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Properties

Label 1150.2.a.q.1.3
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $0$
Dimension $3$
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 = [3,-3,-1,3,0,1,-3,-3,10] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) 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 1.3
Root \(-3.11903\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.11903 q^{3} +1.00000 q^{4} -3.11903 q^{6} -4.50973 q^{7} -1.00000 q^{8} +6.72833 q^{9} +4.33763 q^{11} +3.11903 q^{12} +3.72833 q^{13} +4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} -6.72833 q^{18} +4.50973 q^{19} -14.0660 q^{21} -4.33763 q^{22} +1.00000 q^{23} -3.11903 q^{24} -3.72833 q^{26} +11.6288 q^{27} -4.50973 q^{28} -8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +13.5292 q^{33} +1.11903 q^{34} +6.72833 q^{36} +0.781399 q^{37} -4.50973 q^{38} +11.6288 q^{39} +3.90043 q^{41} +14.0660 q^{42} -8.00000 q^{43} +4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +3.11903 q^{48} +13.3376 q^{49} -3.49027 q^{51} +3.72833 q^{52} +6.00000 q^{53} -11.6288 q^{54} +4.50973 q^{56} +14.0660 q^{57} +8.23805 q^{58} -2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} -30.3429 q^{63} +1.00000 q^{64} -13.5292 q^{66} -2.43720 q^{67} -1.11903 q^{68} +3.11903 q^{69} +7.11903 q^{71} -6.72833 q^{72} +9.45665 q^{73} -0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -11.6288 q^{78} -14.9133 q^{79} +16.0854 q^{81} -3.90043 q^{82} -2.78140 q^{83} -14.0660 q^{84} +8.00000 q^{86} -25.6947 q^{87} -4.33763 q^{88} -7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} +5.39070 q^{93} -11.4567 q^{94} -3.11903 q^{96} +0.642920 q^{97} -13.3376 q^{98} +29.1850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 3 q^{7} - 3 q^{8} + 10 q^{9} + 3 q^{11} - q^{12} + q^{13} + 3 q^{14} + 3 q^{16} + 7 q^{17} - 10 q^{18} + 3 q^{19} - 22 q^{21} - 3 q^{22} + 3 q^{23} + q^{24}+ \cdots + 57 q^{99}+O(q^{100}) \) 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
\(3\) 3.11903 1.80077 0.900385 − 0.435093i \(-0.143285\pi\)
0.900385 + 0.435093i \(0.143285\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.11903 −1.27334
\(7\) −4.50973 −1.70452 −0.852258 − 0.523122i \(-0.824767\pi\)
−0.852258 + 0.523122i \(0.824767\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.72833 2.24278
\(10\) 0 0
\(11\) 4.33763 1.30784 0.653922 − 0.756562i \(-0.273122\pi\)
0.653922 + 0.756562i \(0.273122\pi\)
\(12\) 3.11903 0.900385
\(13\) 3.72833 1.03405 0.517026 − 0.855970i \(-0.327039\pi\)
0.517026 + 0.855970i \(0.327039\pi\)
\(14\) 4.50973 1.20527
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.11903 −0.271404 −0.135702 − 0.990750i \(-0.543329\pi\)
−0.135702 + 0.990750i \(0.543329\pi\)
\(18\) −6.72833 −1.58588
\(19\) 4.50973 1.03460 0.517301 − 0.855803i \(-0.326937\pi\)
0.517301 + 0.855803i \(0.326937\pi\)
\(20\) 0 0
\(21\) −14.0660 −3.06944
\(22\) −4.33763 −0.924785
\(23\) 1.00000 0.208514
\(24\) −3.11903 −0.636669
\(25\) 0 0
\(26\) −3.72833 −0.731185
\(27\) 11.6288 2.23795
\(28\) −4.50973 −0.852258
\(29\) −8.23805 −1.52977 −0.764884 − 0.644168i \(-0.777204\pi\)
−0.764884 + 0.644168i \(0.777204\pi\)
\(30\) 0 0
