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Properties

Label 1150.2.a.o.1.2
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $0$
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 = [2,2,1,2,0,1,-1,2,5] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 1.2
Root \(2.79129\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} +2.79129 q^{6} +1.79129 q^{7} +1.00000 q^{8} +4.79129 q^{9} -0.791288 q^{11} +2.79129 q^{12} -5.79129 q^{13} +1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +4.79129 q^{18} +5.79129 q^{19} +5.00000 q^{21} -0.791288 q^{22} -1.00000 q^{23} +2.79129 q^{24} -5.79129 q^{26} +5.00000 q^{27} +1.79129 q^{28} +7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -2.20871 q^{33} -0.791288 q^{34} +4.79129 q^{36} +4.00000 q^{37} +5.79129 q^{38} -16.1652 q^{39} -6.79129 q^{41} +5.00000 q^{42} -11.1652 q^{43} -0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} +2.79129 q^{48} -3.79129 q^{49} -2.20871 q^{51} -5.79129 q^{52} -6.00000 q^{53} +5.00000 q^{54} +1.79129 q^{56} +16.1652 q^{57} +7.58258 q^{58} -13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +8.58258 q^{63} +1.00000 q^{64} -2.20871 q^{66} -11.1652 q^{67} -0.791288 q^{68} -2.79129 q^{69} +8.37386 q^{71} +4.79129 q^{72} -12.7477 q^{73} +4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} -16.1652 q^{78} +8.00000 q^{79} -0.417424 q^{81} -6.79129 q^{82} +6.00000 q^{83} +5.00000 q^{84} -11.1652 q^{86} +21.1652 q^{87} -0.791288 q^{88} +15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} -9.41742 q^{93} +4.41742 q^{94} +2.79129 q^{96} +7.95644 q^{97} -3.79129 q^{98} -3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} - q^{7} + 2 q^{8} + 5 q^{9} + 3 q^{11} + q^{12} - 7 q^{13} - q^{14} + 2 q^{16} + 3 q^{17} + 5 q^{18} + 7 q^{19} + 10 q^{21} + 3 q^{22} - 2 q^{23} + q^{24} - 7 q^{26}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(3\) 2.79129 1.61155 0.805775 − 0.592221i \(-0.201749\pi\)
0.805775 + 0.592221i \(0.201749\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.79129 1.13954
\(7\) 1.79129 0.677043 0.338522 − 0.940959i \(-0.390073\pi\)
0.338522 + 0.940959i \(0.390073\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 − 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) 2.79129 0.805775
\(13\) −5.79129 −1.60621 −0.803107 − 0.595835i \(-0.796821\pi\)
−0.803107 + 0.595835i \(0.796821\pi\)
\(14\) 1.79129 0.478742
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.791288 −0.191915 −0.0959577 − 0.995385i \(-0.530591\pi\)
−0.0959577 + 0.995385i \(0.530591\pi\)
\(18\) 4.79129 1.12932
\(19\) 5.79129 1.32861 0.664306 − 0.747460i \(-0.268727\pi\)
0.664306 + 0.747460i \(0.268727\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) −0.791288 −0.168703
\(23\) −1.00000 −0.208514
\(24\) 2.79129 0.569769
\(25\) 0 0
\(26\) −5.79129 −1.13576
\(27\) 5.00000 0.962250
\(28\) 1.79129 0.338522
\(29\) 7.58258 1.40805 0.704024 − 0.710176i \(-0.251385\pi\)
0.704024 + 0.710176i \(0.251385\pi\)
\(30\) 0 0
