↑

Properties

Label 363.2.a.j.1.2
Level $363$
Weight $2$
Character 363.1
Self dual yes
Analytic conductor $2.899$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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] = [363,2,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.792287\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.792287 q^{2} +1.00000 q^{3} -1.37228 q^{4} +3.37228 q^{5} -0.792287 q^{6} -2.52434 q^{7} +2.67181 q^{8} +1.00000 q^{9} -2.67181 q^{10} -1.37228 q^{12} +5.84096 q^{13} +2.00000 q^{14} +3.37228 q^{15} +0.627719 q^{16} -2.67181 q^{17} -0.792287 q^{18} +0.939764 q^{19} -4.62772 q^{20} -2.52434 q^{21} +2.00000 q^{23} +2.67181 q^{24} +6.37228 q^{25} -4.62772 q^{26} +1.00000 q^{27} +3.46410 q^{28} +0.792287 q^{29} -2.67181 q^{30} +1.62772 q^{31} -5.84096 q^{32} +2.11684 q^{34} -8.51278 q^{35} -1.37228 q^{36} +5.00000 q^{37} -0.744563 q^{38} +5.84096 q^{39} +9.01011 q^{40} +10.8896 q^{41} +2.00000 q^{42} +6.63325 q^{43} +3.37228 q^{45} -1.58457 q^{46} -12.7446 q^{47} +0.627719 q^{48} -0.627719 q^{49} -5.04868 q^{50} -2.67181 q^{51} -8.01544 q^{52} -4.11684 q^{53} -0.792287 q^{54} -6.74456 q^{56} +0.939764 q^{57} -0.627719 q^{58} -6.00000 q^{59} -4.62772 q^{60} -5.98844 q^{61} -1.28962 q^{62} -2.52434 q^{63} +3.37228 q^{64} +19.6974 q^{65} -1.11684 q^{67} +3.66648 q^{68} +2.00000 q^{69} +6.74456 q^{70} -10.7446 q^{71} +2.67181 q^{72} -9.15759 q^{73} -3.96143 q^{74} +6.37228 q^{75} -1.28962 q^{76} -4.62772 q^{78} -4.10891 q^{79} +2.11684 q^{80} +1.00000 q^{81} -8.62772 q^{82} +1.87953 q^{83} +3.46410 q^{84} -9.01011 q^{85} -5.25544 q^{86} +0.792287 q^{87} -0.627719 q^{89} -2.67181 q^{90} -14.7446 q^{91} -2.74456 q^{92} +1.62772 q^{93} +10.0974 q^{94} +3.16915 q^{95} -5.84096 q^{96} +10.4891 q^{97} +0.497333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 6 q^{4} + 2 q^{5} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 2 q^{15} + 14 q^{16} - 30 q^{20} + 8 q^{23} + 14 q^{25} - 30 q^{26} + 4 q^{27} + 18 q^{31} - 26 q^{34} + 6 q^{36} + 20 q^{37} + 20 q^{38}+ \cdots - 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.792287 −0.560232 −0.280116 − 0.959966i \(-0.590373\pi\)
−0.280116 + 0.959966i \(0.590373\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.37228 −0.686141
\(5\) 3.37228 1.50813 0.754065 − 0.656800i \(-0.228090\pi\)
0.754065 + 0.656800i \(0.228090\pi\)
\(6\) −0.792287 −0.323450
\(7\) −2.52434 −0.954110 −0.477055 − 0.878873i \(-0.658296\pi\)
−0.477055 + 0.878873i \(0.658296\pi\)
\(8\) 2.67181 0.944629
\(9\) 1.00000 0.333333
\(10\) −2.67181 −0.844902
\(11\) 0 0
\(12\) −1.37228 −0.396143
\(13\) 5.84096 1.61999 0.809996 − 0.586436i \(-0.199469\pi\)
0.809996 + 0.586436i \(0.199469\pi\)
\(14\) 2.00000 0.534522
\(15\) 3.37228 0.870719
\(16\) 0.627719 0.156930
\(17\) −2.67181 −0.648010 −0.324005 − 0.946055i \(-0.605030\pi\)
−0.324005 + 0.946055i \(0.605030\pi\)
\(18\) −0.792287 −0.186744
\(19\) 0.939764 0.215597 0.107798 − 0.994173i \(-0.465620\pi\)
0.107798 + 0.994173i \(0.465620\pi\)
\(20\) −4.62772 −1.03479
\(21\) −2.52434 −0.550856
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 2.67181 0.545382
\(25\) 6.37228 1.27446
\(26\) −4.62772 −0.907570
\(27\) 1.00000 0.192450
\(28\) 3.46410 0.654654
\(29\) 0.792287 0.147124 0.0735620 − 0.997291i \(-0.476563\pi\)
0.0735620 + 0.997291i \(0.476563\pi\)
\(30\) −2.67181 −0.487804
\(31\) 1.62772 0.292347 0.146173 − 0.989259i \(-0.453304\pi\)
0.146173 + 0.989259i \(0.453304\pi\)
\(32\) −5.84096 −1.03255
\(33\) 0 0
\(34\) 2.11684 0.363036
\(35\) −8.51278 −1.43892
\(36\) −1.37228 −0.228714
