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Properties

Label 2401.4.a.c.1.7
Level $2401$
Weight $4$
Character 2401.1
Self dual yes
Analytic conductor $141.664$
Analytic rank $1$
Dimension $39$
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] = [2401,4,Mod(1,2401)] 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("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2401 = 7^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2401.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [39,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(141.663585924\)
Analytic rank: \(1\)
Dimension: \(39\)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 2401.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.93571 q^{2} +3.09696 q^{3} +7.48982 q^{4} +18.2977 q^{5} -12.1887 q^{6} +2.00794 q^{8} -17.4089 q^{9} -72.0144 q^{10} +11.1264 q^{11} +23.1956 q^{12} +32.1274 q^{13} +56.6671 q^{15} -67.8212 q^{16} -68.2593 q^{17} +68.5163 q^{18} -113.492 q^{19} +137.046 q^{20} -43.7905 q^{22} +151.295 q^{23} +6.21849 q^{24} +209.805 q^{25} -126.444 q^{26} -137.532 q^{27} -183.747 q^{29} -223.025 q^{30} -167.796 q^{31} +250.861 q^{32} +34.4581 q^{33} +268.649 q^{34} -130.389 q^{36} +255.519 q^{37} +446.673 q^{38} +99.4970 q^{39} +36.7406 q^{40} +238.493 q^{41} -90.2895 q^{43} +83.3350 q^{44} -318.542 q^{45} -595.452 q^{46} -353.761 q^{47} -210.039 q^{48} -825.732 q^{50} -211.396 q^{51} +240.628 q^{52} +231.242 q^{53} +541.287 q^{54} +203.588 q^{55} -351.481 q^{57} +723.174 q^{58} -238.752 q^{59} +424.426 q^{60} -33.1087 q^{61} +660.396 q^{62} -444.747 q^{64} +587.856 q^{65} -135.617 q^{66} -505.936 q^{67} -511.249 q^{68} +468.553 q^{69} +683.159 q^{71} -34.9559 q^{72} -587.218 q^{73} -1005.65 q^{74} +649.757 q^{75} -850.037 q^{76} -391.592 q^{78} -50.1378 q^{79} -1240.97 q^{80} +44.1082 q^{81} -938.639 q^{82} -559.144 q^{83} -1248.99 q^{85} +355.353 q^{86} -569.055 q^{87} +22.3412 q^{88} +1171.00 q^{89} +1253.69 q^{90} +1133.17 q^{92} -519.656 q^{93} +1392.30 q^{94} -2076.65 q^{95} +776.905 q^{96} -5.56045 q^{97} -193.699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.93571 −1.39148 −0.695742 − 0.718292i \(-0.744924\pi\)
−0.695742 + 0.718292i \(0.744924\pi\)
\(3\) 3.09696 0.596009 0.298005 − 0.954564i \(-0.403679\pi\)
0.298005 + 0.954564i \(0.403679\pi\)
\(4\) 7.48982 0.936227
\(5\) 18.2977 1.63659 0.818297 − 0.574795i \(-0.194918\pi\)
0.818297 + 0.574795i \(0.194918\pi\)
\(6\) −12.1887 −0.829337
\(7\) 0 0
\(8\) 2.00794 0.0887390
\(9\) −17.4089 −0.644773
\(10\) −72.0144 −2.27729
\(11\) 11.1264 0.304977 0.152489 − 0.988305i \(-0.451271\pi\)
0.152489 + 0.988305i \(0.451271\pi\)
\(12\) 23.1956 0.558000
\(13\) 32.1274 0.685425 0.342713 − 0.939440i \(-0.388654\pi\)
0.342713 + 0.939440i \(0.388654\pi\)
\(14\) 0 0
\(15\) 56.6671 0.975425
\(16\) −67.8212 −1.05971
\(17\) −68.2593 −0.973842 −0.486921 − 0.873446i \(-0.661880\pi\)
−0.486921 + 0.873446i \(0.661880\pi\)
\(18\) 68.5163 0.897191
\(19\) −113.492 −1.37036 −0.685182 − 0.728372i \(-0.740278\pi\)
−0.685182 + 0.728372i \(0.740278\pi\)
\(20\) 137.046 1.53222
\(21\) 0 0
\(22\) −43.7905 −0.424371
\(23\) 151.295 1.37161 0.685807 − 0.727784i \(-0.259449\pi\)
0.685807 + 0.727784i \(0.259449\pi\)
\(24\) 6.21849 0.0528893
\(25\) 209.805 1.67844