\(31\) −3.37386 −0.605964 −0.302982 − 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.20871 −0.384487
\(34\) −0.791288 −0.135705
\(35\) 0 0
\(36\) 4.79129 0.798548
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 5.79129 0.939471
\(39\) −16.1652 −2.58850
\(40\) 0 0
\(41\) −6.79129 −1.06062 −0.530310 − 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(42\) 5.00000 0.771517
\(43\) −11.1652 −1.70267 −0.851335 − 0.524623i \(-0.824206\pi\)
−0.851335 + 0.524623i \(0.824206\pi\)
\(44\) −0.791288 −0.119291
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.41742 0.644348 0.322174 − 0.946681i \(-0.395586\pi\)
0.322174 + 0.946681i \(0.395586\pi\)
\(48\) 2.79129 0.402888
\(49\) −3.79129 −0.541613
\(50\) 0 0
\(51\) −2.20871 −0.309282
\(52\) −5.79129 −0.803107
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 1.79129 0.239371
\(57\) 16.1652 2.14113
\(58\) 7.58258 0.995641
\(59\) −13.5826 −1.76830 −0.884150 − 0.467202i \(-0.845262\pi\)
−0.884150 + 0.467202i \(0.845262\pi\)
\(60\) 0 0
\(61\) 10.3739 1.32824 0.664119 − 0.747627i \(-0.268807\pi\)
0.664119 + 0.747627i \(0.268807\pi\)
\(62\) −3.37386 −0.428481
\(63\) 8.58258 1.08130
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.20871 −0.271874
\(67\) −11.1652 −1.36404 −0.682020 − 0.731333i \(-0.738898\pi\)
−0.682020 + 0.731333i \(0.738898\pi\)
\(68\) −0.791288 −0.0959577
\(69\) −2.79129 −0.336032
\(70\) 0 0
\(71\) 8.37386 0.993795 0.496897 − 0.867809i \(-0.334473\pi\)
0.496897 + 0.867809i \(0.334473\pi\)
\(72\) 4.79129 0.564659
\(73\) −12.7477 −1.49201 −0.746004 − 0.665941i \(-0.768030\pi\)
−0.746004 + 0.665941i \(0.768030\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 5.79129 0.664306
\(77\) −1.41742 −0.161530
\(78\) −16.1652 −1.83034
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) −6.79129 −0.749972
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 5.00000 0.545545
\(85\) 0 0
\(86\) −11.1652 −1.20397
\(87\) 21.1652 2.26914
\(88\) −0.791288 −0.0843516
\(89\) 15.1652 1.60750 0.803751 − 0.594965i \(-0.202834\pi\)
0.803751 + 0.594965i \(0.202834\pi\)
\(90\) 0 0
\(91\) −10.3739 −1.08748
\(92\) −1.00000 −0.104257
\(93\) −9.41742 −0.976541
\(94\) 4.41742 0.455623
\(95\) 0 0
\(96\) 2.79129 0.284885
\(97\) 7.95644 0.807854 0.403927 − 0.914791i \(-0.367645\pi\)
0.403927 + 0.914791i \(0.367645\pi\)
\(98\) −3.79129 −0.382978
\(99\) −3.79129 −0.381039
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.o.1.2 2
4.3 odd 2 9200.2.a.bs.1.1 2
5.2 odd 4 1150.2.b.g.599.3 4
5.3 odd 4 1150.2.b.g.599.2 4
5.4 even 2 230.2.a.a.1.1 ✓ 2
15.14 odd 2 2070.2.a.x.1.1 2
20.19 odd 2 1840.2.a.n.1.2 2
40.19 odd 2 7360.2.a.bk.1.1 2
40.29 even 2 7360.2.a.bq.1.2 2
115.114 odd 2 5290.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 ✓ 2 5.4 even 2
1150.2.a.o.1.2 2 1.1 even 1 trivial
1150.2.b.g.599.2 4 5.3 odd 4
1150.2.b.g.599.3 4 5.2 odd 4
1840.2.a.n.1.2 2 20.19 odd 2
2070.2.a.x.1.1 2 15.14 odd 2
5290.2.a.e.1.1 2 115.114 odd 2
7360.2.a.bk.1.1 2 40.19 odd 2
7360.2.a.bq.1.2 2 40.29 even 2
9200.2.a.bs.1.1 2 4.3 odd 2