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,-3] 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: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +1.61803 q^{6} +0.618034 q^{7} -1.00000 q^{8} -0.381966 q^{9} -2.85410 q^{11} -1.61803 q^{12} +7.09017 q^{13} -0.618034 q^{14} +1.00000 q^{16} -6.09017 q^{17} +0.381966 q^{18} +1.85410 q^{19} -1.00000 q^{21} +2.85410 q^{22} -1.00000 q^{23} +1.61803 q^{24} -7.09017 q^{26} +5.47214 q^{27} +0.618034 q^{28} -9.23607 q^{29} +9.09017 q^{31} -1.00000 q^{32} +4.61803 q^{33} +6.09017 q^{34} -0.381966 q^{36} -6.47214 q^{37} -1.85410 q^{38} -11.4721 q^{39} +3.32624 q^{41} +1.00000 q^{42} -2.85410 q^{44} +1.00000 q^{46} +3.70820 q^{47} -1.61803 q^{48} -6.61803 q^{49} +9.85410 q^{51} +7.09017 q^{52} -0.472136 q^{53} -5.47214 q^{54} -0.618034 q^{56} -3.00000 q^{57} +9.23607 q^{58} +1.70820 q^{59} -9.32624 q^{61} -9.09017 q^{62} -0.236068 q^{63} +1.00000 q^{64} -4.61803 q^{66} -14.4721 q^{67} -6.09017 q^{68} +1.61803 q^{69} -4.09017 q^{71} +0.381966 q^{72} -3.23607 q^{73} +6.47214 q^{74} +1.85410 q^{76} -1.76393 q^{77} +11.4721 q^{78} +1.52786 q^{79} -7.70820 q^{81} -3.32624 q^{82} +6.94427 q^{83} -1.00000 q^{84} +14.9443 q^{87} +2.85410 q^{88} -10.4721 q^{89} +4.38197 q^{91} -1.00000 q^{92} -14.7082 q^{93} -3.70820 q^{94} +1.61803 q^{96} -12.3820 q^{97} +6.61803 q^{98} +1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{6} - q^{7} - 2 q^{8} - 3 q^{9} + q^{11} - q^{12} + 3 q^{13} + q^{14} + 2 q^{16} - q^{17} + 3 q^{18} - 3 q^{19} - 2 q^{21} - q^{22} - 2 q^{23} + q^{24} - 3 q^{26}+ \cdots - 9 q^{99}+O(q^{100}) \) 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
\(3\) −1.61803 −0.934172 −0.467086 − 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.61803 0.660560
\(7\) 0.618034 0.233595 0.116797 − 0.993156i \(-0.462737\pi\)
0.116797 + 0.993156i \(0.462737\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) −2.85410 −0.860544 −0.430272 − 0.902699i \(-0.641582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(12\) −1.61803 −0.467086
\(13\) 7.09017 1.96646 0.983230 − 0.182372i \(-0.0583774\pi\)
0.983230 + 0.182372i \(0.0583774\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.09017 −1.47708 −0.738542 − 0.674208i \(-0.764485\pi\)
−0.738542 + 0.674208i \(0.764485\pi\)
\(18\) 0.381966 0.0900303
\(19\) 1.85410 0.425360 0.212680 − 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.85410 0.608497
\(23\) −1.00000 −0.208514
\(24\) 1.61803 0.330280
\(25\) 0 0
\(26\) −7.09017 −1.39050
\(27\) 5.47214 1.05311
\(28\) 0.618034 0.116797
\(29\) −9.23607 −1.71509 −0.857547 − 0.514405i \(-0.828013\pi\)
−0.857547 + 0.514405i \(0.828013\pi\)
\(30\) 0 0
\(31\) 9.09017 1.63264 0.816321 − 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.61803 0.803897
\(34\) 6.09017 1.04446
\(35\) 0 0
\(36\) −0.381966 −0.0636610
\(37\) −6.47214 −1.06401 −0.532006 − 0.846740i \(-0.678562\pi\)
−0.532006 + 0.846740i \(0.678562\pi\)
