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Properties

Label 578.6.a.d.1.2
Level $578$
Weight $6$
Character 578.1
Self dual yes
Analytic conductor $92.702$
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] = [578,6,Mod(1,578)] 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("578.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 578.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,12,-4,48,-144] 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: \(92.7018478519\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1505580.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 200x - 480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 34)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-12.1827\) of defining polynomial
Character \(\chi\) \(=\) 578.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -8.14993 q^{3} +16.0000 q^{4} -9.33483 q^{5} -32.5997 q^{6} -162.407 q^{7} +64.0000 q^{8} -176.579 q^{9} -37.3393 q^{10} -297.264 q^{11} -130.399 q^{12} -155.719 q^{13} -649.629 q^{14} +76.0782 q^{15} +256.000 q^{16} -706.315 q^{18} -1777.45 q^{19} -149.357 q^{20} +1323.61 q^{21} -1189.06 q^{22} +384.489 q^{23} -521.596 q^{24} -3037.86 q^{25} -622.874 q^{26} +3419.54 q^{27} -2598.52 q^{28} +5373.87 q^{29} +304.313 q^{30} +1540.65 q^{31} +1024.00 q^{32} +2422.68 q^{33} +1516.04 q^{35} -2825.26 q^{36} -7231.11 q^{37} -7109.81 q^{38} +1269.10 q^{39} -597.429 q^{40} -5943.04 q^{41} +5294.43 q^{42} +9935.17 q^{43} -4756.23 q^{44} +1648.33 q^{45} +1537.95 q^{46} +15112.3 q^{47} -2086.38 q^{48} +9569.11 q^{49} -12151.4 q^{50} -2491.50 q^{52} +34197.6 q^{53} +13678.1 q^{54} +2774.91 q^{55} -10394.1 q^{56} +14486.1 q^{57} +21495.5 q^{58} -20721.4 q^{59} +1217.25 q^{60} -6940.24 q^{61} +6162.58 q^{62} +28677.6 q^{63} +4096.00 q^{64} +1453.61 q^{65} +9690.73 q^{66} -55729.7 q^{67} -3133.56 q^{69} +6064.18 q^{70} +49749.9 q^{71} -11301.0 q^{72} -14627.6 q^{73} -28924.4 q^{74} +24758.4 q^{75} -28439.2 q^{76} +48277.9 q^{77} +5076.38 q^{78} -17017.6 q^{79} -2389.72 q^{80} +15039.6 q^{81} -23772.2 q^{82} +51344.2 q^{83} +21177.7 q^{84} +39740.7 q^{86} -43796.6 q^{87} -19024.9 q^{88} +113166. q^{89} +6593.33 q^{90} +25289.8 q^{91} +6151.82 q^{92} -12556.2 q^{93} +60449.1 q^{94} +16592.2 q^{95} -8345.53 q^{96} +36048.5 q^{97} +38276.4 q^{98} +52490.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} - 4 q^{3} + 48 q^{4} - 144 q^{5} - 16 q^{6} + 18 q^{7} + 192 q^{8} + 883 q^{9} - 576 q^{10} + 180 q^{11} - 64 q^{12} + 318 q^{13} + 72 q^{14} - 2268 q^{15} + 768 q^{16} + 3532 q^{18} + 1848 q^{19}+ \cdots + 253080 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −8.14993 −0.522818 −0.261409 − 0.965228i \(-0.584187\pi\)
−0.261409 + 0.965228i \(0.584187\pi\)
\(4\) 16.0000 0.500000
\(5\) −9.33483 −0.166987 −0.0834933 − 0.996508i \(-0.526608\pi\)
−0.0834933 + 0.996508i \(0.526608\pi\)
\(6\) −32.5997 −0.369688
\(7\) −162.407 −1.25274 −0.626369 − 0.779527i \(-0.715460\pi\)
−0.626369 + 0.779527i \(0.715460\pi\)
\(8\) 64.0000 0.353553
\(9\) −176.579 −0.726661
\(10\) −37.3393 −0.118077
\(11\) −297.264 −0.740732 −0.370366 − 0.928886i \(-0.620768\pi\)
−0.370366 + 0.928886i \(0.620768\pi\)
\(12\) −130.399 −0.261409
\(13\) −155.719 −0.255554 −0.127777 − 0.991803i \(-0.540784\pi\)
−0.127777 + 0.991803i \(0.540784\pi\)
\(14\) −649.629 −0.885820
\(15\) 76.0782 0.0873036
\(16\) 256.000 0.250000
\(17\) 0 0
\(18\) −706.315 −0.513827
\(19\) −1777.45 −1.12957 −0.564786 − 0.825237i \(-0.691041\pi\)
−0.564786 + 0.825237i \(0.691041\pi\)
\(20\) −149.357 −0.0834933
\(21\) 1323.61 0.654954
\(22\) −1189.06 −0.523776
\(23\) 384.489 0.151553 0.0757764 − 0.997125i \(-0.475856\pi\)
