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Properties

Label 1755.2.c.a
Level $1755$
Weight $2$
Character orbit 1755.c
Analytic conductor $14.014$
Analytic rank $0$
Dimension $16$
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] = [1755,2,Mod(1054,1755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1755, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1755.1054");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1755.c (of order \(2\), degree \(1\), 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: \(14.0137455547\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 68 x^{12} - 40 x^{11} + 192 x^{10} - 426 x^{9} + 426 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + ( - \beta_{5} - \beta_{2} - 1) q^{4} + ( - \beta_{13} - \beta_{10}) q^{5} + \beta_1 q^{7} - \beta_{15} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{2} + ( - \beta_{5} - \beta_{2} - 1) q^{4} + ( - \beta_{13} - \beta_{10}) q^{5} + \beta_1 q^{7} - \beta_{15} q^{8} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{3}) q^{10}+ \cdots + (\beta_{15} + 5 \beta_{14} + \cdots + \beta_{12}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{4} + 4 q^{10} - 12 q^{16} + 16 q^{19} - 8 q^{25} - 44 q^{31} + 28 q^{34} + 16 q^{40} - 56 q^{46} + 60 q^{49} + 20 q^{55} - 44 q^{61} + 68 q^{64} + 20 q^{70} - 68 q^{76} + 36 q^{79} + 16 q^{85} - 4 q^{91} - 28 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 68 x^{12} - 40 x^{11} + 192 x^{10} - 426 x^{9} + 426 x^{8} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 23\!\cdots\!51 \nu^{15} + \cdots - 13\!\cdots\!94 ) / 24\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36233983590144 \nu^{15} + 297144292078144 \nu^{14} + \cdots + 23\!\cdots\!13 ) / 10\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!78 \nu^{15} + \cdots - 68\!\cdots\!43 ) / 24\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 492781000839716 \nu^{15} + \cdots + 53\!\cdots\!66 ) / 54\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 119073510980232 \nu^{15} + \cdots - 13\!\cdots\!55 ) / 10\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 130391087646592 \nu^{15} + \cdots - 22\!\cdots\!81 ) / 10\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!59 \nu^{15} + \cdots + 23\!\cdots\!04 ) / 24\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!86 \nu^{15} + \cdots - 15\!\cdots\!91 ) / 27\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39\!\cdots\!48 \nu^{15} + \cdots + 22\!\cdots\!13 ) / 27\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46\!\cdots\!53 \nu^{15} + \cdots + 14\!\cdots\!68 ) / 27\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43\!\cdots\!93 \nu^{15} + \cdots + 42\!\cdots\!58 ) / 24\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 59\!\cdots\!91 \nu^{15} + \cdots - 43\!\cdots\!02 ) / 98\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 60\!\cdots\!02 \nu^{15} + \cdots + 42\!\cdots\!59 ) / 98\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28\!\cdots\!97 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 32\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30\!\cdots\!64 \nu^{15} + \cdots + 19\!\cdots\!46 ) / 32\!\cdots\!71 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{12} + \beta_{10} + \beta_{8} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{7} + 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + \beta_{14} + 5 \beta_{13} + 5 \beta_{12} - 5 \beta_{10} + \beta_{9} - 5 \beta_{8} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{10} + 2\beta_{9} - 8\beta_{8} + \beta_{6} - 2\beta_{4} + 8\beta_{2} - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12 \beta_{15} - 13 \beta_{14} - 32 \beta_{13} - 33 \beta_{12} + 5 \beta_{11} - 33 \beta_{10} + 13 \beta_{9} + \cdots - 59 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26\beta_{15} - 29\beta_{14} - 58\beta_{13} - 61\beta_{12} + 15\beta_{11} - 66\beta_{7} - 116\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 121 \beta_{15} - 141 \beta_{14} - 224 \beta_{13} - 245 \beta_{12} + 70 \beta_{11} + 245 \beta_{10} + \cdots + 435 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 480\beta_{10} - 312\beta_{9} + 424\beta_{8} - 166\beta_{6} - 24\beta_{5} + 260\beta_{4} - 559\beta_{2} + 849 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1127 \beta_{15} + 1393 \beta_{14} + 1642 \beta_{13} + 1936 \beta_{12} - 738 \beta_{11} + 1936 \beta_{10} + \cdots + 3289 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2383 \beta_{15} + 3017 \beta_{14} + 3178 \beta_{13} + 3892 \beta_{12} - 1629 \beta_{11} + \cdots - 346 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10055 \beta_{15} + 13010 \beta_{14} + 12437 \beta_{13} + 15818 \beta_{12} - 7029 \beta_{11} + \cdots - 25541 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 32180 \beta_{10} + 27702 \beta_{9} - 24434 \beta_{8} + 15047 \beta_{6} + 4015 \beta_{5} + \cdots - 50794 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 87589 \beta_{15} - 117352 \beta_{14} - 96660 \beta_{13} - 131498 \beta_{12} + 63765 \beta_{11} + \cdots - 202786 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 181757 \beta_{15} - 246968 \beta_{14} - 192038 \beta_{13} - 269010 \beta_{12} + 134353 \beta_{11} + \cdots + 41524 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 752704 \beta_{15} - 1035316 \beta_{14} - 766720 \beta_{13} - 1102617 \beta_{12} + 563078 \beta_{11} + \cdots + 1637474 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1755\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1054.1
