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Properties

Label 4320.2.h.c
Level $4320$
Weight $2$
Character orbit 4320.h
Analytic conductor $34.495$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4320,2,Mod(2591,4320)] 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("4320.2591"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4320.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(34.4953736732\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 523 x^{12} - 1120 x^{11} + 2214 x^{10} - 3524 x^{9} + \cdots + 2116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{5} + (\beta_{15} - \beta_{11}) q^{7} - \beta_{6} q^{11} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \cdots - 1) q^{13} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{8}) q^{17} + ( - \beta_{14} + \beta_{13} + \cdots - \beta_{8}) q^{19}+ \cdots + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{11} - 8 q^{13} - 32 q^{23} - 16 q^{25} + 8 q^{37} - 16 q^{47} - 8 q^{49} + 32 q^{59} + 8 q^{61} + 32 q^{71} - 8 q^{73} + 32 q^{83} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 523 x^{12} - 1120 x^{11} + 2214 x^{10} - 3524 x^{9} + \cdots + 2116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 91\!\cdots\!98 \nu^{15} + \cdots + 31\!\cdots\!80 ) / 11\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 42\!\cdots\!82 \nu^{15} + \cdots + 55\!\cdots\!46 ) / 31\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 85\!\cdots\!34 \nu^{15} + \cdots - 16\!\cdots\!66 ) / 42\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 57\!\cdots\!29 \nu^{15} + \cdots + 24\!\cdots\!64 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!74 \nu^{15} + \cdots - 17\!\cdots\!34 ) / 34\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 89\!\cdots\!71 \nu^{15} + \cdots - 53\!\cdots\!94 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!60 \nu^{15} + \cdots - 90\!\cdots\!94 ) / 34\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 51\!\cdots\!63 \nu^{15} + \cdots - 10\!\cdots\!41 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63\!\cdots\!80 \nu^{15} + \cdots + 12\!\cdots\!54 ) / 15\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!54 \nu^{15} + \cdots + 45\!\cdots\!94 ) / 60\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65\!\cdots\!18 \nu^{15} + \cdots - 12\!\cdots\!11 ) / 92\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 68\!\cdots\!52 \nu^{15} + \cdots - 12\!\cdots\!66 ) / 80\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 43\!\cdots\!15 \nu^{15} + \cdots + 79\!\cdots\!83 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39\!\cdots\!28 \nu^{15} + \cdots + 72\!\cdots\!52 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 55\!\cdots\!94 \nu^{15} + \cdots + 10\!\cdots\!52 ) / 40\!\cdots\!51 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{12} - 2\beta_{11} - 2\beta_{10} - 3\beta_{9} - \beta_{8} + \beta_{5} - 3\beta_{3} - 2\beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7 \beta_{15} + 3 \beta_{14} + 11 \beta_{13} + 18 \beta_{12} - 4 \beta_{11} - 30 \beta_{10} - 4 \beta_{9} + \cdots - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5 \beta_{15} + 12 \beta_{14} + 6 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - 56 \beta_{10} + \cdots + 16 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{15} - 61 \beta_{14} - 55 \beta_{13} - 120 \beta_{12} - 20 \beta_{11} + 100 \beta_{10} + \cdots - 232 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 82 \beta_{15} - 228 \beta_{14} - 76 \beta_{13} - 143 \beta_{12} - 33 \beta_{11} + 769 \beta_{10} + \cdots - 626 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 466 \beta_{15} + 128 \beta_{14} - 119 \beta_{13} - 164 \beta_{12} + 934 \beta_{11} + 1748 \beta_{10} + \cdots - 190 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1353 \beta_{15} + 2220 \beta_{14} - 816 \beta_{13} - 960 \beta_{12} + 5090 \beta_{11} + 2778 \beta_{10} + \cdots + 5660 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3661 \beta_{15} - 3205 \beta_{14} - 6905 \beta_{13} - 8772 \beta_{12} + 6490 \beta_{11} + \cdots + 46488 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6608 \beta_{15} - 26189 \beta_{14} - 559 \beta_{13} - 539 \beta_{12} - 36465 \beta_{11} + 22303 \beta_{10} + \cdots + 66946 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 123211 \beta_{15} + 24409 \beta_{14} + 107393 \beta_{13} + 147414 \beta_{12} - 215780 \beta_{11} + \cdots + 281092 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 136953 \beta_{15} + 157322 \beta_{14} + 139026 \beta_{13} + 192498 \beta_{12} - 118716 \beta_{11} + \cdots - 331728 \beta_{8} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 803040 \beta_{15} + 160814 \beta_{14} + 678516 \beta_{13} + 947434 \beta_{12} - 1410393 \beta_{11} + \cdots - 1866160 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1048280 \beta_{15} - 2671239 \beta_{14} + 75921 \beta_{13} + 116406 \beta_{12} - 4423754 \beta_{11} + \cdots - 12352992 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 9702271 \beta_{15} - 5560283 \beta_{14} - 11023015 \beta_{13} - 15409874 \beta_{12} + \cdots - 104458028 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times\).

