Properties

Label 1150.4.a.w.1.1
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
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] = [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 = [6,-12,5] 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: \(6\)
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: \( x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.88153\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -6.88153 q^{3} +4.00000 q^{4} +13.7631 q^{6} +23.7447 q^{7} -8.00000 q^{8} +20.3555 q^{9} -70.4223 q^{11} -27.5261 q^{12} -92.5968 q^{13} -47.4894 q^{14} +16.0000 q^{16} +89.8764 q^{17} -40.7110 q^{18} -86.7775 q^{19} -163.400 q^{21} +140.845 q^{22} +23.0000 q^{23} +55.0523 q^{24} +185.194 q^{26} +45.7244 q^{27} +94.9789 q^{28} -20.0659 q^{29} -31.3647 q^{31} -32.0000 q^{32} +484.613 q^{33} -179.753 q^{34} +81.4220 q^{36} +31.3174 q^{37} +173.555 q^{38} +637.208 q^{39} -280.766 q^{41} +326.800 q^{42} -250.791 q^{43} -281.689 q^{44} -46.0000 q^{46} +187.064 q^{47} -110.105 q^{48} +220.811 q^{49} -618.487 q^{51} -370.387 q^{52} -339.092 q^{53} -91.4488 q^{54} -189.958 q^{56} +597.162 q^{57} +40.1318 q^{58} -724.807 q^{59} +58.4222 q^{61} +62.7294 q^{62} +483.335 q^{63} +64.0000 q^{64} -969.226 q^{66} -386.994 q^{67} +359.506 q^{68} -158.275 q^{69} +131.283 q^{71} -162.844 q^{72} +226.305 q^{73} -62.6348 q^{74} -347.110 q^{76} -1672.16 q^{77} -1274.42 q^{78} -425.933 q^{79} -864.252 q^{81} +561.531 q^{82} +1275.71 q^{83} -653.600 q^{84} +501.581 q^{86} +138.084 q^{87} +563.378 q^{88} -1465.16 q^{89} -2198.68 q^{91} +92.0000 q^{92} +215.837 q^{93} -374.127 q^{94} +220.209 q^{96} -48.8911 q^{97} -441.623 q^{98} -1433.48 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{2} + 5 q^{3} + 24 q^{4} - 10 q^{6} + 42 q^{7} - 48 q^{8} + 61 q^{9} - 49 q^{11} + 20 q^{12} + 16 q^{13} - 84 q^{14} + 96 q^{16} + 175 q^{17} - 122 q^{18} - 229 q^{19} + 92 q^{21} + 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −6.88153 −1.32435 −0.662176 − 0.749349i \(-0.730367\pi\)
−0.662176 + 0.749349i \(0.730367\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 13.7631 0.936458
\(7\) 23.7447 1.28209 0.641047 − 0.767502i \(-0.278500\pi\)
0.641047 + 0.767502i \(0.278500\pi\)
\(8\) −8.00000 −0.353553
\(9\) 20.3555 0.753907
\(10\) 0 0
\(11\) −70.4223 −1.93028 −0.965142 − 0.261728i \(-0.915708\pi\)
−0.965142 + 0.261728i \(0.915708\pi\)
\(12\) −27.5261 −0.662176
\(13\) −92.5968 −1.97552 −0.987759 − 0.155987i \(-0.950144\pi\)
−0.987759 + 0.155987i \(0.950144\pi\)
\(14\) −47.4894 −0.906577
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 89.8764 1.28225 0.641124 − 0.767437i \(-0.278468\pi\)
0.641124 + 0.767437i \(0.278468\pi\)
\(18\) −40.7110 −0.533093
\(19\) −86.7775 −1.04780 −0.523898 − 0.851781i \(-0.675523\pi\)
−0.523898 + 0.851781i \(0.675523\pi\)
\(20\) 0 0
\(21\) −163.400 −1.69794
\(22\) 140.845 1.36492
\(23\) 23.0000 0.208514
\(24\) 55.0523 0.468229
\(25\) 0 0
\(26\) 185.194 1.39690
\(27\) 45.7244 0.325913
\(28\) 94.9789 0.641047
\(29\) −20.0659 −0.128488 −0.0642439 − 0.997934i \(-0.520464\pi\)
