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Properties

Label 1155.2.q.k
Level $1155$
Weight $2$
Character orbit 1155.q
Analytic conductor $9.223$
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] = [1155,2,Mod(331,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.331");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.q (of order \(3\), degree \(2\), 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: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 16 x^{16} + 174 x^{14} - 2 x^{13} + 1030 x^{12} - 60 x^{11} + 4430 x^{10} - 236 x^{9} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + 1) q^{3} + (\beta_{15} - 2 \beta_{7} - 2) q^{4} - \beta_{7} q^{5} + ( - \beta_{3} + \beta_1) q^{6} - \beta_{9} q^{7} + (\beta_{16} - \beta_{11} + \cdots - 2 \beta_1) q^{8}+ \cdots + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + 1) q^{3} + (\beta_{15} - 2 \beta_{7} - 2) q^{4} - \beta_{7} q^{5} + ( - \beta_{3} + \beta_1) q^{6} - \beta_{9} q^{7} + (\beta_{16} - \beta_{11} + \cdots - 2 \beta_1) q^{8}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{3} - 14 q^{4} + 9 q^{5} + 3 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{3} - 14 q^{4} + 9 q^{5} + 3 q^{7} - 9 q^{9} + 9 q^{11} + 14 q^{12} - 14 q^{13} + 8 q^{14} + 18 q^{15} - 28 q^{16} + 10 q^{17} + 8 q^{19} - 28 q^{20} + 3 q^{21} + q^{23} - 9 q^{25} + 6 q^{26} - 18 q^{27} + 4 q^{28} + 14 q^{29} + 10 q^{32} - 9 q^{33} - 12 q^{34} + 28 q^{36} - 6 q^{37} - 6 q^{38} - 7 q^{39} - 30 q^{41} + 4 q^{42} + 34 q^{43} + 14 q^{44} + 9 q^{45} - 22 q^{46} + 3 q^{47} - 56 q^{48} + 5 q^{49} - 10 q^{51} + 30 q^{52} + 11 q^{53} + 18 q^{55} + 44 q^{56} + 16 q^{57} + 6 q^{58} + 14 q^{59} - 14 q^{60} + 26 q^{61} - 84 q^{62} + 44 q^{64} - 7 q^{65} - 26 q^{67} + 26 q^{68} + 2 q^{69} + 4 q^{70} - 4 q^{71} + 4 q^{73} - 46 q^{74} + 9 q^{75} - 4 q^{76} + 3 q^{77} + 12 q^{78} - 32 q^{79} + 28 q^{80} - 9 q^{81} + 4 q^{82} - 48 q^{83} - 10 q^{84} + 20 q^{85} + 52 q^{86} + 7 q^{87} - 20 q^{89} + 7 q^{91} - 128 q^{92} + 42 q^{94} - 8 q^{95} - 10 q^{96} + 16 q^{97} + 20 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 16 x^{16} + 174 x^{14} - 2 x^{13} + 1030 x^{12} - 60 x^{11} + 4430 x^{10} - 236 x^{9} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36\!\cdots\!41 \nu^{17} + \cdots + 30\!\cdots\!48 ) / 76\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!57 \nu^{17} + \cdots + 80\!\cdots\!32 ) / 76\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!18 \nu^{17} + \cdots - 96\!\cdots\!00 ) / 16\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 72\!\cdots\!73 \nu^{17} + \cdots + 29\!\cdots\!48 ) / 29\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!81 \nu^{17} + \cdots - 12\!\cdots\!04 ) / 29\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!33 \nu^{17} + \cdots + 57\!\cdots\!52 ) / 30\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!39 \nu^{17} + \cdots - 74\!\cdots\!72 ) / 29\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 75\!\cdots\!65 \nu^{17} + \cdots - 76\!\cdots\!12 ) / 58\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26\!\cdots\!19 \nu^{17} + \cdots - 37\!\cdots\!20 ) / 19\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!55 \nu^{17} + \cdots + 64\!\cdots\!40 ) / 19\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 83\!\cdots\!71 \nu^{17} + \cdots - 16\!\cdots\!56 ) / 58\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 83\!\cdots\!83 \nu^{17} + \cdots - 67\!\cdots\!40 ) / 58\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 87\!\cdots\!39 \nu^{17} + \cdots + 75\!\cdots\!96 ) / 58\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!33 \nu^{17} + \cdots + 31\!\cdots\!88 ) / 76\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 79\!\cdots\!11 \nu^{17} + \cdots + 40\!\cdots\!08 ) / 29\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 66\!\cdots\!91 \nu^{17} + \cdots - 32\!\cdots\!30 ) / 14\!