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Properties

Label 1045.2.b.d.419.11
Level $1045$
Weight $2$
Character 1045.419
Analytic conductor $8.344$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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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] = [1045,2,Mod(419,1045)] 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("1045.419"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1045, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 419.11
Character \(\chi\) \(=\) 1045.419
Dual form 1045.2.b.d.419.12

$q$-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)
 
\(f(q)\) \(=\) \(q-0.104144i q^{2} +0.696995i q^{3} +1.98915 q^{4} +(2.09312 - 0.786662i) q^{5} +0.0725876 q^{6} +3.21657i q^{7} -0.415445i q^{8} +2.51420 q^{9} +(-0.0819259 - 0.217986i) q^{10} -1.00000 q^{11} +1.38643i q^{12} -4.65405i q^{13} +0.334986 q^{14} +(0.548300 + 1.45890i) q^{15} +3.93504 q^{16} -0.00602394i q^{17} -0.261838i q^{18} +1.00000 q^{19} +(4.16354 - 1.56479i) q^{20} -2.24194 q^{21} +0.104144i q^{22} +1.65858i q^{23} +0.289563 q^{24} +(3.76233 - 3.29316i) q^{25} -0.484690 q^{26} +3.84337i q^{27} +6.39826i q^{28} -3.49988 q^{29} +(0.151935 - 0.0571020i) q^{30} -3.34225 q^{31} -1.24070i q^{32} -0.696995i q^{33} -0.000627356 q^{34} +(2.53036 + 6.73268i) q^{35} +5.00113 q^{36} +6.13742i q^{37} -0.104144i q^{38} +3.24385 q^{39} +(-0.326815 - 0.869578i) q^{40} -9.38032 q^{41} +0.233483i q^{42} -8.61740i q^{43} -1.98915 q^{44} +(5.26252 - 1.97782i) q^{45} +0.172731 q^{46} +0.893231i q^{47} +2.74270i q^{48} -3.34635 q^{49} +(-0.342962 - 0.391822i) q^{50} +0.00419866 q^{51} -9.25763i q^{52} -1.78431i q^{53} +0.400263 q^{54} +(-2.09312 + 0.786662i) q^{55} +1.33631 q^{56} +0.696995i q^{57} +0.364491i q^{58} -2.57022 q^{59} +(1.09065 + 2.90197i) q^{60} -3.68861 q^{61} +0.348074i q^{62} +8.08710i q^{63} +7.74087 q^{64} +(-3.66117 - 9.74151i) q^{65} -0.0725876 q^{66} +5.98644i q^{67} -0.0119826i q^{68} -1.15602 q^{69} +(0.701167 - 0.263521i) q^{70} +0.852858 q^{71} -1.04451i q^{72} -4.72786i q^{73} +0.639174 q^{74} +(2.29532 + 2.62232i) q^{75} +1.98915 q^{76} -3.21657i q^{77} -0.337827i q^{78} +4.94098 q^{79} +(8.23653 - 3.09555i) q^{80} +4.86379 q^{81} +0.976902i q^{82} +15.0339i q^{83} -4.45956 q^{84} +(-0.00473881 - 0.0126089i) q^{85} -0.897448 q^{86} -2.43940i q^{87} +0.415445i q^{88} -5.11449 q^{89} +(-0.205978 - 0.548059i) q^{90} +14.9701 q^{91} +3.29917i q^{92} -2.32953i q^{93} +0.0930244 q^{94} +(2.09312 - 0.786662i) q^{95} +0.864762 q^{96} -16.5462i q^{97} +0.348501i q^{98} -2.51420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 32 q^{4} + 7 q^{5} - 12 q^{6} - 34 q^{9} + 2 q^{10} - 22 q^{11} + 8 q^{14} - 23 q^{15} + 40 q^{16} + 22 q^{19} - 22 q^{20} - 22 q^{21} + 22 q^{24} + 13 q^{25} + 16 q^{26} + 10 q^{29} - 22 q^{30}+ \cdots + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.104144i − 0.0736407i −0.999322 − 0.0368204i \(-0.988277\pi\)