\(31\) 1.72833 0.310417 0.155208 − 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(32\) −1.00000 −0.176777
\(33\) 13.5292 2.35513
\(34\) 1.11903 0.191911
\(35\) 0 0
\(36\) 6.72833 1.12139
\(37\) 0.781399 0.128461 0.0642306 − 0.997935i \(-0.479541\pi\)
0.0642306 + 0.997935i \(0.479541\pi\)
\(38\) −4.50973 −0.731574
\(39\) 11.6288 1.86209
\(40\) 0 0
\(41\) 3.90043 0.609144 0.304572 − 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(42\) 14.0660 2.17042
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.33763 0.653922
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 11.4567 1.67112 0.835562 − 0.549396i \(-0.185142\pi\)
0.835562 + 0.549396i \(0.185142\pi\)
\(48\) 3.11903 0.450193
\(49\) 13.3376 1.90538
\(50\) 0 0
\(51\) −3.49027 −0.488736
\(52\) 3.72833 0.517026
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −11.6288 −1.58247
\(55\) 0 0
\(56\) 4.50973 0.602637
\(57\) 14.0660 1.86308
\(58\) 8.23805 1.08171
\(59\) −2.23805 −0.291370 −0.145685 − 0.989331i \(-0.546539\pi\)
−0.145685 + 0.989331i \(0.546539\pi\)
\(60\) 0 0
\(61\) 3.55623 0.455329 0.227664 − 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(62\) −1.72833 −0.219498
\(63\) −30.3429 −3.82285
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −13.5292 −1.66533
\(67\) −2.43720 −0.297752 −0.148876 − 0.988856i \(-0.547565\pi\)
−0.148876 + 0.988856i \(0.547565\pi\)
\(68\) −1.11903 −0.135702
\(69\) 3.11903 0.375487
\(70\) 0 0
\(71\) 7.11903 0.844873 0.422437 − 0.906393i \(-0.361175\pi\)
0.422437 + 0.906393i \(0.361175\pi\)
\(72\) −6.72833 −0.792941
\(73\) 9.45665 1.10682 0.553409 − 0.832910i \(-0.313327\pi\)
0.553409 + 0.832910i \(0.313327\pi\)
\(74\) −0.781399 −0.0908357
\(75\) 0 0
\(76\) 4.50973 0.517301
\(77\) −19.5615 −2.22924
\(78\) −11.6288 −1.31670
\(79\) −14.9133 −1.67788 −0.838939 − 0.544225i \(-0.816824\pi\)
−0.838939 + 0.544225i \(0.816824\pi\)
\(80\) 0 0
\(81\) 16.0854 1.78727
\(82\) −3.90043 −0.430730
\(83\) −2.78140 −0.305298 −0.152649 − 0.988280i \(-0.548780\pi\)
−0.152649 + 0.988280i \(0.548780\pi\)
\(84\) −14.0660 −1.53472
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −25.6947 −2.75476
\(88\) −4.33763 −0.462393
\(89\) −7.69471 −0.815637 −0.407819 − 0.913063i \(-0.633710\pi\)
−0.407819 + 0.913063i \(0.633710\pi\)
\(90\) 0 0
\(91\) −16.8137 −1.76256
\(92\) 1.00000 0.104257
\(93\) 5.39070 0.558989
\(94\) −11.4567 −1.18166
\(95\) 0 0
\(96\) −3.11903 −0.318334
\(97\) 0.642920 0.0652786 0.0326393 − 0.999467i \(-0.489609\pi\)
0.0326393 + 0.999467i \(0.489609\pi\)
\(98\) −13.3376 −1.34730
\(99\) 29.1850 2.93320
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.q.1.3 3
4.3 odd 2 9200.2.a.cf.1.1 3
5.2 odd 4 1150.2.b.j.599.1 6
5.3 odd 4 1150.2.b.j.599.6 6
5.4 even 2 230.2.a.d.1.1 ✓ 3
15.14 odd 2 2070.2.a.z.1.3 3
20.19 odd 2 1840.2.a.r.1.3 3
40.19 odd 2 7360.2.a.ce.1.1 3
40.29 even 2 7360.2.a.bz.1.3 3
115.114 odd 2 5290.2.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 ✓ 3 5.4 even 2
1150.2.a.q.1.3 3 1.1 even 1 trivial
1150.2.b.j.599.1 6 5.2 odd 4
1150.2.b.j.599.6 6 5.3 odd 4
1840.2.a.r.1.3 3 20.19 odd 2
2070.2.a.z.1.3 3 15.14 odd 2
5290.2.a.r.1.1 3 115.114 odd 2
7360.2.a.bz.1.3 3 40.29 even 2
7360.2.a.ce.1.1 3 40.19 odd 2
9200.2.a.cf.1.1 3 4.3 odd 2