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Properties

Label 1755.2.b.c
Level $1755$
Weight $2$
Character orbit 1755.b
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(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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} + 40 x^{13} + 159 x^{12} - 236 x^{11} + 472 x^{10} + 2054 x^{9} + 3381 x^{8} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + ( - \beta_{5} - 1) q^{4} + \beta_1 q^{5} + \beta_{13} q^{7} + ( - \beta_{9} - \beta_{4} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} + ( - \beta_{5} - 1) q^{4} + \beta_1 q^{5} + \beta_{13} q^{7} + ( - \beta_{9} - \beta_{4} + \beta_1) q^{8} - \beta_{7} q^{10} + (\beta_{12} + \beta_{9}) q^{11} + (\beta_{14} - \beta_{7} - \beta_{2}) q^{13} + (2 \beta_{10} + \beta_{3}) q^{14} + ( - \beta_{7} - \beta_{2} - 1) q^{16} + ( - \beta_{8} - \beta_{3}) q^{17} + ( - \beta_{14} - \beta_{6}) q^{19} + (\beta_{9} - \beta_1) q^{20} + (\beta_{7} - 2 \beta_{5} + \beta_{2} - 4) q^{22} + \beta_{11} q^{23} - q^{25} + ( - \beta_{12} + \beta_{10} + \cdots - 3 \beta_1) q^{26}+ \cdots + ( - 5 \beta_{12} - 2 \beta_{9} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{4} + 4 q^{10} + 4 q^{13} - 12 q^{16} - 60 q^{22} - 16 q^{25} - 12 q^{40} - 64 q^{43} - 28 q^{49} + 8 q^{52} - 44 q^{61} + 44 q^{64} + 4 q^{79} + 16 q^{82} + 48 q^{88} - 8 q^{91} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} + 40 x^{13} + 159 x^{12} - 236 x^{11} + 472 x^{10} + 2054 x^{9} + 3381 x^{8} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47\!\cdots\!15 \nu^{15} + \cdots - 50\!\cdots\!68 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!66 \nu^{15} + \cdots + 37\!\cdots\!09 ) / 22\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!91 \nu^{15} + \cdots + 74\!\cdots\!27 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\!\cdots\!50 \nu^{15} + \cdots - 16\!\cdots\!98 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!26 \nu^{15} + \cdots - 93\!\cdots\!71 ) / 22\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!45 \nu^{15} + \cdots - 15\!\cdots\!85 ) / 22\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26\!\cdots\!54 \nu^{15} + \cdots - 11\!\cdots\!18 ) / 22\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37\!\cdots\!51 \nu^{15} + \cdots + 16\!\cdots\!81 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 48\!\cdots\!51 \nu^{15} + \cdots - 34\!\cdots\!84 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 55\!\cdots\!85 \nu^{15} + \cdots - 23\!\cdots\!67 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 58\!\cdots\!64 \nu^{15} + \cdots - 28\!\cdots\!88 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 63\!\cdots\!25 \nu^{15} + \cdots + 43\!\cdots\!69 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 47\!\cdots\!22 \nu^{15} + \cdots - 32\!\cdots\!49 ) / 22\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 69\!\cdots\!63 \nu^{15} + \cdots + 48\!\cdots\!32 ) / 22\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 77\!\cdots\!59 \nu^{15} + \cdots - 54\!\cdots\!10 ) / 22\!\cdots\!53 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} + 2 \beta_{13} + 3 \beta_{12} - \beta_{11} - \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + \cdots + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} + 5\beta_{13} + 6\beta_{12} - 3\beta_{9} - 3\beta_{6} + 3\beta_{4} + 18\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14 \beta_{15} - 29 \beta_{14} + 34 \beta_{13} + 66 \beta_{12} + 14 \beta_{11} + 29 \beta_{10} + \cdots - 84 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 32\beta_{11} + 68\beta_{10} - 106\beta_{8} - 198\beta_{7} - 81\beta_{5} + 72\beta_{3} - 60\beta_{2} - 333 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 277 \beta_{15} + 670 \beta_{14} - 731 \beta_{13} - 1569 \beta_{12} + 277 \beta_{11} + 670 \beta_{10} + \cdots - 2028 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 826 \beta_{15} + 2014 \beta_{14} - 2447 \beta_{13} - 5160 \beta_{12} + 2085 \beta_{9} + \cdots - 7398 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6212 \beta_{15} + 15662 \beta_{14} - 16987 \beta_{13} - 37431 \beta_{12} - 6212 \beta_{11} + \cdots + 49014 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20396 \beta_{11} - 51500 \beta_{10} + 58240 \beta_{8} + 127245 \beta_{7} + 51738 \beta_{5} + \cdots + 173367 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 145768 \beta_{15} - 371680 \beta_{14} + 404462 \beta_{13} + 896559 \beta_{12} - 145768 \beta_{11} + \cdots + 1183281 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 495679 \beta_{15} - 1263766 \beta_{14} + 1397558 \beta_{13} + 3086289 \beta_{12} - 1261125 \beta_{9} + \cdots + 4134978 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3476459 \beta_{15} - 8890385 \beta_{14} + 9701080 \beta_{13} + 21527799 \beta_{12} + 