0.999322 − 0.0368204i \(-0.0117229\pi\)
\(3\) 0.696995i 0.402410i 0.979549 + 0.201205i \(0.0644858\pi\)
−0.979549 + 0.201205i \(0.935514\pi\)
\(4\) 1.98915 0.994577
\(5\) 2.09312 − 0.786662i 0.936073 − 0.351806i
\(6\) 0.0725876 0.0296338
\(7\) 3.21657i 1.21575i 0.794033 + 0.607875i \(0.207978\pi\)
−0.794033 + 0.607875i \(0.792022\pi\)
\(8\) − 0.415445i − 0.146882i
\(9\) 2.51420 0.838066
\(10\) −0.0819259 − 0.217986i −0.0259072 − 0.0689331i
\(11\) −1.00000 −0.301511
\(12\) 1.38643i 0.400228i
\(13\) − 4.65405i − 1.29080i −0.763844 − 0.645401i \(-0.776690\pi\)
0.763844 − 0.645401i \(-0.223310\pi\)
\(14\) 0.334986 0.0895287
\(15\) 0.548300 + 1.45890i 0.141570 + 0.376685i
\(16\) 3.93504 0.983761
\(17\) − 0.00602394i − 0.00146102i −1.00000 0.000730510i \(-0.999767\pi\)
1.00000 0.000730510i \(-0.000232529\pi\)
\(18\) − 0.261838i − 0.0617158i
\(19\) 1.00000 0.229416
\(20\) 4.16354 − 1.56479i 0.930997 − 0.349898i
\(21\) −2.24194 −0.489230
\(22\) 0.104144i 0.0222035i
\(23\) 1.65858i 0.345838i 0.984936 + 0.172919i \(0.0553198\pi\)
−0.984936 + 0.172919i \(0.944680\pi\)
\(24\) 0.289563 0.0591069
\(25\) 3.76233 − 3.29316i 0.752465 − 0.658632i
\(26\) −0.484690 −0.0950556
\(27\) 3.84337i 0.739657i
\(28\) 6.39826i 1.20916i
\(29\) −3.49988 −0.649912 −0.324956 − 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(30\) 0.151935 − 0.0571020i 0.0277394 − 0.0104253i
\(31\) −3.34225 −0.600285 −0.300143 − 0.953894i \(-0.597034\pi\)
−0.300143 + 0.953894i \(0.597034\pi\)
\(32\) − 1.24070i − 0.219327i
\(33\) − 0.696995i − 0.121331i
\(34\) −0.000627356 0 −0.000107591 0
\(35\) 2.53036 + 6.73268i 0.427708 + 1.13803i
\(36\) 5.00113 0.833521
\(37\) 6.13742i 1.00899i 0.863416 + 0.504493i \(0.168320\pi\)
−0.863416 + 0.504493i \(0.831680\pi\)
\(38\) − 0.104144i − 0.0168943i
\(39\) 3.24385 0.519432
\(40\) −0.326815 − 0.869578i −0.0516740 − 0.137492i
\(41\) −9.38032 −1.46496 −0.732480 − 0.680788i \(-0.761637\pi\)
−0.732480 + 0.680788i \(0.761637\pi\)
\(42\) 0.233483i 0.0360273i
\(43\) − 8.61740i − 1.31414i −0.753829 − 0.657071i \(-0.771795\pi\)
0.753829 − 0.657071i \(-0.228205\pi\)
\(44\) −1.98915 −0.299876
\(45\) 5.26252 − 1.97782i 0.784491 − 0.294837i
\(46\) 0.172731 0.0254677
\(47\) 0.893231i 0.130291i 0.997876 + 0.0651456i \(0.0207512\pi\)
−0.997876 + 0.0651456i \(0.979249\pi\)
\(48\) 2.74270i 0.395875i
\(49\) −3.34635 −0.478049
\(50\) −0.342962 − 0.391822i −0.0485021 − 0.0554121i
