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Properties

Label 585.2.i.f
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: \( x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 74 x^{12} - 18 x^{11} + 289 x^{10} - 4 x^{9} + 784 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{14} + \beta_{13} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{14} - \beta_{11} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{14} + \beta_{13} + \cdots - \beta_1) q^{3}+ \cdots + (4 \beta_{15} + 2 \beta_{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - 2 q^{3} - 5 q^{4} - 8 q^{5} - 13 q^{6} + 6 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{2} - 2 q^{3} - 5 q^{4} - 8 q^{5} - 13 q^{6} + 6 q^{7} - 12 q^{8} + 4 q^{9} - 2 q^{10} + 9 q^{11} - 16 q^{12} - 8 q^{13} - 3 q^{14} + q^{15} + 13 q^{16} + 12 q^{17} + 23 q^{18} - 22 q^{19} - 5 q^{20} + 15 q^{21} + 4 q^{22} + 3 q^{23} - 12 q^{24} - 8 q^{25} - 2 q^{26} + 22 q^{27} - 26 q^{28} + 8 q^{29} + 8 q^{30} + 18 q^{31} + 3 q^{32} - 26 q^{33} + 9 q^{34} - 12 q^{35} + 5 q^{36} - 36 q^{37} - 8 q^{38} + q^{39} + 6 q^{40} - 17 q^{41} - 45 q^{42} + 17 q^{43} + 10 q^{44} + q^{45} + 6 q^{46} + 11 q^{47} + 35 q^{48} + 16 q^{49} + q^{50} - 16 q^{51} - 5 q^{52} + 20 q^{53} + 44 q^{54} - 18 q^{55} - q^{56} - 25 q^{57} + 10 q^{58} + 7 q^{59} + 5 q^{60} + 21 q^{61} - 58 q^{62} - 30 q^{63} - 20 q^{64} - 8 q^{65} + 68 q^{66} + 13 q^{67} - 16 q^{68} - 13 q^{69} - 3 q^{70} - 68 q^{71} + 36 q^{72} - 32 q^{73} - 4 q^{74} + q^{75} + 2 q^{76} + 18 q^{77} + 8 q^{78} + 37 q^{79} - 26 q^{80} - 32 q^{81} + 2 q^{82} + 3 q^{83} + 27 q^{84} - 6 q^{85} - 2 q^{86} - 20 q^{87} + 19 q^{88} + 28 q^{89} - 16 q^{90} - 12 q^{91} - 14 q^{92} + 19 q^{93} + 44 q^{94} + 11 q^{95} - 35 q^{96} + 17 q^{97} + 90 q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 482086147342 \nu^{15} - 82999536459994 \nu^{14} + 51742196981879 \nu^{13} + \cdots - 15\!\cdots\!53 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3469225716849 \nu^{15} + 7812059681008 \nu^{14} - 8290755936386 \nu^{13} + \cdots + 12\!\cdots\!15 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6246753964615 \nu^{15} - 59652178722282 \nu^{14} + 157853267854272 \nu^{13} + \cdots + 188168453461782 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10150138176959 \nu^{15} + 69644913861 \nu^{14} - 83292291525414 \nu^{13} + \cdots + 10\!\cdots\!11 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16744116205687 \nu^{15} - 45341940014915 \nu^{14} + 176286820995922 \nu^{13} + \cdots + 34\!\cdots\!58 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26595513341993 \nu^{15} + 73952194191366 \nu^{14} + 112052967333795 \nu^{13} + \cdots + 79\!\cdots\!68 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10058702437115 \nu^{15} - 6203109654385 \nu^{14} + 103630645608798 \nu^{13} + \cdots + 845713476880083 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31322721365929 \nu^{15} + 61498828677274 \nu^{14} - 363159263988374 \nu^{13} + \cdots + 10\!\cdots\!47 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35353445168842 \nu^{15} - 42906025379669 \nu^{14} + 360549986338030 \nu^{13} + \cdots - 18\!\cdots\!67 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47477396037032 \nu^{15} - 55102847496393 \nu^{14} + 436243278002799 \nu^{13} + \cdots - 18\!