\(n\) \(2081\) \(2431\) \(3457\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
2.07925 − 0.273739i
2.86495 − 2.19835i
−0.694273 − 0.904794i
−0.391101 − 2.97071i
1.43545 + 1.87072i
−1.60613 + 1.23243i
0.425176 + 3.22953i
−0.113328 + 0.0149199i
−0.113328 − 0.0149199i
0.425176 − 3.22953i
−1.60613 − 1.23243i
1.43545 − 1.87072i
−0.391101 + 2.97071i
−0.694273 + 0.904794i
2.86495 + 2.19835i
2.07925 + 0.273739i
0 0 0 − 1.00000i 0 − 3.36972i 0 0 0
2591.2 0 0 0 − 1.00000i 0 − 3.18043i 0 0 0
2591.3 0 0 0 − 1.00000i 0 − 1.96845i 0 0 0
2591.4 0 0 0 − 1.00000i 0 − 0.710900i 0 0 0
2591.5 0 0 0 − 1.00000i 0 0.236402i 0 0 0
2591.6 0 0 0 − 1.00000i 0 1.44838i 0 0 0
2591.7 0 0 0 − 1.00000i 0 2.44295i 0 0 0
2591.8 0 0 0 − 1.00000i 0 5.10177i 0 0 0
2591.9 0 0 0 1.00000i 0 − 5.10177i 0 0 0
2591.10 0 0 0 1.00000i 0 − 2.44295i 0 0 0
2591.11 0 0 0 1.00000i 0 − 1.44838i 0 0 0
2591.12 0 0 0 1.00000i 0 − 0.236402i 0 0 0
2591.13 0 0 0 1.00000i 0 0.710900i 0 0 0
2591.14 0 0 0 1.00000i 0 1.96845i 0 0 0
2591.15 0 0 0 1.00000i 0 3.18043i 0 0 0
2591.16 0 0 0 1.00000i 0 3.36972i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4320.2.h.c yes 16
3.b odd 2 1 4320.2.h.b ✓ 16
4.b odd 2 1 4320.2.h.b ✓ 16
12.b even 2 1 inner 4320.2.h.c yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4320.2.h.b ✓ 16 3.b odd 2 1
4320.2.h.b ✓ 16 4.b odd 2 1
4320.2.h.c yes 16 1.a even 1 1 trivial
4320.2.h.c yes 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4320, [\chi])\):

\( T_{7}^{16} + 60 T_{7}^{14} + 1318 T_{7}^{12} + 13884 T_{7}^{10} + 74913 T_{7}^{8} + 201792 T_{7}^{6} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{11}^{8} - 4T_{11}^{7} - 38T_{11}^{6} + 212T_{11}^{5} + 81T_{11}^{4} - 2288T_{11}^{3} + 4840T_{11}^{2} - 3584T_{11} + 784 \) 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
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 60 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + \cdots + 1129)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 132 T^{14} + \cdots + 614656 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 7072137216 \) Copy content Toggle raw display
$23$ \( (T^{8} + 16 T^{7} + \cdots + 400)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 244 T^{14} + \cdots + 6718464 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 2570895616 \) Copy content Toggle raw display
$37$ \( (T^{8} - 4 T^{7} + \cdots + 432400)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 33785380864 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 109047890176 \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{7} + \cdots - 4559600)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 217558810624 \) Copy content Toggle raw display
$59$ \( (T^{8} - 16 T^{7} + \cdots + 116992)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{7} + \cdots + 33424)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1357775104 \) Copy content Toggle raw display
$71$ \( (T^{8} - 16 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 4 T^{7} + \cdots - 25992128)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 30162310929 \) Copy content Toggle raw display
$83$ \( (T^{8} - 16 T^{7} + \cdots - 123392)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23499380883456 \) Copy content Toggle raw display
$97$ \( (T^{8} + 4 T^{7} + \cdots - 3103344)^{2} \) Copy content Toggle raw display
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