\(51\) 0.00419866 0.000587930
\(52\) − 9.25763i − 1.28380i
\(53\) − 1.78431i − 0.245093i −0.992463 − 0.122547i \(-0.960894\pi\)
0.992463 − 0.122547i \(-0.0391061\pi\)
\(54\) 0.400263 0.0544688
\(55\) −2.09312 + 0.786662i −0.282237 + 0.106074i
\(56\) 1.33631 0.178572
\(57\) 0.696995i 0.0923192i
\(58\) 0.364491i 0.0478600i
\(59\) −2.57022 −0.334614 −0.167307 − 0.985905i \(-0.553507\pi\)
−0.167307 + 0.985905i \(0.553507\pi\)
\(60\) 1.09065 + 2.90197i 0.140803 + 0.374643i
\(61\) −3.68861 −0.472278 −0.236139 − 0.971719i \(-0.575882\pi\)
−0.236139 + 0.971719i \(0.575882\pi\)
\(62\) 0.348074i 0.0442054i
\(63\) 8.08710i 1.01888i
\(64\) 7.74087 0.967609
\(65\) −3.66117 − 9.74151i −0.454112 − 1.20829i
\(66\) −0.0725876 −0.00893492
\(67\) 5.98644i 0.731361i 0.930741 + 0.365680i \(0.119164\pi\)
−0.930741 + 0.365680i \(0.880836\pi\)
\(68\) − 0.0119826i − 0.00145310i
\(69\) −1.15602 −0.139169
\(70\) 0.701167 − 0.263521i 0.0838054 − 0.0314968i
\(71\) 0.852858 0.101216 0.0506078 − 0.998719i \(-0.483884\pi\)
0.0506078 + 0.998719i \(0.483884\pi\)
\(72\) − 1.04451i − 0.123097i
\(73\) − 4.72786i − 0.553354i −0.960963 − 0.276677i \(-0.910767\pi\)
0.960963 − 0.276677i \(-0.0892331\pi\)
\(74\) 0.639174 0.0743024
\(75\) 2.29532 + 2.62232i 0.265040 + 0.302800i
\(76\) 1.98915 0.228172
\(77\) − 3.21657i − 0.366563i
\(78\) − 0.337827i − 0.0382513i
\(79\) 4.94098 0.555903 0.277952 − 0.960595i \(-0.410344\pi\)
0.277952 + 0.960595i \(0.410344\pi\)
\(80\) 8.23653 − 3.09555i 0.920872 − 0.346093i
\(81\) 4.86379 0.540421
\(82\) 0.976902i 0.107881i
\(83\) 15.0339i 1.65019i 0.564997 + 0.825093i \(0.308877\pi\)
−0.564997 + 0.825093i \(0.691123\pi\)
\(84\) −4.45956 −0.486577
\(85\) −0.00473881 − 0.0126089i −0.000513996 − 0.00136762i
\(86\) −0.897448 −0.0967743
\(87\) − 2.43940i − 0.261531i
\(88\) 0.415445i 0.0442866i
\(89\) −5.11449 −0.542135 −0.271067 − 0.962560i \(-0.587377\pi\)
−0.271067 + 0.962560i \(0.587377\pi\)
\(90\) −0.205978 − 0.548059i −0.0217120 − 0.0577705i
\(91\) 14.9701 1.56929
\(92\) 3.29917i 0.343962i
\(93\) − 2.32953i − 0.241561i
\(94\) 0.0930244 0.00959474
\(95\) 2.09312 − 0.786662i 0.214750 − 0.0807098i
\(96\) 0.864762 0.0882594
\(97\) − 16.5462i − 1.68001i −0.542578 − 0.840005i \(-0.682552\pi\)
0.542578 − 0.840005i \(-0.317448\pi\)
\(98\) 0.348501i 0.0352039i
\(99\) −2.51420 −0.252686
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 1045.2.b.d.419.11 ✓ 22
5.2 odd 4 5225.2.a.bb.1.12 22
5.3 odd 4 5225.2.a.bb.1.11 22
5.4 even 2 inner 1045.2.b.d.419.12 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.d.419.11 ✓ 22 1.1 even 1 trivial
1045.2.b.d.419.12 yes 22 5.4 even 2 inner
5225.2.a.bb.1.11 22 5.3 odd 4
5225.2.a.bb.1.12 22 5.2 odd 4