\(38\) −1.85410 −0.300775
\(39\) −11.4721 −1.83701
\(40\) 0 0
\(41\) 3.32624 0.519471 0.259736 − 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(42\) 1.00000 0.154303
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.85410 −0.430272
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.70820 0.540897 0.270449 − 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(48\) −1.61803 −0.233543
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) 9.85410 1.37985
\(52\) 7.09017 0.983230
\(53\) −0.472136 −0.0648529 −0.0324264 − 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) −5.47214 −0.744663
\(55\) 0 0
\(56\) −0.618034 −0.0825883
\(57\) −3.00000 −0.397360
\(58\) 9.23607 1.21276
\(59\) 1.70820 0.222389 0.111195 − 0.993799i \(-0.464532\pi\)
0.111195 + 0.993799i \(0.464532\pi\)
\(60\) 0 0
\(61\) −9.32624 −1.19410 −0.597051 − 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) −9.09017 −1.15445
\(63\) −0.236068 −0.0297418
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.61803 −0.568441
\(67\) −14.4721 −1.76805 −0.884026 − 0.467437i \(-0.845177\pi\)
−0.884026 + 0.467437i \(0.845177\pi\)
\(68\) −6.09017 −0.738542
\(69\) 1.61803 0.194788
\(70\) 0 0
\(71\) −4.09017 −0.485414 −0.242707 − 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) 0.381966 0.0450151
\(73\) −3.23607 −0.378753 −0.189377 − 0.981905i \(-0.560647\pi\)
−0.189377 + 0.981905i \(0.560647\pi\)
\(74\) 6.47214 0.752371
\(75\) 0 0
\(76\) 1.85410 0.212680
\(77\) −1.76393 −0.201019
\(78\) 11.4721 1.29896
\(79\) 1.52786 0.171898 0.0859491 − 0.996300i \(-0.472608\pi\)
0.0859491 + 0.996300i \(0.472608\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) −3.32624 −0.367322
\(83\) 6.94427 0.762233 0.381116 − 0.924527i \(-0.375540\pi\)
0.381116 + 0.924527i \(0.375540\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 0 0
\(87\) 14.9443 1.60219
\(88\) 2.85410 0.304248
\(89\) −10.4721 −1.11004 −0.555022 − 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 0 0
\(91\) 4.38197 0.459355
\(92\) −1.00000 −0.104257
\(93\) −14.7082 −1.52517
\(94\) −3.70820 −0.382472
\(95\) 0 0
\(96\) 1.61803 0.165140
\(97\) −12.3820 −1.25720 −0.628599 − 0.777730i \(-0.716371\pi\)
−0.628599 + 0.777730i \(0.716371\pi\)
\(98\) 6.61803 0.668522
\(99\) 1.09017 0.109566
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.j.1.1 2
4.3 odd 2 9200.2.a.bu.1.2 2
5.2 odd 4 1150.2.b.i.599.2 4
5.3 odd 4 1150.2.b.i.599.3 4
5.4 even 2 230.2.a.c.1.2 ✓ 2
15.14 odd 2 2070.2.a.u.1.1 2
20.19 odd 2 1840.2.a.l.1.1 2
40.19 odd 2 7360.2.a.bn.1.2 2
40.29 even 2 7360.2.a.bh.1.1 2
115.114 odd 2 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 ✓ 2 5.4 even 2
1150.2.a.j.1.1 2 1.1 even 1 trivial
1150.2.b.i.599.2 4 5.2 odd 4
1150.2.b.i.599.3 4 5.3 odd 4
1840.2.a.l.1.1 2 20.19 odd 2
2070.2.a.u.1.1 2 15.14 odd 2
5290.2.a.o.1.2 2 115.114 odd 2
7360.2.a.bh.1.1 2 40.29 even 2
7360.2.a.bn.1.2 2 40.19 odd 2
9200.2.a.bu.1.2 2 4.3 odd 2