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Properties

Label 1445.2.a.j.1.1
Level $1445$
Weight $2$
Character 1445.1
Self dual yes
Analytic conductor $11.538$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,2,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1,0,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.5383830921\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.48119 q^{2} +1.67513 q^{3} +0.193937 q^{4} -1.00000 q^{5} -2.48119 q^{6} +1.28726 q^{7} +2.67513 q^{8} -0.193937 q^{9} +1.48119 q^{10} +0.481194 q^{11} +0.324869 q^{12} -2.15633 q^{13} -1.90668 q^{14} -1.67513 q^{15} -4.35026 q^{16} +0.287258 q^{18} -3.35026 q^{19} -0.193937 q^{20} +2.15633 q^{21} -0.712742 q^{22} -8.24965 q^{23} +4.48119 q^{24} +1.00000 q^{25} +3.19394 q^{26} -5.35026 q^{27} +0.249646 q^{28} +0.649738 q^{29} +2.48119 q^{30} +1.83146 q^{31} +1.09332 q^{32} +0.806063 q^{33} -1.28726 q^{35} -0.0376114 q^{36} +4.31265 q^{37} +4.96239 q^{38} -3.61213 q^{39} -2.67513 q^{40} +11.2750 q^{41} -3.19394 q^{42} -8.15633 q^{43} +0.0933212 q^{44} +0.193937 q^{45} +12.2193 q^{46} -6.54420 q^{47} -7.28726 q^{48} -5.34297 q^{49} -1.48119 q^{50} -0.418190 q^{52} +8.57452 q^{53} +7.92478 q^{54} -0.481194 q^{55} +3.44358 q^{56} -5.61213 q^{57} -0.962389 q^{58} -4.96239 q^{59} -0.324869 q^{60} -2.83638 q^{61} -2.71274 q^{62} -0.249646 q^{63} +7.08110 q^{64} +2.15633 q^{65} -1.19394 q^{66} +4.93207 q^{67} -13.8192 q^{69} +1.90668 q^{70} -14.5320 q^{71} -0.518806 q^{72} -13.3503 q^{73} -6.38787 q^{74} +1.67513 q^{75} -0.649738 q^{76} +0.619421 q^{77} +5.35026 q^{78} -9.05571 q^{79} +4.35026 q^{80} -8.38058 q^{81} -16.7005 q^{82} +13.4314 q^{83} +0.418190 q^{84} +12.0811 q^{86} +1.08840 q^{87} +1.28726 q^{88} -16.7816 q^{89} -0.287258 q^{90} -2.77575 q^{91} -1.59991 q^{92} +3.06793 q^{93} +9.69323 q^{94} +3.35026 q^{95} +1.83146 q^{96} -3.66291 q^{97} +7.91397 q^{98} -0.0933212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} + 3 q^{8} - q^{9} - q^{10} - 4 q^{11} + 6 q^{12} + 4 q^{13} - 12 q^{14} - 3 q^{16} - 5 q^{18} - q^{20} - 4 q^{21} - 8 q^{22} - 8 q^{23} + 8 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.48119 −1.04736 −0.523681 − 0.851914i \(-0.675442\pi\)
−0.523681 + 0.851914i \(0.675442\pi\)
\(3\) 1.67513 0.967137 0.483569 − 0.875306i \(-0.339340\pi\)
0.483569 + 0.875306i \(0.339340\pi\)
\(4\) 0.193937 0.0969683
\(5\) −1.00000 −0.447214
\(6\) −2.48119 −1.01294
\(7\) 1.28726 0.486538 0.243269 − 0.969959i \(-0.421780\pi\)
