Properties

Label 95.4.a.d.1.1
Level $95$
Weight $4$
Character 95.1
Self dual yes
Analytic conductor $5.605$
Analytic rank $0$
Dimension $1$
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] = [95,4,Mod(1,95)] 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("95.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,5,4] 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 95.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{2} +4.00000 q^{3} +17.0000 q^{4} +5.00000 q^{5} +20.0000 q^{6} -32.0000 q^{7} +45.0000 q^{8} -11.0000 q^{9} +25.0000 q^{10} -12.0000 q^{11} +68.0000 q^{12} -42.0000 q^{13} -160.000 q^{14} +20.0000 q^{15} +89.0000 q^{16} +114.000 q^{17} -55.0000 q^{18} +19.0000 q^{19} +85.0000 q^{20} -128.000 q^{21} -60.0000 q^{22} +160.000 q^{23} +180.000 q^{24} +25.0000 q^{25} -210.000 q^{26} -152.000 q^{27} -544.000 q^{28} +214.000 q^{29} +100.000 q^{30} -144.000 q^{31} +85.0000 q^{32} -48.0000 q^{33} +570.000 q^{34} -160.000 q^{35} -187.000 q^{36} +94.0000 q^{37} +95.0000 q^{38} -168.000 q^{39} +225.000 q^{40} -6.00000 q^{41} -640.000 q^{42} -308.000 q^{43} -204.000 q^{44} -55.0000 q^{45} +800.000 q^{46} +184.000 q^{47} +356.000 q^{48} +681.000 q^{49} +125.000 q^{50} +456.000 q^{51} -714.000 q^{52} -274.000 q^{53} -760.000 q^{54} -60.0000 q^{55} -1440.00 q^{56} +76.0000 q^{57} +1070.00 q^{58} +276.000 q^{59} +340.000 q^{60} -826.000 q^{61} -720.000 q^{62} +352.000 q^{63} -287.000 q^{64} -210.000 q^{65} -240.000 q^{66} +52.0000 q^{67} +1938.00 q^{68} +640.000 q^{69} -800.000 q^{70} -344.000 q^{71} -495.000 q^{72} -166.000 q^{73} +470.000 q^{74} +100.000 q^{75} +323.000 q^{76} +384.000 q^{77} -840.000 q^{78} -688.000 q^{79} +445.000 q^{80} -311.000 q^{81} -30.0000 q^{82} +996.000 q^{83} -2176.00 q^{84} +570.000 q^{85} -1540.00 q^{86} +856.000 q^{87} -540.000 q^{88} +1578.00 q^{89} -275.000 q^{90} +1344.00 q^{91} +2720.00 q^{92} -576.000 q^{93} +920.000 q^{94} +95.0000 q^{95} +340.000 q^{96} +786.000 q^{97} +3405.00 q^{98} +132.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.00000 1.76777 0.883883 − 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 4.00000 0.769800 0.384900 − 0.922958i \(-0.374236\pi\)
0.384900 + 0.922958i \(0.374236\pi\)
\(4\) 17.0000 2.12500
\(5\) 5.00000 0.447214
\(6\) 20.0000 1.36083
\(7\) −32.0000 −1.72784 −0.863919 − 0.503631i \(-0.831997\pi\)
−0.863919 + 0.503631i \(0.831997\pi\)
\(8\) 45.0000 1.98874
\(9\) −11.0000 −0.407407
\(10\) 25.0000 0.790569
\(11\) −12.0000 −0.328921 −0.164461 − 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 68.0000 1.63583
\(13\) −42.0000 −0.896054 −0.448027 − 0.894020i \(-0.647873\pi\)
−0.448027 + 0.894020i \(0.647873\pi\)
\(14\) −160.000 −3.05441
\(15\) 20.0000 0.344265
\(16\) 89.0000 1.39062
\(17\) 114.000 1.62642 0.813208 − 0.581974i \(-0.197719\pi\)
0.813208 + 0.581974i \(0.197719\pi\)
\(18\) −55.0000 −0.720201
\(19\) 19.0000 0.229416
\(20\) 85.0000 0.950329
\(21\) −128.000 −1.33009
\(22\) −60.0000 −0.581456
\(23\) 160.000 1.45054 0.725268 − 0.688467i \(-0.241716\pi\)
0.725268 + 0.688467i \(0.241716\pi\)
\(24\) 180.000 1.53093
\(25\) 25.0000 0.200000
\(26\) −210.000 −1.58401
\(27\) −152.000 −1.08342
\(28\) −544.000 −3.67165
\(29\) 214.000 1.37030 0.685152 − 0.728400i \(-0.259736\pi\)
0.685152 + 0.728400i \(0.259736\pi\)
\(30\) 100.000 0.608581
