↑

Properties

Label 5733.2.a.bd.1.3
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5733,2,Mod(1,5733)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5733.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5733, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,2,0,6,-5,0,0,6,0,-14,4,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(45.7782354788\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.65544\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.65544 q^{2} +5.05137 q^{4} -3.65544 q^{5} +8.10275 q^{8} -9.70682 q^{10} -0.655442 q^{11} -1.00000 q^{13} +11.4136 q^{16} -2.39593 q^{17} -2.70682 q^{19} -18.4650 q^{20} -1.74049 q^{22} -7.36226 q^{23} +8.36226 q^{25} -2.65544 q^{26} +0.208136 q^{29} -1.13642 q^{31} +14.1027 q^{32} -6.36226 q^{34} -7.44731 q^{37} -7.18780 q^{38} -29.6191 q^{40} +10.2055 q^{41} -3.10275 q^{43} -3.31088 q^{44} -19.5501 q^{46} -4.60407 q^{47} +22.2055 q^{50} -5.05137 q^{52} -5.25951 q^{53} +2.39593 q^{55} +0.552694 q^{58} -8.25951 q^{59} -1.89725 q^{61} -3.01770 q^{62} +14.6218 q^{64} +3.65544 q^{65} -12.8946 q^{67} -12.1027 q^{68} +6.75819 q^{71} +12.5367 q^{73} -19.7759 q^{74} -13.6731 q^{76} -1.51902 q^{79} -41.7219 q^{80} +27.1001 q^{82} -15.7582 q^{83} +8.75819 q^{85} -8.23917 q^{86} -5.31088 q^{88} +14.8096 q^{89} -37.1895 q^{92} -12.2258 q^{94} +9.89461 q^{95} -10.0177 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 6 q^{8} - 14 q^{10} + 4 q^{11} - 3 q^{13} + 4 q^{16} - 4 q^{17} + 7 q^{19} - 16 q^{20} - 8 q^{22} - q^{23} + 4 q^{25} - 2 q^{26} + 7 q^{29} - 3 q^{31} + 24 q^{32} + 2 q^{34}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.65544 1.87768 0.938841 − 0.344352i \(-0.111901\pi\)
0.938841 + 0.344352i \(0.111901\pi\)
\(3\) 0 0
\(4\) 5.05137 2.52569
\(5\) −3.65544 −1.63476 −0.817382 − 0.576096i \(-0.804575\pi\)
−0.817382 + 0.576096i \(0.804575\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.10275 2.86475
\(9\) 0 0
\(10\) −9.70682 −3.06956
\(11\) −0.655442 −0.197623 −0.0988117 − 0.995106i \(-0.531504\pi\)
−0.0988117 + 0.995106i \(0.531504\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 11.4136 2.85341
\(17\) −2.39593 −0.581099 −0.290549 − 0.956860i \(-0.593838\pi\)
−0.290549 + 0.956860i \(0.593838\pi\)
\(18\) 0 0
\(19\) −2.70682 −0.620986 −0.310493 − 0.950576i \(-0.600494\pi\)
−0.310493 + 0.950576i \(0.600494\pi\)
\(20\) −18.4650 −4.12890
\(21\) 0 0
\(22\) −1.74049 −0.371074
\(23\) −7.36226 −1.53514 −0.767569 − 0.640967i \(-0.778534\pi\)
−0.767569 + 0.640967i \(0.778534\pi\)
\(24\) 0 0
\(25\) 8.36226 1.67245
\(26\) −2.65544 −0.520775
\(27\) 0 0
\(28\) 0 0
\(29\) 0.208136 0.0386499 0.0193250 − 0.999813i \(-0.493848\pi\)
0.0193250 + 0.999813i \(0.493848\pi\)
\(30\) 0 0
\(31\) −1.13642 −0.204107 −0.102054 − 0.994779i \(-0.532541\pi\)
−0.102054 + 0.994779i \(0.532541\pi\)
\(32\) 14.1027 2.49304
\(33\) 0 0
\(34\) −6.36226 −1.09112
\(35\) 0 0
\(36\) 0 0
\(37\) −7.44731 −1.22433 −0.612165 − 0.790730i \(-0.709701\pi\)