0.468472 + 0.468472i
0.531528 − 0.531528i
1.72754 + 1.72754i
−0.727539 + 0.727539i
2.05245 − 2.05245i
−1.05245 − 1.05245i
1.54022 + 1.54022i
−0.540223 + 0.540223i
1.54022 − 1.54022i
−0.540223 − 0.540223i
2.05245 + 2.05245i
−1.05245 + 1.05245i
1.72754 − 1.72754i
−0.727539 − 0.727539i
0.468472 − 0.468472i
0.531528 + 0.531528i
− 2.25781i 0 −3.09769 −1.54930 + 1.61235i 0 − 0.400318i 2.47838i 0 3.64038 + 3.49801i
1054.2 − 2.25781i 0 −3.09769 1.54930 + 1.61235i 0 0.400318i 2.47838i 0 3.64038 − 3.49801i
1054.3 − 2.10546i 0 −2.43295 −0.230970 − 2.22411i 0 1.94665i 0.911559i 0 −4.68276 + 0.486298i
1054.4 − 2.10546i 0 −2.43295 0.230970 − 2.22411i 0 − 1.94665i 0.911559i 0 −4.68276 − 0.486298i
1054.5 − 1.05395i 0 0.889179 −1.25260 + 1.85230i 0 − 0.431001i − 3.04506i 0 1.95224 + 1.32018i
1054.6 − 1.05395i 0 0.889179 1.25260 + 1.85230i 0 0.431001i − 3.04506i 0 1.95224 − 1.32018i
1054.7 − 0.598778i 0 1.64146 −2.23099 + 0.150548i 0 − 2.97734i − 2.18043i 0 0.0901446 + 1.33587i
1054.8 − 0.598778i 0 1.64146 2.23099 + 0.150548i 0 2.97734i − 2.18043i 0 0.0901446 − 1.33587i
1054.9 0.598778i 0 1.64146 −2.23099 − 0.150548i 0 2.97734i 2.18043i 0 0.0901446 − 1.33587i
1054.10 0.598778i 0 1.64146 2.23099 − 0.150548i 0 − 2.97734i 2.18043i 0 0.0901446 + 1.33587i
1054.11 1.05395i 0 0.889179 −1.25260 − 1.85230i 0 0.431001i 3.04506i 0 1.95224 − 1.32018i
1054.12 1.05395i 0 0.889179 1.25260 − 1.85230i 0 − 0.431001i 3.04506i 0 1.95224 + 1.32018i
1054.13 2.10546i 0 −2.43295 −0.230970 + 2.22411i 0 − 1.94665i − 0.911559i 0 −4.68276 − 0.486298i
1054.14 2.10546i 0 −2.43295 0.230970 + 2.22411i 0 1.94665i − 0.911559i 0 −4.68276 + 0.486298i
1054.15 2.25781i 0 −3.09769 −1.54930 − 1.61235i 0 0.400318i − 2.47838i 0 3.64038 − 3.49801i
1054.16 2.25781i 0 −3.09769 1.54930 − 1.61235i 0 − 0.400318i − 2.47838i 0 3.64038 + 3.49801i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1054.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1755.2.c.a ✓ 16
3.b odd 2 1 inner 1755.2.c.a ✓ 16
5.b even 2 1 inner 1755.2.c.a ✓ 16
5.c odd 4 1 8775.2.a.cd 8
5.c odd 4 1 8775.2.a.ce 8
15.d odd 2 1 inner 1755.2.c.a ✓ 16
15.e even 4 1 8775.2.a.cd 8
15.e even 4 1 8775.2.a.ce 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1755.2.c.a ✓ 16 1.a even 1 1 trivial
1755.2.c.a ✓ 16 3.b odd 2 1 inner
1755.2.c.a ✓ 16 5.b even 2 1 inner
1755.2.c.a ✓ 16 15.d odd 2 1 inner
8775.2.a.cd 8 5.c odd 4 1
8775.2.a.cd 8 15.e even 4 1
8775.2.a.ce 8 5.c odd 4 1
8775.2.a.ce 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1755, [\chi])\):

\( T_{2}^{8} + 11T_{2}^{6} + 37T_{2}^{4} + 37T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{8} - 12T_{11}^{6} + 48T_{11}^{4} - 71T_{11}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 11 T^{6} + 37 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 13 T^{6} + 38 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 12 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{8} + 36 T^{6} + 345 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} - 19 T^{2} + 25)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 51 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 106 T^{6} + \cdots + 123201)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 11 T^{3} + \cdots + 307)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 137 T^{6} + \cdots + 59049)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 104 T^{6} + \cdots + 8649)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 246 T^{6} + \cdots + 4363921)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 27 T^{6} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 169 T^{6} + \cdots + 57121)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 145 T^{6} + \cdots + 383161)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 11 T^{3} + \cdots + 313)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 272 T^{6} + \cdots + 1050625)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 183 T^{6} + \cdots + 38809)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 321 T^{6} + \cdots + 21613201)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 9 T^{3} + \cdots - 2957)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 267 T^{6} + \cdots + 149769)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 498 T^{6} + \cdots + 15944049)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 121 T^{6} + \cdots + 110889)^{2} \) Copy content Toggle raw display
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