Properties

Label 1150.4.a.h.1.1
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
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] = [1150,4,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,2] 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: \(67.8521965066\)
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 \(\chi\) \(=\) 1150.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{3} +4.00000 q^{4} +4.00000 q^{6} +21.0000 q^{7} +8.00000 q^{8} -23.0000 q^{9} +47.0000 q^{11} +8.00000 q^{12} +57.0000 q^{13} +42.0000 q^{14} +16.0000 q^{16} -84.0000 q^{17} -46.0000 q^{18} -5.00000 q^{19} +42.0000 q^{21} +94.0000 q^{22} +23.0000 q^{23} +16.0000 q^{24} +114.000 q^{26} -100.000 q^{27} +84.0000 q^{28} +285.000 q^{29} +82.0000 q^{31} +32.0000 q^{32} +94.0000 q^{33} -168.000 q^{34} -92.0000 q^{36} -54.0000 q^{37} -10.0000 q^{38} +114.000 q^{39} -53.0000 q^{41} +84.0000 q^{42} +197.000 q^{43} +188.000 q^{44} +46.0000 q^{46} -124.000 q^{47} +32.0000 q^{48} +98.0000 q^{49} -168.000 q^{51} +228.000 q^{52} -148.000 q^{53} -200.000 q^{54} +168.000 q^{56} -10.0000 q^{57} +570.000 q^{58} +30.0000 q^{59} -578.000 q^{61} +164.000 q^{62} -483.000 q^{63} +64.0000 q^{64} +188.000 q^{66} +296.000 q^{67} -336.000 q^{68} +46.0000 q^{69} +422.000 q^{71} -184.000 q^{72} +487.000 q^{73} -108.000 q^{74} -20.0000 q^{76} +987.000 q^{77} +228.000 q^{78} -405.000 q^{79} +421.000 q^{81} -106.000 q^{82} +397.000 q^{83} +168.000 q^{84} +394.000 q^{86} +570.000 q^{87} +376.000 q^{88} +730.000 q^{89} +1197.00 q^{91} +92.0000 q^{92} +164.000 q^{93} -248.000 q^{94} +64.0000 q^{96} -64.0000 q^{97} +196.000 q^{98} -1081.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 2.00000 0.384900 0.192450 − 0.981307i \(-0.438357\pi\)
0.192450 + 0.981307i \(0.438357\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 4.00000 0.272166
\(7\) 21.0000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 8.00000 0.353553
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) 47.0000 1.28828 0.644138 − 0.764909i \(-0.277216\pi\)
0.644138 + 0.764909i \(0.277216\pi\)
\(12\) 8.00000 0.192450
\(13\) 57.0000 1.21607 0.608037 − 0.793909i \(-0.291957\pi\)
0.608037 + 0.793909i \(0.291957\pi\)
\(14\) 42.0000 0.801784
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −84.0000 −1.19841 −0.599206 − 0.800595i \(-0.704517\pi\)
−0.599206 + 0.800595i \(0.704517\pi\)
\(18\) −46.0000 −0.602350
\(19\) −5.00000 −0.0603726 −0.0301863 − 0.999544i \(-0.509610\pi\)
−0.0301863 + 0.999544i \(0.509610\pi\)
\(20\) 0 0
\(21\) 42.0000 0.436436
\(22\) 94.0000 0.910949
\(23\) 23.0000 0.208514
\(24\) 16.0000 0.136083
\(25\) 0 0
\(26\) 114.000 0.859894
\(27\) −100.000 −0.712778
\(28\) 84.0000 0.566947
\(29\) 285.000 1.82494 0.912468 − 0.409147i \(-0.134174\pi\)
0.912468 + 0.409147i \(0.134174\pi\)
\(30\) 0 0
\(31\) 82.0000 0.475085 0.237542 − 0.971377i \(-0.423658\pi\)
0.237542 + 0.971377i \(0.423658\pi\)
\(32\) 32.0000 0.176777
