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Properties

Label 5225.2.a.bb.1.11
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
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] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,32,0,-12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 5225.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.104144 q^{2} -0.696995 q^{3} -1.98915 q^{4} +0.0725876 q^{6} +3.21657 q^{7} +0.415445 q^{8} -2.51420 q^{9} -1.00000 q^{11} +1.38643 q^{12} +4.65405 q^{13} -0.334986 q^{14} +3.93504 q^{16} -0.00602394 q^{17} +0.261838 q^{18} -1.00000 q^{19} -2.24194 q^{21} +0.104144 q^{22} -1.65858 q^{23} -0.289563 q^{24} -0.484690 q^{26} +3.84337 q^{27} -6.39826 q^{28} +3.49988 q^{29} -3.34225 q^{31} -1.24070 q^{32} +0.696995 q^{33} +0.000627356 q^{34} +5.00113 q^{36} +6.13742 q^{37} +0.104144 q^{38} -3.24385 q^{39} -9.38032 q^{41} +0.233483 q^{42} +8.61740 q^{43} +1.98915 q^{44} +0.172731 q^{46} +0.893231 q^{47} -2.74270 q^{48} +3.34635 q^{49} +0.00419866 q^{51} -9.25763 q^{52} +1.78431 q^{53} -0.400263 q^{54} +1.33631 q^{56} +0.696995 q^{57} -0.364491 q^{58} +2.57022 q^{59} -3.68861 q^{61} +0.348074 q^{62} -8.08710 q^{63} -7.74087 q^{64} -0.0725876 q^{66} +5.98644 q^{67} +0.0119826 q^{68} +1.15602 q^{69} +0.852858 q^{71} -1.04451 q^{72} +4.72786 q^{73} -0.639174 q^{74} +1.98915 q^{76} -3.21657 q^{77} +0.337827 q^{78} -4.94098 q^{79} +4.86379 q^{81} +0.976902 q^{82} -15.0339 q^{83} +4.45956 q^{84} -0.897448 q^{86} -2.43940 q^{87} -0.415445 q^{88} +5.11449 q^{89} +14.9701 q^{91} +3.29917 q^{92} +2.32953 q^{93} -0.0930244 q^{94} +0.864762 q^{96} -16.5462 q^{97} -0.348501 q^{98} +2.51420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44}+ \cdots - 34 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.104144 −0.0736407 −0.0368204 − 0.999322i \(-0.511723\pi\)
−0.0368204 + 0.999322i \(0.511723\pi\)
\(3\) −0.696995 −0.402410 −0.201205 − 0.979549i \(-0.564486\pi\)
−0.201205 + 0.979549i \(0.564486\pi\)
\(4\) −1.98915 −0.994577
\(5\) 0 0
\(6\) 0.0725876 0.0296338
\(7\) 3.21657 1.21575 0.607875 − 0.794033i \(-0.292022\pi\)
0.607875 + 0.794033i \(0.292022\pi\)
\(8\) 0.415445 0.146882
\(9\) −2.51420 −0.838066
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.38643 0.400228
\(13\) 4.65405 1.29080 0.645401 − 0.763844i \(-0.276690\pi\)
0.645401 + 0.763844i \(0.276690\pi\)
\(14\) −0.334986 −0.0895287
\(15\) 0 0
\(16\) 3.93504 0.983761
\(17\) −0.00602394 −0.00146102 −0.000730510 − 1.00000i \(-0.500233\pi\)
−0.000730510 1.00000i \(0.500233\pi\)
\(18\) 0.261838 0.0617158
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.24194 −0.489230
\(22\) 0.104144 0.0222035
\(23\) −1.65858 −0.345838 −0.172919 − 0.984936i \(-0.555320\pi\)
−0.172919 + 0.984936i \(0.555320\pi\)
\(24\) −0.289563 −0.0591069
\(25\) 0 0
\(26\) −0.484690 −0.0950556
\(27\) 3.84337 0.739657
\(28\) −6.39826 −1.20916
\(29\) 3.49988 0.649912 0.324956 − 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(30\) 0 0
\(31\) −3.34225 −0.600285 −0.300143 − 0.953894i \(-0.597034\pi\)
