0.500000 + 1.80359i
0 − 1.73205i 0 0 0 − 9.43574i 0 −3.00000 0
1951.2 0 − 1.73205i 0 0 0 − 7.21051i 0 −3.00000 0
1951.3 0 − 1.73205i 0 0 0 − 5.81384i 0 −3.00000 0
1951.4 0 − 1.73205i 0 0 0 2.37003i 0 −3.00000 0
1951.5 0 − 1.73205i 0 0 0 7.06571i 0 −3.00000 0
1951.6 0 − 1.73205i 0 0 0 13.0243i 0 −3.00000 0
1951.7 0 1.73205i 0 0 0 − 13.0243i 0 −3.00000 0
1951.8 0 1.73205i 0 0 0 − 7.06571i 0 −3.00000 0
1951.9 0 1.73205i 0 0 0 − 2.37003i 0 −3.00000 0
1951.10 0 1.73205i 0 0 0 5.81384i 0 −3.00000 0
1951.11 0 1.73205i 0 0 0 7.21051i 0 −3.00000 0
1951.12 0 1.73205i 0 0 0 9.43574i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1951.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.3.e.h 12
4.b odd 2 1 inner 2400.3.e.h 12
5.b even 2 1 2400.3.e.i 12
5.c odd 4 1 480.3.j.a ✓ 12
5.c odd 4 1 480.3.j.b yes 12
15.e even 4 1 1440.3.j.c 12
15.e even 4 1 1440.3.j.d 12
20.d odd 2 1 2400.3.e.i 12
20.e even 4 1 480.3.j.a ✓ 12
20.e even 4 1 480.3.j.b yes 12
40.i odd 4 1 960.3.j.f 12
40.i odd 4 1 960.3.j.g 12
40.k even 4 1 960.3.j.f 12
40.k even 4 1 960.3.j.g 12
60.l odd 4 1 1440.3.j.c 12
60.l odd 4 1 1440.3.j.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.j.a ✓ 12 5.c odd 4 1
480.3.j.a ✓ 12 20.e even 4 1
480.3.j.b yes 12 5.c odd 4 1
480.3.j.b yes 12 20.e even 4 1
960.3.j.f 12 40.i odd 4 1
960.3.j.f 12 40.k even 4 1
960.3.j.g 12 40.i odd 4 1
960.3.j.g 12 40.k even 4 1
1440.3.j.c 12 15.e even 4 1
1440.3.j.c 12 60.l odd 4 1
1440.3.j.d 12 15.e even 4 1
1440.3.j.d 12 60.l odd 4 1
2400.3.e.h 12 1.a even 1 1 trivial
2400.3.e.h 12 4.b odd 2 1 inner
2400.3.e.i 12 5.b even 2 1
2400.3.e.i 12 20.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 \(S_{3}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7}^{12} + 400T_{7}^{10} + 58464T_{7}^{8} + 4015872T_{7}^{6} + 134705408T_{7}^{4} + 1964957696T_{7}^{2} + 7442857984 \) Copy content Toggle raw display
\( T_{13}^{6} - 584T_{13}^{4} - 2080T_{13}^{3} + 92368T_{13}^{2} + 561280T_{13} - 1035008 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 7442857984 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 144769024 \) Copy content Toggle raw display
$13$ \( (T^{6} - 584 T^{4} + \cdots - 1035008)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 40 T^{5} + \cdots + 1401088)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 5879810228224 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{6} - 20 T^{5} + \cdots + 174739456)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 652427275534336 \) Copy content Toggle raw display
$37$ \( (T^{6} + 160 T^{5} + \cdots - 2501136128)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 68 T^{5} + \cdots + 997804096)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 67\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{6} - 88 T^{5} + \cdots - 7496638976)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{6} - 20 T^{5} + \cdots - 24763044800)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 73\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{6} + 224 T^{5} + \cdots + 38911295488)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{6} - 4 T^{5} + \cdots + 34204460608)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 112 T^{5} + \cdots + 321738244096)^{2} \) Copy content Toggle raw display
show more
show less