\(33\) 94.0000 0.495858
\(34\) −168.000 −0.847405
\(35\) 0 0
\(36\) −92.0000 −0.425926
\(37\) −54.0000 −0.239934 −0.119967 − 0.992778i \(-0.538279\pi\)
−0.119967 + 0.992778i \(0.538279\pi\)
\(38\) −10.0000 −0.0426898
\(39\) 114.000 0.468067
\(40\) 0 0
\(41\) −53.0000 −0.201883 −0.100942 − 0.994892i \(-0.532186\pi\)
−0.100942 + 0.994892i \(0.532186\pi\)
\(42\) 84.0000 0.308607
\(43\) 197.000 0.698656 0.349328 − 0.937000i \(-0.386410\pi\)
0.349328 + 0.937000i \(0.386410\pi\)
\(44\) 188.000 0.644138
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) −124.000 −0.384835 −0.192418 − 0.981313i \(-0.561633\pi\)
−0.192418 + 0.981313i \(0.561633\pi\)
\(48\) 32.0000 0.0962250
\(49\) 98.0000 0.285714
\(50\) 0 0
\(51\) −168.000 −0.461269
\(52\) 228.000 0.608037
\(53\) −148.000 −0.383573 −0.191786 − 0.981437i \(-0.561428\pi\)
−0.191786 + 0.981437i \(0.561428\pi\)
\(54\) −200.000 −0.504010
\(55\) 0 0
\(56\) 168.000 0.400892
\(57\) −10.0000 −0.0232374
\(58\) 570.000 1.29043
\(59\) 30.0000 0.0661978 0.0330989 − 0.999452i \(-0.489462\pi\)
0.0330989 + 0.999452i \(0.489462\pi\)
\(60\) 0 0
\(61\) −578.000 −1.21320 −0.606601 − 0.795006i \(-0.707467\pi\)
−0.606601 + 0.795006i \(0.707467\pi\)
\(62\) 164.000 0.335936
\(63\) −483.000 −0.965909
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 188.000 0.350624
\(67\) 296.000 0.539734 0.269867 − 0.962898i \(-0.413020\pi\)
0.269867 + 0.962898i \(0.413020\pi\)
\(68\) −336.000 −0.599206
\(69\) 46.0000 0.0802572
\(70\) 0 0
\(71\) 422.000 0.705383 0.352691 − 0.935740i \(-0.385267\pi\)
0.352691 + 0.935740i \(0.385267\pi\)
\(72\) −184.000 −0.301175
\(73\) 487.000 0.780809 0.390404 − 0.920643i \(-0.372335\pi\)
0.390404 + 0.920643i \(0.372335\pi\)
\(74\) −108.000 −0.169659
\(75\) 0 0
\(76\) −20.0000 −0.0301863
\(77\) 987.000 1.46077
\(78\) 228.000 0.330973
\(79\) −405.000 −0.576786 −0.288393 − 0.957512i \(-0.593121\pi\)
−0.288393 + 0.957512i \(0.593121\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) −106.000 −0.142753
\(83\) 397.000 0.525017 0.262509 − 0.964930i \(-0.415450\pi\)
0.262509 + 0.964930i \(0.415450\pi\)
\(84\) 168.000 0.218218
\(85\) 0 0
\(86\) 394.000 0.494025
\(87\) 570.000 0.702419
\(88\) 376.000 0.455474
\(89\) 730.000 0.869436 0.434718 − 0.900567i \(-0.356848\pi\)
0.434718 + 0.900567i \(0.356848\pi\)
\(90\) 0 0
\(91\) 1197.00 1.37890
\(92\) 92.0000 0.104257
\(93\) 164.000 0.182860
\(94\) −248.000 −0.272120
\(95\) 0 0
\(96\) 64.0000 0.0680414
\(97\) −64.0000 −0.0669919 −0.0334960 − 0.999439i \(-0.510664\pi\)
−0.0334960 + 0.999439i \(0.510664\pi\)
\(98\) 196.000 0.202031
\(99\) −1081.00 −1.09742
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 1150.4.a.h.1.1 yes 1
5.2 odd 4 1150.4.b.g.599.2 2
5.3 odd 4 1150.4.b.g.599.1 2
5.4 even 2 1150.4.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.a.1.1 ✓ 1 5.4 even 2
1150.4.a.h.1.1 yes 1 1.1 even 1 trivial
1150.4.b.g.599.1 2 5.3 odd 4
1150.4.b.g.599.2 2 5.2 odd 4