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Properties

Label 3456.2.p.e
Level $3456$
Weight $2$
Character orbit 3456.p
Analytic conductor $27.596$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3456,2,Mod(575,3456)] 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("3456.575"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3456, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3456 = 2^{7} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3456.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.5962989386\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.9349208943630483456.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{14} - \beta_{10} + \cdots - \beta_{5}) q^{7} + \beta_{9} q^{11} + (\beta_{12} - \beta_{3}) q^{13} + (2 \beta_{15} + \beta_{4}) q^{17} + (2 \beta_{9} + 2 \beta_{8}) q^{19}+ \cdots + ( - 5 \beta_{15} - 5 \beta_{4} + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 72 q^{41} + 16 q^{49} - 72 q^{65} - 64 q^{73} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 56 \nu^{14} + 392 \nu^{13} - 2267 \nu^{12} + 8506 \nu^{11} - 26269 \nu^{10} + 62716 \nu^{9} + \cdots - 2430 ) / 65 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 125460 ) / 17095 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10 \nu^{14} - 70 \nu^{13} + 406 \nu^{12} - 1526 \nu^{11} + 4732 \nu^{10} - 11340 \nu^{9} + 22581 \nu^{8} + \cdots + 670 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54 \nu^{14} + 378 \nu^{13} - 2193 \nu^{12} + 8244 \nu^{11} - 25569 \nu^{10} + 61284 \nu^{9} + \cdots - 3308 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13659 \nu^{15} - 108623 \nu^{14} + 649844 \nu^{13} - 2633126 \nu^{12} + 8564479 \nu^{11} + \cdots - 440990 ) / 17095 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22230 \nu^{15} - 174089 \nu^{14} + 1026678 \nu^{13} - 4107793 \nu^{12} + 13139184 \nu^{11} + \cdots - 678865 ) / 17095 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27318 \nu^{15} - 204885 \nu^{14} + 1213161 \nu^{13} - 4778124 \nu^{12} + 15325041 \nu^{11} + \cdots - 1518440 ) / 17095 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31151 \nu^{15} + 209568 \nu^{14} - 1205575 \nu^{13} + 4413594 \nu^{12} - 13470792 \nu^{11} + \cdots - 318955 ) / 17095 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31151 \nu^{15} - 257697 \nu^{14} + 1542478 \nu^{13} - 6361898 \nu^{12} + 20780877 \nu^{11} + \cdots - 2345235 ) / 17095 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 49663 \nu^{15} + 341833 \nu^{14} - 1975287 \nu^{13} + 7337808 \nu^{12} - 22583912 \nu^{11} + \cdots - 472970 ) / 17095 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 79057 \nu^{15} - 534147 \nu^{14} + 3074382 \nu^{13} - 11283534 \nu^{12} + 34475529 \nu^{11} + \cdots + 988920 ) / 17095 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79890 \nu^{15} + 582080 \nu^{14} - 3409550 \nu^{13} + 13158353 \nu^{12} - 41509888 \nu^{11} + \cdots + 1634075 ) / 17095 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 79057 \nu^{15} + 651708 \nu^{14} - 3897309 \nu^{13} + 16046990 \nu^{12} - 52358214 \nu^{11} + \cdots + 5801740 ) / 17095 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 99326 \nu^{15} - 744945 \nu^{14} + 4379527 \nu^{13} - 17168593 \nu^{12} + 54549297 \nu^{11} + \cdots - 3108205 ) / 17095 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 162552 \nu^{15} + 1254645 \nu^{14} - 7410690 \nu^{13} + 29505615 \nu^{12} - 94503294 \nu^{11} + \cdots + 7071655 ) / 17095 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} + 