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Properties

Label 1755.2.b.e
Level $1755$
Weight $2$
Character orbit 1755.b
Analytic conductor $14.014$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1755,2,Mod(1351,1755)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1755.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1755.b (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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 32 x^{16} + 428 x^{14} + 3114 x^{12} + 13440 x^{10} + 35180 x^{8} + 54641 x^{6} + 46624 x^{4} + \cdots + 1764 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{6} q^{5} + (\beta_{16} + \beta_{6}) q^{7} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{6} q^{5} + (\beta_{16} + \beta_{6}) q^{7} + (\beta_{3} - \beta_1) q^{8} - \beta_{5} q^{10} + ( - \beta_{17} + \beta_{6}) q^{11} + ( - \beta_{12} + \beta_{4} - 1) q^{13} + ( - \beta_{12} + \beta_{11} + \cdots + \beta_{4}) q^{14}+ \cdots + (\beta_{16} - 2 \beta_{15} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 28 q^{4} - 10 q^{13} + 16 q^{14} + 24 q^{16} + 12 q^{22} - 10 q^{23} - 18 q^{25} - 8 q^{26} - 40 q^{29} - 10 q^{35} + 12 q^{38} - 8 q^{43} - 12 q^{49} - 6 q^{52} + 34 q^{53} - 16 q^{55} - 8 q^{56} - 16 q^{61} - 28 q^{62} - 76 q^{64} + 8 q^{65} - 24 q^{68} - 100 q^{74} + 4 q^{77} + 6 q^{79} - 16 q^{82} + 28 q^{88} - 6 q^{91} + 80 q^{92} - 40 q^{94} + 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 32 x^{16} + 428 x^{14} + 3114 x^{12} + 13440 x^{10} + 35180 x^{8} + 54641 x^{6} + 46624 x^{4} + \cdots + 1764 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4 \nu^{16} + 114 \nu^{14} + 1313 \nu^{12} + 7780 \nu^{10} + 24598 \nu^{8} + 36917 \nu^{6} + \cdots - 5712 ) / 966 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65 \nu^{16} - 1933 \nu^{14} - 23389 \nu^{12} - 147999 \nu^{10} - 523929 \nu^{8} - 1029409 \nu^{6} + \cdots - 46284 ) / 1932 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 551 \nu^{17} - 16267 \nu^{15} - 195235 \nu^{13} - 1224645 \nu^{11} - 4297461 \nu^{9} + \cdots - 494088 \nu ) / 40572 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 839 \nu^{17} + 24475 \nu^{15} + 289771 \nu^{13} + 1790601 \nu^{11} + 6187335 \nu^{9} + \cdots + 1004388 \nu ) / 40572 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 110 \nu^{16} - 3296 \nu^{14} - 40374 \nu^{12} - 260479 \nu^{10} - 949662 \nu^{8} - 1945878 \nu^{6} + \cdots - 103740 ) / 966 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 212 \nu^{17} - 57 \nu^{16} - 6364 \nu^{15} - 1866 \nu^{14} - 78283 \nu^{13} - 25110 \nu^{12} + \cdots - 28728 ) / 5796 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 212 \nu^{17} - 57 \nu^{16} + 6364 \nu^{15} - 1866 \nu^{14} + 78283 \nu^{13} - 25110 \nu^{12} + \cdots - 28728 ) / 5796 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 229 \nu^{17} + 1435 \nu^{16} + 7090 \nu^{15} + 42588 \nu^{14} + 90102 \nu^{13} + 515837 \nu^{12} + \cdots + 1156008 ) / 13524 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 229 \nu^{17} - 1435 \nu^{16} + 7090 \nu^{15} - 42588 \nu^{14} + 90102 \nu^{13} - 515837 \nu^{12} + \cdots - 1156008 ) / 13524 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 214 \nu^{16} - 6421 \nu^{14} - 78698 \nu^{12} - 507356 \nu^{10} - 1844717 \nu^{8} - 3759020 \nu^{6} + \cdots - 186102 ) / 966 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 229 \nu^{17} - 2681 \nu^{16} + 7090 \nu^{15} - 79226 \nu^{14} + 90102 \nu^{13} - 953575 \nu^{12} + \cdots - 1973328 ) / 13524 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 541 \nu^{17} - 15982 \nu^{15} - 192516 \nu^{13} - 1218236 \nu^{11} - 4348344 \nu^{9} + \cdots - 502572 \nu ) / 6762 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1898 \nu^{17} - 56347 \nu^{15} - 682186 \nu^{13} - 4335288 \nu^{11} - 15491637 \nu^{9} + \cdots - 1030974 \nu ) / 20286 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 4819 \nu^{17} - 143540 \nu^{15} - 1744970 \nu^{13} - 11149590 \nu^{11} - 40160379 \nu^{9} + \cdots - 3977274 \nu ) / 20286 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - 8\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{16} + 2\beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{6} - 10\beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{14} + 12 \beta_{13} + 13 \beta_{12} - \beta_{11} - 11 \beta_{10} - 11 \beta_{9} + \cdots - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{17} + 13 \beta_{16} - 27 \beta_{15} + 15 \beta_{12} + 15 \beta_{11} - 13 \beta_{10} + \cdots - 183 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 111 \beta_{14} - 108 \beta_{13} - 126 \beta_{12} + 15 \beta_{11} + 95 \beta_{10} + 95 \beta_{9} + \cdots + 1002 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12 \beta_{17} - 126 \beta_{16} + 260 \beta_{15} - 154 \beta_{12} - 154 \beta_{11} + 126 \beta_{10} + \cdots + 1213 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 939 \beta_{14} + 877 \beta_{13} + 1097 \beta_{12} - 158 \beta_{11} - 759 \beta_{10} - 759 \beta_{9} + \cdots - 6971 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 84 \beta_{17} + 1109 \beta_{16} - 2196 \beta_{15} + 1359 \beta_{12} + 1359 \beta_{11} + \cdots - 8289 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7626 \beta_{14} - 6786 \beta_{13} - 9079 \beta_{12} + 1453 \beta_{11} + 5865 \beta_{10} + 5865 \beta_{9} + \cdots + 49310 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 331 \beta_{17} - 9361 \beta_{16} + 17377 \beta_{15} - 11123 \beta_{12} - 11123 \beta_{11} + \cdots + 57799 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 60601 \beta_{14} + 51168 \beta_{13} + 73118 \beta_{12} - 12517 \beta_{11} - 44583 \beta_{10} + \cdots - 352710 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1086 \beta_{17} + 77288 \beta_{16} - 132598 \beta_{15} + 87362 \beta_{12} + 87362 \beta_{11} + \cdots - 408765 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 475407 \beta_{14} - 380155 \beta_{13} - 579491 \beta_{12} + 104084 \beta_{11} + 336005 \beta_{10} + \cdots + 2543161 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 41028 \beta_{17} - 629351 \beta_{16} + 990344 \beta_{15} - 670123 \beta_{12} - 670123 \beta_{11} + \cdots + 2920649 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
− 2.75119i
− 2.61624i
− 2.25571i
− 2.14751i
− 1.60544i
− 1.49951i
− 1.43540i
− 0.926452i
− 0.376265i
0.376265i
0.926452i
1.43540i
1.49951i
1.60544i
2.14751i
2.25571i
2.61624i
2.75119i
− 2.75119i 0 −5.56902 1.00000i 0 3.18887i 9.81905i 0 2.75119
1351.2 − 2.61624i 0 −4.84471 − 1.00000i 0 − 2.49953i 7.44243i 0 −2.61624
1351.3 − 2.25571i 0 −3.08823 − 1.00000i 0 1.04691i 2.45474i 0 −2.25571
1351.4 − 2.14751i 0 −2.61181 1.00000i 0 − 0.873214i 1.31388i 0 2.14751
1351.5 − 1.60544i 0 −0.577439 1.00000i 0 2.10861i − 2.28384i 0 1.60544
