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Properties

Label 1150.4.a.q.1.5
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
Analytic rank $1$
Dimension $5$
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 = [5,-10,-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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(9.27140\) 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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +8.27140 q^{3} +4.00000 q^{4} -16.5428 q^{6} -22.8621 q^{7} -8.00000 q^{8} +41.4161 q^{9} -0.987996 q^{11} +33.0856 q^{12} +4.22276 q^{13} +45.7241 q^{14} +16.0000 q^{16} -73.8741 q^{17} -82.8322 q^{18} +71.1586 q^{19} -189.101 q^{21} +1.97599 q^{22} -23.0000 q^{23} -66.1712 q^{24} -8.44552 q^{26} +119.241 q^{27} -91.4483 q^{28} +27.8984 q^{29} -136.861 q^{31} -32.0000 q^{32} -8.17211 q^{33} +147.748 q^{34} +165.664 q^{36} +201.718 q^{37} -142.317 q^{38} +34.9281 q^{39} -204.140 q^{41} +378.203 q^{42} +54.1024 q^{43} -3.95198 q^{44} +46.0000 q^{46} +41.4490 q^{47} +132.342 q^{48} +179.674 q^{49} -611.042 q^{51} +16.8910 q^{52} -428.612 q^{53} -238.483 q^{54} +182.897 q^{56} +588.581 q^{57} -55.7969 q^{58} -164.858 q^{59} +188.248 q^{61} +273.722 q^{62} -946.857 q^{63} +64.0000 q^{64} +16.3442 q^{66} -932.167 q^{67} -295.496 q^{68} -190.242 q^{69} -263.336 q^{71} -331.329 q^{72} -900.401 q^{73} -403.437 q^{74} +284.634 q^{76} +22.5876 q^{77} -69.8563 q^{78} -956.182 q^{79} -131.942 q^{81} +408.279 q^{82} -194.877 q^{83} -756.405 q^{84} -108.205 q^{86} +230.759 q^{87} +7.90397 q^{88} -213.751 q^{89} -96.5410 q^{91} -92.0000 q^{92} -1132.03 q^{93} -82.8981 q^{94} -264.685 q^{96} -628.762 q^{97} -359.348 q^{98} -40.9189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 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
\(3\) 8.27140 1.59183 0.795916 − 0.605407i \(-0.206990\pi\)
0.795916 + 0.605407i \(0.206990\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −16.5428 −1.12560
\(7\) −22.8621 −1.23444 −0.617218 − 0.786792i \(-0.711740\pi\)
−0.617218 + 0.786792i \(0.711740\pi\)
\(8\) −8.00000 −0.353553
\(9\) 41.4161 1.53393
\(10\) 0 0
\(11\) −0.987996 −0.0270811 −0.0135405 − 0.999908i \(-0.504310\pi\)
−0.0135405 + 0.999908i \(0.504310\pi\)
\(12\) 33.0856 0.795916
\(13\) 4.22276 0.0900910 0.0450455 − 0.998985i \(-0.485657\pi\)
0.0450455 + 0.998985i \(0.485657\pi\)
\(14\) 45.7241 0.872878
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −73.8741 −1.05395 −0.526973 − 0.849882i \(-0.676673\pi\)
−0.526973 + 0.849882i \(0.676673\pi\)
\(18\) −82.8322 −1.08465
\(19\) 71.1586 0.859205 0.429602 − 0.903018i \(-0.358654\pi\)
0.429602 + 0.903018i \(0.358654\pi\)
\(20\) 0 0
\(21\) −189.101 −1.96501
\(22\) 1.97599 0.0191492
\(23\) −23.0000 −0.208514
\(24\) −66.1712 −0.562798
\(25\) 0 0
\(26\) −8.44552 −0.0637039
\(27\) 119.241 0.849926
\(28\) −91.4483 −0.617218
\(29\) 27.8984 0.178642 0.0893209 − 0.996003i \(-0.471530\pi\)
0.0893209 + 0.996003i \(0.471530\pi\)
\(30\) 0 0
\(31\) −136.861 −0.792935 −0.396468 − 0.918049i \(-0.629764\pi\)