3\beta_{14} + 2\beta_{12} + 3\beta_{9} - 3\beta_{8} + \beta_{4} + \beta_{3} + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} + 4 \beta_{14} + 2 \beta_{12} + 2 \beta_{10} - 6 \beta_{8} - 2 \beta_{7} + 4 \beta_{5} + \cdots - 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{15} - 9 \beta_{14} + 3 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} + 3 \beta_{10} - 15 \beta_{9} + \cdots - 24 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{15} - 12 \beta_{14} + 2 \beta_{13} - 12 \beta_{12} - 4 \beta_{11} - 4 \beta_{9} + 20 \beta_{8} + \cdots + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43 \beta_{15} + 26 \beta_{14} - 23 \beta_{13} + 57 \beta_{12} + 13 \beta_{11} - 5 \beta_{10} + \cdots + 183 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 185 \beta_{15} + 126 \beta_{14} - 54 \beta_{13} + 232 \beta_{12} + 84 \beta_{11} - 42 \beta_{10} + \cdots - 30 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 128 \beta_{15} - 105 \beta_{14} + 127 \beta_{13} - 198 \beta_{12} + 13 \beta_{11} - 126 \beta_{10} + \cdots - 1299 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 714 \beta_{15} - 400 \beta_{14} + 256 \beta_{13} - 966 \beta_{12} - 308 \beta_{11} + 112 \beta_{10} + \cdots - 606 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 379 \beta_{15} + 423 \beta_{14} - 522 \beta_{13} - 385 \beta_{12} - 828 \beta_{11} + 1737 \beta_{10} + \cdots + 8202 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10223 \beta_{15} + 5990 \beta_{14} - 4296 \beta_{13} + 14312 \beta_{12} + 3660 \beta_{11} + 448 \beta_{10} + \cdots + 17628 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12443 \beta_{15} + 490 \beta_{14} + 303 \beta_{13} + 16927 \beta_{12} + 10224 \beta_{11} - 14773 \beta_{10} + \cdots - 43491 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 32900 \beta_{15} - 22884 \beta_{14} + 16608 \beta_{13} - 46629 \beta_{12} - 8310 \beta_{11} + \cdots - 89109 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 158381 \beta_{15} - 40595 \beta_{14} + 27473 \beta_{13} - 221934 \beta_{12} - 94774 \beta_{11} + \cdots + 151791 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 352742 \beta_{15} + 322972 \beta_{14} - 232372 \beta_{13} + 501646 \beta_{12} + 28586 \beta_{11} + \cdots + 1521186 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1567315 \beta_{15} + 610197 \beta_{14} - 431475 \beta_{13} + 2212539 \beta_{12} + 753006 \beta_{11} + \cdots + 356820 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(2431\) \(2945\)
\(\chi(n)\) \(-1\) \(-1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
0.500000 + 1.00333i
0.500000 − 0.589118i
0.500000 − 0.410882i
0.500000 − 2.00333i
0.500000 + 1.33108i
0.500000 − 1.74530i
0.500000 − 0.331082i
0.500000 + 2.74530i
0.500000 − 1.00333i
0.500000 + 0.589118i
0.500000 + 0.410882i
0.500000 + 2.00333i
0.500000 − 1.33108i
0.500000 + 1.74530i
0.500000 + 0.331082i
0.500000 − 2.74530i
0 0 0 −1.57313 − 2.72474i 0 −2.21650 − 1.27970i 0 0 0
575.2 0 0 0 −1.57313 − 2.72474i 0 2.21650 + 1.27970i 0 0 0
575.3 0 0 0 −0.158919 − 0.275255i 0 −2.93038 − 1.69185i 0 0 0
575.4 0 0 0 −0.158919 − 0.275255i 0 2.93038 + 1.69185i 0 0 0
575.5 0 0 0 0.158919 + 0.275255i 0 −2.93038 − 1.69185i 0 0 0
575.6 0 0 0 0.158919 + 0.275255i 0 2.93038 + 1.69185i 0 0 0
575.7 0 0 0 1.57313 + 2.72474i 0 −2.21650 − 1.27970i 0 0 0
575.8 0 0 0 1.57313 + 2.72474i 0 2.21650 + 1.27970i 0 0 0
2879.1 0 0 0 −1.57313 + 2.72474i 0 −2.21650 + 1.27970i 0 0 0
2879.2 0 0 0 −1.57313 + 2.72474i 0 2.21650 − 1.27970i 0 0 0
