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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1450,2,Mod(1101,1450)] 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("1450.1101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,-2,4,0,4,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1101.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1450.1101
Dual form 1450.2.c.a.1101.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000i q^{8} +2.00000 q^{9} +5.00000i q^{11} -1.00000i q^{12} +1.00000 q^{13} +2.00000i q^{14} +1.00000 q^{16} -2.00000i q^{17} +2.00000i q^{18} -4.00000i q^{19} +2.00000i q^{21} -5.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000i q^{26} +5.00000i q^{27} -2.00000 q^{28} +(5.00000 - 2.00000i) q^{29} -5.00000i q^{31} +1.00000i q^{32} -5.00000 q^{33} +2.00000 q^{34} -2.00000 q^{36} +8.00000i q^{37} +4.00000 q^{38} +1.00000i q^{39} +10.0000i q^{41} -2.00000 q^{42} -9.00000i q^{43} -5.00000i q^{44} +6.00000i q^{46} +3.00000i q^{47} +1.00000i q^{48} -3.00000 q^{49} +2.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} -5.00000 q^{54} -2.00000i q^{56} +4.00000 q^{57} +(2.00000 + 5.00000i) q^{58} +10.0000 q^{59} +10.0000i q^{61} +5.00000 q^{62} +4.00000 q^{63} -1.00000 q^{64} -5.00000i q^{66} -8.00000 q^{67} +2.00000i q^{68} +6.00000i q^{69} -8.00000 q^{71} -2.00000i q^{72} +16.0000i q^{73} -8.00000 q^{74} +4.00000i q^{76} +10.0000i q^{77} -1.00000 q^{78} +1.00000i q^{79} +1.00000 q^{81} -10.0000 q^{82} -14.0000 q^{83} -2.00000i q^{84} +9.00000 q^{86} +(2.00000 + 5.00000i) q^{87} +5.00000 q^{88} -14.0000i q^{89} +2.00000 q^{91} -6.00000 q^{92} +5.00000 q^{93} -3.00000 q^{94} -1.00000 q^{96} -2.00000i q^{97} -3.00000i q^{98} +10.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{16} - 10 q^{22} + 12 q^{23} + 2 q^{24} - 4 q^{28} + 10 q^{29} - 10 q^{33} + 4 q^{34} - 4 q^{36} + 8 q^{38} - 4 q^{42} - 6 q^{49} + 4 q^{51}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 5.00000i 1.50756i 0.657129 + 0.753778i \(0.271771\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 2.00000i 0.471405i
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) −5.00000 −1.06600
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000i 0.196116i
\(27\) 5.00000i 0.962250i
\(28\) −2.00000 −0.377964
\(29\) 5.00000 − 2.00000i 0.928477 − 0.371391i
\(30\) 0 0
\(31\) − 5.00000i − 0.898027i −0.893525 − 0.449013i \(-0.851776\pi\)
0.893525 − 0.449013i \(-0.148224\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −5.00000 −0.870388
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 4.00000 0.648886
\(39\) 1.00000i 0.160128i
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) −2.00000 −0.308607
\(43\) − 9.00000i − 1.37249i −0.727372 − 0.686244i \(-0.759258\pi\)
0.727372 − 0.686244i \(-0.240742\pi\)
\(44\) − 5.00000i − 0.753778i
\(45\) 0 0
\(46\) 6.00000i 0.884652i
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −1.00000 −0.138675
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) − 2.00000i − 0.267261i
\(57\) 4.00000 0.529813
\(58\) 2.00000 + 5.00000i 0.262613 + 0.656532i
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i 0.768221 + 0.640184i \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 5.00000 0.635001
\(63\) 4.00000 0.503953
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) − 5.00000i − 0.615457i
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 2.00000i − 0.235702i
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 4.00000i 0.458831i
\(77\) 10.0000i 1.13961i
\(78\) −1.00000 −0.113228
\(79\) 1.00000i 0.112509i 0.998416 + 0.0562544i \(0.0179158\pi\)
−0.998416 + 0.0562544i \(0.982084\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) 0 0
\(86\) 9.00000 0.970495
\(87\) 2.00000 + 5.00000i 0.214423 + 0.536056i
\(88\) 5.00000 0.533002
\(89\) − 14.0000i − 1.48400i −0.670402 − 0.741999i \(-0.733878\pi\)
0.670402 − 0.741999i \(-0.266122\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) 5.00000 0.518476
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 10.0000i 1.00504i
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 1450.2.c.a.1101.2 2
5.2 odd 4 1450.2.d.b.1449.2 2
5.3 odd 4 1450.2.d.c.1449.1 2
5.4 even 2 58.2.b.a.57.1 ✓ 2
15.14 odd 2 522.2.d.a.289.2 2
20.19 odd 2 464.2.e.c.289.2 2
29.28 even 2 inner 1450.2.c.a.1101.1 2
40.19 odd 2 1856.2.e.d.1217.1 2
40.29 even 2 1856.2.e.b.1217.2 2
60.59 even 2 4176.2.o.d.289.2 2
145.28 odd 4 1450.2.d.b.1449.1 2
145.57 odd 4 1450.2.d.c.1449.2 2
145.99 odd 4 1682.2.a.g.1.1 1
145.104 odd 4 1682.2.a.c.1.1 1
145.144 even 2 58.2.b.a.57.2 yes 2
435.434 odd 2 522.2.d.a.289.1 2
580.579 odd 2 464.2.e.c.289.1 2
1160.579 odd 2 1856.2.e.d.1217.2 2
1160.869 even 2 1856.2.e.b.1217.1 2
1740.1739 even 2 4176.2.o.d.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.b.a.57.1 ✓ 2 5.4 even 2
58.2.b.a.57.2 yes 2 145.144 even 2
464.2.e.c.289.1 2 580.579 odd 2
464.2.e.c.289.2 2 20.19 odd 2
522.2.d.a.289.1 2 435.434 odd 2
522.2.d.a.289.2 2 15.14 odd 2
1450.2.c.a.1101.1 2 29.28 even 2 inner
1450.2.c.a.1101.2 2 1.1 even 1 trivial
1450.2.d.b.1449.1 2 145.28 odd 4
1450.2.d.b.1449.2 2 5.2 odd 4
1450.2.d.c.1449.1 2 5.3 odd 4
1450.2.d.c.1449.2 2 145.57 odd 4
1682.2.a.c.1.1 1 145.104 odd 4
1682.2.a.g.1.1 1 145.99 odd 4
1856.2.e.b.1217.1 2 1160.869 even 2
1856.2.e.b.1217.2 2 40.29 even 2
1856.2.e.d.1217.1 2 40.19 odd 2
1856.2.e.d.1217.2 2 1160.579 odd 2
4176.2.o.d.289.1 2 1740.1739 even 2
4176.2.o.d.289.2 2 60.59 even 2