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Properties

Label 1150.2.a.m.1.1
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $1$
Dimension $2$
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] = [1150,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-3,2,0,-3,-3,2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.18279623245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} -3.30278 q^{6} +0.302776 q^{7} +1.00000 q^{8} +7.90833 q^{9} -5.30278 q^{11} -3.30278 q^{12} +0.302776 q^{13} +0.302776 q^{14} +1.00000 q^{16} +3.90833 q^{17} +7.90833 q^{18} -4.90833 q^{19} -1.00000 q^{21} -5.30278 q^{22} +1.00000 q^{23} -3.30278 q^{24} +0.302776 q^{26} -16.2111 q^{27} +0.302776 q^{28} +4.60555 q^{29} +2.90833 q^{31} +1.00000 q^{32} +17.5139 q^{33} +3.90833 q^{34} +7.90833 q^{36} -8.00000 q^{37} -4.90833 q^{38} -1.00000 q^{39} -9.90833 q^{41} -1.00000 q^{42} -5.21110 q^{43} -5.30278 q^{44} +1.00000 q^{46} -4.60555 q^{47} -3.30278 q^{48} -6.90833 q^{49} -12.9083 q^{51} +0.302776 q^{52} -3.21110 q^{53} -16.2111 q^{54} +0.302776 q^{56} +16.2111 q^{57} +4.60555 q^{58} -10.6056 q^{59} -6.51388 q^{61} +2.90833 q^{62} +2.39445 q^{63} +1.00000 q^{64} +17.5139 q^{66} +4.00000 q^{67} +3.90833 q^{68} -3.30278 q^{69} -12.6972 q^{71} +7.90833 q^{72} -15.8167 q^{73} -8.00000 q^{74} -4.90833 q^{76} -1.60555 q^{77} -1.00000 q^{78} +14.4222 q^{79} +29.8167 q^{81} -9.90833 q^{82} +3.21110 q^{83} -1.00000 q^{84} -5.21110 q^{86} -15.2111 q^{87} -5.30278 q^{88} +0.0916731 q^{91} +1.00000 q^{92} -9.60555 q^{93} -4.60555 q^{94} -3.30278 q^{96} -2.69722 q^{97} -6.90833 q^{98} -41.9361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 3 q^{7} + 2 q^{8} + 5 q^{9} - 7 q^{11} - 3 q^{12} - 3 q^{13} - 3 q^{14} + 2 q^{16} - 3 q^{17} + 5 q^{18} + q^{19} - 2 q^{21} - 7 q^{22} + 2 q^{23} - 3 q^{24}+ \cdots - 37 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.30278 −1.90686 −0.953429 − 0.301617i \(-0.902474\pi\)
−0.953429 + 0.301617i \(0.902474\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.30278 −1.34835
\(7\) 0.302776 0.114438 0.0572192 − 0.998362i \(-0.481777\pi\)
0.0572192 + 0.998362i \(0.481777\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.90833 2.63611
\(10\) 0 0
\(11\) −5.30278 −1.59885 −0.799424 − 0.600768i \(-0.794862\pi\)
−0.799424 + 0.600768i \(0.794862\pi\)
\(12\) −3.30278 −0.953429
\(13\) 0.302776 0.0839749 0.0419874 − 0.999118i \(-0.486631\pi\)
0.0419874 + 0.999118i \(0.486631\pi\)
\(14\) 0.302776 0.0809202
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.90833 0.947909 0.473954 − 0.880549i \(-0.342826\pi\)
0.473954 + 0.880549i \(0.342826\pi\)
\(18\) 7.90833 1.86401
\(19\) −4.90833 −1.12605 −0.563024 − 0.826441i \(-0.690362\pi\)
−0.563024 + 0.826441i \(0.690362\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −5.30278 −1.13056
\(23\) 1.00000 0.208514
\(24\) −3.30278 −0.674176
\(25\) 0 0
\(26\) 0.302776 0.0593792
\(27\) −16.2111 −3.11983
\(28\) 0.302776 0.0572192
\(29\) 4.60555 0.855229 0.427615 − 0.903961i \(-0.359354\pi\)
0.427615 + 0.903961i \(0.359354\pi\)
\(30\) 0 0
\(31\) 2.90833 0.522351 0.261175 − 0.965291i \(-0.415890\pi\)
