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Properties

Label 123.4.a.d.1.3
Level $123$
Weight $4$
Character 123.1
Self dual yes
Analytic conductor $7.257$
Analytic rank $0$
Dimension $6$
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] = [123,4,Mod(1,123)] 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("123.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(123, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 123.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.25723493071\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 34x^{4} - 26x^{3} + 269x^{2} + 258x - 272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.657015\) of defining polynomial
Character \(\chi\) \(=\) 123.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.342985 q^{2} +3.00000 q^{3} -7.88236 q^{4} +9.86667 q^{5} +1.02895 q^{6} +16.4245 q^{7} -5.44740 q^{8} +9.00000 q^{9} +3.38412 q^{10} -18.0471 q^{11} -23.6471 q^{12} +63.3713 q^{13} +5.63336 q^{14} +29.6000 q^{15} +61.1905 q^{16} +104.257 q^{17} +3.08686 q^{18} -14.7398 q^{19} -77.7727 q^{20} +49.2735 q^{21} -6.18986 q^{22} -47.3678 q^{23} -16.3422 q^{24} -27.6488 q^{25} +21.7354 q^{26} +27.0000 q^{27} -129.464 q^{28} +184.723 q^{29} +10.1523 q^{30} -61.7945 q^{31} +64.5666 q^{32} -54.1412 q^{33} +35.7585 q^{34} +162.055 q^{35} -70.9413 q^{36} -289.624 q^{37} -5.05552 q^{38} +190.114 q^{39} -53.7478 q^{40} +41.0000 q^{41} +16.9001 q^{42} -205.988 q^{43} +142.253 q^{44} +88.8000 q^{45} -16.2464 q^{46} -117.299 q^{47} +183.572 q^{48} -73.2353 q^{49} -9.48311 q^{50} +312.771 q^{51} -499.516 q^{52} -274.992 q^{53} +9.26058 q^{54} -178.064 q^{55} -89.4710 q^{56} -44.2193 q^{57} +63.3570 q^{58} -29.0373 q^{59} -233.318 q^{60} -704.938 q^{61} -21.1946 q^{62} +147.821 q^{63} -467.379 q^{64} +625.264 q^{65} -18.5696 q^{66} +59.1870 q^{67} -821.791 q^{68} -142.104 q^{69} +55.5825 q^{70} -391.749 q^{71} -49.0266 q^{72} +412.828 q^{73} -99.3365 q^{74} -82.9463 q^{75} +116.184 q^{76} -296.414 q^{77} +65.2061 q^{78} +305.287 q^{79} +603.747 q^{80} +81.0000 q^{81} +14.0624 q^{82} -66.1863 q^{83} -388.392 q^{84} +1028.67 q^{85} -70.6508 q^{86} +554.168 q^{87} +98.3096 q^{88} +808.319 q^{89} +30.4570 q^{90} +1040.84 q^{91} +373.370 q^{92} -185.384 q^{93} -40.2317 q^{94} -145.433 q^{95} +193.700 q^{96} -1305.29 q^{97} -25.1186 q^{98} -162.424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 18 q^{3} + 26 q^{4} + 37 q^{5} + 18 q^{6} + 14 q^{7} + 36 q^{8} + 54 q^{9} + 47 q^{10} + 50 q^{11} + 78 q^{12} + 27 q^{13} - 34 q^{14} + 111 q^{15} - 54 q^{16} + 43 q^{17} + 54 q^{18} + 111 q^{19}+ \cdots + 450 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.342985 0.121263 0.0606317 − 0.998160i \(-0.480688\pi\)
0.0606317 + 0.998160i \(0.480688\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.88236 −0.985295
\(5\) 9.86667 0.882502 0.441251 − 0.897384i \(-0.354535\pi\)
0.441251 + 0.897384i \(0.354535\pi\)
\(6\) 1.02895 0.0700114
\(7\) 16.4245 0.886840 0.443420 − 0.896314i \(-0.353765\pi\)
0.443420 + 0.896314i \(0.353765\pi\)
\(8\) −5.44740 −0.240744
\(9\) 9.00000 0.333333
\(10\) 3.38412 0.107015
\(11\) −18.0471 −0.494672 −0.247336 − 0.968930i \(-0.579555\pi\)
−0.247336 + 0.968930i \(0.579555\pi\)
\(12\) −23.6471 −0.568860
\(13\) 63.3713 1.35200 0.676002 − 0.736900i \(-0.263711\pi\)
0.676002 + 0.736900i \(0.263711\pi\)
\(14\) 5.63336 0.107541
\(15\) 29.6000 0.509513
\(16\) 61.1905 0.956102
\(17\) 104.257 1.48741 0.743707 − 0.668506i \(-0.233066\pi\)
0.743707 + 0.668506i \(0.233066\pi\)
\(18\) 3.08686 0.0404211
\(19\) −14.7398 −0.177976 −0.0889878 − 0.996033i \(-0.528363\pi\)