\cdots\!46 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - 4\beta_{7} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{11} - \beta_{9} + 6\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} - 7\beta_{15} + \beta_{12} - \beta_{10} + \beta_{8} + 24\beta_{7} + \beta_{6} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} + \beta_{16} - \beta_{14} + \beta_{13} + 3 \beta_{11} - 10 \beta_{10} + 10 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{16} + 10 \beta_{14} - \beta_{13} - 12 \beta_{12} + 3 \beta_{11} + 2 \beta_{9} - \beta_{5} + \cdots + 155 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{17} - 82 \beta_{16} + 2 \beta_{15} + 13 \beta_{14} - 14 \beta_{13} - 13 \beta_{12} + \cdots + 265 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{17} - 81 \beta_{16} + 303 \beta_{15} - 50 \beta_{14} - 30 \beta_{13} - 44 \beta_{11} + \cdots - 1042 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 507 \beta_{16} - 2 \beta_{14} + 16 \beta_{13} + 128 \beta_{12} - 743 \beta_{11} + 126 \beta_{10} + \cdots - 298 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 128 \beta_{17} + 1029 \beta_{16} - 2025 \beta_{15} - 318 \beta_{14} + 508 \beta_{13} + \cdots - 842 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 919 \beta_{17} + 1139 \beta_{16} - 568 \beta_{15} - 1047 \beta_{14} + 1095 \beta_{13} + 3105 \beta_{11} + \cdots + 3303 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4070 \beta_{16} + 4428 \beta_{14} - 1789 \beta_{13} - 7360 \beta_{12} + 5441 \beta_{11} + \cdots + 50675 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7360 \beta_{17} - 36540 \beta_{16} + 6320 \beta_{15} + 9035 \beta_{14} - 10710 \beta_{13} + \cdots + 94691 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9755 \beta_{17} - 34979 \beta_{16} + 95449 \beta_{15} - 7284 \beta_{14} - 25186 \beta_{13} + \cdots - 362264 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 197423 \beta_{16} - 8708 \beta_{14} + 14894 \beta_{13} + 81258 \beta_{12} - 325657 \beta_{11} + \cdots - 305444 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 81258 \beta_{17} + 584577 \beta_{16} - 672359 \beta_{15} - 207884 \beta_{14} + 334510 \beta_{13} + \cdots - 728832 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 446097 \beta_{17} + 640261 \beta_{16} - 587200 \beta_{15} - 480897 \beta_{14} + 574009 \beta_{13} + \cdots + 2716693 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1
−1.29444 + 2.24203i
−1.24065 + 2.14888i
−0.851736 + 1.47525i
−0.237926 + 0.412100i
−0.0558513 + 0.0967372i
0.325423 − 0.563650i
0.860676 − 1.49074i
1.10216 − 1.90900i
1.39234 − 2.41160i
−1.29444 − 2.24203i
−1.24065 − 2.14888i
−0.851736 − 1.47525i
−0.237926 − 0.412100i
−0.0558513 − 0.0967372i
0.325423 + 0.563650i
0.860676 + 1.49074i
1.10216 + 1.90900i
1.39234 + 2.41160i
−1.29444 + 2.24203i 0.500000 + 0.866025i −2.35112 − 4.07227i 0.500000 − 0.866025i −2.58887 −1.69730 + 2.02958i 6.99577 −0.500000 + 0.866025i 1.29444 + 2.24203i
331.2 −1.24065 + 2.14888i 0.500000 + 0.866025i −2.07844 − 3.59997i 0.500000 − 0.866025i −2.48131 2.38240 − 1.15073i 5.35190 −0.500000 + 0.866025i 1.24065 + 2.14888i
331.3 −0.851736 + 1.47525i 0.500000 + 0.866025i −0.450907 − 0.780993i 0.500000 − 0.866025i −1.70347 −1.15386 − 2.38089i −1.87073 −0.500000 + 0.866025i 0.851736 + 1.47525i
331.4 −0.237926 + 0.412100i 0.500000 + 0.866025i 0.886782 + 1.53595i 0.500000 − 0.866025i −0.475852 −2.64509 − 0.0590459i −1.79566 −0.500000 + 0.866025i 0.237926 + 0.412100i
331.5 −0.0558513 + 0.0967372i 0.500000 + 0.866025i 0.993761 + 1.72125i 0.500000 − 0.866025i −0.111703 2.10571 − 1.60187i −0.445416 −0.500000 + 0.866025i 0.0558513 + 0.0967372i
331.6 0.325423 − 0.563650i 0.500000 + 0.866025i 0.788199 + 1.36520i 0.500000 − 0.866025i 0.650847 2.57473 + 0.608902i 2.32769 −0.500000 + 0.866025i −0.325423 − 0.563650i
331.7 0.860676 − 1.49074i 0.500000 + 0.866025i −0.481527 − 0.834030i 0.500000 − 0.866025i 1.72135 1.66178 + 2.05876i 1.78495 −0.500000 + 0.866025i −0.860676 − 1.49074i
