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Properties

Label 585.2.i.e
Level $585$
Weight $2$
Character orbit 585.i
Analytic conductor $4.671$
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] = [585,2,Mod(196,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, 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("585.196");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.i (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: \(4.67124851824\)
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: \( x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + \cdots + 268 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{14} - \beta_{7}) q^{3} + (\beta_{15} - \beta_{13} + \cdots - \beta_1) q^{4}+ \cdots + ( - \beta_{10} - \beta_{7} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{14} - \beta_{7}) q^{3} + (\beta_{15} - \beta_{13} + \cdots - \beta_1) q^{4}+ \cdots + ( - 3 \beta_{15} - \beta_{14} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} + q^{3} - 9 q^{4} + 8 q^{5} + 8 q^{6} + 11 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{2} + q^{3} - 9 q^{4} + 8 q^{5} + 8 q^{6} + 11 q^{7} - 12 q^{8} + q^{9} - 6 q^{10} - 6 q^{11} + 2 q^{12} + 8 q^{13} - 10 q^{14} + 5 q^{15} - 11 q^{16} - 4 q^{17} + 5 q^{18} - 20 q^{19} + 9 q^{20} + 11 q^{21} - 3 q^{22} - 6 q^{23} - 57 q^{24} - 8 q^{25} - 6 q^{26} - 14 q^{27} - 68 q^{28} - 14 q^{29} + 7 q^{30} + 31 q^{31} - q^{32} - 31 q^{33} + 7 q^{34} + 22 q^{35} + 2 q^{36} + 2 q^{37} - 9 q^{38} + 5 q^{39} - 6 q^{40} + 12 q^{41} - 26 q^{42} - 15 q^{43} + 32 q^{44} - 16 q^{45} - 64 q^{46} + 18 q^{47} - 4 q^{48} - 17 q^{49} - 3 q^{50} + 32 q^{51} + 9 q^{52} + 4 q^{53} + 2 q^{54} - 12 q^{55} - 16 q^{56} + 45 q^{57} + 42 q^{58} - 24 q^{59} - 8 q^{60} + 9 q^{61} - 40 q^{62} + 47 q^{63} - 60 q^{64} - 8 q^{65} + 55 q^{66} + 18 q^{67} + 14 q^{68} - 12 q^{69} + 10 q^{70} + 20 q^{71} - 9 q^{72} + 12 q^{73} + 37 q^{74} + 4 q^{75} + 53 q^{76} + 34 q^{77} + 7 q^{78} + 3 q^{79} - 22 q^{80} + 13 q^{81} - 68 q^{82} + 10 q^{83} + 22 q^{84} - 2 q^{85} - 60 q^{86} + 11 q^{87} + 14 q^{88} - 26 q^{89} + 19 q^{90} + 22 q^{91} - 5 q^{92} + 74 q^{93} - 17 q^{94} - 10 q^{95} + 13 q^{96} + 34 q^{97} - 60 q^{98} - 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + \cdots + 268 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 89 \nu^{15} - 1135 \nu^{14} + 7576 \nu^{13} - 31264 \nu^{12} + 72320 \nu^{11} - 147579 \nu^{10} + \cdots + 194918 ) / 205578 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + 16 \nu^{13} - 28 \nu^{12} + 68 \nu^{11} - 39 \nu^{10} + 139 \nu^{9} + \cdots + 6293 ) / 2187 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53 \nu^{15} + 1221 \nu^{14} - 4880 \nu^{13} + 14280 \nu^{12} - 18634 \nu^{11} + 24185 \nu^{10} + \cdots + 125592 ) / 68526 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 265 \nu^{15} - 851 \nu^{14} - 2404 \nu^{13} + 18478 \nu^{12} - 122780 \nu^{11} + 267189 \nu^{10} + \cdots - 1138394 ) / 205578 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 547 \nu^{15} - 2825 \nu^{14} + 11696 \nu^{13} - 42716 \nu^{12} + 9478 \nu^{11} + 20721 \nu^{10} + \cdots - 568472 ) / 205578 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 685 \nu^{15} - 2275 \nu^{14} + 10492 \nu^{13} - 25438 \nu^{12} + 90950 \nu^{11} - 164457 \nu^{10} + \cdots + 752246 ) / 205578 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 556 \nu^{15} + 910 \nu^{14} + 1274 \nu^{13} - 22988 \nu^{12} + 57808 \nu^{11} - 136806 \nu^{10} + \cdots + 188851 ) / 102789 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1099 \nu^{15} - 4885 \nu^{14} + 28834 \nu^{13} - 92848 \nu^{12} + 290714 \nu^{11} - 524601 \nu^{10} + \cdots + 1087838 ) / 205578 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14 \nu^{15} + 44 \nu^{14} - 53 \nu^{13} - 331 \nu^{12} + 1253 \nu^{11} - 3357 \nu^{10} + \cdots + 3587 ) / 2187 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 195 \nu^{15} - 931 \nu^{14} + 2748 \nu^{13} - 3688 \nu^{12} + 3582 \nu^{11} + 2953 \nu^{10} + \cdots - 138346 ) / 22842 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 938 \nu^{15} + 5045 \nu^{14} - 19022 \nu^{13} + 40454 \nu^{12} - 78373 \nu^{11} + 83322 \nu^{10} + \cdots - 220840 ) / 102789 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 405 \nu^{15} - 2212 \nu^{14} + 6369 \nu^{13} - 7450 \nu^{12} - 3732 \nu^{11} + 40360 \nu^{10} + \cdots - 177730 ) / 34263 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 523 \nu^{15} - 1973 \nu^{14} + 5068 \nu^{13} - 650 \nu^{12} - 10177 \nu^{11} + 56326 \nu^{10} + \cdots - 53165 ) / 34263 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4021 \nu^{15} + 14221 \nu^{14} - 42610 \nu^{13} + 32740 \nu^{12} - 46970 \nu^{11} + \cdots - 512798 ) / 205578 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4721 \nu^{15} - 27797 \nu^{14} + 107414 \nu^{13} - 251234 \nu^{12} + 501802 \nu^{11} + \cdots - 191036 ) / 205578 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 2 \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + \cdots - 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{15} + \beta_{14} + 4 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 7 \beta_{10} - 2 \beta_{9} + \cdots - 31 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 14 \beta_{15} - 10 \beta_{14} - 12 \beta_{13} - 6 \beta_{12} - 21 \beta_{11} + 21 \beta_{10} + \cdots - 56 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 21 \beta_{15} - 23 \beta_{14} - 76 \beta_{13} + 41 \beta_{12} - 78 \beta_{11} + 8 \beta_{10} + \cdots - 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8 \beta_{15} - 14 \beta_{14} - 183 \beta_{13} + 156 \beta_{12} - 153 \beta_{11} - 60 \beta_{10} + \cdots + 299 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 36 \beta_{15} + 16 \beta_{14} - 56 \beta_{13} + 89 \beta_{12} - 45 \beta_{11} - 52 \beta_{10} + \cdots + 368 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 267 \beta_{15} + 200 \beta_{14} + 415 \beta_{13} + 136 \beta_{12} + 309 \beta_{11} - 194 \beta_{10} + \cdots + 2065 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 271 \beta_{15} + 495 \beta_{14} + 2219 \beta_{13} - 718 \beta_{12} + 1797 \beta_{11} - 94 \beta_{10} + \cdots + 966 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 562 \beta_{15} + 819 \beta_{14} + 5284 \beta_{13} - 3086 \beta_{12} + 4665 \beta_{11} + 250 \beta_{10} + \cdots - 7629 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3293 \beta_{15} + 13 \beta_{14} + 6391 \beta_{13} - 7562 \beta_{12} + 6747 \beta_{11} + 1306 \beta_{10} + \cdots - 30547 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 7271 \beta_{15} - 5467 \beta_{14} - 6057 \beta_{13} - 11703 \beta_{12} - 993 \beta_{11} + \cdots - 63437 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 4191 \beta_{15} - 20177 \beta_{14} - 55156 \beta_{13} - 2431 \beta_{12} - 38424 \beta_{11} + \cdots - 56845 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 28025 \beta_{15} - 38420 \beta_{14} - 157983 \beta_{13} + 55743 \beta_{12} - 129234 \beta_{11} + \cdots + 136049 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1
−0.724143 − 0.165319i
−0.628312 + 0.590424i
−0.317019 − 1.12493i
0.172467 − 1.52157i
0.252952 + 1.56266i
0.466399 + 1.64781i
1.48460 − 1.66288i
1.79305 + 1.53983i
−0.724143 + 0.165319i
−0.628312 − 0.590424i
−0.317019 + 1.12493i
0.172467 + 1.52157i
0.252952 − 1.56266i
0.466399 − 1.64781i
1.48460 + 1.66288i
1.79305 − 1.53983i
−1.22414 + 2.12028i −1.71693 − 0.228383i −1.99705 − 3.45900i 0.500000 + 0.866025i 2.58600 − 3.36079i 1.96726 − 3.40740i 4.88214 2.89568 + 0.784233i −2.44829
196.2 −1.12831 + 1.95429i −0.604199 − 1.62325i −1.54617 − 2.67805i 0.500000 + 0.866025i 3.85403 + 0.650751i −0.353285 + 0.611908i 2.46502 −2.26989 + 1.96153i −2.25662
196.3 −0.817019 + 1.41512i 1.40359 + 1.01486i −0.335039 − 0.580304i 0.500000 + 0.866025i −2.58290 + 1.15709i −1.06506 + 1.84473i −2.17314 0.940129 + 2.84889i −1.63404
196.4 −0.327533 + 0.567303i 0.997116 − 1.41625i 0.785445 + 1.36043i 0.500000 + 0.866025i 0.476854 + 1.02954i 0.388592 − 0.673061i −2.33917 −1.01152 − 2.82433i −0.655066
196.5 −0.247048 + 0.427900i 0.674928 + 1.59514i 0.877935 + 1.52063i 0.500000 + 0.866025i −0.849299 − 0.105275i 2.14269 − 3.71125i −1.85576 −2.08895 + 2.15321i −0.494096
196.6 −0.0336011 + 0.0581988i −0.332146 − 1.69991i 0.997742 + 1.72814i 0.500000 + 0.866025i 0.110093 + 0.0377882i −1.23179 + 2.13352i −0.268505 −2.77936 + 1.12923i −0.0672022