\cdots\!59 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 67174987872150 \nu^{15} - 20896415688563 \nu^{14} + 620697587330554 \nu^{13} + \cdots + 27\!\cdots\!48 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 88820209793387 \nu^{15} + 39343350153959 \nu^{14} - 891308516903932 \nu^{13} + \cdots - 71\!\cdots\!75 ) / 91\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 31322721365929 \nu^{15} + 61498828677274 \nu^{14} - 363159263988374 \nu^{13} + \cdots + 10\!\cdots\!47 ) / 30\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 241239132189497 \nu^{15} - 269626419785329 \nu^{14} + \cdots - 19\!\cdots\!11 ) / 91\!\cdots\!73 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + 3\beta_{9} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{10} + 4\beta_{8} - \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{14} - 4\beta_{10} - 12\beta_{9} + 5\beta_{8} - \beta_{7} + 6\beta_{6} - \beta_{5} - \beta_{2} - \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + 8 \beta_{14} - 7 \beta_{13} + \beta_{11} + 8 \beta_{10} - 9 \beta_{9} - 11 \beta_{8} + \cdots - 11 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} + 9 \beta_{14} - 13 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + 28 \beta_{10} + \cdots + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 32 \beta_{14} - 11 \beta_{13} - 10 \beta_{12} + 11 \beta_{11} + \beta_{10} + 63 \beta_{9} - 32 \beta_{8} + \cdots + 63 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 23 \beta_{15} - 170 \beta_{14} + 86 \beta_{13} - 51 \beta_{11} - 110 \beta_{10} + 263 \beta_{9} + \cdots + 23 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 76 \beta_{15} - 148 \beta_{14} + 357 \beta_{13} + 76 \beta_{12} - 173 \beta_{11} - 368 \beta_{10} + \cdots - 403 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 514 \beta_{14} + 198 \beta_{13} + 184 \beta_{12} - 198 \beta_{11} - 222 \beta_{10} - 1358 \beta_{9} + \cdots - 1358 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 519 \beta_{15} + 1886 \beta_{14} - 1638 \beta_{13} + 590 \beta_{11} + 2241 \beta_{10} + \cdots - 966 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1277 \beta_{15} + 2535 \beta_{14} - 5171 \beta_{13} - 1277 \beta_{12} + 3222 \beta_{11} + 6063 \beta_{10} + \cdots + 7316 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4940 \beta_{14} - 3909 \beta_{13} - 3350 \beta_{12} + 3909 \beta_{11} + 526 \beta_{10} + 14858 \beta_{9} + \cdots + 14858 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8265 \beta_{15} - 28774 \beta_{14} + 23287 \beta_{13} - 10699 \beta_{11} - 32526 \beta_{10} + \cdots + 8068 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 20912 \beta_{15} - 37783 \beta_{14} + 83331 \beta_{13} + 20912 \beta_{12} - 47996 \beta_{11} + \cdots - 88279 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(\beta_{9}\) \(1\) \(1\)

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
−0.947115 + 1.64045i
−0.921972 + 1.59690i
−0.624867 + 1.08230i
−0.237622 + 0.411573i
0.316605 − 0.548376i
0.715744 − 1.23970i
0.987387 − 1.71020i
1.21184 − 2.09897i
−0.947115 − 1.64045i
−0.921972 − 1.59690i
−0.624867 − 1.08230i
−0.237622 − 0.411573i
0.316605 + 0.548376i
0.715744 + 1.23970i
0.987387 + 1.71020i
1.21184 + 2.09897i
−0.947115 + 1.64045i 1.31847 − 1.12323i −0.794055 − 1.37534i −0.500000 − 0.866025i 0.593868 + 3.22671i −0.837925 + 1.45133i −0.780216 0.476703 − 2.96188i 1.89423