0.261175 + 0.965291i \(0.415890\pi\)
\(32\) 1.00000 0.176777
\(33\) 17.5139 3.04877
\(34\) 3.90833 0.670273
\(35\) 0 0
\(36\) 7.90833 1.31805
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −4.90833 −0.796236
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −9.90833 −1.54742 −0.773710 − 0.633540i \(-0.781601\pi\)
−0.773710 + 0.633540i \(0.781601\pi\)
\(42\) −1.00000 −0.154303
\(43\) −5.21110 −0.794686 −0.397343 − 0.917670i \(-0.630068\pi\)
−0.397343 + 0.917670i \(0.630068\pi\)
\(44\) −5.30278 −0.799424
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −4.60555 −0.671789 −0.335894 − 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) −3.30278 −0.476715
\(49\) −6.90833 −0.986904
\(50\) 0 0
\(51\) −12.9083 −1.80753
\(52\) 0.302776 0.0419874
\(53\) −3.21110 −0.441079 −0.220539 − 0.975378i \(-0.570782\pi\)
−0.220539 + 0.975378i \(0.570782\pi\)
\(54\) −16.2111 −2.20605
\(55\) 0 0
\(56\) 0.302776 0.0404601
\(57\) 16.2111 2.14721
\(58\) 4.60555 0.604739
\(59\) −10.6056 −1.38073 −0.690363 − 0.723464i \(-0.742549\pi\)
−0.690363 + 0.723464i \(0.742549\pi\)
\(60\) 0 0
\(61\) −6.51388 −0.834017 −0.417008 − 0.908903i \(-0.636921\pi\)
−0.417008 + 0.908903i \(0.636921\pi\)
\(62\) 2.90833 0.369358
\(63\) 2.39445 0.301672
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 17.5139 2.15581
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 3.90833 0.473954
\(69\) −3.30278 −0.397607
\(70\) 0 0
\(71\) −12.6972 −1.50688 −0.753442 − 0.657515i \(-0.771608\pi\)
−0.753442 + 0.657515i \(0.771608\pi\)
\(72\) 7.90833 0.932005
\(73\) −15.8167 −1.85120 −0.925600 − 0.378504i \(-0.876439\pi\)
−0.925600 + 0.378504i \(0.876439\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −4.90833 −0.563024
\(77\) −1.60555 −0.182970
\(78\) −1.00000 −0.113228
\(79\) 14.4222 1.62262 0.811312 − 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) −9.90833 −1.09419
\(83\) 3.21110 0.352464 0.176232 − 0.984349i \(-0.443609\pi\)
0.176232 + 0.984349i \(0.443609\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −5.21110 −0.561928
\(87\) −15.2111 −1.63080
\(88\) −5.30278 −0.565278
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0.0916731 0.00960995
\(92\) 1.00000 0.104257
\(93\) −9.60555 −0.996049
\(94\) −4.60555 −0.475026
\(95\) 0 0
\(96\) −3.30278 −0.337088
\(97\) −2.69722 −0.273862 −0.136931 − 0.990581i \(-0.543724\pi\)
−0.136931 + 0.990581i \(0.543724\pi\)
\(98\) −6.90833 −0.697846
\(99\) −41.9361 −4.21473
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 1150.2.a.m.1.1 2
4.3 odd 2 9200.2.a.ca.1.2 2
5.2 odd 4 1150.2.b.f.599.4 4
5.3 odd 4 1150.2.b.f.599.1 4
5.4 even 2 230.2.a.b.1.2 ✓ 2
15.14 odd 2 2070.2.a.w.1.1 2
20.19 odd 2 1840.2.a.j.1.1 2
40.19 odd 2 7360.2.a.bu.1.2 2
40.29 even 2 7360.2.a.bc.1.1 2
115.114 odd 2 5290.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.2 ✓ 2 5.4 even 2
1150.2.a.m.1.1 2 1.1 even 1 trivial
1150.2.b.f.599.1 4 5.3 odd 4
1150.2.b.f.599.4 4 5.2 odd 4
1840.2.a.j.1.1 2 20.19 odd 2
2070.2.a.w.1.1 2 15.14 odd 2
5290.2.a.j.1.2 2 115.114 odd 2
7360.2.a.bc.1.1 2 40.29 even 2
7360.2.a.bu.1.2 2 40.19 odd 2
9200.2.a.ca.1.2 2 4.3 odd 2