\(31\) −144.000 −0.834296 −0.417148 − 0.908839i \(-0.636970\pi\)
−0.417148 + 0.908839i \(0.636970\pi\)
\(32\) 85.0000 0.469563
\(33\) −48.0000 −0.253204
\(34\) 570.000 2.87512
\(35\) −160.000 −0.772712
\(36\) −187.000 −0.865741
\(37\) 94.0000 0.417662 0.208831 − 0.977952i \(-0.433034\pi\)
0.208831 + 0.977952i \(0.433034\pi\)
\(38\) 95.0000 0.405554
\(39\) −168.000 −0.689783
\(40\) 225.000 0.889391
\(41\) −6.00000 −0.0228547 −0.0114273 − 0.999935i \(-0.503638\pi\)
−0.0114273 + 0.999935i \(0.503638\pi\)
\(42\) −640.000 −2.35129
\(43\) −308.000 −1.09232 −0.546158 − 0.837682i \(-0.683910\pi\)
−0.546158 + 0.837682i \(0.683910\pi\)
\(44\) −204.000 −0.698958
\(45\) −55.0000 −0.182198
\(46\) 800.000 2.56421
\(47\) 184.000 0.571046 0.285523 − 0.958372i \(-0.407833\pi\)
0.285523 + 0.958372i \(0.407833\pi\)
\(48\) 356.000 1.07050
\(49\) 681.000 1.98542
\(50\) 125.000 0.353553
\(51\) 456.000 1.25202
\(52\) −714.000 −1.90412
\(53\) −274.000 −0.710128 −0.355064 − 0.934842i \(-0.615541\pi\)
−0.355064 + 0.934842i \(0.615541\pi\)
\(54\) −760.000 −1.91524
\(55\) −60.0000 −0.147098
\(56\) −1440.00 −3.43622
\(57\) 76.0000 0.176604
\(58\) 1070.00 2.42238
\(59\) 276.000 0.609019 0.304510 − 0.952509i \(-0.401507\pi\)
0.304510 + 0.952509i \(0.401507\pi\)
\(60\) 340.000 0.731564
\(61\) −826.000 −1.73375 −0.866873 − 0.498530i \(-0.833873\pi\)
−0.866873 + 0.498530i \(0.833873\pi\)
\(62\) −720.000 −1.47484
\(63\) 352.000 0.703934
\(64\) −287.000 −0.560547
\(65\) −210.000 −0.400728
\(66\) −240.000 −0.447605
\(67\) 52.0000 0.0948181 0.0474090 − 0.998876i \(-0.484904\pi\)
0.0474090 + 0.998876i \(0.484904\pi\)
\(68\) 1938.00 3.45613
\(69\) 640.000 1.11662
\(70\) −800.000 −1.36598
\(71\) −344.000 −0.575004 −0.287502 − 0.957780i \(-0.592825\pi\)
−0.287502 + 0.957780i \(0.592825\pi\)
\(72\) −495.000 −0.810227
\(73\) −166.000 −0.266148 −0.133074 − 0.991106i \(-0.542485\pi\)
−0.133074 + 0.991106i \(0.542485\pi\)
\(74\) 470.000 0.738330
\(75\) 100.000 0.153960
\(76\) 323.000 0.487508
\(77\) 384.000 0.568323
\(78\) −840.000 −1.21938
\(79\) −688.000 −0.979823 −0.489912 − 0.871772i \(-0.662971\pi\)
−0.489912 + 0.871772i \(0.662971\pi\)
\(80\) 445.000 0.621906
\(81\) −311.000 −0.426612
\(82\) −30.0000 −0.0404018
\(83\) 996.000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2176.00 −2.82644
\(85\) 570.000 0.727355
\(86\) −1540.00 −1.93096
\(87\) 856.000 1.05486
\(88\) −540.000 −0.654139
\(89\) 1578.00 1.87941 0.939706 − 0.341983i \(-0.111099\pi\)
0.939706 + 0.341983i \(0.111099\pi\)
\(90\) −275.000 −0.322084
\(91\) 1344.00 1.54824
\(92\) 2720.00 3.08239
\(93\) −576.000 −0.642241
\(94\) 920.000 1.00948
\(95\) 95.0000 0.102598
\(96\) 340.000 0.361470
\(97\) 786.000 0.822744 0.411372 − 0.911467i \(-0.365050\pi\)
0.411372 + 0.911467i \(0.365050\pi\)
\(98\) 3405.00 3.50976
\(99\) 132.000 0.134005
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 95.4.a.d.1.1 ✓ 1
3.2 odd 2 855.4.a.a.1.1 1
4.3 odd 2 1520.4.a.c.1.1 1
5.2 odd 4 475.4.b.a.324.2 2
5.3 odd 4 475.4.b.a.324.1 2
5.4 even 2 475.4.a.a.1.1 1
19.18 odd 2 1805.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.a.d.1.1 ✓ 1 1.1 even 1 trivial
475.4.a.a.1.1 1 5.4 even 2
475.4.b.a.324.1 2 5.3 odd 4
475.4.b.a.324.2 2 5.2 odd 4
855.4.a.a.1.1 1 3.2 odd 2
1520.4.a.c.1.1 1 4.3 odd 2
1805.4.a.a.1.1 1 19.18 odd 2