0.243269 + 0.969959i \(0.421780\pi\)
\(8\) 2.67513 0.945802
\(9\) −0.193937 −0.0646455
\(10\) 1.48119 0.468395
\(11\) 0.481194 0.145086 0.0725428 − 0.997365i \(-0.476889\pi\)
0.0725428 + 0.997365i \(0.476889\pi\)
\(12\) 0.324869 0.0937816
\(13\) −2.15633 −0.598057 −0.299028 − 0.954244i \(-0.596663\pi\)
−0.299028 + 0.954244i \(0.596663\pi\)
\(14\) −1.90668 −0.509581
\(15\) −1.67513 −0.432517
\(16\) −4.35026 −1.08757
\(17\) 0 0
\(18\) 0.287258 0.0677073
\(19\) −3.35026 −0.768603 −0.384301 − 0.923208i \(-0.625558\pi\)
−0.384301 + 0.923208i \(0.625558\pi\)
\(20\) −0.193937 −0.0433655
\(21\) 2.15633 0.470549
\(22\) −0.712742 −0.151957
\(23\) −8.24965 −1.72017 −0.860085 − 0.510151i \(-0.829590\pi\)
−0.860085 + 0.510151i \(0.829590\pi\)
\(24\) 4.48119 0.914720
\(25\) 1.00000 0.200000
\(26\) 3.19394 0.626382
\(27\) −5.35026 −1.02966
\(28\) 0.249646 0.0471787
\(29\) 0.649738 0.120653 0.0603267 − 0.998179i \(-0.480786\pi\)
0.0603267 + 0.998179i \(0.480786\pi\)
\(30\) 2.48119 0.453002
\(31\) 1.83146 0.328939 0.164470 − 0.986382i \(-0.447409\pi\)
0.164470 + 0.986382i \(0.447409\pi\)
\(32\) 1.09332 0.193274
\(33\) 0.806063 0.140318
\(34\) 0 0
\(35\) −1.28726 −0.217586
\(36\) −0.0376114 −0.00626857
\(37\) 4.31265 0.708995 0.354498 − 0.935057i \(-0.384652\pi\)
0.354498 + 0.935057i \(0.384652\pi\)
\(38\) 4.96239 0.805006
\(39\) −3.61213 −0.578403
\(40\) −2.67513 −0.422975
\(41\) 11.2750 1.76087 0.880433 − 0.474171i \(-0.157252\pi\)
0.880433 + 0.474171i \(0.157252\pi\)
\(42\) −3.19394 −0.492835
\(43\) −8.15633 −1.24383 −0.621914 − 0.783086i \(-0.713645\pi\)
−0.621914 + 0.783086i \(0.713645\pi\)
\(44\) 0.0933212 0.0140687
\(45\) 0.193937 0.0289104
\(46\) 12.2193 1.80164
\(47\) −6.54420 −0.954569 −0.477285 − 0.878749i \(-0.658379\pi\)
−0.477285 + 0.878749i \(0.658379\pi\)
\(48\) −7.28726 −1.05183
\(49\) −5.34297 −0.763281
\(50\) −1.48119 −0.209473
\(51\) 0 0
\(52\) −0.418190 −0.0579926
\(53\) 8.57452 1.17780 0.588900 − 0.808206i \(-0.299561\pi\)
0.588900 + 0.808206i \(0.299561\pi\)
\(54\) 7.92478 1.07843
\(55\) −0.481194 −0.0648842
\(56\) 3.44358 0.460168
\(57\) −5.61213 −0.743344
\(58\) −0.962389 −0.126368
\(59\) −4.96239 −0.646048 −0.323024 − 0.946391i \(-0.604699\pi\)
−0.323024 + 0.946391i \(0.604699\pi\)
\(60\) −0.324869 −0.0419404
\(61\) −2.83638 −0.363161 −0.181581 − 0.983376i \(-0.558121\pi\)
−0.181581 + 0.983376i \(0.558121\pi\)
\(62\) −2.71274 −0.344519
\(63\) −0.249646 −0.0314525
\(64\) 7.08110 0.885138
