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Properties

Label 1755.2.i.e
Level $1755$
Weight $2$
Character orbit 1755.i
Analytic conductor $14.014$
Analytic rank $0$
Dimension $16$
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(586,1755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1755, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1755.586");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1755.i (of order \(3\), degree \(2\), not 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 74 x^{12} - 18 x^{11} + 289 x^{10} - 4 x^{9} + 784 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 585)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{12} - \beta_{4} - \beta_{2} - 1) q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{8} - \beta_{4}) q^{7} + (\beta_{14} + \beta_{9} - \beta_{3} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{12} - \beta_{4} - \beta_{2} - 1) q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{8} - \beta_{4}) q^{7} + (\beta_{14} + \beta_{9} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + (2 \beta_{15} + \beta_{14} + 4 \beta_{13} + \cdots - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - 5 q^{4} + 8 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} - 5 q^{4} + 8 q^{5} + 6 q^{7} + 12 q^{8} - 2 q^{10} - 9 q^{11} - 8 q^{13} + 3 q^{14} + 13 q^{16} - 12 q^{17} - 22 q^{19} + 5 q^{20} + 4 q^{22} - 3 q^{23} - 8 q^{25} + 2 q^{26} - 26 q^{28} - 8 q^{29} + 18 q^{31} - 3 q^{32} + 9 q^{34} + 12 q^{35} - 36 q^{37} + 8 q^{38} + 6 q^{40} + 17 q^{41} + 17 q^{43} - 10 q^{44} + 6 q^{46} - 11 q^{47} + 16 q^{49} - q^{50} - 5 q^{52} - 20 q^{53} - 18 q^{55} + q^{56} + 10 q^{58} - 7 q^{59} + 21 q^{61} + 58 q^{62} - 20 q^{64} + 8 q^{65} + 13 q^{67} + 16 q^{68} - 3 q^{70} + 68 q^{71} - 32 q^{73} + 4 q^{74} + 2 q^{76} - 18 q^{77} + 37 q^{79} + 26 q^{80} + 2 q^{82} - 3 q^{83} - 6 q^{85} + 2 q^{86} + 19 q^{88} - 28 q^{89} - 12 q^{91} + 14 q^{92} + 44 q^{94} - 11 q^{95} + 17 q^{97} - 90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 74 x^{12} - 18 x^{11} + 289 x^{10} - 4 x^{9} + 784 x^{8} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3855592782730 \nu^{15} + 7015081199467 \nu^{14} - 42209590494762 \nu^{13} + \cdots - 83\!\cdots\!58 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10058702437115 \nu^{15} + 6203109654385 \nu^{14} - 103630645608798 \nu^{13} + \cdots - 845713476880083 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31322721365929 \nu^{15} - 61498828677274 \nu^{14} + 363159263988374 \nu^{13} + \cdots - 10\!\cdots\!20 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34156773183877 \nu^{15} - 166584922021666 \nu^{14} + 492186375728540 \nu^{13} + \cdots - 99\!\cdots\!54 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35406075859907 \nu^{15} - 25348858477034 \nu^{14} + 474863545496158 \nu^{13} + \cdots + 55\!\cdots\!97 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12600648760797 \nu^{15} + 14336163612885 \nu^{14} - 119694036026291 \nu^{13} + \cdots - 22\!\cdots\!61 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13841521780766 \nu^{15} + 48364435013822 \nu^{14} - 187738715248315 \nu^{13} + \cdots + 61\!\cdots\!74 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41600199133457 \nu^{15} + 102558204101951 \nu^{14} - 518403254192302 \nu^{13} + \cdots + 18\!\cdots\!85 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14259561929700 \nu^{15} + 8807205216793 \nu^{14} + 136197775034562 \nu^{13} + \cdots - 58\!\cdots\!37 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52504013730712 \nu^{15} - 47783529874105 \nu^{14} + 529907501904716 \nu^{13} + \cdots - 39141864823293 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27467128583199 \nu^{15} - 54483747477807 \nu^{14} + 320949673493612 \nu^{13} + \cdots - 93\!