331.8 1.10216 − 1.90900i 0.500000 + 0.866025i −1.42953 − 2.47602i 0.500000 − 0.866025i 2.20433 −1.38719 + 2.25293i −1.89365 −0.500000 + 0.866025i −1.10216 − 1.90900i
331.9 1.39234 − 2.41160i 0.500000 + 0.866025i −2.87721 − 4.98348i 0.500000 − 0.866025i 2.78468 −0.341183 − 2.62366i −10.4548 −0.500000 + 0.866025i −1.39234 − 2.41160i
991.1 −1.29444 − 2.24203i 0.500000 − 0.866025i −2.35112 + 4.07227i 0.500000 + 0.866025i −2.58887 −1.69730 − 2.02958i 6.99577 −0.500000 − 0.866025i 1.29444 − 2.24203i
991.2 −1.24065 − 2.14888i 0.500000 − 0.866025i −2.07844 + 3.59997i 0.500000 + 0.866025i −2.48131 2.38240 + 1.15073i 5.35190 −0.500000 − 0.866025i 1.24065 − 2.14888i
991.3 −0.851736 − 1.47525i 0.500000 − 0.866025i −0.450907 + 0.780993i 0.500000 + 0.866025i −1.70347 −1.15386 + 2.38089i −1.87073 −0.500000 − 0.866025i 0.851736 − 1.47525i
991.4 −0.237926 − 0.412100i 0.500000 − 0.866025i 0.886782 − 1.53595i 0.500000 + 0.866025i −0.475852 −2.64509 + 0.0590459i −1.79566 −0.500000 − 0.866025i 0.237926 − 0.412100i
991.5 −0.0558513 − 0.0967372i 0.500000 − 0.866025i 0.993761 − 1.72125i 0.500000 + 0.866025i −0.111703 2.10571 + 1.60187i −0.445416 −0.500000 − 0.866025i 0.0558513 − 0.0967372i
991.6 0.325423 + 0.563650i 0.500000 − 0.866025i 0.788199 − 1.36520i 0.500000 + 0.866025i 0.650847 2.57473 − 0.608902i 2.32769 −0.500000 − 0.866025i −0.325423 + 0.563650i
991.7 0.860676 + 1.49074i 0.500000 − 0.866025i −0.481527 + 0.834030i 0.500000 + 0.866025i 1.72135 1.66178 − 2.05876i 1.78495 −0.500000 − 0.866025i −0.860676 + 1.49074i
991.8 1.10216 + 1.90900i 0.500000 − 0.866025i −1.42953 + 2.47602i 0.500000 + 0.866025i 2.20433 −1.38719 − 2.25293i −1.89365 −0.500000 − 0.866025i −1.10216 + 1.90900i
991.9 1.39234 + 2.41160i 0.500000 − 0.866025i −2.87721 + 4.98348i 0.500000 + 0.866025i 2.78468 −0.341183 + 2.62366i −10.4548 −0.500000 − 0.866025i −1.39234 + 2.41160i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.q.k ✓ 18
7.c even 3 1 inner 1155.2.q.k ✓ 18
7.c even 3 1 8085.2.a.cg 9
7.d odd 6 1 8085.2.a.ch 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.k ✓ 18 1.a even 1 1 trivial
1155.2.q.k ✓ 18 7.c even 3 1 inner
8085.2.a.cg 9 7.c even 3 1
8085.2.a.ch 9 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{18} + 16 T_{2}^{16} + 174 T_{2}^{14} - 2 T_{2}^{13} + 1030 T_{2}^{12} - 60 T_{2}^{11} + 4430 T_{2}^{10} + \cdots + 16 \) Copy content Toggle raw display
\( T_{13}^{9} + 7 T_{13}^{8} - 45 T_{13}^{7} - 311 T_{13}^{6} + 707 T_{13}^{5} + 4177 T_{13}^{4} + \cdots + 20 \) Copy content Toggle raw display
\( T_{17}^{18} - 10 T_{17}^{17} + 137 T_{17}^{16} - 766 T_{17}^{15} + 7213 T_{17}^{14} - 32648 T_{17}^{13} + \cdots + 3057647616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 16 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} - 3 T^{17} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$13$ \( (T^{9} + 7 T^{8} - 45 T^{7} + \cdots + 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 3057647616 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 3696640000 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 12734671104 \) Copy content Toggle raw display
$29$ \( (T^{9} - 7 T^{8} + \cdots - 2504784)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 6879707136 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 181268955795456 \) Copy content Toggle raw display
$41$ \( (T^{9} + 15 T^{8} + \cdots + 10160768)^{2} \) Copy content Toggle raw display
$43$ \( (T^{9} - 17 T^{8} + \cdots - 53769)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 48329625600 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 47\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{9} + 2 T^{8} + \cdots + 42480340)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 23514382336 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 61076850868224 \) Copy content Toggle raw display
$83$ \( (T^{9} + 24 T^{8} + \cdots - 1884992)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{9} - 8 T^{8} + \cdots - 6481664)^{2} \) Copy content Toggle raw display
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