−0.0889878 + 0.996033i \(0.528363\pi\)
\(20\) −77.7727 −0.869525
\(21\) 49.2735 0.512018
\(22\) −6.18986 −0.0599856
\(23\) −47.3678 −0.429429 −0.214715 − 0.976677i \(-0.568882\pi\)
−0.214715 + 0.976677i \(0.568882\pi\)
\(24\) −16.3422 −0.138993
\(25\) −27.6488 −0.221190
\(26\) 21.7354 0.163948
\(27\) 27.0000 0.192450
\(28\) −129.464 −0.873800
\(29\) 184.723 1.18283 0.591416 − 0.806367i \(-0.298569\pi\)
0.591416 + 0.806367i \(0.298569\pi\)
\(30\) 10.1523 0.0617852
\(31\) −61.7945 −0.358020 −0.179010 − 0.983847i \(-0.557289\pi\)
−0.179010 + 0.983847i \(0.557289\pi\)
\(32\) 64.5666 0.356684
\(33\) −54.1412 −0.285599
\(34\) 35.7585 0.180369
\(35\) 162.055 0.782638
\(36\) −70.9413 −0.328432
\(37\) −289.624 −1.28686 −0.643431 − 0.765504i \(-0.722490\pi\)
−0.643431 + 0.765504i \(0.722490\pi\)
\(38\) −5.05552 −0.0215819
\(39\) 190.114 0.780579
\(40\) −53.7478 −0.212457
\(41\) 41.0000 0.156174
\(42\) 16.9001 0.0620890
\(43\) −205.988 −0.730533 −0.365266 − 0.930903i \(-0.619022\pi\)
−0.365266 + 0.930903i \(0.619022\pi\)
\(44\) 142.253 0.487398
\(45\) 88.8000 0.294167
\(46\) −16.2464 −0.0520741
\(47\) −117.299 −0.364038 −0.182019 − 0.983295i \(-0.558263\pi\)
−0.182019 + 0.983295i \(0.558263\pi\)
\(48\) 183.572 0.552006
\(49\) −73.2353 −0.213514
\(50\) −9.48311 −0.0268223
\(51\) 312.771 0.858759
\(52\) −499.516 −1.33212
\(53\) −274.992 −0.712699 −0.356349 − 0.934353i \(-0.615979\pi\)
−0.356349 + 0.934353i \(0.615979\pi\)
\(54\) 9.26058 0.0233371
\(55\) −178.064 −0.436549
\(56\) −89.4710 −0.213501
\(57\) −44.2193 −0.102754
\(58\) 63.3570 0.143434
\(59\) −29.0373 −0.0640735 −0.0320367 − 0.999487i \(-0.510199\pi\)
−0.0320367 + 0.999487i \(0.510199\pi\)
\(60\) −233.318 −0.502020
\(61\) −704.938 −1.47964 −0.739820 − 0.672805i \(-0.765089\pi\)
−0.739820 + 0.672805i \(0.765089\pi\)
\(62\) −21.1946 −0.0434147
\(63\) 147.821 0.295613
\(64\) −467.379 −0.912849
\(65\) 625.264 1.19315
\(66\) −18.5696 −0.0346327
\(67\) 59.1870 0.107923 0.0539615 − 0.998543i \(-0.482815\pi\)
0.0539615 + 0.998543i \(0.482815\pi\)
\(68\) −821.791 −1.46554
\(69\) −142.104 −0.247931
\(70\) 55.5825 0.0949054
\(71\) −391.749 −0.654818 −0.327409 − 0.944883i \(-0.606175\pi\)
−0.327409 + 0.944883i \(0.606175\pi\)
\(72\) −49.0266 −0.0802479
\(73\) 412.828 0.661888 0.330944 − 0.943650i \(-0.392633\pi\)
0.330944 + 0.943650i \(0.392633\pi\)
\(74\) −99.3365 −0.156049
\(75\) −82.9463 −0.127704
\(76\) 116.184 0.175359
\(77\) −296.414 −0.438695
\(78\) 65.2061 0.0946557
\(79\) 305.287 0.434778 0.217389 − 0.976085i \(-0.430246\pi\)
0.217389 + 0.976085i \(0.430246\pi\)
\(80\) 603.747 0.843762
\(81\) 81.0000 0.111111
\(82\) 14.0624 0.0189382
\(83\) −66.1863 −0.0875288 −0.0437644 − 0.999042i \(-0.513935\pi\)
−0.0437644 + 0.999042i \(0.513935\pi\)
\(84\) −388.392 −0.504488
\(85\) 1028.67 1.31265
\(86\) −70.6508 −0.0885868
\(87\) 554.168 0.682908
\(88\) 98.3096 0.119089
\(89\) 808.319 0.962715 0.481357 − 0.876524i \(-0.340144\pi\)
0.481357 + 0.876524i \(0.340144\pi\)
\(90\) 30.4570 0.0356717
\(91\) 1040.84 1.19901
\(92\) 373.370 0.423115
\(93\) −185.384 −0.206703
\(94\) −40.2317 −0.0441445
\(95\) −145.433 −0.157064
\(96\) 193.700 0.205931
\(97\) −1305.29 −1.36631 −0.683154 − 0.730274i \(-0.739392\pi\)
−0.683154 + 0.730274i \(0.739392\pi\)
\(98\) −25.1186 −0.0258914
\(99\) −162.424 −0.164891
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 123.4.a.d.1.3 ✓ 6
3.2 odd 2 369.4.a.f.1.4 6
4.3 odd 2 1968.4.a.s.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
123.4.a.d.1.3 ✓ 6 1.1 even 1 trivial
369.4.a.f.1.4 6 3.2 odd 2
1968.4.a.s.1.4 6 4.3 odd 2