196.7 0.984603 − 1.70538i −1.60495 − 0.651266i −0.938888 − 1.62620i 0.500000 + 0.866025i −2.69089 + 2.09581i 1.51414 − 2.62256i 0.240686 2.15171 + 2.09049i 1.96921
196.8 1.29305 − 2.23963i 1.68259 + 0.410979i −2.34397 − 4.05987i 0.500000 + 0.866025i 3.09611 − 3.23696i 2.13745 − 3.70217i −6.95128 2.66219 + 1.38302i 2.58610
391.1 −1.22414 − 2.12028i −1.71693 + 0.228383i −1.99705 + 3.45900i 0.500000 − 0.866025i 2.58600 + 3.36079i 1.96726 + 3.40740i 4.88214 2.89568 − 0.784233i −2.44829
391.2 −1.12831 − 1.95429i −0.604199 + 1.62325i −1.54617 + 2.67805i 0.500000 − 0.866025i 3.85403 − 0.650751i −0.353285 − 0.611908i 2.46502 −2.26989 − 1.96153i −2.25662
391.3 −0.817019 − 1.41512i 1.40359 − 1.01486i −0.335039 + 0.580304i 0.500000 − 0.866025i −2.58290 − 1.15709i −1.06506 − 1.84473i −2.17314 0.940129 − 2.84889i −1.63404
391.4 −0.327533 − 0.567303i 0.997116 + 1.41625i 0.785445 − 1.36043i 0.500000 − 0.866025i 0.476854 − 1.02954i 0.388592 + 0.673061i −2.33917 −1.01152 + 2.82433i −0.655066
391.5 −0.247048 − 0.427900i 0.674928 − 1.59514i 0.877935 − 1.52063i 0.500000 − 0.866025i −0.849299 + 0.105275i 2.14269 + 3.71125i −1.85576 −2.08895 − 2.15321i −0.494096
391.6 −0.0336011 − 0.0581988i −0.332146 + 1.69991i 0.997742 − 1.72814i 0.500000 − 0.866025i 0.110093 − 0.0377882i −1.23179 − 2.13352i −0.268505 −2.77936 − 1.12923i −0.0672022
391.7 0.984603 + 1.70538i −1.60495 + 0.651266i −0.938888 + 1.62620i 0.500000 − 0.866025i −2.69089 − 2.09581i 1.51414 + 2.62256i 0.240686 2.15171 − 2.09049i 1.96921
391.8 1.29305 + 2.23963i 1.68259 − 0.410979i −2.34397 + 4.05987i 0.500000 − 0.866025i 3.09611 + 3.23696i 2.13745 + 3.70217i −6.95128 2.66219 − 1.38302i 2.58610
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 196.8
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 585.2.i.e ✓ 16
3.b odd 2 1 1755.2.i.f 16
9.c even 3 1 inner 585.2.i.e ✓ 16
9.c even 3 1 5265.2.a.bf 8
9.d odd 6 1 1755.2.i.f 16
9.d odd 6 1 5265.2.a.ba 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.e ✓ 16 1.a even 1 1 trivial
585.2.i.e ✓ 16 9.c even 3 1 inner
1755.2.i.f 16 3.b odd 2 1
1755.2.i.f 16 9.d odd 6 1
5265.2.a.ba 8 9.d odd 6 1
5265.2.a.bf 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 \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{16} + 3 T_{2}^{15} + 17 T_{2}^{14} + 38 T_{2}^{13} + 158 T_{2}^{12} + 324 T_{2}^{11} + 875 T_{2}^{10} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{16} - 11 T_{7}^{15} + 97 T_{7}^{14} - 476 T_{7}^{13} + 2127 T_{7}^{12} - 6158 T_{7}^{11} + \cdots + 395641 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} - 11 T^{15} + \cdots + 395641 \) Copy content Toggle raw display
$11$ \( T^{16} + 6 T^{15} + \cdots + 401956 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{8} + 2 T^{7} + \cdots + 892)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} + \cdots - 1584)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 144360225 \) Copy content Toggle raw display
$29$ \( T^{16} + 14 T^{15} + \cdots + 1243225 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 2547422784 \) Copy content Toggle raw display
$37$ \( (T^{8} - T^{7} + \cdots - 33660)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 371525625 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 62415940941376 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 424482219529 \) Copy content Toggle raw display
$53$ \( (T^{8} - 2 T^{7} + \cdots + 111438)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 3524698346724 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 106354483234561 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 974523752673769 \) Copy content Toggle raw display
$71$ \( (T^{8} - 10 T^{7} + \cdots - 127950)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 6 T^{7} + \cdots + 1196532)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{16} - 10 T^{15} + \cdots + 469225 \) Copy content Toggle raw display
$89$ \( (T^{8} + 13 T^{7} + \cdots + 1032219)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 122483314259524 \) Copy content Toggle raw display
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