\(26\) −126.444 −0.953758
\(27\) −137.532 −0.980300
\(28\) 0 0
\(29\) −183.747 −1.17658 −0.588291 − 0.808649i \(-0.700199\pi\)
−0.588291 + 0.808649i \(0.700199\pi\)
\(30\) −223.025 −1.35729
\(31\) −167.796 −0.972162 −0.486081 − 0.873914i \(-0.661574\pi\)
−0.486081 + 0.873914i \(0.661574\pi\)
\(32\) 250.861 1.38582
\(33\) 34.4581 0.181769
\(34\) 268.649 1.35508
\(35\) 0 0
\(36\) −130.389 −0.603654
\(37\) 255.519 1.13533 0.567663 − 0.823261i \(-0.307848\pi\)
0.567663 + 0.823261i \(0.307848\pi\)
\(38\) 446.673 1.90684
\(39\) 99.4970 0.408520
\(40\) 36.7406 0.145230
\(41\) 238.493 0.908447 0.454223 − 0.890888i \(-0.349917\pi\)
0.454223 + 0.890888i \(0.349917\pi\)
\(42\) 0 0
\(43\) −90.2895 −0.320210 −0.160105 − 0.987100i \(-0.551183\pi\)
−0.160105 + 0.987100i \(0.551183\pi\)
\(44\) 83.3350 0.285528
\(45\) −318.542 −1.05523
\(46\) −595.452 −1.90858
\(47\) −353.761 −1.09790 −0.548950 − 0.835855i \(-0.684972\pi\)
−0.548950 + 0.835855i \(0.684972\pi\)
\(48\) −210.039 −0.631595
\(49\) 0 0
\(50\) −825.732 −2.33552
\(51\) −211.396 −0.580419
\(52\) 240.628 0.641714
\(53\) 231.242 0.599311 0.299656 − 0.954047i \(-0.403128\pi\)
0.299656 + 0.954047i \(0.403128\pi\)
\(54\) 541.287 1.36407
\(55\) 203.588 0.499124
\(56\) 0 0
\(57\) −351.481 −0.816750
\(58\) 723.174 1.63720
\(59\) −238.752 −0.526827 −0.263414 − 0.964683i \(-0.584848\pi\)
−0.263414 + 0.964683i \(0.584848\pi\)
\(60\) 424.426 0.913220
\(61\) −33.1087 −0.0694940 −0.0347470 − 0.999396i \(-0.511063\pi\)
−0.0347470 + 0.999396i \(0.511063\pi\)
\(62\) 660.396 1.35275
\(63\) 0 0
\(64\) −444.747 −0.868646
\(65\) 587.856 1.12176
\(66\) −135.617 −0.252929
\(67\) −505.936 −0.922537 −0.461268 − 0.887261i \(-0.652605\pi\)
−0.461268 + 0.887261i \(0.652605\pi\)
\(68\) −511.249 −0.911737
\(69\) 468.553 0.817494
\(70\) 0 0
\(71\) 683.159 1.14192 0.570958 − 0.820979i \(-0.306572\pi\)
0.570958 + 0.820979i \(0.306572\pi\)
\(72\) −34.9559 −0.0572165
\(73\) −587.218 −0.941489 −0.470745 − 0.882269i \(-0.656015\pi\)
−0.470745 + 0.882269i \(0.656015\pi\)
\(74\) −1005.65 −1.57979
\(75\) 649.757 1.00037
\(76\) −850.037 −1.28297
\(77\) 0 0
\(78\) −391.592 −0.568449
\(79\) −50.1378 −0.0714043 −0.0357022 − 0.999362i \(-0.511367\pi\)
−0.0357022 + 0.999362i \(0.511367\pi\)
\(80\) −1240.97 −1.73431
\(81\) 44.1082 0.0605050
\(82\) −938.639 −1.26409
\(83\) −559.144 −0.739446 −0.369723 − 0.929142i \(-0.620547\pi\)
−0.369723 + 0.929142i \(0.620547\pi\)
\(84\) 0 0
\(85\) −1248.99 −1.59378
\(86\) 355.353 0.445566
\(87\) −569.055 −0.701254
\(88\) 22.3412 0.0270634
\(89\) 1171.00 1.39468 0.697338 − 0.716743i \(-0.254368\pi\)
0.697338 + 0.716743i \(0.254368\pi\)
\(90\) 1253.69 1.46834
\(91\) 0 0
\(92\) 1133.17 1.28414
\(93\) −519.656 −0.579418
\(94\) 1392.30 1.52771
\(95\) −2076.65 −2.24273
\(96\) 776.905 0.825964
\(97\) −5.56045 −0.00582039 −0.00291019 − 0.999996i \(-0.500926\pi\)
−0.00291019 + 0.999996i \(0.500926\pi\)
\(98\) 0 0
\(99\) −193.699 −0.196641
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 2401.4.a.c.1.7 39
7.6 odd 2 2401.4.a.d.1.7 39
49.20 odd 14 49.4.e.a.8.3 ✓ 78
49.27 odd 14 49.4.e.a.43.3 yes 78
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.e.a.8.3 ✓ 78 49.20 odd 14
49.4.e.a.43.3 yes 78 49.27 odd 14
2401.4.a.c.1.7 39 1.1 even 1 trivial
2401.4.a.d.1.7 39 7.6 odd 2