\cdots\!05 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 40076188059480 \nu^{15} + 48933383176228 \nu^{14} - 458207314720440 \nu^{13} + \cdots - 38\!\cdots\!55 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 143695299415682 \nu^{15} - 179431434589817 \nu^{14} + \cdots + 46\!\cdots\!63 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 61998452531739 \nu^{15} - 70079195223232 \nu^{14} + 654764354882311 \nu^{13} + \cdots + 26\!\cdots\!40 ) / 30\!\cdots\!91 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - 3\beta_{4} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{9} - 3\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{12} + \beta_{10} - \beta_{9} + \beta_{5} + 12\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + 7 \beta_{14} + \beta_{13} - 8 \beta_{12} - \beta_{7} + \beta_{5} + 9 \beta_{4} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{15} + 10 \beta_{14} + 2 \beta_{13} + \beta_{11} - 8 \beta_{10} + 10 \beta_{9} - \beta_{8} + \cdots + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55 \beta_{12} + 3 \beta_{11} - 12 \beta_{10} + 42 \beta_{9} - 8 \beta_{8} + 10 \beta_{7} + \cdots + 34 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{15} - 74 \beta_{14} - 23 \beta_{13} + 183 \beta_{12} + 23 \beta_{7} - 13 \beta_{6} + \cdots - 263 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 51 \beta_{15} - 245 \beta_{14} - 76 \beta_{13} - 36 \beta_{11} + 97 \beta_{10} - 245 \beta_{9} + \cdots - 403 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1033 \beta_{12} - 112 \beta_{11} + 321 \beta_{10} - 490 \beta_{9} + 86 \beta_{8} - 184 \beta_{7} + \cdots + 15 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 307 \beta_{15} + 1426 \beta_{14} + 519 \beta_{13} - 2182 \beta_{12} - 519 \beta_{7} + 296 \beta_{6} + \cdots + 2473 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 590 \beta_{15} + 3079 \beta_{14} + 1277 \beta_{13} + 815 \beta_{11} - 1945 \beta_{10} + 3079 \beta_{9} + \cdots + 7316 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13205 \beta_{12} + 2092 \beta_{11} - 4356 \beta_{10} + 8323 \beta_{9} - 1817 \beta_{8} + \cdots + 1015 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3797 \beta_{15} - 18819 \beta_{14} - 8265 \beta_{13} + 34216 \beta_{12} + 8265 \beta_{7} - 5442 \beta_{6} + \cdots - 40602 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10699 \beta_{15} - 48712 \beta_{14} - 20912 \beta_{13} - 13707 \beta_{11} + 27084 \beta_{10} + \cdots - 88279 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(326\) \(352\) \(1081\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
586.1
1.21184 − 2.09897i
0.987387 − 1.71020i
0.715744 − 1.23970i
0.316605 − 0.548376i
−0.237622 + 0.411573i
−0.624867 + 1.08230i
−0.921972 + 1.59690i
−0.947115 + 1.64045i
1.21184 + 2.09897i
0.987387 + 1.71020i
0.715744 + 1.23970i
0.316605 + 0.548376i
−0.237622 − 0.411573i
−0.624867 − 1.08230i
−0.921972 − 1.59690i
−0.947115 − 1.64045i
−1.21184 + 2.09897i 0 −1.93711 − 3.35518i 0.500000 + 0.866025i 0 0.905675 − 1.56867i 4.54253 0 −2.42368
586.2 −0.987387 + 1.71020i 0 −0.949865 − 1.64522i 0.500000 + 0.866025i 0 0.124489 − 0.215622i −0.198009 0 −1.97477
586.3 −0.715744 + 1.23970i 0 −0.0245786 − 0.0425715i 0.500000 + 0.866025i 0 0.625355 − 1.08315i −2.79261 0 −1.43149
586.4 −0.316605 + 0.548376i 0 0.799523 + 1.38481i 0.500000 + 0.866025i 0 0.896323 − 1.55248i −2.27895 0 −0.633210
586.5 0.237622 − 0.411573i 0 0.887072 + 1.53645i 0.500000 + 0.866025i 0 −1.24774 + 2.16115i 1.79364 0 0.475244