196.2 −0.921972 + 1.59690i −0.495399 + 1.65969i −0.700064 − 1.21255i −0.500000 − 0.866025i −2.19362 − 2.32129i 2.38510 − 4.13111i −1.10613 −2.50916 − 1.64442i 1.84394
196.3 −0.624867 + 1.08230i 1.28189 + 1.16480i 0.219082 + 0.379462i −0.500000 − 0.866025i −2.06168 + 0.659540i 0.148724 − 0.257597i −3.04706 0.286466 + 2.98629i 1.24973
196.4 −0.237622 + 0.411573i −1.28679 − 1.15939i 0.887072 + 1.53645i −0.500000 − 0.866025i 0.782942 − 0.254110i −1.24774 + 2.16115i −1.79364 0.311632 + 2.98377i 0.475244
196.5 0.316605 − 0.548376i −1.15302 + 1.29250i 0.799523 + 1.38481i −0.500000 − 0.866025i 0.343725 + 1.04150i 0.896323 − 1.55248i 2.27895 −0.341107 − 2.98054i −0.633210
196.6 0.715744 − 1.23970i −1.70726 − 0.291973i −0.0245786 − 0.0425715i −0.500000 − 0.866025i −1.58392 + 1.90753i 0.625355 − 1.08315i 2.79261 2.82950 + 0.996950i −1.43149
196.7 0.987387 − 1.71020i 1.73142 + 0.0465761i −0.949865 − 1.64522i −0.500000 − 0.866025i 1.78924 − 2.91510i 0.124489 − 0.215622i 0.198009 2.99566 + 0.161286i −1.97477
196.8 1.21184 − 2.09897i −0.689312 − 1.58898i −1.93711 − 3.35518i −0.500000 − 0.866025i −4.17055 − 0.478743i 0.905675 − 1.56867i −4.54253 −2.04970 + 2.19060i −2.42368
391.1 −0.947115 − 1.64045i 1.31847 + 1.12323i −0.794055 + 1.37534i −0.500000 + 0.866025i 0.593868 − 3.22671i −0.837925 − 1.45133i −0.780216 0.476703 + 2.96188i 1.89423
391.2 −0.921972 − 1.59690i −0.495399 − 1.65969i −0.700064 + 1.21255i −0.500000 + 0.866025i −2.19362 + 2.32129i 2.38510 + 4.13111i −1.10613 −2.50916 + 1.64442i 1.84394
391.3 −0.624867 − 1.08230i 1.28189 − 1.16480i 0.219082 − 0.379462i −0.500000 + 0.866025i −2.06168 − 0.659540i 0.148724 + 0.257597i −3.04706 0.286466 − 2.98629i 1.24973
391.4 −0.237622 − 0.411573i −1.28679 + 1.15939i 0.887072 − 1.53645i −0.500000 + 0.866025i 0.782942 + 0.254110i −1.24774 − 2.16115i −1.79364 0.311632 − 2.98377i 0.475244
391.5 0.316605 + 0.548376i −1.15302 − 1.29250i 0.799523 − 1.38481i −0.500000 + 0.866025i 0.343725 − 1.04150i 0.896323 + 1.55248i 2.27895 −0.341107 + 2.98054i −0.633210
391.6 0.715744 + 1.23970i −1.70726 + 0.291973i −0.0245786 + 0.0425715i −0.500000 + 0.866025i −1.58392 − 1.90753i 0.625355 + 1.08315i 2.79261 2.82950 − 0.996950i −1.43149
391.7 0.987387 + 1.71020i 1.73142 − 0.0465761i −0.949865 + 1.64522i −0.500000 + 0.866025i 1.78924 + 2.91510i 0.124489 + 0.215622i 0.198009 2.99566 − 0.161286i −1.97477
391.8 1.21184 + 2.09897i −0.689312 + 1.58898i −1.93711 + 3.35518i −0.500000 + 0.866025i −4.17055 + 0.478743i 0.905675 + 1.56867i −4.54253 −2.04970 − 2.19060i −2.42368
\(n\): 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.f ✓ 16
3.b odd 2 1 1755.2.i.e 16
9.c even 3 1 inner 585.2.i.f ✓ 16
9.c even 3 1 5265.2.a.bb 8
9.d odd 6 1 1755.2.i.e 16
9.d odd 6 1 5265.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.f ✓ 16 1.a even 1 1 trivial
585.2.i.f ✓ 16 9.c even 3 1 inner
1755.2.i.e 16 3.b odd 2 1
1755.2.i.e 16 9.d odd 6 1
5265.2.a.bb 8 9.c even 3 1
5265.2.a.be 8 9.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 \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{16} - T_{2}^{15} + 11 T_{2}^{14} - 4 T_{2}^{13} + 74 T_{2}^{12} - 18 T_{2}^{11} + 289 T_{2}^{10} + \cdots + 81 \) 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
$3$ \( T^{16} + 2 T^{15} + \cdots + 6561 \) 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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