−0.0642439 + 0.997934i \(0.520464\pi\)
\(30\) 0 0
\(31\) −31.3647 −0.181718 −0.0908591 − 0.995864i \(-0.528961\pi\)
−0.0908591 + 0.995864i \(0.528961\pi\)
\(32\) −32.0000 −0.176777
\(33\) 484.613 2.55637
\(34\) −179.753 −0.906687
\(35\) 0 0
\(36\) 81.4220 0.376954
\(37\) 31.3174 0.139150 0.0695750 − 0.997577i \(-0.477836\pi\)
0.0695750 + 0.997577i \(0.477836\pi\)
\(38\) 173.555 0.740903
\(39\) 637.208 2.61628
\(40\) 0 0
\(41\) −280.766 −1.06947 −0.534735 − 0.845020i \(-0.679588\pi\)
−0.534735 + 0.845020i \(0.679588\pi\)
\(42\) 326.800 1.20063
\(43\) −250.791 −0.889424 −0.444712 − 0.895674i \(-0.646694\pi\)
−0.444712 + 0.895674i \(0.646694\pi\)
\(44\) −281.689 −0.965142
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 187.064 0.580554 0.290277 − 0.956943i \(-0.406253\pi\)
0.290277 + 0.956943i \(0.406253\pi\)
\(48\) −110.105 −0.331088
\(49\) 220.811 0.643765
\(50\) 0 0
\(51\) −618.487 −1.69815
\(52\) −370.387 −0.987759
\(53\) −339.092 −0.878827 −0.439413 − 0.898285i \(-0.644814\pi\)
−0.439413 + 0.898285i \(0.644814\pi\)
\(54\) −91.4488 −0.230456
\(55\) 0 0
\(56\) −189.958 −0.453289
\(57\) 597.162 1.38765
\(58\) 40.1318 0.0908546
\(59\) −724.807 −1.59935 −0.799677 − 0.600431i \(-0.794996\pi\)
−0.799677 + 0.600431i \(0.794996\pi\)
\(60\) 0 0
\(61\) 58.4222 0.122626 0.0613131 − 0.998119i \(-0.480471\pi\)
0.0613131 + 0.998119i \(0.480471\pi\)
\(62\) 62.7294 0.128494
\(63\) 483.335 0.966580
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −969.226 −1.80763
\(67\) −386.994 −0.705654 −0.352827 − 0.935689i \(-0.614780\pi\)
−0.352827 + 0.935689i \(0.614780\pi\)
\(68\) 359.506 0.641124
\(69\) −158.275 −0.276146
\(70\) 0 0
\(71\) 131.283 0.219443 0.109722 − 0.993962i \(-0.465004\pi\)
0.109722 + 0.993962i \(0.465004\pi\)
\(72\) −162.844 −0.266546
\(73\) 226.305 0.362835 0.181418 − 0.983406i \(-0.441931\pi\)
0.181418 + 0.983406i \(0.441931\pi\)
\(74\) −62.6348 −0.0983939
\(75\) 0 0
\(76\) −347.110 −0.523898
\(77\) −1672.16 −2.47480
\(78\) −1274.42 −1.84999
\(79\) −425.933 −0.606598 −0.303299 − 0.952895i \(-0.598088\pi\)
−0.303299 + 0.952895i \(0.598088\pi\)
\(80\) 0 0
\(81\) −864.252 −1.18553
\(82\) 561.531 0.756229
\(83\) 1275.71 1.68708 0.843541 − 0.537065i \(-0.180467\pi\)
0.843541 + 0.537065i \(0.180467\pi\)
\(84\) −653.600 −0.848972
\(85\) 0 0
\(86\) 501.581 0.628918
\(87\) 138.084 0.170163
\(88\) 563.378 0.682458
\(89\) −1465.16 −1.74502 −0.872509 − 0.488598i \(-0.837509\pi\)
−0.872509 + 0.488598i \(0.837509\pi\)
\(90\) 0 0
\(91\) −2198.68 −2.53280
\(92\) 92.0000 0.104257
\(93\) 215.837 0.240659
\(94\) −374.127 −0.410513
\(95\) 0 0
\(96\) 220.209 0.234115
\(97\) −48.8911 −0.0511767 −0.0255883 − 0.999673i \(-0.508146\pi\)
−0.0255883 + 0.999673i \(0.508146\pi\)
\(98\) −441.623 −0.455211
\(99\) −1433.48 −1.45525
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.w.1.1 ✓ 6
5.2 odd 4 1150.4.b.s.599.6 12
5.3 odd 4 1150.4.b.s.599.7 12
5.4 even 2 1150.4.a.x.1.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.1 ✓ 6 1.1 even 1 trivial
1150.4.a.x.1.6 yes 6 5.4 even 2
1150.4.b.s.599.6 12 5.2 odd 4
1150.4.b.s.599.7 12 5.3 odd 4