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) −0.744563 −0.120784
\(39\) 5.84096 0.935303
\(40\) 9.01011 1.42462
\(41\) 10.8896 1.70068 0.850338 − 0.526237i \(-0.176398\pi\)
0.850338 + 0.526237i \(0.176398\pi\)
\(42\) 2.00000 0.308607
\(43\) 6.63325 1.01156 0.505781 − 0.862662i \(-0.331205\pi\)
0.505781 + 0.862662i \(0.331205\pi\)
\(44\) 0 0
\(45\) 3.37228 0.502710
\(46\) −1.58457 −0.233633
\(47\) −12.7446 −1.85899 −0.929493 − 0.368840i \(-0.879755\pi\)
−0.929493 + 0.368840i \(0.879755\pi\)
\(48\) 0.627719 0.0906034
\(49\) −0.627719 −0.0896741
\(50\) −5.04868 −0.713991
\(51\) −2.67181 −0.374129
\(52\) −8.01544 −1.11154
\(53\) −4.11684 −0.565492 −0.282746 − 0.959195i \(-0.591245\pi\)
−0.282746 + 0.959195i \(0.591245\pi\)
\(54\) −0.792287 −0.107817
\(55\) 0 0
\(56\) −6.74456 −0.901280
\(57\) 0.939764 0.124475
\(58\) −0.627719 −0.0824235
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −4.62772 −0.597436
\(61\) −5.98844 −0.766741 −0.383371 − 0.923595i \(-0.625237\pi\)
−0.383371 + 0.923595i \(0.625237\pi\)
\(62\) −1.28962 −0.163782
\(63\) −2.52434 −0.318037
\(64\) 3.37228 0.421535
\(65\) 19.6974 2.44316
\(66\) 0 0
\(67\) −1.11684 −0.136444 −0.0682221 − 0.997670i \(-0.521733\pi\)
−0.0682221 + 0.997670i \(0.521733\pi\)
\(68\) 3.66648 0.444626
\(69\) 2.00000 0.240772
\(70\) 6.74456 0.806129
\(71\) −10.7446 −1.27514 −0.637572 − 0.770390i \(-0.720061\pi\)
−0.637572 + 0.770390i \(0.720061\pi\)
\(72\) 2.67181 0.314876
\(73\) −9.15759 −1.07181 −0.535907 − 0.844277i \(-0.680030\pi\)
−0.535907 + 0.844277i \(0.680030\pi\)
\(74\) −3.96143 −0.460507
\(75\) 6.37228 0.735808
\(76\) −1.28962 −0.147930
\(77\) 0 0
\(78\) −4.62772 −0.523986
\(79\) −4.10891 −0.462289 −0.231144 − 0.972919i \(-0.574247\pi\)
−0.231144 + 0.972919i \(0.574247\pi\)
\(80\) 2.11684 0.236670
\(81\) 1.00000 0.111111
\(82\) −8.62772 −0.952772
\(83\) 1.87953 0.206305 0.103152 − 0.994666i \(-0.467107\pi\)
0.103152 + 0.994666i \(0.467107\pi\)
\(84\) 3.46410 0.377964
\(85\) −9.01011 −0.977284
\(86\) −5.25544 −0.566708
\(87\) 0.792287 0.0849421
\(88\) 0 0
\(89\) −0.627719 −0.0665380 −0.0332690 − 0.999446i \(-0.510592\pi\)
−0.0332690 + 0.999446i \(0.510592\pi\)
\(90\) −2.67181 −0.281634
\(91\) −14.7446 −1.54565
\(92\) −2.74456 −0.286140
\(93\) 1.62772 0.168787
\(94\) 10.0974 1.04146
\(95\) 3.16915 0.325148
\(96\) −5.84096 −0.596141
\(97\) 10.4891 1.06501 0.532505 − 0.846427i \(-0.321251\pi\)
0.532505 + 0.846427i \(0.321251\pi\)
\(98\) 0.497333 0.0502383
\(99\) 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 363.2.a.j.1.2 ✓ 4
3.2 odd 2 1089.2.a.u.1.3 4
4.3 odd 2 5808.2.a.ck.1.4 4
5.4 even 2 9075.2.a.cv.1.3 4
11.2 odd 10 363.2.e.n.202.2 16
11.3 even 5 363.2.e.n.130.2 16
11.4 even 5 363.2.e.n.148.2 16
11.5 even 5 363.2.e.n.124.3 16
11.6 odd 10 363.2.e.n.124.2 16
11.7 odd 10 363.2.e.n.148.3 16
11.8 odd 10 363.2.e.n.130.3 16
11.9 even 5 363.2.e.n.202.3 16
11.10 odd 2 inner 363.2.a.j.1.3 yes 4
33.32 even 2 1089.2.a.u.1.2 4
44.43 even 2 5808.2.a.ck.1.3 4
55.54 odd 2 9075.2.a.cv.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.a.j.1.2 ✓ 4 1.1 even 1 trivial
363.2.a.j.1.3 yes 4 11.10 odd 2 inner
363.2.e.n.124.2 16 11.6 odd 10
363.2.e.n.124.3 16 11.5 even 5
363.2.e.n.130.2 16 11.3 even 5
363.2.e.n.130.3 16 11.8 odd 10
363.2.e.n.148.2 16 11.4 even 5
363.2.e.n.148.3 16 11.7 odd 10
363.2.e.n.202.2 16 11.2 odd 10
363.2.e.n.202.3 16 11.9 even 5
1089.2.a.u.1.2 4 33.32 even 2
1089.2.a.u.1.3 4 3.2 odd 2
5808.2.a.ck.1.3 4 44.43 even 2
5808.2.a.ck.1.4 4 4.3 odd 2
9075.2.a.cv.1.2 4 55.54 odd 2
9075.2.a.cv.1.3 4 5.4 even 2