−0.300143 + 0.953894i \(0.597034\pi\)
\(32\) −1.24070 −0.219327
\(33\) 0.696995 0.121331
\(34\) 0.000627356 0 0.000107591 0
\(35\) 0 0
\(36\) 5.00113 0.833521
\(37\) 6.13742 1.00899 0.504493 − 0.863416i \(-0.331680\pi\)
0.504493 + 0.863416i \(0.331680\pi\)
\(38\) 0.104144 0.0168943
\(39\) −3.24385 −0.519432
\(40\) 0 0
\(41\) −9.38032 −1.46496 −0.732480 − 0.680788i \(-0.761637\pi\)
−0.732480 + 0.680788i \(0.761637\pi\)
\(42\) 0.233483 0.0360273
\(43\) 8.61740 1.31414 0.657071 − 0.753829i \(-0.271795\pi\)
0.657071 + 0.753829i \(0.271795\pi\)
\(44\) 1.98915 0.299876
\(45\) 0 0
\(46\) 0.172731 0.0254677
\(47\) 0.893231 0.130291 0.0651456 − 0.997876i \(-0.479249\pi\)
0.0651456 + 0.997876i \(0.479249\pi\)
\(48\) −2.74270 −0.395875
\(49\) 3.34635 0.478049
\(50\) 0 0
\(51\) 0.00419866 0.000587930 0
\(52\) −9.25763 −1.28380
\(53\) 1.78431 0.245093 0.122547 − 0.992463i \(-0.460894\pi\)
0.122547 + 0.992463i \(0.460894\pi\)
\(54\) −0.400263 −0.0544688
\(55\) 0 0
\(56\) 1.33631 0.178572
\(57\) 0.696995 0.0923192
\(58\) −0.364491 −0.0478600
\(59\) 2.57022 0.334614 0.167307 − 0.985905i \(-0.446493\pi\)
0.167307 + 0.985905i \(0.446493\pi\)
\(60\) 0 0
\(61\) −3.68861 −0.472278 −0.236139 − 0.971719i \(-0.575882\pi\)
−0.236139 + 0.971719i \(0.575882\pi\)
\(62\) 0.348074 0.0442054
\(63\) −8.08710 −1.01888
\(64\) −7.74087 −0.967609
\(65\) 0 0
\(66\) −0.0725876 −0.00893492
\(67\) 5.98644 0.731361 0.365680 − 0.930741i \(-0.380836\pi\)
0.365680 + 0.930741i \(0.380836\pi\)
\(68\) 0.0119826 0.00145310
\(69\) 1.15602 0.139169
\(70\) 0 0
\(71\) 0.852858 0.101216 0.0506078 − 0.998719i \(-0.483884\pi\)
0.0506078 + 0.998719i \(0.483884\pi\)
\(72\) −1.04451 −0.123097
\(73\) 4.72786 0.553354 0.276677 − 0.960963i \(-0.410767\pi\)
0.276677 + 0.960963i \(0.410767\pi\)
\(74\) −0.639174 −0.0743024
\(75\) 0 0
\(76\) 1.98915 0.228172
\(77\) −3.21657 −0.366563
\(78\) 0.337827 0.0382513
\(79\) −4.94098 −0.555903 −0.277952 − 0.960595i \(-0.589656\pi\)
−0.277952 + 0.960595i \(0.589656\pi\)
\(80\) 0 0
\(81\) 4.86379 0.540421
\(82\) 0.976902 0.107881
\(83\) −15.0339 −1.65019 −0.825093 − 0.564997i \(-0.808877\pi\)
−0.825093 + 0.564997i \(0.808877\pi\)
\(84\) 4.45956 0.486577
\(85\) 0 0
\(86\) −0.897448 −0.0967743
\(87\) −2.43940 −0.261531
\(88\) −0.415445 −0.0442866
\(89\) 5.11449 0.542135 0.271067 − 0.962560i \(-0.412623\pi\)
0.271067 + 0.962560i \(0.412623\pi\)
\(90\) 0 0
\(91\) 14.9701 1.56929
\(92\) 3.29917 0.343962
\(93\) 2.32953 0.241561
\(94\) −0.0930244 −0.00959474
\(95\) 0 0
\(96\) 0.864762 0.0882594
\(97\) −16.5462 −1.68001 −0.840005 − 0.542578i \(-0.817448\pi\)
−0.840005 + 0.542578i \(0.817448\pi\)
\(98\) −0.348501 −0.0352039
\(99\) 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 5225.2.a.bb.1.11 22
5.2 odd 4 1045.2.b.d.419.11 ✓ 22
5.3 odd 4 1045.2.b.d.419.12 yes 22
5.4 even 2 inner 5225.2.a.bb.1.12 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.d.419.11 ✓ 22 5.2 odd 4
1045.2.b.d.419.12 yes 22 5.3 odd 4
5225.2.a.bb.1.11 22 1.1 even 1 trivial
5225.2.a.bb.1.12 22 5.4 even 2 inner