3476459 \beta_{11} + \cdots - 28519989 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 11972234 \beta_{11} + 30609308 \beta_{10} - 33605758 \beta_{8} - 74458563 \beta_{7} - 30502776 \beta_{5} + \cdots - 99199710 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 83386312 \beta_{15} + 213400753 \beta_{14} - 233192102 \beta_{13} - 517531536 \beta_{12} + \cdots - 686742318 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 288479419 \beta_{15} + 738161206 \beta_{14} - 808480700 \beta_{13} - 1793173020 \beta_{12} + \cdots - 2384452503 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2004126953 \beta_{15} + 5129800670 \beta_{14} - 5609115979 \beta_{13} - 12447920865 \beta_{12} + \cdots + 16528813878 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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
−0.643926 − 0.643926i
1.82695 + 1.82695i
3.46848 − 3.46848i
0.281015 − 0.281015i
0.277388 + 0.277388i
−2.09143 − 2.09143i
−0.734983 + 0.734983i
−0.383498 + 0.383498i
−0.383498 − 0.383498i
−0.734983 − 0.734983i
−2.09143 + 2.09143i
0.277388 − 0.277388i
0.281015 + 0.281015i
3.46848 + 3.46848i
1.82695 − 1.82695i
−0.643926 + 0.643926i
− 2.35284i 0 −3.53587 1.00000i 0 − 3.60274i 3.61366i 0 2.35284
1351.2 − 2.35284i 0 −3.53587 1.00000i 0 3.60274i 3.61366i 0 2.35284
1351.3 − 1.94939i 0 −1.80011 − 1.00000i 0 − 3.85858i − 0.389667i 0 −1.94939
1351.4 − 1.94939i 0 −1.80011 − 1.00000i 0 3.85858i − 0.389667i 0 −1.94939
1351.5 − 1.16027i 0 0.653767 1.00000i 0 − 0.541134i − 3.07909i 0 1.16027
1351.6 − 1.16027i 0 0.653767 1.00000i 0 0.541134i − 3.07909i 0 1.16027
1351.7 − 0.563729i 0 1.68221 − 1.00000i 0 − 2.61511i − 2.07577i 0 −0.563729
1351.8 − 0.563729i 0 1.68221 − 1.00000i 0 2.61511i − 2.07577i 0 −0.563729
1351.9 0.563729i 0 1.68221 1.00000i 0 − 2.61511i 2.07577i 0 −0.563729
1351.10 0.563729i 0 1.68221 1.00000i 0 2.61511i 2.07577i 0 −0.563729
1351.11 1.16027i 0 0.653767 − 1.00000i 0 − 0.541134i 3.07909i 0 1.16027
1351.12 1.16027i 0 0.653767 − 1.00000i 0 0.541134i 3.07909i 0 1.16027
1351.13 1.94939i 0 −1.80011 1.00000i 0 − 3.85858i 0.389667i 0 −1.94939
1351.14 1.94939i 0 −1.80011 1.00000i 0 3.85858i 0.389667i 0 −1.94939
1351.15 2.35284i 0 −3.53587 − 1.00000i 0 − 3.60274i − 3.61366i 0 2.35284
1351.16 2.35284i 0 −3.53587 − 1.00000i 0 3.60274i − 3.61366i 0 2.35284
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1351.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.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.b.c ✓ 16
3.b odd 2 1 inner 1755.2.b.c ✓ 16
13.b even 2 1 inner 1755.2.b.c ✓ 16
39.d odd 2 1 inner 1755.2.b.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1755.2.b.c ✓ 16 1.a even 1 1 trivial
1755.2.b.c ✓ 16 3.b odd 2 1 inner
1755.2.b.c ✓ 16 13.b even 2 1 inner
1755.2.b.c ✓ 16 39.d odd 2 1 inner

Hecke kernels

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

\( T_{2}^{8} + 11T_{2}^{6} + 37T_{2}^{4} + 39T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{17}^{8} - 98T_{17}^{6} + 3073T_{17}^{4} - 31860T_{17}^{2} + 31347 \) 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
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 35 T^{6} + \cdots + 387)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 36 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 98 T^{6} + \cdots + 31347)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 80 T^{6} + \cdots + 387)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 95 T^{6} + \cdots + 3483)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 146 T^{6} + \cdots + 3483)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 221 T^{6} + \cdots + 4430763)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 203 T^{6} + \cdots + 139707)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 44 T^{6} + 370 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} + \cdots - 683)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 101 T^{6} + \cdots + 127449)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 177 T^{6} + \cdots + 282123)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 87 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 11 T^{3} + \cdots - 17)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 474 T^{6} + \cdots + 22851963)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 263 T^{6} + \cdots + 154449)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 131 T^{6} + \cdots + 3483)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - T^{3} - 159 T^{2} + \cdots + 3703)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 135 T^{6} + \cdots + 363609)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 324 T^{6} + \cdots + 29241)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 599 T^{6} + \cdots + 142589763)^{2} \) Copy content Toggle raw display
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