\(65\) 2.15633 0.267459
\(66\) −1.19394 −0.146963
\(67\) 4.93207 0.602548 0.301274 − 0.953538i \(-0.402588\pi\)
0.301274 + 0.953538i \(0.402588\pi\)
\(68\) 0 0
\(69\) −13.8192 −1.66364
\(70\) 1.90668 0.227892
\(71\) −14.5320 −1.72463 −0.862314 − 0.506373i \(-0.830986\pi\)
−0.862314 + 0.506373i \(0.830986\pi\)
\(72\) −0.518806 −0.0611418
\(73\) −13.3503 −1.56253 −0.781265 − 0.624200i \(-0.785425\pi\)
−0.781265 + 0.624200i \(0.785425\pi\)
\(74\) −6.38787 −0.742575
\(75\) 1.67513 0.193427
\(76\) −0.649738 −0.0745301
\(77\) 0.619421 0.0705896
\(78\) 5.35026 0.605798
\(79\) −9.05571 −1.01885 −0.509423 − 0.860516i \(-0.670141\pi\)
−0.509423 + 0.860516i \(0.670141\pi\)
\(80\) 4.35026 0.486374
\(81\) −8.38058 −0.931175
\(82\) −16.7005 −1.84426
\(83\) 13.4314 1.47428 0.737142 − 0.675738i \(-0.236175\pi\)
0.737142 + 0.675738i \(0.236175\pi\)
\(84\) 0.418190 0.0456283
\(85\) 0 0
\(86\) 12.0811 1.30274
\(87\) 1.08840 0.116688
\(88\) 1.28726 0.137222
\(89\) −16.7816 −1.77885 −0.889424 − 0.457082i \(-0.848894\pi\)
−0.889424 + 0.457082i \(0.848894\pi\)
\(90\) −0.287258 −0.0302796
\(91\) −2.77575 −0.290977
\(92\) −1.59991 −0.166802
\(93\) 3.06793 0.318129
\(94\) 9.69323 0.999780
\(95\) 3.35026 0.343730
\(96\) 1.83146 0.186922
\(97\) −3.66291 −0.371912 −0.185956 − 0.982558i \(-0.559538\pi\)
−0.185956 + 0.982558i \(0.559538\pi\)
\(98\) 7.91397 0.799432
\(99\) −0.0933212 −0.00937913
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 1445.2.a.j.1.1 3
5.4 even 2 7225.2.a.q.1.3 3
17.4 even 4 85.2.d.a.16.5 ✓ 6
17.13 even 4 85.2.d.a.16.6 yes 6
17.16 even 2 1445.2.a.k.1.1 3
51.38 odd 4 765.2.g.b.271.1 6
51.47 odd 4 765.2.g.b.271.2 6
68.47 odd 4 1360.2.c.f.1121.2 6
68.55 odd 4 1360.2.c.f.1121.5 6
85.4 even 4 425.2.d.c.101.2 6
85.13 odd 4 425.2.c.b.424.2 6
85.38 odd 4 425.2.c.a.424.2 6
85.47 odd 4 425.2.c.a.424.5 6
85.64 even 4 425.2.d.c.101.1 6
85.72 odd 4 425.2.c.b.424.5 6
85.84 even 2 7225.2.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.5 ✓ 6 17.4 even 4
85.2.d.a.16.6 yes 6 17.13 even 4
425.2.c.a.424.2 6 85.38 odd 4
425.2.c.a.424.5 6 85.47 odd 4
425.2.c.b.424.2 6 85.13 odd 4
425.2.c.b.424.5 6 85.72 odd 4
425.2.d.c.101.1 6 85.64 even 4
425.2.d.c.101.2 6 85.4 even 4
765.2.g.b.271.1 6 51.38 odd 4
765.2.g.b.271.2 6 51.47 odd 4
1360.2.c.f.1121.2 6 68.47 odd 4
1360.2.c.f.1121.5 6 68.55 odd 4
1445.2.a.j.1.1 3 1.1 even 1 trivial
1445.2.a.k.1.1 3 17.16 even 2
7225.2.a.q.1.3 3 5.4 even 2
7225.2.a.r.1.3 3 85.84 even 2