−0.396468 + 0.918049i \(0.629764\pi\)
\(32\) −32.0000 −0.176777
\(33\) −8.17211 −0.0431085
\(34\) 147.748 0.745253
\(35\) 0 0
\(36\) 165.664 0.766965
\(37\) 201.718 0.896278 0.448139 − 0.893964i \(-0.352087\pi\)
0.448139 + 0.893964i \(0.352087\pi\)
\(38\) −142.317 −0.607550
\(39\) 34.9281 0.143410
\(40\) 0 0
\(41\) −204.140 −0.777592 −0.388796 − 0.921324i \(-0.627109\pi\)
−0.388796 + 0.921324i \(0.627109\pi\)
\(42\) 378.203 1.38947
\(43\) 54.1024 0.191873 0.0959365 − 0.995387i \(-0.469415\pi\)
0.0959365 + 0.995387i \(0.469415\pi\)
\(44\) −3.95198 −0.0135405
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 41.4490 0.128638 0.0643188 − 0.997929i \(-0.479513\pi\)
0.0643188 + 0.997929i \(0.479513\pi\)
\(48\) 132.342 0.397958
\(49\) 179.674 0.523831
\(50\) 0 0
\(51\) −611.042 −1.67771
\(52\) 16.8910 0.0450455
\(53\) −428.612 −1.11084 −0.555418 − 0.831571i \(-0.687442\pi\)
−0.555418 + 0.831571i \(0.687442\pi\)
\(54\) −238.483 −0.600988
\(55\) 0 0
\(56\) 182.897 0.436439
\(57\) 588.581 1.36771
\(58\) −55.7969 −0.126319
\(59\) −164.858 −0.363774 −0.181887 − 0.983319i \(-0.558221\pi\)
−0.181887 + 0.983319i \(0.558221\pi\)
\(60\) 0 0
\(61\) 188.248 0.395125 0.197563 − 0.980290i \(-0.436697\pi\)
0.197563 + 0.980290i \(0.436697\pi\)
\(62\) 273.722 0.560690
\(63\) −946.857 −1.89354
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 16.3442 0.0304823
\(67\) −932.167 −1.69974 −0.849868 − 0.526995i \(-0.823318\pi\)
−0.849868 + 0.526995i \(0.823318\pi\)
\(68\) −295.496 −0.526973
\(69\) −190.242 −0.331920
\(70\) 0 0
\(71\) −263.336 −0.440173 −0.220086 − 0.975480i \(-0.570634\pi\)
−0.220086 + 0.975480i \(0.570634\pi\)
\(72\) −331.329 −0.542326
\(73\) −900.401 −1.44362 −0.721808 − 0.692094i \(-0.756689\pi\)
−0.721808 + 0.692094i \(0.756689\pi\)
\(74\) −403.437 −0.633764
\(75\) 0 0
\(76\) 284.634 0.429602
\(77\) 22.5876 0.0334299
\(78\) −69.8563 −0.101406
\(79\) −956.182 −1.36176 −0.680879 − 0.732396i \(-0.738402\pi\)
−0.680879 + 0.732396i \(0.738402\pi\)
\(80\) 0 0
\(81\) −131.942 −0.180990
\(82\) 408.279 0.549841
\(83\) −194.877 −0.257717 −0.128858 − 0.991663i \(-0.541131\pi\)
−0.128858 + 0.991663i \(0.541131\pi\)
\(84\) −756.405 −0.982507
\(85\) 0 0
\(86\) −108.205 −0.135675
\(87\) 230.759 0.284368
\(88\) 7.90397 0.00957461
\(89\) −213.751 −0.254580 −0.127290 − 0.991866i \(-0.540628\pi\)
−0.127290 + 0.991866i \(0.540628\pi\)
\(90\) 0 0
\(91\) −96.5410 −0.111211
\(92\) −92.0000 −0.104257
\(93\) −1132.03 −1.26222
\(94\) −82.8981 −0.0909605
\(95\) 0 0
\(96\) −264.685 −0.281399
\(97\) −628.762 −0.658156 −0.329078 − 0.944303i \(-0.606738\pi\)
−0.329078 + 0.944303i \(0.606738\pi\)
\(98\) −359.348 −0.370404
\(99\) −40.9189 −0.0415405
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.q.1.5 ✓ 5
5.2 odd 4 1150.4.b.r.599.1 10
5.3 odd 4 1150.4.b.r.599.10 10
5.4 even 2 1150.4.a.v.1.1 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.q.1.5 ✓ 5 1.1 even 1 trivial
1150.4.a.v.1.1 yes 5 5.4 even 2
1150.4.b.r.599.1 10 5.2 odd 4
1150.4.b.r.599.10 10 5.3 odd 4