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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,2,Mod(599,1150)] 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("1150.599"); 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([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,2,0,0,6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) 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)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.i.599.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.61803i q^{3} -1.00000 q^{4} +1.61803 q^{6} +0.618034i q^{7} +1.00000i q^{8} +0.381966 q^{9} -2.85410 q^{11} -1.61803i q^{12} -7.09017i q^{13} +0.618034 q^{14} +1.00000 q^{16} -6.09017i q^{17} -0.381966i q^{18} -1.85410 q^{19} -1.00000 q^{21} +2.85410i q^{22} +1.00000i q^{23} -1.61803 q^{24} -7.09017 q^{26} +5.47214i q^{27} -0.618034i q^{28} +9.23607 q^{29} +9.09017 q^{31} -1.00000i q^{32} -4.61803i q^{33} -6.09017 q^{34} -0.381966 q^{36} -6.47214i q^{37} +1.85410i q^{38} +11.4721 q^{39} +3.32624 q^{41} +1.00000i q^{42} +2.85410 q^{44} +1.00000 q^{46} +3.70820i q^{47} +1.61803i q^{48} +6.61803 q^{49} +9.85410 q^{51} +7.09017i q^{52} +0.472136i q^{53} +5.47214 q^{54} -0.618034 q^{56} -3.00000i q^{57} -9.23607i q^{58} -1.70820 q^{59} -9.32624 q^{61} -9.09017i q^{62} +0.236068i q^{63} -1.00000 q^{64} -4.61803 q^{66} -14.4721i q^{67} +6.09017i q^{68} -1.61803 q^{69} -4.09017 q^{71} +0.381966i q^{72} +3.23607i q^{73} -6.47214 q^{74} +1.85410 q^{76} -1.76393i q^{77} -11.4721i q^{78} -1.52786 q^{79} -7.70820 q^{81} -3.32624i q^{82} -6.94427i q^{83} +1.00000 q^{84} +14.9443i q^{87} -2.85410i q^{88} +10.4721 q^{89} +4.38197 q^{91} -1.00000i q^{92} +14.7082i q^{93} +3.70820 q^{94} +1.61803 q^{96} -12.3820i q^{97} -6.61803i q^{98} -1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 4 q^{21} - 2 q^{24} - 6 q^{26} + 28 q^{29} + 14 q^{31} - 2 q^{34} - 6 q^{36} + 28 q^{39} - 18 q^{41} - 2 q^{44} + 4 q^{46}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.61803i 0.934172i 0.884212 + 0.467086i \(0.154696\pi\)
−0.884212 + 0.467086i \(0.845304\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.61803 0.660560
\(7\) 0.618034i 0.233595i 0.993156 + 0.116797i \(0.0372628\pi\)
−0.993156 + 0.116797i \(0.962737\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0.381966 0.127322
\(10\) 0 0
\(11\) −2.85410 −0.860544 −0.430272 − 0.902699i \(-0.641582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(12\) − 1.61803i − 0.467086i
\(13\) − 7.09017i − 1.96646i −0.182372 − 0.983230i \(-0.558377\pi\)
0.182372 − 0.983230i \(-0.441623\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.09017i − 1.47708i −0.674208 − 0.738542i \(-0.735515\pi\)
0.674208 − 0.738542i \(-0.264485\pi\)
\(18\) − 0.381966i − 0.0900303i
\(19\) −1.85410 −0.425360 −0.212680 − 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.85410i 0.608497i
\(23\) 1.00000i 0.208514i
\(24\) −1.61803 −0.330280
\(25\) 0 0
\(26\) −7.09017 −1.39050
\(27\) 5.47214i 1.05311i
\(28\) − 0.618034i − 0.116797i
\(29\) 9.23607 1.71509 0.857547 − 0.514405i \(-0.171987\pi\)
0.857547 + 0.514405i \(0.171987\pi\)
\(30\) 0 0