0.0757764 + 0.997125i \(0.475856\pi\)
\(24\) −521.596 −0.184844
\(25\) −3037.86 −0.972115
\(26\) −622.874 −0.180704
\(27\) 3419.54 0.902730
\(28\) −2598.52 −0.626369
\(29\) 5373.87 1.18657 0.593283 − 0.804994i \(-0.297831\pi\)
0.593283 + 0.804994i \(0.297831\pi\)
\(30\) 304.313 0.0617330
\(31\) 1540.65 0.287938 0.143969 − 0.989582i \(-0.454013\pi\)
0.143969 + 0.989582i \(0.454013\pi\)
\(32\) 1024.00 0.176777
\(33\) 2422.68 0.387268
\(34\) 0 0
\(35\) 1516.04 0.209190
\(36\) −2825.26 −0.363331
\(37\) −7231.11 −0.868362 −0.434181 − 0.900826i \(-0.642962\pi\)
−0.434181 + 0.900826i \(0.642962\pi\)
\(38\) −7109.81 −0.798728
\(39\) 1269.10 0.133608
\(40\) −597.429 −0.0590387
\(41\) −5943.04 −0.552140 −0.276070 − 0.961138i \(-0.589032\pi\)
−0.276070 + 0.961138i \(0.589032\pi\)
\(42\) 5294.43 0.463123
\(43\) 9935.17 0.819416 0.409708 − 0.912217i \(-0.365631\pi\)
0.409708 + 0.912217i \(0.365631\pi\)
\(44\) −4756.23 −0.370366
\(45\) 1648.33 0.121343
\(46\) 1537.95 0.107164
\(47\) 15112.3 0.997895 0.498948 − 0.866632i \(-0.333720\pi\)
0.498948 + 0.866632i \(0.333720\pi\)
\(48\) −2086.38 −0.130705
\(49\) 9569.11 0.569353
\(50\) −12151.4 −0.687389
\(51\) 0 0
\(52\) −2491.50 −0.127777
\(53\) 34197.6 1.67227 0.836135 − 0.548523i \(-0.184810\pi\)
0.836135 + 0.548523i \(0.184810\pi\)
\(54\) 13678.1 0.638326
\(55\) 2774.91 0.123692
\(56\) −10394.1 −0.442910
\(57\) 14486.1 0.590561
\(58\) 21495.5 0.839029
\(59\) −20721.4 −0.774977 −0.387489 − 0.921874i \(-0.626657\pi\)
−0.387489 + 0.921874i \(0.626657\pi\)
\(60\) 1217.25 0.0436518
\(61\) −6940.24 −0.238809 −0.119404 − 0.992846i \(-0.538098\pi\)
−0.119404 + 0.992846i \(0.538098\pi\)
\(62\) 6162.58 0.203603
\(63\) 28677.6 0.910316
\(64\) 4096.00 0.125000
\(65\) 1453.61 0.0426740
\(66\) 9690.73 0.273840
\(67\) −55729.7 −1.51670 −0.758351 − 0.651847i \(-0.773995\pi\)
−0.758351 + 0.651847i \(0.773995\pi\)
\(68\) 0 0
\(69\) −3133.56 −0.0792346
\(70\) 6064.18 0.147920
\(71\) 49749.9 1.17124 0.585620 − 0.810586i \(-0.300851\pi\)
0.585620 + 0.810586i \(0.300851\pi\)
\(72\) −11301.0 −0.256913
\(73\) −14627.6 −0.321267 −0.160634 − 0.987014i \(-0.551354\pi\)
−0.160634 + 0.987014i \(0.551354\pi\)
\(74\) −28924.4 −0.614025
\(75\) 24758.4 0.508240
\(76\) −28439.2 −0.564786
\(77\) 48277.9 0.927943
\(78\) 5076.38 0.0944752
\(79\) −17017.6 −0.306782 −0.153391 − 0.988166i \(-0.549019\pi\)
−0.153391 + 0.988166i \(0.549019\pi\)
\(80\) −2389.72 −0.0417466
\(81\) 15039.6 0.254697
\(82\) −23772.2 −0.390422
\(83\) 51344.2 0.818080 0.409040 − 0.912516i \(-0.365864\pi\)
0.409040 + 0.912516i \(0.365864\pi\)
\(84\) 21177.7 0.327477
\(85\) 0 0
\(86\) 39740.7 0.579415
\(87\) −43796.6 −0.620358
\(88\) −19024.9 −0.261888
\(89\) 113166. 1.51440 0.757202 − 0.653180i \(-0.226566\pi\)
0.757202 + 0.653180i \(0.226566\pi\)
\(90\) 6593.33 0.0858022
\(91\) 25289.8 0.320142
\(92\) 6151.82 0.0757764
\(93\) −12556.2 −0.150539
\(94\) 60449.1 0.705618
\(95\) 16592.2 0.188623
\(96\) −8345.53 −0.0924221
\(97\) 36048.5 0.389008 0.194504 − 0.980902i \(-0.437690\pi\)
0.194504 + 0.980902i \(0.437690\pi\)
\(98\) 38276.4 0.402593
\(99\) 52490.5 0.538261
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 578.6.a.d.1.2 3
17.16 even 2 34.6.a.d.1.2 ✓ 3
51.50 odd 2 306.6.a.o.1.3 3
68.67 odd 2 272.6.a.i.1.2 3
85.84 even 2 850.6.a.g.1.2 3
136.67 odd 2 1088.6.a.r.1.2 3
136.101 even 2 1088.6.a.q.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.6.a.d.1.2 ✓ 3 17.16 even 2
272.6.a.i.1.2 3 68.67 odd 2
306.6.a.o.1.3 3 51.50 odd 2
578.6.a.d.1.2 3 1.1 even 1 trivial
850.6.a.g.1.2 3 85.84 even 2
1088.6.a.q.1.2 3 136.101 even 2
1088.6.a.r.1.2 3 136.67 odd 2