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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(289,522)] 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("522.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,-2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
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 289.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 522.289
Dual form 522.2.d.a.289.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000i q^{8} +1.00000i q^{10} +5.00000i q^{11} -1.00000 q^{13} +2.00000i q^{14} +1.00000 q^{16} +2.00000i q^{17} +4.00000i q^{19} +1.00000 q^{20} +5.00000 q^{22} +6.00000 q^{23} -4.00000 q^{25} +1.00000i q^{26} +2.00000 q^{28} +(-5.00000 - 2.00000i) q^{29} +5.00000i q^{31} -1.00000i q^{32} +2.00000 q^{34} +2.00000 q^{35} +8.00000i q^{37} +4.00000 q^{38} -1.00000i q^{40} +10.0000i q^{41} -9.00000i q^{43} -5.00000i q^{44} -6.00000i q^{46} -3.00000i q^{47} -3.00000 q^{49} +4.00000i q^{50} +1.00000 q^{52} +1.00000 q^{53} -5.00000i q^{55} -2.00000i q^{56} +(-2.00000 + 5.00000i) q^{58} -10.0000 q^{59} -10.0000i q^{61} +5.00000 q^{62} -1.00000 q^{64} +1.00000 q^{65} +8.00000 q^{67} -2.00000i q^{68} -2.00000i q^{70} +8.00000 q^{71} +16.0000i q^{73} +8.00000 q^{74} -4.00000i q^{76} -10.0000i q^{77} -1.00000i q^{79} -1.00000 q^{80} +10.0000 q^{82} -14.0000 q^{83} -2.00000i q^{85} -9.00000 q^{86} -5.00000 q^{88} -14.0000i q^{89} +2.00000 q^{91} -6.00000 q^{92} -3.00000 q^{94} -4.00000i q^{95} -2.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5} - 4 q^{7} - 2 q^{13} + 2 q^{16} + 2 q^{20} + 10 q^{22} + 12 q^{23} - 8 q^{25} + 4 q^{28} - 10 q^{29} + 4 q^{34} + 4 q^{35} + 8 q^{38} - 6 q^{49} + 2 q^{52} + 2 q^{53} - 4 q^{58}+ \cdots - 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 1.00000i 0.316228i
\(11\) 5.00000i 1.50756i 0.657129 + 0.753778i \(0.271771\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 1.00000i 0.196116i
\(27\) 0 0
\(28\) 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 \(0.148224\pi\)
−0.893525 + 0.449013i \(0.851776\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) − 1.00000i − 0.158114i
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) − 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 \(-0.929787\pi\)
0.975770 − 0.218797i \(-0.0702134\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 4.00000i 0.565685i
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) − 5.00000i − 0.674200i
\(56\) − 2.00000i − 0.267261i
\(57\) 0 0
\(58\) −2.00000 + 5.00000i −0.262613 + 0.656532i
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) − 10.0000i − 1.28037i −0.768221 − 0.640184i \(-0.778858\pi\)
0.768221 − 0.640184i \(-0.221142\pi\)
\(62\) 5.00000 0.635001
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 0 0
\(70\) − 2.00000i − 0.239046i
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 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
\(78\) 0 0
\(79\) − 1.00000i − 0.112509i −0.998416 − 0.0562544i \(-0.982084\pi\)
0.998416 − 0.0562544i \(-0.0179158\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) − 2.00000i − 0.216930i
\(86\) −9.00000 −0.970495
\(87\) 0 0
\(88\) −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
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) − 4.00000i − 0.410391i
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
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 522.2.d.a.289.1 2
3.2 odd 2 58.2.b.a.57.2 yes 2
4.3 odd 2 4176.2.o.d.289.1 2
12.11 even 2 464.2.e.c.289.1 2
15.2 even 4 1450.2.d.b.1449.1 2
15.8 even 4 1450.2.d.c.1449.2 2
15.14 odd 2 1450.2.c.a.1101.1 2
24.5 odd 2 1856.2.e.b.1217.1 2
24.11 even 2 1856.2.e.d.1217.2 2
29.28 even 2 inner 522.2.d.a.289.2 2
87.17 even 4 1682.2.a.g.1.1 1
87.41 even 4 1682.2.a.c.1.1 1
87.86 odd 2 58.2.b.a.57.1 ✓ 2
116.115 odd 2 4176.2.o.d.289.2 2
348.347 even 2 464.2.e.c.289.2 2
435.173 even 4 1450.2.d.b.1449.2 2
435.347 even 4 1450.2.d.c.1449.1 2
435.434 odd 2 1450.2.c.a.1101.2 2
696.173 odd 2 1856.2.e.b.1217.2 2
696.347 even 2 1856.2.e.d.1217.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.b.a.57.1 ✓ 2 87.86 odd 2
58.2.b.a.57.2 yes 2 3.2 odd 2
464.2.e.c.289.1 2 12.11 even 2
464.2.e.c.289.2 2 348.347 even 2
522.2.d.a.289.1 2 1.1 even 1 trivial
522.2.d.a.289.2 2 29.28 even 2 inner
1450.2.c.a.1101.1 2 15.14 odd 2
1450.2.c.a.1101.2 2 435.434 odd 2
1450.2.d.b.1449.1 2 15.2 even 4
1450.2.d.b.1449.2 2 435.173 even 4
1450.2.d.c.1449.1 2 435.347 even 4
1450.2.d.c.1449.2 2 15.8 even 4
1682.2.a.c.1.1 1 87.41 even 4
1682.2.a.g.1.1 1 87.17 even 4
1856.2.e.b.1217.1 2 24.5 odd 2
1856.2.e.b.1217.2 2 696.173 odd 2
1856.2.e.d.1217.1 2 696.347 even 2
1856.2.e.d.1217.2 2 24.11 even 2
4176.2.o.d.289.1 2 4.3 odd 2
4176.2.o.d.289.2 2 116.115 odd 2