\(31\) 9.09017 1.63264 0.816321 − 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.61803i − 0.803897i
\(34\) −6.09017 −1.04446
\(35\) 0 0
\(36\) −0.381966 −0.0636610
\(37\) − 6.47214i − 1.06401i −0.846740 − 0.532006i \(-0.821438\pi\)
0.846740 − 0.532006i \(-0.178562\pi\)
\(38\) 1.85410i 0.300775i
\(39\) 11.4721 1.83701
\(40\) 0 0
\(41\) 3.32624 0.519471 0.259736 − 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 2.85410 0.430272
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.70820i 0.540897i 0.962734 + 0.270449i \(0.0871720\pi\)
−0.962734 + 0.270449i \(0.912828\pi\)
\(48\) 1.61803i 0.233543i
\(49\) 6.61803 0.945433
\(50\) 0 0
\(51\) 9.85410 1.37985
\(52\) 7.09017i 0.983230i
\(53\) 0.472136i 0.0648529i 0.999474 + 0.0324264i \(0.0103235\pi\)
−0.999474 + 0.0324264i \(0.989677\pi\)
\(54\) 5.47214 0.744663
\(55\) 0 0
\(56\) −0.618034 −0.0825883
\(57\) − 3.00000i − 0.397360i
\(58\) − 9.23607i − 1.21276i
\(59\) −1.70820 −0.222389 −0.111195 − 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) −9.32624 −1.19410 −0.597051 − 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) − 9.09017i − 1.15445i
\(63\) 0.236068i 0.0297418i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.61803 −0.568441
\(67\) − 14.4721i − 1.76805i −0.467437 − 0.884026i \(-0.654823\pi\)
0.467437 − 0.884026i \(-0.345177\pi\)
\(68\) 6.09017i 0.738542i
\(69\) −1.61803 −0.194788
\(70\) 0 0
\(71\) −4.09017 −0.485414 −0.242707 − 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) 0.381966i 0.0450151i
\(73\) 3.23607i 0.378753i 0.981905 + 0.189377i \(0.0606467\pi\)
−0.981905 + 0.189377i \(0.939353\pi\)
\(74\) −6.47214 −0.752371
\(75\) 0 0
\(76\) 1.85410 0.212680
\(77\) − 1.76393i − 0.201019i
\(78\) − 11.4721i − 1.29896i
\(79\) −1.52786 −0.171898 −0.0859491 − 0.996300i \(-0.527392\pi\)
−0.0859491 + 0.996300i \(0.527392\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) − 3.32624i − 0.367322i
\(83\) − 6.94427i − 0.762233i −0.924527 − 0.381116i \(-0.875540\pi\)
0.924527 − 0.381116i \(-0.124460\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 0 0
\(87\) 14.9443i 1.60219i
\(88\) − 2.85410i − 0.304248i
\(89\) 10.4721 1.11004 0.555022 − 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 4.38197 0.459355
\(92\) − 1.00000i − 0.104257i
\(93\) 14.7082i 1.52517i
\(94\) 3.70820 0.382472
\(95\) 0 0
\(96\) 1.61803 0.165140
\(97\) − 12.3820i − 1.25720i −0.777730 − 0.628599i \(-0.783629\pi\)
0.777730 − 0.628599i \(-0.216371\pi\)
\(98\) − 6.61803i − 0.668522i
\(99\) −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.b.i.599.2 4
5.2 odd 4 230.2.a.c.1.2 ✓ 2
5.3 odd 4 1150.2.a.j.1.1 2
5.4 even 2 inner 1150.2.b.i.599.3 4
15.2 even 4 2070.2.a.u.1.1 2
20.3 even 4 9200.2.a.bu.1.2 2
20.7 even 4 1840.2.a.l.1.1 2
40.27 even 4 7360.2.a.bn.1.2 2
40.37 odd 4 7360.2.a.bh.1.1 2
115.22 even 4 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 ✓ 2 5.2 odd 4
1150.2.a.j.1.1 2 5.3 odd 4
1150.2.b.i.599.2 4 1.1 even 1 trivial
1150.2.b.i.599.3 4 5.4 even 2 inner
1840.2.a.l.1.1 2 20.7 even 4
2070.2.a.u.1.1 2 15.2 even 4
5290.2.a.o.1.2 2 115.22 even 4
7360.2.a.bh.1.1 2 40.37 odd 4
7360.2.a.bn.1.2 2 40.27 even 4
9200.2.a.bu.1.2 2 20.3 even 4