−0.612165 + 0.790730i \(0.709701\pi\)
\(38\) −7.18780 −1.16601
\(39\) 0 0
\(40\) −29.6191 −4.68320
\(41\) 10.2055 1.59383 0.796915 − 0.604091i \(-0.206464\pi\)
0.796915 + 0.604091i \(0.206464\pi\)
\(42\) 0 0
\(43\) −3.10275 −0.473165 −0.236582 − 0.971611i \(-0.576027\pi\)
−0.236582 + 0.971611i \(0.576027\pi\)
\(44\) −3.31088 −0.499135
\(45\) 0 0
\(46\) −19.5501 −2.88250
\(47\) −4.60407 −0.671572 −0.335786 − 0.941938i \(-0.609002\pi\)
−0.335786 + 0.941938i \(0.609002\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 22.2055 3.14033
\(51\) 0 0
\(52\) −5.05137 −0.700500
\(53\) −5.25951 −0.722449 −0.361225 − 0.932479i \(-0.617641\pi\)
−0.361225 + 0.932479i \(0.617641\pi\)
\(54\) 0 0
\(55\) 2.39593 0.323067
\(56\) 0 0
\(57\) 0 0
\(58\) 0.552694 0.0725723
\(59\) −8.25951 −1.07530 −0.537648 − 0.843169i \(-0.680687\pi\)
−0.537648 + 0.843169i \(0.680687\pi\)
\(60\) 0 0
\(61\) −1.89725 −0.242918 −0.121459 − 0.992596i \(-0.538757\pi\)
−0.121459 + 0.992596i \(0.538757\pi\)
\(62\) −3.01770 −0.383248
\(63\) 0 0
\(64\) 14.6218 1.82772
\(65\) 3.65544 0.453402
\(66\) 0 0
\(67\) −12.8946 −1.57533 −0.787664 − 0.616105i \(-0.788710\pi\)
−0.787664 + 0.616105i \(0.788710\pi\)
\(68\) −12.1027 −1.46767
\(69\) 0 0
\(70\) 0 0
\(71\) 6.75819 0.802050 0.401025 − 0.916067i \(-0.368654\pi\)
0.401025 + 0.916067i \(0.368654\pi\)
\(72\) 0 0
\(73\) 12.5367 1.46731 0.733656 − 0.679521i \(-0.237812\pi\)
0.733656 + 0.679521i \(0.237812\pi\)
\(74\) −19.7759 −2.29890
\(75\) 0 0
\(76\) −13.6731 −1.56842
\(77\) 0 0
\(78\) 0 0
\(79\) −1.51902 −0.170903 −0.0854516 − 0.996342i \(-0.527233\pi\)
−0.0854516 + 0.996342i \(0.527233\pi\)
\(80\) −41.7219 −4.66465
\(81\) 0 0
\(82\) 27.1001 2.99271
\(83\) −15.7582 −1.72969 −0.864843 − 0.502042i \(-0.832582\pi\)
−0.864843 + 0.502042i \(0.832582\pi\)
\(84\) 0 0
\(85\) 8.75819 0.949959
\(86\) −8.23917 −0.888453
\(87\) 0 0
\(88\) −5.31088 −0.566142
\(89\) 14.8096 1.56981 0.784905 − 0.619616i \(-0.212712\pi\)
0.784905 + 0.619616i \(0.212712\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −37.1895 −3.87728
\(93\) 0 0
\(94\) −12.2258 −1.26100
\(95\) 9.89461 1.01517
\(96\) 0 0
\(97\) −10.0177 −1.01714 −0.508572 − 0.861020i \(-0.669826\pi\)
−0.508572 + 0.861020i \(0.669826\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5733.2.a.bd.1.3 3
3.2 odd 2 637.2.a.i.1.1 yes 3
7.6 odd 2 5733.2.a.be.1.3 3
21.2 odd 6 637.2.e.k.508.3 6
21.5 even 6 637.2.e.l.508.3 6
21.11 odd 6 637.2.e.k.79.3 6
21.17 even 6 637.2.e.l.79.3 6
21.20 even 2 637.2.a.h.1.1 ✓ 3
39.38 odd 2 8281.2.a.bk.1.3 3
273.272 even 2 8281.2.a.bh.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.1 ✓ 3 21.20 even 2
637.2.a.i.1.1 yes 3 3.2 odd 2
637.2.e.k.79.3 6 21.11 odd 6
637.2.e.k.508.3 6 21.2 odd 6
637.2.e.l.79.3 6 21.17 even 6
637.2.e.l.508.3 6 21.5 even 6
5733.2.a.bd.1.3 3 1.1 even 1 trivial
5733.2.a.be.1.3 3 7.6 odd 2
8281.2.a.bh.1.3 3 273.272 even 2
8281.2.a.bk.1.3 3 39.38 odd 2