586.6 0.624867 − 1.08230i 0 0.219082 + 0.379462i 0.500000 + 0.866025i 0 0.148724 − 0.257597i 3.04706 0 1.24973
586.7 0.921972 − 1.59690i 0 −0.700064 − 1.21255i 0.500000 + 0.866025i 0 2.38510 − 4.13111i 1.10613 0 1.84394
586.8 0.947115 − 1.64045i 0 −0.794055 − 1.37534i 0.500000 + 0.866025i 0 −0.837925 + 1.45133i 0.780216 0 1.89423
1171.1 −1.21184 − 2.09897i 0 −1.93711 + 3.35518i 0.500000 − 0.866025i 0 0.905675 + 1.56867i 4.54253 0 −2.42368
1171.2 −0.987387 − 1.71020i 0 −0.949865 + 1.64522i 0.500000 − 0.866025i 0 0.124489 + 0.215622i −0.198009 0 −1.97477
1171.3 −0.715744 − 1.23970i 0 −0.0245786 + 0.0425715i 0.500000 − 0.866025i 0 0.625355 + 1.08315i −2.79261 0 −1.43149
1171.4 −0.316605 − 0.548376i 0 0.799523 − 1.38481i 0.500000 − 0.866025i 0 0.896323 + 1.55248i −2.27895 0 −0.633210
1171.5 0.237622 + 0.411573i 0 0.887072 − 1.53645i 0.500000 − 0.866025i 0 −1.24774 − 2.16115i 1.79364 0 0.475244
1171.6 0.624867 + 1.08230i 0 0.219082 − 0.379462i 0.500000 − 0.866025i 0 0.148724 + 0.257597i 3.04706 0 1.24973
1171.7 0.921972 + 1.59690i 0 −0.700064 + 1.21255i 0.500000 − 0.866025i 0 2.38510 + 4.13111i 1.10613 0 1.84394
1171.8 0.947115 + 1.64045i 0 −0.794055 + 1.37534i 0.500000 − 0.866025i 0 −0.837925 − 1.45133i 0.780216 0 1.89423
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 586.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1755.2.i.e 16
3.b odd 2 1 585.2.i.f ✓ 16
9.c even 3 1 inner 1755.2.i.e 16
9.c even 3 1 5265.2.a.be 8
9.d odd 6 1 585.2.i.f ✓ 16
9.d odd 6 1 5265.2.a.bb 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.f ✓ 16 3.b odd 2 1
585.2.i.f ✓ 16 9.d odd 6 1
1755.2.i.e 16 1.a even 1 1 trivial
1755.2.i.e 16 9.c even 3 1 inner
5265.2.a.bb 8 9.d odd 6 1
5265.2.a.be 8 9.c even 3 1

Hecke kernels

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

\( T_{2}^{16} + T_{2}^{15} + 11 T_{2}^{14} + 4 T_{2}^{13} + 74 T_{2}^{12} + 18 T_{2}^{11} + 289 T_{2}^{10} + \cdots + 81 \) Copy content Toggle raw display
\( T_{7}^{16} - 6 T_{7}^{15} + 38 T_{7}^{14} - 88 T_{7}^{13} + 353 T_{7}^{12} - 763 T_{7}^{11} + 2210 T_{7}^{10} + \cdots + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} - 6 T^{15} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( T^{16} + 9 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{8} + 6 T^{7} + \cdots + 822)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 11 T^{7} + \cdots + 51733)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 898560576 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 310831895529 \) Copy content Toggle raw display
$31$ \( T^{16} - 18 T^{15} + \cdots + 18344089 \) Copy content Toggle raw display
$37$ \( (T^{8} + 18 T^{7} + \cdots + 236259)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 3109623696 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 16841810176 \) Copy content Toggle raw display
$47$ \( T^{16} + 11 T^{15} + \cdots + 10850436 \) Copy content Toggle raw display
$53$ \( (T^{8} + 10 T^{7} + \cdots - 405738)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1642567767129 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 28213508866321 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 9767162560516 \) Copy content Toggle raw display
$71$ \( (T^{8} - 34 T^{7} + \cdots + 1374696)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 16 T^{7} + \cdots + 373998)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 5500298896 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 568679695164036 \) Copy content Toggle raw display
$89$ \( (T^{8} + 14 T^{7} + \cdots + 1052154)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 71550365121 \) Copy content Toggle raw display
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