1351.6 − 1.49951i 0 −0.248529 − 1.00000i 0 − 2.41229i − 2.62635i 0 −1.49951
1351.7 − 1.43540i 0 −0.0603651 − 1.00000i 0 2.58879i − 2.78415i 0 −1.43540
1351.8 − 0.926452i 0 1.14169 1.00000i 0 3.74324i − 2.91062i 0 0.926452
1351.9 − 0.376265i 0 1.85842 1.00000i 0 − 4.44363i − 1.45179i 0 0.376265
1351.10 0.376265i 0 1.85842 − 1.00000i 0 4.44363i 1.45179i 0 0.376265
1351.11 0.926452i 0 1.14169 − 1.00000i 0 − 3.74324i 2.91062i 0 0.926452
1351.12 1.43540i 0 −0.0603651 1.00000i 0 − 2.58879i 2.78415i 0 −1.43540
1351.13 1.49951i 0 −0.248529 1.00000i 0 2.41229i 2.62635i 0 −1.49951
1351.14 1.60544i 0 −0.577439 − 1.00000i 0 − 2.10861i 2.28384i 0 1.60544
1351.15 2.14751i 0 −2.61181 − 1.00000i 0 0.873214i − 1.31388i 0 2.14751
1351.16 2.25571i 0 −3.08823 1.00000i 0 − 1.04691i − 2.45474i 0 −2.25571
1351.17 2.61624i 0 −4.84471 1.00000i 0 2.49953i − 7.44243i 0 −2.61624
1351.18 2.75119i 0 −5.56902 − 1.00000i 0 − 3.18887i − 9.81905i 0 2.75119
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1351.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1755.2.b.e yes 18
3.b odd 2 1 1755.2.b.d ✓ 18
13.b even 2 1 inner 1755.2.b.e yes 18
39.d odd 2 1 1755.2.b.d ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1755.2.b.d ✓ 18 3.b odd 2 1
1755.2.b.d ✓ 18 39.d odd 2 1
1755.2.b.e yes 18 1.a even 1 1 trivial
1755.2.b.e yes 18 13.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 \(S_{2}^{\mathrm{new}}(1755, [\chi])\):

\( T_{2}^{18} + 32 T_{2}^{16} + 428 T_{2}^{14} + 3114 T_{2}^{12} + 13440 T_{2}^{10} + 35180 T_{2}^{8} + \cdots + 1764 \) Copy content Toggle raw display
\( T_{17}^{9} - 85 T_{17}^{7} + 82 T_{17}^{6} + 2171 T_{17}^{5} - 3680 T_{17}^{4} - 17099 T_{17}^{3} + \cdots - 69084 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 32 T^{16} + \cdots + 1764 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} + 69 T^{16} + \cdots + 2547216 \) Copy content Toggle raw display
$11$ \( T^{18} + 126 T^{16} + \cdots + 2985984 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 10604499373 \) Copy content Toggle raw display
$17$ \( (T^{9} - 85 T^{7} + \cdots - 69084)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + 168 T^{16} + \cdots + 9437184 \) Copy content Toggle raw display
$23$ \( (T^{9} + 5 T^{8} + \cdots - 698688)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + 20 T^{8} + \cdots - 256608)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + 344 T^{16} + \cdots + 36578304 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 69493395456 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 86326917747984 \) Copy content Toggle raw display
$43$ \( (T^{9} + 4 T^{8} + \cdots - 393248)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 62258917146624 \) Copy content Toggle raw display
$53$ \( (T^{9} - 17 T^{8} + \cdots + 12546171)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + 449 T^{16} + \cdots + 104976 \) Copy content Toggle raw display
$61$ \( (T^{9} + 8 T^{8} + \cdots + 111839852)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 341067024 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 434182736050704 \) Copy content Toggle raw display
$79$ \( (T^{9} - 3 T^{8} + \cdots - 6168467)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 2538337222656 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 431743613184 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
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