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Properties

Label 1445.2.a.k.1.1
Level $1445$
Weight $2$
Character 1445.1
Self dual yes
Analytic conductor $11.538$
Analytic rank $0$
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: \(0\)
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 \(-0.660660\pi\)
−0.483569 + 0.875306i \(0.660660\pi\)
\(4\) 0.193937 0.0969683
\(5\) 1.00000 0.447214
\(6\) 2.48119 1.01294
\(7\) −1.28726 −0.486538 −0.243269 − 0.969959i \(-0.578220\pi\)
−0.243269 + 0.969959i \(0.578220\pi\)
\(8\) 2.67513 0.945802
\(9\) −0.193937 −0.0646455
\(10\) −1.48119 −0.468395
\(11\) −0.481194 −0.145086 −0.0725428 − 0.997365i \(-0.523111\pi\)
−0.0725428 + 0.997365i \(0.523111\pi\)
\(12\) −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 \(-0.170410\pi\)
0.860085 + 0.510151i \(0.170410\pi\)
\(24\) −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 \(-0.519214\pi\)
−0.0603267 + 0.998179i \(0.519214\pi\)
\(30\) 2.48119 0.453002
\(31\) −1.83146 −0.328939 −0.164470 − 0.986382i \(-0.552591\pi\)
−0.164470 + 0.986382i \(0.552591\pi\)
\(32\) 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 \(-0.615348\pi\)
−0.354498 + 0.935057i \(0.615348\pi\)
\(38\) 4.96239 0.805006
\(39\) 3.61213 0.578403
\(40\) 2.67513 0.422975
\(41\) −11.2750 −1.76087 −0.880433 − 0.474171i \(-0.842748\pi\)
−0.880433 + 0.474171i \(0.842748\pi\)
\(42\) −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 \(-0.441879\pi\)
0.181581 + 0.983376i \(0.441879\pi\)
\(62\) 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 \(-0.169014\pi\)
0.862314 + 0.506373i \(0.169014\pi\)
\(72\) −0.518806 −0.0611418
\(73\) 13.3503 1.56253 0.781265 − 0.624200i \(-0.214575\pi\)
0.781265 + 0.624200i \(0.214575\pi\)
\(74\) 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 \(-0.329859\pi\)
0.509423 + 0.860516i \(0.329859\pi\)
\(80\) −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 \(-0.440462\pi\)
0.185956 + 0.982558i \(0.440462\pi\)
\(98\) 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.k.1.1 3
5.4 even 2 7225.2.a.r.1.3 3
17.4 even 4 85.2.d.a.16.6 yes 6
17.13 even 4 85.2.d.a.16.5 ✓ 6
17.16 even 2 1445.2.a.j.1.1 3
51.38 odd 4 765.2.g.b.271.2 6
51.47 odd 4 765.2.g.b.271.1 6
68.47 odd 4 1360.2.c.f.1121.5 6
68.55 odd 4 1360.2.c.f.1121.2 6
85.4 even 4 425.2.d.c.101.1 6
85.13 odd 4 425.2.c.a.424.2 6
85.38 odd 4 425.2.c.b.424.2 6
85.47 odd 4 425.2.c.b.424.5 6
85.64 even 4 425.2.d.c.101.2 6
85.72 odd 4 425.2.c.a.424.5 6
85.84 even 2 7225.2.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.5 ✓ 6 17.13 even 4
85.2.d.a.16.6 yes 6 17.4 even 4
425.2.c.a.424.2 6 85.13 odd 4
425.2.c.a.424.5 6 85.72 odd 4
425.2.c.b.424.2 6 85.38 odd 4
425.2.c.b.424.5 6 85.47 odd 4
425.2.d.c.101.1 6 85.4 even 4
425.2.d.c.101.2 6 85.64 even 4
765.2.g.b.271.1 6 51.47 odd 4
765.2.g.b.271.2 6 51.38 odd 4
1360.2.c.f.1121.2 6 68.55 odd 4
1360.2.c.f.1121.5 6 68.47 odd 4
1445.2.a.j.1.1 3 17.16 even 2
1445.2.a.k.1.1 3 1.1 even 1 trivial
7225.2.a.q.1.3 3 85.84 even 2
7225.2.a.r.1.3 3 5.4 even 2