2879.3 0 0 0 −0.158919 + 0.275255i 0 −2.93038 + 1.69185i 0 0 0
2879.4 0 0 0 −0.158919 + 0.275255i 0 2.93038 − 1.69185i 0 0 0
2879.5 0 0 0 0.158919 − 0.275255i 0 −2.93038 + 1.69185i 0 0 0
2879.6 0 0 0 0.158919 − 0.275255i 0 2.93038 − 1.69185i 0 0 0
2879.7 0 0 0 1.57313 − 2.72474i 0 −2.21650 + 1.27970i 0 0 0
2879.8 0 0 0 1.57313 − 2.72474i 0 2.21650 − 1.27970i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner
72.j odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3456.2.p.e 16
3.b odd 2 1 1152.2.p.e ✓ 16
4.b odd 2 1 inner 3456.2.p.e 16
8.b even 2 1 inner 3456.2.p.e 16
8.d odd 2 1 inner 3456.2.p.e 16
9.c even 3 1 1152.2.p.e ✓ 16
9.d odd 6 1 inner 3456.2.p.e 16
12.b even 2 1 1152.2.p.e ✓ 16
24.f even 2 1 1152.2.p.e ✓ 16
24.h odd 2 1 1152.2.p.e ✓ 16
36.f odd 6 1 1152.2.p.e ✓ 16
36.h even 6 1 inner 3456.2.p.e 16
72.j odd 6 1 inner 3456.2.p.e 16
72.l even 6 1 inner 3456.2.p.e 16
72.n even 6 1 1152.2.p.e ✓ 16
72.p odd 6 1 1152.2.p.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.p.e ✓ 16 3.b odd 2 1
1152.2.p.e ✓ 16 9.c even 3 1
1152.2.p.e ✓ 16 12.b even 2 1
1152.2.p.e ✓ 16 24.f even 2 1
1152.2.p.e ✓ 16 24.h odd 2 1
1152.2.p.e ✓ 16 36.f odd 6 1
1152.2.p.e ✓ 16 72.n even 6 1
1152.2.p.e ✓ 16 72.p odd 6 1
3456.2.p.e 16 1.a even 1 1 trivial
3456.2.p.e 16 4.b odd 2 1 inner
3456.2.p.e 16 8.b even 2 1 inner
3456.2.p.e 16 8.d odd 2 1 inner
3456.2.p.e 16 9.d odd 6 1 inner
3456.2.p.e 16 36.h even 6 1 inner
3456.2.p.e 16 72.j odd 6 1 inner
3456.2.p.e 16 72.l even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3456, [\chi])\):

\( T_{5}^{8} + 10T_{5}^{6} + 99T_{5}^{4} + 10T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 18T_{7}^{6} + 249T_{7}^{4} - 1350T_{7}^{2} + 5625 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{6} + 33T_{11}^{4} - 18T_{11}^{2} + 9 \) 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
$5$ \( (T^{8} + 10 T^{6} + 99 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 18 T^{6} + \cdots + 5625)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{6} + 33 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 9 T^{2} + 81)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 18)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 72 T^{2} + 432)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 18 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 70 T^{6} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 102 T^{6} + \cdots + 3515625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 54)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 18 T^{3} + \cdots + 81)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 54 T^{6} + \cdots + 59049)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 198 T^{6} + \cdots + 95004009)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 100 T^{2} + 2116)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 18 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 81 T^{2} + 6561)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 162 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 396 T^{2} + 38988)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T + 10)^{8} \) Copy content Toggle raw display
$79$ \( (T^{8} - 198 T^{6} + \cdots + 95004009)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 18 T^{6} + \cdots + 5625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 252 T^{2} + 8100)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + \cdots + 22201)^{4} \) Copy content Toggle raw display
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